LMFDB - Embedded newform 416.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(1,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("416.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.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:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.546295$ of defining polynomial
Character	$\chi$	$=$	416.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.11473 q^{3} -3.70156 q^{5} +4.20732 q^{7} +6.70156 q^{9} -1.09259 q^{11} +1.00000 q^{13} -11.5294 q^{15} +0.298438 q^{17} +1.09259 q^{19} +13.1047 q^{21} +8.70156 q^{25} +11.5294 q^{27} -2.00000 q^{29} -5.13688 q^{31} -3.40312 q^{33} -15.5737 q^{35} -3.70156 q^{37} +3.11473 q^{39} -9.40312 q^{41} -5.29991 q^{43} -24.8062 q^{45} +4.20732 q^{47} +10.7016 q^{49} +0.929554 q^{51} +1.40312 q^{53} +4.04429 q^{55} +3.40312 q^{57} -13.5515 q^{59} +9.40312 q^{61} +28.1956 q^{63} -3.70156 q^{65} -11.3663 q^{67} +8.25161 q^{71} -6.00000 q^{73} +27.1030 q^{75} -4.59688 q^{77} -14.6441 q^{79} +15.8062 q^{81} +7.32206 q^{83} -1.10469 q^{85} -6.22947 q^{87} -6.00000 q^{89} +4.20732 q^{91} -16.0000 q^{93} -4.04429 q^{95} +8.80625 q^{97} -7.32206 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} + 14 q^{9} + 4 q^{13} + 14 q^{17} + 14 q^{21} + 22 q^{25} - 8 q^{29} + 12 q^{33} - 2 q^{37} - 12 q^{41} - 48 q^{45} + 30 q^{49} - 20 q^{53} - 12 q^{57} + 12 q^{61} - 2 q^{65} - 24 q^{73} - 44 q^{77}+ \cdots - 16 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 14 * q^9 + 4 * q^13 + 14 * q^17 + 14 * q^21 + 22 * q^25 - 8 * q^29 + 12 * q^33 - 2 * q^37 - 12 * q^41 - 48 * q^45 + 30 * q^49 - 20 * q^53 - 12 * q^57 + 12 * q^61 - 2 * q^65 - 24 * q^73 - 44 * q^77 + 12 * q^81 + 34 * q^85 - 24 * q^89 - 64 * q^93 - 16 * q^97

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
$3$	3.11473	1.79829	0.899146	−	0.437649i	$-0.144189\pi$
$3$	3.11473	1.79829	0.899146	+	0.437649i	$0.144189\pi$
$4$	0	0
$5$	−3.70156	−1.65539	−0.827694	−	0.561179i	$-0.810348\pi$
$5$	−3.70156	−1.65539	−0.827694	+	0.561179i	$0.810348\pi$
$6$	0	0
$7$	4.20732	1.59022	0.795109	−	0.606466i	$-0.207413\pi$
$7$	4.20732	1.59022	0.795109	+	0.606466i	$0.207413\pi$
$8$	0	0
$9$	6.70156	2.23385
$10$	0	0
$11$	−1.09259	−0.329428	−0.164714	−	0.986341i	$-0.552670\pi$
$11$	−1.09259	−0.329428	−0.164714	+	0.986341i	$0.552670\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−11.5294	−2.97687
$16$	0	0
$17$	0.298438	0.0723818	0.0361909	−	0.999345i	$-0.488478\pi$
$17$	0.298438	0.0723818	0.0361909	+	0.999345i	$0.488478\pi$
$18$	0	0
$19$	1.09259	0.250657	0.125329	−	0.992115i	$-0.460001\pi$
$19$	1.09259	0.250657	0.125329	+	0.992115i	$0.460001\pi$
$20$	0	0
$21$	13.1047	2.85968
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	8.70156	1.74031
$26$	0	0
$27$	11.5294	2.21883
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−5.13688	−0.922610	−0.461305	−	0.887242i	$-0.652619\pi$
$31$	−5.13688	−0.922610	−0.461305	+	0.887242i	$0.652619\pi$
$32$	0	0
$33$	−3.40312	−0.592408
$34$	0	0
$35$	−15.5737	−2.63243
$36$	0	0
$37$	−3.70156	−0.608533	−0.304267	−	0.952587i	$-0.598411\pi$
$37$	−3.70156	−0.608533	−0.304267	+	0.952587i	$0.598411\pi$
$38$	0	0
$39$	3.11473	0.498756
$40$	0	0
$41$	−9.40312	−1.46852	−0.734261	−	0.678868i	$-0.762471\pi$
$41$	−9.40312	−1.46852	−0.734261	+	0.678868i	$0.762471\pi$
$42$	0	0
$43$	−5.29991	−0.808229	−0.404114	−	0.914708i	$-0.632420\pi$
$43$	−5.29991	−0.808229	−0.404114	+	0.914708i	$0.632420\pi$
$44$	0	0
$45$	−24.8062	−3.69790
$46$	0	0
$47$	4.20732	0.613701	0.306851	−	0.951758i	$-0.400725\pi$
$47$	4.20732	0.613701	0.306851	+	0.951758i	$0.400725\pi$
$48$	0	0
$49$	10.7016	1.52879
$50$	0	0
$51$	0.929554	0.130164
$52$	0	0
$53$	1.40312	0.192734	0.0963670	−	0.995346i	$-0.469278\pi$
$53$	1.40312	0.192734	0.0963670	+	0.995346i	$0.469278\pi$
$54$	0	0
$55$	4.04429	0.545332
$56$	0	0
$57$	3.40312	0.450755
$58$	0	0
$59$	−13.5515	−1.76426	−0.882129	−	0.471008i	$-0.843890\pi$
$59$	−13.5515	−1.76426	−0.882129	+	0.471008i	$0.843890\pi$
$60$	0	0
$61$	9.40312	1.20395	0.601973	−	0.798516i	$-0.294381\pi$
$61$	9.40312	1.20395	0.601973	+	0.798516i	$0.294381\pi$
$62$	0	0
$63$	28.1956	3.55232
$64$	0	0
$65$	−3.70156	−0.459122
$66$	0	0
$67$	−11.3663	−1.38862	−0.694310	−	0.719676i	$-0.744290\pi$
$67$	−11.3663	−1.38862	−0.694310	+	0.719676i	$0.744290\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.25161	0.979286	0.489643	−	0.871923i	$-0.337127\pi$
$71$	8.25161	0.979286	0.489643	+	0.871923i	$0.337127\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	27.1030	3.12959
