LMFDB - Embedded newform 3328.2.b.u.1665.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3328,2,Mod(1665,3328)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			1665.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	3328.1665
Dual form			3328.2.b.u.1665.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155i q^{3} -0.561553i q^{5} +0.561553 q^{7} -3.56155 q^{9} -2.00000i q^{11} -1.00000i q^{13} +1.43845 q^{15} -0.561553 q^{17} +6.00000i q^{19} +1.43845i q^{21} +4.68466 q^{25} -1.43845i q^{27} -8.24621i q^{29} +7.12311 q^{31} +5.12311 q^{33} -0.315342i q^{35} +9.68466i q^{37} +2.56155 q^{39} -7.12311 q^{41} +8.80776i q^{43} +2.00000i q^{45} +1.68466 q^{47} -6.68466 q^{49} -1.43845i q^{51} +4.87689i q^{53} -1.12311 q^{55} -15.3693 q^{57} +6.00000i q^{59} +13.3693i q^{61} -2.00000 q^{63} -0.561553 q^{65} +6.00000i q^{67} +1.68466 q^{71} -10.0000 q^{73} +12.0000i q^{75} -1.12311i q^{77} +12.0000 q^{79} -7.00000 q^{81} -17.3693i q^{83} +0.315342i q^{85} +21.1231 q^{87} +8.24621 q^{89} -0.561553i q^{91} +18.2462i q^{93} +3.36932 q^{95} -6.00000 q^{97} +7.12311i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.56155i	1.47891i	0.673204	+	0.739457i	$0.264917\pi$
$3$	2.56155i	1.47891i	−0.673204	+	0.739457i	$0.735083\pi$
$4$	0	0
$5$	− 0.561553i	− 0.251134i	−0.992085	−	0.125567i	$-0.959925\pi$
$5$	− 0.561553i	− 0.251134i	0.992085	−	0.125567i	$-0.0400750\pi$
$6$	0	0
$7$	0.561553	0.212247	0.106124	−	0.994353i	$-0.466156\pi$
$7$	0.561553	0.212247	0.106124	+	0.994353i	$0.466156\pi$
$8$	0	0
$9$	−3.56155	−1.18718
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	1.43845	0.371405
$16$	0	0
$17$	−0.561553	−0.136197	−0.0680983	−	0.997679i	$-0.521693\pi$
$17$	−0.561553	−0.136197	−0.0680983	+	0.997679i	$0.521693\pi$
$18$	0	0
$19$	6.00000i	1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$	6.00000i	1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$	0	0
$21$	1.43845i	0.313895i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	4.68466	0.936932
$26$	0	0
$27$	− 1.43845i	− 0.276829i
$28$	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
$31$	7.12311	1.27935	0.639674	−	0.768647i	$-0.279069\pi$
$31$	7.12311	1.27935	0.639674	+	0.768647i	$0.279069\pi$
$32$	0	0
$33$	5.12311	0.891818
$34$	0	0
$35$	− 0.315342i	− 0.0533025i
$36$	0	0
$37$	9.68466i	1.59215i	0.605199	+	0.796074i	$0.293093\pi$
$37$	9.68466i	1.59215i	−0.605199	+	0.796074i	$0.706907\pi$
$38$	0	0
$39$	2.56155	0.410177
$40$	0	0
$41$	−7.12311	−1.11244	−0.556221	−	0.831034i	$-0.687749\pi$
$41$	−7.12311	−1.11244	−0.556221	+	0.831034i	$0.687749\pi$
$42$	0	0
$43$	8.80776i	1.34317i	0.740927	+	0.671586i	$0.234386\pi$
$43$	8.80776i	1.34317i	−0.740927	+	0.671586i	$0.765614\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	1.68466	0.245733	0.122866	−	0.992423i	$-0.460791\pi$
$47$	1.68466	0.245733	0.122866	+	0.992423i	$0.460791\pi$
$48$	0	0
$49$	−6.68466	−0.954951
$50$	0	0
$51$	− 1.43845i	− 0.201423i
$52$	0	0
$53$	4.87689i	0.669893i	0.942237	+	0.334946i	$0.108718\pi$
$53$	4.87689i	0.669893i	−0.942237	+	0.334946i	$0.891282\pi$
$54$	0	0
$55$	−1.12311	−0.151440
$56$	0	0
$57$	−15.3693	−2.03572
$58$	0	0
$59$	6.00000i	0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$	6.00000i	0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$	0	0
$61$	13.3693i	1.71177i	0.517170	+	0.855883i	$0.326986\pi$
$61$	13.3693i	1.71177i	−0.517170	+	0.855883i	$0.673014\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−0.561553	−0.0696521
$66$	0	0
$67$	6.00000i	0.733017i	0.930415	+	0.366508i	$0.119447\pi$
$67$	6.00000i	0.733017i	−0.930415	+	0.366508i	$0.880553\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.68466	0.199932	0.0999661	−	0.994991i	$-0.468127\pi$
$71$	1.68466	0.199932	0.0999661	+	0.994991i	$0.468127\pi$
$72$	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
$77$	− 1.12311i	− 0.127990i
$78$	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
$83$	− 17.3693i	− 1.90653i	−0.302135	−	0.953265i	$-0.597699\pi$
$83$	− 17.3693i	− 1.90653i	0.302135	−	0.953265i	$-0.402301\pi$
$84$	0	0
$85$	0.315342i	0.0342036i
$86$	0	0
$87$	21.1231	2.26463
$88$	0	0
$89$	8.24621	0.874097	0.437048	−	0.899438i	$-0.356024\pi$
$89$	8.24621	0.874097	0.437048	+	0.899438i	$0.356024\pi$
$90$	0	0
$91$	− 0.561553i	− 0.0588667i
$92$	0	0
$93$	18.2462i	1.89204i
$94$	0	0
$95$	3.36932	0.345685
$96$	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
$99$	7.12311i	0.715899i
$100$	0	0
$101$	7.12311i	0.708776i	0.935099	+	0.354388i	$0.115311\pi$
