LMFDB - Embedded newform 7381.2.a.r.1.34

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7381,2,Mod(1,7381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7381, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7381 = 11^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7381.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:	$58.9375817319$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.34
Character	$\chi$	$=$	7381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.39205 q^{2} -1.88433 q^{3} -0.0621922 q^{4} +3.21283 q^{5} -2.62308 q^{6} -1.93870 q^{7} -2.87068 q^{8} +0.550694 q^{9} +O(q^{10})$	\(q+1.39205 q^{2} -1.88433 q^{3} -0.0621922 q^{4} +3.21283 q^{5} -2.62308 q^{6} -1.93870 q^{7} -2.87068 q^{8} +0.550694 q^{9} +4.47243 q^{10} +0.117191 q^{12} +0.135171 q^{13} -2.69878 q^{14} -6.05403 q^{15} -3.87175 q^{16} -2.10691 q^{17} +0.766595 q^{18} +3.17457 q^{19} -0.199813 q^{20} +3.65315 q^{21} -0.382437 q^{23} +5.40930 q^{24} +5.32229 q^{25} +0.188165 q^{26} +4.61530 q^{27} +0.120572 q^{28} -6.89110 q^{29} -8.42752 q^{30} -6.85274 q^{31} +0.351683 q^{32} -2.93293 q^{34} -6.22873 q^{35} -0.0342489 q^{36} +6.09269 q^{37} +4.41917 q^{38} -0.254707 q^{39} -9.22301 q^{40} -3.84376 q^{41} +5.08538 q^{42} +12.9347 q^{43} +1.76929 q^{45} -0.532372 q^{46} -4.56102 q^{47} +7.29564 q^{48} -3.24143 q^{49} +7.40890 q^{50} +3.97011 q^{51} -0.00840659 q^{52} +5.18338 q^{53} +6.42473 q^{54} +5.56539 q^{56} -5.98193 q^{57} -9.59277 q^{58} -4.69242 q^{59} +0.376514 q^{60} +1.00000 q^{61} -9.53936 q^{62} -1.06763 q^{63} +8.23306 q^{64} +0.434282 q^{65} -12.6269 q^{67} +0.131033 q^{68} +0.720637 q^{69} -8.67071 q^{70} +12.2303 q^{71} -1.58087 q^{72} +5.40683 q^{73} +8.48134 q^{74} -10.0289 q^{75} -0.197434 q^{76} -0.354565 q^{78} +11.4836 q^{79} -12.4393 q^{80} -10.3488 q^{81} -5.35072 q^{82} +14.8726 q^{83} -0.227198 q^{84} -6.76915 q^{85} +18.0057 q^{86} +12.9851 q^{87} -6.19352 q^{89} +2.46294 q^{90} -0.262057 q^{91} +0.0237846 q^{92} +12.9128 q^{93} -6.34917 q^{94} +10.1994 q^{95} -0.662687 q^{96} -4.39546 q^{97} -4.51224 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 10 q^{2} - 2 q^{3} + 50 q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 30 q^{8} + 48 q^{9}+O(q^{10}) $ 50 * q + 10 * q^2 - 2 * q^3 + 50 * q^4 - 2 * q^5 + 12 * q^6 + 8 * q^7 + 30 * q^8 + 48 * q^9	\( 50 q + 10 q^{2} - 2 q^{3} + 50 q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 30 q^{8} + 48 q^{9} + 12 q^{10} - 14 q^{12} + 8 q^{13} - 2 q^{14} - 16 q^{15} + 42 q^{16} + 22 q^{17} + 32 q^{19} - 8 q^{20} + 24 q^{21} + 12 q^{23} - 8 q^{24} + 40 q^{25} + 10 q^{26} - 8 q^{27} + 72 q^{28} + 56 q^{29} + 24 q^{30} + 10 q^{31} + 70 q^{32} - 32 q^{34} + 70 q^{35} + 34 q^{36} - 8 q^{37} - 14 q^{38} + 96 q^{39} - 54 q^{40} + 56 q^{41} - 8 q^{42} + 44 q^{43} - 24 q^{45} - 4 q^{46} - 4 q^{47} - 28 q^{48} + 38 q^{49} + 120 q^{50} + 76 q^{51} + 24 q^{52} + 4 q^{53} + 48 q^{54} - 18 q^{56} + 8 q^{57} + 28 q^{58} + 12 q^{59} - 60 q^{60} + 50 q^{61} + 8 q^{62} + 30 q^{63} + 10 q^{64} + 64 q^{65} + 18 q^{67} - 22 q^{68} - 8 q^{69} + 34 q^{70} + 12 q^{71} + 104 q^{72} - 16 q^{73} + 84 q^{74} - 26 q^{75} + 64 q^{76} + 40 q^{78} + 78 q^{79} - 36 q^{80} + 34 q^{81} + 54 q^{82} + 68 q^{83} - 78 q^{84} - 4 q^{85} + 36 q^{86} + 48 q^{87} + 26 q^{89} - 20 q^{90} + 32 q^{92} + 22 q^{93} + 156 q^{94} + 100 q^{95} - 4 q^{96} - 14 q^{97} + 70 q^{98}+O(q^{100}) \) 50 * q + 10 * q^2 - 2 * q^3 + 50 * q^4 - 2 * q^5 + 12 * q^6 + 8 * q^7 + 30 * q^8 + 48 * q^9 + 12 * q^10 - 14 * q^12 + 8 * q^13 - 2 * q^14 - 16 * q^15 + 42 * q^16 + 22 * q^17 + 32 * q^19 - 8 * q^20 + 24 * q^21 + 12 * q^23 - 8 * q^24 + 40 * q^25 + 10 * q^26 - 8 * q^27 + 72 * q^28 + 56 * q^29 + 24 * q^30 + 10 * q^31 + 70 * q^32 - 32 * q^34 + 70 * q^35 + 34 * q^36 - 8 * q^37 - 14 * q^38 + 96 * q^39 - 54 * q^40 + 56 * q^41 - 8 * q^42 + 44 * q^43 - 24 * q^45 - 4 * q^46 - 4 * q^47 - 28 * q^48 + 38 * q^49 + 120 * q^50 + 76 * q^51 + 24 * q^52 + 4 * q^53 + 48 * q^54 - 18 * q^56 + 8 * q^57 + 28 * q^58 + 12 * q^59 - 60 * q^60 + 50 * q^61 + 8 * q^62 + 30 * q^63 + 10 * q^64 + 64 * q^65 + 18 * q^67 - 22 * q^68 - 8 * q^69 + 34 * q^70 + 12 * q^71 + 104 * q^72 - 16 * q^73 + 84 * q^74 - 26 * q^75 + 64 * q^76 + 40 * q^78 + 78 * q^79 - 36 * q^80 + 34 * q^81 + 54 * q^82 + 68 * q^83 - 78 * q^84 - 4 * q^85 + 36 * q^86 + 48 * q^87 + 26 * q^89 - 20 * q^90 + 32 * q^92 + 22 * q^93 + 156 * q^94 + 100 * q^95 - 4 * q^96 - 14 * q^97 + 70 * 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 $

$2$	1.39205	0.984329	0.492165	−	0.870502i	$-0.336206\pi$
$2$	1.39205	0.984329	0.492165	+	0.870502i	$0.336206\pi$
$3$	−1.88433	−1.08792	−0.543959	−	0.839112i	$-0.683075\pi$
$3$	−1.88433	−1.08792	−0.543959	+	0.839112i	$0.683075\pi$
$4$	−0.0621922	−0.0310961
$5$	3.21283	1.43682	0.718411	−	0.695619i	$-0.244870\pi$
$5$	3.21283	1.43682	0.718411	+	0.695619i	$0.244870\pi$
$6$	−2.62308	−1.07087
$7$	−1.93870	−0.732761	−0.366380	−	0.930465i	$-0.619403\pi$
$7$	−1.93870	−0.732761	−0.366380	+	0.930465i	$0.619403\pi$
$8$	−2.87068	−1.01494
$9$	0.550694	0.183565
$10$	4.47243	1.41431
$11$	0	0
$11$	0	0
$12$	0.117191	0.0338300
$13$	0.135171	0.0374897	0.0187448	−	0.999824i	$-0.494033\pi$
$13$	0.135171	0.0374897	0.0187448	+	0.999824i	$0.494033\pi$
$14$	−2.69878	−0.721278
$15$	−6.05403	−1.56314
$16$	−3.87175	−0.967937
$17$	−2.10691	−0.511001	−0.255500	−	0.966809i	$-0.582240\pi$
$17$	−2.10691	−0.511001	−0.255500	+	0.966809i	$0.582240\pi$
$18$	0.766595	0.180688
$19$	3.17457	0.728296	0.364148	−	0.931341i	$-0.381360\pi$
$19$	3.17457	0.728296	0.364148	+	0.931341i	$0.381360\pi$
$20$	−0.199813	−0.0446796
$21$	3.65315	0.797184
$22$	0	0
$23$	−0.382437	−0.0797436	−0.0398718	−	0.999205i	$-0.512695\pi$
$23$	−0.382437	−0.0797436	−0.0398718	+	0.999205i	$0.512695\pi$
$24$	5.40930	1.10417
$25$	5.32229	1.06446
$26$	0.188165	0.0369022
$27$	4.61530	0.888214
$28$	0.120572	0.0227860
$29$	−6.89110	−1.27965	−0.639823	−	0.768523i	$-0.720992\pi$
$29$	−6.89110	−1.27965	−0.639823	+	0.768523i	$0.720992\pi$
$30$	−8.42752	−1.53865
