LMFDB - Embedded newform 2304.3.e.l.1025.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(1025,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2304.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$62.7794529086$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2304.1025
Dual form			2304.3.e.l.1025.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.24264i q^{5} +8.48528 q^{7} +O(q^{10})$	$q-4.24264i q^{5} +8.48528 q^{7} +4.00000i q^{11} +18.0000 q^{13} -4.24264i q^{17} -16.9706 q^{19} -36.0000i q^{23} +7.00000 q^{25} +12.7279i q^{29} +8.48528 q^{31} -36.0000i q^{35} +36.0000 q^{37} -29.6985i q^{41} +67.8823 q^{43} -36.0000i q^{47} +23.0000 q^{49} +80.6102i q^{53} +16.9706 q^{55} +80.0000i q^{59} -36.0000 q^{61} -76.3675i q^{65} -118.794 q^{67} +108.000i q^{71} +56.0000 q^{73} +33.9411i q^{77} +25.4558 q^{79} -76.0000i q^{83} -18.0000 q^{85} -89.0955i q^{89} +152.735 q^{91} +72.0000i q^{95} +104.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 72 q^{13} + 28 q^{25} + 144 q^{37} + 92 q^{49} - 144 q^{61} + 224 q^{73} - 72 q^{85} + 416 q^{97}+O(q^{100}) $ 4 * q + 72 * q^13 + 28 * q^25 + 144 * q^37 + 92 * q^49 - 144 * q^61 + 224 * q^73 - 72 * q^85 + 416 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$1$	$-1$	$1$

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	0
$3$	0	0
$4$	0	0
$5$	− 4.24264i	− 0.848528i	−0.905539	−	0.424264i	$-0.860533\pi$
$5$	− 4.24264i	− 0.848528i	0.905539	−	0.424264i	$-0.139467\pi$
$6$	0	0
$7$	8.48528	1.21218	0.606092	−	0.795395i	$-0.292737\pi$
$7$	8.48528	1.21218	0.606092	+	0.795395i	$0.292737\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.00000i	0.363636i	0.983332	+	0.181818i	$0.0581982\pi$
$11$	4.00000i	0.363636i	−0.983332	+	0.181818i	$0.941802\pi$
$12$	0	0
$13$	18.0000	1.38462	0.692308	−	0.721602i	$-0.256594\pi$
$13$	18.0000	1.38462	0.692308	+	0.721602i	$0.256594\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.24264i	− 0.249567i	−0.992184	−	0.124784i	$-0.960176\pi$
$17$	− 4.24264i	− 0.249567i	0.992184	−	0.124784i	$-0.0398236\pi$
$18$	0	0
$19$	−16.9706	−0.893188	−0.446594	−	0.894737i	$-0.647363\pi$
$19$	−16.9706	−0.893188	−0.446594	+	0.894737i	$0.647363\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 36.0000i	− 1.56522i	−0.622514	−	0.782609i	$-0.713889\pi$
$23$	− 36.0000i	− 1.56522i	0.622514	−	0.782609i	$-0.286111\pi$
$24$	0	0
$25$	7.00000	0.280000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	12.7279i	0.438894i	0.975624	+	0.219447i	$0.0704253\pi$
$29$	12.7279i	0.438894i	−0.975624	+	0.219447i	$0.929575\pi$
$30$	0	0
$31$	8.48528	0.273719	0.136859	−	0.990590i	$-0.456299\pi$
$31$	8.48528	0.273719	0.136859	+	0.990590i	$0.456299\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 36.0000i	− 1.02857i
$36$	0	0
$37$	36.0000	0.972973	0.486486	−	0.873688i	$-0.338278\pi$
$37$	36.0000	0.972973	0.486486	+	0.873688i	$0.338278\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 29.6985i	− 0.724353i	−0.932109	−	0.362177i	$-0.882034\pi$
$41$	− 29.6985i	− 0.724353i	0.932109	−	0.362177i	$-0.117966\pi$
$42$	0	0
$43$	67.8823	1.57866	0.789328	−	0.613971i	$-0.210429\pi$
$43$	67.8823	1.57866	0.789328	+	0.613971i	$0.210429\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 36.0000i	− 0.765957i	−0.923757	−	0.382979i	$-0.874898\pi$
$47$	− 36.0000i	− 0.765957i	0.923757	−	0.382979i	$-0.125102\pi$
$48$	0	0
$49$	23.0000	0.469388
$50$	0	0
$51$	0	0
$52$	0	0
$53$	80.6102i	1.52095i	0.649369	+	0.760473i	$0.275033\pi$
$53$	80.6102i	1.52095i	−0.649369	+	0.760473i	$0.724967\pi$
$54$	0	0
$55$	16.9706	0.308556
$56$	0	0
$57$	0	0
$58$	0	0
$59$	80.0000i	1.35593i	0.735093	+	0.677966i	$0.237138\pi$
$59$	80.0000i	1.35593i	−0.735093	+	0.677966i	$0.762862\pi$
$60$	0	0
$61$	−36.0000	−0.590164	−0.295082	−	0.955472i	$-0.595347\pi$
$61$	−36.0000	−0.590164	−0.295082	+	0.955472i	$0.595347\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 76.3675i	− 1.17489i
$66$	0	0
$67$	−118.794	−1.77304	−0.886522	−	0.462687i	$-0.846886\pi$
$67$	−118.794	−1.77304	−0.886522	+	0.462687i	$0.846886\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	108.000i	1.52113i	0.649264	+	0.760563i	$0.275077\pi$
$71$	108.000i	1.52113i	−0.649264	+	0.760563i	$0.724923\pi$
$72$	0	0
$73$	56.0000	0.767123	0.383562	−	0.923515i	$-0.374697\pi$
$73$	56.0000	0.767123	0.383562	+	0.923515i	$0.374697\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	33.9411i	0.440794i
$78$	0	0
$79$	25.4558	0.322226	0.161113	−	0.986936i	$-0.448492\pi$
$79$	25.4558	0.322226	0.161113	+	0.986936i	$0.448492\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 76.0000i	− 0.915663i	−0.889039	−	0.457831i	$-0.848626\pi$
