LMFDB - Embedded newform 1620.2.i.n.1081.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(541,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.2.i.n.541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(-0.366025 - 0.633975i) q^{7} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{5} +(-0.366025 - 0.633975i) q^{7} +(0.866025 + 1.50000i) q^{11} +(0.732051 - 1.26795i) q^{13} -1.26795 q^{17} +2.46410 q^{19} +(1.73205 - 3.00000i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(2.13397 + 3.69615i) q^{29} +(3.96410 - 6.86603i) q^{31} -0.732051 q^{35} +4.19615 q^{37} +(0.401924 - 0.696152i) q^{41} +(-3.36603 - 5.83013i) q^{43} +(2.36603 + 4.09808i) q^{47} +(3.23205 - 5.59808i) q^{49} -10.7321 q^{53} +1.73205 q^{55} +(2.13397 - 3.69615i) q^{59} +(2.00000 + 3.46410i) q^{61} +(-0.732051 - 1.26795i) q^{65} +(7.19615 - 12.4641i) q^{67} -0.803848 q^{71} +10.1962 q^{73} +(0.633975 - 1.09808i) q^{77} +(-3.19615 - 5.53590i) q^{79} +(-4.56218 - 7.90192i) q^{83} +(-0.633975 + 1.09808i) q^{85} -5.19615 q^{89} -1.07180 q^{91} +(1.23205 - 2.13397i) q^{95} +(1.36603 + 2.36603i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^5 + 2 * q^7	$ 4 q + 2 q^{5} + 2 q^{7} - 4 q^{13} - 12 q^{17} - 4 q^{19} - 2 q^{25} + 12 q^{29} + 2 q^{31} + 4 q^{35} - 4 q^{37} + 12 q^{41} - 10 q^{43} + 6 q^{47} + 6 q^{49} - 36 q^{53} + 12 q^{59} + 8 q^{61} + 4 q^{65} + 8 q^{67} - 24 q^{71} + 20 q^{73} + 6 q^{77} + 8 q^{79} + 6 q^{83} - 6 q^{85} - 32 q^{91} - 2 q^{95} + 2 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 + 2 * q^7 - 4 * q^13 - 12 * q^17 - 4 * q^19 - 2 * q^25 + 12 * q^29 + 2 * q^31 + 4 * q^35 - 4 * q^37 + 12 * q^41 - 10 * q^43 + 6 * q^47 + 6 * q^49 - 36 * q^53 + 12 * q^59 + 8 * q^61 + 4 * q^65 + 8 * q^67 - 24 * q^71 + 20 * q^73 + 6 * q^77 + 8 * q^79 + 6 * q^83 - 6 * q^85 - 32 * q^91 - 2 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$7$	−0.366025	−	0.633975i	−0.138345	−	0.239620i	0.788526	−	0.615002i	$-0.210845\pi$
$7$	−0.366025	−	0.633975i	−0.138345	−	0.239620i	−0.926870	+	0.375382i	$0.877511\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.866025	+	1.50000i	0.261116	+	0.452267i	0.966539	−	0.256520i	$-0.0825760\pi$
$11$	0.866025	+	1.50000i	0.261116	+	0.452267i	−0.705422	+	0.708787i	$0.749243\pi$
$12$		0			0
$13$	0.732051	−	1.26795i	0.203034	−	0.351666i	−0.746470	−	0.665419i	$-0.768253\pi$
$13$	0.732051	−	1.26795i	0.203034	−	0.351666i	0.949505	+	0.313753i	$0.101586\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.26795			−0.307523			−0.153761	−	0.988108i	$-0.549139\pi$
$17$	−1.26795			−0.307523			−0.153761	+	0.988108i	$0.549139\pi$
$18$		0			0
$19$	2.46410			0.565304			0.282652	−	0.959223i	$-0.408786\pi$
$19$	2.46410			0.565304			0.282652	+	0.959223i	$0.408786\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.73205	−	3.00000i	0.361158	−	0.625543i	−0.626994	−	0.779024i	$-0.715715\pi$
$23$	1.73205	−	3.00000i	0.361158	−	0.625543i	0.988152	+	0.153481i	$0.0490483\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.13397	+	3.69615i	0.396269	+	0.686358i	0.993262	−	0.115888i	$-0.0369714\pi$
$29$	2.13397	+	3.69615i	0.396269	+	0.686358i	−0.596993	+	0.802246i	$0.703638\pi$
$30$		0			0
$31$	3.96410	−	6.86603i	0.711974	−	1.23317i	−0.252142	−	0.967690i	$-0.581135\pi$
$31$	3.96410	−	6.86603i	0.711974	−	1.23317i	0.964115	−	0.265484i	$-0.0855318\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.732051			−0.123739
$36$		0			0
$37$	4.19615			0.689843			0.344922	−	0.938631i	$-0.387905\pi$
$37$	4.19615			0.689843			0.344922	+	0.938631i	$0.387905\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.401924	−	0.696152i	0.0627700	−	0.108721i	−0.832933	−	0.553374i	$-0.813340\pi$
$41$	0.401924	−	0.696152i	0.0627700	−	0.108721i	0.895703	+	0.444654i	$0.146673\pi$
$42$		0			0
$43$	−3.36603	−	5.83013i	−0.513314	−	0.889086i	−0.999881	−	0.0154426i	$-0.995084\pi$
$43$	−3.36603	−	5.83013i	−0.513314	−	0.889086i	0.486567	−	0.873643i	$-0.338249\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.36603	+	4.09808i	0.345120	+	0.597766i	0.985376	−	0.170396i	$-0.0545048\pi$
$47$	2.36603	+	4.09808i	0.345120	+	0.597766i	−0.640255	+	0.768162i	$0.721171\pi$
$48$		0			0
$49$	3.23205	−	5.59808i	0.461722	−	0.799725i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.7321			−1.47416			−0.737080	−	0.675805i	$-0.763796\pi$
$53$	−10.7321			−1.47416			−0.737080	+	0.675805i	$0.763796\pi$
$54$		0			0
$55$	1.73205			0.233550
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.13397	−	3.69615i	0.277820	−	0.481198i	−0.693023	−	0.720916i	$-0.743722\pi$
$59$	2.13397	−	3.69615i	0.277820	−	0.481198i	0.970843	+	0.239718i	$0.0770548\pi$
