LMFDB - Embedded newform 1008.4.bt.d.17.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(17,1008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1008.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.bt (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.11
Character	$\chi$	$=$	1008.17
Dual form			1008.4.bt.d.593.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.56246 - 2.70625i) q^{5} +(-17.1949 + 6.88015i) q^{7} +O(q^{10})$	$q+(-1.56246 - 2.70625i) q^{5} +(-17.1949 + 6.88015i) q^{7} +(-33.2914 - 19.2208i) q^{11} +13.8046i q^{13} +(-47.5817 + 82.4140i) q^{17} +(-9.86718 + 5.69682i) q^{19} +(-23.9992 + 13.8559i) q^{23} +(57.6175 - 99.7964i) q^{25} -44.2317i q^{29} +(119.149 + 68.7907i) q^{31} +(45.4857 + 35.7837i) q^{35} +(70.6631 + 122.392i) q^{37} +337.946 q^{41} -417.510 q^{43} +(-145.043 - 251.222i) q^{47} +(248.327 - 236.607i) q^{49} +(-14.7904 - 8.53925i) q^{53} +120.127i q^{55} +(299.818 - 519.300i) q^{59} +(459.882 - 265.513i) q^{61} +(37.3586 - 21.5690i) q^{65} +(-325.105 + 563.098i) q^{67} -934.128i q^{71} +(787.145 + 454.458i) q^{73} +(704.683 + 101.449i) q^{77} +(397.516 + 688.517i) q^{79} -314.865 q^{83} +297.378 q^{85} +(179.521 + 310.940i) q^{89} +(-94.9775 - 237.368i) q^{91} +(30.8341 + 17.8021i) q^{95} +80.5572i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 24 q^{7}+O(q^{10}) $ 48 * q + 24 * q^7	$ 48 q + 24 q^{7} - 540 q^{19} - 924 q^{25} - 648 q^{31} - 132 q^{37} + 792 q^{43} + 672 q^{49} + 12 q^{67} + 2412 q^{73} - 1680 q^{79} + 480 q^{85} - 1404 q^{91}+O(q^{100}) $ 48 * q + 24 * q^7 - 540 * q^19 - 924 * q^25 - 648 * q^31 - 132 * q^37 + 792 * q^43 + 672 * q^49 + 12 * q^67 + 2412 * q^73 - 1680 * q^79 + 480 * q^85 - 1404 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.56246	−	2.70625i	−0.139750	−	0.242055i	0.787652	−	0.616121i	$-0.211297\pi$
$5$	−1.56246	−	2.70625i	−0.139750	−	0.242055i	−0.927402	+	0.374066i	$0.877963\pi$
$6$		0			0
$7$	−17.1949	+	6.88015i	−0.928436	+	0.371493i
$7$	−17.1949	+	6.88015i	−0.928436	+	0.371493i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−33.2914	−	19.2208i	−0.912521	−	0.526844i	−0.0312793	−	0.999511i	$-0.509958\pi$
$11$	−33.2914	−	19.2208i	−0.912521	−	0.526844i	−0.881241	+	0.472667i	$0.843291\pi$
$12$		0			0
$13$			13.8046i			0.294515i	0.989098	+	0.147258i	$0.0470446\pi$
$13$			13.8046i			0.294515i	−0.989098	+	0.147258i	$0.952955\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−47.5817	+	82.4140i	−0.678839	+	1.17578i	0.296491	+	0.955036i	$0.404183\pi$
$17$	−47.5817	+	82.4140i	−0.678839	+	1.17578i	−0.975331	+	0.220749i	$0.929150\pi$
$18$		0			0
$19$	−9.86718	+	5.69682i	−0.119141	+	0.0687863i	−0.558386	−	0.829581i	$-0.688579\pi$
$19$	−9.86718	+	5.69682i	−0.119141	+	0.0687863i	0.439245	+	0.898367i	$0.355246\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−23.9992	+	13.8559i	−0.217573	+	0.125616i	−0.604826	−	0.796358i	$-0.706757\pi$
$23$	−23.9992	+	13.8559i	−0.217573	+	0.125616i	0.387253	+	0.921973i	$0.373424\pi$
$24$		0			0
$25$	57.6175	−	99.7964i	0.460940	−	0.798371i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	44.2317i		−	0.283228i	−0.989922	−	0.141614i	$-0.954771\pi$
$29$		−	44.2317i		−	0.283228i	0.989922	−	0.141614i	$-0.0452293\pi$
$30$		0			0
$31$	119.149	+	68.7907i	0.690316	+	0.398554i	0.803730	−	0.594994i	$-0.202845\pi$
$31$	119.149	+	68.7907i	0.690316	+	0.398554i	−0.113414	+	0.993548i	$0.536179\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	45.4857	+	35.7837i	0.219671	+	0.172816i
$36$		0			0
$37$	70.6631	+	122.392i	0.313972	+	0.543815i	0.979218	−	0.202809i	$-0.0650070\pi$
$37$	70.6631	+	122.392i	0.313972	+	0.543815i	−0.665247	+	0.746624i	$0.731674\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	337.946			1.28728			0.643639	−	0.765330i	$-0.277424\pi$
$41$	337.946			1.28728			0.643639	+	0.765330i	$0.277424\pi$
$42$		0			0
$43$	−417.510			−1.48069			−0.740346	−	0.672226i	$-0.765338\pi$
$43$	−417.510			−1.48069			−0.740346	+	0.672226i	$0.765338\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−145.043	−	251.222i	−0.450142	−	0.779670i	0.548252	−	0.836313i	$-0.315293\pi$
$47$	−145.043	−	251.222i	−0.450142	−	0.779670i	−0.998394	+	0.0566436i	$0.981960\pi$
$48$		0			0
$49$	248.327	−	236.607i	0.723985	−	0.689815i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−14.7904	−	8.53925i	−0.0383324	−	0.0221312i	0.480711	−	0.876879i	$-0.340379\pi$
$53$	−14.7904	−	8.53925i	−0.0383324	−	0.0221312i	−0.519044	+	0.854748i	$0.673712\pi$
$54$		0			0
$55$			120.127i			0.294507i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	299.818	−	519.300i	0.661576	−	1.14588i	−0.318625	−	0.947881i	$-0.603221\pi$
$59$	299.818	−	519.300i	0.661576	−	1.14588i	0.980201	−	0.198003i	$-0.0634455\pi$
$60$		0			0
$61$	459.882	−	265.513i	0.965276	−	0.557302i	0.0674831	−	0.997720i	$-0.478503\pi$
$61$	459.882	−	265.513i	0.965276	−	0.557302i	0.897793	+	0.440418i	$0.145170\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	37.3586	−	21.5690i	0.0712888	−	0.0411586i
$66$		0			0
$67$	−325.105	+	563.098i	−0.592804	+	1.02677i	0.401048	+	0.916057i	$0.368646\pi$
$67$	−325.105	+	563.098i	−0.592804	+	1.02677i	−0.993853	+	0.110710i	$0.964687\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	934.128i		−	1.56142i	−0.624896	−	0.780708i	$-0.714858\pi$
$71$		−	934.128i		−	1.56142i	0.624896	−	0.780708i	$-0.285142\pi$
$72$		0			0
$73$	787.145	+	454.458i	1.26203	+	0.728634i	0.973467	−	0.228826i	$-0.0734887\pi$
$73$	787.145	+	454.458i	1.26203	+	0.728634i	0.288564	+	0.957460i	$0.406822\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	704.683	+	101.449i	1.04294	+	0.150145i
$78$		0			0
$79$	397.516	+	688.517i	0.566127	+	0.980560i	0.996944	+	0.0781211i	$0.0248921\pi$
$79$	397.516	+	688.517i	0.566127	+	0.980560i	−0.430817	+	0.902439i	$0.641775\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−314.865			−0.416397			−0.208199	−	0.978087i	$-0.566760\pi$
