LMFDB - Embedded newform 2312.2.a.u.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,2,Mod(1,2312)] 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, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2312.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-6,0,-3,0,0,0,6,0,-6,0,-6] 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:	yes
Analytic conductor:	$18.4614129473$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.3418281.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 $ x^6 - 9x^4 - 4x^3 + 18x^2 + 12x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.60714$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.60714 q^{3} -2.19003 q^{5} -4.55776 q^{7} -0.417107 q^{9} +5.18440 q^{11} +5.72305 q^{13} -3.51968 q^{15} +0.750005 q^{19} -7.32495 q^{21} -0.436608 q^{23} -0.203761 q^{25} -5.49176 q^{27} -8.47747 q^{29} -3.22854 q^{31} +8.33205 q^{33} +9.98164 q^{35} -6.25019 q^{37} +9.19774 q^{39} -0.392067 q^{41} -5.78302 q^{43} +0.913477 q^{45} -3.77767 q^{47} +13.7732 q^{49} -9.73222 q^{53} -11.3540 q^{55} +1.20536 q^{57} -1.53405 q^{59} +3.33603 q^{61} +1.90107 q^{63} -12.5337 q^{65} -2.95893 q^{67} -0.701689 q^{69} +5.18265 q^{71} +5.90602 q^{73} -0.327472 q^{75} -23.6292 q^{77} -12.7694 q^{79} -7.57470 q^{81} -9.12073 q^{83} -13.6245 q^{87} +13.0080 q^{89} -26.0843 q^{91} -5.18871 q^{93} -1.64253 q^{95} -16.1089 q^{97} -2.16245 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{5} - 3 q^{7} + 6 q^{11} - 6 q^{13} - 6 q^{15} + 6 q^{19} + 6 q^{21} - 9 q^{23} + 6 q^{25} - 12 q^{27} - 27 q^{29} - 9 q^{31} + 3 q^{33} + 21 q^{35} - 15 q^{37} + 12 q^{39} - 21 q^{41} - 24 q^{45}+ \cdots + 42 q^{99}+O(q^{100}) $ 6 * q - 6 * q^5 - 3 * q^7 + 6 * q^11 - 6 * q^13 - 6 * q^15 + 6 * q^19 + 6 * q^21 - 9 * q^23 + 6 * q^25 - 12 * q^27 - 27 * q^29 - 9 * q^31 + 3 * q^33 + 21 * q^35 - 15 * q^37 + 12 * q^39 - 21 * q^41 - 24 * q^45 + 3 * q^47 + 9 * q^49 + 33 * q^53 - 51 * q^55 - 30 * q^57 - 18 * q^61 - 24 * q^63 + 9 * q^65 - 9 * q^67 - 3 * q^69 - 6 * q^71 - 27 * q^73 + 24 * q^75 - 45 * q^77 - 33 * q^79 - 18 * q^81 - 12 * q^83 + 15 * q^87 + 3 * q^89 - 30 * q^91 - 30 * q^93 - 3 * q^95 - 51 * q^97 + 42 * q^99

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
$3$	1.60714	0.927882	0.463941	−	0.885866i	$-0.346435\pi$
$3$	1.60714	0.927882	0.463941	+	0.885866i	$0.346435\pi$
$4$	0	0
$5$	−2.19003	−0.979412	−0.489706	−	0.871888i	$-0.662896\pi$
$5$	−2.19003	−0.979412	−0.489706	+	0.871888i	$0.662896\pi$
$6$	0	0
$7$	−4.55776	−1.72267	−0.861336	−	0.508036i	$-0.830372\pi$
$7$	−4.55776	−1.72267	−0.861336	+	0.508036i	$0.830372\pi$
$8$	0	0
$9$	−0.417107	−0.139036
$10$	0	0
$11$	5.18440	1.56316	0.781578	−	0.623808i	$-0.214415\pi$
$11$	5.18440	1.56316	0.781578	+	0.623808i	$0.214415\pi$
$12$	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	$-0.472581\pi$
$19$	0.750005	0.172063	0.0860314	+	0.996292i	$0.472581\pi$
$20$	0	0
$21$	−7.32495	−1.59844
$22$	0	0
$23$	−0.436608	−0.0910390	−0.0455195	−	0.998963i	$-0.514494\pi$
$23$	−0.436608	−0.0910390	−0.0455195	+	0.998963i	$0.514494\pi$
$24$	0	0
$25$	−0.203761	−0.0407522
$26$	0	0
$27$	−5.49176	−1.05689
$28$	0	0
$29$	−8.47747	−1.57423	−0.787113	−	0.616808i	$-0.788425\pi$
$29$	−8.47747	−1.57423	−0.787113	+	0.616808i	$0.788425\pi$
$30$	0	0
$31$	−3.22854	−0.579863	−0.289931	−	0.957047i	$-0.593632\pi$
$31$	−3.22854	−0.579863	−0.289931	+	0.957047i	$0.593632\pi$
$32$	0	0
$33$	8.33205	1.45042
$34$	0	0
$35$	9.98164	1.68720
$36$	0	0
$37$	−6.25019	−1.02752	−0.513762	−	0.857933i	$-0.671749\pi$
$37$	−6.25019	−1.02752	−0.513762	+	0.857933i	$0.671749\pi$
$38$	0	0
$39$	9.19774	1.47282
$40$	0	0
$41$	−0.392067	−0.0612306	−0.0306153	−	0.999531i	$-0.509747\pi$
$41$	−0.392067	−0.0612306	−0.0306153	+	0.999531i	$0.509747\pi$
$42$	0	0
$43$	−5.78302	−0.881902	−0.440951	−	0.897531i	$-0.645359\pi$
$43$	−5.78302	−0.881902	−0.440951	+	0.897531i	$0.645359\pi$
$44$	0	0
