LMFDB - Embedded newform 8032.2.a.i.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.12351 q^{3} +0.837743 q^{5} -0.914625 q^{7} -1.73772 q^{9} +O(q^{10})$	$q-1.12351 q^{3} +0.837743 q^{5} -0.914625 q^{7} -1.73772 q^{9} +2.68790 q^{11} +6.48712 q^{13} -0.941213 q^{15} -2.64299 q^{17} +2.38001 q^{19} +1.02759 q^{21} -1.02383 q^{23} -4.29819 q^{25} +5.32288 q^{27} -3.24964 q^{29} +0.564072 q^{31} -3.01988 q^{33} -0.766220 q^{35} -0.959245 q^{37} -7.28836 q^{39} +4.79287 q^{41} +6.94079 q^{43} -1.45576 q^{45} +6.03713 q^{47} -6.16346 q^{49} +2.96943 q^{51} -3.79887 q^{53} +2.25177 q^{55} -2.67396 q^{57} +1.91325 q^{59} +3.45521 q^{61} +1.58936 q^{63} +5.43454 q^{65} +13.9343 q^{67} +1.15029 q^{69} +0.544993 q^{71} +5.94039 q^{73} +4.82906 q^{75} -2.45842 q^{77} -3.81781 q^{79} -0.767154 q^{81} +4.21857 q^{83} -2.21414 q^{85} +3.65101 q^{87} +3.26471 q^{89} -5.93329 q^{91} -0.633742 q^{93} +1.99383 q^{95} -11.8133 q^{97} -4.67082 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.12351	−0.648660	−0.324330	−	0.945944i	$-0.605139\pi$
$3$	−1.12351	−0.648660	−0.324330	+	0.945944i	$0.605139\pi$
$4$	0	0
$5$	0.837743	0.374650	0.187325	−	0.982298i	$-0.440018\pi$
$5$	0.837743	0.374650	0.187325	+	0.982298i	$0.440018\pi$
$6$	0	0
$7$	−0.914625	−0.345696	−0.172848	−	0.984949i	$-0.555297\pi$
$7$	−0.914625	−0.345696	−0.172848	+	0.984949i	$0.555297\pi$
$8$	0	0
$9$	−1.73772	−0.579241
$10$	0	0
$11$	2.68790	0.810431	0.405216	−	0.914221i	$-0.367196\pi$
$11$	2.68790	0.810431	0.405216	+	0.914221i	$0.367196\pi$
$12$	0	0
$13$	6.48712	1.79920	0.899602	−	0.436710i	$-0.143856\pi$
$13$	6.48712	1.79920	0.899602	+	0.436710i	$0.143856\pi$
$14$	0	0
$15$	−0.941213	−0.243020
$16$	0	0
$17$	−2.64299	−0.641019	−0.320509	−	0.947245i	$-0.603854\pi$
$17$	−2.64299	−0.641019	−0.320509	+	0.947245i	$0.603854\pi$
$18$	0	0
$19$	2.38001	0.546011	0.273005	−	0.962012i	$-0.411982\pi$
$19$	2.38001	0.546011	0.273005	+	0.962012i	$0.411982\pi$
$20$	0	0
$21$	1.02759	0.224239
$22$	0	0
$23$	−1.02383	−0.213483	−0.106742	−	0.994287i	$-0.534042\pi$
$23$	−1.02383	−0.213483	−0.106742	+	0.994287i	$0.534042\pi$
$24$	0	0
$25$	−4.29819	−0.859637
$26$	0	0
$27$	5.32288	1.02439
$28$	0	0
$29$	−3.24964	−0.603443	−0.301722	−	0.953396i	$-0.597561\pi$
$29$	−3.24964	−0.603443	−0.301722	+	0.953396i	$0.597561\pi$
$30$	0	0
$31$	0.564072	0.101310	0.0506552	−	0.998716i	$-0.483869\pi$
$31$	0.564072	0.101310	0.0506552	+	0.998716i	$0.483869\pi$
$32$	0	0
$33$	−3.01988	−0.525694
$34$	0	0
$35$	−0.766220	−0.129515
$36$	0	0
$37$	−0.959245	−0.157699	−0.0788494	−	0.996887i	$-0.525125\pi$
$37$	−0.959245	−0.157699	−0.0788494	+	0.996887i	$0.525125\pi$
$38$	0	0
$39$	−7.28836	−1.16707
$40$	0	0
$41$	4.79287	0.748521	0.374261	−	0.927324i	$-0.377897\pi$
$41$	4.79287	0.748521	0.374261	+	0.927324i	$0.377897\pi$
$42$	0	0
$43$	6.94079	1.05846	0.529230	−	0.848478i	$-0.322481\pi$
$43$	6.94079	1.05846	0.529230	+	0.848478i	$0.322481\pi$
$44$	0	0
$45$	−1.45576	−0.217013
$46$	0	0
$47$	6.03713	0.880605	0.440303	−	0.897849i	$-0.354871\pi$
$47$	6.03713	0.880605	0.440303	+	0.897849i	$0.354871\pi$
$48$	0	0
$49$	−6.16346	−0.880494
$50$	0	0
$51$	2.96943	0.415803
$52$	0	0
$53$	−3.79887	−0.521816	−0.260908	−	0.965364i	$-0.584022\pi$
$53$	−3.79887	−0.521816	−0.260908	+	0.965364i	$0.584022\pi$
$54$	0	0
$55$	2.25177	0.303628
$56$	0	0
$57$	−2.67396	−0.354175
$58$	0	0
$59$	1.91325	0.249084	0.124542	−	0.992214i	$-0.460254\pi$
$59$	1.91325	0.249084	0.124542	+	0.992214i	$0.460254\pi$
$60$	0	0
$61$	3.45521	0.442394	0.221197	−	0.975229i	$-0.429004\pi$
$61$	3.45521	0.442394	0.221197	+	0.975229i	$0.429004\pi$
$62$	0	0
$63$	1.58936	0.200241
$64$	0	0
$65$	5.43454	0.674072
$66$	0	0
$67$	13.9343	1.70235	0.851173	−	0.524885i	$-0.175892\pi$
$67$	13.9343	1.70235	0.851173	+	0.524885i	$0.175892\pi$
$68$	0	0
$69$	1.15029	0.138478
$70$	0	0
$71$	0.544993	0.0646788	0.0323394	−	0.999477i	$-0.489704\pi$
$71$	0.544993	0.0646788	0.0323394	+	0.999477i	$0.489704\pi$
$72$	0	0
$73$	5.94039	0.695270	0.347635	−	0.937630i	$-0.386985\pi$
$73$	5.94039	0.695270	0.347635	+	0.937630i	$0.386985\pi$
$74$	0	0
$75$	4.82906	0.557612
$76$	0	0
$77$	−2.45842	−0.280163
$78$	0	0
$79$	−3.81781	−0.429537	−0.214768	−	0.976665i	$-0.568900\pi$
$79$	−3.81781	−0.429537	−0.214768	+	0.976665i	$0.568900\pi$
$80$	0	0
$81$	−0.767154	−0.0852393
$82$	0	0
$83$	4.21857	0.463048	0.231524	−	0.972829i	$-0.425629\pi$
$83$	4.21857	0.463048	0.231524	+	0.972829i	$0.425629\pi$