$60$		0			0
$61$	2.00000	+	3.46410i	0.256074	+	0.443533i	0.965187	−	0.261562i	$-0.0842377\pi$
$61$	2.00000	+	3.46410i	0.256074	+	0.443533i	−0.709113	+	0.705095i	$0.750904\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.732051	−	1.26795i	−0.0907997	−	0.157270i
$66$		0			0
$67$	7.19615	−	12.4641i	0.879150	−	1.52273i	0.0268747	−	0.999639i	$-0.491444\pi$
$67$	7.19615	−	12.4641i	0.879150	−	1.52273i	0.852275	−	0.523094i	$-0.175222\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.803848			−0.0953992			−0.0476996	−	0.998862i	$-0.515189\pi$
$71$	−0.803848			−0.0953992			−0.0476996	+	0.998862i	$0.515189\pi$
$72$		0			0
$73$	10.1962			1.19337			0.596685	−	0.802476i	$-0.296484\pi$
$73$	10.1962			1.19337			0.596685	+	0.802476i	$0.296484\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.633975	−	1.09808i	0.0722481	−	0.125137i
$78$		0			0
$79$	−3.19615	−	5.53590i	−0.359595	−	0.622837i	0.628298	−	0.777973i	$-0.283752\pi$
$79$	−3.19615	−	5.53590i	−0.359595	−	0.622837i	−0.987893	+	0.155136i	$0.950419\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.56218	−	7.90192i	−0.500764	−	0.867349i	−1.00000		0.000882500i	$-0.999719\pi$
$83$	−4.56218	−	7.90192i	−0.500764	−	0.867349i	0.499236	−	0.866466i	$-0.333614\pi$
$84$		0			0
$85$	−0.633975	+	1.09808i	−0.0687642	+	0.119103i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.19615			−0.550791			−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615			−0.550791			−0.275396	+	0.961331i	$0.588809\pi$
$90$		0			0
$91$	−1.07180			−0.112355
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.23205	−	2.13397i	0.126406	−	0.218941i
$96$		0			0
$97$	1.36603	+	2.36603i	0.138699	+	0.240233i	0.927004	−	0.375051i	$-0.122375\pi$
$97$	1.36603	+	2.36603i	0.138699	+	0.240233i	−0.788305	+	0.615284i	$0.789041\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.06218	+	15.6962i	0.901720	+	1.56183i	0.825260	+	0.564753i	$0.191029\pi$
$101$	9.06218	+	15.6962i	0.901720	+	1.56183i	0.0764604	+	0.997073i	$0.475638\pi$
$102$		0			0
$103$	1.19615	−	2.07180i	0.117860	−	0.204140i	−0.801059	−	0.598585i	$-0.795730\pi$
$103$	1.19615	−	2.07180i	0.117860	−	0.204140i	0.918920	+	0.394445i	$0.129063\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.46410			−0.334887			−0.167444	−	0.985882i	$-0.553551\pi$
$107$	−3.46410			−0.334887			−0.167444	+	0.985882i	$0.553551\pi$
$108$		0			0
$109$	5.92820			0.567819			0.283909	−	0.958851i	$-0.408368\pi$
$109$	5.92820			0.567819			0.283909	+	0.958851i	$0.408368\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.83013	−	15.2942i	0.830668	−	1.43876i	−0.0668404	−	0.997764i	$-0.521292\pi$
$113$	8.83013	−	15.2942i	0.830668	−	1.43876i	0.897509	−	0.440996i	$-0.145375\pi$
$114$		0			0
$115$	−1.73205	−	3.00000i	−0.161515	−	0.279751i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.464102	+	0.803848i	0.0425441	+	0.0736886i
$120$		0			0
$121$	4.00000	−	6.92820i	0.363636	−	0.629837i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−6.19615			−0.549820			−0.274910	−	0.961470i	$-0.588648\pi$
$127$	−6.19615			−0.549820			−0.274910	+	0.961470i	$0.588648\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.86603	+	11.8923i	−0.599887	+	1.03904i	0.392950	+	0.919560i	$0.371455\pi$
$131$	−6.86603	+	11.8923i	−0.599887	+	1.03904i	−0.992837	+	0.119476i	$0.961879\pi$
$132$		0			0
$133$	−0.901924	−	1.56218i	−0.0782067	−	0.135458i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.19615	+	14.1962i	0.700245	+	1.21286i	0.968380	+	0.249478i	$0.0802590\pi$
$137$	8.19615	+	14.1962i	0.700245	+	1.21286i	−0.268136	+	0.963381i	$0.586408\pi$
$138$		0			0
$139$	2.69615	−	4.66987i	0.228685	−	0.396093i	−0.728734	−	0.684797i	$-0.759891\pi$
$139$	2.69615	−	4.66987i	0.228685	−	0.396093i	0.957419	+	0.288704i	$0.0932242\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.53590			0.212062
$144$		0			0
$145$	4.26795			0.354434
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	−1.69615	−	2.93782i	−0.138031	−	0.239077i	0.788720	−	0.614752i	$-0.210744\pi$
$151$	−1.69615	−	2.93782i	−0.138031	−	0.239077i	−0.926751	+	0.375676i	$0.877411\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.96410	−	6.86603i	−0.318404	−	0.551492i
$156$		0			0
$157$	−3.36603	+	5.83013i	−0.268638	+	0.465295i	−0.968510	−	0.248973i	$-0.919907\pi$
$157$	−3.36603	+	5.83013i	−0.268638	+	0.465295i	0.699872	+	0.714268i	$0.253240\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.53590			−0.199857
$162$		0			0
$163$	15.2679			1.19588			0.597939	−	0.801542i	$-0.295986\pi$