$83$	−314.865			−0.416397			−0.208199	+	0.978087i	$0.566760\pi$
$84$		0			0
$85$	297.378			0.379472
$86$		0			0
$87$		0			0
$88$		0			0
$89$	179.521	+	310.940i	0.213811	+	0.370332i	0.952904	−	0.303271i	$-0.0980789\pi$
$89$	179.521	+	310.940i	0.213811	+	0.370332i	−0.739093	+	0.673604i	$0.764746\pi$
$90$		0			0
$91$	−94.9775	−	237.368i	−0.109410	−	0.273438i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	30.8341	+	17.8021i	0.0333001	+	0.0192258i
$96$		0			0
$97$			80.5572i			0.0843231i	0.999111	+	0.0421616i	$0.0134244\pi$
$97$			80.5572i			0.0843231i	−0.999111	+	0.0421616i	$0.986576\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	434.184	−	752.028i	0.427752	−	0.740887i	−0.568921	−	0.822392i	$-0.692639\pi$
$101$	434.184	−	752.028i	0.427752	−	0.740887i	0.996673	+	0.0815045i	$0.0259725\pi$
$102$		0			0
$103$	−104.861	+	60.5413i	−0.100313	+	0.0579157i	−0.549317	−	0.835614i	$-0.685112\pi$
$103$	−104.861	+	60.5413i	−0.100313	+	0.0579157i	0.449004	+	0.893530i	$0.351779\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	387.751	−	223.868i	0.350330	−	0.202263i	−0.314501	−	0.949257i	$-0.601837\pi$
$107$	387.751	−	223.868i	0.350330	−	0.202263i	0.664830	+	0.746994i	$0.268504\pi$
$108$		0			0
$109$	507.544	−	879.092i	0.445999	−	0.772493i	−0.552122	−	0.833763i	$-0.686182\pi$
$109$	507.544	−	879.092i	0.445999	−	0.772493i	0.998121	+	0.0612702i	$0.0195151\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	889.812i		−	0.740765i	−0.928879	−	0.370382i	$-0.879227\pi$
$113$		−	889.812i		−	0.740765i	0.928879	−	0.370382i	$-0.120773\pi$
$114$		0			0
$115$	74.9954	+	43.2986i	0.0608118	+	0.0351097i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	251.141	−	1744.47i	0.193463	−	1.34382i
$120$		0			0
$121$	73.3769	+	127.093i	0.0551292	+	0.0954865i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−750.713			−0.537167
$126$		0			0
$127$	−279.668			−0.195406			−0.0977029	−	0.995216i	$-0.531149\pi$
$127$	−279.668			−0.195406			−0.0977029	+	0.995216i	$0.531149\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	81.4798	+	141.127i	0.0543429	+	0.0941247i	0.891917	−	0.452199i	$-0.149360\pi$
$131$	81.4798	+	141.127i	0.0543429	+	0.0941247i	−0.837574	+	0.546324i	$0.816027\pi$
$132$		0			0
$133$	130.470	−	165.844i	0.0850615	−	0.108124i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1004.98	+	580.224i	0.626723	+	0.361839i	0.779482	−	0.626425i	$-0.215482\pi$
$137$	1004.98	+	580.224i	0.626723	+	0.361839i	−0.152759	+	0.988263i	$0.548816\pi$
$138$		0			0
$139$			674.863i			0.411807i	0.978572	+	0.205903i	$0.0660133\pi$
$139$			674.863i			0.411807i	−0.978572	+	0.205903i	$0.933987\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	265.334	−	459.573i	0.155163	−	0.268751i
$144$		0			0
$145$	−119.702	+	69.1101i	−0.0685568	+	0.0395813i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1230.83	−	710.619i	0.676734	−	0.390713i	−0.121889	−	0.992544i	$-0.538895\pi$
$149$	1230.83	−	710.619i	0.676734	−	0.390713i	0.798623	+	0.601831i	$0.205562\pi$
$150$		0			0
$151$	1280.08	−	2217.16i	0.689875	−	1.19490i	−0.282003	−	0.959413i	$-0.590999\pi$
$151$	1280.08	−	2217.16i	0.689875	−	1.19490i	0.971878	−	0.235485i	$-0.0756678\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	429.930i		−	0.222792i
$156$		0			0
$157$	1292.26	+	746.085i	0.656901	+	0.379262i	0.791095	−	0.611693i	$-0.209511\pi$
$157$	1292.26	+	746.085i	0.656901	+	0.379262i	−0.134194	+	0.990955i	$0.542845\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	317.332	−	403.369i	0.155337	−	0.197453i
$162$		0			0
$163$	1068.87	+	1851.34i	0.513623	+	0.889622i	0.999875	+	0.0158029i	$0.00503043\pi$
$163$	1068.87	+	1851.34i	0.513623	+	0.889622i	−0.486252	+	0.873819i	$0.661636\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1154.81			0.535103			0.267552	−	0.963544i	$-0.413785\pi$
$167$	1154.81			0.535103			0.267552	+	0.963544i	$0.413785\pi$
$168$		0			0
$169$	2006.43			0.913261
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1881.14	−	3258.23i	−0.826707	−	1.43190i	−0.900608	−	0.434633i	$-0.856878\pi$
$173$	−1881.14	−	3258.23i	−0.826707	−	1.43190i	0.0739006	−	0.997266i	$-0.476455\pi$
$174$		0			0
$175$	−304.110	+	2112.40i	−0.131363	+	0.912472i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	331.662	+	191.485i	0.138489	+	0.0799568i	0.567644	−	0.823274i	$-0.307855\pi$
$179$	331.662	+	191.485i	0.138489	+	0.0799568i	−0.429155	+	0.903231i	$0.641188\pi$
$180$		0			0
$181$			2689.85i			1.10461i	0.833641	+	0.552307i	$0.186252\pi$
$181$			2689.85i			1.10461i	−0.833641	+	0.552307i	$0.813748\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	220.816	−	382.465i	0.0877553	−	0.151997i
$186$		0			0
$187$	3168.12	−	1829.12i	1.23891	−	0.715285i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	634.579	−	366.374i	0.240401	−	0.138795i	−0.374960	−	0.927041i	$-0.622344\pi$
$191$	634.579	−	366.374i	0.240401	−	0.138795i	0.615361	+	0.788246i	$0.289010\pi$
$192$		0			0
$193$	−2102.20	+	3641.11i	−0.784039	+	1.35800i	0.145532	+	0.989354i	$0.453511\pi$
$193$	−2102.20	+	3641.11i	−0.784039	+	1.35800i	−0.929571	+	0.368642i	$0.879823\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2837.28i		−	1.02613i	−0.858349	−	0.513066i	$-0.828510\pi$
$197$		−	2837.28i		−	1.02613i	0.858349	−	0.513066i	$-0.171490\pi$
$198$		0			0
$199$	1256.70	+	725.558i	0.447665	+	0.258459i	0.706843	−	0.707370i	$-0.250119\pi$
$199$	1256.70	+	725.558i	0.447665	+	0.258459i	−0.259179	+	0.965829i	$0.583452\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	304.321	+	760.559i	0.105217	+	0.262959i
$204$		0			0
$205$	−528.027	−	914.569i	−0.179897	−	0.311591i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	437.989			0.144959
$210$		0			0