$45$	0.913477	0.136173
$46$	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	$-0.733025\pi$
$53$	−9.73222	−1.33682	−0.668412	+	0.743792i	$0.733025\pi$
$54$	0	0
$55$	−11.3540	−1.53097
$56$	0	0
$57$	1.20536	0.159654
$58$	0	0
$59$	−1.53405	−0.199716	−0.0998578	−	0.995002i	$-0.531839\pi$
$59$	−1.53405	−0.199716	−0.0998578	+	0.995002i	$0.531839\pi$
$60$	0	0
$61$	3.33603	0.427135	0.213568	−	0.976928i	$-0.431492\pi$
$61$	3.33603	0.427135	0.213568	+	0.976928i	$0.431492\pi$
$62$	0	0
$63$	1.90107	0.239512
$64$	0	0
$65$	−12.5337	−1.55461
$66$	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
$71$	5.18265	0.615067	0.307534	−	0.951537i	$-0.400496\pi$
$71$	5.18265	0.615067	0.307534	+	0.951537i	$0.400496\pi$
$72$	0	0
$73$	5.90602	0.691248	0.345624	−	0.938373i	$-0.387667\pi$
$73$	5.90602	0.691248	0.345624	+	0.938373i	$0.387667\pi$
$74$	0	0
$75$	−0.327472	−0.0378132
$76$	0	0
$77$	−23.6292	−2.69280
$78$	0	0
$79$	−12.7694	−1.43667	−0.718337	−	0.695695i	$-0.755096\pi$
$79$	−12.7694	−1.43667	−0.718337	+	0.695695i	$0.755096\pi$
$80$	0	0
$81$	−7.57470	−0.841634
$82$	0	0
$83$	−9.12073	−1.00113	−0.500565	−	0.865699i	$-0.666874\pi$
$83$	−9.12073	−1.00113	−0.500565	+	0.865699i	$0.666874\pi$
$84$	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
$91$	−26.0843	−2.73438
$92$	0	0
$93$	−5.18871	−0.538044
$94$	0	0
$95$	−1.64253	−0.168520
$96$	0	0
$97$	−16.1089	−1.63561	−0.817803	−	0.575498i	$-0.804808\pi$
$97$	−16.1089	−1.63561	−0.817803	+	0.575498i	$0.804808\pi$
$98$	0	0
$99$	−2.16245	−0.217334
$100$	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
$105$	16.0419	1.56553
$106$	0	0
$107$	−1.26866	−0.122646	−0.0613232	−	0.998118i	$-0.519532\pi$
$107$	−1.26866	−0.122646	−0.0613232	+	0.998118i	$0.519532\pi$
$108$	0	0
$109$	−20.2648	−1.94102	−0.970510	−	0.241062i	$-0.922504\pi$
$109$	−20.2648	−1.94102	−0.970510	+	0.241062i	$0.922504\pi$
$110$	0	0
$111$	−10.0449	−0.953421
$112$	0	0
$113$	−18.2704	−1.71873	−0.859367	−	0.511359i	$-0.829142\pi$
$113$	−18.2704	−1.71873	−0.859367	+	0.511359i	$0.829142\pi$
$114$	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
$125$	11.3964	1.01933
$126$	0	0
$127$	11.7089	1.03899	0.519497	−	0.854472i	$-0.326119\pi$
$127$	11.7089	1.03899	0.519497	+	0.854472i	$0.326119\pi$
$128$	0	0
$129$	−9.29411	−0.818300
$130$	0	0
$131$	−8.95059	−0.782017	−0.391009	−	0.920387i	$-0.627874\pi$
$131$	−8.95059	−0.782017	−0.391009	+	0.920387i	$0.627874\pi$
$132$	0	0
$133$	−3.41834	−0.296408
$134$	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
$139$	14.4468	1.22536	0.612681	−	0.790330i	$-0.290091\pi$
$139$	14.4468	1.22536	0.612681	+	0.790330i	$0.290091\pi$
$140$	0	0
$141$	−6.07124	−0.511290
$142$	0	0
$143$	29.6706	2.48118
$144$	0	0
$145$	18.5659	1.54182
$146$	0	0
$147$	22.1354	1.82570
$148$	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	$-0.604322\pi$
$151$	−7.91119	−0.643804	−0.321902	+	0.946773i	$0.604322\pi$
$152$	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
$159$	−15.6410	−1.24041
$160$	0	0
$161$	1.98995	0.156830
$162$	0	0
$163$	16.3886	1.28366	0.641829	−	0.766848i	$-0.278176\pi$
$163$	16.3886	1.28366	0.641829	+	0.766848i	$0.278176\pi$
$164$	0	0
$165$	−18.2474	−1.42056
$166$	0	0
$167$	−7.56874	−0.585687	−0.292843	−	0.956160i	$-0.594601\pi$
$167$	−7.56874	−0.585687	−0.292843	+	0.956160i	$0.594601\pi$
$168$	0	0
$169$	19.7533	1.51949
$170$	0	0
$171$	−0.312832	−0.0239228
$172$	0	0
$173$	10.1047	0.768245	0.384123	−	0.923282i	$-0.374504\pi$
$173$	10.1047	0.768245	0.384123	+	0.923282i	$0.374504\pi$
$174$	0	0
$175$	0.928694	0.0702026
$176$	0	0
$177$	−2.46542	−0.185312
$178$	0	0
$179$	2.03063	0.151777	0.0758883	−	0.997116i	$-0.475821\pi$
$179$	2.03063	0.151777	0.0758883	+	0.997116i	$0.475821\pi$
$180$	0	0
$181$	−4.78498	−0.355665	−0.177832	−	0.984061i	$-0.556908\pi$
$181$	−4.78498	−0.355665	−0.177832	+	0.984061i	$0.556908\pi$
$182$	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
