LMFDB - Embedded newform 2432.2.c.i.1217.7

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:	$16$
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} - 4 x^{15} + 8 x^{14} - 8 x^{13} - 3 x^{12} + 20 x^{11} - 24 x^{10} + 28 x^{8} - 96 x^{6} + \cdots + 256 $ x^16 - 4x^15 + 8x^14 - 8x^13 - 3x^12 + 20x^11 - 24x^10 + 28x^8 - 96x^6 + 160x^5 - 48x^4 - 256x^3 + 512x^2 - 512*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.7
Root			$-1.07547 - 0.918353i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.i.1217.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.222191i q^{3} -1.55081i q^{5} -3.06334 q^{7} +2.95063 q^{9} +O(q^{10})$	$q-0.222191i q^{3} -1.55081i q^{5} -3.06334 q^{7} +2.95063 q^{9} +0.950631i q^{11} +3.36113i q^{13} -0.344577 q^{15} +1.82843 q^{17} -1.00000i q^{19} +0.680647i q^{21} -3.36113 q^{23} +2.59499 q^{25} -1.32218i q^{27} +4.95873i q^{29} +3.44620 q^{31} +0.211222 q^{33} +4.75066i q^{35} +2.95889i q^{37} +0.746814 q^{39} +4.55687 q^{41} -1.69373i q^{43} -4.57587i q^{45} +3.39941 q^{47} +2.38404 q^{49} -0.406261i q^{51} -4.47142i q^{53} +1.47425 q^{55} -0.222191 q^{57} +0.395014i q^{59} +5.34158i q^{61} -9.03878 q^{63} +5.21247 q^{65} +6.85064i q^{67} +0.746814i q^{69} +5.15207 q^{71} +1.19998 q^{73} -0.576584i q^{75} -2.91210i q^{77} +14.7249 q^{79} +8.55812 q^{81} +3.35564i q^{83} -2.83554i q^{85} +1.10179 q^{87} +9.64561 q^{89} -10.2963i q^{91} -0.765715i q^{93} -1.55081 q^{95} +3.55562 q^{97} +2.80496i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^9	$ 16 q - 8 q^{9} - 16 q^{17} - 40 q^{25} + 48 q^{33} - 16 q^{41} + 16 q^{49} + 8 q^{57} + 16 q^{65} + 16 q^{73} - 64 q^{81} + 16 q^{89} + 80 q^{97}+O(q^{100}) $ 16 * q - 8 * q^9 - 16 * q^17 - 40 * q^25 + 48 * q^33 - 16 * q^41 + 16 * q^49 + 8 * q^57 + 16 * q^65 + 16 * q^73 - 64 * q^81 + 16 * q^89 + 80 * 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
$3$	− 0.222191i	− 0.128282i	−0.997941	−	0.0641411i	$-0.979569\pi$
$3$	− 0.222191i	− 0.128282i	0.997941	−	0.0641411i	$-0.0204308\pi$
$4$	0	0
$5$	− 1.55081i	− 0.693543i	−0.937950	−	0.346772i	$-0.887278\pi$
$5$	− 1.55081i	− 0.693543i	0.937950	−	0.346772i	$-0.112722\pi$
$6$	0	0
$7$	−3.06334	−1.15783	−0.578917	−	0.815387i	$-0.696524\pi$
$7$	−3.06334	−1.15783	−0.578917	+	0.815387i	$0.696524\pi$
$8$	0	0
$9$	2.95063	0.983544
$10$	0	0
$11$	0.950631i	0.286626i	0.989677	+	0.143313i	$0.0457756\pi$
$11$	0.950631i	0.286626i	−0.989677	+	0.143313i	$0.954224\pi$
$12$	0	0
$13$	3.36113i	0.932209i	0.884730	+	0.466105i	$0.154343\pi$
$13$	3.36113i	0.932209i	−0.884730	+	0.466105i	$0.845657\pi$
$14$	0	0
$15$	−0.344577	−0.0889693
$16$	0	0
$17$	1.82843	0.443459	0.221729	−	0.975108i	$-0.428830\pi$
$17$	1.82843	0.443459	0.221729	+	0.975108i	$0.428830\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0.680647i	0.148529i
$22$	0	0
$23$	−3.36113	−0.700844	−0.350422	−	0.936592i	$-0.613962\pi$
$23$	−3.36113	−0.700844	−0.350422	+	0.936592i	$0.613962\pi$
$24$	0	0
$25$	2.59499	0.518998
$26$	0	0
$27$	− 1.32218i	− 0.254453i
$28$	0	0
$29$	4.95873i	0.920812i	0.887708	+	0.460406i	$0.152296\pi$
$29$	4.95873i	0.920812i	−0.887708	+	0.460406i	$0.847704\pi$
$30$	0	0
$31$	3.44620	0.618955	0.309478	−	0.950907i	$-0.399846\pi$
$31$	3.44620	0.618955	0.309478	+	0.950907i	$0.399846\pi$
$32$	0	0
$33$	0.211222	0.0367690
$34$	0	0
$35$	4.75066i	0.803007i
$36$	0	0
$37$	2.95889i	0.486439i	0.969971	+	0.243219i	$0.0782035\pi$
$37$	2.95889i	0.486439i	−0.969971	+	0.243219i	$0.921797\pi$
$38$	0	0
$39$	0.746814	0.119586
$40$	0	0
$41$	4.55687	0.711663	0.355832	−	0.934550i	$-0.384198\pi$
$41$	4.55687	0.711663	0.355832	+	0.934550i	$0.384198\pi$
$42$	0	0
$43$	− 1.69373i	− 0.258291i	−0.991626	−	0.129145i	$-0.958777\pi$
$43$	− 1.69373i	− 0.258291i	0.991626	−	0.129145i	$-0.0412234\pi$
$44$	0	0
$45$	− 4.57587i	− 0.682130i
$46$	0	0
$47$	3.39941	0.495855	0.247927	−	0.968779i	$-0.420251\pi$
$47$	3.39941	0.495855	0.247927	+	0.968779i	$0.420251\pi$
$48$	0	0
$49$	2.38404	0.340578
$50$	0	0
$51$	− 0.406261i	− 0.0568879i
$52$	0	0
$53$	− 4.47142i	− 0.614197i	−0.951678	−	0.307098i	$-0.900642\pi$
$53$	− 4.47142i	− 0.614197i	0.951678	−	0.307098i	$-0.0993581\pi$
$54$	0	0
$55$	1.47425	0.198788
$56$	0	0
$57$	−0.222191	−0.0294300
$58$	0	0
$59$	0.395014i	0.0514264i	0.999669	+	0.0257132i	$0.00818567\pi$
$59$	0.395014i	0.0514264i	−0.999669	+	0.0257132i	$0.991814\pi$
$60$	0	0
$61$	5.34158i	0.683920i	0.939715	+	0.341960i	$0.111091\pi$
$61$	5.34158i	0.683920i	−0.939715	+	0.341960i	$0.888909\pi$
$62$	0	0
$63$	−9.03878	−1.13878
$64$	0	0
$65$	5.21247	0.646528
$66$	0	0
$67$	6.85064i	0.836939i	0.908231	+	0.418470i	$0.137433\pi$
$67$	6.85064i	0.836939i	−0.908231	+	0.418470i	$0.862567\pi$
$68$	0	0
$69$	0.746814i	0.0899058i
$70$	0	0
$71$	5.15207	0.611438	0.305719	−	0.952122i	$-0.401103\pi$
