LMFDB - Embedded newform 2432.2.c.f.1217.6

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:	$6$
Coefficient field:	6.0.3182656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{3} + 25x^{2} - 10x + 2 $ x^6 - 2x^3 + 25x^2 - 10*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.6
Root			$0.203364 + 0.203364i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.f.1217.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.91729i q^{3} -3.91729i q^{5} -0.593272 q^{7} -5.51056 q^{9} +O(q^{10})$	$q+2.91729i q^{3} -3.91729i q^{5} -0.593272 q^{7} -5.51056 q^{9} +3.51056i q^{11} -2.51056i q^{13} +11.4278 q^{15} +3.00000 q^{17} +1.00000i q^{19} -1.73074i q^{21} -7.32401 q^{23} -10.3451 q^{25} -7.32401i q^{27} -5.73074i q^{29} +4.40673 q^{31} -10.2413 q^{33} +2.32401i q^{35} -3.42784i q^{37} +7.32401 q^{39} +10.2413 q^{41} -4.69710i q^{43} +21.5864i q^{45} +2.08271 q^{47} -6.64803 q^{49} +8.75186i q^{51} +1.08271i q^{53} +13.7519 q^{55} -2.91729 q^{57} -8.51056i q^{59} -7.10383i q^{61} +3.26926 q^{63} -9.83457 q^{65} -12.9173i q^{67} -21.3662i q^{69} -12.8557 q^{71} +16.2288 q^{73} -30.1797i q^{75} -2.08271i q^{77} +4.40673 q^{79} +4.83457 q^{81} -13.4615i q^{83} -11.7519i q^{85} +16.7182 q^{87} -0.241300 q^{89} +1.48944i q^{91} +12.8557i q^{93} +3.91729 q^{95} +15.4615 q^{97} -19.3451i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{7} - 16 q^{9}+O(q^{10}) $ 6 * q - 6 * q^7 - 16 * q^9	$ 6 q - 6 q^{7} - 16 q^{9} + 32 q^{15} + 18 q^{17} - 22 q^{23} - 6 q^{25} + 24 q^{31} - 20 q^{33} + 22 q^{39} + 20 q^{41} + 32 q^{47} + 4 q^{49} + 24 q^{55} + 2 q^{57} + 44 q^{63} - 20 q^{65} - 4 q^{71} + 34 q^{73} + 24 q^{79} - 10 q^{81} + 54 q^{87} + 40 q^{89} + 4 q^{95} + 44 q^{97}+O(q^{100}) $ 6 * q - 6 * q^7 - 16 * q^9 + 32 * q^15 + 18 * q^17 - 22 * q^23 - 6 * q^25 + 24 * q^31 - 20 * q^33 + 22 * q^39 + 20 * q^41 + 32 * q^47 + 4 * q^49 + 24 * q^55 + 2 * q^57 + 44 * q^63 - 20 * q^65 - 4 * q^71 + 34 * q^73 + 24 * q^79 - 10 * q^81 + 54 * q^87 + 40 * q^89 + 4 * q^95 + 44 * 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
$3$	2.91729i	1.68430i	0.539247	+	0.842148i	$0.318709\pi$
$3$	2.91729i	1.68430i	−0.539247	+	0.842148i	$0.681291\pi$
$4$	0	0
$5$	− 3.91729i	− 1.75186i	−0.482435	−	0.875932i	$-0.660248\pi$
$5$	− 3.91729i	− 1.75186i	0.482435	−	0.875932i	$-0.339752\pi$
$6$	0	0
$7$	−0.593272	−0.224236	−0.112118	−	0.993695i	$-0.535763\pi$
$7$	−0.593272	−0.224236	−0.112118	+	0.993695i	$0.535763\pi$
$8$	0	0
$9$	−5.51056	−1.83685
$10$	0	0
$11$	3.51056i	1.05847i	0.848474	+	0.529236i	$0.177522\pi$
$11$	3.51056i	1.05847i	−0.848474	+	0.529236i	$0.822478\pi$
$12$	0	0
$13$	− 2.51056i	− 0.696303i	−0.937438	−	0.348152i	$-0.886809\pi$
$13$	− 2.51056i	− 0.696303i	0.937438	−	0.348152i	$-0.113191\pi$
$14$	0	0
$15$	11.4278	2.95066
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	− 1.73074i	− 0.377679i
$22$	0	0
$23$	−7.32401	−1.52716	−0.763581	−	0.645712i	$-0.776561\pi$
$23$	−7.32401	−1.52716	−0.763581	+	0.645712i	$0.776561\pi$
$24$	0	0
$25$	−10.3451	−2.06903
$26$	0	0
$27$	− 7.32401i	− 1.40951i
$28$	0	0
$29$	− 5.73074i	− 1.06417i	−0.846690	−	0.532086i	$-0.821408\pi$
$29$	− 5.73074i	− 1.06417i	0.846690	−	0.532086i	$-0.178592\pi$
$30$	0	0
$31$	4.40673	0.791472	0.395736	−	0.918364i	$-0.370490\pi$
$31$	4.40673	0.791472	0.395736	+	0.918364i	$0.370490\pi$
$32$	0	0
$33$	−10.2413	−1.78278
$34$	0	0
$35$	2.32401i	0.392830i
$36$	0	0
$37$	− 3.42784i	− 0.563534i	−0.959483	−	0.281767i	$-0.909079\pi$
$37$	− 3.42784i	− 0.563534i	0.959483	−	0.281767i	$-0.0909205\pi$
$38$	0	0
$39$	7.32401	1.17278
$40$	0	0
$41$	10.2413	1.59942	0.799711	−	0.600385i	$-0.204986\pi$
$41$	10.2413	1.59942	0.799711	+	0.600385i	$0.204986\pi$
$42$	0	0
$43$	− 4.69710i	− 0.716301i	−0.933664	−	0.358151i	$-0.883407\pi$
$43$	− 4.69710i	− 0.716301i	0.933664	−	0.358151i	$-0.116593\pi$
$44$	0	0
$45$	21.5864i	3.21791i
$46$	0	0
$47$	2.08271	0.303795	0.151898	−	0.988396i	$-0.451462\pi$
$47$	2.08271	0.303795	0.151898	+	0.988396i	$0.451462\pi$
$48$	0	0
$49$	−6.64803	−0.949718
$50$	0	0
$51$	8.75186i	1.22551i
$52$	0	0
$53$	1.08271i	0.148722i	0.997231	+	0.0743611i	$0.0236917\pi$
$53$	1.08271i	0.148722i	−0.997231	+	0.0743611i	$0.976308\pi$
$54$	0	0
$55$	13.7519	1.85430
$56$	0	0
$57$	−2.91729	−0.386404
$58$	0	0
$59$	− 8.51056i	− 1.10798i	−0.832523	−	0.553990i	$-0.813105\pi$
$59$	− 8.51056i	− 1.10798i	0.832523	−	0.553990i	$-0.186895\pi$
$60$	0	0
$61$	− 7.10383i	− 0.909552i	−0.890606	−	0.454776i	$-0.849719\pi$
$61$	− 7.10383i	− 0.909552i	0.890606	−	0.454776i	$-0.150281\pi$
$62$	0	0
$63$	3.26926	0.411888
$64$	0	0
$65$	−9.83457	−1.21983
$66$	0	0
$67$	− 12.9173i	− 1.57810i	−0.614330	−	0.789049i	$-0.710574\pi$
$67$	− 12.9173i	− 1.57810i	0.614330	−	0.789049i	$-0.289426\pi$
$68$	0	0
$69$	− 21.3662i	− 2.57219i
$70$	0	0
$71$	−12.8557	−1.52569	−0.762845	−	0.646582i	$-0.776198\pi$