$101$	7.12311i	0.708776i	−0.935099	+	0.354388i	$0.884689\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0.807764	0.0788297
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	− 1.68466i	− 0.161361i	−0.996740	−	0.0806805i	$-0.974291\pi$
$109$	− 1.68466i	− 0.161361i	0.996740	−	0.0806805i	$-0.0257093\pi$
$110$	0	0
$111$	−24.8078	−2.35465
$112$	0	0
$113$	8.24621	0.775738	0.387869	−	0.921714i	$-0.373211\pi$
$113$	8.24621	0.775738	0.387869	+	0.921714i	$0.373211\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.56155i	0.329266i
$118$	0	0
$119$	−0.315342	−0.0289073
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	− 18.2462i	− 1.64521i
$124$	0	0
$125$	− 5.43845i	− 0.486430i
$126$	0	0
$127$	2.24621	0.199319	0.0996595	−	0.995022i	$-0.468225\pi$
$127$	2.24621	0.199319	0.0996595	+	0.995022i	$0.468225\pi$
$128$	0	0
$129$	−22.5616	−1.98643
$130$	0	0
$131$	15.6847i	1.37037i	0.728367	+	0.685187i	$0.240280\pi$
$131$	15.6847i	1.37037i	−0.728367	+	0.685187i	$0.759720\pi$
$132$	0	0
$133$	3.36932i	0.292157i
$134$	0	0
$135$	−0.807764	−0.0695213
$136$	0	0
$137$	−4.87689	−0.416661	−0.208331	−	0.978058i	$-0.566803\pi$
$137$	−4.87689	−0.416661	−0.208331	+	0.978058i	$0.566803\pi$
$138$	0	0
$139$	− 7.68466i	− 0.651804i	−0.945404	−	0.325902i	$-0.894332\pi$
$139$	− 7.68466i	− 0.651804i	0.945404	−	0.325902i	$-0.105668\pi$
$140$	0	0
$141$	4.31534i	0.363417i
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−4.63068	−0.384557
$146$	0	0
$147$	− 17.1231i	− 1.41229i
$148$	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
$151$	−10.3153	−0.839451	−0.419725	−	0.907651i	$-0.637874\pi$
$151$	−10.3153	−0.839451	−0.419725	+	0.907651i	$0.637874\pi$
$152$	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	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	−12.4924	−0.990714
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.36932i	0.733862i	0.930248	+	0.366931i	$0.119591\pi$
$163$	9.36932i	0.733862i	−0.930248	+	0.366931i	$0.880409\pi$
$164$	0	0
$165$	− 2.87689i	− 0.223966i
$166$	0	0
$167$	9.36932	0.725020	0.362510	−	0.931980i	$-0.381920\pi$
$167$	9.36932	0.725020	0.362510	+	0.931980i	$0.381920\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 21.3693i	− 1.63415i
$172$	0	0
$173$	21.3693i	1.62468i	0.583185	+	0.812340i	$0.301806\pi$
$173$	21.3693i	1.62468i	−0.583185	+	0.812340i	$0.698194\pi$
$174$	0	0
$175$	2.63068	0.198861
$176$	0	0
$177$	−15.3693	−1.15523
$178$	0	0
$179$	− 19.0540i	− 1.42416i	−0.702098	−	0.712080i	$-0.747753\pi$
$179$	− 19.0540i	− 1.42416i	0.702098	−	0.712080i	$-0.252247\pi$
$180$	0	0
$181$	− 17.3693i	− 1.29105i	−0.763739	−	0.645526i	$-0.776638\pi$
$181$	− 17.3693i	− 1.29105i	0.763739	−	0.645526i	$-0.223362\pi$
$182$	0	0
$183$	−34.2462	−2.53155
$184$	0	0
$185$	5.43845	0.399843
$186$	0	0
$187$	1.12311i	0.0821296i
$188$	0	0
$189$	− 0.807764i	− 0.0587562i
$190$	0	0
$191$	−19.3693	−1.40151	−0.700757	−	0.713400i	$-0.747154\pi$
$191$	−19.3693	−1.40151	−0.700757	+	0.713400i	$0.747154\pi$
$192$	0	0
$193$	1.36932	0.0985656	0.0492828	−	0.998785i	$-0.484306\pi$
$193$	1.36932	0.0985656	0.0492828	+	0.998785i	$0.484306\pi$
$194$	0	0
$195$	− 1.43845i	− 0.103009i
$196$	0	0
$197$	− 13.6847i	− 0.974992i	−0.873125	−	0.487496i	$-0.837910\pi$
$197$	− 13.6847i	− 0.974992i	0.873125	−	0.487496i	$-0.162090\pi$
$198$	0	0
$199$	−15.3693	−1.08950	−0.544751	−	0.838598i	$-0.683376\pi$
$199$	−15.3693	−1.08950	−0.544751	+	0.838598i	$0.683376\pi$
$200$	0	0
$201$	−15.3693	−1.08407
$202$	0	0
$203$	− 4.63068i	− 0.325010i
$204$	0	0
$205$	4.00000i	0.279372i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	3.19224i	0.219763i	0.993945	+	0.109881i	$0.0350471\pi$
$211$	3.19224i	0.219763i	−0.993945	+	0.109881i	$0.964953\pi$
$212$	0	0
$213$	4.31534i	0.295682i
$214$	0	0
$215$	4.94602	0.337316
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	− 25.6155i	− 1.73094i
$220$	0	0
$221$	0.561553i	0.0377741i
$222$	0	0
$223$	26.8078	1.79518	0.897590	−	0.440831i	$-0.145316\pi$
$223$	26.8078	1.79518	0.897590	+	0.440831i	$0.145316\pi$
$224$	0	0
$225$	−16.6847	−1.11231
$226$	0	0
$227$	21.3693i	1.41833i	0.705042	+	0.709166i	$0.250928\pi$
$227$	21.3693i	1.41833i	−0.705042	+	0.709166i	$0.749072\pi$
$228$	0	0
$229$	− 13.6847i	− 0.904308i	−0.891940	−	0.452154i	$-0.850656\pi$
$229$	− 13.6847i	− 0.904308i	0.891940	−	0.452154i	$-0.149344\pi$
$230$	0	0
$231$	2.87689	0.189286
$232$	0	0
$233$	−13.6847	−0.896512	−0.448256	−	0.893905i	$-0.647955\pi$
$233$	−13.6847	−0.896512	−0.448256	+	0.893905i	$0.647955\pi$
