LMFDB - Embedded newform 2304.3.h.k.2177.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(2177,2304)]

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, 1, 1]))

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

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

chi := DirichletCharacter("2304.2177");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.h (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:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
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			2177.2
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2304.2177
Dual form			2304.3.h.k.2177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.87832 q^{5} +11.7980 q^{7} +9.75663 q^{11} +22.6969i q^{13} +22.1988i q^{17} -17.7980i q^{19} +14.1421i q^{23} -1.20204 q^{25} -20.0061 q^{29} +39.7980 q^{31} -57.5542 q^{35} -2.40408i q^{37} +64.3395i q^{41} -3.19184i q^{43} -41.8549i q^{47} +90.1918 q^{49} -55.5043 q^{53} -47.5959 q^{55} -111.423 q^{59} +10.8082i q^{61} -110.723i q^{65} -18.2020i q^{67} +34.7983i q^{71} -87.5959 q^{73} +115.108 q^{77} +151.394 q^{79} -61.8112 q^{83} -108.293i q^{85} -72.9532i q^{89} +267.778i q^{91} +86.8241i q^{95} -87.3735 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7} - 88 q^{25} + 240 q^{31} + 408 q^{49} - 224 q^{55} - 544 q^{73} + 976 q^{79} + 320 q^{97}+O(q^{100}) $ 8 * q + 16 * q^7 - 88 * q^25 + 240 * q^31 + 408 * q^49 - 224 * q^55 - 544 * q^73 + 976 * q^79 + 320 * q^97

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
$5$	−4.87832	−0.975663	−0.487832	−	0.872938i	$-0.662212\pi$
$5$	−4.87832	−0.975663	−0.487832	+	0.872938i	$0.662212\pi$
$6$	0	0
$7$	11.7980	1.68542	0.842711	−	0.538366i	$-0.180958\pi$
$7$	11.7980	1.68542	0.842711	+	0.538366i	$0.180958\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	9.75663	0.886966	0.443483	−	0.896283i	$-0.353743\pi$
$11$	9.75663	0.886966	0.443483	+	0.896283i	$0.353743\pi$
$12$	0	0
$13$	22.6969i	1.74592i	0.487793	+	0.872959i	$0.337802\pi$
$13$	22.6969i	1.74592i	−0.487793	+	0.872959i	$0.662198\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	22.1988i	1.30581i	0.757438	+	0.652907i	$0.226451\pi$
$17$	22.1988i	1.30581i	−0.757438	+	0.652907i	$0.773549\pi$
$18$	0	0
$19$	− 17.7980i	− 0.936735i	−0.883534	−	0.468367i	$-0.844842\pi$
$19$	− 17.7980i	− 0.936735i	0.883534	−	0.468367i	$-0.155158\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	14.1421i	0.614875i	0.951568	+	0.307438i	$0.0994716\pi$
$23$	14.1421i	0.614875i	−0.951568	+	0.307438i	$0.900528\pi$
$24$	0	0
$25$	−1.20204	−0.0480816
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−20.0061	−0.689865	−0.344932	−	0.938628i	$-0.612098\pi$
$29$	−20.0061	−0.689865	−0.344932	+	0.938628i	$0.612098\pi$
$30$	0	0
$31$	39.7980	1.28381	0.641903	−	0.766786i	$-0.278145\pi$
$31$	39.7980	1.28381	0.641903	+	0.766786i	$0.278145\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−57.5542	−1.64440
$36$	0	0
$37$	− 2.40408i	− 0.0649752i	−0.999472	−	0.0324876i	$-0.989657\pi$
$37$	− 2.40408i	− 0.0649752i	0.999472	−	0.0324876i	$-0.0103429\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	64.3395i	1.56926i	0.619967	+	0.784628i	$0.287146\pi$
$41$	64.3395i	1.56926i	−0.619967	+	0.784628i	$0.712854\pi$
$42$	0	0
$43$	− 3.19184i	− 0.0742287i	−0.999311	−	0.0371144i	$-0.988183\pi$
$43$	− 3.19184i	− 0.0742287i	0.999311	−	0.0371144i	$-0.0118166\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 41.8549i	− 0.890531i	−0.895399	−	0.445265i	$-0.853109\pi$
$47$	− 41.8549i	− 0.890531i	0.895399	−	0.445265i	$-0.146891\pi$
$48$	0	0
$49$	90.1918	1.84065
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−55.5043	−1.04725	−0.523625	−	0.851949i	$-0.675421\pi$
$53$	−55.5043	−1.04725	−0.523625	+	0.851949i	$0.675421\pi$
$54$	0	0
$55$	−47.5959	−0.865380
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−111.423	−1.88852	−0.944260	−	0.329200i	$-0.893221\pi$
$59$	−111.423	−1.88852	−0.944260	+	0.329200i	$0.893221\pi$
$60$	0	0
$61$	10.8082i	0.177183i	0.996068	+	0.0885915i	$0.0282366\pi$
$61$	10.8082i	0.177183i	−0.996068	+	0.0885915i	$0.971763\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 110.723i	− 1.70343i
$66$	0	0
$67$	− 18.2020i	− 0.271672i	−0.990731	−	0.135836i	$-0.956628\pi$
$67$	− 18.2020i	− 0.271672i	0.990731	−	0.135836i	$-0.0433721\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	34.7983i	0.490117i	0.969508	+	0.245059i	$0.0788072\pi$
$71$	34.7983i	0.490117i	−0.969508	+	0.245059i	$0.921193\pi$
$72$	0	0
$73$	−87.5959	−1.19994	−0.599972	−	0.800021i	$-0.704822\pi$
$73$	−87.5959	−1.19994	−0.599972	+	0.800021i	$0.704822\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	115.108	1.49491
