LMFDB - Embedded newform 7381.2.a.i.1.16

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:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 $ x^19 - 5x^18 - 18x^17 + 122x^16 + 78x^15 - 1177x^14 + 387x^13 + 5755x^12 - 4673x^11 - 15053x^10 + 16875x^9 + 20141x^8 - 28019x^7 - 11589x^6 + 21077x^5 + 1674x^4 - 6404x^3 + 241x^2 + 643x - 43
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$-1.82302$ of defining polynomial
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)

$f(q)$	$=$	$q+1.82302 q^{2} +3.09920 q^{3} +1.32341 q^{4} -3.15385 q^{5} +5.64991 q^{6} +0.287436 q^{7} -1.23344 q^{8} +6.60505 q^{9} +O(q^{10})$	\(q+1.82302 q^{2} +3.09920 q^{3} +1.32341 q^{4} -3.15385 q^{5} +5.64991 q^{6} +0.287436 q^{7} -1.23344 q^{8} +6.60505 q^{9} -5.74953 q^{10} +4.10152 q^{12} -0.910644 q^{13} +0.524002 q^{14} -9.77441 q^{15} -4.89541 q^{16} -0.621790 q^{17} +12.0412 q^{18} -6.81024 q^{19} -4.17384 q^{20} +0.890822 q^{21} -7.16003 q^{23} -3.82267 q^{24} +4.94675 q^{25} -1.66012 q^{26} +11.1728 q^{27} +0.380396 q^{28} -7.26934 q^{29} -17.8190 q^{30} +5.73531 q^{31} -6.45756 q^{32} -1.13354 q^{34} -0.906529 q^{35} +8.74120 q^{36} +9.51534 q^{37} -12.4152 q^{38} -2.82227 q^{39} +3.89007 q^{40} -6.96842 q^{41} +1.62399 q^{42} -1.62703 q^{43} -20.8313 q^{45} -13.0529 q^{46} -11.4314 q^{47} -15.1718 q^{48} -6.91738 q^{49} +9.01804 q^{50} -1.92705 q^{51} -1.20516 q^{52} +2.14938 q^{53} +20.3682 q^{54} -0.354534 q^{56} -21.1063 q^{57} -13.2522 q^{58} +4.08607 q^{59} -12.9356 q^{60} +1.00000 q^{61} +10.4556 q^{62} +1.89853 q^{63} -1.98147 q^{64} +2.87203 q^{65} +11.2048 q^{67} -0.822884 q^{68} -22.1904 q^{69} -1.65262 q^{70} +4.90057 q^{71} -8.14691 q^{72} -9.71284 q^{73} +17.3467 q^{74} +15.3310 q^{75} -9.01275 q^{76} -5.14506 q^{78} +4.88947 q^{79} +15.4394 q^{80} +14.8115 q^{81} -12.7036 q^{82} +8.93050 q^{83} +1.17892 q^{84} +1.96103 q^{85} -2.96610 q^{86} -22.5292 q^{87} -13.4860 q^{89} -37.9760 q^{90} -0.261752 q^{91} -9.47567 q^{92} +17.7749 q^{93} -20.8398 q^{94} +21.4785 q^{95} -20.0133 q^{96} +3.68860 q^{97} -12.6105 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9}+O(q^{10}) $ 19 * q - 5 * q^2 + 23 * q^4 - q^6 - 9 * q^7 - 9 * q^8 + 29 * q^9	$ 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9} - 7 q^{10} - 4 q^{12} - 8 q^{13} - 11 q^{14} - 5 q^{15} + 31 q^{16} - 9 q^{17} - 10 q^{18} - 17 q^{19} - 6 q^{20} - 18 q^{21} - 10 q^{23} - 26 q^{24} + 45 q^{25} + 5 q^{26} - 33 q^{27} - 36 q^{28} - 27 q^{29} + 30 q^{30} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 38 q^{36} + 20 q^{37} - 37 q^{38} - 24 q^{39} - 10 q^{40} - 19 q^{41} + 21 q^{42} - 20 q^{43} - 32 q^{45} - 41 q^{46} - 19 q^{47} + 5 q^{48} + 42 q^{49} - 36 q^{50} - 47 q^{51} + 28 q^{52} + 3 q^{53} + 33 q^{54} - 44 q^{56} - 11 q^{57} + 23 q^{58} - 28 q^{59} - 96 q^{60} + 19 q^{61} + 11 q^{62} + 32 q^{63} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} - 19 q^{71} - 34 q^{72} - 20 q^{73} + 22 q^{74} - 50 q^{75} + 25 q^{76} - 94 q^{78} - 69 q^{79} - 36 q^{80} + 47 q^{81} - 61 q^{82} - q^{83} + 28 q^{84} - 24 q^{85} - 27 q^{86} + 58 q^{87} - 36 q^{90} + 24 q^{91} - 67 q^{92} - 14 q^{93} - 64 q^{94} + 3 q^{95} + 26 q^{96} + 21 q^{97} + 87 q^{98}+O(q^{100}) $ 19 * q - 5 * q^2 + 23 * q^4 - q^6 - 9 * q^7 - 9 * q^8 + 29 * q^9 - 7 * q^10 - 4 * q^12 - 8 * q^13 - 11 * q^14 - 5 * q^15 + 31 * q^16 - 9 * q^17 - 10 * q^18 - 17 * q^19 - 6 * q^20 - 18 * q^21 - 10 * q^23 - 26 * q^24 + 45 * q^25 + 5 * q^26 - 33 * q^27 - 36 * q^28 - 27 * q^29 + 30 * q^30 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 38 * q^36 + 20 * q^37 - 37 * q^38 - 24 * q^39 - 10 * q^40 - 19 * q^41 + 21 * q^42 - 20 * q^43 - 32 * q^45 - 41 * q^46 - 19 * q^47 + 5 * q^48 + 42 * q^49 - 36 * q^50 - 47 * q^51 + 28 * q^52 + 3 * q^53 + 33 * q^54 - 44 * q^56 - 11 * q^57 + 23 * q^58 - 28 * q^59 - 96 * q^60 + 19 * q^61 + 11 * q^62 + 32 * q^63 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 - 19 * q^71 - 34 * q^72 - 20 * q^73 + 22 * q^74 - 50 * q^75 + 25 * q^76 - 94 * q^78 - 69 * q^79 - 36 * q^80 + 47 * q^81 - 61 * q^82 - q^83 + 28 * q^84 - 24 * q^85 - 27 * q^86 + 58 * q^87 - 36 * q^90 + 24 * q^91 - 67 * q^92 - 14 * q^93 - 64 * q^94 + 3 * q^95 + 26 * q^96 + 21 * q^97 + 87 * 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 $