$234$	0	0
$235$	− 0.946025i	− 0.0617118i
$236$	0	0
$237$	30.7386i	1.99669i
$238$	0	0
$239$	21.6847	1.40266	0.701332	−	0.712835i	$-0.252589\pi$
$239$	21.6847	1.40266	0.701332	+	0.712835i	$0.252589\pi$
$240$	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
$243$	− 22.2462i	− 1.42710i
$244$	0	0
$245$	3.75379i	0.239821i
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	44.4924	2.81959
$250$	0	0
$251$	16.0000i	1.00991i	0.863145	+	0.504956i	$0.168491\pi$
$251$	16.0000i	1.00991i	−0.863145	+	0.504956i	$0.831509\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−0.807764	−0.0505841
$256$	0	0
$257$	15.9309	0.993740	0.496870	−	0.867825i	$-0.334483\pi$
$257$	15.9309	0.993740	0.496870	+	0.867825i	$0.334483\pi$
$258$	0	0
$259$	5.43845i	0.337929i
$260$	0	0
$261$	29.3693i	1.81792i
$262$	0	0
$263$	−26.7386	−1.64877	−0.824387	−	0.566026i	$-0.808480\pi$
$263$	−26.7386	−1.64877	−0.824387	+	0.566026i	$0.808480\pi$
$264$	0	0
$265$	2.73863	0.168233
$266$	0	0
$267$	21.1231i	1.29271i
$268$	0	0
$269$	− 21.3693i	− 1.30291i	−0.758687	−	0.651455i	$-0.774159\pi$
$269$	− 21.3693i	− 1.30291i	0.758687	−	0.651455i	$-0.225841\pi$
$270$	0	0
$271$	1.68466	0.102336	0.0511679	−	0.998690i	$-0.483706\pi$
$271$	1.68466	0.102336	0.0511679	+	0.998690i	$0.483706\pi$
$272$	0	0
$273$	1.43845	0.0870588
$274$	0	0
$275$	− 9.36932i	− 0.564991i
$276$	0	0
$277$	1.36932i	0.0822743i	0.999154	+	0.0411371i	$0.0130981\pi$
$277$	1.36932i	0.0822743i	−0.999154	+	0.0411371i	$0.986902\pi$
$278$	0	0
$279$	−25.3693	−1.51882
$280$	0	0
$281$	−20.2462	−1.20779	−0.603894	−	0.797065i	$-0.706385\pi$
$281$	−20.2462	−1.20779	−0.603894	+	0.797065i	$0.706385\pi$
$282$	0	0
$283$	− 2.24621i	− 0.133523i	−0.997769	−	0.0667617i	$-0.978733\pi$
$283$	− 2.24621i	− 0.133523i	0.997769	−	0.0667617i	$-0.0212667\pi$
$284$	0	0
$285$	8.63068i	0.511238i
$286$	0	0
$287$	−4.00000	−0.236113
$288$	0	0
$289$	−16.6847	−0.981450
$290$	0	0
$291$	− 15.3693i	− 0.900965i
$292$	0	0
$293$	23.4384i	1.36929i	0.728877	+	0.684644i	$0.240042\pi$
$293$	23.4384i	1.36929i	−0.728877	+	0.684644i	$0.759958\pi$
$294$	0	0
$295$	3.36932	0.196169
$296$	0	0
$297$	−2.87689	−0.166934
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.94602i	0.285084i
$302$	0	0
$303$	−18.2462	−1.04822
$304$	0	0
$305$	7.50758	0.429883
$306$	0	0
$307$	− 6.00000i	− 0.342438i	−0.985233	−	0.171219i	$-0.945229\pi$
$307$	− 6.00000i	− 0.342438i	0.985233	−	0.171219i	$-0.0547706\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.3693	0.871514	0.435757	−	0.900064i	$-0.356481\pi$
$311$	15.3693	0.871514	0.435757	+	0.900064i	$0.356481\pi$
$312$	0	0
$313$	−17.0540	−0.963948	−0.481974	−	0.876186i	$-0.660080\pi$
$313$	−17.0540	−0.963948	−0.481974	+	0.876186i	$0.660080\pi$
$314$	0	0
$315$	1.12311i	0.0632798i
$316$	0	0
$317$	− 32.2462i	− 1.81113i	−0.424210	−	0.905564i	$-0.639448\pi$
$317$	− 32.2462i	− 1.81113i	0.424210	−	0.905564i	$-0.360552\pi$
$318$	0	0
$319$	−16.4924	−0.923398
$320$	0	0
$321$	−30.7386	−1.71566
$322$	0	0
$323$	− 3.36932i	− 0.187474i
$324$	0	0
$325$	− 4.68466i	− 0.259858i
$326$	0	0
$327$	4.31534	0.238639
$328$	0	0
$329$	0.946025	0.0521560
$330$	0	0
$331$	− 21.3693i	− 1.17456i	−0.809382	−	0.587282i	$-0.800198\pi$
$331$	− 21.3693i	− 1.17456i	0.809382	−	0.587282i	$-0.199802\pi$
$332$	0	0
$333$	− 34.4924i	− 1.89017i
$334$	0	0
$335$	3.36932	0.184085
$336$	0	0
$337$	1.05398	0.0574137	0.0287068	−	0.999588i	$-0.490861\pi$
$337$	1.05398	0.0574137	0.0287068	+	0.999588i	$0.490861\pi$
$338$	0	0
$339$	21.1231i	1.14725i
$340$	0	0
$341$	− 14.2462i	− 0.771476i
$342$	0	0
$343$	−7.68466	−0.414933
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.3153i	0.875853i	0.899011	+	0.437927i	$0.144287\pi$
$347$	16.3153i	0.875853i	−0.899011	+	0.437927i	$0.855713\pi$
$348$	0	0
$349$	29.0540i	1.55522i	0.628745	+	0.777612i	$0.283569\pi$
$349$	29.0540i	1.55522i	−0.628745	+	0.777612i	$0.716431\pi$
$350$	0	0
$351$	−1.43845	−0.0767786
$352$	0	0
$353$	−25.8617	−1.37648	−0.688241	−	0.725482i	$-0.741617\pi$
$353$	−25.8617	−1.37648	−0.688241	+	0.725482i	$0.741617\pi$
$354$	0	0
$355$	− 0.946025i	− 0.0502098i
$356$	0	0
$357$	− 0.807764i	− 0.0427514i
$358$	0	0
$359$	17.3693	0.916717	0.458359	−	0.888767i	$-0.348437\pi$
$359$	17.3693	0.916717	0.458359	+	0.888767i	$0.348437\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	17.9309i	0.941127i
$364$	0	0
$365$	5.61553i	0.293930i
$366$	0	0
$367$	−27.3693	−1.42867	−0.714333	−	0.699806i	$-0.753270\pi$
$367$	−27.3693	−1.42867	−0.714333	+	0.699806i	$0.753270\pi$
