LMFDB - Embedded newform 3808.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3808,2,Mod(1,3808)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3808, 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("3808.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3808.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:	$30.4070330897$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.804272.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} - 6x^{2} + 3x + 2 $ x^5 - 9x^3 - 6x^2 + 3*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.582493$ of defining polynomial
Character	$\chi$	$=$	3808.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.582493 q^{3} -3.85102 q^{5} -1.00000 q^{7} -2.66070 q^{9} -4.64792 q^{11} -2.54585 q^{13} -2.24319 q^{15} -1.00000 q^{17} -6.76496 q^{19} -0.582493 q^{21} +6.10208 q^{23} +9.83039 q^{25} -3.29732 q^{27} +3.73962 q^{29} -0.0519437 q^{31} -2.70738 q^{33} +3.85102 q^{35} -10.1597 q^{37} -1.48294 q^{39} -5.63662 q^{41} -8.99193 q^{43} +10.2464 q^{45} -6.70550 q^{47} +1.00000 q^{49} -0.582493 q^{51} +8.65356 q^{53} +17.8993 q^{55} -3.94054 q^{57} -0.380646 q^{59} +4.49425 q^{61} +2.66070 q^{63} +9.80413 q^{65} +1.65889 q^{67} +3.55442 q^{69} -4.35876 q^{71} +4.29514 q^{73} +5.72613 q^{75} +4.64792 q^{77} +9.48294 q^{79} +6.06144 q^{81} +15.7429 q^{83} +3.85102 q^{85} +2.17830 q^{87} -10.7562 q^{89} +2.54585 q^{91} -0.0302568 q^{93} +26.0520 q^{95} -9.31901 q^{97} +12.3667 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{5} - 5 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 8 q^{15} - 5 q^{17} - 6 q^{19} + 18 q^{23} + 5 q^{25} + 18 q^{27} - 14 q^{29} + 16 q^{31} - 26 q^{33} + 2 q^{35} + 2 q^{37} + 6 q^{39} - 10 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) $ 5 * q - 2 * q^5 - 5 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 8 * q^15 - 5 * q^17 - 6 * q^19 + 18 * q^23 + 5 * q^25 + 18 * q^27 - 14 * q^29 + 16 * q^31 - 26 * q^33 + 2 * q^35 + 2 * q^37 + 6 * q^39 - 10 * q^41 - 20 * q^43 + 6 * q^45 + 26 * q^47 + 5 * q^49 + 4 * q^55 + 12 * q^57 + 8 * q^61 - 3 * q^63 + 2 * q^65 + 12 * q^67 + 32 * q^69 + 10 * q^71 - 14 * q^73 - 4 * q^75 + 4 * q^77 + 34 * q^79 + 33 * q^81 + 20 * q^83 + 2 * q^85 + 14 * q^87 - 4 * q^89 + 6 * q^91 + 16 * q^93 + 42 * q^95 - 12 * q^97 - 8 * 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
$3$	0.582493	0.336302	0.168151	−	0.985761i	$-0.446220\pi$
$3$	0.582493	0.336302	0.168151	+	0.985761i	$0.446220\pi$
$4$	0	0
$5$	−3.85102	−1.72223	−0.861115	−	0.508410i	$-0.830234\pi$
$5$	−3.85102	−1.72223	−0.861115	+	0.508410i	$0.830234\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.66070	−0.886901
$10$	0	0
$11$	−4.64792	−1.40140	−0.700701	−	0.713455i	$-0.747129\pi$
$11$	−4.64792	−1.40140	−0.700701	+	0.713455i	$0.747129\pi$
$12$	0	0
$13$	−2.54585	−0.706091	−0.353046	−	0.935606i	$-0.614854\pi$
$13$	−2.54585	−0.706091	−0.353046	+	0.935606i	$0.614854\pi$
$14$	0	0
$15$	−2.24319	−0.579190
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.76496	−1.55199	−0.775994	−	0.630740i	$-0.782751\pi$
$19$	−6.76496	−1.55199	−0.775994	+	0.630740i	$0.782751\pi$
$20$	0	0
$21$	−0.582493	−0.127110
$22$	0	0
$23$	6.10208	1.27237	0.636185	−	0.771536i	$-0.280511\pi$
$23$	6.10208	1.27237	0.636185	+	0.771536i	$0.280511\pi$
$24$	0	0
$25$	9.83039	1.96608
$26$	0	0
$27$	−3.29732	−0.634569
$28$	0	0
$29$	3.73962	0.694430	0.347215	−	0.937785i	$-0.387127\pi$
$29$	3.73962	0.694430	0.347215	+	0.937785i	$0.387127\pi$
$30$	0	0
$31$	−0.0519437	−0.00932936	−0.00466468	−	0.999989i	$-0.501485\pi$
$31$	−0.0519437	−0.00932936	−0.00466468	+	0.999989i	$0.501485\pi$
$32$	0	0
$33$	−2.70738	−0.471295
$34$	0	0
$35$	3.85102	0.650942
$36$	0	0
$37$	−10.1597	−1.67024	−0.835118	−	0.550071i	$-0.814601\pi$
$37$	−10.1597	−1.67024	−0.835118	+	0.550071i	$0.814601\pi$
$38$	0	0
$39$	−1.48294	−0.237460
$40$	0	0
$41$	−5.63662	−0.880292	−0.440146	−	0.897926i	$-0.645073\pi$
$41$	−5.63662	−0.880292	−0.440146	+	0.897926i	$0.645073\pi$
$42$	0	0
$43$	−8.99193	−1.37126	−0.685628	−	0.727952i	$-0.740472\pi$
$43$	−8.99193	−1.37126	−0.685628	+	0.727952i	$0.740472\pi$
$44$	0	0
$45$	10.2464	1.52745
$46$	0	0
$47$	−6.70550	−0.978098	−0.489049	−	0.872256i	$-0.662656\pi$
$47$	−6.70550	−0.978098	−0.489049	+	0.872256i	$0.662656\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.582493	−0.0815653
$52$	0	0
$53$	8.65356	1.18866	0.594329	−	0.804222i	$-0.297418\pi$
$53$	8.65356	1.18866	0.594329	+	0.804222i	$0.297418\pi$
$54$	0	0
$55$	17.8993	2.41354
$56$	0	0
$57$	−3.94054	−0.521937
$58$	0	0
$59$	−0.380646	−0.0495558	−0.0247779	−	0.999693i	$-0.507888\pi$
$59$	−0.380646	−0.0495558	−0.0247779	+	0.999693i	$0.507888\pi$
$60$	0	0
