LMFDB - Embedded newform 3808.2.a.g.1.6

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.147697840.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 10x^{3} + 18x^{2} - 16x + 3 $ x^6 - 2x^5 - 10x^4 + 10x^3 + 18x^2 - 16*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.69817$ 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)

$f(q)$	$=$	$q+2.69817 q^{3} -2.06665 q^{5} +1.00000 q^{7} +4.28014 q^{9} -6.03600 q^{11} +2.94465 q^{13} -5.57618 q^{15} +1.00000 q^{17} -6.57383 q^{19} +2.69817 q^{21} -6.75653 q^{23} -0.728963 q^{25} +3.45404 q^{27} -1.85330 q^{29} +10.5543 q^{31} -16.2862 q^{33} -2.06665 q^{35} -4.17170 q^{37} +7.94518 q^{39} -8.24122 q^{41} -7.78952 q^{43} -8.84554 q^{45} +4.43814 q^{47} +1.00000 q^{49} +2.69817 q^{51} +2.60448 q^{53} +12.4743 q^{55} -17.7373 q^{57} -6.62625 q^{59} -3.06899 q^{61} +4.28014 q^{63} -6.08556 q^{65} -7.47640 q^{67} -18.2303 q^{69} -1.11591 q^{71} +10.9169 q^{73} -1.96687 q^{75} -6.03600 q^{77} -3.14091 q^{79} -3.52083 q^{81} +12.9013 q^{83} -2.06665 q^{85} -5.00053 q^{87} -4.25544 q^{89} +2.94465 q^{91} +28.4775 q^{93} +13.5858 q^{95} +11.9761 q^{97} -25.8349 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{13} + 6 q^{17} - 18 q^{19} - 4 q^{21} - 6 q^{23} + 12 q^{25} - 10 q^{27} + 4 q^{29} + 8 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} + 18 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 - 4 * q^5 + 6 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^13 + 6 * q^17 - 18 * q^19 - 4 * q^21 - 6 * q^23 + 12 * q^25 - 10 * q^27 + 4 * q^29 + 8 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 + 18 * q^39 - 4 * q^41 - 20 * q^43 - 24 * q^45 - 6 * q^47 + 6 * q^49 - 4 * q^51 + 14 * q^53 - 4 * q^55 - 4 * q^57 - 20 * q^59 + 2 * q^61 + 8 * q^63 + 6 * q^65 - 28 * q^67 - 4 * q^69 - 18 * q^71 + 4 * q^73 - 20 * q^75 - 8 * q^77 + 2 * q^79 + 34 * q^81 - 20 * q^83 - 4 * q^85 - 22 * q^87 - 6 * q^89 - 4 * q^91 + 40 * q^93 + 6 * q^95 + 30 * q^97 - 32 * 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
$3$	2.69817	1.55779	0.778896	−	0.627154i	$-0.215780\pi$
$3$	2.69817	1.55779	0.778896	+	0.627154i	$0.215780\pi$
$4$	0	0
$5$	−2.06665	−0.924233	−0.462117	−	0.886819i	$-0.652910\pi$
$5$	−2.06665	−0.924233	−0.462117	+	0.886819i	$0.652910\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.28014	1.42671
$10$	0	0
$11$	−6.03600	−1.81992	−0.909962	−	0.414693i	$-0.863889\pi$
$11$	−6.03600	−1.81992	−0.909962	+	0.414693i	$0.863889\pi$
$12$	0	0
$13$	2.94465	0.816699	0.408350	−	0.912826i	$-0.366104\pi$
$13$	2.94465	0.816699	0.408350	+	0.912826i	$0.366104\pi$
$14$	0	0
$15$	−5.57618	−1.43976
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−6.57383	−1.50814	−0.754071	−	0.656793i	$-0.771912\pi$
$19$	−6.57383	−1.50814	−0.754071	+	0.656793i	$0.771912\pi$
$20$	0	0
$21$	2.69817	0.588790
$22$	0	0
$23$	−6.75653	−1.40883	−0.704417	−	0.709786i	$-0.748792\pi$
$23$	−6.75653	−1.40883	−0.704417	+	0.709786i	$0.748792\pi$
$24$	0	0
$25$	−0.728963	−0.145793
$26$	0	0
$27$	3.45404	0.664730
$28$	0	0
$29$	−1.85330	−0.344149	−0.172075	−	0.985084i	$-0.555047\pi$
$29$	−1.85330	−0.344149	−0.172075	+	0.985084i	$0.555047\pi$
$30$	0	0
$31$	10.5543	1.89562	0.947808	−	0.318841i	$-0.103294\pi$
$31$	10.5543	1.89562	0.947808	+	0.318841i	$0.103294\pi$
$32$	0	0
$33$	−16.2862	−2.83506
$34$	0	0
$35$	−2.06665	−0.349327
$36$	0	0
$37$	−4.17170	−0.685823	−0.342912	−	0.939368i	$-0.611413\pi$
$37$	−4.17170	−0.685823	−0.342912	+	0.939368i	$0.611413\pi$
$38$	0	0
$39$	7.94518	1.27225
$40$	0	0
$41$	−8.24122	−1.28706	−0.643531	−	0.765420i	$-0.722531\pi$
$41$	−8.24122	−1.28706	−0.643531	+	0.765420i	$0.722531\pi$
$42$	0	0
$43$	−7.78952	−1.18789	−0.593946	−	0.804505i	$-0.702431\pi$
$43$	−7.78952	−1.18789	−0.593946	+	0.804505i	$0.702431\pi$
$44$	0	0
$45$	−8.84554	−1.31862
$46$	0	0
$47$	4.43814	0.647369	0.323684	−	0.946165i	$-0.395078\pi$
$47$	4.43814	0.647369	0.323684	+	0.946165i	$0.395078\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.69817	0.377820
$52$	0	0
$53$	2.60448	0.357753	0.178877	−	0.983872i	$-0.442754\pi$
$53$	2.60448	0.357753	0.178877	+	0.983872i	$0.442754\pi$
$54$	0	0
$55$	12.4743	1.68203
$56$	0	0
$57$	−17.7373	−2.34937
$58$	0	0
$59$	−6.62625	−0.862665	−0.431332	−	0.902193i	$-0.641956\pi$
$59$	−6.62625	−0.862665	−0.431332	+	0.902193i	$0.641956\pi$
$60$	0	0
$61$	−3.06899	−0.392944	−0.196472	−	0.980509i	$-0.562948\pi$
$61$	−3.06899	−0.392944	−0.196472	+	0.980509i	$0.562948\pi$
$62$	0	0
$63$	4.28014	0.539247
$64$	0	0
$65$	−6.08556	−0.754821
$66$	0	0
$67$	−7.47640	−0.913387	−0.456694	−	0.889624i	$-0.650966\pi$
$67$	−7.47640	−0.913387	−0.456694	+	0.889624i	$0.650966\pi$
$68$	0	0
