LMFDB - Embedded newform 2432.2.c.j.1217.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2432,2,Mod(1217,2432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2432.1217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2432 = 2^{7} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2432.c (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:	$19.4196177716$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 170x^{16} + 6593x^{12} + 64168x^{8} + 95760x^{4} + 4096 $ x^20 + 170x^16 + 6593x^12 + 64168x^8 + 95760x^4 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.3
Root			$1.74565 + 1.74565i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.j.1217.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46871i q^{3} -3.22672i q^{5} -4.04213 q^{7} -3.09455 q^{9} +O(q^{10})$	$q-2.46871i q^{3} -3.22672i q^{5} -4.04213 q^{7} -3.09455 q^{9} -6.41171i q^{11} +0.815411i q^{13} -7.96585 q^{15} -2.62028 q^{17} +1.00000i q^{19} +9.97886i q^{21} -6.16717 q^{23} -5.41171 q^{25} +0.233424i q^{27} +3.85818i q^{29} +7.43670 q^{31} -15.8287 q^{33} +13.0428i q^{35} -5.13742i q^{37} +2.01302 q^{39} +11.5007 q^{41} +0.0837292i q^{43} +9.98525i q^{45} +3.99564 q^{47} +9.33880 q^{49} +6.46871i q^{51} -4.22608i q^{53} -20.6888 q^{55} +2.46871 q^{57} -8.53655i q^{59} +5.72239i q^{61} +12.5086 q^{63} +2.63110 q^{65} -11.4278i q^{67} +15.2250i q^{69} +1.10168 q^{71} +7.25458 q^{73} +13.3600i q^{75} +25.9169i q^{77} -13.9906 q^{79} -8.70740 q^{81} -6.51708i q^{83} +8.45489i q^{85} +9.52474 q^{87} +15.3867 q^{89} -3.29599i q^{91} -18.3591i q^{93} +3.22672 q^{95} -13.2654 q^{97} +19.8414i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 28 q^{9}+O(q^{10}) $ 20 * q - 28 * q^9	$ 20 q - 28 q^{9} + 8 q^{17} - 20 q^{25} - 16 q^{33} + 24 q^{41} + 52 q^{49} - 8 q^{57} - 48 q^{65} - 24 q^{73} + 68 q^{81} + 40 q^{89} - 56 q^{97}+O(q^{100}) $ 20 * q - 28 * q^9 + 8 * q^17 - 20 * q^25 - 16 * q^33 + 24 * q^41 + 52 * q^49 - 8 * q^57 - 48 * q^65 - 24 * q^73 + 68 * q^81 + 40 * q^89 - 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times$.

$n$	$1407$	$1921$	$2053$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 2.46871i	− 1.42531i	−0.701513	−	0.712657i	$-0.747492\pi$
$3$	− 2.46871i	− 1.42531i	0.701513	−	0.712657i	$-0.252508\pi$
$4$	0	0
$5$	− 3.22672i	− 1.44303i	−0.692398	−	0.721516i	$-0.743446\pi$
$5$	− 3.22672i	− 1.44303i	0.692398	−	0.721516i	$-0.256554\pi$
$6$	0	0
$7$	−4.04213	−1.52778	−0.763890	−	0.645346i	$-0.776713\pi$
$7$	−4.04213	−1.52778	−0.763890	+	0.645346i	$0.776713\pi$
$8$	0	0
$9$	−3.09455	−1.03152
$10$	0	0
$11$	− 6.41171i	− 1.93320i	−0.256286	−	0.966601i	$-0.582499\pi$
$11$	− 6.41171i	− 1.93320i	0.256286	−	0.966601i	$-0.417501\pi$
$12$	0	0
$13$	0.815411i	0.226154i	0.993586	+	0.113077i	$0.0360707\pi$
$13$	0.815411i	0.226154i	−0.993586	+	0.113077i	$0.963929\pi$
$14$	0	0
$15$	−7.96585	−2.05677
$16$	0	0
$17$	−2.62028	−0.635510	−0.317755	−	0.948173i	$-0.602929\pi$
$17$	−2.62028	−0.635510	−0.317755	+	0.948173i	$0.602929\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	9.97886i	2.17757i
$22$	0	0
$23$	−6.16717	−1.28594	−0.642972	−	0.765890i	$-0.722299\pi$
$23$	−6.16717	−1.28594	−0.642972	+	0.765890i	$0.722299\pi$
$24$	0	0
$25$	−5.41171	−1.08234
$26$	0	0
$27$	0.233424i	0.0449225i
$28$	0	0
$29$	3.85818i	0.716446i	0.933636	+	0.358223i	$0.116617\pi$
$29$	3.85818i	0.716446i	−0.933636	+	0.358223i	$0.883383\pi$
$30$	0	0
$31$	7.43670	1.33567	0.667835	−	0.744309i	$-0.267221\pi$
$31$	7.43670	1.33567	0.667835	+	0.744309i	$0.267221\pi$
$32$	0	0
$33$	−15.8287	−2.75542
$34$	0	0
$35$	13.0428i	2.20464i
$36$	0	0
$37$	− 5.13742i	− 0.844586i	−0.906459	−	0.422293i	$-0.861225\pi$
$37$	− 5.13742i	− 0.844586i	0.906459	−	0.422293i	$-0.138775\pi$
$38$	0	0
$39$	2.01302	0.322341
$40$	0	0
$41$	11.5007	1.79611	0.898054	−	0.439886i	$-0.144981\pi$
$41$	11.5007	1.79611	0.898054	+	0.439886i	$0.144981\pi$
$42$	0	0
$43$	0.0837292i	0.0127686i	0.999980	+	0.00638429i	$0.00203220\pi$
$43$	0.0837292i	0.0127686i	−0.999980	+	0.00638429i	$0.997968\pi$
$44$	0	0
$45$	9.98525i	1.48851i
$46$	0	0
$47$	3.99564	0.582824	0.291412	−	0.956598i	$-0.405875\pi$
$47$	3.99564	0.582824	0.291412	+	0.956598i	$0.405875\pi$
$48$	0	0
$49$	9.33880	1.33411
$50$	0	0
$51$	6.46871i	0.905801i
$52$	0	0
$53$	− 4.22608i	− 0.580497i	−0.956951	−	0.290248i	$-0.906262\pi$
$53$	− 4.22608i	− 0.580497i	0.956951	−	0.290248i	$-0.0937379\pi$
$54$	0	0
$55$	−20.6888	−2.78967
$56$	0	0
$57$	2.46871	0.326989
$58$	0	0
$59$	− 8.53655i	− 1.11136i	−0.831395	−	0.555682i	$-0.812457\pi$
$59$	− 8.53655i	− 1.11136i	0.831395	−	0.555682i	$-0.187543\pi$
$60$	0	0
$61$	5.72239i	0.732677i	0.930482	+	0.366339i	$0.119389\pi$
$61$	5.72239i	0.732677i	−0.930482	+	0.366339i	$0.880611\pi$
$62$	0	0
$63$	12.5086	1.57593
$64$	0	0
$65$	2.63110	0.326348
$66$	0	0