$368$	0	0
$369$	25.3693	1.32067
$370$	0	0
$371$	2.73863i	0.142183i
$372$	0	0
$373$	13.3693i	0.692237i	0.938191	+	0.346118i	$0.112500\pi$
$373$	13.3693i	0.692237i	−0.938191	+	0.346118i	$0.887500\pi$
$374$	0	0
$375$	13.9309	0.719387
$376$	0	0
$377$	−8.24621	−0.424701
$378$	0	0
$379$	− 16.8769i	− 0.866908i	−0.901176	−	0.433454i	$-0.857295\pi$
$379$	− 16.8769i	− 0.866908i	0.901176	−	0.433454i	$-0.142705\pi$
$380$	0	0
$381$	5.75379i	0.294776i
$382$	0	0
$383$	−1.68466	−0.0860820	−0.0430410	−	0.999073i	$-0.513705\pi$
$383$	−1.68466	−0.0860820	−0.0430410	+	0.999073i	$0.513705\pi$
$384$	0	0
$385$	−0.630683	−0.0321426
$386$	0	0
$387$	− 31.3693i	− 1.59459i
$388$	0	0
$389$	− 20.2462i	− 1.02652i	−0.858232	−	0.513262i	$-0.828437\pi$
$389$	− 20.2462i	− 1.02652i	0.858232	−	0.513262i	$-0.171563\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−40.1771	−2.02667
$394$	0	0
$395$	− 6.73863i	− 0.339057i
$396$	0	0
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	0	0
$399$	−8.63068	−0.432075
$400$	0	0
$401$	31.1231	1.55421	0.777107	−	0.629369i	$-0.216686\pi$
$401$	31.1231	1.55421	0.777107	+	0.629369i	$0.216686\pi$
$402$	0	0
$403$	− 7.12311i	− 0.354827i
$404$	0	0
$405$	3.93087i	0.195326i
$406$	0	0
$407$	19.3693	0.960101
$408$	0	0
$409$	6.63068	0.327866	0.163933	−	0.986471i	$-0.447582\pi$
$409$	6.63068	0.327866	0.163933	+	0.986471i	$0.447582\pi$
$410$	0	0
$411$	− 12.4924i	− 0.616206i
$412$	0	0
$413$	3.36932i	0.165793i
$414$	0	0
$415$	−9.75379	−0.478795
$416$	0	0
$417$	19.6847	0.963962
$418$	0	0
$419$	15.0540i	0.735435i	0.929938	+	0.367717i	$0.119861\pi$
$419$	15.0540i	0.735435i	−0.929938	+	0.367717i	$0.880139\pi$
$420$	0	0
$421$	33.6847i	1.64169i	0.571151	+	0.820845i	$0.306497\pi$
$421$	33.6847i	1.64169i	−0.571151	+	0.820845i	$0.693503\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−2.63068	−0.127607
$426$	0	0
$427$	7.50758i	0.363317i
$428$	0	0
$429$	− 5.12311i	− 0.247346i
$430$	0	0
$431$	5.68466	0.273820	0.136910	−	0.990583i	$-0.456283\pi$
$431$	5.68466	0.273820	0.136910	+	0.990583i	$0.456283\pi$
$432$	0	0
$433$	14.3153	0.687951	0.343976	−	0.938979i	$-0.388226\pi$
$433$	14.3153	0.687951	0.343976	+	0.938979i	$0.388226\pi$
$434$	0	0
$435$	− 11.8617i	− 0.568727i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	41.6155	1.98620	0.993100	−	0.117267i	$-0.0374134\pi$
$439$	41.6155	1.98620	0.993100	+	0.117267i	$0.0374134\pi$
$440$	0	0
$441$	23.8078	1.13370
$442$	0	0
$443$	7.68466i	0.365109i	0.983196	+	0.182555i	$0.0584366\pi$
$443$	7.68466i	0.365109i	−0.983196	+	0.182555i	$0.941563\pi$
$444$	0	0
$445$	− 4.63068i	− 0.219515i
$446$	0	0
$447$	−21.1231	−0.999089
$448$	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
$451$	14.2462i	0.670828i
$452$	0	0
$453$	− 26.4233i	− 1.24147i
$454$	0	0
$455$	−0.315342	−0.0147834
$456$	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
$459$	0.807764i	0.0377032i
$460$	0	0
$461$	− 21.1922i	− 0.987021i	−0.869740	−	0.493510i	$-0.835713\pi$
$461$	− 21.1922i	− 0.987021i	0.869740	−	0.493510i	$-0.164287\pi$
$462$	0	0
$463$	23.6155	1.09751	0.548753	−	0.835984i	$-0.315103\pi$
$463$	23.6155	1.09751	0.548753	+	0.835984i	$0.315103\pi$
$464$	0	0
$465$	10.2462	0.475157
$466$	0	0
$467$	14.7386i	0.682023i	0.940059	+	0.341011i	$0.110769\pi$
$467$	14.7386i	0.682023i	−0.940059	+	0.341011i	$0.889231\pi$
$468$	0	0
$469$	3.36932i	0.155581i
$470$	0	0
$471$	−5.12311	−0.236060
$472$	0	0
$473$	17.6155	0.809963
$474$	0	0
$475$	28.1080i	1.28968i
$476$	0	0
$477$	− 17.3693i	− 0.795286i
$478$	0	0
$479$	33.6847	1.53909	0.769546	−	0.638592i	$-0.220483\pi$
$479$	33.6847	1.53909	0.769546	+	0.638592i	$0.220483\pi$
$480$	0	0
$481$	9.68466	0.441582
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.36932i	0.152993i
$486$	0	0
$487$	19.1231	0.866551	0.433275	−	0.901262i	$-0.357358\pi$
$487$	19.1231	0.866551	0.433275	+	0.901262i	$0.357358\pi$
$488$	0	0
$489$	−24.0000	−1.08532
$490$	0	0
$491$	− 35.6847i	− 1.61043i	−0.592986	−	0.805213i	$-0.702051\pi$
$491$	− 35.6847i	− 1.61043i	0.592986	−	0.805213i	$-0.297949\pi$
$492$	0	0
$493$	4.63068i	0.208555i
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	0.946025	0.0424350
$498$	0	0
$499$	16.8769i	0.755514i	0.925905	+	0.377757i	$0.123304\pi$
$499$	16.8769i	0.755514i	−0.925905	+	0.377757i	$0.876696\pi$
$500$	0	0
$501$	24.0000i	1.07224i
$502$	0	0
$503$	4.63068	0.206472	0.103236	−	0.994657i	$-0.467080\pi$
$503$	4.63068	0.206472	0.103236	+	0.994657i	$0.467080\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	− 2.56155i	− 0.113763i
