LMFDB - Embedded newform 2304.3.e.m.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:	$8$
Coefficient field:	8.0.5473632256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 49x^{4} + 256 $ x^8 + 49*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.2
Root			$1.81129 + 1.81129i$ of defining polynomial
Character	$\chi$	$=$	2304.1025
Dual form			2304.3.e.m.1025.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.83095i q^{5} -2.82843 q^{7} +O(q^{10})$	$q-5.83095i q^{5} -2.82843 q^{7} -16.4924i q^{11} +8.24621 q^{13} -7.07107i q^{17} +23.3238 q^{19} +20.0000i q^{23} -9.00000 q^{25} -29.1548i q^{29} +42.4264 q^{31} +16.4924i q^{35} +49.4773 q^{37} -26.8701i q^{41} -44.0000i q^{47} -41.0000 q^{49} -29.1548i q^{53} -96.1665 q^{55} +65.9697i q^{59} +82.4621 q^{61} -48.0833i q^{65} -116.619 q^{67} +100.000i q^{71} -40.0000 q^{73} +46.6476i q^{77} +127.279 q^{79} -82.4621i q^{83} -41.2311 q^{85} -114.551i q^{89} -23.3238 q^{91} -136.000i q^{95} -40.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 72 q^{25} - 328 q^{49} - 320 q^{73} - 320 q^{97}+O(q^{100}) $ 8 * q - 72 * q^25 - 328 * q^49 - 320 * q^73 - 320 * 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
$5$	− 5.83095i	− 1.16619i	−0.812404	−	0.583095i	$-0.801841\pi$
$5$	− 5.83095i	− 1.16619i	0.812404	−	0.583095i	$-0.198159\pi$
$6$	0	0
$7$	−2.82843	−0.404061	−0.202031	−	0.979379i	$-0.564754\pi$
$7$	−2.82843	−0.404061	−0.202031	+	0.979379i	$0.564754\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 16.4924i	− 1.49931i	−0.661828	−	0.749656i	$-0.730219\pi$
$11$	− 16.4924i	− 1.49931i	0.661828	−	0.749656i	$-0.269781\pi$
$12$	0	0
$13$	8.24621	0.634324	0.317162	−	0.948371i	$-0.397270\pi$
$13$	8.24621	0.634324	0.317162	+	0.948371i	$0.397270\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.07107i	− 0.415945i	−0.978135	−	0.207973i	$-0.933314\pi$
$17$	− 7.07107i	− 0.415945i	0.978135	−	0.207973i	$-0.0666865\pi$
$18$	0	0
$19$	23.3238	1.22757	0.613784	−	0.789474i	$-0.289646\pi$
$19$	23.3238	1.22757	0.613784	+	0.789474i	$0.289646\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	20.0000i	0.869565i	0.900535	+	0.434783i	$0.143175\pi$
$23$	20.0000i	0.869565i	−0.900535	+	0.434783i	$0.856825\pi$
$24$	0	0
$25$	−9.00000	−0.360000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 29.1548i	− 1.00534i	−0.864479	−	0.502668i	$-0.832352\pi$
$29$	− 29.1548i	− 1.00534i	0.864479	−	0.502668i	$-0.167648\pi$
$30$	0	0
$31$	42.4264	1.36859	0.684297	−	0.729204i	$-0.260109\pi$
$31$	42.4264	1.36859	0.684297	+	0.729204i	$0.260109\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	16.4924i	0.471212i
$36$	0	0
$37$	49.4773	1.33722	0.668612	−	0.743612i	$-0.266889\pi$
$37$	49.4773	1.33722	0.668612	+	0.743612i	$0.266889\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 26.8701i	− 0.655367i	−0.944788	−	0.327684i	$-0.893732\pi$
$41$	− 26.8701i	− 0.655367i	0.944788	−	0.327684i	$-0.106268\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 44.0000i	− 0.936170i	−0.883683	−	0.468085i	$-0.844944\pi$
$47$	− 44.0000i	− 0.936170i	0.883683	−	0.468085i	$-0.155056\pi$
$48$	0	0
$49$	−41.0000	−0.836735
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 29.1548i	− 0.550090i	−0.961431	−	0.275045i	$-0.911307\pi$
$53$	− 29.1548i	− 0.550090i	0.961431	−	0.275045i	$-0.0886927\pi$
$54$	0	0
$55$	−96.1665	−1.74848
$56$	0	0
$57$	0	0
$58$	0	0
$59$	65.9697i	1.11813i	0.829124	+	0.559065i	$0.188840\pi$
$59$	65.9697i	1.11813i	−0.829124	+	0.559065i	$0.811160\pi$
$60$	0	0
$61$	82.4621	1.35184	0.675919	−	0.736976i	$-0.263747\pi$
$61$	82.4621	1.35184	0.675919	+	0.736976i	$0.263747\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 48.0833i	− 0.739742i
$66$	0	0
$67$	−116.619	−1.74058	−0.870291	−	0.492537i	$-0.836070\pi$
$67$	−116.619	−1.74058	−0.870291	+	0.492537i	$0.836070\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	100.000i	1.40845i	0.709977	+	0.704225i	$0.248706\pi$
$71$	100.000i	1.40845i	−0.709977	+	0.704225i	$0.751294\pi$
$72$	0	0
$73$	−40.0000	−0.547945	−0.273973	−	0.961737i	$-0.588338\pi$
$73$	−40.0000	−0.547945	−0.273973	+	0.961737i	$0.588338\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	46.6476i	0.605813i
$78$	0	0
$79$	127.279	1.61113	0.805565	−	0.592508i	$-0.201862\pi$
$79$	127.279	1.61113	0.805565	+	0.592508i	$0.201862\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 82.4621i	− 0.993519i	−0.867888	−	0.496760i	$-0.834523\pi$
$83$	− 82.4621i	− 0.993519i	0.867888	−	0.496760i	$-0.165477\pi$
$84$	0	0
$85$	−41.2311	−0.485071