$78$	0	0
$79$	151.394	1.91638	0.958189	−	0.286136i	$-0.0923708\pi$
$79$	151.394	1.91638	0.958189	+	0.286136i	$0.0923708\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−61.8112	−0.744714	−0.372357	−	0.928090i	$-0.621450\pi$
$83$	−61.8112	−0.744714	−0.372357	+	0.928090i	$0.621450\pi$
$84$	0	0
$85$	− 108.293i	− 1.27403i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 72.9532i	− 0.819699i	−0.912153	−	0.409850i	$-0.865581\pi$
$89$	− 72.9532i	− 0.819699i	0.912153	−	0.409850i	$-0.134419\pi$
$90$	0	0
$91$	267.778i	2.94261i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	86.8241i	0.913937i
$96$	0	0
$97$	−87.3735	−0.900757	−0.450379	−	0.892838i	$-0.648711\pi$
$97$	−87.3735	−0.900757	−0.450379	+	0.892838i	$0.648711\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.63573	−0.0161953	−0.00809766	−	0.999967i	$-0.502578\pi$
$101$	−1.63573	−0.0161953	−0.00809766	+	0.999967i	$0.502578\pi$
$102$	0	0
$103$	38.9898	0.378542	0.189271	−	0.981925i	$-0.439388\pi$
$103$	38.9898	0.378542	0.189271	+	0.981925i	$0.439388\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	205.303	1.91872	0.959362	−	0.282179i	$-0.0910573\pi$
$107$	205.303	1.91872	0.959362	+	0.282179i	$0.0910573\pi$
$108$	0	0
$109$	33.3031i	0.305533i	0.988262	+	0.152766i	$0.0488182\pi$
$109$	33.3031i	0.305533i	−0.988262	+	0.152766i	$0.951182\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 93.8951i	− 0.830930i	−0.909609	−	0.415465i	$-0.863619\pi$
$113$	− 93.8951i	− 0.830930i	0.909609	−	0.415465i	$-0.136381\pi$
$114$	0	0
$115$	− 68.9898i	− 0.599911i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	261.901i	2.20085i
$120$	0	0
$121$	−25.8082	−0.213291
$122$	0	0
$123$	0	0
$124$	0	0
$125$	127.822	1.02257
$126$	0	0
$127$	45.8184	0.360775	0.180387	−	0.983596i	$-0.442265\pi$
$127$	45.8184	0.360775	0.180387	+	0.983596i	$0.442265\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−194.304	−1.48324	−0.741619	−	0.670821i	$-0.765942\pi$
$131$	−194.304	−1.48324	−0.741619	+	0.670821i	$0.765942\pi$
$132$	0	0
$133$	− 209.980i	− 1.57879i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	90.3668i	0.659612i	0.944049	+	0.329806i	$0.106983\pi$
$137$	90.3668i	0.659612i	−0.944049	+	0.329806i	$0.893017\pi$
$138$	0	0
$139$	8.22245i	0.0591543i	0.999563	+	0.0295772i	$0.00941608\pi$
$139$	8.22245i	0.0591543i	−0.999563	+	0.0295772i	$0.990584\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	221.446i	1.54857i
$144$	0	0
$145$	97.5959	0.673075
$146$	0	0
$147$	0	0
$148$	0	0
$149$	79.2457	0.531851	0.265925	−	0.963994i	$-0.414323\pi$
$149$	79.2457	0.531851	0.265925	+	0.963994i	$0.414323\pi$
$150$	0	0
$151$	−14.9898	−0.0992702	−0.0496351	−	0.998767i	$-0.515806\pi$
$151$	−14.9898	−0.0992702	−0.0496351	+	0.998767i	$0.515806\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−194.147	−1.25256
$156$	0	0
$157$	267.576i	1.70430i	0.523295	+	0.852151i	$0.324702\pi$
$157$	267.576i	1.70430i	−0.523295	+	0.852151i	$0.675298\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	166.848i	1.03633i
$162$	0	0
$163$	261.171i	1.60228i	0.598478	+	0.801139i	$0.295772\pi$
$163$	261.171i	1.60228i	−0.598478	+	0.801139i	$0.704228\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	300.070i	1.79683i	0.439150	+	0.898414i	$0.355280\pi$
$167$	300.070i	1.79683i	−0.439150	+	0.898414i	$0.644720\pi$
$168$	0	0
$169$	−346.151	−2.04823
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−31.2909	−0.180872	−0.0904362	−	0.995902i	$-0.528826\pi$
$173$	−31.2909	−0.180872	−0.0904362	+	0.995902i	$0.528826\pi$
$174$	0	0
$175$	−14.1816	−0.0810379
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−66.4825	−0.371410	−0.185705	−	0.982606i	$-0.559457\pi$
$179$	−66.4825	−0.371410	−0.185705	+	0.982606i	$0.559457\pi$
$180$	0	0
$181$	235.464i	1.30091i	0.759546	+	0.650454i	$0.225421\pi$
$181$	235.464i	1.30091i	−0.759546	+	0.650454i	$0.774579\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.7279i	0.0633939i
$186$	0	0
$187$	216.586i	1.15821i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 140.535i	− 0.735787i	−0.929868	−	0.367893i	$-0.880079\pi$
$191$	− 140.535i	− 0.735787i	0.929868	−	0.367893i	$-0.119921\pi$
$192$	0	0
$193$	293.151	1.51892	0.759459	−	0.650556i	$-0.225464\pi$
$193$	293.151	1.51892	0.759459	+	0.650556i	$0.225464\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−60.4324	−0.306763	−0.153382	−	0.988167i	$-0.549016\pi$
$197$	−60.4324	−0.306763	−0.153382	+	0.988167i	$0.549016\pi$
$198$	0	0
$199$	−143.757	−0.722398	−0.361199	−	0.932489i	$-0.617632\pi$
$199$	−143.757	−0.722398	−0.361199	+	0.932489i	$0.617632\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−236.031	−1.16271
$204$	0	0
