LMFDB - Embedded newform 4012.2.b.a.237.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.b (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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.5
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.36

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44713i q^{3} -1.66009i q^{5} +3.65944i q^{7} -2.98842 q^{9} +O(q^{10})$	$q-2.44713i q^{3} -1.66009i q^{5} +3.65944i q^{7} -2.98842 q^{9} +1.95066i q^{11} -1.88460 q^{13} -4.06244 q^{15} +(-3.75271 - 1.70796i) q^{17} +4.87170 q^{19} +8.95510 q^{21} -9.29998i q^{23} +2.24411 q^{25} -0.0283342i q^{27} +3.33623i q^{29} -4.53272i q^{31} +4.77351 q^{33} +6.07498 q^{35} -2.82715i q^{37} +4.61185i q^{39} +0.268240i q^{41} -8.42799 q^{43} +4.96104i q^{45} -8.14320 q^{47} -6.39147 q^{49} +(-4.17960 + 9.18336i) q^{51} +2.67756 q^{53} +3.23826 q^{55} -11.9217i q^{57} -1.00000 q^{59} -2.27284i q^{61} -10.9359i q^{63} +3.12860i q^{65} -15.1523 q^{67} -22.7582 q^{69} +6.53554i q^{71} -6.29649i q^{73} -5.49162i q^{75} -7.13831 q^{77} +2.53912i q^{79} -9.03460 q^{81} -2.22158 q^{83} +(-2.83536 + 6.22983i) q^{85} +8.16416 q^{87} -11.3861 q^{89} -6.89657i q^{91} -11.0921 q^{93} -8.08745i q^{95} +14.9275i q^{97} -5.82939i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 36 q^{9}+O(q^{10}) $ 40 * q - 36 * q^9	$ 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) $ 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

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
$3$		−	2.44713i		−	1.41285i	−0.707789	−	0.706424i	$-0.750307\pi$
$3$		−	2.44713i		−	1.41285i	0.707789	−	0.706424i	$-0.249693\pi$
$4$		0			0
$5$		−	1.66009i		−	0.742413i	−0.928550	−	0.371207i	$-0.878944\pi$
$5$		−	1.66009i		−	0.742413i	0.928550	−	0.371207i	$-0.121056\pi$
$6$		0			0
$7$			3.65944i			1.38314i	0.722311	+	0.691568i	$0.243080\pi$
$7$			3.65944i			1.38314i	−0.722311	+	0.691568i	$0.756920\pi$
$8$		0			0
$9$	−2.98842			−0.996140
$10$		0			0
$11$			1.95066i			0.588146i	0.955783	+	0.294073i	$0.0950109\pi$
$11$			1.95066i			0.588146i	−0.955783	+	0.294073i	$0.904989\pi$
$12$		0			0
$13$	−1.88460			−0.522694			−0.261347	−	0.965245i	$-0.584167\pi$
$13$	−1.88460			−0.522694			−0.261347	+	0.965245i	$0.584167\pi$
$14$		0			0
$15$	−4.06244			−1.04892
$16$		0			0
$17$	−3.75271	−	1.70796i	−0.910167	−	0.414242i
$17$	−3.75271	−	1.70796i	−0.910167	−	0.414242i
$18$		0			0
$19$	4.87170			1.11764			0.558822	−	0.829287i	$-0.311253\pi$
$19$	4.87170			1.11764			0.558822	+	0.829287i	$0.311253\pi$
$20$		0			0
$21$	8.95510			1.95416
$22$		0			0
$23$		−	9.29998i		−	1.93918i	−0.244733	−	0.969590i	$-0.578700\pi$
$23$		−	9.29998i		−	1.93918i	0.244733	−	0.969590i	$-0.421300\pi$
$24$		0			0
$25$	2.24411			0.448822
$26$		0			0
$27$		−	0.0283342i		−	0.00545292i
$28$		0			0
$29$			3.33623i			0.619522i	0.950814	+	0.309761i	$0.100249\pi$
$29$			3.33623i			0.619522i	−0.950814	+	0.309761i	$0.899751\pi$
$30$		0			0
$31$		−	4.53272i		−	0.814100i	−0.913406	−	0.407050i	$-0.866558\pi$
$31$		−	4.53272i		−	0.814100i	0.913406	−	0.407050i	$-0.133442\pi$
$32$		0			0
$33$	4.77351			0.830961
$34$		0			0
$35$	6.07498			1.02686
$36$		0			0
$37$		−	2.82715i		−	0.464780i	−0.972623	−	0.232390i	$-0.925345\pi$
$37$		−	2.82715i		−	0.464780i	0.972623	−	0.232390i	$-0.0746546\pi$
$38$		0			0
$39$			4.61185i			0.738487i
$40$		0			0
$41$			0.268240i			0.0418921i	0.999781	+	0.0209460i	$0.00666782\pi$
$41$			0.268240i			0.0418921i	−0.999781	+	0.0209460i	$0.993332\pi$
$42$		0			0
$43$	−8.42799			−1.28526			−0.642628	−	0.766178i	$-0.722156\pi$
$43$	−8.42799			−1.28526			−0.642628	+	0.766178i	$0.722156\pi$
$44$		0			0
$45$			4.96104i			0.739548i
$46$		0			0
$47$	−8.14320			−1.18781			−0.593904	−	0.804536i	$-0.702414\pi$
$47$	−8.14320			−1.18781			−0.593904	+	0.804536i	$0.702414\pi$
$48$		0			0
$49$	−6.39147			−0.913067
$50$		0			0
$51$	−4.17960	+	9.18336i	−0.585260	+	1.28593i
$52$		0			0
$53$	2.67756			0.367792			0.183896	−	0.982946i	$-0.441129\pi$
$53$	2.67756			0.367792			0.183896	+	0.982946i	$0.441129\pi$
$54$		0			0
$55$	3.23826			0.436647
$56$		0			0
$57$		−	11.9217i		−	1.57906i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$		−	2.27284i		−	0.291007i	−0.989358	−	0.145504i	$-0.953520\pi$
$61$		−	2.27284i		−	0.291007i	0.989358	−	0.145504i	$-0.0464803\pi$
$62$		0			0
$63$		−	10.9359i		−	1.37780i
$64$		0			0
$65$			3.12860i			0.388055i
$66$		0			0
$67$	−15.1523			−1.85115			−0.925574	−	0.378566i	$-0.876417\pi$
$67$	−15.1523			−1.85115			−0.925574	+	0.378566i	$0.876417\pi$
$68$		0			0
$69$	−22.7582			−2.73977
$70$		0			0
$71$			6.53554i			0.775626i	0.921738	+	0.387813i	$0.126769\pi$
$71$			6.53554i			0.775626i	−0.921738	+	0.387813i	$0.873231\pi$
$72$		0			0
$73$		−	6.29649i		−	0.736948i	−0.929638	−	0.368474i	$-0.879880\pi$
$73$		−	6.29649i		−	0.736948i	0.929638	−	0.368474i	$-0.120120\pi$
$74$		0			0
$75$		−	5.49162i		−	0.634118i
$76$		0			0
$77$	−7.13831			−0.813486
$78$		0			0
$79$			2.53912i			0.285673i	0.989746	+	0.142837i	$0.0456223\pi$
$79$			2.53912i			0.285673i	−0.989746	+	0.142837i	$0.954378\pi$
$80$		0			0
$81$	−9.03460			−1.00384
$82$		0			0