$71$	5.15207	0.611438	0.305719	+	0.952122i	$0.401103\pi$
$72$	0	0
$73$	1.19998	0.140446	0.0702232	−	0.997531i	$-0.477629\pi$
$73$	1.19998	0.140446	0.0702232	+	0.997531i	$0.477629\pi$
$74$	0	0
$75$	− 0.576584i	− 0.0665782i
$76$	0	0
$77$	− 2.91210i	− 0.331865i
$78$	0	0
$79$	14.7249	1.65669	0.828343	−	0.560222i	$-0.189284\pi$
$79$	14.7249	1.65669	0.828343	+	0.560222i	$0.189284\pi$
$80$	0	0
$81$	8.55812	0.950902
$82$	0	0
$83$	3.35564i	0.368330i	0.982895	+	0.184165i	$0.0589580\pi$
$83$	3.35564i	0.368330i	−0.982895	+	0.184165i	$0.941042\pi$
$84$	0	0
$85$	− 2.83554i	− 0.307558i
$86$	0	0
$87$	1.10179	0.118124
$88$	0	0
$89$	9.64561	1.02243	0.511216	−	0.859452i	$-0.329195\pi$
$89$	9.64561	1.02243	0.511216	+	0.859452i	$0.329195\pi$
$90$	0	0
$91$	− 10.2963i	− 1.07934i
$92$	0	0
$93$	− 0.765715i	− 0.0794010i
$94$	0	0
$95$	−1.55081	−0.159110
$96$	0	0
$97$	3.55562	0.361018	0.180509	−	0.983573i	$-0.442225\pi$
$97$	3.55562	0.361018	0.180509	+	0.983573i	$0.442225\pi$
$98$	0	0
$99$	2.80496i	0.281909i
$100$	0	0
$101$	− 13.5381i	− 1.34709i	−0.739146	−	0.673545i	$-0.764771\pi$
$101$	− 13.5381i	− 1.34709i	0.739146	−	0.673545i	$-0.235229\pi$
$102$	0	0
$103$	6.64570	0.654820	0.327410	−	0.944882i	$-0.393824\pi$
$103$	6.64570	0.654820	0.327410	+	0.944882i	$0.393824\pi$
$104$	0	0
$105$	1.05555	0.103012
$106$	0	0
$107$	− 15.8519i	− 1.53246i	−0.642566	−	0.766230i	$-0.722130\pi$
$107$	− 15.8519i	− 1.53246i	0.642566	−	0.766230i	$-0.277870\pi$
$108$	0	0
$109$	− 13.4213i	− 1.28553i	−0.766064	−	0.642764i	$-0.777788\pi$
$109$	− 13.4213i	− 1.28553i	0.766064	−	0.642764i	$-0.222212\pi$
$110$	0	0
$111$	0.657440	0.0624015
$112$	0	0
$113$	6.65810	0.626342	0.313171	−	0.949697i	$-0.398609\pi$
$113$	6.65810	0.626342	0.313171	+	0.949697i	$0.398609\pi$
$114$	0	0
$115$	5.21247i	0.486065i
$116$	0	0
$117$	9.91745i	0.916869i
$118$	0	0
$119$	−5.60109	−0.513451
$120$	0	0
$121$	10.0963	0.917846
$122$	0	0
$123$	− 1.01250i	− 0.0912937i
$124$	0	0
$125$	− 11.7784i	− 1.05349i
$126$	0	0
$127$	−6.54782	−0.581025	−0.290512	−	0.956871i	$-0.593826\pi$
$127$	−6.54782	−0.581025	−0.290512	+	0.956871i	$0.593826\pi$
$128$	0	0
$129$	−0.376331	−0.0331341
$130$	0	0
$131$	14.2838i	1.24798i	0.781432	+	0.623990i	$0.214489\pi$
$131$	14.2838i	1.24798i	−0.781432	+	0.623990i	$0.785511\pi$
$132$	0	0
$133$	3.06334i	0.265625i
$134$	0	0
$135$	−2.05045	−0.176474
$136$	0	0
$137$	11.9138	1.01786	0.508931	−	0.860808i	$-0.330041\pi$
$137$	11.9138	1.01786	0.508931	+	0.860808i	$0.330041\pi$
$138$	0	0
$139$	− 11.1950i	− 0.949551i	−0.880107	−	0.474775i	$-0.842529\pi$
$139$	− 11.1950i	− 0.949551i	0.880107	−	0.474775i	$-0.157471\pi$
$140$	0	0
$141$	− 0.755320i	− 0.0636094i
$142$	0	0
$143$	−3.19519	−0.267195
$144$	0	0
$145$	7.69004	0.638623
$146$	0	0
$147$	− 0.529714i	− 0.0436901i
$148$	0	0
$149$	7.67749i	0.628964i	0.949263	+	0.314482i	$0.101831\pi$
$149$	7.67749i	0.628964i	−0.949263	+	0.314482i	$0.898169\pi$
$150$	0	0
$151$	3.10162	0.252406	0.126203	−	0.992004i	$-0.459721\pi$
$151$	3.10162	0.252406	0.126203	+	0.992004i	$0.459721\pi$
$152$	0	0
$153$	5.39501	0.436161
$154$	0	0
$155$	− 5.34440i	− 0.429272i
$156$	0	0
$157$	14.8994i	1.18910i	0.804059	+	0.594550i	$0.202670\pi$
$157$	14.8994i	1.18910i	−0.804059	+	0.594550i	$0.797330\pi$
$158$	0	0
$159$	−0.993511	−0.0787905
$160$	0	0
$161$	10.2963	0.811460
$162$	0	0
$163$	18.5481i	1.45280i	0.687272	+	0.726400i	$0.258808\pi$
$163$	18.5481i	1.45280i	−0.687272	+	0.726400i	$0.741192\pi$
$164$	0	0
$165$	− 0.327565i	− 0.0255009i
$166$	0	0
$167$	−11.4532	−0.886274	−0.443137	−	0.896454i	$-0.646135\pi$
$167$	−11.4532	−0.886274	−0.443137	+	0.896454i	$0.646135\pi$
$168$	0	0
$169$	1.70281	0.130986
$170$	0	0
$171$	− 2.95063i	− 0.225640i
$172$	0	0
$173$	− 7.20956i	− 0.548133i	−0.961711	−	0.274066i	$-0.911631\pi$
$173$	− 7.20956i	− 0.548133i	0.961711	−	0.274066i	$-0.0883688\pi$
$174$	0	0
$175$	−7.94933	−0.600913
$176$	0	0
$177$	0.0877686	0.00659710
$178$	0	0
$179$	− 4.90251i	− 0.366431i	−0.983073	−	0.183215i	$-0.941349\pi$
$179$	− 4.90251i	− 0.366431i	0.983073	−	0.183215i	$-0.0586506\pi$
$180$	0	0
$181$	19.1775i	1.42545i	0.701444	+	0.712725i	$0.252539\pi$
$181$	19.1775i	1.42545i	−0.701444	+	0.712725i	$0.747461\pi$
$182$	0	0
$183$	1.18685	0.0877347
$184$	0	0
$185$	4.58868	0.337366
$186$	0	0
$187$	1.73816i	0.127107i
$188$	0	0
$189$	4.05028i	0.294615i
$190$	0	0
$191$	−22.8812	−1.65563	−0.827814	−	0.561003i	$-0.810416\pi$
$191$	−22.8812	−1.65563	−0.827814	+	0.561003i	$0.810416\pi$
$192$	0	0
$193$	−2.11248	−0.152060	−0.0760300	−	0.997106i	$-0.524224\pi$
$193$	−2.11248	−0.152060	−0.0760300	+	0.997106i	$0.524224\pi$
$194$	0	0
$195$	− 1.15817i	− 0.0829380i
$196$	0	0
$197$	− 6.05012i	− 0.431053i	−0.976498	−	0.215526i	$-0.930853\pi$
$197$	− 6.05012i	− 0.431053i	0.976498	−	0.215526i	$-0.0691467\pi$
$198$	0	0
