LMFDB - Embedded newform 5070.2.b.u.1351.1

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			1351.1
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.u.1351.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -4.24698i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -4.24698i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.801938i q^{11} +1.00000 q^{12} -4.24698 q^{14} -1.00000i q^{15} +1.00000 q^{16} -7.43296 q^{17} -1.00000i q^{18} +3.74094i q^{19} -1.00000i q^{20} +4.24698i q^{21} +0.801938 q^{22} +2.54288 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +4.24698i q^{28} +5.63102 q^{29} -1.00000 q^{30} -1.69202i q^{31} -1.00000i q^{32} -0.801938i q^{33} +7.43296i q^{34} +4.24698 q^{35} -1.00000 q^{36} +9.67456i q^{37} +3.74094 q^{38} -1.00000 q^{40} +8.85086i q^{41} +4.24698 q^{42} +6.29590 q^{43} -0.801938i q^{44} +1.00000i q^{45} -2.54288i q^{46} -7.53750i q^{47} -1.00000 q^{48} -11.0368 q^{49} +1.00000i q^{50} +7.43296 q^{51} +13.5646 q^{53} +1.00000i q^{54} -0.801938 q^{55} +4.24698 q^{56} -3.74094i q^{57} -5.63102i q^{58} -9.58211i q^{59} +1.00000i q^{60} -10.7899 q^{61} -1.69202 q^{62} -4.24698i q^{63} -1.00000 q^{64} -0.801938 q^{66} +7.82908i q^{67} +7.43296 q^{68} -2.54288 q^{69} -4.24698i q^{70} -2.04892i q^{71} +1.00000i q^{72} -2.17092i q^{73} +9.67456 q^{74} +1.00000 q^{75} -3.74094i q^{76} +3.40581 q^{77} -16.2349 q^{79} +1.00000i q^{80} +1.00000 q^{81} +8.85086 q^{82} -3.61356i q^{83} -4.24698i q^{84} -7.43296i q^{85} -6.29590i q^{86} -5.63102 q^{87} -0.801938 q^{88} +8.24459i q^{89} +1.00000 q^{90} -2.54288 q^{92} +1.69202i q^{93} -7.53750 q^{94} -3.74094 q^{95} +1.00000i q^{96} +6.07069i q^{97} +11.0368i q^{98} +0.801938i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	− 4.24698i	− 1.60521i	−0.596513	−	0.802604i	$-0.703447\pi$
$7$	− 4.24698i	− 1.60521i	0.596513	−	0.802604i	$-0.296553\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	0.801938i	0.241793i	0.992665	+	0.120897i	$0.0385770\pi$
$11$	0.801938i	0.241793i	−0.992665	+	0.120897i	$0.961423\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−4.24698	−1.13505
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	−7.43296	−1.80276	−0.901379	−	0.433031i	$-0.857444\pi$
$17$	−7.43296	−1.80276	−0.901379	+	0.433031i	$0.857444\pi$
$18$	− 1.00000i	− 0.235702i
$19$	3.74094i	0.858230i	0.903250	+	0.429115i	$0.141175\pi$
$19$	3.74094i	0.858230i	−0.903250	+	0.429115i	$0.858825\pi$
$20$	− 1.00000i	− 0.223607i
$21$	4.24698i	0.926767i
$22$	0.801938	0.170974
$23$	2.54288	0.530226	0.265113	−	0.964217i	$-0.414591\pi$
$23$	2.54288	0.530226	0.265113	+	0.964217i	$0.414591\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	4.24698i	0.802604i
$29$	5.63102	1.04565	0.522827	−	0.852439i	$-0.324877\pi$
$29$	5.63102	1.04565	0.522827	+	0.852439i	$0.324877\pi$
$30$	−1.00000	−0.182574
$31$	− 1.69202i	− 0.303896i	−0.988388	−	0.151948i	$-0.951445\pi$
$31$	− 1.69202i	− 0.303896i	0.988388	−	0.151948i	$-0.0485546\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 0.801938i	− 0.139599i
$34$	7.43296i	1.27474i
$35$	4.24698	0.717871
$36$	−1.00000	−0.166667
$37$	9.67456i	1.59049i	0.606289	+	0.795244i	$0.292657\pi$
$37$	9.67456i	1.59049i	−0.606289	+	0.795244i	$0.707343\pi$
$38$	3.74094	0.606860
$39$	0	0
$40$	−1.00000	−0.158114
$41$	8.85086i	1.38227i	0.722725	+	0.691136i	$0.242889\pi$
$41$	8.85086i	1.38227i	−0.722725	+	0.691136i	$0.757111\pi$
$42$	4.24698	0.655323
$43$	6.29590	0.960115	0.480058	−	0.877237i	$-0.340616\pi$
$43$	6.29590	0.960115	0.480058	+	0.877237i	$0.340616\pi$
$44$	− 0.801938i	− 0.120897i
$45$	1.00000i	0.149071i
$46$	− 2.54288i	− 0.374927i
$47$	− 7.53750i	− 1.09946i	−0.835343	−	0.549729i	$-0.814731\pi$
$47$	− 7.53750i	− 1.09946i	0.835343	−	0.549729i	$-0.185269\pi$
$48$	−1.00000	−0.144338
$49$	−11.0368	−1.57669
$50$	1.00000i	0.141421i
$51$	7.43296	1.04082
$52$	0	0
$53$	13.5646	1.86325	0.931624	−	0.363424i	$-0.118392\pi$
$53$	13.5646	1.86325	0.931624	+	0.363424i	$0.118392\pi$
$54$	1.00000i	0.136083i
$55$	−0.801938	−0.108133
$56$	4.24698	0.567527
$57$	− 3.74094i	− 0.495499i
$58$	− 5.63102i	− 0.739389i
$59$	− 9.58211i	− 1.24748i	−0.781630	−	0.623742i	$-0.785612\pi$
$59$	− 9.58211i	− 1.24748i	0.781630	−	0.623742i	$-0.214388\pi$
$60$	1.00000i	0.129099i
$61$	−10.7899	−1.38150	−0.690750	−	0.723094i	$-0.742719\pi$
$61$	−10.7899	−1.38150	−0.690750	+	0.723094i	$0.742719\pi$
$62$	−1.69202	−0.214887
$63$	− 4.24698i	− 0.535069i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−0.801938	−0.0987117
$67$	7.82908i	0.956475i	0.878231	+	0.478237i	$0.158724\pi$
$67$	7.82908i	0.956475i	−0.878231	+	0.478237i	$0.841276\pi$
$68$	7.43296	0.901379
$69$	−2.54288	−0.306126
$70$	− 4.24698i	− 0.507611i
$71$	− 2.04892i	− 0.243162i	−0.992582	−	0.121581i	$-0.961204\pi$
$71$	− 2.04892i	− 0.243162i	0.992582	−	0.121581i	$-0.0387964\pi$
$72$	1.00000i	0.117851i
$73$	− 2.17092i	− 0.254086i	−0.991897	−	0.127043i	$-0.959451\pi$
$73$	− 2.17092i	− 0.254086i	0.991897	−	0.127043i	$-0.0405487\pi$
$74$	9.67456	1.12465
$75$	1.00000	0.115470
$76$	− 3.74094i	− 0.429115i
$77$	3.40581	0.388128
$78$	0	0
$79$	−16.2349	−1.82657	−0.913284	−	0.407323i	$-0.866462\pi$
$79$	−16.2349	−1.82657	−0.913284	+	0.407323i	$0.866462\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	8.85086	0.977413
$83$	− 3.61356i	− 0.396640i	−0.980137	−	0.198320i	$-0.936451\pi$
$83$	− 3.61356i	− 0.396640i	0.980137	−	0.198320i	$-0.0635486\pi$
$84$	− 4.24698i	− 0.463383i
$85$	− 7.43296i	− 0.806218i
$86$	− 6.29590i	− 0.678904i
$87$	−5.63102	−0.603709
$88$	−0.801938	−0.0854868
