LMFDB - Embedded newform 8664.2.a.bs.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8664,2,Mod(1,8664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8664, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

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

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

chi := DirichletCharacter("8664.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8664 = 2^{3} \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8664.a (trivial)

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:	yes
Analytic conductor:	$69.1823883112$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 37 x^{10} + 52 x^{9} + 526 x^{8} - 414 x^{7} - 3501 x^{6} + 832 x^{5} + 10258 x^{4} + \cdots + 361 $ x^12 - 2x^11 - 37x^10 + 52x^9 + 526x^8 - 414x^7 - 3501x^6 + 832x^5 + 10258x^4 + 1506x^3 - 8617x^2 - 684*x + 361
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$3.82351$ of defining polynomial
Character	$\chi$	$=$	8664.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.82351 q^{5} +4.76113 q^{7} +1.00000 q^{9} -1.67399 q^{11} -1.58781 q^{13} +3.82351 q^{15} +0.353653 q^{17} +4.76113 q^{21} +7.42529 q^{23} +9.61922 q^{25} +1.00000 q^{27} +2.19764 q^{29} +5.77759 q^{31} -1.67399 q^{33} +18.2042 q^{35} -0.763352 q^{37} -1.58781 q^{39} -10.8125 q^{41} -11.9267 q^{43} +3.82351 q^{45} -7.31189 q^{47} +15.6683 q^{49} +0.353653 q^{51} +10.3100 q^{53} -6.40053 q^{55} +0.254293 q^{59} -4.97487 q^{61} +4.76113 q^{63} -6.07102 q^{65} -3.98961 q^{67} +7.42529 q^{69} -11.3801 q^{71} +15.5961 q^{73} +9.61922 q^{75} -7.97009 q^{77} -6.52354 q^{79} +1.00000 q^{81} +15.0766 q^{83} +1.35220 q^{85} +2.19764 q^{87} +8.03077 q^{89} -7.55979 q^{91} +5.77759 q^{93} -15.3927 q^{97} -1.67399 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 2 q^{5} + 4 q^{7} + 12 q^{9} + 14 q^{11} - 4 q^{13} + 2 q^{15} + 4 q^{21} + 26 q^{23} + 18 q^{25} + 12 q^{27} + 10 q^{29} - 8 q^{31} + 14 q^{33} + 30 q^{35} + 8 q^{37} - 4 q^{39} - 18 q^{41}+ \cdots + 14 q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 2 * q^5 + 4 * q^7 + 12 * q^9 + 14 * q^11 - 4 * q^13 + 2 * q^15 + 4 * q^21 + 26 * q^23 + 18 * q^25 + 12 * q^27 + 10 * q^29 - 8 * q^31 + 14 * q^33 + 30 * q^35 + 8 * q^37 - 4 * q^39 - 18 * q^41 - 8 * q^43 + 2 * q^45 + 48 * q^49 + 28 * q^53 + 2 * q^55 - 6 * q^59 + 38 * q^61 + 4 * q^63 - 24 * q^65 + 16 * q^67 + 26 * q^69 + 38 * q^71 + 28 * q^73 + 18 * q^75 + 4 * q^77 + 40 * q^79 + 12 * q^81 + 18 * q^83 + 54 * q^85 + 10 * q^87 - 24 * q^89 - 20 * q^91 - 8 * q^93 - 24 * q^97 + 14 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.82351	1.70993	0.854963	−	0.518690i	$-0.173580\pi$
$5$	3.82351	1.70993	0.854963	+	0.518690i	$0.173580\pi$
$6$	0	0
$7$	4.76113	1.79954	0.899769	−	0.436367i	$-0.143735\pi$
$7$	4.76113	1.79954	0.899769	+	0.436367i	$0.143735\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.67399	−0.504728	−0.252364	−	0.967632i	$-0.581208\pi$
$11$	−1.67399	−0.504728	−0.252364	+	0.967632i	$0.581208\pi$
$12$	0	0
$13$	−1.58781	−0.440380	−0.220190	−	0.975457i	$-0.570668\pi$
$13$	−1.58781	−0.440380	−0.220190	+	0.975457i	$0.570668\pi$
$14$	0	0
$15$	3.82351	0.987226
$16$	0	0
$17$	0.353653	0.0857736	0.0428868	−	0.999080i	$-0.486345\pi$
$17$	0.353653	0.0857736	0.0428868	+	0.999080i	$0.486345\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	4.76113	1.03896
$22$	0	0
$23$	7.42529	1.54828	0.774140	−	0.633015i	$-0.218183\pi$
$23$	7.42529	1.54828	0.774140	+	0.633015i	$0.218183\pi$
$24$	0	0
$25$	9.61922	1.92384
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.19764	0.408091	0.204045	−	0.978961i	$-0.434591\pi$
$29$	2.19764	0.408091	0.204045	+	0.978961i	$0.434591\pi$
$30$	0	0
$31$	5.77759	1.03769	0.518843	−	0.854870i	$-0.326363\pi$
$31$	5.77759	1.03769	0.518843	+	0.854870i	$0.326363\pi$
$32$	0	0
$33$	−1.67399	−0.291405
$34$	0	0
$35$	18.2042	3.07707
$36$	0	0
$37$	−0.763352	−0.125494	−0.0627471	−	0.998029i	$-0.519986\pi$
$37$	−0.763352	−0.125494	−0.0627471	+	0.998029i	$0.519986\pi$
$38$	0	0
$39$	−1.58781	−0.254254
$40$	0	0
$41$	−10.8125	−1.68863	−0.844316	−	0.535845i	$-0.819993\pi$
$41$	−10.8125	−1.68863	−0.844316	+	0.535845i	$0.819993\pi$
$42$	0	0
$43$	−11.9267	−1.81880	−0.909402	−	0.415917i	$-0.863461\pi$
$43$	−11.9267	−1.81880	−0.909402	+	0.415917i	$0.863461\pi$
$44$	0	0
$45$	3.82351	0.569975
$46$	0	0
$47$	−7.31189	−1.06655	−0.533275	−	0.845942i	$-0.679039\pi$
$47$	−7.31189	−1.06655	−0.533275	+	0.845942i	$0.679039\pi$
$48$	0	0
$49$	15.6683	2.23833
$50$	0	0
$51$	0.353653	0.0495214
$52$	0	0
$53$	10.3100	1.41618	0.708091	−	0.706121i	$-0.249557\pi$
$53$	10.3100	1.41618	0.708091	+	0.706121i	$0.249557\pi$
$54$	0	0
$55$	−6.40053	−0.863047
