LMFDB - Embedded newform 2304.2.c.j.2303.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(2303,2304)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2304.2303");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.c (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:	$18.3975326257$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2303.4
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2304.2303
Dual form			2304.2.c.j.2303.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949i q^{5} +2.00000i q^{7} +O(q^{10})$	$q-2.44949i q^{5} +2.00000i q^{7} -4.89898 q^{11} +3.46410 q^{13} +4.24264i q^{17} -6.92820i q^{19} -8.48528 q^{23} -1.00000 q^{25} +7.34847i q^{29} +2.00000i q^{31} +4.89898 q^{35} -6.92820 q^{37} -4.24264i q^{41} -8.48528 q^{47} +3.00000 q^{49} -2.44949i q^{53} +12.0000i q^{55} -6.92820 q^{61} -8.48528i q^{65} +6.92820i q^{67} -8.48528 q^{71} -4.00000 q^{73} -9.79796i q^{77} +14.0000i q^{79} -4.89898 q^{83} +10.3923 q^{85} +4.24264i q^{89} +6.92820i q^{91} -16.9706 q^{95} -16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{25} + 24 q^{49} - 32 q^{73} - 128 q^{97}+O(q^{100}) $ 8 * q - 8 * q^25 + 24 * q^49 - 32 * q^73 - 128 * 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
$5$	− 2.44949i	− 1.09545i	−0.836660	−	0.547723i	$-0.815495\pi$
$5$	− 2.44949i	− 1.09545i	0.836660	−	0.547723i	$-0.184505\pi$
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.24264i	1.02899i	0.857493	+	0.514496i	$0.172021\pi$
$17$	4.24264i	1.02899i	−0.857493	+	0.514496i	$0.827979\pi$
$18$	0	0
$19$	− 6.92820i	− 1.58944i	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	− 6.92820i	− 1.58944i	0.606977	−	0.794719i	$-0.292382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.48528	−1.76930	−0.884652	−	0.466252i	$-0.845604\pi$
$23$	−8.48528	−1.76930	−0.884652	+	0.466252i	$0.845604\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.34847i	1.36458i	0.731083	+	0.682288i	$0.239015\pi$
$29$	7.34847i	1.36458i	−0.731083	+	0.682288i	$0.760985\pi$
$30$	0	0
$31$	2.00000i	0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$	2.00000i	0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.89898	0.828079
$36$	0	0
$37$	−6.92820	−1.13899	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	−6.92820	−1.13899	−0.569495	+	0.821995i	$0.692861\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.24264i	− 0.662589i	−0.943527	−	0.331295i	$-0.892515\pi$
$41$	− 4.24264i	− 0.662589i	0.943527	−	0.331295i	$-0.107485\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.48528	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$47$	−8.48528	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.44949i	− 0.336463i	−0.985747	−	0.168232i	$-0.946194\pi$
$53$	− 2.44949i	− 0.336463i	0.985747	−	0.168232i	$-0.0538057\pi$
$54$	0	0
$55$	12.0000i	1.61808i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−6.92820	−0.887066	−0.443533	−	0.896258i	$-0.646275\pi$
$61$	−6.92820	−0.887066	−0.443533	+	0.896258i	$0.646275\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 8.48528i	− 1.05247i
$66$	0	0
$67$	6.92820i	0.846415i	0.906033	+	0.423207i	$0.139096\pi$
$67$	6.92820i	0.846415i	−0.906033	+	0.423207i	$0.860904\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.48528	−1.00702	−0.503509	−	0.863990i	$-0.667958\pi$
$71$	−8.48528	−1.00702	−0.503509	+	0.863990i	$0.667958\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 9.79796i	− 1.11658i
$78$	0	0
$79$	14.0000i	1.57512i	0.616236	+	0.787562i	$0.288657\pi$
$79$	14.0000i	1.57512i	−0.616236	+	0.787562i	$0.711343\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.89898	−0.537733	−0.268866	−	0.963177i	$-0.586649\pi$
$83$	−4.89898	−0.537733	−0.268866	+	0.963177i	$0.586649\pi$
$84$	0	0
$85$	10.3923	1.12720
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.24264i	0.449719i	0.974391	+	0.224860i	$0.0721923\pi$
$89$	4.24264i	0.449719i	−0.974391	+	0.224860i	$0.927808\pi$
$90$	0	0
$91$	6.92820i	0.726273i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.9706	−1.74114
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.2474i	1.21867i	0.792914	+	0.609333i	$0.208563\pi$
