LMFDB - Embedded newform 2312.2.b.o.577.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,2,Mod(577,2312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2312.577"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,-12,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.4614129473$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 18x^{10} + 117x^{8} + 342x^{6} + 438x^{4} + 180x^{2} + 1 $ x^12 + 18x^10 + 117x^8 + 342x^6 + 438x^4 + 180*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.6
Root			$-0.0750494i$ of defining polynomial
Character	$\chi$	$=$	2312.577
Dual form			2312.2.b.o.577.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0750494i q^{3} +2.06942i q^{5} -2.86317i q^{7} +2.99437 q^{9} +1.65231i q^{11} -3.96428 q^{13} +0.155308 q^{15} +5.57295 q^{19} -0.214879 q^{21} +5.28027i q^{23} +0.717513 q^{25} -0.449874i q^{27} -6.99818i q^{29} -6.39002i q^{31} +0.124005 q^{33} +5.92509 q^{35} +5.70355i q^{37} +0.297517i q^{39} +9.87650i q^{41} +6.67809 q^{43} +6.19660i q^{45} +4.43037 q^{47} -1.19773 q^{49} -8.27111 q^{53} -3.41932 q^{55} -0.418247i q^{57} +2.42912 q^{59} +1.81849i q^{61} -8.57338i q^{63} -8.20375i q^{65} +15.8315 q^{67} +0.396281 q^{69} +8.54703i q^{71} -2.30202i q^{73} -0.0538489i q^{75} +4.73084 q^{77} -14.0262i q^{79} +8.94934 q^{81} +1.00763 q^{83} -0.525209 q^{87} -16.2569 q^{89} +11.3504i q^{91} -0.479567 q^{93} +11.5328i q^{95} -13.8424i q^{97} +4.94762i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{13} + 12 q^{15} - 12 q^{19} + 12 q^{21} - 12 q^{25} + 6 q^{33} + 42 q^{35} + 6 q^{47} - 18 q^{49} - 66 q^{53} - 102 q^{55} - 18 q^{67} - 6 q^{69} + 90 q^{77} - 36 q^{81} + 24 q^{83} - 30 q^{87}+ \cdots + 60 q^{93}+O(q^{100}) $ 12 * q - 12 * q^13 + 12 * q^15 - 12 * q^19 + 12 * q^21 - 12 * q^25 + 6 * q^33 + 42 * q^35 + 6 * q^47 - 18 * q^49 - 66 * q^53 - 102 * q^55 - 18 * q^67 - 6 * q^69 + 90 * q^77 - 36 * q^81 + 24 * q^83 - 30 * q^87 + 6 * q^89 + 60 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 0.0750494i	− 0.0433298i	−0.999765	−	0.0216649i	$-0.993103\pi$
$3$	− 0.0750494i	− 0.0433298i	0.999765	−	0.0216649i	$-0.00689669\pi$
$4$	0	0
$5$	2.06942i	0.925471i	0.886496	+	0.462736i	$0.153132\pi$
$5$	2.06942i	0.925471i	−0.886496	+	0.462736i	$0.846868\pi$
$6$	0	0
$7$	− 2.86317i	− 1.08218i	−0.840966	−	0.541088i	$-0.818013\pi$
$7$	− 2.86317i	− 1.08218i	0.840966	−	0.541088i	$-0.181987\pi$
$8$	0	0
$9$	2.99437	0.998123
$10$	0	0
$11$	1.65231i	0.498190i	0.968479	+	0.249095i	$0.0801332\pi$
$11$	1.65231i	0.498190i	−0.968479	+	0.249095i	$0.919867\pi$
$12$	0	0
$13$	−3.96428	−1.09949	−0.549747	−	0.835331i	$-0.685276\pi$
$13$	−3.96428	−1.09949	−0.549747	+	0.835331i	$0.685276\pi$
$14$	0	0
$15$	0.155308	0.0401005
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	5.57295	1.27852	0.639262	−	0.768989i	$-0.279240\pi$
$19$	5.57295	1.27852	0.639262	+	0.768989i	$0.279240\pi$
$20$	0	0
$21$	−0.214879	−0.0468904
$22$	0	0
$23$	5.28027i	1.10101i	0.834831	+	0.550507i	$0.185565\pi$
$23$	5.28027i	1.10101i	−0.834831	+	0.550507i	$0.814435\pi$
$24$	0	0
$25$	0.717513	0.143503
$26$	0	0
$27$	− 0.449874i	− 0.0865782i
$28$	0	0
$29$	− 6.99818i	− 1.29953i	−0.760135	−	0.649765i	$-0.774867\pi$
$29$	− 6.99818i	− 1.29953i	0.760135	−	0.649765i	$-0.225133\pi$
$30$	0	0
$31$	− 6.39002i	− 1.14768i	−0.818967	−	0.573840i	$-0.805453\pi$
$31$	− 6.39002i	− 1.14768i	0.818967	−	0.573840i	$-0.194547\pi$
$32$	0	0
$33$	0.124005	0.0215865
$34$	0	0
$35$	5.92509	1.00152
$36$	0	0
$37$	5.70355i	0.937658i	0.883289	+	0.468829i	$0.155324\pi$
$37$	5.70355i	0.937658i	−0.883289	+	0.468829i	$0.844676\pi$
$38$	0	0
$39$	0.297517i	0.0476408i
$40$	0	0
$41$	9.87650i	1.54245i	0.636562	+	0.771225i	$0.280356\pi$
$41$	9.87650i	1.54245i	−0.636562	+	0.771225i	$0.719644\pi$
$42$	0	0
$43$	6.67809	1.01840	0.509199	−	0.860649i	$-0.329942\pi$
$43$	6.67809	1.01840	0.509199	+	0.860649i	$0.329942\pi$
$44$	0	0
$45$	6.19660i	0.923734i
$46$	0	0
$47$	4.43037	0.646236	0.323118	−	0.946359i	$-0.395269\pi$
$47$	4.43037	0.646236	0.323118	+	0.946359i	$0.395269\pi$
$48$	0	0
$49$	−1.19773	−0.171104
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.27111	−1.13612	−0.568062	−	0.822985i	$-0.692307\pi$
$53$	−8.27111	−1.13612	−0.568062	+	0.822985i	$0.692307\pi$
$54$	0	0
$55$	−3.41932	−0.461061
$56$	0	0
$57$	− 0.418247i	− 0.0553981i
$58$	0	0
$59$	2.42912	0.316244	0.158122	−	0.987420i	$-0.449456\pi$
$59$	2.42912	0.316244	0.158122	+	0.987420i	$0.449456\pi$
$60$	0	0
$61$	1.81849i	0.232834i	0.993200	+	0.116417i	$0.0371409\pi$
$61$	1.81849i	0.232834i	−0.993200	+	0.116417i	$0.962859\pi$
$62$	0	0
$63$	− 8.57338i	− 1.08014i
$64$	0	0
$65$	− 8.20375i	− 1.01755i
$66$	0	0
$67$	15.8315	1.93413	0.967064	−	0.254535i	$-0.0819224\pi$
$67$	15.8315	1.93413	0.967064	+	0.254535i	$0.0819224\pi$