$69$	−18.2303	−2.19467
$70$	0	0
$71$	−1.11591	−0.132434	−0.0662170	−	0.997805i	$-0.521093\pi$
$71$	−1.11591	−0.132434	−0.0662170	+	0.997805i	$0.521093\pi$
$72$	0	0
$73$	10.9169	1.27772	0.638862	−	0.769322i	$-0.279406\pi$
$73$	10.9169	1.27772	0.638862	+	0.769322i	$0.279406\pi$
$74$	0	0
$75$	−1.96687	−0.227114
$76$	0	0
$77$	−6.03600	−0.687866
$78$	0	0
$79$	−3.14091	−0.353380	−0.176690	−	0.984267i	$-0.556539\pi$
$79$	−3.14091	−0.353380	−0.176690	+	0.984267i	$0.556539\pi$
$80$	0	0
$81$	−3.52083	−0.391203
$82$	0	0
$83$	12.9013	1.41610	0.708050	−	0.706163i	$-0.249575\pi$
$83$	12.9013	1.41610	0.708050	+	0.706163i	$0.249575\pi$
$84$	0	0
$85$	−2.06665	−0.224160
$86$	0	0
$87$	−5.00053	−0.536113
$88$	0	0
$89$	−4.25544	−0.451075	−0.225538	−	0.974234i	$-0.572414\pi$
$89$	−4.25544	−0.451075	−0.225538	+	0.974234i	$0.572414\pi$
$90$	0	0
$91$	2.94465	0.308683
$92$	0	0
$93$	28.4775	2.95297
$94$	0	0
$95$	13.5858	1.39387
$96$	0	0
$97$	11.9761	1.21599	0.607995	−	0.793941i	$-0.291974\pi$
$97$	11.9761	1.21599	0.607995	+	0.793941i	$0.291974\pi$
$98$	0	0
$99$	−25.8349	−2.59651
$100$	0	0
$101$	−3.36487	−0.334817	−0.167409	−	0.985888i	$-0.553540\pi$
$101$	−3.36487	−0.334817	−0.167409	+	0.985888i	$0.553540\pi$
$102$	0	0
$103$	−15.7989	−1.55671	−0.778357	−	0.627822i	$-0.783947\pi$
$103$	−15.7989	−1.55671	−0.778357	+	0.627822i	$0.783947\pi$
$104$	0	0
$105$	−5.57618	−0.544179
$106$	0	0
$107$	−16.2008	−1.56619	−0.783094	−	0.621903i	$-0.786360\pi$
$107$	−16.2008	−1.56619	−0.783094	+	0.621903i	$0.786360\pi$
$108$	0	0
$109$	0.517516	0.0495690	0.0247845	−	0.999693i	$-0.492110\pi$
$109$	0.517516	0.0495690	0.0247845	+	0.999693i	$0.492110\pi$
$110$	0	0
$111$	−11.2560	−1.06837
$112$	0	0
$113$	−3.76248	−0.353944	−0.176972	−	0.984216i	$-0.556630\pi$
$113$	−3.76248	−0.353944	−0.176972	+	0.984216i	$0.556630\pi$
$114$	0	0
$115$	13.9634	1.30209
$116$	0	0
$117$	12.6035	1.16520
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	25.4333	2.31212
$122$	0	0
$123$	−22.2362	−2.00497
$124$	0	0
$125$	11.8398	1.05898
$126$	0	0
$127$	−5.02230	−0.445657	−0.222829	−	0.974858i	$-0.571529\pi$
$127$	−5.02230	−0.445657	−0.222829	+	0.974858i	$0.571529\pi$
$128$	0	0
$129$	−21.0175	−1.85049
$130$	0	0
$131$	−7.43776	−0.649841	−0.324920	−	0.945741i	$-0.605337\pi$
$131$	−7.43776	−0.649841	−0.324920	+	0.945741i	$0.605337\pi$
$132$	0	0
$133$	−6.57383	−0.570024
$134$	0	0
$135$	−7.13828	−0.614365
$136$	0	0
$137$	−12.0284	−1.02765	−0.513827	−	0.857894i	$-0.671773\pi$
$137$	−12.0284	−1.02765	−0.513827	+	0.857894i	$0.671773\pi$
$138$	0	0
$139$	14.6465	1.24230	0.621150	−	0.783691i	$-0.286666\pi$
$139$	14.6465	1.24230	0.621150	+	0.783691i	$0.286666\pi$
$140$	0	0
$141$	11.9749	1.00847
$142$	0	0
$143$	−17.7739	−1.48633
$144$	0	0
$145$	3.83012	0.318074
$146$	0	0
$147$	2.69817	0.222542
$148$	0	0
$149$	8.29364	0.679441	0.339721	−	0.940526i	$-0.389667\pi$
$149$	8.29364	0.679441	0.339721	+	0.940526i	$0.389667\pi$
$150$	0	0
$151$	11.2518	0.915662	0.457831	−	0.889039i	$-0.348626\pi$
$151$	11.2518	0.915662	0.457831	+	0.889039i	$0.348626\pi$
$152$	0	0
$153$	4.28014	0.346029
$154$	0	0
$155$	−21.8121	−1.75199
$156$	0	0
$157$	11.5949	0.925373	0.462686	−	0.886522i	$-0.346886\pi$
$157$	11.5949	0.925373	0.462686	+	0.886522i	$0.346886\pi$
$158$	0	0
$159$	7.02734	0.557304
$160$	0	0
$161$	−6.75653	−0.532490
$162$	0	0
$163$	16.6324	1.30275	0.651377	−	0.758755i	$-0.274192\pi$
$163$	16.6324	1.30275	0.651377	+	0.758755i	$0.274192\pi$
$164$	0	0
$165$	33.6578	2.62026
$166$	0	0
$167$	16.7291	1.29453	0.647267	−	0.762263i	$-0.275912\pi$
$167$	16.7291	1.29453	0.647267	+	0.762263i	$0.275912\pi$
$168$	0	0
$169$	−4.32903	−0.333002
$170$	0	0
$171$	−28.1369	−2.15168
$172$	0	0
$173$	4.64335	0.353028	0.176514	−	0.984298i	$-0.443518\pi$
$173$	4.64335	0.353028	0.176514	+	0.984298i	$0.443518\pi$
$174$	0	0
$175$	−0.728963	−0.0551044
$176$	0	0
$177$	−17.8788	−1.34385
$178$	0	0
$179$	1.43015	0.106894	0.0534472	−	0.998571i	$-0.482979\pi$
$179$	1.43015	0.106894	0.0534472	+	0.998571i	$0.482979\pi$
$180$	0	0
$181$	−15.3718	−1.14258	−0.571288	−	0.820750i	$-0.693556\pi$
$181$	−15.3718	−1.14258	−0.571288	+	0.820750i	$0.693556\pi$
$182$	0	0
$183$	−8.28067	−0.612125
$184$	0	0
$185$	8.62144	0.633861
$186$	0	0
$187$	−6.03600	−0.441396
$188$	0	0
$189$	3.45404	0.251244
$190$	0	0
$191$	−24.9569	−1.80582	−0.902910	−	0.429830i	$-0.858573\pi$
$191$	−24.9569	−1.80582	−0.902910	+	0.429830i	$0.858573\pi$
$192$	0	0
$193$	−16.6404	−1.19780	−0.598902	−	0.800822i	$-0.704396\pi$
$193$	−16.6404	−1.19780	−0.598902	+	0.800822i	$0.704396\pi$
$194$	0	0
$195$	−16.4199	−1.17585
$196$	0	0
$197$	−9.28521	−0.661544	−0.330772	−	0.943711i	$-0.607309\pi$
