LMFDB - Embedded newform 2304.2.c.g.2303.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2304,2,Mod(2303,2304)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,0,0,0,0,0,0,0,8,0,-4,0,0,0,0,0,0,0,0,0,-8,0,6,0,0,0,0, 0,0,0,0,0,-8,0,-16,0,0,0,0,0,0,0,0,0,24,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)

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

Self dual:	no
Analytic conductor:	$18.3975326257$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2303.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	2304.2303
Dual form			2304.2.c.g.2303.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{5} -2.82843i q^{7} +4.00000 q^{11} -2.00000 q^{13} -1.41421i q^{17} -5.65685i q^{19} -4.00000 q^{23} +3.00000 q^{25} -7.07107i q^{29} +8.48528i q^{31} -4.00000 q^{35} -8.00000 q^{37} -4.24264i q^{41} +11.3137i q^{43} +12.0000 q^{47} -1.00000 q^{49} -12.7279i q^{53} -5.65685i q^{55} -8.00000 q^{61} +2.82843i q^{65} -5.65685i q^{67} -4.00000 q^{71} -8.00000 q^{73} -11.3137i q^{77} +2.82843i q^{79} -12.0000 q^{83} -2.00000 q^{85} +15.5563i q^{89} +5.65685i q^{91} -8.00000 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{11} - 4 q^{13} - 8 q^{23} + 6 q^{25} - 8 q^{35} - 16 q^{37} + 24 q^{47} - 2 q^{49} - 16 q^{61} - 8 q^{71} - 16 q^{73} - 24 q^{83} - 4 q^{85} - 16 q^{95}+O(q^{100}) $ 2 * q + 8 * q^11 - 4 * q^13 - 8 * q^23 + 6 * q^25 - 8 * q^35 - 16 * q^37 + 24 * q^47 - 2 * q^49 - 16 * q^61 - 8 * q^71 - 16 * q^73 - 24 * q^83 - 4 * q^85 - 16 * q^95

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
$5$	− 1.41421i	− 0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.41421i	− 0.342997i	−0.985184	−	0.171499i	$-0.945139\pi$
$17$	− 1.41421i	− 0.342997i	0.985184	−	0.171499i	$-0.0548609\pi$
$18$	0	0
$19$	− 5.65685i	− 1.29777i	−0.760886	−	0.648886i	$-0.775235\pi$
$19$	− 5.65685i	− 1.29777i	0.760886	−	0.648886i	$-0.224765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 7.07107i	− 1.31306i	−0.754298	−	0.656532i	$-0.772023\pi$
$29$	− 7.07107i	− 1.31306i	0.754298	−	0.656532i	$-0.227977\pi$
$30$	0	0
$31$	8.48528i	1.52400i	0.647576	+	0.762001i	$0.275783\pi$
$31$	8.48528i	1.52400i	−0.647576	+	0.762001i	$0.724217\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	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
$43$	11.3137i	1.72532i	0.505781	+	0.862662i	$0.331205\pi$
$43$	11.3137i	1.72532i	−0.505781	+	0.862662i	$0.668795\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 12.7279i	− 1.74831i	−0.485643	−	0.874157i	$-0.661414\pi$
$53$	− 12.7279i	− 1.74831i	0.485643	−	0.874157i	$-0.338586\pi$
$54$	0	0
$55$	− 5.65685i	− 0.762770i
$56$	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
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.82843i	0.350823i
$66$	0	0
$67$	− 5.65685i	− 0.691095i	−0.938401	−	0.345547i	$-0.887693\pi$
$67$	− 5.65685i	− 0.691095i	0.938401	−	0.345547i	$-0.112307\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 11.3137i	− 1.28932i
$78$	0	0
$79$	2.82843i	0.318223i	0.987261	+	0.159111i	$0.0508629\pi$
$79$	2.82843i	0.318223i	−0.987261	+	0.159111i	$0.949137\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.5563i	1.64897i	0.565884	+	0.824485i	$0.308535\pi$
$89$	15.5563i	1.64897i	−0.565884	+	0.824485i	$0.691465\pi$
$90$	0	0
$91$	5.65685i	0.592999i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 9.89949i	− 0.985037i	−0.870302	−	0.492518i	$-0.836076\pi$
$101$	− 9.89949i	− 0.985037i	0.870302	−	0.492518i	$-0.163924\pi$
$102$	0	0
$103$	− 8.48528i	− 0.836080i	−0.908429	−	0.418040i	$-0.862717\pi$
