LMFDB - Embedded newform 637.2.c.d.246.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(246,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("637.246");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	637.c (of order $2$, degree $1$, 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:	$5.08647060876$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			246.4
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	637.246
Dual form			637.2.c.d.246.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.688892i q^{2} +2.21432 q^{3} +1.52543 q^{4} -3.21432i q^{5} +1.52543i q^{6} +2.42864i q^{8} +1.90321 q^{9} +O(q^{10})$	\(q+0.688892i q^{2} +2.21432 q^{3} +1.52543 q^{4} -3.21432i q^{5} +1.52543i q^{6} +2.42864i q^{8} +1.90321 q^{9} +2.21432 q^{10} -2.68889i q^{11} +3.37778 q^{12} +(-3.59210 - 0.311108i) q^{13} -7.11753i q^{15} +1.37778 q^{16} +3.59210 q^{17} +1.31111i q^{18} +8.54617i q^{19} -4.90321i q^{20} +1.85236 q^{22} +3.28100 q^{23} +5.37778i q^{24} -5.33185 q^{25} +(0.214320 - 2.47457i) q^{26} -2.42864 q^{27} +2.05086 q^{29} +4.90321 q^{30} -5.83654i q^{31} +5.80642i q^{32} -5.95407i q^{33} +2.47457i q^{34} +2.90321 q^{36} -3.93332i q^{37} -5.88739 q^{38} +(-7.95407 - 0.688892i) q^{39} +7.80642 q^{40} -0.755569i q^{41} -8.80642 q^{43} -4.10171i q^{44} -6.11753i q^{45} +2.26025i q^{46} +1.88247i q^{47} +3.05086 q^{48} -3.67307i q^{50} +7.95407 q^{51} +(-5.47949 - 0.474572i) q^{52} +2.52543 q^{53} -1.67307i q^{54} -8.64296 q^{55} +18.9240i q^{57} +1.41282i q^{58} +7.33185i q^{59} -10.8573i q^{60} -9.05086 q^{61} +4.02074 q^{62} -1.24443 q^{64} +(-1.00000 + 11.5462i) q^{65} +4.10171 q^{66} +0.428639i q^{67} +5.47949 q^{68} +7.26517 q^{69} +8.98418i q^{71} +4.62222i q^{72} +5.79060i q^{73} +2.70964 q^{74} -11.8064 q^{75} +13.0366i q^{76} +(0.474572 - 5.47949i) q^{78} -4.47949 q^{79} -4.42864i q^{80} -11.0874 q^{81} +0.520505 q^{82} +10.8272i q^{83} -11.5462i q^{85} -6.06668i q^{86} +4.54125 q^{87} +6.53035 q^{88} +5.36196i q^{89} +4.21432 q^{90} +5.00492 q^{92} -12.9240i q^{93} -1.29682 q^{94} +27.4701 q^{95} +12.8573i q^{96} -9.62867i q^{97} -5.11753i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{4} - 2 q^{9}+O(q^{10}) $ 6 * q - 4 * q^4 - 2 * q^9	$ 6 q - 4 q^{4} - 2 q^{9} + 20 q^{12} - 8 q^{13} + 8 q^{16} + 8 q^{17} + 24 q^{22} + 6 q^{23} + 8 q^{25} - 12 q^{26} + 12 q^{27} - 14 q^{29} + 16 q^{30} + 4 q^{36} + 4 q^{38} - 8 q^{39} + 20 q^{40} - 26 q^{43} - 8 q^{48} + 8 q^{51} + 20 q^{52} + 2 q^{53} - 12 q^{55} - 28 q^{61} - 16 q^{62} - 8 q^{64} - 6 q^{65} - 28 q^{66} - 20 q^{68} + 4 q^{69} - 24 q^{74} - 44 q^{75} + 16 q^{78} + 26 q^{79} - 26 q^{81} + 56 q^{82} + 40 q^{87} - 40 q^{88} + 12 q^{90} - 36 q^{92} - 20 q^{94} + 58 q^{95}+O(q^{100}) $ 6 * q - 4 * q^4 - 2 * q^9 + 20 * q^12 - 8 * q^13 + 8 * q^16 + 8 * q^17 + 24 * q^22 + 6 * q^23 + 8 * q^25 - 12 * q^26 + 12 * q^27 - 14 * q^29 + 16 * q^30 + 4 * q^36 + 4 * q^38 - 8 * q^39 + 20 * q^40 - 26 * q^43 - 8 * q^48 + 8 * q^51 + 20 * q^52 + 2 * q^53 - 12 * q^55 - 28 * q^61 - 16 * q^62 - 8 * q^64 - 6 * q^65 - 28 * q^66 - 20 * q^68 + 4 * q^69 - 24 * q^74 - 44 * q^75 + 16 * q^78 + 26 * q^79 - 26 * q^81 + 56 * q^82 + 40 * q^87 - 40 * q^88 + 12 * q^90 - 36 * q^92 - 20 * q^94 + 58 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$.

$n$	$197$	$248$
$\chi(n)$	$-1$	$1$

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 $

$2$			0.688892i			0.487120i	0.969886	+	0.243560i	$0.0783153\pi$
$2$			0.688892i			0.487120i	−0.969886	+	0.243560i	$0.921685\pi$
$3$	2.21432			1.27844			0.639219	−	0.769025i	$-0.279258\pi$
$3$	2.21432			1.27844			0.639219	+	0.769025i	$0.279258\pi$
$4$	1.52543			0.762714
$5$		−	3.21432i		−	1.43749i	−0.695275	−	0.718744i	$-0.744717\pi$
$5$		−	3.21432i		−	1.43749i	0.695275	−	0.718744i	$-0.255283\pi$
$6$			1.52543i			0.622753i
$7$		0			0
$7$		0			0
$8$			2.42864i			0.858654i
$9$	1.90321			0.634404
$10$	2.21432			0.700229
$11$		−	2.68889i		−	0.810731i	−0.914155	−	0.405366i	$-0.867144\pi$
$11$		−	2.68889i		−	0.810731i	0.914155	−	0.405366i	$-0.132856\pi$
$12$	3.37778			0.975082
$13$	−3.59210	−	0.311108i	−0.996270	−	0.0862858i
$13$	−3.59210	−	0.311108i	−0.996270	−	0.0862858i
$14$		0			0
$15$		−	7.11753i		−	1.83774i
$16$	1.37778			0.344446
$17$	3.59210			0.871213			0.435607	−	0.900137i	$-0.356534\pi$
$17$	3.59210			0.871213			0.435607	+	0.900137i	$0.356534\pi$
$18$			1.31111i			0.309031i
$19$			8.54617i			1.96063i	0.197449	+	0.980313i	$0.436734\pi$
$19$			8.54617i			1.96063i	−0.197449	+	0.980313i	$0.563266\pi$
$20$		−	4.90321i		−	1.09639i
$21$		0			0
$22$	1.85236			0.394924
$23$	3.28100			0.684135			0.342068	−	0.939675i	$-0.388873\pi$
$23$	3.28100			0.684135			0.342068	+	0.939675i	$0.388873\pi$
$24$			5.37778i			1.09774i
$25$	−5.33185			−1.06637
$26$	0.214320	−	2.47457i	0.0420316	−	0.485304i
$27$	−2.42864			−0.467392
$28$		0			0
$29$	2.05086			0.380834			0.190417	−	0.981703i	$-0.439016\pi$
$29$	2.05086			0.380834			0.190417	+	0.981703i	$0.439016\pi$
$30$	4.90321			0.895200
$31$		−	5.83654i		−	1.04827i	−0.851634	−	0.524136i	$-0.824388\pi$
$31$		−	5.83654i		−	1.04827i	0.851634	−	0.524136i	$-0.175612\pi$
$32$			5.80642i			1.02644i
$33$		−	5.95407i		−	1.03647i
$34$			2.47457i			0.424386i
$35$		0			0