$2$	1.82302	1.28907	0.644536	−	0.764574i	$-0.277051\pi$
$2$	1.82302	1.28907	0.644536	+	0.764574i	$0.277051\pi$
$3$	3.09920	1.78932	0.894662	−	0.446743i	$-0.147416\pi$
$3$	3.09920	1.78932	0.894662	+	0.446743i	$0.147416\pi$
$4$	1.32341	0.661705
$5$	−3.15385	−1.41044	−0.705222	−	0.708987i	$-0.749152\pi$
$5$	−3.15385	−1.41044	−0.705222	+	0.708987i	$0.749152\pi$
$6$	5.64991	2.30657
$7$	0.287436	0.108641	0.0543203	−	0.998524i	$-0.482701\pi$
$7$	0.287436	0.108641	0.0543203	+	0.998524i	$0.482701\pi$
$8$	−1.23344	−0.436086
$9$	6.60505	2.20168
$10$	−5.74953	−1.81816
$11$	0	0
$11$	0	0
$12$	4.10152	1.18401
$13$	−0.910644	−0.252567	−0.126284	−	0.991994i	$-0.540305\pi$
$13$	−0.910644	−0.252567	−0.126284	+	0.991994i	$0.540305\pi$
$14$	0.524002	0.140045
$15$	−9.77441	−2.52374
$16$	−4.89541	−1.22385
$17$	−0.621790	−0.150806	−0.0754031	−	0.997153i	$-0.524024\pi$
$17$	−0.621790	−0.150806	−0.0754031	+	0.997153i	$0.524024\pi$
$18$	12.0412	2.83813
$19$	−6.81024	−1.56238	−0.781188	−	0.624295i	$-0.785386\pi$
$19$	−6.81024	−1.56238	−0.781188	+	0.624295i	$0.785386\pi$
$20$	−4.17384	−0.933298
$21$	0.890822	0.194393
$22$	0	0
$23$	−7.16003	−1.49297	−0.746485	−	0.665402i	$-0.768260\pi$
$23$	−7.16003	−1.49297	−0.746485	+	0.665402i	$0.768260\pi$
$24$	−3.82267	−0.780299
$25$	4.94675	0.989350
$26$	−1.66012	−0.325577
$27$	11.1728	2.15020
$28$	0.380396	0.0718881
$29$	−7.26934	−1.34988	−0.674942	−	0.737871i	$-0.735831\pi$
$29$	−7.26934	−1.34988	−0.674942	+	0.737871i	$0.735831\pi$
$30$	−17.8190	−3.25328
$31$	5.73531	1.03009	0.515046	−	0.857162i	$-0.327775\pi$
$31$	5.73531	1.03009	0.515046	+	0.857162i	$0.327775\pi$
$32$	−6.45756	−1.14155
$33$	0	0
$34$	−1.13354	−0.194400
$35$	−0.906529	−0.153231
$36$	8.74120	1.45687
$37$	9.51534	1.56431	0.782156	−	0.623082i	$-0.214120\pi$
$37$	9.51534	1.56431	0.782156	+	0.623082i	$0.214120\pi$
$38$	−12.4152	−2.01402
$39$	−2.82227	−0.451925
$40$	3.89007	0.615074
$41$	−6.96842	−1.08828	−0.544142	−	0.838993i	$-0.683145\pi$
$41$	−6.96842	−1.08828	−0.544142	+	0.838993i	$0.683145\pi$
$42$	1.62399	0.250587
$43$	−1.62703	−0.248119	−0.124060	−	0.992275i	$-0.539591\pi$
$43$	−1.62703	−0.248119	−0.124060	+	0.992275i	$0.539591\pi$
$44$	0	0
$45$	−20.8313	−3.10535
$46$	−13.0529	−1.92455
$47$	−11.4314	−1.66745	−0.833724	−	0.552181i	$-0.813796\pi$
$47$	−11.4314	−1.66745	−0.833724	+	0.552181i	$0.813796\pi$
$48$	−15.1718	−2.18987
$49$	−6.91738	−0.988197
$50$	9.01804	1.27534
$51$	−1.92705	−0.269841
$52$	−1.20516	−0.167125
$53$	2.14938	0.295240	0.147620	−	0.989044i	$-0.452839\pi$
$53$	2.14938	0.295240	0.147620	+	0.989044i	$0.452839\pi$
$54$	20.3682	2.77176
$55$	0	0
$56$	−0.354534	−0.0473766
$57$	−21.1063	−2.79560
$58$	−13.2522	−1.74010
$59$	4.08607	0.531961	0.265981	−	0.963978i	$-0.414304\pi$
$59$	4.08607	0.531961	0.265981	+	0.963978i	$0.414304\pi$
$60$	−12.9356	−1.66997
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	10.4556	1.32786
$63$	1.89853	0.239192
$64$	−1.98147	−0.247683
$65$	2.87203	0.356232
$66$	0	0
$67$	11.2048	1.36888	0.684442	−	0.729067i	$-0.260046\pi$
$67$	11.2048	1.36888	0.684442	+	0.729067i	$0.260046\pi$
$68$	−0.822884	−0.0997893
$69$	−22.1904	−2.67141
$70$	−1.65262	−0.197526
$71$	4.90057	0.581591	0.290795	−	0.956785i	$-0.406080\pi$
$71$	4.90057	0.581591	0.290795	+	0.956785i	$0.406080\pi$
$72$	−8.14691	−0.960123
$73$	−9.71284	−1.13680	−0.568401	−	0.822752i	$-0.692438\pi$
$73$	−9.71284	−1.13680	−0.568401	+	0.822752i	$0.692438\pi$
$74$	17.3467	2.01651