$71$	−12.8557	−1.52569	−0.762845	+	0.646582i	$0.776198\pi$
$72$	0	0
$73$	16.2288	1.89943	0.949717	−	0.313109i	$-0.101371\pi$
$73$	16.2288	1.89943	0.949717	+	0.313109i	$0.101371\pi$
$74$	0	0
$75$	− 30.1797i	− 3.48485i
$76$	0	0
$77$	− 2.08271i	− 0.237347i
$78$	0	0
$79$	4.40673	0.495796	0.247898	−	0.968786i	$-0.420260\pi$
$79$	4.40673	0.495796	0.247898	+	0.968786i	$0.420260\pi$
$80$	0	0
$81$	4.83457	0.537175
$82$	0	0
$83$	− 13.4615i	− 1.47759i	−0.673930	−	0.738795i	$-0.735395\pi$
$83$	− 13.4615i	− 1.47759i	0.673930	−	0.738795i	$-0.264605\pi$
$84$	0	0
$85$	− 11.7519i	− 1.27467i
$86$	0	0
$87$	16.7182	1.79238
$88$	0	0
$89$	−0.241300	−0.0255778	−0.0127889	−	0.999918i	$-0.504071\pi$
$89$	−0.241300	−0.0255778	−0.0127889	+	0.999918i	$0.504071\pi$
$90$	0	0
$91$	1.48944i	0.156136i
$92$	0	0
$93$	12.8557i	1.33307i
$94$	0	0
$95$	3.91729	0.401905
$96$	0	0
$97$	15.4615	1.56988	0.784938	−	0.619574i	$-0.212695\pi$
$97$	15.4615	1.56988	0.784938	+	0.619574i	$0.212695\pi$
$98$	0	0
$99$	− 19.3451i	− 1.94426i
$100$	0	0
$101$	− 15.8346i	− 1.57560i	−0.615932	−	0.787799i	$-0.711220\pi$
$101$	− 15.8346i	− 1.57560i	0.615932	−	0.787799i	$-0.288780\pi$
$102$	0	0
$103$	−13.4615	−1.32640	−0.663200	−	0.748442i	$-0.730802\pi$
$103$	−13.4615	−1.32640	−0.663200	+	0.748442i	$0.730802\pi$
$104$	0	0
$105$	−6.77981	−0.661642
$106$	0	0
$107$	− 0.302899i	− 0.0292824i	−0.999893	−	0.0146412i	$-0.995339\pi$
$107$	− 0.302899i	− 0.0292824i	0.999893	−	0.0146412i	$-0.00466060\pi$
$108$	0	0
$109$	− 5.29037i	− 0.506726i	−0.967371	−	0.253363i	$-0.918463\pi$
$109$	− 5.29037i	− 0.506726i	0.967371	−	0.253363i	$-0.0815367\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	2.77981	0.261503	0.130751	−	0.991415i	$-0.458261\pi$
$113$	2.77981	0.261503	0.130751	+	0.991415i	$0.458261\pi$
$114$	0	0
$115$	28.6903i	2.67538i
$116$	0	0
$117$	13.8346i	1.27901i
$118$	0	0
$119$	−1.77981	−0.163155
$120$	0	0
$121$	−1.32401	−0.120365
$122$	0	0
$123$	29.8768i	2.69390i
$124$	0	0
$125$	20.9384i	1.87279i
$126$	0	0
$127$	−14.2835	−1.26746	−0.633729	−	0.773555i	$-0.718477\pi$
$127$	−14.2835	−1.26746	−0.633729	+	0.773555i	$0.718477\pi$
$128$	0	0
$129$	13.7028	1.20646
$130$	0	0
$131$	− 16.3662i	− 1.42993i	−0.699163	−	0.714963i	$-0.746444\pi$
$131$	− 16.3662i	− 1.42993i	0.699163	−	0.714963i	$-0.253556\pi$
$132$	0	0
$133$	− 0.593272i	− 0.0514432i
$134$	0	0
$135$	−28.6903	−2.46926
$136$	0	0
$137$	4.66914	0.398912	0.199456	−	0.979907i	$-0.436083\pi$
$137$	4.66914	0.398912	0.199456	+	0.979907i	$0.436083\pi$
$138$	0	0
$139$	− 6.48944i	− 0.550427i	−0.961383	−	0.275214i	$-0.911251\pi$
$139$	− 6.48944i	− 0.550427i	0.961383	−	0.275214i	$-0.0887486\pi$
$140$	0	0
$141$	6.07587i	0.511681i
$142$	0	0
$143$	8.81346	0.737018
$144$	0	0
$145$	−22.4490	−1.86428
$146$	0	0
$147$	− 19.3942i	− 1.59961i
$148$	0	0
$149$	6.73074i	0.551404i	0.961243	+	0.275702i	$0.0889103\pi$
$149$	6.73074i	0.551404i	−0.961243	+	0.275702i	$0.911090\pi$
$150$	0	0
$151$	−18.2077	−1.48172	−0.740859	−	0.671660i	$-0.765581\pi$
$151$	−18.2077	−1.48172	−0.740859	+	0.671660i	$0.765581\pi$
$152$	0	0
$153$	−16.5317	−1.33651
$154$	0	0
$155$	− 17.2624i	− 1.38655i
$156$	0	0
$157$	16.0422i	1.28031i	0.768246	+	0.640155i	$0.221130\pi$
$157$	16.0422i	1.28031i	−0.768246	+	0.640155i	$0.778870\pi$
$158$	0	0
$159$	−3.15859	−0.250492
$160$	0	0
$161$	4.34513	0.342444
$162$	0	0
$163$	24.0422i	1.88313i	0.336827	+	0.941566i	$0.390646\pi$
$163$	24.0422i	1.88313i	−0.336827	+	0.941566i	$0.609354\pi$
$164$	0	0
$165$	40.1181i	3.12319i
$166$	0	0
$167$	19.6355	1.51944	0.759720	−	0.650250i	$-0.225336\pi$
$167$	19.6355	1.51944	0.759720	+	0.650250i	$0.225336\pi$
$168$	0	0
$169$	6.69710	0.515162
$170$	0	0
$171$	− 5.51056i	− 0.421403i
$172$	0	0
$173$	8.37309i	0.636594i	0.947991	+	0.318297i	$0.103111\pi$
$173$	8.37309i	0.636594i	−0.947991	+	0.318297i	$0.896889\pi$
$174$	0	0
$175$	6.13747	0.463949
$176$	0	0
$177$	24.8277	1.86617
$178$	0	0
$179$	− 10.2413i	− 0.765471i	−0.923858	−	0.382735i	$-0.874982\pi$
$179$	− 10.2413i	− 0.765471i	0.923858	−	0.382735i	$-0.125018\pi$
$180$	0	0
$181$	24.5248i	1.82292i	0.411393	+	0.911458i	$0.365042\pi$
$181$	24.5248i	1.82292i	−0.411393	+	0.911458i	$0.634958\pi$
$182$	0	0
$183$	20.7239	1.53195
$184$	0	0
$185$	−13.4278	−0.987235
$186$	0	0
$187$	10.5317i	0.770152i
$188$	0	0
$189$	4.34513i	0.316062i
$190$	0	0
$191$	−9.28353	−0.671733	−0.335866	−	0.941910i	$-0.609029\pi$
$191$	−9.28353	−0.671733	−0.335866	+	0.941910i	$0.609029\pi$
$192$	0	0
$193$	−8.03364	−0.578274	−0.289137	−	0.957288i	$-0.593368\pi$
$193$	−8.03364	−0.578274	−0.289137	+	0.957288i	$0.593368\pi$
$194$	0	0
$195$	− 28.6903i	− 2.05455i
$196$	0	0
$197$	− 5.62691i	− 0.400901i	−0.979704	−	0.200451i	$-0.935759\pi$
$197$	− 5.62691i	− 0.400901i	0.979704	−	0.200451i	$-0.0642406\pi$
$198$	0	0
