LMFDB - Embedded newform 8470.2.a.cf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.381966 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.381966 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} -1.00000 q^{10} +0.381966 q^{12} +2.00000 q^{13} +1.00000 q^{14} -0.381966 q^{15} +1.00000 q^{16} +0.618034 q^{17} -2.85410 q^{18} +0.145898 q^{19} -1.00000 q^{20} +0.381966 q^{21} +6.00000 q^{23} +0.381966 q^{24} +1.00000 q^{25} +2.00000 q^{26} -2.23607 q^{27} +1.00000 q^{28} -1.23607 q^{29} -0.381966 q^{30} +3.23607 q^{31} +1.00000 q^{32} +0.618034 q^{34} -1.00000 q^{35} -2.85410 q^{36} +2.47214 q^{37} +0.145898 q^{38} +0.763932 q^{39} -1.00000 q^{40} +5.32624 q^{41} +0.381966 q^{42} -4.85410 q^{43} +2.85410 q^{45} +6.00000 q^{46} +4.76393 q^{47} +0.381966 q^{48} +1.00000 q^{49} +1.00000 q^{50} +0.236068 q^{51} +2.00000 q^{52} -3.23607 q^{53} -2.23607 q^{54} +1.00000 q^{56} +0.0557281 q^{57} -1.23607 q^{58} -5.38197 q^{59} -0.381966 q^{60} -12.4721 q^{61} +3.23607 q^{62} -2.85410 q^{63} +1.00000 q^{64} -2.00000 q^{65} -5.09017 q^{67} +0.618034 q^{68} +2.29180 q^{69} -1.00000 q^{70} +3.70820 q^{71} -2.85410 q^{72} +7.14590 q^{73} +2.47214 q^{74} +0.381966 q^{75} +0.145898 q^{76} +0.763932 q^{78} +0.472136 q^{79} -1.00000 q^{80} +7.70820 q^{81} +5.32624 q^{82} -5.32624 q^{83} +0.381966 q^{84} -0.618034 q^{85} -4.85410 q^{86} -0.472136 q^{87} +1.90983 q^{89} +2.85410 q^{90} +2.00000 q^{91} +6.00000 q^{92} +1.23607 q^{93} +4.76393 q^{94} -0.145898 q^{95} +0.381966 q^{96} +10.5623 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 2 q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 2 * q^5 + 3 * q^6 + 2 * q^7 + 2 * q^8 + q^9	$ 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 2 q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + q^{9} - 2 q^{10} + 3 q^{12} + 4 q^{13} + 2 q^{14} - 3 q^{15} + 2 q^{16} - q^{17} + q^{18} + 7 q^{19} - 2 q^{20} + 3 q^{21} + 12 q^{23} + 3 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{28} + 2 q^{29} - 3 q^{30} + 2 q^{31} + 2 q^{32} - q^{34} - 2 q^{35} + q^{36} - 4 q^{37} + 7 q^{38} + 6 q^{39} - 2 q^{40} - 5 q^{41} + 3 q^{42} - 3 q^{43} - q^{45} + 12 q^{46} + 14 q^{47} + 3 q^{48} + 2 q^{49} + 2 q^{50} - 4 q^{51} + 4 q^{52} - 2 q^{53} + 2 q^{56} + 18 q^{57} + 2 q^{58} - 13 q^{59} - 3 q^{60} - 16 q^{61} + 2 q^{62} + q^{63} + 2 q^{64} - 4 q^{65} + q^{67} - q^{68} + 18 q^{69} - 2 q^{70} - 6 q^{71} + q^{72} + 21 q^{73} - 4 q^{74} + 3 q^{75} + 7 q^{76} + 6 q^{78} - 8 q^{79} - 2 q^{80} + 2 q^{81} - 5 q^{82} + 5 q^{83} + 3 q^{84} + q^{85} - 3 q^{86} + 8 q^{87} + 15 q^{89} - q^{90} + 4 q^{91} + 12 q^{92} - 2 q^{93} + 14 q^{94} - 7 q^{95} + 3 q^{96} + q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 2 * q^5 + 3 * q^6 + 2 * q^7 + 2 * q^8 + q^9 - 2 * q^10 + 3 * q^12 + 4 * q^13 + 2 * q^14 - 3 * q^15 + 2 * q^16 - q^17 + q^18 + 7 * q^19 - 2 * q^20 + 3 * q^21 + 12 * q^23 + 3 * q^24 + 2 * q^25 + 4 * q^26 + 2 * q^28 + 2 * q^29 - 3 * q^30 + 2 * q^31 + 2 * q^32 - q^34 - 2 * q^35 + q^36 - 4 * q^37 + 7 * q^38 + 6 * q^39 - 2 * q^40 - 5 * q^41 + 3 * q^42 - 3 * q^43 - q^45 + 12 * q^46 + 14 * q^47 + 3 * q^48 + 2 * q^49 + 2 * q^50 - 4 * q^51 + 4 * q^52 - 2 * q^53 + 2 * q^56 + 18 * q^57 + 2 * q^58 - 13 * q^59 - 3 * q^60 - 16 * q^61 + 2 * q^62 + q^63 + 2 * q^64 - 4 * q^65 + q^67 - q^68 + 18 * q^69 - 2 * q^70 - 6 * q^71 + q^72 + 21 * q^73 - 4 * q^74 + 3 * q^75 + 7 * q^76 + 6 * q^78 - 8 * q^79 - 2 * q^80 + 2 * q^81 - 5 * q^82 + 5 * q^83 + 3 * q^84 + q^85 - 3 * q^86 + 8 * q^87 + 15 * q^89 - q^90 + 4 * q^91 + 12 * q^92 - 2 * q^93 + 14 * q^94 - 7 * q^95 + 3 * q^96 + q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.381966	0.220528	0.110264	−	0.993902i	$-0.464830\pi$
$3$	0.381966	0.220528	0.110264	+	0.993902i	$0.464830\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.381966	0.155937
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−2.85410	−0.951367
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	0.381966	0.110264
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	−0.381966	−0.0986232
$16$	1.00000	0.250000
$17$	0.618034	0.149895	0.0749476	−	0.997187i	$-0.476121\pi$
$17$	0.618034	0.149895	0.0749476	+	0.997187i	$0.476121\pi$
$18$	−2.85410	−0.672718
$19$	0.145898	0.0334713	0.0167357	−	0.999860i	$-0.494673\pi$
$19$	0.145898	0.0334713	0.0167357	+	0.999860i	$0.494673\pi$
$20$	−1.00000	−0.223607
$21$	0.381966	0.0833518
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0.381966	0.0779685
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−2.23607	−0.430331
$28$	1.00000	0.188982
$29$	−1.23607	−0.229532	−0.114766	−	0.993393i	$-0.536612\pi$
$29$	−1.23607	−0.229532	−0.114766	+	0.993393i	$0.536612\pi$
$30$	−0.381966	−0.0697371
$31$	3.23607	0.581215	0.290607	−	0.956842i	$-0.406143\pi$
$31$	3.23607	0.581215	0.290607	+	0.956842i	$0.406143\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.618034	0.105992
$35$	−1.00000	−0.169031
$36$	−2.85410	−0.475684
$37$	2.47214	0.406417	0.203208	−	0.979136i	$-0.434863\pi$
$37$	2.47214	0.406417	0.203208	+	0.979136i	$0.434863\pi$
$38$	0.145898	0.0236678
$39$	0.763932	0.122327
$40$	−1.00000	−0.158114
$41$	5.32624	0.831819	0.415909	−	0.909406i	$-0.363463\pi$
$41$	5.32624	0.831819	0.415909	+	0.909406i	$0.363463\pi$
$42$	0.381966	0.0589386
$43$	−4.85410	−0.740244	−0.370122	−	0.928983i	$-0.620684\pi$
$43$	−4.85410	−0.740244	−0.370122	+	0.928983i	$0.620684\pi$
$44$	0	0
$45$	2.85410	0.425464
$46$	6.00000	0.884652
$47$	4.76393	0.694891	0.347445	−	0.937700i	$-0.387049\pi$
$47$	4.76393	0.694891	0.347445	+	0.937700i	$0.387049\pi$
$48$	0.381966	0.0551320
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0.236068	0.0330561
$52$	2.00000	0.277350
$53$	−3.23607	−0.444508	−0.222254	−	0.974989i	$-0.571341\pi$