$31$	−6.85274	−1.23079	−0.615394	−	0.788220i	$-0.711003\pi$
$31$	−6.85274	−1.23079	−0.615394	+	0.788220i	$0.711003\pi$
$32$	0.351683	0.0621694
$33$	0	0
$34$	−2.93293	−0.502993
$35$	−6.22873	−1.05285
$36$	−0.0342489	−0.00570815
$37$	6.09269	1.00163	0.500816	−	0.865554i	$-0.333033\pi$
$37$	6.09269	1.00163	0.500816	+	0.865554i	$0.333033\pi$
$38$	4.41917	0.716883
$39$	−0.254707	−0.0407857
$40$	−9.22301	−1.45829
$41$	−3.84376	−0.600295	−0.300147	−	0.953893i	$-0.597036\pi$
$41$	−3.84376	−0.600295	−0.300147	+	0.953893i	$0.597036\pi$
$42$	5.08538	0.784691
$43$	12.9347	1.97252	0.986260	−	0.165198i	$-0.0528262\pi$
$43$	12.9347	1.97252	0.986260	+	0.165198i	$0.0528262\pi$
$44$	0	0
$45$	1.76929	0.263750
$46$	−0.532372	−0.0784940
$47$	−4.56102	−0.665293	−0.332647	−	0.943052i	$-0.607942\pi$
$47$	−4.56102	−0.665293	−0.332647	+	0.943052i	$0.607942\pi$
$48$	7.29564	1.05304
$49$	−3.24143	−0.463061
$50$	7.40890	1.04778
$51$	3.97011	0.555927
$52$	−0.00840659	−0.00116578
$53$	5.18338	0.711992	0.355996	−	0.934487i	$-0.384142\pi$
$53$	5.18338	0.711992	0.355996	+	0.934487i	$0.384142\pi$
$54$	6.42473	0.874295
$55$	0	0
$56$	5.56539	0.743707
$57$	−5.98193	−0.792327
$58$	−9.59277	−1.25959
$59$	−4.69242	−0.610900	−0.305450	−	0.952208i	$-0.598807\pi$
$59$	−4.69242	−0.610900	−0.305450	+	0.952208i	$0.598807\pi$
$60$	0.376514	0.0486077
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−9.53936	−1.21150
$63$	−1.06763	−0.134509
$64$	8.23306	1.02913
$65$	0.434282	0.0538660
$66$	0	0
$67$	−12.6269	−1.54263	−0.771313	−	0.636456i	$-0.780400\pi$
$67$	−12.6269	−1.54263	−0.771313	+	0.636456i	$0.780400\pi$
$68$	0.131033	0.0158901
$69$	0.720637	0.0867545
$70$	−8.67071	−1.03635
$71$	12.2303	1.45147	0.725734	−	0.687976i	$-0.241500\pi$
$71$	12.2303	1.45147	0.725734	+	0.687976i	$0.241500\pi$
$72$	−1.58087	−0.186307
$73$	5.40683	0.632822	0.316411	−	0.948622i	$-0.397522\pi$
$73$	5.40683	0.632822	0.316411	+	0.948622i	$0.397522\pi$
$74$	8.48134	0.985936
$75$	−10.0289	−1.15804
$76$	−0.197434	−0.0226472
$77$	0	0
$78$	−0.354565	−0.0401466
$79$	11.4836	1.29201	0.646005	−	0.763333i	$-0.276438\pi$
$79$	11.4836	1.29201	0.646005	+	0.763333i	$0.276438\pi$
$80$	−12.4393	−1.39075
$81$	−10.3488	−1.14987
$82$	−5.35072	−0.590888
$83$	14.8726	1.63248	0.816242	−	0.577710i	$-0.196054\pi$
$83$	14.8726	1.63248	0.816242	+	0.577710i	$0.196054\pi$
$84$	−0.227198	−0.0247893
$85$	−6.76915	−0.734217
$86$	18.0057	1.94161
$87$	12.9851	1.39215
$88$	0	0
$89$	−6.19352	−0.656511	−0.328256	−	0.944589i	$-0.606461\pi$
$89$	−6.19352	−0.656511	−0.328256	+	0.944589i	$0.606461\pi$
$90$	2.46294	0.259617
$91$	−0.262057	−0.0274710
$92$	0.0237846	0.00247972
$93$	12.9128	1.33900
$94$	−6.34917	−0.654867
$95$	10.1994	1.04643
$96$	−0.662687	−0.0676352
$97$	−4.39546	−0.446291	−0.223146	−	0.974785i	$-0.571633\pi$
$97$	−4.39546	−0.446291	−0.223146	+	0.974785i	$0.571633\pi$
$98$	−4.51224	−0.455805
$99$	0	0
$100$	−0.331005	−0.0331005
$101$	−0.0959920	−0.00955156	−0.00477578	−	0.999989i	$-0.501520\pi$
$101$	−0.0959920	−0.00955156	−0.00477578	+	0.999989i	$0.501520\pi$
$102$	5.52660	0.547215
$103$	−0.645533	−0.0636063	−0.0318032	−	0.999494i	$-0.510125\pi$
$103$	−0.645533	−0.0636063	−0.0318032	+	0.999494i	$0.510125\pi$
$104$	−0.388032	−0.0380497
$105$	11.7370	1.14541
$106$	7.21553	0.700834
$107$	1.87928	0.181677	0.0908384	−	0.995866i	$-0.471045\pi$
$107$	1.87928	0.181677	0.0908384	+	0.995866i	$0.471045\pi$
$108$	−0.287036	−0.0276200
$109$	7.33716	0.702773	0.351386	−	0.936231i	$-0.385710\pi$
$109$	7.33716	0.702773	0.351386	+	0.936231i	$0.385710\pi$
$110$	0	0
$111$	−11.4806	−1.08969
$112$	7.50617	0.709266
$113$	−0.384242	−0.0361465	−0.0180732	−	0.999837i	$-0.505753\pi$
$113$	−0.384242	−0.0361465	−0.0180732	+	0.999837i	$0.505753\pi$
$114$	−8.32716	−0.779910
$115$	−1.22871	−0.114577
$116$	0.428573	0.0397920
$117$	0.0744379	0.00688179
$118$	−6.53208	−0.601327
$119$	4.08467	0.374441
$120$	17.3792	1.58649
$121$	0	0
$122$	1.39205	0.126030
$123$	7.24291	0.653071
$124$	0.426187	0.0382727
$125$	1.03546	0.0926148
$126$	−1.48620	−0.132401
$127$	0.0822885	0.00730193	0.00365096	−	0.999993i	$-0.498838\pi$
$127$	0.0822885	0.00730193	0.00365096	+	0.999993i	$0.498838\pi$
$128$	10.7575	0.950835
$129$	−24.3732	−2.14594
$130$	0.604543	0.0530219
$131$	−5.22577	−0.456577	−0.228289	−	0.973593i	$-0.573313\pi$
$131$	−5.22577	−0.456577	−0.228289	+	0.973593i	$0.573313\pi$
$132$	0	0
$133$	−6.15455	−0.533667
$134$	−17.5773	−1.51845
$135$	14.8282	1.27621
$136$	6.04826	0.518634
$137$	6.28296	0.536789	0.268395	−	0.963309i	$-0.413507\pi$
$137$	6.28296	0.536789	0.268395	+	0.963309i	$0.413507\pi$
$138$	1.00316	0.0853950
$139$	13.5433	1.14873	0.574364	−	0.818600i	$-0.305249\pi$
$139$	13.5433	1.14873	0.574364	+	0.818600i	$0.305249\pi$
$140$	0.387378	0.0327395
$141$	8.59446	0.723784
$142$	17.0252	1.42872
$143$	0	0
$144$	−2.13215	−0.177679
$145$	−22.1400	−1.83862
$146$	7.52659	0.622905
$147$	6.10792	0.503773
$148$	−0.378918	−0.0311469
$149$	5.47920	0.448873	0.224437	−	0.974489i	$-0.427946\pi$
$149$	5.47920	0.448873	0.224437	+	0.974489i	$0.427946\pi$
$150$	−13.9608	−1.13990
$151$	8.50974	0.692513	0.346256	−	0.938140i	$-0.387453\pi$
$151$	8.50974	0.692513	0.346256	+	0.938140i	$0.387453\pi$
$152$	−9.11317	−0.739176
$153$	−1.16026	−0.0938018
$154$	0	0
$155$	−22.0167	−1.76842
$156$	0.0158408	0.00126828
$157$	2.91298	0.232481	0.116240	−	0.993221i	$-0.462916\pi$
$157$	2.91298	0.232481	0.116240	+	0.993221i	$0.462916\pi$
$158$	15.9858	1.27176
$159$	−9.76719	−0.774589
$160$	1.12990	0.0893264
$161$	0.741432	0.0584330
$162$	−14.4061	−1.13185
$163$	−6.96449	−0.545501	−0.272750	−	0.962085i	$-0.587933\pi$
$163$	−6.96449	−0.545501	−0.272750	+	0.962085i	$0.587933\pi$
$164$	0.239052	0.0186668
$165$	0	0
$166$	20.7035	1.60690
$167$	13.6230	1.05418	0.527091	−	0.849809i	$-0.323283\pi$
$167$	13.6230	1.05418	0.527091	+	0.849809i	$0.323283\pi$
$168$	−10.4870	−0.809092
$169$	−12.9817	−0.998595
$170$	−9.42301	−0.722712
$171$	1.74822	0.133690
$172$	−0.804437	−0.0613377
$173$	−0.458949	−0.0348932	−0.0174466	−	0.999848i	$-0.505554\pi$