$84$	0	0
$85$	−2.21414	−0.240158
$86$	0	0
$87$	3.65101	0.391429
$88$	0	0
$89$	3.26471	0.346059	0.173030	−	0.984917i	$-0.444644\pi$
$89$	3.26471	0.346059	0.173030	+	0.984917i	$0.444644\pi$
$90$	0	0
$91$	−5.93329	−0.621977
$92$	0	0
$93$	−0.633742	−0.0657160
$94$	0	0
$95$	1.99383	0.204563
$96$	0	0
$97$	−11.8133	−1.19946	−0.599729	−	0.800203i	$-0.704725\pi$
$97$	−11.8133	−1.19946	−0.599729	+	0.800203i	$0.704725\pi$
$98$	0	0
$99$	−4.67082	−0.469435
$100$	0	0
$101$	6.27857	0.624741	0.312370	−	0.949960i	$-0.398877\pi$
$101$	6.27857	0.624741	0.312370	+	0.949960i	$0.398877\pi$
$102$	0	0
$103$	−5.57362	−0.549185	−0.274593	−	0.961561i	$-0.588543\pi$
$103$	−5.57362	−0.549185	−0.274593	+	0.961561i	$0.588543\pi$
$104$	0	0
$105$	0.860857	0.0840111
$106$	0	0
$107$	−13.0677	−1.26330	−0.631652	−	0.775252i	$-0.717623\pi$
$107$	−13.0677	−1.26330	−0.631652	+	0.775252i	$0.717623\pi$
$108$	0	0
$109$	−14.2936	−1.36908	−0.684538	−	0.728977i	$-0.739996\pi$
$109$	−14.2936	−1.36908	−0.684538	+	0.728977i	$0.739996\pi$
$110$	0	0
$111$	1.07772	0.102293
$112$	0	0
$113$	−10.0354	−0.944050	−0.472025	−	0.881585i	$-0.656477\pi$
$113$	−10.0354	−0.944050	−0.472025	+	0.881585i	$0.656477\pi$
$114$	0	0
$115$	−0.857707	−0.0799815
$116$	0	0
$117$	−11.2728	−1.04217
$118$	0	0
$119$	2.41734	0.221598
$120$	0	0
$121$	−3.77522	−0.343201
$122$	0	0
$123$	−5.38485	−0.485535
$124$	0	0
$125$	−7.78949	−0.696713
$126$	0	0
$127$	−19.8409	−1.76060	−0.880298	−	0.474421i	$-0.842657\pi$
$127$	−19.8409	−1.76060	−0.880298	+	0.474421i	$0.842657\pi$
$128$	0	0
$129$	−7.79805	−0.686580
$130$	0	0
$131$	19.0221	1.66197	0.830983	−	0.556298i	$-0.187779\pi$
$131$	19.0221	1.66197	0.830983	+	0.556298i	$0.187779\pi$
$132$	0	0
$133$	−2.17681	−0.188754
$134$	0	0
$135$	4.45921	0.383788
$136$	0	0
$137$	10.9489	0.935427	0.467713	−	0.883880i	$-0.345078\pi$
$137$	10.9489	0.935427	0.467713	+	0.883880i	$0.345078\pi$
$138$	0	0
$139$	−3.67341	−0.311574	−0.155787	−	0.987791i	$-0.549791\pi$
$139$	−3.67341	−0.311574	−0.155787	+	0.987791i	$0.549791\pi$
$140$	0	0
$141$	−6.78278	−0.571213
$142$	0	0
$143$	17.4367	1.45813
$144$	0	0
$145$	−2.72236	−0.226080
$146$	0	0
$147$	6.92472	0.571141
$148$	0	0
$149$	12.4910	1.02331	0.511653	−	0.859192i	$-0.329033\pi$
$149$	12.4910	1.02331	0.511653	+	0.859192i	$0.329033\pi$
$150$	0	0
$151$	19.0042	1.54654	0.773270	−	0.634077i	$-0.218620\pi$
$151$	19.0042	1.54654	0.773270	+	0.634077i	$0.218620\pi$
$152$	0	0
$153$	4.59278	0.371304
$154$	0	0
$155$	0.472548	0.0379559
$156$	0	0
$157$	−0.501496	−0.0400238	−0.0200119	−	0.999800i	$-0.506370\pi$
$157$	−0.501496	−0.0400238	−0.0200119	+	0.999800i	$0.506370\pi$
$158$	0	0
$159$	4.26808	0.338481
$160$	0	0
$161$	0.936421	0.0738003
$162$	0	0
$163$	−6.85208	−0.536696	−0.268348	−	0.963322i	$-0.586478\pi$
$163$	−6.85208	−0.536696	−0.268348	+	0.963322i	$0.586478\pi$
$164$	0	0
$165$	−2.52988	−0.196951
$166$	0	0
$167$	12.1821	0.942677	0.471339	−	0.881952i	$-0.343771\pi$
$167$	12.1821	0.942677	0.471339	+	0.881952i	$0.343771\pi$
$168$	0	0
$169$	29.0828	2.23714
$170$	0	0
$171$	−4.13579	−0.316272
$172$	0	0
$173$	8.28011	0.629525	0.314763	−	0.949170i	$-0.398075\pi$
$173$	8.28011	0.629525	0.314763	+	0.949170i	$0.398075\pi$
$174$	0	0
$175$	3.93123	0.297173
$176$	0	0
$177$	−2.14956	−0.161571
$178$	0	0
$179$	−11.4551	−0.856198	−0.428099	−	0.903732i	$-0.640817\pi$
$179$	−11.4551	−0.856198	−0.428099	+	0.903732i	$0.640817\pi$
$180$	0	0
$181$	10.0757	0.748920	0.374460	−	0.927243i	$-0.377828\pi$
$181$	10.0757	0.748920	0.374460	+	0.927243i	$0.377828\pi$
$182$	0	0
$183$	−3.88197	−0.286963
$184$	0	0
$185$	−0.803600	−0.0590819
$186$	0	0
$187$	−7.10408	−0.519502
$188$	0	0
$189$	−4.86844	−0.354127
$190$	0	0
$191$	14.3206	1.03620	0.518102	−	0.855319i	$-0.326639\pi$
$191$	14.3206	1.03620	0.518102	+	0.855319i	$0.326639\pi$
$192$	0	0
$193$	−9.30349	−0.669680	−0.334840	−	0.942275i	$-0.608682\pi$
$193$	−9.30349	−0.669680	−0.334840	+	0.942275i	$0.608682\pi$
$194$	0	0
$195$	−6.10577	−0.437243
$196$	0	0
$197$	−21.2639	−1.51499	−0.757495	−	0.652842i	$-0.773577\pi$
$197$	−21.2639	−1.51499	−0.757495	+	0.652842i	$0.773577\pi$
$198$	0	0
$199$	−16.0729	−1.13938	−0.569689	−	0.821861i	$-0.692936\pi$
$199$	−16.0729	−1.13938	−0.569689	+	0.821861i	$0.692936\pi$
$200$	0	0
$201$	−15.6553	−1.10424
$202$	0	0
$203$	2.97220	0.208608
$204$	0	0
$205$	4.01519	0.280433
$206$	0	0
$207$	1.77913	0.123658
$208$	0	0
$209$	6.39721	0.442504
$210$	0	0
$211$	3.53964	0.243679	0.121839	−	0.992550i	$-0.461121\pi$
