LMFDB - Embedded newform 637.2.c.d.246.1

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			246.1
Root			$0.403032 + 0.403032i$ of defining polynomial
Character	$\chi$	$=$	637.246
Dual form			637.2.c.d.246.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.48119i q^{2} -1.67513 q^{3} -4.15633 q^{4} -0.675131i q^{5} +4.15633i q^{6} +5.35026i q^{8} -0.193937 q^{9} +O(q^{10})$	\(q-2.48119i q^{2} -1.67513 q^{3} -4.15633 q^{4} -0.675131i q^{5} +4.15633i q^{6} +5.35026i q^{8} -0.193937 q^{9} -1.67513 q^{10} +4.48119i q^{11} +6.96239 q^{12} +(-3.28726 - 1.48119i) q^{13} +1.13093i q^{15} +4.96239 q^{16} +3.28726 q^{17} +0.481194i q^{18} +5.21933i q^{19} +2.80606i q^{20} +11.1187 q^{22} +4.76845 q^{23} -8.96239i q^{24} +4.54420 q^{25} +(-3.67513 + 8.15633i) q^{26} +5.35026 q^{27} -9.31265 q^{29} +2.80606 q^{30} -1.63752i q^{31} -1.61213i q^{32} -7.50659i q^{33} -8.15633i q^{34} +0.806063 q^{36} -1.44358i q^{37} +12.9502 q^{38} +(5.50659 + 2.48119i) q^{39} +3.61213 q^{40} +7.92478i q^{41} -4.61213 q^{43} -18.6253i q^{44} +0.130933i q^{45} -11.8315i q^{46} -7.86907i q^{47} -8.31265 q^{48} -11.2750i q^{50} -5.50659 q^{51} +(13.6629 + 6.15633i) q^{52} -3.15633 q^{53} -13.2750i q^{54} +3.02539 q^{55} -8.74306i q^{57} +23.1065i q^{58} +2.54420i q^{59} -4.70052i q^{60} +2.31265 q^{61} -4.06300 q^{62} +5.92478 q^{64} +(-1.00000 + 2.21933i) q^{65} -18.6253 q^{66} +7.35026i q^{67} -13.6629 q^{68} -7.98778 q^{69} +7.75623i q^{71} -1.03761i q^{72} +15.1441i q^{73} -3.58181 q^{74} -7.61213 q^{75} -21.6932i q^{76} +(6.15633 - 13.6629i) q^{78} +14.6629 q^{79} -3.35026i q^{80} -8.38058 q^{81} +19.6629 q^{82} +1.45088i q^{83} -2.21933i q^{85} +11.4436i q^{86} +15.5999 q^{87} -23.9756 q^{88} +7.79384i q^{89} +0.324869 q^{90} -19.8192 q^{92} +2.74306i q^{93} -19.5247 q^{94} +3.52373 q^{95} +2.70052i q^{96} +17.9805i q^{97} -0.869067i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	2.48119i		−	1.75447i	−0.480062	−	0.877235i	$-0.659386\pi$
$2$		−	2.48119i		−	1.75447i	0.480062	−	0.877235i	$-0.340614\pi$
$3$	−1.67513			−0.967137			−0.483569	−	0.875306i	$-0.660660\pi$
$3$	−1.67513			−0.967137			−0.483569	+	0.875306i	$0.660660\pi$
$4$	−4.15633			−2.07816
$5$		−	0.675131i		−	0.301928i	−0.988539	−	0.150964i	$-0.951762\pi$
$5$		−	0.675131i		−	0.301928i	0.988539	−	0.150964i	$-0.0482377\pi$
$6$			4.15633i			1.69681i
$7$		0			0
$7$		0			0
$8$			5.35026i			1.89160i
$9$	−0.193937			−0.0646455
$10$	−1.67513			−0.529723
$11$			4.48119i			1.35113i	0.737300	+	0.675565i	$0.236100\pi$
$11$			4.48119i			1.35113i	−0.737300	+	0.675565i	$0.763900\pi$
$12$	6.96239			2.00987
$13$	−3.28726	−	1.48119i	−0.911721	−	0.410809i
$13$	−3.28726	−	1.48119i	−0.911721	−	0.410809i
$14$		0			0
$15$			1.13093i			0.292006i
$16$	4.96239			1.24060
$17$	3.28726			0.797277			0.398639	−	0.917108i	$-0.369483\pi$
$17$	3.28726			0.797277			0.398639	+	0.917108i	$0.369483\pi$
$18$			0.481194i			0.113419i
$19$			5.21933i			1.19740i	0.800975	+	0.598698i	$0.204315\pi$
$19$			5.21933i			1.19740i	−0.800975	+	0.598698i	$0.795685\pi$
$20$			2.80606i			0.627455i
$21$		0			0
$22$	11.1187			2.37052
$23$	4.76845			0.994291			0.497145	−	0.867667i	$-0.334382\pi$
$23$	4.76845			0.994291			0.497145	+	0.867667i	$0.334382\pi$
$24$		−	8.96239i		−	1.82944i
$25$	4.54420			0.908840
$26$	−3.67513	+	8.15633i	−0.720752	+	1.59959i
$27$	5.35026			1.02966
$28$		0			0
$29$	−9.31265			−1.72932			−0.864658	−	0.502361i	$-0.832465\pi$
$29$	−9.31265			−1.72932			−0.864658	+	0.502361i	$0.832465\pi$
$30$	2.80606			0.512315
$31$		−	1.63752i		−	0.294107i	−0.989129	−	0.147054i	$-0.953021\pi$
$31$		−	1.63752i		−	0.294107i	0.989129	−	0.147054i	$-0.0469790\pi$
$32$		−	1.61213i		−	0.284986i
$33$		−	7.50659i		−	1.30673i
$34$		−	8.15633i		−	1.39880i
$35$		0			0
$36$	0.806063			0.134344
$37$		−	1.44358i		−	0.237324i	−0.992935	−	0.118662i	$-0.962140\pi$
$37$		−	1.44358i		−	0.237324i	0.992935	−	0.118662i	$-0.0378604\pi$
$38$	12.9502			2.10079
$39$	5.50659	+	2.48119i	0.881760	+	0.397309i
$40$	3.61213			0.571127
$41$			7.92478i			1.23764i	0.785532	+	0.618821i	$0.212389\pi$
$41$			7.92478i			1.23764i	−0.785532	+	0.618821i	$0.787611\pi$
$42$		0			0
$43$	−4.61213			−0.703343			−0.351671	−	0.936124i	$-0.614387\pi$
$43$	−4.61213			−0.703343			−0.351671	+	0.936124i	$0.614387\pi$
$44$		−	18.6253i		−	2.80787i
$45$			0.130933i			0.0195183i
$46$		−	11.8315i		−	1.74445i
$47$		−	7.86907i		−	1.14782i	−0.818918	−	0.573911i	$-0.805426\pi$
$47$		−	7.86907i		−	1.14782i	0.818918	−	0.573911i	$-0.194574\pi$
$48$	−8.31265			−1.19983
$49$		0			0
$50$		−	11.2750i		−	1.59453i
$51$	−5.50659			−0.771076
$52$	13.6629	+	6.15633i	1.89471	+	0.853729i
$53$	−3.15633			−0.433555			−0.216777	−	0.976221i	$-0.569555\pi$
$53$	−3.15633			−0.433555			−0.216777	+	0.976221i	$0.569555\pi$
$54$		−	13.2750i		−	1.80650i
$55$	3.02539			0.407944
$56$		0			0
$57$		−	8.74306i		−	1.15805i
$58$			23.1065i			3.03403i