$197$	−9.28521	−0.661544	−0.330772	+	0.943711i	$0.607309\pi$
$198$	0	0
$199$	7.59327	0.538272	0.269136	−	0.963102i	$-0.413262\pi$
$199$	7.59327	0.538272	0.269136	+	0.963102i	$0.413262\pi$
$200$	0	0
$201$	−20.1726	−1.42287
$202$	0	0
$203$	−1.85330	−0.130076
$204$	0	0
$205$	17.0317	1.18955
$206$	0	0
$207$	−28.9189	−2.01000
$208$	0	0
$209$	39.6797	2.74470
$210$	0	0
$211$	−0.478653	−0.0329518	−0.0164759	−	0.999864i	$-0.505245\pi$
$211$	−0.478653	−0.0329518	−0.0164759	+	0.999864i	$0.505245\pi$
$212$	0	0
$213$	−3.01091	−0.206304
$214$	0	0
$215$	16.0982	1.09789
$216$	0	0
$217$	10.5543	0.716476
$218$	0	0
$219$	29.4556	1.99043
$220$	0	0
$221$	2.94465	0.198079
$222$	0	0
$223$	−15.8310	−1.06012	−0.530062	−	0.847959i	$-0.677831\pi$
$223$	−15.8310	−1.06012	−0.530062	+	0.847959i	$0.677831\pi$
$224$	0	0
$225$	−3.12006	−0.208004
$226$	0	0
$227$	−25.8422	−1.71521	−0.857605	−	0.514309i	$-0.828048\pi$
$227$	−25.8422	−1.71521	−0.857605	+	0.514309i	$0.828048\pi$
$228$	0	0
$229$	−20.2618	−1.33894	−0.669468	−	0.742841i	$-0.733478\pi$
$229$	−20.2618	−1.33894	−0.669468	+	0.742841i	$0.733478\pi$
$230$	0	0
$231$	−16.2862	−1.07155
$232$	0	0
$233$	−16.4140	−1.07532	−0.537659	−	0.843162i	$-0.680691\pi$
$233$	−16.4140	−1.07532	−0.537659	+	0.843162i	$0.680691\pi$
$234$	0	0
$235$	−9.17207	−0.598320
$236$	0	0
$237$	−8.47472	−0.550492
$238$	0	0
$239$	5.59025	0.361603	0.180802	−	0.983520i	$-0.442131\pi$
$239$	5.59025	0.361603	0.180802	+	0.983520i	$0.442131\pi$
$240$	0	0
$241$	−21.7544	−1.40133	−0.700663	−	0.713492i	$-0.747112\pi$
$241$	−21.7544	−1.40133	−0.700663	+	0.713492i	$0.747112\pi$
$242$	0	0
$243$	−19.8619	−1.27414
$244$	0	0
$245$	−2.06665	−0.132033
$246$	0	0
$247$	−19.3577	−1.23170
$248$	0	0
$249$	34.8099	2.20599
$250$	0	0
$251$	−1.86504	−0.117720	−0.0588600	−	0.998266i	$-0.518747\pi$
$251$	−1.86504	−0.117720	−0.0588600	+	0.998266i	$0.518747\pi$
$252$	0	0
$253$	40.7825	2.56397
$254$	0	0
$255$	−5.57618	−0.349194
$256$	0	0
$257$	18.1813	1.13412	0.567060	−	0.823676i	$-0.308081\pi$
$257$	18.1813	1.13412	0.567060	+	0.823676i	$0.308081\pi$
$258$	0	0
$259$	−4.17170	−0.259217
$260$	0	0
$261$	−7.93239	−0.491003
$262$	0	0
$263$	−22.6502	−1.39667	−0.698336	−	0.715771i	$-0.746076\pi$
$263$	−22.6502	−1.39667	−0.698336	+	0.715771i	$0.746076\pi$
$264$	0	0
$265$	−5.38255	−0.330647
$266$	0	0
$267$	−11.4819	−0.702681
$268$	0	0
$269$	23.7554	1.44839	0.724197	−	0.689593i	$-0.242211\pi$
$269$	23.7554	1.44839	0.724197	+	0.689593i	$0.242211\pi$
$270$	0	0
$271$	25.9045	1.57359	0.786793	−	0.617217i	$-0.211740\pi$
$271$	25.9045	1.57359	0.786793	+	0.617217i	$0.211740\pi$
$272$	0	0
$273$	7.94518	0.480864
$274$	0	0
$275$	4.40002	0.265331
$276$	0	0
$277$	16.6379	0.999672	0.499836	−	0.866120i	$-0.333394\pi$
$277$	16.6379	0.999672	0.499836	+	0.866120i	$0.333394\pi$
$278$	0	0
$279$	45.1741	2.70450
$280$	0	0
$281$	28.0132	1.67113	0.835564	−	0.549394i	$-0.185141\pi$
$281$	28.0132	1.67113	0.835564	+	0.549394i	$0.185141\pi$
$282$	0	0
$283$	16.3572	0.972336	0.486168	−	0.873865i	$-0.338394\pi$
$283$	16.3572	0.972336	0.486168	+	0.873865i	$0.338394\pi$
$284$	0	0
$285$	36.6569	2.17137
$286$	0	0
$287$	−8.24122	−0.486464
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	32.3136	1.89426
$292$	0	0
$293$	−19.0231	−1.11134	−0.555672	−	0.831402i	$-0.687539\pi$
$293$	−19.0231	−1.11134	−0.555672	+	0.831402i	$0.687539\pi$
$294$	0	0
$295$	13.6941	0.797304
$296$	0	0
$297$	−20.8486	−1.20976
$298$	0	0
$299$	−19.8956	−1.15059
$300$	0	0
$301$	−7.78952	−0.448981
$302$	0	0
$303$	−9.07901	−0.521575
$304$	0	0
$305$	6.34253	0.363172
$306$	0	0
$307$	−8.67806	−0.495283	−0.247641	−	0.968852i	$-0.579655\pi$
$307$	−8.67806	−0.495283	−0.247641	+	0.968852i	$0.579655\pi$
$308$	0	0
$309$	−42.6282	−2.42504
$310$	0	0
$311$	2.79728	0.158619	0.0793096	−	0.996850i	$-0.474728\pi$
$311$	2.79728	0.158619	0.0793096	+	0.996850i	$0.474728\pi$
$312$	0	0
$313$	29.8312	1.68616	0.843081	−	0.537787i	$-0.180740\pi$
$313$	29.8312	1.68616	0.843081	+	0.537787i	$0.180740\pi$
$314$	0	0
$315$	−8.84554	−0.498390
$316$	0	0
$317$	16.1954	0.909622	0.454811	−	0.890588i	$-0.349707\pi$
$317$	16.1954	0.909622	0.454811	+	0.890588i	$0.349707\pi$
$318$	0	0
$319$	11.1865	0.626326
$320$	0	0
$321$	−43.7125	−2.43979
$322$	0	0
$323$	−6.57383	−0.365778
$324$	0	0
$325$	−2.14654	−0.119069
$326$	0	0
$327$	1.39635	0.0772181
$328$	0	0
$329$	4.43814	0.244682
$330$	0	0
$331$	−10.5418	−0.579430	−0.289715	−	0.957113i	$-0.593561\pi$
$331$	−10.5418	−0.579430	−0.289715	+	0.957113i	$0.593561\pi$
$332$	0	0
$333$	−17.8555	−0.978473
$334$	0	0
$335$	15.4511	0.844183
$336$	0	0
$337$	14.3443	0.781386	0.390693	−	0.920521i	$-0.372235\pi$