$83$	−2.22158			−0.243850			−0.121925	−	0.992539i	$-0.538907\pi$
$83$	−2.22158			−0.243850			−0.121925	+	0.992539i	$0.538907\pi$
$84$		0			0
$85$	−2.83536	+	6.22983i	−0.307538	+	0.675720i
$86$		0			0
$87$	8.16416			0.875290
$88$		0			0
$89$	−11.3861			−1.20693			−0.603465	−	0.797390i	$-0.706214\pi$
$89$	−11.3861			−1.20693			−0.603465	+	0.797390i	$0.706214\pi$
$90$		0			0
$91$		−	6.89657i		−	0.722957i
$92$		0			0
$93$	−11.0921			−1.15020
$94$		0			0
$95$		−	8.08745i		−	0.829755i
$96$		0			0
$97$			14.9275i			1.51566i	0.652452	+	0.757830i	$0.273741\pi$
$97$			14.9275i			1.51566i	−0.652452	+	0.757830i	$0.726259\pi$
$98$		0			0
$99$		−	5.82939i		−	0.585876i
$100$		0			0
$101$	−4.48872			−0.446645			−0.223322	−	0.974745i	$-0.571690\pi$
$101$	−4.48872			−0.446645			−0.223322	+	0.974745i	$0.571690\pi$
$102$		0			0
$103$	−17.6969			−1.74373			−0.871865	−	0.489747i	$-0.837089\pi$
$103$	−17.6969			−1.74373			−0.871865	+	0.489747i	$0.837089\pi$
$104$		0			0
$105$		−	14.8662i		−	1.45080i
$106$		0			0
$107$		−	14.0953i		−	1.36265i	−0.731982	−	0.681324i	$-0.761404\pi$
$107$		−	14.0953i		−	1.36265i	0.731982	−	0.681324i	$-0.238596\pi$
$108$		0			0
$109$			7.55639i			0.723770i	0.932223	+	0.361885i	$0.117867\pi$
$109$			7.55639i			0.723770i	−0.932223	+	0.361885i	$0.882133\pi$
$110$		0			0
$111$	−6.91838			−0.656664
$112$		0			0
$113$		−	11.8159i		−	1.11155i	−0.831334	−	0.555773i	$-0.812422\pi$
$113$		−	11.8159i		−	1.11155i	0.831334	−	0.555773i	$-0.187578\pi$
$114$		0			0
$115$	−15.4388			−1.43967
$116$		0			0
$117$	5.63198			0.520676
$118$		0			0
$119$	6.25018	−	13.7328i	0.572953	−	1.25889i
$120$		0			0
$121$	7.19493			0.654084
$122$		0			0
$123$	0.656418			0.0591872
$124$		0			0
$125$		−	12.0259i		−	1.07563i
$126$		0			0
$127$	18.8742			1.67481			0.837407	−	0.546579i	$-0.184070\pi$
$127$	18.8742			1.67481			0.837407	+	0.546579i	$0.184070\pi$
$128$		0			0
$129$			20.6243i			1.81587i
$130$		0			0
$131$			8.77368i			0.766560i	0.923632	+	0.383280i	$0.125205\pi$
$131$			8.77368i			0.766560i	−0.923632	+	0.383280i	$0.874795\pi$
$132$		0			0
$133$			17.8277i			1.54586i
$134$		0			0
$135$	−0.0470372			−0.00404832
$136$		0			0
$137$	9.32087			0.796336			0.398168	−	0.917313i	$-0.369646\pi$
$137$	9.32087			0.796336			0.398168	+	0.917313i	$0.369646\pi$
$138$		0			0
$139$			7.62901i			0.647084i	0.946214	+	0.323542i	$0.104874\pi$
$139$			7.62901i			0.647084i	−0.946214	+	0.323542i	$0.895126\pi$
$140$		0			0
$141$			19.9274i			1.67819i
$142$		0			0
$143$		−	3.67621i		−	0.307420i
$144$		0			0
$145$	5.53843			0.459941
$146$		0			0
$147$			15.6407i			1.29002i
$148$		0			0
$149$	−19.2065			−1.57345			−0.786727	−	0.617301i	$-0.788226\pi$
$149$	−19.2065			−1.57345			−0.786727	+	0.617301i	$0.788226\pi$
$150$		0			0
$151$	2.62269			0.213431			0.106716	−	0.994290i	$-0.465967\pi$
$151$	2.62269			0.213431			0.106716	+	0.994290i	$0.465967\pi$
$152$		0			0
$153$	11.2147	+	5.10411i	0.906654	+	0.412643i
$154$		0			0
$155$	−7.52470			−0.604399
$156$		0			0
$157$	8.77758			0.700528			0.350264	−	0.936651i	$-0.386092\pi$
$157$	8.77758			0.700528			0.350264	+	0.936651i	$0.386092\pi$
$158$		0			0
$159$		−	6.55233i		−	0.519634i
$160$		0			0
$161$	34.0327			2.68215
$162$		0			0
$163$			5.56986i			0.436265i	0.975919	+	0.218133i	$0.0699966\pi$
$163$			5.56986i			0.436265i	−0.975919	+	0.218133i	$0.930003\pi$
$164$		0			0
$165$		−	7.92444i		−	0.616917i
$166$		0			0
$167$		−	11.0444i		−	0.854645i	−0.904099	−	0.427322i	$-0.859457\pi$
$167$		−	11.0444i		−	0.854645i	0.904099	−	0.427322i	$-0.140543\pi$
$168$		0			0
$169$	−9.44829			−0.726791
$170$		0			0
$171$	−14.5587			−1.11333
$172$		0			0
$173$		−	13.3496i		−	1.01495i	−0.861666	−	0.507476i	$-0.830579\pi$
$173$		−	13.3496i		−	1.01495i	0.861666	−	0.507476i	$-0.169421\pi$
$174$		0			0
$175$			8.21218i			0.620783i
$176$		0			0
$177$			2.44713i			0.183937i
$178$		0			0
$179$	−1.14078			−0.0852657			−0.0426328	−	0.999091i	$-0.513575\pi$
$179$	−1.14078			−0.0852657			−0.0426328	+	0.999091i	$0.513575\pi$
$180$		0			0
$181$		−	5.56258i		−	0.413463i	−0.978398	−	0.206732i	$-0.933717\pi$
$181$		−	5.56258i		−	0.413463i	0.978398	−	0.206732i	$-0.0662827\pi$
$182$		0			0
$183$	−5.56192			−0.411149
$184$		0			0
$185$	−4.69331			−0.345059
$186$		0			0
$187$	3.33165	−	7.32027i	0.243634	−	0.535311i
$188$		0			0
$189$	0.103687			0.00754213
$190$		0			0
$191$	3.13657			0.226954			0.113477	−	0.993541i	$-0.463801\pi$
$191$	3.13657			0.226954			0.113477	+	0.993541i	$0.463801\pi$
$192$		0			0
$193$		−	4.66615i		−	0.335877i	−0.985798	−	0.167938i	$-0.946289\pi$
$193$		−	4.66615i		−	0.335877i	0.985798	−	0.167938i	$-0.0537109\pi$
$194$		0			0
$195$	7.65607			0.548263
$196$		0			0
$197$		−	12.7372i		−	0.907485i	−0.891133	−	0.453743i	$-0.850089\pi$
$197$		−	12.7372i		−	0.907485i	0.891133	−	0.453743i	$-0.149911\pi$
$198$		0			0
$199$			10.3918i			0.736653i	0.929696	+	0.368327i	$0.120069\pi$
$199$			10.3918i			0.736653i	−0.929696	+	0.368327i	$0.879931\pi$
$200$		0			0
$201$			37.0796i			2.61539i
$202$		0			0
$203$	−12.2087			−0.856883
$204$		0			0
$205$	0.445302			0.0311013
$206$		0			0
$207$			27.7923i			1.93170i
$208$		0			0
