LMFDB - Embedded newform 448.4.j.b.335.27

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(111,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

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

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

chi := DirichletCharacter("448.111");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.j (of order $4$, 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:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$44$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			335.27
Character	$\chi$	$=$	448.335
Dual form			448.4.j.b.111.27

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.04007 - 2.04007i) q^{3} +(1.81821 - 1.81821i) q^{5} +(10.2362 + 15.4344i) q^{7} +18.6762i q^{9} +O(q^{10})$	\(q+(2.04007 - 2.04007i) q^{3} +(1.81821 - 1.81821i) q^{5} +(10.2362 + 15.4344i) q^{7} +18.6762i q^{9} +(-9.40331 + 9.40331i) q^{11} +(31.8404 + 31.8404i) q^{13} -7.41856i q^{15} -99.6350i q^{17} +(-105.289 + 105.289i) q^{19} +(52.3698 + 10.6047i) q^{21} -196.106 q^{23} +118.388i q^{25} +(93.1827 + 93.1827i) q^{27} +(-0.691326 + 0.691326i) q^{29} -152.952 q^{31} +38.3668i q^{33} +(46.6746 + 9.45147i) q^{35} +(84.9752 + 84.9752i) q^{37} +129.913 q^{39} +321.144 q^{41} +(-149.724 + 149.724i) q^{43} +(33.9574 + 33.9574i) q^{45} +422.042 q^{47} +(-133.441 + 315.978i) q^{49} +(-203.262 - 203.262i) q^{51} +(-449.294 - 449.294i) q^{53} +34.1944i q^{55} +429.595i q^{57} +(537.818 + 537.818i) q^{59} +(352.641 + 352.641i) q^{61} +(-288.256 + 191.173i) q^{63} +115.785 q^{65} +(255.856 + 255.856i) q^{67} +(-400.070 + 400.070i) q^{69} -59.2487 q^{71} +370.065 q^{73} +(241.520 + 241.520i) q^{75} +(-241.388 - 48.8805i) q^{77} -554.493i q^{79} -124.060 q^{81} +(53.1636 - 53.1636i) q^{83} +(-181.158 - 181.158i) q^{85} +2.82071i q^{87} +606.267 q^{89} +(-165.514 + 817.363i) q^{91} +(-312.033 + 312.033i) q^{93} +382.876i q^{95} -588.427i q^{97} +(-175.618 - 175.618i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 4 q^{7}+O(q^{10}) $ 88 * q + 4 * q^7	$ 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77} - 1952 q^{81} + 496 q^{85} + 1964 q^{91} - 112 q^{93} - 7592 q^{99}+O(q^{100}) $ 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$3$	2.04007	−	2.04007i	0.392612	−	0.392612i	−0.483006	−	0.875617i	$-0.660455\pi$
$3$	2.04007	−	2.04007i	0.392612	−	0.392612i	0.875617	+	0.483006i	$0.160455\pi$
$4$		0			0
$5$	1.81821	−	1.81821i	0.162626	−	0.162626i	−0.621103	−	0.783729i	$-0.713315\pi$
$5$	1.81821	−	1.81821i	0.162626	−	0.162626i	0.783729	+	0.621103i	$0.213315\pi$
$6$		0			0
$7$	10.2362	+	15.4344i	0.552702	+	0.833379i
$7$	10.2362	+	15.4344i	0.552702	+	0.833379i
$8$		0			0
$9$			18.6762i			0.691712i
$10$		0			0
$11$	−9.40331	+	9.40331i	−0.257746	+	0.257746i	−0.824137	−	0.566391i	$-0.808339\pi$
$11$	−9.40331	+	9.40331i	−0.257746	+	0.257746i	0.566391	+	0.824137i	$0.308339\pi$
$12$		0			0
$13$	31.8404	+	31.8404i	0.679304	+	0.679304i	0.959843	−	0.280539i	$-0.0905131\pi$
$13$	31.8404	+	31.8404i	0.679304	+	0.679304i	−0.280539	+	0.959843i	$0.590513\pi$
$14$		0			0
$15$		−	7.41856i		−	0.127698i
$16$		0			0
$17$		−	99.6350i		−	1.42147i	−0.703458	−	0.710736i	$-0.748362\pi$
$17$		−	99.6350i		−	1.42147i	0.703458	−	0.710736i	$-0.251638\pi$
$18$		0			0
$19$	−105.289	+	105.289i	−1.27132	+	1.27132i	−0.325917	+	0.945398i	$0.605673\pi$
$19$	−105.289	+	105.289i	−1.27132	+	1.27132i	−0.945398	+	0.325917i	$0.894327\pi$
$20$		0			0
$21$	52.3698	+	10.6047i	0.544192	+	0.110197i
$22$		0			0
$23$	−196.106			−1.77786			−0.888932	−	0.458039i	$-0.848552\pi$
$23$	−196.106			−1.77786			−0.888932	+	0.458039i	$0.848552\pi$
$24$		0			0
$25$			118.388i			0.947106i
$26$		0			0
$27$	93.1827	+	93.1827i	0.664186	+	0.664186i
$28$		0			0
$29$	−0.691326	+	0.691326i	−0.00442676	+	0.00442676i	−0.709317	−	0.704890i	$-0.750996\pi$
$29$	−0.691326	+	0.691326i	−0.00442676	+	0.00442676i	0.704890	+	0.709317i	$0.250996\pi$
$30$		0			0
$31$	−152.952			−0.886161			−0.443081	−	0.896482i	$-0.646114\pi$
$31$	−152.952			−0.886161			−0.443081	+	0.896482i	$0.646114\pi$
$32$		0			0
$33$			38.3668i			0.202388i
$34$		0			0
$35$	46.6746	+	9.45147i	0.225413	+	0.0456454i
$36$		0			0
$37$	84.9752	+	84.9752i	0.377563	+	0.377563i	0.870222	−	0.492659i	$-0.163975\pi$
$37$	84.9752	+	84.9752i	0.377563	+	0.377563i	−0.492659	+	0.870222i	$0.663975\pi$
$38$		0			0
$39$	129.913			0.533405
$40$		0			0
$41$	321.144			1.22328			0.611638	−	0.791138i	$-0.290511\pi$
$41$	321.144			1.22328			0.611638	+	0.791138i	$0.290511\pi$
$42$		0			0
$43$	−149.724	+	149.724i	−0.530993	+	0.530993i	−0.920868	−	0.389875i	$-0.872518\pi$
$43$	−149.724	+	149.724i	−0.530993	+	0.530993i	0.389875	+	0.920868i	$0.372518\pi$
$44$		0			0
$45$	33.9574	+	33.9574i	0.112490	+	0.112490i
$46$		0			0
$47$	422.042			1.30981			0.654905	−	0.755711i	$-0.272709\pi$
$47$	422.042			1.30981			0.654905	+	0.755711i	$0.272709\pi$
$48$		0			0
$49$	−133.441	+	315.978i	−0.389042	+	0.921220i
$50$		0			0
$51$	−203.262	−	203.262i	−0.558087	−	0.558087i
$52$		0			0
$53$	−449.294	−	449.294i	−1.16444	−	1.16444i	−0.983493	−	0.180947i	$-0.942084\pi$
$53$	−449.294	−	449.294i	−1.16444	−	1.16444i	−0.180947	−	0.983493i	$-0.557916\pi$
$54$		0			0
$55$			34.1944i			0.0838323i
$56$		0			0
$57$			429.595i			0.998267i
$58$		0			0
$59$	537.818	+	537.818i	1.18674	+	1.18674i	0.977962	+	0.208782i	$0.0669498\pi$
$59$	537.818	+	537.818i	1.18674	+	1.18674i	0.208782	+	0.977962i	$0.433050\pi$
$60$		0			0
$61$	352.641	+	352.641i	0.740182	+	0.740182i	0.972613	−	0.232431i	$-0.0746681\pi$
$61$	352.641	+	352.641i	0.740182	+	0.740182i	−0.232431	+	0.972613i	$0.574668\pi$
$62$		0			0
$63$	−288.256	+	191.173i	−0.576458	+	0.382310i
$64$		0			0
$65$	115.785			0.220945
$66$		0			0
$67$	255.856	+	255.856i	0.466534	+	0.466534i	0.900790	−	0.434256i	$-0.142989\pi$
$67$	255.856	+	255.856i	0.466534	+	0.466534i	−0.434256	+	0.900790i	$0.642989\pi$
$68$		0			0
$69$	−400.070	+	400.070i	−0.698010	+	0.698010i
$70$		0			0
$71$	−59.2487			−0.0990356			−0.0495178	−	0.998773i	$-0.515768\pi$
$71$	−59.2487			−0.0990356			−0.0495178	+	0.998773i	$0.515768\pi$
$72$		0			0
$73$	370.065			0.593326			0.296663	−	0.954982i	$-0.404126\pi$