$53$	−3.23607	−0.444508	−0.222254	+	0.974989i	$0.571341\pi$
$54$	−2.23607	−0.304290
$55$	0	0
$56$	1.00000	0.133631
$57$	0.0557281	0.00738137
$58$	−1.23607	−0.162304
$59$	−5.38197	−0.700672	−0.350336	−	0.936624i	$-0.613933\pi$
$59$	−5.38197	−0.700672	−0.350336	+	0.936624i	$0.613933\pi$
$60$	−0.381966	−0.0493116
$61$	−12.4721	−1.59689	−0.798447	−	0.602066i	$-0.794345\pi$
$61$	−12.4721	−1.59689	−0.798447	+	0.602066i	$0.794345\pi$
$62$	3.23607	0.410981
$63$	−2.85410	−0.359583
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−5.09017	−0.621863	−0.310932	−	0.950432i	$-0.600641\pi$
$67$	−5.09017	−0.621863	−0.310932	+	0.950432i	$0.600641\pi$
$68$	0.618034	0.0749476
$69$	2.29180	0.275900
$70$	−1.00000	−0.119523
$71$	3.70820	0.440083	0.220041	−	0.975491i	$-0.429381\pi$
$71$	3.70820	0.440083	0.220041	+	0.975491i	$0.429381\pi$
$72$	−2.85410	−0.336359
$73$	7.14590	0.836364	0.418182	−	0.908363i	$-0.362667\pi$
$73$	7.14590	0.836364	0.418182	+	0.908363i	$0.362667\pi$
$74$	2.47214	0.287380
$75$	0.381966	0.0441056
$76$	0.145898	0.0167357
$77$	0	0
$78$	0.763932	0.0864983
$79$	0.472136	0.0531194	0.0265597	−	0.999647i	$-0.491545\pi$
$79$	0.472136	0.0531194	0.0265597	+	0.999647i	$0.491545\pi$
$80$	−1.00000	−0.111803
$81$	7.70820	0.856467
$82$	5.32624	0.588185
$83$	−5.32624	−0.584631	−0.292315	−	0.956322i	$-0.594426\pi$
$83$	−5.32624	−0.584631	−0.292315	+	0.956322i	$0.594426\pi$
$84$	0.381966	0.0416759
$85$	−0.618034	−0.0670352
$86$	−4.85410	−0.523431
$87$	−0.472136	−0.0506183
$88$	0	0
$89$	1.90983	0.202442	0.101221	−	0.994864i	$-0.467725\pi$
$89$	1.90983	0.202442	0.101221	+	0.994864i	$0.467725\pi$
$90$	2.85410	0.300849
$91$	2.00000	0.209657
$92$	6.00000	0.625543
$93$	1.23607	0.128174
$94$	4.76393	0.491362
$95$	−0.145898	−0.0149688
$96$	0.381966	0.0389842
$97$	10.5623	1.07244	0.536220	−	0.844078i	$-0.319852\pi$
$97$	10.5623	1.07244	0.536220	+	0.844078i	$0.319852\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−8.94427	−0.889988	−0.444994	−	0.895533i	$-0.646794\pi$
$101$	−8.94427	−0.889988	−0.444994	+	0.895533i	$0.646794\pi$
$102$	0.236068	0.0233742
$103$	0.944272	0.0930419	0.0465209	−	0.998917i	$-0.485187\pi$
$103$	0.944272	0.0930419	0.0465209	+	0.998917i	$0.485187\pi$
$104$	2.00000	0.196116
$105$	−0.381966	−0.0372761
$106$	−3.23607	−0.314315
$107$	17.6180	1.70320	0.851600	−	0.524192i	$-0.175633\pi$
$107$	17.6180	1.70320	0.851600	+	0.524192i	$0.175633\pi$
$108$	−2.23607	−0.215166
$109$	8.47214	0.811483	0.405742	−	0.913988i	$-0.367013\pi$
$109$	8.47214	0.811483	0.405742	+	0.913988i	$0.367013\pi$
$110$	0	0
$111$	0.944272	0.0896263
$112$	1.00000	0.0944911
$113$	6.32624	0.595122	0.297561	−	0.954703i	$-0.403827\pi$
$113$	6.32624	0.595122	0.297561	+	0.954703i	$0.403827\pi$
$114$	0.0557281	0.00521941
$115$	−6.00000	−0.559503
$116$	−1.23607	−0.114766
$117$	−5.70820	−0.527724
$118$	−5.38197	−0.495450
$119$	0.618034	0.0566551
$120$	−0.381966	−0.0348686
$121$	0	0
$122$	−12.4721	−1.12917
$123$	2.03444	0.183439
$124$	3.23607	0.290607
$125$	−1.00000	−0.0894427
$126$	−2.85410	−0.254264
$127$	15.4164	1.36798	0.683992	−	0.729489i	$-0.260242\pi$
$127$	15.4164	1.36798	0.683992	+	0.729489i	$0.260242\pi$
$128$	1.00000	0.0883883
$129$	−1.85410	−0.163245
$130$	−2.00000	−0.175412
$131$	16.7984	1.46768	0.733840	−	0.679322i	$-0.237726\pi$
$131$	16.7984	1.46768	0.733840	+	0.679322i	$0.237726\pi$
$132$	0	0
$133$	0.145898	0.0126510
$134$	−5.09017	−0.439724
$135$	2.23607	0.192450
$136$	0.618034	0.0529960
$137$	10.0902	0.862061	0.431031	−	0.902337i	$-0.358150\pi$
$137$	10.0902	0.862061	0.431031	+	0.902337i	$0.358150\pi$
$138$	2.29180	0.195091
$139$	9.52786	0.808143	0.404071	−	0.914727i	$-0.367595\pi$
$139$	9.52786	0.808143	0.404071	+	0.914727i	$0.367595\pi$
$140$	−1.00000	−0.0845154
$141$	1.81966	0.153243
$142$	3.70820	0.311186
$143$	0	0
$144$	−2.85410	−0.237842
$145$	1.23607	0.102650
$146$	7.14590	0.591399
$147$	0.381966	0.0315040
$148$	2.47214	0.203208
$149$	1.81966	0.149072	0.0745362	−	0.997218i	$-0.476252\pi$
$149$	1.81966	0.149072	0.0745362	+	0.997218i	$0.476252\pi$
$150$	0.381966	0.0311874
$151$	15.7082	1.27832	0.639158	−	0.769076i	$-0.279283\pi$
$151$	15.7082	1.27832	0.639158	+	0.769076i	$0.279283\pi$
$152$	0.145898	0.0118339
$153$	−1.76393	−0.142605
$154$	0	0
$155$	−3.23607	−0.259927
$156$	0.763932	0.0611635
$157$	13.7082	1.09403	0.547017	−	0.837122i	$-0.315763\pi$
$157$	13.7082	1.09403	0.547017	+	0.837122i	$0.315763\pi$
$158$	0.472136	0.0375611
$159$	−1.23607	−0.0980266
$160$	−1.00000	−0.0790569
$161$	6.00000	0.472866
$162$	7.70820	0.605614
$163$	−3.85410	−0.301877	−0.150938	−	0.988543i	$-0.548229\pi$
$163$	−3.85410	−0.301877	−0.150938	+	0.988543i	$0.548229\pi$
$164$	5.32624	0.415909
$165$	0	0
$166$	−5.32624	−0.413396
$167$	−2.47214	−0.191300	−0.0956498	−	0.995415i	$-0.530493\pi$
$167$	−2.47214	−0.191300	−0.0956498	+	0.995415i	$0.530493\pi$
$168$	0.381966	0.0294693
$169$	−9.00000	−0.692308
$170$	−0.618034	−0.0474010
$171$	−0.416408	−0.0318435
$172$	−4.85410	−0.370122
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	−0.472136	−0.0357925
$175$	1.00000	0.0755929
$176$	0	0
$177$	−2.05573	−0.154518
$178$	1.90983	0.143148
$179$	−8.61803	−0.644142	−0.322071	−	0.946715i	$-0.604379\pi$
$179$	−8.61803	−0.644142	−0.322071	+	0.946715i	$0.604379\pi$
$180$	2.85410	0.212732
$181$	12.4721	0.927047	0.463523	−	0.886085i	$-0.346585\pi$
$181$	12.4721	0.927047	0.463523	+	0.886085i	$0.346585\pi$
$182$	2.00000	0.148250
$183$	−4.76393	−0.352160
$184$	6.00000	0.442326
$185$	−2.47214	−0.181755
$186$	1.23607	0.0906329
$187$	0	0
$188$	4.76393	0.347445
$189$	−2.23607	−0.162650
$190$	−0.145898	−0.0105846
$191$	−10.4721	−0.757737	−0.378869	−	0.925450i	$-0.623687\pi$
$191$	−10.4721	−0.757737	−0.378869	+	0.925450i	$0.623687\pi$