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.254293	0.0331061	0.0165530	−	0.999863i	$-0.494731\pi$
$59$	0.254293	0.0331061	0.0165530	+	0.999863i	$0.494731\pi$
$60$	0	0
$61$	−4.97487	−0.636967	−0.318483	−	0.947928i	$-0.603173\pi$
$61$	−4.97487	−0.636967	−0.318483	+	0.947928i	$0.603173\pi$
$62$	0	0
$63$	4.76113	0.599846
$64$	0	0
$65$	−6.07102	−0.753018
$66$	0	0
$67$	−3.98961	−0.487409	−0.243704	−	0.969850i	$-0.578363\pi$
$67$	−3.98961	−0.487409	−0.243704	+	0.969850i	$0.578363\pi$
$68$	0	0
$69$	7.42529	0.893900
$70$	0	0
$71$	−11.3801	−1.35057	−0.675285	−	0.737557i	$-0.735979\pi$
$71$	−11.3801	−1.35057	−0.675285	+	0.737557i	$0.735979\pi$
$72$	0	0
$73$	15.5961	1.82538	0.912692	−	0.408648i	$-0.134000\pi$
$73$	15.5961	1.82538	0.912692	+	0.408648i	$0.134000\pi$
$74$	0	0
$75$	9.61922	1.11073
$76$	0	0
$77$	−7.97009	−0.908277
$78$	0	0
$79$	−6.52354	−0.733955	−0.366978	−	0.930230i	$-0.619607\pi$
$79$	−6.52354	−0.733955	−0.366978	+	0.930230i	$0.619607\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.0766	1.65487	0.827434	−	0.561563i	$-0.189800\pi$
$83$	15.0766	1.65487	0.827434	+	0.561563i	$0.189800\pi$
$84$	0	0
$85$	1.35220	0.146666
$86$	0	0
$87$	2.19764	0.235611
$88$	0	0
$89$	8.03077	0.851260	0.425630	−	0.904897i	$-0.360052\pi$
$89$	8.03077	0.851260	0.425630	+	0.904897i	$0.360052\pi$
$90$	0	0
$91$	−7.55979	−0.792481
$92$	0	0
$93$	5.77759	0.599108
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.3927	−1.56289	−0.781446	−	0.623973i	$-0.785518\pi$
$97$	−15.3927	−1.56289	−0.781446	+	0.623973i	$0.785518\pi$
$98$	0	0
$99$	−1.67399	−0.168243
$100$	0	0
$101$	−9.79765	−0.974903	−0.487451	−	0.873150i	$-0.662073\pi$
$101$	−9.79765	−0.974903	−0.487451	+	0.873150i	$0.662073\pi$
$102$	0	0
$103$	14.4371	1.42253	0.711266	−	0.702923i	$-0.248122\pi$
$103$	14.4371	1.42253	0.711266	+	0.702923i	$0.248122\pi$
$104$	0	0
$105$	18.2042	1.77655
$106$	0	0
$107$	1.60927	0.155574	0.0777868	−	0.996970i	$-0.475215\pi$
$107$	1.60927	0.155574	0.0777868	+	0.996970i	$0.475215\pi$
$108$	0	0
$109$	−12.3011	−1.17824	−0.589118	−	0.808047i	$-0.700525\pi$
$109$	−12.3011	−1.17824	−0.589118	+	0.808047i	$0.700525\pi$
$110$	0	0
$111$	−0.763352	−0.0724541
$112$	0	0
$113$	−8.81149	−0.828915	−0.414457	−	0.910069i	$-0.636029\pi$
$113$	−8.81149	−0.828915	−0.414457	+	0.910069i	$0.636029\pi$
$114$	0	0
$115$	28.3907	2.64744
$116$	0	0
$117$	−1.58781	−0.146793
$118$	0	0
$119$	1.68379	0.154353
$120$	0	0
$121$	−8.19775	−0.745250
$122$	0	0
$123$	−10.8125	−0.974933
$124$	0	0
$125$	17.6616	1.57970
$126$	0	0
$127$	0.126292	0.0112066	0.00560332	−	0.999984i	$-0.498216\pi$
$127$	0.126292	0.0112066	0.00560332	+	0.999984i	$0.498216\pi$
$128$	0	0
$129$	−11.9267	−1.05009
$130$	0	0
$131$	−5.49654	−0.480235	−0.240117	−	0.970744i	$-0.577186\pi$
$131$	−5.49654	−0.480235	−0.240117	+	0.970744i	$0.577186\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.82351	0.329075
$136$	0	0
$137$	3.32189	0.283808	0.141904	−	0.989880i	$-0.454678\pi$
$137$	3.32189	0.283808	0.141904	+	0.989880i	$0.454678\pi$
$138$	0	0
$139$	13.3753	1.13448	0.567240	−	0.823552i	$-0.308011\pi$
$139$	13.3753	1.13448	0.567240	+	0.823552i	$0.308011\pi$
$140$	0	0
$141$	−7.31189	−0.615772
$142$	0	0
$143$	2.65799	0.222272
$144$	0	0
$145$	8.40268	0.697804
$146$	0	0
$147$	15.6683	1.29230
$148$	0	0
$149$	15.1388	1.24022	0.620109	−	0.784515i	$-0.287088\pi$
$149$	15.1388	1.24022	0.620109	+	0.784515i	$0.287088\pi$
$150$	0	0
$151$	−13.6459	−1.11049	−0.555245	−	0.831687i	$-0.687376\pi$
$151$	−13.6459	−1.11049	−0.555245	+	0.831687i	$0.687376\pi$
$152$	0	0
$153$	0.353653	0.0285912
$154$	0	0
$155$	22.0907	1.77437
$156$	0	0
$157$	5.30408	0.423311	0.211656	−	0.977344i	$-0.432114\pi$
$157$	5.30408	0.423311	0.211656	+	0.977344i	$0.432114\pi$
$158$	0	0
$159$	10.3100	0.817633
$160$	0	0
$161$	35.3528	2.78619
$162$	0	0
$163$	−1.63770	−0.128274	−0.0641371	−	0.997941i	$-0.520430\pi$
$163$	−1.63770	−0.128274	−0.0641371	+	0.997941i	$0.520430\pi$
$164$	0	0
$165$	−6.40053	−0.498280
$166$	0	0
$167$	5.82845	0.451019	0.225510	−	0.974241i	$-0.427595\pi$
$167$	5.82845	0.451019	0.225510	+	0.974241i	$0.427595\pi$
$168$	0	0
$169$	−10.4788	−0.806065
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.31759	0.176203	0.0881014	−	0.996112i	$-0.471920\pi$
$173$	2.31759	0.176203	0.0881014	+	0.996112i	$0.471920\pi$
$174$	0	0
$175$	45.7983	3.46203
$176$	0	0
$177$	0.254293	0.0191138
$178$	0	0
$179$	21.7635	1.62668	0.813341	−	0.581788i	$-0.197647\pi$
$179$	21.7635	1.62668	0.813341	+	0.581788i	$0.197647\pi$
$180$	0	0
$181$	19.3403	1.43755	0.718777	−	0.695241i	$-0.244702\pi$
$181$	19.3403	1.43755	0.718777	+	0.695241i	$0.244702\pi$
$182$	0	0
$183$	−4.97487	−0.367753
$184$	0	0
$185$	−2.91868	−0.214586
