LMFDB - Embedded newform 5070.2.b.y.1351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$, degree $1$, 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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.2
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.y.1351.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -0.643104i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -0.643104i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.40581i q^{11} -1.00000 q^{12} -0.643104 q^{14} +1.00000i q^{15} +1.00000 q^{16} -0.939001 q^{17} -1.00000i q^{18} +2.75302i q^{19} -1.00000i q^{20} -0.643104i q^{21} -4.40581 q^{22} +2.04892 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +0.643104i q^{28} -0.149145 q^{29} +1.00000 q^{30} +2.08815i q^{31} -1.00000i q^{32} -4.40581i q^{33} +0.939001i q^{34} +0.643104 q^{35} -1.00000 q^{36} -6.07069i q^{37} +2.75302 q^{38} -1.00000 q^{40} -5.74094i q^{41} -0.643104 q^{42} -4.69202 q^{43} +4.40581i q^{44} +1.00000i q^{45} -2.04892i q^{46} -7.00000i q^{47} +1.00000 q^{48} +6.58642 q^{49} +1.00000i q^{50} -0.939001 q^{51} +1.68664 q^{53} -1.00000i q^{54} +4.40581 q^{55} +0.643104 q^{56} +2.75302i q^{57} +0.149145i q^{58} -7.85623i q^{59} -1.00000i q^{60} -12.0761 q^{61} +2.08815 q^{62} -0.643104i q^{63} -1.00000 q^{64} -4.40581 q^{66} +1.45712i q^{67} +0.939001 q^{68} +2.04892 q^{69} -0.643104i q^{70} -7.03684i q^{71} +1.00000i q^{72} -14.8605i q^{73} -6.07069 q^{74} -1.00000 q^{75} -2.75302i q^{76} -2.83340 q^{77} +0.929312 q^{79} +1.00000i q^{80} +1.00000 q^{81} -5.74094 q^{82} +11.2959i q^{83} +0.643104i q^{84} -0.939001i q^{85} +4.69202i q^{86} -0.149145 q^{87} +4.40581 q^{88} +7.47650i q^{89} +1.00000 q^{90} -2.04892 q^{92} +2.08815i q^{93} -7.00000 q^{94} -2.75302 q^{95} -1.00000i q^{96} -5.05861i q^{97} -6.58642i q^{98} -4.40581i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	$ 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} - 6 q^{12} - 12 q^{14} + 6 q^{16} + 14 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} - 28 q^{29} + 6 q^{30} + 12 q^{35} - 6 q^{36} + 26 q^{38} - 6 q^{40} - 12 q^{42} - 18 q^{43} + 6 q^{48} - 10 q^{49} + 14 q^{51} + 6 q^{53} + 12 q^{56} - 42 q^{61} + 20 q^{62} - 6 q^{64} - 14 q^{68} - 6 q^{69} - 12 q^{74} - 6 q^{75} + 42 q^{77} + 30 q^{79} + 6 q^{81} - 6 q^{82} - 28 q^{87} + 6 q^{90} + 6 q^{92} - 42 q^{94} - 26 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 - 6 * q^12 - 12 * q^14 + 6 * q^16 + 14 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 - 28 * q^29 + 6 * q^30 + 12 * q^35 - 6 * q^36 + 26 * q^38 - 6 * q^40 - 12 * q^42 - 18 * q^43 + 6 * q^48 - 10 * q^49 + 14 * q^51 + 6 * q^53 + 12 * q^56 - 42 * q^61 + 20 * q^62 - 6 * q^64 - 14 * q^68 - 6 * q^69 - 12 * q^74 - 6 * q^75 + 42 * q^77 + 30 * q^79 + 6 * q^81 - 6 * q^82 - 28 * q^87 + 6 * q^90 + 6 * q^92 - 42 * q^94 - 26 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	− 1.00000i	− 0.408248i
$7$	− 0.643104i	− 0.243071i	−0.992587	−	0.121535i	$-0.961218\pi$
$7$	− 0.643104i	− 0.243071i	0.992587	−	0.121535i	$-0.0387817\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	− 4.40581i	− 1.32840i	−0.747554	−	0.664201i	$-0.768772\pi$
$11$	− 4.40581i	− 1.32840i	0.747554	−	0.664201i	$-0.231228\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−0.643104	−0.171877
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	−0.939001	−0.227741	−0.113871	−	0.993496i	$-0.536325\pi$
$17$	−0.939001	−0.227741	−0.113871	+	0.993496i	$0.536325\pi$
$18$	− 1.00000i	− 0.235702i
$19$	2.75302i	0.631586i	0.948828	+	0.315793i	$0.102271\pi$
$19$	2.75302i	0.631586i	−0.948828	+	0.315793i	$0.897729\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 0.643104i	− 0.140337i
$22$	−4.40581	−0.939323
$23$	2.04892	0.427229	0.213614	−	0.976918i	$-0.431476\pi$
$23$	2.04892	0.427229	0.213614	+	0.976918i	$0.431476\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0.643104i	0.121535i
$29$	−0.149145	−0.0276955	−0.0138478	−	0.999904i	$-0.504408\pi$
$29$	−0.149145	−0.0276955	−0.0138478	+	0.999904i	$0.504408\pi$
$30$	1.00000	0.182574
$31$	2.08815i	0.375042i	0.982261	+	0.187521i	$0.0600453\pi$
$31$	2.08815i	0.375042i	−0.982261	+	0.187521i	$0.939955\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.40581i	− 0.766954i
$34$	0.939001i	0.161037i
$35$	0.643104	0.108704
$36$	−1.00000	−0.166667
$37$	− 6.07069i	− 0.998015i	−0.866598	−	0.499007i	$-0.833698\pi$
$37$	− 6.07069i	− 0.998015i	0.866598	−	0.499007i	$-0.166302\pi$
$38$	2.75302	0.446599
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 5.74094i	− 0.896584i	−0.893887	−	0.448292i	$-0.852032\pi$
$41$	− 5.74094i	− 0.896584i	0.893887	−	0.448292i	$-0.147968\pi$
$42$	−0.643104	−0.0992331
$43$	−4.69202	−0.715527	−0.357763	−	0.933812i	$-0.616461\pi$
$43$	−4.69202	−0.715527	−0.357763	+	0.933812i	$0.616461\pi$
$44$	4.40581i	0.664201i
$45$	1.00000i	0.149071i
$46$	− 2.04892i	− 0.302096i
$47$	− 7.00000i	− 1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$	− 7.00000i	− 1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$	1.00000	0.144338
$49$	6.58642	0.940917
$50$	1.00000i	0.141421i
$51$	−0.939001	−0.131486
$52$	0	0
$53$	1.68664	0.231678	0.115839	−	0.993268i	$-0.463044\pi$
$53$	1.68664	0.231678	0.115839	+	0.993268i	$0.463044\pi$
$54$	− 1.00000i	− 0.136083i
$55$	4.40581	0.594080
$56$	0.643104	0.0859384
$57$	2.75302i	0.364646i
$58$	0.149145i	0.0195837i
$59$	− 7.85623i	− 1.02279i	−0.859344	−	0.511397i	$-0.829128\pi$
$59$	− 7.85623i	− 1.02279i	0.859344	−	0.511397i	$-0.170872\pi$
$60$	− 1.00000i	− 0.129099i
$61$	−12.0761	−1.54618	−0.773091	−	0.634295i	$-0.781290\pi$
$61$	−12.0761	−1.54618	−0.773091	+	0.634295i	$0.781290\pi$
$62$	2.08815	0.265195
$63$	− 0.643104i	− 0.0810235i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.40581	−0.542318
$67$	1.45712i	0.178016i	0.996031	+	0.0890080i	$0.0283697\pi$
$67$	1.45712i	0.178016i	−0.996031	+	0.0890080i	$0.971630\pi$