$199$	18.0970	1.28286	0.641431	−	0.767181i	$-0.278341\pi$
$199$	18.0970	1.28286	0.641431	+	0.767181i	$0.278341\pi$
$200$	0	0
$201$	37.6834	2.65798
$202$	0	0
$203$	3.39989i	0.238625i
$204$	0	0
$205$	− 40.1181i	− 2.80197i
$206$	0	0
$207$	40.3594	2.80517
$208$	0	0
$209$	−3.51056	−0.242830
$210$	0	0
$211$	− 0.302899i	− 0.0208524i	−0.999946	−	0.0104262i	$-0.996681\pi$
$211$	− 0.302899i	− 0.0208524i	0.999946	−	0.0104262i	$-0.00331883\pi$
$212$	0	0
$213$	− 37.5037i	− 2.56971i
$214$	0	0
$215$	−18.3999	−1.25486
$216$	0	0
$217$	−2.61439	−0.177476
$218$	0	0
$219$	47.3440i	3.19921i
$220$	0	0
$221$	− 7.53167i	− 0.506635i
$222$	0	0
$223$	−0.847099	−0.0567259	−0.0283630	−	0.999598i	$-0.509029\pi$
$223$	−0.847099	−0.0567259	−0.0283630	+	0.999598i	$0.509029\pi$
$224$	0	0
$225$	57.0074	3.80050
$226$	0	0
$227$	− 12.5528i	− 0.833158i	−0.909100	−	0.416579i	$-0.863229\pi$
$227$	− 12.5528i	− 0.833158i	0.909100	−	0.416579i	$-0.136771\pi$
$228$	0	0
$229$	14.7730i	0.976226i	0.872781	+	0.488113i	$0.162315\pi$
$229$	14.7730i	0.976226i	−0.872781	+	0.488113i	$0.837685\pi$
$230$	0	0
$231$	6.07587	0.399763
$232$	0	0
$233$	−7.71822	−0.505637	−0.252819	−	0.967514i	$-0.581358\pi$
$233$	−7.71822	−0.505637	−0.252819	+	0.967514i	$0.581358\pi$
$234$	0	0
$235$	− 8.15859i	− 0.532207i
$236$	0	0
$237$	12.8557i	0.835067i
$238$	0	0
$239$	27.6144	1.78623	0.893113	−	0.449832i	$-0.148516\pi$
$239$	27.6144	1.78623	0.893113	+	0.449832i	$0.148516\pi$
$240$	0	0
$241$	−1.42784	−0.0919755	−0.0459877	−	0.998942i	$-0.514644\pi$
$241$	−1.42784	−0.0919755	−0.0459877	+	0.998942i	$0.514644\pi$
$242$	0	0
$243$	− 7.86821i	− 0.504746i
$244$	0	0
$245$	26.0422i	1.66378i
$246$	0	0
$247$	2.51056	0.159743
$248$	0	0
$249$	39.2710	2.48870
$250$	0	0
$251$	20.5317i	1.29595i	0.761663	+	0.647974i	$0.224383\pi$
$251$	20.5317i	1.29595i	−0.761663	+	0.647974i	$0.775617\pi$
$252$	0	0
$253$	− 25.7114i	− 1.61646i
$254$	0	0
$255$	34.2835	2.14692
$256$	0	0
$257$	−25.5037	−1.59088	−0.795439	−	0.606034i	$-0.792760\pi$
$257$	−25.5037	−1.59088	−0.795439	+	0.606034i	$0.792760\pi$
$258$	0	0
$259$	2.03364i	0.126364i
$260$	0	0
$261$	31.5796i	1.95473i
$262$	0	0
$263$	−13.7519	−0.847976	−0.423988	−	0.905668i	$-0.639370\pi$
$263$	−13.7519	−0.847976	−0.423988	+	0.905668i	$0.639370\pi$
$264$	0	0
$265$	4.24130	0.260541
$266$	0	0
$267$	− 0.703942i	− 0.0430806i
$268$	0	0
$269$	− 20.8220i	− 1.26954i	−0.772700	−	0.634771i	$-0.781094\pi$
$269$	− 20.8220i	− 1.26954i	0.772700	−	0.634771i	$-0.218906\pi$
$270$	0	0
$271$	14.8277	0.900720	0.450360	−	0.892847i	$-0.351296\pi$
$271$	14.8277	0.900720	0.450360	+	0.892847i	$0.351296\pi$
$272$	0	0
$273$	−4.34513	−0.262979
$274$	0	0
$275$	− 36.3172i	− 2.19001i
$276$	0	0
$277$	− 15.9173i	− 0.956377i	−0.878257	−	0.478189i	$-0.841294\pi$
$277$	− 15.9173i	− 0.956377i	0.878257	−	0.478189i	$-0.158706\pi$
$278$	0	0
$279$	−24.2835	−1.45382
$280$	0	0
$281$	−4.64803	−0.277278	−0.138639	−	0.990343i	$-0.544273\pi$
$281$	−4.64803	−0.277278	−0.138639	+	0.990343i	$0.544273\pi$
$282$	0	0
$283$	− 3.99316i	− 0.237369i	−0.992932	−	0.118684i	$-0.962132\pi$
$283$	− 3.99316i	− 0.237369i	0.992932	−	0.118684i	$-0.0378676\pi$
$284$	0	0
$285$	11.4278i	0.676927i
$286$	0	0
$287$	−6.07587	−0.358647
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	45.1056i	2.64414i
$292$	0	0
$293$	3.11636i	0.182059i	0.995848	+	0.0910297i	$0.0290158\pi$
$293$	3.11636i	0.182059i	−0.995848	+	0.0910297i	$0.970984\pi$
$294$	0	0
$295$	−33.3383	−1.94103
$296$	0	0
$297$	25.7114	1.49193
$298$	0	0
$299$	18.3874i	1.06337i
$300$	0	0
$301$	2.78666i	0.160620i
$302$	0	0
$303$	46.1940	2.65377
$304$	0	0
$305$	−27.8277	−1.59341
$306$	0	0
$307$	23.7028i	1.35279i	0.736539	+	0.676395i	$0.236459\pi$
$307$	23.7028i	1.35279i	−0.736539	+	0.676395i	$0.763541\pi$
$308$	0	0
$309$	− 39.2710i	− 2.23405i
$310$	0	0
$311$	−1.90301	−0.107910	−0.0539550	−	0.998543i	$-0.517183\pi$
$311$	−1.90301	−0.107910	−0.0539550	+	0.998543i	$0.517183\pi$
$312$	0	0
$313$	−31.6834	−1.79085	−0.895426	−	0.445210i	$-0.853129\pi$
$313$	−31.6834	−1.79085	−0.895426	+	0.445210i	$0.853129\pi$
$314$	0	0
$315$	− 12.8066i	− 0.721571i
$316$	0	0
$317$	− 12.1797i	− 0.684080i	−0.939685	−	0.342040i	$-0.888882\pi$
$317$	− 12.1797i	− 0.684080i	0.939685	−	0.342040i	$-0.111118\pi$
$318$	0	0
$319$	20.1181	1.12640
$320$	0	0
$321$	0.883644	0.0493202
$322$	0	0
$323$	3.00000i	0.166924i
$324$	0	0
$325$	25.9720i	1.44067i
$326$	0	0
$327$	15.4335	0.853476
$328$	0	0
$329$	−1.23562	−0.0681217
$330$	0	0
$331$	7.08271i	0.389301i	0.980873	+	0.194651i	$0.0623573\pi$
$331$	7.08271i	0.389301i	−0.980873	+	0.194651i	$0.937643\pi$
$332$	0	0
$333$	18.8893i	1.03513i
$334$	0	0
$335$	−50.6007	−2.76461
$336$	0	0
$337$	−9.02112	−0.491411	−0.245706	−	0.969344i	$-0.579020\pi$
$337$	−9.02112	−0.491411	−0.245706	+	0.969344i	$0.579020\pi$
$338$	0	0
$339$	8.10951i	0.440448i
