LMFDB - Embedded newform 4140.3.c.a.4049.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,3,Mod(4049,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4140.4049");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4140.c (of order $2$, degree $1$, 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:	$112.806829445$
Analytic rank:	$0$
Dimension:	$88$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.20
Character	$\chi$	$=$	4140.4049
Dual form			4140.3.c.a.4049.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.78150 + 3.27113i) q^{5} +2.52965i q^{7} +O(q^{10})$	$q+(-3.78150 + 3.27113i) q^{5} +2.52965i q^{7} -6.63980i q^{11} -10.3177i q^{13} -25.1606 q^{17} -6.66974 q^{19} +4.79583 q^{23} +(3.59943 - 24.7395i) q^{25} -36.1339i q^{29} +28.9594 q^{31} +(-8.27482 - 9.56587i) q^{35} +42.9777i q^{37} +37.8794i q^{41} -43.2315i q^{43} -19.6822 q^{47} +42.6009 q^{49} +56.7887 q^{53} +(21.7196 + 25.1084i) q^{55} -68.4408i q^{59} -53.2106 q^{61} +(33.7505 + 39.0163i) q^{65} -34.7520i q^{67} +83.8421i q^{71} +101.958i q^{73} +16.7964 q^{77} +153.734 q^{79} -118.533 q^{83} +(95.1446 - 82.3035i) q^{85} -23.7519i q^{89} +26.1002 q^{91} +(25.2216 - 21.8176i) q^{95} -46.4236i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q+O(q^{10}) $ 88 * q	$ 88 q - 16 q^{19} - 48 q^{25} + 272 q^{31} - 600 q^{49} + 112 q^{55} + 448 q^{61} - 32 q^{79} - 264 q^{85} - 16 q^{91}+O(q^{100}) $ 88 * q - 16 * q^19 - 48 * q^25 + 272 * q^31 - 600 * q^49 + 112 * q^55 + 448 * q^61 - 32 * q^79 - 264 * q^85 - 16 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.78150	+	3.27113i	−0.756299	+	0.654226i
$5$	−3.78150	+	3.27113i	−0.756299	+	0.654226i
$6$		0			0
$7$			2.52965i			0.361379i	0.983540	+	0.180689i	$0.0578329\pi$
$7$			2.52965i			0.361379i	−0.983540	+	0.180689i	$0.942167\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	6.63980i		−	0.603618i	−0.953368	−	0.301809i	$-0.902409\pi$
$11$		−	6.63980i		−	0.603618i	0.953368	−	0.301809i	$-0.0975905\pi$
$12$		0			0
$13$		−	10.3177i		−	0.793669i	−0.917890	−	0.396835i	$-0.870109\pi$
$13$		−	10.3177i		−	0.793669i	0.917890	−	0.396835i	$-0.129891\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−25.1606			−1.48003			−0.740017	−	0.672588i	$-0.765183\pi$
$17$	−25.1606			−1.48003			−0.740017	+	0.672588i	$0.765183\pi$
$18$		0			0
$19$	−6.66974			−0.351039			−0.175519	−	0.984476i	$-0.556160\pi$
$19$	−6.66974			−0.351039			−0.175519	+	0.984476i	$0.556160\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.79583			0.208514
$23$	4.79583			0.208514
$24$		0			0
$25$	3.59943	−	24.7395i	0.143977	−	0.989581i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	36.1339i		−	1.24600i	−0.782223	−	0.622998i	$-0.785914\pi$
$29$		−	36.1339i		−	1.24600i	0.782223	−	0.622998i	$-0.214086\pi$
$30$		0			0
$31$	28.9594			0.934173			0.467087	−	0.884212i	$-0.345304\pi$
$31$	28.9594			0.934173			0.467087	+	0.884212i	$0.345304\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.27482	−	9.56587i	−0.236423	−	0.273311i
$36$		0			0
$37$			42.9777i			1.16156i	0.814061	+	0.580780i	$0.197252\pi$
$37$			42.9777i			1.16156i	−0.814061	+	0.580780i	$0.802748\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			37.8794i			0.923889i	0.886909	+	0.461944i	$0.152848\pi$
$41$			37.8794i			0.923889i	−0.886909	+	0.461944i	$0.847152\pi$
$42$		0			0
$43$		−	43.2315i		−	1.00538i	−0.864466	−	0.502691i	$-0.832343\pi$
$43$		−	43.2315i		−	1.00538i	0.864466	−	0.502691i	$-0.167657\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−19.6822			−0.418771			−0.209385	−	0.977833i	$-0.567146\pi$
$47$	−19.6822			−0.418771			−0.209385	+	0.977833i	$0.567146\pi$
$48$		0			0
$49$	42.6009			0.869405
$50$		0			0
$51$		0			0
$52$		0			0
$53$	56.7887			1.07149			0.535743	−	0.844381i	$-0.320032\pi$
$53$	56.7887			1.07149			0.535743	+	0.844381i	$0.320032\pi$
$54$		0			0
$55$	21.7196	+	25.1084i	0.394903	+	0.456516i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	68.4408i		−	1.16001i	−0.814611	−	0.580007i	$-0.803050\pi$
$59$		−	68.4408i		−	1.16001i	0.814611	−	0.580007i	$-0.196950\pi$
$60$		0			0
$61$	−53.2106			−0.872305			−0.436152	−	0.899873i	$-0.643659\pi$
$61$	−53.2106			−0.872305			−0.436152	+	0.899873i	$0.643659\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	33.7505	+	39.0163i	0.519239	+	0.600251i
$66$		0			0
$67$		−	34.7520i		−	0.518687i	−0.965785	−	0.259343i	$-0.916494\pi$
$67$		−	34.7520i		−	0.518687i	0.965785	−	0.259343i	$-0.0835062\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			83.8421i			1.18087i	0.807084	+	0.590437i	$0.201045\pi$
$71$			83.8421i			1.18087i	−0.807084	+	0.590437i	$0.798955\pi$
$72$		0			0
$73$			101.958i			1.39669i	0.715761	+	0.698346i	$0.246080\pi$
$73$			101.958i			1.39669i	−0.715761	+	0.698346i	$0.753920\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	16.7964			0.218135
$78$		0			0
$79$	153.734			1.94600			0.973001	−	0.230800i	$-0.0741343\pi$
$79$	153.734			1.94600			0.973001	+	0.230800i	$0.0741343\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−118.533			−1.42810			−0.714051	−	0.700093i	$-0.753142\pi$