$163$	15.2679			1.19588			0.597939	+	0.801542i	$0.295986\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.56218	+	2.70577i	−0.120885	+	0.209379i	−0.920117	−	0.391644i	$-0.871907\pi$
$167$	−1.56218	+	2.70577i	−0.120885	+	0.209379i	0.799232	+	0.601023i	$0.205240\pi$
$168$		0			0
$169$	5.42820	+	9.40192i	0.417554	+	0.723225i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.1244	−	21.0000i	−0.921798	−	1.59660i	−0.796632	−	0.604465i	$-0.793387\pi$
$173$	−12.1244	−	21.0000i	−0.921798	−	1.59660i	−0.125166	−	0.992136i	$-0.539946\pi$
$174$		0			0
$175$	−0.366025	+	0.633975i	−0.0276689	+	0.0479240i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.1244			0.906217			0.453108	−	0.891455i	$-0.350315\pi$
$179$	12.1244			0.906217			0.453108	+	0.891455i	$0.350315\pi$
$180$		0			0
$181$	−9.53590			−0.708798			−0.354399	−	0.935094i	$-0.615314\pi$
$181$	−9.53590			−0.708798			−0.354399	+	0.935094i	$0.615314\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.09808	−	3.63397i	0.154254	−	0.267175i
$186$		0			0
$187$	−1.09808	−	1.90192i	−0.0802993	−	0.139082i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.52628	+	16.5000i	0.689297	+	1.19390i	0.972065	+	0.234710i	$0.0754140\pi$
$191$	9.52628	+	16.5000i	0.689297	+	1.19390i	−0.282768	+	0.959188i	$0.591253\pi$
$192$		0			0
$193$	−10.2942	+	17.8301i	−0.740995	+	1.28344i	0.211048	+	0.977476i	$0.432312\pi$
$193$	−10.2942	+	17.8301i	−0.740995	+	1.28344i	−0.952043	+	0.305965i	$0.901021\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	13.8564			0.987228			0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564			0.987228			0.493614	+	0.869681i	$0.335676\pi$
$198$		0			0
$199$	−11.8564			−0.840478			−0.420239	−	0.907413i	$-0.638054\pi$
$199$	−11.8564			−0.840478			−0.420239	+	0.907413i	$0.638054\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.56218	−	2.70577i	0.109643	−	0.189908i
$204$		0			0
$205$	−0.401924	−	0.696152i	−0.0280716	−	0.0486214i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.13397	+	3.69615i	0.147610	+	0.255668i
$210$		0			0
$211$	3.03590	−	5.25833i	0.209000	−	0.361998i	−0.742400	−	0.669957i	$-0.766313\pi$
$211$	3.03590	−	5.25833i	0.209000	−	0.361998i	0.951400	+	0.307959i	$0.0996458\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.73205			−0.459122
$216$		0			0
$217$	−5.80385			−0.393991
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.928203	+	1.60770i	−0.0624377	+	0.108145i
$222$		0			0
$223$	−10.9282	−	18.9282i	−0.731807	−	1.26753i	−0.956110	−	0.293007i	$-0.905344\pi$
$223$	−10.9282	−	18.9282i	−0.731807	−	1.26753i	0.224304	−	0.974519i	$-0.427989\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.90192	−	13.6865i	−0.524469	−	0.908407i	−0.999594	−	0.0284888i	$-0.990930\pi$
$227$	−7.90192	−	13.6865i	−0.524469	−	0.908407i	0.475125	−	0.879918i	$-0.342403\pi$
$228$		0			0
$229$	−4.92820	+	8.53590i	−0.325665	+	0.564068i	−0.981647	−	0.190709i	$-0.938921\pi$
$229$	−4.92820	+	8.53590i	−0.325665	+	0.564068i	0.655982	+	0.754777i	$0.272255\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.58846			−0.431624			−0.215812	−	0.976435i	$-0.569240\pi$
$233$	−6.58846			−0.431624			−0.215812	+	0.976435i	$0.569240\pi$
$234$		0			0
$235$	4.73205			0.308685
$236$		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
$241$	12.1603	+	21.0622i	0.783311	+	1.35673i	0.930003	+	0.367552i	$0.119804\pi$
$241$	12.1603	+	21.0622i	0.783311	+	1.35673i	−0.146692	+	0.989182i	$0.546863\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.23205	−	5.59808i	−0.206488	−	0.357648i
$246$		0			0
$247$	1.80385	−	3.12436i	0.114776	−	0.198798i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	27.4641			1.73352			0.866759	−	0.498727i	$-0.166199\pi$
$251$	27.4641			1.73352			0.866759	+	0.498727i	$0.166199\pi$
$252$		0			0
$253$	6.00000			0.377217
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.26795	+	7.39230i	−0.266227	+	0.461119i	−0.967884	−	0.251395i	$-0.919111\pi$
$257$	−4.26795	+	7.39230i	−0.266227	+	0.461119i	0.701657	+	0.712515i	$0.252444\pi$
$258$		0			0
$259$	−1.53590	−	2.66025i	−0.0954361	−	0.165300i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.1244	−	21.0000i	−0.747620	−	1.29492i	−0.948961	−	0.315394i	$-0.897863\pi$
$263$	−12.1244	−	21.0000i	−0.747620	−	1.29492i	0.201341	−	0.979521i	$-0.435470\pi$
$264$		0			0
$265$	−5.36603	+	9.29423i	−0.329632	+	0.570940i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−19.7321			−1.20308			−0.601542	−	0.798841i	$-0.705447\pi$
$269$	−19.7321			−1.20308			−0.601542	+	0.798841i	$0.705447\pi$
$270$		0			0
$271$	−24.7846			−1.50556			−0.752779	−	0.658273i	$-0.771287\pi$
$271$	−24.7846			−1.50556			−0.752779	+	0.658273i	$0.771287\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.866025	−	1.50000i	0.0522233	−	0.0904534i
