LMFDB - Embedded newform 2352.3.f.g.97.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,3,Mod(97,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2352.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.f (of order $2$, degree $1$, 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:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.35911766016.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 $ x^8 - 2x^7 - 7x^6 - 2x^5 + 78x^4 - 18x^3 - 153x^2 - 230*x + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.8
Root			$1.83172 + 0.480194i$ of defining polynomial
Character	$\chi$	$=$	2352.97
Dual form			2352.3.f.g.97.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +6.12627i q^{5} -3.00000 q^{9} +3.78258 q^{11} -9.29319i q^{13} -10.6110 q^{15} -6.75202i q^{17} -6.59962i q^{19} +39.1840 q^{23} -12.5311 q^{25} -5.19615i q^{27} -6.57302 q^{29} -21.9325i q^{31} +6.55161i q^{33} +67.0668 q^{37} +16.0963 q^{39} -53.6570i q^{41} +42.0426 q^{43} -18.3788i q^{45} +57.5392i q^{47} +11.6948 q^{51} -11.4218 q^{53} +23.1731i q^{55} +11.4309 q^{57} +62.4038i q^{59} -119.635i q^{61} +56.9326 q^{65} +55.7480 q^{67} +67.8687i q^{69} +131.158 q^{71} -77.0426i q^{73} -21.7046i q^{75} -149.493 q^{79} +9.00000 q^{81} +39.7649i q^{83} +41.3647 q^{85} -11.3848i q^{87} +56.0011i q^{89} +37.9882 q^{93} +40.4311 q^{95} +142.413i q^{97} -11.3477 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9} - 44 q^{11} + 12 q^{15} + 96 q^{23} - 84 q^{25} + 68 q^{29} + 236 q^{37} - 36 q^{39} + 92 q^{43} + 72 q^{51} - 20 q^{53} + 84 q^{57} + 296 q^{65} + 44 q^{67} + 392 q^{71} + 328 q^{79} + 72 q^{81}+ \cdots + 132 q^{99}+O(q^{100}) $ 8 * q - 24 * q^9 - 44 * q^11 + 12 * q^15 + 96 * q^23 - 84 * q^25 + 68 * q^29 + 236 * q^37 - 36 * q^39 + 92 * q^43 + 72 * q^51 - 20 * q^53 + 84 * q^57 + 296 * q^65 + 44 * q^67 + 392 * q^71 + 328 * q^79 + 72 * q^81 + 200 * q^85 + 120 * q^93 + 132 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	6.12627i	1.22525i	0.790372	+	0.612627i	$0.209887\pi$
$5$	6.12627i	1.22525i	−0.790372	+	0.612627i	$0.790113\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	3.78258	0.343871	0.171935	−	0.985108i	$-0.444998\pi$
$11$	3.78258	0.343871	0.171935	+	0.985108i	$0.444998\pi$
$12$	0	0
$13$	− 9.29319i	− 0.714861i	−0.933940	−	0.357431i	$-0.883653\pi$
$13$	− 9.29319i	− 0.714861i	0.933940	−	0.357431i	$-0.116347\pi$
$14$	0	0
$15$	−10.6110	−0.707400
$16$	0	0
$17$	− 6.75202i	− 0.397178i	−0.980083	−	0.198589i	$-0.936364\pi$
$17$	− 6.75202i	− 0.397178i	0.980083	−	0.198589i	$-0.0636358\pi$
$18$	0	0
$19$	− 6.59962i	− 0.347349i	−0.984803	−	0.173674i	$-0.944436\pi$
$19$	− 6.59962i	− 0.347349i	0.984803	−	0.173674i	$-0.0555640\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	39.1840	1.70365	0.851827	−	0.523824i	$-0.175495\pi$
$23$	39.1840	1.70365	0.851827	+	0.523824i	$0.175495\pi$
$24$	0	0
$25$	−12.5311	−0.501245
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−6.57302	−0.226656	−0.113328	−	0.993558i	$-0.536151\pi$
$29$	−6.57302	−0.226656	−0.113328	+	0.993558i	$0.536151\pi$
$30$	0	0
$31$	− 21.9325i	− 0.707499i	−0.935340	−	0.353750i	$-0.884906\pi$
$31$	− 21.9325i	− 0.707499i	0.935340	−	0.353750i	$-0.115094\pi$
$32$	0	0
$33$	6.55161i	0.198534i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	67.0668	1.81262	0.906309	−	0.422616i	$-0.138888\pi$
$37$	67.0668	1.81262	0.906309	+	0.422616i	$0.138888\pi$
$38$	0	0
$39$	16.0963	0.412725
$40$	0	0
$41$	− 53.6570i	− 1.30871i	−0.756189	−	0.654353i	$-0.772941\pi$
$41$	− 53.6570i	− 1.30871i	0.756189	−	0.654353i	$-0.227059\pi$
$42$	0	0
$43$	42.0426	0.977734	0.488867	−	0.872358i	$-0.337410\pi$
$43$	42.0426	0.977734	0.488867	+	0.872358i	$0.337410\pi$
$44$	0	0
$45$	− 18.3788i	− 0.408418i
$46$	0	0
$47$	57.5392i	1.22424i	0.790765	+	0.612119i	$0.209683\pi$
$47$	57.5392i	1.22424i	−0.790765	+	0.612119i	$0.790317\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	11.6948	0.229311
$52$	0	0
$53$	−11.4218	−0.215506	−0.107753	−	0.994178i	$-0.534366\pi$
$53$	−11.4218	−0.215506	−0.107753	+	0.994178i	$0.534366\pi$
$54$	0	0
$55$	23.1731i	0.421329i
$56$	0	0
$57$	11.4309	0.200542
$58$	0	0
$59$	62.4038i	1.05769i	0.848718	+	0.528846i	$0.177375\pi$
$59$	62.4038i	1.05769i	−0.848718	+	0.528846i	$0.822625\pi$
$60$	0	0
$61$	− 119.635i	− 1.96123i	−0.195936	−	0.980617i	$-0.562775\pi$
$61$	− 119.635i	− 1.96123i	0.195936	−	0.980617i	$-0.437225\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	56.9326	0.875886
$66$	0	0
$67$	55.7480	0.832059	0.416030	−	0.909351i	$-0.363421\pi$
$67$	55.7480	0.832059	0.416030	+	0.909351i	$0.363421\pi$
$68$	0	0
$69$	67.8687i	0.983605i
$70$	0	0
$71$	131.158	1.84730	0.923648	−	0.383243i	$-0.125193\pi$
$71$	131.158	1.84730	0.923648	+	0.383243i	$0.125193\pi$
$72$	0	0
$73$	− 77.0426i	− 1.05538i	−0.849438	−	0.527689i	$-0.823059\pi$
