LMFDB - Embedded newform 2160.2.h.g.431.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,2,Mod(431,2160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2160.431"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1731891456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 $ x^8 - 9x^6 + 65x^4 - 144*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.6
Root			$-1.35234 + 0.780776i$ of defining polynomial
Character	$\chi$	$=$	2160.431
Dual form			2160.2.h.g.431.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} -2.70469i q^{7} -0.972638 q^{11} +1.00000 q^{13} +7.68466i q^{17} +2.70469i q^{19} -0.972638 q^{23} -1.00000 q^{25} -1.68466i q^{29} +0.972638i q^{31} +2.70469 q^{35} -2.68466 q^{37} +6.00000i q^{41} +0.972638i q^{43} +11.3649 q^{47} -0.315342 q^{49} +6.00000i q^{53} -0.972638i q^{55} +1.94528 q^{59} +6.68466 q^{61} +1.00000i q^{65} -11.5782i q^{67} +12.3376 q^{71} +8.68466 q^{73} +2.63068i q^{77} +14.0696i q^{79} +10.3923 q^{83} -7.68466 q^{85} +12.0000i q^{89} -2.70469i q^{91} -2.70469 q^{95} -6.68466 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{13} - 8 q^{25} + 28 q^{37} - 52 q^{49} + 4 q^{61} + 20 q^{73} - 12 q^{85} - 4 q^{97}+O(q^{100}) $ 8 * q + 8 * q^13 - 8 * q^25 + 28 * q^37 - 52 * q^49 + 4 * q^61 + 20 * q^73 - 12 * q^85 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	− 2.70469i	− 1.02228i	−0.859499	−	0.511138i	$-0.829224\pi$
$7$	− 2.70469i	− 1.02228i	0.859499	−	0.511138i	$-0.170776\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.972638	−0.293261	−0.146631	−	0.989191i	$-0.546843\pi$
$11$	−0.972638	−0.293261	−0.146631	+	0.989191i	$0.546843\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.68466i	1.86380i	0.362711	+	0.931902i	$0.381851\pi$
$17$	7.68466i	1.86380i	−0.362711	+	0.931902i	$0.618149\pi$
$18$	0	0
$19$	2.70469i	0.620498i	0.950655	+	0.310249i	$0.100412\pi$
$19$	2.70469i	0.620498i	−0.950655	+	0.310249i	$0.899588\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.972638	−0.202809	−0.101405	−	0.994845i	$-0.532334\pi$
$23$	−0.972638	−0.202809	−0.101405	+	0.994845i	$0.532334\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 1.68466i	− 0.312833i	−0.987691	−	0.156417i	$-0.950006\pi$
$29$	− 1.68466i	− 0.312833i	0.987691	−	0.156417i	$-0.0499942\pi$
$30$	0	0
$31$	0.972638i	0.174691i	0.996178	+	0.0873455i	$0.0278384\pi$
$31$	0.972638i	0.174691i	−0.996178	+	0.0873455i	$0.972162\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.70469	0.457176
$36$	0	0
$37$	−2.68466	−0.441355	−0.220678	−	0.975347i	$-0.570827\pi$
$37$	−2.68466	−0.441355	−0.220678	+	0.975347i	$0.570827\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	0	0
$43$	0.972638i	0.148326i	0.997246	+	0.0741630i	$0.0236285\pi$
$43$	0.972638i	0.148326i	−0.997246	+	0.0741630i	$0.976372\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.3649	1.65775	0.828874	−	0.559436i	$-0.188982\pi$
$47$	11.3649	1.65775	0.828874	+	0.559436i	$0.188982\pi$
$48$	0	0
$49$	−0.315342	−0.0450488
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	− 0.972638i	− 0.131150i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.94528	0.253253	0.126627	−	0.991950i	$-0.459585\pi$
$59$	1.94528	0.253253	0.126627	+	0.991950i	$0.459585\pi$
$60$	0	0
$61$	6.68466	0.855883	0.427941	−	0.903806i	$-0.359239\pi$
$61$	6.68466	0.855883	0.427941	+	0.903806i	$0.359239\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000i	0.124035i
$66$	0	0
$67$	− 11.5782i	− 1.41450i	−0.706964	−	0.707249i	$-0.749936\pi$
$67$	− 11.5782i	− 1.41450i	0.706964	−	0.707249i	$-0.250064\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.3376	1.46420	0.732101	−	0.681196i	$-0.238540\pi$
$71$	12.3376	1.46420	0.732101	+	0.681196i	$0.238540\pi$
$72$	0	0
$73$	8.68466	1.01646	0.508231	−	0.861221i	$-0.330300\pi$
$73$	8.68466	1.01646	0.508231	+	0.861221i	$0.330300\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.63068i	0.299794i
$78$	0	0
$79$	14.0696i	1.58296i	0.611197	+	0.791479i	$0.290688\pi$
$79$	14.0696i	1.58296i	−0.611197	+	0.791479i	$0.709312\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.3923	1.14070	0.570352	−	0.821401i	$-0.306807\pi$
$83$	10.3923	1.14070	0.570352	+	0.821401i	$0.306807\pi$
$84$	0	0
$85$	−7.68466	−0.833518
