LMFDB - Embedded newform 8470.2.a.co.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:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.14896$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -3.14896 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.14896 q^{6} -1.00000 q^{7} -1.00000 q^{8} +6.91596 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.14896 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.14896 q^{6} -1.00000 q^{7} -1.00000 q^{8} +6.91596 q^{9} -1.00000 q^{10} -3.14896 q^{12} +5.43886 q^{13} +1.00000 q^{14} -3.14896 q^{15} +1.00000 q^{16} -0.477092 q^{17} -6.91596 q^{18} -5.91596 q^{19} +1.00000 q^{20} +3.14896 q^{21} -2.65626 q^{23} +3.14896 q^{24} +1.00000 q^{25} -5.43886 q^{26} -12.3312 q^{27} -1.00000 q^{28} -6.56726 q^{29} +3.14896 q^{30} -9.68953 q^{31} -1.00000 q^{32} +0.477092 q^{34} -1.00000 q^{35} +6.91596 q^{36} -4.56726 q^{37} +5.91596 q^{38} -17.1268 q^{39} -1.00000 q^{40} +2.52291 q^{41} -3.14896 q^{42} +11.9003 q^{43} +6.91596 q^{45} +2.65626 q^{46} +9.42632 q^{47} -3.14896 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.50234 q^{51} +5.43886 q^{52} +6.15392 q^{53} +12.3312 q^{54} +1.00000 q^{56} +18.6291 q^{57} +6.56726 q^{58} -0.566092 q^{59} -3.14896 q^{60} +2.04888 q^{61} +9.68953 q^{62} -6.91596 q^{63} +1.00000 q^{64} +5.43886 q^{65} -1.24102 q^{67} -0.477092 q^{68} +8.36447 q^{69} +1.00000 q^{70} -3.62299 q^{71} -6.91596 q^{72} +15.6398 q^{73} +4.56726 q^{74} -3.14896 q^{75} -5.91596 q^{76} +17.1268 q^{78} -3.38007 q^{79} +1.00000 q^{80} +18.0826 q^{81} -2.52291 q^{82} +1.73868 q^{83} +3.14896 q^{84} -0.477092 q^{85} -11.9003 q^{86} +20.6801 q^{87} -9.54364 q^{89} -6.91596 q^{90} -5.43886 q^{91} -2.65626 q^{92} +30.5120 q^{93} -9.42632 q^{94} -5.91596 q^{95} +3.14896 q^{96} -3.93156 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} - 4 q^{10} - 4 q^{12} + 14 q^{13} + 4 q^{14} - 4 q^{15} + 4 q^{16} + 12 q^{17} - 6 q^{18} - 2 q^{19} + 4 q^{20} + 4 q^{21} + 4 q^{24} + 4 q^{25} - 14 q^{26} - 22 q^{27} - 4 q^{28} + 10 q^{29} + 4 q^{30} - 18 q^{31} - 4 q^{32} - 12 q^{34} - 4 q^{35} + 6 q^{36} + 18 q^{37} + 2 q^{38} - 30 q^{39} - 4 q^{40} + 24 q^{41} - 4 q^{42} + 10 q^{43} + 6 q^{45} - 8 q^{47} - 4 q^{48} + 4 q^{49} - 4 q^{50} + 14 q^{52} + 20 q^{53} + 22 q^{54} + 4 q^{56} + 30 q^{57} - 10 q^{58} - 14 q^{59} - 4 q^{60} + 14 q^{61} + 18 q^{62} - 6 q^{63} + 4 q^{64} + 14 q^{65} + 12 q^{68} - 4 q^{69} + 4 q^{70} - 14 q^{71} - 6 q^{72} + 30 q^{73} - 18 q^{74} - 4 q^{75} - 2 q^{76} + 30 q^{78} + 8 q^{79} + 4 q^{80} + 16 q^{81} - 24 q^{82} + 8 q^{83} + 4 q^{84} + 12 q^{85} - 10 q^{86} + 2 q^{87} - 4 q^{89} - 6 q^{90} - 14 q^{91} - 2 q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 10 q^{97} - 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9 - 4 * q^10 - 4 * q^12 + 14 * q^13 + 4 * q^14 - 4 * q^15 + 4 * q^16 + 12 * q^17 - 6 * q^18 - 2 * q^19 + 4 * q^20 + 4 * q^21 + 4 * q^24 + 4 * q^25 - 14 * q^26 - 22 * q^27 - 4 * q^28 + 10 * q^29 + 4 * q^30 - 18 * q^31 - 4 * q^32 - 12 * q^34 - 4 * q^35 + 6 * q^36 + 18 * q^37 + 2 * q^38 - 30 * q^39 - 4 * q^40 + 24 * q^41 - 4 * q^42 + 10 * q^43 + 6 * q^45 - 8 * q^47 - 4 * q^48 + 4 * q^49 - 4 * q^50 + 14 * q^52 + 20 * q^53 + 22 * q^54 + 4 * q^56 + 30 * q^57 - 10 * q^58 - 14 * q^59 - 4 * q^60 + 14 * q^61 + 18 * q^62 - 6 * q^63 + 4 * q^64 + 14 * q^65 + 12 * q^68 - 4 * q^69 + 4 * q^70 - 14 * q^71 - 6 * q^72 + 30 * q^73 - 18 * q^74 - 4 * q^75 - 2 * q^76 + 30 * q^78 + 8 * q^79 + 4 * q^80 + 16 * q^81 - 24 * q^82 + 8 * q^83 + 4 * q^84 + 12 * q^85 - 10 * q^86 + 2 * q^87 - 4 * q^89 - 6 * q^90 - 14 * q^91 - 2 * q^93 + 8 * q^94 - 2 * q^95 + 4 * q^96 - 10 * q^97 - 4 * 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
$3$	−3.14896	−1.81805	−0.909027	−	0.416738i	$-0.863173\pi$
$3$	−3.14896	−1.81805	−0.909027	+	0.416738i	$0.863173\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	3.14896	1.28556
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	6.91596	2.30532
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−3.14896	−0.909027
$13$	5.43886	1.50847	0.754235	−	0.656605i	$-0.228008\pi$
$13$	5.43886	1.50847	0.754235	+	0.656605i	$0.228008\pi$
$14$	1.00000	0.267261
$15$	−3.14896	−0.813058
$16$	1.00000	0.250000
$17$	−0.477092	−0.115712	−0.0578559	−	0.998325i	$-0.518426\pi$
$17$	−0.477092	−0.115712	−0.0578559	+	0.998325i	$0.518426\pi$
$18$	−6.91596	−1.63011
$19$	−5.91596	−1.35721	−0.678607	−	0.734502i	$-0.737416\pi$
$19$	−5.91596	−1.35721	−0.678607	+	0.734502i	$0.737416\pi$
$20$	1.00000	0.223607
$21$	3.14896	0.687160
$22$	0	0
$23$	−2.65626	−0.553869	−0.276934	−	0.960889i	$-0.589318\pi$
$23$	−2.65626	−0.553869	−0.276934	+	0.960889i	$0.589318\pi$
$24$	3.14896	0.642779
$25$	1.00000	0.200000
$26$	−5.43886	−1.06665
$27$	−12.3312	−2.37314
$28$	−1.00000	−0.188982
$29$	−6.56726	−1.21951	−0.609755	−	0.792590i	$-0.708732\pi$
$29$	−6.56726	−1.21951	−0.609755	+	0.792590i	$0.708732\pi$
$30$	3.14896	0.574919
$31$	−9.68953	−1.74029	−0.870146	−	0.492794i	$-0.835976\pi$
$31$	−9.68953	−1.74029	−0.870146	+	0.492794i	$0.835976\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.477092	0.0818206
$35$	−1.00000	−0.169031
$36$	6.91596	1.15266
$37$	−4.56726	−0.750853	−0.375427	−	0.926852i	$-0.622504\pi$
$37$	−4.56726	−0.750853	−0.375427	+	0.926852i	$0.622504\pi$
$38$	5.91596	0.959695
$39$	−17.1268	−2.74248
$40$	−1.00000	−0.158114
$41$	2.52291	0.394012	0.197006	−	0.980402i	$-0.436878\pi$
$41$	2.52291	0.394012	0.197006	+	0.980402i	$0.436878\pi$
$42$	−3.14896	−0.485895
$43$	11.9003	1.81479	0.907393	−	0.420283i	$-0.138069\pi$
$43$	11.9003	1.81479	0.907393	+	0.420283i	$0.138069\pi$
$44$	0	0
$45$	6.91596	1.03097
$46$	2.65626	0.391644
$47$	9.42632	1.37497	0.687485	−	0.726199i	$-0.258715\pi$
$47$	9.42632	1.37497	0.687485	+	0.726199i	$0.258715\pi$