$68$	0.939001	0.113871
$69$	2.04892	0.246661
$70$	− 0.643104i	− 0.0768656i
$71$	− 7.03684i	− 0.835119i	−0.908650	−	0.417559i	$-0.862886\pi$
$71$	− 7.03684i	− 0.835119i	0.908650	−	0.417559i	$-0.137114\pi$
$72$	1.00000i	0.117851i
$73$	− 14.8605i	− 1.73930i	−0.493673	−	0.869648i	$-0.664346\pi$
$73$	− 14.8605i	− 1.73930i	0.493673	−	0.869648i	$-0.335654\pi$
$74$	−6.07069	−0.705703
$75$	−1.00000	−0.115470
$76$	− 2.75302i	− 0.315793i
$77$	−2.83340	−0.322896
$78$	0	0
$79$	0.929312	0.104556	0.0522779	−	0.998633i	$-0.483352\pi$
$79$	0.929312	0.104556	0.0522779	+	0.998633i	$0.483352\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	−5.74094	−0.633981
$83$	11.2959i	1.23989i	0.784647	+	0.619943i	$0.212844\pi$
$83$	11.2959i	1.23989i	−0.784647	+	0.619943i	$0.787156\pi$
$84$	0.643104i	0.0701684i
$85$	− 0.939001i	− 0.101849i
$86$	4.69202i	0.505954i
$87$	−0.149145	−0.0159900
$88$	4.40581	0.469661
$89$	7.47650i	0.792508i	0.918141	+	0.396254i	$0.129690\pi$
$89$	7.47650i	0.792508i	−0.918141	+	0.396254i	$0.870310\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−2.04892	−0.213614
$93$	2.08815i	0.216531i
$94$	−7.00000	−0.721995
$95$	−2.75302	−0.282454
$96$	− 1.00000i	− 0.102062i
$97$	− 5.05861i	− 0.513624i	−0.966461	−	0.256812i	$-0.917328\pi$
$97$	− 5.05861i	− 0.513624i	0.966461	−	0.256812i	$-0.0826721\pi$
$98$	− 6.58642i	− 0.665329i
$99$	− 4.40581i	− 0.442801i
$100$	1.00000	0.100000
$101$	6.81163	0.677782	0.338891	−	0.940826i	$-0.389948\pi$
$101$	6.81163	0.677782	0.338891	+	0.940826i	$0.389948\pi$
$102$	0.939001i	0.0929750i
$103$	12.0858	1.19084	0.595422	−	0.803413i	$-0.296985\pi$
$103$	12.0858	1.19084	0.595422	+	0.803413i	$0.296985\pi$
$104$	0	0
$105$	0.643104	0.0627605
$106$	− 1.68664i	− 0.163821i
$107$	−14.6679	−1.41800	−0.708998	−	0.705211i	$-0.750852\pi$
$107$	−14.6679	−1.41800	−0.708998	+	0.705211i	$0.750852\pi$
$108$	−1.00000	−0.0962250
$109$	8.15883i	0.781475i	0.920502	+	0.390737i	$0.127780\pi$
$109$	8.15883i	0.781475i	−0.920502	+	0.390737i	$0.872220\pi$
$110$	− 4.40581i	− 0.420078i
$111$	− 6.07069i	− 0.576204i
$112$	− 0.643104i	− 0.0607676i
$113$	16.8442	1.58456	0.792282	−	0.610155i	$-0.208893\pi$
$113$	16.8442	1.58456	0.792282	+	0.610155i	$0.208893\pi$
$114$	2.75302	0.257844
$115$	2.04892i	0.191063i
$116$	0.149145	0.0138478
$117$	0	0
$118$	−7.85623	−0.723225
$119$	0.603875i	0.0553572i
$120$	−1.00000	−0.0912871
$121$	−8.41119	−0.764654
$122$	12.0761i	1.09332i
$123$	− 5.74094i	− 0.517643i
$124$	− 2.08815i	− 0.187521i
$125$	− 1.00000i	− 0.0894427i
$126$	−0.643104	−0.0572923
$127$	−4.85086	−0.430444	−0.215222	−	0.976565i	$-0.569047\pi$
$127$	−4.85086	−0.430444	−0.215222	+	0.976565i	$0.569047\pi$
$128$	1.00000i	0.0883883i
$129$	−4.69202	−0.413109
$130$	0	0
$131$	1.52781	0.133485	0.0667427	−	0.997770i	$-0.478739\pi$
$131$	1.52781	0.133485	0.0667427	+	0.997770i	$0.478739\pi$
$132$	4.40581i	0.383477i
$133$	1.77048	0.153520
$134$	1.45712	0.125876
$135$	1.00000i	0.0860663i
$136$	− 0.939001i	− 0.0805187i
$137$	6.41119i	0.547745i	0.961766	+	0.273872i	$0.0883046\pi$
$137$	6.41119i	0.547745i	−0.961766	+	0.273872i	$0.911695\pi$
$138$	− 2.04892i	− 0.174415i
$139$	−8.78017	−0.744724	−0.372362	−	0.928088i	$-0.621452\pi$
$139$	−8.78017	−0.744724	−0.372362	+	0.928088i	$0.621452\pi$
$140$	−0.643104	−0.0543522
$141$	− 7.00000i	− 0.589506i
$142$	−7.03684	−0.590518
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 0.149145i	− 0.0123858i
$146$	−14.8605	−1.22987
$147$	6.58642	0.543239
$148$	6.07069i	0.499007i
$149$	− 22.6504i	− 1.85559i	−0.373087	−	0.927797i	$-0.621701\pi$
$149$	− 22.6504i	− 1.85559i	0.373087	−	0.927797i	$-0.378299\pi$
$150$	1.00000i	0.0816497i
$151$	− 10.2959i	− 0.837868i	−0.908017	−	0.418934i	$-0.862404\pi$
$151$	− 10.2959i	− 0.837868i	0.908017	−	0.418934i	$-0.137596\pi$
$152$	−2.75302	−0.223299
$153$	−0.939001	−0.0759137
$154$	2.83340i	0.228322i
$155$	−2.08815	−0.167724
$156$	0	0
$157$	0.994623	0.0793796	0.0396898	−	0.999212i	$-0.487363\pi$
$157$	0.994623	0.0793796	0.0396898	+	0.999212i	$0.487363\pi$
$158$	− 0.929312i	− 0.0739321i
$159$	1.68664	0.133760
$160$	1.00000	0.0790569
$161$	− 1.31767i	− 0.103847i
$162$	− 1.00000i	− 0.0785674i
$163$	− 21.6015i	− 1.69196i	−0.533216	−	0.845979i	$-0.679017\pi$
$163$	− 21.6015i	− 1.69196i	0.533216	−	0.845979i	$-0.320983\pi$
$164$	5.74094i	0.448292i
$165$	4.40581	0.342992
$166$	11.2959	0.876732
$167$	− 7.96615i	− 0.616439i	−0.951315	−	0.308220i	$-0.900267\pi$
$167$	− 7.96615i	− 0.616439i	0.951315	−	0.308220i	$-0.0997331\pi$
$168$	0.643104	0.0496166
$169$	0	0
$170$	−0.939001	−0.0720181
$171$	2.75302i	0.210529i
$172$	4.69202	0.357763
$173$	−4.11960	−0.313208	−0.156604	−	0.987661i	$-0.550055\pi$
$173$	−4.11960	−0.313208	−0.156604	+	0.987661i	$0.550055\pi$
$174$	0.149145i	0.0113066i
$175$	0.643104i	0.0486141i
$176$	− 4.40581i	− 0.332101i
$177$	− 7.85623i	− 0.590511i
$178$	7.47650	0.560387
$179$	8.78017	0.656261	0.328130	−	0.944632i	$-0.393582\pi$
$179$	8.78017	0.656261	0.328130	+	0.944632i	$0.393582\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−19.4741	−1.44750	−0.723750	−	0.690063i	$-0.757583\pi$
$181$	−19.4741	−1.44750	−0.723750	+	0.690063i	$0.757583\pi$
$182$	0	0
$183$	−12.0761	−0.892688
$184$	2.04892i	0.151048i
$185$	6.07069	0.446326
$186$	2.08815	0.153110
$187$	4.13706i	0.302532i
$188$	7.00000i	0.510527i
$189$	− 0.643104i	− 0.0467789i
$190$	2.75302i	0.199725i
$191$	0.753020	0.0544866	0.0272433	−	0.999629i	$-0.491327\pi$
$191$	0.753020	0.0544866	0.0272433	+	0.999629i	$0.491327\pi$
$192$	−1.00000	−0.0721688
$193$	7.84117i	0.564420i	0.959353	+	0.282210i	$0.0910674\pi$
$193$	7.84117i	0.564420i	−0.959353	+	0.282210i	$0.908933\pi$
$194$	−5.05861	−0.363187
$195$	0	0
$196$	−6.58642	−0.470458
$197$	10.2010i	0.726794i	0.931634	+	0.363397i	$0.118383\pi$
$197$	10.2010i	0.726794i	−0.931634	+	0.363397i	$0.881617\pi$
$198$	−4.40581	−0.313108