$76$	0	0
$77$	−4.59688	−0.523863
$78$	0	0
$79$	−14.6441	−1.64759	−0.823796	−	0.566887i	$-0.808148\pi$
$79$	−14.6441	−1.64759	−0.823796	+	0.566887i	$0.808148\pi$
$80$	0	0
$81$	15.8062	1.75625
$82$	0	0
$83$	7.32206	0.803700	0.401850	−	0.915706i	$-0.368367\pi$
$83$	7.32206	0.803700	0.401850	+	0.915706i	$0.368367\pi$
$84$	0	0
$85$	−1.10469	−0.119820
$86$	0	0
$87$	−6.22947	−0.667869
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	4.20732	0.441047
$92$	0	0
$93$	−16.0000	−1.65912
$94$	0	0
$95$	−4.04429	−0.414935
$96$	0	0
$97$	8.80625	0.894139	0.447070	−	0.894499i	$-0.352468\pi$
$97$	8.80625	0.894139	0.447070	+	0.894499i	$0.352468\pi$
$98$	0	0
$99$	−7.32206	−0.735894
$100$	0	0
$101$	−5.40312	−0.537631	−0.268815	−	0.963192i	$-0.586632\pi$
$101$	−5.40312	−0.537631	−0.268815	+	0.963192i	$0.586632\pi$
$102$	0	0
$103$	12.4589	1.22762	0.613808	−	0.789456i	$-0.289637\pi$
$103$	12.4589	1.22762	0.613808	+	0.789456i	$0.289637\pi$
$104$	0	0
$105$	−48.5078	−4.73388
$106$	0	0
$107$	14.6441	1.41570	0.707850	−	0.706363i	$-0.249665\pi$
$107$	14.6441	1.41570	0.707850	+	0.706363i	$0.249665\pi$
$108$	0	0
$109$	4.29844	0.411716	0.205858	−	0.978582i	$-0.434002\pi$
$109$	4.29844	0.411716	0.205858	+	0.978582i	$0.434002\pi$
$110$	0	0
$111$	−11.5294	−1.09432
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.70156	0.619560
$118$	0	0
$119$	1.25562	0.115103
$120$	0	0
$121$	−9.80625	−0.891477
$122$	0	0
$123$	−29.2882	−2.64083
$124$	0	0
$125$	−13.7016	−1.22550
$126$	0	0
$127$	−16.8293	−1.49336	−0.746679	−	0.665185i	$-0.768353\pi$
$127$	−16.8293	−1.49336	−0.746679	+	0.665185i	$0.768353\pi$
$128$	0	0
$129$	−16.5078	−1.45343
$130$	0	0
$131$	−7.48509	−0.653975	−0.326988	−	0.945029i	$-0.606034\pi$
$131$	−7.48509	−0.653975	−0.326988	+	0.945029i	$0.606034\pi$
$132$	0	0
$133$	4.59688	0.398600
$134$	0	0
$135$	−42.6767	−3.67303
$136$	0	0
$137$	20.2094	1.72660	0.863302	−	0.504688i	$-0.168393\pi$
$137$	20.2094	1.72660	0.863302	+	0.504688i	$0.168393\pi$
$138$	0	0
$139$	−15.5737	−1.32094	−0.660471	−	0.750852i	$-0.729643\pi$
$139$	−15.5737	−1.32094	−0.660471	+	0.750852i	$0.729643\pi$
$140$	0	0
$141$	13.1047	1.10361
$142$	0	0
$143$	−1.09259	−0.0913669
$144$	0	0
$145$	7.40312	0.614796
$146$	0	0
$147$	33.3325	2.74922
$148$	0	0
$149$	12.8062	1.04913	0.524564	−	0.851371i	$-0.324228\pi$
$149$	12.8062	1.04913	0.524564	+	0.851371i	$0.324228\pi$
$150$	0	0
$151$	−2.02214	−0.164560	−0.0822799	−	0.996609i	$-0.526220\pi$
$151$	−2.02214	−0.164560	−0.0822799	+	0.996609i	$0.526220\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	19.0145	1.52728
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	4.37036	0.346592
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.50723	0.744664	0.372332	−	0.928100i	$-0.378558\pi$
$163$	9.50723	0.744664	0.372332	+	0.928100i	$0.378558\pi$
$164$	0	0
$165$	12.5969	0.980665
$166$	0	0
$167$	−11.6924	−0.904786	−0.452393	−	0.891819i	$-0.649430\pi$
$167$	−11.6924	−0.904786	−0.452393	+	0.891819i	$0.649430\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.32206	0.559932
$172$	0	0
$173$	9.40312	0.714906	0.357453	−	0.933931i	$-0.383645\pi$
$173$	9.40312	0.714906	0.357453	+	0.933931i	$0.383645\pi$
$174$	0	0
$175$	36.6103	2.76748
$176$	0	0
$177$	−42.2094	−3.17265
$178$	0	0
$179$	−5.29991	−0.396134	−0.198067	−	0.980188i	$-0.563466\pi$
$179$	−5.29991	−0.396134	−0.198067	+	0.980188i	$0.563466\pi$
$180$	0	0
$181$	2.59688	0.193024	0.0965121	−	0.995332i	$-0.469231\pi$
$181$	2.59688	0.193024	0.0965121	+	0.995332i	$0.469231\pi$
$182$	0	0
$183$	29.2882	2.16505
$184$	0	0
$185$	13.7016	1.00736
$186$	0	0
$187$	−0.326070	−0.0238446
$188$	0	0
$189$	48.5078	3.52842
$190$	0	0
$191$	4.04429	0.292634	0.146317	−	0.989238i	$-0.453258\pi$
$191$	4.04429	0.292634	0.146317	+	0.989238i	$0.453258\pi$
$192$	0	0
$193$	21.4031	1.54063	0.770315	−	0.637663i	$-0.220099\pi$
$193$	21.4031	1.54063	0.770315	+	0.637663i	$0.220099\pi$
$194$	0	0
$195$	−11.5294	−0.825636
$196$	0	0
$197$	−16.2984	−1.16122	−0.580608	−	0.814183i	$-0.697185\pi$
$197$	−16.2984	−1.16122	−0.580608	+	0.814183i	$0.697185\pi$
$198$	0	0
$199$	−18.6884	−1.32479	−0.662393	−	0.749157i	$-0.730459\pi$
$199$	−18.6884	−1.32479	−0.662393	+	0.749157i	$0.730459\pi$
$200$	0	0
$201$	−35.4031	−2.49714
$202$	0	0
$203$	−8.41464	−0.590592
$204$	0	0
$205$	34.8062	2.43097
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.19375	−0.0825735
$210$	0	0
$211$	21.8031	1.50099	0.750495	−	0.660876i	$-0.229815\pi$
