LMFDB - Embedded newform 3328.2.b.u.1665.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1665,3328)] 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("3328.1665"); 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, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-6,0,-6,0,0,0,0,0,14,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 416)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1665.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	3328.1665
Dual form			3328.2.b.u.1665.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{3} -3.56155i q^{5} -3.56155 q^{7} +0.561553 q^{9} +2.00000i q^{11} +1.00000i q^{13} +5.56155 q^{15} +3.56155 q^{17} -6.00000i q^{19} -5.56155i q^{21} -7.68466 q^{25} +5.56155i q^{27} -8.24621i q^{29} -1.12311 q^{31} -3.12311 q^{33} +12.6847i q^{35} +2.68466i q^{37} -1.56155 q^{39} +1.12311 q^{41} +11.8078i q^{43} -2.00000i q^{45} -10.6847 q^{47} +5.68466 q^{49} +5.56155i q^{51} -13.1231i q^{53} +7.12311 q^{55} +9.36932 q^{57} -6.00000i q^{59} +11.3693i q^{61} -2.00000 q^{63} +3.56155 q^{65} -6.00000i q^{67} -10.6847 q^{71} -10.0000 q^{73} -12.0000i q^{75} -7.12311i q^{77} +12.0000 q^{79} -7.00000 q^{81} -7.36932i q^{83} -12.6847i q^{85} +12.8769 q^{87} -8.24621 q^{89} -3.56155i q^{91} -1.75379i q^{93} -21.3693 q^{95} -6.00000 q^{97} +1.12311i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{7} - 6 q^{9} + 14 q^{15} + 6 q^{17} - 6 q^{25} + 12 q^{31} + 4 q^{33} + 2 q^{39} - 12 q^{41} - 18 q^{47} - 2 q^{49} + 12 q^{55} - 12 q^{57} - 8 q^{63} + 6 q^{65} - 18 q^{71} - 40 q^{73} + 48 q^{79}+ \cdots - 24 q^{97}+O(q^{100}) $ 4 * q - 6 * q^7 - 6 * q^9 + 14 * q^15 + 6 * q^17 - 6 * q^25 + 12 * q^31 + 4 * q^33 + 2 * q^39 - 12 * q^41 - 18 * q^47 - 2 * q^49 + 12 * q^55 - 12 * q^57 - 8 * q^63 + 6 * q^65 - 18 * q^71 - 40 * q^73 + 48 * q^79 - 28 * q^81 + 68 * q^87 - 36 * q^95 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.56155i	0.901563i	0.892634	+	0.450781i	$0.148855\pi$
$3$	1.56155i	0.901563i	−0.892634	+	0.450781i	$0.851145\pi$
$4$	0	0
$5$	− 3.56155i	− 1.59277i	−0.604787	−	0.796387i	$-0.706742\pi$
$5$	− 3.56155i	− 1.59277i	0.604787	−	0.796387i	$-0.293258\pi$
$6$	0	0
$7$	−3.56155	−1.34614	−0.673070	−	0.739579i	$-0.735025\pi$
$7$	−3.56155	−1.34614	−0.673070	+	0.739579i	$0.735025\pi$
$8$	0	0
$9$	0.561553	0.187184
$10$	0	0
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	5.56155	1.43599
$16$	0	0
$17$	3.56155	0.863803	0.431902	−	0.901921i	$-0.357843\pi$
$17$	3.56155	0.863803	0.431902	+	0.901921i	$0.357843\pi$
$18$	0	0
$19$	− 6.00000i	− 1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$	− 6.00000i	− 1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$	0	0
$21$	− 5.56155i	− 1.21363i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−7.68466	−1.53693
$26$	0	0
$27$	5.56155i	1.07032i
$28$	0	0
$29$	− 8.24621i	− 1.53128i	−0.643268	−	0.765641i	$-0.722422\pi$
$29$	− 8.24621i	− 1.53128i	0.643268	−	0.765641i	$-0.277578\pi$
$30$	0	0
$31$	−1.12311	−0.201716	−0.100858	−	0.994901i	$-0.532159\pi$
$31$	−1.12311	−0.201716	−0.100858	+	0.994901i	$0.532159\pi$
$32$	0	0
$33$	−3.12311	−0.543663
$34$	0	0
$35$	12.6847i	2.14410i
$36$	0	0
$37$	2.68466i	0.441355i	0.975347	+	0.220678i	$0.0708268\pi$
$37$	2.68466i	0.441355i	−0.975347	+	0.220678i	$0.929173\pi$
$38$	0	0
$39$	−1.56155	−0.250049
$40$	0	0
$41$	1.12311	0.175400	0.0876998	−	0.996147i	$-0.472048\pi$
$41$	1.12311	0.175400	0.0876998	+	0.996147i	$0.472048\pi$
$42$	0	0
$43$	11.8078i	1.80067i	0.435200	+	0.900334i	$0.356677\pi$
$43$	11.8078i	1.80067i	−0.435200	+	0.900334i	$0.643323\pi$
$44$	0	0
$45$	− 2.00000i	− 0.298142i
$46$	0	0
$47$	−10.6847	−1.55852	−0.779259	−	0.626702i	$-0.784404\pi$
$47$	−10.6847	−1.55852	−0.779259	+	0.626702i	$0.784404\pi$
$48$	0	0
$49$	5.68466	0.812094
$50$	0	0
$51$	5.56155i	0.778773i
$52$	0	0
$53$	− 13.1231i	− 1.80260i	−0.433198	−	0.901299i	$-0.642615\pi$
$53$	− 13.1231i	− 1.80260i	0.433198	−	0.901299i	$-0.357385\pi$
$54$	0	0
$55$	7.12311	0.960479
$56$	0	0
$57$	9.36932	1.24100
$58$	0	0
$59$	− 6.00000i	− 0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$	− 6.00000i	− 0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$	0	0
$61$	11.3693i	1.45569i	0.685741	+	0.727846i	$0.259478\pi$
$61$	11.3693i	1.45569i	−0.685741	+	0.727846i	$0.740522\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	3.56155	0.441756
$66$	0	0
$67$	− 6.00000i	− 0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$	− 6.00000i	− 0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.6847	−1.26804	−0.634018	−	0.773318i	$-0.718595\pi$