$186$	0	0
$187$	−0.592013	−0.0432923
$188$	0	0
$189$	4.76113	0.346321
$190$	0	0
$191$	−17.6975	−1.28055	−0.640273	−	0.768148i	$-0.721179\pi$
$191$	−17.6975	−1.28055	−0.640273	+	0.768148i	$0.721179\pi$
$192$	0	0
$193$	−26.7460	−1.92522	−0.962610	−	0.270892i	$-0.912681\pi$
$193$	−26.7460	−1.92522	−0.962610	+	0.270892i	$0.912681\pi$
$194$	0	0
$195$	−6.07102	−0.434755
$196$	0	0
$197$	−7.51553	−0.535460	−0.267730	−	0.963494i	$-0.586273\pi$
$197$	−7.51553	−0.535460	−0.267730	+	0.963494i	$0.586273\pi$
$198$	0	0
$199$	4.05209	0.287245	0.143623	−	0.989633i	$-0.454125\pi$
$199$	4.05209	0.287245	0.143623	+	0.989633i	$0.454125\pi$
$200$	0	0
$201$	−3.98961	−0.281406
$202$	0	0
$203$	10.4632	0.734374
$204$	0	0
$205$	−41.3418	−2.88744
$206$	0	0
$207$	7.42529	0.516093
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.0320	1.37906	0.689528	−	0.724259i	$-0.257818\pi$
$211$	20.0320	1.37906	0.689528	+	0.724259i	$0.257818\pi$
$212$	0	0
$213$	−11.3801	−0.779752
$214$	0	0
$215$	−45.6018	−3.11002
$216$	0	0
$217$	27.5079	1.86735
$218$	0	0
$219$	15.5961	1.05389
$220$	0	0
$221$	−0.561536	−0.0377730
$222$	0	0
$223$	4.32154	0.289392	0.144696	−	0.989476i	$-0.453780\pi$
$223$	4.32154	0.289392	0.144696	+	0.989476i	$0.453780\pi$
$224$	0	0
$225$	9.61922	0.641281
$226$	0	0
$227$	1.50250	0.0997247	0.0498623	−	0.998756i	$-0.484122\pi$
$227$	1.50250	0.0997247	0.0498623	+	0.998756i	$0.484122\pi$
$228$	0	0
$229$	0.345009	0.0227988	0.0113994	−	0.999935i	$-0.496371\pi$
$229$	0.345009	0.0227988	0.0113994	+	0.999935i	$0.496371\pi$
$230$	0	0
$231$	−7.97009	−0.524394
$232$	0	0
$233$	−10.5455	−0.690860	−0.345430	−	0.938444i	$-0.612267\pi$
$233$	−10.5455	−0.690860	−0.345430	+	0.938444i	$0.612267\pi$
$234$	0	0
$235$	−27.9571	−1.82372
$236$	0	0
$237$	−6.52354	−0.423749
$238$	0	0
$239$	−22.6744	−1.46668	−0.733342	−	0.679860i	$-0.762040\pi$
$239$	−22.6744	−1.46668	−0.733342	+	0.679860i	$0.762040\pi$
$240$	0	0
$241$	−27.9433	−1.79999	−0.899993	−	0.435904i	$-0.856429\pi$
$241$	−27.9433	−1.79999	−0.899993	+	0.435904i	$0.856429\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	59.9080	3.82738
$246$	0	0
$247$	0	0
$248$	0	0
$249$	15.0766	0.955438
$250$	0	0
$251$	12.3285	0.778171	0.389085	−	0.921202i	$-0.372791\pi$
$251$	12.3285	0.778171	0.389085	+	0.921202i	$0.372791\pi$
$252$	0	0
$253$	−12.4299	−0.781460
$254$	0	0
$255$	1.35220	0.0846779
$256$	0	0
$257$	20.0019	1.24769	0.623843	−	0.781550i	$-0.285570\pi$
$257$	20.0019	1.24769	0.623843	+	0.781550i	$0.285570\pi$
$258$	0	0
$259$	−3.63441	−0.225832
$260$	0	0
$261$	2.19764	0.136030
$262$	0	0
$263$	7.44697	0.459200	0.229600	−	0.973285i	$-0.426258\pi$
$263$	7.44697	0.459200	0.229600	+	0.973285i	$0.426258\pi$
$264$	0	0
$265$	39.4202	2.42157
$266$	0	0
$267$	8.03077	0.491475
$268$	0	0
$269$	−2.66000	−0.162183	−0.0810915	−	0.996707i	$-0.525841\pi$
$269$	−2.66000	−0.162183	−0.0810915	+	0.996707i	$0.525841\pi$
$270$	0	0
$271$	−16.0348	−0.974044	−0.487022	−	0.873390i	$-0.661917\pi$
$271$	−16.0348	−0.974044	−0.487022	+	0.873390i	$0.661917\pi$
$272$	0	0
$273$	−7.55979	−0.457539
$274$	0	0
$275$	−16.1025	−0.971018
$276$	0	0
$277$	−4.58411	−0.275433	−0.137716	−	0.990472i	$-0.543976\pi$
$277$	−4.58411	−0.275433	−0.137716	+	0.990472i	$0.543976\pi$
$278$	0	0
$279$	5.77759	0.345895
$280$	0	0
$281$	−0.402371	−0.0240034	−0.0120017	−	0.999928i	$-0.503820\pi$
$281$	−0.402371	−0.0240034	−0.0120017	+	0.999928i	$0.503820\pi$
$282$	0	0
$283$	3.96514	0.235703	0.117852	−	0.993031i	$-0.462399\pi$
$283$	3.96514	0.235703	0.117852	+	0.993031i	$0.462399\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−51.4798	−3.03876
$288$	0	0
$289$	−16.8749	−0.992643
$290$	0	0
$291$	−15.3927	−0.902336
$292$	0	0
$293$	−1.54649	−0.0903470	−0.0451735	−	0.998979i	$-0.514384\pi$
$293$	−1.54649	−0.0903470	−0.0451735	+	0.998979i	$0.514384\pi$
$294$	0	0
$295$	0.972290	0.0566089
$296$	0	0
$297$	−1.67399	−0.0971349
$298$	0	0
$299$	−11.7900	−0.681832
$300$	0	0
$301$	−56.7846	−3.27301
$302$	0	0
$303$	−9.79765	−0.562861
$304$	0	0
$305$	−19.0215	−1.08917
$306$	0	0
$307$	21.9797	1.25445	0.627225	−	0.778838i	$-0.284191\pi$
$307$	21.9797	1.25445	0.627225	+	0.778838i	$0.284191\pi$
$308$	0	0
$309$	14.4371	0.821300
$310$	0	0
$311$	−16.2762	−0.922939	−0.461470	−	0.887156i	$-0.652678\pi$
$311$	−16.2762	−0.922939	−0.461470	+	0.887156i	$0.652678\pi$
$312$	0	0
$313$	−5.24925	−0.296705	−0.148352	−	0.988935i	$-0.547397\pi$
$313$	−5.24925	−0.296705	−0.148352	+	0.988935i	$0.547397\pi$
$314$	0	0
$315$	18.2042	1.02569
$316$	0	0
$317$	−22.0068	−1.23603	−0.618014	−	0.786167i	$-0.712062\pi$
$317$	−22.0068	−1.23603	−0.618014	+	0.786167i	$0.712062\pi$
$318$	0	0
$319$	−3.67883	−0.205975
$320$	0	0
$321$	1.60927	0.0898205