$211$	5902.14			1.92569			0.962844	−	0.270058i	$-0.0870430\pi$
$211$	5902.14			1.92569			0.962844	+	0.270058i	$0.0870430\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	652.342	+	1129.89i	0.206927	+	0.358408i
$216$		0			0
$217$	−2522.04	−	363.084i	−0.788974	−	0.113584i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1137.69	−	656.845i	−0.346286	−	0.199928i
$222$		0			0
$223$		−	1445.01i		−	0.433925i	−0.976180	−	0.216962i	$-0.930385\pi$
$223$		−	1445.01i		−	0.433925i	0.976180	−	0.216962i	$-0.0696149\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2754.73	−	4771.33i	0.805453	−	1.39509i	−0.110532	−	0.993873i	$-0.535255\pi$
$227$	2754.73	−	4771.33i	0.805453	−	1.39509i	0.915985	−	0.401213i	$-0.131411\pi$
$228$		0			0
$229$	−4147.74	+	2394.70i	−1.19690	+	0.691032i	−0.959863	−	0.280468i	$-0.909510\pi$
$229$	−4147.74	+	2394.70i	−1.19690	+	0.691032i	−0.237039	+	0.971500i	$0.576177\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3268.04	−	1886.80i	0.918868	−	0.530509i	0.0355944	−	0.999366i	$-0.488668\pi$
$233$	3268.04	−	1886.80i	0.918868	−	0.530509i	0.883274	+	0.468857i	$0.155334\pi$
$234$		0			0
$235$	−453.247	+	785.046i	−0.125815	+	0.217918i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			1246.09i			0.337250i	0.985680	+	0.168625i	$0.0539326\pi$
$239$			1246.09i			0.337250i	−0.985680	+	0.168625i	$0.946067\pi$
$240$		0			0
$241$	821.731	+	474.426i	0.219636	+	0.126807i	0.605782	−	0.795631i	$-0.292860\pi$
$241$	821.731	+	474.426i	0.219636	+	0.126807i	−0.386146	+	0.922438i	$0.626194\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1028.32	−	302.348i	−0.268150	−	0.0788421i
$246$		0			0
$247$	−78.6421	−	136.212i	−0.0202586	−	0.0350889i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1494.83			0.375908			0.187954	−	0.982178i	$-0.439814\pi$
$251$	1494.83			0.375908			0.187954	+	0.982178i	$0.439814\pi$
$252$		0			0
$253$	1065.29			0.264720
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1762.12	−	3052.09i	−0.427697	−	0.740794i	0.568971	−	0.822358i	$-0.307342\pi$
$257$	−1762.12	−	3052.09i	−0.427697	−	0.740794i	−0.996668	+	0.0815642i	$0.974008\pi$
$258$		0			0
$259$	−2057.12	−	1618.34i	−0.493526	−	0.388259i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5514.20	+	3183.62i	1.29285	+	0.746428i	0.979159	−	0.203096i	$-0.0651004\pi$
$263$	5514.20	+	3183.62i	1.29285	+	0.746428i	0.313693	+	0.949524i	$0.398434\pi$
$264$		0			0
$265$			53.3688i			0.0123714i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1738.97	−	3011.98i	0.394152	−	0.682690i	−0.598841	−	0.800868i	$-0.704372\pi$
$269$	1738.97	−	3011.98i	0.394152	−	0.682690i	0.992992	+	0.118178i	$0.0377052\pi$
$270$		0			0
$271$	4121.15	−	2379.35i	0.923772	−	0.533340i	0.0389354	−	0.999242i	$-0.487603\pi$
$271$	4121.15	−	2379.35i	0.923772	−	0.533340i	0.884836	+	0.465902i	$0.154270\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3836.33	+	2214.91i	−0.841234	+	0.485687i
$276$		0			0
$277$	1117.86	−	1936.19i	0.242476	−	0.419980i	−0.718943	−	0.695069i	$-0.755374\pi$
$277$	1117.86	−	1936.19i	0.242476	−	0.419980i	0.961419	+	0.275089i	$0.0887073\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			8109.09i			1.72152i	0.509008	+	0.860762i	$0.330012\pi$
$281$			8109.09i			1.72152i	−0.509008	+	0.860762i	$0.669988\pi$
$282$		0			0
$283$	−2858.65	−	1650.44i	−0.600456	−	0.346673i	0.168765	−	0.985656i	$-0.446022\pi$
$283$	−2858.65	−	1650.44i	−0.600456	−	0.346673i	−0.769221	+	0.638983i	$0.779355\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5810.94	+	2325.12i	−1.19515	+	0.478215i
$288$		0			0
$289$	−2071.55	−	3588.02i	−0.421646	−	0.730312i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3129.58			−0.624001			−0.312001	−	0.950082i	$-0.600999\pi$
$293$	−3129.58			−0.624001			−0.312001	+	0.950082i	$0.600999\pi$
$294$		0			0
$295$	−1873.81			−0.369822
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−191.275	−	331.298i	−0.0369958	−	0.0640785i
$300$		0			0
$301$	7179.04	−	2872.54i	1.37473	−	0.550067i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1437.09	−	829.705i	−0.269795	−	0.155766i
$306$		0			0
$307$		−	9190.88i		−	1.70864i	−0.519751	−	0.854318i	$-0.673975\pi$
$307$		−	9190.88i		−	1.70864i	0.519751	−	0.854318i	$-0.326025\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−953.133	+	1650.88i	−0.173785	+	0.301005i	−0.939740	−	0.341889i	$-0.888933\pi$
$311$	−953.133	+	1650.88i	−0.173785	+	0.301005i	0.765955	+	0.642894i	$0.222267\pi$
$312$		0			0
$313$	−8295.18	+	4789.22i	−1.49799	+	0.864866i	−0.999997	−	0.00231512i	$-0.999263\pi$
$313$	−8295.18	+	4789.22i	−1.49799	+	0.864866i	−0.497994	+	0.867181i	$0.665930\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2310.11	+	1333.74i	−0.409301	+	0.236310i	−0.690489	−	0.723342i	$-0.742605\pi$
$317$	−2310.11	+	1333.74i	−0.409301	+	0.236310i	0.281188	+	0.959653i	$0.409271\pi$
$318$		0			0
$319$	−850.168	+	1472.53i	−0.149217	+	0.258452i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1084.26i		−	0.186779i
$324$		0			0
$325$	1377.64	+	795.384i	0.235132	+	0.135754i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4222.44	+	3321.81i	0.707570	+	0.556648i
$330$		0			0
$331$	3959.19	+	6857.53i	0.657453	+	1.13874i	0.981273	+	0.192624i	$0.0616996\pi$
$331$	3959.19	+	6857.53i	0.657453	+	1.13874i	−0.323819	+	0.946119i	$0.604967\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2031.85			0.331379
$336$		0			0
$337$	−5173.45			−0.836248			−0.418124	−	0.908390i	$-0.637312\pi$
$337$	−5173.45			−0.836248			−0.418124	+	0.908390i	$0.637312\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2644.42	−	4580.28i	−0.419952	−	0.727378i
$342$		0			0
$343$	−2642.06	+	5776.95i	−0.415912	+	0.909405i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−346.881	−	200.272i	−0.0536644	−	0.0309831i	0.472928	−	0.881101i	$-0.343197\pi$