$209$			9.50303i			0.657338i
$210$		0			0
$211$			6.85042i			0.471602i	0.971801	+	0.235801i	$0.0757714\pi$
$211$			6.85042i			0.471602i	−0.971801	+	0.235801i	$0.924229\pi$
$212$		0			0
$213$	15.9933			1.09584
$214$		0			0
$215$			13.9912i			0.954192i
$216$		0			0
$217$	16.5872			1.12601
$218$		0			0
$219$	−15.4083			−1.04120
$220$		0			0
$221$	7.07236	+	3.21882i	0.475739	+	0.216521i
$222$		0			0
$223$	−13.3027			−0.890817			−0.445408	−	0.895328i	$-0.646941\pi$
$223$	−13.3027			−0.890817			−0.445408	+	0.895328i	$0.646941\pi$
$224$		0			0
$225$	−6.70635			−0.447090
$226$		0			0
$227$		−	2.20817i		−	0.146562i	−0.997311	−	0.0732808i	$-0.976653\pi$
$227$		−	2.20817i		−	0.146562i	0.997311	−	0.0732808i	$-0.0233469\pi$
$228$		0			0
$229$	−29.5080			−1.94994			−0.974971	−	0.222333i	$-0.928633\pi$
$229$	−29.5080			−1.94994			−0.974971	+	0.222333i	$0.928633\pi$
$230$		0			0
$231$			17.4683i			1.14933i
$232$		0			0
$233$		−	21.0896i		−	1.38163i	−0.723033	−	0.690813i	$-0.757253\pi$
$233$		−	21.0896i		−	1.38163i	0.723033	−	0.690813i	$-0.242747\pi$
$234$		0			0
$235$			13.5184i			0.881845i
$236$		0			0
$237$	6.21354			0.403613
$238$		0			0
$239$	17.1004			1.10613			0.553067	−	0.833137i	$-0.313457\pi$
$239$	17.1004			1.10613			0.553067	+	0.833137i	$0.313457\pi$
$240$		0			0
$241$		−	5.72180i		−	0.368573i	−0.982873	−	0.184287i	$-0.941003\pi$
$241$		−	5.72180i		−	0.368573i	0.982873	−	0.184287i	$-0.0589975\pi$
$242$		0			0
$243$			22.0238i			1.41283i
$244$		0			0
$245$			10.6104i			0.677873i
$246$		0			0
$247$	−9.18120			−0.584186
$248$		0			0
$249$			5.43649i			0.344524i
$250$		0			0
$251$	−22.2638			−1.40528			−0.702641	−	0.711545i	$-0.747996\pi$
$251$	−22.2638			−1.40528			−0.702641	+	0.711545i	$0.747996\pi$
$252$		0			0
$253$	18.1411			1.14052
$254$		0			0
$255$	15.2452	+	6.93849i	0.954690	+	0.434505i
$256$		0			0
$257$	−14.1030			−0.879720			−0.439860	−	0.898066i	$-0.644972\pi$
$257$	−14.1030			−0.879720			−0.439860	+	0.898066i	$0.644972\pi$
$258$		0			0
$259$	10.3458			0.642855
$260$		0			0
$261$		−	9.97005i		−	0.617131i
$262$		0			0
$263$	12.0980			0.745994			0.372997	−	0.927833i	$-0.378330\pi$
$263$	12.0980			0.745994			0.372997	+	0.927833i	$0.378330\pi$
$264$		0			0
$265$		−	4.44499i		−	0.273054i
$266$		0			0
$267$			27.8633i			1.70521i
$268$		0			0
$269$		−	27.1118i		−	1.65303i	−0.562912	−	0.826517i	$-0.690319\pi$
$269$		−	27.1118i		−	1.65303i	0.562912	−	0.826517i	$-0.309681\pi$
$270$		0			0
$271$	1.82979			0.111152			0.0555760	−	0.998454i	$-0.482300\pi$
$271$	1.82979			0.111152			0.0555760	+	0.998454i	$0.482300\pi$
$272$		0			0
$273$	−16.8768			−1.02143
$274$		0			0
$275$			4.37750i			0.263973i
$276$		0			0
$277$			24.9811i			1.50097i	0.660889	+	0.750484i	$0.270179\pi$
$277$			24.9811i			1.50097i	−0.660889	+	0.750484i	$0.729821\pi$
$278$		0			0
$279$			13.5457i			0.810958i
$280$		0			0
$281$	1.89922			0.113298			0.0566490	−	0.998394i	$-0.481958\pi$
$281$	1.89922			0.113298			0.0566490	+	0.998394i	$0.481958\pi$
$282$		0			0
$283$		−	23.7781i		−	1.41346i	−0.707482	−	0.706731i	$-0.750169\pi$
$283$		−	23.7781i		−	1.41346i	0.707482	−	0.706731i	$-0.249831\pi$
$284$		0			0
$285$	−19.7910			−1.17232
$286$		0			0
$287$	−0.981608			−0.0579425
$288$		0			0
$289$	11.1657	+	12.8190i	0.656808	+	0.754058i
$290$		0			0
$291$	36.5295			2.14140
$292$		0			0
$293$	5.56125			0.324892			0.162446	−	0.986717i	$-0.448062\pi$
$293$	5.56125			0.324892			0.162446	+	0.986717i	$0.448062\pi$
$294$		0			0
$295$			1.66009i			0.0966540i
$296$		0			0
$297$	0.0552703			0.00320711
$298$		0			0
$299$			17.5267i			1.01360i
$300$		0			0
$301$		−	30.8417i		−	1.77769i
$302$		0			0
$303$			10.9845i			0.631041i
$304$		0			0
$305$	−3.77311			−0.216048
$306$		0			0
$307$	−2.01685			−0.115108			−0.0575539	−	0.998342i	$-0.518330\pi$
$307$	−2.01685			−0.115108			−0.0575539	+	0.998342i	$0.518330\pi$
$308$		0			0
$309$			43.3066i			2.46362i
$310$		0			0
$311$			22.2487i			1.26161i	0.775942	+	0.630804i	$0.217275\pi$
$311$			22.2487i			1.26161i	−0.775942	+	0.630804i	$0.782725\pi$
$312$		0			0
$313$		−	29.4934i		−	1.66706i	−0.552472	−	0.833532i	$-0.686315\pi$
$313$		−	29.4934i		−	1.66706i	0.552472	−	0.833532i	$-0.313685\pi$
$314$		0			0
$315$	−18.1546			−1.02290
$316$		0			0
$317$			6.15858i			0.345901i	0.984931	+	0.172950i	$0.0553300\pi$
$317$			6.15858i			0.345901i	−0.984931	+	0.172950i	$0.944670\pi$
$318$		0			0
$319$	−6.50784			−0.364369
$320$		0			0
$321$	−34.4931			−1.92522
$322$		0			0
$323$	−18.2821	−	8.32068i	−1.01724	−	0.462975i
$324$		0			0
$325$	−4.22925			−0.234597
$326$		0			0
$327$	18.4914			1.02258
$328$		0			0
$329$		−	29.7995i		−	1.64290i
$330$		0			0
$331$	−18.6686			−1.02612			−0.513060	−	0.858353i	$-0.671488\pi$
$331$	−18.6686			−1.02612			−0.513060	+	0.858353i	$0.671488\pi$
$332$		0			0
$333$			8.44871i			0.462986i
$334$		0			0
$335$			25.1541i			1.37432i
$336$		0			0
$337$		−	23.7041i		−	1.29125i	−0.763656	−	0.645623i	$-0.776598\pi$
$337$		−	23.7041i		−	1.29125i	0.763656	−	0.645623i	$-0.223402\pi$
$338$		0			0
$339$	−28.9150			−1.57045
$340$		0			0
$341$	8.84178			0.478809
$342$		0			0
$343$			2.22688i			0.120240i