$73$	370.065			0.593326			0.296663	+	0.954982i	$0.404126\pi$
$74$		0			0
$75$	241.520	+	241.520i	0.371845	+	0.371845i
$76$		0			0
$77$	−241.388	−	48.8805i	−0.357256	−	0.0723435i
$78$		0			0
$79$		−	554.493i		−	0.789688i	−0.918748	−	0.394844i	$-0.870799\pi$
$79$		−	554.493i		−	0.789688i	0.918748	−	0.394844i	$-0.129201\pi$
$80$		0			0
$81$	−124.060			−0.170178
$82$		0			0
$83$	53.1636	−	53.1636i	0.0703068	−	0.0703068i	−0.671079	−	0.741386i	$-0.734169\pi$
$83$	53.1636	−	53.1636i	0.0703068	−	0.0703068i	0.741386	+	0.671079i	$0.234169\pi$
$84$		0			0
$85$	−181.158	−	181.158i	−0.231168	−	0.231168i
$86$		0			0
$87$			2.82071i			0.00347600i
$88$		0			0
$89$	606.267			0.722069			0.361035	−	0.932552i	$-0.382424\pi$
$89$	606.267			0.722069			0.361035	+	0.932552i	$0.382424\pi$
$90$		0			0
$91$	−165.514	+	817.363i	−0.190665	+	0.941570i
$92$		0			0
$93$	−312.033	+	312.033i	−0.347917	+	0.347917i
$94$		0			0
$95$			382.876i			0.413498i
$96$		0			0
$97$		−	588.427i		−	0.615935i	−0.951397	−	0.307967i	$-0.900351\pi$
$97$		−	588.427i		−	0.615935i	0.951397	−	0.307967i	$-0.0996488\pi$
$98$		0			0
$99$	−175.618	−	175.618i	−0.178286	−	0.178286i
$100$		0			0
$101$	−568.399	+	568.399i	−0.559978	+	0.559978i	−0.929301	−	0.369323i	$-0.879590\pi$
$101$	−568.399	+	568.399i	−0.559978	+	0.559978i	0.369323	+	0.929301i	$0.379590\pi$
$102$		0			0
$103$		−	867.103i		−	0.829497i	−0.909936	−	0.414748i	$-0.863870\pi$
$103$		−	867.103i		−	0.829497i	0.909936	−	0.414748i	$-0.136130\pi$
$104$		0			0
$105$	114.501	−	75.9377i	0.106421	−	0.0705787i
$106$		0			0
$107$	−1251.28	+	1251.28i	−1.13052	+	1.13052i	−0.140428	+	0.990091i	$0.544848\pi$
$107$	−1251.28	+	1251.28i	−1.13052	+	1.13052i	−0.990091	+	0.140428i	$0.955152\pi$
$108$		0			0
$109$	−6.64025	+	6.64025i	−0.00583505	+	0.00583505i	−0.710018	−	0.704183i	$-0.751313\pi$
$109$	−6.64025	+	6.64025i	−0.00583505	+	0.00583505i	0.704183	+	0.710018i	$0.251313\pi$
$110$		0			0
$111$	346.711			0.296471
$112$		0			0
$113$	588.858			0.490222			0.245111	−	0.969495i	$-0.421176\pi$
$113$	588.858			0.490222			0.245111	+	0.969495i	$0.421176\pi$
$114$		0			0
$115$	−356.562	+	356.562i	−0.289127	+	0.289127i
$116$		0			0
$117$	−594.659	+	594.659i	−0.469883	+	0.469883i
$118$		0			0
$119$	1537.81	−	1019.88i	1.18463	−	0.785650i
$120$		0			0
$121$			1154.16i			0.867134i
$122$		0			0
$123$	655.157	−	655.157i	0.480272	−	0.480272i
$124$		0			0
$125$	442.532	+	442.532i	0.316650	+	0.316650i
$126$		0			0
$127$		−	469.590i		−	0.328105i	−0.986452	−	0.164053i	$-0.947543\pi$
$127$		−	469.590i		−	0.328105i	0.986452	−	0.164053i	$-0.0524567\pi$
$128$		0			0
$129$			610.895i			0.416948i
$130$		0			0
$131$	383.126	−	383.126i	0.255526	−	0.255526i	−0.567706	−	0.823232i	$-0.692169\pi$
$131$	383.126	−	383.126i	0.255526	−	0.255526i	0.823232	+	0.567706i	$0.192169\pi$
$132$		0			0
$133$	−2702.83	−	547.316i	−1.76215	−	0.356830i
$134$		0			0
$135$	338.852			0.216028
$136$		0			0
$137$		−	304.251i		−	0.189736i	−0.995490	−	0.0948682i	$-0.969757\pi$
$137$		−	304.251i		−	0.189736i	0.995490	−	0.0948682i	$-0.0302430\pi$
$138$		0			0
$139$	1448.57	+	1448.57i	0.883927	+	0.883927i	0.993931	−	0.110004i	$-0.0350864\pi$
$139$	1448.57	+	1448.57i	0.883927	+	0.883927i	−0.110004	+	0.993931i	$0.535086\pi$
$140$		0			0
$141$	860.995	−	860.995i	0.514247	−	0.514247i
$142$		0			0
$143$	−598.811			−0.350175
$144$		0			0
$145$			2.51396i			0.00143981i
$146$		0			0
$147$	372.389	+	916.848i	0.208939	+	0.514424i
$148$		0			0
$149$	1217.88	+	1217.88i	0.669616	+	0.669616i	0.957627	−	0.288011i	$-0.0929940\pi$
$149$	1217.88	+	1217.88i	0.669616	+	0.669616i	−0.288011	+	0.957627i	$0.592994\pi$
$150$		0			0
$151$	952.582			0.513378			0.256689	−	0.966494i	$-0.417368\pi$
$151$	952.582			0.513378			0.256689	+	0.966494i	$0.417368\pi$
$152$		0			0
$153$	1860.81			0.983250
$154$		0			0
$155$	−278.099	+	278.099i	−0.144113	+	0.144113i
$156$		0			0
$157$	−2019.97	−	2019.97i	−1.02682	−	1.02682i	−0.999630	−	0.0271905i	$-0.991344\pi$
$157$	−2019.97	−	2019.97i	−1.02682	−	1.02682i	−0.0271905	−	0.999630i	$-0.508656\pi$
$158$		0			0
$159$	−1833.18			−0.914345
$160$		0			0
$161$	−2007.37	−	3026.77i	−0.982629	−	1.48164i
$162$		0			0
$163$	−1401.89	−	1401.89i	−0.673645	−	0.673645i	0.284909	−	0.958555i	$-0.408037\pi$
$163$	−1401.89	−	1401.89i	−0.673645	−	0.673645i	−0.958555	+	0.284909i	$0.908037\pi$
$164$		0			0
$165$	69.7590	+	69.7590i	0.0329135	+	0.0329135i
$166$		0			0
$167$		−	1552.68i		−	0.719462i	−0.933056	−	0.359731i	$-0.882868\pi$
$167$		−	1552.68i		−	0.719462i	0.933056	−	0.359731i	$-0.117132\pi$
$168$		0			0
$169$		−	169.372i		−	0.0770924i
$170$		0			0
$171$	−1966.40	−	1966.40i	−0.879384	−	0.879384i
$172$		0			0
$173$	−1997.09	−	1997.09i	−0.877663	−	0.877663i	0.115630	−	0.993292i	$-0.463111\pi$
$173$	−1997.09	−	1997.09i	−0.877663	−	0.877663i	−0.993292	+	0.115630i	$0.963111\pi$
$174$		0			0
$175$	−1827.25	+	1211.84i	−0.789298	+	0.523467i
$176$		0			0
$177$	2194.37			0.931859
$178$		0			0
$179$	−1233.13	−	1233.13i	−0.514910	−	0.514910i	0.401117	−	0.916027i	$-0.368622\pi$
$179$	−1233.13	−	1233.13i	−0.514910	−	0.514910i	−0.916027	+	0.401117i	$0.868622\pi$
$180$		0			0
$181$	−211.217	+	211.217i	−0.0867383	+	0.0867383i	−0.749145	−	0.662406i	$-0.769535\pi$
$181$	−211.217	+	211.217i	−0.0867383	+	0.0867383i	0.662406	+	0.749145i	$0.269535\pi$
$182$		0			0
$183$	1438.83			0.581208
$184$		0			0
$185$	309.006			0.122803
$186$		0			0
$187$	936.899	+	936.899i	0.366379	+	0.366379i
$188$		0			0
$189$	−484.384	+	2392.05i	−0.186422	+	0.920615i
$190$		0			0
$191$		−	1359.31i		−	0.514956i	−0.966284	−	0.257478i	$-0.917109\pi$
$191$		−	1359.31i		−	0.514956i	0.966284	−	0.257478i	$-0.0828915\pi$
$192$		0			0
$193$	5274.75			1.96728			0.983638	−	0.180154i	$-0.0576595\pi$
$193$	5274.75			1.96728			0.983638	+	0.180154i	$0.0576595\pi$
$194$		0			0
$195$	236.210	−	236.210i	0.0867455	−	0.0867455i
$196$		0			0
$197$	−941.736	−	941.736i	−0.340588	−	0.340588i	0.516000	−	0.856588i	$-0.327420\pi$
$197$	−941.736	−	941.736i	−0.340588	−	0.340588i	−0.856588	+	0.516000i	$0.827420\pi$
$198$		0			0
$199$		−	1628.75i		−	0.580196i	−0.956997	−	0.290098i	$-0.906312\pi$
$199$		−	1628.75i		−	0.580196i	0.956997	−	0.290098i	$-0.0936879\pi$
$200$		0			0
$201$	1043.93			0.366333