$68$	0	0
$69$	0.396281	0.0477067
$70$	0	0
$71$	8.54703i	1.01435i	0.861844	+	0.507173i	$0.169309\pi$
$71$	8.54703i	1.01435i	−0.861844	+	0.507173i	$0.830691\pi$
$72$	0	0
$73$	− 2.30202i	− 0.269431i	−0.990884	−	0.134715i	$-0.956988\pi$
$73$	− 2.30202i	− 0.269431i	0.990884	−	0.134715i	$-0.0430120\pi$
$74$	0	0
$75$	− 0.0538489i	− 0.00621794i
$76$	0	0
$77$	4.73084	0.539129
$78$	0	0
$79$	− 14.0262i	− 1.57807i	−0.614351	−	0.789033i	$-0.710582\pi$
$79$	− 14.0262i	− 1.57807i	0.614351	−	0.789033i	$-0.289418\pi$
$80$	0	0
$81$	8.94934	0.994371
$82$	0	0
$83$	1.00763	0.110601	0.0553006	−	0.998470i	$-0.482388\pi$
$83$	1.00763	0.110601	0.0553006	+	0.998470i	$0.482388\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.525209	−0.0563083
$88$	0	0
$89$	−16.2569	−1.72323	−0.861616	−	0.507561i	$-0.830547\pi$
$89$	−16.2569	−1.72323	−0.861616	+	0.507561i	$0.830547\pi$
$90$	0	0
$91$	11.3504i	1.18985i
$92$	0	0
$93$	−0.479567	−0.0497288
$94$	0	0
$95$	11.5328i	1.18324i
$96$	0	0
$97$	− 13.8424i	− 1.40549i	−0.711443	−	0.702744i	$-0.751958\pi$
$97$	− 13.8424i	− 1.40549i	0.711443	−	0.702744i	$-0.248042\pi$
$98$	0	0
$99$	4.94762i	0.497255i
$100$	0	0
$101$	14.9508	1.48766	0.743832	−	0.668366i	$-0.233006\pi$
$101$	14.9508	1.48766	0.743832	+	0.668366i	$0.233006\pi$
$102$	0	0
$103$	16.1276	1.58910	0.794552	−	0.607196i	$-0.207706\pi$
$103$	16.1276	1.58910	0.794552	+	0.607196i	$0.207706\pi$
$104$	0	0
$105$	− 0.444674i	− 0.0433958i
$106$	0	0
$107$	14.7416i	1.42513i	0.701608	+	0.712563i	$0.252466\pi$
$107$	14.7416i	1.42513i	−0.701608	+	0.712563i	$0.747534\pi$
$108$	0	0
$109$	11.4726i	1.09887i	0.835535	+	0.549437i	$0.185158\pi$
$109$	11.4726i	1.09887i	−0.835535	+	0.549437i	$0.814842\pi$
$110$	0	0
$111$	0.428048	0.0406285
$112$	0	0
$113$	3.33361i	0.313599i	0.987630	+	0.156800i	$0.0501177\pi$
$113$	3.33361i	0.313599i	−0.987630	+	0.156800i	$0.949882\pi$
$114$	0	0
$115$	−10.9271	−1.01896
$116$	0	0
$117$	−11.8705	−1.09743
$118$	0	0
$119$	0	0
$120$	0	0
$121$	8.26987	0.751806
$122$	0	0
$123$	0.741226	0.0668341
$124$	0	0
$125$	11.8319i	1.05828i
$126$	0	0
$127$	−19.8464	−1.76109	−0.880544	−	0.473965i	$-0.842822\pi$
$127$	−19.8464	−1.76109	−0.880544	+	0.473965i	$0.842822\pi$
$128$	0	0
$129$	− 0.501187i	− 0.0441270i
$130$	0	0
$131$	14.8864i	1.30063i	0.759664	+	0.650316i	$0.225364\pi$
$131$	14.8864i	1.30063i	−0.759664	+	0.650316i	$0.774636\pi$
$132$	0	0
$133$	− 15.9563i	− 1.38359i
$134$	0	0
$135$	0.930976	0.0801257
$136$	0	0
$137$	−2.48499	−0.212307	−0.106154	−	0.994350i	$-0.533853\pi$
$137$	−2.48499	−0.212307	−0.106154	+	0.994350i	$0.533853\pi$
$138$	0	0
$139$	3.28018i	0.278221i	0.990277	+	0.139111i	$0.0444244\pi$
$139$	3.28018i	0.278221i	−0.990277	+	0.139111i	$0.955576\pi$
$140$	0	0
$141$	− 0.332497i	− 0.0280013i
$142$	0	0
$143$	− 6.55022i	− 0.547757i
$144$	0	0
$145$	14.4822	1.20268
$146$	0	0
$147$	0.0898887i	0.00741390i
$148$	0	0
$149$	8.18480	0.670525	0.335262	−	0.942125i	$-0.391175\pi$
$149$	8.18480	0.670525	0.335262	+	0.942125i	$0.391175\pi$
$150$	0	0
$151$	15.8084	1.28647	0.643234	−	0.765670i	$-0.277592\pi$
$151$	15.8084	1.28647	0.643234	+	0.765670i	$0.277592\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	13.2236	1.06215
$156$	0	0
$157$	3.84395	0.306781	0.153390	−	0.988166i	$-0.450981\pi$
$157$	3.84395	0.306781	0.153390	+	0.988166i	$0.450981\pi$
$158$	0	0
$159$	0.620742i	0.0492281i
$160$	0	0
$161$	15.1183	1.19149
$162$	0	0
$163$	9.30898i	0.729136i	0.931177	+	0.364568i	$0.118783\pi$
$163$	9.30898i	0.729136i	−0.931177	+	0.364568i	$0.881217\pi$
$164$	0	0
$165$	0.256618i	0.0199777i
$166$	0	0
$167$	− 4.36977i	− 0.338143i	−0.985604	−	0.169071i	$-0.945923\pi$
$167$	− 4.36977i	− 0.338143i	0.985604	−	0.169071i	$-0.0540769\pi$
$168$	0	0
$169$	2.71552	0.208887
$170$	0	0
$171$	16.6875	1.27612
$172$	0	0
$173$	1.58579i	0.120565i	0.998181	+	0.0602826i	$0.0192002\pi$
$173$	1.58579i	0.120565i	−0.998181	+	0.0602826i	$0.980800\pi$
$174$	0	0
$175$	− 2.05436i	− 0.155295i
$176$	0	0
$177$	− 0.182304i	− 0.0137028i
$178$	0	0
$179$	22.0093	1.64505	0.822525	−	0.568729i	$-0.192565\pi$
$179$	22.0093	1.64505	0.822525	+	0.568729i	$0.192565\pi$
$180$	0	0
$181$	14.1859i	1.05443i	0.849731	+	0.527216i	$0.176764\pi$
$181$	14.1859i	1.05443i	−0.849731	+	0.527216i	$0.823236\pi$
$182$	0	0
$183$	0.136477	0.0100886
$184$	0	0
$185$	−11.8030	−0.867776
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.28806	−0.0936928
$190$	0	0
$191$	−9.40915	−0.680823	−0.340411	−	0.940277i	$-0.610566\pi$
$191$	−9.40915	−0.680823	−0.340411	+	0.940277i	$0.610566\pi$
$192$	0	0
$193$	− 7.63065i	− 0.549266i	−0.961549	−	0.274633i	$-0.911444\pi$
$193$	− 7.63065i	− 0.549266i	0.961549	−	0.274633i	$-0.0885564\pi$
$194$	0	0
$195$	−0.615687	−0.0440902
$196$	0	0
$197$	− 9.04547i	− 0.644463i	−0.946661	−	0.322232i	$-0.895567\pi$