$83$	−118.533			−1.42810			−0.714051	+	0.700093i	$0.753142\pi$
$84$		0			0
$85$	95.1446	−	82.3035i	1.11935	−	0.968276i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	23.7519i		−	0.266876i	−0.991057	−	0.133438i	$-0.957398\pi$
$89$		−	23.7519i		−	0.266876i	0.991057	−	0.133438i	$-0.0426017\pi$
$90$		0			0
$91$	26.1002			0.286815
$92$		0			0
$93$		0			0
$94$		0			0
$95$	25.2216	−	21.8176i	0.265490	−	0.229659i
$96$		0			0
$97$		−	46.4236i		−	0.478594i	−0.970946	−	0.239297i	$-0.923083\pi$
$97$		−	46.4236i		−	0.478594i	0.970946	−	0.239297i	$-0.0769169\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			184.371i			1.82545i	0.408570	+	0.912727i	$0.366027\pi$
$101$			184.371i			1.82545i	−0.408570	+	0.912727i	$0.633973\pi$
$102$		0			0
$103$			195.264i			1.89577i	0.318618	+	0.947883i	$0.396781\pi$
$103$			195.264i			1.89577i	−0.318618	+	0.947883i	$0.603219\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−92.7640			−0.866953			−0.433477	−	0.901165i	$-0.642713\pi$
$107$	−92.7640			−0.866953			−0.433477	+	0.901165i	$0.642713\pi$
$108$		0			0
$109$	−154.634			−1.41866			−0.709329	−	0.704877i	$-0.751002\pi$
$109$	−154.634			−1.41866			−0.709329	+	0.704877i	$0.751002\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	70.0293			0.619729			0.309864	−	0.950781i	$-0.399716\pi$
$113$	70.0293			0.619729			0.309864	+	0.950781i	$0.399716\pi$
$114$		0			0
$115$	−18.1354	+	15.6878i	−0.157699	+	0.136416i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	63.6475i		−	0.534853i
$120$		0			0
$121$	76.9131			0.635645
$122$		0			0
$123$		0			0
$124$		0			0
$125$	67.3150	+	105.327i	0.538520	+	0.842613i
$126$		0			0
$127$		−	132.882i		−	1.04632i	−0.852236	−	0.523158i	$-0.824754\pi$
$127$		−	132.882i		−	1.04632i	0.852236	−	0.523158i	$-0.175246\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	247.056i		−	1.88593i	−0.332899	−	0.942963i	$-0.608027\pi$
$131$		−	247.056i		−	1.88593i	0.332899	−	0.942963i	$-0.391973\pi$
$132$		0			0
$133$		−	16.8721i		−	0.126858i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.91281			−0.0431592			−0.0215796	−	0.999767i	$-0.506870\pi$
$137$	−5.91281			−0.0431592			−0.0215796	+	0.999767i	$0.506870\pi$
$138$		0			0
$139$	−0.186655			−0.00134284			−0.000671422	−	1.00000i	$-0.500214\pi$
$139$	−0.186655			−0.00134284			−0.000671422		1.00000i	$0.500214\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−68.5074			−0.479073
$144$		0			0
$145$	118.199	+	136.640i	0.815163	+	0.942346i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			10.1341i			0.0680140i	0.999422	+	0.0340070i	$0.0108269\pi$
$149$			10.1341i			0.0680140i	−0.999422	+	0.0340070i	$0.989173\pi$
$150$		0			0
$151$	111.563			0.738825			0.369413	−	0.929265i	$-0.379559\pi$
$151$	111.563			0.738825			0.369413	+	0.929265i	$0.379559\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−109.510	+	94.7298i	−0.706514	+	0.611160i
$156$		0			0
$157$			75.3594i			0.479996i	0.970773	+	0.239998i	$0.0771468\pi$
$157$			75.3594i			0.479996i	−0.970773	+	0.239998i	$0.922853\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			12.1318i			0.0753527i
$162$		0			0
$163$			53.2019i			0.326392i	0.986594	+	0.163196i	$0.0521803\pi$
$163$			53.2019i			0.326392i	−0.986594	+	0.163196i	$0.947820\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−121.187			−0.725668			−0.362834	−	0.931854i	$-0.618191\pi$
$167$	−121.187			−0.725668			−0.362834	+	0.931854i	$0.618191\pi$
$168$		0			0
$169$	62.5451			0.370089
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−260.120			−1.50358			−0.751791	−	0.659401i	$-0.770810\pi$
$173$	−260.120			−1.50358			−0.751791	+	0.659401i	$0.770810\pi$
$174$		0			0
$175$	62.5824	+	9.10530i	0.357614	+	0.0520303i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			130.612i			0.729677i	0.931071	+	0.364838i	$0.118876\pi$
$179$			130.612i			0.729677i	−0.931071	+	0.364838i	$0.881124\pi$
$180$		0			0
$181$	1.10245			0.00609087			0.00304543	−	0.999995i	$-0.499031\pi$
$181$	1.10245			0.00609087			0.00304543	+	0.999995i	$0.499031\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−140.586	−	162.520i	−0.759923	−	0.878487i
$186$		0			0
$187$			167.061i			0.893375i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	38.9302i		−	0.203823i	−0.994793	−	0.101911i	$-0.967504\pi$
$191$		−	38.9302i		−	0.203823i	0.994793	−	0.101911i	$-0.0324958\pi$
$192$		0			0
$193$		−	187.952i		−	0.973847i	−0.873445	−	0.486923i	$-0.838119\pi$
$193$		−	187.952i		−	0.973847i	0.873445	−	0.486923i	$-0.161881\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	241.741			1.22711			0.613556	−	0.789651i	$-0.289738\pi$
$197$	241.741			1.22711			0.613556	+	0.789651i	$0.289738\pi$
$198$		0			0
$199$	303.325			1.52425			0.762123	−	0.647432i	$-0.224157\pi$
$199$	303.325			1.52425			0.762123	+	0.647432i	$0.224157\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	91.4062			0.450277
$204$		0			0
$205$	−123.909	−	143.241i	−0.604432	−	0.698736i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			44.2857i			0.211893i
