LMFDB - Embedded newform 2304.3.g.m.1279.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(1279,2304)]

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([1, 0, 0]))

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

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

chi := DirichletCharacter("2304.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.g (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:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1279.1
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	2304.1279
Dual form			2304.3.g.m.1279.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{5} -11.3137i q^{7} +O(q^{10})$	$q+4.00000 q^{5} -11.3137i q^{7} -14.1421i q^{11} -20.0000 q^{13} +10.0000 q^{17} -14.1421i q^{19} -11.3137i q^{23} -9.00000 q^{25} +20.0000 q^{29} -45.2548i q^{35} -20.0000 q^{37} -30.0000 q^{41} +2.82843i q^{43} +67.8823i q^{47} -79.0000 q^{49} -60.0000 q^{53} -56.5685i q^{55} +42.4264i q^{59} +28.0000 q^{61} -80.0000 q^{65} -82.0244i q^{67} +56.5685i q^{71} +10.0000 q^{73} -160.000 q^{77} +113.137i q^{79} +25.4558i q^{83} +40.0000 q^{85} +22.0000 q^{89} +226.274i q^{91} -56.5685i q^{95} +150.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{5}+O(q^{10}) $ 2 * q + 8 * q^5	$ 2 q + 8 q^{5} - 40 q^{13} + 20 q^{17} - 18 q^{25} + 40 q^{29} - 40 q^{37} - 60 q^{41} - 158 q^{49} - 120 q^{53} + 56 q^{61} - 160 q^{65} + 20 q^{73} - 320 q^{77} + 80 q^{85} + 44 q^{89} + 300 q^{97}+O(q^{100}) $ 2 * q + 8 * q^5 - 40 * q^13 + 20 * q^17 - 18 * q^25 + 40 * q^29 - 40 * q^37 - 60 * q^41 - 158 * q^49 - 120 * q^53 + 56 * q^61 - 160 * q^65 + 20 * q^73 - 320 * q^77 + 80 * q^85 + 44 * q^89 + 300 * 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
$5$	4.00000	0.800000	0.400000	−	0.916515i	$-0.369010\pi$
$5$	4.00000	0.800000	0.400000	+	0.916515i	$0.369010\pi$
$6$	0	0
$7$	− 11.3137i	− 1.61624i	−0.589015	−	0.808122i	$-0.700484\pi$
$7$	− 11.3137i	− 1.61624i	0.589015	−	0.808122i	$-0.299516\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 14.1421i	− 1.28565i	−0.766014	−	0.642824i	$-0.777763\pi$
$11$	− 14.1421i	− 1.28565i	0.766014	−	0.642824i	$-0.222237\pi$
$12$	0	0
$13$	−20.0000	−1.53846	−0.769231	−	0.638971i	$-0.779360\pi$
$13$	−20.0000	−1.53846	−0.769231	+	0.638971i	$0.779360\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	10.0000	0.588235	0.294118	−	0.955769i	$-0.404974\pi$
$17$	10.0000	0.588235	0.294118	+	0.955769i	$0.404974\pi$
$18$	0	0
$19$	− 14.1421i	− 0.744323i	−0.928168	−	0.372161i	$-0.878617\pi$
$19$	− 14.1421i	− 0.744323i	0.928168	−	0.372161i	$-0.121383\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 11.3137i	− 0.491900i	−0.969282	−	0.245950i	$-0.920900\pi$
$23$	− 11.3137i	− 0.491900i	0.969282	−	0.245950i	$-0.0791000\pi$
$24$	0	0
$25$	−9.00000	−0.360000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	20.0000	0.689655	0.344828	−	0.938666i	$-0.387937\pi$
$29$	20.0000	0.689655	0.344828	+	0.938666i	$0.387937\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 45.2548i	− 1.29300i
$36$	0	0
$37$	−20.0000	−0.540541	−0.270270	−	0.962784i	$-0.587113\pi$
$37$	−20.0000	−0.540541	−0.270270	+	0.962784i	$0.587113\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−30.0000	−0.731707	−0.365854	−	0.930672i	$-0.619223\pi$
$41$	−30.0000	−0.731707	−0.365854	+	0.930672i	$0.619223\pi$
$42$	0	0
$43$	2.82843i	0.0657774i	0.999459	+	0.0328887i	$0.0104707\pi$
$43$	2.82843i	0.0657774i	−0.999459	+	0.0328887i	$0.989529\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	67.8823i	1.44430i	0.691735	+	0.722152i	$0.256847\pi$
$47$	67.8823i	1.44430i	−0.691735	+	0.722152i	$0.743153\pi$
$48$	0	0
$49$	−79.0000	−1.61224
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−60.0000	−1.13208	−0.566038	−	0.824379i	$-0.691524\pi$
$53$	−60.0000	−1.13208	−0.566038	+	0.824379i	$0.691524\pi$
$54$	0	0
$55$	− 56.5685i	− 1.02852i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	42.4264i	0.719092i	0.933127	+	0.359546i	$0.117068\pi$
$59$	42.4264i	0.719092i	−0.933127	+	0.359546i	$0.882932\pi$
$60$	0	0
$61$	28.0000	0.459016	0.229508	−	0.973307i	$-0.426288\pi$
$61$	28.0000	0.459016	0.229508	+	0.973307i	$0.426288\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−80.0000	−1.23077
$66$	0	0
$67$	− 82.0244i	− 1.22424i	−0.790763	−	0.612122i	$-0.790316\pi$
$67$	− 82.0244i	− 1.22424i	0.790763	−	0.612122i	$-0.209684\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	56.5685i	0.796740i	0.917225	+	0.398370i	$0.130424\pi$
$71$	56.5685i	0.796740i	−0.917225	+	0.398370i	$0.869576\pi$
$72$	0	0
$73$	10.0000	0.136986	0.0684932	−	0.997652i	$-0.478181\pi$
