LMFDB - Embedded newform 8470.2.a.ct.1.4

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:	4.4.5225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + x + 11 $ x^4 - x^3 - 8*x^2 + x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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			1.4
Root			$-1.88301$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +3.04678 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.04678 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.28284 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +3.04678 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.04678 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.28284 q^{9} -1.00000 q^{10} +3.04678 q^{12} +0.353057 q^{13} +1.00000 q^{14} -3.04678 q^{15} +1.00000 q^{16} +0.810708 q^{17} +6.28284 q^{18} -2.59251 q^{19} -1.00000 q^{20} +3.04678 q^{21} -2.00000 q^{23} +3.04678 q^{24} +1.00000 q^{25} +0.353057 q^{26} +10.0021 q^{27} +1.00000 q^{28} +9.54782 q^{29} -3.04678 q^{30} -5.70820 q^{31} +1.00000 q^{32} +0.810708 q^{34} -1.00000 q^{35} +6.28284 q^{36} +3.56360 q^{37} -2.59251 q^{38} +1.07569 q^{39} -1.00000 q^{40} +8.25732 q^{41} +3.04678 q^{42} +9.49338 q^{43} -6.28284 q^{45} -2.00000 q^{46} -8.60773 q^{47} +3.04678 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.47005 q^{51} +0.353057 q^{52} -2.79967 q^{53} +10.0021 q^{54} +1.00000 q^{56} -7.89879 q^{57} +9.54782 q^{58} +13.2092 q^{59} -3.04678 q^{60} +6.70611 q^{61} -5.70820 q^{62} +6.28284 q^{63} +1.00000 q^{64} -0.353057 q^{65} -1.40749 q^{67} +0.810708 q^{68} -6.09355 q^{69} -1.00000 q^{70} -12.9187 q^{71} +6.28284 q^{72} +5.15008 q^{73} +3.56360 q^{74} +3.04678 q^{75} -2.59251 q^{76} +1.07569 q^{78} +3.64694 q^{79} -1.00000 q^{80} +11.6256 q^{81} +8.25732 q^{82} +2.66819 q^{83} +3.04678 q^{84} -0.810708 q^{85} +9.49338 q^{86} +29.0901 q^{87} +9.63799 q^{89} -6.28284 q^{90} +0.353057 q^{91} -2.00000 q^{92} -17.3916 q^{93} -8.60773 q^{94} +2.59251 q^{95} +3.04678 q^{96} -9.79837 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{7} + 4 q^{8} + 6 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 + 4 * q^7 + 4 * q^8 + 6 * q^9	$ 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{7} + 4 q^{8} + 6 q^{9} - 4 q^{10} + 2 q^{12} + q^{13} + 4 q^{14} - 2 q^{15} + 4 q^{16} + 2 q^{17} + 6 q^{18} - 3 q^{19} - 4 q^{20} + 2 q^{21} - 8 q^{23} + 2 q^{24} + 4 q^{25} + q^{26} + 14 q^{27} + 4 q^{28} + 15 q^{29} - 2 q^{30} + 4 q^{31} + 4 q^{32} + 2 q^{34} - 4 q^{35} + 6 q^{36} + 2 q^{37} - 3 q^{38} - q^{39} - 4 q^{40} + 11 q^{41} + 2 q^{42} + 7 q^{43} - 6 q^{45} - 8 q^{46} + 5 q^{47} + 2 q^{48} + 4 q^{49} + 4 q^{50} + 18 q^{51} + q^{52} + 10 q^{53} + 14 q^{54} + 4 q^{56} - 34 q^{57} + 15 q^{58} + 13 q^{59} - 2 q^{60} + 26 q^{61} + 4 q^{62} + 6 q^{63} + 4 q^{64} - q^{65} - 13 q^{67} + 2 q^{68} - 4 q^{69} - 4 q^{70} - 13 q^{71} + 6 q^{72} - 18 q^{73} + 2 q^{74} + 2 q^{75} - 3 q^{76} - q^{78} + 15 q^{79} - 4 q^{80} - 8 q^{81} + 11 q^{82} - 2 q^{83} + 2 q^{84} - 2 q^{85} + 7 q^{86} + 26 q^{87} - 7 q^{89} - 6 q^{90} + q^{91} - 8 q^{92} + 2 q^{93} + 5 q^{94} + 3 q^{95} + 2 q^{96} + 10 q^{97} + 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 + 4 * q^7 + 4 * q^8 + 6 * q^9 - 4 * q^10 + 2 * q^12 + q^13 + 4 * q^14 - 2 * q^15 + 4 * q^16 + 2 * q^17 + 6 * q^18 - 3 * q^19 - 4 * q^20 + 2 * q^21 - 8 * q^23 + 2 * q^24 + 4 * q^25 + q^26 + 14 * q^27 + 4 * q^28 + 15 * q^29 - 2 * q^30 + 4 * q^31 + 4 * q^32 + 2 * q^34 - 4 * q^35 + 6 * q^36 + 2 * q^37 - 3 * q^38 - q^39 - 4 * q^40 + 11 * q^41 + 2 * q^42 + 7 * q^43 - 6 * q^45 - 8 * q^46 + 5 * q^47 + 2 * q^48 + 4 * q^49 + 4 * q^50 + 18 * q^51 + q^52 + 10 * q^53 + 14 * q^54 + 4 * q^56 - 34 * q^57 + 15 * q^58 + 13 * q^59 - 2 * q^60 + 26 * q^61 + 4 * q^62 + 6 * q^63 + 4 * q^64 - q^65 - 13 * q^67 + 2 * q^68 - 4 * q^69 - 4 * q^70 - 13 * q^71 + 6 * q^72 - 18 * q^73 + 2 * q^74 + 2 * q^75 - 3 * q^76 - q^78 + 15 * q^79 - 4 * q^80 - 8 * q^81 + 11 * q^82 - 2 * q^83 + 2 * q^84 - 2 * q^85 + 7 * q^86 + 26 * q^87 - 7 * q^89 - 6 * q^90 + q^91 - 8 * q^92 + 2 * q^93 + 5 * q^94 + 3 * q^95 + 2 * 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
$3$	3.04678	1.75906	0.879528	−	0.475846i	$-0.157858\pi$
$3$	3.04678	1.75906	0.879528	+	0.475846i	$0.157858\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	3.04678	1.24384
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	6.28284	2.09428
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	3.04678	0.879528
$13$	0.353057	0.0979204	0.0489602	−	0.998801i	$-0.484409\pi$
$13$	0.353057	0.0979204	0.0489602	+	0.998801i	$0.484409\pi$
$14$	1.00000	0.267261
$15$	−3.04678	−0.786674
$16$	1.00000	0.250000
$17$	0.810708	0.196626	0.0983128	−	0.995156i	$-0.468655\pi$
$17$	0.810708	0.196626	0.0983128	+	0.995156i	$0.468655\pi$
$18$	6.28284	1.48088
$19$	−2.59251	−0.594762	−0.297381	−	0.954759i	$-0.596113\pi$
$19$	−2.59251	−0.594762	−0.297381	+	0.954759i	$0.596113\pi$
$20$	−1.00000	−0.223607
$21$	3.04678	0.664861
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	3.04678	0.621921
$25$	1.00000	0.200000
$26$	0.353057	0.0692401
$27$	10.0021	1.92490
$28$	1.00000	0.188982
$29$	9.54782	1.77299	0.886493	−	0.462742i	$-0.153134\pi$
$29$	9.54782	1.77299	0.886493	+	0.462742i	$0.153134\pi$
$30$	−3.04678	−0.556263
$31$	−5.70820	−1.02522	−0.512612	−	0.858620i	$-0.671322\pi$
$31$	−5.70820	−1.02522	−0.512612	+	0.858620i	$0.671322\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.810708	0.139035
$35$	−1.00000	−0.169031
$36$	6.28284	1.04714
$37$	3.56360	0.585852	0.292926	−	0.956135i	$-0.405371\pi$
$37$	3.56360	0.585852	0.292926	+	0.956135i	$0.405371\pi$
$38$	−2.59251	−0.420560
$39$	1.07569	0.172247
$40$	−1.00000	−0.158114
$41$	8.25732	1.28958	0.644788	−	0.764361i	$-0.276946\pi$
$41$	8.25732	1.28958	0.644788	+	0.764361i	$0.276946\pi$
$42$	3.04678	0.470128
$43$	9.49338	1.44773	0.723864	−	0.689943i	$-0.242364\pi$
