LMFDB - Embedded newform 8470.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.a (trivial)

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:	yes
Analytic conductor:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +1.00000 q^{10} -0.732051 q^{12} +4.00000 q^{13} -1.00000 q^{14} +0.732051 q^{15} +1.00000 q^{16} +7.46410 q^{17} +2.46410 q^{18} +2.73205 q^{19} -1.00000 q^{20} -0.732051 q^{21} -2.19615 q^{23} +0.732051 q^{24} +1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} +9.66025 q^{29} -0.732051 q^{30} +7.46410 q^{31} -1.00000 q^{32} -7.46410 q^{34} -1.00000 q^{35} -2.46410 q^{36} +0.732051 q^{37} -2.73205 q^{38} -2.92820 q^{39} +1.00000 q^{40} -8.19615 q^{41} +0.732051 q^{42} +8.92820 q^{43} +2.46410 q^{45} +2.19615 q^{46} +6.00000 q^{47} -0.732051 q^{48} +1.00000 q^{49} -1.00000 q^{50} -5.46410 q^{51} +4.00000 q^{52} +8.73205 q^{53} -4.00000 q^{54} -1.00000 q^{56} -2.00000 q^{57} -9.66025 q^{58} -12.3923 q^{59} +0.732051 q^{60} -8.00000 q^{61} -7.46410 q^{62} -2.46410 q^{63} +1.00000 q^{64} -4.00000 q^{65} -11.8564 q^{67} +7.46410 q^{68} +1.60770 q^{69} +1.00000 q^{70} +14.9282 q^{71} +2.46410 q^{72} -6.00000 q^{73} -0.732051 q^{74} -0.732051 q^{75} +2.73205 q^{76} +2.92820 q^{78} -9.66025 q^{79} -1.00000 q^{80} +4.46410 q^{81} +8.19615 q^{82} +1.07180 q^{83} -0.732051 q^{84} -7.46410 q^{85} -8.92820 q^{86} -7.07180 q^{87} +4.92820 q^{89} -2.46410 q^{90} +4.00000 q^{91} -2.19615 q^{92} -5.46410 q^{93} -6.00000 q^{94} -2.73205 q^{95} +0.732051 q^{96} -0.732051 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 8 q^{13} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 8 q^{17} - 2 q^{18} + 2 q^{19} - 2 q^{20} + 2 q^{21} + 6 q^{23} - 2 q^{24} + 2 q^{25} - 8 q^{26} + 8 q^{27} + 2 q^{28} + 2 q^{29} + 2 q^{30} + 8 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{35} + 2 q^{36} - 2 q^{37} - 2 q^{38} + 8 q^{39} + 2 q^{40} - 6 q^{41} - 2 q^{42} + 4 q^{43} - 2 q^{45} - 6 q^{46} + 12 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} - 4 q^{51} + 8 q^{52} + 14 q^{53} - 8 q^{54} - 2 q^{56} - 4 q^{57} - 2 q^{58} - 4 q^{59} - 2 q^{60} - 16 q^{61} - 8 q^{62} + 2 q^{63} + 2 q^{64} - 8 q^{65} + 4 q^{67} + 8 q^{68} + 24 q^{69} + 2 q^{70} + 16 q^{71} - 2 q^{72} - 12 q^{73} + 2 q^{74} + 2 q^{75} + 2 q^{76} - 8 q^{78} - 2 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} + 16 q^{83} + 2 q^{84} - 8 q^{85} - 4 q^{86} - 28 q^{87} - 4 q^{89} + 2 q^{90} + 8 q^{91} + 6 q^{92} - 4 q^{93} - 12 q^{94} - 2 q^{95} - 2 q^{96} + 2 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 8 * q^13 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 8 * q^17 - 2 * q^18 + 2 * q^19 - 2 * q^20 + 2 * q^21 + 6 * q^23 - 2 * q^24 + 2 * q^25 - 8 * q^26 + 8 * q^27 + 2 * q^28 + 2 * q^29 + 2 * q^30 + 8 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^35 + 2 * q^36 - 2 * q^37 - 2 * q^38 + 8 * q^39 + 2 * q^40 - 6 * q^41 - 2 * q^42 + 4 * q^43 - 2 * q^45 - 6 * q^46 + 12 * q^47 + 2 * q^48 + 2 * q^49 - 2 * q^50 - 4 * q^51 + 8 * q^52 + 14 * q^53 - 8 * q^54 - 2 * q^56 - 4 * q^57 - 2 * q^58 - 4 * q^59 - 2 * q^60 - 16 * q^61 - 8 * q^62 + 2 * q^63 + 2 * q^64 - 8 * q^65 + 4 * q^67 + 8 * q^68 + 24 * q^69 + 2 * q^70 + 16 * q^71 - 2 * q^72 - 12 * q^73 + 2 * q^74 + 2 * q^75 + 2 * q^76 - 8 * q^78 - 2 * q^79 - 2 * q^80 + 2 * q^81 + 6 * q^82 + 16 * q^83 + 2 * q^84 - 8 * q^85 - 4 * q^86 - 28 * q^87 - 4 * q^89 + 2 * q^90 + 8 * q^91 + 6 * q^92 - 4 * q^93 - 12 * q^94 - 2 * q^95 - 2 * q^96 + 2 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.732051	0.298858
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−2.46410	−0.821367
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−0.732051	−0.211325
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	−1.00000	−0.267261
$15$	0.732051	0.189015
$16$	1.00000	0.250000
$17$	7.46410	1.81031	0.905155	−	0.425081i	$-0.139754\pi$
$17$	7.46410	1.81031	0.905155	+	0.425081i	$0.139754\pi$
$18$	2.46410	0.580794
$19$	2.73205	0.626775	0.313388	−	0.949625i	$-0.398536\pi$
$19$	2.73205	0.626775	0.313388	+	0.949625i	$0.398536\pi$
$20$	−1.00000	−0.223607
$21$	−0.732051	−0.159747
$22$	0	0
$23$	−2.19615	−0.457929	−0.228965	−	0.973435i	$-0.573534\pi$
$23$	−2.19615	−0.457929	−0.228965	+	0.973435i	$0.573534\pi$
$24$	0.732051	0.149429
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	4.00000	0.769800
$28$	1.00000	0.188982
$29$	9.66025	1.79386	0.896932	−	0.442168i	$-0.145791\pi$
$29$	9.66025	1.79386	0.896932	+	0.442168i	$0.145791\pi$
$30$	−0.732051	−0.133654
$31$	7.46410	1.34059	0.670296	−	0.742094i	$-0.266167\pi$
$31$	7.46410	1.34059	0.670296	+	0.742094i	$0.266167\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−7.46410	−1.28008
$35$	−1.00000	−0.169031
$36$	−2.46410	−0.410684
$37$	0.732051	0.120348	0.0601742	−	0.998188i	$-0.480834\pi$
$37$	0.732051	0.120348	0.0601742	+	0.998188i	$0.480834\pi$
$38$	−2.73205	−0.443197
$39$	−2.92820	−0.468888
$40$	1.00000	0.158114
$41$	−8.19615	−1.28002	−0.640012	−	0.768365i	$-0.721071\pi$
$41$	−8.19615	−1.28002	−0.640012	+	0.768365i	$0.721071\pi$
$42$	0.732051	0.112958
$43$	8.92820	1.36154	0.680769	−	0.732498i	$-0.261646\pi$
$43$	8.92820	1.36154	0.680769	+	0.732498i	$0.261646\pi$
$44$	0	0
$45$	2.46410	0.367327
$46$	2.19615	0.323805
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−0.732051	−0.105662
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−5.46410	−0.765127
$52$	4.00000	0.554700
$53$	8.73205	1.19944	0.599720	−	0.800210i	$-0.295279\pi$
$53$	8.73205	1.19944	0.599720	+	0.800210i	$0.295279\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−2.00000	−0.264906
$58$	−9.66025	−1.26845
