LMFDB - Embedded newform 3808.2.a.g.1.3

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			1.3
Root			$0.301346$ 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)

$f(q)$	$=$	$q-0.698654 q^{3} +3.66503 q^{5} +1.00000 q^{7} -2.51188 q^{9} -3.03767 q^{11} -2.40579 q^{13} -2.56059 q^{15} +1.00000 q^{17} -1.71816 q^{19} -0.698654 q^{21} -6.60508 q^{23} +8.43243 q^{25} +3.85090 q^{27} +5.84924 q^{29} +0.381122 q^{31} +2.12228 q^{33} +3.66503 q^{35} -6.48929 q^{37} +1.68081 q^{39} -5.75328 q^{41} -6.74480 q^{43} -9.20612 q^{45} -5.73345 q^{47} +1.00000 q^{49} -0.698654 q^{51} -4.98453 q^{53} -11.1331 q^{55} +1.20040 q^{57} +8.74432 q^{59} +0.822606 q^{61} -2.51188 q^{63} -8.81727 q^{65} -14.1170 q^{67} +4.61467 q^{69} -14.5830 q^{71} +9.22593 q^{73} -5.89135 q^{75} -3.03767 q^{77} -11.2231 q^{79} +4.84520 q^{81} -12.8269 q^{83} +3.66503 q^{85} -4.08660 q^{87} +10.6204 q^{89} -2.40579 q^{91} -0.266273 q^{93} -6.29712 q^{95} -3.08403 q^{97} +7.63027 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.698654	−0.403368	−0.201684	−	0.979451i	$-0.564641\pi$
$3$	−0.698654	−0.403368	−0.201684	+	0.979451i	$0.564641\pi$
$4$	0	0
$5$	3.66503	1.63905	0.819525	−	0.573043i	$-0.194237\pi$
$5$	3.66503	1.63905	0.819525	+	0.573043i	$0.194237\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.51188	−0.837294
$10$	0	0
$11$	−3.03767	−0.915892	−0.457946	−	0.888980i	$-0.651415\pi$
$11$	−3.03767	−0.915892	−0.457946	+	0.888980i	$0.651415\pi$
$12$	0	0
$13$	−2.40579	−0.667245	−0.333623	−	0.942707i	$-0.608271\pi$
$13$	−2.40579	−0.667245	−0.333623	+	0.942707i	$0.608271\pi$
$14$	0	0
$15$	−2.56059	−0.661141
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−1.71816	−0.394174	−0.197087	−	0.980386i	$-0.563148\pi$
$19$	−1.71816	−0.394174	−0.197087	+	0.980386i	$0.563148\pi$
$20$	0	0
$21$	−0.698654	−0.152459
$22$	0	0
$23$	−6.60508	−1.37725	−0.688627	−	0.725115i	$-0.741786\pi$
$23$	−6.60508	−1.37725	−0.688627	+	0.725115i	$0.741786\pi$
$24$	0	0
$25$	8.43243	1.68649
$26$	0	0
$27$	3.85090	0.741106
$28$	0	0
$29$	5.84924	1.08618	0.543089	−	0.839675i	$-0.317255\pi$
$29$	5.84924	1.08618	0.543089	+	0.839675i	$0.317255\pi$
$30$	0	0
$31$	0.381122	0.0684516	0.0342258	−	0.999414i	$-0.489103\pi$
$31$	0.381122	0.0684516	0.0342258	+	0.999414i	$0.489103\pi$
$32$	0	0
$33$	2.12228	0.369442
$34$	0	0
$35$	3.66503	0.619503
$36$	0	0
$37$	−6.48929	−1.06683	−0.533416	−	0.845853i	$-0.679092\pi$
$37$	−6.48929	−1.06683	−0.533416	+	0.845853i	$0.679092\pi$
$38$	0	0
$39$	1.68081	0.269145
$40$	0	0
$41$	−5.75328	−0.898512	−0.449256	−	0.893403i	$-0.648311\pi$
$41$	−5.75328	−0.898512	−0.449256	+	0.893403i	$0.648311\pi$
$42$	0	0
$43$	−6.74480	−1.02857	−0.514286	−	0.857618i	$-0.671943\pi$
$43$	−6.74480	−1.02857	−0.514286	+	0.857618i	$0.671943\pi$
$44$	0	0
$45$	−9.20612	−1.37237
$46$	0	0
$47$	−5.73345	−0.836310	−0.418155	−	0.908376i	$-0.637323\pi$
$47$	−5.73345	−0.836310	−0.418155	+	0.908376i	$0.637323\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.698654	−0.0978312
$52$	0	0
$53$	−4.98453	−0.684678	−0.342339	−	0.939576i	$-0.611219\pi$
$53$	−4.98453	−0.684678	−0.342339	+	0.939576i	$0.611219\pi$
$54$	0	0
$55$	−11.1331	−1.50119
$56$	0	0
$57$	1.20040	0.158997
$58$	0	0
$59$	8.74432	1.13841	0.569207	−	0.822194i	$-0.307250\pi$
$59$	8.74432	1.13841	0.569207	+	0.822194i	$0.307250\pi$
$60$	0	0
$61$	0.822606	0.105324	0.0526619	−	0.998612i	$-0.483229\pi$
$61$	0.822606	0.105324	0.0526619	+	0.998612i	$0.483229\pi$
$62$	0	0
$63$	−2.51188	−0.316467
$64$	0	0
$65$	−8.81727	−1.09365
$66$	0	0
$67$	−14.1170	−1.72466	−0.862331	−	0.506345i	$-0.830996\pi$
$67$	−14.1170	−1.72466	−0.862331	+	0.506345i	$0.830996\pi$
$68$	0	0
$69$	4.61467	0.555541
$70$	0	0
$71$	−14.5830	−1.73069	−0.865343	−	0.501181i	$-0.832899\pi$
$71$	−14.5830	−1.73069	−0.865343	+	0.501181i	$0.832899\pi$
$72$	0	0
$73$	9.22593	1.07981	0.539907	−	0.841725i	$-0.318459\pi$
$73$	9.22593	1.07981	0.539907	+	0.841725i	$0.318459\pi$
$74$	0	0
$75$	−5.89135	−0.680274
$76$	0	0
$77$	−3.03767	−0.346175
$78$	0	0
$79$	−11.2231	−1.26269	−0.631346	−	0.775501i	$-0.717497\pi$
$79$	−11.2231	−1.26269	−0.631346	+	0.775501i	$0.717497\pi$
$80$	0	0
$81$	4.84520	0.538356
$82$	0	0
$83$	−12.8269	−1.40793	−0.703965	−	0.710234i	$-0.748589\pi$
$83$	−12.8269	−1.40793	−0.703965	+	0.710234i	$0.748589\pi$
$84$	0	0
$85$	3.66503	0.397528
$86$	0	0
$87$	−4.08660	−0.438129
$88$	0	0
$89$	10.6204	1.12576	0.562878	−	0.826540i	$-0.309694\pi$
$89$	10.6204	1.12576	0.562878	+	0.826540i	$0.309694\pi$
