LMFDB - Embedded newform 3240.2.q.bg.1081.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3240,2,Mod(1081,3240)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3240, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3240.1081"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-4,0,-2,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$25.8715302549$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.3887771904.9
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 26x^{6} + 217x^{4} + 672x^{2} + 576 $ x^8 + 26x^6 + 217x^4 + 672*x^2 + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$-1.16910i$ of defining polynomial
Character	$\chi$	$=$	3240.1081
Dual form			3240.2.q.bg.2161.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{5} +(-1.95058 - 3.37850i) q^{7} +(3.04760 + 5.27859i) q^{11} +(1.01247 - 1.75365i) q^{13} -2.46199 q^{17} +3.33820 q^{19} +(-0.415449 + 0.719580i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-1.31555 - 2.27859i) q^{29} +(-0.915449 + 1.58560i) q^{31} +3.90115 q^{35} +5.58789 q^{37} +(-3.32802 + 5.76430i) q^{41} +(-5.52600 - 9.57131i) q^{43} +(-0.0834961 - 0.144619i) q^{47} +(-4.10949 + 7.11785i) q^{49} +12.0234 q^{53} -6.09520 q^{55} +(0.866025 - 1.50000i) q^{59} +(6.65815 + 11.5322i) q^{61} +(1.01247 + 1.75365i) q^{65} +(-6.63109 + 11.4854i) q^{67} +10.8829 q^{71} +1.87621 q^{73} +(11.8891 - 20.5926i) q^{77} +(-3.73205 - 6.46410i) q^{79} +(-3.40010 - 5.88914i) q^{83} +(1.23100 - 2.13215i) q^{85} +12.1493 q^{89} -7.89961 q^{91} +(-1.66910 + 2.89097i) q^{95} +(0.331953 + 0.574960i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{5} - 2 q^{7} - 4 q^{11} - 4 q^{17} - 10 q^{23} - 4 q^{25} + 4 q^{29} - 14 q^{31} + 4 q^{35} + 28 q^{37} - 4 q^{41} - 22 q^{43} - 10 q^{49} - 16 q^{53} + 8 q^{55} - 4 q^{61} - 24 q^{67} + 56 q^{71}+ \cdots + 10 q^{97}+O(q^{100}) $ 8 * q - 4 * q^5 - 2 * q^7 - 4 * q^11 - 4 * q^17 - 10 * q^23 - 4 * q^25 + 4 * q^29 - 14 * q^31 + 4 * q^35 + 28 * q^37 - 4 * q^41 - 22 * q^43 - 10 * q^49 - 16 * q^53 + 8 * q^55 - 4 * q^61 - 24 * q^67 + 56 * q^71 + 4 * q^73 + 30 * q^77 - 16 * q^79 - 6 * q^83 + 2 * q^85 - 16 * q^89 + 60 * q^91 + 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	−1.95058	−	3.37850i	−0.737248	−	1.27695i	−0.953730	−	0.300665i	$-0.902791\pi$
$7$	−1.95058	−	3.37850i	−0.737248	−	1.27695i	0.216481	−	0.976287i	$-0.430542\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.04760	+	5.27859i	0.918885	+	1.59156i	0.801111	+	0.598516i	$0.204243\pi$
$11$	3.04760	+	5.27859i	0.918885	+	1.59156i	0.117774	+	0.993040i	$0.462424\pi$
$12$		0			0
$13$	1.01247	−	1.75365i	0.280809	−	0.486375i	−0.690775	−	0.723070i	$-0.742731\pi$
$13$	1.01247	−	1.75365i	0.280809	−	0.486375i	0.971584	+	0.236694i	$0.0760639\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.46199			−0.597121			−0.298560	−	0.954391i	$-0.596506\pi$
$17$	−2.46199			−0.597121			−0.298560	+	0.954391i	$0.596506\pi$
$18$		0			0
$19$	3.33820			0.765836			0.382918	−	0.923782i	$-0.374919\pi$
$19$	3.33820			0.765836			0.382918	+	0.923782i	$0.374919\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.415449	+	0.719580i	−0.0866272	+	0.150043i	−0.906083	−	0.423099i	$-0.860942\pi$
$23$	−0.415449	+	0.719580i	−0.0866272	+	0.150043i	0.819456	+	0.573142i	$0.194276\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.31555	−	2.27859i	−0.244291	−	0.423124i	0.717641	−	0.696413i	$-0.245222\pi$
$29$	−1.31555	−	2.27859i	−0.244291	−	0.423124i	−0.961932	+	0.273289i	$0.911889\pi$
$30$		0			0
$31$	−0.915449	+	1.58560i	−0.164420	+	0.284783i	−0.936449	−	0.350804i	$-0.885908\pi$
$31$	−0.915449	+	1.58560i	−0.164420	+	0.284783i	0.772029	+	0.635587i	$0.219242\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.90115			0.659415
$36$		0			0
$37$	5.58789			0.918644			0.459322	−	0.888270i	$-0.348092\pi$
$37$	5.58789			0.918644			0.459322	+	0.888270i	$0.348092\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.32802	+	5.76430i	−0.519749	+	0.900232i	0.479987	+	0.877275i	$0.340641\pi$
$41$	−3.32802	+	5.76430i	−0.519749	+	0.900232i	−0.999736	+	0.0229565i	$0.992692\pi$
$42$		0			0
$43$	−5.52600	−	9.57131i	−0.842707	−	1.45961i	−0.887598	−	0.460619i	$-0.847628\pi$
$43$	−5.52600	−	9.57131i	−0.842707	−	1.45961i	0.0448914	−	0.998992i	$-0.485706\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.0834961	−	0.144619i	−0.0121792	−	0.0210949i	0.859872	−	0.510510i	$-0.170544\pi$
$47$	−0.0834961	−	0.144619i	−0.0121792	−	0.0210949i	−0.872051	+	0.489415i	$0.837210\pi$
$48$		0			0
$49$	−4.10949	+	7.11785i	−0.587070	+	1.01684i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.0234			1.65154			0.825770	−	0.564006i	$-0.190741\pi$
$53$	12.0234			1.65154			0.825770	+	0.564006i	$0.190741\pi$
$54$		0			0
$55$	−6.09520			−0.821876
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.866025	−	1.50000i	0.112747	−	0.195283i	−0.804130	−	0.594454i	$-0.797368\pi$
$59$	0.866025	−	1.50000i	0.112747	−	0.195283i	0.916877	+	0.399170i	$0.130702\pi$
$60$		0			0
$61$	6.65815	+	11.5322i	0.852488	+	1.47655i	0.878956	+	0.476903i	$0.158241\pi$
$61$	6.65815	+	11.5322i	0.852488	+	1.47655i	−0.0264678	+	0.999650i	$0.508426\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.01247	+	1.75365i	0.125582	+	0.217514i
$66$		0			0
$67$	−6.63109	+	11.4854i	−0.810117	+	1.40316i	0.102665	+	0.994716i	$0.467263\pi$
$67$	−6.63109	+	11.4854i	−0.810117	+	1.40316i	−0.912782	+	0.408448i	$0.866070\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.8829			1.29156			0.645781	−	0.763523i	$-0.276532\pi$
$71$	10.8829			1.29156			0.645781	+	0.763523i	$0.276532\pi$
$72$		0			0
$73$	1.87621			0.219594			0.109797	−	0.993954i	$-0.464980\pi$
$73$	1.87621			0.219594			0.109797	+	0.993954i	$0.464980\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	11.8891	−	20.5926i	1.35489	−	2.34674i