$67$	− 11.4278i	− 1.39613i	−0.716036	−	0.698063i	$-0.754045\pi$
$67$	− 11.4278i	− 1.39613i	0.716036	−	0.698063i	$-0.245955\pi$
$68$	0	0
$69$	15.2250i	1.83287i
$70$	0	0
$71$	1.10168	0.130745	0.0653725	−	0.997861i	$-0.479176\pi$
$71$	1.10168	0.130745	0.0653725	+	0.997861i	$0.479176\pi$
$72$	0	0
$73$	7.25458	0.849085	0.424542	−	0.905408i	$-0.360435\pi$
$73$	7.25458	0.849085	0.424542	+	0.905408i	$0.360435\pi$
$74$	0	0
$75$	13.3600i	1.54268i
$76$	0	0
$77$	25.9169i	2.95351i
$78$	0	0
$79$	−13.9906	−1.57407	−0.787033	−	0.616911i	$-0.788384\pi$
$79$	−13.9906	−1.57407	−0.787033	+	0.616911i	$0.788384\pi$
$80$	0	0
$81$	−8.70740	−0.967489
$82$	0	0
$83$	− 6.51708i	− 0.715343i	−0.933848	−	0.357671i	$-0.883571\pi$
$83$	− 6.51708i	− 0.715343i	0.933848	−	0.357671i	$-0.116429\pi$
$84$	0	0
$85$	8.45489i	0.917062i
$86$	0	0
$87$	9.52474	1.02116
$88$	0	0
$89$	15.3867	1.63098	0.815492	−	0.578768i	$-0.196466\pi$
$89$	15.3867	1.63098	0.815492	+	0.578768i	$0.196466\pi$
$90$	0	0
$91$	− 3.29599i	− 0.345514i
$92$	0	0
$93$	− 18.3591i	− 1.90375i
$94$	0	0
$95$	3.22672	0.331054
$96$	0	0
$97$	−13.2654	−1.34690	−0.673449	−	0.739234i	$-0.735188\pi$
$97$	−13.2654	−1.34690	−0.673449	+	0.739234i	$0.735188\pi$
$98$	0	0
$99$	19.8414i	1.99413i
$100$	0	0
$101$	− 2.76311i	− 0.274940i	−0.990506	−	0.137470i	$-0.956103\pi$
$101$	− 2.76311i	− 0.274940i	0.990506	−	0.137470i	$-0.0438971\pi$
$102$	0	0
$103$	−0.529145	−0.0521382	−0.0260691	−	0.999660i	$-0.508299\pi$
$103$	−0.529145	−0.0521382	−0.0260691	+	0.999660i	$0.508299\pi$
$104$	0	0
$105$	32.1990	3.14230
$106$	0	0
$107$	− 0.211777i	− 0.0204732i	−0.999948	−	0.0102366i	$-0.996742\pi$
$107$	− 0.211777i	− 0.0204732i	0.999948	−	0.0102366i	$-0.00325847\pi$
$108$	0	0
$109$	9.73908i	0.932835i	0.884565	+	0.466417i	$0.154456\pi$
$109$	9.73908i	0.932835i	−0.884565	+	0.466417i	$0.845544\pi$
$110$	0	0
$111$	−12.6828	−1.20380
$112$	0	0
$113$	−6.09269	−0.573152	−0.286576	−	0.958058i	$-0.592517\pi$
$113$	−6.09269	−0.573152	−0.286576	+	0.958058i	$0.592517\pi$
$114$	0	0
$115$	19.8997i	1.85566i
$116$	0	0
$117$	− 2.52333i	− 0.233282i
$118$	0	0
$119$	10.5915	0.970921
$120$	0	0
$121$	−30.1100	−2.73727
$122$	0	0
$123$	− 28.3919i	− 2.56002i
$124$	0	0
$125$	1.32846i	0.118821i
$126$	0	0
$127$	−6.64012	−0.589215	−0.294608	−	0.955618i	$-0.595189\pi$
$127$	−6.64012	−0.589215	−0.294608	+	0.955618i	$0.595189\pi$
$128$	0	0
$129$	0.206704	0.0181992
$130$	0	0
$131$	3.61315i	0.315682i	0.987465	+	0.157841i	$0.0504534\pi$
$131$	3.61315i	0.315682i	−0.987465	+	0.157841i	$0.949547\pi$
$132$	0	0
$133$	− 4.04213i	− 0.350497i
$134$	0	0
$135$	0.753194	0.0648246
$136$	0	0
$137$	11.7468	1.00360	0.501799	−	0.864984i	$-0.332672\pi$
$137$	11.7468	1.00360	0.501799	+	0.864984i	$0.332672\pi$
$138$	0	0
$139$	20.4757i	1.73672i	0.495931	+	0.868362i	$0.334827\pi$
$139$	20.4757i	1.73672i	−0.495931	+	0.868362i	$0.665173\pi$
$140$	0	0
$141$	− 9.86410i	− 0.830707i
$142$	0	0
$143$	5.22817	0.437202
$144$	0	0
$145$	12.4493	1.03385
$146$	0	0
$147$	− 23.0548i	− 1.90153i
$148$	0	0
$149$	20.4213i	1.67298i	0.547982	+	0.836490i	$0.315396\pi$
$149$	20.4213i	1.67298i	−0.547982	+	0.836490i	$0.684604\pi$
$150$	0	0
$151$	13.1686	1.07164	0.535822	−	0.844331i	$-0.320002\pi$
$151$	13.1686	1.07164	0.535822	+	0.844331i	$0.320002\pi$
$152$	0	0
$153$	8.10858	0.655540
$154$	0	0
$155$	− 23.9961i	− 1.92742i
$156$	0	0
$157$	− 15.0668i	− 1.20246i	−0.799074	−	0.601232i	$-0.794677\pi$
$157$	− 15.0668i	− 1.20246i	0.799074	−	0.601232i	$-0.205323\pi$
$158$	0	0
$159$	−10.4330	−0.827389
$160$	0	0
$161$	24.9285	1.96464
$162$	0	0
$163$	11.8860i	0.930982i	0.885053	+	0.465491i	$0.154122\pi$
$163$	11.8860i	0.930982i	−0.885053	+	0.465491i	$0.845878\pi$
$164$	0	0
$165$	51.0747i	3.97616i
$166$	0	0
$167$	19.2419	1.48898	0.744491	−	0.667632i	$-0.232692\pi$
$167$	19.2419	1.48898	0.744491	+	0.667632i	$0.232692\pi$
$168$	0	0
$169$	12.3351	0.948854
$170$	0	0
$171$	− 3.09455i	− 0.236646i
$172$	0	0
$173$	1.53489i	0.116696i	0.998296	+	0.0583478i	$0.0185833\pi$
$173$	1.53489i	0.116696i	−0.998296	+	0.0583478i	$0.981417\pi$
$174$	0	0
$175$	21.8748	1.65358
$176$	0	0
$177$	−21.0743	−1.58404
$178$	0	0
$179$	− 3.82867i	− 0.286169i	−0.989711	−	0.143084i	$-0.954298\pi$
$179$	− 3.82867i	− 0.286169i	0.989711	−	0.143084i	$-0.0457020\pi$
$180$	0	0
$181$	− 5.89663i	− 0.438293i	−0.975692	−	0.219147i	$-0.929673\pi$
$181$	− 5.89663i	− 0.438293i	0.975692	−	0.219147i	$-0.0703273\pi$
$182$	0	0
$183$	14.1270	1.04429
$184$	0	0
$185$	−16.5770	−1.21877
$186$	0	0
$187$	16.8004i	1.22857i
$188$	0	0
$189$	− 0.943531i	− 0.0686318i
$190$	0	0
$191$	19.8427	1.43577	0.717885	−	0.696162i	$-0.245110\pi$
$191$	19.8427	1.43577	0.717885	+	0.696162i	$0.245110\pi$
$192$	0	0
$193$	−3.62263	−0.260763	−0.130381	−	0.991464i	$-0.541620\pi$
$193$	−3.62263	−0.260763	−0.130381	+	0.991464i	$0.541620\pi$
$194$	0	0