$89$	8.24459i	0.873924i	0.899480	+	0.436962i	$0.143946\pi$
$89$	8.24459i	0.873924i	−0.899480	+	0.436962i	$0.856054\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−2.54288	−0.265113
$93$	1.69202i	0.175454i
$94$	−7.53750	−0.777434
$95$	−3.74094	−0.383812
$96$	1.00000i	0.102062i
$97$	6.07069i	0.616385i	0.951324	+	0.308192i	$0.0997241\pi$
$97$	6.07069i	0.616385i	−0.951324	+	0.308192i	$0.900276\pi$
$98$	11.0368i	1.11489i
$99$	0.801938i	0.0805978i
$100$	1.00000	0.100000
$101$	5.38404	0.535732	0.267866	−	0.963456i	$-0.413682\pi$
$101$	5.38404	0.535732	0.267866	+	0.963456i	$0.413682\pi$
$102$	− 7.43296i	− 0.735973i
$103$	8.92154	0.879066	0.439533	−	0.898227i	$-0.355144\pi$
$103$	8.92154	0.879066	0.439533	+	0.898227i	$0.355144\pi$
$104$	0	0
$105$	−4.24698	−0.414463
$106$	− 13.5646i	− 1.31751i
$107$	13.9976	1.35320	0.676600	−	0.736351i	$-0.263453\pi$
$107$	13.9976	1.35320	0.676600	+	0.736351i	$0.263453\pi$
$108$	1.00000	0.0962250
$109$	13.8605i	1.32760i	0.747910	+	0.663800i	$0.231057\pi$
$109$	13.8605i	1.32760i	−0.747910	+	0.663800i	$0.768943\pi$
$110$	0.801938i	0.0764618i
$111$	− 9.67456i	− 0.918269i
$112$	− 4.24698i	− 0.401302i
$113$	2.19806	0.206776	0.103388	−	0.994641i	$-0.467032\pi$
$113$	2.19806	0.206776	0.103388	+	0.994641i	$0.467032\pi$
$114$	−3.74094	−0.350371
$115$	2.54288i	0.237124i
$116$	−5.63102	−0.522827
$117$	0	0
$118$	−9.58211	−0.882104
$119$	31.5676i	2.89380i
$120$	1.00000	0.0912871
$121$	10.3569	0.941536
$122$	10.7899i	0.976868i
$123$	− 8.85086i	− 0.798055i
$124$	1.69202i	0.151948i
$125$	− 1.00000i	− 0.0894427i
$126$	−4.24698	−0.378351
$127$	4.19136	0.371923	0.185961	−	0.982557i	$-0.440460\pi$
$127$	4.19136	0.371923	0.185961	+	0.982557i	$0.440460\pi$
$128$	1.00000i	0.0883883i
$129$	−6.29590	−0.554323
$130$	0	0
$131$	−10.8140	−0.944825	−0.472413	−	0.881378i	$-0.656617\pi$
$131$	−10.8140	−0.944825	−0.472413	+	0.881378i	$0.656617\pi$
$132$	0.801938i	0.0697997i
$133$	15.8877	1.37764
$134$	7.82908	0.676330
$135$	− 1.00000i	− 0.0860663i
$136$	− 7.43296i	− 0.637371i
$137$	− 12.8509i	− 1.09792i	−0.835848	−	0.548961i	$-0.815023\pi$
$137$	− 12.8509i	− 1.09792i	0.835848	−	0.548961i	$-0.184977\pi$
$138$	2.54288i	0.216464i
$139$	16.6233	1.40997	0.704983	−	0.709224i	$-0.250955\pi$
$139$	16.6233	1.40997	0.704983	+	0.709224i	$0.250955\pi$
$140$	−4.24698	−0.358935
$141$	7.53750i	0.634772i
$142$	−2.04892	−0.171941
$143$	0	0
$144$	1.00000	0.0833333
$145$	5.63102i	0.467631i
$146$	−2.17092	−0.179666
$147$	11.0368	0.910303
$148$	− 9.67456i	− 0.795244i
$149$	− 1.74094i	− 0.142623i	−0.997454	−	0.0713116i	$-0.977282\pi$
$149$	− 1.74094i	− 0.142623i	0.997454	−	0.0713116i	$-0.0227185\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	1.75840i	0.143096i	0.997437	+	0.0715482i	$0.0227940\pi$
$151$	1.75840i	0.143096i	−0.997437	+	0.0715482i	$0.977206\pi$
$152$	−3.74094	−0.303430
$153$	−7.43296	−0.600919
$154$	− 3.40581i	− 0.274448i
$155$	1.69202	0.135906
$156$	0	0
$157$	−15.9825	−1.27555	−0.637773	−	0.770225i	$-0.720144\pi$
$157$	−15.9825	−1.27555	−0.637773	+	0.770225i	$0.720144\pi$
$158$	16.2349i	1.29158i
$159$	−13.5646	−1.07575
$160$	1.00000	0.0790569
$161$	− 10.7995i	− 0.851123i
$162$	− 1.00000i	− 0.0785674i
$163$	5.18598i	0.406197i	0.979158	+	0.203099i	$0.0651012\pi$
$163$	5.18598i	0.406197i	−0.979158	+	0.203099i	$0.934899\pi$
$164$	− 8.85086i	− 0.691136i
$165$	0.801938	0.0624308
$166$	−3.61356	−0.280467
$167$	5.29590i	0.409809i	0.978782	+	0.204904i	$0.0656884\pi$
$167$	5.29590i	0.409809i	−0.978782	+	0.204904i	$0.934312\pi$
$168$	−4.24698	−0.327662
$169$	0	0
$170$	−7.43296	−0.570082
$171$	3.74094i	0.286077i
$172$	−6.29590	−0.480058
$173$	19.6558	1.49440	0.747201	−	0.664599i	$-0.231397\pi$
$173$	19.6558	1.49440	0.747201	+	0.664599i	$0.231397\pi$
$174$	5.63102i	0.426887i
$175$	4.24698i	0.321041i
$176$	0.801938i	0.0604483i
$177$	9.58211i	0.720235i
$178$	8.24459	0.617958
$179$	17.9095	1.33862	0.669308	−	0.742985i	$-0.266591\pi$
$179$	17.9095	1.33862	0.669308	+	0.742985i	$0.266591\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	8.74094	0.649709	0.324854	−	0.945764i	$-0.394685\pi$
$181$	8.74094	0.649709	0.324854	+	0.945764i	$0.394685\pi$
$182$	0	0
$183$	10.7899	0.797609
$184$	2.54288i	0.187463i
$185$	−9.67456	−0.711288
$186$	1.69202	0.124065
$187$	− 5.96077i	− 0.435895i
$188$	7.53750i	0.549729i
$189$	4.24698i	0.308922i
$190$	3.74094i	0.271396i
$191$	−3.76510	−0.272433	−0.136217	−	0.990679i	$-0.543494\pi$
$191$	−3.76510	−0.272433	−0.136217	+	0.990679i	$0.543494\pi$
$192$	1.00000	0.0721688
$193$	− 21.9584i	− 1.58060i	−0.612721	−	0.790299i	$-0.709925\pi$
$193$	− 21.9584i	− 1.58060i	0.612721	−	0.790299i	$-0.290075\pi$
$194$	6.07069	0.435850
$195$	0	0
$196$	11.0368	0.788345
$197$	− 18.2446i	− 1.29987i	−0.759989	−	0.649936i	$-0.774795\pi$
$197$	− 18.2446i	− 1.29987i	0.759989	−	0.649936i	$-0.225205\pi$
$198$	0.801938	0.0569912
$199$	1.77479	0.125812	0.0629058	−	0.998019i	$-0.479963\pi$
$199$	1.77479	0.125812	0.0629058	+	0.998019i	$0.479963\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	− 7.82908i	− 0.552221i
$202$	− 5.38404i	− 0.378820i
$203$	− 23.9148i	− 1.67849i
$204$	−7.43296	−0.520411
$205$	−8.85086	−0.618171
$206$	− 8.92154i	− 0.621593i
$207$	2.54288	0.176742
$208$	0	0
$209$	−3.00000	−0.207514
$210$	4.24698i	0.293069i
$211$	0.0978347	0.00673522	0.00336761	−	0.999994i	$-0.498928\pi$
$211$	0.0978347	0.00673522	0.00336761	+	0.999994i	$0.498928\pi$
$212$	−13.5646	−0.931624
$213$	2.04892i	0.140390i
$214$	− 13.9976i	− 0.956857i
$215$	6.29590i	0.429377i
$216$	− 1.00000i	− 0.0680414i
$217$	−7.18598	−0.487816
$218$	13.8605	0.938754