$199$	4.03090	0.285743	0.142872	−	0.989741i	$-0.454366\pi$
$199$	4.03090	0.285743	0.142872	+	0.989741i	$0.454366\pi$
$200$	0	0
$201$	1.52215	0.107364
$202$	0	0
$203$	− 15.1903i	− 1.06615i
$204$	0	0
$205$	− 7.06683i	− 0.493569i
$206$	0	0
$207$	−9.91745	−0.689310
$208$	0	0
$209$	0.950631	0.0657565
$210$	0	0
$211$	1.66532i	0.114646i	0.998356	+	0.0573228i	$0.0182564\pi$
$211$	1.66532i	0.114646i	−0.998356	+	0.0573228i	$0.981744\pi$
$212$	0	0
$213$	− 1.14475i	− 0.0784366i
$214$	0	0
$215$	−2.62665	−0.179136
$216$	0	0
$217$	−10.5569	−0.716647
$218$	0	0
$219$	− 0.266624i	− 0.0180168i
$220$	0	0
$221$	6.14558i	0.413396i
$222$	0	0
$223$	28.9352	1.93764	0.968821	−	0.247761i	$-0.0796948\pi$
$223$	28.9352	1.93764	0.968821	+	0.247761i	$0.0796948\pi$
$224$	0	0
$225$	7.65685	0.510457
$226$	0	0
$227$	20.0519i	1.33089i	0.746447	+	0.665445i	$0.231758\pi$
$227$	20.0519i	1.33089i	−0.746447	+	0.665445i	$0.768242\pi$
$228$	0	0
$229$	14.9953i	0.990919i	0.868631	+	0.495459i	$0.165000\pi$
$229$	14.9953i	0.990919i	−0.868631	+	0.495459i	$0.835000\pi$
$230$	0	0
$231$	−0.647045	−0.0425724
$232$	0	0
$233$	−11.2738	−0.738570	−0.369285	−	0.929316i	$-0.620397\pi$
$233$	−11.2738	−0.738570	−0.369285	+	0.929316i	$0.620397\pi$
$234$	0	0
$235$	− 5.27184i	− 0.343897i
$236$	0	0
$237$	− 3.27176i	− 0.212523i
$238$	0	0
$239$	19.9049	1.28754	0.643770	−	0.765219i	$-0.277369\pi$
$239$	19.9049	1.28754	0.643770	+	0.765219i	$0.277369\pi$
$240$	0	0
$241$	−15.8163	−1.01882	−0.509408	−	0.860525i	$-0.670135\pi$
$241$	−15.8163	−1.01882	−0.509408	+	0.860525i	$0.670135\pi$
$242$	0	0
$243$	− 5.86808i	− 0.376437i
$244$	0	0
$245$	− 3.69720i	− 0.236205i
$246$	0	0
$247$	3.36113	0.213863
$248$	0	0
$249$	0.745595	0.0472501
$250$	0	0
$251$	10.5087i	0.663306i	0.943401	+	0.331653i	$0.107606\pi$
$251$	10.5087i	0.663306i	−0.943401	+	0.331653i	$0.892394\pi$
$252$	0	0
$253$	− 3.19519i	− 0.200880i
$254$	0	0
$255$	−0.630033	−0.0394542
$256$	0	0
$257$	−1.70129	−0.106123	−0.0530617	−	0.998591i	$-0.516898\pi$
$257$	−1.70129	−0.106123	−0.0530617	+	0.998591i	$0.516898\pi$
$258$	0	0
$259$	− 9.06409i	− 0.563215i
$260$	0	0
$261$	14.6314i	0.905659i
$262$	0	0
$263$	−13.8359	−0.853157	−0.426578	−	0.904451i	$-0.640281\pi$
$263$	−13.8359	−0.853157	−0.426578	+	0.904451i	$0.640281\pi$
$264$	0	0
$265$	−6.93432	−0.425972
$266$	0	0
$267$	− 2.14317i	− 0.131160i
$268$	0	0
$269$	14.9655i	0.912466i	0.889860	+	0.456233i	$0.150802\pi$
$269$	14.9655i	0.912466i	−0.889860	+	0.456233i	$0.849198\pi$
$270$	0	0
$271$	−4.12684	−0.250688	−0.125344	−	0.992113i	$-0.540003\pi$
$271$	−4.12684	−0.250688	−0.125344	+	0.992113i	$0.540003\pi$
$272$	0	0
$273$	−2.28774	−0.138461
$274$	0	0
$275$	2.46688i	0.148758i
$276$	0	0
$277$	− 25.7721i	− 1.54849i	−0.632884	−	0.774247i	$-0.718129\pi$
$277$	− 25.7721i	− 1.54849i	0.632884	−	0.774247i	$-0.281871\pi$
$278$	0	0
$279$	10.1685	0.608769
$280$	0	0
$281$	10.2694	0.612621	0.306311	−	0.951932i	$-0.400905\pi$
$281$	10.2694	0.612621	0.306311	+	0.951932i	$0.400905\pi$
$282$	0	0
$283$	6.99506i	0.415813i	0.978149	+	0.207907i	$0.0666650\pi$
$283$	6.99506i	0.415813i	−0.978149	+	0.207907i	$0.933335\pi$
$284$	0	0
$285$	0.344577i	0.0204110i
$286$	0	0
$287$	−13.9592	−0.823987
$288$	0	0
$289$	−13.6569	−0.803344
$290$	0	0
$291$	− 0.790027i	− 0.0463122i
$292$	0	0
$293$	− 31.8434i	− 1.86031i	−0.367162	−	0.930157i	$-0.619670\pi$
$293$	− 31.8434i	− 1.86031i	0.367162	−	0.930157i	$-0.380330\pi$
$294$	0	0
$295$	0.612591	0.0356664
$296$	0	0
$297$	1.25690	0.0729330
$298$	0	0
$299$	− 11.2972i	− 0.653333i
$300$	0	0
$301$	5.18846i	0.299058i
$302$	0	0
$303$	−3.00805	−0.172808
$304$	0	0
$305$	8.28378	0.474328
$306$	0	0
$307$	6.45813i	0.368585i	0.982871	+	0.184292i	$0.0589993\pi$
$307$	6.45813i	0.368585i	−0.982871	+	0.184292i	$0.941001\pi$
$308$	0	0
$309$	− 1.47662i	− 0.0840018i
$310$	0	0
$311$	28.1973	1.59892	0.799462	−	0.600716i	$-0.205118\pi$
$311$	28.1973	1.59892	0.799462	+	0.600716i	$0.205118\pi$
$312$	0	0
$313$	−17.1212	−0.967746	−0.483873	−	0.875138i	$-0.660770\pi$
$313$	−17.1212	−0.967746	−0.483873	+	0.875138i	$0.660770\pi$
$314$	0	0
$315$	14.0174i	0.789793i
$316$	0	0
$317$	29.5841i	1.66161i	0.556564	+	0.830804i	$0.312119\pi$
$317$	29.5841i	1.66161i	−0.556564	+	0.830804i	$0.687881\pi$
$318$	0	0
$319$	−4.71392	−0.263929
$320$	0	0
$321$	−3.52215	−0.196587
$322$	0	0
$323$	− 1.82843i	− 0.101736i
$324$	0	0
$325$	8.72209i	0.483815i
$326$	0	0
$327$	−2.98210	−0.164910
$328$	0	0
$329$	−10.4135	−0.574117
$330$	0	0
$331$	− 20.5519i	− 1.12964i	−0.825215	−	0.564818i	$-0.808946\pi$
$331$	− 20.5519i	− 1.12964i	0.825215	−	0.564818i	$-0.191054\pi$
$332$	0	0
$333$	8.73060i	0.478434i
$334$	0	0
$335$	10.6240	0.580454
$336$	0	0
$337$	−19.1832	−1.04497	−0.522487	−	0.852647i	$-0.674996\pi$
$337$	−19.1832	−1.04497	−0.522487	+	0.852647i	$0.674996\pi$