$75$	15.3310	1.77027
$76$	−9.01275	−1.03383
$77$	0	0
$78$	−5.14506	−0.582563
$79$	4.88947	0.550109	0.275054	−	0.961429i	$-0.411304\pi$
$79$	4.88947	0.550109	0.275054	+	0.961429i	$0.411304\pi$
$80$	15.4394	1.72617
$81$	14.8115	1.64573
$82$	−12.7036	−1.40288
$83$	8.93050	0.980250	0.490125	−	0.871652i	$-0.336951\pi$
$83$	8.93050	0.980250	0.490125	+	0.871652i	$0.336951\pi$
$84$	1.17892	0.128631
$85$	1.96103	0.212704
$86$	−2.96610	−0.319843
$87$	−22.5292	−2.41538
$88$	0	0
$89$	−13.4860	−1.42952	−0.714758	−	0.699372i	$-0.753463\pi$
$89$	−13.4860	−1.42952	−0.714758	+	0.699372i	$0.753463\pi$
$90$	−37.9760	−4.00302
$91$	−0.261752	−0.0274390
$92$	−9.47567	−0.987906
$93$	17.7749	1.84317
$94$	−20.8398	−2.14946
$95$	21.4785	2.20364
$96$	−20.0133	−2.04260
$97$	3.68860	0.374520	0.187260	−	0.982310i	$-0.440039\pi$
$97$	3.68860	0.374520	0.187260	+	0.982310i	$0.440039\pi$
$98$	−12.6105	−1.27386
$99$	0	0
$100$	6.54658	0.654658
$101$	10.0944	1.00443	0.502217	−	0.864742i	$-0.332518\pi$
$101$	10.0944	1.00443	0.502217	+	0.864742i	$0.332518\pi$
$102$	−3.51306	−0.347845
$103$	−10.5365	−1.03819	−0.519097	−	0.854715i	$-0.673732\pi$
$103$	−10.5365	−1.03819	−0.519097	+	0.854715i	$0.673732\pi$
$104$	1.12322	0.110141
$105$	−2.80952	−0.274181
$106$	3.91836	0.380585
$107$	−16.3056	−1.57632	−0.788160	−	0.615470i	$-0.788966\pi$
$107$	−16.3056	−1.57632	−0.788160	+	0.615470i	$0.788966\pi$
$108$	14.7862	1.42280
$109$	−5.49319	−0.526152	−0.263076	−	0.964775i	$-0.584737\pi$
$109$	−5.49319	−0.526152	−0.263076	+	0.964775i	$0.584737\pi$
$110$	0	0
$111$	29.4900	2.79906
$112$	−1.40712	−0.132960
$113$	−11.7663	−1.10688	−0.553442	−	0.832887i	$-0.686686\pi$
$113$	−11.7663	−1.10688	−0.553442	+	0.832887i	$0.686686\pi$
$114$	−38.4773	−3.60373
$115$	22.5816	2.10575
$116$	−9.62033	−0.893225
$117$	−6.01485	−0.556073
$118$	7.44900	0.685736
$119$	−0.178725	−0.0163837
$120$	12.0561	1.10057
$121$	0	0
$122$	1.82302	0.165049
$123$	−21.5965	−1.94729
$124$	7.59017	0.681618
$125$	0.167943	0.0150213
$126$	3.46106	0.308336
$127$	17.6684	1.56781	0.783907	−	0.620878i	$-0.213224\pi$
$127$	17.6684	1.56781	0.783907	+	0.620878i	$0.213224\pi$
$128$	9.30286	0.822265
$129$	−5.04248	−0.443966
$130$	5.23578	0.459208
$131$	6.52874	0.570419	0.285209	−	0.958465i	$-0.407937\pi$
$131$	6.52874	0.570419	0.285209	+	0.958465i	$0.407937\pi$
$132$	0	0
$133$	−1.95751	−0.169738
$134$	20.4266	1.76459
$135$	−35.2372	−3.03274
$136$	0.766939	0.0657645
$137$	−14.2878	−1.22069	−0.610345	−	0.792136i	$-0.708969\pi$
$137$	−14.2878	−1.22069	−0.610345	+	0.792136i	$0.708969\pi$
$138$	−40.4536	−3.44364
$139$	0.596965	0.0506339	0.0253169	−	0.999679i	$-0.491941\pi$
$139$	0.596965	0.0506339	0.0253169	+	0.999679i	$0.491941\pi$
$140$	−1.19971	−0.101394
$141$	−35.4284	−2.98361
$142$	8.93385	0.749712
$143$	0	0
$144$	−32.3344	−2.69453
$145$	22.9264	1.90393
$146$	−17.7067	−1.46542
$147$	−21.4384	−1.76821
$148$	12.5927	1.03511
$149$	−15.4628	−1.26676	−0.633382	−	0.773839i	$-0.718334\pi$
$149$	−15.4628	−1.26676	−0.633382	+	0.773839i	$0.718334\pi$
$150$	27.9487	2.28200
$151$	−7.37707	−0.600338	−0.300169	−	0.953886i	$-0.597043\pi$
$151$	−7.37707	−0.600338	−0.300169	+	0.953886i	$0.597043\pi$
$152$	8.40001	0.681330
$153$	−4.10695	−0.332028
$154$	0	0
$155$	−18.0883	−1.45289
$156$	−3.73502	−0.299041
$157$	7.79705	0.622272	0.311136	−	0.950365i	$-0.399291\pi$
$157$	7.79705	0.622272	0.311136	+	0.950365i	$0.399291\pi$
$158$	8.91362	0.709129
$159$	6.66135	0.528280
$160$	20.3662	1.61009
$161$	−2.05805	−0.162197
$162$	27.0018	2.12146
$163$	−0.591572	−0.0463355	−0.0231677	−	0.999732i	$-0.507375\pi$
$163$	−0.591572	−0.0463355	−0.0231677	+	0.999732i	$0.507375\pi$
$164$	−9.22208	−0.720123
$165$	0	0
$166$	16.2805	1.26361
$167$	−3.67811	−0.284621	−0.142310	−	0.989822i	$-0.545453\pi$
$167$	−3.67811	−0.284621	−0.142310	+	0.989822i	$0.545453\pi$
$168$	−1.09877	−0.0847721
