LMFDB - Embedded newform 8410.2.a.ca.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8410,2,Mod(1,8410)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8410.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8410, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8410 = 2 \cdot 5 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8410.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,18,6,18,18,6,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.1541880999$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 20 x^{16} + 176 x^{15} + 71 x^{14} - 2078 x^{13} + 1135 x^{12} + 12782 x^{11} + \cdots - 5887 $ x^18 - 6x^17 - 20x^16 + 176x^15 + 71x^14 - 2078x^13 + 1135x^12 + 12782x^11 - 12642x^10 - 44478x^9 + 53256x^8 + 90272x^7 - 109242x^6 - 109844x^5 + 104804x^4 + 82768x^3 - 32501x^2 - 32886*x - 5887
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 290)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.603364$ of defining polynomial
Character	$\chi$	$=$	8410.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -0.603364 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.603364 q^{6} +3.60356 q^{7} +1.00000 q^{8} -2.63595 q^{9} +1.00000 q^{10} +3.83198 q^{11} -0.603364 q^{12} +0.122734 q^{13} +3.60356 q^{14} -0.603364 q^{15} +1.00000 q^{16} -2.31987 q^{17} -2.63595 q^{18} -6.64668 q^{19} +1.00000 q^{20} -2.17426 q^{21} +3.83198 q^{22} -4.09077 q^{23} -0.603364 q^{24} +1.00000 q^{25} +0.122734 q^{26} +3.40053 q^{27} +3.60356 q^{28} -0.603364 q^{30} +9.44644 q^{31} +1.00000 q^{32} -2.31208 q^{33} -2.31987 q^{34} +3.60356 q^{35} -2.63595 q^{36} +3.57990 q^{37} -6.64668 q^{38} -0.0740532 q^{39} +1.00000 q^{40} +10.4455 q^{41} -2.17426 q^{42} -9.28941 q^{43} +3.83198 q^{44} -2.63595 q^{45} -4.09077 q^{46} +13.1924 q^{47} -0.603364 q^{48} +5.98561 q^{49} +1.00000 q^{50} +1.39973 q^{51} +0.122734 q^{52} -2.73872 q^{53} +3.40053 q^{54} +3.83198 q^{55} +3.60356 q^{56} +4.01037 q^{57} -0.361609 q^{59} -0.603364 q^{60} -6.29724 q^{61} +9.44644 q^{62} -9.49880 q^{63} +1.00000 q^{64} +0.122734 q^{65} -2.31208 q^{66} -4.42687 q^{67} -2.31987 q^{68} +2.46823 q^{69} +3.60356 q^{70} +10.1456 q^{71} -2.63595 q^{72} +6.51726 q^{73} +3.57990 q^{74} -0.603364 q^{75} -6.64668 q^{76} +13.8088 q^{77} -0.0740532 q^{78} +13.9998 q^{79} +1.00000 q^{80} +5.85609 q^{81} +10.4455 q^{82} -4.87092 q^{83} -2.17426 q^{84} -2.31987 q^{85} -9.28941 q^{86} +3.83198 q^{88} -2.55283 q^{89} -2.63595 q^{90} +0.442278 q^{91} -4.09077 q^{92} -5.69964 q^{93} +13.1924 q^{94} -6.64668 q^{95} -0.603364 q^{96} +8.66831 q^{97} +5.98561 q^{98} -10.1009 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{2} + 6 q^{3} + 18 q^{4} + 18 q^{5} + 6 q^{6} + 8 q^{7} + 18 q^{8} + 22 q^{9} + 18 q^{10} + 22 q^{11} + 6 q^{12} + 16 q^{13} + 8 q^{14} + 6 q^{15} + 18 q^{16} + 4 q^{17} + 22 q^{18} + 20 q^{19}+ \cdots + 86 q^{99}+O(q^{100}) $ 18 * q + 18 * q^2 + 6 * q^3 + 18 * q^4 + 18 * q^5 + 6 * q^6 + 8 * q^7 + 18 * q^8 + 22 * q^9 + 18 * q^10 + 22 * q^11 + 6 * q^12 + 16 * q^13 + 8 * q^14 + 6 * q^15 + 18 * q^16 + 4 * q^17 + 22 * q^18 + 20 * q^19 + 18 * q^20 + 16 * q^21 + 22 * q^22 - 8 * q^23 + 6 * q^24 + 18 * q^25 + 16 * q^26 + 12 * q^27 + 8 * q^28 + 6 * q^30 + 50 * q^31 + 18 * q^32 + 12 * q^33 + 4 * q^34 + 8 * q^35 + 22 * q^36 + 10 * q^37 + 20 * q^38 + 20 * q^39 + 18 * q^40 + 36 * q^41 + 16 * q^42 + 22 * q^43 + 22 * q^44 + 22 * q^45 - 8 * q^46 + 26 * q^47 + 6 * q^48 + 6 * q^49 + 18 * q^50 - 12 * q^51 + 16 * q^52 - 16 * q^53 + 12 * q^54 + 22 * q^55 + 8 * q^56 - 18 * q^57 + 28 * q^59 + 6 * q^60 + 32 * q^61 + 50 * q^62 - 22 * q^63 + 18 * q^64 + 16 * q^65 + 12 * q^66 + 6 * q^67 + 4 * q^68 + 6 * q^69 + 8 * q^70 - 10 * q^71 + 22 * q^72 + 12 * q^73 + 10 * q^74 + 6 * q^75 + 20 * q^76 + 30 * q^77 + 20 * q^78 + 56 * q^79 + 18 * q^80 - 54 * q^81 + 36 * q^82 + 8 * q^83 + 16 * q^84 + 4 * q^85 + 22 * q^86 + 22 * q^88 + 80 * q^89 + 22 * q^90 - 42 * q^91 - 8 * q^92 + 2 * q^93 + 26 * q^94 + 20 * q^95 + 6 * q^96 - 12 * q^97 + 6 * q^98 + 86 * q^99

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
$3$	−0.603364	−0.348353	−0.174176	−	0.984714i	$-0.555726\pi$
$3$	−0.603364	−0.348353	−0.174176	+	0.984714i	$0.555726\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.603364	−0.246322
$7$	3.60356	1.36202	0.681008	−	0.732276i	$-0.261542\pi$
$7$	3.60356	1.36202	0.681008	+	0.732276i	$0.261542\pi$
$8$	1.00000	0.353553
$9$	−2.63595	−0.878650
$10$	1.00000	0.316228
$11$	3.83198	1.15539	0.577693	−	0.816254i	$-0.303953\pi$
$11$	3.83198	1.15539	0.577693	+	0.816254i	$0.303953\pi$
$12$	−0.603364	−0.174176
$13$	0.122734	0.0340402	0.0170201	−	0.999855i	$-0.494582\pi$
$13$	0.122734	0.0340402	0.0170201	+	0.999855i	$0.494582\pi$
$14$	3.60356	0.963091
$15$	−0.603364	−0.155788
$16$	1.00000	0.250000
$17$	−2.31987	−0.562652	−0.281326	−	0.959612i	$-0.590774\pi$
$17$	−2.31987	−0.562652	−0.281326	+	0.959612i	$0.590774\pi$
$18$	−2.63595	−0.621300
$19$	−6.64668	−1.52485	−0.762426	−	0.647075i	$-0.775992\pi$
$19$	−6.64668	−1.52485	−0.762426	+	0.647075i	$0.775992\pi$
$20$	1.00000	0.223607
$21$	−2.17426	−0.474462
$22$	3.83198	0.816981
$23$	−4.09077	−0.852985	−0.426492	−	0.904491i	$-0.640251\pi$
$23$	−4.09077	−0.852985	−0.426492	+	0.904491i	$0.640251\pi$
$24$	−0.603364	−0.123161
$25$	1.00000	0.200000
$26$	0.122734	0.0240701
$27$	3.40053	0.654433
$28$	3.60356	0.681008
$29$	0	0
$29$	0	0
$30$	−0.603364	−0.110159
$31$	9.44644	1.69663	0.848315	−	0.529492i	$-0.177617\pi$
$31$	9.44644	1.69663	0.848315	+	0.529492i	$0.177617\pi$
$32$	1.00000	0.176777
$33$	−2.31208	−0.402482
$34$	−2.31987	−0.397855
$35$	3.60356	0.609112
$36$	−2.63595	−0.439325
$37$	3.57990	0.588532	0.294266	−	0.955724i	$-0.404925\pi$
$37$	3.57990	0.588532	0.294266	+	0.955724i	$0.404925\pi$
$38$	−6.64668	−1.07823
$39$	−0.0740532	−0.0118580
$40$	1.00000	0.158114
$41$	10.4455	1.63131	0.815656	−	0.578537i	$-0.196376\pi$
$41$	10.4455	1.63131	0.815656	+	0.578537i	$0.196376\pi$
$42$	−2.17426	−0.335495
