LMFDB - Embedded newform 180.4.k.e.163.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [180,4,Mod(127,180)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("180.127"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	180.k (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,6,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.6203438010$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096 $ x^12 - 7x^10 + 44x^8 - 156x^6 + 704x^4 - 1792*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			163.2
Root			$-1.83244 + 0.801352i$ of defining polynomial
Character	$\chi$	$=$	180.163
Dual form			180.4.k.e.127.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.03109 - 2.63379i) q^{2} +(-5.87372 + 5.43134i) q^{4} +(10.4994 - 3.84216i) q^{5} +(-1.14202 - 1.14202i) q^{7} +(20.3613 + 9.86997i) q^{8} +(-20.9453 - 23.6917i) q^{10} +27.0350i q^{11} +(40.4777 + 40.4777i) q^{13} +(-1.83032 + 4.18535i) q^{14} +(5.00116 - 63.8043i) q^{16} +(36.2735 - 36.2735i) q^{17} +56.8829 q^{19} +(-40.8026 + 79.5936i) q^{20} +(71.2046 - 27.8754i) q^{22} +(54.9839 - 54.9839i) q^{23} +(95.4757 - 80.6808i) q^{25} +(64.8739 - 148.346i) q^{26} +(12.9106 + 0.505208i) q^{28} -57.1173i q^{29} -190.845i q^{31} +(-173.204 + 52.6158i) q^{32} +(-132.938 - 58.1357i) q^{34} +(-16.3783 - 7.60271i) q^{35} +(-50.4605 + 50.4605i) q^{37} +(-58.6513 - 149.818i) q^{38} +(251.704 + 25.3976i) q^{40} +71.5197 q^{41} +(-66.9381 + 66.9381i) q^{43} +(-146.836 - 158.796i) q^{44} +(-201.509 - 88.1229i) q^{46} +(343.017 + 343.017i) q^{47} -340.392i q^{49} +(-310.940 - 168.274i) q^{50} +(-457.603 - 17.9066i) q^{52} +(-240.148 - 240.148i) q^{53} +(103.873 + 283.852i) q^{55} +(-11.9813 - 34.5247i) q^{56} +(-150.435 + 58.8929i) q^{58} +738.207 q^{59} -187.952 q^{61} +(-502.645 + 196.777i) q^{62} +(317.167 + 401.932i) q^{64} +(580.515 + 269.471i) q^{65} +(576.434 + 576.434i) q^{67} +(-16.0468 + 410.074i) q^{68} +(-3.13647 + 50.9761i) q^{70} +157.380i q^{71} +(180.613 + 180.613i) q^{73} +(184.932 + 80.8733i) q^{74} +(-334.114 + 308.950i) q^{76} +(30.8744 - 30.8744i) q^{77} -55.6778 q^{79} +(-192.637 - 689.123i) q^{80} +(-73.7430 - 188.368i) q^{82} +(-858.601 + 858.601i) q^{83} +(241.482 - 520.219i) q^{85} +(245.320 + 107.282i) q^{86} +(-266.835 + 550.468i) q^{88} +158.689i q^{89} -92.4525i q^{91} +(-24.3239 + 621.596i) q^{92} +(549.756 - 1257.12i) q^{94} +(597.238 - 218.553i) q^{95} +(-1117.12 + 1117.12i) q^{97} +(-896.521 + 350.973i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{2} + 12 q^{8} - 110 q^{10} + 116 q^{13} + 312 q^{16} + 332 q^{17} - 140 q^{20} + 360 q^{22} + 340 q^{25} + 164 q^{26} - 880 q^{28} + 376 q^{32} + 508 q^{37} - 1600 q^{38} + 1420 q^{40} + 656 q^{41}+ \cdots - 1698 q^{98}+O(q^{100}) $ 12 * q + 6 * q^2 + 12 * q^8 - 110 * q^10 + 116 * q^13 + 312 * q^16 + 332 * q^17 - 140 * q^20 + 360 * q^22 + 340 * q^25 + 164 * q^26 - 880 * q^28 + 376 * q^32 + 508 * q^37 - 1600 * q^38 + 1420 * q^40 + 656 * q^41 - 1432 * q^46 + 1570 * q^50 - 932 * q^52 + 644 * q^53 - 2048 * q^56 + 1576 * q^58 - 896 * q^61 - 2440 * q^62 + 2740 * q^65 + 844 * q^68 - 3040 * q^70 + 1436 * q^73 + 800 * q^76 - 3120 * q^77 - 1840 * q^80 - 1352 * q^82 + 500 * q^85 + 2552 * q^86 - 2400 * q^88 + 1840 * q^92 - 4772 * q^97 - 1698 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$.

$n$	$37$	$91$	$101$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.03109	−	2.63379i	−0.364544	−	0.931186i
$2$	−1.03109	−	2.63379i	−0.364544	−	0.931186i
$3$		0			0
$3$		0			0
$4$	−5.87372	+	5.43134i	−0.734215	+	0.678917i
$5$	10.4994	−	3.84216i	0.939097	−	0.343653i
$5$	10.4994	−	3.84216i	0.939097	−	0.343653i
$6$		0			0
$7$	−1.14202	−	1.14202i	−0.0616631	−	0.0616631i	0.675603	−	0.737266i	$-0.263883\pi$
$7$	−1.14202	−	1.14202i	−0.0616631	−	0.0616631i	−0.737266	+	0.675603i	$0.763883\pi$
$8$	20.3613	+	9.86997i	0.899852	+	0.436195i
$9$		0			0
$10$	−20.9453	−	23.6917i	−0.662347	−	0.749197i
$11$			27.0350i			0.741033i	0.928826	+	0.370516i	$0.120819\pi$
$11$			27.0350i			0.741033i	−0.928826	+	0.370516i	$0.879181\pi$
$12$		0			0
$13$	40.4777	+	40.4777i	0.863577	+	0.863577i	0.991752	−	0.128174i	$-0.0409117\pi$
$13$	40.4777	+	40.4777i	0.863577	+	0.863577i	−0.128174	+	0.991752i	$0.540912\pi$
$14$	−1.83032	+	4.18535i	−0.0349409	+	0.0798988i
$15$		0			0
$16$	5.00116	−	63.8043i	0.0781431	−	0.996942i
$17$	36.2735	−	36.2735i	0.517507	−	0.517507i	−0.399309	−	0.916816i	$-0.630750\pi$
$17$	36.2735	−	36.2735i	0.517507	−	0.517507i	0.916816	+	0.399309i	$0.130750\pi$
$18$		0			0
$19$	56.8829			0.686834			0.343417	−	0.939183i	$-0.388416\pi$
$19$	56.8829			0.686834			0.343417	+	0.939183i	$0.388416\pi$
$20$	−40.8026	+	79.5936i	−0.456187	+	0.889884i
$21$		0			0
$22$	71.2046	−	27.8754i	0.690039	−	0.270139i
$23$	54.9839	−	54.9839i	0.498475	−	0.498475i	−0.412488	−	0.910963i	$-0.635340\pi$
$23$	54.9839	−	54.9839i	0.498475	−	0.498475i	0.910963	+	0.412488i	$0.135340\pi$
$24$		0			0
$25$	95.4757	−	80.6808i	0.763805	−	0.645447i
$26$	64.8739	−	148.346i	0.489339	−	1.11896i
$27$		0			0
$28$	12.9106	+	0.505208i	0.0871381	+	0.00340983i
$29$		−	57.1173i		−	0.365739i	−0.983137	−	0.182869i	$-0.941461\pi$
$29$		−	57.1173i		−	0.365739i	0.983137	−	0.182869i	$-0.0585385\pi$
$30$		0			0
$31$		−	190.845i		−	1.10570i	−0.833281	−	0.552850i	$-0.813540\pi$
$31$		−	190.845i		−	1.10570i	0.833281	−	0.552850i	$-0.186460\pi$
$32$	−173.204	+	52.6158i	−0.956825	+	0.290664i
$33$		0			0
$34$	−132.938	−	58.1357i	−0.670549	−	0.293241i
$35$	−16.3783	−	7.60271i	−0.0790983	−	0.0367169i
$36$		0			0
$37$	−50.4605	+	50.4605i	−0.224207	+	0.224207i	−0.810267	−	0.586060i	$-0.800678\pi$
$37$	−50.4605	+	50.4605i	−0.224207	+	0.224207i	0.586060	+	0.810267i	$0.300678\pi$
$38$	−58.6513	−	149.818i	−0.250381	−	0.639570i
$39$		0			0
$40$	251.704	+	25.3976i	0.994948	+	0.100393i
$41$	71.5197			0.272427			0.136213	−	0.990680i	$-0.456507\pi$
$41$	71.5197			0.272427			0.136213	+	0.990680i	$0.456507\pi$
$42$		0			0
$43$	−66.9381	+	66.9381i	−0.237394	+	0.237394i	−0.815770	−	0.578376i	$-0.803687\pi$
$43$	−66.9381	+	66.9381i	−0.237394	+	0.237394i	0.578376	+	0.815770i	$0.303687\pi$
$44$	−146.836	−	158.796i	−0.503100	−	0.544077i
$45$		0			0
$46$	−201.509	−	88.1229i	−0.645889	−	0.282457i
$47$	343.017	+	343.017i	1.06456	+	1.06456i	0.997767	+	0.0667913i	$0.0212762\pi$
$47$	343.017	+	343.017i	1.06456	+	1.06456i	0.0667913	+	0.997767i	$0.478724\pi$
$48$		0			0
$49$		−	340.392i		−	0.992395i
$50$	−310.940	−	168.274i	−0.879472	−	0.475951i
$51$		0			0
$52$	−457.603	−	17.9066i	−1.22035	−	0.0477539i
$53$	−240.148	−	240.148i	−0.622394	−	0.622394i	0.323749	−	0.946143i	$-0.395057\pi$
$53$	−240.148	−	240.148i	−0.622394	−	0.622394i	−0.946143	+	0.323749i	$0.895057\pi$
$54$		0			0
$55$	103.873	+	283.852i	0.254658	+	0.695901i
$56$	−11.9813	−	34.5247i	−0.0285905	−	0.0823848i
$57$		0			0
$58$	−150.435	+	58.8929i	−0.340571	+	0.133328i
$59$	738.207			1.62892			0.814461	−	0.580218i	$-0.197033\pi$
$59$	738.207			1.62892			0.814461	+	0.580218i	$0.197033\pi$
$60$		0			0
$61$	−187.952			−0.394506			−0.197253	−	0.980353i	$-0.563202\pi$
$61$	−187.952			−0.394506			−0.197253	+	0.980353i	$0.563202\pi$
$62$	−502.645	+	196.777i	−1.02961	+	0.403077i
$63$		0			0
$64$	317.167	+	401.932i	0.619467	+	0.785022i
$65$	580.515	+	269.471i	1.10775	+	0.514212i
$66$		0			0
$67$	576.434	+	576.434i	1.05108	+	1.05108i	0.998623	+	0.0524612i	$0.0167066\pi$
$67$	576.434	+	576.434i	1.05108	+	1.05108i	0.0524612	+	0.998623i	$0.483293\pi$