$59$	−12.3923	−1.61334	−0.806670	−	0.591002i	$-0.798733\pi$
$59$	−12.3923	−1.61334	−0.806670	+	0.591002i	$0.798733\pi$
$60$	0.732051	0.0945074
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−7.46410	−0.947942
$63$	−2.46410	−0.310448
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−11.8564	−1.44849	−0.724245	−	0.689542i	$-0.757812\pi$
$67$	−11.8564	−1.44849	−0.724245	+	0.689542i	$0.757812\pi$
$68$	7.46410	0.905155
$69$	1.60770	0.193544
$70$	1.00000	0.119523
$71$	14.9282	1.77165	0.885826	−	0.464018i	$-0.153593\pi$
$71$	14.9282	1.77165	0.885826	+	0.464018i	$0.153593\pi$
$72$	2.46410	0.290397
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−0.732051	−0.0850992
$75$	−0.732051	−0.0845299
$76$	2.73205	0.313388
$77$	0	0
$78$	2.92820	0.331554
$79$	−9.66025	−1.08686	−0.543432	−	0.839453i	$-0.682875\pi$
$79$	−9.66025	−1.08686	−0.543432	+	0.839453i	$0.682875\pi$
$80$	−1.00000	−0.111803
$81$	4.46410	0.496011
$82$	8.19615	0.905114
$83$	1.07180	0.117645	0.0588225	−	0.998268i	$-0.481265\pi$
$83$	1.07180	0.117645	0.0588225	+	0.998268i	$0.481265\pi$
$84$	−0.732051	−0.0798733
$85$	−7.46410	−0.809595
$86$	−8.92820	−0.962753
$87$	−7.07180	−0.758176
$88$	0	0
$89$	4.92820	0.522388	0.261194	−	0.965286i	$-0.415884\pi$
$89$	4.92820	0.522388	0.261194	+	0.965286i	$0.415884\pi$
$90$	−2.46410	−0.259739
$91$	4.00000	0.419314
$92$	−2.19615	−0.228965
$93$	−5.46410	−0.566601
$94$	−6.00000	−0.618853
$95$	−2.73205	−0.280302
$96$	0.732051	0.0747146
$97$	−0.732051	−0.0743285	−0.0371642	−	0.999309i	$-0.511832\pi$
$97$	−0.732051	−0.0743285	−0.0371642	+	0.999309i	$0.511832\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	5.46410	0.541027
$103$	−18.3923	−1.81225	−0.906124	−	0.423013i	$-0.860973\pi$
$103$	−18.3923	−1.81225	−0.906124	+	0.423013i	$0.860973\pi$
$104$	−4.00000	−0.392232
$105$	0.732051	0.0714408
$106$	−8.73205	−0.848132
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	4.00000	0.384900
$109$	−13.2679	−1.27084	−0.635420	−	0.772167i	$-0.719173\pi$
$109$	−13.2679	−1.27084	−0.635420	+	0.772167i	$0.719173\pi$
$110$	0	0
$111$	−0.535898	−0.0508652
$112$	1.00000	0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	2.00000	0.187317
$115$	2.19615	0.204792
$116$	9.66025	0.896932
$117$	−9.85641	−0.911225
$118$	12.3923	1.14080
$119$	7.46410	0.684233
$120$	−0.732051	−0.0668268
$121$	0	0
$122$	8.00000	0.724286
$123$	6.00000	0.541002
$124$	7.46410	0.670296
$125$	−1.00000	−0.0894427
$126$	2.46410	0.219520
$127$	−12.3923	−1.09964	−0.549820	−	0.835283i	$-0.685304\pi$
$127$	−12.3923	−1.09964	−0.549820	+	0.835283i	$0.685304\pi$
$128$	−1.00000	−0.0883883
$129$	−6.53590	−0.575454
$130$	4.00000	0.350823
$131$	−0.196152	−0.0171379	−0.00856896	−	0.999963i	$-0.502728\pi$
$131$	−0.196152	−0.0171379	−0.00856896	+	0.999963i	$0.502728\pi$
$132$	0	0
$133$	2.73205	0.236899
$134$	11.8564	1.02424
$135$	−4.00000	−0.344265
$136$	−7.46410	−0.640041
$137$	19.8564	1.69645	0.848224	−	0.529638i	$-0.177672\pi$
$137$	19.8564	1.69645	0.848224	+	0.529638i	$0.177672\pi$
$138$	−1.60770	−0.136856
$139$	13.6603	1.15865	0.579324	−	0.815097i	$-0.303317\pi$
$139$	13.6603	1.15865	0.579324	+	0.815097i	$0.303317\pi$
$140$	−1.00000	−0.0845154
$141$	−4.39230	−0.369899
$142$	−14.9282	−1.25275
$143$	0	0
$144$	−2.46410	−0.205342
$145$	−9.66025	−0.802240
$146$	6.00000	0.496564
$147$	−0.732051	−0.0603785
$148$	0.732051	0.0601742
$149$	12.1962	0.999148	0.499574	−	0.866271i	$-0.333490\pi$
$149$	12.1962	0.999148	0.499574	+	0.866271i	$0.333490\pi$
$150$	0.732051	0.0597717
$151$	−12.1962	−0.992509	−0.496254	−	0.868177i	$-0.665292\pi$
$151$	−12.1962	−0.992509	−0.496254	+	0.868177i	$0.665292\pi$
$152$	−2.73205	−0.221599
$153$	−18.3923	−1.48693
$154$	0	0
$155$	−7.46410	−0.599531
$156$	−2.92820	−0.234444
$157$	−23.4641	−1.87264	−0.936320	−	0.351149i	$-0.885791\pi$
$157$	−23.4641	−1.87264	−0.936320	+	0.351149i	$0.885791\pi$
$158$	9.66025	0.768529
$159$	−6.39230	−0.506943
$160$	1.00000	0.0790569
$161$	−2.19615	−0.173081
$162$	−4.46410	−0.350733
$163$	−0.928203	−0.0727025	−0.0363512	−	0.999339i	$-0.511574\pi$
$163$	−0.928203	−0.0727025	−0.0363512	+	0.999339i	$0.511574\pi$
$164$	−8.19615	−0.640012
$165$	0	0
$166$	−1.07180	−0.0831876
$167$	18.9282	1.46471	0.732354	−	0.680924i	$-0.238422\pi$
$167$	18.9282	1.46471	0.732354	+	0.680924i	$0.238422\pi$
$168$	0.732051	0.0564789
$169$	3.00000	0.230769
$170$	7.46410	0.572470
$171$	−6.73205	−0.514813
$172$	8.92820	0.680769
$173$	−8.92820	−0.678799	−0.339399	−	0.940642i	$-0.610224\pi$
$173$	−8.92820	−0.678799	−0.339399	+	0.940642i	$0.610224\pi$
$174$	7.07180	0.536112
$175$	1.00000	0.0755929
$176$	0	0
$177$	9.07180	0.681878
$178$	−4.92820	−0.369384
$179$	−14.3923	−1.07573	−0.537866	−	0.843031i	$-0.680769\pi$
$179$	−14.3923	−1.07573	−0.537866	+	0.843031i	$0.680769\pi$
$180$	2.46410	0.183663
$181$	8.92820	0.663628	0.331814	−	0.943345i	$-0.392339\pi$
$181$	8.92820	0.663628	0.331814	+	0.943345i	$0.392339\pi$
$182$	−4.00000	−0.296500
$183$	5.85641	0.432918
$184$	2.19615	0.161903
$185$	−0.732051	−0.0538214
$186$	5.46410	0.400647
$187$	0	0
$188$	6.00000	0.437595
$189$	4.00000	0.290957
$190$	2.73205	0.198204
$191$	0.392305	0.0283862	0.0141931	−	0.999899i	$-0.495482\pi$
$191$	0.392305	0.0283862	0.0141931	+	0.999899i	$0.495482\pi$
$192$	−0.732051	−0.0528312
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0.732051	0.0525582
$195$	2.92820	0.209693
$196$	1.00000	0.0714286
$197$	3.46410	0.246807	0.123404	−	0.992357i	$-0.460619\pi$
$197$	3.46410	0.246807	0.123404	+	0.992357i	$0.460619\pi$
$198$	0	0
$199$	25.8564	1.83291	0.916456	−	0.400135i	$-0.131037\pi$
$199$	25.8564	1.83291	0.916456	+	0.400135i	$0.131037\pi$
$200$	−1.00000	−0.0707107
