LMFDB - Embedded newform 4600.2.e.u.4049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (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:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 28x^{8} + 260x^{6} + 897x^{4} + 1056x^{2} + 256 $ x^10 + 28x^8 + 260x^6 + 897x^4 + 1056x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.2
Root			$-3.30649i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.u.4049.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.30649i q^{3} -2.55040i q^{7} -7.93288 q^{9} +O(q^{10})$	$q-3.30649i q^{3} -2.55040i q^{7} -7.93288 q^{9} -2.72314 q^{11} -7.12637i q^{13} +0.924010i q^{17} -7.51623 q^{19} -8.43286 q^{21} +1.00000i q^{23} +16.3105i q^{27} +2.38248 q^{29} +0.866248 q^{31} +9.00402i q^{33} +0.352855i q^{37} -23.5633 q^{39} +4.34066 q^{41} -13.3239i q^{47} +0.495474 q^{49} +3.05523 q^{51} -3.99262i q^{53} +24.8523i q^{57} +3.84064 q^{59} -9.14262 q^{61} +20.2320i q^{63} -3.15933i q^{67} +3.30649 q^{69} -6.07883 q^{71} +11.3239i q^{73} +6.94508i q^{77} +12.0593 q^{79} +30.1319 q^{81} +6.35285i q^{83} -7.87765i q^{87} +9.71377 q^{89} -18.1751 q^{91} -2.86424i q^{93} +8.76465i q^{97} +21.6023 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 26 q^{9}+O(q^{10}) $ 10 * q - 26 * q^9	$ 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99}+O(q^{100}) $ 10 * q - 26 * q^9 - 2 * q^11 - 14 * q^19 + 12 * q^21 - 8 * q^29 + 38 * q^31 - 38 * q^39 + 50 * q^41 - 50 * q^49 + 38 * q^51 + 2 * q^59 - 10 * q^61 + 2 * q^71 + 4 * q^79 + 114 * q^81 - 12 * q^89 + 22 * q^91 + 130 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 3.30649i	− 1.90900i	−0.298206	−	0.954501i	$-0.596388\pi$
$3$	− 3.30649i	− 1.90900i	0.298206	−	0.954501i	$-0.403612\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.55040i	− 0.963960i	−0.876182	−	0.481980i	$-0.839918\pi$
$7$	− 2.55040i	− 0.963960i	0.876182	−	0.481980i	$-0.160082\pi$
$8$	0	0
$9$	−7.93288	−2.64429
$10$	0	0
$11$	−2.72314	−0.821056	−0.410528	−	0.911848i	$-0.634656\pi$
$11$	−2.72314	−0.821056	−0.410528	+	0.911848i	$0.634656\pi$
$12$	0	0
$13$	− 7.12637i	− 1.97650i	−0.152845	−	0.988250i	$-0.548844\pi$
$13$	− 7.12637i	− 1.97650i	0.152845	−	0.988250i	$-0.451156\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.924010i	0.224105i	0.993702	+	0.112053i	$0.0357426\pi$
$17$	0.924010i	0.224105i	−0.993702	+	0.112053i	$0.964257\pi$
$18$	0	0
$19$	−7.51623	−1.72434	−0.862171	−	0.506617i	$-0.830896\pi$
$19$	−7.51623	−1.72434	−0.862171	+	0.506617i	$0.830896\pi$
$20$	0	0
$21$	−8.43286	−1.84020
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	16.3105i	3.13896i
$28$	0	0
$29$	2.38248	0.442415	0.221208	−	0.975227i	$-0.429000\pi$
$29$	2.38248	0.442415	0.221208	+	0.975227i	$0.429000\pi$
$30$	0	0
$31$	0.866248	0.155583	0.0777913	−	0.996970i	$-0.475213\pi$
$31$	0.866248	0.155583	0.0777913	+	0.996970i	$0.475213\pi$
$32$	0	0
$33$	9.00402i	1.56740i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.352855i	0.0580089i	0.999579	+	0.0290045i	$0.00923370\pi$
$37$	0.352855i	0.0580089i	−0.999579	+	0.0290045i	$0.990766\pi$
$38$	0	0
$39$	−23.5633	−3.77315
$40$	0	0
$41$	4.34066	0.677896	0.338948	−	0.940805i	$-0.389929\pi$
$41$	4.34066	0.677896	0.338948	+	0.940805i	$0.389929\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 13.3239i	− 1.94349i	−0.236027	−	0.971746i	$-0.575845\pi$
$47$	− 13.3239i	− 1.94349i	0.236027	−	0.971746i	$-0.424155\pi$
$48$	0	0
$49$	0.495474	0.0707819
$50$	0	0
$51$	3.05523	0.427818
$52$	0	0
$53$	− 3.99262i	− 0.548429i	−0.961669	−	0.274214i	$-0.911582\pi$
$53$	− 3.99262i	− 0.548429i	0.961669	−	0.274214i	$-0.0884178\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	24.8523i	3.29177i
$58$	0	0
$59$	3.84064	0.500009	0.250004	−	0.968245i	$-0.419568\pi$
$59$	3.84064	0.500009	0.250004	+	0.968245i	$0.419568\pi$
$60$	0	0
$61$	−9.14262	−1.17059	−0.585296	−	0.810820i	$-0.699022\pi$
$61$	−9.14262	−1.17059	−0.585296	+	0.810820i	$0.699022\pi$
$62$	0	0
$63$	20.2320i	2.54899i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.15933i	− 0.385974i	−0.981201	−	0.192987i	$-0.938183\pi$
$67$	− 3.15933i	− 0.385974i	0.981201	−	0.192987i	$-0.0618175\pi$
$68$	0	0
$69$	3.30649	0.398055
$70$	0	0
$71$	−6.07883	−0.721424	−0.360712	−	0.932677i	$-0.617466\pi$
$71$	−6.07883	−0.721424	−0.360712	+	0.932677i	$0.617466\pi$
$72$	0	0
$73$	11.3239i	1.32536i	0.748901	+	0.662682i	$0.230582\pi$
$73$	11.3239i	1.32536i	−0.748901	+	0.662682i	$0.769418\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.94508i	0.791465i
$78$	0	0
$79$	12.0593	1.35677	0.678386	−	0.734706i	$-0.262680\pi$
$79$	12.0593	1.35677	0.678386	+	0.734706i	$0.262680\pi$
$80$	0	0
$81$	30.1319	3.34799
$82$	0	0