$36$	2.90321			0.483869
$37$		−	3.93332i		−	0.646634i	−0.946291	−	0.323317i	$-0.895202\pi$
$37$		−	3.93332i		−	0.646634i	0.946291	−	0.323317i	$-0.104798\pi$
$38$	−5.88739			−0.955061
$39$	−7.95407	−	0.688892i	−1.27367	−	0.110311i
$40$	7.80642			1.23430
$41$		−	0.755569i		−	0.118000i	−0.998258	−	0.0590000i	$-0.981209\pi$
$41$		−	0.755569i		−	0.118000i	0.998258	−	0.0590000i	$-0.0187912\pi$
$42$		0			0
$43$	−8.80642			−1.34297			−0.671484	−	0.741019i	$-0.734343\pi$
$43$	−8.80642			−1.34297			−0.671484	+	0.741019i	$0.734343\pi$
$44$		−	4.10171i		−	0.618356i
$45$		−	6.11753i		−	0.911948i
$46$			2.26025i			0.333256i
$47$			1.88247i			0.274586i	0.990530	+	0.137293i	$0.0438402\pi$
$47$			1.88247i			0.274586i	−0.990530	+	0.137293i	$0.956160\pi$
$48$	3.05086			0.440353
$49$		0			0
$50$		−	3.67307i		−	0.519451i
$51$	7.95407			1.11379
$52$	−5.47949	−	0.474572i	−0.759869	−	0.0658114i
$53$	2.52543			0.346894			0.173447	−	0.984843i	$-0.444509\pi$
$53$	2.52543			0.346894			0.173447	+	0.984843i	$0.444509\pi$
$54$		−	1.67307i		−	0.227676i
$55$	−8.64296			−1.16542
$56$		0			0
$57$			18.9240i			2.50654i
$58$			1.41282i			0.185512i
$59$			7.33185i			0.954526i	0.878761	+	0.477263i	$0.158371\pi$
$59$			7.33185i			0.954526i	−0.878761	+	0.477263i	$0.841629\pi$
$60$		−	10.8573i		−	1.40167i
$61$	−9.05086			−1.15884			−0.579422	−	0.815028i	$-0.696722\pi$
$61$	−9.05086			−1.15884			−0.579422	+	0.815028i	$0.696722\pi$
$62$	4.02074			0.510635
$63$		0			0
$64$	−1.24443			−0.155554
$65$	−1.00000	+	11.5462i	−0.124035	+	1.43213i
$66$	4.10171			0.504886
$67$			0.428639i			0.0523666i	0.999657	+	0.0261833i	$0.00833536\pi$
$67$			0.428639i			0.0523666i	−0.999657	+	0.0261833i	$0.991665\pi$
$68$	5.47949			0.664486
$69$	7.26517			0.874624
$70$		0			0
$71$			8.98418i			1.06623i	0.846044	+	0.533113i	$0.178978\pi$
$71$			8.98418i			1.06623i	−0.846044	+	0.533113i	$0.821022\pi$
$72$			4.62222i			0.544733i
$73$			5.79060i			0.677739i	0.940833	+	0.338869i	$0.110044\pi$
$73$			5.79060i			0.677739i	−0.940833	+	0.338869i	$0.889956\pi$
$74$	2.70964			0.314989
$75$	−11.8064			−1.36329
$76$			13.0366i			1.49540i
$77$		0			0
$78$	0.474572	−	5.47949i	0.0537347	−	0.620431i
$79$	−4.47949			−0.503983			−0.251991	−	0.967730i	$-0.581085\pi$
$79$	−4.47949			−0.503983			−0.251991	+	0.967730i	$0.581085\pi$
$80$		−	4.42864i		−	0.495137i
$81$	−11.0874			−1.23194
$82$	0.520505			0.0574802
$83$			10.8272i			1.18844i	0.804304	+	0.594218i	$0.202538\pi$
$83$			10.8272i			1.18844i	−0.804304	+	0.594218i	$0.797462\pi$
$84$		0			0
$85$		−	11.5462i		−	1.25236i
$86$		−	6.06668i		−	0.654187i
$87$	4.54125			0.486873
$88$	6.53035			0.696138
$89$			5.36196i			0.568367i	0.958770	+	0.284183i	$0.0917225\pi$
$89$			5.36196i			0.568367i	−0.958770	+	0.284183i	$0.908278\pi$
$90$	4.21432			0.444228
$91$		0			0
$92$	5.00492			0.521799
$93$		−	12.9240i		−	1.34015i
$94$	−1.29682			−0.133757
$95$	27.4701			2.81838
$96$			12.8573i			1.31224i
$97$		−	9.62867i		−	0.977643i	−0.872384	−	0.488822i	$-0.837427\pi$
$97$		−	9.62867i		−	0.977643i	0.872384	−	0.488822i	$-0.162573\pi$
$98$		0			0
$99$		−	5.11753i		−	0.514331i
$100$	−8.13335			−0.813335
$101$	13.6938			1.36259			0.681293	−	0.732011i	$-0.261418\pi$
$101$	13.6938			1.36259			0.681293	+	0.732011i	$0.261418\pi$
$102$			5.47949i			0.542551i
$103$	12.2953			1.21149			0.605745	−	0.795659i	$-0.292875\pi$
$103$	12.2953			1.21149			0.605745	+	0.795659i	$0.292875\pi$
$104$	0.755569	−	8.72393i	0.0740896	−	0.855451i
$105$		0			0
$106$			1.73975i			0.168979i
$107$	−18.1891			−1.75841			−0.879205	−	0.476444i	$-0.841925\pi$
$107$	−18.1891			−1.75841			−0.879205	+	0.476444i	$0.841925\pi$
$108$	−3.70471			−0.356486
$109$			8.36196i			0.800931i	0.916312	+	0.400465i	$0.131152\pi$
$109$			8.36196i			0.800931i	−0.916312	+	0.400465i	$0.868848\pi$
$110$		−	5.95407i		−	0.567698i
$111$		−	8.70964i		−	0.826682i
$112$		0			0
$113$	−8.46520			−0.796339			−0.398170	−	0.917312i	$-0.630354\pi$
$113$	−8.46520			−0.796339			−0.398170	+	0.917312i	$0.630354\pi$
$114$	−13.0366			−1.22099
$115$		−	10.5462i		−	0.983436i
$116$	3.12843			0.290468
$117$	−6.83654	−	0.592104i	−0.632038	−	0.0547400i
$118$	−5.05086			−0.464969
$119$		0			0
$120$	17.2859			1.57798
$121$	3.76986			0.342714
$122$		−	6.23506i		−	0.564496i
$123$		−	1.67307i		−	0.150856i
$124$		−	8.90321i		−	0.799532i
$125$			1.06668i			0.0954065i
$126$		0			0
$127$	−4.08742			−0.362700			−0.181350	−	0.983419i	$-0.558047\pi$
$127$	−4.08742			−0.362700			−0.181350	+	0.983419i	$0.558047\pi$
$128$			10.7556i			0.950667i
$129$	−19.5002			−1.71690
$130$	−7.95407	−	0.688892i	−0.697618	−	0.0604198i
$131$	−5.93978			−0.518961			−0.259480	−	0.965748i	$-0.583551\pi$
$131$	−5.93978			−0.518961			−0.259480	+	0.965748i	$0.583551\pi$
$132$		−	9.08250i		−	0.790530i
$133$		0			0
$134$	−0.295286			−0.0255089
$135$			7.80642i			0.671870i
$136$			8.72393i			0.748070i
$137$		−	16.3620i		−	1.39790i	−0.715172	−	0.698948i	$-0.753652\pi$
$137$		−	16.3620i		−	1.39790i	0.715172	−	0.698948i	$-0.246348\pi$
$138$			5.00492i			0.426047i
$139$	−3.03011			−0.257011			−0.128505	−	0.991709i	$-0.541018\pi$
$139$	−3.03011			−0.257011			−0.128505	+	0.991709i	$0.541018\pi$
$140$		0			0
$141$			4.16839i			0.351041i
