LMFDB - Embedded newform 1980.2.z.e.1081.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1980,2,Mod(181,1980)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1980.181"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1980, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1980.z (of order $5$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,2,0,3,0,0,0,3,0,-6,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$15.8103796002$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{5})$
Coefficient field:	8.0.13140625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 $ x^8 - 3x^7 + 5x^6 - 3x^5 + 4x^4 + 3x^3 + 5x^2 + 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 660)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			1081.1
Root			$-0.386111 + 0.280526i$ of defining polynomial
Character	$\chi$	$=$	1980.1081
Dual form			1980.2.z.e.1621.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.809017 + 0.587785i) q^{5} +(0.408851 - 1.25832i) q^{7} +(0.105203 - 3.31496i) q^{11} +(-4.83807 + 3.51506i) q^{13} +(-6.44876 - 4.68530i) q^{17} +(1.37486 + 4.23139i) q^{19} -4.02566 q^{23} +(0.309017 + 0.951057i) q^{25} +(-2.56916 + 7.90707i) q^{29} +(-1.68107 + 1.22137i) q^{31} +(1.07039 - 0.777682i) q^{35} +(-2.83392 + 8.72192i) q^{37} +(-2.41714 - 7.43920i) q^{41} -3.08621 q^{43} +(0.886111 + 2.72717i) q^{47} +(4.24692 + 3.08557i) q^{49} +(3.31118 - 2.40572i) q^{53} +(2.03359 - 2.62002i) q^{55} +(-1.63220 + 5.02341i) q^{59} +(-2.49009 - 1.80916i) q^{61} -5.98018 q^{65} -14.1976 q^{67} +(7.33551 + 5.32956i) q^{71} +(2.83100 - 8.71293i) q^{73} +(-4.12825 - 1.48770i) q^{77} +(8.99857 - 6.53784i) q^{79} +(6.63047 + 4.81732i) q^{83} +(-2.46321 - 7.58097i) q^{85} -9.73424 q^{89} +(2.44501 + 7.52496i) q^{91} +(-1.37486 + 4.23139i) q^{95} +(9.54725 - 6.93648i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{5} + 3 q^{7} + 3 q^{11} - 6 q^{13} - 8 q^{17} - 14 q^{19} - 10 q^{23} - 2 q^{25} - 6 q^{29} - q^{31} + 7 q^{35} - 5 q^{37} + 11 q^{41} + 28 q^{43} + q^{47} + q^{49} + 3 q^{53} + 7 q^{55} + 5 q^{59}+ \cdots + 35 q^{97}+O(q^{100}) $ 8 * q + 2 * q^5 + 3 * q^7 + 3 * q^11 - 6 * q^13 - 8 * q^17 - 14 * q^19 - 10 * q^23 - 2 * q^25 - 6 * q^29 - q^31 + 7 * q^35 - 5 * q^37 + 11 * q^41 + 28 * q^43 + q^47 + q^49 + 3 * q^53 + 7 * q^55 + 5 * q^59 - 3 * q^61 - 14 * q^65 - 50 * q^67 + 11 * q^71 + 33 * q^73 - 7 * q^77 + 2 * q^79 - q^83 - 12 * q^85 + 44 * q^89 + 25 * q^91 + 14 * q^95 + 35 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1980\mathbb{Z}\right)^\times$.

$n$	$397$	$541$	$991$	$1541$
$\chi(n)$	$1$	$e\left(\frac{4}{5}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.809017	+	0.587785i	0.361803	+	0.262866i
$5$	0.809017	+	0.587785i	0.361803	+	0.262866i
$6$		0			0
$7$	0.408851	−	1.25832i	0.154531	−	0.475599i	−0.843582	−	0.537001i	$-0.819557\pi$
$7$	0.408851	−	1.25832i	0.154531	−	0.475599i	0.998113	+	0.0614022i	$0.0195572\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.105203	−	3.31496i	0.0317198	−	0.999497i
$11$	0.105203	−	3.31496i	0.0317198	−	0.999497i
$12$		0			0
$13$	−4.83807	+	3.51506i	−1.34184	+	0.974903i	−0.342465	+	0.939531i	$0.611262\pi$
$13$	−4.83807	+	3.51506i	−1.34184	+	0.974903i	−0.999374	+	0.0353725i	$0.988738\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.44876	−	4.68530i	−1.56405	−	1.13635i	−0.932586	−	0.360949i	$-0.882453\pi$
$17$	−6.44876	−	4.68530i	−1.56405	−	1.13635i	−0.631468	−	0.775402i	$-0.717547\pi$
$18$		0			0
$19$	1.37486	+	4.23139i	0.315415	+	0.970749i	0.975583	+	0.219630i	$0.0704851\pi$
$19$	1.37486	+	4.23139i	0.315415	+	0.970749i	−0.660168	+	0.751118i	$0.729515\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.02566			−0.839409			−0.419704	−	0.907661i	$-0.637866\pi$
$23$	−4.02566			−0.839409			−0.419704	+	0.907661i	$0.637866\pi$
$24$		0			0
$25$	0.309017	+	0.951057i	0.0618034	+	0.190211i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.56916	+	7.90707i	−0.477082	+	1.46831i	0.366048	+	0.930596i	$0.380711\pi$
$29$	−2.56916	+	7.90707i	−0.477082	+	1.46831i	−0.843129	+	0.537711i	$0.819289\pi$
$30$		0			0
$31$	−1.68107	+	1.22137i	−0.301930	+	0.219365i	−0.728426	−	0.685125i	$-0.759748\pi$
$31$	−1.68107	+	1.22137i	−0.301930	+	0.219365i	0.426496	+	0.904489i	$0.359748\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.07039	−	0.777682i	0.180928	−	0.131452i
$36$		0			0
$37$	−2.83392	+	8.72192i	−0.465894	+	1.43388i	0.391960	+	0.919982i	$0.371797\pi$
$37$	−2.83392	+	8.72192i	−0.465894	+	1.43388i	−0.857854	+	0.513893i	$0.828203\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.41714	−	7.43920i	−0.377494	−	1.16181i	−0.941780	−	0.336228i	$-0.890849\pi$
$41$	−2.41714	−	7.43920i	−0.377494	−	1.16181i	0.564286	−	0.825579i	$-0.309151\pi$
$42$		0			0
$43$	−3.08621			−0.470643			−0.235322	−	0.971918i	$-0.575614\pi$
$43$	−3.08621			−0.470643			−0.235322	+	0.971918i	$0.575614\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.886111	+	2.72717i	0.129253	+	0.397799i	0.994652	−	0.103284i	$-0.0329351\pi$
$47$	0.886111	+	2.72717i	0.129253	+	0.397799i	−0.865399	+	0.501083i	$0.832935\pi$
$48$		0			0
$49$	4.24692	+	3.08557i	0.606703	+	0.440796i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.31118	−	2.40572i	0.454826	−	0.330451i	−0.336672	−	0.941622i	$-0.609301\pi$
$53$	3.31118	−	2.40572i	0.454826	−	0.330451i	0.791498	+	0.611171i	$0.209301\pi$
$54$		0			0
$55$	2.03359	−	2.62002i	0.274210	−	0.353283i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.63220	+	5.02341i	−0.212495	+	0.653992i	0.786827	+	0.617174i	$0.211722\pi$
$59$	−1.63220	+	5.02341i	−0.212495	+	0.653992i	−0.999322	+	0.0368186i	$0.988278\pi$
$60$		0			0
$61$	−2.49009	−	1.80916i	−0.318824	−	0.231639i	0.416850	−	0.908975i	$-0.363134\pi$
$61$	−2.49009	−	1.80916i	−0.318824	−	0.231639i	−0.735673	+	0.677337i	$0.763134\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.98018			−0.741750
$66$		0			0
$67$	−14.1976			−1.73451			−0.867256	−	0.497862i	$-0.834119\pi$
$67$	−14.1976			−1.73451			−0.867256	+	0.497862i	$0.834119\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.33551	+	5.32956i	0.870565	+	0.632502i	0.930738	−	0.365686i	$-0.119166\pi$
