LMFDB - Embedded newform 720.2.h.a.431.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,2,Mod(431,720)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("720.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	720.h (of order $2$, degree $1$, 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:	$5.74922894553$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.7
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	720.431
Dual form			720.2.h.a.431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +2.44949i q^{7} +O(q^{10})$	$q+1.00000i q^{5} +2.44949i q^{7} -1.01461 q^{11} +2.24264 q^{13} +4.89898i q^{19} -8.36308 q^{23} -1.00000 q^{25} +6.00000i q^{29} +8.36308i q^{31} -2.44949 q^{35} +6.24264 q^{37} -4.24264i q^{41} -2.02922i q^{43} -2.02922 q^{47} +1.00000 q^{49} +8.48528i q^{53} -1.01461i q^{55} -1.01461 q^{59} +10.4853 q^{61} +2.24264i q^{65} +6.92820i q^{67} +16.7262 q^{71} -10.4853 q^{73} -2.48528i q^{77} -1.43488i q^{79} -14.6969 q^{83} -16.2426i q^{89} +5.49333i q^{91} -4.89898 q^{95} -10.4853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{13} - 8 q^{25} + 16 q^{37} + 8 q^{49} + 16 q^{61} - 16 q^{73} - 16 q^{97}+O(q^{100}) $ 8 * q - 16 * q^13 - 8 * q^25 + 16 * q^37 + 8 * q^49 + 16 * q^61 - 16 * q^73 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$7$	2.44949i	0.925820i	0.886405	+	0.462910i	$0.153195\pi$
$7$	2.44949i	0.925820i	−0.886405	+	0.462910i	$0.846805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.01461	−0.305917	−0.152958	−	0.988233i	$-0.548880\pi$
$11$	−1.01461	−0.305917	−0.152958	+	0.988233i	$0.548880\pi$
$12$	0	0
$13$	2.24264	0.621997	0.310998	−	0.950410i	$-0.399337\pi$
$13$	2.24264	0.621997	0.310998	+	0.950410i	$0.399337\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	4.89898i	1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$	4.89898i	1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.36308	−1.74382	−0.871911	−	0.489664i	$-0.837120\pi$
$23$	−8.36308	−1.74382	−0.871911	+	0.489664i	$0.837120\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	8.36308i	1.50205i	0.660272	+	0.751027i	$0.270441\pi$
$31$	8.36308i	1.50205i	−0.660272	+	0.751027i	$0.729559\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.44949	−0.414039
$36$	0	0
$37$	6.24264	1.02628	0.513142	−	0.858304i	$-0.328481\pi$
$37$	6.24264	1.02628	0.513142	+	0.858304i	$0.328481\pi$
$38$	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
$43$	− 2.02922i	− 0.309454i	−0.987957	−	0.154727i	$-0.950550\pi$
$43$	− 2.02922i	− 0.309454i	0.987957	−	0.154727i	$-0.0494498\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.02922	−0.295993	−0.147996	−	0.988988i	$-0.547282\pi$
$47$	−2.02922	−0.295993	−0.147996	+	0.988988i	$0.547282\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.48528i	1.16554i	0.812636	+	0.582772i	$0.198032\pi$
$53$	8.48528i	1.16554i	−0.812636	+	0.582772i	$0.801968\pi$
$54$	0	0
$55$	− 1.01461i	− 0.136810i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.01461	−0.132091	−0.0660456	−	0.997817i	$-0.521038\pi$
$59$	−1.01461	−0.132091	−0.0660456	+	0.997817i	$0.521038\pi$
$60$	0	0
$61$	10.4853	1.34250	0.671251	−	0.741230i	$-0.265757\pi$
$61$	10.4853	1.34250	0.671251	+	0.741230i	$0.265757\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.24264i	0.278165i
$66$	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
$71$	16.7262	1.98503	0.992515	−	0.122122i	$-0.0389698\pi$
$71$	16.7262	1.98503	0.992515	+	0.122122i	$0.0389698\pi$
$72$	0	0
$73$	−10.4853	−1.22721	−0.613605	−	0.789613i	$-0.710281\pi$
$73$	−10.4853	−1.22721	−0.613605	+	0.789613i	$0.710281\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.48528i	− 0.283224i
$78$	0	0
$79$	− 1.43488i	− 0.161436i	−0.996737	−	0.0807182i	$-0.974279\pi$
$79$	− 1.43488i	− 0.161436i	0.996737	−	0.0807182i	$-0.0257214\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.6969	−1.61320	−0.806599	−	0.591099i	$-0.798694\pi$
$83$	−14.6969	−1.61320	−0.806599	+	0.591099i	$0.798694\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 16.2426i	− 1.72172i	−0.508845	−	0.860858i	$-0.669927\pi$
$89$	− 16.2426i	− 1.72172i	0.508845	−	0.860858i	$-0.330073\pi$
$90$	0	0
$91$	5.49333i	0.575857i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.89898	−0.502625
$96$	0	0
$97$	−10.4853	−1.06462	−0.532310	−	0.846550i	$-0.678676\pi$