$142$	−6.18913			−0.519380
$143$	−0.836535	+	9.65878i	−0.0699546	+	0.807708i
$144$	2.62222			0.218518
$145$		−	6.59210i		−	0.547444i
$146$	−3.98910			−0.330140
$147$		0			0
$148$		−	6.00000i		−	0.493197i
$149$		−	14.5368i		−	1.19090i	−0.803392	−	0.595451i	$-0.796974\pi$
$149$		−	14.5368i		−	1.19090i	0.803392	−	0.595451i	$-0.203026\pi$
$150$		−	8.13335i		−	0.664085i
$151$			19.9748i			1.62553i	0.582594	+	0.812764i	$0.302038\pi$
$151$			19.9748i			1.62553i	−0.582594	+	0.812764i	$0.697962\pi$
$152$	−20.7556			−1.68350
$153$	6.83654			0.552701
$154$		0			0
$155$	−18.7605			−1.50688
$156$	−12.1334	−	1.05086i	−0.971446	−	0.0841357i
$157$	−7.39853			−0.590467			−0.295233	−	0.955425i	$-0.595397\pi$
$157$	−7.39853			−0.590467			−0.295233	+	0.955425i	$0.595397\pi$
$158$		−	3.08589i		−	0.245500i
$159$	5.59210			0.443483
$160$	18.6637			1.47550
$161$		0			0
$162$		−	7.63804i		−	0.600101i
$163$		−	2.32693i		−	0.182259i	−0.995839	−	0.0911296i	$-0.970952\pi$
$163$		−	2.32693i		−	0.182259i	0.995839	−	0.0911296i	$-0.0290477\pi$
$164$		−	1.15257i		−	0.0900002i
$165$	−19.1383			−1.48991
$166$	−7.45875			−0.578911
$167$		−	3.42219i		−	0.264817i	−0.991195	−	0.132408i	$-0.957729\pi$
$167$		−	3.42219i		−	0.264817i	0.991195	−	0.132408i	$-0.0422710\pi$
$168$		0			0
$169$	12.8064	+	2.23506i	0.985110	+	0.171928i
$170$	7.95407			0.610049
$171$			16.2652i			1.24383i
$172$	−13.4336			−1.02430
$173$	8.27454			0.629102			0.314551	−	0.949241i	$-0.398146\pi$
$173$	8.27454			0.629102			0.314551	+	0.949241i	$0.398146\pi$
$174$			3.12843i			0.237166i
$175$		0			0
$176$		−	3.70471i		−	0.279253i
$177$			16.2351i			1.22030i
$178$	−3.69381			−0.276863
$179$	16.2257			1.21277			0.606383	−	0.795173i	$-0.292620\pi$
$179$	16.2257			1.21277			0.606383	+	0.795173i	$0.292620\pi$
$180$		−	9.33185i		−	0.695555i
$181$	−9.20495			−0.684199			−0.342099	−	0.939664i	$-0.611138\pi$
$181$	−9.20495			−0.684199			−0.342099	+	0.939664i	$0.611138\pi$
$182$		0			0
$183$	−20.0415			−1.48151
$184$			7.96836i			0.587435i
$185$	−12.6430			−0.929529
$186$	8.90321			0.652815
$187$		−	9.65878i		−	0.706320i
$188$			2.87157i			0.209431i
$189$		0			0
$190$			18.9240i			1.37289i
$191$	−6.66815			−0.482490			−0.241245	−	0.970464i	$-0.577556\pi$
$191$	−6.66815			−0.482490			−0.241245	+	0.970464i	$0.577556\pi$
$192$	−2.75557			−0.198866
$193$		−	20.6035i		−	1.48307i	−0.670914	−	0.741535i	$-0.734098\pi$
$193$		−	20.6035i		−	1.48307i	0.670914	−	0.741535i	$-0.265902\pi$
$194$	6.63311			0.476230
$195$	−2.21432	+	25.5669i	−0.158571	+	1.83088i
$196$		0			0
$197$		−	8.36842i		−	0.596225i	−0.954531	−	0.298112i	$-0.903643\pi$
$197$		−	8.36842i		−	0.596225i	0.954531	−	0.298112i	$-0.0963571\pi$
$198$	3.52543			0.250541
$199$	−0.601472			−0.0426372			−0.0213186	−	0.999773i	$-0.506786\pi$
$199$	−0.601472			−0.0426372			−0.0213186	+	0.999773i	$0.506786\pi$
$200$		−	12.9491i		−	0.915643i
$201$			0.949145i			0.0669475i
$202$			9.43356i			0.663743i
$203$		0			0
$204$	12.1334			0.849505
$205$	−2.42864			−0.169624
$206$			8.47013i			0.590142i
$207$	6.24443			0.434018
$208$	−4.94914	−	0.428639i	−0.343161	−	0.0297208i
$209$	22.9797			1.58954
$210$		0			0
$211$	−1.90321			−0.131023			−0.0655113	−	0.997852i	$-0.520868\pi$
$211$	−1.90321			−0.131023			−0.0655113	+	0.997852i	$0.520868\pi$
$212$	3.85236			0.264581
$213$			19.8938i			1.36310i
$214$		−	12.5303i		−	0.856557i
$215$			28.3067i			1.93050i
$216$		−	5.89829i		−	0.401328i
$217$		0			0
$218$	−5.76049			−0.390150
$219$			12.8222i			0.866447i
$220$	−13.1842			−0.888879
$221$	−12.9032	−	1.11753i	−0.867964	−	0.0751733i
$222$	6.00000			0.402694
$223$		−	10.6336i		−	0.712078i	−0.934471	−	0.356039i	$-0.884127\pi$
$223$		−	10.6336i		−	0.712078i	0.934471	−	0.356039i	$-0.115873\pi$
$224$		0			0
$225$	−10.1476			−0.676510
$226$		−	5.83161i		−	0.387913i
$227$		−	15.0509i		−	0.998960i	−0.866325	−	0.499480i	$-0.833524\pi$
$227$		−	15.0509i		−	0.998960i	0.866325	−	0.499480i	$-0.166476\pi$
$228$			28.8671i			1.91177i
$229$		−	6.53480i		−	0.431831i	−0.976412	−	0.215916i	$-0.930726\pi$
$229$		−	6.53480i		−	0.431831i	0.976412	−	0.215916i	$-0.0692736\pi$
$230$	7.26517			0.479051
$231$		0			0
$232$			4.98079i			0.327005i
$233$	27.9590			1.83165			0.915827	−	0.401573i	$-0.131536\pi$
$233$	27.9590			1.83165			0.915827	+	0.401573i	$0.131536\pi$
$234$	0.407896	−	4.70964i	0.0266650	−	0.307879i
$235$	6.05086			0.394714
$236$			11.1842i			0.728030i
$237$	−9.91903			−0.644310
$238$		0			0
$239$		−	19.5812i		−	1.26660i	−0.773905	−	0.633301i	$-0.781699\pi$
$239$		−	19.5812i		−	1.26660i	0.773905	−	0.633301i	$-0.218301\pi$
$240$		−	9.80642i		−	0.633002i
$241$		−	13.3575i		−	0.860433i	−0.902726	−	0.430217i	$-0.858437\pi$
$241$		−	13.3575i		−	0.860433i	0.902726	−	0.430217i	$-0.141563\pi$
$242$			2.59703i			0.166943i
$243$	−17.2652			−1.10756
$244$	−13.8064			−0.883866
$245$		0			0
$246$	1.15257			0.0734849
$247$	2.65878	−	30.6987i	0.169174	−	1.95331i
$248$	14.1748			0.900103
$249$			23.9748i			1.51934i
$250$	−0.734825			−0.0464744
$251$	3.29682			0.208093			0.104047	−	0.994572i	$-0.466821\pi$
$251$	3.29682			0.208093			0.104047	+	0.994572i	$0.466821\pi$
$252$		0			0
$253$		−	8.82225i		−	0.554650i
$254$		−	2.81579i		−	0.176678i
$255$		−	25.5669i		−	1.60106i