$103$	− 8.48528i	− 0.836080i	0.908429	−	0.418040i	$-0.137283\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.41421i	− 0.133038i	−0.997785	−	0.0665190i	$-0.978811\pi$
$113$	− 1.41421i	− 0.133038i	0.997785	−	0.0665190i	$-0.0211893\pi$
$114$	0	0
$115$	5.65685i	0.527504i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.3137i	− 1.01193i
$126$	0	0
$127$	− 2.82843i	− 0.250982i	−0.992095	−	0.125491i	$-0.959949\pi$
$127$	− 2.82843i	− 0.250982i	0.992095	−	0.125491i	$-0.0400507\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−16.0000	−1.38738
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.3848i	− 1.57072i	−0.619041	−	0.785359i	$-0.712479\pi$
$137$	− 18.3848i	− 1.57072i	0.619041	−	0.785359i	$-0.287521\pi$
$138$	0	0
$139$	− 16.9706i	− 1.43942i	−0.694273	−	0.719712i	$-0.744274\pi$
$139$	− 16.9706i	− 1.43942i	0.694273	−	0.719712i	$-0.255726\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−10.0000	−0.830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 9.89949i	− 0.810998i	−0.914095	−	0.405499i	$-0.867098\pi$
$149$	− 9.89949i	− 0.810998i	0.914095	−	0.405499i	$-0.132902\pi$
$150$	0	0
$151$	8.48528i	0.690522i	0.938507	+	0.345261i	$0.112210\pi$
$151$	8.48528i	0.690522i	−0.938507	+	0.345261i	$0.887790\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.0000	0.963863
$156$	0	0
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	11.3137i	0.891645i
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 7.07107i	− 0.537603i	−0.963196	−	0.268802i	$-0.913372\pi$
$173$	− 7.07107i	− 0.537603i	0.963196	−	0.268802i	$-0.0866276\pi$
$174$	0	0
$175$	− 8.48528i	− 0.641427i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.3137i	0.831800i
$186$	0	0
$187$	− 5.65685i	− 0.413670i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.41421i	0.100759i	0.998730	+	0.0503793i	$0.0160430\pi$
$197$	1.41421i	0.100759i	−0.998730	+	0.0503793i	$0.983957\pi$
$198$	0	0
$199$	25.4558i	1.80452i	0.431196	+	0.902258i	$0.358092\pi$
$199$	25.4558i	1.80452i	−0.431196	+	0.902258i	$0.641908\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−20.0000	−1.40372
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 22.6274i	− 1.56517i
$210$	0	0
$211$	5.65685i	0.389434i	0.980859	+	0.194717i	$0.0623788\pi$
$211$	5.65685i	0.389434i	−0.980859	+	0.194717i	$0.937621\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	24.0000	1.62923
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.82843i	0.190261i
$222$	0	0
$223$	− 14.1421i	− 0.947027i	−0.880786	−	0.473514i	$-0.842985\pi$
$223$	− 14.1421i	− 0.947027i	0.880786	−	0.473514i	$-0.157015\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 15.5563i	− 1.01913i	−0.860432	−	0.509565i	$-0.829806\pi$
$233$	− 15.5563i	− 1.01913i	0.860432	−	0.509565i	$-0.170194\pi$
$234$	0	0
$235$	− 16.9706i	− 1.10704i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.41421i	0.0903508i
$246$	0	0
$247$	11.3137i	0.719874i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	−16.0000	−1.00591
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.41421i	0.0882162i	0.999027	+	0.0441081i	$0.0140446\pi$
$257$	1.41421i	0.0882162i	−0.999027	+	0.0441081i	$0.985955\pi$
$258$	0	0
$259$	22.6274i	1.40600i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−18.0000	−1.10573
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.07107i	0.431131i	0.976489	+	0.215565i	$0.0691594\pi$
$269$	7.07107i	0.431131i	−0.976489	+	0.215565i	$0.930841\pi$
$270$	0	0
$271$	− 2.82843i	− 0.171815i	−0.996303	−	0.0859074i	$-0.972621\pi$
