LMFDB - Embedded newform 1520.2.q.a.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(881,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("1520.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.q (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:	$12.1372611072$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1520.961
Dual form			1520.2.q.a.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{5} -4.00000 q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-1.00000 + 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{5} -4.00000 q^{7} +(-0.500000 - 0.866025i) q^{9} -1.00000 q^{11} +(2.00000 + 3.46410i) q^{13} +(-1.00000 - 1.73205i) q^{15} +(-2.00000 + 3.46410i) q^{17} +(-3.50000 - 2.59808i) q^{19} +(4.00000 - 6.92820i) q^{21} +(3.00000 + 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} -4.00000 q^{27} +(-2.50000 - 4.33013i) q^{29} +1.00000 q^{31} +(1.00000 - 1.73205i) q^{33} +(2.00000 - 3.46410i) q^{35} -2.00000 q^{37} -8.00000 q^{39} +(3.00000 - 5.19615i) q^{41} +(3.00000 - 5.19615i) q^{43} +1.00000 q^{45} +9.00000 q^{49} +(-4.00000 - 6.92820i) q^{51} +(-5.00000 - 8.66025i) q^{53} +(0.500000 - 0.866025i) q^{55} +(8.00000 - 3.46410i) q^{57} +(1.50000 - 2.59808i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(2.00000 + 3.46410i) q^{63} -4.00000 q^{65} +(2.00000 + 3.46410i) q^{67} -12.0000 q^{69} +(-6.50000 + 11.2583i) q^{71} +(3.00000 - 5.19615i) q^{73} +2.00000 q^{75} +4.00000 q^{77} +(5.50000 - 9.52628i) q^{79} +(5.50000 - 9.52628i) q^{81} +14.0000 q^{83} +(-2.00000 - 3.46410i) q^{85} +10.0000 q^{87} +(3.50000 + 6.06218i) q^{89} +(-8.00000 - 13.8564i) q^{91} +(-1.00000 + 1.73205i) q^{93} +(4.00000 - 1.73205i) q^{95} +(3.00000 - 5.19615i) q^{97} +(0.500000 + 0.866025i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} - 8 q^{7} - q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - q^5 - 8 * q^7 - q^9	$ 2 q - 2 q^{3} - q^{5} - 8 q^{7} - q^{9} - 2 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{17} - 7 q^{19} + 8 q^{21} + 6 q^{23} - q^{25} - 8 q^{27} - 5 q^{29} + 2 q^{31} + 2 q^{33} + 4 q^{35} - 4 q^{37} - 16 q^{39} + 6 q^{41} + 6 q^{43} + 2 q^{45} + 18 q^{49} - 8 q^{51} - 10 q^{53} + q^{55} + 16 q^{57} + 3 q^{59} - 5 q^{61} + 4 q^{63} - 8 q^{65} + 4 q^{67} - 24 q^{69} - 13 q^{71} + 6 q^{73} + 4 q^{75} + 8 q^{77} + 11 q^{79} + 11 q^{81} + 28 q^{83} - 4 q^{85} + 20 q^{87} + 7 q^{89} - 16 q^{91} - 2 q^{93} + 8 q^{95} + 6 q^{97} + q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 - 8 * q^7 - q^9 - 2 * q^11 + 4 * q^13 - 2 * q^15 - 4 * q^17 - 7 * q^19 + 8 * q^21 + 6 * q^23 - q^25 - 8 * q^27 - 5 * q^29 + 2 * q^31 + 2 * q^33 + 4 * q^35 - 4 * q^37 - 16 * q^39 + 6 * q^41 + 6 * q^43 + 2 * q^45 + 18 * q^49 - 8 * q^51 - 10 * q^53 + q^55 + 16 * q^57 + 3 * q^59 - 5 * q^61 + 4 * q^63 - 8 * q^65 + 4 * q^67 - 24 * q^69 - 13 * q^71 + 6 * q^73 + 4 * q^75 + 8 * q^77 + 11 * q^79 + 11 * q^81 + 28 * q^83 - 4 * q^85 + 20 * q^87 + 7 * q^89 - 16 * q^91 - 2 * q^93 + 8 * q^95 + 6 * q^97 + q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$3$	−1.00000	+	1.73205i	−0.577350	+	1.00000i	0.418432	+	0.908248i	$0.362580\pi$
$3$	−1.00000	+	1.73205i	−0.577350	+	1.00000i	−0.995782	+	0.0917517i	$0.970753\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	−4.00000			−1.51186			−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000			−1.51186			−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−1.00000			−0.301511			−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000			−0.301511			−0.150756	+	0.988571i	$0.548171\pi$
$12$		0			0
$13$	2.00000	+	3.46410i	0.554700	+	0.960769i	0.997927	+	0.0643593i	$0.0205004\pi$
$13$	2.00000	+	3.46410i	0.554700	+	0.960769i	−0.443227	+	0.896410i	$0.646166\pi$
$14$		0			0
$15$	−1.00000	−	1.73205i	−0.258199	−	0.447214i
$16$		0			0
$17$	−2.00000	+	3.46410i	−0.485071	+	0.840168i	−0.999853	−	0.0171533i	$-0.994540\pi$
$17$	−2.00000	+	3.46410i	−0.485071	+	0.840168i	0.514782	+	0.857321i	$0.327873\pi$
$18$		0			0
$19$	−3.50000	−	2.59808i	−0.802955	−	0.596040i
$19$	−3.50000	−	2.59808i	−0.802955	−	0.596040i
$20$		0			0
$21$	4.00000	−	6.92820i	0.872872	−	1.51186i
$22$		0			0
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	−4.00000			−0.769800
$28$		0			0
$29$	−2.50000	−	4.33013i	−0.464238	−	0.804084i	0.534928	−	0.844897i	$-0.320339\pi$
$29$	−2.50000	−	4.33013i	−0.464238	−	0.804084i	−0.999167	+	0.0408130i	$0.987005\pi$
$30$		0			0
$31$	1.00000			0.179605			0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000			0.179605			0.0898027	+	0.995960i	$0.471376\pi$
$32$		0			0
$33$	1.00000	−	1.73205i	0.174078	−	0.301511i
$34$		0			0
$35$	2.00000	−	3.46410i	0.338062	−	0.585540i
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$	−8.00000			−1.28103
$40$		0			0
$41$	3.00000	−	5.19615i	0.468521	−	0.811503i	−0.530831	−	0.847477i	$-0.678120\pi$
$41$	3.00000	−	5.19615i	0.468521	−	0.811503i	0.999353	+	0.0359748i	$0.0114536\pi$
$42$		0			0
$43$	3.00000	−	5.19615i	0.457496	−	0.792406i	−0.541332	−	0.840809i	$-0.682080\pi$
$43$	3.00000	−	5.19615i	0.457496	−	0.792406i	0.998828	+	0.0484030i	$0.0154132\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	9.00000			1.28571