$199$	−1.65279	−0.117163	−0.0585817	−	0.998283i	$-0.518658\pi$
$199$	−1.65279	−0.117163	−0.0585817	+	0.998283i	$0.518658\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	1.45712i	0.102778i
$202$	− 6.81163i	− 0.479264i
$203$	0.0959157i	0.00673196i
$204$	0.939001	0.0657432
$205$	5.74094	0.400965
$206$	− 12.0858i	− 0.842054i
$207$	2.04892	0.142410
$208$	0	0
$209$	12.1293	0.839001
$210$	− 0.643104i	− 0.0443784i
$211$	11.7453	0.808576	0.404288	−	0.914632i	$-0.367519\pi$
$211$	11.7453	0.808576	0.404288	+	0.914632i	$0.367519\pi$
$212$	−1.68664	−0.115839
$213$	− 7.03684i	− 0.482156i
$214$	14.6679i	1.00267i
$215$	− 4.69202i	− 0.319993i
$216$	1.00000i	0.0680414i
$217$	1.34290	0.0911617
$218$	8.15883	0.552586
$219$	− 14.8605i	− 1.00418i
$220$	−4.40581	−0.297040
$221$	0	0
$222$	−6.07069	−0.407438
$223$	− 9.73855i	− 0.652141i	−0.945345	−	0.326071i	$-0.894275\pi$
$223$	− 9.73855i	− 0.652141i	0.945345	−	0.326071i	$-0.105725\pi$
$224$	−0.643104	−0.0429692
$225$	−1.00000	−0.0666667
$226$	− 16.8442i	− 1.12046i
$227$	− 25.2500i	− 1.67590i	−0.545748	−	0.837949i	$-0.683754\pi$
$227$	− 25.2500i	− 1.67590i	0.545748	−	0.837949i	$-0.316246\pi$
$228$	− 2.75302i	− 0.182323i
$229$	− 23.7845i	− 1.57172i	−0.618403	−	0.785861i	$-0.712220\pi$
$229$	− 23.7845i	− 1.57172i	0.618403	−	0.785861i	$-0.287780\pi$
$230$	2.04892	0.135102
$231$	−2.83340	−0.186424
$232$	− 0.149145i	− 0.00979184i
$233$	−0.603875	−0.0395612	−0.0197806	−	0.999804i	$-0.506297\pi$
$233$	−0.603875	−0.0395612	−0.0197806	+	0.999804i	$0.506297\pi$
$234$	0	0
$235$	7.00000	0.456630
$236$	7.85623i	0.511397i
$237$	0.929312	0.0603653
$238$	0.603875	0.0391434
$239$	− 28.5579i	− 1.84726i	−0.383286	−	0.923630i	$-0.625208\pi$
$239$	− 28.5579i	− 1.84726i	0.383286	−	0.923630i	$-0.374792\pi$
$240$	1.00000i	0.0645497i
$241$	− 6.31096i	− 0.406525i	−0.979124	−	0.203262i	$-0.934846\pi$
$241$	− 6.31096i	− 0.406525i	0.979124	−	0.203262i	$-0.0651545\pi$
$242$	8.41119i	0.540692i
$243$	1.00000	0.0641500
$244$	12.0761	0.773091
$245$	6.58642i	0.420791i
$246$	−5.74094	−0.366029
$247$	0	0
$248$	−2.08815	−0.132597
$249$	11.2959i	0.715848i
$250$	−1.00000	−0.0632456
$251$	−24.5972	−1.55256	−0.776280	−	0.630388i	$-0.782896\pi$
$251$	−24.5972	−1.55256	−0.776280	+	0.630388i	$0.782896\pi$
$252$	0.643104i	0.0405118i
$253$	− 9.02715i	− 0.567532i
$254$	4.85086i	0.304370i
$255$	− 0.939001i	− 0.0588025i
$256$	1.00000	0.0625000
$257$	26.9463	1.68086	0.840432	−	0.541917i	$-0.182301\pi$
$257$	26.9463	1.68086	0.840432	+	0.541917i	$0.182301\pi$
$258$	4.69202i	0.292112i
$259$	−3.90408	−0.242588
$260$	0	0
$261$	−0.149145	−0.00923184
$262$	− 1.52781i	− 0.0943885i
$263$	−24.0465	−1.48277	−0.741386	−	0.671079i	$-0.765831\pi$
$263$	−24.0465	−1.48277	−0.741386	+	0.671079i	$0.765831\pi$
$264$	4.40581	0.271159
$265$	1.68664i	0.103610i
$266$	− 1.77048i	− 0.108555i
$267$	7.47650i	0.457554i
$268$	− 1.45712i	− 0.0890080i
$269$	−7.17523	−0.437481	−0.218741	−	0.975783i	$-0.570195\pi$
$269$	−7.17523	−0.437481	−0.218741	+	0.975783i	$0.570195\pi$
$270$	1.00000	0.0608581
$271$	− 10.9825i	− 0.667142i	−0.942725	−	0.333571i	$-0.891746\pi$
$271$	− 10.9825i	− 0.667142i	0.942725	−	0.333571i	$-0.108254\pi$
$272$	−0.939001	−0.0569353
$273$	0	0
$274$	6.41119	0.387314
$275$	4.40581i	0.265681i
$276$	−2.04892	−0.123330
$277$	−16.2107	−0.974009	−0.487004	−	0.873400i	$-0.661910\pi$
$277$	−16.2107	−0.974009	−0.487004	+	0.873400i	$0.661910\pi$
$278$	8.78017i	0.526599i
$279$	2.08815i	0.125014i
$280$	0.643104i	0.0384328i
$281$	12.9584i	0.773032i	0.922283	+	0.386516i	$0.126322\pi$
$281$	12.9584i	0.773032i	−0.922283	+	0.386516i	$0.873678\pi$
$282$	−7.00000	−0.416844
$283$	−21.5415	−1.28051	−0.640256	−	0.768162i	$-0.721172\pi$
$283$	−21.5415	−1.28051	−0.640256	+	0.768162i	$0.721172\pi$
$284$	7.03684i	0.417559i
$285$	−2.75302	−0.163075
$286$	0	0
$287$	−3.69202	−0.217933
$288$	− 1.00000i	− 0.0589256i
$289$	−16.1183	−0.948134
$290$	−0.149145	−0.00875809
$291$	− 5.05861i	− 0.296541i
$292$	14.8605i	0.869648i
$293$	− 9.57971i	− 0.559653i	−0.960051	−	0.279826i	$-0.909723\pi$
$293$	− 9.57971i	− 0.559653i	0.960051	−	0.279826i	$-0.0902769\pi$
$294$	− 6.58642i	− 0.384128i
$295$	7.85623	0.457408
$296$	6.07069	0.352852
$297$	− 4.40581i	− 0.255651i
$298$	−22.6504	−1.31210
$299$	0	0
$300$	1.00000	0.0577350
$301$	3.01746i	0.173923i
$302$	−10.2959	−0.592462
$303$	6.81163	0.391318
$304$	2.75302i	0.157897i
$305$	− 12.0761i	− 0.691473i
$306$	0.939001i	0.0536791i
$307$	− 19.0140i	− 1.08519i	−0.839996	−	0.542593i	$-0.817443\pi$
$307$	− 19.0140i	− 1.08519i	0.839996	−	0.542593i	$-0.182557\pi$
$308$	2.83340	0.161448
$309$	12.0858	0.687534
$310$	2.08815i	0.118599i
$311$	3.74764	0.212509	0.106255	−	0.994339i	$-0.466114\pi$
$311$	3.74764	0.212509	0.106255	+	0.994339i	$0.466114\pi$
$312$	0	0
$313$	25.1196	1.41984	0.709922	−	0.704280i	$-0.248730\pi$
$313$	25.1196	1.41984	0.709922	+	0.704280i	$0.248730\pi$
$314$	− 0.994623i	− 0.0561298i
$315$	0.643104	0.0362348
$316$	−0.929312	−0.0522779
$317$	12.6679i	0.711498i	0.934582	+	0.355749i	$0.115774\pi$
$317$	12.6679i	0.711498i	−0.934582	+	0.355749i	$0.884226\pi$
$318$	− 1.68664i	− 0.0945823i
$319$	0.657105i	0.0367908i
$320$	− 1.00000i	− 0.0559017i
$321$	−14.6679	−0.818680
$322$	−1.31767	−0.0734307
$323$	− 2.58509i	− 0.143838i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−21.6015	−1.19640
$327$	8.15883i	0.451185i
$328$	5.74094	0.316990
$329$	−4.50173	−0.248188
$330$	− 4.40581i	− 0.242532i
$331$	− 6.43057i	− 0.353456i	−0.984260	−	0.176728i	$-0.943449\pi$
$331$	− 6.43057i	− 0.353456i	0.984260	−	0.176728i	$-0.0565513\pi$
$332$	− 11.2959i	− 0.619943i
$333$	− 6.07069i	− 0.332672i
$334$	−7.96615	−0.435888
$335$	−1.45712	−0.0796112
$336$	− 0.643104i	− 0.0350842i
$337$	−10.5496	−0.574672	−0.287336	−	0.957830i	$-0.592770\pi$
$337$	−10.5496	−0.574672	−0.287336	+	0.957830i	$0.592770\pi$
$338$	0	0
$339$	16.8442	0.914849