$201$	8.67949	0.612204
$202$	−8.00000	−0.562878
$203$	9.66025	0.678017
$204$	−5.46410	−0.382564
$205$	8.19615	0.572444
$206$	18.3923	1.28145
$207$	5.41154	0.376128
$208$	4.00000	0.277350
$209$	0	0
$210$	−0.732051	−0.0505163
$211$	13.8564	0.953914	0.476957	−	0.878927i	$-0.341740\pi$
$211$	13.8564	0.953914	0.476957	+	0.878927i	$0.341740\pi$
$212$	8.73205	0.599720
$213$	−10.9282	−0.748788
$214$	2.00000	0.136717
$215$	−8.92820	−0.608898
$216$	−4.00000	−0.272166
$217$	7.46410	0.506696
$218$	13.2679	0.898619
$219$	4.39230	0.296804
$220$	0	0
$221$	29.8564	2.00836
$222$	0.535898	0.0359671
$223$	−4.53590	−0.303746	−0.151873	−	0.988400i	$-0.548531\pi$
$223$	−4.53590	−0.303746	−0.151873	+	0.988400i	$0.548531\pi$
$224$	−1.00000	−0.0668153
$225$	−2.46410	−0.164273
$226$	−18.0000	−1.19734
$227$	11.3205	0.751369	0.375684	−	0.926748i	$-0.377408\pi$
$227$	11.3205	0.751369	0.375684	+	0.926748i	$0.377408\pi$
$228$	−2.00000	−0.132453
$229$	−11.3205	−0.748080	−0.374040	−	0.927413i	$-0.622028\pi$
$229$	−11.3205	−0.748080	−0.374040	+	0.927413i	$0.622028\pi$
$230$	−2.19615	−0.144810
$231$	0	0
$232$	−9.66025	−0.634227
$233$	−3.85641	−0.252642	−0.126321	−	0.991989i	$-0.540317\pi$
$233$	−3.85641	−0.252642	−0.126321	+	0.991989i	$0.540317\pi$
$234$	9.85641	0.644333
$235$	−6.00000	−0.391397
$236$	−12.3923	−0.806670
$237$	7.07180	0.459363
$238$	−7.46410	−0.483826
$239$	−18.0526	−1.16772	−0.583861	−	0.811853i	$-0.698459\pi$
$239$	−18.0526	−1.16772	−0.583861	+	0.811853i	$0.698459\pi$
$240$	0.732051	0.0472537
$241$	28.9808	1.86681	0.933407	−	0.358818i	$-0.116820\pi$
$241$	28.9808	1.86681	0.933407	+	0.358818i	$0.116820\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	−8.00000	−0.512148
$245$	−1.00000	−0.0638877
$246$	−6.00000	−0.382546
$247$	10.9282	0.695345
$248$	−7.46410	−0.473971
$249$	−0.784610	−0.0497226
$250$	1.00000	0.0632456
$251$	16.7846	1.05944	0.529718	−	0.848174i	$-0.322298\pi$
$251$	16.7846	1.05944	0.529718	+	0.848174i	$0.322298\pi$
$252$	−2.46410	−0.155224
$253$	0	0
$254$	12.3923	0.777562
$255$	5.46410	0.342175
$256$	1.00000	0.0625000
$257$	−18.1962	−1.13504	−0.567522	−	0.823358i	$-0.692098\pi$
$257$	−18.1962	−1.13504	−0.567522	+	0.823358i	$0.692098\pi$
$258$	6.53590	0.406907
$259$	0.732051	0.0454874
$260$	−4.00000	−0.248069
$261$	−23.8038	−1.47342
$262$	0.196152	0.0121183
$263$	7.60770	0.469111	0.234555	−	0.972103i	$-0.424637\pi$
$263$	7.60770	0.469111	0.234555	+	0.972103i	$0.424637\pi$
$264$	0	0
$265$	−8.73205	−0.536406
$266$	−2.73205	−0.167513
$267$	−3.60770	−0.220787
$268$	−11.8564	−0.724245
$269$	16.3923	0.999456	0.499728	−	0.866182i	$-0.333433\pi$
$269$	16.3923	0.999456	0.499728	+	0.866182i	$0.333433\pi$
$270$	4.00000	0.243432
$271$	11.6077	0.705117	0.352559	−	0.935790i	$-0.385312\pi$
$271$	11.6077	0.705117	0.352559	+	0.935790i	$0.385312\pi$
$272$	7.46410	0.452578
$273$	−2.92820	−0.177223
$274$	−19.8564	−1.19957
$275$	0	0
$276$	1.60770	0.0967719
$277$	17.3205	1.04069	0.520344	−	0.853957i	$-0.325804\pi$
$277$	17.3205	1.04069	0.520344	+	0.853957i	$0.325804\pi$
$278$	−13.6603	−0.819288
$279$	−18.3923	−1.10112
$280$	1.00000	0.0597614
$281$	11.3205	0.675325	0.337662	−	0.941267i	$-0.390364\pi$
$281$	11.3205	0.675325	0.337662	+	0.941267i	$0.390364\pi$
$282$	4.39230	0.261558
$283$	−2.53590	−0.150744	−0.0753718	−	0.997156i	$-0.524014\pi$
$283$	−2.53590	−0.150744	−0.0753718	+	0.997156i	$0.524014\pi$
$284$	14.9282	0.885826
$285$	2.00000	0.118470
$286$	0	0
$287$	−8.19615	−0.483804
$288$	2.46410	0.145199
$289$	38.7128	2.27722
$290$	9.66025	0.567270
$291$	0.535898	0.0314149
$292$	−6.00000	−0.351123
$293$	−27.7128	−1.61900	−0.809500	−	0.587120i	$-0.800262\pi$
$293$	−27.7128	−1.61900	−0.809500	+	0.587120i	$0.800262\pi$
$294$	0.732051	0.0426941
$295$	12.3923	0.721508
$296$	−0.732051	−0.0425496
$297$	0	0
$298$	−12.1962	−0.706504
$299$	−8.78461	−0.508027
$300$	−0.732051	−0.0422650
$301$	8.92820	0.514613
$302$	12.1962	0.701810
$303$	−5.85641	−0.336442
$304$	2.73205	0.156694
$305$	8.00000	0.458079
$306$	18.3923	1.05142
$307$	6.53590	0.373023	0.186512	−	0.982453i	$-0.440282\pi$
$307$	6.53590	0.373023	0.186512	+	0.982453i	$0.440282\pi$
$308$	0	0
$309$	13.4641	0.765946
$310$	7.46410	0.423932
$311$	−17.3205	−0.982156	−0.491078	−	0.871116i	$-0.663397\pi$
$311$	−17.3205	−0.982156	−0.491078	+	0.871116i	$0.663397\pi$
$312$	2.92820	0.165777
$313$	0.732051	0.0413780	0.0206890	−	0.999786i	$-0.493414\pi$
$313$	0.732051	0.0413780	0.0206890	+	0.999786i	$0.493414\pi$
$314$	23.4641	1.32416
$315$	2.46410	0.138836
$316$	−9.66025	−0.543432
$317$	26.5885	1.49336	0.746678	−	0.665185i	$-0.231647\pi$
$317$	26.5885	1.49336	0.746678	+	0.665185i	$0.231647\pi$
$318$	6.39230	0.358463
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	1.46410	0.0817182
$322$	2.19615	0.122387
$323$	20.3923	1.13466
$324$	4.46410	0.248006
$325$	4.00000	0.221880
$326$	0.928203	0.0514084
$327$	9.71281	0.537120
$328$	8.19615	0.452557
$329$	6.00000	0.330791
$330$	0	0
$331$	26.3923	1.45065	0.725326	−	0.688405i	$-0.241689\pi$
$331$	26.3923	1.45065	0.725326	+	0.688405i	$0.241689\pi$
$332$	1.07180	0.0588225
$333$	−1.80385	−0.0988502
$334$	−18.9282	−1.03571
$335$	11.8564	0.647785
$336$	−0.732051	−0.0399366
$337$	0.143594	0.00782204	0.00391102	−	0.999992i	$-0.498755\pi$
$337$	0.143594	0.00782204	0.00391102	+	0.999992i	$0.498755\pi$
$338$	−3.00000	−0.163178
$339$	−13.1769	−0.715672
$340$	−7.46410	−0.404798
$341$	0	0
$342$	6.73205	0.364028
$343$	1.00000	0.0539949
$344$	−8.92820	−0.481376
$345$	−1.60770	−0.0865554
$346$	8.92820	0.479983
$347$	−8.92820	−0.479291	−0.239646	−	0.970860i	$-0.577031\pi$
$347$	−8.92820	−0.479291	−0.239646	+	0.970860i	$0.577031\pi$