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−15.2735	−0.847223
$326$	0	0
$327$	−12.3011	−0.680255
$328$	0	0
$329$	−34.8129	−1.91929
$330$	0	0
$331$	13.9254	0.765408	0.382704	−	0.923871i	$-0.374993\pi$
$331$	13.9254	0.765408	0.382704	+	0.923871i	$0.374993\pi$
$332$	0	0
$333$	−0.763352	−0.0418314
$334$	0	0
$335$	−15.2543	−0.833432
$336$	0	0
$337$	−20.5738	−1.12073	−0.560364	−	0.828246i	$-0.689339\pi$
$337$	−20.5738	−1.12073	−0.560364	+	0.828246i	$0.689339\pi$
$338$	0	0
$339$	−8.81149	−0.478574
$340$	0	0
$341$	−9.67165	−0.523749
$342$	0	0
$343$	41.2711	2.22843
$344$	0	0
$345$	28.3907	1.52850
$346$	0	0
$347$	18.9266	1.01603	0.508017	−	0.861347i	$-0.330379\pi$
$347$	18.9266	1.01603	0.508017	+	0.861347i	$0.330379\pi$
$348$	0	0
$349$	24.4871	1.31077	0.655383	−	0.755297i	$-0.272507\pi$
$349$	24.4871	1.31077	0.655383	+	0.755297i	$0.272507\pi$
$350$	0	0
$351$	−1.58781	−0.0847512
$352$	0	0
$353$	−2.85109	−0.151748	−0.0758742	−	0.997117i	$-0.524175\pi$
$353$	−2.85109	−0.151748	−0.0758742	+	0.997117i	$0.524175\pi$
$354$	0	0
$355$	−43.5119	−2.30937
$356$	0	0
$357$	1.68379	0.0891156
$358$	0	0
$359$	−7.99965	−0.422205	−0.211103	−	0.977464i	$-0.567705\pi$
$359$	−7.99965	−0.422205	−0.211103	+	0.977464i	$0.567705\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−8.19775	−0.430270
$364$	0	0
$365$	59.6318	3.12127
$366$	0	0
$367$	−16.9866	−0.886696	−0.443348	−	0.896350i	$-0.646209\pi$
$367$	−16.9866	−0.886696	−0.443348	+	0.896350i	$0.646209\pi$
$368$	0	0
$369$	−10.8125	−0.562878
$370$	0	0
$371$	49.0871	2.54847
$372$	0	0
$373$	−37.3061	−1.93164	−0.965818	−	0.259221i	$-0.916534\pi$
$373$	−37.3061	−1.93164	−0.965818	+	0.259221i	$0.916534\pi$
$374$	0	0
$375$	17.6616	0.912043
$376$	0	0
$377$	−3.48944	−0.179715
$378$	0	0
$379$	12.7303	0.653913	0.326957	−	0.945039i	$-0.393977\pi$
$379$	12.7303	0.653913	0.326957	+	0.945039i	$0.393977\pi$
$380$	0	0
$381$	0.126292	0.00647016
$382$	0	0
$383$	22.0194	1.12514	0.562568	−	0.826751i	$-0.309813\pi$
$383$	22.0194	1.12514	0.562568	+	0.826751i	$0.309813\pi$
$384$	0	0
$385$	−30.4737	−1.55309
$386$	0	0
$387$	−11.9267	−0.606268
$388$	0	0
$389$	−29.3288	−1.48703	−0.743515	−	0.668719i	$-0.766843\pi$
$389$	−29.3288	−1.48703	−0.743515	+	0.668719i	$0.766843\pi$
$390$	0	0
$391$	2.62598	0.132801
$392$	0	0
$393$	−5.49654	−0.277264
$394$	0	0
$395$	−24.9428	−1.25501
$396$	0	0
$397$	−16.1016	−0.808117	−0.404058	−	0.914733i	$-0.632401\pi$
$397$	−16.1016	−0.808117	−0.404058	+	0.914733i	$0.632401\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−34.6674	−1.73121	−0.865604	−	0.500729i	$-0.833065\pi$
$401$	−34.6674	−1.73121	−0.865604	+	0.500729i	$0.833065\pi$
$402$	0	0
$403$	−9.17374	−0.456977
$404$	0	0
$405$	3.82351	0.189992
$406$	0	0
$407$	1.27785	0.0633404
$408$	0	0
$409$	2.50498	0.123863	0.0619317	−	0.998080i	$-0.480274\pi$
$409$	2.50498	0.123863	0.0619317	+	0.998080i	$0.480274\pi$
$410$	0	0
$411$	3.32189	0.163857
$412$	0	0
$413$	1.21072	0.0595756
$414$	0	0
$415$	57.6454	2.82970
$416$	0	0
$417$	13.3753	0.654993
$418$	0	0
$419$	−10.0915	−0.493004	−0.246502	−	0.969142i	$-0.579281\pi$
$419$	−10.0915	−0.493004	−0.246502	+	0.969142i	$0.579281\pi$
$420$	0	0
$421$	22.2877	1.08624	0.543119	−	0.839656i	$-0.317243\pi$
$421$	22.2877	1.08624	0.543119	+	0.839656i	$0.317243\pi$
$422$	0	0
$423$	−7.31189	−0.355516
$424$	0	0
$425$	3.40187	0.165015
$426$	0	0
$427$	−23.6860	−1.14625
$428$	0	0
$429$	2.65799	0.128329
$430$	0	0
$431$	6.80359	0.327718	0.163859	−	0.986484i	$-0.447606\pi$
$431$	6.80359	0.327718	0.163859	+	0.986484i	$0.447606\pi$
$432$	0	0
$433$	−5.28533	−0.253997	−0.126998	−	0.991903i	$-0.540534\pi$
$433$	−5.28533	−0.253997	−0.126998	+	0.991903i	$0.540534\pi$
$434$	0	0
$435$	8.40268	0.402878
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−17.2492	−0.823258	−0.411629	−	0.911351i	$-0.635040\pi$
$439$	−17.2492	−0.823258	−0.411629	+	0.911351i	$0.635040\pi$
$440$	0	0
$441$	15.6683	0.746112
$442$	0	0
$443$	14.0426	0.667185	0.333593	−	0.942717i	$-0.391739\pi$
$443$	14.0426	0.667185	0.333593	+	0.942717i	$0.391739\pi$
$444$	0	0
$445$	30.7057	1.45559
$446$	0	0
$447$	15.1388	0.716041
$448$	0	0
$449$	25.2243	1.19041	0.595205	−	0.803574i	$-0.297071\pi$
$449$	25.2243	1.19041	0.595205	+	0.803574i	$0.297071\pi$
$450$	0	0
$451$	18.1001	0.852300
$452$	0	0
$453$	−13.6459	−0.641142
$454$	0	0
$455$	−28.9049	−1.35508
$456$	0	0
$457$	−10.5402	−0.493050	−0.246525	−	0.969136i	$-0.579289\pi$
$457$	−10.5402	−0.493050	−0.246525	+	0.969136i	$0.579289\pi$
$458$	0	0
$459$	0.353653	0.0165071
$460$	0	0
$461$	17.0615	0.794635	0.397317	−	0.917681i	$-0.369941\pi$
$461$	17.0615	0.794635	0.397317	+	0.917681i	$0.369941\pi$
$462$	0	0
$463$	−4.60101	−0.213827	−0.106913	−	0.994268i	$-0.534097\pi$