$71$	7.33551	+	5.32956i	0.870565	+	0.632502i	−0.0601738	+	0.998188i	$0.519166\pi$
$72$		0			0
$73$	2.83100	−	8.71293i	0.331344	−	1.01977i	−0.637152	−	0.770739i	$-0.719887\pi$
$73$	2.83100	−	8.71293i	0.331344	−	1.01977i	0.968495	−	0.249032i	$-0.0801125\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.12825	−	1.48770i	−0.470457	−	0.169539i
$78$		0			0
$79$	8.99857	−	6.53784i	1.01242	−	0.735565i	0.0477034	−	0.998862i	$-0.484810\pi$
$79$	8.99857	−	6.53784i	1.01242	−	0.735565i	0.964715	+	0.263297i	$0.0848098\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.63047	+	4.81732i	0.727789	+	0.528769i	0.888863	−	0.458173i	$-0.151496\pi$
$83$	6.63047	+	4.81732i	0.727789	+	0.528769i	−0.161074	+	0.986942i	$0.551496\pi$
$84$		0			0
$85$	−2.46321	−	7.58097i	−0.267172	−	0.822271i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.73424			−1.03183			−0.515914	−	0.856640i	$-0.672548\pi$
$89$	−9.73424			−1.03183			−0.515914	+	0.856640i	$0.672548\pi$
$90$		0			0
$91$	2.44501	+	7.52496i	0.256306	+	0.788830i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.37486	+	4.23139i	−0.141058	+	0.434132i
$96$		0			0
$97$	9.54725	−	6.93648i	0.969377	−	0.704293i	0.0140672	−	0.999901i	$-0.495522\pi$
$97$	9.54725	−	6.93648i	0.969377	−	0.704293i	0.955309	+	0.295608i	$0.0955221\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−11.5092	+	8.36195i	−1.14521	+	0.832045i	−0.987837	−	0.155494i	$-0.950303\pi$
$101$	−11.5092	+	8.36195i	−1.14521	+	0.832045i	−0.157375	+	0.987539i	$0.550303\pi$
$102$		0			0
$103$	−6.12806	+	18.8602i	−0.603816	+	1.85835i	−0.0990798	+	0.995079i	$0.531590\pi$
$103$	−6.12806	+	18.8602i	−0.603816	+	1.85835i	−0.504736	+	0.863274i	$0.668410\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.939127	+	2.89033i	0.0907888	+	0.279419i	0.986133	−	0.165955i	$-0.0530707\pi$
$107$	0.939127	+	2.89033i	0.0907888	+	0.279419i	−0.895345	+	0.445374i	$0.853071\pi$
$108$		0			0
$109$	−1.25740			−0.120437			−0.0602184	−	0.998185i	$-0.519180\pi$
$109$	−1.25740			−0.120437			−0.0602184	+	0.998185i	$0.519180\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.09127	−	18.7470i	−0.573018	−	1.76357i	−0.642831	−	0.766008i	$-0.722240\pi$
$113$	−6.09127	−	18.7470i	−0.573018	−	1.76357i	0.0698130	−	0.997560i	$-0.477760\pi$
$114$		0			0
$115$	−3.25683	−	2.36623i	−0.303701	−	0.220652i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.53216	+	6.19898i	−0.782142	+	0.568260i
$120$		0			0
$121$	−10.9779	−	0.697484i	−0.997988	−	0.0634077i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.309017	+	0.951057i	−0.0276393	+	0.0850651i
$126$		0			0
$127$	−9.65888	−	7.01759i	−0.857087	−	0.622710i	0.0700037	−	0.997547i	$-0.477699\pi$
$127$	−9.65888	−	7.01759i	−0.857087	−	0.622710i	−0.927091	+	0.374836i	$0.877699\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−19.4042			−1.69535			−0.847675	−	0.530516i	$-0.821998\pi$
$131$	−19.4042			−1.69535			−0.847675	+	0.530516i	$0.821998\pi$
$132$		0			0
$133$	5.88654			0.510428
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.67955	+	2.67335i	0.314365	+	0.228399i	0.733767	−	0.679401i	$-0.237760\pi$
$137$	3.67955	+	2.67335i	0.314365	+	0.228399i	−0.419402	+	0.907800i	$0.637760\pi$
$138$		0			0
$139$	5.81853	−	17.9076i	0.493521	−	1.51890i	−0.325727	−	0.945464i	$-0.605609\pi$
$139$	5.81853	−	17.9076i	0.493521	−	1.51890i	0.819249	−	0.573439i	$-0.194391\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	11.1433	+	16.4078i	0.931850	+	1.37209i
$144$		0			0
$145$	−6.72616	+	4.88684i	−0.558577	+	0.405830i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.11669	+	2.26441i	0.255330	+	0.185508i	0.708086	−	0.706127i	$-0.249559\pi$
$149$	3.11669	+	2.26441i	0.255330	+	0.185508i	−0.452756	+	0.891634i	$0.649559\pi$
$150$		0			0
$151$	0.873524	+	2.68843i	0.0710864	+	0.218781i	0.980288	−	0.197575i	$-0.0633067\pi$
$151$	0.873524	+	2.68843i	0.0710864	+	0.218781i	−0.909201	+	0.416357i	$0.863307\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.07792			−0.166903
$156$		0			0
$157$	−2.50631	−	7.71364i	−0.200026	−	0.615615i	−0.999881	−	0.0154227i	$-0.995091\pi$
$157$	−2.50631	−	7.71364i	−0.200026	−	0.615615i	0.799856	−	0.600193i	$-0.204909\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.64590	+	5.06555i	−0.129715	+	0.399222i
$162$		0			0
$163$	−1.94480	+	1.41298i	−0.152328	+	0.110673i	−0.661339	−	0.750087i	$-0.730011\pi$
$163$	−1.94480	+	1.41298i	−0.152328	+	0.110673i	0.509010	+	0.860760i	$0.330011\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.29496	+	3.12047i	−0.332354	+	0.241469i	−0.741429	−	0.671031i	$-0.765852\pi$
$167$	−4.29496	+	3.12047i	−0.332354	+	0.241469i	0.409075	+	0.912501i	$0.365852\pi$
$168$		0			0
$169$	7.03403	−	21.6485i	0.541079	−	1.66527i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.99744	−	12.3028i	−0.303920	−	0.935368i	−0.980078	−	0.198612i	$-0.936357\pi$
$173$	−3.99744	−	12.3028i	−0.303920	−	0.935368i	0.676159	−	0.736756i	$-0.263643\pi$
$174$		0			0
$175$	1.32307			0.100015
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.35505	+	16.4811i	0.400255	+	1.23186i	0.924793	+	0.380471i	$0.124238\pi$
$179$	5.35505	+	16.4811i	0.400255	+	1.23186i	−0.524538	+	0.851387i	$0.675762\pi$
$180$		0			0
$181$	5.81853	+	4.22741i	0.432488	+	0.314221i	0.782643	−	0.622471i	$-0.213871\pi$
$181$	5.81853	+	4.22741i	0.432488	+	0.314221i	−0.350155	+	0.936692i	$0.613871\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.41931	+	5.39044i	−0.545478	+	0.396313i
$186$		0			0
$187$	−16.2100	+	20.8844i	−1.18539	+	1.52722i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.27103	−	6.98953i	0.164326	−	0.505744i	−0.834660	−	0.550766i	$-0.814336\pi$
$191$	2.27103	−	6.98953i	0.164326	−	0.505744i	0.998986	+	0.0450216i	$0.0143357\pi$
$192$		0			0
$193$	14.7801	+	10.7384i	1.06390	+	0.772966i	0.974805	−	0.223058i	$-0.0716038\pi$
$193$	14.7801	+	10.7384i	1.06390	+	0.772966i	0.0890910	+	0.996023i	$0.471604\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.0090			−1.14059			−0.570297	−	0.821438i	$-0.693172\pi$