$338$	0	0
$339$	− 1.47937i	− 0.0803485i
$340$	0	0
$341$	3.27606i	0.177409i
$342$	0	0
$343$	14.1402	0.763501
$344$	0	0
$345$	1.15817	0.0623536
$346$	0	0
$347$	− 10.5507i	− 0.566390i	−0.959062	−	0.283195i	$-0.908606\pi$
$347$	− 10.5507i	− 0.566390i	0.959062	−	0.283195i	$-0.0913944\pi$
$348$	0	0
$349$	− 29.4693i	− 1.57745i	−0.614744	−	0.788727i	$-0.710741\pi$
$349$	− 29.4693i	− 1.57745i	0.614744	−	0.788727i	$-0.289259\pi$
$350$	0	0
$351$	4.44401	0.237204
$352$	0	0
$353$	−3.98903	−0.212315	−0.106157	−	0.994349i	$-0.533855\pi$
$353$	−3.98903	−0.212315	−0.106157	+	0.994349i	$0.533855\pi$
$354$	0	0
$355$	− 7.98988i	− 0.424059i
$356$	0	0
$357$	1.24451i	0.0658667i
$358$	0	0
$359$	14.9377	0.788380	0.394190	−	0.919029i	$-0.371025\pi$
$359$	14.9377	0.788380	0.394190	+	0.919029i	$0.371025\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 2.24331i	− 0.117743i
$364$	0	0
$365$	− 1.86093i	− 0.0974057i
$366$	0	0
$367$	−19.1009	−0.997059	−0.498529	−	0.866873i	$-0.666126\pi$
$367$	−19.1009	−0.997059	−0.498529	+	0.866873i	$0.666126\pi$
$368$	0	0
$369$	13.4456	0.699952
$370$	0	0
$371$	13.6975i	0.711137i
$372$	0	0
$373$	0.461357i	0.0238882i	0.999929	+	0.0119441i	$0.00380201\pi$
$373$	0.461357i	0.0238882i	−0.999929	+	0.0119441i	$0.996198\pi$
$374$	0	0
$375$	−2.61706	−0.135144
$376$	0	0
$377$	−16.6669	−0.858390
$378$	0	0
$379$	29.9622i	1.53906i	0.638613	+	0.769528i	$0.279508\pi$
$379$	29.9622i	1.53906i	−0.638613	+	0.769528i	$0.720492\pi$
$380$	0	0
$381$	1.45487i	0.0745352i
$382$	0	0
$383$	16.9077	0.863944	0.431972	−	0.901887i	$-0.357818\pi$
$383$	16.9077	0.863944	0.431972	+	0.901887i	$0.357818\pi$
$384$	0	0
$385$	−4.51612	−0.230163
$386$	0	0
$387$	− 4.99756i	− 0.254040i
$388$	0	0
$389$	7.92418i	0.401772i	0.979615	+	0.200886i	$0.0643821\pi$
$389$	7.92418i	0.401772i	−0.979615	+	0.200886i	$0.935618\pi$
$390$	0	0
$391$	−6.14558	−0.310795
$392$	0	0
$393$	3.17373	0.160094
$394$	0	0
$395$	− 22.8356i	− 1.14898i
$396$	0	0
$397$	13.6718i	0.686170i	0.939304	+	0.343085i	$0.111472\pi$
$397$	13.6718i	0.686170i	−0.939304	+	0.343085i	$0.888528\pi$
$398$	0	0
$399$	0.680647	0.0340750
$400$	0	0
$401$	12.0107	0.599785	0.299893	−	0.953973i	$-0.403049\pi$
$401$	12.0107	0.599785	0.299893	+	0.953973i	$0.403049\pi$
$402$	0	0
$403$	11.5831i	0.576996i
$404$	0	0
$405$	− 13.2720i	− 0.659492i
$406$	0	0
$407$	−2.81281	−0.139426
$408$	0	0
$409$	−9.47063	−0.468292	−0.234146	−	0.972201i	$-0.575229\pi$
$409$	−9.47063	−0.468292	−0.234146	+	0.972201i	$0.575229\pi$
$410$	0	0
$411$	− 2.64713i	− 0.130574i
$412$	0	0
$413$	− 1.21006i	− 0.0595432i
$414$	0	0
$415$	5.20396	0.255453
$416$	0	0
$417$	−2.48744	−0.121811
$418$	0	0
$419$	28.0843i	1.37201i	0.727598	+	0.686004i	$0.240637\pi$
$419$	28.0843i	1.37201i	−0.727598	+	0.686004i	$0.759363\pi$
$420$	0	0
$421$	− 9.83238i	− 0.479201i	−0.970872	−	0.239601i	$-0.922984\pi$
$421$	− 9.83238i	− 0.479201i	0.970872	−	0.239601i	$-0.0770165\pi$
$422$	0	0
$423$	10.0304	0.487695
$424$	0	0
$425$	4.74475	0.230154
$426$	0	0
$427$	− 16.3631i	− 0.791865i
$428$	0	0
$429$	0.709944i	0.0342764i
$430$	0	0
$431$	8.94284	0.430761	0.215381	−	0.976530i	$-0.430901\pi$
$431$	8.94284	0.430761	0.215381	+	0.976530i	$0.430901\pi$
$432$	0	0
$433$	−29.5274	−1.41900	−0.709499	−	0.704707i	$-0.751079\pi$
$433$	−29.5274	−1.41900	−0.709499	+	0.704707i	$0.751079\pi$
$434$	0	0
$435$	− 1.70866i	− 0.0819240i
$436$	0	0
$437$	3.36113i	0.160785i
$438$	0	0
$439$	−20.5113	−0.978953	−0.489477	−	0.872016i	$-0.662812\pi$
$439$	−20.5113	−0.978953	−0.489477	+	0.872016i	$0.662812\pi$
$440$	0	0
$441$	7.03444	0.334973
$442$	0	0
$443$	13.3188i	0.632794i	0.948627	+	0.316397i	$0.102473\pi$
$443$	13.3188i	0.632794i	−0.948627	+	0.316397i	$0.897527\pi$
$444$	0	0
$445$	− 14.9585i	− 0.709101i
$446$	0	0
$447$	1.70587	0.0806850
$448$	0	0
$449$	10.5138	0.496177	0.248089	−	0.968737i	$-0.420198\pi$
$449$	10.5138	0.496177	0.248089	+	0.968737i	$0.420198\pi$
$450$	0	0
$451$	4.33190i	0.203981i
$452$	0	0
$453$	− 0.689153i	− 0.0323792i
$454$	0	0
$455$	−15.9676	−0.748571
$456$	0	0
$457$	−26.5831	−1.24351	−0.621753	−	0.783214i	$-0.713579\pi$
$457$	−26.5831	−1.24351	−0.621753	+	0.783214i	$0.713579\pi$
$458$	0	0
$459$	− 2.41751i	− 0.112840i
$460$	0	0
$461$	19.2417i	0.896175i	0.893990	+	0.448087i	$0.147895\pi$
$461$	19.2417i	0.896175i	−0.893990	+	0.448087i	$0.852105\pi$
$462$	0	0
$463$	−2.24701	−0.104427	−0.0522137	−	0.998636i	$-0.516628\pi$
$463$	−2.24701	−0.104427	−0.0522137	+	0.998636i	$0.516628\pi$
$464$	0	0
$465$	−1.18748	−0.0550680
$466$	0	0
$467$	27.5682i	1.27570i	0.770159	+	0.637852i	$0.220177\pi$
$467$	27.5682i	1.27570i	−0.770159	+	0.637852i	$0.779823\pi$
$468$	0	0
$469$	− 20.9858i	− 0.969036i
$470$	0	0
$471$	3.31051	0.152540
$472$	0	0
$473$	1.61011	0.0740329
$474$	0	0
$475$	− 2.59499i	− 0.119066i
$476$	0	0