$97$	−10.4853	−1.06462	−0.532310	+	0.846550i	$0.678676\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 2.48528i	− 0.247295i	−0.992326	−	0.123647i	$-0.960541\pi$
$101$	− 2.48528i	− 0.247295i	0.992326	−	0.123647i	$-0.0394592\pi$
$102$	0	0
$103$	− 16.3059i	− 1.60667i	−0.595529	−	0.803334i	$-0.703057\pi$
$103$	− 16.3059i	− 1.60667i	0.595529	−	0.803334i	$-0.296943\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.7554	1.81315	0.906576	−	0.422043i	$-0.138687\pi$
$107$	18.7554	1.81315	0.906576	+	0.422043i	$0.138687\pi$
$108$	0	0
$109$	6.48528	0.621177	0.310589	−	0.950544i	$-0.399474\pi$
$109$	6.48528	0.621177	0.310589	+	0.950544i	$0.399474\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.48528i	0.798228i	0.916901	+	0.399114i	$0.130682\pi$
$113$	8.48528i	0.798228i	−0.916901	+	0.399114i	$0.869318\pi$
$114$	0	0
$115$	− 8.36308i	− 0.779861i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.97056	−0.906415
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	− 7.34847i	− 0.652071i	−0.945357	−	0.326036i	$-0.894287\pi$
$127$	− 7.34847i	− 0.652071i	0.945357	−	0.326036i	$-0.105713\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.6823	1.19543	0.597715	−	0.801709i	$-0.296075\pi$
$131$	13.6823	1.19543	0.597715	+	0.801709i	$0.296075\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.51472i	− 0.300283i	−0.988665	−	0.150141i	$-0.952027\pi$
$137$	− 3.51472i	− 0.300283i	0.988665	−	0.150141i	$-0.0479729\pi$
$138$	0	0
$139$	− 13.2621i	− 1.12487i	−0.826840	−	0.562437i	$-0.809864\pi$
$139$	− 13.2621i	− 1.12487i	0.826840	−	0.562437i	$-0.190136\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.27541	−0.190279
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.48528i	0.203602i	0.994805	+	0.101801i	$0.0324605\pi$
$149$	2.48528i	0.203602i	−0.994805	+	0.101801i	$0.967539\pi$
$150$	0	0
$151$	0.594346i	0.0483672i	0.999708	+	0.0241836i	$0.00769863\pi$
$151$	0.594346i	0.0483672i	−0.999708	+	0.0241836i	$0.992301\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.36308	−0.671739
$156$	0	0
$157$	18.2426	1.45592	0.727961	−	0.685619i	$-0.240468\pi$
$157$	18.2426	1.45592	0.727961	+	0.685619i	$0.240468\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 20.4853i	− 1.61447i
$162$	0	0
$163$	− 21.6251i	− 1.69381i	−0.531743	−	0.846906i	$-0.678463\pi$
$163$	− 21.6251i	− 1.69381i	0.531743	−	0.846906i	$-0.321537\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.36308	0.647155	0.323577	−	0.946202i	$-0.395114\pi$
$167$	8.36308	0.647155	0.323577	+	0.946202i	$0.395114\pi$
$168$	0	0
$169$	−7.97056	−0.613120
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.4853i	1.55747i	0.627355	+	0.778734i	$0.284138\pi$
$173$	20.4853i	1.55747i	−0.627355	+	0.778734i	$0.715862\pi$
$174$	0	0
$175$	− 2.44949i	− 0.185164i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.7700	1.47768	0.738840	−	0.673881i	$-0.235374\pi$
$179$	19.7700	1.47768	0.738840	+	0.673881i	$0.235374\pi$
$180$	0	0
$181$	1.51472	0.112588	0.0562941	−	0.998414i	$-0.482072\pi$
$181$	1.51472	0.112588	0.0562941	+	0.998414i	$0.482072\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.24264i	0.458968i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.6969	1.06343	0.531717	−	0.846922i	$-0.321547\pi$
$191$	14.6969	1.06343	0.531717	+	0.846922i	$0.321547\pi$
$192$	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
$197$	− 22.9706i	− 1.63658i	−0.574802	−	0.818292i	$-0.694921\pi$
$197$	− 22.9706i	− 1.63658i	0.574802	−	0.818292i	$-0.305079\pi$
$198$	0	0
$199$	3.46410i	0.245564i	0.992434	+	0.122782i	$0.0391815\pi$
$199$	3.46410i	0.245564i	−0.992434	+	0.122782i	$0.960818\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−14.6969	−1.03152
$204$	0	0
$205$	4.24264	0.296319
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4.97056i	− 0.343821i
$210$	0	0
$211$	3.46410i	0.238479i	0.992866	+	0.119239i	$0.0380456\pi$
$211$	3.46410i	0.238479i	−0.992866	+	0.119239i	$0.961954\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.02922	0.138392
$216$	0	0
$217$	−20.4853	−1.39063
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 9.37769i	− 0.627977i	−0.949427	−	0.313988i	$-0.898335\pi$
$223$	− 9.37769i	− 0.627977i	0.949427	−	0.313988i	$-0.101665\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.05845	−0.269369	−0.134684	−	0.990889i	$-0.543002\pi$
$227$	−4.05845	−0.269369	−0.134684	+	0.990889i	$0.543002\pi$
$228$	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
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	0	0