$195$	− 6.49544i	− 0.465148i
$196$	0	0
$197$	13.1190i	0.934690i	0.884075	+	0.467345i	$0.154789\pi$
$197$	13.1190i	0.934690i	−0.884075	+	0.467345i	$0.845211\pi$
$198$	0	0
$199$	−15.9156	−1.12823	−0.564113	−	0.825697i	$-0.690782\pi$
$199$	−15.9156	−1.12823	−0.564113	+	0.825697i	$0.690782\pi$
$200$	0	0
$201$	−28.2120	−1.98992
$202$	0	0
$203$	− 15.5953i	− 1.09457i
$204$	0	0
$205$	− 37.1095i	− 2.59184i
$206$	0	0
$207$	19.0846	1.32647
$208$	0	0
$209$	6.41171	0.443507
$210$	0	0
$211$	5.00713i	0.344705i	0.985035	+	0.172352i	$0.0551368\pi$
$211$	5.00713i	0.344705i	−0.985035	+	0.172352i	$0.944863\pi$
$212$	0	0
$213$	− 2.71973i	− 0.186352i
$214$	0	0
$215$	0.270170	0.0184255
$216$	0	0
$217$	−30.0601	−2.04061
$218$	0	0
$219$	− 17.9095i	− 1.21021i
$220$	0	0
$221$	− 2.13660i	− 0.143723i
$222$	0	0
$223$	8.30156	0.555913	0.277957	−	0.960594i	$-0.410343\pi$
$223$	8.30156	0.555913	0.277957	+	0.960594i	$0.410343\pi$
$224$	0	0
$225$	16.7468	1.11645
$226$	0	0
$227$	− 26.8177i	− 1.77995i	−0.456007	−	0.889976i	$-0.650721\pi$
$227$	− 26.8177i	− 1.77995i	0.456007	−	0.889976i	$-0.349279\pi$
$228$	0	0
$229$	− 6.15107i	− 0.406474i	−0.979130	−	0.203237i	$-0.934854\pi$
$229$	− 6.15107i	− 0.406474i	0.979130	−	0.203237i	$-0.0651463\pi$
$230$	0	0
$231$	63.9815	4.20968
$232$	0	0
$233$	−7.79905	−0.510933	−0.255466	−	0.966818i	$-0.582229\pi$
$233$	−7.79905	−0.510933	−0.255466	+	0.966818i	$0.582229\pi$
$234$	0	0
$235$	− 12.8928i	− 0.841034i
$236$	0	0
$237$	34.5388i	2.24354i
$238$	0	0
$239$	−27.7964	−1.79800	−0.898999	−	0.437951i	$-0.855704\pi$
$239$	−27.7964	−1.79800	−0.898999	+	0.437951i	$0.855704\pi$
$240$	0	0
$241$	−5.53347	−0.356442	−0.178221	−	0.983990i	$-0.557034\pi$
$241$	−5.53347	−0.356442	−0.178221	+	0.983990i	$0.557034\pi$
$242$	0	0
$243$	22.1964i	1.42390i
$244$	0	0
$245$	− 30.1337i	− 1.92517i
$246$	0	0
$247$	−0.815411	−0.0518833
$248$	0	0
$249$	−16.0888	−1.01959
$250$	0	0
$251$	14.9928i	0.946334i	0.880973	+	0.473167i	$0.156889\pi$
$251$	14.9928i	0.946334i	−0.880973	+	0.473167i	$0.843111\pi$
$252$	0	0
$253$	39.5421i	2.48599i
$254$	0	0
$255$	20.8727	1.30710
$256$	0	0
$257$	13.6003	0.848365	0.424182	−	0.905577i	$-0.360562\pi$
$257$	13.6003	0.848365	0.424182	+	0.905577i	$0.360562\pi$
$258$	0	0
$259$	20.7661i	1.29034i
$260$	0	0
$261$	− 11.9393i	− 0.739026i
$262$	0	0
$263$	−26.4415	−1.63046	−0.815228	−	0.579140i	$-0.803388\pi$
$263$	−26.4415	−1.63046	−0.815228	+	0.579140i	$0.803388\pi$
$264$	0	0
$265$	−13.6364	−0.837675
$266$	0	0
$267$	− 37.9853i	− 2.32466i
$268$	0	0
$269$	19.6429i	1.19765i	0.800880	+	0.598825i	$0.204365\pi$
$269$	19.6429i	1.19765i	−0.800880	+	0.598825i	$0.795635\pi$
$270$	0	0
$271$	−14.0146	−0.851327	−0.425663	−	0.904882i	$-0.639959\pi$
$271$	−14.0146	−0.851327	−0.425663	+	0.904882i	$0.639959\pi$
$272$	0	0
$273$	−8.13687	−0.492466
$274$	0	0
$275$	34.6983i	2.09238i
$276$	0	0
$277$	− 4.92581i	− 0.295963i	−0.988990	−	0.147982i	$-0.952722\pi$
$277$	− 4.92581i	− 0.295963i	0.988990	−	0.147982i	$-0.0472777\pi$
$278$	0	0
$279$	−23.0133	−1.37777
$280$	0	0
$281$	−4.90466	−0.292587	−0.146294	−	0.989241i	$-0.546734\pi$
$281$	−4.90466	−0.292587	−0.146294	+	0.989241i	$0.546734\pi$
$282$	0	0
$283$	2.07261i	0.123204i	0.998101	+	0.0616018i	$0.0196209\pi$
$283$	2.07261i	0.123204i	−0.998101	+	0.0616018i	$0.980379\pi$
$284$	0	0
$285$	− 7.96585i	− 0.471856i
$286$	0	0
$287$	−46.4873	−2.74406
$288$	0	0
$289$	−10.1342	−0.596127
$290$	0	0
$291$	32.7485i	1.91975i
$292$	0	0
$293$	10.4606i	0.611117i	0.952173	+	0.305559i	$0.0988432\pi$
$293$	10.4606i	0.611117i	−0.952173	+	0.305559i	$0.901157\pi$
$294$	0	0
$295$	−27.5450	−1.60373
$296$	0	0
$297$	1.49665	0.0868443
$298$	0	0
$299$	− 5.02878i	− 0.290822i
$300$	0	0
$301$	− 0.338444i	− 0.0195076i
$302$	0	0
$303$	−6.82134	−0.391875
$304$	0	0
$305$	18.4645	1.05728
$306$	0	0
$307$	5.87873i	0.335517i	0.985828	+	0.167758i	$0.0536528\pi$
$307$	5.87873i	0.335517i	−0.985828	+	0.167758i	$0.946347\pi$
$308$	0	0
$309$	1.30631i	0.0743132i
$310$	0	0
$311$	12.6861	0.719365	0.359683	−	0.933075i	$-0.382885\pi$
$311$	12.6861	0.719365	0.359683	+	0.933075i	$0.382885\pi$
$312$	0	0
$313$	−6.25089	−0.353321	−0.176661	−	0.984272i	$-0.556529\pi$
$313$	−6.25089	−0.353321	−0.176661	+	0.984272i	$0.556529\pi$
$314$	0	0
$315$	− 40.3617i	− 2.27412i
$316$	0	0
$317$	4.72978i	0.265651i	0.991139	+	0.132826i	$0.0424050\pi$
$317$	4.72978i	0.265651i	−0.991139	+	0.132826i	$0.957595\pi$
$318$	0	0
$319$	24.7375	1.38503
$320$	0	0
$321$	−0.522816	−0.0291808
$322$	0	0
$323$	− 2.62028i	− 0.145796i
$324$	0	0
$325$	− 4.41276i	− 0.244776i
$326$	0	0
$327$	24.0430	1.32958
$328$	0	0
$329$	−16.1509	−0.890428
$330$	0	0
$331$	− 19.8060i	− 1.08864i	−0.838879	−	0.544318i	$-0.816788\pi$
$331$	− 19.8060i	− 1.08864i	0.838879	−	0.544318i	$-0.183212\pi$
$332$	0	0
$333$	15.8980i	0.871206i
$334$	0	0
$335$	−36.8743	−2.01466