$219$	2.17092i	0.146697i
$220$	0.801938	0.0540666
$221$	0	0
$222$	−9.67456	−0.649314
$223$	− 19.5080i	− 1.30635i	−0.757207	−	0.653175i	$-0.773437\pi$
$223$	− 19.5080i	− 1.30635i	0.757207	−	0.653175i	$-0.226563\pi$
$224$	−4.24698	−0.283763
$225$	−1.00000	−0.0666667
$226$	− 2.19806i	− 0.146213i
$227$	3.23059i	0.214422i	0.994236	+	0.107211i	$0.0341920\pi$
$227$	3.23059i	0.214422i	−0.994236	+	0.107211i	$0.965808\pi$
$228$	3.74094i	0.247750i
$229$	4.58748i	0.303149i	0.988446	+	0.151575i	$0.0484344\pi$
$229$	4.58748i	0.303149i	−0.988446	+	0.151575i	$0.951566\pi$
$230$	2.54288	0.167672
$231$	−3.40581	−0.224086
$232$	5.63102i	0.369695i
$233$	6.18359	0.405100	0.202550	−	0.979272i	$-0.435077\pi$
$233$	6.18359	0.405100	0.202550	+	0.979272i	$0.435077\pi$
$234$	0	0
$235$	7.53750	0.491692
$236$	9.58211i	0.623742i
$237$	16.2349	1.05457
$238$	31.5676	2.04623
$239$	8.42998i	0.545290i	0.962115	+	0.272645i	$0.0878984\pi$
$239$	8.42998i	0.545290i	−0.962115	+	0.272645i	$0.912102\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	− 20.8485i	− 1.34297i	−0.741019	−	0.671484i	$-0.765657\pi$
$241$	− 20.8485i	− 1.34297i	0.741019	−	0.671484i	$-0.234343\pi$
$242$	− 10.3569i	− 0.665766i
$243$	−1.00000	−0.0641500
$244$	10.7899	0.690750
$245$	− 11.0368i	− 0.705118i
$246$	−8.85086	−0.564310
$247$	0	0
$248$	1.69202	0.107443
$249$	3.61356i	0.229000i
$250$	−1.00000	−0.0632456
$251$	6.39075	0.403380	0.201690	−	0.979449i	$-0.435357\pi$
$251$	6.39075	0.403380	0.201690	+	0.979449i	$0.435357\pi$
$252$	4.24698i	0.267535i
$253$	2.03923i	0.128205i
$254$	− 4.19136i	− 0.262989i
$255$	7.43296i	0.465470i
$256$	1.00000	0.0625000
$257$	−9.78687	−0.610488	−0.305244	−	0.952274i	$-0.598738\pi$
$257$	−9.78687	−0.610488	−0.305244	+	0.952274i	$0.598738\pi$
$258$	6.29590i	0.391965i
$259$	41.0877	2.55306
$260$	0	0
$261$	5.63102	0.348552
$262$	10.8140i	0.668092i
$263$	14.0465	0.866146	0.433073	−	0.901359i	$-0.357429\pi$
$263$	14.0465	0.866146	0.433073	+	0.901359i	$0.357429\pi$
$264$	0.801938	0.0493559
$265$	13.5646i	0.833270i
$266$	− 15.8877i	− 0.974137i
$267$	− 8.24459i	− 0.504561i
$268$	− 7.82908i	− 0.478237i
$269$	−17.2597	−1.05234	−0.526170	−	0.850380i	$-0.676372\pi$
$269$	−17.2597	−1.05234	−0.526170	+	0.850380i	$0.676372\pi$
$270$	−1.00000	−0.0608581
$271$	17.1782i	1.04350i	0.853098	+	0.521751i	$0.174721\pi$
$271$	17.1782i	1.04350i	−0.853098	+	0.521751i	$0.825279\pi$
$272$	−7.43296	−0.450689
$273$	0	0
$274$	−12.8509	−0.776349
$275$	− 0.801938i	− 0.0483587i
$276$	2.54288	0.153063
$277$	21.1444	1.27044	0.635221	−	0.772331i	$-0.280909\pi$
$277$	21.1444	1.27044	0.635221	+	0.772331i	$0.280909\pi$
$278$	− 16.6233i	− 0.996996i
$279$	− 1.69202i	− 0.101299i
$280$	4.24698i	0.253806i
$281$	− 4.09246i	− 0.244136i	−0.992522	−	0.122068i	$-0.961047\pi$
$281$	− 4.09246i	− 0.244136i	0.992522	−	0.122068i	$-0.0389525\pi$
$282$	7.53750	0.448852
$283$	−12.1468	−0.722049	−0.361025	−	0.932556i	$-0.617573\pi$
$283$	−12.1468	−0.722049	−0.361025	+	0.932556i	$0.617573\pi$
$284$	2.04892i	0.121581i
$285$	3.74094	0.221594
$286$	0	0
$287$	37.5894	2.21883
$288$	− 1.00000i	− 0.0589256i
$289$	38.2489	2.24994
$290$	5.63102	0.330665
$291$	− 6.07069i	− 0.355870i
$292$	2.17092i	0.127043i
$293$	− 20.6789i	− 1.20807i	−0.796957	−	0.604036i	$-0.793558\pi$
$293$	− 20.6789i	− 1.20807i	0.796957	−	0.604036i	$-0.206442\pi$
$294$	− 11.0368i	− 0.643681i
$295$	9.58211	0.557892
$296$	−9.67456	−0.562323
$297$	− 0.801938i	− 0.0465331i
$298$	−1.74094	−0.100850
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	− 26.7385i	− 1.54118i
$302$	1.75840	0.101184
$303$	−5.38404	−0.309305
$304$	3.74094i	0.214558i
$305$	− 10.7899i	− 0.617825i
$306$	7.43296i	0.424914i
$307$	− 11.3123i	− 0.645627i	−0.946463	−	0.322813i	$-0.895371\pi$
$307$	− 11.3123i	− 0.645627i	0.946463	−	0.322813i	$-0.104629\pi$
$308$	−3.40581	−0.194064
$309$	−8.92154	−0.507529
$310$	− 1.69202i	− 0.0961004i
$311$	26.2282	1.48726	0.743632	−	0.668589i	$-0.233101\pi$
$311$	26.2282	1.48726	0.743632	+	0.668589i	$0.233101\pi$
$312$	0	0
$313$	29.5023	1.66757	0.833785	−	0.552089i	$-0.186169\pi$
$313$	29.5023	1.66757	0.833785	+	0.552089i	$0.186169\pi$
$314$	15.9825i	0.901947i
$315$	4.24698	0.239290
$316$	16.2349	0.913284
$317$	− 7.06398i	− 0.396753i	−0.980126	−	0.198376i	$-0.936433\pi$
$317$	− 7.06398i	− 0.396753i	0.980126	−	0.198376i	$-0.0635668\pi$
$318$	13.5646i	0.760668i
$319$	4.51573i	0.252832i
$320$	− 1.00000i	− 0.0559017i
$321$	−13.9976	−0.781270
$322$	−10.7995	−0.601835
$323$	− 27.8062i	− 1.54718i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	5.18598	0.287225
$327$	− 13.8605i	− 0.766490i
$328$	−8.85086	−0.488707
$329$	−32.0116	−1.76486
$330$	− 0.801938i	− 0.0441452i
$331$	10.3134i	0.566873i	0.958991	+	0.283437i	$0.0914746\pi$
$331$	10.3134i	0.566873i	−0.958991	+	0.283437i	$0.908525\pi$
$332$	3.61356i	0.198320i
$333$	9.67456i	0.530163i
$334$	5.29590	0.289779
$335$	−7.82908	−0.427749
$336$	4.24698i	0.231692i
$337$	−30.3163	−1.65144	−0.825718	−	0.564083i	$-0.809230\pi$
$337$	−30.3163	−1.65144	−0.825718	+	0.564083i	$0.809230\pi$
$338$	0	0
$339$	−2.19806	−0.119382
$340$	7.43296i	0.403109i
$341$	1.35690	0.0734800
$342$	3.74094	0.202287
$343$	17.1444i	0.925708i
$344$	6.29590i	0.339452i
$345$	− 2.54288i	− 0.136904i
$346$	− 19.6558i	− 1.05670i
$347$	28.8592	1.54924	0.774622	−	0.632425i	$-0.217940\pi$
$347$	28.8592	1.54924	0.774622	+	0.632425i	$0.217940\pi$
$348$	5.63102	0.301854
$349$	27.4566i	1.46972i	0.678218	+	0.734860i	$0.262752\pi$
$349$	27.4566i	1.46972i	−0.678218	+	0.734860i	$0.737248\pi$
$350$	4.24698	0.227011
$351$	0	0