$78$		0			0
$79$	−3.73205	−	6.46410i	−0.419889	−	0.727268i	0.576039	−	0.817422i	$-0.304597\pi$
$79$	−3.73205	−	6.46410i	−0.419889	−	0.727268i	−0.995928	+	0.0901537i	$0.971264\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.40010	−	5.88914i	−0.373209	−	0.646417i	0.616848	−	0.787082i	$-0.288409\pi$
$83$	−3.40010	−	5.88914i	−0.373209	−	0.646417i	−0.990057	+	0.140665i	$0.955076\pi$
$84$		0			0
$85$	1.23100	−	2.13215i	0.133520	−	0.231264i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.1493			1.28782			0.643912	−	0.765100i	$-0.277310\pi$
$89$	12.1493			1.28782			0.643912	+	0.765100i	$0.277310\pi$
$90$		0			0
$91$	−7.89961			−0.828104
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.66910	+	2.89097i	−0.171246	+	0.296607i
$96$		0			0
$97$	0.331953	+	0.574960i	0.0337048	+	0.0583784i	0.882386	−	0.470527i	$-0.155936\pi$
$97$	0.331953	+	0.574960i	0.0337048	+	0.0583784i	−0.848681	+	0.528905i	$0.822603\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.68080	+	16.7676i	0.963276	+	1.66844i	0.714178	+	0.699964i	$0.246801\pi$
$101$	9.68080	+	16.7676i	0.963276	+	1.66844i	0.249097	+	0.968478i	$0.419866\pi$
$102$		0			0
$103$	6.64567	−	11.5106i	0.654818	−	1.13418i	−0.327122	−	0.944982i	$-0.606079\pi$
$103$	6.64567	−	11.5106i	0.654818	−	1.13418i	0.981939	−	0.189195i	$-0.0605879\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.39230			0.617967			0.308984	−	0.951067i	$-0.400011\pi$
$107$	6.39230			0.617967			0.308984	+	0.951067i	$0.400011\pi$
$108$		0			0
$109$	7.29500			0.698734			0.349367	−	0.936986i	$-0.386397\pi$
$109$	7.29500			0.698734			0.349367	+	0.936986i	$0.386397\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.62485	−	2.81431i	0.152853	−	0.264748i	−0.779422	−	0.626499i	$-0.784487\pi$
$113$	1.62485	−	2.81431i	0.152853	−	0.264748i	0.932275	+	0.361750i	$0.117821\pi$
$114$		0			0
$115$	−0.415449	−	0.719580i	−0.0387409	−	0.0671011i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.80230	+	8.31783i	0.440226	+	0.762495i
$120$		0			0
$121$	−13.0757	+	22.6478i	−1.18870	+	2.05889i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	14.8294			1.31589			0.657946	−	0.753065i	$-0.271425\pi$
$127$	14.8294			1.31589			0.657946	+	0.753065i	$0.271425\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.52417	−	11.3002i	0.570019	−	0.987303i	−0.426544	−	0.904467i	$-0.640269\pi$
$131$	6.52417	−	11.3002i	0.570019	−	0.987303i	0.996563	−	0.0828357i	$-0.0263977\pi$
$132$		0			0
$133$	−6.51142	−	11.2781i	−0.564611	−	0.977936i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.90115	+	10.2211i	0.504169	+	0.873247i	0.999988	+	0.00482115i	$0.00153463\pi$
$137$	5.90115	+	10.2211i	0.504169	+	0.873247i	−0.495819	+	0.868426i	$0.665132\pi$
$138$		0			0
$139$	8.35219	−	14.4664i	0.708423	−	1.22703i	−0.257018	−	0.966407i	$-0.582740\pi$
$139$	8.35219	−	14.4664i	0.708423	−	1.22703i	0.965442	−	0.260619i	$-0.0839266\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.3424			1.03213
$144$		0			0
$145$	2.63109			0.218500
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.6332	−	18.4172i	0.871106	−	1.50880i	0.0102520	−	0.999947i	$-0.496737\pi$
$149$	10.6332	−	18.4172i	0.871106	−	1.50880i	0.860854	−	0.508852i	$-0.169930\pi$
$150$		0			0
$151$	5.84154	+	10.1179i	0.475378	+	0.823379i	0.999602	−	0.0282013i	$-0.00897794\pi$
$151$	5.84154	+	10.1179i	0.475378	+	0.823379i	−0.524224	+	0.851580i	$0.675645\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.915449	−	1.58560i	−0.0735307	−	0.127359i
$156$		0			0
$157$	10.0458	−	17.3998i	0.801740	−	1.38865i	−0.116730	−	0.993164i	$-0.537241\pi$
$157$	10.0458	−	17.3998i	0.801740	−	1.38865i	0.918470	−	0.395490i	$-0.129425\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.24146			0.255463
$162$		0			0
$163$	2.99789			0.234813			0.117406	−	0.993084i	$-0.462542\pi$
$163$	2.99789			0.234813			0.117406	+	0.993084i	$0.462542\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.25594	+	9.10355i	−0.406717	+	0.704454i	−0.994520	−	0.104550i	$-0.966660\pi$
$167$	−5.25594	+	9.10355i	−0.406717	+	0.704454i	0.587803	+	0.809004i	$0.299993\pi$
$168$		0			0
$169$	4.44980	+	7.70729i	0.342293	+	0.592868i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.87955	+	10.1837i	0.447014	+	0.774251i	0.998190	−	0.0601380i	$-0.0191541\pi$
$173$	5.87955	+	10.1837i	0.447014	+	0.774251i	−0.551176	+	0.834389i	$0.685821\pi$
$174$		0			0
$175$	−1.95058	+	3.37850i	−0.147450	+	0.255390i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.65604			−0.497496			−0.248748	−	0.968568i	$-0.580019\pi$
$179$	−6.65604			−0.497496			−0.248748	+	0.968568i	$0.580019\pi$
$180$		0			0
$181$	8.50730			0.632343			0.316171	−	0.948702i	$-0.397603\pi$
$181$	8.50730			0.632343			0.316171	+	0.948702i	$0.397603\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.79395	+	4.83926i	−0.205415	+	0.355789i
$186$		0			0
$187$	−7.50316	−	12.9959i	−0.548686	−	0.950351i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.75260	−	15.1599i	−0.633316	−	1.09693i	−0.986869	−	0.161521i	$-0.948360\pi$
$191$	−8.75260	−	15.1599i	−0.633316	−	1.09693i	0.353554	−	0.935414i	$-0.384973\pi$
$192$		0			0
$193$	−13.0582	+	22.6175i	−0.939953	+	1.62805i	−0.174398	+	0.984675i	$0.555798\pi$
$193$	−13.0582	+	22.6175i	−0.939953	+	1.62805i	−0.765555	+	0.643371i	$0.777535\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.9215			−0.920620			−0.460310	−	0.887758i	$-0.652262\pi$
$197$	−12.9215			−0.920620			−0.460310	+	0.887758i	$0.652262\pi$
$198$		0			0
$199$	16.9781			1.20354			0.601772	−	0.798668i	$-0.294461\pi$
$199$	16.9781			1.20354			0.601772	+	0.798668i	$0.294461\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.13215	+	8.88914i	−0.360206	+	0.623895i
$204$		0			0