$61$	4.49425	0.575429	0.287715	−	0.957716i	$-0.407105\pi$
$61$	4.49425	0.575429	0.287715	+	0.957716i	$0.407105\pi$
$62$	0	0
$63$	2.66070	0.335217
$64$	0	0
$65$	9.80413	1.21605
$66$	0	0
$67$	1.65889	0.202666	0.101333	−	0.994853i	$-0.467689\pi$
$67$	1.65889	0.202666	0.101333	+	0.994853i	$0.467689\pi$
$68$	0	0
$69$	3.55442	0.427901
$70$	0	0
$71$	−4.35876	−0.517290	−0.258645	−	0.965972i	$-0.583276\pi$
$71$	−4.35876	−0.517290	−0.258645	+	0.965972i	$0.583276\pi$
$72$	0	0
$73$	4.29514	0.502708	0.251354	−	0.967895i	$-0.419124\pi$
$73$	4.29514	0.502708	0.251354	+	0.967895i	$0.419124\pi$
$74$	0	0
$75$	5.72613	0.661197
$76$	0	0
$77$	4.64792	0.529680
$78$	0	0
$79$	9.48294	1.06691	0.533457	−	0.845827i	$-0.320893\pi$
$79$	9.48294	1.06691	0.533457	+	0.845827i	$0.320893\pi$
$80$	0	0
$81$	6.06144	0.673493
$82$	0	0
$83$	15.7429	1.72800	0.864001	−	0.503489i	$-0.167951\pi$
$83$	15.7429	1.72800	0.864001	+	0.503489i	$0.167951\pi$
$84$	0	0
$85$	3.85102	0.417702
$86$	0	0
$87$	2.17830	0.233539
$88$	0	0
$89$	−10.7562	−1.14015	−0.570076	−	0.821592i	$-0.693086\pi$
$89$	−10.7562	−1.14015	−0.570076	+	0.821592i	$0.693086\pi$
$90$	0	0
$91$	2.54585	0.266877
$92$	0	0
$93$	−0.0302568	−0.00313749
$94$	0	0
$95$	26.0520	2.67288
$96$	0	0
$97$	−9.31901	−0.946202	−0.473101	−	0.881008i	$-0.656865\pi$
$97$	−9.31901	−0.946202	−0.473101	+	0.881008i	$0.656865\pi$
$98$	0	0
$99$	12.3667	1.24290
$100$	0	0
$101$	−8.48458	−0.844247	−0.422124	−	0.906538i	$-0.638715\pi$
$101$	−8.48458	−0.844247	−0.422124	+	0.906538i	$0.638715\pi$
$102$	0	0
$103$	13.0322	1.28410	0.642052	−	0.766661i	$-0.278083\pi$
$103$	13.0322	1.28410	0.642052	+	0.766661i	$0.278083\pi$
$104$	0	0
$105$	2.24319	0.218913
$106$	0	0
$107$	5.38086	0.520188	0.260094	−	0.965583i	$-0.416247\pi$
$107$	5.38086	0.520188	0.260094	+	0.965583i	$0.416247\pi$
$108$	0	0
$109$	17.3264	1.65957	0.829786	−	0.558081i	$-0.188462\pi$
$109$	17.3264	1.65957	0.829786	+	0.558081i	$0.188462\pi$
$110$	0	0
$111$	−5.91793	−0.561705
$112$	0	0
$113$	−6.04698	−0.568852	−0.284426	−	0.958698i	$-0.591803\pi$
$113$	−6.04698	−0.568852	−0.284426	+	0.958698i	$0.591803\pi$
$114$	0	0
$115$	−23.4992	−2.19132
$116$	0	0
$117$	6.77374	0.626233
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	10.6032	0.963928
$122$	0	0
$123$	−3.28329	−0.296044
$124$	0	0
$125$	−18.6020	−1.66381
$126$	0	0
$127$	4.82245	0.427924	0.213962	−	0.976842i	$-0.431363\pi$
$127$	4.82245	0.427924	0.213962	+	0.976842i	$0.431363\pi$
$128$	0	0
$129$	−5.23773	−0.461157
$130$	0	0
$131$	3.54563	0.309783	0.154892	−	0.987931i	$-0.450497\pi$
$131$	3.54563	0.309783	0.154892	+	0.987931i	$0.450497\pi$
$132$	0	0
$133$	6.76496	0.586596
$134$	0	0
$135$	12.6981	1.09287
$136$	0	0
$137$	−14.5820	−1.24583	−0.622914	−	0.782291i	$-0.714051\pi$
$137$	−14.5820	−1.24583	−0.622914	+	0.782291i	$0.714051\pi$
$138$	0	0
$139$	−10.7418	−0.911108	−0.455554	−	0.890208i	$-0.650559\pi$
$139$	−10.7418	−0.911108	−0.455554	+	0.890208i	$0.650559\pi$
$140$	0	0
$141$	−3.90591	−0.328937
$142$	0	0
$143$	11.8329	0.989518
$144$	0	0
$145$	−14.4014	−1.19597
$146$	0	0
$147$	0.582493	0.0480432
$148$	0	0
$149$	5.50899	0.451314	0.225657	−	0.974207i	$-0.427547\pi$
$149$	5.50899	0.451314	0.225657	+	0.974207i	$0.427547\pi$
$150$	0	0
$151$	−0.993783	−0.0808729	−0.0404365	−	0.999182i	$-0.512875\pi$
$151$	−0.993783	−0.0808729	−0.0404365	+	0.999182i	$0.512875\pi$
$152$	0	0
$153$	2.66070	0.215105
$154$	0	0
$155$	0.200036	0.0160673
$156$	0	0
$157$	−19.2889	−1.53942	−0.769710	−	0.638394i	$-0.779599\pi$
$157$	−19.2889	−1.53942	−0.769710	+	0.638394i	$0.779599\pi$
$158$	0	0
$159$	5.04064	0.399748
$160$	0	0
$161$	−6.10208	−0.480911
$162$	0	0
$163$	−21.4931	−1.68347	−0.841734	−	0.539893i	$-0.818465\pi$
$163$	−21.4931	−1.68347	−0.841734	+	0.539893i	$0.818465\pi$
$164$	0	0
$165$	10.4262	0.811679
$166$	0	0
$167$	23.4801	1.81695	0.908473	−	0.417943i	$-0.137249\pi$
$167$	23.4801	1.81695	0.908473	+	0.417943i	$0.137249\pi$
$168$	0	0
$169$	−6.51865	−0.501435
$170$	0	0
$171$	17.9995	1.37646
$172$	0	0
$173$	−8.57694	−0.652093	−0.326046	−	0.945354i	$-0.605717\pi$
$173$	−8.57694	−0.652093	−0.326046	+	0.945354i	$0.605717\pi$
$174$	0	0
$175$	−9.83039	−0.743108
$176$	0	0
$177$	−0.221723	−0.0166657
$178$	0	0
$179$	1.32048	0.0986970	0.0493485	−	0.998782i	$-0.484286\pi$
$179$	1.32048	0.0986970	0.0493485	+	0.998782i	$0.484286\pi$
$180$	0	0
$181$	−10.5640	−0.785219	−0.392609	−	0.919705i	$-0.628428\pi$
$181$	−10.5640	−0.785219	−0.392609	+	0.919705i	$0.628428\pi$
$182$	0	0
$183$	2.61787	0.193518
$184$	0	0
$185$	39.1251	2.87653
$186$	0	0
$187$	4.64792	0.339890
$188$	0	0
$189$	3.29732	0.239845
$190$	0	0
$191$	14.1277	1.02225	0.511123	−	0.859508i	$-0.329230\pi$
$191$	14.1277	1.02225	0.511123	+	0.859508i	$0.329230\pi$