$193$	−20.0225	−1.44125	−0.720626	−	0.693324i	$-0.756146\pi$
$193$	−20.0225	−1.44125	−0.720626	+	0.693324i	$0.756146\pi$
$194$	0	0
$195$	−20.1433	−1.44249
$196$	0	0
$197$	−1.02318	−0.0728987	−0.0364494	−	0.999336i	$-0.511605\pi$
$197$	−1.02318	−0.0728987	−0.0364494	+	0.999336i	$0.511605\pi$
$198$	0	0
$199$	9.12527	0.646873	0.323437	−	0.946250i	$-0.395162\pi$
$199$	9.12527	0.646873	0.323437	+	0.946250i	$0.395162\pi$
$200$	0	0
$201$	−4.75542	−0.335421
$202$	0	0
$203$	38.6383	2.71188
$204$	0	0
$205$	0.858640	0.0599700
$206$	0	0
$207$	0.182112	0.0126576
$208$	0	0
$209$	3.88832	0.268961
$210$	0	0
$211$	1.91333	0.131719	0.0658594	−	0.997829i	$-0.479021\pi$
$211$	1.91333	0.131719	0.0658594	+	0.997829i	$0.479021\pi$
$212$	0	0
$213$	8.32923	0.570710
$214$	0	0
$215$	12.6650	0.863745
$216$	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	$-0.676553\pi$
$223$	−15.7292	−1.05330	−0.526652	+	0.850081i	$0.676553\pi$
$224$	0	0
$225$	0.0849900	0.00566600
$226$	0	0
$227$	13.1228	0.870989	0.435494	−	0.900191i	$-0.356574\pi$
$227$	13.1228	0.870989	0.435494	+	0.900191i	$0.356574\pi$
$228$	0	0
$229$	11.3364	0.749133	0.374566	−	0.927200i	$-0.377792\pi$
$229$	11.3364	0.749133	0.374566	+	0.927200i	$0.377792\pi$
$230$	0	0
$231$	−37.9755	−2.49860
$232$	0	0
$233$	7.59285	0.497424	0.248712	−	0.968577i	$-0.419993\pi$
$233$	7.59285	0.497424	0.248712	+	0.968577i	$0.419993\pi$
$234$	0	0
$235$	8.27322	0.539685
$236$	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
$241$	−18.9917	−1.22336	−0.611682	−	0.791103i	$-0.709507\pi$
$241$	−18.9917	−1.22336	−0.611682	+	0.791103i	$0.709507\pi$
$242$	0	0
$243$	4.30169	0.275954
$244$	0	0
$245$	−30.1637	−1.92709
$246$	0	0
$247$	4.29231	0.273113
$248$	0	0
$249$	−14.6583	−0.928930
$250$	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	$-0.519290\pi$
$257$	−1.94183	−0.121128	−0.0605640	+	0.998164i	$0.519290\pi$
$258$	0	0
$259$	28.4869	1.77009
$260$	0	0
$261$	3.53601	0.218873
$262$	0	0
$263$	31.0706	1.91589	0.957946	−	0.286947i	$-0.0926405\pi$
$263$	31.0706	1.91589	0.957946	+	0.286947i	$0.0926405\pi$
$264$	0	0
$265$	21.3139	1.30930
$266$	0	0
$267$	20.9056	1.27940
$268$	0	0
$269$	5.73515	0.349678	0.174839	−	0.984597i	$-0.444059\pi$
$269$	5.73515	0.349678	0.174839	+	0.984597i	$0.444059\pi$
$270$	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
$275$	−1.05638	−0.0637020
$276$	0	0
$277$	−1.31181	−0.0788189	−0.0394095	−	0.999223i	$-0.512548\pi$
$277$	−1.31181	−0.0788189	−0.0394095	+	0.999223i	$0.512548\pi$
$278$	0	0
$279$	1.34664	0.0806215
$280$	0	0
$281$	−28.7710	−1.71633	−0.858167	−	0.513371i	$-0.828396\pi$
$281$	−28.7710	−1.71633	−0.858167	+	0.513371i	$0.828396\pi$
$282$	0	0
$283$	1.47735	0.0878193	0.0439097	−	0.999036i	$-0.486019\pi$
$283$	1.47735	0.0878193	0.0439097	+	0.999036i	$0.486019\pi$
$284$	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
$295$	3.35961	0.195604
$296$	0	0
$297$	−28.4715	−1.65208
$298$	0	0
$299$	−2.49873	−0.144505
$300$	0	0
$301$	26.3576	1.51923
$302$	0	0
$303$	−0.976599	−0.0561041
$304$	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
$309$	17.9614	1.02179
$310$	0	0
$311$	−16.3635	−0.927892	−0.463946	−	0.885864i	$-0.653567\pi$
$311$	−16.3635	−0.927892	−0.463946	+	0.885864i	$0.653567\pi$
$312$	0	0
$313$	9.55955	0.540338	0.270169	−	0.962813i	$-0.412920\pi$
$313$	9.55955	0.540338	0.270169	+	0.962813i	$0.412920\pi$
$314$	0	0
$315$	−4.16341	−0.234581
$316$	0	0
$317$	−8.82782	−0.495820	−0.247910	−	0.968783i	$-0.579744\pi$
$317$	−8.82782	−0.495820	−0.247910	+	0.968783i	$0.579744\pi$
$318$	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
$329$	17.2177	0.949243
$330$	0	0
$331$	28.8976	1.58836	0.794179	−	0.607684i	$-0.207901\pi$
$331$	28.8976	1.58836	0.794179	+	0.607684i	$0.207901\pi$
$332$	0	0
$333$	2.60699	0.142862
$334$	0	0
$335$	6.48016	0.354049
$336$	0	0
$337$	−2.58235	−0.140669	−0.0703347	−	0.997523i	$-0.522407\pi$