$463$	−4.60101	−0.213827	−0.106913	+	0.994268i	$0.534097\pi$
$464$	0	0
$465$	22.0907	1.02443
$466$	0	0
$467$	21.5496	0.997196	0.498598	−	0.866833i	$-0.333848\pi$
$467$	21.5496	0.997196	0.498598	+	0.866833i	$0.333848\pi$
$468$	0	0
$469$	−18.9951	−0.877110
$470$	0	0
$471$	5.30408	0.244399
$472$	0	0
$473$	19.9652	0.918001
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.3100	0.472061
$478$	0	0
$479$	−32.4671	−1.48346	−0.741730	−	0.670699i	$-0.765994\pi$
$479$	−32.4671	−1.48346	−0.741730	+	0.670699i	$0.765994\pi$
$480$	0	0
$481$	1.21206	0.0552652
$482$	0	0
$483$	35.3528	1.60861
$484$	0	0
$485$	−58.8541	−2.67243
$486$	0	0
$487$	8.23919	0.373353	0.186677	−	0.982421i	$-0.440228\pi$
$487$	8.23919	0.373353	0.186677	+	0.982421i	$0.440228\pi$
$488$	0	0
$489$	−1.63770	−0.0740592
$490$	0	0
$491$	5.93224	0.267718	0.133859	−	0.991000i	$-0.457263\pi$
$491$	5.93224	0.267718	0.133859	+	0.991000i	$0.457263\pi$
$492$	0	0
$493$	0.777201	0.0350034
$494$	0	0
$495$	−6.40053	−0.287682
$496$	0	0
$497$	−54.1821	−2.43040
$498$	0	0
$499$	21.0359	0.941698	0.470849	−	0.882214i	$-0.343948\pi$
$499$	21.0359	0.941698	0.470849	+	0.882214i	$0.343948\pi$
$500$	0	0
$501$	5.82845	0.260396
$502$	0	0
$503$	2.11646	0.0943685	0.0471842	−	0.998886i	$-0.484975\pi$
$503$	2.11646	0.0943685	0.0471842	+	0.998886i	$0.484975\pi$
$504$	0	0
$505$	−37.4614	−1.66701
$506$	0	0
$507$	−10.4788	−0.465382
$508$	0	0
$509$	−25.1963	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$509$	−25.1963	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$510$	0	0
$511$	74.2550	3.28485
$512$	0	0
$513$	0	0
$514$	0	0
$515$	55.2005	2.43242
$516$	0	0
$517$	12.2401	0.538317
$518$	0	0
$519$	2.31759	0.101731
$520$	0	0
$521$	11.5814	0.507392	0.253696	−	0.967284i	$-0.418354\pi$
$521$	11.5814	0.507392	0.253696	+	0.967284i	$0.418354\pi$
$522$	0	0
$523$	−8.30995	−0.363369	−0.181684	−	0.983357i	$-0.558155\pi$
$523$	−8.30995	−0.363369	−0.181684	+	0.983357i	$0.558155\pi$
$524$	0	0
$525$	45.7983	1.99880
$526$	0	0
$527$	2.04326	0.0890060
$528$	0	0
$529$	32.1349	1.39717
$530$	0	0
$531$	0.254293	0.0110354
$532$	0	0
$533$	17.1683	0.743641
$534$	0	0
$535$	6.15305	0.266019
$536$	0	0
$537$	21.7635	0.939165
$538$	0	0
$539$	−26.2287	−1.12975
$540$	0	0
$541$	−14.3093	−0.615206	−0.307603	−	0.951515i	$-0.599527\pi$
$541$	−14.3093	−0.615206	−0.307603	+	0.951515i	$0.599527\pi$
$542$	0	0
$543$	19.3403	0.829972
$544$	0	0
$545$	−47.0335	−2.01470
$546$	0	0
$547$	20.0791	0.858522	0.429261	−	0.903181i	$-0.358774\pi$
$547$	20.0791	0.858522	0.429261	+	0.903181i	$0.358774\pi$
$548$	0	0
$549$	−4.97487	−0.212322
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−31.0594	−1.32078
$554$	0	0
$555$	−2.91868	−0.123891
$556$	0	0
$557$	2.17897	0.0923257	0.0461629	−	0.998934i	$-0.485301\pi$
$557$	2.17897	0.0923257	0.0461629	+	0.998934i	$0.485301\pi$
$558$	0	0
$559$	18.9374	0.800966
$560$	0	0
$561$	−0.592013	−0.0249948
$562$	0	0
$563$	19.1144	0.805574	0.402787	−	0.915294i	$-0.368041\pi$
$563$	19.1144	0.805574	0.402787	+	0.915294i	$0.368041\pi$
$564$	0	0
$565$	−33.6908	−1.41738
$566$	0	0
$567$	4.76113	0.199949
$568$	0	0
$569$	25.0809	1.05144	0.525722	−	0.850656i	$-0.323795\pi$
$569$	25.0809	1.05144	0.525722	+	0.850656i	$0.323795\pi$
$570$	0	0
$571$	9.17104	0.383796	0.191898	−	0.981415i	$-0.438536\pi$
$571$	9.17104	0.383796	0.191898	+	0.981415i	$0.438536\pi$
$572$	0	0
$573$	−17.6975	−0.739323
$574$	0	0
$575$	71.4255	2.97865
$576$	0	0
$577$	38.0492	1.58401	0.792004	−	0.610515i	$-0.209038\pi$
$577$	38.0492	1.58401	0.792004	+	0.610515i	$0.209038\pi$
$578$	0	0
$579$	−26.7460	−1.11153
$580$	0	0
$581$	71.7814	2.97800
$582$	0	0
$583$	−17.2588	−0.714787
$584$	0	0
$585$	−6.07102	−0.251006
$586$	0	0
$587$	−17.6209	−0.727291	−0.363645	−	0.931537i	$-0.618468\pi$
$587$	−17.6209	−0.727291	−0.363645	+	0.931537i	$0.618468\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−7.51553	−0.309148
$592$	0	0
$593$	6.71083	0.275581	0.137790	−	0.990461i	$-0.456000\pi$
$593$	6.71083	0.275581	0.137790	+	0.990461i	$0.456000\pi$
$594$	0	0
$595$	6.43798	0.263932
$596$	0	0
$597$	4.05209	0.165841
$598$	0	0
$599$	−13.5990	−0.555639	−0.277819	−	0.960633i	$-0.589612\pi$
$599$	−13.5990	−0.555639	−0.277819	+	0.960633i	$0.589612\pi$
$600$	0	0
$601$	−18.6352	−0.760147	−0.380073	−	0.924956i	$-0.624101\pi$
$601$	−18.6352	−0.760147	−0.380073	+	0.924956i	$0.624101\pi$
$602$	0	0
$603$	−3.98961	−0.162470
$604$	0	0
$605$	−31.3442	−1.27432
$606$	0	0
$607$	−31.8428	−1.29246	−0.646229	−	0.763143i	$-0.723655\pi$
$607$	−31.8428	−1.29246	−0.646229	+	0.763143i	$0.723655\pi$
$608$	0	0
$609$	10.4632	0.423991
$610$	0	0
$611$	11.6099	0.469687
$612$	0	0
$613$	−25.2112	−1.01827	−0.509136	−	0.860686i	$-0.670035\pi$