$173$	−0.458949	−0.0348932	−0.0174466	+	0.999848i	$0.505554\pi$
$174$	18.0759	1.37033
$175$	−10.3183	−0.779993
$176$	0	0
$177$	8.84205	0.664609
$178$	−8.62169	−0.646223
$179$	15.3843	1.14987	0.574937	−	0.818197i	$-0.305026\pi$
$179$	15.3843	1.14987	0.574937	+	0.818197i	$0.305026\pi$
$180$	−0.110036	−0.00820160
$181$	−2.20059	−0.163569	−0.0817843	−	0.996650i	$-0.526062\pi$
$181$	−2.20059	−0.163569	−0.0817843	+	0.996650i	$0.526062\pi$
$182$	−0.364796	−0.0270405
$183$	−1.88433	−0.139294
$184$	1.09785	0.0809348
$185$	19.5748	1.43917
$186$	17.9753	1.31801
$187$	0	0
$188$	0.283660	0.0206880
$189$	−8.94769	−0.650849
$190$	14.1980	1.03003
$191$	26.5960	1.92442	0.962210	−	0.272309i	$-0.0877874\pi$
$191$	26.5960	1.92442	0.962210	+	0.272309i	$0.0877874\pi$
$192$	−15.5138	−1.11961
$193$	12.2828	0.884137	0.442068	−	0.896981i	$-0.354245\pi$
$193$	12.2828	0.884137	0.442068	+	0.896981i	$0.354245\pi$
$194$	−6.11871	−0.439297
$195$	−0.818330	−0.0586018
$196$	0.201592	0.0143994
$197$	10.7909	0.768823	0.384411	−	0.923162i	$-0.374404\pi$
$197$	10.7909	0.768823	0.384411	+	0.923162i	$0.374404\pi$
$198$	0	0
$199$	20.2530	1.43569	0.717847	−	0.696201i	$-0.245128\pi$
$199$	20.2530	1.43569	0.717847	+	0.696201i	$0.245128\pi$
$200$	−15.2786	−1.08036
$201$	23.7933	1.67825
$202$	−0.133626	−0.00940188
$203$	13.3598	0.937674
$204$	−0.246910	−0.0172872
$205$	−12.3494	−0.862517
$206$	−0.898616	−0.0626095
$207$	−0.210606	−0.0146381
$208$	−0.523348	−0.0362877
$209$	0	0
$210$	16.3385	1.12746
$211$	−11.7789	−0.810890	−0.405445	−	0.914119i	$-0.632883\pi$
$211$	−11.7789	−0.810890	−0.405445	+	0.914119i	$0.632883\pi$
$212$	−0.322366	−0.0221402
$213$	−23.0459	−1.57908
$214$	2.61606	0.178830
$215$	41.5570	2.83416
$216$	−13.2490	−0.901482
$217$	13.2854	0.901873
$218$	10.2137	0.691760
$219$	−10.1882	−0.688458
$220$	0	0
$221$	−0.284793	−0.0191573
$222$	−15.9816	−1.07262
$223$	8.91459	0.596965	0.298483	−	0.954415i	$-0.403519\pi$
$223$	8.91459	0.596965	0.298483	+	0.954415i	$0.403519\pi$
$224$	−0.681810	−0.0455553
$225$	2.93095	0.195397
$226$	−0.534885	−0.0355800
$227$	−0.187933	−0.0124736	−0.00623679	−	0.999981i	$-0.501985\pi$
$227$	−0.187933	−0.0124736	−0.00623679	+	0.999981i	$0.501985\pi$
$228$	0.372030	0.0246383
$229$	4.65905	0.307878	0.153939	−	0.988080i	$-0.450804\pi$
$229$	4.65905	0.307878	0.153939	+	0.988080i	$0.450804\pi$
$230$	−1.71042	−0.112782
$231$	0	0
$232$	19.7821	1.29876
$233$	−10.2655	−0.672516	−0.336258	−	0.941770i	$-0.609161\pi$
$233$	−10.2655	−0.672516	−0.336258	+	0.941770i	$0.609161\pi$
$234$	0.103621	0.00677394
$235$	−14.6538	−0.955908
$236$	0.291832	0.0189966
$237$	−21.6389	−1.40560
$238$	5.68608	0.368574
$239$	9.40716	0.608498	0.304249	−	0.952592i	$-0.401594\pi$
$239$	9.40716	0.608498	0.304249	+	0.952592i	$0.401594\pi$
$240$	23.4397	1.51302
$241$	−7.87335	−0.507167	−0.253584	−	0.967313i	$-0.581609\pi$
$241$	−7.87335	−0.507167	−0.253584	+	0.967313i	$0.581609\pi$
$242$	0	0
$243$	5.65468	0.362748
$244$	−0.0621922	−0.00398145
$245$	−10.4142	−0.665337
$246$	10.0825	0.642837
$247$	0.429110	0.0273036
$248$	19.6720	1.24917
$249$	−28.0249	−1.77601
$250$	1.44142	0.0911634
$251$	28.7097	1.81214	0.906070	−	0.423128i	$-0.139068\pi$
$251$	28.7097	1.81214	0.906070	+	0.423128i	$0.139068\pi$
$252$	0.0663985	0.00418271
$253$	0	0
$254$	0.114550	0.00718750
$255$	12.7553	0.798768
$256$	−1.49115	−0.0931972
$257$	19.7863	1.23423	0.617117	−	0.786872i	$-0.288301\pi$
$257$	19.7863	1.23423	0.617117	+	0.786872i	$0.288301\pi$
$258$	−33.9287	−2.11231
$259$	−11.8119	−0.733957
$260$	−0.0270089	−0.00167502
$261$	−3.79489	−0.234898
$262$	−7.27454	−0.449422
$263$	−0.662567	−0.0408556	−0.0204278	−	0.999791i	$-0.506503\pi$
$263$	−0.662567	−0.0408556	−0.0204278	+	0.999791i	$0.506503\pi$
$264$	0	0
$265$	16.6533	1.02301
$266$	−8.56745	−0.525304
$267$	11.6706	0.714230
$268$	0.785297	0.0479697
$269$	−4.36061	−0.265871	−0.132936	−	0.991125i	$-0.542440\pi$
$269$	−4.36061	−0.265871	−0.132936	+	0.991125i	$0.542440\pi$
$270$	20.6416	1.25621
$271$	−8.38075	−0.509094	−0.254547	−	0.967060i	$-0.581926\pi$
$271$	−8.38075	−0.509094	−0.254547	+	0.967060i	$0.581926\pi$
$272$	8.15743	0.494617
$273$	0.493801	0.0298862
$274$	8.74620	0.528377
$275$	0	0
$276$	−0.0448180	−0.00269773
$277$	26.4623	1.58997	0.794983	−	0.606632i	$-0.207480\pi$
$277$	26.4623	1.58997	0.794983	+	0.606632i	$0.207480\pi$
$278$	18.8530	1.13073
$279$	−3.77376	−0.225929
$280$	17.8807	1.06857
$281$	−7.84803	−0.468174	−0.234087	−	0.972216i	$-0.575210\pi$
$281$	−7.84803	−0.468174	−0.234087	+	0.972216i	$0.575210\pi$
$282$	11.9639	0.712442
$283$	−26.7128	−1.58791	−0.793956	−	0.607975i	$-0.791982\pi$
$283$	−26.7128	−1.58791	−0.793956	+	0.607975i	$0.791982\pi$
$284$	−0.760629	−0.0451350
$285$	−19.2190	−1.13843
$286$	0	0
$287$	7.45192	0.439873
$288$	0.193670	0.0114121
$289$	−12.5609	−0.738878
$290$	−30.8200	−1.80981
$291$	8.28249	0.485528
$292$	−0.336263	−0.0196783
$293$	2.89786	0.169295	0.0846475	−	0.996411i	$-0.473024\pi$
$293$	2.89786	0.169295	0.0846475	+	0.996411i	$0.473024\pi$
$294$	8.50254	0.495878
$295$	−15.0759	−0.877755
$296$	−17.4902	−1.01660
$297$	0	0
$298$	7.62733	0.441839
$299$	−0.0516944	−0.00298956
$300$	0.623722	0.0360106
$301$	−25.0765	−1.44539
$302$	11.8460	0.681661
$303$	0.180881	0.0103913
$304$	−12.2911	−0.704945
$305$	3.21283	0.183966
$306$	−1.61515	−0.0923318
$307$	33.9188	1.93585	0.967924	−	0.251244i	$-0.0808397\pi$
$307$	33.9188	1.93585	0.967924	+	0.251244i	$0.0808397\pi$
$308$	0	0
$309$	1.21640	0.0691984
$310$	−30.6484	−1.74071
$311$	1.71246	0.0971047	0.0485523	−	0.998821i	$-0.484539\pi$
$311$	1.71246	0.0971047	0.0485523	+	0.998821i	$0.484539\pi$
$312$	0.731181	0.0413950
$313$	−27.0869	−1.53104	−0.765522	−	0.643410i	$-0.777519\pi$
$313$	−27.0869	−1.53104	−0.765522	+	0.643410i	$0.777519\pi$
$314$	4.05501	0.228838
$315$	−3.43012	−0.193266
$316$	−0.714193	−0.0401765
$317$	6.50697	0.365468	0.182734	−	0.983162i	$-0.441505\pi$
$317$	6.50697	0.365468	0.182734	+	0.983162i	$0.441505\pi$
$318$	−13.5964	−0.762450
$319$	0	0
$320$	26.4514	1.47868
$321$	−3.54118	−0.197649
