LMFDB - Embedded newform 5184.2.f.a.2591.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(2591,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5184.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.f (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:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 11x^{14} + 85x^{12} + 332x^{10} + 940x^{8} + 1064x^{6} + 880x^{4} + 128x^{2} + 16 $ x^16 + 11x^14 + 85x^12 + 332x^10 + 940x^8 + 1064x^6 + 880x^4 + 128*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.1
Root			$0.192865 - 0.334053i$ of defining polynomial
Character	$\chi$	$=$	5184.2591
Dual form			5184.2.f.a.2591.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.12941 q^{5} -0.331895i q^{7} +O(q^{10})$	$q-4.12941 q^{5} -0.331895i q^{7} -4.18311i q^{11} -4.93844i q^{13} -3.20639i q^{17} +5.42029 q^{19} -2.78876 q^{23} +12.0520 q^{25} +2.07139 q^{29} +4.16649i q^{31} +1.37053i q^{35} -2.21390i q^{37} -1.63157i q^{41} -5.99515 q^{43} +7.95135 q^{47} +6.88985 q^{49} -2.05801 q^{53} +17.2738i q^{55} -6.24112i q^{59} -10.5160i q^{61} +20.3929i q^{65} +11.6732 q^{67} -7.66299 q^{71} -6.21742 q^{73} -1.38835 q^{77} -0.830275i q^{79} -6.88829i q^{83} +13.2405i q^{85} +9.14211i q^{89} -1.63904 q^{91} -22.3826 q^{95} +1.49838 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 12 q^{5}+O(q^{10}) $ 16 * q - 12 * q^5	$ 16 q - 12 q^{5} + 28 q^{25} - 36 q^{29} - 12 q^{49} - 48 q^{53} + 28 q^{73} - 132 q^{77} - 16 q^{97}+O(q^{100}) $ 16 * q - 12 * q^5 + 28 * q^25 - 36 * q^29 - 12 * q^49 - 48 * q^53 + 28 * q^73 - 132 * q^77 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−4.12941	−1.84673	−0.923364	−	0.383926i	$-0.874572\pi$
$5$	−4.12941	−1.84673	−0.923364	+	0.383926i	$0.874572\pi$
$6$	0	0
$7$	− 0.331895i	− 0.125444i	−0.998031	−	0.0627222i	$-0.980022\pi$
$7$	− 0.331895i	− 0.125444i	0.998031	−	0.0627222i	$-0.0199782\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.18311i	− 1.26125i	−0.776086	−	0.630627i	$-0.782798\pi$
$11$	− 4.18311i	− 1.26125i	0.776086	−	0.630627i	$-0.217202\pi$
$12$	0	0
$13$	− 4.93844i	− 1.36968i	−0.728694	−	0.684839i	$-0.759873\pi$
$13$	− 4.93844i	− 1.36968i	0.728694	−	0.684839i	$-0.240127\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.20639i	− 0.777664i	−0.921309	−	0.388832i	$-0.872879\pi$
$17$	− 3.20639i	− 0.777664i	0.921309	−	0.388832i	$-0.127121\pi$
$18$	0	0
$19$	5.42029	1.24350	0.621750	−	0.783215i	$-0.286422\pi$
$19$	5.42029	1.24350	0.621750	+	0.783215i	$0.286422\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.78876	−0.581497	−0.290748	−	0.956800i	$-0.593904\pi$
$23$	−2.78876	−0.581497	−0.290748	+	0.956800i	$0.593904\pi$
$24$	0	0
$25$	12.0520	2.41040
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.07139	0.384648	0.192324	−	0.981331i	$-0.438397\pi$
$29$	2.07139	0.384648	0.192324	+	0.981331i	$0.438397\pi$
$30$	0	0
$31$	4.16649i	0.748323i	0.927364	+	0.374161i	$0.122069\pi$
$31$	4.16649i	0.748323i	−0.927364	+	0.374161i	$0.877931\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.37053i	0.231662i
$36$	0	0
$37$	− 2.21390i	− 0.363963i	−0.983302	−	0.181982i	$-0.941749\pi$
$37$	− 2.21390i	− 0.363963i	0.983302	−	0.181982i	$-0.0582511\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.63157i	− 0.254808i	−0.991851	−	0.127404i	$-0.959336\pi$
$41$	− 1.63157i	− 0.254808i	0.991851	−	0.127404i	$-0.0406645\pi$
$42$	0	0
$43$	−5.99515	−0.914252	−0.457126	−	0.889402i	$-0.651121\pi$
$43$	−5.99515	−0.914252	−0.457126	+	0.889402i	$0.651121\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.95135	1.15982	0.579912	−	0.814679i	$-0.303087\pi$
$47$	7.95135	1.15982	0.579912	+	0.814679i	$0.303087\pi$
$48$	0	0
$49$	6.88985	0.984264
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.05801	−0.282690	−0.141345	−	0.989960i	$-0.545143\pi$
$53$	−2.05801	−0.282690	−0.141345	+	0.989960i	$0.545143\pi$
$54$	0	0
$55$	17.2738i	2.32919i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 6.24112i	− 0.812525i	−0.913756	−	0.406262i	$-0.866832\pi$
$59$	− 6.24112i	− 0.812525i	0.913756	−	0.406262i	$-0.133168\pi$
$60$	0	0
$61$	− 10.5160i	− 1.34643i	−0.739446	−	0.673216i	$-0.764913\pi$
$61$	− 10.5160i	− 1.34643i	0.739446	−	0.673216i	$-0.235087\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	20.3929i	2.52942i
$66$	0	0
$67$	11.6732	1.42610	0.713052	−	0.701112i	$-0.247313\pi$
$67$	11.6732	1.42610	0.713052	+	0.701112i	$0.247313\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.66299	−0.909429	−0.454715	−	0.890637i	$-0.650259\pi$
$71$	−7.66299	−0.909429	−0.454715	+	0.890637i	$0.650259\pi$
$72$	0	0
$73$	−6.21742	−0.727695	−0.363847	−	0.931459i	$-0.618537\pi$
$73$	−6.21742	−0.727695	−0.363847	+	0.931459i	$0.618537\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.38835	−0.158217
$78$	0	0
