LMFDB - Embedded newform 3328.2.a.bi.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-2,0,-2,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13968.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 4 $ x^4 - 2x^3 - 7x^2 + 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 832)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.386509$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.386509 q^{3} +0.386509 q^{5} -4.17452 q^{7} -2.85061 q^{9} +3.78801 q^{11} +1.00000 q^{13} +0.149389 q^{15} -4.49843 q^{17} -5.25211 q^{19} -1.61349 q^{21} +6.11192 q^{23} -4.85061 q^{25} -2.26131 q^{27} +8.88494 q^{29} +4.44911 q^{31} +1.46410 q^{33} -1.61349 q^{35} +3.96254 q^{37} +0.386509 q^{39} -8.11192 q^{41} +12.7356 q^{43} -1.10179 q^{45} +7.40150 q^{47} +10.4266 q^{49} -1.73869 q^{51} -9.04013 q^{53} +1.46410 q^{55} -2.02999 q^{57} -4.56103 q^{59} -1.33891 q^{61} +11.8999 q^{63} +0.386509 q^{65} +6.67296 q^{67} +2.36231 q^{69} +4.59850 q^{71} +11.6880 q^{73} -1.87481 q^{75} -15.8131 q^{77} +2.57916 q^{79} +7.67781 q^{81} -2.67296 q^{83} -1.73869 q^{85} +3.43411 q^{87} +10.3490 q^{89} -4.17452 q^{91} +1.71962 q^{93} -2.02999 q^{95} +0.237120 q^{97} -10.7982 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} + 18 q^{15} - 2 q^{17} + 6 q^{19} - 10 q^{21} + 12 q^{23} - 2 q^{25} - 14 q^{27} + 16 q^{29} + 10 q^{31} - 8 q^{33} - 10 q^{35} - 14 q^{37} - 2 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 + 2 * q^11 + 4 * q^13 + 18 * q^15 - 2 * q^17 + 6 * q^19 - 10 * q^21 + 12 * q^23 - 2 * q^25 - 14 * q^27 + 16 * q^29 + 10 * q^31 - 8 * q^33 - 10 * q^35 - 14 * q^37 - 2 * q^39 - 20 * q^41 + 14 * q^43 - 20 * q^45 + 20 * q^47 - 2 * q^49 - 2 * q^51 + 4 * q^53 - 8 * q^55 + 4 * q^57 + 2 * q^59 + 22 * q^63 - 2 * q^65 - 6 * q^67 - 20 * q^69 + 28 * q^71 + 8 * q^73 - 16 * q^75 - 16 * q^77 + 16 * q^79 + 20 * q^81 + 22 * q^83 - 2 * q^85 + 12 * q^87 + 8 * q^89 + 36 * q^93 + 4 * q^95 - 20 * q^97 - 2 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.386509	0.223151	0.111576	−	0.993756i	$-0.464410\pi$
$3$	0.386509	0.223151	0.111576	+	0.993756i	$0.464410\pi$
$4$	0	0
$5$	0.386509	0.172852	0.0864261	−	0.996258i	$-0.472455\pi$
$5$	0.386509	0.172852	0.0864261	+	0.996258i	$0.472455\pi$
$6$	0	0
$7$	−4.17452	−1.57782	−0.788911	−	0.614508i	$-0.789355\pi$
$7$	−4.17452	−1.57782	−0.788911	+	0.614508i	$0.789355\pi$
$8$	0	0
$9$	−2.85061	−0.950204
$10$	0	0
$11$	3.78801	1.14213	0.571064	−	0.820905i	$-0.306531\pi$
$11$	3.78801	1.14213	0.571064	+	0.820905i	$0.306531\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.149389	0.0385721
$16$	0	0
$17$	−4.49843	−1.09103	−0.545515	−	0.838101i	$-0.683666\pi$
$17$	−4.49843	−1.09103	−0.545515	+	0.838101i	$0.683666\pi$
$18$	0	0
$19$	−5.25211	−1.20492	−0.602459	−	0.798150i	$-0.705812\pi$
$19$	−5.25211	−1.20492	−0.602459	+	0.798150i	$0.705812\pi$
$20$	0	0
$21$	−1.61349	−0.352093
$22$	0	0
$23$	6.11192	1.27442	0.637212	−	0.770688i	$-0.280087\pi$
$23$	6.11192	1.27442	0.637212	+	0.770688i	$0.280087\pi$
$24$	0	0
$25$	−4.85061	−0.970122
$26$	0	0
$27$	−2.26131	−0.435190
$28$	0	0
$29$	8.88494	1.64989	0.824946	−	0.565211i	$-0.191205\pi$
$29$	8.88494	1.64989	0.824946	+	0.565211i	$0.191205\pi$
$30$	0	0
$31$	4.44911	0.799083	0.399542	−	0.916715i	$-0.369169\pi$
$31$	4.44911	0.799083	0.399542	+	0.916715i	$0.369169\pi$
$32$	0	0
$33$	1.46410	0.254867
$34$	0	0
$35$	−1.61349	−0.272730
$36$	0	0
$37$	3.96254	0.651437	0.325718	−	0.945467i	$-0.394394\pi$
$37$	3.96254	0.651437	0.325718	+	0.945467i	$0.394394\pi$
$38$	0	0
$39$	0.386509	0.0618910
$40$	0	0
$41$	−8.11192	−1.26687	−0.633435	−	0.773796i	$-0.718356\pi$
$41$	−8.11192	−1.26687	−0.633435	+	0.773796i	$0.718356\pi$
$42$	0	0
$43$	12.7356	1.94215	0.971077	−	0.238767i	$-0.0767432\pi$
$43$	12.7356	1.94215	0.971077	+	0.238767i	$0.0767432\pi$
$44$	0	0
$45$	−1.10179	−0.164245
$46$	0	0
$47$	7.40150	1.07962	0.539810	−	0.841787i	$-0.318496\pi$
$47$	7.40150	1.07962	0.539810	+	0.841787i	$0.318496\pi$
$48$	0	0
$49$	10.4266	1.48952
$50$	0	0
$51$	−1.73869	−0.243465
$52$	0	0
$53$	−9.04013	−1.24176	−0.620879	−	0.783907i	$-0.713224\pi$
$53$	−9.04013	−1.24176	−0.620879	+	0.783907i	$0.713224\pi$
$54$	0	0
$55$	1.46410	0.197419
$56$	0	0
$57$	−2.02999	−0.268879
$58$	0	0
$59$	−4.56103	−0.593796	−0.296898	−	0.954909i	$-0.595952\pi$
$59$	−4.56103	−0.593796	−0.296898	+	0.954909i	$0.595952\pi$
$60$	0	0
$61$	−1.33891	−0.171429	−0.0857147	−	0.996320i	$-0.527317\pi$
$61$	−1.33891	−0.171429	−0.0857147	+	0.996320i	$0.527317\pi$
$62$	0	0
$63$	11.8999	1.49925
$64$	0	0
$65$	0.386509	0.0479406
$66$	0	0
$67$	6.67296	0.815231	0.407616	−	0.913154i	$-0.366360\pi$
$67$	6.67296	0.815231	0.407616	+	0.913154i	$0.366360\pi$