$101$	12.2474i	1.21867i	−0.792914	+	0.609333i	$0.791437\pi$
$102$	0	0
$103$	− 2.00000i	− 0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$	− 2.00000i	− 0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.5959	1.89441	0.947204	−	0.320630i	$-0.103895\pi$
$107$	19.5959	1.89441	0.947204	+	0.320630i	$0.103895\pi$
$108$	0	0
$109$	10.3923	0.995402	0.497701	−	0.867349i	$-0.334178\pi$
$109$	10.3923	0.995402	0.497701	+	0.867349i	$0.334178\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.24264i	0.399114i	0.979886	+	0.199557i	$0.0639503\pi$
$113$	4.24264i	0.399114i	−0.979886	+	0.199557i	$0.936050\pi$
$114$	0	0
$115$	20.7846i	1.93817i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.48528	−0.777844
$120$	0	0
$121$	13.0000	1.18182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.79796i	− 0.876356i
$126$	0	0
$127$	10.0000i	0.887357i	0.896186	+	0.443678i	$0.146327\pi$
$127$	10.0000i	0.887357i	−0.896186	+	0.443678i	$0.853673\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.79796	−0.856052	−0.428026	−	0.903767i	$-0.640791\pi$
$131$	−9.79796	−0.856052	−0.428026	+	0.903767i	$0.640791\pi$
$132$	0	0
$133$	13.8564	1.20150
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.24264i	0.362473i	0.983440	+	0.181237i	$0.0580100\pi$
$137$	4.24264i	0.362473i	−0.983440	+	0.181237i	$0.941990\pi$
$138$	0	0
$139$	− 6.92820i	− 0.587643i	−0.955860	−	0.293821i	$-0.905073\pi$
$139$	− 6.92820i	− 0.587643i	0.955860	−	0.293821i	$-0.0949270\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−16.9706	−1.41915
$144$	0	0
$145$	18.0000	1.49482
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.44949i	0.200670i	0.994954	+	0.100335i	$0.0319915\pi$
$149$	2.44949i	0.200670i	−0.994954	+	0.100335i	$0.968009\pi$
$150$	0	0
$151$	2.00000i	0.162758i	0.996683	+	0.0813788i	$0.0259324\pi$
$151$	2.00000i	0.162758i	−0.996683	+	0.0813788i	$0.974068\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.89898	0.393496
$156$	0	0
$157$	−20.7846	−1.65879	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	−20.7846	−1.65879	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 16.9706i	− 1.33747i
$162$	0	0
$163$	− 13.8564i	− 1.08532i	−0.839953	−	0.542659i	$-0.817418\pi$
$163$	− 13.8564i	− 1.08532i	0.839953	−	0.542659i	$-0.182582\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.44949i	− 0.186231i	−0.995655	−	0.0931156i	$-0.970317\pi$
$173$	− 2.44949i	− 0.186231i	0.995655	−	0.0931156i	$-0.0296826\pi$
$174$	0	0
$175$	− 2.00000i	− 0.151186i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.79796	0.732334	0.366167	−	0.930549i	$-0.380670\pi$
$179$	9.79796	0.732334	0.366167	+	0.930549i	$0.380670\pi$
$180$	0	0
$181$	−17.3205	−1.28742	−0.643712	−	0.765268i	$-0.722606\pi$
$181$	−17.3205	−1.28742	−0.643712	+	0.765268i	$0.722606\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.9706i	1.24770i
$186$	0	0
$187$	− 20.7846i	− 1.51992i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.44949i	0.174519i	0.996186	+	0.0872595i	$0.0278109\pi$
$197$	2.44949i	0.174519i	−0.996186	+	0.0872595i	$0.972189\pi$
$198$	0	0
$199$	22.0000i	1.55954i	0.626067	+	0.779769i	$0.284664\pi$
$199$	22.0000i	1.55954i	−0.626067	+	0.779769i	$0.715336\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−14.6969	−1.03152
$204$	0	0
$205$	−10.3923	−0.725830
$206$	0	0
$207$	0	0
$208$	0	0
$209$	33.9411i	2.34776i
$210$	0	0
$211$	− 6.92820i	− 0.476957i	−0.971148	−	0.238479i	$-0.923351\pi$
$211$	− 6.92820i	− 0.476957i	0.971148	−	0.238479i	$-0.0766487\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	0	0
$220$	0	0
$221$	14.6969i	0.988623i
$222$	0	0
$223$	10.0000i	0.669650i	0.942280	+	0.334825i	$0.108677\pi$
$223$	10.0000i	0.669650i	−0.942280	+	0.334825i	$0.891323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.6969	−0.975470	−0.487735	−	0.872992i	$-0.662177\pi$
$227$	−14.6969	−0.975470	−0.487735	+	0.872992i	$0.662177\pi$
$228$	0	0
$229$	−10.3923	−0.686743	−0.343371	−	0.939200i	$-0.611569\pi$
$229$	−10.3923	−0.686743	−0.343371	+	0.939200i	$0.611569\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 21.2132i	− 1.38972i	−0.719144	−	0.694862i	$-0.755466\pi$