$211$	3.53964	0.243679	0.121839	+	0.992550i	$0.461121\pi$
$212$	0	0
$213$	−0.612306	−0.0419545
$214$	0	0
$215$	5.81459	0.396552
$216$	0	0
$217$	−0.515915	−0.0350226
$218$	0	0
$219$	−6.67409	−0.450993
$220$	0	0
$221$	−17.1454	−1.15332
$222$	0	0
$223$	0.353039	0.0236412	0.0118206	−	0.999930i	$-0.496237\pi$
$223$	0.353039	0.0236412	0.0118206	+	0.999930i	$0.496237\pi$
$224$	0	0
$225$	7.46906	0.497937
$226$	0	0
$227$	17.8073	1.18191	0.590957	−	0.806703i	$-0.298750\pi$
$227$	17.8073	1.18191	0.590957	+	0.806703i	$0.298750\pi$
$228$	0	0
$229$	4.42091	0.292142	0.146071	−	0.989274i	$-0.453337\pi$
$229$	4.42091	0.292142	0.146071	+	0.989274i	$0.453337\pi$
$230$	0	0
$231$	2.76206	0.181730
$232$	0	0
$233$	−3.22980	−0.211591	−0.105796	−	0.994388i	$-0.533739\pi$
$233$	−3.22980	−0.211591	−0.105796	+	0.994388i	$0.533739\pi$
$234$	0	0
$235$	5.05756	0.329919
$236$	0	0
$237$	4.28935	0.278623
$238$	0	0
$239$	20.7997	1.34542	0.672710	−	0.739907i	$-0.265130\pi$
$239$	20.7997	1.34542	0.672710	+	0.739907i	$0.265130\pi$
$240$	0	0
$241$	28.4909	1.83526	0.917631	−	0.397434i	$-0.130099\pi$
$241$	28.4909	1.83526	0.917631	+	0.397434i	$0.130099\pi$
$242$	0	0
$243$	−15.1067	−0.969098
$244$	0	0
$245$	−5.16340	−0.329877
$246$	0	0
$247$	15.4394	0.982385
$248$	0	0
$249$	−4.73961	−0.300361
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−2.75195	−0.173014
$254$	0	0
$255$	2.48762	0.155781
$256$	0	0
$257$	3.70654	0.231208	0.115604	−	0.993295i	$-0.463120\pi$
$257$	3.70654	0.231208	0.115604	+	0.993295i	$0.463120\pi$
$258$	0	0
$259$	0.877349	0.0545158
$260$	0	0
$261$	5.64697	0.349539
$262$	0	0
$263$	31.2731	1.92838	0.964190	−	0.265212i	$-0.0854421\pi$
$263$	31.2731	1.92838	0.964190	+	0.265212i	$0.0854421\pi$
$264$	0	0
$265$	−3.18248	−0.195498
$266$	0	0
$267$	−3.66794	−0.224475
$268$	0	0
$269$	−6.82685	−0.416240	−0.208120	−	0.978103i	$-0.566735\pi$
$269$	−6.82685	−0.416240	−0.208120	+	0.978103i	$0.566735\pi$
$270$	0	0
$271$	28.0054	1.70120	0.850602	−	0.525810i	$-0.176238\pi$
$271$	28.0054	1.70120	0.850602	+	0.525810i	$0.176238\pi$
$272$	0	0
$273$	6.66611	0.403452
$274$	0	0
$275$	−11.5531	−0.696677
$276$	0	0
$277$	12.7310	0.764930	0.382465	−	0.923970i	$-0.375075\pi$
$277$	12.7310	0.764930	0.382465	+	0.923970i	$0.375075\pi$
$278$	0	0
$279$	−0.980201	−0.0586831
$280$	0	0
$281$	−12.4260	−0.741273	−0.370636	−	0.928778i	$-0.620860\pi$
$281$	−12.4260	−0.741273	−0.370636	+	0.928778i	$0.620860\pi$
$282$	0	0
$283$	14.9540	0.888924	0.444462	−	0.895798i	$-0.353395\pi$
$283$	14.9540	0.888924	0.444462	+	0.895798i	$0.353395\pi$
$284$	0	0
$285$	−2.24009	−0.132692
$286$	0	0
$287$	−4.38368	−0.258761
$288$	0	0
$289$	−10.0146	−0.589095
$290$	0	0
$291$	13.2724	0.778040
$292$	0	0
$293$	1.32019	0.0771261	0.0385630	−	0.999256i	$-0.487722\pi$
$293$	1.32019	0.0771261	0.0385630	+	0.999256i	$0.487722\pi$
$294$	0	0
$295$	1.60281	0.0933193
$296$	0	0
$297$	14.3074	0.830197
$298$	0	0
$299$	−6.64172	−0.384100
$300$	0	0
$301$	−6.34822	−0.365905
$302$	0	0
$303$	−7.05404	−0.405244
$304$	0	0
$305$	2.89458	0.165743
$306$	0	0
$307$	7.14221	0.407628	0.203814	−	0.979010i	$-0.434666\pi$
$307$	7.14221	0.407628	0.203814	+	0.979010i	$0.434666\pi$
$308$	0	0
$309$	6.26203	0.356234
$310$	0	0
$311$	5.35571	0.303694	0.151847	−	0.988404i	$-0.451478\pi$
$311$	5.35571	0.303694	0.151847	+	0.988404i	$0.451478\pi$
$312$	0	0
$313$	21.2503	1.20114	0.600570	−	0.799572i	$-0.294940\pi$
$313$	21.2503	1.20114	0.600570	+	0.799572i	$0.294940\pi$
$314$	0	0
$315$	1.33148	0.0750203
$316$	0	0
$317$	−15.0012	−0.842553	−0.421276	−	0.906932i	$-0.638418\pi$
$317$	−15.0012	−0.842553	−0.421276	+	0.906932i	$0.638418\pi$
$318$	0	0
$319$	−8.73470	−0.489049
$320$	0	0
$321$	14.6817	0.819454
$322$	0	0
$323$	−6.29033	−0.350003
$324$	0	0
$325$	−27.8829	−1.54666
$326$	0	0
$327$	16.0590	0.888064
$328$	0	0
$329$	−5.52171	−0.304421
$330$	0	0
$331$	−0.905269	−0.0497581	−0.0248790	−	0.999690i	$-0.507920\pi$
$331$	−0.905269	−0.0497581	−0.0248790	+	0.999690i	$0.507920\pi$
$332$	0	0
$333$	1.66690	0.0913456
$334$	0	0
$335$	11.6734	0.637784
$336$	0	0
$337$	−12.8204	−0.698370	−0.349185	−	0.937054i	$-0.613541\pi$
$337$	−12.8204	−0.698370	−0.349185	+	0.937054i	$0.613541\pi$
$338$	0	0
$339$	11.2749	0.612367
$340$	0	0
$341$	1.51617	0.0821051
$342$	0	0
$343$	12.0396	0.650079
$344$	0	0
$345$	0.963643	0.0518808
$346$	0	0
$347$	−3.09193	−0.165984	−0.0829919	−	0.996550i	$-0.526448\pi$