$43$	9.49338	1.44773	0.723864	+	0.689943i	$0.242364\pi$
$44$	0	0
$45$	−6.28284	−0.936591
$46$	−2.00000	−0.294884
$47$	−8.60773	−1.25557	−0.627783	−	0.778388i	$-0.716038\pi$
$47$	−8.60773	−1.25557	−0.627783	+	0.778388i	$0.716038\pi$
$48$	3.04678	0.439764
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	2.47005	0.345876
$52$	0.353057	0.0489602
$53$	−2.79967	−0.384564	−0.192282	−	0.981340i	$-0.561589\pi$
$53$	−2.79967	−0.384564	−0.192282	+	0.981340i	$0.561589\pi$
$54$	10.0021	1.36111
$55$	0	0
$56$	1.00000	0.133631
$57$	−7.89879	−1.04622
$58$	9.54782	1.25369
$59$	13.2092	1.71970	0.859849	−	0.510549i	$-0.170558\pi$
$59$	13.2092	1.71970	0.859849	+	0.510549i	$0.170558\pi$
$60$	−3.04678	−0.393337
$61$	6.70611	0.858630	0.429315	−	0.903155i	$-0.358755\pi$
$61$	6.70611	0.858630	0.429315	+	0.903155i	$0.358755\pi$
$62$	−5.70820	−0.724943
$63$	6.28284	0.791564
$64$	1.00000	0.125000
$65$	−0.353057	−0.0437913
$66$	0	0
$67$	−1.40749	−0.171953	−0.0859763	−	0.996297i	$-0.527401\pi$
$67$	−1.40749	−0.171953	−0.0859763	+	0.996297i	$0.527401\pi$
$68$	0.810708	0.0983128
$69$	−6.09355	−0.733577
$70$	−1.00000	−0.119523
$71$	−12.9187	−1.53317	−0.766586	−	0.642141i	$-0.778046\pi$
$71$	−12.9187	−1.53317	−0.766586	+	0.642141i	$0.778046\pi$
$72$	6.28284	0.740440
$73$	5.15008	0.602771	0.301386	−	0.953502i	$-0.402551\pi$
$73$	5.15008	0.602771	0.301386	+	0.953502i	$0.402551\pi$
$74$	3.56360	0.414260
$75$	3.04678	0.351811
$76$	−2.59251	−0.297381
$77$	0	0
$78$	1.07569	0.121797
$79$	3.64694	0.410313	0.205157	−	0.978729i	$-0.434230\pi$
$79$	3.64694	0.410313	0.205157	+	0.978729i	$0.434230\pi$
$80$	−1.00000	−0.111803
$81$	11.6256	1.29173
$82$	8.25732	0.911868
$83$	2.66819	0.292872	0.146436	−	0.989220i	$-0.453220\pi$
$83$	2.66819	0.292872	0.146436	+	0.989220i	$0.453220\pi$
$84$	3.04678	0.332431
$85$	−0.810708	−0.0879336
$86$	9.49338	1.02370
$87$	29.0901	3.11878
$88$	0	0
$89$	9.63799	1.02163	0.510813	−	0.859692i	$-0.329345\pi$
$89$	9.63799	1.02163	0.510813	+	0.859692i	$0.329345\pi$
$90$	−6.28284	−0.662270
$91$	0.353057	0.0370104
$92$	−2.00000	−0.208514
$93$	−17.3916	−1.80343
$94$	−8.60773	−0.887820
$95$	2.59251	0.265986
$96$	3.04678	0.310960
$97$	−9.79837	−0.994874	−0.497437	−	0.867500i	$-0.665725\pi$
$97$	−9.79837	−0.994874	−0.497437	+	0.867500i	$0.665725\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	0.904358	0.0899870	0.0449935	−	0.998987i	$-0.485673\pi$
$101$	0.904358	0.0899870	0.0449935	+	0.998987i	$0.485673\pi$
$102$	2.47005	0.244571
$103$	11.2203	1.10557	0.552784	−	0.833325i	$-0.313565\pi$
$103$	11.2203	1.10557	0.552784	+	0.833325i	$0.313565\pi$
$104$	0.353057	0.0346201
$105$	−3.04678	−0.297335
$106$	−2.79967	−0.271928
$107$	−8.89614	−0.860023	−0.430011	−	0.902823i	$-0.641490\pi$
$107$	−8.89614	−0.860023	−0.430011	+	0.902823i	$0.641490\pi$
$108$	10.0021	0.962452
$109$	−18.4763	−1.76971	−0.884855	−	0.465866i	$-0.845743\pi$
$109$	−18.4763	−1.76971	−0.884855	+	0.465866i	$0.845743\pi$
$110$	0	0
$111$	10.8575	1.03055
$112$	1.00000	0.0944911
$113$	−12.6902	−1.19380	−0.596899	−	0.802317i	$-0.703601\pi$
$113$	−12.6902	−1.19380	−0.596899	+	0.802317i	$0.703601\pi$
$114$	−7.89879	−0.739789
$115$	2.00000	0.186501
$116$	9.54782	0.886493
$117$	2.21820	0.205073
$118$	13.2092	1.21601
$119$	0.810708	0.0743175
$120$	−3.04678	−0.278131
$121$	0	0
$122$	6.70611	0.607143
$123$	25.1582	2.26844
$124$	−5.70820	−0.512612
$125$	−1.00000	−0.0894427
$126$	6.28284	0.559720
$127$	−16.2382	−1.44090	−0.720452	−	0.693505i	$-0.756066\pi$
$127$	−16.2382	−1.44090	−0.720452	+	0.693505i	$0.756066\pi$
$128$	1.00000	0.0883883
$129$	28.9242	2.54664
$130$	−0.353057	−0.0309651
$131$	8.08882	0.706723	0.353362	−	0.935487i	$-0.385039\pi$
$131$	8.08882	0.706723	0.353362	+	0.935487i	$0.385039\pi$
$132$	0	0
$133$	−2.59251	−0.224799
$134$	−1.40749	−0.121589
$135$	−10.0021	−0.860843
$136$	0.810708	0.0695176
$137$	10.4019	0.888696	0.444348	−	0.895854i	$-0.353435\pi$
$137$	10.4019	0.888696	0.444348	+	0.895854i	$0.353435\pi$
$138$	−6.09355	−0.518718
$139$	−0.757168	−0.0642221	−0.0321111	−	0.999484i	$-0.510223\pi$
$139$	−0.757168	−0.0642221	−0.0321111	+	0.999484i	$0.510223\pi$
$140$	−1.00000	−0.0845154
$141$	−26.2258	−2.20861
$142$	−12.9187	−1.08412
$143$	0	0
$144$	6.28284	0.523570
$145$	−9.54782	−0.792903
$146$	5.15008	0.426224
$147$	3.04678	0.251294
$148$	3.56360	0.292926
$149$	8.96980	0.734835	0.367417	−	0.930056i	$-0.380242\pi$
$149$	8.96980	0.734835	0.367417	+	0.930056i	$0.380242\pi$
$150$	3.04678	0.248768
$151$	21.9731	1.78814	0.894072	−	0.447923i	$-0.147836\pi$
$151$	21.9731	1.78814	0.894072	+	0.447923i	$0.147836\pi$
$152$	−2.59251	−0.210280
$153$	5.09355	0.411789
$154$	0	0
$155$	5.70820	0.458494
$156$	1.07569	0.0861237
$157$	0.0668308	0.00533368	0.00266684	−	0.999996i	$-0.499151\pi$
$157$	0.0668308	0.00533368	0.00266684	+	0.999996i	$0.499151\pi$
$158$	3.64694	0.290135
$159$	−8.52995	−0.676469
$160$	−1.00000	−0.0790569
$161$	−2.00000	−0.157622
$162$	11.6256	0.913393
$163$	22.2692	1.74426	0.872128	−	0.489279i	$-0.162740\pi$
$163$	22.2692	1.74426	0.872128	+	0.489279i	$0.162740\pi$
$164$	8.25732	0.644788
$165$	0	0
$166$	2.66819	0.207092
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	3.04678	0.235064
$169$	−12.8754	−0.990412
$170$	−0.810708	−0.0621785
$171$	−16.2883	−1.24560
$172$	9.49338	0.723864
$173$	−19.8698	−1.51067	−0.755336	−	0.655338i	$-0.772526\pi$
$173$	−19.8698	−1.51067	−0.755336	+	0.655338i	$0.772526\pi$
$174$	29.0901	2.20531
$175$	1.00000	0.0755929
$176$	0	0
$177$	40.2456	3.02505
$178$	9.63799	0.722398
$179$	−8.61246	−0.643726	−0.321863	−	0.946786i	$-0.604309\pi$
$179$	−8.61246	−0.643726	−0.321863	+	0.946786i	$0.604309\pi$
$180$	−6.28284	−0.468296
$181$	11.7345	0.872216	0.436108	−	0.899894i	$-0.356357\pi$
$181$	11.7345	0.872216	0.436108	+	0.899894i	$0.356357\pi$
$182$	0.353057	0.0261703
$183$	20.4320	1.51038
$184$	−2.00000	−0.147442
$185$	−3.56360	−0.262001
