LMFDB - Embedded newform 2432.2.c.i.1217.2

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			1217.2
Root			$0.569934 + 1.29429i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.i.1217.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.63640i q^{3} +2.63403i q^{5} +3.88603 q^{7} -3.95063 q^{9} +O(q^{10})$	$q-2.63640i q^{3} +2.63403i q^{5} +3.88603 q^{7} -3.95063 q^{9} +5.95063i q^{11} -4.14940i q^{13} +6.94438 q^{15} +1.82843 q^{17} +1.00000i q^{19} -10.2452i q^{21} -4.14940 q^{23} -1.93813 q^{25} +2.50625i q^{27} +8.19638i q^{29} -1.67631 q^{31} +15.6883 q^{33} +10.2359i q^{35} -8.14452i q^{37} -10.9395 q^{39} +0.514201 q^{41} +10.9631i q^{43} -10.4061i q^{45} +13.3035 q^{47} +8.10124 q^{49} -4.82047i q^{51} +1.62446i q^{53} -15.6742 q^{55} +2.63640 q^{57} +12.2234i q^{59} +5.98666i q^{61} -15.3523 q^{63} +10.9297 q^{65} +4.09328i q^{67} +10.9395i q^{69} +11.8696 q^{71} +9.28531 q^{73} +5.10971i q^{75} +23.1243i q^{77} +2.42124 q^{79} -5.24441 q^{81} -0.987504i q^{83} +4.81614i q^{85} +21.6090 q^{87} +2.25389 q^{89} -16.1247i q^{91} +4.41944i q^{93} -2.63403 q^{95} +9.27281 q^{97} -23.5087i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 2.63640i	− 1.52213i	−0.648676	−	0.761065i	$-0.724677\pi$
$3$	− 2.63640i	− 1.52213i	0.648676	−	0.761065i	$-0.275323\pi$
$4$	0	0
$5$	2.63403i	1.17798i	0.808142	+	0.588988i	$0.200473\pi$
$5$	2.63403i	1.17798i	−0.808142	+	0.588988i	$0.799527\pi$
$6$	0	0
$7$	3.88603	1.46878	0.734391	−	0.678727i	$-0.237468\pi$
$7$	3.88603	1.46878	0.734391	+	0.678727i	$0.237468\pi$
$8$	0	0
$9$	−3.95063	−1.31688
$10$	0	0
$11$	5.95063i	1.79418i	0.441845	+	0.897091i	$0.354324\pi$
$11$	5.95063i	1.79418i	−0.441845	+	0.897091i	$0.645676\pi$
$12$	0	0
$13$	− 4.14940i	− 1.15084i	−0.817859	−	0.575418i	$-0.804839\pi$
$13$	− 4.14940i	− 1.15084i	0.817859	−	0.575418i	$-0.195161\pi$
$14$	0	0
$15$	6.94438	1.79303
$16$	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
$21$	− 10.2452i	− 2.23568i
$22$	0	0
$23$	−4.14940	−0.865210	−0.432605	−	0.901584i	$-0.642406\pi$
$23$	−4.14940	−0.865210	−0.432605	+	0.901584i	$0.642406\pi$
$24$	0	0
$25$	−1.93813	−0.387627
$26$	0	0
$27$	2.50625i	0.482328i
$28$	0	0
$29$	8.19638i	1.52203i	0.648735	+	0.761015i	$0.275298\pi$
$29$	8.19638i	1.52203i	−0.648735	+	0.761015i	$0.724702\pi$
$30$	0	0
$31$	−1.67631	−0.301075	−0.150537	−	0.988604i	$-0.548100\pi$
$31$	−1.67631	−0.301075	−0.150537	+	0.988604i	$0.548100\pi$
$32$	0	0
$33$	15.6883	2.73098
$34$	0	0
$35$	10.2359i	1.73019i
$36$	0	0
$37$	− 8.14452i	− 1.33895i	−0.742834	−	0.669476i	$-0.766519\pi$
$37$	− 8.14452i	− 1.33895i	0.742834	−	0.669476i	$-0.233481\pi$
$38$	0	0
$39$	−10.9395	−1.75172
$40$	0	0
$41$	0.514201	0.0803047	0.0401524	−	0.999194i	$-0.487216\pi$
$41$	0.514201	0.0803047	0.0401524	+	0.999194i	$0.487216\pi$
$42$	0	0
$43$	10.9631i	1.67186i	0.548835	+	0.835931i	$0.315071\pi$
$43$	10.9631i	1.67186i	−0.548835	+	0.835931i	$0.684929\pi$
$44$	0	0
$45$	− 10.4061i	− 1.55125i
$46$	0	0
$47$	13.3035	1.94051	0.970257	−	0.242075i	$-0.0778281\pi$
$47$	13.3035	1.94051	0.970257	+	0.242075i	$0.0778281\pi$
$48$	0	0
$49$	8.10124	1.15732
$50$	0	0
$51$	− 4.82047i	− 0.675001i
$52$	0	0
$53$	1.62446i	0.223137i	0.993757	+	0.111568i	$0.0355874\pi$
$53$	1.62446i	0.223137i	−0.993757	+	0.111568i	$0.964413\pi$
$54$	0	0
$55$	−15.6742	−2.11350
$56$	0	0
$57$	2.63640	0.349200
$58$	0	0
$59$	12.2234i	1.59136i	0.605720	+	0.795678i	$0.292885\pi$
$59$	12.2234i	1.59136i	−0.605720	+	0.795678i	$0.707115\pi$
$60$	0	0
$61$	5.98666i	0.766513i	0.923642	+	0.383257i	$0.125197\pi$
$61$	5.98666i	0.766513i	−0.923642	+	0.383257i	$0.874803\pi$
$62$	0	0
$63$	−15.3523	−1.93420
$64$	0	0
$65$	10.9297	1.35566
$66$	0	0
$67$	4.09328i	0.500074i	0.968236	+	0.250037i	$0.0804428\pi$
$67$	4.09328i	0.500074i	−0.968236	+	0.250037i	$0.919557\pi$
$68$	0	0
$69$	10.9395i	1.31696i
$70$	0	0
$71$	11.8696	1.40866	0.704332	−	0.709871i	$-0.251247\pi$
$71$	11.8696	1.40866	0.704332	+	0.709871i	$0.251247\pi$
$72$	0	0
$73$	9.28531	1.08676	0.543381	−	0.839486i	$-0.317144\pi$
$73$	9.28531	1.08676	0.543381	+	0.839486i	$0.317144\pi$
$74$	0	0
$75$	5.10971i	0.590018i
$76$	0	0
$77$	23.1243i	2.63526i
$78$	0	0
$79$	2.42124	0.272410	0.136205	−	0.990681i	$-0.456509\pi$
$79$	2.42124	0.272410	0.136205	+	0.990681i	$0.456509\pi$
$80$	0	0
$81$	−5.24441	−0.582712
$82$	0	0
$83$	− 0.987504i	− 0.108393i	−0.998530	−	0.0541963i	$-0.982740\pi$
$83$	− 0.987504i	− 0.108393i	0.998530	−	0.0541963i	$-0.0172597\pi$
$84$	0	0
$85$	4.81614i	0.522384i
$86$	0	0
$87$	21.6090	2.31672
$88$	0	0
$89$	2.25389	0.238912	0.119456	−	0.992840i	$-0.461885\pi$
$89$	2.25389	0.238912	0.119456	+	0.992840i	$0.461885\pi$