$83$	6.35285i	0.697316i	0.937250	+	0.348658i	$0.113363\pi$
$83$	6.35285i	0.697316i	−0.937250	+	0.348658i	$0.886637\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 7.87765i	− 0.844572i
$88$	0	0
$89$	9.71377	1.02966	0.514829	−	0.857293i	$-0.327855\pi$
$89$	9.71377	1.02966	0.514829	+	0.857293i	$0.327855\pi$
$90$	0	0
$91$	−18.1751	−1.90527
$92$	0	0
$93$	− 2.86424i	− 0.297008i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.76465i	0.889916i	0.895552	+	0.444958i	$0.146781\pi$
$97$	8.76465i	0.889916i	−0.895552	+	0.444958i	$0.853219\pi$
$98$	0	0
$99$	21.6023	2.17111
$100$	0	0
$101$	4.74794	0.472438	0.236219	−	0.971700i	$-0.424092\pi$
$101$	4.74794	0.472438	0.236219	+	0.971700i	$0.424092\pi$
$102$	0	0
$103$	− 12.6304i	− 1.24451i	−0.782814	−	0.622255i	$-0.786217\pi$
$103$	− 12.6304i	− 1.24451i	0.782814	−	0.622255i	$-0.213783\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.9658i	− 1.25345i	−0.779239	−	0.626727i	$-0.784394\pi$
$107$	− 12.9658i	− 1.25345i	0.779239	−	0.626727i	$-0.215606\pi$
$108$	0	0
$109$	15.0040	1.43712	0.718562	−	0.695463i	$-0.244801\pi$
$109$	15.0040	1.43712	0.718562	+	0.695463i	$0.244801\pi$
$110$	0	0
$111$	1.16671	0.110739
$112$	0	0
$113$	− 2.13496i	− 0.200840i	−0.994945	−	0.100420i	$-0.967981\pi$
$113$	− 2.13496i	− 0.200840i	0.994945	−	0.100420i	$-0.0320187\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	56.5326i	5.22644i
$118$	0	0
$119$	2.35659	0.216029
$120$	0	0
$121$	−3.58454	−0.325867
$122$	0	0
$123$	− 14.3523i	− 1.29411i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 3.54181i	− 0.314285i	−0.987576	−	0.157142i	$-0.949772\pi$
$127$	− 3.54181i	− 0.314285i	0.987576	−	0.157142i	$-0.0502282\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	19.5499	1.70808	0.854042	−	0.520205i	$-0.174144\pi$
$131$	19.5499	1.70808	0.854042	+	0.520205i	$0.174144\pi$
$132$	0	0
$133$	19.1694i	1.66220i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.41544i	0.377236i	0.982051	+	0.188618i	$0.0604008\pi$
$137$	4.41544i	0.377236i	−0.982051	+	0.188618i	$0.939599\pi$
$138$	0	0
$139$	−21.9954	−1.86563	−0.932814	−	0.360358i	$-0.882655\pi$
$139$	−21.9954	−1.86563	−0.932814	+	0.360358i	$0.882655\pi$
$140$	0	0
$141$	−44.0554	−3.71013
$142$	0	0
$143$	19.4061i	1.62282i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.63828i	− 0.135123i
$148$	0	0
$149$	−19.0780	−1.56293	−0.781466	−	0.623947i	$-0.785528\pi$
$149$	−19.0780	−1.56293	−0.781466	+	0.623947i	$0.785528\pi$
$150$	0	0
$151$	−7.75273	−0.630908	−0.315454	−	0.948941i	$-0.602157\pi$
$151$	−7.75273	−0.630908	−0.315454	+	0.948941i	$0.602157\pi$
$152$	0	0
$153$	− 7.33006i	− 0.592600i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.5129i	1.39768i	0.715277	+	0.698841i	$0.246300\pi$
$157$	17.5129i	1.39768i	−0.715277	+	0.698841i	$0.753700\pi$
$158$	0	0
$159$	−13.2016	−1.04695
$160$	0	0
$161$	2.55040	0.200999
$162$	0	0
$163$	2.55491i	0.200116i	0.994982	+	0.100058i	$0.0319028\pi$
$163$	2.55491i	0.200116i	−0.994982	+	0.100058i	$0.968097\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.61301i	− 0.279583i	−0.990181	−	0.139791i	$-0.955357\pi$
$167$	− 3.61301i	− 0.279583i	0.990181	−	0.139791i	$-0.0446432\pi$
$168$	0	0
$169$	−37.7852	−2.90655
$170$	0	0
$171$	59.6253	4.55966
$172$	0	0
$173$	− 12.7824i	− 0.971827i	−0.874007	−	0.485913i	$-0.838487\pi$
$173$	− 12.7824i	− 0.971827i	0.874007	−	0.485913i	$-0.161513\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 12.6990i	− 0.954519i
$178$	0	0
$179$	−10.0149	−0.748546	−0.374273	−	0.927318i	$-0.622108\pi$
$179$	−10.0149	−0.748546	−0.374273	+	0.927318i	$0.622108\pi$
$180$	0	0
$181$	9.96248	0.740505	0.370252	−	0.928931i	$-0.379271\pi$
$181$	9.96248	0.740505	0.370252	+	0.928931i	$0.379271\pi$
$182$	0	0
$183$	30.2300i	2.23466i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.51621i	− 0.184003i
$188$	0	0
$189$	41.5983	3.02583
$190$	0	0
$191$	−6.61298	−0.478498	−0.239249	−	0.970958i	$-0.576901\pi$
$191$	−6.61298	−0.478498	−0.239249	+	0.970958i	$0.576901\pi$
$192$	0	0
$193$	19.0540i	1.37154i	0.727820	+	0.685768i	$0.240534\pi$
$193$	19.0540i	1.37154i	−0.727820	+	0.685768i	$0.759466\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.0886i	− 0.790030i	−0.918675	−	0.395015i	$-0.870739\pi$
$197$	− 11.0886i	− 0.790030i	0.918675	−	0.395015i	$-0.129261\pi$
$198$	0	0
$199$	−3.71377	−0.263263	−0.131631	−	0.991299i	$-0.542021\pi$
$199$	−3.71377	−0.263263	−0.131631	+	0.991299i	$0.542021\pi$
$200$	0	0
$201$	−10.4463	−0.736825
$202$	0	0
$203$	− 6.07627i	− 0.426471i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 7.93288i	− 0.551373i
$208$	0	0
$209$	20.4677	1.41578
$210$	0	0
$211$	−21.7656	−1.49841	−0.749204	−	0.662339i	$-0.769564\pi$