$68$	−16.0468	+	410.074i	−0.0286170	+	0.731305i
$69$		0			0
$70$	−3.13647	+	50.9761i	−0.00535543	+	0.0870402i
$71$			157.380i			0.263064i	0.991312	+	0.131532i	$0.0419896\pi$
$71$			157.380i			0.263064i	−0.991312	+	0.131532i	$0.958010\pi$
$72$		0			0
$73$	180.613	+	180.613i	0.289577	+	0.289577i	0.836913	−	0.547336i	$-0.184358\pi$
$73$	180.613	+	180.613i	0.289577	+	0.289577i	−0.547336	+	0.836913i	$0.684358\pi$
$74$	184.932	+	80.8733i	0.290512	+	0.127045i
$75$		0			0
$76$	−334.114	+	308.950i	−0.504284	+	0.466303i
$77$	30.8744	−	30.8744i	0.0456944	−	0.0456944i
$78$		0			0
$79$	−55.6778			−0.0792942			−0.0396471	−	0.999214i	$-0.512623\pi$
$79$	−55.6778			−0.0792942			−0.0396471	+	0.999214i	$0.512623\pi$
$80$	−192.637	−	689.123i	−0.269218	−	0.963079i
$81$		0			0
$82$	−73.7430	−	188.368i	−0.0993116	−	0.253680i
$83$	−858.601	+	858.601i	−1.13547	+	1.13547i	−0.146213	+	0.989253i	$0.546708\pi$
$83$	−858.601	+	858.601i	−1.13547	+	1.13547i	−0.989253	+	0.146213i	$0.953292\pi$
$84$		0			0
$85$	241.482	−	520.219i	0.308146	−	0.663832i
$86$	245.320	+	107.282i	0.307599	+	0.134518i
$87$		0			0
$88$	−266.835	+	550.468i	−0.323235	+	0.666820i
$89$			158.689i			0.189000i	0.995525	+	0.0944998i	$0.0301252\pi$
$89$			158.689i			0.189000i	−0.995525	+	0.0944998i	$0.969875\pi$
$90$		0			0
$91$		−	92.4525i		−	0.106502i
$92$	−24.3239	+	621.596i	−0.0275646	+	0.704411i
$93$		0			0
$94$	549.756	−	1257.12i	0.603223	−	1.37938i
$95$	597.238	−	218.553i	0.645003	−	0.236032i
$96$		0			0
$97$	−1117.12	+	1117.12i	−1.16935	+	1.16935i	−0.186986	+	0.982363i	$0.559872\pi$
$97$	−1117.12	+	1117.12i	−1.16935	+	1.16935i	−0.982363	+	0.186986i	$0.940128\pi$
$98$	−896.521	+	350.973i	−0.924105	+	0.361772i
$99$		0			0
$100$	−122.592	+	992.457i	−0.122592	+	0.992457i
$101$	−787.780			−0.776109			−0.388055	−	0.921636i	$-0.626853\pi$
$101$	−787.780			−0.776109			−0.388055	+	0.921636i	$0.626853\pi$
$102$		0			0
$103$	−522.455	+	522.455i	−0.499796	+	0.499796i	−0.911374	−	0.411578i	$-0.864978\pi$
$103$	−522.455	+	522.455i	−0.499796	+	0.499796i	0.411578	+	0.911374i	$0.364978\pi$
$104$	424.666	+	1223.69i	0.400403	+	1.15378i
$105$		0			0
$106$	−384.887	+	880.114i	−0.352674	+	0.806455i
$107$	−615.276	−	615.276i	−0.555897	−	0.555897i	0.372240	−	0.928137i	$-0.378590\pi$
$107$	−615.276	−	615.276i	−0.555897	−	0.555897i	−0.928137	+	0.372240i	$0.878590\pi$
$108$		0			0
$109$			398.877i			0.350509i	0.984523	+	0.175254i	$0.0560748\pi$
$109$			398.877i			0.350509i	−0.984523	+	0.175254i	$0.943925\pi$
$110$	640.505	−	566.255i	0.555179	−	0.490821i
$111$		0			0
$112$	−78.5770	+	67.1542i	−0.0662931	+	0.0566560i
$113$	−692.888	−	692.888i	−0.576826	−	0.576826i	0.357201	−	0.934028i	$-0.383731\pi$
$113$	−692.888	−	692.888i	−0.576826	−	0.576826i	−0.934028	+	0.357201i	$0.883731\pi$
$114$		0			0
$115$	366.042	−	788.555i	0.296814	−	0.639419i
$116$	310.223	+	335.491i	0.248306	+	0.268531i
$117$		0			0
$118$	−761.156	−	1944.28i	−0.593814	−	1.51683i
$119$	−82.8499			−0.0638221
$120$		0			0
$121$	600.109			0.450871
$122$	193.795	+	495.028i	0.143815	+	0.367358i
$123$		0			0
$124$	1036.54	+	1120.97i	0.750679	+	0.811821i
$125$	692.451	−	1213.93i	0.495477	−	0.868621i
$126$		0			0
$127$	−498.629	−	498.629i	−0.348395	−	0.348395i	0.511116	−	0.859512i	$-0.329232\pi$
$127$	−498.629	−	498.629i	−0.348395	−	0.348395i	−0.859512	+	0.511116i	$0.829232\pi$
$128$	731.577	−	1249.78i	0.505179	−	0.863015i
$129$		0			0
$130$	111.169	−	1806.80i	0.0750016	−	1.21898i
$131$		−	1747.61i		−	1.16557i	−0.812626	−	0.582785i	$-0.801963\pi$
$131$		−	1747.61i		−	1.16557i	0.812626	−	0.582785i	$-0.198037\pi$
$132$		0			0
$133$	−64.9613	−	64.9613i	−0.0423523	−	0.0423523i
$134$	923.854	−	2112.56i	0.595588	−	1.36192i
$135$		0			0
$136$	1096.59	−	380.558i	0.691413	−	0.239945i
$137$	−124.289	+	124.289i	−0.0775092	+	0.0775092i	−0.744799	−	0.667289i	$-0.767454\pi$
$137$	−124.289	+	124.289i	−0.0775092	+	0.0775092i	0.667289	+	0.744799i	$0.267454\pi$
$138$		0			0
$139$	−9.83873			−0.00600367			−0.00300184	−	0.999995i	$-0.500956\pi$
$139$	−9.83873			−0.00600367			−0.00300184	+	0.999995i	$0.500956\pi$
$140$	137.495	−	44.3000i	0.0830029	−	0.0267431i
$141$		0			0
$142$	414.506	−	162.272i	0.244962	−	0.0958985i
$143$	−1094.32	+	1094.32i	−0.639939	+	0.639939i
$144$		0			0
$145$	−219.454	−	599.699i	−0.125687	−	0.343464i
$146$	289.469	−	661.925i	0.164087	−	0.375214i
$147$		0			0
$148$	22.3229	−	570.459i	0.0123982	−	0.316834i
$149$			2840.41i			1.56171i	0.624711	+	0.780856i	$0.285217\pi$
$149$			2840.41i			1.56171i	−0.624711	+	0.780856i	$0.714783\pi$
$150$		0			0
$151$		−	2913.54i		−	1.57020i	−0.619368	−	0.785101i	$-0.712611\pi$
$151$		−	2913.54i		−	1.57020i	0.619368	−	0.785101i	$-0.287389\pi$
$152$	1158.21	+	561.433i	0.618049	+	0.299594i
$153$		0			0
$154$	−113.151	−	49.4826i	−0.0592076	−	0.0258923i
$155$	−733.255	−	2003.76i	−0.379977	−	1.03836i
$156$		0			0
$157$	1572.29	−	1572.29i	0.799251	−	0.799251i	−0.183727	−	0.982977i	$-0.558816\pi$
$157$	1572.29	−	1572.29i	0.799251	−	0.799251i	0.982977	+	0.183727i	$0.0588161\pi$
$158$	57.4087	+	146.644i	0.0289063	+	0.0738377i
$159$		0			0
$160$	−1616.38	+	1217.91i	−0.798664	+	0.601777i
$161$	−125.585			−0.0614751
$162$		0			0
$163$	−1457.19	+	1457.19i	−0.700222	+	0.700222i	−0.964458	−	0.264236i	$-0.914880\pi$
$163$	−1457.19	+	1457.19i	−0.700222	+	0.700222i	0.264236	+	0.964458i	$0.414880\pi$
$164$	−420.087	+	388.448i	−0.200020	+	0.184955i
$165$		0			0
$166$	3146.67	+	1376.08i	1.47126	+	0.643402i
$167$	801.239	+	801.239i	0.371268	+	0.371268i	0.867939	−	0.496671i	$-0.165444\pi$
$167$	801.239	+	801.239i	0.371268	+	0.371268i	−0.496671	+	0.867939i	$0.665444\pi$
$168$		0			0
$169$			1079.90i			0.491532i
$170$	−1619.14	−	99.6228i	−0.730484	−	0.0449454i
$171$		0			0
$172$	29.6122	−	756.739i	0.0131274	−	0.335470i
$173$	1180.59	+	1180.59i	0.518837	+	0.518837i	0.917220	−	0.398382i	$-0.130428\pi$
$173$	1180.59	+	1180.59i	0.518837	+	0.518837i	−0.398382	+	0.917220i	$0.630428\pi$
$174$		0			0
$175$	−201.174	−	16.8959i	−0.0868989	−	0.00729836i
$176$	1724.95	+	135.206i	0.738767	+	0.0579066i
$177$		0			0
$178$	417.953	−	163.622i	0.175994	−	0.0688987i
$179$	−3724.41			−1.55517			−0.777585	−	0.628778i	$-0.783555\pi$
$179$	−3724.41			−1.55517			−0.777585	+	0.628778i	$0.783555\pi$
$180$		0			0
$181$	545.856			0.224161			0.112081	−	0.993699i	$-0.464249\pi$
$181$	545.856			0.224161			0.112081	+	0.993699i	$0.464249\pi$
$182$	−243.501	+	95.3266i	−0.0991729	+	0.0388246i
$183$		0			0
$184$	1662.23	−	576.855i	0.665986	−	0.231121i
$185$	−335.929	+	723.683i	−0.133503	+	0.287601i
$186$		0			0
$187$	980.654	+	980.654i	0.383489	+	0.383489i
$188$	−3877.83	−	151.745i	−1.50436	−	0.0588677i
$189$		0			0
$190$	−1191.43	−	1347.65i	−0.454922	−	0.514574i
$191$		−	3668.60i		−	1.38980i	−0.719109	−	0.694898i	$-0.755450\pi$
$191$		−	3668.60i		−	1.38980i	0.719109	−	0.694898i	$-0.244550\pi$
$192$		0			0
$193$	−715.028	−	715.028i	−0.266678	−	0.266678i	0.561082	−	0.827760i	$-0.310385\pi$
$193$	−715.028	−	715.028i	−0.266678	−	0.266678i	−0.827760	+	0.561082i	$0.810385\pi$
$194$	4094.13	+	1790.42i	1.51516	+	0.662602i
$195$		0			0
$196$	1848.78	+	1999.36i	0.673754	+	0.728631i
$197$	−272.976	+	272.976i	−0.0987246	+	0.0987246i	−0.754744	−	0.656019i	$-0.772239\pi$
$197$	−272.976	+	272.976i	−0.0987246	+	0.0987246i	0.656019	+	0.754744i	$0.272239\pi$
$198$		0			0
$199$	−4554.16			−1.62229			−0.811146	−	0.584844i	$-0.801156\pi$
$199$	−4554.16			−1.62229			−0.811146	+	0.584844i	$0.801156\pi$