$186$	−17.3916	−1.27522
$187$	0	0
$188$	−8.60773	−0.627783
$189$	10.0021	0.727545
$190$	2.59251	0.188080
$191$	−8.15063	−0.589759	−0.294880	−	0.955534i	$-0.595280\pi$
$191$	−8.15063	−0.589759	−0.294880	+	0.955534i	$0.595280\pi$
$192$	3.04678	0.219882
$193$	−8.74185	−0.629252	−0.314626	−	0.949216i	$-0.601879\pi$
$193$	−8.74185	−0.629252	−0.314626	+	0.949216i	$0.601879\pi$
$194$	−9.79837	−0.703482
$195$	−1.07569	−0.0770314
$196$	1.00000	0.0714286
$197$	−14.4568	−1.03001	−0.515003	−	0.857189i	$-0.672209\pi$
$197$	−14.4568	−1.03001	−0.515003	+	0.857189i	$0.672209\pi$
$198$	0	0
$199$	−11.0446	−0.782930	−0.391465	−	0.920193i	$-0.628032\pi$
$199$	−11.0446	−0.782930	−0.391465	+	0.920193i	$0.628032\pi$
$200$	1.00000	0.0707107
$201$	−4.28832	−0.302475
$202$	0.904358	0.0636304
$203$	9.54782	0.670126
$204$	2.47005	0.172938
$205$	−8.25732	−0.576716
$206$	11.2203	0.781755
$207$	−12.5657	−0.873376
$208$	0.353057	0.0244801
$209$	0	0
$210$	−3.04678	−0.210248
$211$	−23.7102	−1.63228	−0.816139	−	0.577856i	$-0.803889\pi$
$211$	−23.7102	−1.63228	−0.816139	+	0.577856i	$0.803889\pi$
$212$	−2.79967	−0.192282
$213$	−39.3605	−2.69694
$214$	−8.89614	−0.608128
$215$	−9.49338	−0.647443
$216$	10.0021	0.680556
$217$	−5.70820	−0.387498
$218$	−18.4763	−1.25137
$219$	15.6911	1.06031
$220$	0	0
$221$	0.286226	0.0192536
$222$	10.8575	0.728707
$223$	−5.58986	−0.374325	−0.187162	−	0.982329i	$-0.559929\pi$
$223$	−5.58986	−0.374325	−0.187162	+	0.982329i	$0.559929\pi$
$224$	1.00000	0.0668153
$225$	6.28284	0.418856
$226$	−12.6902	−0.844142
$227$	6.32206	0.419610	0.209805	−	0.977743i	$-0.432717\pi$
$227$	6.32206	0.419610	0.209805	+	0.977743i	$0.432717\pi$
$228$	−7.89879	−0.523110
$229$	−3.89113	−0.257133	−0.128566	−	0.991701i	$-0.541038\pi$
$229$	−3.89113	−0.257133	−0.128566	+	0.991701i	$0.541038\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	9.54782	0.626845
$233$	−15.1659	−0.993548	−0.496774	−	0.867880i	$-0.665482\pi$
$233$	−15.1659	−0.993548	−0.496774	+	0.867880i	$0.665482\pi$
$234$	2.21820	0.145008
$235$	8.60773	0.561506
$236$	13.2092	0.859849
$237$	11.1114	0.721764
$238$	0.810708	0.0525504
$239$	21.0949	1.36452	0.682258	−	0.731112i	$-0.260998\pi$
$239$	21.0949	1.36452	0.682258	+	0.731112i	$0.260998\pi$
$240$	−3.04678	−0.196669
$241$	11.9416	0.769228	0.384614	−	0.923078i	$-0.374335\pi$
$241$	11.9416	0.769228	0.384614	+	0.923078i	$0.374335\pi$
$242$	0	0
$243$	5.41432	0.347329
$244$	6.70611	0.429315
$245$	−1.00000	−0.0638877
$246$	25.1582	1.60403
$247$	−0.915302	−0.0582393
$248$	−5.70820	−0.362471
$249$	8.12938	0.515179
$250$	−1.00000	−0.0632456
$251$	2.11834	0.133709	0.0668543	−	0.997763i	$-0.478704\pi$
$251$	2.11834	0.133709	0.0668543	+	0.997763i	$0.478704\pi$
$252$	6.28284	0.395782
$253$	0	0
$254$	−16.2382	−1.01887
$255$	−2.47005	−0.154680
$256$	1.00000	0.0625000
$257$	−20.7137	−1.29208	−0.646042	−	0.763302i	$-0.723577\pi$
$257$	−20.7137	−1.29208	−0.646042	+	0.763302i	$0.723577\pi$
$258$	28.9242	1.80074
$259$	3.56360	0.221431
$260$	−0.353057	−0.0218957
$261$	59.9875	3.71313
$262$	8.08882	0.499729
$263$	−3.89531	−0.240195	−0.120097	−	0.992762i	$-0.538321\pi$
$263$	−3.89531	−0.240195	−0.120097	+	0.992762i	$0.538321\pi$
$264$	0	0
$265$	2.79967	0.171982
$266$	−2.59251	−0.158957
$267$	29.3648	1.79710
$268$	−1.40749	−0.0859763
$269$	−22.8684	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$269$	−22.8684	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$270$	−10.0021	−0.608708
$271$	−20.6082	−1.25186	−0.625929	−	0.779880i	$-0.715280\pi$
$271$	−20.6082	−1.25186	−0.625929	+	0.779880i	$0.715280\pi$
$272$	0.810708	0.0491564
$273$	1.07569	0.0651034
$274$	10.4019	0.628403
$275$	0	0
$276$	−6.09355	−0.366789
$277$	−17.2045	−1.03372	−0.516860	−	0.856070i	$-0.672899\pi$
$277$	−17.2045	−1.03372	−0.516860	+	0.856070i	$0.672899\pi$
$278$	−0.757168	−0.0454119
$279$	−35.8638	−2.14711
$280$	−1.00000	−0.0597614
$281$	−30.4444	−1.81616	−0.908081	−	0.418795i	$-0.862453\pi$
$281$	−30.4444	−1.81616	−0.908081	+	0.418795i	$0.862453\pi$
$282$	−26.2258	−1.56173
$283$	−9.99517	−0.594151	−0.297076	−	0.954854i	$-0.596011\pi$
$283$	−9.99517	−0.594151	−0.297076	+	0.954854i	$0.596011\pi$
$284$	−12.9187	−0.766586
$285$	7.89879	0.467884
$286$	0	0
$287$	8.25732	0.487414
$288$	6.28284	0.370220
$289$	−16.3428	−0.961338
$290$	−9.54782	−0.560667
$291$	−29.8535	−1.75004
$292$	5.15008	0.301386
$293$	6.46796	0.377862	0.188931	−	0.981990i	$-0.439498\pi$
$293$	6.46796	0.377862	0.188931	+	0.981990i	$0.439498\pi$
$294$	3.04678	0.177692
$295$	−13.2092	−0.769072
$296$	3.56360	0.207130
$297$	0	0
$298$	8.96980	0.519607
$299$	−0.706114	−0.0408356
$300$	3.04678	0.175906
$301$	9.49338	0.547190
$302$	21.9731	1.26441
$303$	2.75538	0.158292
$304$	−2.59251	−0.148690
$305$	−6.70611	−0.383991
$306$	5.09355	0.291179
$307$	20.6473	1.17840	0.589202	−	0.807986i	$-0.299442\pi$
$307$	20.6473	1.17840	0.589202	+	0.807986i	$0.299442\pi$
$308$	0	0
$309$	34.1857	1.94476
$310$	5.70820	0.324204
$311$	23.4385	1.32907	0.664537	−	0.747255i	$-0.268629\pi$
$311$	23.4385	1.32907	0.664537	+	0.747255i	$0.268629\pi$
$312$	1.07569	0.0608987
$313$	11.9059	0.672961	0.336480	−	0.941690i	$-0.390763\pi$
$313$	11.9059	0.672961	0.336480	+	0.941690i	$0.390763\pi$
$314$	0.0668308	0.00377148
$315$	−6.28284	−0.353998
$316$	3.64694	0.205157
$317$	−34.4185	−1.93314	−0.966568	−	0.256411i	$-0.917460\pi$
$317$	−34.4185	−1.93314	−0.966568	+	0.256411i	$0.917460\pi$
$318$	−8.52995	−0.478336
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−27.1046	−1.51283
$322$	−2.00000	−0.111456
$323$	−2.10177	−0.116945
$324$	11.6256	0.645866
$325$	0.353057	0.0195841
$326$	22.2692	1.23337
$327$	−56.2932	−3.11302
$328$	8.25732	0.455934
$329$	−8.60773	−0.474559
$330$	0	0
$331$	−11.6770	−0.641829	−0.320914	−	0.947108i	$-0.603990\pi$
$331$	−11.6770	−0.641829	−0.320914	+	0.947108i	$0.603990\pi$
$332$	2.66819	0.146436
$333$	22.3895	1.22694
$334$	12.0000	0.656611