$68$	0	0
$69$	2.36231	0.284389
$70$	0	0
$71$	4.59850	0.545741	0.272871	−	0.962051i	$-0.412027\pi$
$71$	4.59850	0.545741	0.272871	+	0.962051i	$0.412027\pi$
$72$	0	0
$73$	11.6880	1.36797	0.683986	−	0.729495i	$-0.260245\pi$
$73$	11.6880	1.36797	0.683986	+	0.729495i	$0.260245\pi$
$74$	0	0
$75$	−1.87481	−0.216484
$76$	0	0
$77$	−15.8131	−1.80208
$78$	0	0
$79$	2.57916	0.290178	0.145089	−	0.989419i	$-0.453653\pi$
$79$	2.57916	0.290178	0.145089	+	0.989419i	$0.453653\pi$
$80$	0	0
$81$	7.67781	0.853090
$82$	0	0
$83$	−2.67296	−0.293395	−0.146698	−	0.989181i	$-0.546864\pi$
$83$	−2.67296	−0.293395	−0.146698	+	0.989181i	$0.546864\pi$
$84$	0	0
$85$	−1.73869	−0.188587
$86$	0	0
$87$	3.43411	0.368175
$88$	0	0
$89$	10.3490	1.09700	0.548498	−	0.836152i	$-0.315200\pi$
$89$	10.3490	1.09700	0.548498	+	0.836152i	$0.315200\pi$
$90$	0	0
$91$	−4.17452	−0.437609
$92$	0	0
$93$	1.71962	0.178316
$94$	0	0
$95$	−2.02999	−0.208273
$96$	0	0
$97$	0.237120	0.0240759	0.0120379	−	0.999928i	$-0.496168\pi$
$97$	0.237120	0.0240759	0.0120379	+	0.999928i	$0.496168\pi$
$98$	0	0
$99$	−10.7982	−1.08526
$100$	0	0
$101$	−7.15205	−0.711656	−0.355828	−	0.934551i	$-0.615801\pi$
$101$	−7.15205	−0.711656	−0.355828	+	0.934551i	$0.615801\pi$
$102$	0	0
$103$	10.1552	1.00062	0.500310	−	0.865846i	$-0.333219\pi$
$103$	10.1552	1.00062	0.500310	+	0.865846i	$0.333219\pi$
$104$	0	0
$105$	−0.623629	−0.0608600
$106$	0	0
$107$	−2.12519	−0.205450	−0.102725	−	0.994710i	$-0.532756\pi$
$107$	−2.12519	−0.205450	−0.102725	+	0.994710i	$0.532756\pi$
$108$	0	0
$109$	19.1146	1.83085	0.915423	−	0.402494i	$-0.131856\pi$
$109$	19.1146	1.83085	0.915423	+	0.402494i	$0.131856\pi$
$110$	0	0
$111$	1.53156	0.145369
$112$	0	0
$113$	−9.73121	−0.915435	−0.457718	−	0.889098i	$-0.651333\pi$
$113$	−9.73121	−0.915435	−0.457718	+	0.889098i	$0.651333\pi$
$114$	0	0
$115$	2.36231	0.220287
$116$	0	0
$117$	−2.85061	−0.263539
$118$	0	0
$119$	18.7788	1.72145
$120$	0	0
$121$	3.34904	0.304459
$122$	0	0
$123$	−3.13533	−0.282703
$124$	0	0
$125$	−3.80735	−0.340540
$126$	0	0
$127$	3.43411	0.304728	0.152364	−	0.988324i	$-0.451311\pi$
$127$	3.43411	0.304728	0.152364	+	0.988324i	$0.451311\pi$
$128$	0	0
$129$	4.92241	0.433394
$130$	0	0
$131$	13.4699	1.17687	0.588435	−	0.808544i	$-0.299744\pi$
$131$	13.4699	1.17687	0.588435	+	0.808544i	$0.299744\pi$
$132$	0	0
$133$	21.9251	1.90114
$134$	0	0
$135$	−0.874019	−0.0752235
$136$	0	0
$137$	0.497224	0.0424807	0.0212403	−	0.999774i	$-0.493238\pi$
$137$	0.497224	0.0424807	0.0212403	+	0.999774i	$0.493238\pi$
$138$	0	0
$139$	−1.45831	−0.123692	−0.0618459	−	0.998086i	$-0.519699\pi$
$139$	−1.45831	−0.123692	−0.0618459	+	0.998086i	$0.519699\pi$
$140$	0	0
$141$	2.86075	0.240919
$142$	0	0
$143$	3.78801	0.316770
$144$	0	0
$145$	3.43411	0.285187
$146$	0	0
$147$	4.02999	0.332388
$148$	0	0
$149$	9.15205	0.749765	0.374883	−	0.927072i	$-0.377683\pi$
$149$	9.15205	0.749765	0.374883	+	0.927072i	$0.377683\pi$
$150$	0	0
$151$	−2.32971	−0.189589	−0.0947945	−	0.995497i	$-0.530219\pi$
$151$	−2.32971	−0.189589	−0.0947945	+	0.995497i	$0.530219\pi$
$152$	0	0
$153$	12.8233	1.03670
$154$	0	0
$155$	1.71962	0.138123
$156$	0	0
$157$	−3.20400	−0.255707	−0.127853	−	0.991793i	$-0.540809\pi$
$157$	−3.20400	−0.255707	−0.127853	+	0.991793i	$0.540809\pi$
$158$	0	0
$159$	−3.49409	−0.277100
$160$	0	0
$161$	−25.5144	−2.01081
$162$	0	0
$163$	0.635960	0.0498122	0.0249061	−	0.999690i	$-0.492071\pi$
$163$	0.635960	0.0498122	0.0249061	+	0.999690i	$0.492071\pi$
$164$	0	0
$165$	0.565889	0.0440544
$166$	0	0
$167$	3.32704	0.257454	0.128727	−	0.991680i	$-0.458911\pi$
$167$	3.32704	0.257454	0.128727	+	0.991680i	$0.458911\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	14.9717	1.14492
$172$	0	0
$173$	−25.2473	−1.91951	−0.959757	−	0.280833i	$-0.909389\pi$
$173$	−25.2473	−1.91951	−0.959757	+	0.280833i	$0.909389\pi$
$174$	0	0
$175$	20.2490	1.53068
$176$	0	0
$177$	−1.76288	−0.132506
$178$	0	0
$179$	14.1177	1.05521	0.527604	−	0.849490i	$-0.323091\pi$
$179$	14.1177	1.05521	0.527604	+	0.849490i	$0.323091\pi$
$180$	0	0
$181$	−4.84168	−0.359879	−0.179940	−	0.983678i	$-0.557590\pi$
$181$	−4.84168	−0.359879	−0.179940	+	0.983678i	$0.557590\pi$
$182$	0	0
$183$	−0.517500	−0.0382547
$184$	0	0
$185$	1.53156	0.112602
$186$	0	0
$187$	−17.0401	−1.24610
$188$	0	0
$189$	9.43991	0.686652
$190$	0	0
$191$	6.75975	0.489118	0.244559	−	0.969634i	$-0.421357\pi$
$191$	6.75975	0.489118	0.244559	+	0.969634i	$0.421357\pi$
$192$	0	0
$193$	8.66810	0.623943	0.311972	−	0.950091i	$-0.399011\pi$
$193$	8.66810	0.623943	0.311972	+	0.950091i	$0.399011\pi$
$194$	0	0
$195$	0.149389	0.0106980
$196$	0	0
$197$	24.5717	1.75066	0.875330	−	0.483526i	$-0.160644\pi$
$197$	24.5717	1.75066	0.875330	+	0.483526i	$0.160644\pi$