$233$	− 21.2132i	− 1.38972i	0.719144	−	0.694862i	$-0.244534\pi$
$234$	0	0
$235$	20.7846i	1.35584i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.9706	1.09773	0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706	1.09773	0.548867	+	0.835910i	$0.315059\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 7.34847i	− 0.469476i
$246$	0	0
$247$	− 24.0000i	− 1.52708i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.6969	−0.927663	−0.463831	−	0.885924i	$-0.653526\pi$
$251$	−14.6969	−0.927663	−0.463831	+	0.885924i	$0.653526\pi$
$252$	0	0
$253$	41.5692	2.61343
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 21.2132i	− 1.32324i	−0.749838	−	0.661622i	$-0.769869\pi$
$257$	− 21.2132i	− 1.32324i	0.749838	−	0.661622i	$-0.230131\pi$
$258$	0	0
$259$	− 13.8564i	− 0.860995i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.9706	−1.04645	−0.523225	−	0.852195i	$-0.675271\pi$
$263$	−16.9706	−1.04645	−0.523225	+	0.852195i	$0.675271\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.0454i	1.34413i	0.740491	+	0.672066i	$0.234593\pi$
$269$	22.0454i	1.34413i	−0.740491	+	0.672066i	$0.765407\pi$
$270$	0	0
$271$	− 14.0000i	− 0.850439i	−0.905090	−	0.425220i	$-0.860197\pi$
$271$	− 14.0000i	− 0.850439i	0.905090	−	0.425220i	$-0.139803\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.89898	0.295420
$276$	0	0
$277$	10.3923	0.624413	0.312207	−	0.950014i	$-0.398932\pi$
$277$	10.3923	0.624413	0.312207	+	0.950014i	$0.398932\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 12.7279i	− 0.759284i	−0.925133	−	0.379642i	$-0.876047\pi$
$281$	− 12.7279i	− 0.759284i	0.925133	−	0.379642i	$-0.123953\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.48528	0.500870
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 22.0454i	− 1.28791i	−0.765065	−	0.643953i	$-0.777293\pi$
$293$	− 22.0454i	− 1.28791i	0.765065	−	0.643953i	$-0.222707\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−29.3939	−1.69989
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	16.9706i	0.971732i
$306$	0	0
$307$	− 20.7846i	− 1.18624i	−0.805114	−	0.593120i	$-0.797896\pi$
$307$	− 20.7846i	− 1.18624i	0.805114	−	0.593120i	$-0.202104\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.9706	0.962312	0.481156	−	0.876635i	$-0.340217\pi$
$311$	16.9706	0.962312	0.481156	+	0.876635i	$0.340217\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 17.1464i	− 0.963039i	−0.876435	−	0.481520i	$-0.840085\pi$
$317$	− 17.1464i	− 0.963039i	0.876435	−	0.481520i	$-0.159915\pi$
$318$	0	0
$319$	− 36.0000i	− 2.01561i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	29.3939	1.63552
$324$	0	0
$325$	−3.46410	−0.192154
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 16.9706i	− 0.935617i
$330$	0	0
$331$	− 6.92820i	− 0.380808i	−0.981706	−	0.190404i	$-0.939020\pi$
$331$	− 6.92820i	− 0.380808i	0.981706	−	0.190404i	$-0.0609799\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	16.9706	0.927201
$336$	0	0
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 9.79796i	− 0.530589i
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.89898	0.262991	0.131495	−	0.991317i	$-0.458022\pi$
$347$	4.89898	0.262991	0.131495	+	0.991317i	$0.458022\pi$
$348$	0	0
$349$	6.92820	0.370858	0.185429	−	0.982658i	$-0.440632\pi$
$349$	6.92820	0.370858	0.185429	+	0.982658i	$0.440632\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 29.6985i	− 1.58069i	−0.612661	−	0.790345i	$-0.709901\pi$
$353$	− 29.6985i	− 1.58069i	0.612661	−	0.790345i	$-0.290099\pi$
$354$	0	0
$355$	20.7846i	1.10313i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.48528	0.447836	0.223918	−	0.974608i	$-0.428115\pi$
$359$	8.48528	0.447836	0.223918	+	0.974608i	$0.428115\pi$
$360$	0	0
$361$	−29.0000	−1.52632
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.79796i	0.512849i
$366$	0	0
$367$	2.00000i	0.104399i	0.998637	+	0.0521996i	$0.0166232\pi$
$367$	2.00000i	0.104399i	−0.998637	+	0.0521996i	$0.983377\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.89898	0.254342
$372$	0	0