$322$	1.03211	0.0575173
$323$	−6.68854	−0.372160
$324$	0.643616	0.0357564
$325$	0.719419	0.0399062
$326$	−9.69493	−0.536952
$327$	−13.8256	−0.764559
$328$	11.0342	0.609262
$329$	8.84246	0.487501
$330$	0	0
$331$	−31.3390	−1.72255	−0.861274	−	0.508142i	$-0.830333\pi$
$331$	−31.3390	−1.72255	−0.861274	+	0.508142i	$0.830333\pi$
$332$	−0.924962	−0.0507639
$333$	3.35521	0.183864
$334$	18.9640	1.03766
$335$	−40.5682	−2.21648
$336$	−14.1441	−0.771623
$337$	6.24645	0.340266	0.170133	−	0.985421i	$-0.445580\pi$
$337$	6.24645	0.340266	0.170133	+	0.985421i	$0.445580\pi$
$338$	−18.0712	−0.982946
$339$	0.724039	0.0393244
$340$	0.420988	0.0228313
$341$	0	0
$342$	2.43361	0.131595
$343$	19.8551	1.07207
$344$	−37.1313	−2.00199
$345$	2.31529	0.124651
$346$	−0.638880	−0.0343464
$347$	−20.6958	−1.11101	−0.555504	−	0.831514i	$-0.687475\pi$
$347$	−20.6958	−1.11101	−0.555504	+	0.831514i	$0.687475\pi$
$348$	−0.807572	−0.0432904
$349$	−10.6704	−0.571172	−0.285586	−	0.958353i	$-0.592188\pi$
$349$	−10.6704	−0.571172	−0.285586	+	0.958353i	$0.592188\pi$
$350$	−14.3637	−0.767770
$351$	0.623854	0.0332989
$352$	0	0
$353$	−18.7618	−0.998588	−0.499294	−	0.866433i	$-0.666407\pi$
$353$	−18.7618	−0.998588	−0.499294	+	0.866433i	$0.666407\pi$
$354$	12.3086	0.654194
$355$	39.2939	2.08550
$356$	0.385189	0.0204150
$357$	−7.69687	−0.407362
$358$	21.4157	1.13186
$359$	21.9489	1.15842	0.579208	−	0.815180i	$-0.303362\pi$
$359$	21.9489	1.15842	0.579208	+	0.815180i	$0.303362\pi$
$360$	−5.07906	−0.267690
$361$	−8.92210	−0.469584
$362$	−3.06333	−0.161005
$363$	0	0
$364$	0.0162979	0.000854241 0
$365$	17.3712	0.909252
$366$	−2.62308	−0.137111
$367$	−13.2386	−0.691049	−0.345524	−	0.938410i	$-0.612299\pi$
$367$	−13.2386	−0.691049	−0.345524	+	0.938410i	$0.612299\pi$
$368$	1.48070	0.0771868
$369$	−2.11674	−0.110193
$370$	27.2491	1.41662
$371$	−10.0490	−0.521720
$372$	−0.803076	−0.0416376
$373$	−29.1523	−1.50945	−0.754725	−	0.656042i	$-0.772230\pi$
$373$	−29.1523	−1.50945	−0.754725	+	0.656042i	$0.772230\pi$
$374$	0	0
$375$	−1.95116	−0.100757
$376$	13.0932	0.675231
$377$	−0.931477	−0.0479735
$378$	−12.4556	−0.640649
$379$	0.438425	0.0225204	0.0112602	−	0.999937i	$-0.496416\pi$
$379$	0.438425	0.0225204	0.0112602	+	0.999937i	$0.496416\pi$
$380$	−0.634321	−0.0325400
$381$	−0.155059	−0.00794390
$382$	37.0230	1.89426
$383$	19.4449	0.993589	0.496794	−	0.867868i	$-0.334510\pi$
$383$	19.4449	0.993589	0.496794	+	0.867868i	$0.334510\pi$
$384$	−20.2706	−1.03443
$385$	0	0
$386$	17.0983	0.870282
$387$	7.12306	0.362085
$388$	0.273363	0.0138779
$389$	−26.9677	−1.36732	−0.683659	−	0.729801i	$-0.739613\pi$
$389$	−26.9677	−1.36732	−0.683659	+	0.729801i	$0.739613\pi$
$390$	−1.13916	−0.0576835
$391$	0.805761	0.0407491
$392$	9.30510	0.469979
$393$	9.84706	0.496719
$394$	15.0215	0.756774
$395$	36.8950	1.85639
$396$	0	0
$397$	20.8016	1.04400	0.522002	−	0.852944i	$-0.325185\pi$
$397$	20.8016	1.04400	0.522002	+	0.852944i	$0.325185\pi$
$398$	28.1932	1.41320
$399$	11.5972	0.580586
$400$	−20.6066	−1.03033
$401$	6.34061	0.316635	0.158317	−	0.987388i	$-0.449393\pi$
$401$	6.34061	0.316635	0.158317	+	0.987388i	$0.449393\pi$
$402$	33.1215	1.65195
$403$	−0.926292	−0.0461419
$404$	0.00596996	0.000297016 0
$405$	−33.2490	−1.65216
$406$	18.5975	0.922980
$407$	0	0
$408$	−11.3969	−0.564231
$409$	23.9243	1.18298	0.591491	−	0.806312i	$-0.298540\pi$
$409$	23.9243	1.18298	0.591491	+	0.806312i	$0.298540\pi$
$410$	−17.1910	−0.849001
$411$	−11.8392	−0.583983
$412$	0.0401472	0.00197791
$413$	9.09720	0.447644
$414$	−0.293174	−0.0144087
$415$	47.7833	2.34559
$416$	0.0475374	0.00233071
$417$	−25.5200	−1.24972
$418$	0	0
$419$	2.03310	0.0993232	0.0496616	−	0.998766i	$-0.484186\pi$
$419$	2.03310	0.0993232	0.0496616	+	0.998766i	$0.484186\pi$
$420$	−0.729948	−0.0356178
$421$	20.8971	1.01846	0.509230	−	0.860630i	$-0.329930\pi$
$421$	20.8971	1.01846	0.509230	+	0.860630i	$0.329930\pi$
$422$	−16.3968	−0.798183
$423$	−2.51173	−0.122124
$424$	−14.8798	−0.722628
$425$	−11.2136	−0.543939
$426$	−32.0810	−1.55433
$427$	−1.93870	−0.0938204
$428$	−0.116877	−0.00564944
$429$	0	0
$430$	57.8494	2.78975
$431$	8.18955	0.394477	0.197238	−	0.980356i	$-0.436803\pi$
$431$	8.18955	0.394477	0.197238	+	0.980356i	$0.436803\pi$
$432$	−17.8693	−0.859735
$433$	23.6883	1.13839	0.569193	−	0.822204i	$-0.307256\pi$
$433$	23.6883	1.13839	0.569193	+	0.822204i	$0.307256\pi$
$434$	18.4940	0.887740
$435$	41.7189	2.00027
$436$	−0.456314	−0.0218535
$437$	−1.21407	−0.0580770
$438$	−14.1826	−0.677669
$439$	40.5030	1.93310	0.966550	−	0.256477i	$-0.0825619\pi$
$439$	40.5030	1.93310	0.966550	+	0.256477i	$0.0825619\pi$
$440$	0	0
$441$	−1.78504	−0.0850018
$442$	−0.396447	−0.0188571
$443$	20.7139	0.984146	0.492073	−	0.870554i	$-0.336239\pi$
$443$	20.7139	0.984146	0.492073	+	0.870554i	$0.336239\pi$
$444$	0.714006	0.0338852
$445$	−19.8987	−0.943290
$446$	12.4096	0.587610
$447$	−10.3246	−0.488337
$448$	−15.9615	−0.754108
$449$	30.6638	1.44711	0.723556	−	0.690265i	$-0.242506\pi$
$449$	30.6638	1.44711	0.723556	+	0.690265i	$0.242506\pi$
$450$	4.08004	0.192335
$451$	0	0
$452$	0.0238969	0.00112401
$453$	−16.0351	−0.753397
$454$	−0.261613	−0.0122781
$455$	−0.841944	−0.0394709
$456$	17.1722	0.804162
$457$	−9.04240	−0.422986	−0.211493	−	0.977380i	$-0.567833\pi$
$457$	−9.04240	−0.422986	−0.211493	+	0.977380i	$0.567833\pi$
$458$	6.48563	0.303054
$459$	−9.72402	−0.453878
$460$	0.0764160	0.00356291
$461$	6.88267	0.320558	0.160279	−	0.987072i	$-0.448761\pi$
$461$	6.88267	0.320558	0.160279	+	0.987072i	$0.448761\pi$
$462$	0	0
$463$	23.5581	1.09484	0.547420	−	0.836858i	$-0.315610\pi$
$463$	23.5581	1.09484	0.547420	+	0.836858i	$0.315610\pi$
$464$	26.6806	1.23862
$465$	41.4867	1.92390
$466$	−14.2901	−0.661977
$467$	−10.3774	−0.480208	−0.240104	−	0.970747i	$-0.577182\pi$
$467$	−10.3774	−0.480208	−0.240104	+	0.970747i	$0.577182\pi$
$468$	−0.00462946	−0.000213997 0
$469$	24.4799	1.13038
$470$	−20.3988	−0.940928
$471$	−5.48900	−0.252920
$472$	13.4704	0.620026
$473$	0	0
$474$	−30.1225	−1.38357
$475$	16.8960	0.775241
$476$	−0.254035	−0.0116437
$477$	2.85446	0.130697