$73$	− 77.0426i	− 1.05538i	0.849438	−	0.527689i	$-0.176941\pi$
$74$	0	0
$75$	− 21.7046i	− 0.289394i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−149.493	−1.89232	−0.946160	−	0.323699i	$-0.895074\pi$
$79$	−149.493	−1.89232	−0.946160	+	0.323699i	$0.895074\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	39.7649i	0.479096i	0.970885	+	0.239548i	$0.0769992\pi$
$83$	39.7649i	0.479096i	−0.970885	+	0.239548i	$0.923001\pi$
$84$	0	0
$85$	41.3647	0.486643
$86$	0	0
$87$	− 11.3848i	− 0.130860i
$88$	0	0
$89$	56.0011i	0.629226i	0.949220	+	0.314613i	$0.101875\pi$
$89$	56.0011i	0.629226i	−0.949220	+	0.314613i	$0.898125\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	37.9882	0.408475
$94$	0	0
$95$	40.4311	0.425590
$96$	0	0
$97$	142.413i	1.46818i	0.679054	+	0.734089i	$0.262390\pi$
$97$	142.413i	1.46818i	−0.679054	+	0.734089i	$0.737610\pi$
$98$	0	0
$99$	−11.3477	−0.114624
$100$	0	0
$101$	− 146.671i	− 1.45218i	−0.687598	−	0.726092i	$-0.741335\pi$
$101$	− 146.671i	− 1.45218i	0.687598	−	0.726092i	$-0.258665\pi$
$102$	0	0
$103$	7.69922i	0.0747497i	0.999301	+	0.0373748i	$0.0118996\pi$
$103$	7.69922i	0.0747497i	−0.999301	+	0.0373748i	$0.988100\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	139.594	1.30462	0.652308	−	0.757954i	$-0.273801\pi$
$107$	139.594	1.30462	0.652308	+	0.757954i	$0.273801\pi$
$108$	0	0
$109$	−151.612	−1.39093	−0.695467	−	0.718558i	$-0.744802\pi$
$109$	−151.612	−1.39093	−0.695467	+	0.718558i	$0.744802\pi$
$110$	0	0
$111$	116.163i	1.04652i
$112$	0	0
$113$	−40.4809	−0.358238	−0.179119	−	0.983827i	$-0.557325\pi$
$113$	−40.4809	−0.358238	−0.179119	+	0.983827i	$0.557325\pi$
$114$	0	0
$115$	240.052i	2.08741i
$116$	0	0
$117$	27.8796i	0.238287i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−106.692	−0.881753
$122$	0	0
$123$	92.9366	0.755582
$124$	0	0
$125$	76.3876i	0.611101i
$126$	0	0
$127$	88.3367	0.695565	0.347782	−	0.937575i	$-0.386935\pi$
$127$	88.3367	0.695565	0.347782	+	0.937575i	$0.386935\pi$
$128$	0	0
$129$	72.8199i	0.564495i
$130$	0	0
$131$	− 67.6480i	− 0.516397i	−0.966092	−	0.258198i	$-0.916871\pi$
$131$	− 67.6480i	− 0.516397i	0.966092	−	0.258198i	$-0.0831288\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	31.8330	0.235800
$136$	0	0
$137$	−165.238	−1.20612	−0.603058	−	0.797697i	$-0.706051\pi$
$137$	−165.238	−1.20612	−0.603058	+	0.797697i	$0.706051\pi$
$138$	0	0
$139$	73.0610i	0.525619i	0.964848	+	0.262809i	$0.0846490\pi$
$139$	73.0610i	0.525619i	−0.964848	+	0.262809i	$0.915351\pi$
$140$	0	0
$141$	−99.6609	−0.706815
$142$	0	0
$143$	− 35.1522i	− 0.245820i
$144$	0	0
$145$	− 40.2681i	− 0.277711i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	150.588	1.01066	0.505329	−	0.862927i	$-0.331371\pi$
$149$	150.588	1.01066	0.505329	+	0.862927i	$0.331371\pi$
$150$	0	0
$151$	−138.984	−0.920425	−0.460213	−	0.887809i	$-0.652227\pi$
$151$	−138.984	−0.920425	−0.460213	+	0.887809i	$0.652227\pi$
$152$	0	0
$153$	20.2561i	0.132393i
$154$	0	0
$155$	134.364	0.866866
$156$	0	0
$157$	− 157.251i	− 1.00160i	−0.865563	−	0.500800i	$-0.833039\pi$
$157$	− 157.251i	− 1.00160i	0.865563	−	0.500800i	$-0.166961\pi$
$158$	0	0
$159$	− 19.7832i	− 0.124423i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	21.3227	0.130814	0.0654069	−	0.997859i	$-0.479165\pi$
$163$	21.3227	0.130814	0.0654069	+	0.997859i	$0.479165\pi$
$164$	0	0
$165$	−40.1369	−0.243254
$166$	0	0
$167$	177.968i	1.06567i	0.846218	+	0.532837i	$0.178874\pi$
$167$	177.968i	1.06567i	−0.846218	+	0.532837i	$0.821126\pi$
$168$	0	0
$169$	82.6365	0.488974
$170$	0	0
$171$	19.7989i	0.115783i
$172$	0	0
$173$	110.102i	0.636428i	0.948019	+	0.318214i	$0.103083\pi$
$173$	110.102i	0.636428i	−0.948019	+	0.318214i	$0.896917\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−108.087	−0.610659
$178$	0	0
$179$	28.2449	0.157793	0.0788964	−	0.996883i	$-0.474860\pi$
$179$	28.2449	0.157793	0.0788964	+	0.996883i	$0.474860\pi$
$180$	0	0
$181$	206.410i	1.14039i	0.821510	+	0.570195i	$0.193132\pi$
$181$	206.410i	1.14039i	−0.821510	+	0.570195i	$0.806868\pi$
$182$	0	0
$183$	207.214	1.13232
$184$	0	0
$185$	410.869i	2.22092i
$186$	0	0
$187$	− 25.5400i	− 0.136578i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	63.7368	0.333701	0.166850	−	0.985982i	$-0.446640\pi$
$191$	63.7368	0.333701	0.166850	+	0.985982i	$0.446640\pi$
$192$	0	0
$193$	−67.3523	−0.348976	−0.174488	−	0.984659i	$-0.555827\pi$
$193$	−67.3523	−0.348976	−0.174488	+	0.984659i	$0.555827\pi$
$194$	0	0
$195$	98.6101i	0.505693i
$196$	0	0
$197$	96.0707	0.487668	0.243834	−	0.969817i	$-0.421595\pi$
$197$	96.0707	0.487668	0.243834	+	0.969817i	$0.421595\pi$
$198$	0	0
$199$	205.990i	1.03512i	0.855645	+	0.517562i	$0.173160\pi$
$199$	205.990i	1.03512i	−0.855645	+	0.517562i	$0.826840\pi$
$200$	0	0
$201$	96.5583i	0.480390i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	328.717	1.60350