$373$	−6.92820	−0.358729	−0.179364	−	0.983783i	$-0.557404\pi$
$373$	−6.92820	−0.358729	−0.179364	+	0.983783i	$0.557404\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	25.4558i	1.31104i
$378$	0	0
$379$	34.6410i	1.77939i	0.456556	+	0.889695i	$0.349083\pi$
$379$	34.6410i	1.77939i	−0.456556	+	0.889695i	$0.650917\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.9706	0.867155	0.433578	−	0.901116i	$-0.357251\pi$
$383$	16.9706	0.867155	0.433578	+	0.901116i	$0.357251\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 26.9444i	− 1.36613i	−0.730355	−	0.683067i	$-0.760646\pi$
$389$	− 26.9444i	− 1.36613i	0.730355	−	0.683067i	$-0.239354\pi$
$390$	0	0
$391$	− 36.0000i	− 1.82060i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	34.2929	1.72546
$396$	0	0
$397$	−20.7846	−1.04315	−0.521575	−	0.853206i	$-0.674655\pi$
$397$	−20.7846	−1.04315	−0.521575	+	0.853206i	$0.674655\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	38.1838i	1.90681i	0.301699	+	0.953403i	$0.402446\pi$
$401$	38.1838i	1.90681i	−0.301699	+	0.953403i	$0.597554\pi$
$402$	0	0
$403$	6.92820i	0.345118i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	33.9411	1.68240
$408$	0	0
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.0000i	0.589057i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.6969	0.717992	0.358996	−	0.933339i	$-0.383119\pi$
$419$	14.6969	0.717992	0.358996	+	0.933339i	$0.383119\pi$
$420$	0	0
$421$	−3.46410	−0.168830	−0.0844150	−	0.996431i	$-0.526902\pi$
$421$	−3.46410	−0.168830	−0.0844150	+	0.996431i	$0.526902\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 4.24264i	− 0.205798i
$426$	0	0
$427$	− 13.8564i	− 0.670559i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.4558	1.22616	0.613082	−	0.790019i	$-0.289929\pi$
$431$	25.4558	1.22616	0.613082	+	0.790019i	$0.289929\pi$
$432$	0	0
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	58.7878i	2.81220i
$438$	0	0
$439$	− 14.0000i	− 0.668184i	−0.942541	−	0.334092i	$-0.891570\pi$
$439$	− 14.0000i	− 0.668184i	0.942541	−	0.334092i	$-0.108430\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.6969	0.698273	0.349136	−	0.937072i	$-0.386475\pi$
$443$	14.6969	0.698273	0.349136	+	0.937072i	$0.386475\pi$
$444$	0	0
$445$	10.3923	0.492642
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 4.24264i	− 0.200223i	−0.994976	−	0.100111i	$-0.968080\pi$
$449$	− 4.24264i	− 0.200223i	0.994976	−	0.100111i	$-0.0319199\pi$
$450$	0	0
$451$	20.7846i	0.978709i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	16.9706	0.795592
$456$	0	0
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.44949i	0.114084i	0.998372	+	0.0570421i	$0.0181669\pi$
$461$	2.44949i	0.114084i	−0.998372	+	0.0570421i	$0.981833\pi$
$462$	0	0
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−34.2929	−1.58688	−0.793442	−	0.608646i	$-0.791713\pi$
$467$	−34.2929	−1.58688	−0.793442	+	0.608646i	$0.791713\pi$
$468$	0	0
$469$	−13.8564	−0.639829
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.92820i	0.317888i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.48528	0.387702	0.193851	−	0.981031i	$-0.437902\pi$
$479$	8.48528	0.387702	0.193851	+	0.981031i	$0.437902\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	0	0
$483$	0	0
$484$	0	0
$485$	39.1918i	1.77961i
$486$	0	0
$487$	− 10.0000i	− 0.453143i	−0.973995	−	0.226572i	$-0.927248\pi$
$487$	− 10.0000i	− 0.453143i	0.973995	−	0.226572i	$-0.0727517\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.79796	0.442176	0.221088	−	0.975254i	$-0.429039\pi$
$491$	9.79796	0.442176	0.221088	+	0.975254i	$0.429039\pi$
$492$	0	0
$493$	−31.1769	−1.40414
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 16.9706i	− 0.761234i
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.48528	0.378340	0.189170	−	0.981944i	$-0.439420\pi$
$503$	8.48528	0.378340	0.189170	+	0.981944i	$0.439420\pi$
$504$	0	0
$505$	30.0000	1.33498
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 12.2474i	− 0.542859i	−0.962458	−	0.271429i	$-0.912504\pi$
$509$	− 12.2474i	− 0.542859i	0.962458	−	0.271429i	$-0.0874963\pi$
$510$	0	0