$347$	−3.09193	−0.165984	−0.0829919	+	0.996550i	$0.526448\pi$
$348$	0	0
$349$	12.4241	0.665049	0.332525	−	0.943095i	$-0.392100\pi$
$349$	12.4241	0.665049	0.332525	+	0.943095i	$0.392100\pi$
$350$	0	0
$351$	34.5302	1.84309
$352$	0	0
$353$	4.28265	0.227943	0.113971	−	0.993484i	$-0.463643\pi$
$353$	4.28265	0.227943	0.113971	+	0.993484i	$0.463643\pi$
$354$	0	0
$355$	0.456564	0.0242319
$356$	0	0
$357$	−2.71591	−0.143741
$358$	0	0
$359$	28.6478	1.51197	0.755987	−	0.654587i	$-0.227157\pi$
$359$	28.6478	1.51197	0.755987	+	0.654587i	$0.227157\pi$
$360$	0	0
$361$	−13.3356	−0.701872
$362$	0	0
$363$	4.24150	0.222621
$364$	0	0
$365$	4.97652	0.260483
$366$	0	0
$367$	0.571150	0.0298138	0.0149069	−	0.999889i	$-0.495255\pi$
$367$	0.571150	0.0298138	0.0149069	+	0.999889i	$0.495255\pi$
$368$	0	0
$369$	−8.32868	−0.433574
$370$	0	0
$371$	3.47455	0.180389
$372$	0	0
$373$	15.5845	0.806935	0.403468	−	0.914994i	$-0.367805\pi$
$373$	15.5845	0.806935	0.403468	+	0.914994i	$0.367805\pi$
$374$	0	0
$375$	8.75158	0.451930
$376$	0	0
$377$	−21.0808	−1.08572
$378$	0	0
$379$	−21.6281	−1.11096	−0.555480	−	0.831530i	$-0.687466\pi$
$379$	−21.6281	−1.11096	−0.555480	+	0.831530i	$0.687466\pi$
$380$	0	0
$381$	22.2915	1.14203
$382$	0	0
$383$	−18.1681	−0.928344	−0.464172	−	0.885745i	$-0.653648\pi$
$383$	−18.1681	−0.928344	−0.464172	+	0.885745i	$0.653648\pi$
$384$	0	0
$385$	−2.05952	−0.104963
$386$	0	0
$387$	−12.0612	−0.613103
$388$	0	0
$389$	36.9755	1.87473	0.937367	−	0.348344i	$-0.113256\pi$
$389$	36.9755	1.87473	0.937367	+	0.348344i	$0.113256\pi$
$390$	0	0
$391$	2.70597	0.136847
$392$	0	0
$393$	−21.3715	−1.07805
$394$	0	0
$395$	−3.19834	−0.160926
$396$	0	0
$397$	−4.75998	−0.238896	−0.119448	−	0.992840i	$-0.538113\pi$
$397$	−4.75998	−0.238896	−0.119448	+	0.992840i	$0.538113\pi$
$398$	0	0
$399$	2.44567	0.122437
$400$	0	0
$401$	5.37072	0.268201	0.134100	−	0.990968i	$-0.457186\pi$
$401$	5.37072	0.268201	0.134100	+	0.990968i	$0.457186\pi$
$402$	0	0
$403$	3.65921	0.182278
$404$	0	0
$405$	−0.642678	−0.0319349
$406$	0	0
$407$	−2.57835	−0.127804
$408$	0	0
$409$	36.0094	1.78055	0.890275	−	0.455423i	$-0.150512\pi$
$409$	36.0094	1.78055	0.890275	+	0.455423i	$0.150512\pi$
$410$	0	0
$411$	−12.3012	−0.606773
$412$	0	0
$413$	−1.74991	−0.0861073
$414$	0	0
$415$	3.53408	0.173481
$416$	0	0
$417$	4.12712	0.202106
$418$	0	0
$419$	33.2585	1.62478	0.812392	−	0.583112i	$-0.198165\pi$
$419$	33.2585	1.62478	0.812392	+	0.583112i	$0.198165\pi$
$420$	0	0
$421$	−4.70553	−0.229333	−0.114667	−	0.993404i	$-0.536580\pi$
$421$	−4.70553	−0.229333	−0.114667	+	0.993404i	$0.536580\pi$
$422$	0	0
$423$	−10.4908	−0.510082
$424$	0	0
$425$	11.3601	0.551044
$426$	0	0
$427$	−3.16022	−0.152934
$428$	0	0
$429$	−19.5903	−0.945831
$430$	0	0
$431$	−3.34678	−0.161209	−0.0806043	−	0.996746i	$-0.525685\pi$
$431$	−3.34678	−0.161209	−0.0806043	+	0.996746i	$0.525685\pi$
$432$	0	0
$433$	36.9634	1.77635	0.888175	−	0.459506i	$-0.151974\pi$
$433$	36.9634	1.77635	0.888175	+	0.459506i	$0.151974\pi$
$434$	0	0
$435$	3.05861	0.146649
$436$	0	0
$437$	−2.43672	−0.116564
$438$	0	0
$439$	−22.1004	−1.05479	−0.527396	−	0.849619i	$-0.676832\pi$
$439$	−22.1004	−1.05479	−0.527396	+	0.849619i	$0.676832\pi$
$440$	0	0
$441$	10.7104	0.510018
$442$	0	0
$443$	−11.2016	−0.532206	−0.266103	−	0.963945i	$-0.585736\pi$
$443$	−11.2016	−0.532206	−0.266103	+	0.963945i	$0.585736\pi$
$444$	0	0
$445$	2.73499	0.129651
$446$	0	0
$447$	−14.0338	−0.663777
$448$	0	0
$449$	−0.575571	−0.0271629	−0.0135814	−	0.999908i	$-0.504323\pi$
$449$	−0.575571	−0.0271629	−0.0135814	+	0.999908i	$0.504323\pi$
$450$	0	0
$451$	12.8827	0.606625
$452$	0	0
$453$	−21.3514	−1.00318
$454$	0	0
$455$	−4.97057	−0.233024
$456$	0	0
$457$	13.3518	0.624571	0.312286	−	0.949988i	$-0.398905\pi$
$457$	13.3518	0.624571	0.312286	+	0.949988i	$0.398905\pi$
$458$	0	0
$459$	−14.0683	−0.656653
$460$	0	0
$461$	−11.6167	−0.541042	−0.270521	−	0.962714i	$-0.587196\pi$
$461$	−11.6167	−0.541042	−0.270521	+	0.962714i	$0.587196\pi$
$462$	0	0
$463$	−31.2859	−1.45398	−0.726989	−	0.686649i	$-0.759081\pi$
$463$	−31.2859	−1.45398	−0.726989	+	0.686649i	$0.759081\pi$
$464$	0	0
$465$	−0.530913	−0.0246205
$466$	0	0
$467$	18.7554	0.867897	0.433949	−	0.900938i	$-0.357120\pi$
$467$	18.7554	0.867897	0.433949	+	0.900938i	$0.357120\pi$
$468$	0	0
$469$	−12.7447	−0.588494
$470$	0	0
$471$	0.563437	0.0259618
$472$	0	0
$473$	18.6561	0.857809
$474$	0	0
$475$	−10.2297	−0.469371
$476$	0	0
$477$	6.60139	0.302257
$478$	0	0