$90$	0	0
$91$	−2.40579	−0.252195
$92$	0	0
$93$	−0.266273	−0.0276112
$94$	0	0
$95$	−6.29712	−0.646071
$96$	0	0
$97$	−3.08403	−0.313136	−0.156568	−	0.987667i	$-0.550043\pi$
$97$	−3.08403	−0.313136	−0.156568	+	0.987667i	$0.550043\pi$
$98$	0	0
$99$	7.63027	0.766871
$100$	0	0
$101$	−13.6872	−1.36193	−0.680965	−	0.732316i	$-0.738440\pi$
$101$	−13.6872	−1.36193	−0.680965	+	0.732316i	$0.738440\pi$
$102$	0	0
$103$	14.6679	1.44528	0.722638	−	0.691227i	$-0.242929\pi$
$103$	14.6679	1.44528	0.722638	+	0.691227i	$0.242929\pi$
$104$	0	0
$105$	−2.56059	−0.249888
$106$	0	0
$107$	15.3769	1.48654	0.743272	−	0.668990i	$-0.233273\pi$
$107$	15.3769	1.48654	0.743272	+	0.668990i	$0.233273\pi$
$108$	0	0
$109$	7.72529	0.739949	0.369974	−	0.929042i	$-0.379366\pi$
$109$	7.72529	0.739949	0.369974	+	0.929042i	$0.379366\pi$
$110$	0	0
$111$	4.53377	0.430326
$112$	0	0
$113$	7.20610	0.677893	0.338947	−	0.940806i	$-0.389929\pi$
$113$	7.20610	0.677893	0.338947	+	0.940806i	$0.389929\pi$
$114$	0	0
$115$	−24.2078	−2.25739
$116$	0	0
$117$	6.04305	0.558680
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−1.77255	−0.161141
$122$	0	0
$123$	4.01955	0.362431
$124$	0	0
$125$	12.5799	1.12518
$126$	0	0
$127$	3.67319	0.325943	0.162971	−	0.986631i	$-0.447892\pi$
$127$	3.67319	0.325943	0.162971	+	0.986631i	$0.447892\pi$
$128$	0	0
$129$	4.71229	0.414894
$130$	0	0
$131$	12.2574	1.07094	0.535468	−	0.844556i	$-0.320135\pi$
$131$	12.2574	1.07094	0.535468	+	0.844556i	$0.320135\pi$
$132$	0	0
$133$	−1.71816	−0.148984
$134$	0	0
$135$	14.1137	1.21471
$136$	0	0
$137$	−14.0620	−1.20139	−0.600697	−	0.799477i	$-0.705110\pi$
$137$	−14.0620	−1.20139	−0.600697	+	0.799477i	$0.705110\pi$
$138$	0	0
$139$	−0.150965	−0.0128047	−0.00640233	−	0.999980i	$-0.502038\pi$
$139$	−0.150965	−0.0128047	−0.00640233	+	0.999980i	$0.502038\pi$
$140$	0	0
$141$	4.00570	0.337341
$142$	0	0
$143$	7.30799	0.611125
$144$	0	0
$145$	21.4376	1.78030
$146$	0	0
$147$	−0.698654	−0.0576240
$148$	0	0
$149$	−4.70920	−0.385793	−0.192896	−	0.981219i	$-0.561788\pi$
$149$	−4.70920	−0.385793	−0.192896	+	0.981219i	$0.561788\pi$
$150$	0	0
$151$	1.62694	0.132399	0.0661993	−	0.997806i	$-0.478913\pi$
$151$	1.62694	0.132399	0.0661993	+	0.997806i	$0.478913\pi$
$152$	0	0
$153$	−2.51188	−0.203074
$154$	0	0
$155$	1.39682	0.112196
$156$	0	0
$157$	5.13471	0.409794	0.204897	−	0.978783i	$-0.434314\pi$
$157$	5.13471	0.409794	0.204897	+	0.978783i	$0.434314\pi$
$158$	0	0
$159$	3.48247	0.276177
$160$	0	0
$161$	−6.60508	−0.520553
$162$	0	0
$163$	−13.3858	−1.04846	−0.524228	−	0.851578i	$-0.675646\pi$
$163$	−13.3858	−1.04846	−0.524228	+	0.851578i	$0.675646\pi$
$164$	0	0
$165$	7.77822	0.605534
$166$	0	0
$167$	−0.637938	−0.0493651	−0.0246826	−	0.999695i	$-0.507858\pi$
$167$	−0.637938	−0.0493651	−0.0246826	+	0.999695i	$0.507858\pi$
$168$	0	0
$169$	−7.21219	−0.554784
$170$	0	0
$171$	4.31583	0.330040
$172$	0	0
$173$	−5.01784	−0.381499	−0.190750	−	0.981639i	$-0.561092\pi$
$173$	−5.01784	−0.381499	−0.190750	+	0.981639i	$0.561092\pi$
$174$	0	0
$175$	8.43243	0.637431
$176$	0	0
$177$	−6.10926	−0.459200
$178$	0	0
$179$	−13.2331	−0.989091	−0.494545	−	0.869152i	$-0.664665\pi$
$179$	−13.2331	−0.989091	−0.494545	+	0.869152i	$0.664665\pi$
$180$	0	0
$181$	2.53687	0.188564	0.0942819	−	0.995546i	$-0.469945\pi$
$181$	2.53687	0.188564	0.0942819	+	0.995546i	$0.469945\pi$
$182$	0	0
$183$	−0.574717	−0.0424843
$184$	0	0
$185$	−23.7834	−1.74859
$186$	0	0
$187$	−3.03767	−0.222137
$188$	0	0
$189$	3.85090	0.280112
$190$	0	0
$191$	8.88951	0.643223	0.321611	−	0.946872i	$-0.395776\pi$
$191$	8.88951	0.643223	0.321611	+	0.946872i	$0.395776\pi$
$192$	0	0
$193$	−15.6913	−1.12948	−0.564742	−	0.825268i	$-0.691024\pi$
$193$	−15.6913	−1.12948	−0.564742	+	0.825268i	$0.691024\pi$
$194$	0	0
$195$	6.16022	0.441143
$196$	0	0
$197$	−8.29063	−0.590683	−0.295342	−	0.955392i	$-0.595433\pi$
$197$	−8.29063	−0.590683	−0.295342	+	0.955392i	$0.595433\pi$
$198$	0	0
$199$	−6.88404	−0.487997	−0.243998	−	0.969776i	$-0.578459\pi$
$199$	−6.88404	−0.487997	−0.243998	+	0.969776i	$0.578459\pi$
$200$	0	0
$201$	9.86288	0.695674
$202$	0	0
$203$	5.84924	0.410536
$204$	0	0
$205$	−21.0859	−1.47271
$206$	0	0
$207$	16.5912	1.15317
$208$	0	0
$209$	5.21922	0.361021
$210$	0	0
$211$	−26.9297	−1.85392	−0.926959	−	0.375163i	$-0.877587\pi$
$211$	−26.9297	−1.85392	−0.926959	+	0.375163i	$0.877587\pi$
$212$	0	0
$213$	10.1885	0.698103
$214$	0	0
$215$	−24.7199	−1.68588
$216$	0	0
$217$	0.381122	0.0258723
$218$	0	0