$169$	−12.1707	−0.936210
$170$	3.57500	0.274190
$171$	−44.9820	−3.43986
$172$	−2.15322	−0.164182
$173$	12.9607	0.985383	0.492692	−	0.870204i	$-0.336013\pi$
$173$	12.9607	0.985383	0.492692	+	0.870204i	$0.336013\pi$
$174$	−41.0712	−3.11360
$175$	1.42187	0.107484
$176$	0	0
$177$	12.6636	0.951851
$178$	−24.5853	−1.84275
$179$	−10.0940	−0.754458	−0.377229	−	0.926120i	$-0.623123\pi$
$179$	−10.0940	−0.754458	−0.377229	+	0.926120i	$0.623123\pi$
$180$	−27.5684	−2.05483
$181$	−16.4988	−1.22635	−0.613174	−	0.789948i	$-0.710108\pi$
$181$	−16.4988	−1.22635	−0.613174	+	0.789948i	$0.710108\pi$
$182$	−0.477179	−0.0353709
$183$	3.09920	0.229100
$184$	8.83145	0.651063
$185$	−30.0099	−2.20637
$186$	32.4040	2.37598
$187$	0	0
$188$	−15.1285	−1.10336
$189$	3.21146	0.233599
$190$	39.1557	2.84065
$191$	7.51056	0.543445	0.271722	−	0.962376i	$-0.412407\pi$
$191$	7.51056	0.543445	0.271722	+	0.962376i	$0.412407\pi$
$192$	−6.14096	−0.443186
$193$	20.3457	1.46451	0.732257	−	0.681029i	$-0.238467\pi$
$193$	20.3457	1.46451	0.732257	+	0.681029i	$0.238467\pi$
$194$	6.72439	0.482783
$195$	8.90101	0.637414
$196$	−9.15454	−0.653896
$197$	10.2107	0.727481	0.363741	−	0.931500i	$-0.381499\pi$
$197$	10.2107	0.727481	0.363741	+	0.931500i	$0.381499\pi$
$198$	0	0
$199$	9.42808	0.668339	0.334169	−	0.942513i	$-0.391544\pi$
$199$	9.42808	0.668339	0.334169	+	0.942513i	$0.391544\pi$
$200$	−6.10150	−0.431441
$201$	34.7259	2.44938
$202$	18.4024	1.29479
$203$	−2.08947	−0.146652
$204$	−2.55028	−0.178555
$205$	21.9773	1.53496
$206$	−19.2083	−1.33831
$207$	−47.2924	−3.28705
$208$	4.45797	0.309105
$209$	0	0
$210$	−5.12181	−0.353438
$211$	12.0384	0.828761	0.414380	−	0.910104i	$-0.363998\pi$
$211$	12.0384	0.828761	0.414380	+	0.910104i	$0.363998\pi$
$212$	2.84451	0.195362
$213$	15.1879	1.04065
$214$	−29.7254	−2.03199
$215$	5.13139	0.349958
$216$	−13.7809	−0.937673
$217$	1.64853	0.111910
$218$	−10.0142	−0.678248
$219$	−30.1020	−2.03411
$220$	0	0
$221$	0.566229	0.0380887
$222$	53.7609	3.60819
$223$	16.0854	1.07715	0.538577	−	0.842576i	$-0.318962\pi$
$223$	16.0854	1.07715	0.538577	+	0.842576i	$0.318962\pi$
$224$	−1.85613	−0.124018
$225$	32.6735	2.17824
$226$	−21.4503	−1.42685
$227$	−8.88217	−0.589530	−0.294765	−	0.955570i	$-0.595241\pi$
$227$	−8.88217	−0.589530	−0.294765	+	0.955570i	$0.595241\pi$
$228$	−27.9323	−1.84986
$229$	−11.2325	−0.742264	−0.371132	−	0.928580i	$-0.621030\pi$
$229$	−11.2325	−0.742264	−0.371132	+	0.928580i	$0.621030\pi$
$230$	41.1668	2.71446
$231$	0	0
$232$	8.96628	0.588665
$233$	−15.7110	−1.02926	−0.514631	−	0.857412i	$-0.672071\pi$
$233$	−15.7110	−1.02926	−0.514631	+	0.857412i	$0.672071\pi$
$234$	−10.9652	−0.716818
$235$	36.0530	2.35184
$236$	5.40755	0.352002
$237$	15.1535	0.984323
$238$	−0.325819	−0.0211197
$239$	−6.28509	−0.406549	−0.203274	−	0.979122i	$-0.565158\pi$
$239$	−6.28509	−0.406549	−0.203274	+	0.979122i	$0.565158\pi$
$240$	47.8497	3.08868
$241$	−3.70185	−0.238457	−0.119229	−	0.992867i	$-0.538042\pi$
$241$	−3.70185	−0.238457	−0.119229	+	0.992867i	$0.538042\pi$
$242$	0	0
$243$	12.3856	0.794537
$244$	1.32341	0.0847227
$245$	21.8164	1.39380
$246$	−39.3710	−2.51020
$247$	6.20171	0.394605
$248$	−7.07414	−0.449209
$249$	27.6774	1.75399
$250$	0.306165	0.0193636
$251$	7.05635	0.445393	0.222696	−	0.974888i	$-0.428514\pi$
$251$	7.05635	0.445393	0.222696	+	0.974888i	$0.428514\pi$
$252$	2.51253	0.158275
$253$	0	0
$254$	32.2098	2.02102
$255$	6.07763	0.380596
$256$	20.9223	1.30764
$257$	−4.59045	−0.286345	−0.143172	−	0.989698i	$-0.545730\pi$
$257$	−4.59045	−0.286345	−0.143172	+	0.989698i	$0.545730\pi$
$258$	−9.19255	−0.572303
$259$	2.73505	0.169948
$260$	3.80088	0.235720
$261$	−48.0144	−2.97202
$262$	11.9020	0.735310
$263$	−0.961078	−0.0592626	−0.0296313	−	0.999561i	$-0.509433\pi$
$263$	−0.961078	−0.0592626	−0.0296313	+	0.999561i	$0.509433\pi$
$264$	0	0
$265$	−6.77881	−0.416419
$266$	−3.56858	−0.218804