$205$	−3.32802	−	5.76430i	−0.232439	−	0.402596i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.1735	+	17.6210i	0.703716	+	1.21887i
$210$		0			0
$211$	−5.72109	+	9.90923i	−0.393856	+	0.682179i	−0.992954	−	0.118496i	$-0.962193\pi$
$211$	−5.72109	+	9.90923i	−0.393856	+	0.682179i	0.599098	+	0.800676i	$0.295526\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	11.0520			0.753740
$216$		0			0
$217$	7.14261			0.484872
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.49270	+	4.31748i	−0.167677	+	0.290425i
$222$		0			0
$223$	−7.82725	−	13.5572i	−0.524151	−	0.907856i	−0.999605	−	0.0281156i	$-0.991049\pi$
$223$	−7.82725	−	13.5572i	−0.524151	−	0.907856i	0.475454	−	0.879741i	$-0.342284\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.52600	+	13.0354i	0.499518	+	0.865190i	1.00000		0.000556524i	$-0.000177147\pi$
$227$	7.52600	+	13.0354i	0.499518	+	0.865190i	−0.500482	+	0.865747i	$0.666844\pi$
$228$		0			0
$229$	−8.33820	+	14.4422i	−0.551004	+	0.954367i	0.447199	+	0.894435i	$0.352422\pi$
$229$	−8.33820	+	14.4422i	−0.551004	+	0.954367i	−0.998202	+	0.0599320i	$0.980912\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.99789			−0.196398			−0.0981992	−	0.995167i	$-0.531308\pi$
$233$	−2.99789			−0.196398			−0.0981992	+	0.995167i	$0.531308\pi$
$234$		0			0
$235$	0.166992			0.0108934
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7.73205	+	13.3923i	−0.500145	+	0.866276i	0.499855	+	0.866109i	$0.333387\pi$
$239$	−7.73205	+	13.3923i	−0.500145	+	0.866276i	−1.00000		0.000167197i	$0.999947\pi$
$240$		0			0
$241$	−8.15269	−	14.1209i	−0.525161	−	0.909606i	−0.999571	−	0.0293016i	$-0.990672\pi$
$241$	−8.15269	−	14.1209i	−0.525161	−	0.909606i	0.474409	−	0.880304i	$-0.342662\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.10949	−	7.11785i	−0.262546	−	0.454743i
$246$		0			0
$247$	3.37983	−	5.85404i	0.215054	−	0.372484i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	27.6295			1.74396			0.871981	−	0.489540i	$-0.162835\pi$
$251$	27.6295			1.74396			0.871981	+	0.489540i	$0.162835\pi$
$252$		0			0
$253$	−5.06449			−0.318402
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.07391	−	1.86006i	0.0669884	−	0.116027i	−0.830586	−	0.556891i	$-0.811994\pi$
$257$	1.07391	−	1.86006i	0.0669884	−	0.116027i	0.897574	+	0.440863i	$0.145328\pi$
$258$		0			0
$259$	−10.8996	−	18.8787i	−0.677269	−	1.17306i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.80354	+	4.85587i	0.172873	+	0.299426i	0.939423	−	0.342759i	$-0.111361\pi$
$263$	2.80354	+	4.85587i	0.172873	+	0.299426i	−0.766550	+	0.642185i	$0.778028\pi$
$264$		0			0
$265$	−6.01170	+	10.4126i	−0.369296	+	0.639639i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.36469			−0.327091			−0.163545	−	0.986536i	$-0.552293\pi$
$269$	−5.36469			−0.327091			−0.163545	+	0.986536i	$0.552293\pi$
$270$		0			0
$271$	−8.59000			−0.521805			−0.260903	−	0.965365i	$-0.584020\pi$
$271$	−8.59000			−0.521805			−0.260903	+	0.965365i	$0.584020\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.04760	−	5.27859i	0.183777	−	0.318311i
$276$		0			0
$277$	12.3840	+	21.4497i	0.744081	+	1.28879i	0.950623	+	0.310348i	$0.100446\pi$
$277$	12.3840	+	21.4497i	0.744081	+	1.28879i	−0.206542	+	0.978438i	$0.566221\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.7138	−	22.0210i	−0.758443	−	1.31366i	−0.943644	−	0.330961i	$-0.892627\pi$
$281$	−12.7138	−	22.0210i	−0.758443	−	1.31366i	0.185202	−	0.982701i	$-0.440706\pi$
$282$		0			0
$283$	−1.43705	+	2.48904i	−0.0854238	+	0.147958i	−0.905572	−	0.424193i	$-0.860558\pi$
$283$	−1.43705	+	2.48904i	−0.0854238	+	0.147958i	0.820148	+	0.572152i	$0.193891\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	25.9662			1.53274
$288$		0			0
$289$	−10.9386			−0.643447
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.76658	+	8.25597i	−0.278467	+	0.482319i	−0.971004	−	0.239064i	$-0.923160\pi$
$293$	−4.76658	+	8.25597i	−0.278467	+	0.482319i	0.692537	+	0.721382i	$0.256493\pi$
$294$		0			0
$295$	0.866025	+	1.50000i	0.0504219	+	0.0873334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.841261	+	1.45711i	0.0486514	+	0.0842667i
$300$		0			0
$301$	−21.5578	+	37.3391i	−1.24257	+	2.15219i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−13.3163			−0.762489
$306$		0			0
$307$	4.02402			0.229663			0.114831	−	0.993385i	$-0.463367\pi$
$307$	4.02402			0.229663			0.114831	+	0.993385i	$0.463367\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.82861	+	8.36340i	−0.273805	+	0.474245i	−0.969833	−	0.243770i	$-0.921616\pi$
$311$	−4.82861	+	8.36340i	−0.273805	+	0.474245i	0.696028	+	0.718015i	$0.254949\pi$
$312$		0			0
$313$	17.0462	+	29.5249i	0.963510	+	1.66885i	0.713566	+	0.700588i	$0.247079\pi$
$313$	17.0462	+	29.5249i	0.963510	+	1.66885i	0.249944	+	0.968260i	$0.419588\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.25260	+	16.0260i	0.519678	+	0.900108i	0.999738	+	0.0228728i	$0.00728128\pi$
$317$	9.25260	+	16.0260i	0.519678	+	0.900108i	−0.480061	+	0.877235i	$0.659385\pi$
$318$		0			0
$319$	8.01852	−	13.8885i	0.448951	−	0.777605i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.21863			−0.457297
$324$		0			0
$325$	−2.02494			−0.112324
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.325731	+	0.564182i	−0.0179581	+	0.0311044i
$330$		0			0
$331$	−9.16149	−	15.8682i	−0.503561	−	0.872193i	−0.999992	−	0.00411661i	$-0.998690\pi$
$331$	−9.16149	−	15.8682i	−0.503561	−	0.872193i	0.496431	−	0.868076i	$-0.334644\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.63109	−	11.4854i	−0.362295	−	0.627514i
$336$		0			0
$337$	14.0234	−	24.2892i	0.763903	−	1.32312i	−0.176921	−	0.984225i	$-0.556614\pi$
$337$	14.0234	−	24.2892i	0.763903	−	1.32312i	0.940825	−	0.338894i	$-0.110053\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−11.1597			−0.604331
$342$		0			0