$59$			2.54420i			0.331226i	0.986191	+	0.165613i	$0.0529603\pi$
$59$			2.54420i			0.331226i	−0.986191	+	0.165613i	$0.947040\pi$
$60$		−	4.70052i		−	0.606835i
$61$	2.31265			0.296105			0.148052	−	0.988980i	$-0.452700\pi$
$61$	2.31265			0.296105			0.148052	+	0.988980i	$0.452700\pi$
$62$	−4.06300			−0.516002
$63$		0			0
$64$	5.92478			0.740597
$65$	−1.00000	+	2.21933i	−0.124035	+	0.275274i
$66$	−18.6253			−2.29262
$67$			7.35026i			0.897977i	0.893537	+	0.448989i	$0.148216\pi$
$67$			7.35026i			0.897977i	−0.893537	+	0.448989i	$0.851784\pi$
$68$	−13.6629			−1.65687
$69$	−7.98778			−0.961616
$70$		0			0
$71$			7.75623i			0.920496i	0.887791	+	0.460248i	$0.152239\pi$
$71$			7.75623i			0.920496i	−0.887791	+	0.460248i	$0.847761\pi$
$72$		−	1.03761i		−	0.122284i
$73$			15.1441i			1.77248i	0.463223	+	0.886242i	$0.346693\pi$
$73$			15.1441i			1.77248i	−0.463223	+	0.886242i	$0.653307\pi$
$74$	−3.58181			−0.416377
$75$	−7.61213			−0.878973
$76$		−	21.6932i		−	2.48838i
$77$		0			0
$78$	6.15633	−	13.6629i	0.697067	−	1.54702i
$79$	14.6629			1.64971			0.824853	−	0.565347i	$-0.191258\pi$
$79$	14.6629			1.64971			0.824853	+	0.565347i	$0.191258\pi$
$80$		−	3.35026i		−	0.374571i
$81$	−8.38058			−0.931175
$82$	19.6629			2.17141
$83$			1.45088i			0.159254i	0.996825	+	0.0796272i	$0.0253730\pi$
$83$			1.45088i			0.159254i	−0.996825	+	0.0796272i	$0.974627\pi$
$84$		0			0
$85$		−	2.21933i		−	0.240720i
$86$			11.4436i			1.23399i
$87$	15.5999			1.67249
$88$	−23.9756			−2.55580
$89$			7.79384i			0.826146i	0.910698	+	0.413073i	$0.135545\pi$
$89$			7.79384i			0.826146i	−0.910698	+	0.413073i	$0.864455\pi$
$90$	0.324869			0.0342442
$91$		0			0
$92$	−19.8192			−2.06630
$93$			2.74306i			0.284442i
$94$	−19.5247			−2.01382
$95$	3.52373			0.361527
$96$			2.70052i			0.275621i
$97$			17.9805i			1.82564i	0.408360	+	0.912821i	$0.366101\pi$
$97$			17.9805i			1.82564i	−0.408360	+	0.912821i	$0.633899\pi$
$98$		0			0
$99$		−	0.869067i		−	0.0873446i
$100$	−18.8872			−1.88872
$101$	−9.33804			−0.929170			−0.464585	−	0.885529i	$-0.653796\pi$
$101$	−9.33804			−0.929170			−0.464585	+	0.885529i	$0.653796\pi$
$102$			13.6629i			1.35283i
$103$	−6.23743			−0.614592			−0.307296	−	0.951614i	$-0.599424\pi$
$103$	−6.23743			−0.614592			−0.307296	+	0.951614i	$0.599424\pi$
$104$	7.92478	−	17.5877i	0.777088	−	1.72461i
$105$		0			0
$106$			7.83146i			0.760658i
$107$	7.24472			0.700374			0.350187	−	0.936680i	$-0.386118\pi$
$107$	7.24472			0.700374			0.350187	+	0.936680i	$0.386118\pi$
$108$	−22.2374			−2.13980
$109$			4.79384i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$109$			4.79384i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$110$		−	7.50659i		−	0.715725i
$111$			2.41819i			0.229524i
$112$		0			0
$113$	−9.34297			−0.878912			−0.439456	−	0.898264i	$-0.644829\pi$
$113$	−9.34297			−0.878912			−0.439456	+	0.898264i	$0.644829\pi$
$114$	−21.6932			−2.03176
$115$		−	3.21933i		−	0.300204i
$116$	38.7064			3.59380
$117$	0.637519	+	0.287258i	0.0589387	+	0.0265570i
$118$	6.31265			0.581127
$119$		0			0
$120$	−6.05079			−0.552359
$121$	−9.08110			−0.825555
$122$		−	5.73813i		−	0.519506i
$123$		−	13.2750i		−	1.19697i
$124$			6.80606i			0.611203i
$125$		−	6.44358i		−	0.576332i
$126$		0			0
$127$	−1.38058			−0.122507			−0.0612533	−	0.998122i	$-0.519510\pi$
$127$	−1.38058			−0.122507			−0.0612533	+	0.998122i	$0.519510\pi$
$128$		−	17.9248i		−	1.58434i
$129$	7.72592			0.680229
$130$	5.50659	+	2.48119i	0.482960	+	0.217615i
$131$	−12.4993			−1.09207			−0.546034	−	0.837763i	$-0.683863\pi$
$131$	−12.4993			−1.09207			−0.546034	+	0.837763i	$0.683863\pi$
$132$			31.1998i			2.71560i
$133$		0			0
$134$	18.2374			1.57547
$135$		−	3.61213i		−	0.310882i
$136$			17.5877i			1.50813i
$137$			3.20616i			0.273920i	0.990577	+	0.136960i	$0.0437332\pi$
$137$			3.20616i			0.273920i	−0.990577	+	0.136960i	$0.956267\pi$
$138$			19.8192i			1.68713i
$139$	0.249646			0.0211747			0.0105874	−	0.999944i	$-0.496630\pi$
$139$	0.249646			0.0211747			0.0105874	+	0.999944i	$0.496630\pi$
$140$		0			0
$141$			13.1817i			1.11010i
$142$	19.2447			1.61498
$143$	6.63752	−	14.7308i	0.555057	−	1.23185i
$144$	−0.962389			−0.0801991
$145$			6.28726i			0.522128i
$146$	37.5755			3.10977
$147$		0			0
$148$			6.00000i			0.493197i
$149$		−	4.03269i		−	0.330371i	−0.986263	−	0.165185i	$-0.947178\pi$
$149$		−	4.03269i		−	0.330371i	0.986263	−	0.165185i	$-0.0528222\pi$
$150$			18.8872i			1.54213i
$151$			1.56959i			0.127731i	0.997958	+	0.0638657i	$0.0203429\pi$
$151$			1.56959i			0.127731i	−0.997958	+	0.0638657i	$0.979657\pi$
$152$	−27.9248			−2.26500
$153$	−0.637519			−0.0515404
$154$		0			0
$155$	−1.10554			−0.0887991
$156$	−22.8872	−	10.3127i	−1.83244	−	0.825673i
$157$	−2.89938			−0.231396			−0.115698	−	0.993284i	$-0.536910\pi$
$157$	−2.89938			−0.231396			−0.115698	+	0.993284i	$0.536910\pi$
$158$		−	36.3815i		−	2.89436i
$159$	5.28726			0.419307
$160$	−1.08840			−0.0860453