$211$	−21.7656	−1.49841	−0.749204	+	0.662339i	$0.769564\pi$
$212$	0	0
$213$	20.0996i	1.37720i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.20928i	− 0.149975i
$218$	0	0
$219$	37.4424	2.53012
$220$	0	0
$221$	6.58484	0.442944
$222$	0	0
$223$	− 6.90730i	− 0.462547i	−0.972889	−	0.231273i	$-0.925711\pi$
$223$	− 6.90730i	− 0.462547i	0.972889	−	0.231273i	$-0.0742892\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 23.0325i	− 1.52872i	−0.644791	−	0.764359i	$-0.723055\pi$
$227$	− 23.0325i	− 1.52872i	0.644791	−	0.764359i	$-0.276945\pi$
$228$	0	0
$229$	−12.9665	−0.856854	−0.428427	−	0.903576i	$-0.640932\pi$
$229$	−12.9665	−0.856854	−0.428427	+	0.903576i	$0.640932\pi$
$230$	0	0
$231$	22.9638	1.51091
$232$	0	0
$233$	4.80934i	0.315071i	0.987513	+	0.157535i	$0.0503548\pi$
$233$	4.80934i	0.315071i	−0.987513	+	0.157535i	$0.949645\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 39.8738i	− 2.59008i
$238$	0	0
$239$	20.4661	1.32384	0.661922	−	0.749573i	$-0.269741\pi$
$239$	20.4661	1.32384	0.661922	+	0.749573i	$0.269741\pi$
$240$	0	0
$241$	26.8657	1.73057	0.865287	−	0.501277i	$-0.167136\pi$
$241$	26.8657	1.73057	0.865287	+	0.501277i	$0.167136\pi$
$242$	0	0
$243$	− 50.6993i	− 3.25236i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	53.5635i	3.40816i
$248$	0	0
$249$	21.0057	1.33118
$250$	0	0
$251$	−15.9625	−1.00754	−0.503771	−	0.863837i	$-0.668055\pi$
$251$	−15.9625	−1.00754	−0.503771	+	0.863837i	$0.668055\pi$
$252$	0	0
$253$	− 2.72314i	− 0.171202i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.33867i	0.208261i	0.994564	+	0.104130i	$0.0332059\pi$
$257$	3.33867i	0.208261i	−0.994564	+	0.104130i	$0.966794\pi$
$258$	0	0
$259$	0.899919	0.0559183
$260$	0	0
$261$	−18.8999	−1.16988
$262$	0	0
$263$	23.3880i	1.44217i	0.692849	+	0.721083i	$0.256355\pi$
$263$	23.3880i	1.44217i	−0.692849	+	0.721083i	$0.743645\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 32.1185i	− 1.96562i
$268$	0	0
$269$	−14.1378	−0.861995	−0.430998	−	0.902353i	$-0.641838\pi$
$269$	−14.1378	−0.861995	−0.430998	+	0.902353i	$0.641838\pi$
$270$	0	0
$271$	25.9283	1.57503	0.787517	−	0.616293i	$-0.211366\pi$
$271$	25.9283	1.57503	0.787517	+	0.616293i	$0.211366\pi$
$272$	0	0
$273$	60.0957i	3.63716i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.07117i	0.0643603i	0.999482	+	0.0321802i	$0.0102450\pi$
$277$	1.07117i	0.0643603i	−0.999482	+	0.0321802i	$0.989755\pi$
$278$	0	0
$279$	−6.87184	−0.411406
$280$	0	0
$281$	12.2260	0.729341	0.364671	−	0.931137i	$-0.381182\pi$
$281$	12.2260	0.729341	0.364671	+	0.931137i	$0.381182\pi$
$282$	0	0
$283$	30.5454i	1.81573i	0.419259	+	0.907867i	$0.362290\pi$
$283$	30.5454i	1.81573i	−0.419259	+	0.907867i	$0.637710\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 11.0704i	− 0.653465i
$288$	0	0
$289$	16.1462	0.949777
$290$	0	0
$291$	28.9802	1.69885
$292$	0	0
$293$	8.46838i	0.494728i	0.968923	+	0.247364i	$0.0795644\pi$
$293$	8.46838i	0.494728i	−0.968923	+	0.247364i	$0.920436\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 44.4157i	− 2.57726i
$298$	0	0
$299$	7.12637	0.412129
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 15.6990i	− 0.901885i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.32802i	− 0.0757942i	−0.999282	−	0.0378971i	$-0.987934\pi$
$307$	− 1.32802i	− 0.0757942i	0.999282	−	0.0378971i	$-0.0120659\pi$
$308$	0	0
$309$	−41.7623	−2.37578
$310$	0	0
$311$	24.2971	1.37776	0.688882	−	0.724874i	$-0.258102\pi$
$311$	24.2971	1.37776	0.688882	+	0.724874i	$0.258102\pi$
$312$	0	0
$313$	− 20.3478i	− 1.15013i	−0.818109	−	0.575063i	$-0.804977\pi$
$313$	− 20.3478i	− 1.15013i	0.818109	−	0.575063i	$-0.195023\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.57309i	− 0.481513i	−0.970586	−	0.240756i	$-0.922604\pi$
$317$	− 8.57309i	− 0.481513i	0.970586	−	0.240756i	$-0.0773955\pi$
$318$	0	0
$319$	−6.48781	−0.363248
$320$	0	0
$321$	−42.8714	−2.39285
$322$	0	0
$323$	− 6.94508i	− 0.386434i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 49.6106i	− 2.74347i
$328$	0	0
$329$	−33.9813	−1.87345
$330$	0	0
$331$	−20.9767	−1.15299	−0.576493	−	0.817102i	$-0.695579\pi$
$331$	−20.9767	−1.15299	−0.576493	+	0.817102i	$0.695579\pi$
$332$	0	0
$333$	− 2.79915i	− 0.153393i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2.17314i	− 0.118379i	−0.998247	−	0.0591894i	$-0.981148\pi$
$337$	− 2.17314i	− 0.118379i	0.998247	−	0.0591894i	$-0.0188516\pi$
$338$	0	0
$339$	−7.05922	−0.383404
$340$	0	0
$341$	−2.35891	−0.127742
$342$	0	0
$343$	− 19.1164i	− 1.03219i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 32.1335i	− 1.72502i	−0.506042	−	0.862509i	$-0.668892\pi$
$347$	− 32.1335i	− 1.72502i	0.506042	−	0.862509i	$-0.331108\pi$