$205$	− 313.868i	− 1.53107i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 173.648i	− 0.830852i
$210$	0	0
$211$	255.778i	1.21222i	0.795382	+	0.606108i	$0.207270\pi$
$211$	255.778i	1.21222i	−0.795382	+	0.606108i	$0.792730\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	15.5708i	0.0724222i
$216$	0	0
$217$	469.535	2.16375
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−503.845	−2.27984
$222$	0	0
$223$	248.969	1.11645	0.558227	−	0.829688i	$-0.311482\pi$
$223$	248.969	1.11645	0.558227	+	0.829688i	$0.311482\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−277.600	−1.22291	−0.611454	−	0.791280i	$-0.709415\pi$
$227$	−277.600	−1.22291	−0.611454	+	0.791280i	$0.709415\pi$
$228$	0	0
$229$	193.666i	0.845704i	0.906199	+	0.422852i	$0.138971\pi$
$229$	193.666i	0.845704i	−0.906199	+	0.422852i	$0.861029\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	150.435i	0.645643i	0.946460	+	0.322821i	$0.104631\pi$
$233$	150.435i	0.645643i	−0.946460	+	0.322821i	$0.895369\pi$
$234$	0	0
$235$	204.182i	0.868858i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	140.593i	0.588255i	0.955766	+	0.294128i	$0.0950291\pi$
$239$	140.593i	0.588255i	−0.955766	+	0.294128i	$0.904971\pi$
$240$	0	0
$241$	−228.182	−0.946812	−0.473406	−	0.880844i	$-0.656976\pi$
$241$	−228.182	−0.946812	−0.473406	+	0.880844i	$0.656976\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−439.984	−1.79585
$246$	0	0
$247$	403.959	1.63546
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−214.546	−0.854766	−0.427383	−	0.904071i	$-0.640564\pi$
$251$	−214.546	−0.854766	−0.427383	+	0.904071i	$0.640564\pi$
$252$	0	0
$253$	137.980i	0.545374i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	382.652i	1.48892i	0.667669	+	0.744458i	$0.267292\pi$
$257$	382.652i	1.48892i	−0.667669	+	0.744458i	$0.732708\pi$
$258$	0	0
$259$	− 28.3633i	− 0.109511i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 164.563i	− 0.625713i	−0.949800	−	0.312856i	$-0.898714\pi$
$263$	− 164.563i	− 0.625713i	0.949800	−	0.312856i	$-0.101286\pi$
$264$	0	0
$265$	270.767	1.02176
$266$	0	0
$267$	0	0
$268$	0	0
$269$	239.480	0.890262	0.445131	−	0.895465i	$-0.353157\pi$
$269$	239.480	0.890262	0.445131	+	0.895465i	$0.353157\pi$
$270$	0	0
$271$	153.778	0.567445	0.283722	−	0.958906i	$-0.408431\pi$
$271$	153.778	0.567445	0.283722	+	0.958906i	$0.408431\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11.7279	−0.0426468
$276$	0	0
$277$	− 135.242i	− 0.488238i	−0.969745	−	0.244119i	$-0.921501\pi$
$277$	− 135.242i	− 0.488238i	0.969745	−	0.244119i	$-0.0784987\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	63.6396i	0.226475i	0.993568	+	0.113238i	$0.0361222\pi$
$281$	63.6396i	0.226475i	−0.993568	+	0.113238i	$0.963878\pi$
$282$	0	0
$283$	− 333.576i	− 1.17871i	−0.807873	−	0.589356i	$-0.799382\pi$
$283$	− 333.576i	− 1.17871i	0.807873	−	0.589356i	$-0.200618\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	759.075i	2.64486i
$288$	0	0
$289$	−203.788	−0.705148
$290$	0	0
$291$	0	0
$292$	0	0
$293$	278.822	0.951610	0.475805	−	0.879551i	$-0.342157\pi$
$293$	278.822	0.951610	0.475805	+	0.879551i	$0.342157\pi$
$294$	0	0
$295$	543.555	1.84256
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−320.983	−1.07352
$300$	0	0
$301$	− 37.6571i	− 0.125107i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 52.7256i	− 0.172871i
$306$	0	0
$307$	− 99.7367i	− 0.324875i	−0.986719	−	0.162438i	$-0.948064\pi$
$307$	− 99.7367i	− 0.324875i	0.986719	−	0.162438i	$-0.0519356\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 313.927i	− 1.00941i	−0.863292	−	0.504705i	$-0.831601\pi$
$311$	− 313.927i	− 1.00941i	0.863292	−	0.504705i	$-0.168399\pi$
$312$	0	0
$313$	52.4245	0.167490	0.0837452	−	0.996487i	$-0.473312\pi$
$313$	52.4245	0.167490	0.0837452	+	0.996487i	$0.473312\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	68.8888	0.217315	0.108657	−	0.994079i	$-0.465345\pi$
$317$	68.8888	0.217315	0.108657	+	0.994079i	$0.465345\pi$
$318$	0	0
$319$	−195.192	−0.611887
$320$	0	0
$321$	0	0
$322$	0	0
$323$	395.094	1.22320
$324$	0	0
$325$	− 27.2827i	− 0.0839466i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 493.803i	− 1.50092i
$330$	0	0
$331$	− 114.202i	− 0.345021i	−0.985008	−	0.172511i	$-0.944812\pi$
$331$	− 114.202i	− 0.345021i	0.985008	−	0.172511i	$-0.0551879\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	88.7953i	0.265061i
$336$	0	0
$337$	−203.192	−0.602943	−0.301472	−	0.953475i	$-0.597478\pi$
$337$	−203.192	−0.602943	−0.301472	+	0.953475i	$0.597478\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	388.294	1.13869
$342$	0	0
$343$	485.980	1.41685
$344$	0	0
$345$	0	0
$346$	0	0
$347$	79.5524	0.229258	0.114629	−	0.993408i	$-0.463432\pi$