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 114.551i	− 1.28709i	−0.765407	−	0.643547i	$-0.777462\pi$
$89$	− 114.551i	− 1.28709i	0.765407	−	0.643547i	$-0.222538\pi$
$90$	0	0
$91$	−23.3238	−0.256306
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 136.000i	− 1.43158i
$96$	0	0
$97$	−40.0000	−0.412371	−0.206186	−	0.978513i	$-0.566105\pi$
$97$	−40.0000	−0.412371	−0.206186	+	0.978513i	$0.566105\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	29.1548i	0.288661i	0.989530	+	0.144330i	$0.0461028\pi$
$101$	29.1548i	0.288661i	−0.989530	+	0.144330i	$0.953897\pi$
$102$	0	0
$103$	−166.877	−1.62017	−0.810083	−	0.586315i	$-0.800578\pi$
$103$	−166.877	−1.62017	−0.810083	+	0.586315i	$0.800578\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−206.155	−1.89133	−0.945666	−	0.325138i	$-0.894589\pi$
$109$	−206.155	−1.89133	−0.945666	+	0.325138i	$0.894589\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 205.061i	− 1.81470i	−0.420377	−	0.907349i	$-0.638102\pi$
$113$	− 205.061i	− 1.81470i	0.420377	−	0.907349i	$-0.361898\pi$
$114$	0	0
$115$	116.619	1.01408
$116$	0	0
$117$	0	0
$118$	0	0
$119$	20.0000i	0.168067i
$120$	0	0
$121$	−151.000	−1.24793
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 93.2952i	− 0.746362i
$126$	0	0
$127$	144.250	1.13583	0.567913	−	0.823089i	$-0.307751\pi$
$127$	144.250	1.13583	0.567913	+	0.823089i	$0.307751\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	98.9545i	0.755378i	0.925932	+	0.377689i	$0.123281\pi$
$131$	98.9545i	0.755378i	−0.925932	+	0.377689i	$0.876719\pi$
$132$	0	0
$133$	−65.9697	−0.496013
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 120.208i	− 0.877432i	−0.898626	−	0.438716i	$-0.855433\pi$
$137$	− 120.208i	− 0.877432i	0.898626	−	0.438716i	$-0.144567\pi$
$138$	0	0
$139$	116.619	0.838986	0.419493	−	0.907759i	$-0.362208\pi$
$139$	116.619	0.838986	0.419493	+	0.907759i	$0.362208\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 136.000i	− 0.951049i
$144$	0	0
$145$	−170.000	−1.17241
$146$	0	0
$147$	0	0
$148$	0	0
$149$	145.774i	0.978348i	0.872186	+	0.489174i	$0.162702\pi$
$149$	145.774i	0.978348i	−0.872186	+	0.489174i	$0.837298\pi$
$150$	0	0
$151$	−127.279	−0.842909	−0.421454	−	0.906850i	$-0.638480\pi$
$151$	−127.279	−0.842909	−0.421454	+	0.906850i	$0.638480\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 247.386i	− 1.59604i
$156$	0	0
$157$	49.4773	0.315142	0.157571	−	0.987508i	$-0.449634\pi$
$157$	49.4773	0.315142	0.157571	+	0.987508i	$0.449634\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 56.5685i	− 0.351357i
$162$	0	0
$163$	233.238	1.43091	0.715454	−	0.698660i	$-0.246220\pi$
$163$	233.238	1.43091	0.715454	+	0.698660i	$0.246220\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	160.000i	0.958084i	0.877792	+	0.479042i	$0.159016\pi$
$167$	160.000i	0.958084i	−0.877792	+	0.479042i	$0.840984\pi$
$168$	0	0
$169$	−101.000	−0.597633
$170$	0	0
$171$	0	0
$172$	0	0
$173$	320.702i	1.85377i	0.375344	+	0.926885i	$0.377524\pi$
$173$	320.702i	1.85377i	−0.375344	+	0.926885i	$0.622476\pi$
$174$	0	0
$175$	25.4558	0.145462
$176$	0	0
$177$	0	0
$178$	0	0
$179$	98.9545i	0.552819i	0.961040	+	0.276409i	$0.0891445\pi$
$179$	98.9545i	0.552819i	−0.961040	+	0.276409i	$0.910855\pi$
$180$	0	0
$181$	−206.155	−1.13898	−0.569490	−	0.821998i	$-0.692859\pi$
$181$	−206.155	−1.13898	−0.569490	+	0.821998i	$0.692859\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 288.500i	− 1.55946i
$186$	0	0
$187$	−116.619	−0.623631
$188$	0	0
$189$	0	0
$190$	0	0
$191$	160.000i	0.837696i	0.908056	+	0.418848i	$0.137566\pi$
$191$	160.000i	0.837696i	−0.908056	+	0.418848i	$0.862434\pi$
$192$	0	0
$193$	−150.000	−0.777202	−0.388601	−	0.921406i	$-0.627042\pi$
$193$	−150.000	−0.777202	−0.388601	+	0.921406i	$0.627042\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 157.436i	− 0.799166i	−0.916697	−	0.399583i	$-0.869155\pi$
$197$	− 157.436i	− 0.799166i	0.916697	−	0.399583i	$-0.130845\pi$
$198$	0	0
$199$	70.7107	0.355330	0.177665	−	0.984091i	$-0.443146\pi$
$199$	70.7107	0.355330	0.177665	+	0.984091i	$0.443146\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	82.4621i	0.406217i
$204$	0	0
$205$	−156.678	−0.764283
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 384.666i	− 1.84051i
$210$	0	0
$211$	−349.857	−1.65809	−0.829045	−	0.559181i	$-0.811116\pi$
$211$	−349.857	−1.65809	−0.829045	+	0.559181i	$0.811116\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−120.000	−0.552995
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 58.3095i	− 0.263844i
