LMFDB - Embedded newform 5408.2.a.w.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 416)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.31662$ 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)

$f(q)$	$=$	$q-3.31662 q^{3} +3.31662 q^{7} +8.00000 q^{9} +3.31662 q^{11} -3.00000 q^{17} +3.31662 q^{19} -11.0000 q^{21} +3.31662 q^{23} -5.00000 q^{25} -16.5831 q^{27} -5.00000 q^{29} -11.0000 q^{33} -9.00000 q^{37} +3.00000 q^{41} -9.94987 q^{43} -6.63325 q^{47} +4.00000 q^{49} +9.94987 q^{51} -8.00000 q^{53} -11.0000 q^{57} +3.31662 q^{59} -9.00000 q^{61} +26.5330 q^{63} +9.94987 q^{67} -11.0000 q^{69} +9.94987 q^{71} +4.00000 q^{73} +16.5831 q^{75} +11.0000 q^{77} -6.63325 q^{79} +31.0000 q^{81} -13.2665 q^{83} +16.5831 q^{87} +1.00000 q^{89} -7.00000 q^{97} +26.5330 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{9} - 6 q^{17} - 22 q^{21} - 10 q^{25} - 10 q^{29} - 22 q^{33} - 18 q^{37} + 6 q^{41} + 8 q^{49} - 16 q^{53} - 22 q^{57} - 18 q^{61} - 22 q^{69} + 8 q^{73} + 22 q^{77} + 62 q^{81} + 2 q^{89}+ \cdots - 14 q^{97}+O(q^{100}) $ 2 * q + 16 * q^9 - 6 * q^17 - 22 * q^21 - 10 * q^25 - 10 * q^29 - 22 * q^33 - 18 * q^37 + 6 * q^41 + 8 * q^49 - 16 * q^53 - 22 * q^57 - 18 * q^61 - 22 * q^69 + 8 * q^73 + 22 * q^77 + 62 * q^81 + 2 * q^89 - 14 * 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
$3$	−3.31662	−1.91485	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	−3.31662	−1.91485	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	3.31662	1.25357	0.626783	−	0.779194i	$-0.284371\pi$
$7$	3.31662	1.25357	0.626783	+	0.779194i	$0.284371\pi$
$8$	0	0
$9$	8.00000	2.66667
$10$	0	0
$11$	3.31662	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$11$	3.31662	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	3.31662	0.760886	0.380443	−	0.924804i	$-0.375772\pi$
$19$	3.31662	0.760886	0.380443	+	0.924804i	$0.375772\pi$
$20$	0	0
$21$	−11.0000	−2.40040
$22$	0	0
$23$	3.31662	0.691564	0.345782	−	0.938315i	$-0.387614\pi$
$23$	3.31662	0.691564	0.345782	+	0.938315i	$0.387614\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−16.5831	−3.19142
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−11.0000	−1.91485
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	−9.94987	−1.51734	−0.758671	−	0.651474i	$-0.774151\pi$
$43$	−9.94987	−1.51734	−0.758671	+	0.651474i	$0.774151\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.63325	−0.967559	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	−6.63325	−0.967559	−0.483779	+	0.875190i	$0.660736\pi$
$48$	0	0
$49$	4.00000	0.571429
$50$	0	0
$51$	9.94987	1.39326
$52$	0	0
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−11.0000	−1.45699
$58$	0	0
$59$	3.31662	0.431788	0.215894	−	0.976417i	$-0.430733\pi$
$59$	3.31662	0.431788	0.215894	+	0.976417i	$0.430733\pi$
$60$	0	0
$61$	−9.00000	−1.15233	−0.576166	−	0.817333i	$-0.695452\pi$
$61$	−9.00000	−1.15233	−0.576166	+	0.817333i	$0.695452\pi$
$62$	0	0
$63$	26.5330	3.34284
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.94987	1.21557	0.607785	−	0.794101i	$-0.292058\pi$
$67$	9.94987	1.21557	0.607785	+	0.794101i	$0.292058\pi$
$68$	0	0
$69$	−11.0000	−1.32424
$70$	0	0
$71$	9.94987	1.18083	0.590416	−	0.807099i	$-0.298964\pi$
$71$	9.94987	1.18083	0.590416	+	0.807099i	$0.298964\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	16.5831	1.91485
$76$	0	0
$77$	11.0000	1.25357
$78$	0	0
$79$	−6.63325	−0.746299	−0.373149	−	0.927771i	$-0.621722\pi$
$79$	−6.63325	−0.746299	−0.373149	+	0.927771i	$0.621722\pi$
$80$	0	0
$81$	31.0000	3.44444
$82$	0	0
$83$	−13.2665	−1.45619	−0.728094	−	0.685478i	$-0.759593\pi$
$83$	−13.2665	−1.45619	−0.728094	+	0.685478i	$0.759593\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	16.5831	1.77790
$88$	0	0
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	0	0
$99$	26.5330	2.66667
$100$	0	0
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.31662	−0.320630	−0.160315	−	0.987066i	$-0.551251\pi$