$211$	21.8031	1.50099	0.750495	+	0.660876i	$0.229815\pi$
$212$	0	0
$213$	25.7016	1.76104
$214$	0	0
$215$	19.6180	1.33793
$216$	0	0
$217$	−21.6125	−1.46715
$218$	0	0
$219$	−18.6884	−1.26284
$220$	0	0
$221$	0.298438	0.0200751
$222$	0	0
$223$	20.7105	1.38688	0.693440	−	0.720514i	$-0.256094\pi$
$223$	20.7105	1.38688	0.693440	+	0.720514i	$0.256094\pi$
$224$	0	0
$225$	58.3141	3.88760
$226$	0	0
$227$	−5.13688	−0.340946	−0.170473	−	0.985362i	$-0.554530\pi$
$227$	−5.13688	−0.340946	−0.170473	+	0.985362i	$0.554530\pi$
$228$	0	0
$229$	8.89531	0.587819	0.293909	−	0.955833i	$-0.405044\pi$
$229$	8.89531	0.587819	0.293909	+	0.955833i	$0.405044\pi$
$230$	0	0
$231$	−14.3180	−0.942058
$232$	0	0
$233$	23.1047	1.51364	0.756819	−	0.653624i	$-0.226752\pi$
$233$	23.1047	1.51364	0.756819	+	0.653624i	$0.226752\pi$
$234$	0	0
$235$	−15.5737	−1.01591
$236$	0	0
$237$	−45.6125	−2.96285
$238$	0	0
$239$	14.4811	0.936703	0.468351	−	0.883542i	$-0.344848\pi$
$239$	14.4811	0.936703	0.468351	+	0.883542i	$0.344848\pi$
$240$	0	0
$241$	−4.80625	−0.309598	−0.154799	−	0.987946i	$-0.549473\pi$
$241$	−4.80625	−0.309598	−0.154799	+	0.987946i	$0.549473\pi$
$242$	0	0
$243$	14.6441	0.939420
$244$	0	0
$245$	−39.6125	−2.53075
$246$	0	0
$247$	1.09259	0.0695198
$248$	0	0
$249$	22.8062	1.44529
$250$	0	0
$251$	16.8293	1.06226	0.531128	−	0.847292i	$-0.321768\pi$
$251$	16.8293	1.06226	0.531128	+	0.847292i	$0.321768\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3.44080	−0.215471
$256$	0	0
$257$	−14.5078	−0.904972	−0.452486	−	0.891771i	$-0.649463\pi$
$257$	−14.5078	−0.904972	−0.452486	+	0.891771i	$0.649463\pi$
$258$	0	0
$259$	−15.5737	−0.967700
$260$	0	0
$261$	−13.4031	−0.829633
$262$	0	0
$263$	−2.18518	−0.134744	−0.0673719	−	0.997728i	$-0.521461\pi$
$263$	−2.18518	−0.134744	−0.0673719	+	0.997728i	$0.521461\pi$
$264$	0	0
$265$	−5.19375	−0.319050
$266$	0	0
$267$	−18.6884	−1.14371
$268$	0	0
$269$	−12.2094	−0.744419	−0.372209	−	0.928149i	$-0.621400\pi$
$269$	−12.2094	−0.744419	−0.372209	+	0.928149i	$0.621400\pi$
$270$	0	0
$271$	4.20732	0.255577	0.127788	−	0.991801i	$-0.459212\pi$
$271$	4.20732	0.255577	0.127788	+	0.991801i	$0.459212\pi$
$272$	0	0
$273$	13.1047	0.793132
$274$	0	0
$275$	−9.50723	−0.573308
$276$	0	0
$277$	24.2094	1.45460	0.727300	−	0.686320i	$-0.240775\pi$
$277$	24.2094	1.45460	0.727300	+	0.686320i	$0.240775\pi$
$278$	0	0
$279$	−34.4251	−2.06098
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	−12.5969	−0.746175
$286$	0	0
$287$	−39.5620	−2.33527
$288$	0	0
$289$	−16.9109	−0.994761
$290$	0	0
$291$	27.4291	1.60792
$292$	0	0
$293$	19.1047	1.11611	0.558054	−	0.829805i	$-0.311548\pi$
$293$	19.1047	1.11611	0.558054	+	0.829805i	$0.311548\pi$
$294$	0	0
$295$	50.1618	2.92053
$296$	0	0
$297$	−12.5969	−0.730945
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−22.2984	−1.28526
$302$	0	0
$303$	−16.8293	−0.966817
$304$	0	0
$305$	−34.8062	−1.99300
$306$	0	0
$307$	15.7367	0.898141	0.449070	−	0.893496i	$-0.351755\pi$
$307$	15.7367	0.898141	0.449070	+	0.893496i	$0.351755\pi$
$308$	0	0
$309$	38.8062	2.20761
$310$	0	0
$311$	−18.6884	−1.05972	−0.529861	−	0.848085i	$-0.677756\pi$
$311$	−18.6884	−1.05972	−0.529861	+	0.848085i	$0.677756\pi$
$312$	0	0
$313$	2.50781	0.141750	0.0708749	−	0.997485i	$-0.477421\pi$
$313$	2.50781	0.141750	0.0708749	+	0.997485i	$0.477421\pi$
$314$	0	0
$315$	−104.368	−5.88046
$316$	0	0
$317$	−24.8062	−1.39326	−0.696629	−	0.717432i	$-0.745318\pi$
$317$	−24.8062	−1.39326	−0.696629	+	0.717432i	$0.745318\pi$
$318$	0	0
$319$	2.18518	0.122347
$320$	0	0
$321$	45.6125	2.54584
$322$	0	0
$323$	0.326070	0.0181430
$324$	0	0
$325$	8.70156	0.482676
$326$	0	0
$327$	13.3885	0.740385
$328$	0	0
$329$	17.7016	0.975919
$330$	0	0
$331$	24.1513	1.32748	0.663739	−	0.747964i	$-0.268969\pi$
$331$	24.1513	1.32748	0.663739	+	0.747964i	$0.268969\pi$
$332$	0	0
$333$	−24.8062	−1.35937
$334$	0	0
$335$	42.0732	2.29871
$336$	0	0
$337$	−3.10469	−0.169123	−0.0845615	−	0.996418i	$-0.526949\pi$
$337$	−3.10469	−0.169123	−0.0845615	+	0.996418i	$0.526949\pi$
$338$	0	0
$339$	31.1473	1.69169
$340$	0	0
$341$	5.61250	0.303934
$342$	0	0
$343$	15.5737	0.840899
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.97384	0.267010	0.133505	−	0.991048i	$-0.457377\pi$
$347$	4.97384	0.267010	0.133505	+	0.991048i	$0.457377\pi$
$348$	0	0
$349$	33.9109	1.81521	0.907605	−	0.419824i	$-0.137908\pi$
$349$	33.9109	1.81521	0.907605	+	0.419824i	$0.137908\pi$
$350$	0	0
$351$	11.5294	0.615393
$352$	0	0