$271$	− 2.82843i	− 0.171815i	0.996303	−	0.0859074i	$-0.0273789\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.8701i	1.60293i	0.598040	+	0.801467i	$0.295947\pi$
$281$	26.8701i	1.60293i	−0.598040	+	0.801467i	$0.704053\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.41421i	− 0.0826192i	−0.999146	−	0.0413096i	$-0.986847\pi$
$293$	− 1.41421i	− 0.0826192i	0.999146	−	0.0413096i	$-0.0131530\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	32.0000	1.84445
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.3137i	0.647821i
$306$	0	0
$307$	16.9706i	0.968561i	0.874913	+	0.484281i	$0.160919\pi$
$307$	16.9706i	0.968561i	−0.874913	+	0.484281i	$0.839081\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.3848i	1.03259i	0.856410	+	0.516296i	$0.172690\pi$
$317$	18.3848i	1.03259i	−0.856410	+	0.516296i	$0.827310\pi$
$318$	0	0
$319$	− 28.2843i	− 1.58362i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	−6.00000	−0.332820
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 33.9411i	− 1.87123i
$330$	0	0
$331$	5.65685i	0.310929i	0.987841	+	0.155464i	$0.0496874\pi$
$331$	5.65685i	0.310929i	−0.987841	+	0.155464i	$0.950313\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−24.0000	−1.30736	−0.653682	−	0.756770i	$-0.726776\pi$
$337$	−24.0000	−1.30736	−0.653682	+	0.756770i	$0.726776\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.9411i	1.83801i
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	0	0
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.89949i	0.526897i	0.964673	+	0.263448i	$0.0848599\pi$
$353$	9.89949i	0.526897i	−0.964673	+	0.263448i	$0.915140\pi$
$354$	0	0
$355$	5.65685i	0.300235i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.3137i	0.592187i
$366$	0	0
$367$	8.48528i	0.442928i	0.975169	+	0.221464i	$0.0710835\pi$
$367$	8.48528i	0.442928i	−0.975169	+	0.221464i	$0.928916\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−36.0000	−1.86903
$372$	0	0
$373$	24.0000	1.24267	0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000	1.24267	0.621336	+	0.783544i	$0.286590\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.1421i	0.728357i
$378$	0	0
$379$	− 16.9706i	− 0.871719i	−0.900015	−	0.435860i	$-0.856444\pi$
$379$	− 16.9706i	− 0.871719i	0.900015	−	0.435860i	$-0.143556\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.7279i	0.645331i	0.946513	+	0.322666i	$0.104579\pi$
$389$	12.7279i	0.645331i	−0.946513	+	0.322666i	$0.895421\pi$
$390$	0	0
$391$	5.65685i	0.286079i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 24.0416i	− 1.20058i	−0.799782	−	0.600291i	$-0.795051\pi$
$401$	− 24.0416i	− 1.20058i	0.799782	−	0.600291i	$-0.204949\pi$
$402$	0	0
$403$	− 16.9706i	− 0.845364i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−32.0000	−1.58618
$408$	0	0
$409$	16.0000	0.791149	0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000	0.791149	0.395575	+	0.918434i	$0.370545\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	16.9706i	0.833052i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 4.24264i	− 0.205798i
$426$	0	0
$427$	22.6274i	1.09502i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	22.6274i	1.08242i
$438$	0	0
$439$	8.48528i	0.404980i	0.979284	+	0.202490i	$0.0649034\pi$
$439$	8.48528i	0.404980i	−0.979284	+	0.202490i	$0.935097\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	22.0000	1.04290
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.7279i	0.600668i	0.953834	+	0.300334i	$0.0970981\pi$
$449$	12.7279i	0.600668i	−0.953834	+	0.300334i	$0.902902\pi$
$450$	0	0