$210$		0			0
$211$	25.2830			0.119825			0.0599123	−	0.998204i	$-0.480918\pi$
$211$	25.2830			0.119825			0.0599123	+	0.998204i	$0.480918\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	141.416	+	163.480i	0.657747	+	0.760370i
$216$		0			0
$217$			73.2571i			0.337591i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			259.599i			1.17466i
$222$		0			0
$223$			201.264i			0.902529i	0.892390	+	0.451264i	$0.149027\pi$
$223$			201.264i			0.902529i	−0.892390	+	0.451264i	$0.850973\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	100.489			0.442682			0.221341	−	0.975196i	$-0.428957\pi$
$227$	100.489			0.442682			0.221341	+	0.975196i	$0.428957\pi$
$228$		0			0
$229$	−424.886			−1.85540			−0.927700	−	0.373327i	$-0.878217\pi$
$229$	−424.886			−1.85540			−0.927700	+	0.373327i	$0.878217\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−410.888			−1.76347			−0.881734	−	0.471747i	$-0.843624\pi$
$233$	−410.888			−1.76347			−0.881734	+	0.471747i	$0.843624\pi$
$234$		0			0
$235$	74.4283	−	64.3831i	0.316716	−	0.273971i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			153.389i			0.641796i	0.947114	+	0.320898i	$0.103985\pi$
$239$			153.389i			0.641796i	−0.947114	+	0.320898i	$0.896015\pi$
$240$		0			0
$241$	−282.885			−1.17380			−0.586899	−	0.809660i	$-0.699652\pi$
$241$	−282.885			−1.17380			−0.586899	+	0.809660i	$0.699652\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−161.095	+	139.353i	−0.657531	+	0.568787i
$246$		0			0
$247$			68.8164i			0.278609i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			364.013i			1.45025i	0.688618	+	0.725125i	$0.258218\pi$
$251$			364.013i			1.45025i	−0.688618	+	0.725125i	$0.741782\pi$
$252$		0			0
$253$		−	31.8434i		−	0.125863i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−382.140			−1.48692			−0.743462	−	0.668778i	$-0.766818\pi$
$257$	−382.140			−1.48692			−0.743462	+	0.668778i	$0.766818\pi$
$258$		0			0
$259$	−108.719			−0.419763
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−493.158			−1.87513			−0.937563	−	0.347817i	$-0.886923\pi$
$263$	−493.158			−1.87513			−0.937563	+	0.347817i	$0.886923\pi$
$264$		0			0
$265$	−214.746	+	185.763i	−0.810364	+	0.700994i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			428.480i			1.59286i	0.604729	+	0.796431i	$0.293281\pi$
$269$			428.480i			1.59286i	−0.604729	+	0.796431i	$0.706719\pi$
$270$		0			0
$271$	−360.011			−1.32845			−0.664226	−	0.747532i	$-0.731239\pi$
$271$	−360.011			−1.32845			−0.664226	+	0.747532i	$0.731239\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−164.265	−	23.8995i	−0.597329	−	0.0869072i
$276$		0			0
$277$			21.9241i			0.0791482i	0.999217	+	0.0395741i	$0.0126001\pi$
$277$			21.9241i			0.0791482i	−0.999217	+	0.0395741i	$0.987400\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			407.669i			1.45078i	0.688339	+	0.725389i	$0.258340\pi$
$281$			407.669i			1.45078i	−0.688339	+	0.725389i	$0.741660\pi$
$282$		0			0
$283$			474.423i			1.67641i	0.545358	+	0.838203i	$0.316394\pi$
$283$			474.423i			1.67641i	−0.545358	+	0.838203i	$0.683606\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−95.8218			−0.333874
$288$		0			0
$289$	344.054			1.19050
$290$		0			0
$291$		0			0
$292$		0			0
$293$	170.989			0.583582			0.291791	−	0.956482i	$-0.405749\pi$
$293$	170.989			0.583582			0.291791	+	0.956482i	$0.405749\pi$
$294$		0			0
$295$	223.879	+	258.809i	0.758911	+	0.877318i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	49.4819i		−	0.165491i
$300$		0			0
$301$	109.361			0.363324
$302$		0			0
$303$		0			0
$304$		0			0
$305$	201.216	−	174.059i	0.659723	−	0.570684i
$306$		0			0
$307$			25.7118i			0.0837516i	0.999123	+	0.0418758i	$0.0133334\pi$
$307$			25.7118i			0.0837516i	−0.999123	+	0.0418758i	$0.986667\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			272.089i			0.874886i	0.899246	+	0.437443i	$0.144116\pi$
$311$			272.089i			0.874886i	−0.899246	+	0.437443i	$0.855884\pi$
$312$		0			0
$313$			45.3852i			0.145000i	0.997368	+	0.0725002i	$0.0230978\pi$
$313$			45.3852i			0.145000i	−0.997368	+	0.0725002i	$0.976902\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−71.1046			−0.224305			−0.112152	−	0.993691i	$-0.535774\pi$
$317$	−71.1046			−0.224305			−0.112152	+	0.993691i	$0.535774\pi$
$318$		0			0
$319$	−239.922			−0.752106
$320$		0			0
$321$		0			0
$322$		0			0
$323$	167.814			0.519549
$324$		0			0
$325$	−255.255	−	37.1378i	−0.785400	−	0.114270i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	49.7892i		−	0.151335i
$330$		0			0
$331$	479.901			1.44985			0.724926	−	0.688826i	$-0.241874\pi$
$331$	479.901			1.44985			0.724926	+	0.688826i	$0.241874\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	113.678	+	131.415i	0.339338	+	0.392283i
$336$		0			0
$337$			268.661i			0.797213i	0.917122	+	0.398607i	$0.130506\pi$
$337$			268.661i			0.797213i	−0.917122	+	0.398607i	$0.869494\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	192.284i		−	0.563884i
$342$		0			0
$343$			231.718i			0.675564i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	160.969			0.463889			0.231944	−	0.972729i	$-0.425491\pi$