$477$	− 13.1935i	− 0.604089i
$478$	0	0
$479$	−10.5447	−0.481802	−0.240901	−	0.970550i	$-0.577443\pi$
$479$	−10.5447	−0.481802	−0.240901	+	0.970550i	$0.577443\pi$
$480$	0	0
$481$	−9.94521	−0.453463
$482$	0	0
$483$	− 2.28774i	− 0.104096i
$484$	0	0
$485$	− 5.51409i	− 0.250382i
$486$	0	0
$487$	−39.7294	−1.80031	−0.900154	−	0.435571i	$-0.856546\pi$
$487$	−39.7294	−1.80031	−0.900154	+	0.435571i	$0.856546\pi$
$488$	0	0
$489$	4.12123	0.186369
$490$	0	0
$491$	− 31.0150i	− 1.39969i	−0.714296	−	0.699844i	$-0.753253\pi$
$491$	− 31.0150i	− 1.39969i	0.714296	−	0.699844i	$-0.246747\pi$
$492$	0	0
$493$	9.06667i	0.408342i
$494$	0	0
$495$	4.34996	0.195516
$496$	0	0
$497$	−15.7825	−0.707943
$498$	0	0
$499$	− 4.90620i	− 0.219632i	−0.993952	−	0.109816i	$-0.964974\pi$
$499$	− 4.90620i	− 0.219632i	0.993952	−	0.109816i	$-0.0350261\pi$
$500$	0	0
$501$	2.54480i	0.113693i
$502$	0	0
$503$	−14.6847	−0.654759	−0.327380	−	0.944893i	$-0.606166\pi$
$503$	−14.6847	−0.654759	−0.327380	+	0.944893i	$0.606166\pi$
$504$	0	0
$505$	−20.9950	−0.934265
$506$	0	0
$507$	− 0.378351i	− 0.0168031i
$508$	0	0
$509$	− 20.3988i	− 0.904159i	−0.891978	−	0.452080i	$-0.850682\pi$
$509$	− 20.3988i	− 0.904159i	0.891978	−	0.452080i	$-0.149318\pi$
$510$	0	0
$511$	−3.67593	−0.162614
$512$	0	0
$513$	−1.32218	−0.0583756
$514$	0	0
$515$	− 10.3062i	− 0.454146i
$516$	0	0
$517$	3.23158i	0.142125i
$518$	0	0
$519$	−1.60190	−0.0703157
$520$	0	0
$521$	−22.4356	−0.982923	−0.491462	−	0.870899i	$-0.663537\pi$
$521$	−22.4356	−0.982923	−0.491462	+	0.870899i	$0.663537\pi$
$522$	0	0
$523$	9.71531i	0.424821i	0.977181	+	0.212410i	$0.0681314\pi$
$523$	9.71531i	0.424821i	−0.977181	+	0.212410i	$0.931869\pi$
$524$	0	0
$525$	1.76627i	0.0770865i
$526$	0	0
$527$	6.30112	0.274481
$528$	0	0
$529$	−11.7028	−0.508818
$530$	0	0
$531$	1.16554i	0.0505801i
$532$	0	0
$533$	15.3162i	0.663419i
$534$	0	0
$535$	−24.5833	−1.06283
$536$	0	0
$537$	−1.08930	−0.0470066
$538$	0	0
$539$	2.26635i	0.0976185i
$540$	0	0
$541$	− 14.7230i	− 0.632991i	−0.948594	−	0.316496i	$-0.897494\pi$
$541$	− 14.7230i	− 0.632991i	0.948594	−	0.316496i	$-0.102506\pi$
$542$	0	0
$543$	4.26107	0.182860
$544$	0	0
$545$	−20.8139	−0.891569
$546$	0	0
$547$	− 10.9906i	− 0.469922i	−0.972005	−	0.234961i	$-0.924504\pi$
$547$	− 10.9906i	− 0.469922i	0.972005	−	0.234961i	$-0.0754963\pi$
$548$	0	0
$549$	15.7610i	0.672665i
$550$	0	0
$551$	4.95873	0.211249
$552$	0	0
$553$	−45.1075	−1.91817
$554$	0	0
$555$	− 1.01956i	− 0.0432781i
$556$	0	0
$557$	− 25.9252i	− 1.09849i	−0.835663	−	0.549243i	$-0.814916\pi$
$557$	− 25.9252i	− 1.09849i	0.835663	−	0.549243i	$-0.185084\pi$
$558$	0	0
$559$	5.69283	0.240781
$560$	0	0
$561$	0.386204	0.0163055
$562$	0	0
$563$	− 21.0775i	− 0.888310i	−0.895950	−	0.444155i	$-0.853504\pi$
$563$	− 21.0775i	− 0.888310i	0.895950	−	0.444155i	$-0.146496\pi$
$564$	0	0
$565$	− 10.3255i	− 0.434395i
$566$	0	0
$567$	−26.2164	−1.10099
$568$	0	0
$569$	11.2369	0.471076	0.235538	−	0.971865i	$-0.424315\pi$
$569$	11.2369	0.471076	0.235538	+	0.971865i	$0.424315\pi$
$570$	0	0
$571$	42.5037i	1.77872i	0.457204	+	0.889362i	$0.348851\pi$
$571$	42.5037i	1.77872i	−0.457204	+	0.889362i	$0.651149\pi$
$572$	0	0
$573$	5.08401i	0.212388i
$574$	0	0
$575$	−8.72209	−0.363736
$576$	0	0
$577$	−26.6503	−1.10947	−0.554733	−	0.832029i	$-0.687179\pi$
$577$	−26.6503	−1.10947	−0.554733	+	0.832029i	$0.687179\pi$
$578$	0	0
$579$	0.469376i	0.0195066i
$580$	0	0
$581$	− 10.2795i	− 0.426464i
$582$	0	0
$583$	4.25067	0.176045
$584$	0	0
$585$	15.3801	0.635888
$586$	0	0
$587$	− 33.2195i	− 1.37111i	−0.728019	−	0.685557i	$-0.759559\pi$
$587$	− 33.2195i	− 1.37111i	0.728019	−	0.685557i	$-0.240441\pi$
$588$	0	0
$589$	− 3.44620i	− 0.141998i
$590$	0	0
$591$	−1.34428	−0.0552964
$592$	0	0
$593$	−25.4149	−1.04367	−0.521833	−	0.853047i	$-0.674752\pi$
$593$	−25.4149	−1.04367	−0.521833	+	0.853047i	$0.674752\pi$
$594$	0	0
$595$	8.68623i	0.356101i
$596$	0	0
$597$	− 0.895632i	− 0.0366558i
$598$	0	0
$599$	46.6431	1.90578	0.952892	−	0.303310i	$-0.0980918\pi$
$599$	46.6431	1.90578	0.952892	+	0.303310i	$0.0980918\pi$
$600$	0	0
$601$	−47.4644	−1.93611	−0.968056	−	0.250734i	$-0.919328\pi$
$601$	−47.4644	−1.93611	−0.968056	+	0.250734i	$0.919328\pi$
$602$	0	0
$603$	20.2137i	0.823166i
$604$	0	0
$605$	− 15.6574i	− 0.636566i
$606$	0	0
$607$	24.4897	0.994006	0.497003	−	0.867749i	$-0.334434\pi$
$607$	24.4897	0.994006	0.497003	+	0.867749i	$0.334434\pi$
$608$	0	0
$609$	−3.37514	−0.136768
$610$	0	0
$611$	11.4259i	0.462241i
$612$	0	0
$613$	18.9950i	0.767201i	0.923499	+	0.383600i	$0.125316\pi$
$613$	18.9950i	0.767201i	−0.923499	+	0.383600i	$0.874684\pi$
$614$	0	0
$615$	−1.57019	−0.0633162
$616$	0	0
$617$	0.219907	0.00885311	0.00442656	−	0.999990i	$-0.498591\pi$
$617$	0.219907	0.00885311	0.00442656	+	0.999990i	$0.498591\pi$
$618$	0	0
