LMFDB - Embedded newform 5070.2.b.x.1351.4

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			1351.4
Root			$-0.445042i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.x.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -2.24698i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -2.24698i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.69202i q^{11} -1.00000 q^{12} +2.24698 q^{14} +1.00000i q^{15} +1.00000 q^{16} -0.445042 q^{17} +1.00000i q^{18} -1.74094i q^{19} -1.00000i q^{20} -2.24698i q^{21} +1.69202 q^{22} -4.04892 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +2.24698i q^{28} +0.643104 q^{29} -1.00000 q^{30} +4.80194i q^{31} +1.00000i q^{32} -1.69202i q^{33} -0.445042i q^{34} +2.24698 q^{35} -1.00000 q^{36} -0.862937i q^{37} +1.74094 q^{38} +1.00000 q^{40} -5.74094i q^{41} +2.24698 q^{42} +1.58211 q^{43} +1.69202i q^{44} +1.00000i q^{45} -4.04892i q^{46} +5.49396i q^{47} +1.00000 q^{48} +1.95108 q^{49} -1.00000i q^{50} -0.445042 q^{51} +0.137063 q^{53} +1.00000i q^{54} +1.69202 q^{55} -2.24698 q^{56} -1.74094i q^{57} +0.643104i q^{58} -12.6136i q^{59} -1.00000i q^{60} -0.472189 q^{61} -4.80194 q^{62} -2.24698i q^{63} -1.00000 q^{64} +1.69202 q^{66} -4.54288i q^{67} +0.445042 q^{68} -4.04892 q^{69} +2.24698i q^{70} -12.6407i q^{71} -1.00000i q^{72} -11.1588i q^{73} +0.862937 q^{74} -1.00000 q^{75} +1.74094i q^{76} -3.80194 q^{77} -6.57673 q^{79} +1.00000i q^{80} +1.00000 q^{81} +5.74094 q^{82} -1.91185i q^{83} +2.24698i q^{84} -0.445042i q^{85} +1.58211i q^{86} +0.643104 q^{87} -1.69202 q^{88} -11.9269i q^{89} -1.00000 q^{90} +4.04892 q^{92} +4.80194i q^{93} -5.49396 q^{94} +1.74094 q^{95} +1.00000i q^{96} -7.55257i q^{97} +1.95108i q^{98} -1.69202i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{14} + 6 q^{16} - 2 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} + 12 q^{29} - 6 q^{30} + 4 q^{35} - 6 q^{36} - 18 q^{38} + 6 q^{40} + 4 q^{42} - 2 q^{43} + 6 q^{48} + 30 q^{49} - 2 q^{51} - 10 q^{53} - 4 q^{56} + 10 q^{61} - 20 q^{62} - 6 q^{64} + 2 q^{68} - 6 q^{69} + 16 q^{74} - 6 q^{75} - 14 q^{77} - 34 q^{79} + 6 q^{81} + 6 q^{82} + 12 q^{87} - 6 q^{90} + 6 q^{92} - 14 q^{94} - 18 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 - 6 * q^12 + 4 * q^14 + 6 * q^16 - 2 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 + 12 * q^29 - 6 * q^30 + 4 * q^35 - 6 * q^36 - 18 * q^38 + 6 * q^40 + 4 * q^42 - 2 * q^43 + 6 * q^48 + 30 * q^49 - 2 * q^51 - 10 * q^53 - 4 * q^56 + 10 * q^61 - 20 * q^62 - 6 * q^64 + 2 * q^68 - 6 * q^69 + 16 * q^74 - 6 * q^75 - 14 * q^77 - 34 * q^79 + 6 * q^81 + 6 * q^82 + 12 * q^87 - 6 * q^90 + 6 * q^92 - 14 * q^94 - 18 * 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
$7$	− 2.24698i	− 0.849278i	−0.905363	−	0.424639i	$-0.860401\pi$
$7$	− 2.24698i	− 0.849278i	0.905363	−	0.424639i	$-0.139599\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	− 1.69202i	− 0.510164i	−0.966919	−	0.255082i	$-0.917898\pi$
$11$	− 1.69202i	− 0.510164i	0.966919	−	0.255082i	$-0.0821024\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	2.24698	0.600531
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	−0.445042	−0.107939	−0.0539693	−	0.998543i	$-0.517187\pi$
$17$	−0.445042	−0.107939	−0.0539693	+	0.998543i	$0.517187\pi$
$18$	1.00000i	0.235702i
$19$	− 1.74094i	− 0.399399i	−0.979857	−	0.199699i	$-0.936003\pi$
$19$	− 1.74094i	− 0.399399i	0.979857	−	0.199699i	$-0.0639966\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 2.24698i	− 0.490331i
$22$	1.69202	0.360740
$23$	−4.04892	−0.844258	−0.422129	−	0.906536i	$-0.638717\pi$
$23$	−4.04892	−0.844258	−0.422129	+	0.906536i	$0.638717\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	2.24698i	0.424639i
$29$	0.643104	0.119421	0.0597107	−	0.998216i	$-0.480982\pi$
$29$	0.643104	0.119421	0.0597107	+	0.998216i	$0.480982\pi$
$30$	−1.00000	−0.182574
$31$	4.80194i	0.862453i	0.902244	+	0.431227i	$0.141919\pi$
$31$	4.80194i	0.862453i	−0.902244	+	0.431227i	$0.858081\pi$
$32$	1.00000i	0.176777i
$33$	− 1.69202i	− 0.294543i
$34$	− 0.445042i	− 0.0763241i
$35$	2.24698	0.379809
$36$	−1.00000	−0.166667
$37$	− 0.862937i	− 0.141866i	−0.997481	−	0.0709330i	$-0.977402\pi$
$37$	− 0.862937i	− 0.141866i	0.997481	−	0.0709330i	$-0.0225976\pi$
$38$	1.74094	0.282418
$39$	0	0
$40$	1.00000	0.158114
$41$	− 5.74094i	− 0.896584i	−0.893887	−	0.448292i	$-0.852032\pi$
$41$	− 5.74094i	− 0.896584i	0.893887	−	0.448292i	$-0.147968\pi$
$42$	2.24698	0.346716
$43$	1.58211	0.241269	0.120634	−	0.992697i	$-0.461507\pi$
$43$	1.58211	0.241269	0.120634	+	0.992697i	$0.461507\pi$
$44$	1.69202i	0.255082i
$45$	1.00000i	0.149071i
$46$	− 4.04892i	− 0.596980i
$47$	5.49396i	0.801376i	0.916214	+	0.400688i	$0.131229\pi$
$47$	5.49396i	0.801376i	−0.916214	+	0.400688i	$0.868771\pi$
$48$	1.00000	0.144338
$49$	1.95108	0.278726
$50$	− 1.00000i	− 0.141421i
$51$	−0.445042	−0.0623183
$52$	0	0
$53$	0.137063	0.0188271	0.00941355	−	0.999956i	$-0.497004\pi$
$53$	0.137063	0.0188271	0.00941355	+	0.999956i	$0.497004\pi$
$54$	1.00000i	0.136083i
$55$	1.69202	0.228152
$56$	−2.24698	−0.300265
$57$	− 1.74094i	− 0.230593i
$58$	0.643104i	0.0844437i
$59$	− 12.6136i	− 1.64215i	−0.570823	−	0.821073i	$-0.693376\pi$
$59$	− 12.6136i	− 1.64215i	0.570823	−	0.821073i	$-0.306624\pi$
$60$	− 1.00000i	− 0.129099i
$61$	−0.472189	−0.0604576	−0.0302288	−	0.999543i	$-0.509624\pi$
$61$	−0.472189	−0.0604576	−0.0302288	+	0.999543i	$0.509624\pi$
$62$	−4.80194	−0.609847
$63$	− 2.24698i	− 0.283093i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	1.69202	0.208273
$67$	− 4.54288i	− 0.555001i	−0.960726	−	0.277500i	$-0.910494\pi$
$67$	− 4.54288i	− 0.555001i	0.960726	−	0.277500i	$-0.0895060\pi$
$68$	0.445042	0.0539693
$69$	−4.04892	−0.487432
$70$	2.24698i	0.268565i
$71$	− 12.6407i	− 1.50018i	−0.661338	−	0.750088i	$-0.730011\pi$