$79$	− 0.830275i	− 0.0934132i	−0.998909	−	0.0467066i	$-0.985127\pi$
$79$	− 0.830275i	− 0.0934132i	0.998909	−	0.0467066i	$-0.0148726\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6.88829i	− 0.756088i	−0.925788	−	0.378044i	$-0.876597\pi$
$83$	− 6.88829i	− 0.756088i	0.925788	−	0.378044i	$-0.123403\pi$
$84$	0	0
$85$	13.2405i	1.43613i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.14211i	0.969061i	0.874774	+	0.484531i	$0.161010\pi$
$89$	9.14211i	0.969061i	−0.874774	+	0.484531i	$0.838990\pi$
$90$	0	0
$91$	−1.63904	−0.171818
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−22.3826	−2.29641
$96$	0	0
$97$	1.49838	0.152137	0.0760687	−	0.997103i	$-0.475763\pi$
$97$	1.49838	0.152137	0.0760687	+	0.997103i	$0.475763\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.64717	−0.959929	−0.479965	−	0.877288i	$-0.659350\pi$
$101$	−9.64717	−0.959929	−0.479965	+	0.877288i	$0.659350\pi$
$102$	0	0
$103$	4.99568i	0.492239i	0.969239	+	0.246120i	$0.0791556\pi$
$103$	4.99568i	0.492239i	−0.969239	+	0.246120i	$0.920844\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.1294i	− 1.26927i	−0.772813	−	0.634634i	$-0.781151\pi$
$107$	− 13.1294i	− 1.26927i	0.772813	−	0.634634i	$-0.218849\pi$
$108$	0	0
$109$	10.7401i	1.02872i	0.857576	+	0.514358i	$0.171970\pi$
$109$	10.7401i	1.02872i	−0.857576	+	0.514358i	$0.828030\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4.65195i	− 0.437618i	−0.975768	−	0.218809i	$-0.929783\pi$
$113$	− 4.65195i	− 0.437618i	0.975768	−	0.218809i	$-0.0702172\pi$
$114$	0	0
$115$	11.5159	1.07387
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.06418	−0.0975537
$120$	0	0
$121$	−6.49838	−0.590762
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−29.1207	−2.60463
$126$	0	0
$127$	0.663789i	0.0589018i	0.999566	+	0.0294509i	$0.00937587\pi$
$127$	0.663789i	0.0589018i	−0.999566	+	0.0294509i	$0.990624\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.51332i	0.481701i	0.970562	+	0.240850i	$0.0774263\pi$
$131$	5.51332i	0.481701i	−0.970562	+	0.240850i	$0.922574\pi$
$132$	0	0
$133$	− 1.79897i	− 0.155990i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.94595i	0.337125i	0.985691	+	0.168563i	$0.0539126\pi$
$137$	3.94595i	0.337125i	−0.985691	+	0.168563i	$0.946087\pi$
$138$	0	0
$139$	−18.1864	−1.54255	−0.771276	−	0.636501i	$-0.780381\pi$
$139$	−18.1864	−1.54255	−0.771276	+	0.636501i	$0.780381\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−20.6580	−1.72751
$144$	0	0
$145$	−8.55364	−0.710341
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.6610	−1.20108	−0.600539	−	0.799595i	$-0.705047\pi$
$149$	−14.6610	−1.20108	−0.600539	+	0.799595i	$0.705047\pi$
$150$	0	0
$151$	− 21.2738i	− 1.73123i	−0.500707	−	0.865617i	$-0.666927\pi$
$151$	− 21.2738i	− 1.73123i	0.500707	−	0.865617i	$-0.333073\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 17.2051i	− 1.38195i
$156$	0	0
$157$	− 11.2507i	− 0.897907i	−0.893555	−	0.448954i	$-0.851797\pi$
$157$	− 11.2507i	− 0.897907i	0.893555	−	0.448954i	$-0.148203\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.925575i	0.0729455i
$162$	0	0
$163$	13.3410	1.04495	0.522473	−	0.852656i	$-0.325009\pi$
$163$	13.3410	1.04495	0.522473	+	0.852656i	$0.325009\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.7564	1.91571	0.957855	−	0.287251i	$-0.0927414\pi$
$167$	24.7564	1.91571	0.957855	+	0.287251i	$0.0927414\pi$
$168$	0	0
$169$	−11.3882	−0.876017
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.27220	0.628924	0.314462	−	0.949270i	$-0.398176\pi$
$173$	8.27220	0.628924	0.314462	+	0.949270i	$0.398176\pi$
$174$	0	0
$175$	− 4.00000i	− 0.302372i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 12.3748i	− 0.924939i	−0.886635	−	0.462470i	$-0.846963\pi$
$179$	− 12.3748i	− 0.924939i	0.886635	−	0.462470i	$-0.153037\pi$
$180$	0	0
$181$	2.41487i	0.179496i	0.995965	+	0.0897478i	$0.0286061\pi$
$181$	2.41487i	0.179496i	−0.995965	+	0.0897478i	$0.971394\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.14211i	0.672141i
$186$	0	0
$187$	−13.4127	−0.980832
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.4313	−1.04421	−0.522105	−	0.852881i	$-0.674853\pi$
$191$	−14.4313	−1.04421	−0.522105	+	0.852881i	$0.674853\pi$
$192$	0	0
$193$	−17.2781	−1.24370	−0.621851	−	0.783135i	$-0.713619\pi$
$193$	−17.2781	−1.24370	−0.621851	+	0.783135i	$0.713619\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.1427	1.07887	0.539435	−	0.842027i	$-0.318638\pi$
$197$	15.1427	1.07887	0.539435	+	0.842027i	$0.318638\pi$
$198$	0	0
$199$	− 21.6057i	− 1.53158i	−0.643088	−	0.765792i	$-0.722347\pi$
$199$	− 21.6057i	− 1.53158i	0.643088	−	0.765792i	$-0.277653\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 0.687485i	− 0.0482520i
$204$	0	0
$205$	6.73741i	0.470561i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 22.6737i	− 1.56837i