$43$	−9.28941	−1.41662	−0.708310	−	0.705901i	$-0.750542\pi$
$43$	−9.28941	−1.41662	−0.708310	+	0.705901i	$0.750542\pi$
$44$	3.83198	0.577693
$45$	−2.63595	−0.392944
$46$	−4.09077	−0.603151
$47$	13.1924	1.92430	0.962152	−	0.272512i	$-0.0878543\pi$
$47$	13.1924	1.92430	0.962152	+	0.272512i	$0.0878543\pi$
$48$	−0.603364	−0.0870881
$49$	5.98561	0.855087
$50$	1.00000	0.141421
$51$	1.39973	0.196001
$52$	0.122734	0.0170201
$53$	−2.73872	−0.376192	−0.188096	−	0.982151i	$-0.560232\pi$
$53$	−2.73872	−0.376192	−0.188096	+	0.982151i	$0.560232\pi$
$54$	3.40053	0.462754
$55$	3.83198	0.516704
$56$	3.60356	0.481545
$57$	4.01037	0.531186
$58$	0	0
$59$	−0.361609	−0.0470775	−0.0235388	−	0.999723i	$-0.507493\pi$
$59$	−0.361609	−0.0470775	−0.0235388	+	0.999723i	$0.507493\pi$
$60$	−0.603364	−0.0778940
$61$	−6.29724	−0.806279	−0.403140	−	0.915138i	$-0.632081\pi$
$61$	−6.29724	−0.806279	−0.403140	+	0.915138i	$0.632081\pi$
$62$	9.44644	1.19970
$63$	−9.49880	−1.19674
$64$	1.00000	0.125000
$65$	0.122734	0.0152233
$66$	−2.31208	−0.284597
$67$	−4.42687	−0.540828	−0.270414	−	0.962744i	$-0.587161\pi$
$67$	−4.42687	−0.540828	−0.270414	+	0.962744i	$0.587161\pi$
$68$	−2.31987	−0.281326
$69$	2.46823	0.297139
$70$	3.60356	0.430707
$71$	10.1456	1.20407	0.602033	−	0.798471i	$-0.294357\pi$
$71$	10.1456	1.20407	0.602033	+	0.798471i	$0.294357\pi$
$72$	−2.63595	−0.310650
$73$	6.51726	0.762787	0.381394	−	0.924413i	$-0.375444\pi$
$73$	6.51726	0.762787	0.381394	+	0.924413i	$0.375444\pi$
$74$	3.57990	0.416155
$75$	−0.603364	−0.0696705
$76$	−6.64668	−0.762426
$77$	13.8088	1.57365
$78$	−0.0740532	−0.00838488
$79$	13.9998	1.57510	0.787549	−	0.616252i	$-0.211350\pi$
$79$	13.9998	1.57510	0.787549	+	0.616252i	$0.211350\pi$
$80$	1.00000	0.111803
$81$	5.85609	0.650677
$82$	10.4455	1.15351
$83$	−4.87092	−0.534653	−0.267327	−	0.963606i	$-0.586140\pi$
$83$	−4.87092	−0.534653	−0.267327	+	0.963606i	$0.586140\pi$
$84$	−2.17426	−0.237231
$85$	−2.31987	−0.251626
$86$	−9.28941	−1.00170
$87$	0	0
$88$	3.83198	0.408491
$89$	−2.55283	−0.270600	−0.135300	−	0.990805i	$-0.543200\pi$
$89$	−2.55283	−0.270600	−0.135300	+	0.990805i	$0.543200\pi$
$90$	−2.63595	−0.277854
$91$	0.442278	0.0463634
$92$	−4.09077	−0.426492
$93$	−5.69964	−0.591025
$94$	13.1924	1.36069
$95$	−6.64668	−0.681935
$96$	−0.603364	−0.0615806
$97$	8.66831	0.880134	0.440067	−	0.897965i	$-0.354955\pi$
$97$	8.66831	0.880134	0.440067	+	0.897965i	$0.354955\pi$
$98$	5.98561	0.604638
$99$	−10.1009	−1.01518
$100$	1.00000	0.100000
$101$	0.170291	0.0169446	0.00847230	−	0.999964i	$-0.497303\pi$
$101$	0.170291	0.0169446	0.00847230	+	0.999964i	$0.497303\pi$
$102$	1.39973	0.138594
$103$	11.9876	1.18118	0.590589	−	0.806973i	$-0.298896\pi$
$103$	11.9876	1.18118	0.590589	+	0.806973i	$0.298896\pi$
$104$	0.122734	0.0120350
$105$	−2.17426	−0.212186
$106$	−2.73872	−0.266008
$107$	−3.61602	−0.349573	−0.174787	−	0.984606i	$-0.555924\pi$
$107$	−3.61602	−0.349573	−0.174787	+	0.984606i	$0.555924\pi$
$108$	3.40053	0.327216
$109$	16.3476	1.56581	0.782907	−	0.622138i	$-0.213736\pi$
$109$	16.3476	1.56581	0.782907	+	0.622138i	$0.213736\pi$
$110$	3.83198	0.365365
$111$	−2.15998	−0.205017
$112$	3.60356	0.340504
$113$	11.5901	1.09031	0.545154	−	0.838336i	$-0.316471\pi$
$113$	11.5901	1.09031	0.545154	+	0.838336i	$0.316471\pi$
$114$	4.01037	0.375605
$115$	−4.09077	−0.381466
$116$	0	0
$117$	−0.323520	−0.0299095
$118$	−0.361609	−0.0332888
$119$	−8.35979	−0.766341
$120$	−0.603364	−0.0550794
$121$	3.68408	0.334916
$122$	−6.29724	−0.570126
$123$	−6.30244	−0.568272
$124$	9.44644	0.848315
$125$	1.00000	0.0894427
$126$	−9.49880	−0.846220
$127$	20.0845	1.78221	0.891106	−	0.453794i	$-0.149930\pi$
$127$	20.0845	1.78221	0.891106	+	0.453794i	$0.149930\pi$
$128$	1.00000	0.0883883
$129$	5.60490	0.493484
$130$	0.122734	0.0107645
$131$	5.25645	0.459258	0.229629	−	0.973278i	$-0.426249\pi$
$131$	5.25645	0.459258	0.229629	+	0.973278i	$0.426249\pi$
$132$	−2.31208	−0.201241
$133$	−23.9517	−2.07687
$134$	−4.42687	−0.382423
$135$	3.40053	0.292671
$136$	−2.31987	−0.198928
$137$	−1.27995	−0.109354	−0.0546769	−	0.998504i	$-0.517413\pi$
$137$	−1.27995	−0.109354	−0.0546769	+	0.998504i	$0.517413\pi$
$138$	2.46823	0.210109
$139$	9.99988	0.848179	0.424089	−	0.905620i	$-0.360594\pi$
$139$	9.99988	0.848179	0.424089	+	0.905620i	$0.360594\pi$
$140$	3.60356	0.304556
$141$	−7.95981	−0.670337
$142$	10.1456	0.851404
$143$	0.470314	0.0393296
$144$	−2.63595	−0.219663
$145$	0	0
$146$	6.51726	0.539372
$147$	−3.61150	−0.297872
$148$	3.57990	0.294266
$149$	−11.7392	−0.961716	−0.480858	−	0.876799i	$-0.659675\pi$
$149$	−11.7392	−0.961716	−0.480858	+	0.876799i	$0.659675\pi$
$150$	−0.603364	−0.0492645
$151$	0.0354882	0.00288799	0.00144400	−	0.999999i	$-0.499540\pi$
$151$	0.0354882	0.00288799	0.00144400	+	0.999999i	$0.499540\pi$
$152$	−6.64668	−0.539117
$153$	6.11508	0.494375
$154$	13.8088	1.11274
$155$	9.44644	0.758756
$156$	−0.0740532	−0.00592900
$157$	−20.3105	−1.62096	−0.810479	−	0.585768i	$-0.800793\pi$
$157$	−20.3105	−1.62096	−0.810479	+	0.585768i	$0.800793\pi$
$158$	13.9998	1.11376
$159$	1.65245	0.131048
$160$	1.00000	0.0790569
$161$	−14.7413	−1.16178
$162$	5.85609	0.460098
$163$	15.0038	1.17519	0.587595	−	0.809155i	$-0.300075\pi$
$163$	15.0038	1.17519	0.587595	+	0.809155i	$0.300075\pi$
$164$	10.4455	0.815656
$165$	−2.31208	−0.179995
$166$	−4.87092	−0.378057
$167$	9.97196	0.771653	0.385827	−	0.922571i	$-0.373916\pi$
$167$	9.97196	0.771653	0.385827	+	0.922571i	$0.373916\pi$
$168$	−2.17426	−0.167748
$169$	−12.9849	−0.998841
$170$	−2.31987	−0.177926
$171$	17.5203	1.33981
$172$	−9.28941	−0.708310
$173$	−6.48842	−0.493305	−0.246653	−	0.969104i	$-0.579331\pi$
$173$	−6.48842	−0.493305	−0.246653	+	0.969104i	$0.579331\pi$
$174$	0	0
$175$	3.60356	0.272403
$176$	3.83198	0.288846
$177$	0.218182	0.0163996
$178$	−2.55283	−0.191343
$179$	−5.08224	−0.379865	−0.189932	−	0.981797i	$-0.560827\pi$
$179$	−5.08224	−0.379865	−0.189932	+	0.981797i	$0.560827\pi$