$347$	160.969			0.463889			0.231944	+	0.972729i	$0.425491\pi$
$348$		0			0
$349$	678.825			1.94506			0.972529	−	0.232781i	$-0.0747825\pi$
$349$	678.825			1.94506			0.972529	+	0.232781i	$0.0747825\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	579.066			1.64041			0.820207	−	0.572067i	$-0.193858\pi$
$353$	579.066			1.64041			0.820207	+	0.572067i	$0.193858\pi$
$354$		0			0
$355$	−274.258	−	317.048i	−0.772558	−	0.893094i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			125.963i			0.350873i	0.984491	+	0.175436i	$0.0561336\pi$
$359$			125.963i			0.350873i	−0.984491	+	0.175436i	$0.943866\pi$
$360$		0			0
$361$	−316.515			−0.876772
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−333.519	−	385.555i	−0.913751	−	1.05632i
$366$		0			0
$367$		−	506.717i		−	1.38070i	−0.723475	−	0.690350i	$-0.757456\pi$
$367$		−	506.717i		−	1.38070i	0.723475	−	0.690350i	$-0.242544\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			143.656i			0.387212i
$372$		0			0
$373$			320.677i			0.859725i	0.902894	+	0.429863i	$0.141438\pi$
$373$			320.677i			0.859725i	−0.902894	+	0.429863i	$0.858562\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−372.819			−0.988909
$378$		0			0
$379$	355.339			0.937571			0.468786	−	0.883312i	$-0.344692\pi$
$379$	355.339			0.937571			0.468786	+	0.883312i	$0.344692\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	60.8363			0.158842			0.0794208	−	0.996841i	$-0.474693\pi$
$383$	60.8363			0.158842			0.0794208	+	0.996841i	$0.474693\pi$
$384$		0			0
$385$	−63.5155	+	54.9431i	−0.164975	+	0.142709i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			167.454i			0.430474i	0.976562	+	0.215237i	$0.0690523\pi$
$389$			167.454i			0.430474i	−0.976562	+	0.215237i	$0.930948\pi$
$390$		0			0
$391$	−120.666			−0.308608
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−581.345	+	502.884i	−1.47176	+	1.27313i
$396$		0			0
$397$			317.115i			0.798779i	0.916781	+	0.399390i	$0.130778\pi$
$397$			317.115i			0.798779i	−0.916781	+	0.399390i	$0.869222\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			399.801i			0.997010i	0.866887	+	0.498505i	$0.166117\pi$
$401$			399.801i			0.997010i	−0.866887	+	0.498505i	$0.833883\pi$
$402$		0			0
$403$		−	298.794i		−	0.741424i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	285.363			0.701139
$408$		0			0
$409$	514.705			1.25845			0.629224	−	0.777224i	$-0.283373\pi$
$409$	514.705			1.25845			0.629224	+	0.777224i	$0.283373\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	173.132			0.419205
$414$		0			0
$415$	448.230	−	387.735i	1.08007	−	0.934302i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			681.258i			1.62591i	0.582324	+	0.812957i	$0.302143\pi$
$419$			681.258i			1.62591i	−0.582324	+	0.812957i	$0.697857\pi$
$420$		0			0
$421$	512.870			1.21822			0.609110	−	0.793086i	$-0.291527\pi$
$421$	512.870			1.21822			0.609110	+	0.793086i	$0.291527\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−90.5637	+	622.461i	−0.213091	+	1.46461i
$426$		0			0
$427$		−	134.604i		−	0.315233i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	639.489i		−	1.48373i	−0.670547	−	0.741867i	$-0.733941\pi$
$431$		−	639.489i		−	1.48373i	0.670547	−	0.741867i	$-0.266059\pi$
$432$		0			0
$433$		−	815.884i		−	1.88426i	−0.335250	−	0.942129i	$-0.608821\pi$
$433$		−	815.884i		−	1.88426i	0.335250	−	0.942129i	$-0.391179\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−31.9869			−0.0731967
$438$		0			0
$439$	610.064			1.38967			0.694834	−	0.719170i	$-0.255478\pi$
$439$	610.064			1.38967			0.694834	+	0.719170i	$0.255478\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	241.249			0.544581			0.272291	−	0.962215i	$-0.412219\pi$
$443$	241.249			0.544581			0.272291	+	0.962215i	$0.412219\pi$
$444$		0			0
$445$	77.6956	+	89.8178i	0.174597	+	0.201838i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	104.846i		−	0.233511i	−0.993161	−	0.116755i	$-0.962751\pi$
$449$		−	104.846i		−	0.233511i	0.993161	−	0.116755i	$-0.0372494\pi$
$450$		0			0
$451$	251.512			0.557676
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−98.6978	+	85.3771i	−0.216918	+	0.187642i
$456$		0			0
$457$		−	61.9262i		−	0.135506i	−0.997702	−	0.0677529i	$-0.978417\pi$
$457$		−	61.9262i		−	0.135506i	0.997702	−	0.0677529i	$-0.0215830\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	328.418i		−	0.712404i	−0.934409	−	0.356202i	$-0.884071\pi$
$461$		−	328.418i		−	0.712404i	0.934409	−	0.356202i	$-0.115929\pi$
$462$		0			0
$463$			710.865i			1.53535i	0.640842	+	0.767673i	$0.278585\pi$
$463$			710.865i			1.53535i	−0.640842	+	0.767673i	$0.721415\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−629.909			−1.34884			−0.674420	−	0.738348i	$-0.735606\pi$
$467$	−629.909			−1.34884			−0.674420	+	0.738348i	$0.735606\pi$
$468$		0			0
$469$	87.9106			0.187443
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−287.048			−0.606867
$474$		0			0
$475$	−24.0072	+	165.006i	−0.0505416	+	0.347381i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			864.710i			1.80524i	0.430438	+	0.902620i	$0.358359\pi$