$340$	0.939001i	0.0509245i
$341$	9.19998	0.498207
$342$	2.75302	0.148866
$343$	− 8.73748i	− 0.471780i
$344$	− 4.69202i	− 0.252977i
$345$	2.04892i	0.110310i
$346$	4.11960i	0.221471i
$347$	9.87130	0.529919	0.264960	−	0.964260i	$-0.414641\pi$
$347$	9.87130	0.529919	0.264960	+	0.964260i	$0.414641\pi$
$348$	0.149145	0.00799501
$349$	30.5338i	1.63444i	0.576329	+	0.817218i	$0.304485\pi$
$349$	30.5338i	1.63444i	−0.576329	+	0.817218i	$0.695515\pi$
$350$	0.643104	0.0343754
$351$	0	0
$352$	−4.40581	−0.234831
$353$	− 24.7047i	− 1.31490i	−0.753499	−	0.657449i	$-0.771635\pi$
$353$	− 24.7047i	− 1.31490i	0.753499	−	0.657449i	$-0.228365\pi$
$354$	−7.85623	−0.417554
$355$	7.03684	0.373476
$356$	− 7.47650i	− 0.396254i
$357$	0.603875i	0.0319605i
$358$	− 8.78017i	− 0.464046i
$359$	11.5888i	0.611634i	0.952090	+	0.305817i	$0.0989296\pi$
$359$	11.5888i	0.611634i	−0.952090	+	0.305817i	$0.901070\pi$
$360$	−1.00000	−0.0527046
$361$	11.4209	0.601099
$362$	19.4741i	1.02354i
$363$	−8.41119	−0.441473
$364$	0	0
$365$	14.8605	0.777836
$366$	12.0761i	0.631226i
$367$	−20.5579	−1.07312	−0.536558	−	0.843863i	$-0.680276\pi$
$367$	−20.5579	−1.07312	−0.536558	+	0.843863i	$0.680276\pi$
$368$	2.04892	0.106807
$369$	− 5.74094i	− 0.298861i
$370$	− 6.07069i	− 0.315600i
$371$	− 1.08469i	− 0.0563142i
$372$	− 2.08815i	− 0.108265i
$373$	1.31037	0.0678485	0.0339242	−	0.999424i	$-0.489200\pi$
$373$	1.31037	0.0678485	0.0339242	+	0.999424i	$0.489200\pi$
$374$	4.13706	0.213922
$375$	− 1.00000i	− 0.0516398i
$376$	7.00000	0.360997
$377$	0	0
$378$	−0.643104	−0.0330777
$379$	28.1551i	1.44623i	0.690727	+	0.723115i	$0.257291\pi$
$379$	28.1551i	1.44623i	−0.690727	+	0.723115i	$0.742709\pi$
$380$	2.75302	0.141227
$381$	−4.85086	−0.248517
$382$	− 0.753020i	− 0.0385279i
$383$	− 18.6679i	− 0.953883i	−0.878935	−	0.476941i	$-0.841745\pi$
$383$	− 18.6679i	− 0.953883i	0.878935	−	0.476941i	$-0.158255\pi$
$384$	1.00000i	0.0510310i
$385$	− 2.83340i	− 0.144403i
$386$	7.84117	0.399105
$387$	−4.69202	−0.238509
$388$	5.05861i	0.256812i
$389$	29.6752	1.50459	0.752295	−	0.658826i	$-0.228947\pi$
$389$	29.6752	1.50459	0.752295	+	0.658826i	$0.228947\pi$
$390$	0	0
$391$	−1.92394	−0.0972976
$392$	6.58642i	0.332664i
$393$	1.52781	0.0770679
$394$	10.2010	0.513921
$395$	0.929312i	0.0467588i
$396$	4.40581i	0.221400i
$397$	28.7536i	1.44310i	0.692361	+	0.721551i	$0.256571\pi$
$397$	28.7536i	1.44310i	−0.692361	+	0.721551i	$0.743429\pi$
$398$	1.65279i	0.0828470i
$399$	1.77048	0.0886348
$400$	−1.00000	−0.0500000
$401$	− 18.6993i	− 0.933799i	−0.884310	−	0.466900i	$-0.845371\pi$
$401$	− 18.6993i	− 0.933799i	0.884310	−	0.466900i	$-0.154629\pi$
$402$	1.45712	0.0726747
$403$	0	0
$404$	−6.81163	−0.338891
$405$	1.00000i	0.0496904i
$406$	0.0959157	0.00476022
$407$	−26.7463	−1.32577
$408$	− 0.939001i	− 0.0464875i
$409$	10.6233i	0.525286i	0.964893	+	0.262643i	$0.0845942\pi$
$409$	10.6233i	0.525286i	−0.964893	+	0.262643i	$0.915406\pi$
$410$	− 5.74094i	− 0.283525i
$411$	6.41119i	0.316241i
$412$	−12.0858	−0.595422
$413$	−5.05238	−0.248611
$414$	− 2.04892i	− 0.100699i
$415$	−11.2959	−0.554494
$416$	0	0
$417$	−8.78017	−0.429967
$418$	− 12.1293i	− 0.593263i
$419$	19.4198	0.948720	0.474360	−	0.880331i	$-0.342680\pi$
$419$	19.4198	0.948720	0.474360	+	0.880331i	$0.342680\pi$
$420$	−0.643104	−0.0313803
$421$	20.5719i	1.00262i	0.865269	+	0.501308i	$0.167147\pi$
$421$	20.5719i	1.00262i	−0.865269	+	0.501308i	$0.832853\pi$
$422$	− 11.7453i	− 0.571750i
$423$	− 7.00000i	− 0.340352i
$424$	1.68664i	0.0819107i
$425$	0.939001	0.0455482
$426$	−7.03684	−0.340936
$427$	7.76617i	0.375831i
$428$	14.6679	0.708998
$429$	0	0
$430$	−4.69202	−0.226269
$431$	33.8079i	1.62847i	0.580536	+	0.814235i	$0.302843\pi$
$431$	33.8079i	1.62847i	−0.580536	+	0.814235i	$0.697157\pi$
$432$	1.00000	0.0481125
$433$	−27.2838	−1.31118	−0.655588	−	0.755119i	$-0.727579\pi$
$433$	−27.2838	−1.31118	−0.655588	+	0.755119i	$0.727579\pi$
$434$	− 1.34290i	− 0.0644610i
$435$	− 0.149145i	− 0.00715095i
$436$	− 8.15883i	− 0.390737i
$437$	5.64071i	0.269832i
$438$	−14.8605	−0.710064
$439$	1.56943	0.0749049	0.0374525	−	0.999298i	$-0.488076\pi$
$439$	1.56943	0.0749049	0.0374525	+	0.999298i	$0.488076\pi$
$440$	4.40581i	0.210039i
$441$	6.58642	0.313639
$442$	0	0
$443$	38.0489	1.80776	0.903879	−	0.427788i	$-0.140707\pi$
$443$	38.0489	1.80776	0.903879	+	0.427788i	$0.140707\pi$
$444$	6.07069i	0.288102i
$445$	−7.47650	−0.354420
$446$	−9.73855	−0.461134
$447$	− 22.6504i	− 1.07133i
$448$	0.643104i	0.0303838i
$449$	24.2373i	1.14383i	0.820313	+	0.571914i	$0.193799\pi$
$449$	24.2373i	1.14383i	−0.820313	+	0.571914i	$0.806201\pi$
$450$	1.00000i	0.0471405i
$451$	−25.2935	−1.19102
$452$	−16.8442	−0.792282
$453$	− 10.2959i	− 0.483743i
$454$	−25.2500	−1.18504
$455$	0	0
$456$	−2.75302	−0.128922
$457$	9.75063i	0.456115i	0.973648	+	0.228058i	$0.0732374\pi$
$457$	9.75063i	0.456115i	−0.973648	+	0.228058i	$0.926763\pi$
$458$	−23.7845	−1.11138
$459$	−0.939001	−0.0438288
$460$	− 2.04892i	− 0.0955313i
$461$	31.0358i	1.44548i	0.691120	+	0.722740i	$0.257118\pi$
$461$	31.0358i	1.44548i	−0.691120	+	0.722740i	$0.742882\pi$
$462$	2.83340i	0.131822i
$463$	− 33.1618i	− 1.54116i	−0.637343	−	0.770580i	$-0.719967\pi$
$463$	− 33.1618i	− 1.54116i	0.637343	−	0.770580i	$-0.280033\pi$
$464$	−0.149145	−0.00692388
$465$	−2.08815	−0.0968355
$466$	0.603875i	0.0279740i
$467$	16.1153	0.745727	0.372863	−	0.927886i	$-0.378376\pi$
$467$	16.1153	0.745727	0.372863	+	0.927886i	$0.378376\pi$
$468$	0	0
$469$	0.937082	0.0432704
$470$	− 7.00000i	− 0.322886i
$471$	0.994623	0.0458298
$472$	7.85623	0.361612
$473$	20.6722i	0.950507i
$474$	− 0.929312i	− 0.0426847i
$475$	− 2.75302i	− 0.126317i
$476$	− 0.603875i	− 0.0276786i
$477$	1.68664	0.0772262
$478$	−28.5579	−1.30621
$479$	30.6474i	1.40032i	0.713988	+	0.700158i	$0.246887\pi$
$479$	30.6474i	1.40032i	−0.713988	+	0.700158i	$0.753113\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−6.31096	−0.287456