$71$	−10.6847	−1.26804	−0.634018	+	0.773318i	$0.718595\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	− 12.0000i	− 1.38564i
$76$	0	0
$77$	− 7.12311i	− 0.811753i
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 7.36932i	− 0.808888i	−0.914563	−	0.404444i	$-0.867465\pi$
$83$	− 7.36932i	− 0.808888i	0.914563	−	0.404444i	$-0.132535\pi$
$84$	0	0
$85$	− 12.6847i	− 1.37584i
$86$	0	0
$87$	12.8769	1.38055
$88$	0	0
$89$	−8.24621	−0.874097	−0.437048	−	0.899438i	$-0.643976\pi$
$89$	−8.24621	−0.874097	−0.437048	+	0.899438i	$0.643976\pi$
$90$	0	0
$91$	− 3.56155i	− 0.373352i
$92$	0	0
$93$	− 1.75379i	− 0.181859i
$94$	0	0
$95$	−21.3693	−2.19245
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	1.12311i	0.112876i
$100$	0	0
$101$	1.12311i	0.111753i	0.998438	+	0.0558766i	$0.0177953\pi$
$101$	1.12311i	0.111753i	−0.998438	+	0.0558766i	$0.982205\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−19.8078	−1.93304
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	− 10.6847i	− 1.02340i	−0.859163	−	0.511702i	$-0.829015\pi$
$109$	− 10.6847i	− 1.02340i	0.859163	−	0.511702i	$-0.170985\pi$
$110$	0	0
$111$	−4.19224	−0.397909
$112$	0	0
$113$	−8.24621	−0.775738	−0.387869	−	0.921714i	$-0.626789\pi$
$113$	−8.24621	−0.775738	−0.387869	+	0.921714i	$0.626789\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.561553i	0.0519156i
$118$	0	0
$119$	−12.6847	−1.16280
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	1.75379i	0.158134i
$124$	0	0
$125$	9.56155i	0.855211i
$126$	0	0
$127$	−14.2462	−1.26415	−0.632073	−	0.774909i	$-0.717796\pi$
$127$	−14.2462	−1.26415	−0.632073	+	0.774909i	$0.717796\pi$
$128$	0	0
$129$	−18.4384	−1.62341
$130$	0	0
$131$	− 3.31534i	− 0.289663i	−0.989456	−	0.144831i	$-0.953736\pi$
$131$	− 3.31534i	− 0.289663i	0.989456	−	0.144831i	$-0.0462640\pi$
$132$	0	0
$133$	21.3693i	1.85295i
$134$	0	0
$135$	19.8078	1.70478
$136$	0	0
$137$	−13.1231	−1.12118	−0.560591	−	0.828093i	$-0.689426\pi$
$137$	−13.1231	−1.12118	−0.560591	+	0.828093i	$0.689426\pi$
$138$	0	0
$139$	− 4.68466i	− 0.397348i	−0.980066	−	0.198674i	$-0.936337\pi$
$139$	− 4.68466i	− 0.397348i	0.980066	−	0.198674i	$-0.0636634\pi$
$140$	0	0
$141$	− 16.6847i	− 1.40510i
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−29.3693	−2.43899
$146$	0	0
$147$	8.87689i	0.732154i
$148$	0	0
$149$	8.24621i	0.675556i	0.941226	+	0.337778i	$0.109675\pi$
$149$	8.24621i	0.675556i	−0.941226	+	0.337778i	$0.890325\pi$
$150$	0	0
$151$	−22.6847	−1.84605	−0.923026	−	0.384738i	$-0.874292\pi$
$151$	−22.6847	−1.84605	−0.923026	+	0.384738i	$0.874292\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	4.00000i	0.321288i
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	20.4924	1.62515
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.3693i	1.20382i	0.798565	+	0.601909i	$0.205593\pi$
$163$	15.3693i	1.20382i	−0.798565	+	0.601909i	$0.794407\pi$
$164$	0	0
$165$	11.1231i	0.865933i
$166$	0	0
$167$	−15.3693	−1.18931	−0.594657	−	0.803980i	$-0.702712\pi$
$167$	−15.3693	−1.18931	−0.594657	+	0.803980i	$0.702712\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 3.36932i	− 0.257658i
$172$	0	0
$173$	3.36932i	0.256164i	0.991764	+	0.128082i	$0.0408821\pi$
$173$	3.36932i	0.256164i	−0.991764	+	0.128082i	$0.959118\pi$
$174$	0	0
$175$	27.3693	2.06893
$176$	0	0
$177$	9.36932	0.704241
$178$	0	0
$179$	− 18.0540i	− 1.34942i	−0.738084	−	0.674709i	$-0.764269\pi$
$179$	− 18.0540i	− 1.34942i	0.738084	−	0.674709i	$-0.235731\pi$
$180$	0	0
$181$	− 7.36932i	− 0.547757i	−0.961764	−	0.273879i	$-0.911693\pi$
$181$	− 7.36932i	− 0.547757i	0.961764	−	0.273879i	$-0.0883066\pi$
$182$	0	0
$183$	−17.7538	−1.31240
$184$	0	0
$185$	9.56155	0.702979
$186$	0	0
$187$	7.12311i	0.520893i
$188$	0	0
$189$	− 19.8078i	− 1.44080i
$190$	0	0
$191$	5.36932	0.388510	0.194255	−	0.980951i	$-0.437771\pi$
$191$	5.36932	0.388510	0.194255	+	0.980951i	$0.437771\pi$
$192$	0	0
$193$	−23.3693	−1.68216	−0.841080	−	0.540911i	$-0.818080\pi$
$193$	−23.3693	−1.68216	−0.841080	+	0.540911i	$0.818080\pi$
$194$	0	0
$195$	5.56155i	0.398271i
$196$	0	0
$197$	1.31534i	0.0937142i	0.998902	+	0.0468571i	$0.0149205\pi$
$197$	1.31534i	0.0937142i	−0.998902	+	0.0468571i	$0.985079\pi$
$198$	0	0
$199$	9.36932	0.664173	0.332087	−	0.943249i	$-0.392247\pi$
$199$	9.36932	0.664173	0.332087	+	0.943249i	$0.392247\pi$
$200$	0	0
$201$	9.36932	0.660861
$202$	0	0
$203$	29.3693i	2.06132i
$204$	0	0
$205$	− 4.00000i	− 0.279372i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	− 23.8078i	− 1.63899i	−0.573083	−	0.819497i	$-0.694253\pi$
$211$	− 23.8078i	− 1.63899i	0.573083	−	0.819497i	$-0.305747\pi$
$212$	0	0
$213$	− 16.6847i	− 1.14321i
$214$	0	0
$215$	42.0540	2.86806