$479$			864.710i			1.80524i	−0.430438	+	0.902620i	$0.641641\pi$
$480$		0			0
$481$	443.431			0.921894
$482$		0			0
$483$		0			0
$484$		0			0
$485$	151.858	+	175.551i	0.313108	+	0.361960i
$486$		0			0
$487$			401.392i			0.824213i	0.911136	+	0.412106i	$0.135207\pi$
$487$			401.392i			0.824213i	−0.911136	+	0.412106i	$0.864793\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	692.612i		−	1.41061i	−0.708902	−	0.705307i	$-0.750809\pi$
$491$		−	692.612i		−	1.41061i	0.708902	−	0.705307i	$-0.249191\pi$
$492$		0			0
$493$			909.150i			1.84412i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−212.091			−0.426743
$498$		0			0
$499$	178.801			0.358318			0.179159	−	0.983820i	$-0.442662\pi$
$499$	178.801			0.358318			0.179159	+	0.983820i	$0.442662\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−348.997			−0.693832			−0.346916	−	0.937896i	$-0.612771\pi$
$503$	−348.997			−0.693832			−0.346916	+	0.937896i	$0.612771\pi$
$504$		0			0
$505$	−603.101	−	697.198i	−1.19426	−	1.38059i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	358.797i		−	0.704905i	−0.935830	−	0.352453i	$-0.885348\pi$
$509$		−	358.797i		−	0.704905i	0.935830	−	0.352453i	$-0.114652\pi$
$510$		0			0
$511$	−257.919			−0.504735
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−638.734	−	738.390i	−1.24026	−	1.43377i
$516$		0			0
$517$			130.686i			0.252778i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	29.9171i		−	0.0574225i	−0.999588	−	0.0287112i	$-0.990860\pi$
$521$		−	29.9171i		−	0.0574225i	0.999588	−	0.0287112i	$-0.00914033\pi$
$522$		0			0
$523$			321.370i			0.614474i	0.951633	+	0.307237i	$0.0994044\pi$
$523$			321.370i			0.614474i	−0.951633	+	0.307237i	$0.900596\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−728.634			−1.38261
$528$		0			0
$529$	23.0000			0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$	390.829			0.733262
$534$		0			0
$535$	350.787	−	303.443i	0.655676	−	0.567183i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	282.861i		−	0.524789i
$540$		0			0
$541$	205.949			0.380682			0.190341	−	0.981718i	$-0.439041\pi$
$541$	205.949			0.380682			0.190341	+	0.981718i	$0.439041\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	584.747	−	505.827i	1.07293	−	0.928123i
$546$		0			0
$547$			123.852i			0.226420i	0.993571	+	0.113210i	$0.0361133\pi$
$547$			123.852i			0.226420i	−0.993571	+	0.113210i	$0.963887\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			241.004i			0.437393i
$552$		0			0
$553$			388.894i			0.703244i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−704.217			−1.26430			−0.632151	−	0.774845i	$-0.717828\pi$
$557$	−704.217			−1.26430			−0.632151	+	0.774845i	$0.717828\pi$
$558$		0			0
$559$	−446.049			−0.797941
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−856.939			−1.52209			−0.761047	−	0.648696i	$-0.775315\pi$
$563$	−856.939			−1.52209			−0.761047	+	0.648696i	$0.775315\pi$
$564$		0			0
$565$	−264.816	+	229.075i	−0.468700	+	0.405442i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			643.361i			1.13069i	0.824855	+	0.565344i	$0.191257\pi$
$569$			643.361i			1.13069i	−0.824855	+	0.565344i	$0.808743\pi$
$570$		0			0
$571$	687.753			1.20447			0.602235	−	0.798319i	$-0.294277\pi$
$571$	687.753			1.20447			0.602235	+	0.798319i	$0.294277\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	17.2622	−	118.647i	0.0300213	−	0.206342i
$576$		0			0
$577$		−	353.930i		−	0.613398i	−0.951807	−	0.306699i	$-0.900776\pi$
$577$		−	353.930i		−	0.613398i	0.951807	−	0.306699i	$-0.0992244\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	299.846i		−	0.516086i
$582$		0			0
$583$		−	377.066i		−	0.646768i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	359.397			0.612260			0.306130	−	0.951990i	$-0.400966\pi$
$587$	359.397			0.612260			0.306130	+	0.951990i	$0.400966\pi$
$588$		0			0
$589$	−193.151			−0.327931
$590$		0			0
$591$		0			0
$592$		0			0
$593$	945.671			1.59472			0.797362	−	0.603502i	$-0.206228\pi$
$593$	945.671			1.59472			0.797362	+	0.603502i	$0.206228\pi$
$594$		0			0
$595$	208.199	+	240.683i	0.349915	+	0.404509i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			206.299i			0.344406i	0.985061	+	0.172203i	$0.0550885\pi$
$599$			206.299i			0.344406i	−0.985061	+	0.172203i	$0.944911\pi$
$600$		0			0
$601$	313.469			0.521578			0.260789	−	0.965396i	$-0.416017\pi$
$601$	313.469			0.521578			0.260789	+	0.965396i	$0.416017\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−290.846	+	251.593i	−0.480738	+	0.415856i
$606$		0			0
$607$			1098.52i			1.80975i	0.425677	+	0.904875i	$0.360036\pi$
$607$			1098.52i			1.80975i	−0.425677	+	0.904875i	$0.639964\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			203.075i			0.332366i
$612$		0			0
$613$			240.449i			0.392249i	0.980579	+	0.196125i	$0.0628357\pi$
$613$			240.449i			0.392249i	−0.980579	+	0.196125i	$0.937164\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	379.924			0.615760			0.307880	−	0.951425i	$-0.400380\pi$
