LMFDB - Embedded newform 3920.2.k.d.2351.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3920,2,Mod(2351,3920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3920.2351"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$31.3013575923$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 43 x^{10} - 160 x^{9} + 572 x^{8} - 1394 x^{7} + 3039 x^{6} - 4844 x^{5} + \cdots + 657 $ x^12 - 6x^11 + 43x^10 - 160x^9 + 572x^8 - 1394x^7 + 3039x^6 - 4844x^5 + 6526x^4 - 6318x^3 + 4737x^2 - 2196*x + 657
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2351.4
Root			$0.500000 - 1.58132i$ of defining polynomial
Character	$\chi$	$=$	3920.2351
Dual form			3920.2.k.d.2351.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44735 q^{3} +1.00000i q^{5} +2.98952 q^{9} -6.47077i q^{11} -2.30812i q^{13} -2.44735i q^{15} -5.43548i q^{17} +0.989524 q^{19} -5.97765i q^{23} -1.00000 q^{25} +0.0256392 q^{27} +3.02424 q^{29} -5.25569 q^{31} +15.8362i q^{33} +0.423932 q^{37} +5.64878i q^{39} +9.84894i q^{41} -5.22517i q^{43} +2.98952i q^{45} -3.92999 q^{47} +13.3025i q^{51} +9.01155 q^{53} +6.47077 q^{55} -2.42171 q^{57} -8.22097 q^{59} +6.63385i q^{61} +2.30812 q^{65} -14.5901i q^{67} +14.6294i q^{69} +2.28875i q^{71} +12.0727i q^{73} +2.44735 q^{75} +8.11688i q^{79} -9.03132 q^{81} +2.94129 q^{83} +5.43548 q^{85} -7.40139 q^{87} -17.6222i q^{89} +12.8625 q^{93} +0.989524i q^{95} +4.96528i q^{97} -19.3445i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 20 q^{9} - 4 q^{19} - 12 q^{25} + 20 q^{27} + 16 q^{29} - 8 q^{31} + 8 q^{37} - 24 q^{47} + 24 q^{53} + 24 q^{55} + 16 q^{57} - 48 q^{59} + 4 q^{65} + 4 q^{75} + 12 q^{81} + 36 q^{83} - 16 q^{85}+ \cdots - 56 q^{93}+O(q^{100}) $ 12 * q - 4 * q^3 + 20 * q^9 - 4 * q^19 - 12 * q^25 + 20 * q^27 + 16 * q^29 - 8 * q^31 + 8 * q^37 - 24 * q^47 + 24 * q^53 + 24 * q^55 + 16 * q^57 - 48 * q^59 + 4 * q^65 + 4 * q^75 + 12 * q^81 + 36 * q^83 - 16 * q^85 + 60 * q^87 - 56 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times$.

$n$	$981$	$1471$	$3041$	$3137$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−2.44735	−1.41298	−0.706489	−	0.707724i	$-0.749722\pi$
$3$	−2.44735	−1.41298	−0.706489	+	0.707724i	$0.749722\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.98952	0.996508
$10$	0	0
$11$	− 6.47077i	− 1.95101i	−0.219977	−	0.975505i	$-0.570598\pi$
$11$	− 6.47077i	− 1.95101i	0.219977	−	0.975505i	$-0.429402\pi$
$12$	0	0
$13$	− 2.30812i	− 0.640157i	−0.947391	−	0.320079i	$-0.896291\pi$
$13$	− 2.30812i	− 0.640157i	0.947391	−	0.320079i	$-0.103709\pi$
$14$	0	0
$15$	− 2.44735i	− 0.631903i
$16$	0	0
$17$	− 5.43548i	− 1.31830i	−0.752013	−	0.659149i	$-0.770917\pi$
$17$	− 5.43548i	− 1.31830i	0.752013	−	0.659149i	$-0.229083\pi$
$18$	0	0
$19$	0.989524	0.227012	0.113506	−	0.993537i	$-0.463792\pi$
$19$	0.989524	0.227012	0.113506	+	0.993537i	$0.463792\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 5.97765i	− 1.24643i	−0.782052	−	0.623213i	$-0.785827\pi$
$23$	− 5.97765i	− 1.24643i	0.782052	−	0.623213i	$-0.214173\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0.0256392	0.00493426
$28$	0	0
$29$	3.02424	0.561588	0.280794	−	0.959768i	$-0.409402\pi$
$29$	3.02424	0.561588	0.280794	+	0.959768i	$0.409402\pi$
$30$	0	0
$31$	−5.25569	−0.943949	−0.471975	−	0.881612i	$-0.656459\pi$
$31$	−5.25569	−0.943949	−0.471975	+	0.881612i	$0.656459\pi$
$32$	0	0
$33$	15.8362i	2.75674i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.423932	0.0696940	0.0348470	−	0.999393i	$-0.488906\pi$
$37$	0.423932	0.0696940	0.0348470	+	0.999393i	$0.488906\pi$
$38$	0	0
$39$	5.64878i	0.904528i
$40$	0	0
$41$	9.84894i	1.53815i	0.639161	+	0.769073i	$0.279282\pi$
$41$	9.84894i	1.53815i	−0.639161	+	0.769073i	$0.720718\pi$
$42$	0	0
$43$	− 5.22517i	− 0.796830i	−0.917205	−	0.398415i	$-0.869560\pi$
$43$	− 5.22517i	− 0.796830i	0.917205	−	0.398415i	$-0.130440\pi$
$44$	0	0
$45$	2.98952i	0.445652i
$46$	0	0
$47$	−3.92999	−0.573248	−0.286624	−	0.958043i	$-0.592533\pi$
$47$	−3.92999	−0.573248	−0.286624	+	0.958043i	$0.592533\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	13.3025i	1.86273i
$52$	0	0
$53$	9.01155	1.23783	0.618916	−	0.785457i	$-0.287572\pi$
$53$	9.01155	1.23783	0.618916	+	0.785457i	$0.287572\pi$
$54$	0	0
$55$	6.47077	0.872518
$56$	0	0
$57$	−2.42171	−0.320763
$58$	0	0
$59$	−8.22097	−1.07028	−0.535139	−	0.844764i	$-0.679741\pi$
$59$	−8.22097	−1.07028	−0.535139	+	0.844764i	$0.679741\pi$
$60$	0	0
$61$	6.63385i	0.849378i	0.905339	+	0.424689i	$0.139617\pi$
$61$	6.63385i	0.849378i	−0.905339	+	0.424689i	$0.860383\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.30812	0.286287
$66$	0	0
$67$	− 14.5901i	− 1.78246i	−0.453551	−	0.891230i	$-0.649843\pi$
$67$	− 14.5901i	− 1.78246i	0.453551	−	0.891230i	$-0.350157\pi$
$68$	0	0