$206$	0	0
$207$	−117.552	−0.567884
$208$	0	0
$209$	− 24.9636i	− 0.119443i
$210$	0	0
$211$	121.942	0.577925	0.288963	−	0.957340i	$-0.406690\pi$
$211$	121.942	0.577925	0.288963	+	0.957340i	$0.406690\pi$
$212$	0	0
$213$	227.172i	1.06654i
$214$	0	0
$215$	257.564i	1.19797i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	133.442	0.609323
$220$	0	0
$221$	−62.7478	−0.283927
$222$	0	0
$223$	18.5026i	0.0829712i	0.999139	+	0.0414856i	$0.0132091\pi$
$223$	18.5026i	0.0829712i	−0.999139	+	0.0414856i	$0.986791\pi$
$224$	0	0
$225$	37.5934	0.167082
$226$	0	0
$227$	− 147.049i	− 0.647791i	−0.946093	−	0.323896i	$-0.895007\pi$
$227$	− 147.049i	− 0.647791i	0.946093	−	0.323896i	$-0.104993\pi$
$228$	0	0
$229$	− 359.915i	− 1.57168i	−0.618430	−	0.785840i	$-0.712231\pi$
$229$	− 359.915i	− 1.57168i	0.618430	−	0.785840i	$-0.287769\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	44.0681	0.189133	0.0945667	−	0.995519i	$-0.469853\pi$
$233$	44.0681	0.189133	0.0945667	+	0.995519i	$0.469853\pi$
$234$	0	0
$235$	−352.501	−1.50000
$236$	0	0
$237$	− 258.930i	− 1.09253i
$238$	0	0
$239$	−165.077	−0.690698	−0.345349	−	0.938474i	$-0.612239\pi$
$239$	−165.077	−0.690698	−0.345349	+	0.938474i	$0.612239\pi$
$240$	0	0
$241$	133.746i	0.554964i	0.960731	+	0.277482i	$0.0894999\pi$
$241$	133.746i	0.554964i	−0.960731	+	0.277482i	$0.910500\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−61.3316	−0.248306
$248$	0	0
$249$	−68.8749	−0.276606
$250$	0	0
$251$	− 403.639i	− 1.60812i	−0.594545	−	0.804062i	$-0.702668\pi$
$251$	− 403.639i	− 1.60812i	0.594545	−	0.804062i	$-0.297332\pi$
$252$	0	0
$253$	148.217	0.585836
$254$	0	0
$255$	71.6457i	0.280963i
$256$	0	0
$257$	279.920i	1.08918i	0.838701	+	0.544592i	$0.183315\pi$
$257$	279.920i	1.08918i	−0.838701	+	0.544592i	$0.816685\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	19.7191	0.0755519
$262$	0	0
$263$	141.892	0.539515	0.269758	−	0.962928i	$-0.413056\pi$
$263$	141.892	0.539515	0.269758	+	0.962928i	$0.413056\pi$
$264$	0	0
$265$	− 69.9731i	− 0.264050i
$266$	0	0
$267$	−96.9968	−0.363284
$268$	0	0
$269$	− 397.803i	− 1.47882i	−0.673254	−	0.739411i	$-0.735104\pi$
$269$	− 397.803i	− 1.47882i	0.673254	−	0.739411i	$-0.264896\pi$
$270$	0	0
$271$	437.126i	1.61301i	0.591227	+	0.806505i	$0.298644\pi$
$271$	437.126i	1.61301i	−0.591227	+	0.806505i	$0.701356\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−47.4000	−0.172364
$276$	0	0
$277$	5.30807	0.0191627	0.00958136	−	0.999954i	$-0.496950\pi$
$277$	5.30807	0.0191627	0.00958136	+	0.999954i	$0.496950\pi$
$278$	0	0
$279$	65.7974i	0.235833i
$280$	0	0
$281$	297.662	1.05930	0.529648	−	0.848217i	$-0.322324\pi$
$281$	297.662	1.05930	0.529648	+	0.848217i	$0.322324\pi$
$282$	0	0
$283$	205.878i	0.727484i	0.931500	+	0.363742i	$0.118501\pi$
$283$	205.878i	0.727484i	−0.931500	+	0.363742i	$0.881499\pi$
$284$	0	0
$285$	70.0286i	0.245715i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	243.410	0.842250
$290$	0	0
$291$	−246.667	−0.847652
$292$	0	0
$293$	138.398i	0.472347i	0.971711	+	0.236173i	$0.0758933\pi$
$293$	138.398i	0.472347i	−0.971711	+	0.236173i	$0.924107\pi$
$294$	0	0
$295$	−382.302	−1.29594
$296$	0	0
$297$	− 19.6548i	− 0.0661779i
$298$	0	0
$299$	− 364.145i	− 1.21788i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	254.041	0.838419
$304$	0	0
$305$	732.917	2.40301
$306$	0	0
$307$	157.473i	0.512941i	0.966552	+	0.256471i	$0.0825597\pi$
$307$	157.473i	0.512941i	−0.966552	+	0.256471i	$0.917440\pi$
$308$	0	0
$309$	−13.3354	−0.0431567
$310$	0	0
$311$	− 274.812i	− 0.883638i	−0.897104	−	0.441819i	$-0.854333\pi$
$311$	− 274.812i	− 0.883638i	0.897104	−	0.441819i	$-0.145667\pi$
$312$	0	0
$313$	176.210i	0.562970i	0.959566	+	0.281485i	$0.0908270\pi$
$313$	176.210i	0.562970i	−0.959566	+	0.281485i	$0.909173\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−292.367	−0.922293	−0.461146	−	0.887324i	$-0.652562\pi$
$317$	−292.367	−0.922293	−0.461146	+	0.887324i	$0.652562\pi$
$318$	0	0
$319$	−24.8629	−0.0779403
$320$	0	0
$321$	241.784i	0.753221i
$322$	0	0
$323$	−44.5608	−0.137959
$324$	0	0
$325$	116.454i	0.358321i
$326$	0	0
$327$	− 262.599i	− 0.803056i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−142.597	−0.430806	−0.215403	−	0.976525i	$-0.569107\pi$
$331$	−142.597	−0.430806	−0.215403	+	0.976525i	$0.569107\pi$
$332$	0	0
$333$	−201.201	−0.604206
$334$	0	0
$335$	341.527i	1.01948i
$336$	0	0
$337$	477.413	1.41666	0.708328	−	0.705884i	$-0.249450\pi$
$337$	477.413	1.41666	0.708328	+	0.705884i	$0.249450\pi$
$338$	0	0
$339$	− 70.1149i	− 0.206829i
$340$	0	0
$341$	− 82.9613i	− 0.243288i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−415.782	−1.20516
$346$	0	0
$347$	453.373	1.30655	0.653275	−	0.757121i	$-0.273395\pi$
$347$	453.373	1.30655	0.653275	+	0.757121i	$0.273395\pi$
$348$	0	0
$349$	55.8211i	0.159946i	0.996797	+	0.0799730i	$0.0254834\pi$