$336$	0	0
$337$	9.40107	0.512109	0.256054	−	0.966662i	$-0.417577\pi$
$337$	9.40107	0.512109	0.256054	+	0.966662i	$0.417577\pi$
$338$	0	0
$339$	15.0411i	0.816921i
$340$	0	0
$341$	− 47.6819i	− 2.58212i
$342$	0	0
$343$	−9.45373	−0.510454
$344$	0	0
$345$	49.1267	2.64489
$346$	0	0
$347$	25.2991i	1.35813i	0.734080	+	0.679063i	$0.237614\pi$
$347$	25.2991i	1.35813i	−0.734080	+	0.679063i	$0.762386\pi$
$348$	0	0
$349$	− 28.9343i	− 1.54882i	−0.632687	−	0.774408i	$-0.718048\pi$
$349$	− 28.9343i	− 1.54882i	0.632687	−	0.774408i	$-0.281952\pi$
$350$	0	0
$351$	−0.190337	−0.0101594
$352$	0	0
$353$	−22.4897	−1.19701	−0.598503	−	0.801121i	$-0.704238\pi$
$353$	−22.4897	−1.19701	−0.598503	+	0.801121i	$0.704238\pi$
$354$	0	0
$355$	− 3.55480i	− 0.188669i
$356$	0	0
$357$	− 26.1474i	− 1.38387i
$358$	0	0
$359$	−25.7745	−1.36033	−0.680164	−	0.733060i	$-0.738091\pi$
$359$	−25.7745	−1.36033	−0.680164	+	0.733060i	$0.738091\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	74.3330i	3.90147i
$364$	0	0
$365$	− 23.4085i	− 1.22526i
$366$	0	0
$367$	−3.91733	−0.204483	−0.102242	−	0.994760i	$-0.532601\pi$
$367$	−3.91733	−0.204483	−0.102242	+	0.994760i	$0.532601\pi$
$368$	0	0
$369$	−35.5895	−1.85272
$370$	0	0
$371$	17.0824i	0.886871i
$372$	0	0
$373$	− 7.17543i	− 0.371530i	−0.982594	−	0.185765i	$-0.940524\pi$
$373$	− 7.17543i	− 0.371530i	0.982594	−	0.185765i	$-0.0594762\pi$
$374$	0	0
$375$	3.27959	0.169357
$376$	0	0
$377$	−3.14600	−0.162027
$378$	0	0
$379$	9.11028i	0.467964i	0.972241	+	0.233982i	$0.0751756\pi$
$379$	9.11028i	0.467964i	−0.972241	+	0.233982i	$0.924824\pi$
$380$	0	0
$381$	16.3926i	0.839816i
$382$	0	0
$383$	−28.4774	−1.45513	−0.727564	−	0.686040i	$-0.759348\pi$
$383$	−28.4774	−1.45513	−0.727564	+	0.686040i	$0.759348\pi$
$384$	0	0
$385$	83.6266	4.26201
$386$	0	0
$387$	− 0.259104i	− 0.0131710i
$388$	0	0
$389$	− 27.5101i	− 1.39482i	−0.716674	−	0.697409i	$-0.754336\pi$
$389$	− 27.5101i	− 1.39482i	0.716674	−	0.697409i	$-0.245664\pi$
$390$	0	0
$391$	16.1597	0.817231
$392$	0	0
$393$	8.91983	0.449946
$394$	0	0
$395$	45.1437i	2.27143i
$396$	0	0
$397$	− 32.7251i	− 1.64242i	−0.570624	−	0.821212i	$-0.693298\pi$
$397$	− 32.7251i	− 1.64242i	0.570624	−	0.821212i	$-0.306702\pi$
$398$	0	0
$399$	−9.97886	−0.499568
$400$	0	0
$401$	13.5472	0.676515	0.338257	−	0.941054i	$-0.390163\pi$
$401$	13.5472	0.676515	0.338257	+	0.941054i	$0.390163\pi$
$402$	0	0
$403$	6.06397i	0.302068i
$404$	0	0
$405$	28.0963i	1.39612i
$406$	0	0
$407$	−32.9396	−1.63276
$408$	0	0
$409$	−27.0909	−1.33956	−0.669779	−	0.742561i	$-0.733611\pi$
$409$	−27.0909	−1.33956	−0.669779	+	0.742561i	$0.733611\pi$
$410$	0	0
$411$	− 28.9995i	− 1.43044i
$412$	0	0
$413$	34.5058i	1.69792i
$414$	0	0
$415$	−21.0288	−1.03226
$416$	0	0
$417$	50.5486	2.47538
$418$	0	0
$419$	12.0175i	0.587092i	0.955945	+	0.293546i	$0.0948353\pi$
$419$	12.0175i	0.587092i	−0.955945	+	0.293546i	$0.905165\pi$
$420$	0	0
$421$	− 21.6253i	− 1.05395i	−0.849880	−	0.526977i	$-0.823325\pi$
$421$	− 21.6253i	− 1.05395i	0.849880	−	0.526977i	$-0.176675\pi$
$422$	0	0
$423$	−12.3647	−0.601194
$424$	0	0
$425$	14.1802	0.687839
$426$	0	0
$427$	− 23.1306i	− 1.11937i
$428$	0	0
$429$	− 12.9069i	− 0.623150i
$430$	0	0
$431$	−0.466340	−0.0224628	−0.0112314	−	0.999937i	$-0.503575\pi$
$431$	−0.466340	−0.0224628	−0.0112314	+	0.999937i	$0.503575\pi$
$432$	0	0
$433$	−12.1789	−0.585281	−0.292640	−	0.956223i	$-0.594534\pi$
$433$	−12.1789	−0.585281	−0.292640	+	0.956223i	$0.594534\pi$
$434$	0	0
$435$	− 30.7337i	− 1.47357i
$436$	0	0
$437$	− 6.16717i	− 0.295016i
$438$	0	0
$439$	−22.9016	−1.09303	−0.546517	−	0.837448i	$-0.684047\pi$
$439$	−22.9016	−1.09303	−0.546517	+	0.837448i	$0.684047\pi$
$440$	0	0
$441$	−28.8994	−1.37616
$442$	0	0
$443$	− 4.15864i	− 0.197583i	−0.995108	−	0.0987914i	$-0.968502\pi$
$443$	− 4.15864i	− 0.197583i	0.995108	−	0.0987914i	$-0.0314976\pi$
$444$	0	0
$445$	− 49.6485i	− 2.35356i
$446$	0	0
$447$	50.4144	2.38452
$448$	0	0
$449$	−13.4154	−0.633112	−0.316556	−	0.948574i	$-0.602527\pi$
$449$	−13.4154	−0.633112	−0.316556	+	0.948574i	$0.602527\pi$
$450$	0	0
$451$	− 73.7391i	− 3.47224i
$452$	0	0
$453$	− 32.5095i	− 1.52743i
$454$	0	0
$455$	−10.6352	−0.498588
$456$	0	0
$457$	23.9359	1.11968	0.559838	−	0.828602i	$-0.310864\pi$
$457$	23.9359	1.11968	0.559838	+	0.828602i	$0.310864\pi$
$458$	0	0
$459$	− 0.611636i	− 0.0285487i
$460$	0	0
$461$	− 31.0335i	− 1.44537i	−0.691176	−	0.722686i	$-0.742907\pi$
$461$	− 31.0335i	− 1.44537i	0.691176	−	0.722686i	$-0.257093\pi$
$462$	0	0
$463$	31.3472	1.45683	0.728415	−	0.685136i	$-0.240257\pi$
$463$	31.3472	1.45683	0.728415	+	0.685136i	$0.240257\pi$
$464$	0	0
$465$	−59.2396	−2.74717
$466$	0	0
$467$	7.14257i	0.330519i	0.986250	+	0.165259i	$0.0528461\pi$
$467$	7.14257i	0.330519i	−0.986250	+	0.165259i	$0.947154\pi$
$468$	0	0
$469$	46.1926i	2.13298i
$470$	0	0