$90$	0	0
$91$	− 16.1247i	− 1.69033i
$92$	0	0
$93$	4.41944i	0.458274i
$94$	0	0
$95$	−2.63403	−0.270246
$96$	0	0
$97$	9.27281	0.941511	0.470756	−	0.882264i	$-0.343981\pi$
$97$	9.27281	0.941511	0.470756	+	0.882264i	$0.343981\pi$
$98$	0	0
$99$	− 23.5087i	− 2.36272i
$100$	0	0
$101$	− 13.3620i	− 1.32957i	−0.747034	−	0.664785i	$-0.768523\pi$
$101$	− 13.3620i	− 1.32957i	0.747034	−	0.664785i	$-0.231477\pi$
$102$	0	0
$103$	−10.0094	−0.986255	−0.493127	−	0.869957i	$-0.664146\pi$
$103$	−10.0094	−0.986255	−0.493127	+	0.869957i	$0.664146\pi$
$104$	0	0
$105$	26.9861	2.63357
$106$	0	0
$107$	− 4.85189i	− 0.469050i	−0.972110	−	0.234525i	$-0.924646\pi$
$107$	− 4.85189i	− 0.469050i	0.972110	−	0.234525i	$-0.0753535\pi$
$108$	0	0
$109$	− 15.1198i	− 1.44822i	−0.689687	−	0.724108i	$-0.742252\pi$
$109$	− 15.1198i	− 1.44822i	0.689687	−	0.724108i	$-0.257748\pi$
$110$	0	0
$111$	−21.4723	−2.03806
$112$	0	0
$113$	−3.10175	−0.291789	−0.145894	−	0.989300i	$-0.546606\pi$
$113$	−3.10175	−0.291789	−0.145894	+	0.989300i	$0.546606\pi$
$114$	0	0
$115$	− 10.9297i	− 1.01920i
$116$	0	0
$117$	16.3928i	1.51551i
$118$	0	0
$119$	7.10532	0.651344
$120$	0	0
$121$	−24.4100	−2.21909
$122$	0	0
$123$	− 1.35564i	− 0.122234i
$124$	0	0
$125$	8.06506i	0.721361i
$126$	0	0
$127$	−3.59176	−0.318717	−0.159358	−	0.987221i	$-0.550943\pi$
$127$	−3.59176	−0.318717	−0.159358	+	0.987221i	$0.550943\pi$
$128$	0	0
$129$	28.9032	2.54479
$130$	0	0
$131$	9.76906i	0.853527i	0.904363	+	0.426763i	$0.140346\pi$
$131$	9.76906i	0.853527i	−0.904363	+	0.426763i	$0.859654\pi$
$132$	0	0
$133$	3.88603i	0.336962i
$134$	0	0
$135$	−6.60154	−0.568170
$136$	0	0
$137$	−4.25690	−0.363692	−0.181846	−	0.983327i	$-0.558207\pi$
$137$	−4.25690	−0.363692	−0.181846	+	0.983327i	$0.558207\pi$
$138$	0	0
$139$	− 9.50875i	− 0.806521i	−0.915085	−	0.403261i	$-0.867877\pi$
$139$	− 9.50875i	− 0.806521i	0.915085	−	0.403261i	$-0.132123\pi$
$140$	0	0
$141$	− 35.0734i	− 2.95371i
$142$	0	0
$143$	24.6916	2.06481
$144$	0	0
$145$	−21.5895	−1.79291
$146$	0	0
$147$	− 21.3581i	− 1.76159i
$148$	0	0
$149$	5.13803i	0.420924i	0.977602	+	0.210462i	$0.0674968\pi$
$149$	5.13803i	0.420924i	−0.977602	+	0.210462i	$0.932503\pi$
$150$	0	0
$151$	5.26807	0.428709	0.214355	−	0.976756i	$-0.431235\pi$
$151$	5.26807	0.428709	0.214355	+	0.976756i	$0.431235\pi$
$152$	0	0
$153$	−7.22344	−0.583981
$154$	0	0
$155$	− 4.41546i	− 0.354659i
$156$	0	0
$157$	− 7.12828i	− 0.568899i	−0.958691	−	0.284449i	$-0.908189\pi$
$157$	− 7.12828i	− 0.568899i	0.958691	−	0.284449i	$-0.0918108\pi$
$158$	0	0
$159$	4.28273	0.339643
$160$	0	0
$161$	−16.1247	−1.27080
$162$	0	0
$163$	12.4060i	0.971712i	0.874039	+	0.485856i	$0.161492\pi$
$163$	12.4060i	0.971712i	−0.874039	+	0.485856i	$0.838508\pi$
$164$	0	0
$165$	41.3234i	3.21703i
$166$	0	0
$167$	−8.80460	−0.681320	−0.340660	−	0.940187i	$-0.610651\pi$
$167$	−8.80460	−0.681320	−0.340660	+	0.940187i	$0.610651\pi$
$168$	0	0
$169$	−4.21753	−0.324426
$170$	0	0
$171$	− 3.95063i	− 0.302112i
$172$	0	0
$173$	− 1.52203i	− 0.115718i	−0.998325	−	0.0578590i	$-0.981573\pi$
$173$	− 1.52203i	− 0.115718i	0.998325	−	0.0578590i	$-0.0184274\pi$
$174$	0	0
$175$	−7.53165	−0.569339
$176$	0	0
$177$	32.2259	2.42225
$178$	0	0
$179$	− 18.6599i	− 1.39470i	−0.716729	−	0.697352i	$-0.754361\pi$
$179$	− 18.6599i	− 1.39470i	0.716729	−	0.697352i	$-0.245639\pi$
$180$	0	0
$181$	11.3132i	0.840907i	0.907314	+	0.420453i	$0.138129\pi$
$181$	11.3132i	0.840907i	−0.907314	+	0.420453i	$0.861871\pi$
$182$	0	0
$183$	15.7833	1.16673
$184$	0	0
$185$	21.4530	1.57725
$186$	0	0
$187$	10.8803i	0.795646i
$188$	0	0
$189$	9.73936i	0.708434i
$190$	0	0
$191$	−11.5963	−0.839078	−0.419539	−	0.907737i	$-0.637808\pi$
$191$	−11.5963	−0.839078	−0.419539	+	0.907737i	$0.637808\pi$
$192$	0	0
$193$	−3.78701	−0.272595	−0.136298	−	0.990668i	$-0.543520\pi$
$193$	−3.78701	−0.272595	−0.136298	+	0.990668i	$0.543520\pi$
$194$	0	0
$195$	− 28.8150i	− 2.06349i
$196$	0	0
$197$	− 26.0803i	− 1.85814i	−0.369902	−	0.929071i	$-0.620609\pi$
$197$	− 26.0803i	− 1.85814i	0.369902	−	0.929071i	$-0.379391\pi$
$198$	0	0
$199$	−3.53452	−0.250555	−0.125278	−	0.992122i	$-0.539982\pi$
$199$	−3.53452	−0.250555	−0.125278	+	0.992122i	$0.539982\pi$
$200$	0	0
$201$	10.7916	0.761178
$202$	0	0
$203$	31.8514i	2.23553i
$204$	0	0
$205$	1.35442i	0.0945970i
$206$	0	0
$207$	16.3928	1.13938
$208$	0	0
$209$	−5.95063	−0.411614
$210$	0	0
$211$	− 2.84939i	− 0.196160i	−0.995179	−	0.0980802i	$-0.968730\pi$
$211$	− 2.84939i	− 0.196160i	0.995179	−	0.0980802i	$-0.0312702\pi$
$212$	0	0
$213$	− 31.2931i	− 2.14417i
$214$	0	0
$215$	−28.8772	−1.96941
$216$	0	0
$217$	−6.51420	−0.442213