$192$	0.381966	0.0275660
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	10.5623	0.758329
$195$	−0.763932	−0.0547063
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	7.41641	0.525735	0.262868	−	0.964832i	$-0.415332\pi$
$199$	7.41641	0.525735	0.262868	+	0.964832i	$0.415332\pi$
$200$	1.00000	0.0707107
$201$	−1.94427	−0.137138
$202$	−8.94427	−0.629317
$203$	−1.23607	−0.0867550
$204$	0.236068	0.0165281
$205$	−5.32624	−0.372001
$206$	0.944272	0.0657905
$207$	−17.1246	−1.19024
$208$	2.00000	0.138675
$209$	0	0
$210$	−0.381966	−0.0263582
$211$	−22.2705	−1.53317	−0.766583	−	0.642146i	$-0.778044\pi$
$211$	−22.2705	−1.53317	−0.766583	+	0.642146i	$0.778044\pi$
$212$	−3.23607	−0.222254
$213$	1.41641	0.0970507
$214$	17.6180	1.20434
$215$	4.85410	0.331047
$216$	−2.23607	−0.152145
$217$	3.23607	0.219679
$218$	8.47214	0.573805
$219$	2.72949	0.184442
$220$	0	0
$221$	1.23607	0.0831469
$222$	0.944272	0.0633754
$223$	19.7082	1.31976	0.659879	−	0.751371i	$-0.270607\pi$
$223$	19.7082	1.31976	0.659879	+	0.751371i	$0.270607\pi$
$224$	1.00000	0.0668153
$225$	−2.85410	−0.190273
$226$	6.32624	0.420815
$227$	9.27051	0.615305	0.307653	−	0.951499i	$-0.400457\pi$
$227$	9.27051	0.615305	0.307653	+	0.951499i	$0.400457\pi$
$228$	0.0557281	0.00369068
$229$	−29.1246	−1.92461	−0.962304	−	0.271975i	$-0.912323\pi$
$229$	−29.1246	−1.92461	−0.962304	+	0.271975i	$0.912323\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	−1.23607	−0.0811518
$233$	22.0902	1.44718	0.723588	−	0.690233i	$-0.242492\pi$
$233$	22.0902	1.44718	0.723588	+	0.690233i	$0.242492\pi$
$234$	−5.70820	−0.373157
$235$	−4.76393	−0.310765
$236$	−5.38197	−0.350336
$237$	0.180340	0.0117143
$238$	0.618034	0.0400612
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−0.381966	−0.0246558
$241$	27.2705	1.75665	0.878324	−	0.478066i	$-0.158662\pi$
$241$	27.2705	1.75665	0.878324	+	0.478066i	$0.158662\pi$
$242$	0	0
$243$	9.65248	0.619207
$244$	−12.4721	−0.798447
$245$	−1.00000	−0.0638877
$246$	2.03444	0.129711
$247$	0.291796	0.0185665
$248$	3.23607	0.205491
$249$	−2.03444	−0.128928
$250$	−1.00000	−0.0632456
$251$	−26.8328	−1.69367	−0.846836	−	0.531854i	$-0.821496\pi$
$251$	−26.8328	−1.69367	−0.846836	+	0.531854i	$0.821496\pi$
$252$	−2.85410	−0.179792
$253$	0	0
$254$	15.4164	0.967311
$255$	−0.236068	−0.0147832
$256$	1.00000	0.0625000
$257$	9.43769	0.588707	0.294354	−	0.955697i	$-0.404896\pi$
$257$	9.43769	0.588707	0.294354	+	0.955697i	$0.404896\pi$
$258$	−1.85410	−0.115431
$259$	2.47214	0.153611
$260$	−2.00000	−0.124035
$261$	3.52786	0.218369
$262$	16.7984	1.03781
$263$	−2.18034	−0.134446	−0.0672228	−	0.997738i	$-0.521414\pi$
$263$	−2.18034	−0.134446	−0.0672228	+	0.997738i	$0.521414\pi$
$264$	0	0
$265$	3.23607	0.198790
$266$	0.145898	0.00894558
$267$	0.729490	0.0446441
$268$	−5.09017	−0.310932
$269$	10.1803	0.620706	0.310353	−	0.950621i	$-0.399553\pi$
$269$	10.1803	0.620706	0.310353	+	0.950621i	$0.399553\pi$
$270$	2.23607	0.136083
$271$	17.4164	1.05797	0.528986	−	0.848631i	$-0.322573\pi$
$271$	17.4164	1.05797	0.528986	+	0.848631i	$0.322573\pi$
$272$	0.618034	0.0374738
$273$	0.763932	0.0462353
$274$	10.0902	0.609569
$275$	0	0
$276$	2.29180	0.137950
$277$	19.7082	1.18415	0.592076	−	0.805882i	$-0.298309\pi$
$277$	19.7082	1.18415	0.592076	+	0.805882i	$0.298309\pi$
$278$	9.52786	0.571443
$279$	−9.23607	−0.552949
$280$	−1.00000	−0.0597614
$281$	5.03444	0.300330	0.150165	−	0.988661i	$-0.452020\pi$
$281$	5.03444	0.300330	0.150165	+	0.988661i	$0.452020\pi$
$282$	1.81966	0.108359
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	3.70820	0.220041
$285$	−0.0557281	−0.00330105
$286$	0	0
$287$	5.32624	0.314398
$288$	−2.85410	−0.168180
$289$	−16.6180	−0.977531
$290$	1.23607	0.0725844
$291$	4.03444	0.236503
$292$	7.14590	0.418182
$293$	7.52786	0.439783	0.219891	−	0.975524i	$-0.429430\pi$
$293$	7.52786	0.439783	0.219891	+	0.975524i	$0.429430\pi$
$294$	0.381966	0.0222767
$295$	5.38197	0.313350
$296$	2.47214	0.143690
$297$	0	0
$298$	1.81966	0.105410
$299$	12.0000	0.693978
$300$	0.381966	0.0220528
$301$	−4.85410	−0.279786
$302$	15.7082	0.903906
$303$	−3.41641	−0.196268
$304$	0.145898	0.00836783
$305$	12.4721	0.714152
$306$	−1.76393	−0.100837
$307$	−7.43769	−0.424492	−0.212246	−	0.977216i	$-0.568078\pi$
$307$	−7.43769	−0.424492	−0.212246	+	0.977216i	$0.568078\pi$
$308$	0	0
$309$	0.360680	0.0205184
$310$	−3.23607	−0.183796
$311$	11.1246	0.630819	0.315409	−	0.948956i	$-0.397858\pi$
$311$	11.1246	0.630819	0.315409	+	0.948956i	$0.397858\pi$
$312$	0.763932	0.0432491
$313$	5.85410	0.330893	0.165447	−	0.986219i	$-0.447093\pi$
$313$	5.85410	0.330893	0.165447	+	0.986219i	$0.447093\pi$
$314$	13.7082	0.773599
$315$	2.85410	0.160810
$316$	0.472136	0.0265597
$317$	−25.2361	−1.41740	−0.708699	−	0.705511i	$-0.750718\pi$
$317$	−25.2361	−1.41740	−0.708699	+	0.705511i	$0.750718\pi$
$318$	−1.23607	−0.0693153
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	6.72949	0.375604
$322$	6.00000	0.334367
$323$	0.0901699	0.00501719
$324$	7.70820	0.428234
$325$	2.00000	0.110940
$326$	−3.85410	−0.213459
$327$	3.23607	0.178955
$328$	5.32624	0.294092
$329$	4.76393	0.262644
$330$	0	0
$331$	−23.0344	−1.26609	−0.633044	−	0.774116i	$-0.718195\pi$
$331$	−23.0344	−1.26609	−0.633044	+	0.774116i	$0.718195\pi$
$332$	−5.32624	−0.292315
$333$	−7.05573	−0.386652
$334$	−2.47214	−0.135269
$335$	5.09017	0.278106
$336$	0.381966	0.0208380
$337$	−13.3820	−0.728962	−0.364481	−	0.931211i	$-0.618754\pi$
$337$	−13.3820	−0.728962	−0.364481	+	0.931211i	$0.618754\pi$
$338$	−9.00000	−0.489535
$339$	2.41641	0.131241
$340$	−0.618034	−0.0335176
$341$	0	0
$342$	−0.416408	−0.0225168
$343$	1.00000	0.0539949
$344$	−4.85410	−0.261716
$345$	−2.29180	−0.123386
$346$	12.0000	0.645124