$511$	− 8.00000i	− 0.353899i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.89898	−0.215875
$516$	0	0
$517$	41.5692	1.82821
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21.2132i	0.929367i	0.885477	+	0.464684i	$0.153832\pi$
$521$	21.2132i	0.929367i	−0.885477	+	0.464684i	$0.846168\pi$
$522$	0	0
$523$	− 6.92820i	− 0.302949i	−0.988461	−	0.151475i	$-0.951598\pi$
$523$	− 6.92820i	− 0.302949i	0.988461	−	0.151475i	$-0.0484022\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.48528	−0.369625
$528$	0	0
$529$	49.0000	2.13043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 14.6969i	− 0.636595i
$534$	0	0
$535$	− 48.0000i	− 2.07522i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−14.6969	−0.633042
$540$	0	0
$541$	−17.3205	−0.744667	−0.372333	−	0.928099i	$-0.621442\pi$
$541$	−17.3205	−0.744667	−0.372333	+	0.928099i	$0.621442\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 25.4558i	− 1.09041i
$546$	0	0
$547$	27.7128i	1.18491i	0.805602	+	0.592457i	$0.201842\pi$
$547$	27.7128i	1.18491i	−0.805602	+	0.592457i	$0.798158\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	50.9117	2.16891
$552$	0	0
$553$	−28.0000	−1.19068
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 26.9444i	− 1.14167i	−0.821065	−	0.570835i	$-0.806620\pi$
$557$	− 26.9444i	− 1.14167i	0.821065	−	0.570835i	$-0.193380\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.4949	−1.03234	−0.516168	−	0.856487i	$-0.672642\pi$
$563$	−24.4949	−1.03234	−0.516168	+	0.856487i	$0.672642\pi$
$564$	0	0
$565$	10.3923	0.437208
$566$	0	0
$567$	0	0
$568$	0	0
$569$	38.1838i	1.60075i	0.599502	+	0.800373i	$0.295365\pi$
$569$	38.1838i	1.60075i	−0.599502	+	0.800373i	$0.704635\pi$
$570$	0	0
$571$	− 13.8564i	− 0.579873i	−0.957046	−	0.289936i	$-0.906366\pi$
$571$	− 13.8564i	− 0.579873i	0.957046	−	0.289936i	$-0.0936341\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.48528	0.353861
$576$	0	0
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 9.79796i	− 0.406488i
$582$	0	0
$583$	12.0000i	0.496989i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.5959	0.808810	0.404405	−	0.914580i	$-0.367479\pi$
$587$	19.5959	0.808810	0.404405	+	0.914580i	$0.367479\pi$
$588$	0	0
$589$	13.8564	0.570943
$590$	0	0
$591$	0	0
$592$	0	0
$593$	29.6985i	1.21957i	0.792567	+	0.609785i	$0.208744\pi$
$593$	29.6985i	1.21957i	−0.792567	+	0.609785i	$0.791256\pi$
$594$	0	0
$595$	20.7846i	0.852086i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.48528	−0.346699	−0.173350	−	0.984860i	$-0.555459\pi$
$599$	−8.48528	−0.346699	−0.173350	+	0.984860i	$0.555459\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 31.8434i	− 1.29462i
$606$	0	0
$607$	− 38.0000i	− 1.54237i	−0.636610	−	0.771186i	$-0.719664\pi$
$607$	− 38.0000i	− 1.54237i	0.636610	−	0.771186i	$-0.280336\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−29.3939	−1.18915
$612$	0	0
$613$	6.92820	0.279827	0.139914	−	0.990164i	$-0.455317\pi$
$613$	6.92820	0.279827	0.139914	+	0.990164i	$0.455317\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 12.7279i	− 0.512407i	−0.966623	−	0.256203i	$-0.917528\pi$
$617$	− 12.7279i	− 0.512407i	0.966623	−	0.256203i	$-0.0824717\pi$
$618$	0	0
$619$	− 13.8564i	− 0.556936i	−0.960446	−	0.278468i	$-0.910173\pi$
$619$	− 13.8564i	− 0.556936i	0.960446	−	0.278468i	$-0.0898266\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.48528	−0.339956
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 29.3939i	− 1.17201i
$630$	0	0
$631$	14.0000i	0.557331i	0.960388	+	0.278666i	$0.0898921\pi$
$631$	14.0000i	0.557331i	−0.960388	+	0.278666i	$0.910108\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	24.4949	0.972050
$636$	0	0
$637$	10.3923	0.411758
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 12.7279i	− 0.502723i	−0.967893	−	0.251361i	$-0.919122\pi$
$641$	− 12.7279i	− 0.502723i	0.967893	−	0.251361i	$-0.0808782\pi$
$642$	0	0
$643$	− 41.5692i	− 1.63933i	−0.572843	−	0.819665i	$-0.694160\pi$
$643$	− 41.5692i	− 1.63933i	0.572843	−	0.819665i	$-0.305840\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.4264	1.66795	0.833977	−	0.551799i	$-0.186058\pi$
