LMFDB - Embedded newform 5070.2.b.u.1351.5

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			1351.5
Root			$-0.445042i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.u.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +2.55496i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +2.55496i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.24698i q^{11} +1.00000 q^{12} -2.55496 q^{14} +1.00000i q^{15} +1.00000 q^{16} -2.02177 q^{17} +1.00000i q^{18} +1.33513i q^{19} +1.00000i q^{20} -2.55496i q^{21} -2.24698 q^{22} -5.58211 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.55496i q^{28} +3.26875 q^{29} -1.00000 q^{30} +1.35690i q^{31} +1.00000i q^{32} -2.24698i q^{33} -2.02177i q^{34} +2.55496 q^{35} -1.00000 q^{36} +9.64071i q^{37} -1.33513 q^{38} -1.00000 q^{40} -1.06100i q^{41} +2.55496 q^{42} -0.137063 q^{43} -2.24698i q^{44} -1.00000i q^{45} -5.58211i q^{46} -12.7995i q^{47} -1.00000 q^{48} +0.472189 q^{49} -1.00000i q^{50} +2.02177 q^{51} -3.03684 q^{53} -1.00000i q^{54} +2.24698 q^{55} +2.55496 q^{56} -1.33513i q^{57} +3.26875i q^{58} +11.9608i q^{59} -1.00000i q^{60} -0.972853 q^{61} -1.35690 q^{62} +2.55496i q^{63} -1.00000 q^{64} +2.24698 q^{66} -8.51573i q^{67} +2.02177 q^{68} +5.58211 q^{69} +2.55496i q^{70} -2.69202i q^{71} -1.00000i q^{72} +1.48427i q^{73} -9.64071 q^{74} +1.00000 q^{75} -1.33513i q^{76} -5.74094 q^{77} -7.77479 q^{79} -1.00000i q^{80} +1.00000 q^{81} +1.06100 q^{82} -17.7289i q^{83} +2.55496i q^{84} +2.02177i q^{85} -0.137063i q^{86} -3.26875 q^{87} +2.24698 q^{88} +15.4601i q^{89} +1.00000 q^{90} +5.58211 q^{92} -1.35690i q^{93} +12.7995 q^{94} +1.33513 q^{95} -1.00000i q^{96} +7.14675i q^{97} +0.472189i q^{98} +2.24698i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 16 q^{14} + 6 q^{16} - 6 q^{17} - 4 q^{22} - 22 q^{23} - 6 q^{25} - 6 q^{27} + 4 q^{29} - 6 q^{30} + 16 q^{35} - 6 q^{36} - 6 q^{38} - 6 q^{40} + 16 q^{42} + 10 q^{43} - 6 q^{48} - 10 q^{49} + 6 q^{51} + 38 q^{53} + 4 q^{55} + 16 q^{56} - 18 q^{61} - 6 q^{64} + 4 q^{66} + 6 q^{68} + 22 q^{69} + 16 q^{74} + 6 q^{75} - 6 q^{77} - 50 q^{79} + 6 q^{81} + 26 q^{82} - 4 q^{87} + 4 q^{88} + 6 q^{90} + 22 q^{92} - 14 q^{94} + 6 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 + 6 * q^12 - 16 * q^14 + 6 * q^16 - 6 * q^17 - 4 * q^22 - 22 * q^23 - 6 * q^25 - 6 * q^27 + 4 * q^29 - 6 * q^30 + 16 * q^35 - 6 * q^36 - 6 * q^38 - 6 * q^40 + 16 * q^42 + 10 * q^43 - 6 * q^48 - 10 * q^49 + 6 * q^51 + 38 * q^53 + 4 * q^55 + 16 * q^56 - 18 * q^61 - 6 * q^64 + 4 * q^66 + 6 * q^68 + 22 * q^69 + 16 * q^74 + 6 * q^75 - 6 * q^77 - 50 * q^79 + 6 * q^81 + 26 * q^82 - 4 * q^87 + 4 * q^88 + 6 * q^90 + 22 * q^92 - 14 * q^94 + 6 * 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
$7$	2.55496i	0.965683i	0.875708	+	0.482842i	$0.160395\pi$
$7$	2.55496i	0.965683i	−0.875708	+	0.482842i	$0.839605\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	2.24698i	0.677490i	0.940878	+	0.338745i	$0.110002\pi$
$11$	2.24698i	0.677490i	−0.940878	+	0.338745i	$0.889998\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−2.55496	−0.682841
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	−2.02177	−0.490351	−0.245176	−	0.969479i	$-0.578846\pi$
$17$	−2.02177	−0.490351	−0.245176	+	0.969479i	$0.578846\pi$
$18$	1.00000i	0.235702i
$19$	1.33513i	0.306299i	0.988203	+	0.153149i	$0.0489416\pi$
$19$	1.33513i	0.306299i	−0.988203	+	0.153149i	$0.951058\pi$
$20$	1.00000i	0.223607i
$21$	− 2.55496i	− 0.557538i
$22$	−2.24698	−0.479058
$23$	−5.58211	−1.16395	−0.581975	−	0.813207i	$-0.697720\pi$
$23$	−5.58211	−1.16395	−0.581975	+	0.813207i	$0.697720\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	− 2.55496i	− 0.482842i
$29$	3.26875	0.606992	0.303496	−	0.952833i	$-0.401846\pi$
$29$	3.26875	0.606992	0.303496	+	0.952833i	$0.401846\pi$
$30$	−1.00000	−0.182574
$31$	1.35690i	0.243706i	0.992548	+	0.121853i	$0.0388836\pi$
$31$	1.35690i	0.243706i	−0.992548	+	0.121853i	$0.961116\pi$
$32$	1.00000i	0.176777i
$33$	− 2.24698i	− 0.391149i
$34$	− 2.02177i	− 0.346731i
$35$	2.55496	0.431867
$36$	−1.00000	−0.166667
$37$	9.64071i	1.58492i	0.609922	+	0.792462i	$0.291201\pi$
$37$	9.64071i	1.58492i	−0.609922	+	0.792462i	$0.708799\pi$
$38$	−1.33513	−0.216586
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 1.06100i	− 0.165700i	−0.996562	−	0.0828501i	$-0.973598\pi$
$41$	− 1.06100i	− 0.165700i	0.996562	−	0.0828501i	$-0.0264023\pi$
$42$	2.55496	0.394239
$43$	−0.137063	−0.0209020	−0.0104510	−	0.999945i	$-0.503327\pi$
$43$	−0.137063	−0.0209020	−0.0104510	+	0.999945i	$0.503327\pi$
$44$	− 2.24698i	− 0.338745i
$45$	− 1.00000i	− 0.149071i
$46$	− 5.58211i	− 0.823037i
$47$	− 12.7995i	− 1.86701i	−0.358570	−	0.933503i	$-0.616736\pi$
$47$	− 12.7995i	− 1.86701i	0.358570	−	0.933503i	$-0.383264\pi$
$48$	−1.00000	−0.144338
$49$	0.472189	0.0674556
$50$	− 1.00000i	− 0.141421i
$51$	2.02177	0.283104
$52$	0	0
$53$	−3.03684	−0.417141	−0.208571	−	0.978007i	$-0.566881\pi$
$53$	−3.03684	−0.417141	−0.208571	+	0.978007i	$0.566881\pi$
$54$	− 1.00000i	− 0.136083i
$55$	2.24698	0.302983
$56$	2.55496	0.341421
$57$	− 1.33513i	− 0.176842i
$58$	3.26875i	0.429208i
$59$	11.9608i	1.55716i	0.627545	+	0.778580i	$0.284060\pi$
$59$	11.9608i	1.55716i	−0.627545	+	0.778580i	$0.715940\pi$
$60$	− 1.00000i	− 0.129099i
$61$	−0.972853	−0.124561	−0.0622805	−	0.998059i	$-0.519837\pi$
$61$	−0.972853	−0.124561	−0.0622805	+	0.998059i	$0.519837\pi$
$62$	−1.35690	−0.172326
$63$	2.55496i	0.321894i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	2.24698	0.276584
$67$	− 8.51573i	− 1.04036i	−0.854056	−	0.520181i	$-0.825864\pi$
$67$	− 8.51573i	− 1.04036i	0.854056	−	0.520181i	$-0.174136\pi$
$68$	2.02177	0.245176
$69$	5.58211	0.672006
$70$	2.55496i	0.305376i
$71$	− 2.69202i	− 0.319484i	−0.987159	−	0.159742i	$-0.948934\pi$