$276$		0			0
$277$	9.56218	+	16.5622i	0.574536	+	0.995125i	0.996092	+	0.0883226i	$0.0281506\pi$
$277$	9.56218	+	16.5622i	0.574536	+	0.995125i	−0.421556	+	0.906802i	$0.638516\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.66025	+	4.60770i	0.158697	+	0.274872i	0.934399	−	0.356228i	$-0.115937\pi$
$281$	2.66025	+	4.60770i	0.158697	+	0.274872i	−0.775702	+	0.631100i	$0.782604\pi$
$282$		0			0
$283$	−11.7321	+	20.3205i	−0.697398	+	1.20793i	0.271968	+	0.962306i	$0.412326\pi$
$283$	−11.7321	+	20.3205i	−0.697398	+	1.20793i	−0.969366	+	0.245622i	$0.921008\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.588457			−0.0347355
$288$		0			0
$289$	−15.3923			−0.905430
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.29423	−	5.70577i	0.192451	−	0.333335i	−0.753611	−	0.657321i	$-0.771690\pi$
$293$	3.29423	−	5.70577i	0.192451	−	0.333335i	0.946062	+	0.323986i	$0.105023\pi$
$294$		0			0
$295$	−2.13397	−	3.69615i	−0.124245	−	0.215198i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.53590	−	4.39230i	−0.146655	−	0.254014i
$300$		0			0
$301$	−2.46410	+	4.26795i	−0.142028	+	0.246001i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.00000			0.229039
$306$		0			0
$307$	−12.1962			−0.696071			−0.348036	−	0.937481i	$-0.613151\pi$
$307$	−12.1962			−0.696071			−0.348036	+	0.937481i	$0.613151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−14.2583	+	24.6962i	−0.808516	+	1.40039i	0.105376	+	0.994432i	$0.466395\pi$
$311$	−14.2583	+	24.6962i	−0.808516	+	1.40039i	−0.913892	+	0.405958i	$0.866938\pi$
$312$		0			0
$313$	−0.535898	−	0.928203i	−0.0302908	−	0.0524651i	0.850483	−	0.526003i	$-0.176310\pi$
$313$	−0.535898	−	0.928203i	−0.0302908	−	0.0524651i	−0.880773	+	0.473538i	$0.842977\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.5622	+	18.2942i	0.593231	+	1.02751i	0.993794	+	0.111237i	$0.0354813\pi$
$317$	10.5622	+	18.2942i	0.593231	+	1.02751i	−0.400563	+	0.916269i	$0.631185\pi$
$318$		0			0
$319$	−3.69615	+	6.40192i	−0.206945	+	0.358439i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.12436			−0.173844
$324$		0			0
$325$	−1.46410			−0.0812137
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.73205	−	3.00000i	0.0954911	−	0.165395i
$330$		0			0
$331$	4.30385	+	7.45448i	0.236561	+	0.409735i	0.959725	−	0.280941i	$-0.0906464\pi$
$331$	4.30385	+	7.45448i	0.236561	+	0.409735i	−0.723164	+	0.690676i	$0.757313\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.19615	−	12.4641i	−0.393168	−	0.680987i
$336$		0			0
$337$	−7.12436	+	12.3397i	−0.388088	+	0.672189i	−0.992192	−	0.124717i	$-0.960198\pi$
$337$	−7.12436	+	12.3397i	−0.388088	+	0.672189i	0.604104	+	0.796905i	$0.293531\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	13.7321			0.743632
$342$		0			0
$343$	−9.85641			−0.532196
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−16.0981	+	27.8827i	−0.864190	+	1.49682i	0.00365919	+	0.999993i	$0.498835\pi$
$347$	−16.0981	+	27.8827i	−0.864190	+	1.49682i	−0.867849	+	0.496828i	$0.834498\pi$
$348$		0			0
$349$	12.5000	+	21.6506i	0.669110	+	1.15893i	0.978153	+	0.207884i	$0.0666577\pi$
$349$	12.5000	+	21.6506i	0.669110	+	1.15893i	−0.309044	+	0.951048i	$0.600009\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.4904	−	19.9019i	−0.611571	−	1.05927i	−0.990976	−	0.134042i	$-0.957204\pi$
$353$	−11.4904	−	19.9019i	−0.611571	−	1.05927i	0.379404	−	0.925231i	$-0.376129\pi$
$354$		0			0
$355$	−0.401924	+	0.696152i	−0.0213319	+	0.0369479i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	26.6603			1.40707			0.703537	−	0.710658i	$-0.251603\pi$
$359$	26.6603			1.40707			0.703537	+	0.710658i	$0.251603\pi$
$360$		0			0
$361$	−12.9282			−0.680432
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.09808	−	8.83013i	0.266846	−	0.462190i
$366$		0			0
$367$	2.80385	+	4.85641i	0.146360	+	0.253502i	0.929879	−	0.367865i	$-0.119911\pi$
$367$	2.80385	+	4.85641i	0.146360	+	0.253502i	−0.783520	+	0.621367i	$0.786578\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.92820	+	6.80385i	0.203942	+	0.353238i
$372$		0			0
$373$	−12.0263	+	20.8301i	−0.622697	+	1.07854i	0.366284	+	0.930503i	$0.380630\pi$
$373$	−12.0263	+	20.8301i	−0.622697	+	1.07854i	−0.988981	+	0.148040i	$0.952704\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.24871			0.321825
$378$		0			0
$379$	11.4641			0.588871			0.294436	−	0.955671i	$-0.404868\pi$
$379$	11.4641			0.588871			0.294436	+	0.955671i	$0.404868\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	15.1244	−	26.1962i	0.772818	−	1.33856i	−0.163194	−	0.986594i	$-0.552180\pi$