$349$	55.8211i	0.159946i	−0.996797	+	0.0799730i	$0.974517\pi$
$350$	0	0
$351$	−48.2889	−0.137575
$352$	0	0
$353$	228.213i	0.646496i	0.946314	+	0.323248i	$0.104775\pi$
$353$	228.213i	0.646496i	−0.946314	+	0.323248i	$0.895225\pi$
$354$	0	0
$355$	803.509i	2.26340i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	467.810	1.30309	0.651547	−	0.758609i	$-0.274120\pi$
$359$	467.810	1.30309	0.651547	+	0.758609i	$0.274120\pi$
$360$	0	0
$361$	317.445	0.879349
$362$	0	0
$363$	− 184.796i	− 0.509080i
$364$	0	0
$365$	471.983	1.29311
$366$	0	0
$367$	− 305.649i	− 0.832830i	−0.909175	−	0.416415i	$-0.863286\pi$
$367$	− 305.649i	− 0.832830i	0.909175	−	0.416415i	$-0.136714\pi$
$368$	0	0
$369$	160.971i	0.436235i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	262.433	0.703574	0.351787	−	0.936080i	$-0.385574\pi$
$373$	262.433	0.703574	0.351787	+	0.936080i	$0.385574\pi$
$374$	0	0
$375$	−132.307	−0.352819
$376$	0	0
$377$	61.0843i	0.162027i
$378$	0	0
$379$	489.503	1.29156	0.645782	−	0.763522i	$-0.276531\pi$
$379$	489.503	1.29156	0.645782	+	0.763522i	$0.276531\pi$
$380$	0	0
$381$	153.004i	0.401585i
$382$	0	0
$383$	− 333.734i	− 0.871368i	−0.900100	−	0.435684i	$-0.856506\pi$
$383$	− 333.734i	− 0.871368i	0.900100	−	0.435684i	$-0.143494\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−126.128	−0.325911
$388$	0	0
$389$	596.084	1.53235	0.766174	−	0.642633i	$-0.222158\pi$
$389$	596.084	1.53235	0.766174	+	0.642633i	$0.222158\pi$
$390$	0	0
$391$	− 264.571i	− 0.676653i
$392$	0	0
$393$	117.170	0.298142
$394$	0	0
$395$	− 915.836i	− 2.31857i
$396$	0	0
$397$	309.915i	0.780641i	0.920679	+	0.390321i	$0.127636\pi$
$397$	309.915i	0.780641i	−0.920679	+	0.390321i	$0.872364\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−335.146	−0.835776	−0.417888	−	0.908498i	$-0.637230\pi$
$401$	−335.146	−0.835776	−0.417888	+	0.908498i	$0.637230\pi$
$402$	0	0
$403$	−203.823	−0.505764
$404$	0	0
$405$	55.1364i	0.136139i
$406$	0	0
$407$	253.685	0.623306
$408$	0	0
$409$	349.246i	0.853902i	0.904275	+	0.426951i	$0.140412\pi$
$409$	349.246i	0.853902i	−0.904275	+	0.426951i	$0.859588\pi$
$410$	0	0
$411$	− 286.201i	− 0.696352i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−243.611	−0.587013
$416$	0	0
$417$	−126.545	−0.303466
$418$	0	0
$419$	359.369i	0.857683i	0.903380	+	0.428841i	$0.141078\pi$
$419$	359.369i	0.857683i	−0.903380	+	0.428841i	$0.858922\pi$
$420$	0	0
$421$	−831.803	−1.97578	−0.987890	−	0.155159i	$-0.950411\pi$
$421$	−831.803	−1.97578	−0.987890	+	0.155159i	$0.950411\pi$
$422$	0	0
$423$	− 172.618i	− 0.408080i
$424$	0	0
$425$	84.6104i	0.199083i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	60.8854	0.141924
$430$	0	0
$431$	−322.840	−0.749048	−0.374524	−	0.927217i	$-0.622194\pi$
$431$	−322.840	−0.749048	−0.374524	+	0.927217i	$0.622194\pi$
$432$	0	0
$433$	523.962i	1.21007i	0.796197	+	0.605037i	$0.206842\pi$
$433$	523.962i	1.21007i	−0.796197	+	0.605037i	$0.793158\pi$
$434$	0	0
$435$	69.7463	0.160336
$436$	0	0
$437$	− 258.600i	− 0.591762i
$438$	0	0
$439$	− 700.766i	− 1.59628i	−0.602474	−	0.798138i	$-0.705818\pi$
$439$	− 700.766i	− 1.59628i	0.602474	−	0.798138i	$-0.294182\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	96.8893	0.218712	0.109356	−	0.994003i	$-0.465121\pi$
$443$	96.8893	0.218712	0.109356	+	0.994003i	$0.465121\pi$
$444$	0	0
$445$	−343.078	−0.770962
$446$	0	0
$447$	260.826i	0.583504i
$448$	0	0
$449$	−499.063	−1.11150	−0.555749	−	0.831350i	$-0.687568\pi$
$449$	−499.063	−1.11150	−0.555749	+	0.831350i	$0.687568\pi$
$450$	0	0
$451$	− 202.962i	− 0.450026i
$452$	0	0
$453$	− 240.728i	− 0.531408i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	316.568	0.692708	0.346354	−	0.938104i	$-0.387420\pi$
$457$	316.568	0.692708	0.346354	+	0.938104i	$0.387420\pi$
$458$	0	0
$459$	−35.0845	−0.0764369
$460$	0	0
$461$	− 387.287i	− 0.840103i	−0.907500	−	0.420051i	$-0.862012\pi$
$461$	− 387.287i	− 0.840103i	0.907500	−	0.420051i	$-0.137988\pi$
$462$	0	0
$463$	−909.661	−1.96471	−0.982355	−	0.187027i	$-0.940115\pi$
$463$	−909.661	−1.96471	−0.982355	+	0.187027i	$0.940115\pi$
$464$	0	0
$465$	232.726i	0.500485i
$466$	0	0
$467$	44.3876i	0.0950483i	0.998870	+	0.0475242i	$0.0151331\pi$
$467$	44.3876i	0.0950483i	−0.998870	+	0.0475242i	$0.984867\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	272.367	0.578274
$472$	0	0
$473$	159.029	0.336214
$474$	0	0
$475$	82.7008i	0.174107i
$476$	0	0
$477$	34.2655	0.0718354
$478$	0	0
$479$	149.548i	0.312209i	0.987741	+	0.156105i	$0.0498937\pi$
$479$	149.548i	0.312209i	−0.987741	+	0.156105i	$0.950106\pi$
$480$	0	0
$481$	− 623.265i	− 1.29577i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−872.461	−1.79889
$486$	0	0
$487$	−57.0282	−0.117101	−0.0585505	−	0.998284i	$-0.518648\pi$
$487$	−57.0282	−0.117101	−0.0585505	+	0.998284i	$0.518648\pi$
$488$	0	0