$344$		0			0
$345$			37.7806i			2.03404i
$346$		0			0
$347$			27.3088i			1.46602i	0.680220	+	0.733008i	$0.261884\pi$
$347$			27.3088i			1.46602i	−0.680220	+	0.733008i	$0.738116\pi$
$348$		0			0
$349$	−19.7076			−1.05492			−0.527462	−	0.849579i	$-0.676856\pi$
$349$	−19.7076			−1.05492			−0.527462	+	0.849579i	$0.676856\pi$
$350$		0			0
$351$			0.0533986i			0.00285020i
$352$		0			0
$353$	27.1815			1.44673			0.723363	−	0.690468i	$-0.242595\pi$
$353$	27.1815			1.44673			0.723363	+	0.690468i	$0.242595\pi$
$354$		0			0
$355$	10.8496			0.575835
$356$		0			0
$357$	−33.6059	−	15.2950i	−1.77861	−	0.809495i
$358$		0			0
$359$	−20.4096			−1.07718			−0.538588	−	0.842569i	$-0.681042\pi$
$359$	−20.4096			−1.07718			−0.538588	+	0.842569i	$0.681042\pi$
$360$		0			0
$361$	4.73347			0.249130
$362$		0			0
$363$		−	17.6069i		−	0.924122i
$364$		0			0
$365$	−10.4527			−0.547120
$366$		0			0
$367$			22.9589i			1.19845i	0.800582	+	0.599224i	$0.204524\pi$
$367$			22.9589i			1.19845i	−0.800582	+	0.599224i	$0.795476\pi$
$368$		0			0
$369$		−	0.801615i		−	0.0417304i
$370$		0			0
$371$			9.79837i			0.508706i
$372$		0			0
$373$	30.2437			1.56596			0.782979	−	0.622048i	$-0.213699\pi$
$373$	30.2437			1.56596			0.782979	+	0.622048i	$0.213699\pi$
$374$		0			0
$375$	−29.4288			−1.51970
$376$		0			0
$377$		−	6.28745i		−	0.323820i
$378$		0			0
$379$		−	24.1637i		−	1.24120i	−0.784126	−	0.620602i	$-0.786888\pi$
$379$		−	24.1637i		−	1.24120i	0.784126	−	0.620602i	$-0.213112\pi$
$380$		0			0
$381$		−	46.1875i		−	2.36626i
$382$		0			0
$383$	10.9208			0.558025			0.279012	−	0.960287i	$-0.409993\pi$
$383$	10.9208			0.558025			0.279012	+	0.960287i	$0.409993\pi$
$384$		0			0
$385$			11.8502i			0.603943i
$386$		0			0
$387$	25.1864			1.28030
$388$		0			0
$389$	30.7751			1.56036			0.780181	−	0.625554i	$-0.215127\pi$
$389$	30.7751			1.56036			0.780181	+	0.625554i	$0.215127\pi$
$390$		0			0
$391$	−15.8840	+	34.9002i	−0.803289	+	1.76498i
$392$		0			0
$393$	21.4703			1.08303
$394$		0			0
$395$	4.21516			0.212087
$396$		0			0
$397$		−	4.32581i		−	0.217106i	−0.994091	−	0.108553i	$-0.965378\pi$
$397$		−	4.32581i		−	0.217106i	0.994091	−	0.108553i	$-0.0346217\pi$
$398$		0			0
$399$	43.6266			2.18406
$400$		0			0
$401$		−	14.1274i		−	0.705488i	−0.935720	−	0.352744i	$-0.885249\pi$
$401$		−	14.1274i		−	0.705488i	0.935720	−	0.352744i	$-0.114751\pi$
$402$		0			0
$403$			8.54235i			0.425525i
$404$		0			0
$405$			14.9982i			0.745268i
$406$		0			0
$407$	5.51480			0.273359
$408$		0			0
$409$	17.8864			0.884427			0.442213	−	0.896910i	$-0.354193\pi$
$409$	17.8864			0.884427			0.442213	+	0.896910i	$0.354193\pi$
$410$		0			0
$411$		−	22.8093i		−	1.12510i
$412$		0			0
$413$		−	3.65944i		−	0.180069i
$414$		0			0
$415$			3.68802i			0.181038i
$416$		0			0
$417$	18.6691			0.914232
$418$		0			0
$419$		−	21.8574i		−	1.06780i	−0.845547	−	0.533901i	$-0.820726\pi$
$419$		−	21.8574i		−	1.06780i	0.845547	−	0.533901i	$-0.179274\pi$
$420$		0			0
$421$	35.8778			1.74858			0.874290	−	0.485405i	$-0.161328\pi$
$421$	35.8778			1.74858			0.874290	+	0.485405i	$0.161328\pi$
$422$		0			0
$423$	24.3353			1.18322
$424$		0			0
$425$	−8.42151	−	3.83286i	−0.408503	−	0.185921i
$426$		0			0
$427$	8.31731			0.402503
$428$		0			0
$429$	−8.99615			−0.434338
$430$		0			0
$431$			22.1694i			1.06786i	0.845528	+	0.533931i	$0.179286\pi$
$431$			22.1694i			1.06786i	−0.845528	+	0.533931i	$0.820714\pi$
$432$		0			0
$433$	−16.4783			−0.791895			−0.395948	−	0.918273i	$-0.629584\pi$
$433$	−16.4783			−0.791895			−0.395948	+	0.918273i	$0.629584\pi$
$434$		0			0
$435$		−	13.5532i		−	0.649827i
$436$		0			0
$437$		−	45.3067i		−	2.16732i
$438$		0			0
$439$			19.0168i			0.907621i	0.891098	+	0.453810i	$0.149936\pi$
$439$			19.0168i			0.907621i	−0.891098	+	0.453810i	$0.850064\pi$
$440$		0			0
$441$	19.1004			0.909543
$442$		0			0
$443$	−18.7287			−0.889829			−0.444915	−	0.895573i	$-0.646766\pi$
$443$	−18.7287			−0.889829			−0.444915	+	0.895573i	$0.646766\pi$
$444$		0			0
$445$			18.9020i			0.896041i
$446$		0			0
$447$			47.0006i			2.22305i
$448$		0			0
$449$		−	16.5835i		−	0.782626i	−0.920258	−	0.391313i	$-0.872021\pi$
$449$		−	16.5835i		−	0.782626i	0.920258	−	0.391313i	$-0.127979\pi$
$450$		0			0
$451$	−0.523245			−0.0246387
$452$		0			0
$453$		−	6.41805i		−	0.301546i
$454$		0			0
$455$	−11.4489			−0.536733
$456$		0			0
$457$	22.6547			1.05974			0.529872	−	0.848078i	$-0.322240\pi$
$457$	22.6547			1.05974			0.529872	+	0.848078i	$0.322240\pi$
$458$		0			0
$459$	−0.0483937	+	0.106330i	−0.00225882	+	0.00496306i
$460$		0			0
$461$	15.0852			0.702587			0.351294	−	0.936265i	$-0.385742\pi$
$461$	15.0852			0.702587			0.351294	+	0.936265i	$0.385742\pi$
$462$		0			0
$463$	−7.19632			−0.334441			−0.167221	−	0.985919i	$-0.553479\pi$
$463$	−7.19632			−0.334441			−0.167221	+	0.985919i	$0.553479\pi$
$464$		0			0
$465$			18.4139i			0.853924i
$466$		0			0
$467$	−36.1274			−1.67178			−0.835889	−	0.548899i	$-0.815047\pi$
$467$	−36.1274			−1.67178			−0.835889	+	0.548899i	$0.815047\pi$
$468$		0			0
$469$		−	55.4489i		−	2.56039i
$470$		0			0
$471$		−	21.4798i		−	0.989739i
$472$		0			0
$473$		−	16.4401i		−	0.755918i
$474$		0			0