$107$	−3.31662	−0.320630	−0.160315	+	0.987066i	$0.551251\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	29.8496	2.83320
$112$	0	0
$113$	−11.0000	−1.03479	−0.517396	−	0.855746i	$-0.673099\pi$
$113$	−11.0000	−1.03479	−0.517396	+	0.855746i	$0.673099\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.94987	−0.912103
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−9.94987	−0.897150
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.94987	0.882909	0.441454	−	0.897284i	$-0.354463\pi$
$127$	9.94987	0.882909	0.441454	+	0.897284i	$0.354463\pi$
$128$	0	0
$129$	33.0000	2.90549
$130$	0	0
$131$	−13.2665	−1.15910	−0.579550	−	0.814937i	$-0.696772\pi$
$131$	−13.2665	−1.15910	−0.579550	+	0.814937i	$0.696772\pi$
$132$	0	0
$133$	11.0000	0.953821
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	−3.31662	−0.281312	−0.140656	−	0.990058i	$-0.544921\pi$
$139$	−3.31662	−0.281312	−0.140656	+	0.990058i	$0.544921\pi$
$140$	0	0
$141$	22.0000	1.85273
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−13.2665	−1.09420
$148$	0	0
$149$	1.00000	0.0819232	0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000	0.0819232	0.0409616	+	0.999161i	$0.486958\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−24.0000	−1.94029
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	0	0
$159$	26.5330	2.10420
$160$	0	0
$161$	11.0000	0.866921
$162$	0	0
$163$	−16.5831	−1.29889	−0.649445	−	0.760408i	$-0.724999\pi$
$163$	−16.5831	−1.29889	−0.649445	+	0.760408i	$0.724999\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.31662	−0.256648	−0.128324	−	0.991732i	$-0.540960\pi$
$167$	−3.31662	−0.256648	−0.128324	+	0.991732i	$0.540960\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	26.5330	2.02903
$172$	0	0
$173$	7.00000	0.532200	0.266100	−	0.963945i	$-0.414265\pi$
$173$	7.00000	0.532200	0.266100	+	0.963945i	$0.414265\pi$
$174$	0	0
$175$	−16.5831	−1.25357
$176$	0	0
$177$	−11.0000	−0.826811
$178$	0	0
$179$	−3.31662	−0.247896	−0.123948	−	0.992289i	$-0.539556\pi$
$179$	−3.31662	−0.247896	−0.123948	+	0.992289i	$0.539556\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	29.8496	2.20655
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−9.94987	−0.727607
$188$	0	0
$189$	−55.0000	−4.00066
$190$	0	0
$191$	23.2164	1.67988	0.839939	−	0.542681i	$-0.182591\pi$
$191$	23.2164	1.67988	0.839939	+	0.542681i	$0.182591\pi$
$192$	0	0
$193$	−13.0000	−0.935760	−0.467880	−	0.883792i	$-0.654982\pi$
$193$	−13.0000	−0.935760	−0.467880	+	0.883792i	$0.654982\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.0000	0.926212	0.463106	−	0.886303i	$-0.346735\pi$
$197$	13.0000	0.926212	0.463106	+	0.886303i	$0.346735\pi$
$198$	0	0
$199$	−9.94987	−0.705328	−0.352664	−	0.935750i	$-0.614724\pi$
$199$	−9.94987	−0.705328	−0.352664	+	0.935750i	$0.614724\pi$
$200$	0	0
$201$	−33.0000	−2.32764
$202$	0	0
$203$	−16.5831	−1.16391
$204$	0	0
$205$	0	0
$206$	0	0
$207$	26.5330	1.84417
$208$	0	0
$209$	11.0000	0.760886
$210$	0	0
$211$	−3.31662	−0.228326	−0.114163	−	0.993462i	$-0.536419\pi$
$211$	−3.31662	−0.228326	−0.114163	+	0.993462i	$0.536419\pi$
$212$	0	0
$213$	−33.0000	−2.26112
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−13.2665	−0.896467
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−3.31662	−0.222098	−0.111049	−	0.993815i	$-0.535421\pi$
$223$	−3.31662	−0.222098	−0.111049	+	0.993815i	$0.535421\pi$
$224$	0	0
$225$	−40.0000	−2.66667
$226$	0	0
$227$	3.31662	0.220132	0.110066	−	0.993924i	$-0.464894\pi$
$227$	3.31662	0.220132	0.110066	+	0.993924i	$0.464894\pi$
$228$	0	0
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	0	0
$231$	−36.4829	−2.40040
$232$	0	0
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	22.0000	1.42905
$238$	0	0
$239$	26.5330	1.71628	0.858138	−	0.513418i	$-0.171621\pi$
$239$	26.5330	1.71628	0.858138	+	0.513418i	$0.171621\pi$
$240$	0	0
$241$	−13.0000	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$242$	0	0