$451$	− 16.9706i	− 0.799113i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.6985i	1.38320i	0.722282	+	0.691598i	$0.243093\pi$
$461$	29.6985i	1.38320i	−0.722282	+	0.691598i	$0.756907\pi$
$462$	0	0
$463$	25.4558i	1.18303i	0.806293	+	0.591517i	$0.201471\pi$
$463$	25.4558i	1.18303i	−0.806293	+	0.591517i	$0.798529\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	0	0
$473$	45.2548i	2.08082i
$474$	0	0
$475$	− 16.9706i	− 0.778663i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8.48528i	− 0.384505i	−0.981346	−	0.192252i	$-0.938421\pi$
$487$	− 8.48528i	− 0.384505i	0.981346	−	0.192252i	$-0.0615792\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	−10.0000	−0.450377
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.3137i	0.507489i
$498$	0	0
$499$	− 22.6274i	− 1.01294i	−0.862257	−	0.506471i	$-0.830950\pi$
$499$	− 22.6274i	− 1.01294i	0.862257	−	0.506471i	$-0.169050\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	0	0
$508$	0	0
$509$	38.1838i	1.69247i	0.532813	+	0.846233i	$0.321135\pi$
$509$	38.1838i	1.69247i	−0.532813	+	0.846233i	$0.678865\pi$
$510$	0	0
$511$	22.6274i	1.00098i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.0000	−0.528783
$516$	0	0
$517$	48.0000	2.11104
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 29.6985i	− 1.30111i	−0.759457	−	0.650557i	$-0.774535\pi$
$521$	− 29.6985i	− 1.30111i	0.759457	−	0.650557i	$-0.225465\pi$
$522$	0	0
$523$	39.5980i	1.73150i	0.500478	+	0.865749i	$0.333158\pi$
$523$	39.5980i	1.73150i	−0.500478	+	0.865749i	$0.666842\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.48528i	0.367538i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.00000	−0.172292
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	19.7990i	0.848096i
$546$	0	0
$547$	− 33.9411i	− 1.45122i	−0.688107	−	0.725609i	$-0.741558\pi$
$547$	− 33.9411i	− 1.45122i	0.688107	−	0.725609i	$-0.258442\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−40.0000	−1.70406
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.3848i	0.778988i	0.921029	+	0.389494i	$0.127350\pi$
$557$	18.3848i	0.778988i	−0.921029	+	0.389494i	$0.872650\pi$
$558$	0	0
$559$	− 22.6274i	− 0.957038i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 7.07107i	− 0.296435i	−0.988955	−	0.148217i	$-0.952646\pi$
$569$	− 7.07107i	− 0.296435i	0.988955	−	0.148217i	$-0.0473535\pi$
$570$	0	0
$571$	33.9411i	1.42039i	0.704004	+	0.710196i	$0.251394\pi$
$571$	33.9411i	1.42039i	−0.704004	+	0.710196i	$0.748606\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	33.9411i	1.40812i
$582$	0	0
$583$	− 50.9117i	− 2.10855i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.0000	1.32078	0.660391	−	0.750922i	$-0.270391\pi$
$587$	32.0000	1.32078	0.660391	+	0.750922i	$0.270391\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 43.8406i	− 1.80032i	−0.435561	−	0.900159i	$-0.643450\pi$
$593$	− 43.8406i	− 1.80032i	0.435561	−	0.900159i	$-0.356550\pi$
$594$	0	0
$595$	5.65685i	0.231908i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	46.0000	1.87638	0.938190	−	0.346122i	$-0.112502\pi$
$601$	46.0000	1.87638	0.938190	+	0.346122i	$0.112502\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 7.07107i	− 0.287480i
$606$	0	0
$607$	19.7990i	0.803616i	0.915724	+	0.401808i	$0.131618\pi$
$607$	19.7990i	0.803616i	−0.915724	+	0.401808i	$0.868382\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−24.0000	−0.970936
$612$	0	0
$613$	−24.0000	−0.969351	−0.484675	−	0.874694i	$-0.661062\pi$
$613$	−24.0000	−0.969351	−0.484675	+	0.874694i	$0.661062\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7.07107i	− 0.284670i	−0.989819	−	0.142335i	$-0.954539\pi$
