LMFDB - Embedded newform 3360.2.z.c.1231.26

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(1231,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 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("3360.1231"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.z (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [28,0,0,0,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1231.26
Character	$\chi$	$=$	3360.1231
Dual form			3360.2.z.c.1231.25

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.00000 q^{5} +(-2.64060 - 0.165018i) q^{7} -1.00000 q^{9} -0.264307 q^{11} +6.07875 q^{13} -1.00000i q^{15} -1.85661i q^{17} -3.89210i q^{19} +(0.165018 - 2.64060i) q^{21} +2.09615i q^{23} +1.00000 q^{25} -1.00000i q^{27} +7.65773i q^{29} -7.19786 q^{31} -0.264307i q^{33} +(2.64060 + 0.165018i) q^{35} +8.95826i q^{37} +6.07875i q^{39} +2.28921i q^{41} +1.71132 q^{43} +1.00000 q^{45} -11.7259 q^{47} +(6.94554 + 0.871493i) q^{49} +1.85661 q^{51} -4.04788i q^{53} +0.264307 q^{55} +3.89210 q^{57} +4.59290i q^{59} -9.96739 q^{61} +(2.64060 + 0.165018i) q^{63} -6.07875 q^{65} -2.89202 q^{67} -2.09615 q^{69} +8.61213i q^{71} -4.16846i q^{73} +1.00000i q^{75} +(0.697928 + 0.0436153i) q^{77} -10.3007i q^{79} +1.00000 q^{81} -5.71691i q^{83} +1.85661i q^{85} -7.65773 q^{87} -13.3494i q^{89} +(-16.0515 - 1.00310i) q^{91} -7.19786i q^{93} +3.89210i q^{95} -11.1183i q^{97} +0.264307 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 28 q^{5} + 4 q^{7} - 28 q^{9} + 8 q^{13} + 28 q^{25} + 24 q^{31} - 4 q^{35} - 24 q^{43} + 28 q^{45} - 24 q^{47} + 28 q^{49} + 8 q^{57} + 16 q^{61} - 4 q^{63} - 8 q^{65} + 8 q^{67} + 24 q^{69} - 16 q^{77}+ \cdots - 48 q^{91}+O(q^{100}) $ 28 * q - 28 * q^5 + 4 * q^7 - 28 * q^9 + 8 * q^13 + 28 * q^25 + 24 * q^31 - 4 * q^35 - 24 * q^43 + 28 * q^45 - 24 * q^47 + 28 * q^49 + 8 * q^57 + 16 * q^61 - 4 * q^63 - 8 * q^65 + 8 * q^67 + 24 * q^69 - 16 * q^77 + 28 * q^81 + 16 * q^87 - 48 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$-1$	$1$	$-1$	$-1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−2.64060	−	0.165018i	−0.998053	−	0.0623709i
$7$	−2.64060	−	0.165018i	−0.998053	−	0.0623709i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−0.264307			−0.0796915			−0.0398457	−	0.999206i	$-0.512687\pi$
$11$	−0.264307			−0.0796915			−0.0398457	+	0.999206i	$0.512687\pi$
$12$		0			0
$13$	6.07875			1.68594			0.842971	−	0.537959i	$-0.180805\pi$
$13$	6.07875			1.68594			0.842971	+	0.537959i	$0.180805\pi$
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$		−	1.85661i		−	0.450294i	−0.974325	−	0.225147i	$-0.927714\pi$
$17$		−	1.85661i		−	0.450294i	0.974325	−	0.225147i	$-0.0722863\pi$
$18$		0			0
$19$		−	3.89210i		−	0.892910i	−0.894806	−	0.446455i	$-0.852686\pi$
$19$		−	3.89210i		−	0.892910i	0.894806	−	0.446455i	$-0.147314\pi$
$20$		0			0
$21$	0.165018	−	2.64060i	0.0360099	−	0.576226i
$22$		0			0
$23$			2.09615i			0.437078i	0.975828	+	0.218539i	$0.0701291\pi$
$23$			2.09615i			0.437078i	−0.975828	+	0.218539i	$0.929871\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$			7.65773i			1.42200i	0.703190	+	0.711002i	$0.251758\pi$
$29$			7.65773i			1.42200i	−0.703190	+	0.711002i	$0.748242\pi$
$30$		0			0
$31$	−7.19786			−1.29277			−0.646387	−	0.763009i	$-0.723721\pi$
$31$	−7.19786			−1.29277			−0.646387	+	0.763009i	$0.723721\pi$
$32$		0			0
$33$		−	0.264307i		−	0.0460099i
$34$		0			0
$35$	2.64060	+	0.165018i	0.446343	+	0.0278931i
$36$		0			0
$37$			8.95826i			1.47273i	0.676585	+	0.736365i	$0.263459\pi$
$37$			8.95826i			1.47273i	−0.676585	+	0.736365i	$0.736541\pi$
$38$		0			0
$39$			6.07875i			0.973379i
$40$		0			0
$41$			2.28921i			0.357514i	0.983893	+	0.178757i	$0.0572076\pi$
$41$			2.28921i			0.357514i	−0.983893	+	0.178757i	$0.942792\pi$
$42$		0			0
$43$	1.71132			0.260974			0.130487	−	0.991450i	$-0.458346\pi$
$43$	1.71132			0.260974			0.130487	+	0.991450i	$0.458346\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$	−11.7259			−1.71040			−0.855201	−	0.518297i	$-0.826566\pi$
$47$	−11.7259			−1.71040			−0.855201	+	0.518297i	$0.826566\pi$
$48$		0			0
$49$	6.94554	+	0.871493i	0.992220	+	0.124499i
$50$		0			0
$51$	1.85661			0.259977
$52$		0			0
$53$		−	4.04788i		−	0.556019i	−0.960578	−	0.278009i	$-0.910325\pi$
$53$		−	4.04788i		−	0.556019i	0.960578	−	0.278009i	$-0.0896746\pi$
$54$		0			0
$55$	0.264307			0.0356391
$56$		0			0
$57$	3.89210			0.515522
$58$		0			0
$59$			4.59290i			0.597945i	0.954262	+	0.298973i	$0.0966439\pi$
$59$			4.59290i			0.597945i	−0.954262	+	0.298973i	$0.903356\pi$
$60$		0			0
$61$	−9.96739			−1.27619			−0.638097	−	0.769956i	$-0.720278\pi$
$61$	−9.96739			−1.27619			−0.638097	+	0.769956i	$0.720278\pi$
$62$		0			0
$63$	2.64060	+	0.165018i	0.332684	+	0.0207903i
$64$		0			0
$65$	−6.07875			−0.753976
$66$		0			0
$67$	−2.89202			−0.353317			−0.176658	−	0.984272i	$-0.556529\pi$
$67$	−2.89202			−0.353317			−0.176658	+	0.984272i	$0.556529\pi$
$68$		0			0
$69$	−2.09615			−0.252347
$70$		0			0
$71$			8.61213i			1.02207i	0.859559	+	0.511036i	$0.170738\pi$
$71$			8.61213i			1.02207i	−0.859559	+	0.511036i	$0.829262\pi$
$72$		0			0
$73$		−	4.16846i		−	0.487881i	−0.969790	−	0.243940i	$-0.921560\pi$
$73$		−	4.16846i		−	0.487881i	0.969790	−	0.243940i	$-0.0784401\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	0.697928	+	0.0436153i	0.0795363	+	0.00497043i
$78$		0			0
$79$		−	10.3007i		−	1.15891i	−0.815003	−	0.579457i	$-0.803265\pi$