$192$	0	0
$193$	1.91288	0.137692	0.0688461	−	0.997627i	$-0.478068\pi$
$193$	1.91288	0.137692	0.0688461	+	0.997627i	$0.478068\pi$
$194$	0	0
$195$	5.71083	0.408961
$196$	0	0
$197$	−13.2364	−0.943054	−0.471527	−	0.881852i	$-0.656297\pi$
$197$	−13.2364	−0.943054	−0.471527	+	0.881852i	$0.656297\pi$
$198$	0	0
$199$	17.5021	1.24069	0.620344	−	0.784330i	$-0.286993\pi$
$199$	17.5021	1.24069	0.620344	+	0.784330i	$0.286993\pi$
$200$	0	0
$201$	0.966292	0.0681570
$202$	0	0
$203$	−3.73962	−0.262470
$204$	0	0
$205$	21.7068	1.51607
$206$	0	0
$207$	−16.2358	−1.12847
$208$	0	0
$209$	31.4430	2.17496
$210$	0	0
$211$	−7.21316	−0.496574	−0.248287	−	0.968687i	$-0.579868\pi$
$211$	−7.21316	−0.496574	−0.248287	+	0.968687i	$0.579868\pi$
$212$	0	0
$213$	−2.53895	−0.173966
$214$	0	0
$215$	34.6281	2.36162
$216$	0	0
$217$	0.0519437	0.00352617
$218$	0	0
$219$	2.50189	0.169062
$220$	0	0
$221$	2.54585	0.171252
$222$	0	0
$223$	−1.21747	−0.0815279	−0.0407639	−	0.999169i	$-0.512979\pi$
$223$	−1.21747	−0.0815279	−0.0407639	+	0.999169i	$0.512979\pi$
$224$	0	0
$225$	−26.1557	−1.74372
$226$	0	0
$227$	22.9496	1.52322	0.761608	−	0.648037i	$-0.224410\pi$
$227$	22.9496	1.52322	0.761608	+	0.648037i	$0.224410\pi$
$228$	0	0
$229$	−13.2440	−0.875187	−0.437594	−	0.899173i	$-0.644169\pi$
$229$	−13.2440	−0.875187	−0.437594	+	0.899173i	$0.644169\pi$
$230$	0	0
$231$	2.70738	0.178133
$232$	0	0
$233$	−1.90483	−0.124789	−0.0623946	−	0.998052i	$-0.519874\pi$
$233$	−1.90483	−0.124789	−0.0623946	+	0.998052i	$0.519874\pi$
$234$	0	0
$235$	25.8230	1.68451
$236$	0	0
$237$	5.52375	0.358806
$238$	0	0
$239$	−19.0815	−1.23428	−0.617141	−	0.786853i	$-0.711709\pi$
$239$	−19.0815	−1.23428	−0.617141	+	0.786853i	$0.711709\pi$
$240$	0	0
$241$	−11.1006	−0.715050	−0.357525	−	0.933904i	$-0.616379\pi$
$241$	−11.1006	−0.715050	−0.357525	+	0.933904i	$0.616379\pi$
$242$	0	0
$243$	13.4227	0.861067
$244$	0	0
$245$	−3.85102	−0.246033
$246$	0	0
$247$	17.2226	1.09585
$248$	0	0
$249$	9.17010	0.581132
$250$	0	0
$251$	22.8290	1.44095	0.720477	−	0.693479i	$-0.243923\pi$
$251$	22.8290	1.44095	0.720477	+	0.693479i	$0.243923\pi$
$252$	0	0
$253$	−28.3620	−1.78310
$254$	0	0
$255$	2.24319	0.140474
$256$	0	0
$257$	−0.707384	−0.0441254	−0.0220627	−	0.999757i	$-0.507023\pi$
$257$	−0.707384	−0.0441254	−0.0220627	+	0.999757i	$0.507023\pi$
$258$	0	0
$259$	10.1597	0.631290
$260$	0	0
$261$	−9.95002	−0.615891
$262$	0	0
$263$	−3.29835	−0.203385	−0.101693	−	0.994816i	$-0.532426\pi$
$263$	−3.29835	−0.203385	−0.101693	+	0.994816i	$0.532426\pi$
$264$	0	0
$265$	−33.3251	−2.04714
$266$	0	0
$267$	−6.26540	−0.383436
$268$	0	0
$269$	−13.2226	−0.806194	−0.403097	−	0.915157i	$-0.632066\pi$
$269$	−13.2226	−0.806194	−0.403097	+	0.915157i	$0.632066\pi$
$270$	0	0
$271$	−17.6640	−1.07301	−0.536506	−	0.843896i	$-0.680256\pi$
$271$	−17.6640	−1.07301	−0.536506	+	0.843896i	$0.680256\pi$
$272$	0	0
$273$	1.48294	0.0897516
$274$	0	0
$275$	−45.6909	−2.75527
$276$	0	0
$277$	27.8416	1.67284	0.836421	−	0.548088i	$-0.184644\pi$
$277$	27.8416	1.67284	0.836421	+	0.548088i	$0.184644\pi$
$278$	0	0
$279$	0.138207	0.00827422
$280$	0	0
$281$	15.5162	0.925621	0.462810	−	0.886457i	$-0.346841\pi$
$281$	15.5162	0.925621	0.462810	+	0.886457i	$0.346841\pi$
$282$	0	0
$283$	12.6717	0.753254	0.376627	−	0.926365i	$-0.377084\pi$
$283$	12.6717	0.753254	0.376627	+	0.926365i	$0.377084\pi$
$284$	0	0
$285$	15.1751	0.898897
$286$	0	0
$287$	5.63662	0.332719
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−5.42826	−0.318210
$292$	0	0
$293$	7.65569	0.447250	0.223625	−	0.974675i	$-0.428211\pi$
$293$	7.65569	0.447250	0.223625	+	0.974675i	$0.428211\pi$
$294$	0	0
$295$	1.46588	0.0853466
$296$	0	0
$297$	15.3257	0.889287
$298$	0	0
$299$	−15.5350	−0.898410
$300$	0	0
$301$	8.99193	0.518286
$302$	0	0
$303$	−4.94221	−0.283922
$304$	0	0
$305$	−17.3075	−0.991022
$306$	0	0
$307$	−33.6476	−1.92037	−0.960186	−	0.279362i	$-0.909877\pi$
$307$	−33.6476	−1.92037	−0.960186	+	0.279362i	$0.909877\pi$
$308$	0	0
$309$	7.59119	0.431848
$310$	0	0
$311$	−21.5180	−1.22017	−0.610086	−	0.792335i	$-0.708865\pi$
$311$	−21.5180	−1.22017	−0.610086	+	0.792335i	$0.708865\pi$
$312$	0	0
$313$	−19.2044	−1.08550	−0.542750	−	0.839894i	$-0.682617\pi$
$313$	−19.2044	−1.08550	−0.542750	+	0.839894i	$0.682617\pi$
$314$	0	0
$315$	−10.2464	−0.577321
$316$	0	0
$317$	14.7492	0.828400	0.414200	−	0.910186i	$-0.364061\pi$
$317$	14.7492	0.828400	0.414200	+	0.910186i	$0.364061\pi$
$318$	0	0
$319$	−17.3815	−0.973176
$320$	0	0
$321$	3.13431	0.174940
$322$	0	0
$323$	6.76496	0.376412
$324$	0	0
$325$	−25.0267	−1.38823
$326$	0	0
$327$	10.0925	0.558118
$328$	0	0
$329$	6.70550	0.369686
$330$	0	0
$331$	−22.0766	−1.21344	−0.606720	−	0.794916i	$-0.707515\pi$