$197$	−16.0090			−1.14059			−0.570297	+	0.821438i	$0.693172\pi$
$198$		0			0
$199$	−15.4438			−1.09478			−0.547391	−	0.836877i	$-0.684379\pi$
$199$	−15.4438			−1.09478			−0.547391	+	0.836877i	$0.684379\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.89919	+	6.46564i	0.624600	+	0.453799i
$204$		0			0
$205$	2.41714	−	7.43920i	0.168821	−	0.519576i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	14.1715	−	4.11246i	0.980265	−	0.284465i
$210$		0			0
$211$	−1.29610	+	0.941669i	−0.0892269	+	0.0648271i	−0.631504	−	0.775372i	$-0.717562\pi$
$211$	−1.29610	+	0.941669i	−0.0892269	+	0.0648271i	0.542277	+	0.840200i	$0.317562\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.49680	−	1.81403i	−0.170280	−	0.123716i
$216$		0			0
$217$	0.849561	+	2.61468i	0.0576720	+	0.177496i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	47.6686			3.20654
$222$		0			0
$223$	0.973942	+	2.99749i	0.0652200	+	0.200727i	0.978356	−	0.206928i	$-0.0663467\pi$
$223$	0.973942	+	2.99749i	0.0652200	+	0.200727i	−0.913136	+	0.407655i	$0.866347\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.145142	+	0.446702i	−0.00963343	+	0.0296487i	−0.955758	−	0.294155i	$-0.904962\pi$
$227$	−0.145142	+	0.446702i	−0.00963343	+	0.0296487i	0.946124	+	0.323804i	$0.104962\pi$
$228$		0			0
$229$	20.5047	−	14.8975i	1.35499	−	0.984457i	0.356242	−	0.934394i	$-0.384058\pi$
$229$	20.5047	−	14.8975i	1.35499	−	0.984457i	0.998746	−	0.0500631i	$-0.0159422\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.78597	+	1.29758i	−0.117003	+	0.0850076i	−0.644748	−	0.764395i	$-0.723038\pi$
$233$	−1.78597	+	1.29758i	−0.117003	+	0.0850076i	0.527745	+	0.849403i	$0.323038\pi$
$234$		0			0
$235$	−0.886111	+	2.72717i	−0.0578035	+	0.177901i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.522006	−	1.60657i	−0.0337657	−	0.103920i	0.932753	−	0.360516i	$-0.117399\pi$
$239$	−0.522006	−	1.60657i	−0.0337657	−	0.103920i	−0.966519	+	0.256595i	$0.917399\pi$
$240$		0			0
$241$	−13.6730			−0.880754			−0.440377	−	0.897813i	$-0.645155\pi$
$241$	−13.6730			−0.880754			−0.440377	+	0.897813i	$0.645155\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.62218	+	4.99255i	0.103637	+	0.318963i
$246$		0			0
$247$	−21.5253	−	15.6390i	−1.36962	−	0.995089i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.9407	−	7.94889i	0.690571	−	0.501730i	−0.186276	−	0.982497i	$-0.559642\pi$
$251$	10.9407	−	7.94889i	0.690571	−	0.501730i	0.876848	+	0.480768i	$0.159642\pi$
$252$		0			0
$253$	−0.423510	+	13.3449i	−0.0266259	+	0.838986i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.42596	+	7.46632i	−0.151327	+	0.465737i	−0.997770	−	0.0667424i	$-0.978739\pi$
$257$	−2.42596	+	7.46632i	−0.151327	+	0.465737i	0.846443	+	0.532479i	$0.178739\pi$
$258$		0			0
$259$	9.81627	+	7.13194i	0.609954	+	0.443157i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.4132			−0.765433			−0.382717	−	0.923866i	$-0.625011\pi$
$263$	−12.4132			−0.765433			−0.382717	+	0.923866i	$0.625011\pi$
$264$		0			0
$265$	4.09285			0.251422
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.2873	−	9.65377i	−0.810140	−	0.588601i	0.103732	−	0.994605i	$-0.466922\pi$
$269$	−13.2873	−	9.65377i	−0.810140	−	0.588601i	−0.913871	+	0.406004i	$0.866922\pi$
$270$		0			0
$271$	5.32895	−	16.4008i	0.323711	−	0.996279i	−0.648309	−	0.761378i	$-0.724523\pi$
$271$	5.32895	−	16.4008i	0.323711	−	0.996279i	0.972019	−	0.234901i	$-0.0754767\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.18522	−	0.924324i	0.192076	−	0.0557388i
$276$		0			0
$277$	3.67199	−	2.66786i	0.220629	−	0.160296i	−0.471981	−	0.881609i	$-0.656461\pi$
$277$	3.67199	−	2.66786i	0.220629	−	0.160296i	0.692610	+	0.721313i	$0.256461\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.96448	−	7.23962i	−0.594431	−	0.431880i	0.249467	−	0.968383i	$-0.419745\pi$
$281$	−9.96448	−	7.23962i	−0.594431	−	0.431880i	−0.843898	+	0.536504i	$0.819745\pi$
$282$		0			0
$283$	5.74487	+	17.6809i	0.341497	+	1.05102i	0.963432	+	0.267951i	$0.0863467\pi$
$283$	5.74487	+	17.6809i	0.341497	+	1.05102i	−0.621935	+	0.783069i	$0.713653\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.3491			−0.610889
$288$		0			0
$289$	14.3812	+	44.2607i	0.845952	+	2.60357i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.23345	−	16.1069i	0.305741	−	0.940975i	−0.673658	−	0.739043i	$-0.735278\pi$
$293$	5.23345	−	16.1069i	0.305741	−	0.940975i	0.979400	−	0.201932i	$-0.0647221\pi$
$294$		0			0
$295$	−4.27317	+	3.10464i	−0.248793	+	0.180759i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	19.4764	−	14.1505i	1.12635	−	0.818342i
$300$		0			0
$301$	−1.26180	+	3.88343i	−0.0727291	+	0.223837i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.951130	−	2.92728i	−0.0544616	−	0.167615i
$306$		0			0
$307$	−11.1953			−0.638949			−0.319475	−	0.947595i	$-0.603506\pi$
$307$	−11.1953			−0.638949			−0.319475	+	0.947595i	$0.603506\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.32004	+	4.06267i	0.0748527	+	0.230373i	0.981482	−	0.191556i	$-0.0613533\pi$
$311$	1.32004	+	4.06267i	0.0748527	+	0.230373i	−0.906629	+	0.421929i	$0.861353\pi$
$312$		0			0
$313$	−22.4356	−	16.3004i	−1.26813	−	0.921353i	−0.269008	−	0.963138i	$-0.586696\pi$
$313$	−22.4356	−	16.3004i	−1.26813	−	0.921353i	−0.999127	+	0.0417845i	$0.986696\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.15608	−	5.19919i	0.401925	−	0.292016i	−0.368399	−	0.929668i	$-0.620094\pi$
$317$	7.15608	−	5.19919i	0.401925	−	0.292016i	0.770325	+	0.637652i	$0.220094\pi$
$318$		0			0
$319$	25.9413	+	9.34851i	1.45244	+	0.523416i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10.9592	−	33.7289i	0.609785	−	1.87673i
$324$		0			0
$325$	−4.83807	−	3.51506i	−0.268368	−	0.194981i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.79393			0.209166
$330$		0			0
$331$	29.2241			1.60630			0.803150	−	0.595777i	$-0.203156\pi$
$331$	29.2241			1.60630			0.803150	+	0.595777i	$0.203156\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.4861	−	8.34514i	−0.627553	−	0.455944i
$336$		0			0