$337$	14.3443	0.781386	0.390693	+	0.920521i	$0.372235\pi$
$338$	0	0
$339$	−10.1518	−0.551371
$340$	0	0
$341$	−63.7060	−3.44988
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	37.6756	2.02839
$346$	0	0
$347$	23.9604	1.28626	0.643131	−	0.765756i	$-0.277635\pi$
$347$	23.9604	1.28626	0.643131	+	0.765756i	$0.277635\pi$
$348$	0	0
$349$	−7.85213	−0.420315	−0.210157	−	0.977668i	$-0.567398\pi$
$349$	−7.85213	−0.420315	−0.210157	+	0.977668i	$0.567398\pi$
$350$	0	0
$351$	10.1709	0.542884
$352$	0	0
$353$	7.45251	0.396657	0.198329	−	0.980136i	$-0.436449\pi$
$353$	7.45251	0.396657	0.198329	+	0.980136i	$0.436449\pi$
$354$	0	0
$355$	2.30619	0.122400
$356$	0	0
$357$	2.69817	0.142802
$358$	0	0
$359$	−21.9980	−1.16101	−0.580505	−	0.814257i	$-0.697145\pi$
$359$	−21.9980	−1.16101	−0.580505	+	0.814257i	$0.697145\pi$
$360$	0	0
$361$	24.2153	1.27449
$362$	0	0
$363$	68.6235	3.60180
$364$	0	0
$365$	−22.5613	−1.18091
$366$	0	0
$367$	11.6430	0.607757	0.303879	−	0.952711i	$-0.401718\pi$
$367$	11.6430	0.607757	0.303879	+	0.952711i	$0.401718\pi$
$368$	0	0
$369$	−35.2736	−1.83627
$370$	0	0
$371$	2.60448	0.135218
$372$	0	0
$373$	30.0513	1.55600	0.777999	−	0.628266i	$-0.216235\pi$
$373$	30.0513	1.55600	0.777999	+	0.628266i	$0.216235\pi$
$374$	0	0
$375$	31.9457	1.64967
$376$	0	0
$377$	−5.45733	−0.281067
$378$	0	0
$379$	6.81796	0.350215	0.175108	−	0.984549i	$-0.443973\pi$
$379$	6.81796	0.350215	0.175108	+	0.984549i	$0.443973\pi$
$380$	0	0
$381$	−13.5510	−0.694241
$382$	0	0
$383$	−2.26730	−0.115854	−0.0579269	−	0.998321i	$-0.518449\pi$
$383$	−2.26730	−0.115854	−0.0579269	+	0.998321i	$0.518449\pi$
$384$	0	0
$385$	12.4743	0.635749
$386$	0	0
$387$	−33.3402	−1.69478
$388$	0	0
$389$	1.05212	0.0533446	0.0266723	−	0.999644i	$-0.491509\pi$
$389$	1.05212	0.0533446	0.0266723	+	0.999644i	$0.491509\pi$
$390$	0	0
$391$	−6.75653	−0.341693
$392$	0	0
$393$	−20.0684	−1.01232
$394$	0	0
$395$	6.49116	0.326605
$396$	0	0
$397$	16.8593	0.846146	0.423073	−	0.906096i	$-0.360951\pi$
$397$	16.8593	0.846146	0.423073	+	0.906096i	$0.360951\pi$
$398$	0	0
$399$	−17.7373	−0.887978
$400$	0	0
$401$	31.2778	1.56194	0.780970	−	0.624568i	$-0.214725\pi$
$401$	31.2778	1.56194	0.780970	+	0.624568i	$0.214725\pi$
$402$	0	0
$403$	31.0789	1.54815
$404$	0	0
$405$	7.27631	0.361563
$406$	0	0
$407$	25.1804	1.24815
$408$	0	0
$409$	−4.18136	−0.206755	−0.103377	−	0.994642i	$-0.532965\pi$
$409$	−4.18136	−0.206755	−0.103377	+	0.994642i	$0.532965\pi$
$410$	0	0
$411$	−32.4547	−1.60087
$412$	0	0
$413$	−6.62625	−0.326057
$414$	0	0
$415$	−26.6624	−1.30881
$416$	0	0
$417$	39.5188	1.93525
$418$	0	0
$419$	−16.0859	−0.785849	−0.392924	−	0.919571i	$-0.628537\pi$
$419$	−16.0859	−0.785849	−0.392924	+	0.919571i	$0.628537\pi$
$420$	0	0
$421$	−23.4639	−1.14356	−0.571780	−	0.820407i	$-0.693747\pi$
$421$	−23.4639	−1.14356	−0.571780	+	0.820407i	$0.693747\pi$
$422$	0	0
$423$	18.9958	0.923610
$424$	0	0
$425$	−0.728963	−0.0353599
$426$	0	0
$427$	−3.06899	−0.148519
$428$	0	0
$429$	−47.9571	−2.31539
$430$	0	0
$431$	−2.65126	−0.127706	−0.0638532	−	0.997959i	$-0.520339\pi$
$431$	−2.65126	−0.127706	−0.0638532	+	0.997959i	$0.520339\pi$
$432$	0	0
$433$	−10.8144	−0.519706	−0.259853	−	0.965648i	$-0.583674\pi$
$433$	−10.8144	−0.519706	−0.259853	+	0.965648i	$0.583674\pi$
$434$	0	0
$435$	10.3343	0.495494
$436$	0	0
$437$	44.4163	2.12472
$438$	0	0
$439$	−17.7704	−0.848136	−0.424068	−	0.905630i	$-0.639398\pi$
$439$	−17.7704	−0.848136	−0.424068	+	0.905630i	$0.639398\pi$
$440$	0	0
$441$	4.28014	0.203816
$442$	0	0
$443$	−16.5600	−0.786789	−0.393394	−	0.919370i	$-0.628699\pi$
$443$	−16.5600	−0.786789	−0.393394	+	0.919370i	$0.628699\pi$
$444$	0	0
$445$	8.79449	0.416899
$446$	0	0
$447$	22.3777	1.05843
$448$	0	0
$449$	10.1671	0.479814	0.239907	−	0.970796i	$-0.422883\pi$
$449$	10.1671	0.479814	0.239907	+	0.970796i	$0.422883\pi$
$450$	0	0
$451$	49.7440	2.34235
$452$	0	0
$453$	30.3594	1.42641
$454$	0	0
$455$	−6.08556	−0.285295
$456$	0	0
$457$	−17.3416	−0.811206	−0.405603	−	0.914049i	$-0.632938\pi$
$457$	−17.3416	−0.811206	−0.405603	+	0.914049i	$0.632938\pi$
$458$	0	0
$459$	3.45404	0.161221
$460$	0	0
$461$	−9.37885	−0.436817	−0.218408	−	0.975857i	$-0.570086\pi$
$461$	−9.37885	−0.436817	−0.218408	+	0.975857i	$0.570086\pi$
$462$	0	0
$463$	8.43849	0.392170	0.196085	−	0.980587i	$-0.437177\pi$
$463$	8.43849	0.392170	0.196085	+	0.980587i	$0.437177\pi$
$464$	0	0
$465$	−58.8529	−2.72924
$466$	0	0
$467$	−31.8747	−1.47499	−0.737493	−	0.675355i	$-0.763991\pi$
$467$	−31.8747	−1.47499	−0.737493	+	0.675355i	$0.763991\pi$
$468$	0	0
$469$	−7.47640	−0.345228
$470$	0	0
$471$	31.2850	1.44154
$472$	0	0
$473$	47.0176	2.16187
$474$	0	0
$475$	4.79208	0.219876