$202$		0			0
$203$	−17.7467	−	3.59367i	−0.00613585	−	0.00124249i
$204$		0			0
$205$	583.908	−	583.908i	0.198936	−	0.198936i
$206$		0			0
$207$		−	3662.52i		−	1.22977i
$208$		0			0
$209$		−	1980.13i		−	0.655352i
$210$		0			0
$211$	1419.37	+	1419.37i	0.463097	+	0.463097i	0.899669	−	0.436572i	$-0.143808\pi$
$211$	1419.37	+	1419.37i	0.463097	+	0.463097i	−0.436572	+	0.899669i	$0.643808\pi$
$212$		0			0
$213$	−120.872	+	120.872i	−0.0388825	+	0.0388825i
$214$		0			0
$215$			544.460i			0.172706i
$216$		0			0
$217$	−1565.64	−	2360.72i	−0.489783	−	0.738508i
$218$		0			0
$219$	754.958	−	754.958i	0.232947	−	0.232947i
$220$		0			0
$221$	3172.42	−	3172.42i	0.965612	−	0.965612i
$222$		0			0
$223$	−3437.98			−1.03240			−0.516198	−	0.856469i	$-0.672653\pi$
$223$	−3437.98			−1.03240			−0.516198	+	0.856469i	$0.672653\pi$
$224$		0			0
$225$	−2211.04			−0.655124
$226$		0			0
$227$	241.799	−	241.799i	0.0706995	−	0.0706995i	−0.670873	−	0.741572i	$-0.734080\pi$
$227$	241.799	−	241.799i	0.0706995	−	0.0706995i	0.741572	+	0.670873i	$0.234080\pi$
$228$		0			0
$229$	2812.47	−	2812.47i	0.811585	−	0.811585i	−0.173286	−	0.984871i	$-0.555439\pi$
$229$	2812.47	−	2812.47i	0.811585	−	0.811585i	0.984871	+	0.173286i	$0.0554387\pi$
$230$		0			0
$231$	−592.169	+	392.730i	−0.168666	+	0.111860i
$232$		0			0
$233$			4660.18i			1.31029i	0.755502	+	0.655147i	$0.227393\pi$
$233$			4660.18i			1.31029i	−0.755502	+	0.655147i	$0.772607\pi$
$234$		0			0
$235$	767.362	−	767.362i	0.213009	−	0.213009i
$236$		0			0
$237$	−1131.21	−	1131.21i	−0.310041	−	0.310041i
$238$		0			0
$239$		−	2269.24i		−	0.614162i	−0.951683	−	0.307081i	$-0.900648\pi$
$239$		−	2269.24i		−	0.614162i	0.951683	−	0.307081i	$-0.0993523\pi$
$240$		0			0
$241$		−	3867.21i		−	1.03365i	−0.856092	−	0.516824i	$-0.827114\pi$
$241$		−	3867.21i		−	1.03365i	0.856092	−	0.516824i	$-0.172886\pi$
$242$		0			0
$243$	−2769.02	+	2769.02i	−0.731000	+	0.731000i
$244$		0			0
$245$	331.891	+	817.141i	0.0865460	+	0.213083i
$246$		0			0
$247$	−6704.91			−1.72722
$248$		0			0
$249$		−	216.915i		−	0.0552065i
$250$		0			0
$251$	−193.623	−	193.623i	−0.0486909	−	0.0486909i	0.682342	−	0.731033i	$-0.260961\pi$
$251$	−193.623	−	193.623i	−0.0486909	−	0.0486909i	−0.731033	+	0.682342i	$0.760961\pi$
$252$		0			0
$253$	1844.04	−	1844.04i	0.458237	−	0.458237i
$254$		0			0
$255$	−739.149			−0.181519
$256$		0			0
$257$			1379.66i			0.334867i	0.985883	+	0.167434i	$0.0535480\pi$
$257$			1379.66i			0.334867i	−0.985883	+	0.167434i	$0.946452\pi$
$258$		0			0
$259$	−441.720	+	2181.36i	−0.105973	+	0.523333i
$260$		0			0
$261$	−12.9114	−	12.9114i	−0.00306204	−	0.00306204i
$262$		0			0
$263$	4976.15			1.16670			0.583351	−	0.812220i	$-0.301741\pi$
$263$	4976.15			1.16670			0.583351	+	0.812220i	$0.301741\pi$
$264$		0			0
$265$	−1633.82			−0.378736
$266$		0			0
$267$	1236.83	−	1236.83i	0.283493	−	0.283493i
$268$		0			0
$269$	3118.78	+	3118.78i	0.706898	+	0.706898i	0.965882	−	0.258983i	$-0.0833875\pi$
$269$	3118.78	+	3118.78i	0.706898	+	0.706898i	−0.258983	+	0.965882i	$0.583387\pi$
$270$		0			0
$271$	6633.06			1.48683			0.743413	−	0.668833i	$-0.233206\pi$
$271$	6633.06			1.48683			0.743413	+	0.668833i	$0.233206\pi$
$272$		0			0
$273$	1329.82	+	2005.14i	0.294814	+	0.444529i
$274$		0			0
$275$	−1113.24	−	1113.24i	−0.244112	−	0.244112i
$276$		0			0
$277$	610.232	+	610.232i	0.132366	+	0.132366i	0.770186	−	0.637820i	$-0.220164\pi$
$277$	610.232	+	610.232i	0.132366	+	0.132366i	−0.637820	+	0.770186i	$0.720164\pi$
$278$		0			0
$279$		−	2856.57i		−	0.612968i
$280$		0			0
$281$		−	3180.56i		−	0.675217i	−0.941287	−	0.337609i	$-0.890382\pi$
$281$		−	3180.56i		−	0.675217i	0.941287	−	0.337609i	$-0.109618\pi$
$282$		0			0
$283$	1065.15	+	1065.15i	0.223733	+	0.223733i	0.810068	−	0.586335i	$-0.199430\pi$
$283$	1065.15	+	1065.15i	0.223733	+	0.223733i	−0.586335	+	0.810068i	$0.699430\pi$
$284$		0			0
$285$	781.095	+	781.095i	0.162344	+	0.162344i
$286$		0			0
$287$	3287.29	+	4956.67i	0.676106	+	1.01945i
$288$		0			0
$289$	−5014.13			−1.02059
$290$		0			0
$291$	−1200.43	−	1200.43i	−0.241823	−	0.241823i
$292$		0			0
$293$	36.2480	−	36.2480i	0.00722742	−	0.00722742i	−0.703484	−	0.710711i	$-0.748373\pi$
$293$	36.2480	−	36.2480i	0.00722742	−	0.00722742i	0.710711	+	0.703484i	$0.248373\pi$
$294$		0			0
$295$	1955.73			0.385991
$296$		0			0
$297$	−1752.45			−0.342382
$298$		0			0
$299$	−6244.10	−	6244.10i	−1.20771	−	1.20771i
$300$		0			0
$301$	−3843.50	−	778.298i	−0.735999	−	0.149038i
$302$		0			0
$303$			2319.15i			0.439708i
$304$		0			0
$305$	1282.35			0.240745
$306$		0			0
$307$	4186.02	−	4186.02i	0.778204	−	0.778204i	−0.201321	−	0.979525i	$-0.564523\pi$
$307$	4186.02	−	4186.02i	0.778204	−	0.778204i	0.979525	+	0.201321i	$0.0645234\pi$
$308$		0			0
$309$	−1768.95	−	1768.95i	−0.325670	−	0.325670i
$310$		0			0
$311$			2066.23i			0.376736i	0.982099	+	0.188368i	$0.0603198\pi$
$311$			2066.23i			0.376736i	−0.982099	+	0.188368i	$0.939680\pi$
$312$		0			0
$313$	−10741.6			−1.93978			−0.969889	−	0.243549i	$-0.921688\pi$
$313$	−10741.6			−1.93978			−0.969889	+	0.243549i	$0.921688\pi$
$314$		0			0
$315$	−176.518	+	871.705i	−0.0315735	+	0.155921i
$316$		0			0
$317$	−3469.39	+	3469.39i	−0.614702	+	0.614702i	−0.944167	−	0.329466i	$-0.893131\pi$
$317$	−3469.39	+	3469.39i	−0.614702	+	0.614702i	0.329466	+	0.944167i	$0.393131\pi$
$318$		0			0
$319$		−	13.0015i		−	0.00228196i
$320$		0			0
$321$			5105.39i			0.887710i
$322$		0			0
$323$	10490.5	+	10490.5i	1.80714	+	1.80714i
$324$		0			0
$325$	−3769.53	+	3769.53i	−0.643373	+	0.643373i
$326$		0			0
$327$			27.0932i			0.00458182i
$328$		0			0
$329$	4320.09	+	6513.96i	0.723934	+	1.09157i
$330$		0			0
$331$	−4176.68	+	4176.68i	−0.693569	+	0.693569i	−0.963015	−	0.269447i	$-0.913159\pi$
$331$	−4176.68	+	4176.68i	−0.693569	+	0.693569i	0.269447	+	0.963015i	$0.413159\pi$
$332$		0			0
$333$	−1587.02	+	1587.02i	−0.261165	+	0.261165i
$334$		0			0
$335$	930.401			0.151741
$336$		0			0
$337$	−8694.46			−1.40539			−0.702697	−	0.711490i	$-0.748021\pi$
$337$	−8694.46			−1.40539			−0.702697	+	0.711490i	$0.748021\pi$
$338$		0			0
$339$	1201.31	−	1201.31i	0.192467	−	0.192467i
$340$		0			0
$341$	1438.25	−	1438.25i	0.228404	−	0.228404i
$342$		0			0