$161$		0			0
$162$			20.7938i			1.63372i
$163$			17.2750i			1.35309i	0.736403	+	0.676543i	$0.236523\pi$
$163$			17.2750i			1.35309i	−0.736403	+	0.676543i	$0.763477\pi$
$164$		−	32.9380i		−	2.57202i
$165$	−5.06793			−0.394538
$166$	3.59991			0.279407
$167$		−	16.2931i		−	1.26080i	−0.776270	−	0.630400i	$-0.782891\pi$
$167$		−	16.2931i		−	1.26080i	0.776270	−	0.630400i	$-0.217109\pi$
$168$		0			0
$169$	8.61213	+	9.73813i	0.662471	+	0.749087i
$170$	−5.50659			−0.422336
$171$		−	1.01222i		−	0.0774063i
$172$	19.1695			1.46166
$173$	−2.17442			−0.165318			−0.0826592	−	0.996578i	$-0.526341\pi$
$173$	−2.17442			−0.165318			−0.0826592	+	0.996578i	$0.526341\pi$
$174$		−	38.7064i		−	2.93432i
$175$		0			0
$176$			22.2374i			1.67621i
$177$		−	4.26187i		−	0.320341i
$178$	19.3380			1.44945
$179$	−0.551493			−0.0412205			−0.0206102	−	0.999788i	$-0.506561\pi$
$179$	−0.551493			−0.0412205			−0.0206102	+	0.999788i	$0.506561\pi$
$180$		−	0.544198i		−	0.0405621i
$181$	−0.511511			−0.0380203			−0.0190102	−	0.999819i	$-0.506051\pi$
$181$	−0.511511			−0.0380203			−0.0190102	+	0.999819i	$0.506051\pi$
$182$		0			0
$183$	−3.87399			−0.286374
$184$			25.5125i			1.88080i
$185$	−0.974607			−0.0716546
$186$	6.80606			0.499045
$187$			14.7308i			1.07723i
$188$			32.7064i			2.38536i
$189$		0			0
$190$		−	8.74306i		−	0.634288i
$191$	−16.5442			−1.19710			−0.598548	−	0.801087i	$-0.704255\pi$
$191$	−16.5442			−1.19710			−0.598548	+	0.801087i	$0.704255\pi$
$192$	−9.92478			−0.716259
$193$			7.41090i			0.533448i	0.963773	+	0.266724i	$0.0859412\pi$
$193$			7.41090i			0.533448i	−0.963773	+	0.266724i	$0.914059\pi$
$194$	44.6131			3.20303
$195$	1.67513	−	3.71767i	0.119959	−	0.266228i
$196$		0			0
$197$			7.14903i			0.509347i	0.967027	+	0.254674i	$0.0819681\pi$
$197$			7.14903i			0.509347i	−0.967027	+	0.254674i	$0.918032\pi$
$198$	−2.15633			−0.153243
$199$	−5.10062			−0.361573			−0.180787	−	0.983522i	$-0.557864\pi$
$199$	−5.10062			−0.361573			−0.180787	+	0.983522i	$0.557864\pi$
$200$			24.3127i			1.71916i
$201$		−	12.3127i		−	0.868467i
$202$			23.1695i			1.63020i
$203$		0			0
$204$	22.8872			1.60242
$205$	5.35026			0.373678
$206$			15.4763i			1.07828i
$207$	−0.924777			−0.0642765
$208$	−16.3127	−	7.35026i	−1.13108	−	0.509649i
$209$	−23.3888			−1.61784
$210$		0			0
$211$	0.193937			0.0133511			0.00667557	−	0.999978i	$-0.497875\pi$
$211$	0.193937			0.0133511			0.00667557	+	0.999978i	$0.497875\pi$
$212$	13.1187			0.900997
$213$		−	12.9927i		−	0.890246i
$214$		−	17.9756i		−	1.22878i
$215$			3.11379i			0.212359i
$216$			28.6253i			1.94771i
$217$		0			0
$218$	11.8945			0.805594
$219$		−	25.3684i		−	1.71423i
$220$	−12.5745			−0.847774
$221$	−10.8061	−	4.86907i	−0.726894	−	0.327529i
$222$	6.00000			0.402694
$223$		−	5.83875i		−	0.390992i	−0.980705	−	0.195496i	$-0.937368\pi$
$223$		−	5.83875i		−	0.390992i	0.980705	−	0.195496i	$-0.0626316\pi$
$224$		0			0
$225$	−0.881286			−0.0587524
$226$			23.1817i			1.54202i
$227$			3.68735i			0.244738i	0.992485	+	0.122369i	$0.0390491\pi$
$227$			3.68735i			0.244738i	−0.992485	+	0.122369i	$0.960951\pi$
$228$			36.3390i			2.40661i
$229$			5.65703i			0.373827i	0.982376	+	0.186914i	$0.0598484\pi$
$229$			5.65703i			0.373827i	−0.982376	+	0.186914i	$0.940152\pi$
$230$	−7.98778			−0.526699
$231$		0			0
$232$		−	49.8251i		−	3.27118i
$233$	−10.3258			−0.676467			−0.338234	−	0.941062i	$-0.609829\pi$
$233$	−10.3258			−0.676467			−0.338234	+	0.941062i	$0.609829\pi$
$234$	0.712742	−	1.58181i	0.0465934	−	0.103406i
$235$	−5.31265			−0.346559
$236$		−	10.5745i		−	0.688342i
$237$	−24.5623			−1.59549
$238$		0			0
$239$		−	22.2882i		−	1.44170i	−0.693089	−	0.720852i	$-0.743751\pi$
$239$		−	22.2882i		−	1.44170i	0.693089	−	0.720852i	$-0.256249\pi$
$240$			5.61213i			0.362261i
$241$		−	29.4264i		−	1.89552i	−0.318979	−	0.947762i	$-0.603340\pi$
$241$		−	29.4264i		−	1.89552i	0.318979	−	0.947762i	$-0.396660\pi$
$242$			22.5320i			1.44841i
$243$	−2.01222			−0.129084
$244$	−9.61213			−0.615353
$245$		0			0
$246$	−32.9380			−2.10005
$247$	7.73084	−	17.1573i	0.491902	−	1.09169i
$248$	8.76116			0.556334
$249$		−	2.43041i		−	0.154021i
$250$	−15.9878			−1.01116
$251$	21.5247			1.35863			0.679313	−	0.733849i	$-0.262278\pi$
$251$	21.5247			1.35863			0.679313	+	0.733849i	$0.262278\pi$
$252$		0			0
$253$			21.3684i			1.34342i
$254$			3.42548i			0.214934i
$255$			3.71767i			0.232809i
$256$	−32.6253			−2.03908
$257$	−0.661957			−0.0412917			−0.0206459	−	0.999787i	$-0.506572\pi$
$257$	−0.661957			−0.0412917			−0.0206459	+	0.999787i	$0.506572\pi$
$258$		−	19.1695i		−	1.19344i
$259$		0			0
$260$	4.15633	−	9.22425i	0.257764	−	0.572064i
$261$	1.80606			0.111793
$262$			31.0132i			1.91600i
$263$	5.18664			0.319822			0.159911	−	0.987131i	$-0.448879\pi$
$263$	5.18664			0.319822			0.159911	+	0.987131i	$0.448879\pi$
$264$	40.1622			2.47181
$265$			2.13093i			0.130902i
$266$		0			0
$267$		−	13.0557i		−	0.798996i