$218$	0	0
$219$	− 24.4798i	− 1.65419i
$220$	0	0
$221$	− 7.58688i	− 0.510349i
$222$	0	0
$223$	23.4383	1.56954	0.784772	−	0.619785i	$-0.212780\pi$
$223$	23.4383	1.56954	0.784772	+	0.619785i	$0.212780\pi$
$224$	0	0
$225$	7.65685	0.510457
$226$	0	0
$227$	− 7.43341i	− 0.493373i	−0.969095	−	0.246686i	$-0.920658\pi$
$227$	− 7.43341i	− 0.493373i	0.969095	−	0.246686i	$-0.0793418\pi$
$228$	0	0
$229$	− 19.2316i	− 1.27086i	−0.772157	−	0.635431i	$-0.780822\pi$
$229$	− 19.2316i	− 1.27086i	0.772157	−	0.635431i	$-0.219178\pi$
$230$	0	0
$231$	60.9651	4.01121
$232$	0	0
$233$	29.9306	1.96082	0.980411	−	0.196965i	$-0.0631084\pi$
$233$	29.9306	1.96082	0.980411	+	0.196965i	$0.0631084\pi$
$234$	0	0
$235$	35.0419i	2.28588i
$236$	0	0
$237$	− 6.38336i	− 0.414644i
$238$	0	0
$239$	−16.0479	−1.03805	−0.519026	−	0.854758i	$-0.673705\pi$
$239$	−16.0479	−1.03805	−0.519026	+	0.854758i	$0.673705\pi$
$240$	0	0
$241$	23.9168	1.54061	0.770307	−	0.637673i	$-0.220103\pi$
$241$	23.9168	1.54061	0.770307	+	0.637673i	$0.220103\pi$
$242$	0	0
$243$	21.3451i	1.36929i
$244$	0	0
$245$	21.3389i	1.36329i
$246$	0	0
$247$	4.14940	0.264020
$248$	0	0
$249$	−2.60346	−0.164988
$250$	0	0
$251$	10.1950i	0.643505i	0.946824	+	0.321753i	$0.104272\pi$
$251$	10.1950i	0.643505i	−0.946824	+	0.321753i	$0.895728\pi$
$252$	0	0
$253$	− 24.6916i	− 1.55234i
$254$	0	0
$255$	12.6973	0.795135
$256$	0	0
$257$	20.1866	1.25920	0.629602	−	0.776918i	$-0.283218\pi$
$257$	20.1866	1.25920	0.629602	+	0.776918i	$0.283218\pi$
$258$	0	0
$259$	− 31.6499i	− 1.96663i
$260$	0	0
$261$	− 32.3809i	− 2.00432i
$262$	0	0
$263$	5.32659	0.328451	0.164226	−	0.986423i	$-0.447487\pi$
$263$	5.32659	0.328451	0.164226	+	0.986423i	$0.447487\pi$
$264$	0	0
$265$	−4.27888	−0.262850
$266$	0	0
$267$	− 5.94216i	− 0.363654i
$268$	0	0
$269$	14.0564i	0.857032i	0.903534	+	0.428516i	$0.140963\pi$
$269$	14.0564i	0.857032i	−0.903534	+	0.428516i	$0.859037\pi$
$270$	0	0
$271$	−8.56884	−0.520520	−0.260260	−	0.965539i	$-0.583808\pi$
$271$	−8.56884	−0.520520	−0.260260	+	0.965539i	$0.583808\pi$
$272$	0	0
$273$	−42.5112	−2.57290
$274$	0	0
$275$	− 11.5331i	− 0.695474i
$276$	0	0
$277$	− 22.7013i	− 1.36399i	−0.731357	−	0.681995i	$-0.761113\pi$
$277$	− 22.7013i	− 1.36399i	0.731357	−	0.681995i	$-0.238887\pi$
$278$	0	0
$279$	6.62249	0.396478
$280$	0	0
$281$	−8.26940	−0.493311	−0.246656	−	0.969103i	$-0.579332\pi$
$281$	−8.26940	−0.493311	−0.246656	+	0.969103i	$0.579332\pi$
$282$	0	0
$283$	21.7941i	1.29552i	0.761844	+	0.647761i	$0.224294\pi$
$283$	21.7941i	1.29552i	−0.761844	+	0.647761i	$0.775706\pi$
$284$	0	0
$285$	6.94438i	0.411350i
$286$	0	0
$287$	1.99820	0.117950
$288$	0	0
$289$	−13.6569	−0.803344
$290$	0	0
$291$	− 24.4469i	− 1.43310i
$292$	0	0
$293$	8.64036i	0.504775i	0.967626	+	0.252388i	$0.0812158\pi$
$293$	8.64036i	0.504775i	−0.967626	+	0.252388i	$0.918784\pi$
$294$	0	0
$295$	−32.1970	−1.87458
$296$	0	0
$297$	−14.9138	−0.865384
$298$	0	0
$299$	17.2175i	0.995716i
$300$	0	0
$301$	42.6031i	2.45560i
$302$	0	0
$303$	−35.2277	−2.02378
$304$	0	0
$305$	−15.7691	−0.902934
$306$	0	0
$307$	11.3871i	0.649894i	0.945732	+	0.324947i	$0.105346\pi$
$307$	11.3871i	0.649894i	−0.945732	+	0.324947i	$0.894654\pi$
$308$	0	0
$309$	26.3888i	1.50121i
$310$	0	0
$311$	−11.3199	−0.641893	−0.320947	−	0.947097i	$-0.604001\pi$
$311$	−11.3199	−0.641893	−0.320947	+	0.947097i	$0.604001\pi$
$312$	0	0
$313$	−18.3052	−1.03467	−0.517337	−	0.855782i	$-0.673076\pi$
$313$	−18.3052	−1.03467	−0.517337	+	0.855782i	$0.673076\pi$
$314$	0	0
$315$	− 40.4384i	− 2.27845i
$316$	0	0
$317$	− 26.0999i	− 1.46592i	−0.680273	−	0.732959i	$-0.738139\pi$
$317$	− 26.0999i	− 1.46592i	0.680273	−	0.732959i	$-0.261861\pi$
$318$	0	0
$319$	−48.7736	−2.73080
$320$	0	0
$321$	−12.7916	−0.713955
$322$	0	0
$323$	1.82843i	0.101736i
$324$	0	0
$325$	8.04210i	0.446095i
$326$	0	0
$327$	−39.8619	−2.20437
$328$	0	0
$329$	51.6978	2.85019
$330$	0	0
$331$	− 12.2799i	− 0.674962i	−0.941332	−	0.337481i	$-0.890425\pi$
$331$	− 12.2799i	− 0.674962i	0.941332	−	0.337481i	$-0.109575\pi$
$332$	0	0
$333$	32.1760i	1.76323i
$334$	0	0
$335$	−10.7818	−0.589075
$336$	0	0
$337$	15.5263	0.845772	0.422886	−	0.906183i	$-0.361017\pi$
$337$	15.5263	0.845772	0.422886	+	0.906183i	$0.361017\pi$
$338$	0	0
$339$	8.17748i	0.444140i
$340$	0	0
$341$	− 9.97512i	− 0.540183i
$342$	0	0
$343$	4.27944	0.231068
$344$	0	0
$345$	−28.8150	−1.55135
$346$	0	0
$347$	− 12.5212i	− 0.672176i	−0.941831	−	0.336088i	$-0.890896\pi$
$347$	− 12.5212i	− 0.672176i	0.941831	−	0.336088i	$-0.109104\pi$
$348$	0	0
$349$	− 1.36237i	− 0.0729259i	−0.999335	−	0.0364630i	$-0.988391\pi$
$349$	− 1.36237i	− 0.0729259i	0.999335	−	0.0364630i	$-0.0116091\pi$