$476$	0	0
$477$	11.1475	0.510411
$478$	0	0
$479$	16.8008	0.767646	0.383823	−	0.923407i	$-0.374607\pi$
$479$	16.8008	0.767646	0.383823	+	0.923407i	$0.374607\pi$
$480$	0	0
$481$	−12.2842	−0.560111
$482$	0	0
$483$	−18.2303	−0.829507
$484$	0	0
$485$	−24.7504	−1.12386
$486$	0	0
$487$	39.5498	1.79217	0.896087	−	0.443878i	$-0.146398\pi$
$487$	39.5498	1.79217	0.896087	+	0.443878i	$0.146398\pi$
$488$	0	0
$489$	44.8772	2.02942
$490$	0	0
$491$	−28.1655	−1.27109	−0.635545	−	0.772064i	$-0.719225\pi$
$491$	−28.1655	−1.27109	−0.635545	+	0.772064i	$0.719225\pi$
$492$	0	0
$493$	−1.85330	−0.0834685
$494$	0	0
$495$	53.3917	2.39978
$496$	0	0
$497$	−1.11591	−0.0500553
$498$	0	0
$499$	−35.4626	−1.58753	−0.793763	−	0.608227i	$-0.791881\pi$
$499$	−35.4626	−1.58753	−0.793763	+	0.608227i	$0.791881\pi$
$500$	0	0
$501$	45.1379	2.01661
$502$	0	0
$503$	−12.3445	−0.550415	−0.275208	−	0.961385i	$-0.588747\pi$
$503$	−12.3445	−0.550415	−0.275208	+	0.961385i	$0.588747\pi$
$504$	0	0
$505$	6.95401	0.309449
$506$	0	0
$507$	−11.6805	−0.518748
$508$	0	0
$509$	−11.7911	−0.522633	−0.261317	−	0.965253i	$-0.584157\pi$
$509$	−11.7911	−0.522633	−0.261317	+	0.965253i	$0.584157\pi$
$510$	0	0
$511$	10.9169	0.482934
$512$	0	0
$513$	−22.7063	−1.00251
$514$	0	0
$515$	32.6508	1.43877
$516$	0	0
$517$	−26.7886	−1.17816
$518$	0	0
$519$	12.5286	0.549943
$520$	0	0
$521$	27.5332	1.20625	0.603125	−	0.797646i	$-0.293922\pi$
$521$	27.5332	1.20625	0.603125	+	0.797646i	$0.293922\pi$
$522$	0	0
$523$	−7.48162	−0.327148	−0.163574	−	0.986531i	$-0.552302\pi$
$523$	−7.48162	−0.327148	−0.163574	+	0.986531i	$0.552302\pi$
$524$	0	0
$525$	−1.96687	−0.0858412
$526$	0	0
$527$	10.5543	0.459755
$528$	0	0
$529$	22.6508	0.984816
$530$	0	0
$531$	−28.3613	−1.23078
$532$	0	0
$533$	−24.2675	−1.05114
$534$	0	0
$535$	33.4813	1.44752
$536$	0	0
$537$	3.85879	0.166519
$538$	0	0
$539$	−6.03600	−0.259989
$540$	0	0
$541$	12.1165	0.520927	0.260464	−	0.965484i	$-0.416125\pi$
$541$	12.1165	0.520927	0.260464	+	0.965484i	$0.416125\pi$
$542$	0	0
$543$	−41.4757	−1.77989
$544$	0	0
$545$	−1.06952	−0.0458133
$546$	0	0
$547$	−22.0333	−0.942074	−0.471037	−	0.882113i	$-0.656120\pi$
$547$	−22.0333	−0.942074	−0.471037	+	0.882113i	$0.656120\pi$
$548$	0	0
$549$	−13.1357	−0.560618
$550$	0	0
$551$	12.1833	0.519026
$552$	0	0
$553$	−3.14091	−0.133565
$554$	0	0
$555$	23.2621	0.987422
$556$	0	0
$557$	42.6927	1.80895	0.904474	−	0.426529i	$-0.140264\pi$
$557$	42.6927	1.80895	0.904474	+	0.426529i	$0.140264\pi$
$558$	0	0
$559$	−22.9374	−0.970150
$560$	0	0
$561$	−16.2862	−0.687603
$562$	0	0
$563$	−11.9348	−0.502990	−0.251495	−	0.967859i	$-0.580922\pi$
$563$	−11.9348	−0.502990	−0.251495	+	0.967859i	$0.580922\pi$
$564$	0	0
$565$	7.77573	0.327127
$566$	0	0
$567$	−3.52083	−0.147861
$568$	0	0
$569$	−21.5972	−0.905401	−0.452700	−	0.891663i	$-0.649539\pi$
$569$	−21.5972	−0.905401	−0.452700	+	0.891663i	$0.649539\pi$
$570$	0	0
$571$	−2.27788	−0.0953264	−0.0476632	−	0.998863i	$-0.515177\pi$
$571$	−2.27788	−0.0953264	−0.0476632	+	0.998863i	$0.515177\pi$
$572$	0	0
$573$	−67.3381	−2.81309
$574$	0	0
$575$	4.92526	0.205398
$576$	0	0
$577$	22.9129	0.953875	0.476938	−	0.878937i	$-0.341747\pi$
$577$	22.9129	0.953875	0.476938	+	0.878937i	$0.341747\pi$
$578$	0	0
$579$	−44.8987	−1.86593
$580$	0	0
$581$	12.9013	0.535235
$582$	0	0
$583$	−15.7207	−0.651083
$584$	0	0
$585$	−26.0470	−1.07691
$586$	0	0
$587$	10.0619	0.415299	0.207650	−	0.978203i	$-0.433419\pi$
$587$	10.0619	0.415299	0.207650	+	0.978203i	$0.433419\pi$
$588$	0	0
$589$	−69.3825	−2.85886
$590$	0	0
$591$	−25.0531	−1.03055
$592$	0	0
$593$	31.3563	1.28765	0.643824	−	0.765174i	$-0.277347\pi$
$593$	31.3563	1.28765	0.643824	+	0.765174i	$0.277347\pi$
$594$	0	0
$595$	−2.06665	−0.0847243
$596$	0	0
$597$	20.4879	0.838516
$598$	0	0
$599$	−1.95870	−0.0800304	−0.0400152	−	0.999199i	$-0.512741\pi$
$599$	−1.95870	−0.0800304	−0.0400152	+	0.999199i	$0.512741\pi$
$600$	0	0
$601$	15.1805	0.619227	0.309613	−	0.950863i	$-0.399800\pi$
$601$	15.1805	0.619227	0.309613	+	0.950863i	$0.399800\pi$
$602$	0	0
$603$	−32.0000	−1.30314
$604$	0	0
$605$	−52.5617	−2.13694
$606$	0	0
$607$	41.6186	1.68925	0.844623	−	0.535361i	$-0.179824\pi$
$607$	41.6186	1.68925	0.844623	+	0.535361i	$0.179824\pi$
$608$	0	0
$609$	−5.00053	−0.202632
$610$	0	0
$611$	13.0688	0.528706
$612$	0	0
$613$	−18.8221	−0.760219	−0.380110	−	0.924941i	$-0.624114\pi$
$613$	−18.8221	−0.760219	−0.380110	+	0.924941i	$0.624114\pi$
$614$	0	0
$615$	45.9545	1.85306
$616$	0	0
$617$	9.04376	0.364088	0.182044	−	0.983290i	$-0.441729\pi$
$617$	9.04376	0.364088	0.182044	+	0.983290i	$0.441729\pi$
$618$	0	0
$619$	38.2588	1.53775	0.768875	−	0.639399i	$-0.220817\pi$