$48$	−3.14896	−0.454513
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	1.50234	0.210370
$52$	5.43886	0.754235
$53$	6.15392	0.845306	0.422653	−	0.906292i	$-0.361099\pi$
$53$	6.15392	0.845306	0.422653	+	0.906292i	$0.361099\pi$
$54$	12.3312	1.67806
$55$	0	0
$56$	1.00000	0.133631
$57$	18.6291	2.46749
$58$	6.56726	0.862324
$59$	−0.566092	−0.0736989	−0.0368495	−	0.999321i	$-0.511732\pi$
$59$	−0.566092	−0.0736989	−0.0368495	+	0.999321i	$0.511732\pi$
$60$	−3.14896	−0.406529
$61$	2.04888	0.262332	0.131166	−	0.991360i	$-0.458128\pi$
$61$	2.04888	0.262332	0.131166	+	0.991360i	$0.458128\pi$
$62$	9.68953	1.23057
$63$	−6.91596	−0.871329
$64$	1.00000	0.125000
$65$	5.43886	0.674608
$66$	0	0
$67$	−1.24102	−0.151615	−0.0758076	−	0.997122i	$-0.524153\pi$
$67$	−1.24102	−0.151615	−0.0758076	+	0.997122i	$0.524153\pi$
$68$	−0.477092	−0.0578559
$69$	8.36447	1.00696
$70$	1.00000	0.119523
$71$	−3.62299	−0.429970	−0.214985	−	0.976617i	$-0.568970\pi$
$71$	−3.62299	−0.429970	−0.214985	+	0.976617i	$0.568970\pi$
$72$	−6.91596	−0.815053
$73$	15.6398	1.83050	0.915248	−	0.402891i	$-0.131994\pi$
$73$	15.6398	1.83050	0.915248	+	0.402891i	$0.131994\pi$
$74$	4.56726	0.530933
$75$	−3.14896	−0.363611
$76$	−5.91596	−0.678607
$77$	0	0
$78$	17.1268	1.93923
$79$	−3.38007	−0.380288	−0.190144	−	0.981756i	$-0.560895\pi$
$79$	−3.38007	−0.380288	−0.190144	+	0.981756i	$0.560895\pi$
$80$	1.00000	0.111803
$81$	18.0826	2.00918
$82$	−2.52291	−0.278609
$83$	1.73868	0.190845	0.0954224	−	0.995437i	$-0.469580\pi$
$83$	1.73868	0.190845	0.0954224	+	0.995437i	$0.469580\pi$
$84$	3.14896	0.343580
$85$	−0.477092	−0.0517479
$86$	−11.9003	−1.28325
$87$	20.6801	2.21713
$88$	0	0
$89$	−9.54364	−1.01162	−0.505812	−	0.862644i	$-0.668807\pi$
$89$	−9.54364	−1.01162	−0.505812	+	0.862644i	$0.668807\pi$
$90$	−6.91596	−0.729006
$91$	−5.43886	−0.570148
$92$	−2.65626	−0.276934
$93$	30.5120	3.16394
$94$	−9.42632	−0.972251
$95$	−5.91596	−0.606964
$96$	3.14896	0.321390
$97$	−3.93156	−0.399190	−0.199595	−	0.979879i	$-0.563963\pi$
$97$	−3.93156	−0.399190	−0.199595	+	0.979879i	$0.563963\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−2.90794	−0.289351	−0.144675	−	0.989479i	$-0.546214\pi$
$101$	−2.90794	−0.289351	−0.144675	+	0.989479i	$0.546214\pi$
$102$	−1.50234	−0.148754
$103$	−15.9287	−1.56950	−0.784749	−	0.619814i	$-0.787208\pi$
$103$	−15.9287	−1.56950	−0.784749	+	0.619814i	$0.787208\pi$
$104$	−5.43886	−0.533325
$105$	3.14896	0.307307
$106$	−6.15392	−0.597721
$107$	−6.17232	−0.596701	−0.298350	−	0.954456i	$-0.596436\pi$
$107$	−6.17232	−0.596701	−0.298350	+	0.954456i	$0.596436\pi$
$108$	−12.3312	−1.18657
$109$	−14.9240	−1.42946	−0.714729	−	0.699402i	$-0.753450\pi$
$109$	−14.9240	−1.42946	−0.714729	+	0.699402i	$0.753450\pi$
$110$	0	0
$111$	14.3821	1.36509
$112$	−1.00000	−0.0944911
$113$	19.0906	1.79589	0.897946	−	0.440105i	$-0.145059\pi$
$113$	19.0906	1.79589	0.897946	+	0.440105i	$0.145059\pi$
$114$	−18.6291	−1.74478
$115$	−2.65626	−0.247698
$116$	−6.56726	−0.609755
$117$	37.6149	3.47750
$118$	0.566092	0.0521130
$119$	0.477092	0.0437350
$120$	3.14896	0.287460
$121$	0	0
$122$	−2.04888	−0.185497
$123$	−7.94454	−0.716335
$124$	−9.68953	−0.870146
$125$	1.00000	0.0894427
$126$	6.91596	0.616122
$127$	16.3815	1.45362	0.726812	−	0.686836i	$-0.241001\pi$
$127$	16.3815	1.45362	0.726812	+	0.686836i	$0.241001\pi$
$128$	−1.00000	−0.0883883
$129$	−37.4737	−3.29938
$130$	−5.43886	−0.477020
$131$	5.60812	0.489984	0.244992	−	0.969525i	$-0.421215\pi$
$131$	5.60812	0.489984	0.244992	+	0.969525i	$0.421215\pi$
$132$	0	0
$133$	5.91596	0.512978
$134$	1.24102	0.107208
$135$	−12.3312	−1.06130
$136$	0.477092	0.0409103
$137$	−2.50874	−0.214336	−0.107168	−	0.994241i	$-0.534178\pi$
$137$	−2.50874	−0.214336	−0.107168	+	0.994241i	$0.534178\pi$
$138$	−8.36447	−0.712031
$139$	3.30261	0.280124	0.140062	−	0.990143i	$-0.455270\pi$
$139$	3.30261	0.280124	0.140062	+	0.990143i	$0.455270\pi$
$140$	−1.00000	−0.0845154
$141$	−29.6831	−2.49977
$142$	3.62299	0.304035
$143$	0	0
$144$	6.91596	0.576330
$145$	−6.56726	−0.545381
$146$	−15.6398	−1.29436
$147$	−3.14896	−0.259722
$148$	−4.56726	−0.375427
$149$	6.83048	0.559574	0.279787	−	0.960062i	$-0.409736\pi$
$149$	6.83048	0.559574	0.279787	+	0.960062i	$0.409736\pi$
$150$	3.14896	0.257112
$151$	−10.8189	−0.880433	−0.440216	−	0.897892i	$-0.645098\pi$
$151$	−10.8189	−0.880433	−0.440216	+	0.897892i	$0.645098\pi$
$152$	5.91596	0.479847
$153$	−3.29955	−0.266753
$154$	0	0
$155$	−9.68953	−0.778282
$156$	−17.1268	−1.37124
$157$	8.01766	0.639879	0.319940	−	0.947438i	$-0.396337\pi$
$157$	8.01766	0.639879	0.319940	+	0.947438i	$0.396337\pi$
$158$	3.38007	0.268904
$159$	−19.3784	−1.53681
$160$	−1.00000	−0.0790569
$161$	2.65626	0.209343
$162$	−18.0826	−1.42070
$163$	−18.2991	−1.43330	−0.716648	−	0.697435i	$-0.754324\pi$
$163$	−18.2991	−1.43330	−0.716648	+	0.697435i	$0.754324\pi$
$164$	2.52291	0.197006
$165$	0	0
$166$	−1.73868	−0.134948
$167$	−17.9260	−1.38716	−0.693579	−	0.720381i	$-0.743967\pi$
$167$	−17.9260	−1.38716	−0.693579	+	0.720381i	$0.743967\pi$
$168$	−3.14896	−0.242948
$169$	16.5812	1.27548
$170$	0.477092	0.0365913
$171$	−40.9145	−3.12881
$172$	11.9003	0.907393
$173$	−7.68791	−0.584501	−0.292250	−	0.956342i	$-0.594404\pi$
$173$	−7.68791	−0.584501	−0.292250	+	0.956342i	$0.594404\pi$
$174$	−20.6801	−1.56775
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	1.78260	0.133989
$178$	9.54364	0.715326
$179$	22.7197	1.69815	0.849076	−	0.528271i	$-0.177160\pi$
$179$	22.7197	1.69815	0.849076	+	0.528271i	$0.177160\pi$
$180$	6.91596	0.515485
$181$	−13.6567	−1.01509	−0.507547	−	0.861624i	$-0.669448\pi$
$181$	−13.6567	−1.01509	−0.507547	+	0.861624i	$0.669448\pi$
$182$	5.43886	0.403155
$183$	−6.45184	−0.476934
$184$	2.65626	0.195822
$185$	−4.56726	−0.335792
$186$	−30.5120	−2.23725
$187$	0	0
$188$	9.42632	0.687485