$71$	− 2.69202i	− 0.319484i	0.987159	−	0.159742i	$-0.0510663\pi$
$72$	− 1.00000i	− 0.117851i
$73$	1.48427i	0.173721i	0.996220	+	0.0868604i	$0.0276834\pi$
$73$	1.48427i	0.173721i	−0.996220	+	0.0868604i	$0.972317\pi$
$74$	−9.64071	−1.12071
$75$	1.00000	0.115470
$76$	− 1.33513i	− 0.153149i
$77$	−5.74094	−0.654241
$78$	0	0
$79$	−7.77479	−0.874732	−0.437366	−	0.899284i	$-0.644089\pi$
$79$	−7.77479	−0.874732	−0.437366	+	0.899284i	$0.644089\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	1.06100	0.117168
$83$	− 17.7289i	− 1.94599i	−0.230816	−	0.972997i	$-0.574140\pi$
$83$	− 17.7289i	− 1.94599i	0.230816	−	0.972997i	$-0.425860\pi$
$84$	2.55496i	0.278769i
$85$	2.02177i	0.219292i
$86$	− 0.137063i	− 0.0147799i
$87$	−3.26875	−0.350447
$88$	2.24698	0.239529
$89$	15.4601i	1.63877i	0.573245	+	0.819384i	$0.305684\pi$
$89$	15.4601i	1.63877i	−0.573245	+	0.819384i	$0.694316\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	5.58211	0.581975
$93$	− 1.35690i	− 0.140704i
$94$	12.7995	1.32017
$95$	1.33513	0.136981
$96$	− 1.00000i	− 0.102062i
$97$	7.14675i	0.725643i	0.931859	+	0.362821i	$0.118186\pi$
$97$	7.14675i	0.725643i	−0.931859	+	0.362821i	$0.881814\pi$
$98$	0.472189i	0.0476983i
$99$	2.24698i	0.225830i
$100$	1.00000	0.100000
$101$	4.71379	0.469040	0.234520	−	0.972111i	$-0.424648\pi$
$101$	4.71379	0.469040	0.234520	+	0.972111i	$0.424648\pi$
$102$	2.02177i	0.200185i
$103$	−12.0858	−1.19084	−0.595422	−	0.803413i	$-0.703015\pi$
$103$	−12.0858	−1.19084	−0.595422	+	0.803413i	$0.703015\pi$
$104$	0	0
$105$	−2.55496	−0.249338
$106$	− 3.03684i	− 0.294964i
$107$	−8.01507	−0.774846	−0.387423	−	0.921902i	$-0.626635\pi$
$107$	−8.01507	−0.774846	−0.387423	+	0.921902i	$0.626635\pi$
$108$	1.00000	0.0962250
$109$	9.17390i	0.878700i	0.898316	+	0.439350i	$0.144791\pi$
$109$	9.17390i	0.878700i	−0.898316	+	0.439350i	$0.855209\pi$
$110$	2.24698i	0.214241i
$111$	− 9.64071i	− 0.915056i
$112$	2.55496i	0.241421i
$113$	5.24698	0.493594	0.246797	−	0.969067i	$-0.420622\pi$
$113$	5.24698	0.493594	0.246797	+	0.969067i	$0.420622\pi$
$114$	1.33513	0.125046
$115$	5.58211i	0.520534i
$116$	−3.26875	−0.303496
$117$	0	0
$118$	−11.9608	−1.10108
$119$	− 5.16554i	− 0.473524i
$120$	1.00000	0.0912871
$121$	5.95108	0.541008
$122$	− 0.972853i	− 0.0880780i
$123$	1.06100i	0.0956671i
$124$	− 1.35690i	− 0.121853i
$125$	1.00000i	0.0894427i
$126$	−2.55496	−0.227614
$127$	12.6843	1.12555	0.562773	−	0.826612i	$-0.309735\pi$
$127$	12.6843	1.12555	0.562773	+	0.826612i	$0.309735\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0.137063	0.0120678
$130$	0	0
$131$	−14.5332	−1.26977	−0.634885	−	0.772606i	$-0.718953\pi$
$131$	−14.5332	−1.26977	−0.634885	+	0.772606i	$0.718953\pi$
$132$	2.24698i	0.195574i
$133$	−3.41119	−0.295788
$134$	8.51573	0.735647
$135$	1.00000i	0.0860663i
$136$	2.02177i	0.173365i
$137$	5.06100i	0.432390i	0.976350	+	0.216195i	$0.0693647\pi$
$137$	5.06100i	0.432390i	−0.976350	+	0.216195i	$0.930635\pi$
$138$	5.58211i	0.475180i
$139$	−19.9638	−1.69330	−0.846652	−	0.532147i	$-0.821385\pi$
$139$	−19.9638	−1.69330	−0.846652	+	0.532147i	$0.821385\pi$
$140$	−2.55496	−0.215933
$141$	12.7995i	1.07792i
$142$	2.69202	0.225909
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 3.26875i	− 0.271455i
$146$	−1.48427	−0.122839
$147$	−0.472189	−0.0389455
$148$	− 9.64071i	− 0.792462i
$149$	− 3.33513i	− 0.273224i	−0.990625	−	0.136612i	$-0.956379\pi$
$149$	− 3.33513i	− 0.273224i	0.990625	−	0.136612i	$-0.0436214\pi$
$150$	1.00000i	0.0816497i
$151$	− 15.6625i	− 1.27459i	−0.770618	−	0.637297i	$-0.780052\pi$
$151$	− 15.6625i	− 1.27459i	0.770618	−	0.637297i	$-0.219948\pi$
$152$	1.33513	0.108293
$153$	−2.02177	−0.163450
$154$	− 5.74094i	− 0.462618i
$155$	1.35690	0.108988
$156$	0	0
$157$	2.99761	0.239235	0.119618	−	0.992820i	$-0.461833\pi$
$157$	2.99761	0.239235	0.119618	+	0.992820i	$0.461833\pi$
$158$	− 7.77479i	− 0.618529i
$159$	3.03684	0.240837
$160$	1.00000	0.0790569
$161$	− 14.2620i	− 1.12401i
$162$	1.00000i	0.0785674i
$163$	− 1.46681i	− 0.114890i	−0.998349	−	0.0574448i	$-0.981705\pi$
$163$	− 1.46681i	− 0.114890i	0.998349	−	0.0574448i	$-0.0182953\pi$
$164$	1.06100i	0.0828501i
$165$	−2.24698	−0.174927
$166$	17.7289	1.37603
$167$	1.13706i	0.0879886i	0.999032	+	0.0439943i	$0.0140083\pi$
$167$	1.13706i	0.0879886i	−0.999032	+	0.0439943i	$0.985992\pi$
$168$	−2.55496	−0.197119
$169$	0	0
$170$	−2.02177	−0.155063
$171$	1.33513i	0.102100i
$172$	0.137063	0.0104510
$173$	−0.983607	−0.0747822	−0.0373911	−	0.999301i	$-0.511905\pi$
$173$	−0.983607	−0.0747822	−0.0373911	+	0.999301i	$0.511905\pi$
$174$	− 3.26875i	− 0.247803i
$175$	− 2.55496i	− 0.193137i
$176$	2.24698i	0.169372i
$177$	− 11.9608i	− 0.899027i
$178$	−15.4601	−1.15878
$179$	−9.86592	−0.737414	−0.368707	−	0.929546i	$-0.620199\pi$
$179$	−9.86592	−0.737414	−0.368707	+	0.929546i	$0.620199\pi$
$180$	1.00000i	0.0745356i
$181$	3.66487	0.272408	0.136204	−	0.990681i	$-0.456510\pi$
$181$	3.66487	0.272408	0.136204	+	0.990681i	$0.456510\pi$
$182$	0	0
$183$	0.972853	0.0719154
$184$	5.58211i	0.411518i
$185$	9.64071	0.708799
$186$	1.35690	0.0994924
$187$	− 4.54288i	− 0.332208i
$188$	12.7995i	0.933503i
$189$	− 2.55496i	− 0.185846i
$190$	1.33513i	0.0968602i
$191$	−12.2252	−0.884585	−0.442293	−	0.896871i	$-0.645835\pi$
$191$	−12.2252	−0.884585	−0.442293	+	0.896871i	$0.645835\pi$
$192$	1.00000	0.0721688
$193$	− 10.5579i	− 0.759977i	−0.924991	−	0.379989i	$-0.875928\pi$
$193$	− 10.5579i	− 0.759977i	0.924991	−	0.379989i	$-0.124072\pi$
$194$	−7.14675	−0.513107
$195$	0	0
$196$	−0.472189	−0.0337278
$197$	− 5.46011i	− 0.389017i	−0.980901	−	0.194508i	$-0.937689\pi$
$197$	− 5.46011i	− 0.389017i	0.980901	−	0.194508i	$-0.0623111\pi$
$198$	−2.24698	−0.159686
$199$	−5.00969	−0.355127	−0.177564	−	0.984109i	$-0.556822\pi$
$199$	−5.00969	−0.355127	−0.177564	+	0.984109i	$0.556822\pi$