$337$	−1.77586	+	5.46553i	−0.0967371	+	0.297726i	−0.987702	−	0.156345i	$-0.950029\pi$
$337$	−1.77586	+	5.46553i	−0.0967371	+	0.297726i	0.890965	+	0.454071i	$0.150029\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.87194	+	5.70118i	0.209677	+	0.308736i
$342$		0			0
$343$	13.1117	−	9.52620i	0.707965	−	0.514366i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.45382	−	4.68897i	−0.346459	−	0.251717i	0.400923	−	0.916112i	$-0.368690\pi$
$347$	−6.45382	−	4.68897i	−0.346459	−	0.251717i	−0.747382	+	0.664394i	$0.768690\pi$
$348$		0			0
$349$	−2.80783	−	8.64161i	−0.150300	−	0.462575i	0.847355	−	0.531027i	$-0.178194\pi$
$349$	−2.80783	−	8.64161i	−0.150300	−	0.462575i	−0.997654	+	0.0684526i	$0.978194\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	17.0715			0.908625			0.454313	−	0.890842i	$-0.349885\pi$
$353$	17.0715			0.908625			0.454313	+	0.890842i	$0.349885\pi$
$354$		0			0
$355$	2.80191	+	8.62341i	0.148710	+	0.457683i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−11.3507	+	34.9338i	−0.599066	+	1.84374i	−0.0657196	+	0.997838i	$0.520934\pi$
$359$	−11.3507	+	34.9338i	−0.599066	+	1.84374i	−0.533346	+	0.845897i	$0.679066\pi$
$360$		0			0
$361$	−0.643130	+	0.467261i	−0.0338489	+	0.0245927i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.41166	−	5.38488i	0.387944	−	0.281858i
$366$		0			0
$367$	9.82154	−	30.2276i	0.512680	−	1.57787i	−0.274784	−	0.961506i	$-0.588606\pi$
$367$	9.82154	−	30.2276i	0.512680	−	1.57787i	0.787464	−	0.616361i	$-0.211394\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.67337	−	5.15009i	−0.0868769	−	0.267380i
$372$		0			0
$373$	9.17548			0.475088			0.237544	−	0.971377i	$-0.423658\pi$
$373$	9.17548			0.475088			0.237544	+	0.971377i	$0.423658\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−15.3641	−	47.2858i	−0.791290	−	2.43534i
$378$		0			0
$379$	0.640018	+	0.465001i	0.0328755	+	0.0238855i	0.604102	−	0.796907i	$-0.293532\pi$
$379$	0.640018	+	0.465001i	0.0328755	+	0.0238855i	−0.571226	+	0.820793i	$0.693532\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.67065	+	4.11997i	−0.289757	+	0.210521i	−0.723162	−	0.690679i	$-0.757312\pi$
$383$	−5.67065	+	4.11997i	−0.289757	+	0.210521i	0.433405	+	0.901199i	$0.357312\pi$
$384$		0			0
$385$	−2.46537	−	3.63010i	−0.125647	−	0.185007i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.99979	−	12.3101i	0.202798	−	0.624147i	−0.796999	−	0.603981i	$-0.793580\pi$
$389$	3.99979	−	12.3101i	0.202798	−	0.624147i	0.999797	−	0.0201662i	$-0.00641955\pi$
$390$		0			0
$391$	25.9605	+	18.8614i	1.31288	+	0.953863i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	11.1228			0.559651
$396$		0			0
$397$	−34.5780			−1.73542			−0.867710	−	0.497071i	$-0.834409\pi$
$397$	−34.5780			−1.73542			−0.867710	+	0.497071i	$0.834409\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.21570	−	0.883259i	−0.0607092	−	0.0441078i	0.557017	−	0.830501i	$-0.311946\pi$
$401$	−1.21570	−	0.883259i	−0.0607092	−	0.0441078i	−0.617726	+	0.786393i	$0.711946\pi$
$402$		0			0
$403$	3.83995	−	11.8182i	0.191282	−	0.588705i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	28.6146	+	10.3119i	1.41838	+	0.511142i
$408$		0			0
$409$	1.94299	−	1.41167i	0.0960748	−	0.0698025i	−0.538711	−	0.842491i	$-0.681088\pi$
$409$	1.94299	−	1.41167i	0.0960748	−	0.0698025i	0.634786	+	0.772688i	$0.281088\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.65370	+	4.10766i	0.278201	+	0.202125i
$414$		0			0
$415$	2.53261	+	7.79459i	0.124321	+	0.382621i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	34.5961			1.69013			0.845065	−	0.534664i	$-0.179562\pi$
$419$	34.5961			1.69013			0.845065	+	0.534664i	$0.179562\pi$
$420$		0			0
$421$	9.70713	+	29.8755i	0.473097	+	1.45604i	0.848508	+	0.529183i	$0.177502\pi$
$421$	9.70713	+	29.8755i	0.473097	+	1.45604i	−0.375411	+	0.926859i	$0.622498\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.46321	−	7.58097i	0.119483	−	0.367731i
$426$		0			0
$427$	−3.29457	+	2.39364i	−0.159435	+	0.115837i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7.11519	−	5.16949i	0.342727	−	0.249006i	−0.403085	−	0.915163i	$-0.632062\pi$
$431$	7.11519	−	5.16949i	0.342727	−	0.249006i	0.745811	+	0.666157i	$0.232062\pi$
$432$		0			0
$433$	−1.42435	+	4.38370i	−0.0684500	+	0.210667i	−0.979430	−	0.201782i	$-0.935327\pi$
$433$	−1.42435	+	4.38370i	−0.0684500	+	0.210667i	0.910980	+	0.412450i	$0.135327\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.53474	−	17.0342i	−0.264762	−	0.814855i
$438$		0			0
$439$	37.8748			1.80767			0.903834	−	0.427884i	$-0.140741\pi$
$439$	37.8748			1.80767			0.903834	+	0.427884i	$0.140741\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.92666	+	15.1627i	0.234072	+	0.720401i	0.997243	+	0.0742040i	$0.0236416\pi$
$443$	4.92666	+	15.1627i	0.234072	+	0.720401i	−0.763171	+	0.646197i	$0.776358\pi$
$444$		0			0
$445$	−7.87517	−	5.72164i	−0.373319	−	0.271232i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	15.4511	−	11.2259i	0.729181	−	0.529781i	−0.160123	−	0.987097i	$-0.551189\pi$
$449$	15.4511	−	11.2259i	0.729181	−	0.529781i	0.889304	+	0.457316i	$0.151189\pi$
$450$		0			0
$451$	−24.9149	+	7.23010i	−1.17320	+	0.340452i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.44501	+	7.52496i	−0.114624	+	0.352775i
$456$		0			0
$457$	−10.2623	−	7.45599i	−0.480050	−	0.348777i	0.321295	−	0.946979i	$-0.395882\pi$
$457$	−10.2623	−	7.45599i	−0.480050	−	0.348777i	−0.801345	+	0.598202i	$0.795882\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.87253			0.133787			0.0668935	−	0.997760i	$-0.478691\pi$
$461$	2.87253			0.133787			0.0668935	+	0.997760i	$0.478691\pi$
$462$		0			0
$463$	−6.29101			−0.292368			−0.146184	−	0.989257i	$-0.546699\pi$
$463$	−6.29101			−0.292368			−0.146184	+	0.989257i	$0.546699\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.74814	−	1.99664i	−0.127169	−	0.0923936i	0.522383	−	0.852711i	$-0.325043\pi$
$467$	−2.74814	−	1.99664i	−0.127169	−	0.0923936i	−0.649551	+	0.760318i	$0.725043\pi$
$468$		0			0
$469$	−5.80471	+	17.8651i	−0.268037	+	0.824932i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.324678	+	10.2307i	−0.0149287	+	0.470406i