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.0000i	1.27200i	0.771690	+	0.635999i	$0.219412\pi$
$89$	12.0000i	1.27200i	−0.771690	+	0.635999i	$0.780588\pi$
$90$	0	0
$91$	− 2.70469i	− 0.283528i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.70469	−0.277495
$96$	0	0
$97$	−6.68466	−0.678724	−0.339362	−	0.940656i	$-0.610211\pi$
$97$	−6.68466	−0.678724	−0.339362	+	0.940656i	$0.610211\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.68466i	0.764652i	0.924027	+	0.382326i	$0.124877\pi$
$101$	7.68466i	0.764652i	−0.924027	+	0.382326i	$0.875123\pi$
$102$	0	0
$103$	− 0.759413i	− 0.0748272i	−0.999300	−	0.0374136i	$-0.988088\pi$
$103$	− 0.759413i	− 0.0748272i	0.999300	−	0.0374136i	$-0.0119119\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.3376	−1.19272	−0.596359	−	0.802717i	$-0.703387\pi$
$107$	−12.3376	−1.19272	−0.596359	+	0.802717i	$0.703387\pi$
$108$	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
$113$	13.6847i	1.28734i	0.765301	+	0.643672i	$0.222590\pi$
$113$	13.6847i	1.28734i	−0.765301	+	0.643672i	$0.777410\pi$
$114$	0	0
$115$	− 0.972638i	− 0.0906990i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	20.7846	1.90532
$120$	0	0
$121$	−10.0540	−0.913998
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	10.3923i	0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$	10.3923i	0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.3649	−0.992960	−0.496480	−	0.868048i	$-0.665374\pi$
$131$	−11.3649	−0.992960	−0.496480	+	0.868048i	$0.665374\pi$
$132$	0	0
$133$	7.31534	0.634321
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	9.63289i	0.817051i	0.912747	+	0.408526i	$0.133957\pi$
$139$	9.63289i	0.817051i	−0.912747	+	0.408526i	$0.866043\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.972638	−0.0813361
$144$	0	0
$145$	1.68466	0.139903
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.0540i	1.88866i	0.329007	+	0.944328i	$0.393286\pi$
$149$	23.0540i	1.88866i	−0.329007	+	0.944328i	$0.606714\pi$
$150$	0	0
$151$	− 10.6055i	− 0.863066i	−0.902097	−	0.431533i	$-0.857973\pi$
$151$	− 10.6055i	− 0.863066i	0.902097	−	0.431533i	$-0.142027\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.972638	−0.0781242
$156$	0	0
$157$	13.0540	1.04182	0.520910	−	0.853611i	$-0.325593\pi$
$157$	13.0540	1.04182	0.520910	+	0.853611i	$0.325593\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.63068i	0.207327i
$162$	0	0
$163$	− 12.1244i	− 0.949653i	−0.880079	−	0.474826i	$-0.842511\pi$
$163$	− 12.1244i	− 0.949653i	0.880079	−	0.474826i	$-0.157489\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.94528	−0.150530	−0.0752650	−	0.997164i	$-0.523980\pi$
$167$	−1.94528	−0.150530	−0.0752650	+	0.997164i	$0.523980\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	2.70469i	0.204455i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.2829	−1.06755	−0.533775	−	0.845626i	$-0.679227\pi$
$179$	−14.2829	−1.06755	−0.533775	+	0.845626i	$0.679227\pi$
$180$	0	0
$181$	18.0540	1.34194	0.670971	−	0.741484i	$-0.265878\pi$
$181$	18.0540	1.34194	0.670971	+	0.741484i	$0.265878\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.68466i	− 0.197380i
$186$	0	0
$187$	− 7.47439i	− 0.546582i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.7846	1.50392	0.751961	−	0.659208i	$-0.229108\pi$
$191$	20.7846	1.50392	0.751961	+	0.659208i	$0.229108\pi$
$192$	0	0
$193$	24.6847	1.77684	0.888420	−	0.459031i	$-0.151803\pi$
$193$	24.6847	1.77684	0.888420	+	0.459031i	$0.151803\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	− 1.73205i	− 0.122782i	−0.998114	−	0.0613909i	$-0.980446\pi$
$199$	− 1.73205i	− 0.122782i	0.998114	−	0.0613909i	$-0.0195536\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.55648	−0.319802
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 2.63068i	− 0.181968i
$210$	0	0
$211$	11.5782i	0.797074i	0.917152	+	0.398537i	$0.130482\pi$
$211$	11.5782i	0.797074i	−0.917152	+	0.398537i	$0.869518\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.972638	−0.0663334
$216$	0	0
$217$	2.63068	0.178582
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.68466i	0.516926i
$222$	0	0
$223$	− 12.3376i	− 0.826186i	−0.910689	−	0.413093i	$-0.864449\pi$
$223$	− 12.3376i	− 0.826186i	0.910689	−	0.413093i	$-0.135551\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.3376	0.818874	0.409437	−	0.912338i	$-0.365725\pi$