$210$	0	0
$211$	−7.40258	−0.509615	−0.254807	−	0.966992i	$-0.582012\pi$
$211$	−7.40258	−0.509615	−0.254807	+	0.966992i	$0.582012\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	24.7564	1.68837
$216$	0	0
$217$	1.38283	0.0938729
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−15.8346	−1.06515
$222$	0	0
$223$	− 3.00432i	− 0.201184i	−0.994928	−	0.100592i	$-0.967926\pi$
$223$	− 3.00432i	− 0.201184i	0.994928	−	0.100592i	$-0.0320737\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.40653i	− 0.557961i	−0.960297	−	0.278980i	$-0.910004\pi$
$227$	− 8.40653i	− 0.557961i	0.960297	−	0.278980i	$-0.0899965\pi$
$228$	0	0
$229$	3.17283i	0.209666i	0.994490	+	0.104833i	$0.0334308\pi$
$229$	3.17283i	0.209666i	−0.994490	+	0.104833i	$0.966569\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 10.0341i	− 0.657357i	−0.944442	−	0.328678i	$-0.893397\pi$
$233$	− 10.0341i	− 0.657357i	0.944442	−	0.328678i	$-0.106603\pi$
$234$	0	0
$235$	−32.8344	−2.14188
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.91793	−0.641538	−0.320769	−	0.947158i	$-0.603941\pi$
$239$	−9.91793	−0.641538	−0.320769	+	0.947158i	$0.603941\pi$
$240$	0	0
$241$	−0.669181	−0.0431057	−0.0215529	−	0.999768i	$-0.506861\pi$
$241$	−0.669181	−0.0431057	−0.0215529	+	0.999768i	$0.506861\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−28.4510	−1.81767
$246$	0	0
$247$	− 26.7678i	− 1.70320i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.38823i	0.213863i	0.994266	+	0.106931i	$0.0341025\pi$
$251$	3.38823i	0.213863i	−0.994266	+	0.106931i	$0.965897\pi$
$252$	0	0
$253$	11.6657i	0.733415i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.5780i	1.40837i	0.710014	+	0.704187i	$0.248688\pi$
$257$	22.5780i	1.40837i	−0.710014	+	0.704187i	$0.751312\pi$
$258$	0	0
$259$	−0.734782	−0.0456571
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.41753	0.457384	0.228692	−	0.973499i	$-0.426555\pi$
$263$	7.41753	0.457384	0.228692	+	0.973499i	$0.426555\pi$
$264$	0	0
$265$	8.49838	0.522051
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−19.9774	−1.21804	−0.609021	−	0.793154i	$-0.708438\pi$
$269$	−19.9774	−1.21804	−0.609021	+	0.793154i	$0.708438\pi$
$270$	0	0
$271$	− 26.1040i	− 1.58571i	−0.609412	−	0.792853i	$-0.708595\pi$
$271$	− 26.1040i	− 1.58571i	0.609412	−	0.792853i	$-0.291405\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 50.4149i	− 3.04013i
$276$	0	0
$277$	24.5342i	1.47411i	0.675830	+	0.737057i	$0.263785\pi$
$277$	24.5342i	1.47411i	−0.675830	+	0.737057i	$0.736215\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	27.5220i	1.64183i	0.571053	+	0.820913i	$0.306535\pi$
$281$	27.5220i	1.64183i	−0.571053	+	0.820913i	$0.693465\pi$
$282$	0	0
$283$	0.961839	0.0571754	0.0285877	−	0.999591i	$-0.490899\pi$
$283$	0.961839	0.0571754	0.0285877	+	0.999591i	$0.490899\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.541509	−0.0319643
$288$	0	0
$289$	6.71904	0.395238
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.72768	−0.159353	−0.0796763	−	0.996821i	$-0.525389\pi$
$293$	−2.72768	−0.159353	−0.0796763	+	0.996821i	$0.525389\pi$
$294$	0	0
$295$	25.7721i	1.50051i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.7721i	0.796463i
$300$	0	0
$301$	1.98976i	0.114688i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	43.4247i	2.48649i
$306$	0	0
$307$	−5.29186	−0.302022	−0.151011	−	0.988532i	$-0.548253\pi$
$307$	−5.29186	−0.302022	−0.151011	+	0.988532i	$0.548253\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.3415	−1.38028	−0.690140	−	0.723676i	$-0.742451\pi$
$311$	−24.3415	−1.38028	−0.690140	+	0.723676i	$0.742451\pi$
$312$	0	0
$313$	−30.4316	−1.72010	−0.860048	−	0.510213i	$-0.829567\pi$
$313$	−30.4316	−1.72010	−0.860048	+	0.510213i	$0.829567\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−29.7769	−1.67244	−0.836220	−	0.548395i	$-0.815239\pi$
$317$	−29.7769	−1.67244	−0.836220	+	0.548395i	$0.815239\pi$
$318$	0	0
$319$	− 8.66486i	− 0.485139i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 17.3796i	− 0.967026i
$324$	0	0
$325$	− 59.5182i	− 3.30148i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 2.63901i	− 0.145493i
$330$	0	0
$331$	−19.0784	−1.04865	−0.524323	−	0.851520i	$-0.675681\pi$
$331$	−19.0784	−1.04865	−0.524323	+	0.851520i	$0.675681\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−48.2032	−2.63362
$336$	0	0
$337$	2.66055	0.144929	0.0724647	−	0.997371i	$-0.476914\pi$
$337$	2.66055	0.144929	0.0724647	+	0.997371i	$0.476914\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	17.4289	0.943825
$342$	0	0
$343$	− 4.60997i	− 0.248915i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 22.1025i	− 1.18652i	−0.805010	−	0.593261i	$-0.797840\pi$
$347$	− 22.1025i	− 1.18652i	0.805010	−	0.593261i	$-0.202160\pi$
$348$	0	0