$222$	0	0
$223$	−2.82843	−0.0126835	−0.00634176	−	0.999980i	$-0.502019\pi$
$223$	−2.82843	−0.0126835	−0.00634176	+	0.999980i	$0.502019\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 247.386i	− 1.08981i	−0.838499	−	0.544904i	$-0.816566\pi$
$227$	− 247.386i	− 1.08981i	0.838499	−	0.544904i	$-0.183434\pi$
$228$	0	0
$229$	206.155	0.900241	0.450121	−	0.892968i	$-0.351381\pi$
$229$	206.155	0.900241	0.450121	+	0.892968i	$0.351381\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 190.919i	− 0.819394i	−0.912222	−	0.409697i	$-0.865634\pi$
$233$	− 190.919i	− 0.819394i	0.912222	−	0.409697i	$-0.134366\pi$
$234$	0	0
$235$	−256.562	−1.09175
$236$	0	0
$237$	0	0
$238$	0	0
$239$	40.0000i	0.167364i	0.996493	+	0.0836820i	$0.0266680\pi$
$239$	40.0000i	0.167364i	−0.996493	+	0.0836820i	$0.973332\pi$
$240$	0	0
$241$	−240.000	−0.995851	−0.497925	−	0.867220i	$-0.665905\pi$
$241$	−240.000	−0.995851	−0.497925	+	0.867220i	$0.665905\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	239.069i	0.975792i
$246$	0	0
$247$	192.333	0.778676
$248$	0	0
$249$	0	0
$250$	0	0
$251$	346.341i	1.37984i	0.723884	+	0.689922i	$0.242355\pi$
$251$	346.341i	1.37984i	−0.723884	+	0.689922i	$0.757645\pi$
$252$	0	0
$253$	329.848	1.30375
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 21.2132i	− 0.0825416i	−0.999148	−	0.0412708i	$-0.986859\pi$
$257$	− 21.2132i	− 0.0825416i	0.999148	−	0.0412708i	$-0.0131406\pi$
$258$	0	0
$259$	−139.943	−0.540320
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 360.000i	− 1.36882i	−0.729097	−	0.684411i	$-0.760060\pi$
$263$	− 360.000i	− 1.36882i	0.729097	−	0.684411i	$-0.239940\pi$
$264$	0	0
$265$	−170.000	−0.641509
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 437.321i	− 1.62573i	−0.582452	−	0.812865i	$-0.697907\pi$
$269$	− 437.321i	− 1.62573i	0.582452	−	0.812865i	$-0.302093\pi$
$270$	0	0
$271$	−240.416	−0.887145	−0.443573	−	0.896238i	$-0.646289\pi$
$271$	−240.416	−0.887145	−0.443573	+	0.896238i	$0.646289\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	148.432i	0.539752i
$276$	0	0
$277$	321.602	1.16102	0.580509	−	0.814254i	$-0.302853\pi$
$277$	321.602	1.16102	0.580509	+	0.814254i	$0.302853\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 227.688i	− 0.810279i	−0.914255	−	0.405139i	$-0.867223\pi$
$281$	− 227.688i	− 0.810279i	0.914255	−	0.405139i	$-0.132777\pi$
$282$	0	0
$283$	466.476	1.64833	0.824163	−	0.566353i	$-0.191646\pi$
$283$	466.476	1.64833	0.824163	+	0.566353i	$0.191646\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	76.0000i	0.264808i
$288$	0	0
$289$	239.000	0.826990
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 215.745i	− 0.736332i	−0.929760	−	0.368166i	$-0.879986\pi$
$293$	− 215.745i	− 0.736332i	0.929760	−	0.368166i	$-0.120014\pi$
$294$	0	0
$295$	384.666	1.30395
$296$	0	0
$297$	0	0
$298$	0	0
$299$	164.924i	0.551586i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 480.833i	− 1.57650i
$306$	0	0
$307$	−116.619	−0.379867	−0.189933	−	0.981797i	$-0.560827\pi$
$307$	−116.619	−0.379867	−0.189933	+	0.981797i	$0.560827\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	40.0000i	0.128617i	0.997930	+	0.0643087i	$0.0204842\pi$
$311$	40.0000i	0.128617i	−0.997930	+	0.0643087i	$0.979516\pi$
$312$	0	0
$313$	−250.000	−0.798722	−0.399361	−	0.916794i	$-0.630768\pi$
$313$	−250.000	−0.798722	−0.399361	+	0.916794i	$0.630768\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 17.4929i	− 0.0551825i	−0.999619	−	0.0275913i	$-0.991216\pi$
$317$	− 17.4929i	− 0.0551825i	0.999619	−	0.0275913i	$-0.00878368\pi$
$318$	0	0
$319$	−480.833	−1.50731
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 164.924i	− 0.510601i
$324$	0	0
$325$	−74.2159	−0.228357
$326$	0	0
$327$	0	0
$328$	0	0
$329$	124.451i	0.378270i
$330$	0	0
$331$	23.3238	0.0704647	0.0352323	−	0.999379i	$-0.488783\pi$
$331$	23.3238	0.0704647	0.0352323	+	0.999379i	$0.488783\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	680.000i	2.02985i
$336$	0	0
$337$	−520.000	−1.54303	−0.771513	−	0.636213i	$-0.780500\pi$
$337$	−520.000	−1.54303	−0.771513	+	0.636213i	$0.780500\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 699.714i	− 2.05195i
$342$	0	0
$343$	254.558	0.742153
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 247.386i	− 0.712929i	−0.934309	−	0.356464i	$-0.883982\pi$
$347$	− 247.386i	− 0.712929i	0.934309	−	0.356464i	$-0.116018\pi$
$348$	0	0
$349$	82.4621	0.236281	0.118141	−	0.992997i	$-0.462307\pi$