$268$		−	30.5501i		−	1.86614i
$269$	−27.2506			−1.66150			−0.830749	−	0.556647i	$-0.812088\pi$
$269$	−27.2506			−1.66150			−0.830749	+	0.556647i	$0.812088\pi$
$270$	−8.96239			−0.545434
$271$			18.7612i			1.13966i	0.821763	+	0.569830i	$0.192991\pi$
$271$			18.7612i			1.13966i	−0.821763	+	0.569830i	$0.807009\pi$
$272$	16.3127			0.989100
$273$		0			0
$274$	7.95509			0.480585
$275$			20.3634i			1.22796i
$276$	33.1998			1.99839
$277$	−15.4241			−0.926743			−0.463371	−	0.886164i	$-0.653360\pi$
$277$	−15.4241			−0.926743			−0.463371	+	0.886164i	$0.653360\pi$
$278$		−	0.619421i		−	0.0371504i
$279$			0.317575i			0.0190127i
$280$		0			0
$281$		−	24.8446i		−	1.48211i	−0.671446	−	0.741053i	$-0.734327\pi$
$281$		−	24.8446i		−	1.48211i	0.671446	−	0.741053i	$-0.265673\pi$
$282$	32.7064			1.94764
$283$	22.8872			1.36050			0.680250	−	0.732980i	$-0.261871\pi$
$283$	22.8872			1.36050			0.680250	+	0.732980i	$0.261871\pi$
$284$		−	32.2374i		−	1.91294i
$285$	−5.90271			−0.349646
$286$	−36.5501	−	16.4690i	−2.16125	−	0.973831i
$287$		0			0
$288$			0.312650i			0.0184231i
$289$	−6.19394			−0.364349
$290$	15.5999			0.916058
$291$		−	30.1197i		−	1.76565i
$292$		−	62.9438i		−	3.68351i
$293$		−	25.2193i		−	1.47333i	−0.676258	−	0.736664i	$-0.736400\pi$
$293$		−	25.2193i		−	1.47333i	0.676258	−	0.736664i	$-0.263600\pi$
$294$		0			0
$295$	1.71767			0.100006
$296$	7.72355			0.448922
$297$			23.9756i			1.39120i
$298$	−10.0059			−0.579625
$299$	−15.6751	−	7.06300i	−0.906516	−	0.408464i
$300$	31.6385			1.82665
$301$		0			0
$302$	3.89446			0.224101
$303$	15.6424			0.898635
$304$			25.9003i			1.48549i
$305$		−	1.56134i		−	0.0894022i
$306$			1.58181i			0.0904260i
$307$		−	7.24965i		−	0.413759i	−0.978366	−	0.206880i	$-0.933669\pi$
$307$		−	7.24965i		−	0.413759i	0.978366	−	0.206880i	$-0.0663308\pi$
$308$		0			0
$309$	10.4485			0.594395
$310$			2.74306i			0.155795i
$311$	−20.2398			−1.14769			−0.573847	−	0.818963i	$-0.694550\pi$
$311$	−20.2398			−1.14769			−0.573847	+	0.818963i	$0.694550\pi$
$312$	−13.2750	+	29.4617i	−0.751551	+	1.66794i
$313$	33.1368			1.87300			0.936502	−	0.350663i	$-0.114044\pi$
$313$	33.1368			1.87300			0.936502	+	0.350663i	$0.114044\pi$
$314$			7.19394i			0.405977i
$315$		0			0
$316$	−60.9438			−3.42836
$317$			17.0132i			0.955555i	0.878481	+	0.477778i	$0.158557\pi$
$317$			17.0132i			0.955555i	−0.878481	+	0.477778i	$0.841443\pi$
$318$		−	13.1187i		−	0.735661i
$319$		−	41.7318i		−	2.33653i
$320$		−	4.00000i		−	0.223607i
$321$	−12.1359			−0.677357
$322$		0			0
$323$			17.1573i			0.954657i
$324$	34.8324			1.93513
$325$	−14.9380	−	6.73084i	−0.828608	−	0.373360i
$326$	42.8627			2.37395
$327$		−	8.03032i		−	0.444078i
$328$	−42.3996			−2.34113
$329$		0			0
$330$			12.5745i			0.692204i
$331$		−	10.8364i		−	0.595621i	−0.954625	−	0.297811i	$-0.903744\pi$
$331$		−	10.8364i		−	0.595621i	0.954625	−	0.297811i	$-0.0962564\pi$
$332$		−	6.03032i		−	0.330957i
$333$			0.279964i			0.0153419i
$334$	−40.4264			−2.21204
$335$	4.96239			0.271124
$336$		0			0
$337$	−2.96968			−0.161769			−0.0808845	−	0.996723i	$-0.525774\pi$
$337$	−2.96968			−0.161769			−0.0808845	+	0.996723i	$0.525774\pi$
$338$	24.1622	−	21.3684i	1.31425	−	1.16229i
$339$	15.6507			0.850029
$340$			9.22425i			0.500255i
$341$	7.33804			0.397377
$342$	−2.51151			−0.135807
$343$		0			0
$344$		−	24.6761i		−	1.33045i
$345$			5.39280i			0.290338i
$346$			5.39517i			0.290046i
$347$	30.7367			1.65003			0.825017	−	0.565108i	$-0.191166\pi$
$347$	30.7367			1.65003			0.825017	+	0.565108i	$0.191166\pi$
$348$	−64.8383			−3.47570
$349$		−	2.00492i		−	0.107321i	−0.998559	−	0.0536606i	$-0.982911\pi$
$349$		−	2.00492i		−	0.107321i	0.998559	−	0.0536606i	$-0.0170889\pi$
$350$		0			0
$351$	−17.5877	−	7.92478i	−0.938761	−	0.422993i
$352$	7.22425			0.385054
$353$			17.2546i			0.918368i	0.888341	+	0.459184i	$0.151858\pi$
$353$			17.2546i			0.918368i	−0.888341	+	0.459184i	$0.848142\pi$
$354$	−10.5745			−0.562029
$355$	5.23647			0.277923
$356$		−	32.3938i		−	1.71687i
$357$		0			0
$358$			1.36836i			0.0723201i
$359$		−	7.10650i		−	0.375066i	−0.982258	−	0.187533i	$-0.939951\pi$
$359$		−	7.10650i		−	0.375066i	0.982258	−	0.187533i	$-0.0600492\pi$
$360$	−0.700523			−0.0369208
$361$	−8.24140			−0.433758
$362$			1.26916i			0.0667055i
$363$	15.2120			0.798425
$364$		0			0
$365$	10.2243			0.535162
$366$			9.61213i			0.502434i
$367$	27.2628			1.42311			0.711554	−	0.702632i	$-0.247992\pi$
$367$	27.2628			1.42311			0.711554	+	0.702632i	$0.247992\pi$
$368$	23.6629			1.23351
$369$		−	1.53690i		−	0.0800080i
$370$			2.41819i			0.125716i
$371$		0			0
$372$		−	11.4010i		−	0.591117i
$373$	−5.91890			−0.306469			−0.153234	−	0.988190i	$-0.548969\pi$
$373$	−5.91890			−0.306469			−0.153234	+	0.988190i	$0.548969\pi$
$374$	36.5501			1.88996
$375$			10.7938i			0.557392i
$376$	42.1016			2.17122
$377$	30.6131	+	13.7938i	1.57665	+	0.710419i
$378$		0			0
$379$			28.9706i			1.48812i	0.668112	+	0.744061i	$0.267103\pi$