$200$	2740.33	−	700.427i	0.968853	−	0.247638i
$201$		0			0
$202$	812.269	+	2074.85i	0.282926	+	0.722702i
$203$	−65.2289	+	65.2289i	−0.0225526	+	0.0225526i
$204$		0			0
$205$	750.915	−	274.790i	0.255835	−	0.0936203i
$206$	1914.73	+	837.341i	0.647601	+	0.283205i
$207$		0			0
$208$	2785.09	−	2380.22i	0.928419	−	0.793454i
$209$			1537.83i			0.508966i
$210$		0			0
$211$		−	4250.89i		−	1.38694i	−0.720487	−	0.693468i	$-0.756082\pi$
$211$		−	4250.89i		−	1.38694i	0.720487	−	0.693468i	$-0.243918\pi$
$212$	2714.89	+	106.237i	0.879525	+	0.0344170i
$213$		0			0
$214$	−986.105	+	2254.91i	−0.314994	+	0.720292i
$215$	−445.624	+	959.998i	−0.141355	+	0.304518i
$216$		0			0
$217$	−217.948	+	217.948i	−0.0681809	+	0.0681809i
$218$	1050.56	−	411.276i	0.326389	−	0.127776i
$219$		0			0
$220$	−2151.81	−	1103.10i	−0.659433	−	0.338049i
$221$	2936.54			0.893814
$222$		0			0
$223$	258.746	−	258.746i	0.0776991	−	0.0776991i	−0.667189	−	0.744888i	$-0.732503\pi$
$223$	258.746	−	258.746i	0.0776991	−	0.0776991i	0.744888	+	0.667189i	$0.232503\pi$
$224$	257.890	+	137.714i	0.0769241	+	0.0410776i
$225$		0			0
$226$	−1110.49	+	2539.35i	−0.326854	+	0.747412i
$227$	2127.58	+	2127.58i	0.622080	+	0.622080i	0.946063	−	0.323983i	$-0.105022\pi$
$227$	2127.58	+	2127.58i	0.622080	+	0.622080i	−0.323983	+	0.946063i	$0.605022\pi$
$228$		0			0
$229$			4654.98i			1.34328i	0.740880	+	0.671638i	$0.234409\pi$
$229$			4654.98i			1.34328i	−0.740880	+	0.671638i	$0.765591\pi$
$230$	−2454.31	−	151.010i	−0.703620	−	0.0432925i
$231$		0			0
$232$	563.746	−	1162.98i	0.159533	−	0.329111i
$233$	−3392.53	−	3392.53i	−0.953871	−	0.953871i	0.0451108	−	0.998982i	$-0.485636\pi$
$233$	−3392.53	−	3392.53i	−0.953871	−	0.953871i	−0.998982	+	0.0451108i	$0.985636\pi$
$234$		0			0
$235$	4919.41	+	2283.56i	1.36556	+	0.633885i
$236$	−4336.02	+	4009.45i	−1.19598	+	1.10590i
$237$		0			0
$238$	85.4254	+	218.209i	0.0232660	+	0.0594303i
$239$	1434.32			0.388195			0.194098	−	0.980982i	$-0.437822\pi$
$239$	1434.32			0.388195			0.194098	+	0.980982i	$0.437822\pi$
$240$		0			0
$241$	−6438.62			−1.72094			−0.860472	−	0.509497i	$-0.829831\pi$
$241$	−6438.62			−1.72094			−0.860472	+	0.509497i	$0.829831\pi$
$242$	−618.764	−	1580.56i	−0.164362	−	0.419845i
$243$		0			0
$244$	1103.98	−	1020.83i	0.289652	−	0.267837i
$245$	−1307.84	−	3573.91i	−0.341040	−	0.931955i
$246$		0			0
$247$	2302.49	+	2302.49i	0.593134	+	0.593134i
$248$	1883.63	−	3885.85i	0.482301	−	0.994966i
$249$		0			0
$250$	−3911.23	−	572.099i	−0.989471	−	0.144731i
$251$			1877.58i			0.472159i	0.971734	+	0.236079i	$0.0758626\pi$
$251$			1877.58i			0.472159i	−0.971734	+	0.236079i	$0.924137\pi$
$252$		0			0
$253$	1486.49	+	1486.49i	0.369386	+	0.369386i
$254$	−799.156	+	1827.42i	−0.197415	+	0.451426i
$255$		0			0
$256$	−4045.98	−	638.191i	−0.987787	−	0.155808i
$257$	−1940.33	+	1940.33i	−0.470952	+	0.470952i	−0.902223	−	0.431270i	$-0.858066\pi$
$257$	−1940.33	+	1940.33i	−0.470952	+	0.470952i	0.431270	+	0.902223i	$0.358066\pi$
$258$		0			0
$259$	115.254			0.0276506
$260$	−4873.37	+	1570.17i	−1.16244	+	0.374531i
$261$		0			0
$262$	−4602.85	+	1801.94i	−1.08536	+	0.424902i
$263$	−1004.94	+	1004.94i	−0.235616	+	0.235616i	−0.815032	−	0.579416i	$-0.803281\pi$
$263$	−1004.94	+	1004.94i	−0.235616	+	0.235616i	0.579416	+	0.815032i	$0.303281\pi$
$264$		0			0
$265$	−3444.10	−	1598.73i	−0.798376	−	0.370601i
$266$	−104.114	+	238.075i	−0.0239986	+	0.0548772i
$267$		0			0
$268$	−6516.62	−	255.004i	−1.48532	−	0.0581227i
$269$		−	6594.94i		−	1.49480i	−0.664376	−	0.747398i	$-0.731303\pi$
$269$		−	6594.94i		−	1.49480i	0.664376	−	0.747398i	$-0.268697\pi$
$270$		0			0
$271$			4781.49i			1.07179i	0.844285	+	0.535895i	$0.180026\pi$
$271$			4781.49i			1.07179i	−0.844285	+	0.535895i	$0.819974\pi$
$272$	−2133.00	−	2495.81i	−0.475485	−	0.556364i
$273$		0			0
$274$	455.506	+	199.199i	0.100431	+	0.0439200i
$275$	2181.21	+	2581.18i	0.478297	+	0.566005i
$276$		0			0
$277$	5490.27	−	5490.27i	1.19090	−	1.19090i	0.214082	−	0.976816i	$-0.431324\pi$
$277$	5490.27	−	5490.27i	1.19090	−	1.19090i	0.976816	−	0.214082i	$-0.0686760\pi$
$278$	10.1446	+	25.9132i	0.00218860	+	0.00559053i
$279$		0			0
$280$	−258.446	−	316.455i	−0.0551610	−	0.0675421i
$281$	−3046.29			−0.646714			−0.323357	−	0.946277i	$-0.604811\pi$
$281$	−3046.29			−0.646714			−0.323357	+	0.946277i	$0.604811\pi$
$282$		0			0
$283$	−5858.21	+	5858.21i	−1.23051	+	1.23051i	−0.266743	+	0.963768i	$0.585947\pi$
$283$	−5858.21	+	5858.21i	−1.23051	+	1.23051i	−0.963768	+	0.266743i	$0.914053\pi$
$284$	−854.783	−	924.405i	−0.178599	−	0.193146i
$285$		0			0
$286$	4010.53	+	1753.87i	0.829188	+	0.362616i
$287$	−81.6767	−	81.6767i	−0.0167987	−	0.0167987i
$288$		0			0
$289$			2281.47i			0.464374i
$290$	−1353.21	+	1196.34i	−0.274010	+	0.242246i
$291$		0			0
$292$	−2041.84	−	79.9000i	−0.409211	−	0.0160130i
$293$	5371.12	+	5371.12i	1.07094	+	1.07094i	0.997284	+	0.0736515i	$0.0234653\pi$
$293$	5371.12	+	5371.12i	1.07094	+	1.07094i	0.0736515	+	0.997284i	$0.476535\pi$
$294$		0			0
$295$	7750.75	−	2836.31i	1.52972	−	0.559784i
$296$	−1525.49	+	529.399i	−0.299551	+	0.103955i
$297$		0			0
$298$	7481.04	−	2928.71i	1.45425	−	0.569314i
$299$	4451.25			0.860944
$300$		0			0
$301$	152.889			0.0292770
$302$	−7673.66	+	3004.11i	−1.46215	+	0.572408i
$303$		0			0
$304$	284.481	−	3629.38i	0.0536713	−	0.684734i
$305$	−1973.39	+	722.143i	−0.370479	+	0.135573i
$306$		0			0
$307$	1464.75	+	1464.75i	0.272306	+	0.272306i	0.830028	−	0.557722i	$-0.188324\pi$
$307$	1464.75	+	1464.75i	0.272306	+	0.272306i	−0.557722	+	0.830028i	$0.688324\pi$
$308$	−13.6583	+	349.037i	−0.00252680	+	0.0645722i
$309$		0			0
$310$	−4521.43	+	3997.29i	−0.828387	+	0.732357i
$311$			1381.23i			0.251840i	0.992040	+	0.125920i	$0.0401882\pi$
$311$			1381.23i			0.251840i	−0.992040	+	0.125920i	$0.959812\pi$
$312$		0			0
$313$	−1989.95	−	1989.95i	−0.359356	−	0.359356i	0.504220	−	0.863575i	$-0.331780\pi$
$313$	−1989.95	−	1989.95i	−0.359356	−	0.359356i	−0.863575	+	0.504220i	$0.831780\pi$
$314$	−5762.25	−	2519.92i	−1.03561	−	0.452889i
$315$		0			0
$316$	327.036	−	302.405i	0.0582190	−	0.0538342i
$317$	2078.83	−	2078.83i	0.368325	−	0.368325i	−0.498541	−	0.866866i	$-0.666131\pi$
$317$	2078.83	−	2078.83i	0.368325	−	0.368325i	0.866866	+	0.498541i	$0.166131\pi$
$318$		0			0
$319$	1544.17			0.271024
$320$	4874.36	+	3001.44i	0.851515	+	0.524330i
$321$		0			0
$322$	129.489	+	330.765i	0.0224104	+	0.0572447i
$323$	2063.34	−	2063.34i	0.355441	−	0.355441i
$324$		0			0
$325$	7130.42	+	598.861i	1.21700	+	0.102212i
$326$	5340.43	+	2335.45i	0.907298	+	0.396775i
$327$		0			0
$328$	1456.24	+	705.898i	0.245144	+	0.118831i
$329$		−	783.463i		−	0.131288i
$330$		0			0
$331$		−	8633.95i		−	1.43373i	−0.697211	−	0.716866i	$-0.745576\pi$
$331$		−	8633.95i		−	1.43373i	0.697211	−	0.716866i	$-0.254424\pi$
$332$	379.830	−	9706.53i	0.0627888	−	1.60456i
$333$		0			0
$334$	1284.15	−	2936.44i	0.210376	−	0.481063i
$335$	8266.97	+	3837.47i	1.34828	+	0.625862i
$336$		0			0
$337$	−3115.08	+	3115.08i	−0.503528	+	0.503528i	−0.912532	−	0.409004i	$-0.865876\pi$
$337$	−3115.08	+	3115.08i	−0.503528	+	0.503528i	0.409004	+	0.912532i	$0.365876\pi$
$338$	2844.22	−	1113.47i	0.457708	−	0.179185i
$339$		0			0
$340$	1407.09	+	4367.19i	0.224441	+	0.696601i
$341$	5159.48			0.819360
$342$		0			0
$343$	−780.445	+	780.445i	−0.122857	+	0.122857i
$344$	−2023.62	+	702.271i	−0.317170	+	0.110070i
$345$		0			0
$346$	1892.14	−	4326.73i	0.293995	−	0.672273i