$349$	20.6793i	1.10694i	0.832869	+	0.553470i	$0.186697\pi$
$349$	20.6793i	1.10694i	−0.832869	+	0.553470i	$0.813303\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 13.6219i	− 0.725019i	−0.931980	−	0.362510i	$-0.881920\pi$
$353$	− 13.6219i	− 0.725019i	0.931980	−	0.362510i	$-0.118080\pi$
$354$	0	0
$355$	31.6436	1.67947
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.3818	−0.917377	−0.458688	−	0.888597i	$-0.651681\pi$
$359$	−17.3818	−0.917377	−0.458688	+	0.888597i	$0.651681\pi$
$360$	0	0
$361$	10.3796	0.546294
$362$	0	0
$363$	0	0
$364$	0	0
$365$	25.6743	1.34385
$366$	0	0
$367$	18.9343i	0.988363i	0.869359	+	0.494181i	$0.164532\pi$
$367$	18.9343i	0.988363i	−0.869359	+	0.494181i	$0.835468\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.683044i	0.0354619i
$372$	0	0
$373$	27.9703i	1.44825i	0.689670	+	0.724124i	$0.257756\pi$
$373$	27.9703i	1.44825i	−0.689670	+	0.724124i	$0.742244\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 10.2295i	− 0.526844i
$378$	0	0
$379$	11.7997	0.606111	0.303056	−	0.952973i	$-0.401993\pi$
$379$	11.7997	0.606111	0.303056	+	0.952973i	$0.401993\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.7556	1.00947	0.504733	−	0.863276i	$-0.331591\pi$
$383$	19.7556	1.00947	0.504733	+	0.863276i	$0.331591\pi$
$384$	0	0
$385$	5.73307	0.292184
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.53114	−0.280440	−0.140220	−	0.990120i	$-0.544781\pi$
$389$	−5.53114	−0.280440	−0.140220	+	0.990120i	$0.544781\pi$
$390$	0	0
$391$	8.94186i	0.452209i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.42854i	0.172509i
$396$	0	0
$397$	31.5247i	1.58218i	0.611700	+	0.791090i	$0.290486\pi$
$397$	31.5247i	1.58218i	−0.611700	+	0.791090i	$0.709514\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.7510i	1.03626i	0.855303	+	0.518129i	$0.173371\pi$
$401$	20.7510i	1.03626i	−0.855303	+	0.518129i	$0.826629\pi$
$402$	0	0
$403$	20.5760	1.02496
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.26099	−0.459050
$408$	0	0
$409$	11.9968	0.593202	0.296601	−	0.955002i	$-0.404147\pi$
$409$	11.9968	0.593202	0.296601	+	0.955002i	$0.404147\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.07139	−0.101927
$414$	0	0
$415$	28.4446i	1.39629i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.0890i	1.17682i	0.808562	+	0.588412i	$0.200246\pi$
$419$	24.0890i	1.17682i	−0.808562	+	0.588412i	$0.799754\pi$
$420$	0	0
$421$	− 30.4707i	− 1.48505i	−0.669818	−	0.742526i	$-0.733628\pi$
$421$	− 30.4707i	− 1.48505i	0.669818	−	0.742526i	$-0.266372\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 38.6435i	− 1.87448i
$426$	0	0
$427$	−3.49019	−0.168902
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−0.902398	−0.0434670	−0.0217335	−	0.999764i	$-0.506919\pi$
$431$	−0.902398	−0.0434670	−0.0217335	+	0.999764i	$0.506919\pi$
$432$	0	0
$433$	0.559027	0.0268651	0.0134326	−	0.999910i	$-0.495724\pi$
$433$	0.559027	0.0268651	0.0134326	+	0.999910i	$0.495724\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15.1159	−0.723092
$438$	0	0
$439$	23.2651i	1.11038i	0.831722	+	0.555192i	$0.187355\pi$
$439$	23.2651i	1.11038i	−0.831722	+	0.555192i	$0.812645\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	25.4998i	1.21153i	0.795643	+	0.605766i	$0.207133\pi$
$443$	25.4998i	1.21153i	−0.795643	+	0.605766i	$0.792867\pi$
$444$	0	0
$445$	− 37.7515i	− 1.78959i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 16.6758i	− 0.786981i	−0.919329	−	0.393490i	$-0.871268\pi$
$449$	− 16.6758i	− 0.786981i	0.919329	−	0.393490i	$-0.128732\pi$
$450$	0	0
$451$	−6.82502	−0.321378
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.76828	0.317302
$456$	0	0
$457$	−6.60565	−0.308999	−0.154500	−	0.987993i	$-0.549377\pi$
$457$	−6.60565	−0.308999	−0.154500	+	0.987993i	$0.549377\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.8038	0.875781	0.437891	−	0.899028i	$-0.355726\pi$
$461$	18.8038	0.875781	0.437891	+	0.899028i	$0.355726\pi$
$462$	0	0
$463$	24.9408i	1.15910i	0.814938	+	0.579548i	$0.196771\pi$
$463$	24.9408i	1.15910i	−0.814938	+	0.579548i	$0.803229\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.2003i	0.472015i	0.971751	+	0.236007i	$0.0758389\pi$
$467$	10.2003i	0.472015i	−0.971751	+	0.236007i	$0.924161\pi$
$468$	0	0
$469$	− 3.87426i	− 0.178897i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	25.0784i	1.15310i
$474$	0	0
$475$	65.3255	2.99734
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.8667	−0.496511	−0.248256	−	0.968695i	$-0.579857\pi$
$479$	−10.8667	−0.496511	−0.248256	+	0.968695i	$0.579857\pi$
$480$	0	0
$481$	−10.9332	−0.498512
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.18742	−0.280956
$486$	0	0
$487$	− 21.1073i	− 0.956462i	−0.878234	−	0.478231i	$-0.841278\pi$