$347$	−10.3262	−0.554341	−0.277171	−	0.960821i	$-0.589397\pi$
$347$	−10.3262	−0.554341	−0.277171	+	0.960821i	$0.589397\pi$
$348$	−0.472136	−0.0253091
$349$	14.4721	0.774676	0.387338	−	0.921938i	$-0.373395\pi$
$349$	14.4721	0.774676	0.387338	+	0.921938i	$0.373395\pi$
$350$	1.00000	0.0534522
$351$	−4.47214	−0.238705
$352$	0	0
$353$	27.4508	1.46106	0.730531	−	0.682880i	$-0.239273\pi$
$353$	27.4508	1.46106	0.730531	+	0.682880i	$0.239273\pi$
$354$	−2.05573	−0.109261
$355$	−3.70820	−0.196811
$356$	1.90983	0.101221
$357$	0.236068	0.0124940
$358$	−8.61803	−0.455477
$359$	−12.6525	−0.667772	−0.333886	−	0.942613i	$-0.608360\pi$
$359$	−12.6525	−0.667772	−0.333886	+	0.942613i	$0.608360\pi$
$360$	2.85410	0.150424
$361$	−18.9787	−0.998880
$362$	12.4721	0.655521
$363$	0	0
$364$	2.00000	0.104828
$365$	−7.14590	−0.374033
$366$	−4.76393	−0.249015
$367$	5.81966	0.303784	0.151892	−	0.988397i	$-0.451463\pi$
$367$	5.81966	0.303784	0.151892	+	0.988397i	$0.451463\pi$
$368$	6.00000	0.312772
$369$	−15.2016	−0.791365
$370$	−2.47214	−0.128520
$371$	−3.23607	−0.168008
$372$	1.23607	0.0640871
$373$	−15.5279	−0.804002	−0.402001	−	0.915639i	$-0.631685\pi$
$373$	−15.5279	−0.804002	−0.402001	+	0.915639i	$0.631685\pi$
$374$	0	0
$375$	−0.381966	−0.0197246
$376$	4.76393	0.245681
$377$	−2.47214	−0.127321
$378$	−2.23607	−0.115011
$379$	−21.1459	−1.08619	−0.543096	−	0.839671i	$-0.682748\pi$
$379$	−21.1459	−1.08619	−0.543096	+	0.839671i	$0.682748\pi$
$380$	−0.145898	−0.00748441
$381$	5.88854	0.301679
$382$	−10.4721	−0.535801
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0.381966	0.0194921
$385$	0	0
$386$	−2.00000	−0.101797
$387$	13.8541	0.704244
$388$	10.5623	0.536220
$389$	20.8328	1.05627	0.528133	−	0.849162i	$-0.322892\pi$
$389$	20.8328	1.05627	0.528133	+	0.849162i	$0.322892\pi$
$390$	−0.763932	−0.0386832
$391$	3.70820	0.187532
$392$	1.00000	0.0505076
$393$	6.41641	0.323665
$394$	6.00000	0.302276
$395$	−0.472136	−0.0237557
$396$	0	0
$397$	0.944272	0.0473916	0.0236958	−	0.999719i	$-0.492457\pi$
$397$	0.944272	0.0473916	0.0236958	+	0.999719i	$0.492457\pi$
$398$	7.41641	0.371751
$399$	0.0557281	0.00278989
$400$	1.00000	0.0500000
$401$	11.5066	0.574611	0.287306	−	0.957839i	$-0.407240\pi$
$401$	11.5066	0.574611	0.287306	+	0.957839i	$0.407240\pi$
$402$	−1.94427	−0.0969715
$403$	6.47214	0.322400
$404$	−8.94427	−0.444994
$405$	−7.70820	−0.383024
$406$	−1.23607	−0.0613450
$407$	0	0
$408$	0.236068	0.0116871
$409$	−7.88854	−0.390063	−0.195032	−	0.980797i	$-0.562481\pi$
$409$	−7.88854	−0.390063	−0.195032	+	0.980797i	$0.562481\pi$
$410$	−5.32624	−0.263044
$411$	3.85410	0.190109
$412$	0.944272	0.0465209
$413$	−5.38197	−0.264829
$414$	−17.1246	−0.841629
$415$	5.32624	0.261455
$416$	2.00000	0.0980581
$417$	3.63932	0.178218
$418$	0	0
$419$	−3.32624	−0.162497	−0.0812487	−	0.996694i	$-0.525891\pi$
$419$	−3.32624	−0.162497	−0.0812487	+	0.996694i	$0.525891\pi$
$420$	−0.381966	−0.0186380
$421$	38.3607	1.86959	0.934793	−	0.355194i	$-0.115585\pi$
$421$	38.3607	1.86959	0.934793	+	0.355194i	$0.115585\pi$
$422$	−22.2705	−1.08411
$423$	−13.5967	−0.661096
$424$	−3.23607	−0.157157
$425$	0.618034	0.0299791
$426$	1.41641	0.0686252
$427$	−12.4721	−0.603569
$428$	17.6180	0.851600
$429$	0	0
$430$	4.85410	0.234086
$431$	26.7639	1.28917	0.644587	−	0.764531i	$-0.277030\pi$
$431$	26.7639	1.28917	0.644587	+	0.764531i	$0.277030\pi$
$432$	−2.23607	−0.107583
$433$	−39.4508	−1.89589	−0.947943	−	0.318439i	$-0.896841\pi$
$433$	−39.4508	−1.89589	−0.947943	+	0.318439i	$0.896841\pi$
$434$	3.23607	0.155336
$435$	0.472136	0.0226372
$436$	8.47214	0.405742
$437$	0.875388	0.0418755
$438$	2.72949	0.130420
$439$	2.65248	0.126596	0.0632979	−	0.997995i	$-0.479838\pi$
$439$	2.65248	0.126596	0.0632979	+	0.997995i	$0.479838\pi$
$440$	0	0
$441$	−2.85410	−0.135910
$442$	1.23607	0.0587938
$443$	−1.03444	−0.0491478	−0.0245739	−	0.999698i	$-0.507823\pi$
$443$	−1.03444	−0.0491478	−0.0245739	+	0.999698i	$0.507823\pi$
$444$	0.944272	0.0448132
$445$	−1.90983	−0.0905346
$446$	19.7082	0.933211
$447$	0.695048	0.0328747
$448$	1.00000	0.0472456
$449$	−31.4508	−1.48426	−0.742129	−	0.670257i	$-0.766184\pi$
$449$	−31.4508	−1.48426	−0.742129	+	0.670257i	$0.766184\pi$
$450$	−2.85410	−0.134544
$451$	0	0
$452$	6.32624	0.297561
$453$	6.00000	0.281905
$454$	9.27051	0.435087
$455$	−2.00000	−0.0937614
$456$	0.0557281	0.00260971
$457$	2.27051	0.106210	0.0531050	−	0.998589i	$-0.483088\pi$
$457$	2.27051	0.106210	0.0531050	+	0.998589i	$0.483088\pi$
$458$	−29.1246	−1.36090
$459$	−1.38197	−0.0645046
$460$	−6.00000	−0.279751
$461$	−10.9443	−0.509726	−0.254863	−	0.966977i	$-0.582030\pi$
$461$	−10.9443	−0.509726	−0.254863	+	0.966977i	$0.582030\pi$
$462$	0	0
$463$	19.4164	0.902357	0.451178	−	0.892434i	$-0.351004\pi$
$463$	19.4164	0.902357	0.451178	+	0.892434i	$0.351004\pi$
$464$	−1.23607	−0.0573830
$465$	−1.23607	−0.0573213
$466$	22.0902	1.02331
$467$	17.8885	0.827783	0.413892	−	0.910326i	$-0.364169\pi$
$467$	17.8885	0.827783	0.413892	+	0.910326i	$0.364169\pi$
$468$	−5.70820	−0.263862
$469$	−5.09017	−0.235042
$470$	−4.76393	−0.219744
$471$	5.23607	0.241265
$472$	−5.38197	−0.247725
$473$	0	0
$474$	0.180340	0.00828329
$475$	0.145898	0.00669426
$476$	0.618034	0.0283275
$477$	9.23607	0.422891
$478$	0	0
$479$	−2.11146	−0.0964749	−0.0482374	−	0.998836i	$-0.515360\pi$
$479$	−2.11146	−0.0964749	−0.0482374	+	0.998836i	$0.515360\pi$
$480$	−0.381966	−0.0174343
$481$	4.94427	0.225439
$482$	27.2705	1.24214
$483$	2.29180	0.104280
$484$	0	0
$485$	−10.5623	−0.479610
$486$	9.65248	0.437845
$487$	26.8328	1.21591	0.607955	−	0.793971i	$-0.291990\pi$
$487$	26.8328	1.21591	0.607955	+	0.793971i	$0.291990\pi$
$488$	−12.4721	−0.564587
$489$	−1.47214	−0.0665723
$490$	−1.00000	−0.0451754
$491$	5.50658	0.248508	0.124254	−	0.992250i	$-0.460346\pi$