$71$	− 12.6407i	− 1.50018i	0.661338	−	0.750088i	$-0.269989\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 11.1588i	− 1.30604i	−0.757340	−	0.653021i	$-0.773501\pi$
$73$	− 11.1588i	− 1.30604i	0.757340	−	0.653021i	$-0.226499\pi$
$74$	0.862937	0.100314
$75$	−1.00000	−0.115470
$76$	1.74094i	0.199699i
$77$	−3.80194	−0.433271
$78$	0	0
$79$	−6.57673	−0.739940	−0.369970	−	0.929044i	$-0.620632\pi$
$79$	−6.57673	−0.739940	−0.369970	+	0.929044i	$0.620632\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	5.74094	0.633981
$83$	− 1.91185i	− 0.209853i	−0.994480	−	0.104927i	$-0.966539\pi$
$83$	− 1.91185i	− 0.209853i	0.994480	−	0.104927i	$-0.0334608\pi$
$84$	2.24698i	0.245166i
$85$	− 0.445042i	− 0.0482716i
$86$	1.58211i	0.170603i
$87$	0.643104	0.0689480
$88$	−1.69202	−0.180370
$89$	− 11.9269i	− 1.26425i	−0.774866	−	0.632125i	$-0.782183\pi$
$89$	− 11.9269i	− 1.26425i	0.774866	−	0.632125i	$-0.217817\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	4.04892	0.422129
$93$	4.80194i	0.497938i
$94$	−5.49396	−0.566659
$95$	1.74094	0.178617
$96$	1.00000i	0.102062i
$97$	− 7.55257i	− 0.766847i	−0.923573	−	0.383423i	$-0.874745\pi$
$97$	− 7.55257i	− 0.766847i	0.923573	−	0.383423i	$-0.125255\pi$
$98$	1.95108i	0.197089i
$99$	− 1.69202i	− 0.170055i
$100$	1.00000	0.100000
$101$	−9.03146	−0.898664	−0.449332	−	0.893365i	$-0.648338\pi$
$101$	−9.03146	−0.898664	−0.449332	+	0.893365i	$0.648338\pi$
$102$	− 0.445042i	− 0.0440657i
$103$	5.81163	0.572637	0.286318	−	0.958135i	$-0.407569\pi$
$103$	5.81163	0.572637	0.286318	+	0.958135i	$0.407569\pi$
$104$	0	0
$105$	2.24698	0.219283
$106$	0.137063i	0.0133128i
$107$	−3.00969	−0.290958	−0.145479	−	0.989361i	$-0.546472\pi$
$107$	−3.00969	−0.290958	−0.145479	+	0.989361i	$0.546472\pi$
$108$	−1.00000	−0.0962250
$109$	18.9269i	1.81287i	0.422345	+	0.906435i	$0.361207\pi$
$109$	18.9269i	1.81287i	−0.422345	+	0.906435i	$0.638793\pi$
$110$	1.69202i	0.161328i
$111$	− 0.862937i	− 0.0819063i
$112$	− 2.24698i	− 0.212320i
$113$	−0.472189	−0.0444198	−0.0222099	−	0.999753i	$-0.507070\pi$
$113$	−0.472189	−0.0444198	−0.0222099	+	0.999753i	$0.507070\pi$
$114$	1.74094	0.163054
$115$	− 4.04892i	− 0.377563i
$116$	−0.643104	−0.0597107
$117$	0	0
$118$	12.6136	1.16117
$119$	1.00000i	0.0916698i
$120$	1.00000	0.0912871
$121$	8.13706	0.739733
$122$	− 0.472189i	− 0.0427500i
$123$	− 5.74094i	− 0.517643i
$124$	− 4.80194i	− 0.431227i
$125$	− 1.00000i	− 0.0894427i
$126$	2.24698	0.200177
$127$	−10.8944	−0.966721	−0.483361	−	0.875421i	$-0.660584\pi$
$127$	−10.8944	−0.966721	−0.483361	+	0.875421i	$0.660584\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	1.58211	0.139297
$130$	0	0
$131$	−13.0532	−1.14047	−0.570233	−	0.821483i	$-0.693147\pi$
$131$	−13.0532	−1.14047	−0.570233	+	0.821483i	$0.693147\pi$
$132$	1.69202i	0.147272i
$133$	−3.91185	−0.339201
$134$	4.54288	0.392445
$135$	1.00000i	0.0860663i
$136$	0.445042i	0.0381620i
$137$	− 16.4306i	− 1.40376i	−0.712296	−	0.701879i	$-0.752345\pi$
$137$	− 16.4306i	− 1.40376i	0.712296	−	0.701879i	$-0.247655\pi$
$138$	− 4.04892i	− 0.344667i
$139$	−5.39612	−0.457693	−0.228847	−	0.973462i	$-0.573495\pi$
$139$	−5.39612	−0.457693	−0.228847	+	0.973462i	$0.573495\pi$
$140$	−2.24698	−0.189904
$141$	5.49396i	0.462675i
$142$	12.6407	1.06078
$143$	0	0
$144$	1.00000	0.0833333
$145$	0.643104i	0.0534069i
$146$	11.1588	0.923512
$147$	1.95108	0.160923
$148$	0.862937i	0.0709330i
$149$	− 21.1444i	− 1.73221i	−0.499859	−	0.866107i	$-0.666615\pi$
$149$	− 21.1444i	− 1.73221i	0.499859	−	0.866107i	$-0.333385\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	− 21.8213i	− 1.77579i	−0.460043	−	0.887897i	$-0.652166\pi$
$151$	− 21.8213i	− 1.77579i	0.460043	−	0.887897i	$-0.347834\pi$
$152$	−1.74094	−0.141209
$153$	−0.445042	−0.0359795
$154$	− 3.80194i	− 0.306369i
$155$	−4.80194	−0.385701
$156$	0	0
$157$	−6.28083	−0.501265	−0.250632	−	0.968082i	$-0.580639\pi$
$157$	−6.28083	−0.501265	−0.250632	+	0.968082i	$0.580639\pi$
$158$	− 6.57673i	− 0.523216i
$159$	0.137063	0.0108698
$160$	−1.00000	−0.0790569
$161$	9.09783i	0.717010i
$162$	1.00000i	0.0785674i
$163$	8.11960i	0.635977i	0.948095	+	0.317988i	$0.103007\pi$
$163$	8.11960i	0.635977i	−0.948095	+	0.317988i	$0.896993\pi$
$164$	5.74094i	0.448292i
$165$	1.69202	0.131724
$166$	1.91185	0.148389
$167$	− 0.362273i	− 0.0280335i	−0.999902	−	0.0140168i	$-0.995538\pi$
$167$	− 0.362273i	− 0.0280335i	0.999902	−	0.0140168i	$-0.00446182\pi$
$168$	−2.24698	−0.173358
$169$	0	0
$170$	0.445042	0.0341332
$171$	− 1.74094i	− 0.133133i
$172$	−1.58211	−0.120634
$173$	24.4480	1.85875	0.929374	−	0.369138i	$-0.120347\pi$
$173$	24.4480	1.85875	0.929374	+	0.369138i	$0.120347\pi$
$174$	0.643104i	0.0487536i
$175$	2.24698i	0.169856i
$176$	− 1.69202i	− 0.127541i
$177$	− 12.6136i	− 0.948094i
$178$	11.9269	0.893960
$179$	−3.81163	−0.284894	−0.142447	−	0.989802i	$-0.545497\pi$
$179$	−3.81163	−0.284894	−0.142447	+	0.989802i	$0.545497\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	15.7181	1.16832	0.584159	−	0.811639i	$-0.301425\pi$
$181$	15.7181	1.16832	0.584159	+	0.811639i	$0.301425\pi$
$182$	0	0
$183$	−0.472189	−0.0349052
$184$	4.04892i	0.298490i
$185$	0.862937	0.0634444
$186$	−4.80194	−0.352095
$187$	0.753020i	0.0550663i
$188$	− 5.49396i	− 0.400688i
$189$	− 2.24698i	− 0.163444i
$190$	1.74094i	0.126301i
$191$	−18.7832	−1.35910	−0.679551	−	0.733629i	$-0.737825\pi$
$191$	−18.7832	−1.35910	−0.679551	+	0.733629i	$0.737825\pi$
$192$	−1.00000	−0.0721688
$193$	1.99330i	0.143481i	0.997423	+	0.0717403i	$0.0228553\pi$
$193$	1.99330i	0.143481i	−0.997423	+	0.0717403i	$0.977145\pi$
$194$	7.55257	0.542243
$195$	0	0
$196$	−1.95108	−0.139363
$197$	− 8.26875i	− 0.589124i	−0.955632	−	0.294562i	$-0.904826\pi$
$197$	− 8.26875i	− 0.589124i	0.955632	−	0.294562i	$-0.0951738\pi$
$198$	1.69202	0.120247
$199$	15.7071	1.11345	0.556723	−	0.830698i	$-0.312059\pi$
$199$	15.7071	1.11345	0.556723	+	0.830698i	$0.312059\pi$