$348$	0	0
$349$	−13.5157	−0.723479	−0.361740	−	0.932279i	$-0.617817\pi$
$349$	−13.5157	−0.723479	−0.361740	+	0.932279i	$0.617817\pi$
$350$	0	0
$351$	116.235	6.20415
$352$	0	0
$353$	28.0645i	1.49372i	0.664981	+	0.746861i	$0.268440\pi$
$353$	28.0645i	1.49372i	−0.664981	+	0.746861i	$0.731560\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 7.79205i	− 0.412399i
$358$	0	0
$359$	−20.2340	−1.06791	−0.533956	−	0.845513i	$-0.679295\pi$
$359$	−20.2340	−1.06791	−0.533956	+	0.845513i	$0.679295\pi$
$360$	0	0
$361$	37.4937	1.97336
$362$	0	0
$363$	11.8522i	0.622081i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.76666i	0.248818i	0.992231	+	0.124409i	$0.0397034\pi$
$367$	4.76666i	0.248818i	−0.992231	+	0.124409i	$0.960297\pi$
$368$	0	0
$369$	−34.4339	−1.79256
$370$	0	0
$371$	−10.1828	−0.528663
$372$	0	0
$373$	29.1510i	1.50938i	0.656082	+	0.754690i	$0.272213\pi$
$373$	29.1510i	1.50938i	−0.656082	+	0.754690i	$0.727787\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 16.9784i	− 0.874434i
$378$	0	0
$379$	−4.20856	−0.216179	−0.108090	−	0.994141i	$-0.534473\pi$
$379$	−4.20856	−0.216179	−0.108090	+	0.994141i	$0.534473\pi$
$380$	0	0
$381$	−11.7110	−0.599971
$382$	0	0
$383$	− 3.55206i	− 0.181502i	−0.995874	−	0.0907508i	$-0.971073\pi$
$383$	− 3.55206i	− 0.181502i	0.995874	−	0.0907508i	$-0.0289267\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	29.2515	1.48311	0.741554	−	0.670893i	$-0.234089\pi$
$389$	29.2515	1.48311	0.741554	+	0.670893i	$0.234089\pi$
$390$	0	0
$391$	−0.924010	−0.0467292
$392$	0	0
$393$	− 64.6416i	− 3.26074i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 17.8000i	− 0.893356i	−0.894695	−	0.446678i	$-0.852607\pi$
$397$	− 17.8000i	− 0.893356i	0.894695	−	0.446678i	$-0.147393\pi$
$398$	0	0
$399$	63.3834	3.17314
$400$	0	0
$401$	−11.6210	−0.580327	−0.290164	−	0.956977i	$-0.593710\pi$
$401$	−11.6210	−0.580327	−0.290164	+	0.956977i	$0.593710\pi$
$402$	0	0
$403$	− 6.17320i	− 0.307509i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 0.960871i	− 0.0476286i
$408$	0	0
$409$	13.5710	0.671041	0.335521	−	0.942033i	$-0.391088\pi$
$409$	13.5710	0.671041	0.335521	+	0.942033i	$0.391088\pi$
$410$	0	0
$411$	14.5996	0.720145
$412$	0	0
$413$	− 9.79516i	− 0.481988i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	72.7277i	3.56149i
$418$	0	0
$419$	−13.4959	−0.659317	−0.329658	−	0.944100i	$-0.606934\pi$
$419$	−13.4959	−0.659317	−0.329658	+	0.944100i	$0.606934\pi$
$420$	0	0
$421$	37.6495	1.83492	0.917462	−	0.397824i	$-0.130234\pi$
$421$	37.6495	1.83492	0.917462	+	0.397824i	$0.130234\pi$
$422$	0	0
$423$	105.697i	5.13916i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	23.3173i	1.12840i
$428$	0	0
$429$	64.1660	3.09796
$430$	0	0
$431$	10.0568	0.484421	0.242210	−	0.970224i	$-0.422128\pi$
$431$	10.0568	0.484421	0.242210	+	0.970224i	$0.422128\pi$
$432$	0	0
$433$	− 2.30001i	− 0.110531i	−0.998472	−	0.0552657i	$-0.982399\pi$
$433$	− 2.30001i	− 0.110531i	0.998472	−	0.0552657i	$-0.0176006\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.51623i	− 0.359550i
$438$	0	0
$439$	−12.4273	−0.593124	−0.296562	−	0.955014i	$-0.595840\pi$
$439$	−12.4273	−0.593124	−0.296562	+	0.955014i	$0.595840\pi$
$440$	0	0
$441$	−3.93053	−0.187168
$442$	0	0
$443$	− 2.18268i	− 0.103702i	−0.998655	−	0.0518510i	$-0.983488\pi$
$443$	− 2.18268i	− 0.103702i	0.998655	−	0.0518510i	$-0.0165121\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	63.0813i	2.98364i
$448$	0	0
$449$	−3.09603	−0.146111	−0.0730554	−	0.997328i	$-0.523275\pi$
$449$	−3.09603	−0.146111	−0.0730554	+	0.997328i	$0.523275\pi$
$450$	0	0
$451$	−11.8202	−0.556591
$452$	0	0
$453$	25.6343i	1.20441i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 0.320363i	− 0.0149860i	−0.999972	−	0.00749298i	$-0.997615\pi$
$457$	− 0.320363i	− 0.0149860i	0.999972	−	0.00749298i	$-0.00238511\pi$
$458$	0	0
$459$	−15.0711	−0.703458
$460$	0	0
$461$	−15.3239	−0.713706	−0.356853	−	0.934161i	$-0.616150\pi$
$461$	−15.3239	−0.713706	−0.356853	+	0.934161i	$0.616150\pi$
$462$	0	0
$463$	− 6.81556i	− 0.316746i	−0.987379	−	0.158373i	$-0.949375\pi$
$463$	− 6.81556i	− 0.316746i	0.987379	−	0.158373i	$-0.0506248\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.2444i	1.26072i	0.776303	+	0.630360i	$0.217093\pi$
$467$	27.2444i	1.26072i	−0.776303	+	0.630360i	$0.782907\pi$
$468$	0	0
$469$	−8.05755	−0.372063
$470$	0	0
$471$	57.9062	2.66818
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	31.6730i	1.45021i
$478$	0	0
$479$	−23.5031	−1.07388	−0.536942	−	0.843619i	$-0.680421\pi$
$479$	−23.5031	−1.07388	−0.536942	+	0.843619i	$0.680421\pi$
$480$	0	0
$481$	2.51457	0.114655
$482$	0	0