$489$	36.9319i	0.0755254i
$490$	0	0
$491$	563.958	1.14859	0.574296	−	0.818648i	$-0.305276\pi$
$491$	563.958	1.14859	0.574296	+	0.818648i	$0.305276\pi$
$492$	0	0
$493$	44.3811i	0.0900226i
$494$	0	0
$495$	− 69.5192i	− 0.140443i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	388.037	0.777629	0.388815	−	0.921316i	$-0.372885\pi$
$499$	388.037	0.777629	0.388815	+	0.921316i	$0.372885\pi$
$500$	0	0
$501$	−308.249	−0.615267
$502$	0	0
$503$	− 712.944i	− 1.41738i	−0.705518	−	0.708692i	$-0.749286\pi$
$503$	− 712.944i	− 1.41738i	0.705518	−	0.708692i	$-0.250714\pi$
$504$	0	0
$505$	898.543	1.77929
$506$	0	0
$507$	143.131i	0.282309i
$508$	0	0
$509$	515.042i	1.01187i	0.862571	+	0.505935i	$0.168853\pi$
$509$	515.042i	1.01187i	−0.862571	+	0.505935i	$0.831147\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−34.2927	−0.0668473
$514$	0	0
$515$	−47.1674	−0.0915873
$516$	0	0
$517$	217.647i	0.420980i
$518$	0	0
$519$	−190.702	−0.367442
$520$	0	0
$521$	− 31.9925i	− 0.0614059i	−0.999529	−	0.0307030i	$-0.990225\pi$
$521$	− 31.9925i	− 0.0614059i	0.999529	−	0.0307030i	$-0.00977459\pi$
$522$	0	0
$523$	− 318.149i	− 0.608316i	−0.952622	−	0.304158i	$-0.901625\pi$
$523$	− 318.149i	− 0.608316i	0.952622	−	0.304158i	$-0.0983750\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−148.089	−0.281003
$528$	0	0
$529$	1006.39	1.90243
$530$	0	0
$531$	− 187.211i	− 0.352564i
$532$	0	0
$533$	−498.645	−0.935543
$534$	0	0
$535$	855.190i	1.59849i
$536$	0	0
$537$	48.9216i	0.0911018i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	239.017	0.441806	0.220903	−	0.975296i	$-0.429100\pi$
$541$	239.017	0.441806	0.220903	+	0.975296i	$0.429100\pi$
$542$	0	0
$543$	−357.513	−0.658404
$544$	0	0
$545$	− 928.814i	− 1.70425i
$546$	0	0
$547$	−436.346	−0.797707	−0.398854	−	0.917015i	$-0.630592\pi$
$547$	−436.346	−0.797707	−0.398854	+	0.917015i	$0.630592\pi$
$548$	0	0
$549$	358.906i	0.653744i
$550$	0	0
$551$	43.3795i	0.0787286i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−711.647	−1.28225
$556$	0	0
$557$	−885.577	−1.58991	−0.794953	−	0.606672i	$-0.792504\pi$
$557$	−885.577	−1.58991	−0.794953	+	0.606672i	$0.792504\pi$
$558$	0	0
$559$	− 390.710i	− 0.698944i
$560$	0	0
$561$	44.2366	0.0788532
$562$	0	0
$563$	− 178.076i	− 0.316299i	−0.987415	−	0.158150i	$-0.949447\pi$
$563$	− 178.076i	− 0.316299i	0.987415	−	0.158150i	$-0.0505528\pi$
$564$	0	0
$565$	− 247.996i	− 0.438932i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	902.576	1.58625	0.793125	−	0.609059i	$-0.208453\pi$
$569$	902.576	1.58625	0.793125	+	0.609059i	$0.208453\pi$
$570$	0	0
$571$	−80.0281	−0.140154	−0.0700771	−	0.997542i	$-0.522325\pi$
$571$	−80.0281	−0.140154	−0.0700771	+	0.997542i	$0.522325\pi$
$572$	0	0
$573$	110.395i	0.192662i
$574$	0	0
$575$	−491.020	−0.853948
$576$	0	0
$577$	− 503.848i	− 0.873220i	−0.899651	−	0.436610i	$-0.856179\pi$
$577$	− 503.848i	− 0.873220i	0.899651	−	0.436610i	$-0.143821\pi$
$578$	0	0
$579$	− 116.658i	− 0.201481i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−43.2039	−0.0741062
$584$	0	0
$585$	−170.798	−0.291962
$586$	0	0
$587$	− 458.274i	− 0.780706i	−0.920665	−	0.390353i	$-0.872353\pi$
$587$	− 458.274i	− 0.780706i	0.920665	−	0.390353i	$-0.127647\pi$
$588$	0	0
$589$	−144.746	−0.245749
$590$	0	0
$591$	166.399i	0.281556i
$592$	0	0
$593$	− 670.468i	− 1.13064i	−0.824873	−	0.565318i	$-0.808753\pi$
$593$	− 670.468i	− 1.13064i	0.824873	−	0.565318i	$-0.191247\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−356.785	−0.597630
$598$	0	0
$599$	440.129	0.734773	0.367387	−	0.930068i	$-0.380253\pi$
$599$	440.129	0.734773	0.367387	+	0.930068i	$0.380253\pi$
$600$	0	0
$601$	732.160i	1.21824i	0.793080	+	0.609118i	$0.208476\pi$
$601$	732.160i	1.21824i	−0.793080	+	0.609118i	$0.791524\pi$
$602$	0	0
$603$	−167.244	−0.277353
$604$	0	0
$605$	− 653.624i	− 1.08037i
$606$	0	0
$607$	326.897i	0.538546i	0.963064	+	0.269273i	$0.0867834\pi$
$607$	326.897i	0.538546i	−0.963064	+	0.269273i	$0.913217\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	534.723	0.875161
$612$	0	0
$613$	−988.663	−1.61283	−0.806414	−	0.591352i	$-0.798594\pi$
$613$	−988.663	−1.61283	−0.806414	+	0.591352i	$0.798594\pi$
$614$	0	0
$615$	569.354i	0.925779i
$616$	0	0
$617$	280.536	0.454678	0.227339	−	0.973816i	$-0.426997\pi$
$617$	280.536	0.454678	0.227339	+	0.973816i	$0.426997\pi$
$618$	0	0
$619$	1032.81i	1.66851i	0.551377	+	0.834256i	$0.314103\pi$
$619$	1032.81i	1.66851i	−0.551377	+	0.834256i	$0.685897\pi$
$620$	0	0
$621$	− 203.606i	− 0.327868i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−781.249	−1.25000
$626$	0	0
$627$	43.2382	0.0689604
$628$	0	0
$629$	− 452.837i	− 0.719931i
$630$	0	0
$631$	−178.252	−0.282491	−0.141245	−	0.989975i	$-0.545111\pi$
$631$	−178.252	−0.282491	−0.141245	+	0.989975i	$0.545111\pi$
$632$	0	0
$633$	211.210i	0.333665i
$634$	0	0