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	− 15.6155i	− 1.05520i
$220$	0	0
$221$	3.56155i	0.239576i
$222$	0	0
$223$	6.19224	0.414663	0.207331	−	0.978271i	$-0.433522\pi$
$223$	6.19224	0.414663	0.207331	+	0.978271i	$0.433522\pi$
$224$	0	0
$225$	−4.31534	−0.287689
$226$	0	0
$227$	3.36932i	0.223629i	0.993729	+	0.111815i	$0.0356663\pi$
$227$	3.36932i	0.223629i	−0.993729	+	0.111815i	$0.964334\pi$
$228$	0	0
$229$	1.31534i	0.0869202i	0.999055	+	0.0434601i	$0.0138381\pi$
$229$	1.31534i	0.0869202i	−0.999055	+	0.0434601i	$0.986162\pi$
$230$	0	0
$231$	11.1231	0.731847
$232$	0	0
$233$	−1.31534	−0.0861709	−0.0430854	−	0.999071i	$-0.513719\pi$
$233$	−1.31534	−0.0861709	−0.0430854	+	0.999071i	$0.513719\pi$
$234$	0	0
$235$	38.0540i	2.48237i
$236$	0	0
$237$	18.7386i	1.21721i
$238$	0	0
$239$	9.31534	0.602559	0.301280	−	0.953536i	$-0.402586\pi$
$239$	9.31534	0.602559	0.301280	+	0.953536i	$0.402586\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	5.75379i	0.369106i
$244$	0	0
$245$	− 20.2462i	− 1.29348i
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	11.5076	0.729263
$250$	0	0
$251$	− 16.0000i	− 1.00991i	−0.863145	−	0.504956i	$-0.831509\pi$
$251$	− 16.0000i	− 1.00991i	0.863145	−	0.504956i	$-0.168491\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	19.8078	1.24041
$256$	0	0
$257$	−12.9309	−0.806605	−0.403303	−	0.915067i	$-0.632138\pi$
$257$	−12.9309	−0.806605	−0.403303	+	0.915067i	$0.632138\pi$
$258$	0	0
$259$	− 9.56155i	− 0.594126i
$260$	0	0
$261$	− 4.63068i	− 0.286632i
$262$	0	0
$263$	22.7386	1.40212	0.701062	−	0.713100i	$-0.252710\pi$
$263$	22.7386	1.40212	0.701062	+	0.713100i	$0.252710\pi$
$264$	0	0
$265$	−46.7386	−2.87113
$266$	0	0
$267$	− 12.8769i	− 0.788053i
$268$	0	0
$269$	− 3.36932i	− 0.205431i	−0.994711	−	0.102715i	$-0.967247\pi$
$269$	− 3.36932i	− 0.205431i	0.994711	−	0.102715i	$-0.0327531\pi$
$270$	0	0
$271$	−10.6847	−0.649047	−0.324523	−	0.945878i	$-0.605204\pi$
$271$	−10.6847	−0.649047	−0.324523	+	0.945878i	$0.605204\pi$
$272$	0	0
$273$	5.56155	0.336600
$274$	0	0
$275$	− 15.3693i	− 0.926805i
$276$	0	0
$277$	23.3693i	1.40413i	0.712115	+	0.702063i	$0.247738\pi$
$277$	23.3693i	1.40413i	−0.712115	+	0.702063i	$0.752262\pi$
$278$	0	0
$279$	−0.630683	−0.0377580
$280$	0	0
$281$	−3.75379	−0.223932	−0.111966	−	0.993712i	$-0.535715\pi$
$281$	−3.75379	−0.223932	−0.111966	+	0.993712i	$0.535715\pi$
$282$	0	0
$283$	− 14.2462i	− 0.846849i	−0.905931	−	0.423425i	$-0.860828\pi$
$283$	− 14.2462i	− 0.846849i	0.905931	−	0.423425i	$-0.139172\pi$
$284$	0	0
$285$	− 33.3693i	− 1.97663i
$286$	0	0
$287$	−4.00000	−0.236113
$288$	0	0
$289$	−4.31534	−0.253844
$290$	0	0
$291$	− 9.36932i	− 0.549239i
$292$	0	0
$293$	− 27.5616i	− 1.61016i	−0.593164	−	0.805082i	$-0.702121\pi$
$293$	− 27.5616i	− 1.61016i	0.593164	−	0.805082i	$-0.297879\pi$
$294$	0	0
$295$	−21.3693	−1.24417
$296$	0	0
$297$	−11.1231	−0.645428
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 42.0540i	− 2.42395i
$302$	0	0
$303$	−1.75379	−0.100753
$304$	0	0
$305$	40.4924	2.31859
$306$	0	0
$307$	6.00000i	0.342438i	0.985233	+	0.171219i	$0.0547706\pi$
$307$	6.00000i	0.342438i	−0.985233	+	0.171219i	$0.945229\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.36932	−0.531285	−0.265643	−	0.964072i	$-0.585584\pi$
$311$	−9.36932	−0.531285	−0.265643	+	0.964072i	$0.585584\pi$
$312$	0	0
$313$	20.0540	1.13352	0.566759	−	0.823884i	$-0.308197\pi$
$313$	20.0540	1.13352	0.566759	+	0.823884i	$0.308197\pi$
$314$	0	0
$315$	7.12311i	0.401342i
$316$	0	0
$317$	15.7538i	0.884821i	0.896813	+	0.442410i	$0.145877\pi$
$317$	15.7538i	0.884821i	−0.896813	+	0.442410i	$0.854123\pi$
$318$	0	0
$319$	16.4924	0.923398
$320$	0	0
$321$	18.7386	1.04589
$322$	0	0
$323$	− 21.3693i	− 1.18902i
$324$	0	0
$325$	− 7.68466i	− 0.426268i
$326$	0	0
$327$	16.6847	0.922664
$328$	0	0
$329$	38.0540	2.09798
$330$	0	0
$331$	− 3.36932i	− 0.185194i	−0.995704	−	0.0925972i	$-0.970483\pi$
$331$	− 3.36932i	− 0.185194i	0.995704	−	0.0925972i	$-0.0295169\pi$
$332$	0	0
$333$	1.50758i	0.0826147i
$334$	0	0
$335$	−21.3693	−1.16753
$336$	0	0
$337$	−36.0540	−1.96399	−0.981993	−	0.188919i	$-0.939501\pi$
$337$	−36.0540	−1.96399	−0.981993	+	0.188919i	$0.939501\pi$
$338$	0	0
$339$	− 12.8769i	− 0.699377i
$340$	0	0
$341$	− 2.24621i	− 0.121639i
$342$	0	0
$343$	4.68466	0.252948
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 28.6847i	− 1.53987i	−0.638120	−	0.769937i	$-0.720288\pi$
$347$	− 28.6847i	− 1.53987i	0.638120	−	0.769937i	$-0.279712\pi$
$348$	0	0
$349$	8.05398i	0.431119i	0.976491	+	0.215560i	$0.0691576\pi$
$349$	8.05398i	0.431119i	−0.976491	+	0.215560i	$0.930842\pi$
$350$	0	0
$351$	−5.56155	−0.296854
$352$	0	0