$219$	−6.44574	−0.435563
$220$	0	0
$221$	−2.40579	−0.161831
$222$	0	0
$223$	16.0856	1.07717	0.538587	−	0.842570i	$-0.318958\pi$
$223$	16.0856	1.07717	0.538587	+	0.842570i	$0.318958\pi$
$224$	0	0
$225$	−21.1813	−1.41208
$226$	0	0
$227$	−25.3913	−1.68528	−0.842639	−	0.538478i	$-0.818999\pi$
$227$	−25.3913	−1.68528	−0.842639	+	0.538478i	$0.818999\pi$
$228$	0	0
$229$	19.6992	1.30176	0.650880	−	0.759180i	$-0.274400\pi$
$229$	19.6992	1.30176	0.650880	+	0.759180i	$0.274400\pi$
$230$	0	0
$231$	2.12228	0.139636
$232$	0	0
$233$	21.3725	1.40016	0.700080	−	0.714064i	$-0.253148\pi$
$233$	21.3725	1.40016	0.700080	+	0.714064i	$0.253148\pi$
$234$	0	0
$235$	−21.0133	−1.37075
$236$	0	0
$237$	7.84104	0.509330
$238$	0	0
$239$	17.7954	1.15109	0.575545	−	0.817770i	$-0.304790\pi$
$239$	17.7954	1.15109	0.575545	+	0.817770i	$0.304790\pi$
$240$	0	0
$241$	8.64503	0.556875	0.278438	−	0.960454i	$-0.410183\pi$
$241$	8.64503	0.556875	0.278438	+	0.960454i	$0.410183\pi$
$242$	0	0
$243$	−14.9378	−0.958261
$244$	0	0
$245$	3.66503	0.234150
$246$	0	0
$247$	4.13354	0.263011
$248$	0	0
$249$	8.96154	0.567914
$250$	0	0
$251$	17.0342	1.07519	0.537596	−	0.843203i	$-0.319333\pi$
$251$	17.0342	1.07519	0.537596	+	0.843203i	$0.319333\pi$
$252$	0	0
$253$	20.0641	1.26142
$254$	0	0
$255$	−2.56059	−0.160350
$256$	0	0
$257$	20.8027	1.29764	0.648818	−	0.760944i	$-0.275264\pi$
$257$	20.8027	1.29764	0.648818	+	0.760944i	$0.275264\pi$
$258$	0	0
$259$	−6.48929	−0.403225
$260$	0	0
$261$	−14.6926	−0.909450
$262$	0	0
$263$	22.7749	1.40436	0.702180	−	0.711999i	$-0.252210\pi$
$263$	22.7749	1.40436	0.702180	+	0.711999i	$0.252210\pi$
$264$	0	0
$265$	−18.2685	−1.12222
$266$	0	0
$267$	−7.41996	−0.454094
$268$	0	0
$269$	−31.4642	−1.91841	−0.959204	−	0.282714i	$-0.908765\pi$
$269$	−31.4642	−1.91841	−0.959204	+	0.282714i	$0.908765\pi$
$270$	0	0
$271$	9.97927	0.606197	0.303099	−	0.952959i	$-0.401979\pi$
$271$	9.97927	0.606197	0.303099	+	0.952959i	$0.401979\pi$
$272$	0	0
$273$	1.68081	0.101727
$274$	0	0
$275$	−25.6149	−1.54464
$276$	0	0
$277$	−23.2836	−1.39897	−0.699487	−	0.714645i	$-0.746588\pi$
$277$	−23.2836	−1.39897	−0.699487	+	0.714645i	$0.746588\pi$
$278$	0	0
$279$	−0.957334	−0.0573141
$280$	0	0
$281$	−15.6648	−0.934483	−0.467241	−	0.884130i	$-0.654752\pi$
$281$	−15.6648	−0.934483	−0.467241	+	0.884130i	$0.654752\pi$
$282$	0	0
$283$	1.95968	0.116491	0.0582454	−	0.998302i	$-0.481449\pi$
$283$	1.95968	0.116491	0.0582454	+	0.998302i	$0.481449\pi$
$284$	0	0
$285$	4.39951	0.260605
$286$	0	0
$287$	−5.75328	−0.339605
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	2.15467	0.126309
$292$	0	0
$293$	4.05503	0.236897	0.118449	−	0.992960i	$-0.462208\pi$
$293$	4.05503	0.236897	0.118449	+	0.992960i	$0.462208\pi$
$294$	0	0
$295$	32.0482	1.86592
$296$	0	0
$297$	−11.6978	−0.678773
$298$	0	0
$299$	15.8904	0.918966
$300$	0	0
$301$	−6.74480	−0.388764
$302$	0	0
$303$	9.56264	0.549359
$304$	0	0
$305$	3.01487	0.172631
$306$	0	0
$307$	−26.2543	−1.49841	−0.749207	−	0.662336i	$-0.769565\pi$
$307$	−26.2543	−1.49841	−0.749207	+	0.662336i	$0.769565\pi$
$308$	0	0
$309$	−10.2478	−0.582978
$310$	0	0
$311$	−6.31056	−0.357839	−0.178919	−	0.983864i	$-0.557260\pi$
$311$	−6.31056	−0.357839	−0.178919	+	0.983864i	$0.557260\pi$
$312$	0	0
$313$	26.9937	1.52577	0.762886	−	0.646533i	$-0.223782\pi$
$313$	26.9937	1.52577	0.762886	+	0.646533i	$0.223782\pi$
$314$	0	0
$315$	−9.20612	−0.518706
$316$	0	0
$317$	−5.47914	−0.307739	−0.153870	−	0.988091i	$-0.549174\pi$
$317$	−5.47914	−0.307739	−0.153870	+	0.988091i	$0.549174\pi$
$318$	0	0
$319$	−17.7681	−0.994822
$320$	0	0
$321$	−10.7432	−0.599624
$322$	0	0
$323$	−1.71816	−0.0956013
$324$	0	0
$325$	−20.2866	−1.12530
$326$	0	0
$327$	−5.39731	−0.298472
$328$	0	0
$329$	−5.73345	−0.316095
$330$	0	0
$331$	28.0056	1.53933	0.769663	−	0.638450i	$-0.220424\pi$
$331$	28.0056	1.53933	0.769663	+	0.638450i	$0.220424\pi$
$332$	0	0
$333$	16.3003	0.893252
$334$	0	0
$335$	−51.7391	−2.82681
$336$	0	0
$337$	−12.7267	−0.693269	−0.346634	−	0.938000i	$-0.612676\pi$
$337$	−12.7267	−0.693269	−0.346634	+	0.938000i	$0.612676\pi$
$338$	0	0
$339$	−5.03457	−0.273441
$340$	0	0
$341$	−1.15772	−0.0626943
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	16.9129	0.910559
$346$	0	0
$347$	−27.8442	−1.49476	−0.747378	−	0.664399i	$-0.768688\pi$
$347$	−27.8442	−1.49476	−0.747378	+	0.664399i	$0.768688\pi$
$348$	0	0
$349$	−30.2329	−1.61833	−0.809165	−	0.587582i	$-0.800080\pi$
$349$	−30.2329	−1.61833	−0.809165	+	0.587582i	$0.800080\pi$