$348$	−7.07180	−0.379088
$349$	−1.46410	−0.0783716	−0.0391858	−	0.999232i	$-0.512476\pi$
$349$	−1.46410	−0.0783716	−0.0391858	+	0.999232i	$0.512476\pi$
$350$	−1.00000	−0.0534522
$351$	16.0000	0.854017
$352$	0	0
$353$	−26.9808	−1.43604	−0.718021	−	0.696022i	$-0.754952\pi$
$353$	−26.9808	−1.43604	−0.718021	+	0.696022i	$0.754952\pi$
$354$	−9.07180	−0.482161
$355$	−14.9282	−0.792307
$356$	4.92820	0.261194
$357$	−5.46410	−0.289191
$358$	14.3923	0.760657
$359$	−34.4449	−1.81793	−0.908965	−	0.416872i	$-0.863126\pi$
$359$	−34.4449	−1.81793	−0.908965	+	0.416872i	$0.863126\pi$
$360$	−2.46410	−0.129870
$361$	−11.5359	−0.607153
$362$	−8.92820	−0.469256
$363$	0	0
$364$	4.00000	0.209657
$365$	6.00000	0.314054
$366$	−5.85641	−0.306119
$367$	6.39230	0.333676	0.166838	−	0.985984i	$-0.446644\pi$
$367$	6.39230	0.333676	0.166838	+	0.985984i	$0.446644\pi$
$368$	−2.19615	−0.114482
$369$	20.1962	1.05137
$370$	0.732051	0.0380575
$371$	8.73205	0.453345
$372$	−5.46410	−0.283300
$373$	−1.32051	−0.0683733	−0.0341867	−	0.999415i	$-0.510884\pi$
$373$	−1.32051	−0.0683733	−0.0341867	+	0.999415i	$0.510884\pi$
$374$	0	0
$375$	0.732051	0.0378029
$376$	−6.00000	−0.309426
$377$	38.6410	1.99011
$378$	−4.00000	−0.205738
$379$	2.14359	0.110109	0.0550545	−	0.998483i	$-0.482467\pi$
$379$	2.14359	0.110109	0.0550545	+	0.998483i	$0.482467\pi$
$380$	−2.73205	−0.140151
$381$	9.07180	0.464762
$382$	−0.392305	−0.0200721
$383$	−3.46410	−0.177007	−0.0885037	−	0.996076i	$-0.528208\pi$
$383$	−3.46410	−0.177007	−0.0885037	+	0.996076i	$0.528208\pi$
$384$	0.732051	0.0373573
$385$	0	0
$386$	16.0000	0.814379
$387$	−22.0000	−1.11832
$388$	−0.732051	−0.0371642
$389$	5.60770	0.284321	0.142161	−	0.989844i	$-0.454595\pi$
$389$	5.60770	0.284321	0.142161	+	0.989844i	$0.454595\pi$
$390$	−2.92820	−0.148275
$391$	−16.3923	−0.828994
$392$	−1.00000	−0.0505076
$393$	0.143594	0.00724334
$394$	−3.46410	−0.174519
$395$	9.66025	0.486060
$396$	0	0
$397$	0.535898	0.0268960	0.0134480	−	0.999910i	$-0.495719\pi$
$397$	0.535898	0.0268960	0.0134480	+	0.999910i	$0.495719\pi$
$398$	−25.8564	−1.29606
$399$	−2.00000	−0.100125
$400$	1.00000	0.0500000
$401$	−0.392305	−0.0195908	−0.00979538	−	0.999952i	$-0.503118\pi$
$401$	−0.392305	−0.0195908	−0.00979538	+	0.999952i	$0.503118\pi$
$402$	−8.67949	−0.432894
$403$	29.8564	1.48725
$404$	8.00000	0.398015
$405$	−4.46410	−0.221823
$406$	−9.66025	−0.479430
$407$	0	0
$408$	5.46410	0.270513
$409$	19.8038	0.979237	0.489619	−	0.871937i	$-0.337136\pi$
$409$	19.8038	0.979237	0.489619	+	0.871937i	$0.337136\pi$
$410$	−8.19615	−0.404779
$411$	−14.5359	−0.717003
$412$	−18.3923	−0.906124
$413$	−12.3923	−0.609785
$414$	−5.41154	−0.265963
$415$	−1.07180	−0.0526124
$416$	−4.00000	−0.196116
$417$	−10.0000	−0.489702
$418$	0	0
$419$	10.1436	0.495547	0.247773	−	0.968818i	$-0.420301\pi$
$419$	10.1436	0.495547	0.247773	+	0.968818i	$0.420301\pi$
$420$	0.732051	0.0357204
$421$	−24.2487	−1.18181	−0.590905	−	0.806741i	$-0.701229\pi$
$421$	−24.2487	−1.18181	−0.590905	+	0.806741i	$0.701229\pi$
$422$	−13.8564	−0.674519
$423$	−14.7846	−0.718852
$424$	−8.73205	−0.424066
$425$	7.46410	0.362062
$426$	10.9282	0.529473
$427$	−8.00000	−0.387147
$428$	−2.00000	−0.0966736
$429$	0	0
$430$	8.92820	0.430556
$431$	−13.2679	−0.639095	−0.319547	−	0.947570i	$-0.603531\pi$
$431$	−13.2679	−0.639095	−0.319547	+	0.947570i	$0.603531\pi$
$432$	4.00000	0.192450
$433$	−32.7321	−1.57300	−0.786501	−	0.617589i	$-0.788110\pi$
$433$	−32.7321	−1.57300	−0.786501	+	0.617589i	$0.788110\pi$
$434$	−7.46410	−0.358288
$435$	7.07180	0.339067
$436$	−13.2679	−0.635420
$437$	−6.00000	−0.287019
$438$	−4.39230	−0.209872
$439$	−22.5359	−1.07558	−0.537790	−	0.843079i	$-0.680741\pi$
$439$	−22.5359	−1.07558	−0.537790	+	0.843079i	$0.680741\pi$
$440$	0	0
$441$	−2.46410	−0.117338
$442$	−29.8564	−1.42012
$443$	−15.0718	−0.716083	−0.358041	−	0.933706i	$-0.616555\pi$
$443$	−15.0718	−0.716083	−0.358041	+	0.933706i	$0.616555\pi$
$444$	−0.535898	−0.0254326
$445$	−4.92820	−0.233619
$446$	4.53590	0.214781
$447$	−8.92820	−0.422290
$448$	1.00000	0.0472456
$449$	38.2487	1.80507	0.902534	−	0.430618i	$-0.141704\pi$
$449$	38.2487	1.80507	0.902534	+	0.430618i	$0.141704\pi$
$450$	2.46410	0.116159
$451$	0	0
$452$	18.0000	0.846649
$453$	8.92820	0.419484
$454$	−11.3205	−0.531298
$455$	−4.00000	−0.187523
$456$	2.00000	0.0936586
$457$	11.8564	0.554619	0.277310	−	0.960781i	$-0.410557\pi$
$457$	11.8564	0.554619	0.277310	+	0.960781i	$0.410557\pi$
$458$	11.3205	0.528973
$459$	29.8564	1.39358
$460$	2.19615	0.102396
$461$	−39.3205	−1.83134	−0.915669	−	0.401932i	$-0.868339\pi$
$461$	−39.3205	−1.83134	−0.915669	+	0.401932i	$0.868339\pi$
$462$	0	0
$463$	38.5885	1.79336	0.896679	−	0.442682i	$-0.145973\pi$
$463$	38.5885	1.79336	0.896679	+	0.442682i	$0.145973\pi$
$464$	9.66025	0.448466
$465$	5.46410	0.253392
$466$	3.85641	0.178645
$467$	9.41154	0.435514	0.217757	−	0.976003i	$-0.430126\pi$
$467$	9.41154	0.435514	0.217757	+	0.976003i	$0.430126\pi$
$468$	−9.85641	−0.455613
$469$	−11.8564	−0.547478
$470$	6.00000	0.276759
$471$	17.1769	0.791470
$472$	12.3923	0.570402
$473$	0	0
$474$	−7.07180	−0.324818
$475$	2.73205	0.125355
$476$	7.46410	0.342117
$477$	−21.5167	−0.985180
$478$	18.0526	0.825705
$479$	35.7128	1.63176	0.815880	−	0.578221i	$-0.196253\pi$
$479$	35.7128	1.63176	0.815880	+	0.578221i	$0.196253\pi$
$480$	−0.732051	−0.0334134
$481$	2.92820	0.133515
$482$	−28.9808	−1.32004
$483$	1.60770	0.0731527
$484$	0	0
$485$	0.732051	0.0332407
$486$	15.2679	0.692568
$487$	8.05256	0.364896	0.182448	−	0.983215i	$-0.441598\pi$
$487$	8.05256	0.364896	0.182448	+	0.983215i	$0.441598\pi$
$488$	8.00000	0.362143
$489$	0.679492	0.0307277
$490$	1.00000	0.0451754