$491$	5.50658	0.248508	0.124254	+	0.992250i	$0.460346\pi$
$492$	2.03444	0.0917197
$493$	−0.763932	−0.0344058
$494$	0.291796	0.0131285
$495$	0	0
$496$	3.23607	0.145304
$497$	3.70820	0.166336
$498$	−2.03444	−0.0911655
$499$	−39.7984	−1.78162	−0.890810	−	0.454376i	$-0.849862\pi$
$499$	−39.7984	−1.78162	−0.890810	+	0.454376i	$0.849862\pi$
$500$	−1.00000	−0.0447214
$501$	−0.944272	−0.0421870
$502$	−26.8328	−1.19761
$503$	23.1246	1.03108	0.515538	−	0.856867i	$-0.327592\pi$
$503$	23.1246	1.03108	0.515538	+	0.856867i	$0.327592\pi$
$504$	−2.85410	−0.127132
$505$	8.94427	0.398015
$506$	0	0
$507$	−3.43769	−0.152673
$508$	15.4164	0.683992
$509$	−34.4721	−1.52795	−0.763975	−	0.645246i	$-0.776755\pi$
$509$	−34.4721	−1.52795	−0.763975	+	0.645246i	$0.776755\pi$
$510$	−0.236068	−0.0104533
$511$	7.14590	0.316116
$512$	1.00000	0.0441942
$513$	−0.326238	−0.0144038
$514$	9.43769	0.416279
$515$	−0.944272	−0.0416096
$516$	−1.85410	−0.0816223
$517$	0	0
$518$	2.47214	0.108619
$519$	4.58359	0.201197
$520$	−2.00000	−0.0877058
$521$	11.2016	0.490752	0.245376	−	0.969428i	$-0.421089\pi$
$521$	11.2016	0.490752	0.245376	+	0.969428i	$0.421089\pi$
$522$	3.52786	0.154410
$523$	−40.9787	−1.79187	−0.895937	−	0.444181i	$-0.853495\pi$
$523$	−40.9787	−1.79187	−0.895937	+	0.444181i	$0.853495\pi$
$524$	16.7984	0.733840
$525$	0.381966	0.0166704
$526$	−2.18034	−0.0950673
$527$	2.00000	0.0871214
$528$	0	0
$529$	13.0000	0.565217
$530$	3.23607	0.140566
$531$	15.3607	0.666597
$532$	0.145898	0.00632548
$533$	10.6525	0.461410
$534$	0.729490	0.0315681
$535$	−17.6180	−0.761694
$536$	−5.09017	−0.219862
$537$	−3.29180	−0.142051
$538$	10.1803	0.438906
$539$	0	0
$540$	2.23607	0.0962250
$541$	−38.8328	−1.66955	−0.834777	−	0.550589i	$-0.814403\pi$
$541$	−38.8328	−1.66955	−0.834777	+	0.550589i	$0.814403\pi$
$542$	17.4164	0.748099
$543$	4.76393	0.204440
$544$	0.618034	0.0264980
$545$	−8.47214	−0.362906
$546$	0.763932	0.0326933
$547$	−29.0902	−1.24381	−0.621903	−	0.783094i	$-0.713640\pi$
$547$	−29.0902	−1.24381	−0.621903	+	0.783094i	$0.713640\pi$
$548$	10.0902	0.431031
$549$	35.5967	1.51923
$550$	0	0
$551$	−0.180340	−0.00768274
$552$	2.29180	0.0975453
$553$	0.472136	0.0200773
$554$	19.7082	0.837321
$555$	−0.944272	−0.0400821
$556$	9.52786	0.404071
$557$	15.8197	0.670301	0.335150	−	0.942165i	$-0.391213\pi$
$557$	15.8197	0.670301	0.335150	+	0.942165i	$0.391213\pi$
$558$	−9.23607	−0.390994
$559$	−9.70820	−0.410613
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	5.03444	0.212365
$563$	41.5066	1.74929	0.874647	−	0.484761i	$-0.161093\pi$
$563$	41.5066	1.74929	0.874647	+	0.484761i	$0.161093\pi$
$564$	1.81966	0.0766215
$565$	−6.32624	−0.266147
$566$	−12.0000	−0.504398
$567$	7.70820	0.323714
$568$	3.70820	0.155593
$569$	3.09017	0.129547	0.0647733	−	0.997900i	$-0.479368\pi$
$569$	3.09017	0.129547	0.0647733	+	0.997900i	$0.479368\pi$
$570$	−0.0557281	−0.00233419
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	−4.00000	−0.167102
$574$	5.32624	0.222313
$575$	6.00000	0.250217
$576$	−2.85410	−0.118921
$577$	−26.2148	−1.09134	−0.545668	−	0.838002i	$-0.683724\pi$
$577$	−26.2148	−1.09134	−0.545668	+	0.838002i	$0.683724\pi$
$578$	−16.6180	−0.691219
$579$	−0.763932	−0.0317479
$580$	1.23607	0.0513249
$581$	−5.32624	−0.220970
$582$	4.03444	0.167233
$583$	0	0
$584$	7.14590	0.295699
$585$	5.70820	0.236005
$586$	7.52786	0.310973
$587$	−39.7426	−1.64035	−0.820177	−	0.572109i	$-0.806125\pi$
$587$	−39.7426	−1.64035	−0.820177	+	0.572109i	$0.806125\pi$
$588$	0.381966	0.0157520
$589$	0.472136	0.0194540
$590$	5.38197	0.221572
$591$	2.29180	0.0942719
$592$	2.47214	0.101604
$593$	0.909830	0.0373622	0.0186811	−	0.999825i	$-0.494053\pi$
$593$	0.909830	0.0373622	0.0186811	+	0.999825i	$0.494053\pi$
$594$	0	0
$595$	−0.618034	−0.0253369
$596$	1.81966	0.0745362
$597$	2.83282	0.115939
$598$	12.0000	0.490716
$599$	−20.7639	−0.848391	−0.424196	−	0.905571i	$-0.639443\pi$
$599$	−20.7639	−0.848391	−0.424196	+	0.905571i	$0.639443\pi$
$600$	0.381966	0.0155937
$601$	−17.9098	−0.730557	−0.365279	−	0.930898i	$-0.619026\pi$
$601$	−17.9098	−0.730557	−0.365279	+	0.930898i	$0.619026\pi$
$602$	−4.85410	−0.197838
$603$	14.5279	0.591620
$604$	15.7082	0.639158
$605$	0	0
$606$	−3.41641	−0.138782
$607$	19.5967	0.795407	0.397704	−	0.917514i	$-0.369807\pi$
$607$	19.5967	0.795407	0.397704	+	0.917514i	$0.369807\pi$
$608$	0.145898	0.00591695
$609$	−0.472136	−0.0191319
$610$	12.4721	0.504982
$611$	9.52786	0.385456
$612$	−1.76393	−0.0713027
$613$	−10.5836	−0.427467	−0.213734	−	0.976892i	$-0.568562\pi$
$613$	−10.5836	−0.427467	−0.213734	+	0.976892i	$0.568562\pi$
$614$	−7.43769	−0.300161
$615$	−2.03444	−0.0820366
$616$	0	0
$617$	−6.03444	−0.242937	−0.121469	−	0.992595i	$-0.538760\pi$
$617$	−6.03444	−0.242937	−0.121469	+	0.992595i	$0.538760\pi$
$618$	0.360680	0.0145087
$619$	34.3262	1.37969	0.689844	−	0.723958i	$-0.257679\pi$
$619$	34.3262	1.37969	0.689844	+	0.723958i	$0.257679\pi$
$620$	−3.23607	−0.129964
$621$	−13.4164	−0.538382
$622$	11.1246	0.446056
$623$	1.90983	0.0765157
$624$	0.763932	0.0305818
$625$	1.00000	0.0400000
$626$	5.85410	0.233977
$627$	0	0
$628$	13.7082	0.547017
$629$	1.52786	0.0609199
$630$	2.85410	0.113710
$631$	−10.2918	−0.409710	−0.204855	−	0.978792i	$-0.565672\pi$
$631$	−10.2918	−0.409710	−0.204855	+	0.978792i	$0.565672\pi$
$632$	0.472136	0.0187806
$633$	−8.50658	−0.338106
$634$	−25.2361	−1.00225
$635$	−15.4164	−0.611781
$636$	−1.23607	−0.0490133
$637$	2.00000	0.0792429
$638$	0	0
$639$	−10.5836	−0.418680
$640$	−1.00000	−0.0395285
$641$	24.6738	0.974555	0.487278	−	0.873247i	$-0.337990\pi$
$641$	24.6738	0.974555	0.487278	+	0.873247i	$0.337990\pi$
$642$	6.72949	0.265592
$643$	31.9787	1.26112	0.630559	−	0.776142i	$-0.282826\pi$
$643$	31.9787	1.26112	0.630559	+	0.776142i	$0.282826\pi$