$197$	− 9.04547i	− 0.644463i	0.946661	−	0.322232i	$-0.104433\pi$
$198$	0	0
$199$	− 11.4553i	− 0.812047i	−0.913863	−	0.406023i	$-0.866915\pi$
$199$	− 11.4553i	− 0.812047i	0.913863	−	0.406023i	$-0.133085\pi$
$200$	0	0
$201$	− 1.18815i	− 0.0838053i
$202$	0	0
$203$	−20.0370	−1.40632
$204$	0	0
$205$	−20.4386	−1.42749
$206$	0	0
$207$	15.8111i	1.09895i
$208$	0	0
$209$	9.20825i	0.636948i
$210$	0	0
$211$	9.18283i	0.632172i	0.948731	+	0.316086i	$0.102369\pi$
$211$	9.18283i	0.632172i	−0.948731	+	0.316086i	$0.897631\pi$
$212$	0	0
$213$	0.641449	0.0439514
$214$	0	0
$215$	13.8198i	0.942499i
$216$	0	0
$217$	−18.2957	−1.24199
$218$	0	0
$219$	−0.172765	−0.0116744
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.34273	0.491705	0.245853	−	0.969307i	$-0.420932\pi$
$223$	7.34273	0.491705	0.245853	+	0.969307i	$0.420932\pi$
$224$	0	0
$225$	2.14850	0.143233
$226$	0	0
$227$	11.7173i	0.777703i	0.921300	+	0.388852i	$0.127128\pi$
$227$	11.7173i	0.777703i	−0.921300	+	0.388852i	$0.872872\pi$
$228$	0	0
$229$	8.45705	0.558857	0.279429	−	0.960166i	$-0.409855\pi$
$229$	8.45705	0.558857	0.279429	+	0.960166i	$0.409855\pi$
$230$	0	0
$231$	− 0.355047i	− 0.0233604i
$232$	0	0
$233$	− 18.7135i	− 1.22596i	−0.790099	−	0.612980i	$-0.789971\pi$
$233$	− 18.7135i	− 1.22596i	0.790099	−	0.612980i	$-0.210029\pi$
$234$	0	0
$235$	9.16829i	0.598073i
$236$	0	0
$237$	−1.05265	−0.0683772
$238$	0	0
$239$	7.40650	0.479086	0.239543	−	0.970886i	$-0.423002\pi$
$239$	7.40650	0.479086	0.239543	+	0.970886i	$0.423002\pi$
$240$	0	0
$241$	− 22.6670i	− 1.46011i	−0.683390	−	0.730054i	$-0.739495\pi$
$241$	− 22.6670i	− 1.46011i	0.683390	−	0.730054i	$-0.260505\pi$
$242$	0	0
$243$	− 2.02126i	− 0.129664i
$244$	0	0
$245$	− 2.47860i	− 0.158352i
$246$	0	0
$247$	−22.0928	−1.40573
$248$	0	0
$249$	− 0.0756217i	− 0.00479233i
$250$	0	0
$251$	−5.67032	−0.357907	−0.178954	−	0.983857i	$-0.557271\pi$
$251$	−5.67032	−0.357907	−0.178954	+	0.983857i	$0.557271\pi$
$252$	0	0
$253$	−8.72465	−0.548514
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.2181	−1.76020	−0.880100	−	0.474789i	$-0.842524\pi$
$257$	−28.2181	−1.76020	−0.880100	+	0.474789i	$0.842524\pi$
$258$	0	0
$259$	16.3302	1.01471
$260$	0	0
$261$	− 20.9551i	− 1.29709i
$262$	0	0
$263$	4.06551	0.250690	0.125345	−	0.992113i	$-0.459996\pi$
$263$	4.06551	0.250690	0.125345	+	0.992113i	$0.459996\pi$
$264$	0	0
$265$	− 17.1164i	− 1.05145i
$266$	0	0
$267$	1.22007i	0.0746673i
$268$	0	0
$269$	− 1.95914i	− 0.119451i	−0.998215	−	0.0597255i	$-0.980977\pi$
$269$	− 1.95914i	− 0.119451i	0.998215	−	0.0597255i	$-0.0190225\pi$
$270$	0	0
$271$	6.00392	0.364712	0.182356	−	0.983233i	$-0.441628\pi$
$271$	6.00392	0.364712	0.182356	+	0.983233i	$0.441628\pi$
$272$	0	0
$273$	0.851841	0.0515557
$274$	0	0
$275$	1.18555i	0.0714916i
$276$	0	0
$277$	− 5.74968i	− 0.345465i	−0.984969	−	0.172732i	$-0.944740\pi$
$277$	− 5.74968i	− 0.345465i	0.984969	−	0.172732i	$-0.0552596\pi$
$278$	0	0
$279$	− 19.1341i	− 1.14553i
$280$	0	0
$281$	−20.4528	−1.22011	−0.610056	−	0.792359i	$-0.708853\pi$
$281$	−20.4528	−1.22011	−0.610056	+	0.792359i	$0.708853\pi$
$282$	0	0
$283$	− 26.9965i	− 1.60478i	−0.596801	−	0.802389i	$-0.703562\pi$
$283$	− 26.9965i	− 1.60478i	0.596801	−	0.802389i	$-0.296438\pi$
$284$	0	0
$285$	0.865527	0.0512694
$286$	0	0
$287$	28.2781	1.66920
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−1.03887	−0.0608995
$292$	0	0
$293$	−5.55130	−0.324310	−0.162155	−	0.986765i	$-0.551845\pi$
$293$	−5.55130	−0.324310	−0.162155	+	0.986765i	$0.551845\pi$
$294$	0	0
$295$	5.02686i	0.292675i
$296$	0	0
$297$	0.743331	0.0431324
$298$	0	0
$299$	− 20.9325i	− 1.21056i
$300$	0	0
$301$	− 19.1205i	− 1.10209i
$302$	0	0
$303$	− 1.12205i	− 0.0644602i
$304$	0	0
$305$	−3.76322	−0.215481
$306$	0	0
$307$	−16.1215	−0.920104	−0.460052	−	0.887892i	$-0.652169\pi$
$307$	−16.1215	−0.920104	−0.460052	+	0.887892i	$0.652169\pi$
$308$	0	0
$309$	− 1.21037i	− 0.0688555i
$310$	0	0
$311$	− 18.2242i	− 1.03340i	−0.856167	−	0.516699i	$-0.827161\pi$
$311$	− 18.2242i	− 1.03340i	0.856167	−	0.516699i	$-0.172839\pi$
$312$	0	0
$313$	9.98052i	0.564132i	0.959395	+	0.282066i	$0.0910198\pi$
$313$	9.98052i	0.564132i	−0.959395	+	0.282066i	$0.908980\pi$
$314$	0	0
$315$	17.7419	0.999642
$316$	0	0
$317$	− 4.47278i	− 0.251216i	−0.992080	−	0.125608i	$-0.959912\pi$
$317$	− 4.47278i	− 0.251216i	0.992080	−	0.125608i	$-0.0400882\pi$
$318$	0	0
$319$	11.5632	0.647413
$320$	0	0
$321$	1.10635	0.0617504
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−2.84442	−0.157780
$326$	0	0
$327$	0.861010	0.0476140
$328$	0	0
$329$	− 12.6849i	− 0.699341i
$330$	0	0
$331$	−0.514782	−0.0282950	−0.0141475	−	0.999900i	$-0.504503\pi$
$331$	−0.514782	−0.0282950	−0.0141475	+	0.999900i	$0.504503\pi$
$332$	0	0
$333$	17.0785i	0.935898i
$334$	0	0
$335$	32.7620i	1.78998i