$350$	0	0
$351$	10.3994	0.555081
$352$	0	0
$353$	−22.3247	−1.18822	−0.594111	−	0.804383i	$-0.702496\pi$
$353$	−22.3247	−1.18822	−0.594111	+	0.804383i	$0.702496\pi$
$354$	0	0
$355$	31.2650i	1.65937i
$356$	0	0
$357$	− 18.7325i	− 0.991430i
$358$	0	0
$359$	16.2824	0.859351	0.429676	−	0.902983i	$-0.358628\pi$
$359$	16.2824	0.859351	0.429676	+	0.902983i	$0.358628\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	64.3547i	3.37774i
$364$	0	0
$365$	24.4578i	1.28018i
$366$	0	0
$367$	29.6214	1.54623	0.773113	−	0.634268i	$-0.218698\pi$
$367$	29.6214	1.54623	0.773113	+	0.634268i	$0.218698\pi$
$368$	0	0
$369$	−2.03142	−0.105751
$370$	0	0
$371$	6.31270i	0.327739i
$372$	0	0
$373$	5.18659i	0.268551i	0.990944	+	0.134276i	$0.0428708\pi$
$373$	5.18659i	0.268551i	−0.990944	+	0.134276i	$0.957129\pi$
$374$	0	0
$375$	21.2628	1.09800
$376$	0	0
$377$	34.0101	1.75161
$378$	0	0
$379$	11.6485i	0.598344i	0.954199	+	0.299172i	$0.0967104\pi$
$379$	11.6485i	0.598344i	−0.954199	+	0.299172i	$0.903290\pi$
$380$	0	0
$381$	9.46932i	0.485128i
$382$	0	0
$383$	−33.3465	−1.70393	−0.851964	−	0.523600i	$-0.824589\pi$
$383$	−33.3465	−1.70393	−0.851964	+	0.523600i	$0.824589\pi$
$384$	0	0
$385$	−60.9103	−3.10428
$386$	0	0
$387$	− 43.3113i	− 2.20164i
$388$	0	0
$389$	− 1.51874i	− 0.0770033i	−0.999259	−	0.0385016i	$-0.987742\pi$
$389$	− 1.51874i	− 0.0770033i	0.999259	−	0.0385016i	$-0.0122585\pi$
$390$	0	0
$391$	−7.58688	−0.383685
$392$	0	0
$393$	25.7552	1.29918
$394$	0	0
$395$	6.37762i	0.320893i
$396$	0	0
$397$	− 29.4592i	− 1.47852i	−0.673422	−	0.739258i	$-0.735176\pi$
$397$	− 29.4592i	− 1.47852i	0.673422	−	0.739258i	$-0.264824\pi$
$398$	0	0
$399$	10.2452	0.512899
$400$	0	0
$401$	−32.0523	−1.60062	−0.800308	−	0.599589i	$-0.795331\pi$
$401$	−32.0523	−1.60062	−0.800308	+	0.599589i	$0.795331\pi$
$402$	0	0
$403$	6.95569i	0.346488i
$404$	0	0
$405$	− 13.8139i	− 0.686421i
$406$	0	0
$407$	48.4651	2.40232
$408$	0	0
$409$	10.7427	0.531192	0.265596	−	0.964084i	$-0.414431\pi$
$409$	10.7427	0.531192	0.265596	+	0.964084i	$0.414431\pi$
$410$	0	0
$411$	11.2229i	0.553586i
$412$	0	0
$413$	47.5007i	2.33736i
$414$	0	0
$415$	2.60112	0.127684
$416$	0	0
$417$	−25.0689	−1.22763
$418$	0	0
$419$	− 19.9990i	− 0.977013i	−0.872560	−	0.488507i	$-0.837542\pi$
$419$	− 19.9990i	− 0.977013i	0.872560	−	0.488507i	$-0.162458\pi$
$420$	0	0
$421$	− 10.5670i	− 0.515006i	−0.966277	−	0.257503i	$-0.917100\pi$
$421$	− 10.5670i	− 0.515006i	0.966277	−	0.257503i	$-0.0828997\pi$
$422$	0	0
$423$	−52.5572	−2.55542
$424$	0	0
$425$	−3.54374	−0.171897
$426$	0	0
$427$	23.2643i	1.12584i
$428$	0	0
$429$	− 65.0969i	− 3.14291i
$430$	0	0
$431$	3.24892	0.156495	0.0782474	−	0.996934i	$-0.475068\pi$
$431$	3.24892	0.156495	0.0782474	+	0.996934i	$0.475068\pi$
$432$	0	0
$433$	−25.4848	−1.22472	−0.612360	−	0.790579i	$-0.709780\pi$
$433$	−25.4848	−1.22472	−0.612360	+	0.790579i	$0.709780\pi$
$434$	0	0
$435$	56.9188i	2.72905i
$436$	0	0
$437$	− 4.14940i	− 0.198493i
$438$	0	0
$439$	−17.9520	−0.856804	−0.428402	−	0.903588i	$-0.640923\pi$
$439$	−17.9520	−0.856804	−0.428402	+	0.903588i	$0.640923\pi$
$440$	0	0
$441$	−32.0050	−1.52405
$442$	0	0
$443$	− 1.68123i	− 0.0798777i	−0.999202	−	0.0399388i	$-0.987284\pi$
$443$	− 1.68123i	− 0.0798777i	0.999202	−	0.0399388i	$-0.0127163\pi$
$444$	0	0
$445$	5.93682i	0.281432i
$446$	0	0
$447$	13.5459	0.640700
$448$	0	0
$449$	−21.8275	−1.03010	−0.515052	−	0.857159i	$-0.672227\pi$
$449$	−21.8275	−1.03010	−0.515052	+	0.857159i	$0.672227\pi$
$450$	0	0
$451$	3.05982i	0.144081i
$452$	0	0
$453$	− 13.8888i	− 0.652551i
$454$	0	0
$455$	42.4730	1.99117
$456$	0	0
$457$	−8.04431	−0.376297	−0.188148	−	0.982141i	$-0.560249\pi$
$457$	−8.04431	−0.376297	−0.188148	+	0.982141i	$0.560249\pi$
$458$	0	0
$459$	4.58249i	0.213892i
$460$	0	0
$461$	6.79339i	0.316400i	0.987407	+	0.158200i	$0.0505690\pi$
$461$	6.79339i	0.316400i	−0.987407	+	0.158200i	$0.949431\pi$
$462$	0	0
$463$	31.2479	1.45221	0.726107	−	0.687582i	$-0.241328\pi$
$463$	31.2479	1.45221	0.726107	+	0.687582i	$0.241328\pi$
$464$	0	0
$465$	−11.6409	−0.539836
$466$	0	0
$467$	36.9113i	1.70805i	0.520230	+	0.854026i	$0.325846\pi$
$467$	36.9113i	1.70805i	−0.520230	+	0.854026i	$0.674154\pi$
$468$	0	0
$469$	15.9066i	0.734500i
$470$	0	0
$471$	−18.7930	−0.865938
$472$	0	0
$473$	−65.2375	−2.99962
$474$	0	0
$475$	− 1.93813i	− 0.0889277i
$476$	0	0
$477$	− 6.41764i	− 0.293844i
$478$	0	0
$479$	−7.26161	−0.331792	−0.165896	−	0.986143i	$-0.553052\pi$
$479$	−7.26161	−0.331792	−0.165896	+	0.986143i	$0.553052\pi$
$480$	0	0
$481$	−33.7949	−1.54091
$482$	0	0
$483$	42.5112i	1.93433i
$484$	0	0
$485$	24.4249i	1.10908i