$200$	1.00000i	0.0707107i
$201$	8.51573i	0.600653i
$202$	4.71379i	0.331661i
$203$	8.35152i	0.586162i
$204$	−2.02177	−0.141552
$205$	−1.06100	−0.0741034
$206$	− 12.0858i	− 0.842054i
$207$	−5.58211	−0.387983
$208$	0	0
$209$	−3.00000	−0.207514
$210$	− 2.55496i	− 0.176309i
$211$	−9.38404	−0.646024	−0.323012	−	0.946395i	$-0.604695\pi$
$211$	−9.38404	−0.646024	−0.323012	+	0.946395i	$0.604695\pi$
$212$	3.03684	0.208571
$213$	2.69202i	0.184454i
$214$	− 8.01507i	− 0.547899i
$215$	0.137063i	0.00934764i
$216$	1.00000i	0.0680414i
$217$	−3.46681	−0.235343
$218$	−9.17390	−0.621335
$219$	− 1.48427i	− 0.100298i
$220$	−2.24698	−0.151491
$221$	0	0
$222$	9.64071	0.647042
$223$	− 26.5773i	− 1.77975i	−0.456205	−	0.889874i	$-0.650792\pi$
$223$	− 26.5773i	− 1.77975i	0.456205	−	0.889874i	$-0.349208\pi$
$224$	−2.55496	−0.170710
$225$	−1.00000	−0.0666667
$226$	5.24698i	0.349024i
$227$	− 22.2271i	− 1.47527i	−0.675202	−	0.737633i	$-0.735943\pi$
$227$	− 22.2271i	− 1.47527i	0.675202	−	0.737633i	$-0.264057\pi$
$228$	1.33513i	0.0884209i
$229$	− 19.1782i	− 1.26733i	−0.773607	−	0.633666i	$-0.781549\pi$
$229$	− 19.1782i	− 1.26733i	0.773607	−	0.633666i	$-0.218451\pi$
$230$	−5.58211	−0.368073
$231$	5.74094	0.377726
$232$	− 3.26875i	− 0.214604i
$233$	−19.5483	−1.28065	−0.640324	−	0.768105i	$-0.721200\pi$
$233$	−19.5483	−1.28065	−0.640324	+	0.768105i	$0.721200\pi$
$234$	0	0
$235$	−12.7995	−0.834950
$236$	− 11.9608i	− 0.778580i
$237$	7.77479	0.505027
$238$	5.16554	0.334832
$239$	− 12.8194i	− 0.829218i	−0.910000	−	0.414609i	$-0.863918\pi$
$239$	− 12.8194i	− 0.829218i	0.910000	−	0.414609i	$-0.136082\pi$
$240$	1.00000i	0.0645497i
$241$	− 8.95407i	− 0.576782i	−0.957513	−	0.288391i	$-0.906880\pi$
$241$	− 8.95407i	− 0.576782i	0.957513	−	0.288391i	$-0.0931203\pi$
$242$	5.95108i	0.382550i
$243$	−1.00000	−0.0641500
$244$	0.972853	0.0622805
$245$	− 0.472189i	− 0.0301670i
$246$	−1.06100	−0.0676468
$247$	0	0
$248$	1.35690	0.0861630
$249$	17.7289i	1.12352i
$250$	−1.00000	−0.0632456
$251$	0.276520	0.0174538	0.00872688	−	0.999962i	$-0.497222\pi$
$251$	0.276520	0.0174538	0.00872688	+	0.999962i	$0.497222\pi$
$252$	− 2.55496i	− 0.160947i
$253$	− 12.5429i	− 0.788564i
$254$	12.6843i	0.795881i
$255$	− 2.02177i	− 0.126608i
$256$	1.00000	0.0625000
$257$	−9.77048	−0.609466	−0.304733	−	0.952438i	$-0.598567\pi$
$257$	−9.77048	−0.609466	−0.304733	+	0.952438i	$0.598567\pi$
$258$	0.137063i	0.00853319i
$259$	−24.6316	−1.53053
$260$	0	0
$261$	3.26875	0.202331
$262$	− 14.5332i	− 0.897863i
$263$	−12.7071	−0.783553	−0.391776	−	0.920061i	$-0.628139\pi$
$263$	−12.7071	−0.783553	−0.391776	+	0.920061i	$0.628139\pi$
$264$	−2.24698	−0.138292
$265$	3.03684i	0.186551i
$266$	− 3.41119i	− 0.209153i
$267$	− 15.4601i	− 0.946143i
$268$	8.51573i	0.520181i
$269$	9.47757	0.577857	0.288929	−	0.957351i	$-0.406701\pi$
$269$	9.47757	0.577857	0.288929	+	0.957351i	$0.406701\pi$
$270$	−1.00000	−0.0608581
$271$	20.7657i	1.26143i	0.776016	+	0.630713i	$0.217237\pi$
$271$	20.7657i	1.26143i	−0.776016	+	0.630713i	$0.782763\pi$
$272$	−2.02177	−0.122588
$273$	0	0
$274$	−5.06100	−0.305746
$275$	− 2.24698i	− 0.135498i
$276$	−5.58211	−0.336003
$277$	−15.0911	−0.906738	−0.453369	−	0.891323i	$-0.649778\pi$
$277$	−15.0911	−0.906738	−0.453369	+	0.891323i	$0.649778\pi$
$278$	− 19.9638i	− 1.19735i
$279$	1.35690i	0.0812352i
$280$	− 2.55496i	− 0.152688i
$281$	− 17.6015i	− 1.05002i	−0.851097	−	0.525008i	$-0.824062\pi$
$281$	− 17.6015i	− 1.05002i	0.851097	−	0.525008i	$-0.175938\pi$
$282$	−12.7995	−0.762202
$283$	2.07606	0.123409	0.0617046	−	0.998094i	$-0.480346\pi$
$283$	2.07606	0.123409	0.0617046	+	0.998094i	$0.480346\pi$
$284$	2.69202i	0.159742i
$285$	−1.33513	−0.0790860
$286$	0	0
$287$	2.71081	0.160014
$288$	1.00000i	0.0589256i
$289$	−12.9124	−0.759556
$290$	3.26875	0.191948
$291$	− 7.14675i	− 0.418950i
$292$	− 1.48427i	− 0.0868604i
$293$	− 26.0930i	− 1.52437i	−0.647358	−	0.762186i	$-0.724126\pi$
$293$	− 26.0930i	− 1.52437i	0.647358	−	0.762186i	$-0.275874\pi$
$294$	− 0.472189i	− 0.0275386i
$295$	11.9608	0.696383
$296$	9.64071	0.560355
$297$	− 2.24698i	− 0.130383i
$298$	3.33513	0.193199
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	− 0.350191i	− 0.0201847i
$302$	15.6625	0.901275
$303$	−4.71379	−0.270800
$304$	1.33513i	0.0765747i
$305$	0.972853i	0.0557054i
$306$	− 2.02177i	− 0.115577i
$307$	− 15.8092i	− 0.902281i	−0.892453	−	0.451140i	$-0.851017\pi$
$307$	− 15.8092i	− 0.902281i	0.892453	−	0.451140i	$-0.148983\pi$
$308$	5.74094	0.327120
$309$	12.0858	0.687534
$310$	1.35690i	0.0770665i
$311$	23.2121	1.31624	0.658118	−	0.752915i	$-0.271353\pi$
$311$	23.2121	1.31624	0.658118	+	0.752915i	$0.271353\pi$
$312$	0	0
$313$	28.5297	1.61260	0.806298	−	0.591510i	$-0.201468\pi$
$313$	28.5297	1.61260	0.806298	+	0.591510i	$0.201468\pi$
$314$	2.99761i	0.169165i
$315$	2.55496	0.143956
$316$	7.77479	0.437366
$317$	− 0.709480i	− 0.0398484i	−0.999801	−	0.0199242i	$-0.993658\pi$
$317$	− 0.709480i	− 0.0398484i	0.999801	−	0.0199242i	$-0.00634248\pi$
$318$	3.03684i	0.170297i
$319$	7.34481i	0.411231i
$320$	1.00000i	0.0559017i
$321$	8.01507	0.447357
$322$	14.2620	0.794793
$323$	− 2.69932i	− 0.150194i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	1.46681	0.0812392
$327$	− 9.17390i	− 0.507318i
$328$	−1.06100	−0.0585839
$329$	32.7023	1.80294
$330$	− 2.24698i	− 0.123692i
$331$	− 22.8605i	− 1.25653i	−0.778000	−	0.628265i	$-0.783766\pi$
$331$	− 22.8605i	− 1.25653i	0.778000	−	0.628265i	$-0.216234\pi$
$332$	17.7289i	0.972997i
$333$	9.64071i	0.528308i
$334$	−1.13706	−0.0622173
$335$	−8.51573	−0.465264
$336$	− 2.55496i	− 0.139384i
$337$	−33.0629	−1.80105	−0.900526	−	0.434802i	$-0.856818\pi$
$337$	−33.0629	−1.80105	−0.900526	+	0.434802i	$0.856818\pi$
$338$	0	0
$339$	−5.24698	−0.284977
$340$	− 2.02177i	− 0.109646i
$341$	−3.04892	−0.165108