$379$			28.9706i			1.48812i	−0.668112	+	0.744061i	$0.732897\pi$
$380$	−14.6458			−0.751312
$381$	2.31265			0.118481
$382$			41.0494i			2.10027i
$383$			23.2243i			1.18670i	0.804943	+	0.593352i	$0.202196\pi$
$383$			23.2243i			1.18670i	−0.804943	+	0.593352i	$0.797804\pi$
$384$			30.0263i			1.53228i
$385$		0			0
$386$	18.3879			0.935918
$387$	0.894460			0.0454680
$388$		−	74.7328i		−	3.79398i
$389$	−16.1319			−0.817919			−0.408960	−	0.912552i	$-0.634108\pi$
$389$	−16.1319			−0.817919			−0.408960	+	0.912552i	$0.634108\pi$
$390$	−9.22425	−	4.15633i	−0.467088	−	0.210464i
$391$	15.6751			0.792725
$392$		0			0
$393$	20.9380			1.05618
$394$	17.7381			0.893634
$395$		−	9.89938i		−	0.498092i
$396$			3.61213i			0.181516i
$397$		−	24.0557i		−	1.20732i	−0.797241	−	0.603661i	$-0.793708\pi$
$397$		−	24.0557i		−	1.20732i	0.797241	−	0.603661i	$-0.206292\pi$
$398$			12.6556i			0.634369i
$399$		0			0
$400$	22.5501			1.12750
$401$			2.77575i			0.138614i	0.997595	+	0.0693071i	$0.0220788\pi$
$401$			2.77575i			0.138614i	−0.997595	+	0.0693071i	$0.977921\pi$
$402$	−30.5501			−1.52370
$403$	−2.42548	+	5.38295i	−0.120822	+	0.268144i
$404$	38.8119			1.93097
$405$			5.65799i			0.281148i
$406$		0			0
$407$	6.46898			0.320655
$408$		−	29.4617i		−	1.45857i
$409$		−	20.1309i		−	0.995411i	−0.867346	−	0.497705i	$-0.834176\pi$
$409$		−	20.1309i		−	0.995411i	0.867346	−	0.497705i	$-0.165824\pi$
$410$		−	13.2750i		−	0.655607i
$411$		−	5.37073i		−	0.264919i
$412$	25.9248			1.27722
$413$		0			0
$414$			2.29455i			0.112771i
$415$	0.979532			0.0480833
$416$	−2.38787	+	5.29948i	−0.117075	+	0.259828i
$417$	−0.418190			−0.0204789
$418$			58.0322i			2.83845i
$419$	0.385503			0.0188331			0.00941654	−	0.999956i	$-0.497003\pi$
$419$	0.385503			0.0188331			0.00941654	+	0.999956i	$0.497003\pi$
$420$		0			0
$421$		−	15.6810i		−	0.764246i	−0.924112	−	0.382123i	$-0.875193\pi$
$421$		−	15.6810i		−	0.764246i	0.924112	−	0.382123i	$-0.124807\pi$
$422$		−	0.481194i		−	0.0234242i
$423$			1.52610i			0.0742015i
$424$		−	16.8872i		−	0.820113i
$425$	14.9380			0.724597
$426$	−32.2374			−1.56191
$427$		0			0
$428$	−30.1114			−1.45549
$429$	−11.1187	+	24.6761i	−0.536817	+	1.19137i
$430$	7.72592			0.372577
$431$		−	1.53690i		−	0.0740301i	−0.999315	−	0.0370150i	$-0.988215\pi$
$431$		−	1.53690i		−	0.0740301i	0.999315	−	0.0370150i	$-0.0117849\pi$
$432$	26.5501			1.27739
$433$	−26.0362			−1.25122			−0.625610	−	0.780136i	$-0.715150\pi$
$433$	−26.0362			−1.25122			−0.625610	+	0.780136i	$0.715150\pi$
$434$		0			0
$435$		−	10.5320i		−	0.504970i
$436$		−	19.9248i		−	0.954224i
$437$			24.8881i			1.19056i
$438$	−62.9438			−3.00757
$439$	19.1246			0.912767			0.456384	−	0.889783i	$-0.349145\pi$
$439$	19.1246			0.912767			0.456384	+	0.889783i	$0.349145\pi$
$440$			16.1866i			0.771668i
$441$		0			0
$442$	−12.0811	+	26.8119i	−0.574639	+	1.27531i
$443$	−12.6180			−0.599500			−0.299750	−	0.954018i	$-0.596903\pi$
$443$	−12.6180			−0.599500			−0.299750	+	0.954018i	$0.596903\pi$
$444$		−	10.0508i		−	0.476989i
$445$	5.26187			0.249436
$446$	−14.4871			−0.685983
$447$			6.75528i			0.319514i
$448$		0			0
$449$		−	1.02302i		−	0.0482794i	−0.999709	−	0.0241397i	$-0.992315\pi$
$449$		−	1.02302i		−	0.0482794i	0.999709	−	0.0241397i	$-0.00768466\pi$
$450$			2.18664i			0.103079i
$451$	−35.5125			−1.67222
$452$	38.8324			1.82652
$453$		−	2.62927i		−	0.123534i
$454$	9.14903			0.429385
$455$		0			0
$456$	46.7777			2.19056
$457$		−	28.5320i		−	1.33467i	−0.744758	−	0.667335i	$-0.767435\pi$
$457$		−	28.5320i		−	1.33467i	0.744758	−	0.667335i	$-0.232565\pi$
$458$	14.0362			0.655868
$459$	17.5877			0.820923
$460$			13.3806i			0.623873i
$461$			25.3503i			1.18068i	0.807155	+	0.590340i	$0.201006\pi$
$461$			25.3503i			1.18068i	−0.807155	+	0.590340i	$0.798994\pi$
$462$		0			0
$463$			39.6810i			1.84413i	0.387032	+	0.922066i	$0.373500\pi$
$463$			39.6810i			1.84413i	−0.387032	+	0.922066i	$0.626500\pi$
$464$	−46.2130			−2.14538
$465$	1.85192			0.0858809
$466$			25.6204i			1.18684i
$467$	1.95158			0.0903086			0.0451543	−	0.998980i	$-0.485622\pi$
$467$	1.95158			0.0903086			0.0451543	+	0.998980i	$0.485622\pi$
$468$	−2.64974	−	1.19394i	−0.122484	−	0.0551897i
$469$		0			0
$470$			13.1817i			0.608027i
$471$	4.85685			0.223792
$472$	−13.6121			−0.626549
$473$		−	20.6678i		−	0.950308i
$474$			60.9438i			2.79924i
$475$			23.7177i			1.08824i
$476$		0			0
$477$	0.612127			0.0280274
$478$	−55.3014			−2.52943
$479$			26.1368i			1.19422i	0.802159	+	0.597111i	$0.203685\pi$
$479$			26.1368i			1.19422i	−0.802159	+	0.597111i	$0.796315\pi$
$480$	1.82321			0.0832176
$481$	−2.13823	+	4.74543i	−0.0974947	+	0.216373i
$482$	−73.0127			−3.32564
$483$		0			0
$484$	37.7440			1.71564
$485$	12.1392			0.551212
$486$			4.99271i			0.226474i
$487$			20.9624i			0.949896i	0.880014	+	0.474948i	$0.157533\pi$
$487$			20.9624i			0.949896i	−0.880014	+	0.474948i	$0.842467\pi$