$267$	−41.7959	−2.55787
$268$	14.8286	0.905799
$269$	13.6257	0.830773	0.415386	−	0.909645i	$-0.363646\pi$
$269$	13.6257	0.830773	0.415386	+	0.909645i	$0.363646\pi$
$270$	−64.2382	−3.90942
$271$	29.0169	1.76265	0.881327	−	0.472507i	$-0.156651\pi$
$271$	29.0169	1.76265	0.881327	+	0.472507i	$0.156651\pi$
$272$	3.04391	0.184564
$273$	−0.811222	−0.0490974
$274$	−26.0470	−1.57356
$275$	0	0
$276$	−29.3670	−1.76769
$277$	16.5957	0.997137	0.498569	−	0.866850i	$-0.333859\pi$
$277$	16.5957	0.997137	0.498569	+	0.866850i	$0.333859\pi$
$278$	1.08828	0.0652707
$279$	37.8820	2.26794
$280$	1.11815	0.0668220
$281$	−7.06643	−0.421548	−0.210774	−	0.977535i	$-0.567598\pi$
$281$	−7.06643	−0.421548	−0.210774	+	0.977535i	$0.567598\pi$
$282$	−64.5867	−3.84608
$283$	14.4028	0.856160	0.428080	−	0.903741i	$-0.359190\pi$
$283$	14.4028	0.856160	0.428080	+	0.903741i	$0.359190\pi$
$284$	6.48547	0.384842
$285$	66.5661	3.94303
$286$	0	0
$287$	−2.00297	−0.118232
$288$	−42.6525	−2.51332
$289$	−16.6134	−0.977257
$290$	41.7953	2.45431
$291$	11.4317	0.670138
$292$	−12.8541	−0.752228
$293$	28.4883	1.66431	0.832153	−	0.554547i	$-0.187108\pi$
$293$	28.4883	1.66431	0.832153	+	0.554547i	$0.187108\pi$
$294$	−39.0826	−2.27934
$295$	−12.8868	−0.750301
$296$	−11.7366	−0.682175
$297$	0	0
$298$	−28.1891	−1.63295
$299$	6.52024	0.377075
$300$	20.2892	1.17140
$301$	−0.467666	−0.0269558
$302$	−13.4486	−0.773878
$303$	31.2847	1.79726
$304$	33.3389	1.91212
$305$	−3.15385	−0.180589
$306$	−7.48707	−0.428007
$307$	−10.5356	−0.601297	−0.300649	−	0.953735i	$-0.597203\pi$
$307$	−10.5356	−0.601297	−0.300649	+	0.953735i	$0.597203\pi$
$308$	0	0
$309$	−32.6548	−1.85767
$310$	−32.9754	−1.87287
$311$	16.5963	0.941087	0.470544	−	0.882377i	$-0.344058\pi$
$311$	16.5963	0.941087	0.470544	+	0.882377i	$0.344058\pi$
$312$	3.48109	0.197078
$313$	9.85264	0.556904	0.278452	−	0.960450i	$-0.410179\pi$
$313$	9.85264	0.556904	0.278452	+	0.960450i	$0.410179\pi$
$314$	14.2142	0.802153
$315$	−5.98767	−0.337367
$316$	6.47078	0.364010
$317$	−29.8035	−1.67393	−0.836964	−	0.547257i	$-0.815672\pi$
$317$	−29.8035	−1.67393	−0.836964	+	0.547257i	$0.815672\pi$
$318$	12.1438	0.680990
$319$	0	0
$320$	6.24924	0.349343
$321$	−50.5343	−2.82055
$322$	−3.75187	−0.209084
$323$	4.23454	0.235616
$324$	19.6017	1.08899
$325$	−4.50473	−0.249877
$326$	−1.07845	−0.0597297
$327$	−17.0245	−0.941457
$328$	8.59510	0.474585
$329$	−3.28581	−0.181153
$330$	0	0
$331$	−33.2914	−1.82986	−0.914931	−	0.403611i	$-0.867755\pi$
$331$	−33.2914	−1.82986	−0.914931	+	0.403611i	$0.867755\pi$
$332$	11.8187	0.648637
$333$	62.8493	3.44412
$334$	−6.70528	−0.366897
$335$	−35.3382	−1.93073
$336$	−4.36093	−0.237908
$337$	−17.5453	−0.955753	−0.477876	−	0.878427i	$-0.658593\pi$
$337$	−17.5453	−0.955753	−0.477876	+	0.878427i	$0.658593\pi$
$338$	−22.1875	−1.20684
$339$	−36.4663	−1.98058
$340$	2.59525	0.140747
$341$	0	0
$342$	−82.0032	−4.43422
$343$	−4.00035	−0.215999
$344$	2.00683	0.108201
$345$	69.9851	3.76787
$346$	23.6276	1.27023
$347$	−19.7792	−1.06180	−0.530902	−	0.847433i	$-0.678147\pi$
$347$	−19.7792	−1.06180	−0.530902	+	0.847433i	$0.678147\pi$
$348$	−29.8153	−1.59827
$349$	19.0217	1.01821	0.509103	−	0.860705i	$-0.329977\pi$
$349$	19.0217	1.01821	0.509103	+	0.860705i	$0.329977\pi$
$350$	2.59211	0.138554
$351$	−10.1744	−0.543070
$352$	0	0
$353$	−28.2882	−1.50563	−0.752814	−	0.658233i	$-0.771304\pi$
$353$	−28.2882	−1.50563	−0.752814	+	0.658233i	$0.771304\pi$
$354$	23.0860	1.22700
$355$	−15.4557	−0.820301
$356$	−17.8476	−0.945918
$357$	−0.553904	−0.0293157
$358$	−18.4015	−0.972551
$359$	19.0153	1.00359	0.501796	−	0.864986i	$-0.332673\pi$
$359$	19.0153	1.00359	0.501796	+	0.864986i	$0.332673\pi$
$360$	25.6941	1.35420
$361$	27.3794	1.44102
$362$	−30.0777	−1.58085
$363$	0	0
$364$	−0.346405	−0.0181566
$365$	30.6328	1.60339
$366$	5.64991	0.295326
$367$	1.78686	0.0932731	0.0466365	−	0.998912i	$-0.485150\pi$