$478$	13.0952	0.598963
$479$	−5.76519	−0.263418	−0.131709	−	0.991288i	$-0.542046\pi$
$479$	−5.76519	−0.263418	−0.131709	+	0.991288i	$0.542046\pi$
$480$	−2.12910	−0.0971798
$481$	0.823556	0.0375509
$482$	−10.9601	−0.499220
$483$	−1.39710	−0.0635703
$484$	0	0
$485$	−14.1219	−0.641241
$486$	7.87161	0.357064
$487$	26.8157	1.21513	0.607567	−	0.794268i	$-0.292145\pi$
$487$	26.8157	1.21513	0.607567	+	0.794268i	$0.292145\pi$
$488$	−2.87068	−0.129949
$489$	13.1234	0.593460
$490$	−14.4971	−0.654911
$491$	−5.31262	−0.239755	−0.119878	−	0.992789i	$-0.538250\pi$
$491$	−5.31262	−0.239755	−0.119878	+	0.992789i	$0.538250\pi$
$492$	−0.450453	−0.0203080
$493$	14.5189	0.653900
$494$	0.597343	0.0268757
$495$	0	0
$496$	26.5321	1.19133
$497$	−23.7109	−1.06358
$498$	−39.0122	−1.74818
$499$	−20.7949	−0.930908	−0.465454	−	0.885072i	$-0.654109\pi$
$499$	−20.7949	−0.930908	−0.465454	+	0.885072i	$0.654109\pi$
$500$	−0.0643978	−0.00287996
$501$	−25.6703	−1.14686
$502$	39.9654	1.78374
$503$	37.5699	1.67516	0.837580	−	0.546315i	$-0.183970\pi$
$503$	37.5699	1.67516	0.837580	+	0.546315i	$0.183970\pi$
$504$	3.06483	0.136518
$505$	−0.308406	−0.0137239
$506$	0	0
$507$	24.4618	1.08639
$508$	−0.00511771	−0.000227062 0
$509$	38.5380	1.70817	0.854084	−	0.520135i	$-0.174118\pi$
$509$	38.5380	1.70817	0.854084	+	0.520135i	$0.174118\pi$
$510$	17.7560	0.786251
$511$	−10.4822	−0.463707
$512$	−23.5907	−1.04257
$513$	14.6516	0.646883
$514$	27.5435	1.21489
$515$	−2.07399	−0.0913909
$516$	1.51582	0.0667304
$517$	0	0
$518$	−16.4428	−0.722456
$519$	0.864810	0.0379610
$520$	−1.24668	−0.0546707
$521$	−12.7519	−0.558671	−0.279335	−	0.960194i	$-0.590114\pi$
$521$	−12.7519	−0.558671	−0.279335	+	0.960194i	$0.590114\pi$
$522$	−5.28268	−0.231217
$523$	−31.3283	−1.36989	−0.684945	−	0.728595i	$-0.740174\pi$
$523$	−31.3283	−1.36989	−0.684945	+	0.728595i	$0.740174\pi$
$524$	0.325002	0.0141978
$525$	19.4431	0.848568
$526$	−0.922328	−0.0402154
$527$	14.4381	0.628934
$528$	0	0
$529$	−22.8537	−0.993641
$530$	23.1823	1.00697
$531$	−2.58409	−0.112140
$532$	0.382765	0.0165950
$533$	−0.519565	−0.0225049
$534$	16.2461	0.703038
$535$	6.03781	0.261037
$536$	36.2479	1.56567
$537$	−28.9890	−1.25097
$538$	−6.07019	−0.261705
$539$	0	0
$540$	−0.922197	−0.0396850
$541$	42.1196	1.81086	0.905431	−	0.424494i	$-0.139548\pi$
$541$	42.1196	1.81086	0.905431	+	0.424494i	$0.139548\pi$
$542$	−11.6664	−0.501116
$543$	4.14663	0.177949
$544$	−0.740966	−0.0317686
$545$	23.5731	1.00976
$546$	0.687396	0.0294178
$547$	36.6787	1.56827	0.784133	−	0.620593i	$-0.213108\pi$
$547$	36.6787	1.56827	0.784133	+	0.620593i	$0.213108\pi$
$548$	−0.390751	−0.0166921
$549$	0.550694	0.0235031
$550$	0	0
$551$	−21.8763	−0.931961
$552$	−2.06872	−0.0880504
$553$	−22.2634	−0.946735
$554$	36.8369	1.56505
$555$	−36.8854	−1.56570
$556$	−0.842288	−0.0357210
$557$	19.8394	0.840621	0.420311	−	0.907380i	$-0.361921\pi$
$557$	19.8394	0.840621	0.420311	+	0.907380i	$0.361921\pi$
$558$	−5.25327	−0.222389
$559$	1.74839	0.0739492
$560$	24.1161	1.01909
$561$	0	0
$562$	−10.9249	−0.460837
$563$	−0.472027	−0.0198936	−0.00994678	−	0.999951i	$-0.503166\pi$
$563$	−0.472027	−0.0198936	−0.00994678	+	0.999951i	$0.503166\pi$
$564$	−0.534509	−0.0225069
$565$	−1.23451	−0.0519360
$566$	−37.1856	−1.56303
$567$	20.0633	0.842579
$568$	−35.1092	−1.47315
$569$	26.4478	1.10875	0.554376	−	0.832267i	$-0.312957\pi$
$569$	26.4478	1.10875	0.554376	+	0.832267i	$0.312957\pi$
$570$	−26.7538	−1.12059
$571$	−1.20173	−0.0502910	−0.0251455	−	0.999684i	$-0.508005\pi$
$571$	−1.20173	−0.0502910	−0.0251455	+	0.999684i	$0.508005\pi$
$572$	0	0
$573$	−50.1156	−2.09361
$574$	10.3735	0.432979
$575$	−2.03544	−0.0848838
$576$	4.53390	0.188912
$577$	15.2465	0.634722	0.317361	−	0.948305i	$-0.397203\pi$
$577$	15.2465	0.634722	0.317361	+	0.948305i	$0.397203\pi$
$578$	−17.4855	−0.727299
$579$	−23.1449	−0.961868
$580$	1.37693	0.0571740
$581$	−28.8336	−1.19622
$582$	11.5297	0.477919
$583$	0	0
$584$	−15.5213	−0.642275
$585$	0.239157	0.00988790
$586$	4.03398	0.166642
$587$	−11.2350	−0.463717	−0.231859	−	0.972750i	$-0.574481\pi$
$587$	−11.2350	−0.463717	−0.231859	+	0.972750i	$0.574481\pi$
$588$	−0.379865	−0.0156654
$589$	−21.7545	−0.896379
$590$	−20.9865	−0.864000
$591$	−20.3337	−0.836416
$592$	−23.5894	−0.969517
$593$	−10.1471	−0.416690	−0.208345	−	0.978055i	$-0.566808\pi$
$593$	−10.1471	−0.416690	−0.208345	+	0.978055i	$0.566808\pi$
$594$	0	0
$595$	13.1234	0.538006
$596$	−0.340763	−0.0139582
$597$	−38.1632	−1.56192
$598$	−0.0719613	−0.00294272
$599$	−34.1529	−1.39545	−0.697724	−	0.716367i	$-0.745804\pi$
$599$	−34.1529	−1.39545	−0.697724	+	0.716367i	$0.745804\pi$
$600$	28.7899	1.17534
$601$	25.2278	1.02906	0.514532	−	0.857471i	$-0.327966\pi$
$601$	25.2278	1.02906	0.514532	+	0.857471i	$0.327966\pi$
$602$	−34.9078	−1.42274
$603$	−6.95358	−0.283172
$604$	−0.529240	−0.0215345
$605$	0	0
$606$	0.251795	0.0102285
$607$	−11.8743	−0.481962	−0.240981	−	0.970530i	$-0.577469\pi$
$607$	−11.8743	−0.481962	−0.240981	+	0.970530i	$0.577469\pi$
$608$	1.11644	0.0452778
$609$	−25.1743	−1.02011
$610$	4.47243	0.181083
$611$	−0.616518	−0.0249416
$612$	0.0721594	0.00291687
$613$	27.4436	1.10844	0.554218	−	0.832372i	$-0.313018\pi$
$613$	27.4436	1.10844	0.554218	+	0.832372i	$0.313018\pi$
$614$	47.2167	1.90551
$615$	23.2703	0.938347
$616$	0	0
$617$	16.5315	0.665531	0.332766	−	0.943010i	$-0.392018\pi$
$617$	16.5315	0.665531	0.332766	+	0.943010i	$0.392018\pi$
$618$	1.69329	0.0681140
$619$	1.08267	0.0435162	0.0217581	−	0.999763i	$-0.493074\pi$
$619$	1.08267	0.0435162	0.0217581	+	0.999763i	$0.493074\pi$
$620$	1.36927	0.0549911
$621$	−1.76506	−0.0708294
$622$	2.38383	0.0955830
$623$	12.0074	0.481066
$624$	0.986160	0.0394780
$625$	−23.2847	−0.931387
$626$	−37.7064	−1.50705
$627$	0	0
$628$	−0.181164	−0.00722925
$629$	−12.8368	−0.511835
$630$	−4.77491	−0.190237
$631$	1.82472	0.0726411	0.0363205	−	0.999340i	$-0.488436\pi$
$631$	1.82472	0.0726411	0.0363205	+	0.999340i	$0.488436\pi$
$632$	−32.9658	−1.31131
$633$	22.1952	0.882182
$634$	9.05803	0.359740
$635$	0.264379	0.0104916