$343$	4.75545			0.256770
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.03906	−	1.79971i	0.0557798	−	0.0966134i	−0.836787	−	0.547528i	$-0.815569\pi$
$347$	1.03906	−	1.79971i	0.0557798	−	0.0966134i	0.892567	+	0.450915i	$0.148902\pi$
$348$		0			0
$349$	−7.69070	−	13.3207i	−0.411674	−	0.713040i	0.583399	−	0.812186i	$-0.301722\pi$
$349$	−7.69070	−	13.3207i	−0.411674	−	0.713040i	−0.995073	+	0.0991456i	$0.968389\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.27631	+	2.21063i	0.0679310	+	0.117660i	0.897990	−	0.440015i	$-0.145027\pi$
$353$	1.27631	+	2.21063i	0.0679310	+	0.117660i	−0.830059	+	0.557675i	$0.811694\pi$
$354$		0			0
$355$	−5.44145	+	9.42486i	−0.288802	+	0.500220i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.57699			0.241564			0.120782	−	0.992679i	$-0.461460\pi$
$359$	4.57699			0.241564			0.120782	+	0.992679i	$0.461460\pi$
$360$		0			0
$361$	−7.85641			−0.413495
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.938105	+	1.62485i	−0.0491026	+	0.0850483i
$366$		0			0
$367$	−11.2460	−	19.4787i	−0.587038	−	1.01678i	−0.994618	−	0.103611i	$-0.966960\pi$
$367$	−11.2460	−	19.4787i	−0.587038	−	1.01678i	0.407580	−	0.913170i	$-0.366373\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−23.4526	−	40.6210i	−1.21760	−	2.10894i
$372$		0			0
$373$	15.5655	−	26.9603i	0.805953	−	1.39595i	−0.109692	−	0.993966i	$-0.534986\pi$
$373$	15.5655	−	26.9603i	0.805953	−	1.39595i	0.915646	−	0.401987i	$-0.131680\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.32781			−0.274396
$378$		0			0
$379$	−35.4136			−1.81907			−0.909537	−	0.415623i	$-0.863564\pi$
$379$	−35.4136			−1.81907			−0.909537	+	0.415623i	$0.863564\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.43674	−	14.6129i	0.431097	−	0.746682i	−0.565871	−	0.824494i	$-0.691460\pi$
$383$	8.43674	−	14.6129i	0.431097	−	0.746682i	0.996968	+	0.0778116i	$0.0247933\pi$
$384$		0			0
$385$	11.8891	+	20.5926i	0.605927	+	1.04950i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.23935	+	9.07483i	0.265646	+	0.460112i	0.967733	−	0.251979i	$-0.0810815\pi$
$389$	5.23935	+	9.07483i	0.265646	+	0.460112i	−0.702087	+	0.712091i	$0.747748\pi$
$390$		0			0
$391$	1.02283	−	1.77160i	0.0517269	−	0.0895936i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.46410			0.375560
$396$		0			0
$397$	−1.55781			−0.0781843			−0.0390921	−	0.999236i	$-0.512447\pi$
$397$	−1.55781			−0.0781843			−0.0390921	+	0.999236i	$0.512447\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.32784	+	5.76399i	−0.166184	+	0.287840i	−0.937075	−	0.349127i	$-0.886478\pi$
$401$	−3.32784	+	5.76399i	−0.166184	+	0.287840i	0.770891	+	0.636967i	$0.219811\pi$
$402$		0			0
$403$	1.85373	+	3.21076i	0.0923410	+	0.159939i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	17.0296	+	29.4962i	0.844128	+	1.46207i
$408$		0			0
$409$	17.8452	−	30.9088i	0.882388	−	1.52834i	0.0337097	−	0.999432i	$-0.489268\pi$
$409$	17.8452	−	30.9088i	0.882388	−	1.52834i	0.848678	−	0.528909i	$-0.177399\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.75699			−0.332490
$414$		0			0
$415$	6.80019			0.333808
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.43705	−	11.1493i	0.314470	−	0.544679i	−0.664854	−	0.746973i	$-0.731506\pi$
$419$	6.43705	−	11.1493i	0.314470	−	0.544679i	0.979325	+	0.202294i	$0.0648397\pi$
$420$		0			0
$421$	7.08666	+	12.2745i	0.345383	+	0.598220i	0.985423	−	0.170121i	$-0.0544158\pi$
$421$	7.08666	+	12.2745i	0.345383	+	0.598220i	−0.640041	+	0.768341i	$0.721082\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.23100	+	2.13215i	0.0597121	+	0.103424i
$426$		0			0
$427$	25.9744	−	44.9890i	1.25699	−	2.17717i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.786211			0.0378704			0.0189352	−	0.999821i	$-0.493972\pi$
$431$	0.786211			0.0378704			0.0189352	+	0.999821i	$0.493972\pi$
$432$		0			0
$433$	31.2184			1.50026			0.750129	−	0.661291i	$-0.229991\pi$
$433$	31.2184			1.50026			0.750129	+	0.661291i	$0.229991\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.38685	+	2.40210i	−0.0663422	+	0.114908i
$438$		0			0
$439$	14.5179	+	25.1458i	0.692904	+	1.20014i	0.970882	+	0.239557i	$0.0770023\pi$
$439$	14.5179	+	25.1458i	0.692904	+	1.20014i	−0.277978	+	0.960587i	$0.589664\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	8.54426	+	14.7991i	0.405950	+	0.703126i	0.994432	−	0.105385i	$-0.0336074\pi$
$443$	8.54426	+	14.7991i	0.405950	+	0.703126i	−0.588482	+	0.808511i	$0.700274\pi$
$444$		0			0
$445$	−6.07465	+	10.5216i	−0.287966	+	0.498772i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.5082			0.873457			0.436729	−	0.899593i	$-0.356137\pi$
$449$	18.5082			0.873457			0.436729	+	0.899593i	$0.356137\pi$
$450$		0			0
$451$	−40.5698			−1.91036
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.94980	−	6.84126i	0.185170	−	0.320723i
$456$		0			0
$457$	−0.0618953	−	0.107206i	−0.00289534	−	0.00501488i	0.864574	−	0.502505i	$-0.167588\pi$
$457$	−0.0618953	−	0.107206i	−0.00289534	−	0.00501488i	−0.867469	+	0.497490i	$0.834255\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.43687	−	14.6131i	−0.392944	−	0.680599i	0.599892	−	0.800081i	$-0.295210\pi$
$461$	−8.43687	−	14.6131i	−0.392944	−	0.680599i	−0.992836	+	0.119481i	$0.961877\pi$
$462$		0			0
$463$	2.28253	−	3.95346i	0.106078	−	0.183733i	−0.808100	−	0.589045i	$-0.799504\pi$
$463$	2.28253	−	3.95346i	0.106078	−	0.183733i	0.914178	+	0.405313i	$0.132837\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−34.8793			−1.61402			−0.807011	−	0.590537i	$-0.798916\pi$
$467$	−34.8793			−1.61402			−0.807011	+	0.590537i	$0.798916\pi$
$468$		0			0
$469$	51.7378			2.38903
$470$		0			0
$471$		0			0
$472$		0			0
$473$	33.6820	−	58.3390i	1.54870	−	2.68243i
$474$		0			0