$189$	12.3312	0.896962
$190$	5.91596	0.429189
$191$	9.42182	0.681739	0.340869	−	0.940111i	$-0.389279\pi$
$191$	9.42182	0.681739	0.340869	+	0.940111i	$0.389279\pi$
$192$	−3.14896	−0.227257
$193$	−5.09919	−0.367048	−0.183524	−	0.983015i	$-0.558751\pi$
$193$	−5.09919	−0.367048	−0.183524	+	0.983015i	$0.558751\pi$
$194$	3.93156	0.282270
$195$	−17.1268	−1.22647
$196$	1.00000	0.0714286
$197$	17.1284	1.22035	0.610174	−	0.792268i	$-0.291100\pi$
$197$	17.1284	1.22035	0.610174	+	0.792268i	$0.291100\pi$
$198$	0	0
$199$	11.9225	0.845166	0.422583	−	0.906324i	$-0.361123\pi$
$199$	11.9225	0.845166	0.422583	+	0.906324i	$0.361123\pi$
$200$	−1.00000	−0.0707107
$201$	3.90794	0.275645
$202$	2.90794	0.204602
$203$	6.56726	0.460931
$204$	1.50234	0.105185
$205$	2.52291	0.176208
$206$	15.9287	1.10980
$207$	−18.3706	−1.27684
$208$	5.43886	0.377117
$209$	0	0
$210$	−3.14896	−0.217299
$211$	22.8331	1.57189	0.785947	−	0.618294i	$-0.212176\pi$
$211$	22.8331	1.57189	0.785947	+	0.618294i	$0.212176\pi$
$212$	6.15392	0.422653
$213$	11.4087	0.781708
$214$	6.17232	0.421931
$215$	11.9003	0.811597
$216$	12.3312	0.839031
$217$	9.68953	0.657768
$218$	14.9240	1.01078
$219$	−49.2490	−3.32794
$220$	0	0
$221$	−2.59484	−0.174548
$222$	−14.3821	−0.965265
$223$	−15.8782	−1.06328	−0.531640	−	0.846970i	$-0.678424\pi$
$223$	−15.8782	−1.06328	−0.531640	+	0.846970i	$0.678424\pi$
$224$	1.00000	0.0668153
$225$	6.91596	0.461064
$226$	−19.0906	−1.26989
$227$	−8.41830	−0.558742	−0.279371	−	0.960183i	$-0.590126\pi$
$227$	−8.41830	−0.558742	−0.279371	+	0.960183i	$0.590126\pi$
$228$	18.6291	1.23374
$229$	22.2844	1.47259	0.736296	−	0.676659i	$-0.236573\pi$
$229$	22.2844	1.47259	0.736296	+	0.676659i	$0.236573\pi$
$230$	2.65626	0.175149
$231$	0	0
$232$	6.56726	0.431162
$233$	−12.4011	−0.812421	−0.406210	−	0.913780i	$-0.633150\pi$
$233$	−12.4011	−0.812421	−0.406210	+	0.913780i	$0.633150\pi$
$234$	−37.6149	−2.45897
$235$	9.42632	0.614905
$236$	−0.566092	−0.0368495
$237$	10.6437	0.691384
$238$	−0.477092	−0.0309253
$239$	19.3571	1.25211	0.626055	−	0.779779i	$-0.284669\pi$
$239$	19.3571	1.25211	0.626055	+	0.779779i	$0.284669\pi$
$240$	−3.14896	−0.203265
$241$	−5.12445	−0.330095	−0.165047	−	0.986286i	$-0.552778\pi$
$241$	−5.12445	−0.330095	−0.165047	+	0.986286i	$0.552778\pi$
$242$	0	0
$243$	−19.9478	−1.27965
$244$	2.04888	0.131166
$245$	1.00000	0.0638877
$246$	7.94454	0.506525
$247$	−32.1761	−2.04732
$248$	9.68953	0.615286
$249$	−5.47503	−0.346966
$250$	−1.00000	−0.0632456
$251$	−8.99152	−0.567540	−0.283770	−	0.958892i	$-0.591585\pi$
$251$	−8.99152	−0.567540	−0.283770	+	0.958892i	$0.591585\pi$
$252$	−6.91596	−0.435664
$253$	0	0
$254$	−16.3815	−1.02787
$255$	1.50234	0.0940805
$256$	1.00000	0.0625000
$257$	−17.9476	−1.11954	−0.559770	−	0.828648i	$-0.689111\pi$
$257$	−17.9476	−1.11954	−0.559770	+	0.828648i	$0.689111\pi$
$258$	37.4737	2.33301
$259$	4.56726	0.283796
$260$	5.43886	0.337304
$261$	−45.4189	−2.81136
$262$	−5.60812	−0.346471
$263$	27.3312	1.68531	0.842657	−	0.538451i	$-0.180990\pi$
$263$	27.3312	1.68531	0.842657	+	0.538451i	$0.180990\pi$
$264$	0	0
$265$	6.15392	0.378032
$266$	−5.91596	−0.362731
$267$	30.0525	1.83919
$268$	−1.24102	−0.0758076
$269$	−12.0775	−0.736376	−0.368188	−	0.929751i	$-0.620022\pi$
$269$	−12.0775	−0.736376	−0.368188	+	0.929751i	$0.620022\pi$
$270$	12.3312	0.750453
$271$	25.8325	1.56921	0.784607	−	0.619993i	$-0.212865\pi$
$271$	25.8325	1.56921	0.784607	+	0.619993i	$0.212865\pi$
$272$	−0.477092	−0.0289280
$273$	17.1268	1.03656
$274$	2.50874	0.151558
$275$	0	0
$276$	8.36447	0.503482
$277$	−18.2240	−1.09497	−0.547486	−	0.836815i	$-0.684415\pi$
$277$	−18.2240	−1.09497	−0.547486	+	0.836815i	$0.684415\pi$
$278$	−3.30261	−0.198077
$279$	−67.0124	−4.01193
$280$	1.00000	0.0597614
$281$	−4.05953	−0.242171	−0.121086	−	0.992642i	$-0.538638\pi$
$281$	−4.05953	−0.242171	−0.121086	+	0.992642i	$0.538638\pi$
$282$	29.6831	1.76760
$283$	−3.16574	−0.188184	−0.0940918	−	0.995564i	$-0.529995\pi$
$283$	−3.16574	−0.188184	−0.0940918	+	0.995564i	$0.529995\pi$
$284$	−3.62299	−0.214985
$285$	18.6291	1.10349
$286$	0	0
$287$	−2.52291	−0.148923
$288$	−6.91596	−0.407527
$289$	−16.7724	−0.986611
$290$	6.56726	0.385643
$291$	12.3803	0.725748
$292$	15.6398	0.915248
$293$	10.2517	0.598909	0.299455	−	0.954111i	$-0.403195\pi$
$293$	10.2517	0.598909	0.299455	+	0.954111i	$0.403195\pi$
$294$	3.14896	0.183651
$295$	−0.566092	−0.0329592
$296$	4.56726	0.265467
$297$	0	0
$298$	−6.83048	−0.395679
$299$	−14.4470	−0.835494
$300$	−3.14896	−0.181805
$301$	−11.9003	−0.685925
$302$	10.8189	0.622560
$303$	9.15698	0.526055
$304$	−5.91596	−0.339303
$305$	2.04888	0.117318
$306$	3.29955	0.188623
$307$	−3.38972	−0.193461	−0.0967307	−	0.995311i	$-0.530839\pi$
$307$	−3.38972	−0.193461	−0.0967307	+	0.995311i	$0.530839\pi$
$308$	0	0
$309$	50.1587	2.85343
$310$	9.68953	0.550329
$311$	−29.4523	−1.67009	−0.835043	−	0.550185i	$-0.814557\pi$
$311$	−29.4523	−1.67009	−0.835043	+	0.550185i	$0.814557\pi$
$312$	17.1268	0.969613
$313$	−22.0645	−1.24716	−0.623579	−	0.781761i	$-0.714322\pi$
$313$	−22.0645	−1.24716	−0.623579	+	0.781761i	$0.714322\pi$
$314$	−8.01766	−0.452463
$315$	−6.91596	−0.389670
$316$	−3.38007	−0.190144
$317$	21.1159	1.18598	0.592992	−	0.805208i	$-0.297946\pi$
$317$	21.1159	1.18598	0.592992	+	0.805208i	$0.297946\pi$
$318$	19.3784	1.08669
$319$	0	0
$320$	1.00000	0.0559017
$321$	19.4364	1.08483
$322$	−2.65626	−0.148028
$323$	2.82246	0.157046
$324$	18.0826	1.00459
$325$	5.43886	0.301694
$326$	18.2991	1.01349
$327$	46.9950	2.59883
$328$	−2.52291	−0.139304
$329$	−9.42632	−0.519690
$330$	0	0
$331$	−31.1409	−1.71166	−0.855830	−	0.517258i	$-0.826953\pi$
$331$	−31.1409	−1.71166	−0.855830	+	0.517258i	$0.826953\pi$
$332$	1.73868	0.0954224
$333$	−31.5870	−1.73096
$334$	17.9260	0.980869
$335$	−1.24102	−0.0678044
