LMFDB - Embedded newform 2312.2.b.o.577.4

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			577.4
Root			$-1.60714i$ of defining polynomial
Character	$\chi$	$=$	2312.577
Dual form			2312.2.b.o.577.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.60714i q^{3} +2.19003i q^{5} -4.55776i q^{7} +0.417107 q^{9} +5.18440i q^{11} +5.72305 q^{13} +3.51968 q^{15} -0.750005 q^{19} -7.32495 q^{21} -0.436608i q^{23} +0.203761 q^{25} -5.49176i q^{27} +8.47747i q^{29} +3.22854i q^{31} +8.33205 q^{33} +9.98164 q^{35} +6.25019i q^{37} -9.19774i q^{39} -0.392067i q^{41} +5.78302 q^{43} +0.913477i q^{45} -3.77767 q^{47} -13.7732 q^{49} +9.73222 q^{53} -11.3540 q^{55} +1.20536i q^{57} +1.53405 q^{59} +3.33603i q^{61} -1.90107i q^{63} +12.5337i q^{65} -2.95893 q^{67} -0.701689 q^{69} -5.18265i q^{71} -5.90602i q^{73} -0.327472i q^{75} +23.6292 q^{77} -12.7694i q^{79} -7.57470 q^{81} +9.12073 q^{83} +13.6245 q^{87} +13.0080 q^{89} -26.0843i q^{91} +5.18871 q^{93} -1.64253i q^{95} +16.1089i q^{97} +2.16245i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 1.60714i	− 0.927882i	−0.885866	−	0.463941i	$-0.846435\pi$
$3$	− 1.60714i	− 0.927882i	0.885866	−	0.463941i	$-0.153565\pi$
$4$	0	0
$5$	2.19003i	0.979412i	0.871888	+	0.489706i	$0.162896\pi$
$5$	2.19003i	0.979412i	−0.871888	+	0.489706i	$0.837104\pi$
$6$	0	0
$7$	− 4.55776i	− 1.72267i	−0.508036	−	0.861336i	$-0.669628\pi$
$7$	− 4.55776i	− 1.72267i	0.508036	−	0.861336i	$-0.330372\pi$
$8$	0	0
$9$	0.417107	0.139036
$10$	0	0
$11$	5.18440i	1.56316i	0.623808	+	0.781578i	$0.285585\pi$
$11$	5.18440i	1.56316i	−0.623808	+	0.781578i	$0.714415\pi$
$12$	0	0
$13$	5.72305	1.58729	0.793644	−	0.608382i	$-0.208181\pi$
$13$	5.72305	1.58729	0.793644	+	0.608382i	$0.208181\pi$
$14$	0	0
$15$	3.51968	0.908778
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−0.750005	−0.172063	−0.0860314	−	0.996292i	$-0.527419\pi$
$19$	−0.750005	−0.172063	−0.0860314	+	0.996292i	$0.527419\pi$
$20$	0	0
$21$	−7.32495	−1.59844
$22$	0	0
$23$	− 0.436608i	− 0.0910390i	−0.998963	−	0.0455195i	$-0.985506\pi$
$23$	− 0.436608i	− 0.0910390i	0.998963	−	0.0455195i	$-0.0144943\pi$
$24$	0	0
$25$	0.203761	0.0407522
$26$	0	0
$27$	− 5.49176i	− 1.05689i
$28$	0	0
$29$	8.47747i	1.57423i	0.616808	+	0.787113i	$0.288425\pi$
$29$	8.47747i	1.57423i	−0.616808	+	0.787113i	$0.711575\pi$
$30$	0	0
$31$	3.22854i	0.579863i	0.957047	+	0.289931i	$0.0936325\pi$
$31$	3.22854i	0.579863i	−0.957047	+	0.289931i	$0.906368\pi$
$32$	0	0
$33$	8.33205	1.45042
$34$	0	0
$35$	9.98164	1.68720
$36$	0	0
$37$	6.25019i	1.02752i	0.857933	+	0.513762i	$0.171749\pi$
$37$	6.25019i	1.02752i	−0.857933	+	0.513762i	$0.828251\pi$
$38$	0	0
$39$	− 9.19774i	− 1.47282i
$40$	0	0
$41$	− 0.392067i	− 0.0612306i	−0.999531	−	0.0306153i	$-0.990253\pi$
$41$	− 0.392067i	− 0.0612306i	0.999531	−	0.0306153i	$-0.00974668\pi$
$42$	0	0
$43$	5.78302	0.881902	0.440951	−	0.897531i	$-0.354641\pi$
$43$	5.78302	0.881902	0.440951	+	0.897531i	$0.354641\pi$
$44$	0	0
$45$	0.913477i	0.136173i
$46$	0	0
$47$	−3.77767	−0.551030	−0.275515	−	0.961297i	$-0.588848\pi$
$47$	−3.77767	−0.551030	−0.275515	+	0.961297i	$0.588848\pi$
$48$	0	0
$49$	−13.7732	−1.96760
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.73222	1.33682	0.668412	−	0.743792i	$-0.266975\pi$
$53$	9.73222	1.33682	0.668412	+	0.743792i	$0.266975\pi$
$54$	0	0
$55$	−11.3540	−1.53097
$56$	0	0
$57$	1.20536i	0.159654i
$58$	0	0
$59$	1.53405	0.199716	0.0998578	−	0.995002i	$-0.468161\pi$
$59$	1.53405	0.199716	0.0998578	+	0.995002i	$0.468161\pi$
$60$	0	0
$61$	3.33603i	0.427135i	0.976928	+	0.213568i	$0.0685083\pi$
$61$	3.33603i	0.427135i	−0.976928	+	0.213568i	$0.931492\pi$
$62$	0	0
$63$	− 1.90107i	− 0.239512i
$64$	0	0
$65$	12.5337i	1.55461i
$66$	0	0
$67$	−2.95893	−0.361491	−0.180746	−	0.983530i	$-0.557851\pi$
$67$	−2.95893	−0.361491	−0.180746	+	0.983530i	$0.557851\pi$
$68$	0	0
$69$	−0.701689	−0.0844734
$70$	0	0
$71$	− 5.18265i	− 0.615067i	−0.951537	−	0.307534i	$-0.900496\pi$
$71$	− 5.18265i	− 0.615067i	0.951537	−	0.307534i	$-0.0995037\pi$
$72$	0	0
$73$	− 5.90602i	− 0.691248i	−0.938373	−	0.345624i	$-0.887667\pi$
$73$	− 5.90602i	− 0.691248i	0.938373	−	0.345624i	$-0.112333\pi$
$74$	0	0
$75$	− 0.327472i	− 0.0378132i
$76$	0	0
$77$	23.6292	2.69280
$78$	0	0
$79$	− 12.7694i	− 1.43667i	−0.695695	−	0.718337i	$-0.744904\pi$
$79$	− 12.7694i	− 1.43667i	0.695695	−	0.718337i	$-0.255096\pi$
$80$	0	0
$81$	−7.57470	−0.841634
$82$	0	0
$83$	9.12073	1.00113	0.500565	−	0.865699i	$-0.333126\pi$
$83$	9.12073	1.00113	0.500565	+	0.865699i	$0.333126\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	13.6245	1.46070
$88$	0	0