$636$	0.607443	0.0240867
$637$	−0.438147	−0.0173600
$638$	0	0
$639$	6.73515	0.266438
$640$	34.5620	1.36618
$641$	−42.6112	−1.68304	−0.841520	−	0.540226i	$-0.818339\pi$
$641$	−42.6112	−1.68304	−0.841520	+	0.540226i	$0.818339\pi$
$642$	−4.92951	−0.194552
$643$	−0.997904	−0.0393535	−0.0196767	−	0.999806i	$-0.506264\pi$
$643$	−0.997904	−0.0393535	−0.0196767	+	0.999806i	$0.506264\pi$
$644$	−0.0461113	−0.00181704
$645$	−78.3070	−3.08333
$646$	−9.31079	−0.366328
$647$	22.7881	0.895892	0.447946	−	0.894061i	$-0.352156\pi$
$647$	22.7881	0.895892	0.447946	+	0.894061i	$0.352156\pi$
$648$	29.7081	1.16705
$649$	0	0
$650$	1.00147	0.0392808
$651$	−25.0341	−0.981164
$652$	0.433137	0.0169630
$653$	−15.1312	−0.592128	−0.296064	−	0.955168i	$-0.595674\pi$
$653$	−15.1312	−0.592128	−0.296064	+	0.955168i	$0.595674\pi$
$654$	−19.2460	−0.752578
$655$	−16.7895	−0.656021
$656$	14.8821	0.581048
$657$	2.97751	0.116164
$658$	12.3092	0.479861
$659$	−2.14024	−0.0833719	−0.0416860	−	0.999131i	$-0.513273\pi$
$659$	−2.14024	−0.0833719	−0.0416860	+	0.999131i	$0.513273\pi$
$660$	0	0
$661$	−36.0987	−1.40408	−0.702039	−	0.712139i	$-0.747727\pi$
$661$	−36.0987	−1.40408	−0.702039	+	0.712139i	$0.747727\pi$
$662$	−43.6255	−1.69555
$663$	0.536644	0.0208415
$664$	−42.6945	−1.65687
$665$	−19.7735	−0.766785
$666$	4.67063	0.180983
$667$	2.63541	0.102044
$668$	−0.847247	−0.0327810
$669$	−16.7980	−0.649449
$670$	−56.4731	−2.18175
$671$	0	0
$672$	1.28475	0.0495604
$673$	5.83488	0.224918	0.112459	−	0.993656i	$-0.464127\pi$
$673$	5.83488	0.224918	0.112459	+	0.993656i	$0.464127\pi$
$674$	8.69538	0.334934
$675$	24.5639	0.945467
$676$	0.807363	0.0310524
$677$	−35.5652	−1.36688	−0.683441	−	0.730006i	$-0.739517\pi$
$677$	−35.5652	−1.36688	−0.683441	+	0.730006i	$0.739517\pi$
$678$	1.00790	0.0387081
$679$	8.52149	0.327025
$680$	19.4321	0.745185
$681$	0.354128	0.0135702
$682$	0	0
$683$	−33.1268	−1.26756	−0.633781	−	0.773512i	$-0.718498\pi$
$683$	−33.1268	−1.26756	−0.633781	+	0.773512i	$0.718498\pi$
$684$	−0.108726	−0.00415723
$685$	20.1861	0.771271
$686$	27.6393	1.05527
$687$	−8.77918	−0.334946
$688$	−50.0798	−1.90928
$689$	0.700643	0.0266924
$690$	3.22300	0.122697
$691$	3.02838	0.115205	0.0576025	−	0.998340i	$-0.481654\pi$
$691$	3.02838	0.115205	0.0576025	+	0.998340i	$0.481654\pi$
$692$	0.0285430	0.00108504
$693$	0	0
$694$	−28.8096	−1.09360
$695$	43.5123	1.65052
$696$	−37.2760	−1.41294
$697$	8.09846	0.306751
$698$	−14.8537	−0.562221
$699$	19.3436	0.731642
$700$	0.641721	0.0242548
$701$	−46.5106	−1.75668	−0.878340	−	0.478036i	$-0.841349\pi$
$701$	−46.5106	−1.75668	−0.878340	+	0.478036i	$0.841349\pi$
$702$	0.868438	0.0327771
$703$	19.3417	0.729486
$704$	0	0
$705$	27.6126	1.03995
$706$	−26.1173	−0.982939
$707$	0.186100	0.00699901
$708$	−0.549907	−0.0206668
$709$	25.1723	0.945367	0.472684	−	0.881232i	$-0.343285\pi$
$709$	25.1723	0.945367	0.472684	+	0.881232i	$0.343285\pi$
$710$	54.6991	2.05282
$711$	6.32397	0.237168
$712$	17.7796	0.666318
$713$	2.62074	0.0981475
$714$	−10.7144	−0.400978
$715$	0	0
$716$	−0.956782	−0.0357566
$717$	−17.7262	−0.661996
$718$	30.5540	1.14026
$719$	48.2035	1.79769	0.898844	−	0.438269i	$-0.144408\pi$
$719$	48.2035	1.79769	0.898844	+	0.438269i	$0.144408\pi$
$720$	−6.85024	−0.255293
$721$	1.25150	0.0466082
$722$	−12.4200	−0.462225
$723$	14.8360	0.551756
$724$	0.136860	0.00508634
$725$	−36.6764	−1.36213
$726$	0	0
$727$	−41.1988	−1.52798	−0.763989	−	0.645229i	$-0.776762\pi$
$727$	−41.1988	−1.52798	−0.763989	+	0.645229i	$0.776762\pi$
$728$	0.752280	0.0278813
$729$	20.3912	0.755229
$730$	24.1817	0.895004
$731$	−27.2522	−1.00796
$732$	0.117191	0.00433149
$733$	−21.6584	−0.799972	−0.399986	−	0.916521i	$-0.630985\pi$
$733$	−21.6584	−0.799972	−0.399986	+	0.916521i	$0.630985\pi$
$734$	−18.4288	−0.680219
$735$	19.6237	0.723832
$736$	−0.134497	−0.00495762
$737$	0	0
$738$	−2.94661	−0.108466
$739$	14.4628	0.532025	0.266012	−	0.963970i	$-0.414294\pi$
$739$	14.4628	0.532025	0.266012	+	0.963970i	$0.414294\pi$
$740$	−1.21740	−0.0447525
$741$	−0.808584	−0.0297041
$742$	−13.9888	−0.513544
$743$	−13.8176	−0.506919	−0.253460	−	0.967346i	$-0.581568\pi$
$743$	−13.8176	−0.506919	−0.253460	+	0.967346i	$0.581568\pi$
$744$	−37.0685	−1.35900
$745$	17.6037	0.644951
$746$	−40.5815	−1.48579
$747$	8.19028	0.299666
$748$	0	0
$749$	−3.64337	−0.133126
$750$	−2.71611	−0.0991783
$751$	1.11653	0.0407428	0.0203714	−	0.999792i	$-0.493515\pi$
$751$	1.11653	0.0407428	0.0203714	+	0.999792i	$0.493515\pi$
$752$	17.6591	0.643962
$753$	−54.0985	−1.97146
$754$	−1.29666	−0.0472217
$755$	27.3404	0.995018
$756$	0.556477	0.0202389
$757$	−3.03140	−0.110178	−0.0550891	−	0.998481i	$-0.517544\pi$
$757$	−3.03140	−0.110178	−0.0550891	+	0.998481i	$0.517544\pi$
$758$	0.610310	0.0221675
$759$	0	0
$760$	−29.2791	−1.06206
$761$	44.7062	1.62060	0.810300	−	0.586016i	$-0.199304\pi$
$761$	44.7062	1.62060	0.810300	+	0.586016i	$0.199304\pi$
$762$	−0.215850	−0.00781941
$763$	−14.2246	−0.514964
$764$	−1.65406	−0.0598420
$765$	−3.72773	−0.134776
$766$	27.0683	0.978019
$767$	−0.634279	−0.0229025
$768$	2.80983	0.101391
$769$	−0.666530	−0.0240357	−0.0120179	−	0.999928i	$-0.503825\pi$
$769$	−0.666530	−0.0240357	−0.0120179	+	0.999928i	$0.503825\pi$
$770$	0	0
$771$	−37.2838	−1.34274
$772$	−0.763896	−0.0274932
$773$	−23.9036	−0.859752	−0.429876	−	0.902888i	$-0.641443\pi$
$773$	−23.9036	−0.859752	−0.429876	+	0.902888i	$0.641443\pi$
$774$	9.91566	0.356411
$775$	−36.4723	−1.31012
$776$	12.6179	0.452958
$777$	22.2575	0.798485
$778$	−37.5405	−1.34589
$779$	−12.2023	−0.437193
$780$	0.0508937	0.00182229
$781$	0	0
$782$	1.12166	0.0401105
$783$	−31.8045	−1.13660
$784$	12.5500	0.448214
$785$	9.35890	0.334034
$786$	13.7076	0.488935
$787$	28.8138	1.02710	0.513550	−	0.858060i	$-0.328330\pi$
$787$	28.8138	1.02710	0.513550	+	0.858060i	$0.328330\pi$
$788$	−0.671112	−0.0239074
$789$	1.24849	0.0444476
$790$	51.3597	1.82730
$791$	0.744932	0.0264867
$792$	0	0
$793$	0.135171	0.00480006
$794$	28.9570	1.02764
$795$	−31.3803	−1.11295
$796$	−1.25958	−0.0446445
$797$	21.9840	0.778712	0.389356	−	0.921087i	$-0.372698\pi$