$491$	−22.9282	−1.03474	−0.517368	−	0.855763i	$-0.673088\pi$
$491$	−22.9282	−1.03474	−0.517368	+	0.855763i	$0.673088\pi$
$492$	6.00000	0.270501
$493$	72.1051	3.24745
$494$	−10.9282	−0.491683
$495$	0	0
$496$	7.46410	0.335148
$497$	14.9282	0.669621
$498$	0.784610	0.0351592
$499$	−1.60770	−0.0719703	−0.0359852	−	0.999352i	$-0.511457\pi$
$499$	−1.60770	−0.0719703	−0.0359852	+	0.999352i	$0.511457\pi$
$500$	−1.00000	−0.0447214
$501$	−13.8564	−0.619059
$502$	−16.7846	−0.749134
$503$	−37.8564	−1.68793	−0.843967	−	0.536395i	$-0.819786\pi$
$503$	−37.8564	−1.68793	−0.843967	+	0.536395i	$0.819786\pi$
$504$	2.46410	0.109760
$505$	−8.00000	−0.355995
$506$	0	0
$507$	−2.19615	−0.0975346
$508$	−12.3923	−0.549820
$509$	12.9282	0.573033	0.286516	−	0.958075i	$-0.407503\pi$
$509$	12.9282	0.573033	0.286516	+	0.958075i	$0.407503\pi$
$510$	−5.46410	−0.241954
$511$	−6.00000	−0.265424
$512$	−1.00000	−0.0441942
$513$	10.9282	0.482492
$514$	18.1962	0.802598
$515$	18.3923	0.810462
$516$	−6.53590	−0.287727
$517$	0	0
$518$	−0.732051	−0.0321645
$519$	6.53590	0.286894
$520$	4.00000	0.175412
$521$	24.2487	1.06236	0.531178	−	0.847260i	$-0.321750\pi$
$521$	24.2487	1.06236	0.531178	+	0.847260i	$0.321750\pi$
$522$	23.8038	1.04187
$523$	−21.4641	−0.938560	−0.469280	−	0.883050i	$-0.655486\pi$
$523$	−21.4641	−0.938560	−0.469280	+	0.883050i	$0.655486\pi$
$524$	−0.196152	−0.00856896
$525$	−0.732051	−0.0319493
$526$	−7.60770	−0.331711
$527$	55.7128	2.42689
$528$	0	0
$529$	−18.1769	−0.790301
$530$	8.73205	0.379296
$531$	30.5359	1.32515
$532$	2.73205	0.118449
$533$	−32.7846	−1.42006
$534$	3.60770	0.156120
$535$	2.00000	0.0864675
$536$	11.8564	0.512119
$537$	10.5359	0.454658
$538$	−16.3923	−0.706722
$539$	0	0
$540$	−4.00000	−0.172133
$541$	−9.26795	−0.398460	−0.199230	−	0.979953i	$-0.563844\pi$
$541$	−9.26795	−0.398460	−0.199230	+	0.979953i	$0.563844\pi$
$542$	−11.6077	−0.498593
$543$	−6.53590	−0.280482
$544$	−7.46410	−0.320021
$545$	13.2679	0.568337
$546$	2.92820	0.125316
$547$	−31.7128	−1.35594	−0.677971	−	0.735089i	$-0.737141\pi$
$547$	−31.7128	−1.35594	−0.677971	+	0.735089i	$0.737141\pi$
$548$	19.8564	0.848224
$549$	19.7128	0.841322
$550$	0	0
$551$	26.3923	1.12435
$552$	−1.60770	−0.0684280
$553$	−9.66025	−0.410796
$554$	−17.3205	−0.735878
$555$	0.535898	0.0227476
$556$	13.6603	0.579324
$557$	16.9282	0.717271	0.358635	−	0.933478i	$-0.383242\pi$
$557$	16.9282	0.717271	0.358635	+	0.933478i	$0.383242\pi$
$558$	18.3923	0.778608
$559$	35.7128	1.51049
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−11.3205	−0.477527
$563$	−15.7128	−0.662216	−0.331108	−	0.943593i	$-0.607422\pi$
$563$	−15.7128	−0.662216	−0.331108	+	0.943593i	$0.607422\pi$
$564$	−4.39230	−0.184949
$565$	−18.0000	−0.757266
$566$	2.53590	0.106592
$567$	4.46410	0.187475
$568$	−14.9282	−0.626373
$569$	−6.14359	−0.257553	−0.128776	−	0.991674i	$-0.541105\pi$
$569$	−6.14359	−0.257553	−0.128776	+	0.991674i	$0.541105\pi$
$570$	−2.00000	−0.0837708
$571$	−15.3205	−0.641143	−0.320572	−	0.947224i	$-0.603875\pi$
$571$	−15.3205	−0.641143	−0.320572	+	0.947224i	$0.603875\pi$
$572$	0	0
$573$	−0.287187	−0.0119974
$574$	8.19615	0.342101
$575$	−2.19615	−0.0915859
$576$	−2.46410	−0.102671
$577$	4.73205	0.196998	0.0984989	−	0.995137i	$-0.468596\pi$
$577$	4.73205	0.196998	0.0984989	+	0.995137i	$0.468596\pi$
$578$	−38.7128	−1.61024
$579$	11.7128	0.486768
$580$	−9.66025	−0.401120
$581$	1.07180	0.0444656
$582$	−0.535898	−0.0222137
$583$	0	0
$584$	6.00000	0.248282
$585$	9.85641	0.407512
$586$	27.7128	1.14481
$587$	23.2679	0.960371	0.480186	−	0.877167i	$-0.340569\pi$
$587$	23.2679	0.960371	0.480186	+	0.877167i	$0.340569\pi$
$588$	−0.732051	−0.0301893
$589$	20.3923	0.840250
$590$	−12.3923	−0.510183
$591$	−2.53590	−0.104313
$592$	0.732051	0.0300871
$593$	40.6410	1.66893	0.834463	−	0.551064i	$-0.185778\pi$
$593$	40.6410	1.66893	0.834463	+	0.551064i	$0.185778\pi$
$594$	0	0
$595$	−7.46410	−0.305998
$596$	12.1962	0.499574
$597$	−18.9282	−0.774680
$598$	8.78461	0.359229
$599$	−33.0718	−1.35128	−0.675638	−	0.737233i	$-0.736132\pi$
$599$	−33.0718	−1.35128	−0.675638	+	0.737233i	$0.736132\pi$
$600$	0.732051	0.0298858
$601$	27.1244	1.10643	0.553213	−	0.833040i	$-0.313402\pi$
$601$	27.1244	1.10643	0.553213	+	0.833040i	$0.313402\pi$
$602$	−8.92820	−0.363886
$603$	29.2154	1.18974
$604$	−12.1962	−0.496254
$605$	0	0
$606$	5.85641	0.237900
$607$	10.0000	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$608$	−2.73205	−0.110799
$609$	−7.07180	−0.286564
$610$	−8.00000	−0.323911
$611$	24.0000	0.970936
$612$	−18.3923	−0.743465
$613$	−13.7128	−0.553855	−0.276928	−	0.960891i	$-0.589316\pi$
$613$	−13.7128	−0.553855	−0.276928	+	0.960891i	$0.589316\pi$
$614$	−6.53590	−0.263767
$615$	−6.00000	−0.241943
$616$	0	0
$617$	17.6077	0.708859	0.354430	−	0.935083i	$-0.384675\pi$
$617$	17.6077	0.708859	0.354430	+	0.935083i	$0.384675\pi$
$618$	−13.4641	−0.541606
$619$	18.9282	0.760789	0.380394	−	0.924824i	$-0.375788\pi$
$619$	18.9282	0.760789	0.380394	+	0.924824i	$0.375788\pi$
$620$	−7.46410	−0.299766
$621$	−8.78461	−0.352514
$622$	17.3205	0.694489
$623$	4.92820	0.197444
$624$	−2.92820	−0.117222
$625$	1.00000	0.0400000
$626$	−0.732051	−0.0292586
$627$	0	0
$628$	−23.4641	−0.936320
$629$	5.46410	0.217868
$630$	−2.46410	−0.0981722
$631$	1.85641	0.0739024	0.0369512	−	0.999317i	$-0.488235\pi$
$631$	1.85641	0.0739024	0.0369512	+	0.999317i	$0.488235\pi$
$632$	9.66025	0.384264
$633$	−10.1436	−0.403172
$634$	−26.5885	−1.05596
$635$	12.3923	0.491774
$636$	−6.39230	−0.253471
$637$	4.00000	0.158486
$638$	0	0
$639$	−36.7846	−1.45518
$640$	1.00000	0.0395285
$641$	3.85641	0.152319	0.0761594	−	0.997096i	$-0.475734\pi$