$353$	14.5969	0.776913	0.388457	−	0.921467i	$-0.373008\pi$
$353$	14.5969	0.776913	0.388457	+	0.921467i	$0.373008\pi$
$354$	0	0
$355$	−30.5438	−1.62110
$356$	0	0
$357$	3.91093	0.206989
$358$	0	0
$359$	−2.95170	−0.155785	−0.0778923	−	0.996962i	$-0.524819\pi$
$359$	−2.95170	−0.155785	−0.0778923	+	0.996962i	$0.524819\pi$
$360$	0	0
$361$	−17.8062	−0.937171
$362$	0	0
$363$	−30.5438	−1.60314
$364$	0	0
$365$	22.2094	1.16249
$366$	0	0
$367$	−28.9622	−1.51181	−0.755906	−	0.654680i	$-0.772803\pi$
$367$	−28.9622	−1.51181	−0.755906	+	0.654680i	$0.772803\pi$
$368$	0	0
$369$	−63.0156	−3.28046
$370$	0	0
$371$	5.90340	0.306489
$372$	0	0
$373$	16.2094	0.839290	0.419645	−	0.907688i	$-0.362155\pi$
$373$	16.2094	0.839290	0.419645	+	0.907688i	$0.362155\pi$
$374$	0	0
$375$	−42.6767	−2.20382
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	27.8696	1.43156	0.715782	−	0.698324i	$-0.246071\pi$
$379$	27.8696	1.43156	0.715782	+	0.698324i	$0.246071\pi$
$380$	0	0
$381$	−52.4187	−2.68549
$382$	0	0
$383$	−8.57768	−0.438299	−0.219149	−	0.975691i	$-0.570328\pi$
$383$	−8.57768	−0.438299	−0.219149	+	0.975691i	$0.570328\pi$
$384$	0	0
$385$	17.0156	0.867196
$386$	0	0
$387$	−35.5177	−1.80547
$388$	0	0
$389$	−32.8062	−1.66334	−0.831671	−	0.555268i	$-0.812616\pi$
$389$	−32.8062	−1.66334	−0.831671	+	0.555268i	$0.812616\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−23.3141	−1.17604
$394$	0	0
$395$	54.2061	2.72740
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	14.3180	0.716799
$400$	0	0
$401$	−24.2094	−1.20896	−0.604479	−	0.796621i	$-0.706619\pi$
$401$	−24.2094	−1.20896	−0.604479	+	0.796621i	$0.706619\pi$
$402$	0	0
$403$	−5.13688	−0.255886
$404$	0	0
$405$	−58.5078	−2.90728
$406$	0	0
$407$	4.04429	0.200468
$408$	0	0
$409$	−9.40312	−0.464955	−0.232477	−	0.972602i	$-0.574683\pi$
$409$	−9.40312	−0.464955	−0.232477	+	0.972602i	$0.574683\pi$
$410$	0	0
$411$	62.9468	3.10494
$412$	0	0
$413$	−57.0156	−2.80556
$414$	0	0
$415$	−27.1030	−1.33044
$416$	0	0
$417$	−48.5078	−2.37544
$418$	0	0
$419$	32.4030	1.58299	0.791494	−	0.611177i	$-0.209304\pi$
$419$	32.4030	1.58299	0.791494	+	0.611177i	$0.209304\pi$
$420$	0	0
$421$	19.1047	0.931105	0.465553	−	0.885020i	$-0.345856\pi$
$421$	19.1047	0.931105	0.465553	+	0.885020i	$0.345856\pi$
$422$	0	0
$423$	28.1956	1.37092
$424$	0	0
$425$	2.59688	0.125967
$426$	0	0
$427$	39.5620	1.91454
$428$	0	0
$429$	−3.40312	−0.164304
$430$	0	0
$431$	18.8514	0.908042	0.454021	−	0.890991i	$-0.349989\pi$
$431$	18.8514	0.908042	0.454021	+	0.890991i	$0.349989\pi$
$432$	0	0
$433$	4.89531	0.235254	0.117627	−	0.993058i	$-0.462471\pi$
$433$	4.89531	0.235254	0.117627	+	0.993058i	$0.462471\pi$
$434$	0	0
$435$	23.0588	1.10558
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−27.4291	−1.30912	−0.654560	−	0.756010i	$-0.727146\pi$
$439$	−27.4291	−1.30912	−0.654560	+	0.756010i	$0.727146\pi$
$440$	0	0
$441$	71.7172	3.41510
$442$	0	0
$443$	11.8554	0.563269	0.281635	−	0.959522i	$-0.409123\pi$
$443$	11.8554	0.563269	0.281635	+	0.959522i	$0.409123\pi$
$444$	0	0
$445$	22.2094	1.05283
$446$	0	0
$447$	39.8880	1.88664
$448$	0	0
$449$	24.8062	1.17068	0.585340	−	0.810788i	$-0.300961\pi$
$449$	24.8062	1.17068	0.585340	+	0.810788i	$0.300961\pi$
$450$	0	0
$451$	10.2738	0.483772
$452$	0	0
$453$	−6.29844	−0.295926
$454$	0	0
$455$	−15.5737	−0.730105
$456$	0	0
$457$	0.806248	0.0377147	0.0188574	−	0.999822i	$-0.493997\pi$
$457$	0.806248	0.0377147	0.0188574	+	0.999822i	$0.493997\pi$
$458$	0	0
$459$	3.44080	0.160603
$460$	0	0
$461$	−11.7016	−0.544996	−0.272498	−	0.962156i	$-0.587850\pi$
$461$	−11.7016	−0.544996	−0.272498	+	0.962156i	$0.587850\pi$
$462$	0	0
$463$	−32.2399	−1.49832	−0.749158	−	0.662391i	$-0.769542\pi$
$463$	−32.2399	−1.49832	−0.749158	+	0.662391i	$0.769542\pi$
$464$	0	0
$465$	59.2250	2.74649
$466$	0	0
$467$	−29.2882	−1.35530	−0.677649	−	0.735386i	$-0.737001\pi$
$467$	−29.2882	−1.35530	−0.677649	+	0.735386i	$0.737001\pi$
$468$	0	0
$469$	−47.8219	−2.20821
$470$	0	0
$471$	−6.22947	−0.287039
$472$	0	0
$473$	5.79063	0.266253
$474$	0	0
$475$	9.50723	0.436222
$476$	0	0
$477$	9.40312	0.430539
$478$	0	0
$479$	−25.0809	−1.14598	−0.572988	−	0.819564i	$-0.694216\pi$
$479$	−25.0809	−1.14598	−0.572988	+	0.819564i	$0.694216\pi$
$480$	0	0
$481$	−3.70156	−0.168777
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−32.5969	−1.48015
$486$	0	0
$487$	2.95170	0.133754	0.0668771	−	0.997761i	$-0.478696\pi$
$487$	2.95170	0.133754	0.0668771	+	0.997761i	$0.478696\pi$
$488$	0	0
$489$	29.6125	1.33912
$490$	0	0