$483$	− 1.31767i	− 0.0599559i
$484$	8.41119	0.382327
$485$	5.05861	0.229699
$486$	− 1.00000i	− 0.0453609i
$487$	17.6829i	0.801290i	0.916233	+	0.400645i	$0.131214\pi$
$487$	17.6829i	0.801290i	−0.916233	+	0.400645i	$0.868786\pi$
$488$	− 12.0761i	− 0.546658i
$489$	− 21.6015i	− 0.976853i
$490$	6.58642	0.297544
$491$	28.5435	1.28815	0.644074	−	0.764963i	$-0.277243\pi$
$491$	28.5435	1.28815	0.644074	+	0.764963i	$0.277243\pi$
$492$	5.74094i	0.258822i
$493$	0.140047	0.00630741
$494$	0	0
$495$	4.40581	0.198027
$496$	2.08815i	0.0937605i
$497$	−4.52542	−0.202993
$498$	11.2959	0.506181
$499$	− 27.9409i	− 1.25081i	−0.780301	−	0.625404i	$-0.784934\pi$
$499$	− 27.9409i	− 1.25081i	0.780301	−	0.625404i	$-0.215066\pi$
$500$	1.00000i	0.0447214i
$501$	− 7.96615i	− 0.355901i
$502$	24.5972i	1.09783i
$503$	31.0901	1.38624	0.693119	−	0.720823i	$-0.256236\pi$
$503$	31.0901	1.38624	0.693119	+	0.720823i	$0.256236\pi$
$504$	0.643104	0.0286461
$505$	6.81163i	0.303113i
$506$	−9.02715	−0.401306
$507$	0	0
$508$	4.85086	0.215222
$509$	0.0217703i	0 0.000964950i	1.00000		0.000482475i	$0.000153577\pi$
$509$	0.0217703i	0 0.000964950i	−1.00000		0.000482475i	$0.999846\pi$
$510$	−0.939001	−0.0415797
$511$	−9.55688	−0.422771
$512$	− 1.00000i	− 0.0441942i
$513$	2.75302i	0.121549i
$514$	− 26.9463i	− 1.18855i
$515$	12.0858i	0.532562i
$516$	4.69202	0.206555
$517$	−30.8407	−1.35637
$518$	3.90408i	0.171536i
$519$	−4.11960	−0.180831
$520$	0	0
$521$	18.0747	0.791869	0.395934	−	0.918279i	$-0.370421\pi$
$521$	18.0747	0.791869	0.395934	+	0.918279i	$0.370421\pi$
$522$	0.149145i	0.00652790i
$523$	−15.3793	−0.672488	−0.336244	−	0.941775i	$-0.609157\pi$
$523$	−15.3793	−0.672488	−0.336244	+	0.941775i	$0.609157\pi$
$524$	−1.52781	−0.0667427
$525$	0.643104i	0.0280674i
$526$	24.0465i	1.04848i
$527$	− 1.96077i	− 0.0854125i
$528$	− 4.40581i	− 0.191738i
$529$	−18.8019	−0.817476
$530$	1.68664	0.0732632
$531$	− 7.85623i	− 0.340931i
$532$	−1.77048	−0.0767600
$533$	0	0
$534$	7.47650	0.323540
$535$	− 14.6679i	− 0.634147i
$536$	−1.45712	−0.0629381
$537$	8.78017	0.378892
$538$	7.17523i	0.309346i
$539$	− 29.0185i	− 1.24992i
$540$	− 1.00000i	− 0.0430331i
$541$	− 5.06339i	− 0.217692i	−0.994059	−	0.108846i	$-0.965284\pi$
$541$	− 5.06339i	− 0.217692i	0.994059	−	0.108846i	$-0.0347156\pi$
$542$	−10.9825	−0.471741
$543$	−19.4741	−0.835714
$544$	0.939001i	0.0402593i
$545$	−8.15883	−0.349486
$546$	0	0
$547$	8.77107	0.375024	0.187512	−	0.982262i	$-0.439958\pi$
$547$	8.77107	0.375024	0.187512	+	0.982262i	$0.439958\pi$
$548$	− 6.41119i	− 0.273872i
$549$	−12.0761	−0.515394
$550$	4.40581	0.187865
$551$	− 0.410599i	− 0.0174921i
$552$	2.04892i	0.0872077i
$553$	− 0.597645i	− 0.0254144i
$554$	16.2107i	0.688728i
$555$	6.07069	0.257686
$556$	8.78017	0.372362
$557$	8.10215i	0.343299i	0.985158	+	0.171649i	$0.0549097\pi$
$557$	8.10215i	0.343299i	−0.985158	+	0.171649i	$0.945090\pi$
$558$	2.08815	0.0883983
$559$	0	0
$560$	0.643104	0.0271761
$561$	4.13706i	0.174667i
$562$	12.9584	0.546616
$563$	6.86699	0.289409	0.144704	−	0.989475i	$-0.453777\pi$
$563$	6.86699	0.289409	0.144704	+	0.989475i	$0.453777\pi$
$564$	7.00000i	0.294753i
$565$	16.8442i	0.708639i
$566$	21.5415i	0.905459i
$567$	− 0.643104i	− 0.0270078i
$568$	7.03684	0.295259
$569$	−6.93171	−0.290592	−0.145296	−	0.989388i	$-0.546413\pi$
$569$	−6.93171	−0.290592	−0.145296	+	0.989388i	$0.546413\pi$
$570$	2.75302i	0.115311i
$571$	24.2760	1.01592	0.507960	−	0.861380i	$-0.330400\pi$
$571$	24.2760	1.01592	0.507960	+	0.861380i	$0.330400\pi$
$572$	0	0
$573$	0.753020	0.0314579
$574$	3.69202i	0.154102i
$575$	−2.04892	−0.0854458
$576$	−1.00000	−0.0416667
$577$	13.8189i	0.575289i	0.957737	+	0.287645i	$0.0928722\pi$
$577$	13.8189i	0.575289i	−0.957737	+	0.287645i	$0.907128\pi$
$578$	16.1183i	0.670432i
$579$	7.84117i	0.325868i
$580$	0.149145i	0.00619291i
$581$	7.26444	0.301380
$582$	−5.05861	−0.209686
$583$	− 7.43104i	− 0.307762i
$584$	14.8605	0.614934
$585$	0	0
$586$	−9.57971	−0.395734
$587$	27.6165i	1.13986i	0.821694	+	0.569928i	$0.193029\pi$
$587$	27.6165i	1.13986i	−0.821694	+	0.569928i	$0.806971\pi$
$588$	−6.58642	−0.271619
$589$	−5.74871	−0.236871
$590$	− 7.85623i	− 0.323436i
$591$	10.2010i	0.419615i
$592$	− 6.07069i	− 0.249504i
$593$	30.3913i	1.24802i	0.781415	+	0.624011i	$0.214498\pi$
$593$	30.3913i	1.24802i	−0.781415	+	0.624011i	$0.785502\pi$
$594$	−4.40581	−0.180773
$595$	−0.603875	−0.0247565
$596$	22.6504i	0.927797i
$597$	−1.65279	−0.0676443
$598$	0	0
$599$	10.6480	0.435066	0.217533	−	0.976053i	$-0.430199\pi$
$599$	10.6480	0.435066	0.217533	+	0.976053i	$0.430199\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	42.2495	1.72339	0.861696	−	0.507424i	$-0.169402\pi$
$601$	42.2495	1.72339	0.861696	+	0.507424i	$0.169402\pi$
$602$	3.01746	0.122982
$603$	1.45712i	0.0593387i
$604$	10.2959i	0.418934i
$605$	− 8.41119i	− 0.341964i
$606$	− 6.81163i	− 0.276703i
$607$	−17.6209	−0.715209	−0.357604	−	0.933873i	$-0.616406\pi$
$607$	−17.6209	−0.715209	−0.357604	+	0.933873i	$0.616406\pi$
$608$	2.75302	0.111650
$609$	0.0959157i	0.00388670i
$610$	−12.0761	−0.488946
$611$	0	0
$612$	0.939001	0.0379569
$613$	− 27.9527i	− 1.12900i	−0.825433	−	0.564500i	$-0.809069\pi$
$613$	− 27.9527i	− 1.12900i	0.825433	−	0.564500i	$-0.190931\pi$
$614$	−19.0140	−0.767343
$615$	5.74094	0.231497
$616$	− 2.83340i	− 0.114161i
$617$	14.9065i	0.600112i	0.953922	+	0.300056i	$0.0970054\pi$
$617$	14.9065i	0.600112i	−0.953922	+	0.300056i	$0.902995\pi$
$618$	− 12.0858i	− 0.486160i
$619$	33.7493i	1.35650i	0.734832	+	0.678249i	$0.237261\pi$
$619$	33.7493i	1.35650i	−0.734832	+	0.678249i	$0.762739\pi$
$620$	2.08815	0.0838620
$621$	2.04892	0.0822202
$622$	− 3.74764i	− 0.150267i
$623$	4.80817	0.192635
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 25.1196i	− 1.00398i
$627$	12.1293	0.484397
$628$	−0.994623	−0.0396898
$629$	5.70038i	0.227289i