$180$	−2.63595	−0.196472
$181$	−6.30332	−0.468522	−0.234261	−	0.972174i	$-0.575267\pi$
$181$	−6.30332	−0.468522	−0.234261	+	0.972174i	$0.575267\pi$
$182$	0.442278	0.0327838
$183$	3.79953	0.280870
$184$	−4.09077	−0.301576
$185$	3.57990	0.263199
$186$	−5.69964	−0.417918
$187$	−8.88971	−0.650080
$188$	13.1924	0.962152
$189$	12.2540	0.891348
$190$	−6.64668	−0.482201
$191$	−16.2449	−1.17544	−0.587719	−	0.809065i	$-0.699974\pi$
$191$	−16.2449	−1.17544	−0.587719	+	0.809065i	$0.699974\pi$
$192$	−0.603364	−0.0435441
$193$	−1.52191	−0.109549	−0.0547747	−	0.998499i	$-0.517444\pi$
$193$	−1.52191	−0.109549	−0.0547747	+	0.998499i	$0.517444\pi$
$194$	8.66831	0.622349
$195$	−0.0740532	−0.00530306
$196$	5.98561	0.427544
$197$	−25.1520	−1.79201	−0.896005	−	0.444045i	$-0.853543\pi$
$197$	−25.1520	−1.79201	−0.896005	+	0.444045i	$0.853543\pi$
$198$	−10.1009	−0.717841
$199$	−5.81104	−0.411934	−0.205967	−	0.978559i	$-0.566034\pi$
$199$	−5.81104	−0.411934	−0.205967	+	0.978559i	$0.566034\pi$
$200$	1.00000	0.0707107
$201$	2.67101	0.188399
$202$	0.170291	0.0119816
$203$	0	0
$204$	1.39973	0.0980007
$205$	10.4455	0.729545
$206$	11.9876	0.835219
$207$	10.7831	0.749476
$208$	0.122734	0.00851006
$209$	−25.4699	−1.76179
$210$	−2.17426	−0.150038
$211$	−4.07697	−0.280670	−0.140335	−	0.990104i	$-0.544818\pi$
$211$	−4.07697	−0.280670	−0.140335	+	0.990104i	$0.544818\pi$
$212$	−2.73872	−0.188096
$213$	−6.12152	−0.419440
$214$	−3.61602	−0.247186
$215$	−9.28941	−0.633532
$216$	3.40053	0.231377
$217$	34.0407	2.31084
$218$	16.3476	1.10720
$219$	−3.93228	−0.265719
$220$	3.83198	0.258352
$221$	−0.284727	−0.0191528
$222$	−2.15998	−0.144969
$223$	4.31633	0.289043	0.144521	−	0.989502i	$-0.453836\pi$
$223$	4.31633	0.289043	0.144521	+	0.989502i	$0.453836\pi$
$224$	3.60356	0.240773
$225$	−2.63595	−0.175730
$226$	11.5901	0.770964
$227$	8.07974	0.536271	0.268136	−	0.963381i	$-0.413592\pi$
$227$	8.07974	0.536271	0.268136	+	0.963381i	$0.413592\pi$
$228$	4.01037	0.265593
$229$	−20.0722	−1.32641	−0.663203	−	0.748439i	$-0.730804\pi$
$229$	−20.0722	−1.32641	−0.663203	+	0.748439i	$0.730804\pi$
$230$	−4.09077	−0.269738
$231$	−8.33171	−0.548186
$232$	0	0
$233$	3.90478	0.255811	0.127905	−	0.991786i	$-0.459175\pi$
$233$	3.90478	0.255811	0.127905	+	0.991786i	$0.459175\pi$
$234$	−0.323520	−0.0211492
$235$	13.1924	0.860575
$236$	−0.361609	−0.0235388
$237$	−8.44697	−0.548690
$238$	−8.35979	−0.541885
$239$	−26.6702	−1.72515	−0.862576	−	0.505928i	$-0.831150\pi$
$239$	−26.6702	−1.72515	−0.862576	+	0.505928i	$0.831150\pi$
$240$	−0.603364	−0.0389470
$241$	−25.4064	−1.63657	−0.818286	−	0.574811i	$-0.805076\pi$
$241$	−25.4064	−1.63657	−0.818286	+	0.574811i	$0.805076\pi$
$242$	3.68408	0.236822
$243$	−13.7350	−0.881098
$244$	−6.29724	−0.403140
$245$	5.98561	0.382407
$246$	−6.30244	−0.401829
$247$	−0.815772	−0.0519063
$248$	9.44644	0.599849
$249$	2.93894	0.186248
$250$	1.00000	0.0632456
$251$	17.7013	1.11730	0.558649	−	0.829404i	$-0.311320\pi$
$251$	17.7013	1.11730	0.558649	+	0.829404i	$0.311320\pi$
$252$	−9.49880	−0.598368
$253$	−15.6758	−0.985527
$254$	20.0845	1.26021
$255$	1.39973	0.0876545
$256$	1.00000	0.0625000
$257$	−4.20376	−0.262224	−0.131112	−	0.991368i	$-0.541855\pi$
$257$	−4.20376	−0.262224	−0.131112	+	0.991368i	$0.541855\pi$
$258$	5.60490	0.348946
$259$	12.9004	0.801589
$260$	0.122734	0.00761163
$261$	0	0
$262$	5.25645	0.324745
$263$	−5.67802	−0.350122	−0.175061	−	0.984558i	$-0.556012\pi$
$263$	−5.67802	−0.350122	−0.175061	+	0.984558i	$0.556012\pi$
$264$	−2.31208	−0.142299
$265$	−2.73872	−0.168238
$266$	−23.9517	−1.46857
$267$	1.54029	0.0942642
$268$	−4.42687	−0.270414
$269$	20.6000	1.25600	0.628002	−	0.778212i	$-0.283873\pi$
$269$	20.6000	1.25600	0.628002	+	0.778212i	$0.283873\pi$
$270$	3.40053	0.206950
$271$	4.53160	0.275275	0.137638	−	0.990483i	$-0.456049\pi$
$271$	4.53160	0.275275	0.137638	+	0.990483i	$0.456049\pi$
$272$	−2.31987	−0.140663
$273$	−0.266855	−0.0161508
$274$	−1.27995	−0.0773249
$275$	3.83198	0.231077
$276$	2.46823	0.148570
$277$	0.117400	0.00705390	0.00352695	−	0.999994i	$-0.498877\pi$
$277$	0.117400	0.00705390	0.00352695	+	0.999994i	$0.498877\pi$
$278$	9.99988	0.599753
$279$	−24.9003	−1.49074
$280$	3.60356	0.215354
$281$	25.9075	1.54551	0.772757	−	0.634702i	$-0.218877\pi$
$281$	25.9075	1.54551	0.772757	+	0.634702i	$0.218877\pi$
$282$	−7.95981	−0.474000
$283$	−0.0238487	−0.00141766	−0.000708830	−	1.00000i	$-0.500226\pi$
$283$	−0.0238487	−0.00141766	−0.000708830		1.00000i	$0.500226\pi$
$284$	10.1456	0.602033
$285$	4.01037	0.237554
$286$	0.470314	0.0278102
$287$	37.6409	2.22187
$288$	−2.63595	−0.155325
$289$	−11.6182	−0.683423
$290$	0	0
$291$	−5.23015	−0.306597
$292$	6.51726	0.381394
$293$	−20.4762	−1.19623	−0.598117	−	0.801409i	$-0.704084\pi$
$293$	−20.4762	−1.19623	−0.598117	+	0.801409i	$0.704084\pi$
$294$	−3.61150	−0.210627
$295$	−0.361609	−0.0210537
$296$	3.57990	0.208077
$297$	13.0308	0.756122
$298$	−11.7392	−0.680036
$299$	−0.502076	−0.0290358
$300$	−0.603364	−0.0348353
$301$	−33.4749	−1.92946
$302$	0.0354882	0.00204212
$303$	−0.102748	−0.00590269
$304$	−6.64668	−0.381213
$305$	−6.29724	−0.360579
$306$	6.11508	0.349576
$307$	−4.82792	−0.275544	−0.137772	−	0.990464i	$-0.543994\pi$
$307$	−4.82792	−0.275544	−0.137772	+	0.990464i	$0.543994\pi$
$308$	13.8088	0.786827
$309$	−7.23292	−0.411466
$310$	9.44644	0.536521
$311$	3.69623	0.209594	0.104797	−	0.994494i	$-0.466581\pi$
$311$	3.69623	0.209594	0.104797	+	0.994494i	$0.466581\pi$
$312$	−0.0740532	−0.00419244
$313$	18.8606	1.06606	0.533032	−	0.846095i	$-0.321053\pi$
$313$	18.8606	1.06606	0.533032	+	0.846095i	$0.321053\pi$
$314$	−20.3105	−1.14619
$315$	−9.49880	−0.535197
$316$	13.9998	0.787549
$317$	−21.8144	−1.22522	−0.612610	−	0.790385i	$-0.709880\pi$
$317$	−21.8144	−1.22522	−0.612610	+	0.790385i	$0.709880\pi$
$318$	1.65245	0.0926646
$319$	0	0
$320$	1.00000	0.0559017
$321$	2.18177	0.121775
$322$	−14.7413	−0.821502
$323$	15.4195	0.857961
$324$	5.85609	0.325339