$227$	12.3376	0.818874	0.409437	+	0.912338i	$0.365725\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3.36932i	− 0.220731i	−0.993891	−	0.110366i	$-0.964798\pi$
$233$	− 3.36932i	− 0.220731i	0.993891	−	0.110366i	$-0.0352022\pi$
$234$	0	0
$235$	11.3649i	0.741367i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.3376	−0.798052	−0.399026	−	0.916940i	$-0.630652\pi$
$239$	−12.3376	−0.798052	−0.399026	+	0.916940i	$0.630652\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 0.315342i	− 0.0201464i
$246$	0	0
$247$	2.70469i	0.172095i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−21.7572	−1.37331	−0.686653	−	0.726986i	$-0.740921\pi$
$251$	−21.7572	−1.37331	−0.686653	+	0.726986i	$0.740921\pi$
$252$	0	0
$253$	0.946025	0.0594761
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.68466i	0.479356i	0.970852	+	0.239678i	$0.0770418\pi$
$257$	7.68466i	0.479356i	−0.970852	+	0.239678i	$0.922958\pi$
$258$	0	0
$259$	7.26117i	0.451187i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	22.7299	1.40158	0.700792	−	0.713365i	$-0.252830\pi$
$263$	22.7299	1.40158	0.700792	+	0.713365i	$0.252830\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 23.0540i	− 1.40563i	−0.711375	−	0.702813i	$-0.751927\pi$
$269$	− 23.0540i	− 1.40563i	0.711375	−	0.702813i	$-0.248073\pi$
$270$	0	0
$271$	− 20.0252i	− 1.21644i	−0.793767	−	0.608222i	$-0.791883\pi$
$271$	− 20.0252i	− 1.21644i	0.793767	−	0.608222i	$-0.208117\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.972638	0.0586523
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 27.3693i	− 1.63272i	−0.577546	−	0.816358i	$-0.695990\pi$
$281$	− 27.3693i	− 1.63272i	0.577546	−	0.816358i	$-0.304010\pi$
$282$	0	0
$283$	− 14.2829i	− 0.849028i	−0.905421	−	0.424514i	$-0.860445\pi$
$283$	− 14.2829i	− 0.849028i	0.905421	−	0.424514i	$-0.139555\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.2281	0.957916
$288$	0	0
$289$	−42.0540	−2.47376
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 9.36932i	− 0.547361i	−0.961821	−	0.273681i	$-0.911759\pi$
$293$	− 9.36932i	− 0.547361i	0.961821	−	0.273681i	$-0.0882411\pi$
$294$	0	0
$295$	1.94528i	0.113258i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.972638	−0.0562491
$300$	0	0
$301$	2.63068	0.151630
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.68466i	0.382762i
$306$	0	0
$307$	25.6478i	1.46380i	0.681414	+	0.731899i	$0.261366\pi$
$307$	25.6478i	1.46380i	−0.681414	+	0.731899i	$0.738634\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.94528	0.110306	0.0551532	−	0.998478i	$-0.482435\pi$
$311$	1.94528	0.110306	0.0551532	+	0.998478i	$0.482435\pi$
$312$	0	0
$313$	28.0540	1.58570	0.792852	−	0.609414i	$-0.208595\pi$
$313$	28.0540	1.58570	0.792852	+	0.609414i	$0.208595\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.36932i	0.189240i	0.995513	+	0.0946198i	$0.0301636\pi$
$317$	3.36932i	0.189240i	−0.995513	+	0.0946198i	$0.969836\pi$
$318$	0	0
$319$	1.63856i	0.0917419i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−20.7846	−1.15649
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 30.7386i	− 1.69468i
$330$	0	0
$331$	9.63289i	0.529472i	0.964321	+	0.264736i	$0.0852848\pi$
$331$	9.63289i	0.529472i	−0.964321	+	0.264736i	$0.914715\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.5782	0.632583
$336$	0	0
$337$	12.0540	0.656622	0.328311	−	0.944570i	$-0.393521\pi$
$337$	12.0540	0.656622	0.328311	+	0.944570i	$0.393521\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 0.946025i	− 0.0512301i
$342$	0	0
$343$	− 18.0799i	− 0.976224i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−31.1769	−1.67366	−0.836832	−	0.547459i	$-0.815595\pi$
$347$	−31.1769	−1.67366	−0.836832	+	0.547459i	$0.815595\pi$
$348$	0	0
$349$	−20.6847	−1.10722	−0.553612	−	0.832775i	$-0.686751\pi$
$349$	−20.6847	−1.10722	−0.553612	+	0.832775i	$0.686751\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 31.6847i	− 1.68640i	−0.537597	−	0.843202i	$-0.680668\pi$
$353$	− 31.6847i	− 1.68640i	0.537597	−	0.843202i	$-0.319332\pi$
$354$	0	0
$355$	12.3376i	0.654811i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.2829	0.753820	0.376910	−	0.926250i	$-0.376987\pi$
$359$	14.2829	0.753820	0.376910	+	0.926250i	$0.376987\pi$
$360$	0	0
$361$	11.6847	0.614982