$619$	15.9136i	0.639623i	0.947481	+	0.319811i	$0.103620\pi$
$619$	15.9136i	0.639623i	−0.947481	+	0.319811i	$0.896380\pi$
$620$	0	0
$621$	4.44401i	0.178332i
$622$	0	0
$623$	−29.5478	−1.18381
$624$	0	0
$625$	−5.29109	−0.211644
$626$	0	0
$627$	− 0.211222i	− 0.00843539i
$628$	0	0
$629$	5.41012i	0.215716i
$630$	0	0
$631$	−27.6207	−1.09956	−0.549781	−	0.835309i	$-0.685289\pi$
$631$	−27.6207	−1.09956	−0.549781	+	0.835309i	$0.685289\pi$
$632$	0	0
$633$	0.370021	0.0147070
$634$	0	0
$635$	10.1544i	0.402966i
$636$	0	0
$637$	8.01308i	0.317490i
$638$	0	0
$639$	15.2019	0.601376
$640$	0	0
$641$	−4.17066	−0.164731	−0.0823656	−	0.996602i	$-0.526248\pi$
$641$	−4.17066	−0.164731	−0.0823656	+	0.996602i	$0.526248\pi$
$642$	0	0
$643$	− 20.8643i	− 0.822806i	−0.911453	−	0.411403i	$-0.865039\pi$
$643$	− 20.8643i	− 0.822806i	0.911453	−	0.411403i	$-0.134961\pi$
$644$	0	0
$645$	0.583619i	0.0229800i
$646$	0	0
$647$	31.7924	1.24989	0.624943	−	0.780670i	$-0.285122\pi$
$647$	31.7924	1.24989	0.624943	+	0.780670i	$0.285122\pi$
$648$	0	0
$649$	−0.375512	−0.0147401
$650$	0	0
$651$	2.34564i	0.0919331i
$652$	0	0
$653$	27.2099i	1.06481i	0.846491	+	0.532403i	$0.178711\pi$
$653$	27.2099i	1.06481i	−0.846491	+	0.532403i	$0.821289\pi$
$654$	0	0
$655$	22.1514	0.865528
$656$	0	0
$657$	3.54068	0.138135
$658$	0	0
$659$	− 32.8090i	− 1.27806i	−0.769182	−	0.639030i	$-0.779336\pi$
$659$	− 32.8090i	− 1.27806i	0.769182	−	0.639030i	$-0.220664\pi$
$660$	0	0
$661$	12.5824i	0.489398i	0.969599	+	0.244699i	$0.0786891\pi$
$661$	12.5824i	0.489398i	−0.969599	+	0.244699i	$0.921311\pi$
$662$	0	0
$663$	1.36549	0.0530314
$664$	0	0
$665$	4.75066	0.184223
$666$	0	0
$667$	− 16.6669i	− 0.645345i
$668$	0	0
$669$	− 6.42915i	− 0.248565i
$670$	0	0
$671$	−5.07787	−0.196029
$672$	0	0
$673$	−2.58049	−0.0994704	−0.0497352	−	0.998762i	$-0.515838\pi$
$673$	−2.58049	−0.0994704	−0.0497352	+	0.998762i	$0.515838\pi$
$674$	0	0
$675$	− 3.43104i	− 0.132061i
$676$	0	0
$677$	5.86443i	0.225388i	0.993630	+	0.112694i	$0.0359480\pi$
$677$	5.86443i	0.225388i	−0.993630	+	0.112694i	$0.964052\pi$
$678$	0	0
$679$	−10.8921	−0.417999
$680$	0	0
$681$	4.45535	0.170729
$682$	0	0
$683$	13.3162i	0.509531i	0.967003	+	0.254765i	$0.0819982\pi$
$683$	13.3162i	0.509531i	−0.967003	+	0.254765i	$0.918002\pi$
$684$	0	0
$685$	− 18.4760i	− 0.705931i
$686$	0	0
$687$	3.33183	0.127117
$688$	0	0
$689$	15.0290	0.572560
$690$	0	0
$691$	− 34.9751i	− 1.33051i	−0.746614	−	0.665257i	$-0.768322\pi$
$691$	− 34.9751i	− 1.33051i	0.746614	−	0.665257i	$-0.231678\pi$
$692$	0	0
$693$	− 8.59255i	− 0.326404i
$694$	0	0
$695$	−17.3614	−0.658555
$696$	0	0
$697$	8.33190	0.315593
$698$	0	0
$699$	2.50494i	0.0947454i
$700$	0	0
$701$	− 4.91851i	− 0.185769i	−0.995677	−	0.0928847i	$-0.970391\pi$
$701$	− 4.91851i	− 0.185769i	0.995677	−	0.0928847i	$-0.0296088\pi$
$702$	0	0
$703$	2.95889	0.111597
$704$	0	0
$705$	−1.17136	−0.0441159
$706$	0	0
$707$	41.4717i	1.55971i
$708$	0	0
$709$	27.4930i	1.03252i	0.856431	+	0.516261i	$0.172676\pi$
$709$	27.4930i	1.03252i	−0.856431	+	0.516261i	$0.827324\pi$
$710$	0	0
$711$	43.4479	1.62942
$712$	0	0
$713$	−11.5831	−0.433791
$714$	0	0
$715$	4.95514i	0.185312i
$716$	0	0
$717$	− 4.42270i	− 0.165169i
$718$	0	0
$719$	−10.8746	−0.405553	−0.202777	−	0.979225i	$-0.564996\pi$
$719$	−10.8746	−0.405553	−0.202777	+	0.979225i	$0.564996\pi$
$720$	0	0
$721$	−20.3580	−0.758172
$722$	0	0
$723$	3.51424i	0.130696i
$724$	0	0
$725$	12.8678i	0.477899i
$726$	0	0
$727$	−1.13818	−0.0422127	−0.0211063	−	0.999777i	$-0.506719\pi$
$727$	−1.13818	−0.0422127	−0.0211063	+	0.999777i	$0.506719\pi$
$728$	0	0
$729$	24.3705	0.902612
$730$	0	0
$731$	− 3.09686i	− 0.114541i
$732$	0	0
$733$	27.9950i	1.03402i	0.855980	+	0.517010i	$0.172955\pi$
$733$	27.9950i	1.03402i	−0.855980	+	0.517010i	$0.827045\pi$
$734$	0	0
$735$	−0.821486	−0.0303010
$736$	0	0
$737$	−6.51243	−0.239889
$738$	0	0
$739$	− 44.2375i	− 1.62730i	−0.581352	−	0.813652i	$-0.697476\pi$
$739$	− 44.2375i	− 1.62730i	0.581352	−	0.813652i	$-0.302524\pi$
$740$	0	0
$741$	− 0.746814i	− 0.0274349i
$742$	0	0
$743$	49.4810	1.81528	0.907640	−	0.419749i	$-0.137882\pi$
$743$	49.4810	1.81528	0.907640	+	0.419749i	$0.137882\pi$
$744$	0	0
$745$	11.9063	0.436214
$746$	0	0
$747$	9.90126i	0.362268i
$748$	0	0
$749$	48.5597i	1.77433i
$750$	0	0
$751$	−25.9833	−0.948145	−0.474073	−	0.880486i	$-0.657217\pi$
$751$	−25.9833	−0.948145	−0.474073	+	0.880486i	$0.657217\pi$
$752$	0	0
$753$	2.33495	0.0850904
$754$	0	0
$755$	− 4.81002i	− 0.175055i
$756$	0	0
$757$	43.1605i	1.56869i	0.620322	+	0.784347i	$0.287002\pi$
$757$	43.1605i	1.56869i	−0.620322	+	0.784347i	$0.712998\pi$
$758$	0	0
$759$	−0.709944	−0.0257693
$760$	0	0
$761$	−4.16498	−0.150981	−0.0754903	−	0.997147i	$-0.524052\pi$
$761$	−4.16498	−0.150981	−0.0754903	+	0.997147i	$0.524052\pi$
$762$	0	0
$763$	41.1140i	1.48843i