$89$	13.0080	1.37884	0.689421	−	0.724361i	$-0.257865\pi$
$89$	13.0080	1.37884	0.689421	+	0.724361i	$0.257865\pi$
$90$	0	0
$91$	− 26.0843i	− 2.73438i
$92$	0	0
$93$	5.18871	0.538044
$94$	0	0
$95$	− 1.64253i	− 0.168520i
$96$	0	0
$97$	16.1089i	1.63561i	0.575498	+	0.817803i	$0.304808\pi$
$97$	16.1089i	1.63561i	−0.575498	+	0.817803i	$0.695192\pi$
$98$	0	0
$99$	2.16245i	0.217334i
$100$	0	0
$101$	−0.607663	−0.0604647	−0.0302324	−	0.999543i	$-0.509625\pi$
$101$	−0.607663	−0.0604647	−0.0302324	+	0.999543i	$0.509625\pi$
$102$	0	0
$103$	11.1760	1.10121	0.550603	−	0.834767i	$-0.314398\pi$
$103$	11.1760	1.10121	0.550603	+	0.834767i	$0.314398\pi$
$104$	0	0
$105$	− 16.0419i	− 1.56553i
$106$	0	0
$107$	1.26866i	0.122646i	0.998118	+	0.0613232i	$0.0195320\pi$
$107$	1.26866i	0.122646i	−0.998118	+	0.0613232i	$0.980468\pi$
$108$	0	0
$109$	− 20.2648i	− 1.94102i	−0.241062	−	0.970510i	$-0.577496\pi$
$109$	− 20.2648i	− 1.94102i	0.241062	−	0.970510i	$-0.422504\pi$
$110$	0	0
$111$	10.0449	0.953421
$112$	0	0
$113$	− 18.2704i	− 1.71873i	−0.511359	−	0.859367i	$-0.670858\pi$
$113$	− 18.2704i	− 1.71873i	0.511359	−	0.859367i	$-0.329142\pi$
$114$	0	0
$115$	0.956184	0.0891646
$116$	0	0
$117$	2.38712	0.220690
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−15.8780	−1.44345
$122$	0	0
$123$	−0.630106	−0.0568148
$124$	0	0
$125$	11.3964i	1.01933i
$126$	0	0
$127$	−11.7089	−1.03899	−0.519497	−	0.854472i	$-0.673881\pi$
$127$	−11.7089	−1.03899	−0.519497	+	0.854472i	$0.673881\pi$
$128$	0	0
$129$	− 9.29411i	− 0.818300i
$130$	0	0
$131$	8.95059i	0.782017i	0.920387	+	0.391009i	$0.127874\pi$
$131$	8.95059i	0.782017i	−0.920387	+	0.391009i	$0.872126\pi$
$132$	0	0
$133$	3.41834i	0.296408i
$134$	0	0
$135$	12.0271	1.03513
$136$	0	0
$137$	4.24376	0.362569	0.181284	−	0.983431i	$-0.441975\pi$
$137$	4.24376	0.362569	0.181284	+	0.983431i	$0.441975\pi$
$138$	0	0
$139$	− 14.4468i	− 1.22536i	−0.790330	−	0.612681i	$-0.790091\pi$
$139$	− 14.4468i	− 1.22536i	0.790330	−	0.612681i	$-0.209909\pi$
$140$	0	0
$141$	6.07124i	0.511290i
$142$	0	0
$143$	29.6706i	2.48118i
$144$	0	0
$145$	−18.5659	−1.54182
$146$	0	0
$147$	22.1354i	1.82570i
$148$	0	0
$149$	−5.83245	−0.477813	−0.238906	−	0.971043i	$-0.576789\pi$
$149$	−5.83245	−0.477813	−0.238906	+	0.971043i	$0.576789\pi$
$150$	0	0
$151$	7.91119	0.643804	0.321902	−	0.946773i	$-0.395678\pi$
$151$	7.91119	0.643804	0.321902	+	0.946773i	$0.395678\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−7.07060	−0.567924
$156$	0	0
$157$	−12.3692	−0.987172	−0.493586	−	0.869697i	$-0.664314\pi$
$157$	−12.3692	−0.987172	−0.493586	+	0.869697i	$0.664314\pi$
$158$	0	0
$159$	− 15.6410i	− 1.24041i
$160$	0	0
$161$	−1.98995	−0.156830
$162$	0	0
$163$	16.3886i	1.28366i	0.766848	+	0.641829i	$0.221824\pi$
$163$	16.3886i	1.28366i	−0.766848	+	0.641829i	$0.778176\pi$
$164$	0	0
$165$	18.2474i	1.42056i
$166$	0	0
$167$	7.56874i	0.585687i	0.956160	+	0.292843i	$0.0946015\pi$
$167$	7.56874i	0.585687i	−0.956160	+	0.292843i	$0.905399\pi$
$168$	0	0
$169$	19.7533	1.51949
$170$	0	0
$171$	−0.312832	−0.0239228
$172$	0	0
$173$	− 10.1047i	− 0.768245i	−0.923282	−	0.384123i	$-0.874504\pi$
$173$	− 10.1047i	− 0.768245i	0.923282	−	0.384123i	$-0.125496\pi$
$174$	0	0
$175$	− 0.928694i	− 0.0702026i
$176$	0	0
$177$	− 2.46542i	− 0.185312i
$178$	0	0
$179$	−2.03063	−0.151777	−0.0758883	−	0.997116i	$-0.524179\pi$
$179$	−2.03063	−0.151777	−0.0758883	+	0.997116i	$0.524179\pi$
$180$	0	0
$181$	− 4.78498i	− 0.355665i	−0.984061	−	0.177832i	$-0.943092\pi$
$181$	− 4.78498i	− 0.355665i	0.984061	−	0.177832i	$-0.0569085\pi$
$182$	0	0
$183$	5.36146	0.396331
$184$	0	0
$185$	−13.6881	−1.00637
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−25.0301	−1.82067
$190$	0	0
$191$	9.31083	0.673708	0.336854	−	0.941557i	$-0.390637\pi$
$191$	9.31083	0.673708	0.336854	+	0.941557i	$0.390637\pi$
$192$	0	0
$193$	− 20.0225i	− 1.44125i	−0.693324	−	0.720626i	$-0.743854\pi$
$193$	− 20.0225i	− 1.44125i	0.693324	−	0.720626i	$-0.256146\pi$
$194$	0	0
$195$	20.1433	1.44249
$196$	0	0
$197$	− 1.02318i	− 0.0728987i	−0.999336	−	0.0364494i	$-0.988395\pi$
$197$	− 1.02318i	− 0.0728987i	0.999336	−	0.0364494i	$-0.0116048\pi$
$198$	0	0
$199$	− 9.12527i	− 0.646873i	−0.946250	−	0.323437i	$-0.895162\pi$
$199$	− 9.12527i	− 0.646873i	0.946250	−	0.323437i	$-0.104838\pi$
$200$	0	0
$201$	4.75542i	0.335421i
$202$	0	0
$203$	38.6383	2.71188
$204$	0	0
$205$	0.858640	0.0599700
$206$	0	0
$207$	− 0.182112i	− 0.0126576i
$208$	0	0
$209$	− 3.88832i	− 0.268961i
$210$	0	0
$211$	1.91333i	0.131719i	0.997829	+	0.0658594i	$0.0209789\pi$
$211$	1.91333i	0.131719i	−0.997829	+	0.0658594i	$0.979021\pi$