$617$	− 7.07107i	− 0.284670i	0.989819	−	0.142335i	$-0.0454611\pi$
$618$	0	0
$619$	− 33.9411i	− 1.36421i	−0.731255	−	0.682105i	$-0.761065\pi$
$619$	− 33.9411i	− 1.36421i	0.731255	−	0.682105i	$-0.238935\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	44.0000	1.76282
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.3137i	0.451107i
$630$	0	0
$631$	− 42.4264i	− 1.68897i	−0.535580	−	0.844484i	$-0.679907\pi$
$631$	− 42.4264i	− 1.68897i	0.535580	−	0.844484i	$-0.320093\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.41421i	− 0.0558581i	−0.999610	−	0.0279290i	$-0.991109\pi$
$641$	− 1.41421i	− 0.0558581i	0.999610	−	0.0279290i	$-0.00889125\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 29.6985i	− 1.16219i	−0.813835	−	0.581096i	$-0.802624\pi$
$653$	− 29.6985i	− 1.16219i	0.813835	−	0.581096i	$-0.197376\pi$
$654$	0	0
$655$	− 11.3137i	− 0.442063i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	22.6274i	0.877454i
$666$	0	0
$667$	28.2843i	1.09517i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−32.0000	−1.23535
$672$	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
$677$	21.2132i	0.815290i	0.913141	+	0.407645i	$0.133650\pi$
$677$	21.2132i	0.815290i	−0.913141	+	0.407645i	$0.866350\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−26.0000	−0.993409
$686$	0	0
$687$	0	0
$688$	0	0
$689$	25.4558i	0.969790i
$690$	0	0
$691$	− 22.6274i	− 0.860788i	−0.902641	−	0.430394i	$-0.858375\pi$
$691$	− 22.6274i	− 0.860788i	0.902641	−	0.430394i	$-0.141625\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24.0000	−0.910372
$696$	0	0
$697$	−6.00000	−0.227266
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.8701i	1.01487i	0.861691	+	0.507434i	$0.169406\pi$
$701$	26.8701i	1.01487i	−0.861691	+	0.507434i	$0.830594\pi$
$702$	0	0
$703$	45.2548i	1.70682i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−28.0000	−1.05305
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 33.9411i	− 1.27111i
$714$	0	0
$715$	11.3137i	0.423109i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 21.2132i	− 0.787839i
$726$	0	0
$727$	14.1421i	0.524503i	0.965000	+	0.262251i	$0.0844650\pi$
$727$	14.1421i	0.524503i	−0.965000	+	0.262251i	$0.915535\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 22.6274i	− 0.833492i
$738$	0	0
$739$	33.9411i	1.24854i	0.781207	+	0.624272i	$0.214604\pi$
$739$	33.9411i	1.24854i	−0.781207	+	0.624272i	$0.785396\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	19.7990i	0.722475i	0.932474	+	0.361238i	$0.117646\pi$
$751$	19.7990i	0.722475i	−0.932474	+	0.361238i	$0.882354\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 26.8701i	− 0.974039i	−0.873391	−	0.487019i	$-0.838084\pi$
$761$	− 26.8701i	− 0.974039i	0.873391	−	0.487019i	$-0.161916\pi$
$762$	0	0
$763$	39.5980i	1.43354i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 35.3553i	− 1.27164i	−0.771836	−	0.635822i	$-0.780661\pi$
$773$	− 35.3553i	− 1.27164i	0.771836	−	0.635822i	$-0.219339\pi$
$774$	0	0
$775$	25.4558i	0.914401i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 11.3137i	− 0.403804i
$786$	0	0
$787$	28.2843i	1.00823i	0.863638	+	0.504113i	$0.168180\pi$
$787$	28.2843i	1.00823i	−0.863638	+	0.504113i	$0.831820\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	16.0000	0.568177
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.5563i	0.551034i	0.961296	+	0.275517i	$0.0888491\pi$
$797$	15.5563i	0.551034i	−0.961296	+	0.275517i	$0.911151\pi$
$798$	0	0