$641$	3.85641	0.152319	0.0761594	+	0.997096i	$0.475734\pi$
$642$	−1.46410	−0.0577835
$643$	−6.19615	−0.244352	−0.122176	−	0.992508i	$-0.538987\pi$
$643$	−6.19615	−0.244352	−0.122176	+	0.992508i	$0.538987\pi$
$644$	−2.19615	−0.0865405
$645$	6.53590	0.257351
$646$	−20.3923	−0.802325
$647$	−26.3923	−1.03759	−0.518794	−	0.854899i	$-0.673619\pi$
$647$	−26.3923	−1.03759	−0.518794	+	0.854899i	$0.673619\pi$
$648$	−4.46410	−0.175366
$649$	0	0
$650$	−4.00000	−0.156893
$651$	−5.46410	−0.214155
$652$	−0.928203	−0.0363512
$653$	−31.2679	−1.22361	−0.611805	−	0.791009i	$-0.709556\pi$
$653$	−31.2679	−1.22361	−0.611805	+	0.791009i	$0.709556\pi$
$654$	−9.71281	−0.379801
$655$	0.196152	0.00766431
$656$	−8.19615	−0.320006
$657$	14.7846	0.576803
$658$	−6.00000	−0.233904
$659$	19.6077	0.763807	0.381904	−	0.924202i	$-0.375269\pi$
$659$	19.6077	0.763807	0.381904	+	0.924202i	$0.375269\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	−26.3923	−1.02577
$663$	−21.8564	−0.848832
$664$	−1.07180	−0.0415938
$665$	−2.73205	−0.105944
$666$	1.80385	0.0698977
$667$	−21.2154	−0.821463
$668$	18.9282	0.732354
$669$	3.32051	0.128378
$670$	−11.8564	−0.458053
$671$	0	0
$672$	0.732051	0.0282395
$673$	−16.1436	−0.622290	−0.311145	−	0.950362i	$-0.600712\pi$
$673$	−16.1436	−0.622290	−0.311145	+	0.950362i	$0.600712\pi$
$674$	−0.143594	−0.00553102
$675$	4.00000	0.153960
$676$	3.00000	0.115385
$677$	0.928203	0.0356737	0.0178369	−	0.999841i	$-0.494322\pi$
$677$	0.928203	0.0356737	0.0178369	+	0.999841i	$0.494322\pi$
$678$	13.1769	0.506056
$679$	−0.732051	−0.0280935
$680$	7.46410	0.286235
$681$	−8.28719	−0.317566
$682$	0	0
$683$	22.3923	0.856818	0.428409	−	0.903585i	$-0.359074\pi$
$683$	22.3923	0.856818	0.428409	+	0.903585i	$0.359074\pi$
$684$	−6.73205	−0.257406
$685$	−19.8564	−0.758674
$686$	−1.00000	−0.0381802
$687$	8.28719	0.316176
$688$	8.92820	0.340385
$689$	34.9282	1.33066
$690$	1.60770	0.0612039
$691$	−1.07180	−0.0407731	−0.0203865	−	0.999792i	$-0.506490\pi$
$691$	−1.07180	−0.0407731	−0.0203865	+	0.999792i	$0.506490\pi$
$692$	−8.92820	−0.339399
$693$	0	0
$694$	8.92820	0.338910
$695$	−13.6603	−0.518163
$696$	7.07180	0.268056
$697$	−61.1769	−2.31724
$698$	1.46410	0.0554171
$699$	2.82309	0.106779
$700$	1.00000	0.0377964
$701$	34.7321	1.31181	0.655906	−	0.754843i	$-0.272287\pi$
$701$	34.7321	1.31181	0.655906	+	0.754843i	$0.272287\pi$
$702$	−16.0000	−0.603881
$703$	2.00000	0.0754314
$704$	0	0
$705$	4.39230	0.165424
$706$	26.9808	1.01543
$707$	8.00000	0.300871
$708$	9.07180	0.340939
$709$	23.8564	0.895946	0.447973	−	0.894047i	$-0.352146\pi$
$709$	23.8564	0.895946	0.447973	+	0.894047i	$0.352146\pi$
$710$	14.9282	0.560245
$711$	23.8038	0.892714
$712$	−4.92820	−0.184692
$713$	−16.3923	−0.613897
$714$	5.46410	0.204489
$715$	0	0
$716$	−14.3923	−0.537866
$717$	13.2154	0.493538
$718$	34.4449	1.28547
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	2.46410	0.0918316
$721$	−18.3923	−0.684965
$722$	11.5359	0.429322
$723$	−21.2154	−0.789009
$724$	8.92820	0.331814
$725$	9.66025	0.358773
$726$	0	0
$727$	9.71281	0.360228	0.180114	−	0.983646i	$-0.442353\pi$
$727$	9.71281	0.360228	0.180114	+	0.983646i	$0.442353\pi$
$728$	−4.00000	−0.148250
$729$	−2.21539	−0.0820515
$730$	−6.00000	−0.222070
$731$	66.6410	2.46481
$732$	5.85641	0.216459
$733$	24.1436	0.891764	0.445882	−	0.895092i	$-0.352890\pi$
$733$	24.1436	0.891764	0.445882	+	0.895092i	$0.352890\pi$
$734$	−6.39230	−0.235944
$735$	0.732051	0.0270021
$736$	2.19615	0.0809513
$737$	0	0
$738$	−20.1962	−0.743431
$739$	17.8564	0.656859	0.328429	−	0.944529i	$-0.393481\pi$
$739$	17.8564	0.656859	0.328429	+	0.944529i	$0.393481\pi$
$740$	−0.732051	−0.0269107
$741$	−8.00000	−0.293887
$742$	−8.73205	−0.320564
$743$	−24.3923	−0.894867	−0.447433	−	0.894317i	$-0.647662\pi$
$743$	−24.3923	−0.894867	−0.447433	+	0.894317i	$0.647662\pi$
$744$	5.46410	0.200324
$745$	−12.1962	−0.446832
$746$	1.32051	0.0483472
$747$	−2.64102	−0.0966297
$748$	0	0
$749$	−2.00000	−0.0730784
$750$	−0.732051	−0.0267307
$751$	49.8564	1.81929	0.909643	−	0.415391i	$-0.136355\pi$
$751$	49.8564	1.81929	0.909643	+	0.415391i	$0.136355\pi$
$752$	6.00000	0.218797
$753$	−12.2872	−0.447770
$754$	−38.6410	−1.40722
$755$	12.1962	0.443863
$756$	4.00000	0.145479
$757$	−27.6603	−1.00533	−0.502665	−	0.864482i	$-0.667647\pi$
$757$	−27.6603	−1.00533	−0.502665	+	0.864482i	$0.667647\pi$
$758$	−2.14359	−0.0778588
$759$	0	0
$760$	2.73205	0.0991019
$761$	−3.41154	−0.123668	−0.0618342	−	0.998086i	$-0.519695\pi$
$761$	−3.41154	−0.123668	−0.0618342	+	0.998086i	$0.519695\pi$
$762$	−9.07180	−0.328637
$763$	−13.2679	−0.480332
$764$	0.392305	0.0141931
$765$	18.3923	0.664975
$766$	3.46410	0.125163
$767$	−49.5692	−1.78984
$768$	−0.732051	−0.0264156
$769$	6.73205	0.242764	0.121382	−	0.992606i	$-0.461267\pi$
$769$	6.73205	0.242764	0.121382	+	0.992606i	$0.461267\pi$
$770$	0	0
$771$	13.3205	0.479726
$772$	−16.0000	−0.575853
$773$	9.71281	0.349346	0.174673	−	0.984627i	$-0.444113\pi$
$773$	9.71281	0.349346	0.174673	+	0.984627i	$0.444113\pi$
$774$	22.0000	0.790774
$775$	7.46410	0.268118
$776$	0.732051	0.0262791
$777$	−0.535898	−0.0192252
$778$	−5.60770	−0.201046
$779$	−22.3923	−0.802288
$780$	2.92820	0.104846
$781$	0	0
$782$	16.3923	0.586188
$783$	38.6410	1.38092
$784$	1.00000	0.0357143
$785$	23.4641	0.837470
$786$	−0.143594	−0.00512181
$787$	−36.7846	−1.31123	−0.655615	−	0.755095i	$-0.727590\pi$
$787$	−36.7846	−1.31123	−0.655615	+	0.755095i	$0.727590\pi$
$788$	3.46410	0.123404
$789$	−5.56922	−0.198269
$790$	−9.66025	−0.343696
$791$	18.0000	0.640006
$792$	0	0
$793$	−32.0000	−1.13635
$794$	−0.535898	−0.0190183
$795$	6.39230	0.226712
$796$	25.8564	0.916456