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.68466i	0.454576i
$366$	0	0
$367$	1.18586i	0.0619016i	0.999521	+	0.0309508i	$0.00985351\pi$
$367$	1.18586i	0.0619016i	−0.999521	+	0.0309508i	$0.990146\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	16.2281	0.842523
$372$	0	0
$373$	1.63068	0.0844336	0.0422168	−	0.999108i	$-0.486558\pi$
$373$	1.63068	0.0844336	0.0422168	+	0.999108i	$0.486558\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.68466i	− 0.0867643i
$378$	0	0
$379$	− 21.9705i	− 1.12855i	−0.825588	−	0.564274i	$-0.809156\pi$
$379$	− 21.9705i	− 1.12855i	0.825588	−	0.564274i	$-0.190844\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.2555	−0.779519	−0.389760	−	0.920917i	$-0.627442\pi$
$383$	−15.2555	−0.779519	−0.389760	+	0.920917i	$0.627442\pi$
$384$	0	0
$385$	−2.63068	−0.134072
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 19.6847i	− 0.998052i	−0.866587	−	0.499026i	$-0.833691\pi$
$389$	− 19.6847i	− 0.998052i	0.866587	−	0.499026i	$-0.166309\pi$
$390$	0	0
$391$	− 7.47439i	− 0.377996i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14.0696	−0.707920
$396$	0	0
$397$	25.0540	1.25742	0.628711	−	0.777639i	$-0.283583\pi$
$397$	25.0540	1.25742	0.628711	+	0.777639i	$0.283583\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 6.00000i	− 0.299626i	−0.988714	−	0.149813i	$-0.952133\pi$
$401$	− 6.00000i	− 0.299626i	0.988714	−	0.149813i	$-0.0478671\pi$
$402$	0	0
$403$	0.972638i	0.0484505i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.61120	0.129432
$408$	0	0
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5.26137i	− 0.258895i
$414$	0	0
$415$	10.3923i	0.510138i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−34.0948	−1.66564	−0.832821	−	0.553543i	$-0.813276\pi$
$419$	−34.0948	−1.66564	−0.832821	+	0.553543i	$0.813276\pi$
$420$	0	0
$421$	14.6847	0.715686	0.357843	−	0.933782i	$-0.383512\pi$
$421$	14.6847	0.715686	0.357843	+	0.933782i	$0.383512\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 7.68466i	− 0.372761i
$426$	0	0
$427$	− 18.0799i	− 0.874949i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−38.9580	−1.87654	−0.938271	−	0.345901i	$-0.887573\pi$
$431$	−38.9580	−1.87654	−0.938271	+	0.345901i	$0.887573\pi$
$432$	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
$437$	− 2.63068i	− 0.125843i
$438$	0	0
$439$	35.0675i	1.67368i	0.547448	+	0.836839i	$0.315599\pi$
$439$	35.0675i	1.67368i	−0.547448	+	0.836839i	$0.684401\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.5110	−1.44962	−0.724810	−	0.688948i	$-0.758073\pi$
$443$	−30.5110	−1.44962	−0.724810	+	0.688948i	$0.758073\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.7386i	0.884331i	0.896934	+	0.442165i	$0.145790\pi$
$449$	18.7386i	0.884331i	−0.896934	+	0.442165i	$0.854210\pi$
$450$	0	0
$451$	− 5.83583i	− 0.274798i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.70469	0.126798
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 27.3693i	− 1.27472i	−0.770568	−	0.637358i	$-0.780027\pi$
$461$	− 27.3693i	− 1.27472i	0.770568	−	0.637358i	$-0.219973\pi$
$462$	0	0
$463$	− 0.759413i	− 0.0352929i	−0.999844	−	0.0176465i	$-0.994383\pi$
$463$	− 0.759413i	− 0.0352929i	0.999844	−	0.0176465i	$-0.00561733\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.94528	0.0900166	0.0450083	−	0.998987i	$-0.485669\pi$
$467$	1.94528	0.0900166	0.0450083	+	0.998987i	$0.485669\pi$
$468$	0	0
$469$	−31.3153	−1.44601
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 0.946025i	− 0.0434983i
$474$	0	0
$475$	− 2.70469i	− 0.124100i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.3923	−0.474837	−0.237418	−	0.971408i	$-0.576301\pi$
$479$	−10.3923	−0.474837	−0.237418	+	0.971408i	$0.576301\pi$
$480$	0	0
$481$	−2.68466	−0.122410
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 6.68466i	− 0.303535i
$486$	0	0
$487$	42.7551i	1.93742i	0.248200	+	0.968709i	$0.420161\pi$
$487$	42.7551i	1.93742i	−0.248200	+	0.968709i	$0.579839\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10.3923	−0.468998	−0.234499	−	0.972116i	$-0.575345\pi$
$491$	−10.3923	−0.468998	−0.234499	+	0.972116i	$0.575345\pi$
$492$	0	0
$493$	12.9460	0.583060
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 33.3693i	− 1.49682i
$498$	0	0
$499$	38.9580i	1.74400i	0.489505	+	0.872000i	$0.337177\pi$