$347$	−346.881	−	200.272i	−0.0536644	−	0.0309831i	−0.526592	+	0.850118i	$0.676530\pi$
$348$		0			0
$349$		−	6360.10i		−	0.975496i	−0.872984	−	0.487748i	$-0.837818\pi$
$349$		−	6360.10i		−	0.975496i	0.872984	−	0.487748i	$-0.162182\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6154.02	+	10659.1i	−0.927891	+	1.60715i	−0.141047	+	0.990003i	$0.545047\pi$
$353$	−6154.02	+	10659.1i	−0.927891	+	1.60715i	−0.786844	+	0.617152i	$0.788286\pi$
$354$		0			0
$355$	−2527.99	+	1459.53i	−0.377948	+	0.218209i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7649.34	+	4416.35i	−1.12456	+	0.649264i	−0.942561	−	0.334035i	$-0.891590\pi$
$359$	−7649.34	+	4416.35i	−1.12456	+	0.649264i	−0.181998	+	0.983299i	$0.558256\pi$
$360$		0			0
$361$	−3364.59	+	5827.65i	−0.490537	+	0.849635i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	2840.28i		−	0.407308i
$366$		0			0
$367$	−3655.01	−	2110.22i	−0.519864	−	0.300144i	0.217015	−	0.976168i	$-0.430368\pi$
$367$	−3655.01	−	2110.22i	−0.519864	−	0.300144i	−0.736879	+	0.676025i	$0.763701\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	313.070	+	45.0709i	0.0438108	+	0.00630718i
$372$		0			0
$373$	−2165.80	−	3751.28i	−0.300646	−	0.520734i	0.675637	−	0.737235i	$-0.263869\pi$
$373$	−2165.80	−	3751.28i	−0.300646	−	0.520734i	−0.976282	+	0.216501i	$0.930536\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	610.599			0.0834150
$378$		0			0
$379$	−9160.86			−1.24159			−0.620794	−	0.783974i	$-0.713190\pi$
$379$	−9160.86			−1.24159			−0.620794	+	0.783974i	$0.713190\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	382.423	+	662.376i	0.0510206	+	0.0883703i	0.890408	−	0.455164i	$-0.150419\pi$
$383$	382.423	+	662.376i	0.0510206	+	0.0883703i	−0.839387	+	0.543534i	$0.817086\pi$
$384$		0			0
$385$	−826.489	−	2065.56i	−0.109407	−	0.273430i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5806.30	+	3352.27i	0.756789	+	0.436932i	0.828142	−	0.560519i	$-0.189398\pi$
$389$	5806.30	+	3352.27i	0.756789	+	0.436932i	−0.0713526	+	0.997451i	$0.522732\pi$
$390$		0			0
$391$		−	2637.16i		−	0.341092i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1242.20	−	2151.56i	0.158233	−	0.274067i
$396$		0			0
$397$	4026.96	−	2324.97i	0.509087	−	0.293921i	−0.223371	−	0.974733i	$-0.571706\pi$
$397$	4026.96	−	2324.97i	0.509087	−	0.293921i	0.732458	+	0.680812i	$0.238373\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3072.03	−	1773.64i	0.382568	−	0.220876i	−0.296367	−	0.955074i	$-0.595775\pi$
$401$	3072.03	−	1773.64i	0.382568	−	0.220876i	0.678935	+	0.734198i	$0.262442\pi$
$402$		0			0
$403$	−949.626	+	1644.80i	−0.117380	+	0.203309i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	5432.80i		−	0.661656i
$408$		0			0
$409$	11353.0	+	6554.64i	1.37254	+	0.792436i	0.991247	−	0.132019i	$-0.0421460\pi$
$409$	11353.0	+	6554.64i	1.37254	+	0.792436i	0.381292	+	0.924455i	$0.375479\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1582.47	+	10992.1i	−0.188543	+	1.30965i
$414$		0			0
$415$	491.963	+	852.106i	0.0581916	+	0.100791i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5937.56			−0.692288			−0.346144	−	0.938181i	$-0.612509\pi$
$419$	−5937.56			−0.692288			−0.346144	+	0.938181i	$0.612509\pi$
$420$		0			0
$421$	12210.2			1.41351			0.706757	−	0.707456i	$-0.250157\pi$
$421$	12210.2			1.41351			0.706757	+	0.707456i	$0.250157\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5483.08	+	9496.97i	0.625808	+	1.08393i
$426$		0			0
$427$	−6080.84	+	7729.52i	−0.689162	+	0.876013i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2782.80	−	1606.65i	−0.311004	−	0.179558i	0.336372	−	0.941729i	$-0.390800\pi$
$431$	−2782.80	−	1606.65i	−0.311004	−	0.179558i	−0.647376	+	0.762171i	$0.724134\pi$
$432$		0			0
$433$		−	6616.31i		−	0.734318i	−0.930158	−	0.367159i	$-0.880331\pi$
$433$		−	6616.31i		−	0.734318i	0.930158	−	0.367159i	$-0.119669\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	157.870	−	273.438i	0.0172813	−	0.0299321i
$438$		0			0
$439$	2084.48	−	1203.48i	0.226622	−	0.130840i	−0.382391	−	0.924001i	$-0.624899\pi$
$439$	2084.48	−	1203.48i	0.226622	−	0.130840i	0.609013	+	0.793161i	$0.291566\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9059.31	+	5230.40i	−0.971605	+	0.560956i	−0.899725	−	0.436457i	$-0.856233\pi$
$443$	−9059.31	+	5230.40i	−0.971605	+	0.560956i	−0.0718798	+	0.997413i	$0.522900\pi$
$444$		0			0
$445$	560.988	−	971.660i	0.0597604	−	0.103508i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			7633.63i			0.802346i	0.916002	+	0.401173i	$0.131397\pi$
$449$			7633.63i			0.802346i	−0.916002	+	0.401173i	$0.868603\pi$
$450$		0			0
$451$	−11250.7	−	6495.59i	−1.17467	−	0.678194i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−493.979	+	627.910i	−0.0508969	+	0.0646964i
$456$		0			0
$457$	−7723.99	−	13378.3i	−0.790620	−	1.36939i	−0.925584	−	0.378543i	$-0.876425\pi$
$457$	−7723.99	−	13378.3i	−0.790620	−	1.36939i	0.134964	−	0.990850i	$-0.456908\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19428.0			−1.96280			−0.981401	−	0.191970i	$-0.938512\pi$
$461$	−19428.0			−1.96280			−0.981401	+	0.191970i	$0.938512\pi$
$462$		0			0
$463$	−8196.41			−0.822720			−0.411360	−	0.911473i	$-0.634946\pi$
$463$	−8196.41			−0.822720			−0.411360	+	0.911473i	$0.634946\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7344.66	+	12721.3i	0.727774	+	1.26054i	0.957822	+	0.287362i	$0.0927784\pi$
$467$	7344.66	+	12721.3i	0.727774	+	1.26054i	−0.230048	+	0.973179i	$0.573888\pi$
$468$		0			0
$469$	1715.93	−	11919.2i	0.168943	−	1.17351i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	13899.5	+	8024.88i	1.35116	+	0.780093i
$474$		0			0
$475$			1312.95i			0.126825i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8549.24	−	14807.7i	0.815500	−	1.41249i	−0.0934678	−	0.995622i	$-0.529795\pi$
$479$	8549.24	−	14807.7i	0.815500	−	1.41249i	0.908968	−	0.416866i	$-0.136871\pi$