$764$	0	0
$765$	− 8.36664i	− 0.302497i
$766$	0	0
$767$	−1.32769	−0.0479402
$768$	0	0
$769$	5.24441	0.189118	0.0945591	−	0.995519i	$-0.469856\pi$
$769$	5.24441	0.189118	0.0945591	+	0.995519i	$0.469856\pi$
$770$	0	0
$771$	0.378011i	0.0136137i
$772$	0	0
$773$	3.91186i	0.140700i	0.997522	+	0.0703499i	$0.0224116\pi$
$773$	3.91186i	0.140700i	−0.997522	+	0.0703499i	$0.977588\pi$
$774$	0	0
$775$	8.94284	0.321236
$776$	0	0
$777$	−2.01396	−0.0722505
$778$	0	0
$779$	− 4.55687i	− 0.163267i
$780$	0	0
$781$	4.89772i	0.175254i
$782$	0	0
$783$	6.55632	0.234304
$784$	0	0
$785$	23.1061	0.824692
$786$	0	0
$787$	− 16.8869i	− 0.601952i	−0.953632	−	0.300976i	$-0.902688\pi$
$787$	− 16.8869i	− 0.601952i	0.953632	−	0.300976i	$-0.0973125\pi$
$788$	0	0
$789$	3.07421i	0.109445i
$790$	0	0
$791$	−20.3960	−0.725199
$792$	0	0
$793$	−17.9537	−0.637556
$794$	0	0
$795$	1.54075i	0.0546447i
$796$	0	0
$797$	− 25.3722i	− 0.898729i	−0.893348	−	0.449365i	$-0.851650\pi$
$797$	− 25.3722i	− 0.898729i	0.893348	−	0.449365i	$-0.148350\pi$
$798$	0	0
$799$	6.21557	0.219891
$800$	0	0
$801$	28.4606	1.00561
$802$	0	0
$803$	1.14073i	0.0402556i
$804$	0	0
$805$	− 15.9676i	− 0.562783i
$806$	0	0
$807$	3.32522	0.117053
$808$	0	0
$809$	31.5640	1.10973	0.554866	−	0.831940i	$-0.312770\pi$
$809$	31.5640	1.10973	0.554866	+	0.831940i	$0.312770\pi$
$810$	0	0
$811$	− 44.1203i	− 1.54927i	−0.632407	−	0.774636i	$-0.717933\pi$
$811$	− 44.1203i	− 1.54927i	0.632407	−	0.774636i	$-0.282067\pi$
$812$	0	0
$813$	0.916949i	0.0321588i
$814$	0	0
$815$	28.7646	1.00758
$816$	0	0
$817$	−1.69373	−0.0592560
$818$	0	0
$819$	− 30.3805i	− 1.06158i
$820$	0	0
$821$	3.81015i	0.132975i	0.997787	+	0.0664876i	$0.0211793\pi$
$821$	3.81015i	0.132975i	−0.997787	+	0.0664876i	$0.978821\pi$
$822$	0	0
$823$	34.3301	1.19667	0.598336	−	0.801245i	$-0.295829\pi$
$823$	34.3301	1.19667	0.598336	+	0.801245i	$0.295829\pi$
$824$	0	0
$825$	0.548119	0.0190830
$826$	0	0
$827$	30.7484i	1.06923i	0.845097	+	0.534613i	$0.179543\pi$
$827$	30.7484i	1.06923i	−0.845097	+	0.534613i	$0.820457\pi$
$828$	0	0
$829$	10.1917i	0.353971i	0.984213	+	0.176986i	$0.0566346\pi$
$829$	10.1917i	0.353971i	−0.984213	+	0.176986i	$0.943365\pi$
$830$	0	0
$831$	−5.72633	−0.198644
$832$	0	0
$833$	4.35905	0.151032
$834$	0	0
$835$	17.7617i	0.614669i
$836$	0	0
$837$	− 4.55649i	− 0.157495i
$838$	0	0
$839$	−7.81123	−0.269674	−0.134837	−	0.990868i	$-0.543051\pi$
$839$	−7.81123	−0.269674	−0.134837	+	0.990868i	$0.543051\pi$
$840$	0	0
$841$	4.41104	0.152105
$842$	0	0
$843$	− 2.28177i	− 0.0785884i
$844$	0	0
$845$	− 2.64074i	− 0.0908442i
$846$	0	0
$847$	−30.9284	−1.06271
$848$	0	0
$849$	1.55424	0.0533415
$850$	0	0
$851$	− 9.94521i	− 0.340918i
$852$	0	0
$853$	− 46.9706i	− 1.60824i	−0.594465	−	0.804122i	$-0.702636\pi$
$853$	− 46.9706i	− 1.60824i	0.594465	−	0.804122i	$-0.297364\pi$
$854$	0	0
$855$	−4.57587	−0.156491
$856$	0	0
$857$	−2.24191	−0.0765821	−0.0382911	−	0.999267i	$-0.512191\pi$
$857$	−2.24191	−0.0765821	−0.0382911	+	0.999267i	$0.512191\pi$
$858$	0	0
$859$	49.9456i	1.70412i	0.523442	+	0.852061i	$0.324648\pi$
$859$	49.9456i	1.70412i	−0.523442	+	0.852061i	$0.675352\pi$
$860$	0	0
$861$	3.10162i	0.105703i
$862$	0	0
$863$	10.4322	0.355115	0.177557	−	0.984110i	$-0.443180\pi$
$863$	10.4322	0.355115	0.177557	+	0.984110i	$0.443180\pi$
$864$	0	0
$865$	−11.1807	−0.380154
$866$	0	0
$867$	3.03444i	0.103055i
$868$	0	0
$869$	13.9980i	0.474849i
$870$	0	0
$871$	−23.0259	−0.780203
$872$	0	0
$873$	10.4913	0.355077
$874$	0	0
$875$	36.0812i	1.21977i
$876$	0	0
$877$	− 23.6385i	− 0.798215i	−0.916904	−	0.399107i	$-0.869320\pi$
$877$	− 23.6385i	− 0.798215i	0.916904	−	0.399107i	$-0.130680\pi$
$878$	0	0
$879$	−7.07534	−0.238645
$880$	0	0
$881$	−29.2713	−0.986175	−0.493087	−	0.869980i	$-0.664132\pi$
$881$	−29.2713	−0.986175	−0.493087	+	0.869980i	$0.664132\pi$
$882$	0	0
$883$	− 42.1875i	− 1.41972i	−0.704341	−	0.709862i	$-0.748757\pi$
$883$	− 42.1875i	− 1.41972i	0.704341	−	0.709862i	$-0.251243\pi$
$884$	0	0
$885$	− 0.136112i	− 0.00457537i
$886$	0	0
$887$	1.32254	0.0444064	0.0222032	−	0.999753i	$-0.492932\pi$
$887$	1.32254	0.0444064	0.0222032	+	0.999753i	$0.492932\pi$
$888$	0	0
$889$	20.0582	0.672730
$890$	0	0
$891$	8.13561i	0.272553i
$892$	0	0
$893$	− 3.39941i	− 0.113757i
$894$	0	0
$895$	−7.60286	−0.254136
$896$	0	0
$897$	−2.51014	−0.0838110
$898$	0	0
$899$	17.0887i	0.569941i
$900$	0	0
$901$	− 8.17567i	− 0.272371i
$902$	0	0
$903$	1.15283	0.0383638
$904$	0	0
$905$	29.7406	0.988611
$906$	0	0
$907$	3.58771i	0.119128i	0.998224	+	0.0595639i	$0.0189710\pi$
$907$	3.58771i	0.119128i	−0.998224	+	0.0595639i	$0.981029\pi$
$908$	0	0
$909$	− 39.9459i	− 1.32492i
$910$	0	0
$911$	−14.6828	−0.486464	−0.243232	−	0.969968i	$-0.578208\pi$
$911$	−14.6828	−0.486464	−0.243232	+	0.969968i	$0.578208\pi$
$912$	0	0
$913$	−3.18998	−0.105573