$617$	379.924			0.615760			0.307880	+	0.951425i	$0.400380\pi$
$618$		0			0
$619$	−477.508			−0.771419			−0.385709	−	0.922620i	$-0.626043\pi$
$619$	−477.508			−0.771419			−0.385709	+	0.922620i	$0.626043\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	60.0841			0.0964432
$624$		0			0
$625$	−599.088	−	178.096i	−0.958541	−	0.284954i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1081.34i		−	1.71915i
$630$		0			0
$631$	928.732			1.47184			0.735921	−	0.677068i	$-0.236750\pi$
$631$	928.732			1.47184			0.735921	+	0.677068i	$0.236750\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	434.675	+	502.493i	0.684527	+	0.791328i
$636$		0			0
$637$		−	439.543i		−	0.690020i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			876.704i			1.36771i	0.729617	+	0.683856i	$0.239698\pi$
$641$			876.704i			1.36771i	−0.729617	+	0.683856i	$0.760302\pi$
$642$		0			0
$643$		−	119.924i		−	0.186508i	−0.995642	−	0.0932538i	$-0.970273\pi$
$643$		−	119.924i		−	0.186508i	0.995642	−	0.0932538i	$-0.0297268\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−70.3339			−0.108708			−0.0543539	−	0.998522i	$-0.517310\pi$
$647$	−70.3339			−0.108708			−0.0543539	+	0.998522i	$0.517310\pi$
$648$		0			0
$649$	−454.433			−0.700205
$650$		0			0
$651$		0			0
$652$		0			0
$653$	769.935			1.17907			0.589537	−	0.807742i	$-0.299310\pi$
$653$	769.935			1.17907			0.589537	+	0.807742i	$0.299310\pi$
$654$		0			0
$655$	808.153	+	934.242i	1.23382	+	1.42632i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			910.897i			1.38224i	0.722740	+	0.691120i	$0.242883\pi$
$659$			910.897i			1.38224i	−0.722740	+	0.691120i	$0.757117\pi$
$660$		0			0
$661$	454.616			0.687771			0.343885	−	0.939012i	$-0.388257\pi$
$661$	454.616			0.687771			0.343885	+	0.939012i	$0.388257\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	55.1909	+	63.8019i	0.0829938	+	0.0959427i
$666$		0			0
$667$		−	173.292i		−	0.259808i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			353.308i			0.526539i
$672$		0			0
$673$			849.996i			1.26300i	0.775378	+	0.631498i	$0.217559\pi$
$673$			849.996i			1.26300i	−0.775378	+	0.631498i	$0.782441\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−801.631			−1.18409			−0.592047	−	0.805904i	$-0.701680\pi$
$677$	−801.631			−1.18409			−0.592047	+	0.805904i	$0.701680\pi$
$678$		0			0
$679$	117.436			0.172954
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−415.671			−0.608596			−0.304298	−	0.952577i	$-0.598422\pi$
$683$	−415.671			−0.608596			−0.304298	+	0.952577i	$0.598422\pi$
$684$		0			0
$685$	22.3593	−	19.3416i	0.0326413	−	0.0282359i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	585.929i		−	0.850405i
$690$		0			0
$691$	−438.558			−0.634671			−0.317335	−	0.948313i	$-0.602788\pi$
$691$	−438.558			−0.634671			−0.317335	+	0.948313i	$0.602788\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.705836	−	0.610573i	0.00101559	−	0.000878523i
$696$		0			0
$697$		−	953.068i		−	1.36739i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1240.03i		−	1.76895i	−0.466589	−	0.884474i	$-0.654517\pi$
$701$		−	1240.03i		−	1.76895i	0.466589	−	0.884474i	$-0.345483\pi$
$702$		0			0
$703$		−	286.650i		−	0.407753i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−466.394			−0.659681
$708$		0			0
$709$	−1205.29			−1.69998			−0.849991	−	0.526796i	$-0.823393\pi$
$709$	−1205.29			−1.69998			−0.849991	+	0.526796i	$0.823393\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	138.884			0.194789
$714$		0			0
$715$	259.061	−	224.097i	0.362323	−	0.313422i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	775.513i		−	1.07860i	−0.842114	−	0.539300i	$-0.818689\pi$
$719$		−	775.513i		−	1.07860i	0.842114	−	0.539300i	$-0.181311\pi$
$720$		0			0
$721$	−493.950			−0.685090
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−893.935	−	130.061i	−1.23301	−	0.179395i
$726$		0			0
$727$		−	925.819i		−	1.27348i	−0.771079	−	0.636740i	$-0.780283\pi$
$727$		−	925.819i		−	1.27348i	0.771079	−	0.636740i	$-0.219717\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			1087.73i			1.48800i
$732$		0			0
$733$		−	184.459i		−	0.251650i	−0.992052	−	0.125825i	$-0.959842\pi$
$733$		−	184.459i		−	0.251650i	0.992052	−	0.125825i	$-0.0401577\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−230.746			−0.313089
$738$		0			0
$739$	294.428			0.398414			0.199207	−	0.979957i	$-0.436163\pi$
$739$	294.428			0.398414			0.199207	+	0.979957i	$0.436163\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	875.086			1.17777			0.588887	−	0.808215i	$-0.299566\pi$
$743$	875.086			1.17777			0.588887	+	0.808215i	$0.299566\pi$
$744$		0			0
$745$	−33.1499	−	38.3220i	−0.0444965	−	0.0514390i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	234.661i		−	0.313299i
$750$		0			0
$751$	241.009			0.320918			0.160459	−	0.987043i	$-0.448703\pi$
$751$	241.009			0.320918			0.160459	+	0.987043i	$0.448703\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−421.874	+	364.936i	−0.558773	+	0.483359i
$756$		0			0
$757$			477.154i			0.630322i	0.949038	+	0.315161i	$0.102059\pi$