$487$	− 21.1073i	− 0.956462i	0.878234	−	0.478231i	$-0.158722\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.3838i	0.965037i	0.875886	+	0.482518i	$0.160278\pi$
$491$	21.3838i	0.965037i	−0.875886	+	0.482518i	$0.839722\pi$
$492$	0	0
$493$	− 6.64171i	− 0.299127i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.54330i	0.114083i
$498$	0	0
$499$	−25.9962	−1.16375	−0.581876	−	0.813278i	$-0.697681\pi$
$499$	−25.9962	−1.16375	−0.581876	+	0.813278i	$0.697681\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	28.4476	1.26842	0.634208	−	0.773163i	$-0.281326\pi$
$503$	28.4476	1.26842	0.634208	+	0.773163i	$0.281326\pi$
$504$	0	0
$505$	39.8371	1.77273
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.85660	0.0822925	0.0411462	−	0.999153i	$-0.486899\pi$
$509$	1.85660	0.0822925	0.0411462	+	0.999153i	$0.486899\pi$
$510$	0	0
$511$	2.06353i	0.0912852i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 20.6292i	− 0.909032i
$516$	0	0
$517$	− 33.2613i	− 1.46283i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	40.2415i	1.76301i	0.472173	+	0.881506i	$0.343470\pi$
$521$	40.2415i	1.76301i	−0.472173	+	0.881506i	$0.656530\pi$
$522$	0	0
$523$	21.0040	0.918440	0.459220	−	0.888323i	$-0.348129\pi$
$523$	21.0040	0.918440	0.459220	+	0.888323i	$0.348129\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.3594	0.581944
$528$	0	0
$529$	−15.2228	−0.661862
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.05741	−0.349005
$534$	0	0
$535$	54.2167i	2.34399i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 28.8210i	− 1.24141i
$540$	0	0
$541$	− 5.33020i	− 0.229163i	−0.993414	−	0.114582i	$-0.963447\pi$
$541$	− 5.33020i	− 0.229163i	0.993414	−	0.114582i	$-0.0365528\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 44.3503i	− 1.89976i
$546$	0	0
$547$	2.18325	0.0933490	0.0466745	−	0.998910i	$-0.485138\pi$
$547$	2.18325	0.0933490	0.0466745	+	0.998910i	$0.485138\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11.2276	0.478311
$552$	0	0
$553$	−0.275564	−0.0117182
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.6744	−0.621775	−0.310887	−	0.950447i	$-0.600626\pi$
$557$	−14.6744	−0.621775	−0.310887	+	0.950447i	$0.600626\pi$
$558$	0	0
$559$	29.6067i	1.25223i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.34328i	0.225193i	0.993641	+	0.112596i	$0.0359167\pi$
$563$	5.34328i	0.225193i	−0.993641	+	0.112596i	$0.964083\pi$
$564$	0	0
$565$	19.2098i	0.808162i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.04650i	0.0857939i	0.999080	+	0.0428969i	$0.0136587\pi$
$569$	2.04650i	0.0857939i	−0.999080	+	0.0428969i	$0.986341\pi$
$570$	0	0
$571$	−8.59604	−0.359733	−0.179866	−	0.983691i	$-0.557567\pi$
$571$	−8.59604	−0.359733	−0.179866	+	0.983691i	$0.557567\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−33.6102	−1.40164
$576$	0	0
$577$	26.6609	1.10991	0.554954	−	0.831881i	$-0.312736\pi$
$577$	26.6609	1.10991	0.554954	+	0.831881i	$0.312736\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.28619	−0.0948470
$582$	0	0
$583$	8.60889i	0.356544i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 46.1602i	− 1.90524i	−0.304168	−	0.952618i	$-0.598378\pi$
$587$	− 46.1602i	− 1.90524i	0.304168	−	0.952618i	$-0.401622\pi$
$588$	0	0
$589$	22.5836i	0.930540i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 15.7409i	− 0.646402i	−0.946330	−	0.323201i	$-0.895241\pi$
$593$	− 15.7409i	− 0.646402i	0.946330	−	0.323201i	$-0.104759\pi$
$594$	0	0
$595$	4.39445	0.180155
$596$	0	0
$597$	0	0
$598$	0	0
$599$	43.0406	1.75859	0.879297	−	0.476275i	$-0.158013\pi$
$599$	43.0406	1.75859	0.879297	+	0.476275i	$0.158013\pi$
$600$	0	0
$601$	39.3735	1.60608	0.803039	−	0.595927i	$-0.203215\pi$
$601$	39.3735	1.60608	0.803039	+	0.595927i	$0.203215\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	26.8345	1.09098
$606$	0	0
$607$	− 29.7721i	− 1.20841i	−0.796828	−	0.604207i	$-0.793490\pi$
$607$	− 29.7721i	− 1.20841i	0.796828	−	0.604207i	$-0.206510\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 39.2673i	− 1.58858i
$612$	0	0
$613$	0.615900i	0.0248760i	0.999923	+	0.0124380i	$0.00395924\pi$
$613$	0.615900i	0.0248760i	−0.999923	+	0.0124380i	$0.996041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.34795i	− 0.0945250i	−0.998883	−	0.0472625i	$-0.984950\pi$
$617$	− 2.34795i	− 0.0945250i	0.998883	−	0.0472625i	$-0.0150497\pi$
$618$	0	0
$619$	−19.1744	−0.770682	−0.385341	−	0.922774i	$-0.625916\pi$
$619$	−19.1744	−0.770682	−0.385341	+	0.922774i	$0.625916\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.03422	0.121563
$624$	0	0
$625$	59.9910	2.39964
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.09864	−0.283041
$630$	0	0
$631$	3.77969i	0.150467i	0.997166	+	0.0752336i	$0.0239702\pi$
$631$	3.77969i	0.150467i	−0.997166	+	0.0752336i	$0.976030\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 2.74106i	− 0.108776i
$636$	0	0