$337$	−2.58235	−0.140669	−0.0703347	+	0.997523i	$0.522407\pi$
$338$	0	0
$339$	−29.3631	−1.59478
$340$	0	0
$341$	−16.7380	−0.906415
$342$	0	0
$343$	−30.8705	−1.66685
$344$	0	0
$345$	1.53672	0.0827342
$346$	0	0
$347$	−10.2310	−0.549229	−0.274615	−	0.961554i	$-0.588550\pi$
$347$	−10.2310	−0.549229	−0.274615	+	0.961554i	$0.588550\pi$
$348$	0	0
$349$	−0.672835	−0.0360160	−0.0180080	−	0.999838i	$-0.505732\pi$
$349$	−0.672835	−0.0360160	−0.0180080	+	0.999838i	$0.505732\pi$
$350$	0	0
$351$	−31.4296	−1.67759
$352$	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	$-0.314764\pi$
$359$	20.8285	1.09929	0.549643	+	0.835400i	$0.314764\pi$
$360$	0	0
$361$	−18.4375	−0.970394
$362$	0	0
$363$	25.5181	1.33935
$364$	0	0
$365$	−12.9344	−0.677016
$366$	0	0
$367$	0.287929	0.0150298	0.00751489	−	0.999972i	$-0.497608\pi$
$367$	0.287929	0.0150298	0.00751489	+	0.999972i	$0.497608\pi$
$368$	0	0
$369$	0.163534	0.00851323
$370$	0	0
$371$	44.3571	2.30291
$372$	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
$377$	−48.5170	−2.49875
$378$	0	0
$379$	20.3610	1.04588	0.522938	−	0.852371i	$-0.324836\pi$
$379$	20.3610	1.04588	0.522938	+	0.852371i	$0.324836\pi$
$380$	0	0
$381$	18.8178	0.964064
$382$	0	0
$383$	30.6482	1.56605	0.783025	−	0.621990i	$-0.213676\pi$
$383$	30.6482	1.56605	0.783025	+	0.621990i	$0.213676\pi$
$384$	0	0
$385$	51.7488	2.63736
$386$	0	0
$387$	2.41213	0.122616
$388$	0	0
$389$	−1.61640	−0.0819546	−0.0409773	−	0.999160i	$-0.513047\pi$
$389$	−1.61640	−0.0819546	−0.0409773	+	0.999160i	$0.513047\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−14.3848	−0.725619
$394$	0	0
$395$	27.9655	1.40710
$396$	0	0
$397$	5.29986	0.265993	0.132996	−	0.991117i	$-0.457540\pi$
$397$	5.29986	0.265993	0.132996	+	0.991117i	$0.457540\pi$
$398$	0	0
$399$	−5.49375	−0.275031
$400$	0	0
$401$	9.37263	0.468047	0.234023	−	0.972231i	$-0.424811\pi$
$401$	9.37263	0.468047	0.234023	+	0.972231i	$0.424811\pi$
$402$	0	0
$403$	−18.4771	−0.920409
$404$	0	0
$405$	16.5888	0.824306
$406$	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
$411$	6.82031	0.336421
$412$	0	0
$413$	6.99181	0.344044
$414$	0	0
$415$	19.9747	0.980519
$416$	0	0
$417$	23.2180	1.13699
$418$	0	0
$419$	33.0196	1.61311	0.806557	−	0.591156i	$-0.201328\pi$
$419$	33.0196	1.61311	0.806557	+	0.591156i	$0.201328\pi$
$420$	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
$431$	32.3623	1.55884	0.779419	−	0.626503i	$-0.215514\pi$
$431$	32.3623	1.55884	0.779419	+	0.626503i	$0.215514\pi$
$432$	0	0
$433$	−24.7892	−1.19129	−0.595645	−	0.803248i	$-0.703104\pi$
$433$	−24.7892	−1.19129	−0.595645	+	0.803248i	$0.703104\pi$
$434$	0	0
$435$	29.8380	1.43062
$436$	0	0
$437$	−0.327458	−0.0156644
$438$	0	0
$439$	−5.18883	−0.247649	−0.123825	−	0.992304i	$-0.539516\pi$
$439$	−5.18883	−0.247649	−0.123825	+	0.992304i	$0.539516\pi$
$440$	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
$445$	−28.4879	−1.35045
$446$	0	0
$447$	−9.37355	−0.443354
$448$	0	0
$449$	24.2953	1.14657	0.573283	−	0.819357i	$-0.305669\pi$
$449$	24.2953	1.14657	0.573283	+	0.819357i	$0.305669\pi$
$450$	0	0
$451$	−2.03263	−0.0957130
$452$	0	0
$453$	−12.7144	−0.597374
$454$	0	0
$455$	57.1254	2.67808
$456$	0	0
$457$	−20.5422	−0.960923	−0.480462	−	0.877016i	$-0.659531\pi$
$457$	−20.5422	−0.960923	−0.480462	+	0.877016i	$0.659531\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	24.8856	1.15904	0.579519	−	0.814959i	$-0.303240\pi$
$461$	24.8856	1.15904	0.579519	+	0.814959i	$0.303240\pi$
$462$	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
$465$	11.3634	0.526967
$466$	0	0
$467$	−33.2698	−1.53954	−0.769772	−	0.638319i	$-0.779630\pi$
$467$	−33.2698	−1.53954	−0.769772	+	0.638319i	$0.779630\pi$
$468$	0	0
$469$	13.4861	0.622731
$470$	0	0
$471$	−19.8791	−0.915979
$472$	0	0
$473$	−29.9815	−1.37855
$474$	0	0
$475$	−0.152822	−0.00701194
$476$	0	0
$477$	4.05937	0.185866
$478$	0	0
$479$	10.6962	0.488723	0.244362	−	0.969684i	$-0.421422\pi$