$488$			12.3733i			0.560112i
$489$		−	28.9380i		−	1.30862i
$490$		0			0
$491$	2.95651			0.133425			0.0667127	−	0.997772i	$-0.478749\pi$
$491$	2.95651			0.133425			0.0667127	+	0.997772i	$0.478749\pi$
$492$			55.1754i			2.48750i
$493$	−30.6131			−1.37874
$494$	−42.5705	−	19.1817i	−1.91534	−	0.863026i
$495$	−0.586734			−0.0263717
$496$		−	8.12601i		−	0.364869i
$497$		0			0
$498$	−6.03032			−0.270225
$499$			2.85448i			0.127784i	0.997957	+	0.0638920i	$0.0203513\pi$
$499$			2.85448i			0.127784i	−0.997957	+	0.0638920i	$0.979649\pi$
$500$			26.7816i			1.19771i
$501$			27.2931i			1.21937i
$502$		−	53.4069i		−	2.38367i
$503$	−23.8641			−1.06405			−0.532025	−	0.846729i	$-0.678569\pi$
$503$	−23.8641			−1.06405			−0.532025	+	0.846729i	$0.678569\pi$
$504$		0			0
$505$			6.30440i			0.280542i
$506$	53.0191			2.35698
$507$	−14.4264	−	16.3127i	−0.640701	−	0.724470i
$508$	5.73813			0.254589
$509$		−	11.1949i		−	0.496205i	−0.968734	−	0.248102i	$-0.920193\pi$
$509$		−	11.1949i		−	0.496205i	0.968734	−	0.248102i	$-0.0798070\pi$
$510$	9.22425			0.408457
$511$		0			0
$512$			45.1002i			1.99316i
$513$			27.9248i			1.23291i
$514$			1.64244i			0.0724451i
$515$			4.21108i			0.185562i
$516$	−32.1114			−1.41363
$517$	35.2628			1.55086
$518$		0			0
$519$	3.64244			0.159886
$520$	−11.8740	−	5.35026i	−0.520709	−	0.234624i
$521$	26.4894			1.16052			0.580262	−	0.814430i	$-0.302950\pi$
$521$	26.4894			1.16052			0.580262	+	0.814430i	$0.302950\pi$
$522$		−	4.48119i		−	0.196137i
$523$	12.9525			0.566375			0.283188	−	0.959065i	$-0.408608\pi$
$523$	12.9525			0.566375			0.283188	+	0.959065i	$0.408608\pi$
$524$	51.9511			2.26950
$525$		0			0
$526$		−	12.8691i		−	0.561118i
$527$		−	5.38295i		−	0.234485i
$528$		−	37.2506i		−	1.62112i
$529$	−0.261865			−0.0113854
$530$	5.28726			0.229664
$531$		−	0.493413i		−	0.0214123i
$532$		0			0
$533$	11.7381	−	26.0508i	0.508435	−	1.12838i
$534$	−32.3938			−1.40181
$535$		−	4.89114i		−	0.211462i
$536$	−39.3258			−1.69862
$537$	0.923822			0.0398659
$538$			67.6140i			2.91505i
$539$		0			0
$540$			15.0132i			0.646064i
$541$		−	28.0933i		−	1.20783i	−0.797050	−	0.603913i	$-0.793607\pi$
$541$		−	28.0933i		−	1.20783i	0.797050	−	0.603913i	$-0.206393\pi$
$542$	46.5501			1.99950
$543$	0.856849			0.0367709
$544$		−	5.29948i		−	0.227213i
$545$	3.23647			0.138635
$546$		0			0
$547$	14.8192			0.633625			0.316812	−	0.948488i	$-0.397387\pi$
$547$	14.8192			0.633625			0.316812	+	0.948488i	$0.397387\pi$
$548$		−	13.3258i		−	0.569251i
$549$	−0.448507			−0.0191418
$550$	50.5256			2.15442
$551$		−	48.6058i		−	2.07068i
$552$		−	42.7367i		−	1.81900i
$553$		0			0
$554$			38.2701i			1.62594i
$555$	1.63259			0.0692998
$556$	−1.03761			−0.0440045
$557$			18.3879i			0.779119i	0.921001	+	0.389560i	$0.127373\pi$
$557$			18.3879i			0.779119i	−0.921001	+	0.389560i	$0.872627\pi$
$558$	0.787965			0.0333572
$559$	15.1612	+	6.83146i	0.641253	+	0.288940i
$560$		0			0
$561$		−	24.6761i		−	1.04183i
$562$	−61.6444			−2.60031
$563$	−15.3357			−0.646322			−0.323161	−	0.946344i	$-0.604745\pi$
$563$	−15.3357			−0.646322			−0.323161	+	0.946344i	$0.604745\pi$
$564$		−	54.7875i		−	2.30697i
$565$			6.30773i			0.265368i
$566$		−	56.7875i		−	2.38696i
$567$		0			0
$568$	−41.4979			−1.74121
$569$	−1.32250			−0.0554421			−0.0277210	−	0.999616i	$-0.508825\pi$
$569$	−1.32250			−0.0554421			−0.0277210	+	0.999616i	$0.508825\pi$
$570$			14.6458i			0.613444i
$571$	37.9175			1.58680			0.793399	−	0.608702i	$-0.208310\pi$
$571$	37.9175			1.58680			0.793399	+	0.608702i	$0.208310\pi$
$572$	−27.5877	+	61.2262i	−1.15350	+	2.55999i
$573$	27.7137			1.15776
$574$		0			0
$575$	21.6688			0.903651
$576$	−1.14903			−0.0478763
$577$		−	12.8627i		−	0.535482i	−0.963491	−	0.267741i	$-0.913723\pi$
$577$		−	12.8627i		−	0.535482i	0.963491	−	0.267741i	$-0.0862772\pi$
$578$			15.3684i			0.639240i
$579$		−	12.4142i		−	0.515917i
$580$		−	26.1319i		−	1.08507i
$581$		0			0
$582$	−74.7328			−3.09777
$583$		−	14.1441i		−	0.585789i
$584$	−81.0249			−3.35284
$585$	0.193937	−	0.430409i	0.00801829	−	0.0177952i
$586$	−62.5741			−2.58491
$587$		−	31.0240i		−	1.28050i	−0.768168	−	0.640248i	$-0.778831\pi$
$587$		−	31.0240i		−	1.28050i	0.768168	−	0.640248i	$-0.221169\pi$
$588$		0			0
$589$	8.54675			0.352163
$590$		−	4.26187i		−	0.175458i
$591$		−	11.9756i		−	0.492609i
$592$		−	7.16362i		−	0.294423i
$593$			11.4119i			0.468629i	0.972161	+	0.234314i	$0.0752845\pi$
$593$			11.4119i			0.468629i	−0.972161	+	0.234314i	$0.924716\pi$
$594$	59.4880			2.44082
$595$		0			0
$596$			16.7612i			0.686564i
$597$	8.54420			0.349691
$598$	−17.5247	+	38.8930i	−0.716638	+	1.59045i
$599$	7.37328			0.301264			0.150632	−	0.988590i	$-0.451869\pi$
$599$	7.37328			0.301264			0.150632	+	0.988590i	$0.451869\pi$
$600$		−	40.7269i		−	1.66267i
$601$	−17.2144			−0.702190			−0.351095	−	0.936340i	$-0.614191\pi$
$601$	−17.2144			−0.702190			−0.351095	+	0.936340i	$0.614191\pi$
$602$		0			0