$347$	646.980	+	646.980i	0.100091	+	0.100091i	0.755379	−	0.655288i	$-0.227453\pi$
$347$	646.980	+	646.980i	0.100091	+	0.100091i	−0.655288	+	0.755379i	$0.727453\pi$
$348$		0			0
$349$		−	9611.76i		−	1.47423i	−0.675768	−	0.737114i	$-0.736188\pi$
$349$		−	9611.76i		−	1.47423i	0.675768	−	0.737114i	$-0.263812\pi$
$350$	162.927	+	547.271i	0.0248824	+	0.0835796i
$351$		0			0
$352$	−1422.47	−	4682.57i	−0.215391	−	0.709039i
$353$	5085.16	+	5085.16i	0.766730	+	0.766730i	0.977529	−	0.210799i	$-0.0676066\pi$
$353$	5085.16	+	5085.16i	0.766730	+	0.766730i	−0.210799	+	0.977529i	$0.567607\pi$
$354$		0			0
$355$	604.678	+	1652.40i	0.0904027	+	0.247043i
$356$	−861.892	−	932.093i	−0.128315	−	0.138766i
$357$		0			0
$358$	3840.19	+	9809.32i	0.566928	+	1.44815i
$359$	−5598.06			−0.822992			−0.411496	−	0.911411i	$-0.634994\pi$
$359$	−5598.06			−0.822992			−0.411496	+	0.911411i	$0.634994\pi$
$360$		0			0
$361$	−3623.33			−0.528259
$362$	−562.825	−	1437.67i	−0.0817167	−	0.208736i
$363$		0			0
$364$	502.141	+	543.040i	0.0723059	+	0.0781952i
$365$	2590.28	+	1202.39i	0.371455	+	0.172427i
$366$		0			0
$367$	−3676.76	−	3676.76i	−0.522957	−	0.522957i	0.395506	−	0.918463i	$-0.370569\pi$
$367$	−3676.76	−	3676.76i	−0.522957	−	0.522957i	−0.918463	+	0.395506i	$0.870569\pi$
$368$	−3233.22	−	3783.19i	−0.457999	−	0.535903i
$369$		0			0
$370$	2252.40	+	138.587i	0.316478	+	0.0194724i
$371$			548.506i			0.0767575i
$372$		0			0
$373$	6801.10	+	6801.10i	0.944096	+	0.944096i	0.998518	−	0.0544219i	$-0.0173316\pi$
$373$	6801.10	+	6801.10i	0.944096	+	0.944096i	−0.0544219	+	0.998518i	$0.517332\pi$
$374$	1571.70	−	3593.98i	0.217301	−	0.496899i
$375$		0			0
$376$	3598.72	+	10369.9i	0.493590	+	1.42230i
$377$	2311.98	−	2311.98i	0.315844	−	0.315844i
$378$		0			0
$379$	−9992.48			−1.35430			−0.677150	−	0.735845i	$-0.736785\pi$
$379$	−9992.48			−1.35430			−0.677150	+	0.735845i	$0.736785\pi$
$380$	−2320.97	+	4527.52i	−0.313325	+	0.611202i
$381$		0			0
$382$	−9662.34	+	3782.65i	−1.29416	+	0.506642i
$383$	1910.84	−	1910.84i	0.254932	−	0.254932i	−0.568057	−	0.822989i	$-0.692305\pi$
$383$	1910.84	−	1910.84i	0.254932	−	0.254932i	0.822989	+	0.568057i	$0.192305\pi$
$384$		0			0
$385$	205.539	−	442.788i	0.0272084	−	0.0586144i
$386$	−1145.98	+	2620.49i	−0.151111	+	0.345543i
$387$		0			0
$388$	494.196	−	12629.2i	0.0646624	−	1.65244i
$389$			152.974i			0.0199385i	0.999950	+	0.00996927i	$0.00317337\pi$
$389$			152.974i			0.0199385i	−0.999950	+	0.00996927i	$0.996827\pi$
$390$		0			0
$391$		−	3988.91i		−	0.515929i
$392$	3359.66	−	6930.82i	0.432878	−	0.893009i
$393$		0			0
$394$	1000.42	+	437.500i	0.127920	+	0.0559415i
$395$	−584.585	+	213.923i	−0.0744649	+	0.0272497i
$396$		0			0
$397$	3823.37	−	3823.37i	0.483349	−	0.483349i	−0.422850	−	0.906199i	$-0.638970\pi$
$397$	3823.37	−	3823.37i	0.483349	−	0.483349i	0.906199	+	0.422850i	$0.138970\pi$
$398$	4695.74	+	11994.7i	0.591397	+	1.51065i
$399$		0			0
$400$	−4670.30	−	6495.26i	−0.583787	−	0.811907i
$401$	−10939.9			−1.36238			−0.681190	−	0.732106i	$-0.738537\pi$
$401$	−10939.9			−1.36238			−0.681190	+	0.732106i	$0.738537\pi$
$402$		0			0
$403$	7724.96	−	7724.96i	0.954858	−	0.954858i
$404$	4627.20	−	4278.70i	0.569831	−	0.526914i
$405$		0			0
$406$	239.056	+	104.543i	0.0292221	+	0.0127792i
$407$	−1364.20	−	1364.20i	−0.166145	−	0.166145i
$408$		0			0
$409$		−	2982.65i		−	0.360593i	−0.983612	−	0.180296i	$-0.942294\pi$
$409$		−	2982.65i		−	0.360593i	0.983612	−	0.180296i	$-0.0577057\pi$
$410$	−1498.00	−	1694.42i	−0.180441	−	0.204101i
$411$		0			0
$412$	231.125	−	5906.38i	0.0276376	−	0.706278i
$413$	−843.045	−	843.045i	−0.100444	−	0.100444i
$414$		0			0
$415$	−5715.93	+	12313.7i	−0.676106	+	1.45652i
$416$	−9140.67	−	4881.13i	−1.07730	−	0.575282i
$417$		0			0
$418$	4050.32	−	1585.64i	0.473942	−	0.185541i
$419$	−2828.22			−0.329755			−0.164878	−	0.986314i	$-0.552723\pi$
$419$	−2828.22			−0.329755			−0.164878	+	0.986314i	$0.552723\pi$
$420$		0			0
$421$	739.946			0.0856597			0.0428299	−	0.999082i	$-0.486363\pi$
$421$	739.946			0.0856597			0.0428299	+	0.999082i	$0.486363\pi$
$422$	−11196.0	+	4383.04i	−1.29150	+	0.505600i
$423$		0			0
$424$	−2519.48	−	7259.99i	−0.288577	−	0.831548i
$425$	536.660	−	6389.81i	0.0612514	−	0.729297i
$426$		0			0
$427$	214.645	+	214.645i	0.0243265	+	0.0243265i
$428$	6955.73	+	272.187i	0.785555	+	0.0307399i
$429$		0			0
$430$	2987.91	+	183.841i	0.335093	+	0.0206177i
$431$		−	7074.45i		−	0.790636i	−0.918544	−	0.395318i	$-0.870634\pi$
$431$		−	7074.45i		−	0.790636i	0.918544	−	0.395318i	$-0.129366\pi$
$432$		0			0
$433$	2645.06	+	2645.06i	0.293564	+	0.293564i	0.838487	−	0.544922i	$-0.183441\pi$
$433$	2645.06	+	2645.06i	0.293564	+	0.293564i	−0.544922	+	0.838487i	$0.683441\pi$
$434$	798.752	+	349.306i	0.0883441	+	0.0386341i
$435$		0			0
$436$	−2166.43	−	2342.89i	−0.237966	−	0.257349i
$437$	3127.64	−	3127.64i	0.342370	−	0.342370i
$438$		0			0
$439$	12903.4			1.40284			0.701419	−	0.712750i	$-0.252550\pi$
$439$	12903.4			1.40284			0.701419	+	0.712750i	$0.252550\pi$
$440$	−686.624	+	6804.82i	−0.0743944	+	0.737289i
$441$		0			0
$442$	−3027.83	−	7734.23i	−0.325835	−	0.832307i
$443$	2108.72	−	2108.72i	0.226159	−	0.226159i	−0.584927	−	0.811086i	$-0.698877\pi$
$443$	2108.72	−	2108.72i	0.226159	−	0.226159i	0.811086	+	0.584927i	$0.198877\pi$
$444$		0			0
$445$	609.707	+	1666.14i	0.0649503	+	0.177489i
$446$	−948.272	−	414.693i	−0.100677	−	0.0440275i
$447$		0			0
$448$	96.8022	−	821.223i	0.0102087	−	0.0866052i
$449$			136.127i			0.0143079i	0.999974	+	0.00715394i	$0.00227719\pi$
$449$			136.127i			0.0143079i	−0.999974	+	0.00715394i	$0.997723\pi$
$450$		0			0
$451$			1933.53i			0.201877i
$452$	7833.13	+	306.521i	0.815132	+	0.0318972i
$453$		0			0
$454$	3409.88	−	7797.31i	0.352497	−	0.806048i
$455$	−355.217	−	970.698i	−0.0365996	−	0.100015i
$456$		0			0
$457$	−6660.47	+	6660.47i	−0.681759	+	0.681759i	−0.960396	−	0.278638i	$-0.910117\pi$
$457$	−6660.47	+	6660.47i	−0.681759	+	0.681759i	0.278638	+	0.960396i	$0.410117\pi$
$458$	12260.3	−	4799.69i	1.25084	−	0.489683i
$459$		0			0
$460$	2132.88	+	6619.85i	0.216187	+	0.670983i
$461$	−9556.54			−0.965493			−0.482747	−	0.875760i	$-0.660361\pi$
$461$	−9556.54			−0.965493			−0.482747	+	0.875760i	$0.660361\pi$
$462$		0			0
$463$	914.613	−	914.613i	0.0918049	−	0.0918049i	−0.659713	−	0.751518i	$-0.729322\pi$
$463$	914.613	−	914.613i	0.0918049	−	0.0918049i	0.751518	+	0.659713i	$0.229322\pi$
$464$	−3644.33	−	285.653i	−0.364620	−	0.0285799i
$465$		0			0
$466$	−5437.22	+	12433.2i	−0.540503	+	1.23596i
$467$	−541.819	−	541.819i	−0.0536882	−	0.0536882i	0.679753	−	0.733441i	$-0.262087\pi$
$467$	−541.819	−	541.819i	−0.0536882	−	0.0536882i	−0.733441	+	0.679753i	$0.762087\pi$
$468$		0			0
$469$		−	1316.59i		−	0.129626i
$470$	942.075	−	15311.3i	0.0924568	−	1.50267i
$471$		0			0
$472$	15030.9	+	7286.09i	1.46579	+	0.710528i
$473$	−1809.67	−	1809.67i	−0.175917	−	0.175917i
$474$		0			0
$475$	5430.94	−	4589.36i	0.524607	−	0.443315i
$476$	486.637	−	449.986i	0.0468592	−	0.0433299i
$477$		0			0
$478$	−1478.91	−	3777.71i	−0.141514	−	0.361482i
$479$	−19357.9			−1.84652			−0.923260	−	0.384176i	$-0.874486\pi$
$479$	−19357.9			−1.84652			−0.923260	+	0.384176i	$0.874486\pi$
$480$		0			0
$481$	−4085.06			−0.387240
$482$	6638.77	+	16958.0i	0.627361	+	1.60252i
$483$		0			0
$484$	−3524.87	+	3259.39i	−0.331036	+	0.306104i
$485$	−7436.99	+	16021.3i	−0.696281	+	1.49998i
$486$		0			0