$243$	−53.0660	−3.40419
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	44.0000	2.78839
$250$	0	0
$251$	16.5831	1.04672	0.523359	−	0.852112i	$-0.324679\pi$
$251$	16.5831	1.04672	0.523359	+	0.852112i	$0.324679\pi$
$252$	0	0
$253$	11.0000	0.691564
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	0	0
$259$	−29.8496	−1.85477
$260$	0	0
$261$	−40.0000	−2.47594
$262$	0	0
$263$	−23.2164	−1.43158	−0.715791	−	0.698314i	$-0.753934\pi$
$263$	−23.2164	−1.43158	−0.715791	+	0.698314i	$0.753934\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.31662	−0.202974
$268$	0	0
$269$	29.0000	1.76816	0.884081	−	0.467334i	$-0.154786\pi$
$269$	29.0000	1.76816	0.884081	+	0.467334i	$0.154786\pi$
$270$	0	0
$271$	−29.8496	−1.81324	−0.906618	−	0.421953i	$-0.861345\pi$
$271$	−29.8496	−1.81324	−0.906618	+	0.421953i	$0.861345\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−16.5831	−1.00000
$276$	0	0
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	23.2164	1.38007	0.690035	−	0.723776i	$-0.257595\pi$
$283$	23.2164	1.38007	0.690035	+	0.723776i	$0.257595\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.94987	0.587323
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	23.2164	1.36097
$292$	0	0
$293$	−7.00000	−0.408944	−0.204472	−	0.978872i	$-0.565548\pi$
$293$	−7.00000	−0.408944	−0.204472	+	0.978872i	$0.565548\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−55.0000	−3.19142
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−33.0000	−1.90209
$302$	0	0
$303$	49.7494	2.85803
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.2665	0.757159	0.378580	−	0.925569i	$-0.376413\pi$
$307$	13.2665	0.757159	0.378580	+	0.925569i	$0.376413\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.5330	−1.50455	−0.752274	−	0.658850i	$-0.771043\pi$
$311$	−26.5330	−1.50455	−0.752274	+	0.658850i	$0.771043\pi$
$312$	0	0
$313$	4.00000	0.226093	0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000	0.226093	0.113047	+	0.993590i	$0.463939\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−16.5831	−0.928477
$320$	0	0
$321$	11.0000	0.613960
$322$	0	0
$323$	−9.94987	−0.553626
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−53.0660	−2.93456
$328$	0	0
$329$	−22.0000	−1.21290
$330$	0	0
$331$	9.94987	0.546895	0.273447	−	0.961887i	$-0.411836\pi$
$331$	9.94987	0.546895	0.273447	+	0.961887i	$0.411836\pi$
$332$	0	0
$333$	−72.0000	−3.94558
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	0	0
$339$	36.4829	1.98148
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−9.94987	−0.537243
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.5831	0.890229	0.445114	−	0.895474i	$-0.353163\pi$
$347$	16.5831	0.890229	0.445114	+	0.895474i	$0.353163\pi$
$348$	0	0
$349$	−1.00000	−0.0535288	−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000	−0.0535288	−0.0267644	+	0.999642i	$0.508520\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−31.0000	−1.64996	−0.824982	−	0.565159i	$-0.808815\pi$
$353$	−31.0000	−1.64996	−0.824982	+	0.565159i	$0.808815\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	33.0000	1.74655
$358$	0	0
$359$	−26.5330	−1.40036	−0.700179	−	0.713967i	$-0.746897\pi$
$359$	−26.5330	−1.40036	−0.700179	+	0.713967i	$0.746897\pi$
$360$	0	0
$361$	−8.00000	−0.421053
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−23.2164	−1.21188	−0.605942	−	0.795509i	$-0.707204\pi$
$367$	−23.2164	−1.21188	−0.605942	+	0.795509i	$0.707204\pi$
$368$	0	0
$369$	24.0000	1.24939
$370$	0	0
$371$	−26.5330	−1.37752
$372$	0	0
$373$	13.0000	0.673114	0.336557	−	0.941663i	$-0.390737\pi$
$373$	13.0000	0.673114	0.336557	+	0.941663i	$0.390737\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	29.8496	1.53327	0.766636	−	0.642082i	$-0.221929\pi$
$379$	29.8496	1.53327	0.766636	+	0.642082i	$0.221929\pi$
$380$	0	0
$381$	−33.0000	−1.69064
$382$	0	0
$383$	3.31662	0.169472	0.0847358	−	0.996403i	$-0.472995\pi$