$347$	79.5524	0.229258	0.114629	+	0.993408i	$0.463432\pi$
$348$	0	0
$349$	− 479.898i	− 1.37507i	−0.726153	−	0.687533i	$-0.758694\pi$
$349$	− 479.898i	− 1.37507i	0.726153	−	0.687533i	$-0.241306\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 132.922i	− 0.376549i	−0.982116	−	0.188274i	$-0.939711\pi$
$353$	− 132.922i	− 0.376549i	0.982116	−	0.188274i	$-0.0602894\pi$
$354$	0	0
$355$	− 169.757i	− 0.478189i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	660.308i	1.83930i	0.392742	+	0.919649i	$0.371527\pi$
$359$	660.308i	1.83930i	−0.392742	+	0.919649i	$0.628473\pi$
$360$	0	0
$361$	44.2327	0.122528
$362$	0	0
$363$	0	0
$364$	0	0
$365$	427.320	1.17074
$366$	0	0
$367$	−281.373	−0.766685	−0.383343	−	0.923606i	$-0.625227\pi$
$367$	−281.373	−0.766685	−0.383343	+	0.923606i	$0.625227\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−654.837	−1.76506
$372$	0	0
$373$	617.959i	1.65673i	0.560191	+	0.828364i	$0.310728\pi$
$373$	617.959i	1.65673i	−0.560191	+	0.828364i	$0.689272\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 454.077i	− 1.20445i
$378$	0	0
$379$	− 695.292i	− 1.83454i	−0.398262	−	0.917272i	$-0.630387\pi$
$379$	− 695.292i	− 1.83454i	0.398262	−	0.917272i	$-0.369613\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 315.269i	− 0.823156i	−0.911375	−	0.411578i	$-0.864978\pi$
$383$	− 315.269i	− 0.823156i	0.911375	−	0.411578i	$-0.135022\pi$
$384$	0	0
$385$	−561.535	−1.45853
$386$	0	0
$387$	0	0
$388$	0	0
$389$	539.364	1.38654	0.693270	−	0.720677i	$-0.256169\pi$
$389$	539.364	1.38654	0.693270	+	0.720677i	$0.256169\pi$
$390$	0	0
$391$	−313.939	−0.802912
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−738.547	−1.86974
$396$	0	0
$397$	− 490.404i	− 1.23527i	−0.786463	−	0.617637i	$-0.788090\pi$
$397$	− 490.404i	− 1.23527i	0.786463	−	0.617637i	$-0.211910\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	460.891i	1.14935i	0.818380	+	0.574677i	$0.194872\pi$
$401$	460.891i	1.14935i	−0.818380	+	0.574677i	$0.805128\pi$
$402$	0	0
$403$	903.292i	2.24142i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 23.4557i	− 0.0576308i
$408$	0	0
$409$	−535.292	−1.30878	−0.654391	−	0.756156i	$-0.727075\pi$
$409$	−535.292	−1.30878	−0.654391	+	0.756156i	$0.727075\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1314.56	−3.18296
$414$	0	0
$415$	301.535	0.726590
$416$	0	0
$417$	0	0
$418$	0	0
$419$	405.993	0.968958	0.484479	−	0.874803i	$-0.339009\pi$
$419$	405.993	0.968958	0.484479	+	0.874803i	$0.339009\pi$
$420$	0	0
$421$	70.6969i	0.167926i	0.996469	+	0.0839631i	$0.0267578\pi$
$421$	70.6969i	0.167926i	−0.996469	+	0.0839631i	$0.973242\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 26.6839i	− 0.0627856i
$426$	0	0
$427$	127.514i	0.298628i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 334.239i	− 0.775497i	−0.921765	−	0.387749i	$-0.873253\pi$
$431$	− 334.239i	− 0.775497i	0.921765	−	0.387749i	$-0.126747\pi$
$432$	0	0
$433$	88.3429	0.204025	0.102013	−	0.994783i	$-0.467472\pi$
$433$	88.3429	0.204025	0.102013	+	0.994783i	$0.467472\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	251.701	0.575975
$438$	0	0
$439$	−231.353	−0.527000	−0.263500	−	0.964659i	$-0.584877\pi$
$439$	−231.353	−0.527000	−0.263500	+	0.964659i	$0.584877\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	715.091	1.61420	0.807101	−	0.590414i	$-0.201035\pi$
$443$	715.091	1.61420	0.807101	+	0.590414i	$0.201035\pi$
$444$	0	0
$445$	355.889i	0.799750i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 576.455i	− 1.28386i	−0.766761	−	0.641932i	$-0.778133\pi$
$449$	− 576.455i	− 1.28386i	0.766761	−	0.641932i	$-0.221867\pi$
$450$	0	0
$451$	627.737i	1.39188i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1306.30i	− 2.87100i
$456$	0	0
$457$	168.990	0.369781	0.184890	−	0.982759i	$-0.440807\pi$
$457$	168.990	0.369781	0.184890	+	0.982759i	$0.440807\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	772.653	1.67604	0.838019	−	0.545641i	$-0.183714\pi$
$461$	772.653	1.67604	0.838019	+	0.545641i	$0.183714\pi$
$462$	0	0
$463$	−461.818	−0.997448	−0.498724	−	0.866761i	$-0.666198\pi$
$463$	−461.818	−0.997448	−0.498724	+	0.866761i	$0.666198\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	516.016	1.10496	0.552480	−	0.833526i	$-0.313682\pi$
$467$	516.016	1.10496	0.552480	+	0.833526i	$0.313682\pi$
$468$	0	0
$469$	− 214.747i	− 0.457883i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 31.1416i	− 0.0658384i
$474$	0	0
$475$	21.3939i	0.0450397i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 175.706i	− 0.366818i	−0.983037	−	0.183409i	$-0.941287\pi$
$479$	− 175.706i	− 0.366818i	0.983037	−	0.183409i	$-0.0587133\pi$
$480$	0	0
$481$	54.5653	0.113441
$482$	0	0
$483$	0	0