$79$		−	10.3007i		−	1.15891i	0.815003	−	0.579457i	$-0.196735\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	5.71691i		−	0.627512i	−0.949504	−	0.313756i	$-0.898413\pi$
$83$		−	5.71691i		−	0.627512i	0.949504	−	0.313756i	$-0.101587\pi$
$84$		0			0
$85$			1.85661i			0.201378i
$86$		0			0
$87$	−7.65773			−0.820994
$88$		0			0
$89$		−	13.3494i		−	1.41503i	−0.706696	−	0.707517i	$-0.749815\pi$
$89$		−	13.3494i		−	1.41503i	0.706696	−	0.707517i	$-0.250185\pi$
$90$		0			0
$91$	−16.0515	−	1.00310i	−1.68266	−	0.105154i
$92$		0			0
$93$		−	7.19786i		−	0.746384i
$94$		0			0
$95$			3.89210i			0.399321i
$96$		0			0
$97$		−	11.1183i		−	1.12889i	−0.825470	−	0.564446i	$-0.809090\pi$
$97$		−	11.1183i		−	1.12889i	0.825470	−	0.564446i	$-0.190910\pi$
$98$		0			0
$99$	0.264307			0.0265638
$100$		0			0
$101$	−4.37247			−0.435077			−0.217539	−	0.976052i	$-0.569803\pi$
$101$	−4.37247			−0.435077			−0.217539	+	0.976052i	$0.569803\pi$
$102$		0			0
$103$	−4.82084			−0.475012			−0.237506	−	0.971386i	$-0.576330\pi$
$103$	−4.82084			−0.475012			−0.237506	+	0.971386i	$0.576330\pi$
$104$		0			0
$105$	−0.165018	+	2.64060i	−0.0161041	+	0.257696i
$106$		0			0
$107$	6.14307			0.593873			0.296937	−	0.954897i	$-0.404035\pi$
$107$	6.14307			0.593873			0.296937	+	0.954897i	$0.404035\pi$
$108$		0			0
$109$		−	15.4612i		−	1.48091i	−0.672105	−	0.740456i	$-0.734610\pi$
$109$		−	15.4612i		−	1.48091i	0.672105	−	0.740456i	$-0.265390\pi$
$110$		0			0
$111$	−8.95826			−0.850281
$112$		0			0
$113$	−16.7857			−1.57907			−0.789534	−	0.613707i	$-0.789678\pi$
$113$	−16.7857			−1.57907			−0.789534	+	0.613707i	$0.789678\pi$
$114$		0			0
$115$		−	2.09615i		−	0.195467i
$116$		0			0
$117$	−6.07875			−0.561981
$118$		0			0
$119$	−0.306374	+	4.90257i	−0.0280853	+	0.449417i
$120$		0			0
$121$	−10.9301			−0.993649
$122$		0			0
$123$	−2.28921			−0.206411
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$		−	17.0682i		−	1.51456i	−0.653089	−	0.757281i	$-0.726527\pi$
$127$		−	17.0682i		−	1.51456i	0.653089	−	0.757281i	$-0.273473\pi$
$128$		0			0
$129$			1.71132i			0.150674i
$130$		0			0
$131$		−	1.99977i		−	0.174721i	−0.996177	−	0.0873605i	$-0.972157\pi$
$131$		−	1.99977i		−	0.174721i	0.996177	−	0.0873605i	$-0.0278432\pi$
$132$		0			0
$133$	−0.642267	+	10.2775i	−0.0556916	+	0.891171i
$134$		0			0
$135$			1.00000i			0.0860663i
$136$		0			0
$137$	−16.9556			−1.44862			−0.724308	−	0.689476i	$-0.757841\pi$
$137$	−16.9556			−1.44862			−0.724308	+	0.689476i	$0.757841\pi$
$138$		0			0
$139$			19.9469i			1.69188i	0.533280	+	0.845939i	$0.320959\pi$
$139$			19.9469i			1.69188i	−0.533280	+	0.845939i	$0.679041\pi$
$140$		0			0
$141$		−	11.7259i		−	0.987501i
$142$		0			0
$143$	−1.60665			−0.134355
$144$		0			0
$145$		−	7.65773i		−	0.635939i
$146$		0			0
$147$	−0.871493	+	6.94554i	−0.0718795	+	0.572858i
$148$		0			0
$149$			3.68761i			0.302101i	0.988526	+	0.151050i	$0.0482656\pi$
$149$			3.68761i			0.302101i	−0.988526	+	0.151050i	$0.951734\pi$
$150$		0			0
$151$			14.4831i			1.17862i	0.807907	+	0.589309i	$0.200600\pi$
$151$			14.4831i			1.17862i	−0.807907	+	0.589309i	$0.799400\pi$
$152$		0			0
$153$			1.85661i			0.150098i
$154$		0			0
$155$	7.19786			0.578146
$156$		0			0
$157$	−23.1101			−1.84439			−0.922193	−	0.386730i	$-0.873604\pi$
$157$	−23.1101			−1.84439			−0.922193	+	0.386730i	$0.873604\pi$
$158$		0			0
$159$	4.04788			0.321018
$160$		0			0
$161$	0.345903	−	5.53510i	0.0272609	−	0.436227i
$162$		0			0
$163$	−13.9465			−1.09238			−0.546188	−	0.837663i	$-0.683922\pi$
$163$	−13.9465			−1.09238			−0.546188	+	0.837663i	$0.683922\pi$
$164$		0			0
$165$			0.264307i			0.0205762i
$166$		0			0
$167$	−5.00401			−0.387222			−0.193611	−	0.981078i	$-0.562020\pi$
$167$	−5.00401			−0.387222			−0.193611	+	0.981078i	$0.562020\pi$
$168$		0			0
$169$	23.9512			1.84240
$170$		0			0
$171$			3.89210i			0.297637i
$172$		0			0
$173$	0.0398406			0.00302903			0.00151451	−	0.999999i	$-0.499518\pi$
$173$	0.0398406			0.00302903			0.00151451	+	0.999999i	$0.499518\pi$
$174$		0			0
$175$	−2.64060	−	0.165018i	−0.199611	−	0.0124742i
$176$		0			0
$177$	−4.59290			−0.345224
$178$		0			0
$179$	−0.662470			−0.0495153			−0.0247577	−	0.999693i	$-0.507881\pi$
$179$	−0.662470			−0.0495153			−0.0247577	+	0.999693i	$0.507881\pi$
$180$		0			0
$181$	8.47656			0.630058			0.315029	−	0.949082i	$-0.397986\pi$
$181$	8.47656			0.630058			0.315029	+	0.949082i	$0.397986\pi$
$182$		0			0
$183$		−	9.96739i		−	0.736810i
$184$		0			0
$185$		−	8.95826i		−	0.658625i
$186$		0			0
$187$			0.490714i			0.0358846i
$188$		0			0
$189$	−0.165018	+	2.64060i	−0.0120033	+	0.192075i
$190$		0			0
$191$		−	11.6714i		−	0.844513i	−0.906476	−	0.422257i	$-0.861238\pi$
$191$		−	11.6714i		−	0.844513i	0.906476	−	0.422257i	$-0.138762\pi$
$192$		0			0
$193$	2.29670			0.165320			0.0826600	−	0.996578i	$-0.473658\pi$
$193$	2.29670			0.165320			0.0826600	+	0.996578i	$0.473658\pi$
$194$		0			0
$195$		−	6.07875i		−	0.435308i
$196$		0			0
$197$			23.3822i			1.66591i	0.553339	+	0.832956i	$0.313354\pi$
$197$			23.3822i			1.66591i	−0.553339	+	0.832956i	$0.686646\pi$
$198$		0			0
$199$	−9.36275			−0.663708			−0.331854	−	0.943331i	$-0.607674\pi$
$199$	−9.36275			−0.663708			−0.331854	+	0.943331i	$0.607674\pi$
$200$		0			0
$201$		−	2.89202i		−	0.203987i
$202$		0			0
$203$	1.26366	−	20.2210i	0.0886917	−	1.41924i
$204$		0			0
$205$		−	2.28921i		−	0.159885i
$206$		0			0
$207$		−	2.09615i		−	0.145693i