$73$	10.0000	0.136986	0.0684932	+	0.997652i	$0.478181\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−160.000	−2.07792
$78$	0	0
$79$	113.137i	1.43211i	0.698041	+	0.716057i	$0.254055\pi$
$79$	113.137i	1.43211i	−0.698041	+	0.716057i	$0.745945\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	25.4558i	0.306697i	0.988172	+	0.153348i	$0.0490057\pi$
$83$	25.4558i	0.306697i	−0.988172	+	0.153348i	$0.950994\pi$
$84$	0	0
$85$	40.0000	0.470588
$86$	0	0
$87$	0	0
$88$	0	0
$89$	22.0000	0.247191	0.123596	−	0.992333i	$-0.460557\pi$
$89$	22.0000	0.247191	0.123596	+	0.992333i	$0.460557\pi$
$90$	0	0
$91$	226.274i	2.48653i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 56.5685i	− 0.595458i
$96$	0	0
$97$	150.000	1.54639	0.773196	−	0.634167i	$-0.218657\pi$
$97$	150.000	1.54639	0.773196	+	0.634167i	$0.218657\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−140.000	−1.38614	−0.693069	−	0.720871i	$-0.743742\pi$
$101$	−140.000	−1.38614	−0.693069	+	0.720871i	$0.743742\pi$
$102$	0	0
$103$	− 101.823i	− 0.988576i	−0.869298	−	0.494288i	$-0.835429\pi$
$103$	− 101.823i	− 0.988576i	0.869298	−	0.494288i	$-0.164571\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	144.250i	1.34813i	0.738673	+	0.674064i	$0.235453\pi$
$107$	144.250i	1.34813i	−0.738673	+	0.674064i	$0.764547\pi$
$108$	0	0
$109$	−68.0000	−0.623853	−0.311927	−	0.950106i	$-0.600974\pi$
$109$	−68.0000	−0.623853	−0.311927	+	0.950106i	$0.600974\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	190.000	1.68142	0.840708	−	0.541489i	$-0.182139\pi$
$113$	190.000	1.68142	0.840708	+	0.541489i	$0.182139\pi$
$114$	0	0
$115$	− 45.2548i	− 0.393520i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 113.137i	− 0.950732i
$120$	0	0
$121$	−79.0000	−0.652893
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−136.000	−1.08800
$126$	0	0
$127$	− 45.2548i	− 0.356337i	−0.984000	−	0.178169i	$-0.942983\pi$
$127$	− 45.2548i	− 0.356337i	0.984000	−	0.178169i	$-0.0570173\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.1421i	0.107955i	0.998542	+	0.0539776i	$0.0171900\pi$
$131$	14.1421i	0.107955i	−0.998542	+	0.0539776i	$0.982810\pi$
$132$	0	0
$133$	−160.000	−1.20301
$134$	0	0
$135$	0	0
$136$	0	0
$137$	130.000	0.948905	0.474453	−	0.880281i	$-0.342646\pi$
$137$	130.000	0.948905	0.474453	+	0.880281i	$0.342646\pi$
$138$	0	0
$139$	− 42.4264i	− 0.305226i	−0.988286	−	0.152613i	$-0.951231\pi$
$139$	− 42.4264i	− 0.305226i	0.988286	−	0.152613i	$-0.0487688\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	282.843i	1.97792i
$144$	0	0
$145$	80.0000	0.551724
$146$	0	0
$147$	0	0
$148$	0	0
$149$	52.0000	0.348993	0.174497	−	0.984658i	$-0.444170\pi$
$149$	52.0000	0.348993	0.174497	+	0.984658i	$0.444170\pi$
$150$	0	0
$151$	− 169.706i	− 1.12388i	−0.827179	−	0.561939i	$-0.810055\pi$
$151$	− 169.706i	− 1.12388i	0.827179	−	0.561939i	$-0.189945\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−100.000	−0.636943	−0.318471	−	0.947932i	$-0.603169\pi$
$157$	−100.000	−0.636943	−0.318471	+	0.947932i	$0.603169\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−128.000	−0.795031
$162$	0	0
$163$	110.309i	0.676740i	0.941013	+	0.338370i	$0.109876\pi$
$163$	110.309i	0.676740i	−0.941013	+	0.338370i	$0.890124\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 124.451i	− 0.745214i	−0.927989	−	0.372607i	$-0.878464\pi$
$167$	− 124.451i	− 0.745214i	0.927989	−	0.372607i	$-0.121536\pi$
$168$	0	0
$169$	231.000	1.36686
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.0000	0.115607	0.0578035	−	0.998328i	$-0.481590\pi$
$173$	20.0000	0.115607	0.0578035	+	0.998328i	$0.481590\pi$
$174$	0	0
$175$	101.823i	0.581848i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 42.4264i	− 0.237019i	−0.992953	−	0.118510i	$-0.962188\pi$
$179$	− 42.4264i	− 0.237019i	0.992953	−	0.118510i	$-0.0378116\pi$
$180$	0	0
$181$	−180.000	−0.994475	−0.497238	−	0.867614i	$-0.665652\pi$
$181$	−180.000	−0.994475	−0.497238	+	0.867614i	$0.665652\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−80.0000	−0.432432
$186$	0	0
$187$	− 141.421i	− 0.756264i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 226.274i	− 1.18468i	−0.805688	−	0.592341i	$-0.798204\pi$
$191$	− 226.274i	− 1.18468i	0.805688	−	0.592341i	$-0.201796\pi$
$192$	0	0
$193$	−170.000	−0.880829	−0.440415	−	0.897795i	$-0.645169\pi$
$193$	−170.000	−0.880829	−0.440415	+	0.897795i	$0.645169\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−220.000	−1.11675	−0.558376	−	0.829588i	$-0.688575\pi$
$197$	−220.000	−1.11675	−0.558376	+	0.829588i	$0.688575\pi$
$198$	0	0
$199$	169.706i	0.852792i	0.904537	+	0.426396i	$0.140217\pi$
$199$	169.706i	0.852792i	−0.904537	+	0.426396i	$0.859783\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 226.274i	− 1.11465i