$383$	15.1244	−	26.1962i	0.772818	−	1.33856i	0.936012	−	0.351967i	$-0.114487\pi$
$384$		0			0
$385$	−0.633975	−	1.09808i	−0.0323103	−	0.0559631i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.2679	−	17.7846i	−0.520606	−	0.901716i	−0.999713	−	0.0239591i	$-0.992373\pi$
$389$	−10.2679	−	17.7846i	−0.520606	−	0.901716i	0.479107	−	0.877756i	$-0.340960\pi$
$390$		0			0
$391$	−2.19615	+	3.80385i	−0.111064	+	0.192369i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.39230			−0.321632
$396$		0			0
$397$	2.67949			0.134480			0.0672399	−	0.997737i	$-0.478581\pi$
$397$	2.67949			0.134480			0.0672399	+	0.997737i	$0.478581\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−13.3923	+	23.1962i	−0.668780	+	1.15836i	0.309466	+	0.950911i	$0.399850\pi$
$401$	−13.3923	+	23.1962i	−0.668780	+	1.15836i	−0.978246	+	0.207450i	$0.933484\pi$
$402$		0			0
$403$	−5.80385	−	10.0526i	−0.289110	−	0.500754i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.63397	+	6.29423i	0.180129	+	0.311993i
$408$		0			0
$409$	−4.92820	+	8.53590i	−0.243684	+	0.422073i	−0.961761	−	0.273891i	$-0.911689\pi$
$409$	−4.92820	+	8.53590i	−0.243684	+	0.422073i	0.718077	+	0.695964i	$0.245023\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.12436			−0.153739
$414$		0			0
$415$	−9.12436			−0.447897
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.26795	+	12.5885i	−0.355063	+	0.614986i	−0.987129	−	0.159928i	$-0.948874\pi$
$419$	−7.26795	+	12.5885i	−0.355063	+	0.614986i	0.632066	+	0.774914i	$0.282207\pi$
$420$		0			0
$421$	13.8923	+	24.0622i	0.677070	+	1.17272i	0.975859	+	0.218400i	$0.0700837\pi$
$421$	13.8923	+	24.0622i	0.677070	+	1.17272i	−0.298790	+	0.954319i	$0.596583\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.633975	+	1.09808i	0.0307523	+	0.0532645i
$426$		0			0
$427$	1.46410	−	2.53590i	0.0708528	−	0.122721i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.41154			−0.116160			−0.0580800	−	0.998312i	$-0.518498\pi$
$431$	−2.41154			−0.116160			−0.0580800	+	0.998312i	$0.518498\pi$
$432$		0			0
$433$	4.53590			0.217981			0.108991	−	0.994043i	$-0.465238\pi$
$433$	4.53590			0.217981			0.108991	+	0.994043i	$0.465238\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.26795	−	7.39230i	0.204164	−	0.353622i
$438$		0			0
$439$	−13.6962	−	23.7224i	−0.653682	−	1.13221i	−0.982223	−	0.187720i	$-0.939890\pi$
$439$	−13.6962	−	23.7224i	−0.653682	−	1.13221i	0.328541	−	0.944490i	$-0.393443\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−20.3660	−	35.2750i	−0.967619	−	1.67597i	−0.702407	−	0.711775i	$-0.747891\pi$
$443$	−20.3660	−	35.2750i	−0.967619	−	1.67597i	−0.265212	−	0.964190i	$-0.585442\pi$
$444$		0			0
$445$	−2.59808	+	4.50000i	−0.123161	+	0.213320i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.124356			−0.00586871			−0.00293435	−	0.999996i	$-0.500934\pi$
$449$	−0.124356			−0.00586871			−0.00293435	+	0.999996i	$0.500934\pi$
$450$		0			0
$451$	1.39230			0.0655611
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.535898	+	0.928203i	−0.0251233	+	0.0435148i
$456$		0			0
$457$	−5.90192	−	10.2224i	−0.276080	−	0.478185i	0.694327	−	0.719660i	$-0.255702\pi$
$457$	−5.90192	−	10.2224i	−0.276080	−	0.478185i	−0.970407	+	0.241475i	$0.922369\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.7942	+	18.6962i	0.502737	+	0.870767i	0.999995	+	0.00316371i	$0.00100704\pi$
$461$	10.7942	+	18.6962i	0.502737	+	0.870767i	−0.497258	+	0.867603i	$0.665660\pi$
$462$		0			0
$463$	−9.19615	+	15.9282i	−0.427381	+	0.740246i	−0.996640	−	0.0819125i	$-0.973897\pi$
$463$	−9.19615	+	15.9282i	−0.427381	+	0.740246i	0.569258	+	0.822159i	$0.307231\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	15.8038			0.731315			0.365657	−	0.930750i	$-0.380844\pi$
$467$	15.8038			0.731315			0.365657	+	0.930750i	$0.380844\pi$
$468$		0			0
$469$	−10.5359			−0.486503
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.83013	−	10.0981i	0.268070	−	0.464310i
$474$		0			0
$475$	−1.23205	−	2.13397i	−0.0565304	−	0.0979135i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.3301	+	17.8923i	0.471996	+	0.817520i	0.999487	−	0.0320403i	$-0.0102005\pi$
$479$	10.3301	+	17.8923i	0.471996	+	0.817520i	−0.527491	+	0.849561i	$0.676867\pi$
$480$		0			0
$481$	3.07180	−	5.32051i	0.140062	−	0.242594i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.73205			0.124056
$486$		0			0
$487$	−37.4641			−1.69766			−0.848830	−	0.528666i	$-0.822693\pi$
$487$	−37.4641			−1.69766			−0.848830	+	0.528666i	$0.822693\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.86603	−	6.69615i	0.174471	−	0.302193i	−0.765507	−	0.643428i	$-0.777512\pi$
$491$	3.86603	−	6.69615i	0.174471	−	0.302193i	0.939978	+	0.341235i	$0.110845\pi$