$342$	−1.33513	−0.0721953
$343$	19.0911i	1.03082i
$344$	0.137063i	0.00738996i
$345$	− 5.58211i	− 0.300530i
$346$	− 0.983607i	− 0.0528790i
$347$	23.4808	1.26052	0.630258	−	0.776386i	$-0.282949\pi$
$347$	23.4808	1.26052	0.630258	+	0.776386i	$0.282949\pi$
$348$	3.26875	0.175223
$349$	35.9004i	1.92170i	0.277065	+	0.960851i	$0.410638\pi$
$349$	35.9004i	1.92170i	−0.277065	+	0.960851i	$0.589362\pi$
$350$	2.55496	0.136568
$351$	0	0
$352$	−2.24698	−0.119764
$353$	− 31.0562i	− 1.65296i	−0.562969	−	0.826478i	$-0.690341\pi$
$353$	− 31.0562i	− 1.65296i	0.562969	−	0.826478i	$-0.309659\pi$
$354$	11.9608	0.635708
$355$	−2.69202	−0.142878
$356$	− 15.4601i	− 0.819384i
$357$	5.16554i	0.273389i
$358$	− 9.86592i	− 0.521430i
$359$	− 6.14675i	− 0.324413i	−0.986757	−	0.162207i	$-0.948139\pi$
$359$	− 6.14675i	− 0.324413i	0.986757	−	0.162207i	$-0.0518611\pi$
$360$	−1.00000	−0.0527046
$361$	17.2174	0.906181
$362$	3.66487i	0.192622i
$363$	−5.95108	−0.312351
$364$	0	0
$365$	1.48427	0.0776903
$366$	0.972853i	0.0508518i
$367$	−26.5163	−1.38414	−0.692070	−	0.721830i	$-0.743301\pi$
$367$	−26.5163	−1.38414	−0.692070	+	0.721830i	$0.743301\pi$
$368$	−5.58211	−0.290987
$369$	− 1.06100i	− 0.0552334i
$370$	9.64071i	0.501197i
$371$	− 7.75899i	− 0.402827i
$372$	1.35690i	0.0703518i
$373$	−27.0616	−1.40120	−0.700598	−	0.713556i	$-0.747083\pi$
$373$	−27.0616	−1.40120	−0.700598	+	0.713556i	$0.747083\pi$
$374$	4.54288	0.234907
$375$	− 1.00000i	− 0.0516398i
$376$	−12.7995	−0.660086
$377$	0	0
$378$	2.55496	0.131413
$379$	16.7313i	0.859427i	0.902965	+	0.429713i	$0.141385\pi$
$379$	16.7313i	0.859427i	−0.902965	+	0.429713i	$0.858615\pi$
$380$	−1.33513	−0.0684905
$381$	−12.6843	−0.649834
$382$	− 12.2252i	− 0.625496i
$383$	− 34.3569i	− 1.75556i	−0.479068	−	0.877778i	$-0.659025\pi$
$383$	− 34.3569i	− 1.75556i	0.479068	−	0.877778i	$-0.340975\pi$
$384$	1.00000i	0.0510310i
$385$	5.74094i	0.292585i
$386$	10.5579	0.537385
$387$	−0.137063	−0.00696732
$388$	− 7.14675i	− 0.362821i
$389$	−11.6732	−0.591857	−0.295928	−	0.955210i	$-0.595629\pi$
$389$	−11.6732	−0.591857	−0.295928	+	0.955210i	$0.595629\pi$
$390$	0	0
$391$	11.2857	0.570744
$392$	− 0.472189i	− 0.0238491i
$393$	14.5332	0.733102
$394$	5.46011	0.275076
$395$	7.77479i	0.391192i
$396$	− 2.24698i	− 0.112915i
$397$	12.6431i	0.634539i	0.948335	+	0.317270i	$0.102766\pi$
$397$	12.6431i	0.634539i	−0.948335	+	0.317270i	$0.897234\pi$
$398$	− 5.00969i	− 0.251113i
$399$	3.41119	0.170773
$400$	−1.00000	−0.0500000
$401$	29.7724i	1.48676i	0.668868	+	0.743381i	$0.266779\pi$
$401$	29.7724i	1.48676i	−0.668868	+	0.743381i	$0.733221\pi$
$402$	−8.51573	−0.424726
$403$	0	0
$404$	−4.71379	−0.234520
$405$	− 1.00000i	− 0.0496904i
$406$	−8.35152	−0.414479
$407$	−21.6625	−1.07377
$408$	− 2.02177i	− 0.100093i
$409$	− 5.61117i	− 0.277455i	−0.990331	−	0.138727i	$-0.955699\pi$
$409$	− 5.61117i	− 0.277455i	0.990331	−	0.138727i	$-0.0443011\pi$
$410$	− 1.06100i	− 0.0523990i
$411$	− 5.06100i	− 0.249641i
$412$	12.0858	0.595422
$413$	−30.5593	−1.50372
$414$	− 5.58211i	− 0.274346i
$415$	−17.7289	−0.870275
$416$	0	0
$417$	19.9638	0.977629
$418$	− 3.00000i	− 0.146735i
$419$	−1.24937	−0.0610358	−0.0305179	−	0.999534i	$-0.509716\pi$
$419$	−1.24937	−0.0610358	−0.0305179	+	0.999534i	$0.509716\pi$
$420$	2.55496	0.124669
$421$	− 2.76749i	− 0.134879i	−0.997723	−	0.0674397i	$-0.978517\pi$
$421$	− 2.76749i	− 0.134879i	0.997723	−	0.0674397i	$-0.0214830\pi$
$422$	− 9.38404i	− 0.456808i
$423$	− 12.7995i	− 0.622335i
$424$	3.03684i	0.147482i
$425$	2.02177	0.0980703
$426$	−2.69202	−0.130429
$427$	− 2.48560i	− 0.120287i
$428$	8.01507	0.387423
$429$	0	0
$430$	−0.137063	−0.00660978
$431$	− 19.1793i	− 0.923833i	−0.886923	−	0.461917i	$-0.847162\pi$
$431$	− 19.1793i	− 0.923833i	0.886923	−	0.461917i	$-0.152838\pi$
$432$	−1.00000	−0.0481125
$433$	−36.9051	−1.77355	−0.886774	−	0.462203i	$-0.847059\pi$
$433$	−36.9051	−1.77355	−0.886774	+	0.462203i	$0.847059\pi$
$434$	− 3.46681i	− 0.166412i
$435$	3.26875i	0.156725i
$436$	− 9.17390i	− 0.439350i
$437$	− 7.45281i	− 0.356516i
$438$	1.48427	0.0709212
$439$	−14.6213	−0.697838	−0.348919	−	0.937153i	$-0.613451\pi$
$439$	−14.6213	−0.697838	−0.348919	+	0.937153i	$0.613451\pi$
$440$	− 2.24698i	− 0.107121i
$441$	0.472189	0.0224852
$442$	0	0
$443$	−35.1377	−1.66944	−0.834720	−	0.550674i	$-0.814371\pi$
$443$	−35.1377	−1.66944	−0.834720	+	0.550674i	$0.814371\pi$
$444$	9.64071i	0.457528i
$445$	15.4601	0.732879
$446$	26.5773	1.25847
$447$	3.33513i	0.157746i
$448$	− 2.55496i	− 0.120710i
$449$	− 6.67994i	− 0.315246i	−0.987499	−	0.157623i	$-0.949617\pi$
$449$	− 6.67994i	− 0.315246i	0.987499	−	0.157623i	$-0.0503831\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	2.38404	0.112260
$452$	−5.24698	−0.246797
$453$	15.6625i	0.735888i
$454$	22.2271	1.04317
$455$	0	0
$456$	−1.33513	−0.0625230
$457$	22.9963i	1.07572i	0.843034	+	0.537860i	$0.180767\pi$
$457$	22.9963i	1.07572i	−0.843034	+	0.537860i	$0.819233\pi$
$458$	19.1782	0.896139
$459$	2.02177	0.0943682
$460$	− 5.58211i	− 0.260267i
$461$	− 33.9705i	− 1.58216i	−0.611711	−	0.791081i	$-0.709519\pi$
$461$	− 33.9705i	− 1.58216i	0.611711	−	0.791081i	$-0.290481\pi$
$462$	5.74094i	0.267093i
$463$	26.9594i	1.25291i	0.779457	+	0.626456i	$0.215495\pi$
$463$	26.9594i	1.25291i	−0.779457	+	0.626456i	$0.784505\pi$
$464$	3.26875	0.151748
$465$	−1.35690	−0.0629245
$466$	− 19.5483i	− 0.905555i
$467$	25.0411	1.15877	0.579383	−	0.815055i	$-0.303294\pi$
$467$	25.0411	1.15877	0.579383	+	0.815055i	$0.303294\pi$
$468$	0	0
$469$	21.7573	1.00466
$470$	− 12.7995i	− 0.590399i
$471$	−2.99761	−0.138122
$472$	11.9608	0.550539
$473$	− 0.307979i	− 0.0141609i
$474$	7.77479i	0.357108i
$475$	− 1.33513i	− 0.0612598i
$476$	5.16554i	0.236762i
$477$	−3.03684	−0.139047
$478$	12.8194	0.586346
$479$	7.21983i	0.329883i	0.986303	+	0.164941i	$0.0527435\pi$