$69$	14.6294i	1.76117i
$70$	0	0
$71$	2.28875i	0.271625i	0.990735	+	0.135813i	$0.0433645\pi$
$71$	2.28875i	0.271625i	−0.990735	+	0.135813i	$0.956635\pi$
$72$	0	0
$73$	12.0727i	1.41300i	0.707713	+	0.706500i	$0.249727\pi$
$73$	12.0727i	1.41300i	−0.707713	+	0.706500i	$0.750273\pi$
$74$	0	0
$75$	2.44735	0.282596
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.11688i	0.913221i	0.889667	+	0.456610i	$0.150937\pi$
$79$	8.11688i	0.913221i	−0.889667	+	0.456610i	$0.849063\pi$
$80$	0	0
$81$	−9.03132	−1.00348
$82$	0	0
$83$	2.94129	0.322849	0.161424	−	0.986885i	$-0.448391\pi$
$83$	2.94129	0.322849	0.161424	+	0.986885i	$0.448391\pi$
$84$	0	0
$85$	5.43548	0.589561
$86$	0	0
$87$	−7.40139	−0.793512
$88$	0	0
$89$	− 17.6222i	− 1.86795i	−0.357335	−	0.933976i	$-0.616315\pi$
$89$	− 17.6222i	− 1.86795i	0.357335	−	0.933976i	$-0.383685\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	12.8625	1.33378
$94$	0	0
$95$	0.989524i	0.101523i
$96$	0	0
$97$	4.96528i	0.504148i	0.967708	+	0.252074i	$0.0811126\pi$
$97$	4.96528i	0.504148i	−0.967708	+	0.252074i	$0.918887\pi$
$98$	0	0
$99$	− 19.3445i	− 1.94420i
$100$	0	0
$101$	− 6.39028i	− 0.635857i	−0.948115	−	0.317928i	$-0.897013\pi$
$101$	− 6.39028i	− 0.635857i	0.948115	−	0.317928i	$-0.102987\pi$
$102$	0	0
$103$	13.5289	1.33304	0.666521	−	0.745486i	$-0.267782\pi$
$103$	13.5289	1.33304	0.666521	+	0.745486i	$0.267782\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.24382i	− 0.410265i	−0.978734	−	0.205133i	$-0.934237\pi$
$107$	− 4.24382i	− 0.410265i	0.978734	−	0.205133i	$-0.0657626\pi$
$108$	0	0
$109$	−9.38079	−0.898517	−0.449258	−	0.893402i	$-0.648312\pi$
$109$	−9.38079	−0.898517	−0.449258	+	0.893402i	$0.648312\pi$
$110$	0	0
$111$	−1.03751	−0.0984761
$112$	0	0
$113$	−7.45643	−0.701442	−0.350721	−	0.936480i	$-0.614063\pi$
$113$	−7.45643	−0.701442	−0.350721	+	0.936480i	$0.614063\pi$
$114$	0	0
$115$	5.97765	0.557419
$116$	0	0
$117$	− 6.90018i	− 0.637922i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−30.8708	−2.80644
$122$	0	0
$123$	− 24.1038i	− 2.17337i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	− 13.8065i	− 1.22513i	−0.790420	−	0.612565i	$-0.790138\pi$
$127$	− 13.8065i	− 1.22513i	0.790420	−	0.612565i	$-0.209862\pi$
$128$	0	0
$129$	12.7878i	1.12590i
$130$	0	0
$131$	2.55796	0.223490	0.111745	−	0.993737i	$-0.464356\pi$
$131$	2.55796	0.223490	0.111745	+	0.993737i	$0.464356\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.0256392i	0.00220667i
$136$	0	0
$137$	0.312750	0.0267200	0.0133600	−	0.999911i	$-0.495747\pi$
$137$	0.312750	0.0267200	0.0133600	+	0.999911i	$0.495747\pi$
$138$	0	0
$139$	0.186858	0.0158491	0.00792453	−	0.999969i	$-0.497478\pi$
$139$	0.186858	0.0158491	0.00792453	+	0.999969i	$0.497478\pi$
$140$	0	0
$141$	9.61806	0.809987
$142$	0	0
$143$	−14.9353	−1.24895
$144$	0	0
$145$	3.02424i	0.251150i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.1592	1.15997	0.579984	−	0.814628i	$-0.303059\pi$
$149$	14.1592	1.15997	0.579984	+	0.814628i	$0.303059\pi$
$150$	0	0
$151$	21.0004i	1.70899i	0.519461	+	0.854494i	$0.326133\pi$
$151$	21.0004i	1.70899i	−0.519461	+	0.854494i	$0.673867\pi$
$152$	0	0
$153$	− 16.2495i	− 1.31369i
$154$	0	0
$155$	− 5.25569i	− 0.422147i
$156$	0	0
$157$	6.81540i	0.543928i	0.962307	+	0.271964i	$0.0876731\pi$
$157$	6.81540i	0.543928i	−0.962307	+	0.271964i	$0.912327\pi$
$158$	0	0
$159$	−22.0544	−1.74903
$160$	0	0
$161$	0	0
$162$	0	0
$163$	7.62234i	0.597028i	0.954405	+	0.298514i	$0.0964909\pi$
$163$	7.62234i	0.597028i	−0.954405	+	0.298514i	$0.903509\pi$
$164$	0	0
$165$	−15.8362	−1.23285
$166$	0	0
$167$	−5.01130	−0.387786	−0.193893	−	0.981023i	$-0.562112\pi$
$167$	−5.01130	−0.387786	−0.193893	+	0.981023i	$0.562112\pi$
$168$	0	0
$169$	7.67259	0.590199
$170$	0	0
$171$	2.95820	0.226220
$172$	0	0
$173$	− 12.0765i	− 0.918161i	−0.888395	−	0.459080i	$-0.848179\pi$
$173$	− 12.0765i	− 0.918161i	0.888395	−	0.459080i	$-0.151821\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.1196	1.51228
$178$	0	0
$179$	− 10.1820i	− 0.761039i	−0.924773	−	0.380520i	$-0.875745\pi$
$179$	− 10.1820i	− 0.761039i	0.924773	−	0.380520i	$-0.124255\pi$
$180$	0	0
$181$	− 6.35399i	− 0.472288i	−0.971718	−	0.236144i	$-0.924116\pi$
$181$	− 6.35399i	− 0.472288i	0.971718	−	0.236144i	$-0.0758838\pi$
$182$	0	0
$183$	− 16.2354i	− 1.20015i
$184$	0	0
$185$	0.423932i	0.0311681i
$186$	0	0
$187$	−35.1717	−2.57201
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.88258i	0.425649i	0.977090	+	0.212824i	$0.0682662\pi$
$191$	5.88258i	0.425649i	−0.977090	+	0.212824i	$0.931734\pi$
$192$	0	0
$193$	−8.41326	−0.605600	−0.302800	−	0.953054i	$-0.597921\pi$
$193$	−8.41326	−0.605600	−0.302800	+	0.953054i	$0.597921\pi$
$194$	0	0
$195$	−5.64878	−0.404517
$196$	0	0
$197$	−15.8091	−1.12635	−0.563174	−	0.826338i	$-0.690420\pi$