$352$	0.801938	0.0427434
$353$	0.885772i	0.0471449i	0.999722	+	0.0235724i	$0.00750404\pi$
$353$	0.885772i	0.0471449i	−0.999722	+	0.0235724i	$0.992496\pi$
$354$	9.58211	0.509283
$355$	2.04892	0.108745
$356$	− 8.24459i	− 0.436962i
$357$	− 31.5676i	− 1.67074i
$358$	− 17.9095i	− 0.946544i
$359$	− 7.07069i	− 0.373177i	−0.982438	−	0.186588i	$-0.940257\pi$
$359$	− 7.07069i	− 0.373177i	0.982438	−	0.186588i	$-0.0597430\pi$
$360$	−1.00000	−0.0527046
$361$	5.00538	0.263441
$362$	− 8.74094i	− 0.459414i
$363$	−10.3569	−0.543596
$364$	0	0
$365$	2.17092	0.113631
$366$	− 10.7899i	− 0.563995i
$367$	27.3588	1.42812	0.714059	−	0.700085i	$-0.246855\pi$
$367$	27.3588	1.42812	0.714059	+	0.700085i	$0.246855\pi$
$368$	2.54288	0.132557
$369$	8.85086i	0.460757i
$370$	9.67456i	0.502957i
$371$	− 57.6088i	− 2.99090i
$372$	− 1.69202i	− 0.0877272i
$373$	18.3370	0.949456	0.474728	−	0.880133i	$-0.342546\pi$
$373$	18.3370	0.949456	0.474728	+	0.880133i	$0.342546\pi$
$374$	−5.96077	−0.308224
$375$	1.00000i	0.0516398i
$376$	7.53750	0.388717
$377$	0	0
$378$	4.24698	0.218441
$379$	− 14.3690i	− 0.738085i	−0.929413	−	0.369042i	$-0.879686\pi$
$379$	− 14.3690i	− 0.738085i	0.929413	−	0.369042i	$-0.120314\pi$
$380$	3.74094	0.191906
$381$	−4.19136	−0.214730
$382$	3.76510i	0.192639i
$383$	34.6920i	1.77268i	0.463035	+	0.886340i	$0.346760\pi$
$383$	34.6920i	1.77268i	−0.463035	+	0.886340i	$0.653240\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	3.40581i	0.173576i
$386$	−21.9584	−1.11765
$387$	6.29590	0.320038
$388$	− 6.07069i	− 0.308192i
$389$	32.6872	1.65731	0.828654	−	0.559761i	$-0.189107\pi$
$389$	32.6872	1.65731	0.828654	+	0.559761i	$0.189107\pi$
$390$	0	0
$391$	−18.9011	−0.955870
$392$	− 11.0368i	− 0.557444i
$393$	10.8140	0.545495
$394$	−18.2446	−0.919149
$395$	− 16.2349i	− 0.816866i
$396$	− 0.801938i	− 0.0402989i
$397$	− 12.3080i	− 0.617720i	−0.951108	−	0.308860i	$-0.900053\pi$
$397$	− 12.3080i	− 0.617720i	0.951108	−	0.308860i	$-0.0999474\pi$
$398$	− 1.77479i	− 0.0889622i
$399$	−15.8877	−0.795379
$400$	−1.00000	−0.0500000
$401$	− 19.2524i	− 0.961417i	−0.876881	−	0.480708i	$-0.840380\pi$
$401$	− 19.2524i	− 0.961417i	0.876881	−	0.480708i	$-0.159620\pi$
$402$	−7.82908	−0.390479
$403$	0	0
$404$	−5.38404	−0.267866
$405$	1.00000i	0.0496904i
$406$	−23.9148	−1.18687
$407$	−7.75840	−0.384569
$408$	7.43296i	0.367986i
$409$	− 22.8672i	− 1.13071i	−0.824847	−	0.565356i	$-0.808739\pi$
$409$	− 22.8672i	− 1.13071i	0.824847	−	0.565356i	$-0.191261\pi$
$410$	8.85086i	0.437113i
$411$	12.8509i	0.633886i
$412$	−8.92154	−0.439533
$413$	−40.6950	−2.00247
$414$	− 2.54288i	− 0.124976i
$415$	3.61356	0.177383
$416$	0	0
$417$	−16.6233	−0.814044
$418$	3.00000i	0.146735i
$419$	−17.1806	−0.839327	−0.419664	−	0.907680i	$-0.637852\pi$
$419$	−17.1806	−0.839327	−0.419664	+	0.907680i	$0.637852\pi$
$420$	4.24698	0.207231
$421$	36.9922i	1.80289i	0.432893	+	0.901445i	$0.357493\pi$
$421$	36.9922i	1.80289i	−0.432893	+	0.901445i	$0.642507\pi$
$422$	− 0.0978347i	− 0.00476252i
$423$	− 7.53750i	− 0.366486i
$424$	13.5646i	0.658757i
$425$	7.43296	0.360552
$426$	2.04892	0.0992704
$427$	45.8243i	2.21759i
$428$	−13.9976	−0.676600
$429$	0	0
$430$	6.29590	0.303615
$431$	− 19.0834i	− 0.919213i	−0.888123	−	0.459607i	$-0.847990\pi$
$431$	− 19.0834i	− 0.919213i	0.888123	−	0.459607i	$-0.152010\pi$
$432$	−1.00000	−0.0481125
$433$	−11.5084	−0.553060	−0.276530	−	0.961005i	$-0.589185\pi$
$433$	−11.5084	−0.553060	−0.276530	+	0.961005i	$0.589185\pi$
$434$	7.18598i	0.344938i
$435$	− 5.63102i	− 0.269987i
$436$	− 13.8605i	− 0.663800i
$437$	9.51275i	0.455056i
$438$	2.17092	0.103730
$439$	−8.87502	−0.423582	−0.211791	−	0.977315i	$-0.567930\pi$
$439$	−8.87502	−0.423582	−0.211791	+	0.977315i	$0.567930\pi$
$440$	− 0.801938i	− 0.0382309i
$441$	−11.0368	−0.525564
$442$	0	0
$443$	24.4838	1.16326	0.581630	−	0.813453i	$-0.302415\pi$
$443$	24.4838	1.16326	0.581630	+	0.813453i	$0.302415\pi$
$444$	9.67456i	0.459134i
$445$	−8.24459	−0.390831
$446$	−19.5080	−0.923729
$447$	1.74094i	0.0823436i
$448$	4.24698i	0.200651i
$449$	− 10.2567i	− 0.484042i	−0.970271	−	0.242021i	$-0.922190\pi$
$449$	− 10.2567i	− 0.484042i	0.970271	−	0.242021i	$-0.0778103\pi$
$450$	1.00000i	0.0471405i
$451$	−7.09783	−0.334224
$452$	−2.19806	−0.103388
$453$	− 1.75840i	− 0.0826167i
$454$	3.23059	0.151619
$455$	0	0
$456$	3.74094	0.175186
$457$	38.6359i	1.80731i	0.428259	+	0.903656i	$0.359127\pi$
$457$	38.6359i	1.80731i	−0.428259	+	0.903656i	$0.640873\pi$
$458$	4.58748	0.214359
$459$	7.43296	0.346941
$460$	− 2.54288i	− 0.118562i
$461$	24.8073i	1.15539i	0.816252	+	0.577696i	$0.196048\pi$
$461$	24.8073i	1.15539i	−0.816252	+	0.577696i	$0.803952\pi$
$462$	3.40581i	0.158453i
$463$	18.0713i	0.839844i	0.907560	+	0.419922i	$0.137942\pi$
$463$	18.0713i	0.839844i	−0.907560	+	0.419922i	$0.862058\pi$
$464$	5.63102	0.261414
$465$	−1.69202	−0.0784656
$466$	− 6.18359i	− 0.286449i
$467$	16.8834	0.781270	0.390635	−	0.920546i	$-0.372256\pi$
$467$	16.8834	0.781270	0.390635	+	0.920546i	$0.372256\pi$
$468$	0	0
$469$	33.2500	1.53534
$470$	− 7.53750i	− 0.347679i
$471$	15.9825	0.736437
$472$	9.58211	0.441052
$473$	5.04892i	0.232149i
$474$	− 16.2349i	− 0.745694i
$475$	− 3.74094i	− 0.171646i
$476$	− 31.5676i	− 1.44690i
$477$	13.5646	0.621083
$478$	8.42998	0.385578
$479$	− 13.9879i	− 0.639124i	−0.947565	−	0.319562i	$-0.896464\pi$
$479$	− 13.9879i	− 0.639124i	0.947565	−	0.319562i	$-0.103536\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−20.8485	−0.949621
$483$	10.7995i	0.491396i
$484$	−10.3569	−0.470768
$485$	−6.07069	−0.275656
$486$	1.00000i	0.0453609i
