LMFDB - Embedded newform 5408.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5408,2,Mod(1,5408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5408, 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("5408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5408 = 2^{5} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5408.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:	$43.1830974131$
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:	no (minimal twist has level 416)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.546295$ of defining polynomial
Character	$\chi$	$=$	5408.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.11473 q^{3} +3.70156 q^{5} +4.20732 q^{7} +6.70156 q^{9} -1.09259 q^{11} -11.5294 q^{15} +0.298438 q^{17} +1.09259 q^{19} -13.1047 q^{21} +8.70156 q^{25} -11.5294 q^{27} -2.00000 q^{29} -5.13688 q^{31} +3.40312 q^{33} +15.5737 q^{35} +3.70156 q^{37} +9.40312 q^{41} +5.29991 q^{43} +24.8062 q^{45} +4.20732 q^{47} +10.7016 q^{49} -0.929554 q^{51} +1.40312 q^{53} -4.04429 q^{55} -3.40312 q^{57} -13.5515 q^{59} +9.40312 q^{61} +28.1956 q^{63} -11.3663 q^{67} +8.25161 q^{71} +6.00000 q^{73} -27.1030 q^{75} -4.59688 q^{77} +14.6441 q^{79} +15.8062 q^{81} +7.32206 q^{83} +1.10469 q^{85} +6.22947 q^{87} +6.00000 q^{89} +16.0000 q^{93} +4.04429 q^{95} -8.80625 q^{97} -7.32206 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} + 14 q^{9} + 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} + 24 q^{73} - 44 q^{77} + 12 q^{81}+ \cdots + 16 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 + 14 * q^9 + 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 + 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	$-0.855811\pi$
$3$	−3.11473	−1.79829	−0.899146	+	0.437649i	$0.855811\pi$
$4$	0	0
$5$	3.70156	1.65539	0.827694	−	0.561179i	$-0.189652\pi$
$5$	3.70156	1.65539	0.827694	+	0.561179i	$0.189652\pi$
$6$	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
$13$	0	0
$13$	0	0
$14$	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	$-0.401589\pi$
$37$	3.70156	0.608533	0.304267	+	0.952587i	$0.401589\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.40312	1.46852	0.734261	−	0.678868i	$-0.237529\pi$
$41$	9.40312	1.46852	0.734261	+	0.678868i	$0.237529\pi$
$42$	0	0
$43$	5.29991	0.808229	0.404114	−	0.914708i	$-0.367580\pi$
$43$	5.29991	0.808229	0.404114	+	0.914708i	$0.367580\pi$
$44$	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
$65$	0	0
$66$	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	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	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	$-0.191852\pi$
$79$	14.6441	1.64759	0.823796	+	0.566887i	$0.191852\pi$
$80$	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	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	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	$-0.647532\pi$
$97$	−8.80625	−0.894139	−0.447070	+	0.894499i	$0.647532\pi$
$98$	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	$-0.710363\pi$
$103$	−12.4589	−1.22762	−0.613808	+	0.789456i	$0.710363\pi$
$104$	0	0
$105$	−48.5078	−4.73388
$106$	0	0
$107$	−14.6441	−1.41570	−0.707850	−	0.706363i	$-0.750335\pi$
$107$	−14.6441	−1.41570	−0.707850	+	0.706363i	$0.750335\pi$
$108$	0	0
$109$	−4.29844	−0.411716	−0.205858	−	0.978582i	$-0.565998\pi$
$109$	−4.29844	−0.411716	−0.205858	+	0.978582i	$0.565998\pi$
$110$	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
$117$	0	0
$118$	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	$-0.231647\pi$
$127$	16.8293	1.49336	0.746679	+	0.665185i	$0.231647\pi$
$128$	0	0
$129$	−16.5078	−1.45343
$130$	0	0
$131$	7.48509	0.653975	0.326988	−	0.945029i	$-0.393966\pi$
$131$	7.48509	0.653975	0.326988	+	0.945029i	$0.393966\pi$
$132$	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	$-0.831607\pi$
$137$	−20.2094	−1.72660	−0.863302	+	0.504688i	$0.831607\pi$
$138$	0	0
$139$	15.5737	1.32094	0.660471	−	0.750852i	$-0.270357\pi$
$139$	15.5737	1.32094	0.660471	+	0.750852i	$0.270357\pi$
$140$	0	0
$141$	−13.1047	−1.10361
$142$	0	0
$143$	0	0
$144$	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	$-0.675772\pi$