$197$	−15.8091	−1.12635	−0.563174	+	0.826338i	$0.690420\pi$
$198$	0	0
$199$	−12.3714	−0.876982	−0.438491	−	0.898736i	$-0.644487\pi$
$199$	−12.3714	−0.876982	−0.438491	+	0.898736i	$0.644487\pi$
$200$	0	0
$201$	35.7070i	2.51858i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.84894	−0.687879
$206$	0	0
$207$	− 17.8703i	− 1.24207i
$208$	0	0
$209$	− 6.40298i	− 0.442903i
$210$	0	0
$211$	− 3.49347i	− 0.240500i	−0.992744	−	0.120250i	$-0.961630\pi$
$211$	− 3.49347i	− 0.240500i	0.992744	−	0.120250i	$-0.0383697\pi$
$212$	0	0
$213$	− 5.60138i	− 0.383801i
$214$	0	0
$215$	5.22517	0.356353
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 29.5461i	− 1.99654i
$220$	0	0
$221$	−12.5457	−0.843917
$222$	0	0
$223$	−22.6778	−1.51862	−0.759309	−	0.650731i	$-0.774463\pi$
$223$	−22.6778	−1.51862	−0.759309	+	0.650731i	$0.774463\pi$
$224$	0	0
$225$	−2.98952	−0.199302
$226$	0	0
$227$	7.55455	0.501413	0.250706	−	0.968063i	$-0.419337\pi$
$227$	7.55455	0.501413	0.250706	+	0.968063i	$0.419337\pi$
$228$	0	0
$229$	− 12.2910i	− 0.812213i	−0.913826	−	0.406106i	$-0.866886\pi$
$229$	− 12.2910i	− 0.812213i	0.913826	−	0.406106i	$-0.133114\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.16464	−0.207322	−0.103661	−	0.994613i	$-0.533056\pi$
$233$	−3.16464	−0.207322	−0.103661	+	0.994613i	$0.533056\pi$
$234$	0	0
$235$	− 3.92999i	− 0.256364i
$236$	0	0
$237$	− 19.8649i	− 1.29036i
$238$	0	0
$239$	27.0004i	1.74651i	0.487263	+	0.873255i	$0.337995\pi$
$239$	27.0004i	1.74651i	−0.487263	+	0.873255i	$0.662005\pi$
$240$	0	0
$241$	13.5551i	0.873162i	0.899665	+	0.436581i	$0.143811\pi$
$241$	13.5551i	0.873162i	−0.899665	+	0.436581i	$0.856189\pi$
$242$	0	0
$243$	22.0259	1.41296
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.28394i	− 0.145324i
$248$	0	0
$249$	−7.19837	−0.456178
$250$	0	0
$251$	−14.2304	−0.898215	−0.449108	−	0.893478i	$-0.648258\pi$
$251$	−14.2304	−0.898215	−0.449108	+	0.893478i	$0.648258\pi$
$252$	0	0
$253$	−38.6800	−2.43179
$254$	0	0
$255$	−13.3025	−0.833036
$256$	0	0
$257$	− 19.1249i	− 1.19298i	−0.802621	−	0.596489i	$-0.796562\pi$
$257$	− 19.1249i	− 1.19298i	0.802621	−	0.596489i	$-0.203438\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	9.04105	0.559627
$262$	0	0
$263$	− 6.94670i	− 0.428352i	−0.976795	−	0.214176i	$-0.931293\pi$
$263$	− 6.94670i	− 0.428352i	0.976795	−	0.214176i	$-0.0687066\pi$
$264$	0	0
$265$	9.01155i	0.553575i
$266$	0	0
$267$	43.1278i	2.63938i
$268$	0	0
$269$	7.73750i	0.471764i	0.971782	+	0.235882i	$0.0757978\pi$
$269$	7.73750i	0.471764i	−0.971782	+	0.235882i	$0.924202\pi$
$270$	0	0
$271$	−2.52196	−0.153198	−0.0765990	−	0.997062i	$-0.524406\pi$
$271$	−2.52196	−0.153198	−0.0765990	+	0.997062i	$0.524406\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.47077i	0.390202i
$276$	0	0
$277$	29.6640	1.78234	0.891169	−	0.453671i	$-0.149886\pi$
$277$	29.6640	1.78234	0.891169	+	0.453671i	$0.149886\pi$
$278$	0	0
$279$	−15.7120	−0.940653
$280$	0	0
$281$	−15.3527	−0.915862	−0.457931	−	0.888988i	$-0.651409\pi$
$281$	−15.3527	−0.915862	−0.457931	+	0.888988i	$0.651409\pi$
$282$	0	0
$283$	−9.02461	−0.536458	−0.268229	−	0.963355i	$-0.586438\pi$
$283$	−9.02461	−0.536458	−0.268229	+	0.963355i	$0.586438\pi$
$284$	0	0
$285$	− 2.42171i	− 0.143450i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−12.5444	−0.737908
$290$	0	0
$291$	− 12.1518i	− 0.712350i
$292$	0	0
$293$	0.815396i	0.0476359i	0.999716	+	0.0238180i	$0.00758221\pi$
$293$	0.815396i	0.0476359i	−0.999716	+	0.0238180i	$0.992418\pi$
$294$	0	0
$295$	− 8.22097i	− 0.478643i
$296$	0	0
$297$	− 0.165905i	− 0.00962679i
$298$	0	0
$299$	−13.7971	−0.797909
$300$	0	0
$301$	0	0
$302$	0	0
$303$	15.6393i	0.898452i
$304$	0	0
$305$	−6.63385	−0.379853
$306$	0	0
$307$	−8.42073	−0.480597	−0.240298	−	0.970699i	$-0.577245\pi$
$307$	−8.42073	−0.480597	−0.240298	+	0.970699i	$0.577245\pi$
$308$	0	0
$309$	−33.1100	−1.88356
$310$	0	0
$311$	−20.2441	−1.14794	−0.573968	−	0.818878i	$-0.694597\pi$
$311$	−20.2441	−1.14794	−0.573968	+	0.818878i	$0.694597\pi$
$312$	0	0
$313$	− 24.7225i	− 1.39740i	−0.715415	−	0.698700i	$-0.753762\pi$
$313$	− 24.7225i	− 1.39740i	0.715415	−	0.698700i	$-0.246238\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.6062	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$317$	18.6062	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$318$	0	0
$319$	− 19.5692i	− 1.09566i
$320$	0	0
$321$	10.3861i	0.579696i
$322$	0	0
$323$	− 5.37854i	− 0.299270i
$324$	0	0
$325$	2.30812i	0.128031i
$326$	0	0
$327$	22.9581	1.26958
$328$	0	0
$329$	0	0
$330$	0	0
$331$	21.3900i	1.17570i	0.808969	+	0.587851i	$0.200026\pi$
$331$	21.3900i	1.17570i	−0.808969	+	0.587851i	$0.799974\pi$
$332$	0	0
$333$	1.26735	0.0694506
$334$	0	0
$335$	14.5901	0.797141
$336$	0	0
$337$	−11.2762	−0.614252	−0.307126	−	0.951669i	$-0.599367\pi$