$198$	0	0
$199$	22.8533	1.62003	0.810013	−	0.586412i	$-0.199460\pi$
$199$	22.8533	1.62003	0.810013	+	0.586412i	$0.199460\pi$
$200$	0	0
$201$	2.57916	0.181920
$202$	0	0
$203$	−37.0904	−2.60324
$204$	0	0
$205$	−3.13533	−0.218981
$206$	0	0
$207$	−17.4227	−1.21096
$208$	0	0
$209$	−19.8951	−1.37617
$210$	0	0
$211$	−8.38651	−0.577351	−0.288676	−	0.957427i	$-0.593215\pi$
$211$	−8.38651	−0.577351	−0.288676	+	0.957427i	$0.593215\pi$
$212$	0	0
$213$	1.77736	0.121783
$214$	0	0
$215$	4.92241	0.335705
$216$	0	0
$217$	−18.5729	−1.26081
$218$	0	0
$219$	4.51750	0.305264
$220$	0	0
$221$	−4.49843	−0.302597
$222$	0	0
$223$	−13.6819	−0.916207	−0.458103	−	0.888899i	$-0.651471\pi$
$223$	−13.6819	−0.916207	−0.458103	+	0.888899i	$0.651471\pi$
$224$	0	0
$225$	13.8272	0.921814
$226$	0	0
$227$	15.4892	1.02806	0.514028	−	0.857773i	$-0.328153\pi$
$227$	15.4892	1.02806	0.514028	+	0.857773i	$0.328153\pi$
$228$	0	0
$229$	−24.0128	−1.58681	−0.793405	−	0.608694i	$-0.791694\pi$
$229$	−24.0128	−1.58681	−0.793405	+	0.608694i	$0.791694\pi$
$230$	0	0
$231$	−6.11192	−0.402135
$232$	0	0
$233$	13.0027	0.851833	0.425916	−	0.904763i	$-0.359952\pi$
$233$	13.0027	0.851833	0.425916	+	0.904763i	$0.359952\pi$
$234$	0	0
$235$	2.86075	0.186615
$236$	0	0
$237$	0.996868	0.0647535
$238$	0	0
$239$	13.0341	0.843103	0.421552	−	0.906804i	$-0.361486\pi$
$239$	13.0341	0.843103	0.421552	+	0.906804i	$0.361486\pi$
$240$	0	0
$241$	−24.3693	−1.56977	−0.784883	−	0.619644i	$-0.787277\pi$
$241$	−24.3693	−1.56977	−0.784883	+	0.619644i	$0.787277\pi$
$242$	0	0
$243$	9.75149	0.625558
$244$	0	0
$245$	4.02999	0.257467
$246$	0	0
$247$	−5.25211	−0.334184
$248$	0	0
$249$	−1.03312	−0.0654714
$250$	0	0
$251$	−4.37904	−0.276402	−0.138201	−	0.990404i	$-0.544132\pi$
$251$	−4.37904	−0.276402	−0.138201	+	0.990404i	$0.544132\pi$
$252$	0	0
$253$	23.1521	1.45556
$254$	0	0
$255$	−0.672018	−0.0420834
$256$	0	0
$257$	−12.0058	−0.748901	−0.374450	−	0.927247i	$-0.622169\pi$
$257$	−12.0058	−0.748901	−0.374450	+	0.927247i	$0.622169\pi$
$258$	0	0
$259$	−16.5417	−1.02785
$260$	0	0
$261$	−25.3275	−1.56773
$262$	0	0
$263$	19.3089	1.19064	0.595319	−	0.803489i	$-0.297026\pi$
$263$	19.3089	1.19064	0.595319	+	0.803489i	$0.297026\pi$
$264$	0	0
$265$	−3.49409	−0.214640
$266$	0	0
$267$	4.00000	0.244796
$268$	0	0
$269$	−16.5862	−1.01128	−0.505638	−	0.862746i	$-0.668743\pi$
$269$	−16.5862	−1.01128	−0.505638	+	0.862746i	$0.668743\pi$
$270$	0	0
$271$	−32.6125	−1.98107	−0.990534	−	0.137265i	$-0.956169\pi$
$271$	−32.6125	−1.98107	−0.990534	+	0.137265i	$0.956169\pi$
$272$	0	0
$273$	−1.61349	−0.0976529
$274$	0	0
$275$	−18.3742	−1.10800
$276$	0	0
$277$	−9.72663	−0.584416	−0.292208	−	0.956355i	$-0.594390\pi$
$277$	−9.72663	−0.584416	−0.292208	+	0.956355i	$0.594390\pi$
$278$	0	0
$279$	−12.6827	−0.759292
$280$	0	0
$281$	−18.7967	−1.12132	−0.560660	−	0.828046i	$-0.689452\pi$
$281$	−18.7967	−1.12132	−0.560660	+	0.828046i	$0.689452\pi$
$282$	0	0
$283$	19.2207	1.14255	0.571277	−	0.820758i	$-0.306448\pi$
$283$	19.2207	1.14255	0.571277	+	0.820758i	$0.306448\pi$
$284$	0	0
$285$	−0.784610	−0.0464763
$286$	0	0
$287$	33.8634	1.99889
$288$	0	0
$289$	3.23591	0.190348
$290$	0	0
$291$	0.0916490	0.00537256
$292$	0	0
$293$	−11.6638	−0.681404	−0.340702	−	0.940171i	$-0.610665\pi$
$293$	−11.6638	−0.681404	−0.340702	+	0.940171i	$0.610665\pi$
$294$	0	0
$295$	−1.76288	−0.102639
$296$	0	0
$297$	−8.56589	−0.497043
$298$	0	0
$299$	6.11192	0.353462
$300$	0	0
$301$	−53.1649	−3.06437
$302$	0	0
$303$	−2.76433	−0.158807
$304$	0	0
$305$	−0.517500	−0.0296319
$306$	0	0
$307$	−9.29537	−0.530515	−0.265258	−	0.964178i	$-0.585457\pi$
$307$	−9.29537	−0.530515	−0.265258	+	0.964178i	$0.585457\pi$
$308$	0	0
$309$	3.92507	0.223290
$310$	0	0
$311$	14.7643	0.837209	0.418604	−	0.908169i	$-0.362519\pi$
$311$	14.7643	0.837209	0.418604	+	0.908169i	$0.362519\pi$
$312$	0	0
$313$	13.8806	0.784578	0.392289	−	0.919842i	$-0.371683\pi$
$313$	13.8806	0.784578	0.392289	+	0.919842i	$0.371683\pi$
$314$	0	0
$315$	4.59943	0.259149
$316$	0	0
$317$	30.3606	1.70522	0.852612	−	0.522545i	$-0.175017\pi$
$317$	30.3606	1.70522	0.852612	+	0.522545i	$0.175017\pi$
$318$	0	0
$319$	33.6563	1.88439
$320$	0	0
$321$	−0.821407	−0.0458465
$322$	0	0
$323$	23.6263	1.31460
$324$	0	0
$325$	−4.85061	−0.269063
$326$	0	0
$327$	7.38796	0.408555
$328$	0	0
$329$	−30.8977	−1.70345
$330$	0	0
$331$	−9.35077	−0.513965	−0.256982	−	0.966416i	$-0.582728\pi$
$331$	−9.35077	−0.513965	−0.256982	+	0.966416i	$0.582728\pi$
$332$	0	0
$333$	−11.2956	−0.618998
$334$	0	0
$335$	2.57916	0.140914
$336$	0	0
$337$	17.8806	0.974018	0.487009	−	0.873397i	$-0.338088\pi$
$337$	17.8806	0.974018	0.487009	+	0.873397i	$0.338088\pi$