$474$		0			0
$475$	−3.59944	+	2.61515i	−0.165154	+	0.119991i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.3946	+	9.00524i	0.566326	+	0.411460i	0.833769	−	0.552114i	$-0.186179\pi$
$479$	12.3946	+	9.00524i	0.566326	+	0.411460i	−0.267443	+	0.963574i	$0.586179\pi$
$480$		0			0
$481$	−16.9474	−	52.1587i	−0.772734	−	2.37823i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	11.8011			0.535858
$486$		0			0
$487$	5.84044	+	17.9750i	0.264656	+	0.814527i	0.991773	+	0.128013i	$0.0408599\pi$
$487$	5.84044	+	17.9750i	0.264656	+	0.814527i	−0.727117	+	0.686514i	$0.759140\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.64555	−	8.14216i	0.119392	−	0.367450i	−0.873446	−	0.486921i	$-0.838120\pi$
$491$	2.64555	−	8.14216i	0.119392	−	0.367450i	0.992838	+	0.119471i	$0.0381198\pi$
$492$		0			0
$493$	53.6149	−	38.9535i	2.41469	−	1.75438i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.70540	−	7.05138i	0.435347	−	0.316298i
$498$		0			0
$499$	−11.0689	+	34.0664i	−0.495510	+	1.52502i	0.320651	+	0.947197i	$0.396098\pi$
$499$	−11.0689	+	34.0664i	−0.495510	+	1.52502i	−0.816161	+	0.577825i	$0.803902\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.11424	−	6.50696i	−0.0942693	−	0.290131i	0.892793	−	0.450467i	$-0.148742\pi$
$503$	−2.11424	−	6.50696i	−0.0942693	−	0.290131i	−0.987063	+	0.160336i	$0.948742\pi$
$504$		0			0
$505$	−14.2262			−0.633057
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.567708	−	1.74723i	−0.0251632	−	0.0774445i	0.937686	−	0.347483i	$-0.112964\pi$
$509$	−0.567708	−	1.74723i	−0.0251632	−	0.0774445i	−0.962850	+	0.270039i	$0.912964\pi$
$510$		0			0
$511$	−9.80615	−	7.12458i	−0.433799	−	0.315173i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.0435	+	11.6563i	−0.706960	+	0.513636i
$516$		0			0
$517$	9.13367	−	2.65051i	0.401698	−	0.116570i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5.56920	+	17.1402i	−0.243991	+	0.750928i	0.751810	+	0.659380i	$0.229181\pi$
$521$	−5.56920	+	17.1402i	−0.243991	+	0.750928i	−0.995801	+	0.0915473i	$0.970819\pi$
$522$		0			0
$523$	11.0172	+	8.00444i	0.481747	+	0.350010i	0.802002	−	0.597322i	$-0.203768\pi$
$523$	11.0172	+	8.00444i	0.481747	+	0.350010i	−0.320255	+	0.947331i	$0.603768\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	16.5633			0.721510
$528$		0			0
$529$	−6.79404			−0.295393
$530$		0			0
$531$		0			0
$532$		0			0
$533$	37.8436	+	27.4950i	1.63919	+	1.19094i
$534$		0			0
$535$	−0.939127	+	2.89033i	−0.0406020	+	0.124960i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	10.6753	−	13.7537i	0.459818	−	0.592416i
$540$		0			0
$541$	−1.18837	+	0.863398i	−0.0510918	+	0.0371204i	−0.613038	−	0.790053i	$-0.710053\pi$
$541$	−1.18837	+	0.863398i	−0.0510918	+	0.0371204i	0.561946	+	0.827174i	$0.310053\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.01726	−	0.739079i	−0.0435744	−	0.0316587i
$546$		0			0
$547$	−7.74300	−	23.8305i	−0.331067	−	1.01892i	−0.968627	−	0.248518i	$-0.920056\pi$
$547$	−7.74300	−	23.8305i	−0.331067	−	1.01892i	0.637561	−	0.770400i	$-0.279944\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−36.9902			−1.57584
$552$		0			0
$553$	−4.54759	−	13.9960i	−0.193383	−	0.595172i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.23801	+	3.81021i	−0.0524563	+	0.161444i	−0.973853	−	0.227180i	$-0.927049\pi$
$557$	−1.23801	+	3.81021i	−0.0524563	+	0.161444i	0.921397	+	0.388624i	$0.127049\pi$
$558$		0			0
$559$	14.9313	−	10.8482i	0.631527	−	0.458831i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−32.7914	+	23.8244i	−1.38199	+	1.00408i	−0.385303	+	0.922790i	$0.625903\pi$
$563$	−32.7914	+	23.8244i	−1.38199	+	1.00408i	−0.996691	+	0.0812875i	$0.974097\pi$
$564$		0			0
$565$	6.09127	−	18.7470i	0.256261	−	0.788692i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.322604	−	0.992873i	−0.0135243	−	0.0416234i	0.944067	−	0.329755i	$-0.106966\pi$
$569$	−0.322604	−	0.992873i	−0.0135243	−	0.0416234i	−0.957591	+	0.288131i	$0.906966\pi$
$570$		0			0
$571$	13.3111			0.557054			0.278527	−	0.960428i	$-0.410154\pi$
$571$	13.3111			0.557054			0.278527	+	0.960428i	$0.410154\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.24400	−	3.82863i	−0.0518783	−	0.159665i
$576$		0			0
$577$	−6.11389	−	4.44200i	−0.254525	−	0.184923i	0.453205	−	0.891406i	$-0.350281\pi$
$577$	−6.11389	−	4.44200i	−0.254525	−	0.184923i	−0.707730	+	0.706483i	$0.750281\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.77258	−	6.37366i	0.363948	−	0.264424i
$582$		0			0
$583$	−7.62650	−	11.2295i	−0.315857	−	0.465079i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.6185	+	35.7580i	−0.479545	+	1.47589i	0.360183	+	0.932882i	$0.382714\pi$
$587$	−11.6185	+	35.7580i	−0.479545	+	1.47589i	−0.839728	+	0.543007i	$0.817286\pi$
$588$		0			0
$589$	−7.47935	−	5.43407i	−0.308181	−	0.223907i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.36380			−0.0560047			−0.0280024	−	0.999608i	$-0.508915\pi$
$593$	−1.36380			−0.0560047			−0.0280024	+	0.999608i	$0.508915\pi$
$594$		0			0
$595$	−10.5463			−0.432358
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.12176	−	5.90080i	−0.331846	−	0.241100i	0.409367	−	0.912370i	$-0.365749\pi$
$599$	−8.12176	−	5.90080i	−0.331846	−	0.241100i	−0.741214	+	0.671269i	$0.765749\pi$
$600$		0			0
$601$	6.79845	−	20.9235i	0.277314	−	0.853486i	−0.711283	−	0.702906i	$-0.751886\pi$
$601$	6.79845	−	20.9235i	0.277314	−	0.853486i	0.988598	−	0.150581i	$-0.0481143\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8.47131	−	7.01690i	−0.344408	−	0.285278i
$606$		0			0
$607$	17.1112	−	12.4320i	0.694521	−	0.504599i	−0.183622	−	0.982997i	$-0.558782\pi$
$607$	17.1112	−	12.4320i	0.694521	−	0.504599i	0.878143	+	0.478398i	$0.158782\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−13.8732	−	10.0795i	−0.561251	−	0.407773i
$612$		0			0
$613$	0.833203	+	2.56433i	0.0336527	+	0.103572i	0.966472	−	0.256772i	$-0.0826590\pi$
$613$	0.833203	+	2.56433i	0.0336527	+	0.103572i	−0.932819	+	0.360345i	$0.882659\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.26666			0.373062			0.186531	−	0.982449i	$-0.440276\pi$