$499$	38.9580i	1.74400i	−0.489505	+	0.872000i	$0.662823\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	23.7025	1.05684	0.528422	−	0.848982i	$-0.322784\pi$
$503$	23.7025	1.05684	0.528422	+	0.848982i	$0.322784\pi$
$504$	0	0
$505$	−7.68466	−0.341963
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.3153i	0.457219i	0.973518	+	0.228610i	$0.0734180\pi$
$509$	10.3153i	0.457219i	−0.973518	+	0.228610i	$0.926582\pi$
$510$	0	0
$511$	− 23.4893i	− 1.03911i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.759413	0.0334637
$516$	0	0
$517$	−11.0540	−0.486153
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.7386i	1.08382i	0.840437	+	0.541910i	$0.182298\pi$
$521$	24.7386i	1.08382i	−0.840437	+	0.541910i	$0.817702\pi$
$522$	0	0
$523$	− 20.9978i	− 0.918171i	−0.888392	−	0.459086i	$-0.848177\pi$
$523$	− 20.9978i	− 0.918171i	0.888392	−	0.459086i	$-0.151823\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.47439	−0.325590
$528$	0	0
$529$	−22.0540	−0.958868
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.00000i	0.259889i
$534$	0	0
$535$	− 12.3376i	− 0.533400i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.306713	0.0132111
$540$	0	0
$541$	−0.684658	−0.0294358	−0.0147179	−	0.999892i	$-0.504685\pi$
$541$	−0.684658	−0.0294358	−0.0147179	+	0.999892i	$0.504685\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 14.0000i	− 0.599694i
$546$	0	0
$547$	− 34.8542i	− 1.49026i	−0.666919	−	0.745130i	$-0.732387\pi$
$547$	− 34.8542i	− 1.49026i	0.666919	−	0.745130i	$-0.267613\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.55648	0.194112
$552$	0	0
$553$	38.0540	1.61822
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 39.3693i	− 1.66813i	−0.551665	−	0.834066i	$-0.686007\pi$
$557$	− 39.3693i	− 1.66813i	0.551665	−	0.834066i	$-0.313993\pi$
$558$	0	0
$559$	0.972638i	0.0411382i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.5657	−1.20390	−0.601951	−	0.798533i	$-0.705610\pi$
$563$	−28.5657	−1.20390	−0.601951	+	0.798533i	$0.705610\pi$
$564$	0	0
$565$	−13.6847	−0.575718
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 12.0000i	− 0.503066i	−0.967849	−	0.251533i	$-0.919065\pi$
$569$	− 12.0000i	− 0.503066i	0.967849	−	0.251533i	$-0.0809347\pi$
$570$	0	0
$571$	− 3.13114i	− 0.131034i	−0.997851	−	0.0655170i	$-0.979130\pi$
$571$	− 3.13114i	− 0.131034i	0.997851	−	0.0655170i	$-0.0208697\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.972638	0.0405618
$576$	0	0
$577$	36.6847	1.52720	0.763601	−	0.645688i	$-0.223429\pi$
$577$	36.6847	1.52720	0.763601	+	0.645688i	$0.223429\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 28.1080i	− 1.16611i
$582$	0	0
$583$	− 5.83583i	− 0.241695i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−31.1769	−1.28681	−0.643404	−	0.765526i	$-0.722479\pi$
$587$	−31.1769	−1.28681	−0.643404	+	0.765526i	$0.722479\pi$
$588$	0	0
$589$	−2.63068	−0.108395
$590$	0	0
$591$	0	0
$592$	0	0
$593$	13.6847i	0.561962i	0.959713	+	0.280981i	$0.0906597\pi$
$593$	13.6847i	0.561962i	−0.959713	+	0.280981i	$0.909340\pi$
$594$	0	0
$595$	20.7846i	0.852086i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	37.0127	1.51230	0.756150	−	0.654399i	$-0.227078\pi$
$599$	37.0127	1.51230	0.756150	+	0.654399i	$0.227078\pi$
$600$	0	0
$601$	2.94602	0.120171	0.0600854	−	0.998193i	$-0.480863\pi$
$601$	2.94602	0.120171	0.0600854	+	0.998193i	$0.480863\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 10.0540i	− 0.408752i
$606$	0	0
$607$	− 0.759413i	− 0.0308236i	−0.999881	−	0.0154118i	$-0.995094\pi$
$607$	− 0.759413i	− 0.0308236i	0.999881	−	0.0154118i	$-0.00490592\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11.3649	0.459776
$612$	0	0
$613$	20.3693	0.822709	0.411354	−	0.911475i	$-0.365056\pi$
$613$	20.3693	0.822709	0.411354	+	0.911475i	$0.365056\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 10.3153i	− 0.415280i	−0.978205	−	0.207640i	$-0.933422\pi$
$617$	− 10.3153i	− 0.415280i	0.978205	−	0.207640i	$-0.0665783\pi$
$618$	0	0
$619$	− 3.13114i	− 0.125851i	−0.998018	−	0.0629256i	$-0.979957\pi$
$619$	− 3.13114i	− 0.125851i	0.998018	−	0.0629256i	$-0.0200431\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	32.4563	1.30033
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 20.6307i	− 0.822599i