$349$	82.4621	0.236281	0.118141	+	0.992997i	$0.462307\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	360.624i	1.02160i	0.859700	+	0.510800i	$0.170651\pi$
$353$	360.624i	1.02160i	−0.859700	+	0.510800i	$0.829349\pi$
$354$	0	0
$355$	583.095	1.64252
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 340.000i	− 0.947075i	−0.880774	−	0.473538i	$-0.842977\pi$
$359$	− 340.000i	− 0.947075i	0.880774	−	0.473538i	$-0.157023\pi$
$360$	0	0
$361$	183.000	0.506925
$362$	0	0
$363$	0	0
$364$	0	0
$365$	233.238i	0.639008i
$366$	0	0
$367$	−364.867	−0.994188	−0.497094	−	0.867697i	$-0.665600\pi$
$367$	−364.867	−0.994188	−0.497094	+	0.867697i	$0.665600\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	82.4621i	0.222270i
$372$	0	0
$373$	−49.4773	−0.132647	−0.0663234	−	0.997798i	$-0.521127\pi$
$373$	−49.4773	−0.132647	−0.0663234	+	0.997798i	$0.521127\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 240.416i	− 0.637709i
$378$	0	0
$379$	−116.619	−0.307702	−0.153851	−	0.988094i	$-0.549168\pi$
$379$	−116.619	−0.307702	−0.153851	+	0.988094i	$0.549168\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 184.000i	− 0.480418i	−0.970721	−	0.240209i	$-0.922784\pi$
$383$	− 184.000i	− 0.480418i	0.970721	−	0.240209i	$-0.0772159\pi$
$384$	0	0
$385$	272.000	0.706494
$386$	0	0
$387$	0	0
$388$	0	0
$389$	379.012i	0.974324i	0.873312	+	0.487162i	$0.161968\pi$
$389$	379.012i	0.974324i	−0.873312	+	0.487162i	$0.838032\pi$
$390$	0	0
$391$	141.421	0.361691
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 742.159i	− 1.87888i
$396$	0	0
$397$	544.250	1.37091	0.685453	−	0.728117i	$-0.259604\pi$
$397$	544.250	1.37091	0.685453	+	0.728117i	$0.259604\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 267.286i	− 0.666550i	−0.942830	−	0.333275i	$-0.891846\pi$
$401$	− 267.286i	− 0.666550i	0.942830	−	0.333275i	$-0.108154\pi$
$402$	0	0
$403$	349.857	0.868132
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 816.000i	− 2.00491i
$408$	0	0
$409$	−232.000	−0.567237	−0.283619	−	0.958937i	$-0.591535\pi$
$409$	−232.000	−0.567237	−0.283619	+	0.958937i	$0.591535\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 186.590i	− 0.451793i
$414$	0	0
$415$	−480.833	−1.15863
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 16.4924i	− 0.0393614i	−0.999806	−	0.0196807i	$-0.993735\pi$
$419$	− 16.4924i	− 0.0393614i	0.999806	−	0.0196807i	$-0.00626496\pi$
$420$	0	0
$421$	123.693	0.293808	0.146904	−	0.989151i	$-0.453069\pi$
$421$	123.693	0.293808	0.146904	+	0.989151i	$0.453069\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	63.6396i	0.149740i
$426$	0	0
$427$	−233.238	−0.546225
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 140.000i	− 0.324826i	−0.986723	−	0.162413i	$-0.948072\pi$
$431$	− 140.000i	− 0.324826i	0.986723	−	0.162413i	$-0.0519277\pi$
$432$	0	0
$433$	−190.000	−0.438799	−0.219400	−	0.975635i	$-0.570410\pi$
$433$	−190.000	−0.438799	−0.219400	+	0.975635i	$0.570410\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	466.476i	1.06745i
$438$	0	0
$439$	664.680	1.51408	0.757039	−	0.653370i	$-0.226645\pi$
$439$	664.680	1.51408	0.757039	+	0.653370i	$0.226645\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 82.4621i	− 0.186145i	−0.995659	−	0.0930724i	$-0.970331\pi$
$443$	− 82.4621i	− 0.186145i	0.995659	−	0.0930724i	$-0.0296688\pi$
$444$	0	0
$445$	−667.943	−1.50100
$446$	0	0
$447$	0	0
$448$	0	0
$449$	436.992i	0.973256i	0.873609	+	0.486628i	$0.161773\pi$
$449$	436.992i	0.973256i	−0.873609	+	0.486628i	$0.838227\pi$
$450$	0	0
$451$	−443.152	−0.982599
$452$	0	0
$453$	0	0
$454$	0	0
$455$	136.000i	0.298901i
$456$	0	0
$457$	600.000	1.31291	0.656455	−	0.754365i	$-0.272055\pi$
$457$	600.000	1.31291	0.656455	+	0.754365i	$0.272055\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 87.4643i	− 0.189727i	−0.995490	−	0.0948636i	$-0.969758\pi$
$461$	− 87.4643i	− 0.189727i	0.995490	−	0.0948636i	$-0.0302415\pi$
$462$	0	0
$463$	−53.7401	−0.116069	−0.0580347	−	0.998315i	$-0.518483\pi$
$463$	−53.7401	−0.116069	−0.0580347	+	0.998315i	$0.518483\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	82.4621i	0.176578i	0.996095	+	0.0882892i	$0.0281400\pi$
$467$	82.4621i	0.176578i	−0.996095	+	0.0882892i	$0.971860\pi$
$468$	0	0
$469$	329.848	0.703302
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−209.914	−0.441925
$476$	0	0
$477$	0	0
$478$	0	0
$479$	860.000i	1.79541i	0.440600	+	0.897704i	$0.354766\pi$
$479$	860.000i	1.79541i	−0.440600	+	0.897704i	$0.645234\pi$
$480$	0	0
$481$	408.000	0.848233
$482$	0	0
$483$	0	0
$484$	0	0
$485$	233.238i	0.480903i
$486$	0	0