$337$	−11.2762	−0.614252	−0.307126	+	0.951669i	$0.599367\pi$
$338$	0	0
$339$	18.2485	0.991123
$340$	0	0
$341$	34.0083i	1.84165i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−14.6294	−0.787621
$346$	0	0
$347$	25.9372i	1.39238i	0.717855	+	0.696192i	$0.245124\pi$
$347$	25.9372i	1.39238i	−0.717855	+	0.696192i	$0.754876\pi$
$348$	0	0
$349$	− 15.3031i	− 0.819159i	−0.912274	−	0.409579i	$-0.865675\pi$
$349$	− 15.3031i	− 0.819159i	0.912274	−	0.409579i	$-0.134325\pi$
$350$	0	0
$351$	− 0.0591782i	− 0.00315870i
$352$	0	0
$353$	3.97059i	0.211333i	0.994402	+	0.105667i	$0.0336977\pi$
$353$	3.97059i	0.211333i	−0.994402	+	0.105667i	$0.966302\pi$
$354$	0	0
$355$	−2.28875	−0.121474
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.8260i	1.52138i	0.649116	+	0.760689i	$0.275139\pi$
$359$	28.8260i	1.52138i	−0.649116	+	0.760689i	$0.724861\pi$
$360$	0	0
$361$	−18.0208	−0.948465
$362$	0	0
$363$	75.5518	3.96544
$364$	0	0
$365$	−12.0727	−0.631912
$366$	0	0
$367$	−10.2693	−0.536053	−0.268026	−	0.963412i	$-0.586371\pi$
$367$	−10.2693	−0.536053	−0.268026	+	0.963412i	$0.586371\pi$
$368$	0	0
$369$	29.4436i	1.53277i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−34.4220	−1.78230	−0.891151	−	0.453706i	$-0.850102\pi$
$373$	−34.4220	−1.78230	−0.891151	+	0.453706i	$0.850102\pi$
$374$	0	0
$375$	2.44735i	0.126381i
$376$	0	0
$377$	− 6.98032i	− 0.359505i
$378$	0	0
$379$	− 10.7999i	− 0.554754i	−0.960761	−	0.277377i	$-0.910535\pi$
$379$	− 10.7999i	− 0.554754i	0.960761	−	0.277377i	$-0.0894651\pi$
$380$	0	0
$381$	33.7894i	1.73108i
$382$	0	0
$383$	−24.9586	−1.27533	−0.637664	−	0.770315i	$-0.720099\pi$
$383$	−24.9586	−1.27533	−0.637664	+	0.770315i	$0.720099\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 15.6208i	− 0.794048i
$388$	0	0
$389$	29.2842	1.48477	0.742384	−	0.669974i	$-0.233695\pi$
$389$	29.2842	1.48477	0.742384	+	0.669974i	$0.233695\pi$
$390$	0	0
$391$	−32.4914	−1.64316
$392$	0	0
$393$	−6.26022	−0.315787
$394$	0	0
$395$	−8.11688	−0.408405
$396$	0	0
$397$	− 12.9125i	− 0.648059i	−0.946047	−	0.324029i	$-0.894962\pi$
$397$	− 12.9125i	− 0.648059i	0.946047	−	0.324029i	$-0.105038\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.76191	−0.0879856	−0.0439928	−	0.999032i	$-0.514008\pi$
$401$	−1.76191	−0.0879856	−0.0439928	+	0.999032i	$0.514008\pi$
$402$	0	0
$403$	12.1307i	0.604276i
$404$	0	0
$405$	− 9.03132i	− 0.448770i
$406$	0	0
$407$	− 2.74316i	− 0.135974i
$408$	0	0
$409$	− 40.1105i	− 1.98333i	−0.128824	−	0.991667i	$-0.541120\pi$
$409$	− 40.1105i	− 1.98333i	0.128824	−	0.991667i	$-0.458880\pi$
$410$	0	0
$411$	−0.765408	−0.0377548
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2.94129i	0.144382i
$416$	0	0
$417$	−0.457306	−0.0223944
$418$	0	0
$419$	12.0471	0.588539	0.294269	−	0.955723i	$-0.404924\pi$
$419$	12.0471	0.588539	0.294269	+	0.955723i	$0.404924\pi$
$420$	0	0
$421$	28.6654	1.39707	0.698533	−	0.715578i	$-0.253836\pi$
$421$	28.6654	1.39707	0.698533	+	0.715578i	$0.253836\pi$
$422$	0	0
$423$	−11.7488	−0.571246
$424$	0	0
$425$	5.43548i	0.263660i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	36.5519	1.76474
$430$	0	0
$431$	17.1305i	0.825146i	0.910925	+	0.412573i	$0.135370\pi$
$431$	17.1305i	0.825146i	−0.910925	+	0.412573i	$0.864630\pi$
$432$	0	0
$433$	− 12.4485i	− 0.598235i	−0.954216	−	0.299117i	$-0.903308\pi$
$433$	− 12.4485i	− 0.598235i	0.954216	−	0.299117i	$-0.0966923\pi$
$434$	0	0
$435$	− 7.40139i	− 0.354869i
$436$	0	0
$437$	− 5.91503i	− 0.282954i
$438$	0	0
$439$	10.8004	0.515474	0.257737	−	0.966215i	$-0.417023\pi$
$439$	10.8004	0.515474	0.257737	+	0.966215i	$0.417023\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 0.0960518i	− 0.00456356i	−0.999997	−	0.00228178i	$-0.999274\pi$
$443$	− 0.0960518i	− 0.00456356i	0.999997	−	0.00228178i	$-0.000726313\pi$
$444$	0	0
$445$	17.6222	0.835374
$446$	0	0
$447$	−34.6525	−1.63901
$448$	0	0
$449$	13.7048	0.646770	0.323385	−	0.946268i	$-0.395179\pi$
$449$	13.7048	0.646770	0.323385	+	0.946268i	$0.395179\pi$
$450$	0	0
$451$	63.7302	3.00094
$452$	0	0
$453$	− 51.3953i	− 2.41476i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	21.4880	1.00517	0.502583	−	0.864529i	$-0.332383\pi$
$457$	21.4880	1.00517	0.502583	+	0.864529i	$0.332383\pi$
$458$	0	0
$459$	− 0.139361i	− 0.00650482i
$460$	0	0
$461$	27.2004i	1.26685i	0.773804	+	0.633425i	$0.218351\pi$
$461$	27.2004i	1.26685i	−0.773804	+	0.633425i	$0.781649\pi$
$462$	0	0
$463$	− 30.2578i	− 1.40620i	−0.711092	−	0.703099i	$-0.751799\pi$
$463$	− 30.2578i	− 1.40620i	0.711092	−	0.703099i	$-0.248201\pi$
$464$	0	0
$465$	12.8625i	0.596484i
$466$	0	0
$467$	−16.7747	−0.776243	−0.388121	−	0.921608i	$-0.626876\pi$
$467$	−16.7747	−0.776243	−0.388121	+	0.921608i	$0.626876\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 16.6797i	− 0.768558i