$619$	38.2588	1.53775	0.768875	+	0.639399i	$0.220817\pi$
$620$	0	0
$621$	−23.3373	−0.936494
$622$	0	0
$623$	−4.25544	−0.170490
$624$	0	0
$625$	−20.8238	−0.832952
$626$	0	0
$627$	107.063	4.27567
$628$	0	0
$629$	−4.17170	−0.166337
$630$	0	0
$631$	−42.6093	−1.69625	−0.848126	−	0.529795i	$-0.822269\pi$
$631$	−42.6093	−1.69625	−0.848126	+	0.529795i	$0.822269\pi$
$632$	0	0
$633$	−1.29149	−0.0513320
$634$	0	0
$635$	10.3793	0.411891
$636$	0	0
$637$	2.94465	0.116671
$638$	0	0
$639$	−4.77624	−0.188945
$640$	0	0
$641$	−38.7679	−1.53124	−0.765620	−	0.643294i	$-0.777567\pi$
$641$	−38.7679	−1.53124	−0.765620	+	0.643294i	$0.777567\pi$
$642$	0	0
$643$	10.9046	0.430037	0.215019	−	0.976610i	$-0.431019\pi$
$643$	10.9046	0.430037	0.215019	+	0.976610i	$0.431019\pi$
$644$	0	0
$645$	43.4358	1.71028
$646$	0	0
$647$	−45.0312	−1.77036	−0.885179	−	0.465251i	$-0.845964\pi$
$647$	−45.0312	−1.77036	−0.885179	+	0.465251i	$0.845964\pi$
$648$	0	0
$649$	39.9961	1.56998
$650$	0	0
$651$	28.4775	1.11612
$652$	0	0
$653$	−23.4045	−0.915890	−0.457945	−	0.888981i	$-0.651414\pi$
$653$	−23.4045	−0.915890	−0.457945	+	0.888981i	$0.651414\pi$
$654$	0	0
$655$	15.3712	0.600604
$656$	0	0
$657$	46.7257	1.82294
$658$	0	0
$659$	10.8385	0.422209	0.211105	−	0.977463i	$-0.432294\pi$
$659$	10.8385	0.422209	0.211105	+	0.977463i	$0.432294\pi$
$660$	0	0
$661$	−17.2153	−0.669599	−0.334800	−	0.942289i	$-0.608669\pi$
$661$	−17.2153	−0.669599	−0.334800	+	0.942289i	$0.608669\pi$
$662$	0	0
$663$	7.94518	0.308565
$664$	0	0
$665$	13.5858	0.526835
$666$	0	0
$667$	12.5219	0.484850
$668$	0	0
$669$	−42.7149	−1.65145
$670$	0	0
$671$	18.5244	0.715128
$672$	0	0
$673$	−16.0173	−0.617422	−0.308711	−	0.951156i	$-0.599898\pi$
$673$	−16.0173	−0.617422	−0.308711	+	0.951156i	$0.599898\pi$
$674$	0	0
$675$	−2.51786	−0.0969127
$676$	0	0
$677$	50.4523	1.93904	0.969520	−	0.245011i	$-0.0787917\pi$
$677$	50.4523	1.93904	0.969520	+	0.245011i	$0.0787917\pi$
$678$	0	0
$679$	11.9761	0.459601
$680$	0	0
$681$	−69.7268	−2.67194
$682$	0	0
$683$	−8.09388	−0.309704	−0.154852	−	0.987938i	$-0.549490\pi$
$683$	−8.09388	−0.309704	−0.154852	+	0.987938i	$0.549490\pi$
$684$	0	0
$685$	24.8585	0.949793
$686$	0	0
$687$	−54.6698	−2.08578
$688$	0	0
$689$	7.66929	0.292177
$690$	0	0
$691$	−44.8400	−1.70580	−0.852898	−	0.522078i	$-0.825157\pi$
$691$	−44.8400	−1.70580	−0.852898	+	0.522078i	$0.825157\pi$
$692$	0	0
$693$	−25.8349	−0.981388
$694$	0	0
$695$	−30.2692	−1.14818
$696$	0	0
$697$	−8.24122	−0.312158
$698$	0	0
$699$	−44.2879	−1.67512
$700$	0	0
$701$	3.17476	0.119909	0.0599546	−	0.998201i	$-0.480904\pi$
$701$	3.17476	0.119909	0.0599546	+	0.998201i	$0.480904\pi$
$702$	0	0
$703$	27.4241	1.03432
$704$	0	0
$705$	−24.7478	−0.932058
$706$	0	0
$707$	−3.36487	−0.126549
$708$	0	0
$709$	−43.5824	−1.63677	−0.818385	−	0.574670i	$-0.805130\pi$
$709$	−43.5824	−1.63677	−0.818385	+	0.574670i	$0.805130\pi$
$710$	0	0
$711$	−13.4435	−0.504172
$712$	0	0
$713$	−71.3108	−2.67061
$714$	0	0
$715$	36.7325	1.37372
$716$	0	0
$717$	15.0835	0.563302
$718$	0	0
$719$	−48.2140	−1.79808	−0.899039	−	0.437869i	$-0.855733\pi$
$719$	−48.2140	−1.79808	−0.899039	+	0.437869i	$0.855733\pi$
$720$	0	0
$721$	−15.7989	−0.588383
$722$	0	0
$723$	−58.6972	−2.18297
$724$	0	0
$725$	1.35099	0.0501744
$726$	0	0
$727$	21.7931	0.808261	0.404130	−	0.914701i	$-0.367574\pi$
$727$	21.7931	0.808261	0.404130	+	0.914701i	$0.367574\pi$
$728$	0	0
$729$	−43.0284	−1.59364
$730$	0	0
$731$	−7.78952	−0.288106
$732$	0	0
$733$	31.3972	1.15968	0.579840	−	0.814730i	$-0.303115\pi$
$733$	31.3972	1.15968	0.579840	+	0.814730i	$0.303115\pi$
$734$	0	0
$735$	−5.57618	−0.205680
$736$	0	0
$737$	45.1275	1.66229
$738$	0	0
$739$	−42.7648	−1.57313	−0.786564	−	0.617508i	$-0.788142\pi$
$739$	−42.7648	−1.57313	−0.786564	+	0.617508i	$0.788142\pi$
$740$	0	0
$741$	−52.2303	−1.91873
$742$	0	0
$743$	14.3880	0.527845	0.263922	−	0.964544i	$-0.414984\pi$
$743$	14.3880	0.527845	0.263922	+	0.964544i	$0.414984\pi$
$744$	0	0
$745$	−17.1400	−0.627962
$746$	0	0
$747$	55.2193	2.02037
$748$	0	0
$749$	−16.2008	−0.591963
$750$	0	0
$751$	1.70839	0.0623399	0.0311699	−	0.999514i	$-0.490077\pi$
$751$	1.70839	0.0623399	0.0311699	+	0.999514i	$0.490077\pi$
$752$	0	0
$753$	−5.03219	−0.183383
$754$	0	0
$755$	−23.2536	−0.846286
$756$	0	0
$757$	48.4062	1.75935	0.879676	−	0.475574i	$-0.157760\pi$
$757$	48.4062	1.75935	0.879676	+	0.475574i	$0.157760\pi$
$758$	0	0
$759$	110.038	3.99413
$760$	0	0
$761$	31.2822	1.13398	0.566990	−	0.823725i	$-0.308108\pi$
$761$	31.2822	1.13398	0.566990	+	0.823725i	$0.308108\pi$
$762$	0	0
$763$	0.517516	0.0187353
$764$	0	0
$765$	−8.84554	−0.319811
$766$	0	0
$767$	−19.5120	−0.704538