$204$	0	0
$205$	−120.000	−0.585366
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−200.000	−0.956938
$210$	0	0
$211$	− 410.122i	− 1.94371i	−0.235588	−	0.971853i	$-0.575702\pi$
$211$	− 410.122i	− 1.94371i	0.235588	−	0.971853i	$-0.424298\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.3137i	0.0526219i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−200.000	−0.904977
$222$	0	0
$223$	181.019i	0.811746i	0.913930	+	0.405873i	$0.133032\pi$
$223$	181.019i	0.811746i	−0.913930	+	0.405873i	$0.866968\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 223.446i	− 0.984342i	−0.870498	−	0.492171i	$-0.836203\pi$
$227$	− 223.446i	− 0.984342i	0.870498	−	0.492171i	$-0.163797\pi$
$228$	0	0
$229$	−260.000	−1.13537	−0.567686	−	0.823245i	$-0.692161\pi$
$229$	−260.000	−1.13537	−0.567686	+	0.823245i	$0.692161\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.0429185	−0.0214592	−	0.999770i	$-0.506831\pi$
$233$	−10.0000	−0.0429185	−0.0214592	+	0.999770i	$0.506831\pi$
$234$	0	0
$235$	271.529i	1.15544i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 339.411i	− 1.42013i	−0.704136	−	0.710065i	$-0.748665\pi$
$239$	− 339.411i	− 1.42013i	0.704136	−	0.710065i	$-0.251335\pi$
$240$	0	0
$241$	−10.0000	−0.0414938	−0.0207469	−	0.999785i	$-0.506604\pi$
$241$	−10.0000	−0.0414938	−0.0207469	+	0.999785i	$0.506604\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−316.000	−1.28980
$246$	0	0
$247$	282.843i	1.14511i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 183.848i	− 0.732461i	−0.930524	−	0.366231i	$-0.880648\pi$
$251$	− 183.848i	− 0.732461i	0.930524	−	0.366231i	$-0.119352\pi$
$252$	0	0
$253$	−160.000	−0.632411
$254$	0	0
$255$	0	0
$256$	0	0
$257$	350.000	1.36187	0.680934	−	0.732345i	$-0.261574\pi$
$257$	350.000	1.36187	0.680934	+	0.732345i	$0.261574\pi$
$258$	0	0
$259$	226.274i	0.873645i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	237.588i	0.903376i	0.892176	+	0.451688i	$0.149178\pi$
$263$	237.588i	0.903376i	−0.892176	+	0.451688i	$0.850822\pi$
$264$	0	0
$265$	−240.000	−0.905660
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−188.000	−0.698885	−0.349442	−	0.936958i	$-0.613629\pi$
$269$	−188.000	−0.698885	−0.349442	+	0.936958i	$0.613629\pi$
$270$	0	0
$271$	− 113.137i	− 0.417480i	−0.977971	−	0.208740i	$-0.933064\pi$
$271$	− 113.137i	− 0.417480i	0.977971	−	0.208740i	$-0.0669362\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	127.279i	0.462834i
$276$	0	0
$277$	540.000	1.94946	0.974729	−	0.223390i	$-0.0717122\pi$
$277$	540.000	1.94946	0.974729	+	0.223390i	$0.0717122\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−490.000	−1.74377	−0.871886	−	0.489709i	$-0.837103\pi$
$281$	−490.000	−1.74377	−0.871886	+	0.489709i	$0.837103\pi$
$282$	0	0
$283$	− 483.661i	− 1.70905i	−0.519411	−	0.854525i	$-0.673848\pi$
$283$	− 483.661i	− 1.70905i	0.519411	−	0.854525i	$-0.326152\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	339.411i	1.18262i
$288$	0	0
$289$	−189.000	−0.653979
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−380.000	−1.29693	−0.648464	−	0.761245i	$-0.724588\pi$
$293$	−380.000	−1.29693	−0.648464	+	0.761245i	$0.724588\pi$
$294$	0	0
$295$	169.706i	0.575273i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	226.274i	0.756770i
$300$	0	0
$301$	32.0000	0.106312
$302$	0	0
$303$	0	0
$304$	0	0
$305$	112.000	0.367213
$306$	0	0
$307$	483.661i	1.57544i	0.616031	+	0.787722i	$0.288739\pi$
$307$	483.661i	1.57544i	−0.616031	+	0.787722i	$0.711261\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	169.706i	0.545677i	0.962060	+	0.272839i	$0.0879625\pi$
$311$	169.706i	0.545677i	−0.962060	+	0.272839i	$0.912037\pi$
$312$	0	0
$313$	−130.000	−0.415335	−0.207668	−	0.978199i	$-0.566587\pi$
$313$	−130.000	−0.415335	−0.207668	+	0.978199i	$0.566587\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−460.000	−1.45110	−0.725552	−	0.688167i	$-0.758416\pi$
$317$	−460.000	−1.45110	−0.725552	+	0.688167i	$0.758416\pi$
$318$	0	0
$319$	− 282.843i	− 0.886654i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 141.421i	− 0.437837i
$324$	0	0
$325$	180.000	0.553846
$326$	0	0
$327$	0	0
$328$	0	0
$329$	768.000	2.33435
$330$	0	0
$331$	− 325.269i	− 0.982686i	−0.870966	−	0.491343i	$-0.836506\pi$
$331$	− 325.269i	− 0.982686i	0.870966	−	0.491343i	$-0.163494\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 328.098i	− 0.979396i
$336$	0	0
$337$	450.000	1.33531	0.667656	−	0.744470i	$-0.267298\pi$
$337$	450.000	1.33531	0.667656	+	0.744470i	$0.267298\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	339.411i	0.989537i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 115.966i	− 0.334195i	−0.985940	−	0.167097i	$-0.946561\pi$