$484$	0	0
$485$	426.235	0.878836
$486$	0	0
$487$	694.100	1.42526	0.712628	−	0.701542i	$-0.247505\pi$
$487$	694.100	1.42526	0.712628	+	0.701542i	$0.247505\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−512.173	−1.04312	−0.521561	−	0.853214i	$-0.674650\pi$
$491$	−512.173	−1.04312	−0.521561	+	0.853214i	$0.674650\pi$
$492$	0	0
$493$	− 444.111i	− 0.900834i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	410.549i	0.826054i
$498$	0	0
$499$	− 132.849i	− 0.266230i	−0.991101	−	0.133115i	$-0.957502\pi$
$499$	− 132.849i	− 0.266230i	0.991101	−	0.133115i	$-0.0424980\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	243.986i	0.485063i	0.970144	+	0.242531i	$0.0779777\pi$
$503$	243.986i	0.485063i	−0.970144	+	0.242531i	$0.922022\pi$
$504$	0	0
$505$	7.97959	0.0158012
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−396.530	−0.779038	−0.389519	−	0.921018i	$-0.627359\pi$
$509$	−396.530	−0.779038	−0.389519	+	0.921018i	$0.627359\pi$
$510$	0	0
$511$	−1033.45	−2.02241
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−190.205	−0.369329
$516$	0	0
$517$	− 408.363i	− 0.789871i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	76.4527i	0.146742i	0.997305	+	0.0733711i	$0.0233757\pi$
$521$	76.4527i	0.146742i	−0.997305	+	0.0733711i	$0.976624\pi$
$522$	0	0
$523$	298.524i	0.570793i	0.958410	+	0.285396i	$0.0921252\pi$
$523$	298.524i	0.570793i	−0.958410	+	0.285396i	$0.907875\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	883.468i	1.67641i
$528$	0	0
$529$	329.000	0.621928
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1460.31	−2.73979
$534$	0	0
$535$	−1001.53	−1.87203
$536$	0	0
$537$	0	0
$538$	0	0
$539$	879.968	1.63259
$540$	0	0
$541$	566.070i	1.04634i	0.852228	+	0.523170i	$0.175251\pi$
$541$	566.070i	1.04634i	−0.852228	+	0.523170i	$0.824749\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 162.463i	− 0.298097i
$546$	0	0
$547$	− 500.727i	− 0.915405i	−0.889105	−	0.457702i	$-0.848672\pi$
$547$	− 500.727i	− 0.915405i	0.889105	−	0.457702i	$-0.151328\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	356.067i	0.646220i
$552$	0	0
$553$	1786.14	3.22991
$554$	0	0
$555$	0	0
$556$	0	0
$557$	224.568	0.403174	0.201587	−	0.979471i	$-0.435390\pi$
$557$	224.568	0.403174	0.201587	+	0.979471i	$0.435390\pi$
$558$	0	0
$559$	72.4449	0.129597
$560$	0	0
$561$	0	0
$562$	0	0
$563$	708.092	1.25771	0.628856	−	0.777521i	$-0.283523\pi$
$563$	708.092	1.25771	0.628856	+	0.777521i	$0.283523\pi$
$564$	0	0
$565$	458.050i	0.810708i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	627.453i	1.10273i	0.834264	+	0.551365i	$0.185893\pi$
$569$	627.453i	1.10273i	−0.834264	+	0.551365i	$0.814107\pi$
$570$	0	0
$571$	− 25.2531i	− 0.0442260i	−0.999755	−	0.0221130i	$-0.992961\pi$
$571$	− 25.2531i	− 0.0442260i	0.999755	−	0.0221130i	$-0.00703937\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 16.9994i	− 0.0295642i
$576$	0	0
$577$	158.000	0.273830	0.136915	−	0.990583i	$-0.456281\pi$
$577$	158.000	0.273830	0.136915	+	0.990583i	$0.456281\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−729.246	−1.25516
$582$	0	0
$583$	−541.535	−0.928876
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1060.63	1.80687	0.903434	−	0.428728i	$-0.141038\pi$
$587$	1060.63	1.80687	0.903434	+	0.428728i	$0.141038\pi$
$588$	0	0
$589$	− 708.322i	− 1.20258i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1051.50i	1.77319i	0.462545	+	0.886596i	$0.346936\pi$
$593$	1051.50i	1.77319i	−0.462545	+	0.886596i	$0.653064\pi$
$594$	0	0
$595$	− 1277.63i	− 2.14729i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 52.0835i	− 0.0869507i	−0.999054	−	0.0434754i	$-0.986157\pi$
$599$	− 52.0835i	− 0.0869507i	0.999054	−	0.0434754i	$-0.0138430\pi$
$600$	0	0
$601$	−66.0000	−0.109817	−0.0549085	−	0.998491i	$-0.517487\pi$
$601$	−66.0000	−0.109817	−0.0549085	+	0.998491i	$0.517487\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	125.900	0.208100
$606$	0	0
$607$	−210.627	−0.346996	−0.173498	−	0.984834i	$-0.555507\pi$
$607$	−210.627	−0.346996	−0.173498	+	0.984834i	$0.555507\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	949.979	1.55479
$612$	0	0
$613$	320.706i	0.523175i	0.965180	+	0.261587i	$0.0842460\pi$
$613$	320.706i	0.523175i	−0.965180	+	0.261587i	$0.915754\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 233.316i	− 0.378146i	−0.981963	−	0.189073i	$-0.939452\pi$
$617$	− 233.316i	− 0.378146i	0.981963	−	0.189073i	$-0.0605484\pi$
$618$	0	0
$619$	206.020i	0.332828i	0.986056	+	0.166414i	$0.0532188\pi$
$619$	206.020i	0.332828i	−0.986056	+	0.166414i	$0.946781\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 860.699i	− 1.38154i
$624$	0	0
$625$	−593.504	−0.949607
$626$	0	0
$627$	0	0
$628$	0	0