$479$	7.21983i	0.329883i	−0.986303	+	0.164941i	$0.947257\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	8.95407	0.407847
$483$	14.2620i	0.648946i
$484$	−5.95108	−0.270504
$485$	7.14675	0.324517
$486$	− 1.00000i	− 0.0453609i
$487$	24.1497i	1.09433i	0.837025	+	0.547164i	$0.184293\pi$
$487$	24.1497i	1.09433i	−0.837025	+	0.547164i	$0.815707\pi$
$488$	0.972853i	0.0440390i
$489$	1.46681i	0.0663315i
$490$	0.472189	0.0213313
$491$	19.7259	0.890216	0.445108	−	0.895477i	$-0.353165\pi$
$491$	19.7259	0.890216	0.445108	+	0.895477i	$0.353165\pi$
$492$	− 1.06100i	− 0.0478335i
$493$	−6.60866	−0.297639
$494$	0	0
$495$	2.24698	0.100994
$496$	1.35690i	0.0609264i
$497$	6.87800	0.308521
$498$	−17.7289	−0.794449
$499$	35.3599i	1.58293i	0.611217	+	0.791463i	$0.290680\pi$
$499$	35.3599i	1.58293i	−0.611217	+	0.791463i	$0.709320\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 1.13706i	− 0.0508002i
$502$	0.276520i	0.0123417i
$503$	14.3558	0.640095	0.320047	−	0.947402i	$-0.396301\pi$
$503$	14.3558	0.640095	0.320047	+	0.947402i	$0.396301\pi$
$504$	2.55496	0.113807
$505$	− 4.71379i	− 0.209761i
$506$	12.5429	0.557599
$507$	0	0
$508$	−12.6843	−0.562773
$509$	− 21.4426i	− 0.950429i	−0.879870	−	0.475214i	$-0.842371\pi$
$509$	− 21.4426i	− 0.950429i	0.879870	−	0.475214i	$-0.157629\pi$
$510$	2.02177	0.0895255
$511$	−3.79225	−0.167759
$512$	1.00000i	0.0441942i
$513$	− 1.33513i	− 0.0589472i
$514$	− 9.77048i	− 0.430957i
$515$	12.0858i	0.532562i
$516$	−0.137063	−0.00603388
$517$	28.7603	1.26488
$518$	− 24.6316i	− 1.08225i
$519$	0.983607	0.0431755
$520$	0	0
$521$	9.58748	0.420035	0.210018	−	0.977698i	$-0.432648\pi$
$521$	9.58748	0.420035	0.210018	+	0.977698i	$0.432648\pi$
$522$	3.26875i	0.143069i
$523$	32.5241	1.42218	0.711090	−	0.703101i	$-0.248202\pi$
$523$	32.5241	1.42218	0.711090	+	0.703101i	$0.248202\pi$
$524$	14.5332	0.634885
$525$	2.55496i	0.111508i
$526$	− 12.7071i	− 0.554055i
$527$	− 2.74333i	− 0.119501i
$528$	− 2.24698i	− 0.0977872i
$529$	8.15990	0.354778
$530$	−3.03684	−0.131912
$531$	11.9608i	0.519053i
$532$	3.41119	0.147894
$533$	0	0
$534$	15.4601	0.669024
$535$	8.01507i	0.346521i
$536$	−8.51573	−0.367823
$537$	9.86592	0.425746
$538$	9.47757i	0.408607i
$539$	1.06100i	0.0457005i
$540$	− 1.00000i	− 0.0430331i
$541$	27.5381i	1.18395i	0.805954	+	0.591977i	$0.201653\pi$
$541$	27.5381i	1.18395i	−0.805954	+	0.591977i	$0.798347\pi$
$542$	−20.7657	−0.891963
$543$	−3.66487	−0.157275
$544$	− 2.02177i	− 0.0866827i
$545$	9.17390	0.392967
$546$	0	0
$547$	−18.0670	−0.772488	−0.386244	−	0.922397i	$-0.626228\pi$
$547$	−18.0670	−0.772488	−0.386244	+	0.922397i	$0.626228\pi$
$548$	− 5.06100i	− 0.216195i
$549$	−0.972853	−0.0415204
$550$	2.24698	0.0958115
$551$	4.36419i	0.185921i
$552$	− 5.58211i	− 0.237590i
$553$	− 19.8643i	− 0.844714i
$554$	− 15.0911i	− 0.641161i
$555$	−9.64071	−0.409225
$556$	19.9638	0.846652
$557$	− 33.5609i	− 1.42202i	−0.703181	−	0.711011i	$-0.748238\pi$
$557$	− 33.5609i	− 1.42202i	0.703181	−	0.711011i	$-0.251762\pi$
$558$	−1.35690	−0.0574420
$559$	0	0
$560$	2.55496	0.107967
$561$	4.54288i	0.191800i
$562$	17.6015	0.742474
$563$	−12.6974	−0.535132	−0.267566	−	0.963540i	$-0.586219\pi$
$563$	−12.6974	−0.535132	−0.267566	+	0.963540i	$0.586219\pi$
$564$	− 12.7995i	− 0.538958i
$565$	− 5.24698i	− 0.220742i
$566$	2.07606i	0.0872635i
$567$	2.55496i	0.107298i
$568$	−2.69202	−0.112955
$569$	−30.1317	−1.26319	−0.631593	−	0.775300i	$-0.717599\pi$
$569$	−30.1317	−1.26319	−0.631593	+	0.775300i	$0.717599\pi$
$570$	− 1.33513i	− 0.0559223i
$571$	−0.693349	−0.0290158	−0.0145079	−	0.999895i	$-0.504618\pi$
$571$	−0.693349	−0.0290158	−0.0145079	+	0.999895i	$0.504618\pi$
$572$	0	0
$573$	12.2252	0.510715
$574$	2.71081i	0.113147i
$575$	5.58211	0.232790
$576$	−1.00000	−0.0416667
$577$	− 34.6983i	− 1.44451i	−0.691628	−	0.722254i	$-0.743106\pi$
$577$	− 34.6983i	− 1.44451i	0.691628	−	0.722254i	$-0.256894\pi$
$578$	− 12.9124i	− 0.537087i
$579$	10.5579i	0.438773i
$580$	3.26875i	0.135727i
$581$	45.2965	1.87921
$582$	7.14675	0.296242
$583$	− 6.82371i	− 0.282609i
$584$	1.48427	0.0614196
$585$	0	0
$586$	26.0930	1.07789
$587$	− 0.891149i	− 0.0367816i	−0.999831	−	0.0183908i	$-0.994146\pi$
$587$	− 0.891149i	− 0.0367816i	0.999831	−	0.0183908i	$-0.00585431\pi$
$588$	0.472189	0.0194727
$589$	−1.81163	−0.0746468
$590$	11.9608i	0.492417i
$591$	5.46011i	0.224599i
$592$	9.64071i	0.396231i
$593$	34.7439i	1.42676i	0.700776	+	0.713381i	$0.252837\pi$
$593$	34.7439i	1.42676i	−0.700776	+	0.713381i	$0.747163\pi$
$594$	2.24698	0.0921947
$595$	−5.16554	−0.211766
$596$	3.33513i	0.136612i
$597$	5.00969	0.205033
$598$	0	0
$599$	10.4843	0.428376	0.214188	−	0.976792i	$-0.431290\pi$
$599$	10.4843	0.428376	0.214188	+	0.976792i	$0.431290\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	−2.28813	−0.0933347	−0.0466673	−	0.998910i	$-0.514860\pi$
$601$	−2.28813	−0.0933347	−0.0466673	+	0.998910i	$0.514860\pi$
$602$	0.350191	0.0142727
$603$	− 8.51573i	− 0.346787i
$604$	15.6625i	0.637297i
$605$	− 5.95108i	− 0.241946i
$606$	− 4.71379i	− 0.191485i
$607$	0.0499823	0.00202872	0.00101436	−	0.999999i	$-0.499677\pi$
$607$	0.0499823	0.00202872	0.00101436	+	0.999999i	$0.499677\pi$
$608$	−1.33513	−0.0541465
$609$	− 8.35152i	− 0.338421i
$610$	−0.972853	−0.0393897
$611$	0	0
$612$	2.02177	0.0817252
$613$	41.2737i	1.66703i	0.552499	+	0.833514i	$0.313674\pi$
$613$	41.2737i	1.66703i	−0.552499	+	0.833514i	$0.686326\pi$
$614$	15.8092	0.638009
$615$	1.06100	0.0427836
$616$	5.74094i	0.231309i
$617$	− 2.35450i	− 0.0947887i	−0.998876	−	0.0473944i	$-0.984908\pi$
$617$	− 2.35450i	− 0.0947887i	0.998876	−	0.0473944i	$-0.0150917\pi$
$618$	12.0858i	0.486160i
$619$	− 30.0097i	− 1.20619i	−0.797669	−	0.603096i	$-0.793934\pi$
$619$	− 30.0097i	− 1.20619i	0.797669	−	0.603096i	$-0.206066\pi$
$620$	−1.35690	−0.0544942