$471$	−37.1957	−1.71389
$472$	0	0
$473$	0.536847	0.0246843
$474$	0	0
$475$	− 5.41171i	− 0.248306i
$476$	0	0
$477$	13.0778i	0.598792i
$478$	0	0
$479$	26.5447	1.21286	0.606430	−	0.795137i	$-0.292601\pi$
$479$	26.5447	1.21286	0.606430	+	0.795137i	$0.292601\pi$
$480$	0	0
$481$	4.18911	0.191007
$482$	0	0
$483$	− 61.5413i	− 2.80023i
$484$	0	0
$485$	42.8037i	1.94362i
$486$	0	0
$487$	9.74569	0.441619	0.220810	−	0.975317i	$-0.429130\pi$
$487$	9.74569	0.441619	0.220810	+	0.975317i	$0.429130\pi$
$488$	0	0
$489$	29.3431	1.32694
$490$	0	0
$491$	− 25.6150i	− 1.15599i	−0.816040	−	0.577995i	$-0.803835\pi$
$491$	− 25.6150i	− 1.15599i	0.816040	−	0.577995i	$-0.196165\pi$
$492$	0	0
$493$	− 10.1095i	− 0.455309i
$494$	0	0
$495$	64.0225	2.87760
$496$	0	0
$497$	−4.45312	−0.199750
$498$	0	0
$499$	− 5.48801i	− 0.245677i	−0.992427	−	0.122838i	$-0.960800\pi$
$499$	− 5.48801i	− 0.245677i	0.992427	−	0.122838i	$-0.0391997\pi$
$500$	0	0
$501$	− 47.5028i	− 2.12227i
$502$	0	0
$503$	−9.00013	−0.401296	−0.200648	−	0.979663i	$-0.564305\pi$
$503$	−9.00013	−0.401296	−0.200648	+	0.979663i	$0.564305\pi$
$504$	0	0
$505$	−8.91578	−0.396747
$506$	0	0
$507$	− 30.4519i	− 1.35241i
$508$	0	0
$509$	28.2818i	1.25357i	0.779193	+	0.626784i	$0.215629\pi$
$509$	28.2818i	1.25357i	−0.779193	+	0.626784i	$0.784371\pi$
$510$	0	0
$511$	−29.3240	−1.29722
$512$	0	0
$513$	−0.233424	−0.0103059
$514$	0	0
$515$	1.70740i	0.0752371i
$516$	0	0
$517$	− 25.6189i	− 1.12672i
$518$	0	0
$519$	3.78921	0.166328
$520$	0	0
$521$	4.19618	0.183838	0.0919190	−	0.995766i	$-0.470700\pi$
$521$	4.19618	0.183838	0.0919190	+	0.995766i	$0.470700\pi$
$522$	0	0
$523$	33.6631i	1.47198i	0.676990	+	0.735992i	$0.263284\pi$
$523$	33.6631i	1.47198i	−0.676990	+	0.735992i	$0.736716\pi$
$524$	0	0
$525$	− 54.0027i	− 2.35687i
$526$	0	0
$527$	−19.4862	−0.848833
$528$	0	0
$529$	15.0340	0.653651
$530$	0	0
$531$	26.4168i	1.14639i
$532$	0	0
$533$	9.37779i	0.406197i
$534$	0	0
$535$	−0.683344	−0.0295435
$536$	0	0
$537$	−9.45191	−0.407880
$538$	0	0
$539$	− 59.8776i	− 2.57911i
$540$	0	0
$541$	− 18.3995i	− 0.791056i	−0.918454	−	0.395528i	$-0.870562\pi$
$541$	− 18.3995i	− 0.791056i	0.918454	−	0.395528i	$-0.129438\pi$
$542$	0	0
$543$	−14.5571	−0.624705
$544$	0	0
$545$	31.4253	1.34611
$546$	0	0
$547$	− 5.83156i	− 0.249340i	−0.992198	−	0.124670i	$-0.960213\pi$
$547$	− 5.83156i	− 0.249340i	0.992198	−	0.124670i	$-0.0397872\pi$
$548$	0	0
$549$	− 17.7082i	− 0.755770i
$550$	0	0
$551$	−3.85818	−0.164364
$552$	0	0
$553$	56.5518	2.40483
$554$	0	0
$555$	40.9239i	1.73712i
$556$	0	0
$557$	− 30.9517i	− 1.31146i	−0.754993	−	0.655732i	$-0.772360\pi$
$557$	− 30.9517i	− 1.31146i	0.754993	−	0.655732i	$-0.227640\pi$
$558$	0	0
$559$	−0.0682737	−0.00288767
$560$	0	0
$561$	41.4755	1.75110
$562$	0	0
$563$	43.3007i	1.82490i	0.409183	+	0.912452i	$0.365814\pi$
$563$	43.3007i	1.82490i	−0.409183	+	0.912452i	$0.634186\pi$
$564$	0	0
$565$	19.6594i	0.827076i
$566$	0	0
$567$	35.1964	1.47811
$568$	0	0
$569$	8.32798	0.349127	0.174563	−	0.984646i	$-0.444149\pi$
$569$	8.32798	0.349127	0.174563	+	0.984646i	$0.444149\pi$
$570$	0	0
$571$	− 21.5676i	− 0.902574i	−0.892379	−	0.451287i	$-0.850965\pi$
$571$	− 21.5676i	− 0.902574i	0.892379	−	0.451287i	$-0.149035\pi$
$572$	0	0
$573$	− 48.9861i	− 2.04642i
$574$	0	0
$575$	33.3749	1.39183
$576$	0	0
$577$	−31.5716	−1.31434	−0.657172	−	0.753741i	$-0.728247\pi$
$577$	−31.5716	−1.31434	−0.657172	+	0.753741i	$0.728247\pi$
$578$	0	0
$579$	8.94324i	0.371669i
$580$	0	0
$581$	26.3429i	1.09289i
$582$	0	0
$583$	−27.0964	−1.12222
$584$	0	0
$585$	−8.14208	−0.336634
$586$	0	0
$587$	− 28.9679i	− 1.19563i	−0.801633	−	0.597817i	$-0.796035\pi$
$587$	− 28.9679i	− 1.19563i	0.801633	−	0.597817i	$-0.203965\pi$
$588$	0	0
$589$	7.43670i	0.306424i
$590$	0	0
$591$	32.3871	1.33223
$592$	0	0
$593$	23.8431	0.979117	0.489559	−	0.871970i	$-0.337158\pi$
$593$	23.8431	0.979117	0.489559	+	0.871970i	$0.337158\pi$
$594$	0	0
$595$	− 34.1758i	− 1.40107i
$596$	0	0
$597$	39.2911i	1.60808i
$598$	0	0
$599$	24.0716	0.983537	0.491769	−	0.870726i	$-0.336351\pi$
$599$	24.0716	0.983537	0.491769	+	0.870726i	$0.336351\pi$
$600$	0	0
$601$	1.72226	0.0702523	0.0351262	−	0.999383i	$-0.488817\pi$
$601$	1.72226	0.0702523	0.0351262	+	0.999383i	$0.488817\pi$
$602$	0	0
$603$	35.3639i	1.44013i
$604$	0	0
$605$	97.1564i	3.94997i
$606$	0	0
$607$	−17.2198	−0.698928	−0.349464	−	0.936950i	$-0.613636\pi$
$607$	−17.2198	−0.698928	−0.349464	+	0.936950i	$0.613636\pi$
$608$	0	0
$609$	−38.5002	−1.56011
$610$	0	0
$611$	3.25809i	0.131808i
$612$	0	0
$613$	− 32.5924i	− 1.31639i	−0.752845	−	0.658197i	$-0.771319\pi$
$613$	− 32.5924i	− 1.31639i	0.752845	−	0.658197i	$-0.228681\pi$
$614$	0	0
$615$	−91.6128	−3.69418
$616$	0	0
$617$	−6.44448	−0.259445	−0.129722	−	0.991550i	$-0.541409\pi$
$617$	−6.44448	−0.259445	−0.129722	+	0.991550i	$0.541409\pi$