$613$	−25.2112	−1.01827	−0.509136	+	0.860686i	$0.670035\pi$
$614$	0	0
$615$	−41.3418	−1.66706
$616$	0	0
$617$	7.93215	0.319336	0.159668	−	0.987171i	$-0.448958\pi$
$617$	7.93215	0.319336	0.159668	+	0.987171i	$0.448958\pi$
$618$	0	0
$619$	−37.5519	−1.50934	−0.754669	−	0.656106i	$-0.772203\pi$
$619$	−37.5519	−1.50934	−0.754669	+	0.656106i	$0.772203\pi$
$620$	0	0
$621$	7.42529	0.297967
$622$	0	0
$623$	38.2355	1.53187
$624$	0	0
$625$	19.4333	0.777333
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.269962	−0.0107641
$630$	0	0
$631$	19.1872	0.763831	0.381915	−	0.924197i	$-0.375265\pi$
$631$	19.1872	0.763831	0.381915	+	0.924197i	$0.375265\pi$
$632$	0	0
$633$	20.0320	0.796199
$634$	0	0
$635$	0.482880	0.0191625
$636$	0	0
$637$	−24.8784	−0.985719
$638$	0	0
$639$	−11.3801	−0.450190
$640$	0	0
$641$	32.2608	1.27422	0.637112	−	0.770771i	$-0.280129\pi$
$641$	32.2608	1.27422	0.637112	+	0.770771i	$0.280129\pi$
$642$	0	0
$643$	−1.90975	−0.0753130	−0.0376565	−	0.999291i	$-0.511989\pi$
$643$	−1.90975	−0.0753130	−0.0376565	+	0.999291i	$0.511989\pi$
$644$	0	0
$645$	−45.6018	−1.79557
$646$	0	0
$647$	2.81194	0.110549	0.0552744	−	0.998471i	$-0.482397\pi$
$647$	2.81194	0.110549	0.0552744	+	0.998471i	$0.482397\pi$
$648$	0	0
$649$	−0.425684	−0.0167096
$650$	0	0
$651$	27.5079	1.07812
$652$	0	0
$653$	25.8339	1.01096	0.505479	−	0.862839i	$-0.331316\pi$
$653$	25.8339	1.01096	0.505479	+	0.862839i	$0.331316\pi$
$654$	0	0
$655$	−21.0161	−0.821166
$656$	0	0
$657$	15.5961	0.608461
$658$	0	0
$659$	−19.4189	−0.756455	−0.378227	−	0.925713i	$-0.623466\pi$
$659$	−19.4189	−0.756455	−0.378227	+	0.925713i	$0.623466\pi$
$660$	0	0
$661$	1.99501	0.0775967	0.0387984	−	0.999247i	$-0.487647\pi$
$661$	1.99501	0.0775967	0.0387984	+	0.999247i	$0.487647\pi$
$662$	0	0
$663$	−0.561536	−0.0218082
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.3181	0.631838
$668$	0	0
$669$	4.32154	0.167081
$670$	0	0
$671$	8.32789	0.321495
$672$	0	0
$673$	−37.1665	−1.43266	−0.716331	−	0.697761i	$-0.754180\pi$
$673$	−37.1665	−1.43266	−0.716331	+	0.697761i	$0.754180\pi$
$674$	0	0
$675$	9.61922	0.370244
$676$	0	0
$677$	26.1933	1.00669	0.503345	−	0.864086i	$-0.332102\pi$
$677$	26.1933	1.00669	0.503345	+	0.864086i	$0.332102\pi$
$678$	0	0
$679$	−73.2866	−2.81248
$680$	0	0
$681$	1.50250	0.0575761
$682$	0	0
$683$	−12.7168	−0.486596	−0.243298	−	0.969952i	$-0.578229\pi$
$683$	−12.7168	−0.486596	−0.243298	+	0.969952i	$0.578229\pi$
$684$	0	0
$685$	12.7013	0.485291
$686$	0	0
$687$	0.345009	0.0131629
$688$	0	0
$689$	−16.3703	−0.623659
$690$	0	0
$691$	33.5596	1.27667	0.638333	−	0.769760i	$-0.279624\pi$
$691$	33.5596	1.27667	0.638333	+	0.769760i	$0.279624\pi$
$692$	0	0
$693$	−7.97009	−0.302759
$694$	0	0
$695$	51.1407	1.93988
$696$	0	0
$697$	−3.82389	−0.144840
$698$	0	0
$699$	−10.5455	−0.398868
$700$	0	0
$701$	−28.1443	−1.06300	−0.531498	−	0.847059i	$-0.678371\pi$
$701$	−28.1443	−1.06300	−0.531498	+	0.847059i	$0.678371\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−27.9571	−1.05292
$706$	0	0
$707$	−46.6479	−1.75437
$708$	0	0
$709$	7.48684	0.281174	0.140587	−	0.990068i	$-0.455101\pi$
$709$	7.48684	0.281174	0.140587	+	0.990068i	$0.455101\pi$
$710$	0	0
$711$	−6.52354	−0.244652
$712$	0	0
$713$	42.9003	1.60663
$714$	0	0
$715$	10.1628	0.380069
$716$	0	0
$717$	−22.6744	−0.846790
$718$	0	0
$719$	32.2294	1.20195	0.600976	−	0.799267i	$-0.294779\pi$
$719$	32.2294	1.20195	0.600976	+	0.799267i	$0.294779\pi$
$720$	0	0
$721$	68.7370	2.55990
$722$	0	0
$723$	−27.9433	−1.03922
$724$	0	0
$725$	21.1395	0.785103
$726$	0	0
$727$	8.16204	0.302713	0.151357	−	0.988479i	$-0.451636\pi$
$727$	8.16204	0.302713	0.151357	+	0.988479i	$0.451636\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.21792	−0.156005
$732$	0	0
$733$	28.3944	1.04877	0.524386	−	0.851481i	$-0.324295\pi$
$733$	28.3944	1.04877	0.524386	+	0.851481i	$0.324295\pi$
$734$	0	0
$735$	59.9080	2.20974
$736$	0	0
$737$	6.67858	0.246009
$738$	0	0
$739$	2.91255	0.107140	0.0535699	−	0.998564i	$-0.482940\pi$
$739$	2.91255	0.107140	0.0535699	+	0.998564i	$0.482940\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0.622047	0.0228207	0.0114104	−	0.999935i	$-0.496368\pi$
$743$	0.622047	0.0228207	0.0114104	+	0.999935i	$0.496368\pi$
$744$	0	0
$745$	57.8833	2.12068
$746$	0	0
$747$	15.0766	0.551622
$748$	0	0
$749$	7.66192	0.279961
$750$	0	0
$751$	−1.60410	−0.0585343	−0.0292671	−	0.999572i	$-0.509317\pi$
$751$	−1.60410	−0.0585343	−0.0292671	+	0.999572i	$0.509317\pi$
$752$	0	0
$753$	12.3285	0.449277
$754$	0	0
$755$	−52.1754	−1.89886
$756$	0	0
$757$	13.7349	0.499202	0.249601	−	0.968349i	$-0.419701\pi$
$757$	13.7349	0.499202	0.249601	+	0.968349i	$0.419701\pi$
$758$	0	0
$759$	−12.4299	−0.451176