$256$	−9.89829			−0.618643
$257$	−23.6938			−1.47798			−0.738990	−	0.673717i	$-0.764697\pi$
$257$	−23.6938			−1.47798			−0.738990	+	0.673717i	$0.764697\pi$
$258$		−	13.4336i		−	0.836337i
$259$		0			0
$260$	−1.52543	+	17.6128i	−0.0946030	+	1.09230i
$261$	3.90321			0.241603
$262$		−	4.09187i		−	0.252796i
$263$	9.99063			0.616049			0.308024	−	0.951378i	$-0.400332\pi$
$263$	9.99063			0.616049			0.308024	+	0.951378i	$0.400332\pi$
$264$	14.4603			0.889969
$265$		−	8.11753i		−	0.498656i
$266$		0			0
$267$			11.8731i			0.726622i
$268$			0.653858i			0.0399408i
$269$	18.2034			1.10988			0.554941	−	0.831890i	$-0.312741\pi$
$269$	18.2034			1.10988			0.554941	+	0.831890i	$0.312741\pi$
$270$	−5.37778			−0.327282
$271$		−	24.1748i		−	1.46852i	−0.678870	−	0.734258i	$-0.737530\pi$
$271$		−	24.1748i		−	1.46852i	0.678870	−	0.734258i	$-0.262470\pi$
$272$	4.94914			0.300086
$273$		0			0
$274$	11.2716			0.680944
$275$			14.3368i			0.864540i
$276$	11.0825			0.667088
$277$	−1.69535			−0.101863			−0.0509317	−	0.998702i	$-0.516219\pi$
$277$	−1.69535			−0.101863			−0.0509317	+	0.998702i	$0.516219\pi$
$278$		−	2.08742i		−	0.125195i
$279$		−	11.1082i		−	0.665028i
$280$		0			0
$281$		−	11.6479i		−	0.694854i	−0.937707	−	0.347427i	$-0.887055\pi$
$281$		−	11.6479i		−	0.694854i	0.937707	−	0.347427i	$-0.112945\pi$
$282$	−2.87157			−0.170999
$283$	12.1334			0.721253			0.360626	−	0.932710i	$-0.382563\pi$
$283$	12.1334			0.721253			0.360626	+	0.932710i	$0.382563\pi$
$284$			13.7047i			0.813225i
$285$	60.8276			3.60312
$286$	−6.65386	−	0.576283i	−0.393451	−	0.0340763i
$287$		0			0
$288$			11.0509i			0.651178i
$289$	−4.09679			−0.240988
$290$	4.54125			0.266671
$291$		−	21.3210i		−	1.24986i
$292$			8.83314i			0.516921i
$293$			11.4538i			0.669140i	0.942371	+	0.334570i	$0.108591\pi$
$293$			11.4538i			0.669140i	−0.942371	+	0.334570i	$0.891409\pi$
$294$		0			0
$295$	23.5669			1.37212
$296$	9.55262			0.555235
$297$			6.53035i			0.378929i
$298$	10.0143			0.580112
$299$	−11.7857	−	1.02074i	−0.681583	−	0.0590311i
$300$	−18.0098			−1.03980
$301$		0			0
$302$	−13.7605			−0.791827
$303$	30.3225			1.74198
$304$			11.7748i			0.675330i
$305$			29.0923i			1.66582i
$306$			4.70964i			0.269232i
$307$			3.96989i			0.226574i	0.993562	+	0.113287i	$0.0361379\pi$
$307$			3.96989i			0.226574i	−0.993562	+	0.113287i	$0.963862\pi$
$308$		0			0
$309$	27.2257			1.54882
$310$		−	12.9240i		−	0.734031i
$311$	27.3481			1.55077			0.775386	−	0.631488i	$-0.217556\pi$
$311$	27.3481			1.55077			0.775386	+	0.631488i	$0.217556\pi$
$312$	1.67307	−	19.3176i	0.0947190	−	1.09364i
$313$	19.1032			1.07978			0.539890	−	0.841736i	$-0.318466\pi$
$313$	19.1032			1.07978			0.539890	+	0.841736i	$0.318466\pi$
$314$		−	5.09679i		−	0.287628i
$315$		0			0
$316$	−6.83314			−0.384394
$317$			9.90813i			0.556496i	0.960509	+	0.278248i	$0.0897538\pi$
$317$			9.90813i			0.556496i	−0.960509	+	0.278248i	$0.910246\pi$
$318$			3.85236i			0.216029i
$319$		−	5.51453i		−	0.308754i
$320$			4.00000i			0.223607i
$321$	−40.2766			−2.24802
$322$		0			0
$323$			30.6987i			1.70812i
$324$	−16.9131			−0.939614
$325$	19.1526	+	1.65878i	1.06239	+	0.0920126i
$326$	1.60300			0.0887821
$327$			18.5161i			1.02394i
$328$	1.83500			0.101321
$329$		0			0
$330$		−	13.1842i		−	0.725767i
$331$			23.4193i			1.28724i	0.765345	+	0.643620i	$0.222568\pi$
$331$			23.4193i			1.28724i	−0.765345	+	0.643620i	$0.777432\pi$
$332$			16.5161i			0.906437i
$333$		−	7.48595i		−	0.410227i
$334$	2.35752			0.128998
$335$	1.37778			0.0752764
$336$		0			0
$337$	7.51606			0.409426			0.204713	−	0.978822i	$-0.434374\pi$
$337$	7.51606			0.409426			0.204713	+	0.978822i	$0.434374\pi$
$338$	−1.53972	+	8.82225i	−0.0837496	+	0.479867i
$339$	−18.7447			−1.01807
$340$		−	17.6128i		−	0.955191i
$341$	−15.6938			−0.849868
$342$	−11.2050			−0.605894
$343$		0			0
$344$		−	21.3876i		−	1.15314i
$345$		−	23.3526i		−	1.25726i
$346$			5.70027i			0.306448i
$347$	5.64449			0.303012			0.151506	−	0.988456i	$-0.451588\pi$
$347$	5.64449			0.303012			0.151506	+	0.988456i	$0.451588\pi$
$348$	6.92735			0.371345
$349$			24.1590i			1.29320i	0.762828	+	0.646601i	$0.223810\pi$
$349$			24.1590i			1.29320i	−0.762828	+	0.646601i	$0.776190\pi$
$350$		0			0
$351$	8.72393	+	0.755569i	0.465649	+	0.0403293i
$352$	15.6128			0.832168
$353$		−	36.1289i		−	1.92295i	−0.274896	−	0.961474i	$-0.588644\pi$
$353$		−	36.1289i		−	1.92295i	0.274896	−	0.961474i	$-0.411356\pi$
$354$	−11.1842			−0.594434
$355$	28.8780			1.53269
$356$			8.17929i			0.433501i
$357$		0			0
$358$			11.1778i			0.590763i
$359$		−	17.4128i		−	0.919013i	−0.888174	−	0.459507i	$-0.848026\pi$
$359$		−	17.4128i		−	0.919013i	0.888174	−	0.459507i	$-0.151974\pi$
$360$	14.8573			0.783047
$361$	−54.0370			−2.84405
$362$		−	6.34122i		−	0.333287i
$363$	8.34767			0.438139
$364$		0			0
$365$	18.6128			0.974241
$366$		−	13.8064i		−	0.721673i
$367$	−2.93825			−0.153375			−0.0766876	−	0.997055i	$-0.524434\pi$
$367$	−2.93825			−0.153375			−0.0766876	+	0.997055i	$0.524434\pi$
$368$	4.52051			0.235648
$369$		−	1.43801i		−	0.0748597i
$370$		−	8.70964i		−	0.452792i
$371$		0			0
$372$		−	19.7146i		−	1.02215i
$373$	−18.7699			−0.971866			−0.485933	−	0.873996i	$-0.661520\pi$
$373$	−18.7699			−0.971866			−0.485933	+	0.873996i	$0.661520\pi$