$492$		0			0
$493$	−2.70577	−	4.68653i	−0.121862	−	0.211071i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.294229	+	0.509619i	0.0131980	+	0.0228595i
$498$		0			0
$499$	−3.30385	+	5.72243i	−0.147901	+	0.256171i	−0.930451	−	0.366416i	$-0.880585\pi$
$499$	−3.30385	+	5.72243i	−0.147901	+	0.256171i	0.782551	+	0.622587i	$0.213918\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.679492			−0.0302970			−0.0151485	−	0.999885i	$-0.504822\pi$
$503$	−0.679492			−0.0302970			−0.0151485	+	0.999885i	$0.504822\pi$
$504$		0			0
$505$	18.1244			0.806523
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.39230	+	12.8038i	−0.327658	+	0.567521i	−0.982047	−	0.188638i	$-0.939593\pi$
$509$	−7.39230	+	12.8038i	−0.327658	+	0.567521i	0.654389	+	0.756158i	$0.272926\pi$
$510$		0			0
$511$	−3.73205	−	6.46410i	−0.165096	−	0.285955i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.19615	−	2.07180i	−0.0527088	−	0.0912943i
$516$		0			0
$517$	−4.09808	+	7.09808i	−0.180233	+	0.312173i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10.3923			0.455295			0.227648	−	0.973744i	$-0.426897\pi$
$521$	10.3923			0.455295			0.227648	+	0.973744i	$0.426897\pi$
$522$		0			0
$523$	24.3923			1.06660			0.533301	−	0.845926i	$-0.320952\pi$
$523$	24.3923			1.06660			0.533301	+	0.845926i	$0.320952\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.02628	+	8.70577i	−0.218948	+	0.379229i
$528$		0			0
$529$	5.50000	+	9.52628i	0.239130	+	0.414186i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.588457	−	1.01924i	−0.0254889	−	0.0441481i
$534$		0			0
$535$	−1.73205	+	3.00000i	−0.0748831	+	0.129701i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	11.1962			0.482252
$540$		0			0
$541$	−4.46410			−0.191927			−0.0959634	−	0.995385i	$-0.530593\pi$
$541$	−4.46410			−0.191927			−0.0959634	+	0.995385i	$0.530593\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.96410	−	5.13397i	0.126968	−	0.219915i
$546$		0			0
$547$	18.3923	+	31.8564i	0.786398	+	1.36208i	0.928160	+	0.372181i	$0.121390\pi$
$547$	18.3923	+	31.8564i	0.786398	+	1.36208i	−0.141762	+	0.989901i	$0.545277\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.25833	+	9.10770i	0.224012	+	0.388001i
$552$		0			0
$553$	−2.33975	+	4.05256i	−0.0994961	+	0.172332i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.3923			0.694564			0.347282	−	0.937761i	$-0.387105\pi$
$557$	16.3923			0.694564			0.347282	+	0.937761i	$0.387105\pi$
$558$		0			0
$559$	−9.85641			−0.416882
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9.63397	+	16.6865i	−0.406024	+	0.703254i	−0.994440	−	0.105305i	$-0.966418\pi$
$563$	−9.63397	+	16.6865i	−0.406024	+	0.703254i	0.588416	+	0.808558i	$0.299752\pi$
$564$		0			0
$565$	−8.83013	−	15.2942i	−0.371486	−	0.643433i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.47372	+	4.28461i	0.103704	+	0.179620i	0.913208	−	0.407494i	$-0.133597\pi$
$569$	2.47372	+	4.28461i	0.103704	+	0.179620i	−0.809504	+	0.587114i	$0.800264\pi$
$570$		0			0
$571$	13.4282	−	23.2583i	0.561953	−	0.973331i	−0.435373	−	0.900250i	$-0.643384\pi$
$571$	13.4282	−	23.2583i	0.561953	−	0.973331i	0.997326	−	0.0730808i	$-0.0232831\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.46410			−0.144463
$576$		0			0
$577$	22.1962			0.924038			0.462019	−	0.886870i	$-0.347125\pi$
$577$	22.1962			0.924038			0.462019	+	0.886870i	$0.347125\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.33975	+	5.78461i	−0.138556	+	0.239986i
$582$		0			0
$583$	−9.29423	−	16.0981i	−0.384928	−	0.666714i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.09808	−	12.2942i	−0.292969	−	0.507437i	0.681541	−	0.731780i	$-0.261310\pi$
$587$	−7.09808	−	12.2942i	−0.292969	−	0.507437i	−0.974510	+	0.224342i	$0.927977\pi$
$588$		0			0
$589$	9.76795	−	16.9186i	0.402481	−	0.697118i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−36.9282			−1.51646			−0.758230	−	0.651987i	$-0.773935\pi$
$593$	−36.9282			−1.51646			−0.758230	+	0.651987i	$0.773935\pi$
$594$		0			0
$595$	0.928203			0.0380526
$596$		0			0
$597$		0			0
$598$		0			0
$599$	22.7942	−	39.4808i	0.931347	−	1.61314i	0.150325	−	0.988637i	$-0.451968\pi$
$599$	22.7942	−	39.4808i	0.931347	−	1.61314i	0.781022	−	0.624504i	$-0.214699\pi$
$600$		0			0
$601$	−5.16025	−	8.93782i	−0.210491	−	0.364581i	0.741377	−	0.671089i	$-0.234173\pi$
$601$	−5.16025	−	8.93782i	−0.210491	−	0.364581i	−0.951868	+	0.306507i	$0.900840\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.00000	−	6.92820i	−0.162623	−	0.281672i
$606$		0			0
$607$	14.2942	−	24.7583i	0.580185	−	1.00491i	−0.415272	−	0.909697i	$-0.636314\pi$