$760$	0	0
$761$	9.67261	0.350632	0.175316	−	0.984512i	$-0.443905\pi$
$761$	9.67261	0.350632	0.175316	+	0.984512i	$0.443905\pi$
$762$	0	0
$763$	−58.5673	−2.12028
$764$	0	0
$765$	1.35220	0.0488888
$766$	0	0
$767$	−0.403769	−0.0145793
$768$	0	0
$769$	−13.6514	−0.492281	−0.246140	−	0.969234i	$-0.579162\pi$
$769$	−13.6514	−0.492281	−0.246140	+	0.969234i	$0.579162\pi$
$770$	0	0
$771$	20.0019	0.720352
$772$	0	0
$773$	26.0616	0.937371	0.468685	−	0.883365i	$-0.344728\pi$
$773$	26.0616	0.937371	0.468685	+	0.883365i	$0.344728\pi$
$774$	0	0
$775$	55.5759	1.99635
$776$	0	0
$777$	−3.63441	−0.130384
$778$	0	0
$779$	0	0
$780$	0	0
$781$	19.0502	0.681670
$782$	0	0
$783$	2.19764	0.0785371
$784$	0	0
$785$	20.2802	0.723831
$786$	0	0
$787$	−31.0347	−1.10627	−0.553134	−	0.833092i	$-0.686568\pi$
$787$	−31.0347	−1.10627	−0.553134	+	0.833092i	$0.686568\pi$
$788$	0	0
$789$	7.44697	0.265119
$790$	0	0
$791$	−41.9526	−1.49166
$792$	0	0
$793$	7.89917	0.280508
$794$	0	0
$795$	39.4202	1.39809
$796$	0	0
$797$	−0.191379	−0.00677899	−0.00338950	−	0.999994i	$-0.501079\pi$
$797$	−0.191379	−0.00677899	−0.00338950	+	0.999994i	$0.501079\pi$
$798$	0	0
$799$	−2.58588	−0.0914817
$800$	0	0
$801$	8.03077	0.283753
$802$	0	0
$803$	−26.1077	−0.921322
$804$	0	0
$805$	135.172	4.76417
$806$	0	0
$807$	−2.66000	−0.0936364
$808$	0	0
$809$	−1.87416	−0.0658919	−0.0329460	−	0.999457i	$-0.510489\pi$
$809$	−1.87416	−0.0658919	−0.0329460	+	0.999457i	$0.510489\pi$
$810$	0	0
$811$	30.1680	1.05934	0.529672	−	0.848203i	$-0.322315\pi$
$811$	30.1680	1.05934	0.529672	+	0.848203i	$0.322315\pi$
$812$	0	0
$813$	−16.0348	−0.562365
$814$	0	0
$815$	−6.26175	−0.219339
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−7.55979	−0.264160
$820$	0	0
$821$	−23.2065	−0.809912	−0.404956	−	0.914336i	$-0.632713\pi$
$821$	−23.2065	−0.809912	−0.404956	+	0.914336i	$0.632713\pi$
$822$	0	0
$823$	−13.6392	−0.475433	−0.237716	−	0.971335i	$-0.576399\pi$
$823$	−13.6392	−0.475433	−0.237716	+	0.971335i	$0.576399\pi$
$824$	0	0
$825$	−16.1025	−0.560617
$826$	0	0
$827$	45.8336	1.59379	0.796895	−	0.604118i	$-0.206475\pi$
$827$	45.8336	1.59379	0.796895	+	0.604118i	$0.206475\pi$
$828$	0	0
$829$	−5.91274	−0.205358	−0.102679	−	0.994715i	$-0.532741\pi$
$829$	−5.91274	−0.205358	−0.102679	+	0.994715i	$0.532741\pi$
$830$	0	0
$831$	−4.58411	−0.159021
$832$	0	0
$833$	5.54116	0.191990
$834$	0	0
$835$	22.2851	0.771209
$836$	0	0
$837$	5.77759	0.199703
$838$	0	0
$839$	19.7781	0.682817	0.341408	−	0.939915i	$-0.389096\pi$
$839$	19.7781	0.682817	0.341408	+	0.939915i	$0.389096\pi$
$840$	0	0
$841$	−24.1704	−0.833462
$842$	0	0
$843$	−0.402371	−0.0138584
$844$	0	0
$845$	−40.0660	−1.37831
$846$	0	0
$847$	−39.0305	−1.34110
$848$	0	0
$849$	3.96514	0.136083
$850$	0	0
$851$	−5.66811	−0.194300
$852$	0	0
$853$	−1.89746	−0.0649679	−0.0324839	−	0.999472i	$-0.510342\pi$
$853$	−1.89746	−0.0649679	−0.0324839	+	0.999472i	$0.510342\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.7599	1.56313	0.781565	−	0.623824i	$-0.214422\pi$
$857$	45.7599	1.56313	0.781565	+	0.623824i	$0.214422\pi$
$858$	0	0
$859$	−46.7021	−1.59345	−0.796727	−	0.604340i	$-0.793437\pi$
$859$	−46.7021	−1.59345	−0.796727	+	0.604340i	$0.793437\pi$
$860$	0	0
$861$	−51.4798	−1.75443
$862$	0	0
$863$	14.0945	0.479781	0.239890	−	0.970800i	$-0.422888\pi$
$863$	14.0945	0.479781	0.239890	+	0.970800i	$0.422888\pi$
$864$	0	0
$865$	8.86131	0.301294
$866$	0	0
$867$	−16.8749	−0.573103
$868$	0	0
$869$	10.9204	0.370448
$870$	0	0
$871$	6.33476	0.214645
$872$	0	0
$873$	−15.3927	−0.520964
$874$	0	0
$875$	84.0893	2.84274
$876$	0	0
$877$	−2.38775	−0.0806286	−0.0403143	−	0.999187i	$-0.512836\pi$
$877$	−2.38775	−0.0806286	−0.0403143	+	0.999187i	$0.512836\pi$
$878$	0	0
$879$	−1.54649	−0.0521619
$880$	0	0
$881$	6.15032	0.207210	0.103605	−	0.994619i	$-0.466962\pi$
$881$	6.15032	0.207210	0.103605	+	0.994619i	$0.466962\pi$
$882$	0	0
$883$	41.2923	1.38960	0.694798	−	0.719205i	$-0.255494\pi$
$883$	41.2923	1.38960	0.694798	+	0.719205i	$0.255494\pi$
$884$	0	0
$885$	0.972290	0.0326832
$886$	0	0
$887$	52.6453	1.76766	0.883829	−	0.467811i	$-0.154957\pi$
$887$	52.6453	1.76766	0.883829	+	0.467811i	$0.154957\pi$
$888$	0	0
$889$	0.601295	0.0201668
$890$	0	0
$891$	−1.67399	−0.0560809
$892$	0	0
$893$	0	0
$894$	0	0
$895$	83.2130	2.78150
$896$	0	0
$897$	−11.7900	−0.393656
$898$	0	0
$899$	12.6970	0.423470
$900$	0	0
$901$	3.64615	0.121471
$902$	0	0
$903$	−56.7846	−1.88967
$904$	0	0
$905$	73.9479	2.45811
$906$	0	0
$907$	−0.306681	−0.0101832	−0.00509159	−	0.999987i	$-0.501621\pi$
$907$	−0.306681	−0.0101832	−0.00509159	+	0.999987i	$0.501621\pi$
$908$	0	0
$909$	−9.79765	−0.324968
$910$	0	0
$911$	13.7960	0.457081	0.228540	−	0.973534i	$-0.426605\pi$