$83$	− 76.0000i	− 0.915663i	0.889039	−	0.457831i	$-0.151374\pi$
$84$	0	0
$85$	−18.0000	−0.211765
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 89.0955i	− 1.00107i	−0.865716	−	0.500536i	$-0.833136\pi$
$89$	− 89.0955i	− 1.00107i	0.865716	−	0.500536i	$-0.166864\pi$
$90$	0	0
$91$	152.735	1.67841
$92$	0	0
$93$	0	0
$94$	0	0
$95$	72.0000i	0.757895i
$96$	0	0
$97$	104.000	1.07216	0.536082	−	0.844166i	$-0.319904\pi$
$97$	104.000	1.07216	0.536082	+	0.844166i	$0.319904\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 148.492i	− 1.47022i	−0.677947	−	0.735111i	$-0.737130\pi$
$101$	− 148.492i	− 1.47022i	0.677947	−	0.735111i	$-0.262870\pi$
$102$	0	0
$103$	−178.191	−1.73001	−0.865004	−	0.501764i	$-0.832684\pi$
$103$	−178.191	−1.73001	−0.865004	+	0.501764i	$0.832684\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	128.000i	1.19626i	0.801398	+	0.598131i	$0.204090\pi$
$107$	128.000i	1.19626i	−0.801398	+	0.598131i	$0.795910\pi$
$108$	0	0
$109$	126.000	1.15596	0.577982	−	0.816050i	$-0.303841\pi$
$109$	126.000	1.15596	0.577982	+	0.816050i	$0.303841\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 89.0955i	− 0.788455i	−0.919013	−	0.394228i	$-0.871012\pi$
$113$	− 89.0955i	− 0.788455i	0.919013	−	0.394228i	$-0.128988\pi$
$114$	0	0
$115$	−152.735	−1.32813
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 36.0000i	− 0.302521i
$120$	0	0
$121$	105.000	0.867769
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 135.765i	− 1.08612i
$126$	0	0
$127$	−161.220	−1.26945	−0.634726	−	0.772737i	$-0.718887\pi$
$127$	−161.220	−1.26945	−0.634726	+	0.772737i	$0.718887\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 152.000i	− 1.16031i	−0.814508	−	0.580153i	$-0.802993\pi$
$131$	− 152.000i	− 1.16031i	0.814508	−	0.580153i	$-0.197007\pi$
$132$	0	0
$133$	−144.000	−1.08271
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 207.889i	− 1.51744i	−0.651416	−	0.758720i	$-0.725825\pi$
$137$	− 207.889i	− 1.51744i	0.651416	−	0.758720i	$-0.274175\pi$
$138$	0	0
$139$	118.794	0.854633	0.427316	−	0.904102i	$-0.359459\pi$
$139$	118.794	0.854633	0.427316	+	0.904102i	$0.359459\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	72.0000i	0.503497i
$144$	0	0
$145$	54.0000	0.372414
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 97.5807i	− 0.654904i	−0.944868	−	0.327452i	$-0.893810\pi$
$149$	− 97.5807i	− 0.654904i	0.944868	−	0.327452i	$-0.106190\pi$
$150$	0	0
$151$	246.073	1.62962	0.814812	−	0.579726i	$-0.196840\pi$
$151$	246.073	1.62962	0.814812	+	0.579726i	$0.196840\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 36.0000i	− 0.232258i
$156$	0	0
$157$	−252.000	−1.60510	−0.802548	−	0.596588i	$-0.796523\pi$
$157$	−252.000	−1.60510	−0.802548	+	0.596588i	$0.796523\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 305.470i	− 1.89733i
$162$	0	0
$163$	−101.823	−0.624683	−0.312342	−	0.949970i	$-0.601113\pi$
$163$	−101.823	−0.624683	−0.312342	+	0.949970i	$0.601113\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	155.000	0.917160
$170$	0	0
$171$	0	0
$172$	0	0
$173$	29.6985i	0.171668i	0.996309	+	0.0858338i	$0.0273554\pi$
$173$	29.6985i	0.171668i	−0.996309	+	0.0858338i	$0.972645\pi$
$174$	0	0
$175$	59.3970	0.339411
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 56.0000i	− 0.312849i	−0.987690	−	0.156425i	$-0.950003\pi$
$179$	− 56.0000i	− 0.312849i	0.987690	−	0.156425i	$-0.0499968\pi$
$180$	0	0
$181$	126.000	0.696133	0.348066	−	0.937470i	$-0.386838\pi$
$181$	126.000	0.696133	0.348066	+	0.937470i	$0.386838\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 152.735i	− 0.825595i
$186$	0	0
$187$	16.9706	0.0907517
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	298.000	1.54404	0.772021	−	0.635597i	$-0.219246\pi$
$193$	298.000	1.54404	0.772021	+	0.635597i	$0.219246\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	326.683i	1.65829i	0.559033	+	0.829146i	$0.311173\pi$
$197$	326.683i	1.65829i	−0.559033	+	0.829146i	$0.688827\pi$
$198$	0	0
$199$	59.3970	0.298477	0.149239	−	0.988801i	$-0.452318\pi$
$199$	59.3970	0.298477	0.149239	+	0.988801i	$0.452318\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	108.000i	0.532020i
$204$	0	0
$205$	−126.000	−0.614634
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 67.8823i	− 0.324795i
$210$	0	0
$211$	50.9117	0.241288	0.120644	−	0.992696i	$-0.461504\pi$
$211$	50.9117	0.241288	0.120644	+	0.992696i	$0.461504\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 288.000i	− 1.33953i
$216$	0	0
$217$	72.0000	0.331797