$353$	31.8617	1.69583	0.847915	−	0.530133i	$-0.177858\pi$
$353$	31.8617	1.69583	0.847915	+	0.530133i	$0.177858\pi$
$354$	0	0
$355$	38.0540i	2.01970i
$356$	0	0
$357$	− 19.8078i	− 1.04834i
$358$	0	0
$359$	−7.36932	−0.388938	−0.194469	−	0.980909i	$-0.562298\pi$
$359$	−7.36932	−0.388938	−0.194469	+	0.980909i	$0.562298\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	10.9309i	0.573722i
$364$	0	0
$365$	35.6155i	1.86420i
$366$	0	0
$367$	−2.63068	−0.137321	−0.0686603	−	0.997640i	$-0.521872\pi$
$367$	−2.63068	−0.137321	−0.0686603	+	0.997640i	$0.521872\pi$
$368$	0	0
$369$	0.630683	0.0328321
$370$	0	0
$371$	46.7386i	2.42655i
$372$	0	0
$373$	11.3693i	0.588681i	0.955701	+	0.294340i	$0.0951000\pi$
$373$	11.3693i	0.588681i	−0.955701	+	0.294340i	$0.904900\pi$
$374$	0	0
$375$	−14.9309	−0.771027
$376$	0	0
$377$	8.24621	0.424701
$378$	0	0
$379$	25.1231i	1.29049i	0.763977	+	0.645244i	$0.223244\pi$
$379$	25.1231i	1.29049i	−0.763977	+	0.645244i	$0.776756\pi$
$380$	0	0
$381$	− 22.2462i	− 1.13971i
$382$	0	0
$383$	10.6847	0.545961	0.272980	−	0.962020i	$-0.411991\pi$
$383$	10.6847	0.545961	0.272980	+	0.962020i	$0.411991\pi$
$384$	0	0
$385$	−25.3693	−1.29294
$386$	0	0
$387$	6.63068i	0.337057i
$388$	0	0
$389$	3.75379i	0.190325i	0.995462	+	0.0951623i	$0.0303370\pi$
$389$	3.75379i	0.190325i	−0.995462	+	0.0951623i	$0.969663\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	5.17708	0.261149
$394$	0	0
$395$	− 42.7386i	− 2.15041i
$396$	0	0
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	0	0
$399$	−33.3693	−1.67056
$400$	0	0
$401$	22.8769	1.14242	0.571209	−	0.820805i	$-0.306475\pi$
$401$	22.8769	1.14242	0.571209	+	0.820805i	$0.306475\pi$
$402$	0	0
$403$	− 1.12311i	− 0.0559459i
$404$	0	0
$405$	24.9309i	1.23882i
$406$	0	0
$407$	−5.36932	−0.266147
$408$	0	0
$409$	31.3693	1.55111	0.775556	−	0.631278i	$-0.217469\pi$
$409$	31.3693	1.55111	0.775556	+	0.631278i	$0.217469\pi$
$410$	0	0
$411$	− 20.4924i	− 1.01082i
$412$	0	0
$413$	21.3693i	1.05152i
$414$	0	0
$415$	−26.2462	−1.28838
$416$	0	0
$417$	7.31534	0.358234
$418$	0	0
$419$	22.0540i	1.07741i	0.842495	+	0.538704i	$0.181086\pi$
$419$	22.0540i	1.07741i	−0.842495	+	0.538704i	$0.818914\pi$
$420$	0	0
$421$	− 21.3153i	− 1.03885i	−0.854517	−	0.519423i	$-0.826147\pi$
$421$	− 21.3153i	− 1.03885i	0.854517	−	0.519423i	$-0.173853\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−27.3693	−1.32761
$426$	0	0
$427$	− 40.4924i	− 1.95957i
$428$	0	0
$429$	− 3.12311i	− 0.150785i
$430$	0	0
$431$	−6.68466	−0.321989	−0.160994	−	0.986955i	$-0.551470\pi$
$431$	−6.68466	−0.321989	−0.160994	+	0.986955i	$0.551470\pi$
$432$	0	0
$433$	26.6847	1.28238	0.641191	−	0.767381i	$-0.278440\pi$
$433$	26.6847	1.28238	0.641191	+	0.767381i	$0.278440\pi$
$434$	0	0
$435$	− 45.8617i	− 2.19890i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0.384472	0.0183498	0.00917492	−	0.999958i	$-0.497079\pi$
$439$	0.384472	0.0183498	0.00917492	+	0.999958i	$0.497079\pi$
$440$	0	0
$441$	3.19224	0.152011
$442$	0	0
$443$	4.68466i	0.222575i	0.993788	+	0.111287i	$0.0354974\pi$
$443$	4.68466i	0.222575i	−0.993788	+	0.111287i	$0.964503\pi$
$444$	0	0
$445$	29.3693i	1.39224i
$446$	0	0
$447$	−12.8769	−0.609056
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	2.24621i	0.105770i
$452$	0	0
$453$	− 35.4233i	− 1.66433i
$454$	0	0
$455$	−12.6847	−0.594666
$456$	0	0
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	0	0
$459$	19.8078i	0.924547i
$460$	0	0
$461$	41.8078i	1.94718i	0.228300	+	0.973591i	$0.426683\pi$
$461$	41.8078i	1.94718i	−0.228300	+	0.973591i	$0.573317\pi$
$462$	0	0
$463$	−17.6155	−0.818663	−0.409332	−	0.912386i	$-0.634238\pi$
$463$	−17.6155	−0.818663	−0.409332	+	0.912386i	$0.634238\pi$
$464$	0	0
$465$	−6.24621	−0.289661
$466$	0	0
$467$	34.7386i	1.60751i	0.594959	+	0.803756i	$0.297168\pi$
$467$	34.7386i	1.60751i	−0.594959	+	0.803756i	$0.702832\pi$
$468$	0	0
$469$	21.3693i	0.986743i
$470$	0	0
$471$	3.12311	0.143905
$472$	0	0
$473$	−23.6155	−1.08584
$474$	0	0
$475$	46.1080i	2.11558i
$476$	0	0
$477$	− 7.36932i	− 0.337418i
$478$	0	0
$479$	21.3153	0.973923	0.486961	−	0.873423i	$-0.338105\pi$
$479$	21.3153	0.973923	0.486961	+	0.873423i	$0.338105\pi$
$480$	0	0
$481$	−2.68466	−0.122410
$482$	0	0
$483$	0	0
$484$	0	0
$485$	21.3693i	0.970331i
$486$	0	0
$487$	10.8769	0.492879	0.246440	−	0.969158i	$-0.420739\pi$
$487$	10.8769	0.492879	0.246440	+	0.969158i	$0.420739\pi$
$488$	0	0
$489$	−24.0000	−1.08532
$490$	0	0
$491$	23.3153i	1.05221i	0.850421	+	0.526103i	$0.176347\pi$
$491$	23.3153i	1.05221i	−0.850421	+	0.526103i	$0.823653\pi$
$492$	0	0
$493$	− 29.3693i	− 1.32273i
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	38.0540	1.70695