$603$		−	1.42548i		−	0.0580502i
$604$		−	6.52373i		−	0.265447i
$605$			6.13093i			0.249258i
$606$		−	38.8119i		−	1.57663i
$607$	−32.4264			−1.31615			−0.658074	−	0.752953i	$-0.728629\pi$
$607$	−32.4264			−1.31615			−0.658074	+	0.752953i	$0.728629\pi$
$608$	8.41422			0.341242
$609$		0			0
$610$	−3.87399			−0.156853
$611$	−11.6556	+	25.8677i	−0.471536	+	1.04649i
$612$	2.64974			0.107109
$613$			35.6991i			1.44187i	0.693001	+	0.720937i	$0.256288\pi$
$613$			35.6991i			1.44187i	−0.693001	+	0.720937i	$0.743712\pi$
$614$	−17.9878			−0.725928
$615$	−8.96239			−0.361398
$616$		0			0
$617$		−	20.3733i		−	0.820198i	−0.912041	−	0.410099i	$-0.865494\pi$
$617$		−	20.3733i		−	0.820198i	0.912041	−	0.410099i	$-0.134506\pi$
$618$		−	25.9248i		−	1.04285i
$619$			40.0118i			1.60821i	0.594488	+	0.804104i	$0.297355\pi$
$619$			40.0118i			1.60821i	−0.594488	+	0.804104i	$0.702645\pi$
$620$	4.59498			0.184539
$621$	25.5125			1.02378
$622$			50.2189i			2.01359i
$623$		0			0
$624$	27.3258	+	12.3127i	1.09391	+	0.492900i
$625$	18.3707			0.734829
$626$		−	82.2189i		−	3.28613i
$627$	39.1793			1.56467
$628$	12.0508			0.480879
$629$		−	4.74543i		−	0.189213i
$630$		0			0
$631$		−	16.3879i		−	0.652391i	−0.945302	−	0.326195i	$-0.894233\pi$
$631$		−	16.3879i		−	0.652391i	0.945302	−	0.326195i	$-0.105767\pi$
$632$			78.4504i			3.12059i
$633$	−0.324869			−0.0129124
$634$	42.2130			1.67649
$635$			0.932071i			0.0369881i
$636$	−21.9756			−0.871388
$637$		0			0
$638$	−103.545			−4.09937
$639$		−	1.50422i		−	0.0595059i
$640$	−12.1016			−0.478357
$641$	8.23884			0.325415			0.162707	−	0.986674i	$-0.447977\pi$
$641$	8.23884			0.325415			0.162707	+	0.986674i	$0.447977\pi$
$642$			30.1114i			1.18840i
$643$		−	30.4847i		−	1.20220i	−0.799174	−	0.601100i	$-0.794729\pi$
$643$		−	30.4847i		−	1.20220i	0.799174	−	0.601100i	$-0.205271\pi$
$644$		0			0
$645$		−	5.21600i		−	0.205380i
$646$	42.5705			1.67492
$647$	28.8388			1.13377			0.566884	−	0.823798i	$-0.308149\pi$
$647$	28.8388			1.13377			0.566884	+	0.823798i	$0.308149\pi$
$648$		−	44.8383i		−	1.76141i
$649$	−11.4010			−0.447530
$650$	−16.7005	+	37.0640i	−0.655048	+	1.45377i
$651$		0			0
$652$		−	71.8007i		−	2.81193i
$653$	21.4518			0.839475			0.419738	−	0.907646i	$-0.362122\pi$
$653$	21.4518			0.839475			0.419738	+	0.907646i	$0.362122\pi$
$654$	−19.9248			−0.779120
$655$			8.43866i			0.329726i
$656$			39.3258i			1.53542i
$657$		−	2.93700i		−	0.114583i
$658$		0			0
$659$	−50.6589			−1.97339			−0.986696	−	0.162575i	$-0.948020\pi$
$659$	−50.6589			−1.97339			−0.986696	+	0.162575i	$0.948020\pi$
$660$	21.0640			0.819913
$661$			15.5477i			0.604736i	0.953191	+	0.302368i	$0.0977771\pi$
$661$			15.5477i			0.604736i	−0.953191	+	0.302368i	$0.902223\pi$
$662$	−26.8872			−1.04500
$663$	18.1016	+	8.15633i	0.703007	+	0.316765i
$664$	−7.76257			−0.301246
$665$		0			0
$666$	0.694644			0.0269169
$667$	−44.4069			−1.71944
$668$			67.7196i			2.62015i
$669$			9.78067i			0.378143i
$670$		−	12.3127i		−	0.475679i
$671$			10.3634i			0.400076i
$672$		0			0
$673$	26.8700			1.03576			0.517882	−	0.855452i	$-0.326721\pi$
$673$	26.8700			1.03576			0.517882	+	0.855452i	$0.326721\pi$
$674$			7.36836i			0.283819i
$675$	24.3127			0.935794
$676$	−35.7948	−	40.4749i	−1.37672	−	1.55673i
$677$	−16.3757			−0.629368			−0.314684	−	0.949197i	$-0.601898\pi$
$677$	−16.3757			−0.629368			−0.314684	+	0.949197i	$0.601898\pi$
$678$		−	38.8324i		−	1.49135i
$679$		0			0
$680$	11.8740			0.455347
$681$		−	6.17679i		−	0.236695i
$682$		−	18.2071i		−	0.697186i
$683$		−	15.7988i		−	0.604523i	−0.953225	−	0.302262i	$-0.902258\pi$
$683$		−	15.7988i		−	0.604523i	0.953225	−	0.302262i	$-0.0977416\pi$
$684$			4.20711i			0.160863i
$685$	2.16457			0.0827041
$686$		0			0
$687$		−	9.47627i		−	0.361542i
$688$	−22.8872			−0.872565
$689$	10.3757	+	4.67513i	0.395281	+	0.178108i
$690$	13.3806			0.509390
$691$			7.56818i			0.287907i	0.989584	+	0.143953i	$0.0459816\pi$
$691$			7.56818i			0.287907i	−0.989584	+	0.143953i	$0.954018\pi$
$692$	9.03761			0.343558
$693$		0			0
$694$		−	76.2638i		−	2.89493i
$695$		−	0.168544i		−	0.00639324i
$696$			83.4636i			3.16368i
$697$			26.0508i			0.986744i
$698$	−4.97461			−0.188292
$699$	17.2971			0.654237
$700$		0			0
$701$	−32.6629			−1.23366			−0.616831	−	0.787096i	$-0.711584\pi$
$701$	−32.6629			−1.23366			−0.616831	+	0.787096i	$0.711584\pi$
$702$	−19.6629	+	43.6385i	−0.742129	+	1.64703i
$703$	7.53453			0.284170
$704$			26.5501i			1.00064i
$705$	8.89938			0.335170
$706$	42.8119			1.61125
$707$		0			0
$708$			17.7137i			0.665722i
$709$			25.0214i			0.939699i	0.882746	+	0.469850i	$0.155692\pi$
$709$			25.0214i			0.939699i	−0.882746	+	0.469850i	$0.844308\pi$
$710$		−	12.9927i		−	0.487608i
$711$	−2.84367			−0.106646
$712$	−41.6991			−1.56274
$713$		−	7.80843i		−	0.292428i
$714$		0			0
$715$	−9.94525	−	4.48119i	−0.371931	−	0.167587i
$716$	2.29218			0.0856629
$717$			37.3357i			1.39433i
$718$	−17.6326			−0.658043