$50$		0			0
$51$	−4.00000	−	6.92820i	−0.560112	−	0.970143i
$52$		0			0
$53$	−5.00000	−	8.66025i	−0.686803	−	1.18958i	−0.972867	−	0.231367i	$-0.925680\pi$
$53$	−5.00000	−	8.66025i	−0.686803	−	1.18958i	0.286064	−	0.958211i	$-0.407653\pi$
$54$		0			0
$55$	0.500000	−	0.866025i	0.0674200	−	0.116775i
$56$		0			0
$57$	8.00000	−	3.46410i	1.05963	−	0.458831i
$58$		0			0
$59$	1.50000	−	2.59808i	0.195283	−	0.338241i	−0.751710	−	0.659494i	$-0.770771\pi$
$59$	1.50000	−	2.59808i	0.195283	−	0.338241i	0.946993	+	0.321253i	$0.104104\pi$
$60$		0			0
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	0.660415	−	0.750901i	$-0.270381\pi$
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	−0.980507	+	0.196485i	$0.937047\pi$
$62$		0			0
$63$	2.00000	+	3.46410i	0.251976	+	0.436436i
$64$		0			0
$65$	−4.00000			−0.496139
$66$		0			0
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	0.961946	−	0.273241i	$-0.0880957\pi$
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	−0.717607	+	0.696449i	$0.754762\pi$
$68$		0			0
$69$	−12.0000			−1.44463
$70$		0			0
$71$	−6.50000	+	11.2583i	−0.771408	+	1.33612i	0.165383	+	0.986229i	$0.447114\pi$
$71$	−6.50000	+	11.2583i	−0.771408	+	1.33612i	−0.936791	+	0.349889i	$0.886219\pi$
$72$		0			0
$73$	3.00000	−	5.19615i	0.351123	−	0.608164i	−0.635323	−	0.772246i	$-0.719133\pi$
$73$	3.00000	−	5.19615i	0.351123	−	0.608164i	0.986447	+	0.164083i	$0.0524664\pi$
$74$		0			0
$75$	2.00000			0.230940
$76$		0			0
$77$	4.00000			0.455842
$78$		0			0
$79$	5.50000	−	9.52628i	0.618798	−	1.07179i	−0.370907	−	0.928670i	$-0.620953\pi$
$79$	5.50000	−	9.52628i	0.618798	−	1.07179i	0.989705	−	0.143120i	$-0.0457135\pi$
$80$		0			0
$81$	5.50000	−	9.52628i	0.611111	−	1.05848i
$82$		0			0
$83$	14.0000			1.53670			0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000			1.53670			0.768350	+	0.640030i	$0.221078\pi$
$84$		0			0
$85$	−2.00000	−	3.46410i	−0.216930	−	0.375735i
$86$		0			0
$87$	10.0000			1.07211
$88$		0			0
$89$	3.50000	+	6.06218i	0.370999	+	0.642590i	0.989720	−	0.143022i	$-0.0456819\pi$
$89$	3.50000	+	6.06218i	0.370999	+	0.642590i	−0.618720	+	0.785611i	$0.712349\pi$
$90$		0			0
$91$	−8.00000	−	13.8564i	−0.838628	−	1.45255i
$92$		0			0
$93$	−1.00000	+	1.73205i	−0.103695	+	0.179605i
$94$		0			0
$95$	4.00000	−	1.73205i	0.410391	−	0.177705i
$96$		0			0
$97$	3.00000	−	5.19615i	0.304604	−	0.527589i	−0.672569	−	0.740034i	$-0.734809\pi$
$97$	3.00000	−	5.19615i	0.304604	−	0.527589i	0.977173	+	0.212445i	$0.0681426\pi$
$98$		0			0
$99$	0.500000	+	0.866025i	0.0502519	+	0.0870388i
$100$		0			0
$101$	2.50000	+	4.33013i	0.248759	+	0.430864i	0.963182	−	0.268851i	$-0.0866439\pi$
$101$	2.50000	+	4.33013i	0.248759	+	0.430864i	−0.714423	+	0.699715i	$0.753311\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$	4.00000	+	6.92820i	0.390360	+	0.676123i
$106$		0			0
$107$	−6.00000			−0.580042			−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000			−0.580042			−0.290021	+	0.957020i	$0.593662\pi$
$108$		0			0
$109$	−9.50000	+	16.4545i	−0.909935	+	1.57605i	−0.0957826	+	0.995402i	$0.530535\pi$
$109$	−9.50000	+	16.4545i	−0.909935	+	1.57605i	−0.814152	+	0.580651i	$0.802798\pi$
$110$		0			0
$111$	2.00000	−	3.46410i	0.189832	−	0.328798i
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	−6.00000			−0.559503
$116$		0			0
$117$	2.00000	−	3.46410i	0.184900	−	0.320256i
$118$		0			0
$119$	8.00000	−	13.8564i	0.733359	−	1.27021i
$120$		0			0
$121$	−10.0000			−0.909091
$122$		0			0
$123$	6.00000	+	10.3923i	0.541002	+	0.937043i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	6.00000	+	10.3923i	0.532414	+	0.922168i	0.999284	+	0.0378419i	$0.0120483\pi$
$127$	6.00000	+	10.3923i	0.532414	+	0.922168i	−0.466870	+	0.884326i	$0.654618\pi$
$128$		0			0
$129$	6.00000	+	10.3923i	0.528271	+	0.914991i
$130$		0			0
$131$	6.00000	−	10.3923i	0.524222	−	0.907980i	−0.475380	−	0.879781i	$-0.657689\pi$
$131$	6.00000	−	10.3923i	0.524222	−	0.907980i	0.999602	−	0.0281993i	$-0.00897729\pi$
$132$		0			0
$133$	14.0000	+	10.3923i	1.21395	+	0.901127i
$134$		0			0
$135$	2.00000	−	3.46410i	0.172133	−	0.298142i
$136$		0			0
$137$	−10.0000	−	17.3205i	−0.854358	−	1.47979i	−0.877240	−	0.480053i	$-0.840618\pi$
$137$	−10.0000	−	17.3205i	−0.854358	−	1.47979i	0.0228820	−	0.999738i	$-0.492716\pi$
$138$		0			0
$139$	−6.00000	−	10.3923i	−0.508913	−	0.881464i	−0.999947	−	0.0103230i	$-0.996714\pi$
$139$	−6.00000	−	10.3923i	−0.508913	−	0.881464i	0.491033	−	0.871141i	$-0.336619\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.00000	−	3.46410i	−0.167248	−	0.289683i
$144$		0			0
$145$	5.00000			0.415227
$146$		0			0
$147$	−9.00000	+	15.5885i	−0.742307	+	1.28571i
$148$		0			0
$149$	−3.50000	+	6.06218i	−0.286731	+	0.496633i	−0.973028	−	0.230689i	$-0.925902\pi$
$149$	−3.50000	+	6.06218i	−0.286731	+	0.496633i	0.686296	+	0.727322i	$0.259235\pi$
$150$		0			0
$151$	−3.00000			−0.244137			−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000			−0.244137			−0.122068	+	0.992522i	$0.538953\pi$
$152$		0			0
$153$	4.00000			0.323381
$154$		0			0
$155$	−0.500000	+	0.866025i	−0.0401610	+	0.0695608i