$343$	−6242.87	+	1174.83i	−0.982750	+	0.184941i
$344$		0			0
$345$			1454.82i			0.227029i
$346$		0			0
$347$	5755.26	−	5755.26i	0.890370	−	0.890370i	−0.104188	−	0.994558i	$-0.533224\pi$
$347$	5755.26	−	5755.26i	0.890370	−	0.890370i	0.994558	+	0.104188i	$0.0332243\pi$
$348$		0			0
$349$	4727.84	+	4727.84i	0.725144	+	0.725144i	0.969648	−	0.244504i	$-0.0786252\pi$
$349$	4727.84	+	4727.84i	0.725144	+	0.725144i	−0.244504	+	0.969648i	$0.578625\pi$
$350$		0			0
$351$			5933.96i			0.902368i
$352$		0			0
$353$			9966.21i			1.50268i	0.659913	+	0.751342i	$0.270593\pi$
$353$			9966.21i			1.50268i	−0.659913	+	0.751342i	$0.729407\pi$
$354$		0			0
$355$	−107.727	+	107.727i	−0.0161058	+	0.0161058i
$356$		0			0
$357$	1056.60	−	5217.86i	0.156642	−	0.773554i
$358$		0			0
$359$	−5687.50			−0.836141			−0.418071	−	0.908414i	$-0.637294\pi$
$359$	−5687.50			−0.836141			−0.418071	+	0.908414i	$0.637294\pi$
$360$		0			0
$361$		−	15312.6i		−	2.23249i
$362$		0			0
$363$	2354.56	+	2354.56i	0.340447	+	0.340447i
$364$		0			0
$365$	672.857	−	672.857i	0.0964902	−	0.0964902i
$366$		0			0
$367$	4470.48			0.635850			0.317925	−	0.948116i	$-0.397014\pi$
$367$	4470.48			0.635850			0.317925	+	0.948116i	$0.397014\pi$
$368$		0			0
$369$			5997.76i			0.846154i
$370$		0			0
$371$	2335.53	−	11533.6i	0.326832	−	1.61401i
$372$		0			0
$373$	3699.43	+	3699.43i	0.513537	+	0.513537i	0.915608	−	0.402071i	$-0.131710\pi$
$373$	3699.43	+	3699.43i	0.513537	+	0.513537i	−0.402071	+	0.915608i	$0.631710\pi$
$374$		0			0
$375$	1805.59			0.248641
$376$		0			0
$377$	−44.0243			−0.00601423
$378$		0			0
$379$	1853.29	−	1853.29i	0.251179	−	0.251179i	−0.570275	−	0.821454i	$-0.693163\pi$
$379$	1853.29	−	1853.29i	0.251179	−	0.251179i	0.821454	+	0.570275i	$0.193163\pi$
$380$		0			0
$381$	−957.996	−	957.996i	−0.128818	−	0.128818i
$382$		0			0
$383$	4335.44			0.578409			0.289204	−	0.957267i	$-0.406609\pi$
$383$	4335.44			0.578409			0.289204	+	0.957267i	$0.406609\pi$
$384$		0			0
$385$	−527.770	+	350.020i	−0.0698641	+	0.0463342i
$386$		0			0
$387$	−2796.28	−	2796.28i	−0.367294	−	0.367294i
$388$		0			0
$389$	8332.19	+	8332.19i	1.08601	+	1.08601i	0.995935	+	0.0900779i	$0.0287116\pi$
$389$	8332.19	+	8332.19i	1.08601	+	1.08601i	0.0900779	+	0.995935i	$0.471288\pi$
$390$		0			0
$391$			19539.0i			2.52719i
$392$		0			0
$393$		−	1563.21i		−	0.200645i
$394$		0			0
$395$	−1008.19	−	1008.19i	−0.128424	−	0.128424i
$396$		0			0
$397$	2142.20	+	2142.20i	0.270816	+	0.270816i	0.829429	−	0.558613i	$-0.188666\pi$
$397$	2142.20	+	2142.20i	0.270816	+	0.270816i	−0.558613	+	0.829429i	$0.688666\pi$
$398$		0			0
$399$	−6630.54	+	4397.41i	−0.831935	+	0.551744i
$400$		0			0
$401$	1030.83			0.128372			0.0641860	−	0.997938i	$-0.479555\pi$
$401$	1030.83			0.128372			0.0641860	+	0.997938i	$0.479555\pi$
$402$		0			0
$403$	−4870.06	−	4870.06i	−0.601973	−	0.601973i
$404$		0			0
$405$	−225.567	+	225.567i	−0.0276753	+	0.0276753i
$406$		0			0
$407$	−1598.10			−0.194631
$408$		0			0
$409$	767.913			0.0928382			0.0464191	−	0.998922i	$-0.485219\pi$
$409$	767.913			0.0928382			0.0464191	+	0.998922i	$0.485219\pi$
$410$		0			0
$411$	−620.692	−	620.692i	−0.0744927	−	0.0744927i
$412$		0			0
$413$	−2795.69	+	13806.1i	−0.333092	+	1.64492i
$414$		0			0
$415$		−	193.325i		−	0.0228674i
$416$		0			0
$417$	5910.36			0.694080
$418$		0			0
$419$	−4201.23	+	4201.23i	−0.489841	+	0.489841i	−0.908256	−	0.418415i	$-0.862586\pi$
$419$	−4201.23	+	4201.23i	−0.489841	+	0.489841i	0.418415	+	0.908256i	$0.362586\pi$
$420$		0			0
$421$	1071.52	+	1071.52i	0.124044	+	0.124044i	0.766404	−	0.642359i	$-0.222044\pi$
$421$	1071.52	+	1071.52i	0.124044	+	0.124044i	−0.642359	+	0.766404i	$0.722044\pi$
$422$		0			0
$423$			7882.15i			0.906012i
$424$		0			0
$425$	11795.6			1.34628
$426$		0			0
$427$	−1833.11	+	9052.50i	−0.207752	+	1.02595i
$428$		0			0
$429$	−1221.62	+	1221.62i	−0.137483	+	0.137483i
$430$		0			0
$431$			1446.43i			0.161652i	0.996728	+	0.0808258i	$0.0257558\pi$
$431$			1446.43i			0.161652i	−0.996728	+	0.0808258i	$0.974244\pi$
$432$		0			0
$433$		−	9272.67i		−	1.02914i	−0.857449	−	0.514568i	$-0.827952\pi$
$433$		−	9272.67i		−	1.02914i	0.857449	−	0.514568i	$-0.172048\pi$
$434$		0			0
$435$	5.12865	+	5.12865i	0.000565287	+	0.000565287i
$436$		0			0
$437$	20647.8	−	20647.8i	2.26023	−	2.26023i
$438$		0			0
$439$			10245.7i			1.11390i	0.830547	+	0.556949i	$0.188028\pi$
$439$			10245.7i			1.11390i	−0.830547	+	0.556949i	$0.811972\pi$
$440$		0			0
$441$	−5901.29	−	2492.18i	−0.637219	−	0.269105i
$442$		0			0
$443$	−3739.21	+	3739.21i	−0.401027	+	0.401027i	−0.878595	−	0.477568i	$-0.841519\pi$
$443$	−3739.21	+	3739.21i	−0.401027	+	0.401027i	0.477568	+	0.878595i	$0.341519\pi$
$444$		0			0
$445$	1102.32	−	1102.32i	0.117427	−	0.117427i
$446$		0			0
$447$	4969.13			0.525798
$448$		0			0
$449$	−8246.73			−0.866787			−0.433393	−	0.901205i	$-0.642684\pi$
$449$	−8246.73			−0.866787			−0.433393	+	0.901205i	$0.642684\pi$
$450$		0			0
$451$	−3019.82	+	3019.82i	−0.315294	+	0.315294i
$452$		0			0
$453$	1943.33	−	1943.33i	0.201558	−	0.201558i
$454$		0			0
$455$	1185.20	+	1787.08i	0.122117	+	0.184131i
$456$		0			0
$457$			6987.70i			0.715253i	0.933865	+	0.357626i	$0.116414\pi$
$457$			6987.70i			0.715253i	−0.933865	+	0.357626i	$0.883586\pi$
$458$		0			0
$459$	9284.26	−	9284.26i	0.944122	−	0.944122i
$460$		0			0
$461$	−2670.38	−	2670.38i	−0.269788	−	0.269788i	0.559227	−	0.829015i	$-0.311098\pi$
$461$	−2670.38	−	2670.38i	−0.269788	−	0.269788i	−0.829015	+	0.559227i	$0.811098\pi$
$462$		0			0
$463$		−	14094.3i		−	1.41473i	−0.706848	−	0.707365i	$-0.749884\pi$
$463$		−	14094.3i		−	1.41473i	0.706848	−	0.707365i	$-0.250116\pi$
$464$		0			0
$465$			1134.68i			0.113161i
$466$		0			0
$467$	686.206	−	686.206i	0.0679953	−	0.0679953i	−0.672291	−	0.740287i	$-0.734690\pi$
$467$	686.206	−	686.206i	0.0679953	−	0.0679953i	0.740287	+	0.672291i	$0.234690\pi$
$468$		0			0
$469$	−1330.00	+	6567.97i	−0.130946	+	0.646654i
$470$		0			0
$471$	−8241.75			−0.806284
$472$		0			0
$473$		−	2815.80i		−	0.273722i
$474$		0			0
$475$	−12465.0	−	12465.0i	−1.20407	−	1.20407i
$476$		0			0
$477$	8391.12	−	8391.12i	0.805457	−	0.805457i
$478$		0			0
$479$	8950.40			0.853766			0.426883	−	0.904307i	$-0.359612\pi$