$325$	0.122734	0.00680805
$326$	15.0038	0.830985
$327$	−9.86355	−0.545456
$328$	10.4455	0.576756
$329$	47.5394	2.62093
$330$	−2.31208	−0.127276
$331$	−17.8206	−0.979511	−0.489755	−	0.871860i	$-0.662914\pi$
$331$	−17.8206	−0.979511	−0.489755	+	0.871860i	$0.662914\pi$
$332$	−4.87092	−0.267327
$333$	−9.43644	−0.517114
$334$	9.97196	0.545641
$335$	−4.42687	−0.241866
$336$	−2.17426	−0.118615
$337$	2.63549	0.143564	0.0717822	−	0.997420i	$-0.477131\pi$
$337$	2.63549	0.143564	0.0717822	+	0.997420i	$0.477131\pi$
$338$	−12.9849	−0.706287
$339$	−6.99307	−0.379812
$340$	−2.31987	−0.125813
$341$	36.1986	1.96026
$342$	17.5203	0.947390
$343$	−3.65541	−0.197374
$344$	−9.28941	−0.500851
$345$	2.46823	0.132885
$346$	−6.48842	−0.348820
$347$	−0.194460	−0.0104392	−0.00521958	−	0.999986i	$-0.501661\pi$
$347$	−0.194460	−0.0104392	−0.00521958	+	0.999986i	$0.501661\pi$
$348$	0	0
$349$	25.8228	1.38226	0.691130	−	0.722730i	$-0.257113\pi$
$349$	25.8228	1.38226	0.691130	+	0.722730i	$0.257113\pi$
$350$	3.60356	0.192618
$351$	0.417360	0.0222771
$352$	3.83198	0.204245
$353$	1.95990	0.104315	0.0521575	−	0.998639i	$-0.483390\pi$
$353$	1.95990	0.104315	0.0521575	+	0.998639i	$0.483390\pi$
$354$	0.218182	0.0115963
$355$	10.1456	0.538475
$356$	−2.55283	−0.135300
$357$	5.04400	0.266957
$358$	−5.08224	−0.268605
$359$	0.296858	0.0156676	0.00783379	−	0.999969i	$-0.497506\pi$
$359$	0.296858	0.0156676	0.00783379	+	0.999969i	$0.497506\pi$
$360$	−2.63595	−0.138927
$361$	25.1783	1.32517
$362$	−6.30332	−0.331295
$363$	−2.22284	−0.116669
$364$	0.442278	0.0231817
$365$	6.51726	0.341129
$366$	3.79953	0.198605
$367$	26.4923	1.38288	0.691442	−	0.722432i	$-0.256976\pi$
$367$	26.4923	1.38288	0.691442	+	0.722432i	$0.256976\pi$
$368$	−4.09077	−0.213246
$369$	−27.5338	−1.43335
$370$	3.57990	0.186110
$371$	−9.86913	−0.512380
$372$	−5.69964	−0.295513
$373$	0.354533	0.0183570	0.00917852	−	0.999958i	$-0.497078\pi$
$373$	0.354533	0.0183570	0.00917852	+	0.999958i	$0.497078\pi$
$374$	−8.88971	−0.459676
$375$	−0.603364	−0.0311576
$376$	13.1924	0.680345
$377$	0	0
$378$	12.2540	0.630278
$379$	−30.3765	−1.56034	−0.780168	−	0.625570i	$-0.784867\pi$
$379$	−30.3765	−1.56034	−0.780168	+	0.625570i	$0.784867\pi$
$380$	−6.64668	−0.340967
$381$	−12.1183	−0.620838
$382$	−16.2449	−0.831160
$383$	13.1239	0.670601	0.335300	−	0.942111i	$-0.391162\pi$
$383$	13.1239	0.670601	0.335300	+	0.942111i	$0.391162\pi$
$384$	−0.603364	−0.0307903
$385$	13.8088	0.703759
$386$	−1.52191	−0.0774631
$387$	24.4864	1.24471
$388$	8.66831	0.440067
$389$	27.2570	1.38198	0.690992	−	0.722863i	$-0.257174\pi$
$389$	27.2570	1.38198	0.690992	+	0.722863i	$0.257174\pi$
$390$	−0.0740532	−0.00374983
$391$	9.49008	0.479934
$392$	5.98561	0.302319
$393$	−3.17156	−0.159984
$394$	−25.1520	−1.26714
$395$	13.9998	0.704405
$396$	−10.1009	−0.507590
$397$	−6.14121	−0.308218	−0.154109	−	0.988054i	$-0.549251\pi$
$397$	−6.14121	−0.308218	−0.154109	+	0.988054i	$0.549251\pi$
$398$	−5.81104	−0.291281
$399$	14.4516	0.723484
$400$	1.00000	0.0500000
$401$	8.87415	0.443154	0.221577	−	0.975143i	$-0.428880\pi$
$401$	8.87415	0.443154	0.221577	+	0.975143i	$0.428880\pi$
$402$	2.67101	0.133218
$403$	1.15940	0.0577537
$404$	0.170291	0.00847230
$405$	5.85609	0.290992
$406$	0	0
$407$	13.7181	0.679981
$408$	1.39973	0.0692969
$409$	19.6980	0.974005	0.487003	−	0.873400i	$-0.338090\pi$
$409$	19.6980	0.974005	0.487003	+	0.873400i	$0.338090\pi$
$410$	10.4455	0.515866
$411$	0.772279	0.0380937
$412$	11.9876	0.590589
$413$	−1.30308	−0.0641204
$414$	10.7831	0.529959
$415$	−4.87092	−0.239104
$416$	0.122734	0.00601752
$417$	−6.03357	−0.295465
$418$	−25.4699	−1.24578
$419$	25.9470	1.26759	0.633796	−	0.773500i	$-0.281496\pi$
$419$	25.9470	1.26759	0.633796	+	0.773500i	$0.281496\pi$
$420$	−2.17426	−0.106093
$421$	15.1158	0.736701	0.368350	−	0.929687i	$-0.379923\pi$
$421$	15.1158	0.736701	0.368350	+	0.929687i	$0.379923\pi$
$422$	−4.07697	−0.198464
$423$	−34.7744	−1.69079
$424$	−2.73872	−0.133004
$425$	−2.31987	−0.112530
$426$	−6.12152	−0.296589
$427$	−22.6925	−1.09817
$428$	−3.61602	−0.174787
$429$	−0.283771	−0.0137006
$430$	−9.28941	−0.447975
$431$	25.2941	1.21837	0.609186	−	0.793027i	$-0.291496\pi$
$431$	25.2941	1.21837	0.609186	+	0.793027i	$0.291496\pi$
$432$	3.40053	0.163608
$433$	41.1212	1.97616	0.988080	−	0.153943i	$-0.0491972\pi$
$433$	41.1212	1.97616	0.988080	+	0.153943i	$0.0491972\pi$
$434$	34.0407	1.63401
$435$	0	0
$436$	16.3476	0.782907
$437$	27.1900	1.30068
$438$	−3.93228	−0.187892
$439$	−35.9038	−1.71360	−0.856798	−	0.515652i	$-0.827550\pi$
$439$	−35.9038	−1.71360	−0.856798	+	0.515652i	$0.827550\pi$
$440$	3.83198	0.182683
$441$	−15.7778	−0.751323
$442$	−0.284727	−0.0135431
$443$	17.1974	0.817073	0.408536	−	0.912742i	$-0.366039\pi$
$443$	17.1974	0.817073	0.408536	+	0.912742i	$0.366039\pi$
$444$	−2.15998	−0.102508
$445$	−2.55283	−0.121016
$446$	4.31633	0.204384
$447$	7.08304	0.335016
$448$	3.60356	0.170252
$449$	20.9010	0.986381	0.493191	−	0.869921i	$-0.335831\pi$
$449$	20.9010	0.986381	0.493191	+	0.869921i	$0.335831\pi$
$450$	−2.63595	−0.124260
$451$	40.0269	1.88479
$452$	11.5901	0.545154
$453$	−0.0214123	−0.00100604
$454$	8.07974	0.379201
$455$	0.442278	0.0207343
$456$	4.01037	0.187803
$457$	2.10802	0.0986089	0.0493045	−	0.998784i	$-0.484300\pi$
$457$	2.10802	0.0986089	0.0493045	+	0.998784i	$0.484300\pi$
$458$	−20.0722	−0.937911
$459$	−7.88881	−0.368218
$460$	−4.09077	−0.190733
$461$	0.540464	0.0251719	0.0125860	−	0.999921i	$-0.495994\pi$
$461$	0.540464	0.0251719	0.0125860	+	0.999921i	$0.495994\pi$
$462$	−8.33171	−0.387626
$463$	−19.6759	−0.914418	−0.457209	−	0.889359i	$-0.651151\pi$
$463$	−19.6759	−0.914418	−0.457209	+	0.889359i	$0.651151\pi$
$464$	0	0
$465$	−5.69964	−0.264315
$466$	3.90478	0.180885
$467$	−38.8360	−1.79711	−0.898557	−	0.438856i	$-0.855384\pi$
$467$	−38.8360	−1.79711	−0.898557	+	0.438856i	$0.855384\pi$
$468$	−0.323520	−0.0149547
$469$	−15.9525	−0.736616
$470$	13.1924	0.608519
$471$	12.2547	0.564665