$475$	−1.66910	−	2.89097i	−0.0765836	−	0.132647i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−21.2152	−	36.7457i	−0.969345	−	1.67896i	−0.697458	−	0.716626i	$-0.745685\pi$
$479$	−21.2152	−	36.7457i	−0.969345	−	1.67896i	−0.271888	−	0.962329i	$-0.587648\pi$
$480$		0			0
$481$	5.65758	−	9.79922i	0.257963	−	0.446806i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.663907			−0.0301465
$486$		0			0
$487$	18.1331			0.821691			0.410846	−	0.911705i	$-0.365233\pi$
$487$	18.1331			0.821691			0.410846	+	0.911705i	$0.365233\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.25411	+	16.0286i	−0.417632	+	0.723360i	−0.995701	−	0.0926279i	$-0.970473\pi$
$491$	−9.25411	+	16.0286i	−0.417632	+	0.723360i	0.578069	+	0.815988i	$0.303807\pi$
$492$		0			0
$493$	3.23887	+	5.60988i	0.145871	+	0.252656i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−21.2279	−	36.7678i	−0.952202	−	1.64926i
$498$		0			0
$499$	1.08332	−	1.87636i	0.0484960	−	0.0839975i	−0.840758	−	0.541410i	$-0.817891\pi$
$499$	1.08332	−	1.87636i	0.0484960	−	0.0839975i	0.889254	+	0.457413i	$0.151224\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−21.6088			−0.963490			−0.481745	−	0.876311i	$-0.659997\pi$
$503$	−21.6088			−0.963490			−0.481745	+	0.876311i	$0.659997\pi$
$504$		0			0
$505$	−19.3616			−0.861580
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.67640	+	2.90362i	−0.0743053	+	0.128701i	−0.900784	−	0.434267i	$-0.857007\pi$
$509$	−1.67640	+	2.90362i	−0.0743053	+	0.128701i	0.826479	+	0.562968i	$0.190341\pi$
$510$		0			0
$511$	−3.65969	−	6.33877i	−0.161895	−	0.280411i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.64567	+	11.5106i	0.292843	+	0.507220i
$516$		0			0
$517$	0.508925	−	0.881484i	0.0223825	−	0.0387676i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	33.4775			1.46668			0.733338	−	0.679864i	$-0.237961\pi$
$521$	33.4775			1.46668			0.733338	+	0.679864i	$0.237961\pi$
$522$		0			0
$523$	−32.8230			−1.43525			−0.717624	−	0.696431i	$-0.754770\pi$
$523$	−32.8230			−1.43525			−0.717624	+	0.696431i	$0.754770\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.25383	−	3.90375i	0.0981784	−	0.170050i
$528$		0			0
$529$	11.1548	+	19.3207i	0.484991	+	0.840030i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.73905	+	11.6724i	0.291900	+	0.505586i
$534$		0			0
$535$	−3.19615	+	5.53590i	−0.138182	+	0.239338i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−50.0963			−2.15780
$540$		0			0
$541$	−24.9038			−1.07070			−0.535350	−	0.844631i	$-0.679820\pi$
$541$	−24.9038			−1.07070			−0.535350	+	0.844631i	$0.679820\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.64750	+	6.31766i	−0.156242	+	0.270619i
$546$		0			0
$547$	−20.2628	−	35.0961i	−0.866373	−	1.50060i	−0.865678	−	0.500602i	$-0.833112\pi$
$547$	−20.2628	−	35.0961i	−0.866373	−	1.50060i	−0.000695095	−	1.00000i	$-0.500221\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.39156	−	7.60641i	−0.187087	−	0.324044i
$552$		0			0
$553$	−14.5593	+	25.2174i	−0.619124	+	1.07235i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25.8673			−1.09603			−0.548016	−	0.836468i	$-0.684617\pi$
$557$	−25.8673			−1.09603			−0.548016	+	0.836468i	$0.684617\pi$
$558$		0			0
$559$	−22.3797			−0.946558
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.01504	+	6.95425i	−0.169214	+	0.293087i	−0.938144	−	0.346246i	$-0.887456\pi$
$563$	−4.01504	+	6.95425i	−0.169214	+	0.293087i	0.768930	+	0.639333i	$0.220790\pi$
$564$		0			0
$565$	1.62485	+	2.81431i	0.0683577	+	0.118399i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	16.6933	+	28.9136i	0.699818	+	1.21212i	0.968529	+	0.248900i	$0.0800691\pi$
$569$	16.6933	+	28.9136i	0.699818	+	1.21212i	−0.268711	+	0.963221i	$0.586598\pi$
$570$		0			0
$571$	−13.9659	+	24.1896i	−0.584455	+	1.01231i	0.410488	+	0.911866i	$0.365358\pi$
$571$	−13.9659	+	24.1896i	−0.584455	+	1.01231i	−0.994943	+	0.100439i	$0.967975\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.830899			0.0346509
$576$		0			0
$577$	−2.74187			−0.114146			−0.0570729	−	0.998370i	$-0.518177\pi$
$577$	−2.74187			−0.114146			−0.0570729	+	0.998370i	$0.518177\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−13.2643	+	22.9744i	−0.550296	+	0.953140i
$582$		0			0
$583$	36.6425	+	63.4666i	1.51758	+	2.62852i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.261700	+	0.453278i	0.0108015	+	0.0187088i	0.871376	−	0.490616i	$-0.163228\pi$
$587$	0.261700	+	0.453278i	0.0108015	+	0.0187088i	−0.860574	+	0.509325i	$0.829895\pi$
$588$		0			0
$589$	−3.05596	+	5.29307i	−0.125918	+	0.218097i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−39.5973			−1.62607			−0.813033	−	0.582217i	$-0.802185\pi$
$593$	−39.5973			−1.62607			−0.813033	+	0.582217i	$0.802185\pi$
$594$		0			0
$595$	−9.60461			−0.393751
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.2936	+	19.5611i	−0.461445	+	0.799247i	−0.999033	−	0.0439608i	$-0.986002\pi$
$599$	−11.2936	+	19.5611i	−0.461445	+	0.799247i	0.537588	+	0.843208i	$0.319336\pi$
$600$		0			0
$601$	−6.88598	−	11.9269i	−0.280885	−	0.486507i	0.690718	−	0.723124i	$-0.257295\pi$
$601$	−6.88598	−	11.9269i	−0.280885	−	0.486507i	−0.971603	+	0.236617i	$0.923961\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−13.0757	−	22.6478i	−0.531603	−	0.920763i
$606$		0			0
$607$	13.0874	−	22.6680i	0.531201	−	0.920068i	−0.468136	−	0.883657i	$-0.655074\pi$
$607$	13.0874	−	22.6680i	0.531201	−	0.920068i	0.999337	−	0.0364110i	$-0.0115925\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.338150			−0.0136801
$612$		0			0
$613$	1.67943			0.0678317			0.0339159	−	0.999425i	$-0.489202\pi$
$613$	1.67943			0.0678317			0.0339159	+	0.999425i	$0.489202\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.953146	+	1.65090i	−0.0383722	+	0.0664626i	−0.884574	−	0.466400i	$-0.845551\pi$
$617$	−0.953146	+	1.65090i	−0.0383722	+	0.0664626i	0.846201	+	0.532863i	$0.178884\pi$