$156$		0			0
$157$	−11.0000	+	19.0526i	−0.877896	+	1.52056i	−0.0242497	+	0.999706i	$0.507720\pi$
$157$	−11.0000	+	19.0526i	−0.877896	+	1.52056i	−0.853646	+	0.520854i	$0.825614\pi$
$158$		0			0
$159$	20.0000			1.58610
$160$		0			0
$161$	−12.0000	−	20.7846i	−0.945732	−	1.63806i
$162$		0			0
$163$	−18.0000			−1.40987			−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000			−1.40987			−0.704934	+	0.709273i	$0.749024\pi$
$164$		0			0
$165$	1.00000	+	1.73205i	0.0778499	+	0.134840i
$166$		0			0
$167$	−10.0000	−	17.3205i	−0.773823	−	1.34030i	−0.935454	−	0.353450i	$-0.885009\pi$
$167$	−10.0000	−	17.3205i	−0.773823	−	1.34030i	0.161630	−	0.986851i	$-0.448325\pi$
$168$		0			0
$169$	−1.50000	+	2.59808i	−0.115385	+	0.199852i
$170$		0			0
$171$	−0.500000	+	4.33013i	−0.0382360	+	0.331133i
$172$		0			0
$173$	−4.00000	+	6.92820i	−0.304114	+	0.526742i	−0.977064	−	0.212947i	$-0.931694\pi$
$173$	−4.00000	+	6.92820i	−0.304114	+	0.526742i	0.672949	+	0.739689i	$0.265027\pi$
$174$		0			0
$175$	2.00000	+	3.46410i	0.151186	+	0.261861i
$176$		0			0
$177$	3.00000	+	5.19615i	0.225494	+	0.390567i
$178$		0			0
$179$	−21.0000			−1.56961			−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000			−1.56961			−0.784807	+	0.619740i	$0.787238\pi$
$180$		0			0
$181$	−11.0000	−	19.0526i	−0.817624	−	1.41617i	−0.907429	−	0.420206i	$-0.861958\pi$
$181$	−11.0000	−	19.0526i	−0.817624	−	1.41617i	0.0898051	−	0.995959i	$-0.471376\pi$
$182$		0			0
$183$	10.0000			0.739221
$184$		0			0
$185$	1.00000	−	1.73205i	0.0735215	−	0.127343i
$186$		0			0
$187$	2.00000	−	3.46410i	0.146254	−	0.253320i
$188$		0			0
$189$	16.0000			1.16383
$190$		0			0
$191$	5.00000			0.361787			0.180894	−	0.983503i	$-0.442101\pi$
$191$	5.00000			0.361787			0.180894	+	0.983503i	$0.442101\pi$
$192$		0			0
$193$	4.00000	−	6.92820i	0.287926	−	0.498703i	−0.685388	−	0.728178i	$-0.740368\pi$
$193$	4.00000	−	6.92820i	0.287926	−	0.498703i	0.973315	+	0.229475i	$0.0737008\pi$
$194$		0			0
$195$	4.00000	−	6.92820i	0.286446	−	0.496139i
$196$		0			0
$197$	−16.0000			−1.13995			−0.569976	−	0.821661i	$-0.693048\pi$
$197$	−16.0000			−1.13995			−0.569976	+	0.821661i	$0.693048\pi$
$198$		0			0
$199$	−2.50000	−	4.33013i	−0.177220	−	0.306955i	0.763707	−	0.645563i	$-0.223377\pi$
$199$	−2.50000	−	4.33013i	−0.177220	−	0.306955i	−0.940927	+	0.338608i	$0.890044\pi$
$200$		0			0
$201$	−8.00000			−0.564276
$202$		0			0
$203$	10.0000	+	17.3205i	0.701862	+	1.21566i
$204$		0			0
$205$	3.00000	+	5.19615i	0.209529	+	0.362915i
$206$		0			0
$207$	3.00000	−	5.19615i	0.208514	−	0.361158i
$208$		0			0
$209$	3.50000	+	2.59808i	0.242100	+	0.179713i
$210$		0			0
$211$	11.5000	−	19.9186i	0.791693	−	1.37125i	−0.133226	−	0.991086i	$-0.542533\pi$
$211$	11.5000	−	19.9186i	0.791693	−	1.37125i	0.924918	−	0.380166i	$-0.124133\pi$
$212$		0			0
$213$	−13.0000	−	22.5167i	−0.890745	−	1.54282i
$214$		0			0
$215$	3.00000	+	5.19615i	0.204598	+	0.354375i
$216$		0			0
$217$	−4.00000			−0.271538
$218$		0			0
$219$	6.00000	+	10.3923i	0.405442	+	0.702247i
$220$		0			0
$221$	−16.0000			−1.07628
$222$		0			0
$223$	−13.0000	+	22.5167i	−0.870544	+	1.50783i	−0.00910984	+	0.999959i	$0.502900\pi$
$223$	−13.0000	+	22.5167i	−0.870544	+	1.50783i	−0.861435	+	0.507869i	$0.830434\pi$
$224$		0			0
$225$	−0.500000	+	0.866025i	−0.0333333	+	0.0577350i
$226$		0			0
$227$	26.0000			1.72568			0.862840	−	0.505477i	$-0.168683\pi$
$227$	26.0000			1.72568			0.862840	+	0.505477i	$0.168683\pi$
$228$		0			0
$229$	−15.0000			−0.991228			−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000			−0.991228			−0.495614	+	0.868543i	$0.665057\pi$
$230$		0			0
$231$	−4.00000	+	6.92820i	−0.263181	+	0.455842i
$232$		0			0
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	0.404674	+	0.914461i	$0.367385\pi$
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	−0.994283	+	0.106773i	$0.965948\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	11.0000	+	19.0526i	0.714527	+	1.23760i
$238$		0			0
$239$	21.0000			1.35838			0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000			1.35838			0.679189	+	0.733964i	$0.262332\pi$
$240$		0			0
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	0.774202	−	0.632938i	$-0.218151\pi$
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	−0.935242	+	0.354010i	$0.884818\pi$
$242$		0			0
$243$	5.00000	+	8.66025i	0.320750	+	0.555556i
$244$		0			0
$245$	−4.50000	+	7.79423i	−0.287494	+	0.497955i
$246$		0			0
$247$	2.00000	−	17.3205i	0.127257	−	1.10208i
$248$		0			0
$249$	−14.0000	+	24.2487i	−0.887214	+	1.53670i
$250$		0			0
$251$	−8.50000	−	14.7224i	−0.536515	−	0.929272i	−0.999088	−	0.0426905i	$-0.986407\pi$
$251$	−8.50000	−	14.7224i	−0.536515	−	0.929272i	0.462573	−	0.886581i	$-0.346926\pi$
$252$		0			0
$253$	−3.00000	−	5.19615i	−0.188608	−	0.326679i
$254$		0			0
$255$	8.00000			0.500979
$256$		0			0
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$		0			0
$261$	−2.50000	+	4.33013i	−0.154746	+	0.268028i
$262$		0			0
$263$	−13.0000	+	22.5167i	−0.801614	+	1.38844i	0.116939	+	0.993139i	$0.462692\pi$
$263$	−13.0000	+	22.5167i	−0.801614	+	1.38844i	−0.918553	+	0.395298i	$0.870641\pi$