$483$	− 8.43286i	− 0.383709i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.63058i	0.345774i	0.984942	+	0.172887i	$0.0553096\pi$
$487$	7.63058i	0.345774i	−0.984942	+	0.172887i	$0.944690\pi$
$488$	0	0
$489$	8.44779	0.382022
$490$	0	0
$491$	−19.1206	−0.862902	−0.431451	−	0.902136i	$-0.641998\pi$
$491$	−19.1206	−0.862902	−0.431451	+	0.902136i	$0.641998\pi$
$492$	0	0
$493$	2.20144i	0.0991477i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.5034i	0.695424i
$498$	0	0
$499$	25.6581	1.14861	0.574306	−	0.818641i	$-0.305272\pi$
$499$	25.6581	1.14861	0.574306	+	0.818641i	$0.305272\pi$
$500$	0	0
$501$	−11.9464	−0.533725
$502$	0	0
$503$	8.42282i	0.375555i	0.982212	+	0.187777i	$0.0601284\pi$
$503$	8.42282i	0.375555i	−0.982212	+	0.187777i	$0.939872\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	124.936i	5.54862i
$508$	0	0
$509$	−18.2328	−0.808153	−0.404076	−	0.914725i	$-0.632407\pi$
$509$	−18.2328	−0.808153	−0.404076	+	0.914725i	$0.632407\pi$
$510$	0	0
$511$	28.8805	1.27760
$512$	0	0
$513$	− 122.594i	− 5.41264i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	36.2828i	1.59572i
$518$	0	0
$519$	−42.2648	−1.85522
$520$	0	0
$521$	19.8738	0.870689	0.435345	−	0.900264i	$-0.356627\pi$
$521$	19.8738	0.870689	0.435345	+	0.900264i	$0.356627\pi$
$522$	0	0
$523$	19.7228i	0.862418i	0.902252	+	0.431209i	$0.141913\pi$
$523$	19.7228i	0.862418i	−0.902252	+	0.431209i	$0.858087\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.800422i	0.0348669i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−30.4673	−1.32217
$532$	0	0
$533$	− 30.9331i	− 1.33986i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	33.1141i	1.42898i
$538$	0	0
$539$	−1.34924	−0.0581160
$540$	0	0
$541$	−16.4662	−0.707938	−0.353969	−	0.935257i	$-0.615168\pi$
$541$	−16.4662	−0.707938	−0.353969	+	0.935257i	$0.615168\pi$
$542$	0	0
$543$	− 32.9408i	− 1.41363i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 38.3286i	− 1.63881i	−0.573215	−	0.819405i	$-0.694304\pi$
$547$	− 38.3286i	− 1.63881i	0.573215	−	0.819405i	$-0.305696\pi$
$548$	0	0
$549$	72.5273	3.09539
$550$	0	0
$551$	−17.9073	−0.762875
$552$	0	0
$553$	− 30.7559i	− 1.30787i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 20.3973i	− 0.864263i	−0.901811	−	0.432132i	$-0.857762\pi$
$557$	− 20.3973i	− 0.864263i	0.901811	−	0.432132i	$-0.142238\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−8.31981	−0.351263
$562$	0	0
$563$	1.72270i	0.0726032i	0.999341	+	0.0363016i	$0.0115577\pi$
$563$	1.72270i	0.0726032i	−0.999341	+	0.0363016i	$0.988442\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 76.8483i	− 3.22733i
$568$	0	0
$569$	−25.3283	−1.06182	−0.530909	−	0.847429i	$-0.678150\pi$
$569$	−25.3283	−1.06182	−0.530909	+	0.847429i	$0.678150\pi$
$570$	0	0
$571$	−5.68968	−0.238106	−0.119053	−	0.992888i	$-0.537986\pi$
$571$	−5.68968	−0.238106	−0.119053	+	0.992888i	$0.537986\pi$
$572$	0	0
$573$	21.8658i	0.913455i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 18.0377i	− 0.750918i	−0.926839	−	0.375459i	$-0.877485\pi$
$577$	− 18.0377i	− 0.750918i	0.926839	−	0.375459i	$-0.122515\pi$
$578$	0	0
$579$	63.0019	2.61827
$580$	0	0
$581$	16.2023	0.672185
$582$	0	0
$583$	10.8724i	0.450291i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.1894i	0.544383i	0.962243	+	0.272192i	$0.0877485\pi$
$587$	13.1894i	0.544383i	−0.962243	+	0.272192i	$0.912252\pi$
$588$	0	0
$589$	−6.51092	−0.268278
$590$	0	0
$591$	−36.6643	−1.50817
$592$	0	0
$593$	0.326755i	0.0134182i	0.999977	+	0.00670911i	$0.00213559\pi$
$593$	0.326755i	0.0134182i	−0.999977	+	0.00670911i	$0.997864\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.2796i	0.502569i
$598$	0	0
$599$	7.49619	0.306286	0.153143	−	0.988204i	$-0.451061\pi$
$599$	7.49619	0.306286	0.153143	+	0.988204i	$0.451061\pi$
$600$	0	0
$601$	0.121724	0.00496523	0.00248262	−	0.999997i	$-0.499210\pi$
$601$	0.121724	0.00496523	0.00248262	+	0.999997i	$0.499210\pi$
$602$	0	0
$603$	25.0626i	1.02063i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.52992i	0.0620975i	0.999518	+	0.0310488i	$0.00988471\pi$
$607$	1.52992i	0.0620975i	−0.999518	+	0.0310488i	$0.990115\pi$
$608$	0	0
$609$	−20.0911	−0.814134
$610$	0	0
$611$	−94.9512	−3.84131
$612$	0	0
$613$	− 24.3927i	− 0.985211i	−0.870253	−	0.492605i	$-0.836045\pi$
$613$	− 24.3927i	− 0.985211i	0.870253	−	0.492605i	$-0.163955\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.8906i	1.12283i	0.827534	+	0.561416i	$0.189743\pi$
$617$	27.8906i	1.12283i	−0.827534	+	0.561416i	$0.810257\pi$
$618$	0	0
$619$	−12.8617	−0.516957	−0.258478	−	0.966017i	$-0.583221\pi$
$619$	−12.8617	−0.516957	−0.258478	+	0.966017i	$0.583221\pi$
$620$	0	0
$621$	−16.3105	−0.654518
$622$	0	0