$336$	0	0
$337$	3.35929i	0.182992i	0.995805	+	0.0914960i	$0.0291649\pi$
$337$	3.35929i	0.182992i	−0.995805	+	0.0914960i	$0.970835\pi$
$338$	0	0
$339$	0.250185	0.0135882
$340$	0	0
$341$	10.5583	0.571764
$342$	0	0
$343$	− 16.6129i	− 0.897011i
$344$	0	0
$345$	0.820071i	0.0441512i
$346$	0	0
$347$	− 23.5337i	− 1.26336i	−0.775231	−	0.631678i	$-0.782366\pi$
$347$	− 23.5337i	− 1.26336i	0.775231	−	0.631678i	$-0.217634\pi$
$348$	0	0
$349$	−25.3356	−1.35619	−0.678093	−	0.734976i	$-0.737193\pi$
$349$	−25.3356	−1.35619	−0.678093	+	0.734976i	$0.737193\pi$
$350$	0	0
$351$	1.78343i	0.0951922i
$352$	0	0
$353$	17.0119	0.905454	0.452727	−	0.891649i	$-0.350451\pi$
$353$	17.0119	0.905454	0.452727	+	0.891649i	$0.350451\pi$
$354$	0	0
$355$	−17.6874	−0.938748
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.00460	0.316911	0.158455	−	0.987366i	$-0.449349\pi$
$359$	6.00460	0.316911	0.158455	+	0.987366i	$0.449349\pi$
$360$	0	0
$361$	12.0578	0.634621
$362$	0	0
$363$	− 0.620649i	− 0.0325756i
$364$	0	0
$365$	4.76383	0.249350
$366$	0	0
$367$	7.43083i	0.387886i	0.981013	+	0.193943i	$0.0621277\pi$
$367$	7.43083i	0.387886i	−0.981013	+	0.193943i	$0.937872\pi$
$368$	0	0
$369$	29.5739i	1.53955i
$370$	0	0
$371$	23.6816i	1.22949i
$372$	0	0
$373$	−20.8566	−1.07991	−0.539957	−	0.841692i	$-0.681560\pi$
$373$	−20.8566	−1.07991	−0.539957	+	0.841692i	$0.681560\pi$
$374$	0	0
$375$	0.887978	0.0458550
$376$	0	0
$377$	27.7428i	1.42882i
$378$	0	0
$379$	− 2.34051i	− 0.120224i	−0.998192	−	0.0601119i	$-0.980854\pi$
$379$	− 2.34051i	− 0.120224i	0.998192	−	0.0601119i	$-0.0191457\pi$
$380$	0	0
$381$	1.48946i	0.0763075i
$382$	0	0
$383$	−31.7087	−1.62024	−0.810119	−	0.586265i	$-0.800598\pi$
$383$	−31.7087	−1.62024	−0.810119	+	0.586265i	$0.800598\pi$
$384$	0	0
$385$	9.79008i	0.498949i
$386$	0	0
$387$	19.9967	1.01649
$388$	0	0
$389$	10.8275	0.548975	0.274487	−	0.961591i	$-0.411492\pi$
$389$	10.8275	0.548975	0.274487	+	0.961591i	$0.411492\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1.11722	0.0563561
$394$	0	0
$395$	29.0260	1.46045
$396$	0	0
$397$	5.82927i	0.292562i	0.989243	+	0.146281i	$0.0467304\pi$
$397$	5.82927i	0.292562i	−0.989243	+	0.146281i	$0.953270\pi$
$398$	0	0
$399$	−1.19751	−0.0599505
$400$	0	0
$401$	25.5361i	1.27521i	0.770363	+	0.637605i	$0.220075\pi$
$401$	25.5361i	1.27521i	−0.770363	+	0.637605i	$0.779925\pi$
$402$	0	0
$403$	25.3318i	1.26187i
$404$	0	0
$405$	18.5199i	0.920262i
$406$	0	0
$407$	−9.42404	−0.467132
$408$	0	0
$409$	−19.4417	−0.961329	−0.480664	−	0.876905i	$-0.659604\pi$
$409$	−19.4417	−0.961329	−0.480664	+	0.876905i	$0.659604\pi$
$410$	0	0
$411$	0.186497i	0.00919922i
$412$	0	0
$413$	− 6.95497i	− 0.342232i
$414$	0	0
$415$	2.08520i	0.102358i
$416$	0	0
$417$	0.246176	0.0120553
$418$	0	0
$419$	− 37.6604i	− 1.83983i	−0.392116	−	0.919916i	$-0.628257\pi$
$419$	− 37.6604i	− 1.83983i	0.392116	−	0.919916i	$-0.371743\pi$
$420$	0	0
$421$	−13.2597	−0.646238	−0.323119	−	0.946358i	$-0.604731\pi$
$421$	−13.2597	−0.646238	−0.323119	+	0.946358i	$0.604731\pi$
$422$	0	0
$423$	13.2662	0.645023
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.20664	0.251967
$428$	0	0
$429$	−0.491590	−0.0237342
$430$	0	0
$431$	− 31.8730i	− 1.53527i	−0.640887	−	0.767635i	$-0.721433\pi$
$431$	− 31.8730i	− 1.53527i	0.640887	−	0.767635i	$-0.278567\pi$
$432$	0	0
$433$	−37.8807	−1.82043	−0.910215	−	0.414137i	$-0.864083\pi$
$433$	−37.8807	−1.82043	−0.910215	+	0.414137i	$0.864083\pi$
$434$	0	0
$435$	− 1.08688i	− 0.0521118i
$436$	0	0
$437$	29.4267i	1.40767i
$438$	0	0
$439$	4.95029i	0.236264i	0.992998	+	0.118132i	$0.0376907\pi$
$439$	4.95029i	0.236264i	−0.992998	+	0.118132i	$0.962309\pi$
$440$	0	0
$441$	−3.58644	−0.170783
$442$	0	0
$443$	−25.7002	−1.22106	−0.610528	−	0.791995i	$-0.709043\pi$
$443$	−25.7002	−1.22106	−0.610528	+	0.791995i	$0.709043\pi$
$444$	0	0
$445$	− 33.6424i	− 1.59480i
$446$	0	0
$447$	− 0.614264i	− 0.0290537i
$448$	0	0
$449$	13.9403i	0.657882i	0.944350	+	0.328941i	$0.106692\pi$
$449$	13.9403i	0.657882i	−0.944350	+	0.328941i	$0.893308\pi$
$450$	0	0
$451$	−16.3190	−0.768434
$452$	0	0
$453$	− 1.18641i	− 0.0557424i
$454$	0	0
$455$	−23.4887	−1.10117
$456$	0	0
$457$	−7.21894	−0.337688	−0.168844	−	0.985643i	$-0.554003\pi$
$457$	−7.21894	−0.337688	−0.168844	+	0.985643i	$0.554003\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.0248950	0.00115948	0.000579739	−	1.00000i	$-0.499815\pi$
$461$	0.0248950	0.00115948	0.000579739		1.00000i	$0.499815\pi$
$462$	0	0
$463$	−41.3218	−1.92039	−0.960194	−	0.279334i	$-0.909886\pi$
$463$	−41.3218	−1.92039	−0.960194	+	0.279334i	$0.909886\pi$
$464$	0	0
$465$	− 0.992424i	− 0.0460226i
$466$	0	0
$467$	2.01187	0.0930985	0.0465492	−	0.998916i	$-0.485178\pi$
$467$	2.01187	0.0930985	0.0465492	+	0.998916i	$0.485178\pi$
$468$	0	0
$469$	− 45.3283i	− 2.09307i