$335$	1.40749	0.0768996
$336$	3.04678	0.166215
$337$	10.3337	0.562915	0.281457	−	0.959574i	$-0.409182\pi$
$337$	10.3337	0.562915	0.281457	+	0.959574i	$0.409182\pi$
$338$	−12.8754	−0.700327
$339$	−38.6643	−2.09996
$340$	−0.810708	−0.0439668
$341$	0	0
$342$	−16.2883	−0.880771
$343$	1.00000	0.0539949
$344$	9.49338	0.511849
$345$	6.09355	0.328066
$346$	−19.8698	−1.06821
$347$	18.3202	0.983480	0.491740	−	0.870742i	$-0.336361\pi$
$347$	18.3202	0.983480	0.491740	+	0.870742i	$0.336361\pi$
$348$	29.0901	1.55939
$349$	29.9216	1.60167	0.800833	−	0.598888i	$-0.204391\pi$
$349$	29.9216	1.60167	0.800833	+	0.598888i	$0.204391\pi$
$350$	1.00000	0.0534522
$351$	3.53131	0.188487
$352$	0	0
$353$	−8.93024	−0.475309	−0.237654	−	0.971350i	$-0.576379\pi$
$353$	−8.93024	−0.475309	−0.237654	+	0.971350i	$0.576379\pi$
$354$	40.2456	2.13903
$355$	12.9187	0.685656
$356$	9.63799	0.510813
$357$	2.47005	0.130729
$358$	−8.61246	−0.455183
$359$	29.7915	1.57234	0.786169	−	0.618012i	$-0.212062\pi$
$359$	29.7915	1.57234	0.786169	+	0.618012i	$0.212062\pi$
$360$	−6.28284	−0.331135
$361$	−12.2789	−0.646258
$362$	11.7345	0.616750
$363$	0	0
$364$	0.353057	0.0185052
$365$	−5.15008	−0.269567
$366$	20.4320	1.06800
$367$	−24.7535	−1.29212	−0.646062	−	0.763285i	$-0.723585\pi$
$367$	−24.7535	−1.29212	−0.646062	+	0.763285i	$0.723585\pi$
$368$	−2.00000	−0.104257
$369$	51.8794	2.70074
$370$	−3.56360	−0.185263
$371$	−2.79967	−0.145351
$372$	−17.3916	−0.901713
$373$	9.75070	0.504872	0.252436	−	0.967614i	$-0.418768\pi$
$373$	9.75070	0.504872	0.252436	+	0.967614i	$0.418768\pi$
$374$	0	0
$375$	−3.04678	−0.157335
$376$	−8.60773	−0.443910
$377$	3.37092	0.173611
$378$	10.0021	0.514452
$379$	21.0454	1.08103	0.540514	−	0.841335i	$-0.318230\pi$
$379$	21.0454	1.08103	0.540514	+	0.841335i	$0.318230\pi$
$380$	2.59251	0.132993
$381$	−49.4740	−2.53463
$382$	−8.15063	−0.417023
$383$	−34.8828	−1.78243	−0.891214	−	0.453582i	$-0.850146\pi$
$383$	−34.8828	−1.78243	−0.891214	+	0.453582i	$0.850146\pi$
$384$	3.04678	0.155480
$385$	0	0
$386$	−8.74185	−0.444948
$387$	59.6455	3.03195
$388$	−9.79837	−0.497437
$389$	−12.7069	−0.644263	−0.322131	−	0.946695i	$-0.604399\pi$
$389$	−12.7069	−0.644263	−0.322131	+	0.946695i	$0.604399\pi$
$390$	−1.07569	−0.0544694
$391$	−1.62142	−0.0819985
$392$	1.00000	0.0505076
$393$	24.6448	1.24317
$394$	−14.4568	−0.728324
$395$	−3.64694	−0.183498
$396$	0	0
$397$	3.33036	0.167146	0.0835729	−	0.996502i	$-0.473367\pi$
$397$	3.33036	0.167146	0.0835729	+	0.996502i	$0.473367\pi$
$398$	−11.0446	−0.553615
$399$	−7.89879	−0.395434
$400$	1.00000	0.0500000
$401$	−9.27248	−0.463046	−0.231523	−	0.972829i	$-0.574371\pi$
$401$	−9.27248	−0.463046	−0.231523	+	0.972829i	$0.574371\pi$
$402$	−4.28832	−0.213882
$403$	−2.01532	−0.100390
$404$	0.904358	0.0449935
$405$	−11.6256	−0.577681
$406$	9.54782	0.473850
$407$	0	0
$408$	2.47005	0.122285
$409$	0.349614	0.0172873	0.00864366	−	0.999963i	$-0.497249\pi$
$409$	0.349614	0.0172873	0.00864366	+	0.999963i	$0.497249\pi$
$410$	−8.25732	−0.407800
$411$	31.6923	1.56327
$412$	11.2203	0.552784
$413$	13.2092	0.649985
$414$	−12.5657	−0.617570
$415$	−2.66819	−0.130976
$416$	0.353057	0.0173100
$417$	−2.30692	−0.112970
$418$	0	0
$419$	27.9340	1.36466	0.682332	−	0.731043i	$-0.260966\pi$
$419$	27.9340	1.36466	0.682332	+	0.731043i	$0.260966\pi$
$420$	−3.04678	−0.148667
$421$	8.07221	0.393415	0.196708	−	0.980462i	$-0.436975\pi$
$421$	8.07221	0.393415	0.196708	+	0.980462i	$0.436975\pi$
$422$	−23.7102	−1.15419
$423$	−54.0810	−2.62951
$424$	−2.79967	−0.135964
$425$	0.810708	0.0393251
$426$	−39.3605	−1.90702
$427$	6.70611	0.324532
$428$	−8.89614	−0.430011
$429$	0	0
$430$	−9.49338	−0.457812
$431$	−20.0088	−0.963791	−0.481895	−	0.876229i	$-0.660051\pi$
$431$	−20.0088	−0.963791	−0.481895	+	0.876229i	$0.660051\pi$
$432$	10.0021	0.481226
$433$	−38.7447	−1.86195	−0.930977	−	0.365079i	$-0.881042\pi$
$433$	−38.7447	−1.86195	−0.930977	+	0.365079i	$0.881042\pi$
$434$	−5.70820	−0.274003
$435$	−29.0901	−1.39476
$436$	−18.4763	−0.884855
$437$	5.18501	0.248033
$438$	15.6911	0.749751
$439$	−5.62560	−0.268495	−0.134248	−	0.990948i	$-0.542862\pi$
$439$	−5.62560	−0.268495	−0.134248	+	0.990948i	$0.542862\pi$
$440$	0	0
$441$	6.28284	0.299183
$442$	0.286226	0.0136144
$443$	−15.6763	−0.744804	−0.372402	−	0.928072i	$-0.621466\pi$
$443$	−15.6763	−0.744804	−0.372402	+	0.928072i	$0.621466\pi$
$444$	10.8575	0.515273
$445$	−9.63799	−0.456885
$446$	−5.58986	−0.264688
$447$	27.3290	1.29262
$448$	1.00000	0.0472456
$449$	−26.1032	−1.23189	−0.615943	−	0.787791i	$-0.711225\pi$
$449$	−26.1032	−1.23189	−0.615943	+	0.787791i	$0.711225\pi$
$450$	6.28284	0.296176
$451$	0	0
$452$	−12.6902	−0.596899
$453$	66.9471	3.14545
$454$	6.32206	0.296709
$455$	−0.353057	−0.0165516
$456$	−7.89879	−0.369895
$457$	35.5911	1.66488	0.832441	−	0.554114i	$-0.186943\pi$
$457$	35.5911	1.66488	0.832441	+	0.554114i	$0.186943\pi$
$458$	−3.89113	−0.181820
$459$	8.10877	0.378485
$460$	2.00000	0.0932505
$461$	9.66571	0.450177	0.225088	−	0.974338i	$-0.427733\pi$
$461$	9.66571	0.450177	0.225088	+	0.974338i	$0.427733\pi$
$462$	0	0
$463$	−29.9726	−1.39295	−0.696473	−	0.717583i	$-0.745248\pi$
$463$	−29.9726	−1.39295	−0.696473	+	0.717583i	$0.745248\pi$
$464$	9.54782	0.443246
$465$	17.3916	0.806517
$466$	−15.1659	−0.702545
$467$	30.5086	1.41177	0.705885	−	0.708326i	$-0.250550\pi$
$467$	30.5086	1.41177	0.705885	+	0.708326i	$0.250550\pi$
$468$	2.21820	0.102536
$469$	−1.40749	−0.0649920
$470$	8.60773	0.397045
$471$	0.203619	0.00938225
$472$	13.2092	0.608005
$473$	0	0
$474$	11.1114	0.510364
$475$	−2.59251	−0.118952
$476$	0.810708	0.0371587
$477$	−17.5899	−0.805384
$478$	21.0949	0.964858
$479$	38.7127	1.76883	0.884414	−	0.466704i	$-0.154559\pi$
$479$	38.7127	1.76883	0.884414	+	0.466704i	$0.154559\pi$
$480$	−3.04678	−0.139066
$481$	1.25815	0.0573668
$482$	11.9416	0.543926
$483$	−6.09355	−0.277266
$484$	0	0
$485$	9.79837	0.444921