$637$	− 34.0251i	− 1.34812i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	14.0422i	0.554634i	0.960779	+	0.277317i	$0.0894452\pi$
$641$	14.0422i	0.554634i	−0.960779	+	0.277317i	$0.910555\pi$
$642$	0	0
$643$	−8.79700	−0.346920	−0.173460	−	0.984841i	$-0.555495\pi$
$643$	−8.79700	−0.346920	−0.173460	+	0.984841i	$0.555495\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29.5118	1.16023	0.580114	−	0.814535i	$-0.303008\pi$
$647$	29.5118	1.16023	0.580114	+	0.814535i	$0.303008\pi$
$648$	0	0
$649$	−26.1073	−1.02480
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.0133810	−0.000523638 0	−0.000261819	−	1.00000i	$-0.500083\pi$
$653$	−0.0133810	−0.000523638 0	−0.000261819		1.00000i	$0.500083\pi$
$654$	0	0
$655$	− 22.7667i	− 0.889570i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	39.6345i	1.54394i	0.635660	+	0.771970i	$0.280728\pi$
$659$	39.6345i	1.54394i	−0.635660	+	0.771970i	$0.719272\pi$
$660$	0	0
$661$	13.6264i	0.530007i	0.964247	+	0.265003i	$0.0853731\pi$
$661$	13.6264i	0.530007i	−0.964247	+	0.265003i	$0.914627\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.42867i	0.288071i
$666$	0	0
$667$	−5.77662	−0.223672
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−43.9894	−1.69819
$672$	0	0
$673$	6.44312	0.248364	0.124182	−	0.992259i	$-0.460369\pi$
$673$	6.44312	0.248364	0.124182	+	0.992259i	$0.460369\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.3441	−0.589721	−0.294860	−	0.955540i	$-0.595273\pi$
$677$	−15.3441	−0.589721	−0.294860	+	0.955540i	$0.595273\pi$
$678$	0	0
$679$	− 0.497304i	− 0.0190848i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	17.2454i	0.659878i	0.944002	+	0.329939i	$0.107028\pi$
$683$	17.2454i	0.659878i	−0.944002	+	0.329939i	$0.892972\pi$
$684$	0	0
$685$	− 16.2945i	− 0.622579i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10.1634i	0.387194i
$690$	0	0
$691$	−19.9417	−0.758616	−0.379308	−	0.925270i	$-0.623838\pi$
$691$	−19.9417	−0.758616	−0.379308	+	0.925270i	$0.623838\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	75.0992	2.84867
$696$	0	0
$697$	−5.23145	−0.198155
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.45975	0.130673	0.0653364	−	0.997863i	$-0.479188\pi$
$701$	3.45975	0.130673	0.0653364	+	0.997863i	$0.479188\pi$
$702$	0	0
$703$	− 12.0000i	− 0.452589i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.20184i	0.120418i
$708$	0	0
$709$	− 39.5403i	− 1.48497i	−0.669865	−	0.742483i	$-0.733648\pi$
$709$	− 39.5403i	− 1.48497i	0.669865	−	0.742483i	$-0.266352\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 11.6193i	− 0.435147i
$714$	0	0
$715$	85.3055	3.19024
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.62877	−0.172624	−0.0863120	−	0.996268i	$-0.527508\pi$
$719$	−4.62877	−0.172624	−0.0863120	+	0.996268i	$0.527508\pi$
$720$	0	0
$721$	1.65804	0.0617487
$722$	0	0
$723$	0	0
$724$	0	0
$725$	24.9645	0.927158
$726$	0	0
$727$	− 6.15785i	− 0.228382i	−0.993459	−	0.114191i	$-0.963572\pi$
$727$	− 6.15785i	− 0.228382i	0.993459	−	0.114191i	$-0.0364276\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	19.2228i	0.710982i
$732$	0	0
$733$	21.8720i	0.807860i	0.914790	+	0.403930i	$0.132356\pi$
$733$	21.8720i	0.807860i	−0.914790	+	0.403930i	$0.867644\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 48.8301i	− 1.79868i
$738$	0	0
$739$	−24.5677	−0.903738	−0.451869	−	0.892084i	$-0.649243\pi$
$739$	−24.5677	−0.903738	−0.451869	+	0.892084i	$0.649243\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.04895	−0.258601	−0.129300	−	0.991605i	$-0.541273\pi$
$743$	−7.04895	−0.258601	−0.129300	+	0.991605i	$0.541273\pi$
$744$	0	0
$745$	60.5414	2.21806
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.35758	−0.159223
$750$	0	0
$751$	− 24.2221i	− 0.883877i	−0.897045	−	0.441938i	$-0.854291\pi$
$751$	− 24.2221i	− 0.883877i	0.897045	−	0.441938i	$-0.145709\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	87.8480i	3.19712i
$756$	0	0
$757$	29.8780i	1.08593i	0.839754	+	0.542967i	$0.182699\pi$
$757$	29.8780i	1.08593i	−0.839754	+	0.542967i	$0.817301\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 41.8731i	− 1.51790i	−0.651150	−	0.758949i	$-0.725713\pi$
$761$	− 41.8731i	− 1.51790i	0.651150	−	0.758949i	$-0.274287\pi$
$762$	0	0
$763$	3.56458	0.129047
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−30.8214	−1.11290
$768$	0	0
$769$	−12.6172	−0.454987	−0.227493	−	0.973780i	$-0.573053\pi$
$769$	−12.6172	−0.454987	−0.227493	+	0.973780i	$0.573053\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28.6390	1.03007	0.515037	−	0.857168i	$-0.327778\pi$
$773$	28.6390	1.03007	0.515037	+	0.857168i	$0.327778\pi$
$774$	0	0
$775$	50.2145i	1.80376i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 8.84358i	− 0.316854i
$780$	0	0
$781$	32.0551i	1.14702i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	46.4589i	1.65819i