$479$	10.6962	0.488723	0.244362	+	0.969684i	$0.421422\pi$
$480$	0	0
$481$	−35.7701	−1.63098
$482$	0	0
$483$	3.19813	0.145520
$484$	0	0
$485$	35.2789	1.60193
$486$	0	0
$487$	19.1335	0.867020	0.433510	−	0.901149i	$-0.357275\pi$
$487$	19.1335	0.867020	0.433510	+	0.901149i	$0.357275\pi$
$488$	0	0
$489$	26.3388	1.19108
$490$	0	0
$491$	6.56476	0.296263	0.148132	−	0.988968i	$-0.452674\pi$
$491$	6.56476	0.296263	0.148132	+	0.988968i	$0.452674\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.73583	0.212860
$496$	0	0
$497$	−23.6213	−1.05956
$498$	0	0
$499$	−25.9028	−1.15957	−0.579784	−	0.814770i	$-0.696863\pi$
$499$	−25.9028	−1.15957	−0.579784	+	0.814770i	$0.696863\pi$
$500$	0	0
$501$	−12.1640	−0.543448
$502$	0	0
$503$	−2.02620	−0.0903437	−0.0451719	−	0.998979i	$-0.514384\pi$
$503$	−2.02620	−0.0903437	−0.0451719	+	0.998979i	$0.514384\pi$
$504$	0	0
$505$	1.33080	0.0592199
$506$	0	0
$507$	31.7463	1.40990
$508$	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
$513$	−4.11885	−0.181852
$514$	0	0
$515$	−24.4758	−1.07853
$516$	0	0
$517$	−19.5849	−0.861345
$518$	0	0
$519$	16.2396	0.712841
$520$	0	0
$521$	−4.72853	−0.207161	−0.103580	−	0.994621i	$-0.533030\pi$
$521$	−4.72853	−0.207161	−0.103580	+	0.994621i	$0.533030\pi$
$522$	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
$533$	−2.24382	−0.0971907
$534$	0	0
$535$	2.77841	0.120121
$536$	0	0
$537$	3.26351	0.140831
$538$	0	0
$539$	71.4056	3.07566
$540$	0	0
$541$	−13.9762	−0.600883	−0.300442	−	0.953800i	$-0.597134\pi$
$541$	−13.9762	−0.600883	−0.300442	+	0.953800i	$0.597134\pi$
$542$	0	0
$543$	−7.69012	−0.330015
$544$	0	0
$545$	44.3806	1.90106
$546$	0	0
$547$	−8.41081	−0.359620	−0.179810	−	0.983701i	$-0.557548\pi$
$547$	−8.41081	−0.359620	−0.179810	+	0.983701i	$0.557548\pi$
$548$	0	0
$549$	−1.39148	−0.0593869
$550$	0	0
$551$	−6.35814	−0.270866
$552$	0	0
$553$	58.2000	2.47492
$554$	0	0
$555$	21.9987	0.933792
$556$	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	$-0.291671\pi$
$563$	28.8884	1.21750	0.608750	+	0.793362i	$0.291671\pi$
$564$	0	0
$565$	40.0128	1.68335
$566$	0	0
$567$	34.5237	1.44986
$568$	0	0
$569$	17.0273	0.713822	0.356911	−	0.934138i	$-0.383830\pi$
$569$	17.0273	0.713822	0.356911	+	0.934138i	$0.383830\pi$
$570$	0	0
$571$	22.9391	0.959972	0.479986	−	0.877276i	$-0.340642\pi$
$571$	22.9391	0.959972	0.479986	+	0.877276i	$0.340642\pi$
$572$	0	0
$573$	14.9638	0.625121
$574$	0	0
$575$	0.0889636	0.00371004
$576$	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
$581$	41.5701	1.72462
$582$	0	0
$583$	−50.4557	−2.08966
$584$	0	0
$585$	5.22787	0.216146
$586$	0	0
$587$	−32.2870	−1.33263	−0.666313	−	0.745672i	$-0.732128\pi$
$587$	−32.2870	−1.33263	−0.666313	+	0.745672i	$0.732128\pi$
$588$	0	0
$589$	−2.42142	−0.0997728
$590$	0	0
$591$	−1.64440	−0.0676414
$592$	0	0
$593$	25.8513	1.06158	0.530792	−	0.847502i	$-0.321895\pi$
$593$	25.8513	1.06158	0.530792	+	0.847502i	$0.321895\pi$
$594$	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
$601$	39.0351	1.59228	0.796138	−	0.605115i	$-0.206873\pi$
$601$	39.0351	1.59228	0.796138	+	0.605115i	$0.206873\pi$
$602$	0	0
$603$	1.23419	0.0502601
$604$	0	0
$605$	−34.7733	−1.41374
$606$	0	0
$607$	−17.5713	−0.713198	−0.356599	−	0.934258i	$-0.616064\pi$
$607$	−17.5713	−0.713198	−0.356599	+	0.934258i	$0.616064\pi$
$608$	0	0
$609$	62.0971	2.51630
$610$	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
$615$	1.37995	0.0556451
$616$	0	0
$617$	40.6511	1.63655	0.818275	−	0.574827i	$-0.194931\pi$
$617$	40.6511	1.63655	0.818275	+	0.574827i	$0.194931\pi$
$618$	0	0
$619$	−28.0856	−1.12885	−0.564427	−	0.825483i	$-0.690903\pi$
$619$	−28.0856	−1.12885	−0.564427	+	0.825483i	$0.690903\pi$
$620$	0	0
$621$	2.39774	0.0962182
$622$	0	0
$623$	−59.2872	−2.37529
$624$	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	$-0.439573\pi$
$631$	9.48010	0.377397	0.188698	+	0.982035i	$0.439573\pi$
$632$	0	0
$633$	3.07498	0.122220