$340$	0	0
$341$	15.4701i	0.837751i
$342$	0	0
$343$	8.09699	0.437196
$344$	0	0
$345$	−83.6977	−4.50613
$346$	0	0
$347$	− 28.0354i	− 1.50502i	−0.658582	−	0.752509i	$-0.728843\pi$
$347$	− 28.0354i	− 1.50502i	0.658582	−	0.752509i	$-0.271157\pi$
$348$	0	0
$349$	− 4.39989i	− 0.235521i	−0.993042	−	0.117760i	$-0.962429\pi$
$349$	− 4.39989i	− 0.235521i	0.993042	−	0.117760i	$-0.0375714\pi$
$350$	0	0
$351$	−18.3874	−0.981445
$352$	0	0
$353$	−0.950928	−0.0506128	−0.0253064	−	0.999680i	$-0.508056\pi$
$353$	−0.950928	−0.0506128	−0.0253064	+	0.999680i	$0.508056\pi$
$354$	0	0
$355$	50.3594i	2.67280i
$356$	0	0
$357$	− 5.19223i	− 0.274802i
$358$	0	0
$359$	−12.3856	−0.653688	−0.326844	−	0.945078i	$-0.605985\pi$
$359$	−12.3856	−0.653688	−0.326844	+	0.945078i	$0.605985\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 3.86253i	− 0.202730i
$364$	0	0
$365$	− 63.5727i	− 3.32755i
$366$	0	0
$367$	−20.4826	−1.06918	−0.534592	−	0.845111i	$-0.679535\pi$
$367$	−20.4826	−1.06918	−0.534592	+	0.845111i	$0.679535\pi$
$368$	0	0
$369$	−56.4353	−2.93790
$370$	0	0
$371$	− 0.642343i	− 0.0333488i
$372$	0	0
$373$	− 4.88364i	− 0.252865i	−0.991975	−	0.126433i	$-0.959647\pi$
$373$	− 4.88364i	− 0.252865i	0.991975	−	0.126433i	$-0.0403528\pi$
$374$	0	0
$375$	−61.0833	−3.15433
$376$	0	0
$377$	−14.3874	−0.740987
$378$	0	0
$379$	− 0.179702i	− 0.00923065i	−0.999989	−	0.00461532i	$-0.998531\pi$
$379$	− 0.179702i	− 0.00923065i	0.999989	−	0.00461532i	$-0.00146911\pi$
$380$	0	0
$381$	− 41.6691i	− 2.13477i
$382$	0	0
$383$	7.86821	0.402047	0.201023	−	0.979586i	$-0.435573\pi$
$383$	7.86821	0.402047	0.201023	+	0.979586i	$0.435573\pi$
$384$	0	0
$385$	−8.15859	−0.415800
$386$	0	0
$387$	25.8836i	1.31574i
$388$	0	0
$389$	31.8614i	1.61544i	0.589569	+	0.807718i	$0.299298\pi$
$389$	31.8614i	1.61544i	−0.589569	+	0.807718i	$0.700702\pi$
$390$	0	0
$391$	−21.9720	−1.11117
$392$	0	0
$393$	47.7450	2.40842
$394$	0	0
$395$	− 17.2624i	− 0.868566i
$396$	0	0
$397$	− 2.93840i	− 0.147474i	−0.997278	−	0.0737371i	$-0.976507\pi$
$397$	− 2.93840i	− 0.147474i	0.997278	−	0.0737371i	$-0.0234926\pi$
$398$	0	0
$399$	1.73074	0.0866455
$400$	0	0
$401$	4.77981	0.238693	0.119346	−	0.992853i	$-0.461920\pi$
$401$	4.77981	0.238693	0.119346	+	0.992853i	$0.461920\pi$
$402$	0	0
$403$	− 11.0633i	− 0.551104i
$404$	0	0
$405$	− 18.9384i	− 0.941057i
$406$	0	0
$407$	12.0336	0.596485
$408$	0	0
$409$	17.8009	0.880199	0.440100	−	0.897949i	$-0.354943\pi$
$409$	17.8009	0.880199	0.440100	+	0.897949i	$0.354943\pi$
$410$	0	0
$411$	13.6212i	0.671886i
$412$	0	0
$413$	5.04907i	0.248449i
$414$	0	0
$415$	−52.7325	−2.58854
$416$	0	0
$417$	18.9316	0.927082
$418$	0	0
$419$	− 16.9230i	− 0.826741i	−0.910563	−	0.413371i	$-0.864351\pi$
$419$	− 16.9230i	− 0.826741i	0.910563	−	0.413371i	$-0.135649\pi$
$420$	0	0
$421$	11.9384i	0.581842i	0.956747	+	0.290921i	$0.0939617\pi$
$421$	11.9384i	0.581842i	−0.956747	+	0.290921i	$0.906038\pi$
$422$	0	0
$423$	−11.4769	−0.558027
$424$	0	0
$425$	−31.0354	−1.50544
$426$	0	0
$427$	4.21450i	0.203954i
$428$	0	0
$429$	25.7114i	1.24136i
$430$	0	0
$431$	−29.1729	−1.40521	−0.702604	−	0.711581i	$-0.747979\pi$
$431$	−29.1729	−1.40521	−0.702604	+	0.711581i	$0.747979\pi$
$432$	0	0
$433$	−5.91044	−0.284038	−0.142019	−	0.989864i	$-0.545359\pi$
$433$	−5.91044	−0.284038	−0.142019	+	0.989864i	$0.545359\pi$
$434$	0	0
$435$	− 65.4900i	− 3.14001i
$436$	0	0
$437$	− 7.32401i	− 0.350355i
$438$	0	0
$439$	37.7450	1.80147	0.900736	−	0.434368i	$-0.143028\pi$
$439$	37.7450	1.80147	0.900736	+	0.434368i	$0.143028\pi$
$440$	0	0
$441$	36.6343	1.74449
$442$	0	0
$443$	9.95093i	0.472783i	0.971658	+	0.236391i	$0.0759648\pi$
$443$	9.95093i	0.472783i	−0.971658	+	0.236391i	$0.924035\pi$
$444$	0	0
$445$	0.945243i	0.0448088i
$446$	0	0
$447$	−19.6355	−0.928727
$448$	0	0
$449$	21.6269	1.02064	0.510319	−	0.859985i	$-0.329527\pi$
$449$	21.6269	1.02064	0.510319	+	0.859985i	$0.329527\pi$
$450$	0	0
$451$	35.9527i	1.69295i
$452$	0	0
$453$	− 53.1169i	− 2.49565i
$454$	0	0
$455$	5.83457	0.273529
$456$	0	0
$457$	−5.49628	−0.257105	−0.128553	−	0.991703i	$-0.541033\pi$
$457$	−5.49628	−0.257105	−0.128553	+	0.991703i	$0.541033\pi$
$458$	0	0
$459$	− 21.9720i	− 1.02557i
$460$	0	0
$461$	− 5.42100i	− 0.252481i	−0.992000	−	0.126241i	$-0.959709\pi$
$461$	− 5.42100i	− 0.252481i	0.992000	−	0.126241i	$-0.0402911\pi$
$462$	0	0
$463$	7.31149	0.339794	0.169897	−	0.985462i	$-0.445657\pi$
$463$	7.31149	0.339794	0.169897	+	0.985462i	$0.445657\pi$
$464$	0	0
$465$	50.3594	2.33536
$466$	0	0
$467$	6.97204i	0.322628i	0.986903	+	0.161314i	$0.0515731\pi$
$467$	6.97204i	0.322628i	−0.986903	+	0.161314i	$0.948427\pi$
$468$	0	0
$469$	7.66346i	0.353866i
$470$	0	0
$471$	−46.7998	−2.15642
$472$	0	0
$473$	16.4894	0.758185
$474$	0	0
$475$	− 10.3451i	− 0.474667i
$476$	0	0
$477$	− 5.96636i	− 0.273181i
$478$	0	0