$383$	3.31662	0.169472	0.0847358	+	0.996403i	$0.472995\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−79.5990	−4.04624
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−9.94987	−0.503187
$392$	0	0
$393$	44.0000	2.21951
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5.00000	−0.250943	−0.125471	−	0.992097i	$-0.540044\pi$
$397$	−5.00000	−0.250943	−0.125471	+	0.992097i	$0.540044\pi$
$398$	0	0
$399$	−36.4829	−1.82643
$400$	0	0
$401$	11.0000	0.549314	0.274657	−	0.961542i	$-0.411436\pi$
$401$	11.0000	0.549314	0.274657	+	0.961542i	$0.411436\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−29.8496	−1.47959
$408$	0	0
$409$	−13.0000	−0.642809	−0.321404	−	0.946942i	$-0.604155\pi$
$409$	−13.0000	−0.642809	−0.321404	+	0.946942i	$0.604155\pi$
$410$	0	0
$411$	−29.8496	−1.47237
$412$	0	0
$413$	11.0000	0.541275
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.0000	0.538672
$418$	0	0
$419$	16.5831	0.810139	0.405069	−	0.914286i	$-0.367247\pi$
$419$	16.5831	0.810139	0.405069	+	0.914286i	$0.367247\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	−53.0660	−2.58016
$424$	0	0
$425$	15.0000	0.727607
$426$	0	0
$427$	−29.8496	−1.44452
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.5831	0.798781	0.399390	−	0.916781i	$-0.369222\pi$
$431$	16.5831	0.798781	0.399390	+	0.916781i	$0.369222\pi$
$432$	0	0
$433$	21.0000	1.00920	0.504598	−	0.863355i	$-0.331641\pi$
$433$	21.0000	1.00920	0.504598	+	0.863355i	$0.331641\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11.0000	0.526201
$438$	0	0
$439$	−16.5831	−0.791469	−0.395735	−	0.918365i	$-0.629510\pi$
$439$	−16.5831	−0.791469	−0.395735	+	0.918365i	$0.629510\pi$
$440$	0	0
$441$	32.0000	1.52381
$442$	0	0
$443$	13.2665	0.630310	0.315155	−	0.949040i	$-0.397943\pi$
$443$	13.2665	0.630310	0.315155	+	0.949040i	$0.397943\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.31662	−0.156871
$448$	0	0
$449$	3.00000	0.141579	0.0707894	−	0.997491i	$-0.477448\pi$
$449$	3.00000	0.141579	0.0707894	+	0.997491i	$0.477448\pi$
$450$	0	0
$451$	9.94987	0.468521
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	0	0
$459$	49.7494	2.32210
$460$	0	0
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	−26.5330	−1.23309	−0.616547	−	0.787318i	$-0.711469\pi$
$463$	−26.5330	−1.23309	−0.616547	+	0.787318i	$0.711469\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.8997	−0.920851	−0.460425	−	0.887698i	$-0.652303\pi$
$467$	−19.8997	−0.920851	−0.460425	+	0.887698i	$0.652303\pi$
$468$	0	0
$469$	33.0000	1.52380
$470$	0	0
$471$	26.5330	1.22258
$472$	0	0
$473$	−33.0000	−1.51734
$474$	0	0
$475$	−16.5831	−0.760886
$476$	0	0
$477$	−64.0000	−2.93036
$478$	0	0
$479$	−9.94987	−0.454621	−0.227311	−	0.973822i	$-0.572993\pi$
$479$	−9.94987	−0.454621	−0.227311	+	0.973822i	$0.572993\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−36.4829	−1.66003
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9.94987	0.450872	0.225436	−	0.974258i	$-0.427619\pi$
$487$	9.94987	0.450872	0.225436	+	0.974258i	$0.427619\pi$
$488$	0	0
$489$	55.0000	2.48719
$490$	0	0
$491$	16.5831	0.748386	0.374193	−	0.927351i	$-0.377920\pi$
$491$	16.5831	0.748386	0.374193	+	0.927351i	$0.377920\pi$
$492$	0	0
$493$	15.0000	0.675566
$494$	0	0
$495$	0	0
$496$	0	0
$497$	33.0000	1.48025
$498$	0	0
$499$	−13.2665	−0.593890	−0.296945	−	0.954895i	$-0.595968\pi$
$499$	−13.2665	−0.593890	−0.296945	+	0.954895i	$0.595968\pi$
$500$	0	0
$501$	11.0000	0.491444
$502$	0	0
$503$	−3.31662	−0.147881	−0.0739405	−	0.997263i	$-0.523557\pi$
$503$	−3.31662	−0.147881	−0.0739405	+	0.997263i	$0.523557\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	39.0000	1.72864	0.864322	−	0.502938i	$-0.167748\pi$
$509$	39.0000	1.72864	0.864322	+	0.502938i	$0.167748\pi$
$510$	0	0
$511$	13.2665	0.586875
$512$	0	0
$513$	−55.0000	−2.42831
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−22.0000	−0.967559
$518$	0	0
$519$	−23.2164	−1.01909
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	16.5831	0.725129	0.362565	−	0.931959i	$-0.381901\pi$