$644$	6.00000	0.236433
$645$	1.85410	0.0730052
$646$	0.0901699	0.00354769
$647$	−22.1803	−0.871999	−0.436000	−	0.899947i	$-0.643605\pi$
$647$	−22.1803	−0.871999	−0.436000	+	0.899947i	$0.643605\pi$
$648$	7.70820	0.302807
$649$	0	0
$650$	2.00000	0.0784465
$651$	1.23607	0.0484453
$652$	−3.85410	−0.150938
$653$	−26.8328	−1.05005	−0.525025	−	0.851087i	$-0.675944\pi$
$653$	−26.8328	−1.05005	−0.525025	+	0.851087i	$0.675944\pi$
$654$	3.23607	0.126540
$655$	−16.7984	−0.656367
$656$	5.32624	0.207955
$657$	−20.3951	−0.795689
$658$	4.76393	0.185717
$659$	−7.09017	−0.276194	−0.138097	−	0.990419i	$-0.544099\pi$
$659$	−7.09017	−0.276194	−0.138097	+	0.990419i	$0.544099\pi$
$660$	0	0
$661$	−10.1115	−0.393290	−0.196645	−	0.980475i	$-0.563005\pi$
$661$	−10.1115	−0.393290	−0.196645	+	0.980475i	$0.563005\pi$
$662$	−23.0344	−0.895259
$663$	0.472136	0.0183362
$664$	−5.32624	−0.206698
$665$	−0.145898	−0.00565768
$666$	−7.05573	−0.273404
$667$	−7.41641	−0.287164
$668$	−2.47214	−0.0956498
$669$	7.52786	0.291044
$670$	5.09017	0.196650
$671$	0	0
$672$	0.381966	0.0147347
$673$	−32.1459	−1.23913	−0.619567	−	0.784944i	$-0.712692\pi$
$673$	−32.1459	−1.23913	−0.619567	+	0.784944i	$0.712692\pi$
$674$	−13.3820	−0.515454
$675$	−2.23607	−0.0860663
$676$	−9.00000	−0.346154
$677$	35.5967	1.36809	0.684047	−	0.729438i	$-0.260218\pi$
$677$	35.5967	1.36809	0.684047	+	0.729438i	$0.260218\pi$
$678$	2.41641	0.0928016
$679$	10.5623	0.405344
$680$	−0.618034	−0.0237005
$681$	3.54102	0.135692
$682$	0	0
$683$	−48.9443	−1.87280	−0.936400	−	0.350934i	$-0.885864\pi$
$683$	−48.9443	−1.87280	−0.936400	+	0.350934i	$0.885864\pi$
$684$	−0.416408	−0.0159218
$685$	−10.0902	−0.385526
$686$	1.00000	0.0381802
$687$	−11.1246	−0.424430
$688$	−4.85410	−0.185061
$689$	−6.47214	−0.246569
$690$	−2.29180	−0.0872472
$691$	27.8541	1.05962	0.529810	−	0.848116i	$-0.322263\pi$
$691$	27.8541	1.05962	0.529810	+	0.848116i	$0.322263\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	−10.3262	−0.391979
$695$	−9.52786	−0.361412
$696$	−0.472136	−0.0178963
$697$	3.29180	0.124686
$698$	14.4721	0.547778
$699$	8.43769	0.319143
$700$	1.00000	0.0377964
$701$	10.1803	0.384506	0.192253	−	0.981345i	$-0.438421\pi$
$701$	10.1803	0.384506	0.192253	+	0.981345i	$0.438421\pi$
$702$	−4.47214	−0.168790
$703$	0.360680	0.0136033
$704$	0	0
$705$	−1.81966	−0.0685324
$706$	27.4508	1.03313
$707$	−8.94427	−0.336384
$708$	−2.05573	−0.0772590
$709$	−7.05573	−0.264983	−0.132492	−	0.991184i	$-0.542298\pi$
$709$	−7.05573	−0.264983	−0.132492	+	0.991184i	$0.542298\pi$
$710$	−3.70820	−0.139166
$711$	−1.34752	−0.0505361
$712$	1.90983	0.0715739
$713$	19.4164	0.727150
$714$	0.236068	0.00883462
$715$	0	0
$716$	−8.61803	−0.322071
$717$	0	0
$718$	−12.6525	−0.472186
$719$	−12.3607	−0.460976	−0.230488	−	0.973075i	$-0.574032\pi$
$719$	−12.3607	−0.460976	−0.230488	+	0.973075i	$0.574032\pi$
$720$	2.85410	0.106366
$721$	0.944272	0.0351665
$722$	−18.9787	−0.706315
$723$	10.4164	0.387390
$724$	12.4721	0.463523
$725$	−1.23607	−0.0459064
$726$	0	0
$727$	−14.8328	−0.550119	−0.275059	−	0.961427i	$-0.588698\pi$
$727$	−14.8328	−0.550119	−0.275059	+	0.961427i	$0.588698\pi$
$728$	2.00000	0.0741249
$729$	−19.4377	−0.719915
$730$	−7.14590	−0.264482
$731$	−3.00000	−0.110959
$732$	−4.76393	−0.176080
$733$	−8.76393	−0.323703	−0.161852	−	0.986815i	$-0.551747\pi$
$733$	−8.76393	−0.323703	−0.161852	+	0.986815i	$0.551747\pi$
$734$	5.81966	0.214808
$735$	−0.381966	−0.0140890
$736$	6.00000	0.221163
$737$	0	0
$738$	−15.2016	−0.559580
$739$	−40.3262	−1.48342	−0.741712	−	0.670718i	$-0.765986\pi$
$739$	−40.3262	−1.48342	−0.741712	+	0.670718i	$0.765986\pi$
$740$	−2.47214	−0.0908775
$741$	0.111456	0.00409445
$742$	−3.23607	−0.118800
$743$	−47.8885	−1.75686	−0.878430	−	0.477871i	$-0.841409\pi$
$743$	−47.8885	−1.75686	−0.878430	+	0.477871i	$0.841409\pi$
$744$	1.23607	0.0453165
$745$	−1.81966	−0.0666672
$746$	−15.5279	−0.568515
$747$	15.2016	0.556198
$748$	0	0
$749$	17.6180	0.643749
$750$	−0.381966	−0.0139474
$751$	−1.34752	−0.0491719	−0.0245859	−	0.999698i	$-0.507827\pi$
$751$	−1.34752	−0.0491719	−0.0245859	+	0.999698i	$0.507827\pi$
$752$	4.76393	0.173723
$753$	−10.2492	−0.373502
$754$	−2.47214	−0.0900299
$755$	−15.7082	−0.571680
$756$	−2.23607	−0.0813250
$757$	−15.8885	−0.577479	−0.288739	−	0.957408i	$-0.593236\pi$
$757$	−15.8885	−0.577479	−0.288739	+	0.957408i	$0.593236\pi$
$758$	−21.1459	−0.768054
$759$	0	0
$760$	−0.145898	−0.00529228
$761$	−7.20163	−0.261059	−0.130529	−	0.991444i	$-0.541668\pi$
$761$	−7.20163	−0.261059	−0.130529	+	0.991444i	$0.541668\pi$
$762$	5.88854	0.213319
$763$	8.47214	0.306712
$764$	−10.4721	−0.378869
$765$	1.76393	0.0637751
$766$	−24.0000	−0.867155
$767$	−10.7639	−0.388663
$768$	0.381966	0.0137830
$769$	27.8885	1.00569	0.502843	−	0.864378i	$-0.332287\pi$
$769$	27.8885	1.00569	0.502843	+	0.864378i	$0.332287\pi$
$770$	0	0
$771$	3.60488	0.129827
$772$	−2.00000	−0.0719816
$773$	6.65248	0.239273	0.119636	−	0.992818i	$-0.461827\pi$
$773$	6.65248	0.239273	0.119636	+	0.992818i	$0.461827\pi$
$774$	13.8541	0.497975
$775$	3.23607	0.116243
$776$	10.5623	0.379165
$777$	0.944272	0.0338756
$778$	20.8328	0.746893
$779$	0.777088	0.0278421
$780$	−0.763932	−0.0273532
$781$	0	0
$782$	3.70820	0.132605
$783$	2.76393	0.0987749
$784$	1.00000	0.0357143
$785$	−13.7082	−0.489267
$786$	6.41641	0.228866
$787$	−36.1033	−1.28694	−0.643472	−	0.765469i	$-0.722507\pi$
$787$	−36.1033	−1.28694	−0.643472	+	0.765469i	$0.722507\pi$
$788$	6.00000	0.213741
$789$	−0.832816	−0.0296490
$790$	−0.472136	−0.0167978
$791$	6.32624	0.224935
$792$	0	0
$793$	−24.9443	−0.885797
$794$	0.944272	0.0335110
$795$	1.23607	0.0438388
$796$	7.41641	0.262868
$797$	38.3607	1.35880	0.679402	−	0.733766i	$-0.262239\pi$