$235$	− 2.02922i	− 0.132372i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.02922	−0.131260	−0.0656298	−	0.997844i	$-0.520906\pi$
$239$	−2.02922	−0.131260	−0.0656298	+	0.997844i	$0.520906\pi$
$240$	0	0
$241$	20.9706	1.35083	0.675416	−	0.737437i	$-0.263964\pi$
$241$	20.9706	1.35083	0.675416	+	0.737437i	$0.263964\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000i	0.0638877i
$246$	0	0
$247$	10.9867i	0.699064i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.7700	1.24787	0.623936	−	0.781476i	$-0.285533\pi$
$251$	19.7700	1.24787	0.623936	+	0.781476i	$0.285533\pi$
$252$	0	0
$253$	8.48528	0.533465
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.4853i	1.27784i	0.769275	+	0.638918i	$0.220618\pi$
$257$	20.4853i	1.27784i	−0.769275	+	0.638918i	$0.779382\pi$
$258$	0	0
$259$	15.2913i	0.950154i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−29.1477	−1.79732	−0.898662	−	0.438642i	$-0.855460\pi$
$263$	−29.1477	−1.79732	−0.898662	+	0.438642i	$0.855460\pi$
$264$	0	0
$265$	−8.48528	−0.521247
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.9706i	1.40054i	0.713878	+	0.700270i	$0.246937\pi$
$269$	22.9706i	1.40054i	−0.713878	+	0.700270i	$0.753063\pi$
$270$	0	0
$271$	13.2621i	0.805613i	0.915285	+	0.402806i	$0.131965\pi$
$271$	13.2621i	0.805613i	−0.915285	+	0.402806i	$0.868035\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.01461	0.0611834
$276$	0	0
$277$	5.75736	0.345926	0.172963	−	0.984928i	$-0.444666\pi$
$277$	5.75736	0.345926	0.172963	+	0.984928i	$0.444666\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 4.24264i	− 0.253095i	−0.991961	−	0.126547i	$-0.959610\pi$
$281$	− 4.24264i	− 0.253095i	0.991961	−	0.126547i	$-0.0403896\pi$
$282$	0	0
$283$	− 4.05845i	− 0.241250i	−0.992698	−	0.120625i	$-0.961510\pi$
$283$	− 4.05845i	− 0.241250i	0.992698	−	0.120625i	$-0.0384898\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.3923	0.613438
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.9706i	0.640907i	0.947264	+	0.320454i	$0.103835\pi$
$293$	10.9706i	0.640907i	−0.947264	+	0.320454i	$0.896165\pi$
$294$	0	0
$295$	− 1.01461i	− 0.0590730i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−18.7554	−1.08465
$300$	0	0
$301$	4.97056	0.286498
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.4853i	0.600385i
$306$	0	0
$307$	− 4.05845i	− 0.231628i	−0.993271	−	0.115814i	$-0.963052\pi$
$307$	− 4.05845i	− 0.231628i	0.993271	−	0.115814i	$-0.0369476\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.7554	−1.06352	−0.531760	−	0.846895i	$-0.678469\pi$
$311$	−18.7554	−1.06352	−0.531760	+	0.846895i	$0.678469\pi$
$312$	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
$317$	1.02944i	0.0578190i	0.999582	+	0.0289095i	$0.00920345\pi$
$317$	1.02944i	0.0578190i	−0.999582	+	0.0289095i	$0.990797\pi$
$318$	0	0
$319$	− 6.08767i	− 0.340844i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−2.24264	−0.124399
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 4.97056i	− 0.274036i
$330$	0	0
$331$	11.8272i	0.650081i	0.945700	+	0.325040i	$0.105378\pi$
$331$	11.8272i	0.650081i	−0.945700	+	0.325040i	$0.894622\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.92820	−0.378528
$336$	0	0
$337$	23.4558	1.27772	0.638861	−	0.769322i	$-0.279406\pi$
$337$	23.4558	1.27772	0.638861	+	0.769322i	$0.279406\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 8.48528i	− 0.459504i
$342$	0	0
$343$	19.5959i	1.05808i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.7846	−1.11578	−0.557888	−	0.829916i	$-0.688388\pi$
$347$	−20.7846	−1.11578	−0.557888	+	0.829916i	$0.688388\pi$
$348$	0	0
$349$	−1.51472	−0.0810810	−0.0405405	−	0.999178i	$-0.512908\pi$
$349$	−1.51472	−0.0810810	−0.0405405	+	0.999178i	$0.512908\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.0000i	0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$	12.0000i	0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$	0	0
$355$	16.7262i	0.887733i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.7554	0.989871	0.494936	−	0.868930i	$-0.335192\pi$
$359$	18.7554	0.989871	0.494936	+	0.868930i	$0.335192\pi$
$360$	0	0
$361$	−5.00000	−0.263158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 10.4853i	− 0.548825i
$366$	0	0
$367$	15.1172i	0.789112i	0.918872	+	0.394556i	$0.129102\pi$
$367$	15.1172i	0.789112i	−0.918872	+	0.394556i	$0.870898\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−20.7846	−1.07908