$212$	0	0
$213$	−8.32923	−0.570710
$214$	0	0
$215$	12.6650i	0.863745i
$216$	0	0
$217$	14.7149	0.998913
$218$	0	0
$219$	−9.49180	−0.641396
$220$	0	0
$221$	0	0
$222$	0	0
$223$	15.7292	1.05330	0.526652	−	0.850081i	$-0.323447\pi$
$223$	15.7292	1.05330	0.526652	+	0.850081i	$0.323447\pi$
$224$	0	0
$225$	0.0849900	0.00566600
$226$	0	0
$227$	13.1228i	0.870989i	0.900191	+	0.435494i	$0.143426\pi$
$227$	13.1228i	0.870989i	−0.900191	+	0.435494i	$0.856574\pi$
$228$	0	0
$229$	−11.3364	−0.749133	−0.374566	−	0.927200i	$-0.622208\pi$
$229$	−11.3364	−0.749133	−0.374566	+	0.927200i	$0.622208\pi$
$230$	0	0
$231$	− 37.9755i	− 2.49860i
$232$	0	0
$233$	− 7.59285i	− 0.497424i	−0.968577	−	0.248712i	$-0.919993\pi$
$233$	− 7.59285i	− 0.497424i	0.968577	−	0.248712i	$-0.0800073\pi$
$234$	0	0
$235$	− 8.27322i	− 0.539685i
$236$	0	0
$237$	−20.5223	−1.33306
$238$	0	0
$239$	2.11104	0.136552	0.0682760	−	0.997666i	$-0.478250\pi$
$239$	2.11104	0.136552	0.0682760	+	0.997666i	$0.478250\pi$
$240$	0	0
$241$	18.9917i	1.22336i	0.791103	+	0.611682i	$0.209507\pi$
$241$	18.9917i	1.22336i	−0.791103	+	0.611682i	$0.790493\pi$
$242$	0	0
$243$	− 4.30169i	− 0.275954i
$244$	0	0
$245$	− 30.1637i	− 1.92709i
$246$	0	0
$247$	−4.29231	−0.273113
$248$	0	0
$249$	− 14.6583i	− 0.928930i
$250$	0	0
$251$	13.3230	0.840942	0.420471	−	0.907306i	$-0.361865\pi$
$251$	13.3230	0.840942	0.420471	+	0.907306i	$0.361865\pi$
$252$	0	0
$253$	2.26355	0.142308
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.94183	0.121128	0.0605640	−	0.998164i	$-0.480710\pi$
$257$	1.94183	0.121128	0.0605640	+	0.998164i	$0.480710\pi$
$258$	0	0
$259$	28.4869	1.77009
$260$	0	0
$261$	3.53601i	0.218873i
$262$	0	0
$263$	−31.0706	−1.91589	−0.957946	−	0.286947i	$-0.907360\pi$
$263$	−31.0706	−1.91589	−0.957946	+	0.286947i	$0.907360\pi$
$264$	0	0
$265$	21.3139i	1.30930i
$266$	0	0
$267$	− 20.9056i	− 1.27940i
$268$	0	0
$269$	− 5.73515i	− 0.349678i	−0.984597	−	0.174839i	$-0.944059\pi$
$269$	− 5.73515i	− 0.349678i	0.984597	−	0.174839i	$-0.0559405\pi$
$270$	0	0
$271$	−21.1400	−1.28416	−0.642082	−	0.766636i	$-0.721929\pi$
$271$	−21.1400	−1.28416	−0.642082	+	0.766636i	$0.721929\pi$
$272$	0	0
$273$	−41.9211	−2.53718
$274$	0	0
$275$	1.05638i	0.0637020i
$276$	0	0
$277$	1.31181i	0.0788189i	0.999223	+	0.0394095i	$0.0125477\pi$
$277$	1.31181i	0.0788189i	−0.999223	+	0.0394095i	$0.987452\pi$
$278$	0	0
$279$	1.34664i	0.0806215i
$280$	0	0
$281$	28.7710	1.71633	0.858167	−	0.513371i	$-0.171604\pi$
$281$	28.7710	1.71633	0.858167	+	0.513371i	$0.171604\pi$
$282$	0	0
$283$	1.47735i	0.0878193i	0.999036	+	0.0439097i	$0.0139814\pi$
$283$	1.47735i	0.0878193i	−0.999036	+	0.0439097i	$0.986019\pi$
$284$	0	0
$285$	−2.63978	−0.156367
$286$	0	0
$287$	−1.78695	−0.105480
$288$	0	0
$289$	0	0
$290$	0	0
$291$	25.8892	1.51765
$292$	0	0
$293$	−21.8020	−1.27369	−0.636843	−	0.770994i	$-0.719760\pi$
$293$	−21.8020	−1.27369	−0.636843	+	0.770994i	$0.719760\pi$
$294$	0	0
$295$	3.35961i	0.195604i
$296$	0	0
$297$	28.4715	1.65208
$298$	0	0
$299$	− 2.49873i	− 0.144505i
$300$	0	0
$301$	− 26.3576i	− 1.51923i
$302$	0	0
$303$	0.976599i	0.0561041i
$304$	0	0
$305$	−7.30601	−0.418341
$306$	0	0
$307$	21.4219	1.22261	0.611306	−	0.791395i	$-0.290645\pi$
$307$	21.4219	1.22261	0.611306	+	0.791395i	$0.290645\pi$
$308$	0	0
$309$	− 17.9614i	− 1.02179i
$310$	0	0
$311$	16.3635i	0.927892i	0.885864	+	0.463946i	$0.153567\pi$
$311$	16.3635i	0.927892i	−0.885864	+	0.463946i	$0.846433\pi$
$312$	0	0
$313$	9.55955i	0.540338i	0.962813	+	0.270169i	$0.0870795\pi$
$313$	9.55955i	0.540338i	−0.962813	+	0.270169i	$0.912920\pi$
$314$	0	0
$315$	4.16341	0.234581
$316$	0	0
$317$	− 8.82782i	− 0.495820i	−0.968783	−	0.247910i	$-0.920256\pi$
$317$	− 8.82782i	− 0.495820i	0.968783	−	0.247910i	$-0.0797437\pi$
$318$	0	0
$319$	−43.9506	−2.46076
$320$	0	0
$321$	2.03892	0.113801
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1.16613	0.0646855
$326$	0	0
$327$	−32.5684	−1.80104
$328$	0	0
$329$	17.2177i	0.949243i
$330$	0	0
$331$	−28.8976	−1.58836	−0.794179	−	0.607684i	$-0.792099\pi$
$331$	−28.8976	−1.58836	−0.794179	+	0.607684i	$0.792099\pi$
$332$	0	0
$333$	2.60699i	0.142862i
$334$	0	0
$335$	− 6.48016i	− 0.354049i
$336$	0	0
$337$	2.58235i	0.140669i	0.997523	+	0.0703347i	$0.0224067\pi$
$337$	2.58235i	0.140669i	−0.997523	+	0.0703347i	$0.977593\pi$
$338$	0	0
$339$	−29.3631	−1.59478
$340$	0	0
$341$	−16.7380	−0.906415
$342$	0	0
$343$	30.8705i	1.66685i
$344$	0	0
$345$	− 1.53672i	− 0.0827342i
$346$	0	0
$347$	− 10.2310i	− 0.549229i	−0.961554	−	0.274615i	$-0.911450\pi$
$347$	− 10.2310i	− 0.549229i	0.961554	−	0.274615i	$-0.0885503\pi$
$348$	0	0