$338$	0	0
$339$	−3.76120	−0.204280
$340$	0	0
$341$	16.8533	0.912656
$342$	0	0
$343$	−14.3046	−0.772374
$344$	0	0
$345$	0.913056	0.0491573
$346$	0	0
$347$	−2.84047	−0.152485	−0.0762423	−	0.997089i	$-0.524292\pi$
$347$	−2.84047	−0.152485	−0.0762423	+	0.997089i	$0.524292\pi$
$348$	0	0
$349$	19.3147	1.03389	0.516946	−	0.856018i	$-0.327069\pi$
$349$	19.3147	1.03389	0.516946	+	0.856018i	$0.327069\pi$
$350$	0	0
$351$	−2.26131	−0.120700
$352$	0	0
$353$	8.61783	0.458681	0.229340	−	0.973346i	$-0.426343\pi$
$353$	8.61783	0.458681	0.229340	+	0.973346i	$0.426343\pi$
$354$	0	0
$355$	1.77736	0.0943325
$356$	0	0
$357$	7.25818	0.384144
$358$	0	0
$359$	0.672957	0.0355173	0.0177586	−	0.999842i	$-0.494347\pi$
$359$	0.672957	0.0355173	0.0177586	+	0.999842i	$0.494347\pi$
$360$	0	0
$361$	8.58471	0.451827
$362$	0	0
$363$	1.29444	0.0679403
$364$	0	0
$365$	4.51750	0.236457
$366$	0	0
$367$	14.1189	0.737002	0.368501	−	0.929627i	$-0.379871\pi$
$367$	14.1189	0.737002	0.368501	+	0.929627i	$0.379871\pi$
$368$	0	0
$369$	23.1239	1.20378
$370$	0	0
$371$	37.7382	1.95927
$372$	0	0
$373$	−5.22698	−0.270643	−0.135321	−	0.990802i	$-0.543207\pi$
$373$	−5.22698	−0.270643	−0.135321	+	0.990802i	$0.543207\pi$
$374$	0	0
$375$	−1.47158	−0.0759918
$376$	0	0
$377$	8.88494	0.457598
$378$	0	0
$379$	34.5426	1.77434	0.887168	−	0.461447i	$-0.152669\pi$
$379$	34.5426	1.77434	0.887168	+	0.461447i	$0.152669\pi$
$380$	0	0
$381$	1.32732	0.0680004
$382$	0	0
$383$	−22.1296	−1.13077	−0.565384	−	0.824828i	$-0.691272\pi$
$383$	−22.1296	−1.13077	−0.565384	+	0.824828i	$0.691272\pi$
$384$	0	0
$385$	−6.11192	−0.311493
$386$	0	0
$387$	−36.3041	−1.84544
$388$	0	0
$389$	34.4426	1.74631	0.873154	−	0.487445i	$-0.162071\pi$
$389$	34.4426	1.74631	0.873154	+	0.487445i	$0.162071\pi$
$390$	0	0
$391$	−27.4941	−1.39044
$392$	0	0
$393$	5.20624	0.262620
$394$	0	0
$395$	0.996868	0.0501579
$396$	0	0
$397$	16.5042	0.828324	0.414162	−	0.910203i	$-0.364075\pi$
$397$	16.5042	0.828324	0.414162	+	0.910203i	$0.364075\pi$
$398$	0	0
$399$	8.47424	0.424243
$400$	0	0
$401$	19.5530	0.976432	0.488216	−	0.872723i	$-0.337648\pi$
$401$	19.5530	0.976432	0.488216	+	0.872723i	$0.337648\pi$
$402$	0	0
$403$	4.44911	0.221626
$404$	0	0
$405$	2.96754	0.147458
$406$	0	0
$407$	15.0101	0.744025
$408$	0	0
$409$	−3.36545	−0.166411	−0.0832053	−	0.996532i	$-0.526516\pi$
$409$	−3.36545	−0.166411	−0.0832053	+	0.996532i	$0.526516\pi$
$410$	0	0
$411$	0.192181	0.00947961
$412$	0	0
$413$	19.0401	0.936903
$414$	0	0
$415$	−1.03312	−0.0507140
$416$	0	0
$417$	−0.563648	−0.0276020
$418$	0	0
$419$	−28.0877	−1.37218	−0.686088	−	0.727519i	$-0.740673\pi$
$419$	−28.0877	−1.37218	−0.686088	+	0.727519i	$0.740673\pi$
$420$	0	0
$421$	−2.31784	−0.112965	−0.0564824	−	0.998404i	$-0.517988\pi$
$421$	−2.31784	−0.112965	−0.0564824	+	0.998404i	$0.517988\pi$
$422$	0	0
$423$	−21.0988	−1.02586
$424$	0	0
$425$	21.8202	1.05843
$426$	0	0
$427$	5.58930	0.270485
$428$	0	0
$429$	1.46410	0.0706875
$430$	0	0
$431$	−28.9591	−1.39491	−0.697456	−	0.716627i	$-0.745685\pi$
$431$	−28.9591	−1.39491	−0.697456	+	0.716627i	$0.745685\pi$
$432$	0	0
$433$	27.1578	1.30512	0.652561	−	0.757736i	$-0.273694\pi$
$433$	27.1578	1.30512	0.652561	+	0.757736i	$0.273694\pi$
$434$	0	0
$435$	1.32732	0.0636399
$436$	0	0
$437$	−32.1005	−1.53558
$438$	0	0
$439$	−19.3592	−0.923963	−0.461982	−	0.886889i	$-0.652861\pi$
$439$	−19.3592	−0.923963	−0.461982	+	0.886889i	$0.652861\pi$
$440$	0	0
$441$	−29.7223	−1.41535
$442$	0	0
$443$	6.21638	0.295349	0.147674	−	0.989036i	$-0.452821\pi$
$443$	6.21638	0.295349	0.147674	+	0.989036i	$0.452821\pi$
$444$	0	0
$445$	4.00000	0.189618
$446$	0	0
$447$	3.53735	0.167311
$448$	0	0
$449$	16.2971	0.769108	0.384554	−	0.923103i	$-0.374355\pi$
$449$	16.2971	0.769108	0.384554	+	0.923103i	$0.374355\pi$
$450$	0	0
$451$	−30.7281	−1.44693
$452$	0	0
$453$	−0.900453	−0.0423070
$454$	0	0
$455$	−1.61349	−0.0756416
$456$	0	0
$457$	−24.1552	−1.12993	−0.564966	−	0.825114i	$-0.691111\pi$
$457$	−24.1552	−1.12993	−0.564966	+	0.825114i	$0.691111\pi$
$458$	0	0
$459$	10.1724	0.474806
$460$	0	0
$461$	16.4102	0.764301	0.382150	−	0.924100i	$-0.375184\pi$
$461$	16.4102	0.764301	0.382150	+	0.924100i	$0.375184\pi$
$462$	0	0
$463$	18.7029	0.869200	0.434600	−	0.900624i	$-0.356890\pi$
$463$	18.7029	0.869200	0.434600	+	0.900624i	$0.356890\pi$
$464$	0	0
$465$	0.664649	0.0308224
$466$	0	0
$467$	−41.2410	−1.90841	−0.954203	−	0.299160i	$-0.903293\pi$
$467$	−41.2410	−1.90841	−0.954203	+	0.299160i	$0.903293\pi$
$468$	0	0
$469$	−27.8564	−1.28629
$470$	0	0
$471$	−1.23837	−0.0570613
$472$	0	0
$473$	48.2424	2.21819
$474$	0	0
$475$	25.4760	1.16892
$476$	0	0
$477$	25.7699	1.17992
$478$	0	0
$479$	28.7474	1.31350	0.656752	−	0.754107i	$-0.271930\pi$