$799$	− 16.9706i	− 0.600375i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−32.0000	−1.12926
$804$	0	0
$805$	16.0000	0.563926
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.3848i	0.646374i	0.946335	+	0.323187i	$0.104754\pi$
$809$	18.3848i	0.646374i	−0.946335	+	0.323187i	$0.895246\pi$
$810$	0	0
$811$	− 16.9706i	− 0.595917i	−0.954579	−	0.297959i	$-0.903694\pi$
$811$	− 16.9706i	− 0.595917i	0.954579	−	0.297959i	$-0.0963057\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	64.0000	2.23908
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.41421i	0.0493564i	0.999695	+	0.0246782i	$0.00785611\pi$
$821$	1.41421i	0.0493564i	−0.999695	+	0.0246782i	$0.992144\pi$
$822$	0	0
$823$	8.48528i	0.295778i	0.989004	+	0.147889i	$0.0472479\pi$
$823$	8.48528i	0.295778i	−0.989004	+	0.147889i	$0.952752\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	0	0
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.41421i	0.0489996i
$834$	0	0
$835$	− 22.6274i	− 0.783054i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	52.0000	1.79524	0.897620	−	0.440771i	$-0.145295\pi$
$839$	52.0000	1.79524	0.897620	+	0.440771i	$0.145295\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.7279i	0.437854i
$846$	0	0
$847$	− 14.1421i	− 0.485930i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	32.0000	1.09695
$852$	0	0
$853$	−24.0000	−0.821744	−0.410872	−	0.911693i	$-0.634776\pi$
$853$	−24.0000	−0.821744	−0.410872	+	0.911693i	$0.634776\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	52.3259i	1.78742i	0.448646	+	0.893709i	$0.351906\pi$
$857$	52.3259i	1.78742i	−0.448646	+	0.893709i	$0.648094\pi$
$858$	0	0
$859$	− 33.9411i	− 1.15806i	−0.815308	−	0.579028i	$-0.803432\pi$
$859$	− 33.9411i	− 1.15806i	0.815308	−	0.579028i	$-0.196568\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.3137i	0.383791i
$870$	0	0
$871$	11.3137i	0.383350i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−32.0000	−1.08180
$876$	0	0
$877$	8.00000	0.270141	0.135070	−	0.990836i	$-0.456874\pi$
$877$	8.00000	0.270141	0.135070	+	0.990836i	$0.456874\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.7279i	0.428815i	0.976744	+	0.214407i	$0.0687820\pi$
$881$	12.7279i	0.428815i	−0.976744	+	0.214407i	$0.931218\pi$
$882$	0	0
$883$	− 33.9411i	− 1.14221i	−0.820877	−	0.571105i	$-0.806515\pi$
$883$	− 33.9411i	− 1.14221i	0.820877	−	0.571105i	$-0.193485\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 67.8823i	− 2.27159i
$894$	0	0
$895$	− 33.9411i	− 1.13453i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	60.0000	2.00111
$900$	0	0
$901$	−18.0000	−0.599667
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.48528i	0.282060i
$906$	0	0
$907$	− 33.9411i	− 1.12700i	−0.826117	−	0.563498i	$-0.809455\pi$
$907$	− 33.9411i	− 1.12700i	0.826117	−	0.563498i	$-0.190545\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	−48.0000	−1.58857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 22.6274i	− 0.747223i
$918$	0	0
$919$	31.1127i	1.02631i	0.858295	+	0.513157i	$0.171524\pi$
$919$	31.1127i	1.02631i	−0.858295	+	0.513157i	$0.828476\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.00000	0.263323
$924$	0	0
$925$	−24.0000	−0.789115
$926$	0	0
$927$	0	0
$928$	0	0
$929$	32.5269i	1.06717i	0.845745	+	0.533587i	$0.179156\pi$
$929$	32.5269i	1.06717i	−0.845745	+	0.533587i	$0.820844\pi$
$930$	0	0
$931$	5.65685i	0.185396i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 15.5563i	− 0.507122i	−0.967319	−	0.253561i	$-0.918398\pi$
$941$	− 15.5563i	− 0.507122i	0.967319	−	0.253561i	$-0.0816019\pi$
$942$	0	0