$480$		0			0
$481$	−1689.57	+	975.473i	−0.160162	+	0.0924693i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	218.008	−	125.867i	0.0204108	−	0.0117842i
$486$		0			0
$487$	−2428.25	+	4205.85i	−0.225944	+	0.391346i	−0.956602	−	0.291397i	$-0.905880\pi$
$487$	−2428.25	+	4205.85i	−0.225944	+	0.391346i	0.730659	+	0.682743i	$0.239213\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			3844.24i			0.353336i	0.984270	+	0.176668i	$0.0565320\pi$
$491$			3844.24i			0.353336i	−0.984270	+	0.176668i	$0.943468\pi$
$492$		0			0
$493$	3645.31	+	2104.62i	0.333016	+	0.192267i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6426.94	+	16062.2i	0.580056	+	1.44967i
$498$		0			0
$499$	2307.84	+	3997.30i	0.207041	+	0.358605i	0.950781	−	0.309864i	$-0.100283\pi$
$499$	2307.84	+	3997.30i	0.207041	+	0.358605i	−0.743740	+	0.668469i	$0.766950\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−11692.2			−1.03644			−0.518219	−	0.855248i	$-0.673405\pi$
$503$	−11692.2			−1.03644			−0.518219	+	0.855248i	$0.673405\pi$
$504$		0			0
$505$	−2713.57			−0.239114
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10752.4	−	18623.7i	−0.936327	−	1.62177i	−0.772250	−	0.635319i	$-0.780869\pi$
$509$	−10752.4	−	18623.7i	−0.936327	−	1.62177i	−0.164077	−	0.986447i	$-0.552465\pi$
$510$		0			0
$511$	−16661.6	−	2398.67i	−1.44240	−	0.207654i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	327.680	+	189.186i	0.0280375	+	0.0161875i
$516$		0			0
$517$			11151.4i			0.948619i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−184.719	+	319.942i	−0.0155330	+	0.0269039i	−0.873687	−	0.486488i	$-0.838278\pi$
$521$	−184.719	+	319.942i	−0.0155330	+	0.0269039i	0.858154	+	0.513392i	$0.171611\pi$
$522$		0			0
$523$	13219.0	−	7631.98i	1.10521	−	0.638094i	0.167626	−	0.985851i	$-0.446390\pi$
$523$	13219.0	−	7631.98i	1.10521	−	0.638094i	0.937585	+	0.347757i	$0.113057\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11338.6	+	6546.37i	−0.937228	+	0.541109i
$528$		0			0
$529$	−5699.53	+	9871.87i	−0.468441	+	0.811364i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			4665.20i			0.379122i
$534$		0			0
$535$	−1211.69	−	699.568i	−0.0979173	−	0.0565326i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12814.9	+	3103.92i	−1.02408	+	0.248043i
$540$		0			0
$541$	−8163.08	−	14138.9i	−0.648721	−	1.12362i	−0.983429	−	0.181296i	$-0.941971\pi$
$541$	−8163.08	−	14138.9i	−0.648721	−	1.12362i	0.334707	−	0.942322i	$-0.391363\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3172.06			−0.249314
$546$		0			0
$547$	−13065.7			−1.02130			−0.510649	−	0.859789i	$-0.670595\pi$
$547$	−13065.7			−1.02130			−0.510649	+	0.859789i	$0.670595\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	251.980	+	436.442i	0.0194822	+	0.0337442i
$552$		0			0
$553$	−11572.3	−	9104.00i	−0.889884	−	0.700075i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9145.80	−	5280.33i	−0.695727	−	0.401678i	0.110027	−	0.993929i	$-0.464906\pi$
$557$	−9145.80	−	5280.33i	−0.695727	−	0.401678i	−0.805754	+	0.592250i	$0.798240\pi$
$558$		0			0
$559$		−	5763.55i		−	0.436086i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6474.13	+	11213.5i	−0.484640	+	0.839420i	−0.999844	−	0.0176468i	$-0.994383\pi$
$563$	−6474.13	+	11213.5i	−0.484640	+	0.839420i	0.515205	+	0.857067i	$0.327716\pi$
$564$		0			0
$565$	−2408.06	+	1390.29i	−0.179306	+	0.103522i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12555.1	+	7248.69i	−0.925021	+	0.534061i	−0.885233	−	0.465147i	$-0.846001\pi$
$569$	−12555.1	+	7248.69i	−0.925021	+	0.534061i	−0.0397876	+	0.999208i	$0.512668\pi$
$570$		0			0
$571$	−1966.46	+	3406.00i	−0.144122	+	0.249626i	−0.929045	−	0.369967i	$-0.879369\pi$
$571$	−1966.46	+	3406.00i	−0.144122	+	0.249626i	0.784923	+	0.619593i	$0.212702\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			3193.38i			0.231605i
$576$		0			0
$577$	7970.48	+	4601.76i	0.575070	+	0.332017i	0.759172	−	0.650890i	$-0.225604\pi$
$577$	7970.48	+	4601.76i	0.575070	+	0.332017i	−0.184101	+	0.982907i	$0.558938\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5414.07	−	2166.32i	0.386598	−	0.154689i
$582$		0			0
$583$	328.262	+	568.566i	0.0233194	+	0.0403904i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4778.34			0.335985			0.167992	−	0.985788i	$-0.446272\pi$
$587$	4778.34			0.335985			0.167992	+	0.985788i	$0.446272\pi$
$588$		0			0
$589$	−1567.55			−0.109660
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7388.21	+	12796.8i	0.511631	+	0.886172i	0.999909	+	0.0134834i	$0.00429202\pi$
$593$	7388.21	+	12796.8i	0.511631	+	0.886172i	−0.488278	+	0.872688i	$0.662375\pi$
$594$		0			0
$595$	−5113.37	+	2046.00i	−0.352315	+	0.140971i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20384.4	+	11769.0i	1.39046	+	0.802783i	0.993366	−	0.114995i	$-0.0366852\pi$
$599$	20384.4	+	11769.0i	1.39046	+	0.802783i	0.397094	+	0.917778i	$0.370019\pi$
$600$		0			0
$601$			13020.6i			0.883727i	0.897082	+	0.441864i	$0.145682\pi$
$601$			13020.6i			0.883727i	−0.897082	+	0.441864i	$0.854318\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	229.296	−	397.153i	0.0154086	−	0.0266885i
$606$		0			0
$607$	15450.9	−	8920.59i	1.03317	−	0.596500i	0.115278	−	0.993333i	$-0.463224\pi$
$607$	15450.9	−	8920.59i	1.03317	−	0.596500i	0.917891	+	0.396833i	$0.129891\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3468.01	−	2002.25i	0.229624	−	0.132574i
$612$		0			0
$613$	8589.91	−	14878.2i	0.565976	−	0.980299i	−0.430982	−	0.902361i	$-0.641833\pi$
$613$	8589.91	−	14878.2i	0.565976	−	0.980299i	0.996958	−	0.0779389i	$-0.0248339\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			7147.22i			0.466347i	0.972435	+	0.233174i	$0.0749111\pi$
$617$			7147.22i			0.466347i	−0.972435	+	0.233174i	$0.925089\pi$
$618$		0			0
$619$	5170.09	+	2984.95i	0.335708	+	0.193821i	0.658373	−	0.752692i	$-0.271245\pi$