$487$	−31.1127	−0.0638864	−0.0319432	−	0.999490i	$-0.510170\pi$
$487$	−31.1127	−0.0638864	−0.0319432	+	0.999490i	$0.510170\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	428.803i	0.873326i	0.899625	+	0.436663i	$0.143840\pi$
$491$	428.803i	0.873326i	−0.899625	+	0.436663i	$0.856160\pi$
$492$	0	0
$493$	−206.155	−0.418165
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 282.843i	− 0.569100i
$498$	0	0
$499$	93.2952	0.186964	0.0934822	−	0.995621i	$-0.470200\pi$
$499$	93.2952	0.186964	0.0934822	+	0.995621i	$0.470200\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	556.000i	1.10537i	0.833391	+	0.552684i	$0.186396\pi$
$503$	556.000i	1.10537i	−0.833391	+	0.552684i	$0.813604\pi$
$504$	0	0
$505$	170.000	0.336634
$506$	0	0
$507$	0	0
$508$	0	0
$509$	670.559i	1.31741i	0.752403	+	0.658703i	$0.228895\pi$
$509$	670.559i	1.31741i	−0.752403	+	0.658703i	$0.771105\pi$
$510$	0	0
$511$	113.137	0.221403
$512$	0	0
$513$	0	0
$514$	0	0
$515$	973.053i	1.88942i
$516$	0	0
$517$	−725.667	−1.40361
$518$	0	0
$519$	0	0
$520$	0	0
$521$	920.653i	1.76709i	0.468348	+	0.883544i	$0.344849\pi$
$521$	920.653i	1.76709i	−0.468348	+	0.883544i	$0.655151\pi$
$522$	0	0
$523$	583.095	1.11490	0.557452	−	0.830209i	$-0.311779\pi$
$523$	583.095	1.11490	0.557452	+	0.830209i	$0.311779\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 300.000i	− 0.569260i
$528$	0	0
$529$	129.000	0.243856
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 221.576i	− 0.415715i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	676.189i	1.25453i
$540$	0	0
$541$	−865.852	−1.60047	−0.800233	−	0.599689i	$-0.795291\pi$
$541$	−865.852	−1.60047	−0.800233	+	0.599689i	$0.795291\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1202.08i	2.20565i
$546$	0	0
$547$	−932.952	−1.70558	−0.852790	−	0.522254i	$-0.825091\pi$
$547$	−932.952	−1.70558	−0.852790	+	0.522254i	$0.825091\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 680.000i	− 1.23412i
$552$	0	0
$553$	−360.000	−0.650995
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 87.4643i	− 0.157027i	−0.996913	−	0.0785137i	$-0.974983\pi$
$557$	− 87.4643i	− 0.157027i	0.996913	−	0.0785137i	$-0.0250174\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 1072.01i	− 1.90410i	−0.305945	−	0.952049i	$-0.598972\pi$
$563$	− 1072.01i	− 1.90410i	0.305945	−	0.952049i	$-0.401028\pi$
$564$	0	0
$565$	−1195.70	−2.11628
$566$	0	0
$567$	0	0
$568$	0	0
$569$	72.1249i	0.126757i	0.997990	+	0.0633786i	$0.0201876\pi$
$569$	72.1249i	0.126757i	−0.997990	+	0.0633786i	$0.979812\pi$
$570$	0	0
$571$	−793.009	−1.38881	−0.694404	−	0.719585i	$-0.744332\pi$
$571$	−793.009	−1.38881	−0.694404	+	0.719585i	$0.744332\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 180.000i	− 0.313043i
$576$	0	0
$577$	890.000	1.54246	0.771231	−	0.636556i	$-0.219642\pi$
$577$	890.000	1.54246	0.771231	+	0.636556i	$0.219642\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	233.238i	0.401442i
$582$	0	0
$583$	−480.833	−0.824756
$584$	0	0
$585$	0	0
$586$	0	0
$587$	989.545i	1.68577i	0.538096	+	0.842884i	$0.319144\pi$
$587$	989.545i	1.68577i	−0.538096	+	0.842884i	$0.680856\pi$
$588$	0	0
$589$	989.545	1.68004
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 473.762i	− 0.798923i	−0.916750	−	0.399462i	$-0.869197\pi$
$593$	− 473.762i	− 0.798923i	0.916750	−	0.399462i	$-0.130803\pi$
$594$	0	0
$595$	116.619	0.195998
$596$	0	0
$597$	0	0
$598$	0	0
$599$	740.000i	1.23539i	0.786417	+	0.617696i	$0.211934\pi$
$599$	740.000i	1.23539i	−0.786417	+	0.617696i	$0.788066\pi$
$600$	0	0
$601$	310.000	0.515807	0.257903	−	0.966171i	$-0.416968\pi$
$601$	310.000	0.515807	0.257903	+	0.966171i	$0.416968\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	880.474i	1.45533i
$606$	0	0
$607$	370.524	0.610418	0.305209	−	0.952285i	$-0.401274\pi$
$607$	370.524	0.610418	0.305209	+	0.952285i	$0.401274\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 362.833i	− 0.593835i
$612$	0	0
$613$	280.371	0.457376	0.228688	−	0.973500i	$-0.426557\pi$
$613$	280.371	0.457376	0.228688	+	0.973500i	$0.426557\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	586.899i	0.951213i	0.879658	+	0.475607i	$0.157771\pi$
$617$	586.899i	0.951213i	−0.879658	+	0.475607i	$0.842229\pi$
$618$	0	0
$619$	699.714	1.13039	0.565197	−	0.824956i	$-0.308800\pi$
$619$	699.714	1.13039	0.565197	+	0.824956i	$0.308800\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	324.000i	0.520064i
$624$	0	0
$625$	−769.000	−1.23040
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 349.857i	− 0.556212i
$630$	0	0