$372$	0	0
$373$	−22.7279	−1.17681	−0.588404	−	0.808567i	$-0.700243\pi$
$373$	−22.7279	−1.17681	−0.588404	+	0.808567i	$0.700243\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.4558i	0.693011i
$378$	0	0
$379$	17.3205i	0.889695i	0.895606	+	0.444847i	$0.146742\pi$
$379$	17.3205i	0.889695i	−0.895606	+	0.444847i	$0.853258\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.02922	−0.103688	−0.0518442	−	0.998655i	$-0.516510\pi$
$383$	−2.02922	−0.103688	−0.0518442	+	0.998655i	$0.516510\pi$
$384$	0	0
$385$	2.48528	0.126662
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.9706i	0.556230i	0.960548	+	0.278115i	$0.0897096\pi$
$389$	10.9706i	0.556230i	−0.960548	+	0.278115i	$0.910290\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.43488	0.0721965
$396$	0	0
$397$	14.7279	0.739173	0.369587	−	0.929196i	$-0.379499\pi$
$397$	14.7279	0.739173	0.369587	+	0.929196i	$0.379499\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	19.7574i	0.986635i	0.869849	+	0.493318i	$0.164216\pi$
$401$	19.7574i	0.986635i	−0.869849	+	0.493318i	$0.835784\pi$
$402$	0	0
$403$	18.7554i	0.934272i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.33386	−0.313958
$408$	0	0
$409$	14.9706	0.740247	0.370123	−	0.928983i	$-0.379315\pi$
$409$	14.9706	0.740247	0.370123	+	0.928983i	$0.379315\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2.48528i	− 0.122293i
$414$	0	0
$415$	− 14.6969i	− 0.721444i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−17.7408	−0.866694	−0.433347	−	0.901227i	$-0.642667\pi$
$419$	−17.7408	−0.866694	−0.433347	+	0.901227i	$0.642667\pi$
$420$	0	0
$421$	−6.97056	−0.339724	−0.169862	−	0.985468i	$-0.554332\pi$
$421$	−6.97056	−0.339724	−0.169862	+	0.985468i	$0.554332\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	25.6836i	1.24292i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.02922	−0.0977443	−0.0488721	−	0.998805i	$-0.515563\pi$
$431$	−2.02922	−0.0977443	−0.0488721	+	0.998805i	$0.515563\pi$
$432$	0	0
$433$	27.4558	1.31944	0.659722	−	0.751510i	$-0.270674\pi$
$433$	27.4558	1.31944	0.659722	+	0.751510i	$0.270674\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 40.9706i	− 1.95989i
$438$	0	0
$439$	23.0600i	1.10059i	0.834969	+	0.550297i	$0.185486\pi$
$439$	23.0600i	1.10059i	−0.834969	+	0.550297i	$0.814514\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.7262	0.794684	0.397342	−	0.917671i	$-0.369933\pi$
$443$	16.7262	0.794684	0.397342	+	0.917671i	$0.369933\pi$
$444$	0	0
$445$	16.2426	0.769975
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 24.7279i	− 1.16698i	−0.812119	−	0.583491i	$-0.801686\pi$
$449$	− 24.7279i	− 1.16698i	0.812119	−	0.583491i	$-0.198314\pi$
$450$	0	0
$451$	4.30463i	0.202697i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.49333	−0.257531
$456$	0	0
$457$	−34.4853	−1.61315	−0.806577	−	0.591129i	$-0.798682\pi$
$457$	−34.4853	−1.61315	−0.806577	+	0.591129i	$0.798682\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	31.4558i	1.46504i	0.680743	+	0.732522i	$0.261657\pi$
$461$	31.4558i	1.46504i	−0.680743	+	0.732522i	$0.738343\pi$
$462$	0	0
$463$	4.47871i	0.208143i	0.994570	+	0.104072i	$0.0331872\pi$
$463$	4.47871i	0.208143i	−0.994570	+	0.104072i	$0.966813\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.02922	0.0939013	0.0469506	−	0.998897i	$-0.485050\pi$
$467$	2.02922	0.0939013	0.0469506	+	0.998897i	$0.485050\pi$
$468$	0	0
$469$	−16.9706	−0.783628
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.05887i	0.0946672i
$474$	0	0
$475$	− 4.89898i	− 0.224781i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.7846	−0.949673	−0.474837	−	0.880074i	$-0.657493\pi$
$479$	−20.7846	−0.949673	−0.474837	+	0.880074i	$0.657493\pi$
$480$	0	0
$481$	14.0000	0.638345
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 10.4853i	− 0.476112i
$486$	0	0
$487$	− 20.3643i	− 0.922796i	−0.887193	−	0.461398i	$-0.847348\pi$
$487$	− 20.3643i	− 0.922796i	0.887193	−	0.461398i	$-0.152652\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.04384	0.137366	0.0686832	−	0.997639i	$-0.478120\pi$
$491$	3.04384	0.137366	0.0686832	+	0.997639i	$0.478120\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.9706i	1.83778i
$498$	0	0
$499$	− 39.5400i	− 1.77005i	−0.465540	−	0.885027i	$-0.654140\pi$
$499$	− 39.5400i	− 1.77005i	0.465540	−	0.885027i	$-0.345860\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	35.4815	1.58204	0.791022	−	0.611788i	$-0.209549\pi$