$487$	141.166	+	141.166i	0.0131352	+	0.0131352i	0.713644	−	0.700509i	$-0.247044\pi$
$487$	141.166	+	141.166i	0.0131352	+	0.0131352i	−0.700509	+	0.713644i	$0.747044\pi$
$488$	−3826.96	−	1855.09i	−0.354997	−	0.172082i
$489$		0			0
$490$	−8064.45	+	7129.59i	−0.743500	+	0.657310i
$491$		−	929.849i		−	0.0854654i	−0.999087	−	0.0427327i	$-0.986394\pi$
$491$		−	929.849i		−	0.0854654i	0.999087	−	0.0427327i	$-0.0136064\pi$
$492$		0			0
$493$	−2071.84	−	2071.84i	−0.189272	−	0.189272i
$494$	3690.22	−	8438.36i	0.336095	−	0.768542i
$495$		0			0
$496$	−12176.7	−	954.444i	−1.10232	−	0.0864028i
$497$	179.730	−	179.730i	0.0162213	−	0.0162213i
$498$		0			0
$499$	−13526.6			−1.21349			−0.606747	−	0.794895i	$-0.707526\pi$
$499$	−13526.6			−1.21349			−0.606747	+	0.794895i	$0.707526\pi$
$500$	2526.03	+	10891.2i	0.225935	+	0.974142i
$501$		0			0
$502$	4945.16	−	1935.95i	0.439668	−	0.172123i
$503$	−4770.81	+	4770.81i	−0.422902	+	0.422902i	−0.886202	−	0.463300i	$-0.846665\pi$
$503$	−4770.81	+	4770.81i	−0.422902	+	0.422902i	0.463300	+	0.886202i	$0.346665\pi$
$504$		0			0
$505$	−8271.23	+	3026.77i	−0.728842	+	0.266712i
$506$	2382.40	−	5447.80i	0.209310	−	0.478625i
$507$		0			0
$508$	5637.03	+	220.585i	0.492328	+	0.0192655i
$509$		−	8188.23i		−	0.713039i	−0.934288	−	0.356520i	$-0.883963\pi$
$509$		−	8188.23i		−	0.713039i	0.934288	−	0.356520i	$-0.116037\pi$
$510$		0			0
$511$		−	412.526i		−	0.0357125i
$512$	2490.89	+	11314.3i	0.215006	+	0.976613i
$513$		0			0
$514$	7111.09	+	3109.78i	0.610227	+	0.266861i
$515$	−3478.12	+	7492.83i	−0.297600	+	0.641113i
$516$		0			0
$517$	−9273.48	+	9273.48i	−0.788872	+	0.788872i
$518$	−118.836	−	303.554i	−0.0100799	−	0.0257479i
$519$		0			0
$520$	9160.38	+	11216.5i	0.772518	+	0.945912i
$521$	5465.70			0.459610			0.229805	−	0.973237i	$-0.426191\pi$
$521$	5465.70			0.459610			0.229805	+	0.973237i	$0.426191\pi$
$522$		0			0
$523$	−9877.72	+	9877.72i	−0.825856	+	0.825856i	−0.986941	−	0.161085i	$-0.948501\pi$
$523$	−9877.72	+	9877.72i	−0.825856	+	0.825856i	0.161085	+	0.986941i	$0.448501\pi$
$524$	9491.88	+	10265.0i	0.791326	+	0.855780i
$525$		0			0
$526$	3682.97	+	1610.62i	0.305295	+	0.133510i
$527$	−6922.60	−	6922.60i	−0.572207	−	0.572207i
$528$		0			0
$529$			6120.55i			0.503045i
$530$	−659.551	+	10719.5i	−0.0540549	+	0.878537i
$531$		0			0
$532$	734.391	+	28.7377i	0.0598494	+	0.00234199i
$533$	2894.96	+	2894.96i	0.235262	+	0.235262i
$534$		0			0
$535$	−8824.02	−	4096.05i	−0.713076	−	0.331005i
$536$	6047.57	+	17426.4i	0.487342	+	1.40430i
$537$		0			0
$538$	−17369.7	+	6799.95i	−1.39193	+	0.544920i
$539$	9202.49			0.735397
$540$		0			0
$541$	15069.4			1.19757			0.598784	−	0.800910i	$-0.295651\pi$
$541$	15069.4			1.19757			0.598784	+	0.800910i	$0.295651\pi$
$542$	12593.5	−	4930.13i	0.998035	−	0.390715i
$543$		0			0
$544$	−4374.15	+	8191.27i	−0.344743	+	0.645584i
$545$	1532.55	+	4187.97i	0.120453	+	0.329162i
$546$		0			0
$547$	−4573.04	−	4573.04i	−0.357457	−	0.357457i	0.505418	−	0.862875i	$-0.331338\pi$
$547$	−4573.04	−	4573.04i	−0.357457	−	0.357457i	−0.862875	+	0.505418i	$0.831338\pi$
$548$	54.9835	−	1405.10i	0.00428609	−	0.109531i
$549$		0			0
$550$	4549.29	−	8406.27i	0.352695	−	0.651717i
$551$		−	3249.00i		−	0.251202i
$552$		0			0
$553$	63.5850	+	63.5850i	0.00488953	+	0.00488953i
$554$	−20121.2	−	8799.29i	−1.54308	−	0.674812i
$555$		0			0
$556$	57.7899	−	53.4375i	0.00440798	−	0.00407599i
$557$	−1915.65	+	1915.65i	−0.145725	+	0.145725i	−0.776205	−	0.630480i	$-0.782858\pi$
$557$	−1915.65	+	1915.65i	−0.145725	+	0.145725i	0.630480	+	0.776205i	$0.282858\pi$
$558$		0			0
$559$	−5419.00			−0.410017
$560$	−566.996	+	1006.98i	−0.0427856	+	0.0759873i
$561$		0			0
$562$	3140.99	+	8023.30i	0.235756	+	0.602211i
$563$	1561.15	−	1561.15i	0.116864	−	0.116864i	−0.646256	−	0.763120i	$-0.723666\pi$
$563$	1561.15	−	1561.15i	0.116864	−	0.116864i	0.763120	+	0.646256i	$0.223666\pi$
$564$		0			0
$565$	−9937.10	−	4612.74i	−0.739924	−	0.343468i
$566$	21469.6	+	9388.98i	1.59441	+	0.697259i
$567$		0			0
$568$	−1553.33	+	3204.46i	−0.114747	+	0.236719i
$569$			21457.5i			1.58093i	0.612510	+	0.790463i	$0.290160\pi$
$569$			21457.5i			1.58093i	−0.612510	+	0.790463i	$0.709840\pi$
$570$		0			0
$571$			5553.22i			0.406997i	0.979075	+	0.203498i	$0.0652312\pi$
$571$			5553.22i			0.406997i	−0.979075	+	0.203498i	$0.934769\pi$
$572$	484.106	−	12371.3i	0.0353872	−	0.904318i
$573$		0			0
$574$	−130.904	+	299.335i	−0.00951884	+	0.0217666i
$575$	813.477	−	9685.77i	0.0589988	−	0.702477i
$576$		0			0
$577$	762.522	−	762.522i	0.0550160	−	0.0550160i	−0.679064	−	0.734079i	$-0.737614\pi$
$577$	762.522	−	762.522i	0.0550160	−	0.0550160i	0.734079	+	0.679064i	$0.237614\pi$
$578$	6008.91	−	2352.39i	0.432418	−	0.169285i
$579$		0			0
$580$	4546.17	+	2330.53i	0.325465	+	0.166845i
$581$	1961.07			0.140033
$582$		0			0
$583$	6492.40	−	6492.40i	0.461214	−	0.461214i
$584$	1894.87	+	5460.17i	0.134265	+	0.386889i
$585$		0			0
$586$	8608.32	−	19684.5i	0.606837	−	1.38764i
$587$	−15138.4	−	15138.4i	−1.06445	−	1.06445i	−0.997775	−	0.0666722i	$-0.978762\pi$
$587$	−15138.4	−	15138.4i	−1.06445	−	1.06445i	−0.0666722	−	0.997775i	$-0.521238\pi$
$588$		0			0
$589$		−	10855.8i		−	0.759432i
$590$	−15461.9	−	17489.4i	−1.07891	−	1.22038i
$591$		0			0
$592$	2967.24	+	3471.96i	0.206001	+	0.241042i
$593$	−1637.51	−	1637.51i	−0.113397	−	0.113397i	0.648131	−	0.761529i	$-0.275551\pi$
$593$	−1637.51	−	1637.51i	−0.113397	−	0.113397i	−0.761529	+	0.648131i	$0.775551\pi$
$594$		0			0
$595$	−869.876	+	318.322i	−0.0599352	+	0.0219327i
$596$	−15427.2	−	16683.8i	−1.06027	−	1.14663i
$597$		0			0
$598$	−4589.62	−	11723.7i	−0.313852	−	0.801699i
$599$	10193.4			0.695312			0.347656	−	0.937622i	$-0.386978\pi$
$599$	10193.4			0.695312			0.347656	+	0.937622i	$0.386978\pi$
$600$		0			0
$601$	−10688.6			−0.725454			−0.362727	−	0.931895i	$-0.618154\pi$
$601$	−10688.6			−0.725454			−0.362727	+	0.931895i	$0.618154\pi$
$602$	−157.642	−	402.677i	−0.0106727	−	0.0272623i
$603$		0			0
$604$	15824.4	+	17113.3i	1.06604	+	1.15287i
$605$	6300.80	−	2305.71i	0.423411	−	0.154943i
$606$		0			0
$607$	12050.0	+	12050.0i	0.805759	+	0.805759i	0.983989	−	0.178230i	$-0.0570372\pi$
$607$	12050.0	+	12050.0i	0.805759	+	0.805759i	−0.178230	+	0.983989i	$0.557037\pi$
$608$	−9852.35	+	2992.94i	−0.657180	+	0.199638i
$609$		0			0
$610$	3936.71	+	4452.91i	0.261300	+	0.295563i
$611$			27769.1i			1.83866i
$612$		0			0
$613$	3893.39	+	3893.39i	0.256529	+	0.256529i	0.823641	−	0.567112i	$-0.191939\pi$
$613$	3893.39	+	3893.39i	0.256529	+	0.256529i	−0.567112	+	0.823641i	$0.691939\pi$
$614$	2347.57	−	5368.15i	0.154300	−	0.352835i
$615$		0			0
$616$	933.374	−	323.914i	0.0610498	−	0.0211865i
$617$	−4210.58	+	4210.58i	−0.274735	+	0.274735i	−0.831003	−	0.556268i	$-0.812233\pi$
$617$	−4210.58	+	4210.58i	−0.274735	+	0.274735i	0.556268	+	0.831003i	$0.312233\pi$
$618$		0			0
$619$	6990.42			0.453907			0.226954	−	0.973906i	$-0.427123\pi$
$619$	6990.42			0.453907			0.226954	+	0.973906i	$0.427123\pi$
$620$	15190.0	+	7786.96i	0.983945	+	0.504406i
$621$		0			0
$622$	3637.86	−	1424.16i	0.234510	−	0.0918068i
$623$	181.225	−	181.225i	0.0116543	−	0.0116543i
$624$		0			0
$625$	2606.21	−	15406.1i	0.166797	−	0.985991i
$626$	−3189.30	+	7292.91i	−0.203626	+	0.465628i
$627$		0			0
$628$	−695.553	+	17774.8i	−0.0441968	+	1.12945i
$629$			3660.76i			0.232057i
$630$		0			0
$631$			16801.3i			1.05998i	0.848003	+	0.529991i	$0.177805\pi$
$631$			16801.3i			1.05998i	−0.848003	+	0.529991i	$0.822195\pi$