$618$	0	0
$619$	31.8637i	1.28071i	0.768078	+	0.640356i	$0.221213\pi$
$619$	31.8637i	1.28071i	−0.768078	+	0.640356i	$0.778787\pi$
$620$	0	0
$621$	− 1.43957i	− 0.0577678i
$622$	0	0
$623$	−62.1949	−2.49179
$624$	0	0
$625$	−22.7720	−0.910879
$626$	0	0
$627$	− 15.8287i	− 0.632136i
$628$	0	0
$629$	13.4615i	0.536743i
$630$	0	0
$631$	−7.36348	−0.293135	−0.146568	−	0.989201i	$-0.546823\pi$
$631$	−7.36348	−0.293135	−0.146568	+	0.989201i	$0.546823\pi$
$632$	0	0
$633$	12.3612	0.491312
$634$	0	0
$635$	21.4258i	0.850257i
$636$	0	0
$637$	7.61496i	0.301716i
$638$	0	0
$639$	−3.40920	−0.134866
$640$	0	0
$641$	39.4715	1.55903	0.779514	−	0.626384i	$-0.215466\pi$
$641$	39.4715	1.55903	0.779514	+	0.626384i	$0.215466\pi$
$642$	0	0
$643$	15.7411i	0.620768i	0.950611	+	0.310384i	$0.100458\pi$
$643$	15.7411i	0.620768i	−0.950611	+	0.310384i	$0.899542\pi$
$644$	0	0
$645$	− 0.666974i	− 0.0262621i
$646$	0	0
$647$	17.8689	0.702499	0.351250	−	0.936282i	$-0.385757\pi$
$647$	17.8689	0.702499	0.351250	+	0.936282i	$0.385757\pi$
$648$	0	0
$649$	−54.7338	−2.14849
$650$	0	0
$651$	74.2098i	2.90851i
$652$	0	0
$653$	1.83114i	0.0716581i	0.999358	+	0.0358291i	$0.0114072\pi$
$653$	1.83114i	0.0716581i	−0.999358	+	0.0358291i	$0.988593\pi$
$654$	0	0
$655$	11.6586	0.455540
$656$	0	0
$657$	−22.4497	−0.875846
$658$	0	0
$659$	17.9446i	0.699021i	0.936932	+	0.349510i	$0.113652\pi$
$659$	17.9446i	0.699021i	−0.936932	+	0.349510i	$0.886348\pi$
$660$	0	0
$661$	0.256100i	0.00996114i	0.999988	+	0.00498057i	$0.00158537\pi$
$661$	0.256100i	0.00996114i	−0.999988	+	0.00498057i	$0.998415\pi$
$662$	0	0
$663$	−5.27466	−0.204851
$664$	0	0
$665$	−13.0428	−0.505778
$666$	0	0
$667$	− 23.7940i	− 0.921309i
$668$	0	0
$669$	− 20.4942i	− 0.792350i
$670$	0	0
$671$	36.6903	1.41641
$672$	0	0
$673$	−3.46127	−0.133422	−0.0667111	−	0.997772i	$-0.521251\pi$
$673$	−3.46127	−0.133422	−0.0667111	+	0.997772i	$0.521251\pi$
$674$	0	0
$675$	− 1.26322i	− 0.0486215i
$676$	0	0
$677$	13.7402i	0.528080i	0.964512	+	0.264040i	$0.0850551\pi$
$677$	13.7402i	0.528080i	−0.964512	+	0.264040i	$0.914945\pi$
$678$	0	0
$679$	53.6205	2.05777
$680$	0	0
$681$	−66.2052	−2.53699
$682$	0	0
$683$	5.31127i	0.203230i	0.994824	+	0.101615i	$0.0324010\pi$
$683$	5.31127i	0.203230i	−0.994824	+	0.101615i	$0.967599\pi$
$684$	0	0
$685$	− 37.9036i	− 1.44822i
$686$	0	0
$687$	−15.1852	−0.579353
$688$	0	0
$689$	3.44599	0.131282
$690$	0	0
$691$	14.0583i	0.534802i	0.963585	+	0.267401i	$0.0861648\pi$
$691$	14.0583i	0.534802i	−0.963585	+	0.267401i	$0.913835\pi$
$692$	0	0
$693$	− 80.2013i	− 3.04660i
$694$	0	0
$695$	66.0692	2.50615
$696$	0	0
$697$	−30.1350	−1.14144
$698$	0	0
$699$	19.2536i	0.728239i
$700$	0	0
$701$	− 47.7193i	− 1.80233i	−0.433473	−	0.901167i	$-0.642712\pi$
$701$	− 47.7193i	− 1.80233i	0.433473	−	0.901167i	$-0.357288\pi$
$702$	0	0
$703$	5.13742	0.193761
$704$	0	0
$705$	−31.8287	−1.19874
$706$	0	0
$707$	11.1689i	0.420048i
$708$	0	0
$709$	− 12.8747i	− 0.483519i	−0.970336	−	0.241759i	$-0.922276\pi$
$709$	− 12.8747i	− 0.483519i	0.970336	−	0.241759i	$-0.0777245\pi$
$710$	0	0
$711$	43.2947	1.62368
$712$	0	0
$713$	−45.8634	−1.71760
$714$	0	0
$715$	− 16.8698i	− 0.630896i
$716$	0	0
$717$	68.6213i	2.56271i
$718$	0	0
$719$	6.30671	0.235201	0.117600	−	0.993061i	$-0.462480\pi$
$719$	6.30671	0.235201	0.117600	+	0.993061i	$0.462480\pi$
$720$	0	0
$721$	2.13887	0.0796557
$722$	0	0
$723$	13.6606i	0.508042i
$724$	0	0
$725$	− 20.8793i	− 0.775439i
$726$	0	0
$727$	−24.8092	−0.920123	−0.460061	−	0.887887i	$-0.652173\pi$
$727$	−24.8092	−0.920123	−0.460061	+	0.887887i	$0.652173\pi$
$728$	0	0
$729$	28.6743	1.06201
$730$	0	0
$731$	− 0.219394i	− 0.00811457i
$732$	0	0
$733$	− 20.8961i	− 0.771816i	−0.922537	−	0.385908i	$-0.873888\pi$
$733$	− 20.8961i	− 0.771816i	0.922537	−	0.385908i	$-0.126112\pi$
$734$	0	0
$735$	−74.3914	−2.74397
$736$	0	0
$737$	−73.2716	−2.69900
$738$	0	0
$739$	9.57101i	0.352075i	0.984383	+	0.176038i	$0.0563281\pi$
$739$	9.57101i	0.352075i	−0.984383	+	0.176038i	$0.943672\pi$
$740$	0	0
$741$	2.01302i	0.0739500i
$742$	0	0
$743$	−10.3553	−0.379900	−0.189950	−	0.981794i	$-0.560833\pi$
$743$	−10.3553	−0.379900	−0.189950	+	0.981794i	$0.560833\pi$
$744$	0	0
$745$	65.8939	2.41416
$746$	0	0
$747$	20.1675i	0.737889i
$748$	0	0
$749$	0.856029i	0.0312786i
$750$	0	0
$751$	30.4868	1.11248	0.556239	−	0.831022i	$-0.312244\pi$
$751$	30.4868	1.11248	0.556239	+	0.831022i	$0.312244\pi$
$752$	0	0
$753$	37.0128	1.34882
$754$	0	0
$755$	− 42.4913i	− 1.54642i
$756$	0	0
$757$	− 13.0841i	− 0.475549i	−0.971320	−	0.237774i	$-0.923582\pi$
$757$	− 13.0841i	− 0.475549i	0.971320	−	0.237774i	$-0.0764179\pi$
$758$	0	0
$759$	97.6181	3.54331
$760$	0	0
$761$	−32.3638	−1.17319	−0.586594	−	0.809881i	$-0.699532\pi$
$761$	−32.3638	−1.17319	−0.586594	+	0.809881i	$0.699532\pi$
$762$	0	0
$763$	− 39.3666i	− 1.42517i
$764$	0	0