$336$	3.14896	0.171790
$337$	−1.60099	−0.0872115	−0.0436057	−	0.999049i	$-0.513885\pi$
$337$	−1.60099	−0.0872115	−0.0436057	+	0.999049i	$0.513885\pi$
$338$	−16.5812	−0.901901
$339$	−60.1156	−3.26503
$340$	−0.477092	−0.0258740
$341$	0	0
$342$	40.9145	2.21240
$343$	−1.00000	−0.0539949
$344$	−11.9003	−0.641624
$345$	8.36447	0.450328
$346$	7.68791	0.413304
$347$	19.7193	1.05859	0.529293	−	0.848439i	$-0.322457\pi$
$347$	19.7193	1.05859	0.529293	+	0.848439i	$0.322457\pi$
$348$	20.6801	1.10857
$349$	15.3791	0.823223	0.411611	−	0.911359i	$-0.364966\pi$
$349$	15.3791	0.823223	0.411611	+	0.911359i	$0.364966\pi$
$350$	1.00000	0.0534522
$351$	−67.0677	−3.57981
$352$	0	0
$353$	−10.3793	−0.552436	−0.276218	−	0.961095i	$-0.589081\pi$
$353$	−10.3793	−0.552436	−0.276218	+	0.961095i	$0.589081\pi$
$354$	−1.78260	−0.0947442
$355$	−3.62299	−0.192288
$356$	−9.54364	−0.505812
$357$	−1.50234	−0.0795125
$358$	−22.7197	−1.20077
$359$	23.8319	1.25780	0.628900	−	0.777486i	$-0.283505\pi$
$359$	23.8319	1.25780	0.628900	+	0.777486i	$0.283505\pi$
$360$	−6.91596	−0.364503
$361$	15.9985	0.842028
$362$	13.6567	0.717780
$363$	0	0
$364$	−5.43886	−0.285074
$365$	15.6398	0.818623
$366$	6.45184	0.337243
$367$	−19.3945	−1.01238	−0.506192	−	0.862421i	$-0.668947\pi$
$367$	−19.3945	−1.01238	−0.506192	+	0.862421i	$0.668947\pi$
$368$	−2.65626	−0.138467
$369$	17.4483	0.908323
$370$	4.56726	0.237441
$371$	−6.15392	−0.319495
$372$	30.5120	1.58197
$373$	10.9015	0.564459	0.282230	−	0.959347i	$-0.408926\pi$
$373$	10.9015	0.564459	0.282230	+	0.959347i	$0.408926\pi$
$374$	0	0
$375$	−3.14896	−0.162612
$376$	−9.42632	−0.486125
$377$	−35.7184	−1.83959
$378$	−12.3312	−0.634248
$379$	6.80767	0.349686	0.174843	−	0.984596i	$-0.444058\pi$
$379$	6.80767	0.349686	0.174843	+	0.984596i	$0.444058\pi$
$380$	−5.91596	−0.303482
$381$	−51.5847	−2.64277
$382$	−9.42182	−0.482062
$383$	−24.2311	−1.23815	−0.619075	−	0.785332i	$-0.712492\pi$
$383$	−24.2311	−1.23815	−0.619075	+	0.785332i	$0.712492\pi$
$384$	3.14896	0.160695
$385$	0	0
$386$	5.09919	0.259542
$387$	82.3023	4.18366
$388$	−3.93156	−0.199595
$389$	−18.2552	−0.925574	−0.462787	−	0.886470i	$-0.653150\pi$
$389$	−18.2552	−0.925574	−0.462787	+	0.886470i	$0.653150\pi$
$390$	17.1268	0.867248
$391$	1.26728	0.0640892
$392$	−1.00000	−0.0505076
$393$	−17.6598	−0.890817
$394$	−17.1284	−0.862916
$395$	−3.38007	−0.170070
$396$	0	0
$397$	17.6742	0.887043	0.443522	−	0.896264i	$-0.353729\pi$
$397$	17.6742	0.887043	0.443522	+	0.896264i	$0.353729\pi$
$398$	−11.9225	−0.597623
$399$	−18.6291	−0.932622
$400$	1.00000	0.0500000
$401$	30.4187	1.51904	0.759520	−	0.650485i	$-0.225434\pi$
$401$	30.4187	1.51904	0.759520	+	0.650485i	$0.225434\pi$
$402$	−3.90794	−0.194910
$403$	−52.7001	−2.62518
$404$	−2.90794	−0.144675
$405$	18.0826	0.898531
$406$	−6.56726	−0.325928
$407$	0	0
$408$	−1.50234	−0.0743771
$409$	15.4065	0.761802	0.380901	−	0.924616i	$-0.375614\pi$
$409$	15.4065	0.761802	0.380901	+	0.924616i	$0.375614\pi$
$410$	−2.52291	−0.124598
$411$	7.89992	0.389674
$412$	−15.9287	−0.784749
$413$	0.566092	0.0278556
$414$	18.3706	0.902865
$415$	1.73868	0.0853484
$416$	−5.43886	−0.266662
$417$	−10.3998	−0.509280
$418$	0	0
$419$	3.21914	0.157265	0.0786327	−	0.996904i	$-0.474945\pi$
$419$	3.21914	0.157265	0.0786327	+	0.996904i	$0.474945\pi$
$420$	3.14896	0.153654
$421$	13.1605	0.641402	0.320701	−	0.947180i	$-0.396081\pi$
$421$	13.1605	0.641402	0.320701	+	0.947180i	$0.396081\pi$
$422$	−22.8331	−1.11150
$423$	65.1920	3.16974
$424$	−6.15392	−0.298861
$425$	−0.477092	−0.0231424
$426$	−11.4087	−0.552751
$427$	−2.04888	−0.0991522
$428$	−6.17232	−0.298350
$429$	0	0
$430$	−11.9003	−0.573886
$431$	2.38429	0.114847	0.0574236	−	0.998350i	$-0.481711\pi$
$431$	2.38429	0.114847	0.0574236	+	0.998350i	$0.481711\pi$
$432$	−12.3312	−0.593285
$433$	9.02568	0.433747	0.216873	−	0.976200i	$-0.430414\pi$
$433$	9.02568	0.433747	0.216873	+	0.976200i	$0.430414\pi$
$434$	−9.68953	−0.465113
$435$	20.6801	0.991533
$436$	−14.9240	−0.714729
$437$	15.7143	0.751718
$438$	49.2490	2.35321
$439$	23.1241	1.10366	0.551828	−	0.833958i	$-0.313931\pi$
$439$	23.1241	1.10366	0.551828	+	0.833958i	$0.313931\pi$
$440$	0	0
$441$	6.91596	0.329331
$442$	2.59484	0.123424
$443$	−6.78799	−0.322507	−0.161254	−	0.986913i	$-0.551554\pi$
$443$	−6.78799	−0.322507	−0.161254	+	0.986913i	$0.551554\pi$
$444$	14.3821	0.682546
$445$	−9.54364	−0.452412
$446$	15.8782	0.751853
$447$	−21.5089	−1.01734
$448$	−1.00000	−0.0472456
$449$	13.1114	0.618767	0.309383	−	0.950937i	$-0.399877\pi$
$449$	13.1114	0.618767	0.309383	+	0.950937i	$0.399877\pi$
$450$	−6.91596	−0.326021
$451$	0	0
$452$	19.0906	0.897946
$453$	34.0684	1.60067
$454$	8.41830	0.395090
$455$	−5.43886	−0.254978
$456$	−18.6291	−0.872388
$457$	−19.0200	−0.889718	−0.444859	−	0.895601i	$-0.646746\pi$
$457$	−19.0200	−0.889718	−0.444859	+	0.895601i	$0.646746\pi$
$458$	−22.2844	−1.04128
$459$	5.88312	0.274600
$460$	−2.65626	−0.123849
$461$	18.8767	0.879177	0.439588	−	0.898199i	$-0.355124\pi$
$461$	18.8767	0.879177	0.439588	+	0.898199i	$0.355124\pi$
$462$	0	0
$463$	13.6194	0.632945	0.316473	−	0.948602i	$-0.397501\pi$
$463$	13.6194	0.632945	0.316473	+	0.948602i	$0.397501\pi$
$464$	−6.56726	−0.304877
$465$	30.5120	1.41496
$466$	12.4011	0.574468
$467$	−18.9801	−0.878296	−0.439148	−	0.898415i	$-0.644720\pi$
$467$	−18.9801	−0.878296	−0.439148	+	0.898415i	$0.644720\pi$
$468$	37.6149	1.73875
$469$	1.24102	0.0573052
$470$	−9.42632	−0.434804
$471$	−25.2473	−1.16333
$472$	0.566092	0.0260565
$473$	0	0
$474$	−10.6437	−0.488882
$475$	−5.91596	−0.271443
$476$	0.477092	0.0218675
$477$	42.5602	1.94870
$478$	−19.3571	−0.885375
$479$	−5.51532	−0.252001	−0.126001	−	0.992030i	$-0.540214\pi$
$479$	−5.51532	−0.252001	−0.126001	+	0.992030i	$0.540214\pi$
$480$	3.14896	0.143730
$481$	−24.8407	−1.13264
$482$	5.12445	0.233412