$347$	− 115.966i	− 0.334195i	0.985940	−	0.167097i	$-0.0534394\pi$
$348$	0	0
$349$	140.000	0.401146	0.200573	−	0.979679i	$-0.435720\pi$
$349$	140.000	0.401146	0.200573	+	0.979679i	$0.435720\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−130.000	−0.368272	−0.184136	−	0.982901i	$-0.558949\pi$
$353$	−130.000	−0.368272	−0.184136	+	0.982901i	$0.558949\pi$
$354$	0	0
$355$	226.274i	0.637392i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 622.254i	− 1.73330i	−0.498918	−	0.866649i	$-0.666269\pi$
$359$	− 622.254i	− 1.73330i	0.498918	−	0.866649i	$-0.333731\pi$
$360$	0	0
$361$	161.000	0.445983
$362$	0	0
$363$	0	0
$364$	0	0
$365$	40.0000	0.109589
$366$	0	0
$367$	− 610.940i	− 1.66469i	−0.554260	−	0.832344i	$-0.686999\pi$
$367$	− 610.940i	− 1.66469i	0.554260	−	0.832344i	$-0.313001\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	678.823i	1.82971i
$372$	0	0
$373$	−420.000	−1.12601	−0.563003	−	0.826455i	$-0.690354\pi$
$373$	−420.000	−1.12601	−0.563003	+	0.826455i	$0.690354\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−400.000	−1.06101
$378$	0	0
$379$	296.985i	0.783601i	0.920050	+	0.391801i	$0.128148\pi$
$379$	296.985i	0.783601i	−0.920050	+	0.391801i	$0.871852\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 181.019i	− 0.472635i	−0.971676	−	0.236318i	$-0.924059\pi$
$383$	− 181.019i	− 0.472635i	0.971676	−	0.236318i	$-0.0759406\pi$
$384$	0	0
$385$	−640.000	−1.66234
$386$	0	0
$387$	0	0
$388$	0	0
$389$	372.000	0.956298	0.478149	−	0.878279i	$-0.341308\pi$
$389$	372.000	0.956298	0.478149	+	0.878279i	$0.341308\pi$
$390$	0	0
$391$	− 113.137i	− 0.289353i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	452.548i	1.14569i
$396$	0	0
$397$	−100.000	−0.251889	−0.125945	−	0.992037i	$-0.540196\pi$
$397$	−100.000	−0.251889	−0.125945	+	0.992037i	$0.540196\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−502.000	−1.25187	−0.625935	−	0.779875i	$-0.715283\pi$
$401$	−502.000	−1.25187	−0.625935	+	0.779875i	$0.715283\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	282.843i	0.694945i
$408$	0	0
$409$	190.000	0.464548	0.232274	−	0.972650i	$-0.425383\pi$
$409$	190.000	0.464548	0.232274	+	0.972650i	$0.425383\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	480.000	1.16223
$414$	0	0
$415$	101.823i	0.245358i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	806.102i	1.92387i	0.273278	+	0.961935i	$0.411892\pi$
$419$	806.102i	1.92387i	−0.273278	+	0.961935i	$0.588108\pi$
$420$	0	0
$421$	−292.000	−0.693587	−0.346793	−	0.937942i	$-0.612729\pi$
$421$	−292.000	−0.693587	−0.346793	+	0.937942i	$0.612729\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−90.0000	−0.211765
$426$	0	0
$427$	− 316.784i	− 0.741883i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 339.411i	− 0.787497i	−0.919218	−	0.393749i	$-0.871178\pi$
$431$	− 339.411i	− 0.787497i	0.919218	−	0.393749i	$-0.128822\pi$
$432$	0	0
$433$	−10.0000	−0.0230947	−0.0115473	−	0.999933i	$-0.503676\pi$
$433$	−10.0000	−0.0230947	−0.0115473	+	0.999933i	$0.503676\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−160.000	−0.366133
$438$	0	0
$439$	− 169.706i	− 0.386573i	−0.981142	−	0.193287i	$-0.938085\pi$
$439$	− 169.706i	− 0.386573i	0.981142	−	0.193287i	$-0.0619147\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 534.573i	− 1.20671i	−0.797473	−	0.603355i	$-0.793830\pi$
$443$	− 534.573i	− 1.20671i	0.797473	−	0.603355i	$-0.206170\pi$
$444$	0	0
$445$	88.0000	0.197753
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−150.000	−0.334076	−0.167038	−	0.985950i	$-0.553420\pi$
$449$	−150.000	−0.334076	−0.167038	+	0.985950i	$0.553420\pi$
$450$	0	0
$451$	424.264i	0.940719i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	905.097i	1.98922i
$456$	0	0
$457$	−290.000	−0.634573	−0.317287	−	0.948330i	$-0.602772\pi$
$457$	−290.000	−0.634573	−0.317287	+	0.948330i	$0.602772\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−60.0000	−0.130152	−0.0650759	−	0.997880i	$-0.520729\pi$
$461$	−60.0000	−0.130152	−0.0650759	+	0.997880i	$0.520729\pi$
$462$	0	0
$463$	− 67.8823i	− 0.146614i	−0.997309	−	0.0733070i	$-0.976645\pi$
$463$	− 67.8823i	− 0.146614i	0.997309	−	0.0733070i	$-0.0233553\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 280.014i	− 0.599602i	−0.954002	−	0.299801i	$-0.903080\pi$
$467$	− 280.014i	− 0.599602i	0.954002	−	0.299801i	$-0.0969203\pi$
$468$	0	0
$469$	−928.000	−1.97868
$470$	0	0
$471$	0	0
$472$	0	0
$473$	40.0000	0.0845666
$474$	0	0
$475$	127.279i	0.267956i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	905.097i	1.88955i	0.327713	+	0.944777i	$0.393722\pi$
$479$	905.097i	1.88955i	−0.327713	+	0.944777i	$0.606278\pi$