$765$	− 26.1641i	− 0.945965i
$766$	0	0
$767$	6.96079	0.251340
$768$	0	0
$769$	10.9234	0.393908	0.196954	−	0.980413i	$-0.436895\pi$
$769$	10.9234	0.393908	0.196954	+	0.980413i	$0.436895\pi$
$770$	0	0
$771$	− 33.5753i	− 1.20919i
$772$	0	0
$773$	− 34.9008i	− 1.25530i	−0.778497	−	0.627648i	$-0.784018\pi$
$773$	− 34.9008i	− 1.25530i	0.778497	−	0.627648i	$-0.215982\pi$
$774$	0	0
$775$	−40.2452	−1.44565
$776$	0	0
$777$	51.2656	1.83914
$778$	0	0
$779$	11.5007i	0.412055i
$780$	0	0
$781$	− 7.06363i	− 0.252756i
$782$	0	0
$783$	−0.900592	−0.0321845
$784$	0	0
$785$	−48.6164	−1.73519
$786$	0	0
$787$	− 5.72739i	− 0.204159i	−0.994776	−	0.102080i	$-0.967450\pi$
$787$	− 5.72739i	− 0.204159i	0.994776	−	0.102080i	$-0.0325497\pi$
$788$	0	0
$789$	65.2766i	2.32391i
$790$	0	0
$791$	24.6274	0.875650
$792$	0	0
$793$	−4.66610	−0.165698
$794$	0	0
$795$	33.6643i	1.19395i
$796$	0	0
$797$	− 45.1622i	− 1.59973i	−0.600183	−	0.799863i	$-0.704906\pi$
$797$	− 45.1622i	− 1.59973i	0.600183	−	0.799863i	$-0.295094\pi$
$798$	0	0
$799$	−10.4697	−0.370391
$800$	0	0
$801$	−47.6149	−1.68239
$802$	0	0
$803$	− 46.5143i	− 1.64145i
$804$	0	0
$805$	− 80.4372i	− 2.83504i
$806$	0	0
$807$	48.4928	1.70703
$808$	0	0
$809$	−35.4870	−1.24766	−0.623828	−	0.781562i	$-0.714423\pi$
$809$	−35.4870	−1.24766	−0.623828	+	0.781562i	$0.714423\pi$
$810$	0	0
$811$	20.2334i	0.710492i	0.934773	+	0.355246i	$0.115603\pi$
$811$	20.2334i	0.710492i	−0.934773	+	0.355246i	$0.884397\pi$
$812$	0	0
$813$	34.5981i	1.21341i
$814$	0	0
$815$	38.3527	1.34344
$816$	0	0
$817$	−0.0837292	−0.00292931
$818$	0	0
$819$	10.1996i	0.356404i
$820$	0	0
$821$	52.0305i	1.81588i	0.419104	+	0.907938i	$0.362344\pi$
$821$	52.0305i	1.81588i	−0.419104	+	0.907938i	$0.637656\pi$
$822$	0	0
$823$	−5.60310	−0.195312	−0.0976559	−	0.995220i	$-0.531134\pi$
$823$	−5.60310	−0.195312	−0.0976559	+	0.995220i	$0.531134\pi$
$824$	0	0
$825$	85.6601	2.98230
$826$	0	0
$827$	− 28.3875i	− 0.987129i	−0.869709	−	0.493565i	$-0.835694\pi$
$827$	− 28.3875i	− 0.987129i	0.869709	−	0.493565i	$-0.164306\pi$
$828$	0	0
$829$	− 32.5931i	− 1.13201i	−0.824403	−	0.566003i	$-0.808489\pi$
$829$	− 32.5931i	− 1.13201i	0.824403	−	0.566003i	$-0.191511\pi$
$830$	0	0
$831$	−12.1604	−0.421841
$832$	0	0
$833$	−24.4702	−0.847844
$834$	0	0
$835$	− 62.0882i	− 2.14865i
$836$	0	0
$837$	1.73591i	0.0600017i
$838$	0	0
$839$	−45.0055	−1.55376	−0.776881	−	0.629647i	$-0.783199\pi$
$839$	−45.0055	−1.55376	−0.776881	+	0.629647i	$0.783199\pi$
$840$	0	0
$841$	14.1145	0.486706
$842$	0	0
$843$	12.1082i	0.417029i
$844$	0	0
$845$	− 39.8019i	− 1.36923i
$846$	0	0
$847$	121.708	4.18195
$848$	0	0
$849$	5.11667	0.175604
$850$	0	0
$851$	31.6833i	1.08609i
$852$	0	0
$853$	− 21.6381i	− 0.740875i	−0.928857	−	0.370438i	$-0.879208\pi$
$853$	− 21.6381i	− 0.740875i	0.928857	−	0.370438i	$-0.120792\pi$
$854$	0	0
$855$	−9.98525	−0.341488
$856$	0	0
$857$	11.4432	0.390892	0.195446	−	0.980714i	$-0.437384\pi$
$857$	11.4432	0.390892	0.195446	+	0.980714i	$0.437384\pi$
$858$	0	0
$859$	− 7.57422i	− 0.258429i	−0.991617	−	0.129215i	$-0.958754\pi$
$859$	− 7.57422i	− 0.258429i	0.991617	−	0.129215i	$-0.0412456\pi$
$860$	0	0
$861$	114.764i	3.91114i
$862$	0	0
$863$	−34.6063	−1.17801	−0.589007	−	0.808128i	$-0.700481\pi$
$863$	−34.6063	−1.17801	−0.589007	+	0.808128i	$0.700481\pi$
$864$	0	0
$865$	4.95266	0.168396
$866$	0	0
$867$	25.0183i	0.849667i
$868$	0	0
$869$	89.7036i	3.04299i
$870$	0	0
$871$	9.31834	0.315740
$872$	0	0
$873$	41.0505	1.38935
$874$	0	0
$875$	− 5.36981i	− 0.181533i
$876$	0	0
$877$	29.4493i	0.994434i	0.867626	+	0.497217i	$0.165645\pi$
$877$	29.4493i	0.994434i	−0.867626	+	0.497217i	$0.834355\pi$
$878$	0	0
$879$	25.8243	0.871033
$880$	0	0
$881$	51.5583	1.73704	0.868522	−	0.495651i	$-0.165070\pi$
$881$	51.5583	1.73704	0.868522	+	0.495651i	$0.165070\pi$
$882$	0	0
$883$	25.2991i	0.851382i	0.904869	+	0.425691i	$0.139969\pi$
$883$	25.2991i	0.851382i	−0.904869	+	0.425691i	$0.860031\pi$
$884$	0	0
$885$	68.0008i	2.28582i
$886$	0	0
$887$	1.92340	0.0645815	0.0322908	−	0.999479i	$-0.489720\pi$
$887$	1.92340	0.0645815	0.0322908	+	0.999479i	$0.489720\pi$
$888$	0	0
$889$	26.8402	0.900192
$890$	0	0
$891$	55.8293i	1.87035i
$892$	0	0
$893$	3.99564i	0.133709i
$894$	0	0
$895$	−12.3541	−0.412950
$896$	0	0
$897$	−12.4146	−0.414512
$898$	0	0
$899$	28.6921i	0.956936i
$900$	0	0
$901$	11.0735i	0.368912i
$902$	0	0
$903$	−0.835522	−0.0278044
$904$	0	0
$905$	−19.0268	−0.632471
$906$	0	0
$907$	27.7093i	0.920071i	0.887900	+	0.460036i	$0.152163\pi$
$907$	27.7093i	0.920071i	−0.887900	+	0.460036i	$0.847837\pi$
$908$	0	0
$909$	8.55060i	0.283605i
$910$	0	0
$911$	51.4735	1.70539	0.852696	−	0.522407i	$-0.174966\pi$
$911$	51.4735	1.70539	0.852696	+	0.522407i	$0.174966\pi$
$912$	0	0
$913$	−41.7856	−1.38290
$914$	0	0
$915$	− 45.5837i	− 1.50695i
$916$	0	0
$917$	− 14.6048i	− 0.482293i