$503$	35.4815	1.58204	0.791022	+	0.611788i	$0.209549\pi$
$504$	0	0
$505$	2.48528	0.110594
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.48528i	0.110158i	0.998482	+	0.0550791i	$0.0175411\pi$
$509$	2.48528i	0.110158i	−0.998482	+	0.0550791i	$0.982459\pi$
$510$	0	0
$511$	− 25.6836i	− 1.13618i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.3059	0.718524
$516$	0	0
$517$	2.05887	0.0905492
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.21320i	0.403638i	0.979423	+	0.201819i	$0.0646852\pi$
$521$	9.21320i	0.403638i	−0.979423	+	0.201819i	$0.935315\pi$
$522$	0	0
$523$	25.6836i	1.12306i	0.827455	+	0.561532i	$0.189788\pi$
$523$	25.6836i	1.12306i	−0.827455	+	0.561532i	$0.810212\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	46.9411	2.04092
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 9.51472i	− 0.412128i
$534$	0	0
$535$	18.7554i	0.810866i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.01461	−0.0437024
$540$	0	0
$541$	−22.4853	−0.966718	−0.483359	−	0.875422i	$-0.660583\pi$
$541$	−22.4853	−0.966718	−0.483359	+	0.875422i	$0.660583\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.48528i	0.277799i
$546$	0	0
$547$	− 38.3513i	− 1.63978i	−0.572519	−	0.819892i	$-0.694034\pi$
$547$	− 38.3513i	− 1.63978i	0.572519	−	0.819892i	$-0.305966\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−29.3939	−1.25222
$552$	0	0
$553$	3.51472	0.149461
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8.48528i	− 0.359533i	−0.983709	−	0.179766i	$-0.942466\pi$
$557$	− 8.48528i	− 0.359533i	0.983709	−	0.179766i	$-0.0575342\pi$
$558$	0	0
$559$	− 4.55082i	− 0.192479i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.6677	0.533881	0.266940	−	0.963713i	$-0.413987\pi$
$563$	12.6677	0.533881	0.266940	+	0.963713i	$0.413987\pi$
$564$	0	0
$565$	−8.48528	−0.356978
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 19.7574i	− 0.828272i	−0.910215	−	0.414136i	$-0.864084\pi$
$569$	− 19.7574i	− 0.828272i	0.910215	−	0.414136i	$-0.135916\pi$
$570$	0	0
$571$	− 15.8856i	− 0.664793i	−0.943140	−	0.332396i	$-0.892143\pi$
$571$	− 15.8856i	− 0.664793i	0.943140	−	0.332396i	$-0.107857\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.36308	0.348765
$576$	0	0
$577$	−13.5147	−0.562625	−0.281313	−	0.959616i	$-0.590770\pi$
$577$	−13.5147	−0.562625	−0.281313	+	0.959616i	$0.590770\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 36.0000i	− 1.49353i
$582$	0	0
$583$	− 8.60927i	− 0.356559i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.6969	−0.606608	−0.303304	−	0.952894i	$-0.598090\pi$
$587$	−14.6969	−0.606608	−0.303304	+	0.952894i	$0.598090\pi$
$588$	0	0
$589$	−40.9706	−1.68816
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.51472i	0.144332i	0.997393	+	0.0721661i	$0.0229912\pi$
$593$	3.51472i	0.144332i	−0.997393	+	0.0721661i	$0.977009\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.08767	0.248736	0.124368	−	0.992236i	$-0.460310\pi$
$599$	6.08767	0.248736	0.124368	+	0.992236i	$0.460310\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 9.97056i	− 0.405361i
$606$	0	0
$607$	25.2633i	1.02541i	0.858566	+	0.512703i	$0.171356\pi$
$607$	25.2633i	1.02541i	−0.858566	+	0.512703i	$0.828644\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.55082	−0.184106
$612$	0	0
$613$	−19.2132	−0.776014	−0.388007	−	0.921656i	$-0.626836\pi$
$613$	−19.2132	−0.776014	−0.388007	+	0.921656i	$0.626836\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.9411i	1.84952i	0.380551	+	0.924760i	$0.375734\pi$
$617$	45.9411i	1.84952i	−0.380551	+	0.924760i	$0.624266\pi$
$618$	0	0
$619$	− 20.1903i	− 0.811515i	−0.913981	−	0.405758i	$-0.867008\pi$
$619$	− 20.1903i	− 0.811515i	0.913981	−	0.405758i	$-0.132992\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	39.7862	1.59400
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 18.1610i	− 0.722980i	−0.932376	−	0.361490i	$-0.882268\pi$
$631$	− 18.1610i	− 0.722980i	0.932376	−	0.361490i	$-0.117732\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.34847	0.291615
$636$	0	0
$637$	2.24264	0.0888567
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 0.727922i	− 0.0287512i	−0.999897	−	0.0143756i	$-0.995424\pi$
$641$	− 0.727922i	− 0.0287512i	0.999897	−	0.0143756i	$-0.00457605\pi$
$642$	0	0
$643$	− 14.6969i	− 0.579591i	−0.957089	−	0.289795i	$-0.906413\pi$
$643$	− 14.6969i	− 0.579591i	0.957089	−	0.289795i	$-0.0935872\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.02922	−0.0797770	−0.0398885	−	0.999204i	$-0.512700\pi$