$479$	28.7474	1.31350	0.656752	+	0.754107i	$0.271930\pi$
$480$	0	0
$481$	3.96254	0.180676
$482$	0	0
$483$	−9.86154	−0.448715
$484$	0	0
$485$	0.0916490	0.00416157
$486$	0	0
$487$	12.1625	0.551134	0.275567	−	0.961282i	$-0.411134\pi$
$487$	12.1625	0.551134	0.275567	+	0.961282i	$0.411134\pi$
$488$	0	0
$489$	0.245804	0.0111157
$490$	0	0
$491$	−32.4102	−1.46265	−0.731327	−	0.682027i	$-0.761099\pi$
$491$	−32.4102	−1.46265	−0.731327	+	0.682027i	$0.761099\pi$
$492$	0	0
$493$	−39.9683	−1.80008
$494$	0	0
$495$	−4.17358	−0.187589
$496$	0	0
$497$	−19.1965	−0.861082
$498$	0	0
$499$	21.1726	0.947816	0.473908	−	0.880574i	$-0.342843\pi$
$499$	21.1726	0.947816	0.473908	+	0.880574i	$0.342843\pi$
$500$	0	0
$501$	1.28593	0.0574512
$502$	0	0
$503$	15.5760	0.694501	0.347250	−	0.937772i	$-0.387115\pi$
$503$	15.5760	0.694501	0.347250	+	0.937772i	$0.387115\pi$
$504$	0	0
$505$	−2.76433	−0.123011
$506$	0	0
$507$	0.386509	0.0171655
$508$	0	0
$509$	4.35773	0.193153	0.0965765	−	0.995326i	$-0.469211\pi$
$509$	4.35773	0.193153	0.0965765	+	0.995326i	$0.469211\pi$
$510$	0	0
$511$	−48.7916	−2.15841
$512$	0	0
$513$	11.8767	0.524368
$514$	0	0
$515$	3.92507	0.172959
$516$	0	0
$517$	28.0370	1.23307
$518$	0	0
$519$	−9.75829	−0.428342
$520$	0	0
$521$	17.6568	0.773556	0.386778	−	0.922173i	$-0.373588\pi$
$521$	17.6568	0.773556	0.386778	+	0.922173i	$0.373588\pi$
$522$	0	0
$523$	−10.3490	−0.452532	−0.226266	−	0.974066i	$-0.572652\pi$
$523$	−10.3490	−0.452532	−0.226266	+	0.974066i	$0.572652\pi$
$524$	0	0
$525$	7.82642	0.341573
$526$	0	0
$527$	−20.0140	−0.871824
$528$	0	0
$529$	14.3556	0.624158
$530$	0	0
$531$	13.0017	0.564227
$532$	0	0
$533$	−8.11192	−0.351366
$534$	0	0
$535$	−0.821407	−0.0355125
$536$	0	0
$537$	5.45663	0.235471
$538$	0	0
$539$	39.4962	1.70122
$540$	0	0
$541$	20.0375	0.861478	0.430739	−	0.902477i	$-0.358253\pi$
$541$	20.0375	0.861478	0.430739	+	0.902477i	$0.358253\pi$
$542$	0	0
$543$	−1.87135	−0.0803075
$544$	0	0
$545$	7.38796	0.316466
$546$	0	0
$547$	29.1446	1.24613	0.623066	−	0.782169i	$-0.285887\pi$
$547$	29.1446	1.24613	0.623066	+	0.782169i	$0.285887\pi$
$548$	0	0
$549$	3.81670	0.162893
$550$	0	0
$551$	−46.6647	−1.98798
$552$	0	0
$553$	−10.7668	−0.457849
$554$	0	0
$555$	0.591960	0.0251273
$556$	0	0
$557$	−23.1411	−0.980521	−0.490261	−	0.871576i	$-0.663098\pi$
$557$	−23.1411	−0.980521	−0.490261	+	0.871576i	$0.663098\pi$
$558$	0	0
$559$	12.7356	0.538657
$560$	0	0
$561$	−6.58616	−0.278068
$562$	0	0
$563$	1.19578	0.0503962	0.0251981	−	0.999682i	$-0.491978\pi$
$563$	1.19578	0.0503962	0.0251981	+	0.999682i	$0.491978\pi$
$564$	0	0
$565$	−3.76120	−0.158235
$566$	0	0
$567$	−32.0512	−1.34602
$568$	0	0
$569$	−34.6087	−1.45087	−0.725436	−	0.688290i	$-0.758362\pi$
$569$	−34.6087	−1.45087	−0.725436	+	0.688290i	$0.758362\pi$
$570$	0	0
$571$	−27.6372	−1.15658	−0.578291	−	0.815831i	$-0.696280\pi$
$571$	−27.6372	−1.15658	−0.578291	+	0.815831i	$0.696280\pi$
$572$	0	0
$573$	2.61270	0.109147
$574$	0	0
$575$	−29.6466	−1.23635
$576$	0	0
$577$	−2.65241	−0.110421	−0.0552106	−	0.998475i	$-0.517583\pi$
$577$	−2.65241	−0.110421	−0.0552106	+	0.998475i	$0.517583\pi$
$578$	0	0
$579$	3.35030	0.139234
$580$	0	0
$581$	11.1583	0.462925
$582$	0	0
$583$	−34.2441	−1.41825
$584$	0	0
$585$	−1.10179	−0.0455533
$586$	0	0
$587$	−18.7302	−0.773079	−0.386540	−	0.922273i	$-0.626330\pi$
$587$	−18.7302	−0.773079	−0.386540	+	0.922273i	$0.626330\pi$
$588$	0	0
$589$	−23.3672	−0.962829
$590$	0	0
$591$	9.49718	0.390662
$592$	0	0
$593$	−4.74761	−0.194961	−0.0974806	−	0.995237i	$-0.531078\pi$
$593$	−4.74761	−0.194961	−0.0974806	+	0.995237i	$0.531078\pi$
$594$	0	0
$595$	7.25818	0.297556
$596$	0	0
$597$	8.83300	0.361511
$598$	0	0
$599$	−19.0217	−0.777207	−0.388603	−	0.921405i	$-0.627042\pi$
$599$	−19.0217	−0.777207	−0.388603	+	0.921405i	$0.627042\pi$
$600$	0	0
$601$	−34.0585	−1.38927	−0.694637	−	0.719360i	$-0.744435\pi$
$601$	−34.0585	−1.38927	−0.694637	+	0.719360i	$0.744435\pi$
$602$	0	0
$603$	−19.0220	−0.774636
$604$	0	0
$605$	1.29444	0.0526263
$606$	0	0
$607$	−40.2908	−1.63535	−0.817677	−	0.575677i	$-0.804739\pi$
$607$	−40.2908	−1.63535	−0.817677	+	0.575677i	$0.804739\pi$
$608$	0	0
$609$	−14.3358	−0.580915
$610$	0	0
$611$	7.40150	0.299433
$612$	0	0
$613$	1.01472	0.0409843	0.0204921	−	0.999790i	$-0.493477\pi$
$613$	1.01472	0.0409843	0.0204921	+	0.999790i	$0.493477\pi$
$614$	0	0
$615$	−1.21183	−0.0488659
$616$	0	0
$617$	8.74303	0.351981	0.175991	−	0.984392i	$-0.443687\pi$
$617$	8.74303	0.351981	0.175991	+	0.984392i	$0.443687\pi$
$618$	0	0
$619$	−14.1433	−0.568468	−0.284234	−	0.958755i	$-0.591739\pi$
$619$	−14.1433	−0.568468	−0.284234	+	0.958755i	$0.591739\pi$
$620$	0	0
$621$	−13.8210	−0.554617