$918$	0	0
$919$	25.4158	0.838390	0.419195	−	0.907896i	$-0.362312\pi$
$919$	25.4158	0.838390	0.419195	+	0.907896i	$0.362312\pi$
$920$	0	0
$921$	14.5129	0.478216
$922$	0	0
$923$	0.898319i	0.0295685i
$924$	0	0
$925$	27.8022i	0.914131i
$926$	0	0
$927$	1.63747	0.0537815
$928$	0	0
$929$	−41.2485	−1.35332	−0.676661	−	0.736295i	$-0.736574\pi$
$929$	−41.2485	−1.35332	−0.676661	+	0.736295i	$0.736574\pi$
$930$	0	0
$931$	9.33880i	0.306067i
$932$	0	0
$933$	− 31.3185i	− 1.02532i
$934$	0	0
$935$	54.2103	1.77287
$936$	0	0
$937$	36.1849	1.18211	0.591054	−	0.806632i	$-0.298712\pi$
$937$	36.1849	1.18211	0.591054	+	0.806632i	$0.298712\pi$
$938$	0	0
$939$	15.4317i	0.503593i
$940$	0	0
$941$	17.2262i	0.561559i	0.959772	+	0.280779i	$0.0905929\pi$
$941$	17.2262i	0.561559i	−0.959772	+	0.280779i	$0.909407\pi$
$942$	0	0
$943$	−70.9267	−2.30969
$944$	0	0
$945$	−3.04451	−0.0990378
$946$	0	0
$947$	17.7823i	0.577847i	0.957352	+	0.288924i	$0.0932974\pi$
$947$	17.7823i	0.577847i	−0.957352	+	0.288924i	$0.906703\pi$
$948$	0	0
$949$	5.91546i	0.192024i
$950$	0	0
$951$	11.6765	0.378636
$952$	0	0
$953$	47.3478	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$953$	47.3478	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$954$	0	0
$955$	− 64.0269i	− 2.07186i
$956$	0	0
$957$	− 61.0698i	− 1.97411i
$958$	0	0
$959$	−47.4821	−1.53328
$960$	0	0
$961$	24.3045	0.784017
$962$	0	0
$963$	0.655354i	0.0211185i
$964$	0	0
$965$	11.6892i	0.376289i
$966$	0	0
$967$	25.3504	0.815213	0.407606	−	0.913158i	$-0.366364\pi$
$967$	25.3504	0.815213	0.407606	+	0.913158i	$0.366364\pi$
$968$	0	0
$969$	−6.46871	−0.207805
$970$	0	0
$971$	− 45.3901i	− 1.45664i	−0.685238	−	0.728319i	$-0.740302\pi$
$971$	− 45.3901i	− 1.45664i	0.685238	−	0.728319i	$-0.259698\pi$
$972$	0	0
$973$	− 82.7653i	− 2.65333i
$974$	0	0
$975$	−10.8939	−0.348883
$976$	0	0
$977$	27.3398	0.874679	0.437339	−	0.899297i	$-0.355921\pi$
$977$	27.3398	0.874679	0.437339	+	0.899297i	$0.355921\pi$
$978$	0	0
$979$	− 98.6549i	− 3.15302i
$980$	0	0
$981$	− 30.1381i	− 0.962236i
$982$	0	0
$983$	4.84750	0.154611	0.0773056	−	0.997007i	$-0.475368\pi$
$983$	4.84750	0.154611	0.0773056	+	0.997007i	$0.475368\pi$
$984$	0	0
$985$	42.3313	1.34879
$986$	0	0
$987$	39.8720i	1.26914i
$988$	0	0
$989$	− 0.516372i	− 0.0164197i
$990$	0	0
$991$	28.1022	0.892696	0.446348	−	0.894859i	$-0.352724\pi$
$991$	28.1022	0.892696	0.446348	+	0.894859i	$0.352724\pi$
$992$	0	0
$993$	−48.8954	−1.55165
$994$	0	0
$995$	51.3551i	1.62807i
$996$	0	0
$997$	− 6.99262i	− 0.221458i	−0.993851	−	0.110729i	$-0.964681\pi$
$997$	− 6.99262i	− 0.221458i	0.993851	−	0.110729i	$-0.0353186\pi$
$998$	0	0
$999$	1.19920	0.0379409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.j.1217.3	✓	20
4.3	odd	2	inner	2432.2.c.j.1217.17	yes	20
8.3	odd	2	inner	2432.2.c.j.1217.4	yes	20
8.5	even	2	inner	2432.2.c.j.1217.18	yes	20
16.3	odd	4		4864.2.a.bt.1.1		10
16.5	even	4		4864.2.a.bt.1.2		10
16.11	odd	4		4864.2.a.bs.1.10		10
16.13	even	4		4864.2.a.bs.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.j.1217.3	✓	20	1.1	even	1	trivial
2432.2.c.j.1217.4	yes	20	8.3	odd	2	inner
2432.2.c.j.1217.17	yes	20	4.3	odd	2	inner
2432.2.c.j.1217.18	yes	20	8.5	even	2	inner
4864.2.a.bs.1.9		10	16.13	even	4
4864.2.a.bs.1.10		10	16.11	odd	4
4864.2.a.bt.1.1		10	16.3	odd	4
4864.2.a.bt.1.2		10	16.5	even	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:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.46871i q^{3} -3.22672i q^{5} -4.04213 q^{7} -3.09455 q^{9} +O(q^{10})\)	\(q-2.46871i q^{3} -3.22672i q^{5} -4.04213 q^{7} -3.09455 q^{9} -6.41171i q^{11} +0.815411i q^{13} -7.96585 q^{15} -2.62028 q^{17} +1.00000i q^{19} +9.97886i q^{21} -6.16717 q^{23} -5.41171 q^{25} +0.233424i q^{27} +3.85818i q^{29} +7.43670 q^{31} -15.8287 q^{33} +13.0428i q^{35} -5.13742i q^{37} +2.01302 q^{39} +11.5007 q^{41} +0.0837292i q^{43} +9.98525i q^{45} +3.99564 q^{47} +9.33880 q^{49} +6.46871i q^{51} -4.22608i q^{53} -20.6888 q^{55} +2.46871 q^{57} -8.53655i q^{59} +5.72239i q^{61} +12.5086 q^{63} +2.63110 q^{65} -11.4278i q^{67} +15.2250i q^{69} +1.10168 q^{71} +7.25458 q^{73} +13.3600i q^{75} +25.9169i q^{77} -13.9906 q^{79} -8.70740 q^{81} -6.51708i q^{83} +8.45489i q^{85} +9.52474 q^{87} +15.3867 q^{89} -3.29599i q^{91} -18.3591i q^{93} +3.22672 q^{95} -13.2654 q^{97} +19.8414i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 28 q^{9}+O(q^{10}) \) 20 * q - 28 * q^9	\( 20 q - 28 q^{9} + 8 q^{17} - 20 q^{25} - 16 q^{33} + 24 q^{41} + 52 q^{49} - 8 q^{57} - 48 q^{65} - 24 q^{73} + 68 q^{81} + 40 q^{89} - 56 q^{97}+O(q^{100}) \) 20 * q - 28 * q^9 + 8 * q^17 - 20 * q^25 - 16 * q^33 + 24 * q^41 + 52 * q^49 - 8 * q^57 - 48 * q^65 - 24 * q^73 + 68 * q^81 + 40 * q^89 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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