$647$	−2.02922	−0.0797770	−0.0398885	+	0.999204i	$0.512700\pi$
$648$	0	0
$649$	1.02944	0.0404089
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 46.9706i	− 1.83810i	−0.394141	−	0.919050i	$-0.628958\pi$
$653$	− 46.9706i	− 1.83810i	0.394141	−	0.919050i	$-0.371042\pi$
$654$	0	0
$655$	13.6823i	0.534613i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	32.4377	1.26359	0.631797	−	0.775134i	$-0.282318\pi$
$659$	32.4377	1.26359	0.631797	+	0.775134i	$0.282318\pi$
$660$	0	0
$661$	−30.4853	−1.18574	−0.592870	−	0.805298i	$-0.702005\pi$
$661$	−30.4853	−1.18574	−0.592870	+	0.805298i	$0.702005\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 12.0000i	− 0.465340i
$666$	0	0
$667$	− 50.1785i	− 1.94292i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.6385	−0.410694
$672$	0	0
$673$	17.5147	0.675143	0.337571	−	0.941300i	$-0.390395\pi$
$673$	17.5147	0.675143	0.337571	+	0.941300i	$0.390395\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 39.9411i	− 1.53506i	−0.641012	−	0.767531i	$-0.721485\pi$
$677$	− 39.9411i	− 1.53506i	0.641012	−	0.767531i	$-0.278515\pi$
$678$	0	0
$679$	− 25.6836i	− 0.985646i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.3647	1.04708	0.523540	−	0.852001i	$-0.324611\pi$
$683$	27.3647	1.04708	0.523540	+	0.852001i	$0.324611\pi$
$684$	0	0
$685$	3.51472	0.134290
$686$	0	0
$687$	0	0
$688$	0	0
$689$	19.0294i	0.724964i
$690$	0	0
$691$	− 14.6969i	− 0.559098i	−0.960131	−	0.279549i	$-0.909815\pi$
$691$	− 14.6969i	− 0.559098i	0.960131	−	0.279549i	$-0.0901849\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.2621	0.503059
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31.4558i	1.18807i	0.804439	+	0.594035i	$0.202466\pi$
$701$	31.4558i	1.18807i	−0.804439	+	0.594035i	$0.797534\pi$
$702$	0	0
$703$	30.5826i	1.15344i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.08767	0.228950
$708$	0	0
$709$	−35.9411	−1.34980	−0.674899	−	0.737910i	$-0.735813\pi$
$709$	−35.9411	−1.34980	−0.674899	+	0.737910i	$0.735813\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 69.9411i	− 2.61932i
$714$	0	0
$715$	− 2.27541i	− 0.0850955i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.6385	−0.396749	−0.198374	−	0.980126i	$-0.563566\pi$
$719$	−10.6385	−0.396749	−0.198374	+	0.980126i	$0.563566\pi$
$720$	0	0
$721$	39.9411	1.48749
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 6.00000i	− 0.222834i
$726$	0	0
$727$	− 44.8592i	− 1.66374i	−0.554973	−	0.831869i	$-0.687271\pi$
$727$	− 44.8592i	− 1.66374i	0.554973	−	0.831869i	$-0.312729\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	26.2426	0.969294	0.484647	−	0.874710i	$-0.338948\pi$
$733$	26.2426	0.969294	0.484647	+	0.874710i	$0.338948\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7.02944i	− 0.258933i
$738$	0	0
$739$	35.4815i	1.30521i	0.757698	+	0.652605i	$0.226324\pi$
$739$	35.4815i	1.30521i	−0.757698	+	0.652605i	$0.773676\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.4231	1.15280	0.576401	−	0.817167i	$-0.304457\pi$
$743$	31.4231	1.15280	0.576401	+	0.817167i	$0.304457\pi$
$744$	0	0
$745$	−2.48528	−0.0910537
$746$	0	0
$747$	0	0
$748$	0	0
$749$	45.9411i	1.67865i
$750$	0	0
$751$	− 51.6134i	− 1.88340i	−0.336456	−	0.941699i	$-0.609228\pi$
$751$	− 51.6134i	− 1.88340i	0.336456	−	0.941699i	$-0.390772\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−0.594346	−0.0216305
$756$	0	0
$757$	−35.2132	−1.27985	−0.639923	−	0.768439i	$-0.721034\pi$
$757$	−35.2132	−1.27985	−0.639923	+	0.768439i	$0.721034\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41.6985i	1.51157i	0.654820	+	0.755784i	$0.272744\pi$
$761$	41.6985i	1.51157i	−0.654820	+	0.755784i	$0.727256\pi$
$762$	0	0
$763$	15.8856i	0.575098i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.27541	−0.0821603
$768$	0	0
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.9706i	0.394584i	0.980345	+	0.197292i	$0.0632147\pi$
$773$	10.9706i	0.394584i	−0.980345	+	0.197292i	$0.936785\pi$
$774$	0	0
$775$	− 8.36308i	− 0.300411i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20.7846	0.744686
$780$	0	0
$781$	−16.9706	−0.607254
$782$	0	0
$783$	0	0
$784$	0	0
$785$	18.2426i	0.651108i
$786$	0	0
$787$	46.4682i	1.65641i	0.560423	+	0.828206i	$0.310638\pi$
$787$	46.4682i	1.65641i	−0.560423	+	0.828206i	$0.689362\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−20.7846	−0.739016