$719$	−1.85192			−0.0690651			−0.0345326	−	0.999404i	$-0.510994\pi$
$719$	−1.85192			−0.0690651			−0.0345326	+	0.999404i	$0.510994\pi$
$720$			0.649738i			0.0242143i
$721$		0			0
$722$			20.4485i			0.761015i
$723$			49.2931i			1.83323i
$724$	2.12601			0.0790125
$725$	−42.3185			−1.57167
$726$		−	37.7440i		−	1.40081i
$727$	10.8265			0.401534			0.200767	−	0.979639i	$-0.435657\pi$
$727$	10.8265			0.401534			0.200767	+	0.979639i	$0.435657\pi$
$728$		0			0
$729$	28.5125			1.05602
$730$		−	25.3684i		−	0.938925i
$731$	−15.1612			−0.560759
$732$	16.1016			0.595131
$733$		−	23.4264i		−	0.865275i	−0.901568	−	0.432638i	$-0.857583\pi$
$733$		−	23.4264i		−	0.865275i	0.901568	−	0.432638i	$-0.142417\pi$
$734$		−	67.6444i		−	2.49680i
$735$		0			0
$736$		−	7.68735i		−	0.283359i
$737$	−32.9380			−1.21329
$738$	−3.81336			−0.140372
$739$			0.481194i			0.0177010i	0.999961	+	0.00885051i	$0.00281724\pi$
$739$			0.481194i			0.0177010i	−0.999961	+	0.00885051i	$0.997183\pi$
$740$	4.05079			0.148910
$741$	−12.9502	+	28.7407i	−0.475736	+	1.05582i
$742$		0			0
$743$		−	13.7889i		−	0.505866i	−0.967484	−	0.252933i	$-0.918605\pi$
$743$		−	13.7889i		−	0.505866i	0.967484	−	0.252933i	$-0.0813953\pi$
$744$	−14.6761			−0.538051
$745$	−2.72259			−0.0997480
$746$			14.6859i			0.537690i
$747$		−	0.281378i		−	0.0102951i
$748$		−	61.2262i		−	2.23865i
$749$		0			0
$750$	26.7816			0.977927
$751$	26.4617			0.965600			0.482800	−	0.875731i	$-0.339620\pi$
$751$	26.4617			0.965600			0.482800	+	0.875731i	$0.339620\pi$
$752$		−	39.0494i		−	1.42398i
$753$	−36.0567			−1.31398
$754$	34.2252	−	75.9570i	1.24641	−	2.76619i
$755$	1.05968			0.0385657
$756$		0			0
$757$	−21.1114			−0.767308			−0.383654	−	0.923477i	$-0.625334\pi$
$757$	−21.1114			−0.767308			−0.383654	+	0.923477i	$0.625334\pi$
$758$	71.8818			2.61086
$759$		−	35.7948i		−	1.29927i
$760$			18.8529i			0.683866i
$761$			28.5247i			1.03402i	0.855980	+	0.517010i	$0.172955\pi$
$761$			28.5247i			1.03402i	−0.855980	+	0.517010i	$0.827045\pi$
$762$		−	5.73813i		−	0.207871i
$763$		0			0
$764$	68.7631			2.48776
$765$			0.430409i			0.0155615i
$766$	57.6239			2.08204
$767$	3.76845	−	8.36344i	0.136071	−	0.301986i
$768$	54.6516			1.97207
$769$			25.8388i			0.931769i	0.884845	+	0.465885i	$0.154264\pi$
$769$			25.8388i			0.931769i	−0.884845	+	0.465885i	$0.845736\pi$
$770$		0			0
$771$	1.10886			0.0399348
$772$		−	30.8021i		−	1.10859i
$773$			27.4010i			0.985547i	0.870158	+	0.492774i	$0.164017\pi$
$773$			27.4010i			0.985547i	−0.870158	+	0.492774i	$0.835983\pi$
$774$		−	2.21933i		−	0.0797721i
$775$		−	7.44121i		−	0.267296i
$776$	−96.2003			−3.45339
$777$		0			0
$778$			40.0263i			1.43501i
$779$	−41.3620			−1.48195
$780$	−6.96239	+	15.4518i	−0.249294	+	0.553264i
$781$	−34.7572			−1.24371
$782$		−	38.8930i		−	1.39081i
$783$	−49.8251			−1.78060
$784$		0			0
$785$			1.95746i			0.0698649i
$786$		−	51.9511i		−	1.85304i
$787$		−	31.0240i		−	1.10589i	−0.833219	−	0.552943i	$-0.813505\pi$
$787$		−	31.0240i		−	1.10589i	0.833219	−	0.552943i	$-0.186495\pi$
$788$

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)

\(f(q)\)	\(=\)	\(q-2.48119i q^{2} -1.67513 q^{3} -4.15633 q^{4} -0.675131i q^{5} +4.15633i q^{6} +5.35026i q^{8} -0.193937 q^{9} +O(q^{10})\)	\(q-2.48119i q^{2} -1.67513 q^{3} -4.15633 q^{4} -0.675131i q^{5} +4.15633i q^{6} +5.35026i q^{8} -0.193937 q^{9} -1.67513 q^{10} +4.48119i q^{11} +6.96239 q^{12} +(-3.28726 - 1.48119i) q^{13} +1.13093i q^{15} +4.96239 q^{16} +3.28726 q^{17} +0.481194i q^{18} +5.21933i q^{19} +2.80606i q^{20} +11.1187 q^{22} +4.76845 q^{23} -8.96239i q^{24} +4.54420 q^{25} +(-3.67513 + 8.15633i) q^{26} +5.35026 q^{27} -9.31265 q^{29} +2.80606 q^{30} -1.63752i q^{31} -1.61213i q^{32} -7.50659i q^{33} -8.15633i q^{34} +0.806063 q^{36} -1.44358i q^{37} +12.9502 q^{38} +(5.50659 + 2.48119i) q^{39} +3.61213 q^{40} +7.92478i q^{41} -4.61213 q^{43} -18.6253i q^{44} +0.130933i q^{45} -11.8315i q^{46} -7.86907i q^{47} -8.31265 q^{48} -11.2750i q^{50} -5.50659 q^{51} +(13.6629 + 6.15633i) q^{52} -3.15633 q^{53} -13.2750i q^{54} +3.02539 q^{55} -8.74306i q^{57} +23.1065i q^{58} +2.54420i q^{59} -4.70052i q^{60} +2.31265 q^{61} -4.06300 q^{62} +5.92478 q^{64} +(-1.00000 + 2.21933i) q^{65} -18.6253 q^{66} +7.35026i q^{67} -13.6629 q^{68} -7.98778 q^{69} +7.75623i q^{71} -1.03761i q^{72} +15.1441i q^{73} -3.58181 q^{74} -7.61213 q^{75} -21.6932i q^{76} +(6.15633 - 13.6629i) q^{78} +14.6629 q^{79} -3.35026i q^{80} -8.38058 q^{81} +19.6629 q^{82} +1.45088i q^{83} -2.21933i q^{85} +11.4436i q^{86} +15.5999 q^{87} -23.9756 q^{88} +7.79384i q^{89} +0.324869 q^{90} -19.8192 q^{92} +2.74306i q^{93} -19.5247 q^{94} +3.52373 q^{95} +2.70052i q^{96} +17.9805i q^{97} -0.869067i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{4} - 2 q^{9}+O(q^{10}) \) 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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