$479$	8950.40			0.853766			0.426883	+	0.904307i	$0.359612\pi$
$480$		0			0
$481$			5411.30i			0.512960i
$482$		0			0
$483$	−10270.0	−	2079.65i	−0.967499	−	0.195916i
$484$		0			0
$485$	−1069.88	−	1069.88i	−0.100167	−	0.100167i
$486$		0			0
$487$	1046.06			0.0973336			0.0486668	−	0.998815i	$-0.484503\pi$
$487$	1046.06			0.0973336			0.0486668	+	0.998815i	$0.484503\pi$
$488$		0			0
$489$	−5719.89			−0.528962
$490$		0			0
$491$	699.475	−	699.475i	0.0642910	−	0.0642910i	−0.674230	−	0.738521i	$-0.735524\pi$
$491$	699.475	−	699.475i	0.0642910	−	0.0642910i	0.738521	+	0.674230i	$0.235524\pi$
$492$		0			0
$493$	68.8803	+	68.8803i	0.00629252	+	0.00629252i
$494$		0			0
$495$	−638.623			−0.0579878
$496$		0			0
$497$	−606.480	−	914.468i	−0.0547372	−	0.0825342i
$498$		0			0
$499$	10971.6	+	10971.6i	0.984284	+	0.984284i	0.999878	−	0.0155943i	$-0.00496403\pi$
$499$	10971.6	+	10971.6i	0.984284	+	0.984284i	−0.0155943	+	0.999878i	$0.504964\pi$
$500$		0			0
$501$	−3167.58	−	3167.58i	−0.282469	−	0.282469i
$502$		0			0
$503$		−	19231.7i		−	1.70477i	−0.522916	−	0.852384i	$-0.675156\pi$
$503$		−	19231.7i		−	1.70477i	0.522916	−	0.852384i	$-0.324844\pi$
$504$		0			0
$505$			2066.94i			0.182134i
$506$		0			0
$507$	−345.531	−	345.531i	−0.0302674	−	0.0302674i
$508$		0			0
$509$	−114.316	−	114.316i	−0.00995477	−	0.00995477i	0.702112	−	0.712067i	$-0.252241\pi$
$509$	−114.316	−	114.316i	−0.00995477	−	0.00995477i	−0.712067	+	0.702112i	$0.752241\pi$
$510$		0			0
$511$	3788.05	+	5711.73i	0.327932	+	0.494466i
$512$		0			0
$513$	−19622.3			−1.68878
$514$		0			0
$515$	−1576.58	−	1576.58i	−0.134898	−	0.134898i
$516$		0			0
$517$	−3968.59	+	3968.59i	−0.337598	+	0.337598i
$518$		0			0
$519$	−8148.39			−0.689161
$520$		0			0
$521$	−6072.62			−0.510645			−0.255323	−	0.966856i	$-0.582182\pi$
$521$	−6072.62			−0.510645			−0.255323	+	0.966856i	$0.582182\pi$
$522$		0			0
$523$	387.898	+	387.898i	0.0324313	+	0.0324313i	0.723136	−	0.690705i	$-0.242700\pi$
$523$	387.898	+	387.898i	0.0324313	+	0.0324313i	−0.690705	+	0.723136i	$0.742700\pi$
$524$		0			0
$525$	−1255.48	+	6199.96i	−0.104368	+	0.515407i
$526$		0			0
$527$			15239.4i			1.25965i
$528$		0			0
$529$	26290.5			2.16080
$530$		0			0
$531$	−10044.4	+	10044.4i	−0.820885	+	0.820885i
$532$		0			0
$533$	10225.4	+	10225.4i	0.830976	+	0.830976i
$534$		0			0
$535$			4550.18i			0.367703i
$536$		0			0
$537$	−5031.36			−0.404319
$538$		0			0
$539$	−1716.45	−	4226.03i	−0.137167	−	0.337714i
$540$		0			0
$541$	−13620.7	+	13620.7i	−1.08244	+	1.08244i	−0.0861554	+	0.996282i	$0.527458\pi$
$541$	−13620.7	+	13620.7i	−1.08244	+	1.08244i	−0.996282	+	0.0861554i	$0.972542\pi$
$542$		0			0
$543$			861.794i			0.0681089i
$544$		0			0
$545$			24.1468i			0.00189786i
$546$		0			0
$547$	1706.16	+	1706.16i	0.133364	+	0.133364i	0.770638	−	0.637274i	$-0.219938\pi$
$547$	1706.16	+	1706.16i	0.133364	+	0.133364i	−0.637274	+	0.770638i	$0.719938\pi$
$548$		0			0
$549$	−6586.01	+	6586.01i	−0.511993	+	0.511993i
$550$		0			0
$551$		−	145.578i		−	0.0112556i
$552$		0			0
$553$	8558.27	−	5675.89i	0.658110	−	0.436462i
$554$		0			0
$555$	630.394	−	630.394i	0.0482139	−	0.0482139i
$556$		0			0
$557$	7968.29	−	7968.29i	0.606153	−	0.606153i	−0.335785	−	0.941939i	$-0.609002\pi$
$557$	7968.29	−	7968.29i	0.606153	−	0.606153i	0.941939	+	0.335785i	$0.109002\pi$
$558$		0			0
$559$	−9534.55			−0.721411
$560$		0			0
$561$	3822.68			0.287689
$562$		0			0
$563$	6820.50	−	6820.50i	0.510568	−	0.510568i	−0.404132	−	0.914700i	$-0.632427\pi$
$563$	6820.50	−	6820.50i	0.510568	−	0.510568i	0.914700	+	0.404132i	$0.132427\pi$
$564$		0			0
$565$	1070.67	−	1070.67i	0.0797228	−	0.0797228i
$566$		0			0
$567$	−1269.90	−	1914.79i	−0.0940575	−	0.141823i
$568$		0			0
$569$		−	12933.9i		−	0.952929i	−0.879194	−	0.476464i	$-0.841918\pi$
$569$		−	12933.9i		−	0.952929i	0.879194	−	0.476464i	$-0.158082\pi$
$570$		0			0
$571$	2895.27	−	2895.27i	0.212195	−	0.212195i	−0.593004	−	0.805199i	$-0.702058\pi$
$571$	2895.27	−	2895.27i	0.212195	−	0.212195i	0.805199	+	0.593004i	$0.202058\pi$
$572$		0			0
$573$	−2773.10	−	2773.10i	−0.202178	−	0.202178i
$574$		0			0
$575$		−	23216.6i		−	1.68383i
$576$		0			0
$577$		−	1294.59i		−	0.0934046i	−0.998909	−	0.0467023i	$-0.985129\pi$
$577$		−	1294.59i		−	0.0934046i	0.998909	−	0.0467023i	$-0.0148712\pi$
$578$		0			0
$579$	10760.9	−	10760.9i	0.772376	−	0.772376i
$580$		0			0
$581$	1364.74	+	276.356i	0.0974508	+	0.0197335i
$582$		0			0
$583$	8449.70			0.600259
$584$		0			0
$585$			2162.43i			0.152830i
$586$		0			0
$587$	3839.28	+	3839.28i	0.269956	+	0.269956i	0.829082	−	0.559127i	$-0.188863\pi$
$587$	3839.28	+	3839.28i	0.269956	+	0.269956i	−0.559127	+	0.829082i	$0.688863\pi$
$588$		0			0
$589$	16104.2	−	16104.2i	1.12659	−	1.12659i
$590$		0			0
$591$	−3842.41			−0.267438
$592$		0			0
$593$		−	7080.01i		−	0.490289i	−0.969487	−	0.245144i	$-0.921165\pi$
$593$		−	7080.01i		−	0.490289i	0.969487	−	0.245144i	$-0.0788354\pi$
$594$		0			0
$595$	941.697	−	4650.42i	0.0648838	−	0.320418i
$596$		0			0
$597$	−3322.76	−	3322.76i	−0.227792	−	0.227792i
$598$		0			0
$599$	28147.2			1.91997			0.959986	−	0.280049i	$-0.0903507\pi$
$599$	28147.2			1.91997			0.959986	+	0.280049i	$0.0903507\pi$
$600$		0			0
$601$	−17306.7			−1.17464			−0.587318	−	0.809356i	$-0.699816\pi$
$601$	−17306.7			−1.17464			−0.587318	+	0.809356i	$0.699816\pi$
$602$		0			0
$603$	−4778.42	+	4778.42i	−0.322707	+	0.322707i
$604$		0			0
$605$	2098.50	+	2098.50i	0.141018	+	0.141018i
$606$		0			0
$607$	−20620.7			−1.37886			−0.689430	−	0.724353i	$-0.742139\pi$
$607$	−20620.7			−1.37886			−0.689430	+	0.724353i	$0.742139\pi$
$608$		0			0
$609$	−43.5359	+	28.8733i	−0.00289682	+	0.00192119i
$610$		0			0
$611$	13438.0	+	13438.0i	0.889759	+	0.889759i
$612$		0			0
$613$	−19231.6	−	19231.6i	−1.26714	−	1.26714i	−0.947558	−	0.319583i	$-0.896457\pi$
$613$	−19231.6	−	19231.6i	−1.26714	−	1.26714i	−0.319583	−	0.947558i	$-0.603543\pi$
$614$		0			0
$615$		−	2382.43i		−	0.156209i
$616$		0			0
$617$			29293.4i			1.91136i	0.294409	+	0.955679i	$0.404877\pi$
$617$			29293.4i			1.91136i	−0.294409	+	0.955679i	$0.595123\pi$
$618$		0			0
$619$	−2166.48	−	2166.48i	−0.140675	−	0.140675i	0.633262	−	0.773937i	$-0.281716\pi$