$635$	541.174i	0.852243i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−393.474	−0.615765
$640$	0	0
$641$	668.526	1.04294	0.521471	−	0.853269i	$-0.325383\pi$
$641$	668.526	1.04294	0.521471	+	0.853269i	$0.325383\pi$
$642$	0	0
$643$	313.584i	0.487690i	0.969814	+	0.243845i	$0.0784088\pi$
$643$	313.584i	0.487690i	−0.969814	+	0.243845i	$0.921591\pi$
$644$	0	0
$645$	−446.114	−0.691649
$646$	0	0
$647$	− 337.531i	− 0.521686i	−0.965381	−	0.260843i	$-0.916000\pi$
$647$	− 337.531i	− 0.521686i	0.965381	−	0.260843i	$-0.0840005\pi$
$648$	0	0
$649$	236.047i	0.363709i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	903.354	1.38339	0.691695	−	0.722189i	$-0.256864\pi$
$653$	903.354	1.38339	0.691695	+	0.722189i	$0.256864\pi$
$654$	0	0
$655$	414.430	0.632717
$656$	0	0
$657$	231.128i	0.351793i
$658$	0	0
$659$	−768.092	−1.16554	−0.582771	−	0.812637i	$-0.698031\pi$
$659$	−768.092	−1.16554	−0.582771	+	0.812637i	$0.698031\pi$
$660$	0	0
$661$	− 411.223i	− 0.622123i	−0.950390	−	0.311062i	$-0.899315\pi$
$661$	− 411.223i	− 0.622123i	0.950390	−	0.311062i	$-0.100685\pi$
$662$	0	0
$663$	− 108.682i	− 0.163925i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−257.557	−0.386143
$668$	0	0
$669$	−32.0474	−0.0479034
$670$	0	0
$671$	− 452.529i	− 0.674410i
$672$	0	0
$673$	−818.448	−1.21612	−0.608060	−	0.793891i	$-0.708052\pi$
$673$	−818.448	−1.21612	−0.608060	+	0.793891i	$0.708052\pi$
$674$	0	0
$675$	65.1137i	0.0964647i
$676$	0	0
$677$	− 55.5247i	− 0.0820158i	−0.999159	−	0.0410079i	$-0.986943\pi$
$677$	− 55.5247i	− 0.0820158i	0.999159	−	0.0410079i	$-0.0130569\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	254.696	0.374003
$682$	0	0
$683$	−76.2838	−0.111689	−0.0558447	−	0.998439i	$-0.517785\pi$
$683$	−76.2838	−0.111689	−0.0558447	+	0.998439i	$0.517785\pi$
$684$	0	0
$685$	− 1012.29i	− 1.47780i
$686$	0	0
$687$	623.391	0.907410
$688$	0	0
$689$	106.145i	0.154057i
$690$	0	0
$691$	22.4912i	0.0325488i	0.999868	+	0.0162744i	$0.00518053\pi$
$691$	22.4912i	0.0325488i	−0.999868	+	0.0162744i	$0.994819\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−447.591	−0.644016
$696$	0	0
$697$	−362.293	−0.519789
$698$	0	0
$699$	76.3282i	0.109196i
$700$	0	0
$701$	−854.197	−1.21854	−0.609270	−	0.792963i	$-0.708538\pi$
$701$	−854.197	−1.21854	−0.609270	+	0.792963i	$0.708538\pi$
$702$	0	0
$703$	− 442.616i	− 0.629610i
$704$	0	0
$705$	− 610.549i	− 0.866027i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−158.602	−0.223698	−0.111849	−	0.993725i	$-0.535677\pi$
$709$	−158.602	−0.223698	−0.111849	+	0.993725i	$0.535677\pi$
$710$	0	0
$711$	448.480	0.630773
$712$	0	0
$713$	− 859.403i	− 1.20533i
$714$	0	0
$715$	215.352	0.301191
$716$	0	0
$717$	− 285.921i	− 0.398775i
$718$	0	0
$719$	380.118i	0.528675i	0.964430	+	0.264338i	$0.0851533\pi$
$719$	380.118i	0.528675i	−0.964430	+	0.264338i	$0.914847\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−231.656	−0.320409
$724$	0	0
$725$	82.3674	0.113610
$726$	0	0
$727$	159.283i	0.219096i	0.993981	+	0.109548i	$0.0349404\pi$
$727$	159.283i	0.219096i	−0.993981	+	0.109548i	$0.965060\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 283.872i	− 0.388334i
$732$	0	0
$733$	− 178.179i	− 0.243082i	−0.992586	−	0.121541i	$-0.961216\pi$
$733$	− 178.179i	− 0.243082i	0.992586	−	0.121541i	$-0.0387836\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	210.871	0.286121
$738$	0	0
$739$	−317.386	−0.429481	−0.214740	−	0.976671i	$-0.568891\pi$
$739$	−317.386	−0.429481	−0.214740	+	0.976671i	$0.568891\pi$
$740$	0	0
$741$	− 106.229i	− 0.143360i
$742$	0	0
$743$	1055.73	1.42090	0.710450	−	0.703748i	$-0.248492\pi$
$743$	1055.73	1.42090	0.710450	+	0.703748i	$0.248492\pi$
$744$	0	0
$745$	922.542i	1.23831i
$746$	0	0
$747$	− 119.295i	− 0.159699i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−307.209	−0.409067	−0.204533	−	0.978860i	$-0.565568\pi$
$751$	−307.209	−0.409067	−0.204533	+	0.978860i	$0.565568\pi$
$752$	0	0
$753$	699.124	0.928451
$754$	0	0
$755$	− 851.454i	− 1.12775i
$756$	0	0
$757$	−711.179	−0.939470	−0.469735	−	0.882808i	$-0.655651\pi$
$757$	−711.179	−0.939470	−0.469735	+	0.882808i	$0.655651\pi$
$758$	0	0
$759$	256.719i	0.338233i
$760$	0	0
$761$	527.953i	0.693763i	0.937909	+	0.346881i	$0.112759\pi$
$761$	527.953i	0.693763i	−0.937909	+	0.346881i	$0.887241\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−124.094	−0.162214
$766$	0	0
$767$	579.931	0.756103
$768$	0	0
$769$	− 831.961i	− 1.08187i	−0.841063	−	0.540937i	$-0.818070\pi$
$769$	− 831.961i	− 1.08187i	0.841063	−	0.540937i	$-0.181930\pi$
$770$	0	0
$771$	−484.836	−0.628840
$772$	0	0
$773$	− 758.707i	− 0.981510i	−0.871298	−	0.490755i	$-0.836721\pi$
$773$	− 758.707i	− 0.981510i	0.871298	−	0.490755i	$-0.163279\pi$
$774$	0	0
$775$	274.839i	0.354631i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−354.116	−0.454577
$780$	0	0
$781$	496.115	0.635231