$486$	5.41432	0.245598
$487$	33.8375	1.53332	0.766662	−	0.642052i	$-0.221916\pi$
$487$	33.8375	1.53332	0.766662	+	0.642052i	$0.221916\pi$
$488$	6.70611	0.303572
$489$	67.8491	3.06824
$490$	−1.00000	−0.0451754
$491$	18.5755	0.838301	0.419150	−	0.907917i	$-0.362328\pi$
$491$	18.5755	0.838301	0.419150	+	0.907917i	$0.362328\pi$
$492$	25.1582	1.13422
$493$	7.74050	0.348614
$494$	−0.915302	−0.0411814
$495$	0	0
$496$	−5.70820	−0.256306
$497$	−12.9187	−0.579485
$498$	8.12938	0.364286
$499$	−14.7069	−0.658372	−0.329186	−	0.944265i	$-0.606774\pi$
$499$	−14.7069	−0.658372	−0.329186	+	0.944265i	$0.606774\pi$
$500$	−1.00000	−0.0447214
$501$	36.5613	1.63344
$502$	2.11834	0.0945462
$503$	−5.50234	−0.245337	−0.122669	−	0.992448i	$-0.539145\pi$
$503$	−5.50234	−0.245337	−0.122669	+	0.992448i	$0.539145\pi$
$504$	6.28284	0.279860
$505$	−0.904358	−0.0402434
$506$	0	0
$507$	−39.2283	−1.74219
$508$	−16.2382	−0.720452
$509$	24.1045	1.06841	0.534207	−	0.845354i	$-0.320610\pi$
$509$	24.1045	1.06841	0.534207	+	0.845354i	$0.320610\pi$
$510$	−2.47005	−0.109375
$511$	5.15008	0.227826
$512$	1.00000	0.0441942
$513$	−25.9305	−1.14486
$514$	−20.7137	−0.913641
$515$	−11.2203	−0.494425
$516$	28.9242	1.27332
$517$	0	0
$518$	3.56360	0.156575
$519$	−60.5388	−2.65736
$520$	−0.353057	−0.0154826
$521$	8.73293	0.382597	0.191298	−	0.981532i	$-0.438730\pi$
$521$	8.73293	0.382597	0.191298	+	0.981532i	$0.438730\pi$
$522$	59.9875	2.62558
$523$	−44.2475	−1.93481	−0.967405	−	0.253235i	$-0.918505\pi$
$523$	−44.2475	−1.93481	−0.967405	+	0.253235i	$0.918505\pi$
$524$	8.08882	0.353362
$525$	3.04678	0.132972
$526$	−3.89531	−0.169843
$527$	−4.62769	−0.201585
$528$	0	0
$529$	−19.0000	−0.826087
$530$	2.79967	0.121610
$531$	82.9917	3.60153
$532$	−2.59251	−0.112399
$533$	2.91530	0.126276
$534$	29.3648	1.27074
$535$	8.89614	0.384614
$536$	−1.40749	−0.0607944
$537$	−26.2402	−1.13235
$538$	−22.8684	−0.985928
$539$	0	0
$540$	−10.0021	−0.430421
$541$	−14.0407	−0.603655	−0.301827	−	0.953363i	$-0.597597\pi$
$541$	−14.0407	−0.603655	−0.301827	+	0.953363i	$0.597597\pi$
$542$	−20.6082	−0.885197
$543$	35.7523	1.53428
$544$	0.810708	0.0347588
$545$	18.4763	0.791438
$546$	1.07569	0.0460351
$547$	−6.48093	−0.277105	−0.138552	−	0.990355i	$-0.544245\pi$
$547$	−6.48093	−0.277105	−0.138552	+	0.990355i	$0.544245\pi$
$548$	10.4019	0.444348
$549$	42.1335	1.79821
$550$	0	0
$551$	−24.7528	−1.05450
$552$	−6.09355	−0.259359
$553$	3.64694	0.155084
$554$	−17.2045	−0.730950
$555$	−10.8575	−0.460875
$556$	−0.757168	−0.0321111
$557$	−10.0109	−0.424177	−0.212089	−	0.977250i	$-0.568027\pi$
$557$	−10.0109	−0.424177	−0.212089	+	0.977250i	$0.568027\pi$
$558$	−35.8638	−1.51823
$559$	3.35170	0.141762
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−30.4444	−1.28422
$563$	−3.32206	−0.140008	−0.0700040	−	0.997547i	$-0.522301\pi$
$563$	−3.32206	−0.140008	−0.0700040	+	0.997547i	$0.522301\pi$
$564$	−26.2258	−1.10431
$565$	12.6902	0.533882
$566$	−9.99517	−0.420128
$567$	11.6256	0.488229
$568$	−12.9187	−0.542058
$569$	−30.5120	−1.27913	−0.639564	−	0.768738i	$-0.720885\pi$
$569$	−30.5120	−1.27913	−0.639564	+	0.768738i	$0.720885\pi$
$570$	7.89879	0.330844
$571$	−2.37904	−0.0995597	−0.0497799	−	0.998760i	$-0.515852\pi$
$571$	−2.37904	−0.0995597	−0.0497799	+	0.998760i	$0.515852\pi$
$572$	0	0
$573$	−24.8332	−1.03742
$574$	8.25732	0.344654
$575$	−2.00000	−0.0834058
$576$	6.28284	0.261785
$577$	−12.5384	−0.521981	−0.260990	−	0.965341i	$-0.584049\pi$
$577$	−12.5384	−0.521981	−0.260990	+	0.965341i	$0.584049\pi$
$578$	−16.3428	−0.679769
$579$	−26.6345	−1.10689
$580$	−9.54782	−0.396452
$581$	2.66819	0.110695
$582$	−29.8535	−1.23747
$583$	0	0
$584$	5.15008	0.213112
$585$	−2.21820	−0.0917113
$586$	6.46796	0.267189
$587$	43.5800	1.79874	0.899370	−	0.437189i	$-0.144026\pi$
$587$	43.5800	1.79874	0.899370	+	0.437189i	$0.144026\pi$
$588$	3.04678	0.125647
$589$	14.7986	0.609764
$590$	−13.2092	−0.543816
$591$	−44.0467	−1.81184
$592$	3.56360	0.146463
$593$	−0.0535401	−0.00219863	−0.00109931	−	0.999999i	$-0.500350\pi$
$593$	−0.0535401	−0.00219863	−0.00109931	+	0.999999i	$0.500350\pi$
$594$	0	0
$595$	−0.810708	−0.0332358
$596$	8.96980	0.367417
$597$	−33.6504	−1.37722
$598$	−0.706114	−0.0288751
$599$	28.9504	1.18288	0.591440	−	0.806349i	$-0.298560\pi$
$599$	28.9504	1.18288	0.591440	+	0.806349i	$0.298560\pi$
$600$	3.04678	0.124384
$601$	−31.0406	−1.26617	−0.633085	−	0.774082i	$-0.718212\pi$
$601$	−31.0406	−1.26617	−0.633085	+	0.774082i	$0.718212\pi$
$602$	9.49338	0.386921
$603$	−8.84306	−0.360117
$604$	21.9731	0.894072
$605$	0	0
$606$	2.75538	0.111929
$607$	−8.20288	−0.332945	−0.166472	−	0.986046i	$-0.553238\pi$
$607$	−8.20288	−0.332945	−0.166472	+	0.986046i	$0.553238\pi$
$608$	−2.59251	−0.105140
$609$	29.0901	1.17879
$610$	−6.70611	−0.271523
$611$	−3.03902	−0.122946
$612$	5.09355	0.205895
$613$	21.6035	0.872558	0.436279	−	0.899812i	$-0.356296\pi$
$613$	21.6035	0.872558	0.436279	+	0.899812i	$0.356296\pi$
$614$	20.6473	0.833257
$615$	−25.1582	−1.01448
$616$	0	0
$617$	16.5772	0.667372	0.333686	−	0.942684i	$-0.391707\pi$
$617$	16.5772	0.667372	0.333686	+	0.942684i	$0.391707\pi$
$618$	34.1857	1.37515
$619$	−17.8133	−0.715975	−0.357988	−	0.933726i	$-0.616537\pi$
$619$	−17.8133	−0.715975	−0.357988	+	0.933726i	$0.616537\pi$
$620$	5.70820	0.229247
$621$	−20.0042	−0.802740
$622$	23.4385	0.939798
$623$	9.63799	0.386138
$624$	1.07569	0.0430619
$625$	1.00000	0.0400000
$626$	11.9059	0.475855
$627$	0	0
$628$	0.0668308	0.00266684
$629$	2.88904	0.115193
$630$	−6.28284	−0.250314
$631$	13.5121	0.537908	0.268954	−	0.963153i	$-0.413322\pi$
$631$	13.5121	0.537908	0.268954	+	0.963153i	$0.413322\pi$
$632$	3.64694	0.145068
$633$	−72.2397	−2.87127
$634$	−34.4185	−1.36693
$635$	16.2382	0.644392
$636$	−8.52995	−0.338235
$637$	0.353057	0.0139886
$638$	0	0
$639$	−81.1665	−3.21090
$640$	−1.00000	−0.0395285
$641$	44.3116	1.75020	0.875102	−	0.483939i	$-0.160794\pi$