$475$	10.9326			0.501624
$476$		0			0
$477$	−8.00169			−0.366372
$478$		0			0
$479$		−	24.4261i		−	1.11606i	−0.829821	−	0.558029i	$-0.811558\pi$
$479$		−	24.4261i		−	1.11606i	0.829821	−	0.558029i	$-0.188442\pi$
$480$		0			0
$481$			5.32804i			0.242938i
$482$		0			0
$483$		−	83.2823i		−	3.78947i
$484$		0			0
$485$	24.7810			1.12525
$486$		0			0
$487$			26.9195i			1.21984i	0.792463	+	0.609920i	$0.208798\pi$
$487$			26.9195i			1.21984i	−0.792463	+	0.609920i	$0.791202\pi$
$488$		0			0
$489$	13.6302			0.616377
$490$		0			0
$491$	−27.4229			−1.23758			−0.618789	−	0.785557i	$-0.712377\pi$
$491$	−27.4229			−1.23758			−0.618789	+	0.785557i	$0.712377\pi$
$492$		0			0
$493$	5.69815	−	12.5199i	0.256632	−	0.563868i
$494$		0			0
$495$	−9.67730			−0.434962
$496$		0			0
$497$	−23.9164			−1.07280
$498$		0			0
$499$		−	39.6948i		−	1.77698i	−0.458893	−	0.888492i	$-0.651754\pi$
$499$		−	39.6948i		−	1.77698i	0.458893	−	0.888492i	$-0.348246\pi$
$500$		0			0
$501$	−27.0271			−1.20748
$502$		0			0
$503$			11.1816i			0.498562i	0.968431	+	0.249281i	$0.0801943\pi$
$503$			11.1816i			0.498562i	−0.968431	+	0.249281i	$0.919806\pi$
$504$		0			0
$505$			7.45167i			0.331595i
$506$		0			0
$507$			23.1211i			1.02685i
$508$		0			0
$509$	−4.98016			−0.220742			−0.110371	−	0.993890i	$-0.535204\pi$
$509$	−4.98016			−0.220742			−0.110371	+	0.993890i	$0.535204\pi$
$510$		0			0
$511$	23.0416			1.01930
$512$		0			0
$513$		−	0.138036i		−	0.00609442i
$514$		0			0
$515$			29.3784i			1.29457i
$516$		0			0
$517$		−	15.8846i		−	0.698605i
$518$		0			0
$519$	−32.6682			−1.43397
$520$		0			0
$521$			5.28507i			0.231543i	0.993276	+	0.115772i	$0.0369341\pi$
$521$			5.28507i			0.231543i	−0.993276	+	0.115772i	$0.963066\pi$
$522$		0			0
$523$	−21.1559			−0.925082			−0.462541	−	0.886598i	$-0.653062\pi$
$523$	−21.1559			−0.925082			−0.462541	+	0.886598i	$0.653062\pi$
$524$		0			0
$525$	20.0962			0.877072
$526$		0			0
$527$	−7.74171	+	17.0100i	−0.337234	+	0.740967i
$528$		0			0
$529$	−63.4897			−2.76042
$530$		0			0
$531$	2.98842			0.129686
$532$		0			0
$533$		−	0.505525i		−	0.0218967i
$534$		0			0
$535$	−23.3995			−1.01165
$536$		0			0
$537$			2.79162i			0.120467i
$538$		0			0
$539$		−	12.4676i		−	0.537017i
$540$		0			0
$541$		−	3.17993i		−	0.136716i	−0.997661	−	0.0683579i	$-0.978224\pi$
$541$		−	3.17993i		−	0.136716i	0.997661	−	0.0683579i	$-0.0217760\pi$
$542$		0			0
$543$	−13.6123			−0.584161
$544$		0			0
$545$	12.5443			0.537337
$546$		0			0
$547$		−	7.26644i		−	0.310690i	−0.987860	−	0.155345i	$-0.950351\pi$
$547$		−	7.26644i		−	0.310690i	0.987860	−	0.155345i	$-0.0496490\pi$
$548$		0			0
$549$			6.79220i			0.289884i
$550$		0			0
$551$			16.2531i			0.692405i
$552$		0			0
$553$	−9.29174			−0.395125
$554$		0			0
$555$			11.4851i			0.487516i
$556$		0			0
$557$	−30.7461			−1.30275			−0.651377	−	0.758754i	$-0.725808\pi$
$557$	−30.7461			−1.30275			−0.651377	+	0.758754i	$0.725808\pi$
$558$		0			0
$559$	15.8834			0.671795
$560$		0			0
$561$	−17.9136	−	8.15297i	−0.756313	−	0.344219i
$562$		0			0
$563$	−7.47020			−0.314831			−0.157416	−	0.987532i	$-0.550316\pi$
$563$	−7.47020			−0.314831			−0.157416	+	0.987532i	$0.550316\pi$
$564$		0			0
$565$	−19.6154			−0.825227
$566$		0			0
$567$		−	33.0615i		−	1.38845i
$568$		0			0
$569$	8.66982			0.363458			0.181729	−	0.983349i	$-0.441831\pi$
$569$	8.66982			0.363458			0.181729	+	0.983349i	$0.441831\pi$
$570$		0			0
$571$		−	14.6724i		−	0.614021i	−0.951706	−	0.307010i	$-0.900671\pi$
$571$		−	14.6724i		−	0.614021i	0.951706	−	0.307010i	$-0.0993286\pi$
$572$		0			0
$573$		−	7.67558i		−	0.320652i
$574$		0			0
$575$		−	20.8702i		−	0.870348i
$576$		0			0
$577$	−7.38842			−0.307584			−0.153792	−	0.988103i	$-0.549149\pi$
$577$	−7.38842			−0.307584			−0.153792	+	0.988103i	$0.549149\pi$
$578$		0			0
$579$	−11.4186			−0.474543
$580$		0			0
$581$		−	8.12974i		−	0.337279i
$582$		0			0
$583$			5.22302i			0.216315i
$584$		0			0
$585$		−	9.34957i		−	0.386557i
$586$		0			0
$587$	35.8463			1.47954			0.739768	−	0.672862i	$-0.234935\pi$
$587$	35.8463			1.47954			0.739768	+	0.672862i	$0.234935\pi$
$588$		0			0
$589$		−	22.0820i		−	0.909874i
$590$		0			0
$591$	−31.1694			−1.28214
$592$		0			0
$593$	−39.0091			−1.60191			−0.800957	−	0.598722i	$-0.795675\pi$
$593$	−39.0091			−1.60191			−0.800957	+	0.598722i	$0.795675\pi$
$594$		0			0
$595$	−22.7977	−	10.3758i	−0.934613	−	0.425368i
$596$		0			0
$597$	25.4300			1.04078
$598$		0			0
$599$	31.2132			1.27533			0.637667	−	0.770312i	$-0.279899\pi$
$599$	31.2132			1.27533			0.637667	+	0.770312i	$0.279899\pi$
$600$		0			0
$601$			17.1812i			0.700838i	0.936593	+	0.350419i	$0.113961\pi$
$601$			17.1812i			0.700838i	−0.936593	+	0.350419i	$0.886039\pi$
$602$		0			0
$603$	45.2815			1.84400
$604$		0			0
$605$		−	11.9442i		−	0.485601i
$606$		0			0
$607$			42.2530i			1.71500i	0.514486	+	0.857499i	$0.327983\pi$
$607$			42.2530i			1.71500i	−0.514486	+	0.857499i	$0.672017\pi$
$608$		0			0
$609$			29.8762i			1.21065i
$610$		0			0
$611$	15.3467			0.620860
$612$		0			0
$613$	19.1640			0.774026			0.387013	−	0.922074i	$-0.373507\pi$
$613$	19.1640			0.774026			0.387013	+	0.922074i	$0.373507\pi$
$614$		0			0