$607$	14.2942	−	24.7583i	0.580185	−	1.00491i	0.995457	−	0.0952124i	$-0.0303530\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.92820			0.280285
$612$		0			0
$613$	3.60770			0.145713			0.0728567	−	0.997342i	$-0.476788\pi$
$613$	3.60770			0.145713			0.0728567	+	0.997342i	$0.476788\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.00000	−	10.3923i	0.241551	−	0.418378i	−0.719605	−	0.694383i	$-0.755677\pi$
$617$	6.00000	−	10.3923i	0.241551	−	0.418378i	0.961156	+	0.276005i	$0.0890106\pi$
$618$		0			0
$619$	5.00000	+	8.66025i	0.200967	+	0.348085i	0.948840	−	0.315757i	$-0.102258\pi$
$619$	5.00000	+	8.66025i	0.200967	+	0.348085i	−0.747873	+	0.663842i	$0.768925\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.90192	+	3.29423i	0.0761990	+	0.131980i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.32051			−0.212143
$630$		0			0
$631$	−13.9282			−0.554473			−0.277237	−	0.960802i	$-0.589419\pi$
$631$	−13.9282			−0.554473			−0.277237	+	0.960802i	$0.589419\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.09808	+	5.36603i	−0.122943	+	0.212944i
$636$		0			0
$637$	−4.73205	−	8.19615i	−0.187491	−	0.324743i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	11.5981	+	20.0885i	0.458096	+	0.793446i	0.998860	−	0.0477281i	$-0.0151981\pi$
$641$	11.5981	+	20.0885i	0.458096	+	0.793446i	−0.540764	+	0.841174i	$0.681865\pi$
$642$		0			0
$643$	8.29423	−	14.3660i	0.327092	−	0.566541i	−0.654841	−	0.755767i	$-0.727264\pi$
$643$	8.29423	−	14.3660i	0.327092	−	0.566541i	0.981934	+	0.189226i	$0.0605978\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−48.2487			−1.89685			−0.948426	−	0.316998i	$-0.897325\pi$
$647$	−48.2487			−1.89685			−0.948426	+	0.316998i	$0.897325\pi$
$648$		0			0
$649$	7.39230			0.290173
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.73205	−	8.19615i	0.185179	−	0.320740i	−0.758458	−	0.651722i	$-0.774047\pi$
$653$	4.73205	−	8.19615i	0.185179	−	0.320740i	0.943637	+	0.330982i	$0.107380\pi$
$654$		0			0
$655$	6.86603	+	11.8923i	0.268278	+	0.464671i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.73205	+	8.19615i	0.184335	+	0.319277i	0.943352	−	0.331793i	$-0.107654\pi$
$659$	4.73205	+	8.19615i	0.184335	+	0.319277i	−0.759018	+	0.651070i	$0.774320\pi$
$660$		0			0
$661$	2.69615	−	4.66987i	0.104868	−	0.181637i	−0.808816	−	0.588062i	$-0.799891\pi$
$661$	2.69615	−	4.66987i	0.104868	−	0.181637i	0.913684	+	0.406425i	$0.133225\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.80385			−0.0699502
$666$		0			0
$667$	14.7846			0.572462
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.46410	+	6.00000i	−0.133730	+	0.231627i
$672$		0			0
$673$	8.80385	+	15.2487i	0.339363	+	0.587795i	0.984313	−	0.176430i	$-0.0564550\pi$
$673$	8.80385	+	15.2487i	0.339363	+	0.587795i	−0.644950	+	0.764225i	$0.723122\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.3205	+	24.8038i	0.550382	+	0.953289i	0.998247	+	0.0591881i	$0.0188512\pi$
$677$	14.3205	+	24.8038i	0.550382	+	0.953289i	−0.447865	+	0.894101i	$0.647815\pi$
$678$		0			0
$679$	1.00000	−	1.73205i	0.0383765	−	0.0664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	14.5359			0.556201			0.278100	−	0.960552i	$-0.410295\pi$
$683$	14.5359			0.556201			0.278100	+	0.960552i	$0.410295\pi$
$684$		0			0
$685$	16.3923			0.626318
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.85641	+	13.6077i	−0.299305	+	0.518412i
$690$		0			0
$691$	5.00000	+	8.66025i	0.190209	+	0.329452i	0.945319	−	0.326146i	$-0.105750\pi$
$691$	5.00000	+	8.66025i	0.190209	+	0.329452i	−0.755110	+	0.655598i	$0.772417\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.69615	−	4.66987i	−0.102271	−	0.177138i
$696$		0			0
$697$	−0.509619	+	0.882686i	−0.0193032	+	0.0334341i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−42.1244			−1.59101			−0.795507	−	0.605944i	$-0.792796\pi$
$701$	−42.1244			−1.59101			−0.795507	+	0.605944i	$0.792796\pi$
$702$		0			0
$703$	10.3397			0.389971
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.63397	−	11.4904i	0.249496	−	0.432140i
$708$		0			0
$709$	−8.73205	−	15.1244i	−0.327939	−	0.568007i	0.654164	−	0.756353i	$-0.273021\pi$
$709$	−8.73205	−	15.1244i	−0.327939	−	0.568007i	−0.982103	+	0.188346i	$0.939687\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13.7321	−	23.7846i	−0.514269	−	0.890741i
$714$		0			0
$715$	1.26795	−	2.19615i	0.0474186	−	0.0821314i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.80385			−0.253741			−0.126870	−	0.991919i	$-0.540493\pi$
$719$	−6.80385			−0.253741			−0.126870	+	0.991919i	$0.540493\pi$
$720$		0			0
$721$	−1.75129			−0.0652214
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.13397	−	3.69615i	0.0792538	−	0.137272i