$480$	0	0
$481$	400.000	0.831601
$482$	0	0
$483$	0	0
$484$	0	0
$485$	600.000	1.23711
$486$	0	0
$487$	− 237.588i	− 0.487860i	−0.969793	−	0.243930i	$-0.921563\pi$
$487$	− 237.588i	− 0.487860i	0.969793	−	0.243930i	$-0.0784367\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 749.533i	− 1.52654i	−0.646077	−	0.763272i	$-0.723592\pi$
$491$	− 749.533i	− 1.52654i	0.646077	−	0.763272i	$-0.276408\pi$
$492$	0	0
$493$	200.000	0.405680
$494$	0	0
$495$	0	0
$496$	0	0
$497$	640.000	1.28773
$498$	0	0
$499$	155.563i	0.311750i	0.987777	+	0.155875i	$0.0498198\pi$
$499$	155.563i	0.311750i	−0.987777	+	0.155875i	$0.950180\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 237.588i	− 0.472342i	−0.971712	−	0.236171i	$-0.924107\pi$
$503$	− 237.588i	− 0.472342i	0.971712	−	0.236171i	$-0.0758925\pi$
$504$	0	0
$505$	−560.000	−1.10891
$506$	0	0
$507$	0	0
$508$	0	0
$509$	900.000	1.76817	0.884086	−	0.467323i	$-0.154782\pi$
$509$	900.000	1.76817	0.884086	+	0.467323i	$0.154782\pi$
$510$	0	0
$511$	− 113.137i	− 0.221403i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 407.294i	− 0.790861i
$516$	0	0
$517$	960.000	1.85687
$518$	0	0
$519$	0	0
$520$	0	0
$521$	642.000	1.23225	0.616123	−	0.787650i	$-0.288702\pi$
$521$	642.000	1.23225	0.616123	+	0.787650i	$0.288702\pi$
$522$	0	0
$523$	− 596.798i	− 1.14111i	−0.821261	−	0.570553i	$-0.806729\pi$
$523$	− 596.798i	− 1.14111i	0.821261	−	0.570553i	$-0.193271\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	401.000	0.758034
$530$	0	0
$531$	0	0
$532$	0	0
$533$	600.000	1.12570
$534$	0	0
$535$	576.999i	1.07850i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1117.23i	2.07278i
$540$	0	0
$541$	−580.000	−1.07209	−0.536044	−	0.844190i	$-0.680082\pi$
$541$	−580.000	−1.07209	−0.536044	+	0.844190i	$0.680082\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−272.000	−0.499083
$546$	0	0
$547$	− 398.808i	− 0.729083i	−0.931187	−	0.364541i	$-0.881226\pi$
$547$	− 398.808i	− 0.729083i	0.931187	−	0.364541i	$-0.118774\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 282.843i	− 0.513326i
$552$	0	0
$553$	1280.00	2.31465
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−460.000	−0.825853	−0.412926	−	0.910764i	$-0.635493\pi$
$557$	−460.000	−0.825853	−0.412926	+	0.910764i	$0.635493\pi$
$558$	0	0
$559$	− 56.5685i	− 0.101196i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 619.426i	− 1.10022i	−0.835091	−	0.550111i	$-0.814585\pi$
$563$	− 619.426i	− 1.10022i	0.835091	−	0.550111i	$-0.185415\pi$
$564$	0	0
$565$	760.000	1.34513
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−670.000	−1.17750	−0.588752	−	0.808314i	$-0.700381\pi$
$569$	−670.000	−1.17750	−0.588752	+	0.808314i	$0.700381\pi$
$570$	0	0
$571$	127.279i	0.222906i	0.993770	+	0.111453i	$0.0355504\pi$
$571$	127.279i	0.222906i	−0.993770	+	0.111453i	$0.964450\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	101.823i	0.177084i
$576$	0	0
$577$	610.000	1.05719	0.528596	−	0.848873i	$-0.322719\pi$
$577$	610.000	1.05719	0.528596	+	0.848873i	$0.322719\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	288.000	0.495697
$582$	0	0
$583$	848.528i	1.45545i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 511.945i	− 0.872139i	−0.899913	−	0.436069i	$-0.856370\pi$
$587$	− 511.945i	− 0.872139i	0.899913	−	0.436069i	$-0.143630\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	670.000	1.12985	0.564924	−	0.825143i	$-0.308905\pi$
$593$	670.000	1.12985	0.564924	+	0.825143i	$0.308905\pi$
$594$	0	0
$595$	− 452.548i	− 0.760585i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	848.528i	1.41657i	0.705924	+	0.708287i	$0.250532\pi$
$599$	848.528i	1.41657i	−0.705924	+	0.708287i	$0.749468\pi$
$600$	0	0
$601$	−470.000	−0.782030	−0.391015	−	0.920384i	$-0.627876\pi$
$601$	−470.000	−0.782030	−0.391015	+	0.920384i	$0.627876\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−316.000	−0.522314
$606$	0	0
$607$	45.2548i	0.0745549i	0.999305	+	0.0372775i	$0.0118685\pi$
$607$	45.2548i	0.0745549i	−0.999305	+	0.0372775i	$0.988131\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1357.65i	− 2.22200i
$612$	0	0
$613$	540.000	0.880914	0.440457	−	0.897774i	$-0.354817\pi$
$613$	540.000	0.880914	0.440457	+	0.897774i	$0.354817\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	950.000	1.53971	0.769854	−	0.638220i	$-0.220329\pi$
$617$	950.000	1.53971	0.769854	+	0.638220i	$0.220329\pi$
$618$	0	0
$619$	466.690i	0.753943i	0.926225	+	0.376971i	$0.123034\pi$
$619$	466.690i	0.753943i	−0.926225	+	0.376971i	$0.876966\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 248.902i	− 0.399521i
$624$	0	0
$625$	−319.000	−0.510400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−200.000	−0.317965