$374$	6.65386			0.344063
$375$			2.36196i			0.121971i
$376$	−4.57184			−0.235774
$377$	−7.36689	−	0.638037i	−0.379414	−	0.0328606i
$378$		0			0
$379$			23.6894i			1.21684i	0.793615	+	0.608421i	$0.208197\pi$
$379$			23.6894i			1.21684i	−0.793615	+	0.608421i	$0.791803\pi$
$380$	41.9037			2.14961
$381$	−9.05086			−0.463689
$382$		−	4.59364i		−	0.235031i
$383$		−	31.6128i		−	1.61534i	−0.589634	−	0.807671i	$-0.700728\pi$
$383$		−	31.6128i		−	1.61534i	0.589634	−	0.807671i	$-0.299272\pi$
$384$			23.8163i			1.21537i
$385$		0			0
$386$	14.1936			0.722434
$387$	−16.7605			−0.851984
$388$		−	14.6878i		−	0.745662i
$389$	20.0558			1.01687			0.508434	−	0.861101i	$-0.330225\pi$
$389$	20.0558			1.01687			0.508434	+	0.861101i	$0.330225\pi$
$390$	−17.6128	−	1.52543i	−0.891861	−	0.0772430i
$391$	11.7857			0.596027
$392$		0			0
$393$	−13.1526			−0.663459
$394$	5.76494			0.290433
$395$			14.3985i			0.724469i
$396$		−	7.80642i		−	0.392288i
$397$			22.8731i			1.14797i	0.818867	+	0.573984i	$0.194603\pi$
$397$			22.8731i			1.14797i	−0.818867	+	0.573984i	$0.805397\pi$
$398$		−	0.414349i		−	0.0207695i
$399$		0			0
$400$	−7.34614			−0.367307
$401$			5.61285i			0.280292i	0.990131	+	0.140146i	$0.0447572\pi$
$401$			5.61285i			0.280292i	−0.990131	+	0.140146i	$0.955243\pi$
$402$	−0.653858			−0.0326115
$403$	−1.81579	+	20.9654i	−0.0904510	+	1.04436i
$404$	20.8889			1.03926
$405$			35.6385i			1.77089i
$406$		0			0
$407$	−10.5763			−0.524247
$408$			19.3176i			0.956362i
$409$			26.1175i			1.29143i	0.763579	+	0.645714i	$0.223440\pi$
$409$			26.1175i			1.29143i	−0.763579	+	0.645714i	$0.776560\pi$
$410$		−	1.67307i		−	0.0826271i
$411$		−	36.2306i		−	1.78712i
$412$	18.7556			0.924021
$413$		0			0
$414$			4.30174i			0.211419i
$415$	34.8020			1.70836
$416$	1.80642	−	20.8573i	0.0885672	−	1.02261i
$417$	−6.70964			−0.328572
$418$			15.8306i			0.774298i
$419$	25.2464			1.23337			0.616685	−	0.787210i	$-0.288475\pi$
$419$	25.2464			1.23337			0.616685	+	0.787210i	$0.288475\pi$
$420$		0			0
$421$		−	8.22861i		−	0.401038i	−0.979690	−	0.200519i	$-0.935737\pi$
$421$		−	8.22861i		−	0.401038i	0.979690	−	0.200519i	$-0.0642628\pi$
$422$		−	1.31111i		−	0.0638237i
$423$			3.58274i			0.174199i
$424$			6.13335i			0.297862i
$425$	−19.1526			−0.929036
$426$	−13.7047			−0.663996
$427$		0			0
$428$	−27.7462			−1.34116
$429$	−1.85236	+	21.3876i	−0.0894326	+	1.03260i
$430$	−19.5002			−0.940385
$431$		−	1.43801i		−	0.0692664i	−0.999400	−	0.0346332i	$-0.988974\pi$
$431$		−	1.43801i		−	0.0692664i	0.999400	−	0.0346332i	$-0.0110263\pi$
$432$	−3.34614			−0.160991
$433$	−16.5018			−0.793024			−0.396512	−	0.918029i	$-0.629780\pi$
$433$	−16.5018			−0.793024			−0.396512	+	0.918029i	$0.629780\pi$
$434$		0			0
$435$		−	14.5970i		−	0.699874i
$436$			12.7556i			0.610881i
$437$			28.0400i			1.34133i
$438$	−8.83314			−0.422064
$439$	−10.1619			−0.485003			−0.242501	−	0.970151i	$-0.577968\pi$
$439$	−10.1619			−0.485003			−0.242501	+	0.970151i	$0.577968\pi$
$440$		−	20.9906i		−	1.00069i
$441$		0			0
$442$	0.769859	−	8.88892i	0.0366185	−	0.422803i
$443$	3.20787			0.152410			0.0762052	−	0.997092i	$-0.475720\pi$
$443$	3.20787			0.152410			0.0762052	+	0.997092i	$0.475720\pi$
$444$		−	13.2859i		−	0.630522i
$445$	17.2351			0.817020
$446$	7.32540			0.346868
$447$		−	32.1891i		−	1.52249i
$448$		0			0
$449$			18.4099i			0.868817i	0.900716	+	0.434409i	$0.143043\pi$
$449$			18.4099i			0.868817i	−0.900716	+	0.434409i	$0.856957\pi$
$450$		−	6.99063i		−	0.329542i
$451$	−2.03164			−0.0956663
$452$	−12.9131			−0.607379
$453$			44.2306i			2.07814i
$454$	10.3684			0.486614
$455$		0			0
$456$	−45.9595			−2.15225
$457$			3.40297i			0.159184i	0.996828	+	0.0795922i	$0.0253618\pi$
$457$			3.40297i			0.159184i	−0.996828	+	0.0795922i	$0.974638\pi$
$458$	4.50177			0.210354
$459$	−8.72393			−0.407198
$460$		−	16.0874i		−	0.750080i
$461$		−	17.5714i		−	0.818380i	−0.912449	−	0.409190i	$-0.865811\pi$
$461$		−	17.5714i		−	0.818380i	0.912449	−	0.409190i	$-0.134189\pi$
$462$		0			0
$463$		−	15.7714i		−	0.732959i	−0.930426	−	0.366479i	$-0.880563\pi$
$463$		−	15.7714i		−	0.732959i	0.930426	−	0.366479i	$-0.119437\pi$
$464$	2.82564			0.131177
$465$	−41.5417			−1.92645
$466$			19.2607i			0.892236i
$467$	−3.76694			−0.174313			−0.0871567	−	0.996195i	$-0.527778\pi$
$467$	−3.76694			−0.174313			−0.0871567	+	0.996195i	$0.527778\pi$
$468$	−10.4286	−	0.903212i	−0.482064	−	0.0417510i
$469$		0			0
$470$			4.16839i			0.192273i
$471$	−16.3827			−0.754875
$472$	−17.8064			−0.819607
$473$			23.6795i			1.08879i
$474$		−	6.83314i		−	0.313857i
$475$		−	45.5669i		−	2.09075i
$476$		0			0
$477$	4.80642			0.220071
$478$	13.4893			0.616988
$479$		−	12.1032i		−	0.553011i	−0.961012	−	0.276506i	$-0.910824\pi$
$479$		−	12.1032i		−	0.553011i	0.961012	−	0.276506i	$-0.0891764\pi$
$480$	41.3274			1.88633
$481$	−1.22369	+	14.1289i	−0.0557954	+	0.644223i
$482$	9.20189			0.419135
$483$		0			0
$484$	5.75065			0.261393
$485$	−30.9496			−1.40535
$486$		−	11.8938i		−	0.539516i
$487$		−	17.3778i		−	0.787463i	−0.919226	−	0.393731i	$-0.871184\pi$
$487$		−	17.3778i		−	0.787463i	0.919226	−	0.393731i	$-0.128816\pi$
$488$		−	21.9813i		−	0.995045i
$489$		−	5.15257i		−	0.233007i
$490$		0			0