$149$	−12.8062	−1.04913	−0.524564	+	0.851371i	$0.675772\pi$
$150$	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
$169$	0	0
$170$	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	$-0.436534\pi$
$179$	5.29991	0.396134	0.198067	+	0.980188i	$0.436534\pi$
$180$	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	$-0.546742\pi$
$191$	−4.04429	−0.292634	−0.146317	+	0.989238i	$0.546742\pi$
$192$	0	0
$193$	−21.4031	−1.54063	−0.770315	−	0.637663i	$-0.779901\pi$
$193$	−21.4031	−1.54063	−0.770315	+	0.637663i	$0.779901\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.2984	1.16122	0.580608	−	0.814183i	$-0.302815\pi$
$197$	16.2984	1.16122	0.580608	+	0.814183i	$0.302815\pi$
$198$	0	0
$199$	18.6884	1.32479	0.662393	−	0.749157i	$-0.269541\pi$
$199$	18.6884	1.32479	0.662393	+	0.749157i	$0.269541\pi$
$200$	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	$-0.770185\pi$
$211$	−21.8031	−1.50099	−0.750495	+	0.660876i	$0.770185\pi$
$212$	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	0
$222$	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	$-0.594956\pi$
$229$	−8.89531	−0.587819	−0.293909	+	0.955833i	$0.594956\pi$
$230$	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	$-0.450527\pi$
$241$	4.80625	0.309598	0.154799	+	0.987946i	$0.450527\pi$
$242$	0	0
$243$	−14.6441	−0.939420
$244$	0	0
$245$	39.6125	2.53075
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−22.8062	−1.44529
$250$	0	0
$251$	−16.8293	−1.06226	−0.531128	−	0.847292i	$-0.678232\pi$
$251$	−16.8293	−1.06226	−0.531128	+	0.847292i	$0.678232\pi$
$252$	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	$-0.478539\pi$
$263$	2.18518	0.134744	0.0673719	+	0.997728i	$0.478539\pi$
$264$	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
$273$	0	0
$274$	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	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	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	$-0.688452\pi$
$293$	−19.1047	−1.11611	−0.558054	+	0.829805i	$0.688452\pi$
$294$	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	$-0.322244\pi$
$311$	18.6884	1.05972	0.529861	+	0.848085i	$0.322244\pi$
$312$	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	$-0.254682\pi$
$317$	24.8062	1.39326	0.696629	+	0.717432i	$0.254682\pi$
$318$	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
$325$	0	0
$326$	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	$-0.542623\pi$
$347$	−4.97384	−0.267010	−0.133505	+	0.991048i	$0.542623\pi$
$348$	0	0
$349$	−33.9109	−1.81521	−0.907605	−	0.419824i	$-0.862092\pi$
$349$	−33.9109	−1.81521	−0.907605	+	0.419824i	$0.862092\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.5969	−0.776913	−0.388457	−	0.921467i	$-0.626992\pi$
$353$	−14.5969	−0.776913	−0.388457	+	0.921467i	$0.626992\pi$
$354$	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	$-0.227197\pi$
$367$	28.9622	1.51181	0.755906	+	0.654680i	$0.227197\pi$
$368$	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
$377$	0	0
$378$	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	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	−14.3180	−0.716799
$400$	0	0
$401$	24.2094	1.20896	0.604479	−	0.796621i	$-0.293381\pi$
$401$	24.2094	1.20896	0.604479	+	0.796621i	$0.293381\pi$
$402$	0	0
$403$	0	0
$404$	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	$-0.425317\pi$
$409$	9.40312	0.464955	0.232477	+	0.972602i	$0.425317\pi$
$410$	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	$-0.790696\pi$
$419$	−32.4030	−1.58299	−0.791494	+	0.611177i	$0.790696\pi$
$420$	0	0
$421$	−19.1047	−0.931105	−0.465553	−	0.885020i	$-0.654144\pi$
$421$	−19.1047	−0.931105	−0.465553	+	0.885020i	$0.654144\pi$
$422$	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
$429$	0	0
$430$	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	$-0.272854\pi$
$439$	27.4291	1.30912	0.654560	+	0.756010i	$0.272854\pi$
$440$	0	0
$441$	71.7172	3.41510
$442$	0	0
$443$	−11.8554	−0.563269	−0.281635	−	0.959522i	$-0.590877\pi$
$443$	−11.8554	−0.563269	−0.281635	+	0.959522i	$0.590877\pi$
$444$	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	$-0.699039\pi$
$449$	−24.8062	−1.17068	−0.585340	+	0.810788i	$0.699039\pi$
$450$	0	0