$797$	−2.39230	−0.0847398	−0.0423699	−	0.999102i	$-0.513491\pi$
$797$	−2.39230	−0.0847398	−0.0423699	+	0.999102i	$0.513491\pi$
$798$	2.00000	0.0707992
$799$	44.7846	1.58437
$800$	−1.00000	−0.0353553
$801$	−12.1436	−0.429073
$802$	0.392305	0.0138528
$803$	0	0
$804$	8.67949	0.306102
$805$	2.19615	0.0774042
$806$	−29.8564	−1.05165
$807$	−12.0000	−0.422420
$808$	−8.00000	−0.281439
$809$	32.1051	1.12876	0.564378	−	0.825517i	$-0.309116\pi$
$809$	32.1051	1.12876	0.564378	+	0.825517i	$0.309116\pi$
$810$	4.46410	0.156853
$811$	6.73205	0.236394	0.118197	−	0.992990i	$-0.462289\pi$
$811$	6.73205	0.236394	0.118197	+	0.992990i	$0.462289\pi$
$812$	9.66025	0.339008
$813$	−8.49742	−0.298018
$814$	0	0
$815$	0.928203	0.0325135
$816$	−5.46410	−0.191282
$817$	24.3923	0.853379
$818$	−19.8038	−0.692425
$819$	−9.85641	−0.344411
$820$	8.19615	0.286222
$821$	49.3731	1.72313	0.861566	−	0.507646i	$-0.169484\pi$
$821$	49.3731	1.72313	0.861566	+	0.507646i	$0.169484\pi$
$822$	14.5359	0.506998
$823$	40.0526	1.39614	0.698072	−	0.716027i	$-0.254041\pi$
$823$	40.0526	1.39614	0.698072	+	0.716027i	$0.254041\pi$
$824$	18.3923	0.640726
$825$	0	0
$826$	12.3923	0.431183
$827$	20.7846	0.722752	0.361376	−	0.932420i	$-0.382307\pi$
$827$	20.7846	0.722752	0.361376	+	0.932420i	$0.382307\pi$
$828$	5.41154	0.188064
$829$	24.3923	0.847180	0.423590	−	0.905854i	$-0.360770\pi$
$829$	24.3923	0.847180	0.423590	+	0.905854i	$0.360770\pi$
$830$	1.07180	0.0372026
$831$	−12.6795	−0.439847
$832$	4.00000	0.138675
$833$	7.46410	0.258616
$834$	10.0000	0.346272
$835$	−18.9282	−0.655037
$836$	0	0
$837$	29.8564	1.03199
$838$	−10.1436	−0.350405
$839$	20.2487	0.699063	0.349532	−	0.936925i	$-0.386341\pi$
$839$	20.2487	0.699063	0.349532	+	0.936925i	$0.386341\pi$
$840$	−0.732051	−0.0252582
$841$	64.3205	2.21795
$842$	24.2487	0.835666
$843$	−8.28719	−0.285426
$844$	13.8564	0.476957
$845$	−3.00000	−0.103203
$846$	14.7846	0.508305
$847$	0	0
$848$	8.73205	0.299860
$849$	1.85641	0.0637117
$850$	−7.46410	−0.256017
$851$	−1.60770	−0.0551111
$852$	−10.9282	−0.374394
$853$	44.6410	1.52848	0.764240	−	0.644932i	$-0.223114\pi$
$853$	44.6410	1.52848	0.764240	+	0.644932i	$0.223114\pi$
$854$	8.00000	0.273754
$855$	6.73205	0.230231
$856$	2.00000	0.0683586
$857$	−11.0718	−0.378205	−0.189103	−	0.981957i	$-0.560558\pi$
$857$	−11.0718	−0.378205	−0.189103	+	0.981957i	$0.560558\pi$
$858$	0	0
$859$	26.9282	0.918778	0.459389	−	0.888235i	$-0.348068\pi$
$859$	26.9282	0.918778	0.459389	+	0.888235i	$0.348068\pi$
$860$	−8.92820	−0.304449
$861$	6.00000	0.204479
$862$	13.2679	0.451908
$863$	−11.6603	−0.396920	−0.198460	−	0.980109i	$-0.563594\pi$
$863$	−11.6603	−0.396920	−0.198460	+	0.980109i	$0.563594\pi$
$864$	−4.00000	−0.136083
$865$	8.92820	0.303568
$866$	32.7321	1.11228
$867$	−28.3397	−0.962468
$868$	7.46410	0.253348
$869$	0	0
$870$	−7.07180	−0.239756
$871$	−47.4256	−1.60696
$872$	13.2679	0.449309
$873$	1.80385	0.0610510
$874$	6.00000	0.202953
$875$	−1.00000	−0.0338062
$876$	4.39230	0.148402
$877$	−17.6077	−0.594570	−0.297285	−	0.954789i	$-0.596081\pi$
$877$	−17.6077	−0.594570	−0.297285	+	0.954789i	$0.596081\pi$
$878$	22.5359	0.760550
$879$	20.2872	0.684270
$880$	0	0
$881$	57.7128	1.94439	0.972197	−	0.234164i	$-0.0752354\pi$
$881$	57.7128	1.94439	0.972197	+	0.234164i	$0.0752354\pi$
$882$	2.46410	0.0829706
$883$	−22.6795	−0.763226	−0.381613	−	0.924322i	$-0.624631\pi$
$883$	−22.6795	−0.763226	−0.381613	+	0.924322i	$0.624631\pi$
$884$	29.8564	1.00418
$885$	−9.07180	−0.304945
$886$	15.0718	0.506347
$887$	22.6410	0.760211	0.380105	−	0.924943i	$-0.375888\pi$
$887$	22.6410	0.760211	0.380105	+	0.924943i	$0.375888\pi$
$888$	0.535898	0.0179836
$889$	−12.3923	−0.415625
$890$	4.92820	0.165194
$891$	0	0
$892$	−4.53590	−0.151873
$893$	16.3923	0.548548
$894$	8.92820	0.298604
$895$	14.3923	0.481082
$896$	−1.00000	−0.0334077
$897$	6.43078	0.214718
$898$	−38.2487	−1.27638
$899$	72.1051	2.40484
$900$	−2.46410	−0.0821367
$901$	65.1769	2.17136
$902$	0	0
$903$	−6.53590	−0.217501
$904$	−18.0000	−0.598671
$905$	−8.92820	−0.296784
$906$	−8.92820	−0.296620
$907$	25.3205	0.840754	0.420377	−	0.907350i	$-0.361898\pi$
$907$	25.3205	0.840754	0.420377	+	0.907350i	$0.361898\pi$
$908$	11.3205	0.375684
$909$	−19.7128	−0.653833
$910$	4.00000	0.132599
$911$	−47.0333	−1.55828	−0.779142	−	0.626848i	$-0.784345\pi$
$911$	−47.0333	−1.55828	−0.779142	+	0.626848i	$0.784345\pi$
$912$	−2.00000	−0.0662266
$913$	0	0
$914$	−11.8564	−0.392175
$915$	−5.85641	−0.193607
$916$	−11.3205	−0.374040
$917$	−0.196152	−0.00647752
$918$	−29.8564	−0.985408
$919$	37.3731	1.23282	0.616412	−	0.787424i	$-0.288586\pi$
$919$	37.3731	1.23282	0.616412	+	0.787424i	$0.288586\pi$
$920$	−2.19615	−0.0724050
$921$	−4.78461	−0.157658
$922$	39.3205	1.29495
$923$	59.7128	1.96547
$924$	0	0
$925$	0.732051	0.0240697
$926$	−38.5885	−1.26810
$927$	45.3205	1.48852
$928$	−9.66025	−0.317113
$929$	12.2487	0.401867	0.200934	−	0.979605i	$-0.435602\pi$
$929$	12.2487	0.401867	0.200934	+	0.979605i	$0.435602\pi$
$930$	−5.46410	−0.179175
$931$	2.73205	0.0895393
$932$	−3.85641	−0.126321
$933$	12.6795	0.415108
$934$	−9.41154	−0.307955
$935$	0	0
$936$	9.85641	0.322167
$937$	−41.3205	−1.34988	−0.674941	−	0.737872i	$-0.735831\pi$
$937$	−41.3205	−1.34988	−0.674941	+	0.737872i	$0.735831\pi$
$938$	11.8564	0.387125
$939$	−0.535898	−0.0174884
$940$	−6.00000	−0.195698
$941$	6.92820	0.225853	0.112926	−	0.993603i	$-0.463978\pi$
$941$	6.92820	0.225853	0.112926	+	0.993603i	$0.463978\pi$
$942$	−17.1769	−0.559654
$943$	18.0000	0.586161
$944$	−12.3923	−0.403335
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−18.3923	−0.597670	−0.298835	−	0.954305i	$-0.596598\pi$