$621$	5.58211	0.224002
$622$	23.2121i	0.930719i
$623$	−39.4999	−1.58253
$624$	0	0
$625$	1.00000	0.0400000
$626$	28.5297i	1.14028i
$627$	3.00000	0.119808
$628$	−2.99761	−0.119618
$629$	− 19.4913i	− 0.777169i
$630$	2.55496i	0.101792i
$631$	35.6209i	1.41804i	0.705186	+	0.709022i	$0.250863\pi$
$631$	35.6209i	1.41804i	−0.705186	+	0.709022i	$0.749137\pi$
$632$	7.77479i	0.309265i
$633$	9.38404	0.372982
$634$	0.709480	0.0281770
$635$	− 12.6843i	− 0.503359i
$636$	−3.03684	−0.120418
$637$	0	0
$638$	−7.34481	−0.290784
$639$	− 2.69202i	− 0.106495i
$640$	−1.00000	−0.0395285
$641$	1.93230	0.0763211	0.0381606	−	0.999272i	$-0.487850\pi$
$641$	1.93230	0.0763211	0.0381606	+	0.999272i	$0.487850\pi$
$642$	8.01507i	0.316329i
$643$	31.8812i	1.25727i	0.777699	+	0.628637i	$0.216387\pi$
$643$	31.8812i	1.25727i	−0.777699	+	0.628637i	$0.783613\pi$
$644$	14.2620i	0.562003i
$645$	− 0.137063i	− 0.00539686i
$646$	2.69932	0.106203
$647$	−34.1527	−1.34268	−0.671341	−	0.741149i	$-0.734281\pi$
$647$	−34.1527	−1.34268	−0.671341	+	0.741149i	$0.734281\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−26.8756	−1.05496
$650$	0	0
$651$	3.46681	0.135875
$652$	1.46681i	0.0574448i
$653$	11.7909	0.461414	0.230707	−	0.973023i	$-0.425896\pi$
$653$	11.7909	0.461414	0.230707	+	0.973023i	$0.425896\pi$
$654$	9.17390	0.358728
$655$	14.5332i	0.567859i
$656$	− 1.06100i	− 0.0414250i
$657$	1.48427i	0.0579069i
$658$	32.7023i	1.27487i
$659$	−1.19806	−0.0466699	−0.0233349	−	0.999728i	$-0.507428\pi$
$659$	−1.19806	−0.0466699	−0.0233349	+	0.999728i	$0.507428\pi$
$660$	2.24698	0.0874636
$661$	2.95348i	0.114877i	0.998349	+	0.0574384i	$0.0182933\pi$
$661$	2.95348i	0.114877i	−0.998349	+	0.0574384i	$0.981707\pi$
$662$	22.8605	0.888500
$663$	0	0
$664$	−17.7289	−0.688013
$665$	3.41119i	0.132280i
$666$	−9.64071	−0.373570
$667$	−18.2465	−0.706508
$668$	− 1.13706i	− 0.0439943i
$669$	26.5773i	1.02754i
$670$	− 8.51573i	− 0.328991i
$671$	− 2.18598i	− 0.0843888i
$672$	2.55496	0.0985596
$673$	−32.4771	−1.25190	−0.625950	−	0.779863i	$-0.715289\pi$
$673$	−32.4771	−1.25190	−0.625950	+	0.779863i	$0.715289\pi$
$674$	− 33.0629i	− 1.27354i
$675$	1.00000	0.0384900
$676$	0	0
$677$	11.8418	0.455116	0.227558	−	0.973765i	$-0.426926\pi$
$677$	11.8418	0.455116	0.227558	+	0.973765i	$0.426926\pi$
$678$	− 5.24698i	− 0.201509i
$679$	−18.2597	−0.700741
$680$	2.02177	0.0775314
$681$	22.2271i	0.851745i
$682$	− 3.04892i	− 0.116749i
$683$	7.34481i	0.281042i	0.990078	+	0.140521i	$0.0448777\pi$
$683$	7.34481i	0.281042i	−0.990078	+	0.140521i	$0.955122\pi$
$684$	− 1.33513i	− 0.0510498i
$685$	5.06100	0.193371
$686$	−19.0911	−0.728903
$687$	19.1782i	0.731694i
$688$	−0.137063	−0.00522549
$689$	0	0
$690$	5.58211	0.212507
$691$	32.8678i	1.25035i	0.780484	+	0.625176i	$0.214973\pi$
$691$	32.8678i	1.25035i	−0.780484	+	0.625176i	$0.785027\pi$
$692$	0.983607	0.0373911
$693$	−5.74094	−0.218080
$694$	23.4808i	0.891319i
$695$	19.9638i	0.757268i
$696$	3.26875i	0.123902i
$697$	2.14510i	0.0812513i
$698$	−35.9004	−1.35885
$699$	19.5483	0.739383
$700$	2.55496i	0.0965683i
$701$	48.7133	1.83988	0.919938	−	0.392063i	$-0.128239\pi$
$701$	48.7133	1.83988	0.919938	+	0.392063i	$0.128239\pi$
$702$	0	0
$703$	−12.8716	−0.485460
$704$	− 2.24698i	− 0.0846862i
$705$	12.7995	0.482059
$706$	31.0562	1.16882
$707$	12.0435i	0.452944i
$708$	11.9608i	0.449513i
$709$	− 29.6209i	− 1.11243i	−0.831037	−	0.556217i	$-0.812252\pi$
$709$	− 29.6209i	− 1.11243i	0.831037	−	0.556217i	$-0.187748\pi$
$710$	− 2.69202i	− 0.101030i
$711$	−7.77479	−0.291577
$712$	15.4601	0.579392
$713$	− 7.57434i	− 0.283661i
$714$	−5.16554	−0.193315
$715$	0	0
$716$	9.86592	0.368707
$717$	12.8194i	0.478749i
$718$	6.14675	0.229395
$719$	33.5599	1.25157	0.625786	−	0.779995i	$-0.284778\pi$
$719$	33.5599	1.25157	0.625786	+	0.779995i	$0.284778\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	− 30.8786i	− 1.14998i
$722$	17.2174i	0.640767i
$723$	8.95407i	0.333005i
$724$	−3.66487	−0.136204
$725$	−3.26875	−0.121398
$726$	− 5.95108i	− 0.220865i
$727$	−29.8950	−1.10874	−0.554372	−	0.832269i	$-0.687041\pi$
$727$	−29.8950	−1.10874	−0.554372	+	0.832269i	$0.687041\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	1.48427i	0.0549353i
$731$	0.277111	0.0102493
$732$	−0.972853	−0.0359577
$733$	− 19.8812i	− 0.734331i	−0.930156	−	0.367165i	$-0.880328\pi$
$733$	− 19.8812i	− 0.734331i	0.930156	−	0.367165i	$-0.119672\pi$
$734$	− 26.5163i	− 0.978735i
$735$	0.472189i	0.0174170i
$736$	− 5.58211i	− 0.205759i
$737$	19.1347	0.704835
$738$	1.06100	0.0390559
$739$	− 35.4144i	− 1.30274i	−0.758760	−	0.651371i	$-0.774194\pi$
$739$	− 35.4144i	− 1.30274i	0.758760	−	0.651371i	$-0.225806\pi$
$740$	−9.64071	−0.354400
$741$	0	0
$742$	7.75899	0.284841
$743$	− 14.9815i	− 0.549617i	−0.961499	−	0.274809i	$-0.911386\pi$
$743$	− 14.9815i	− 0.549617i	0.961499	−	0.274809i	$-0.0886145\pi$
$744$	−1.35690	−0.0497462
$745$	−3.33513	−0.122190
$746$	− 27.0616i	− 0.990795i
$747$	− 17.7289i	− 0.648665i
$748$	4.54288i	0.166104i
$749$	− 20.4782i	− 0.748256i
$750$	1.00000	0.0365148
$751$	34.3599	1.25381	0.626905	−	0.779096i	$-0.284321\pi$
$751$	34.3599	1.25381	0.626905	+	0.779096i	$0.284321\pi$
$752$	− 12.7995i	− 0.466751i
$753$	−0.276520	−0.0100769
$754$	0	0
$755$	−15.6625	−0.570016
$756$	2.55496i	0.0929229i
$757$	−20.8653	−0.758363	−0.379182	−	0.925322i	$-0.623794\pi$
$757$	−20.8653	−0.758363	−0.379182	+	0.925322i	$0.623794\pi$
$758$	−16.7313	−0.607706
$759$	12.5429i	0.455278i
$760$	− 1.33513i	− 0.0484301i
$761$	48.0974i	1.74353i	0.489926	+	0.871764i	$0.337024\pi$
$761$	48.0974i	1.74353i	−0.489926	+	0.871764i	$0.662976\pi$
$762$	− 12.6843i	− 0.459502i
$763$	−23.4389	−0.848546
$764$	12.2252	0.442293
$765$	2.02177i	0.0730973i
$766$	34.3569	1.24137
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	15.5235i	0.559792i	0.960030	+	0.279896i	$0.0902999\pi$