$350$	0	0
$351$	−9.26444	−0.494499
$352$	0	0
$353$	−0.967067	−0.0514718	−0.0257359	−	0.999669i	$-0.508193\pi$
$353$	−0.967067	−0.0514718	−0.0257359	+	0.999669i	$0.508193\pi$
$354$	0	0
$355$	−53.4471	−2.83668
$356$	0	0
$357$	−0.698654	−0.0369767
$358$	0	0
$359$	16.4556	0.868495	0.434247	−	0.900794i	$-0.357014\pi$
$359$	16.4556	0.868495	0.434247	+	0.900794i	$0.357014\pi$
$360$	0	0
$361$	−16.0479	−0.844627
$362$	0	0
$363$	1.23840	0.0649992
$364$	0	0
$365$	33.8133	1.76987
$366$	0	0
$367$	7.27078	0.379532	0.189766	−	0.981829i	$-0.439227\pi$
$367$	7.27078	0.379532	0.189766	+	0.981829i	$0.439227\pi$
$368$	0	0
$369$	14.4516	0.752319
$370$	0	0
$371$	−4.98453	−0.258784
$372$	0	0
$373$	−10.9544	−0.567198	−0.283599	−	0.958943i	$-0.591528\pi$
$373$	−10.9544	−0.567198	−0.283599	+	0.958943i	$0.591528\pi$
$374$	0	0
$375$	−8.78902	−0.453863
$376$	0	0
$377$	−14.0720	−0.724747
$378$	0	0
$379$	5.57491	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$379$	5.57491	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$380$	0	0
$381$	−2.56629	−0.131475
$382$	0	0
$383$	19.4301	0.992834	0.496417	−	0.868084i	$-0.334649\pi$
$383$	19.4301	0.992834	0.496417	+	0.868084i	$0.334649\pi$
$384$	0	0
$385$	−11.1331	−0.567398
$386$	0	0
$387$	16.9422	0.861218
$388$	0	0
$389$	4.00596	0.203110	0.101555	−	0.994830i	$-0.467618\pi$
$389$	4.00596	0.203110	0.101555	+	0.994830i	$0.467618\pi$
$390$	0	0
$391$	−6.60508	−0.334033
$392$	0	0
$393$	−8.56370	−0.431981
$394$	0	0
$395$	−41.1328	−2.06962
$396$	0	0
$397$	−2.45964	−0.123446	−0.0617230	−	0.998093i	$-0.519660\pi$
$397$	−2.45964	−0.123446	−0.0617230	+	0.998093i	$0.519660\pi$
$398$	0	0
$399$	1.20040	0.0600953
$400$	0	0
$401$	6.90723	0.344931	0.172465	−	0.985016i	$-0.444827\pi$
$401$	6.90723	0.344931	0.172465	+	0.985016i	$0.444827\pi$
$402$	0	0
$403$	−0.916899	−0.0456740
$404$	0	0
$405$	17.7578	0.882392
$406$	0	0
$407$	19.7123	0.977104
$408$	0	0
$409$	−16.7759	−0.829513	−0.414756	−	0.909932i	$-0.636133\pi$
$409$	−16.7759	−0.829513	−0.414756	+	0.909932i	$0.636133\pi$
$410$	0	0
$411$	9.82445	0.484604
$412$	0	0
$413$	8.74432	0.430280
$414$	0	0
$415$	−47.0108	−2.30767
$416$	0	0
$417$	0.105472	0.00516499
$418$	0	0
$419$	−35.7475	−1.74638	−0.873191	−	0.487379i	$-0.837953\pi$
$419$	−35.7475	−1.74638	−0.873191	+	0.487379i	$0.837953\pi$
$420$	0	0
$421$	21.4145	1.04368	0.521839	−	0.853044i	$-0.325246\pi$
$421$	21.4145	1.04368	0.521839	+	0.853044i	$0.325246\pi$
$422$	0	0
$423$	14.4018	0.700237
$424$	0	0
$425$	8.43243	0.409033
$426$	0	0
$427$	0.822606	0.0398087
$428$	0	0
$429$	−5.10576	−0.246508
$430$	0	0
$431$	18.1043	0.872052	0.436026	−	0.899934i	$-0.356386\pi$
$431$	18.1043	0.872052	0.436026	+	0.899934i	$0.356386\pi$
$432$	0	0
$433$	−24.5870	−1.18158	−0.590789	−	0.806826i	$-0.701183\pi$
$433$	−24.5870	−1.18158	−0.590789	+	0.806826i	$0.701183\pi$
$434$	0	0
$435$	−14.9775	−0.718116
$436$	0	0
$437$	11.3486	0.542878
$438$	0	0
$439$	−4.54640	−0.216988	−0.108494	−	0.994097i	$-0.534603\pi$
$439$	−4.54640	−0.216988	−0.108494	+	0.994097i	$0.534603\pi$
$440$	0	0
$441$	−2.51188	−0.119613
$442$	0	0
$443$	−9.03838	−0.429426	−0.214713	−	0.976677i	$-0.568882\pi$
$443$	−9.03838	−0.429426	−0.214713	+	0.976677i	$0.568882\pi$
$444$	0	0
$445$	38.9239	1.84517
$446$	0	0
$447$	3.29010	0.155617
$448$	0	0
$449$	27.6166	1.30331	0.651653	−	0.758517i	$-0.274076\pi$
$449$	27.6166	1.30331	0.651653	+	0.758517i	$0.274076\pi$
$450$	0	0
$451$	17.4766	0.822940
$452$	0	0
$453$	−1.13667	−0.0534054
$454$	0	0
$455$	−8.81727	−0.413360
$456$	0	0
$457$	30.8579	1.44347	0.721736	−	0.692168i	$-0.243344\pi$
$457$	30.8579	1.44347	0.721736	+	0.692168i	$0.243344\pi$
$458$	0	0
$459$	3.85090	0.179745
$460$	0	0
$461$	−35.7213	−1.66371	−0.831853	−	0.554996i	$-0.812720\pi$
$461$	−35.7213	−1.66371	−0.831853	+	0.554996i	$0.812720\pi$
$462$	0	0
$463$	−0.262644	−0.0122061	−0.00610304	−	0.999981i	$-0.501943\pi$
$463$	−0.262644	−0.0122061	−0.00610304	+	0.999981i	$0.501943\pi$
$464$	0	0
$465$	−0.975897	−0.0452561
$466$	0	0
$467$	27.0367	1.25111	0.625555	−	0.780180i	$-0.284873\pi$
$467$	27.0367	1.25111	0.625555	+	0.780180i	$0.284873\pi$
$468$	0	0
$469$	−14.1170	−0.651861
$470$	0	0
$471$	−3.58739	−0.165298
$472$	0	0
$473$	20.4885	0.942062
$474$	0	0
$475$	−14.4883	−0.664769
$476$	0	0
$477$	12.5206	0.573277
$478$	0	0
$479$	14.3121	0.653939	0.326969	−	0.945035i	$-0.393973\pi$
$479$	14.3121	0.653939	0.326969	+	0.945035i	$0.393973\pi$
$480$	0	0
$481$	15.6118	0.711839
$482$	0	0