$782$	0	0
$783$	34.1544i	0.0436199i
$784$	0	0
$785$	963.363	1.22721
$786$	0	0
$787$	− 687.187i	− 0.873173i	−0.899662	−	0.436586i	$-0.856187\pi$
$787$	− 687.187i	− 0.873173i	0.899662	−	0.436586i	$-0.143813\pi$
$788$	0	0
$789$	245.765i	0.311489i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1111.79	−1.40201
$794$	0	0
$795$	121.197	0.152449
$796$	0	0
$797$	− 716.069i	− 0.898456i	−0.893417	−	0.449228i	$-0.851699\pi$
$797$	− 716.069i	− 0.898456i	0.893417	−	0.449228i	$-0.148301\pi$
$798$	0	0
$799$	388.506	0.486240
$800$	0	0
$801$	− 168.003i	− 0.209742i
$802$	0	0
$803$	− 291.419i	− 0.362913i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	689.015	0.853798
$808$	0	0
$809$	−136.167	−0.168315	−0.0841577	−	0.996452i	$-0.526820\pi$
$809$	−136.167	−0.168315	−0.0841577	+	0.996452i	$0.526820\pi$
$810$	0	0
$811$	421.599i	0.519851i	0.965629	+	0.259926i	$0.0836981\pi$
$811$	421.599i	0.519851i	−0.965629	+	0.259926i	$0.916302\pi$
$812$	0	0
$813$	−757.124	−0.931272
$814$	0	0
$815$	130.628i	0.160280i
$816$	0	0
$817$	− 277.465i	− 0.339615i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	79.6476	0.0970130	0.0485065	−	0.998823i	$-0.484554\pi$
$821$	79.6476	0.0970130	0.0485065	+	0.998823i	$0.484554\pi$
$822$	0	0
$823$	1332.30	1.61883	0.809416	−	0.587236i	$-0.199784\pi$
$823$	1332.30	1.61883	0.809416	+	0.587236i	$0.199784\pi$
$824$	0	0
$825$	− 82.0991i	− 0.0995141i
$826$	0	0
$827$	−468.548	−0.566564	−0.283282	−	0.959037i	$-0.591423\pi$
$827$	−468.548	−0.566564	−0.283282	+	0.959037i	$0.591423\pi$
$828$	0	0
$829$	− 492.032i	− 0.593524i	−0.954951	−	0.296762i	$-0.904093\pi$
$829$	− 492.032i	− 0.593524i	0.954951	−	0.296762i	$-0.0959068\pi$
$830$	0	0
$831$	9.19385i	0.0110636i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1090.28	−1.30572
$836$	0	0
$837$	−113.965	−0.136158
$838$	0	0
$839$	− 1045.41i	− 1.24602i	−0.782214	−	0.623010i	$-0.785910\pi$
$839$	− 1045.41i	− 1.24602i	0.782214	−	0.623010i	$-0.214090\pi$
$840$	0	0
$841$	−797.795	−0.948627
$842$	0	0
$843$	515.566i	0.611585i
$844$	0	0
$845$	506.253i	0.599116i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−356.591	−0.420013
$850$	0	0
$851$	2627.95	3.08807
$852$	0	0
$853$	− 339.686i	− 0.398225i	−0.979977	−	0.199113i	$-0.936194\pi$
$853$	− 339.686i	− 0.398225i	0.979977	−	0.199113i	$-0.0638060\pi$
$854$	0	0
$855$	−121.293	−0.141863
$856$	0	0
$857$	− 1404.99i	− 1.63943i	−0.572769	−	0.819717i	$-0.694131\pi$
$857$	− 1404.99i	− 1.63943i	0.572769	−	0.819717i	$-0.305869\pi$
$858$	0	0
$859$	− 1209.22i	− 1.40771i	−0.710346	−	0.703853i	$-0.751462\pi$
$859$	− 1209.22i	− 1.40771i	0.710346	−	0.703853i	$-0.248538\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−994.257	−1.15209	−0.576047	−	0.817416i	$-0.695406\pi$
$863$	−994.257	−1.15209	−0.576047	+	0.817416i	$0.695406\pi$
$864$	0	0
$865$	−674.514	−0.779785
$866$	0	0
$867$	421.599i	0.486273i
$868$	0	0
$869$	−565.470	−0.650713
$870$	0	0
$871$	− 518.077i	− 0.594807i
$872$	0	0
$873$	− 427.240i	− 0.489392i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1554.74	−1.77279	−0.886396	−	0.462928i	$-0.846799\pi$
$877$	−1554.74	−1.77279	−0.886396	+	0.462928i	$0.846799\pi$
$878$	0	0
$879$	−239.712	−0.272710
$880$	0	0
$881$	921.015i	1.04542i	0.852511	+	0.522710i	$0.175079\pi$
$881$	921.015i	1.04542i	−0.852511	+	0.522710i	$0.824921\pi$
$882$	0	0
$883$	156.823	0.177603	0.0888014	−	0.996049i	$-0.471696\pi$
$883$	156.823	0.177603	0.0888014	+	0.996049i	$0.471696\pi$
$884$	0	0
$885$	− 662.167i	− 0.748212i
$886$	0	0
$887$	1618.72i	1.82494i	0.409141	+	0.912471i	$0.365828\pi$
$887$	1618.72i	1.82494i	−0.409141	+	0.912471i	$0.634172\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	34.0432	0.0382078
$892$	0	0
$893$	379.737	0.425238
$894$	0	0
$895$	173.036i	0.193336i
$896$	0	0
$897$	630.717	0.703141
$898$	0	0
$899$	144.163i	0.160359i
$900$	0	0
$901$	77.1204i	0.0855942i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1264.53	−1.39727
$906$	0	0
$907$	558.690	0.615975	0.307988	−	0.951390i	$-0.400344\pi$
$907$	558.690	0.615975	0.307988	+	0.951390i	$0.400344\pi$
$908$	0	0
$909$	440.012i	0.484061i
$910$	0	0
$911$	−510.774	−0.560674	−0.280337	−	0.959902i	$-0.590446\pi$
$911$	−510.774	−0.560674	−0.280337	+	0.959902i	$0.590446\pi$
$912$	0	0
$913$	150.414i	0.164747i
$914$	0	0
$915$	1269.45i	1.38738i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−667.354	−0.726174	−0.363087	−	0.931755i	$-0.618277\pi$
$919$	−667.354	−0.726174	−0.363087	+	0.931755i	$0.618277\pi$
$920$	0	0
$921$	−272.751	−0.296147
$922$	0	0
$923$	− 1218.88i	− 1.32056i
$924$	0	0
$925$	−840.424	−0.908566
$926$	0	0
$927$	− 23.0977i	− 0.0249166i
$928$	0	0
$929$	877.672i	0.944749i	0.881398	+	0.472375i	$0.156603\pi$
$929$	877.672i	0.944749i	−0.881398	+	0.472375i	$0.843397\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	475.988	0.510169