$491$	19.3921			0.875152			0.437576	−	0.899181i	$-0.355837\pi$
$491$	19.3921			0.875152			0.437576	+	0.899181i	$0.355837\pi$
$492$		−	2.55215i		−	0.115060i
$493$	7.36689			0.331788
$494$	21.1481	+	1.83161i	0.951499	+	0.0824082i
$495$	−16.4494			−0.739345
$496$		−	8.04149i		−	0.361073i
$497$		0			0
$498$	−16.5161			−0.740102
$499$		−	10.6702i		−	0.477662i	−0.971061	−	0.238831i	$-0.923236\pi$
$499$		−	10.6702i		−	0.477662i	0.971061	−	0.238831i	$-0.0767642\pi$
$500$			1.62714i			0.0727678i
$501$		−	7.57781i		−	0.338552i
$502$			2.27115i			0.101366i
$503$	4.27655			0.190682			0.0953410	−	0.995445i	$-0.469606\pi$
$503$	4.27655			0.190682			0.0953410	+	0.995445i	$0.469606\pi$
$504$		0			0
$505$		−	44.0163i		−	1.95870i
$506$	6.07758			0.270181
$507$	28.3575	+	4.94914i	1.25940	+	0.219799i
$508$	−6.23506			−0.276636
$509$		−	33.0765i		−	1.46609i	−0.680180	−	0.733046i	$-0.738098\pi$
$509$		−	33.0765i		−	1.46609i	0.680180	−	0.733046i	$-0.261902\pi$
$510$	17.6128			0.779910
$511$		0			0
$512$			14.6923i			0.649313i
$513$		−	20.7556i		−	0.916381i
$514$		−	16.3225i		−	0.719954i
$515$		−	39.5210i		−	1.74150i
$516$	−29.7462			−1.30950
$517$	5.06175			0.222616
$518$		0			0
$519$	18.3225			0.804268
$520$	−28.0415	−	2.42864i	−1.22970	−	0.106503i
$521$	−24.3783			−1.06803			−0.534015	−	0.845475i	$-0.679318\pi$
$521$	−24.3783			−1.06803			−0.534015	+	0.845475i	$0.679318\pi$
$522$			2.68889i			0.117690i
$523$	−34.9403			−1.52783			−0.763915	−	0.645317i	$-0.776725\pi$
$523$	−34.9403			−1.52783			−0.763915	+	0.645317i	$0.776725\pi$
$524$	−9.06070			−0.395818
$525$		0			0
$526$			6.88247i			0.300090i
$527$		−	20.9654i		−	0.913269i
$528$		−	8.20342i		−	0.357008i
$529$	−12.2351			−0.531959
$530$	5.59210			0.242905
$531$			13.9541i			0.605555i
$532$		0			0
$533$	−0.235063	+	2.71408i	−0.0101817	+	0.117560i
$534$	−8.17929			−0.353952
$535$			58.4657i			2.52769i
$536$	−1.04101			−0.0449648
$537$	35.9289			1.55045
$538$			12.5402i			0.540646i
$539$		0			0
$540$			11.9081i			0.512445i
$541$			30.4953i			1.31110i	0.755153	+	0.655548i	$0.227562\pi$
$541$			30.4953i			1.31110i	−0.755153	+	0.655548i	$0.772438\pi$
$542$	16.6539			0.715344
$543$	−20.3827			−0.874706
$544$			20.8573i			0.894248i
$545$	26.8780			1.15133
$546$		0			0
$547$	−10.0049			−0.427780			−0.213890	−	0.976858i	$-0.568613\pi$
$547$	−10.0049			−0.427780			−0.213890	+	0.976858i	$0.568613\pi$
$548$		−	24.9590i		−	1.06620i
$549$	−17.2257			−0.735175
$550$	−9.87649			−0.421135
$551$			17.5270i			0.746674i
$552$			17.6445i			0.750999i
$553$		0			0
$554$		−	1.16791i		−	0.0496198i
$555$	−27.9956			−1.18835
$556$	−4.62222			−0.196026
$557$		−	14.1936i		−	0.601401i	−0.953719	−	0.300701i	$-0.902780\pi$
$557$		−	14.1936i		−	0.601401i	0.953719	−	0.300701i	$-0.0972205\pi$
$558$	7.65233			0.323949
$559$	31.6336	+	2.73975i	1.33796	+	0.115879i
$560$		0			0
$561$		−	21.3876i		−	0.902986i
$562$	8.02413			0.338478
$563$	−21.3590			−0.900177			−0.450088	−	0.892984i	$-0.648607\pi$
$563$	−21.3590			−0.900177			−0.450088	+	0.892984i	$0.648607\pi$
$564$			6.35857i			0.267744i
$565$			27.2099i			1.14473i
$566$			8.35857i			0.351337i
$567$		0			0
$568$	−21.8193			−0.915519
$569$	−34.2672			−1.43656			−0.718278	−	0.695757i	$-0.755069\pi$
$569$	−34.2672			−1.43656			−0.718278	+	0.695757i	$0.755069\pi$
$570$			41.9037i			1.75515i
$571$	37.6494			1.57558			0.787789	−	0.615945i	$-0.211226\pi$
$571$	37.6494			1.57558			0.787789	+	0.615945i	$0.211226\pi$
$572$	−1.27607	+	14.7338i	−0.0533553	+	0.616050i
$573$	−14.7654			−0.616834
$574$		0			0
$575$	−17.4938			−0.729541
$576$	−2.36842			−0.0986840
$577$		−	28.3970i		−	1.18218i	−0.806605	−	0.591091i	$-0.798697\pi$
$577$		−	28.3970i		−	1.18218i	0.806605	−	0.591091i	$-0.201303\pi$
$578$		−	2.82225i		−	0.117390i
$579$		−	45.6227i		−	1.89601i
$580$		−	10.0558i		−	0.417543i
$581$		0			0
$582$	14.6878			0.608830
$583$		−	6.79060i		−	0.281238i
$584$	−14.0633			−0.581943
$585$	−1.90321	+	21.9748i	−0.0786881	+	0.908547i
$586$	−7.89045			−0.325952
$587$			6.23659i			0.257412i	0.991683	+	0.128706i	$0.0410823\pi$
$587$			6.23659i			0.257412i	−0.991683	+	0.128706i	$0.958918\pi$
$588$		0			0
$589$	49.8800			2.05527
$590$			16.2351i			0.668387i
$591$		−	18.5303i		−	0.762237i
$592$		−	5.41927i		−	0.222731i
$593$			17.5698i			0.721506i	0.932661	+	0.360753i	$0.117480\pi$
$593$			17.5698i			0.721506i	−0.932661	+	0.360753i	$0.882520\pi$
$594$	−4.49871			−0.184584
$595$		0			0
$596$		−	22.1748i		−	0.908317i
$597$	−1.33185			−0.0545090
$598$	0.703182	−	8.11906i	0.0287553	−	0.332013i
$599$	16.9813			0.693836			0.346918	−	0.937896i	$-0.387228\pi$
$599$	16.9813			0.693836			0.346918	+	0.937896i	$0.387228\pi$
$600$		−	28.6735i		−	1.17059i
$601$	18.7052			0.763001			0.381500	−	0.924369i	$-0.375408\pi$
$601$	18.7052			0.763001			0.381500	+	0.924369i	$0.375408\pi$
$602$		0			0
$603$			0.815792i			0.0332216i
$604$			30.4701i			1.23981i
$605$		−	12.1175i		−	0.492648i
$606$			20.8889i			0.848554i
$607$	10.3575			0.420399			0.210199	−	0.977659i	$-0.432589\pi$
$607$	10.3575			0.420399			0.210199	+	0.977659i	$0.432589\pi$
$608$	−49.6227			−2.01247
$609$		0			0
$610$	−20.0415			−0.811456
$611$	0.585651	−	6.76202i	0.0236929	−	0.273562i
$612$	10.4286			0.421553