$483$	−8.36447	−0.380596
$484$	0	0
$485$	−3.93156	−0.178523
$486$	19.9478	0.904849
$487$	28.0086	1.26919	0.634596	−	0.772844i	$-0.281167\pi$
$487$	28.0086	1.26919	0.634596	+	0.772844i	$0.281167\pi$
$488$	−2.04888	−0.0927484
$489$	57.6231	2.60581
$490$	−1.00000	−0.0451754
$491$	−12.9545	−0.584626	−0.292313	−	0.956323i	$-0.594425\pi$
$491$	−12.9545	−0.584626	−0.292313	+	0.956323i	$0.594425\pi$
$492$	−7.94454	−0.358167
$493$	3.13319	0.141112
$494$	32.1761	1.44767
$495$	0	0
$496$	−9.68953	−0.435073
$497$	3.62299	0.162513
$498$	5.47503	0.245342
$499$	3.93985	0.176372	0.0881859	−	0.996104i	$-0.471893\pi$
$499$	3.93985	0.176372	0.0881859	+	0.996104i	$0.471893\pi$
$500$	1.00000	0.0447214
$501$	56.4484	2.52193
$502$	8.99152	0.401311
$503$	28.7158	1.28037	0.640186	−	0.768220i	$-0.278857\pi$
$503$	28.7158	1.28037	0.640186	+	0.768220i	$0.278857\pi$
$504$	6.91596	0.308061
$505$	−2.90794	−0.129401
$506$	0	0
$507$	−52.2137	−2.31889
$508$	16.3815	0.726812
$509$	6.73822	0.298666	0.149333	−	0.988787i	$-0.452287\pi$
$509$	6.73822	0.298666	0.149333	+	0.988787i	$0.452287\pi$
$510$	−1.50234	−0.0665249
$511$	−15.6398	−0.691863
$512$	−1.00000	−0.0441942
$513$	72.9508	3.22086
$514$	17.9476	0.791635
$515$	−15.9287	−0.701901
$516$	−37.4737	−1.64969
$517$	0	0
$518$	−4.56726	−0.200674
$519$	24.2089	1.06265
$520$	−5.43886	−0.238510
$521$	32.2003	1.41072	0.705361	−	0.708848i	$-0.250785\pi$
$521$	32.2003	1.41072	0.705361	+	0.708848i	$0.250785\pi$
$522$	45.4189	1.98793
$523$	−28.4244	−1.24291	−0.621457	−	0.783449i	$-0.713459\pi$
$523$	−28.4244	−1.24291	−0.621457	+	0.783449i	$0.713459\pi$
$524$	5.60812	0.244992
$525$	3.14896	0.137432
$526$	−27.3312	−1.19170
$527$	4.62280	0.201372
$528$	0	0
$529$	−15.9443	−0.693229
$530$	−6.15392	−0.267309
$531$	−3.91507	−0.169899
$532$	5.91596	0.256489
$533$	13.7218	0.594355
$534$	−30.0525	−1.30050
$535$	−6.17232	−0.266853
$536$	1.24102	0.0536041
$537$	−71.5435	−3.08733
$538$	12.0775	0.520696
$539$	0	0
$540$	−12.3312	−0.530650
$541$	21.6078	0.928993	0.464496	−	0.885575i	$-0.346235\pi$
$541$	21.6078	0.928993	0.464496	+	0.885575i	$0.346235\pi$
$542$	−25.8325	−1.10960
$543$	43.0044	1.84550
$544$	0.477092	0.0204552
$545$	−14.9240	−0.639273
$546$	−17.1268	−0.732958
$547$	31.9393	1.36562	0.682812	−	0.730594i	$-0.260757\pi$
$547$	31.9393	1.36562	0.682812	+	0.730594i	$0.260757\pi$
$548$	−2.50874	−0.107168
$549$	14.1700	0.604759
$550$	0	0
$551$	38.8516	1.65514
$552$	−8.36447	−0.356015
$553$	3.38007	0.143735
$554$	18.2240	0.774262
$555$	14.3821	0.610487
$556$	3.30261	0.140062
$557$	18.6958	0.792169	0.396084	−	0.918214i	$-0.370369\pi$
$557$	18.6958	0.792169	0.396084	+	0.918214i	$0.370369\pi$
$558$	67.0124	2.83686
$559$	64.7244	2.73755
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	4.05953	0.171241
$563$	41.8680	1.76453	0.882263	−	0.470758i	$-0.156019\pi$
$563$	41.8680	1.76453	0.882263	+	0.470758i	$0.156019\pi$
$564$	−29.6831	−1.24988
$565$	19.0906	0.803148
$566$	3.16574	0.133066
$567$	−18.0826	−0.759397
$568$	3.62299	0.152017
$569$	14.7563	0.618618	0.309309	−	0.950962i	$-0.399902\pi$
$569$	14.7563	0.618618	0.309309	+	0.950962i	$0.399902\pi$
$570$	−18.6291	−0.780288
$571$	−43.4272	−1.81737	−0.908686	−	0.417481i	$-0.862913\pi$
$571$	−43.4272	−1.81737	−0.908686	+	0.417481i	$0.862913\pi$
$572$	0	0
$573$	−29.6689	−1.23944
$574$	2.52291	0.105304
$575$	−2.65626	−0.110774
$576$	6.91596	0.288165
$577$	−19.1867	−0.798754	−0.399377	−	0.916787i	$-0.630774\pi$
$577$	−19.1867	−0.798754	−0.399377	+	0.916787i	$0.630774\pi$
$578$	16.7724	0.697639
$579$	16.0572	0.667313
$580$	−6.56726	−0.272691
$581$	−1.73868	−0.0721326
$582$	−12.3803	−0.513182
$583$	0	0
$584$	−15.6398	−0.647178
$585$	37.6149	1.55519
$586$	−10.2517	−0.423493
$587$	3.76583	0.155432	0.0777161	−	0.996976i	$-0.475237\pi$
$587$	3.76583	0.155432	0.0777161	+	0.996976i	$0.475237\pi$
$588$	−3.14896	−0.129861
$589$	57.3229	2.36195
$590$	0.566092	0.0233056
$591$	−53.9367	−2.21866
$592$	−4.56726	−0.187713
$593$	37.8401	1.55391	0.776954	−	0.629558i	$-0.216764\pi$
$593$	37.8401	1.55391	0.776954	+	0.629558i	$0.216764\pi$
$594$	0	0
$595$	0.477092	0.0195589
$596$	6.83048	0.279787
$597$	−37.5436	−1.53656
$598$	14.4470	0.590784
$599$	−32.2167	−1.31634	−0.658169	−	0.752870i	$-0.728669\pi$
$599$	−32.2167	−1.31634	−0.658169	+	0.752870i	$0.728669\pi$
$600$	3.14896	0.128556
$601$	3.26376	0.133132	0.0665658	−	0.997782i	$-0.478796\pi$
$601$	3.26376	0.133132	0.0665658	+	0.997782i	$0.478796\pi$
$602$	11.9003	0.485022
$603$	−8.58287	−0.349521
$604$	−10.8189	−0.440216
$605$	0	0
$606$	−9.15698	−0.371977
$607$	19.3916	0.787081	0.393541	−	0.919307i	$-0.371250\pi$
$607$	19.3916	0.787081	0.393541	+	0.919307i	$0.371250\pi$
$608$	5.91596	0.239924
$609$	−20.6801	−0.837998
$610$	−2.04888	−0.0829567
$611$	51.2685	2.07410
$612$	−3.29955	−0.133376
$613$	29.8111	1.20406	0.602029	−	0.798474i	$-0.294359\pi$
$613$	29.8111	1.20406	0.602029	+	0.798474i	$0.294359\pi$
$614$	3.38972	0.136798
$615$	−7.94454	−0.320355
$616$	0	0
$617$	−3.60999	−0.145333	−0.0726664	−	0.997356i	$-0.523151\pi$
$617$	−3.60999	−0.145333	−0.0726664	+	0.997356i	$0.523151\pi$
$618$	−50.1587	−2.01768
$619$	26.7648	1.07577	0.537884	−	0.843019i	$-0.319224\pi$
$619$	26.7648	1.07577	0.537884	+	0.843019i	$0.319224\pi$
$620$	−9.68953	−0.389141
$621$	32.7549	1.31441
$622$	29.4523	1.18093
$623$	9.54364	0.382358
$624$	−17.1268	−0.685620
$625$	1.00000	0.0400000
$626$	22.0645	0.881873
$627$	0	0
$628$	8.01766	0.319940
$629$	2.17900	0.0868826
$630$	6.91596	0.275538
$631$	−42.4651	−1.69051	−0.845254	−	0.534364i	$-0.820551\pi$
$631$	−42.4651	−1.69051	−0.845254	+	0.534364i	$0.820551\pi$
$632$	3.38007	0.134452
$633$	−71.9005	−2.85779
$634$	−21.1159	−0.838618
$635$	16.3815	0.650080
$636$	−19.3784	−0.768405
$637$	5.43886	0.215496
$638$	0	0
$639$	−25.0564	−0.991218