$619$	5170.09	+	2984.95i	0.335708	+	0.193821i	−0.322664	+	0.946514i	$0.604578\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5226.16	−	4111.44i	−0.336086	−	0.264400i
$624$		0			0
$625$	−6029.23	−	10442.9i	−0.385870	−	0.668347i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−13449.1			−0.852545
$630$		0			0
$631$	15148.2			0.955693			0.477846	−	0.878443i	$-0.341418\pi$
$631$	15148.2			0.955693			0.477846	+	0.878443i	$0.341418\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	436.969	+	756.853i	0.0273080	+	0.0472989i
$636$		0			0
$637$	3266.25	+	3428.04i	0.203161	+	0.213225i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4316.39	+	2492.07i	0.265971	+	0.153558i	0.627055	−	0.778975i	$-0.284260\pi$
$641$	4316.39	+	2492.07i	0.265971	+	0.153558i	−0.361085	+	0.932533i	$0.617593\pi$
$642$		0			0
$643$		−	6255.66i		−	0.383669i	−0.981427	−	0.191834i	$-0.938556\pi$
$643$		−	6255.66i		−	0.383669i	0.981427	−	0.191834i	$-0.0614436\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12254.3	−	21225.1i	0.744618	−	1.28972i	−0.205755	−	0.978603i	$-0.565965\pi$
$647$	12254.3	−	21225.1i	0.744618	−	1.28972i	0.950373	−	0.311112i	$-0.100702\pi$
$648$		0			0
$649$	−19962.7	+	11525.5i	−1.20740	+	0.697095i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	26910.6	−	15536.8i	1.61270	−	0.931093i	0.623959	−	0.781457i	$-0.285523\pi$
$653$	26910.6	−	15536.8i	1.61270	−	0.931093i	0.988741	−	0.149636i	$-0.0478101\pi$
$654$		0			0
$655$	254.617	−	441.010i	0.0151889	−	0.0263079i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	13489.0i		−	0.797357i	−0.917091	−	0.398678i	$-0.869469\pi$
$659$		−	13489.0i		−	0.797357i	0.917091	−	0.398678i	$-0.130531\pi$
$660$		0			0
$661$	16896.0	+	9754.92i	0.994219	+	0.574013i	0.906533	−	0.422135i	$-0.138719\pi$
$661$	16896.0	+	9754.92i	0.994219	+	0.574013i	0.0876864	+	0.996148i	$0.472053\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−652.669	−	93.9609i	−0.0380593	−	0.00547917i
$666$		0			0
$667$	612.872	+	1061.53i	0.0355780	+	0.0616229i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−20413.5			−1.17445
$672$		0			0
$673$	−7967.95			−0.456377			−0.228189	−	0.973617i	$-0.573280\pi$
$673$	−7967.95			−0.456377			−0.228189	+	0.973617i	$0.573280\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13315.2	−	23062.5i	−0.755898	−	1.30925i	−0.944926	−	0.327283i	$-0.893867\pi$
$677$	−13315.2	−	23062.5i	−0.755898	−	1.30925i	0.189028	−	0.981972i	$-0.439466\pi$
$678$		0			0
$679$	−554.246	−	1385.17i	−0.0313255	−	0.0782886i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	26596.3	+	15355.4i	1.49001	+	0.860259i	0.999935	−	0.0114194i	$-0.00363498\pi$
$683$	26596.3	+	15355.4i	1.49001	+	0.860259i	0.490078	+	0.871679i	$0.336968\pi$
$684$		0			0
$685$		−	3626.30i		−	0.202268i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	117.881	−	204.175i	0.00651798	−	0.0112895i
$690$		0			0
$691$	25217.6	−	14559.4i	1.38831	−	0.801541i	0.395185	−	0.918602i	$-0.370680\pi$
$691$	25217.6	−	14559.4i	1.38831	−	0.801541i	0.993125	+	0.117061i	$0.0373471\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1826.35	−	1054.44i	0.0996798	−	0.0575502i
$696$		0			0
$697$	−16080.1	+	27851.5i	−0.873854	+	1.51356i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	30038.9i		−	1.61848i	−0.587480	−	0.809239i	$-0.699880\pi$
$701$		−	30038.9i		−	1.61848i	0.587480	−	0.809239i	$-0.300120\pi$
$702$		0			0
$703$	−1394.49	−	805.110i	−0.0748140	−	0.0431939i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2291.66	+	15918.3i	−0.121905	+	0.846773i
$708$		0			0
$709$	−4914.07	−	8511.42i	−0.260299	−	0.450851i	0.706023	−	0.708189i	$-0.250488\pi$
$709$	−4914.07	−	8511.42i	−0.260299	−	0.450851i	−0.966321	+	0.257339i	$0.917154\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3812.64			−0.200259
$714$		0			0
$715$	−1658.29			−0.0867366
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12838.1	+	22236.2i	0.665897	+	1.15337i	0.979041	+	0.203662i	$0.0652843\pi$
$719$	12838.1	+	22236.2i	0.665897	+	1.15337i	−0.313144	+	0.949706i	$0.601382\pi$
$720$		0			0
$721$	1386.53	−	1762.46i	0.0716188	−	0.0910366i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4414.17	−	2548.52i	−0.226121	−	0.130551i
$726$		0			0
$727$		−	9770.68i		−	0.498452i	−0.968445	−	0.249226i	$-0.919824\pi$
$727$		−	9770.68i		−	0.498452i	0.968445	−	0.249226i	$-0.0801761\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	19865.9	−	34408.7i	1.00515	−	1.74097i
$732$		0			0
$733$	1265.95	−	730.896i	0.0637912	−	0.0368298i	−0.467765	−	0.883853i	$-0.654941\pi$
$733$	1265.95	−	730.896i	0.0637912	−	0.0368298i	0.531556	+	0.847023i	$0.321607\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	21646.4	−	12497.5i	1.08189	−	0.624631i
$738$		0			0
$739$	6722.13	−	11643.1i	0.334611	−	0.579563i	−0.648799	−	0.760960i	$-0.724728\pi$
$739$	6722.13	−	11643.1i	0.334611	−	0.579563i	0.983410	+	0.181397i	$0.0580618\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			13757.2i			0.679278i	0.940556	+	0.339639i	$0.110305\pi$
$743$			13757.2i			0.679278i	−0.940556	+	0.339639i	$0.889695\pi$
$744$		0			0
$745$	−3846.23	−	2220.62i	−0.189148	−	0.109204i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5127.07	+	6517.16i	−0.250119	+	0.317933i
$750$		0			0
$751$	14594.6	+	25278.6i	0.709142	+	1.22827i	0.965176	+	0.261602i	$0.0842508\pi$
$751$	14594.6	+	25278.6i	0.709142	+	1.22827i	−0.256034	+	0.966668i	$0.582416\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8000.25			−0.385641
$756$		0			0
$757$	8236.25			0.395445			0.197722	−	0.980258i	$-0.436646\pi$
$757$	8236.25			0.395445			0.197722	+	0.980258i	$0.436646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16521.8	−	28616.6i	−0.787011	−	1.36314i	−0.927790	−	0.373102i	$-0.878294\pi$
$761$	−16521.8	−	28616.6i	−0.787011	−	1.36314i	0.140779	−	0.990041i	$-0.455039\pi$