$200$	1.00000i	0.0707107i
$201$	− 4.54288i	− 0.320430i
$202$	− 9.03146i	− 0.635451i
$203$	− 1.44504i	− 0.101422i
$204$	0.445042	0.0311592
$205$	5.74094	0.400965
$206$	5.81163i	0.404915i
$207$	−4.04892	−0.281419
$208$	0	0
$209$	−2.94571	−0.203759
$210$	2.24698i	0.155056i
$211$	17.2814	1.18970	0.594851	−	0.803836i	$-0.297211\pi$
$211$	17.2814	1.18970	0.594851	+	0.803836i	$0.297211\pi$
$212$	−0.137063	−0.00941355
$213$	− 12.6407i	− 0.866127i
$214$	− 3.00969i	− 0.205738i
$215$	1.58211i	0.107899i
$216$	− 1.00000i	− 0.0680414i
$217$	10.7899	0.732463
$218$	−18.9269	−1.28189
$219$	− 11.1588i	− 0.754044i
$220$	−1.69202	−0.114076
$221$	0	0
$222$	0.862937	0.0579165
$223$	22.8049i	1.52713i	0.645731	+	0.763565i	$0.276553\pi$
$223$	22.8049i	1.52713i	−0.645731	+	0.763565i	$0.723447\pi$
$224$	2.24698	0.150133
$225$	−1.00000	−0.0666667
$226$	− 0.472189i	− 0.0314095i
$227$	− 11.5483i	− 0.766484i	−0.923648	−	0.383242i	$-0.874807\pi$
$227$	− 11.5483i	− 0.766484i	0.923648	−	0.383242i	$-0.125193\pi$
$228$	1.74094i	0.115296i
$229$	− 26.1129i	− 1.72559i	−0.505555	−	0.862795i	$-0.668712\pi$
$229$	− 26.1129i	− 1.72559i	0.505555	−	0.862795i	$-0.331288\pi$
$230$	4.04892	0.266978
$231$	−3.80194	−0.250149
$232$	− 0.643104i	− 0.0422219i
$233$	10.1293	0.663592	0.331796	−	0.943351i	$-0.392345\pi$
$233$	10.1293	0.663592	0.331796	+	0.943351i	$0.392345\pi$
$234$	0	0
$235$	−5.49396	−0.358386
$236$	12.6136i	0.821073i
$237$	−6.57673	−0.427204
$238$	−1.00000	−0.0648204
$239$	16.3080i	1.05488i	0.849594	+	0.527438i	$0.176847\pi$
$239$	16.3080i	1.05488i	−0.849594	+	0.527438i	$0.823153\pi$
$240$	1.00000i	0.0645497i
$241$	2.20237i	0.141867i	0.997481	+	0.0709337i	$0.0225979\pi$
$241$	2.20237i	0.141867i	−0.997481	+	0.0709337i	$0.977402\pi$
$242$	8.13706i	0.523070i
$243$	1.00000	0.0641500
$244$	0.472189	0.0302288
$245$	1.95108i	0.124650i
$246$	5.74094	0.366029
$247$	0	0
$248$	4.80194	0.304923
$249$	− 1.91185i	− 0.121159i
$250$	1.00000	0.0632456
$251$	7.33513	0.462989	0.231495	−	0.972836i	$-0.425638\pi$
$251$	7.33513	0.462989	0.231495	+	0.972836i	$0.425638\pi$
$252$	2.24698i	0.141546i
$253$	6.85086i	0.430710i
$254$	− 10.8944i	− 0.683575i
$255$	− 0.445042i	− 0.0278696i
$256$	1.00000	0.0625000
$257$	−10.4523	−0.651999	−0.325999	−	0.945370i	$-0.605701\pi$
$257$	−10.4523	−0.651999	−0.325999	+	0.945370i	$0.605701\pi$
$258$	1.58211i	0.0984976i
$259$	−1.93900	−0.120484
$260$	0	0
$261$	0.643104	0.0398071
$262$	− 13.0532i	− 0.806431i
$263$	13.0694	0.805891	0.402946	−	0.915224i	$-0.367986\pi$
$263$	13.0694	0.805891	0.402946	+	0.915224i	$0.367986\pi$
$264$	−1.69202	−0.104137
$265$	0.137063i	0.00841973i
$266$	− 3.91185i	− 0.239851i
$267$	− 11.9269i	− 0.729916i
$268$	4.54288i	0.277500i
$269$	0.119605	0.00729244	0.00364622	−	0.999993i	$-0.498839\pi$
$269$	0.119605	0.00729244	0.00364622	+	0.999993i	$0.498839\pi$
$270$	−1.00000	−0.0608581
$271$	27.3545i	1.66167i	0.556521	+	0.830834i	$0.312136\pi$
$271$	27.3545i	1.66167i	−0.556521	+	0.830834i	$0.687864\pi$
$272$	−0.445042	−0.0269846
$273$	0	0
$274$	16.4306	0.992607
$275$	1.69202i	0.102033i
$276$	4.04892	0.243716
$277$	3.14914	0.189214	0.0946069	−	0.995515i	$-0.469841\pi$
$277$	3.14914	0.189214	0.0946069	+	0.995515i	$0.469841\pi$
$278$	− 5.39612i	− 0.323638i
$279$	4.80194i	0.287484i
$280$	− 2.24698i	− 0.134283i
$281$	− 16.0683i	− 0.958554i	−0.877664	−	0.479277i	$-0.840899\pi$
$281$	− 16.0683i	− 0.958554i	0.877664	−	0.479277i	$-0.159101\pi$
$282$	−5.49396	−0.327161
$283$	−15.8834	−0.944169	−0.472084	−	0.881553i	$-0.656498\pi$
$283$	−15.8834	−0.944169	−0.472084	+	0.881553i	$0.656498\pi$
$284$	12.6407i	0.750088i
$285$	1.74094	0.103124
$286$	0	0
$287$	−12.8998	−0.761449
$288$	1.00000i	0.0589256i
$289$	−16.8019	−0.988349
$290$	−0.643104	−0.0377644
$291$	− 7.55257i	− 0.442739i
$292$	11.1588i	0.653021i
$293$	15.6039i	0.911588i	0.890085	+	0.455794i	$0.150645\pi$
$293$	15.6039i	0.911588i	−0.890085	+	0.455794i	$0.849355\pi$
$294$	1.95108i	0.113789i
$295$	12.6136	0.734390
$296$	−0.862937	−0.0501572
$297$	− 1.69202i	− 0.0981810i
$298$	21.1444	1.22486
$299$	0	0
$300$	1.00000	0.0577350
$301$	− 3.55496i	− 0.204904i
$302$	21.8213	1.25568
$303$	−9.03146	−0.518844
$304$	− 1.74094i	− 0.0998497i
$305$	− 0.472189i	− 0.0270375i
$306$	− 0.445042i	− 0.0254414i
$307$	− 18.3002i	− 1.04445i	−0.852808	−	0.522224i	$-0.825102\pi$
$307$	− 18.3002i	− 1.04445i	0.852808	−	0.522224i	$-0.174898\pi$
$308$	3.80194	0.216636
$309$	5.81163	0.330612
$310$	− 4.80194i	− 0.272732i
$311$	−14.7899	−0.838656	−0.419328	−	0.907835i	$-0.637734\pi$
$311$	−14.7899	−0.838656	−0.419328	+	0.907835i	$0.637734\pi$
$312$	0	0
$313$	4.05323	0.229102	0.114551	−	0.993417i	$-0.463457\pi$
$313$	4.05323	0.229102	0.114551	+	0.993417i	$0.463457\pi$
$314$	− 6.28083i	− 0.354448i
$315$	2.24698	0.126603
$316$	6.57673	0.369970
$317$	− 9.52781i	− 0.535135i	−0.963539	−	0.267568i	$-0.913780\pi$
$317$	− 9.52781i	− 0.535135i	0.963539	−	0.267568i	$-0.0862199\pi$
$318$	0.137063i	0.00768613i
$319$	− 1.08815i	− 0.0609245i
$320$	− 1.00000i	− 0.0559017i
$321$	−3.00969	−0.167984
$322$	−9.09783	−0.507003
$323$	0.774791i	0.0431105i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−8.11960	−0.449703
$327$	18.9269i	1.04666i
$328$	−5.74094	−0.316990
$329$	12.3448	0.680592
$330$	1.69202i	0.0931427i
$331$	10.2107i	0.561233i	0.959820	+	0.280616i	$0.0905389\pi$
$331$	10.2107i	0.561233i	−0.959820	+	0.280616i	$0.909461\pi$
$332$	1.91185i	0.104927i
$333$	− 0.862937i	− 0.0472886i
$334$	0.362273	0.0198227
$335$	4.54288	0.248204
$336$	− 2.24698i	− 0.122583i
$337$	21.6353	1.17855	0.589276	−	0.807932i	$-0.299413\pi$
$337$	21.6353	1.17855	0.589276	+	0.807932i	$0.299413\pi$
$338$	0	0
$339$	−0.472189	−0.0256458
$340$	0.445042i	0.0241358i
$341$	8.12498	0.439992
$342$	1.74094	0.0941392
$343$	− 20.1129i	− 1.08599i
$344$	− 1.58211i	− 0.0853014i
$345$	− 4.04892i	− 0.217986i
$346$	24.4480i	1.31433i