$523$	16.5831	0.725129	0.362565	+	0.931959i	$0.381901\pi$
$524$	0	0
$525$	55.0000	2.40040
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12.0000	−0.521739
$530$	0	0
$531$	26.5330	1.15143
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11.0000	0.474685
$538$	0	0
$539$	13.2665	0.571429
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	26.5330	1.13864
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13.2665	0.567235	0.283617	−	0.958938i	$-0.408465\pi$
$547$	13.2665	0.567235	0.283617	+	0.958938i	$0.408465\pi$
$548$	0	0
$549$	−72.0000	−3.07289
$550$	0	0
$551$	−16.5831	−0.706465
$552$	0	0
$553$	−22.0000	−0.935535
$554$	0	0
$555$	0	0
$556$	0	0
$557$	37.0000	1.56774	0.783870	−	0.620925i	$-0.213243\pi$
$557$	37.0000	1.56774	0.783870	+	0.620925i	$0.213243\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	33.0000	1.39326
$562$	0	0
$563$	−3.31662	−0.139779	−0.0698895	−	0.997555i	$-0.522265\pi$
$563$	−3.31662	−0.139779	−0.0698895	+	0.997555i	$0.522265\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	102.815	4.31784
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	−13.2665	−0.555186	−0.277593	−	0.960699i	$-0.589537\pi$
$571$	−13.2665	−0.555186	−0.277593	+	0.960699i	$0.589537\pi$
$572$	0	0
$573$	−77.0000	−3.21672
$574$	0	0
$575$	−16.5831	−0.691564
$576$	0	0
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	0	0
$579$	43.1161	1.79184
$580$	0	0
$581$	−44.0000	−1.82543
$582$	0	0
$583$	−26.5330	−1.09888
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−43.1161	−1.77959	−0.889796	−	0.456358i	$-0.849154\pi$
$587$	−43.1161	−1.77959	−0.889796	+	0.456358i	$0.849154\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−43.1161	−1.77356
$592$	0	0
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	33.0000	1.35060
$598$	0	0
$599$	33.1662	1.35514	0.677568	−	0.735460i	$-0.263034\pi$
$599$	33.1662	1.35514	0.677568	+	0.735460i	$0.263034\pi$
$600$	0	0
$601$	29.0000	1.18293	0.591467	−	0.806329i	$-0.298549\pi$
$601$	29.0000	1.18293	0.591467	+	0.806329i	$0.298549\pi$
$602$	0	0
$603$	79.5990	3.24152
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9.94987	−0.403853	−0.201926	−	0.979401i	$-0.564720\pi$
$607$	−9.94987	−0.403853	−0.201926	+	0.979401i	$0.564720\pi$
$608$	0	0
$609$	55.0000	2.22871
$610$	0	0
$611$	0	0
$612$	0	0
$613$	7.00000	0.282727	0.141364	−	0.989958i	$-0.454851\pi$
$613$	7.00000	0.282727	0.141364	+	0.989958i	$0.454851\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.0000	0.684394	0.342197	−	0.939628i	$-0.388829\pi$
$617$	17.0000	0.684394	0.342197	+	0.939628i	$0.388829\pi$
$618$	0	0
$619$	−39.7995	−1.59968	−0.799838	−	0.600215i	$-0.795082\pi$
$619$	−39.7995	−1.59968	−0.799838	+	0.600215i	$0.795082\pi$
$620$	0	0
$621$	−55.0000	−2.20707
$622$	0	0
$623$	3.31662	0.132878
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	−36.4829	−1.45699
$628$	0	0
$629$	27.0000	1.07656
$630$	0	0
$631$	9.94987	0.396098	0.198049	−	0.980192i	$-0.436539\pi$
$631$	9.94987	0.396098	0.198049	+	0.980192i	$0.436539\pi$
$632$	0	0
$633$	11.0000	0.437211
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	79.5990	3.14889
$640$	0	0
$641$	21.0000	0.829450	0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000	0.829450	0.414725	+	0.909947i	$0.363878\pi$
$642$	0	0
$643$	36.4829	1.43874	0.719372	−	0.694625i	$-0.244430\pi$
$643$	36.4829	1.43874	0.719372	+	0.694625i	$0.244430\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.94987	0.391170	0.195585	−	0.980687i	$-0.437339\pi$
$647$	9.94987	0.391170	0.195585	+	0.980687i	$0.437339\pi$
$648$	0	0
$649$	11.0000	0.431788
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15.0000	−0.586995	−0.293498	−	0.955960i	$-0.594819\pi$
$653$	−15.0000	−0.586995	−0.293498	+	0.955960i	$0.594819\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	32.0000	1.24844
$658$	0	0
$659$	43.1161	1.67957	0.839783	−	0.542922i	$-0.182682\pi$