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 76.3675i	− 0.345554i
$222$	0	0
$223$	280.014	1.25567	0.627835	−	0.778347i	$-0.283941\pi$
$223$	280.014	1.25567	0.627835	+	0.778347i	$0.283941\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	284.000i	1.25110i	0.780184	+	0.625551i	$0.215126\pi$
$227$	284.000i	1.25110i	−0.780184	+	0.625551i	$0.784874\pi$
$228$	0	0
$229$	−126.000	−0.550218	−0.275109	−	0.961413i	$-0.588714\pi$
$229$	−126.000	−0.550218	−0.275109	+	0.961413i	$0.588714\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 284.257i	− 1.21999i	−0.792406	−	0.609993i	$-0.791172\pi$
$233$	− 284.257i	− 1.21999i	0.792406	−	0.609993i	$-0.208828\pi$
$234$	0	0
$235$	−152.735	−0.649936
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 360.000i	− 1.50628i	−0.657862	−	0.753138i	$-0.728539\pi$
$239$	− 360.000i	− 1.50628i	0.657862	−	0.753138i	$-0.271461\pi$
$240$	0	0
$241$	−32.0000	−0.132780	−0.0663900	−	0.997794i	$-0.521148\pi$
$241$	−32.0000	−0.132780	−0.0663900	+	0.997794i	$0.521148\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 97.5807i	− 0.398289i
$246$	0	0
$247$	−305.470	−1.23672
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 20.0000i	− 0.0796813i	−0.999206	−	0.0398406i	$-0.987315\pi$
$251$	− 20.0000i	− 0.0796813i	0.999206	−	0.0398406i	$-0.0126850\pi$
$252$	0	0
$253$	144.000	0.569170
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 386.080i	− 1.50226i	−0.660155	−	0.751129i	$-0.729510\pi$
$257$	− 386.080i	− 1.50226i	0.660155	−	0.751129i	$-0.270490\pi$
$258$	0	0
$259$	305.470	1.17942
$260$	0	0
$261$	0	0
$262$	0	0
$263$	72.0000i	0.273764i	0.990587	+	0.136882i	$0.0437082\pi$
$263$	72.0000i	0.273764i	−0.990587	+	0.136882i	$0.956292\pi$
$264$	0	0
$265$	342.000	1.29057
$266$	0	0
$267$	0	0
$268$	0	0
$269$	394.566i	1.46679i	0.679805	+	0.733393i	$0.262065\pi$
$269$	394.566i	1.46679i	−0.679805	+	0.733393i	$0.737935\pi$
$270$	0	0
$271$	−93.3381	−0.344421	−0.172211	−	0.985060i	$-0.555091\pi$
$271$	−93.3381	−0.344421	−0.172211	+	0.985060i	$0.555091\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	28.0000i	0.101818i
$276$	0	0
$277$	126.000	0.454874	0.227437	−	0.973793i	$-0.426965\pi$
$277$	126.000	0.454874	0.227437	+	0.973793i	$0.426965\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 89.0955i	− 0.317066i	−0.987354	−	0.158533i	$-0.949324\pi$
$281$	− 89.0955i	− 0.317066i	0.987354	−	0.158533i	$-0.0506764\pi$
$282$	0	0
$283$	−203.647	−0.719600	−0.359800	−	0.933029i	$-0.617155\pi$
$283$	−203.647	−0.719600	−0.359800	+	0.933029i	$0.617155\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 252.000i	− 0.878049i
$288$	0	0
$289$	271.000	0.937716
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 190.919i	− 0.651600i	−0.945439	−	0.325800i	$-0.894366\pi$
$293$	− 190.919i	− 0.651600i	0.945439	−	0.325800i	$-0.105634\pi$
$294$	0	0
$295$	339.411	1.15055
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 648.000i	− 2.16722i
$300$	0	0
$301$	576.000	1.91362
$302$	0	0
$303$	0	0
$304$	0	0
$305$	152.735i	0.500771i
$306$	0	0
$307$	288.500	0.939738	0.469869	−	0.882736i	$-0.344301\pi$
$307$	288.500	0.939738	0.469869	+	0.882736i	$0.344301\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	504.000i	1.62058i	0.586030	+	0.810289i	$0.300690\pi$
$311$	504.000i	1.62058i	−0.586030	+	0.810289i	$0.699310\pi$
$312$	0	0
$313$	−58.0000	−0.185304	−0.0926518	−	0.995699i	$-0.529534\pi$
$313$	−58.0000	−0.185304	−0.0926518	+	0.995699i	$0.529534\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 182.434i	− 0.575500i	−0.957706	−	0.287750i	$-0.907093\pi$
$317$	− 182.434i	− 0.575500i	0.957706	−	0.287750i	$-0.0929072\pi$
$318$	0	0
$319$	−50.9117	−0.159598
$320$	0	0
$321$	0	0
$322$	0	0
$323$	72.0000i	0.222910i
$324$	0	0
$325$	126.000	0.387692
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 305.470i	− 0.928481i
$330$	0	0
$331$	−356.382	−1.07668	−0.538341	−	0.842727i	$-0.680949\pi$
$331$	−356.382	−1.07668	−0.538341	+	0.842727i	$0.680949\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	504.000i	1.50448i
$336$	0	0
$337$	−8.00000	−0.0237389	−0.0118694	−	0.999930i	$-0.503778\pi$
$337$	−8.00000	−0.0237389	−0.0118694	+	0.999930i	$0.503778\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.9411i	0.0995341i
$342$	0	0
$343$	−220.617	−0.643199
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.0000i	0.0806916i	0.999186	+	0.0403458i	$0.0128460\pi$
$347$	28.0000i	0.0806916i	−0.999186	+	0.0403458i	$0.987154\pi$
$348$	0	0
$349$	252.000	0.722063	0.361032	−	0.932554i	$-0.382425\pi$