$347$	−24.1347	−1.29562	−0.647808	−	0.761803i	$-0.724314\pi$
$347$	−24.1347	−1.29562	−0.647808	+	0.761803i	$0.724314\pi$
$348$	−0.643104	−0.0344740
$349$	− 14.0532i	− 0.752252i	−0.926569	−	0.376126i	$-0.877256\pi$
$349$	− 14.0532i	− 0.752252i	0.926569	−	0.376126i	$-0.122744\pi$
$350$	−2.24698	−0.120106
$351$	0	0
$352$	1.69202	0.0901850
$353$	30.8810i	1.64363i	0.569755	+	0.821815i	$0.307038\pi$
$353$	30.8810i	1.64363i	−0.569755	+	0.821815i	$0.692962\pi$
$354$	12.6136	0.670403
$355$	12.6407	0.670899
$356$	11.9269i	0.632125i
$357$	1.00000i	0.0529256i
$358$	− 3.81163i	− 0.201451i
$359$	28.5827i	1.50854i	0.656566	+	0.754269i	$0.272008\pi$
$359$	28.5827i	1.50854i	−0.656566	+	0.754269i	$0.727992\pi$
$360$	1.00000	0.0527046
$361$	15.9691	0.840481
$362$	15.7181i	0.826125i
$363$	8.13706	0.427085
$364$	0	0
$365$	11.1588	0.584080
$366$	− 0.472189i	− 0.0246817i
$367$	−5.83340	−0.304501	−0.152250	−	0.988342i	$-0.548652\pi$
$367$	−5.83340	−0.304501	−0.152250	+	0.988342i	$0.548652\pi$
$368$	−4.04892	−0.211064
$369$	− 5.74094i	− 0.298861i
$370$	0.862937i	0.0448619i
$371$	− 0.307979i	− 0.0159894i
$372$	− 4.80194i	− 0.248969i
$373$	36.1172	1.87008	0.935039	−	0.354544i	$-0.115364\pi$
$373$	36.1172	1.87008	0.935039	+	0.354544i	$0.115364\pi$
$374$	−0.753020	−0.0389378
$375$	− 1.00000i	− 0.0516398i
$376$	5.49396	0.283329
$377$	0	0
$378$	2.24698	0.115572
$379$	18.7651i	0.963899i	0.876199	+	0.481949i	$0.160071\pi$
$379$	18.7651i	0.963899i	−0.876199	+	0.481949i	$0.839929\pi$
$380$	−1.74094	−0.0893083
$381$	−10.8944	−0.558137
$382$	− 18.7832i	− 0.961030i
$383$	− 15.4709i	− 0.790524i	−0.918568	−	0.395262i	$-0.870654\pi$
$383$	− 15.4709i	− 0.790524i	0.918568	−	0.395262i	$-0.129346\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	− 3.80194i	− 0.193765i
$386$	−1.99330	−0.101456
$387$	1.58211	0.0804229
$388$	7.55257i	0.383423i
$389$	−5.67994	−0.287984	−0.143992	−	0.989579i	$-0.545994\pi$
$389$	−5.67994	−0.287984	−0.143992	+	0.989579i	$0.545994\pi$
$390$	0	0
$391$	1.80194	0.0911279
$392$	− 1.95108i	− 0.0985446i
$393$	−13.0532	−0.658448
$394$	8.26875	0.416574
$395$	− 6.57673i	− 0.330911i
$396$	1.69202i	0.0850273i
$397$	− 19.2282i	− 0.965035i	−0.875886	−	0.482518i	$-0.839722\pi$
$397$	− 19.2282i	− 0.965035i	0.875886	−	0.482518i	$-0.160278\pi$
$398$	15.7071i	0.787325i
$399$	−3.91185	−0.195838
$400$	−1.00000	−0.0500000
$401$	27.5593i	1.37624i	0.725595	+	0.688122i	$0.241565\pi$
$401$	27.5593i	1.37624i	−0.725595	+	0.688122i	$0.758435\pi$
$402$	4.54288	0.226578
$403$	0	0
$404$	9.03146	0.449332
$405$	1.00000i	0.0496904i
$406$	1.44504	0.0717162
$407$	−1.46011	−0.0723748
$408$	0.445042i	0.0220329i
$409$	17.7681i	0.878575i	0.898346	+	0.439288i	$0.144769\pi$
$409$	17.7681i	0.878575i	−0.898346	+	0.439288i	$0.855231\pi$
$410$	5.74094i	0.283525i
$411$	− 16.4306i	− 0.810460i
$412$	−5.81163	−0.286318
$413$	−28.3424	−1.39464
$414$	− 4.04892i	− 0.198993i
$415$	1.91185	0.0938492
$416$	0	0
$417$	−5.39612	−0.264249
$418$	− 2.94571i	− 0.144079i
$419$	22.4426	1.09640	0.548198	−	0.836349i	$-0.315314\pi$
$419$	22.4426	1.09640	0.548198	+	0.836349i	$0.315314\pi$
$420$	−2.24698	−0.109641
$421$	− 1.24698i	− 0.0607741i	−0.999538	−	0.0303870i	$-0.990326\pi$
$421$	− 1.24698i	− 0.0607741i	0.999538	−	0.0303870i	$-0.00967398\pi$
$422$	17.2814i	0.841246i
$423$	5.49396i	0.267125i
$424$	− 0.137063i	− 0.00665638i
$425$	0.445042	0.0215877
$426$	12.6407	0.612444
$427$	1.06100i	0.0513453i
$428$	3.00969	0.145479
$429$	0	0
$430$	−1.58211	−0.0762959
$431$	− 5.97823i	− 0.287961i	−0.989581	−	0.143981i	$-0.954010\pi$
$431$	− 5.97823i	− 0.287961i	0.989581	−	0.143981i	$-0.0459903\pi$
$432$	1.00000	0.0481125
$433$	14.2717	0.685856	0.342928	−	0.939362i	$-0.388581\pi$
$433$	14.2717	0.685856	0.342928	+	0.939362i	$0.388581\pi$
$434$	10.7899i	0.517930i
$435$	0.643104i	0.0308345i
$436$	− 18.9269i	− 0.906435i
$437$	7.04892i	0.337195i
$438$	11.1588	0.533190
$439$	12.2349	0.583940	0.291970	−	0.956428i	$-0.405689\pi$
$439$	12.2349	0.583940	0.291970	+	0.956428i	$0.405689\pi$
$440$	− 1.69202i	− 0.0806640i
$441$	1.95108	0.0929087
$442$	0	0
$443$	−19.4631	−0.924719	−0.462360	−	0.886692i	$-0.652997\pi$
$443$	−19.4631	−0.924719	−0.462360	+	0.886692i	$0.652997\pi$
$444$	0.862937i	0.0409532i
$445$	11.9269	0.565390
$446$	−22.8049	−1.07984
$447$	− 21.1444i	− 1.00009i
$448$	2.24698i	0.106160i
$449$	− 10.4222i	− 0.491854i	−0.969288	−	0.245927i	$-0.920908\pi$
$449$	− 10.4222i	− 0.491854i	0.969288	−	0.245927i	$-0.0790924\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	−9.71379	−0.457405
$452$	0.472189	0.0222099
$453$	− 21.8213i	− 1.02525i
$454$	11.5483	0.541986
$455$	0	0
$456$	−1.74094	−0.0815269
$457$	− 12.9148i	− 0.604131i	−0.953287	−	0.302065i	$-0.902324\pi$
$457$	− 12.9148i	− 0.604131i	0.953287	−	0.302065i	$-0.0976761\pi$
$458$	26.1129	1.22018
$459$	−0.445042	−0.0207728
$460$	4.04892i	0.188782i
$461$	− 27.7409i	− 1.29202i	−0.763327	−	0.646012i	$-0.776436\pi$
$461$	− 27.7409i	− 1.29202i	0.763327	−	0.646012i	$-0.223564\pi$
$462$	− 3.80194i	− 0.176882i
$463$	31.4795i	1.46298i	0.681854	+	0.731488i	$0.261174\pi$
$463$	31.4795i	1.46298i	−0.681854	+	0.731488i	$0.738826\pi$
$464$	0.643104	0.0298554
$465$	−4.80194	−0.222685
$466$	10.1293i	0.469230i
$467$	−27.2097	−1.25911	−0.629557	−	0.776955i	$-0.716763\pi$
$467$	−27.2097	−1.25911	−0.629557	+	0.776955i	$0.716763\pi$
$468$	0	0
$469$	−10.2078	−0.471350
$470$	− 5.49396i	− 0.253417i
$471$	−6.28083	−0.289405
$472$	−12.6136	−0.580586
$473$	− 2.67696i	− 0.123087i
$474$	− 6.57673i	− 0.302079i
$475$	1.74094i	0.0798798i
$476$	− 1.00000i	− 0.0458349i
$477$	0.137063	0.00627570
$478$	−16.3080	−0.745910
$479$	− 20.2728i	− 0.926288i	−0.886283	−	0.463144i	$-0.846721\pi$
$479$	− 20.2728i	− 0.926288i	0.886283	−	0.463144i	$-0.153279\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−2.20237	−0.100315
$483$	9.09783i	0.413966i