$483$	4.61467	0.209975
$484$	0	0
$485$	−11.3031	−0.513245
$486$	0	0
$487$	−17.6507	−0.799828	−0.399914	−	0.916553i	$-0.630960\pi$
$487$	−17.6507	−0.799828	−0.399914	+	0.916553i	$0.630960\pi$
$488$	0	0
$489$	9.35204	0.422914
$490$	0	0
$491$	−4.47987	−0.202174	−0.101087	−	0.994878i	$-0.532232\pi$
$491$	−4.47987	−0.202174	−0.101087	+	0.994878i	$0.532232\pi$
$492$	0	0
$493$	5.84924	0.263437
$494$	0	0
$495$	27.9652	1.25694
$496$	0	0
$497$	−14.5830	−0.654138
$498$	0	0
$499$	−29.6589	−1.32772	−0.663858	−	0.747859i	$-0.731082\pi$
$499$	−29.6589	−1.32772	−0.663858	+	0.747859i	$0.731082\pi$
$500$	0	0
$501$	0.445698	0.0199123
$502$	0	0
$503$	19.9897	0.891295	0.445647	−	0.895209i	$-0.352974\pi$
$503$	19.9897	0.891295	0.445647	+	0.895209i	$0.352974\pi$
$504$	0	0
$505$	−50.1641	−2.23227
$506$	0	0
$507$	5.03883	0.223782
$508$	0	0
$509$	−18.7636	−0.831680	−0.415840	−	0.909438i	$-0.636512\pi$
$509$	−18.7636	−0.831680	−0.415840	+	0.909438i	$0.636512\pi$
$510$	0	0
$511$	9.22593	0.408131
$512$	0	0
$513$	−6.61648	−0.292125
$514$	0	0
$515$	53.7584	2.36888
$516$	0	0
$517$	17.4163	0.765970
$518$	0	0
$519$	3.50574	0.153885
$520$	0	0
$521$	−2.72185	−0.119246	−0.0596231	−	0.998221i	$-0.518990\pi$
$521$	−2.72185	−0.119246	−0.0596231	+	0.998221i	$0.518990\pi$
$522$	0	0
$523$	−9.48211	−0.414624	−0.207312	−	0.978275i	$-0.566471\pi$
$523$	−9.48211	−0.414624	−0.207312	+	0.978275i	$0.566471\pi$
$524$	0	0
$525$	−5.89135	−0.257120
$526$	0	0
$527$	0.381122	0.0166019
$528$	0	0
$529$	20.6271	0.896830
$530$	0	0
$531$	−21.9647	−0.953187
$532$	0	0
$533$	13.8412	0.599527
$534$	0	0
$535$	56.3568	2.43652
$536$	0	0
$537$	9.24538	0.398968
$538$	0	0
$539$	−3.03767	−0.130842
$540$	0	0
$541$	0.877989	0.0377477	0.0188738	−	0.999822i	$-0.493992\pi$
$541$	0.877989	0.0377477	0.0188738	+	0.999822i	$0.493992\pi$
$542$	0	0
$543$	−1.77239	−0.0760606
$544$	0	0
$545$	28.3134	1.21281
$546$	0	0
$547$	19.4730	0.832605	0.416303	−	0.909226i	$-0.363326\pi$
$547$	19.4730	0.832605	0.416303	+	0.909226i	$0.363326\pi$
$548$	0	0
$549$	−2.06629	−0.0881870
$550$	0	0
$551$	−10.0500	−0.428143
$552$	0	0
$553$	−11.2231	−0.477253
$554$	0	0
$555$	16.6164	0.705326
$556$	0	0
$557$	17.8746	0.757369	0.378685	−	0.925526i	$-0.376377\pi$
$557$	17.8746	0.757369	0.378685	+	0.925526i	$0.376377\pi$
$558$	0	0
$559$	16.2266	0.686310
$560$	0	0
$561$	2.12228	0.0896028
$562$	0	0
$563$	−0.137928	−0.00581295	−0.00290648	−	0.999996i	$-0.500925\pi$
$563$	−0.137928	−0.00581295	−0.00290648	+	0.999996i	$0.500925\pi$
$564$	0	0
$565$	26.4106	1.11110
$566$	0	0
$567$	4.84520	0.203479
$568$	0	0
$569$	19.9320	0.835591	0.417796	−	0.908541i	$-0.362803\pi$
$569$	19.9320	0.835591	0.417796	+	0.908541i	$0.362803\pi$
$570$	0	0
$571$	24.3246	1.01795	0.508977	−	0.860780i	$-0.330024\pi$
$571$	24.3246	1.01795	0.508977	+	0.860780i	$0.330024\pi$
$572$	0	0
$573$	−6.21070	−0.259456
$574$	0	0
$575$	−55.6969	−2.32272
$576$	0	0
$577$	30.7677	1.28088	0.640439	−	0.768009i	$-0.278753\pi$
$577$	30.7677	1.28088	0.640439	+	0.768009i	$0.278753\pi$
$578$	0	0
$579$	10.9628	0.455598
$580$	0	0
$581$	−12.8269	−0.532148
$582$	0	0
$583$	15.1414	0.627092
$584$	0	0
$585$	22.1480	0.915705
$586$	0	0
$587$	10.2883	0.424643	0.212321	−	0.977200i	$-0.431898\pi$
$587$	10.2883	0.424643	0.212321	+	0.977200i	$0.431898\pi$
$588$	0	0
$589$	−0.654831	−0.0269818
$590$	0	0
$591$	5.79229	0.238263
$592$	0	0
$593$	−28.5600	−1.17282	−0.586409	−	0.810015i	$-0.699459\pi$
$593$	−28.5600	−1.17282	−0.586409	+	0.810015i	$0.699459\pi$
$594$	0	0
$595$	3.66503	0.150251
$596$	0	0
$597$	4.80956	0.196842
$598$	0	0
$599$	−41.2052	−1.68360	−0.841799	−	0.539791i	$-0.818503\pi$
$599$	−41.2052	−1.68360	−0.841799	+	0.539791i	$0.818503\pi$
$600$	0	0
$601$	24.7782	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$601$	24.7782	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$602$	0	0
$603$	35.4602	1.44405
$604$	0	0
$605$	−6.49645	−0.264118
$606$	0	0
$607$	−7.45090	−0.302423	−0.151211	−	0.988501i	$-0.548317\pi$
$607$	−7.45090	−0.302423	−0.151211	+	0.988501i	$0.548317\pi$
$608$	0	0
$609$	−4.08660	−0.165597
$610$	0	0
$611$	13.7935	0.558024
$612$	0	0
$613$	35.5222	1.43473	0.717364	−	0.696699i	$-0.245349\pi$
$613$	35.5222	1.43473	0.717364	+	0.696699i	$0.245349\pi$
$614$	0	0
$615$	14.7318	0.594043
$616$	0	0
$617$	−43.9746	−1.77035	−0.885175	−	0.465259i	$-0.845961\pi$
$617$	−43.9746	−1.77035	−0.885175	+	0.465259i	$0.845961\pi$
$618$	0	0
$619$	−0.523028	−0.0210223	−0.0105111	−	0.999945i	$-0.503346\pi$