$498$	0	0
$499$	− 25.1231i	− 1.12466i	−0.826911	−	0.562332i	$-0.809904\pi$
$499$	− 25.1231i	− 1.12466i	0.826911	−	0.562332i	$-0.190096\pi$
$500$	0	0
$501$	− 24.0000i	− 1.07224i
$502$	0	0
$503$	29.3693	1.30951	0.654757	−	0.755840i	$-0.272771\pi$
$503$	29.3693	1.30951	0.654757	+	0.755840i	$0.272771\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	− 1.56155i	− 0.0693510i
$508$	0	0
$509$	− 22.4924i	− 0.996959i	−0.866901	−	0.498480i	$-0.833892\pi$
$509$	− 22.4924i	− 0.996959i	0.866901	−	0.498480i	$-0.166108\pi$
$510$	0	0
$511$	35.6155	1.57554
$512$	0	0
$513$	33.3693	1.47329
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 21.3693i	− 0.939821i
$518$	0	0
$519$	−5.26137	−0.230948
$520$	0	0
$521$	−8.43845	−0.369695	−0.184848	−	0.982767i	$-0.559179\pi$
$521$	−8.43845	−0.369695	−0.184848	+	0.982767i	$0.559179\pi$
$522$	0	0
$523$	− 7.50758i	− 0.328283i	−0.986437	−	0.164142i	$-0.947515\pi$
$523$	− 7.50758i	− 0.328283i	0.986437	−	0.164142i	$-0.0524854\pi$
$524$	0	0
$525$	42.7386i	1.86527i
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 3.36932i	− 0.146216i
$532$	0	0
$533$	1.12311i	0.0486471i
$534$	0	0
$535$	−42.7386	−1.84775
$536$	0	0
$537$	28.1922	1.21658
$538$	0	0
$539$	11.3693i	0.489711i
$540$	0	0
$541$	30.6847i	1.31924i	0.751601	+	0.659618i	$0.229282\pi$
$541$	30.6847i	1.31924i	−0.751601	+	0.659618i	$0.770718\pi$
$542$	0	0
$543$	11.5076	0.493837
$544$	0	0
$545$	−38.0540	−1.63005
$546$	0	0
$547$	14.0540i	0.600905i	0.953797	+	0.300452i	$0.0971376\pi$
$547$	14.0540i	0.600905i	−0.953797	+	0.300452i	$0.902862\pi$
$548$	0	0
$549$	6.38447i	0.272483i
$550$	0	0
$551$	−49.4773	−2.10780
$552$	0	0
$553$	−42.7386	−1.81743
$554$	0	0
$555$	14.9309i	0.633780i
$556$	0	0
$557$	− 10.3002i	− 0.436433i	−0.975900	−	0.218216i	$-0.929976\pi$
$557$	− 10.3002i	− 0.436433i	0.975900	−	0.218216i	$-0.0700239\pi$
$558$	0	0
$559$	−11.8078	−0.499415
$560$	0	0
$561$	−11.1231	−0.469618
$562$	0	0
$563$	− 31.3153i	− 1.31978i	−0.751360	−	0.659892i	$-0.770602\pi$
$563$	− 31.3153i	− 1.31978i	0.751360	−	0.659892i	$-0.229398\pi$
$564$	0	0
$565$	29.3693i	1.23558i
$566$	0	0
$567$	24.9309	1.04700
$568$	0	0
$569$	13.3153	0.558208	0.279104	−	0.960261i	$-0.409963\pi$
$569$	13.3153	0.558208	0.279104	+	0.960261i	$0.409963\pi$
$570$	0	0
$571$	− 21.5616i	− 0.902323i	−0.892442	−	0.451161i	$-0.851010\pi$
$571$	− 21.5616i	− 0.902323i	0.892442	−	0.451161i	$-0.148990\pi$
$572$	0	0
$573$	8.38447i	0.350266i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	44.7386	1.86249	0.931247	−	0.364389i	$-0.118722\pi$
$577$	44.7386	1.86249	0.931247	+	0.364389i	$0.118722\pi$
$578$	0	0
$579$	− 36.4924i	− 1.51657i
$580$	0	0
$581$	26.2462i	1.08888i
$582$	0	0
$583$	26.2462	1.08701
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	15.3693i	0.634360i	0.948365	+	0.317180i	$0.102736\pi$
$587$	15.3693i	0.634360i	−0.948365	+	0.317180i	$0.897264\pi$
$588$	0	0
$589$	6.73863i	0.277661i
$590$	0	0
$591$	−2.05398	−0.0844893
$592$	0	0
$593$	3.75379	0.154150	0.0770748	−	0.997025i	$-0.475442\pi$
$593$	3.75379	0.154150	0.0770748	+	0.997025i	$0.475442\pi$
$594$	0	0
$595$	45.1771i	1.85208i
$596$	0	0
$597$	14.6307i	0.598794i
$598$	0	0
$599$	−10.6307	−0.434358	−0.217179	−	0.976132i	$-0.569686\pi$
$599$	−10.6307	−0.434358	−0.217179	+	0.976132i	$0.569686\pi$
$600$	0	0
$601$	−16.0540	−0.654855	−0.327428	−	0.944876i	$-0.606182\pi$
$601$	−16.0540	−0.654855	−0.327428	+	0.944876i	$0.606182\pi$
$602$	0	0
$603$	− 3.36932i	− 0.137209i
$604$	0	0
$605$	− 24.9309i	− 1.01358i
$606$	0	0
$607$	−25.8617	−1.04970	−0.524848	−	0.851196i	$-0.675878\pi$
$607$	−25.8617	−1.04970	−0.524848	+	0.851196i	$0.675878\pi$
$608$	0	0
$609$	−45.8617	−1.85841
$610$	0	0
$611$	− 10.6847i	− 0.432255i
$612$	0	0
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	0	0
$615$	6.24621	0.251872
$616$	0	0
$617$	−17.6155	−0.709174	−0.354587	−	0.935023i	$-0.615379\pi$
$617$	−17.6155	−0.709174	−0.354587	+	0.935023i	$0.615379\pi$
$618$	0	0
$619$	− 15.3693i	− 0.617745i	−0.951103	−	0.308873i	$-0.900048\pi$
$619$	− 15.3693i	− 0.617745i	0.951103	−	0.308873i	$-0.0999516\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	29.3693	1.17666
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	18.7386i	0.748349i
$628$	0	0
$629$	9.56155i	0.381244i
$630$	0	0
$631$	−3.94602	−0.157089	−0.0785444	−	0.996911i	$-0.525027\pi$
$631$	−3.94602	−0.157089	−0.0785444	+	0.996911i	$0.525027\pi$
$632$	0	0
$633$	37.1771	1.47766
$634$	0	0
$635$	50.7386i	2.01350i
$636$	0	0
$637$	5.68466i	0.225234i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−22.4924	−0.888397	−0.444199	−	0.895928i	$-0.646512\pi$