$769$	15.5235i	0.559792i	−0.960030	+	0.279896i	$0.909700\pi$
$770$	−5.74094	−0.206889
$771$	9.77048	0.351875
$772$	10.5579i	0.379989i
$773$	− 15.0043i	− 0.539668i	−0.962907	−	0.269834i	$-0.913031\pi$
$773$	− 15.0043i	− 0.539668i	0.962907	−	0.269834i	$-0.0869688\pi$
$774$	− 0.137063i	− 0.00492664i
$775$	− 1.35690i	− 0.0487411i
$776$	7.14675	0.256553
$777$	24.6316	0.883654
$778$	− 11.6732i	− 0.418506i
$779$	1.41657	0.0507538
$780$	0	0
$781$	6.04892	0.216447
$782$	11.2857i	0.403577i
$783$	−3.26875	−0.116816
$784$	0.472189	0.0168639
$785$	− 2.99761i	− 0.106989i
$786$	14.5332i	0.518382i
$787$	18.2024i	0.648845i	0.945912	+	0.324422i	$0.105170\pi$
$787$	18.2024i	0.648845i	−0.945912	+	0.324422i	$0.894830\pi$
$788$	5.46011i	0.194508i
$789$	12.7071	0.452384
$790$	−7.77479	−0.276615
$791$	13.4058i	0.476656i
$792$	2.24698	0.0798429
$793$	0	0
$794$	−12.6431	−0.448687
$795$	− 3.03684i	− 0.107705i
$796$	5.00969	0.177564
$797$	36.0079	1.27546	0.637732	−	0.770258i	$-0.279873\pi$
$797$	36.0079	1.27546	0.637732	+	0.770258i	$0.279873\pi$
$798$	3.41119i	0.120755i
$799$	25.8777i	0.915489i
$800$	− 1.00000i	− 0.0353553i
$801$	15.4601i	0.546256i
$802$	−29.7724	−1.05130
$803$	−3.33513	−0.117694
$804$	− 8.51573i	− 0.300327i
$805$	−14.2620	−0.502671
$806$	0	0
$807$	−9.47757	−0.333626
$808$	− 4.71379i	− 0.165831i
$809$	21.6832	0.762340	0.381170	−	0.924505i	$-0.375521\pi$
$809$	21.6832	0.762340	0.381170	+	0.924505i	$0.375521\pi$
$810$	1.00000	0.0351364
$811$	16.0301i	0.562894i	0.959577	+	0.281447i	$0.0908144\pi$
$811$	16.0301i	0.562894i	−0.959577	+	0.281447i	$0.909186\pi$
$812$	− 8.35152i	− 0.293081i
$813$	− 20.7657i	− 0.728285i
$814$	− 21.6625i	− 0.759270i
$815$	−1.46681	−0.0513802
$816$	2.02177	0.0707761
$817$	− 0.182997i	− 0.00640225i
$818$	5.61117	0.196190
$819$	0	0
$820$	1.06100	0.0370517
$821$	10.2543i	0.357877i	0.983860	+	0.178938i	$0.0572663\pi$
$821$	10.2543i	0.357877i	−0.983860	+	0.178938i	$0.942734\pi$
$822$	5.06100	0.176523
$823$	−3.81807	−0.133089	−0.0665447	−	0.997783i	$-0.521198\pi$
$823$	−3.81807	−0.133089	−0.0665447	+	0.997783i	$0.521198\pi$
$824$	12.0858i	0.421027i
$825$	2.24698i	0.0782298i
$826$	− 30.5593i	− 1.06329i
$827$	− 29.9172i	− 1.04032i	−0.854068	−	0.520162i	$-0.825872\pi$
$827$	− 29.9172i	− 1.04032i	0.854068	−	0.520162i	$-0.174128\pi$
$828$	5.58211	0.193992
$829$	−48.3105	−1.67789	−0.838946	−	0.544214i	$-0.816828\pi$
$829$	−48.3105	−1.67789	−0.838946	+	0.544214i	$0.816828\pi$
$830$	− 17.7289i	− 0.615378i
$831$	15.0911	0.523505
$832$	0	0
$833$	−0.954658	−0.0330769
$834$	19.9638i	0.691288i
$835$	1.13706	0.0393497
$836$	3.00000	0.103757
$837$	− 1.35690i	− 0.0469012i
$838$	− 1.24937i	− 0.0431589i
$839$	28.9275i	0.998689i	0.866404	+	0.499344i	$0.166426\pi$
$839$	28.9275i	0.998689i	−0.866404	+	0.499344i	$0.833574\pi$
$840$	2.55496i	0.0881544i
$841$	−18.3153	−0.631561
$842$	2.76749	0.0953742
$843$	17.6015i	0.606227i
$844$	9.38404	0.323012
$845$	0	0
$846$	12.7995	0.440057
$847$	15.2048i	0.522442i
$848$	−3.03684	−0.104285
$849$	−2.07606	−0.0712503
$850$	2.02177i	0.0693461i
$851$	− 53.8155i	− 1.84477i
$852$	− 2.69202i	− 0.0922271i
$853$	54.7193i	1.87355i	0.349929	+	0.936776i	$0.386206\pi$
$853$	54.7193i	1.87355i	−0.349929	+	0.936776i	$0.613794\pi$
$854$	2.48560	0.0850554
$855$	1.33513	0.0456603
$856$	8.01507i	0.273949i
$857$	57.7904	1.97408	0.987042	−	0.160462i	$-0.0512984\pi$
$857$	57.7904	1.97408	0.987042	+	0.160462i	$0.0512984\pi$
$858$	0	0
$859$	−11.0261	−0.376205	−0.188103	−	0.982149i	$-0.560234\pi$
$859$	−11.0261	−0.376205	−0.188103	+	0.982149i	$0.560234\pi$
$860$	− 0.137063i	− 0.00467382i
$861$	−2.71081	−0.0923841
$862$	19.1793	0.653249
$863$	47.8501i	1.62884i	0.580278	+	0.814418i	$0.302944\pi$
$863$	47.8501i	1.62884i	−0.580278	+	0.814418i	$0.697056\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	0.983607i	0.0334436i
$866$	− 36.9051i	− 1.25409i
$867$	12.9124	0.438530
$868$	3.46681	0.117671
$869$	− 17.4698i	− 0.592622i
$870$	−3.26875	−0.110821
$871$	0	0
$872$	9.17390	0.310667
$873$	7.14675i	0.241881i
$874$	7.45281	0.252095
$875$	−2.55496	−0.0863733
$876$	1.48427i	0.0501489i
$877$	− 1.76569i	− 0.0596232i	−0.999556	−	0.0298116i	$-0.990509\pi$
$877$	− 1.76569i	− 0.0596232i	0.999556	−	0.0298116i	$-0.00949074\pi$
$878$	− 14.6213i	− 0.493446i
$879$	26.0930i	0.880097i
$880$	2.24698	0.0757457
$881$	−46.8394	−1.57806	−0.789029	−	0.614356i	$-0.789416\pi$
$881$	−46.8394	−1.57806	−0.789029	+	0.614356i	$0.789416\pi$
$882$	0.472189i	0.0158994i
$883$	11.3730	0.382733	0.191366	−	0.981519i	$-0.438708\pi$
$883$	11.3730	0.382733	0.191366	+	0.981519i	$0.438708\pi$
$884$	0	0
$885$	−11.9608	−0.402057
$886$	− 35.1377i	− 1.18047i
$887$	−53.1245	−1.78375	−0.891873	−	0.452286i	$-0.850609\pi$
$887$	−53.1245	−1.78375	−0.891873	+	0.452286i	$0.850609\pi$
$888$	−9.64071	−0.323521
$889$	32.4077i	1.08692i
$890$	15.4601i	0.518224i
$891$	2.24698i	0.0752766i
$892$	26.5773i	0.889874i
$893$	17.0890	0.571862
$894$	−3.33513	−0.111543
$895$	9.86592i	0.329781i
$896$	2.55496	0.0853552
$897$	0	0
$898$	6.67994	0.222912
$899$	4.43535i	0.147927i
$900$	1.00000	0.0333333
$901$	6.13978	0.204546
$902$	2.38404i	0.0793799i
$903$	0.350191i	0.0116536i
$904$	− 5.24698i	− 0.174512i
$905$	− 3.66487i	− 0.121825i
$906$	−15.6625	−0.520351
$907$	0.871560	0.0289397	0.0144698	−	0.999895i	$-0.495394\pi$
$907$	0.871560	0.0289397	0.0144698	+	0.999895i	$0.495394\pi$
$908$	22.2271i	0.737633i
$909$	4.71379	0.156347
$910$	0	0
$911$	−34.8431	−1.15440	−0.577201	−	0.816602i	$-0.695855\pi$
$911$	−34.8431	−1.15440	−0.577201	+	0.816602i	$0.695855\pi$
$912$	− 1.33513i	− 0.0442104i
$913$	39.8364	1.31839
$914$	−22.9963	−0.760649
$915$	− 0.972853i	− 0.0321615i
$916$	19.1782i	0.633666i
$917$	− 37.1317i	− 1.22620i
$918$	2.02177i	0.0667284i