$487$	23.2760i	1.05474i	0.849636	+	0.527369i	$0.176822\pi$
$487$	23.2760i	1.05474i	−0.849636	+	0.527369i	$0.823178\pi$
$488$	− 10.7899i	− 0.488434i
$489$	− 5.18598i	− 0.234518i
$490$	−11.0368	−0.498593
$491$	32.5918	1.47085	0.735424	−	0.677608i	$-0.236983\pi$
$491$	32.5918	1.47085	0.735424	+	0.677608i	$0.236983\pi$
$492$	8.85086i	0.399027i
$493$	−41.8552	−1.88506
$494$	0	0
$495$	−0.801938	−0.0360444
$496$	− 1.69202i	− 0.0759740i
$497$	−8.70171	−0.390325
$498$	3.61356	0.161928
$499$	− 1.48666i	− 0.0665522i	−0.999446	−	0.0332761i	$-0.989406\pi$
$499$	− 1.48666i	− 0.0665522i	0.999446	−	0.0332761i	$-0.0105941\pi$
$500$	1.00000i	0.0447214i
$501$	− 5.29590i	− 0.236603i
$502$	− 6.39075i	− 0.285233i
$503$	38.3629	1.71052	0.855258	−	0.518203i	$-0.173399\pi$
$503$	38.3629	1.71052	0.855258	+	0.518203i	$0.173399\pi$
$504$	4.24698	0.189176
$505$	5.38404i	0.239587i
$506$	2.03923	0.0906548
$507$	0	0
$508$	−4.19136	−0.185961
$509$	0.770479i	0.0341509i	0.999854	+	0.0170754i	$0.00543554\pi$
$509$	0.770479i	0.0341509i	−0.999854	+	0.0170754i	$0.994564\pi$
$510$	7.43296	0.329137
$511$	−9.21983	−0.407861
$512$	− 1.00000i	− 0.0441942i
$513$	− 3.74094i	− 0.165166i
$514$	9.78687i	0.431680i
$515$	8.92154i	0.393130i
$516$	6.29590	0.277161
$517$	6.04461	0.265841
$518$	− 41.0877i	− 1.80529i
$519$	−19.6558	−0.862793
$520$	0	0
$521$	−13.7657	−0.603086	−0.301543	−	0.953453i	$-0.597502\pi$
$521$	−13.7657	−0.603086	−0.301543	+	0.953453i	$0.597502\pi$
$522$	− 5.63102i	− 0.246463i
$523$	−17.5991	−0.769554	−0.384777	−	0.923009i	$-0.625722\pi$
$523$	−17.5991	−0.769554	−0.384777	+	0.923009i	$0.625722\pi$
$524$	10.8140	0.472413
$525$	− 4.24698i	− 0.185353i
$526$	− 14.0465i	− 0.612458i
$527$	12.5767i	0.547851i
$528$	− 0.801938i	− 0.0348999i
$529$	−16.5338	−0.718860
$530$	13.5646	0.589211
$531$	− 9.58211i	− 0.415828i
$532$	−15.8877	−0.688819
$533$	0	0
$534$	−8.24459	−0.356778
$535$	13.9976i	0.605169i
$536$	−7.82908	−0.338165
$537$	−17.9095	−0.772850
$538$	17.2597i	0.744116i
$539$	− 8.85086i	− 0.381233i
$540$	1.00000i	0.0430331i
$541$	20.9259i	0.899673i	0.893111	+	0.449836i	$0.148518\pi$
$541$	20.9259i	0.899673i	−0.893111	+	0.449836i	$0.851482\pi$
$542$	17.1782	0.737867
$543$	−8.74094	−0.375110
$544$	7.43296i	0.318686i
$545$	−13.8605	−0.593720
$546$	0	0
$547$	42.5599	1.81973	0.909864	−	0.414906i	$-0.136186\pi$
$547$	42.5599	1.81973	0.909864	+	0.414906i	$0.136186\pi$
$548$	12.8509i	0.548961i
$549$	−10.7899	−0.460500
$550$	−0.801938	−0.0341947
$551$	21.0653i	0.897412i
$552$	− 2.54288i	− 0.108232i
$553$	68.9493i	2.93202i
$554$	− 21.1444i	− 0.898338i
$555$	9.67456	0.410662
$556$	−16.6233	−0.704983
$557$	− 33.1637i	− 1.40519i	−0.711589	−	0.702596i	$-0.752024\pi$
$557$	− 33.1637i	− 1.40519i	0.711589	−	0.702596i	$-0.247976\pi$
$558$	−1.69202	−0.0716290
$559$	0	0
$560$	4.24698	0.179468
$561$	5.96077i	0.251664i
$562$	−4.09246	−0.172630
$563$	7.27173	0.306467	0.153234	−	0.988190i	$-0.451031\pi$
$563$	7.27173	0.306467	0.153234	+	0.988190i	$0.451031\pi$
$564$	− 7.53750i	− 0.317386i
$565$	2.19806i	0.0924732i
$566$	12.1468i	0.510566i
$567$	− 4.24698i	− 0.178356i
$568$	2.04892	0.0859707
$569$	−38.9269	−1.63190	−0.815951	−	0.578122i	$-0.803786\pi$
$569$	−38.9269	−1.63190	−0.815951	+	0.578122i	$0.803786\pi$
$570$	− 3.74094i	− 0.156691i
$571$	−38.6045	−1.61555	−0.807773	−	0.589493i	$-0.799327\pi$
$571$	−38.6045	−1.61555	−0.807773	+	0.589493i	$0.799327\pi$
$572$	0	0
$573$	3.76510	0.157289
$574$	− 37.5894i	− 1.56895i
$575$	−2.54288	−0.106045
$576$	−1.00000	−0.0416667
$577$	27.8646i	1.16002i	0.814610	+	0.580009i	$0.196951\pi$
$577$	27.8646i	1.16002i	−0.814610	+	0.580009i	$0.803049\pi$
$578$	− 38.2489i	− 1.59094i
$579$	21.9584i	0.912559i
$580$	− 5.63102i	− 0.233815i
$581$	−15.3467	−0.636690
$582$	−6.07069	−0.251638
$583$	10.8780i	0.450521i
$584$	2.17092	0.0898331
$585$	0	0
$586$	−20.6789	−0.854236
$587$	− 26.1648i	− 1.07994i	−0.841685	−	0.539968i	$-0.818436\pi$
$587$	− 26.1648i	− 1.07994i	0.841685	−	0.539968i	$-0.181564\pi$
$588$	−11.0368	−0.455151
$589$	6.32975	0.260813
$590$	− 9.58211i	− 0.394489i
$591$	18.2446i	0.750482i
$592$	9.67456i	0.397622i
$593$	8.61117i	0.353618i	0.984245	+	0.176809i	$0.0565776\pi$
$593$	8.61117i	0.353618i	−0.984245	+	0.176809i	$0.943422\pi$
$594$	−0.801938	−0.0329039
$595$	−31.5676	−1.29415
$596$	1.74094i	0.0713116i
$597$	−1.77479	−0.0726373
$598$	0	0
$599$	11.1709	0.456431	0.228216	−	0.973611i	$-0.426711\pi$
$599$	11.1709	0.456431	0.228216	+	0.973611i	$0.426711\pi$
$600$	1.00000i	0.0408248i
$601$	8.91856	0.363796	0.181898	−	0.983317i	$-0.441776\pi$
$601$	8.91856	0.363796	0.181898	+	0.983317i	$0.441776\pi$
$602$	−26.7385	−1.08978
$603$	7.82908i	0.318825i
$604$	− 1.75840i	− 0.0715482i
$605$	10.3569i	0.421068i
$606$	5.38404i	0.218712i
$607$	−28.0277	−1.13761	−0.568805	−	0.822472i	$-0.692594\pi$
$607$	−28.0277	−1.13761	−0.568805	+	0.822472i	$0.692594\pi$
$608$	3.74094	0.151715
$609$	23.9148i	0.969078i
$610$	−10.7899	−0.436869
$611$	0	0
$612$	7.43296	0.300460
$613$	1.10885i	0.0447861i	0.999749	+	0.0223930i	$0.00712852\pi$
$613$	1.10885i	0.0447861i	−0.999749	+	0.0223930i	$0.992871\pi$
$614$	−11.3123	−0.456527
$615$	8.85086	0.356901
$616$	3.40581i	0.137224i
$617$	− 16.2905i	− 0.655832i	−0.944707	−	0.327916i	$-0.893654\pi$
$617$	− 16.2905i	− 0.655832i	0.944707	−	0.327916i	$-0.106346\pi$
$618$	8.92154i	0.358877i
$619$	23.2252i	0.933500i	0.884389	+	0.466750i	$0.154575\pi$
$619$	23.2252i	0.933500i	−0.884389	+	0.466750i	$0.845425\pi$
$620$	−1.69202	−0.0679532
$621$	−2.54288	−0.102042
$622$	− 26.2282i	− 1.05165i
$623$	35.0146	1.40283