$632$	−1133.67	−	549.539i	−0.0713531	−	0.0345878i
$633$		0			0
$634$	−7618.67	−	3331.76i	−0.477249	−	0.208708i
$635$	−7151.13	−	3319.51i	−0.446904	−	0.207450i
$636$		0			0
$637$	13778.3	−	13778.3i	0.857010	−	0.857010i
$638$	−1592.17	−	4067.01i	−0.0988003	−	0.252374i
$639$		0			0
$640$	2879.29	−	15932.8i	0.177834	−	0.984060i
$641$	−7637.22			−0.470596			−0.235298	−	0.971923i	$-0.575607\pi$
$641$	−7637.22			−0.470596			−0.235298	+	0.971923i	$0.575607\pi$
$642$		0			0
$643$	13378.3	−	13378.3i	0.820509	−	0.820509i	−0.165672	−	0.986181i	$-0.552979\pi$
$643$	13378.3	−	13378.3i	0.820509	−	0.820509i	0.986181	+	0.165672i	$0.0529793\pi$
$644$	737.651	−	682.094i	0.0451359	−	0.0417365i
$645$		0			0
$646$	−7561.90	−	3306.93i	−0.460556	−	0.201408i
$647$	9497.02	+	9497.02i	0.577073	+	0.577073i	0.934096	−	0.357023i	$-0.116208\pi$
$647$	9497.02	+	9497.02i	0.577073	+	0.577073i	−0.357023	+	0.934096i	$0.616208\pi$
$648$		0			0
$649$			19957.4i			1.20708i
$650$	−5774.80	−	19397.5i	−0.348472	−	1.17051i
$651$		0			0
$652$	644.636	−	16473.6i	0.0387207	−	0.989506i
$653$	−12420.7	−	12420.7i	−0.744346	−	0.744346i	0.229065	−	0.973411i	$-0.426433\pi$
$653$	−12420.7	−	12420.7i	−0.744346	−	0.744346i	−0.973411	+	0.229065i	$0.926433\pi$
$654$		0			0
$655$	−6714.61	−	18348.9i	−0.400552	−	1.09458i
$656$	357.681	−	4563.26i	0.0212883	−	0.271594i
$657$		0			0
$658$	−2063.48	+	807.819i	−0.122254	+	0.0478603i
$659$	5899.85			0.348749			0.174374	−	0.984679i	$-0.444210\pi$
$659$	5899.85			0.348749			0.174374	+	0.984679i	$0.444210\pi$
$660$		0			0
$661$	25892.6			1.52361			0.761804	−	0.647808i	$-0.224314\pi$
$661$	25892.6			1.52361			0.761804	+	0.647808i	$0.224314\pi$
$662$	−22740.0	+	8902.36i	−1.33507	+	0.522658i
$663$		0			0
$664$	−25956.6	+	9007.88i	−1.51704	+	0.526466i
$665$	−931.647	−	432.464i	−0.0543274	−	0.0252184i
$666$		0			0
$667$	−3140.53	−	3140.53i	−0.182312	−	0.182312i
$668$	−9058.05	−	354.454i	−0.524650	−	0.0205303i
$669$		0			0
$670$	1583.14	−	25730.3i	0.0912865	−	1.48365i
$671$		−	5081.29i		−	0.292342i
$672$		0			0
$673$	−19481.3	−	19481.3i	−1.11582	−	1.11582i	−0.992348	−	0.123475i	$-0.960596\pi$
$673$	−19481.3	−	19481.3i	−1.11582	−	1.11582i	−0.123475	−	0.992348i	$-0.539404\pi$
$674$	11416.4	+	4992.55i	0.652437	+	0.285320i
$675$		0			0
$676$	−5865.28	−	6343.00i	−0.333709	−	0.360890i
$677$	17811.4	−	17811.4i	1.01115	−	1.01115i	0.0112114	−	0.999937i	$-0.496431\pi$
$677$	17811.4	−	17811.4i	1.01115	−	1.01115i	0.999937	−	0.0112114i	$-0.00356878\pi$
$678$		0			0
$679$	2551.55			0.144211
$680$	10051.4	−	8208.93i	0.566846	−	0.462938i
$681$		0			0
$682$	−5319.88	−	13589.0i	−0.298693	−	0.762976i
$683$	13579.6	−	13579.6i	0.760775	−	0.760775i	−0.215687	−	0.976463i	$-0.569199\pi$
$683$	13579.6	−	13579.6i	0.760775	−	0.760775i	0.976463	+	0.215687i	$0.0691991\pi$
$684$		0			0
$685$	−827.428	+	1782.51i	−0.0461524	+	0.0994249i
$686$	2860.24	+	1250.82i	0.159190	+	0.0696161i
$687$		0			0
$688$	3936.17	+	4605.70i	0.218118	+	0.255219i
$689$		−	19441.3i		−	1.07497i
$690$		0			0
$691$		−	12840.8i		−	0.706928i	−0.935448	−	0.353464i	$-0.885004\pi$
$691$		−	12840.8i		−	0.706928i	0.935448	−	0.353464i	$-0.114996\pi$
$692$	−13346.7	−	522.274i	−0.733186	−	0.0286906i
$693$		0			0
$694$	1036.92	−	2371.10i	0.0567159	−	0.129691i
$695$	−103.301	+	37.8019i	−0.00563803	+	0.00206318i
$696$		0			0
$697$	2594.27	−	2594.27i	0.140983	−	0.140983i
$698$	−25315.4	+	9910.56i	−1.37278	+	0.537422i
$699$		0			0
$700$	1273.41	−	993.400i	0.0687574	−	0.0536386i
$701$	23786.6			1.28161			0.640805	−	0.767704i	$-0.278601\pi$
$701$	23786.6			1.28161			0.640805	+	0.767704i	$0.278601\pi$
$702$		0			0
$703$	−2870.34	+	2870.34i	−0.153993	+	0.153993i
$704$	−10866.2	+	8574.62i	−0.581727	+	0.459045i
$705$		0			0
$706$	8150.01	−	18636.5i	0.434461	−	0.993476i
$707$	899.658	+	899.658i	0.0478573	+	0.0478573i
$708$		0			0
$709$			21270.0i			1.12667i	0.826227	+	0.563337i	$0.190483\pi$
$709$			21270.0i			1.12667i	−0.826227	+	0.563337i	$0.809517\pi$
$710$	3728.59	−	3296.36i	0.197087	−	0.174240i
$711$		0			0
$712$	−1566.25	+	3231.11i	−0.0824408	+	0.170072i
$713$	−10493.4	−	10493.4i	−0.551164	−	0.551164i
$714$		0			0
$715$	−7285.15	+	15694.2i	−0.381048	+	0.820882i
$716$	21876.1	−	20228.5i	1.14183	−	1.05583i
$717$		0			0
$718$	5772.09	+	14744.1i	0.300017	+	0.766359i
$719$	22553.8			1.16984			0.584921	−	0.811090i	$-0.301126\pi$
$719$	22553.8			1.16984			0.584921	+	0.811090i	$0.301126\pi$
$720$		0			0
$721$	1193.30			0.0616380
$722$	3735.97	+	9543.10i	0.192574	+	0.491908i
$723$		0			0
$724$	−3206.21	+	2964.73i	−0.164583	+	0.152187i
$725$	−4608.27	−	5453.31i	−0.236065	−	0.279353i
$726$		0			0
$727$	−19795.0	−	19795.0i	−1.00984	−	1.00984i	−0.999951	−	0.00989382i	$-0.996851\pi$
$727$	−19795.0	−	19795.0i	−1.00984	−	1.00984i	−0.00989382	−	0.999951i	$-0.503149\pi$
$728$	912.504	−	1882.46i	0.0464556	−	0.0958358i
$729$		0			0
$730$	496.042	−	8062.01i	0.0251498	−	0.408751i
$731$			4856.16i			0.245706i
$732$		0			0
$733$	10191.1	+	10191.1i	0.513528	+	0.513528i	0.915606	−	0.402078i	$-0.131712\pi$
$733$	10191.1	+	10191.1i	0.513528	+	0.513528i	−0.402078	+	0.915606i	$0.631712\pi$
$734$	−5892.76	+	13474.9i	−0.296329	+	0.677612i
$735$		0			0
$736$	−6630.40	+	12416.4i	−0.332065	+	0.621842i
$737$	−15583.9	+	15583.9i	−0.778888	+	0.778888i
$738$		0			0
$739$	28579.3			1.42261			0.711304	−	0.702885i	$-0.248105\pi$
$739$	28579.3			1.42261			0.711304	+	0.702885i	$0.248105\pi$
$740$	−1957.42	−	6075.26i	−0.0972379	−	0.301799i
$741$		0			0
$742$	1444.65	−	565.558i	0.0714755	−	0.0279815i
$743$	−20848.1	+	20848.1i	−1.02940	+	1.02940i	−0.0298445	+	0.999555i	$0.509501\pi$
$743$	−20848.1	+	20848.1i	−1.02940	+	1.02940i	−0.999555	+	0.0298445i	$0.990499\pi$
$744$		0			0
$745$	10913.3	+	29822.6i	0.536687	+	1.46660i
$746$	10900.2	−	24925.2i	0.534964	−	1.22329i
$747$		0			0
$748$	−11086.3	−	433.824i	−0.541921	−	0.0212061i
$749$			1405.31i			0.0685566i
$750$		0			0
$751$			13588.4i			0.660252i	0.943937	+	0.330126i	$0.107091\pi$
$751$			13588.4i			0.660252i	−0.943937	+	0.330126i	$0.892909\pi$
$752$	23601.5	−	20170.5i	1.14449	−	0.978115i
$753$		0			0
$754$	−8473.13	−	3705.42i	−0.409248	−	0.178970i
$755$	−11194.3	−	30590.5i	−0.539605	−	1.47457i
$756$		0			0
$757$	−1630.23	+	1630.23i	−0.0782719	+	0.0782719i	−0.745159	−	0.666887i	$-0.767626\pi$
$757$	−1630.23	+	1630.23i	−0.0782719	+	0.0782719i	0.666887	+	0.745159i	$0.267626\pi$
$758$	10303.1	+	26318.1i	0.493702	+	1.26110i
$759$		0			0
$760$	14317.7	+	1444.69i	0.683364	+	0.0689532i
$761$	33489.9			1.59528			0.797640	−	0.603134i	$-0.206081\pi$
$761$	33489.9			1.59528			0.797640	+	0.603134i	$0.206081\pi$
$762$		0			0
$763$	455.524	−	455.524i	0.0216135	−	0.0216135i
$764$	19925.4	+	21548.4i	0.943556	+	1.02041i
$765$		0			0
$766$	−7002.98	−	3062.51i	−0.330324	−	0.144455i
$767$	29881.0	+	29881.0i	1.40670	+	1.40670i
$768$		0			0
$769$		−	26755.6i		−	1.25466i	−0.778755	−	0.627328i	$-0.784149\pi$
$769$		−	26755.6i		−	1.25466i	0.778755	−	0.627328i	$-0.215851\pi$
$770$	−1378.14	−	84.7946i	−0.0644996	−	0.00396855i
$771$		0			0
$772$	8083.43	+	316.316i	0.376851	+	0.0147467i
$773$	6155.27	+	6155.27i	0.286403	+	0.286403i	0.835656	−	0.549253i	$-0.185088\pi$
$773$	6155.27	+	6155.27i	0.286403	+	0.286403i	−0.549253	+	0.835656i	$0.685088\pi$
$774$		0			0
$775$	−15397.5	−	18221.0i	−0.713670	−	0.844540i
$776$	−33772.1	+	11720.1i	−1.56231	+	0.542176i
$777$		0			0
$778$	402.902	−	157.729i	0.0185665	−	0.00726848i
$779$	4068.25			0.187112