$786$	0	0
$787$	−39.5040	−1.40817	−0.704083	−	0.710118i	$-0.748642\pi$
$787$	−39.5040	−1.40817	−0.704083	+	0.710118i	$0.748642\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.54396	−0.0548968
$792$	0	0
$793$	−51.9325	−1.84418
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.2507	0.965268	0.482634	−	0.875822i	$-0.339680\pi$
$797$	27.2507	0.965268	0.482634	+	0.875822i	$0.339680\pi$
$798$	0	0
$799$	− 25.4951i	− 0.901953i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	26.0081i	0.917808i
$804$	0	0
$805$	− 3.82208i	− 0.134710i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 32.7182i	− 1.15031i	−0.818045	−	0.575155i	$-0.804942\pi$
$809$	− 32.7182i	− 1.15031i	0.818045	−	0.575155i	$-0.195058\pi$
$810$	0	0
$811$	23.3901	0.821336	0.410668	−	0.911785i	$-0.365296\pi$
$811$	23.3901	0.821336	0.410668	+	0.911785i	$0.365296\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−55.0904	−1.92973
$816$	0	0
$817$	−32.4955	−1.13687
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.9906	1.08158	0.540790	−	0.841157i	$-0.318125\pi$
$821$	30.9906	1.08158	0.540790	+	0.841157i	$0.318125\pi$
$822$	0	0
$823$	− 48.3897i	− 1.68676i	−0.537320	−	0.843379i	$-0.680563\pi$
$823$	− 48.3897i	− 1.68676i	0.537320	−	0.843379i	$-0.319437\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	53.6957i	1.86718i	0.358340	+	0.933591i	$0.383343\pi$
$827$	53.6957i	1.86718i	−0.358340	+	0.933591i	$0.616657\pi$
$828$	0	0
$829$	− 14.1371i	− 0.491001i	−0.969396	−	0.245500i	$-0.921048\pi$
$829$	− 14.1371i	− 0.491001i	0.969396	−	0.245500i	$-0.0789523\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 22.0916i	− 0.765427i
$834$	0	0
$835$	−102.229	−3.53780
$836$	0	0
$837$	0	0
$838$	0	0
$839$	42.3914	1.46351	0.731757	−	0.681566i	$-0.238701\pi$
$839$	42.3914	1.46351	0.731757	+	0.681566i	$0.238701\pi$
$840$	0	0
$841$	−24.7093	−0.852046
$842$	0	0
$843$	0	0
$844$	0	0
$845$	47.0266	1.61777
$846$	0	0
$847$	2.15678i	0.0741078i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.17404i	0.211643i
$852$	0	0
$853$	23.9908i	0.821429i	0.911764	+	0.410714i	$0.134721\pi$
$853$	23.9908i	0.821429i	−0.911764	+	0.410714i	$0.865279\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.6665i	0.569316i	0.958629	+	0.284658i	$0.0918801\pi$
$857$	16.6665i	0.569316i	−0.958629	+	0.284658i	$0.908120\pi$
$858$	0	0
$859$	−19.5651	−0.667551	−0.333776	−	0.942653i	$-0.608323\pi$
$859$	−19.5651	−0.667551	−0.333776	+	0.942653i	$0.608323\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.5794	−1.04094	−0.520468	−	0.853881i	$-0.674242\pi$
$863$	−30.5794	−1.04094	−0.520468	+	0.853881i	$0.674242\pi$
$864$	0	0
$865$	−34.1593	−1.16145
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.47313	−0.117818
$870$	0	0
$871$	− 57.6472i	− 1.95330i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.66499i	0.326736i
$876$	0	0
$877$	− 39.2501i	− 1.32538i	−0.748894	−	0.662690i	$-0.769415\pi$
$877$	− 39.2501i	− 1.32538i	0.748894	−	0.662690i	$-0.230585\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.2971i	0.683828i	0.939731	+	0.341914i	$0.111075\pi$
$881$	20.2971i	0.683828i	−0.939731	+	0.341914i	$0.888925\pi$
$882$	0	0
$883$	−13.2658	−0.446431	−0.223216	−	0.974769i	$-0.571655\pi$
$883$	−13.2658	−0.446431	−0.223216	+	0.974769i	$0.571655\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.5947	0.825808	0.412904	−	0.910775i	$-0.364515\pi$
$887$	24.5947	0.825808	0.412904	+	0.910775i	$0.364515\pi$
$888$	0	0
$889$	0.220308	0.00738890
$890$	0	0
$891$	0	0
$892$	0	0
$893$	43.0986	1.44224
$894$	0	0
$895$	51.1008i	1.70811i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.63044i	0.287841i
$900$	0	0
$901$	6.59880i	0.219838i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 9.97197i	− 0.331480i
$906$	0	0
$907$	8.43425	0.280055	0.140027	−	0.990148i	$-0.455281\pi$
$907$	8.43425	0.280055	0.140027	+	0.990148i	$0.455281\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	14.7119	0.487428	0.243714	−	0.969847i	$-0.421634\pi$
$911$	14.7119	0.487428	0.243714	+	0.969847i	$0.421634\pi$
$912$	0	0
$913$	−28.8144	−0.953619
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.82984	0.0604267
$918$	0	0
$919$	− 38.4521i	− 1.26842i	−0.773162	−	0.634209i	$-0.781326\pi$
$919$	− 38.4521i	− 1.26842i	0.773162	−	0.634209i	$-0.218674\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	37.8432i	1.24562i
$924$	0	0
$925$	− 26.6820i	− 0.877298i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 4.49016i	− 0.147317i	−0.997284	−	0.0736587i	$-0.976532\pi$
$929$	− 4.49016i	− 0.147317i	0.997284	−	0.0736587i	$-0.0234675\pi$
$930$	0	0
$931$	37.3450	1.22393
$932$	0	0
$933$	0	0
$934$	0	0
$935$	55.3864	1.81133
$936$	0	0
$937$	−22.9267	−0.748984	−0.374492	−	0.927230i	$-0.622183\pi$