$349$	252.000	0.722063	0.361032	+	0.932554i	$0.382425\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 564.271i	− 1.59850i	−0.600997	−	0.799251i	$-0.705230\pi$
$353$	− 564.271i	− 1.59850i	0.600997	−	0.799251i	$-0.294770\pi$
$354$	0	0
$355$	458.205	1.29072
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 252.000i	− 0.701950i	−0.936385	−	0.350975i	$-0.885850\pi$
$359$	− 252.000i	− 0.701950i	0.936385	−	0.350975i	$-0.114150\pi$
$360$	0	0
$361$	−73.0000	−0.202216
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 237.588i	− 0.650926i
$366$	0	0
$367$	8.48528	0.0231207	0.0115603	−	0.999933i	$-0.496320\pi$
$367$	8.48528	0.0231207	0.0115603	+	0.999933i	$0.496320\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	684.000i	1.84367i
$372$	0	0
$373$	252.000	0.675603	0.337802	−	0.941217i	$-0.390317\pi$
$373$	252.000	0.675603	0.337802	+	0.941217i	$0.390317\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	229.103i	0.607699i
$378$	0	0
$379$	−593.970	−1.56720	−0.783601	−	0.621264i	$-0.786619\pi$
$379$	−593.970	−1.56720	−0.783601	+	0.621264i	$0.786619\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	504.000i	1.31593i	0.753050	+	0.657963i	$0.228582\pi$
$383$	504.000i	1.31593i	−0.753050	+	0.657963i	$0.771418\pi$
$384$	0	0
$385$	144.000	0.374026
$386$	0	0
$387$	0	0
$388$	0	0
$389$	207.889i	0.534420i	0.963638	+	0.267210i	$0.0861017\pi$
$389$	207.889i	0.534420i	−0.963638	+	0.267210i	$0.913898\pi$
$390$	0	0
$391$	−152.735	−0.390627
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 108.000i	− 0.273418i
$396$	0	0
$397$	−756.000	−1.90428	−0.952141	−	0.305659i	$-0.901123\pi$
$397$	−756.000	−1.90428	−0.952141	+	0.305659i	$0.901123\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	504.874i	1.25904i	0.776985	+	0.629519i	$0.216748\pi$
$401$	504.874i	1.25904i	−0.776985	+	0.629519i	$0.783252\pi$
$402$	0	0
$403$	152.735	0.378995
$404$	0	0
$405$	0	0
$406$	0	0
$407$	144.000i	0.353808i
$408$	0	0
$409$	136.000	0.332518	0.166259	−	0.986082i	$-0.446831\pi$
$409$	136.000	0.332518	0.166259	+	0.986082i	$0.446831\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	678.823i	1.64364i
$414$	0	0
$415$	−322.441	−0.776966
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 28.0000i	− 0.0668258i	−0.999442	−	0.0334129i	$-0.989362\pi$
$419$	− 28.0000i	− 0.0668258i	0.999442	−	0.0334129i	$-0.0106376\pi$
$420$	0	0
$421$	270.000	0.641330	0.320665	−	0.947193i	$-0.396094\pi$
$421$	270.000	0.641330	0.320665	+	0.947193i	$0.396094\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 29.6985i	− 0.0698788i
$426$	0	0
$427$	−305.470	−0.715387
$428$	0	0
$429$	0	0
$430$	0	0
$431$	252.000i	0.584687i	0.956313	+	0.292343i	$0.0944350\pi$
$431$	252.000i	0.584687i	−0.956313	+	0.292343i	$0.905565\pi$
$432$	0	0
$433$	−574.000	−1.32564	−0.662818	−	0.748781i	$-0.730640\pi$
$433$	−574.000	−1.32564	−0.662818	+	0.748781i	$0.730640\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	610.940i	1.39803i
$438$	0	0
$439$	−93.3381	−0.212615	−0.106308	−	0.994333i	$-0.533903\pi$
$439$	−93.3381	−0.212615	−0.106308	+	0.994333i	$0.533903\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 716.000i	− 1.61625i	−0.589009	−	0.808126i	$-0.700482\pi$
$443$	− 716.000i	− 1.61625i	0.589009	−	0.808126i	$-0.299518\pi$
$444$	0	0
$445$	−378.000	−0.849438
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 199.404i	− 0.444107i	−0.975034	−	0.222054i	$-0.928724\pi$
$449$	− 199.404i	− 0.444107i	0.975034	−	0.222054i	$-0.0712760\pi$
$450$	0	0
$451$	118.794	0.263401
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 648.000i	− 1.42418i
$456$	0	0
$457$	40.0000	0.0875274	0.0437637	−	0.999042i	$-0.486065\pi$
$457$	40.0000	0.0875274	0.0437637	+	0.999042i	$0.486065\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 742.462i	− 1.61055i	−0.592904	−	0.805273i	$-0.702018\pi$
$461$	− 742.462i	− 1.61055i	0.592904	−	0.805273i	$-0.297982\pi$
$462$	0	0
$463$	−653.367	−1.41116	−0.705580	−	0.708631i	$-0.749313\pi$
$463$	−653.367	−1.41116	−0.705580	+	0.708631i	$0.749313\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 52.0000i	− 0.111349i	−0.998449	−	0.0556745i	$-0.982269\pi$
$467$	− 52.0000i	− 0.111349i	0.998449	−	0.0556745i	$-0.0177309\pi$
$468$	0	0
$469$	−1008.00	−2.14925
$470$	0	0
$471$	0	0
$472$	0	0
$473$	271.529i	0.574057i
$474$	0	0
$475$	−118.794	−0.250093
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 684.000i	− 1.42797i	−0.700158	−	0.713987i	$-0.746887\pi$
$479$	− 684.000i	− 1.42797i	0.700158	−	0.713987i	$-0.253113\pi$
$480$	0	0
$481$	648.000	1.34719
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 441.235i	− 0.909762i