$623$	− 24.7740i	− 0.992549i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 67.6763i	− 2.70273i
$628$	0	0
$629$	−0.326041	−0.0130001
$630$	0	0
$631$	−11.9518	−0.475794	−0.237897	−	0.971290i	$-0.576458\pi$
$631$	−11.9518	−0.475794	−0.237897	+	0.971290i	$0.576458\pi$
$632$	0	0
$633$	71.9679i	2.86047i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.53093i	− 0.139901i
$638$	0	0
$639$	48.2226	1.90766
$640$	0	0
$641$	−23.9139	−0.944544	−0.472272	−	0.881453i	$-0.656566\pi$
$641$	−23.9139	−0.944544	−0.472272	+	0.881453i	$0.656566\pi$
$642$	0	0
$643$	− 3.87313i	− 0.152741i	−0.997079	−	0.0763707i	$-0.975667\pi$
$643$	− 3.87313i	− 0.152741i	0.997079	−	0.0763707i	$-0.0243333\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 10.3330i	− 0.406231i	−0.979155	−	0.203115i	$-0.934893\pi$
$647$	− 10.3330i	− 0.406231i	0.979155	−	0.203115i	$-0.0651067\pi$
$648$	0	0
$649$	−10.4586	−0.410535
$650$	0	0
$651$	−7.30495	−0.286303
$652$	0	0
$653$	− 21.6319i	− 0.846522i	−0.906008	−	0.423261i	$-0.860885\pi$
$653$	− 21.6319i	− 0.846522i	0.906008	−	0.423261i	$-0.139115\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 89.8312i	− 3.50465i
$658$	0	0
$659$	−15.7168	−0.612238	−0.306119	−	0.951993i	$-0.599030\pi$
$659$	−15.7168	−0.612238	−0.306119	+	0.951993i	$0.599030\pi$
$660$	0	0
$661$	−40.3002	−1.56750	−0.783749	−	0.621078i	$-0.786695\pi$
$661$	−40.3002	−1.56750	−0.783749	+	0.621078i	$0.786695\pi$
$662$	0	0
$663$	− 21.7727i	− 0.845582i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.38248i	0.0922500i
$668$	0	0
$669$	−22.8389	−0.883003
$670$	0	0
$671$	24.8966	0.961122
$672$	0	0
$673$	6.69888i	0.258223i	0.991630	+	0.129111i	$0.0412125\pi$
$673$	6.69888i	0.258223i	−0.991630	+	0.129111i	$0.958788\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 22.5812i	− 0.867866i	−0.900945	−	0.433933i	$-0.857125\pi$
$677$	− 22.5812i	− 0.867866i	0.900945	−	0.433933i	$-0.142875\pi$
$678$	0	0
$679$	22.3533	0.857843
$680$	0	0
$681$	−76.1566	−2.91833
$682$	0	0
$683$	1.42797i	0.0546398i	0.999627	+	0.0273199i	$0.00869728\pi$
$683$	1.42797i	0.0546398i	−0.999627	+	0.0273199i	$0.991303\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	42.8738i	1.63574i
$688$	0	0
$689$	−28.4529	−1.08397
$690$	0	0
$691$	−1.50255	−0.0571595	−0.0285798	−	0.999592i	$-0.509098\pi$
$691$	−1.50255	−0.0571595	−0.0285798	+	0.999592i	$0.509098\pi$
$692$	0	0
$693$	− 55.0944i	− 2.09286i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.01081i	0.151920i
$698$	0	0
$699$	15.9020	0.601471
$700$	0	0
$701$	18.7341	0.707577	0.353789	−	0.935325i	$-0.384893\pi$
$701$	18.7341	0.707577	0.353789	+	0.935325i	$0.384893\pi$
$702$	0	0
$703$	− 2.65214i	− 0.100027i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 12.1091i	− 0.455411i
$708$	0	0
$709$	13.5052	0.507199	0.253599	−	0.967309i	$-0.418385\pi$
$709$	13.5052	0.507199	0.253599	+	0.967309i	$0.418385\pi$
$710$	0	0
$711$	−95.6646	−3.58770
$712$	0	0
$713$	0.866248i	0.0324412i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 67.6711i	− 2.52722i
$718$	0	0
$719$	−31.7496	−1.18406	−0.592030	−	0.805916i	$-0.701673\pi$
$719$	−31.7496	−1.18406	−0.592030	+	0.805916i	$0.701673\pi$
$720$	0	0
$721$	−32.2126	−1.19966
$722$	0	0
$723$	− 88.8313i	− 3.30367i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21.6531i	0.803070i	0.915844	+	0.401535i	$0.131523\pi$
$727$	21.6531i	0.803070i	−0.915844	+	0.401535i	$0.868477\pi$
$728$	0	0
$729$	−77.2411	−2.86078
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 0.640507i	− 0.0236577i	−0.999930	−	0.0118288i	$-0.996235\pi$
$733$	− 0.640507i	− 0.0236577i	0.999930	−	0.0118288i	$-0.00376533\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.60329i	0.316906i
$738$	0	0
$739$	−16.8191	−0.618700	−0.309350	−	0.950948i	$-0.600111\pi$
$739$	−16.8191	−0.618700	−0.309350	+	0.950948i	$0.600111\pi$
$740$	0	0
$741$	177.107	6.50619
$742$	0	0
$743$	15.9691i	0.585851i	0.956135	+	0.292925i	$0.0946288\pi$
$743$	15.9691i	0.585851i	−0.956135	+	0.292925i	$0.905371\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 50.3964i	− 1.84391i
$748$	0	0
$749$	−33.0680	−1.20828
$750$	0	0
$751$	−8.09841	−0.295515	−0.147758	−	0.989024i	$-0.547206\pi$
$751$	−8.09841	−0.295515	−0.147758	+	0.989024i	$0.547206\pi$
$752$	0	0
$753$	52.7798i	1.92340i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.5691i	0.747598i	0.927510	+	0.373799i	$0.121945\pi$
$757$	20.5691i	0.747598i	−0.927510	+	0.373799i	$0.878055\pi$
$758$	0	0
$759$	−9.00402	−0.326825
$760$	0	0
$761$	−17.1840	−0.622919	−0.311459	−	0.950259i	$-0.600818\pi$
$761$	−17.1840	−0.622919	−0.311459	+	0.950259i	$0.600818\pi$
$762$	0	0