$472$	−0.361609	−0.0166444
$473$	−35.5968	−1.63674
$474$	−8.44697	−0.387982
$475$	−6.64668	−0.304970
$476$	−8.35979	−0.383171
$477$	7.21914	0.330542
$478$	−26.6702	−1.21987
$479$	−14.4617	−0.660774	−0.330387	−	0.943846i	$-0.607179\pi$
$479$	−14.4617	−0.660774	−0.330387	+	0.943846i	$0.607179\pi$
$480$	−0.603364	−0.0275397
$481$	0.439375	0.0200338
$482$	−25.4064	−1.15723
$483$	8.89439	0.404709
$484$	3.68408	0.167458
$485$	8.66831	0.393608
$486$	−13.7350	−0.623030
$487$	41.4896	1.88007	0.940037	−	0.341072i	$-0.110790\pi$
$487$	41.4896	1.88007	0.940037	+	0.341072i	$0.110790\pi$
$488$	−6.29724	−0.285063
$489$	−9.05278	−0.409381
$490$	5.98561	0.270402
$491$	37.3413	1.68519	0.842595	−	0.538548i	$-0.181027\pi$
$491$	37.3413	1.68519	0.842595	+	0.538548i	$0.181027\pi$
$492$	−6.30244	−0.284136
$493$	0	0
$494$	−0.815772	−0.0367033
$495$	−10.1009	−0.454002
$496$	9.44644	0.424157
$497$	36.5604	1.63996
$498$	2.93894	0.131697
$499$	−3.12421	−0.139859	−0.0699294	−	0.997552i	$-0.522277\pi$
$499$	−3.12421	−0.139859	−0.0699294	+	0.997552i	$0.522277\pi$
$500$	1.00000	0.0447214
$501$	−6.01672	−0.268807
$502$	17.7013	0.790049
$503$	10.5499	0.470397	0.235199	−	0.971947i	$-0.424426\pi$
$503$	10.5499	0.470397	0.235199	+	0.971947i	$0.424426\pi$
$504$	−9.49880	−0.423110
$505$	0.170291	0.00757785
$506$	−15.6758	−0.696873
$507$	7.83465	0.347949
$508$	20.0845	0.891106
$509$	12.5371	0.555699	0.277850	−	0.960625i	$-0.410378\pi$
$509$	12.5371	0.555699	0.277850	+	0.960625i	$0.410378\pi$
$510$	1.39973	0.0619811
$511$	23.4853	1.03893
$512$	1.00000	0.0441942
$513$	−22.6022	−0.997913
$514$	−4.20376	−0.185420
$515$	11.9876	0.528239
$516$	5.60490	0.246742
$517$	50.5529	2.22331
$518$	12.9004	0.566809
$519$	3.91488	0.171844
$520$	0.122734	0.00538224
$521$	−24.0757	−1.05478	−0.527388	−	0.849625i	$-0.676829\pi$
$521$	−24.0757	−1.05478	−0.527388	+	0.849625i	$0.676829\pi$
$522$	0	0
$523$	−15.4390	−0.675102	−0.337551	−	0.941307i	$-0.609598\pi$
$523$	−15.4390	−0.675102	−0.337551	+	0.941307i	$0.609598\pi$
$524$	5.25645	0.229629
$525$	−2.17426	−0.0948923
$526$	−5.67802	−0.247574
$527$	−21.9145	−0.954612
$528$	−2.31208	−0.100620
$529$	−6.26559	−0.272417
$530$	−2.73872	−0.118962
$531$	0.953185	0.0413647
$532$	−23.9517	−1.03844
$533$	1.28202	0.0555303
$534$	1.54029	0.0666548
$535$	−3.61602	−0.156334
$536$	−4.42687	−0.191211
$537$	3.06644	0.132327
$538$	20.6000	0.888129
$539$	22.9367	0.987955
$540$	3.40053	0.146336
$541$	−1.92205	−0.0826353	−0.0413176	−	0.999146i	$-0.513156\pi$
$541$	−1.92205	−0.0826353	−0.0413176	+	0.999146i	$0.513156\pi$
$542$	4.53160	0.194649
$543$	3.80320	0.163211
$544$	−2.31987	−0.0994638
$545$	16.3476	0.700254
$546$	−0.266855	−0.0114203
$547$	−0.779603	−0.0333334	−0.0166667	−	0.999861i	$-0.505305\pi$
$547$	−0.779603	−0.0333334	−0.0166667	+	0.999861i	$0.505305\pi$
$548$	−1.27995	−0.0546769
$549$	16.5992	0.708438
$550$	3.83198	0.163396
$551$	0	0
$552$	2.46823	0.105055
$553$	50.4490	2.14531
$554$	0.117400	0.00498786
$555$	−2.15998	−0.0916862
$556$	9.99988	0.424089
$557$	23.2478	0.985040	0.492520	−	0.870301i	$-0.336076\pi$
$557$	23.2478	0.985040	0.492520	+	0.870301i	$0.336076\pi$
$558$	−24.9003	−1.05412
$559$	−1.14012	−0.0482221
$560$	3.60356	0.152278
$561$	5.36374	0.226457
$562$	25.9075	1.09284
$563$	−22.6546	−0.954778	−0.477389	−	0.878692i	$-0.658417\pi$
$563$	−22.6546	−0.954778	−0.477389	+	0.878692i	$0.658417\pi$
$564$	−7.95981	−0.335168
$565$	11.5901	0.487600
$566$	−0.0238487	−0.00100244
$567$	21.1028	0.886233
$568$	10.1456	0.425702
$569$	1.48366	0.0621981	0.0310990	−	0.999516i	$-0.490099\pi$
$569$	1.48366	0.0621981	0.0310990	+	0.999516i	$0.490099\pi$
$570$	4.01037	0.167976
$571$	−3.43685	−0.143828	−0.0719139	−	0.997411i	$-0.522911\pi$
$571$	−3.43685	−0.143828	−0.0719139	+	0.997411i	$0.522911\pi$
$572$	0.470314	0.0196648
$573$	9.80158	0.409467
$574$	37.6409	1.57110
$575$	−4.09077	−0.170597
$576$	−2.63595	−0.109831
$577$	−22.3286	−0.929554	−0.464777	−	0.885428i	$-0.653865\pi$
$577$	−22.3286	−0.929554	−0.464777	+	0.885428i	$0.653865\pi$
$578$	−11.6182	−0.483253
$579$	0.918266	0.0381618
$580$	0	0
$581$	−17.5526	−0.728206
$582$	−5.23015	−0.216797
$583$	−10.4947	−0.434647
$584$	6.51726	0.269686
$585$	−0.323520	−0.0133759
$586$	−20.4762	−0.845865
$587$	−8.88067	−0.366544	−0.183272	−	0.983062i	$-0.558669\pi$
$587$	−8.88067	−0.366544	−0.183272	+	0.983062i	$0.558669\pi$
$588$	−3.61150	−0.148936
$589$	−62.7874	−2.58711
$590$	−0.361609	−0.0148872
$591$	15.1759	0.624251
$592$	3.57990	0.147133
$593$	−40.6401	−1.66889	−0.834444	−	0.551092i	$-0.814211\pi$
$593$	−40.6401	−1.66889	−0.834444	+	0.551092i	$0.814211\pi$
$594$	13.0308	0.534659
$595$	−8.35979	−0.342718
$596$	−11.7392	−0.480858
$597$	3.50617	0.143498
$598$	−0.502076	−0.0205314
$599$	10.9687	0.448168	0.224084	−	0.974570i	$-0.428061\pi$
$599$	10.9687	0.448168	0.224084	+	0.974570i	$0.428061\pi$
$600$	−0.603364	−0.0246322
$601$	−25.8168	−1.05309	−0.526544	−	0.850148i	$-0.676513\pi$
$601$	−25.8168	−1.05309	−0.526544	+	0.850148i	$0.676513\pi$
$602$	−33.4749	−1.36433
$603$	11.6690	0.475199
$604$	0.0354882	0.00144400
$605$	3.68408	0.149779
$606$	−0.102748	−0.00417383
$607$	8.62429	0.350049	0.175025	−	0.984564i	$-0.443999\pi$
$607$	8.62429	0.350049	0.175025	+	0.984564i	$0.443999\pi$
$608$	−6.64668	−0.269558
$609$	0	0
$610$	−6.29724	−0.254968
$611$	1.61915	0.0655038
$612$	6.11508	0.247187
$613$	−31.2508	−1.26221	−0.631105	−	0.775698i	$-0.717398\pi$
$613$	−31.2508	−1.26221	−0.631105	+	0.775698i	$0.717398\pi$
$614$	−4.82792	−0.194839
$615$	−6.30244	−0.254139
$616$	13.8088	0.556371
$617$	−44.9370	−1.80910	−0.904548	−	0.426372i	$-0.859792\pi$
$617$	−44.9370	−1.80910	−0.904548	+	0.426372i	$0.859792\pi$
$618$	−7.23292	−0.290951
$619$	−5.56321	−0.223604	−0.111802	−	0.993730i	$-0.535662\pi$
$619$	−5.56321	−0.223604	−0.111802	+	0.993730i	$0.535662\pi$
$620$	9.44644	0.379378
$621$	−13.9108	−0.558221
$622$	3.69623	0.148205
$623$	−9.19928	−0.368561
$624$	−0.0740532	−0.00296450