$629$	53.3678	0.0848455
$630$	0	0
$631$	42.5857	0.0674892	0.0337446	−	0.999430i	$-0.489257\pi$
$631$	42.5857	0.0674892	0.0337446	+	0.999430i	$0.489257\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−223.516	−0.351994
$636$	0	0
$637$	2047.08i	3.21362i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 160.021i	− 0.249643i	−0.992179	−	0.124821i	$-0.960164\pi$
$641$	− 160.021i	− 0.249643i	0.992179	−	0.124821i	$-0.0398358\pi$
$642$	0	0
$643$	− 997.980i	− 1.55207i	−0.630691	−	0.776034i	$-0.717229\pi$
$643$	− 997.980i	− 1.55207i	0.630691	−	0.776034i	$-0.282771\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 552.628i	− 0.854140i	−0.904219	−	0.427070i	$-0.859546\pi$
$647$	− 552.628i	− 0.854140i	0.904219	−	0.427070i	$-0.140454\pi$
$648$	0	0
$649$	−1087.11	−1.67505
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−374.272	−0.573158	−0.286579	−	0.958057i	$-0.592518\pi$
$653$	−374.272	−0.573158	−0.286579	+	0.958057i	$0.592518\pi$
$654$	0	0
$655$	947.878	1.44714
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−251.759	−0.382032	−0.191016	−	0.981587i	$-0.561178\pi$
$659$	−251.759	−0.382032	−0.191016	+	0.981587i	$0.561178\pi$
$660$	0	0
$661$	339.535i	0.513668i	0.966456	+	0.256834i	$0.0826794\pi$
$661$	339.535i	0.513668i	−0.966456	+	0.256834i	$0.917321\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1024.35i	1.54037i
$666$	0	0
$667$	− 282.929i	− 0.424181i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	105.451i	0.157155i
$672$	0	0
$673$	1049.03	1.55873	0.779367	−	0.626567i	$-0.215541\pi$
$673$	1049.03	1.55873	0.779367	+	0.626567i	$0.215541\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	310.390	0.458479	0.229240	−	0.973370i	$-0.426376\pi$
$677$	310.390	0.458479	0.229240	+	0.973370i	$0.426376\pi$
$678$	0	0
$679$	−1030.83	−1.51816
$680$	0	0
$681$	0	0
$682$	0	0
$683$	746.233	1.09258	0.546291	−	0.837596i	$-0.316039\pi$
$683$	746.233	1.09258	0.546291	+	0.837596i	$0.316039\pi$
$684$	0	0
$685$	− 440.838i	− 0.643559i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1259.78i	− 1.82841i
$690$	0	0
$691$	776.241i	1.12336i	0.827355	+	0.561679i	$0.189845\pi$
$691$	776.241i	1.12336i	−0.827355	+	0.561679i	$0.810155\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 40.1117i	− 0.0577147i
$696$	0	0
$697$	−1428.26	−2.04916
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−801.223	−1.14297	−0.571486	−	0.820612i	$-0.693633\pi$
$701$	−801.223	−1.14297	−0.571486	+	0.820612i	$0.693633\pi$
$702$	0	0
$703$	−42.7878	−0.0608645
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−19.2982	−0.0272959
$708$	0	0
$709$	− 1198.25i	− 1.69006i	−0.534719	−	0.845030i	$-0.679583\pi$
$709$	− 1198.25i	− 1.69006i	0.534719	−	0.845030i	$-0.320417\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	562.828i	0.789380i
$714$	0	0
$715$	− 1080.28i	− 1.51088i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1206.22i	− 1.67764i	−0.544409	−	0.838820i	$-0.683246\pi$
$719$	− 1206.22i	− 1.67764i	0.544409	−	0.838820i	$-0.316754\pi$
$720$	0	0
$721$	460.000	0.638003
$722$	0	0
$723$	0	0
$724$	0	0
$725$	24.0481	0.0331698
$726$	0	0
$727$	−503.394	−0.692426	−0.346213	−	0.938156i	$-0.612533\pi$
$727$	−503.394	−0.692426	−0.346213	+	0.938156i	$0.612533\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	70.8550	0.0969289
$732$	0	0
$733$	− 380.736i	− 0.519421i	−0.965687	−	0.259711i	$-0.916373\pi$
$733$	− 380.736i	− 0.519421i	0.965687	−	0.259711i	$-0.0836272\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 177.591i	− 0.240964i
$738$	0	0
$739$	717.775i	0.971279i	0.874159	+	0.485640i	$0.161413\pi$
$739$	717.775i	0.971279i	−0.874159	+	0.485640i	$0.838587\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 239.675i	− 0.322577i	−0.986907	−	0.161288i	$-0.948435\pi$
$743$	− 239.675i	− 0.322577i	0.986907	−	0.161288i	$-0.0515649\pi$
$744$	0	0
$745$	−386.586	−0.518907
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2422.16	3.23386
$750$	0	0
$751$	−481.010	−0.640493	−0.320246	−	0.947334i	$-0.603766\pi$
$751$	−481.010	−0.640493	−0.320246	+	0.947334i	$0.603766\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	73.1249	0.0968542
$756$	0	0
$757$	− 338.656i	− 0.447366i	−0.974662	−	0.223683i	$-0.928192\pi$
$757$	− 338.656i	− 0.447366i	0.974662	−	0.223683i	$-0.0718080\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 525.945i	− 0.691123i	−0.938396	−	0.345561i	$-0.887688\pi$
$761$	− 525.945i	− 0.691123i	0.938396	−	0.345561i	$-0.112312\pi$
$762$	0	0
$763$	392.908i	0.514952i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2528.95i	− 3.29720i
$768$	0	0
$769$	248.506	0.323155	0.161577	−	0.986860i	$-0.448342\pi$