$486$	0	0
$487$	−1.00500	−0.0455411	−0.0227705	−	0.999741i	$-0.507249\pi$
$487$	−1.00500	−0.0455411	−0.0227705	+	0.999741i	$0.507249\pi$
$488$	0	0
$489$	32.7072	1.47907
$490$	0	0
$491$	9.12714i	0.411902i	0.978562	+	0.205951i	$0.0660288\pi$
$491$	9.12714i	0.411902i	−0.978562	+	0.205951i	$0.933971\pi$
$492$	0	0
$493$	14.9865i	0.674957i
$494$	0	0
$495$	61.9228	2.78322
$496$	0	0
$497$	46.1257	2.06902
$498$	0	0
$499$	19.8928i	0.890524i	0.895400	+	0.445262i	$0.146889\pi$
$499$	19.8928i	0.890524i	−0.895400	+	0.445262i	$0.853111\pi$
$500$	0	0
$501$	23.2125i	1.03706i
$502$	0	0
$503$	−18.9716	−0.845904	−0.422952	−	0.906152i	$-0.639006\pi$
$503$	−18.9716	−0.845904	−0.422952	+	0.906152i	$0.639006\pi$
$504$	0	0
$505$	35.1960	1.56620
$506$	0	0
$507$	11.1191i	0.493818i
$508$	0	0
$509$	− 24.2981i	− 1.07700i	−0.842627	−	0.538498i	$-0.818992\pi$
$509$	− 24.2981i	− 1.07700i	0.842627	−	0.538498i	$-0.181008\pi$
$510$	0	0
$511$	36.0830	1.59622
$512$	0	0
$513$	−2.50625	−0.110654
$514$	0	0
$515$	− 26.3651i	− 1.16178i
$516$	0	0
$517$	79.1642i	3.48164i
$518$	0	0
$519$	−4.01270	−0.176138
$520$	0	0
$521$	10.1930	0.446563	0.223282	−	0.974754i	$-0.428323\pi$
$521$	10.1930	0.446563	0.223282	+	0.974754i	$0.428323\pi$
$522$	0	0
$523$	− 1.42683i	− 0.0623907i	−0.999513	−	0.0311954i	$-0.990069\pi$
$523$	− 1.42683i	− 0.0623907i	0.999513	−	0.0311954i	$-0.00993140\pi$
$524$	0	0
$525$	19.8565i	0.866608i
$526$	0	0
$527$	−3.06501	−0.133514
$528$	0	0
$529$	−5.78247	−0.251412
$530$	0	0
$531$	− 48.2903i	− 2.09562i
$532$	0	0
$533$	− 2.13363i	− 0.0924176i
$534$	0	0
$535$	12.7801	0.552530
$536$	0	0
$537$	−49.1950	−2.12292
$538$	0	0
$539$	48.2075i	2.07644i
$540$	0	0
$541$	28.1257i	1.20922i	0.796522	+	0.604610i	$0.206671\pi$
$541$	28.1257i	1.20922i	−0.796522	+	0.604610i	$0.793329\pi$
$542$	0	0
$543$	29.8263	1.27997
$544$	0	0
$545$	39.8261	1.70596
$546$	0	0
$547$	45.2937i	1.93662i	0.249753	+	0.968310i	$0.419651\pi$
$547$	45.2937i	1.93662i	−0.249753	+	0.968310i	$0.580349\pi$
$548$	0	0
$549$	− 23.6511i	− 1.00940i
$550$	0	0
$551$	−8.19638	−0.349177
$552$	0	0
$553$	9.40900	0.400111
$554$	0	0
$555$	− 56.5587i	− 2.40078i
$556$	0	0
$557$	13.9151i	0.589601i	0.955559	+	0.294801i	$0.0952533\pi$
$557$	13.9151i	0.589601i	−0.955559	+	0.294801i	$0.904747\pi$
$558$	0	0
$559$	45.4904	1.92404
$560$	0	0
$561$	28.6849	1.21108
$562$	0	0
$563$	10.3367i	0.435641i	0.975989	+	0.217820i	$0.0698947\pi$
$563$	10.3367i	0.435641i	−0.975989	+	0.217820i	$0.930105\pi$
$564$	0	0
$565$	− 8.17013i	− 0.343720i
$566$	0	0
$567$	−20.3799	−0.855877
$568$	0	0
$569$	−39.2369	−1.64490	−0.822448	−	0.568840i	$-0.807392\pi$
$569$	−39.2369	−1.64490	−0.822448	+	0.568840i	$0.807392\pi$
$570$	0	0
$571$	− 33.4374i	− 1.39931i	−0.714479	−	0.699657i	$-0.753336\pi$
$571$	− 33.4374i	− 1.39931i	0.714479	−	0.699657i	$-0.246664\pi$
$572$	0	0
$573$	30.5725i	1.27718i
$574$	0	0
$575$	8.04210	0.335379
$576$	0	0
$577$	−22.3203	−0.929206	−0.464603	−	0.885519i	$-0.653803\pi$
$577$	−22.3203	−0.929206	−0.464603	+	0.885519i	$0.653803\pi$
$578$	0	0
$579$	9.98409i	0.414925i
$580$	0	0
$581$	− 3.83747i	− 0.159205i
$582$	0	0
$583$	−9.66656	−0.400348
$584$	0	0
$585$	−43.1791	−1.78523
$586$	0	0
$587$	− 43.6753i	− 1.80267i	−0.433119	−	0.901337i	$-0.642587\pi$
$587$	− 43.6753i	− 1.80267i	0.433119	−	0.901337i	$-0.357413\pi$
$588$	0	0
$589$	− 1.67631i	− 0.0690712i
$590$	0	0
$591$	−68.7581	−2.82833
$592$	0	0
$593$	−19.6978	−0.808890	−0.404445	−	0.914562i	$-0.632535\pi$
$593$	−19.6978	−0.808890	−0.404445	+	0.914562i	$0.632535\pi$
$594$	0	0
$595$	18.7157i	0.767267i
$596$	0	0
$597$	9.31842i	0.381377i
$598$	0	0
$599$	−34.2570	−1.39970	−0.699851	−	0.714289i	$-0.746750\pi$
$599$	−34.2570	−1.39970	−0.699851	+	0.714289i	$0.746750\pi$
$600$	0	0
$601$	19.1801	0.782373	0.391186	−	0.920311i	$-0.372065\pi$
$601$	19.1801	0.782373	0.391186	+	0.920311i	$0.372065\pi$
$602$	0	0
$603$	− 16.1711i	− 0.658536i
$604$	0	0
$605$	− 64.2968i	− 2.61404i
$606$	0	0
$607$	17.1796	0.697297	0.348649	−	0.937253i	$-0.386641\pi$
$607$	17.1796	0.697297	0.348649	+	0.937253i	$0.386641\pi$
$608$	0	0
$609$	83.9731	3.40276
$610$	0	0
$611$	− 55.2016i	− 2.23322i
$612$	0	0
$613$	13.4502i	0.543247i	0.962404	+	0.271623i	$0.0875605\pi$
$613$	13.4502i	0.543247i	−0.962404	+	0.271623i	$0.912439\pi$
$614$	0	0
$615$	3.57081	0.143989
$616$	0	0
$617$	15.2065	0.612191	0.306095	−	0.952001i	$-0.400977\pi$
$617$	15.2065	0.612191	0.306095	+	0.952001i	$0.400977\pi$
$618$	0	0
$619$	− 40.1696i	− 1.61455i	−0.590173	−	0.807277i	$-0.700940\pi$
$619$	− 40.1696i	− 1.61455i	0.590173	−	0.807277i	$-0.299060\pi$
$620$	0	0
$621$	− 10.3994i	− 0.417315i
$622$	0	0