$797$	38.3607	1.35880	0.679402	+	0.733766i	$0.262239\pi$
$798$	0.0557281	0.00197275
$799$	2.94427	0.104161
$800$	1.00000	0.0353553
$801$	−5.45085	−0.192596
$802$	11.5066	0.406311
$803$	0	0
$804$	−1.94427	−0.0685692
$805$	−6.00000	−0.211472
$806$	6.47214	0.227971
$807$	3.88854	0.136883
$808$	−8.94427	−0.314658
$809$	51.8115	1.82160	0.910798	−	0.412852i	$-0.135467\pi$
$809$	51.8115	1.82160	0.910798	+	0.412852i	$0.135467\pi$
$810$	−7.70820	−0.270839
$811$	34.9230	1.22631	0.613156	−	0.789962i	$-0.289900\pi$
$811$	34.9230	1.22631	0.613156	+	0.789962i	$0.289900\pi$
$812$	−1.23607	−0.0433775
$813$	6.65248	0.233313
$814$	0	0
$815$	3.85410	0.135003
$816$	0.236068	0.00826403
$817$	−0.708204	−0.0247769
$818$	−7.88854	−0.275816
$819$	−5.70820	−0.199461
$820$	−5.32624	−0.186000
$821$	48.9443	1.70817	0.854083	−	0.520136i	$-0.174119\pi$
$821$	48.9443	1.70817	0.854083	+	0.520136i	$0.174119\pi$
$822$	3.85410	0.134427
$823$	50.8328	1.77192	0.885960	−	0.463761i	$-0.153500\pi$
$823$	50.8328	1.77192	0.885960	+	0.463761i	$0.153500\pi$
$824$	0.944272	0.0328953
$825$	0	0
$826$	−5.38197	−0.187263
$827$	1.74265	0.0605977	0.0302989	−	0.999541i	$-0.490354\pi$
$827$	1.74265	0.0605977	0.0302989	+	0.999541i	$0.490354\pi$
$828$	−17.1246	−0.595121
$829$	11.8197	0.410514	0.205257	−	0.978708i	$-0.434197\pi$
$829$	11.8197	0.410514	0.205257	+	0.978708i	$0.434197\pi$
$830$	5.32624	0.184876
$831$	7.52786	0.261139
$832$	2.00000	0.0693375
$833$	0.618034	0.0214136
$834$	3.63932	0.126019
$835$	2.47214	0.0855518
$836$	0	0
$837$	−7.23607	−0.250115
$838$	−3.32624	−0.114903
$839$	−4.47214	−0.154395	−0.0771976	−	0.997016i	$-0.524597\pi$
$839$	−4.47214	−0.154395	−0.0771976	+	0.997016i	$0.524597\pi$
$840$	−0.381966	−0.0131791
$841$	−27.4721	−0.947315
$842$	38.3607	1.32200
$843$	1.92299	0.0662311
$844$	−22.2705	−0.766583
$845$	9.00000	0.309609
$846$	−13.5967	−0.467466
$847$	0	0
$848$	−3.23607	−0.111127
$849$	−4.58359	−0.157308
$850$	0.618034	0.0211984
$851$	14.8328	0.508462
$852$	1.41641	0.0485253
$853$	−17.7082	−0.606317	−0.303159	−	0.952940i	$-0.598041\pi$
$853$	−17.7082	−0.606317	−0.303159	+	0.952940i	$0.598041\pi$
$854$	−12.4721	−0.426788
$855$	0.416408	0.0142408
$856$	17.6180	0.602172
$857$	24.2705	0.829065	0.414532	−	0.910035i	$-0.363945\pi$
$857$	24.2705	0.829065	0.414532	+	0.910035i	$0.363945\pi$
$858$	0	0
$859$	−27.5623	−0.940414	−0.470207	−	0.882556i	$-0.655821\pi$
$859$	−27.5623	−0.940414	−0.470207	+	0.882556i	$0.655821\pi$
$860$	4.85410	0.165524
$861$	2.03444	0.0693336
$862$	26.7639	0.911583
$863$	11.2361	0.382480	0.191240	−	0.981543i	$-0.438749\pi$
$863$	11.2361	0.382480	0.191240	+	0.981543i	$0.438749\pi$
$864$	−2.23607	−0.0760726
$865$	−12.0000	−0.408012
$866$	−39.4508	−1.34059
$867$	−6.34752	−0.215573
$868$	3.23607	0.109839
$869$	0	0
$870$	0.472136	0.0160069
$871$	−10.1803	−0.344948
$872$	8.47214	0.286903
$873$	−30.1459	−1.02028
$874$	0.875388	0.0296104
$875$	−1.00000	−0.0338062
$876$	2.72949	0.0922209
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	2.65248	0.0895167
$879$	2.87539	0.0969844
$880$	0	0
$881$	10.8541	0.365684	0.182842	−	0.983142i	$-0.441470\pi$
$881$	10.8541	0.365684	0.182842	+	0.983142i	$0.441470\pi$
$882$	−2.85410	−0.0961026
$883$	22.5623	0.759282	0.379641	−	0.925134i	$-0.376048\pi$
$883$	22.5623	0.759282	0.379641	+	0.925134i	$0.376048\pi$
$884$	1.23607	0.0415735
$885$	2.05573	0.0691025
$886$	−1.03444	−0.0347528
$887$	40.3607	1.35518	0.677589	−	0.735440i	$-0.263025\pi$
$887$	40.3607	1.35518	0.677589	+	0.735440i	$0.263025\pi$
$888$	0.944272	0.0316877
$889$	15.4164	0.517050
$890$	−1.90983	−0.0640176
$891$	0	0
$892$	19.7082	0.659879
$893$	0.695048	0.0232589
$894$	0.695048	0.0232459
$895$	8.61803	0.288069
$896$	1.00000	0.0334077
$897$	4.58359	0.153042
$898$	−31.4508	−1.04953
$899$	−4.00000	−0.133407
$900$	−2.85410	−0.0951367
$901$	−2.00000	−0.0666297
$902$	0	0
$903$	−1.85410	−0.0617007
$904$	6.32624	0.210408
$905$	−12.4721	−0.414588
$906$	6.00000	0.199337
$907$	−11.2705	−0.374231	−0.187116	−	0.982338i	$-0.559914\pi$
$907$	−11.2705	−0.374231	−0.187116	+	0.982338i	$0.559914\pi$
$908$	9.27051	0.307653
$909$	25.5279	0.846706
$910$	−2.00000	−0.0662994
$911$	−2.40325	−0.0796233	−0.0398116	−	0.999207i	$-0.512676\pi$
$911$	−2.40325	−0.0796233	−0.0398116	+	0.999207i	$0.512676\pi$
$912$	0.0557281	0.00184534
$913$	0	0
$914$	2.27051	0.0751018
$915$	4.76393	0.157491
$916$	−29.1246	−0.962304
$917$	16.7984	0.554731
$918$	−1.38197	−0.0456117
$919$	44.8328	1.47890	0.739449	−	0.673213i	$-0.235086\pi$
$919$	44.8328	1.47890	0.739449	+	0.673213i	$0.235086\pi$
$920$	−6.00000	−0.197814
$921$	−2.84095	−0.0936124
$922$	−10.9443	−0.360430
$923$	7.41641	0.244114
$924$	0	0
$925$	2.47214	0.0812833
$926$	19.4164	0.638063
$927$	−2.69505	−0.0885170
$928$	−1.23607	−0.0405759
$929$	−45.7426	−1.50077	−0.750384	−	0.661002i	$-0.770131\pi$
$929$	−45.7426	−1.50077	−0.750384	+	0.661002i	$0.770131\pi$
$930$	−1.23607	−0.0405323
$931$	0.145898	0.00478161
$932$	22.0902	0.723588
$933$	4.24922	0.139113
$934$	17.8885	0.585331
$935$	0	0
$936$	−5.70820	−0.186578
$937$	−40.3951	−1.31965	−0.659826	−	0.751419i	$-0.729370\pi$
$937$	−40.3951	−1.31965	−0.659826	+	0.751419i	$0.729370\pi$
$938$	−5.09017	−0.166200
$939$	2.23607	0.0729713
$940$	−4.76393	−0.155382
$941$	11.8197	0.385310	0.192655	−	0.981267i	$-0.438290\pi$
$941$	11.8197	0.385310	0.192655	+	0.981267i	$0.438290\pi$
$942$	5.23607	0.170600
$943$	31.9574	1.04068
$944$	−5.38197	−0.175168
$945$	2.23607	0.0727393
$946$	0	0
$947$	−53.0902	−1.72520	−0.862599	−	0.505888i	$-0.831165\pi$
$947$	−53.0902	−1.72520	−0.862599	+	0.505888i	$0.831165\pi$
$948$	0.180340	0.00585717
$949$	14.2918	0.463931
$950$	0.145898	0.00473356
$951$	−9.63932	−0.312576
$952$	0.618034	0.0200306