$631$	−381.838	−0.605131	−0.302566	−	0.953129i	$-0.597843\pi$
$631$	−381.838	−0.605131	−0.302566	+	0.953129i	$0.597843\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 841.114i	− 1.32459i
$636$	0	0
$637$	−338.095	−0.530761
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1203.50i	1.87753i	0.344560	+	0.938764i	$0.388028\pi$
$641$	1203.50i	1.87753i	−0.344560	+	0.938764i	$0.611972\pi$
$642$	0	0
$643$	233.238	0.362734	0.181367	−	0.983415i	$-0.441948\pi$
$643$	233.238	0.362734	0.181367	+	0.983415i	$0.441948\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	844.000i	1.30448i	0.758012	+	0.652241i	$0.226171\pi$
$647$	844.000i	1.30448i	−0.758012	+	0.652241i	$0.773829\pi$
$648$	0	0
$649$	1088.00	1.67643
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1032.08i	− 1.58052i	−0.612773	−	0.790259i	$-0.709946\pi$
$653$	− 1032.08i	− 1.58052i	0.612773	−	0.790259i	$-0.290054\pi$
$654$	0	0
$655$	576.999	0.880915
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 263.879i	− 0.400423i	−0.979753	−	0.200212i	$-0.935837\pi$
$659$	− 263.879i	− 0.400423i	0.979753	−	0.200212i	$-0.0641629\pi$
$660$	0	0
$661$	907.083	1.37229	0.686145	−	0.727465i	$-0.259302\pi$
$661$	907.083	1.37229	0.686145	+	0.727465i	$0.259302\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	384.666i	0.578445i
$666$	0	0
$667$	583.095	0.874206
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1360.00i	− 2.02683i
$672$	0	0
$673$	1070.00	1.58990	0.794948	−	0.606678i	$-0.207498\pi$
$673$	1070.00	1.58990	0.794948	+	0.606678i	$0.207498\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	40.8167i	0.0602905i	0.999546	+	0.0301452i	$0.00959698\pi$
$677$	40.8167i	0.0602905i	−0.999546	+	0.0301452i	$0.990403\pi$
$678$	0	0
$679$	113.137	0.166623
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1072.01i	− 1.56956i	−0.619776	−	0.784779i	$-0.712777\pi$
$683$	− 1072.01i	− 1.56956i	0.619776	−	0.784779i	$-0.287223\pi$
$684$	0	0
$685$	−700.928	−1.02325
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 240.416i	− 0.348935i
$690$	0	0
$691$	326.533	0.472552	0.236276	−	0.971686i	$-0.424073\pi$
$691$	326.533	0.472552	0.236276	+	0.971686i	$0.424073\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 680.000i	− 0.978417i
$696$	0	0
$697$	−190.000	−0.272597
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 495.631i	− 0.707034i	−0.935428	−	0.353517i	$-0.884986\pi$
$701$	− 495.631i	− 0.707034i	0.935428	−	0.353517i	$-0.115014\pi$
$702$	0	0
$703$	1154.00	1.64153
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 82.4621i	− 0.116637i
$708$	0	0
$709$	−123.693	−0.174461	−0.0872307	−	0.996188i	$-0.527802\pi$
$709$	−123.693	−0.174461	−0.0872307	+	0.996188i	$0.527802\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	848.528i	1.19008i
$714$	0	0
$715$	−793.009	−1.10910
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 300.000i	− 0.417246i	−0.977996	−	0.208623i	$-0.933102\pi$
$719$	− 300.000i	− 0.417246i	0.977996	−	0.208623i	$-0.0668982\pi$
$720$	0	0
$721$	472.000	0.654646
$722$	0	0
$723$	0	0
$724$	0	0
$725$	262.393i	0.361921i
$726$	0	0
$727$	−1298.25	−1.78576	−0.892880	−	0.450294i	$-0.851319\pi$
$727$	−1298.25	−1.78576	−0.892880	+	0.450294i	$0.851319\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	74.2159	0.101250	0.0506248	−	0.998718i	$-0.483879\pi$
$733$	74.2159	0.101250	0.0506248	+	0.998718i	$0.483879\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1923.33i	2.60967i
$738$	0	0
$739$	139.943	0.189368	0.0946839	−	0.995507i	$-0.469816\pi$
$739$	139.943	0.189368	0.0946839	+	0.995507i	$0.469816\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1264.00i	1.70121i	0.525804	+	0.850606i	$0.323764\pi$
$743$	1264.00i	1.70121i	−0.525804	+	0.850606i	$0.676236\pi$
$744$	0	0
$745$	850.000	1.14094
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	664.680	0.885060	0.442530	−	0.896754i	$-0.354081\pi$
$751$	664.680	0.885060	0.442530	+	0.896754i	$0.354081\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	742.159i	0.982992i
$756$	0	0
$757$	733.913	0.969502	0.484751	−	0.874652i	$-0.338910\pi$
$757$	733.913	0.969502	0.484751	+	0.874652i	$0.338910\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 72.1249i	− 0.0947765i	−0.998877	−	0.0473882i	$-0.984910\pi$
$761$	− 72.1249i	− 0.0947765i	0.998877	−	0.0473882i	$-0.0150898\pi$
$762$	0	0
$763$	583.095	0.764214
$764$	0	0
$765$	0	0
$766$	0	0
$767$	544.000i	0.709257i
$768$	0	0
$769$	−10.0000	−0.0130039	−0.00650195	−	0.999979i	$-0.502070\pi$