$726$		0			0
$727$	−16.5885	−	28.7321i	−0.615232	−	1.06561i	−0.990344	−	0.138633i	$-0.955729\pi$
$727$	−16.5885	−	28.7321i	−0.615232	−	1.06561i	0.375112	−	0.926980i	$-0.377604\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.26795	+	7.39230i	0.157856	+	0.273414i
$732$		0			0
$733$	19.7846	−	34.2679i	0.730761	−	1.26572i	−0.225797	−	0.974174i	$-0.572499\pi$
$733$	19.7846	−	34.2679i	0.730761	−	1.26572i	0.956558	−	0.291541i	$-0.0941680\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.9282			0.918242
$738$		0			0
$739$	36.1769			1.33079			0.665395	−	0.746492i	$-0.268263\pi$
$739$	36.1769			1.33079			0.665395	+	0.746492i	$0.268263\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.2942	−	31.6865i	0.671150	−	1.16247i	−0.306428	−	0.951894i	$-0.599134\pi$
$743$	18.2942	−	31.6865i	0.671150	−	1.16247i	0.977578	−	0.210572i	$-0.0675328\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.26795	+	2.19615i	0.0463299	+	0.0802457i
$750$		0			0
$751$	0.392305	−	0.679492i	0.0143154	−	0.0247950i	−0.858779	−	0.512346i	$-0.828776\pi$
$751$	0.392305	−	0.679492i	0.0143154	−	0.0247950i	0.873094	+	0.487551i	$0.162110\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.39230			−0.123459
$756$		0			0
$757$	0.392305			0.0142586			0.00712928	−	0.999975i	$-0.497731\pi$
$757$	0.392305			0.0142586			0.00712928	+	0.999975i	$0.497731\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.93782	−	5.08846i	0.106496	−	0.184456i	−0.807852	−	0.589385i	$-0.799370\pi$
$761$	2.93782	−	5.08846i	0.106496	−	0.184456i	0.914348	+	0.404928i	$0.132704\pi$
$762$		0			0
$763$	−2.16987	−	3.75833i	−0.0785547	−	0.136061i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.12436	−	5.41154i	−0.112814	−	0.195399i
$768$		0			0
$769$	6.62436	−	11.4737i	0.238880	−	0.413753i	−0.721513	−	0.692401i	$-0.756553\pi$
$769$	6.62436	−	11.4737i	0.238880	−	0.413753i	0.960393	+	0.278648i	$0.0898863\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.339746			0.0122198			0.00610991	−	0.999981i	$-0.498055\pi$
$773$	0.339746			0.0122198			0.00610991	+	0.999981i	$0.498055\pi$
$774$		0			0
$775$	−7.92820			−0.284789
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.990381	−	1.71539i	0.0354841	−	0.0614602i
$780$		0			0
$781$	−0.696152	−	1.20577i	−0.0249103	−	0.0431459i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3.36603	+	5.83013i	0.120139	+	0.208086i
$786$		0			0
$787$	12.9019	−	22.3468i	0.459904	−	0.796577i	−0.539051	−	0.842273i	$-0.681217\pi$
$787$	12.9019	−	22.3468i	0.459904	−	0.796577i	0.998955	+	0.0456959i	$0.0145505\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.9282			−0.459674
$792$		0			0
$793$	5.85641			0.207967
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.66025	−	9.80385i	0.200496	−	0.347270i	−0.748192	−	0.663482i	$-0.769078\pi$
$797$	5.66025	−	9.80385i	0.200496	−	0.347270i	0.948688

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(-0.366025 - 0.633975i) q^{7} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{5} +(-0.366025 - 0.633975i) q^{7} +(0.866025 + 1.50000i) q^{11} +(0.732051 - 1.26795i) q^{13} -1.26795 q^{17} +2.46410 q^{19} +(1.73205 - 3.00000i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(2.13397 + 3.69615i) q^{29} +(3.96410 - 6.86603i) q^{31} -0.732051 q^{35} +4.19615 q^{37} +(0.401924 - 0.696152i) q^{41} +(-3.36603 - 5.83013i) q^{43} +(2.36603 + 4.09808i) q^{47} +(3.23205 - 5.59808i) q^{49} -10.7321 q^{53} +1.73205 q^{55} +(2.13397 - 3.69615i) q^{59} +(2.00000 + 3.46410i) q^{61} +(-0.732051 - 1.26795i) q^{65} +(7.19615 - 12.4641i) q^{67} -0.803848 q^{71} +10.1962 q^{73} +(0.633975 - 1.09808i) q^{77} +(-3.19615 - 5.53590i) q^{79} +(-4.56218 - 7.90192i) q^{83} +(-0.633975 + 1.09808i) q^{85} -5.19615 q^{89} -1.07180 q^{91} +(1.23205 - 2.13397i) q^{95} +(1.36603 + 2.36603i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^5 + 2 * q^7	\( 4 q + 2 q^{5} + 2 q^{7} - 4 q^{13} - 12 q^{17} - 4 q^{19} - 2 q^{25} + 12 q^{29} + 2 q^{31} + 4 q^{35} - 4 q^{37} + 12 q^{41} - 10 q^{43} + 6 q^{47} + 6 q^{49} - 36 q^{53} + 12 q^{59} + 8 q^{61} + 4 q^{65} + 8 q^{67} - 24 q^{71} + 20 q^{73} + 6 q^{77} + 8 q^{79} + 6 q^{83} - 6 q^{85} - 32 q^{91} - 2 q^{95} + 2 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 + 2 * q^7 - 4 * q^13 - 12 * q^17 - 4 * q^19 - 2 * q^25 + 12 * q^29 + 2 * q^31 + 4 * q^35 - 4 * q^37 + 12 * q^41 - 10 * q^43 + 6 * q^47 + 6 * q^49 - 36 * q^53 + 12 * q^59 + 8 * q^61 + 4 * q^65 + 8 * q^67 - 24 * q^71 + 20 * q^73 + 6 * q^77 + 8 * q^79 + 6 * q^83 - 6 * q^85 - 32 * q^91 - 2 * q^95 + 2 * q^97

Introduction

L-functions

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