$470$	0	0
$471$	− 0.288486i	− 0.0132927i
$472$	0	0
$473$	11.0343i	0.507357i
$474$	0	0
$475$	3.99867	0.183471
$476$	0	0
$477$	−24.7668	−1.13399
$478$	0	0
$479$	− 40.6285i	− 1.85637i	−0.372124	−	0.928183i	$-0.621371\pi$
$479$	− 40.6285i	− 1.85637i	0.372124	−	0.928183i	$-0.378629\pi$
$480$	0	0
$481$	− 22.6105i	− 1.03095i
$482$	0	0
$483$	− 1.13462i	− 0.0516270i
$484$	0	0
$485$	28.6458	1.30074
$486$	0	0
$487$	30.6041i	1.38680i	0.720552	+	0.693401i	$0.243889\pi$
$487$	30.6041i	1.38680i	−0.720552	+	0.693401i	$0.756111\pi$
$488$	0	0
$489$	0.698633	0.0315933
$490$	0	0
$491$	−15.3815	−0.694157	−0.347078	−	0.937836i	$-0.612826\pi$
$491$	−15.3815	−0.694157	−0.347078	+	0.937836i	$0.612826\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−10.2387	−0.460195
$496$	0	0
$497$	24.4716	1.09770
$498$	0	0
$499$	12.2998i	0.550617i	0.961356	+	0.275308i	$0.0887799\pi$
$499$	12.2998i	0.550617i	−0.961356	+	0.275308i	$0.911220\pi$
$500$	0	0
$501$	−0.327949	−0.0146517
$502$	0	0
$503$	23.2615i	1.03718i	0.855024	+	0.518589i	$0.173543\pi$
$503$	23.2615i	1.03718i	−0.855024	+	0.518589i	$0.826457\pi$
$504$	0	0
$505$	30.9395i	1.37679i
$506$	0	0
$507$	− 0.203799i	− 0.00905101i
$508$	0	0
$509$	−17.9307	−0.794762	−0.397381	−	0.917654i	$-0.630081\pi$
$509$	−17.9307	−0.794762	−0.397381	+	0.917654i	$0.630081\pi$
$510$	0	0
$511$	−6.59106	−0.291571
$512$	0	0
$513$	− 2.50712i	− 0.110692i
$514$	0	0
$515$	33.3748i	1.47067i
$516$	0	0
$517$	7.32035i	0.321949i
$518$	0	0
$519$	0.119012	0.00522407
$520$	0	0
$521$	12.5178i	0.548413i	0.961671	+	0.274206i	$0.0884151\pi$
$521$	12.5178i	0.548413i	−0.961671	+	0.274206i	$0.911585\pi$
$522$	0	0
$523$	−10.5810	−0.462677	−0.231338	−	0.972873i	$-0.574310\pi$
$523$	−10.5810	−0.462677	−0.231338	+	0.972873i	$0.574310\pi$
$524$	0	0
$525$	−0.154179	−0.00672890
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−4.88129	−0.212230
$530$	0	0
$531$	7.27367	0.315651
$532$	0	0
$533$	− 39.1532i	− 1.69591i
$534$	0	0
$535$	−30.5066	−1.31891
$536$	0	0
$537$	− 1.65178i	− 0.0712797i
$538$	0	0
$539$	− 1.97902i	− 0.0852423i
$540$	0	0
$541$	31.4314i	1.35134i	0.737203	+	0.675671i	$0.236146\pi$
$541$	31.4314i	1.35134i	−0.737203	+	0.675671i	$0.763854\pi$
$542$	0	0
$543$	1.06465	0.0456883
$544$	0	0
$545$	−23.7415	−1.01698
$546$	0	0
$547$	− 4.84354i	− 0.207095i	−0.994625	−	0.103547i	$-0.966981\pi$
$547$	− 4.84354i	− 0.207095i	0.994625	−	0.103547i	$-0.0330193\pi$
$548$	0	0
$549$	5.44523i	0.232397i
$550$	0	0
$551$	− 39.0005i	− 1.66148i
$552$	0	0
$553$	−40.1592	−1.70774
$554$	0	0
$555$	0.885810i	0.0376005i
$556$	0	0
$557$	13.3327	0.564927	0.282463	−	0.959278i	$-0.408848\pi$
$557$	13.3327	0.564927	0.282463	+	0.959278i	$0.408848\pi$
$558$	0	0
$559$	−26.4738	−1.11972
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−30.2597	−1.27529	−0.637647	−	0.770329i	$-0.720092\pi$
$563$	−30.2597	−1.27529	−0.637647	+	0.770329i	$0.720092\pi$
$564$	0	0
$565$	−6.89862	−0.290227
$566$	0	0
$567$	− 25.6235i	− 1.07608i
$568$	0	0
$569$	30.4438	1.27627	0.638136	−	0.769924i	$-0.279706\pi$
$569$	30.4438	1.27627	0.638136	+	0.769924i	$0.279706\pi$
$570$	0	0
$571$	− 37.5161i	− 1.57000i	−0.619496	−	0.785000i	$-0.712663\pi$
$571$	− 37.5161i	− 1.57000i	0.619496	−	0.785000i	$-0.287337\pi$
$572$	0	0
$573$	0.706151i	0.0294999i
$574$	0	0
$575$	3.78867i	0.157998i
$576$	0	0
$577$	−24.2745	−1.01056	−0.505280	−	0.862955i	$-0.668611\pi$
$577$	−24.2745	−1.01056	−0.505280	+	0.862955i	$0.668611\pi$
$578$	0	0
$579$	−0.572676	−0.0237996
$580$	0	0
$581$	− 2.88500i	− 0.119690i
$582$	0	0
$583$	− 13.6664i	− 0.566007i
$584$	0	0
$585$	− 24.5650i	− 1.01564i
$586$	0	0
$587$	−18.4536	−0.761661	−0.380831	−	0.924645i	$-0.624362\pi$
$587$	−18.4536	−0.761661	−0.380831	+	0.924645i	$0.624362\pi$
$588$	0	0
$589$	− 35.6113i	− 1.46734i
$590$	0	0
$591$	−0.678857	−0.0279245
$592$	0	0
$593$	12.0405	0.494444	0.247222	−	0.968959i	$-0.420482\pi$
$593$	12.0405	0.494444	0.247222	+	0.968959i	$0.420482\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.859715	−0.0351858
$598$	0	0
$599$	21.0725	0.860999	0.430499	−	0.902591i	$-0.358337\pi$
$599$	21.0725	0.860999	0.430499	+	0.902591i	$0.358337\pi$
$600$	0	0
$601$	19.5781i	0.798609i	0.916818	+	0.399305i	$0.130748\pi$
$601$	19.5781i	0.798609i	−0.916818	+	0.399305i	$0.869252\pi$
$602$	0	0
$603$	47.4054	1.93050
$604$	0	0
$605$	17.1138i	0.695775i
$606$	0	0
$607$	− 7.14287i	− 0.289920i	−0.989438	−	0.144960i	$-0.953695\pi$
$607$	− 7.14287i	− 0.289920i	0.989438	−	0.144960i	$-0.0463054\pi$
$608$	0	0
$609$	1.50376i	0.0609355i
$610$	0	0
$611$	−17.5632	−0.710533
$612$	0	0
$613$	9.73218	0.393079	0.196539	−	0.980496i	$-0.437030\pi$
$613$	9.73218	0.393079	0.196539	+	0.980496i	$0.437030\pi$
$614$	0	0
$615$	1.53390i	0.0618530i
$616$	0	0
$617$	− 15.7203i	− 0.632875i	−0.948613	−	0.316437i	$-0.897513\pi$