$769$	−10.0000	−0.0130039	−0.00650195	+	0.999979i	$0.502070\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	250.731i	0.324361i	0.986761	+	0.162180i	$0.0518527\pi$
$773$	250.731i	0.324361i	−0.986761	+	0.162180i	$0.948147\pi$
$774$	0	0
$775$	−381.838	−0.492694
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 626.712i	− 0.804508i
$780$	0	0
$781$	1649.24	2.11171
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 288.500i	− 0.367515i
$786$	0	0
$787$	816.333	1.03727	0.518636	−	0.854995i	$-0.326440\pi$
$787$	816.333	1.03727	0.518636	+	0.854995i	$0.326440\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	580.000i	0.733249i
$792$	0	0
$793$	680.000	0.857503
$794$	0	0
$795$	0	0
$796$	0	0
$797$	320.702i	0.402387i	0.979552	+	0.201193i	$0.0644820\pi$
$797$	320.702i	0.402387i	−0.979552	+	0.201193i	$0.935518\pi$
$798$	0	0
$799$	−311.127	−0.389395
$800$	0	0
$801$	0	0
$802$	0	0
$803$	659.697i	0.821540i
$804$	0	0
$805$	−329.848	−0.409750
$806$	0	0
$807$	0	0
$808$	0	0
$809$	550.129i	0.680011i	0.940423	+	0.340006i	$0.110429\pi$
$809$	550.129i	0.680011i	−0.940423	+	0.340006i	$0.889571\pi$
$810$	0	0
$811$	−116.619	−0.143797	−0.0718983	−	0.997412i	$-0.522906\pi$
$811$	−116.619	−0.143797	−0.0718983	+	0.997412i	$0.522906\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1360.00i	− 1.66871i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 204.083i	− 0.248579i	−0.992246	−	0.124289i	$-0.960335\pi$
$821$	− 204.083i	− 0.248579i	0.992246	−	0.124289i	$-0.0396651\pi$
$822$	0	0
$823$	−1439.67	−1.74929	−0.874647	−	0.484760i	$-0.838907\pi$
$823$	−1439.67	−1.74929	−0.874647	+	0.484760i	$0.838907\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	494.773i	0.598274i	0.954210	+	0.299137i	$0.0966988\pi$
$827$	494.773i	0.598274i	−0.954210	+	0.299137i	$0.903301\pi$
$828$	0	0
$829$	1113.24	1.34287	0.671435	−	0.741064i	$-0.265678\pi$
$829$	1113.24	1.34287	0.671435	+	0.741064i	$0.265678\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	289.914i	0.348036i
$834$	0	0
$835$	932.952	1.11731
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 820.000i	− 0.977354i	−0.872465	−	0.488677i	$-0.837480\pi$
$839$	− 820.000i	− 0.977354i	0.872465	−	0.488677i	$-0.162520\pi$
$840$	0	0
$841$	−9.00000	−0.0107015
$842$	0	0
$843$	0	0
$844$	0	0
$845$	588.926i	0.696954i
$846$	0	0
$847$	427.092	0.504241
$848$	0	0
$849$	0	0
$850$	0	0
$851$	989.545i	1.16280i
$852$	0	0
$853$	1104.99	1.29542	0.647709	−	0.761887i	$-0.275727\pi$
$853$	1104.99	1.29542	0.647709	+	0.761887i	$0.275727\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	742.462i	0.866350i	0.901310	+	0.433175i	$0.142607\pi$
$857$	742.462i	0.866350i	−0.901310	+	0.433175i	$0.857393\pi$
$858$	0	0
$859$	−932.952	−1.08609	−0.543046	−	0.839703i	$-0.682729\pi$
$859$	−932.952	−1.08609	−0.543046	+	0.839703i	$0.682729\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1360.00i	− 1.57590i	−0.615740	−	0.787949i	$-0.711143\pi$
$863$	− 1360.00i	− 1.57590i	0.615740	−	0.787949i	$-0.288857\pi$
$864$	0	0
$865$	1870.00	2.16185
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2099.14i	− 2.41558i
$870$	0	0
$871$	−961.665	−1.10409
$872$	0	0
$873$	0	0
$874$	0	0
$875$	263.879i	0.301576i
$876$	0	0
$877$	−379.326	−0.432526	−0.216263	−	0.976335i	$-0.569387\pi$
$877$	−379.326	−0.432526	−0.216263	+	0.976335i	$0.569387\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 111.723i	− 0.126814i	−0.997988	−	0.0634069i	$-0.979803\pi$
$881$	− 111.723i	− 0.126814i	0.997988	−	0.0634069i	$-0.0201966\pi$
$882$	0	0
$883$	−1399.43	−1.58486	−0.792428	−	0.609965i	$-0.791183\pi$
$883$	−1399.43	−1.58486	−0.792428	+	0.609965i	$0.791183\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	840.000i	0.947012i	0.880790	+	0.473506i	$0.157012\pi$
$887$	840.000i	0.947012i	−0.880790	+	0.473506i	$0.842988\pi$
$888$	0	0
$889$	−408.000	−0.458943
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1026.25i	− 1.14921i
$894$	0	0
$895$	576.999	0.644692
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1236.93i	− 1.37590i
$900$	0	0
$901$	−206.155	−0.228807
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1202.08i	1.32827i
$906$	0	0
$907$	−932.952	−1.02861	−0.514307	−	0.857606i	$-0.671951\pi$
$907$	−932.952	−1.02861	−0.514307	+	0.857606i	$0.671951\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	560.000i	0.614709i	0.951595	+	0.307355i	$0.0994438\pi$
$911$	560.000i	0.614709i	−0.951595	+	0.307355i	$0.900556\pi$
$912$	0	0
$913$	−1360.00	−1.48959
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 279.886i	− 0.305219i