$630$	0	0
$631$	− 16.1346i	− 0.642310i	−0.947027	−	0.321155i	$-0.895929\pi$
$631$	− 16.1346i	− 0.642310i	0.947027	−	0.321155i	$-0.104071\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−10.3923	−0.412406
$636$	0	0
$637$	−0.315342	−0.0124943
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.73863i	0.266160i	0.991105	+	0.133080i	$0.0424867\pi$
$641$	6.73863i	0.266160i	−0.991105	+	0.133080i	$0.957513\pi$
$642$	0	0
$643$	− 17.8667i	− 0.704594i	−0.935888	−	0.352297i	$-0.885401\pi$
$643$	− 17.8667i	− 0.704594i	0.935888	−	0.352297i	$-0.114599\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.1734	−0.714470	−0.357235	−	0.934014i	$-0.616281\pi$
$647$	−18.1734	−0.714470	−0.357235	+	0.934014i	$0.616281\pi$
$648$	0	0
$649$	−1.89205	−0.0742694
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 3.36932i	− 0.131852i	−0.997825	−	0.0659258i	$-0.979000\pi$
$653$	− 3.36932i	− 0.131852i	0.997825	−	0.0659258i	$-0.0210001\pi$
$654$	0	0
$655$	− 11.3649i	− 0.444065i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26.6204	1.03699	0.518493	−	0.855082i	$-0.326493\pi$
$659$	26.6204	1.03699	0.518493	+	0.855082i	$0.326493\pi$
$660$	0	0
$661$	37.4233	1.45560	0.727799	−	0.685791i	$-0.240544\pi$
$661$	37.4233	1.45560	0.727799	+	0.685791i	$0.240544\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.31534i	0.283677i
$666$	0	0
$667$	1.63856i	0.0634454i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.50175	−0.250997
$672$	0	0
$673$	21.3153	0.821646	0.410823	−	0.911715i	$-0.365241\pi$
$673$	21.3153	0.821646	0.410823	+	0.911715i	$0.365241\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6.73863i	− 0.258987i	−0.991580	−	0.129493i	$-0.958665\pi$
$677$	− 6.73863i	− 0.258987i	0.991580	−	0.129493i	$-0.0413351\pi$
$678$	0	0
$679$	18.0799i	0.693844i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.50175	−0.248783	−0.124391	−	0.992233i	$-0.539698\pi$
$683$	−6.50175	−0.248783	−0.124391	+	0.992233i	$0.539698\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.00000i	0.228582i
$690$	0	0
$691$	37.0127i	1.40803i	0.710185	+	0.704016i	$0.248611\pi$
$691$	37.0127i	1.40803i	−0.710185	+	0.704016i	$0.751389\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.63289	−0.365396
$696$	0	0
$697$	−46.1080	−1.74646
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 17.0540i	− 0.644120i	−0.946719	−	0.322060i	$-0.895625\pi$
$701$	− 17.0540i	− 0.644120i	0.946719	−	0.322060i	$-0.104375\pi$
$702$	0	0
$703$	− 7.26117i	− 0.273860i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.7846	0.781686
$708$	0	0
$709$	21.4233	0.804569	0.402284	−	0.915515i	$-0.368216\pi$
$709$	21.4233	0.804569	0.402284	+	0.915515i	$0.368216\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 0.946025i	− 0.0354289i
$714$	0	0
$715$	− 0.972638i	− 0.0363746i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−18.1734	−0.677754	−0.338877	−	0.940831i	$-0.610047\pi$
$719$	−18.1734	−0.677754	−0.338877	+	0.940831i	$0.610047\pi$
$720$	0	0
$721$	−2.05398	−0.0764940
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.68466i	0.0625666i
$726$	0	0
$727$	− 38.9580i	− 1.44487i	−0.691437	−	0.722436i	$-0.743022\pi$
$727$	− 38.9580i	− 1.44487i	0.691437	−	0.722436i	$-0.256978\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.47439	−0.276450
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.2614i	0.414818i
$738$	0	0
$739$	− 4.55648i	− 0.167613i	−0.996482	−	0.0838064i	$-0.973292\pi$
$739$	− 4.55648i	− 0.167613i	0.996482	−	0.0838064i	$-0.0267077\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.0401	1.32218	0.661092	−	0.750305i	$-0.270093\pi$
$743$	36.0401	1.32218	0.661092	+	0.750305i	$0.270093\pi$
$744$	0	0
$745$	−23.0540	−0.844632
$746$	0	0
$747$	0	0
$748$	0	0
$749$	33.3693i	1.21929i
$750$	0	0
$751$	− 33.3354i	− 1.21643i	−0.793774	−	0.608213i	$-0.791887\pi$
$751$	− 33.3354i	− 1.21643i	0.793774	−	0.608213i	$-0.208113\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10.6055	0.385975
$756$	0	0
$757$	−5.73863	−0.208574	−0.104287	−	0.994547i	$-0.533256\pi$
$757$	−5.73863	−0.208574	−0.104287	+	0.994547i	$0.533256\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.738634i	0.0267755i	0.999910	+	0.0133877i	$0.00426157\pi$
$761$	0.738634i	0.0267755i	−0.999910	+	0.0133877i	$0.995738\pi$