$647$	42.4264	1.66795	0.833977	+	0.551799i	$0.186058\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.44949i	− 0.0958559i	−0.998851	−	0.0479280i	$-0.984738\pi$
$653$	− 2.44949i	− 0.0958559i	0.998851	−	0.0479280i	$-0.0152618\pi$
$654$	0	0
$655$	24.0000i	0.937758i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	39.1918	1.52670	0.763349	−	0.645987i	$-0.223554\pi$
$659$	39.1918	1.52670	0.763349	+	0.645987i	$0.223554\pi$
$660$	0	0
$661$	6.92820	0.269476	0.134738	−	0.990881i	$-0.456981\pi$
$661$	6.92820	0.269476	0.134738	+	0.990881i	$0.456981\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 33.9411i	− 1.31618i
$666$	0	0
$667$	− 62.3538i	− 2.41435i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	33.9411	1.31028
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.7423i	1.41212i	0.708150	+	0.706062i	$0.249530\pi$
$677$	36.7423i	1.41212i	−0.708150	+	0.706062i	$0.750470\pi$
$678$	0	0
$679$	− 32.0000i	− 1.22805i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−44.0908	−1.68709	−0.843544	−	0.537060i	$-0.819535\pi$
$683$	−44.0908	−1.68709	−0.843544	+	0.537060i	$0.819535\pi$
$684$	0	0
$685$	10.3923	0.397070
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 8.48528i	− 0.323263i
$690$	0	0
$691$	− 41.5692i	− 1.58137i	−0.612225	−	0.790684i	$-0.709725\pi$
$691$	− 41.5692i	− 1.58137i	0.612225	−	0.790684i	$-0.290275\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.9706	−0.643730
$696$	0	0
$697$	18.0000	0.681799
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17.1464i	0.647612i	0.946124	+	0.323806i	$0.104962\pi$
$701$	17.1464i	0.647612i	−0.946124	+	0.323806i	$0.895038\pi$
$702$	0	0
$703$	48.0000i	1.81035i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−24.4949	−0.921225
$708$	0	0
$709$	−24.2487	−0.910679	−0.455340	−	0.890318i	$-0.650482\pi$
$709$	−24.2487	−0.910679	−0.455340	+	0.890318i	$0.650482\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 16.9706i	− 0.635553i
$714$	0	0
$715$	41.5692i	1.55460i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.4558	0.949343	0.474671	−	0.880163i	$-0.342567\pi$
$719$	25.4558	0.949343	0.474671	+	0.880163i	$0.342567\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 7.34847i	− 0.272915i
$726$	0	0
$727$	14.0000i	0.519231i	0.965712	+	0.259616i	$0.0835959\pi$
$727$	14.0000i	0.519231i	−0.965712	+	0.259616i	$0.916404\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−10.3923	−0.383849	−0.191924	−	0.981410i	$-0.561473\pi$
$733$	−10.3923	−0.383849	−0.191924	+	0.981410i	$0.561473\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 33.9411i	− 1.25024i
$738$	0	0
$739$	− 41.5692i	− 1.52915i	−0.644536	−	0.764574i	$-0.722949\pi$
$739$	− 41.5692i	− 1.52915i	0.644536	−	0.764574i	$-0.277051\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.9411	−1.24518	−0.622590	−	0.782549i	$-0.713919\pi$
$743$	−33.9411	−1.24518	−0.622590	+	0.782549i	$0.713919\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	39.1918i	1.43204i
$750$	0	0
$751$	− 46.0000i	− 1.67856i	−0.543696	−	0.839282i	$-0.682976\pi$
$751$	− 46.0000i	− 1.67856i	0.543696	−	0.839282i	$-0.317024\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.89898	0.178292
$756$	0	0
$757$	17.3205	0.629525	0.314762	−	0.949171i	$-0.398075\pi$
$757$	17.3205	0.629525	0.314762	+	0.949171i	$0.398075\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.7279i	0.461387i	0.973026	+	0.230693i	$0.0740994\pi$
$761$	12.7279i	0.461387i	−0.973026	+	0.230693i	$0.925901\pi$
$762$	0	0
$763$	20.7846i	0.752453i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.34847i	0.264306i	0.991229	+	0.132153i	$0.0421890\pi$
$773$	7.34847i	0.264306i	−0.991229	+	0.132153i	$0.957811\pi$
$774$	0	0
$775$	− 2.00000i	− 0.0718421i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−29.3939	−1.05314
$780$	0	0
$781$	41.5692	1.48746
$782$	0	0
$783$	0	0
$784$	0	0
$785$	50.9117i	1.81712i
$786$	0	0
$787$	34.6410i	1.23482i	0.786642	+	0.617409i	$0.211818\pi$
$787$	34.6410i	1.23482i	−0.786642	+	0.617409i	$0.788182\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.48528	−0.301702