$918$	0	0
$919$	777.817	0.846374	0.423187	−	0.906042i	$-0.360911\pi$
$919$	777.817	0.846374	0.423187	+	0.906042i	$0.360911\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	824.621i	0.893414i
$924$	0	0
$925$	−445.295	−0.481400
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1497.65i	1.61211i	0.591839	+	0.806056i	$0.298402\pi$
$929$	1497.65i	1.61211i	−0.591839	+	0.806056i	$0.701598\pi$
$930$	0	0
$931$	−956.276	−1.02715
$932$	0	0
$933$	0	0
$934$	0	0
$935$	680.000i	0.727273i
$936$	0	0
$937$	510.000	0.544290	0.272145	−	0.962256i	$-0.412267\pi$
$937$	510.000	0.544290	0.272145	+	0.962256i	$0.412267\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 553.940i	− 0.588672i	−0.955702	−	0.294336i	$-0.904902\pi$
$941$	− 553.940i	− 0.588672i	0.955702	−	0.294336i	$-0.0950985\pi$
$942$	0	0
$943$	537.401	0.569885
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	−329.848	−0.347575
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 912.168i	− 0.957154i	−0.878046	−	0.478577i	$-0.841153\pi$
$953$	− 912.168i	− 0.957154i	0.878046	−	0.478577i	$-0.158847\pi$
$954$	0	0
$955$	932.952	0.976913
$956$	0	0
$957$	0	0
$958$	0	0
$959$	340.000i	0.354536i
$960$	0	0
$961$	839.000	0.873049
$962$	0	0
$963$	0	0
$964$	0	0
$965$	874.643i	0.906366i
$966$	0	0
$967$	285.671	0.295420	0.147710	−	0.989031i	$-0.452810\pi$
$967$	285.671	0.295420	0.147710	+	0.989031i	$0.452810\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	643.204i	0.662414i	0.943558	+	0.331207i	$0.107456\pi$
$971$	643.204i	0.662414i	−0.943558	+	0.331207i	$0.892544\pi$
$972$	0	0
$973$	−329.848	−0.339001
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 388.909i	− 0.398064i	−0.979993	−	0.199032i	$-0.936220\pi$
$977$	− 388.909i	− 0.398064i	0.979993	−	0.199032i	$-0.0637798\pi$
$978$	0	0
$979$	−1889.23	−1.92975
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 984.000i	− 1.00102i	−0.865732	−	0.500509i	$-0.833146\pi$
$983$	− 984.000i	− 1.00102i	0.865732	−	0.500509i	$-0.166854\pi$
$984$	0	0
$985$	−918.000	−0.931980
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	890.955	0.899046	0.449523	−	0.893269i	$-0.351594\pi$
$991$	890.955	0.899046	0.449523	+	0.893269i	$0.351594\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 412.311i	− 0.414382i
$996$	0	0
$997$	940.068	0.942897	0.471448	−	0.881894i	$-0.343731\pi$
$997$	940.068	0.942897	0.471448	+	0.881894i	$0.343731\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.e.m.1025.2		8
3.2	odd	2	inner	2304.3.e.m.1025.6		8
4.3	odd	2	inner	2304.3.e.m.1025.4		8
8.3	odd	2	inner	2304.3.e.m.1025.7		8
8.5	even	2	inner	2304.3.e.m.1025.5		8
12.11	even	2	inner	2304.3.e.m.1025.8		8
16.3	odd	4		1152.3.h.f.449.2	yes	8
16.5	even	4		1152.3.h.f.449.7	yes	8
16.11	odd	4		1152.3.h.f.449.5	yes	8
16.13	even	4		1152.3.h.f.449.4	yes	8
24.5	odd	2	inner	2304.3.e.m.1025.1		8
24.11	even	2	inner	2304.3.e.m.1025.3		8
48.5	odd	4		1152.3.h.f.449.3	yes	8
48.11	even	4		1152.3.h.f.449.1	✓	8
48.29	odd	4		1152.3.h.f.449.8	yes	8
48.35	even	4		1152.3.h.f.449.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.h.f.449.1	✓	8	48.11	even	4
1152.3.h.f.449.2	yes	8	16.3	odd	4
1152.3.h.f.449.3	yes	8	48.5	odd	4
1152.3.h.f.449.4	yes	8	16.13	even	4
1152.3.h.f.449.5	yes	8	16.11	odd	4
1152.3.h.f.449.6	yes	8	48.35	even	4
1152.3.h.f.449.7	yes	8	16.5	even	4
1152.3.h.f.449.8	yes	8	48.29	odd	4
2304.3.e.m.1025.1		8	24.5	odd	2	inner
2304.3.e.m.1025.2		8	1.1	even	1	trivial
2304.3.e.m.1025.3		8	24.11	even	2	inner
2304.3.e.m.1025.4		8	4.3	odd	2	inner
2304.3.e.m.1025.5		8	8.5	even	2	inner
2304.3.e.m.1025.6		8	3.2	odd	2	inner
2304.3.e.m.1025.7		8	8.3	odd	2	inner
2304.3.e.m.1025.8		8	12.11	even	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)

\(f(q)\)	\(=\)	\(q-5.83095i q^{5} -2.82843 q^{7} +O(q^{10})\)	\(q-5.83095i q^{5} -2.82843 q^{7} -16.4924i q^{11} +8.24621 q^{13} -7.07107i q^{17} +23.3238 q^{19} +20.0000i q^{23} -9.00000 q^{25} -29.1548i q^{29} +42.4264 q^{31} +16.4924i q^{35} +49.4773 q^{37} -26.8701i q^{41} -44.0000i q^{47} -41.0000 q^{49} -29.1548i q^{53} -96.1665 q^{55} +65.9697i q^{59} +82.4621 q^{61} -48.0833i q^{65} -116.619 q^{67} +100.000i q^{71} -40.0000 q^{73} +46.6476i q^{77} +127.279 q^{79} -82.4621i q^{83} -41.2311 q^{85} -114.551i q^{89} -23.3238 q^{91} -136.000i q^{95} -40.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 72 q^{25} - 328 q^{49} - 320 q^{73} - 320 q^{97}+O(q^{100}) \) 8 * q - 72 * q^25 - 328 * q^49 - 320 * q^73 - 320 * q^97

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