$630$	− 0.643104i	− 0.0256219i
$631$	37.2519i	1.48297i	0.670967	+	0.741487i	$0.265879\pi$
$631$	37.2519i	1.48297i	−0.670967	+	0.741487i	$0.734121\pi$
$632$	0.929312i	0.0369661i
$633$	11.7453	0.466832
$634$	12.6679	0.503105
$635$	− 4.85086i	− 0.192500i
$636$	−1.68664	−0.0668798
$637$	0	0
$638$	0.657105	0.0260150
$639$	− 7.03684i	− 0.278373i
$640$	−1.00000	−0.0395285
$641$	−24.8203	−0.980341	−0.490170	−	0.871627i	$-0.663065\pi$
$641$	−24.8203	−0.980341	−0.490170	+	0.871627i	$0.663065\pi$
$642$	14.6679i	0.578894i
$643$	− 5.66248i	− 0.223306i	−0.993747	−	0.111653i	$-0.964385\pi$
$643$	− 5.66248i	− 0.223306i	0.993747	−	0.111653i	$-0.0356146\pi$
$644$	1.31767i	0.0519234i
$645$	− 4.69202i	− 0.184748i
$646$	−2.58509	−0.101709
$647$	27.0465	1.06331	0.531654	−	0.846961i	$-0.321571\pi$
$647$	27.0465	1.06331	0.531654	+	0.846961i	$0.321571\pi$
$648$	1.00000i	0.0392837i
$649$	−34.6131	−1.35868
$650$	0	0
$651$	1.34290	0.0526322
$652$	21.6015i	0.845979i
$653$	42.7741	1.67388	0.836939	−	0.547296i	$-0.184343\pi$
$653$	42.7741	1.67388	0.836939	+	0.547296i	$0.184343\pi$
$654$	8.15883	0.319036
$655$	1.52781i	0.0596965i
$656$	− 5.74094i	− 0.224146i
$657$	− 14.8605i	− 0.579765i
$658$	4.50173i	0.175496i
$659$	21.5623	0.839946	0.419973	−	0.907537i	$-0.362040\pi$
$659$	21.5623	0.839946	0.419973	+	0.907537i	$0.362040\pi$
$660$	−4.40581	−0.171496
$661$	− 48.5002i	− 1.88644i	−0.332170	−	0.943219i	$-0.607781\pi$
$661$	− 48.5002i	− 1.88644i	0.332170	−	0.943219i	$-0.392219\pi$
$662$	−6.43057	−0.249931
$663$	0	0
$664$	−11.2959	−0.438366
$665$	1.77048i	0.0686562i
$666$	−6.07069	−0.235234
$667$	−0.305586	−0.0118323
$668$	7.96615i	0.308220i
$669$	− 9.73855i	− 0.376514i
$670$	1.45712i	0.0562936i
$671$	53.2049i	2.05395i
$672$	−0.643104	−0.0248083
$673$	−30.4711	−1.17458	−0.587288	−	0.809378i	$-0.699804\pi$
$673$	−30.4711	−1.17458	−0.587288	+	0.809378i	$0.699804\pi$
$674$	10.5496i	0.406355i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−8.57971	−0.329745	−0.164873	−	0.986315i	$-0.552721\pi$
$677$	−8.57971	−0.329745	−0.164873	+	0.986315i	$0.552721\pi$
$678$	− 16.8442i	− 0.646896i
$679$	−3.25321	−0.124847
$680$	0.939001	0.0360090
$681$	− 25.2500i	− 0.967581i
$682$	− 9.19998i	− 0.352285i
$683$	− 27.8890i	− 1.06714i	−0.845755	−	0.533572i	$-0.820849\pi$
$683$	− 27.8890i	− 1.06714i	0.845755	−	0.533572i	$-0.179151\pi$
$684$	− 2.75302i	− 0.105264i
$685$	−6.41119	−0.244959
$686$	−8.73748	−0.333599
$687$	− 23.7845i	− 0.907434i
$688$	−4.69202	−0.178882
$689$	0	0
$690$	2.04892	0.0780010
$691$	39.4198i	1.49960i	0.661664	+	0.749800i	$0.269850\pi$
$691$	39.4198i	1.49960i	−0.661664	+	0.749800i	$0.730150\pi$
$692$	4.11960	0.156604
$693$	−2.83340	−0.107632
$694$	− 9.87130i	− 0.374709i
$695$	− 8.78017i	− 0.333051i
$696$	− 0.149145i	− 0.00565332i
$697$	5.39075i	0.204189i
$698$	30.5338	1.15572
$699$	−0.603875	−0.0228407
$700$	− 0.643104i	− 0.0243071i
$701$	51.7101	1.95306	0.976531	−	0.215376i	$-0.0690977\pi$
$701$	51.7101	1.95306	0.976531	+	0.215376i	$0.0690977\pi$
$702$	0	0
$703$	16.7127	0.630332
$704$	4.40581i	0.166050i
$705$	7.00000	0.263635
$706$	−24.7047	−0.929773
$707$	− 4.38059i	− 0.164749i
$708$	7.85623i	0.295255i
$709$	12.7278i	0.478002i	0.971019	+	0.239001i	$0.0768200\pi$
$709$	12.7278i	0.478002i	−0.971019	+	0.239001i	$0.923180\pi$
$710$	− 7.03684i	− 0.264088i
$711$	0.929312	0.0348519
$712$	−7.47650	−0.280194
$713$	4.27844i	0.160229i
$714$	0.603875	0.0225995
$715$	0	0
$716$	−8.78017	−0.328130
$717$	− 28.5579i	− 1.06652i
$718$	11.5888	0.432491
$719$	11.7342	0.437613	0.218807	−	0.975768i	$-0.429784\pi$
$719$	11.7342	0.437613	0.218807	+	0.975768i	$0.429784\pi$
$720$	1.00000i	0.0372678i
$721$	− 7.77240i	− 0.289459i
$722$	− 11.4209i	− 0.425041i
$723$	− 6.31096i	− 0.234707i
$724$	19.4741	0.723750
$725$	0.149145	0.00553910
$726$	8.41119i	0.312169i
$727$	6.84979	0.254045	0.127022	−	0.991900i	$-0.459458\pi$
$727$	6.84979	0.254045	0.127022	+	0.991900i	$0.459458\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 14.8605i	− 0.550013i
$731$	4.40581	0.162955
$732$	12.0761	0.446344
$733$	− 0.224144i	− 0.00827896i	−0.999991	−	0.00413948i	$-0.998682\pi$
$733$	− 0.224144i	− 0.00827896i	0.999991	−	0.00413948i	$-0.00131764\pi$
$734$	20.5579i	0.758807i
$735$	6.58642i	0.242944i
$736$	− 2.04892i	− 0.0755241i
$737$	6.41981	0.236477
$738$	−5.74094	−0.211327
$739$	2.39075i	0.0879451i	0.999033	+	0.0439725i	$0.0140014\pi$
$739$	2.39075i	0.0879451i	−0.999033	+	0.0439725i	$0.985999\pi$
$740$	−6.07069	−0.223163
$741$	0	0
$742$	−1.08469	−0.0398202
$743$	− 25.5144i	− 0.936033i	−0.883720	−	0.468016i	$-0.844969\pi$
$743$	− 25.5144i	− 0.936033i	0.883720	−	0.468016i	$-0.155031\pi$
$744$	−2.08815	−0.0765551
$745$	22.6504	0.829846
$746$	− 1.31037i	− 0.0479761i
$747$	11.2959i	0.413295i
$748$	− 4.13706i	− 0.151266i
$749$	9.43296i	0.344673i
$750$	−1.00000	−0.0365148
$751$	−18.2755	−0.666881	−0.333440	−	0.942771i	$-0.608210\pi$
$751$	−18.2755	−0.666881	−0.333440	+	0.942771i	$0.608210\pi$
$752$	− 7.00000i	− 0.255264i
$753$	−24.5972	−0.896371
$754$	0	0
$755$	10.2959	0.374706
$756$	0.643104i	0.0233895i
$757$	−20.2301	−0.735276	−0.367638	−	0.929969i	$-0.619833\pi$
$757$	−20.2301	−0.735276	−0.367638	+	0.929969i	$0.619833\pi$
$758$	28.1551	1.02264
$759$	− 9.02715i	− 0.327665i
$760$	− 2.75302i	− 0.0998625i
$761$	0.217440i	0.00788218i	0.999992	+	0.00394109i	$0.00125449\pi$
$761$	0.217440i	0.00788218i	−0.999992	+	0.00394109i	$0.998746\pi$
$762$	4.85086i	0.175728i
$763$	5.24698	0.189953
$764$	−0.753020	−0.0272433
$765$	− 0.939001i	− 0.0339497i
$766$	−18.6679	−0.674497
$767$	0	0
$768$	1.00000	0.0360844
$769$	0.124982i	0.00450696i	0.999997	+	0.00225348i	$0.000717305\pi$
$769$	0.124982i	0.00450696i	−0.999997	+	0.00225348i	$0.999283\pi$
$770$	−2.83340	−0.102109
$771$	26.9463	0.970447
$772$	− 7.84117i	− 0.282210i
$773$	11.7028i	0.420920i	0.977603	+	0.210460i	$0.0674961\pi$