$762$	0	0
$763$	37.8656i	1.37083i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.94528	0.0702398
$768$	0	0
$769$	−35.1080	−1.26603	−0.633013	−	0.774142i	$-0.718182\pi$
$769$	−35.1080	−1.26603	−0.633013	+	0.774142i	$0.718182\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 40.1080i	− 1.44258i	−0.692632	−	0.721291i	$-0.743549\pi$
$773$	− 40.1080i	− 1.44258i	0.692632	−	0.721291i	$-0.256451\pi$
$774$	0	0
$775$	− 0.972638i	− 0.0349382i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−16.2281	−0.581433
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	13.0540i	0.465916i
$786$	0	0
$787$	− 41.7824i	− 1.48938i	−0.667409	−	0.744692i	$-0.732597\pi$
$787$	− 41.7824i	− 1.48938i	0.667409	−	0.744692i	$-0.267403\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	37.0127	1.31602
$792$	0	0
$793$	6.68466	0.237379
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.7386i	0.663756i	0.943322	+	0.331878i	$0.107682\pi$
$797$	18.7386i	0.663756i	−0.943322	+	0.331878i	$0.892318\pi$
$798$	0	0
$799$	87.3357i	3.08972i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8.44703	−0.298089
$804$	0	0
$805$	−2.63068	−0.0927194
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20.6307i	0.725336i	0.931918	+	0.362668i	$0.118134\pi$
$809$	20.6307i	0.725336i	−0.931918	+	0.362668i	$0.881866\pi$
$810$	0	0
$811$	44.7938i	1.57292i	0.617638	+	0.786462i	$0.288090\pi$
$811$	44.7938i	1.57292i	−0.617638	+	0.786462i	$0.711910\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12.1244	0.424698
$816$	0	0
$817$	−2.63068	−0.0920360
$818$	0	0
$819$	0	0
$820$	0	0
$821$	51.3693i	1.79280i	0.443244	+	0.896401i	$0.353827\pi$
$821$	51.3693i	1.79280i	−0.443244	+	0.896401i	$0.646173\pi$
$822$	0	0
$823$	15.0423i	0.524341i	0.965022	+	0.262170i	$0.0844382\pi$
$823$	15.0423i	0.524341i	−0.965022	+	0.262170i	$0.915562\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.89055	−0.135288	−0.0676439	−	0.997710i	$-0.521548\pi$
$827$	−3.89055	−0.135288	−0.0676439	+	0.997710i	$0.521548\pi$
$828$	0	0
$829$	−43.4233	−1.50815	−0.754077	−	0.656786i	$-0.771915\pi$
$829$	−43.4233	−1.50815	−0.754077	+	0.656786i	$0.771915\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.42329i	− 0.0839621i
$834$	0	0
$835$	− 1.94528i	− 0.0673191i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.78110	−0.268634	−0.134317	−	0.990938i	$-0.542884\pi$
$839$	−7.78110	−0.268634	−0.134317	+	0.990938i	$0.542884\pi$
$840$	0	0
$841$	26.1619	0.902135
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 12.0000i	− 0.412813i
$846$	0	0
$847$	27.1929i	0.934358i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.61120	0.0895108
$852$	0	0
$853$	39.1080	1.33903	0.669515	−	0.742798i	$-0.266502\pi$
$853$	39.1080	1.33903	0.669515	+	0.742798i	$0.266502\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	36.7386i	1.25497i	0.778630	+	0.627484i	$0.215915\pi$
$857$	36.7386i	1.25497i	−0.778630	+	0.627484i	$0.784085\pi$
$858$	0	0
$859$	15.4687i	0.527786i	0.964552	+	0.263893i	$0.0850066\pi$
$859$	15.4687i	0.527786i	−0.964552	+	0.263893i	$0.914993\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−39.9307	−1.35926	−0.679628	−	0.733557i	$-0.737859\pi$
$863$	−39.9307	−1.35926	−0.679628	+	0.733557i	$0.737859\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 13.6847i	− 0.464220i
$870$	0	0
$871$	− 11.5782i	− 0.392311i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.70469	−0.0914352
$876$	0	0
$877$	−10.3693	−0.350147	−0.175073	−	0.984555i	$-0.556016\pi$
$877$	−10.3693	−0.350147	−0.175073	+	0.984555i	$0.556016\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 24.7386i	− 0.833466i	−0.909029	−	0.416733i	$-0.863175\pi$
$881$	− 24.7386i	− 0.833466i	0.909029	−	0.416733i	$-0.136825\pi$
$882$	0	0
$883$	− 50.1097i	− 1.68633i	−0.537657	−	0.843163i	$-0.680691\pi$
$883$	− 50.1097i	− 1.68633i	0.537657	−	0.843163i	$-0.319309\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41.8759	1.40606	0.703028	−	0.711162i	$-0.251831\pi$
$887$	41.8759	1.40606	0.703028	+	0.711162i	$0.251831\pi$
$888$	0	0
$889$	28.1080	0.942710
$890$	0	0
$891$	0	0
$892$	0	0
$893$	30.7386i	1.02863i
$894$	0	0
$895$	− 14.2829i	− 0.477423i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.63856	0.0546491
$900$	0	0
$901$	−46.1080	−1.53608