$618$		0			0
$619$	−5.49239	−	9.51309i	−0.220758	−	0.382363i	0.734281	−	0.678846i	$-0.237520\pi$
$619$	−5.49239	−	9.51309i	−0.220758	−	0.382363i	−0.955038	+	0.296483i	$0.904186\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−23.6981	−	41.0464i	−0.949445	−	1.64449i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−13.7573			−0.548541
$630$		0			0
$631$	−0.318756			−0.0126895			−0.00634473	−	0.999980i	$-0.502020\pi$
$631$	−0.318756			−0.0126895			−0.00634473	+	0.999980i	$0.502020\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.41468	+	12.8426i	−0.294243	+	0.509643i
$636$		0			0
$637$	8.32149	+	14.4132i	0.329709	+	0.571073i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.56735	−	6.17882i	−0.140902	−	0.244049i	0.786935	−	0.617036i	$-0.211667\pi$
$641$	−3.56735	−	6.17882i	−0.140902	−	0.244049i	−0.927836	+	0.372987i	$0.878333\pi$
$642$		0			0
$643$	−5.79029	+	10.0291i	−0.228347	+	0.395508i	−0.957318	−	0.289036i	$-0.906665\pi$
$643$	−5.79029	+	10.0291i	−0.228347	+	0.395508i	0.728971	+	0.684544i	$0.239999\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.68555			−0.341464			−0.170732	−	0.985317i	$-0.554613\pi$
$647$	−8.68555			−0.341464			−0.170732	+	0.985317i	$0.554613\pi$
$648$		0			0
$649$	10.5572			0.414406
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−22.6092	+	39.1603i	−0.884766	+	1.53246i	−0.0387837	+	0.999248i	$0.512348\pi$
$653$	−22.6092	+	39.1603i	−0.884766	+	1.53246i	−0.845982	+	0.533211i	$0.820985\pi$
$654$		0			0
$655$	6.52417	+	11.3002i	0.254920	+	0.441535i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5.49786	−	9.52258i	−0.214166	−	0.370947i	0.738848	−	0.673872i	$-0.235370\pi$
$659$	−5.49786	−	9.52258i	−0.214166	−	0.370947i	−0.953014	+	0.302925i	$0.902037\pi$
$660$		0			0
$661$	−11.5553	+	20.0144i	−0.449450	+	0.778471i	−0.998350	−	0.0574171i	$-0.981714\pi$
$661$	−11.5553	+	20.0144i	−0.449450	+	0.778471i	0.548900	+	0.835888i	$0.315047\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	13.0228			0.505004
$666$		0			0
$667$	2.18617			0.0846490
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−40.5827	+	70.2913i	−1.56668	+	2.71357i
$672$		0			0
$673$	2.99424	+	5.18617i	0.115419	+	0.199912i	0.917947	−	0.396702i	$-0.129845\pi$
$673$	2.99424	+	5.18617i	0.115419	+	0.199912i	−0.802528	+	0.596615i	$0.796512\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−8.77705	−	15.2023i	−0.337329	−	0.584272i	0.646600	−	0.762829i	$-0.276190\pi$
$677$	−8.77705	−	15.2023i	−0.337329	−	0.584272i	−0.983929	+	0.178557i	$0.942857\pi$
$678$		0			0
$679$	1.29500	−	2.24301i	0.0496976	−	0.0860787i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.9458			−0.954524			−0.477262	−	0.878761i	$-0.658371\pi$
$683$	−24.9458			−0.954524			−0.477262	+	0.878761i	$0.658371\pi$
$684$		0			0
$685$	−11.8023			−0.450943
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12.1733	−	21.0849i	0.463768	−	0.803269i
$690$		0			0
$691$	17.5356	+	30.3725i	0.667085	+	1.15543i	0.978715	+	0.205222i	$0.0657916\pi$
$691$	17.5356	+	30.3725i	0.667085	+	1.15543i	−0.311630	+	0.950203i	$0.600875\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.35219	+	14.4664i	0.316817	+	0.548742i
$696$		0			0
$697$	8.19356	−	14.1917i	0.310353	−	0.537547i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	11.6530			0.440128			0.220064	−	0.975485i	$-0.429373\pi$
$701$	11.6530			0.440128			0.220064	+	0.975485i	$0.429373\pi$
$702$		0			0
$703$	18.6535			0.703531
$704$		0			0
$705$		0			0
$706$		0			0
$707$	37.7663	−	65.4131i	1.42035	−	2.46011i
$708$		0			0
$709$	−10.7795	−	18.6706i	−0.404832	−	0.701189i	0.589470	−	0.807790i	$-0.299336\pi$
$709$	−10.7795	−	18.6706i	−0.404832	−	0.701189i	−0.994302	+	0.106601i	$0.966003\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.760646	−	1.31748i	−0.0284864	−	0.0493399i
$714$		0			0
$715$	−6.17121	+	10.6888i	−0.230790	+	0.399740i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−27.4199			−1.02259			−0.511295	−	0.859405i	$-0.670834\pi$
$719$	−27.4199			−1.02259			−0.511295	+	0.859405i	$0.670834\pi$
$720$		0			0
$721$	−51.8516			−1.93105
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.31555	+	2.27859i	−0.0488582	+	0.0846249i
$726$		0			0
$727$	−6.42247	−	11.1240i	−0.238196	−	0.412568i	0.722001	−	0.691893i	$-0.243223\pi$
$727$	−6.42247	−	11.1240i	−0.238196	−	0.412568i	−0.960197	+	0.279325i	$0.909889\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	13.6050	+	23.5645i	0.503198	+	0.871564i
$732$		0			0
$733$	−3.29500	+	5.70711i	−0.121704	+	0.210797i	−0.920440	−	0.390885i	$-0.872169\pi$
$733$	−3.29500	+	5.70711i	−0.121704	+	0.210797i	0.798736	+	0.601682i	$0.205502\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−80.8356			−2.97762
$738$		0			0
$739$	33.5238			1.23319			0.616596	−	0.787280i	$-0.288511\pi$
$739$	33.5238			1.23319			0.616596	+	0.787280i	$0.288511\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−18.3204	+	31.7319i	−0.672111	+	1.16413i	0.305193	+	0.952291i	$0.401279\pi$
$743$	−18.3204	+	31.7319i	−0.672111	+	1.16413i	−0.977304	+	0.211841i	$0.932054\pi$
$744$		0			0
$745$	10.6332	+	18.4172i	0.389570	+	0.674756i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.4687	−	21.5964i	−0.455596	−	0.789115i
$750$		0			0
$751$	0.826681	−	1.43185i	0.0301660	−	0.0522491i	−0.850548	−	0.525897i	$-0.823730\pi$
$751$	0.826681	−	1.43185i	0.0301660	−	0.0522491i	0.880714	+	0.473648i	$0.157063\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−11.6831			−0.425191
$756$		0			0
$757$	−24.9447			−0.906631			−0.453315	−	0.891350i	$-0.649759\pi$
$757$	−24.9447			−0.906631			−0.453315	+	0.891350i	$0.649759\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.86392	+	17.0848i	−0.357567	+	0.619324i	−0.987554	−	0.157282i	$-0.949727\pi$