$659$	43.1161	1.67957	0.839783	+	0.542922i	$0.182682\pi$
$660$	0	0
$661$	−17.0000	−0.661223	−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000	−0.661223	−0.330612	+	0.943767i	$0.607255\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−16.5831	−0.642101
$668$	0	0
$669$	11.0000	0.425285
$670$	0	0
$671$	−29.8496	−1.15233
$672$	0	0
$673$	−49.0000	−1.88881	−0.944406	−	0.328783i	$-0.893362\pi$
$673$	−49.0000	−1.88881	−0.944406	+	0.328783i	$0.893362\pi$
$674$	0	0
$675$	82.9156	3.19142
$676$	0	0
$677$	−24.0000	−0.922395	−0.461197	−	0.887298i	$-0.652580\pi$
$677$	−24.0000	−0.922395	−0.461197	+	0.887298i	$0.652580\pi$
$678$	0	0
$679$	−23.2164	−0.890963
$680$	0	0
$681$	−11.0000	−0.421521
$682$	0	0
$683$	29.8496	1.14216	0.571082	−	0.820893i	$-0.306524\pi$
$683$	29.8496	1.14216	0.571082	+	0.820893i	$0.306524\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	26.5330	1.01230
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−23.2164	−0.883192	−0.441596	−	0.897214i	$-0.645588\pi$
$691$	−23.2164	−0.883192	−0.441596	+	0.897214i	$0.645588\pi$
$692$	0	0
$693$	88.0000	3.34284
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−9.00000	−0.340899
$698$	0	0
$699$	−13.2665	−0.501785
$700$	0	0
$701$	−32.0000	−1.20862	−0.604312	−	0.796748i	$-0.706552\pi$
$701$	−32.0000	−1.20862	−0.604312	+	0.796748i	$0.706552\pi$
$702$	0	0
$703$	−29.8496	−1.12580
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−49.7494	−1.87102
$708$	0	0
$709$	−29.0000	−1.08912	−0.544559	−	0.838723i	$-0.683303\pi$
$709$	−29.0000	−1.08912	−0.544559	+	0.838723i	$0.683303\pi$
$710$	0	0
$711$	−53.0660	−1.99013
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−88.0000	−3.28642
$718$	0	0
$719$	−29.8496	−1.11320	−0.556602	−	0.830780i	$-0.687895\pi$
$719$	−29.8496	−1.11320	−0.556602	+	0.830780i	$0.687895\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	43.1161	1.60351
$724$	0	0
$725$	25.0000	0.928477
$726$	0	0
$727$	26.5330	0.984054	0.492027	−	0.870580i	$-0.336256\pi$
$727$	26.5330	0.984054	0.492027	+	0.870580i	$0.336256\pi$
$728$	0	0
$729$	83.0000	3.07407
$730$	0	0
$731$	29.8496	1.10403
$732$	0	0
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.0000	1.21557
$738$	0	0
$739$	9.94987	0.366012	0.183006	−	0.983112i	$-0.441417\pi$
$739$	9.94987	0.366012	0.183006	+	0.983112i	$0.441417\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	43.1161	1.58178	0.790889	−	0.611960i	$-0.209619\pi$
$743$	43.1161	1.58178	0.790889	+	0.611960i	$0.209619\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−106.132	−3.88317
$748$	0	0
$749$	−11.0000	−0.401931
$750$	0	0
$751$	29.8496	1.08923	0.544614	−	0.838687i	$-0.316676\pi$
$751$	29.8496	1.08923	0.544614	+	0.838687i	$0.316676\pi$
$752$	0	0
$753$	−55.0000	−2.00431
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−7.00000	−0.254419	−0.127210	−	0.991876i	$-0.540602\pi$
$757$	−7.00000	−0.254419	−0.127210	+	0.991876i	$0.540602\pi$
$758$	0	0
$759$	−36.4829	−1.32424
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	53.0660	1.92112
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	27.0000	0.973645	0.486822	−	0.873501i	$-0.338156\pi$
$769$	27.0000	0.973645	0.486822	+	0.873501i	$0.338156\pi$
$770$	0	0
$771$	−43.1161	−1.55279
$772$	0	0
$773$	−21.0000	−0.755318	−0.377659	−	0.925945i	$-0.623271\pi$
$773$	−21.0000	−0.755318	−0.377659	+	0.925945i	$0.623271\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	99.0000	3.55161
$778$	0	0
$779$	9.94987	0.356491
$780$	0	0
$781$	33.0000	1.18083
$782$	0	0
$783$	82.9156	2.96316
$784$	0	0
$785$	0	0
$786$	0	0
$787$	9.94987	0.354675	0.177337	−	0.984150i	$-0.443252\pi$
$787$	9.94987	0.354675	0.177337	+	0.984150i	$0.443252\pi$
$788$	0	0
$789$	77.0000	2.74127
$790$	0	0
$791$	−36.4829	−1.29718
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33.0000	−1.16892	−0.584460	−	0.811423i	$-0.698694\pi$
$797$	−33.0000	−1.16892	−0.584460	+	0.811423i	$0.698694\pi$
$798$	0	0
$799$	19.8997	0.704003