$615$		−	1.08971i		−	0.0439414i
$616$		0			0
$617$		−	7.69430i		−	0.309761i	−0.987933	−	0.154880i	$-0.950501\pi$
$617$		−	7.69430i		−	0.309761i	0.987933	−	0.154880i	$-0.0494993\pi$
$618$		0			0
$619$			25.9688i			1.04377i	0.853015	+	0.521887i	$0.174772\pi$
$619$			25.9688i			1.04377i	−0.853015	+	0.521887i	$0.825228\pi$
$620$		0			0
$621$	−0.263507			−0.0105742
$622$		0			0
$623$		−	41.6669i		−	1.66935i
$624$		0			0
$625$	−8.74341			−0.349736
$626$		0			0
$627$	23.2551			0.928719
$628$		0			0
$629$	−4.82866	+	10.6095i	−0.192531	+	0.423028i
$630$		0			0
$631$	−16.1095			−0.641308			−0.320654	−	0.947196i	$-0.603903\pi$
$631$	−16.1095			−0.641308			−0.320654	+	0.947196i	$0.603903\pi$
$632$		0			0
$633$	16.7638			0.666302
$634$		0			0
$635$		−	31.3328i		−	1.24341i
$636$		0			0
$637$	12.0454			0.477254
$638$		0			0
$639$		−	19.5310i		−	0.772632i
$640$		0			0
$641$		−	4.41509i		−	0.174385i	−0.996191	−	0.0871927i	$-0.972210\pi$
$641$		−	4.41509i		−	0.174385i	0.996191	−	0.0871927i	$-0.0277896\pi$
$642$		0			0
$643$			9.71802i			0.383241i	0.981469	+	0.191621i	$0.0613743\pi$
$643$			9.71802i			0.383241i	−0.981469	+	0.191621i	$0.938626\pi$
$644$		0			0
$645$	34.2382			1.34813
$646$		0			0
$647$	3.78118			0.148654			0.0743268	−	0.997234i	$-0.476319\pi$
$647$	3.78118			0.148654			0.0743268	+	0.997234i	$0.476319\pi$
$648$		0			0
$649$		−	1.95066i		−	0.0765701i
$650$		0			0
$651$		−	40.5909i		−	1.59088i
$652$		0			0
$653$		−	28.3967i		−	1.11125i	−0.831434	−	0.555624i	$-0.812480\pi$
$653$		−	28.3967i		−	1.11125i	0.831434	−	0.555624i	$-0.187520\pi$
$654$		0			0
$655$	14.5651			0.569104
$656$		0			0
$657$			18.8166i			0.734104i
$658$		0			0
$659$	28.8732			1.12474			0.562371	−	0.826885i	$-0.309889\pi$
$659$	28.8732			1.12474			0.562371	+	0.826885i	$0.309889\pi$
$660$		0			0
$661$	16.4655			0.640434			0.320217	−	0.947344i	$-0.396244\pi$
$661$	16.4655			0.640434			0.320217	+	0.947344i	$0.396244\pi$
$662$		0			0
$663$	7.87686	−	17.3070i	0.305912	−	0.672146i
$664$		0			0
$665$	29.5955			1.14766
$666$		0			0
$667$	31.0269			1.20136
$668$		0			0
$669$			32.5534i			1.25859i
$670$		0			0
$671$	4.43354			0.171155
$672$		0			0
$673$		−	21.8052i		−	0.840530i	−0.907401	−	0.420265i	$-0.861937\pi$
$673$		−	21.8052i		−	0.840530i	0.907401	−	0.420265i	$-0.138063\pi$
$674$		0			0
$675$		−	0.0635851i		−	0.00244739i
$676$		0			0
$677$			42.5692i			1.63607i	0.575171	+	0.818033i	$0.304936\pi$
$677$			42.5692i			1.63607i	−0.575171	+	0.818033i	$0.695064\pi$
$678$		0			0
$679$	−54.6263			−2.09637
$680$		0			0
$681$	−5.40367			−0.207069
$682$		0			0
$683$		−	10.3756i		−	0.397013i	−0.980100	−	0.198507i	$-0.936391\pi$
$683$		−	10.3756i		−	0.397013i	0.980100	−	0.198507i	$-0.0636091\pi$
$684$		0			0
$685$		−	15.4735i		−	0.591210i
$686$		0			0
$687$			72.2097i			2.75497i
$688$		0			0
$689$	−5.04613			−0.192242
$690$		0			0
$691$		−	51.4201i		−	1.95611i	−0.208341	−	0.978056i	$-0.566806\pi$
$691$		−	51.4201i		−	1.95611i	0.208341	−	0.978056i	$-0.433194\pi$
$692$		0			0
$693$	21.3323			0.810346
$694$		0			0
$695$	12.6648			0.480404
$696$		0			0
$697$	0.458144	−	1.00663i	0.0173534	−	0.0381288i
$698$		0			0
$699$	−51.6089			−1.95203
$700$		0			0
$701$	28.1674			1.06387			0.531934	−	0.846786i	$-0.321465\pi$
$701$	28.1674			1.06387			0.531934	+	0.846786i	$0.321465\pi$
$702$		0			0
$703$		−	13.7730i		−	0.519459i
$704$		0			0
$705$	33.0813			1.24591
$706$		0			0
$707$		−	16.4262i		−	0.617771i
$708$		0			0
$709$			3.88214i			0.145797i	0.997339	+	0.0728985i	$0.0232249\pi$
$709$			3.88214i			0.145797i	−0.997339	+	0.0728985i	$0.976775\pi$
$710$		0			0
$711$		−	7.58795i		−	0.284570i
$712$		0			0
$713$	−42.1542			−1.57869
$714$		0			0
$715$	−6.10283			−0.228233
$716$		0			0
$717$		−	41.8468i		−	1.56280i
$718$		0			0
$719$			14.1855i			0.529031i	0.964381	+	0.264516i	$0.0852120\pi$
$719$			14.1855i			0.529031i	−0.964381	+	0.264516i	$0.914788\pi$
$720$		0			0
$721$		−	64.7607i		−	2.41182i
$722$		0			0
$723$	−14.0020			−0.520738
$724$		0			0
$725$			7.48687i			0.278055i
$726$		0			0
$727$	−16.3698			−0.607123			−0.303561	−	0.952812i	$-0.598176\pi$
$727$	−16.3698			−0.607123			−0.303561	+	0.952812i	$0.598176\pi$
$728$		0			0
$729$	26.7912			0.992266
$730$		0			0
$731$	31.6278	+	14.3947i	1.16980	+	0.532407i
$732$		0			0
$733$	−30.5715			−1.12918			−0.564591	−	0.825371i	$-0.690966\pi$
$733$	−30.5715			−1.12918			−0.564591	+	0.825371i	$0.690966\pi$
$734$		0			0
$735$	25.9650			0.957732
$736$		0			0
$737$		−	29.5570i		−	1.08875i
$738$		0			0
$739$	34.7769			1.27929			0.639644	−	0.768671i	$-0.279082\pi$
$739$	34.7769			1.27929			0.639644	+	0.768671i	$0.279082\pi$
$740$		0			0
$741$			22.4676i			0.825366i
$742$		0			0
$743$		−	42.2206i		−	1.54892i	−0.632621	−	0.774461i	$-0.718021\pi$
$743$		−	42.2206i		−	1.54892i	0.632621	−	0.774461i	$-0.281979\pi$
$744$		0			0
$745$			31.8844i			1.16815i
$746$		0			0
$747$	6.63903			0.242909
$748$		0			0
$749$	51.5810			1.88473
$750$		0			0
$751$			42.3310i			1.54468i	0.635210	+	0.772340i	$0.280914\pi$
$751$			42.3310i			1.54468i	−0.635210	+	0.772340i	$0.719086\pi$
$752$		0			0
$753$			54.4824i			1.98545i
$754$		0			0