$208$		0			0
$209$			1.02871i			0.0711573i
$210$		0			0
$211$	15.8913			1.09400			0.547002	−	0.837132i	$-0.315769\pi$
$211$	15.8913			1.09400			0.547002	+	0.837132i	$0.315769\pi$
$212$		0			0
$213$	−8.61213			−0.590093
$214$		0			0
$215$	−1.71132			−0.116711
$216$		0			0
$217$	19.0067	+	1.18778i	1.29026	+	0.0806316i
$218$		0			0
$219$	4.16846			0.281678
$220$		0			0
$221$		−	11.2859i		−	0.759170i
$222$		0			0
$223$	−11.3286			−0.758619			−0.379309	−	0.925270i	$-0.623838\pi$
$223$	−11.3286			−0.758619			−0.379309	+	0.925270i	$0.623838\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$		−	4.60858i		−	0.305882i	−0.988235	−	0.152941i	$-0.951126\pi$
$227$		−	4.60858i		−	0.305882i	0.988235	−	0.152941i	$-0.0488745\pi$
$228$		0			0
$229$	1.54270			0.101945			0.0509723	−	0.998700i	$-0.483768\pi$
$229$	1.54270			0.101945			0.0509723	+	0.998700i	$0.483768\pi$
$230$		0			0
$231$	−0.0436153	+	0.697928i	−0.00286968	+	0.0459203i
$232$		0			0
$233$	−3.72016			−0.243716			−0.121858	−	0.992548i	$-0.538885\pi$
$233$	−3.72016			−0.243716			−0.121858	+	0.992548i	$0.538885\pi$
$234$		0			0
$235$	11.7259			0.764915
$236$		0			0
$237$	10.3007			0.669100
$238$		0			0
$239$			28.0309i			1.81317i	0.422027	+	0.906583i	$0.361319\pi$
$239$			28.0309i			1.81317i	−0.422027	+	0.906583i	$0.638681\pi$
$240$		0			0
$241$			8.95756i			0.577007i	0.957479	+	0.288504i	$0.0931577\pi$
$241$			8.95756i			0.577007i	−0.957479	+	0.288504i	$0.906842\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−6.94554	−	0.871493i	−0.443734	−	0.0556776i
$246$		0			0
$247$		−	23.6591i		−	1.50539i
$248$		0			0
$249$	5.71691			0.362294
$250$		0			0
$251$			14.7968i			0.933965i	0.884266	+	0.466983i	$0.154659\pi$
$251$			14.7968i			0.933965i	−0.884266	+	0.466983i	$0.845341\pi$
$252$		0			0
$253$		−	0.554027i		−	0.0348314i
$254$		0			0
$255$	−1.85661			−0.116265
$256$		0			0
$257$			12.8851i			0.803754i	0.915694	+	0.401877i	$0.131642\pi$
$257$			12.8851i			0.803754i	−0.915694	+	0.401877i	$0.868358\pi$
$258$		0			0
$259$	1.47827	−	23.6552i	0.0918555	−	1.46986i
$260$		0			0
$261$		−	7.65773i		−	0.474001i
$262$		0			0
$263$			7.26011i			0.447677i	0.974626	+	0.223839i	$0.0718589\pi$
$263$			7.26011i			0.447677i	−0.974626	+	0.223839i	$0.928141\pi$
$264$		0			0
$265$			4.04788i			0.248659i
$266$		0			0
$267$	13.3494			0.816971
$268$		0			0
$269$	−12.0068			−0.732067			−0.366034	−	0.930602i	$-0.619285\pi$
$269$	−12.0068			−0.732067			−0.366034	+	0.930602i	$0.619285\pi$
$270$		0			0
$271$	18.8132			1.14282			0.571411	−	0.820664i	$-0.306396\pi$
$271$	18.8132			1.14282			0.571411	+	0.820664i	$0.306396\pi$
$272$		0			0
$273$	1.00310	−	16.0515i	0.0607105	−	0.971484i
$274$		0			0
$275$	−0.264307			−0.0159383
$276$		0			0
$277$		−	9.57593i		−	0.575362i	−0.957726	−	0.287681i	$-0.907116\pi$
$277$		−	9.57593i		−	0.575362i	0.957726	−	0.287681i	$-0.0928842\pi$
$278$		0			0
$279$	7.19786			0.430925
$280$		0			0
$281$	6.08335			0.362902			0.181451	−	0.983400i	$-0.441921\pi$
$281$	6.08335			0.362902			0.181451	+	0.983400i	$0.441921\pi$
$282$		0			0
$283$			8.00209i			0.475675i	0.971305	+	0.237838i	$0.0764386\pi$
$283$			8.00209i			0.475675i	−0.971305	+	0.237838i	$0.923561\pi$
$284$		0			0
$285$	−3.89210			−0.230548
$286$		0			0
$287$	0.377760	−	6.04488i	0.0222985	−	0.356818i
$288$		0			0
$289$	13.5530			0.797235
$290$		0			0
$291$	11.1183			0.651766
$292$		0			0
$293$	−14.1818			−0.828509			−0.414255	−	0.910161i	$-0.635958\pi$
$293$	−14.1818			−0.828509			−0.414255	+	0.910161i	$0.635958\pi$
$294$		0			0
$295$		−	4.59290i		−	0.267409i
$296$		0			0
$297$			0.264307i			0.0153366i
$298$		0			0
$299$			12.7420i			0.736888i
$300$		0			0
$301$	−4.51892	−	0.282399i	−0.260466	−	0.0162772i
$302$		0			0
$303$		−	4.37247i		−	0.251192i
$304$		0			0
$305$	9.96739			0.570731
$306$		0			0
$307$		−	20.0654i		−	1.14519i	−0.819837	−	0.572597i	$-0.805936\pi$
$307$		−	20.0654i		−	1.14519i	0.819837	−	0.572597i	$-0.194064\pi$
$308$		0			0
$309$		−	4.82084i		−	0.274248i
$310$		0			0
$311$	−15.0389			−0.852777			−0.426389	−	0.904540i	$-0.640214\pi$
$311$	−15.0389			−0.852777			−0.426389	+	0.904540i	$0.640214\pi$
$312$		0			0
$313$		−	22.2502i		−	1.25766i	−0.777545	−	0.628828i	$-0.783535\pi$
$313$		−	22.2502i		−	1.25766i	0.777545	−	0.628828i	$-0.216465\pi$
$314$		0			0
$315$	−2.64060	−	0.165018i	−0.148781	−	0.00929771i
$316$		0			0
$317$			33.4862i			1.88077i	0.340112	+	0.940385i	$0.389535\pi$
$317$			33.4862i			1.88077i	−0.340112	+	0.940385i	$0.610465\pi$
$318$		0			0
$319$		−	2.02399i		−	0.113322i
$320$		0			0
$321$			6.14307i			0.342873i
$322$		0			0
$323$	−7.22612			−0.402072
$324$		0			0
$325$	6.07875			0.337188
$326$		0			0
$327$	15.4612			0.855004
$328$		0			0
$329$	30.9635	+	1.93499i	1.70707	+	0.106679i
$330$		0			0
$331$	−23.5161			−1.29256			−0.646280	−	0.763100i	$-0.723676\pi$
$331$	−23.5161			−1.29256			−0.646280	+	0.763100i	$0.723676\pi$
$332$		0			0
$333$		−	8.95826i		−	0.490910i
$334$		0			0
$335$	2.89202			0.158008
$336$		0			0
$337$	−18.0622			−0.983911			−0.491955	−	0.870620i	$-0.663718\pi$
$337$	−18.0622			−0.983911			−0.491955	+	0.870620i	$0.663718\pi$
$338$		0			0
$339$		−	16.7857i		−	0.911675i
$340$		0			0
$341$	1.90244			0.103023
$342$		0			0
$343$	−18.1966	−	3.44740i	−0.982523	−	0.186142i
$344$		0			0
$345$	2.09615			0.112853
$346$		0			0
$347$	−14.1722			−0.760803			−0.380401	−	0.924822i	$-0.624214\pi$