$472$	0	0
$473$	−33.8108	−1.55462
$474$	0	0
$475$	−0.989524	−0.0454025
$476$	0	0
$477$	26.9402	1.23351
$478$	0	0
$479$	17.1399	0.783142	0.391571	−	0.920148i	$-0.371932\pi$
$479$	17.1399	0.783142	0.391571	+	0.920148i	$0.371932\pi$
$480$	0	0
$481$	− 0.978485i	− 0.0446151i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.96528	−0.225462
$486$	0	0
$487$	− 11.2878i	− 0.511501i	−0.966743	−	0.255750i	$-0.917677\pi$
$487$	− 11.2878i	− 0.511501i	0.966743	−	0.255750i	$-0.0823225\pi$
$488$	0	0
$489$	− 18.6545i	− 0.843587i
$490$	0	0
$491$	25.5317i	1.15223i	0.817369	+	0.576115i	$0.195432\pi$
$491$	25.5317i	1.15223i	−0.817369	+	0.576115i	$0.804568\pi$
$492$	0	0
$493$	− 16.4382i	− 0.740340i
$494$	0	0
$495$	19.3445	0.869471
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 17.0501i	− 0.763265i	−0.924314	−	0.381633i	$-0.875362\pi$
$499$	− 17.0501i	− 0.763265i	0.924314	−	0.381633i	$-0.124638\pi$
$500$	0	0
$501$	12.2644	0.547934
$502$	0	0
$503$	29.6588	1.32242	0.661211	−	0.750200i	$-0.270043\pi$
$503$	29.6588	1.32242	0.661211	+	0.750200i	$0.270043\pi$
$504$	0	0
$505$	6.39028	0.284364
$506$	0	0
$507$	−18.7775	−0.833938
$508$	0	0
$509$	− 23.6328i	− 1.04751i	−0.851870	−	0.523753i	$-0.824531\pi$
$509$	− 23.6328i	− 1.04751i	0.851870	−	0.523753i	$-0.175469\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.0253706	0.00112014
$514$	0	0
$515$	13.5289i	0.596155i
$516$	0	0
$517$	25.4301i	1.11841i
$518$	0	0
$519$	29.5555i	1.29734i
$520$	0	0
$521$	− 17.6387i	− 0.772766i	−0.922338	−	0.386383i	$-0.873724\pi$
$521$	− 17.6387i	− 0.772766i	0.922338	−	0.386383i	$-0.126276\pi$
$522$	0	0
$523$	−21.9280	−0.958846	−0.479423	−	0.877584i	$-0.659154\pi$
$523$	−21.9280	−0.958846	−0.479423	+	0.877584i	$0.659154\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.5672i	1.24441i
$528$	0	0
$529$	−12.7323	−0.553580
$530$	0	0
$531$	−24.5768	−1.06654
$532$	0	0
$533$	22.7325	0.984655
$534$	0	0
$535$	4.24382	0.183476
$536$	0	0
$537$	24.9190i	1.07533i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.07940	0.0894005	0.0447002	−	0.999000i	$-0.485767\pi$
$541$	2.07940	0.0894005	0.0447002	+	0.999000i	$0.485767\pi$
$542$	0	0
$543$	15.5504i	0.667333i
$544$	0	0
$545$	− 9.38079i	− 0.401829i
$546$	0	0
$547$	− 9.15708i	− 0.391528i	−0.980651	−	0.195764i	$-0.937281\pi$
$547$	− 9.15708i	− 0.391528i	0.980651	−	0.195764i	$-0.0627187\pi$
$548$	0	0
$549$	19.8321i	0.846412i
$550$	0	0
$551$	2.99256	0.127487
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 1.03751i	− 0.0440398i
$556$	0	0
$557$	−15.7794	−0.668596	−0.334298	−	0.942467i	$-0.608499\pi$
$557$	−15.7794	−0.668596	−0.334298	+	0.942467i	$0.608499\pi$
$558$	0	0
$559$	−12.0603	−0.510097
$560$	0	0
$561$	86.0776	3.63420
$562$	0	0
$563$	−4.80198	−0.202379	−0.101190	−	0.994867i	$-0.532265\pi$
$563$	−4.80198	−0.202379	−0.101190	+	0.994867i	$0.532265\pi$
$564$	0	0
$565$	− 7.45643i	− 0.313694i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.7219	−1.07832	−0.539158	−	0.842204i	$-0.681258\pi$
$569$	−25.7219	−1.07832	−0.539158	+	0.842204i	$0.681258\pi$
$570$	0	0
$571$	17.8290i	0.746122i	0.927807	+	0.373061i	$0.121692\pi$
$571$	17.8290i	0.746122i	−0.927807	+	0.373061i	$0.878308\pi$
$572$	0	0
$573$	− 14.3967i	− 0.601433i
$574$	0	0
$575$	5.97765i	0.249285i
$576$	0	0
$577$	33.0316i	1.37513i	0.726125	+	0.687563i	$0.241319\pi$
$577$	33.0316i	1.37513i	−0.726125	+	0.687563i	$0.758681\pi$
$578$	0	0
$579$	20.5902	0.855699
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 58.3116i	− 2.41502i
$584$	0	0
$585$	6.90018	0.285287
$586$	0	0
$587$	−14.6086	−0.602960	−0.301480	−	0.953472i	$-0.597481\pi$
$587$	−14.6086	−0.602960	−0.301480	+	0.953472i	$0.597481\pi$
$588$	0	0
$589$	−5.20063	−0.214288
$590$	0	0
$591$	38.6903	1.59151
$592$	0	0
$593$	10.6780i	0.438493i	0.975669	+	0.219247i	$0.0703600\pi$
$593$	10.6780i	0.438493i	−0.975669	+	0.219247i	$0.929640\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	30.2770	1.23916
$598$	0	0
$599$	− 14.5038i	− 0.592610i	−0.955093	−	0.296305i	$-0.904245\pi$
$599$	− 14.5038i	− 0.592610i	0.955093	−	0.296305i	$-0.0957546\pi$
$600$	0	0
$601$	38.6998i	1.57860i	0.614008	+	0.789299i	$0.289556\pi$
$601$	38.6998i	1.57860i	−0.614008	+	0.789299i	$0.710444\pi$
$602$	0	0
$603$	− 43.6174i	− 1.77624i
$604$	0	0
$605$	− 30.8708i	− 1.25508i
$606$	0	0
$607$	−17.4589	−0.708635	−0.354317	−	0.935125i	$-0.615287\pi$
$607$	−17.4589	−0.708635	−0.354317	+	0.935125i	$0.615287\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.07088i	0.366969i
$612$	0	0
$613$	−17.9479	−0.724909	−0.362455	−	0.932001i	$-0.618061\pi$
$613$	−17.9479	−0.724909	−0.362455	+	0.932001i	$0.618061\pi$
$614$	0	0
$615$	24.1038	0.971959
$616$	0	0
$617$	47.2698	1.90301	0.951506	−	0.307629i	$-0.0995357\pi$
$617$	47.2698	1.90301	0.951506	+	0.307629i	$0.0995357\pi$