$761$	−9.86392	+	17.0848i	−0.357567	+	0.619324i	0.629987	+	0.776606i	$0.283060\pi$
$762$		0			0
$763$	−14.2295	−	24.6461i	−0.515141	−	0.892250i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.75365	−	3.03741i	−0.0633207	−	0.109675i
$768$		0			0
$769$	23.1849	−	40.1574i	0.836068	−	1.44811i	−0.0570893	−	0.998369i	$-0.518182\pi$
$769$	23.1849	−	40.1574i	0.836068	−	1.44811i	0.893158	−	0.449744i	$-0.148485\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	24.2163			0.870998			0.435499	−	0.900189i	$-0.356572\pi$
$773$	24.2163			0.870998			0.435499	+	0.900189i	$0.356572\pi$
$774$		0			0
$775$	1.83090			0.0657678
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−11.1096	+	19.2424i	−0.398043	+	0.689430i
$780$		0			0
$781$	33.1667	+	57.4464i	1.18680	+	2.05559i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10.0458	+	17.3998i	0.358549	+	0.621025i
$786$		0			0
$787$	1.72004	−	2.97920i	0.0613128	−	0.106197i	−0.833740	−	0.552158i	$-0.813805\pi$
$787$	1.72004	−	2.97920i	0.0613128	−	0.106197i	0.895052	+	0.445961i	$0.147138\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.6775			−0.450761
$792$		0			0
$793$	26.9647			0.957545
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−23.4875	+	40.6815i	−0.831970	+	1.44101i	0.0645039	+	0.997917i	$0.479454\pi$
$797$	−23.4875	+	40.6815i	−0.831970	+	1.44101i	−0.896474	+	0.443097i	$0.853880\pi$
$798$		0			0
$799$	0.205567	+	0.356052i	0.00727243	+	0.0125962i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5.71793	+	9.90375i	0.201781	+	0.349496i
$804$		0			0
$805$	−1.62073	+	2.80719i	−0.0571233	+	0.0989404i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−24.8829			−0.874838			−0.437419	−	0.899258i	$-0.644107\pi$
$809$	−24.8829			−0.874838			−0.437419	+	0.899258i	$0.644107\pi$
$810$		0			0
$811$	−40.1794			−1.41089			−0.705444	−	0.708765i	$-0.749253\pi$
$811$	−40.1794			−1.41089			−0.705444	+	0.708765i	$0.749253\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.49895	+	2.59625i	−0.0525058	+	0.0909427i
$816$		0			0
$817$	−18.4469	−	31.9510i	−0.645375	−	1.11782i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.783301	−	1.35672i	−0.0273374	−	0.0473498i	0.852033	−	0.523488i	$-0.175370\pi$
$821$	−0.783301	−	1.35672i	−0.0273374	−	0.0473498i	−0.879370	+	0.476138i	$0.842036\pi$
$822$		0			0
$823$	15.7488	−	27.2777i	0.548967	−	0.950840i	−0.449378	−	0.893342i	$-0.648354\pi$
$823$	15.7488	−	27.2777i	0.548967	−	0.950840i	0.998346	−	0.0574979i	$-0.0183123\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	19.7013			0.685081			0.342540	−	0.939503i	$-0.388713\pi$
$827$	19.7013			0.685081			0.342540	+	0.939503i	$0.388713\pi$
$828$		0			0
$829$	−15.2950			−0.531217			−0.265609	−	0.964081i	$-0.585573\pi$
$829$	−15.2950			−0.531217			−0.265609	+	0.964081i	$0.585573\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	10.1175	−	17.5241i	0.350552	−	0.607174i
$834$		0			0
$835$	−5.25594	−	9.10355i	−0.181889	−	0.315041i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−17.5600	−	30.4149i	−0.606240	−	1.05004i	−0.991854	−	0.127378i	$-0.959344\pi$
$839$	−17.5600	−	30.4149i	−0.606240	−	1.05004i	0.385614	−	0.922660i	$-0.373990\pi$
$840$		0			0
$841$	11.0387	−	19.1195i	0.380644	−	0.659295i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.89961			−0.306156
$846$		0			0
$847$	102.021			3.50547
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.32149	+	4.02093i	−0.0795795	+	0.137836i
$852$		0			0
$853$	−12.8152	−	22.1966i	−0.438785	−	0.759998i	0.558811	−	0.829295i	$-0.311258\pi$
$853$	−12.8152	−	22.1966i	−0.438785	−	0.759998i	−0.997596	+	0.0692967i	$0.977924\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.14719	−	14.1113i	−0.278303	−	0.482034i	0.692660	−	0.721264i	$-0.256438\pi$
$857$	−8.14719	−	14.1113i	−0.278303	−	0.482034i	−0.970963	+	0.239230i	$0.923105\pi$
$858$		0			0
$859$	11.5523	−	20.0092i	0.394160	−	0.682704i	−0.598834	−	0.800873i	$-0.704369\pi$
$859$	11.5523	−	20.0092i	0.394160	−	0.682704i	0.992994	+	0.118169i	$0.0377024\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	18.3148			0.623443			0.311722	−	0.950173i	$-0.399094\pi$
$863$	18.3148			0.623443			0.311722	+	0.950173i	$0.399094\pi$
$864$		0			0
$865$	−11.7591			−0.399821
$866$		0			0
$867$		0			0
$868$		0			0
$869$	22.7476	−	39.4000i	0.771659	−	1.33655i
$870$		0			0
$871$	13.4276	+	23.2573i	0.454976	+	0.788042i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.95058	−	3.37850i	−0.0659415	−	0.114214i
$876$		0			0
$877$	3.91573	−	6.78225i	0.132225	−	0.229020i	−0.792309	−	0.610120i	$-0.791121\pi$
$877$	3.91573	−	6.78225i	0.132225	−	0.229020i	0.924534	+	0.381100i	$0.124455\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	3.17430			0.106945			0.0534724	−	0.998569i	$-0.482971\pi$
$881$	3.17430			0.106945			0.0534724	+	0.998569i	$0.482971\pi$
$882$		0			0
$883$	−34.4132			−1.15810			−0.579049	−	0.815293i	$-0.696576\pi$
$883$	−34.4132			−1.15810			−0.579049	+	0.815293i	$0.696576\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27.8353	−	48.2121i	0.934618	−	1.61881i	0.159303	−	0.987230i	$-0.449075\pi$
$887$	27.8353	−	48.2121i	0.934618	−	1.61881i	0.775314	−	0.631576i	$-0.217591\pi$
$888$		0			0
$889$	−28.9258	−	50.1009i	−0.970140	−	1.68033i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.278727	−	0.482769i	−0.00932724	−	0.0161552i
$894$		0			0
$895$	3.32802	−	5.76430i	0.111243	−	0.192679i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.81727			0.160665
$900$		0			0
$901$	−29.6015			−0.986170
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.25365	+	7.36754i	−0.141396	+	0.244905i
$906$		0			0
$907$	14.0146	+	24.2740i	0.465347	+	0.806005i	0.999217	−	0.0395614i	$-0.0125961\pi$