$630$	0	0
$631$	− 848.528i	− 1.34474i	−0.740217	−	0.672368i	$-0.765277\pi$
$631$	− 848.528i	− 1.34474i	0.740217	−	0.672368i	$-0.234723\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 181.019i	− 0.285070i
$636$	0	0
$637$	1580.00	2.48038
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−470.000	−0.733229	−0.366615	−	0.930373i	$-0.619483\pi$
$641$	−470.000	−0.733229	−0.366615	+	0.930373i	$0.619483\pi$
$642$	0	0
$643$	− 534.573i	− 0.831373i	−0.909508	−	0.415686i	$-0.863541\pi$
$643$	− 534.573i	− 0.831373i	0.909508	−	0.415686i	$-0.136459\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 667.509i	− 1.03170i	−0.856679	−	0.515849i	$-0.827476\pi$
$647$	− 667.509i	− 1.03170i	0.856679	−	0.515849i	$-0.172524\pi$
$648$	0	0
$649$	600.000	0.924499
$650$	0	0
$651$	0	0
$652$	0	0
$653$	900.000	1.37825	0.689127	−	0.724640i	$-0.257994\pi$
$653$	900.000	1.37825	0.689127	+	0.724640i	$0.257994\pi$
$654$	0	0
$655$	56.5685i	0.0863642i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	523.259i	0.794020i	0.917814	+	0.397010i	$0.129952\pi$
$659$	523.259i	0.794020i	−0.917814	+	0.397010i	$0.870048\pi$
$660$	0	0
$661$	172.000	0.260212	0.130106	−	0.991500i	$-0.458468\pi$
$661$	172.000	0.260212	0.130106	+	0.991500i	$0.458468\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−640.000	−0.962406
$666$	0	0
$667$	− 226.274i	− 0.339242i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 395.980i	− 0.590134i
$672$	0	0
$673$	−170.000	−0.252600	−0.126300	−	0.991992i	$-0.540310\pi$
$673$	−170.000	−0.252600	−0.126300	+	0.991992i	$0.540310\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−300.000	−0.443131	−0.221566	−	0.975145i	$-0.571117\pi$
$677$	−300.000	−0.443131	−0.221566	+	0.975145i	$0.571117\pi$
$678$	0	0
$679$	− 1697.06i	− 2.49935i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	110.309i	0.161506i	0.996734	+	0.0807530i	$0.0257325\pi$
$683$	110.309i	0.161506i	−0.996734	+	0.0807530i	$0.974267\pi$
$684$	0	0
$685$	520.000	0.759124
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1200.00	1.74165
$690$	0	0
$691$	− 183.848i	− 0.266060i	−0.991112	−	0.133030i	$-0.957529\pi$
$691$	− 183.848i	− 0.266060i	0.991112	−	0.133030i	$-0.0424707\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 169.706i	− 0.244181i
$696$	0	0
$697$	−300.000	−0.430416
$698$	0	0
$699$	0	0
$700$	0	0
$701$	548.000	0.781740	0.390870	−	0.920446i	$-0.372174\pi$
$701$	548.000	0.781740	0.390870	+	0.920446i	$0.372174\pi$
$702$	0	0
$703$	282.843i	0.402337i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1583.92i	2.24034i
$708$	0	0
$709$	460.000	0.648801	0.324401	−	0.945920i	$-0.394837\pi$
$709$	460.000	0.648801	0.324401	+	0.945920i	$0.394837\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	1131.37i	1.58234i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1244.51i	1.73089i	0.501006	+	0.865444i	$0.332963\pi$
$719$	1244.51i	1.73089i	−0.501006	+	0.865444i	$0.667037\pi$
$720$	0	0
$721$	−1152.00	−1.59778
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−180.000	−0.248276
$726$	0	0
$727$	− 1029.55i	− 1.41616i	−0.706133	−	0.708079i	$-0.749562\pi$
$727$	− 1029.55i	− 1.41616i	0.706133	−	0.708079i	$-0.250438\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	28.2843i	0.0386926i
$732$	0	0
$733$	−660.000	−0.900409	−0.450205	−	0.892925i	$-0.648649\pi$
$733$	−660.000	−0.900409	−0.450205	+	0.892925i	$0.648649\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1160.00	−1.57395
$738$	0	0
$739$	− 636.396i	− 0.861158i	−0.902553	−	0.430579i	$-0.858309\pi$
$739$	− 636.396i	− 0.861158i	0.902553	−	0.430579i	$-0.141691\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	237.588i	0.319768i	0.987136	+	0.159884i	$0.0511121\pi$
$743$	237.588i	0.319768i	−0.987136	+	0.159884i	$0.948888\pi$
$744$	0	0
$745$	208.000	0.279195
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1632.00	2.17891
$750$	0	0
$751$	791.960i	1.05454i	0.849698	+	0.527270i	$0.176784\pi$
$751$	791.960i	1.05454i	−0.849698	+	0.527270i	$0.823216\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 678.823i	− 0.899103i
$756$	0	0
$757$	−660.000	−0.871863	−0.435931	−	0.899980i	$-0.643581\pi$
$757$	−660.000	−0.871863	−0.435931	+	0.899980i	$0.643581\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1022.00	−1.34297	−0.671485	−	0.741018i	$-0.734343\pi$
$761$	−1022.00	−1.34297	−0.671485	+	0.741018i	$0.734343\pi$
$762$	0	0
$763$	769.332i	1.00830i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 848.528i	− 1.10629i
$768$	0	0
$769$	−362.000	−0.470741	−0.235371	−	0.971906i	$-0.575630\pi$
$769$	−362.000	−0.470741	−0.235371	+	0.971906i	$0.575630\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	820.000	1.06080	0.530401	−	0.847747i	$-0.322041\pi$