$486$	0	0
$487$	229.103	0.470437	0.235218	−	0.971943i	$-0.424420\pi$
$487$	229.103	0.470437	0.235218	+	0.971943i	$0.424420\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	440.000i	0.896130i	0.894001	+	0.448065i	$0.147887\pi$
$491$	440.000i	0.896130i	−0.894001	+	0.448065i	$0.852113\pi$
$492$	0	0
$493$	54.0000	0.109533
$494$	0	0
$495$	0	0
$496$	0	0
$497$	916.410i	1.84388i
$498$	0	0
$499$	475.176	0.952256	0.476128	−	0.879376i	$-0.342040\pi$
$499$	475.176	0.952256	0.476128	+	0.879376i	$0.342040\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	900.000i	1.78926i	0.446803	+	0.894632i	$0.352562\pi$
$503$	900.000i	1.78926i	−0.446803	+	0.894632i	$0.647438\pi$
$504$	0	0
$505$	−630.000	−1.24752
$506$	0	0
$507$	0	0
$508$	0	0
$509$	182.434i	0.358416i	0.983811	+	0.179208i	$0.0573534\pi$
$509$	182.434i	0.358416i	−0.983811	+	0.179208i	$0.942647\pi$
$510$	0	0
$511$	475.176	0.929894
$512$	0	0
$513$	0	0
$514$	0	0
$515$	756.000i	1.46796i
$516$	0	0
$517$	144.000	0.278530
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 4.24264i	− 0.00814326i	−0.999992	−	0.00407163i	$-0.998704\pi$
$521$	− 4.24264i	− 0.00814326i	0.999992	−	0.00407163i	$-0.00129604\pi$
$522$	0	0
$523$	254.558	0.486727	0.243364	−	0.969935i	$-0.421749\pi$
$523$	254.558	0.486727	0.243364	+	0.969935i	$0.421749\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 36.0000i	− 0.0683112i
$528$	0	0
$529$	−767.000	−1.44991
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 534.573i	− 1.00295i
$534$	0	0
$535$	543.058	1.01506
$536$	0	0
$537$	0	0
$538$	0	0
$539$	92.0000i	0.170686i
$540$	0	0
$541$	414.000	0.765250	0.382625	−	0.923904i	$-0.375020\pi$
$541$	414.000	0.765250	0.382625	+	0.923904i	$0.375020\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 534.573i	− 0.980867i
$546$	0	0
$547$	−67.8823	−0.124099	−0.0620496	−	0.998073i	$-0.519764\pi$
$547$	−67.8823	−0.124099	−0.0620496	+	0.998073i	$0.519764\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 216.000i	− 0.392015i
$552$	0	0
$553$	216.000	0.390597
$554$	0	0
$555$	0	0
$556$	0	0
$557$	275.772i	0.495102i	0.968875	+	0.247551i	$0.0796257\pi$
$557$	275.772i	0.495102i	−0.968875	+	0.247551i	$0.920374\pi$
$558$	0	0
$559$	1221.88	2.18583
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 892.000i	− 1.58437i	−0.610281	−	0.792185i	$-0.708944\pi$
$563$	− 892.000i	− 1.58437i	0.610281	−	0.792185i	$-0.291056\pi$
$564$	0	0
$565$	−378.000	−0.669027
$566$	0	0
$567$	0	0
$568$	0	0
$569$	844.285i	1.48381i	0.670507	+	0.741903i	$0.266076\pi$
$569$	844.285i	1.48381i	−0.670507	+	0.741903i	$0.733924\pi$
$570$	0	0
$571$	441.235	0.772740	0.386370	−	0.922344i	$-0.373729\pi$
$571$	441.235	0.772740	0.386370	+	0.922344i	$0.373729\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 252.000i	− 0.438261i
$576$	0	0
$577$	−326.000	−0.564991	−0.282496	−	0.959269i	$-0.591162\pi$
$577$	−326.000	−0.564991	−0.282496	+	0.959269i	$0.591162\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 644.881i	− 1.10995i
$582$	0	0
$583$	−322.441	−0.553072
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 400.000i	− 0.681431i	−0.940166	−	0.340716i	$-0.889331\pi$
$587$	− 400.000i	− 0.681431i	0.940166	−	0.340716i	$-0.110669\pi$
$588$	0	0
$589$	−144.000	−0.244482
$590$	0	0
$591$	0	0
$592$	0	0
$593$	903.682i	1.52392i	0.647626	+	0.761958i	$0.275762\pi$
$593$	903.682i	1.52392i	−0.647626	+	0.761958i	$0.724238\pi$
$594$	0	0
$595$	−152.735	−0.256698
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 756.000i	− 1.26210i	−0.775741	−	0.631052i	$-0.782624\pi$
$599$	− 756.000i	− 1.26210i	0.775741	−	0.631052i	$-0.217376\pi$
$600$	0	0
$601$	−778.000	−1.29451	−0.647255	−	0.762274i	$-0.724083\pi$
$601$	−778.000	−1.29451	−0.647255	+	0.762274i	$0.724083\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 445.477i	− 0.736326i
$606$	0	0
$607$	381.838	0.629057	0.314529	−	0.949248i	$-0.398154\pi$
$607$	381.838	0.629057	0.314529	+	0.949248i	$0.398154\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 648.000i	− 1.06056i
$612$	0	0
$613$	−180.000	−0.293638	−0.146819	−	0.989163i	$-0.546903\pi$
$613$	−180.000	−0.293638	−0.146819	+	0.989163i	$0.546903\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	453.963i	0.735758i	0.929874	+	0.367879i	$0.119916\pi$
$617$	453.963i	0.735758i	−0.929874	+	0.367879i	$0.880084\pi$
$618$	0	0
$619$	33.9411	0.0548322	0.0274161	−	0.999624i	$-0.491272\pi$
$619$	33.9411	0.0548322	0.0274161	+	0.999624i	$0.491272\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 756.000i	− 1.21348i
$624$	0	0
$625$	−401.000	−0.641600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 152.735i	− 0.242822i