$619$	−2166.48	−	2166.48i	−0.140675	−	0.140675i	−0.773937	+	0.633262i	$0.781716\pi$
$620$		0			0
$621$	−18273.7	−	18273.7i	−1.18083	−	1.18083i
$622$		0			0
$623$	6205.86	+	9357.37i	0.399089	+	0.601758i
$624$		0			0
$625$	−13189.3			−0.844115
$626$		0			0
$627$	−4039.61	−	4039.61i	−0.257299	−	0.257299i
$628$		0			0
$629$	8466.50	−	8466.50i	0.536696	−	0.536696i
$630$		0			0
$631$	−18780.9			−1.18487			−0.592437	−	0.805616i	$-0.701834\pi$
$631$	−18780.9			−1.18487			−0.592437	+	0.805616i	$0.701834\pi$
$632$		0			0
$633$	5791.23			0.363635
$634$		0			0
$635$	−853.814	−	853.814i	−0.0533584	−	0.0533584i
$636$		0			0
$637$	−14309.7	+	5812.06i	−0.890066	+	0.361511i
$638$		0			0
$639$		−	1106.54i		−	0.0685041i
$640$		0			0
$641$	10827.5			0.667174			0.333587	−	0.942719i	$-0.391741\pi$
$641$	10827.5			0.667174			0.333587	+	0.942719i	$0.391741\pi$
$642$		0			0
$643$	15036.9	−	15036.9i	0.922234	−	0.922234i	−0.0749530	−	0.997187i	$-0.523881\pi$
$643$	15036.9	−	15036.9i	0.922234	−	0.922234i	0.997187	+	0.0749530i	$0.0238807\pi$
$644$		0			0
$645$	1110.74	+	1110.74i	0.0678065	+	0.0678065i
$646$		0			0
$647$		−	29547.7i		−	1.79542i	−0.440584	−	0.897712i	$-0.645228\pi$
$647$		−	29547.7i		−	1.79542i	0.440584	−	0.897712i	$-0.354772\pi$
$648$		0			0
$649$	−10114.5			−0.611756
$650$		0			0
$651$	−8010.06	−	1622.02i	−0.482241	−	0.0976526i
$652$		0			0
$653$	16704.0	−	16704.0i	1.00104	−	1.00104i	0.00104004	−	0.999999i	$-0.499669\pi$
$653$	16704.0	−	16704.0i	1.00104	−	1.00104i	0.999999	−	0.00104004i	$-0.000331055\pi$
$654$		0			0
$655$		−	1393.21i		−	0.0831103i
$656$		0			0
$657$			6911.41i			0.410411i
$658$		0			0
$659$	3471.93	+	3471.93i	0.205231	+	0.205231i	0.802237	−	0.597006i	$-0.203643\pi$
$659$	3471.93	+	3471.93i	0.205231	+	0.205231i	−0.597006	+	0.802237i	$0.703643\pi$
$660$		0			0
$661$	−15545.0	+	15545.0i	−0.914718	+	0.914718i	−0.996639	−	0.0819205i	$-0.973895\pi$
$661$	−15545.0	+	15545.0i	−0.914718	+	0.914718i	0.0819205	+	0.996639i	$0.473895\pi$
$662$		0			0
$663$		−	12943.9i		−	0.758221i
$664$		0			0
$665$	−5909.47	+	3919.19i	−0.344600	+	0.228541i
$666$		0			0
$667$	135.573	−	135.573i	0.00787018	−	0.00787018i
$668$		0			0
$669$	−7013.72	+	7013.72i	−0.405331	+	0.405331i
$670$		0			0
$671$	−6631.99			−0.381557
$672$		0			0
$673$	5612.91			0.321488			0.160744	−	0.986996i	$-0.448611\pi$
$673$	5612.91			0.321488			0.160744	+	0.986996i	$0.448611\pi$
$674$		0			0
$675$	−11031.7	+	11031.7i	−0.629054	+	0.629054i
$676$		0			0
$677$	−10323.2	+	10323.2i	−0.586046	+	0.586046i	−0.936558	−	0.350512i	$-0.886007\pi$
$677$	−10323.2	+	10323.2i	−0.586046	+	0.586046i	0.350512	+	0.936558i	$0.386007\pi$
$678$		0			0
$679$	9082.01	−	6023.24i	0.513307	−	0.340428i
$680$		0			0
$681$		−	986.576i		−	0.0555149i
$682$		0			0
$683$	19895.3	−	19895.3i	1.11460	−	1.11460i	0.122079	−	0.992520i	$-0.461044\pi$
$683$	19895.3	−	19895.3i	1.11460	−	1.11460i	0.992520	−	0.122079i	$-0.0389560\pi$
$684$		0			0
$685$	−553.192	−	553.192i	−0.0308560	−	0.0308560i
$686$		0			0
$687$		−	11475.3i		−	0.637276i
$688$		0			0
$689$		−	28611.5i		−	1.58202i
$690$		0			0
$691$	−12897.5	+	12897.5i	−0.710048	+	0.710048i	−0.966545	−	0.256497i	$-0.917432\pi$
$691$	−12897.5	+	12897.5i	−0.710048	+	0.710048i	0.256497	+	0.966545i	$0.417432\pi$
$692$		0			0
$693$	912.903	−	4508.22i	0.0500408	−	0.247119i
$694$		0			0
$695$	5267.61			0.287499
$696$		0			0
$697$		−	31997.2i		−	1.73885i
$698$		0			0
$699$	9507.09	+	9507.09i	0.514437	+	0.514437i
$700$		0			0
$701$	17016.9	−	17016.9i	0.916862	−	0.916862i	−0.0799374	−	0.996800i	$-0.525472\pi$
$701$	17016.9	−	17016.9i	0.916862	−	0.916862i	0.996800	+	0.0799374i	$0.0254720\pi$
$702$		0			0
$703$	−17893.9			−0.960004
$704$		0			0
$705$		−	3130.94i		−	0.167260i
$706$		0			0
$707$	−14591.1	−	2954.66i	−0.776175	−	0.157173i
$708$		0			0
$709$	−12305.1	−	12305.1i	−0.651802	−	0.651802i	0.301625	−	0.953427i	$-0.402471\pi$
$709$	−12305.1	−	12305.1i	−0.651802	−	0.651802i	−0.953427	+	0.301625i	$0.902471\pi$
$710$		0			0
$711$	10355.8			0.546237
$712$		0			0
$713$	29994.8			1.57547
$714$		0			0
$715$	−1088.77	+	1088.77i	−0.0569476	+	0.0569476i
$716$		0			0
$717$	−4629.40	−	4629.40i	−0.241127	−	0.241127i
$718$		0			0
$719$	19494.4			1.01115			0.505575	−	0.862783i	$-0.331280\pi$
$719$	19494.4			1.01115			0.505575	+	0.862783i	$0.331280\pi$
$720$		0			0
$721$	13383.2	−	8875.82i	0.691285	−	0.458464i
$722$		0			0
$723$	−7889.38	−	7889.38i	−0.405822	−	0.405822i
$724$		0			0
$725$	−81.8449	−	81.8449i	−0.00419261	−	0.00419261i
$726$		0			0
$727$		−	264.898i		−	0.0135138i	−0.999977	−	0.00675689i	$-0.997849\pi$
$727$		−	264.898i		−	0.0135138i	0.999977	−	0.00675689i	$-0.00215080\pi$
$728$		0			0
$729$			7948.40i			0.403820i
$730$		0			0
$731$	14917.7	+	14917.7i	0.754792	+	0.754792i
$732$		0			0
$733$	20566.1	+	20566.1i	1.03632	+	1.03632i	0.999315	+	0.0370078i	$0.0117826\pi$
$733$	20566.1	+	20566.1i	1.03632	+	1.03632i	0.0370078	+	0.999315i	$0.488217\pi$
$734$		0			0
$735$	2344.11	+	989.943i	0.117638	+	0.0496797i
$736$		0			0
$737$	−4811.78			−0.240494
$738$		0			0
$739$	5171.06	+	5171.06i	0.257403	+	0.257403i	0.823997	−	0.566594i	$-0.191739\pi$
$739$	5171.06	+	5171.06i	0.257403	+	0.257403i	−0.566594	+	0.823997i	$0.691739\pi$
$740$		0			0
$741$	−13678.5	+	13678.5i	−0.678127	+	0.678127i
$742$		0			0
$743$	−11114.9			−0.548811			−0.274406	−	0.961614i	$-0.588481\pi$
$743$	−11114.9			−0.548811			−0.274406	+	0.961614i	$0.588481\pi$
$744$		0			0
$745$	4428.74			0.217794
$746$		0			0
$747$	992.895	+	992.895i	0.0486320	+	0.0486320i
$748$		0			0
$749$	−32121.0	−	6504.42i	−1.56699	−	0.317311i
$750$		0			0
$751$		−	26003.3i		−	1.26348i	−0.775180	−	0.631740i	$-0.782341\pi$
$751$		−	26003.3i		−	1.26348i	0.775180	−	0.631740i	$-0.217659\pi$
$752$		0			0
$753$	−790.011			−0.0382332
$754$		0			0
$755$	1732.00	−	1732.00i	0.0834885	−	0.0834885i
$756$		0			0
$757$	16959.5	+	16959.5i	0.814273	+	0.814273i	0.985271	−	0.170998i	$-0.0546992\pi$
$757$	16959.5	+	16959.5i	0.814273	+	0.814273i	−0.170998	+	0.985271i	$0.554699\pi$
$758$		0			0
$759$		−	7523.95i		−	0.359818i
$760$		0			0
$761$	24676.8			1.17547			0.587735	−	0.809053i	$-0.300020\pi$
$761$	24676.8			1.17547			0.587735	+	0.809053i	$0.300020\pi$
$762$		0			0