$634$	0	0
$635$	−25.6428	−1.01760
$636$	0	0
$637$	78.8246	3.12314
$638$	0	0
$639$	−2.16172	−0.0855162
$640$	0	0
$641$	15.8059	0.624295	0.312147	−	0.950034i	$-0.398952\pi$
$641$	15.8059	0.624295	0.312147	+	0.950034i	$0.398952\pi$
$642$	0	0
$643$	−28.9071	−1.13998	−0.569992	−	0.821650i	$-0.693054\pi$
$643$	−28.9071	−1.13998	−0.569992	+	0.821650i	$0.693054\pi$
$644$	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
$649$	−7.95310	−0.312187
$650$	0	0
$651$	23.6489	0.926873
$652$	0	0
$653$	46.3129	1.81237	0.906183	−	0.422887i	$-0.138983\pi$
$653$	46.3129	1.81237	0.906183	+	0.422887i	$0.138983\pi$
$654$	0	0
$655$	19.6021	0.765917
$656$	0	0
$657$	−2.46344	−0.0961080
$658$	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	$-0.561956\pi$
$661$	−9.94533	−0.386829	−0.193414	+	0.981117i	$0.561956\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.48628	0.290305
$666$	0	0
$667$	3.70133	0.143316
$668$	0	0
$669$	−25.2790	−0.977342
$670$	0	0
$671$	17.2953	0.667678
$672$	0	0
$673$	−35.2526	−1.35889	−0.679444	−	0.733727i	$-0.737779\pi$
$673$	−35.2526	−1.35889	−0.679444	+	0.733727i	$0.737779\pi$
$674$	0	0
$675$	1.11901	0.0430706
$676$	0	0
$677$	46.4301	1.78445	0.892226	−	0.451588i	$-0.149142\pi$
$677$	46.4301	1.78445	0.892226	+	0.451588i	$0.149142\pi$
$678$	0	0
$679$	73.4203	2.81761
$680$	0	0
$681$	21.0901	0.808174
$682$	0	0
$683$	−35.6740	−1.36503	−0.682514	−	0.730873i	$-0.739113\pi$
$683$	−35.6740	−1.36503	−0.682514	+	0.730873i	$0.739113\pi$
$684$	0	0
$685$	−9.29397	−0.355104
$686$	0	0
$687$	18.2192	0.695106
$688$	0	0
$689$	−55.6980	−2.12192
$690$	0	0
$691$	−10.0109	−0.380832	−0.190416	−	0.981703i	$-0.560984\pi$
$691$	−10.0109	−0.380832	−0.190416	+	0.981703i	$0.560984\pi$
$692$	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
$703$	−4.68767	−0.176799
$704$	0	0
$705$	13.2962	0.500764
$706$	0	0
$707$	2.76958	0.104161
$708$	0	0
$709$	27.7345	1.04159	0.520795	−	0.853682i	$-0.325635\pi$
$709$	27.7345	1.04159	0.520795	+	0.853682i	$0.325635\pi$
$710$	0	0
$711$	5.32622	0.199749
$712$	0	0
$713$	1.40960	0.0527901
$714$	0	0
$715$	−64.9795	−2.43010
$716$	0	0
$717$	3.39274	0.126704
$718$	0	0
$719$	21.8968	0.816613	0.408307	−	0.912845i	$-0.366119\pi$
$719$	21.8968	0.816613	0.408307	+	0.912845i	$0.366119\pi$
$720$	0	0
$721$	−50.9376	−1.89701
$722$	0	0
$723$	−30.5223	−1.13514
$724$	0	0
$725$	1.72738	0.0641532
$726$	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	$-0.525778\pi$
$733$	−4.38032	−0.161791	−0.0808954	+	0.996723i	$0.525778\pi$
$734$	0	0
$735$	−48.4772	−1.78811
$736$	0	0
$737$	−15.3403	−0.565067
$738$	0	0
$739$	−36.8554	−1.35575	−0.677875	−	0.735177i	$-0.737099\pi$
$739$	−36.8554	−1.35575	−0.677875	+	0.735177i	$0.737099\pi$
$740$	0	0
$741$	6.89834	0.253417
$742$	0	0
$743$	37.9459	1.39210	0.696051	−	0.717993i	$-0.254939\pi$
$743$	37.9459	1.39210	0.696051	+	0.717993i	$0.254939\pi$
$744$	0	0
$745$	12.7732	0.467976
$746$	0	0
$747$	3.80432	0.139193
$748$	0	0
$749$	5.78226	0.211279
$750$	0	0
$751$	−9.08102	−0.331371	−0.165686	−	0.986179i	$-0.552984\pi$
$751$	−9.08102	−0.331371	−0.165686	+	0.986179i	$0.552984\pi$
$752$	0	0
$753$	21.4119	0.780295
$754$	0	0
$755$	17.3258	0.630549
$756$	0	0
$757$	−13.9433	−0.506779	−0.253389	−	0.967364i	$-0.581545\pi$
$757$	−13.9433	−0.506779	−0.253389	+	0.967364i	$0.581545\pi$
$758$	0	0
$759$	−3.63783	−0.132045
$760$	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
$771$	−3.12079	−0.112392
$772$	0	0
$773$	−22.6810	−0.815779	−0.407890	−	0.913031i	$-0.633735\pi$
$773$	−22.6810	−0.815779	−0.407890	+	0.913031i	$0.633735\pi$
$774$	0	0
$775$	0.657850	0.0236307
$776$	0	0
$777$	45.7823	1.64243
$778$	0	0
$779$	−0.294052	−0.0105355
$780$	0	0
$781$	26.8689	0.961446
$782$	0	0
$783$	46.5563	1.66379
$784$	0	0
$785$	27.0890	0.966848
$786$	0	0
$787$	14.7018	0.524064	0.262032	−	0.965059i	$-0.415607\pi$
$787$	14.7018	0.524064	0.262032	+	0.965059i	$0.415607\pi$
$788$	0	0