$617$	9.26666			0.373062			0.186531	+	0.982449i	$0.440276\pi$
$618$		0			0
$619$	4.75896	+	14.6466i	0.191279	+	0.588695i	1.00000		0.000503118i	$0.000160147\pi$
$619$	4.75896	+	14.6466i	0.191279	+	0.588695i	−0.808721	+	0.588192i	$0.799840\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.97986	+	12.2487i	−0.159450	+	0.490736i
$624$		0			0
$625$	−0.809017	+	0.587785i	−0.0323607	+	0.0235114i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	59.1401	−	42.9678i	2.35807	−	1.71324i
$630$		0			0
$631$	−6.94137	+	21.3633i	−0.276332	+	0.850461i	0.712532	+	0.701639i	$0.247548\pi$
$631$	−6.94137	+	21.3633i	−0.276332	+	0.850461i	−0.988864	+	0.148822i	$0.952452\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.68937	−	11.3547i	−0.146408	−	0.450597i
$636$		0			0
$637$	−31.3929			−1.24383
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.32794	+	19.4754i	0.249939	+	0.769232i	0.994785	+	0.101996i	$0.0325228\pi$
$641$	6.32794	+	19.4754i	0.249939	+	0.769232i	−0.744846	+	0.667236i	$0.767477\pi$
$642$		0			0
$643$	4.72609	+	3.43370i	0.186379	+	0.135412i	0.677062	−	0.735926i	$-0.263253\pi$
$643$	4.72609	+	3.43370i	0.186379	+	0.135412i	−0.490683	+	0.871338i	$0.663253\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4.00539	+	2.91009i	−0.157468	+	0.114407i	−0.663729	−	0.747973i	$-0.731027\pi$
$647$	−4.00539	+	2.91009i	−0.157468	+	0.114407i	0.506261	+	0.862380i	$0.331027\pi$
$648$		0			0
$649$	16.4807	+	5.93916i	0.646923	+	0.233133i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.37985	+	28.8682i	−0.367062	+	1.12970i	0.581618	+	0.813462i	$0.302420\pi$
$653$	−9.37985	+	28.8682i	−0.367062	+	1.12970i	−0.948680	+	0.316238i	$0.897580\pi$
$654$		0			0
$655$	−15.6983	−	11.4055i	−0.613383	−	0.445649i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.9400			−0.504071			−0.252036	−	0.967718i	$-0.581100\pi$
$659$	−12.9400			−0.504071			−0.252036	+	0.967718i	$0.581100\pi$
$660$		0			0
$661$	−5.41592			−0.210655			−0.105327	−	0.994438i	$-0.533589\pi$
$661$	−5.41592			−0.210655			−0.105327	+	0.994438i	$0.533589\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.76231	+	3.46002i	0.184675	+	0.134174i
$666$		0			0
$667$	10.3426	−	31.8312i	0.400467	−	1.23251i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6.25924	+	8.06421i	−0.241635	+	0.311316i
$672$		0			0
$673$	33.6903	−	24.4774i	1.29867	−	0.943536i	0.298724	−	0.954339i	$-0.403439\pi$
$673$	33.6903	−	24.4774i	1.29867	−	0.943536i	0.999942	+	0.0108032i	$0.00343884\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−37.1606	−	26.9988i	−1.42820	−	1.03765i	−0.990349	−	0.138599i	$-0.955740\pi$
$677$	−37.1606	−	26.9988i	−1.42820	−	1.03765i	−0.437850	−	0.899048i	$-0.644260\pi$
$678$		0			0
$679$	−4.82488	−	14.8494i	−0.185162	−	0.569869i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.462944			0.0177141			0.00885703	−	0.999961i	$-0.497181\pi$
$683$	0.462944			0.0177141			0.00885703	+	0.999961i	$0.497181\pi$
$684$		0			0
$685$	1.40546	+	4.32557i	0.0536999	+	0.165271i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.56349	+	23.2780i	−0.288146	+	0.886823i
$690$		0			0
$691$	26.6269	−	19.3455i	1.01293	−	0.735939i	0.0481110	−	0.998842i	$-0.484680\pi$
$691$	26.6269	−	19.3455i	1.01293	−	0.735939i	0.964822	+	0.262903i	$0.0846799\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	15.2331	−	11.0675i	0.577825	−	0.419814i
$696$		0			0
$697$	−19.2673	+	59.2986i	−0.729801	+	2.24610i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	9.87750	+	30.3998i	0.373068	+	1.14819i	0.944773	+	0.327726i	$0.106282\pi$
$701$	9.87750	+	30.3998i	0.373068	+	1.14819i	−0.571705	+	0.820459i	$0.693718\pi$
$702$		0			0
$703$	−40.8022			−1.53888
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.81640	+	17.9010i	0.218748	+	0.673238i
$708$		0			0
$709$	24.1808	+	17.5684i	0.908129	+	0.659794i	0.940541	−	0.339680i	$-0.110319\pi$
$709$	24.1808	+	17.5684i	0.908129	+	0.659794i	−0.0324120	+	0.999475i	$0.510319\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.76744	−	4.91683i	0.253443	−	0.184137i
$714$		0			0
$715$	−0.629131	+	19.8240i	−0.0235282	+	0.741377i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.37070	+	19.6070i	−0.237587	+	0.731217i	0.759181	+	0.650880i	$0.225600\pi$
$719$	−6.37070	+	19.6070i	−0.237587	+	0.731217i	−0.996768	+	0.0803375i	$0.974400\pi$
$720$		0			0
$721$	21.2266	+	15.4221i	0.790522	+	0.574348i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.31399			−0.308774
$726$		0			0
$727$	26.4641			0.981499			0.490749	−	0.871301i	$-0.336723\pi$
$727$	26.4641			0.981499			0.490749	+	0.871301i	$0.336723\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	19.9022	+	14.4598i	0.736111	+	0.534816i
$732$		0			0
$733$	−5.62538	+	17.3131i	−0.207778	+	0.639475i	0.791810	+	0.610768i	$0.209139\pi$
$733$	−5.62538	+	17.3131i	−0.207778	+	0.639475i	−0.999588	+	0.0287076i	$0.990861\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.49363	+	47.0644i	−0.0550184	+	1.73364i
$738$		0			0
$739$	−32.8386	+	23.8586i	−1.20799	+	0.877653i	−0.995046	−	0.0994120i	$-0.968304\pi$
$739$	−32.8386	+	23.8586i	−1.20799	+	0.877653i	−0.212940	+	0.977065i	$0.568304\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	26.3393	+	19.1366i	0.966296	+	0.702055i	0.954604	−	0.297877i	$-0.0962784\pi$
$743$	26.3393	+	19.1366i	0.966296	+	0.702055i	0.0116914	+	0.999932i	$0.496278\pi$
$744$		0			0
$745$	1.19047	+	3.66389i	0.0436155	+	0.134235i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.02092			0.146921
$750$		0			0
$751$	1.53589	+	4.72698i	0.0560453	+	0.172490i	0.975161	−	0.221499i	$-0.0710949\pi$
$751$	1.53589	+	4.72698i	0.0560453	+	0.172490i	−0.919115	+	0.393989i	$0.871095\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.873524	+	2.68843i	−0.0317908	+	0.0978420i
$756$		0			0
$757$	−20.3767	+	14.8045i	−0.740602	+	0.538079i	−0.892900	−	0.450256i	$-0.851333\pi$
$757$	−20.3767	+	14.8045i	−0.740602	+	0.538079i	0.152297	+	0.988335i	$0.451333\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.05151	+	6.57631i	−0.328117	+	0.238391i	−0.739631	−	0.673012i	$-0.765000\pi$