$630$	0	0
$631$	−755.190	−1.19681	−0.598407	−	0.801192i	$-0.704200\pi$
$631$	−755.190	−1.19681	−0.598407	+	0.801192i	$0.704200\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	684.000i	1.07717i
$636$	0	0
$637$	414.000	0.649922
$638$	0	0
$639$	0	0
$640$	0	0
$641$	233.345i	0.364033i	0.983295	+	0.182017i	$0.0582625\pi$
$641$	233.345i	0.364033i	−0.983295	+	0.182017i	$0.941738\pi$
$642$	0	0
$643$	−1187.94	−1.84750	−0.923748	−	0.383002i	$-0.874890\pi$
$643$	−1187.94	−1.84750	−0.923748	+	0.383002i	$0.874890\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	324.000i	0.500773i	0.968146	+	0.250386i	$0.0805577\pi$
$647$	324.000i	0.500773i	−0.968146	+	0.250386i	$0.919442\pi$
$648$	0	0
$649$	−320.000	−0.493066
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 886.712i	− 1.35790i	−0.734182	−	0.678952i	$-0.762434\pi$
$653$	− 886.712i	− 1.35790i	0.734182	−	0.678952i	$-0.237566\pi$
$654$	0	0
$655$	−644.881	−0.984552
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1120.00i	1.69954i	0.527150	+	0.849772i	$0.323261\pi$
$659$	1120.00i	1.69954i	−0.527150	+	0.849772i	$0.676739\pi$
$660$	0	0
$661$	180.000	0.272315	0.136157	−	0.990687i	$-0.456525\pi$
$661$	180.000	0.272315	0.136157	+	0.990687i	$0.456525\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	610.940i	0.918707i
$666$	0	0
$667$	458.205	0.686964
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 144.000i	− 0.214605i
$672$	0	0
$673$	302.000	0.448737	0.224368	−	0.974504i	$-0.427968\pi$
$673$	302.000	0.448737	0.224368	+	0.974504i	$0.427968\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	776.403i	1.14683i	0.819265	+	0.573415i	$0.194382\pi$
$677$	776.403i	1.14683i	−0.819265	+	0.573415i	$0.805618\pi$
$678$	0	0
$679$	882.469	1.29966
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1060.00i	1.55198i	0.630747	+	0.775988i	$0.282748\pi$
$683$	1060.00i	1.55198i	−0.630747	+	0.775988i	$0.717252\pi$
$684$	0	0
$685$	−882.000	−1.28759
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1450.98i	2.10593i
$690$	0	0
$691$	−712.764	−1.03150	−0.515748	−	0.856740i	$-0.672486\pi$
$691$	−712.764	−1.03150	−0.515748	+	0.856740i	$0.672486\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 504.000i	− 0.725180i
$696$	0	0
$697$	−126.000	−0.180775
$698$	0	0
$699$	0	0
$700$	0	0
$701$	284.257i	0.405502i	0.979230	+	0.202751i	$0.0649882\pi$
$701$	284.257i	0.405502i	−0.979230	+	0.202751i	$0.935012\pi$
$702$	0	0
$703$	−610.940	−0.869047
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1260.00i	− 1.78218i
$708$	0	0
$709$	882.000	1.24401	0.622003	−	0.783015i	$-0.286319\pi$
$709$	882.000	1.24401	0.622003	+	0.783015i	$0.286319\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 305.470i	− 0.428429i
$714$	0	0
$715$	305.470	0.427231
$716$	0	0
$717$	0	0
$718$	0	0
$719$	252.000i	0.350487i	0.984525	+	0.175243i	$0.0560712\pi$
$719$	252.000i	0.350487i	−0.984525	+	0.175243i	$0.943929\pi$
$720$	0	0
$721$	−1512.00	−2.09709
$722$	0	0
$723$	0	0
$724$	0	0
$725$	89.0955i	0.122890i
$726$	0	0
$727$	−42.4264	−0.0583582	−0.0291791	−	0.999574i	$-0.509289\pi$
$727$	−42.4264	−0.0583582	−0.0291791	+	0.999574i	$0.509289\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 288.000i	− 0.393981i
$732$	0	0
$733$	162.000	0.221010	0.110505	−	0.993876i	$-0.464753\pi$
$733$	162.000	0.221010	0.110505	+	0.993876i	$0.464753\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 475.176i	− 0.644743i
$738$	0	0
$739$	−237.588	−0.321499	−0.160750	−	0.986995i	$-0.551391\pi$
$739$	−237.588	−0.321499	−0.160750	+	0.986995i	$0.551391\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1008.00i	1.35666i	0.734756	+	0.678331i	$0.237296\pi$
$743$	1008.00i	1.35666i	−0.734756	+	0.678331i	$0.762704\pi$
$744$	0	0
$745$	−414.000	−0.555705
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1086.12i	1.45009i
$750$	0	0
$751$	−500.632	−0.666620	−0.333310	−	0.942817i	$-0.608166\pi$
$751$	−500.632	−0.666620	−0.333310	+	0.942817i	$0.608166\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1044.00i	− 1.38278i
$756$	0	0
$757$	1026.00	1.35535	0.677675	−	0.735362i	$-0.262988\pi$
$757$	1026.00	1.35535	0.677675	+	0.735362i	$0.262988\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1260.06i	1.65580i	0.560875	+	0.827900i	$0.310465\pi$
$761$	1260.06i	1.65580i	−0.560875	+	0.827900i	$0.689535\pi$
$762$	0	0
$763$	1069.15	1.40124
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1440.00i	1.87744i
$768$	0	0
$769$	694.000	0.902471	0.451235	−	0.892405i	$-0.350983\pi$