$367$	1.78686	0.0932731	0.0466365	+	0.998912i	$0.485150\pi$
$368$	35.0513	1.82717
$369$	−46.0267	−2.39606
$370$	−54.7088	−2.84417
$371$	0.617808	0.0320750
$372$	23.5235	1.21964
$373$	−19.1615	−0.992146	−0.496073	−	0.868281i	$-0.665225\pi$
$373$	−19.1615	−0.992146	−0.496073	+	0.868281i	$0.665225\pi$
$374$	0	0
$375$	0.520491	0.0268780
$376$	14.1000	0.727151
$377$	6.61979	0.340936
$378$	5.85456	0.301126
$379$	−22.8534	−1.17390	−0.586949	−	0.809624i	$-0.699671\pi$
$379$	−22.8534	−1.17390	−0.586949	+	0.809624i	$0.699671\pi$
$380$	28.4248	1.45816
$381$	54.7578	2.80533
$382$	13.6919	0.700539
$383$	7.28326	0.372157	0.186079	−	0.982535i	$-0.440422\pi$
$383$	7.28326	0.372157	0.186079	+	0.982535i	$0.440422\pi$
$384$	28.8314	1.47130
$385$	0	0
$386$	37.0906	1.88786
$387$	−10.7466	−0.546280
$388$	4.88153	0.247822
$389$	20.4440	1.03655	0.518276	−	0.855214i	$-0.326574\pi$
$389$	20.4440	1.03655	0.518276	+	0.855214i	$0.326574\pi$
$390$	16.2267	0.821673
$391$	4.45204	0.225149
$392$	8.53215	0.430939
$393$	20.2339	1.02066
$394$	18.6143	0.937775
$395$	−15.4206	−0.775897
$396$	0	0
$397$	15.9369	0.799852	0.399926	−	0.916547i	$-0.369036\pi$
$397$	15.9369	0.799852	0.399926	+	0.916547i	$0.369036\pi$
$398$	17.1876	0.861537
$399$	−6.06671	−0.303716
$400$	−24.2163	−1.21082
$401$	37.1941	1.85738	0.928692	−	0.370852i	$-0.120934\pi$
$401$	37.1941	1.85738	0.928692	+	0.370852i	$0.120934\pi$
$402$	63.3062	3.15743
$403$	−5.22283	−0.260168
$404$	13.3591	0.664640
$405$	−46.7133	−2.32120
$406$	−3.80915	−0.189045
$407$	0	0
$408$	2.37690	0.117674
$409$	−33.4964	−1.65629	−0.828145	−	0.560513i	$-0.810604\pi$
$409$	−33.4964	−1.65629	−0.828145	+	0.560513i	$0.810604\pi$
$410$	40.0651	1.97868
$411$	−44.2808	−2.18421
$412$	−13.9441	−0.686978
$413$	1.17448	0.0577926
$414$	−86.2151	−4.23724
$415$	−28.1654	−1.38259
$416$	5.88054	0.288317
$417$	1.85011	0.0906005
$418$	0	0
$419$	−10.6606	−0.520803	−0.260402	−	0.965500i	$-0.583855\pi$
$419$	−10.6606	−0.520803	−0.260402	+	0.965500i	$0.583855\pi$
$420$	−3.71814	−0.181427
$421$	34.4306	1.67804	0.839021	−	0.544098i	$-0.183128\pi$
$421$	34.4306	1.67804	0.839021	+	0.544098i	$0.183128\pi$
$422$	21.9464	1.06833
$423$	−75.5053	−3.67119
$424$	−2.65112	−0.128750
$425$	−3.07584	−0.149200
$426$	27.6878	1.34148
$427$	0.287436	0.0139100
$428$	−21.5790	−1.04306
$429$	0	0
$430$	9.35464	0.451121
$431$	10.8437	0.522324	0.261162	−	0.965295i	$-0.415894\pi$
$431$	10.8437	0.522324	0.261162	+	0.965295i	$0.415894\pi$
$432$	−54.6953	−2.63153
$433$	−1.14549	−0.0550485	−0.0275243	−	0.999621i	$-0.508762\pi$
$433$	−1.14549	−0.0550485	−0.0275243	+	0.999621i	$0.508762\pi$
$434$	3.00532	0.144260
$435$	71.0535	3.40676
$436$	−7.26975	−0.348158
$437$	48.7616	2.33258
$438$	−54.8767	−2.62211
$439$	−30.5933	−1.46014	−0.730069	−	0.683373i	$-0.760512\pi$
$439$	−30.5933	−1.46014	−0.730069	+	0.683373i	$0.760512\pi$
$440$	0	0
$441$	−45.6896	−2.17570
$442$	1.03225	0.0490991
$443$	−9.53948	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$443$	−9.53948	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$444$	39.0273	1.85216
$445$	42.5329	2.01625
$446$	29.3240	1.38853
$447$	−47.9224	−2.26665
$448$	−0.569545	−0.0269085
$449$	16.6546	0.785979	0.392990	−	0.919543i	$-0.371441\pi$
$449$	16.6546	0.785979	0.392990	+	0.919543i	$0.371441\pi$
$450$	59.5646	2.80790
$451$	0	0
$452$	−15.5717	−0.732432
$453$	−22.8630	−1.07420
$454$	−16.1924	−0.759947
$455$	0.825525	0.0387012
$456$	26.0333	1.21912
$457$	−6.60614	−0.309022	−0.154511	−	0.987991i	$-0.549380\pi$
$457$	−6.60614	−0.309022	−0.154511	+	0.987991i	$0.549380\pi$
$458$	−20.4771	−0.956832
$459$	−6.94712	−0.324264
$460$	29.8848	1.39339
$461$	−9.91877	−0.461963	−0.230982	−	0.972958i	$-0.574194\pi$
$461$	−9.91877	−0.461963	−0.230982	+	0.972958i	$0.574194\pi$
$462$	0	0
$463$	−19.7305	−0.916953	−0.458477	−	0.888707i	$-0.651605\pi$