$618$	0	0
$619$	−46.9231	−1.88600	−0.942999	−	0.332795i	$-0.892008\pi$
$619$	−46.9231	−1.88600	−0.942999	+	0.332795i	$0.892008\pi$
$620$	0	0
$621$	− 0.153262i	− 0.00615019i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	15.6703i	0.625813i
$628$	0	0
$629$	− 2.30427i	− 0.0918774i
$630$	0	0
$631$	26.1192i	1.03979i	0.854231	+	0.519894i	$0.174028\pi$
$631$	26.1192i	1.03979i	−0.854231	+	0.519894i	$0.825972\pi$
$632$	0	0
$633$	8.54975i	0.339822i
$634$	0	0
$635$	13.8065	0.547895
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.84229i	0.270677i
$640$	0	0
$641$	−17.9508	−0.709013	−0.354506	−	0.935054i	$-0.615351\pi$
$641$	−17.9508	−0.709013	−0.354506	+	0.935054i	$0.615351\pi$
$642$	0	0
$643$	28.8535	1.13787	0.568936	−	0.822382i	$-0.307355\pi$
$643$	28.8535	1.13787	0.568936	+	0.822382i	$0.307355\pi$
$644$	0	0
$645$	−12.7878	−0.503520
$646$	0	0
$647$	31.8722	1.25302	0.626512	−	0.779412i	$-0.284482\pi$
$647$	31.8722	1.25302	0.626512	+	0.779412i	$0.284482\pi$
$648$	0	0
$649$	53.1960i	2.08812i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.06801	0.0417943	0.0208972	−	0.999782i	$-0.493348\pi$
$653$	1.06801	0.0417943	0.0208972	+	0.999782i	$0.493348\pi$
$654$	0	0
$655$	2.55796i	0.0999478i
$656$	0	0
$657$	36.0915i	1.40806i
$658$	0	0
$659$	− 27.6777i	− 1.07817i	−0.842252	−	0.539085i	$-0.818770\pi$
$659$	− 27.6777i	− 1.07817i	0.842252	−	0.539085i	$-0.181230\pi$
$660$	0	0
$661$	− 29.1653i	− 1.13440i	−0.823580	−	0.567200i	$-0.808027\pi$
$661$	− 29.1653i	− 1.13440i	0.823580	−	0.567200i	$-0.191973\pi$
$662$	0	0
$663$	30.7038	1.19244
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 18.0779i	− 0.699979i
$668$	0	0
$669$	55.5005	2.14577
$670$	0	0
$671$	42.9261	1.65714
$672$	0	0
$673$	5.57642	0.214955	0.107478	−	0.994208i	$-0.465723\pi$
$673$	5.57642	0.214955	0.107478	+	0.994208i	$0.465723\pi$
$674$	0	0
$675$	−0.0256392	−0.000986852 0
$676$	0	0
$677$	− 11.8097i	− 0.453883i	−0.973908	−	0.226942i	$-0.927127\pi$
$677$	− 11.8097i	− 0.453883i	0.973908	−	0.226942i	$-0.0728726\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−18.4886	−0.708486
$682$	0	0
$683$	− 19.7758i	− 0.756698i	−0.925663	−	0.378349i	$-0.876492\pi$
$683$	− 19.7758i	− 0.756698i	0.925663	−	0.378349i	$-0.123508\pi$
$684$	0	0
$685$	0.312750i	0.0119496i
$686$	0	0
$687$	30.0804i	1.14764i
$688$	0	0
$689$	− 20.7997i	− 0.792406i
$690$	0	0
$691$	−10.1384	−0.385682	−0.192841	−	0.981230i	$-0.561770\pi$
$691$	−10.1384	−0.385682	−0.192841	+	0.981230i	$0.561770\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.186858i	0.00708791i
$696$	0	0
$697$	53.5337	2.02773
$698$	0	0
$699$	7.74497	0.292942
$700$	0	0
$701$	−0.203655	−0.00769196	−0.00384598	−	0.999993i	$-0.501224\pi$
$701$	−0.203655	−0.00769196	−0.00384598	+	0.999993i	$0.501224\pi$
$702$	0	0
$703$	0.419491	0.0158214
$704$	0	0
$705$	9.61806i	0.362237i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−14.5411	−0.546104	−0.273052	−	0.961999i	$-0.588033\pi$
$709$	−14.5411	−0.546104	−0.273052	+	0.961999i	$0.588033\pi$
$710$	0	0
$711$	24.2656i	0.910032i
$712$	0	0
$713$	31.4167i	1.17656i
$714$	0	0
$715$	− 14.9353i	− 0.558549i
$716$	0	0
$717$	− 66.0794i	− 2.46778i
$718$	0	0
$719$	40.5903	1.51376	0.756881	−	0.653553i	$-0.226722\pi$
$719$	40.5903	1.51376	0.756881	+	0.653553i	$0.226722\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 33.1741i	− 1.23376i
$724$	0	0
$725$	−3.02424	−0.112318
$726$	0	0
$727$	7.75162	0.287491	0.143746	−	0.989615i	$-0.454085\pi$
$727$	7.75162	0.287491	0.143746	+	0.989615i	$0.454085\pi$
$728$	0	0
$729$	−26.8111	−0.993004
$730$	0	0
$731$	−28.4013	−1.05046
$732$	0	0
$733$	− 6.66887i	− 0.246320i	−0.992387	−	0.123160i	$-0.960697\pi$
$733$	− 6.66887i	− 0.246320i	0.992387	−	0.123160i	$-0.0393029\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−94.4090	−3.47760
$738$	0	0
$739$	28.8170i	1.06005i	0.847982	+	0.530026i	$0.177818\pi$
$739$	28.8170i	1.06005i	−0.847982	+	0.530026i	$0.822182\pi$
$740$	0	0
$741$	5.58960i	0.205339i
$742$	0	0
$743$	− 17.9238i	− 0.657560i	−0.944406	−	0.328780i	$-0.893362\pi$
$743$	− 17.9238i	− 0.657560i	0.944406	−	0.328780i	$-0.106638\pi$
$744$	0	0
$745$	14.1592i	0.518753i
$746$	0	0
$747$	8.79306	0.321721
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 22.8565i	− 0.834045i	−0.908896	−	0.417023i	$-0.863074\pi$
$751$	− 22.8565i	− 0.834045i	0.908896	−	0.417023i	$-0.136926\pi$
$752$	0	0
$753$	34.8268	1.26916
$754$	0	0
$755$	−21.0004	−0.764283
$756$	0	0
$757$	−42.7423	−1.55349	−0.776747	−	0.629812i	$-0.783132\pi$
$757$	−42.7423	−1.55349	−0.776747	+	0.629812i	$0.783132\pi$
$758$	0	0
$759$	94.6635	3.43607
$760$	0	0
$761$	33.5084i	1.21468i	0.794443	+	0.607339i	$0.207763\pi$
$761$	33.5084i	1.21468i	−0.794443	+	0.607339i	$0.792237\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	16.2495	0.587502