$769$	694.000	0.902471	0.451235	+	0.892405i	$0.350983\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1438.26i	1.86061i	0.366781	+	0.930307i	$0.380460\pi$
$773$	1438.26i	1.86061i	−0.366781	+	0.930307i	$0.619540\pi$
$774$	0	0
$775$	59.3970	0.0766413
$776$	0	0
$777$	0	0
$778$	0	0
$779$	504.000i	0.646983i
$780$	0	0
$781$	−432.000	−0.553137
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1069.15i	1.36197i
$786$	0	0
$787$	−865.499	−1.09974	−0.549872	−	0.835249i	$-0.685324\pi$
$787$	−865.499	−1.09974	−0.549872	+	0.835249i	$0.685324\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 756.000i	− 0.955752i
$792$	0	0
$793$	−648.000	−0.817150
$794$	0	0
$795$	0	0
$796$	0	0
$797$	640.639i	0.803813i	0.915681	+	0.401906i	$0.131652\pi$
$797$	640.639i	0.803813i	−0.915681	+	0.401906i	$0.868348\pi$
$798$	0	0
$799$	−152.735	−0.191158
$800$	0	0
$801$	0	0
$802$	0	0
$803$	224.000i	0.278954i
$804$	0	0
$805$	−1296.00	−1.60994
$806$	0	0
$807$	0	0
$808$	0	0
$809$	683.065i	0.844333i	0.906518	+	0.422166i	$0.138730\pi$
$809$	683.065i	0.844333i	−0.906518	+	0.422166i	$0.861270\pi$
$810$	0	0
$811$	1306.73	1.61126	0.805631	−	0.592418i	$-0.201827\pi$
$811$	1306.73	1.61126	0.805631	+	0.592418i	$0.201827\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	432.000i	0.530061i
$816$	0	0
$817$	−1152.00	−1.41004
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 114.551i	− 0.139527i	−0.997564	−	0.0697633i	$-0.977776\pi$
$821$	− 114.551i	− 0.139527i	0.997564	−	0.0697633i	$-0.0222244\pi$
$822$	0	0
$823$	653.367	0.793884	0.396942	−	0.917844i	$-0.370071\pi$
$823$	653.367	0.793884	0.396942	+	0.917844i	$0.370071\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	776.000i	0.938331i	0.883110	+	0.469166i	$0.155445\pi$
$827$	776.000i	0.938331i	−0.883110	+	0.469166i	$0.844555\pi$
$828$	0	0
$829$	702.000	0.846803	0.423402	−	0.905942i	$-0.360836\pi$
$829$	702.000	0.846803	0.423402	+	0.905942i	$0.360836\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 97.5807i	− 0.117144i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	612.000i	0.729440i	0.931117	+	0.364720i	$0.118835\pi$
$839$	612.000i	0.729440i	−0.931117	+	0.364720i	$0.881165\pi$
$840$	0	0
$841$	679.000	0.807372
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 657.609i	− 0.778236i
$846$	0	0
$847$	890.955	1.05189
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1296.00i	− 1.52291i
$852$	0	0
$853$	−252.000	−0.295428	−0.147714	−	0.989030i	$-0.547191\pi$
$853$	−252.000	−0.295428	−0.147714	+	0.989030i	$0.547191\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	649.124i	0.757438i	0.925512	+	0.378719i	$0.123635\pi$
$857$	649.124i	0.757438i	−0.925512	+	0.378719i	$0.876365\pi$
$858$	0	0
$859$	−339.411	−0.395124	−0.197562	−	0.980290i	$-0.563302\pi$
$859$	−339.411	−0.395124	−0.197562	+	0.980290i	$0.563302\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1008.00i	1.16802i	0.811747	+	0.584009i	$0.198517\pi$
$863$	1008.00i	1.16802i	−0.811747	+	0.584009i	$0.801483\pi$
$864$	0	0
$865$	126.000	0.145665
$866$	0	0
$867$	0	0
$868$	0	0
$869$	101.823i	0.117173i
$870$	0	0
$871$	−2138.29	−2.45498
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1152.00i	− 1.31657i
$876$	0	0
$877$	1260.00	1.43672	0.718358	−	0.695674i	$-0.244894\pi$
$877$	1260.00	1.43672	0.718358	+	0.695674i	$0.244894\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1514.62i	1.71921i	0.510960	+	0.859604i	$0.329290\pi$
$881$	1514.62i	1.71921i	−0.510960	+	0.859604i	$0.670710\pi$
$882$	0	0
$883$	−950.352	−1.07628	−0.538138	−	0.842857i	$-0.680872\pi$
$883$	−950.352	−1.07628	−0.538138	+	0.842857i	$0.680872\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	216.000i	0.243517i	0.992560	+	0.121759i	$0.0388534\pi$
$887$	216.000i	0.243517i	−0.992560	+	0.121759i	$0.961147\pi$
$888$	0	0
$889$	−1368.00	−1.53881
$890$	0	0
$891$	0	0
$892$	0	0
$893$	610.940i	0.684144i
$894$	0	0
$895$	−237.588	−0.265461
$896$	0	0
$897$	0	0
$898$	0	0
$899$	108.000i	0.120133i
$900$	0	0
$901$	342.000	0.379578
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 534.573i	− 0.590688i
$906$	0	0
$907$	746.705	0.823269	0.411634	−	0.911349i	$-0.364958\pi$
$907$	746.705	0.823269	0.411634	+	0.911349i	$0.364958\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 144.000i	− 0.158068i	−0.996872	−	0.0790340i	$-0.974816\pi$
$911$	− 144.000i	− 0.158068i	0.996872	−	0.0790340i	$-0.0251836\pi$
$912$	0	0
$913$	304.000	0.332968
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1289.76i	− 1.40650i
$918$	0	0
$919$	653.367	0.710954	0.355477	−	0.934685i	$-0.384318\pi$