$773$	820.000	1.06080	0.530401	+	0.847747i	$0.322041\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	424.264i	0.544627i
$780$	0	0
$781$	800.000	1.02433
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−400.000	−0.509554
$786$	0	0
$787$	766.504i	0.973956i	0.873414	+	0.486978i	$0.161901\pi$
$787$	766.504i	0.973956i	−0.873414	+	0.486978i	$0.838099\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 2149.60i	− 2.71758i
$792$	0	0
$793$	−560.000	−0.706179
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−860.000	−1.07905	−0.539523	−	0.841971i	$-0.681395\pi$
$797$	−860.000	−1.07905	−0.539523	+	0.841971i	$0.681395\pi$
$798$	0	0
$799$	678.823i	0.849590i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 141.421i	− 0.176116i
$804$	0	0
$805$	−512.000	−0.636025
$806$	0	0
$807$	0	0
$808$	0	0
$809$	482.000	0.595797	0.297899	−	0.954598i	$-0.403714\pi$
$809$	482.000	0.595797	0.297899	+	0.954598i	$0.403714\pi$
$810$	0	0
$811$	− 890.955i	− 1.09859i	−0.835629	−	0.549294i	$-0.814897\pi$
$811$	− 890.955i	− 1.09859i	0.835629	−	0.549294i	$-0.185103\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	441.235i	0.541392i
$816$	0	0
$817$	40.0000	0.0489596
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1508.00	1.83678	0.918392	−	0.395671i	$-0.129488\pi$
$821$	1508.00	1.83678	0.918392	+	0.395671i	$0.129488\pi$
$822$	0	0
$823$	916.410i	1.11350i	0.830680	+	0.556750i	$0.187952\pi$
$823$	916.410i	1.11350i	−0.830680	+	0.556750i	$0.812048\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	845.700i	1.02261i	0.859399	+	0.511306i	$0.170838\pi$
$827$	845.700i	1.02261i	−0.859399	+	0.511306i	$0.829162\pi$
$828$	0	0
$829$	−1092.00	−1.31725	−0.658625	−	0.752471i	$-0.728862\pi$
$829$	−1092.00	−1.31725	−0.658625	+	0.752471i	$0.728862\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−790.000	−0.948379
$834$	0	0
$835$	− 497.803i	− 0.596171i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	282.843i	0.337119i	0.985691	+	0.168559i	$0.0539115\pi$
$839$	282.843i	0.337119i	−0.985691	+	0.168559i	$0.946088\pi$
$840$	0	0
$841$	−441.000	−0.524376
$842$	0	0
$843$	0	0
$844$	0	0
$845$	924.000	1.09349
$846$	0	0
$847$	893.783i	1.05523i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	226.274i	0.265892i
$852$	0	0
$853$	1500.00	1.75850	0.879250	−	0.476361i	$-0.158045\pi$
$853$	1500.00	1.75850	0.879250	+	0.476361i	$0.158045\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	290.000	0.338390	0.169195	−	0.985583i	$-0.445883\pi$
$857$	290.000	0.338390	0.169195	+	0.985583i	$0.445883\pi$
$858$	0	0
$859$	975.807i	1.13598i	0.823035	+	0.567990i	$0.192279\pi$
$859$	975.807i	1.13598i	−0.823035	+	0.567990i	$0.807721\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	633.568i	0.734146i	0.930192	+	0.367073i	$0.119640\pi$
$863$	633.568i	0.734146i	−0.930192	+	0.367073i	$0.880360\pi$
$864$	0	0
$865$	80.0000	0.0924855
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1600.00	1.84120
$870$	0	0
$871$	1640.49i	1.88345i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1538.66i	1.75847i
$876$	0	0
$877$	−100.000	−0.114025	−0.0570125	−	0.998373i	$-0.518158\pi$
$877$	−100.000	−0.114025	−0.0570125	+	0.998373i	$0.518158\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	510.000	0.578888	0.289444	−	0.957195i	$-0.406530\pi$
$881$	510.000	0.578888	0.289444	+	0.957195i	$0.406530\pi$
$882$	0	0
$883$	− 398.808i	− 0.451651i	−0.974168	−	0.225826i	$-0.927492\pi$
$883$	− 398.808i	− 0.451651i	0.974168	−	0.225826i	$-0.0725080\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	350.725i	0.395406i	0.980262	+	0.197703i	$0.0633481\pi$
$887$	350.725i	0.395406i	−0.980262	+	0.197703i	$0.936652\pi$
$888$	0	0
$889$	−512.000	−0.575928
$890$	0	0
$891$	0	0
$892$	0	0
$893$	960.000	1.07503
$894$	0	0
$895$	− 169.706i	− 0.189615i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−600.000	−0.665927
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−720.000	−0.795580
$906$	0	0
$907$	1383.10i	1.52492i	0.647036	+	0.762459i	$0.276008\pi$
$907$	1383.10i	1.52492i	−0.647036	+	0.762459i	$0.723992\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	565.685i	0.620950i	0.950582	+	0.310475i	$0.100488\pi$
$911$	565.685i	0.620950i	−0.950582	+	0.310475i	$0.899512\pi$
$912$	0	0
$913$	360.000	0.394304
$914$	0	0
$915$	0	0
$916$	0	0
$917$	160.000	0.174482
$918$	0	0
$919$	− 622.254i	− 0.677099i	−0.940949	−	0.338549i	$-0.890064\pi$
$919$	− 622.254i	− 0.677099i	0.940949	−	0.338549i	$-0.109936\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1131.37i	− 1.22575i
$924$	0	0
$925$	180.000	0.194595
$926$	0	0