$349$	0.672835	0.0360160	0.0180080	−	0.999838i	$-0.494268\pi$
$349$	0.672835	0.0360160	0.0180080	+	0.999838i	$0.494268\pi$
$350$	0	0
$351$	− 31.4296i	− 1.67759i
$352$	0	0
$353$	8.63396	0.459540	0.229770	−	0.973245i	$-0.426203\pi$
$353$	8.63396	0.459540	0.229770	+	0.973245i	$0.426203\pi$
$354$	0	0
$355$	11.3502	0.602404
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.8285	−1.09929	−0.549643	−	0.835400i	$-0.685236\pi$
$359$	−20.8285	−1.09929	−0.549643	+	0.835400i	$0.685236\pi$
$360$	0	0
$361$	−18.4375	−0.970394
$362$	0	0
$363$	25.5181i	1.33935i
$364$	0	0
$365$	12.9344	0.677016
$366$	0	0
$367$	0.287929i	0.0150298i	0.999972	+	0.00751489i	$0.00239209\pi$
$367$	0.287929i	0.0150298i	−0.999972	+	0.00751489i	$0.997608\pi$
$368$	0	0
$369$	− 0.163534i	− 0.00851323i
$370$	0	0
$371$	− 44.3571i	− 2.30291i
$372$	0	0
$373$	−10.2872	−0.532652	−0.266326	−	0.963883i	$-0.585810\pi$
$373$	−10.2872	−0.532652	−0.266326	+	0.963883i	$0.585810\pi$
$374$	0	0
$375$	18.3156	0.945813
$376$	0	0
$377$	48.5170i	2.49875i
$378$	0	0
$379$	− 20.3610i	− 1.04588i	−0.852371	−	0.522938i	$-0.824836\pi$
$379$	− 20.3610i	− 1.04588i	0.852371	−	0.522938i	$-0.175164\pi$
$380$	0	0
$381$	18.8178i	0.964064i
$382$	0	0
$383$	−30.6482	−1.56605	−0.783025	−	0.621990i	$-0.786324\pi$
$383$	−30.6482	−1.56605	−0.783025	+	0.621990i	$0.786324\pi$
$384$	0	0
$385$	51.7488i	2.63736i
$386$	0	0
$387$	2.41213	0.122616
$388$	0	0
$389$	1.61640	0.0819546	0.0409773	−	0.999160i	$-0.486953\pi$
$389$	1.61640	0.0819546	0.0409773	+	0.999160i	$0.486953\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	14.3848	0.725619
$394$	0	0
$395$	27.9655	1.40710
$396$	0	0
$397$	5.29986i	0.265993i	0.991117	+	0.132996i	$0.0424598\pi$
$397$	5.29986i	0.265993i	−0.991117	+	0.132996i	$0.957540\pi$
$398$	0	0
$399$	5.49375	0.275031
$400$	0	0
$401$	9.37263i	0.468047i	0.972231	+	0.234023i	$0.0751892\pi$
$401$	9.37263i	0.468047i	−0.972231	+	0.234023i	$0.924811\pi$
$402$	0	0
$403$	18.4771i	0.920409i
$404$	0	0
$405$	− 16.5888i	− 0.824306i
$406$	0	0
$407$	−32.4035	−1.60618
$408$	0	0
$409$	−8.23755	−0.407321	−0.203660	−	0.979042i	$-0.565284\pi$
$409$	−8.23755	−0.407321	−0.203660	+	0.979042i	$0.565284\pi$
$410$	0	0
$411$	− 6.82031i	− 0.336421i
$412$	0	0
$413$	− 6.99181i	− 0.344044i
$414$	0	0
$415$	19.9747i	0.980519i
$416$	0	0
$417$	−23.2180	−1.13699
$418$	0	0
$419$	33.0196i	1.61311i	0.591156	+	0.806557i	$0.298672\pi$
$419$	33.0196i	1.61311i	−0.591156	+	0.806557i	$0.701328\pi$
$420$	0	0
$421$	−28.0885	−1.36895	−0.684476	−	0.729036i	$-0.739969\pi$
$421$	−28.0885	−1.36895	−0.684476	+	0.729036i	$0.739969\pi$
$422$	0	0
$423$	−1.57569	−0.0766127
$424$	0	0
$425$	0	0
$426$	0	0
$427$	15.2048	0.735813
$428$	0	0
$429$	47.6847	2.30224
$430$	0	0
$431$	32.3623i	1.55884i	0.626503	+	0.779419i	$0.284486\pi$
$431$	32.3623i	1.55884i	−0.626503	+	0.779419i	$0.715514\pi$
$432$	0	0
$433$	24.7892	1.19129	0.595645	−	0.803248i	$-0.296896\pi$
$433$	24.7892	1.19129	0.595645	+	0.803248i	$0.296896\pi$
$434$	0	0
$435$	29.8380i	1.43062i
$436$	0	0
$437$	0.327458i	0.0156644i
$438$	0	0
$439$	5.18883i	0.247649i	0.992304	+	0.123825i	$0.0395160\pi$
$439$	5.18883i	0.247649i	−0.992304	+	0.123825i	$0.960484\pi$
$440$	0	0
$441$	−5.74488	−0.273566
$442$	0	0
$443$	−7.08823	−0.336772	−0.168386	−	0.985721i	$-0.553855\pi$
$443$	−7.08823	−0.336772	−0.168386	+	0.985721i	$0.553855\pi$
$444$	0	0
$445$	28.4879i	1.35045i
$446$	0	0
$447$	9.37355i	0.443354i
$448$	0	0
$449$	24.2953i	1.14657i	0.819357	+	0.573283i	$0.194331\pi$
$449$	24.2953i	1.14657i	−0.819357	+	0.573283i	$0.805669\pi$
$450$	0	0
$451$	2.03263	0.0957130
$452$	0	0
$453$	− 12.7144i	− 0.597374i
$454$	0	0
$455$	57.1254	2.67808
$456$	0	0
$457$	20.5422	0.960923	0.480462	−	0.877016i	$-0.340469\pi$
$457$	20.5422	0.960923	0.480462	+	0.877016i	$0.340469\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−24.8856	−1.15904	−0.579519	−	0.814959i	$-0.696760\pi$
$461$	−24.8856	−1.15904	−0.579519	+	0.814959i	$0.696760\pi$
$462$	0	0
$463$	15.2279	0.707703	0.353851	−	0.935302i	$-0.384872\pi$
$463$	15.2279	0.707703	0.353851	+	0.935302i	$0.384872\pi$
$464$	0	0
$465$	11.3634i	0.526967i
$466$	0	0
$467$	33.2698	1.53954	0.769772	−	0.638319i	$-0.220370\pi$
$467$	33.2698	1.53954	0.769772	+	0.638319i	$0.220370\pi$
$468$	0	0
$469$	13.4861i	0.622731i
$470$	0	0
$471$	19.8791i	0.915979i
$472$	0	0
$473$	29.9815i	1.37855i
$474$	0	0
$475$	−0.152822	−0.00701194
$476$	0	0
$477$	4.05937	0.185866
$478$	0	0
$479$	− 10.6962i	− 0.488723i	−0.969684	−	0.244362i	$-0.921422\pi$
$479$	− 10.6962i	− 0.488723i	0.969684	−	0.244362i	$-0.0785784\pi$
$480$	0	0
$481$	35.7701i	1.63098i
$482$	0	0