$347$	−14.1722			−0.760803			−0.380401	+	0.924822i	$0.624214\pi$
$348$		0			0
$349$	−29.8810			−1.59949			−0.799746	−	0.600339i	$-0.795032\pi$
$349$	−29.8810			−1.59949			−0.799746	+	0.600339i	$0.795032\pi$
$350$		0			0
$351$		−	6.07875i		−	0.324460i
$352$		0			0
$353$			17.7999i			0.947392i	0.880688	+	0.473696i	$0.157080\pi$
$353$			17.7999i			0.947392i	−0.880688	+	0.473696i	$0.842920\pi$
$354$		0			0
$355$		−	8.61213i		−	0.457084i
$356$		0			0
$357$	−4.90257	−	0.306374i	−0.259471	−	0.0162150i
$358$		0			0
$359$			16.1540i			0.852576i	0.904588	+	0.426288i	$0.140179\pi$
$359$			16.1540i			0.852576i	−0.904588	+	0.426288i	$0.859821\pi$
$360$		0			0
$361$	3.85153			0.202712
$362$		0			0
$363$		−	10.9301i		−	0.573684i
$364$		0			0
$365$			4.16846i			0.218187i
$366$		0			0
$367$	33.5356			1.75054			0.875272	−	0.483631i	$-0.160682\pi$
$367$	33.5356			1.75054			0.875272	+	0.483631i	$0.160682\pi$
$368$		0			0
$369$		−	2.28921i		−	0.119171i
$370$		0			0
$371$	−0.667973	+	10.6888i	−0.0346794	+	0.554936i
$372$		0			0
$373$		−	20.3929i		−	1.05590i	−0.849275	−	0.527951i	$-0.822960\pi$
$373$		−	20.3929i		−	1.05590i	0.849275	−	0.527951i	$-0.177040\pi$
$374$		0			0
$375$		−	1.00000i		−	0.0516398i
$376$		0			0
$377$			46.5494i			2.39742i
$378$		0			0
$379$	11.2350			0.577101			0.288550	−	0.957465i	$-0.406827\pi$
$379$	11.2350			0.577101			0.288550	+	0.957465i	$0.406827\pi$
$380$		0			0
$381$	17.0682			0.874433
$382$		0			0
$383$	8.65284			0.442140			0.221070	−	0.975258i	$-0.429045\pi$
$383$	8.65284			0.442140			0.221070	+	0.975258i	$0.429045\pi$
$384$		0			0
$385$	−0.697928	−	0.0436153i	−0.0355697	−	0.00222284i
$386$		0			0
$387$	−1.71132			−0.0869915
$388$		0			0
$389$			5.18569i			0.262925i	0.991321	+	0.131462i	$0.0419673\pi$
$389$			5.18569i			0.262925i	−0.991321	+	0.131462i	$0.958033\pi$
$390$		0			0
$391$	3.89174			0.196814
$392$		0			0
$393$	1.99977			0.100875
$394$		0			0
$395$			10.3007i			0.518282i
$396$		0			0
$397$	6.42434			0.322428			0.161214	−	0.986919i	$-0.448459\pi$
$397$	6.42434			0.322428			0.161214	+	0.986919i	$0.448459\pi$
$398$		0			0
$399$	−10.2775	−	0.642267i	−0.514518	−	0.0321536i
$400$		0			0
$401$	−36.9183			−1.84361			−0.921806	−	0.387651i	$-0.873286\pi$
$401$	−36.9183			−1.84361			−0.921806	+	0.387651i	$0.873286\pi$
$402$		0			0
$403$	−43.7540			−2.17954
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$		−	2.36773i		−	0.117364i
$408$		0			0
$409$			31.5872i			1.56189i	0.624602	+	0.780943i	$0.285261\pi$
$409$			31.5872i			1.56189i	−0.624602	+	0.780943i	$0.714739\pi$
$410$		0			0
$411$		−	16.9556i		−	0.836359i
$412$		0			0
$413$	0.757912	−	12.1280i	0.0372944	−	0.596781i
$414$		0			0
$415$			5.71691i			0.280632i
$416$		0			0
$417$	−19.9469			−0.976806
$418$		0			0
$419$		−	25.7821i		−	1.25954i	−0.776782	−	0.629770i	$-0.783149\pi$
$419$		−	25.7821i		−	1.25954i	0.776782	−	0.629770i	$-0.216851\pi$
$420$		0			0
$421$			1.53513i			0.0748174i	0.999300	+	0.0374087i	$0.0119103\pi$
$421$			1.53513i			0.0748174i	−0.999300	+	0.0374087i	$0.988090\pi$
$422$		0			0
$423$	11.7259			0.570134
$424$		0			0
$425$		−	1.85661i		−	0.0900588i
$426$		0			0
$427$	26.3199	+	1.64480i	1.27371	+	0.0795973i
$428$		0			0
$429$		−	1.60665i		−	0.0775700i
$430$		0			0
$431$			31.2356i			1.50457i	0.658840	+	0.752283i	$0.271047\pi$
$431$			31.2356i			1.50457i	−0.658840	+	0.752283i	$0.728953\pi$
$432$		0			0
$433$		−	6.85246i		−	0.329308i	−0.986351	−	0.164654i	$-0.947349\pi$
$433$		−	6.85246i		−	0.329308i	0.986351	−	0.164654i	$-0.0526508\pi$
$434$		0			0
$435$	7.65773			0.367160
$436$		0			0
$437$	8.15844			0.390271
$438$		0			0
$439$	0.157801			0.00753144			0.00376572	−	0.999993i	$-0.498801\pi$
$439$	0.157801			0.00753144			0.00376572	+	0.999993i	$0.498801\pi$
$440$		0			0
$441$	−6.94554	−	0.871493i	−0.330740	−	0.0414997i
$442$		0			0
$443$	−39.7323			−1.88774			−0.943868	−	0.330323i	$-0.892842\pi$
$443$	−39.7323			−1.88774			−0.943868	+	0.330323i	$0.892842\pi$
$444$		0			0
$445$			13.3494i			0.632823i
$446$		0			0
$447$	−3.68761			−0.174418
$448$		0			0
$449$	−9.45801			−0.446351			−0.223176	−	0.974778i	$-0.571642\pi$
$449$	−9.45801			−0.446351			−0.223176	+	0.974778i	$0.571642\pi$
$450$		0			0
$451$		−	0.605053i		−	0.0284908i
$452$		0			0
$453$	−14.4831			−0.680476
$454$		0			0
$455$	16.0515	+	1.00310i	0.752508	+	0.0470262i
$456$		0			0
$457$	−25.6849			−1.20149			−0.600745	−	0.799441i	$-0.705129\pi$
$457$	−25.6849			−1.20149			−0.600745	+	0.799441i	$0.705129\pi$
$458$		0			0
$459$	−1.85661			−0.0866591
$460$		0			0
$461$	−27.4268			−1.27739			−0.638697	−	0.769459i	$-0.720526\pi$
$461$	−27.4268			−1.27739			−0.638697	+	0.769459i	$0.720526\pi$
$462$		0			0
$463$		−	16.9982i		−	0.789972i	−0.918687	−	0.394986i	$-0.870749\pi$
$463$		−	16.9982i		−	0.789972i	0.918687	−	0.394986i	$-0.129251\pi$
$464$		0			0
$465$			7.19786i			0.333793i
$466$		0			0
$467$		−	8.95474i		−	0.414376i	−0.978301	−	0.207188i	$-0.933569\pi$
$467$		−	8.95474i		−	0.414376i	0.978301	−	0.207188i	$-0.0664312\pi$
$468$		0			0
$469$	7.63667	+	0.477235i	0.352629	+	0.0220367i
$470$		0			0
$471$		−	23.1101i		−	1.06486i
$472$		0			0
$473$	−0.452314			−0.0207974
$474$		0			0
$475$		−	3.89210i		−	0.178582i
$476$		0			0
$477$			4.04788i			0.185340i
$478$		0			0
$479$	4.76512			0.217724			0.108862	−	0.994057i	$-0.465279\pi$
$479$	4.76512			0.217724			0.108862	+	0.994057i	$0.465279\pi$
$480$		0			0
$481$			54.4550i			2.48294i