$479$	18.8888	0.863052	0.431526	−	0.902100i	$-0.357975\pi$
$479$	18.8888	0.863052	0.431526	+	0.902100i	$0.357975\pi$
$480$	0	0
$481$	−6.22274	−0.283733
$482$	0	0
$483$	−1.05208	−0.0478713
$484$	0	0
$485$	−9.89650	−0.449377
$486$	0	0
$487$	41.5019	1.88063	0.940315	−	0.340306i	$-0.110531\pi$
$487$	41.5019	1.88063	0.940315	+	0.340306i	$0.110531\pi$
$488$	0	0
$489$	7.69839	0.348133
$490$	0	0
$491$	8.75278	0.395007	0.197504	−	0.980302i	$-0.436717\pi$
$491$	8.75278	0.395007	0.197504	+	0.980302i	$0.436717\pi$
$492$	0	0
$493$	8.58876	0.386818
$494$	0	0
$495$	−3.91294	−0.175874
$496$	0	0
$497$	−0.498464	−0.0223592
$498$	0	0
$499$	−30.8347	−1.38035	−0.690176	−	0.723642i	$-0.742467\pi$
$499$	−30.8347	−1.38035	−0.690176	+	0.723642i	$0.742467\pi$
$500$	0	0
$501$	−13.6867	−0.611477
$502$	0	0
$503$	21.5937	0.962815	0.481408	−	0.876497i	$-0.340126\pi$
$503$	21.5937	0.962815	0.481408	+	0.876497i	$0.340126\pi$
$504$	0	0
$505$	5.25982	0.234059
$506$	0	0
$507$	−32.6748	−1.45114
$508$	0	0
$509$	0.236848	0.0104981	0.00524905	−	0.999986i	$-0.498329\pi$
$509$	0.236848	0.0104981	0.00524905	+	0.999986i	$0.498329\pi$
$510$	0	0
$511$	−5.43323	−0.240352
$512$	0	0
$513$	12.6685	0.559328
$514$	0	0
$515$	−4.66926	−0.205752
$516$	0	0
$517$	16.2272	0.713670
$518$	0	0
$519$	−9.30280	−0.408348
$520$	0	0
$521$	20.9509	0.917874	0.458937	−	0.888469i	$-0.348230\pi$
$521$	20.9509	0.917874	0.458937	+	0.888469i	$0.348230\pi$
$522$	0	0
$523$	28.6823	1.25419	0.627094	−	0.778944i	$-0.284244\pi$
$523$	28.6823	1.25419	0.627094	+	0.778944i	$0.284244\pi$
$524$	0	0
$525$	−4.41678	−0.192764
$526$	0	0
$527$	−1.49084	−0.0649419
$528$	0	0
$529$	−21.9518	−0.954425
$530$	0	0
$531$	−3.32470	−0.144280
$532$	0	0
$533$	31.0920	1.34674
$534$	0	0
$535$	−10.9474	−0.473297
$536$	0	0
$537$	12.8700	0.555381
$538$	0	0
$539$	−16.5667	−0.713580
$540$	0	0
$541$	17.1991	0.739446	0.369723	−	0.929142i	$-0.379453\pi$
$541$	17.1991	0.739446	0.369723	+	0.929142i	$0.379453\pi$
$542$	0	0
$543$	−11.3201	−0.485794
$544$	0	0
$545$	−11.9743	−0.512924
$546$	0	0
$547$	−27.1830	−1.16226	−0.581130	−	0.813811i	$-0.697389\pi$
$547$	−27.1830	−1.16226	−0.581130	+	0.813811i	$0.697389\pi$
$548$	0	0
$549$	−6.00419	−0.256253
$550$	0	0
$551$	−7.73417	−0.329487
$552$	0	0
$553$	3.49186	0.148489
$554$	0	0
$555$	0.902854	0.0383240
$556$	0	0
$557$	22.5830	0.956872	0.478436	−	0.878122i	$-0.341204\pi$
$557$	22.5830	0.956872	0.478436	+	0.878122i	$0.341204\pi$
$558$	0	0
$559$	45.0257	1.90439
$560$	0	0
$561$	7.98151	0.336980
$562$	0	0
$563$	29.0940	1.22617	0.613083	−	0.790019i	$-0.289929\pi$
$563$	29.0940	1.22617	0.613083	+	0.790019i	$0.289929\pi$
$564$	0	0
$565$	−8.40707	−0.353688
$566$	0	0
$567$	0.701658	0.0294669
$568$	0	0
$569$	−46.7632	−1.96041	−0.980207	−	0.197977i	$-0.936563\pi$
$569$	−46.7632	−1.96041	−0.980207	+	0.197977i	$0.936563\pi$
$570$	0	0
$571$	−23.0896	−0.966270	−0.483135	−	0.875546i	$-0.660502\pi$
$571$	−23.0896	−0.966270	−0.483135	+	0.875546i	$0.660502\pi$
$572$	0	0
$573$	−16.0894	−0.672144
$574$	0	0
$575$	4.40061	0.183518
$576$	0	0
$577$	−3.73647	−0.155551	−0.0777756	−	0.996971i	$-0.524782\pi$
$577$	−3.73647	−0.155551	−0.0777756	+	0.996971i	$0.524782\pi$
$578$	0	0
$579$	10.4526	0.434394
$580$	0	0
$581$	−3.85841	−0.160074
$582$	0	0
$583$	−10.2110	−0.422896
$584$	0	0
$585$	−9.44373	−0.390450
$586$	0	0
$587$	39.5144	1.63094	0.815468	−	0.578803i	$-0.196480\pi$
$587$	39.5144	1.63094	0.815468	+	0.578803i	$0.196480\pi$
$588$	0	0
$589$	1.34250	0.0553166
$590$	0	0
$591$	23.8902	0.982712
$592$	0	0
$593$	−23.8975	−0.981353	−0.490677	−	0.871342i	$-0.663250\pi$
$593$	−23.8975	−0.981353	−0.490677	+	0.871342i	$0.663250\pi$
$594$	0	0
$595$	2.02511	0.0830215
$596$	0	0
$597$	18.0581	0.739068
$598$	0	0
$599$	−12.3427	−0.504310	−0.252155	−	0.967687i	$-0.581139\pi$
$599$	−12.3427	−0.504310	−0.252155	+	0.967687i	$0.581139\pi$
$600$	0	0
$601$	−15.3314	−0.625381	−0.312690	−	0.949855i	$-0.601230\pi$
$601$	−15.3314	−0.625381	−0.312690	+	0.949855i	$0.601230\pi$
$602$	0	0
$603$	−24.2139	−0.986068
$604$	0	0
$605$	−3.16266	−0.128580
$606$	0	0
$607$	−3.95851	−0.160671	−0.0803354	−	0.996768i	$-0.525599\pi$
$607$	−3.95851	−0.160671	−0.0803354	+	0.996768i	$0.525599\pi$
$608$	0	0
$609$	−3.33930	−0.135315
$610$	0	0
$611$	39.1636	1.58439
$612$	0	0
$613$	36.5554	1.47646	0.738229	−	0.674551i	$-0.235663\pi$
$613$	36.5554	1.47646	0.738229	+	0.674551i	$0.235663\pi$
$614$	0	0
$615$	−4.51112	−0.181906
$616$	0	0