$937$	−22.9267	−0.748984	−0.374492	+	0.927230i	$0.622183\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−55.2632	−1.80153	−0.900764	−	0.434308i	$-0.856993\pi$
$941$	−55.2632	−1.80153	−0.900764	+	0.434308i	$0.856993\pi$
$942$	0	0
$943$	4.55005i	0.148170i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 23.6116i	− 0.767274i	−0.923484	−	0.383637i	$-0.874671\pi$
$947$	− 23.6116i	− 0.767274i	0.923484	−	0.383637i	$-0.125329\pi$
$948$	0	0
$949$	30.7044i	0.996707i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 0.892008i	− 0.0288950i	−0.999896	−	0.0144475i	$-0.995401\pi$
$953$	− 0.892008i	− 0.0288950i	0.999896	−	0.0144475i	$-0.00459894\pi$
$954$	0	0
$955$	59.5926	1.92837
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.30964	0.0422905
$960$	0	0
$961$	13.6404	0.440013
$962$	0	0
$963$	0	0
$964$	0	0
$965$	71.3482	2.29678
$966$	0	0
$967$	− 36.1542i	− 1.16264i	−0.813674	−	0.581321i	$-0.802536\pi$
$967$	− 36.1542i	− 1.16264i	0.813674	−	0.581321i	$-0.197464\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 42.0395i	− 1.34911i	−0.738223	−	0.674556i	$-0.764335\pi$
$971$	− 42.0395i	− 1.34911i	0.738223	−	0.674556i	$-0.235665\pi$
$972$	0	0
$973$	6.03598i	0.193505i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.3353i	0.490620i	0.969445	+	0.245310i	$0.0788897\pi$
$977$	15.3353i	0.490620i	−0.969445	+	0.245310i	$0.921110\pi$
$978$	0	0
$979$	38.2424	1.22223
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.81953	0.185614	0.0928071	−	0.995684i	$-0.470416\pi$
$983$	5.81953	0.185614	0.0928071	+	0.995684i	$0.470416\pi$
$984$	0	0
$985$	−62.5302	−1.99238
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.7190	0.531635
$990$	0	0
$991$	− 42.7700i	− 1.35863i	−0.733845	−	0.679316i	$-0.762276\pi$
$991$	− 42.7700i	− 1.35863i	0.733845	−	0.679316i	$-0.237724\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	89.2186i	2.82842i
$996$	0	0
$997$	51.9901i	1.64654i	0.567648	+	0.823271i	$0.307854\pi$
$997$	51.9901i	1.64654i	−0.567648	+	0.823271i	$0.692146\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.f.a.2591.1		16
3.2	odd	2		5184.2.f.f.2591.13		16
4.3	odd	2	inner	5184.2.f.a.2591.3		16
8.3	odd	2		5184.2.f.f.2591.16		16
8.5	even	2		5184.2.f.f.2591.14		16
9.2	odd	6		1728.2.p.a.287.1		16
9.4	even	3		1728.2.p.c.1439.8		16
9.5	odd	6		576.2.p.a.479.7	yes	16
9.7	even	3		576.2.p.c.95.7	yes	16
12.11	even	2		5184.2.f.f.2591.15		16
24.5	odd	2	inner	5184.2.f.a.2591.2		16
24.11	even	2	inner	5184.2.f.a.2591.4		16
36.7	odd	6		576.2.p.c.95.2	yes	16
36.11	even	6		1728.2.p.a.287.2		16
36.23	even	6		576.2.p.a.479.2	yes	16
36.31	odd	6		1728.2.p.c.1439.7		16
72.5	odd	6		576.2.p.c.479.2	yes	16
72.11	even	6		1728.2.p.c.287.8		16
72.13	even	6		1728.2.p.a.1439.2		16
72.29	odd	6		1728.2.p.c.287.7		16
72.43	odd	6		576.2.p.a.95.7	yes	16
72.59	even	6		576.2.p.c.479.7	yes	16
72.61	even	6		576.2.p.a.95.2	✓	16
72.67	odd	6		1728.2.p.a.1439.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.p.a.95.2	✓	16	72.61	even	6
576.2.p.a.95.7	yes	16	72.43	odd	6
576.2.p.a.479.2	yes	16	36.23	even	6
576.2.p.a.479.7	yes	16	9.5	odd	6
576.2.p.c.95.2	yes	16	36.7	odd	6
576.2.p.c.95.7	yes	16	9.7	even	3
576.2.p.c.479.2	yes	16	72.5	odd	6
576.2.p.c.479.7	yes	16	72.59	even	6
1728.2.p.a.287.1		16	9.2	odd	6
1728.2.p.a.287.2		16	36.11	even	6
1728.2.p.a.1439.1		16	72.67	odd	6
1728.2.p.a.1439.2		16	72.13	even	6
1728.2.p.c.287.7		16	72.29	odd	6
1728.2.p.c.287.8		16	72.11	even	6
1728.2.p.c.1439.7		16	36.31	odd	6
1728.2.p.c.1439.8		16	9.4	even	3
5184.2.f.a.2591.1		16	1.1	even	1	trivial
5184.2.f.a.2591.2		16	24.5	odd	2	inner
5184.2.f.a.2591.3		16	4.3	odd	2	inner
5184.2.f.a.2591.4		16	24.11	even	2	inner
5184.2.f.f.2591.13		16	3.2	odd	2
5184.2.f.f.2591.14		16	8.5	even	2
5184.2.f.f.2591.15		16	12.11	even	2
5184.2.f.f.2591.16		16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.12941 q^{5} -0.331895i q^{7} +O(q^{10})\)	\(q-4.12941 q^{5} -0.331895i q^{7} -4.18311i q^{11} -4.93844i q^{13} -3.20639i q^{17} +5.42029 q^{19} -2.78876 q^{23} +12.0520 q^{25} +2.07139 q^{29} +4.16649i q^{31} +1.37053i q^{35} -2.21390i q^{37} -1.63157i q^{41} -5.99515 q^{43} +7.95135 q^{47} +6.88985 q^{49} -2.05801 q^{53} +17.2738i q^{55} -6.24112i q^{59} -10.5160i q^{61} +20.3929i q^{65} +11.6732 q^{67} -7.66299 q^{71} -6.21742 q^{73} -1.38835 q^{77} -0.830275i q^{79} -6.88829i q^{83} +13.2405i q^{85} +9.14211i q^{89} -1.63904 q^{91} -22.3826 q^{95} +1.49838 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 12 q^{5}+O(q^{10}) \) 16 * q - 12 * q^5	\( 16 q - 12 q^{5} + 28 q^{25} - 36 q^{29} - 12 q^{49} - 48 q^{53} + 28 q^{73} - 132 q^{77} - 16 q^{97}+O(q^{100}) \) 16 * q - 12 * q^5 + 28 * q^25 - 36 * q^29 - 12 * q^49 - 48 * q^53 + 28 * q^73 - 132 * q^77 - 16 * q^97

Introduction

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