$641$	44.3116	1.75020	0.875102	+	0.483939i	$0.160794\pi$
$642$	−27.1046	−1.06973
$643$	34.2852	1.35208	0.676039	−	0.736866i	$-0.263695\pi$
$643$	34.2852	1.35208	0.676039	+	0.736866i	$0.263695\pi$
$644$	−2.00000	−0.0788110
$645$	−28.9242	−1.13889
$646$	−2.10177	−0.0826929
$647$	27.8742	1.09585	0.547925	−	0.836528i	$-0.315418\pi$
$647$	27.8742	1.09585	0.547925	+	0.836528i	$0.315418\pi$
$648$	11.6256	0.456697
$649$	0	0
$650$	0.353057	0.0138480
$651$	−17.3916	−0.681631
$652$	22.2692	0.872128
$653$	−27.5831	−1.07941	−0.539705	−	0.841854i	$-0.681464\pi$
$653$	−27.5831	−1.07941	−0.539705	+	0.841854i	$0.681464\pi$
$654$	−56.2932	−2.20124
$655$	−8.08882	−0.316056
$656$	8.25732	0.322394
$657$	32.3571	1.26237
$658$	−8.60773	−0.335564
$659$	−2.49969	−0.0973742	−0.0486871	−	0.998814i	$-0.515504\pi$
$659$	−2.49969	−0.0973742	−0.0486871	+	0.998814i	$0.515504\pi$
$660$	0	0
$661$	10.1499	0.394785	0.197393	−	0.980325i	$-0.436753\pi$
$661$	10.1499	0.394785	0.197393	+	0.980325i	$0.436753\pi$
$662$	−11.6770	−0.453841
$663$	0.872067	0.0338683
$664$	2.66819	0.103546
$665$	2.59251	0.100533
$666$	22.3895	0.867577
$667$	−19.0956	−0.739386
$668$	12.0000	0.464294
$669$	−17.0311	−0.658459
$670$	1.40749	0.0543762
$671$	0	0
$672$	3.04678	0.117532
$673$	−36.9488	−1.42427	−0.712136	−	0.702042i	$-0.752272\pi$
$673$	−36.9488	−1.42427	−0.712136	+	0.702042i	$0.752272\pi$
$674$	10.3337	0.398041
$675$	10.0021	0.384981
$676$	−12.8754	−0.495206
$677$	−8.83753	−0.339654	−0.169827	−	0.985474i	$-0.554321\pi$
$677$	−8.83753	−0.339654	−0.169827	+	0.985474i	$0.554321\pi$
$678$	−38.6643	−1.48489
$679$	−9.79837	−0.376027
$680$	−0.810708	−0.0310892
$681$	19.2619	0.738117
$682$	0	0
$683$	33.0803	1.26578	0.632892	−	0.774240i	$-0.281868\pi$
$683$	33.0803	1.26578	0.632892	+	0.774240i	$0.281868\pi$
$684$	−16.2883	−0.622799
$685$	−10.4019	−0.397437
$686$	1.00000	0.0381802
$687$	−11.8554	−0.452311
$688$	9.49338	0.361932
$689$	−0.988441	−0.0376566
$690$	6.09355	0.231978
$691$	−9.22130	−0.350795	−0.175397	−	0.984498i	$-0.556121\pi$
$691$	−9.22130	−0.350795	−0.175397	+	0.984498i	$0.556121\pi$
$692$	−19.8698	−0.755336
$693$	0	0
$694$	18.3202	0.695426
$695$	0.757168	0.0287210
$696$	29.0901	1.10266
$697$	6.69427	0.253564
$698$	29.9216	1.13255
$699$	−46.2070	−1.74771
$700$	1.00000	0.0377964
$701$	−26.2969	−0.993220	−0.496610	−	0.867974i	$-0.665422\pi$
$701$	−26.2969	−0.993220	−0.496610	+	0.867974i	$0.665422\pi$
$702$	3.53131	0.133281
$703$	−9.23865	−0.348442
$704$	0	0
$705$	26.2258	0.987722
$706$	−8.93024	−0.336094
$707$	0.904358	0.0340119
$708$	40.2456	1.51252
$709$	−34.6228	−1.30029	−0.650143	−	0.759812i	$-0.725291\pi$
$709$	−34.6228	−1.30029	−0.650143	+	0.759812i	$0.725291\pi$
$710$	12.9187	0.484832
$711$	22.9132	0.859311
$712$	9.63799	0.361199
$713$	11.4164	0.427548
$714$	2.47005	0.0924391
$715$	0	0
$716$	−8.61246	−0.321863
$717$	64.2714	2.40026
$718$	29.7915	1.11181
$719$	−37.1839	−1.38673	−0.693363	−	0.720589i	$-0.743872\pi$
$719$	−37.1839	−1.38673	−0.693363	+	0.720589i	$0.743872\pi$
$720$	−6.28284	−0.234148
$721$	11.2203	0.417865
$722$	−12.2789	−0.456974
$723$	36.3835	1.35312
$724$	11.7345	0.436108
$725$	9.54782	0.354597
$726$	0	0
$727$	16.8666	0.625547	0.312773	−	0.949828i	$-0.398742\pi$
$727$	16.8666	0.625547	0.312773	+	0.949828i	$0.398742\pi$
$728$	0.353057	0.0130852
$729$	−18.3806	−0.680762
$730$	−5.15008	−0.190613
$731$	7.69636	0.284660
$732$	20.4320	0.755189
$733$	−35.0947	−1.29625	−0.648126	−	0.761533i	$-0.724447\pi$
$733$	−35.0947	−1.29625	−0.648126	+	0.761533i	$0.724447\pi$
$734$	−24.7535	−0.913669
$735$	−3.04678	−0.112382
$736$	−2.00000	−0.0737210
$737$	0	0
$738$	51.8794	1.90971
$739$	8.98641	0.330570	0.165285	−	0.986246i	$-0.447146\pi$
$739$	8.98641	0.330570	0.165285	+	0.986246i	$0.447146\pi$
$740$	−3.56360	−0.131000
$741$	−2.78872	−0.102446
$742$	−2.79967	−0.102779
$743$	9.59663	0.352066	0.176033	−	0.984384i	$-0.443673\pi$
$743$	9.59663	0.352066	0.176033	+	0.984384i	$0.443673\pi$
$744$	−17.3916	−0.637608
$745$	−8.96980	−0.328628
$746$	9.75070	0.356999
$747$	16.7638	0.613357
$748$	0	0
$749$	−8.89614	−0.325058
$750$	−3.04678	−0.111253
$751$	−14.1491	−0.516309	−0.258154	−	0.966104i	$-0.583114\pi$
$751$	−14.1491	−0.516309	−0.258154	+	0.966104i	$0.583114\pi$
$752$	−8.60773	−0.313892
$753$	6.45411	0.235201
$754$	3.37092	0.122762
$755$	−21.9731	−0.799682
$756$	10.0021	0.363773
$757$	45.6765	1.66014	0.830070	−	0.557660i	$-0.188301\pi$
$757$	45.6765	1.66014	0.830070	+	0.557660i	$0.188301\pi$
$758$	21.0454	0.764403
$759$	0	0
$760$	2.59251	0.0940401
$761$	16.5237	0.598985	0.299493	−	0.954099i	$-0.403183\pi$
$761$	16.5237	0.598985	0.299493	+	0.954099i	$0.403183\pi$
$762$	−49.4740	−1.79226
$763$	−18.4763	−0.668888
$764$	−8.15063	−0.294880
$765$	−5.09355	−0.184158
$766$	−34.8828	−1.26037
$767$	4.66362	0.168393
$768$	3.04678	0.109941
$769$	−26.5929	−0.958963	−0.479482	−	0.877552i	$-0.659175\pi$
$769$	−26.5929	−0.958963	−0.479482	+	0.877552i	$0.659175\pi$
$770$	0	0
$771$	−63.1099	−2.27285
$772$	−8.74185	−0.314626
$773$	35.4777	1.27604	0.638022	−	0.770019i	$-0.279753\pi$
$773$	35.4777	1.27604	0.638022	+	0.770019i	$0.279753\pi$
$774$	59.6455	2.14391
$775$	−5.70820	−0.205045
$776$	−9.79837	−0.351741
$777$	10.8575	0.389510
$778$	−12.7069	−0.455563
$779$	−21.4072	−0.766991
$780$	−1.07569	−0.0385157
$781$	0	0
$782$	−1.62142	−0.0579817
$783$	95.4982	3.41283
$784$	1.00000	0.0357143
$785$	−0.0668308	−0.00238529
$786$	24.6448	0.879051
$787$	16.0990	0.573868	0.286934	−	0.957950i	$-0.407364\pi$
$787$	16.0990	0.573868	0.286934	+	0.957950i	$0.407364\pi$
$788$	−14.4568	−0.515003
$789$	−11.8681	−0.422517
$790$	−3.64694	−0.129752
$791$	−12.6902	−0.451213
$792$	0	0
$793$	2.36764	0.0840773
$794$	3.33036	0.118190
$795$	8.52995	0.302526
$796$	−11.0446	−0.391465
$797$	38.0457	1.34765	0.673825	−	0.738891i	$-0.264650\pi$
$797$	38.0457	1.34765	0.673825	+	0.738891i	$0.264650\pi$