$619$	−0.523028	−0.0210223	−0.0105111	+	0.999945i	$0.503346\pi$
$620$	0	0
$621$	−25.4355	−1.02069
$622$	0	0
$623$	10.6204	0.425496
$624$	0	0
$625$	3.94367	0.157747
$626$	0	0
$627$	−3.64643	−0.145624
$628$	0	0
$629$	−6.48929	−0.258745
$630$	0	0
$631$	22.0261	0.876845	0.438423	−	0.898769i	$-0.355537\pi$
$631$	22.0261	0.876845	0.438423	+	0.898769i	$0.355537\pi$
$632$	0	0
$633$	18.8146	0.747811
$634$	0	0
$635$	13.4623	0.534236
$636$	0	0
$637$	−2.40579	−0.0953207
$638$	0	0
$639$	36.6308	1.44909
$640$	0	0
$641$	−28.9669	−1.14412	−0.572062	−	0.820210i	$-0.693856\pi$
$641$	−28.9669	−1.14412	−0.572062	+	0.820210i	$0.693856\pi$
$642$	0	0
$643$	18.0983	0.713729	0.356864	−	0.934156i	$-0.383846\pi$
$643$	18.0983	0.713729	0.356864	+	0.934156i	$0.383846\pi$
$644$	0	0
$645$	17.2707	0.680031
$646$	0	0
$647$	−10.7062	−0.420904	−0.210452	−	0.977604i	$-0.567493\pi$
$647$	−10.7062	−0.420904	−0.210452	+	0.977604i	$0.567493\pi$
$648$	0	0
$649$	−26.5624	−1.04266
$650$	0	0
$651$	−0.266273	−0.0104360
$652$	0	0
$653$	5.07346	0.198540	0.0992700	−	0.995061i	$-0.468349\pi$
$653$	5.07346	0.198540	0.0992700	+	0.995061i	$0.468349\pi$
$654$	0	0
$655$	44.9238	1.75532
$656$	0	0
$657$	−23.1745	−0.904122
$658$	0	0
$659$	48.4215	1.88623	0.943117	−	0.332461i	$-0.107879\pi$
$659$	48.4215	1.88623	0.943117	+	0.332461i	$0.107879\pi$
$660$	0	0
$661$	−21.2675	−0.827209	−0.413605	−	0.910457i	$-0.635730\pi$
$661$	−21.2675	−0.827209	−0.413605	+	0.910457i	$0.635730\pi$
$662$	0	0
$663$	1.68081	0.0652774
$664$	0	0
$665$	−6.29712	−0.244192
$666$	0	0
$667$	−38.6347	−1.49594
$668$	0	0
$669$	−11.2383	−0.434497
$670$	0	0
$671$	−2.49881	−0.0964653
$672$	0	0
$673$	−33.6347	−1.29652	−0.648260	−	0.761419i	$-0.724503\pi$
$673$	−33.6347	−1.29652	−0.648260	+	0.761419i	$0.724503\pi$
$674$	0	0
$675$	32.4724	1.24986
$676$	0	0
$677$	30.9105	1.18799	0.593994	−	0.804469i	$-0.297550\pi$
$677$	30.9105	1.18799	0.593994	+	0.804469i	$0.297550\pi$
$678$	0	0
$679$	−3.08403	−0.118354
$680$	0	0
$681$	17.7397	0.679788
$682$	0	0
$683$	−2.49537	−0.0954828	−0.0477414	−	0.998860i	$-0.515202\pi$
$683$	−2.49537	−0.0954828	−0.0477414	+	0.998860i	$0.515202\pi$
$684$	0	0
$685$	−51.5375	−1.96915
$686$	0	0
$687$	−13.7629	−0.525089
$688$	0	0
$689$	11.9917	0.456848
$690$	0	0
$691$	35.9955	1.36933	0.684667	−	0.728856i	$-0.259948\pi$
$691$	35.9955	1.36933	0.684667	+	0.728856i	$0.259948\pi$
$692$	0	0
$693$	7.63027	0.289850
$694$	0	0
$695$	−0.553290	−0.0209875
$696$	0	0
$697$	−5.75328	−0.217921
$698$	0	0
$699$	−14.9320	−0.564780
$700$	0	0
$701$	−23.4220	−0.884638	−0.442319	−	0.896858i	$-0.645844\pi$
$701$	−23.4220	−0.884638	−0.442319	+	0.896858i	$0.645844\pi$
$702$	0	0
$703$	11.1497	0.420518
$704$	0	0
$705$	14.6810	0.552918
$706$	0	0
$707$	−13.6872	−0.514761
$708$	0	0
$709$	41.5410	1.56011	0.780053	−	0.625713i	$-0.215192\pi$
$709$	41.5410	1.56011	0.780053	+	0.625713i	$0.215192\pi$
$710$	0	0
$711$	28.1910	1.05725
$712$	0	0
$713$	−2.51734	−0.0942753
$714$	0	0
$715$	26.7840	1.00166
$716$	0	0
$717$	−12.4328	−0.464313
$718$	0	0
$719$	−14.9152	−0.556242	−0.278121	−	0.960546i	$-0.589712\pi$
$719$	−14.9152	−0.556242	−0.278121	+	0.960546i	$0.589712\pi$
$720$	0	0
$721$	14.6679	0.546263
$722$	0	0
$723$	−6.03989	−0.224626
$724$	0	0
$725$	49.3233	1.83182
$726$	0	0
$727$	−6.06227	−0.224837	−0.112419	−	0.993661i	$-0.535860\pi$
$727$	−6.06227	−0.224837	−0.112419	+	0.993661i	$0.535860\pi$
$728$	0	0
$729$	−4.09923	−0.151823
$730$	0	0
$731$	−6.74480	−0.249466
$732$	0	0
$733$	−20.4569	−0.755592	−0.377796	−	0.925889i	$-0.623318\pi$
$733$	−20.4569	−0.755592	−0.377796	+	0.925889i	$0.623318\pi$
$734$	0	0
$735$	−2.56059	−0.0944487
$736$	0	0
$737$	42.8827	1.57960
$738$	0	0
$739$	−2.95860	−0.108834	−0.0544170	−	0.998518i	$-0.517330\pi$
$739$	−2.95860	−0.108834	−0.0544170	+	0.998518i	$0.517330\pi$
$740$	0	0
$741$	−2.88791	−0.106090
$742$	0	0
$743$	1.63220	0.0598798	0.0299399	−	0.999552i	$-0.490468\pi$
$743$	1.63220	0.0598798	0.0299399	+	0.999552i	$0.490468\pi$
$744$	0	0
$745$	−17.2594	−0.632334
$746$	0	0
$747$	32.2196	1.17885
$748$	0	0
$749$	15.3769	0.561861
$750$	0	0
$751$	46.5443	1.69843	0.849213	−	0.528051i	$-0.177077\pi$
$751$	46.5443	1.69843	0.849213	+	0.528051i	$0.177077\pi$
$752$	0	0
$753$	−11.9010	−0.433698
$754$	0	0
$755$	5.96278	0.217008
$756$	0	0
$757$	30.5528	1.11046	0.555231	−	0.831696i	$-0.312630\pi$
$757$	30.5528	1.11046	0.555231	+	0.831696i	$0.312630\pi$
$758$	0	0
$759$	−14.0178	−0.508816
$760$	0	0
$761$	−42.4738	−1.53967	−0.769836	−	0.638241i	$-0.779662\pi$