$762$		0			0
$763$	−2678.86	+	18607.9i	−0.127105	+	0.882896i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7168.71	+	4138.86i	0.337480	+	0.194844i
$768$		0			0
$769$			23591.7i			1.10629i	0.833084	+	0.553146i	$0.186573\pi$
$769$			23591.7i			1.10629i	−0.833084	+	0.553146i	$0.813427\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	15883.6	−	27511.2i	0.739060	−	1.28009i	−0.213858	−	0.976865i	$-0.568603\pi$
$773$	15883.6	−	27511.2i	0.739060	−	1.28009i	0.952919	−	0.303225i	$-0.0980635\pi$
$774$		0			0
$775$	13730.1	−	7927.10i	0.636388	−	0.367419i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3334.58	+	1925.22i	−0.153368	+	0.0885471i
$780$		0			0
$781$	−17954.7	+	31098.4i	−0.822623	+	1.42482i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	4662.90i		−	0.212008i
$786$		0			0
$787$	34978.6	+	20194.9i	1.58431	+	0.914702i	0.994220	+	0.107366i	$0.0342417\pi$
$787$	34978.6	+	20194.9i	1.58431	+	0.914702i	0.590092	+	0.807336i	$0.299092\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6122.04	+	15300.2i	0.275189	+	0.687752i
$792$		0			0
$793$	3665.29	+	6348.46i	0.164134	+	0.284288i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−14289.7			−0.635092			−0.317546	−	0.948243i	$-0.602859\pi$
$797$	−14289.7			−0.635092			−0.317546	+	0.948243i	$0.602859\pi$
$798$		0			0
$799$	27605.6			1.22230
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−17470.1	−	30259.1i	−0.767753	−	1.32979i
$804$		0			0
$805$	−1587.44	−	228.534i	−0.0695029	−	0.0100059i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	28576.0	+	16498.3i	1.24188	+	0.716997i	0.969476	−	0.245187i	$-0.0788493\pi$
$809$	28576.0	+	16498.3i	1.24188	+	0.716997i	0.272400	+	0.962184i	$0.412183\pi$
$810$		0			0
$811$		−	36278.0i		−	1.57077i	−0.619008	−	0.785384i	$-0.712465\pi$
$811$		−	36278.0i		−	1.57077i	0.619008	−	0.785384i	$-0.287535\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3340.14	−	5785.28i	0.143558	−	0.248650i
$816$		0			0
$817$	4119.65	−	2378.48i	0.176412	−	0.101851i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−33219.3	+	19179.2i	−1.41213	+	0.815296i	−0.995589	−	0.0938191i	$-0.970092\pi$
$821$	−33219.3	+	19179.2i	−1.41213	+	0.815296i	−0.416545	+	0.909115i	$0.636759\pi$
$822$		0			0
$823$	−18987.0	+	32886.5i	−0.804186	+	1.39289i	0.112652	+	0.993634i	$0.464065\pi$
$823$	−18987.0	+	32886.5i	−0.804186	+	1.39289i	−0.916839	+	0.399257i	$0.869268\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	21864.0i		−	0.919331i	−0.888092	−	0.459666i	$-0.847969\pi$
$827$		−	21864.0i		−	0.919331i	0.888092	−	0.459666i	$-0.152031\pi$
$828$		0			0
$829$	16368.9	+	9450.57i	0.685783	+	0.395937i	0.802030	−	0.597283i	$-0.203753\pi$
$829$	16368.9	+	9450.57i	0.685783	+	0.395937i	−0.116247	+	0.993220i	$0.537087\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	7683.87	+	31723.8i	0.319604	+	1.31952i
$834$		0			0
$835$	−1804.35	−	3125.22i	−0.0747809	−	0.129524i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	26571.8			1.09340			0.546699	−	0.837329i	$-0.315884\pi$
$839$	26571.8			1.09340			0.546699	+	0.837329i	$0.315884\pi$
$840$		0			0
$841$	22432.6			0.919782
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3134.97	−	5429.92i	−0.127629	−	0.221059i
$846$		0			0
$847$	−2136.12	−	1680.49i	−0.0866565	−	0.0681729i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3391.72	−	1958.21i	−0.136623	−	0.0788796i
$852$		0			0
$853$		−	43958.5i		−	1.76449i	−0.470789	−	0.882246i	$-0.656031\pi$
$853$		−	43958.5i		−	1.76449i	0.470789	−	0.882246i	$-0.343969\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3199.70	−	5542.04i	0.127537	−	0.220901i	−0.795185	−	0.606367i	$-0.792626\pi$
$857$	3199.70	−	5542.04i	0.127537	−	0.220901i	0.922722	+	0.385466i	$0.125959\pi$
$858$		0			0
$859$	−6804.26	+	3928.44i	−0.270266	+	0.156038i	−0.629009	−	0.777398i	$-0.716539\pi$
$859$	−6804.26	+	3928.44i	−0.270266	+	0.156038i	0.358743	+	0.933437i	$0.383206\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−27014.2	+	15596.7i	−1.06556	+	0.615199i	−0.926964	−	0.375150i	$-0.877591\pi$
$863$	−27014.2	+	15596.7i	−1.06556	+	0.615199i	−0.138592	+	0.990349i	$0.544258\pi$
$864$		0			0
$865$	−5878.39	+	10181.7i	−0.231065	+	0.400217i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	30562.3i		−	1.19304i
$870$		0			0
$871$	−7773.32	−	4487.93i	−0.302398	−	0.174590i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	12908.4	−	5165.02i	0.498725	−	0.199554i
$876$		0			0
$877$	5510.34	+	9544.19i	0.212168	+	0.367485i	0.952393	−	0.304874i	$-0.0986144\pi$
$877$	5510.34	+	9544.19i	0.212168	+	0.367485i	−0.740225	+	0.672359i	$0.765281\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	751.266			0.0287296			0.0143648	−	0.999897i	$-0.495427\pi$
$881$	751.266			0.0287296			0.0143648	+	0.999897i	$0.495427\pi$
$882$		0			0
$883$	1345.17			0.0512669			0.0256334	−	0.999671i	$-0.491840\pi$
$883$	1345.17			0.0512669			0.0256334	+	0.999671i	$0.491840\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−845.523	−	1464.49i	−0.0320066	−	0.0554371i	0.849578	−	0.527462i	$-0.176856\pi$
$887$	−845.523	−	1464.49i	−0.0320066	−	0.0554371i	−0.881585	+	0.472025i	$0.843523\pi$
$888$		0			0
$889$	4808.86	−	1924.16i	0.181422	−	0.0725920i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2862.33	+	1652.57i	0.107261	+	0.0619273i
$894$		0			0
$895$		−	1196.75i		−	0.0446960i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3042.73	−	5270.17i	0.112882	−	0.195517i
$900$		0			0
$901$	1407.51	−	812.624i	0.0520431	−	0.0300471i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7279.43	−	4202.78i	0.267377	−	0.154370i
$906$		0			0
$907$	−5003.83	+	8666.89i	−0.183186	+	0.317287i	−0.942964	−	0.332896i	$-0.891974\pi$