$623$	8.75868	0.350909
$624$	0	0
$625$	−30.9343	−1.23737
$626$	0	0
$627$	15.6883i	0.626529i
$628$	0	0
$629$	− 14.8917i	− 0.593770i
$630$	0	0
$631$	12.0318	0.478980	0.239490	−	0.970899i	$-0.423020\pi$
$631$	12.0318	0.478980	0.239490	+	0.970899i	$0.423020\pi$
$632$	0	0
$633$	−7.51216	−0.298581
$634$	0	0
$635$	− 9.46081i	− 0.375441i
$636$	0	0
$637$	− 33.6153i	− 1.33189i
$638$	0	0
$639$	−46.8925	−1.85504
$640$	0	0
$641$	28.1707	1.11267	0.556337	−	0.830957i	$-0.312206\pi$
$641$	28.1707	1.11267	0.556337	+	0.830957i	$0.312206\pi$
$642$	0	0
$643$	38.2190i	1.50721i	0.657327	+	0.753605i	$0.271687\pi$
$643$	38.2190i	1.50721i	−0.657327	+	0.753605i	$0.728313\pi$
$644$	0	0
$645$	76.1321i	2.99770i
$646$	0	0
$647$	7.26170	0.285487	0.142743	−	0.989760i	$-0.454408\pi$
$647$	7.26170	0.285487	0.142743	+	0.989760i	$0.454408\pi$
$648$	0	0
$649$	−72.7372	−2.85518
$650$	0	0
$651$	17.1741i	0.673105i
$652$	0	0
$653$	− 16.0972i	− 0.629932i	−0.949103	−	0.314966i	$-0.898007\pi$
$653$	− 16.0972i	− 0.629932i	0.949103	−	0.314966i	$-0.101993\pi$
$654$	0	0
$655$	−25.7320	−1.00543
$656$	0	0
$657$	−36.6828	−1.43113
$658$	0	0
$659$	− 17.8679i	− 0.696035i	−0.937488	−	0.348018i	$-0.886855\pi$
$659$	− 17.8679i	− 0.696035i	0.937488	−	0.348018i	$-0.113145\pi$
$660$	0	0
$661$	− 21.6386i	− 0.841644i	−0.907143	−	0.420822i	$-0.861742\pi$
$661$	− 21.6386i	− 0.841644i	0.907143	−	0.420822i	$-0.138258\pi$
$662$	0	0
$663$	−20.0021	−0.776817
$664$	0	0
$665$	−10.2359	−0.396933
$666$	0	0
$667$	− 34.0101i	− 1.31687i
$668$	0	0
$669$	− 61.7928i	− 2.38905i
$670$	0	0
$671$	−35.6244	−1.37526
$672$	0	0
$673$	−43.9881	−1.69561	−0.847807	−	0.530304i	$-0.822078\pi$
$673$	−43.9881	−1.69561	−0.847807	+	0.530304i	$0.822078\pi$
$674$	0	0
$675$	− 4.85745i	− 0.186963i
$676$	0	0
$677$	− 29.6983i	− 1.14140i	−0.821160	−	0.570699i	$-0.806672\pi$
$677$	− 29.6983i	− 1.14140i	0.821160	−	0.570699i	$-0.193328\pi$
$678$	0	0
$679$	36.0344	1.38287
$680$	0	0
$681$	−19.5975	−0.750977
$682$	0	0
$683$	6.20351i	0.237371i	0.992932	+	0.118685i	$0.0378680\pi$
$683$	6.20351i	0.237371i	−0.992932	+	0.118685i	$0.962132\pi$
$684$	0	0
$685$	− 11.2128i	− 0.428420i
$686$	0	0
$687$	−50.7024	−1.93442
$688$	0	0
$689$	6.74053	0.256794
$690$	0	0
$691$	− 28.1172i	− 1.06963i	−0.844970	−	0.534814i	$-0.820382\pi$
$691$	− 28.1172i	− 1.06963i	0.844970	−	0.534814i	$-0.179618\pi$
$692$	0	0
$693$	− 91.3557i	− 3.47032i
$694$	0	0
$695$	25.0464	0.950063
$696$	0	0
$697$	0.940179	0.0356118
$698$	0	0
$699$	− 78.9093i	− 2.98462i
$700$	0	0
$701$	8.35403i	0.315527i	0.987477	+	0.157764i	$0.0504284\pi$
$701$	8.35403i	0.315527i	−0.987477	+	0.157764i	$0.949572\pi$
$702$	0	0
$703$	8.14452	0.307177
$704$	0	0
$705$	92.3846	3.47940
$706$	0	0
$707$	− 51.9252i	− 1.95285i
$708$	0	0
$709$	31.7187i	1.19122i	0.803274	+	0.595610i	$0.203090\pi$
$709$	31.7187i	1.19122i	−0.803274	+	0.595610i	$0.796910\pi$
$710$	0	0
$711$	−9.56541	−0.358731
$712$	0	0
$713$	6.95569	0.260493
$714$	0	0
$715$	65.0384i	2.43230i
$716$	0	0
$717$	42.3088i	1.58005i
$718$	0	0
$719$	−33.7972	−1.26042	−0.630211	−	0.776424i	$-0.717032\pi$
$719$	−33.7972	−1.26042	−0.630211	+	0.776424i	$0.717032\pi$
$720$	0	0
$721$	−38.8968	−1.44859
$722$	0	0
$723$	− 63.0543i	− 2.34501i
$724$	0	0
$725$	− 15.8857i	− 0.589979i
$726$	0	0
$727$	32.8637	1.21885	0.609423	−	0.792845i	$-0.291401\pi$
$727$	32.8637	1.21885	0.609423	+	0.792845i	$0.291401\pi$
$728$	0	0
$729$	40.5412	1.50153
$730$	0	0
$731$	20.0453i	0.741401i
$732$	0	0
$733$	− 14.3118i	− 0.528618i	−0.964438	−	0.264309i	$-0.914856\pi$
$733$	− 14.3118i	− 0.528618i	0.964438	−	0.264309i	$-0.0851439\pi$
$734$	0	0
$735$	56.2581	2.07511
$736$	0	0
$737$	−24.3576	−0.897225
$738$	0	0
$739$	− 22.6101i	− 0.831726i	−0.909427	−	0.415863i	$-0.863480\pi$
$739$	− 22.6101i	− 0.831726i	0.909427	−	0.415863i	$-0.136520\pi$
$740$	0	0
$741$	− 10.9395i	− 0.401873i
$742$	0	0
$743$	12.6222	0.463062	0.231531	−	0.972828i	$-0.425627\pi$
$743$	12.6222	0.463062	0.231531	+	0.972828i	$0.425627\pi$
$744$	0	0
$745$	−13.5337	−0.495838
$746$	0	0
$747$	3.90126i	0.142740i
$748$	0	0
$749$	− 18.8546i	− 0.688932i
$750$	0	0
$751$	4.69943	0.171485	0.0857423	−	0.996317i	$-0.472674\pi$
$751$	4.69943	0.171485	0.0857423	+	0.996317i	$0.472674\pi$
$752$	0	0
$753$	26.8783	0.979498
$754$	0	0
$755$	13.8763i	0.505009i
$756$	0	0
$757$	− 21.8920i	− 0.795679i	−0.917455	−	0.397839i	$-0.869760\pi$
$757$	− 21.8920i	− 0.795679i	0.917455	−	0.397839i	$-0.130240\pi$
$758$	0	0
$759$	−65.0969	−2.36287
$760$	0	0
$761$	0.164985	0.00598068	0.00299034	−	0.999996i	$-0.499048\pi$
$761$	0.164985	0.00598068	0.00299034	+	0.999996i	$0.499048\pi$
$762$	0	0
$763$	− 58.7560i	− 2.12711i