$763$	− 38.2662i	− 1.38533i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 27.3698i	− 0.988268i
$768$	0	0
$769$	−41.8648	−1.50968	−0.754841	−	0.655908i	$-0.772286\pi$
$769$	−41.8648	−1.50968	−0.754841	+	0.655908i	$0.772286\pi$
$770$	0	0
$771$	11.0393	0.397570
$772$	0	0
$773$	35.7639i	1.28634i	0.765724	+	0.643170i	$0.222381\pi$
$773$	35.7639i	1.28634i	−0.765724	+	0.643170i	$0.777619\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.97557i	− 0.106748i
$778$	0	0
$779$	−32.6254	−1.16893
$780$	0	0
$781$	16.5535	0.592330
$782$	0	0
$783$	38.8595i	1.38872i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.9835i	0.961858i	0.876759	+	0.480929i	$0.159701\pi$
$787$	26.9835i	0.961858i	−0.876759	+	0.480929i	$0.840299\pi$
$788$	0	0
$789$	77.3321	2.75310
$790$	0	0
$791$	−5.44500	−0.193602
$792$	0	0
$793$	65.1537i	2.31368i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.7226i	0.946561i	0.880912	+	0.473281i	$0.156930\pi$
$797$	26.7226i	0.946561i	−0.880912	+	0.473281i	$0.843070\pi$
$798$	0	0
$799$	12.3114	0.435547
$800$	0	0
$801$	−77.0582	−2.72272
$802$	0	0
$803$	− 30.8366i	− 1.08820i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	46.7464i	1.64555i
$808$	0	0
$809$	−50.0319	−1.75903	−0.879513	−	0.475874i	$-0.842132\pi$
$809$	−50.0319	−1.75903	−0.879513	+	0.475874i	$0.842132\pi$
$810$	0	0
$811$	33.5774	1.17906	0.589531	−	0.807746i	$-0.299313\pi$
$811$	33.5774	1.17906	0.589531	+	0.807746i	$0.299313\pi$
$812$	0	0
$813$	− 85.7318i	− 3.00675i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	144.181	5.03808
$820$	0	0
$821$	27.1429	0.947295	0.473647	−	0.880715i	$-0.342937\pi$
$821$	27.1429	0.947295	0.473647	+	0.880715i	$0.342937\pi$
$822$	0	0
$823$	− 32.6392i	− 1.13773i	−0.822431	−	0.568866i	$-0.807382\pi$
$823$	− 32.6392i	− 1.13773i	0.822431	−	0.568866i	$-0.192618\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 37.0828i	− 1.28949i	−0.764396	−	0.644747i	$-0.776963\pi$
$827$	− 37.0828i	− 1.28949i	0.764396	−	0.644747i	$-0.223037\pi$
$828$	0	0
$829$	−33.9405	−1.17880	−0.589401	−	0.807841i	$-0.700636\pi$
$829$	−33.9405	−1.17880	−0.589401	+	0.807841i	$0.700636\pi$
$830$	0	0
$831$	3.54181	0.122864
$832$	0	0
$833$	0.457823i	0.0158626i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	14.1289i	0.488368i
$838$	0	0
$839$	1.36929	0.0472730	0.0236365	−	0.999721i	$-0.492476\pi$
$839$	1.36929	0.0472730	0.0236365	+	0.999721i	$0.492476\pi$
$840$	0	0
$841$	−23.3238	−0.804269
$842$	0	0
$843$	− 40.4251i	− 1.39231i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.14199i	0.314122i
$848$	0	0
$849$	100.998	3.46624
$850$	0	0
$851$	−0.352855	−0.0120957
$852$	0	0
$853$	− 9.21382i	− 0.315475i	−0.987481	−	0.157738i	$-0.949580\pi$
$853$	− 9.21382i	− 0.315475i	0.987481	−	0.157738i	$-0.0504200\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.3564i	0.490405i	0.969472	+	0.245202i	$0.0788544\pi$
$857$	14.3564i	0.490405i	−0.969472	+	0.245202i	$0.921146\pi$
$858$	0	0
$859$	−31.3758	−1.07053	−0.535264	−	0.844685i	$-0.679788\pi$
$859$	−31.3758	−1.07053	−0.535264	+	0.844685i	$0.679788\pi$
$860$	0	0
$861$	−36.6042	−1.24747
$862$	0	0
$863$	− 26.1214i	− 0.889182i	−0.895734	−	0.444591i	$-0.853349\pi$
$863$	− 26.1214i	− 0.889182i	0.895734	−	0.444591i	$-0.146651\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 53.3873i	− 1.81313i
$868$	0	0
$869$	−32.8390	−1.11399
$870$	0	0
$871$	−22.5146	−0.762877
$872$	0	0
$873$	− 69.5289i	− 2.35320i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 11.5530i	− 0.390118i	−0.980792	−	0.195059i	$-0.937510\pi$
$877$	− 11.5530i	− 0.390118i	0.980792	−	0.195059i	$-0.0624898\pi$
$878$	0	0
$879$	28.0006	0.944437
$880$	0	0
$881$	−34.3559	−1.15748	−0.578740	−	0.815512i	$-0.696456\pi$
$881$	−34.3559	−1.15748	−0.578740	+	0.815512i	$0.696456\pi$
$882$	0	0
$883$	− 12.8316i	− 0.431818i	−0.976414	−	0.215909i	$-0.930729\pi$
$883$	− 12.8316i	− 0.431818i	0.976414	−	0.215909i	$-0.0692714\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.46724i	− 0.0492652i	−0.999697	−	0.0246326i	$-0.992158\pi$
$887$	− 1.46724i	− 0.0492652i	0.999697	−	0.0246326i	$-0.00784159\pi$
$888$	0	0
$889$	−9.03303	−0.302958
$890$	0	0
$891$	−82.0533	−2.74889
$892$	0	0
$893$	100.146i	3.35125i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 23.5633i	− 0.786755i
$898$	0	0
$899$	2.06382	0.0688322
$900$	0	0
$901$	3.68922	0.122906
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.44558i	0.147613i	0.997273	+	0.0738066i	$0.0235147\pi$
$907$	4.44558i	0.147613i	−0.997273	+	0.0738066i	$0.976485\pi$
$908$	0	0
$909$	−37.6648	−1.24926
$910$	0	0
$911$	40.1697	1.33088	0.665440	−	0.746451i	$-0.268244\pi$
$911$	40.1697	1.33088	0.665440	+	0.746451i	$0.268244\pi$
$912$	0	0