$792$	0	0
$793$	23.5147	0.835032
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 27.5147i	− 0.974621i	−0.873229	−	0.487311i	$-0.837978\pi$
$797$	− 27.5147i	− 0.974621i	0.873229	−	0.487311i	$-0.162022\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.6385	0.375424
$804$	0	0
$805$	20.4853	0.722011
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 17.6985i	− 0.622246i	−0.950370	−	0.311123i	$-0.899295\pi$
$809$	− 17.6985i	− 0.622246i	0.950370	−	0.311123i	$-0.100705\pi$
$810$	0	0
$811$	− 43.8446i	− 1.53959i	−0.638289	−	0.769797i	$-0.720358\pi$
$811$	− 43.8446i	− 1.53959i	0.638289	−	0.769797i	$-0.279642\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	21.6251	0.757496
$816$	0	0
$817$	9.94113	0.347796
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 18.0000i	− 0.628204i	−0.949389	−	0.314102i	$-0.898297\pi$
$821$	− 18.0000i	− 0.628204i	0.949389	−	0.314102i	$-0.101703\pi$
$822$	0	0
$823$	− 0.420266i	− 0.0146496i	−0.999973	−	0.00732478i	$-0.997668\pi$
$823$	− 0.420266i	− 0.0146496i	0.999973	−	0.00732478i	$-0.00233157\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.6677	−0.440500	−0.220250	−	0.975444i	$-0.570687\pi$
$827$	−12.6677	−0.440500	−0.220250	+	0.975444i	$0.570687\pi$
$828$	0	0
$829$	6.48528	0.225243	0.112622	−	0.993638i	$-0.464075\pi$
$829$	6.48528	0.225243	0.112622	+	0.993638i	$0.464075\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8.36308i	0.289416i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	18.7554	0.647508	0.323754	−	0.946141i	$-0.395055\pi$
$839$	18.7554	0.647508	0.323754	+	0.946141i	$0.395055\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 7.97056i	− 0.274196i
$846$	0	0
$847$	− 24.4228i	− 0.839177i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−52.2077	−1.78966
$852$	0	0
$853$	−26.2426	−0.898531	−0.449265	−	0.893398i	$-0.648314\pi$
$853$	−26.2426	−0.898531	−0.449265	+	0.893398i	$0.648314\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.9706i	0.989616i	0.869002	+	0.494808i	$0.164762\pi$
$857$	28.9706i	0.989616i	−0.869002	+	0.494808i	$0.835238\pi$
$858$	0	0
$859$	− 4.65279i	− 0.158751i	−0.996845	−	0.0793756i	$-0.974707\pi$
$859$	− 4.65279i	− 0.158751i	0.996845	−	0.0793756i	$-0.0252927\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.30463	0.146531	0.0732657	−	0.997312i	$-0.476658\pi$
$863$	4.30463	0.146531	0.0732657	+	0.997312i	$0.476658\pi$
$864$	0	0
$865$	−20.4853	−0.696520
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.45584i	0.0493861i
$870$	0	0
$871$	15.5375i	0.526467i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.44949	0.0828079
$876$	0	0
$877$	21.7574	0.734694	0.367347	−	0.930084i	$-0.380266\pi$
$877$	21.7574	0.734694	0.367347	+	0.930084i	$0.380266\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 9.21320i	− 0.310401i	−0.987883	−	0.155200i	$-0.950398\pi$
$881$	− 9.21320i	− 0.310401i	0.987883	−	0.155200i	$-0.0496023\pi$
$882$	0	0
$883$	15.8856i	0.534594i	0.963614	+	0.267297i	$0.0861305\pi$
$883$	15.8856i	0.534594i	−0.963614	+	0.267297i	$0.913870\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29.1477	−0.978684	−0.489342	−	0.872092i	$-0.662763\pi$
$887$	−29.1477	−0.978684	−0.489342	+	0.872092i	$0.662763\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 9.94113i	− 0.332667i
$894$	0	0
$895$	19.7700i	0.660838i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−50.1785	−1.67355
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.51472i	0.0503510i
$906$	0	0
$907$	− 3.21792i	− 0.106849i	−0.998572	−	0.0534246i	$-0.982986\pi$
$907$	− 3.21792i	− 0.106849i	0.998572	−	0.0534246i	$-0.0170137\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25.3354	−0.839400	−0.419700	−	0.907663i	$-0.637865\pi$
$911$	−25.3354	−0.839400	−0.419700	+	0.907663i	$0.637865\pi$
$912$	0	0
$913$	14.9117	0.493505
$914$	0	0
$915$	0	0
$916$	0	0
$917$	33.5147i	1.10675i
$918$	0	0
$919$	− 8.36308i	− 0.275873i	−0.990441	−	0.137936i	$-0.955953\pi$
$919$	− 8.36308i	− 0.275873i	0.990441	−	0.137936i	$-0.0440469\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	37.5108	1.23468
$924$	0	0
$925$	−6.24264	−0.205257
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 24.7279i	− 0.811297i	−0.914029	−	0.405648i	$-0.867046\pi$
$929$	− 24.7279i	− 0.811297i	0.914029	−	0.405648i	$-0.132954\pi$
$930$	0	0
$931$	4.89898i	0.160558i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−38.9706	−1.27311	−0.636556	−	0.771230i	$-0.719642\pi$