$483$	3.19813i	0.145520i
$484$	0	0
$485$	−35.2789	−1.60193
$486$	0	0
$487$	19.1335i	0.867020i	0.901149	+	0.433510i	$0.142725\pi$
$487$	19.1335i	0.867020i	−0.901149	+	0.433510i	$0.857275\pi$
$488$	0	0
$489$	26.3388	1.19108
$490$	0	0
$491$	−6.56476	−0.296263	−0.148132	−	0.988968i	$-0.547326\pi$
$491$	−6.56476	−0.296263	−0.148132	+	0.988968i	$0.547326\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−4.73583	−0.212860
$496$	0	0
$497$	−23.6213	−1.05956
$498$	0	0
$499$	− 25.9028i	− 1.15957i	−0.814770	−	0.579784i	$-0.803137\pi$
$499$	− 25.9028i	− 1.15957i	0.814770	−	0.579784i	$-0.196863\pi$
$500$	0	0
$501$	12.1640	0.543448
$502$	0	0
$503$	− 2.02620i	− 0.0903437i	−0.998979	−	0.0451719i	$-0.985616\pi$
$503$	− 2.02620i	− 0.0903437i	0.998979	−	0.0451719i	$-0.0143835\pi$
$504$	0	0
$505$	− 1.33080i	− 0.0592199i
$506$	0	0
$507$	− 31.7463i	− 1.40990i
$508$	0	0
$509$	−1.82543	−0.0809106	−0.0404553	−	0.999181i	$-0.512881\pi$
$509$	−1.82543	−0.0809106	−0.0404553	+	0.999181i	$0.512881\pi$
$510$	0	0
$511$	−26.9182	−1.19079
$512$	0	0
$513$	4.11885i	0.181852i
$514$	0	0
$515$	24.4758i	1.07853i
$516$	0	0
$517$	− 19.5849i	− 0.861345i
$518$	0	0
$519$	−16.2396	−0.712841
$520$	0	0
$521$	− 4.72853i	− 0.207161i	−0.994621	−	0.103580i	$-0.966970\pi$
$521$	− 4.72853i	− 0.207161i	0.994621	−	0.103580i	$-0.0330299\pi$
$522$	0	0
$523$	−27.9297	−1.22128	−0.610640	−	0.791908i	$-0.709088\pi$
$523$	−27.9297	−1.22128	−0.610640	+	0.791908i	$0.709088\pi$
$524$	0	0
$525$	−1.49254	−0.0651397
$526$	0	0
$527$	0	0
$528$	0	0
$529$	22.8094	0.991712
$530$	0	0
$531$	0.639860	0.0277676
$532$	0	0
$533$	− 2.24382i	− 0.0971907i
$534$	0	0
$535$	−2.77841	−0.120121
$536$	0	0
$537$	3.26351i	0.140831i
$538$	0	0
$539$	− 71.4056i	− 3.07566i
$540$	0	0
$541$	13.9762i	0.600883i	0.953800	+	0.300442i	$0.0971341\pi$
$541$	13.9762i	0.600883i	−0.953800	+	0.300442i	$0.902866\pi$
$542$	0	0
$543$	−7.69012	−0.330015
$544$	0	0
$545$	44.3806	1.90106
$546$	0	0
$547$	8.41081i	0.359620i	0.983701	+	0.179810i	$0.0575483\pi$
$547$	8.41081i	0.359620i	−0.983701	+	0.179810i	$0.942452\pi$
$548$	0	0
$549$	1.39148i	0.0593869i
$550$	0	0
$551$	− 6.35814i	− 0.270866i
$552$	0	0
$553$	−58.2000	−2.47492
$554$	0	0
$555$	21.9987i	0.933792i
$556$	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
$559$	33.0965	1.39983
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.8884	−1.21750	−0.608750	−	0.793362i	$-0.708329\pi$
$563$	−28.8884	−1.21750	−0.608750	+	0.793362i	$0.708329\pi$
$564$	0	0
$565$	40.0128	1.68335
$566$	0	0
$567$	34.5237i	1.44986i
$568$	0	0
$569$	−17.0273	−0.713822	−0.356911	−	0.934138i	$-0.616170\pi$
$569$	−17.0273	−0.713822	−0.356911	+	0.934138i	$0.616170\pi$
$570$	0	0
$571$	22.9391i	0.959972i	0.877276	+	0.479986i	$0.159358\pi$
$571$	22.9391i	0.959972i	−0.877276	+	0.479986i	$0.840642\pi$
$572$	0	0
$573$	− 14.9638i	− 0.625121i
$574$	0	0
$575$	− 0.0889636i	− 0.00371004i
$576$	0	0
$577$	−41.3828	−1.72279	−0.861393	−	0.507938i	$-0.830408\pi$
$577$	−41.3828	−1.72279	−0.861393	+	0.507938i	$0.830408\pi$
$578$	0	0
$579$	−32.1790	−1.33731
$580$	0	0
$581$	− 41.5701i	− 1.72462i
$582$	0	0
$583$	50.4557i	2.08966i
$584$	0	0
$585$	5.22787i	0.216146i
$586$	0	0
$587$	32.2870	1.33263	0.666313	−	0.745672i	$-0.267872\pi$
$587$	32.2870	1.33263	0.666313	+	0.745672i	$0.267872\pi$
$588$	0	0
$589$	− 2.42142i	− 0.0997728i
$590$	0	0
$591$	−1.64440	−0.0676414
$592$	0	0
$593$	−25.8513	−1.06158	−0.530792	−	0.847502i	$-0.678105\pi$
$593$	−25.8513	−1.06158	−0.530792	+	0.847502i	$0.678105\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−14.6656	−0.600222
$598$	0	0
$599$	−13.4549	−0.549751	−0.274876	−	0.961480i	$-0.588637\pi$
$599$	−13.4549	−0.549751	−0.274876	+	0.961480i	$0.588637\pi$
$600$	0	0
$601$	39.0351i	1.59228i	0.605115	+	0.796138i	$0.293127\pi$
$601$	39.0351i	1.59228i	−0.605115	+	0.796138i	$0.706873\pi$
$602$	0	0
$603$	−1.23419	−0.0502601
$604$	0	0
$605$	− 34.7733i	− 1.41374i
$606$	0	0
$607$	17.5713i	0.713198i	0.934258	+	0.356599i	$0.116064\pi$
$607$	17.5713i	0.713198i	−0.934258	+	0.356599i	$0.883936\pi$
$608$	0	0
$609$	− 62.0971i	− 2.51630i
$610$	0	0
$611$	−21.6198	−0.874643
$612$	0	0
$613$	−35.9993	−1.45400	−0.726999	−	0.686638i	$-0.759086\pi$
$613$	−35.9993	−1.45400	−0.726999	+	0.686638i	$0.759086\pi$
$614$	0	0
$615$	− 1.37995i	− 0.0556451i
$616$	0	0
$617$	− 40.6511i	− 1.63655i	−0.574827	−	0.818275i	$-0.694931\pi$
$617$	− 40.6511i	− 1.63655i	0.574827	−	0.818275i	$-0.305069\pi$
$618$	0	0
$619$	− 28.0856i	− 1.12885i	−0.825483	−	0.564427i	$-0.809097\pi$
$619$	− 28.0856i	− 1.12885i	0.825483	−	0.564427i	$-0.190903\pi$
$620$	0	0
$621$	−2.39774	−0.0962182