$479$	25.2288	1.15273	0.576366	−	0.817192i	$-0.304470\pi$
$479$	25.2288	1.15273	0.576366	+	0.817192i	$0.304470\pi$
$480$	0	0
$481$	−8.60580	−0.392391
$482$	0	0
$483$	12.6760i	0.576777i
$484$	0	0
$485$	− 60.5671i	− 2.75021i
$486$	0	0
$487$	1.06335	0.0481848	0.0240924	−	0.999710i	$-0.492330\pi$
$487$	1.06335	0.0481848	0.0240924	+	0.999710i	$0.492330\pi$
$488$	0	0
$489$	−70.1381	−3.17175
$490$	0	0
$491$	18.6903i	0.843480i	0.906717	+	0.421740i	$0.138580\pi$
$491$	18.6903i	0.843480i	−0.906717	+	0.421740i	$0.861420\pi$
$492$	0	0
$493$	− 17.1922i	− 0.774299i
$494$	0	0
$495$	−75.7804	−3.40608
$496$	0	0
$497$	7.62691	0.342114
$498$	0	0
$499$	15.8836i	0.711050i	0.934667	+	0.355525i	$0.115698\pi$
$499$	15.8836i	0.711050i	−0.934667	+	0.355525i	$0.884302\pi$
$500$	0	0
$501$	57.2824i	2.55919i
$502$	0	0
$503$	21.2430	0.947181	0.473590	−	0.880745i	$-0.342958\pi$
$503$	21.2430	0.947181	0.473590	+	0.880745i	$0.342958\pi$
$504$	0	0
$505$	−62.0285	−2.76023
$506$	0	0
$507$	19.5374i	0.867685i
$508$	0	0
$509$	− 20.6230i	− 0.914097i	−0.889442	−	0.457049i	$-0.848907\pi$
$509$	− 20.6230i	− 0.914097i	0.889442	−	0.457049i	$-0.151093\pi$
$510$	0	0
$511$	−9.62807	−0.425921
$512$	0	0
$513$	7.32401	0.323363
$514$	0	0
$515$	52.7325i	2.32367i
$516$	0	0
$517$	7.31149i	0.321559i
$518$	0	0
$519$	−24.4267	−1.07221
$520$	0	0
$521$	5.26242	0.230551	0.115275	−	0.993334i	$-0.463225\pi$
$521$	5.26242	0.230551	0.115275	+	0.993334i	$0.463225\pi$
$522$	0	0
$523$	− 2.67599i	− 0.117013i	−0.998287	−	0.0585063i	$-0.981366\pi$
$523$	− 2.67599i	− 0.117013i	0.998287	−	0.0585063i	$-0.0186338\pi$
$524$	0	0
$525$	17.9048i	0.781428i
$526$	0	0
$527$	13.2202	0.575880
$528$	0	0
$529$	30.6412	1.33223
$530$	0	0
$531$	46.8979i	2.03520i
$532$	0	0
$533$	− 25.7114i	− 1.11368i
$534$	0	0
$535$	−1.18654	−0.0512987
$536$	0	0
$537$	29.8768	1.28928
$538$	0	0
$539$	− 23.3383i	− 1.00525i
$540$	0	0
$541$	22.5653i	0.970159i	0.874470	+	0.485079i	$0.161209\pi$
$541$	22.5653i	0.970159i	−0.874470	+	0.485079i	$0.838791\pi$
$542$	0	0
$543$	−71.5459	−3.07033
$544$	0	0
$545$	−20.7239	−0.887714
$546$	0	0
$547$	− 8.08446i	− 0.345667i	−0.984951	−	0.172833i	$-0.944708\pi$
$547$	− 8.08446i	− 0.345667i	0.984951	−	0.172833i	$-0.0552922\pi$
$548$	0	0
$549$	39.1461i	1.67071i
$550$	0	0
$551$	5.73074	0.244138
$552$	0	0
$553$	−2.61439	−0.111175
$554$	0	0
$555$	− 39.1729i	− 1.66280i
$556$	0	0
$557$	− 1.12888i	− 0.0478323i	−0.999714	−	0.0239162i	$-0.992387\pi$
$557$	− 1.12888i	− 0.0478323i	0.999714	−	0.0239162i	$-0.00761348\pi$
$558$	0	0
$559$	−11.7923	−0.498763
$560$	0	0
$561$	−30.7239	−1.29716
$562$	0	0
$563$	− 35.4701i	− 1.49489i	−0.664326	−	0.747443i	$-0.731281\pi$
$563$	− 35.4701i	− 1.49489i	0.664326	−	0.747443i	$-0.268719\pi$
$564$	0	0
$565$	− 10.8893i	− 0.458118i
$566$	0	0
$567$	−2.86821	−0.120454
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	− 12.2499i	− 0.512642i	−0.966592	−	0.256321i	$-0.917490\pi$
$571$	− 12.2499i	− 0.512642i	0.966592	−	0.256321i	$-0.0825104\pi$
$572$	0	0
$573$	− 27.0827i	− 1.13140i
$574$	0	0
$575$	75.7679	3.15974
$576$	0	0
$577$	−22.8768	−0.952374	−0.476187	−	0.879344i	$-0.657981\pi$
$577$	−22.8768	−0.952374	−0.476187	+	0.879344i	$0.657981\pi$
$578$	0	0
$579$	− 23.4364i	− 0.973985i
$580$	0	0
$581$	7.98632i	0.331328i
$582$	0	0
$583$	−3.80093	−0.157418
$584$	0	0
$585$	54.1940	2.24065
$586$	0	0
$587$	− 29.2219i	− 1.20612i	−0.797697	−	0.603059i	$-0.793948\pi$
$587$	− 29.2219i	− 1.20612i	0.797697	−	0.603059i	$-0.206052\pi$
$588$	0	0
$589$	4.40673i	0.181576i
$590$	0	0
$591$	16.4153	0.675236
$592$	0	0
$593$	−1.35197	−0.0555188	−0.0277594	−	0.999615i	$-0.508837\pi$
$593$	−1.35197	−0.0555188	−0.0277594	+	0.999615i	$0.508837\pi$
$594$	0	0
$595$	6.97204i	0.285826i
$596$	0	0
$597$	52.7941i	2.16072i
$598$	0	0
$599$	−41.1814	−1.68263	−0.841314	−	0.540546i	$-0.818218\pi$
$599$	−41.1814	−1.68263	−0.841314	+	0.540546i	$0.818218\pi$
$600$	0	0
$601$	5.25383	0.214308	0.107154	−	0.994242i	$-0.465826\pi$
$601$	5.25383	0.214308	0.107154	+	0.994242i	$0.465826\pi$
$602$	0	0
$603$	71.1814i	2.89873i
$604$	0	0
$605$	5.18654i	0.210863i
$606$	0	0
$607$	9.17286	0.372315	0.186157	−	0.982520i	$-0.440397\pi$
$607$	9.17286	0.372315	0.186157	+	0.982520i	$0.440397\pi$
$608$	0	0
$609$	−9.91844	−0.401916
$610$	0	0
$611$	− 5.22877i	− 0.211534i
$612$	0	0
$613$	− 28.3577i	− 1.14535i	−0.819781	−	0.572677i	$-0.805905\pi$
$613$	− 28.3577i	− 1.14535i	0.819781	−	0.572677i	$-0.194095\pi$
$614$	0	0
$615$	117.036	4.71935
$616$	0	0
$617$	−19.4546	−0.783214	−0.391607	−	0.920132i	$-0.628081\pi$
$617$	−19.4546	−0.783214	−0.391607	+	0.920132i	$0.628081\pi$
$618$	0	0
$619$	− 25.2538i	− 1.01504i	−0.861641	−	0.507519i	$-0.830563\pi$
$619$	− 25.2538i	− 1.01504i	0.861641	−	0.507519i	$-0.169437\pi$
$620$	0	0
$621$	53.6412i	2.15255i
$622$	0	0
$623$	0.143157	0.00573545
$624$	0	0