$769$	248.506	0.323155	0.161577	+	0.986860i	$0.448342\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	402.701	0.520958	0.260479	−	0.965479i	$-0.416119\pi$
$773$	402.701	0.520958	0.260479	+	0.965479i	$0.416119\pi$
$774$	0	0
$775$	−47.8388	−0.0617275
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1145.11	1.46998
$780$	0	0
$781$	339.514i	0.434717i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1305.32i	− 1.66283i
$786$	0	0
$787$	− 1258.97i	− 1.59971i	−0.600195	−	0.799853i	$-0.704910\pi$
$787$	− 1258.97i	− 1.59971i	0.600195	−	0.799853i	$-0.295090\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1107.77i	− 1.40047i
$792$	0	0
$793$	−245.312	−0.309347
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−878.734	−1.10255	−0.551276	−	0.834323i	$-0.685859\pi$
$797$	−878.734	−1.10255	−0.551276	+	0.834323i	$0.685859\pi$
$798$	0	0
$799$	929.131	1.16287
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−854.641	−1.06431
$804$	0	0
$805$	− 813.939i	− 1.01110i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1199.39i	− 1.48256i	−0.671195	−	0.741281i	$-0.734218\pi$
$809$	− 1199.39i	− 1.48256i	0.671195	−	0.741281i	$-0.265782\pi$
$810$	0	0
$811$	30.2837i	0.0373412i	0.999826	+	0.0186706i	$0.00594338\pi$
$811$	30.2837i	0.0373412i	−0.999826	+	0.0186706i	$0.994057\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1274.08i	− 1.56328i
$816$	0	0
$817$	−56.8082	−0.0695326
$818$	0	0
$819$	0	0
$820$	0	0
$821$	144.441	0.175933	0.0879665	−	0.996123i	$-0.471963\pi$
$821$	144.441	0.175933	0.0879665	+	0.996123i	$0.471963\pi$
$822$	0	0
$823$	−795.839	−0.966997	−0.483499	−	0.875345i	$-0.660634\pi$
$823$	−795.839	−0.966997	−0.483499	+	0.875345i	$0.660634\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−567.541	−0.686265	−0.343133	−	0.939287i	$-0.611488\pi$
$827$	−567.541	−0.686265	−0.343133	+	0.939287i	$0.611488\pi$
$828$	0	0
$829$	1188.54i	1.43370i	0.697228	+	0.716849i	$0.254416\pi$
$829$	1188.54i	1.43370i	−0.697228	+	0.716849i	$0.745584\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2002.15i	2.40354i
$834$	0	0
$835$	− 1463.84i	− 1.75310i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	598.256i	0.713058i	0.934284	+	0.356529i	$0.116040\pi$
$839$	598.256i	0.713058i	−0.934284	+	0.356529i	$0.883960\pi$
$840$	0	0
$841$	−440.757	−0.524087
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1688.63	1.99838
$846$	0	0
$847$	−304.484	−0.359485
$848$	0	0
$849$	0	0
$850$	0	0
$851$	33.9989	0.0399517
$852$	0	0
$853$	337.192i	0.395301i	0.980273	+	0.197651i	$0.0633311\pi$
$853$	337.192i	0.395301i	−0.980273	+	0.197651i	$0.936669\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	913.497i	1.06592i	0.846139	+	0.532962i	$0.178921\pi$
$857$	913.497i	1.06592i	−0.846139	+	0.532962i	$0.821079\pi$
$858$	0	0
$859$	748.645i	0.871531i	0.900060	+	0.435765i	$0.143522\pi$
$859$	748.645i	0.871531i	−0.900060	+	0.435765i	$0.856478\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	460.632i	0.533757i	0.963730	+	0.266879i	$0.0859923\pi$
$863$	460.632i	0.533757i	−0.963730	+	0.266879i	$0.914008\pi$
$864$	0	0
$865$	152.647	0.176470
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1477.09	1.69976
$870$	0	0
$871$	413.131	0.474318
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1508.04	1.72347
$876$	0	0
$877$	− 1066.32i	− 1.21588i	−0.793985	−	0.607938i	$-0.791997\pi$
$877$	− 1066.32i	− 1.21588i	0.793985	−	0.607938i	$-0.208003\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 586.899i	− 0.666173i	−0.942896	−	0.333087i	$-0.891910\pi$
$881$	− 586.899i	− 0.666173i	0.942896	−	0.333087i	$-0.108090\pi$
$882$	0	0
$883$	− 1242.95i	− 1.40764i	−0.710378	−	0.703820i	$-0.751476\pi$
$883$	− 1242.95i	− 1.40764i	0.710378	−	0.703820i	$-0.248524\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.8259i	0.0426447i	0.999773	+	0.0213224i	$0.00678763\pi$
$887$	37.8259i	0.0426447i	−0.999773	+	0.0213224i	$0.993212\pi$
$888$	0	0
$889$	540.563	0.608058
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−744.933	−0.834191
$894$	0	0
$895$	324.322	0.362371
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−796.201	−0.885652
$900$	0	0
$901$	− 1232.13i	− 1.36751i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1148.67i	− 1.26925i
$906$	0	0
$907$	− 763.837i	− 0.842157i	−0.907024	−	0.421079i	$-0.861652\pi$
$907$	− 763.837i	− 0.842157i	0.907024	−	0.421079i	$-0.138348\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	378.008i	0.414937i	0.978242	+	0.207469i	$0.0665225\pi$
$911$	378.008i	0.414937i	−0.978242	+	0.207469i	$0.933478\pi$
$912$	0	0
$913$	−603.069	−0.660536
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2292.39	−2.49988
$918$	0	0
$919$	206.141	0.224310	0.112155	−	0.993691i	$-0.464225\pi$