$761$	−42.4738	−1.53967	−0.769836	+	0.638241i	$0.779662\pi$
$762$	0	0
$763$	7.72529	0.279674
$764$	0	0
$765$	−9.20612	−0.332848
$766$	0	0
$767$	−21.0370	−0.759601
$768$	0	0
$769$	−23.2387	−0.838009	−0.419004	−	0.907984i	$-0.637621\pi$
$769$	−23.2387	−0.838009	−0.419004	+	0.907984i	$0.637621\pi$
$770$	0	0
$771$	−14.5339	−0.523425
$772$	0	0
$773$	−32.3609	−1.16394	−0.581970	−	0.813211i	$-0.697718\pi$
$773$	−32.3609	−1.16394	−0.581970	+	0.813211i	$0.697718\pi$
$774$	0	0
$775$	3.21379	0.115443
$776$	0	0
$777$	4.53377	0.162648
$778$	0	0
$779$	9.88509	0.354170
$780$	0	0
$781$	44.2984	1.58512
$782$	0	0
$783$	22.5249	0.804973
$784$	0	0
$785$	18.8189	0.671674
$786$	0	0
$787$	35.2401	1.25618	0.628088	−	0.778143i	$-0.283838\pi$
$787$	35.2401	1.25618	0.628088	+	0.778143i	$0.283838\pi$
$788$	0	0
$789$	−15.9118	−0.566474
$790$	0	0
$791$	7.20610	0.256220
$792$	0	0
$793$	−1.97901	−0.0702768
$794$	0	0
$795$	12.7633	0.452669
$796$	0	0
$797$	−13.3278	−0.472094	−0.236047	−	0.971742i	$-0.575852\pi$
$797$	−13.3278	−0.472094	−0.236047	+	0.971742i	$0.575852\pi$
$798$	0	0
$799$	−5.73345	−0.202835
$800$	0	0
$801$	−26.6771	−0.942589
$802$	0	0
$803$	−28.0254	−0.988993
$804$	0	0
$805$	−24.2078	−0.853213
$806$	0	0
$807$	21.9826	0.773825
$808$	0	0
$809$	24.0642	0.846052	0.423026	−	0.906118i	$-0.360968\pi$
$809$	24.0642	0.846052	0.423026	+	0.906118i	$0.360968\pi$
$810$	0	0
$811$	33.4787	1.17559	0.587797	−	0.809008i	$-0.299995\pi$
$811$	33.4787	1.17559	0.587797	+	0.809008i	$0.299995\pi$
$812$	0	0
$813$	−6.97206	−0.244521
$814$	0	0
$815$	−49.0593	−1.71847
$816$	0	0
$817$	11.5887	0.405437
$818$	0	0
$819$	6.04305	0.211161
$820$	0	0
$821$	47.0944	1.64361	0.821803	−	0.569772i	$-0.192968\pi$
$821$	47.0944	1.64361	0.821803	+	0.569772i	$0.192968\pi$
$822$	0	0
$823$	33.9685	1.18407	0.592035	−	0.805913i	$-0.298325\pi$
$823$	33.9685	1.18407	0.592035	+	0.805913i	$0.298325\pi$
$824$	0	0
$825$	17.8960	0.623058
$826$	0	0
$827$	7.72016	0.268456	0.134228	−	0.990950i	$-0.457145\pi$
$827$	7.72016	0.268456	0.134228	+	0.990950i	$0.457145\pi$
$828$	0	0
$829$	−5.77689	−0.200640	−0.100320	−	0.994955i	$-0.531987\pi$
$829$	−5.77689	−0.200640	−0.100320	+	0.994955i	$0.531987\pi$
$830$	0	0
$831$	16.2672	0.564302
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−2.33806	−0.0809119
$836$	0	0
$837$	1.46766	0.0507299
$838$	0	0
$839$	27.9051	0.963392	0.481696	−	0.876338i	$-0.340021\pi$
$839$	27.9051	0.963392	0.481696	+	0.876338i	$0.340021\pi$
$840$	0	0
$841$	5.21366	0.179781
$842$	0	0
$843$	10.9443	0.376941
$844$	0	0
$845$	−26.4329	−0.909319
$846$	0	0
$847$	−1.77255	−0.0609056
$848$	0	0
$849$	−1.36914	−0.0469887
$850$	0	0
$851$	42.8623	1.46930
$852$	0	0
$853$	−15.8720	−0.543448	−0.271724	−	0.962375i	$-0.587594\pi$
$853$	−15.8720	−0.543448	−0.271724	+	0.962375i	$0.587594\pi$
$854$	0	0
$855$	15.8176	0.540951
$856$	0	0
$857$	−51.7170	−1.76662	−0.883309	−	0.468791i	$-0.844690\pi$
$857$	−51.7170	−1.76662	−0.883309	+	0.468791i	$0.844690\pi$
$858$	0	0
$859$	−4.55167	−0.155301	−0.0776506	−	0.996981i	$-0.524742\pi$
$859$	−4.55167	−0.155301	−0.0776506	+	0.996981i	$0.524742\pi$
$860$	0	0
$861$	4.01955	0.136986
$862$	0	0
$863$	−16.4356	−0.559474	−0.279737	−	0.960077i	$-0.590247\pi$
$863$	−16.4356	−0.559474	−0.279737	+	0.960077i	$0.590247\pi$
$864$	0	0
$865$	−18.3905	−0.625297
$866$	0	0
$867$	−0.698654	−0.0237275
$868$	0	0
$869$	34.0920	1.15649
$870$	0	0
$871$	33.9624	1.15077
$872$	0	0
$873$	7.74672	0.262187
$874$	0	0
$875$	12.5799	0.425279
$876$	0	0
$877$	−4.47537	−0.151122	−0.0755612	−	0.997141i	$-0.524075\pi$
$877$	−4.47537	−0.151122	−0.0755612	+	0.997141i	$0.524075\pi$
$878$	0	0
$879$	−2.83306	−0.0955568
$880$	0	0
$881$	21.3197	0.718280	0.359140	−	0.933284i	$-0.383070\pi$
$881$	21.3197	0.718280	0.359140	+	0.933284i	$0.383070\pi$
$882$	0	0
$883$	32.6860	1.09997	0.549986	−	0.835174i	$-0.314633\pi$
$883$	32.6860	1.09997	0.549986	+	0.835174i	$0.314633\pi$
$884$	0	0
$885$	−22.3906	−0.752651
$886$	0	0
$887$	−36.1878	−1.21507	−0.607533	−	0.794294i	$-0.707841\pi$
$887$	−36.1878	−1.21507	−0.607533	+	0.794294i	$0.707841\pi$
$888$	0	0
$889$	3.67319	0.123195
$890$	0	0
$891$	−14.7181	−0.493076
$892$	0	0
$893$	9.85102	0.329652
$894$	0	0
$895$	−48.4998	−1.62117
$896$	0	0
$897$	−11.1019	−0.370682
$898$	0	0
$899$	2.22928	0.0743506
$900$	0	0
$901$	−4.98453	−0.166059
$902$	0	0
$903$	4.71229	0.156815
$904$	0	0
$905$	9.29768	0.309065
$906$	0	0
$907$	3.20778	0.106513	0.0532563	−	0.998581i	$-0.483040\pi$