$780$		0			0
$781$	−4254.76			−0.194939
$782$	−10506.0	+	4112.92i	−0.480425	+	0.188079i
$783$		0			0
$784$	−21718.4	−	1702.35i	−0.989361	−	0.0775489i
$785$	10467.1	−	22549.1i	0.475909	−	1.02524i
$786$		0			0
$787$	11524.6	+	11524.6i	0.521993	+	0.521993i	0.918173	−	0.396180i	$-0.129664\pi$
$787$	11524.6	+	11524.6i	0.521993	+	0.521993i	−0.396180	+	0.918173i	$0.629664\pi$
$788$	120.760	−	3086.01i	0.00545925	−	0.139511i
$789$		0			0
$790$	1166.19	+	1319.10i	0.0525203	+	0.0594070i
$791$			1582.58i			0.0711378i
$792$		0			0
$793$	−7607.89	−	7607.89i	−0.340686	−	0.340686i
$794$	−14012.2	−	6127.74i	−0.626290	−	0.273886i
$795$		0			0
$796$	26749.9	−	24735.2i	1.19111	−	1.10140i
$797$	−656.794	+	656.794i	−0.0291905	+	0.0291905i	−0.721551	−	0.692361i	$-0.756571\pi$
$797$	−656.794	+	656.794i	−0.0291905	+	0.0291905i	0.692361	+	0.721551i	$0.256571\pi$
$798$		0			0
$799$	24884.9			1.10183
$800$	−12291.7	+	18997.8i	−0.543220	+	0.839590i
$801$		0			0
$802$	11280.0	+	28813.5i	0.496648	+	1.26863i
$803$	−4882.87	+	4882.87i	−0.214586	+	0.214586i
$804$		0			0
$805$	−1318.57	+	482.517i	−0.0577310	+	0.0211261i
$806$	−28311.0	−	12380.8i	−1.23724	−	0.541062i
$807$		0			0
$808$	−16040.2	−	7775.37i	−0.698383	−	0.338535i
$809$			14179.8i			0.616238i	0.951348	+	0.308119i	$0.0996994\pi$
$809$			14179.8i			0.616238i	−0.951348	+	0.308119i	$0.900301\pi$
$810$		0			0
$811$			29366.2i			1.27150i	0.771896	+	0.635749i	$0.219309\pi$
$811$			29366.2i			1.27150i	−0.771896	+	0.635749i	$0.780691\pi$
$812$	28.8561	−	737.417i	0.00124711	−	0.0318698i
$813$		0			0
$814$	−2186.41	+	4999.63i	−0.0941445	+	0.215279i
$815$	−9700.92	+	20898.4i	−0.416943	+	0.898209i
$816$		0			0
$817$	−3807.63	+	3807.63i	−0.163050	+	0.163050i
$818$	−7855.67	+	3075.37i	−0.335779	+	0.131452i
$819$		0			0
$820$	−2918.19	+	5692.51i	−0.124278	+	0.242428i
$821$	−8845.67			−0.376025			−0.188012	−	0.982167i	$-0.560204\pi$
$821$	−8845.67			−0.376025			−0.188012	+	0.982167i	$0.560204\pi$
$822$		0			0
$823$	16257.0	−	16257.0i	0.688559	−	0.688559i	−0.273355	−	0.961913i	$-0.588133\pi$
$823$	16257.0	−	16257.0i	0.688559	−	0.688559i	0.961913	+	0.273355i	$0.0881332\pi$
$824$	−15794.5	+	5481.26i	−0.667751	+	0.231734i
$825$		0			0
$826$	−1351.15	+	3089.66i	−0.0569160	+	0.130149i
$827$	19183.3	+	19183.3i	0.806612	+	0.806612i	0.984119	−	0.177507i	$-0.0568034\pi$
$827$	19183.3	+	19183.3i	0.806612	+	0.806612i	−0.177507	+	0.984119i	$0.556803\pi$
$828$		0			0
$829$		−	27097.5i		−	1.13527i	−0.823282	−	0.567633i	$-0.807859\pi$
$829$		−	27097.5i		−	1.13527i	0.823282	−	0.567633i	$-0.192141\pi$
$830$	38325.3	+	2358.09i	1.60276	+	0.0986151i
$831$		0			0
$832$	−3431.07	+	29107.5i	−0.142970	+	1.21289i
$833$	−12347.2	−	12347.2i	−0.513571	−	0.513571i
$834$		0			0
$835$	11491.0	+	5334.06i	0.476244	+	0.221069i
$836$	−8352.47	−	9032.78i	−0.345546	−	0.373691i
$837$		0			0
$838$	2916.14	+	7448.93i	0.120210	+	0.307063i
$839$	13726.1			0.564814			0.282407	−	0.959295i	$-0.408867\pi$
$839$	13726.1			0.564814			0.282407	+	0.959295i	$0.408867\pi$
$840$		0			0
$841$	21126.6			0.866235
$842$	−762.948	−	1948.86i	−0.0312268	−	0.0797651i
$843$		0			0
$844$	23088.0	+	24968.6i	0.941615	+	1.01831i
$845$	4149.13	+	11338.3i	0.168916	+	0.461596i
$846$		0			0
$847$	−685.334	−	685.334i	−0.0278021	−	0.0278021i
$848$	−16523.5	+	14121.5i	−0.669127	+	0.571855i
$849$		0			0
$850$	−17382.8	+	5175.00i	−0.701440	+	0.208825i
$851$			5549.03i			0.223523i
$852$		0			0
$853$	−20720.5	−	20720.5i	−0.831720	−	0.831720i	0.156032	−	0.987752i	$-0.450130\pi$
$853$	−20720.5	−	20720.5i	−0.831720	−	0.831720i	−0.987752	+	0.156032i	$0.950130\pi$
$854$	344.012	−	786.648i	0.0137844	−	0.0315205i
$855$		0			0
$856$	−6455.07	−	18600.6i	−0.257745	−	0.742704i
$857$	−26011.0	+	26011.0i	−1.03678	+	1.03678i	−0.0374797	+	0.999297i	$0.511933\pi$
$857$	−26011.0	+	26011.0i	−1.03678	+	1.03678i	−0.999297	+	0.0374797i	$0.988067\pi$
$858$		0			0
$859$	4857.84			0.192954			0.0964769	−	0.995335i	$-0.469243\pi$
$859$	4857.84			0.192954			0.0964769	+	0.995335i	$0.469243\pi$
$860$	−2596.60	−	8059.09i	−0.102957	−	0.319550i
$861$		0			0
$862$	−18632.6	+	7294.37i	−0.736229	+	0.288222i
$863$	23423.0	−	23423.0i	0.923904	−	0.923904i	−0.0733988	−	0.997303i	$-0.523385\pi$
$863$	23423.0	−	23423.0i	0.923904	−	0.923904i	0.997303	+	0.0733988i	$0.0233846\pi$
$864$		0			0
$865$	16931.6	+	7859.52i	0.665539	+	0.308939i
$866$	4239.25	−	9693.81i	0.166346	−	0.380380i
$867$		0			0
$868$	96.4163	−	2463.91i	0.00377025	−	0.0963486i
$869$		−	1505.25i		−	0.0587596i
$870$		0			0
$871$			46665.5i			1.81539i
$872$	−3936.90	+	8121.66i	−0.152890	+	0.315406i
$873$		0			0
$874$	−11462.4	−	5012.69i	−0.443619	−	0.194001i
$875$	−2177.12	+	595.543i	−0.0841145	+	0.0230092i
$876$		0			0
$877$	−19062.6	+	19062.6i	−0.733976	+	0.733976i	−0.971405	−	0.237429i	$-0.923695\pi$
$877$	−19062.6	+	19062.6i	−0.733976	+	0.733976i	0.237429	+	0.971405i	$0.423695\pi$
$878$	−13304.5	−	33984.9i	−0.511396	−	1.30630i
$879$		0			0
$880$	18630.4	−	5207.94i	0.713673	−	0.199499i
$881$	6538.78			0.250053			0.125027	−	0.992153i	$-0.460098\pi$
$881$	6538.78			0.250053			0.125027	+	0.992153i	$0.460098\pi$
$882$		0			0
$883$	−23546.4	+	23546.4i	−0.897395	+	0.897395i	−0.995205	−	0.0978097i	$-0.968816\pi$
$883$	−23546.4	+	23546.4i	−0.897395	+	0.897395i	0.0978097	+	0.995205i	$0.468816\pi$
$884$	−17248.4	+	15949.3i	−0.656252	+	0.606826i
$885$		0			0
$886$	−7728.22	−	3379.66i	−0.293041	−	0.128151i
$887$	−13448.5	−	13448.5i	−0.509082	−	0.509082i	0.405162	−	0.914245i	$-0.367215\pi$
$887$	−13448.5	−	13448.5i	−0.509082	−	0.509082i	−0.914245	+	0.405162i	$0.867215\pi$
$888$		0			0
$889$			1138.89i			0.0429663i
$890$	3759.60	−	3323.78i	0.141598	−	0.125183i
$891$		0			0
$892$	−114.465	+	2925.13i	−0.00429659	+	0.109799i
$893$	19511.8	+	19511.8i	0.731175	+	0.731175i
$894$		0			0
$895$	−39104.1	+	14309.8i	−1.46045	+	0.534439i
$896$	−2262.74	+	591.795i	−0.0843671	+	0.0220653i
$897$		0			0
$898$	358.531	−	140.359i	0.0133233	−	0.00521585i
$899$	−10900.5			−0.404397
$900$		0			0
$901$	−17422.0			−0.644186
$902$	5092.53	−	1993.64i	0.187985	−	0.0735932i
$903$		0			0
$904$	−7269.33	−	20946.9i	−0.267449	−	0.770667i
$905$	5731.17	−	2097.27i	0.210509	−	0.0770337i
$906$		0			0
$907$	−200.288	−	200.288i	−0.00733237	−	0.00733237i	0.703431	−	0.710763i	$-0.251650\pi$
$907$	−200.288	−	200.288i	−0.00733237	−	0.00733237i	−0.710763	+	0.703431i	$0.751650\pi$
$908$	−24052.4	−	941.202i	−0.879081	−	0.0343997i
$909$		0			0
$910$	−2190.36	+	1936.44i	−0.0797908	+	0.0705411i
$911$		−	7662.84i		−	0.278684i	−0.990244	−	0.139342i	$-0.955501\pi$
$911$		−	7662.84i		−	0.278684i	0.990244	−	0.139342i	$-0.0444987\pi$
$912$		0			0
$913$	−23212.3	−	23212.3i	−0.841417	−	0.841417i
$914$	24409.8	+	10674.8i	0.883375	+	0.386313i
$915$		0			0
$916$	−25282.8	−	27342.1i	−0.911973	−	0.986253i
$917$	−1995.81	+	1995.81i	−0.0718727	+	0.0718727i
$918$		0			0
$919$	17176.1			0.616527			0.308263	−	0.951301i	$-0.400252\pi$
$919$	17176.1			0.616527			0.308263	+	0.951301i	$0.400252\pi$
$920$	15236.1	−	12443.2i	0.546000	−	0.445914i
$921$		0			0
$922$	9853.62	+	25169.9i	0.351965	+	0.899054i
$923$	−6370.38	+	6370.38i	−0.227176	+	0.227176i
$924$		0			0
$925$	−746.555	+	8888.95i	−0.0265368	+	0.315964i
$926$	−3351.95	−	1465.85i	−0.118954	−	0.0520205i
$927$		0			0
$928$	3005.27	+	9892.94i	0.106307	+	0.349948i
$929$		−	27097.0i		−	0.956968i	−0.878096	−	0.478484i	$-0.841186\pi$
$929$		−	27097.0i		−	0.956968i	0.878096	−	0.478484i	$-0.158814\pi$