$613$		−	7.02227i		−	0.283627i	−0.989893	−	0.141814i	$-0.954707\pi$
$613$		−	7.02227i		−	0.283627i	0.989893	−	0.141814i	$-0.0452933\pi$
$614$	−2.73483			−0.110369
$615$	−5.37778			−0.216853
$616$		0			0
$617$			29.9813i			1.20700i	0.797363	+	0.603500i	$0.206228\pi$
$617$			29.9813i			1.20700i	−0.797363	+	0.603500i	$0.793772\pi$
$618$			18.7556i			0.754460i
$619$			0.0285802i			0.00114874i	1.00000		0.000574368i	$0.000182827\pi$
$619$			0.0285802i			0.00114874i	−1.00000		0.000574368i	$0.999817\pi$
$620$	−28.6178			−1.14932
$621$	−7.96836			−0.319759
$622$			18.8399i			0.755412i
$623$		0			0
$624$	−10.9590	−	0.949145i	−0.438711	−	0.0379962i
$625$	−23.2306			−0.929225
$626$			13.1601i			0.525982i
$627$	50.8845			2.03213
$628$	−11.2859			−0.450357
$629$		−	14.1289i		−	0.563356i
$630$		0			0
$631$			12.1936i			0.485419i	0.970099	+	0.242709i	$0.0780361\pi$
$631$			12.1936i			0.485419i	−0.970099	+	0.242709i	$0.921964\pi$
$632$		−	10.8791i		−	0.432746i
$633$	−4.21432			−0.167504
$634$	−6.82564			−0.271081
$635$			13.1383i			0.521377i
$636$	8.53035			0.338250
$637$		0			0
$638$	3.79892			0.150401
$639$			17.0988i			0.676418i
$640$	34.5718			1.36657
$641$	2.82516			0.111587			0.0557935	−	0.998442i	$-0.482231\pi$
$641$	2.82516			0.111587			0.0557935	+	0.998442i	$0.482231\pi$
$642$		−	27.7462i		−	1.09506i
$643$			37.7275i			1.48783i	0.668276	+	0.743913i	$0.267032\pi$
$643$			37.7275i			1.48783i	−0.668276	+	0.743913i	$0.732968\pi$
$644$		0			0
$645$			62.6800i			2.46802i
$646$	−21.1481			−0.832062
$647$	12.3664			0.486174			0.243087	−	0.970005i	$-0.421840\pi$
$647$	12.3664			0.486174			0.243087	+	0.970005i	$0.421840\pi$
$648$		−	26.9273i		−	1.05781i
$649$	19.7146			0.773864
$650$	−1.14272	+	13.1941i	−0.0448212	+	0.517513i
$651$		0			0
$652$		−	3.54956i		−	0.139012i
$653$	−33.0005			−1.29141			−0.645704	−	0.763588i	$-0.723436\pi$
$653$	−33.0005			−1.29141			−0.645704	+	0.763588i	$0.723436\pi$
$654$	−12.7556			−0.498782
$655$			19.0923i			0.746000i
$656$		−	1.04101i		−	0.0406446i
$657$			11.0207i			0.429960i
$658$		0			0
$659$	32.8118			1.27817			0.639084	−	0.769137i	$-0.279314\pi$
$659$	32.8118			1.27817			0.639084	+	0.769137i	$0.279314\pi$
$660$	−29.1941			−1.13638
$661$		−	14.7067i		−	0.572025i	−0.958226	−	0.286013i	$-0.907670\pi$
$661$		−	14.7067i		−	0.572025i	0.958226	−	0.286013i	$-0.0923299\pi$
$662$	−16.1334			−0.627041
$663$	−28.5718	−	2.47457i	−1.10964	−	0.0961044i
$664$	−26.2953			−1.02046
$665$		0			0
$666$	5.15701			0.199830
$667$	6.72885			0.260542
$668$		−	5.22030i		−	0.201979i
$669$		−	23.5462i		−	0.910348i
$670$			0.949145i			0.0366687i
$671$			24.3368i			0.939511i
$672$		0			0
$673$	−21.2908			−0.820702			−0.410351	−	0.911928i	$-0.634594\pi$
$673$	−21.2908			−0.820702			−0.410351	+	0.911928i	$0.634594\pi$
$674$			5.17775i			0.199440i
$675$	12.9491			0.498413
$676$	19.5353	+	3.40943i	0.751357	+	0.131132i
$677$	3.07160			0.118051			0.0590256	−	0.998256i	$-0.481201\pi$
$677$	3.07160			0.118051			0.0590256	+	0.998256i	$0.481201\pi$
$678$		−	12.9131i		−	0.495923i
$679$		0			0
$680$	28.0415			1.07534
$681$		−	33.3274i		−	1.27711i
$682$		−	10.8113i		−	0.413988i
$683$			24.7971i			0.948833i	0.880301	+	0.474416i	$0.157341\pi$
$683$			24.7971i			0.948833i	−0.880301	+	0.474416i	$0.842659\pi$
$684$			24.8113i			0.948686i
$685$	−52.5926			−2.00946
$686$		0			0
$687$		−	14.4701i		−	0.552070i
$688$	−12.1334			−0.462580
$689$	−9.07160	−	0.785680i	−0.345600	−	0.0299320i
$690$	16.0874			0.612438
$691$			27.0953i			1.03075i	0.856964	+	0.515376i	$0.172348\pi$
$691$			27.0953i			1.03075i	−0.856964	+	0.515376i	$0.827652\pi$
$692$	12.6222			0.479825
$693$		0			0
$694$			3.88845i			0.147603i
$695$			9.73975i			0.369450i
$696$			11.0291i			0.418055i
$697$		−	2.71408i		−	0.102803i
$698$	−16.6430			−0.629945
$699$	61.9101			2.34166
$700$		0			0
$701$	−13.5205			−0.510662			−0.255331	−	0.966854i	$-0.582185\pi$
$701$	−13.5205			−0.510662			−0.255331	+	0.966854i	$0.582185\pi$
$702$	−0.520505	+	6.00984i	−0.0196452	+	0.226827i
$703$	33.6149			1.26781
$704$			3.34614i			0.126112i
$705$	13.3985			0.504618
$706$	24.8889			0.936707
$707$		0			0
$708$			24.7654i			0.930741i
$709$			50.9753i			1.91442i	0.289399	+	0.957209i	$0.406545\pi$
$709$			50.9753i			1.91442i	−0.289399	+	0.957209i	$0.593455\pi$
$710$			19.8938i			0.746603i
$711$	−8.52543			−0.319729
$712$	−13.0223			−0.488030
$713$		−	19.1497i		−	0.717160i
$714$		0			0
$715$	31.0464	+	2.68889i	1.16107	+	0.100559i
$716$	24.7511			0.924993
$717$		−	43.3590i		−	1.61927i
$718$	11.9956			0.447670
$719$	41.5417			1.54924			0.774622	−	0.632424i	$-0.217940\pi$
$719$	41.5417			1.54924			0.774622	+	0.632424i	$0.217940\pi$
$720$		−	8.42864i		−	0.314117i
$721$		0			0
$722$		−	37.2257i		−	1.38540i
$723$		−	29.5778i		−	1.10001i
$724$	−14.0415			−0.521848
$725$	−10.9349			−0.406110
$726$			5.75065i			0.213427i
$727$	−20.8988			−0.775092			−0.387546	−	0.921850i	$-0.626677\pi$
$727$	−20.8988			−0.775092			−0.387546	+	0.921850i	$0.626677\pi$
$728$		0			0
$729$	−4.96836			−0.184013
$730$			12.8222i			0.474573i
$731$	−31.6336			−1.17001
$732$	−30.5718			−1.12997
$733$		−	19.3575i		−	0.714986i	−0.933916	−	0.357493i	$-0.883632\pi$
$733$		−	19.3575i		−	0.714986i	0.933916	−	0.357493i	$-0.116368\pi$