$640$	−1.00000	−0.0395285
$641$	−12.7639	−0.504144	−0.252072	−	0.967708i	$-0.581112\pi$
$641$	−12.7639	−0.504144	−0.252072	+	0.967708i	$0.581112\pi$
$642$	−19.4364	−0.767093
$643$	1.15422	0.0455181	0.0227591	−	0.999741i	$-0.492755\pi$
$643$	1.15422	0.0455181	0.0227591	+	0.999741i	$0.492755\pi$
$644$	2.65626	0.104671
$645$	−37.4737	−1.47553
$646$	−2.82246	−0.111048
$647$	−11.1492	−0.438321	−0.219161	−	0.975689i	$-0.570332\pi$
$647$	−11.1492	−0.438321	−0.219161	+	0.975689i	$0.570332\pi$
$648$	−18.0826	−0.710351
$649$	0	0
$650$	−5.43886	−0.213330
$651$	−30.5120	−1.19586
$652$	−18.2991	−0.716648
$653$	16.0342	0.627468	0.313734	−	0.949511i	$-0.398420\pi$
$653$	16.0342	0.627468	0.313734	+	0.949511i	$0.398420\pi$
$654$	−46.9950	−1.83765
$655$	5.60812	0.219127
$656$	2.52291	0.0985030
$657$	108.164	4.21988
$658$	9.42632	0.367476
$659$	29.5362	1.15057	0.575283	−	0.817955i	$-0.304892\pi$
$659$	29.5362	1.15057	0.575283	+	0.817955i	$0.304892\pi$
$660$	0	0
$661$	9.49650	0.369371	0.184686	−	0.982798i	$-0.440873\pi$
$661$	9.49650	0.369371	0.184686	+	0.982798i	$0.440873\pi$
$662$	31.1409	1.21033
$663$	8.17105	0.317337
$664$	−1.73868	−0.0674739
$665$	5.91596	0.229411
$666$	31.5870	1.22397
$667$	17.4444	0.675449
$668$	−17.9260	−0.693579
$669$	49.9997	1.93310
$670$	1.24102	0.0479449
$671$	0	0
$672$	−3.14896	−0.121474
$673$	11.5388	0.444788	0.222394	−	0.974957i	$-0.428613\pi$
$673$	11.5388	0.444788	0.222394	+	0.974957i	$0.428613\pi$
$674$	1.60099	0.0616678
$675$	−12.3312	−0.474628
$676$	16.5812	0.637740
$677$	−48.8523	−1.87755	−0.938774	−	0.344533i	$-0.888037\pi$
$677$	−48.8523	−1.87755	−0.938774	+	0.344533i	$0.888037\pi$
$678$	60.1156	2.30872
$679$	3.93156	0.150880
$680$	0.477092	0.0182956
$681$	26.5089	1.01582
$682$	0	0
$683$	35.4736	1.35736	0.678679	−	0.734435i	$-0.262553\pi$
$683$	35.4736	1.35736	0.678679	+	0.734435i	$0.262553\pi$
$684$	−40.9145	−1.56440
$685$	−2.50874	−0.0958539
$686$	1.00000	0.0381802
$687$	−70.1726	−2.67725
$688$	11.9003	0.453697
$689$	33.4703	1.27512
$690$	−8.36447	−0.318430
$691$	−30.1855	−1.14831	−0.574156	−	0.818746i	$-0.694670\pi$
$691$	−30.1855	−1.14831	−0.574156	+	0.818746i	$0.694670\pi$
$692$	−7.68791	−0.292250
$693$	0	0
$694$	−19.7193	−0.748534
$695$	3.30261	0.125275
$696$	−20.6801	−0.783875
$697$	−1.20366	−0.0455919
$698$	−15.3791	−0.582106
$699$	39.0505	1.47702
$700$	−1.00000	−0.0377964
$701$	−11.2689	−0.425621	−0.212810	−	0.977094i	$-0.568262\pi$
$701$	−11.2689	−0.425621	−0.212810	+	0.977094i	$0.568262\pi$
$702$	67.0677	2.53131
$703$	27.0197	1.01907
$704$	0	0
$705$	−29.6831	−1.11793
$706$	10.3793	0.390631
$707$	2.90794	0.109364
$708$	1.78260	0.0669943
$709$	−43.0830	−1.61802	−0.809008	−	0.587797i	$-0.799995\pi$
$709$	−43.0830	−1.61802	−0.809008	+	0.587797i	$0.799995\pi$
$710$	3.62299	0.135968
$711$	−23.3764	−0.876685
$712$	9.54364	0.357663
$713$	25.7379	0.963893
$714$	1.50234	0.0562238
$715$	0	0
$716$	22.7197	0.849076
$717$	−60.9549	−2.27640
$718$	−23.8319	−0.889399
$719$	27.1078	1.01095	0.505475	−	0.862841i	$-0.331317\pi$
$719$	27.1078	1.01095	0.505475	+	0.862841i	$0.331317\pi$
$720$	6.91596	0.257742
$721$	15.9287	0.593214
$722$	−15.9985	−0.595404
$723$	16.1367	0.600130
$724$	−13.6567	−0.507547
$725$	−6.56726	−0.243902
$726$	0	0
$727$	45.5273	1.68851	0.844257	−	0.535939i	$-0.180042\pi$
$727$	45.5273	1.68851	0.844257	+	0.535939i	$0.180042\pi$
$728$	5.43886	0.201578
$729$	8.56700	0.317296
$730$	−15.6398	−0.578854
$731$	−5.67756	−0.209992
$732$	−6.45184	−0.238467
$733$	−23.4004	−0.864312	−0.432156	−	0.901799i	$-0.642247\pi$
$733$	−23.4004	−0.864312	−0.432156	+	0.901799i	$0.642247\pi$
$734$	19.3945	0.715864
$735$	−3.14896	−0.116151
$736$	2.65626	0.0979111
$737$	0	0
$738$	−17.4483	−0.642282
$739$	41.0880	1.51145	0.755723	−	0.654892i	$-0.227286\pi$
$739$	41.0880	1.51145	0.755723	+	0.654892i	$0.227286\pi$
$740$	−4.56726	−0.167896
$741$	101.321	3.72213
$742$	6.15392	0.225917
$743$	16.0312	0.588128	0.294064	−	0.955786i	$-0.404992\pi$
$743$	16.0312	0.588128	0.294064	+	0.955786i	$0.404992\pi$
$744$	−30.5120	−1.11862
$745$	6.83048	0.250249
$746$	−10.9015	−0.399133
$747$	12.0246	0.439958
$748$	0	0
$749$	6.17232	0.225532
$750$	3.14896	0.114984
$751$	5.53050	0.201811	0.100905	−	0.994896i	$-0.467826\pi$
$751$	5.53050	0.201811	0.100905	+	0.994896i	$0.467826\pi$
$752$	9.42632	0.343743
$753$	28.3140	1.03182
$754$	35.7184	1.30079
$755$	−10.8189	−0.393741
$756$	12.3312	0.448481
$757$	25.0946	0.912077	0.456039	−	0.889960i	$-0.349268\pi$
$757$	25.0946	0.912077	0.456039	+	0.889960i	$0.349268\pi$
$758$	−6.80767	−0.247266
$759$	0	0
$760$	5.91596	0.214594
$761$	3.21785	0.116647	0.0583236	−	0.998298i	$-0.481424\pi$
$761$	3.21785	0.116647	0.0583236	+	0.998298i	$0.481424\pi$
$762$	51.5847	1.86872
$763$	14.9240	0.540284
$764$	9.42182	0.340869
$765$	−3.29955	−0.119295
$766$	24.2311	0.875505
$767$	−3.07890	−0.111173
$768$	−3.14896	−0.113628
$769$	9.89520	0.356830	0.178415	−	0.983955i	$-0.442903\pi$
$769$	9.89520	0.356830	0.178415	+	0.983955i	$0.442903\pi$
$770$	0	0
$771$	56.5163	2.03538
$772$	−5.09919	−0.183524
$773$	18.3508	0.660033	0.330017	−	0.943975i	$-0.392946\pi$
$773$	18.3508	0.660033	0.330017	+	0.943975i	$0.392946\pi$
$774$	−82.3023	−2.95829
$775$	−9.68953	−0.348058
$776$	3.93156	0.141135
$777$	−14.3821	−0.515956
$778$	18.2552	0.654480
$779$	−14.9254	−0.534758
$780$	−17.1268	−0.613237
$781$	0	0
$782$	−1.26728	−0.0453179
$783$	80.9822	2.89407
$784$	1.00000	0.0357143
$785$	8.01766	0.286163
$786$	17.6598	0.629903
$787$	6.04479	0.215473	0.107737	−	0.994179i	$-0.465640\pi$
$787$	6.04479	0.215473	0.107737	+	0.994179i	$0.465640\pi$
$788$	17.1284	0.610174
$789$	−86.0649	−3.06399
$790$	3.38007	0.120258
$791$	−19.0906	−0.678784
$792$	0	0
$793$	11.1436	0.395720
$794$	−17.6742	−0.627234
$795$	−19.3784	−0.687283
$796$	11.9225	0.422583