$331$	−22.0766	−1.21344	−0.606720	+	0.794916i	$0.707515\pi$
$332$	0	0
$333$	27.0318	1.48133
$334$	0	0
$335$	−6.38843	−0.349037
$336$	0	0
$337$	−19.8836	−1.08313	−0.541564	−	0.840660i	$-0.682168\pi$
$337$	−19.8836	−1.08313	−0.541564	+	0.840660i	$0.682168\pi$
$338$	0	0
$339$	−3.52232	−0.191306
$340$	0	0
$341$	0.241430	0.0130742
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−13.6881	−0.736945
$346$	0	0
$347$	10.9728	0.589050	0.294525	−	0.955644i	$-0.404839\pi$
$347$	10.9728	0.589050	0.294525	+	0.955644i	$0.404839\pi$
$348$	0	0
$349$	−1.84028	−0.0985077	−0.0492539	−	0.998786i	$-0.515684\pi$
$349$	−1.84028	−0.0985077	−0.0492539	+	0.998786i	$0.515684\pi$
$350$	0	0
$351$	8.39448	0.448064
$352$	0	0
$353$	23.1182	1.23046	0.615228	−	0.788349i	$-0.289064\pi$
$353$	23.1182	1.23046	0.615228	+	0.788349i	$0.289064\pi$
$354$	0	0
$355$	16.7857	0.890892
$356$	0	0
$357$	0.582493	0.0308288
$358$	0	0
$359$	26.3800	1.39228	0.696142	−	0.717904i	$-0.254898\pi$
$359$	26.3800	1.39228	0.696142	+	0.717904i	$0.254898\pi$
$360$	0	0
$361$	26.7647	1.40867
$362$	0	0
$363$	6.17629	0.324171
$364$	0	0
$365$	−16.5407	−0.865779
$366$	0	0
$367$	−23.4895	−1.22614	−0.613072	−	0.790027i	$-0.710066\pi$
$367$	−23.4895	−1.22614	−0.613072	+	0.790027i	$0.710066\pi$
$368$	0	0
$369$	14.9974	0.780731
$370$	0	0
$371$	−8.65356	−0.449270
$372$	0	0
$373$	−18.8168	−0.974296	−0.487148	−	0.873319i	$-0.661963\pi$
$373$	−18.8168	−0.974296	−0.487148	+	0.873319i	$0.661963\pi$
$374$	0	0
$375$	−10.8355	−0.559543
$376$	0	0
$377$	−9.52051	−0.490331
$378$	0	0
$379$	20.5866	1.05746	0.528731	−	0.848790i	$-0.322668\pi$
$379$	20.5866	1.05746	0.528731	+	0.848790i	$0.322668\pi$
$380$	0	0
$381$	2.80905	0.143912
$382$	0	0
$383$	−14.6293	−0.747521	−0.373760	−	0.927525i	$-0.621932\pi$
$383$	−14.6293	−0.747521	−0.373760	+	0.927525i	$0.621932\pi$
$384$	0	0
$385$	−17.8993	−0.912231
$386$	0	0
$387$	23.9248	1.21617
$388$	0	0
$389$	11.4040	0.578207	0.289103	−	0.957298i	$-0.406643\pi$
$389$	11.4040	0.578207	0.289103	+	0.957298i	$0.406643\pi$
$390$	0	0
$391$	−6.10208	−0.308595
$392$	0	0
$393$	2.06531	0.104181
$394$	0	0
$395$	−36.5190	−1.83747
$396$	0	0
$397$	−28.4747	−1.42911	−0.714553	−	0.699581i	$-0.753370\pi$
$397$	−28.4747	−1.42911	−0.714553	+	0.699581i	$0.753370\pi$
$398$	0	0
$399$	3.94054	0.197274
$400$	0	0
$401$	21.1048	1.05392	0.526961	−	0.849889i	$-0.323331\pi$
$401$	21.1048	1.05392	0.526961	+	0.849889i	$0.323331\pi$
$402$	0	0
$403$	0.132241	0.00658738
$404$	0	0
$405$	−23.3428	−1.15991
$406$	0	0
$407$	47.2213	2.34067
$408$	0	0
$409$	33.2177	1.64251	0.821254	−	0.570562i	$-0.193275\pi$
$409$	33.2177	1.64251	0.821254	+	0.570562i	$0.193275\pi$
$410$	0	0
$411$	−8.49393	−0.418975
$412$	0	0
$413$	0.380646	0.0187303
$414$	0	0
$415$	−60.6261	−2.97602
$416$	0	0
$417$	−6.25702	−0.306408
$418$	0	0
$419$	26.2462	1.28221	0.641105	−	0.767454i	$-0.278476\pi$
$419$	26.2462	1.28221	0.641105	+	0.767454i	$0.278476\pi$
$420$	0	0
$421$	3.79079	0.184752	0.0923759	−	0.995724i	$-0.470554\pi$
$421$	3.79079	0.184752	0.0923759	+	0.995724i	$0.470554\pi$
$422$	0	0
$423$	17.8413	0.867475
$424$	0	0
$425$	−9.83039	−0.476844
$426$	0	0
$427$	−4.49425	−0.217492
$428$	0	0
$429$	6.89259	0.332777
$430$	0	0
$431$	37.0751	1.78584	0.892922	−	0.450211i	$-0.148651\pi$
$431$	37.0751	1.78584	0.892922	+	0.450211i	$0.148651\pi$
$432$	0	0
$433$	−18.3813	−0.883349	−0.441675	−	0.897175i	$-0.645615\pi$
$433$	−18.3813	−0.883349	−0.441675	+	0.897175i	$0.645615\pi$
$434$	0	0
$435$	−8.38870	−0.402207
$436$	0	0
$437$	−41.2803	−1.97470
$438$	0	0
$439$	−12.9794	−0.619475	−0.309738	−	0.950822i	$-0.600241\pi$
$439$	−12.9794	−0.619475	−0.309738	+	0.950822i	$0.600241\pi$
$440$	0	0
$441$	−2.66070	−0.126700
$442$	0	0
$443$	−21.0667	−1.00091	−0.500455	−	0.865762i	$-0.666834\pi$
$443$	−21.0667	−1.00091	−0.500455	+	0.865762i	$0.666834\pi$
$444$	0	0
$445$	41.4223	1.96360
$446$	0	0
$447$	3.20895	0.151778
$448$	0	0
$449$	−20.4265	−0.963986	−0.481993	−	0.876175i	$-0.660087\pi$
$449$	−20.4265	−0.963986	−0.481993	+	0.876175i	$0.660087\pi$
$450$	0	0
$451$	26.1986	1.23364
$452$	0	0
$453$	−0.578872	−0.0271978
$454$	0	0
$455$	−9.80413	−0.459625
$456$	0	0
$457$	−15.8962	−0.743593	−0.371796	−	0.928314i	$-0.621258\pi$
$457$	−15.8962	−0.743593	−0.371796	+	0.928314i	$0.621258\pi$
$458$	0	0
$459$	3.29732	0.153906
$460$	0	0
$461$	−22.0933	−1.02899	−0.514494	−	0.857494i	$-0.672020\pi$
$461$	−22.0933	−1.02899	−0.514494	+	0.857494i	$0.672020\pi$
$462$	0	0
$463$	−33.7663	−1.56925	−0.784627	−	0.619968i	$-0.787146\pi$
$463$	−33.7663	−1.56925	−0.784627	+	0.619968i	$0.787146\pi$
$464$	0	0
$465$	0.116520	0.00540348
$466$	0	0
$467$	24.4025	1.12921	0.564607	−	0.825360i	$-0.309028\pi$