$482$		0			0
$483$	5.53510	+	0.345903i	0.251856	+	0.0157391i
$484$		0			0
$485$			11.1183i			0.504856i
$486$		0			0
$487$		−	22.2240i		−	1.00707i	−0.863975	−	0.503534i	$-0.832033\pi$
$487$		−	22.2240i		−	1.00707i	0.863975	−	0.503534i	$-0.167967\pi$
$488$		0			0
$489$		−	13.9465i		−	0.630684i
$490$		0			0
$491$	29.6305			1.33721			0.668604	−	0.743619i	$-0.266892\pi$
$491$	29.6305			1.33721			0.668604	+	0.743619i	$0.266892\pi$
$492$		0			0
$493$	14.2174			0.640320
$494$		0			0
$495$	−0.264307			−0.0118797
$496$		0			0
$497$	1.42116	−	22.7412i	0.0637475	−	1.02008i
$498$		0			0
$499$	17.4850			0.782738			0.391369	−	0.920234i	$-0.372002\pi$
$499$	17.4850			0.782738			0.391369	+	0.920234i	$0.372002\pi$
$500$		0			0
$501$		−	5.00401i		−	0.223562i
$502$		0			0
$503$	26.5951			1.18582			0.592908	−	0.805270i	$-0.297980\pi$
$503$	26.5951			1.18582			0.592908	+	0.805270i	$0.297980\pi$
$504$		0			0
$505$	4.37247			0.194572
$506$		0			0
$507$			23.9512i			1.06371i
$508$		0			0
$509$	41.2008			1.82620			0.913098	−	0.407741i	$-0.133683\pi$
$509$	41.2008			1.82620			0.913098	+	0.407741i	$0.133683\pi$
$510$		0			0
$511$	−0.687870	+	11.0072i	−0.0304296	+	0.486931i
$512$		0			0
$513$	−3.89210			−0.171841
$514$		0			0
$515$	4.82084			0.212432
$516$		0			0
$517$	3.09924			0.136304
$518$		0			0
$519$			0.0398406i			0.00174881i
$520$		0			0
$521$		−	4.12574i		−	0.180752i	−0.995908	−	0.0903761i	$-0.971193\pi$
$521$		−	4.12574i		−	0.180752i	0.995908	−	0.0903761i	$-0.0288069\pi$
$522$		0			0
$523$		−	14.9470i		−	0.653586i	−0.945096	−	0.326793i	$-0.894032\pi$
$523$		−	14.9470i		−	0.653586i	0.945096	−	0.326793i	$-0.105968\pi$
$524$		0			0
$525$	0.165018	−	2.64060i	0.00720197	−	0.115245i
$526$		0			0
$527$			13.3636i			0.582129i
$528$		0			0
$529$	18.6061			0.808963
$530$		0			0
$531$		−	4.59290i		−	0.199315i
$532$		0			0
$533$			13.9155i			0.602748i
$534$		0			0
$535$	−6.14307			−0.265588
$536$		0			0
$537$		−	0.662470i		−	0.0285877i
$538$		0			0
$539$	−1.83575	−	0.230341i	−0.0790714	−	0.00992151i
$540$		0			0
$541$			8.25685i			0.354990i	0.984122	+	0.177495i	$0.0567993\pi$
$541$			8.25685i			0.354990i	−0.984122	+	0.177495i	$0.943201\pi$
$542$		0			0
$543$			8.47656i			0.363764i
$544$		0			0
$545$			15.4612i			0.662284i
$546$		0			0
$547$	23.1912			0.991586			0.495793	−	0.868441i	$-0.334877\pi$
$547$	23.1912			0.991586			0.495793	+	0.868441i	$0.334877\pi$
$548$		0			0
$549$	9.96739			0.425398
$550$		0			0
$551$	29.8047			1.26972
$552$		0			0
$553$	−1.69979	+	27.1999i	−0.0722826	+	1.15666i
$554$		0			0
$555$	8.95826			0.380257
$556$		0			0
$557$		−	24.5841i		−	1.04166i	−0.853660	−	0.520830i	$-0.825622\pi$
$557$		−	24.5841i		−	1.04166i	0.853660	−	0.520830i	$-0.174378\pi$
$558$		0			0
$559$	10.4027			0.439988
$560$		0			0
$561$	−0.490714			−0.0207180
$562$		0			0
$563$		−	11.9197i		−	0.502354i	−0.967941	−	0.251177i	$-0.919182\pi$
$563$		−	11.9197i		−	0.502354i	0.967941	−	0.251177i	$-0.0808175\pi$
$564$		0			0
$565$	16.7857			0.706180
$566$		0			0
$567$	−2.64060	−	0.165018i	−0.110895	−	0.00693010i
$568$		0			0
$569$	17.2754			0.724221			0.362111	−	0.932135i	$-0.382056\pi$
$569$	17.2754			0.724221			0.362111	+	0.932135i	$0.382056\pi$
$570$		0			0
$571$	15.3607			0.642823			0.321412	−	0.946940i	$-0.395843\pi$
$571$	15.3607			0.642823			0.321412	+	0.946940i	$0.395843\pi$
$572$		0			0
$573$	11.6714			0.487580
$574$		0			0
$575$			2.09615i			0.0874156i
$576$		0			0
$577$		−	18.5279i		−	0.771326i	−0.922640	−	0.385663i	$-0.873973\pi$
$577$		−	18.5279i		−	0.771326i	0.922640	−	0.385663i	$-0.126027\pi$
$578$		0			0
$579$			2.29670i			0.0954475i
$580$		0			0
$581$	−0.943392	+	15.0961i	−0.0391385	+	0.626290i
$582$		0			0
$583$			1.06988i			0.0443099i
$584$		0			0
$585$	6.07875			0.251325
$586$		0			0
$587$			20.7809i			0.857720i	0.903371	+	0.428860i	$0.141085\pi$
$587$			20.7809i			0.857720i	−0.903371	+	0.428860i	$0.858915\pi$
$588$		0			0
$589$			28.0148i			1.15433i
$590$		0			0
$591$	−23.3822			−0.961815
$592$		0			0
$593$			1.59413i			0.0654632i	0.999464	+	0.0327316i	$0.0104206\pi$
$593$			1.59413i			0.0654632i	−0.999464	+	0.0327316i	$0.989579\pi$
$594$		0			0
$595$	0.306374	−	4.90257i	0.0125601	−	0.200986i
$596$		0			0
$597$		−	9.36275i		−	0.383192i
$598$		0			0
$599$			23.8540i			0.974650i	0.873221	+	0.487325i	$0.162027\pi$
$599$			23.8540i			0.974650i	−0.873221	+	0.487325i	$0.837973\pi$
$600$		0			0
$601$		−	28.1413i		−	1.14791i	−0.818887	−	0.573954i	$-0.805409\pi$
$601$		−	28.1413i		−	1.14791i	0.818887	−	0.573954i	$-0.194591\pi$
$602$		0			0
$603$	2.89202			0.117772
$604$		0			0
$605$	10.9301			0.444373
$606$		0			0
$607$	−9.91696			−0.402517			−0.201258	−	0.979538i	$-0.564503\pi$
$607$	−9.91696			−0.402517			−0.201258	+	0.979538i	$0.564503\pi$
$608$		0			0
$609$	20.2210	+	1.26366i	0.819396	+	0.0512062i
$610$		0			0
$611$	−71.2789			−2.88364
$612$		0			0
$613$		−	36.6198i		−	1.47906i	−0.673123	−	0.739531i	$-0.735047\pi$
$613$		−	36.6198i		−	1.47906i	0.673123	−	0.739531i	$-0.264953\pi$
$614$		0			0
$615$	2.28921			0.0923098
$616$		0			0
$617$	−21.8103			−0.878050			−0.439025	−	0.898475i	$-0.644676\pi$
$617$	−21.8103			−0.878050			−0.439025	+	0.898475i	$0.644676\pi$
$618$		0			0
$619$			31.3205i			1.25888i	0.777050	+	0.629439i	$0.216715\pi$
$619$			31.3205i			1.25888i	−0.777050	+	0.629439i	$0.783285\pi$
$620$		0			0
$621$	2.09615			0.0841157
$622$		0			0