$622$	0	0
$623$	− 59.2872i	− 2.37529i
$624$	0	0
$625$	−23.9397	−0.957587
$626$	0	0
$627$	−6.24907	−0.249564
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−9.48010	−0.377397	−0.188698	−	0.982035i	$-0.560427\pi$
$631$	−9.48010	−0.377397	−0.188698	+	0.982035i	$0.560427\pi$
$632$	0	0
$633$	3.07498	0.122220
$634$	0	0
$635$	− 25.6428i	− 1.01760i
$636$	0	0
$637$	−78.8246	−3.12314
$638$	0	0
$639$	− 2.16172i	− 0.0855162i
$640$	0	0
$641$	− 15.8059i	− 0.624295i	−0.950034	−	0.312147i	$-0.898952\pi$
$641$	− 15.8059i	− 0.624295i	0.950034	−	0.312147i	$-0.101048\pi$
$642$	0	0
$643$	28.9071i	1.13998i	0.821650	+	0.569992i	$0.193054\pi$
$643$	28.9071i	1.13998i	−0.821650	+	0.569992i	$0.806946\pi$
$644$	0	0
$645$	20.3544	0.801453
$646$	0	0
$647$	−44.3941	−1.74531	−0.872655	−	0.488337i	$-0.837604\pi$
$647$	−44.3941	−1.74531	−0.872655	+	0.488337i	$0.837604\pi$
$648$	0	0
$649$	7.95310i	0.312187i
$650$	0	0
$651$	− 23.6489i	− 0.926873i
$652$	0	0
$653$	46.3129i	1.81237i	0.422887	+	0.906183i	$0.361017\pi$
$653$	46.3129i	1.81237i	−0.422887	+	0.906183i	$0.638983\pi$
$654$	0	0
$655$	−19.6021	−0.765917
$656$	0	0
$657$	− 2.46344i	− 0.0961080i
$658$	0	0
$659$	−17.8181	−0.694093	−0.347047	−	0.937848i	$-0.612815\pi$
$659$	−17.8181	−0.694093	−0.347047	+	0.937848i	$0.612815\pi$
$660$	0	0
$661$	9.94533	0.386829	0.193414	−	0.981117i	$-0.438044\pi$
$661$	9.94533	0.386829	0.193414	+	0.981117i	$0.438044\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.48628	−0.290305
$666$	0	0
$667$	3.70133	0.143316
$668$	0	0
$669$	− 25.2790i	− 0.977342i
$670$	0	0
$671$	−17.2953	−0.667678
$672$	0	0
$673$	− 35.2526i	− 1.35889i	−0.733727	−	0.679444i	$-0.762221\pi$
$673$	− 35.2526i	− 1.35889i	0.733727	−	0.679444i	$-0.237779\pi$
$674$	0	0
$675$	− 1.11901i	− 0.0430706i
$676$	0	0
$677$	− 46.4301i	− 1.78445i	−0.451588	−	0.892226i	$-0.649142\pi$
$677$	− 46.4301i	− 1.78445i	0.451588	−	0.892226i	$-0.350858\pi$
$678$	0	0
$679$	73.4203	2.81761
$680$	0	0
$681$	21.0901	0.808174
$682$	0	0
$683$	35.6740i	1.36503i	0.730873	+	0.682514i	$0.239113\pi$
$683$	35.6740i	1.36503i	−0.730873	+	0.682514i	$0.760887\pi$
$684$	0	0
$685$	9.29397i	0.355104i
$686$	0	0
$687$	18.2192i	0.695106i
$688$	0	0
$689$	55.6980	2.12192
$690$	0	0
$691$	− 10.0109i	− 0.380832i	−0.981703	−	0.190416i	$-0.939016\pi$
$691$	− 10.0109i	− 0.380832i	0.981703	−	0.190416i	$-0.0609837\pi$
$692$	0	0
$693$	9.85591	0.374395
$694$	0	0
$695$	31.6390	1.20013
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−12.2028	−0.461551
$700$	0	0
$701$	1.31324	0.0496002	0.0248001	−	0.999692i	$-0.492105\pi$
$701$	1.31324	0.0496002	0.0248001	+	0.999692i	$0.492105\pi$
$702$	0	0
$703$	− 4.68767i	− 0.176799i
$704$	0	0
$705$	−13.2962	−0.500764
$706$	0	0
$707$	2.76958i	0.104161i
$708$	0	0
$709$	− 27.7345i	− 1.04159i	−0.853682	−	0.520795i	$-0.825635\pi$
$709$	− 27.7345i	− 1.04159i	0.853682	−	0.520795i	$-0.174365\pi$
$710$	0	0
$711$	− 5.32622i	− 0.199749i
$712$	0	0
$713$	1.40960	0.0527901
$714$	0	0
$715$	−64.9795	−2.43010
$716$	0	0
$717$	− 3.39274i	− 0.126704i
$718$	0	0
$719$	− 21.8968i	− 0.816613i	−0.912845	−	0.408307i	$-0.866119\pi$
$719$	− 21.8968i	− 0.816613i	0.912845	−	0.408307i	$-0.133881\pi$
$720$	0	0
$721$	− 50.9376i	− 1.89701i
$722$	0	0
$723$	30.5223	1.13514
$724$	0	0
$725$	1.72738i	0.0641532i
$726$	0	0
$727$	−20.1780	−0.748362	−0.374181	−	0.927356i	$-0.622076\pi$
$727$	−20.1780	−0.748362	−0.374181	+	0.927356i	$0.622076\pi$
$728$	0	0
$729$	−29.6375	−1.09769
$730$	0	0
$731$	0	0
$732$	0	0
$733$	4.38032	0.161791	0.0808954	−	0.996723i	$-0.474222\pi$
$733$	4.38032	0.161791	0.0808954	+	0.996723i	$0.474222\pi$
$734$	0	0
$735$	−48.4772	−1.78811
$736$	0	0
$737$	− 15.3403i	− 0.565067i
$738$	0	0
$739$	36.8554	1.35575	0.677875	−	0.735177i	$-0.262901\pi$
$739$	36.8554	1.35575	0.677875	+	0.735177i	$0.262901\pi$
$740$	0	0
$741$	6.89834i	0.253417i
$742$	0	0
$743$	− 37.9459i	− 1.39210i	−0.717993	−	0.696051i	$-0.754939\pi$
$743$	− 37.9459i	− 1.39210i	0.717993	−	0.696051i	$-0.245061\pi$
$744$	0	0
$745$	− 12.7732i	− 0.467976i
$746$	0	0
$747$	3.80432	0.139193
$748$	0	0
$749$	5.78226	0.211279
$750$	0	0
$751$	9.08102i	0.331371i	0.986179	+	0.165686i	$0.0529837\pi$
$751$	9.08102i	0.331371i	−0.986179	+	0.165686i	$0.947016\pi$
$752$	0	0
$753$	− 21.4119i	− 0.780295i
$754$	0	0
$755$	17.3258i	0.630549i
$756$	0	0
$757$	13.9433	0.506779	0.253389	−	0.967364i	$-0.418455\pi$
$757$	13.9433	0.506779	0.253389	+	0.967364i	$0.418455\pi$
$758$	0	0
$759$	− 3.63783i	− 0.132045i
$760$	0	0
$761$	−19.0549	−0.690740	−0.345370	−	0.938467i	$-0.612247\pi$
$761$	−19.0549	−0.690740	−0.345370	+	0.938467i	$0.612247\pi$