$508$	0	0
$509$	− 10.4924i	− 0.465068i	−0.972588	−	0.232534i	$-0.925298\pi$
$509$	− 10.4924i	− 0.465068i	0.972588	−	0.232534i	$-0.0747018\pi$
$510$	0	0
$511$	−5.61553	−0.248416
$512$	0	0
$513$	8.63068	0.381054
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 3.36932i	− 0.148182i
$518$	0	0
$519$	−54.7386	−2.40276
$520$	0	0
$521$	−12.5616	−0.550332	−0.275166	−	0.961397i	$-0.588733\pi$
$521$	−12.5616	−0.550332	−0.275166	+	0.961397i	$0.588733\pi$
$522$	0	0
$523$	40.4924i	1.77061i	0.465011	+	0.885305i	$0.346050\pi$
$523$	40.4924i	1.77061i	−0.465011	+	0.885305i	$0.653950\pi$
$524$	0	0
$525$	6.73863i	0.294098i
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 21.3693i	− 0.927349i
$532$	0	0
$533$	7.12311i	0.308536i
$534$	0	0
$535$	6.73863	0.291337
$536$	0	0
$537$	48.8078	2.10621
$538$	0	0
$539$	13.3693i	0.575857i
$540$	0	0
$541$	− 18.3153i	− 0.787438i	−0.919231	−	0.393719i	$-0.871188\pi$
$541$	− 18.3153i	− 0.787438i	0.919231	−	0.393719i	$-0.128812\pi$
$542$	0	0
$543$	44.4924	1.90935
$544$	0	0
$545$	−0.946025	−0.0405232
$546$	0	0
$547$	23.0540i	0.985717i	0.870110	+	0.492858i	$0.164048\pi$
$547$	23.0540i	0.985717i	−0.870110	+	0.492858i	$0.835952\pi$
$548$	0	0
$549$	− 47.6155i	− 2.03218i
$550$	0	0
$551$	49.4773	2.10780
$552$	0	0
$553$	6.73863	0.286556
$554$	0	0
$555$	13.9309i	0.591332i
$556$	0	0
$557$	− 43.3002i	− 1.83469i	−0.398096	−	0.917344i	$-0.630329\pi$
$557$	− 43.3002i	− 1.83469i	0.398096	−	0.917344i	$-0.369671\pi$
$558$	0	0
$559$	8.80776	0.372529
$560$	0	0
$561$	−2.87689	−0.121463
$562$	0	0
$563$	43.6847i	1.84109i	0.390638	+	0.920544i	$0.372254\pi$
$563$	43.6847i	1.84109i	−0.390638	+	0.920544i	$0.627746\pi$
$564$	0	0
$565$	− 4.63068i	− 0.194814i
$566$	0	0
$567$	−3.93087	−0.165081
$568$	0	0
$569$	25.6847	1.07676	0.538378	−	0.842703i	$-0.319037\pi$
$569$	25.6847	1.07676	0.538378	+	0.842703i	$0.319037\pi$
$570$	0	0
$571$	17.4384i	0.729776i	0.931051	+	0.364888i	$0.118893\pi$
$571$	17.4384i	0.729776i	−0.931051	+	0.364888i	$0.881107\pi$
$572$	0	0
$573$	− 49.6155i	− 2.07272i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.73863	−0.197272	−0.0986360	−	0.995124i	$-0.531448\pi$
$577$	−4.73863	−0.197272	−0.0986360	+	0.995124i	$0.531448\pi$
$578$	0	0
$579$	3.50758i	0.145770i
$580$	0	0
$581$	− 9.75379i	− 0.404655i
$582$	0	0
$583$	9.75379	0.403961
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	9.36932i	0.386713i	0.981129	+	0.193357i	$0.0619374\pi$
$587$	9.36932i	0.386713i	−0.981129	+	0.193357i	$0.938063\pi$
$588$	0	0
$589$	42.7386i	1.76101i
$590$	0	0
$591$	35.0540	1.44193
$592$	0	0
$593$	20.2462	0.831412	0.415706	−	0.909499i	$-0.363534\pi$
$593$	20.2462	0.831412	0.415706	+	0.909499i	$0.363534\pi$
$594$	0	0
$595$	0.177081i	0.00725961i
$596$	0	0
$597$	− 39.3693i	− 1.61128i
$598$	0	0
$599$	−35.3693	−1.44515	−0.722576	−	0.691292i	$-0.757042\pi$
$599$	−35.3693	−1.44515	−0.722576	+	0.691292i	$0.757042\pi$
$600$	0	0
$601$	21.0540	0.858810	0.429405	−	0.903112i	$-0.358723\pi$
$601$	21.0540	0.858810	0.429405	+	0.903112i	$0.358723\pi$
$602$	0	0
$603$	− 21.3693i	− 0.870226i
$604$	0	0
$605$	− 3.93087i	− 0.159813i
$606$	0	0
$607$	31.8617	1.29323	0.646614	−	0.762817i	$-0.276184\pi$
$607$	31.8617	1.29323	0.646614	+	0.762817i	$0.276184\pi$
$608$	0	0
$609$	11.8617	0.480662
$610$	0	0
$611$	− 1.68466i	− 0.0681540i
$612$	0	0
$613$	− 10.0000i	− 0.403896i	−0.979396	−	0.201948i	$-0.935273\pi$
$613$	− 10.0000i	− 0.403896i	0.979396	−	0.201948i	$-0.0647272\pi$
$614$	0	0
$615$	−10.2462	−0.413167
$616$	0	0
$617$	23.6155	0.950725	0.475363	−	0.879790i	$-0.342317\pi$
$617$	23.6155	0.950725	0.475363	+	0.879790i	$0.342317\pi$
$618$	0	0
$619$	− 9.36932i	− 0.376585i	−0.982113	−	0.188292i	$-0.939705\pi$
$619$	− 9.36932i	− 0.376585i	0.982113	−	0.188292i	$-0.0602952\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.63068	0.185524
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	30.7386i	1.22758i
$628$	0	0
$629$	− 5.43845i	− 0.216845i
$630$	0	0
$631$	−41.0540	−1.63433	−0.817166	−	0.576402i	$-0.804456\pi$
$631$	−41.0540	−1.63433	−0.817166	+	0.576402i	$0.804456\pi$
$632$	0	0
$633$	−8.17708	−0.325010
$634$	0	0
$635$	− 1.26137i	− 0.0500558i
$636$	0	0
$637$	6.68466i	0.264856i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	10.4924	0.414426	0.207213	−	0.978296i	$-0.433561\pi$
$641$	10.4924	0.414426	0.207213	+	0.978296i	$0.433561\pi$
$642$	0	0
$643$	− 18.0000i	− 0.709851i	−0.934895	−	0.354925i	$-0.884506\pi$