$622$	0	0
$623$	−43.2023	−1.73086
$624$	0	0
$625$	22.7815	0.911259
$626$	0	0
$627$	−7.68963	−0.307094
$628$	0	0
$629$	−17.8252	−0.710737
$630$	0	0
$631$	−22.2697	−0.886544	−0.443272	−	0.896387i	$-0.646182\pi$
$631$	−22.2697	−0.886544	−0.443272	+	0.896387i	$0.646182\pi$
$632$	0	0
$633$	−3.24146	−0.128837
$634$	0	0
$635$	1.32732	0.0526729
$636$	0	0
$637$	10.4266	0.413118
$638$	0	0
$639$	−13.1085	−0.518565
$640$	0	0
$641$	−41.8317	−1.65225	−0.826127	−	0.563484i	$-0.809461\pi$
$641$	−41.8317	−1.65225	−0.826127	+	0.563484i	$0.809461\pi$
$642$	0	0
$643$	10.7692	0.424695	0.212348	−	0.977194i	$-0.431889\pi$
$643$	10.7692	0.424695	0.212348	+	0.977194i	$0.431889\pi$
$644$	0	0
$645$	1.90256	0.0749130
$646$	0	0
$647$	28.6548	1.12654	0.563269	−	0.826274i	$-0.309544\pi$
$647$	28.6548	1.12654	0.563269	+	0.826274i	$0.309544\pi$
$648$	0	0
$649$	−17.2772	−0.678191
$650$	0	0
$651$	−7.17859	−0.281351
$652$	0	0
$653$	−11.9048	−0.465871	−0.232935	−	0.972492i	$-0.574833\pi$
$653$	−11.9048	−0.465871	−0.232935	+	0.972492i	$0.574833\pi$
$654$	0	0
$655$	5.20624	0.203425
$656$	0	0
$657$	−33.3178	−1.29985
$658$	0	0
$659$	−33.6915	−1.31243	−0.656217	−	0.754572i	$-0.727845\pi$
$659$	−33.6915	−1.31243	−0.656217	+	0.754572i	$0.727845\pi$
$660$	0	0
$661$	−11.5963	−0.451044	−0.225522	−	0.974238i	$-0.572409\pi$
$661$	−11.5963	−0.451044	−0.225522	+	0.974238i	$0.572409\pi$
$662$	0	0
$663$	−1.73869	−0.0675250
$664$	0	0
$665$	8.47424	0.328617
$666$	0	0
$667$	54.3041	2.10266
$668$	0	0
$669$	−5.28817	−0.204453
$670$	0	0
$671$	−5.07180	−0.195795
$672$	0	0
$673$	−18.4414	−0.710862	−0.355431	−	0.934702i	$-0.615666\pi$
$673$	−18.4414	−0.710862	−0.355431	+	0.934702i	$0.615666\pi$
$674$	0	0
$675$	10.9688	0.422188
$676$	0	0
$677$	9.47111	0.364004	0.182002	−	0.983298i	$-0.441742\pi$
$677$	9.47111	0.364004	0.182002	+	0.983298i	$0.441742\pi$
$678$	0	0
$679$	−0.989862	−0.0379874
$680$	0	0
$681$	5.98673	0.229412
$682$	0	0
$683$	−5.61087	−0.214694	−0.107347	−	0.994222i	$-0.534236\pi$
$683$	−5.61087	−0.214694	−0.107347	+	0.994222i	$0.534236\pi$
$684$	0	0
$685$	0.192181	0.00734288
$686$	0	0
$687$	−9.28117	−0.354099
$688$	0	0
$689$	−9.04013	−0.344401
$690$	0	0
$691$	3.96618	0.150881	0.0754403	−	0.997150i	$-0.475964\pi$
$691$	3.96618	0.150881	0.0754403	+	0.997150i	$0.475964\pi$
$692$	0	0
$693$	45.0771	1.71234
$694$	0	0
$695$	−0.563648	−0.0213804
$696$	0	0
$697$	36.4910	1.38219
$698$	0	0
$699$	5.02565	0.190087
$700$	0	0
$701$	−15.5893	−0.588800	−0.294400	−	0.955682i	$-0.595120\pi$
$701$	−15.5893	−0.588800	−0.294400	+	0.955682i	$0.595120\pi$
$702$	0	0
$703$	−20.8117	−0.784928
$704$	0	0
$705$	1.10571	0.0416433
$706$	0	0
$707$	29.8564	1.12287
$708$	0	0
$709$	−1.82642	−0.0685925	−0.0342963	−	0.999412i	$-0.510919\pi$
$709$	−1.82642	−0.0685925	−0.0342963	+	0.999412i	$0.510919\pi$
$710$	0	0
$711$	−7.35218	−0.275728
$712$	0	0
$713$	27.1926	1.01837
$714$	0	0
$715$	1.46410	0.0547543
$716$	0	0
$717$	5.03778	0.188139
$718$	0	0
$719$	43.9991	1.64089	0.820444	−	0.571727i	$-0.193726\pi$
$719$	43.9991	1.64089	0.820444	+	0.571727i	$0.193726\pi$
$720$	0	0
$721$	−42.3930	−1.57880
$722$	0	0
$723$	−9.41896	−0.350295
$724$	0	0
$725$	−43.0974	−1.60060
$726$	0	0
$727$	33.4878	1.24199	0.620997	−	0.783813i	$-0.286728\pi$
$727$	33.4878	1.24199	0.620997	+	0.783813i	$0.286728\pi$
$728$	0	0
$729$	−19.2644	−0.713496
$730$	0	0
$731$	−57.2901	−2.11895
$732$	0	0
$733$	10.4817	0.387151	0.193575	−	0.981085i	$-0.437992\pi$
$733$	10.4817	0.387151	0.193575	+	0.981085i	$0.437992\pi$
$734$	0	0
$735$	1.55763	0.0574540
$736$	0	0
$737$	25.2772	0.931099
$738$	0	0
$739$	−36.8854	−1.35685	−0.678426	−	0.734669i	$-0.737338\pi$
$739$	−36.8854	−1.35685	−0.678426	+	0.734669i	$0.737338\pi$
$740$	0	0
$741$	−2.02999	−0.0745736
$742$	0	0
$743$	30.3800	1.11453	0.557267	−	0.830334i	$-0.311850\pi$
$743$	30.3800	1.11453	0.557267	+	0.830334i	$0.311850\pi$
$744$	0	0
$745$	3.53735	0.129599
$746$	0	0
$747$	7.61956	0.278785
$748$	0	0
$749$	8.87167	0.324164
$750$	0	0
$751$	−10.0230	−0.365744	−0.182872	−	0.983137i	$-0.558539\pi$
$751$	−10.0230	−0.365744	−0.182872	+	0.983137i	$0.558539\pi$
$752$	0	0
$753$	−1.69254	−0.0616795
$754$	0	0
$755$	−0.900453	−0.0327708
$756$	0	0
$757$	−15.8218	−0.575054	−0.287527	−	0.957772i	$-0.592833\pi$
$757$	−15.8218	−0.575054	−0.287527	+	0.957772i	$0.592833\pi$
$758$	0	0
$759$	8.94848	0.324809
$760$	0	0
$761$	18.2787	0.662602	0.331301	−	0.943525i	$-0.392512\pi$
$761$	18.2787	0.662602	0.331301	+	0.943525i	$0.392512\pi$
$762$	0	0
$763$	−79.7943	−2.88875
$764$	0	0
$765$	4.95632	0.179196
$766$	0	0
$767$	−4.56103	−0.164689
$768$	0	0
$769$	45.0674	1.62517	0.812586	−	0.582841i	$-0.198059\pi$
$769$	45.0674	1.62517	0.812586	+	0.582841i	$0.198059\pi$