$625$	1.00000	0.0400000
$626$	18.8606	0.753821
$627$	15.3677	0.613725
$628$	−20.3105	−0.810479
$629$	−8.30491	−0.331139
$630$	−9.49880	−0.378441
$631$	−18.5487	−0.738411	−0.369205	−	0.929348i	$-0.620370\pi$
$631$	−18.5487	−0.738411	−0.369205	+	0.929348i	$0.620370\pi$
$632$	13.9998	0.556881
$633$	2.45990	0.0977721
$634$	−21.8144	−0.866361
$635$	20.0845	0.797030
$636$	1.65245	0.0655238
$637$	0.734637	0.0291074
$638$	0	0
$639$	−26.7434	−1.05795
$640$	1.00000	0.0395285
$641$	−15.4052	−0.608467	−0.304234	−	0.952597i	$-0.598400\pi$
$641$	−15.4052	−0.608467	−0.304234	+	0.952597i	$0.598400\pi$
$642$	2.18177	0.0861078
$643$	−30.9993	−1.22249	−0.611247	−	0.791440i	$-0.709332\pi$
$643$	−30.9993	−1.22249	−0.611247	+	0.791440i	$0.709332\pi$
$644$	−14.7413	−0.580889
$645$	5.60490	0.220693
$646$	15.4195	0.606670
$647$	−47.0809	−1.85094	−0.925470	−	0.378820i	$-0.876330\pi$
$647$	−47.0809	−1.85094	−0.925470	+	0.378820i	$0.876330\pi$
$648$	5.85609	0.230049
$649$	−1.38568	−0.0543927
$650$	0.122734	0.00481402
$651$	−20.5390	−0.804986
$652$	15.0038	0.587595
$653$	33.5439	1.31267	0.656336	−	0.754468i	$-0.272105\pi$
$653$	33.5439	1.31267	0.656336	+	0.754468i	$0.272105\pi$
$654$	−9.86355	−0.385695
$655$	5.25645	0.205387
$656$	10.4455	0.407828
$657$	−17.1792	−0.670223
$658$	47.5394	1.85328
$659$	4.28214	0.166809	0.0834043	−	0.996516i	$-0.473421\pi$
$659$	4.28214	0.166809	0.0834043	+	0.996516i	$0.473421\pi$
$660$	−2.31208	−0.0899976
$661$	−30.7677	−1.19673	−0.598363	−	0.801225i	$-0.704182\pi$
$661$	−30.7677	−1.19673	−0.598363	+	0.801225i	$0.704182\pi$
$662$	−17.8206	−0.692619
$663$	0.171794	0.00667193
$664$	−4.87092	−0.189028
$665$	−23.9517	−0.928806
$666$	−9.43644	−0.365655
$667$	0	0
$668$	9.97196	0.385827
$669$	−2.60432	−0.100689
$670$	−4.42687	−0.171025
$671$	−24.1309	−0.931564
$672$	−2.17426	−0.0838738
$673$	27.7149	1.06833	0.534165	−	0.845380i	$-0.320626\pi$
$673$	27.7149	1.06833	0.534165	+	0.845380i	$0.320626\pi$
$674$	2.63549	0.101515
$675$	3.40053	0.130887
$676$	−12.9849	−0.499421
$677$	−18.0761	−0.694723	−0.347361	−	0.937731i	$-0.612922\pi$
$677$	−18.0761	−0.694723	−0.347361	+	0.937731i	$0.612922\pi$
$678$	−6.99307	−0.268567
$679$	31.2367	1.19876
$680$	−2.31987	−0.0889631
$681$	−4.87503	−0.186812
$682$	36.1986	1.38611
$683$	−35.1214	−1.34388	−0.671942	−	0.740604i	$-0.734540\pi$
$683$	−35.1214	−1.34388	−0.671942	+	0.740604i	$0.734540\pi$
$684$	17.5203	0.669906
$685$	−1.27995	−0.0489045
$686$	−3.65541	−0.139564
$687$	12.1108	0.462057
$688$	−9.28941	−0.354155
$689$	−0.336134	−0.0128057
$690$	2.46823	0.0939638
$691$	26.7629	1.01811	0.509055	−	0.860734i	$-0.329995\pi$
$691$	26.7629	1.01811	0.509055	+	0.860734i	$0.329995\pi$
$692$	−6.48842	−0.246653
$693$	−36.3992	−1.38269
$694$	−0.194460	−0.00738160
$695$	9.99988	0.379317
$696$	0	0
$697$	−24.2322	−0.917861
$698$	25.8228	0.977406
$699$	−2.35600	−0.0891123
$700$	3.60356	0.136202
$701$	−25.6224	−0.967743	−0.483872	−	0.875139i	$-0.660770\pi$
$701$	−25.6224	−0.967743	−0.483872	+	0.875139i	$0.660770\pi$
$702$	0.417360	0.0157523
$703$	−23.7944	−0.897424
$704$	3.83198	0.144423
$705$	−7.95981	−0.299784
$706$	1.95990	0.0737618
$707$	0.613653	0.0230788
$708$	0.218182	0.00819979
$709$	23.6702	0.888954	0.444477	−	0.895790i	$-0.353390\pi$
$709$	23.6702	0.888954	0.444477	+	0.895790i	$0.353390\pi$
$710$	10.1456	0.380759
$711$	−36.9027	−1.38396
$712$	−2.55283	−0.0956715
$713$	−38.6432	−1.44720
$714$	5.04400	0.188767
$715$	0.470314	0.0175887
$716$	−5.08224	−0.189932
$717$	16.0918	0.600961
$718$	0.296858	0.0110787
$719$	0.902040	0.0336404	0.0168202	−	0.999859i	$-0.494646\pi$
$719$	0.902040	0.0336404	0.0168202	+	0.999859i	$0.494646\pi$
$720$	−2.63595	−0.0982361
$721$	43.1981	1.60878
$722$	25.1783	0.937040
$723$	15.3293	0.570104
$724$	−6.30332	−0.234261
$725$	0	0
$726$	−2.22284	−0.0824974
$727$	−38.7704	−1.43792	−0.718958	−	0.695054i	$-0.755381\pi$
$727$	−38.7704	−1.43792	−0.718958	+	0.695054i	$0.755381\pi$
$728$	0.442278	0.0163919
$729$	−9.28110	−0.343744
$730$	6.51726	0.241215
$731$	21.5503	0.797065
$732$	3.79953	0.140435
$733$	−6.37964	−0.235637	−0.117819	−	0.993035i	$-0.537590\pi$
$733$	−6.37964	−0.235637	−0.117819	+	0.993035i	$0.537590\pi$
$734$	26.4923	0.977847
$735$	−3.61150	−0.133212
$736$	−4.09077	−0.150788
$737$	−16.9637	−0.624865
$738$	−27.5338	−1.01353
$739$	23.8248	0.876411	0.438205	−	0.898875i	$-0.355614\pi$
$739$	23.8248	0.876411	0.438205	+	0.898875i	$0.355614\pi$
$740$	3.57990	0.131600
$741$	0.492208	0.0180817
$742$	−9.86913	−0.362307
$743$	15.9546	0.585318	0.292659	−	0.956217i	$-0.405460\pi$
$743$	15.9546	0.585318	0.292659	+	0.956217i	$0.405460\pi$
$744$	−5.69964	−0.208959
$745$	−11.7392	−0.430092
$746$	0.354533	0.0129804
$747$	12.8395	0.469773
$748$	−8.88971	−0.325040
$749$	−13.0305	−0.476124
$750$	−0.603364	−0.0220318
$751$	0.789105	0.0287948	0.0143974	−	0.999896i	$-0.495417\pi$
$751$	0.789105	0.0287948	0.0143974	+	0.999896i	$0.495417\pi$
$752$	13.1924	0.481076
$753$	−10.6804	−0.389214
$754$	0	0
$755$	0.0354882	0.00129155
$756$	12.2540	0.445674
$757$	20.6059	0.748934	0.374467	−	0.927240i	$-0.377826\pi$
$757$	20.6059	0.748934	0.374467	+	0.927240i	$0.377826\pi$
$758$	−30.3765	−1.10332
$759$	9.45820	0.343311
$760$	−6.64668	−0.241100
$761$	−45.3956	−1.64559	−0.822794	−	0.568339i	$-0.807586\pi$
$761$	−45.3956	−1.64559	−0.822794	+	0.568339i	$0.807586\pi$
$762$	−12.1183	−0.438999
$763$	58.9094	2.13266
$764$	−16.2449	−0.587719
$765$	6.11508	0.221091
$766$	13.1239	0.474186
$767$	−0.0443817	−0.00160253
$768$	−0.603364	−0.0217720
$769$	−4.77005	−0.172012	−0.0860061	−	0.996295i	$-0.527410\pi$
$769$	−4.77005	−0.172012	−0.0860061	+	0.996295i	$0.527410\pi$
$770$	13.8088	0.497633
$771$	2.53640	0.0913462
$772$	−1.52191	−0.0547747
$773$	40.0731	1.44133	0.720665	−	0.693284i	$-0.243837\pi$
$773$	40.0731	1.44133	0.720665	+	0.693284i	$0.243837\pi$
$774$	24.4864	0.880146
$775$	9.44644	0.339326
$776$	8.66831	0.311174
$777$	−7.78362	−0.279236
$778$	27.2570	0.977210