$264$		0			0
$265$	10.0000			0.614295
$266$		0			0
$267$	−14.0000			−0.856786
$268$		0			0
$269$	6.50000	−	11.2583i	0.396312	−	0.686433i	−0.596956	−	0.802274i	$-0.703623\pi$
$269$	6.50000	−	11.2583i	0.396312	−	0.686433i	0.993268	+	0.115842i	$0.0369565\pi$
$270$		0			0
$271$	−11.5000	+	19.9186i	−0.698575	+	1.20997i	0.270385	+	0.962752i	$0.412849\pi$
$271$	−11.5000	+	19.9186i	−0.698575	+	1.20997i	−0.968960	+	0.247216i	$0.920484\pi$
$272$		0			0
$273$	32.0000			1.93673
$274$		0			0
$275$	0.500000	+	0.866025i	0.0301511	+	0.0522233i
$276$		0			0
$277$	−28.0000			−1.68236			−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000			−1.68236			−0.841178	+	0.540758i	$0.818138\pi$
$278$		0			0
$279$	−0.500000	−	0.866025i	−0.0299342	−	0.0518476i
$280$		0			0
$281$	11.0000	+	19.0526i	0.656205	+	1.13658i	0.981590	+	0.190999i	$0.0611727\pi$
$281$	11.0000	+	19.0526i	0.656205	+	1.13658i	−0.325385	+	0.945582i	$0.605494\pi$
$282$		0			0
$283$	−8.00000	+	13.8564i	−0.475551	+	0.823678i	−0.999608	−	0.0280052i	$-0.991084\pi$
$283$	−8.00000	+	13.8564i	−0.475551	+	0.823678i	0.524057	+	0.851683i	$0.324418\pi$
$284$		0			0
$285$	−1.00000	+	8.66025i	−0.0592349	+	0.512989i
$286$		0			0
$287$	−12.0000	+	20.7846i	−0.708338	+	1.22688i
$288$		0			0
$289$	0.500000	+	0.866025i	0.0294118	+	0.0509427i
$290$		0			0
$291$	6.00000	+	10.3923i	0.351726	+	0.609208i
$292$		0			0
$293$	12.0000			0.701047			0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000			0.701047			0.350524	+	0.936554i	$0.386004\pi$
$294$		0			0
$295$	1.50000	+	2.59808i	0.0873334	+	0.151266i
$296$		0			0
$297$	4.00000			0.232104
$298$		0			0
$299$	−12.0000	+	20.7846i	−0.693978	+	1.20201i
$300$		0			0
$301$	−12.0000	+	20.7846i	−0.691669	+	1.19800i
$302$		0			0
$303$	−10.0000			−0.574485
$304$		0			0
$305$	5.00000			0.286299
$306$		0			0
$307$	−13.0000	+	22.5167i	−0.741949	+	1.28509i	0.209657	+	0.977775i	$0.432765\pi$
$307$	−13.0000	+	22.5167i	−0.741949	+	1.28509i	−0.951607	+	0.307319i	$0.900568\pi$
$308$		0			0
$309$	−4.00000	+	6.92820i	−0.227552	+	0.394132i
$310$		0			0
$311$	−4.00000			−0.226819			−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000			−0.226819			−0.113410	+	0.993548i	$0.536177\pi$
$312$		0			0
$313$	−8.00000	−	13.8564i	−0.452187	−	0.783210i	0.546335	−	0.837567i	$-0.316023\pi$
$313$	−8.00000	−	13.8564i	−0.452187	−	0.783210i	−0.998522	+	0.0543564i	$0.982689\pi$
$314$		0			0
$315$	−4.00000			−0.225374
$316$		0			0
$317$	−7.00000	−	12.1244i	−0.393159	−	0.680972i	0.599705	−	0.800221i	$-0.295285\pi$
$317$	−7.00000	−	12.1244i	−0.393159	−	0.680972i	−0.992864	+	0.119249i	$0.961951\pi$
$318$		0			0
$319$	2.50000	+	4.33013i	0.139973	+	0.242441i
$320$		0			0
$321$	6.00000	−	10.3923i	0.334887	−	0.580042i
$322$		0			0
$323$	16.0000	−	6.92820i	0.890264	−	0.385496i
$324$		0			0
$325$	2.00000	−	3.46410i	0.110940	−	0.192154i
$326$		0			0
$327$	−19.0000	−	32.9090i	−1.05070	−	1.81987i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−28.0000			−1.53902			−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000			−1.53902			−0.769510	+	0.638635i	$0.779499\pi$
$332$		0			0
$333$	1.00000	+	1.73205i	0.0547997	+	0.0949158i
$334$		0			0
$335$	−4.00000			−0.218543
$336$		0			0
$337$	1.00000	−	1.73205i	0.0544735	−	0.0943508i	−0.837503	−	0.546433i	$-0.815985\pi$
$337$	1.00000	−	1.73205i	0.0544735	−	0.0943508i	0.891976	+	0.452082i	$0.149319\pi$
$338$		0			0
$339$	−6.00000	+	10.3923i	−0.325875	+	0.564433i
$340$		0			0
$341$	−1.00000			−0.0541530
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$	6.00000	−	10.3923i	0.323029	−	0.559503i
$346$		0			0
$347$	4.00000	−	6.92820i	0.214731	−	0.371925i	−0.738458	−	0.674299i	$-0.764446\pi$
$347$	4.00000	−	6.92820i	0.214731	−	0.371925i	0.953189	+	0.302374i	$0.0977791\pi$
$348$		0			0
$349$	34.0000			1.81998			0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000			1.81998			0.909989	+	0.414632i	$0.136090\pi$
$350$		0			0
$351$	−8.00000	−	13.8564i	−0.427008	−	0.739600i
$352$		0			0
$353$	24.0000			1.27739			0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000			1.27739			0.638696	+	0.769460i	$0.279474\pi$
$354$		0			0
$355$	−6.50000	−	11.2583i	−0.344984	−	0.597530i
$356$		0			0
$357$	16.0000	+	27.7128i	0.846810	+	1.46672i
$358$		0			0
$359$	8.00000	−	13.8564i	0.422224	−	0.731313i	−0.573933	−	0.818902i	$-0.694583\pi$
$359$	8.00000	−	13.8564i	0.422224	−	0.731313i	0.996157	+	0.0875892i	$0.0279163\pi$
$360$		0			0
$361$	5.50000	+	18.1865i	0.289474	+	0.957186i
$362$		0			0
$363$	10.0000	−	17.3205i	0.524864	−	0.909091i
$364$		0			0
$365$	3.00000	+	5.19615i	0.157027	+	0.271979i
$366$		0			0
$367$	6.00000	+	10.3923i	0.313197	+	0.542474i	0.979053	−	0.203607i	$-0.0652665\pi$
$367$	6.00000	+	10.3923i	0.313197	+	0.542474i	−0.665855	+	0.746081i	$0.731933\pi$
$368$		0			0
$369$	−6.00000			−0.312348
$370$		0			0
$371$	20.0000	+	34.6410i	1.03835	+	1.79847i
$372$		0			0
$373$	12.0000			0.621336			0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000			0.621336			0.310668	+	0.950518i	$0.399447\pi$
$374$		0			0
$375$	−1.00000	+	1.73205i	−0.0516398	+	0.0894427i
$376$		0			0