$798$	−7.89879	−0.279614
$799$	−6.97836	−0.246876
$800$	1.00000	0.0353553
$801$	60.5540	2.13957
$802$	−9.27248	−0.327423
$803$	0	0
$804$	−4.28832	−0.151237
$805$	2.00000	0.0704907
$806$	−2.01532	−0.0709866
$807$	−69.6750	−2.45268
$808$	0.904358	0.0318152
$809$	−14.7984	−0.520283	−0.260142	−	0.965570i	$-0.583769\pi$
$809$	−14.7984	−0.520283	−0.260142	+	0.965570i	$0.583769\pi$
$810$	−11.6256	−0.408482
$811$	0.702276	0.0246602	0.0123301	−	0.999924i	$-0.496075\pi$
$811$	0.702276	0.0246602	0.0123301	+	0.999924i	$0.496075\pi$
$812$	9.54782	0.335063
$813$	−62.7885	−2.20209
$814$	0	0
$815$	−22.2692	−0.780055
$816$	2.47005	0.0864689
$817$	−24.6117	−0.861053
$818$	0.349614	0.0122240
$819$	2.21820	0.0775102
$820$	−8.25732	−0.288358
$821$	1.50680	0.0525877	0.0262938	−	0.999654i	$-0.491629\pi$
$821$	1.50680	0.0525877	0.0262938	+	0.999654i	$0.491629\pi$
$822$	31.6923	1.10540
$823$	−30.8989	−1.07707	−0.538534	−	0.842604i	$-0.681021\pi$
$823$	−30.8989	−1.07707	−0.538534	+	0.842604i	$0.681021\pi$
$824$	11.2203	0.390877
$825$	0	0
$826$	13.2092	0.459609
$827$	52.7658	1.83485	0.917423	−	0.397914i	$-0.130266\pi$
$827$	52.7658	1.83485	0.917423	+	0.397914i	$0.130266\pi$
$828$	−12.5657	−0.436688
$829$	−27.7575	−0.964057	−0.482028	−	0.876156i	$-0.660100\pi$
$829$	−27.7575	−0.964057	−0.482028	+	0.876156i	$0.660100\pi$
$830$	−2.66819	−0.0926143
$831$	−52.4183	−1.81837
$832$	0.353057	0.0122400
$833$	0.810708	0.0280894
$834$	−2.30692	−0.0798821
$835$	−12.0000	−0.415277
$836$	0	0
$837$	−57.0940	−1.97346
$838$	27.9340	0.964963
$839$	−48.0237	−1.65796	−0.828981	−	0.559277i	$-0.811079\pi$
$839$	−48.0237	−1.65796	−0.828981	+	0.559277i	$0.811079\pi$
$840$	−3.04678	−0.105124
$841$	62.1609	2.14348
$842$	8.07221	0.278187
$843$	−92.7573	−3.19473
$844$	−23.7102	−0.816139
$845$	12.8754	0.442926
$846$	−54.0810	−1.85934
$847$	0	0
$848$	−2.79967	−0.0961409
$849$	−30.4530	−1.04515
$850$	0.810708	0.0278071
$851$	−7.12720	−0.244317
$852$	−39.3605	−1.34847
$853$	−36.0186	−1.23325	−0.616626	−	0.787256i	$-0.711501\pi$
$853$	−36.0186	−1.23325	−0.616626	+	0.787256i	$0.711501\pi$
$854$	6.70611	0.229478
$855$	16.2883	0.557049
$856$	−8.89614	−0.304064
$857$	5.73142	0.195782	0.0978908	−	0.995197i	$-0.468790\pi$
$857$	5.73142	0.195782	0.0978908	+	0.995197i	$0.468790\pi$
$858$	0	0
$859$	33.4569	1.14153	0.570767	−	0.821112i	$-0.306646\pi$
$859$	33.4569	1.14153	0.570767	+	0.821112i	$0.306646\pi$
$860$	−9.49338	−0.323722
$861$	25.1582	0.857389
$862$	−20.0088	−0.681503
$863$	10.8522	0.369413	0.184707	−	0.982794i	$-0.440867\pi$
$863$	10.8522	0.369413	0.184707	+	0.982794i	$0.440867\pi$
$864$	10.0021	0.340278
$865$	19.8698	0.675593
$866$	−38.7447	−1.31660
$867$	−49.7927	−1.69105
$868$	−5.70820	−0.193749
$869$	0	0
$870$	−29.0901	−0.986246
$871$	−0.496925	−0.0168377
$872$	−18.4763	−0.625687
$873$	−61.5617	−2.08355
$874$	5.18501	0.175386
$875$	−1.00000	−0.0338062
$876$	15.6911	0.530154
$877$	−21.1287	−0.713465	−0.356732	−	0.934207i	$-0.616109\pi$
$877$	−21.1287	−0.713465	−0.356732	+	0.934207i	$0.616109\pi$
$878$	−5.62560	−0.189855
$879$	19.7064	0.664681
$880$	0	0
$881$	24.5805	0.828137	0.414069	−	0.910246i	$-0.364107\pi$
$881$	24.5805	0.828137	0.414069	+	0.910246i	$0.364107\pi$
$882$	6.28284	0.211554
$883$	−15.4728	−0.520700	−0.260350	−	0.965514i	$-0.583838\pi$
$883$	−15.4728	−0.520700	−0.260350	+	0.965514i	$0.583838\pi$
$884$	0.286226	0.00962682
$885$	−40.2456	−1.35284
$886$	−15.6763	−0.526656
$887$	−20.4358	−0.686165	−0.343083	−	0.939305i	$-0.611471\pi$
$887$	−20.4358	−0.686165	−0.343083	+	0.939305i	$0.611471\pi$
$888$	10.8575	0.364353
$889$	−16.2382	−0.544610
$890$	−9.63799	−0.323066
$891$	0	0
$892$	−5.58986	−0.187162
$893$	22.3156	0.746763
$894$	27.3290	0.914017
$895$	8.61246	0.287883
$896$	1.00000	0.0334077
$897$	−2.15137	−0.0718322
$898$	−26.1032	−0.871075
$899$	−54.5009	−1.81771
$900$	6.28284	0.209428
$901$	−2.26971	−0.0756151
$902$	0	0
$903$	28.9242	0.962538
$904$	−12.6902	−0.422071
$905$	−11.7345	−0.390067
$906$	66.9471	2.22417
$907$	30.9777	1.02860	0.514298	−	0.857611i	$-0.328052\pi$
$907$	30.9777	1.02860	0.514298	+	0.857611i	$0.328052\pi$
$908$	6.32206	0.209805
$909$	5.68194	0.188458
$910$	−0.353057	−0.0117037
$911$	−29.0687	−0.963090	−0.481545	−	0.876421i	$-0.659924\pi$
$911$	−29.0687	−0.963090	−0.481545	+	0.876421i	$0.659924\pi$
$912$	−7.89879	−0.261555
$913$	0	0
$914$	35.5911	1.17725
$915$	−20.4320	−0.675462
$916$	−3.89113	−0.128566
$917$	8.08882	0.267116
$918$	8.10877	0.267629
$919$	−38.2375	−1.26134	−0.630669	−	0.776052i	$-0.717220\pi$
$919$	−38.2375	−1.26134	−0.630669	+	0.776052i	$0.717220\pi$
$920$	2.00000	0.0659380
$921$	62.9077	2.07288
$922$	9.66571	0.318323
$923$	−4.56105	−0.150129
$924$	0	0
$925$	3.56360	0.117170
$926$	−29.9726	−0.984962
$927$	70.4953	2.31537
$928$	9.54782	0.313423
$929$	38.7575	1.27159	0.635797	−	0.771857i	$-0.280672\pi$
$929$	38.7575	1.27159	0.635797	+	0.771857i	$0.280672\pi$
$930$	17.3916	0.570294
$931$	−2.59251	−0.0849660
$932$	−15.1659	−0.496774
$933$	71.4118	2.33792
$934$	30.5086	0.998272
$935$	0	0
$936$	2.21820	0.0725042
$937$	−10.4997	−0.343010	−0.171505	−	0.985183i	$-0.554863\pi$
$937$	−10.4997	−0.343010	−0.171505	+	0.985183i	$0.554863\pi$
$938$	−1.40749	−0.0459563
$939$	36.2746	1.18378
$940$	8.60773	0.280753
$941$	10.1957	0.332369	0.166185	−	0.986095i	$-0.446855\pi$
$941$	10.1957	0.332369	0.166185	+	0.986095i	$0.446855\pi$
$942$	0.203619	0.00663425
$943$	−16.5146	−0.537790
$944$	13.2092	0.429924
$945$	−10.0021	−0.325368
$946$	0	0
$947$	−47.2023	−1.53387	−0.766934	−	0.641726i	$-0.778219\pi$
$947$	−47.2023	−1.53387	−0.766934	+	0.641726i	$0.778219\pi$
$948$	11.1114	0.360882
$949$	1.81827	0.0590236
$950$	−2.59251	−0.0841120
$951$	−104.865	−3.40050
$952$	0.810708	0.0262752
$953$	−5.18794	−0.168054	−0.0840269	−	0.996463i	$-0.526778\pi$
$953$	−5.18794	−0.168054	−0.0840269	+	0.996463i	$0.526778\pi$
$954$	−17.5899	−0.569493