$755$		−	4.35389i		−	0.158454i
$756$		0			0
$757$	40.5379			1.47337			0.736687	−	0.676234i	$-0.236389\pi$
$757$	40.5379			1.47337			0.736687	+	0.676234i	$0.236389\pi$
$758$		0			0
$759$		−	44.3935i		−	1.61138i
$760$		0			0
$761$	48.5505			1.75995			0.879977	−	0.475016i	$-0.157558\pi$
$761$	48.5505			1.75995			0.879977	+	0.475016i	$0.157558\pi$
$762$		0			0
$763$	−27.6521			−1.00107
$764$		0			0
$765$	8.47326	−	18.6174i	0.306352	−	0.673112i
$766$		0			0
$767$	1.88460			0.0680489
$768$		0			0
$769$	3.93593			0.141933			0.0709666	−	0.997479i	$-0.477392\pi$
$769$	3.93593			0.141933			0.0709666	+	0.997479i	$0.477392\pi$
$770$		0			0
$771$			34.5117i			1.24291i
$772$		0			0
$773$	25.6453			0.922396			0.461198	−	0.887297i	$-0.347420\pi$
$773$	25.6453			0.922396			0.461198	+	0.887297i	$0.347420\pi$
$774$		0			0
$775$		−	10.1719i		−	0.365386i
$776$		0			0
$777$		−	25.3174i		−	0.908256i
$778$		0			0
$779$			1.30679i			0.0468205i
$780$		0			0
$781$	−12.7486			−0.456181
$782$		0			0
$783$	0.0945293			0.00337820
$784$		0			0
$785$		−	14.5716i		−	0.520081i
$786$		0			0
$787$			15.3766i			0.548117i	0.961713	+	0.274059i	$0.0883662\pi$
$787$			15.3766i			0.548117i	−0.961713	+	0.274059i	$0.911634\pi$
$788$		0			0
$789$		−	29.6053i		−	1.05398i
$790$		0			0
$791$	43.2395			1.53742
$792$		0			0
$793$			4.28339i			0.152108i
$794$		0			0
$795$	−10.8774			−0.385783
$796$		0			0
$797$	−19.0948			−0.676371			−0.338185	−	0.941079i	$-0.609813\pi$
$797$	−19.0948			−0.676371			−0.338185	+	0.941079i	$0.609813\pi$
$798$		0			0
$799$	30.5591	+	13.9083i	1.08110	+	0.492040i
$800$		0			0
$801$	34.0266			1.20227
$802$		0			0
$803$	12.2823			0.433433
$804$		0			0
$805$		−	56.4972i		−	1.99127i
$806$		0			0
$807$	−66.3459			−2.33549
$808$		0			0
$809$			19.6778i			0.691836i	0.938265	+	0.345918i	$0.112432\pi$
$809$			19.6778i			0.691836i	−0.938265	+	0.345918i	$0.887568\pi$
$810$		0			0
$811$		−	33.1887i		−	1.16541i	−0.812683	−	0.582706i	$-0.801994\pi$
$811$		−	33.1887i		−	1.16541i	0.812683	−	0.582706i	$-0.198006\pi$
$812$		0			0
$813$		−	4.47773i		−	0.157041i
$814$		0			0
$815$	9.24646			0.323889
$816$		0			0
$817$	−41.0586			−1.43646
$818$		0			0
$819$			20.6099i			0.720167i
$820$		0			0
$821$			8.75011i			0.305381i	0.988274	+	0.152691i	$0.0487938\pi$
$821$			8.75011i			0.305381i	−0.988274	+	0.152691i	$0.951206\pi$
$822$		0			0
$823$			21.5943i			0.752730i	0.926471	+	0.376365i	$0.122826\pi$
$823$			21.5943i			0.752730i	−0.926471	+	0.376365i	$0.877174\pi$
$824$		0			0
$825$	10.7123			0.372954
$826$		0			0
$827$		−	42.6129i		−	1.48180i	−0.671617	−	0.740899i	$-0.734400\pi$
$827$		−	42.6129i		−	1.48180i	0.671617	−	0.740899i	$-0.265600\pi$
$828$		0			0
$829$	17.2293			0.598399			0.299199	−	0.954191i	$-0.403280\pi$
$829$	17.2293			0.598399			0.299199	+	0.954191i	$0.403280\pi$
$830$		0			0
$831$	61.1318			2.12064
$832$		0			0
$833$	23.9854	+	10.9164i	0.831043	+	0.378230i
$834$		0			0
$835$	−18.3347			−0.634500
$836$		0			0
$837$	−0.128431			−0.00443922
$838$		0			0
$839$		−	3.47333i		−	0.119913i	−0.998201	−	0.0599564i	$-0.980904\pi$
$839$		−	3.47333i		−	0.119913i	0.998201	−	0.0599564i	$-0.0190962\pi$
$840$		0			0
$841$	17.8696			0.616193
$842$		0			0
$843$		−	4.64763i		−	0.160073i
$844$		0			0
$845$			15.6850i			0.539580i
$846$		0			0
$847$			26.3294i			0.904688i
$848$		0			0
$849$	−58.1880			−1.99701
$850$		0			0
$851$	−26.2924			−0.901293
$852$		0			0
$853$		−	0.242761i		−	0.00831199i	−0.999991	−	0.00415600i	$-0.998677\pi$
$853$		−	0.242761i		−	0.00831199i	0.999991	−	0.00415600i	$-0.00132290\pi$
$854$		0			0
$855$			24.1687i			0.826552i
$856$		0			0
$857$			40.7514i			1.39204i	0.718022	+	0.696021i	$0.245048\pi$
$857$			40.7514i			1.39204i	−0.718022	+	0.696021i	$0.754952\pi$
$858$		0			0
$859$	34.1939			1.16668			0.583340	−	0.812228i	$-0.301746\pi$
$859$	34.1939			1.16668			0.583340	+	0.812228i	$0.301746\pi$
$860$		0			0
$861$			2.40212i			0.0818639i
$862$		0			0
$863$	23.8791			0.812853			0.406426	−	0.913684i	$-0.366775\pi$
$863$	23.8791			0.812853			0.406426	+	0.913684i	$0.366775\pi$
$864$		0			0
$865$	−22.1615			−0.753514
$866$		0			0
$867$	31.3697	−	27.3239i	1.06537	−	0.927970i
$868$		0			0
$869$	−4.95295			−0.168017
$870$		0			0
$871$	28.5560			0.967584
$872$		0			0
$873$		−	44.6097i		−	1.50981i
$874$		0			0
$875$	44.0078			1.48774
$876$		0			0
$877$			26.3774i			0.890701i	0.895356	+	0.445350i	$0.146921\pi$
$877$			26.3774i			0.890701i	−0.895356	+	0.445350i	$0.853079\pi$
$878$		0			0
$879$		−	13.6091i		−	0.459022i
$880$		0			0
$881$		−	39.7274i		−	1.33845i	−0.743059	−	0.669226i	$-0.766626\pi$
$881$		−	39.7274i		−	1.33845i	0.743059	−	0.669226i	$-0.233374\pi$
$882$		0			0
$883$	−32.4719			−1.09277			−0.546383	−	0.837535i	$-0.683996\pi$
$883$	−32.4719			−1.09277			−0.546383	+	0.837535i	$0.683996\pi$
$884$		0			0
$885$	4.06244			0.136557
$886$		0			0
$887$			24.3417i			0.817313i	0.912688	+	0.408657i	$0.134003\pi$
$887$			24.3417i			0.817313i	−0.912688	+	0.408657i	$0.865997\pi$
$888$		0			0
$889$			69.0689i			2.31650i
$890$		0			0
$891$		−	17.6234i		−	0.590407i
$892$		0			0
$893$	−39.6712			−1.32755
$894$		0			0