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 29.5023i	− 1.17915i
$627$	3.00000	0.119808
$628$	15.9825	0.637773
$629$	− 71.9106i	− 2.86727i
$630$	− 4.24698i	− 0.169204i
$631$	− 0.357961i	− 0.0142502i	−0.999975	−	0.00712510i	$-0.997732\pi$
$631$	− 0.357961i	− 0.0142502i	0.999975	−	0.00712510i	$-0.00226801\pi$
$632$	− 16.2349i	− 0.645790i
$633$	−0.0978347	−0.00388858
$634$	−7.06398	−0.280547
$635$	4.19136i	0.166329i
$636$	13.5646	0.537873
$637$	0	0
$638$	4.51573	0.178779
$639$	− 2.04892i	− 0.0810539i
$640$	−1.00000	−0.0395285
$641$	−33.2814	−1.31454	−0.657269	−	0.753656i	$-0.728288\pi$
$641$	−33.2814	−1.31454	−0.657269	+	0.753656i	$0.728288\pi$
$642$	13.9976i	0.552441i
$643$	− 48.4172i	− 1.90939i	−0.297592	−	0.954693i	$-0.596183\pi$
$643$	− 48.4172i	− 1.90939i	0.297592	−	0.954693i	$-0.403817\pi$
$644$	10.7995i	0.425562i
$645$	− 6.29590i	− 0.247901i
$646$	−27.8062	−1.09402
$647$	47.4814	1.86669	0.933343	−	0.358985i	$-0.116877\pi$
$647$	47.4814	1.86669	0.933343	+	0.358985i	$0.116877\pi$
$648$	1.00000i	0.0392837i
$649$	7.68425	0.301633
$650$	0	0
$651$	7.18598	0.281641
$652$	− 5.18598i	− 0.203099i
$653$	−25.4336	−0.995292	−0.497646	−	0.867380i	$-0.665802\pi$
$653$	−25.4336	−0.995292	−0.497646	+	0.867380i	$0.665802\pi$
$654$	−13.8605	−0.541990
$655$	− 10.8140i	− 0.422539i
$656$	8.85086i	0.345568i
$657$	− 2.17092i	− 0.0846955i
$658$	32.0116i	1.24794i
$659$	−2.55496	−0.0995270	−0.0497635	−	0.998761i	$-0.515847\pi$
$659$	−2.55496	−0.0995270	−0.0497635	+	0.998761i	$0.515847\pi$
$660$	−0.801938	−0.0312154
$661$	− 26.3394i	− 1.02449i	−0.858841	−	0.512243i	$-0.828815\pi$
$661$	− 26.3394i	− 1.02449i	0.858841	−	0.512243i	$-0.171185\pi$
$662$	10.3134	0.400840
$663$	0	0
$664$	3.61356	0.140233
$665$	15.8877i	0.616098i
$666$	9.67456	0.374882
$667$	14.3190	0.554434
$668$	− 5.29590i	− 0.204904i
$669$	19.5080i	0.754221i
$670$	7.82908i	0.302464i
$671$	− 8.65279i	− 0.334037i
$672$	4.24698	0.163831
$673$	23.7767	0.916525	0.458262	−	0.888817i	$-0.348472\pi$
$673$	23.7767	0.916525	0.458262	+	0.888817i	$0.348472\pi$
$674$	30.3163i	1.16774i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−40.3250	−1.54981	−0.774907	−	0.632075i	$-0.782203\pi$
$677$	−40.3250	−1.54981	−0.774907	+	0.632075i	$0.782203\pi$
$678$	2.19806i	0.0844161i
$679$	25.7821	0.989426
$680$	7.43296	0.285041
$681$	− 3.23059i	− 0.123796i
$682$	− 1.35690i	− 0.0519582i
$683$	4.51573i	0.172790i	0.996261	+	0.0863948i	$0.0275346\pi$
$683$	4.51573i	0.172790i	−0.996261	+	0.0863948i	$0.972465\pi$
$684$	− 3.74094i	− 0.143038i
$685$	12.8509	0.491006
$686$	17.1444	0.654575
$687$	− 4.58748i	− 0.175023i
$688$	6.29590	0.240029
$689$	0	0
$690$	−2.54288	−0.0968057
$691$	5.44398i	0.207099i	0.994624	+	0.103549i	$0.0330200\pi$
$691$	5.44398i	0.207099i	−0.994624	+	0.103549i	$0.966980\pi$
$692$	−19.6558	−0.747201
$693$	3.40581	0.129376
$694$	− 28.8592i	− 1.09548i
$695$	16.6233i	0.630556i
$696$	− 5.63102i	− 0.213443i
$697$	− 65.7881i	− 2.49190i
$698$	27.4566	1.03925
$699$	−6.18359	−0.233885
$700$	− 4.24698i	− 0.160521i
$701$	19.8670	0.750366	0.375183	−	0.926951i	$-0.377580\pi$
$701$	19.8670	0.750366	0.375183	+	0.926951i	$0.377580\pi$
$702$	0	0
$703$	−36.1919	−1.36501
$704$	− 0.801938i	− 0.0302242i
$705$	−7.53750	−0.283879
$706$	0.885772	0.0333365
$707$	− 22.8659i	− 0.859961i
$708$	− 9.58211i	− 0.360118i
$709$	− 5.64204i	− 0.211891i	−0.994372	−	0.105946i	$-0.966213\pi$
$709$	− 5.64204i	− 0.211891i	0.994372	−	0.105946i	$-0.0337869\pi$
$710$	− 2.04892i	− 0.0768945i
$711$	−16.2349	−0.608856
$712$	−8.24459	−0.308979
$713$	− 4.30260i	− 0.161134i
$714$	−31.5676	−1.18139
$715$	0	0
$716$	−17.9095	−0.669308
$717$	− 8.42998i	− 0.314823i
$718$	−7.07069	−0.263876
$719$	−9.49289	−0.354025	−0.177013	−	0.984209i	$-0.556643\pi$
$719$	−9.49289	−0.354025	−0.177013	+	0.984209i	$0.556643\pi$
$720$	1.00000i	0.0372678i
$721$	− 37.8896i	− 1.41108i
$722$	− 5.00538i	− 0.186281i
$723$	20.8485i	0.775363i
$724$	−8.74094	−0.324854
$725$	−5.63102	−0.209131
$726$	10.3569i	0.384380i
$727$	18.2338	0.676255	0.338128	−	0.941100i	$-0.390206\pi$
$727$	18.2338	0.676255	0.338128	+	0.941100i	$0.390206\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 2.17092i	− 0.0803492i
$731$	−46.7972	−1.73086
$732$	−10.7899	−0.398805
$733$	36.4172i	1.34510i	0.740052	+	0.672549i	$0.234801\pi$
$733$	36.4172i	1.34510i	−0.740052	+	0.672549i	$0.765199\pi$
$734$	− 27.3588i	− 1.00983i
$735$	11.0368i	0.407100i
$736$	− 2.54288i	− 0.0937317i
$737$	−6.27844	−0.231269
$738$	8.85086	0.325804
$739$	48.2312i	1.77421i	0.461565	+	0.887106i	$0.347288\pi$
$739$	48.2312i	1.77421i	−0.461565	+	0.887106i	$0.652712\pi$
$740$	9.67456	0.355644
$741$	0	0
$742$	−57.6088	−2.11488
$743$	41.6859i	1.52931i	0.644441	+	0.764654i	$0.277090\pi$
$743$	41.6859i	1.52931i	−0.644441	+	0.764654i	$0.722910\pi$
$744$	−1.69202	−0.0620325
$745$	1.74094	0.0637831
$746$	− 18.3370i	− 0.671367i
$747$	− 3.61356i	− 0.132213i
$748$	5.96077i	0.217947i
$749$	− 59.4476i	− 2.17217i
$750$	1.00000	0.0365148
$751$	0.486663	0.0177586	0.00887930	−	0.999961i	$-0.497174\pi$
$751$	0.486663	0.0177586	0.00887930	+	0.999961i	$0.497174\pi$
$752$	− 7.53750i	− 0.274864i
$753$	−6.39075	−0.232892
$754$	0	0
$755$	−1.75840	−0.0639946
$756$	− 4.24698i	− 0.154461i
$757$	−46.2784	−1.68202	−0.841009	−	0.541021i	$-0.818038\pi$
$757$	−46.2784	−1.68202	−0.841009	+	0.541021i	$0.818038\pi$
$758$	−14.3690	−0.521905
$759$	− 2.03923i	− 0.0740193i
$760$	− 3.74094i	− 0.135698i
$761$	− 9.76915i	− 0.354131i	−0.984199	−	0.177066i	$-0.943339\pi$
$761$	− 9.76915i	− 0.354131i	0.984199	−	0.177066i	$-0.0566605\pi$
$762$	4.19136i	0.151837i