$955$	8.15063	0.263748
$956$	21.0949	0.682258
$957$	0	0
$958$	38.7127	1.25075
$959$	10.4019	0.335896
$960$	−3.04678	−0.0983343
$961$	1.58359	0.0510836
$962$	1.25815	0.0405645
$963$	−55.8931	−1.80113
$964$	11.9416	0.384614
$965$	8.74185	0.281410
$966$	−6.09355	−0.196057
$967$	−18.9841	−0.610486	−0.305243	−	0.952274i	$-0.598738\pi$
$967$	−18.9841	−0.610486	−0.305243	+	0.952274i	$0.598738\pi$
$968$	0	0
$969$	−6.40361	−0.205714
$970$	9.79837	0.314607
$971$	−41.9080	−1.34489	−0.672447	−	0.740146i	$-0.734757\pi$
$971$	−41.9080	−1.34489	−0.672447	+	0.740146i	$0.734757\pi$
$972$	5.41432	0.173664
$973$	−0.757168	−0.0242737
$974$	33.8375	1.08422
$975$	1.07569	0.0344495
$976$	6.70611	0.214657
$977$	8.12073	0.259805	0.129903	−	0.991527i	$-0.458534\pi$
$977$	8.12073	0.259805	0.129903	+	0.991527i	$0.458534\pi$
$978$	67.8491	2.16958
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−116.084	−3.70627
$982$	18.5755	0.592768
$983$	−38.8143	−1.23799	−0.618993	−	0.785397i	$-0.712459\pi$
$983$	−38.8143	−1.23799	−0.618993	+	0.785397i	$0.712459\pi$
$984$	25.1582	0.802014
$985$	14.4568	0.460632
$986$	7.74050	0.246508
$987$	−26.2258	−0.834777
$988$	−0.915302	−0.0291196
$989$	−18.9868	−0.603744
$990$	0	0
$991$	1.30139	0.0413400	0.0206700	−	0.999786i	$-0.493420\pi$
$991$	1.30139	0.0413400	0.0206700	+	0.999786i	$0.493420\pi$
$992$	−5.70820	−0.181236
$993$	−35.5773	−1.12901
$994$	−12.9187	−0.409758
$995$	11.0446	0.350137
$996$	8.12938	0.257589
$997$	−14.8018	−0.468777	−0.234389	−	0.972143i	$-0.575309\pi$
$997$	−14.8018	−0.468777	−0.234389	+	0.972143i	$0.575309\pi$
$998$	−14.7069	−0.465539
$999$	35.6434	1.12771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.ct.1.4		4
11.2	odd	10		770.2.n.e.631.2	yes	8
11.6	odd	10		770.2.n.e.421.2	✓	8
11.10	odd	2		8470.2.a.cq.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.e.421.2	✓	8	11.6	odd	10
770.2.n.e.631.2	yes	8	11.2	odd	10
8470.2.a.cq.1.4		4	11.10	odd	2
8470.2.a.ct.1.4		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.04678 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.04678 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.28284 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +3.04678 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.04678 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.28284 q^{9} -1.00000 q^{10} +3.04678 q^{12} +0.353057 q^{13} +1.00000 q^{14} -3.04678 q^{15} +1.00000 q^{16} +0.810708 q^{17} +6.28284 q^{18} -2.59251 q^{19} -1.00000 q^{20} +3.04678 q^{21} -2.00000 q^{23} +3.04678 q^{24} +1.00000 q^{25} +0.353057 q^{26} +10.0021 q^{27} +1.00000 q^{28} +9.54782 q^{29} -3.04678 q^{30} -5.70820 q^{31} +1.00000 q^{32} +0.810708 q^{34} -1.00000 q^{35} +6.28284 q^{36} +3.56360 q^{37} -2.59251 q^{38} +1.07569 q^{39} -1.00000 q^{40} +8.25732 q^{41} +3.04678 q^{42} +9.49338 q^{43} -6.28284 q^{45} -2.00000 q^{46} -8.60773 q^{47} +3.04678 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.47005 q^{51} +0.353057 q^{52} -2.79967 q^{53} +10.0021 q^{54} +1.00000 q^{56} -7.89879 q^{57} +9.54782 q^{58} +13.2092 q^{59} -3.04678 q^{60} +6.70611 q^{61} -5.70820 q^{62} +6.28284 q^{63} +1.00000 q^{64} -0.353057 q^{65} -1.40749 q^{67} +0.810708 q^{68} -6.09355 q^{69} -1.00000 q^{70} -12.9187 q^{71} +6.28284 q^{72} +5.15008 q^{73} +3.56360 q^{74} +3.04678 q^{75} -2.59251 q^{76} +1.07569 q^{78} +3.64694 q^{79} -1.00000 q^{80} +11.6256 q^{81} +8.25732 q^{82} +2.66819 q^{83} +3.04678 q^{84} -0.810708 q^{85} +9.49338 q^{86} +29.0901 q^{87} +9.63799 q^{89} -6.28284 q^{90} +0.353057 q^{91} -2.00000 q^{92} -17.3916 q^{93} -8.60773 q^{94} +2.59251 q^{95} +3.04678 q^{96} -9.79837 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{7} + 4 q^{8} + 6 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 + 4 * q^7 + 4 * q^8 + 6 * q^9	\( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{7} + 4 q^{8} + 6 q^{9} - 4 q^{10} + 2 q^{12} + q^{13} + 4 q^{14} - 2 q^{15} + 4 q^{16} + 2 q^{17} + 6 q^{18} - 3 q^{19} - 4 q^{20} + 2 q^{21} - 8 q^{23} + 2 q^{24} + 4 q^{25} + q^{26} + 14 q^{27} + 4 q^{28} + 15 q^{29} - 2 q^{30} + 4 q^{31} + 4 q^{32} + 2 q^{34} - 4 q^{35} + 6 q^{36} + 2 q^{37} - 3 q^{38} - q^{39} - 4 q^{40} + 11 q^{41} + 2 q^{42} + 7 q^{43} - 6 q^{45} - 8 q^{46} + 5 q^{47} + 2 q^{48} + 4 q^{49} + 4 q^{50} + 18 q^{51} + q^{52} + 10 q^{53} + 14 q^{54} + 4 q^{56} - 34 q^{57} + 15 q^{58} + 13 q^{59} - 2 q^{60} + 26 q^{61} + 4 q^{62} + 6 q^{63} + 4 q^{64} - q^{65} - 13 q^{67} + 2 q^{68} - 4 q^{69} - 4 q^{70} - 13 q^{71} + 6 q^{72} - 18 q^{73} + 2 q^{74} + 2 q^{75} - 3 q^{76} - q^{78} + 15 q^{79} - 4 q^{80} - 8 q^{81} + 11 q^{82} - 2 q^{83} + 2 q^{84} - 2 q^{85} + 7 q^{86} + 26 q^{87} - 7 q^{89} - 6 q^{90} + q^{91} - 8 q^{92} + 2 q^{93} + 5 q^{94} + 3 q^{95} + 2 q^{96} + 10 q^{97} + 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 + 4 * q^7 + 4 * q^8 + 6 * q^9 - 4 * q^10 + 2 * q^12 + q^13 + 4 * q^14 - 2 * q^15 + 4 * q^16 + 2 * q^17 + 6 * q^18 - 3 * q^19 - 4 * q^20 + 2 * q^21 - 8 * q^23 + 2 * q^24 + 4 * q^25 + q^26 + 14 * q^27 + 4 * q^28 + 15 * q^29 - 2 * q^30 + 4 * q^31 + 4 * q^32 + 2 * q^34 - 4 * q^35 + 6 * q^36 + 2 * q^37 - 3 * q^38 - q^39 - 4 * q^40 + 11 * q^41 + 2 * q^42 + 7 * q^43 - 6 * q^45 - 8 * q^46 + 5 * q^47 + 2 * q^48 + 4 * q^49 + 4 * q^50 + 18 * q^51 + q^52 + 10 * q^53 + 14 * q^54 + 4 * q^56 - 34 * q^57 + 15 * q^58 + 13 * q^59 - 2 * q^60 + 26 * q^61 + 4 * q^62 + 6 * q^63 + 4 * q^64 - q^65 - 13 * q^67 + 2 * q^68 - 4 * q^69 - 4 * q^70 - 13 * q^71 + 6 * q^72 - 18 * q^73 + 2 * q^74 + 2 * q^75 - 3 * q^76 - q^78 + 15 * q^79 - 4 * q^80 - 8 * q^81 + 11 * q^82 - 2 * q^83 + 2 * q^84 - 2 * q^85 + 7 * q^86 + 26 * q^87 - 7 * q^89 - 6 * q^90 + q^91 - 8 * q^92 + 2 * q^93 + 5 * q^94 + 3 * q^95 + 2 * q^96 + 10 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

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Database

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