$779$	−69.4278	−2.48751
$780$	−0.0740532	−0.00265153
$781$	38.8779	1.39116
$782$	9.49008	0.339364
$783$	0	0
$784$	5.98561	0.213772
$785$	−20.3105	−0.724914
$786$	−3.17156	−0.113126
$787$	−17.6384	−0.628742	−0.314371	−	0.949300i	$-0.601794\pi$
$787$	−17.6384	−0.628742	−0.314371	+	0.949300i	$0.601794\pi$
$788$	−25.1520	−0.896005
$789$	3.42592	0.121966
$790$	13.9998	0.498090
$791$	41.7657	1.48502
$792$	−10.1009	−0.358920
$793$	−0.772885	−0.0274459
$794$	−6.14121	−0.217943
$795$	1.65245	0.0586063
$796$	−5.81104	−0.205967
$797$	−1.63486	−0.0579098	−0.0289549	−	0.999581i	$-0.509218\pi$
$797$	−1.63486	−0.0579098	−0.0289549	+	0.999581i	$0.509218\pi$
$798$	14.4516	0.511580
$799$	−30.6046	−1.08271
$800$	1.00000	0.0353553
$801$	6.72915	0.237763
$802$	8.87415	0.313357
$803$	24.9740	0.881314
$804$	2.67101	0.0941994
$805$	−14.7413	−0.519563
$806$	1.15940	0.0408380
$807$	−12.4293	−0.437532
$808$	0.170291	0.00599082
$809$	34.8204	1.22422	0.612111	−	0.790772i	$-0.290321\pi$
$809$	34.8204	1.22422	0.612111	+	0.790772i	$0.290321\pi$
$810$	5.85609	0.205762
$811$	6.86089	0.240918	0.120459	−	0.992718i	$-0.461563\pi$
$811$	6.86089	0.240918	0.120459	+	0.992718i	$0.461563\pi$
$812$	0	0
$813$	−2.73421	−0.0958929
$814$	13.7181	0.480819
$815$	15.0038	0.525561
$816$	1.39973	0.0490003
$817$	61.7437	2.16014
$818$	19.6980	0.688726
$819$	−1.16582	−0.0407372
$820$	10.4455	0.364772
$821$	54.4518	1.90038	0.950190	−	0.311671i	$-0.100889\pi$
$821$	54.4518	1.90038	0.950190	+	0.311671i	$0.100889\pi$
$822$	0.772279	0.0269363
$823$	−22.9091	−0.798561	−0.399281	−	0.916829i	$-0.630740\pi$
$823$	−22.9091	−0.798561	−0.399281	+	0.916829i	$0.630740\pi$
$824$	11.9876	0.417609
$825$	−2.31208	−0.0804963
$826$	−1.30308	−0.0453399
$827$	−2.08475	−0.0724939	−0.0362469	−	0.999343i	$-0.511540\pi$
$827$	−2.08475	−0.0724939	−0.0362469	+	0.999343i	$0.511540\pi$
$828$	10.7831	0.374738
$829$	6.08168	0.211226	0.105613	−	0.994407i	$-0.466320\pi$
$829$	6.08168	0.211226	0.105613	+	0.994407i	$0.466320\pi$
$830$	−4.87092	−0.169072
$831$	−0.0708352	−0.00245725
$832$	0.122734	0.00425503
$833$	−13.8859	−0.481117
$834$	−6.03357	−0.208925
$835$	9.97196	0.345094
$836$	−25.4699	−0.880896
$837$	32.1229	1.11033
$838$	25.9470	0.896323
$839$	−10.9715	−0.378780	−0.189390	−	0.981902i	$-0.560651\pi$
$839$	−10.9715	−0.378780	−0.189390	+	0.981902i	$0.560651\pi$
$840$	−2.17426	−0.0750190
$841$	0	0
$842$	15.1158	0.520926
$843$	−15.6317	−0.538384
$844$	−4.07697	−0.140335
$845$	−12.9849	−0.446695
$846$	−34.7744	−1.19557
$847$	13.2758	0.456161
$848$	−2.73872	−0.0940481
$849$	0.0143895	0.000493846 0
$850$	−2.31987	−0.0795710
$851$	−14.6445	−0.502009
$852$	−6.12152	−0.209720
$853$	37.6867	1.29037	0.645183	−	0.764028i	$-0.276781\pi$
$853$	37.6867	1.29037	0.645183	+	0.764028i	$0.276781\pi$
$854$	−22.6925	−0.776520
$855$	17.5203	0.599182
$856$	−3.61602	−0.123593
$857$	33.7203	1.15186	0.575932	−	0.817498i	$-0.304639\pi$
$857$	33.7203	1.15186	0.575932	+	0.817498i	$0.304639\pi$
$858$	−0.283771	−0.00968777
$859$	−26.4372	−0.902024	−0.451012	−	0.892518i	$-0.648937\pi$
$859$	−26.4372	−0.902024	−0.451012	+	0.892518i	$0.648937\pi$
$860$	−9.28941	−0.316766
$861$	−22.7112	−0.773995
$862$	25.2941	0.861520
$863$	−11.5573	−0.393414	−0.196707	−	0.980462i	$-0.563025\pi$
$863$	−11.5573	−0.393414	−0.196707	+	0.980462i	$0.563025\pi$
$864$	3.40053	0.115688
$865$	−6.48842	−0.220613
$866$	41.1212	1.39736
$867$	7.01000	0.238072
$868$	34.0407	1.15542
$869$	53.6469	1.81985
$870$	0	0
$871$	−0.543326	−0.0184099
$872$	16.3476	0.553599
$873$	−22.8493	−0.773330
$874$	27.1900	0.919717
$875$	3.60356	0.121822
$876$	−3.93228	−0.132859
$877$	−10.7417	−0.362723	−0.181361	−	0.983416i	$-0.558050\pi$
$877$	−10.7417	−0.362723	−0.181361	+	0.983416i	$0.558050\pi$
$878$	−35.9038	−1.21170
$879$	12.3546	0.416711
$880$	3.83198	0.129176
$881$	6.57916	0.221657	0.110829	−	0.993840i	$-0.464650\pi$
$881$	6.57916	0.221657	0.110829	+	0.993840i	$0.464650\pi$
$882$	−15.7778	−0.531265
$883$	24.3827	0.820544	0.410272	−	0.911963i	$-0.365434\pi$
$883$	24.3827	0.820544	0.410272	+	0.911963i	$0.365434\pi$
$884$	−0.284727	−0.00957641
$885$	0.218182	0.00733412
$886$	17.1974	0.577758
$887$	12.6067	0.423291	0.211645	−	0.977347i	$-0.432118\pi$
$887$	12.6067	0.423291	0.211645	+	0.977347i	$0.432118\pi$
$888$	−2.15998	−0.0724843
$889$	72.3757	2.42740
$890$	−2.55283	−0.0855712
$891$	22.4404	0.751783
$892$	4.31633	0.144521
$893$	−87.6854	−2.93428
$894$	7.08304	0.236892
$895$	−5.08224	−0.169881
$896$	3.60356	0.120386
$897$	0.302935	0.0101147
$898$	20.9010	0.697477
$899$	0	0
$900$	−2.63595	−0.0878650
$901$	6.35349	0.211665
$902$	40.0269	1.33275
$903$	20.1976	0.672132
$904$	11.5901	0.385482
$905$	−6.30332	−0.209529
$906$	−0.0214123	−0.000711377 0
$907$	−36.1775	−1.20126	−0.600628	−	0.799529i	$-0.705083\pi$
$907$	−36.1775	−1.20126	−0.600628	+	0.799529i	$0.705083\pi$
$908$	8.07974	0.268136
$909$	−0.448879	−0.0148884
$910$	0.442278	0.0146614
$911$	−4.96412	−0.164469	−0.0822344	−	0.996613i	$-0.526206\pi$
$911$	−4.96412	−0.164469	−0.0822344	+	0.996613i	$0.526206\pi$
$912$	4.01037	0.132797
$913$	−18.6653	−0.617731
$914$	2.10802	0.0697271
$915$	3.79953	0.125609
$916$	−20.0722	−0.663203
$917$	18.9419	0.625517
$918$	−7.88881	−0.260369
$919$	−56.4943	−1.86358	−0.931788	−	0.363003i	$-0.881751\pi$
$919$	−56.4943	−1.86358	−0.931788	+	0.363003i	$0.881751\pi$
$920$	−4.09077	−0.134869
$921$	2.91299	0.0959864
$922$	0.540464	0.0177992
$923$	1.24521	0.0409867
$924$	−8.33171	−0.274093
$925$	3.57990	0.117706
$926$	−19.6759	−0.646591
$927$	−31.5988	−1.03784
$928$	0	0
$929$	−10.0874	−0.330958	−0.165479	−	0.986213i	$-0.552917\pi$
$929$	−10.0874	−0.330958	−0.165479	+	0.986213i	$0.552917\pi$
$930$	−5.69964	−0.186899
$931$	−39.7844	−1.30388
$932$	3.90478	0.127905
$933$	−2.23017	−0.0730125
$934$	−38.8360	−1.27075
$935$	−8.88971	−0.290725
$936$	−0.323520	−0.0105746
$937$	−32.4711	−1.06079	−0.530393	−	0.847752i	$-0.677956\pi$
$937$	−32.4711	−1.06079	−0.530393	+	0.847752i	$0.677956\pi$