$953$	−26.4508	−0.856827	−0.428414	−	0.903583i	$-0.640927\pi$
$953$	−26.4508	−0.856827	−0.428414	+	0.903583i	$0.640927\pi$
$954$	9.23607	0.299029
$955$	10.4721	0.338870
$956$	0	0
$957$	0	0
$958$	−2.11146	−0.0682181
$959$	10.0902	0.325829
$960$	−0.381966	−0.0123279
$961$	−20.5279	−0.662189
$962$	4.94427	0.159410
$963$	−50.2837	−1.62037
$964$	27.2705	0.878324
$965$	2.00000	0.0643823
$966$	2.29180	0.0737373
$967$	−49.0132	−1.57616	−0.788078	−	0.615575i	$-0.788924\pi$
$967$	−49.0132	−1.57616	−0.788078	+	0.615575i	$0.788924\pi$
$968$	0	0
$969$	0.0344419	0.00110643
$970$	−10.5623	−0.339135
$971$	26.8328	0.861106	0.430553	−	0.902565i	$-0.358319\pi$
$971$	26.8328	0.861106	0.430553	+	0.902565i	$0.358319\pi$
$972$	9.65248	0.309603
$973$	9.52786	0.305449
$974$	26.8328	0.859779
$975$	0.763932	0.0244654
$976$	−12.4721	−0.399223
$977$	−19.3050	−0.617620	−0.308810	−	0.951124i	$-0.599931\pi$
$977$	−19.3050	−0.617620	−0.308810	+	0.951124i	$0.599931\pi$
$978$	−1.47214	−0.0470737
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−24.1803	−0.772019
$982$	5.50658	0.175722
$983$	16.6525	0.531131	0.265566	−	0.964093i	$-0.414441\pi$
$983$	16.6525	0.531131	0.265566	+	0.964093i	$0.414441\pi$
$984$	2.03444	0.0648556
$985$	−6.00000	−0.191176
$986$	−0.763932	−0.0243286
$987$	1.81966	0.0579204
$988$	0.291796	0.00928327
$989$	−29.1246	−0.926109
$990$	0	0
$991$	−10.6525	−0.338387	−0.169194	−	0.985583i	$-0.554116\pi$
$991$	−10.6525	−0.338387	−0.169194	+	0.985583i	$0.554116\pi$
$992$	3.23607	0.102745
$993$	−8.79837	−0.279208
$994$	3.70820	0.117617
$995$	−7.41641	−0.235116
$996$	−2.03444	−0.0644638
$997$	17.1246	0.542342	0.271171	−	0.962531i	$-0.412589\pi$
$997$	17.1246	0.542342	0.271171	+	0.962531i	$0.412589\pi$
$998$	−39.7984	−1.25980
$999$	−5.52786	−0.174894

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cf.1.1		2
11.2	odd	10		770.2.n.b.631.1	yes	4
11.6	odd	10		770.2.n.b.421.1	✓	4
11.10	odd	2		8470.2.a.bt.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.b.421.1	✓	4	11.6	odd	10
770.2.n.b.631.1	yes	4	11.2	odd	10
8470.2.a.bt.1.1		2	11.10	odd	2
8470.2.a.cf.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.381966 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.381966 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} -1.00000 q^{10} +0.381966 q^{12} +2.00000 q^{13} +1.00000 q^{14} -0.381966 q^{15} +1.00000 q^{16} +0.618034 q^{17} -2.85410 q^{18} +0.145898 q^{19} -1.00000 q^{20} +0.381966 q^{21} +6.00000 q^{23} +0.381966 q^{24} +1.00000 q^{25} +2.00000 q^{26} -2.23607 q^{27} +1.00000 q^{28} -1.23607 q^{29} -0.381966 q^{30} +3.23607 q^{31} +1.00000 q^{32} +0.618034 q^{34} -1.00000 q^{35} -2.85410 q^{36} +2.47214 q^{37} +0.145898 q^{38} +0.763932 q^{39} -1.00000 q^{40} +5.32624 q^{41} +0.381966 q^{42} -4.85410 q^{43} +2.85410 q^{45} +6.00000 q^{46} +4.76393 q^{47} +0.381966 q^{48} +1.00000 q^{49} +1.00000 q^{50} +0.236068 q^{51} +2.00000 q^{52} -3.23607 q^{53} -2.23607 q^{54} +1.00000 q^{56} +0.0557281 q^{57} -1.23607 q^{58} -5.38197 q^{59} -0.381966 q^{60} -12.4721 q^{61} +3.23607 q^{62} -2.85410 q^{63} +1.00000 q^{64} -2.00000 q^{65} -5.09017 q^{67} +0.618034 q^{68} +2.29180 q^{69} -1.00000 q^{70} +3.70820 q^{71} -2.85410 q^{72} +7.14590 q^{73} +2.47214 q^{74} +0.381966 q^{75} +0.145898 q^{76} +0.763932 q^{78} +0.472136 q^{79} -1.00000 q^{80} +7.70820 q^{81} +5.32624 q^{82} -5.32624 q^{83} +0.381966 q^{84} -0.618034 q^{85} -4.85410 q^{86} -0.472136 q^{87} +1.90983 q^{89} +2.85410 q^{90} +2.00000 q^{91} +6.00000 q^{92} +1.23607 q^{93} +4.76393 q^{94} -0.145898 q^{95} +0.381966 q^{96} +10.5623 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 2 q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 2 * q^5 + 3 * q^6 + 2 * q^7 + 2 * q^8 + q^9	\( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 2 q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + q^{9} - 2 q^{10} + 3 q^{12} + 4 q^{13} + 2 q^{14} - 3 q^{15} + 2 q^{16} - q^{17} + q^{18} + 7 q^{19} - 2 q^{20} + 3 q^{21} + 12 q^{23} + 3 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{28} + 2 q^{29} - 3 q^{30} + 2 q^{31} + 2 q^{32} - q^{34} - 2 q^{35} + q^{36} - 4 q^{37} + 7 q^{38} + 6 q^{39} - 2 q^{40} - 5 q^{41} + 3 q^{42} - 3 q^{43} - q^{45} + 12 q^{46} + 14 q^{47} + 3 q^{48} + 2 q^{49} + 2 q^{50} - 4 q^{51} + 4 q^{52} - 2 q^{53} + 2 q^{56} + 18 q^{57} + 2 q^{58} - 13 q^{59} - 3 q^{60} - 16 q^{61} + 2 q^{62} + q^{63} + 2 q^{64} - 4 q^{65} + q^{67} - q^{68} + 18 q^{69} - 2 q^{70} - 6 q^{71} + q^{72} + 21 q^{73} - 4 q^{74} + 3 q^{75} + 7 q^{76} + 6 q^{78} - 8 q^{79} - 2 q^{80} + 2 q^{81} - 5 q^{82} + 5 q^{83} + 3 q^{84} + q^{85} - 3 q^{86} + 8 q^{87} + 15 q^{89} - q^{90} + 4 q^{91} + 12 q^{92} - 2 q^{93} + 14 q^{94} - 7 q^{95} + 3 q^{96} + q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 2 * q^5 + 3 * q^6 + 2 * q^7 + 2 * q^8 + q^9 - 2 * q^10 + 3 * q^12 + 4 * q^13 + 2 * q^14 - 3 * q^15 + 2 * q^16 - q^17 + q^18 + 7 * q^19 - 2 * q^20 + 3 * q^21 + 12 * q^23 + 3 * q^24 + 2 * q^25 + 4 * q^26 + 2 * q^28 + 2 * q^29 - 3 * q^30 + 2 * q^31 + 2 * q^32 - q^34 - 2 * q^35 + q^36 - 4 * q^37 + 7 * q^38 + 6 * q^39 - 2 * q^40 - 5 * q^41 + 3 * q^42 - 3 * q^43 - q^45 + 12 * q^46 + 14 * q^47 + 3 * q^48 + 2 * q^49 + 2 * q^50 - 4 * q^51 + 4 * q^52 - 2 * q^53 + 2 * q^56 + 18 * q^57 + 2 * q^58 - 13 * q^59 - 3 * q^60 - 16 * q^61 + 2 * q^62 + q^63 + 2 * q^64 - 4 * q^65 + q^67 - q^68 + 18 * q^69 - 2 * q^70 - 6 * q^71 + q^72 + 21 * q^73 - 4 * q^74 + 3 * q^75 + 7 * q^76 + 6 * q^78 - 8 * q^79 - 2 * q^80 + 2 * q^81 - 5 * q^82 + 5 * q^83 + 3 * q^84 + q^85 - 3 * q^86 + 8 * q^87 + 15 * q^89 - q^90 + 4 * q^91 + 12 * q^92 - 2 * q^93 + 14 * q^94 - 7 * q^95 + 3 * q^96 + q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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