$484$	−8.13706	−0.369867
$485$	7.55257	0.342944
$486$	1.00000i	0.0453609i
$487$	− 22.3424i	− 1.01243i	−0.862407	−	0.506216i	$-0.831044\pi$
$487$	− 22.3424i	− 1.01243i	0.862407	−	0.506216i	$-0.168956\pi$
$488$	0.472189i	0.0213750i
$489$	8.11960i	0.367181i
$490$	−1.95108	−0.0881409
$491$	−1.57971	−0.0712914	−0.0356457	−	0.999364i	$-0.511349\pi$
$491$	−1.57971	−0.0712914	−0.0356457	+	0.999364i	$0.511349\pi$
$492$	5.74094i	0.258822i
$493$	−0.286208	−0.0128902
$494$	0	0
$495$	1.69202	0.0760507
$496$	4.80194i	0.215613i
$497$	−28.4034	−1.27407
$498$	1.91185	0.0856722
$499$	− 5.76941i	− 0.258274i	−0.991627	−	0.129137i	$-0.958779\pi$
$499$	− 5.76941i	− 0.258274i	0.991627	−	0.129137i	$-0.0412208\pi$
$500$	1.00000i	0.0447214i
$501$	− 0.362273i	− 0.0161851i
$502$	7.33513i	0.327383i
$503$	6.00431	0.267719	0.133860	−	0.991000i	$-0.457263\pi$
$503$	6.00431	0.267719	0.133860	+	0.991000i	$0.457263\pi$
$504$	−2.24698	−0.100088
$505$	− 9.03146i	− 0.401895i
$506$	−6.85086	−0.304558
$507$	0	0
$508$	10.8944	0.483361
$509$	− 3.58211i	− 0.158774i	−0.996844	−	0.0793870i	$-0.974704\pi$
$509$	− 3.58211i	− 0.158774i	0.996844	−	0.0793870i	$-0.0252963\pi$
$510$	0.445042	0.0197068
$511$	−25.0737	−1.10919
$512$	1.00000i	0.0441942i
$513$	− 1.74094i	− 0.0768643i
$514$	− 10.4523i	− 0.461033i
$515$	5.81163i	0.256091i
$516$	−1.58211	−0.0696483
$517$	9.29590	0.408833
$518$	− 1.93900i	− 0.0851948i
$519$	24.4480	1.07315
$520$	0	0
$521$	−26.0532	−1.14141	−0.570706	−	0.821154i	$-0.693331\pi$
$521$	−26.0532	−1.14141	−0.570706	+	0.821154i	$0.693331\pi$
$522$	0.643104i	0.0281479i
$523$	22.5483	0.985966	0.492983	−	0.870039i	$-0.335906\pi$
$523$	22.5483	0.985966	0.492983	+	0.870039i	$0.335906\pi$
$524$	13.0532	0.570233
$525$	2.24698i	0.0980662i
$526$	13.0694i	0.569851i
$527$	− 2.13706i	− 0.0930919i
$528$	− 1.69202i	− 0.0736358i
$529$	−6.60627	−0.287229
$530$	−0.137063	−0.00595365
$531$	− 12.6136i	− 0.547382i
$532$	3.91185	0.169600
$533$	0	0
$534$	11.9269	0.516128
$535$	− 3.00969i	− 0.130120i
$536$	−4.54288	−0.196222
$537$	−3.81163	−0.164484
$538$	0.119605i	0.00515654i
$539$	− 3.30127i	− 0.142196i
$540$	− 1.00000i	− 0.0430331i
$541$	− 25.7375i	− 1.10654i	−0.833002	−	0.553270i	$-0.813380\pi$
$541$	− 25.7375i	− 1.10654i	0.833002	−	0.553270i	$-0.186620\pi$
$542$	−27.3545	−1.17498
$543$	15.7181	0.674528
$544$	− 0.445042i	− 0.0190810i
$545$	−18.9269	−0.810740
$546$	0	0
$547$	−45.2844	−1.93622	−0.968111	−	0.250523i	$-0.919397\pi$
$547$	−45.2844	−1.93622	−0.968111	+	0.250523i	$0.919397\pi$
$548$	16.4306i	0.701879i
$549$	−0.472189	−0.0201525
$550$	−1.69202	−0.0721480
$551$	− 1.11960i	− 0.0476968i
$552$	4.04892i	0.172333i
$553$	14.7778i	0.628415i
$554$	3.14914i	0.133794i
$555$	0.862937	0.0366296
$556$	5.39612	0.228847
$557$	21.8388i	0.925339i	0.886531	+	0.462669i	$0.153108\pi$
$557$	21.8388i	0.925339i	−0.886531	+	0.462669i	$0.846892\pi$
$558$	−4.80194	−0.203282
$559$	0	0
$560$	2.24698	0.0949522
$561$	0.753020i	0.0317925i
$562$	16.0683	0.677800
$563$	14.2537	0.600721	0.300361	−	0.953826i	$-0.402893\pi$
$563$	14.2537	0.600721	0.300361	+	0.953826i	$0.402893\pi$
$564$	− 5.49396i	− 0.231337i
$565$	− 0.472189i	− 0.0198651i
$566$	− 15.8834i	− 0.667628i
$567$	− 2.24698i	− 0.0943643i
$568$	−12.6407	−0.530392
$569$	8.35929	0.350440	0.175220	−	0.984529i	$-0.443936\pi$
$569$	8.35929	0.350440	0.175220	+	0.984529i	$0.443936\pi$
$570$	1.74094i	0.0729199i
$571$	−7.06829	−0.295799	−0.147899	−	0.989002i	$-0.547251\pi$
$571$	−7.06829	−0.295799	−0.147899	+	0.989002i	$0.547251\pi$
$572$	0	0
$573$	−18.7832	−0.784677
$574$	− 12.8998i	− 0.538426i
$575$	4.04892	0.168852
$576$	−1.00000	−0.0416667
$577$	16.4397i	0.684392i	0.939629	+	0.342196i	$0.111171\pi$
$577$	16.4397i	0.684392i	−0.939629	+	0.342196i	$0.888829\pi$
$578$	− 16.8019i	− 0.698868i
$579$	1.99330i	0.0828385i
$580$	− 0.643104i	− 0.0267034i
$581$	−4.29590	−0.178224
$582$	7.55257	0.313064
$583$	− 0.231914i	− 0.00960490i
$584$	−11.1588	−0.461756
$585$	0	0
$586$	−15.6039	−0.644590
$587$	20.3437i	0.839676i	0.907599	+	0.419838i	$0.137913\pi$
$587$	20.3437i	0.839676i	−0.907599	+	0.419838i	$0.862087\pi$
$588$	−1.95108	−0.0804613
$589$	8.35988	0.344463
$590$	12.6136i	0.519292i
$591$	− 8.26875i	− 0.340131i
$592$	− 0.862937i	− 0.0354665i
$593$	24.2586i	0.996181i	0.867125	+	0.498090i	$0.165965\pi$
$593$	24.2586i	0.996181i	−0.867125	+	0.498090i	$0.834035\pi$
$594$	1.69202	0.0694245
$595$	−1.00000	−0.0409960
$596$	21.1444i	0.866107i
$597$	15.7071	0.642848
$598$	0	0
$599$	15.2553	0.623316	0.311658	−	0.950194i	$-0.399116\pi$
$599$	15.2553	0.623316	0.311658	+	0.950194i	$0.399116\pi$
$600$	1.00000i	0.0408248i
$601$	−9.10560	−0.371425	−0.185713	−	0.982604i	$-0.559459\pi$
$601$	−9.10560	−0.371425	−0.185713	+	0.982604i	$0.559459\pi$
$602$	3.55496	0.144889
$603$	− 4.54288i	− 0.185000i
$604$	21.8213i	0.887897i
$605$	8.13706i	0.330819i
$606$	− 9.03146i	− 0.366878i
$607$	13.2838	0.539173	0.269587	−	0.962976i	$-0.413113\pi$
$607$	13.2838	0.539173	0.269587	+	0.962976i	$0.413113\pi$
$608$	1.74094	0.0706044
$609$	− 1.44504i	− 0.0585561i
$610$	0.472189	0.0191184
$611$	0	0
$612$	0.445042	0.0179898
$613$	18.7101i	0.755693i	0.925868	+	0.377846i	$0.123335\pi$
$613$	18.7101i	0.755693i	−0.925868	+	0.377846i	$0.876665\pi$
$614$	18.3002	0.738536
$615$	5.74094	0.231497
$616$	3.80194i	0.153184i
$617$	9.45473i	0.380633i	0.981723	+	0.190317i	$0.0609514\pi$
$617$	9.45473i	0.380633i	−0.981723	+	0.190317i	$0.939049\pi$
$618$	5.81163i	0.233778i
$619$	− 29.5429i	− 1.18743i	−0.804676	−	0.593714i	$-0.797661\pi$
$619$	− 29.5429i	− 1.18743i	0.804676	−	0.593714i	$-0.202339\pi$
$620$	4.80194	0.192850
$621$	−4.04892	−0.162477
$622$	− 14.7899i	− 0.593019i
$623$	−26.7995	−1.07370
$624$	0	0
$625$	1.00000	0.0400000
$626$	4.05323i	0.162000i
$627$	−2.94571	−0.117640
$628$	6.28083	0.250632
$629$	0.384043i	0.0153128i
$630$	2.24698i	0.0895218i