$625$	30.2961	1.21184
$626$	0	0
$627$	− 10.2413i	− 0.408998i
$628$	0	0
$629$	− 10.2835i	− 0.410031i
$630$	0	0
$631$	23.6287	0.940642	0.470321	−	0.882495i	$-0.344138\pi$
$631$	23.6287	0.940642	0.470321	+	0.882495i	$0.344138\pi$
$632$	0	0
$633$	0.883644	0.0351217
$634$	0	0
$635$	55.9527i	2.22041i
$636$	0	0
$637$	16.6903i	0.661292i
$638$	0	0
$639$	70.8420	2.80247
$640$	0	0
$641$	26.9230	1.06339	0.531697	−	0.846935i	$-0.321555\pi$
$641$	26.9230	1.06339	0.531697	+	0.846935i	$0.321555\pi$
$642$	0	0
$643$	13.4124i	0.528934i	0.964395	+	0.264467i	$0.0851960\pi$
$643$	13.4124i	0.528934i	−0.964395	+	0.264467i	$0.914804\pi$
$644$	0	0
$645$	− 53.6777i	− 2.11356i
$646$	0	0
$647$	−6.55104	−0.257548	−0.128774	−	0.991674i	$-0.541104\pi$
$647$	−6.55104	−0.257548	−0.128774	+	0.991674i	$0.541104\pi$
$648$	0	0
$649$	29.8768	1.17277
$650$	0	0
$651$	− 7.62691i	− 0.298922i
$652$	0	0
$653$	2.52308i	0.0987359i	0.998781	+	0.0493680i	$0.0157207\pi$
$653$	2.52308i	0.0987359i	−0.998781	+	0.0493680i	$0.984279\pi$
$654$	0	0
$655$	−64.1113	−2.50503
$656$	0	0
$657$	−89.4296	−3.48898
$658$	0	0
$659$	− 5.17227i	− 0.201483i	−0.994913	−	0.100742i	$-0.967878\pi$
$659$	− 5.17227i	− 0.201483i	0.994913	−	0.100742i	$-0.0321215\pi$
$660$	0	0
$661$	41.6412i	1.61965i	0.586668	+	0.809827i	$0.300439\pi$
$661$	41.6412i	1.61965i	−0.586668	+	0.809827i	$0.699561\pi$
$662$	0	0
$663$	21.9720	0.853323
$664$	0	0
$665$	−2.32401	−0.0901214
$666$	0	0
$667$	41.9720i	1.62516i
$668$	0	0
$669$	− 2.47123i	− 0.0955433i
$670$	0	0
$671$	24.9384	0.962736
$672$	0	0
$673$	−16.7325	−0.644990	−0.322495	−	0.946571i	$-0.604522\pi$
$673$	−16.7325	−0.644990	−0.322495	+	0.946571i	$0.604522\pi$
$674$	0	0
$675$	75.7679i	2.91631i
$676$	0	0
$677$	1.45580i	0.0559510i	0.999609	+	0.0279755i	$0.00890603\pi$
$677$	1.45580i	0.0559510i	−0.999609	+	0.0279755i	$0.991094\pi$
$678$	0	0
$679$	−9.17286	−0.352022
$680$	0	0
$681$	36.6201	1.40328
$682$	0	0
$683$	8.81346i	0.337238i	0.985681	+	0.168619i	$0.0539307\pi$
$683$	8.81346i	0.337238i	−0.985681	+	0.168619i	$0.946069\pi$
$684$	0	0
$685$	− 18.2904i	− 0.698839i
$686$	0	0
$687$	−43.0970	−1.64425
$688$	0	0
$689$	2.71822	0.103556
$690$	0	0
$691$	− 12.6834i	− 0.482500i	−0.970463	−	0.241250i	$-0.922443\pi$
$691$	− 12.6834i	− 0.482500i	0.970463	−	0.241250i	$-0.0775574\pi$
$692$	0	0
$693$	11.4769i	0.435972i
$694$	0	0
$695$	−25.4210	−0.964274
$696$	0	0
$697$	30.7239	1.16375
$698$	0	0
$699$	− 22.5162i	− 0.851643i
$700$	0	0
$701$	1.46149i	0.0551996i	0.999619	+	0.0275998i	$0.00878640\pi$
$701$	1.46149i	0.0551996i	−0.999619	+	0.0275998i	$0.991214\pi$
$702$	0	0
$703$	3.42784	0.129284
$704$	0	0
$705$	23.8009	0.896395
$706$	0	0
$707$	9.39420i	0.353305i
$708$	0	0
$709$	− 19.7114i	− 0.740276i	−0.928977	−	0.370138i	$-0.879310\pi$
$709$	− 19.7114i	− 0.740276i	0.928977	−	0.370138i	$-0.120690\pi$
$710$	0	0
$711$	−24.2835	−0.910704
$712$	0	0
$713$	−32.2749	−1.20871
$714$	0	0
$715$	− 34.5248i	− 1.29116i
$716$	0	0
$717$	80.5591i	3.00853i
$718$	0	0
$719$	20.5933	0.767999	0.384000	−	0.923333i	$-0.374546\pi$
$719$	20.5933	0.767999	0.384000	+	0.923333i	$0.374546\pi$
$720$	0	0
$721$	7.98632	0.297426
$722$	0	0
$723$	− 4.16543i	− 0.154914i
$724$	0	0
$725$	59.2853i	2.20180i
$726$	0	0
$727$	−4.66056	−0.172850	−0.0864252	−	0.996258i	$-0.527544\pi$
$727$	−4.66056	−0.172850	−0.0864252	+	0.996258i	$0.527544\pi$
$728$	0	0
$729$	37.4575	1.38732
$730$	0	0
$731$	− 14.0913i	− 0.521186i
$732$	0	0
$733$	− 34.4826i	− 1.27364i	−0.771011	−	0.636822i	$-0.780249\pi$
$733$	− 34.4826i	− 1.27364i	0.771011	−	0.636822i	$-0.219751\pi$
$734$	0	0
$735$	−75.9726	−2.80229
$736$	0	0
$737$	45.3469	1.67037
$738$	0	0
$739$	12.4222i	0.456956i	0.973549	+	0.228478i	$0.0733750\pi$
$739$	12.4222i	0.456956i	−0.973549	+	0.228478i	$0.926625\pi$
$740$	0	0
$741$	7.32401i	0.269054i
$742$	0	0
$743$	−6.77981	−0.248727	−0.124364	−	0.992237i	$-0.539689\pi$
$743$	−6.77981	−0.248727	−0.124364	+	0.992237i	$0.539689\pi$
$744$	0	0
$745$	26.3662	0.965984
$746$	0	0
$747$	74.1803i	2.71411i
$748$	0	0
$749$	0.179702i	0.00656615i
$750$	0	0
$751$	−40.4826	−1.47723	−0.738616	−	0.674127i	$-0.764520\pi$
$751$	−40.4826	−1.47723	−0.738616	+	0.674127i	$0.764520\pi$
$752$	0	0
$753$	−59.8968	−2.18276
$754$	0	0
$755$	71.3246i	2.59577i
$756$	0	0
$757$	− 23.5191i	− 0.854818i	−0.904058	−	0.427409i	$-0.859426\pi$
$757$	− 23.5191i	− 0.854818i	0.904058	−	0.427409i	$-0.140574\pi$
$758$	0	0
$759$	75.0074	2.72260
$760$	0	0
$761$	2.32692	0.0843507	0.0421753	−	0.999110i	$-0.486571\pi$
$761$	2.32692	0.0843507	0.0421753	+	0.999110i	$0.486571\pi$
$762$	0	0
$763$	3.13863i	0.113626i
$764$	0	0
$765$	64.7593i	2.34138i
$766$	0	0
$767$	−21.3662	−0.771490
$768$	0	0
$769$	34.8768	1.25769	0.628845	−	0.777531i	$-0.283528\pi$
$769$	34.8768	1.25769	0.628845	+	0.777531i	$0.283528\pi$
$770$	0	0
$771$	− 74.4016i	− 2.67951i