$451$	−10.2738	−0.483772
$452$	0	0
$453$	6.29844	0.295926
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.806248	−0.0377147	−0.0188574	−	0.999822i	$-0.506003\pi$
$457$	−0.806248	−0.0377147	−0.0188574	+	0.999822i	$0.506003\pi$
$458$	0	0
$459$	−3.44080	−0.160603
$460$	0	0
$461$	11.7016	0.544996	0.272498	−	0.962156i	$-0.412150\pi$
$461$	11.7016	0.544996	0.272498	+	0.962156i	$0.412150\pi$
$462$	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	$-0.262999\pi$
$467$	29.2882	1.35530	0.677649	+	0.735386i	$0.262999\pi$
$468$	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
$481$	0	0
$482$	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	$-0.214980\pi$
$491$	34.5881	1.56094	0.780470	+	0.625193i	$0.214980\pi$
$492$	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	$-0.439931\pi$
$503$	8.41464	0.375190	0.187595	+	0.982246i	$0.439931\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	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	$-0.636471\pi$
$523$	−19.0145	−0.831445	−0.415722	+	0.909492i	$0.636471\pi$
$524$	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
$533$	0	0
$534$	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	$-0.442912\pi$
$541$	8.29844	0.356778	0.178389	+	0.983960i	$0.442912\pi$
$542$	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	$-0.345652\pi$
$547$	21.8031	0.932235	0.466117	+	0.884723i	$0.345652\pi$
$548$	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	$-0.552170\pi$
$557$	−7.70156	−0.326326	−0.163163	+	0.986599i	$0.552170\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.01562	0.0428796
$562$	0	0
$563$	9.01813	0.380069	0.190034	−	0.981777i	$-0.439140\pi$
$563$	9.01813	0.380069	0.190034	+	0.981777i	$0.439140\pi$
$564$	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	$-0.535372\pi$
$571$	−5.29991	−0.221794	−0.110897	+	0.993832i	$0.535372\pi$
$572$	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	$-0.452156\pi$
$577$	7.19375	0.299480	0.149740	+	0.988725i	$0.452156\pi$
$578$	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
$585$	0	0
$586$	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	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	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	$-0.738439\pi$
$599$	−33.3325	−1.36193	−0.680965	+	0.732316i	$0.738439\pi$
$600$	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	$-0.420809\pi$
$607$	12.1329	0.492458	0.246229	+	0.969212i	$0.420809\pi$
$608$	0	0
$609$	26.2094	1.06206
$610$	0	0
$611$	0	0
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	−108.412	−4.37160
$616$	0	0
$617$	−6.59688	−0.265580	−0.132790	−	0.991144i	$-0.542394\pi$
$617$	−6.59688	−0.265580	−0.132790	+	0.991144i	$0.542394\pi$
$618$	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
$637$	0	0
$638$	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	$-0.380262\pi$
$647$	18.6884	0.734717	0.367358	+	0.930079i	$0.380262\pi$
$648$	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	$-0.620762\pi$
$659$	−19.0145	−0.740699	−0.370349	+	0.928893i	$0.620762\pi$
$660$	0	0
$661$	−28.8062	−1.12043	−0.560217	−	0.828346i	$-0.689282\pi$
$661$	−28.8062	−1.12043	−0.560217	+	0.828346i	$0.689282\pi$
$662$	0	0
$663$	0	0
$664$	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
$689$	0	0
$690$	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	$-0.480899\pi$
$709$	3.19375	0.119944	0.0599719	+	0.998200i	$0.480899\pi$
$710$	0	0
$711$	98.1384	3.68048
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−45.1047	−1.68447
$718$	0	0
$719$	31.1473	1.16160	0.580800	−	0.814046i	$-0.302740\pi$
$719$	31.1473	1.16160	0.580800	+	0.814046i	$0.302740\pi$
$720$	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	$-0.514829\pi$
$727$	−2.51125	−0.0931371	−0.0465685	+	0.998915i	$0.514829\pi$
$728$	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	$-0.253123\pi$
$733$	37.9109	1.40027	0.700136	+	0.714009i	$0.253123\pi$
$734$	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
$741$	0	0
$742$	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	$-0.863356\pi$
$751$	−49.8357	−1.81853	−0.909266	+	0.416216i	$0.863356\pi$
$752$	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	$-0.640773\pi$