$797$	−6.98324	−0.247359	−0.123679	−	0.992322i	$-0.539469\pi$
$797$	−6.98324	−0.247359	−0.123679	+	0.992322i	$0.539469\pi$
$798$	18.6291	0.659464
$799$	−4.49722	−0.159100
$800$	−1.00000	−0.0353553
$801$	−66.0034	−2.33211
$802$	−30.4187	−1.07412
$803$	0	0
$804$	3.90794	0.137822
$805$	2.65626	0.0936209
$806$	52.7001	1.85628
$807$	38.0315	1.33877
$808$	2.90794	0.102301
$809$	−12.1699	−0.427870	−0.213935	−	0.976848i	$-0.568628\pi$
$809$	−12.1699	−0.427870	−0.213935	+	0.976848i	$0.568628\pi$
$810$	−18.0826	−0.635357
$811$	−45.0840	−1.58311	−0.791557	−	0.611096i	$-0.790729\pi$
$811$	−45.0840	−1.58311	−0.791557	+	0.611096i	$0.790729\pi$
$812$	6.56726	0.230466
$813$	−81.3456	−2.85292
$814$	0	0
$815$	−18.2991	−0.640989
$816$	1.50234	0.0525926
$817$	−70.4019	−2.46305
$818$	−15.4065	−0.538675
$819$	−37.6149	−1.31437
$820$	2.52291	0.0881038
$821$	3.79471	0.132436	0.0662182	−	0.997805i	$-0.478907\pi$
$821$	3.79471	0.132436	0.0662182	+	0.997805i	$0.478907\pi$
$822$	−7.89992	−0.275541
$823$	−39.5469	−1.37852	−0.689259	−	0.724515i	$-0.742064\pi$
$823$	−39.5469	−1.37852	−0.689259	+	0.724515i	$0.742064\pi$
$824$	15.9287	0.554901
$825$	0	0
$826$	−0.566092	−0.0196969
$827$	21.2066	0.737424	0.368712	−	0.929544i	$-0.379799\pi$
$827$	21.2066	0.737424	0.368712	+	0.929544i	$0.379799\pi$
$828$	−18.3706	−0.638422
$829$	−16.6338	−0.577716	−0.288858	−	0.957372i	$-0.593276\pi$
$829$	−16.6338	−0.577716	−0.288858	+	0.957372i	$0.593276\pi$
$830$	−1.73868	−0.0603504
$831$	57.3865	1.99072
$832$	5.43886	0.188559
$833$	−0.477092	−0.0165303
$834$	10.3998	0.360115
$835$	−17.9260	−0.620356
$836$	0	0
$837$	119.484	4.12995
$838$	−3.21914	−0.111203
$839$	44.0630	1.52122	0.760612	−	0.649207i	$-0.224899\pi$
$839$	44.0630	1.52122	0.760612	+	0.649207i	$0.224899\pi$
$840$	−3.14896	−0.108649
$841$	14.1289	0.487205
$842$	−13.1605	−0.453540
$843$	12.7833	0.440280
$844$	22.8331	0.785947
$845$	16.5812	0.570412
$846$	−65.1920	−2.24135
$847$	0	0
$848$	6.15392	0.211326
$849$	9.96879	0.342128
$850$	0.477092	0.0163641
$851$	12.1318	0.415874
$852$	11.4087	0.390854
$853$	8.75092	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$853$	8.75092	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$854$	2.04888	0.0701112
$855$	−40.9145	−1.39925
$856$	6.17232	0.210966
$857$	0.827106	0.0282534	0.0141267	−	0.999900i	$-0.495503\pi$
$857$	0.827106	0.0282534	0.0141267	+	0.999900i	$0.495503\pi$
$858$	0	0
$859$	11.5033	0.392488	0.196244	−	0.980555i	$-0.437125\pi$
$859$	11.5033	0.392488	0.196244	+	0.980555i	$0.437125\pi$
$860$	11.9003	0.405799
$861$	7.94454	0.270749
$862$	−2.38429	−0.0812093
$863$	13.9927	0.476318	0.238159	−	0.971226i	$-0.423456\pi$
$863$	13.9927	0.476318	0.238159	+	0.971226i	$0.423456\pi$
$864$	12.3312	0.419516
$865$	−7.68791	−0.261397
$866$	−9.02568	−0.306705
$867$	52.8156	1.79371
$868$	9.68953	0.328884
$869$	0	0
$870$	−20.6801	−0.701119
$871$	−6.74976	−0.228707
$872$	14.9240	0.505390
$873$	−27.1905	−0.920260
$874$	−15.7143	−0.531545
$875$	−1.00000	−0.0338062
$876$	−49.2490	−1.66397
$877$	−6.87711	−0.232223	−0.116112	−	0.993236i	$-0.537043\pi$
$877$	−6.87711	−0.232223	−0.116112	+	0.993236i	$0.537043\pi$
$878$	−23.1241	−0.780402
$879$	−32.2821	−1.08885
$880$	0	0
$881$	−40.9746	−1.38047	−0.690235	−	0.723585i	$-0.742493\pi$
$881$	−40.9746	−1.38047	−0.690235	+	0.723585i	$0.742493\pi$
$882$	−6.91596	−0.232872
$883$	22.6100	0.760887	0.380443	−	0.924804i	$-0.375771\pi$
$883$	22.6100	0.760887	0.380443	+	0.924804i	$0.375771\pi$
$884$	−2.59484	−0.0872739
$885$	1.78260	0.0599215
$886$	6.78799	0.228047
$887$	3.14606	0.105634	0.0528172	−	0.998604i	$-0.483180\pi$
$887$	3.14606	0.105634	0.0528172	+	0.998604i	$0.483180\pi$
$888$	−14.3821	−0.482633
$889$	−16.3815	−0.549418
$890$	9.54364	0.319903
$891$	0	0
$892$	−15.8782	−0.531640
$893$	−55.7657	−1.86613
$894$	21.5089	0.719365
$895$	22.7197	0.759436
$896$	1.00000	0.0334077
$897$	45.4932	1.51897
$898$	−13.1114	−0.437534
$899$	63.6337	2.12230
$900$	6.91596	0.230532
$901$	−2.93599	−0.0978119
$902$	0	0
$903$	37.4737	1.24705
$904$	−19.0906	−0.634944
$905$	−13.6567	−0.453964
$906$	−34.0684	−1.13185
$907$	6.57950	0.218469	0.109234	−	0.994016i	$-0.465160\pi$
$907$	6.57950	0.218469	0.109234	+	0.994016i	$0.465160\pi$
$908$	−8.41830	−0.279371
$909$	−20.1112	−0.667045
$910$	5.43886	0.180297
$911$	−10.0045	−0.331464	−0.165732	−	0.986171i	$-0.552999\pi$
$911$	−10.0045	−0.331464	−0.165732	+	0.986171i	$0.552999\pi$
$912$	18.6291	0.616872
$913$	0	0
$914$	19.0200	0.629125
$915$	−6.45184	−0.213291
$916$	22.2844	0.736296
$917$	−5.60812	−0.185196
$918$	−5.88312	−0.194172
$919$	5.27944	0.174153	0.0870764	−	0.996202i	$-0.472248\pi$
$919$	5.27944	0.174153	0.0870764	+	0.996202i	$0.472248\pi$
$920$	2.65626	0.0875744
$921$	10.6741	0.351723
$922$	−18.8767	−0.621672
$923$	−19.7050	−0.648596
$924$	0	0
$925$	−4.56726	−0.150171
$926$	−13.6194	−0.447560
$927$	−110.162	−3.61819
$928$	6.56726	0.215581
$929$	−19.7726	−0.648717	−0.324359	−	0.945934i	$-0.605148\pi$
$929$	−19.7726	−0.648717	−0.324359	+	0.945934i	$0.605148\pi$
$930$	−30.5120	−1.00053
$931$	−5.91596	−0.193888
$932$	−12.4011	−0.406210
$933$	92.7441	3.03630
$934$	18.9801	0.621049
$935$	0	0
$936$	−37.6149	−1.22948
$937$	−46.0026	−1.50284	−0.751419	−	0.659825i	$-0.770630\pi$
$937$	−46.0026	−1.50284	−0.751419	+	0.659825i	$0.770630\pi$
$938$	−1.24102	−0.0405209
$939$	69.4801	2.26740
$940$	9.42632	0.307453
$941$	−37.3066	−1.21616	−0.608081	−	0.793875i	$-0.708060\pi$
$941$	−37.3066	−1.21616	−0.608081	+	0.793875i	$0.708060\pi$
$942$	25.2473	0.822602
$943$	−6.70150	−0.218231
$944$	−0.566092	−0.0184247
$945$	12.3312	0.401134
$946$	0	0
$947$	49.7593	1.61696	0.808479	−	0.588524i	$-0.200291\pi$
$947$	49.7593	1.61696	0.808479	+	0.588524i	$0.200291\pi$
$948$	10.6437	0.345692
$949$	85.0626	2.76125
$950$	5.91596	0.191939