$641$	−22.4924	−0.888397	−0.444199	+	0.895928i	$0.646512\pi$
$642$	0	0
$643$	18.0000i	0.709851i	0.934895	+	0.354925i	$0.115494\pi$
$643$	18.0000i	0.709851i	−0.934895	+	0.354925i	$0.884506\pi$
$644$	0	0
$645$	65.6695i	2.58573i
$646$	0	0
$647$	−44.1080	−1.73406	−0.867031	−	0.498254i	$-0.833975\pi$
$647$	−44.1080	−1.73406	−0.867031	+	0.498254i	$0.833975\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	6.24621i	0.244808i
$652$	0	0
$653$	6.38447i	0.249844i	0.992167	+	0.124922i	$0.0398680\pi$
$653$	6.38447i	0.249844i	−0.992167	+	0.124922i	$0.960132\pi$
$654$	0	0
$655$	−11.8078	−0.461368
$656$	0	0
$657$	−5.61553	−0.219083
$658$	0	0
$659$	− 12.0000i	− 0.467454i	−0.972302	−	0.233727i	$-0.924908\pi$
$659$	− 12.0000i	− 0.467454i	0.972302	−	0.233727i	$-0.0750921\pi$
$660$	0	0
$661$	28.7386i	1.11780i	0.829234	+	0.558902i	$0.188777\pi$
$661$	28.7386i	1.11780i	−0.829234	+	0.558902i	$0.811223\pi$
$662$	0	0
$663$	−5.56155	−0.215993
$664$	0	0
$665$	76.1080	2.95134
$666$	0	0
$667$	0	0
$668$	0	0
$669$	9.66950i	0.373845i
$670$	0	0
$671$	−22.7386	−0.877815
$672$	0	0
$673$	16.0540	0.618835	0.309418	−	0.950926i	$-0.399866\pi$
$673$	16.0540	0.618835	0.309418	+	0.950926i	$0.399866\pi$
$674$	0	0
$675$	− 42.7386i	− 1.64501i
$676$	0	0
$677$	− 3.36932i	− 0.129493i	−0.997902	−	0.0647467i	$-0.979376\pi$
$677$	− 3.36932i	− 0.129493i	0.997902	−	0.0647467i	$-0.0206239\pi$
$678$	0	0
$679$	21.3693	0.820079
$680$	0	0
$681$	−5.26137	−0.201616
$682$	0	0
$683$	19.3693i	0.741146i	0.928803	+	0.370573i	$0.120839\pi$
$683$	19.3693i	0.741146i	−0.928803	+	0.370573i	$0.879161\pi$
$684$	0	0
$685$	46.7386i	1.78579i
$686$	0	0
$687$	−2.05398	−0.0783640
$688$	0	0
$689$	13.1231	0.499951
$690$	0	0
$691$	13.1231i	0.499226i	0.968346	+	0.249613i	$0.0803035\pi$
$691$	13.1231i	0.499226i	−0.968346	+	0.249613i	$0.919697\pi$
$692$	0	0
$693$	− 4.00000i	− 0.151947i
$694$	0	0
$695$	−16.6847	−0.632885
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	− 2.05398i	− 0.0776885i
$700$	0	0
$701$	15.3693i	0.580491i	0.956952	+	0.290246i	$0.0937370\pi$
$701$	15.3693i	0.580491i	−0.956952	+	0.290246i	$0.906263\pi$
$702$	0	0
$703$	16.1080	0.607523
$704$	0	0
$705$	−59.4233	−2.23801
$706$	0	0
$707$	− 4.00000i	− 0.150435i
$708$	0	0
$709$	− 32.7386i	− 1.22953i	−0.788712	−	0.614763i	$-0.789252\pi$
$709$	− 32.7386i	− 1.22953i	0.788712	−	0.614763i	$-0.210748\pi$
$710$	0	0
$711$	6.73863	0.252719
$712$	0	0
$713$	0	0
$714$	0	0
$715$	7.12311i	0.266389i
$716$	0	0
$717$	14.5464i	0.543245i
$718$	0	0
$719$	9.36932	0.349417	0.174708	−	0.984620i	$-0.444102\pi$
$719$	9.36932	0.349417	0.174708	+	0.984620i	$0.444102\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.12311i	0.116150i
$724$	0	0
$725$	63.3693i	2.35348i
$726$	0	0
$727$	0.384472	0.0142593	0.00712964	−	0.999975i	$-0.497731\pi$
$727$	0.384472	0.0142593	0.00712964	+	0.999975i	$0.497731\pi$
$728$	0	0
$729$	−29.9848	−1.11055
$730$	0	0
$731$	42.0540i	1.55542i
$732$	0	0
$733$	− 12.0540i	− 0.445224i	−0.974907	−	0.222612i	$-0.928542\pi$
$733$	− 12.0540i	− 0.445224i	0.974907	−	0.222612i	$-0.0714583\pi$
$734$	0	0
$735$	31.6155	1.16616
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	− 22.1080i	− 0.813254i	−0.913594	−	0.406627i	$-0.866705\pi$
$739$	− 22.1080i	− 0.813254i	0.913594	−	0.406627i	$-0.133295\pi$
$740$	0	0
$741$	9.36932i	0.344190i
$742$	0	0
$743$	−1.31534	−0.0482552	−0.0241276	−	0.999709i	$-0.507681\pi$
$743$	−1.31534	−0.0482552	−0.0241276	+	0.999709i	$0.507681\pi$
$744$	0	0
$745$	29.3693	1.07601
$746$	0	0
$747$	− 4.13826i	− 0.151411i
$748$	0	0
$749$	42.7386i	1.56164i
$750$	0	0
$751$	−9.75379	−0.355921	−0.177960	−	0.984038i	$-0.556950\pi$
$751$	−9.75379	−0.355921	−0.177960	+	0.984038i	$0.556950\pi$
$752$	0	0
$753$	24.9848	0.910498
$754$	0	0
$755$	80.7926i	2.94034i
$756$	0	0
$757$	15.3693i	0.558607i	0.960203	+	0.279304i	$0.0901036\pi$
$757$	15.3693i	0.558607i	−0.960203	+	0.279304i	$0.909896\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	38.0540i	1.37765i
$764$	0	0
$765$	− 7.12311i	− 0.257536i
$766$	0	0
$767$	6.00000	0.216647
$768$	0	0
$769$	−15.3693	−0.554232	−0.277116	−	0.960837i	$-0.589379\pi$
$769$	−15.3693	−0.554232	−0.277116	+	0.960837i	$0.589379\pi$
$770$	0	0
$771$	− 20.1922i	− 0.727206i
$772$	0	0
$773$	− 13.3153i	− 0.478920i	−0.970906	−	0.239460i	$-0.923030\pi$
$773$	− 13.3153i	− 0.478920i	0.970906	−	0.239460i	$-0.0769703\pi$
$774$	0	0
$775$	8.63068	0.310023
$776$	0	0
$777$	14.9309	0.535642
$778$	0	0
$779$	− 6.73863i	− 0.241437i
$780$	0	0
$781$	− 21.3693i	− 0.764654i
$782$	0	0
$783$	45.8617	1.63896
$784$	0	0
$785$	−7.12311	−0.254235
$786$	0	0
$787$	5.61553i	0.200172i	0.994979	+	0.100086i	$0.0319118\pi$