$927$	0	0
$928$	0	0
$929$	490.000	0.527449	0.263724	−	0.964598i	$-0.415049\pi$
$929$	490.000	0.527449	0.263724	+	0.964598i	$0.415049\pi$
$930$	0	0
$931$	1117.23i	1.20003i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 565.685i	− 0.605011i
$936$	0	0
$937$	10.0000	0.0106724	0.00533618	−	0.999986i	$-0.498301\pi$
$937$	10.0000	0.0106724	0.00533618	+	0.999986i	$0.498301\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−700.000	−0.743889	−0.371945	−	0.928255i	$-0.621309\pi$
$941$	−700.000	−0.743889	−0.371945	+	0.928255i	$0.621309\pi$
$942$	0	0
$943$	339.411i	0.359927i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 709.935i	− 0.749668i	−0.927092	−	0.374834i	$-0.877700\pi$
$947$	− 709.935i	− 0.749668i	0.927092	−	0.374834i	$-0.122300\pi$
$948$	0	0
$949$	−200.000	−0.210748
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−670.000	−0.703043	−0.351522	−	0.936180i	$-0.614336\pi$
$953$	−670.000	−0.703043	−0.351522	+	0.936180i	$0.614336\pi$
$954$	0	0
$955$	− 905.097i	− 0.947745i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1470.78i	− 1.53366i
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−680.000	−0.704663
$966$	0	0
$967$	− 1006.92i	− 1.04128i	−0.853776	−	0.520641i	$-0.825693\pi$
$967$	− 1006.92i	− 1.04128i	0.853776	−	0.520641i	$-0.174307\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	608.112i	0.626274i	0.949708	+	0.313137i	$0.101380\pi$
$971$	608.112i	0.626274i	−0.949708	+	0.313137i	$0.898620\pi$
$972$	0	0
$973$	−480.000	−0.493320
$974$	0	0
$975$	0	0
$976$	0	0
$977$	330.000	0.337769	0.168884	−	0.985636i	$-0.445984\pi$
$977$	330.000	0.337769	0.168884	+	0.985636i	$0.445984\pi$
$978$	0	0
$979$	− 311.127i	− 0.317801i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	576.999i	0.586978i	0.955962	+	0.293489i	$0.0948164\pi$
$983$	576.999i	0.586978i	−0.955962	+	0.293489i	$0.905184\pi$
$984$	0	0
$985$	−880.000	−0.893401
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32.0000	0.0323559
$990$	0	0
$991$	− 226.274i	− 0.228329i	−0.993462	−	0.114165i	$-0.963581\pi$
$991$	− 226.274i	− 0.228329i	0.993462	−	0.114165i	$-0.0364191\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	678.823i	0.682234i
$996$	0	0
$997$	620.000	0.621866	0.310933	−	0.950432i	$-0.399359\pi$
$997$	620.000	0.621866	0.310933	+	0.950432i	$0.399359\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.g.m.1279.1		2
3.2	odd	2		256.3.c.c.255.1		2
4.3	odd	2	inner	2304.3.g.m.1279.2		2
8.3	odd	2		2304.3.g.h.1279.2		2
8.5	even	2		2304.3.g.h.1279.1		2
12.11	even	2		256.3.c.c.255.2		2
16.3	odd	4		1152.3.b.g.703.1		4
16.5	even	4		1152.3.b.g.703.4		4
16.11	odd	4		1152.3.b.g.703.3		4
16.13	even	4		1152.3.b.g.703.2		4
24.5	odd	2		256.3.c.f.255.2		2
24.11	even	2		256.3.c.f.255.1		2
48.5	odd	4		128.3.d.c.63.1	✓	4
48.11	even	4		128.3.d.c.63.3	yes	4
48.29	odd	4		128.3.d.c.63.4	yes	4
48.35	even	4		128.3.d.c.63.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.d.c.63.1	✓	4	48.5	odd	4
128.3.d.c.63.2	yes	4	48.35	even	4
128.3.d.c.63.3	yes	4	48.11	even	4
128.3.d.c.63.4	yes	4	48.29	odd	4
256.3.c.c.255.1		2	3.2	odd	2
256.3.c.c.255.2		2	12.11	even	2
256.3.c.f.255.1		2	24.11	even	2
256.3.c.f.255.2		2	24.5	odd	2
1152.3.b.g.703.1		4	16.3	odd	4
1152.3.b.g.703.2		4	16.13	even	4
1152.3.b.g.703.3		4	16.11	odd	4
1152.3.b.g.703.4		4	16.5	even	4
2304.3.g.h.1279.1		2	8.5	even	2
2304.3.g.h.1279.2		2	8.3	odd	2
2304.3.g.m.1279.1		2	1.1	even	1	trivial
2304.3.g.m.1279.2		2	4.3	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.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000 q^{5} -11.3137i q^{7} +O(q^{10})\)	\(q+4.00000 q^{5} -11.3137i q^{7} -14.1421i q^{11} -20.0000 q^{13} +10.0000 q^{17} -14.1421i q^{19} -11.3137i q^{23} -9.00000 q^{25} +20.0000 q^{29} -45.2548i q^{35} -20.0000 q^{37} -30.0000 q^{41} +2.82843i q^{43} +67.8823i q^{47} -79.0000 q^{49} -60.0000 q^{53} -56.5685i q^{55} +42.4264i q^{59} +28.0000 q^{61} -80.0000 q^{65} -82.0244i q^{67} +56.5685i q^{71} +10.0000 q^{73} -160.000 q^{77} +113.137i q^{79} +25.4558i q^{83} +40.0000 q^{85} +22.0000 q^{89} +226.274i q^{91} -56.5685i q^{95} +150.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{5}+O(q^{10}) \) 2 * q + 8 * q^5	\( 2 q + 8 q^{5} - 40 q^{13} + 20 q^{17} - 18 q^{25} + 40 q^{29} - 40 q^{37} - 60 q^{41} - 158 q^{49} - 120 q^{53} + 56 q^{61} - 160 q^{65} + 20 q^{73} - 320 q^{77} + 80 q^{85} + 44 q^{89} + 300 q^{97}+O(q^{100}) \) 2 * q + 8 * q^5 - 40 * q^13 + 20 * q^17 - 18 * q^25 + 40 * q^29 - 40 * q^37 - 60 * q^41 - 158 * q^49 - 120 * q^53 + 56 * q^61 - 160 * q^65 + 20 * q^73 - 320 * q^77 + 80 * q^85 + 44 * q^89 + 300 * q^97

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