$764$	0	0
$765$	− 19.0268i	− 0.687915i
$766$	0	0
$767$	50.7200	1.83139
$768$	0	0
$769$	−8.55812	−0.308614	−0.154307	−	0.988023i	$-0.549314\pi$
$769$	−8.55812	−0.308614	−0.154307	+	0.988023i	$0.549314\pi$
$770$	0	0
$771$	− 53.2200i	− 1.91667i
$772$	0	0
$773$	− 9.49557i	− 0.341532i	−0.985312	−	0.170766i	$-0.945376\pi$
$773$	− 9.49557i	− 0.341532i	0.985312	−	0.170766i	$-0.0546242\pi$
$774$	0	0
$775$	3.24892	0.116705
$776$	0	0
$777$	−83.4419	−2.99346
$778$	0	0
$779$	0.514201i	0.0184232i
$780$	0	0
$781$	70.6317i	2.52740i
$782$	0	0
$783$	−20.5422	−0.734117
$784$	0	0
$785$	18.7761	0.670149
$786$	0	0
$787$	8.59840i	0.306500i	0.988187	+	0.153250i	$0.0489739\pi$
$787$	8.59840i	0.306500i	−0.988187	+	0.153250i	$0.951026\pi$
$788$	0	0
$789$	− 14.0430i	− 0.499945i
$790$	0	0
$791$	−12.0535	−0.428574
$792$	0	0
$793$	24.8410	0.882131
$794$	0	0
$795$	11.2809i	0.400091i
$796$	0	0
$797$	23.3568i	0.827340i	0.910427	+	0.413670i	$0.135753\pi$
$797$	23.3568i	0.827340i	−0.910427	+	0.413670i	$0.864247\pi$
$798$	0	0
$799$	24.3245	0.860538
$800$	0	0
$801$	−8.90428	−0.314617
$802$	0	0
$803$	55.2534i	1.94985i
$804$	0	0
$805$	− 42.4730i	− 1.49698i
$806$	0	0
$807$	37.0583	1.30451
$808$	0	0
$809$	11.0634	0.388968	0.194484	−	0.980906i	$-0.437697\pi$
$809$	11.0634	0.388968	0.194484	+	0.980906i	$0.437697\pi$
$810$	0	0
$811$	12.9630i	0.455193i	0.973756	+	0.227596i	$0.0730866\pi$
$811$	12.9630i	0.455193i	−0.973756	+	0.227596i	$0.926913\pi$
$812$	0	0
$813$	22.5909i	0.792299i
$814$	0	0
$815$	−32.6778	−1.14465
$816$	0	0
$817$	−10.9631	−0.383551
$818$	0	0
$819$	63.7028i	2.22595i
$820$	0	0
$821$	14.8255i	0.517415i	0.965956	+	0.258707i	$0.0832965\pi$
$821$	14.8255i	0.517415i	−0.965956	+	0.258707i	$0.916703\pi$
$822$	0	0
$823$	4.04240	0.140909	0.0704547	−	0.997515i	$-0.477555\pi$
$823$	4.04240	0.140909	0.0704547	+	0.997515i	$0.477555\pi$
$824$	0	0
$825$	−30.4060	−1.05860
$826$	0	0
$827$	− 33.6070i	− 1.16863i	−0.811527	−	0.584314i	$-0.801364\pi$
$827$	− 33.6070i	− 1.16863i	0.811527	−	0.584314i	$-0.198636\pi$
$828$	0	0
$829$	− 38.3399i	− 1.33160i	−0.746130	−	0.665800i	$-0.768090\pi$
$829$	− 38.3399i	− 1.33160i	0.746130	−	0.665800i	$-0.231910\pi$
$830$	0	0
$831$	−59.8498	−2.07617
$832$	0	0
$833$	14.8125	0.513223
$834$	0	0
$835$	− 23.1916i	− 0.802579i
$836$	0	0
$837$	− 4.20125i	− 0.145217i
$838$	0	0
$839$	−37.6832	−1.30097	−0.650484	−	0.759520i	$-0.725434\pi$
$839$	−37.6832	−1.30097	−0.650484	+	0.759520i	$0.725434\pi$
$840$	0	0
$841$	−38.1806	−1.31657
$842$	0	0
$843$	21.8015i	0.750883i
$844$	0	0
$845$	− 11.1091i	− 0.382165i
$846$	0	0
$847$	−94.8581	−3.25936
$848$	0	0
$849$	57.4580	1.97195
$850$	0	0
$851$	33.7949i	1.15847i
$852$	0	0
$853$	7.06647i	0.241951i	0.992655	+	0.120976i	$0.0386023\pi$
$853$	7.06647i	0.241951i	−0.992655	+	0.120976i	$0.961398\pi$
$854$	0	0
$855$	10.4061	0.355881
$856$	0	0
$857$	−7.95910	−0.271878	−0.135939	−	0.990717i	$-0.543405\pi$
$857$	−7.95910	−0.271878	−0.135939	+	0.990717i	$0.543405\pi$
$858$	0	0
$859$	13.1466i	0.448558i	0.974525	+	0.224279i	$0.0720026\pi$
$859$	13.1466i	0.448558i	−0.974525	+	0.224279i	$0.927997\pi$
$860$	0	0
$861$	− 5.26807i	− 0.179535i
$862$	0	0
$863$	−34.9886	−1.19102	−0.595512	−	0.803346i	$-0.703051\pi$
$863$	−34.9886	−1.19102	−0.595512	+	0.803346i	$0.703051\pi$
$864$	0	0
$865$	4.00909	0.136313
$866$	0	0
$867$	36.0050i	1.22279i
$868$	0	0
$869$	14.4079i	0.488754i
$870$	0	0
$871$	16.9847	0.575504
$872$	0	0
$873$	−36.6335	−1.23985
$874$	0	0
$875$	31.3411i	1.05952i
$876$	0	0
$877$	− 49.1816i	− 1.66075i	−0.557208	−	0.830373i	$-0.688127\pi$
$877$	− 49.1816i	− 1.66075i	0.557208	−	0.830373i	$-0.311873\pi$
$878$	0	0
$879$	22.7795	0.768333
$880$	0	0
$881$	−7.58658	−0.255598	−0.127799	−	0.991800i	$-0.540791\pi$
$881$	−7.58658	−0.255598	−0.127799	+	0.991800i	$0.540791\pi$
$882$	0	0
$883$	− 15.1875i	− 0.511101i	−0.966796	−	0.255551i	$-0.917743\pi$
$883$	− 15.1875i	− 0.511101i	0.966796	−	0.255551i	$-0.0822568\pi$
$884$	0	0
$885$	84.8842i	2.85335i
$886$	0	0
$887$	32.9000	1.10467	0.552337	−	0.833621i	$-0.313736\pi$
$887$	32.9000	1.10467	0.552337	+	0.833621i	$0.313736\pi$
$888$	0	0
$889$	−13.9577	−0.468125
$890$	0	0
$891$	− 31.2075i	− 1.04549i
$892$	0	0
$893$	13.3035i	0.445185i
$894$	0	0
$895$	49.1507	1.64293
$896$	0	0
$897$	45.3924	1.51561
$898$	0	0
$899$	− 13.7397i	− 0.458244i
$900$	0	0
$901$	2.97021i	0.0989519i
$902$	0	0
$903$	112.319	3.73774
$904$	0	0
$905$	−29.7995	−0.990568
$906$	0	0
$907$	− 55.9392i	− 1.85743i	−0.370793	−	0.928715i	$-0.620914\pi$
$907$	− 55.9392i	− 1.85743i	0.370793	−	0.928715i	$-0.379086\pi$
$908$	0	0
$909$	52.7884i	1.75088i
$910$	0	0