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.9444i	0.954419i	0.878790	+	0.477210i	$0.158352\pi$
$797$	26.9444i	0.954419i	−0.878790	+	0.477210i	$0.841648\pi$
$798$	0	0
$799$	− 36.0000i	− 1.27359i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	19.5959	0.691525
$804$	0	0
$805$	−41.5692	−1.46512
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 38.1838i	− 1.34247i	−0.741245	−	0.671235i	$-0.765764\pi$
$809$	− 38.1838i	− 1.34247i	0.741245	−	0.671235i	$-0.234236\pi$
$810$	0	0
$811$	− 20.7846i	− 0.729846i	−0.931038	−	0.364923i	$-0.881095\pi$
$811$	− 20.7846i	− 0.729846i	0.931038	−	0.364923i	$-0.118905\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−33.9411	−1.18891
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 7.34847i	− 0.256463i	−0.991744	−	0.128232i	$-0.959070\pi$
$821$	− 7.34847i	− 0.256463i	0.991744	−	0.128232i	$-0.0409301\pi$
$822$	0	0
$823$	26.0000i	0.906303i	0.891434	+	0.453152i	$0.149700\pi$
$823$	26.0000i	0.906303i	−0.891434	+	0.453152i	$0.850300\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29.3939	−1.02213	−0.511063	−	0.859543i	$-0.670748\pi$
$827$	−29.3939	−1.02213	−0.511063	+	0.859543i	$0.670748\pi$
$828$	0	0
$829$	−45.0333	−1.56407	−0.782036	−	0.623233i	$-0.785819\pi$
$829$	−45.0333	−1.56407	−0.782036	+	0.623233i	$0.785819\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	12.7279i	0.440996i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25.4558	−0.878833	−0.439417	−	0.898283i	$-0.644815\pi$
$839$	−25.4558	−0.878833	−0.439417	+	0.898283i	$0.644815\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.44949i	0.0842650i
$846$	0	0
$847$	26.0000i	0.893371i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	58.7878	2.01522
$852$	0	0
$853$	−48.4974	−1.66052	−0.830260	−	0.557376i	$-0.811808\pi$
$853$	−48.4974	−1.66052	−0.830260	+	0.557376i	$0.811808\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4.24264i	− 0.144926i	−0.997371	−	0.0724629i	$-0.976914\pi$
$857$	− 4.24264i	− 0.144926i	0.997371	−	0.0724629i	$-0.0230859\pi$
$858$	0	0
$859$	27.7128i	0.945549i	0.881183	+	0.472774i	$0.156747\pi$
$859$	27.7128i	0.945549i	−0.881183	+	0.472774i	$0.843253\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−33.9411	−1.15537	−0.577685	−	0.816260i	$-0.696044\pi$
$863$	−33.9411	−1.15537	−0.577685	+	0.816260i	$0.696044\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 68.5857i	− 2.32661i
$870$	0	0
$871$	24.0000i	0.813209i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	19.5959	0.662463
$876$	0	0
$877$	6.92820	0.233949	0.116974	−	0.993135i	$-0.462680\pi$
$877$	6.92820	0.233949	0.116974	+	0.993135i	$0.462680\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 21.2132i	− 0.714691i	−0.933972	−	0.357345i	$-0.883682\pi$
$881$	− 21.2132i	− 0.714691i	0.933972	−	0.357345i	$-0.116318\pi$
$882$	0	0
$883$	27.7128i	0.932610i	0.884624	+	0.466305i	$0.154415\pi$
$883$	27.7128i	0.932610i	−0.884624	+	0.466305i	$0.845585\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.9706	0.569816	0.284908	−	0.958555i	$-0.408037\pi$
$887$	16.9706	0.569816	0.284908	+	0.958555i	$0.408037\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	0	0
$892$	0	0
$893$	58.7878i	1.96726i
$894$	0	0
$895$	− 24.0000i	− 0.802232i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14.6969	−0.490170
$900$	0	0
$901$	10.3923	0.346218
$902$	0	0
$903$	0	0
$904$	0	0
$905$	42.4264i	1.41030i
$906$	0	0
$907$	55.4256i	1.84038i	0.391475	+	0.920189i	$0.371965\pi$
$907$	55.4256i	1.84038i	−0.391475	+	0.920189i	$0.628035\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	33.9411	1.12452	0.562260	−	0.826961i	$-0.309932\pi$
$911$	33.9411	1.12452	0.562260	+	0.826961i	$0.309932\pi$
$912$	0	0
$913$	24.0000	0.794284
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 19.5959i	− 0.647114i
$918$	0	0
$919$	26.0000i	0.857661i	0.903385	+	0.428830i	$0.141074\pi$
$919$	26.0000i	0.857661i	−0.903385	+	0.428830i	$0.858926\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−29.3939	−0.967511
$924$	0	0
$925$	6.92820	0.227798
$926$	0	0
$927$	0	0
$928$	0	0
$929$	38.1838i	1.25277i	0.779514	+	0.626384i	$0.215466\pi$