$643$	− 18.0000i	− 0.709851i	0.934895	−	0.354925i	$-0.115494\pi$
$644$	0	0
$645$	12.6695i	0.498861i
$646$	0	0
$647$	30.1080	1.18367	0.591833	−	0.806061i	$-0.298405\pi$
$647$	30.1080	1.18367	0.591833	+	0.806061i	$0.298405\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	10.2462i	0.401581i
$652$	0	0
$653$	− 47.6155i	− 1.86334i	−0.363306	−	0.931670i	$-0.618352\pi$
$653$	− 47.6155i	− 1.86334i	0.363306	−	0.931670i	$-0.381648\pi$
$654$	0	0
$655$	8.80776	0.344148
$656$	0	0
$657$	35.6155	1.38949
$658$	0	0
$659$	12.0000i	0.467454i	0.972302	+	0.233727i	$0.0750921\pi$
$659$	12.0000i	0.467454i	−0.972302	+	0.233727i	$0.924908\pi$
$660$	0	0
$661$	20.7386i	0.806639i	0.915059	+	0.403320i	$0.132144\pi$
$661$	20.7386i	0.806639i	−0.915059	+	0.403320i	$0.867856\pi$
$662$	0	0
$663$	−1.43845	−0.0558647
$664$	0	0
$665$	1.89205	0.0733705
$666$	0	0
$667$	0	0
$668$	0	0
$669$	68.6695i	2.65492i
$670$	0	0
$671$	26.7386	1.03223
$672$	0	0
$673$	−21.0540	−0.811571	−0.405786	−	0.913968i	$-0.633002\pi$
$673$	−21.0540	−0.811571	−0.405786	+	0.913968i	$0.633002\pi$
$674$	0	0
$675$	− 6.73863i	− 0.259370i
$676$	0	0
$677$	− 21.3693i	− 0.821290i	−0.911795	−	0.410645i	$-0.865304\pi$
$677$	− 21.3693i	− 0.821290i	0.911795	−	0.410645i	$-0.134696\pi$
$678$	0	0
$679$	−3.36932	−0.129303
$680$	0	0
$681$	−54.7386	−2.09759
$682$	0	0
$683$	5.36932i	0.205451i	0.994710	+	0.102726i	$0.0327564\pi$
$683$	5.36932i	0.205451i	−0.994710	+	0.102726i	$0.967244\pi$
$684$	0	0
$685$	2.73863i	0.104638i
$686$	0	0
$687$	35.0540	1.33739
$688$	0	0
$689$	4.87689	0.185795
$690$	0	0
$691$	− 4.87689i	− 0.185526i	−0.995688	−	0.0927629i	$-0.970430\pi$
$691$	− 4.87689i	− 0.185526i	0.995688	−	0.0927629i	$-0.0295699\pi$
$692$	0	0
$693$	4.00000i	0.151947i
$694$	0	0
$695$	−4.31534	−0.163690
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	− 35.0540i	− 1.32586i
$700$	0	0
$701$	9.36932i	0.353874i	0.984222	+	0.176937i	$0.0566189\pi$
$701$	9.36932i	0.353874i	−0.984222	+	0.176937i	$0.943381\pi$
$702$	0	0
$703$	−58.1080	−2.19158
$704$	0	0
$705$	2.42329	0.0912665
$706$	0	0
$707$	4.00000i	0.150435i
$708$	0	0
$709$	− 16.7386i	− 0.628633i	−0.949318	−	0.314316i	$-0.898225\pi$
$709$	− 16.7386i	− 0.628633i	0.949318	−	0.314316i	$-0.101775\pi$
$710$	0	0
$711$	−42.7386	−1.60282
$712$	0	0
$713$	0	0
$714$	0	0
$715$	1.12311i	0.0420018i
$716$	0	0
$717$	55.5464i	2.07442i
$718$	0	0
$719$	−15.3693	−0.573179	−0.286589	−	0.958054i	$-0.592522\pi$
$719$	−15.3693	−0.573179	−0.286589	+	0.958054i	$0.592522\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	5.12311i	0.190530i
$724$	0	0
$725$	− 38.6307i	− 1.43471i
$726$	0	0
$727$	41.6155	1.54343	0.771717	−	0.635966i	$-0.219398\pi$
$727$	41.6155	1.54343	0.771717	+	0.635966i	$0.219398\pi$
$728$	0	0
$729$	35.9848	1.33277
$730$	0	0
$731$	− 4.94602i	− 0.182935i
$732$	0	0
$733$	− 25.0540i	− 0.925390i	−0.886518	−	0.462695i	$-0.846883\pi$
$733$	− 25.0540i	− 0.925390i	0.886518	−	0.462695i	$-0.153117\pi$
$734$	0	0
$735$	−9.61553	−0.354674
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	− 52.1080i	− 1.91682i	−0.285392	−	0.958411i	$-0.592124\pi$
$739$	− 52.1080i	− 1.91682i	0.285392	−	0.958411i	$-0.407876\pi$
$740$	0	0
$741$	15.3693i	0.564606i
$742$	0	0
$743$	−13.6847	−0.502041	−0.251021	−	0.967982i	$-0.580766\pi$
$743$	−13.6847	−0.502041	−0.251021	+	0.967982i	$0.580766\pi$
$744$	0	0
$745$	4.63068	0.169655
$746$	0	0
$747$	61.8617i	2.26340i
$748$	0	0
$749$	6.73863i	0.246224i
$750$	0	0
$751$	−26.2462	−0.957738	−0.478869	−	0.877886i	$-0.658953\pi$
$751$	−26.2462	−0.957738	−0.478869	+	0.877886i	$0.658953\pi$
$752$	0	0
$753$	−40.9848	−1.49357
$754$	0	0
$755$	5.79261i	0.210815i
$756$	0	0
$757$	9.36932i	0.340534i	0.985398	+	0.170267i	$0.0544630\pi$
$757$	9.36932i	0.340534i	−0.985398	+	0.170267i	$0.945537\pi$
$758$	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
$763$	− 0.946025i	− 0.0342484i
$764$	0	0
$765$	− 1.12311i	− 0.0406060i
$766$	0	0
$767$	6.00000	0.216647
$768$	0	0
$769$	9.36932	0.337866	0.168933	−	0.985628i	$-0.445968\pi$
$769$	9.36932	0.337866	0.168933	+	0.985628i	$0.445968\pi$
$770$	0	0
$771$	40.8078i	1.46966i
$772$	0	0
$773$	25.6847i	0.923813i	0.886929	+	0.461906i	$0.152834\pi$
$773$	25.6847i	0.923813i	−0.886929	+	0.461906i	$0.847166\pi$
$774$	0	0
$775$	33.3693	1.19866
$776$	0	0
$777$	−13.9309	−0.499767
$778$	0	0
$779$	− 42.7386i	− 1.53127i
$780$	0	0
$781$	− 3.36932i	− 0.120564i
$782$	0	0
$783$	−11.8617	−0.423904
$784$	0	0
$785$	1.12311	0.0400854
$786$	0	0
$787$	35.6155i	1.26956i	0.772694	+	0.634778i	$0.218909\pi$