$631$	− 5.08516i	− 0.202437i	−0.994864	−	0.101219i	$-0.967726\pi$
$631$	− 5.08516i	− 0.202437i	0.994864	−	0.101219i	$-0.0322741\pi$
$632$	6.57673i	0.261608i
$633$	17.2814	0.686875
$634$	9.52781	0.378398
$635$	− 10.8944i	− 0.432331i
$636$	−0.137063	−0.00543491
$637$	0	0
$638$	1.08815	0.0430801
$639$	− 12.6407i	− 0.500059i
$640$	1.00000	0.0395285
$641$	14.1220	0.557785	0.278893	−	0.960322i	$-0.410033\pi$
$641$	14.1220	0.557785	0.278893	+	0.960322i	$0.410033\pi$
$642$	− 3.00969i	− 0.118783i
$643$	30.9439i	1.22031i	0.792283	+	0.610154i	$0.208893\pi$
$643$	30.9439i	1.22031i	−0.792283	+	0.610154i	$0.791107\pi$
$644$	− 9.09783i	− 0.358505i
$645$	1.58211i	0.0622953i
$646$	−0.774791	−0.0304837
$647$	37.1685	1.46125	0.730623	−	0.682781i	$-0.239230\pi$
$647$	37.1685	1.46125	0.730623	+	0.682781i	$0.239230\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−21.3424	−0.837763
$650$	0	0
$651$	10.7899	0.422888
$652$	− 8.11960i	− 0.317988i
$653$	−14.6655	−0.573904	−0.286952	−	0.957945i	$-0.592642\pi$
$653$	−14.6655	−0.573904	−0.286952	+	0.957945i	$0.592642\pi$
$654$	−18.9269	−0.740101
$655$	− 13.0532i	− 0.510032i
$656$	− 5.74094i	− 0.224146i
$657$	− 11.1588i	− 0.435348i
$658$	12.3448i	0.481251i
$659$	−39.5488	−1.54060	−0.770302	−	0.637679i	$-0.779895\pi$
$659$	−39.5488	−1.54060	−0.770302	+	0.637679i	$0.779895\pi$
$660$	−1.69202	−0.0658618
$661$	26.3937i	1.02660i	0.858210	+	0.513299i	$0.171577\pi$
$661$	26.3937i	1.02660i	−0.858210	+	0.513299i	$0.828423\pi$
$662$	−10.2107	−0.396851
$663$	0	0
$664$	−1.91185	−0.0741943
$665$	− 3.91185i	− 0.151695i
$666$	0.862937	0.0334381
$667$	−2.60388	−0.100822
$668$	0.362273i	0.0140168i
$669$	22.8049i	0.881689i
$670$	4.54288i	0.175507i
$671$	0.798954i	0.0308433i
$672$	2.24698	0.0866791
$673$	1.36121	0.0524707	0.0262354	−	0.999656i	$-0.491648\pi$
$673$	1.36121	0.0524707	0.0262354	+	0.999656i	$0.491648\pi$
$674$	21.6353i	0.833362i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−21.0301	−0.808254	−0.404127	−	0.914703i	$-0.632425\pi$
$677$	−21.0301	−0.808254	−0.404127	+	0.914703i	$0.632425\pi$
$678$	− 0.472189i	− 0.0181343i
$679$	−16.9705	−0.651266
$680$	−0.445042	−0.0170666
$681$	− 11.5483i	− 0.442530i
$682$	8.12498i	0.311122i
$683$	50.8635i	1.94624i	0.230301	+	0.973119i	$0.426029\pi$
$683$	50.8635i	1.94624i	−0.230301	+	0.973119i	$0.573971\pi$
$684$	1.74094i	0.0665665i
$685$	16.4306	0.627780
$686$	20.1129	0.767914
$687$	− 26.1129i	− 0.996269i
$688$	1.58211	0.0603172
$689$	0	0
$690$	4.04892	0.154140
$691$	12.9065i	0.490986i	0.969398	+	0.245493i	$0.0789498\pi$
$691$	12.9065i	0.490986i	−0.969398	+	0.245493i	$0.921050\pi$
$692$	−24.4480	−0.929374
$693$	−3.80194	−0.144424
$694$	− 24.1347i	− 0.916140i
$695$	− 5.39612i	− 0.204687i
$696$	− 0.643104i	− 0.0243768i
$697$	2.55496i	0.0967759i
$698$	14.0532	0.531923
$699$	10.1293	0.383125
$700$	− 2.24698i	− 0.0849278i
$701$	9.07739	0.342848	0.171424	−	0.985197i	$-0.445163\pi$
$701$	9.07739	0.342848	0.171424	+	0.985197i	$0.445163\pi$
$702$	0	0
$703$	−1.50232	−0.0566611
$704$	1.69202i	0.0637705i
$705$	−5.49396	−0.206914
$706$	−30.8810	−1.16222
$707$	20.2935i	0.763216i
$708$	12.6136i	0.474047i
$709$	37.1191i	1.39404i	0.717053	+	0.697019i	$0.245491\pi$
$709$	37.1191i	1.39404i	−0.717053	+	0.697019i	$0.754509\pi$
$710$	12.6407i	0.474397i
$711$	−6.57673	−0.246647
$712$	−11.9269	−0.446980
$713$	− 19.4426i	− 0.728133i
$714$	−1.00000	−0.0374241
$715$	0	0
$716$	3.81163	0.142447
$717$	16.3080i	0.609033i
$718$	−28.5827	−1.06670
$719$	19.5907	0.730611	0.365305	−	0.930888i	$-0.380965\pi$
$719$	19.5907	0.730611	0.365305	+	0.930888i	$0.380965\pi$
$720$	1.00000i	0.0372678i
$721$	− 13.0586i	− 0.486328i
$722$	15.9691i	0.594310i
$723$	2.20237i	0.0819072i
$724$	−15.7181	−0.584159
$725$	−0.643104	−0.0238843
$726$	8.13706i	0.301995i
$727$	−9.30021	−0.344926	−0.172463	−	0.985016i	$-0.555172\pi$
$727$	−9.30021	−0.344926	−0.172463	+	0.985016i	$0.555172\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	11.1588i	0.413007i
$731$	−0.704103	−0.0260422
$732$	0.472189	0.0174526
$733$	25.3159i	0.935063i	0.883976	+	0.467531i	$0.154857\pi$
$733$	25.3159i	0.935063i	−0.883976	+	0.467531i	$0.845143\pi$
$734$	− 5.83340i	− 0.215315i
$735$	1.95108i	0.0719668i
$736$	− 4.04892i	− 0.149245i
$737$	−7.68664	−0.283141
$738$	5.74094	0.211327
$739$	− 36.2398i	− 1.33310i	−0.745459	−	0.666551i	$-0.767770\pi$
$739$	− 36.2398i	− 1.33310i	0.745459	−	0.666551i	$-0.232230\pi$
$740$	−0.862937	−0.0317222
$741$	0	0
$742$	0.307979	0.0113062
$743$	31.8273i	1.16763i	0.811887	+	0.583815i	$0.198441\pi$
$743$	31.8273i	1.16763i	−0.811887	+	0.583815i	$0.801559\pi$
$744$	4.80194	0.176048
$745$	21.1444	0.774669
$746$	36.1172i	1.32235i
$747$	− 1.91185i	− 0.0699511i
$748$	− 0.753020i	− 0.0275332i
$749$	6.76271i	0.247104i
$750$	1.00000	0.0365148
$751$	52.3309	1.90958	0.954791	−	0.297277i	$-0.0960784\pi$
$751$	52.3309	1.90958	0.954791	+	0.297277i	$0.0960784\pi$
$752$	5.49396i	0.200344i
$753$	7.33513	0.267307
$754$	0	0
$755$	21.8213	0.794159
$756$	2.24698i	0.0817219i
$757$	2.72886	0.0991820	0.0495910	−	0.998770i	$-0.484208\pi$
$757$	2.72886	0.0991820	0.0495910	+	0.998770i	$0.484208\pi$
$758$	−18.7651	−0.681579
$759$	6.85086i	0.248670i
$760$	− 1.74094i	− 0.0631505i
$761$	8.26577i	0.299634i	0.988714	+	0.149817i	$0.0478684\pi$
$761$	8.26577i	0.299634i	−0.988714	+	0.149817i	$0.952132\pi$
$762$	− 10.8944i	− 0.394662i
$763$	42.5284	1.53963
$764$	18.7832	0.679551
$765$	− 0.445042i	− 0.0160905i
$766$	15.4709	0.558985
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 46.8447i	− 1.68926i	−0.535347	−	0.844632i	$-0.679819\pi$
$769$	− 46.8447i	− 1.68926i	0.535347	−	0.844632i	$-0.320181\pi$
$770$	3.80194	0.137012
$771$	−10.4523	−0.376432
$772$	− 1.99330i	− 0.0717403i
$773$	7.23085i	0.260076i	0.991509	+	0.130038i	$0.0415099\pi$
$773$	7.23085i	0.260076i	−0.991509	+	0.130038i	$0.958490\pi$