$617$	18.8829	0.760196	0.380098	−	0.924946i	$-0.375890\pi$
$617$	18.8829	0.760196	0.380098	+	0.924946i	$0.375890\pi$
$618$	0	0
$619$	34.1093	1.37097	0.685485	−	0.728087i	$-0.259590\pi$
$619$	34.1093	1.37097	0.685485	+	0.728087i	$0.259590\pi$
$620$	0	0
$621$	−5.44973	−0.218690
$622$	0	0
$623$	−2.98599	−0.119631
$624$	0	0
$625$	14.9653	0.598614
$626$	0	0
$627$	−7.18734	−0.287035
$628$	0	0
$629$	2.53527	0.101088
$630$	0	0
$631$	32.5357	1.29523	0.647613	−	0.761970i	$-0.275767\pi$
$631$	32.5357	1.29523	0.647613	+	0.761970i	$0.275767\pi$
$632$	0	0
$633$	−3.97682	−0.158064
$634$	0	0
$635$	−16.6216	−0.659607
$636$	0	0
$637$	−39.9831	−1.58419
$638$	0	0
$639$	−0.947046	−0.0374646
$640$	0	0
$641$	−8.62567	−0.340693	−0.170347	−	0.985384i	$-0.554489\pi$
$641$	−8.62567	−0.340693	−0.170347	+	0.985384i	$0.554489\pi$
$642$	0	0
$643$	−23.9906	−0.946099	−0.473049	−	0.881036i	$-0.656847\pi$
$643$	−23.9906	−0.946099	−0.473049	+	0.881036i	$0.656847\pi$
$644$	0	0
$645$	−6.53276	−0.257227
$646$	0	0
$647$	15.5477	0.611244	0.305622	−	0.952153i	$-0.401136\pi$
$647$	15.5477	0.611244	0.305622	+	0.952153i	$0.401136\pi$
$648$	0	0
$649$	5.14262	0.201865
$650$	0	0
$651$	0.579636	0.0227177
$652$	0	0
$653$	16.3626	0.640319	0.320159	−	0.947364i	$-0.396264\pi$
$653$	16.3626	0.640319	0.320159	+	0.947364i	$0.396264\pi$
$654$	0	0
$655$	15.9356	0.622656
$656$	0	0
$657$	−10.3227	−0.402729
$658$	0	0
$659$	−10.2467	−0.399153	−0.199577	−	0.979882i	$-0.563957\pi$
$659$	−10.2467	−0.399153	−0.199577	+	0.979882i	$0.563957\pi$
$660$	0	0
$661$	3.86675	0.150399	0.0751995	−	0.997169i	$-0.476041\pi$
$661$	3.86675	0.150399	0.0751995	+	0.997169i	$0.476041\pi$
$662$	0	0
$663$	19.2630	0.748115
$664$	0	0
$665$	−1.82361	−0.0707165
$666$	0	0
$667$	3.32708	0.128825
$668$	0	0
$669$	−0.396643	−0.0153351
$670$	0	0
$671$	9.28724	0.358530
$672$	0	0
$673$	−25.5164	−0.983583	−0.491792	−	0.870713i	$-0.663658\pi$
$673$	−25.5164	−0.983583	−0.491792	+	0.870713i	$0.663658\pi$
$674$	0	0
$675$	−22.8788	−0.880604
$676$	0	0
$677$	−12.7382	−0.489569	−0.244784	−	0.969578i	$-0.578717\pi$
$677$	−12.7382	−0.489569	−0.244784	+	0.969578i	$0.578717\pi$
$678$	0	0
$679$	10.8047	0.414648
$680$	0	0
$681$	−20.0068	−0.766660
$682$	0	0
$683$	−14.5051	−0.555021	−0.277511	−	0.960723i	$-0.589509\pi$
$683$	−14.5051	−0.555021	−0.277511	+	0.960723i	$0.589509\pi$
$684$	0	0
$685$	9.17235	0.350458
$686$	0	0
$687$	−4.96694	−0.189501
$688$	0	0
$689$	−24.6438	−0.938853
$690$	0	0
$691$	−27.4368	−1.04375	−0.521873	−	0.853023i	$-0.674767\pi$
$691$	−27.4368	−1.04375	−0.521873	+	0.853023i	$0.674767\pi$
$692$	0	0
$693$	4.27205	0.162282
$694$	0	0
$695$	−3.07737	−0.116731
$696$	0	0
$697$	−12.6675	−0.479816
$698$	0	0
$699$	3.62872	0.137251
$700$	0	0
$701$	4.45900	0.168414	0.0842070	−	0.996448i	$-0.473164\pi$
$701$	4.45900	0.168414	0.0842070	+	0.996448i	$0.473164\pi$
$702$	0	0
$703$	−2.28301	−0.0861053
$704$	0	0
$705$	−5.68222	−0.214005
$706$	0	0
$707$	−5.74253	−0.215970
$708$	0	0
$709$	−28.8867	−1.08486	−0.542432	−	0.840100i	$-0.682496\pi$
$709$	−28.8867	−1.08486	−0.542432	+	0.840100i	$0.682496\pi$
$710$	0	0
$711$	6.63429	0.248805
$712$	0	0
$713$	−0.577515	−0.0216281
$714$	0	0
$715$	14.6075	0.546289
$716$	0	0
$717$	−23.3687	−0.872719
$718$	0	0
$719$	−19.5059	−0.727446	−0.363723	−	0.931507i	$-0.618495\pi$
$719$	−19.5059	−0.727446	−0.363723	+	0.931507i	$0.618495\pi$
$720$	0	0
$721$	5.09777	0.189851
$722$	0	0
$723$	−32.0099	−1.19046
$724$	0	0
$725$	13.9676	0.518742
$726$	0	0
$727$	35.1180	1.30245	0.651227	−	0.758883i	$-0.274254\pi$
$727$	35.1180	1.30245	0.651227	+	0.758883i	$0.274254\pi$
$728$	0	0
$729$	19.2741	0.713854
$730$	0	0
$731$	−18.3444	−0.678493
$732$	0	0
$733$	9.30475	0.343679	0.171839	−	0.985125i	$-0.445029\pi$
$733$	9.30475	0.343679	0.171839	+	0.985125i	$0.445029\pi$
$734$	0	0
$735$	5.80113	0.213978
$736$	0	0
$737$	37.4540	1.37963
$738$	0	0
$739$	4.50063	0.165558	0.0827792	−	0.996568i	$-0.473620\pi$
$739$	4.50063	0.165558	0.0827792	+	0.996568i	$0.473620\pi$
$740$	0	0
$741$	−17.3463	−0.637234
$742$	0	0
$743$	−32.0965	−1.17751	−0.588753	−	0.808313i	$-0.700381\pi$
$743$	−32.0965	−1.17751	−0.588753	+	0.808313i	$0.700381\pi$
$744$	0	0
$745$	10.4643	0.383382
$746$	0	0
$747$	−7.33070	−0.268216
$748$	0	0
$749$	11.9521	0.436719
$750$	0	0
$751$	−39.2904	−1.43373	−0.716863	−	0.697214i	$-0.754423\pi$
$751$	−39.2904	−1.43373	−0.716863	+	0.697214i	$0.754423\pi$
$752$	0	0
$753$	−1.12351	−0.0409430
$754$	0	0
$755$	15.9206	0.579411