$757$			477.154i			0.630322i	−0.949038	+	0.315161i	$0.897941\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			137.825i			0.181110i	0.995891	+	0.0905549i	$0.0288641\pi$
$761$			137.825i			0.181110i	−0.995891	+	0.0905549i	$0.971136\pi$
$762$		0			0
$763$		−	391.170i		−	0.512673i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−706.152			−0.920667
$768$		0			0
$769$	129.263			0.168092			0.0840460	−	0.996462i	$-0.473216\pi$
$769$	129.263			0.168092			0.0840460	+	0.996462i	$0.473216\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−534.650			−0.691656			−0.345828	−	0.938298i	$-0.612402\pi$
$773$	−534.650			−0.691656			−0.345828	+	0.938298i	$0.612402\pi$
$774$		0			0
$775$	104.237	−	716.441i	0.134500	−	0.924440i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	252.646i		−	0.324321i
$780$		0			0
$781$	556.694			0.712797
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−246.510	−	284.971i	−0.314026	−	0.363021i
$786$		0			0
$787$		−	268.277i		−	0.340885i	−0.985368	−	0.170443i	$-0.945480\pi$
$787$		−	268.277i		−	0.340885i	0.985368	−	0.170443i	$-0.0545198\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			177.150i			0.223957i
$792$		0			0
$793$			549.011i			0.692321i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1406.13			1.76428			0.882138	−	0.470991i	$-0.156104\pi$
$797$	1406.13			1.76428			0.882138	+	0.470991i	$0.156104\pi$
$798$		0			0
$799$	495.216			0.619795
$800$		0			0
$801$		0			0
$802$		0			0
$803$	676.984			0.843068
$804$		0			0
$805$	−39.6846	−	45.8763i	−0.0492977	−	0.0569892i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			433.288i			0.535585i	0.963477	+	0.267793i	$0.0862942\pi$
$809$			433.288i			0.535585i	−0.963477	+	0.267793i	$0.913706\pi$
$810$		0			0
$811$	−962.990			−1.18741			−0.593705	−	0.804683i	$-0.702335\pi$
$811$	−962.990			−1.18741			−0.593705	+	0.804683i	$0.702335\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−174.030	−	201.183i	−0.213534	−	0.246850i
$816$		0			0
$817$			288.343i			0.352928i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	450.949i		−	0.549268i	−0.961549	−	0.274634i	$-0.911443\pi$
$821$		−	450.949i		−	0.549268i	0.961549	−	0.274634i	$-0.0885567\pi$
$822$		0			0
$823$		−	98.4679i		−	0.119645i	−0.998209	−	0.0598225i	$-0.980947\pi$
$823$		−	98.4679i		−	0.119645i	0.998209	−	0.0598225i	$-0.0190535\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	964.200			1.16590			0.582951	−	0.812508i	$-0.301898\pi$
$827$	964.200			1.16590			0.582951	+	0.812508i	$0.301898\pi$
$828$		0			0
$829$	−860.925			−1.03851			−0.519255	−	0.854619i	$-0.673791\pi$
$829$	−860.925			−1.03851			−0.519255	+	0.854619i	$0.673791\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1071.86			−1.28675
$834$		0			0
$835$	458.267	−	396.417i	0.548822	−	0.474751i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			559.111i			0.666402i	0.942856	+	0.333201i	$0.108129\pi$
$839$			559.111i			0.666402i	−0.942856	+	0.333201i	$0.891871\pi$
$840$		0			0
$841$	−464.659			−0.552507
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−236.514	+	204.593i	−0.279898	+	0.242122i
$846$		0			0
$847$			194.563i			0.229709i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			206.114i			0.242202i
$852$		0			0
$853$		−	327.474i		−	0.383909i	−0.981404	−	0.191954i	$-0.938517\pi$
$853$		−	327.474i		−	0.383909i	0.981404	−	0.191954i	$-0.0614826\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1419.98			1.65692			0.828459	−	0.560049i	$-0.189218\pi$
$857$	1419.98			1.65692			0.828459	+	0.560049i	$0.189218\pi$
$858$		0			0
$859$	−640.440			−0.745565			−0.372782	−	0.927919i	$-0.621596\pi$
$859$	−640.440			−0.745565			−0.372782	+	0.927919i	$0.621596\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	142.734			0.165393			0.0826965	−	0.996575i	$-0.473647\pi$
$863$	142.734			0.165393			0.0826965	+	0.996575i	$0.473647\pi$
$864$		0			0
$865$	983.642	−	850.885i	1.13716	−	0.983682i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1020.76i		−	1.17464i
$870$		0			0
$871$	−358.561			−0.411666
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−266.440	+	170.283i	−0.304503	+	0.194610i
$876$		0			0
$877$		−	1396.06i		−	1.59186i	−0.605390	−	0.795929i	$-0.706983\pi$
$877$		−	1396.06i		−	1.59186i	0.605390	−	0.795929i	$-0.293017\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			871.130i			0.988796i	0.869236	+	0.494398i	$0.164612\pi$
$881$			871.130i			0.988796i	−0.869236	+	0.494398i	$0.835388\pi$
$882$		0			0
$883$			1185.43i			1.34250i	0.741232	+	0.671249i	$0.234242\pi$
$883$			1185.43i			1.34250i	−0.741232	+	0.671249i	$0.765758\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1152.89			1.29976			0.649882	−	0.760036i	$-0.274818\pi$
$887$	1152.89			1.29976			0.649882	+	0.760036i	$0.274818\pi$
$888$		0			0
$889$	336.146			0.378117
$890$		0			0
$891$		0			0
$892$		0			0
$893$	131.275			0.147005
$894$		0			0
$895$	−427.249	−	493.909i	−0.477373	−	0.551854i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1046.41i		−	1.16398i
$900$		0			0
$901$	−1428.84			−1.58584