$774$	1.58211i	0.0568676i
$775$	− 4.80194i	− 0.172491i
$776$	−7.55257	−0.271121
$777$	−1.93900	−0.0695613
$778$	− 5.67994i	− 0.203636i
$779$	−9.99462	−0.358095
$780$	0	0
$781$	−21.3884	−0.765336
$782$	1.80194i	0.0644372i
$783$	0.643104	0.0229827
$784$	1.95108	0.0696815
$785$	− 6.28083i	− 0.224172i
$786$	− 13.0532i	− 0.465593i
$787$	4.73019i	0.168613i	0.996440	+	0.0843064i	$0.0268675\pi$
$787$	4.73019i	0.168613i	−0.996440	+	0.0843064i	$0.973133\pi$
$788$	8.26875i	0.294562i
$789$	13.0694	0.465282
$790$	6.57673	0.233989
$791$	1.06100i	0.0377248i
$792$	−1.69202	−0.0601234
$793$	0	0
$794$	19.2282	0.682383
$795$	0.137063i	0.00486114i
$796$	−15.7071	−0.556723
$797$	−4.79523	−0.169856	−0.0849279	−	0.996387i	$-0.527066\pi$
$797$	−4.79523	−0.169856	−0.0849279	+	0.996387i	$0.527066\pi$
$798$	− 3.91185i	− 0.138478i
$799$	− 2.44504i	− 0.0864994i
$800$	− 1.00000i	− 0.0353553i
$801$	− 11.9269i	− 0.421417i
$802$	−27.5593	−0.973152
$803$	−18.8810	−0.666296
$804$	4.54288i	0.160215i
$805$	−9.09783	−0.320657
$806$	0	0
$807$	0.119605	0.00421029
$808$	9.03146i	0.317726i
$809$	10.5171	0.369760	0.184880	−	0.982761i	$-0.440810\pi$
$809$	10.5171	0.369760	0.184880	+	0.982761i	$0.440810\pi$
$810$	−1.00000	−0.0351364
$811$	− 43.8383i	− 1.53937i	−0.638423	−	0.769685i	$-0.720413\pi$
$811$	− 43.8383i	− 1.53937i	0.638423	−	0.769685i	$-0.279587\pi$
$812$	1.44504i	0.0507110i
$813$	27.3545i	0.959364i
$814$	− 1.46011i	− 0.0511767i
$815$	−8.11960	−0.284417
$816$	−0.445042	−0.0155796
$817$	− 2.75435i	− 0.0963625i
$818$	−17.7681	−0.621247
$819$	0	0
$820$	−5.74094	−0.200482
$821$	− 48.2476i	− 1.68385i	−0.539593	−	0.841926i	$-0.681422\pi$
$821$	− 48.2476i	− 1.68385i	0.539593	−	0.841926i	$-0.318578\pi$
$822$	16.4306	0.573082
$823$	−51.1739	−1.78381	−0.891905	−	0.452223i	$-0.850631\pi$
$823$	−51.1739	−1.78381	−0.891905	+	0.452223i	$0.850631\pi$
$824$	− 5.81163i	− 0.202458i
$825$	1.69202i	0.0589086i
$826$	− 28.3424i	− 0.986159i
$827$	39.7864i	1.38351i	0.722133	+	0.691754i	$0.243162\pi$
$827$	39.7864i	1.38351i	−0.722133	+	0.691754i	$0.756838\pi$
$828$	4.04892	0.140710
$829$	38.3666	1.33253	0.666263	−	0.745717i	$-0.267893\pi$
$829$	38.3666	1.33253	0.666263	+	0.745717i	$0.267893\pi$
$830$	1.91185i	0.0663614i
$831$	3.14914	0.109243
$832$	0	0
$833$	−0.868313	−0.0300853
$834$	− 5.39612i	− 0.186853i
$835$	0.362273	0.0125370
$836$	2.94571	0.101879
$837$	4.80194i	0.165979i
$838$	22.4426i	0.775268i
$839$	− 35.8745i	− 1.23853i	−0.785183	−	0.619263i	$-0.787431\pi$
$839$	− 35.8745i	− 1.23853i	0.785183	−	0.619263i	$-0.212569\pi$
$840$	− 2.24698i	− 0.0775282i
$841$	−28.5864	−0.985739
$842$	1.24698	0.0429738
$843$	− 16.0683i	− 0.553421i
$844$	−17.2814	−0.594851
$845$	0	0
$846$	−5.49396	−0.188886
$847$	− 18.2838i	− 0.628239i
$848$	0.137063	0.00470677
$849$	−15.8834	−0.545116
$850$	0.445042i	0.0152648i
$851$	3.49396i	0.119771i
$852$	12.6407i	0.433064i
$853$	24.8829i	0.851974i	0.904729	+	0.425987i	$0.140073\pi$
$853$	24.8829i	0.851974i	−0.904729	+	0.425987i	$0.859927\pi$
$854$	−1.06100	−0.0363066
$855$	1.74094	0.0595389
$856$	3.00969i	0.102869i
$857$	−13.8407	−0.472789	−0.236395	−	0.971657i	$-0.575966\pi$
$857$	−13.8407	−0.472789	−0.236395	+	0.971657i	$0.575966\pi$
$858$	0	0
$859$	9.39314	0.320490	0.160245	−	0.987077i	$-0.448772\pi$
$859$	9.39314	0.320490	0.160245	+	0.987077i	$0.448772\pi$
$860$	− 1.58211i	− 0.0539493i
$861$	−12.8998	−0.439623
$862$	5.97823	0.203619
$863$	1.29888i	0.0442144i	0.999756	+	0.0221072i	$0.00703752\pi$
$863$	1.29888i	0.0442144i	−0.999756	+	0.0221072i	$0.992962\pi$
$864$	1.00000i	0.0340207i
$865$	24.4480i	0.831258i
$866$	14.2717i	0.484973i
$867$	−16.8019	−0.570624
$868$	−10.7899	−0.366232
$869$	11.1280i	0.377490i
$870$	−0.643104	−0.0218033
$871$	0	0
$872$	18.9269	0.640946
$873$	− 7.55257i	− 0.255616i
$874$	−7.04892	−0.238433
$875$	−2.24698	−0.0759618
$876$	11.1588i	0.377022i
$877$	12.0476i	0.406818i	0.979094	+	0.203409i	$0.0652021\pi$
$877$	12.0476i	0.406818i	−0.979094	+	0.203409i	$0.934798\pi$
$878$	12.2349i	0.412908i
$879$	15.6039i	0.526306i
$880$	1.69202	0.0570380
$881$	−3.24591	−0.109358	−0.0546788	−	0.998504i	$-0.517413\pi$
$881$	−3.24591	−0.109358	−0.0546788	+	0.998504i	$0.517413\pi$
$882$	1.95108i	0.0656964i
$883$	−10.2494	−0.344919	−0.172459	−	0.985017i	$-0.555171\pi$
$883$	−10.2494	−0.344919	−0.172459	+	0.985017i	$0.555171\pi$
$884$	0	0
$885$	12.6136	0.424000
$886$	− 19.4631i	− 0.653875i
$887$	−46.6413	−1.56606	−0.783031	−	0.621983i	$-0.786327\pi$
$887$	−46.6413	−1.56606	−0.783031	+	0.621983i	$0.786327\pi$
$888$	−0.862937	−0.0289583
$889$	24.4795i	0.821016i
$890$	11.9269i	0.399791i
$891$	− 1.69202i	− 0.0566849i
$892$	− 22.8049i	− 0.763565i
$893$	9.56465	0.320069
$894$	21.1444	0.707173
$895$	− 3.81163i	− 0.127409i
$896$	−2.24698	−0.0750663
$897$	0	0
$898$	10.4222	0.347794
$899$	3.08815i	0.102995i
$900$	1.00000	0.0333333
$901$	−0.0609989	−0.00203217
$902$	− 9.71379i	− 0.323434i
$903$	− 3.55496i	− 0.118302i
$904$	0.472189i	0.0157048i
$905$	15.7181i	0.522487i
$906$	21.8213	0.724965
$907$	30.9734	1.02846	0.514228	−	0.857653i	$-0.328078\pi$
$907$	30.9734	1.02846	0.514228	+	0.857653i	$0.328078\pi$
$908$	11.5483i	0.383242i
$909$	−9.03146	−0.299555
$910$	0	0
$911$	3.96854	0.131484	0.0657418	−	0.997837i	$-0.479059\pi$
$911$	3.96854	0.131484	0.0657418	+	0.997837i	$0.479059\pi$
$912$	− 1.74094i	− 0.0576482i
$913$	−3.23490	−0.107059
$914$	12.9148	0.427185
$915$	− 0.472189i	− 0.0156101i
$916$	26.1129i	0.862795i
$917$	29.3303i	0.968573i
$918$	− 0.445042i	− 0.0146886i
$919$	14.3907	0.474707	0.237353	−	0.971423i	$-0.423720\pi$
$919$	14.3907	0.474707	0.237353	+	0.971423i	$0.423720\pi$
$920$	−4.04892	−0.133489
$921$	− 18.3002i	− 0.603012i
$922$	27.7409	0.913599
$923$	0	0
$924$	3.80194	0.125075
$925$	0.862937i	0.0283732i
$926$	−31.4795	−1.03448
$927$	5.81163	0.190879
$928$	0.643104i	0.0211109i