$766$	0	0
$767$	18.9750i	0.685146i
$768$	0	0
$769$	29.5287i	1.06483i	0.846483	+	0.532415i	$0.178716\pi$
$769$	29.5287i	1.06483i	−0.846483	+	0.532415i	$0.821284\pi$
$770$	0	0
$771$	46.8053i	1.68565i
$772$	0	0
$773$	41.7845i	1.50288i	0.659800	+	0.751442i	$0.270641\pi$
$773$	41.7845i	1.50288i	−0.659800	+	0.751442i	$0.729359\pi$
$774$	0	0
$775$	5.25569	0.188790
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.74576i	0.349178i
$780$	0	0
$781$	14.8100	0.529944
$782$	0	0
$783$	0.0775391	0.00277102
$784$	0	0
$785$	−6.81540	−0.243252
$786$	0	0
$787$	11.0208	0.392851	0.196425	−	0.980519i	$-0.437067\pi$
$787$	11.0208	0.392851	0.196425	+	0.980519i	$0.437067\pi$
$788$	0	0
$789$	17.0010i	0.605252i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	15.3117	0.543735
$794$	0	0
$795$	− 22.0544i	− 0.782190i
$796$	0	0
$797$	− 12.9488i	− 0.458669i	−0.973348	−	0.229334i	$-0.926345\pi$
$797$	− 12.9488i	− 0.458669i	0.973348	−	0.229334i	$-0.0736549\pi$
$798$	0	0
$799$	21.3614i	0.755711i
$800$	0	0
$801$	− 52.6821i	− 1.86143i
$802$	0	0
$803$	78.1195	2.75678
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 18.9364i	− 0.666592i
$808$	0	0
$809$	25.5225	0.897323	0.448661	−	0.893702i	$-0.351901\pi$
$809$	25.5225	0.897323	0.448661	+	0.893702i	$0.351901\pi$
$810$	0	0
$811$	−6.60292	−0.231860	−0.115930	−	0.993257i	$-0.536985\pi$
$811$	−6.60292	−0.231860	−0.115930	+	0.993257i	$0.536985\pi$
$812$	0	0
$813$	6.17212	0.216466
$814$	0	0
$815$	−7.62234	−0.266999
$816$	0	0
$817$	− 5.17043i	− 0.180890i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.76517	0.201206	0.100603	−	0.994927i	$-0.467923\pi$
$821$	5.76517	0.201206	0.100603	+	0.994927i	$0.467923\pi$
$822$	0	0
$823$	− 3.13521i	− 0.109287i	−0.998506	−	0.0546433i	$-0.982598\pi$
$823$	− 3.13521i	− 0.109287i	0.998506	−	0.0546433i	$-0.0174022\pi$
$824$	0	0
$825$	− 15.8362i	− 0.551347i
$826$	0	0
$827$	22.0174i	0.765621i	0.923827	+	0.382811i	$0.125044\pi$
$827$	22.0174i	0.765621i	−0.923827	+	0.382811i	$0.874956\pi$
$828$	0	0
$829$	− 46.7718i	− 1.62445i	−0.583343	−	0.812226i	$-0.698256\pi$
$829$	− 46.7718i	− 1.62445i	0.583343	−	0.812226i	$-0.301744\pi$
$830$	0	0
$831$	−72.5983	−2.51841
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 5.01130i	− 0.173423i
$836$	0	0
$837$	−0.134751	−0.00465769
$838$	0	0
$839$	36.3872	1.25622	0.628112	−	0.778123i	$-0.283828\pi$
$839$	36.3872	1.25622	0.628112	+	0.778123i	$0.283828\pi$
$840$	0	0
$841$	−19.8539	−0.684619
$842$	0	0
$843$	37.5733	1.29409
$844$	0	0
$845$	7.67259i	0.263945i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	22.0864	0.758003
$850$	0	0
$851$	− 2.53412i	− 0.0868684i
$852$	0	0
$853$	5.80981i	0.198924i	0.995041	+	0.0994621i	$0.0317122\pi$
$853$	5.80981i	0.198924i	−0.995041	+	0.0994621i	$0.968288\pi$
$854$	0	0
$855$	2.95820i	0.101168i
$856$	0	0
$857$	− 0.327118i	− 0.0111741i	−0.999984	−	0.00558706i	$-0.998222\pi$
$857$	− 0.327118i	− 0.0111741i	0.999984	−	0.00558706i	$-0.00177843\pi$
$858$	0	0
$859$	−9.62729	−0.328479	−0.164239	−	0.986420i	$-0.552517\pi$
$859$	−9.62729	−0.328479	−0.164239	+	0.986420i	$0.552517\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.5081i	0.698104i	0.937103	+	0.349052i	$0.113496\pi$
$863$	20.5081i	0.698104i	−0.937103	+	0.349052i	$0.886504\pi$
$864$	0	0
$865$	12.0765	0.410614
$866$	0	0
$867$	30.7006	1.04265
$868$	0	0
$869$	52.5225	1.78170
$870$	0	0
$871$	−33.6756	−1.14105
$872$	0	0
$873$	14.8438i	0.502387i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17.4104	0.587909	0.293954	−	0.955819i	$-0.405029\pi$
$877$	17.4104	0.587909	0.293954	+	0.955819i	$0.405029\pi$
$878$	0	0
$879$	− 1.99556i	− 0.0673085i
$880$	0	0
$881$	− 25.3913i	− 0.855455i	−0.903908	−	0.427727i	$-0.859314\pi$
$881$	− 25.3913i	− 0.855455i	0.903908	−	0.427727i	$-0.140686\pi$
$882$	0	0
$883$	24.0260i	0.808541i	0.914639	+	0.404270i	$0.132475\pi$
$883$	24.0260i	0.808541i	−0.914639	+	0.404270i	$0.867525\pi$
$884$	0	0
$885$	20.1196i	0.676312i
$886$	0	0
$887$	4.56644	0.153326	0.0766631	−	0.997057i	$-0.475573\pi$
$887$	4.56644	0.153326	0.0766631	+	0.997057i	$0.475573\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	58.4396i	1.95780i
$892$	0	0
$893$	−3.88882	−0.130134
$894$	0	0
$895$	10.1820	0.340347
$896$	0	0
$897$	33.7664	1.12743
$898$	0	0
$899$	−15.8945	−0.530111
$900$	0	0
$901$	− 48.9821i	− 1.63183i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.35399	0.211214
$906$	0	0
$907$	41.8509i	1.38963i	0.719186	+	0.694817i	$0.244515\pi$
$907$	41.8509i	1.38963i	−0.719186	+	0.694817i	$0.755485\pi$
$908$	0	0
$909$	− 19.1039i	− 0.633636i
$910$	0	0
$911$	33.5744i	1.11237i	0.831059	+	0.556184i	$0.187735\pi$
$911$	33.5744i	1.11237i	−0.831059	+	0.556184i	$0.812265\pi$
$912$	0	0
$913$	− 19.0324i	− 0.629881i
$914$	0	0