$770$	0	0
$771$	−4.64035	−0.167118
$772$	0	0
$773$	24.8873	0.895134	0.447567	−	0.894251i	$-0.352291\pi$
$773$	24.8873	0.895134	0.447567	+	0.894251i	$0.352291\pi$
$774$	0	0
$775$	−21.5809	−0.775208
$776$	0	0
$777$	−6.39352	−0.229366
$778$	0	0
$779$	42.6048	1.52647
$780$	0	0
$781$	17.4192	0.623307
$782$	0	0
$783$	−20.0916	−0.718017
$784$	0	0
$785$	−1.23837	−0.0441995
$786$	0	0
$787$	41.2857	1.47167	0.735837	−	0.677158i	$-0.236789\pi$
$787$	41.2857	1.47167	0.735837	+	0.677158i	$0.236789\pi$
$788$	0	0
$789$	7.46307	0.265692
$790$	0	0
$791$	40.6232	1.44439
$792$	0	0
$793$	−1.33891	−0.0475460
$794$	0	0
$795$	−1.35050	−0.0478972
$796$	0	0
$797$	−7.69496	−0.272569	−0.136285	−	0.990670i	$-0.543516\pi$
$797$	−7.69496	−0.272569	−0.136285	+	0.990670i	$0.543516\pi$
$798$	0	0
$799$	−33.2952	−1.17790
$800$	0	0
$801$	−29.5011	−1.04237
$802$	0	0
$803$	44.2741	1.56240
$804$	0	0
$805$	−9.86154	−0.347573
$806$	0	0
$807$	−6.41070	−0.225667
$808$	0	0
$809$	−21.7457	−0.764538	−0.382269	−	0.924051i	$-0.624857\pi$
$809$	−21.7457	−0.764538	−0.382269	+	0.924051i	$0.624857\pi$
$810$	0	0
$811$	37.7501	1.32558	0.662792	−	0.748803i	$-0.269371\pi$
$811$	37.7501	1.32558	0.662792	+	0.748803i	$0.269371\pi$
$812$	0	0
$813$	−12.6050	−0.442078
$814$	0	0
$815$	0.245804	0.00861015
$816$	0	0
$817$	−66.8886	−2.34014
$818$	0	0
$819$	11.8999	0.415817
$820$	0	0
$821$	41.7827	1.45823	0.729113	−	0.684393i	$-0.239933\pi$
$821$	41.7827	1.45823	0.729113	+	0.684393i	$0.239933\pi$
$822$	0	0
$823$	−26.8990	−0.937639	−0.468819	−	0.883294i	$-0.655320\pi$
$823$	−26.8990	−0.937639	−0.468819	+	0.883294i	$0.655320\pi$
$824$	0	0
$825$	−7.10179	−0.247252
$826$	0	0
$827$	−28.2939	−0.983876	−0.491938	−	0.870630i	$-0.663711\pi$
$827$	−28.2939	−0.983876	−0.491938	+	0.870630i	$0.663711\pi$
$828$	0	0
$829$	−11.9251	−0.414175	−0.207087	−	0.978322i	$-0.566398\pi$
$829$	−11.9251	−0.414175	−0.207087	+	0.978322i	$0.566398\pi$
$830$	0	0
$831$	−3.75943	−0.130413
$832$	0	0
$833$	−46.9035	−1.62511
$834$	0	0
$835$	1.28593	0.0445015
$836$	0	0
$837$	−10.0608	−0.347753
$838$	0	0
$839$	−12.8465	−0.443512	−0.221756	−	0.975102i	$-0.571179\pi$
$839$	−12.8465	−0.443512	−0.221756	+	0.975102i	$0.571179\pi$
$840$	0	0
$841$	49.9422	1.72215
$842$	0	0
$843$	−7.26511	−0.250224
$844$	0	0
$845$	0.386509	0.0132963
$846$	0	0
$847$	−13.9807	−0.480381
$848$	0	0
$849$	7.42898	0.254962
$850$	0	0
$851$	24.2187	0.830207
$852$	0	0
$853$	37.8311	1.29531	0.647656	−	0.761933i	$-0.275750\pi$
$853$	37.8311	1.29531	0.647656	+	0.761933i	$0.275750\pi$
$854$	0	0
$855$	5.78671	0.197901
$856$	0	0
$857$	16.3041	0.556938	0.278469	−	0.960445i	$-0.410173\pi$
$857$	16.3041	0.556938	0.278469	+	0.960445i	$0.410173\pi$
$858$	0	0
$859$	−45.3512	−1.54736	−0.773682	−	0.633574i	$-0.781587\pi$
$859$	−45.3512	−1.54736	−0.773682	+	0.633574i	$0.781587\pi$
$860$	0	0
$861$	13.0885	0.446055
$862$	0	0
$863$	49.2816	1.67757	0.838783	−	0.544466i	$-0.183267\pi$
$863$	49.2816	1.67757	0.838783	+	0.544466i	$0.183267\pi$
$864$	0	0
$865$	−9.75829	−0.331792
$866$	0	0
$867$	1.25071	0.0424763
$868$	0	0
$869$	9.76989	0.331421
$870$	0	0
$871$	6.67296	0.226105
$872$	0	0
$873$	−0.675936	−0.0228770
$874$	0	0
$875$	15.8939	0.537311
$876$	0	0
$877$	−39.2687	−1.32601	−0.663006	−	0.748614i	$-0.730719\pi$
$877$	−39.2687	−1.32601	−0.663006	+	0.748614i	$0.730719\pi$
$878$	0	0
$879$	−4.50815	−0.152056
$880$	0	0
$881$	46.5627	1.56874	0.784369	−	0.620295i	$-0.212987\pi$
$881$	46.5627	1.56874	0.784369	+	0.620295i	$0.212987\pi$
$882$	0	0
$883$	−5.74737	−0.193414	−0.0967072	−	0.995313i	$-0.530831\pi$
$883$	−5.74737	−0.193414	−0.0967072	+	0.995313i	$0.530831\pi$
$884$	0	0
$885$	−0.681369	−0.0229040
$886$	0	0
$887$	−30.0007	−1.00733	−0.503663	−	0.863900i	$-0.668015\pi$
$887$	−30.0007	−1.00733	−0.503663	+	0.863900i	$0.668015\pi$
$888$	0	0
$889$	−14.3358	−0.480806
$890$	0	0
$891$	29.0837	0.974339
$892$	0	0
$893$	−38.8736	−1.30085
$894$	0	0
$895$	5.45663	0.182395
$896$	0	0
$897$	2.36231	0.0788754
$898$	0	0
$899$	39.5301	1.31840
$900$	0	0
$901$	40.6664	1.35479
$902$	0	0
$903$	−20.5487	−0.683818
$904$	0	0
$905$	−1.87135	−0.0622059
$906$	0	0
$907$	−1.37124	−0.0455313	−0.0227657	−	0.999741i	$-0.507247\pi$
$907$	−1.37124	−0.0455313	−0.0227657	+	0.999741i	$0.507247\pi$
$908$	0	0
$909$	20.3877	0.676218
$910$	0	0
$911$	18.5959	0.616109	0.308054	−	0.951369i	$-0.400322\pi$
$911$	18.5959	0.616109	0.308054	+	0.951369i	$0.400322\pi$
$912$	0	0
$913$	−10.1252	−0.335095
$914$	0	0
$915$	−0.200018	−0.00661240
$916$	0	0
$917$	−56.2304	−1.85689
$918$	0	0
$919$	−51.3927	−1.69529	−0.847645	−	0.530564i	$-0.821980\pi$
$919$	−51.3927	−1.69529	−0.847645	+	0.530564i	$0.821980\pi$
$920$	0	0
$921$	−3.59275	−0.118385