$763$	−170.459	−	34.5175i	−0.00808786	−	0.00163777i
$764$		0			0
$765$	3383.34	−	3383.34i	0.159902	−	0.159902i
$766$		0			0
$767$			34248.7i			1.61232i
$768$		0			0
$769$			4733.15i			0.221953i	0.993823	+	0.110976i	$0.0353978\pi$
$769$			4733.15i			0.221953i	−0.993823	+	0.110976i	$0.964602\pi$
$770$		0			0
$771$	2814.60	+	2814.60i	0.131473	+	0.131473i
$772$		0			0
$773$	−14530.1	+	14530.1i	−0.676082	+	0.676082i	−0.959111	−	0.283030i	$-0.908661\pi$
$773$	−14530.1	+	14530.1i	−0.676082	+	0.676082i	0.283030	+	0.959111i	$0.408661\pi$
$774$		0			0
$775$		−	18107.7i		−	0.839288i
$776$		0			0
$777$	3548.99	+	5351.27i	0.163860	+	0.247073i
$778$		0			0
$779$	−33813.0	+	33813.0i	−1.55517	+	1.55517i
$780$		0			0
$781$	557.134	−	557.134i	0.0255260	−	0.0255260i
$782$		0			0
$783$	−128.839			−0.00588039
$784$		0			0
$785$	−7345.46			−0.333975
$786$		0			0
$787$	5871.53	−	5871.53i	0.265943	−	0.265943i	−0.561520	−	0.827463i	$-0.689783\pi$
$787$	5871.53	−	5871.53i	0.265943	−	0.265943i	0.827463	+	0.561520i	$0.189783\pi$
$788$		0			0
$789$	10151.7	−	10151.7i	0.458061	−	0.458061i
$790$		0			0
$791$	6027.65	+	9088.67i	0.270947	+	0.408541i
$792$		0			0
$793$			22456.5i			1.00562i
$794$		0			0
$795$	−3333.12	+	3333.12i	−0.148696	+	0.148696i
$796$		0			0
$797$	−5006.58	−	5006.58i	−0.222512	−	0.222512i	0.587043	−	0.809556i	$-0.300292\pi$
$797$	−5006.58	−	5006.58i	−0.222512	−	0.222512i	−0.809556	+	0.587043i	$0.800292\pi$
$798$		0			0
$799$		−	42050.1i		−	1.86186i
$800$		0			0
$801$			11322.8i			0.499464i
$802$		0			0
$803$	−3479.83	+	3479.83i	−0.152927	+	0.152927i
$804$		0			0
$805$	−9153.15	−	1853.49i	−0.400753	−	0.0811514i
$806$		0			0
$807$	12725.1			0.555073
$808$		0			0
$809$			4215.92i			0.183218i	0.995795	+	0.0916092i	$0.0292011\pi$
$809$			4215.92i			0.183218i	−0.995795	+	0.0916092i	$0.970799\pi$
$810$		0			0
$811$	8853.78	+	8853.78i	0.383352	+	0.383352i	0.872308	−	0.488956i	$-0.162622\pi$
$811$	8853.78	+	8853.78i	0.383352	+	0.383352i	−0.488956	+	0.872308i	$0.662622\pi$
$812$		0			0
$813$	13531.9	−	13531.9i	0.583745	−	0.583745i
$814$		0			0
$815$	−5097.85			−0.219104
$816$		0			0
$817$		−	31528.6i		−	1.35012i
$818$		0			0
$819$	−15265.2	−	3091.17i	−0.651295	−	0.131886i
$820$		0			0
$821$	−11488.3	−	11488.3i	−0.488362	−	0.488362i	0.419427	−	0.907789i	$-0.362231\pi$
$821$	−11488.3	−	11488.3i	−0.488362	−	0.488362i	−0.907789	+	0.419427i	$0.862231\pi$
$822$		0			0
$823$	19461.0			0.824263			0.412131	−	0.911124i	$-0.364784\pi$
$823$	19461.0			0.824263			0.412131	+	0.911124i	$0.364784\pi$
$824$		0			0
$825$	−4542.18			−0.191683
$826$		0			0
$827$	−1802.10	+	1802.10i	−0.0757743	+	0.0757743i	−0.743978	−	0.668204i	$-0.767063\pi$
$827$	−1802.10	+	1802.10i	−0.0757743	+	0.0757743i	0.668204	+	0.743978i	$0.267063\pi$
$828$		0			0
$829$	23538.7	+	23538.7i	0.986168	+	0.986168i	0.999906	−	0.0137374i	$-0.00437287\pi$
$829$	23538.7	+	23538.7i	0.986168	+	0.986168i	−0.0137374	+	0.999906i	$0.504373\pi$
$830$		0			0
$831$	2489.83			0.103937
$832$		0			0
$833$	31482.5	+	13295.4i	1.30949	+	0.553012i
$834$		0			0
$835$	−2823.11	−	2823.11i	−0.117003	−	0.117003i
$836$		0			0
$837$	−14252.5	−	14252.5i	−0.588576	−	0.588576i
$838$		0			0
$839$		−	3944.19i		−	0.162298i	−0.996702	−	0.0811492i	$-0.974141\pi$
$839$		−	3944.19i		−	0.162298i	0.996702	−	0.0811492i	$-0.0258590\pi$
$840$		0			0
$841$			24388.0i			0.999961i
$842$		0			0
$843$	−6488.56	−	6488.56i	−0.265098	−	0.265098i
$844$		0			0
$845$	−307.954	−	307.954i	−0.0125372	−	0.0125372i
$846$		0			0
$847$	−17813.7	+	11814.1i	−0.722652	+	0.479267i
$848$		0			0
$849$	4345.95			0.175680
$850$		0			0
$851$	−16664.1	−	16664.1i	−0.671256	−	0.671256i
$852$		0			0
$853$	12245.7	−	12245.7i	0.491543	−	0.491543i	−0.417249	−	0.908792i	$-0.637006\pi$
$853$	12245.7	−	12245.7i	0.491543	−	0.491543i	0.908792	+	0.417249i	$0.137006\pi$
$854$		0			0
$855$	−7150.68			−0.286021
$856$		0			0
$857$	−3074.93			−0.122564			−0.0612822	−	0.998120i	$-0.519519\pi$
$857$	−3074.93			−0.122564			−0.0612822	+	0.998120i	$0.519519\pi$
$858$		0			0
$859$	24969.5	+	24969.5i	0.991792	+	0.991792i	0.999967	−	0.00817484i	$-0.00260216\pi$
$859$	24969.5	+	24969.5i	0.991792	+	0.991792i	−0.00817484	+	0.999967i	$0.502602\pi$
$860$		0			0
$861$	16818.2	+	3405.65i	0.665696	+	0.134802i
$862$		0			0
$863$			19941.4i			0.786573i	0.919416	+	0.393287i	$0.128662\pi$
$863$			19941.4i			0.786573i	−0.919416	+	0.393287i	$0.871338\pi$
$864$		0			0
$865$	−7262.25			−0.285461
$866$		0			0
$867$	−10229.2	+	10229.2i	−0.400694	+	0.400694i
$868$		0			0
$869$	5214.07	+	5214.07i	0.203539	+	0.203539i
$870$		0			0
$871$			16293.1i			0.633837i
$872$		0			0
$873$	10989.6			0.426049
$874$		0			0
$875$	−2300.38	+	11360.0i	−0.0888765	+	0.438902i
$876$		0			0
$877$	−14771.5	+	14771.5i	−0.568753	+	0.568753i	−0.931779	−	0.363026i	$-0.881744\pi$
$877$	−14771.5	+	14771.5i	−0.568753	+	0.568753i	0.363026	+	0.931779i	$0.381744\pi$
$878$		0			0
$879$		−	147.897i		−	0.00567514i
$880$		0			0
$881$		−	3567.23i		−	0.136417i	−0.997671	−	0.0682083i	$-0.978272\pi$
$881$		−	3567.23i		−	0.136417i	0.997671	−	0.0682083i	$-0.0217283\pi$
$882$		0			0
$883$	30325.4	+	30325.4i	1.15576	+	1.15576i	0.985379	+	0.170377i	$0.0544985\pi$
$883$	30325.4	+	30325.4i	1.15576	+	1.15576i	0.170377	+	0.985379i	$0.445501\pi$
$884$		0			0
$885$	3989.83	−	3989.83i	0.151544	−	0.151544i
$886$		0			0
$887$			11845.7i			0.448409i	0.974542	+	0.224204i	$0.0719784\pi$
$887$			11845.7i			0.448409i	−0.974542	+	0.224204i	$0.928022\pi$
$888$		0			0
$889$	7247.84	−	4806.80i	0.273436	−	0.181344i
$890$		0			0
$891$	1166.57	−	1166.57i	0.0438626	−	0.0438626i
$892$		0			0
$893$	−44436.4	+	44436.4i	−1.66518	+	1.66518i
$894$		0			0
$895$	−4484.20			−0.167475
$896$		0			0
$897$	−25476.8			−0.948322
$898$		0			0
$899$	105.740	−	105.740i	0.00392282	−	0.00392282i
$900$		0			0
$901$	−44765.4	+	44765.4i	−1.65522	+	1.65522i
$902$		0			0
$903$	−9428.79	+	6253.23i	−0.347476	+	0.230448i
$904$		0			0
$905$			768.074i			0.0282118i
$906$		0			0
$907$	−18959.1	+	18959.1i	−0.694074	+	0.694074i	−0.963126	−	0.269052i	$-0.913290\pi$
$907$	−18959.1	+	18959.1i	−0.694074	+	0.694074i	0.269052	+	0.963126i	$0.413290\pi$
$908$		0			0
$909$	−10615.6	−	10615.6i	−0.387344	−	0.387344i