$787$	5.61553i	0.200172i	−0.994979	+	0.100086i	$0.968088\pi$
$788$	0	0
$789$	35.5076i	1.26410i
$790$	0	0
$791$	29.3693	1.04425
$792$	0	0
$793$	−11.3693	−0.403736
$794$	0	0
$795$	− 72.9848i	− 2.58851i
$796$	0	0
$797$	− 36.7386i	− 1.30135i	−0.759357	−	0.650675i	$-0.774486\pi$
$797$	− 36.7386i	− 1.30135i	0.759357	−	0.650675i	$-0.225514\pi$
$798$	0	0
$799$	−38.0540	−1.34625
$800$	0	0
$801$	−4.63068	−0.163617
$802$	0	0
$803$	− 20.0000i	− 0.705785i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.26137	0.185209
$808$	0	0
$809$	11.0691	0.389170	0.194585	−	0.980886i	$-0.437664\pi$
$809$	11.0691	0.389170	0.194585	+	0.980886i	$0.437664\pi$
$810$	0	0
$811$	− 31.8617i	− 1.11882i	−0.828892	−	0.559408i	$-0.811028\pi$
$811$	− 31.8617i	− 1.11882i	0.828892	−	0.559408i	$-0.188972\pi$
$812$	0	0
$813$	− 16.6847i	− 0.585157i
$814$	0	0
$815$	54.7386	1.91741
$816$	0	0
$817$	70.8466	2.47861
$818$	0	0
$819$	− 2.00000i	− 0.0698857i
$820$	0	0
$821$	− 32.0540i	− 1.11869i	−0.828934	−	0.559346i	$-0.811052\pi$
$821$	− 32.0540i	− 1.11869i	0.828934	−	0.559346i	$-0.188948\pi$
$822$	0	0
$823$	14.6307	0.509994	0.254997	−	0.966942i	$-0.417926\pi$
$823$	14.6307	0.509994	0.254997	+	0.966942i	$0.417926\pi$
$824$	0	0
$825$	24.0000	0.835573
$826$	0	0
$827$	12.7386i	0.442966i	0.975164	+	0.221483i	$0.0710897\pi$
$827$	12.7386i	0.442966i	−0.975164	+	0.221483i	$0.928910\pi$
$828$	0	0
$829$	− 16.7386i	− 0.581357i	−0.956821	−	0.290678i	$-0.906119\pi$
$829$	− 16.7386i	− 0.581357i	0.956821	−	0.290678i	$-0.0938810\pi$
$830$	0	0
$831$	−36.4924	−1.26591
$832$	0	0
$833$	20.2462	0.701490
$834$	0	0
$835$	54.7386i	1.89431i
$836$	0	0
$837$	− 6.24621i	− 0.215901i
$838$	0	0
$839$	38.1080	1.31563	0.657816	−	0.753178i	$-0.271480\pi$
$839$	38.1080	1.31563	0.657816	+	0.753178i	$0.271480\pi$
$840$	0	0
$841$	−39.0000	−1.34483
$842$	0	0
$843$	− 5.86174i	− 0.201889i
$844$	0	0
$845$	3.56155i	0.122521i
$846$	0	0
$847$	−24.9309	−0.856635
$848$	0	0
$849$	22.2462	0.763488
$850$	0	0
$851$	0	0
$852$	0	0
$853$	9.31534i	0.318951i	0.987202	+	0.159476i	$0.0509803\pi$
$853$	9.31534i	0.318951i	−0.987202	+	0.159476i	$0.949020\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	0	0
$857$	15.7538	0.538139	0.269070	−	0.963121i	$-0.413284\pi$
$857$	15.7538	0.538139	0.269070	+	0.963121i	$0.413284\pi$
$858$	0	0
$859$	− 36.0000i	− 1.22830i	−0.789188	−	0.614152i	$-0.789498\pi$
$859$	− 36.0000i	− 1.22830i	0.789188	−	0.614152i	$-0.210502\pi$
$860$	0	0
$861$	− 6.24621i	− 0.212870i
$862$	0	0
$863$	12.0540	0.410322	0.205161	−	0.978728i	$-0.434228\pi$
$863$	12.0540	0.410322	0.205161	+	0.978728i	$0.434228\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	− 6.73863i	− 0.228856i
$868$	0	0
$869$	24.0000i	0.814144i
$870$	0	0
$871$	6.00000	0.203302
$872$	0	0
$873$	−3.36932	−0.114034
$874$	0	0
$875$	− 34.0540i	− 1.15123i
$876$	0	0
$877$	− 54.7926i	− 1.85021i	−0.379705	−	0.925107i	$-0.623975\pi$
$877$	− 54.7926i	− 1.85021i	0.379705	−	0.925107i	$-0.376025\pi$
$878$	0	0
$879$	43.0388	1.45166
$880$	0	0
$881$	34.3002	1.15560	0.577801	−	0.816177i	$-0.303911\pi$
$881$	34.3002	1.15560	0.577801	+	0.816177i	$0.303911\pi$
$882$	0	0
$883$	21.5616i	0.725604i	0.931866	+	0.362802i	$0.118180\pi$
$883$	21.5616i	0.725604i	−0.931866	+	0.362802i	$0.881820\pi$
$884$	0	0
$885$	− 33.3693i	− 1.12170i
$886$	0	0
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	50.7386	1.70172
$890$	0	0
$891$	− 14.0000i	− 0.469018i
$892$	0	0
$893$	64.1080i	2.14529i
$894$	0	0
$895$	−64.3002	−2.14932
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.26137i	0.308884i
$900$	0	0
$901$	− 46.7386i	− 1.55709i
$902$	0	0
$903$	65.6695	2.18534
$904$	0	0
$905$	−26.2462	−0.872454
$906$	0	0
$907$	− 35.4233i	− 1.17621i	−0.808784	−	0.588106i	$-0.799874\pi$
$907$	− 35.4233i	− 1.17621i	0.808784	−	0.588106i	$-0.200126\pi$
$908$	0	0
$909$	0.630683i	0.0209184i
$910$	0	0
$911$	−2.63068	−0.0871584	−0.0435792	−	0.999050i	$-0.513876\pi$
$911$	−2.63068	−0.0871584	−0.0435792	+	0.999050i	$0.513876\pi$
$912$	0	0
$913$	14.7386	0.487778
$914$	0	0
$915$	63.2311i	2.09035i
$916$	0	0
$917$	11.8078i	0.389927i
$918$	0	0
$919$	−50.2462	−1.65747	−0.828735	−	0.559642i	$-0.810939\pi$
$919$	−50.2462	−1.65747	−0.828735	+	0.559642i	$0.810939\pi$
$920$	0	0
$921$	−9.36932	−0.308729
$922$	0	0
$923$	− 10.6847i	− 0.351690i
$924$	0	0
$925$	− 20.6307i	− 0.678333i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−5.61553	−0.184240	−0.0921198	−	0.995748i	$-0.529364\pi$
$929$	−5.61553	−0.184240	−0.0921198	+	0.995748i	$0.529364\pi$
$930$	0	0
$931$	− 34.1080i	− 1.11784i
$932$	0	0
$933$	− 14.6307i	− 0.478987i
$934$	0	0