$947$	−18.3923	−0.597670	−0.298835	+	0.954305i	$0.596598\pi$
$948$	7.07180	0.229681
$949$	−24.0000	−0.779073
$950$	−2.73205	−0.0886394
$951$	−19.4641	−0.631167
$952$	−7.46410	−0.241913
$953$	−46.6410	−1.51085	−0.755425	−	0.655235i	$-0.772570\pi$
$953$	−46.6410	−1.51085	−0.755425	+	0.655235i	$0.772570\pi$
$954$	21.5167	0.696628
$955$	−0.392305	−0.0126947
$956$	−18.0526	−0.583861
$957$	0	0
$958$	−35.7128	−1.15383
$959$	19.8564	0.641197
$960$	0.732051	0.0236268
$961$	24.7128	0.797188
$962$	−2.92820	−0.0944091
$963$	4.92820	0.158809
$964$	28.9808	0.933407
$965$	16.0000	0.515058
$966$	−1.60770	−0.0517267
$967$	15.6077	0.501910	0.250955	−	0.967999i	$-0.419255\pi$
$967$	15.6077	0.501910	0.250955	+	0.967999i	$0.419255\pi$
$968$	0	0
$969$	−14.9282	−0.479563
$970$	−0.732051	−0.0235047
$971$	24.3923	0.782786	0.391393	−	0.920224i	$-0.371993\pi$
$971$	24.3923	0.782786	0.391393	+	0.920224i	$0.371993\pi$
$972$	−15.2679	−0.489720
$973$	13.6603	0.437928
$974$	−8.05256	−0.258021
$975$	−2.92820	−0.0937776
$976$	−8.00000	−0.256074
$977$	4.14359	0.132565	0.0662827	−	0.997801i	$-0.478886\pi$
$977$	4.14359	0.132565	0.0662827	+	0.997801i	$0.478886\pi$
$978$	−0.679492	−0.0217278
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	32.6936	1.04383
$982$	22.9282	0.731668
$983$	20.5359	0.654993	0.327497	−	0.944852i	$-0.393795\pi$
$983$	20.5359	0.654993	0.327497	+	0.944852i	$0.393795\pi$
$984$	−6.00000	−0.191273
$985$	−3.46410	−0.110375
$986$	−72.1051	−2.29629
$987$	−4.39230	−0.139809
$988$	10.9282	0.347672
$989$	−19.6077	−0.623488
$990$	0	0
$991$	−24.3923	−0.774847	−0.387424	−	0.921902i	$-0.626635\pi$
$991$	−24.3923	−0.774847	−0.387424	+	0.921902i	$0.626635\pi$
$992$	−7.46410	−0.236985
$993$	−19.3205	−0.613118
$994$	−14.9282	−0.473494
$995$	−25.8564	−0.819703
$996$	−0.784610	−0.0248613
$997$	−3.85641	−0.122134	−0.0610668	−	0.998134i	$-0.519450\pi$
$997$	−3.85641	−0.122134	−0.0610668	+	0.998134i	$0.519450\pi$
$998$	1.60770	0.0508907
$999$	2.92820	0.0926443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.bs.1.1	✓	2
11.10	odd	2		8470.2.a.cd.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.bs.1.1	✓	2	1.1	even	1	trivial
8470.2.a.cd.1.1	yes	2	11.10	odd	2

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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +1.00000 q^{10} -0.732051 q^{12} +4.00000 q^{13} -1.00000 q^{14} +0.732051 q^{15} +1.00000 q^{16} +7.46410 q^{17} +2.46410 q^{18} +2.73205 q^{19} -1.00000 q^{20} -0.732051 q^{21} -2.19615 q^{23} +0.732051 q^{24} +1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} +9.66025 q^{29} -0.732051 q^{30} +7.46410 q^{31} -1.00000 q^{32} -7.46410 q^{34} -1.00000 q^{35} -2.46410 q^{36} +0.732051 q^{37} -2.73205 q^{38} -2.92820 q^{39} +1.00000 q^{40} -8.19615 q^{41} +0.732051 q^{42} +8.92820 q^{43} +2.46410 q^{45} +2.19615 q^{46} +6.00000 q^{47} -0.732051 q^{48} +1.00000 q^{49} -1.00000 q^{50} -5.46410 q^{51} +4.00000 q^{52} +8.73205 q^{53} -4.00000 q^{54} -1.00000 q^{56} -2.00000 q^{57} -9.66025 q^{58} -12.3923 q^{59} +0.732051 q^{60} -8.00000 q^{61} -7.46410 q^{62} -2.46410 q^{63} +1.00000 q^{64} -4.00000 q^{65} -11.8564 q^{67} +7.46410 q^{68} +1.60770 q^{69} +1.00000 q^{70} +14.9282 q^{71} +2.46410 q^{72} -6.00000 q^{73} -0.732051 q^{74} -0.732051 q^{75} +2.73205 q^{76} +2.92820 q^{78} -9.66025 q^{79} -1.00000 q^{80} +4.46410 q^{81} +8.19615 q^{82} +1.07180 q^{83} -0.732051 q^{84} -7.46410 q^{85} -8.92820 q^{86} -7.07180 q^{87} +4.92820 q^{89} -2.46410 q^{90} +4.00000 q^{91} -2.19615 q^{92} -5.46410 q^{93} -6.00000 q^{94} -2.73205 q^{95} +0.732051 q^{96} -0.732051 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 8 q^{13} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 8 q^{17} - 2 q^{18} + 2 q^{19} - 2 q^{20} + 2 q^{21} + 6 q^{23} - 2 q^{24} + 2 q^{25} - 8 q^{26} + 8 q^{27} + 2 q^{28} + 2 q^{29} + 2 q^{30} + 8 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{35} + 2 q^{36} - 2 q^{37} - 2 q^{38} + 8 q^{39} + 2 q^{40} - 6 q^{41} - 2 q^{42} + 4 q^{43} - 2 q^{45} - 6 q^{46} + 12 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} - 4 q^{51} + 8 q^{52} + 14 q^{53} - 8 q^{54} - 2 q^{56} - 4 q^{57} - 2 q^{58} - 4 q^{59} - 2 q^{60} - 16 q^{61} - 8 q^{62} + 2 q^{63} + 2 q^{64} - 8 q^{65} + 4 q^{67} + 8 q^{68} + 24 q^{69} + 2 q^{70} + 16 q^{71} - 2 q^{72} - 12 q^{73} + 2 q^{74} + 2 q^{75} + 2 q^{76} - 8 q^{78} - 2 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} + 16 q^{83} + 2 q^{84} - 8 q^{85} - 4 q^{86} - 28 q^{87} - 4 q^{89} + 2 q^{90} + 8 q^{91} + 6 q^{92} - 4 q^{93} - 12 q^{94} - 2 q^{95} - 2 q^{96} + 2 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 8 * q^13 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 8 * q^17 - 2 * q^18 + 2 * q^19 - 2 * q^20 + 2 * q^21 + 6 * q^23 - 2 * q^24 + 2 * q^25 - 8 * q^26 + 8 * q^27 + 2 * q^28 + 2 * q^29 + 2 * q^30 + 8 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^35 + 2 * q^36 - 2 * q^37 - 2 * q^38 + 8 * q^39 + 2 * q^40 - 6 * q^41 - 2 * q^42 + 4 * q^43 - 2 * q^45 - 6 * q^46 + 12 * q^47 + 2 * q^48 + 2 * q^49 - 2 * q^50 - 4 * q^51 + 8 * q^52 + 14 * q^53 - 8 * q^54 - 2 * q^56 - 4 * q^57 - 2 * q^58 - 4 * q^59 - 2 * q^60 - 16 * q^61 - 8 * q^62 + 2 * q^63 + 2 * q^64 - 8 * q^65 + 4 * q^67 + 8 * q^68 + 24 * q^69 + 2 * q^70 + 16 * q^71 - 2 * q^72 - 12 * q^73 + 2 * q^74 + 2 * q^75 + 2 * q^76 - 8 * q^78 - 2 * q^79 - 2 * q^80 + 2 * q^81 + 6 * q^82 + 16 * q^83 + 2 * q^84 - 8 * q^85 - 4 * q^86 - 28 * q^87 - 4 * q^89 + 2 * q^90 + 8 * q^91 + 6 * q^92 - 4 * q^93 - 12 * q^94 - 2 * q^95 - 2 * q^96 + 2 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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