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.0540i	0.600134i
$906$	0	0
$907$	− 43.7277i	− 1.45196i	−0.687718	−	0.725978i	$-0.741388\pi$
$907$	− 43.7277i	− 1.45196i	0.687718	−	0.725978i	$-0.258612\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−43.5145	−1.44170	−0.720850	−	0.693091i	$-0.756248\pi$
$911$	−43.5145	−1.44170	−0.720850	+	0.693091i	$0.756248\pi$
$912$	0	0
$913$	−10.1080	−0.334524
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30.7386i	1.01508i
$918$	0	0
$919$	− 48.3777i	− 1.59583i	−0.602768	−	0.797916i	$-0.705936\pi$
$919$	− 48.3777i	− 1.59583i	0.602768	−	0.797916i	$-0.294064\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.3376	0.406096
$924$	0	0
$925$	2.68466	0.0882710
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 24.7386i	− 0.811648i	−0.913951	−	0.405824i	$-0.866985\pi$
$929$	− 24.7386i	− 0.811648i	0.913951	−	0.405824i	$-0.133015\pi$
$930$	0	0
$931$	− 0.852901i	− 0.0279527i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.47439	0.244439
$936$	0	0
$937$	36.6847	1.19844	0.599218	−	0.800586i	$-0.295478\pi$
$937$	36.6847	1.19844	0.599218	+	0.800586i	$0.295478\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	56.4233i	1.83935i	0.392684	+	0.919673i	$0.371547\pi$
$941$	56.4233i	1.83935i	−0.392684	+	0.919673i	$0.628453\pi$
$942$	0	0
$943$	− 5.83583i	− 0.190041i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.4050	1.54046	0.770229	−	0.637768i	$-0.220142\pi$
$947$	47.4050	1.54046	0.770229	+	0.637768i	$0.220142\pi$
$948$	0	0
$949$	8.68466	0.281916
$950$	0	0
$951$	0	0
$952$	0	0
$953$	43.6847i	1.41508i	0.706671	+	0.707542i	$0.250196\pi$
$953$	43.6847i	1.41508i	−0.706671	+	0.707542i	$0.749804\pi$
$954$	0	0
$955$	20.7846i	0.672574i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16.2281	−0.524034
$960$	0	0
$961$	30.0540	0.969483
$962$	0	0
$963$	0	0
$964$	0	0
$965$	24.6847i	0.794627i
$966$	0	0
$967$	11.1517i	0.358615i	0.983793	+	0.179308i	$0.0573857\pi$
$967$	11.1517i	0.358615i	−0.983793	+	0.179308i	$0.942614\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	15.2555	0.489572	0.244786	−	0.969577i	$-0.421282\pi$
$971$	15.2555	0.489572	0.244786	+	0.969577i	$0.421282\pi$
$972$	0	0
$973$	26.0540	0.835252
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.68466i	− 0.0538970i	−0.999637	−	0.0269485i	$-0.991421\pi$
$977$	− 1.68466i	− 0.0538970i	0.999637	−	0.0269485i	$-0.00857901\pi$
$978$	0	0
$979$	− 11.6717i	− 0.373028i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.6478	−0.818038	−0.409019	−	0.912526i	$-0.634129\pi$
$983$	−25.6478	−0.818038	−0.409019	+	0.912526i	$0.634129\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 0.946025i	− 0.0300818i
$990$	0	0
$991$	6.04905i	0.192155i	0.995374	+	0.0960773i	$0.0306296\pi$
$991$	6.04905i	0.192155i	−0.995374	+	0.0960773i	$0.969370\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.73205	0.0549097
$996$	0	0
$997$	−45.0540	−1.42687	−0.713437	−	0.700720i	$-0.752862\pi$
$997$	−45.0540	−1.42687	−0.713437	+	0.700720i	$0.752862\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.h.g.431.6	yes	8
3.2	odd	2	inner	2160.2.h.g.431.2	✓	8
4.3	odd	2	inner	2160.2.h.g.431.7	yes	8
12.11	even	2	inner	2160.2.h.g.431.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2160.2.h.g.431.2	✓	8	3.2	odd	2	inner
2160.2.h.g.431.3	yes	8	12.11	even	2	inner
2160.2.h.g.431.6	yes	8	1.1	even	1	trivial
2160.2.h.g.431.7	yes	8	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} -2.70469i q^{7} -0.972638 q^{11} +1.00000 q^{13} +7.68466i q^{17} +2.70469i q^{19} -0.972638 q^{23} -1.00000 q^{25} -1.68466i q^{29} +0.972638i q^{31} +2.70469 q^{35} -2.68466 q^{37} +6.00000i q^{41} +0.972638i q^{43} +11.3649 q^{47} -0.315342 q^{49} +6.00000i q^{53} -0.972638i q^{55} +1.94528 q^{59} +6.68466 q^{61} +1.00000i q^{65} -11.5782i q^{67} +12.3376 q^{71} +8.68466 q^{73} +2.63068i q^{77} +14.0696i q^{79} +10.3923 q^{83} -7.68466 q^{85} +12.0000i q^{89} -2.70469i q^{91} -2.70469 q^{95} -6.68466 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{13} - 8 q^{25} + 28 q^{37} - 52 q^{49} + 4 q^{61} + 20 q^{73} - 12 q^{85} - 4 q^{97}+O(q^{100}) \) 8 * q + 8 * q^13 - 8 * q^25 + 28 * q^37 - 52 * q^49 + 4 * q^61 + 20 * q^73 - 12 * q^85 - 4 * q^97

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