$773$	11.7028i	0.420920i	−0.977603	+	0.210460i	$0.932504\pi$
$774$	4.69202i	0.168651i
$775$	− 2.08815i	− 0.0750084i
$776$	5.05861	0.181593
$777$	−3.90408	−0.140058
$778$	− 29.6752i	− 1.06391i
$779$	15.8049	0.566270
$780$	0	0
$781$	−31.0030	−1.10937
$782$	1.92394i	0.0687998i
$783$	−0.149145	−0.00533000
$784$	6.58642	0.235229
$785$	0.994623i	0.0354996i
$786$	− 1.52781i	− 0.0544952i
$787$	50.6926i	1.80700i	0.428592	+	0.903498i	$0.359010\pi$
$787$	50.6926i	1.80700i	−0.428592	+	0.903498i	$0.640990\pi$
$788$	− 10.2010i	− 0.363397i
$789$	−24.0465	−0.856079
$790$	0.929312	0.0330635
$791$	− 10.8325i	− 0.385161i
$792$	4.40581	0.156554
$793$	0	0
$794$	28.7536	1.02043
$795$	1.68664i	0.0598191i
$796$	1.65279	0.0585817
$797$	−45.7977	−1.62224	−0.811120	−	0.584880i	$-0.801141\pi$
$797$	−45.7977	−1.62224	−0.811120	+	0.584880i	$0.801141\pi$
$798$	− 1.77048i	− 0.0626743i
$799$	6.57301i	0.232536i
$800$	1.00000i	0.0353553i
$801$	7.47650i	0.264169i
$802$	−18.6993	−0.660296
$803$	−65.4728	−2.31048
$804$	− 1.45712i	− 0.0513888i
$805$	1.31767	0.0464417
$806$	0	0
$807$	−7.17523	−0.252580
$808$	6.81163i	0.239632i
$809$	22.5579	0.793095	0.396548	−	0.918014i	$-0.370208\pi$
$809$	22.5579	0.793095	0.396548	+	0.918014i	$0.370208\pi$
$810$	1.00000	0.0351364
$811$	19.2728i	0.676759i	0.941010	+	0.338380i	$0.109879\pi$
$811$	19.2728i	0.676759i	−0.941010	+	0.338380i	$0.890121\pi$
$812$	− 0.0959157i	− 0.00336598i
$813$	− 10.9825i	− 0.385175i
$814$	26.7463i	0.937458i
$815$	21.6015	0.756667
$816$	−0.939001	−0.0328716
$817$	− 12.9172i	− 0.451917i
$818$	10.6233	0.371433
$819$	0	0
$820$	−5.74094	−0.200482
$821$	11.5036i	0.401480i	0.979645	+	0.200740i	$0.0643347\pi$
$821$	11.5036i	0.401480i	−0.979645	+	0.200740i	$0.935665\pi$
$822$	6.41119	0.223616
$823$	25.2610	0.880542	0.440271	−	0.897865i	$-0.354882\pi$
$823$	25.2610	0.880542	0.440271	+	0.897865i	$0.354882\pi$
$824$	12.0858i	0.421027i
$825$	4.40581i	0.153391i
$826$	5.05238i	0.175795i
$827$	7.09677i	0.246779i	0.992358	+	0.123389i	$0.0393764\pi$
$827$	7.09677i	0.246779i	−0.992358	+	0.123389i	$0.960624\pi$
$828$	−2.04892	−0.0712048
$829$	−10.1226	−0.351572	−0.175786	−	0.984428i	$-0.556247\pi$
$829$	−10.1226	−0.351572	−0.175786	+	0.984428i	$0.556247\pi$
$830$	11.2959i	0.392086i
$831$	−16.2107	−0.562344
$832$	0	0
$833$	−6.18465	−0.214286
$834$	8.78017i	0.304032i
$835$	7.96615	0.275680
$836$	−12.1293	−0.419500
$837$	2.08815i	0.0721769i
$838$	− 19.4198i	− 0.670846i
$839$	43.0844i	1.48744i	0.668492	+	0.743720i	$0.266940\pi$
$839$	43.0844i	1.48744i	−0.668492	+	0.743720i	$0.733060\pi$
$840$	0.643104i	0.0221892i
$841$	−28.9778	−0.999233
$842$	20.5719	0.708956
$843$	12.9584i	0.446310i
$844$	−11.7453	−0.404288
$845$	0	0
$846$	−7.00000	−0.240665
$847$	5.40927i	0.185865i
$848$	1.68664	0.0579196
$849$	−21.5415	−0.739304
$850$	− 0.939001i	− 0.0322075i
$851$	− 12.4383i	− 0.426381i
$852$	7.03684i	0.241078i
$853$	− 43.7837i	− 1.49913i	−0.661933	−	0.749563i	$-0.730263\pi$
$853$	− 43.7837i	− 1.49913i	0.661933	−	0.749563i	$-0.269737\pi$
$854$	7.76617	0.265753
$855$	−2.75302	−0.0941513
$856$	− 14.6679i	− 0.501337i
$857$	16.3067	0.557025	0.278512	−	0.960433i	$-0.410159\pi$
$857$	16.3067	0.557025	0.278512	+	0.960433i	$0.410159\pi$
$858$	0	0
$859$	−31.8629	−1.08715	−0.543575	−	0.839361i	$-0.682930\pi$
$859$	−31.8629	−1.08715	−0.543575	+	0.839361i	$0.682930\pi$
$860$	4.69202i	0.159997i
$861$	−3.69202	−0.125824
$862$	33.8079	1.15150
$863$	− 0.511418i	− 0.0174089i	−0.999962	−	0.00870443i	$-0.997229\pi$
$863$	− 0.511418i	− 0.0174089i	0.999962	−	0.00870443i	$-0.00277074\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	− 4.11960i	− 0.140071i
$866$	27.2838i	0.927142i
$867$	−16.1183	−0.547405
$868$	−1.34290	−0.0455808
$869$	− 4.09438i	− 0.138892i
$870$	−0.149145	−0.00505649
$871$	0	0
$872$	−8.15883	−0.276293
$873$	− 5.05861i	− 0.171208i
$874$	5.64071	0.190800
$875$	−0.643104	−0.0217409
$876$	14.8605i	0.502091i
$877$	3.83638i	0.129545i	0.997900	+	0.0647727i	$0.0206322\pi$
$877$	3.83638i	0.129545i	−0.997900	+	0.0647727i	$0.979368\pi$
$878$	− 1.56943i	− 0.0529658i
$879$	− 9.57971i	− 0.323116i
$880$	4.40581	0.148520
$881$	0.499336	0.0168231	0.00841153	−	0.999965i	$-0.497322\pi$
$881$	0.499336	0.0168231	0.00841153	+	0.999965i	$0.497322\pi$
$882$	− 6.58642i	− 0.221776i
$883$	38.8993	1.30907	0.654533	−	0.756034i	$-0.272865\pi$
$883$	38.8993	1.30907	0.654533	+	0.756034i	$0.272865\pi$
$884$	0	0
$885$	7.85623	0.264084
$886$	− 38.0489i	− 1.27828i
$887$	−8.32245	−0.279441	−0.139720	−	0.990191i	$-0.544620\pi$
$887$	−8.32245	−0.279441	−0.139720	+	0.990191i	$0.544620\pi$
$888$	6.07069	0.203719
$889$	3.11960i	0.104628i
$890$	7.47650i	0.250613i
$891$	− 4.40581i	− 0.147600i
$892$	9.73855i	0.326071i
$893$	19.2711	0.644884
$894$	−22.6504	−0.757543
$895$	8.78017i	0.293489i
$896$	0.643104	0.0214846
$897$	0	0
$898$	24.2373	0.808809
$899$	− 0.311436i	− 0.0103870i
$900$	1.00000	0.0333333
$901$	−1.58376	−0.0527627
$902$	25.2935i	0.842182i
$903$	3.01746i	0.100415i
$904$	16.8442i	0.560228i
$905$	− 19.4741i	− 0.647341i
$906$	−10.2959	−0.342058
$907$	−27.4566	−0.911683	−0.455842	−	0.890061i	$-0.650662\pi$
$907$	−27.4566	−0.911683	−0.455842	+	0.890061i	$0.650662\pi$
$908$	25.2500i	0.837949i
$909$	6.81163	0.225927
$910$	0	0
$911$	−29.7224	−0.984748	−0.492374	−	0.870384i	$-0.663871\pi$
$911$	−29.7224	−0.984748	−0.492374	+	0.870384i	$0.663871\pi$
$912$	2.75302i	0.0911616i
$913$	49.7676	1.64707
$914$	9.75063	0.322522
$915$	− 12.0761i	− 0.399222i
$916$	23.7845i	0.785861i
$917$	− 0.982542i	− 0.0324464i
$918$	0.939001i	0.0309917i
$919$	−2.02954	−0.0669483	−0.0334742	−	0.999440i	$-0.510657\pi$
$919$	−2.02954	−0.0669483	−0.0334742	+	0.999440i	$0.510657\pi$
$920$	−2.04892	−0.0675508
$921$	− 19.0140i	− 0.626533i
$922$	31.0358	1.02211
$923$	0	0
$924$	2.83340	0.0932119
$925$	6.07069i	0.199603i
$926$	−33.1618	−1.08976
$927$	12.0858	0.396948
$928$	0.149145i	0.00489592i