$463$	−19.7305	−0.916953	−0.458477	+	0.888707i	$0.651605\pi$
$464$	35.5864	1.65206
$465$	−56.0593	−2.59969
$466$	−28.6415	−1.32679
$467$	24.2804	1.12356	0.561782	−	0.827285i	$-0.310116\pi$
$467$	24.2804	1.12356	0.561782	+	0.827285i	$0.310116\pi$
$468$	−7.96012	−0.367957
$469$	3.22066	0.148716
$470$	65.7255	3.03169
$471$	24.1646	1.11345
$472$	−5.03991	−0.231981
$473$	0	0
$474$	27.6251	1.26886
$475$	−33.6886	−1.54574
$476$	−0.236526	−0.0108412
$477$	14.1967	0.650024
$478$	−11.4579	−0.524070
$479$	−3.15535	−0.144172	−0.0720859	−	0.997398i	$-0.522966\pi$
$479$	−3.15535	−0.144172	−0.0720859	+	0.997398i	$0.522966\pi$
$480$	63.1188	2.88097
$481$	−8.66509	−0.395094
$482$	−6.74856	−0.307388
$483$	−6.37831	−0.290223
$484$	0	0
$485$	−11.6333	−0.528239
$486$	22.5792	1.02421
$487$	−37.8261	−1.71406	−0.857031	−	0.515265i	$-0.827694\pi$
$487$	−37.8261	−1.71406	−0.857031	+	0.515265i	$0.827694\pi$
$488$	−1.23344	−0.0558351
$489$	−1.83340	−0.0829092
$490$	39.7717	1.79670
$491$	−32.9576	−1.48736	−0.743678	−	0.668538i	$-0.766920\pi$
$491$	−32.9576	−1.48736	−0.743678	+	0.668538i	$0.766920\pi$
$492$	−28.5811	−1.28853
$493$	4.52001	0.203571
$494$	11.3059	0.508674
$495$	0	0
$496$	−28.0767	−1.26068
$497$	1.40860	0.0631844
$498$	50.4566	2.26101
$499$	−16.4934	−0.738345	−0.369173	−	0.929361i	$-0.620359\pi$
$499$	−16.4934	−0.738345	−0.369173	+	0.929361i	$0.620359\pi$
$500$	0.222258	0.00993969
$501$	−11.3992	−0.509279
$502$	12.8639	0.574143
$503$	−27.8051	−1.23977	−0.619883	−	0.784694i	$-0.712820\pi$
$503$	−27.8051	−1.23977	−0.619883	+	0.784694i	$0.712820\pi$
$504$	−2.34172	−0.104308
$505$	−31.8363	−1.41670
$506$	0	0
$507$	−37.7195	−1.67518
$508$	23.3825	1.03743
$509$	22.1434	0.981487	0.490744	−	0.871304i	$-0.336725\pi$
$509$	22.1434	0.981487	0.490744	+	0.871304i	$0.336725\pi$
$510$	11.0797	0.490615
$511$	−2.79182	−0.123503
$512$	19.5360	0.863378
$513$	−76.0893	−3.35943
$514$	−8.36850	−0.369119
$515$	33.2306	1.46431
$516$	−6.67327	−0.293774
$517$	0	0
$518$	4.98606	0.219075
$519$	40.1678	1.76317
$520$	−3.54247	−0.155348
$521$	5.33123	0.233565	0.116783	−	0.993157i	$-0.462742\pi$
$521$	5.33123	0.233565	0.116783	+	0.993157i	$0.462742\pi$
$522$	−87.5313	−3.83114
$523$	−14.4681	−0.632645	−0.316322	−	0.948652i	$-0.602448\pi$
$523$	−14.4681	−0.632645	−0.316322	+	0.948652i	$0.602448\pi$
$524$	8.64020	0.377449
$525$	4.40667	0.192323
$526$	−1.75207	−0.0763938
$527$	−3.56616	−0.155344
$528$	0	0
$529$	28.2661	1.22896
$530$	−12.3579	−0.536794
$531$	26.9887	1.17121
$532$	−2.59059	−0.112316
$533$	6.34575	0.274865
$534$	−76.1949	−3.29728
$535$	51.4253	2.22331
$536$	−13.8204	−0.596951
$537$	−31.2832	−1.34997
$538$	24.8399	1.07093
$539$	0	0
$540$	−46.6333	−2.00678
$541$	−18.4464	−0.793073	−0.396536	−	0.918019i	$-0.629788\pi$
$541$	−18.4464	−0.793073	−0.396536	+	0.918019i	$0.629788\pi$
$542$	52.8985	2.27219
$543$	−51.1332	−2.19434
$544$	4.01525	0.172152
$545$	17.3247	0.742108
$546$	−1.47888	−0.0632900
$547$	36.5079	1.56097	0.780483	−	0.625177i	$-0.214973\pi$
$547$	36.5079	1.56097	0.780483	+	0.625177i	$0.214973\pi$
$548$	−18.9086	−0.807737
$549$	6.60505	0.281897
$550$	0	0
$551$	49.5060	2.10903
$552$	27.3704	1.16496
$553$	1.40541	0.0597641
$554$	30.2543	1.28538
$555$	−93.0068	−3.94792
$556$	0.790030	0.0335047
$557$	−15.5872	−0.660449	−0.330225	−	0.943902i	$-0.607124\pi$
$557$	−15.5872	−0.660449	−0.330225	+	0.943902i	$0.607124\pi$
$558$	69.0598	2.92353
$559$	1.48164	0.0626667
$560$	4.43783	0.187532
$561$	0	0
$562$	−12.8823	−0.543406
$563$	30.8511	1.30022	0.650110	−	0.759840i	$-0.274723\pi$
$563$	30.8511	1.30022	0.650110	+	0.759840i	$0.274723\pi$
$564$	−46.8863	−1.97427
$565$	37.1093	1.56120
$566$	26.2567	1.10365
$567$	4.25737	0.178793
$568$	−6.04455	−0.253624
$569$	−40.2875	−1.68894	−0.844469	−	0.535604i	$-0.820084\pi$
$569$	−40.2875	−1.68894	−0.844469	+	0.535604i	$0.820084\pi$
$570$	121.351	5.08285