$929$	13.6082i	0.446470i	0.974765	+	0.223235i	$0.0716618\pi$
$929$	13.6082i	0.446470i	−0.974765	+	0.223235i	$0.928338\pi$
$930$	− 4.80194i	− 0.157462i
$931$	− 3.39672i	− 0.111323i
$932$	−10.1293	−0.331796
$933$	−14.7899	−0.484198
$934$	− 27.2097i	− 0.890328i
$935$	−0.753020	−0.0246264
$936$	0	0
$937$	48.6781	1.59025	0.795123	−	0.606449i	$-0.207406\pi$
$937$	48.6781	1.59025	0.795123	+	0.606449i	$0.207406\pi$
$938$	− 10.2078i	− 0.333295i
$939$	4.05323	0.132272
$940$	5.49396	0.179193
$941$	8.94092i	0.291466i	0.989324	+	0.145733i	$0.0465540\pi$
$941$	8.94092i	0.291466i	−0.989324	+	0.145733i	$0.953446\pi$
$942$	− 6.28083i	− 0.204641i
$943$	23.2446i	0.756948i
$944$	− 12.6136i	− 0.410537i
$945$	2.24698	0.0730943
$946$	2.67696	0.0870353
$947$	− 15.2107i	− 0.494282i	−0.968979	−	0.247141i	$-0.920509\pi$
$947$	− 15.2107i	− 0.494282i	0.968979	−	0.247141i	$-0.0794912\pi$
$948$	6.57673	0.213602
$949$	0	0
$950$	−1.74094	−0.0564835
$951$	− 9.52781i	− 0.308960i
$952$	1.00000	0.0324102
$953$	59.0283	1.91212	0.956058	−	0.293179i	$-0.0947132\pi$
$953$	59.0283	1.91212	0.956058	+	0.293179i	$0.0947132\pi$
$954$	0.137063i	0.00443759i
$955$	− 18.7832i	− 0.607809i
$956$	− 16.3080i	− 0.527438i
$957$	− 1.08815i	− 0.0351748i
$958$	20.2728	0.654984
$959$	−36.9191	−1.19218
$960$	− 1.00000i	− 0.0322749i
$961$	7.94139	0.256174
$962$	0	0
$963$	−3.00969	−0.0969859
$964$	− 2.20237i	− 0.0709337i
$965$	−1.99330	−0.0641664
$966$	−9.09783	−0.292718
$967$	7.78986i	0.250505i	0.992125	+	0.125252i	$0.0399741\pi$
$967$	7.78986i	0.250505i	−0.992125	+	0.125252i	$0.960026\pi$
$968$	− 8.13706i	− 0.261535i
$969$	0.774791i	0.0248899i
$970$	7.55257i	0.242498i
$971$	−39.3521	−1.26287	−0.631435	−	0.775429i	$-0.717534\pi$
$971$	−39.3521	−1.26287	−0.631435	+	0.775429i	$0.717534\pi$
$972$	−1.00000	−0.0320750
$973$	12.1250i	0.388709i
$974$	22.3424	0.715897
$975$	0	0
$976$	−0.472189	−0.0151144
$977$	− 49.7154i	− 1.59054i	−0.606257	−	0.795269i	$-0.707330\pi$
$977$	− 49.7154i	− 1.59054i	0.606257	−	0.795269i	$-0.292670\pi$
$978$	−8.11960	−0.259636
$979$	−20.1806	−0.644975
$980$	− 1.95108i	− 0.0623250i
$981$	18.9269i	0.604290i
$982$	− 1.57971i	− 0.0504106i
$983$	56.5889i	1.80491i	0.430788	+	0.902453i	$0.358236\pi$
$983$	56.5889i	1.80491i	−0.430788	+	0.902453i	$0.641764\pi$
$984$	−5.74094	−0.183014
$985$	8.26875	0.263464
$986$	− 0.286208i	− 0.00911473i
$987$	12.3448	0.392940
$988$	0	0
$989$	−6.40581	−0.203693
$990$	1.69202i	0.0537760i
$991$	38.5153	1.22348	0.611739	−	0.791060i	$-0.290470\pi$
$991$	38.5153	1.22348	0.611739	+	0.791060i	$0.290470\pi$
$992$	−4.80194	−0.152462
$993$	10.2107i	0.324028i
$994$	− 28.4034i	− 0.900902i
$995$	15.7071i	0.497948i
$996$	1.91185i	0.0605794i
$997$	39.1457	1.23976	0.619878	−	0.784698i	$-0.287182\pi$
$997$	39.1457	1.23976	0.619878	+	0.784698i	$0.287182\pi$
$998$	5.76941	0.182628
$999$	− 0.862937i	− 0.0273021i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.x.1351.4		6
13.5	odd	4		5070.2.a.by.1.3	yes	3
13.8	odd	4		5070.2.a.bn.1.1	✓	3
13.12	even	2	inner	5070.2.b.x.1351.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bn.1.1	✓	3	13.8	odd	4
5070.2.a.by.1.3	yes	3	13.5	odd	4
5070.2.b.x.1351.3		6	13.12	even	2	inner
5070.2.b.x.1351.4		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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -2.24698i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -2.24698i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.69202i q^{11} -1.00000 q^{12} +2.24698 q^{14} +1.00000i q^{15} +1.00000 q^{16} -0.445042 q^{17} +1.00000i q^{18} -1.74094i q^{19} -1.00000i q^{20} -2.24698i q^{21} +1.69202 q^{22} -4.04892 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +2.24698i q^{28} +0.643104 q^{29} -1.00000 q^{30} +4.80194i q^{31} +1.00000i q^{32} -1.69202i q^{33} -0.445042i q^{34} +2.24698 q^{35} -1.00000 q^{36} -0.862937i q^{37} +1.74094 q^{38} +1.00000 q^{40} -5.74094i q^{41} +2.24698 q^{42} +1.58211 q^{43} +1.69202i q^{44} +1.00000i q^{45} -4.04892i q^{46} +5.49396i q^{47} +1.00000 q^{48} +1.95108 q^{49} -1.00000i q^{50} -0.445042 q^{51} +0.137063 q^{53} +1.00000i q^{54} +1.69202 q^{55} -2.24698 q^{56} -1.74094i q^{57} +0.643104i q^{58} -12.6136i q^{59} -1.00000i q^{60} -0.472189 q^{61} -4.80194 q^{62} -2.24698i q^{63} -1.00000 q^{64} +1.69202 q^{66} -4.54288i q^{67} +0.445042 q^{68} -4.04892 q^{69} +2.24698i q^{70} -12.6407i q^{71} -1.00000i q^{72} -11.1588i q^{73} +0.862937 q^{74} -1.00000 q^{75} +1.74094i q^{76} -3.80194 q^{77} -6.57673 q^{79} +1.00000i q^{80} +1.00000 q^{81} +5.74094 q^{82} -1.91185i q^{83} +2.24698i q^{84} -0.445042i q^{85} +1.58211i q^{86} +0.643104 q^{87} -1.69202 q^{88} -11.9269i q^{89} -1.00000 q^{90} +4.04892 q^{92} +4.80194i q^{93} -5.49396 q^{94} +1.74094 q^{95} +1.00000i q^{96} -7.55257i q^{97} +1.95108i q^{98} -1.69202i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{10} - 6 q^{12} + 4 q^{14} + 6 q^{16} - 2 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} + 12 q^{29} - 6 q^{30} + 4 q^{35} - 6 q^{36} - 18 q^{38} + 6 q^{40} + 4 q^{42} - 2 q^{43} + 6 q^{48} + 30 q^{49} - 2 q^{51} - 10 q^{53} - 4 q^{56} + 10 q^{61} - 20 q^{62} - 6 q^{64} + 2 q^{68} - 6 q^{69} + 16 q^{74} - 6 q^{75} - 14 q^{77} - 34 q^{79} + 6 q^{81} + 6 q^{82} + 12 q^{87} - 6 q^{90} + 6 q^{92} - 14 q^{94} - 18 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 - 6 * q^12 + 4 * q^14 + 6 * q^16 - 2 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 + 12 * q^29 - 6 * q^30 + 4 * q^35 - 6 * q^36 - 18 * q^38 + 6 * q^40 + 4 * q^42 - 2 * q^43 + 6 * q^48 + 30 * q^49 - 2 * q^51 - 10 * q^53 - 4 * q^56 + 10 * q^61 - 20 * q^62 - 6 * q^64 + 2 * q^68 - 6 * q^69 + 16 * q^74 - 6 * q^75 - 14 * q^77 - 34 * q^79 + 6 * q^81 + 6 * q^82 + 12 * q^87 - 6 * q^90 + 6 * q^92 - 14 * q^94 - 18 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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