$617$	− 15.7203i	− 0.632875i	0.948613	−	0.316437i	$-0.102487\pi$
$618$	0	0
$619$	46.8385i	1.88260i	0.337573	+	0.941299i	$0.390394\pi$
$619$	46.8385i	1.88260i	−0.337573	+	0.941299i	$0.609606\pi$
$620$	0	0
$621$	2.37546	0.0953238
$622$	0	0
$623$	46.5463i	1.86484i
$624$	0	0
$625$	−20.8976	−0.835904
$626$	0	0
$627$	0.691073	0.0275988
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0.994345	0.0395843	0.0197921	−	0.999804i	$-0.493700\pi$
$631$	0.994345	0.0395843	0.0197921	+	0.999804i	$0.493700\pi$
$632$	0	0
$633$	0.689166	0.0273919
$634$	0	0
$635$	− 41.0706i	− 1.62984i
$636$	0	0
$637$	4.74813	0.188128
$638$	0	0
$639$	25.5929i	1.01244i
$640$	0	0
$641$	− 12.9583i	− 0.511821i	−0.966700	−	0.255911i	$-0.917625\pi$
$641$	− 12.9583i	− 0.511821i	0.966700	−	0.255911i	$-0.0823753\pi$
$642$	0	0
$643$	5.44186i	0.214606i	0.994226	+	0.107303i	$0.0342215\pi$
$643$	5.44186i	0.214606i	−0.994226	+	0.107303i	$0.965778\pi$
$644$	0	0
$645$	1.03716	0.0408383
$646$	0	0
$647$	−13.4339	−0.528143	−0.264072	−	0.964503i	$-0.585065\pi$
$647$	−13.4339	−0.528143	−0.264072	+	0.964503i	$0.585065\pi$
$648$	0	0
$649$	4.01366i	0.157550i
$650$	0	0
$651$	1.37308i	0.0538153i
$652$	0	0
$653$	− 29.8838i	− 1.16944i	−0.811234	−	0.584722i	$-0.801203\pi$
$653$	− 29.8838i	− 1.16944i	0.811234	−	0.584722i	$-0.198797\pi$
$654$	0	0
$655$	−30.8062	−1.20370
$656$	0	0
$657$	− 6.89309i	− 0.268925i
$658$	0	0
$659$	26.0153	1.01341	0.506707	−	0.862119i	$-0.330863\pi$
$659$	26.0153	1.01341	0.506707	+	0.862119i	$0.330863\pi$
$660$	0	0
$661$	29.6768	1.15430	0.577148	−	0.816640i	$-0.304166\pi$
$661$	29.6768	1.15430	0.577148	+	0.816640i	$0.304166\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	33.0202	1.28047
$666$	0	0
$667$	36.9523	1.43080
$668$	0	0
$669$	− 0.551067i	− 0.0213055i
$670$	0	0
$671$	−3.00471	−0.115996
$672$	0	0
$673$	− 27.1439i	− 1.04632i	−0.852235	−	0.523159i	$-0.824753\pi$
$673$	− 27.1439i	− 1.04632i	0.852235	−	0.523159i	$-0.175247\pi$
$674$	0	0
$675$	− 0.322790i	− 0.0124242i
$676$	0	0
$677$	− 22.8605i	− 0.878602i	−0.898340	−	0.439301i	$-0.855226\pi$
$677$	− 22.8605i	− 0.878602i	0.898340	−	0.439301i	$-0.144774\pi$
$678$	0	0
$679$	−39.6332	−1.52098
$680$	0	0
$681$	0.879375	0.0336977
$682$	0	0
$683$	9.49776i	0.363422i	0.983352	+	0.181711i	$0.0581635\pi$
$683$	9.49776i	0.363422i	−0.983352	+	0.181711i	$0.941837\pi$
$684$	0	0
$685$	− 5.14248i	− 0.196484i
$686$	0	0
$687$	− 0.634696i	− 0.0242152i
$688$	0	0
$689$	32.7890	1.24916
$690$	0	0
$691$	− 34.6226i	− 1.31711i	−0.752534	−	0.658554i	$-0.771169\pi$
$691$	− 34.6226i	− 1.31711i	0.752534	−	0.658554i	$-0.228831\pi$
$692$	0	0
$693$	14.1659	0.538117
$694$	0	0
$695$	−6.78806	−0.257486
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−1.40443	−0.0531206
$700$	0	0
$701$	−5.15517	−0.194708	−0.0973541	−	0.995250i	$-0.531038\pi$
$701$	−5.15517	−0.194708	−0.0973541	+	0.995250i	$0.531038\pi$
$702$	0	0
$703$	31.7856i	1.19882i
$704$	0	0
$705$	0.688075	0.0259144
$706$	0	0
$707$	− 42.8068i	− 1.60991i
$708$	0	0
$709$	2.45568i	0.0922250i	0.998936	+	0.0461125i	$0.0146833\pi$
$709$	2.45568i	0.0922250i	−0.998936	+	0.0461125i	$0.985317\pi$
$710$	0	0
$711$	− 41.9995i	− 1.57510i
$712$	0	0
$713$	33.7410	1.26361
$714$	0	0
$715$	13.5551	0.506934
$716$	0	0
$717$	− 0.555853i	− 0.0207587i
$718$	0	0
$719$	− 35.6674i	− 1.33017i	−0.746767	−	0.665086i	$-0.768395\pi$
$719$	− 35.6674i	− 1.33017i	0.746767	−	0.665086i	$-0.231605\pi$
$720$	0	0
$721$	− 46.1761i	− 1.71969i
$722$	0	0
$723$	−1.70114	−0.0632661
$724$	0	0
$725$	− 5.02129i	− 0.186486i
$726$	0	0
$727$	−14.4438	−0.535692	−0.267846	−	0.963462i	$-0.586312\pi$
$727$	−14.4438	−0.535692	−0.267846	+	0.963462i	$0.586312\pi$
$728$	0	0
$729$	26.6963	0.988753
$730$	0	0
$731$	0	0
$732$	0	0
$733$	22.8514	0.844036	0.422018	−	0.906587i	$-0.361322\pi$
$733$	22.8514	0.844036	0.422018	+	0.906587i	$0.361322\pi$
$734$	0	0
$735$	−0.186017	−0.00686135
$736$	0	0
$737$	26.1586i	0.963563i
$738$	0	0
$739$	13.1134	0.482383	0.241192	−	0.970478i	$-0.422462\pi$
$739$	13.1134	0.482383	0.241192	+	0.970478i	$0.422462\pi$
$740$	0	0
$741$	1.65805i	0.0609099i
$742$	0	0
$743$	− 5.86298i	− 0.215092i	−0.994200	−	0.107546i	$-0.965701\pi$
$743$	− 5.86298i	− 0.215092i	0.994200	−	0.107546i	$-0.0342993\pi$
$744$	0	0
$745$	16.9378i	0.620551i
$746$	0	0
$747$	3.01720	0.110394
$748$	0	0
$749$	42.2077	1.54224
$750$	0	0
$751$	− 10.5603i	− 0.385351i	−0.981262	−	0.192676i	$-0.938283\pi$
$751$	− 10.5603i	− 0.385351i	0.981262	−	0.192676i	$-0.0617165\pi$
$752$	0	0
$753$	0.425554i	0.0155081i
$754$	0	0
$755$	32.7141i	1.19059i
$756$	0	0
$757$	−52.3557	−1.90290	−0.951451	−	0.307801i	$-0.900407\pi$
$757$	−52.3557	−1.90290	−0.951451	+	0.307801i	$0.900407\pi$
$758$	0	0
$759$	0.654780i	0.0237670i
$760$	0	0
$761$	−1.60283	−0.0581027	−0.0290513	−	0.999578i	$-0.509249\pi$