$919$	206.141	0.224310	0.112155	+	0.993691i	$0.464225\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−789.815	−0.855704
$924$	0	0
$925$	2.88981i	0.00312411i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1793.22i	− 1.93027i	−0.261752	−	0.965135i	$-0.584300\pi$
$929$	− 1793.22i	− 1.93027i	0.261752	−	0.965135i	$-0.415700\pi$
$930$	0	0
$931$	− 1605.23i	− 1.72420i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1056.57i	− 1.13002i
$936$	0	0
$937$	−322.767	−0.344469	−0.172234	−	0.985056i	$-0.555099\pi$
$937$	−322.767	−0.344469	−0.172234	+	0.985056i	$0.555099\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	805.965	0.856499	0.428249	−	0.903661i	$-0.359130\pi$
$941$	805.965	0.856499	0.428249	+	0.903661i	$0.359130\pi$
$942$	0	0
$943$	−909.898	−0.964897
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1197.31	1.26432	0.632158	−	0.774839i	$-0.282169\pi$
$947$	1197.31	1.26432	0.632158	+	0.774839i	$0.282169\pi$
$948$	0	0
$949$	− 1988.16i	− 2.09500i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1109.71i	− 1.16444i	−0.813030	−	0.582222i	$-0.802184\pi$
$953$	− 1109.71i	− 1.16444i	0.813030	−	0.582222i	$-0.197816\pi$
$954$	0	0
$955$	685.576i	0.717880i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1066.14i	1.11172i
$960$	0	0
$961$	622.878	0.648156
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1430.08	−1.48195
$966$	0	0
$967$	−1075.72	−1.11243	−0.556213	−	0.831040i	$-0.687746\pi$
$967$	−1075.72	−1.11243	−0.556213	+	0.831040i	$0.687746\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	958.163	0.986779	0.493390	−	0.869808i	$-0.335758\pi$
$971$	958.163	0.986779	0.493390	+	0.869808i	$0.335758\pi$
$972$	0	0
$973$	97.0081i	0.0997000i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 116.792i	− 0.119542i	−0.998212	−	0.0597709i	$-0.980963\pi$
$977$	− 116.792i	− 0.119542i	0.998212	−	0.0597709i	$-0.0190370\pi$
$978$	0	0
$979$	− 711.778i	− 0.727046i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	209.817i	0.213446i	0.994289	+	0.106723i	$0.0340358\pi$
$983$	209.817i	0.213446i	−0.994289	+	0.106723i	$0.965964\pi$
$984$	0	0
$985$	294.808	0.299298
$986$	0	0
$987$	0	0
$988$	0	0
$989$	45.1394	0.0456414
$990$	0	0
$991$	359.798	0.363066	0.181533	−	0.983385i	$-0.441894\pi$
$991$	359.798	0.363066	0.181533	+	0.983385i	$0.441894\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	701.293	0.704817
$996$	0	0
$997$	1628.06i	1.63296i	0.577377	+	0.816478i	$0.304076\pi$
$997$	1628.06i	1.63296i	−0.577377	+	0.816478i	$0.695924\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.h.k.2177.2		8
3.2	odd	2	inner	2304.3.h.k.2177.8		8
4.3	odd	2		2304.3.h.i.2177.2		8
8.3	odd	2		2304.3.h.i.2177.7		8
8.5	even	2	inner	2304.3.h.k.2177.7		8
12.11	even	2		2304.3.h.i.2177.8		8
16.3	odd	4		1152.3.e.f.1025.4	yes	4
16.5	even	4		1152.3.e.d.1025.1	yes	4
16.11	odd	4		1152.3.e.h.1025.1	yes	4
16.13	even	4		1152.3.e.b.1025.4	yes	4
24.5	odd	2	inner	2304.3.h.k.2177.1		8
24.11	even	2		2304.3.h.i.2177.1		8
48.5	odd	4		1152.3.e.d.1025.4	yes	4
48.11	even	4		1152.3.e.h.1025.4	yes	4
48.29	odd	4		1152.3.e.b.1025.1	✓	4
48.35	even	4		1152.3.e.f.1025.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.e.b.1025.1	✓	4	48.29	odd	4
1152.3.e.b.1025.4	yes	4	16.13	even	4
1152.3.e.d.1025.1	yes	4	16.5	even	4
1152.3.e.d.1025.4	yes	4	48.5	odd	4
1152.3.e.f.1025.1	yes	4	48.35	even	4
1152.3.e.f.1025.4	yes	4	16.3	odd	4
1152.3.e.h.1025.1	yes	4	16.11	odd	4
1152.3.e.h.1025.4	yes	4	48.11	even	4
2304.3.h.i.2177.1		8	24.11	even	2
2304.3.h.i.2177.2		8	4.3	odd	2
2304.3.h.i.2177.7		8	8.3	odd	2
2304.3.h.i.2177.8		8	12.11	even	2
2304.3.h.k.2177.1		8	24.5	odd	2	inner
2304.3.h.k.2177.2		8	1.1	even	1	trivial
2304.3.h.k.2177.7		8	8.5	even	2	inner
2304.3.h.k.2177.8		8	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.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.87832 q^{5} +11.7980 q^{7} +9.75663 q^{11} +22.6969i q^{13} +22.1988i q^{17} -17.7980i q^{19} +14.1421i q^{23} -1.20204 q^{25} -20.0061 q^{29} +39.7980 q^{31} -57.5542 q^{35} -2.40408i q^{37} +64.3395i q^{41} -3.19184i q^{43} -41.8549i q^{47} +90.1918 q^{49} -55.5043 q^{53} -47.5959 q^{55} -111.423 q^{59} +10.8082i q^{61} -110.723i q^{65} -18.2020i q^{67} +34.7983i q^{71} -87.5959 q^{73} +115.108 q^{77} +151.394 q^{79} -61.8112 q^{83} -108.293i q^{85} -72.9532i q^{89} +267.778i q^{91} +86.8241i q^{95} -87.3735 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7} - 88 q^{25} + 240 q^{31} + 408 q^{49} - 224 q^{55} - 544 q^{73} + 976 q^{79} + 320 q^{97}+O(q^{100}) \) 8 * q + 16 * q^7 - 88 * q^25 + 240 * q^31 + 408 * q^49 - 224 * q^55 - 544 * q^73 + 976 * q^79 + 320 * q^97

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