$800$	0	0
$801$	8.00000	0.282666
$802$	0	0
$803$	13.2665	0.468165
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−96.1821	−3.38577
$808$	0	0
$809$	−19.0000	−0.668004	−0.334002	−	0.942572i	$-0.608399\pi$
$809$	−19.0000	−0.668004	−0.334002	+	0.942572i	$0.608399\pi$
$810$	0	0
$811$	−6.63325	−0.232925	−0.116462	−	0.993195i	$-0.537155\pi$
$811$	−6.63325	−0.232925	−0.116462	+	0.993195i	$0.537155\pi$
$812$	0	0
$813$	99.0000	3.47208
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−33.0000	−1.15452
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.0000	−0.942306	−0.471153	−	0.882051i	$-0.656162\pi$
$821$	−27.0000	−0.942306	−0.471153	+	0.882051i	$0.656162\pi$
$822$	0	0
$823$	−36.4829	−1.27171	−0.635856	−	0.771807i	$-0.719353\pi$
$823$	−36.4829	−1.27171	−0.635856	+	0.771807i	$0.719353\pi$
$824$	0	0
$825$	55.0000	1.91485
$826$	0	0
$827$	−13.2665	−0.461321	−0.230661	−	0.973034i	$-0.574089\pi$
$827$	−13.2665	−0.461321	−0.230661	+	0.973034i	$0.574089\pi$
$828$	0	0
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	−16.5831	−0.575262
$832$	0	0
$833$	−12.0000	−0.415775
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.94987	0.343508	0.171754	−	0.985140i	$-0.445057\pi$
$839$	9.94987	0.343508	0.171754	+	0.985140i	$0.445057\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	66.3325	2.28461
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−77.0000	−2.64263
$850$	0	0
$851$	−29.8496	−1.02323
$852$	0	0
$853$	−56.0000	−1.91740	−0.958702	−	0.284413i	$-0.908201\pi$
$853$	−56.0000	−1.91740	−0.958702	+	0.284413i	$0.908201\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.0000	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$857$	28.0000	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$858$	0	0
$859$	13.2665	0.452647	0.226324	−	0.974052i	$-0.427329\pi$
$859$	13.2665	0.452647	0.226324	+	0.974052i	$0.427329\pi$
$860$	0	0
$861$	−33.0000	−1.12464
$862$	0	0
$863$	53.0660	1.80639	0.903194	−	0.429233i	$-0.141216\pi$
$863$	53.0660	1.80639	0.903194	+	0.429233i	$0.141216\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.5330	0.901108
$868$	0	0
$869$	−22.0000	−0.746299
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−56.0000	−1.89531
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−15.0000	−0.506514	−0.253257	−	0.967399i	$-0.581502\pi$
$877$	−15.0000	−0.506514	−0.253257	+	0.967399i	$0.581502\pi$
$878$	0	0
$879$	23.2164	0.783069
$880$	0	0
$881$	5.00000	0.168454	0.0842271	−	0.996447i	$-0.473158\pi$
$881$	5.00000	0.168454	0.0842271	+	0.996447i	$0.473158\pi$
$882$	0	0
$883$	−13.2665	−0.446453	−0.223227	−	0.974767i	$-0.571659\pi$
$883$	−13.2665	−0.446453	−0.223227	+	0.974767i	$0.571659\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−43.1161	−1.44770	−0.723849	−	0.689959i	$-0.757629\pi$
$887$	−43.1161	−1.44770	−0.723849	+	0.689959i	$0.757629\pi$
$888$	0	0
$889$	33.0000	1.10678
$890$	0	0
$891$	102.815	3.44444
$892$	0	0
$893$	−22.0000	−0.736202
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	109.449	3.64222
$904$	0	0
$905$	0	0
$906$	0	0
$907$	49.7494	1.65190	0.825950	−	0.563743i	$-0.190639\pi$
$907$	49.7494	1.65190	0.825950	+	0.563743i	$0.190639\pi$
$908$	0	0
$909$	−120.000	−3.98015
$910$	0	0
$911$	−26.5330	−0.879077	−0.439539	−	0.898224i	$-0.644858\pi$
$911$	−26.5330	−0.879077	−0.439539	+	0.898224i	$0.644858\pi$
$912$	0	0
$913$	−44.0000	−1.45619
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−44.0000	−1.45301
$918$	0	0
$919$	−3.31662	−0.109405	−0.0547027	−	0.998503i	$-0.517421\pi$
$919$	−3.31662	−0.109405	−0.0547027	+	0.998503i	$0.517421\pi$
$920$	0	0
$921$	−44.0000	−1.44985
$922$	0	0
$923$	0	0
$924$	0	0
$925$	45.0000	1.47959
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13.0000	−0.426516	−0.213258	−	0.976996i	$-0.568408\pi$
$929$	−13.0000	−0.426516	−0.213258	+	0.976996i	$0.568408\pi$
$930$	0	0
$931$	13.2665	0.434792
$932$	0	0
$933$	88.0000	2.88099
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	0	0