$756$	0	0
$757$	48.2225	1.75267	0.876337	−	0.481698i	$-0.159980\pi$
$757$	48.2225	1.75267	0.876337	+	0.481698i	$0.159980\pi$
$758$	0	0
$759$	3.09185	0.112227
$760$	0	0
$761$	−15.1518	−0.549253	−0.274626	−	0.961551i	$-0.588554\pi$
$761$	−15.1518	−0.549253	−0.274626	+	0.961551i	$0.588554\pi$
$762$	0	0
$763$	13.0733	0.473284
$764$	0	0
$765$	3.84757	0.139109
$766$	0	0
$767$	12.4115	0.448153
$768$	0	0
$769$	−21.6348	−0.780171	−0.390085	−	0.920779i	$-0.627554\pi$
$769$	−21.6348	−0.780171	−0.390085	+	0.920779i	$0.627554\pi$
$770$	0	0
$771$	−4.16434	−0.149975
$772$	0	0
$773$	6.56620	0.236170	0.118085	−	0.993004i	$-0.462325\pi$
$773$	6.56620	0.236170	0.118085	+	0.993004i	$0.462325\pi$
$774$	0	0
$775$	−2.42449	−0.0870902
$776$	0	0
$777$	−0.985712	−0.0353622
$778$	0	0
$779$	11.4071	0.408701
$780$	0	0
$781$	1.46488	0.0524177
$782$	0	0
$783$	−17.2975	−0.618161
$784$	0	0
$785$	−0.420125	−0.0149949
$786$	0	0
$787$	24.8098	0.884374	0.442187	−	0.896923i	$-0.354203\pi$
$787$	24.8098	0.884374	0.442187	+	0.896923i	$0.354203\pi$
$788$	0	0
$789$	−35.1356	−1.25086
$790$	0	0
$791$	9.17861	0.326354
$792$	0	0
$793$	22.4144	0.795958
$794$	0	0
$795$	3.57555	0.126812
$796$	0	0
$797$	28.0966	0.995232	0.497616	−	0.867397i	$-0.334209\pi$
$797$	28.0966	0.995232	0.497616	+	0.867397i	$0.334209\pi$
$798$	0	0
$799$	−15.9561	−0.564485
$800$	0	0
$801$	−5.67317	−0.200452
$802$	0	0
$803$	15.9671	0.563468
$804$	0	0
$805$	0.784480	0.0276493
$806$	0	0
$807$	7.67004	0.269998
$808$	0	0
$809$	−2.74367	−0.0964623	−0.0482311	−	0.998836i	$-0.515358\pi$
$809$	−2.74367	−0.0964623	−0.0482311	+	0.998836i	$0.515358\pi$
$810$	0	0
$811$	−7.65863	−0.268931	−0.134465	−	0.990918i	$-0.542932\pi$
$811$	−7.65863	−0.268931	−0.134465	+	0.990918i	$0.542932\pi$
$812$	0	0
$813$	−31.4643	−1.10350
$814$	0	0
$815$	−5.74028	−0.201073
$816$	0	0
$817$	16.5191	0.577931
$818$	0	0
$819$	10.3104	0.360275
$820$	0	0
$821$	−19.3509	−0.675352	−0.337676	−	0.941262i	$-0.609641\pi$
$821$	−19.3509	−0.675352	−0.337676	+	0.941262i	$0.609641\pi$
$822$	0	0
$823$	25.8145	0.899839	0.449919	−	0.893069i	$-0.351453\pi$
$823$	25.8145	0.899839	0.449919	+	0.893069i	$0.351453\pi$
$824$	0	0
$825$	12.9800	0.451906
$826$	0	0
$827$	1.30389	0.0453409	0.0226704	−	0.999743i	$-0.492783\pi$
$827$	1.30389	0.0453409	0.0226704	+	0.999743i	$0.492783\pi$
$828$	0	0
$829$	−7.38214	−0.256392	−0.128196	−	0.991749i	$-0.540919\pi$
$829$	−7.38214	−0.256392	−0.128196	+	0.991749i	$0.540919\pi$
$830$	0	0
$831$	−14.3034	−0.496179
$832$	0	0
$833$	16.2900	0.564414
$834$	0	0
$835$	10.2054	0.353174
$836$	0	0
$837$	3.00249	0.103781
$838$	0	0
$839$	36.1387	1.24765	0.623823	−	0.781565i	$-0.285578\pi$
$839$	36.1387	1.24765	0.623823	+	0.781565i	$0.285578\pi$
$840$	0	0
$841$	−18.4398	−0.635856
$842$	0	0
$843$	13.9607	0.480834
$844$	0	0
$845$	24.3639	0.838143
$846$	0	0
$847$	3.45291	0.118643
$848$	0	0
$849$	−16.8010	−0.576609
$850$	0	0
$851$	0.982104	0.0336661
$852$	0	0
$853$	21.9567	0.751782	0.375891	−	0.926664i	$-0.377337\pi$
$853$	21.9567	0.751782	0.375891	+	0.926664i	$0.377337\pi$
$854$	0	0
$855$	−3.46473	−0.118491
$856$	0	0
$857$	41.5956	1.42088	0.710439	−	0.703759i	$-0.248496\pi$
$857$	41.5956	1.42088	0.710439	+	0.703759i	$0.248496\pi$
$858$	0	0
$859$	−30.9788	−1.05698	−0.528491	−	0.848939i	$-0.677242\pi$
$859$	−30.9788	−1.05698	−0.528491	+	0.848939i	$0.677242\pi$
$860$	0	0
$861$	4.92512	0.167847
$862$	0	0
$863$	−14.6703	−0.499384	−0.249692	−	0.968325i	$-0.580329\pi$
$863$	−14.6703	−0.499384	−0.249692	+	0.968325i	$0.580329\pi$
$864$	0	0
$865$	6.93661	0.235852
$866$	0	0
$867$	11.2515	0.382122
$868$	0	0
$869$	−10.2619	−0.348110
$870$	0	0
$871$	90.3936	3.06287
$872$	0	0
$873$	20.5282	0.694775
$874$	0	0
$875$	7.12446	0.240851
$876$	0	0
$877$	12.7697	0.431203	0.215601	−	0.976481i	$-0.430829\pi$
$877$	12.7697	0.431203	0.215601	+	0.976481i	$0.430829\pi$
$878$	0	0
$879$	−1.48324	−0.0500286
$880$	0	0
$881$	−36.6925	−1.23620	−0.618100	−	0.786099i	$-0.712097\pi$
$881$	−36.6925	−1.23620	−0.618100	+	0.786099i	$0.712097\pi$
$882$	0	0
$883$	−11.8974	−0.400379	−0.200189	−	0.979757i	$-0.564156\pi$
$883$	−11.8974	−0.400379	−0.200189	+	0.979757i	$0.564156\pi$
$884$	0	0
$885$	−1.80078	−0.0605325
$886$	0	0
$887$	−13.1527	−0.441624	−0.220812	−	0.975316i	$-0.570871\pi$
$887$	−13.1527	−0.441624	−0.220812	+	0.975316i	$0.570871\pi$
$888$	0	0
$889$	18.1470	0.608630
$890$	0	0
$891$	−2.06203	−0.0690806
$892$	0	0
$893$	14.3684	0.480820
$894$	0	0