$467$	24.4025	1.12921	0.564607	+	0.825360i	$0.309028\pi$
$468$	0	0
$469$	−1.65889	−0.0766005
$470$	0	0
$471$	−11.2356	−0.517711
$472$	0	0
$473$	41.7938	1.92168
$474$	0	0
$475$	−66.5022	−3.05133
$476$	0	0
$477$	−23.0245	−1.05422
$478$	0	0
$479$	4.38153	0.200197	0.100099	−	0.994978i	$-0.468084\pi$
$479$	4.38153	0.200197	0.100099	+	0.994978i	$0.468084\pi$
$480$	0	0
$481$	25.8649	1.17934
$482$	0	0
$483$	−3.55442	−0.161732
$484$	0	0
$485$	35.8877	1.62958
$486$	0	0
$487$	37.4055	1.69500	0.847502	−	0.530792i	$-0.178105\pi$
$487$	37.4055	1.69500	0.847502	+	0.530792i	$0.178105\pi$
$488$	0	0
$489$	−12.5196	−0.566154
$490$	0	0
$491$	9.91229	0.447336	0.223668	−	0.974665i	$-0.428197\pi$
$491$	9.91229	0.447336	0.223668	+	0.974665i	$0.428197\pi$
$492$	0	0
$493$	−3.73962	−0.168424
$494$	0	0
$495$	−47.6246	−2.14057
$496$	0	0
$497$	4.35876	0.195517
$498$	0	0
$499$	23.0416	1.03149	0.515743	−	0.856743i	$-0.327516\pi$
$499$	23.0416	1.03149	0.515743	+	0.856743i	$0.327516\pi$
$500$	0	0
$501$	13.6770	0.611044
$502$	0	0
$503$	17.0649	0.760885	0.380443	−	0.924805i	$-0.375772\pi$
$503$	17.0649	0.760885	0.380443	+	0.924805i	$0.375772\pi$
$504$	0	0
$505$	32.6743	1.45399
$506$	0	0
$507$	−3.79707	−0.168634
$508$	0	0
$509$	−5.28706	−0.234345	−0.117173	−	0.993112i	$-0.537383\pi$
$509$	−5.28706	−0.234345	−0.117173	+	0.993112i	$0.537383\pi$
$510$	0	0
$511$	−4.29514	−0.190006
$512$	0	0
$513$	22.3062	0.984844
$514$	0	0
$515$	−50.1875	−2.21152
$516$	0	0
$517$	31.1667	1.37071
$518$	0	0
$519$	−4.99601	−0.219300
$520$	0	0
$521$	28.1368	1.23269	0.616347	−	0.787474i	$-0.288612\pi$
$521$	28.1368	1.23269	0.616347	+	0.787474i	$0.288612\pi$
$522$	0	0
$523$	−24.1414	−1.05563	−0.527816	−	0.849359i	$-0.676989\pi$
$523$	−24.1414	−1.05563	−0.527816	+	0.849359i	$0.676989\pi$
$524$	0	0
$525$	−5.72613	−0.249909
$526$	0	0
$527$	0.0519437	0.00226270
$528$	0	0
$529$	14.2353	0.618928
$530$	0	0
$531$	1.01278	0.0439511
$532$	0	0
$533$	14.3500	0.621566
$534$	0	0
$535$	−20.7218	−0.895883
$536$	0	0
$537$	0.769168	0.0331920
$538$	0	0
$539$	−4.64792	−0.200200
$540$	0	0
$541$	29.3315	1.26106	0.630531	−	0.776164i	$-0.282837\pi$
$541$	29.3315	1.26106	0.630531	+	0.776164i	$0.282837\pi$
$542$	0	0
$543$	−6.15348	−0.264071
$544$	0	0
$545$	−66.7246	−2.85817
$546$	0	0
$547$	−20.6034	−0.880937	−0.440468	−	0.897768i	$-0.645188\pi$
$547$	−20.6034	−0.880937	−0.440468	+	0.897768i	$0.645188\pi$
$548$	0	0
$549$	−11.9579	−0.510349
$550$	0	0
$551$	−25.2984	−1.07775
$552$	0	0
$553$	−9.48294	−0.403256
$554$	0	0
$555$	22.7901	0.967385
$556$	0	0
$557$	−42.5588	−1.80327	−0.901636	−	0.432495i	$-0.857633\pi$
$557$	−42.5588	−1.80327	−0.901636	+	0.432495i	$0.857633\pi$
$558$	0	0
$559$	22.8921	0.968232
$560$	0	0
$561$	2.70738	0.114306
$562$	0	0
$563$	−38.6153	−1.62744	−0.813721	−	0.581256i	$-0.802562\pi$
$563$	−38.6153	−1.62744	−0.813721	+	0.581256i	$0.802562\pi$
$564$	0	0
$565$	23.2871	0.979694
$566$	0	0
$567$	−6.06144	−0.254557
$568$	0	0
$569$	9.65111	0.404596	0.202298	−	0.979324i	$-0.435159\pi$
$569$	9.65111	0.404596	0.202298	+	0.979324i	$0.435159\pi$
$570$	0	0
$571$	−22.6533	−0.948013	−0.474006	−	0.880521i	$-0.657193\pi$
$571$	−22.6533	−0.948013	−0.474006	+	0.880521i	$0.657193\pi$
$572$	0	0
$573$	8.22929	0.343784
$574$	0	0
$575$	59.9858	2.50158
$576$	0	0
$577$	3.15762	0.131454	0.0657268	−	0.997838i	$-0.479063\pi$
$577$	3.15762	0.131454	0.0657268	+	0.997838i	$0.479063\pi$
$578$	0	0
$579$	1.11424	0.0463062
$580$	0	0
$581$	−15.7429	−0.653124
$582$	0	0
$583$	−40.2211	−1.66579
$584$	0	0
$585$	−26.0859	−1.07852
$586$	0	0
$587$	21.9850	0.907420	0.453710	−	0.891149i	$-0.350100\pi$
$587$	21.9850	0.907420	0.453710	+	0.891149i	$0.350100\pi$
$588$	0	0
$589$	0.351397	0.0144791
$590$	0	0
$591$	−7.71011	−0.317151
$592$	0	0
$593$	−14.4951	−0.595243	−0.297621	−	0.954684i	$-0.596193\pi$
$593$	−14.4951	−0.595243	−0.297621	+	0.954684i	$0.596193\pi$
$594$	0	0
$595$	−3.85102	−0.157877
$596$	0	0
$597$	10.1948	0.417246
$598$	0	0
$599$	27.3702	1.11832	0.559159	−	0.829061i	$-0.311124\pi$
$599$	27.3702	1.11832	0.559159	+	0.829061i	$0.311124\pi$
$600$	0	0
$601$	−12.9688	−0.529009	−0.264505	−	0.964384i	$-0.585208\pi$
$601$	−12.9688	−0.529009	−0.264505	+	0.964384i	$0.585208\pi$
$602$	0	0
$603$	−4.41381	−0.179744
$604$	0	0
$605$	−40.8332	−1.66011
$606$	0	0
$607$	12.8438	0.521315	0.260658	−	0.965431i	$-0.416061\pi$
$607$	12.8438	0.521315	0.260658	+	0.965431i	$0.416061\pi$
$608$	0	0
$609$	−2.17830	−0.0882693
$610$	0	0
$611$	17.0712	0.690626
$612$	0	0
$613$	−42.0422	−1.69807	−0.849034	−	0.528338i	$-0.822815\pi$
$613$	−42.0422	−1.69807	−0.849034	+	0.528338i	$0.822815\pi$
$614$	0	0
$615$	12.6440	0.509857
$616$	0	0
$617$	−35.5895	−1.43278	−0.716390	−	0.697700i	$-0.754207\pi$