$377$	10.0000	−	17.3205i	0.515026	−	0.892052i
$378$		0			0
$379$	−25.0000			−1.28416			−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000			−1.28416			−0.642082	+	0.766636i	$0.721929\pi$
$380$		0			0
$381$	−24.0000			−1.22956
$382$		0			0
$383$	−2.00000	+	3.46410i	−0.102195	+	0.177007i	−0.912589	−	0.408879i	$-0.865920\pi$
$383$	−2.00000	+	3.46410i	−0.102195	+	0.177007i	0.810394	+	0.585886i	$0.199253\pi$
$384$		0			0
$385$	−2.00000	+	3.46410i	−0.101929	+	0.176547i
$386$		0			0
$387$	−6.00000			−0.304997
$388$		0			0
$389$	−1.50000	−	2.59808i	−0.0760530	−	0.131728i	0.825491	−	0.564416i	$-0.190898\pi$
$389$	−1.50000	−	2.59808i	−0.0760530	−	0.131728i	−0.901544	+	0.432688i	$0.857565\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$		0			0
$393$	12.0000	+	20.7846i	0.605320	+	1.04844i
$394$		0			0
$395$	5.50000	+	9.52628i	0.276735	+	0.479319i
$396$		0			0
$397$	−10.0000	+	17.3205i	−0.501886	+	0.869291i	0.498112	+	0.867113i	$0.334027\pi$
$397$	−10.0000	+	17.3205i	−0.501886	+	0.869291i	−0.999998	+	0.00217869i	$0.999307\pi$
$398$		0			0
$399$	−32.0000	+	13.8564i	−1.60200	+	0.693688i
$400$		0			0
$401$	2.50000	−	4.33013i	0.124844	−	0.216236i	−0.796828	−	0.604206i	$-0.793490\pi$
$401$	2.50000	−	4.33013i	0.124844	−	0.216236i	0.921672	+	0.387970i	$0.126824\pi$
$402$		0			0
$403$	2.00000	+	3.46410i	0.0996271	+	0.172559i
$404$		0			0
$405$	5.50000	+	9.52628i	0.273297	+	0.473365i
$406$		0			0
$407$	2.00000			0.0991363
$408$		0			0
$409$	2.50000	+	4.33013i	0.123617	+	0.214111i	0.921192	−	0.389109i	$-0.127217\pi$
$409$	2.50000	+	4.33013i	0.123617	+	0.214111i	−0.797574	+	0.603220i	$0.793884\pi$
$410$		0			0
$411$	40.0000			1.97305
$412$		0			0
$413$	−6.00000	+	10.3923i	−0.295241	+	0.511372i
$414$		0			0
$415$	−7.00000	+	12.1244i	−0.343616	+	0.595161i
$416$		0			0
$417$	24.0000			1.17529
$418$		0			0
$419$	9.00000			0.439679			0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000			0.439679			0.219839	+	0.975536i	$0.429447\pi$
$420$		0			0
$421$	9.50000	−	16.4545i	0.463002	−	0.801942i	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	9.50000	−	16.4545i	0.463002	−	0.801942i	0.999109	+	0.0422075i	$0.0134391\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.00000			0.194029
$426$		0			0
$427$	10.0000	+	17.3205i	0.483934	+	0.838198i
$428$		0			0
$429$	8.00000			0.386244
$430$		0			0
$431$	3.50000	+	6.06218i	0.168589	+	0.292005i	0.937924	−	0.346841i	$-0.112746\pi$
$431$	3.50000	+	6.06218i	0.168589	+	0.292005i	−0.769335	+	0.638846i	$0.779412\pi$
$432$		0			0
$433$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$433$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$434$		0			0
$435$	−5.00000	+	8.66025i	−0.239732	+	0.415227i
$436$		0			0
$437$	3.00000	−	25.9808i	0.143509	−	1.24283i
$438$		0			0
$439$	4.50000	−	7.79423i	0.214773	−	0.371998i	−0.738429	−	0.674331i	$-0.764432\pi$
$439$	4.50000	−	7.79423i	0.214773	−	0.371998i	0.953202	+	0.302333i	$0.0977654\pi$
$440$		0			0
$441$	−4.50000	−	7.79423i	−0.214286	−	0.371154i
$442$		0			0
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	0.687557	−	0.726130i	$-0.258683\pi$
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	−0.972626	+	0.232377i	$0.925350\pi$
$444$		0			0
$445$	−7.00000			−0.331832
$446$		0			0
$447$	−7.00000	−	12.1244i	−0.331089	−	0.573462i
$448$		0			0
$449$	−29.0000			−1.36859			−0.684297	−	0.729203i	$-0.739891\pi$
$449$	−29.0000			−1.36859			−0.684297	+	0.729203i	$0.739891\pi$
$450$		0			0
$451$	−3.00000	+	5.19615i	−0.141264	+	0.244677i
$452$		0			0
$453$	3.00000	−	5.19615i	0.140952	−	0.244137i
$454$		0			0
$455$	16.0000			0.750092
$456$		0			0
$457$	−28.0000			−1.30978			−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000			−1.30978			−0.654892	+	0.755722i	$0.727286\pi$
$458$		0			0
$459$	8.00000	−	13.8564i	0.373408	−	0.646762i
$460$		0			0
$461$	−11.5000	+	19.9186i	−0.535608	+	0.927701i	0.463525	+	0.886084i	$0.346584\pi$
$461$	−11.5000	+	19.9186i	−0.535608	+	0.927701i	−0.999134	+	0.0416172i	$0.986749\pi$
$462$		0			0
$463$	−28.0000			−1.30127			−0.650635	−	0.759390i	$-0.725497\pi$
$463$	−28.0000			−1.30127			−0.650635	+	0.759390i	$0.725497\pi$
$464$		0			0
$465$	−1.00000	−	1.73205i	−0.0463739	−	0.0803219i
$466$		0			0
$467$	30.0000			1.38823			0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000			1.38823			0.694117	+	0.719862i	$0.255795\pi$
$468$		0			0
$469$	−8.00000	−	13.8564i	−0.369406	−	0.639829i
$470$		0			0
$471$	−22.0000	−	38.1051i	−1.01371	−	1.75579i
$472$		0			0
$473$	−3.00000	+	5.19615i	−0.137940	+	0.238919i
$474$		0			0
$475$	−0.500000	+	4.33013i	−0.0229416	+	0.198680i
$476$		0			0
$477$	−5.00000	+	8.66025i	−0.228934	+	0.396526i
$478$		0			0
$479$	−7.50000	−	12.9904i	−0.342684	−	0.593546i	0.642246	−	0.766498i	$-0.278003\pi$
$479$	−7.50000	−	12.9904i	−0.342684	−	0.593546i	−0.984930	+	0.172953i	$0.944669\pi$
$480$		0			0
$481$	−4.00000	−	6.92820i	−0.182384	−	0.315899i
$482$		0			0
$483$	48.0000			2.18408
$484$		0			0
$485$	3.00000	+	5.19615i	0.136223	+	0.235945i
$486$		0			0
$487$	−4.00000			−0.181257			−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000			−0.181257			−0.0906287	+	0.995885i	$0.528888\pi$