$907$	14.0146	+	24.2740i	0.465347	+	0.806005i	−0.533870	+	0.845567i	$0.679263\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.4618	−	21.5845i	−0.412878	−	0.715126i	0.582325	−	0.812956i	$-0.302143\pi$
$911$	−12.4618	−	21.5845i	−0.412878	−	0.715126i	−0.995203	+	0.0978300i	$0.968810\pi$
$912$		0			0
$913$	20.7243	−	35.8955i	0.685873	−	1.18797i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−50.9036			−1.68098
$918$		0			0
$919$	13.2512			0.437116			0.218558	−	0.975824i	$-0.429865\pi$
$919$	13.2512			0.437116			0.218558	+	0.975824i	$0.429865\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11.0186	−	19.0848i	0.362682	−	0.628184i
$924$		0			0
$925$	−2.79395	−	4.83926i	−0.0918644	−	0.159114i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	21.3456	+	36.9717i	0.700327	+	1.21300i	0.968352	+	0.249590i	$0.0802960\pi$
$929$	21.3456	+	36.9717i	0.700327	+	1.21300i	−0.268024	+	0.963412i	$0.586371\pi$
$930$		0			0
$931$	−13.7183	+	23.7608i	−0.449600	+	0.778730i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	15.0063			0.490759
$936$		0			0
$937$	−11.5620			−0.377715			−0.188857	−	0.982005i	$-0.560478\pi$
$937$	−11.5620			−0.377715			−0.188857	+	0.982005i	$0.560478\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.3865	−	33.5785i	0.631983	−	1.09463i	−0.355163	−	0.934805i	$-0.615574\pi$
$941$	19.3865	−	33.5785i	0.631983	−	1.09463i	0.987146	−	0.159823i	$-0.0510922\pi$
$942$		0			0
$943$	−2.76525	−	4.78955i	−0.0900488	−	0.155969i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	25.1697	+	43.5951i	0.817904	+	1.41665i	0.907224	+	0.420649i	$0.138197\pi$
$947$	25.1697	+	43.5951i	0.817904	+	1.41665i	−0.0893194	+	0.996003i	$0.528469\pi$
$948$		0			0
$949$	1.89961	−	3.29022i	0.0616639	−	0.106805i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−11.4118			−0.369666			−0.184833	−	0.982770i	$-0.559174\pi$
$953$	−11.4118			−0.369666			−0.184833	+	0.982770i	$0.559174\pi$
$954$		0			0
$955$	17.5052			0.566455
$956$		0			0
$957$		0			0
$958$		0			0
$959$	23.0213	−	39.8740i	0.743396	−	1.28760i
$960$		0			0
$961$	13.8239	+	23.9437i	0.445932	+	0.772378i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−13.0582	−	22.6175i	−0.420360	−	0.728084i
$966$		0			0
$967$	3.81642	−	6.61024i	0.122728	−	0.212571i	−0.798115	−	0.602506i	$-0.794169\pi$
$967$	3.81642	−	6.61024i	0.122728	−	0.212571i	0.920843	+	0.389935i	$0.127502\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20.5124			0.658276			0.329138	−	0.944282i	$-0.393242\pi$
$971$	20.5124			0.658276			0.329138	+	0.944282i	$0.393242\pi$
$972$		0			0
$973$	−65.1663			−2.08914
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.96930	−	6.87502i	0.126989	−	0.219951i	−0.795520	−	0.605928i	$-0.792802\pi$
$977$	3.96930	−	6.87502i	0.126989	−	0.219951i	0.922509	+	0.385976i	$0.126135\pi$
$978$		0			0
$979$	37.0262	+	64.1312i	1.18336	+	2.04964i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−11.8627	−	20.5467i	−0.378360	−	0.655339i	0.612464	−	0.790499i	$-0.290179\pi$
$983$	−11.8627	−	20.5467i	−0.378360	−	0.655339i	−0.990824	+	0.135160i	$0.956845\pi$
$984$		0			0
$985$	6.46076	−	11.1904i	0.205857	−	0.356555i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9.18309			0.292005
$990$		0			0
$991$	4.04618			0.128531			0.0642656	−	0.997933i	$-0.479530\pi$
$991$	4.04618			0.128531			0.0642656	+	0.997933i	$0.479530\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.48904	+	14.7035i	−0.269121	+	0.466131i
$996$		0			0
$997$	−4.54058	−	7.86451i	−0.143802	−	0.249072i	0.785124	−	0.619339i	$-0.212599\pi$
$997$	−4.54058	−	7.86451i	−0.143802	−	0.249072i	−0.928925	+	0.370268i	$0.879266\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.q.bg.1081.1		8
3.2	odd	2		3240.2.q.bh.1081.1		8
9.2	odd	6		3240.2.q.bh.2161.1		8
9.4	even	3		3240.2.a.v.1.4	yes	4
9.5	odd	6		3240.2.a.t.1.4	✓	4
9.7	even	3	inner	3240.2.q.bg.2161.1		8
36.23	even	6		6480.2.a.by.1.1		4
36.31	odd	6		6480.2.a.ca.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.t.1.4	✓	4	9.5	odd	6
3240.2.a.v.1.4	yes	4	9.4	even	3
3240.2.q.bg.1081.1		8	1.1	even	1	trivial
3240.2.q.bg.2161.1		8	9.7	even	3	inner
3240.2.q.bh.1081.1		8	3.2	odd	2
3240.2.q.bh.2161.1		8	9.2	odd	6
6480.2.a.by.1.1		4	36.23	even	6
6480.2.a.ca.1.1		4	36.31	odd	6

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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{5} +(-1.95058 - 3.37850i) q^{7} +(3.04760 + 5.27859i) q^{11} +(1.01247 - 1.75365i) q^{13} -2.46199 q^{17} +3.33820 q^{19} +(-0.415449 + 0.719580i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-1.31555 - 2.27859i) q^{29} +(-0.915449 + 1.58560i) q^{31} +3.90115 q^{35} +5.58789 q^{37} +(-3.32802 + 5.76430i) q^{41} +(-5.52600 - 9.57131i) q^{43} +(-0.0834961 - 0.144619i) q^{47} +(-4.10949 + 7.11785i) q^{49} +12.0234 q^{53} -6.09520 q^{55} +(0.866025 - 1.50000i) q^{59} +(6.65815 + 11.5322i) q^{61} +(1.01247 + 1.75365i) q^{65} +(-6.63109 + 11.4854i) q^{67} +10.8829 q^{71} +1.87621 q^{73} +(11.8891 - 20.5926i) q^{77} +(-3.73205 - 6.46410i) q^{79} +(-3.40010 - 5.88914i) q^{83} +(1.23100 - 2.13215i) q^{85} +12.1493 q^{89} -7.89961 q^{91} +(-1.66910 + 2.89097i) q^{95} +(0.331953 + 0.574960i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{5} - 2 q^{7} - 4 q^{11} - 4 q^{17} - 10 q^{23} - 4 q^{25} + 4 q^{29} - 14 q^{31} + 4 q^{35} + 28 q^{37} - 4 q^{41} - 22 q^{43} - 10 q^{49} - 16 q^{53} + 8 q^{55} - 4 q^{61} - 24 q^{67} + 56 q^{71}+ \cdots + 10 q^{97}+O(q^{100}) \) 8 * q - 4 * q^5 - 2 * q^7 - 4 * q^11 - 4 * q^17 - 10 * q^23 - 4 * q^25 + 4 * q^29 - 14 * q^31 + 4 * q^35 + 28 * q^37 - 4 * q^41 - 22 * q^43 - 10 * q^49 - 16 * q^53 + 8 * q^55 - 4 * q^61 - 24 * q^67 + 56 * q^71 + 4 * q^73 + 30 * q^77 - 16 * q^79 - 6 * q^83 + 2 * q^85 - 16 * q^89 + 60 * q^91 + 10 * q^97

Introduction

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