$623$	−2.20289	+	35.2505i	−0.0882570	+	1.41228i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−1.02871			−0.0410827
$628$		0			0
$629$	16.6320			0.663161
$630$		0			0
$631$		−	28.9853i		−	1.15389i	−0.816785	−	0.576943i	$-0.804246\pi$
$631$		−	28.9853i		−	1.15389i	0.816785	−	0.576943i	$-0.195754\pi$
$632$		0			0
$633$			15.8913i			0.631623i
$634$		0			0
$635$			17.0682i			0.677333i
$636$		0			0
$637$	42.2202	+	5.29759i	1.67282	+	0.209898i
$638$		0			0
$639$		−	8.61213i		−	0.340690i
$640$		0			0
$641$	−5.84700			−0.230943			−0.115471	−	0.993311i	$-0.536838\pi$
$641$	−5.84700			−0.230943			−0.115471	+	0.993311i	$0.536838\pi$
$642$		0			0
$643$			47.7829i			1.88437i	0.335087	+	0.942187i	$0.391234\pi$
$643$			47.7829i			1.88437i	−0.335087	+	0.942187i	$0.608766\pi$
$644$		0			0
$645$		−	1.71132i		−	0.0673833i
$646$		0			0
$647$	−1.20494			−0.0473713			−0.0236856	−	0.999719i	$-0.507540\pi$
$647$	−1.20494			−0.0473713			−0.0236856	+	0.999719i	$0.507540\pi$
$648$		0			0
$649$		−	1.21393i		−	0.0476511i
$650$		0			0
$651$	−1.18778	+	19.0067i	−0.0465526	+	0.744931i
$652$		0			0
$653$		−	2.29865i		−	0.0899531i	−0.998988	−	0.0449765i	$-0.985679\pi$
$653$		−	2.29865i		−	0.0899531i	0.998988	−	0.0449765i	$-0.0143213\pi$
$654$		0			0
$655$			1.99977i			0.0781376i
$656$		0			0
$657$			4.16846i			0.162627i
$658$		0			0
$659$	31.7367			1.23629			0.618143	−	0.786066i	$-0.287885\pi$
$659$	31.7367			1.23629			0.618143	+	0.786066i	$0.287885\pi$
$660$		0			0
$661$	−29.3254			−1.14063			−0.570313	−	0.821427i	$-0.693178\pi$
$661$	−29.3254			−1.14063			−0.570313	+	0.821427i	$0.693178\pi$
$662$		0			0
$663$	11.2859			0.438307
$664$		0			0
$665$	0.642267	−	10.2775i	0.0249060	−	0.398544i
$666$		0			0
$667$	−16.0518			−0.621526
$668$		0			0
$669$		−	11.3286i		−	0.437989i
$670$		0			0
$671$	2.63445			0.101702
$672$		0			0
$673$	−5.35156			−0.206287			−0.103144	−	0.994666i	$-0.532890\pi$
$673$	−5.35156			−0.206287			−0.103144	+	0.994666i	$0.532890\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$	38.3482			1.47384			0.736921	−	0.675979i	$-0.236279\pi$
$677$	38.3482			1.47384			0.736921	+	0.675979i	$0.236279\pi$
$678$		0			0
$679$	−1.83472	+	29.3590i	−0.0704100	+	1.12669i
$680$		0			0
$681$	4.60858			0.176601
$682$		0			0
$683$	−40.9876			−1.56835			−0.784174	−	0.620541i	$-0.786913\pi$
$683$	−40.9876			−1.56835			−0.784174	+	0.620541i	$0.786913\pi$
$684$		0			0
$685$	16.9556			0.647841
$686$		0			0
$687$			1.54270i			0.0588577i
$688$		0			0
$689$		−	24.6060i		−	0.937415i
$690$		0			0
$691$		−	12.3363i		−	0.469297i	−0.972080	−	0.234648i	$-0.924606\pi$
$691$		−	12.3363i		−	0.469297i	0.972080	−	0.234648i	$-0.0753938\pi$
$692$		0			0
$693$	−0.697928	−	0.0436153i	−0.0265121	−	0.00165681i
$694$		0			0
$695$		−	19.9469i		−	0.756631i
$696$		0			0
$697$	4.25017			0.160987
$698$		0			0
$699$		−	3.72016i		−	0.140710i
$700$		0			0
$701$			30.1199i			1.13761i	0.822472	+	0.568806i	$0.192594\pi$
$701$			30.1199i			1.13761i	−0.822472	+	0.568806i	$0.807406\pi$
$702$		0			0
$703$	34.8665			1.31501
$704$		0			0
$705$			11.7259i			0.441624i
$706$		0			0
$707$	11.5459	+	0.721536i	0.434230	+	0.0271362i
$708$		0			0
$709$		−	1.69009i		−	0.0634728i	−0.999496	−	0.0317364i	$-0.989896\pi$
$709$		−	1.69009i		−	0.0634728i	0.999496	−	0.0317364i	$-0.0101037\pi$
$710$		0			0
$711$			10.3007i			0.386305i
$712$		0			0
$713$		−	15.0878i		−	0.565043i
$714$		0			0
$715$	1.60665			0.0600855
$716$		0			0
$717$	−28.0309			−1.04683
$718$		0			0
$719$	−43.0068			−1.60388			−0.801942	−	0.597402i	$-0.796200\pi$
$719$	−43.0068			−1.60388			−0.801942	+	0.597402i	$0.796200\pi$
$720$		0			0
$721$	12.7299	+	0.795526i	0.474087	+	0.0296269i
$722$		0			0
$723$	−8.95756			−0.333135
$724$		0			0
$725$			7.65773i			0.284401i
$726$		0			0
$727$	13.9472			0.517273			0.258636	−	0.965975i	$-0.416727\pi$
$727$	13.9472			0.517273			0.258636	+	0.965975i	$0.416727\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	3.17726i		−	0.117515i
$732$		0			0
$733$	13.6799			0.505280			0.252640	−	0.967560i	$-0.418701\pi$
$733$	13.6799			0.505280			0.252640	+	0.967560i	$0.418701\pi$
$734$		0			0
$735$	0.871493	−	6.94554i	0.0321455	−	0.256190i
$736$		0			0
$737$	0.764380			0.0281563
$738$		0			0
$739$	7.98250			0.293641			0.146820	−	0.989163i	$-0.453096\pi$
$739$	7.98250			0.293641			0.146820	+	0.989163i	$0.453096\pi$
$740$		0			0
$741$	23.6591			0.869140
$742$		0			0
$743$			16.4647i			0.604030i	0.953303	+	0.302015i	$0.0976593\pi$
$743$			16.4647i			0.604030i	−0.953303	+	0.302015i	$0.902341\pi$
$744$		0			0
$745$		−	3.68761i		−	0.135104i
$746$		0			0
$747$			5.71691i			0.209171i
$748$		0			0
$749$	−16.2214	−	1.01372i	−0.592717	−	0.0370404i
$750$		0			0
$751$		−	25.8761i		−	0.944234i	−0.881536	−	0.472117i	$-0.843490\pi$
$751$		−	25.8761i		−	0.944234i	0.881536	−	0.472117i	$-0.156510\pi$
$752$		0			0
$753$	−14.7968			−0.539225
$754$		0			0
$755$		−	14.4831i		−	0.527094i
$756$		0			0
$757$		−	0.192056i		−	0.00698040i	−0.999994	−	0.00349020i	$-0.998889\pi$
$757$		−	0.192056i		−	0.00698040i	0.999994	−	0.00349020i	$-0.00111097\pi$
$758$		0			0
$759$	0.554027			0.0201099
$760$		0			0
$761$		−	50.4508i		−	1.82884i	−0.404766	−	0.914420i	$-0.632647\pi$
$761$		−	50.4508i		−	1.82884i	0.404766	−	0.914420i	$-0.367353\pi$
$762$		0			0
$763$	−2.55137	+	40.8268i	−0.0923658	+	1.47803i
$764$		0			0
$765$		−	1.85661i		−	0.0671259i
$766$		0			0
$767$			27.9191i			1.00810i
$768$		0			0