$617$	−35.5895	−1.43278	−0.716390	+	0.697700i	$0.754207\pi$
$618$	0	0
$619$	15.7453	0.632858	0.316429	−	0.948616i	$-0.397516\pi$
$619$	15.7453	0.632858	0.316429	+	0.948616i	$0.397516\pi$
$620$	0	0
$621$	−20.1205	−0.807408
$622$	0	0
$623$	10.7562	0.430937
$624$	0	0
$625$	22.4846	0.899385
$626$	0	0
$627$	18.3153	0.731444
$628$	0	0
$629$	10.1597	0.405092
$630$	0	0
$631$	8.70746	0.346639	0.173319	−	0.984866i	$-0.444551\pi$
$631$	8.70746	0.346639	0.173319	+	0.984866i	$0.444551\pi$
$632$	0	0
$633$	−4.20161	−0.166999
$634$	0	0
$635$	−18.5714	−0.736983
$636$	0	0
$637$	−2.54585	−0.100870
$638$	0	0
$639$	11.5974	0.458784
$640$	0	0
$641$	18.0314	0.712198	0.356099	−	0.934448i	$-0.384107\pi$
$641$	18.0314	0.712198	0.356099	+	0.934448i	$0.384107\pi$
$642$	0	0
$643$	−29.5341	−1.16471	−0.582355	−	0.812934i	$-0.697869\pi$
$643$	−29.5341	−1.16471	−0.582355	+	0.812934i	$0.697869\pi$
$644$	0	0
$645$	20.1706	0.794218
$646$	0	0
$647$	−11.2951	−0.444056	−0.222028	−	0.975040i	$-0.571268\pi$
$647$	−11.2951	−0.444056	−0.222028	+	0.975040i	$0.571268\pi$
$648$	0	0
$649$	1.76921	0.0694476
$650$	0	0
$651$	0.0302568	0.00118586
$652$	0	0
$653$	21.3164	0.834174	0.417087	−	0.908867i	$-0.363051\pi$
$653$	21.3164	0.834174	0.417087	+	0.908867i	$0.363051\pi$
$654$	0	0
$655$	−13.6543	−0.533518
$656$	0	0
$657$	−11.4281	−0.445852
$658$	0	0
$659$	−27.4874	−1.07076	−0.535379	−	0.844612i	$-0.679831\pi$
$659$	−27.4874	−1.07076	−0.535379	+	0.844612i	$0.679831\pi$
$660$	0	0
$661$	−23.6276	−0.919008	−0.459504	−	0.888176i	$-0.651973\pi$
$661$	−23.6276	−0.919008	−0.459504	+	0.888176i	$0.651973\pi$
$662$	0	0
$663$	1.48294	0.0575926
$664$	0	0
$665$	−26.0520	−1.01025
$666$	0	0
$667$	22.8195	0.883573
$668$	0	0
$669$	−0.709168	−0.0274180
$670$	0	0
$671$	−20.8889	−0.806408
$672$	0	0
$673$	29.7545	1.14695	0.573476	−	0.819222i	$-0.305595\pi$
$673$	29.7545	1.14695	0.573476	+	0.819222i	$0.305595\pi$
$674$	0	0
$675$	−32.4139	−1.24761
$676$	0	0
$677$	38.3681	1.47461	0.737303	−	0.675562i	$-0.236099\pi$
$677$	38.3681	1.47461	0.737303	+	0.675562i	$0.236099\pi$
$678$	0	0
$679$	9.31901	0.357631
$680$	0	0
$681$	13.3680	0.512262
$682$	0	0
$683$	31.8142	1.21734	0.608668	−	0.793425i	$-0.291704\pi$
$683$	31.8142	1.21734	0.608668	+	0.793425i	$0.291704\pi$
$684$	0	0
$685$	56.1558	2.14560
$686$	0	0
$687$	−7.71453	−0.294328
$688$	0	0
$689$	−22.0306	−0.839301
$690$	0	0
$691$	13.1861	0.501623	0.250812	−	0.968036i	$-0.419303\pi$
$691$	13.1861	0.501623	0.250812	+	0.968036i	$0.419303\pi$
$692$	0	0
$693$	−12.3667	−0.469774
$694$	0	0
$695$	41.3669	1.56914
$696$	0	0
$697$	5.63662	0.213502
$698$	0	0
$699$	−1.10955	−0.0419669
$700$	0	0
$701$	26.0297	0.983127	0.491564	−	0.870842i	$-0.336426\pi$
$701$	26.0297	0.983127	0.491564	+	0.870842i	$0.336426\pi$
$702$	0	0
$703$	68.7296	2.59219
$704$	0	0
$705$	15.0417	0.566505
$706$	0	0
$707$	8.48458	0.319095
$708$	0	0
$709$	−45.6490	−1.71438	−0.857191	−	0.514998i	$-0.827793\pi$
$709$	−45.6490	−1.71438	−0.857191	+	0.514998i	$0.827793\pi$
$710$	0	0
$711$	−25.2313	−0.946247
$712$	0	0
$713$	−0.316964	−0.0118704
$714$	0	0
$715$	−45.5688	−1.70418
$716$	0	0
$717$	−11.1149	−0.415092
$718$	0	0
$719$	−21.8166	−0.813621	−0.406810	−	0.913513i	$-0.633359\pi$
$719$	−21.8166	−0.813621	−0.406810	+	0.913513i	$0.633359\pi$
$720$	0	0
$721$	−13.0322	−0.485346
$722$	0	0
$723$	−6.46600	−0.240473
$724$	0	0
$725$	36.7619	1.36530
$726$	0	0
$727$	−44.3721	−1.64567	−0.822834	−	0.568281i	$-0.807608\pi$
$727$	−44.3721	−1.64567	−0.822834	+	0.568281i	$0.807608\pi$
$728$	0	0
$729$	−10.3657	−0.383914
$730$	0	0
$731$	8.99193	0.332578
$732$	0	0
$733$	26.5767	0.981634	0.490817	−	0.871263i	$-0.336698\pi$
$733$	26.5767	0.981634	0.490817	+	0.871263i	$0.336698\pi$
$734$	0	0
$735$	−2.24319	−0.0827415
$736$	0	0
$737$	−7.71040	−0.284016
$738$	0	0
$739$	17.2009	0.632744	0.316372	−	0.948635i	$-0.397535\pi$
$739$	17.2009	0.632744	0.316372	+	0.948635i	$0.397535\pi$
$740$	0	0
$741$	10.0320	0.368536
$742$	0	0
$743$	29.8491	1.09506	0.547528	−	0.836787i	$-0.315569\pi$
$743$	29.8491	1.09506	0.547528	+	0.836787i	$0.315569\pi$
$744$	0	0
$745$	−21.2152	−0.777266
$746$	0	0
$747$	−41.8870	−1.53257
$748$	0	0
$749$	−5.38086	−0.196612
$750$	0	0
$751$	−9.81591	−0.358188	−0.179094	−	0.983832i	$-0.557317\pi$
$751$	−9.81591	−0.358188	−0.179094	+	0.983832i	$0.557317\pi$
$752$	0	0
$753$	13.2977	0.484596
$754$	0	0
$755$	3.82708	0.139282
$756$	0	0
$757$	49.8321	1.81118	0.905589	−	0.424157i	$-0.139429\pi$
$757$	49.8321	1.81118	0.905589	+	0.424157i	$0.139429\pi$
$758$	0	0
$759$	−16.5207	−0.599662
$760$	0	0
$761$	−37.2833	−1.35152	−0.675760	−	0.737122i	$-0.736184\pi$
$761$	−37.2833	−1.35152	−0.675760	+	0.737122i	$0.736184\pi$
$762$	0	0
$763$	−17.3264	−0.627260
$764$	0	0