$938$	−15.9525	−0.520866
$939$	−11.3798	−0.371366
$940$	13.1924	0.430288
$941$	13.3717	0.435906	0.217953	−	0.975959i	$-0.430062\pi$
$941$	13.3717	0.435906	0.217953	+	0.975959i	$0.430062\pi$
$942$	12.2547	0.399278
$943$	−42.7301	−1.39148
$944$	−0.361609	−0.0117694
$945$	12.2540	0.398623
$946$	−35.5968	−1.15735
$947$	−0.0979024	−0.00318140	−0.00159070	−	0.999999i	$-0.500506\pi$
$947$	−0.0979024	−0.00318140	−0.00159070	+	0.999999i	$0.500506\pi$
$948$	−8.44697	−0.274345
$949$	0.799888	0.0259655
$950$	−6.64668	−0.215647
$951$	13.1620	0.426809
$952$	−8.35979	−0.270942
$953$	47.1826	1.52839	0.764197	−	0.644983i	$-0.223135\pi$
$953$	47.1826	1.52839	0.764197	+	0.644983i	$0.223135\pi$
$954$	7.21914	0.233728
$955$	−16.2449	−0.525672
$956$	−26.6702	−0.862576
$957$	0	0
$958$	−14.4617	−0.467238
$959$	−4.61239	−0.148942
$960$	−0.603364	−0.0194735
$961$	58.2351	1.87855
$962$	0.439375	0.0141660
$963$	9.53164	0.307153
$964$	−25.4064	−0.818286
$965$	−1.52191	−0.0489920
$966$	8.89439	0.286172
$967$	−22.6980	−0.729919	−0.364959	−	0.931023i	$-0.618917\pi$
$967$	−22.6980	−0.729919	−0.364959	+	0.931023i	$0.618917\pi$
$968$	3.68408	0.118411
$969$	−9.30355	−0.298873
$970$	8.66831	0.278323
$971$	4.15521	0.133347	0.0666735	−	0.997775i	$-0.478761\pi$
$971$	4.15521	0.133347	0.0666735	+	0.997775i	$0.478761\pi$
$972$	−13.7350	−0.440549
$973$	36.0351	1.15523
$974$	41.4896	1.32941
$975$	−0.0740532	−0.00237160
$976$	−6.29724	−0.201570
$977$	50.6645	1.62090	0.810450	−	0.585807i	$-0.199222\pi$
$977$	50.6645	1.62090	0.810450	+	0.585807i	$0.199222\pi$
$978$	−9.05278	−0.289476
$979$	−9.78241	−0.312647
$980$	5.98561	0.191203
$981$	−43.0914	−1.37580
$982$	37.3413	1.19161
$983$	−56.8414	−1.81296	−0.906479	−	0.422250i	$-0.861240\pi$
$983$	−56.8414	−1.81296	−0.906479	+	0.422250i	$0.861240\pi$
$984$	−6.30244	−0.200914
$985$	−25.1520	−0.801411
$986$	0	0
$987$	−28.6836	−0.913009
$988$	−0.815772	−0.0259532
$989$	38.0008	1.20836
$990$	−10.1009	−0.321028
$991$	−0.787146	−0.0250045	−0.0125023	−	0.999922i	$-0.503980\pi$
$991$	−0.787146	−0.0250045	−0.0125023	+	0.999922i	$0.503980\pi$
$992$	9.44644	0.299925
$993$	10.7523	0.341215
$994$	36.5604	1.15963
$995$	−5.81104	−0.184222
$996$	2.93894	0.0931239
$997$	−49.1329	−1.55605	−0.778027	−	0.628231i	$-0.783779\pi$
$997$	−49.1329	−1.55605	−0.778027	+	0.628231i	$0.783779\pi$
$998$	−3.12421	−0.0988952
$999$	12.1736	0.385154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8410.2.a.ca.1.7		18
29.19	odd	28		290.2.m.b.71.2	✓	36
29.26	odd	28		290.2.m.b.241.2	yes	36
29.28	even	2		8410.2.a.bz.1.12		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
290.2.m.b.71.2	✓	36	29.19	odd	28
290.2.m.b.241.2	yes	36	29.26	odd	28
8410.2.a.bz.1.12		18	29.28	even	2
8410.2.a.ca.1.7		18	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8410 = 2 \cdot 5 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8410.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.603364 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.603364 q^{6} +3.60356 q^{7} +1.00000 q^{8} -2.63595 q^{9} +1.00000 q^{10} +3.83198 q^{11} -0.603364 q^{12} +0.122734 q^{13} +3.60356 q^{14} -0.603364 q^{15} +1.00000 q^{16} -2.31987 q^{17} -2.63595 q^{18} -6.64668 q^{19} +1.00000 q^{20} -2.17426 q^{21} +3.83198 q^{22} -4.09077 q^{23} -0.603364 q^{24} +1.00000 q^{25} +0.122734 q^{26} +3.40053 q^{27} +3.60356 q^{28} -0.603364 q^{30} +9.44644 q^{31} +1.00000 q^{32} -2.31208 q^{33} -2.31987 q^{34} +3.60356 q^{35} -2.63595 q^{36} +3.57990 q^{37} -6.64668 q^{38} -0.0740532 q^{39} +1.00000 q^{40} +10.4455 q^{41} -2.17426 q^{42} -9.28941 q^{43} +3.83198 q^{44} -2.63595 q^{45} -4.09077 q^{46} +13.1924 q^{47} -0.603364 q^{48} +5.98561 q^{49} +1.00000 q^{50} +1.39973 q^{51} +0.122734 q^{52} -2.73872 q^{53} +3.40053 q^{54} +3.83198 q^{55} +3.60356 q^{56} +4.01037 q^{57} -0.361609 q^{59} -0.603364 q^{60} -6.29724 q^{61} +9.44644 q^{62} -9.49880 q^{63} +1.00000 q^{64} +0.122734 q^{65} -2.31208 q^{66} -4.42687 q^{67} -2.31987 q^{68} +2.46823 q^{69} +3.60356 q^{70} +10.1456 q^{71} -2.63595 q^{72} +6.51726 q^{73} +3.57990 q^{74} -0.603364 q^{75} -6.64668 q^{76} +13.8088 q^{77} -0.0740532 q^{78} +13.9998 q^{79} +1.00000 q^{80} +5.85609 q^{81} +10.4455 q^{82} -4.87092 q^{83} -2.17426 q^{84} -2.31987 q^{85} -9.28941 q^{86} +3.83198 q^{88} -2.55283 q^{89} -2.63595 q^{90} +0.442278 q^{91} -4.09077 q^{92} -5.69964 q^{93} +13.1924 q^{94} -6.64668 q^{95} -0.603364 q^{96} +8.66831 q^{97} +5.98561 q^{98} -10.1009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{2} + 6 q^{3} + 18 q^{4} + 18 q^{5} + 6 q^{6} + 8 q^{7} + 18 q^{8} + 22 q^{9} + 18 q^{10} + 22 q^{11} + 6 q^{12} + 16 q^{13} + 8 q^{14} + 6 q^{15} + 18 q^{16} + 4 q^{17} + 22 q^{18} + 20 q^{19}+ \cdots + 86 q^{99}+O(q^{100}) \) 18 * q + 18 * q^2 + 6 * q^3 + 18 * q^4 + 18 * q^5 + 6 * q^6 + 8 * q^7 + 18 * q^8 + 22 * q^9 + 18 * q^10 + 22 * q^11 + 6 * q^12 + 16 * q^13 + 8 * q^14 + 6 * q^15 + 18 * q^16 + 4 * q^17 + 22 * q^18 + 20 * q^19 + 18 * q^20 + 16 * q^21 + 22 * q^22 - 8 * q^23 + 6 * q^24 + 18 * q^25 + 16 * q^26 + 12 * q^27 + 8 * q^28 + 6 * q^30 + 50 * q^31 + 18 * q^32 + 12 * q^33 + 4 * q^34 + 8 * q^35 + 22 * q^36 + 10 * q^37 + 20 * q^38 + 20 * q^39 + 18 * q^40 + 36 * q^41 + 16 * q^42 + 22 * q^43 + 22 * q^44 + 22 * q^45 - 8 * q^46 + 26 * q^47 + 6 * q^48 + 6 * q^49 + 18 * q^50 - 12 * q^51 + 16 * q^52 - 16 * q^53 + 12 * q^54 + 22 * q^55 + 8 * q^56 - 18 * q^57 + 28 * q^59 + 6 * q^60 + 32 * q^61 + 50 * q^62 - 22 * q^63 + 18 * q^64 + 16 * q^65 + 12 * q^66 + 6 * q^67 + 4 * q^68 + 6 * q^69 + 8 * q^70 - 10 * q^71 + 22 * q^72 + 12 * q^73 + 10 * q^74 + 6 * q^75 + 20 * q^76 + 30 * q^77 + 20 * q^78 + 56 * q^79 + 18 * q^80 - 54 * q^81 + 36 * q^82 + 8 * q^83 + 16 * q^84 + 4 * q^85 + 22 * q^86 + 22 * q^88 + 80 * q^89 + 22 * q^90 - 42 * q^91 - 8 * q^92 + 2 * q^93 + 26 * q^94 + 20 * q^95 + 6 * q^96 - 12 * q^97 + 6 * q^98 + 86 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more