$915$	16.2354	0.536725
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 36.0584i	− 1.18946i	−0.803927	−	0.594728i	$-0.797260\pi$
$919$	− 36.0584i	− 1.18946i	0.803927	−	0.594728i	$-0.202740\pi$
$920$	0	0
$921$	20.6085	0.679073
$922$	0	0
$923$	5.28272	0.173883
$924$	0	0
$925$	−0.423932	−0.0139388
$926$	0	0
$927$	40.4450	1.32839
$928$	0	0
$929$	− 1.40370i	− 0.0460539i	−0.999735	−	0.0230269i	$-0.992670\pi$
$929$	− 1.40370i	− 0.0460539i	0.999735	−	0.0230269i	$-0.00733035\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	49.5443	1.62201
$934$	0	0
$935$	− 35.1717i	− 1.15024i
$936$	0	0
$937$	58.7549i	1.91944i	0.280960	+	0.959720i	$0.409347\pi$
$937$	58.7549i	1.91944i	−0.280960	+	0.959720i	$0.590653\pi$
$938$	0	0
$939$	60.5047i	1.97450i
$940$	0	0
$941$	− 28.7596i	− 0.937537i	−0.883321	−	0.468768i	$-0.844698\pi$
$941$	− 28.7596i	− 0.937537i	0.883321	−	0.468768i	$-0.155302\pi$
$942$	0	0
$943$	58.8735	1.91719
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 16.5821i	− 0.538844i	−0.963022	−	0.269422i	$-0.913167\pi$
$947$	− 16.5821i	− 0.538844i	0.963022	−	0.269422i	$-0.0868327\pi$
$948$	0	0
$949$	27.8652	0.904541
$950$	0	0
$951$	−45.5359	−1.47660
$952$	0	0
$953$	−6.47062	−0.209604	−0.104802	−	0.994493i	$-0.533421\pi$
$953$	−6.47062	−0.209604	−0.104802	+	0.994493i	$0.533421\pi$
$954$	0	0
$955$	−5.88258	−0.190356
$956$	0	0
$957$	47.8927i	1.54815i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−3.37776	−0.108960
$962$	0	0
$963$	− 12.6870i	− 0.408832i
$964$	0	0
$965$	− 8.41326i	− 0.270832i
$966$	0	0
$967$	7.89312i	0.253826i	0.991914	+	0.126913i	$0.0405069\pi$
$967$	7.89312i	0.253826i	−0.991914	+	0.126913i	$0.959493\pi$
$968$	0	0
$969$	13.1632i	0.422862i
$970$	0	0
$971$	−18.7936	−0.603114	−0.301557	−	0.953448i	$-0.597506\pi$
$971$	−18.7936	−0.603114	−0.301557	+	0.953448i	$0.597506\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 5.64878i	− 0.180906i
$976$	0	0
$977$	−9.67640	−0.309575	−0.154788	−	0.987948i	$-0.549469\pi$
$977$	−9.67640	−0.309575	−0.154788	+	0.987948i	$0.549469\pi$
$978$	0	0
$979$	−114.029	−3.64439
$980$	0	0
$981$	−28.0441	−0.895379
$982$	0	0
$983$	36.0310	1.14921	0.574606	−	0.818430i	$-0.305155\pi$
$983$	36.0310	1.14921	0.574606	+	0.818430i	$0.305155\pi$
$984$	0	0
$985$	− 15.8091i	− 0.503718i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−31.2342	−0.993191
$990$	0	0
$991$	36.3805i	1.15566i	0.816155	+	0.577832i	$0.196101\pi$
$991$	36.3805i	1.15566i	−0.816155	+	0.577832i	$0.803899\pi$
$992$	0	0
$993$	− 52.3489i	− 1.66124i
$994$	0	0
$995$	− 12.3714i	− 0.392198i
$996$	0	0
$997$	− 23.9444i	− 0.758327i	−0.925330	−	0.379164i	$-0.876212\pi$
$997$	− 23.9444i	− 0.758327i	0.925330	−	0.379164i	$-0.123788\pi$
$998$	0	0
$999$	0.0108693	0.000343888 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.k.d.2351.4		12
4.3	odd	2		3920.2.k.e.2351.10		12
7.4	even	3		560.2.bs.c.271.5	yes	12
7.5	odd	6		560.2.bs.b.31.2	✓	12
7.6	odd	2		3920.2.k.e.2351.9		12
28.11	odd	6		560.2.bs.b.271.2	yes	12
28.19	even	6		560.2.bs.c.31.5	yes	12
28.27	even	2	inner	3920.2.k.d.2351.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.bs.b.31.2	✓	12	7.5	odd	6
560.2.bs.b.271.2	yes	12	28.11	odd	6
560.2.bs.c.31.5	yes	12	28.19	even	6
560.2.bs.c.271.5	yes	12	7.4	even	3
3920.2.k.d.2351.3		12	28.27	even	2	inner
3920.2.k.d.2351.4		12	1.1	even	1	trivial
3920.2.k.e.2351.9		12	7.6	odd	2
3920.2.k.e.2351.10		12	4.3	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.44735 q^{3} +1.00000i q^{5} +2.98952 q^{9} -6.47077i q^{11} -2.30812i q^{13} -2.44735i q^{15} -5.43548i q^{17} +0.989524 q^{19} -5.97765i q^{23} -1.00000 q^{25} +0.0256392 q^{27} +3.02424 q^{29} -5.25569 q^{31} +15.8362i q^{33} +0.423932 q^{37} +5.64878i q^{39} +9.84894i q^{41} -5.22517i q^{43} +2.98952i q^{45} -3.92999 q^{47} +13.3025i q^{51} +9.01155 q^{53} +6.47077 q^{55} -2.42171 q^{57} -8.22097 q^{59} +6.63385i q^{61} +2.30812 q^{65} -14.5901i q^{67} +14.6294i q^{69} +2.28875i q^{71} +12.0727i q^{73} +2.44735 q^{75} +8.11688i q^{79} -9.03132 q^{81} +2.94129 q^{83} +5.43548 q^{85} -7.40139 q^{87} -17.6222i q^{89} +12.8625 q^{93} +0.989524i q^{95} +4.96528i q^{97} -19.3445i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 20 q^{9} - 4 q^{19} - 12 q^{25} + 20 q^{27} + 16 q^{29} - 8 q^{31} + 8 q^{37} - 24 q^{47} + 24 q^{53} + 24 q^{55} + 16 q^{57} - 48 q^{59} + 4 q^{65} + 4 q^{75} + 12 q^{81} + 36 q^{83} - 16 q^{85}+ \cdots - 56 q^{93}+O(q^{100}) \) 12 * q - 4 * q^3 + 20 * q^9 - 4 * q^19 - 12 * q^25 + 20 * q^27 + 16 * q^29 - 8 * q^31 + 8 * q^37 - 24 * q^47 + 24 * q^53 + 24 * q^55 + 16 * q^57 - 48 * q^59 + 4 * q^65 + 4 * q^75 + 12 * q^81 + 36 * q^83 - 16 * q^85 + 60 * q^87 - 56 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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