$929$	0.815938i	0.0267701i	0.999910	+	0.0133850i	$0.00426072\pi$
$929$	0.815938i	0.0267701i	−0.999910	+	0.0133850i	$0.995739\pi$
$930$	2.08815i	0.0684730i
$931$	18.1325i	0.594270i
$932$	0.603875	0.0197806
$933$	3.74764	0.122692
$934$	− 16.1153i	− 0.527308i
$935$	−4.13706	−0.135296
$936$	0	0
$937$	−29.5864	−0.966546	−0.483273	−	0.875470i	$-0.660552\pi$
$937$	−29.5864	−0.966546	−0.483273	+	0.875470i	$0.660552\pi$
$938$	− 0.937082i	− 0.0305968i
$939$	25.1196	0.819747
$940$	−7.00000	−0.228315
$941$	30.2573i	0.986358i	0.869928	+	0.493179i	$0.164165\pi$
$941$	30.2573i	0.986358i	−0.869928	+	0.493179i	$0.835835\pi$
$942$	− 0.994623i	− 0.0324066i
$943$	− 11.7627i	− 0.383047i
$944$	− 7.85623i	− 0.255699i
$945$	0.643104	0.0209202
$946$	20.6722	0.672110
$947$	− 7.12152i	− 0.231418i	−0.993283	−	0.115709i	$-0.963086\pi$
$947$	− 7.12152i	− 0.231418i	0.993283	−	0.115709i	$-0.0369141\pi$
$948$	−0.929312	−0.0301827
$949$	0	0
$950$	−2.75302	−0.0893198
$951$	12.6679i	0.410783i
$952$	−0.603875	−0.0195717
$953$	38.3279	1.24156	0.620782	−	0.783983i	$-0.286815\pi$
$953$	38.3279	1.24156	0.620782	+	0.783983i	$0.286815\pi$
$954$	− 1.68664i	− 0.0546071i
$955$	0.753020i	0.0243672i
$956$	28.5579i	0.923630i
$957$	0.657105i	0.0212412i
$958$	30.6474	0.990173
$959$	4.12306	0.133141
$960$	− 1.00000i	− 0.0322749i
$961$	26.6396	0.859343
$962$	0	0
$963$	−14.6679	−0.472665
$964$	6.31096i	0.203262i
$965$	−7.84117	−0.252416
$966$	−1.31767	−0.0423952
$967$	− 50.6015i	− 1.62723i	−0.581401	−	0.813617i	$-0.697495\pi$
$967$	− 50.6015i	− 1.62723i	0.581401	−	0.813617i	$-0.302505\pi$
$968$	− 8.41119i	− 0.270346i
$969$	− 2.58509i	− 0.0830450i
$970$	− 5.05861i	− 0.162422i
$971$	−27.3026	−0.876182	−0.438091	−	0.898931i	$-0.644345\pi$
$971$	−27.3026	−0.876182	−0.438091	+	0.898931i	$0.644345\pi$
$972$	−1.00000	−0.0320750
$973$	5.64656i	0.181020i
$974$	17.6829	0.566597
$975$	0	0
$976$	−12.0761	−0.386545
$977$	− 46.5491i	− 1.48924i	−0.667490	−	0.744619i	$-0.732631\pi$
$977$	− 46.5491i	− 1.48924i	0.667490	−	0.744619i	$-0.267369\pi$
$978$	−21.6015	−0.690739
$979$	32.9401	1.05277
$980$	− 6.58642i	− 0.210395i
$981$	8.15883i	0.260492i
$982$	− 28.5435i	− 0.910859i
$983$	− 1.10321i	− 0.0351870i	−0.999845	−	0.0175935i	$-0.994400\pi$
$983$	− 1.10321i	− 0.0351870i	0.999845	−	0.0175935i	$-0.00560047\pi$
$984$	5.74094	0.183014
$985$	−10.2010	−0.325032
$986$	− 0.140047i	− 0.00446001i
$987$	−4.50173	−0.143292
$988$	0	0
$989$	−9.61356	−0.305694
$990$	− 4.40581i	− 0.140026i
$991$	−51.4752	−1.63516	−0.817581	−	0.575813i	$-0.804686\pi$
$991$	−51.4752	−1.63516	−0.817581	+	0.575813i	$0.804686\pi$
$992$	2.08815	0.0662987
$993$	− 6.43057i	− 0.204068i
$994$	4.52542i	0.143538i
$995$	− 1.65279i	− 0.0523971i
$996$	− 11.2959i	− 0.357924i
$997$	−28.9715	−0.917537	−0.458769	−	0.888556i	$-0.651709\pi$
$997$	−28.9715	−0.917537	−0.458769	+	0.888556i	$0.651709\pi$
$998$	−27.9409	−0.884454
$999$	− 6.07069i	− 0.192068i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.y.1351.2		6
13.5	odd	4		5070.2.a.bq.1.2	✓	3
13.8	odd	4		5070.2.a.bv.1.2	yes	3
13.12	even	2	inner	5070.2.b.y.1351.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bq.1.2	✓	3	13.5	odd	4
5070.2.a.bv.1.2	yes	3	13.8	odd	4
5070.2.b.y.1351.2		6	1.1	even	1	trivial
5070.2.b.y.1351.5		6	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -0.643104i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -0.643104i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.40581i q^{11} -1.00000 q^{12} -0.643104 q^{14} +1.00000i q^{15} +1.00000 q^{16} -0.939001 q^{17} -1.00000i q^{18} +2.75302i q^{19} -1.00000i q^{20} -0.643104i q^{21} -4.40581 q^{22} +2.04892 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +0.643104i q^{28} -0.149145 q^{29} +1.00000 q^{30} +2.08815i q^{31} -1.00000i q^{32} -4.40581i q^{33} +0.939001i q^{34} +0.643104 q^{35} -1.00000 q^{36} -6.07069i q^{37} +2.75302 q^{38} -1.00000 q^{40} -5.74094i q^{41} -0.643104 q^{42} -4.69202 q^{43} +4.40581i q^{44} +1.00000i q^{45} -2.04892i q^{46} -7.00000i q^{47} +1.00000 q^{48} +6.58642 q^{49} +1.00000i q^{50} -0.939001 q^{51} +1.68664 q^{53} -1.00000i q^{54} +4.40581 q^{55} +0.643104 q^{56} +2.75302i q^{57} +0.149145i q^{58} -7.85623i q^{59} -1.00000i q^{60} -12.0761 q^{61} +2.08815 q^{62} -0.643104i q^{63} -1.00000 q^{64} -4.40581 q^{66} +1.45712i q^{67} +0.939001 q^{68} +2.04892 q^{69} -0.643104i q^{70} -7.03684i q^{71} +1.00000i q^{72} -14.8605i q^{73} -6.07069 q^{74} -1.00000 q^{75} -2.75302i q^{76} -2.83340 q^{77} +0.929312 q^{79} +1.00000i q^{80} +1.00000 q^{81} -5.74094 q^{82} +11.2959i q^{83} +0.643104i q^{84} -0.939001i q^{85} +4.69202i q^{86} -0.149145 q^{87} +4.40581 q^{88} +7.47650i q^{89} +1.00000 q^{90} -2.04892 q^{92} +2.08815i q^{93} -7.00000 q^{94} -2.75302 q^{95} -1.00000i q^{96} -5.05861i q^{97} -6.58642i q^{98} -4.40581i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} - 6 q^{12} - 12 q^{14} + 6 q^{16} + 14 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} - 28 q^{29} + 6 q^{30} + 12 q^{35} - 6 q^{36} + 26 q^{38} - 6 q^{40} - 12 q^{42} - 18 q^{43} + 6 q^{48} - 10 q^{49} + 14 q^{51} + 6 q^{53} + 12 q^{56} - 42 q^{61} + 20 q^{62} - 6 q^{64} - 14 q^{68} - 6 q^{69} - 12 q^{74} - 6 q^{75} + 42 q^{77} + 30 q^{79} + 6 q^{81} - 6 q^{82} - 28 q^{87} + 6 q^{90} + 6 q^{92} - 42 q^{94} - 26 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 - 6 * q^12 - 12 * q^14 + 6 * q^16 + 14 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 - 28 * q^29 + 6 * q^30 + 12 * q^35 - 6 * q^36 + 26 * q^38 - 6 * q^40 - 12 * q^42 - 18 * q^43 + 6 * q^48 - 10 * q^49 + 14 * q^51 + 6 * q^53 + 12 * q^56 - 42 * q^61 + 20 * q^62 - 6 * q^64 - 14 * q^68 - 6 * q^69 - 12 * q^74 - 6 * q^75 + 42 * q^77 + 30 * q^79 + 6 * q^81 - 6 * q^82 - 28 * q^87 + 6 * q^90 + 6 * q^92 - 42 * q^94 - 26 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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