$911$	44.6551	1.47949	0.739745	−	0.672887i	$-0.234946\pi$
$911$	44.6551	1.47949	0.739745	+	0.672887i	$0.234946\pi$
$912$	0	0
$913$	5.87627	0.194476
$914$	0	0
$915$	41.5736i	1.37438i
$916$	0	0
$917$	37.9629i	1.25364i
$918$	0	0
$919$	7.52507	0.248229	0.124115	−	0.992268i	$-0.460391\pi$
$919$	7.52507	0.248229	0.124115	+	0.992268i	$0.460391\pi$
$920$	0	0
$921$	30.0209	0.989222
$922$	0	0
$923$	− 49.2518i	− 1.62114i
$924$	0	0
$925$	15.7852i	0.519014i
$926$	0	0
$927$	39.5434	1.29878
$928$	0	0
$929$	−10.4360	−0.342395	−0.171198	−	0.985237i	$-0.554764\pi$
$929$	−10.4360	−0.342395	−0.171198	+	0.985237i	$0.554764\pi$
$930$	0	0
$931$	8.10124i	0.265507i
$932$	0	0
$933$	29.8439i	0.977044i
$934$	0	0
$935$	−28.6591	−0.937252
$936$	0	0
$937$	19.1073	0.624207	0.312104	−	0.950048i	$-0.398966\pi$
$937$	19.1073	0.624207	0.312104	+	0.950048i	$0.398966\pi$
$938$	0	0
$939$	48.2600i	1.57491i
$940$	0	0
$941$	− 43.2146i	− 1.40875i	−0.709826	−	0.704377i	$-0.751226\pi$
$941$	− 43.2146i	− 1.40875i	0.709826	−	0.704377i	$-0.248774\pi$
$942$	0	0
$943$	−2.13363	−0.0694804
$944$	0	0
$945$	−25.6538	−0.834518
$946$	0	0
$947$	− 1.25385i	− 0.0407446i	−0.999792	−	0.0203723i	$-0.993515\pi$
$947$	− 1.25385i	− 0.0407446i	0.999792	−	0.0203723i	$-0.00648516\pi$
$948$	0	0
$949$	− 38.5285i	− 1.25069i
$950$	0	0
$951$	−68.8100	−2.23132
$952$	0	0
$953$	35.0739	1.13616	0.568078	−	0.822975i	$-0.307687\pi$
$953$	35.0739	1.13616	0.568078	+	0.822975i	$0.307687\pi$
$954$	0	0
$955$	− 30.5450i	− 0.988413i
$956$	0	0
$957$	128.587i	4.15663i
$958$	0	0
$959$	−16.5425	−0.534184
$960$	0	0
$961$	−28.1900	−0.909354
$962$	0	0
$963$	19.1680i	0.617681i
$964$	0	0
$965$	− 9.97512i	− 0.321110i
$966$	0	0
$967$	1.66764	0.0536275	0.0268138	−	0.999640i	$-0.491464\pi$
$967$	1.66764	0.0536275	0.0268138	+	0.999640i	$0.491464\pi$
$968$	0	0
$969$	4.82047	0.154856
$970$	0	0
$971$	16.1136i	0.517110i	0.965996	+	0.258555i	$0.0832464\pi$
$971$	16.1136i	0.517110i	−0.965996	+	0.258555i	$0.916754\pi$
$972$	0	0
$973$	− 36.9513i	− 1.18460i
$974$	0	0
$975$	21.2022	0.679015
$976$	0	0
$977$	33.8857	1.08410	0.542051	−	0.840346i	$-0.317648\pi$
$977$	33.8857	1.08410	0.542051	+	0.840346i	$0.317648\pi$
$978$	0	0
$979$	13.4121i	0.428651i
$980$	0	0
$981$	59.7328i	1.90712i
$982$	0	0
$983$	9.89492	0.315599	0.157799	−	0.987471i	$-0.449560\pi$
$983$	9.89492	0.315599	0.157799	+	0.987471i	$0.449560\pi$
$984$	0	0
$985$	68.6963	2.18885
$986$	0	0
$987$	− 136.296i	− 4.33836i
$988$	0	0
$989$	− 45.4904i	− 1.44651i
$990$	0	0
$991$	31.3693	0.996478	0.498239	−	0.867040i	$-0.333980\pi$
$991$	31.3693	0.996478	0.498239	+	0.867040i	$0.333980\pi$
$992$	0	0
$993$	−32.3747	−1.02738
$994$	0	0
$995$	− 9.31004i	− 0.295148i
$996$	0	0
$997$	4.55606i	0.144292i	0.997394	+	0.0721459i	$0.0229847\pi$
$997$	4.55606i	0.144292i	−0.997394	+	0.0721459i	$0.977015\pi$
$998$	0	0
$999$	20.4122	0.645813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.i.1217.2	yes	16
4.3	odd	2	inner	2432.2.c.i.1217.16	yes	16
8.3	odd	2	inner	2432.2.c.i.1217.1	✓	16
8.5	even	2	inner	2432.2.c.i.1217.15	yes	16
16.3	odd	4		4864.2.a.bm.1.2		8
16.5	even	4		4864.2.a.bm.1.1		8
16.11	odd	4		4864.2.a.br.1.7		8
16.13	even	4		4864.2.a.br.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.i.1217.1	✓	16	8.3	odd	2	inner
2432.2.c.i.1217.2	yes	16	1.1	even	1	trivial
2432.2.c.i.1217.15	yes	16	8.5	even	2	inner
2432.2.c.i.1217.16	yes	16	4.3	odd	2	inner
4864.2.a.bm.1.1		8	16.5	even	4
4864.2.a.bm.1.2		8	16.3	odd	4
4864.2.a.br.1.7		8	16.11	odd	4
4864.2.a.br.1.8		8	16.13	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)

\(f(q)\)	\(=\)	\(q-2.63640i q^{3} +2.63403i q^{5} +3.88603 q^{7} -3.95063 q^{9} +O(q^{10})\)	\(q-2.63640i q^{3} +2.63403i q^{5} +3.88603 q^{7} -3.95063 q^{9} +5.95063i q^{11} -4.14940i q^{13} +6.94438 q^{15} +1.82843 q^{17} +1.00000i q^{19} -10.2452i q^{21} -4.14940 q^{23} -1.93813 q^{25} +2.50625i q^{27} +8.19638i q^{29} -1.67631 q^{31} +15.6883 q^{33} +10.2359i q^{35} -8.14452i q^{37} -10.9395 q^{39} +0.514201 q^{41} +10.9631i q^{43} -10.4061i q^{45} +13.3035 q^{47} +8.10124 q^{49} -4.82047i q^{51} +1.62446i q^{53} -15.6742 q^{55} +2.63640 q^{57} +12.2234i q^{59} +5.98666i q^{61} -15.3523 q^{63} +10.9297 q^{65} +4.09328i q^{67} +10.9395i q^{69} +11.8696 q^{71} +9.28531 q^{73} +5.10971i q^{75} +23.1243i q^{77} +2.42124 q^{79} -5.24441 q^{81} -0.987504i q^{83} +4.81614i q^{85} +21.6090 q^{87} +2.25389 q^{89} -16.1247i q^{91} +4.41944i q^{93} -2.63403 q^{95} +9.27281 q^{97} -23.5087i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 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

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