$943$	16.9706i	0.552638i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.0000	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$947$	16.0000	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 41.0122i	− 1.32852i	−0.747504	−	0.664258i	$-0.768748\pi$
$953$	− 41.0122i	− 1.32852i	0.747504	−	0.664258i	$-0.231252\pi$
$954$	0	0
$955$	− 22.6274i	− 0.732206i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−52.0000	−1.67917
$960$	0	0
$961$	−41.0000	−1.32258
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 14.1421i	− 0.455251i
$966$	0	0
$967$	− 31.1127i	− 1.00052i	−0.865876	−	0.500258i	$-0.833238\pi$
$967$	− 31.1127i	− 1.00052i	0.865876	−	0.500258i	$-0.166762\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	−48.0000	−1.53881
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 21.2132i	− 0.678671i	−0.940666	−	0.339335i	$-0.889798\pi$
$977$	− 21.2132i	− 0.678671i	0.940666	−	0.339335i	$-0.110202\pi$
$978$	0	0
$979$	62.2254i	1.98873i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 45.2548i	− 1.43902i
$990$	0	0
$991$	− 48.0833i	− 1.52742i	−0.645562	−	0.763708i	$-0.723377\pi$
$991$	− 48.0833i	− 1.52742i	0.645562	−	0.763708i	$-0.276623\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	36.0000	1.14128
$996$	0	0
$997$	−40.0000	−1.26681	−0.633406	−	0.773819i	$-0.718344\pi$
$997$	−40.0000	−1.26681	−0.633406	+	0.773819i	$0.718344\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.c.g.2303.1		2
3.2	odd	2		2304.2.c.a.2303.2		2
4.3	odd	2		2304.2.c.a.2303.1		2
8.3	odd	2		2304.2.c.h.2303.2		2
8.5	even	2		2304.2.c.b.2303.2		2
12.11	even	2	inner	2304.2.c.g.2303.2		2
16.3	odd	4		1152.2.f.a.575.1	✓	4
16.5	even	4		1152.2.f.d.575.4	yes	4
16.11	odd	4		1152.2.f.a.575.3	yes	4
16.13	even	4		1152.2.f.d.575.2	yes	4
24.5	odd	2		2304.2.c.h.2303.1		2
24.11	even	2		2304.2.c.b.2303.1		2
48.5	odd	4		1152.2.f.a.575.2	yes	4
48.11	even	4		1152.2.f.d.575.1	yes	4
48.29	odd	4		1152.2.f.a.575.4	yes	4
48.35	even	4		1152.2.f.d.575.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.f.a.575.1	✓	4	16.3	odd	4
1152.2.f.a.575.2	yes	4	48.5	odd	4
1152.2.f.a.575.3	yes	4	16.11	odd	4
1152.2.f.a.575.4	yes	4	48.29	odd	4
1152.2.f.d.575.1	yes	4	48.11	even	4
1152.2.f.d.575.2	yes	4	16.13	even	4
1152.2.f.d.575.3	yes	4	48.35	even	4
1152.2.f.d.575.4	yes	4	16.5	even	4
2304.2.c.a.2303.1		2	4.3	odd	2
2304.2.c.a.2303.2		2	3.2	odd	2
2304.2.c.b.2303.1		2	24.11	even	2
2304.2.c.b.2303.2		2	8.5	even	2
2304.2.c.g.2303.1		2	1.1	even	1	trivial
2304.2.c.g.2303.2		2	12.11	even	2	inner
2304.2.c.h.2303.1		2	24.5	odd	2
2304.2.c.h.2303.2		2	8.3	odd	2

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.41421i q^{5} -2.82843i q^{7} +4.00000 q^{11} -2.00000 q^{13} -1.41421i q^{17} -5.65685i q^{19} -4.00000 q^{23} +3.00000 q^{25} -7.07107i q^{29} +8.48528i q^{31} -4.00000 q^{35} -8.00000 q^{37} -4.24264i q^{41} +11.3137i q^{43} +12.0000 q^{47} -1.00000 q^{49} -12.7279i q^{53} -5.65685i q^{55} -8.00000 q^{61} +2.82843i q^{65} -5.65685i q^{67} -4.00000 q^{71} -8.00000 q^{73} -11.3137i q^{77} +2.82843i q^{79} -12.0000 q^{83} -2.00000 q^{85} +15.5563i q^{89} +5.65685i q^{91} -8.00000 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{11} - 4 q^{13} - 8 q^{23} + 6 q^{25} - 8 q^{35} - 16 q^{37} + 24 q^{47} - 2 q^{49} - 16 q^{61} - 8 q^{71} - 16 q^{73} - 24 q^{83} - 4 q^{85} - 16 q^{95}+O(q^{100}) \) 2 * q + 8 * q^11 - 4 * q^13 - 8 * q^23 + 6 * q^25 - 8 * q^35 - 16 * q^37 + 24 * q^47 - 2 * q^49 - 16 * q^61 - 8 * q^71 - 16 * q^73 - 24 * q^83 - 4 * q^85 - 16 * q^95

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