$765$	−10.2464	−0.370460
$766$	0	0
$767$	0.969066	0.0349909
$768$	0	0
$769$	5.08067	0.183214	0.0916068	−	0.995795i	$-0.470800\pi$
$769$	5.08067	0.183214	0.0916068	+	0.995795i	$0.470800\pi$
$770$	0	0
$771$	−0.412046	−0.0148395
$772$	0	0
$773$	17.5543	0.631384	0.315692	−	0.948862i	$-0.397763\pi$
$773$	17.5543	0.631384	0.315692	+	0.948862i	$0.397763\pi$
$774$	0	0
$775$	−0.510627	−0.0183423
$776$	0	0
$777$	5.91793	0.212304
$778$	0	0
$779$	38.1315	1.36620
$780$	0	0
$781$	20.2592	0.724931
$782$	0	0
$783$	−12.3307	−0.440664
$784$	0	0
$785$	74.2819	2.65124
$786$	0	0
$787$	37.9118	1.35141	0.675706	−	0.737172i	$-0.263839\pi$
$787$	37.9118	1.35141	0.675706	+	0.737172i	$0.263839\pi$
$788$	0	0
$789$	−1.92127	−0.0683990
$790$	0	0
$791$	6.04698	0.215006
$792$	0	0
$793$	−11.4417	−0.406306
$794$	0	0
$795$	−19.4116	−0.688459
$796$	0	0
$797$	−16.4612	−0.583086	−0.291543	−	0.956558i	$-0.594169\pi$
$797$	−16.4612	−0.583086	−0.291543	+	0.956558i	$0.594169\pi$
$798$	0	0
$799$	6.70550	0.237224
$800$	0	0
$801$	28.6190	1.01120
$802$	0	0
$803$	−19.9635	−0.704496
$804$	0	0
$805$	23.4992	0.828240
$806$	0	0
$807$	−7.70205	−0.271125
$808$	0	0
$809$	−43.6942	−1.53621	−0.768103	−	0.640326i	$-0.778799\pi$
$809$	−43.6942	−1.53621	−0.768103	+	0.640326i	$0.778799\pi$
$810$	0	0
$811$	−16.9380	−0.594774	−0.297387	−	0.954757i	$-0.596115\pi$
$811$	−16.9380	−0.594774	−0.297387	+	0.954757i	$0.596115\pi$
$812$	0	0
$813$	−10.2892	−0.360857
$814$	0	0
$815$	82.7704	2.89932
$816$	0	0
$817$	60.8300	2.12817
$818$	0	0
$819$	−6.77374	−0.236694
$820$	0	0
$821$	9.90439	0.345666	0.172833	−	0.984951i	$-0.444708\pi$
$821$	9.90439	0.345666	0.172833	+	0.984951i	$0.444708\pi$
$822$	0	0
$823$	38.2148	1.33209	0.666043	−	0.745914i	$-0.267987\pi$
$823$	38.2148	1.33209	0.666043	+	0.745914i	$0.267987\pi$
$824$	0	0
$825$	−26.6146	−0.926603
$826$	0	0
$827$	47.0218	1.63511	0.817554	−	0.575851i	$-0.195329\pi$
$827$	47.0218	1.63511	0.817554	+	0.575851i	$0.195329\pi$
$828$	0	0
$829$	−9.95734	−0.345833	−0.172916	−	0.984937i	$-0.555319\pi$
$829$	−9.95734	−0.345833	−0.172916	+	0.984937i	$0.555319\pi$
$830$	0	0
$831$	16.2176	0.562581
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−90.4225	−3.12920
$836$	0	0
$837$	0.171275	0.00592013
$838$	0	0
$839$	30.9370	1.06807	0.534033	−	0.845464i	$-0.320676\pi$
$839$	30.9370	1.06807	0.534033	+	0.845464i	$0.320676\pi$
$840$	0	0
$841$	−15.0152	−0.517766
$842$	0	0
$843$	9.03810	0.311289
$844$	0	0
$845$	25.1035	0.863587
$846$	0	0
$847$	−10.6032	−0.364330
$848$	0	0
$849$	7.38118	0.253321
$850$	0	0
$851$	−61.9950	−2.12516
$852$	0	0
$853$	−22.1629	−0.758843	−0.379421	−	0.925224i	$-0.623877\pi$
$853$	−22.1629	−0.758843	−0.379421	+	0.925224i	$0.623877\pi$
$854$	0	0
$855$	−69.3167	−2.37058
$856$	0	0
$857$	−27.4914	−0.939088	−0.469544	−	0.882909i	$-0.655582\pi$
$857$	−27.4914	−0.939088	−0.469544	+	0.882909i	$0.655582\pi$
$858$	0	0
$859$	1.90545	0.0650130	0.0325065	−	0.999472i	$-0.489651\pi$
$859$	1.90545	0.0650130	0.0325065	+	0.999472i	$0.489651\pi$
$860$	0	0
$861$	3.28329	0.111894
$862$	0	0
$863$	−0.423210	−0.0144062	−0.00720312	−	0.999974i	$-0.502293\pi$
$863$	−0.423210	−0.0144062	−0.00720312	+	0.999974i	$0.502293\pi$
$864$	0	0
$865$	33.0300	1.12305
$866$	0	0
$867$	0.582493	0.0197825
$868$	0	0
$869$	−44.0760	−1.49518
$870$	0	0
$871$	−4.22329	−0.143101
$872$	0	0
$873$	24.7951	0.839187
$874$	0	0
$875$	18.6020	0.628861
$876$	0	0
$877$	11.6562	0.393603	0.196802	−	0.980443i	$-0.436945\pi$
$877$	11.6562	0.393603	0.196802	+	0.980443i	$0.436945\pi$
$878$	0	0
$879$	4.45939	0.150411
$880$	0	0
$881$	22.4703	0.757043	0.378522	−	0.925592i	$-0.376433\pi$
$881$	22.4703	0.757043	0.378522	+	0.925592i	$0.376433\pi$
$882$	0	0
$883$	−40.3687	−1.35851	−0.679257	−	0.733900i	$-0.737698\pi$
$883$	−40.3687	−1.35851	−0.679257	+	0.733900i	$0.737698\pi$
$884$	0	0
$885$	0.853862	0.0287023
$886$	0	0
$887$	−23.8461	−0.800673	−0.400337	−	0.916368i	$-0.631107\pi$
$887$	−23.8461	−0.800673	−0.400337	+	0.916368i	$0.631107\pi$
$888$	0	0
$889$	−4.82245	−0.161740
$890$	0	0
$891$	−28.1731	−0.943835
$892$	0	0
$893$	45.3624	1.51800
$894$	0	0
$895$	−5.08518	−0.169979
$896$	0	0
$897$	−9.04901	−0.302138
$898$	0	0
$899$	−0.194250	−0.00647859
$900$	0	0
$901$	−8.65356	−0.288292
$902$	0	0
$903$	5.23773	0.174301
$904$	0	0
$905$	40.6824	1.35233
$906$	0	0
$907$	22.0212	0.731202	0.365601	−	0.930772i	$-0.380864\pi$
$907$	22.0212	0.731202	0.365601	+	0.930772i	$0.380864\pi$
$908$	0	0
$909$	22.5749	0.748763
$910$	0	0
$911$	34.7769	1.15221	0.576105	−	0.817376i	$-0.304572\pi$
$911$	34.7769	1.15221	0.576105	+	0.817376i	$0.304572\pi$
$912$	0	0
$913$	−73.1716	−2.42163
$914$	0	0
$915$	−10.0815	−0.333283
$916$	0	0
$917$	−3.54563	−0.117087
$918$	0	0