$769$		−	26.3495i		−	0.950186i	−0.879936	−	0.475093i	$-0.842414\pi$
$769$		−	26.3495i		−	0.950186i	0.879936	−	0.475093i	$-0.157586\pi$
$770$		0			0
$771$	−12.8851			−0.464047
$772$		0			0
$773$	45.1896			1.62536			0.812678	−	0.582712i	$-0.198009\pi$
$773$	45.1896			1.62536			0.812678	+	0.582712i	$0.198009\pi$
$774$		0			0
$775$	−7.19786			−0.258555
$776$		0			0
$777$	23.6552	+	1.47827i	0.848625	+	0.0530328i
$778$		0			0
$779$	8.90983			0.319228
$780$		0			0
$781$		−	2.27624i		−	0.0814504i
$782$		0			0
$783$	7.65773			0.273665
$784$		0			0
$785$	23.1101			0.824834
$786$		0			0
$787$			19.9506i			0.711161i	0.934646	+	0.355580i	$0.115717\pi$
$787$			19.9506i			0.711161i	−0.934646	+	0.355580i	$0.884283\pi$
$788$		0			0
$789$	−7.26011			−0.258467
$790$		0			0
$791$	44.3244	+	2.76994i	1.57599	+	0.0984879i
$792$		0			0
$793$	−60.5892			−2.15159
$794$		0			0
$795$	−4.04788			−0.143563
$796$		0			0
$797$	18.4653			0.654075			0.327037	−	0.945011i	$-0.393950\pi$
$797$	18.4653			0.654075			0.327037	+	0.945011i	$0.393950\pi$
$798$		0			0
$799$			21.7705i			0.770184i
$800$		0			0
$801$			13.3494i			0.471678i
$802$		0			0
$803$			1.10175i			0.0388799i
$804$		0			0
$805$	−0.345903	+	5.53510i	−0.0121915	+	0.195087i
$806$		0			0
$807$		−	12.0068i		−	0.422659i
$808$		0			0
$809$	38.2137			1.34352			0.671762	−	0.740767i	$-0.265538\pi$
$809$	38.2137			1.34352			0.671762	+	0.740767i	$0.265538\pi$
$810$		0			0
$811$			2.49948i			0.0877686i	0.999037	+	0.0438843i	$0.0139733\pi$
$811$			2.49948i			0.0877686i	−0.999037	+	0.0438843i	$0.986027\pi$
$812$		0			0
$813$			18.8132i			0.659809i
$814$		0			0
$815$	13.9465			0.488525
$816$		0			0
$817$		−	6.66065i		−	0.233027i
$818$		0			0
$819$	16.0515	+	1.00310i	0.560886	+	0.0350513i
$820$		0			0
$821$		−	45.2255i		−	1.57838i	−0.614149	−	0.789190i	$-0.710501\pi$
$821$		−	45.2255i		−	1.57838i	0.614149	−	0.789190i	$-0.289499\pi$
$822$		0			0
$823$		−	29.9944i		−	1.04554i	−0.852474	−	0.522770i	$-0.824899\pi$
$823$		−	29.9944i		−	1.04554i	0.852474	−	0.522770i	$-0.175101\pi$
$824$		0			0
$825$		−	0.264307i		−	0.00920198i
$826$		0			0
$827$	12.7955			0.444944			0.222472	−	0.974939i	$-0.428587\pi$
$827$	12.7955			0.444944			0.222472	+	0.974939i	$0.428587\pi$
$828$		0			0
$829$	−15.8675			−0.551103			−0.275551	−	0.961286i	$-0.588860\pi$
$829$	−15.8675			−0.551103			−0.275551	+	0.961286i	$0.588860\pi$
$830$		0			0
$831$	9.57593			0.332185
$832$		0			0
$833$	1.61802	−	12.8952i	0.0560612	−	0.446791i
$834$		0			0
$835$	5.00401			0.173171
$836$		0			0
$837$			7.19786i			0.248795i
$838$		0			0
$839$	7.71085			0.266208			0.133104	−	0.991102i	$-0.457506\pi$
$839$	7.71085			0.266208			0.133104	+	0.991102i	$0.457506\pi$
$840$		0			0
$841$	−29.6408			−1.02210
$842$		0			0
$843$			6.08335i			0.209522i
$844$		0			0
$845$	−23.9512			−0.823946
$846$		0			0
$847$	28.8621	+	1.80367i	0.991715	+	0.0619748i
$848$		0			0
$849$	−8.00209			−0.274631
$850$		0			0
$851$	−18.7779			−0.643697
$852$		0			0
$853$	3.27559			0.112154			0.0560771	−	0.998426i	$-0.482141\pi$
$853$	3.27559			0.112154			0.0560771	+	0.998426i	$0.482141\pi$
$854$		0			0
$855$		−	3.89210i		−	0.133107i
$856$		0			0
$857$			14.2127i			0.485497i	0.970089	+	0.242749i	$0.0780490\pi$
$857$			14.2127i			0.485497i	−0.970089	+	0.242749i	$0.921951\pi$
$858$		0			0
$859$			48.8464i			1.66662i	0.552807	+	0.833309i	$0.313557\pi$
$859$			48.8464i			1.66662i	−0.552807	+	0.833309i	$0.686443\pi$
$860$		0			0
$861$	6.04488	+	0.377760i	0.206009	+	0.0128740i
$862$		0			0
$863$		−	46.2121i		−	1.57308i	−0.617540	−	0.786539i	$-0.711871\pi$
$863$		−	46.2121i		−	1.57308i	0.617540	−	0.786539i	$-0.288129\pi$
$864$		0			0
$865$	−0.0398406			−0.00135462
$866$		0			0
$867$			13.5530i			0.460284i
$868$		0			0
$869$			2.72253i			0.0923556i
$870$		0			0
$871$	−17.5799			−0.595671
$872$		0			0
$873$			11.1183i			0.376297i
$874$		0			0
$875$	2.64060	+	0.165018i	0.0892686	+	0.00557863i
$876$		0			0
$877$		−	19.3145i		−	0.652205i	−0.945334	−	0.326103i	$-0.894265\pi$
$877$		−	19.3145i		−	0.652205i	0.945334	−	0.326103i	$-0.105735\pi$
$878$		0			0
$879$		−	14.1818i		−	0.478340i
$880$		0			0
$881$		−	5.01092i		−	0.168822i	−0.996431	−	0.0844110i	$-0.973099\pi$
$881$		−	5.01092i		−	0.168822i	0.996431	−	0.0844110i	$-0.0269009\pi$
$882$		0			0
$883$	46.2754			1.55729			0.778646	−	0.627463i	$-0.215907\pi$
$883$	46.2754			1.55729			0.778646	+	0.627463i	$0.215907\pi$
$884$		0			0
$885$	4.59290			0.154389
$886$		0			0
$887$	−29.4098			−0.987484			−0.493742	−	0.869608i	$-0.664371\pi$
$887$	−29.4098			−0.987484			−0.493742	+	0.869608i	$0.664371\pi$
$888$		0			0
$889$	−2.81657	+	45.0704i	−0.0944646	+	1.51161i
$890$		0			0
$891$	−0.264307			−0.00885461
$892$		0			0
$893$			45.6385i			1.52723i
$894$		0			0
$895$	0.662470			0.0221439
$896$		0			0
$897$	−12.7420			−0.425442
$898$		0			0
$899$		−	55.1193i		−	1.83833i
$900$		0			0
$901$	−7.51533			−0.250372
$902$		0			0
$903$	0.282399	−	4.51892i	0.00939766	−	0.150380i
$904$		0			0
$905$	−8.47656			−0.281770
$906$		0			0
$907$	−12.0290			−0.399417			−0.199709	−	0.979855i	$-0.564000\pi$
$907$	−12.0290			−0.399417			−0.199709	+	0.979855i	$0.564000\pi$
$908$		0			0
$909$	4.37247			0.145026
$910$		0			0
$911$		−	57.5846i		−	1.90786i	−0.300023	−	0.953932i	$-0.596994\pi$
$911$		−	57.5846i		−	1.90786i	0.300023	−	0.953932i	$-0.403006\pi$
$912$		0			0
$913$			1.51102i			0.0500073i
$914$		0			0
$915$			9.96739i			0.329512i