$919$	36.6765	1.20985	0.604923	−	0.796284i	$-0.293204\pi$
$919$	36.6765	1.20985	0.604923	+	0.796284i	$0.293204\pi$
$920$	5.58211	0.184037
$921$	15.8092i	0.520932i
$922$	33.9705	1.11876
$923$	0	0
$924$	−5.74094	−0.188863
$925$	− 9.64071i	− 0.316985i
$926$	−26.9594	−0.885942
$927$	−12.0858	−0.396948
$928$	3.26875i	0.107302i
$929$	20.2446i	0.664203i	0.943244	+	0.332102i	$0.107758\pi$
$929$	20.2446i	0.664203i	−0.943244	+	0.332102i	$0.892242\pi$
$930$	− 1.35690i	− 0.0444944i
$931$	0.630432i	0.0206616i
$932$	19.5483	0.640324
$933$	−23.2121	−0.759929
$934$	25.0411i	0.819371i
$935$	−4.54288	−0.148568
$936$	0	0
$937$	7.66189	0.250303	0.125152	−	0.992138i	$-0.460058\pi$
$937$	7.66189	0.250303	0.125152	+	0.992138i	$0.460058\pi$
$938$	21.7573i	0.710402i
$939$	−28.5297	−0.931033
$940$	12.7995	0.417475
$941$	21.4107i	0.697969i	0.937128	+	0.348985i	$0.113473\pi$
$941$	21.4107i	0.697969i	−0.937128	+	0.348985i	$0.886527\pi$
$942$	− 2.99761i	− 0.0976673i
$943$	5.92261i	0.192867i
$944$	11.9608i	0.389290i
$945$	−2.55496	−0.0831128
$946$	0.307979	0.0100132
$947$	− 36.3631i	− 1.18164i	−0.806802	−	0.590821i	$-0.798804\pi$
$947$	− 36.3631i	− 1.18164i	0.806802	−	0.590821i	$-0.201196\pi$
$948$	−7.77479	−0.252513
$949$	0	0
$950$	1.33513	0.0433172
$951$	0.709480i	0.0230065i
$952$	−5.16554	−0.167416
$953$	−48.8256	−1.58162	−0.790809	−	0.612064i	$-0.790340\pi$
$953$	−48.8256	−1.58162	−0.790809	+	0.612064i	$0.790340\pi$
$954$	− 3.03684i	− 0.0983212i
$955$	12.2252i	0.395598i
$956$	12.8194i	0.414609i
$957$	− 7.34481i	− 0.237424i
$958$	−7.21983	−0.233262
$959$	−12.9306	−0.417552
$960$	− 1.00000i	− 0.0322749i
$961$	29.1588	0.940608
$962$	0	0
$963$	−8.01507	−0.258282
$964$	8.95407i	0.288391i
$965$	−10.5579	−0.339872
$966$	−14.2620	−0.458874
$967$	4.23623i	0.136228i	0.997678	+	0.0681139i	$0.0216981\pi$
$967$	4.23623i	0.136228i	−0.997678	+	0.0681139i	$0.978302\pi$
$968$	− 5.95108i	− 0.191275i
$969$	2.69932i	0.0867146i
$970$	7.14675i	0.229468i
$971$	−25.7735	−0.827110	−0.413555	−	0.910479i	$-0.635713\pi$
$971$	−25.7735	−0.827110	−0.413555	+	0.910479i	$0.635713\pi$
$972$	1.00000	0.0320750
$973$	− 51.0066i	− 1.63520i
$974$	−24.1497	−0.773807
$975$	0	0
$976$	−0.972853	−0.0311403
$977$	33.2881i	1.06498i	0.846436	+	0.532491i	$0.178744\pi$
$977$	33.2881i	1.06498i	−0.846436	+	0.532491i	$0.821256\pi$
$978$	−1.46681	−0.0469035
$979$	−34.7385	−1.11025
$980$	0.472189i	0.0150835i
$981$	9.17390i	0.292900i
$982$	19.7259i	0.629478i
$983$	45.1605i	1.44040i	0.693769	+	0.720198i	$0.255949\pi$
$983$	45.1605i	1.44040i	−0.693769	+	0.720198i	$0.744051\pi$
$984$	1.06100	0.0338234
$985$	−5.46011	−0.173973
$986$	− 6.60866i	− 0.210463i
$987$	−32.7023	−1.04093
$988$	0	0
$989$	0.765102	0.0243288
$990$	2.24698i	0.0714137i
$991$	−24.5364	−0.779426	−0.389713	−	0.920936i	$-0.627426\pi$
$991$	−24.5364	−0.779426	−0.389713	+	0.920936i	$0.627426\pi$
$992$	−1.35690	−0.0430815
$993$	22.8605i	0.725457i
$994$	6.87800i	0.218157i
$995$	5.00969i	0.158818i
$996$	− 17.7289i	− 0.561760i
$997$	20.6286	0.653315	0.326658	−	0.945143i	$-0.394078\pi$
$997$	20.6286	0.653315	0.326658	+	0.945143i	$0.394078\pi$
$998$	−35.3599	−1.11930
$999$	− 9.64071i	− 0.305019i

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +2.55496i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +2.55496i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.24698i q^{11} +1.00000 q^{12} -2.55496 q^{14} +1.00000i q^{15} +1.00000 q^{16} -2.02177 q^{17} +1.00000i q^{18} +1.33513i q^{19} +1.00000i q^{20} -2.55496i q^{21} -2.24698 q^{22} -5.58211 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.55496i q^{28} +3.26875 q^{29} -1.00000 q^{30} +1.35690i q^{31} +1.00000i q^{32} -2.24698i q^{33} -2.02177i q^{34} +2.55496 q^{35} -1.00000 q^{36} +9.64071i q^{37} -1.33513 q^{38} -1.00000 q^{40} -1.06100i q^{41} +2.55496 q^{42} -0.137063 q^{43} -2.24698i q^{44} -1.00000i q^{45} -5.58211i q^{46} -12.7995i q^{47} -1.00000 q^{48} +0.472189 q^{49} -1.00000i q^{50} +2.02177 q^{51} -3.03684 q^{53} -1.00000i q^{54} +2.24698 q^{55} +2.55496 q^{56} -1.33513i q^{57} +3.26875i q^{58} +11.9608i q^{59} -1.00000i q^{60} -0.972853 q^{61} -1.35690 q^{62} +2.55496i q^{63} -1.00000 q^{64} +2.24698 q^{66} -8.51573i q^{67} +2.02177 q^{68} +5.58211 q^{69} +2.55496i q^{70} -2.69202i q^{71} -1.00000i q^{72} +1.48427i q^{73} -9.64071 q^{74} +1.00000 q^{75} -1.33513i q^{76} -5.74094 q^{77} -7.77479 q^{79} -1.00000i q^{80} +1.00000 q^{81} +1.06100 q^{82} -17.7289i q^{83} +2.55496i q^{84} +2.02177i q^{85} -0.137063i q^{86} -3.26875 q^{87} +2.24698 q^{88} +15.4601i q^{89} +1.00000 q^{90} +5.58211 q^{92} -1.35690i q^{93} +12.7995 q^{94} +1.33513 q^{95} -1.00000i q^{96} +7.14675i q^{97} +0.472189i q^{98} +2.24698i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} + 6 q^{12} - 16 q^{14} + 6 q^{16} - 6 q^{17} - 4 q^{22} - 22 q^{23} - 6 q^{25} - 6 q^{27} + 4 q^{29} - 6 q^{30} + 16 q^{35} - 6 q^{36} - 6 q^{38} - 6 q^{40} + 16 q^{42} + 10 q^{43} - 6 q^{48} - 10 q^{49} + 6 q^{51} + 38 q^{53} + 4 q^{55} + 16 q^{56} - 18 q^{61} - 6 q^{64} + 4 q^{66} + 6 q^{68} + 22 q^{69} + 16 q^{74} + 6 q^{75} - 6 q^{77} - 50 q^{79} + 6 q^{81} + 26 q^{82} - 4 q^{87} + 4 q^{88} + 6 q^{90} + 22 q^{92} - 14 q^{94} + 6 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 + 6 * q^12 - 16 * q^14 + 6 * q^16 - 6 * q^17 - 4 * q^22 - 22 * q^23 - 6 * q^25 - 6 * q^27 + 4 * q^29 - 6 * q^30 + 16 * q^35 - 6 * q^36 - 6 * q^38 - 6 * q^40 + 16 * q^42 + 10 * q^43 - 6 * q^48 - 10 * q^49 + 6 * q^51 + 38 * q^53 + 4 * q^55 + 16 * q^56 - 18 * q^61 - 6 * q^64 + 4 * q^66 + 6 * q^68 + 22 * q^69 + 16 * q^74 + 6 * q^75 - 6 * q^77 - 50 * q^79 + 6 * q^81 + 26 * q^82 - 4 * q^87 + 4 * q^88 + 6 * q^90 + 22 * q^92 - 14 * q^94 + 6 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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