$922$	0	0
$923$	4.59850	0.151361
$924$	0	0
$925$	−19.2207	−0.631973
$926$	0	0
$927$	−28.9485	−0.950793
$928$	0	0
$929$	18.4090	0.603981	0.301990	−	0.953311i	$-0.402349\pi$
$929$	18.4090	0.603981	0.301990	+	0.953311i	$0.402349\pi$
$930$	0	0
$931$	−54.7619	−1.79475
$932$	0	0
$933$	5.70655	0.186824
$934$	0	0
$935$	−6.58616	−0.215391
$936$	0	0
$937$	−14.4143	−0.470893	−0.235447	−	0.971887i	$-0.575655\pi$
$937$	−14.4143	−0.470893	−0.235447	+	0.971887i	$0.575655\pi$
$938$	0	0
$939$	5.36498	0.175079
$940$	0	0
$941$	−45.4458	−1.48149	−0.740745	−	0.671786i	$-0.765527\pi$
$941$	−45.4458	−1.48149	−0.740745	+	0.671786i	$0.765527\pi$
$942$	0	0
$943$	−49.5795	−1.61453
$944$	0	0
$945$	3.64861	0.118689
$946$	0	0
$947$	−53.9688	−1.75375	−0.876875	−	0.480718i	$-0.840376\pi$
$947$	−53.9688	−1.75375	−0.876875	+	0.480718i	$0.840376\pi$
$948$	0	0
$949$	11.6880	0.379407
$950$	0	0
$951$	11.7347	0.380522
$952$	0	0
$953$	18.3732	0.595168	0.297584	−	0.954696i	$-0.403819\pi$
$953$	18.3732	0.595168	0.297584	+	0.954696i	$0.403819\pi$
$954$	0	0
$955$	2.61270	0.0845451
$956$	0	0
$957$	13.0085	0.420504
$958$	0	0
$959$	−2.07567	−0.0670269
$960$	0	0
$961$	−11.2055	−0.361466
$962$	0	0
$963$	6.05810	0.195220
$964$	0	0
$965$	3.35030	0.107850
$966$	0	0
$967$	21.2966	0.684852	0.342426	−	0.939545i	$-0.388751\pi$
$967$	21.2966	0.684852	0.342426	+	0.939545i	$0.388751\pi$
$968$	0	0
$969$	9.13178	0.293355
$970$	0	0
$971$	9.81949	0.315122	0.157561	−	0.987509i	$-0.449637\pi$
$971$	9.81949	0.315122	0.157561	+	0.987509i	$0.449637\pi$
$972$	0	0
$973$	6.08773	0.195164
$974$	0	0
$975$	−1.87481	−0.0600418
$976$	0	0
$977$	55.6703	1.78105	0.890525	−	0.454934i	$-0.150337\pi$
$977$	55.6703	1.78105	0.890525	+	0.454934i	$0.150337\pi$
$978$	0	0
$979$	39.2023	1.25291
$980$	0	0
$981$	−54.4883	−1.73968
$982$	0	0
$983$	18.2697	0.582714	0.291357	−	0.956614i	$-0.405893\pi$
$983$	18.2697	0.582714	0.291357	+	0.956614i	$0.405893\pi$
$984$	0	0
$985$	9.49718	0.302605
$986$	0	0
$987$	−11.9423	−0.380126
$988$	0	0
$989$	77.8388	2.47513
$990$	0	0
$991$	−28.0503	−0.891046	−0.445523	−	0.895270i	$-0.646982\pi$
$991$	−28.0503	−0.891046	−0.445523	+	0.895270i	$0.646982\pi$
$992$	0	0
$993$	−3.61416	−0.114692
$994$	0	0
$995$	8.83300	0.280025
$996$	0	0
$997$	−22.9739	−0.727590	−0.363795	−	0.931479i	$-0.618519\pi$
$997$	−22.9739	−0.727590	−0.363795	+	0.931479i	$0.618519\pi$
$998$	0	0
$999$	−8.96054	−0.283499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.bi.1.3		4
4.3	odd	2		3328.2.a.bm.1.2		4
8.3	odd	2		3328.2.a.bj.1.3		4
8.5	even	2		3328.2.a.bn.1.2		4
16.3	odd	4		832.2.b.c.417.4	✓	8
16.5	even	4		832.2.b.d.417.4	yes	8
16.11	odd	4		832.2.b.c.417.5	yes	8
16.13	even	4		832.2.b.d.417.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
832.2.b.c.417.4	✓	8	16.3	odd	4
832.2.b.c.417.5	yes	8	16.11	odd	4
832.2.b.d.417.4	yes	8	16.5	even	4
832.2.b.d.417.5	yes	8	16.13	even	4
3328.2.a.bi.1.3		4	1.1	even	1	trivial
3328.2.a.bj.1.3		4	8.3	odd	2
3328.2.a.bm.1.2		4	4.3	odd	2
3328.2.a.bn.1.2		4	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q+0.386509 q^{3} +0.386509 q^{5} -4.17452 q^{7} -2.85061 q^{9} +3.78801 q^{11} +1.00000 q^{13} +0.149389 q^{15} -4.49843 q^{17} -5.25211 q^{19} -1.61349 q^{21} +6.11192 q^{23} -4.85061 q^{25} -2.26131 q^{27} +8.88494 q^{29} +4.44911 q^{31} +1.46410 q^{33} -1.61349 q^{35} +3.96254 q^{37} +0.386509 q^{39} -8.11192 q^{41} +12.7356 q^{43} -1.10179 q^{45} +7.40150 q^{47} +10.4266 q^{49} -1.73869 q^{51} -9.04013 q^{53} +1.46410 q^{55} -2.02999 q^{57} -4.56103 q^{59} -1.33891 q^{61} +11.8999 q^{63} +0.386509 q^{65} +6.67296 q^{67} +2.36231 q^{69} +4.59850 q^{71} +11.6880 q^{73} -1.87481 q^{75} -15.8131 q^{77} +2.57916 q^{79} +7.67781 q^{81} -2.67296 q^{83} -1.73869 q^{85} +3.43411 q^{87} +10.3490 q^{89} -4.17452 q^{91} +1.71962 q^{93} -2.02999 q^{95} +0.237120 q^{97} -10.7982 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} + 18 q^{15} - 2 q^{17} + 6 q^{19} - 10 q^{21} + 12 q^{23} - 2 q^{25} - 14 q^{27} + 16 q^{29} + 10 q^{31} - 8 q^{33} - 10 q^{35} - 14 q^{37} - 2 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 + 2 * q^11 + 4 * q^13 + 18 * q^15 - 2 * q^17 + 6 * q^19 - 10 * q^21 + 12 * q^23 - 2 * q^25 - 14 * q^27 + 16 * q^29 + 10 * q^31 - 8 * q^33 - 10 * q^35 - 14 * q^37 - 2 * q^39 - 20 * q^41 + 14 * q^43 - 20 * q^45 + 20 * q^47 - 2 * q^49 - 2 * q^51 + 4 * q^53 - 8 * q^55 + 4 * q^57 + 2 * q^59 + 22 * q^63 - 2 * q^65 - 6 * q^67 - 20 * q^69 + 28 * q^71 + 8 * q^73 - 16 * q^75 - 16 * q^77 + 16 * q^79 + 20 * q^81 + 22 * q^83 - 2 * q^85 + 12 * q^87 + 8 * q^89 + 36 * q^93 + 4 * q^95 - 20 * q^97 - 2 * q^99

Introduction

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