$914$	0	0
$915$	− 1.84058i	− 0.0608478i
$916$	0	0
$917$	− 43.7561i	− 1.44495i
$918$	0	0
$919$	38.2168	1.26066	0.630328	−	0.776329i	$-0.282920\pi$
$919$	38.2168	1.26066	0.630328	+	0.776329i	$0.282920\pi$
$920$	0	0
$921$	1.43494	0.0472829
$922$	0	0
$923$	17.3168i	0.569988i
$924$	0	0
$925$	7.67829i	0.252461i
$926$	0	0
$927$	19.6090	0.644044
$928$	0	0
$929$	36.8919	1.21038	0.605192	−	0.796080i	$-0.293096\pi$
$929$	36.8919	1.21038	0.605192	+	0.796080i	$0.293096\pi$
$930$	0	0
$931$	− 2.38404i	− 0.0781339i
$932$	0	0
$933$	− 6.26521i	− 0.205114i
$934$	0	0
$935$	2.69555	0.0881541
$936$	0	0
$937$	−33.7347	−1.10206	−0.551032	−	0.834484i	$-0.685766\pi$
$937$	−33.7347	−1.10206	−0.551032	+	0.834484i	$0.685766\pi$
$938$	0	0
$939$	3.80418i	0.124145i
$940$	0	0
$941$	− 3.91029i	− 0.127472i	−0.997967	−	0.0637360i	$-0.979698\pi$
$941$	− 3.91029i	− 0.127472i	0.997967	−	0.0637360i	$-0.0203015\pi$
$942$	0	0
$943$	−15.3162	−0.498765
$944$	0	0
$945$	6.28122	0.204328
$946$	0	0
$947$	− 46.8518i	− 1.52248i	−0.648470	−	0.761240i	$-0.724591\pi$
$947$	− 46.8518i	− 1.52248i	0.648470	−	0.761240i	$-0.275409\pi$
$948$	0	0
$949$	4.03327i	0.130925i
$950$	0	0
$951$	6.57333	0.213155
$952$	0	0
$953$	−26.5470	−0.859941	−0.429971	−	0.902843i	$-0.641476\pi$
$953$	−26.5470	−0.859941	−0.429971	+	0.902843i	$0.641476\pi$
$954$	0	0
$955$	35.4844i	1.14825i
$956$	0	0
$957$	1.04739i	0.0338574i
$958$	0	0
$959$	−36.4959	−1.17851
$960$	0	0
$961$	−19.1237	−0.616895
$962$	0	0
$963$	− 46.7731i	− 1.50724i
$964$	0	0
$965$	3.27606i	0.105460i
$966$	0	0
$967$	−6.80878	−0.218956	−0.109478	−	0.993989i	$-0.534918\pi$
$967$	−6.80878	−0.218956	−0.109478	+	0.993989i	$0.534918\pi$
$968$	0	0
$969$	−0.406261	−0.0130510
$970$	0	0
$971$	− 48.4549i	− 1.55499i	−0.628887	−	0.777496i	$-0.716489\pi$
$971$	− 48.4549i	− 1.55499i	0.628887	−	0.777496i	$-0.283511\pi$
$972$	0	0
$973$	34.2942i	1.09942i
$974$	0	0
$975$	1.93797	0.0620648
$976$	0	0
$977$	46.0137	1.47211	0.736055	−	0.676922i	$-0.236686\pi$
$977$	46.0137	1.47211	0.736055	+	0.676922i	$0.236686\pi$
$978$	0	0
$979$	9.16941i	0.293056i
$980$	0	0
$981$	− 39.6013i	− 1.26437i
$982$	0	0
$983$	−27.8031	−0.886782	−0.443391	−	0.896328i	$-0.646225\pi$
$983$	−27.8031	−0.886782	−0.443391	+	0.896328i	$0.646225\pi$
$984$	0	0
$985$	−9.38258	−0.298954
$986$	0	0
$987$	2.31380i	0.0736490i
$988$	0	0
$989$	5.69283i	0.181022i
$990$	0	0
$991$	5.16951	0.164215	0.0821074	−	0.996623i	$-0.473835\pi$
$991$	5.16951	0.164215	0.0821074	+	0.996623i	$0.473835\pi$
$992$	0	0
$993$	−4.56646	−0.144912
$994$	0	0
$995$	− 6.25116i	− 0.198175i
$996$	0	0
$997$	18.7227i	0.592953i	0.955040	+	0.296476i	$0.0958116\pi$
$997$	18.7227i	0.592953i	−0.955040	+	0.296476i	$0.904188\pi$
$998$	0	0
$999$	3.91218	0.123776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.i.1217.7	✓	16
4.3	odd	2	inner	2432.2.c.i.1217.9	yes	16
8.3	odd	2	inner	2432.2.c.i.1217.8	yes	16
8.5	even	2	inner	2432.2.c.i.1217.10	yes	16
16.3	odd	4		4864.2.a.br.1.3		8
16.5	even	4		4864.2.a.br.1.4		8
16.11	odd	4		4864.2.a.bm.1.6		8
16.13	even	4		4864.2.a.bm.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.i.1217.7	✓	16	1.1	even	1	trivial
2432.2.c.i.1217.8	yes	16	8.3	odd	2	inner
2432.2.c.i.1217.9	yes	16	4.3	odd	2	inner
2432.2.c.i.1217.10	yes	16	8.5	even	2	inner
4864.2.a.bm.1.5		8	16.13	even	4
4864.2.a.bm.1.6		8	16.11	odd	4
4864.2.a.br.1.3		8	16.3	odd	4
4864.2.a.br.1.4		8	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)

\(f(q)\)	\(=\)	\(q-0.222191i q^{3} -1.55081i q^{5} -3.06334 q^{7} +2.95063 q^{9} +O(q^{10})\)	\(q-0.222191i q^{3} -1.55081i q^{5} -3.06334 q^{7} +2.95063 q^{9} +0.950631i q^{11} +3.36113i q^{13} -0.344577 q^{15} +1.82843 q^{17} -1.00000i q^{19} +0.680647i q^{21} -3.36113 q^{23} +2.59499 q^{25} -1.32218i q^{27} +4.95873i q^{29} +3.44620 q^{31} +0.211222 q^{33} +4.75066i q^{35} +2.95889i q^{37} +0.746814 q^{39} +4.55687 q^{41} -1.69373i q^{43} -4.57587i q^{45} +3.39941 q^{47} +2.38404 q^{49} -0.406261i q^{51} -4.47142i q^{53} +1.47425 q^{55} -0.222191 q^{57} +0.395014i q^{59} +5.34158i q^{61} -9.03878 q^{63} +5.21247 q^{65} +6.85064i q^{67} +0.746814i q^{69} +5.15207 q^{71} +1.19998 q^{73} -0.576584i q^{75} -2.91210i q^{77} +14.7249 q^{79} +8.55812 q^{81} +3.35564i q^{83} -2.83554i q^{85} +1.10179 q^{87} +9.64561 q^{89} -10.2963i q^{91} -0.765715i q^{93} -1.55081 q^{95} +3.55562 q^{97} +2.80496i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^9	\( 16 q - 8 q^{9} - 16 q^{17} - 40 q^{25} + 48 q^{33} - 16 q^{41} + 16 q^{49} + 8 q^{57} + 16 q^{65} + 16 q^{73} - 64 q^{81} + 16 q^{89} + 80 q^{97}+O(q^{100}) \) 16 * q - 8 * q^9 - 16 * q^17 - 40 * q^25 + 48 * q^33 - 16 * q^41 + 16 * q^49 + 8 * q^57 + 16 * q^65 + 16 * q^73 - 64 * q^81 + 16 * q^89 + 80 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more