$787$	35.6155i	1.26956i	−0.772694	+	0.634778i	$0.781091\pi$
$788$	0	0
$789$	− 68.4924i	− 2.43839i
$790$	0	0
$791$	4.63068	0.164648
$792$	0	0
$793$	13.3693	0.474758
$794$	0	0
$795$	7.01515i	0.248802i
$796$	0	0
$797$	− 12.7386i	− 0.451226i	−0.974217	−	0.225613i	$-0.927562\pi$
$797$	− 12.7386i	− 0.451226i	0.974217	−	0.225613i	$-0.0724384\pi$
$798$	0	0
$799$	−0.946025	−0.0334679
$800$	0	0
$801$	−29.3693	−1.03771
$802$	0	0
$803$	20.0000i	0.705785i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	54.7386	1.92689
$808$	0	0
$809$	39.9309	1.40389	0.701947	−	0.712229i	$-0.252314\pi$
$809$	39.9309	1.40389	0.701947	+	0.712229i	$0.252314\pi$
$810$	0	0
$811$	− 25.8617i	− 0.908128i	−0.890969	−	0.454064i	$-0.849974\pi$
$811$	− 25.8617i	− 0.908128i	0.890969	−	0.454064i	$-0.150026\pi$
$812$	0	0
$813$	4.31534i	0.151346i
$814$	0	0
$815$	5.26137	0.184298
$816$	0	0
$817$	−52.8466	−1.84887
$818$	0	0
$819$	2.00000i	0.0698857i
$820$	0	0
$821$	− 5.05398i	− 0.176385i	−0.996103	−	0.0881925i	$-0.971891\pi$
$821$	− 5.05398i	− 0.176385i	0.996103	−	0.0881925i	$-0.0281091\pi$
$822$	0	0
$823$	39.3693	1.37233	0.686164	−	0.727447i	$-0.259293\pi$
$823$	39.3693	1.37233	0.686164	+	0.727447i	$0.259293\pi$
$824$	0	0
$825$	24.0000	0.835573
$826$	0	0
$827$	36.7386i	1.27753i	0.769403	+	0.638764i	$0.220554\pi$
$827$	36.7386i	1.27753i	−0.769403	+	0.638764i	$0.779446\pi$
$828$	0	0
$829$	− 32.7386i	− 1.13706i	−0.822663	−	0.568530i	$-0.807512\pi$
$829$	− 32.7386i	− 1.13706i	0.822663	−	0.568530i	$-0.192488\pi$
$830$	0	0
$831$	−3.50758	−0.121677
$832$	0	0
$833$	3.75379	0.130061
$834$	0	0
$835$	− 5.26137i	− 0.182077i
$836$	0	0
$837$	− 10.2462i	− 0.354161i
$838$	0	0
$839$	−36.1080	−1.24658	−0.623292	−	0.781989i	$-0.714205\pi$
$839$	−36.1080	−1.24658	−0.623292	+	0.781989i	$0.714205\pi$
$840$	0	0
$841$	−39.0000	−1.34483
$842$	0	0
$843$	− 51.8617i	− 1.78621i
$844$	0	0
$845$	0.561553i	0.0193180i
$846$	0	0
$847$	3.93087	0.135066
$848$	0	0
$849$	5.75379	0.197470
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 21.6847i	− 0.742469i	−0.928539	−	0.371234i	$-0.878935\pi$
$853$	− 21.6847i	− 0.742469i	0.928539	−	0.371234i	$-0.121065\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	0	0
$857$	32.2462	1.10151	0.550755	−	0.834667i	$-0.314340\pi$
$857$	32.2462	1.10151	0.550755	+	0.834667i	$0.314340\pi$
$858$	0	0
$859$	36.0000i	1.22830i	0.789188	+	0.614152i	$0.210502\pi$
$859$	36.0000i	1.22830i	−0.789188	+	0.614152i	$0.789498\pi$
$860$	0	0
$861$	− 10.2462i	− 0.349190i
$862$	0	0
$863$	−25.0540	−0.852847	−0.426424	−	0.904524i	$-0.640227\pi$
$863$	−25.0540	−0.852847	−0.426424	+	0.904524i	$0.640227\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	− 42.7386i	− 1.45148i
$868$	0	0
$869$	− 24.0000i	− 0.814144i
$870$	0	0
$871$	6.00000	0.203302
$872$	0	0
$873$	21.3693	0.723242
$874$	0	0
$875$	− 3.05398i	− 0.103243i
$876$	0	0
$877$	− 31.7926i	− 1.07356i	−0.843722	−	0.536780i	$-0.819640\pi$
$877$	− 31.7926i	− 1.07356i	0.843722	−	0.536780i	$-0.180360\pi$
$878$	0	0
$879$	−60.0388	−2.02506
$880$	0	0
$881$	−19.3002	−0.650240	−0.325120	−	0.945673i	$-0.605405\pi$
$881$	−19.3002	−0.650240	−0.325120	+	0.945673i	$0.605405\pi$
$882$	0	0
$883$	− 17.4384i	− 0.586850i	−0.955982	−	0.293425i	$-0.905205\pi$
$883$	− 17.4384i	− 0.586850i	0.955982	−	0.293425i	$-0.0947952\pi$
$884$	0	0
$885$	8.63068i	0.290117i
$886$	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
$889$	1.26137	0.0423049
$890$	0	0
$891$	14.0000i	0.469018i
$892$	0	0
$893$	10.1080i	0.338250i
$894$	0	0
$895$	−10.6998	−0.357655
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 58.7386i	− 1.95904i
$900$	0	0
$901$	− 2.73863i	− 0.0912371i
$902$	0	0
$903$	−12.6695	−0.421615
$904$	0	0
$905$	−9.75379	−0.324227
$906$	0	0
$907$	− 26.4233i	− 0.877371i	−0.898641	−	0.438686i	$-0.855444\pi$
$907$	− 26.4233i	− 0.877371i	0.898641	−	0.438686i	$-0.144556\pi$
$908$	0	0
$909$	− 25.3693i	− 0.841447i
$910$	0	0
$911$	−27.3693	−0.906786	−0.453393	−	0.891311i	$-0.649787\pi$
$911$	−27.3693	−0.906786	−0.453393	+	0.891311i	$0.649787\pi$
$912$	0	0
$913$	−34.7386	−1.14968
$914$	0	0
$915$	19.2311i	0.635759i
$916$	0	0
$917$	8.80776i	0.290858i
$918$	0	0
$919$	−33.7538	−1.11343	−0.556717	−	0.830702i	$-0.687939\pi$
$919$	−33.7538	−1.11343	−0.556717	+	0.830702i	$0.687939\pi$
$920$	0	0
$921$	15.3693	0.506436
$922$	0	0
$923$	− 1.68466i	− 0.0554512i
$924$	0	0
$925$	45.3693i	1.49173i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	35.6155	1.16851	0.584254	−	0.811571i	$-0.301387\pi$