$491$	−34.5881	−1.56094	−0.780470	−	0.625193i	$-0.785020\pi$
$491$	−34.5881	−1.56094	−0.780470	+	0.625193i	$0.785020\pi$
$492$	0	0
$493$	−0.596876	−0.0268819
$494$	0	0
$495$	27.1030	1.21819
$496$	0	0
$497$	34.7172	1.55728
$498$	0	0
$499$	−38.7955	−1.73672	−0.868362	−	0.495932i	$-0.834827\pi$
$499$	−38.7955	−1.73672	−0.868362	+	0.495932i	$0.834827\pi$
$500$	0	0
$501$	−36.4187	−1.62707
$502$	0	0
$503$	−8.41464	−0.375190	−0.187595	−	0.982246i	$-0.560069\pi$
$503$	−8.41464	−0.375190	−0.187595	+	0.982246i	$0.560069\pi$
$504$	0	0
$505$	20.0000	0.889988
$506$	0	0
$507$	3.11473	0.138330
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	−25.2439	−1.11673
$512$	0	0
$513$	12.5969	0.556166
$514$	0	0
$515$	−46.1175	−2.03218
$516$	0	0
$517$	−4.59688	−0.202170
$518$	0	0
$519$	29.2882	1.28561
$520$	0	0
$521$	18.5078	0.810842	0.405421	−	0.914130i	$-0.367125\pi$
$521$	18.5078	0.810842	0.405421	+	0.914130i	$0.367125\pi$
$522$	0	0
$523$	19.0145	0.831445	0.415722	−	0.909492i	$-0.363529\pi$
$523$	19.0145	0.831445	0.415722	+	0.909492i	$0.363529\pi$
$524$	0	0
$525$	114.031	4.97673
$526$	0	0
$527$	−1.53304	−0.0667802
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−90.8164	−3.94109
$532$	0	0
$533$	−9.40312	−0.407295
$534$	0	0
$535$	−54.2061	−2.34353
$536$	0	0
$537$	−16.5078	−0.712365
$538$	0	0
$539$	−11.6924	−0.503628
$540$	0	0
$541$	−8.29844	−0.356778	−0.178389	−	0.983960i	$-0.557088\pi$
$541$	−8.29844	−0.356778	−0.178389	+	0.983960i	$0.557088\pi$
$542$	0	0
$543$	8.08857	0.347114
$544$	0	0
$545$	−15.9109	−0.681550
$546$	0	0
$547$	−21.8031	−0.932235	−0.466117	−	0.884723i	$-0.654348\pi$
$547$	−21.8031	−0.932235	−0.466117	+	0.884723i	$0.654348\pi$
$548$	0	0
$549$	63.0156	2.68944
$550$	0	0
$551$	−2.18518	−0.0930917
$552$	0	0
$553$	−61.6125	−2.62003
$554$	0	0
$555$	42.6767	1.81153
$556$	0	0
$557$	7.70156	0.326326	0.163163	−	0.986599i	$-0.447830\pi$
$557$	7.70156	0.326326	0.163163	+	0.986599i	$0.447830\pi$
$558$	0	0
$559$	−5.29991	−0.224162
$560$	0	0
$561$	−1.01562	−0.0428796
$562$	0	0
$563$	−9.01813	−0.380069	−0.190034	−	0.981777i	$-0.560860\pi$
$563$	−9.01813	−0.380069	−0.190034	+	0.981777i	$0.560860\pi$
$564$	0	0
$565$	−37.0156	−1.55726
$566$	0	0
$567$	66.5020	2.79282
$568$	0	0
$569$	35.7016	1.49669	0.748344	−	0.663311i	$-0.230849\pi$
$569$	35.7016	1.49669	0.748344	+	0.663311i	$0.230849\pi$
$570$	0	0
$571$	5.29991	0.221794	0.110897	−	0.993832i	$-0.464628\pi$
$571$	5.29991	0.221794	0.110897	+	0.993832i	$0.464628\pi$
$572$	0	0
$573$	12.5969	0.526242
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.19375	−0.299480	−0.149740	−	0.988725i	$-0.547844\pi$
$577$	−7.19375	−0.299480	−0.149740	+	0.988725i	$0.547844\pi$
$578$	0	0
$579$	66.6650	2.77050
$580$	0	0
$581$	30.8062	1.27806
$582$	0	0
$583$	−1.53304	−0.0634920
$584$	0	0
$585$	−24.8062	−1.02561
$586$	0	0
$587$	−9.50723	−0.392406	−0.196203	−	0.980563i	$-0.562861\pi$
$587$	−9.50723	−0.392406	−0.196203	+	0.980563i	$0.562861\pi$
$588$	0	0
$589$	−5.61250	−0.231259
$590$	0	0
$591$	−50.7653	−2.08820
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	−4.64777	−0.190540
$596$	0	0
$597$	−58.2094	−2.38235
$598$	0	0
$599$	33.3325	1.36193	0.680965	−	0.732316i	$-0.261561\pi$
$599$	33.3325	1.36193	0.680965	+	0.732316i	$0.261561\pi$
$600$	0	0
$601$	24.2984	0.991154	0.495577	−	0.868564i	$-0.334957\pi$
$601$	24.2984	0.991154	0.495577	+	0.868564i	$0.334957\pi$
$602$	0	0
$603$	−76.1723	−3.10197
$604$	0	0
$605$	36.2984	1.47574
$606$	0	0
$607$	−12.1329	−0.492458	−0.246229	−	0.969212i	$-0.579191\pi$
$607$	−12.1329	−0.492458	−0.246229	+	0.969212i	$0.579191\pi$
$608$	0	0
$609$	−26.2094	−1.06206
$610$	0	0
$611$	4.20732	0.170210
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	108.412	4.37160
$616$	0	0
$617$	6.59688	0.265580	0.132790	−	0.991144i	$-0.457606\pi$
$617$	6.59688	0.265580	0.132790	+	0.991144i	$0.457606\pi$
$618$	0	0
$619$	−36.6103	−1.47149	−0.735746	−	0.677258i	$-0.763168\pi$
$619$	−36.6103	−1.47149	−0.735746	+	0.677258i	$0.763168\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−25.2439	−1.01138
$624$	0	0
$625$	7.20937	0.288375
$626$	0	0
$627$	−3.71822	−0.148491
$628$	0	0
$629$	−1.10469	−0.0440467
$630$	0	0
$631$	1.69607	0.0675196	0.0337598	−	0.999430i	$-0.489252\pi$
$631$	1.69607	0.0675196	0.0337598	+	0.999430i	$0.489252\pi$
$632$	0	0
$633$	67.9109	2.69922
$634$	0	0
$635$	62.2947	2.47209
$636$	0	0
$637$	10.7016	0.424011
$638$	0	0
$639$	55.2987	2.18758
$640$	0	0
$641$	32.8062	1.29577	0.647884	−	0.761739i	$-0.275654\pi$