$910$		0			0
$911$		−	12256.5i		−	0.445749i	−0.974847	−	0.222875i	$-0.928456\pi$
$911$		−	12256.5i		−	0.445749i	0.974847	−	0.222875i	$-0.0715440\pi$
$912$		0			0
$913$			999.827i			0.0362425i
$914$		0			0
$915$	2616.09	−	2616.09i	0.0945195	−	0.0945195i
$916$		0			0
$917$	9835.08	+	1991.58i	0.354180	+	0.0717204i
$918$		0			0
$919$	78.9425			0.00283359			0.00141680	−	0.999999i	$-0.499549\pi$
$919$	78.9425			0.00283359			0.00141680	+	0.999999i	$0.499549\pi$
$920$		0			0
$921$		−	17079.5i		−	0.611064i
$922$		0			0
$923$	−1886.51	−	1886.51i	−0.0672753	−	0.0672753i
$924$		0			0
$925$	−10060.1	+	10060.1i	−0.357592	+	0.357592i
$926$		0			0
$927$	16194.2			0.573773
$928$		0			0
$929$			6902.55i			0.243773i	0.992544	+	0.121887i	$0.0388944\pi$
$929$			6902.55i			0.243773i	−0.992544	+	0.121887i	$0.961106\pi$
$930$		0			0
$931$	−19219.2	−	47319.0i	−0.676567	−	1.66576i
$932$		0			0
$933$	4215.25	+	4215.25i	0.147911	+	0.147911i
$934$		0			0
$935$	3406.96			0.119165
$936$		0			0
$937$	52087.0			1.81602			0.908008	−	0.418953i	$-0.137603\pi$
$937$	52087.0			1.81602			0.908008	+	0.418953i	$0.137603\pi$
$938$		0			0
$939$	−21913.6	+	21913.6i	−0.761579	+	0.761579i
$940$		0			0
$941$	−36488.1	−	36488.1i	−1.26406	−	1.26406i	−0.949108	−	0.314950i	$-0.898012\pi$
$941$	−36488.1	−	36488.1i	−1.26406	−	1.26406i	−0.314950	−	0.949108i	$-0.601988\pi$
$942$		0			0
$943$	−62978.2			−2.17482
$944$		0			0
$945$	3468.55	+	5229.98i	0.119399	+	0.180033i
$946$		0			0
$947$	21506.1	+	21506.1i	0.737966	+	0.737966i	0.972184	−	0.234218i	$-0.0752528\pi$
$947$	21506.1	+	21506.1i	0.737966	+	0.737966i	−0.234218	+	0.972184i	$0.575253\pi$
$948$		0			0
$949$	11783.0	+	11783.0i	0.403049	+	0.403049i
$950$		0			0
$951$			14155.6i			0.482678i
$952$		0			0
$953$			6230.83i			0.211790i	0.994377	+	0.105895i	$0.0337708\pi$
$953$			6230.83i			0.211790i	−0.994377	+	0.105895i	$0.966229\pi$
$954$		0			0
$955$	−2471.52	−	2471.52i	−0.0837451	−	0.0837451i
$956$		0			0
$957$	−26.5240	−	26.5240i	−0.000895924	−	0.000895924i
$958$		0			0
$959$	4695.92	−	3114.36i	0.158122	−	0.104868i
$960$		0			0
$961$	−6396.68			−0.214718
$962$		0			0
$963$	−23369.1	−	23369.1i	−0.781994	−	0.781994i
$964$		0			0
$965$	9590.61	−	9590.61i	0.319930	−	0.319930i
$966$		0			0
$967$	−19540.5			−0.649825			−0.324912	−	0.945744i	$-0.605335\pi$
$967$	−19540.5			−0.649825			−0.324912	+	0.945744i	$0.605335\pi$
$968$		0			0
$969$	42802.7			1.41901
$970$		0			0
$971$	30508.0	+	30508.0i	1.00829	+	1.00829i	0.999965	+	0.00832220i	$0.00264907\pi$
$971$	30508.0	+	30508.0i	1.00829	+	1.00829i	0.00832220	+	0.999965i	$0.497351\pi$
$972$		0			0
$973$	−7529.97	+	37185.6i	−0.248099	+	1.22519i
$974$		0			0
$975$			15380.2i			0.505191i
$976$		0			0
$977$	39679.8			1.29936			0.649678	−	0.760210i	$-0.274904\pi$
$977$	39679.8			1.29936			0.649678	+	0.760210i	$0.274904\pi$
$978$		0			0
$979$	−5700.91	+	5700.91i	−0.186110	+	0.186110i
$980$		0			0
$981$	−124.015	−	124.015i	−0.00403618	−	0.00403618i
$982$		0			0
$983$			11032.7i			0.357974i	0.983851	+	0.178987i	$0.0572820\pi$
$983$			11032.7i			0.357974i	−0.983851	+	0.178987i	$0.942718\pi$
$984$		0			0
$985$	−3424.55			−0.110777
$986$		0			0
$987$	22102.2	+	4475.64i	0.712788	+	0.144338i
$988$		0			0
$989$	29361.7	−	29361.7i	0.944033	−	0.944033i
$990$		0			0
$991$			20583.1i			0.659782i	0.944019	+	0.329891i	$0.107012\pi$
$991$			20583.1i			0.659782i	−0.944019	+	0.329891i	$0.892988\pi$
$992$		0			0
$993$			17041.5i			0.544606i
$994$		0			0
$995$	−2961.41	−	2961.41i	−0.0943549	−	0.0943549i
$996$		0			0
$997$	25414.8	−	25414.8i	0.807318	−	0.807318i	−0.176909	−	0.984227i	$-0.556610\pi$
$997$	25414.8	−	25414.8i	0.807318	−	0.807318i	0.984227	+	0.176909i	$0.0566098\pi$
$998$		0			0
$999$			15836.4i			0.501544i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.j.b.335.27		88
4.3	odd	2		112.4.j.b.27.3	✓	88
7.6	odd	2	inner	448.4.j.b.335.18		88
16.3	odd	4	inner	448.4.j.b.111.18		88
16.13	even	4		112.4.j.b.83.4	yes	88
28.27	even	2		112.4.j.b.27.4	yes	88
112.13	odd	4		112.4.j.b.83.3	yes	88
112.83	even	4	inner	448.4.j.b.111.27		88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.j.b.27.3	✓	88	4.3	odd	2
112.4.j.b.27.4	yes	88	28.27	even	2
112.4.j.b.83.3	yes	88	112.13	odd	4
112.4.j.b.83.4	yes	88	16.13	even	4
448.4.j.b.111.18		88	16.3	odd	4	inner
448.4.j.b.111.27		88	112.83	even	4	inner
448.4.j.b.335.18		88	7.6	odd	2	inner
448.4.j.b.335.27		88	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.04007 - 2.04007i) q^{3} +(1.81821 - 1.81821i) q^{5} +(10.2362 + 15.4344i) q^{7} +18.6762i q^{9} +O(q^{10})\)	\(q+(2.04007 - 2.04007i) q^{3} +(1.81821 - 1.81821i) q^{5} +(10.2362 + 15.4344i) q^{7} +18.6762i q^{9} +(-9.40331 + 9.40331i) q^{11} +(31.8404 + 31.8404i) q^{13} -7.41856i q^{15} -99.6350i q^{17} +(-105.289 + 105.289i) q^{19} +(52.3698 + 10.6047i) q^{21} -196.106 q^{23} +118.388i q^{25} +(93.1827 + 93.1827i) q^{27} +(-0.691326 + 0.691326i) q^{29} -152.952 q^{31} +38.3668i q^{33} +(46.6746 + 9.45147i) q^{35} +(84.9752 + 84.9752i) q^{37} +129.913 q^{39} +321.144 q^{41} +(-149.724 + 149.724i) q^{43} +(33.9574 + 33.9574i) q^{45} +422.042 q^{47} +(-133.441 + 315.978i) q^{49} +(-203.262 - 203.262i) q^{51} +(-449.294 - 449.294i) q^{53} +34.1944i q^{55} +429.595i q^{57} +(537.818 + 537.818i) q^{59} +(352.641 + 352.641i) q^{61} +(-288.256 + 191.173i) q^{63} +115.785 q^{65} +(255.856 + 255.856i) q^{67} +(-400.070 + 400.070i) q^{69} -59.2487 q^{71} +370.065 q^{73} +(241.520 + 241.520i) q^{75} +(-241.388 - 48.8805i) q^{77} -554.493i q^{79} -124.060 q^{81} +(53.1636 - 53.1636i) q^{83} +(-181.158 - 181.158i) q^{85} +2.82071i q^{87} +606.267 q^{89} +(-165.514 + 817.363i) q^{91} +(-312.033 + 312.033i) q^{93} +382.876i q^{95} -588.427i q^{97} +(-175.618 - 175.618i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 4 q^{7}+O(q^{10}) \) 88 * q + 4 * q^7	\( 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77} - 1952 q^{81} + 496 q^{85} + 1964 q^{91} - 112 q^{93} - 7592 q^{99}+O(q^{100}) \) 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Introduction

L-functions

Modular forms

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Database

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