$789$	49.9347	1.77772
$790$	0	0
$791$	83.2721	2.96081
$792$	0	0
$793$	19.0923	0.677987
$794$	0	0
$795$	34.2543	1.21488
$796$	0	0
$797$	−8.21691	−0.291058	−0.145529	−	0.989354i	$-0.546488\pi$
$797$	−8.21691	−0.291058	−0.145529	+	0.989354i	$0.546488\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−5.42571	−0.191708
$802$	0	0
$803$	30.6192	1.08053
$804$	0	0
$805$	−4.35806	−0.153601
$806$	0	0
$807$	9.21718	0.324460
$808$	0	0
$809$	35.5939	1.25142	0.625708	−	0.780057i	$-0.284810\pi$
$809$	35.5939	1.25142	0.625708	+	0.780057i	$0.284810\pi$
$810$	0	0
$811$	−5.67292	−0.199203	−0.0996015	−	0.995027i	$-0.531757\pi$
$811$	−5.67292	−0.199203	−0.0996015	+	0.995027i	$0.531757\pi$
$812$	0	0
$813$	−33.9749	−1.19155
$814$	0	0
$815$	−35.8916	−1.25723
$816$	0	0
$817$	−4.33729	−0.151742
$818$	0	0
$819$	10.8799	0.380176
$820$	0	0
$821$	−3.95355	−0.137980	−0.0689899	−	0.997617i	$-0.521978\pi$
$821$	−3.95355	−0.137980	−0.0689899	+	0.997617i	$0.521978\pi$
$822$	0	0
$823$	4.31091	0.150269	0.0751344	−	0.997173i	$-0.476061\pi$
$823$	4.31091	0.150269	0.0751344	+	0.997173i	$0.476061\pi$
$824$	0	0
$825$	−1.69775	−0.0591079
$826$	0	0
$827$	−41.8738	−1.45609	−0.728047	−	0.685527i	$-0.759572\pi$
$827$	−41.8738	−1.45609	−0.728047	+	0.685527i	$0.759572\pi$
$828$	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
$839$	−12.0055	−0.414475	−0.207237	−	0.978291i	$-0.566447\pi$
$839$	−12.0055	−0.414475	−0.207237	+	0.978291i	$0.566447\pi$
$840$	0	0
$841$	42.8675	1.47819
$842$	0	0
$843$	−46.2390	−1.59255
$844$	0	0
$845$	−43.2604	−1.48820
$846$	0	0
$847$	−72.3681	−2.48660
$848$	0	0
$849$	2.37430	0.0814859
$850$	0	0
$851$	2.72888	0.0935448
$852$	0	0
$853$	−11.2288	−0.384465	−0.192233	−	0.981349i	$-0.561573\pi$
$853$	−11.2288	−0.384465	−0.192233	+	0.981349i	$0.561573\pi$
$854$	0	0
$855$	0.685112	0.0234303
$856$	0	0
$857$	−6.73481	−0.230057	−0.115028	−	0.993362i	$-0.536696\pi$
$857$	−6.73481	−0.230057	−0.115028	+	0.993362i	$0.536696\pi$
$858$	0	0
$859$	−7.84826	−0.267779	−0.133890	−	0.990996i	$-0.542747\pi$
$859$	−7.84826	−0.267779	−0.133890	+	0.990996i	$0.542747\pi$
$860$	0	0
$861$	2.87187	0.0978732
$862$	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
$873$	6.71911	0.227407
$874$	0	0
$875$	−51.9421	−1.75596
$876$	0	0
$877$	18.2845	0.617424	0.308712	−	0.951156i	$-0.400102\pi$
$877$	18.2845	0.617424	0.308712	+	0.951156i	$0.400102\pi$
$878$	0	0
$879$	−35.0388	−1.18183
$880$	0	0
$881$	−21.8193	−0.735110	−0.367555	−	0.930002i	$-0.619805\pi$
$881$	−21.8193	−0.735110	−0.367555	+	0.930002i	$0.619805\pi$
$882$	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
$887$	44.8589	1.50621	0.753107	−	0.657898i	$-0.228554\pi$
$887$	44.8589	1.50621	0.753107	+	0.657898i	$0.228554\pi$
$888$	0	0
$889$	−53.3662	−1.78985
$890$	0	0
$891$	−39.2703	−1.31560
$892$	0	0
$893$	−2.83327	−0.0948117
$894$	0	0
$895$	−4.44715	−0.148652
$896$	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
$907$	−2.95342	−0.0980666	−0.0490333	−	0.998797i	$-0.515614\pi$
$907$	−2.95342	−0.0980666	−0.0490333	+	0.998797i	$0.515614\pi$
$908$	0	0
$909$	0.253460	0.00840675
$910$	0	0
$911$	11.5101	0.381346	0.190673	−	0.981654i	$-0.438933\pi$
$911$	11.5101	0.381346	0.190673	+	0.981654i	$0.438933\pi$
$912$	0	0
$913$	−47.2855	−1.56492
$914$	0	0
$915$	−11.7418	−0.388171
$916$	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
$921$	34.4279	1.13444
$922$	0	0
$923$	29.6606	0.976290
$924$	0	0
$925$	1.27354	0.0418739
$926$	0	0
$927$	−4.66159	−0.153107
$928$	0	0
$929$	33.3053	1.09271	0.546355	−	0.837553i	$-0.316015\pi$
$929$	33.3053	1.09271	0.546355	+	0.837553i	$0.316015\pi$
$930$	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	$-0.666155\pi$
$937$	−30.5252	−0.997213	−0.498607	+	0.866828i	$0.666155\pi$
$938$	0	0
$939$	15.3635	0.501369
$940$	0	0
$941$	−37.8296	−1.23321	−0.616604	−	0.787273i	$-0.711492\pi$
$941$	−37.8296	−1.23321	−0.616604	+	0.787273i	$0.711492\pi$