$763$	58.8654	2.13107
$764$	3.76510	0.136217
$765$	− 7.43296i	− 0.268739i
$766$	34.6920	1.25347
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 18.5888i	− 0.670330i	−0.942160	−	0.335165i	$-0.891208\pi$
$769$	− 18.5888i	− 0.670330i	0.942160	−	0.335165i	$-0.108792\pi$
$770$	3.40581	0.122737
$771$	9.78687	0.352466
$772$	21.9584i	0.790299i
$773$	23.4480i	0.843367i	0.906743	+	0.421683i	$0.138561\pi$
$773$	23.4480i	0.843367i	−0.906743	+	0.421683i	$0.861439\pi$
$774$	− 6.29590i	− 0.226301i
$775$	1.69202i	0.0607792i
$776$	−6.07069	−0.217925
$777$	−41.0877	−1.47401
$778$	− 32.6872i	− 1.17189i
$779$	−33.1105	−1.18631
$780$	0	0
$781$	1.64310	0.0587949
$782$	18.9011i	0.675902i
$783$	−5.63102	−0.201236
$784$	−11.0368	−0.394173
$785$	− 15.9825i	− 0.570441i
$786$	− 10.8140i	− 0.385723i
$787$	− 28.0030i	− 0.998199i	−0.866545	−	0.499099i	$-0.833664\pi$
$787$	− 28.0030i	− 0.998199i	0.866545	−	0.499099i	$-0.166336\pi$
$788$	18.2446i	0.649936i
$789$	−14.0465	−0.500070
$790$	−16.2349	−0.577612
$791$	− 9.33513i	− 0.331919i
$792$	−0.801938	−0.0284956
$793$	0	0
$794$	−12.3080	−0.436794
$795$	− 13.5646i	− 0.481088i
$796$	−1.77479	−0.0629058
$797$	−42.9452	−1.52120	−0.760599	−	0.649222i	$-0.775095\pi$
$797$	−42.9452	−1.52120	−0.760599	+	0.649222i	$0.775095\pi$
$798$	15.8877i	0.562418i
$799$	56.0259i	1.98206i
$800$	1.00000i	0.0353553i
$801$	8.24459i	0.291308i
$802$	−19.2524	−0.679824
$803$	1.74094	0.0614364
$804$	7.82908i	0.276111i
$805$	10.7995	0.380634
$806$	0	0
$807$	17.2597	0.607569
$808$	5.38404i	0.189410i
$809$	36.8622	1.29601	0.648003	−	0.761638i	$-0.275604\pi$
$809$	36.8622	1.29601	0.648003	+	0.761638i	$0.275604\pi$
$810$	1.00000	0.0351364
$811$	27.9952i	0.983045i	0.870865	+	0.491522i	$0.163559\pi$
$811$	27.9952i	0.983045i	−0.870865	+	0.491522i	$0.836441\pi$
$812$	23.9148i	0.839246i
$813$	− 17.1782i	− 0.602466i
$814$	7.75840i	0.271932i
$815$	−5.18598	−0.181657
$816$	7.43296	0.260206
$817$	23.5526i	0.824000i
$818$	−22.8672	−0.799535
$819$	0	0
$820$	8.85086	0.309085
$821$	18.5593i	0.647723i	0.946105	+	0.323861i	$0.104981\pi$
$821$	18.5593i	0.647723i	−0.946105	+	0.323861i	$0.895019\pi$
$822$	12.8509	0.448225
$823$	43.2234	1.50667	0.753337	−	0.657635i	$-0.228443\pi$
$823$	43.2234	1.50667	0.753337	+	0.657635i	$0.228443\pi$
$824$	8.92154i	0.310797i
$825$	0.801938i	0.0279199i
$826$	40.6950i	1.41596i
$827$	16.7162i	0.581279i	0.956833	+	0.290639i	$0.0938680\pi$
$827$	16.7162i	0.581279i	−0.956833	+	0.290639i	$0.906132\pi$
$828$	−2.54288	−0.0883711
$829$	10.6735	0.370706	0.185353	−	0.982672i	$-0.440657\pi$
$829$	10.6735	0.370706	0.185353	+	0.982672i	$0.440657\pi$
$830$	− 3.61356i	− 0.125429i
$831$	−21.1444	−0.733490
$832$	0	0
$833$	82.0364	2.84239
$834$	16.6233i	0.575616i
$835$	−5.29590	−0.183272
$836$	3.00000	0.103757
$837$	1.69202i	0.0584848i
$838$	17.1806i	0.593494i
$839$	44.2465i	1.52756i	0.645477	+	0.763780i	$0.276659\pi$
$839$	44.2465i	1.52756i	−0.645477	+	0.763780i	$0.723341\pi$
$840$	− 4.24698i	− 0.146535i
$841$	2.70841	0.0933936
$842$	36.9922	1.27484
$843$	4.09246i	0.140952i
$844$	−0.0978347	−0.00336761
$845$	0	0
$846$	−7.53750	−0.259145
$847$	− 43.9855i	− 1.51136i
$848$	13.5646	0.465812
$849$	12.1468	0.416875
$850$	− 7.43296i	− 0.254948i
$851$	24.6012i	0.843319i
$852$	− 2.04892i	− 0.0701948i
$853$	42.5437i	1.45667i	0.685221	+	0.728335i	$0.259705\pi$
$853$	42.5437i	1.45667i	−0.685221	+	0.728335i	$0.740295\pi$
$854$	45.8243	1.56808
$855$	−3.74094	−0.127937
$856$	13.9976i	0.478428i
$857$	−8.95061	−0.305747	−0.152873	−	0.988246i	$-0.548853\pi$
$857$	−8.95061	−0.305747	−0.152873	+	0.988246i	$0.548853\pi$
$858$	0	0
$859$	−24.8810	−0.848928	−0.424464	−	0.905445i	$-0.639538\pi$
$859$	−24.8810	−0.848928	−0.424464	+	0.905445i	$0.639538\pi$
$860$	− 6.29590i	− 0.214688i
$861$	−37.5894	−1.28104
$862$	−19.0834	−0.649982
$863$	53.7531i	1.82978i	0.403705	+	0.914889i	$0.367722\pi$
$863$	53.7531i	1.82978i	−0.403705	+	0.914889i	$0.632278\pi$
$864$	1.00000i	0.0340207i
$865$	19.6558i	0.668316i
$866$	11.5084i	0.391072i
$867$	−38.2489	−1.29900
$868$	7.18598	0.243908
$869$	− 13.0194i	− 0.441652i
$870$	−5.63102	−0.190910
$871$	0	0
$872$	−13.8605	−0.469377
$873$	6.07069i	0.205462i
$874$	9.51275	0.321773
$875$	−4.24698	−0.143574
$876$	− 2.17092i	− 0.0733484i
$877$	− 36.1782i	− 1.22165i	−0.791765	−	0.610826i	$-0.790838\pi$
$877$	− 36.1782i	− 1.22165i	0.791765	−	0.610826i	$-0.209162\pi$
$878$	8.87502i	0.299517i
$879$	20.6789i	0.697481i
$880$	−0.801938	−0.0270333
$881$	24.3075	0.818941	0.409470	−	0.912323i	$-0.365714\pi$
$881$	24.3075	0.818941	0.409470	+	0.912323i	$0.365714\pi$
$882$	11.0368i	0.371630i
$883$	−33.9764	−1.14340	−0.571699	−	0.820463i	$-0.693715\pi$
$883$	−33.9764	−1.14340	−0.571699	+	0.820463i	$0.693715\pi$
$884$	0	0
$885$	−9.58211	−0.322099
$886$	− 24.4838i	− 0.822549i
$887$	−4.97929	−0.167188	−0.0835942	−	0.996500i	$-0.526640\pi$
$887$	−4.97929	−0.167188	−0.0835942	+	0.996500i	$0.526640\pi$
$888$	9.67456	0.324657
$889$	− 17.8006i	− 0.597013i
$890$	8.24459i	0.276359i
$891$	0.801938i	0.0268659i
$892$	19.5080i	0.653175i
$893$	28.1973	0.943588
$894$	1.74094	0.0582257
$895$	17.9095i	0.598647i
$896$	4.24698	0.141882
$897$	0	0
$898$	−10.2567	−0.342269
$899$	− 9.52781i	− 0.317770i
$900$	1.00000	0.0333333
$901$	−100.825	−3.35898
$902$	7.09783i	0.236332i
$903$	26.7385i	0.889803i
$904$	2.19806i	0.0731065i
$905$	8.74094i	0.290559i
$906$	−1.75840	−0.0584188
$907$	24.1919	0.803280	0.401640	−	0.915798i	$-0.368440\pi$
$907$	24.1919	0.803280	0.401640	+	0.915798i	$0.368440\pi$
$908$	− 3.23059i	− 0.107211i
$909$	5.38404	0.178577
$910$	0	0