$895$			1.89379i			0.0633024i
$896$		0			0
$897$	42.8901			1.43206
$898$		0			0
$899$	15.1222			0.504353
$900$		0			0
$901$	−10.0481	−	4.57318i	−0.334752	−	0.152355i
$902$		0			0
$903$	−75.4735			−2.51160
$904$		0			0
$905$	−9.23436			−0.306961
$906$		0			0
$907$		−	18.1651i		−	0.603163i	−0.953440	−	0.301581i	$-0.902485\pi$
$907$		−	18.1651i		−	0.603163i	0.953440	−	0.301581i	$-0.0975145\pi$
$908$		0			0
$909$	13.4142			0.444921
$910$		0			0
$911$			2.77676i			0.0919981i	0.998941	+	0.0459991i	$0.0146471\pi$
$911$			2.77676i			0.0919981i	−0.998941	+	0.0459991i	$0.985353\pi$
$912$		0			0
$913$		−	4.33355i		−	0.143420i
$914$		0			0
$915$			9.23328i			0.305243i
$916$		0			0
$917$	−32.1067			−1.06026
$918$		0			0
$919$	−35.7442			−1.17909			−0.589546	−	0.807735i	$-0.700693\pi$
$919$	−35.7442			−1.17909			−0.589546	+	0.807735i	$0.700693\pi$
$920$		0			0
$921$			4.93549i			0.162630i
$922$		0			0
$923$		−	12.3169i		−	0.405415i
$924$		0			0
$925$		−	6.34443i		−	0.208604i
$926$		0			0
$927$	52.8858			1.73700
$928$		0			0
$929$		−	30.3989i		−	0.997355i	−0.866788	−	0.498677i	$-0.833819\pi$
$929$		−	30.3989i		−	0.997355i	0.866788	−	0.498677i	$-0.166181\pi$
$930$		0			0
$931$	−31.1373			−1.02048
$932$		0			0
$933$	54.4453			1.78246
$934$		0			0
$935$	−12.1523	−	5.53083i	−0.397422	−	0.180878i
$936$		0			0
$937$	−42.7091			−1.39524			−0.697622	−	0.716466i	$-0.745759\pi$
$937$	−42.7091			−1.39524			−0.697622	+	0.716466i	$0.745759\pi$
$938$		0			0
$939$	−72.1739			−2.35531
$940$		0			0
$941$		−	31.3912i		−	1.02332i	−0.859187	−	0.511662i	$-0.829030\pi$
$941$		−	31.3912i		−	1.02332i	0.859187	−	0.511662i	$-0.170970\pi$
$942$		0			0
$943$	2.49463			0.0812363
$944$		0			0
$945$		−	0.172130i		−	0.00559938i
$946$		0			0
$947$		−	48.4051i		−	1.57295i	−0.617620	−	0.786477i	$-0.711903\pi$
$947$		−	48.4051i		−	1.57295i	0.617620	−	0.786477i	$-0.288097\pi$
$948$		0			0
$949$			11.8664i			0.385198i
$950$		0			0
$951$	15.0708			0.488705
$952$		0			0
$953$	−12.7349			−0.412524			−0.206262	−	0.978497i	$-0.566130\pi$
$953$	−12.7349			−0.412524			−0.206262	+	0.978497i	$0.566130\pi$
$954$		0			0
$955$		−	5.20698i		−	0.168494i
$956$		0			0
$957$			15.9255i			0.514798i
$958$		0			0
$959$			34.1091i			1.10144i
$960$		0			0
$961$	10.4545			0.337241
$962$		0			0
$963$			42.1228i			1.35739i
$964$		0			0
$965$	−7.74621			−0.249359
$966$		0			0
$967$	24.7652			0.796395			0.398197	−	0.917300i	$-0.369636\pi$
$967$	24.7652			0.796395			0.398197	+	0.917300i	$0.369636\pi$
$968$		0			0
$969$	−20.3617	+	44.7386i	−0.654113	+	1.43721i
$970$		0			0
$971$	−2.97708			−0.0955389			−0.0477694	−	0.998858i	$-0.515211\pi$
$971$	−2.97708			−0.0955389			−0.0477694	+	0.998858i	$0.515211\pi$
$972$		0			0
$973$	−27.9179			−0.895006
$974$		0			0
$975$			10.3495i			0.331449i
$976$		0			0
$977$	37.1271			1.18780			0.593901	−	0.804538i	$-0.297587\pi$
$977$	37.1271			1.18780			0.593901	+	0.804538i	$0.297587\pi$
$978$		0			0
$979$		−	22.2105i		−	0.709851i
$980$		0			0
$981$		−	22.5817i		−	0.720977i
$982$		0			0
$983$		−	13.2382i		−	0.422234i	−0.977461	−	0.211117i	$-0.932290\pi$
$983$		−	13.2382i		−	0.422234i	0.977461	−	0.211117i	$-0.0677101\pi$
$984$		0			0
$985$	−21.1448			−0.673729
$986$		0			0
$987$	−72.9232			−2.32117
$988$		0			0
$989$			78.3802i			2.49234i
$990$		0			0
$991$			23.6257i			0.750496i	0.926924	+	0.375248i	$0.122443\pi$
$991$			23.6257i			0.750496i	−0.926924	+	0.375248i	$0.877557\pi$
$992$		0			0
$993$			45.6845i			1.44975i
$994$		0			0
$995$	17.2512			0.546901
$996$		0			0
$997$		−	22.6973i		−	0.718830i	−0.933178	−	0.359415i	$-0.882976\pi$
$997$		−	22.6973i		−	0.718830i	0.933178	−	0.359415i	$-0.117024\pi$
$998$		0			0
$999$	−0.0801049			−0.00253441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.5	✓	40
17.16	even	2	inner	4012.2.b.a.237.36	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.5	✓	40	1.1	even	1	trivial
4012.2.b.a.237.36	yes	40	17.16	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.44713i q^{3} -1.66009i q^{5} +3.65944i q^{7} -2.98842 q^{9} +O(q^{10})\)	\(q-2.44713i q^{3} -1.66009i q^{5} +3.65944i q^{7} -2.98842 q^{9} +1.95066i q^{11} -1.88460 q^{13} -4.06244 q^{15} +(-3.75271 - 1.70796i) q^{17} +4.87170 q^{19} +8.95510 q^{21} -9.29998i q^{23} +2.24411 q^{25} -0.0283342i q^{27} +3.33623i q^{29} -4.53272i q^{31} +4.77351 q^{33} +6.07498 q^{35} -2.82715i q^{37} +4.61185i q^{39} +0.268240i q^{41} -8.42799 q^{43} +4.96104i q^{45} -8.14320 q^{47} -6.39147 q^{49} +(-4.17960 + 9.18336i) q^{51} +2.67756 q^{53} +3.23826 q^{55} -11.9217i q^{57} -1.00000 q^{59} -2.27284i q^{61} -10.9359i q^{63} +3.12860i q^{65} -15.1523 q^{67} -22.7582 q^{69} +6.53554i q^{71} -6.29649i q^{73} -5.49162i q^{75} -7.13831 q^{77} +2.53912i q^{79} -9.03460 q^{81} -2.22158 q^{83} +(-2.83536 + 6.22983i) q^{85} +8.16416 q^{87} -11.3861 q^{89} -6.89657i q^{91} -11.0921 q^{93} -8.08745i q^{95} +14.9275i q^{97} -5.82939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 40 * q - 36 * q^9	\( 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) \) 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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