$935$	25.3693	0.829665
$936$	0	0
$937$	8.73863	0.285479	0.142739	−	0.989760i	$-0.454409\pi$
$937$	8.73863	0.285479	0.142739	+	0.989760i	$0.454409\pi$
$938$	0	0
$939$	31.3153i	1.02194i
$940$	0	0
$941$	− 25.3153i	− 0.825257i	−0.910900	−	0.412628i	$-0.864611\pi$
$941$	− 25.3153i	− 0.825257i	0.910900	−	0.412628i	$-0.135389\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−70.5464	−2.29487
$946$	0	0
$947$	− 2.00000i	− 0.0649913i	−0.999472	−	0.0324956i	$-0.989654\pi$
$947$	− 2.00000i	− 0.0649913i	0.999472	−	0.0324956i	$-0.0103455\pi$
$948$	0	0
$949$	− 10.0000i	− 0.324614i
$950$	0	0
$951$	−24.6004	−0.797722
$952$	0	0
$953$	41.8078	1.35429	0.677143	−	0.735851i	$-0.263218\pi$
$953$	41.8078	1.35429	0.677143	+	0.735851i	$0.263218\pi$
$954$	0	0
$955$	− 19.1231i	− 0.618809i
$956$	0	0
$957$	25.7538i	0.832502i
$958$	0	0
$959$	46.7386	1.50927
$960$	0	0
$961$	−29.7386	−0.959311
$962$	0	0
$963$	− 6.73863i	− 0.217149i
$964$	0	0
$965$	83.2311i	2.67930i
$966$	0	0
$967$	−53.4233	−1.71798	−0.858989	−	0.511995i	$-0.828907\pi$
$967$	−53.4233	−1.71798	−0.858989	+	0.511995i	$0.828907\pi$
$968$	0	0
$969$	33.3693	1.07198
$970$	0	0
$971$	− 3.31534i	− 0.106394i	−0.998584	−	0.0531972i	$-0.983059\pi$
$971$	− 3.31534i	− 0.106394i	0.998584	−	0.0531972i	$-0.0169412\pi$
$972$	0	0
$973$	16.6847i	0.534886i
$974$	0	0
$975$	12.0000	0.384308
$976$	0	0
$977$	−14.9848	−0.479408	−0.239704	−	0.970846i	$-0.577050\pi$
$977$	−14.9848	−0.479408	−0.239704	+	0.970846i	$0.577050\pi$
$978$	0	0
$979$	− 16.4924i	− 0.527100i
$980$	0	0
$981$	− 6.00000i	− 0.191565i
$982$	0	0
$983$	21.4233	0.683297	0.341648	−	0.939828i	$-0.389015\pi$
$983$	21.4233	0.683297	0.341648	+	0.939828i	$0.389015\pi$
$984$	0	0
$985$	4.68466	0.149266
$986$	0	0
$987$	59.4233i	1.89146i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−21.3693	−0.678819	−0.339409	−	0.940639i	$-0.610227\pi$
$991$	−21.3693	−0.678819	−0.339409	+	0.940639i	$0.610227\pi$
$992$	0	0
$993$	5.26137	0.166964
$994$	0	0
$995$	− 33.3693i	− 1.05788i
$996$	0	0
$997$	− 4.73863i	− 0.150074i	−0.997181	−	0.0750370i	$-0.976093\pi$
$997$	− 4.73863i	− 0.150074i	0.997181	−	0.0750370i	$-0.0239075\pi$
$998$	0	0
$999$	−14.9309	−0.472392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.b.u.1665.3		4
4.3	odd	2		3328.2.b.ba.1665.2		4
8.3	odd	2		3328.2.b.ba.1665.3		4
8.5	even	2	inner	3328.2.b.u.1665.2		4
16.3	odd	4		416.2.a.c.1.2	✓	2
16.5	even	4		832.2.a.l.1.2		2
16.11	odd	4		832.2.a.o.1.1		2
16.13	even	4		416.2.a.e.1.1	yes	2
48.5	odd	4		7488.2.a.cf.1.1		2
48.11	even	4		7488.2.a.ce.1.1		2
48.29	odd	4		3744.2.a.y.1.2		2
48.35	even	4		3744.2.a.x.1.2		2
208.51	odd	4		5408.2.a.p.1.2		2
208.77	even	4		5408.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.a.c.1.2	✓	2	16.3	odd	4
416.2.a.e.1.1	yes	2	16.13	even	4
832.2.a.l.1.2		2	16.5	even	4
832.2.a.o.1.1		2	16.11	odd	4
3328.2.b.u.1665.2		4	8.5	even	2	inner
3328.2.b.u.1665.3		4	1.1	even	1	trivial
3328.2.b.ba.1665.2		4	4.3	odd	2
3328.2.b.ba.1665.3		4	8.3	odd	2
3744.2.a.x.1.2		2	48.35	even	4
3744.2.a.y.1.2		2	48.29	odd	4
5408.2.a.p.1.2		2	208.51	odd	4
5408.2.a.bd.1.1		2	208.77	even	4
7488.2.a.ce.1.1		2	48.11	even	4
7488.2.a.cf.1.1		2	48.5	odd	4

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{3} -3.56155i q^{5} -3.56155 q^{7} +0.561553 q^{9} +2.00000i q^{11} +1.00000i q^{13} +5.56155 q^{15} +3.56155 q^{17} -6.00000i q^{19} -5.56155i q^{21} -7.68466 q^{25} +5.56155i q^{27} -8.24621i q^{29} -1.12311 q^{31} -3.12311 q^{33} +12.6847i q^{35} +2.68466i q^{37} -1.56155 q^{39} +1.12311 q^{41} +11.8078i q^{43} -2.00000i q^{45} -10.6847 q^{47} +5.68466 q^{49} +5.56155i q^{51} -13.1231i q^{53} +7.12311 q^{55} +9.36932 q^{57} -6.00000i q^{59} +11.3693i q^{61} -2.00000 q^{63} +3.56155 q^{65} -6.00000i q^{67} -10.6847 q^{71} -10.0000 q^{73} -12.0000i q^{75} -7.12311i q^{77} +12.0000 q^{79} -7.00000 q^{81} -7.36932i q^{83} -12.6847i q^{85} +12.8769 q^{87} -8.24621 q^{89} -3.56155i q^{91} -1.75379i q^{93} -21.3693 q^{95} -6.00000 q^{97} +1.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{7} - 6 q^{9} + 14 q^{15} + 6 q^{17} - 6 q^{25} + 12 q^{31} + 4 q^{33} + 2 q^{39} - 12 q^{41} - 18 q^{47} - 2 q^{49} + 12 q^{55} - 12 q^{57} - 8 q^{63} + 6 q^{65} - 18 q^{71} - 40 q^{73} + 48 q^{79}+ \cdots - 24 q^{97}+O(q^{100}) \) 4 * q - 6 * q^7 - 6 * q^9 + 14 * q^15 + 6 * q^17 - 6 * q^25 + 12 * q^31 + 4 * q^33 + 2 * q^39 - 12 * q^41 - 18 * q^47 - 2 * q^49 + 12 * q^55 - 12 * q^57 - 8 * q^63 + 6 * q^65 - 18 * q^71 - 40 * q^73 + 48 * q^79 - 28 * q^81 + 68 * q^87 - 36 * q^95 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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