$641$	32.8062	1.29577	0.647884	+	0.761739i	$0.275654\pi$
$642$	0	0
$643$	−20.1071	−0.792945	−0.396472	−	0.918047i	$-0.629766\pi$
$643$	−20.1071	−0.792945	−0.396472	+	0.918047i	$0.629766\pi$
$644$	0	0
$645$	61.1047	2.40599
$646$	0	0
$647$	−18.6884	−0.734717	−0.367358	−	0.930079i	$-0.619738\pi$
$647$	−18.6884	−0.734717	−0.367358	+	0.930079i	$0.619738\pi$
$648$	0	0
$649$	14.8062	0.581196
$650$	0	0
$651$	−67.3172	−2.63837
$652$	0	0
$653$	47.0156	1.83986	0.919932	−	0.392079i	$-0.128244\pi$
$653$	47.0156	1.83986	0.919932	+	0.392079i	$0.128244\pi$
$654$	0	0
$655$	27.7065	1.08258
$656$	0	0
$657$	−40.2094	−1.56872
$658$	0	0
$659$	19.0145	0.740699	0.370349	−	0.928893i	$-0.379238\pi$
$659$	19.0145	0.740699	0.370349	+	0.928893i	$0.379238\pi$
$660$	0	0
$661$	28.8062	1.12043	0.560217	−	0.828346i	$-0.310718\pi$
$661$	28.8062	1.12043	0.560217	+	0.828346i	$0.310718\pi$
$662$	0	0
$663$	0.929554	0.0361009
$664$	0	0
$665$	−17.0156	−0.659837
$666$	0	0
$667$	0	0
$668$	0	0
$669$	64.5078	2.49402
$670$	0	0
$671$	−10.2738	−0.396614
$672$	0	0
$673$	−29.3141	−1.12997	−0.564987	−	0.825100i	$-0.691119\pi$
$673$	−29.3141	−1.12997	−0.564987	+	0.825100i	$0.691119\pi$
$674$	0	0
$675$	100.324	3.86146
$676$	0	0
$677$	17.4031	0.668856	0.334428	−	0.942421i	$-0.391457\pi$
$677$	17.4031	0.668856	0.334428	+	0.942421i	$0.391457\pi$
$678$	0	0
$679$	37.0507	1.42188
$680$	0	0
$681$	−16.0000	−0.613121
$682$	0	0
$683$	−49.0692	−1.87758	−0.938791	−	0.344488i	$-0.888052\pi$
$683$	−49.0692	−1.87758	−0.938791	+	0.344488i	$0.888052\pi$
$684$	0	0
$685$	−74.8062	−2.85820
$686$	0	0
$687$	27.7065	1.05707
$688$	0	0
$689$	1.40312	0.0534548
$690$	0	0
$691$	28.5217	1.08502	0.542508	−	0.840050i	$-0.317475\pi$
$691$	28.5217	1.08502	0.542508	+	0.840050i	$0.317475\pi$
$692$	0	0
$693$	−30.8062	−1.17023
$694$	0	0
$695$	57.6469	2.18667
$696$	0	0
$697$	−2.80625	−0.106294
$698$	0	0
$699$	71.9649	2.72196
$700$	0	0
$701$	17.4031	0.657307	0.328653	−	0.944451i	$-0.393405\pi$
$701$	17.4031	0.657307	0.328653	+	0.944451i	$0.393405\pi$
$702$	0	0
$703$	−4.04429	−0.152533
$704$	0	0
$705$	−48.5078	−1.82691
$706$	0	0
$707$	−22.7327	−0.854951
$708$	0	0
$709$	−3.19375	−0.119944	−0.0599719	−	0.998200i	$-0.519101\pi$
$709$	−3.19375	−0.119944	−0.0599719	+	0.998200i	$0.519101\pi$
$710$	0	0
$711$	−98.1384	−3.68048
$712$	0	0
$713$	0	0
$714$	0	0
$715$	4.04429	0.151248
$716$	0	0
$717$	45.1047	1.68447
$718$	0	0
$719$	−31.1473	−1.16160	−0.580800	−	0.814046i	$-0.697260\pi$
$719$	−31.1473	−1.16160	−0.580800	+	0.814046i	$0.697260\pi$
$720$	0	0
$721$	52.4187	1.95218
$722$	0	0
$723$	−14.9702	−0.556747
$724$	0	0
$725$	−17.4031	−0.646336
$726$	0	0
$727$	2.51125	0.0931371	0.0465685	−	0.998915i	$-0.485171\pi$
$727$	2.51125	0.0931371	0.0465685	+	0.998915i	$0.485171\pi$
$728$	0	0
$729$	−1.80625	−0.0668981
$730$	0	0
$731$	−1.58169	−0.0585011
$732$	0	0
$733$	−37.9109	−1.40027	−0.700136	−	0.714009i	$-0.746877\pi$
$733$	−37.9109	−1.40027	−0.700136	+	0.714009i	$0.746877\pi$
$734$	0	0
$735$	−123.382	−4.55103
$736$	0	0
$737$	12.4187	0.457450
$738$	0	0
$739$	17.5958	0.647272	0.323636	−	0.946182i	$-0.395095\pi$
$739$	17.5958	0.647272	0.323636	+	0.946182i	$0.395095\pi$
$740$	0	0
$741$	3.40312	0.125017
$742$	0	0
$743$	2.02214	0.0741853	0.0370926	−	0.999312i	$-0.488190\pi$
$743$	2.02214	0.0741853	0.0370926	+	0.999312i	$0.488190\pi$
$744$	0	0
$745$	−47.4031	−1.73672
$746$	0	0
$747$	49.0692	1.79535
$748$	0	0
$749$	61.6125	2.25127
$750$	0	0
$751$	49.8357	1.81853	0.909266	−	0.416216i	$-0.136644\pi$
$751$	49.8357	1.81853	0.909266	+	0.416216i	$0.136644\pi$
$752$	0	0
$753$	52.4187	1.91025
$754$	0	0
$755$	7.48509	0.272410
$756$	0	0
$757$	−12.2094	−0.443757	−0.221879	−	0.975074i	$-0.571219\pi$
$757$	−12.2094	−0.443757	−0.221879	+	0.975074i	$0.571219\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23.6125	0.855952	0.427976	−	0.903790i	$-0.359227\pi$
$761$	23.6125	0.855952	0.427976	+	0.903790i	$0.359227\pi$
$762$	0	0
$763$	18.0849	0.654718
$764$	0	0
$765$	−7.40312	−0.267661
$766$	0	0
$767$	−13.5515	−0.489317
$768$	0	0
$769$	12.2094	0.440281	0.220141	−	0.975468i	$-0.429348\pi$
$769$	12.2094	0.440281	0.220141	+	0.975468i	$0.429348\pi$
$770$	0	0
$771$	−45.1880	−1.62740
$772$	0	0
$773$	3.10469	0.111668	0.0558339	−	0.998440i	$-0.482218\pi$
$773$	3.10469	0.111668	0.0558339	+	0.998440i	$0.482218\pi$
$774$	0	0
$775$	−44.6989	−1.60563
$776$	0	0
$777$	−48.5078	−1.74021
$778$	0	0
$779$	−10.2738	−0.368095
$780$	0	0
$781$	−9.01562	−0.322604
$782$	0	0