$911$	−25.3284	−0.839168	−0.419584	−	0.907717i	$-0.637824\pi$
$911$	−25.3284	−0.839168	−0.419584	+	0.907717i	$0.637824\pi$
$912$	− 3.74094i	− 0.123875i
$913$	2.89785	0.0959050
$914$	38.6359	1.27796
$915$	10.7899i	0.356702i
$916$	− 4.58748i	− 0.151575i
$917$	45.9269i	1.51664i
$918$	− 7.43296i	− 0.245324i
$919$	24.4316	0.805925	0.402963	−	0.915216i	$-0.367980\pi$
$919$	24.4316	0.805925	0.402963	+	0.915216i	$0.367980\pi$
$920$	−2.54288	−0.0838362
$921$	11.3123i	0.372753i
$922$	24.8073	0.816985
$923$	0	0
$924$	3.40581	0.112043
$925$	− 9.67456i	− 0.318098i
$926$	18.0713	0.593859
$927$	8.92154	0.293022
$928$	− 5.63102i	− 0.184847i
$929$	1.78448i	0.0585469i	0.999571	+	0.0292734i	$0.00931935\pi$
$929$	1.78448i	0.0585469i	−0.999571	+	0.0292734i	$0.990681\pi$
$930$	1.69202i	0.0554836i
$931$	− 41.2881i	− 1.35316i
$932$	−6.18359	−0.202550
$933$	−26.2282	−0.858672
$934$	− 16.8834i	− 0.552441i
$935$	5.96077	0.194938
$936$	0	0
$937$	46.9463	1.53367	0.766834	−	0.641845i	$-0.221831\pi$
$937$	46.9463	1.53367	0.766834	+	0.641845i	$0.221831\pi$
$938$	− 33.2500i	− 1.08565i
$939$	−29.5023	−0.962772
$940$	−7.53750	−0.245846
$941$	27.4047i	0.893369i	0.894691	+	0.446685i	$0.147395\pi$
$941$	27.4047i	0.893369i	−0.894691	+	0.446685i	$0.852605\pi$
$942$	− 15.9825i	− 0.520739i
$943$	22.5066i	0.732917i
$944$	− 9.58211i	− 0.311871i
$945$	−4.24698	−0.138154
$946$	5.04892	0.164154
$947$	34.6055i	1.12453i	0.826958	+	0.562264i	$0.190070\pi$
$947$	34.6055i	1.12453i	−0.826958	+	0.562264i	$0.809930\pi$
$948$	−16.2349	−0.527285
$949$	0	0
$950$	−3.74094	−0.121372
$951$	7.06398i	0.229065i
$952$	−31.5676	−1.02311
$953$	−42.3435	−1.37164	−0.685820	−	0.727771i	$-0.740556\pi$
$953$	−42.3435	−1.37164	−0.685820	+	0.727771i	$0.740556\pi$
$954$	− 13.5646i	− 0.439172i
$955$	− 3.76510i	− 0.121836i
$956$	− 8.42998i	− 0.272645i
$957$	− 4.51573i	− 0.145973i
$958$	−13.9879	−0.451929
$959$	−54.5773	−1.76239
$960$	1.00000i	0.0322749i
$961$	28.1371	0.907647
$962$	0	0
$963$	13.9976	0.451067
$964$	20.8485i	0.671484i
$965$	21.9584	0.706865
$966$	10.7995	0.347470
$967$	− 31.6437i	− 1.01759i	−0.860887	−	0.508796i	$-0.830091\pi$
$967$	− 31.6437i	− 1.01759i	0.860887	−	0.508796i	$-0.169909\pi$
$968$	10.3569i	0.332883i
$969$	27.8062i	0.893265i
$970$	6.07069i	0.194918i
$971$	8.41849	0.270162	0.135081	−	0.990835i	$-0.456871\pi$
$971$	8.41849	0.270162	0.135081	+	0.990835i	$0.456871\pi$
$972$	1.00000	0.0320750
$973$	− 70.5986i	− 2.26329i
$974$	23.2760	0.745813
$975$	0	0
$976$	−10.7899	−0.345375
$977$	− 22.0814i	− 0.706448i	−0.935539	−	0.353224i	$-0.885085\pi$
$977$	− 22.0814i	− 0.706448i	0.935539	−	0.353224i	$-0.114915\pi$
$978$	−5.18598	−0.165829
$979$	−6.61165	−0.211309
$980$	11.0368i	0.352559i
$981$	13.8605i	0.442533i
$982$	− 32.5918i	− 1.04005i
$983$	32.7217i	1.04366i	0.853050	+	0.521830i	$0.174750\pi$
$983$	32.7217i	1.04366i	−0.853050	+	0.521830i	$0.825250\pi$
$984$	8.85086	0.282155
$985$	18.2446	0.581321
$986$	41.8552i	1.33294i
$987$	32.0116	1.01894
$988$	0	0
$989$	16.0097	0.509078
$990$	0.801938i	0.0254873i
$991$	−52.9329	−1.68147	−0.840734	−	0.541448i	$-0.817876\pi$
$991$	−52.9329	−1.68147	−0.840734	+	0.541448i	$0.817876\pi$
$992$	−1.69202	−0.0537217
$993$	− 10.3134i	− 0.327284i
$994$	8.70171i	0.276001i
$995$	1.77479i	0.0562646i
$996$	− 3.61356i	− 0.114500i
$997$	−10.8823	−0.344646	−0.172323	−	0.985040i	$-0.555127\pi$
$997$	−10.8823	−0.344646	−0.172323	+	0.985040i	$0.555127\pi$
$998$	−1.48666	−0.0470595
$999$	− 9.67456i	− 0.306090i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.u.1351.1		6
13.5	odd	4		5070.2.a.bm.1.3	✓	3
13.8	odd	4		5070.2.a.br.1.1	yes	3
13.12	even	2	inner	5070.2.b.u.1351.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bm.1.3	✓	3	13.5	odd	4
5070.2.a.br.1.1	yes	3	13.8	odd	4
5070.2.b.u.1351.1		6	1.1	even	1	trivial
5070.2.b.u.1351.6		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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -4.24698i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -4.24698i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.801938i q^{11} +1.00000 q^{12} -4.24698 q^{14} -1.00000i q^{15} +1.00000 q^{16} -7.43296 q^{17} -1.00000i q^{18} +3.74094i q^{19} -1.00000i q^{20} +4.24698i q^{21} +0.801938 q^{22} +2.54288 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +4.24698i q^{28} +5.63102 q^{29} -1.00000 q^{30} -1.69202i q^{31} -1.00000i q^{32} -0.801938i q^{33} +7.43296i q^{34} +4.24698 q^{35} -1.00000 q^{36} +9.67456i q^{37} +3.74094 q^{38} -1.00000 q^{40} +8.85086i q^{41} +4.24698 q^{42} +6.29590 q^{43} -0.801938i q^{44} +1.00000i q^{45} -2.54288i q^{46} -7.53750i q^{47} -1.00000 q^{48} -11.0368 q^{49} +1.00000i q^{50} +7.43296 q^{51} +13.5646 q^{53} +1.00000i q^{54} -0.801938 q^{55} +4.24698 q^{56} -3.74094i q^{57} -5.63102i q^{58} -9.58211i q^{59} +1.00000i q^{60} -10.7899 q^{61} -1.69202 q^{62} -4.24698i q^{63} -1.00000 q^{64} -0.801938 q^{66} +7.82908i q^{67} +7.43296 q^{68} -2.54288 q^{69} -4.24698i q^{70} -2.04892i q^{71} +1.00000i q^{72} -2.17092i q^{73} +9.67456 q^{74} +1.00000 q^{75} -3.74094i q^{76} +3.40581 q^{77} -16.2349 q^{79} +1.00000i q^{80} +1.00000 q^{81} +8.85086 q^{82} -3.61356i q^{83} -4.24698i q^{84} -7.43296i q^{85} -6.29590i q^{86} -5.63102 q^{87} -0.801938 q^{88} +8.24459i q^{89} +1.00000 q^{90} -2.54288 q^{92} +1.69202i q^{93} -7.53750 q^{94} -3.74094 q^{95} +1.00000i q^{96} +6.07069i q^{97} +11.0368i q^{98} +0.801938i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 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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