$488$		0			0
$489$	18.0000	−	31.1769i	0.813988	−	1.40987i
$490$		0			0
$491$	−2.50000	+	4.33013i	−0.112823	+	0.195416i	−0.916908	−	0.399100i	$-0.869323\pi$
$491$	−2.50000	+	4.33013i	−0.112823	+	0.195416i	0.804084	+	0.594515i	$0.202656\pi$
$492$		0			0
$493$	20.0000			0.900755
$494$		0			0
$495$	−1.00000			−0.0449467
$496$		0			0
$497$	26.0000	−	45.0333i	1.16626	−	2.02002i
$498$		0			0
$499$	6.00000	−	10.3923i	0.268597	−	0.465223i	−0.699903	−	0.714238i	$-0.746773\pi$
$499$	6.00000	−	10.3923i	0.268597	−	0.465223i	0.968500	+	0.249015i	$0.0801067\pi$
$500$		0			0
$501$	40.0000			1.78707
$502$		0			0
$503$	10.0000	+	17.3205i	0.445878	+	0.772283i	0.998113	−	0.0614052i	$-0.0195582\pi$
$503$	10.0000	+	17.3205i	0.445878	+	0.772283i	−0.552235	+	0.833689i	$0.686225\pi$
$504$		0			0
$505$	−5.00000			−0.222497
$506$		0			0
$507$	−3.00000	−	5.19615i	−0.133235	−	0.230769i
$508$		0			0
$509$	−11.0000	−	19.0526i	−0.487566	−	0.844490i	0.512331	−	0.858788i	$-0.328782\pi$
$509$	−11.0000	−	19.0526i	−0.487566	−	0.844490i	−0.999898	+	0.0142980i	$0.995449\pi$
$510$		0			0
$511$	−12.0000	+	20.7846i	−0.530849	+	0.919457i
$512$		0			0
$513$	14.0000	+	10.3923i	0.618115	+	0.458831i
$514$		0			0
$515$	−2.00000	+	3.46410i	−0.0881305	+	0.152647i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−8.00000	−	13.8564i	−0.351161	−	0.608229i
$520$		0			0
$521$	15.0000			0.657162			0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000			0.657162			0.328581	+	0.944476i	$0.393430\pi$
$522$		0			0
$523$	1.00000	+	1.73205i	0.0437269	+	0.0757373i	0.887061	−	0.461653i	$-0.152744\pi$
$523$	1.00000	+	1.73205i	0.0437269	+	0.0757373i	−0.843334	+	0.537390i	$0.819410\pi$
$524$		0			0
$525$	−8.00000			−0.349149
$526$		0			0
$527$	−2.00000	+	3.46410i	−0.0871214	+	0.150899i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$	−3.00000			−0.130189
$532$		0			0
$533$	24.0000			1.03956
$534$		0			0
$535$	3.00000	−	5.19615i	0.129701	−	0.224649i
$536$		0			0
$537$	21.0000	−	36.3731i	0.906217	−	1.56961i
$538$		0			0
$539$	−9.00000			−0.387657
$540$		0			0
$541$	6.50000	+	11.2583i	0.279457	+	0.484033i	0.971250	−	0.238062i	$-0.0765123\pi$
$541$	6.50000	+	11.2583i	0.279457	+	0.484033i	−0.691793	+	0.722096i	$0.743179\pi$
$542$		0			0
$543$	44.0000			1.88822
$544$		0			0
$545$	−9.50000	−	16.4545i	−0.406935	−	0.704833i
$546$		0			0
$547$	−19.0000	−	32.9090i	−0.812381	−	1.40709i	−0.911193	−	0.411980i	$-0.864837\pi$
$547$	−19.0000	−	32.9090i	−0.812381	−	1.40709i	0.0988117	−	0.995106i	$-0.468496\pi$
$548$		0			0
$549$	−2.50000	+	4.33013i	−0.106697	+	0.184805i
$550$		0			0
$551$	−2.50000	+	21.6506i	−0.106504	+	0.922348i
$552$		0			0
$553$	−22.0000	+	38.1051i	−0.935535	+	1.62039i
$554$		0			0
$555$	2.00000	+	3.46410i	0.0848953	+	0.147043i
$556$		0			0
$557$	−21.0000	−	36.3731i	−0.889799	−	1.54118i	−0.840113	−	0.542411i	$-0.817511\pi$
$557$	−21.0000	−	36.3731i	−0.889799	−	1.54118i	−0.0496855	−	0.998765i	$-0.515822\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$	4.00000	+	6.92820i	0.168880	+	0.292509i
$562$		0			0
$563$	38.0000			1.60151			0.800755	−	0.598993i	$-0.204432\pi$
$563$	38.0000			1.60151			0.800755	+	0.598993i	$0.204432\pi$
$564$		0			0
$565$	−3.00000	+	5.19615i	−0.126211	+	0.218604i
$566$		0			0
$567$	−22.0000	+	38.1051i	−0.923913	+	1.60026i
$568$		0			0
$569$	39.0000			1.63497			0.817483	−	0.575953i	$-0.195369\pi$
$569$	39.0000			1.63497			0.817483	+	0.575953i	$0.195369\pi$
$570$		0			0
$571$	−19.0000			−0.795125			−0.397563	−	0.917575i	$-0.630144\pi$
$571$	−19.0000			−0.795125			−0.397563	+	0.917575i	$0.630144\pi$
$572$		0			0
$573$	−5.00000	+	8.66025i	−0.208878	+	0.361787i
$574$		0			0
$575$	3.00000	−	5.19615i	0.125109	−	0.216695i
$576$		0			0
$577$	−30.0000			−1.24892			−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000			−1.24892			−0.624458	+	0.781058i	$0.714680\pi$
$578$		0			0
$579$	8.00000	+	13.8564i	0.332469	+	0.575853i
$580$		0			0
$581$	−56.0000			−2.32327
$582$		0			0
$583$	5.00000	+	8.66025i	0.207079	+	0.358671i
$584$		0			0
$585$	2.00000	+	3.46410i	0.0826898	+	0.143223i
$586$		0			0
$587$	16.0000	−	27.7128i	0.660391	−	1.14383i	−0.320122	−	0.947376i	$-0.603724\pi$
$587$	16.0000	−	27.7128i	0.660391	−	1.14383i	0.980513	−	0.196454i	$-0.0629426\pi$
$588$		0			0
$589$	−3.50000	−	2.59808i	−0.144215	−	0.107052i
$590$		0			0
$591$	16.0000	−	27.7128i	0.658152	−	1.13995i
$592$		0			0
$593$	−14.0000	−	24.2487i	−0.574911	−	0.995775i	−0.996051	−	0.0887797i	$-0.971703\pi$
$593$	−14.0000	−	24.2487i	−0.574911	−	0.995775i	0.421140	−	0.906996i	$-0.361630\pi$
$594$		0			0
$595$	8.00000	+	13.8564i	0.327968	+	0.568057i
$596$		0			0
$597$	10.0000			0.409273
$598$		0			0
$599$	−4.00000	−	6.92820i	−0.163436	−	0.283079i	0.772663	−	0.634816i	$-0.218924\pi$
$599$	−4.00000	−	6.92820i	−0.163436	−	0.283079i	−0.936099	+	0.351738i	$0.885591\pi$
$600$		0			0
$601$	45.0000			1.83559			0.917794	−	0.397057i	$-0.129968\pi$
$601$	45.0000			1.83559			0.917794	+	0.397057i	$0.129968\pi$
$602$		0			0
$603$	2.00000	−	3.46410i	0.0814463	−	0.141069i
$604$		0			0