$772$	0	0
$773$	− 5.94699i	− 0.213898i	−0.994264	−	0.106949i	$-0.965892\pi$
$773$	− 5.94699i	− 0.213898i	0.994264	−	0.106949i	$-0.0341082\pi$
$774$	0	0
$775$	−45.5882	−1.63758
$776$	0	0
$777$	−5.93272	−0.212835
$778$	0	0
$779$	10.2413i	0.366933i
$780$	0	0
$781$	− 45.1306i	− 1.61490i
$782$	0	0
$783$	−41.9720	−1.49996
$784$	0	0
$785$	62.8420	2.24293
$786$	0	0
$787$	0.277845i	0.00990411i	0.999988	+	0.00495206i	$0.00157630\pi$
$787$	0.277845i	0.00990411i	−0.999988	+	0.00495206i	$0.998424\pi$
$788$	0	0
$789$	− 40.1181i	− 1.42824i
$790$	0	0
$791$	−1.64919	−0.0586383
$792$	0	0
$793$	−17.8346	−0.633324
$794$	0	0
$795$	12.3731i	0.438828i
$796$	0	0
$797$	− 5.01543i	− 0.177656i	−0.996047	−	0.0888278i	$-0.971688\pi$
$797$	− 5.01543i	− 0.177656i	0.996047	−	0.0888278i	$-0.0283121\pi$
$798$	0	0
$799$	6.24814	0.221043
$800$	0	0
$801$	1.32970	0.0469826
$802$	0	0
$803$	56.9720i	2.01050i
$804$	0	0
$805$	− 17.0211i	− 0.599915i
$806$	0	0
$807$	60.7439	2.13829
$808$	0	0
$809$	18.9190	0.665158	0.332579	−	0.943075i	$-0.392081\pi$
$809$	18.9190	0.665158	0.332579	+	0.943075i	$0.392081\pi$
$810$	0	0
$811$	45.3103i	1.59106i	0.605914	+	0.795530i	$0.292808\pi$
$811$	45.3103i	1.59106i	−0.605914	+	0.795530i	$0.707192\pi$
$812$	0	0
$813$	43.2567i	1.51708i
$814$	0	0
$815$	94.1803	3.29899
$816$	0	0
$817$	4.69710	0.164331
$818$	0	0
$819$	− 8.20766i	− 0.286799i
$820$	0	0
$821$	12.6498i	0.441480i	0.975333	+	0.220740i	$0.0708473\pi$
$821$	12.6498i	0.441480i	−0.975333	+	0.220740i	$0.929153\pi$
$822$	0	0
$823$	−43.9065	−1.53048	−0.765242	−	0.643742i	$-0.777381\pi$
$823$	−43.9065	−1.53048	−0.765242	+	0.643742i	$0.777381\pi$
$824$	0	0
$825$	105.948	3.68862
$826$	0	0
$827$	− 13.0405i	− 0.453462i	−0.973957	−	0.226731i	$-0.927196\pi$
$827$	− 13.0405i	− 0.453462i	0.973957	−	0.226731i	$-0.0728038\pi$
$828$	0	0
$829$	49.4507i	1.71749i	0.512400	+	0.858747i	$0.328757\pi$
$829$	49.4507i	1.71749i	−0.512400	+	0.858747i	$0.671243\pi$
$830$	0	0
$831$	46.4353	1.61082
$832$	0	0
$833$	−19.9441	−0.691022
$834$	0	0
$835$	− 76.9179i	− 2.66185i
$836$	0	0
$837$	− 32.2749i	− 1.11559i
$838$	0	0
$839$	15.2288	0.525756	0.262878	−	0.964829i	$-0.415328\pi$
$839$	15.2288	0.525756	0.262878	+	0.964829i	$0.415328\pi$
$840$	0	0
$841$	−3.84141	−0.132463
$842$	0	0
$843$	− 13.5596i	− 0.467018i
$844$	0	0
$845$	− 26.2345i	− 0.902493i
$846$	0	0
$847$	0.785500	0.0269901
$848$	0	0
$849$	11.6492	0.399799
$850$	0	0
$851$	25.1056i	0.860608i
$852$	0	0
$853$	15.4193i	0.527945i	0.964530	+	0.263973i	$0.0850329\pi$
$853$	15.4193i	0.527945i	−0.964530	+	0.263973i	$0.914967\pi$
$854$	0	0
$855$	−21.5864	−0.738240
$856$	0	0
$857$	14.3422	0.489921	0.244961	−	0.969533i	$-0.421225\pi$
$857$	14.3422	0.489921	0.244961	+	0.969533i	$0.421225\pi$
$858$	0	0
$859$	46.2681i	1.57865i	0.613977	+	0.789324i	$0.289569\pi$
$859$	46.2681i	1.57865i	−0.613977	+	0.789324i	$0.710431\pi$
$860$	0	0
$861$	− 17.7251i	− 0.604068i
$862$	0	0
$863$	25.9527	0.883439	0.441720	−	0.897153i	$-0.354369\pi$
$863$	25.9527	0.883439	0.441720	+	0.897153i	$0.354369\pi$
$864$	0	0
$865$	32.7998	1.11523
$866$	0	0
$867$	− 23.3383i	− 0.792610i
$868$	0	0
$869$	15.4701i	0.524786i
$870$	0	0
$871$	−32.4296	−1.09883
$872$	0	0
$873$	−85.2014	−2.88363
$874$	0	0
$875$	− 12.4222i	− 0.419946i
$876$	0	0
$877$	21.3526i	0.721025i	0.932754	+	0.360512i	$0.117398\pi$
$877$	21.3526i	0.721025i	−0.932754	+	0.360512i	$0.882602\pi$
$878$	0	0
$879$	−9.09130	−0.306642
$880$	0	0
$881$	53.2642	1.79452	0.897258	−	0.441507i	$-0.145556\pi$
$881$	53.2642	1.79452	0.897258	+	0.441507i	$0.145556\pi$
$882$	0	0
$883$	38.1586i	1.28414i	0.766647	+	0.642069i	$0.221924\pi$
$883$	38.1586i	1.28414i	−0.766647	+	0.642069i	$0.778076\pi$
$884$	0	0
$885$	− 97.2573i	− 3.26927i
$886$	0	0
$887$	11.1615	0.374766	0.187383	−	0.982287i	$-0.439999\pi$
$887$	11.1615	0.374766	0.187383	+	0.982287i	$0.439999\pi$
$888$	0	0
$889$	8.47401	0.284209
$890$	0	0
$891$	16.9720i	0.568585i
$892$	0	0
$893$	2.08271i	0.0696954i
$894$	0	0
$895$	−40.1181	−1.34100
$896$	0	0
$897$	−53.6412	−1.79103
$898$	0	0
$899$	− 25.2538i	− 0.842262i
$900$	0	0
$901$	3.24814i	0.108211i
$902$	0	0
$903$	−8.12947	−0.270532
$904$	0	0
$905$	96.0708	3.19350
$906$	0	0
$907$	32.7941i	1.08891i	0.838790	+	0.544455i	$0.183263\pi$
$907$	32.7941i	1.08891i	−0.838790	+	0.544455i	$0.816737\pi$
$908$	0	0
$909$	87.2573i	2.89414i
$910$	0	0
$911$	46.8979	1.55380	0.776899	−	0.629626i	$-0.216792\pi$
$911$	46.8979	1.55380	0.776899	+	0.629626i	$0.216792\pi$
$912$	0	0
$913$	47.2573	1.56399
$914$	0	0
$915$	− 81.1814i	− 2.68378i
$916$	0	0
$917$	9.70963i	0.320640i
$918$	0	0
$919$	45.1871	1.49059	0.745293	−	0.666737i	$-0.232310\pi$
$919$	45.1871	1.49059	0.745293	+	0.666737i	$0.232310\pi$
$920$	0	0
$921$	−69.1478	−2.27850
$922$	0	0
$923$	32.2749i	1.06234i
$924$	0	0
$925$	35.4615i	1.16597i
$926$	0	0