$768$	0	0
$769$	−32.1505	−1.15938	−0.579688	−	0.814839i	$-0.696825\pi$
$769$	−32.1505	−1.15938	−0.579688	+	0.814839i	$0.696825\pi$
$770$	0	0
$771$	49.0564	1.76672
$772$	0	0
$773$	−9.17979	−0.330174	−0.165087	−	0.986279i	$-0.552791\pi$
$773$	−9.17979	−0.330174	−0.165087	+	0.986279i	$0.552791\pi$
$774$	0	0
$775$	−7.69373	−0.276367
$776$	0	0
$777$	−11.2560	−0.403806
$778$	0	0
$779$	54.1764	1.94107
$780$	0	0
$781$	6.73563	0.241020
$782$	0	0
$783$	−6.40137	−0.228766
$784$	0	0
$785$	−23.9626	−0.855260
$786$	0	0
$787$	−11.1887	−0.398834	−0.199417	−	0.979915i	$-0.563905\pi$
$787$	−11.1887	−0.398834	−0.199417	+	0.979915i	$0.563905\pi$
$788$	0	0
$789$	−61.1142	−2.17572
$790$	0	0
$791$	−3.76248	−0.133778
$792$	0	0
$793$	−9.03711	−0.320917
$794$	0	0
$795$	−14.5230	−0.515079
$796$	0	0
$797$	3.64731	0.129194	0.0645971	−	0.997911i	$-0.479424\pi$
$797$	3.64731	0.129194	0.0645971	+	0.997911i	$0.479424\pi$
$798$	0	0
$799$	4.43814	0.157010
$800$	0	0
$801$	−18.2139	−0.643555
$802$	0	0
$803$	−65.8943	−2.32536
$804$	0	0
$805$	13.9634	0.492145
$806$	0	0
$807$	64.0962	2.25629
$808$	0	0
$809$	22.9897	0.808273	0.404137	−	0.914699i	$-0.367572\pi$
$809$	22.9897	0.808273	0.404137	+	0.914699i	$0.367572\pi$
$810$	0	0
$811$	−4.61114	−0.161919	−0.0809594	−	0.996717i	$-0.525798\pi$
$811$	−4.61114	−0.161919	−0.0809594	+	0.996717i	$0.525798\pi$
$812$	0	0
$813$	69.8948	2.45132
$814$	0	0
$815$	−34.3734	−1.20405
$816$	0	0
$817$	51.2070	1.79151
$818$	0	0
$819$	12.6035	0.440403
$820$	0	0
$821$	−35.6653	−1.24473	−0.622364	−	0.782728i	$-0.713828\pi$
$821$	−35.6653	−1.24473	−0.622364	+	0.782728i	$0.713828\pi$
$822$	0	0
$823$	−37.8241	−1.31847	−0.659233	−	0.751939i	$-0.729119\pi$
$823$	−37.8241	−1.31847	−0.659233	+	0.751939i	$0.729119\pi$
$824$	0	0
$825$	11.8720	0.413331
$826$	0	0
$827$	35.0659	1.21936	0.609680	−	0.792648i	$-0.291298\pi$
$827$	35.0659	1.21936	0.609680	+	0.792648i	$0.291298\pi$
$828$	0	0
$829$	−2.11269	−0.0733769	−0.0366885	−	0.999327i	$-0.511681\pi$
$829$	−2.11269	−0.0733769	−0.0366885	+	0.999327i	$0.511681\pi$
$830$	0	0
$831$	44.8918	1.55728
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−34.5731	−1.19645
$836$	0	0
$837$	36.4551	1.26007
$838$	0	0
$839$	22.6688	0.782613	0.391306	−	0.920261i	$-0.372023\pi$
$839$	22.6688	0.782613	0.391306	+	0.920261i	$0.372023\pi$
$840$	0	0
$841$	−25.5653	−0.881561
$842$	0	0
$843$	75.5845	2.60327
$844$	0	0
$845$	8.94658	0.307772
$846$	0	0
$847$	25.4333	0.873899
$848$	0	0
$849$	44.1346	1.51470
$850$	0	0
$851$	28.1862	0.966211
$852$	0	0
$853$	2.76785	0.0947693	0.0473847	−	0.998877i	$-0.484911\pi$
$853$	2.76785	0.0947693	0.0473847	+	0.998877i	$0.484911\pi$
$854$	0	0
$855$	58.1491	1.98866
$856$	0	0
$857$	10.6206	0.362792	0.181396	−	0.983410i	$-0.441938\pi$
$857$	10.6206	0.362792	0.181396	+	0.983410i	$0.441938\pi$
$858$	0	0
$859$	−34.1180	−1.16409	−0.582045	−	0.813157i	$-0.697747\pi$
$859$	−34.1180	−1.16409	−0.582045	+	0.813157i	$0.697747\pi$
$860$	0	0
$861$	−22.2362	−0.757809
$862$	0	0
$863$	38.6533	1.31578	0.657888	−	0.753116i	$-0.271450\pi$
$863$	38.6533	1.31578	0.657888	+	0.753116i	$0.271450\pi$
$864$	0	0
$865$	−9.59618	−0.326280
$866$	0	0
$867$	2.69817	0.0916348
$868$	0	0
$869$	18.9585	0.643124
$870$	0	0
$871$	−22.0154	−0.745963
$872$	0	0
$873$	51.2594	1.73487
$874$	0	0
$875$	11.8398	0.400257
$876$	0	0
$877$	−14.6546	−0.494850	−0.247425	−	0.968907i	$-0.579584\pi$
$877$	−14.6546	−0.494850	−0.247425	+	0.968907i	$0.579584\pi$
$878$	0	0
$879$	−51.3277	−1.73124
$880$	0	0
$881$	−32.1168	−1.08204	−0.541021	−	0.841009i	$-0.681962\pi$
$881$	−32.1168	−1.08204	−0.541021	+	0.841009i	$0.681962\pi$
$882$	0	0
$883$	−9.35225	−0.314728	−0.157364	−	0.987541i	$-0.550300\pi$
$883$	−9.35225	−0.314728	−0.157364	+	0.987541i	$0.550300\pi$
$884$	0	0
$885$	36.9492	1.24203
$886$	0	0
$887$	−57.6954	−1.93722	−0.968611	−	0.248581i	$-0.920036\pi$
$887$	−57.6954	−1.93722	−0.968611	+	0.248581i	$0.920036\pi$
$888$	0	0
$889$	−5.02230	−0.168443
$890$	0	0
$891$	21.2517	0.711960
$892$	0	0
$893$	−29.1756	−0.976324
$894$	0	0
$895$	−2.95562	−0.0987954
$896$	0	0
$897$	−53.6819	−1.79239
$898$	0	0
$899$	−19.5604	−0.652375
$900$	0	0
$901$	2.60448	0.0867679
$902$	0	0
$903$	−21.0175	−0.699418
$904$	0	0
$905$	31.7681	1.05601
$906$	0	0
$907$	29.5931	0.982622	0.491311	−	0.870984i	$-0.336518\pi$
$907$	29.5931	0.982622	0.491311	+	0.870984i	$0.336518\pi$
$908$	0	0
$909$	−14.4021	−0.477688
$910$	0	0
$911$	15.9598	0.528771	0.264385	−	0.964417i	$-0.414831\pi$
$911$	15.9598	0.528771	0.264385	+	0.964417i	$0.414831\pi$
$912$	0	0
$913$	−77.8721	−2.57719
$914$	0	0
$915$	17.1132	0.565746
$916$	0	0
$917$	−7.43776	−0.245617
$918$	0	0