$942$	0	0
$943$	0.171180	0.00557437
$944$	0	0
$945$	−54.8168	−1.78319
$946$	0	0
$947$	−32.0402	−1.04117	−0.520583	−	0.853811i	$-0.674285\pi$
$947$	−32.0402	−1.04117	−0.520583	+	0.853811i	$0.674285\pi$
$948$	0	0
$949$	33.8005	1.09721
$950$	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
$955$	−20.3910	−0.659838
$956$	0	0
$957$	−70.6347	−2.28330
$958$	0	0
$959$	−19.3420	−0.624587
$960$	0	0
$961$	−20.5765	−0.663759
$962$	0	0
$963$	0.529168	0.0170522
$964$	0	0
$965$	43.8500	1.41158
$966$	0	0
$967$	−18.2854	−0.588018	−0.294009	−	0.955803i	$-0.594989\pi$
$967$	−18.2854	−0.588018	−0.294009	+	0.955803i	$0.594989\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.4541	0.399672	0.199836	−	0.979829i	$-0.435959\pi$
$971$	12.4541	0.399672	0.199836	+	0.979829i	$0.435959\pi$
$972$	0	0
$973$	−65.8451	−2.11090
$974$	0	0
$975$	−1.87414	−0.0600205
$976$	0	0
$977$	−29.1828	−0.933639	−0.466819	−	0.884353i	$-0.654600\pi$
$977$	−29.1828	−0.933639	−0.466819	+	0.884353i	$0.654600\pi$
$978$	0	0
$979$	67.4385	2.15534
$980$	0	0
$981$	8.45260	0.269871
$982$	0	0
$983$	−25.0005	−0.797392	−0.398696	−	0.917083i	$-0.630537\pi$
$983$	−25.0005	−0.797392	−0.398696	+	0.917083i	$0.630537\pi$
$984$	0	0
$985$	2.24080	0.0713979
$986$	0	0
$987$	27.6712	0.880785
$988$	0	0
$989$	2.52491	0.0802874
$990$	0	0
$991$	20.7858	0.660282	0.330141	−	0.943932i	$-0.392904\pi$
$991$	20.7858	0.660282	0.330141	+	0.943932i	$0.392904\pi$
$992$	0	0
$993$	46.4425	1.47381
$994$	0	0
$995$	−19.9846	−0.633555
$996$	0	0
$997$	5.83306	0.184735	0.0923675	−	0.995725i	$-0.470557\pi$
$997$	5.83306	0.184735	0.0923675	+	0.995725i	$0.470557\pi$
$998$	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.a.u.1.5	✓	6
4.3	odd	2		4624.2.a.br.1.2		6
17.4	even	4		2312.2.b.o.577.4		12
17.13	even	4		2312.2.b.o.577.9		12
17.16	even	2		2312.2.a.v.1.2	yes	6
68.67	odd	2		4624.2.a.bs.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.2.a.u.1.5	✓	6	1.1	even	1	trivial
2312.2.a.v.1.2	yes	6	17.16	even	2
2312.2.b.o.577.4		12	17.4	even	4
2312.2.b.o.577.9		12	17.13	even	4
4624.2.a.br.1.2		6	4.3	odd	2
4624.2.a.bs.1.5		6	68.67	odd	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q+1.60714 q^{3} -2.19003 q^{5} -4.55776 q^{7} -0.417107 q^{9} +5.18440 q^{11} +5.72305 q^{13} -3.51968 q^{15} +0.750005 q^{19} -7.32495 q^{21} -0.436608 q^{23} -0.203761 q^{25} -5.49176 q^{27} -8.47747 q^{29} -3.22854 q^{31} +8.33205 q^{33} +9.98164 q^{35} -6.25019 q^{37} +9.19774 q^{39} -0.392067 q^{41} -5.78302 q^{43} +0.913477 q^{45} -3.77767 q^{47} +13.7732 q^{49} -9.73222 q^{53} -11.3540 q^{55} +1.20536 q^{57} -1.53405 q^{59} +3.33603 q^{61} +1.90107 q^{63} -12.5337 q^{65} -2.95893 q^{67} -0.701689 q^{69} +5.18265 q^{71} +5.90602 q^{73} -0.327472 q^{75} -23.6292 q^{77} -12.7694 q^{79} -7.57470 q^{81} -9.12073 q^{83} -13.6245 q^{87} +13.0080 q^{89} -26.0843 q^{91} -5.18871 q^{93} -1.64253 q^{95} -16.1089 q^{97} -2.16245 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{5} - 3 q^{7} + 6 q^{11} - 6 q^{13} - 6 q^{15} + 6 q^{19} + 6 q^{21} - 9 q^{23} + 6 q^{25} - 12 q^{27} - 27 q^{29} - 9 q^{31} + 3 q^{33} + 21 q^{35} - 15 q^{37} + 12 q^{39} - 21 q^{41} - 24 q^{45}+ \cdots + 42 q^{99}+O(q^{100}) \) 6 * q - 6 * q^5 - 3 * q^7 + 6 * q^11 - 6 * q^13 - 6 * q^15 + 6 * q^19 + 6 * q^21 - 9 * q^23 + 6 * q^25 - 12 * q^27 - 27 * q^29 - 9 * q^31 + 3 * q^33 + 21 * q^35 - 15 * q^37 + 12 * q^39 - 21 * q^41 - 24 * q^45 + 3 * q^47 + 9 * q^49 + 33 * q^53 - 51 * q^55 - 30 * q^57 - 18 * q^61 - 24 * q^63 + 9 * q^65 - 9 * q^67 - 3 * q^69 - 6 * q^71 - 27 * q^73 + 24 * q^75 - 45 * q^77 - 33 * q^79 - 18 * q^81 - 12 * q^83 + 15 * q^87 + 3 * q^89 - 30 * q^91 - 30 * q^93 - 3 * q^95 - 51 * q^97 + 42 * q^99

Introduction

L-functions

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Representations

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Database

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