$783$	−23.0588	−0.824053
$784$	0	0
$785$	7.40312	0.264229
$786$	0	0
$787$	0.766519	0.0273235	0.0136617	−	0.999907i	$-0.495651\pi$
$787$	0.766519	0.0273235	0.0136617	+	0.999907i	$0.495651\pi$
$788$	0	0
$789$	−6.80625	−0.242309
$790$	0	0
$791$	42.0732	1.49595
$792$	0	0
$793$	9.40312	0.333915
$794$	0	0
$795$	−16.1771	−0.573744
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	1.25562	0.0444208
$800$	0	0
$801$	−40.2094	−1.42073
$802$	0	0
$803$	6.55554	0.231340
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−38.0289	−1.33868
$808$	0	0
$809$	−27.1047	−0.952950	−0.476475	−	0.879188i	$-0.658086\pi$
$809$	−27.1047	−0.952950	−0.476475	+	0.879188i	$0.658086\pi$
$810$	0	0
$811$	16.0628	0.564040	0.282020	−	0.959409i	$-0.408996\pi$
$811$	16.0628	0.564040	0.282020	+	0.959409i	$0.408996\pi$
$812$	0	0
$813$	13.1047	0.459601
$814$	0	0
$815$	−35.1916	−1.23271
$816$	0	0
$817$	−5.79063	−0.202588
$818$	0	0
$819$	28.1956	0.985235
$820$	0	0
$821$	25.9109	0.904298	0.452149	−	0.891942i	$-0.350658\pi$
$821$	25.9109	0.904298	0.452149	+	0.891942i	$0.350658\pi$
$822$	0	0
$823$	31.7995	1.10846	0.554230	−	0.832364i	$-0.313013\pi$
$823$	31.7995	1.10846	0.554230	+	0.832364i	$0.313013\pi$
$824$	0	0
$825$	−29.6125	−1.03097
$826$	0	0
$827$	11.3663	0.395246	0.197623	−	0.980278i	$-0.436678\pi$
$827$	11.3663	0.395246	0.197623	+	0.980278i	$0.436678\pi$
$828$	0	0
$829$	51.6125	1.79258	0.896288	−	0.443472i	$-0.146254\pi$
$829$	51.6125	1.79258	0.896288	+	0.443472i	$0.146254\pi$
$830$	0	0
$831$	75.4057	2.61580
$832$	0	0
$833$	3.19375	0.110657
$834$	0	0
$835$	43.2802	1.49777
$836$	0	0
$837$	−59.2250	−2.04712
$838$	0	0
$839$	−32.2399	−1.11305	−0.556523	−	0.830832i	$-0.687865\pi$
$839$	−32.2399	−1.11305	−0.556523	+	0.830832i	$0.687865\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	31.1473	1.07277
$844$	0	0
$845$	−3.70156	−0.127338
$846$	0	0
$847$	−41.2580	−1.41764
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−23.1047	−0.791089	−0.395545	−	0.918447i	$-0.629444\pi$
$853$	−23.1047	−0.791089	−0.395545	+	0.918447i	$0.629444\pi$
$854$	0	0
$855$	−27.1030	−0.926905
$856$	0	0
$857$	−12.8062	−0.437453	−0.218727	−	0.975786i	$-0.570190\pi$
$857$	−12.8062	−0.437453	−0.218727	+	0.975786i	$0.570190\pi$
$858$	0	0
$859$	−14.6441	−0.499651	−0.249825	−	0.968291i	$-0.580373\pi$
$859$	−14.6441	−0.499651	−0.249825	+	0.968291i	$0.580373\pi$
$860$	0	0
$861$	−123.225	−4.19950
$862$	0	0
$863$	−27.2661	−0.928148	−0.464074	−	0.885796i	$-0.653613\pi$
$863$	−27.2661	−0.928148	−0.464074	+	0.885796i	$0.653613\pi$
$864$	0	0
$865$	−34.8062	−1.18345
$866$	0	0
$867$	−52.6730	−1.78887
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	−11.3663	−0.385134
$872$	0	0
$873$	59.0156	1.99738
$874$	0	0
$875$	−57.6469	−1.94882
$876$	0	0
$877$	−44.7172	−1.50999	−0.754996	−	0.655729i	$-0.772361\pi$
$877$	−44.7172	−1.50999	−0.754996	+	0.655729i	$0.772361\pi$
$878$	0	0
$879$	59.5060	2.00709
$880$	0	0
$881$	−55.7016	−1.87663	−0.938317	−	0.345777i	$-0.887615\pi$
$881$	−55.7016	−1.87663	−0.938317	+	0.345777i	$0.887615\pi$
$882$	0	0
$883$	−40.8176	−1.37362	−0.686811	−	0.726836i	$-0.740990\pi$
$883$	−40.8176	−1.37362	−0.686811	+	0.726836i	$0.740990\pi$
$884$	0	0
$885$	156.241	5.25197
$886$	0	0
$887$	19.0145	0.638443	0.319222	−	0.947680i	$-0.396579\pi$
$887$	19.0145	0.638443	0.319222	+	0.947680i	$0.396579\pi$
$888$	0	0
$889$	−70.8062	−2.37477
$890$	0	0
$891$	−17.2697	−0.578558
$892$	0	0
$893$	4.59688	0.153829
$894$	0	0
$895$	19.6180	0.655756
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.2738	0.342649
$900$	0	0
$901$	0.418745	0.0139504
$902$	0	0
$903$	−69.4537	−2.31127
$904$	0	0
$905$	−9.61250	−0.319530
$906$	0	0
$907$	−17.7588	−0.589673	−0.294836	−	0.955548i	$-0.595265\pi$
$907$	−17.7588	−0.589673	−0.294836	+	0.955548i	$0.595265\pi$
$908$	0	0
$909$	−36.2094	−1.20099
$910$	0	0
$911$	−25.2439	−0.836369	−0.418184	−	0.908362i	$-0.637333\pi$
$911$	−25.2439	−0.836369	−0.418184	+	0.908362i	$0.637333\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	−108.412	−3.58400
$916$	0	0
$917$	−31.4922	−1.03996
$918$	0	0
$919$	−14.6441	−0.483065	−0.241532	−	0.970393i	$-0.577650\pi$
$919$	−14.6441	−0.483065	−0.241532	+	0.970393i	$0.577650\pi$
$920$	0	0
$921$	49.0156	1.61512
$922$	0	0
$923$	8.25161	0.271605
$924$	0	0
$925$	−32.2094	−1.05904
$926$	0	0
$927$	83.4943	2.74231
$928$	0	0
$929$	5.40312	0.177271	0.0886354	−	0.996064i	$-0.471749\pi$
$929$	5.40312	0.177271	0.0886354	+	0.996064i	$0.471749\pi$
$930$	0	0
$931$	11.6924	0.383203