$761$	−23.6125	−0.855952	−0.427976	+	0.903790i	$0.640773\pi$
$762$	0	0
$763$	−18.0849	−0.654718
$764$	0	0
$765$	7.40312	0.267661
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−12.2094	−0.440281	−0.220141	−	0.975468i	$-0.570652\pi$
$769$	−12.2094	−0.440281	−0.220141	+	0.975468i	$0.570652\pi$
$770$	0	0
$771$	45.1880	1.62740
$772$	0	0
$773$	−3.10469	−0.111668	−0.0558339	−	0.998440i	$-0.517782\pi$
$773$	−3.10469	−0.111668	−0.0558339	+	0.998440i	$0.517782\pi$
$774$	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
$793$	0	0
$794$	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
$819$	0	0
$820$	0	0
$821$	−25.9109	−0.904298	−0.452149	−	0.891942i	$-0.649342\pi$
$821$	−25.9109	−0.904298	−0.452149	+	0.891942i	$0.649342\pi$
$822$	0	0
$823$	−31.7995	−1.10846	−0.554230	−	0.832364i	$-0.686987\pi$
$823$	−31.7995	−1.10846	−0.554230	+	0.832364i	$0.686987\pi$
$824$	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
$845$	0	0
$846$	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	$-0.370556\pi$
$853$	23.1047	0.791089	0.395545	+	0.918447i	$0.370556\pi$
$854$	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	$-0.419627\pi$
$859$	14.6441	0.499651	0.249825	+	0.968291i	$0.419627\pi$
$860$	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
$871$	0	0
$872$	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	$-0.227639\pi$
$877$	44.7172	1.50999	0.754996	+	0.655729i	$0.227639\pi$
$878$	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	$-0.259010\pi$
$883$	40.8176	1.37362	0.686811	+	0.726836i	$0.259010\pi$
$884$	0	0
$885$	156.241	5.25197
$886$	0	0
$887$	−19.0145	−0.638443	−0.319222	−	0.947680i	$-0.603421\pi$
$887$	−19.0145	−0.638443	−0.319222	+	0.947680i	$0.603421\pi$
$888$	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	$-0.404735\pi$
$907$	17.7588	0.589673	0.294836	+	0.955548i	$0.404735\pi$
$908$	0	0
$909$	−36.2094	−1.20099
$910$	0	0
$911$	25.2439	0.836369	0.418184	−	0.908362i	$-0.362667\pi$
$911$	25.2439	0.836369	0.418184	+	0.908362i	$0.362667\pi$
$912$	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	$-0.422350\pi$
$919$	14.6441	0.483065	0.241532	+	0.970393i	$0.422350\pi$
$920$	0	0
$921$	−49.0156	−1.61512
$922$	0	0
$923$	0	0
$924$	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	$-0.528251\pi$
$929$	−5.40312	−0.177271	−0.0886354	+	0.996064i	$0.528251\pi$
$930$	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	$-0.588806\pi$
$941$	−16.8953	−0.550771	−0.275386	+	0.961334i	$0.588806\pi$
$942$	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
$949$	0	0
$950$	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	$-0.409758\pi$
$971$	17.4328	0.559444	0.279722	+	0.960081i	$0.409758\pi$
$972$	0	0
$973$	65.5234	2.10058
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	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	$-0.607567\pi$
$991$	−20.8736	−0.663071	−0.331536	+	0.943443i	$0.607567\pi$
$992$	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	5408.2.a.bj.1.1		4
4.3	odd	2	inner	5408.2.a.bj.1.4		4
13.12	even	2		416.2.a.f.1.1	✓	4
39.38	odd	2		3744.2.a.be.1.3		4
52.51	odd	2		416.2.a.f.1.4	yes	4
104.51	odd	2		832.2.a.p.1.1		4
104.77	even	2		832.2.a.p.1.4		4
156.155	even	2		3744.2.a.be.1.4		4
208.51	odd	4		3328.2.b.bb.1665.8		8
208.77	even	4		3328.2.b.bb.1665.2		8
208.155	odd	4		3328.2.b.bb.1665.1		8
208.181	even	4		3328.2.b.bb.1665.7		8
312.77	odd	2		7488.2.a.da.1.1		4
312.155	even	2		7488.2.a.da.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.a.f.1.1	✓	4	13.12	even	2
416.2.a.f.1.4	yes	4	52.51	odd	2
832.2.a.p.1.1		4	104.51	odd	2
832.2.a.p.1.4		4	104.77	even	2
3328.2.b.bb.1665.1		8	208.155	odd	4
3328.2.b.bb.1665.2		8	208.77	even	4
3328.2.b.bb.1665.7		8	208.181	even	4
3328.2.b.bb.1665.8		8	208.51	odd	4
3744.2.a.be.1.3		4	39.38	odd	2
3744.2.a.be.1.4		4	156.155	even	2
5408.2.a.bj.1.1		4	1.1	even	1	trivial
5408.2.a.bj.1.4		4	4.3	odd	2	inner
7488.2.a.da.1.1		4	312.77	odd	2
7488.2.a.da.1.2		4	312.155	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5408 = 2^{5} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5408.a (trivial)

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