$762$	0	0
$763$	−92.3623	−3.34374
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.77942	0.317006
$768$	0	0
$769$	41.8070	1.50760	0.753799	−	0.657105i	$-0.228219\pi$
$769$	41.8070	1.50760	0.753799	+	0.657105i	$0.228219\pi$
$770$	0	0
$771$	− 3.12079i	− 0.112392i
$772$	0	0
$773$	22.6810	0.815779	0.407890	−	0.913031i	$-0.366265\pi$
$773$	22.6810	0.815779	0.407890	+	0.913031i	$0.366265\pi$
$774$	0	0
$775$	0.657850i	0.0236307i
$776$	0	0
$777$	− 45.7823i	− 1.64243i
$778$	0	0
$779$	0.294052i	0.0105355i
$780$	0	0
$781$	26.8689	0.961446
$782$	0	0
$783$	46.5563	1.66379
$784$	0	0
$785$	− 27.0890i	− 0.966848i
$786$	0	0
$787$	− 14.7018i	− 0.524064i	−0.965059	−	0.262032i	$-0.915607\pi$
$787$	− 14.7018i	− 0.524064i	0.965059	−	0.262032i	$-0.0843925\pi$
$788$	0	0
$789$	49.9347i	1.77772i
$790$	0	0
$791$	−83.2721	−2.96081
$792$	0	0
$793$	19.0923i	0.677987i
$794$	0	0
$795$	34.2543	1.21488
$796$	0	0
$797$	8.21691	0.291058	0.145529	−	0.989354i	$-0.453512\pi$
$797$	8.21691	0.291058	0.145529	+	0.989354i	$0.453512\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	5.42571	0.191708
$802$	0	0
$803$	30.6192	1.08053
$804$	0	0
$805$	− 4.35806i	− 0.153601i
$806$	0	0
$807$	−9.21718	−0.324460
$808$	0	0
$809$	35.5939i	1.25142i	0.780057	+	0.625708i	$0.215190\pi$
$809$	35.5939i	1.25142i	−0.780057	+	0.625708i	$0.784810\pi$
$810$	0	0
$811$	5.67292i	0.199203i	0.995027	+	0.0996015i	$0.0317568\pi$
$811$	5.67292i	0.199203i	−0.995027	+	0.0996015i	$0.968243\pi$
$812$	0	0
$813$	33.9749i	1.19155i
$814$	0	0
$815$	−35.8916	−1.25723
$816$	0	0
$817$	−4.33729	−0.151742
$818$	0	0
$819$	− 10.8799i	− 0.380176i
$820$	0	0
$821$	3.95355i	0.137980i	0.997617	+	0.0689899i	$0.0219776\pi$
$821$	3.95355i	0.137980i	−0.997617	+	0.0689899i	$0.978022\pi$
$822$	0	0
$823$	4.31091i	0.150269i	0.997173	+	0.0751344i	$0.0239386\pi$
$823$	4.31091i	0.150269i	−0.997173	+	0.0751344i	$0.976061\pi$
$824$	0	0
$825$	1.69775	0.0591079
$826$	0	0
$827$	− 41.8738i	− 1.45609i	−0.685527	−	0.728047i	$-0.740428\pi$
$827$	− 41.8738i	− 1.45609i	0.685527	−	0.728047i	$-0.259572\pi$
$828$	0	0
$829$	−22.4808	−0.780792	−0.390396	−	0.920647i	$-0.627662\pi$
$829$	−22.4808	−0.780792	−0.390396	+	0.920647i	$0.627662\pi$
$830$	0	0
$831$	2.10826	0.0731346
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−16.5758	−0.573629
$836$	0	0
$837$	17.7304	0.612851
$838$	0	0
$839$	− 12.0055i	− 0.414475i	−0.978291	−	0.207237i	$-0.933553\pi$
$839$	− 12.0055i	− 0.414475i	0.978291	−	0.207237i	$-0.0664473\pi$
$840$	0	0
$841$	−42.8675	−1.47819
$842$	0	0
$843$	− 46.2390i	− 1.59255i
$844$	0	0
$845$	43.2604i	1.48820i
$846$	0	0
$847$	72.3681i	2.48660i
$848$	0	0
$849$	2.37430	0.0814859
$850$	0	0
$851$	2.72888	0.0935448
$852$	0	0
$853$	11.2288i	0.384465i	0.981349	+	0.192233i	$0.0615728\pi$
$853$	11.2288i	0.384465i	−0.981349	+	0.192233i	$0.938427\pi$
$854$	0	0
$855$	− 0.685112i	− 0.0234303i
$856$	0	0
$857$	− 6.73481i	− 0.230057i	−0.993362	−	0.115028i	$-0.963304\pi$
$857$	− 6.73481i	− 0.230057i	0.993362	−	0.115028i	$-0.0366959\pi$
$858$	0	0
$859$	7.84826	0.267779	0.133890	−	0.990996i	$-0.457253\pi$
$859$	7.84826	0.267779	0.133890	+	0.990996i	$0.457253\pi$
$860$	0	0
$861$	2.87187i	0.0978732i
$862$	0	0
$863$	21.0758	0.717430	0.358715	−	0.933447i	$-0.383215\pi$
$863$	21.0758	0.717430	0.358715	+	0.933447i	$0.383215\pi$
$864$	0	0
$865$	22.1296	0.752428
$866$	0	0
$867$	0	0
$868$	0	0
$869$	66.2019	2.24574
$870$	0	0
$871$	−16.9341	−0.573791
$872$	0	0
$873$	6.71911i	0.227407i
$874$	0	0
$875$	51.9421	1.75596
$876$	0	0
$877$	18.2845i	0.617424i	0.951156	+	0.308712i	$0.0998979\pi$
$877$	18.2845i	0.617424i	−0.951156	+	0.308712i	$0.900102\pi$
$878$	0	0
$879$	35.0388i	1.18183i
$880$	0	0
$881$	21.8193i	0.735110i	0.930002	+	0.367555i	$0.119805\pi$
$881$	21.8193i	0.735110i	−0.930002	+	0.367555i	$0.880195\pi$
$882$	0	0
$883$	21.8187	0.734257	0.367128	−	0.930170i	$-0.380341\pi$
$883$	21.8187	0.734257	0.367128	+	0.930170i	$0.380341\pi$
$884$	0	0
$885$	5.39935	0.181497
$886$	0	0
$887$	− 44.8589i	− 1.50621i	−0.657898	−	0.753107i	$-0.728554\pi$
$887$	− 44.8589i	− 1.50621i	0.657898	−	0.753107i	$-0.271446\pi$
$888$	0	0
$889$	53.3662i	1.78985i
$890$	0	0
$891$	− 39.2703i	− 1.31560i
$892$	0	0
$893$	2.83327	0.0948117
$894$	0	0
$895$	− 4.44715i	− 0.148652i
$896$	0	0
$897$	−4.01580	−0.134084
$898$	0	0
$899$	−27.3698	−0.912835
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−42.3603	−1.40966
$904$	0	0
$905$	10.4793	0.348342
$906$	0	0
$907$	− 2.95342i	− 0.0980666i	−0.998797	−	0.0490333i	$-0.984386\pi$
$907$	− 2.95342i	− 0.0980666i	0.998797	−	0.0490333i	$-0.0156140\pi$
$908$	0	0
$909$	−0.253460	−0.00840675