$761$	−1.60283	−0.0581027	−0.0290513	+	0.999578i	$0.509249\pi$
$762$	0	0
$763$	32.8479	1.18917
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.62971	−0.347709
$768$	0	0
$769$	19.0060	0.685373	0.342686	−	0.939450i	$-0.388663\pi$
$769$	19.0060	0.685373	0.342686	+	0.939450i	$0.388663\pi$
$770$	0	0
$771$	2.11775i	0.0762691i
$772$	0	0
$773$	−42.9719	−1.54559	−0.772795	−	0.634655i	$-0.781142\pi$
$773$	−42.9719	−1.54559	−0.772795	+	0.634655i	$0.781142\pi$
$774$	0	0
$775$	− 4.58492i	− 0.164695i
$776$	0	0
$777$	− 1.22557i	− 0.0439672i
$778$	0	0
$779$	55.0413i	1.97206i
$780$	0	0
$781$	−14.1223	−0.505337
$782$	0	0
$783$	−3.14830	−0.112511
$784$	0	0
$785$	7.95474i	0.283917i
$786$	0	0
$787$	− 15.6985i	− 0.559592i	−0.960059	−	0.279796i	$-0.909733\pi$
$787$	− 15.6985i	− 0.559592i	0.960059	−	0.279796i	$-0.0902669\pi$
$788$	0	0
$789$	− 0.305114i	− 0.0108623i
$790$	0	0
$791$	9.54468	0.339370
$792$	0	0
$793$	− 7.20901i	− 0.255999i
$794$	0	0
$795$	−1.28457	−0.0455592
$796$	0	0
$797$	−10.2757	−0.363984	−0.181992	−	0.983300i	$-0.558255\pi$
$797$	−10.2757	−0.363984	−0.181992	+	0.983300i	$0.558255\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−48.6792	−1.72000
$802$	0	0
$803$	3.80365	0.134228
$804$	0	0
$805$	31.2861i	1.10269i
$806$	0	0
$807$	−0.147032	−0.00517578
$808$	0	0
$809$	7.07102i	0.248604i	0.992244	+	0.124302i	$0.0396691\pi$
$809$	7.07102i	0.248604i	−0.992244	+	0.124302i	$0.960331\pi$
$810$	0	0
$811$	− 25.2175i	− 0.885506i	−0.896644	−	0.442753i	$-0.854002\pi$
$811$	− 25.2175i	− 0.885506i	0.896644	−	0.442753i	$-0.145998\pi$
$812$	0	0
$813$	− 0.450591i	− 0.0158029i
$814$	0	0
$815$	−19.2642	−0.674794
$816$	0	0
$817$	37.2167	1.30205
$818$	0	0
$819$	33.9873i	1.18761i
$820$	0	0
$821$	17.1538i	0.598672i	0.954148	+	0.299336i	$0.0967652\pi$
$821$	17.1538i	0.598672i	−0.954148	+	0.299336i	$0.903235\pi$
$822$	0	0
$823$	19.1448i	0.667345i	0.942689	+	0.333673i	$0.108288\pi$
$823$	19.1448i	0.667345i	−0.942689	+	0.333673i	$0.891712\pi$
$824$	0	0
$825$	0.0889752	0.00309772
$826$	0	0
$827$	− 12.2442i	− 0.425772i	−0.977077	−	0.212886i	$-0.931714\pi$
$827$	− 12.2442i	− 0.425772i	0.977077	−	0.212886i	$-0.0682863\pi$
$828$	0	0
$829$	−22.8247	−0.792734	−0.396367	−	0.918092i	$-0.629729\pi$
$829$	−22.8247	−0.792734	−0.396367	+	0.918092i	$0.629729\pi$
$830$	0	0
$831$	−0.431510	−0.0149689
$832$	0	0
$833$	0	0
$834$	0	0
$835$	9.04287	0.312942
$836$	0	0
$837$	−2.87470	−0.0993642
$838$	0	0
$839$	− 42.9945i	− 1.48434i	−0.670214	−	0.742168i	$-0.733798\pi$
$839$	− 42.9945i	− 1.48434i	0.670214	−	0.742168i	$-0.266202\pi$
$840$	0	0
$841$	−19.9745	−0.688777
$842$	0	0
$843$	1.53497i	0.0528672i
$844$	0	0
$845$	5.61955i	0.193319i
$846$	0	0
$847$	− 23.6780i	− 0.813586i
$848$	0	0
$849$	−2.02607	−0.0695347
$850$	0	0
$851$	−30.1163	−1.03237
$852$	0	0
$853$	− 6.75339i	− 0.231232i	−0.993294	−	0.115616i	$-0.963116\pi$
$853$	− 6.75339i	− 0.231232i	0.993294	−	0.115616i	$-0.0368842\pi$
$854$	0	0
$855$	34.5333i	1.18102i
$856$	0	0
$857$	− 22.3142i	− 0.762237i	−0.924526	−	0.381119i	$-0.875539\pi$
$857$	− 22.3142i	− 0.762237i	0.924526	−	0.381119i	$-0.124461\pi$
$858$	0	0
$859$	49.3533	1.68391	0.841956	−	0.539546i	$-0.181404\pi$
$859$	49.3533	1.68391	0.841956	+	0.539546i	$0.181404\pi$
$860$	0	0
$861$	− 2.12225i	− 0.0723262i
$862$	0	0
$863$	9.00113	0.306402	0.153201	−	0.988195i	$-0.451042\pi$
$863$	9.00113	0.306402	0.153201	+	0.988195i	$0.451042\pi$
$864$	0	0
$865$	−3.28166	−0.111580
$866$	0	0
$867$	0	0
$868$	0	0
$869$	23.1756	0.786177
$870$	0	0
$871$	−62.7606	−2.12656
$872$	0	0
$873$	− 41.4494i	− 1.40285i
$874$	0	0
$875$	33.8768	1.14524
$876$	0	0
$877$	55.1621i	1.86269i	0.364134	+	0.931346i	$0.381365\pi$
$877$	55.1621i	1.86269i	−0.364134	+	0.931346i	$0.618635\pi$
$878$	0	0
$879$	0.416622i	0.0140523i
$880$	0	0
$881$	− 8.00493i	− 0.269693i	−0.990867	−	0.134846i	$-0.956946\pi$
$881$	− 8.00493i	− 0.269693i	0.990867	−	0.134846i	$-0.0430541\pi$
$882$	0	0
$883$	49.4099	1.66277	0.831387	−	0.555693i	$-0.187547\pi$
$883$	49.4099	1.66277	0.831387	+	0.555693i	$0.187547\pi$
$884$	0	0
$885$	0.377263	0.0126815
$886$	0	0
$887$	3.38687i	0.113720i	0.998382	+	0.0568600i	$0.0181089\pi$
$887$	3.38687i	0.113720i	−0.998382	+	0.0568600i	$0.981891\pi$
$888$	0	0
$889$	56.8237i	1.90581i
$890$	0	0
$891$	14.7871i	0.495386i
$892$	0	0
$893$	24.6903	0.826228
$894$	0	0
$895$	45.5464i	1.52245i
$896$	0	0
$897$	−1.57097	−0.0524532
$898$	0	0
$899$	−44.7185	−1.49145
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−1.43498	−0.0477532
$904$	0	0
$905$	−29.3566	−0.975846
$906$	0	0
$907$	35.6952i	1.18524i	0.805483	+	0.592619i	$0.201906\pi$
$907$	35.6952i	1.18524i	−0.805483	+	0.592619i	$0.798094\pi$
$908$	0	0
$909$	44.7683	1.48487
$910$	0	0
$911$	30.1013i	0.997301i	0.866803	+	0.498650i	$0.166171\pi$