$907$	−5003.83	+	8666.89i	−0.183186	+	0.317287i	0.759778	+	0.650183i	$0.225308\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	35217.1i		−	1.28078i	−0.768048	−	0.640392i	$-0.778772\pi$
$911$		−	35217.1i		−	1.28078i	0.768048	−	0.640392i	$-0.221228\pi$
$912$		0			0
$913$	10482.3	+	6051.96i	0.379971	+	0.219376i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2372.01	−	1866.07i	−0.0854206	−	0.0672007i
$918$		0			0
$919$	−8680.34	−	15034.8i	−0.311576	−	0.539665i	0.667128	−	0.744943i	$-0.267523\pi$
$919$	−8680.34	−	15034.8i	−0.311576	−	0.539665i	−0.978704	+	0.205278i	$0.934190\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12895.2			0.459861
$924$		0			0
$925$	16285.7			0.578888
$926$		0			0
$927$		0			0
$928$		0			0
$929$	931.872	+	1614.05i	0.0329104	+	0.0570024i	0.882011	−	0.471228i	$-0.156189\pi$
$929$	931.872	+	1614.05i	0.0329104	+	0.0570024i	−0.849101	+	0.528230i	$0.822856\pi$
$930$		0			0
$931$	−1102.38	+	3749.32i	−0.0388068	+	0.131986i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−9900.11	−	5715.83i	−0.346276	−	0.199923i
$936$		0			0
$937$		−	44596.5i		−	1.55486i	−0.628970	−	0.777429i	$-0.716523\pi$
$937$		−	44596.5i		−	1.55486i	0.628970	−	0.777429i	$-0.283477\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23617.3	−	40906.4i	0.818175	−	1.41712i	−0.0888512	−	0.996045i	$-0.528320\pi$
$941$	23617.3	−	40906.4i	0.818175	−	1.41712i	0.907026	−	0.421075i	$-0.138347\pi$
$942$		0			0
$943$	−8110.44	+	4682.57i	−0.280077	+	0.161702i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8008.25	−	4623.57i	0.274797	−	0.158654i	−0.356268	−	0.934384i	$-0.615951\pi$
$947$	8008.25	−	4623.57i	0.274797	−	0.158654i	0.631066	+	0.775729i	$0.282618\pi$
$948$		0			0
$949$	−6273.59	+	10866.2i	−0.214594	+	0.371687i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1138.35i			0.0386932i	0.999813	+	0.0193466i	$0.00615860\pi$
$953$			1138.35i			0.0386932i	−0.999813	+	0.0193466i	$0.993841\pi$
$954$		0			0
$955$	−1983.00	−	1144.89i	−0.0671921	−	0.0387934i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−21272.5	−	3062.48i	−0.716293	−	0.103120i
$960$		0			0
$961$	−5431.17	−	9407.06i	−0.182309	−	0.315769i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	13138.4			0.438279
$966$		0			0
$967$	−5284.04			−0.175722			−0.0878610	−	0.996133i	$-0.528003\pi$
$967$	−5284.04			−0.175722			−0.0878610	+	0.996133i	$0.528003\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11258.4	−	19500.1i	−0.372089	−	0.644478i	0.617797	−	0.786337i	$-0.288025\pi$
$971$	−11258.4	−	19500.1i	−0.372089	−	0.644478i	−0.989887	+	0.141860i	$0.954692\pi$
$972$		0			0
$973$	−4643.16	−	11604.2i	−0.152984	−	0.382336i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	32273.4	+	18633.1i	1.05683	+	0.610158i	0.924552	−	0.381055i	$-0.124439\pi$
$977$	32273.4	+	18633.1i	1.05683	+	0.610158i	0.132273	+	0.991213i	$0.457773\pi$
$978$		0			0
$979$		−	13802.2i		−	0.450581i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20156.0	−	34911.3i	0.653995	−	1.13275i	−0.328149	−	0.944626i	$-0.606425\pi$
$983$	20156.0	−	34911.3i	0.653995	−	1.13275i	0.982145	−	0.188127i	$-0.0602418\pi$
$984$		0			0
$985$	−7678.40	+	4433.13i	−0.248380	+	0.143402i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	10019.9	−	5785.00i	0.322159	−	0.185998i
$990$		0			0
$991$	2601.46	−	4505.86i	0.0833885	−	0.144433i	−0.821315	−	0.570475i	$-0.806759\pi$
$991$	2601.46	−	4505.86i	0.0833885	−	0.144433i	0.904703	+	0.426042i	$0.140092\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	4534.61i		−	0.144479i
$996$		0			0
$997$	54015.6	+	31185.9i	1.71584	+	0.990640i	0.926169	+	0.377108i	$0.123082\pi$
$997$	54015.6	+	31185.9i	1.71584	+	0.990640i	0.789670	+	0.613532i	$0.210252\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.bt.d.17.11		48
3.2	odd	2	inner	1008.4.bt.d.17.14		48
4.3	odd	2		504.4.bl.a.17.11	✓	48
7.5	odd	6	inner	1008.4.bt.d.593.14		48
12.11	even	2		504.4.bl.a.17.14	yes	48
21.5	even	6	inner	1008.4.bt.d.593.11		48
28.19	even	6		504.4.bl.a.89.14	yes	48
84.47	odd	6		504.4.bl.a.89.11	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.11	✓	48	4.3	odd	2
504.4.bl.a.17.14	yes	48	12.11	even	2
504.4.bl.a.89.11	yes	48	84.47	odd	6
504.4.bl.a.89.14	yes	48	28.19	even	6
1008.4.bt.d.17.11		48	1.1	even	1	trivial
1008.4.bt.d.17.14		48	3.2	odd	2	inner
1008.4.bt.d.593.11		48	21.5	even	6	inner
1008.4.bt.d.593.14		48	7.5	odd	6	inner

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.bt (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.56246 - 2.70625i) q^{5} +(-17.1949 + 6.88015i) q^{7} +O(q^{10})\)	\(q+(-1.56246 - 2.70625i) q^{5} +(-17.1949 + 6.88015i) q^{7} +(-33.2914 - 19.2208i) q^{11} +13.8046i q^{13} +(-47.5817 + 82.4140i) q^{17} +(-9.86718 + 5.69682i) q^{19} +(-23.9992 + 13.8559i) q^{23} +(57.6175 - 99.7964i) q^{25} -44.2317i q^{29} +(119.149 + 68.7907i) q^{31} +(45.4857 + 35.7837i) q^{35} +(70.6631 + 122.392i) q^{37} +337.946 q^{41} -417.510 q^{43} +(-145.043 - 251.222i) q^{47} +(248.327 - 236.607i) q^{49} +(-14.7904 - 8.53925i) q^{53} +120.127i q^{55} +(299.818 - 519.300i) q^{59} +(459.882 - 265.513i) q^{61} +(37.3586 - 21.5690i) q^{65} +(-325.105 + 563.098i) q^{67} -934.128i q^{71} +(787.145 + 454.458i) q^{73} +(704.683 + 101.449i) q^{77} +(397.516 + 688.517i) q^{79} -314.865 q^{83} +297.378 q^{85} +(179.521 + 310.940i) q^{89} +(-94.9775 - 237.368i) q^{91} +(30.8341 + 17.8021i) q^{95} +80.5572i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 24 q^{7}+O(q^{10}) \) 48 * q + 24 * q^7	\( 48 q + 24 q^{7} - 540 q^{19} - 924 q^{25} - 648 q^{31} - 132 q^{37} + 792 q^{43} + 672 q^{49} + 12 q^{67} + 2412 q^{73} - 1680 q^{79} + 480 q^{85} - 1404 q^{91}+O(q^{100}) \) 48 * q + 24 * q^7 - 540 * q^19 - 924 * q^25 - 648 * q^31 - 132 * q^37 + 792 * q^43 + 672 * q^49 + 12 * q^67 + 2412 * q^73 - 1680 * q^79 + 480 * q^85 - 1404 * q^91

Introduction

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