$929$	35.6155	1.16851	0.584254	+	0.811571i	$0.301387\pi$
$930$	0	0
$931$	− 40.1080i	− 1.31448i
$932$	0	0
$933$	39.3693i	1.28889i
$934$	0	0
$935$	0.630683	0.0206255
$936$	0	0
$937$	−40.7386	−1.33087	−0.665437	−	0.746454i	$-0.731755\pi$
$937$	−40.7386	−1.33087	−0.665437	+	0.746454i	$0.731755\pi$
$938$	0	0
$939$	− 43.6847i	− 1.42559i
$940$	0	0
$941$	37.6847i	1.22848i	0.789117	+	0.614242i	$0.210538\pi$
$941$	37.6847i	1.22848i	−0.789117	+	0.614242i	$0.789462\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−0.453602	−0.0147557
$946$	0	0
$947$	2.00000i	0.0649913i	0.999472	+	0.0324956i	$0.0103455\pi$
$947$	2.00000i	0.0649913i	−0.999472	+	0.0324956i	$0.989654\pi$
$948$	0	0
$949$	10.0000i	0.324614i
$950$	0	0
$951$	82.6004	2.67850
$952$	0	0
$953$	21.1922	0.686484	0.343242	−	0.939247i	$-0.388475\pi$
$953$	21.1922	0.686484	0.343242	+	0.939247i	$0.388475\pi$
$954$	0	0
$955$	10.8769i	0.351968i
$956$	0	0
$957$	− 42.2462i	− 1.36563i
$958$	0	0
$959$	−2.73863	−0.0884351
$960$	0	0
$961$	19.7386	0.636730
$962$	0	0
$963$	− 42.7386i	− 1.37723i
$964$	0	0
$965$	− 0.768944i	− 0.0247532i
$966$	0	0
$967$	8.42329	0.270875	0.135437	−	0.990786i	$-0.456756\pi$
$967$	8.42329	0.270875	0.135437	+	0.990786i	$0.456756\pi$
$968$	0	0
$969$	8.63068	0.277257
$970$	0	0
$971$	15.6847i	0.503345i	0.967813	+	0.251672i	$0.0809805\pi$
$971$	15.6847i	0.503345i	−0.967813	+	0.251672i	$0.919019\pi$
$972$	0	0
$973$	− 4.31534i	− 0.138343i
$974$	0	0
$975$	12.0000	0.384308
$976$	0	0
$977$	50.9848	1.63115	0.815575	−	0.578652i	$-0.196421\pi$
$977$	50.9848	1.63115	0.815575	+	0.578652i	$0.196421\pi$
$978$	0	0
$979$	− 16.4924i	− 0.527100i
$980$	0	0
$981$	6.00000i	0.191565i
$982$	0	0
$983$	−40.4233	−1.28930	−0.644651	−	0.764477i	$-0.722997\pi$
$983$	−40.4233	−1.28930	−0.644651	+	0.764477i	$0.722997\pi$
$984$	0	0
$985$	−7.68466	−0.244854
$986$	0	0
$987$	2.42329i	0.0771342i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	3.36932	0.107030	0.0535149	−	0.998567i	$-0.482958\pi$
$991$	3.36932	0.107030	0.0535149	+	0.998567i	$0.482958\pi$
$992$	0	0
$993$	54.7386	1.73708
$994$	0	0
$995$	8.63068i	0.273611i
$996$	0	0
$997$	− 44.7386i	− 1.41689i	−0.705768	−	0.708443i	$-0.749398\pi$
$997$	− 44.7386i	− 1.41689i	0.705768	−	0.708443i	$-0.250602\pi$
$998$	0	0
$999$	13.9309	0.440753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.b.u.1665.4		4
4.3	odd	2		3328.2.b.ba.1665.1		4
8.3	odd	2		3328.2.b.ba.1665.4		4
8.5	even	2	inner	3328.2.b.u.1665.1		4
16.3	odd	4		832.2.a.o.1.2		2
16.5	even	4		416.2.a.e.1.2	yes	2
16.11	odd	4		416.2.a.c.1.1	✓	2
16.13	even	4		832.2.a.l.1.1		2
48.5	odd	4		3744.2.a.y.1.1		2
48.11	even	4		3744.2.a.x.1.1		2
48.29	odd	4		7488.2.a.cf.1.2		2
48.35	even	4		7488.2.a.ce.1.2		2
208.155	odd	4		5408.2.a.p.1.1		2
208.181	even	4		5408.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.a.c.1.1	✓	2	16.11	odd	4
416.2.a.e.1.2	yes	2	16.5	even	4
832.2.a.l.1.1		2	16.13	even	4
832.2.a.o.1.2		2	16.3	odd	4
3328.2.b.u.1665.1		4	8.5	even	2	inner
3328.2.b.u.1665.4		4	1.1	even	1	trivial
3328.2.b.ba.1665.1		4	4.3	odd	2
3328.2.b.ba.1665.4		4	8.3	odd	2
3744.2.a.x.1.1		2	48.11	even	4
3744.2.a.y.1.1		2	48.5	odd	4
5408.2.a.p.1.1		2	208.155	odd	4
5408.2.a.bd.1.2		2	208.181	even	4
7488.2.a.ce.1.2		2	48.35	even	4
7488.2.a.cf.1.2		2	48.29	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.56155i q^{3} -0.561553i q^{5} +0.561553 q^{7} -3.56155 q^{9} -2.00000i q^{11} -1.00000i q^{13} +1.43845 q^{15} -0.561553 q^{17} +6.00000i q^{19} +1.43845i q^{21} +4.68466 q^{25} -1.43845i q^{27} -8.24621i q^{29} +7.12311 q^{31} +5.12311 q^{33} -0.315342i q^{35} +9.68466i q^{37} +2.56155 q^{39} -7.12311 q^{41} +8.80776i q^{43} +2.00000i q^{45} +1.68466 q^{47} -6.68466 q^{49} -1.43845i q^{51} +4.87689i q^{53} -1.12311 q^{55} -15.3693 q^{57} +6.00000i q^{59} +13.3693i q^{61} -2.00000 q^{63} -0.561553 q^{65} +6.00000i q^{67} +1.68466 q^{71} -10.0000 q^{73} +12.0000i q^{75} -1.12311i q^{77} +12.0000 q^{79} -7.00000 q^{81} -17.3693i q^{83} +0.315342i q^{85} +21.1231 q^{87} +8.24621 q^{89} -0.561553i q^{91} +18.2462i q^{93} +3.36932 q^{95} -6.00000 q^{97} +7.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{7} - 6 q^{9} + 14 q^{15} + 6 q^{17} - 6 q^{25} + 12 q^{31} + 4 q^{33} + 2 q^{39} - 12 q^{41} - 18 q^{47} - 2 q^{49} + 12 q^{55} - 12 q^{57} - 8 q^{63} + 6 q^{65} - 18 q^{71} - 40 q^{73} + 48 q^{79}+ \cdots - 24 q^{97}+O(q^{100}) \) 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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