$937$	−38.9706	−1.27311	−0.636556	+	0.771230i	$0.719642\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	22.9706i	0.748819i	0.927263	+	0.374409i	$0.122155\pi$
$941$	22.9706i	0.748819i	−0.927263	+	0.374409i	$0.877845\pi$
$942$	0	0
$943$	35.4815i	1.15544i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.7846	0.675409	0.337705	−	0.941252i	$-0.390350\pi$
$947$	20.7846	0.675409	0.337705	+	0.941252i	$0.390350\pi$
$948$	0	0
$949$	−23.5147	−0.763320
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 30.4264i	− 0.985608i	−0.870140	−	0.492804i	$-0.835972\pi$
$953$	− 30.4264i	− 0.985608i	0.870140	−	0.492804i	$-0.164028\pi$
$954$	0	0
$955$	14.6969i	0.475582i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.60927	0.278008
$960$	0	0
$961$	−38.9411	−1.25617
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 10.0000i	− 0.321911i
$966$	0	0
$967$	51.7874i	1.66537i	0.553745	+	0.832686i	$0.313198\pi$
$967$	51.7874i	1.66537i	−0.553745	+	0.832686i	$0.686802\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−57.2808	−1.83823	−0.919114	−	0.393992i	$-0.871094\pi$
$971$	−57.2808	−1.83823	−0.919114	+	0.393992i	$0.871094\pi$
$972$	0	0
$973$	32.4853	1.04143
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.9706i	0.926850i	0.886136	+	0.463425i	$0.153380\pi$
$977$	28.9706i	0.926850i	−0.886136	+	0.463425i	$0.846620\pi$
$978$	0	0
$979$	16.4800i	0.526702i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.02922	−0.0647222	−0.0323611	−	0.999476i	$-0.510303\pi$
$983$	−2.02922	−0.0647222	−0.0323611	+	0.999476i	$0.510303\pi$
$984$	0	0
$985$	22.9706	0.731903
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.9706i	0.539633i
$990$	0	0
$991$	49.5841i	1.57509i	0.616256	+	0.787546i	$0.288649\pi$
$991$	49.5841i	1.57509i	−0.616256	+	0.787546i	$0.711351\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.46410	−0.109819
$996$	0	0
$997$	47.2132	1.49526	0.747629	−	0.664117i	$-0.231192\pi$
$997$	47.2132	1.49526	0.747629	+	0.664117i	$0.231192\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.2.h.a.431.7	yes	8
3.2	odd	2	inner	720.2.h.a.431.4	yes	8
4.3	odd	2	inner	720.2.h.a.431.6	yes	8
5.2	odd	4		3600.2.o.c.3599.1		8
5.3	odd	4		3600.2.o.d.3599.6		8
5.4	even	2		3600.2.h.j.1151.2		8
8.3	odd	2		2880.2.h.f.1151.1		8
8.5	even	2		2880.2.h.f.1151.4		8
12.11	even	2	inner	720.2.h.a.431.1	✓	8
15.2	even	4		3600.2.o.d.3599.3		8
15.8	even	4		3600.2.o.c.3599.8		8
15.14	odd	2		3600.2.h.j.1151.3		8
20.3	even	4		3600.2.o.d.3599.4		8
20.7	even	4		3600.2.o.c.3599.7		8
20.19	odd	2		3600.2.h.j.1151.7		8
24.5	odd	2		2880.2.h.f.1151.7		8
24.11	even	2		2880.2.h.f.1151.6		8
60.23	odd	4		3600.2.o.c.3599.2		8
60.47	odd	4		3600.2.o.d.3599.5		8
60.59	even	2		3600.2.h.j.1151.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.h.a.431.1	✓	8	12.11	even	2	inner
720.2.h.a.431.4	yes	8	3.2	odd	2	inner
720.2.h.a.431.6	yes	8	4.3	odd	2	inner
720.2.h.a.431.7	yes	8	1.1	even	1	trivial
2880.2.h.f.1151.1		8	8.3	odd	2
2880.2.h.f.1151.4		8	8.5	even	2
2880.2.h.f.1151.6		8	24.11	even	2
2880.2.h.f.1151.7		8	24.5	odd	2
3600.2.h.j.1151.2		8	5.4	even	2
3600.2.h.j.1151.3		8	15.14	odd	2
3600.2.h.j.1151.6		8	60.59	even	2
3600.2.h.j.1151.7		8	20.19	odd	2
3600.2.o.c.3599.1		8	5.2	odd	4
3600.2.o.c.3599.2		8	60.23	odd	4
3600.2.o.c.3599.7		8	20.7	even	4
3600.2.o.c.3599.8		8	15.8	even	4
3600.2.o.d.3599.3		8	15.2	even	4
3600.2.o.d.3599.4		8	20.3	even	4
3600.2.o.d.3599.5		8	60.47	odd	4
3600.2.o.d.3599.6		8	5.3	odd	4

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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +2.44949i q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} +2.44949i q^{7} -1.01461 q^{11} +2.24264 q^{13} +4.89898i q^{19} -8.36308 q^{23} -1.00000 q^{25} +6.00000i q^{29} +8.36308i q^{31} -2.44949 q^{35} +6.24264 q^{37} -4.24264i q^{41} -2.02922i q^{43} -2.02922 q^{47} +1.00000 q^{49} +8.48528i q^{53} -1.01461i q^{55} -1.01461 q^{59} +10.4853 q^{61} +2.24264i q^{65} +6.92820i q^{67} +16.7262 q^{71} -10.4853 q^{73} -2.48528i q^{77} -1.43488i q^{79} -14.6969 q^{83} -16.2426i q^{89} +5.49333i q^{91} -4.89898 q^{95} -10.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{13} - 8 q^{25} + 16 q^{37} + 8 q^{49} + 16 q^{61} - 16 q^{73} - 16 q^{97}+O(q^{100}) \) 8 * q - 16 * q^13 - 8 * q^25 + 16 * q^37 + 8 * q^49 + 16 * q^61 - 16 * q^73 - 16 * q^97

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