$919$	653.367	0.710954	0.355477	+	0.934685i	$0.384318\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1944.00i	2.10618i
$924$	0	0
$925$	252.000	0.272432
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1260.06i	− 1.35637i	−0.734893	−	0.678183i	$-0.762768\pi$
$929$	− 1260.06i	− 1.35637i	0.734893	−	0.678183i	$-0.237232\pi$
$930$	0	0
$931$	−390.323	−0.419251
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 72.0000i	− 0.0770053i
$936$	0	0
$937$	−1154.00	−1.23159	−0.615795	−	0.787906i	$-0.711165\pi$
$937$	−1154.00	−1.23159	−0.615795	+	0.787906i	$0.711165\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1090.36i	1.15872i	0.815071	+	0.579362i	$0.196698\pi$
$941$	1090.36i	1.15872i	−0.815071	+	0.579362i	$0.803302\pi$
$942$	0	0
$943$	−1069.15	−1.13377
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1120.00i	1.18268i	0.806422	+	0.591341i	$0.201401\pi$
$947$	1120.00i	1.18268i	−0.806422	+	0.591341i	$0.798599\pi$
$948$	0	0
$949$	1008.00	1.06217
$950$	0	0
$951$	0	0
$952$	0	0
$953$	335.169i	0.351698i	0.984417	+	0.175849i	$0.0562671\pi$
$953$	335.169i	0.351698i	−0.984417	+	0.175849i	$0.943733\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1764.00i	− 1.83942i
$960$	0	0
$961$	−889.000	−0.925078
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1264.31i	− 1.31016i
$966$	0	0
$967$	−313.955	−0.324670	−0.162335	−	0.986736i	$-0.551902\pi$
$967$	−313.955	−0.324670	−0.162335	+	0.986736i	$0.551902\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	772.000i	0.795057i	0.917590	+	0.397528i	$0.130132\pi$
$971$	772.000i	0.795057i	−0.917590	+	0.397528i	$0.869868\pi$
$972$	0	0
$973$	1008.00	1.03597
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1192.18i	1.22025i	0.792306	+	0.610124i	$0.208880\pi$
$977$	1192.18i	1.22025i	−0.792306	+	0.610124i	$0.791120\pi$
$978$	0	0
$979$	356.382	0.364026
$980$	0	0
$981$	0	0
$982$	0	0
$983$	504.000i	0.512716i	0.966582	+	0.256358i	$0.0825226\pi$
$983$	504.000i	0.512716i	−0.966582	+	0.256358i	$0.917477\pi$
$984$	0	0
$985$	1386.00	1.40711
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 2443.76i	− 2.47094i
$990$	0	0
$991$	−364.867	−0.368181	−0.184090	−	0.982909i	$-0.558934\pi$
$991$	−364.867	−0.368181	−0.184090	+	0.982909i	$0.558934\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 252.000i	− 0.253266i
$996$	0	0
$997$	−756.000	−0.758275	−0.379137	−	0.925340i	$-0.623779\pi$
$997$	−756.000	−0.758275	−0.379137	+	0.925340i	$0.623779\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.e.l.1025.2		4
3.2	odd	2	inner	2304.3.e.l.1025.4		4
4.3	odd	2	inner	2304.3.e.l.1025.1		4
8.3	odd	2		2304.3.e.e.1025.3		4
8.5	even	2		2304.3.e.e.1025.4		4
12.11	even	2	inner	2304.3.e.l.1025.3		4
16.3	odd	4		1152.3.h.a.449.2	yes	4
16.5	even	4		1152.3.h.a.449.3	yes	4
16.11	odd	4		1152.3.h.d.449.3	yes	4
16.13	even	4		1152.3.h.d.449.2	yes	4
24.5	odd	2		2304.3.e.e.1025.2		4
24.11	even	2		2304.3.e.e.1025.1		4
48.5	odd	4		1152.3.h.d.449.1	yes	4
48.11	even	4		1152.3.h.a.449.1	✓	4
48.29	odd	4		1152.3.h.a.449.4	yes	4
48.35	even	4		1152.3.h.d.449.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.h.a.449.1	✓	4	48.11	even	4
1152.3.h.a.449.2	yes	4	16.3	odd	4
1152.3.h.a.449.3	yes	4	16.5	even	4
1152.3.h.a.449.4	yes	4	48.29	odd	4
1152.3.h.d.449.1	yes	4	48.5	odd	4
1152.3.h.d.449.2	yes	4	16.13	even	4
1152.3.h.d.449.3	yes	4	16.11	odd	4
1152.3.h.d.449.4	yes	4	48.35	even	4
2304.3.e.e.1025.1		4	24.11	even	2
2304.3.e.e.1025.2		4	24.5	odd	2
2304.3.e.e.1025.3		4	8.3	odd	2
2304.3.e.e.1025.4		4	8.5	even	2
2304.3.e.l.1025.1		4	4.3	odd	2	inner
2304.3.e.l.1025.2		4	1.1	even	1	trivial
2304.3.e.l.1025.3		4	12.11	even	2	inner
2304.3.e.l.1025.4		4	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.24264i q^{5} +8.48528 q^{7} +O(q^{10})\)	\(q-4.24264i q^{5} +8.48528 q^{7} +4.00000i q^{11} +18.0000 q^{13} -4.24264i q^{17} -16.9706 q^{19} -36.0000i q^{23} +7.00000 q^{25} +12.7279i q^{29} +8.48528 q^{31} -36.0000i q^{35} +36.0000 q^{37} -29.6985i q^{41} +67.8823 q^{43} -36.0000i q^{47} +23.0000 q^{49} +80.6102i q^{53} +16.9706 q^{55} +80.0000i q^{59} -36.0000 q^{61} -76.3675i q^{65} -118.794 q^{67} +108.000i q^{71} +56.0000 q^{73} +33.9411i q^{77} +25.4558 q^{79} -76.0000i q^{83} -18.0000 q^{85} -89.0955i q^{89} +152.735 q^{91} +72.0000i q^{95} +104.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 72 q^{13} + 28 q^{25} + 144 q^{37} + 92 q^{49} - 144 q^{61} + 224 q^{73} - 72 q^{85} + 416 q^{97}+O(q^{100}) \) 4 * q + 72 * q^13 + 28 * q^25 + 144 * q^37 + 92 * q^49 - 144 * q^61 + 224 * q^73 - 72 * q^85 + 416 * q^97

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