$913$	− 17.2997i	− 0.572536i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 49.8600i	− 1.64652i
$918$	0	0
$919$	−46.8063	−1.54400	−0.771999	−	0.635624i	$-0.780743\pi$
$919$	−46.8063	−1.54400	−0.771999	+	0.635624i	$0.780743\pi$
$920$	0	0
$921$	−4.39109	−0.144691
$922$	0	0
$923$	43.3200i	1.42590i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	100.195i	3.29085i
$928$	0	0
$929$	−8.92145	−0.292703	−0.146352	−	0.989233i	$-0.546753\pi$
$929$	−8.92145	−0.292703	−0.146352	+	0.989233i	$0.546753\pi$
$930$	0	0
$931$	−3.72409	−0.122052
$932$	0	0
$933$	− 80.3382i	− 2.63016i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.70070i	0.251571i	0.992057	+	0.125785i	$0.0401451\pi$
$937$	7.70070i	0.251571i	−0.992057	+	0.125785i	$0.959855\pi$
$938$	0	0
$939$	−67.2799	−2.19560
$940$	0	0
$941$	−16.8979	−0.550856	−0.275428	−	0.961322i	$-0.588820\pi$
$941$	−16.8979	−0.550856	−0.275428	+	0.961322i	$0.588820\pi$
$942$	0	0
$943$	4.34066i	0.141351i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 43.7150i	− 1.42055i	−0.703927	−	0.710273i	$-0.748572\pi$
$947$	− 43.7150i	− 1.42055i	0.703927	−	0.710273i	$-0.251428\pi$
$948$	0	0
$949$	80.6985	2.61958
$950$	0	0
$951$	−28.3469	−0.919210
$952$	0	0
$953$	− 30.2353i	− 0.979418i	−0.871886	−	0.489709i	$-0.837103\pi$
$953$	− 30.2353i	− 0.979418i	0.871886	−	0.489709i	$-0.162897\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	21.4519i	0.693441i
$958$	0	0
$959$	11.2611	0.363641
$960$	0	0
$961$	−30.2496	−0.975794
$962$	0	0
$963$	102.856i	3.31450i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16.2727i	0.523296i	0.965163	+	0.261648i	$0.0842659\pi$
$967$	16.2727i	0.523296i	−0.965163	+	0.261648i	$0.915734\pi$
$968$	0	0
$969$	−22.9638	−0.737704
$970$	0	0
$971$	6.21424	0.199424	0.0997122	−	0.995016i	$-0.468208\pi$
$971$	6.21424	0.199424	0.0997122	+	0.995016i	$0.468208\pi$
$972$	0	0
$973$	56.0971i	1.79839i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.6705i	0.821271i	0.911800	+	0.410636i	$0.134693\pi$
$977$	25.6705i	0.821271i	−0.911800	+	0.410636i	$0.865307\pi$
$978$	0	0
$979$	−26.4519	−0.845407
$980$	0	0
$981$	−119.025	−3.80018
$982$	0	0
$983$	30.7124i	0.979574i	0.871842	+	0.489787i	$0.162925\pi$
$983$	30.7124i	0.979574i	−0.871842	+	0.489787i	$0.837075\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	112.359i	3.57642i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	29.4052	0.934086	0.467043	−	0.884235i	$-0.345319\pi$
$991$	29.4052	0.934086	0.467043	+	0.884235i	$0.345319\pi$
$992$	0	0
$993$	69.3594i	2.20105i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.59825i	0.177298i	0.996063	+	0.0886492i	$0.0282550\pi$
$997$	5.59825i	0.177298i	−0.996063	+	0.0886492i	$0.971745\pi$
$998$	0	0
$999$	−5.75524	−0.182088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.u.4049.2		10
5.2	odd	4		4600.2.a.be.1.1		5
5.3	odd	4		920.2.a.j.1.5	✓	5
5.4	even	2	inner	4600.2.e.u.4049.9		10
15.8	even	4		8280.2.a.bs.1.2		5
20.3	even	4		1840.2.a.v.1.1		5
20.7	even	4		9200.2.a.cu.1.5		5
40.3	even	4		7360.2.a.cp.1.5		5
40.13	odd	4		7360.2.a.co.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.5	✓	5	5.3	odd	4
1840.2.a.v.1.1		5	20.3	even	4
4600.2.a.be.1.1		5	5.2	odd	4
4600.2.e.u.4049.2		10	1.1	even	1	trivial
4600.2.e.u.4049.9		10	5.4	even	2	inner
7360.2.a.co.1.1		5	40.13	odd	4
7360.2.a.cp.1.5		5	40.3	even	4
8280.2.a.bs.1.2		5	15.8	even	4
9200.2.a.cu.1.5		5	20.7	even	4

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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.30649i q^{3} -2.55040i q^{7} -7.93288 q^{9} +O(q^{10})\)	\(q-3.30649i q^{3} -2.55040i q^{7} -7.93288 q^{9} -2.72314 q^{11} -7.12637i q^{13} +0.924010i q^{17} -7.51623 q^{19} -8.43286 q^{21} +1.00000i q^{23} +16.3105i q^{27} +2.38248 q^{29} +0.866248 q^{31} +9.00402i q^{33} +0.352855i q^{37} -23.5633 q^{39} +4.34066 q^{41} -13.3239i q^{47} +0.495474 q^{49} +3.05523 q^{51} -3.99262i q^{53} +24.8523i q^{57} +3.84064 q^{59} -9.14262 q^{61} +20.2320i q^{63} -3.15933i q^{67} +3.30649 q^{69} -6.07883 q^{71} +11.3239i q^{73} +6.94508i q^{77} +12.0593 q^{79} +30.1319 q^{81} +6.35285i q^{83} -7.87765i q^{87} +9.71377 q^{89} -18.1751 q^{91} -2.86424i q^{93} +8.76465i q^{97} +21.6023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 26 q^{9}+O(q^{10}) \) 10 * q - 26 * q^9	\( 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99}+O(q^{100}) \) 10 * q - 26 * q^9 - 2 * q^11 - 14 * q^19 + 12 * q^21 - 8 * q^29 + 38 * q^31 - 38 * q^39 + 50 * q^41 - 50 * q^49 + 38 * q^51 + 2 * q^59 - 10 * q^61 + 2 * q^71 + 4 * q^79 + 114 * q^81 - 12 * q^89 + 22 * q^91 + 130 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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