$927$	74.1803	2.43640
$928$	0	0
$929$	−51.0776	−1.67580	−0.837901	−	0.545822i	$-0.816217\pi$
$929$	−51.0776	−1.67580	−0.837901	+	0.545822i	$0.816217\pi$
$930$	0	0
$931$	− 6.64803i	− 0.217880i
$932$	0	0
$933$	− 5.55163i	− 0.181752i
$934$	0	0
$935$	41.2556	1.34920
$936$	0	0
$937$	−24.5037	−0.800502	−0.400251	−	0.916406i	$-0.631077\pi$
$937$	−24.5037	−0.800502	−0.400251	+	0.916406i	$0.631077\pi$
$938$	0	0
$939$	− 92.4296i	− 3.01633i
$940$	0	0
$941$	55.1871i	1.79905i	0.436870	+	0.899525i	$0.356087\pi$
$941$	55.1871i	1.79905i	−0.436870	+	0.899525i	$0.643913\pi$
$942$	0	0
$943$	−75.0074	−2.44258
$944$	0	0
$945$	17.0211	0.553697
$946$	0	0
$947$	− 29.9019i	− 0.971680i	−0.874048	−	0.485840i	$-0.838514\pi$
$947$	− 29.9019i	− 0.971680i	0.874048	−	0.485840i	$-0.161486\pi$
$948$	0	0
$949$	− 40.7433i	− 1.32258i
$950$	0	0
$951$	35.5317	1.15219
$952$	0	0
$953$	−26.0086	−0.842501	−0.421250	−	0.906944i	$-0.638409\pi$
$953$	−26.0086	−0.842501	−0.421250	+	0.906944i	$0.638409\pi$
$954$	0	0
$955$	36.3662i	1.17678i
$956$	0	0
$957$	58.6903i	1.89719i
$958$	0	0
$959$	−2.77007	−0.0894502
$960$	0	0
$961$	−11.5807	−0.373572
$962$	0	0
$963$	1.66914i	0.0537874i
$964$	0	0
$965$	31.4701i	1.01306i
$966$	0	0
$967$	−50.8420	−1.63497	−0.817484	−	0.575951i	$-0.804632\pi$
$967$	−50.8420	−1.63497	−0.817484	+	0.575951i	$0.804632\pi$
$968$	0	0
$969$	−8.75186	−0.281150
$970$	0	0
$971$	22.1267i	0.710079i	0.934851	+	0.355040i	$0.115533\pi$
$971$	22.1267i	0.710079i	−0.934851	+	0.355040i	$0.884467\pi$
$972$	0	0
$973$	3.85000i	0.123425i
$974$	0	0
$975$	−75.7679	−2.42651
$976$	0	0
$977$	43.1814	1.38150	0.690748	−	0.723095i	$-0.257281\pi$
$977$	43.1814	1.38150	0.690748	+	0.723095i	$0.257281\pi$
$978$	0	0
$979$	− 0.847099i	− 0.0270734i
$980$	0	0
$981$	29.1529i	0.930780i
$982$	0	0
$983$	−34.8842	−1.11263	−0.556317	−	0.830970i	$-0.687786\pi$
$983$	−34.8842	−1.11263	−0.556317	+	0.830970i	$0.687786\pi$
$984$	0	0
$985$	−22.0422	−0.702324
$986$	0	0
$987$	− 3.60464i	− 0.114737i
$988$	0	0
$989$	34.4016i	1.09391i
$990$	0	0
$991$	7.36056	0.233816	0.116908	−	0.993143i	$-0.462702\pi$
$991$	7.36056	0.233816	0.116908	+	0.993143i	$0.462702\pi$
$992$	0	0
$993$	−20.6623	−0.655698
$994$	0	0
$995$	− 70.8911i	− 2.24740i
$996$	0	0
$997$	− 37.2978i	− 1.18123i	−0.806952	−	0.590617i	$-0.798885\pi$
$997$	− 37.2978i	− 1.18123i	0.806952	−	0.590617i	$-0.201115\pi$
$998$	0	0
$999$	−25.1056	−0.794305

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.f.1217.6	yes	6
4.3	odd	2		2432.2.c.g.1217.1	yes	6
8.3	odd	2		2432.2.c.g.1217.6	yes	6
8.5	even	2	inner	2432.2.c.f.1217.1	✓	6
16.3	odd	4		4864.2.a.bc.1.3		3
16.5	even	4		4864.2.a.bd.1.3		3
16.11	odd	4		4864.2.a.bf.1.1		3
16.13	even	4		4864.2.a.be.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.f.1217.1	✓	6	8.5	even	2	inner
2432.2.c.f.1217.6	yes	6	1.1	even	1	trivial
2432.2.c.g.1217.1	yes	6	4.3	odd	2
2432.2.c.g.1217.6	yes	6	8.3	odd	2
4864.2.a.bc.1.3		3	16.3	odd	4
4864.2.a.bd.1.3		3	16.5	even	4
4864.2.a.be.1.1		3	16.13	even	4
4864.2.a.bf.1.1		3	16.11	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:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.91729i q^{3} -3.91729i q^{5} -0.593272 q^{7} -5.51056 q^{9} +O(q^{10})\)	\(q+2.91729i q^{3} -3.91729i q^{5} -0.593272 q^{7} -5.51056 q^{9} +3.51056i q^{11} -2.51056i q^{13} +11.4278 q^{15} +3.00000 q^{17} +1.00000i q^{19} -1.73074i q^{21} -7.32401 q^{23} -10.3451 q^{25} -7.32401i q^{27} -5.73074i q^{29} +4.40673 q^{31} -10.2413 q^{33} +2.32401i q^{35} -3.42784i q^{37} +7.32401 q^{39} +10.2413 q^{41} -4.69710i q^{43} +21.5864i q^{45} +2.08271 q^{47} -6.64803 q^{49} +8.75186i q^{51} +1.08271i q^{53} +13.7519 q^{55} -2.91729 q^{57} -8.51056i q^{59} -7.10383i q^{61} +3.26926 q^{63} -9.83457 q^{65} -12.9173i q^{67} -21.3662i q^{69} -12.8557 q^{71} +16.2288 q^{73} -30.1797i q^{75} -2.08271i q^{77} +4.40673 q^{79} +4.83457 q^{81} -13.4615i q^{83} -11.7519i q^{85} +16.7182 q^{87} -0.241300 q^{89} +1.48944i q^{91} +12.8557i q^{93} +3.91729 q^{95} +15.4615 q^{97} -19.3451i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{7} - 16 q^{9}+O(q^{10}) \) 6 * q - 6 * q^7 - 16 * q^9	\( 6 q - 6 q^{7} - 16 q^{9} + 32 q^{15} + 18 q^{17} - 22 q^{23} - 6 q^{25} + 24 q^{31} - 20 q^{33} + 22 q^{39} + 20 q^{41} + 32 q^{47} + 4 q^{49} + 24 q^{55} + 2 q^{57} + 44 q^{63} - 20 q^{65} - 4 q^{71} + 34 q^{73} + 24 q^{79} - 10 q^{81} + 54 q^{87} + 40 q^{89} + 4 q^{95} + 44 q^{97}+O(q^{100}) \) 6 * q - 6 * q^7 - 16 * q^9 + 32 * q^15 + 18 * q^17 - 22 * q^23 - 6 * q^25 + 24 * q^31 - 20 * q^33 + 22 * q^39 + 20 * q^41 + 32 * q^47 + 4 * q^49 + 24 * q^55 + 2 * q^57 + 44 * q^63 - 20 * q^65 - 4 * q^71 + 34 * q^73 + 24 * q^79 - 10 * q^81 + 54 * q^87 + 40 * q^89 + 4 * q^95 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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