$951$	−66.4930	−2.15618
$952$	−0.477092	−0.0154626
$953$	20.5931	0.667076	0.333538	−	0.942737i	$-0.391757\pi$
$953$	20.5931	0.667076	0.333538	+	0.942737i	$0.391757\pi$
$954$	−42.5602	−1.37794
$955$	9.42182	0.304883
$956$	19.3571	0.626055
$957$	0	0
$958$	5.51532	0.178192
$959$	2.50874	0.0810114
$960$	−3.14896	−0.101632
$961$	62.8871	2.02862
$962$	24.8407	0.800897
$963$	−42.6875	−1.37559
$964$	−5.12445	−0.165047
$965$	−5.09919	−0.164149
$966$	8.36447	0.269122
$967$	−36.0618	−1.15967	−0.579834	−	0.814735i	$-0.696883\pi$
$967$	−36.0618	−1.15967	−0.579834	+	0.814735i	$0.696883\pi$
$968$	0	0
$969$	−8.88781	−0.285517
$970$	3.93156	0.126235
$971$	4.29180	0.137730	0.0688651	−	0.997626i	$-0.478062\pi$
$971$	4.29180	0.137730	0.0688651	+	0.997626i	$0.478062\pi$
$972$	−19.9478	−0.639825
$973$	−3.30261	−0.105877
$974$	−28.0086	−0.897454
$975$	−17.1268	−0.548496
$976$	2.04888	0.0655830
$977$	36.2757	1.16056	0.580281	−	0.814416i	$-0.302943\pi$
$977$	36.2757	1.16056	0.580281	+	0.814416i	$0.302943\pi$
$978$	−57.6231	−1.84258
$979$	0	0
$980$	1.00000	0.0319438
$981$	−103.214	−3.29536
$982$	12.9545	0.413393
$983$	18.9549	0.604567	0.302284	−	0.953218i	$-0.402251\pi$
$983$	18.9549	0.604567	0.302284	+	0.953218i	$0.402251\pi$
$984$	7.94454	0.253263
$985$	17.1284	0.545756
$986$	−3.13319	−0.0997811
$987$	29.6831	0.944824
$988$	−32.1761	−1.02366
$989$	−31.6104	−1.00515
$990$	0	0
$991$	−7.43374	−0.236141	−0.118070	−	0.993005i	$-0.537671\pi$
$991$	−7.43374	−0.236141	−0.118070	+	0.993005i	$0.537671\pi$
$992$	9.68953	0.307643
$993$	98.0615	3.11189
$994$	−3.62299	−0.114914
$995$	11.9225	0.377970
$996$	−5.47503	−0.173483
$997$	21.9062	0.693777	0.346888	−	0.937906i	$-0.387238\pi$
$997$	21.9062	0.693777	0.346888	+	0.937906i	$0.387238\pi$
$998$	−3.93985	−0.124714
$999$	56.3198	1.78188

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.co.1.1		4
11.3	even	5		770.2.n.f.141.2	yes	8
11.4	even	5		770.2.n.f.71.2	✓	8
11.10	odd	2		8470.2.a.cs.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.f.71.2	✓	8	11.4	even	5
770.2.n.f.141.2	yes	8	11.3	even	5
8470.2.a.co.1.1		4	1.1	even	1	trivial
8470.2.a.cs.1.1		4	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.14896 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.14896 q^{6} -1.00000 q^{7} -1.00000 q^{8} +6.91596 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.14896 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.14896 q^{6} -1.00000 q^{7} -1.00000 q^{8} +6.91596 q^{9} -1.00000 q^{10} -3.14896 q^{12} +5.43886 q^{13} +1.00000 q^{14} -3.14896 q^{15} +1.00000 q^{16} -0.477092 q^{17} -6.91596 q^{18} -5.91596 q^{19} +1.00000 q^{20} +3.14896 q^{21} -2.65626 q^{23} +3.14896 q^{24} +1.00000 q^{25} -5.43886 q^{26} -12.3312 q^{27} -1.00000 q^{28} -6.56726 q^{29} +3.14896 q^{30} -9.68953 q^{31} -1.00000 q^{32} +0.477092 q^{34} -1.00000 q^{35} +6.91596 q^{36} -4.56726 q^{37} +5.91596 q^{38} -17.1268 q^{39} -1.00000 q^{40} +2.52291 q^{41} -3.14896 q^{42} +11.9003 q^{43} +6.91596 q^{45} +2.65626 q^{46} +9.42632 q^{47} -3.14896 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.50234 q^{51} +5.43886 q^{52} +6.15392 q^{53} +12.3312 q^{54} +1.00000 q^{56} +18.6291 q^{57} +6.56726 q^{58} -0.566092 q^{59} -3.14896 q^{60} +2.04888 q^{61} +9.68953 q^{62} -6.91596 q^{63} +1.00000 q^{64} +5.43886 q^{65} -1.24102 q^{67} -0.477092 q^{68} +8.36447 q^{69} +1.00000 q^{70} -3.62299 q^{71} -6.91596 q^{72} +15.6398 q^{73} +4.56726 q^{74} -3.14896 q^{75} -5.91596 q^{76} +17.1268 q^{78} -3.38007 q^{79} +1.00000 q^{80} +18.0826 q^{81} -2.52291 q^{82} +1.73868 q^{83} +3.14896 q^{84} -0.477092 q^{85} -11.9003 q^{86} +20.6801 q^{87} -9.54364 q^{89} -6.91596 q^{90} -5.43886 q^{91} -2.65626 q^{92} +30.5120 q^{93} -9.42632 q^{94} -5.91596 q^{95} +3.14896 q^{96} -3.93156 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} - 4 q^{10} - 4 q^{12} + 14 q^{13} + 4 q^{14} - 4 q^{15} + 4 q^{16} + 12 q^{17} - 6 q^{18} - 2 q^{19} + 4 q^{20} + 4 q^{21} + 4 q^{24} + 4 q^{25} - 14 q^{26} - 22 q^{27} - 4 q^{28} + 10 q^{29} + 4 q^{30} - 18 q^{31} - 4 q^{32} - 12 q^{34} - 4 q^{35} + 6 q^{36} + 18 q^{37} + 2 q^{38} - 30 q^{39} - 4 q^{40} + 24 q^{41} - 4 q^{42} + 10 q^{43} + 6 q^{45} - 8 q^{47} - 4 q^{48} + 4 q^{49} - 4 q^{50} + 14 q^{52} + 20 q^{53} + 22 q^{54} + 4 q^{56} + 30 q^{57} - 10 q^{58} - 14 q^{59} - 4 q^{60} + 14 q^{61} + 18 q^{62} - 6 q^{63} + 4 q^{64} + 14 q^{65} + 12 q^{68} - 4 q^{69} + 4 q^{70} - 14 q^{71} - 6 q^{72} + 30 q^{73} - 18 q^{74} - 4 q^{75} - 2 q^{76} + 30 q^{78} + 8 q^{79} + 4 q^{80} + 16 q^{81} - 24 q^{82} + 8 q^{83} + 4 q^{84} + 12 q^{85} - 10 q^{86} + 2 q^{87} - 4 q^{89} - 6 q^{90} - 14 q^{91} - 2 q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 10 q^{97} - 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9 - 4 * q^10 - 4 * q^12 + 14 * q^13 + 4 * q^14 - 4 * q^15 + 4 * q^16 + 12 * q^17 - 6 * q^18 - 2 * q^19 + 4 * q^20 + 4 * q^21 + 4 * q^24 + 4 * q^25 - 14 * q^26 - 22 * q^27 - 4 * q^28 + 10 * q^29 + 4 * q^30 - 18 * q^31 - 4 * q^32 - 12 * q^34 - 4 * q^35 + 6 * q^36 + 18 * q^37 + 2 * q^38 - 30 * q^39 - 4 * q^40 + 24 * q^41 - 4 * q^42 + 10 * q^43 + 6 * q^45 - 8 * q^47 - 4 * q^48 + 4 * q^49 - 4 * q^50 + 14 * q^52 + 20 * q^53 + 22 * q^54 + 4 * q^56 + 30 * q^57 - 10 * q^58 - 14 * q^59 - 4 * q^60 + 14 * q^61 + 18 * q^62 - 6 * q^63 + 4 * q^64 + 14 * q^65 + 12 * q^68 - 4 * q^69 + 4 * q^70 - 14 * q^71 - 6 * q^72 + 30 * q^73 - 18 * q^74 - 4 * q^75 - 2 * q^76 + 30 * q^78 + 8 * q^79 + 4 * q^80 + 16 * q^81 - 24 * q^82 + 8 * q^83 + 4 * q^84 + 12 * q^85 - 10 * q^86 + 2 * q^87 - 4 * q^89 - 6 * q^90 - 14 * q^91 - 2 * q^93 + 8 * q^94 - 2 * q^95 + 4 * q^96 - 10 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

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Database

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