$605$	5.00000	−	8.66025i	0.203279	−	0.352089i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$	−40.0000			−1.62088
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−23.0000	+	39.8372i	−0.928961	+	1.60901i	−0.143898	+	0.989593i	$0.545964\pi$
$613$	−23.0000	+	39.8372i	−0.928961	+	1.60901i	−0.785063	+	0.619416i	$0.787370\pi$
$614$		0			0
$615$	−12.0000			−0.483887
$616$		0			0
$617$	−5.00000	−	8.66025i	−0.201292	−	0.348649i	0.747653	−	0.664090i	$-0.231181\pi$
$617$	−5.00000	−	8.66025i	−0.201292	−	0.348649i	−0.948945	+	0.315441i	$0.897847\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$		0			0
$621$	−12.0000	−	20.7846i	−0.481543	−	0.834058i
$622$		0			0
$623$	−14.0000	−	24.2487i	−0.560898	−	0.971504i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−8.00000	+	3.46410i	−0.319489	+	0.138343i
$628$		0			0
$629$	4.00000	−	6.92820i	0.159490	−	0.276246i
$630$		0			0
$631$	1.50000	+	2.59808i	0.0597141	+	0.103428i	0.894337	−	0.447394i	$-0.147648\pi$
$631$	1.50000	+	2.59808i	0.0597141	+	0.103428i	−0.834623	+	0.550822i	$0.814314\pi$
$632$		0			0
$633$	23.0000	+	39.8372i	0.914168	+	1.58339i
$634$		0			0
$635$	−12.0000			−0.476205
$636$		0			0
$637$	18.0000	+	31.1769i	0.713186	+	1.23527i
$638$		0			0
$639$	13.0000			0.514272
$640$		0			0
$641$	−14.5000	+	25.1147i	−0.572716	+	0.991972i	0.423570	+	0.905863i	$0.360777\pi$
$641$	−14.5000	+	25.1147i	−0.572716	+	0.991972i	−0.996286	+	0.0861092i	$0.972557\pi$
$642$		0			0
$643$	−2.00000	+	3.46410i	−0.0788723	+	0.136611i	−0.902764	−	0.430137i	$-0.858465\pi$
$643$	−2.00000	+	3.46410i	−0.0788723	+	0.136611i	0.823891	+	0.566748i	$0.191799\pi$
$644$		0			0
$645$	−12.0000			−0.472500
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	−1.50000	+	2.59808i	−0.0588802	+	0.101983i
$650$		0			0
$651$	4.00000	−	6.92820i	0.156772	−	0.271538i
$652$		0			0
$653$	−14.0000			−0.547862			−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000			−0.547862			−0.273931	+	0.961749i	$0.588324\pi$
$654$		0			0
$655$	6.00000	+	10.3923i	0.234439	+	0.406061i
$656$		0			0
$657$	−6.00000			−0.234082
$658$		0			0
$659$	8.00000	+	13.8564i	0.311636	+	0.539769i	0.978717	−	0.205216i	$-0.0657898\pi$
$659$	8.00000	+	13.8564i	0.311636	+	0.539769i	−0.667081	+	0.744985i	$0.732456\pi$
$660$		0			0
$661$	−8.50000	−	14.7224i	−0.330612	−	0.572636i	0.652020	−	0.758202i	$-0.273922\pi$
$661$	−8.50000	−	14.7224i	−0.330612	−	0.572636i	−0.982632	+	0.185565i	$0.940588\pi$
$662$		0			0
$663$	16.0000	−	27.7128i	0.621389	−	1.07628i
$664$		0			0
$665$	−16.0000	+	6.92820i	−0.620453	+	0.268664i
$666$		0			0
$667$	15.0000	−	25.9808i	0.580802	−	1.00598i
$668$		0			0
$669$	−26.0000	−	45.0333i	−1.00522	−	1.74109i
$670$		0			0
$671$	2.50000	+	4.33013i	0.0965114	+	0.167163i
$672$		0			0
$673$	−44.0000			−1.69608			−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000			−1.69608			−0.848038	+	0.529936i	$0.822216\pi$
$674$		0			0
$675$	2.00000	+	3.46410i	0.0769800	+	0.133333i
$676$		0			0
$677$	12.0000			0.461197			0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000			0.461197			0.230599	+	0.973049i	$0.425932\pi$
$678$		0			0
$679$	−12.0000	+	20.7846i	−0.460518	+	0.797640i
$680$		0			0
$681$	−26.0000	+	45.0333i	−0.996322	+	1.72568i
$682$		0			0
$683$	−4.00000			−0.153056			−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000			−0.153056			−0.0765279	+	0.997067i	$0.524383\pi$
$684$		0			0
$685$	20.0000			0.764161
$686$		0			0
$687$	15.0000	−	25.9808i	0.572286	−	0.991228i
$688$		0			0
$689$	20.0000	−	34.6410i	0.761939	−	1.31972i
$690$		0			0
$691$	−33.0000			−1.25538			−0.627690	−	0.778464i	$-0.715999\pi$
$691$	−33.0000			−1.25538			−0.627690	+	0.778464i	$0.715999\pi$
$692$		0			0
$693$	−2.00000	−	3.46410i	−0.0759737	−	0.131590i
$694$		0			0
$695$	12.0000			0.455186
$696$		0			0
$697$	12.0000	+	20.7846i	0.454532	+	0.787273i
$698$		0			0
$699$	−18.0000	−	31.1769i	−0.680823	−	1.17922i
$700$		0			0
$701$	−1.00000	+	1.73205i	−0.0377695	+	0.0654187i	−0.884292	−	0.466934i	$-0.845359\pi$
$701$	−1.00000	+	1.73205i	−0.0377695	+	0.0654187i	0.846523	+	0.532353i	$0.178692\pi$
$702$		0			0
$703$	7.00000	+	5.19615i	0.264010	+	0.195977i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.0000	−	17.3205i	−0.376089	−	0.651405i
$708$		0			0
$709$	−14.5000	−	25.1147i	−0.544559	−	0.943204i	−0.998635	−	0.0522406i	$-0.983364\pi$
$709$	−14.5000	−	25.1147i	−0.544559	−	0.943204i	0.454076	−	0.890963i	$-0.349970\pi$
$710$		0			0
$711$	−11.0000			−0.412532
$712$		0			0
$713$	3.00000	+	5.19615i	0.112351	+	0.194597i
$714$		0			0
$715$	4.00000			0.149592
$716$		0			0
$717$	−21.0000	+	36.3731i	−0.784259	+	1.35838i
$718$		0			0
$719$	10.5000	−	18.1865i	0.391584	−	0.678243i	−0.601075	−	0.799193i	$-0.705261\pi$
$719$	10.5000	−	18.1865i	0.391584	−	0.678243i	0.992659	+	0.120950i	$0.0385939\pi$
$720$		0			0
$721$	−16.0000			−0.595871
$722$		0			0
$723$	10.0000			0.371904
$724$		0			0
$725$	−2.50000	+	4.33013i	−0.0928477	+	0.160817i
$726$		0			0
$727$	−7.00000	+	12.1244i	−0.259616	+	0.449667i	−0.966139	−	0.258022i	$-0.916929\pi$
$727$	−7.00000	+	12.1244i	−0.259616	+	0.449667i	0.706523	+	0.707690i	$0.250263\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	12.0000	+	20.7846i	0.443836	+	0.768747i