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.16890	+	3.60625i	−0.00460652	+	0.00398480i
$906$		0			0
$907$			1394.43i			1.53741i	0.639602	+	0.768706i	$0.279099\pi$
$907$			1394.43i			1.53741i	−0.639602	+	0.768706i	$0.720901\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	251.770i		−	0.276367i	−0.990407	−	0.138184i	$-0.955874\pi$
$911$		−	251.770i		−	0.276367i	0.990407	−	0.138184i	$-0.0441264\pi$
$912$		0			0
$913$			787.032i			0.862029i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	624.966			0.681534
$918$		0			0
$919$	313.138			0.340738			0.170369	−	0.985380i	$-0.445504\pi$
$919$	313.138			0.340738			0.170369	+	0.985380i	$0.445504\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	865.057			0.937223
$924$		0			0
$925$	1063.25	+	154.695i	1.14946	+	0.167238i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			245.899i			0.264692i	0.991204	+	0.132346i	$0.0422510\pi$
$929$			245.899i			0.264692i	−0.991204	+	0.132346i	$0.957749\pi$
$930$		0			0
$931$	−284.137			−0.305195
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−546.479	−	631.741i	−0.584469	−	0.675659i
$936$		0			0
$937$		−	1068.83i		−	1.14069i	−0.821404	−	0.570347i	$-0.806809\pi$
$937$		−	1068.83i		−	1.14069i	0.821404	−	0.570347i	$-0.193191\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1366.22i		−	1.45188i	−0.687757	−	0.725941i	$-0.741405\pi$
$941$		−	1366.22i		−	1.45188i	0.687757	−	0.725941i	$-0.258595\pi$
$942$		0			0
$943$			181.663i			0.192644i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1568.17			−1.65593			−0.827967	−	0.560777i	$-0.810503\pi$
$947$	−1568.17			−1.65593			−0.827967	+	0.560777i	$0.810503\pi$
$948$		0			0
$949$	1051.98			1.10851
$950$		0			0
$951$		0			0
$952$		0			0
$953$	247.407			0.259609			0.129805	−	0.991540i	$-0.458565\pi$
$953$	247.407			0.259609			0.129805	+	0.991540i	$0.458565\pi$
$954$		0			0
$955$	127.346	+	147.214i	0.133346	+	0.154151i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	14.9574i		−	0.0155968i
$960$		0			0
$961$	−122.355			−0.127321
$962$		0			0
$963$		0			0
$964$		0			0
$965$	614.817	+	710.741i	0.637116	+	0.736519i
$966$		0			0
$967$		−	9.22545i		−	0.00954028i	−0.999989	−	0.00477014i	$-0.998482\pi$
$967$		−	9.22545i		−	0.00954028i	0.999989	−	0.00477014i	$-0.00151839\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1142.30i			1.17641i	0.808711	+	0.588207i	$0.200166\pi$
$971$			1142.30i			1.17641i	−0.808711	+	0.588207i	$0.799834\pi$
$972$		0			0
$973$		−	0.472173i		−	0.000485275i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1171.54			−1.19912			−0.599559	−	0.800330i	$-0.704657\pi$
$977$	−1171.54			−1.19912			−0.599559	+	0.800330i	$0.704657\pi$
$978$		0			0
$979$	−157.708			−0.161091
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1303.33			1.32586			0.662932	−	0.748679i	$-0.269312\pi$
$983$	1303.33			1.32586			0.662932	+	0.748679i	$0.269312\pi$
$984$		0			0
$985$	−914.143	+	790.766i	−0.928064	+	0.802809i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	207.331i		−	0.209637i
$990$		0			0
$991$	−887.523			−0.895583			−0.447792	−	0.894138i	$-0.647789\pi$
$991$	−887.523			−0.895583			−0.447792	+	0.894138i	$0.647789\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1147.02	+	992.215i	−1.15279	+	0.997201i
$996$		0			0
$997$		−	577.503i		−	0.579241i	−0.957142	−	0.289620i	$-0.906471\pi$
$997$		−	577.503i		−	0.579241i	0.957142	−	0.289620i	$-0.0935290\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.c.a.4049.20	yes	88
3.2	odd	2	inner	4140.3.c.a.4049.69	yes	88
5.4	even	2	inner	4140.3.c.a.4049.70	yes	88
15.14	odd	2	inner	4140.3.c.a.4049.19	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.3.c.a.4049.19	✓	88	15.14	odd	2	inner
4140.3.c.a.4049.20	yes	88	1.1	even	1	trivial
4140.3.c.a.4049.69	yes	88	3.2	odd	2	inner
4140.3.c.a.4049.70	yes	88	5.4	even	2	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4140.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.78150 + 3.27113i) q^{5} +2.52965i q^{7} +O(q^{10})\)	\(q+(-3.78150 + 3.27113i) q^{5} +2.52965i q^{7} -6.63980i q^{11} -10.3177i q^{13} -25.1606 q^{17} -6.66974 q^{19} +4.79583 q^{23} +(3.59943 - 24.7395i) q^{25} -36.1339i q^{29} +28.9594 q^{31} +(-8.27482 - 9.56587i) q^{35} +42.9777i q^{37} +37.8794i q^{41} -43.2315i q^{43} -19.6822 q^{47} +42.6009 q^{49} +56.7887 q^{53} +(21.7196 + 25.1084i) q^{55} -68.4408i q^{59} -53.2106 q^{61} +(33.7505 + 39.0163i) q^{65} -34.7520i q^{67} +83.8421i q^{71} +101.958i q^{73} +16.7964 q^{77} +153.734 q^{79} -118.533 q^{83} +(95.1446 - 82.3035i) q^{85} -23.7519i q^{89} +26.1002 q^{91} +(25.2216 - 21.8176i) q^{95} -46.4236i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q+O(q^{10}) \) 88 * q	\( 88 q - 16 q^{19} - 48 q^{25} + 272 q^{31} - 600 q^{49} + 112 q^{55} + 448 q^{61} - 32 q^{79} - 264 q^{85} - 16 q^{91}+O(q^{100}) \) 88 * q - 16 * q^19 - 48 * q^25 + 272 * q^31 - 600 * q^49 + 112 * q^55 + 448 * q^61 - 32 * q^79 - 264 * q^85 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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