$919$	16.4561	0.542837	0.271418	−	0.962461i	$-0.412507\pi$
$919$	16.4561	0.542837	0.271418	+	0.962461i	$0.412507\pi$
$920$	0	0
$921$	−23.4149	−0.771547
$922$	0	0
$923$	−3.28596	−0.108159
$924$	0	0
$925$	3.04101	0.0999879
$926$	0	0
$927$	−67.6216	−2.22098
$928$	0	0
$929$	24.2400	0.795290	0.397645	−	0.917539i	$-0.369828\pi$
$929$	24.2400	0.795290	0.397645	+	0.917539i	$0.369828\pi$
$930$	0	0
$931$	−6.57383	−0.215449
$932$	0	0
$933$	7.54755	0.247096
$934$	0	0
$935$	12.4743	0.407953
$936$	0	0
$937$	−28.5568	−0.932909	−0.466454	−	0.884545i	$-0.654469\pi$
$937$	−28.5568	−0.932909	−0.466454	+	0.884545i	$0.654469\pi$
$938$	0	0
$939$	80.4898	2.62669
$940$	0	0
$941$	24.7694	0.807461	0.403730	−	0.914878i	$-0.367713\pi$
$941$	24.7694	0.807461	0.403730	+	0.914878i	$0.367713\pi$
$942$	0	0
$943$	55.6821	1.81326
$944$	0	0
$945$	−7.13828	−0.232208
$946$	0	0
$947$	−0.960336	−0.0312067	−0.0156034	−	0.999878i	$-0.504967\pi$
$947$	−0.960336	−0.0312067	−0.0156034	+	0.999878i	$0.504967\pi$
$948$	0	0
$949$	32.1464	1.04352
$950$	0	0
$951$	43.6979	1.41700
$952$	0	0
$953$	−37.5957	−1.21784	−0.608922	−	0.793230i	$-0.708398\pi$
$953$	−37.5957	−1.21784	−0.608922	+	0.793230i	$0.708398\pi$
$954$	0	0
$955$	51.5772	1.66900
$956$	0	0
$957$	30.1832	0.975684
$958$	0	0
$959$	−12.0284	−0.388417
$960$	0	0
$961$	80.3942	2.59336
$962$	0	0
$963$	−69.3416	−2.23450
$964$	0	0
$965$	34.3899	1.10705
$966$	0	0
$967$	51.1922	1.64623	0.823115	−	0.567874i	$-0.192234\pi$
$967$	51.1922	1.64623	0.823115	+	0.567874i	$0.192234\pi$
$968$	0	0
$969$	−17.7373	−0.569806
$970$	0	0
$971$	52.9062	1.69784	0.848921	−	0.528520i	$-0.177253\pi$
$971$	52.9062	1.69784	0.848921	+	0.528520i	$0.177253\pi$
$972$	0	0
$973$	14.6465	0.469546
$974$	0	0
$975$	−5.79174	−0.185484
$976$	0	0
$977$	13.7743	0.440678	0.220339	−	0.975423i	$-0.429284\pi$
$977$	13.7743	0.440678	0.220339	+	0.975423i	$0.429284\pi$
$978$	0	0
$979$	25.6858	0.820923
$980$	0	0
$981$	2.21504	0.0707207
$982$	0	0
$983$	55.5522	1.77184	0.885920	−	0.463837i	$-0.153528\pi$
$983$	55.5522	1.77184	0.885920	+	0.463837i	$0.153528\pi$
$984$	0	0
$985$	19.1893	0.611421
$986$	0	0
$987$	11.9749	0.381164
$988$	0	0
$989$	52.6302	1.67354
$990$	0	0
$991$	26.9638	0.856533	0.428267	−	0.903652i	$-0.359124\pi$
$991$	26.9638	0.856533	0.428267	+	0.903652i	$0.359124\pi$
$992$	0	0
$993$	−28.4436	−0.902631
$994$	0	0
$995$	−15.6926	−0.497489
$996$	0	0
$997$	−49.0462	−1.55331	−0.776654	−	0.629927i	$-0.783085\pi$
$997$	−49.0462	−1.55331	−0.776654	+	0.629927i	$0.783085\pi$
$998$	0	0
$999$	−14.4092	−0.455887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.g.1.6	✓	6
4.3	odd	2		3808.2.a.o.1.1	yes	6
8.3	odd	2		7616.2.a.bv.1.6		6
8.5	even	2		7616.2.a.cd.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.g.1.6	✓	6	1.1	even	1	trivial
3808.2.a.o.1.1	yes	6	4.3	odd	2
7616.2.a.bv.1.6		6	8.3	odd	2
7616.2.a.cd.1.1		6	8.5	even	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)

\(f(q)\)	\(=\)	\(q+2.69817 q^{3} -2.06665 q^{5} +1.00000 q^{7} +4.28014 q^{9} -6.03600 q^{11} +2.94465 q^{13} -5.57618 q^{15} +1.00000 q^{17} -6.57383 q^{19} +2.69817 q^{21} -6.75653 q^{23} -0.728963 q^{25} +3.45404 q^{27} -1.85330 q^{29} +10.5543 q^{31} -16.2862 q^{33} -2.06665 q^{35} -4.17170 q^{37} +7.94518 q^{39} -8.24122 q^{41} -7.78952 q^{43} -8.84554 q^{45} +4.43814 q^{47} +1.00000 q^{49} +2.69817 q^{51} +2.60448 q^{53} +12.4743 q^{55} -17.7373 q^{57} -6.62625 q^{59} -3.06899 q^{61} +4.28014 q^{63} -6.08556 q^{65} -7.47640 q^{67} -18.2303 q^{69} -1.11591 q^{71} +10.9169 q^{73} -1.96687 q^{75} -6.03600 q^{77} -3.14091 q^{79} -3.52083 q^{81} +12.9013 q^{83} -2.06665 q^{85} -5.00053 q^{87} -4.25544 q^{89} +2.94465 q^{91} +28.4775 q^{93} +13.5858 q^{95} +11.9761 q^{97} -25.8349 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{13} + 6 q^{17} - 18 q^{19} - 4 q^{21} - 6 q^{23} + 12 q^{25} - 10 q^{27} + 4 q^{29} + 8 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} + 18 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 - 4 * q^5 + 6 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^13 + 6 * q^17 - 18 * q^19 - 4 * q^21 - 6 * q^23 + 12 * q^25 - 10 * q^27 + 4 * q^29 + 8 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 + 18 * q^39 - 4 * q^41 - 20 * q^43 - 24 * q^45 - 6 * q^47 + 6 * q^49 - 4 * q^51 + 14 * q^53 - 4 * q^55 - 4 * q^57 - 20 * q^59 + 2 * q^61 + 8 * q^63 + 6 * q^65 - 28 * q^67 - 4 * q^69 - 18 * q^71 + 4 * q^73 - 20 * q^75 - 8 * q^77 + 2 * q^79 + 34 * q^81 - 20 * q^83 - 4 * q^85 - 22 * q^87 - 6 * q^89 - 4 * q^91 + 40 * q^93 + 6 * q^95 + 30 * q^97 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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