$732$		0			0
$733$	−30.0000			−1.10808			−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000			−1.10808			−0.554038	+	0.832492i	$0.686914\pi$
$734$		0			0
$735$	−9.00000	−	15.5885i	−0.331970	−	0.574989i
$736$		0			0
$737$	−2.00000	−	3.46410i	−0.0736709	−	0.127602i
$738$		0			0
$739$	−23.5000	+	40.7032i	−0.864461	+	1.49729i	0.00311943	+	0.999995i	$0.499007\pi$
$739$	−23.5000	+	40.7032i	−0.864461	+	1.49729i	−0.867581	+	0.497296i	$0.834326\pi$
$740$		0			0
$741$	28.0000	+	20.7846i	1.02861	+	0.763542i
$742$		0			0
$743$	19.0000	−	32.9090i	0.697042	−	1.20731i	−0.272445	−	0.962171i	$-0.587832\pi$
$743$	19.0000	−	32.9090i	0.697042	−	1.20731i	0.969487	−	0.245141i	$-0.0788344\pi$
$744$		0			0
$745$	−3.50000	−	6.06218i	−0.128230	−	0.222101i
$746$		0			0
$747$	−7.00000	−	12.1244i	−0.256117	−	0.443607i
$748$		0			0
$749$	24.0000			0.876941
$750$		0			0
$751$	5.50000	+	9.52628i	0.200698	+	0.347619i	0.948753	−	0.316017i	$-0.102346\pi$
$751$	5.50000	+	9.52628i	0.200698	+	0.347619i	−0.748056	+	0.663636i	$0.769012\pi$
$752$		0			0
$753$	34.0000			1.23903
$754$		0			0
$755$	1.50000	−	2.59808i	0.0545906	−	0.0945537i
$756$		0			0
$757$	22.0000	−	38.1051i	0.799604	−	1.38495i	−0.120271	−	0.992741i	$-0.538376\pi$
$757$	22.0000	−	38.1051i	0.799604	−	1.38495i	0.919874	−	0.392213i	$-0.128290\pi$
$758$		0			0
$759$	12.0000			0.435572
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$	38.0000	−	65.8179i	1.37569	−	2.38277i
$764$		0			0
$765$	−2.00000	+	3.46410i	−0.0723102	+	0.125245i
$766$		0			0
$767$	12.0000			0.433295
$768$		0			0
$769$	11.5000	+	19.9186i	0.414701	+	0.718283i	0.995397	−	0.0958377i	$-0.0305530\pi$
$769$	11.5000	+	19.9186i	0.414701	+	0.718283i	−0.580696	+	0.814120i	$0.697220\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	−0.500000	−	0.866025i	−0.0179605	−	0.0311086i
$776$		0			0
$777$	−8.00000	+	13.8564i	−0.286998	+	0.497096i
$778$		0			0
$779$	−24.0000	+	10.3923i	−0.859889	+	0.372343i
$780$		0			0
$781$	6.50000	−	11.2583i	0.232588	−	0.402855i
$782$		0			0
$783$	10.0000	+	17.3205i	0.357371	+	0.618984i
$784$		0			0
$785$	−11.0000	−	19.0526i	−0.392607	−	0.680015i
$786$		0			0
$787$	4.00000			0.142585			0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000			0.142585			0.0712923	+	0.997455i	$0.477288\pi$
$788$		0			0
$789$	−26.0000	−	45.0333i	−0.925625	−	1.60323i
$790$		0			0

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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{5} -4.00000 q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{5} -4.00000 q^{7} +(-0.500000 - 0.866025i) q^{9} -1.00000 q^{11} +(2.00000 + 3.46410i) q^{13} +(-1.00000 - 1.73205i) q^{15} +(-2.00000 + 3.46410i) q^{17} +(-3.50000 - 2.59808i) q^{19} +(4.00000 - 6.92820i) q^{21} +(3.00000 + 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} -4.00000 q^{27} +(-2.50000 - 4.33013i) q^{29} +1.00000 q^{31} +(1.00000 - 1.73205i) q^{33} +(2.00000 - 3.46410i) q^{35} -2.00000 q^{37} -8.00000 q^{39} +(3.00000 - 5.19615i) q^{41} +(3.00000 - 5.19615i) q^{43} +1.00000 q^{45} +9.00000 q^{49} +(-4.00000 - 6.92820i) q^{51} +(-5.00000 - 8.66025i) q^{53} +(0.500000 - 0.866025i) q^{55} +(8.00000 - 3.46410i) q^{57} +(1.50000 - 2.59808i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(2.00000 + 3.46410i) q^{63} -4.00000 q^{65} +(2.00000 + 3.46410i) q^{67} -12.0000 q^{69} +(-6.50000 + 11.2583i) q^{71} +(3.00000 - 5.19615i) q^{73} +2.00000 q^{75} +4.00000 q^{77} +(5.50000 - 9.52628i) q^{79} +(5.50000 - 9.52628i) q^{81} +14.0000 q^{83} +(-2.00000 - 3.46410i) q^{85} +10.0000 q^{87} +(3.50000 + 6.06218i) q^{89} +(-8.00000 - 13.8564i) q^{91} +(-1.00000 + 1.73205i) q^{93} +(4.00000 - 1.73205i) q^{95} +(3.00000 - 5.19615i) q^{97} +(0.500000 + 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} - 8 q^{7} - q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - q^5 - 8 * q^7 - q^9	\( 2 q - 2 q^{3} - q^{5} - 8 q^{7} - q^{9} - 2 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{17} - 7 q^{19} + 8 q^{21} + 6 q^{23} - q^{25} - 8 q^{27} - 5 q^{29} + 2 q^{31} + 2 q^{33} + 4 q^{35} - 4 q^{37} - 16 q^{39} + 6 q^{41} + 6 q^{43} + 2 q^{45} + 18 q^{49} - 8 q^{51} - 10 q^{53} + q^{55} + 16 q^{57} + 3 q^{59} - 5 q^{61} + 4 q^{63} - 8 q^{65} + 4 q^{67} - 24 q^{69} - 13 q^{71} + 6 q^{73} + 4 q^{75} + 8 q^{77} + 11 q^{79} + 11 q^{81} + 28 q^{83} - 4 q^{85} + 20 q^{87} + 7 q^{89} - 16 q^{91} - 2 q^{93} + 8 q^{95} + 6 q^{97} + q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 - 8 * q^7 - q^9 - 2 * q^11 + 4 * q^13 - 2 * q^15 - 4 * q^17 - 7 * q^19 + 8 * q^21 + 6 * q^23 - q^25 - 8 * q^27 - 5 * q^29 + 2 * q^31 + 2 * q^33 + 4 * q^35 - 4 * q^37 - 16 * q^39 + 6 * q^41 + 6 * q^43 + 2 * q^45 + 18 * q^49 - 8 * q^51 - 10 * q^53 + q^55 + 16 * q^57 + 3 * q^59 - 5 * q^61 + 4 * q^63 - 8 * q^65 + 4 * q^67 - 24 * q^69 - 13 * q^71 + 6 * q^73 + 4 * q^75 + 8 * q^77 + 11 * q^79 + 11 * q^81 + 28 * q^83 - 4 * q^85 + 20 * q^87 + 7 * q^89 - 16 * q^91 - 2 * q^93 + 8 * q^95 + 6 * q^97 + q^99

Introduction

L-functions

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