$929$	38.1838i	1.25277i	−0.779514	+	0.626384i	$0.784534\pi$
$930$	0	0
$931$	− 20.7846i	− 0.681188i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−50.9117	−1.66499
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 56.3383i	− 1.83657i	−0.395914	−	0.918287i	$-0.629572\pi$
$941$	− 56.3383i	− 1.83657i	0.395914	−	0.918287i	$-0.370428\pi$
$942$	0	0
$943$	36.0000i	1.17232i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.5959	−0.636782	−0.318391	−	0.947960i	$-0.603142\pi$
$947$	−19.5959	−0.636782	−0.318391	+	0.947960i	$0.603142\pi$
$948$	0	0
$949$	−13.8564	−0.449798
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.2132i	0.687163i	0.939123	+	0.343582i	$0.111640\pi$
$953$	21.2132i	0.687163i	−0.939123	+	0.343582i	$0.888360\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.48528	−0.274004
$960$	0	0
$961$	27.0000	0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	24.4949i	0.788519i
$966$	0	0
$967$	− 26.0000i	− 0.836104i	−0.908423	−	0.418052i	$-0.862713\pi$
$967$	− 26.0000i	− 0.836104i	0.908423	−	0.418052i	$-0.137287\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.89898	−0.157216	−0.0786079	−	0.996906i	$-0.525048\pi$
$971$	−4.89898	−0.157216	−0.0786079	+	0.996906i	$0.525048\pi$
$972$	0	0
$973$	13.8564	0.444216
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.6985i	0.950139i	0.879948	+	0.475069i	$0.157577\pi$
$977$	29.6985i	0.950139i	−0.879948	+	0.475069i	$0.842423\pi$
$978$	0	0
$979$	− 20.7846i	− 0.664279i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−50.9117	−1.62383	−0.811915	−	0.583775i	$-0.801575\pi$
$983$	−50.9117	−1.62383	−0.811915	+	0.583775i	$0.801575\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	50.0000i	1.58830i	0.607720	+	0.794151i	$0.292084\pi$
$991$	50.0000i	1.58830i	−0.607720	+	0.794151i	$0.707916\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	53.8888	1.70839
$996$	0	0
$997$	20.7846	0.658255	0.329128	−	0.944285i	$-0.393245\pi$
$997$	20.7846	0.658255	0.329128	+	0.944285i	$0.393245\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.c.j.2303.4		8
3.2	odd	2	inner	2304.2.c.j.2303.8		8
4.3	odd	2	inner	2304.2.c.j.2303.2		8
8.3	odd	2	inner	2304.2.c.j.2303.5		8
8.5	even	2	inner	2304.2.c.j.2303.7		8
12.11	even	2	inner	2304.2.c.j.2303.6		8
16.3	odd	4		576.2.f.a.287.4	yes	8
16.5	even	4		576.2.f.a.287.5	yes	8
16.11	odd	4		576.2.f.a.287.7	yes	8
16.13	even	4		576.2.f.a.287.2	yes	8
24.5	odd	2	inner	2304.2.c.j.2303.3		8
24.11	even	2	inner	2304.2.c.j.2303.1		8
48.5	odd	4		576.2.f.a.287.1	✓	8
48.11	even	4		576.2.f.a.287.3	yes	8
48.29	odd	4		576.2.f.a.287.6	yes	8
48.35	even	4		576.2.f.a.287.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.f.a.287.1	✓	8	48.5	odd	4
576.2.f.a.287.2	yes	8	16.13	even	4
576.2.f.a.287.3	yes	8	48.11	even	4
576.2.f.a.287.4	yes	8	16.3	odd	4
576.2.f.a.287.5	yes	8	16.5	even	4
576.2.f.a.287.6	yes	8	48.29	odd	4
576.2.f.a.287.7	yes	8	16.11	odd	4
576.2.f.a.287.8	yes	8	48.35	even	4
2304.2.c.j.2303.1		8	24.11	even	2	inner
2304.2.c.j.2303.2		8	4.3	odd	2	inner
2304.2.c.j.2303.3		8	24.5	odd	2	inner
2304.2.c.j.2303.4		8	1.1	even	1	trivial
2304.2.c.j.2303.5		8	8.3	odd	2	inner
2304.2.c.j.2303.6		8	12.11	even	2	inner
2304.2.c.j.2303.7		8	8.5	even	2	inner
2304.2.c.j.2303.8		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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.44949i q^{5} +2.00000i q^{7} +O(q^{10})\)	\(q-2.44949i q^{5} +2.00000i q^{7} -4.89898 q^{11} +3.46410 q^{13} +4.24264i q^{17} -6.92820i q^{19} -8.48528 q^{23} -1.00000 q^{25} +7.34847i q^{29} +2.00000i q^{31} +4.89898 q^{35} -6.92820 q^{37} -4.24264i q^{41} -8.48528 q^{47} +3.00000 q^{49} -2.44949i q^{53} +12.0000i q^{55} -6.92820 q^{61} -8.48528i q^{65} +6.92820i q^{67} -8.48528 q^{71} -4.00000 q^{73} -9.79796i q^{77} +14.0000i q^{79} -4.89898 q^{83} +10.3923 q^{85} +4.24264i q^{89} +6.92820i q^{91} -16.9706 q^{95} -16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{25} + 24 q^{49} - 32 q^{73} - 128 q^{97}+O(q^{100}) \) 8 * q - 8 * q^25 + 24 * q^49 - 32 * q^73 - 128 * q^97

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