$916$		0			0
$917$	−0.329998	+	5.28060i	−0.0108975	+	0.174381i
$918$		0			0
$919$			21.4808i			0.708585i	0.935135	+	0.354293i	$0.115278\pi$
$919$			21.4808i			0.708585i	−0.935135	+	0.354293i	$0.884722\pi$
$920$		0			0
$921$	20.0654			0.661178
$922$		0			0
$923$			52.3510i			1.72315i
$924$		0			0
$925$			8.95826i			0.294546i
$926$		0			0
$927$	4.82084			0.158337
$928$		0			0
$929$			11.2995i			0.370723i	0.982670	+	0.185362i	$0.0593457\pi$
$929$			11.2995i			0.370723i	−0.982670	+	0.185362i	$0.940654\pi$
$930$		0			0
$931$	3.39194	−	27.0328i	0.111166	−	0.885963i
$932$		0			0
$933$		−	15.0389i		−	0.492351i
$934$		0			0
$935$		−	0.490714i		−	0.0160481i
$936$		0			0
$937$		−	24.5392i		−	0.801660i	−0.916152	−	0.400830i	$-0.868722\pi$
$937$		−	24.5392i		−	0.801660i	0.916152	−	0.400830i	$-0.131278\pi$
$938$		0			0
$939$	22.2502			0.726108
$940$		0			0
$941$	51.5447			1.68031			0.840154	−	0.542348i	$-0.182464\pi$
$941$	51.5447			1.68031			0.840154	+	0.542348i	$0.182464\pi$
$942$		0			0
$943$	−4.79853			−0.156262
$944$		0			0
$945$	0.165018	−	2.64060i	0.00536803	−	0.0858987i
$946$		0			0
$947$	−16.0492			−0.521530			−0.260765	−	0.965402i	$-0.583975\pi$
$947$	−16.0492			−0.521530			−0.260765	+	0.965402i	$0.583975\pi$
$948$		0			0
$949$		−	25.3390i		−	0.822539i
$950$		0			0
$951$	−33.4862			−1.08586
$952$		0			0
$953$	14.3868			0.466035			0.233018	−	0.972472i	$-0.425140\pi$
$953$	14.3868			0.466035			0.233018	+	0.972472i	$0.425140\pi$
$954$		0			0
$955$			11.6714i			0.377678i
$956$		0			0
$957$	2.02399			0.0654262
$958$		0			0
$959$	44.7730	+	2.79798i	1.44580	+	0.0903515i
$960$		0			0
$961$	20.8093			0.671266
$962$		0			0
$963$	−6.14307			−0.197958
$964$		0			0
$965$	−2.29670			−0.0739333
$966$		0			0
$967$			37.2171i			1.19682i	0.801189	+	0.598411i	$0.204201\pi$
$967$			37.2171i			1.19682i	−0.801189	+	0.598411i	$0.795799\pi$
$968$		0			0
$969$		−	7.22612i		−	0.232136i
$970$		0			0
$971$			43.0613i			1.38190i	0.722902	+	0.690951i	$0.242808\pi$
$971$			43.0613i			1.38190i	−0.722902	+	0.690951i	$0.757192\pi$
$972$		0			0
$973$	3.29160	−	52.6719i	0.105524	−	1.68858i
$974$		0			0
$975$			6.07875i			0.194676i
$976$		0			0
$977$	−5.91287			−0.189169			−0.0945847	−	0.995517i	$-0.530152\pi$
$977$	−5.91287			−0.189169			−0.0945847	+	0.995517i	$0.530152\pi$
$978$		0			0
$979$			3.52834i			0.112766i
$980$		0			0
$981$			15.4612i			0.493637i
$982$		0			0
$983$	−1.42530			−0.0454599			−0.0227300	−	0.999742i	$-0.507236\pi$
$983$	−1.42530			−0.0454599			−0.0227300	+	0.999742i	$0.507236\pi$
$984$		0			0
$985$		−	23.3822i		−	0.745019i
$986$		0			0
$987$	−1.93499	+	30.9635i	−0.0615913	+	0.985578i
$988$		0			0
$989$			3.58719i			0.114066i
$990$		0			0
$991$		−	25.8732i		−	0.821889i	−0.911660	−	0.410945i	$-0.865199\pi$
$991$		−	25.8732i		−	0.821889i	0.911660	−	0.410945i	$-0.134801\pi$
$992$		0			0
$993$		−	23.5161i		−	0.746260i
$994$		0			0
$995$	9.36275			0.296819
$996$		0			0
$997$	−24.4403			−0.774031			−0.387015	−	0.922073i	$-0.626494\pi$
$997$	−24.4403			−0.774031			−0.387015	+	0.922073i	$0.626494\pi$
$998$		0			0
$999$	8.95826			0.283427

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.z.c.1231.26		28
4.3	odd	2		840.2.z.c.811.18	yes	28
7.6	odd	2		3360.2.z.d.1231.3		28
8.3	odd	2		3360.2.z.d.1231.4		28
8.5	even	2		840.2.z.d.811.17	yes	28
28.27	even	2		840.2.z.d.811.18	yes	28
56.13	odd	2		840.2.z.c.811.17	✓	28
56.27	even	2	inner	3360.2.z.c.1231.25		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.z.c.811.17	✓	28	56.13	odd	2
840.2.z.c.811.18	yes	28	4.3	odd	2
840.2.z.d.811.17	yes	28	8.5	even	2
840.2.z.d.811.18	yes	28	28.27	even	2
3360.2.z.c.1231.25		28	56.27	even	2	inner
3360.2.z.c.1231.26		28	1.1	even	1	trivial
3360.2.z.d.1231.3		28	7.6	odd	2
3360.2.z.d.1231.4		28	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.z (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.00000 q^{5} +(-2.64060 - 0.165018i) q^{7} -1.00000 q^{9} -0.264307 q^{11} +6.07875 q^{13} -1.00000i q^{15} -1.85661i q^{17} -3.89210i q^{19} +(0.165018 - 2.64060i) q^{21} +2.09615i q^{23} +1.00000 q^{25} -1.00000i q^{27} +7.65773i q^{29} -7.19786 q^{31} -0.264307i q^{33} +(2.64060 + 0.165018i) q^{35} +8.95826i q^{37} +6.07875i q^{39} +2.28921i q^{41} +1.71132 q^{43} +1.00000 q^{45} -11.7259 q^{47} +(6.94554 + 0.871493i) q^{49} +1.85661 q^{51} -4.04788i q^{53} +0.264307 q^{55} +3.89210 q^{57} +4.59290i q^{59} -9.96739 q^{61} +(2.64060 + 0.165018i) q^{63} -6.07875 q^{65} -2.89202 q^{67} -2.09615 q^{69} +8.61213i q^{71} -4.16846i q^{73} +1.00000i q^{75} +(0.697928 + 0.0436153i) q^{77} -10.3007i q^{79} +1.00000 q^{81} -5.71691i q^{83} +1.85661i q^{85} -7.65773 q^{87} -13.3494i q^{89} +(-16.0515 - 1.00310i) q^{91} -7.19786i q^{93} +3.89210i q^{95} -11.1183i q^{97} +0.264307 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 28 q^{5} + 4 q^{7} - 28 q^{9} + 8 q^{13} + 28 q^{25} + 24 q^{31} - 4 q^{35} - 24 q^{43} + 28 q^{45} - 24 q^{47} + 28 q^{49} + 8 q^{57} + 16 q^{61} - 4 q^{63} - 8 q^{65} + 8 q^{67} + 24 q^{69} - 16 q^{77}+ \cdots - 48 q^{91}+O(q^{100}) \) 28 * q - 28 * q^5 + 4 * q^7 - 28 * q^9 + 8 * q^13 + 28 * q^25 + 24 * q^31 - 4 * q^35 - 24 * q^43 + 28 * q^45 - 24 * q^47 + 28 * q^49 + 8 * q^57 + 16 * q^61 - 4 * q^63 - 8 * q^65 + 8 * q^67 + 24 * q^69 - 16 * q^77 + 28 * q^81 + 16 * q^87 - 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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