$761$	−9.05151	+	6.57631i	−0.328117	+	0.238391i	0.411514	+	0.911403i	$0.365000\pi$
$762$		0			0
$763$	−0.514088	+	1.58220i	−0.0186113	+	0.0572795i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.76088	−	30.0409i	−0.352445	−	1.08471i
$768$		0			0
$769$	−53.3116			−1.92246			−0.961232	−	0.275741i	$-0.911077\pi$
$769$	−53.3116			−1.92246			−0.961232	+	0.275741i	$0.911077\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.30689	+	16.3329i	0.190876	+	0.587455i	1.00000		0.000182505i	$-5.80931e-5\pi$
$773$	5.30689	+	16.3329i	0.190876	+	0.587455i	−0.809124	+	0.587638i	$0.800058\pi$
$774$		0			0
$775$	−1.68107	−	1.22137i	−0.0603860	−	0.0438730i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28.1549	−	20.4558i	1.00876	−	0.732904i
$780$		0			0
$781$	18.4390	−	23.7562i	0.659798	−	0.850064i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.50631	−	7.71364i	0.0894541	−	0.275311i
$786$		0			0
$787$	−3.48717	−	2.53358i	−0.124304	−	0.0903122i	0.523896	−	0.851782i	$-0.324478\pi$
$787$	−3.48717	−	2.53358i	−0.124304	−	0.0903122i	−0.648200	+	0.761470i	$0.724478\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−26.0800			−0.927300
$792$		0			0
$793$	18.4065			0.653635
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13.1444	−	9.54996i	−0.465598	−	0.338277i	0.330125	−	0.943937i	$-0.392909\pi$
$797$	−13.1444	−	9.54996i	−0.465598	−	0.338277i	−0.795723	+	0.605660i	$0.792909\pi$
$798$		0			0
$799$	7.06328	−	21.7386i	0.249881	−	0.769055i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−28.5851	−	10.3013i	−1.00875	−	0.363524i
$804$		0			0
$805$	−4.30902	+	3.13068i	−0.151873	+	0.110342i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−12.7235	−	9.24413i	−0.447333	−	0.325006i	0.341209	−	0.939987i	$-0.389164\pi$
$809$	−12.7235	−	9.24413i	−0.447333	−	0.325006i	−0.788542	+	0.614981i	$0.789164\pi$
$810$		0			0
$811$	5.02578	+	15.4678i	0.176479	+	0.543146i	0.999698	−	0.0245776i	$-0.00782407\pi$
$811$	5.02578	+	15.4678i	0.176479	+	0.543146i	−0.823219	+	0.567724i	$0.807824\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.40390			−0.0842051
$816$		0			0
$817$	−4.24312	−	13.0590i	−0.148448	−	0.456876i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.81964	−	11.7556i	0.133306	−	0.410274i	−0.862016	−	0.506880i	$-0.830799\pi$
$821$	3.81964	−	11.7556i	0.133306	−	0.410274i	0.995323	+	0.0966058i	$0.0307986\pi$
$822$		0			0
$823$	7.90757	−	5.74518i	0.275640	−	0.200265i	−0.441373	−	0.897324i	$-0.645508\pi$
$823$	7.90757	−	5.74518i	0.275640	−	0.200265i	0.717014	+	0.697059i	$0.245508\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−20.7322	+	15.0629i	−0.720931	+	0.523787i	−0.886681	−	0.462381i	$-0.846995\pi$
$827$	−20.7322	+	15.0629i	−0.720931	+	0.523787i	0.165751	+	0.986168i	$0.446995\pi$
$828$		0			0
$829$	10.0035	−	30.7877i	0.347437	−	1.06930i	−0.612829	−	0.790216i	$-0.709968\pi$
$829$	10.0035	−	30.7877i	0.347437	−	1.06930i	0.960266	−	0.279086i	$-0.0900316\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.9306	−	39.7962i	−0.448017	−	1.37886i
$834$		0			0
$835$	−5.30887			−0.183721
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3.09372	−	9.52149i	−0.106807	−	0.328718i	0.883343	−	0.468727i	$-0.155287\pi$
$839$	−3.09372	−	9.52149i	−0.106807	−	0.328718i	−0.990150	+	0.140008i	$0.955287\pi$
$840$		0			0
$841$	−32.4597	−	23.5834i	−1.11930	−	0.813220i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	18.4153	−	13.3795i	0.633506	−	0.460269i
$846$		0			0
$847$	−5.36597	+	13.5284i	−0.184377	+	0.464843i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11.4084	−	35.1115i	0.391076	−	1.20361i
$852$		0			0
$853$	8.16802	+	5.93441i	0.279668	+	0.203190i	0.718772	−	0.695245i	$-0.244704\pi$
$853$	8.16802	+	5.93441i	0.279668	+	0.203190i	−0.439105	+	0.898436i	$0.644704\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−26.9504			−0.920609			−0.460305	−	0.887761i	$-0.652260\pi$
$857$	−26.9504			−0.920609			−0.460305	+	0.887761i	$0.652260\pi$
$858$		0			0
$859$	−23.4354			−0.799606			−0.399803	−	0.916601i	$-0.630921\pi$
$859$	−23.4354			−0.799606			−0.399803	+	0.916601i	$0.630921\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	10.8203	+	7.86141i	0.368327	+	0.267606i	0.756517	−	0.653974i	$-0.226899\pi$
$863$	10.8203	+	7.86141i	0.368327	+	0.267606i	−0.388190	+	0.921580i	$0.626899\pi$
$864$		0			0
$865$	3.99744	−	12.3028i	0.135917	−	0.418309i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−20.7260	−	30.5177i	−0.703081	−	1.03524i
$870$		0			0
$871$	68.6890	−	49.9055i	2.32744	−	1.69098i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.07039	+	0.777682i	0.0361857	+	0.0262904i
$876$		0			0
$877$	−3.92517	−	12.0804i	−0.132543	−	0.407927i	0.862656	−	0.505790i	$-0.168799\pi$
$877$	−3.92517	−	12.0804i	−0.132543	−	0.407927i	−0.995200	+	0.0978637i	$0.968799\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−22.5811			−0.760778			−0.380389	−	0.924827i	$-0.624210\pi$
$881$	−22.5811			−0.760778			−0.380389	+	0.924827i	$0.624210\pi$
$882$		0			0
$883$	8.87708	+	27.3208i	0.298737	+	0.919419i	0.981940	+	0.189191i	$0.0605864\pi$
$883$	8.87708	+	27.3208i	0.298737	+	0.919419i	−0.683203	+	0.730228i	$0.739414\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2.35720	−	7.25473i	0.0791471	−	0.243590i	−0.903652	−	0.428267i	$-0.859124\pi$
$887$	2.35720	−	7.25473i	0.0791471	−	0.243590i	0.982799	+	0.184678i	$0.0591241\pi$
$888$		0			0
$889$	−12.7794	+	9.28477i	−0.428607	+	0.311401i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.3215	+	7.49897i	−0.345394	+	0.250944i
$894$		0			0
$895$	−5.35505	+	16.4811i	−0.178999	+	0.550904i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5.33852	−	16.4303i	−0.178050	−	0.547981i
$900$		0			0
$901$	−32.6245			−1.08688
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.22248	+	6.84009i	0.0738778	+	0.227372i
$906$		0			0
$907$	−15.0288	−	10.9191i	−0.499023	−	0.362562i	0.309620	−	0.950860i	$-0.399798\pi$
$907$	−15.0288	−	10.9191i	−0.499023	−	0.362562i	−0.808644	+	0.588298i	$0.799798\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	26.1152	−	18.9738i	0.865234	−	0.628630i	−0.0640696	−	0.997945i	$-0.520408\pi$