$734$		−	2.02413i		−	0.0747122i
$735$		0			0
$736$			19.0509i			0.702224i
$737$	1.15257			0.0424553
$738$	0.990632			0.0364657
$739$			1.31111i			0.0482299i	0.999709	+	0.0241149i	$0.00767677\pi$
$739$			1.31111i			0.0482299i	−0.999709	+	0.0241149i	$0.992323\pi$
$740$	−19.2859			−0.708964
$741$	5.88739	−	67.9768i	0.216279	−	2.49719i
$742$		0			0
$743$		−	21.5210i		−	0.789528i	−0.918783	−	0.394764i	$-0.870826\pi$
$743$		−	21.5210i		−	0.789528i	0.918783	−	0.394764i	$-0.129174\pi$
$744$	31.3876			1.15073
$745$	−46.7259			−1.71191
$746$		−	12.9304i		−	0.473416i
$747$			20.6064i			0.753949i
$748$		−	14.7338i		−	0.538720i
$749$		0			0
$750$	−1.62714			−0.0594147
$751$	16.3176			0.595436			0.297718	−	0.954654i	$-0.403774\pi$
$751$	16.3176			0.595436			0.297718	+	0.954654i	$0.403774\pi$
$752$			2.59364i			0.0945802i
$753$	7.30021			0.266034
$754$	0.439539	−	5.07499i	0.0160071	−	0.184820i
$755$	64.2054			2.33667
$756$		0			0
$757$	−18.7462			−0.681342			−0.340671	−	0.940183i	$-0.610654\pi$
$757$	−18.7462			−0.681342			−0.340671	+	0.940183i	$0.610654\pi$
$758$	−16.3194			−0.592748
$759$		−	19.5353i		−	0.709085i
$760$			66.7150i			2.42001i
$761$		−	10.2968i		−	0.373259i	−0.982430	−	0.186630i	$-0.940244\pi$
$761$		−	10.2968i		−	0.373259i	0.982430	−	0.186630i	$-0.0597564\pi$
$762$		−	6.23506i		−	0.225873i
$763$		0			0
$764$	−10.1718			−0.368002
$765$		−	21.9748i		−	0.794501i
$766$	21.7778			0.786865
$767$	2.28100	−	26.3368i	0.0823620	−	0.950966i
$768$	−21.9180			−0.790897
$769$		−	9.36641i		−	0.337761i	−0.985637	−	0.168881i	$-0.945985\pi$
$769$		−	9.36641i		−	0.337761i	0.985637	−	0.168881i	$-0.0540153\pi$
$770$		0			0
$771$	−52.4657			−1.88951
$772$		−	31.4291i		−	1.13116i
$773$			3.71456i			0.133603i	0.997766	+	0.0668017i	$0.0212795\pi$
$773$			3.71456i			0.133603i	−0.997766	+	0.0668017i	$0.978721\pi$
$774$		−	11.5462i		−	0.415019i
$775$			31.1195i			1.11785i
$776$	23.3846			0.839457
$777$		0			0
$778$			13.8163i			0.495337i
$779$	6.45722			0.231354
$780$	−3.37778	+	39.0005i	−0.120944	+	1.39644i
$781$	24.1575			0.864423
$782$			8.11906i			0.290337i
$783$	−4.98079			−0.177999
$784$		0			0
$785$			23.7812i			0.848789i
$786$		−	9.06070i		−	0.323184i
$787$			6.23659i			0.222311i	0.993803	+	0.111155i	$0.0354551\pi$
$787$			6.23659i			0.222311i	−0.993803	+	0.111155i	$0.964545\pi$

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:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.688892i q^{2} +2.21432 q^{3} +1.52543 q^{4} -3.21432i q^{5} +1.52543i q^{6} +2.42864i q^{8} +1.90321 q^{9} +O(q^{10})\)	\(q+0.688892i q^{2} +2.21432 q^{3} +1.52543 q^{4} -3.21432i q^{5} +1.52543i q^{6} +2.42864i q^{8} +1.90321 q^{9} +2.21432 q^{10} -2.68889i q^{11} +3.37778 q^{12} +(-3.59210 - 0.311108i) q^{13} -7.11753i q^{15} +1.37778 q^{16} +3.59210 q^{17} +1.31111i q^{18} +8.54617i q^{19} -4.90321i q^{20} +1.85236 q^{22} +3.28100 q^{23} +5.37778i q^{24} -5.33185 q^{25} +(0.214320 - 2.47457i) q^{26} -2.42864 q^{27} +2.05086 q^{29} +4.90321 q^{30} -5.83654i q^{31} +5.80642i q^{32} -5.95407i q^{33} +2.47457i q^{34} +2.90321 q^{36} -3.93332i q^{37} -5.88739 q^{38} +(-7.95407 - 0.688892i) q^{39} +7.80642 q^{40} -0.755569i q^{41} -8.80642 q^{43} -4.10171i q^{44} -6.11753i q^{45} +2.26025i q^{46} +1.88247i q^{47} +3.05086 q^{48} -3.67307i q^{50} +7.95407 q^{51} +(-5.47949 - 0.474572i) q^{52} +2.52543 q^{53} -1.67307i q^{54} -8.64296 q^{55} +18.9240i q^{57} +1.41282i q^{58} +7.33185i q^{59} -10.8573i q^{60} -9.05086 q^{61} +4.02074 q^{62} -1.24443 q^{64} +(-1.00000 + 11.5462i) q^{65} +4.10171 q^{66} +0.428639i q^{67} +5.47949 q^{68} +7.26517 q^{69} +8.98418i q^{71} +4.62222i q^{72} +5.79060i q^{73} +2.70964 q^{74} -11.8064 q^{75} +13.0366i q^{76} +(0.474572 - 5.47949i) q^{78} -4.47949 q^{79} -4.42864i q^{80} -11.0874 q^{81} +0.520505 q^{82} +10.8272i q^{83} -11.5462i q^{85} -6.06668i q^{86} +4.54125 q^{87} +6.53035 q^{88} +5.36196i q^{89} +4.21432 q^{90} +5.00492 q^{92} -12.9240i q^{93} -1.29682 q^{94} +27.4701 q^{95} +12.8573i q^{96} -9.62867i q^{97} -5.11753i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{4} - 2 q^{9}+O(q^{10}) \) 6 * q - 4 * q^4 - 2 * q^9	\( 6 q - 4 q^{4} - 2 q^{9} + 20 q^{12} - 8 q^{13} + 8 q^{16} + 8 q^{17} + 24 q^{22} + 6 q^{23} + 8 q^{25} - 12 q^{26} + 12 q^{27} - 14 q^{29} + 16 q^{30} + 4 q^{36} + 4 q^{38} - 8 q^{39} + 20 q^{40} - 26 q^{43} - 8 q^{48} + 8 q^{51} + 20 q^{52} + 2 q^{53} - 12 q^{55} - 28 q^{61} - 16 q^{62} - 8 q^{64} - 6 q^{65} - 28 q^{66} - 20 q^{68} + 4 q^{69} - 24 q^{74} - 44 q^{75} + 16 q^{78} + 26 q^{79} - 26 q^{81} + 56 q^{82} + 40 q^{87} - 40 q^{88} + 12 q^{90} - 36 q^{92} - 20 q^{94} + 58 q^{95}+O(q^{100}) \) 6 * q - 4 * q^4 - 2 * q^9 + 20 * q^12 - 8 * q^13 + 8 * q^16 + 8 * q^17 + 24 * q^22 + 6 * q^23 + 8 * q^25 - 12 * q^26 + 12 * q^27 - 14 * q^29 + 16 * q^30 + 4 * q^36 + 4 * q^38 - 8 * q^39 + 20 * q^40 - 26 * q^43 - 8 * q^48 + 8 * q^51 + 20 * q^52 + 2 * q^53 - 12 * q^55 - 28 * q^61 - 16 * q^62 - 8 * q^64 - 6 * q^65 - 28 * q^66 - 20 * q^68 + 4 * q^69 - 24 * q^74 - 44 * q^75 + 16 * q^78 + 26 * q^79 - 26 * q^81 + 56 * q^82 + 40 * q^87 - 40 * q^88 + 12 * q^90 - 36 * q^92 - 20 * q^94 + 58 * q^95

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