$919$	56.0414	1.84864	0.924318	−	0.381623i	$-0.124635\pi$
$919$	56.0414	1.84864	0.924318	+	0.381623i	$0.124635\pi$
$920$	0	0
$921$	−19.5995	−0.645826
$922$	0	0
$923$	11.0967	0.365254
$924$	0	0
$925$	−99.8733	−3.28382
$926$	0	0
$927$	−34.6749	−1.13887
$928$	0	0
$929$	16.2756	0.533986	0.266993	−	0.963698i	$-0.413970\pi$
$929$	16.2756	0.533986	0.266993	+	0.963698i	$0.413970\pi$
$930$	0	0
$931$	−6.76496	−0.221713
$932$	0	0
$933$	−12.5341	−0.410347
$934$	0	0
$935$	−17.8993	−0.585369
$936$	0	0
$937$	−11.3951	−0.372263	−0.186131	−	0.982525i	$-0.559595\pi$
$937$	−11.3951	−0.372263	−0.186131	+	0.982525i	$0.559595\pi$
$938$	0	0
$939$	−11.1865	−0.365056
$940$	0	0
$941$	−31.8138	−1.03710	−0.518550	−	0.855047i	$-0.673528\pi$
$941$	−31.8138	−1.03710	−0.518550	+	0.855047i	$0.673528\pi$
$942$	0	0
$943$	−34.3951	−1.12006
$944$	0	0
$945$	−12.6981	−0.413068
$946$	0	0
$947$	3.89310	0.126509	0.0632544	−	0.997997i	$-0.479852\pi$
$947$	3.89310	0.126509	0.0632544	+	0.997997i	$0.479852\pi$
$948$	0	0
$949$	−10.9348	−0.354958
$950$	0	0
$951$	8.59133	0.278593
$952$	0	0
$953$	−30.7331	−0.995544	−0.497772	−	0.867308i	$-0.665848\pi$
$953$	−30.7331	−0.995544	−0.497772	+	0.867308i	$0.665848\pi$
$954$	0	0
$955$	−54.4062	−1.76054
$956$	0	0
$957$	−10.1246	−0.327282
$958$	0	0
$959$	14.5820	0.470878
$960$	0	0
$961$	−30.9973	−0.999913
$962$	0	0
$963$	−14.3169	−0.461355
$964$	0	0
$965$	−7.36655	−0.237138
$966$	0	0
$967$	−23.3237	−0.750038	−0.375019	−	0.927017i	$-0.622364\pi$
$967$	−23.3237	−0.750038	−0.375019	+	0.927017i	$0.622364\pi$
$968$	0	0
$969$	3.94054	0.126588
$970$	0	0
$971$	−10.6257	−0.340993	−0.170497	−	0.985358i	$-0.554537\pi$
$971$	−10.6257	−0.340993	−0.170497	+	0.985358i	$0.554537\pi$
$972$	0	0
$973$	10.7418	0.344366
$974$	0	0
$975$	−14.5779	−0.466865
$976$	0	0
$977$	24.2395	0.775489	0.387745	−	0.921767i	$-0.373254\pi$
$977$	24.2395	0.775489	0.387745	+	0.921767i	$0.373254\pi$
$978$	0	0
$979$	49.9939	1.59781
$980$	0	0
$981$	−46.1005	−1.47188
$982$	0	0
$983$	2.70430	0.0862539	0.0431269	−	0.999070i	$-0.486268\pi$
$983$	2.70430	0.0862539	0.0431269	+	0.999070i	$0.486268\pi$
$984$	0	0
$985$	50.9737	1.62416
$986$	0	0
$987$	3.90591	0.124326
$988$	0	0
$989$	−54.8694	−1.74475
$990$	0	0
$991$	−1.72466	−0.0547858	−0.0273929	−	0.999625i	$-0.508721\pi$
$991$	−1.72466	−0.0547858	−0.0273929	+	0.999625i	$0.508721\pi$
$992$	0	0
$993$	−12.8595	−0.408083
$994$	0	0
$995$	−67.4009	−2.13675
$996$	0	0
$997$	−2.21823	−0.0702520	−0.0351260	−	0.999383i	$-0.511183\pi$
$997$	−2.21823	−0.0702520	−0.0351260	+	0.999383i	$0.511183\pi$
$998$	0	0
$999$	33.4996	1.05988

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.e.1.4	✓	5
4.3	odd	2		3808.2.a.f.1.2	yes	5
8.3	odd	2		7616.2.a.bs.1.4		5
8.5	even	2		7616.2.a.br.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.e.1.4	✓	5	1.1	even	1	trivial
3808.2.a.f.1.2	yes	5	4.3	odd	2
7616.2.a.br.1.2		5	8.5	even	2
7616.2.a.bs.1.4		5	8.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)

\(f(q)\)	\(=\)	\(q+0.582493 q^{3} -3.85102 q^{5} -1.00000 q^{7} -2.66070 q^{9} -4.64792 q^{11} -2.54585 q^{13} -2.24319 q^{15} -1.00000 q^{17} -6.76496 q^{19} -0.582493 q^{21} +6.10208 q^{23} +9.83039 q^{25} -3.29732 q^{27} +3.73962 q^{29} -0.0519437 q^{31} -2.70738 q^{33} +3.85102 q^{35} -10.1597 q^{37} -1.48294 q^{39} -5.63662 q^{41} -8.99193 q^{43} +10.2464 q^{45} -6.70550 q^{47} +1.00000 q^{49} -0.582493 q^{51} +8.65356 q^{53} +17.8993 q^{55} -3.94054 q^{57} -0.380646 q^{59} +4.49425 q^{61} +2.66070 q^{63} +9.80413 q^{65} +1.65889 q^{67} +3.55442 q^{69} -4.35876 q^{71} +4.29514 q^{73} +5.72613 q^{75} +4.64792 q^{77} +9.48294 q^{79} +6.06144 q^{81} +15.7429 q^{83} +3.85102 q^{85} +2.17830 q^{87} -10.7562 q^{89} +2.54585 q^{91} -0.0302568 q^{93} +26.0520 q^{95} -9.31901 q^{97} +12.3667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{5} - 5 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 8 q^{15} - 5 q^{17} - 6 q^{19} + 18 q^{23} + 5 q^{25} + 18 q^{27} - 14 q^{29} + 16 q^{31} - 26 q^{33} + 2 q^{35} + 2 q^{37} + 6 q^{39} - 10 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) \) 5 * q - 2 * q^5 - 5 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 8 * q^15 - 5 * q^17 - 6 * q^19 + 18 * q^23 + 5 * q^25 + 18 * q^27 - 14 * q^29 + 16 * q^31 - 26 * q^33 + 2 * q^35 + 2 * q^37 + 6 * q^39 - 10 * q^41 - 20 * q^43 + 6 * q^45 + 26 * q^47 + 5 * q^49 + 4 * q^55 + 12 * q^57 + 8 * q^61 - 3 * q^63 + 2 * q^65 + 12 * q^67 + 32 * q^69 + 10 * q^71 - 14 * q^73 - 4 * q^75 + 4 * q^77 + 34 * q^79 + 33 * q^81 + 20 * q^83 + 2 * q^85 + 14 * q^87 - 4 * q^89 + 6 * q^91 + 16 * q^93 + 42 * q^95 - 12 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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