$895$	−9.59647	−0.320775
$896$	0	0
$897$	7.46204	0.249150
$898$	0	0
$899$	−1.83303	−0.0611351
$900$	0	0
$901$	10.0404	0.334494
$902$	0	0
$903$	7.13229	0.237348
$904$	0	0
$905$	8.44083	0.280583
$906$	0	0
$907$	57.3827	1.90536	0.952680	−	0.303975i	$-0.0983141\pi$
$907$	57.3827	1.90536	0.952680	+	0.303975i	$0.0983141\pi$
$908$	0	0
$909$	−10.9104	−0.361875
$910$	0	0
$911$	39.1353	1.29661	0.648306	−	0.761380i	$-0.275478\pi$
$911$	39.1353	1.29661	0.648306	+	0.761380i	$0.275478\pi$
$912$	0	0
$913$	11.3391	0.375269
$914$	0	0
$915$	−3.25209	−0.107511
$916$	0	0
$917$	−17.3981	−0.574535
$918$	0	0
$919$	−47.2548	−1.55879	−0.779396	−	0.626532i	$-0.784474\pi$
$919$	−47.2548	−1.55879	−0.779396	+	0.626532i	$0.784474\pi$
$920$	0	0
$921$	−8.02436	−0.264412
$922$	0	0
$923$	3.53544	0.116370
$924$	0	0
$925$	4.12301	0.135564
$926$	0	0
$927$	9.68541	0.318111
$928$	0	0
$929$	−48.5513	−1.59292	−0.796459	−	0.604693i	$-0.793296\pi$
$929$	−48.5513	−1.59292	−0.796459	+	0.604693i	$0.793296\pi$
$930$	0	0
$931$	−14.6691	−0.480760
$932$	0	0
$933$	−6.01720	−0.196994
$934$	0	0
$935$	−5.95139	−0.194631
$936$	0	0
$937$	24.1197	0.787957	0.393978	−	0.919120i	$-0.371098\pi$
$937$	24.1197	0.787957	0.393978	+	0.919120i	$0.371098\pi$
$938$	0	0
$939$	−23.8750	−0.779131
$940$	0	0
$941$	15.2531	0.497237	0.248619	−	0.968601i	$-0.420023\pi$
$941$	15.2531	0.497237	0.248619	+	0.968601i	$0.420023\pi$
$942$	0	0
$943$	−4.90709	−0.159797
$944$	0	0
$945$	−4.07850	−0.132674
$946$	0	0
$947$	−28.5900	−0.929049	−0.464525	−	0.885560i	$-0.653775\pi$
$947$	−28.5900	−0.929049	−0.464525	+	0.885560i	$0.653775\pi$
$948$	0	0
$949$	38.5360	1.25093
$950$	0	0
$951$	16.8540	0.546530
$952$	0	0
$953$	−5.02081	−0.162640	−0.0813200	−	0.996688i	$-0.525914\pi$
$953$	−5.02081	−0.162640	−0.0813200	+	0.996688i	$0.525914\pi$
$954$	0	0
$955$	11.9970	0.388214
$956$	0	0
$957$	9.81353	0.317226
$958$	0	0
$959$	−10.0141	−0.323373
$960$	0	0
$961$	−30.6818	−0.989736
$962$	0	0
$963$	22.7081	0.731757
$964$	0	0
$965$	−7.79393	−0.250895
$966$	0	0
$967$	18.1791	0.584601	0.292300	−	0.956327i	$-0.405579\pi$
$967$	18.1791	0.584601	0.292300	+	0.956327i	$0.405579\pi$
$968$	0	0
$969$	7.06726	0.227033
$970$	0	0
$971$	−21.5831	−0.692634	−0.346317	−	0.938118i	$-0.612568\pi$
$971$	−21.5831	−0.692634	−0.346317	+	0.938118i	$0.612568\pi$
$972$	0	0
$973$	3.35979	0.107710
$974$	0	0
$975$	31.3267	1.00326
$976$	0	0
$977$	14.6723	0.469409	0.234705	−	0.972067i	$-0.424588\pi$
$977$	14.6723	0.469409	0.234705	+	0.972067i	$0.424588\pi$
$978$	0	0
$979$	8.77521	0.280457
$980$	0	0
$981$	24.8383	0.793025
$982$	0	0
$983$	−1.66628	−0.0531459	−0.0265730	−	0.999647i	$-0.508459\pi$
$983$	−1.66628	−0.0531459	−0.0265730	+	0.999647i	$0.508459\pi$
$984$	0	0
$985$	−17.8137	−0.567591
$986$	0	0
$987$	6.20370	0.197466
$988$	0	0
$989$	−7.10619	−0.225964
$990$	0	0
$991$	57.6294	1.83066	0.915329	−	0.402707i	$-0.131931\pi$
$991$	57.6294	1.83066	0.915329	+	0.402707i	$0.131931\pi$
$992$	0	0
$993$	1.01708	0.0322761
$994$	0	0
$995$	−13.4649	−0.426868
$996$	0	0
$997$	33.5092	1.06125	0.530624	−	0.847607i	$-0.321958\pi$
$997$	33.5092	1.06125	0.530624	+	0.847607i	$0.321958\pi$
$998$	0	0
$999$	−5.10595	−0.161545

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.i.1.11	yes	30
4.3	odd	2		8032.2.a.h.1.20	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.20	✓	30	4.3	odd	2
8032.2.a.i.1.11	yes	30	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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-1.12351 q^{3} +0.837743 q^{5} -0.914625 q^{7} -1.73772 q^{9} +O(q^{10})\)	\(q-1.12351 q^{3} +0.837743 q^{5} -0.914625 q^{7} -1.73772 q^{9} +2.68790 q^{11} +6.48712 q^{13} -0.941213 q^{15} -2.64299 q^{17} +2.38001 q^{19} +1.02759 q^{21} -1.02383 q^{23} -4.29819 q^{25} +5.32288 q^{27} -3.24964 q^{29} +0.564072 q^{31} -3.01988 q^{33} -0.766220 q^{35} -0.959245 q^{37} -7.28836 q^{39} +4.79287 q^{41} +6.94079 q^{43} -1.45576 q^{45} +6.03713 q^{47} -6.16346 q^{49} +2.96943 q^{51} -3.79887 q^{53} +2.25177 q^{55} -2.67396 q^{57} +1.91325 q^{59} +3.45521 q^{61} +1.58936 q^{63} +5.43454 q^{65} +13.9343 q^{67} +1.15029 q^{69} +0.544993 q^{71} +5.94039 q^{73} +4.82906 q^{75} -2.45842 q^{77} -3.81781 q^{79} -0.767154 q^{81} +4.21857 q^{83} -2.21414 q^{85} +3.65101 q^{87} +3.26471 q^{89} -5.93329 q^{91} -0.633742 q^{93} +1.99383 q^{95} -11.8133 q^{97} -4.67082 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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