$907$	3.20778	0.106513	0.0532563	+	0.998581i	$0.483040\pi$
$908$	0	0
$909$	34.3807	1.14034
$910$	0	0
$911$	33.1202	1.09732	0.548660	−	0.836045i	$-0.315138\pi$
$911$	33.1202	1.09732	0.548660	+	0.836045i	$0.315138\pi$
$912$	0	0
$913$	38.9638	1.28951
$914$	0	0
$915$	−2.10635	−0.0696339
$916$	0	0
$917$	12.2574	0.404776
$918$	0	0
$919$	2.78562	0.0918893	0.0459446	−	0.998944i	$-0.485370\pi$
$919$	2.78562	0.0918893	0.0459446	+	0.998944i	$0.485370\pi$
$920$	0	0
$921$	18.3427	0.604412
$922$	0	0
$923$	35.0836	1.15479
$924$	0	0
$925$	−54.7204	−1.79920
$926$	0	0
$927$	−36.8442	−1.21012
$928$	0	0
$929$	36.3760	1.19346	0.596728	−	0.802443i	$-0.296467\pi$
$929$	36.3760	1.19346	0.596728	+	0.802443i	$0.296467\pi$
$930$	0	0
$931$	−1.71816	−0.0563106
$932$	0	0
$933$	4.40890	0.144341
$934$	0	0
$935$	−11.1331	−0.364093
$936$	0	0
$937$	27.7698	0.907199	0.453600	−	0.891206i	$-0.350140\pi$
$937$	27.7698	0.907199	0.453600	+	0.891206i	$0.350140\pi$
$938$	0	0
$939$	−18.8592	−0.615448
$940$	0	0
$941$	28.9593	0.944047	0.472023	−	0.881586i	$-0.343524\pi$
$941$	28.9593	0.944047	0.472023	+	0.881586i	$0.343524\pi$
$942$	0	0
$943$	38.0009	1.23748
$944$	0	0
$945$	14.1137	0.459117
$946$	0	0
$947$	−1.67668	−0.0544849	−0.0272425	−	0.999629i	$-0.508673\pi$
$947$	−1.67668	−0.0544849	−0.0272425	+	0.999629i	$0.508673\pi$
$948$	0	0
$949$	−22.1956	−0.720500
$950$	0	0
$951$	3.82802	0.124132
$952$	0	0
$953$	−48.0897	−1.55778	−0.778889	−	0.627162i	$-0.784217\pi$
$953$	−48.0897	−1.55778	−0.778889	+	0.627162i	$0.784217\pi$
$954$	0	0
$955$	32.5803	1.05427
$956$	0	0
$957$	12.4137	0.401279
$958$	0	0
$959$	−14.0620	−0.454084
$960$	0	0
$961$	−30.8547	−0.995314
$962$	0	0
$963$	−38.6250	−1.24467
$964$	0	0
$965$	−57.5090	−1.85128
$966$	0	0
$967$	18.2687	0.587480	0.293740	−	0.955885i	$-0.405100\pi$
$967$	18.2687	0.587480	0.293740	+	0.955885i	$0.405100\pi$
$968$	0	0
$969$	1.20040	0.0385625
$970$	0	0
$971$	−34.3321	−1.10177	−0.550885	−	0.834581i	$-0.685710\pi$
$971$	−34.3321	−1.10177	−0.550885	+	0.834581i	$0.685710\pi$
$972$	0	0
$973$	−0.150965	−0.00483971
$974$	0	0
$975$	14.1733	0.453910
$976$	0	0
$977$	31.1256	0.995795	0.497897	−	0.867236i	$-0.334106\pi$
$977$	31.1256	0.995795	0.497897	+	0.867236i	$0.334106\pi$
$978$	0	0
$979$	−32.2612	−1.03107
$980$	0	0
$981$	−19.4050	−0.619555
$982$	0	0
$983$	−15.5467	−0.495863	−0.247931	−	0.968778i	$-0.579751\pi$
$983$	−15.5467	−0.495863	−0.247931	+	0.968778i	$0.579751\pi$
$984$	0	0
$985$	−30.3854	−0.968159
$986$	0	0
$987$	4.00570	0.127503
$988$	0	0
$989$	44.5500	1.41661
$990$	0	0
$991$	−15.5023	−0.492446	−0.246223	−	0.969213i	$-0.579190\pi$
$991$	−15.5023	−0.492446	−0.246223	+	0.969213i	$0.579190\pi$
$992$	0	0
$993$	−19.5662	−0.620915
$994$	0	0
$995$	−25.2302	−0.799851
$996$	0	0
$997$	−41.9042	−1.32712	−0.663560	−	0.748123i	$-0.730955\pi$
$997$	−41.9042	−1.32712	−0.663560	+	0.748123i	$0.730955\pi$
$998$	0	0
$999$	−24.9896	−0.790636

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.g.1.3	✓	6
4.3	odd	2		3808.2.a.o.1.4	yes	6
8.3	odd	2		7616.2.a.bv.1.3		6
8.5	even	2		7616.2.a.cd.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.g.1.3	✓	6	1.1	even	1	trivial
3808.2.a.o.1.4	yes	6	4.3	odd	2
7616.2.a.bv.1.3		6	8.3	odd	2
7616.2.a.cd.1.4		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)

\(f(q)\)	\(=\)	\(q-0.698654 q^{3} +3.66503 q^{5} +1.00000 q^{7} -2.51188 q^{9} -3.03767 q^{11} -2.40579 q^{13} -2.56059 q^{15} +1.00000 q^{17} -1.71816 q^{19} -0.698654 q^{21} -6.60508 q^{23} +8.43243 q^{25} +3.85090 q^{27} +5.84924 q^{29} +0.381122 q^{31} +2.12228 q^{33} +3.66503 q^{35} -6.48929 q^{37} +1.68081 q^{39} -5.75328 q^{41} -6.74480 q^{43} -9.20612 q^{45} -5.73345 q^{47} +1.00000 q^{49} -0.698654 q^{51} -4.98453 q^{53} -11.1331 q^{55} +1.20040 q^{57} +8.74432 q^{59} +0.822606 q^{61} -2.51188 q^{63} -8.81727 q^{65} -14.1170 q^{67} +4.61467 q^{69} -14.5830 q^{71} +9.22593 q^{73} -5.89135 q^{75} -3.03767 q^{77} -11.2231 q^{79} +4.84520 q^{81} -12.8269 q^{83} +3.66503 q^{85} -4.08660 q^{87} +10.6204 q^{89} -2.40579 q^{91} -0.266273 q^{93} -6.29712 q^{95} -3.08403 q^{97} +7.63027 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{13} + 6 q^{17} - 18 q^{19} - 4 q^{21} - 6 q^{23} + 12 q^{25} - 10 q^{27} + 4 q^{29} + 8 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} + 18 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) \) 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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