$911$	13.7960	0.457081	0.228540	+	0.973534i	$0.426605\pi$
$912$	0	0
$913$	−25.2381	−0.835258
$914$	0	0
$915$	−19.0215	−0.628830
$916$	0	0
$917$	−26.1697	−0.864200
$918$	0	0
$919$	−15.9866	−0.527348	−0.263674	−	0.964612i	$-0.584934\pi$
$919$	−15.9866	−0.527348	−0.263674	+	0.964612i	$0.584934\pi$
$920$	0	0
$921$	21.9797	0.724257
$922$	0	0
$923$	18.0695	0.594764
$924$	0	0
$925$	−7.34285	−0.241431
$926$	0	0
$927$	14.4371	0.474178
$928$	0	0
$929$	−16.2033	−0.531614	−0.265807	−	0.964026i	$-0.585638\pi$
$929$	−16.2033	−0.531614	−0.265807	+	0.964026i	$0.585638\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−16.2762	−0.532859
$934$	0	0
$935$	−2.26357	−0.0740266
$936$	0	0
$937$	3.01980	0.0986526	0.0493263	−	0.998783i	$-0.484293\pi$
$937$	3.01980	0.0986526	0.0493263	+	0.998783i	$0.484293\pi$
$938$	0	0
$939$	−5.24925	−0.171303
$940$	0	0
$941$	−23.6675	−0.771537	−0.385769	−	0.922596i	$-0.626064\pi$
$941$	−23.6675	−0.771537	−0.385769	+	0.922596i	$0.626064\pi$
$942$	0	0
$943$	−80.2861	−2.61448
$944$	0	0
$945$	18.2042	0.592183
$946$	0	0
$947$	24.2901	0.789323	0.394662	−	0.918827i	$-0.370862\pi$
$947$	24.2901	0.789323	0.394662	+	0.918827i	$0.370862\pi$
$948$	0	0
$949$	−24.7637	−0.803863
$950$	0	0
$951$	−22.0068	−0.713621
$952$	0	0
$953$	−55.6335	−1.80214	−0.901072	−	0.433669i	$-0.857219\pi$
$953$	−55.6335	−1.80214	−0.901072	+	0.433669i	$0.857219\pi$
$954$	0	0
$955$	−67.6665	−2.18964
$956$	0	0
$957$	−3.67883	−0.118920
$958$	0	0
$959$	15.8159	0.510723
$960$	0	0
$961$	2.38056	0.0767921
$962$	0	0
$963$	1.60927	0.0518579
$964$	0	0
$965$	−102.264	−3.29198
$966$	0	0
$967$	40.3589	1.29786	0.648928	−	0.760850i	$-0.275218\pi$
$967$	40.3589	1.29786	0.648928	+	0.760850i	$0.275218\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−53.0229	−1.70159	−0.850793	−	0.525501i	$-0.823878\pi$
$971$	−53.0229	−1.70159	−0.850793	+	0.525501i	$0.823878\pi$
$972$	0	0
$973$	63.6817	2.04154
$974$	0	0
$975$	−15.2735	−0.489145
$976$	0	0
$977$	10.4167	0.333260	0.166630	−	0.986020i	$-0.446712\pi$
$977$	10.4167	0.333260	0.166630	+	0.986020i	$0.446712\pi$
$978$	0	0
$979$	−13.4435	−0.429655
$980$	0	0
$981$	−12.3011	−0.392745
$982$	0	0
$983$	47.8416	1.52591	0.762955	−	0.646452i	$-0.223748\pi$
$983$	47.8416	1.52591	0.762955	+	0.646452i	$0.223748\pi$
$984$	0	0
$985$	−28.7357	−0.915596
$986$	0	0
$987$	−34.8129	−1.10811
$988$	0	0
$989$	−88.5592	−2.81602
$990$	0	0
$991$	−41.6088	−1.32175	−0.660873	−	0.750497i	$-0.729814\pi$
$991$	−41.6088	−1.32175	−0.660873	+	0.750497i	$0.729814\pi$
$992$	0	0
$993$	13.9254	0.441909
$994$	0	0
$995$	15.4932	0.491168
$996$	0	0
$997$	−20.0117	−0.633776	−0.316888	−	0.948463i	$-0.602638\pi$
$997$	−20.0117	−0.633776	−0.316888	+	0.948463i	$0.602638\pi$
$998$	0	0
$999$	−0.763352	−0.0241514

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8664.2.a.bs.1.11	yes	12
19.18	odd	2		8664.2.a.br.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8664.2.a.br.1.11	✓	12	19.18	odd	2
8664.2.a.bs.1.11	yes	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8664.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.82351 q^{5} +4.76113 q^{7} +1.00000 q^{9} -1.67399 q^{11} -1.58781 q^{13} +3.82351 q^{15} +0.353653 q^{17} +4.76113 q^{21} +7.42529 q^{23} +9.61922 q^{25} +1.00000 q^{27} +2.19764 q^{29} +5.77759 q^{31} -1.67399 q^{33} +18.2042 q^{35} -0.763352 q^{37} -1.58781 q^{39} -10.8125 q^{41} -11.9267 q^{43} +3.82351 q^{45} -7.31189 q^{47} +15.6683 q^{49} +0.353653 q^{51} +10.3100 q^{53} -6.40053 q^{55} +0.254293 q^{59} -4.97487 q^{61} +4.76113 q^{63} -6.07102 q^{65} -3.98961 q^{67} +7.42529 q^{69} -11.3801 q^{71} +15.5961 q^{73} +9.61922 q^{75} -7.97009 q^{77} -6.52354 q^{79} +1.00000 q^{81} +15.0766 q^{83} +1.35220 q^{85} +2.19764 q^{87} +8.03077 q^{89} -7.55979 q^{91} +5.77759 q^{93} -15.3927 q^{97} -1.67399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 2 q^{5} + 4 q^{7} + 12 q^{9} + 14 q^{11} - 4 q^{13} + 2 q^{15} + 4 q^{21} + 26 q^{23} + 18 q^{25} + 12 q^{27} + 10 q^{29} - 8 q^{31} + 14 q^{33} + 30 q^{35} + 8 q^{37} - 4 q^{39} - 18 q^{41}+ \cdots + 14 q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 2 * q^5 + 4 * q^7 + 12 * q^9 + 14 * q^11 - 4 * q^13 + 2 * q^15 + 4 * q^21 + 26 * q^23 + 18 * q^25 + 12 * q^27 + 10 * q^29 - 8 * q^31 + 14 * q^33 + 30 * q^35 + 8 * q^37 - 4 * q^39 - 18 * q^41 - 8 * q^43 + 2 * q^45 + 48 * q^49 + 28 * q^53 + 2 * q^55 - 6 * q^59 + 38 * q^61 + 4 * q^63 - 24 * q^65 + 16 * q^67 + 26 * q^69 + 38 * q^71 + 28 * q^73 + 18 * q^75 + 4 * q^77 + 40 * q^79 + 12 * q^81 + 18 * q^83 + 54 * q^85 + 10 * q^87 - 24 * q^89 - 20 * q^91 - 8 * q^93 - 24 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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