$797$	21.9840	0.778712	0.389356	+	0.921087i	$0.372698\pi$
$798$	16.1439	0.571488
$799$	9.60966	0.339965
$800$	1.87176	0.0661767
$801$	−3.41073	−0.120512
$802$	8.82645	0.311673
$803$	0	0
$804$	−1.47976	−0.0521870
$805$	2.38210	0.0839579
$806$	−1.28945	−0.0454188
$807$	8.21682	0.289246
$808$	0.275562	0.00969424
$809$	−27.5492	−0.968578	−0.484289	−	0.874908i	$-0.660922\pi$
$809$	−27.5492	−0.968578	−0.484289	+	0.874908i	$0.660922\pi$
$810$	−46.2843	−1.62627
$811$	−30.9368	−1.08634	−0.543169	−	0.839624i	$-0.682776\pi$
$811$	−30.9368	−1.08634	−0.543169	+	0.839624i	$0.682776\pi$
$812$	−0.830876	−0.0291580
$813$	15.7921	0.553852
$814$	0	0
$815$	−22.3757	−0.783788
$816$	−15.3713	−0.538102
$817$	41.0621	1.43658
$818$	33.3039	1.16444
$819$	−0.144313	−0.00504270
$820$	0.768034	0.0268209
$821$	18.4096	0.642501	0.321250	−	0.946994i	$-0.395897\pi$
$821$	18.4096	0.642501	0.321250	+	0.946994i	$0.395897\pi$
$822$	−16.4807	−0.574831
$823$	−46.2931	−1.61368	−0.806838	−	0.590773i	$-0.798823\pi$
$823$	−46.2931	−1.61368	−0.806838	+	0.590773i	$0.798823\pi$
$824$	1.85312	0.0645565
$825$	0	0
$826$	12.6638	0.440629
$827$	−2.39708	−0.0833546	−0.0416773	−	0.999131i	$-0.513270\pi$
$827$	−2.39708	−0.0833546	−0.0416773	+	0.999131i	$0.513270\pi$
$828$	0.0130980	0.000455189 0
$829$	−0.994747	−0.0345490	−0.0172745	−	0.999851i	$-0.505499\pi$
$829$	−0.994747	−0.0345490	−0.0172745	+	0.999851i	$0.505499\pi$
$830$	66.5168	2.30883
$831$	−49.8637	−1.72975
$832$	1.11287	0.0385819
$833$	6.82940	0.236625
$834$	−35.5252	−1.23014
$835$	43.7685	1.51467
$836$	0	0
$837$	−31.6274	−1.09320
$838$	2.83017	0.0977667
$839$	−41.9093	−1.44687	−0.723434	−	0.690393i	$-0.757437\pi$
$839$	−41.9093	−1.44687	−0.723434	+	0.690393i	$0.757437\pi$
$840$	−33.6931	−1.16252
$841$	18.4873	0.637492
$842$	29.0898	1.00250
$843$	14.7883	0.509335
$844$	0.732553	0.0252155
$845$	−41.7081	−1.43480
$846$	−3.49645	−0.120211
$847$	0	0
$848$	−20.0687	−0.689163
$849$	50.3357	1.72752
$850$	−15.6099	−0.535415
$851$	−2.33007	−0.0798738
$852$	1.43327	0.0491032
$853$	−53.6835	−1.83809	−0.919044	−	0.394154i	$-0.871038\pi$
$853$	−53.6835	−1.83809	−0.919044	+	0.394154i	$0.871038\pi$
$854$	−2.69878	−0.0923502
$855$	5.61673	0.192088
$856$	−5.39481	−0.184391
$857$	39.7268	1.35704	0.678521	−	0.734581i	$-0.262621\pi$
$857$	39.7268	1.35704	0.678521	+	0.734581i	$0.262621\pi$
$858$	0	0
$859$	31.6246	1.07902	0.539508	−	0.841980i	$-0.318610\pi$
$859$	31.6246	1.07902	0.539508	+	0.841980i	$0.318610\pi$
$860$	−2.58452	−0.0881314
$861$	−14.0419	−0.478545
$862$	11.4003	0.388295
$863$	−10.2697	−0.349585	−0.174792	−	0.984605i	$-0.555925\pi$
$863$	−10.2697	−0.349585	−0.174792	+	0.984605i	$0.555925\pi$
$864$	1.62312	0.0552198
$865$	−1.47453	−0.0501354
$866$	32.9753	1.12055
$867$	23.6689	0.803838
$868$	−0.826250	−0.0280448
$869$	0	0
$870$	58.0749	1.96892
$871$	−1.70680	−0.0578326
$872$	−21.0626	−0.713271
$873$	−2.42055	−0.0819233
$874$	−1.69005	−0.0571669
$875$	−2.00746	−0.0678645
$876$	0.633630	0.0214084
$877$	32.2242	1.08813	0.544067	−	0.839042i	$-0.316884\pi$
$877$	32.2242	1.08813	0.544067	+	0.839042i	$0.316884\pi$
$878$	56.3822	1.90281
$879$	−5.46053	−0.184179
$880$	0	0
$881$	10.7453	0.362017	0.181008	−	0.983482i	$-0.442064\pi$
$881$	10.7453	0.362017	0.181008	+	0.983482i	$0.442064\pi$
$882$	−2.48486	−0.0836697
$883$	−49.8434	−1.67736	−0.838682	−	0.544622i	$-0.816673\pi$
$883$	−49.8434	−1.67736	−0.838682	+	0.544622i	$0.816673\pi$
$884$	0.0177119	0.000595717 0
$885$	28.4080	0.954926
$886$	28.8348	0.968723
$887$	−3.61128	−0.121255	−0.0606275	−	0.998160i	$-0.519310\pi$
$887$	−3.61128	−0.121255	−0.0606275	+	0.998160i	$0.519310\pi$
$888$	32.9572	1.10597
$889$	−0.159533	−0.00535057
$890$	−27.7001	−0.928508
$891$	0	0
$892$	−0.554418	−0.0185633
$893$	−14.4793	−0.484531
$894$	−14.3724	−0.480685
$895$	49.4271	1.65217
$896$	−20.8555	−0.696735
$897$	0.0974093	0.00325240
$898$	42.6855	1.42443
$899$	47.2229	1.57497
$900$	−0.182283	−0.00607609
$901$	−10.9209	−0.363829
$902$	0	0
$903$	47.2524	1.57246
$904$	1.10304	0.0366864
$905$	−7.07012	−0.235019
$906$	−22.3218	−0.741591
$907$	41.2558	1.36988	0.684939	−	0.728601i	$-0.259829\pi$
$907$	41.2558	1.36988	0.684939	+	0.728601i	$0.259829\pi$
$908$	0.0116880	0.000387880 0
$909$	−0.0528623	−0.00175333
$910$	−1.17203	−0.0388524
$911$	−6.65208	−0.220393	−0.110197	−	0.993910i	$-0.535148\pi$
$911$	−6.65208	−0.220393	−0.110197	+	0.993910i	$0.535148\pi$
$912$	23.1605	0.766922
$913$	0	0
$914$	−12.5875	−0.416357
$915$	−6.05403	−0.200140
$916$	−0.289756	−0.00957382
$917$	10.1312	0.334562
$918$	−13.5363	−0.446766
$919$	−52.1247	−1.71944	−0.859718	−	0.510769i	$-0.829361\pi$
$919$	−52.1247	−1.71944	−0.859718	+	0.510769i	$0.829361\pi$
$920$	3.52722	0.116289
$921$	−63.9141	−2.10604
$922$	9.58104	0.315535
$923$	1.65318	0.0544151
$924$	0	0
$925$	32.4271	1.06620
$926$	32.7942	1.07768
$927$	−0.355492	−0.0116759
$928$	−2.42349	−0.0795548
$929$	10.6691	0.350041	0.175021	−	0.984565i	$-0.444001\pi$
$929$	10.6691	0.350041	0.175021	+	0.984565i	$0.444001\pi$
$930$	57.7516	1.89375
$931$	−10.2901	−0.337246
$932$	0.638435	0.0209126
$933$	−3.22684	−0.105642
$934$	−14.4459	−0.472683
$935$	0	0
$936$	−0.213687	−0.00698459
$937$	−10.2877	−0.336084	−0.168042	−	0.985780i	$-0.553744\pi$
$937$	−10.2877	−0.336084	−0.168042	+	0.985780i	$0.553744\pi$
$938$	34.0773	1.11266
$939$	51.0407	1.66565
$940$	0.911352	0.0297250
$941$	−59.4431	−1.93779	−0.968894	−	0.247475i	$-0.920399\pi$
$941$	−59.4431	−1.93779	−0.968894	+	0.247475i	$0.920399\pi$
$942$	−7.64098	−0.248956
$943$	1.47000	0.0478697
$944$	18.1678	0.591313
$945$	−28.7474	−0.935154
$946$	0	0
$947$	5.71500	0.185713	0.0928563	−	0.995680i	$-0.470400\pi$
$947$	5.71500	0.185713	0.0928563	+	0.995680i	$0.470400\pi$
$948$	1.34577	0.0437087
$949$	0.730847	0.0237243
$950$	23.5201	0.763092
$951$	−12.2613	−0.397599
$952$	−11.7258	−0.380035
$953$	−3.81804	−0.123679	−0.0618393	−	0.998086i	$-0.519697\pi$
$953$	−3.81804	−0.123679	−0.0618393	+	0.998086i	$0.519697\pi$
$954$	3.97355	0.128649
$955$	85.4485	2.76505
$956$	−0.585052	−0.0189219