$911$	30.1013i	0.997301i	−0.866803	+	0.498650i	$0.833829\pi$
$912$	0	0
$913$	1.66491i	0.0551005i
$914$	0	0
$915$	0.282427i	0.00933675i
$916$	0	0
$917$	42.6223	1.40751
$918$	0	0
$919$	−35.6600	−1.17632	−0.588158	−	0.808746i	$-0.700147\pi$
$919$	−35.6600	−1.17632	−0.588158	+	0.808746i	$0.700147\pi$
$920$	0	0
$921$	1.20991i	0.0398679i
$922$	0	0
$923$	− 33.8828i	− 1.11527i
$924$	0	0
$925$	4.09237i	0.134556i
$926$	0	0
$927$	48.2921	1.58612
$928$	0	0
$929$	− 27.7694i	− 0.911086i	−0.890214	−	0.455543i	$-0.849445\pi$
$929$	− 27.7694i	− 0.911086i	0.890214	−	0.455543i	$-0.150555\pi$
$930$	0	0
$931$	−6.67488	−0.218760
$932$	0	0
$933$	−1.36771	−0.0447769
$934$	0	0
$935$	0	0
$936$	0	0
$937$	34.9960	1.14327	0.571635	−	0.820508i	$-0.306309\pi$
$937$	34.9960	1.14327	0.571635	+	0.820508i	$0.306309\pi$
$938$	0	0
$939$	0.749032	0.0244437
$940$	0	0
$941$	20.3155i	0.662267i	0.943584	+	0.331134i	$0.107431\pi$
$941$	20.3155i	0.662267i	−0.943584	+	0.331134i	$0.892569\pi$
$942$	0	0
$943$	−52.1506	−1.69826
$944$	0	0
$945$	− 2.66554i	− 0.0867100i
$946$	0	0
$947$	− 18.7350i	− 0.608807i	−0.952543	−	0.304403i	$-0.901543\pi$
$947$	− 18.7350i	− 0.608807i	0.952543	−	0.304403i	$-0.0984571\pi$
$948$	0	0
$949$	9.12584i	0.296237i
$950$	0	0
$951$	−0.335679	−0.0108852
$952$	0	0
$953$	23.4719	0.760330	0.380165	−	0.924919i	$-0.375867\pi$
$953$	23.4719	0.760330	0.380165	+	0.924919i	$0.375867\pi$
$954$	0	0
$955$	− 19.4715i	− 0.630082i
$956$	0	0
$957$	− 0.867809i	− 0.0280523i
$958$	0	0
$959$	7.11494i	0.229753i
$960$	0	0
$961$	−9.83232	−0.317172
$962$	0	0
$963$	44.1418i	1.42245i
$964$	0	0
$965$	15.7910	0.508330
$966$	0	0
$967$	−7.08838	−0.227947	−0.113974	−	0.993484i	$-0.536358\pi$
$967$	−7.08838	−0.227947	−0.113974	+	0.993484i	$0.536358\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−46.1859	−1.48218	−0.741088	−	0.671408i	$-0.765690\pi$
$971$	−46.1859	−1.48218	−0.741088	+	0.671408i	$0.765690\pi$
$972$	0	0
$973$	9.39171	0.301084
$974$	0	0
$975$	0.213472i	0.00683659i
$976$	0	0
$977$	−21.3334	−0.682516	−0.341258	−	0.939970i	$-0.610853\pi$
$977$	−21.3334	−0.682516	−0.341258	+	0.939970i	$0.610853\pi$
$978$	0	0
$979$	− 26.8615i	− 0.858497i
$980$	0	0
$981$	34.3531i	1.09681i
$982$	0	0
$983$	− 46.3727i	− 1.47906i	−0.673124	−	0.739530i	$-0.735048\pi$
$983$	− 46.3727i	− 1.47906i	0.673124	−	0.739530i	$-0.264952\pi$
$984$	0	0
$985$	18.7189	0.596432
$986$	0	0
$987$	−0.951994	−0.0303023
$988$	0	0
$989$	35.2621i	1.12127i
$990$	0	0
$991$	21.5269i	0.683824i	0.939732	+	0.341912i	$0.111074\pi$
$991$	21.5269i	0.683824i	−0.939732	+	0.341912i	$0.888926\pi$
$992$	0	0
$993$	0.0386341i	0.00122602i
$994$	0	0
$995$	23.7058	0.751526
$996$	0	0
$997$	− 5.98544i	− 0.189561i	−0.995498	−	0.0947804i	$-0.969785\pi$
$997$	− 5.98544i	− 0.189561i	0.995498	−	0.0947804i	$-0.0302149\pi$
$998$	0	0
$999$	2.56588	0.0811808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.b.o.577.6		12
17.4	even	4		2312.2.a.u.1.3	✓	6
17.13	even	4		2312.2.a.v.1.4	yes	6
17.16	even	2	inner	2312.2.b.o.577.7		12
68.47	odd	4		4624.2.a.bs.1.3		6
68.55	odd	4		4624.2.a.br.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.2.a.u.1.3	✓	6	17.4	even	4
2312.2.a.v.1.4	yes	6	17.13	even	4
2312.2.b.o.577.6		12	1.1	even	1	trivial
2312.2.b.o.577.7		12	17.16	even	2	inner
4624.2.a.br.1.4		6	68.55	odd	4
4624.2.a.bs.1.3		6	68.47	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.0750494i q^{3} +2.06942i q^{5} -2.86317i q^{7} +2.99437 q^{9} +1.65231i q^{11} -3.96428 q^{13} +0.155308 q^{15} +5.57295 q^{19} -0.214879 q^{21} +5.28027i q^{23} +0.717513 q^{25} -0.449874i q^{27} -6.99818i q^{29} -6.39002i q^{31} +0.124005 q^{33} +5.92509 q^{35} +5.70355i q^{37} +0.297517i q^{39} +9.87650i q^{41} +6.67809 q^{43} +6.19660i q^{45} +4.43037 q^{47} -1.19773 q^{49} -8.27111 q^{53} -3.41932 q^{55} -0.418247i q^{57} +2.42912 q^{59} +1.81849i q^{61} -8.57338i q^{63} -8.20375i q^{65} +15.8315 q^{67} +0.396281 q^{69} +8.54703i q^{71} -2.30202i q^{73} -0.0538489i q^{75} +4.73084 q^{77} -14.0262i q^{79} +8.94934 q^{81} +1.00763 q^{83} -0.525209 q^{87} -16.2569 q^{89} +11.3504i q^{91} -0.479567 q^{93} +11.5328i q^{95} -13.8424i q^{97} +4.94762i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{13} + 12 q^{15} - 12 q^{19} + 12 q^{21} - 12 q^{25} + 6 q^{33} + 42 q^{35} + 6 q^{47} - 18 q^{49} - 66 q^{53} - 102 q^{55} - 18 q^{67} - 6 q^{69} + 90 q^{77} - 36 q^{81} + 24 q^{83} - 30 q^{87}+ \cdots + 60 q^{93}+O(q^{100}) \) 12 * q - 12 * q^13 + 12 * q^15 - 12 * q^19 + 12 * q^21 - 12 * q^25 + 6 * q^33 + 42 * q^35 + 6 * q^47 - 18 * q^49 - 66 * q^53 - 102 * q^55 - 18 * q^67 - 6 * q^69 + 90 * q^77 - 36 * q^81 + 24 * q^83 - 30 * q^87 + 6 * q^89 + 60 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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