$911$	26.1152	−	18.9738i	0.865234	−	0.628630i	0.929304	+	0.369316i	$0.120408\pi$
$912$		0			0
$913$	16.6667	−	21.4729i	0.551589	−	0.710650i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7.93342	+	24.4166i	−0.261985	+	0.806306i
$918$		0			0
$919$	−23.9513	−	17.4017i	−0.790082	−	0.574028i	0.117906	−	0.993025i	$-0.462382\pi$
$919$	−23.9513	−	17.4017i	−0.790082	−	0.574028i	−0.907988	+	0.418997i	$0.862382\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−54.2234			−1.78479
$924$		0			0
$925$	−9.17077			−0.301533
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−29.6500	−	21.5420i	−0.972784	−	0.706769i	−0.0166993	−	0.999861i	$-0.505316\pi$
$929$	−29.6500	−	21.5420i	−0.972784	−	0.706769i	−0.956084	+	0.293092i	$0.905316\pi$
$930$		0			0
$931$	−7.21732	+	22.2126i	−0.236538	+	0.727990i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−25.3897	+	7.36788i	−0.830332	+	0.240955i
$936$		0			0
$937$	−11.2165	+	8.14929i	−0.366428	+	0.266226i	−0.755728	−	0.654885i	$-0.772717\pi$
$937$	−11.2165	+	8.14929i	−0.366428	+	0.266226i	0.389300	+	0.921111i	$0.372717\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.8062	+	17.2962i	0.776059	+	0.563840i	0.903794	−	0.427969i	$-0.140771\pi$
$941$	23.8062	+	17.2962i	0.776059	+	0.563840i	−0.127735	+	0.991808i	$0.540771\pi$
$942$		0			0
$943$	9.73060	+	29.9477i	0.316872	+	0.975232i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	21.3123			0.692557			0.346278	−	0.938132i	$-0.387445\pi$
$947$	21.3123			0.692557			0.346278	+	0.938132i	$0.387445\pi$
$948$		0			0
$949$	16.9299	+	52.1049i	0.549568	+	1.69140i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.17738	−	6.70130i	0.0705323	−	0.217076i	−0.909577	−	0.415536i	$-0.863594\pi$
$953$	2.17738	−	6.70130i	0.0705323	−	0.217076i	0.980109	+	0.198460i	$0.0635940\pi$
$954$		0			0
$955$	5.94565	−	4.31976i	0.192397	−	0.139784i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.86830	−	3.53703i	0.157206	−	0.114217i
$960$		0			0
$961$	−8.24527	+	25.3763i	−0.265976	+	0.818591i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.64550	+	17.3751i	0.181735	+	0.559323i
$966$		0			0
$967$	−16.4122			−0.527781			−0.263891	−	0.964553i	$-0.585006\pi$
$967$	−16.4122			−0.527781			−0.263891	+	0.964553i	$0.585006\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.72502	−	11.4644i	−0.119542	−	0.367911i	0.873326	−	0.487137i	$-0.161959\pi$
$971$	−3.72502	−	11.4644i	−0.119542	−	0.367911i	−0.992867	+	0.119226i	$0.961959\pi$
$972$		0			0
$973$	−20.1545	−	14.6431i	−0.646123	−	0.469436i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−38.2903	+	27.8195i	−1.22501	+	0.890025i	−0.996506	−	0.0835173i	$-0.973385\pi$
$977$	−38.2903	+	27.8195i	−1.22501	+	0.890025i	−0.228508	+	0.973542i	$0.573385\pi$
$978$		0			0
$979$	−1.02407	+	32.2686i	−0.0327294	+	1.03131i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	0.439405	−	1.35235i	0.0140148	−	0.0431333i	−0.943805	−	0.330504i	$-0.892781\pi$
$983$	0.439405	−	1.35235i	0.0140148	−	0.0431333i	0.957819	+	0.287371i	$0.0927812\pi$
$984$		0			0
$985$	−12.9516	−	9.40986i	−0.412671	−	0.299823i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.4241			0.395062
$990$		0			0
$991$	−41.5050			−1.31845			−0.659225	−	0.751946i	$-0.729116\pi$
$991$	−41.5050			−1.31845			−0.659225	+	0.751946i	$0.729116\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−12.4943	−	9.07763i	−0.396096	−	0.287780i
$996$		0			0
$997$	1.08772	−	3.34767i	0.0344486	−	0.106022i	−0.932354	−	0.361548i	$-0.882249\pi$
$997$	1.08772	−	3.34767i	0.0344486	−	0.106022i	0.966802	+	0.255526i	$0.0822485\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1980.2.z.e.1081.1		8
3.2	odd	2		660.2.y.a.421.1	yes	8
11.4	even	5	inner	1980.2.z.e.1621.1		8
33.2	even	10		7260.2.a.bh.1.3		4
33.20	odd	10		7260.2.a.bj.1.2		4
33.26	odd	10		660.2.y.a.301.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
660.2.y.a.301.1	✓	8	33.26	odd	10
660.2.y.a.421.1	yes	8	3.2	odd	2
1980.2.z.e.1081.1		8	1.1	even	1	trivial
1980.2.z.e.1621.1		8	11.4	even	5	inner
7260.2.a.bh.1.3		4	33.2	even	10
7260.2.a.bj.1.2		4	33.20	odd	10

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:	\( N \)	\(=\)	\( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1980.z (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.809017 + 0.587785i) q^{5} +(0.408851 - 1.25832i) q^{7} +(0.105203 - 3.31496i) q^{11} +(-4.83807 + 3.51506i) q^{13} +(-6.44876 - 4.68530i) q^{17} +(1.37486 + 4.23139i) q^{19} -4.02566 q^{23} +(0.309017 + 0.951057i) q^{25} +(-2.56916 + 7.90707i) q^{29} +(-1.68107 + 1.22137i) q^{31} +(1.07039 - 0.777682i) q^{35} +(-2.83392 + 8.72192i) q^{37} +(-2.41714 - 7.43920i) q^{41} -3.08621 q^{43} +(0.886111 + 2.72717i) q^{47} +(4.24692 + 3.08557i) q^{49} +(3.31118 - 2.40572i) q^{53} +(2.03359 - 2.62002i) q^{55} +(-1.63220 + 5.02341i) q^{59} +(-2.49009 - 1.80916i) q^{61} -5.98018 q^{65} -14.1976 q^{67} +(7.33551 + 5.32956i) q^{71} +(2.83100 - 8.71293i) q^{73} +(-4.12825 - 1.48770i) q^{77} +(8.99857 - 6.53784i) q^{79} +(6.63047 + 4.81732i) q^{83} +(-2.46321 - 7.58097i) q^{85} -9.73424 q^{89} +(2.44501 + 7.52496i) q^{91} +(-1.37486 + 4.23139i) q^{95} +(9.54725 - 6.93648i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{5} + 3 q^{7} + 3 q^{11} - 6 q^{13} - 8 q^{17} - 14 q^{19} - 10 q^{23} - 2 q^{25} - 6 q^{29} - q^{31} + 7 q^{35} - 5 q^{37} + 11 q^{41} + 28 q^{43} + q^{47} + q^{49} + 3 q^{53} + 7 q^{55} + 5 q^{59}+ \cdots + 35 q^{97}+O(q^{100}) \) 8 * q + 2 * q^5 + 3 * q^7 + 3 * q^11 - 6 * q^13 - 8 * q^17 - 14 * q^19 - 10 * q^23 - 2 * q^25 - 6 * q^29 - q^31 + 7 * q^35 - 5 * q^37 + 11 * q^41 + 28 * q^43 + q^47 + q^49 + 3 * q^53 + 7 * q^55 + 5 * q^59 - 3 * q^61 - 14 * q^65 - 50 * q^67 + 11 * q^71 + 33 * q^73 - 7 * q^77 + 2 * q^79 - q^83 - 12 * q^85 + 44 * q^89 + 25 * q^91 + 14 * q^95 + 35 * q^97

Introduction

L-functions

Modular forms

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Database

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