$930$		0			0
$931$		−	19362.5i		−	0.681611i
$932$	38352.7	+	1500.80i	1.34795	+	0.0527470i
$933$		0			0
$934$	−868.376	+	1985.70i	−0.0304220	+	0.0695654i
$935$	14064.1	+	6528.47i	0.491921	+	0.228346i
$936$		0			0
$937$	−4000.15	+	4000.15i	−0.139465	+	0.139465i	−0.773393	−	0.633927i	$-0.781442\pi$
$937$	−4000.15	+	4000.15i	−0.139465	+	0.139465i	0.633927	+	0.773393i	$0.281442\pi$
$938$	−3467.64	+	1357.52i	−0.120706	+	0.0472545i
$939$		0			0
$940$	−41298.0	+	13306.0i	−1.43297	+	0.461696i
$941$	−37957.6			−1.31496			−0.657482	−	0.753470i	$-0.728379\pi$
$941$	−37957.6			−1.31496			−0.657482	+	0.753470i	$0.728379\pi$
$942$		0			0
$943$	3932.43	−	3932.43i	0.135798	−	0.135798i
$944$	3691.89	−	47100.8i	0.127289	−	1.62394i
$945$		0			0
$946$	−2900.37	+	6632.22i	−0.0996819	+	0.227941i
$947$	−13560.7	−	13560.7i	−0.465327	−	0.465327i	0.435070	−	0.900397i	$-0.356724\pi$
$947$	−13560.7	−	13560.7i	−0.465327	−	0.465327i	−0.900397	+	0.435070i	$0.856724\pi$
$948$		0			0
$949$			14621.6i			0.500145i
$950$	−17687.2	−	9571.92i	−0.604051	−	0.326899i
$951$		0			0
$952$	−1686.93	−	817.726i	−0.0574305	−	0.0278389i
$953$	−15366.0	−	15366.0i	−0.522302	−	0.522302i	0.395964	−	0.918266i	$-0.370411\pi$
$953$	−15366.0	−	15366.0i	−0.522302	−	0.522302i	−0.918266	+	0.395964i	$0.870411\pi$
$954$		0			0
$955$	−14095.4	−	38518.2i	−0.477607	−	1.30515i
$956$	−8424.81	+	7790.29i	−0.285019	+	0.263552i
$957$		0			0
$958$	19959.6	+	50984.6i	0.673138	+	1.71945i
$959$	283.881			0.00955892
$960$		0			0
$961$	−6630.66			−0.222573
$962$	4212.05	+	10759.2i	0.141166	+	0.360593i
$963$		0			0
$964$	37818.6	−	34970.3i	1.26354	−	1.16838i
$965$	−10254.6	−	4760.13i	−0.342081	−	0.158792i
$966$		0			0
$967$	−14792.8	−	14792.8i	−0.491939	−	0.491939i	0.416978	−	0.908917i	$-0.363089\pi$
$967$	−14792.8	−	14792.8i	−0.491939	−	0.491939i	−0.908917	+	0.416978i	$0.863089\pi$
$968$	12219.0	+	5923.06i	0.405717	+	0.196668i
$969$		0			0
$970$	49865.0	+	3068.11i	1.65059	+	0.101558i
$971$		−	14139.9i		−	0.467322i	−0.972318	−	0.233661i	$-0.924929\pi$
$971$		−	14139.9i		−	0.467322i	0.972318	−	0.233661i	$-0.0750706\pi$
$972$		0			0
$973$	11.2360	+	11.2360i	0.000370205	+	0.000370205i
$974$	226.247	−	517.355i	0.00744294	−	0.0170196i
$975$		0			0
$976$	−939.980	+	11992.2i	−0.0308279	+	0.393299i
$977$	−2288.77	+	2288.77i	−0.0749481	+	0.0749481i	−0.743587	−	0.668639i	$-0.766877\pi$
$977$	−2288.77	+	2288.77i	−0.0749481	+	0.0749481i	0.668639	+	0.743587i	$0.266877\pi$
$978$		0			0
$979$	−4290.15			−0.140055
$980$	27093.0	+	13888.9i	0.883117	+	0.452718i
$981$		0			0
$982$	−2449.03	+	958.755i	−0.0795842	+	0.0311559i
$983$	−6757.28	+	6757.28i	−0.219251	+	0.219251i	−0.808183	−	0.588932i	$-0.799549\pi$
$983$	−6757.28	+	6757.28i	−0.219251	+	0.219251i	0.588932	+	0.808183i	$0.299549\pi$
$984$		0			0
$985$	−1817.27	+	3914.91i	−0.0587849	+	0.126639i
$986$	−3320.56	+	7593.06i	−0.107250	+	0.245246i
$987$		0			0
$988$	−26029.8	−	1018.58i	−0.838177	−	0.0327990i
$989$			7361.03i			0.236670i
$990$		0			0
$991$		−	18118.1i		−	0.580767i	−0.956910	−	0.290384i	$-0.906217\pi$
$991$		−	18118.1i		−	0.580767i	0.956910	−	0.290384i	$-0.0937829\pi$
$992$	10041.4	+	33055.0i	0.321387	+	1.05796i
$993$		0			0
$994$	−658.690	−	288.055i	−0.0210185	−	0.00919169i
$995$	−47816.1	+	17497.8i	−1.52349	+	0.557505i
$996$		0			0
$997$	18211.2	−	18211.2i	0.578491	−	0.578491i	−0.355996	−	0.934487i	$-0.615858\pi$
$997$	18211.2	−	18211.2i	0.578491	−	0.578491i	0.934487	+	0.355996i	$0.115858\pi$
$998$	13947.1	+	35626.2i	0.442372	+	1.12999i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.4.k.e.163.2		12
3.2	odd	2		20.4.e.b.3.5	yes	12
4.3	odd	2	inner	180.4.k.e.163.5		12
5.2	odd	4	inner	180.4.k.e.127.5		12
12.11	even	2		20.4.e.b.3.2	✓	12
15.2	even	4		20.4.e.b.7.2	yes	12
15.8	even	4		100.4.e.e.7.5		12
15.14	odd	2		100.4.e.e.43.2		12
20.7	even	4	inner	180.4.k.e.127.2		12
24.5	odd	2		320.4.n.k.63.1		12
24.11	even	2		320.4.n.k.63.6		12
60.23	odd	4		100.4.e.e.7.2		12
60.47	odd	4		20.4.e.b.7.5	yes	12
60.59	even	2		100.4.e.e.43.5		12
120.77	even	4		320.4.n.k.127.6		12
120.107	odd	4		320.4.n.k.127.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.4.e.b.3.2	✓	12	12.11	even	2
20.4.e.b.3.5	yes	12	3.2	odd	2
20.4.e.b.7.2	yes	12	15.2	even	4
20.4.e.b.7.5	yes	12	60.47	odd	4
100.4.e.e.7.2		12	60.23	odd	4
100.4.e.e.7.5		12	15.8	even	4
100.4.e.e.43.2		12	15.14	odd	2
100.4.e.e.43.5		12	60.59	even	2
180.4.k.e.127.2		12	20.7	even	4	inner
180.4.k.e.127.5		12	5.2	odd	4	inner
180.4.k.e.163.2		12	1.1	even	1	trivial
180.4.k.e.163.5		12	4.3	odd	2	inner
320.4.n.k.63.1		12	24.5	odd	2
320.4.n.k.63.6		12	24.11	even	2
320.4.n.k.127.1		12	120.107	odd	4
320.4.n.k.127.6		12	120.77	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	180.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.03109 - 2.63379i) q^{2} +(-5.87372 + 5.43134i) q^{4} +(10.4994 - 3.84216i) q^{5} +(-1.14202 - 1.14202i) q^{7} +(20.3613 + 9.86997i) q^{8} +(-20.9453 - 23.6917i) q^{10} +27.0350i q^{11} +(40.4777 + 40.4777i) q^{13} +(-1.83032 + 4.18535i) q^{14} +(5.00116 - 63.8043i) q^{16} +(36.2735 - 36.2735i) q^{17} +56.8829 q^{19} +(-40.8026 + 79.5936i) q^{20} +(71.2046 - 27.8754i) q^{22} +(54.9839 - 54.9839i) q^{23} +(95.4757 - 80.6808i) q^{25} +(64.8739 - 148.346i) q^{26} +(12.9106 + 0.505208i) q^{28} -57.1173i q^{29} -190.845i q^{31} +(-173.204 + 52.6158i) q^{32} +(-132.938 - 58.1357i) q^{34} +(-16.3783 - 7.60271i) q^{35} +(-50.4605 + 50.4605i) q^{37} +(-58.6513 - 149.818i) q^{38} +(251.704 + 25.3976i) q^{40} +71.5197 q^{41} +(-66.9381 + 66.9381i) q^{43} +(-146.836 - 158.796i) q^{44} +(-201.509 - 88.1229i) q^{46} +(343.017 + 343.017i) q^{47} -340.392i q^{49} +(-310.940 - 168.274i) q^{50} +(-457.603 - 17.9066i) q^{52} +(-240.148 - 240.148i) q^{53} +(103.873 + 283.852i) q^{55} +(-11.9813 - 34.5247i) q^{56} +(-150.435 + 58.8929i) q^{58} +738.207 q^{59} -187.952 q^{61} +(-502.645 + 196.777i) q^{62} +(317.167 + 401.932i) q^{64} +(580.515 + 269.471i) q^{65} +(576.434 + 576.434i) q^{67} +(-16.0468 + 410.074i) q^{68} +(-3.13647 + 50.9761i) q^{70} +157.380i q^{71} +(180.613 + 180.613i) q^{73} +(184.932 + 80.8733i) q^{74} +(-334.114 + 308.950i) q^{76} +(30.8744 - 30.8744i) q^{77} -55.6778 q^{79} +(-192.637 - 689.123i) q^{80} +(-73.7430 - 188.368i) q^{82} +(-858.601 + 858.601i) q^{83} +(241.482 - 520.219i) q^{85} +(245.320 + 107.282i) q^{86} +(-266.835 + 550.468i) q^{88} +158.689i q^{89} -92.4525i q^{91} +(-24.3239 + 621.596i) q^{92} +(549.756 - 1257.12i) q^{94} +(597.238 - 218.553i) q^{95} +(-1117.12 + 1117.12i) q^{97} +(-896.521 + 350.973i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} + 12 q^{8} - 110 q^{10} + 116 q^{13} + 312 q^{16} + 332 q^{17} - 140 q^{20} + 360 q^{22} + 340 q^{25} + 164 q^{26} - 880 q^{28} + 376 q^{32} + 508 q^{37} - 1600 q^{38} + 1420 q^{40} + 656 q^{41}+ \cdots - 1698 q^{98}+O(q^{100}) \) 12 * q + 6 * q^2 + 12 * q^8 - 110 * q^10 + 116 * q^13 + 312 * q^16 + 332 * q^17 - 140 * q^20 + 360 * q^22 + 340 * q^25 + 164 * q^26 - 880 * q^28 + 376 * q^32 + 508 * q^37 - 1600 * q^38 + 1420 * q^40 + 656 * q^41 - 1432 * q^46 + 1570 * q^50 - 932 * q^52 + 644 * q^53 - 2048 * q^56 + 1576 * q^58 - 896 * q^61 - 2440 * q^62 + 2740 * q^65 + 844 * q^68 - 3040 * q^70 + 1436 * q^73 + 800 * q^76 - 3120 * q^77 - 1840 * q^80 - 1352 * q^82 + 500 * q^85 + 2552 * q^86 - 2400 * q^88 + 1840 * q^92 - 4772 * q^97 - 1698 * q^98

Introduction

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