$957$	0	0
$958$	−8.02544	−0.259290
$959$	−12.1808	−0.393338
$960$	−49.8432	−1.60868
$961$	15.9600	0.514839
$962$	1.14643	0.0369625
$963$	1.03491	0.0333495
$964$	0.489661	0.0157709
$965$	39.4626	1.27035
$966$	−1.94484	−0.0625741
$967$	−6.79452	−0.218497	−0.109249	−	0.994014i	$-0.534844\pi$
$967$	−6.79452	−0.218497	−0.109249	+	0.994014i	$0.534844\pi$
$968$	0	0
$969$	12.6034	0.404880
$970$	−19.6584	−0.631192
$971$	−34.1222	−1.09503	−0.547516	−	0.836795i	$-0.684427\pi$
$971$	−34.1222	−1.09503	−0.547516	+	0.836795i	$0.684427\pi$
$972$	−0.351677	−0.0112801
$973$	−26.2564	−0.841743
$974$	37.3288	1.19609
$975$	−1.35562	−0.0434147
$976$	−3.87175	−0.123932
$977$	−4.22267	−0.135095	−0.0675476	−	0.997716i	$-0.521517\pi$
$977$	−4.22267	−0.135095	−0.0675476	+	0.997716i	$0.521517\pi$
$978$	18.2684	0.584160
$979$	0	0
$980$	0.647680	0.0206894
$981$	4.04053	0.129004
$982$	−7.39544	−0.235998
$983$	−39.9846	−1.27531	−0.637656	−	0.770322i	$-0.720096\pi$
$983$	−39.9846	−1.27531	−0.637656	+	0.770322i	$0.720096\pi$
$984$	−20.7921	−0.662827
$985$	34.6695	1.10466
$986$	20.2111	0.643653
$987$	−16.6621	−0.530361
$988$	−0.0266873	−0.000849036 0
$989$	−4.94670	−0.157296
$990$	0	0
$991$	7.35892	0.233764	0.116882	−	0.993146i	$-0.462710\pi$
$991$	7.35892	0.233764	0.116882	+	0.993146i	$0.462710\pi$
$992$	−2.40999	−0.0765174
$993$	59.0530	1.87399
$994$	−33.0068	−1.04691
$995$	65.0694	2.06284
$996$	1.74293	0.0552269
$997$	−56.3232	−1.78377	−0.891887	−	0.452259i	$-0.850618\pi$
$997$	−56.3232	−1.78377	−0.891887	+	0.452259i	$0.850618\pi$
$998$	−28.9476	−0.916320
$999$	28.1196	0.889664

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7381.2.a.r.1.34	yes	50
11.10	odd	2		7381.2.a.q.1.17	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7381.2.a.q.1.17	✓	50	11.10	odd	2
7381.2.a.r.1.34	yes	50	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:	\( N \)	\(=\)	\( 7381 = 11^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7381.a (trivial)

\(f(q)\)	\(=\)	\(q+1.39205 q^{2} -1.88433 q^{3} -0.0621922 q^{4} +3.21283 q^{5} -2.62308 q^{6} -1.93870 q^{7} -2.87068 q^{8} +0.550694 q^{9} +O(q^{10})\)	\(q+1.39205 q^{2} -1.88433 q^{3} -0.0621922 q^{4} +3.21283 q^{5} -2.62308 q^{6} -1.93870 q^{7} -2.87068 q^{8} +0.550694 q^{9} +4.47243 q^{10} +0.117191 q^{12} +0.135171 q^{13} -2.69878 q^{14} -6.05403 q^{15} -3.87175 q^{16} -2.10691 q^{17} +0.766595 q^{18} +3.17457 q^{19} -0.199813 q^{20} +3.65315 q^{21} -0.382437 q^{23} +5.40930 q^{24} +5.32229 q^{25} +0.188165 q^{26} +4.61530 q^{27} +0.120572 q^{28} -6.89110 q^{29} -8.42752 q^{30} -6.85274 q^{31} +0.351683 q^{32} -2.93293 q^{34} -6.22873 q^{35} -0.0342489 q^{36} +6.09269 q^{37} +4.41917 q^{38} -0.254707 q^{39} -9.22301 q^{40} -3.84376 q^{41} +5.08538 q^{42} +12.9347 q^{43} +1.76929 q^{45} -0.532372 q^{46} -4.56102 q^{47} +7.29564 q^{48} -3.24143 q^{49} +7.40890 q^{50} +3.97011 q^{51} -0.00840659 q^{52} +5.18338 q^{53} +6.42473 q^{54} +5.56539 q^{56} -5.98193 q^{57} -9.59277 q^{58} -4.69242 q^{59} +0.376514 q^{60} +1.00000 q^{61} -9.53936 q^{62} -1.06763 q^{63} +8.23306 q^{64} +0.434282 q^{65} -12.6269 q^{67} +0.131033 q^{68} +0.720637 q^{69} -8.67071 q^{70} +12.2303 q^{71} -1.58087 q^{72} +5.40683 q^{73} +8.48134 q^{74} -10.0289 q^{75} -0.197434 q^{76} -0.354565 q^{78} +11.4836 q^{79} -12.4393 q^{80} -10.3488 q^{81} -5.35072 q^{82} +14.8726 q^{83} -0.227198 q^{84} -6.76915 q^{85} +18.0057 q^{86} +12.9851 q^{87} -6.19352 q^{89} +2.46294 q^{90} -0.262057 q^{91} +0.0237846 q^{92} +12.9128 q^{93} -6.34917 q^{94} +10.1994 q^{95} -0.662687 q^{96} -4.39546 q^{97} -4.51224 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} - 2 q^{3} + 50 q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 30 q^{8} + 48 q^{9}+O(q^{10}) \) 50 * q + 10 * q^2 - 2 * q^3 + 50 * q^4 - 2 * q^5 + 12 * q^6 + 8 * q^7 + 30 * q^8 + 48 * q^9	\( 50 q + 10 q^{2} - 2 q^{3} + 50 q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 30 q^{8} + 48 q^{9} + 12 q^{10} - 14 q^{12} + 8 q^{13} - 2 q^{14} - 16 q^{15} + 42 q^{16} + 22 q^{17} + 32 q^{19} - 8 q^{20} + 24 q^{21} + 12 q^{23} - 8 q^{24} + 40 q^{25} + 10 q^{26} - 8 q^{27} + 72 q^{28} + 56 q^{29} + 24 q^{30} + 10 q^{31} + 70 q^{32} - 32 q^{34} + 70 q^{35} + 34 q^{36} - 8 q^{37} - 14 q^{38} + 96 q^{39} - 54 q^{40} + 56 q^{41} - 8 q^{42} + 44 q^{43} - 24 q^{45} - 4 q^{46} - 4 q^{47} - 28 q^{48} + 38 q^{49} + 120 q^{50} + 76 q^{51} + 24 q^{52} + 4 q^{53} + 48 q^{54} - 18 q^{56} + 8 q^{57} + 28 q^{58} + 12 q^{59} - 60 q^{60} + 50 q^{61} + 8 q^{62} + 30 q^{63} + 10 q^{64} + 64 q^{65} + 18 q^{67} - 22 q^{68} - 8 q^{69} + 34 q^{70} + 12 q^{71} + 104 q^{72} - 16 q^{73} + 84 q^{74} - 26 q^{75} + 64 q^{76} + 40 q^{78} + 78 q^{79} - 36 q^{80} + 34 q^{81} + 54 q^{82} + 68 q^{83} - 78 q^{84} - 4 q^{85} + 36 q^{86} + 48 q^{87} + 26 q^{89} - 20 q^{90} + 32 q^{92} + 22 q^{93} + 156 q^{94} + 100 q^{95} - 4 q^{96} - 14 q^{97} + 70 q^{98}+O(q^{100}) \) 50 * q + 10 * q^2 - 2 * q^3 + 50 * q^4 - 2 * q^5 + 12 * q^6 + 8 * q^7 + 30 * q^8 + 48 * q^9 + 12 * q^10 - 14 * q^12 + 8 * q^13 - 2 * q^14 - 16 * q^15 + 42 * q^16 + 22 * q^17 + 32 * q^19 - 8 * q^20 + 24 * q^21 + 12 * q^23 - 8 * q^24 + 40 * q^25 + 10 * q^26 - 8 * q^27 + 72 * q^28 + 56 * q^29 + 24 * q^30 + 10 * q^31 + 70 * q^32 - 32 * q^34 + 70 * q^35 + 34 * q^36 - 8 * q^37 - 14 * q^38 + 96 * q^39 - 54 * q^40 + 56 * q^41 - 8 * q^42 + 44 * q^43 - 24 * q^45 - 4 * q^46 - 4 * q^47 - 28 * q^48 + 38 * q^49 + 120 * q^50 + 76 * q^51 + 24 * q^52 + 4 * q^53 + 48 * q^54 - 18 * q^56 + 8 * q^57 + 28 * q^58 + 12 * q^59 - 60 * q^60 + 50 * q^61 + 8 * q^62 + 30 * q^63 + 10 * q^64 + 64 * q^65 + 18 * q^67 - 22 * q^68 - 8 * q^69 + 34 * q^70 + 12 * q^71 + 104 * q^72 - 16 * q^73 + 84 * q^74 - 26 * q^75 + 64 * q^76 + 40 * q^78 + 78 * q^79 - 36 * q^80 + 34 * q^81 + 54 * q^82 + 68 * q^83 - 78 * q^84 - 4 * q^85 + 36 * q^86 + 48 * q^87 + 26 * q^89 - 20 * q^90 + 32 * q^92 + 22 * q^93 + 156 * q^94 + 100 * q^95 - 4 * q^96 - 14 * q^97 + 70 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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