$910$	0	0
$911$	11.5101i	0.381346i	0.981654	+	0.190673i	$0.0610671\pi$
$911$	11.5101i	0.381346i	−0.981654	+	0.190673i	$0.938933\pi$
$912$	0	0
$913$	47.2855i	1.56492i
$914$	0	0
$915$	11.7418i	0.388171i
$916$	0	0
$917$	40.7947	1.34716
$918$	0	0
$919$	12.1680	0.401384	0.200692	−	0.979654i	$-0.435681\pi$
$919$	12.1680	0.401384	0.200692	+	0.979654i	$0.435681\pi$
$920$	0	0
$921$	− 34.4279i	− 1.13444i
$922$	0	0
$923$	− 29.6606i	− 0.976290i
$924$	0	0
$925$	1.27354i	0.0418739i
$926$	0	0
$927$	4.66159	0.153107
$928$	0	0
$929$	33.3053i	1.09271i	0.837553	+	0.546355i	$0.183985\pi$
$929$	33.3053i	1.09271i	−0.837553	+	0.546355i	$0.816015\pi$
$930$	0	0
$931$	10.3299	0.338550
$932$	0	0
$933$	26.2985	0.860974
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30.5252	0.997213	0.498607	−	0.866828i	$-0.333845\pi$
$937$	30.5252	0.997213	0.498607	+	0.866828i	$0.333845\pi$
$938$	0	0
$939$	15.3635	0.501369
$940$	0	0
$941$	− 37.8296i	− 1.23321i	−0.787273	−	0.616604i	$-0.788508\pi$
$941$	− 37.8296i	− 1.23321i	0.787273	−	0.616604i	$-0.211492\pi$
$942$	0	0
$943$	−0.171180	−0.00557437
$944$	0	0
$945$	− 54.8168i	− 1.78319i
$946$	0	0
$947$	32.0402i	1.04117i	0.853811	+	0.520583i	$0.174285\pi$
$947$	32.0402i	1.04117i	−0.853811	+	0.520583i	$0.825715\pi$
$948$	0	0
$949$	− 33.8005i	− 1.09721i
$950$	0	0
$951$	−14.1875	−0.460062
$952$	0	0
$953$	53.1556	1.72188	0.860940	−	0.508707i	$-0.169876\pi$
$953$	53.1556	1.72188	0.860940	+	0.508707i	$0.169876\pi$
$954$	0	0
$955$	20.3910i	0.659838i
$956$	0	0
$957$	70.6347i	2.28330i
$958$	0	0
$959$	− 19.3420i	− 0.624587i
$960$	0	0
$961$	20.5765	0.663759
$962$	0	0
$963$	0.529168i	0.0170522i
$964$	0	0
$965$	43.8500	1.41158
$966$	0	0
$967$	18.2854	0.588018	0.294009	−	0.955803i	$-0.405011\pi$
$967$	18.2854	0.588018	0.294009	+	0.955803i	$0.405011\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.4541	−0.399672	−0.199836	−	0.979829i	$-0.564041\pi$
$971$	−12.4541	−0.399672	−0.199836	+	0.979829i	$0.564041\pi$
$972$	0	0
$973$	−65.8451	−2.11090
$974$	0	0
$975$	− 1.87414i	− 0.0600205i
$976$	0	0
$977$	29.1828	0.933639	0.466819	−	0.884353i	$-0.345400\pi$
$977$	29.1828	0.933639	0.466819	+	0.884353i	$0.345400\pi$
$978$	0	0
$979$	67.4385i	2.15534i
$980$	0	0
$981$	− 8.45260i	− 0.269871i
$982$	0	0
$983$	25.0005i	0.797392i	0.917083	+	0.398696i	$0.130537\pi$
$983$	25.0005i	0.797392i	−0.917083	+	0.398696i	$0.869463\pi$
$984$	0	0
$985$	2.24080	0.0713979
$986$	0	0
$987$	27.6712	0.880785
$988$	0	0
$989$	− 2.52491i	− 0.0802874i
$990$	0	0
$991$	− 20.7858i	− 0.660282i	−0.943932	−	0.330141i	$-0.892904\pi$
$991$	− 20.7858i	− 0.660282i	0.943932	−	0.330141i	$-0.107096\pi$
$992$	0	0
$993$	46.4425i	1.47381i
$994$	0	0
$995$	19.9846	0.633555
$996$	0	0
$997$	5.83306i	0.184735i	0.995725	+	0.0923675i	$0.0294434\pi$
$997$	5.83306i	0.184735i	−0.995725	+	0.0923675i	$0.970557\pi$
$998$	0	0
$999$	34.3245	1.08598

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.b.o.577.4		12
17.4	even	4		2312.2.a.v.1.2	yes	6
17.13	even	4		2312.2.a.u.1.5	✓	6
17.16	even	2	inner	2312.2.b.o.577.9		12
68.47	odd	4		4624.2.a.br.1.2		6
68.55	odd	4		4624.2.a.bs.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.2.a.u.1.5	✓	6	17.13	even	4
2312.2.a.v.1.2	yes	6	17.4	even	4
2312.2.b.o.577.4		12	1.1	even	1	trivial
2312.2.b.o.577.9		12	17.16	even	2	inner
4624.2.a.br.1.2		6	68.47	odd	4
4624.2.a.bs.1.5		6	68.55	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)

\(f(q)\)	\(=\)	\(q-1.60714i q^{3} +2.19003i q^{5} -4.55776i q^{7} +0.417107 q^{9} +5.18440i q^{11} +5.72305 q^{13} +3.51968 q^{15} -0.750005 q^{19} -7.32495 q^{21} -0.436608i q^{23} +0.203761 q^{25} -5.49176i q^{27} +8.47747i q^{29} +3.22854i q^{31} +8.33205 q^{33} +9.98164 q^{35} +6.25019i q^{37} -9.19774i q^{39} -0.392067i q^{41} +5.78302 q^{43} +0.913477i q^{45} -3.77767 q^{47} -13.7732 q^{49} +9.73222 q^{53} -11.3540 q^{55} +1.20536i q^{57} +1.53405 q^{59} +3.33603i q^{61} -1.90107i q^{63} +12.5337i q^{65} -2.95893 q^{67} -0.701689 q^{69} -5.18265i q^{71} -5.90602i q^{73} -0.327472i q^{75} +23.6292 q^{77} -12.7694i q^{79} -7.57470 q^{81} +9.12073 q^{83} +13.6245 q^{87} +13.0080 q^{89} -26.0843i q^{91} +5.18871 q^{93} -1.64253i q^{95} +16.1089i q^{97} +2.16245i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{13} + 12 q^{15} - 12 q^{19} + 12 q^{21} - 12 q^{25} + 6 q^{33} + 42 q^{35} + 6 q^{47} - 18 q^{49} - 66 q^{53} - 102 q^{55} - 18 q^{67} - 6 q^{69} + 90 q^{77} - 36 q^{81} + 24 q^{83} - 30 q^{87}+ \cdots + 60 q^{93}+O(q^{100}) \) 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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