$939$	−13.2665	−0.432936
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	9.94987	0.324012
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.31662	0.107776	0.0538879	−	0.998547i	$-0.482839\pi$
$947$	3.31662	0.107776	0.0538879	+	0.998547i	$0.482839\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	59.6992	1.93588
$952$	0	0
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	55.0000	1.77790
$958$	0	0
$959$	29.8496	0.963895
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−26.5330	−0.855014
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−26.5330	−0.853244	−0.426622	−	0.904430i	$-0.640296\pi$
$967$	−26.5330	−0.853244	−0.426622	+	0.904430i	$0.640296\pi$
$968$	0	0
$969$	33.0000	1.06011
$970$	0	0
$971$	16.5831	0.532178	0.266089	−	0.963948i	$-0.414269\pi$
$971$	16.5831	0.532178	0.266089	+	0.963948i	$0.414269\pi$
$972$	0	0
$973$	−11.0000	−0.352644
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.00000	−0.223950	−0.111975	−	0.993711i	$-0.535718\pi$
$977$	−7.00000	−0.223950	−0.111975	+	0.993711i	$0.535718\pi$
$978$	0	0
$979$	3.31662	0.106000
$980$	0	0
$981$	128.000	4.08673
$982$	0	0
$983$	53.0660	1.69254	0.846271	−	0.532752i	$-0.178842\pi$
$983$	53.0660	1.69254	0.846271	+	0.532752i	$0.178842\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	72.9657	2.32253
$988$	0	0
$989$	−33.0000	−1.04934
$990$	0	0
$991$	−43.1161	−1.36963	−0.684814	−	0.728718i	$-0.740117\pi$
$991$	−43.1161	−1.36963	−0.684814	+	0.728718i	$0.740117\pi$
$992$	0	0
$993$	−33.0000	−1.04722
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−17.0000	−0.538395	−0.269198	−	0.963085i	$-0.586759\pi$
$997$	−17.0000	−0.538395	−0.269198	+	0.963085i	$0.586759\pi$
$998$	0	0
$999$	149.248	4.72200

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5408.2.a.w.1.1		2
4.3	odd	2	inner	5408.2.a.w.1.2		2
13.4	even	6		416.2.i.d.289.2	yes	4
13.10	even	6		416.2.i.d.321.2	yes	4
13.12	even	2		5408.2.a.x.1.1		2
52.23	odd	6		416.2.i.d.321.1	yes	4
52.43	odd	6		416.2.i.d.289.1	✓	4
52.51	odd	2		5408.2.a.x.1.2		2
104.43	odd	6		832.2.i.n.705.2		4
104.69	even	6		832.2.i.n.705.1		4
104.75	odd	6		832.2.i.n.321.2		4
104.101	even	6		832.2.i.n.321.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.i.d.289.1	✓	4	52.43	odd	6
416.2.i.d.289.2	yes	4	13.4	even	6
416.2.i.d.321.1	yes	4	52.23	odd	6
416.2.i.d.321.2	yes	4	13.10	even	6
832.2.i.n.321.1		4	104.101	even	6
832.2.i.n.321.2		4	104.75	odd	6
832.2.i.n.705.1		4	104.69	even	6
832.2.i.n.705.2		4	104.43	odd	6
5408.2.a.w.1.1		2	1.1	even	1	trivial
5408.2.a.w.1.2		2	4.3	odd	2	inner
5408.2.a.x.1.1		2	13.12	even	2
5408.2.a.x.1.2		2	52.51	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:	\( N \)	\(=\)	\( 5408 = 2^{5} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5408.a (trivial)

\(f(q)\)	\(=\)	\(q-3.31662 q^{3} +3.31662 q^{7} +8.00000 q^{9} +3.31662 q^{11} -3.00000 q^{17} +3.31662 q^{19} -11.0000 q^{21} +3.31662 q^{23} -5.00000 q^{25} -16.5831 q^{27} -5.00000 q^{29} -11.0000 q^{33} -9.00000 q^{37} +3.00000 q^{41} -9.94987 q^{43} -6.63325 q^{47} +4.00000 q^{49} +9.94987 q^{51} -8.00000 q^{53} -11.0000 q^{57} +3.31662 q^{59} -9.00000 q^{61} +26.5330 q^{63} +9.94987 q^{67} -11.0000 q^{69} +9.94987 q^{71} +4.00000 q^{73} +16.5831 q^{75} +11.0000 q^{77} -6.63325 q^{79} +31.0000 q^{81} -13.2665 q^{83} +16.5831 q^{87} +1.00000 q^{89} -7.00000 q^{97} +26.5330 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{9} - 6 q^{17} - 22 q^{21} - 10 q^{25} - 10 q^{29} - 22 q^{33} - 18 q^{37} + 6 q^{41} + 8 q^{49} - 16 q^{53} - 22 q^{57} - 18 q^{61} - 22 q^{69} + 8 q^{73} + 22 q^{77} + 62 q^{81} + 2 q^{89}+ \cdots - 14 q^{97}+O(q^{100}) \) 2 * q + 16 * q^9 - 6 * q^17 - 22 * q^21 - 10 * q^25 - 10 * q^29 - 22 * q^33 - 18 * q^37 + 6 * q^41 + 8 * q^49 - 16 * q^53 - 22 * q^57 - 18 * q^61 - 22 * q^69 + 8 * q^73 + 22 * q^77 + 62 * q^81 + 2 * q^89 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more