$932$	0	0
$933$	−58.2094	−1.90569
$934$	0	0
$935$	1.20697	0.0394721
$936$	0	0
$937$	32.8062	1.07173	0.535867	−	0.844303i	$-0.319985\pi$
$937$	32.8062	1.07173	0.535867	+	0.844303i	$0.319985\pi$
$938$	0	0
$939$	7.81116	0.254908
$940$	0	0
$941$	16.8953	0.550771	0.275386	−	0.961334i	$-0.411194\pi$
$941$	16.8953	0.550771	0.275386	+	0.961334i	$0.411194\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−179.555	−5.84091
$946$	0	0
$947$	19.4549	0.632200	0.316100	−	0.948726i	$-0.397627\pi$
$947$	19.4549	0.632200	0.316100	+	0.948726i	$0.397627\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	−77.2648	−2.50548
$952$	0	0
$953$	49.3141	1.59744	0.798720	−	0.601704i	$-0.205511\pi$
$953$	49.3141	1.59744	0.798720	+	0.601704i	$0.205511\pi$
$954$	0	0
$955$	−14.9702	−0.484424
$956$	0	0
$957$	6.80625	0.220015
$958$	0	0
$959$	85.0273	2.74568
$960$	0	0
$961$	−4.61250	−0.148790
$962$	0	0
$963$	98.1384	3.16247
$964$	0	0
$965$	−79.2250	−2.55034
$966$	0	0
$967$	0.163035	0.00524285	0.00262143	−	0.999997i	$-0.499166\pi$
$967$	0.163035	0.00524285	0.00262143	+	0.999997i	$0.499166\pi$
$968$	0	0
$969$	1.01562	0.0326265
$970$	0	0
$971$	−17.4328	−0.559444	−0.279722	−	0.960081i	$-0.590242\pi$
$971$	−17.4328	−0.559444	−0.279722	+	0.960081i	$0.590242\pi$
$972$	0	0
$973$	−65.5234	−2.10058
$974$	0	0
$975$	27.1030	0.867992
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	6.55554	0.209516
$980$	0	0
$981$	28.8062	0.919713
$982$	0	0
$983$	−49.9988	−1.59471	−0.797356	−	0.603509i	$-0.793769\pi$
$983$	−49.9988	−1.59471	−0.797356	+	0.603509i	$0.793769\pi$
$984$	0	0
$985$	60.3297	1.92226
$986$	0	0
$987$	55.1356	1.75499
$988$	0	0
$989$	0	0
$990$	0	0
$991$	20.8736	0.663071	0.331536	−	0.943443i	$-0.392433\pi$
$991$	20.8736	0.663071	0.331536	+	0.943443i	$0.392433\pi$
$992$	0	0
$993$	75.2250	2.38719
$994$	0	0
$995$	69.1763	2.19304
$996$	0	0
$997$	4.80625	0.152215	0.0761077	−	0.997100i	$-0.475751\pi$
$997$	4.80625	0.152215	0.0761077	+	0.997100i	$0.475751\pi$
$998$	0	0
$999$	−42.6767	−1.35023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.a.f.1.4	yes	4
3.2	odd	2		3744.2.a.be.1.4		4
4.3	odd	2	inner	416.2.a.f.1.1	✓	4
8.3	odd	2		832.2.a.p.1.4		4
8.5	even	2		832.2.a.p.1.1		4
12.11	even	2		3744.2.a.be.1.3		4
13.12	even	2		5408.2.a.bj.1.4		4
16.3	odd	4		3328.2.b.bb.1665.2		8
16.5	even	4		3328.2.b.bb.1665.1		8
16.11	odd	4		3328.2.b.bb.1665.7		8
16.13	even	4		3328.2.b.bb.1665.8		8
24.5	odd	2		7488.2.a.da.1.2		4
24.11	even	2		7488.2.a.da.1.1		4
52.51	odd	2		5408.2.a.bj.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.a.f.1.1	✓	4	4.3	odd	2	inner
416.2.a.f.1.4	yes	4	1.1	even	1	trivial
832.2.a.p.1.1		4	8.5	even	2
832.2.a.p.1.4		4	8.3	odd	2
3328.2.b.bb.1665.1		8	16.5	even	4
3328.2.b.bb.1665.2		8	16.3	odd	4
3328.2.b.bb.1665.7		8	16.11	odd	4
3328.2.b.bb.1665.8		8	16.13	even	4
3744.2.a.be.1.3		4	12.11	even	2
3744.2.a.be.1.4		4	3.2	odd	2
5408.2.a.bj.1.1		4	52.51	odd	2
5408.2.a.bj.1.4		4	13.12	even	2
7488.2.a.da.1.1		4	24.11	even	2
7488.2.a.da.1.2		4	24.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.a (trivial)

\(f(q)\)	\(=\)	\(q+3.11473 q^{3} -3.70156 q^{5} +4.20732 q^{7} +6.70156 q^{9} -1.09259 q^{11} +1.00000 q^{13} -11.5294 q^{15} +0.298438 q^{17} +1.09259 q^{19} +13.1047 q^{21} +8.70156 q^{25} +11.5294 q^{27} -2.00000 q^{29} -5.13688 q^{31} -3.40312 q^{33} -15.5737 q^{35} -3.70156 q^{37} +3.11473 q^{39} -9.40312 q^{41} -5.29991 q^{43} -24.8062 q^{45} +4.20732 q^{47} +10.7016 q^{49} +0.929554 q^{51} +1.40312 q^{53} +4.04429 q^{55} +3.40312 q^{57} -13.5515 q^{59} +9.40312 q^{61} +28.1956 q^{63} -3.70156 q^{65} -11.3663 q^{67} +8.25161 q^{71} -6.00000 q^{73} +27.1030 q^{75} -4.59688 q^{77} -14.6441 q^{79} +15.8062 q^{81} +7.32206 q^{83} -1.10469 q^{85} -6.22947 q^{87} -6.00000 q^{89} +4.20732 q^{91} -16.0000 q^{93} -4.04429 q^{95} +8.80625 q^{97} -7.32206 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 14 q^{9} + 4 q^{13} + 14 q^{17} + 14 q^{21} + 22 q^{25} - 8 q^{29} + 12 q^{33} - 2 q^{37} - 12 q^{41} - 48 q^{45} + 30 q^{49} - 20 q^{53} - 12 q^{57} + 12 q^{61} - 2 q^{65} - 24 q^{73} - 44 q^{77}+ \cdots - 16 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 14 * q^9 + 4 * q^13 + 14 * q^17 + 14 * q^21 + 22 * q^25 - 8 * q^29 + 12 * q^33 - 2 * q^37 - 12 * q^41 - 48 * q^45 + 30 * q^49 - 20 * q^53 - 12 * q^57 + 12 * q^61 - 2 * q^65 - 24 * q^73 - 44 * q^77 + 12 * q^81 + 34 * q^85 - 24 * q^89 - 64 * q^93 - 16 * q^97

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