$934$	0	0
$935$	156.465	0.167342
$936$	0	0
$937$	− 211.107i	− 0.225301i	−0.993635	−	0.112650i	$-0.964066\pi$
$937$	− 211.107i	− 0.225301i	0.993635	−	0.112650i	$-0.0359340\pi$
$938$	0	0
$939$	−305.204	−0.325031
$940$	0	0
$941$	− 1526.19i	− 1.62188i	−0.585132	−	0.810938i	$-0.698957\pi$
$941$	− 1526.19i	− 1.62188i	0.585132	−	0.810938i	$-0.301043\pi$
$942$	0	0
$943$	− 2102.50i	− 2.22958i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	642.310	0.678257	0.339129	−	0.940740i	$-0.389868\pi$
$947$	642.310	0.678257	0.339129	+	0.940740i	$0.389868\pi$
$948$	0	0
$949$	−715.972	−0.754449
$950$	0	0
$951$	− 506.394i	− 0.532486i
$952$	0	0
$953$	208.041	0.218302	0.109151	−	0.994025i	$-0.465187\pi$
$953$	208.041	0.218302	0.109151	+	0.994025i	$0.465187\pi$
$954$	0	0
$955$	390.469i	0.408868i
$956$	0	0
$957$	− 43.0639i	− 0.0449988i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	479.966	0.499445
$962$	0	0
$963$	−418.782	−0.434872
$964$	0	0
$965$	− 412.618i	− 0.427584i
$966$	0	0
$967$	1485.38	1.53607	0.768034	−	0.640409i	$-0.221235\pi$
$967$	1485.38	1.53607	0.768034	+	0.640409i	$0.221235\pi$
$968$	0	0
$969$	− 77.1815i	− 0.0796507i
$970$	0	0
$971$	− 342.856i	− 0.353096i	−0.984292	−	0.176548i	$-0.943507\pi$
$971$	− 342.856i	− 0.353096i	0.984292	−	0.176548i	$-0.0564931\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−201.705	−0.206877
$976$	0	0
$977$	1670.18	1.70950	0.854748	−	0.519043i	$-0.173711\pi$
$977$	1670.18	1.70950	0.854748	+	0.519043i	$0.173711\pi$
$978$	0	0
$979$	211.829i	0.216372i
$980$	0	0
$981$	454.835	0.463644
$982$	0	0
$983$	805.192i	0.819117i	0.912284	+	0.409558i	$0.134317\pi$
$983$	805.192i	0.819117i	−0.912284	+	0.409558i	$0.865683\pi$
$984$	0	0
$985$	588.555i	0.597517i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1647.40	1.66572
$990$	0	0
$991$	−1503.00	−1.51665	−0.758326	−	0.651875i	$-0.773983\pi$
$991$	−1503.00	−1.51665	−0.758326	+	0.651875i	$0.773983\pi$
$992$	0	0
$993$	− 246.985i	− 0.248726i
$994$	0	0
$995$	−1261.95	−1.26829
$996$	0	0
$997$	− 704.037i	− 0.706156i	−0.935594	−	0.353078i	$-0.885135\pi$
$997$	− 704.037i	− 0.706156i	0.935594	−	0.353078i	$-0.114865\pi$
$998$	0	0
$999$	− 348.490i	− 0.348838i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.f.g.97.8		8
4.3	odd	2		1176.3.f.c.97.4		8
7.2	even	3		336.3.bh.g.241.4		8
7.3	odd	6		336.3.bh.g.145.4		8
7.6	odd	2	inner	2352.3.f.g.97.1		8
12.11	even	2		3528.3.f.b.2449.2		8
21.2	odd	6		1008.3.cg.p.577.1		8
21.17	even	6		1008.3.cg.p.145.1		8
28.3	even	6		168.3.z.b.145.4	yes	8
28.11	odd	6		1176.3.z.c.313.1		8
28.19	even	6		1176.3.z.c.913.1		8
28.23	odd	6		168.3.z.b.73.4	✓	8
28.27	even	2		1176.3.f.c.97.5		8
84.23	even	6		504.3.by.c.73.1		8
84.59	odd	6		504.3.by.c.145.1		8
84.83	odd	2		3528.3.f.b.2449.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.z.b.73.4	✓	8	28.23	odd	6
168.3.z.b.145.4	yes	8	28.3	even	6
336.3.bh.g.145.4		8	7.3	odd	6
336.3.bh.g.241.4		8	7.2	even	3
504.3.by.c.73.1		8	84.23	even	6
504.3.by.c.145.1		8	84.59	odd	6
1008.3.cg.p.145.1		8	21.17	even	6
1008.3.cg.p.577.1		8	21.2	odd	6
1176.3.f.c.97.4		8	4.3	odd	2
1176.3.f.c.97.5		8	28.27	even	2
1176.3.z.c.313.1		8	28.11	odd	6
1176.3.z.c.913.1		8	28.19	even	6
2352.3.f.g.97.1		8	7.6	odd	2	inner
2352.3.f.g.97.8		8	1.1	even	1	trivial
3528.3.f.b.2449.2		8	12.11	even	2
3528.3.f.b.2449.7		8	84.83	odd	2

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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +6.12627i q^{5} -3.00000 q^{9} +3.78258 q^{11} -9.29319i q^{13} -10.6110 q^{15} -6.75202i q^{17} -6.59962i q^{19} +39.1840 q^{23} -12.5311 q^{25} -5.19615i q^{27} -6.57302 q^{29} -21.9325i q^{31} +6.55161i q^{33} +67.0668 q^{37} +16.0963 q^{39} -53.6570i q^{41} +42.0426 q^{43} -18.3788i q^{45} +57.5392i q^{47} +11.6948 q^{51} -11.4218 q^{53} +23.1731i q^{55} +11.4309 q^{57} +62.4038i q^{59} -119.635i q^{61} +56.9326 q^{65} +55.7480 q^{67} +67.8687i q^{69} +131.158 q^{71} -77.0426i q^{73} -21.7046i q^{75} -149.493 q^{79} +9.00000 q^{81} +39.7649i q^{83} +41.3647 q^{85} -11.3848i q^{87} +56.0011i q^{89} +37.9882 q^{93} +40.4311 q^{95} +142.413i q^{97} -11.3477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9} - 44 q^{11} + 12 q^{15} + 96 q^{23} - 84 q^{25} + 68 q^{29} + 236 q^{37} - 36 q^{39} + 92 q^{43} + 72 q^{51} - 20 q^{53} + 84 q^{57} + 296 q^{65} + 44 q^{67} + 392 q^{71} + 328 q^{79} + 72 q^{81}+ \cdots + 132 q^{99}+O(q^{100}) \) 8 * q - 24 * q^9 - 44 * q^11 + 12 * q^15 + 96 * q^23 - 84 * q^25 + 68 * q^29 + 236 * q^37 - 36 * q^39 + 92 * q^43 + 72 * q^51 - 20 * q^53 + 84 * q^57 + 296 * q^65 + 44 * q^67 + 392 * q^71 + 328 * q^79 + 72 * q^81 + 200 * q^85 + 120 * q^93 + 132 * q^99

Introduction

L-functions

Modular forms

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Database

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