$571$	34.7792	1.45546	0.727732	−	0.685861i	$-0.240574\pi$
$571$	34.7792	1.45546	0.727732	+	0.685861i	$0.240574\pi$
$572$	0	0
$573$	23.2767	0.972400
$574$	−3.65146	−0.152409
$575$	−35.4189	−1.47707
$576$	−13.0877	−0.545320
$577$	−3.33773	−0.138951	−0.0694757	−	0.997584i	$-0.522133\pi$
$577$	−3.33773	−0.138951	−0.0694757	+	0.997584i	$0.522133\pi$
$578$	−30.2866	−1.25975
$579$	63.0553	2.62049
$580$	30.3410	1.25984
$581$	2.56695	0.106495
$582$	20.8403	0.863856
$583$	0	0
$584$	11.9802	0.495743
$585$	18.9699	0.784309
$586$	51.9348	2.14541
$587$	−27.5509	−1.13715	−0.568575	−	0.822632i	$-0.692505\pi$
$587$	−27.5509	−1.13715	−0.568575	+	0.822632i	$0.692505\pi$
$588$	−28.3718	−1.17003
$589$	−39.0589	−1.60939
$590$	−23.4930	−0.967192
$591$	31.6450	1.30170
$592$	−46.5815	−1.91449
$593$	−0.856831	−0.0351858	−0.0175929	−	0.999845i	$-0.505600\pi$
$593$	−0.856831	−0.0351858	−0.0175929	+	0.999845i	$0.505600\pi$
$594$	0	0
$595$	0.563671	0.0231082
$596$	−20.4637	−0.838225
$597$	29.2195	1.19588
$598$	11.8865	0.486077
$599$	−18.1092	−0.739920	−0.369960	−	0.929048i	$-0.620629\pi$
$599$	−18.1092	−0.739920	−0.369960	+	0.929048i	$0.620629\pi$
$600$	−18.9098	−0.771989
$601$	−18.2819	−0.745736	−0.372868	−	0.927884i	$-0.621625\pi$
$601$	−18.2819	−0.745736	−0.372868	+	0.927884i	$0.621625\pi$
$602$	−0.852565	−0.0347479
$603$	74.0083	3.01385
$604$	−9.76290	−0.397247
$605$	0	0
$606$	57.0327	2.31680
$607$	−6.78910	−0.275561	−0.137781	−	0.990463i	$-0.543997\pi$
$607$	−6.78910	−0.275561	−0.137781	+	0.990463i	$0.543997\pi$
$608$	43.9776	1.78353
$609$	−6.47569	−0.262408
$610$	−5.74953	−0.232792
$611$	10.4100	0.421143
$612$	−5.43519	−0.219704
$613$	25.5339	1.03131	0.515653	−	0.856797i	$-0.327549\pi$
$613$	25.5339	1.03131	0.515653	+	0.856797i	$0.327549\pi$
$614$	−19.2066	−0.775115
$615$	68.1121	2.74655
$616$	0	0
$617$	0.104760	0.00421748	0.00210874	−	0.999998i	$-0.499329\pi$
$617$	0.104760	0.00421748	0.00210874	+	0.999998i	$0.499329\pi$
$618$	−59.5304	−2.39466
$619$	26.2158	1.05370	0.526852	−	0.849957i	$-0.323372\pi$
$619$	26.2158	1.05370	0.526852	+	0.849957i	$0.323372\pi$
$620$	−23.9382	−0.961383
$621$	−79.9974	−3.21019
$622$	30.2553	1.21313
$623$	−3.87637	−0.155303
$624$	13.8162	0.553089
$625$	−25.2634	−1.01054
$626$	17.9616	0.717889
$627$	0	0
$628$	10.3187	0.411761
$629$	−5.91655	−0.235908
$630$	−10.9157	−0.434890
$631$	20.1814	0.803408	0.401704	−	0.915769i	$-0.368418\pi$
$631$	20.1814	0.803408	0.401704	+	0.915769i	$0.368418\pi$
$632$	−6.03086	−0.239895
$633$	37.3096	1.48292
$634$	−54.3324	−2.15781
$635$	−55.7233	−2.21131
$636$	8.81571	0.349566
$637$	6.29927	0.249586
$638$	0	0
$639$	32.3685	1.28048
$640$	−29.3398	−1.15976
$641$	3.60690	0.142464	0.0712319	−	0.997460i	$-0.477307\pi$
$641$	3.60690	0.142464	0.0712319	+	0.997460i	$0.477307\pi$
$642$	−92.1251	−3.63589
$643$	−33.7876	−1.33245	−0.666226	−	0.745750i	$-0.732091\pi$
$643$	−33.7876	−1.33245	−0.666226	+	0.745750i	$0.732091\pi$
$644$	−2.72365	−0.107327
$645$	15.9032	0.626188
$646$	7.71966	0.303726
$647$	−0.768454	−0.0302110	−0.0151055	−	0.999886i	$-0.504808\pi$
$647$	−0.768454	−0.0302110	−0.0151055	+	0.999886i	$0.504808\pi$
$648$	−18.2691	−0.717678
$649$	0	0
$650$	−8.21222	−0.322110
$651$	5.10914	0.200243
$652$	−0.782893	−0.0306604
$653$	−36.4030	−1.42456	−0.712280	−	0.701896i	$-0.752337\pi$
$653$	−36.4030	−1.42456	−0.712280	+	0.701896i	$0.752337\pi$
$654$	−31.0360	−1.21361
$655$	−20.5906	−0.804543
$656$	34.1132	1.33190
$657$	−64.1538	−2.50288
$658$	−5.99010	−0.233519
$659$	13.4552	0.524142	0.262071	−	0.965049i	$-0.415595\pi$
$659$	13.4552	0.524142	0.262071	+	0.965049i	$0.415595\pi$
$660$	0	0
$661$	−28.1101	−1.09335	−0.546677	−	0.837343i	$-0.684108\pi$
$661$	−28.1101	−1.09335	−0.546677	+	0.837343i	$0.684108\pi$
$662$	−60.6910	−2.35882
$663$	1.75486	0.0681531
$664$	−11.0152	−0.427473
$665$	6.17368	0.239405
$666$	114.576	4.43972
$667$	52.0487	2.01534
$668$	−4.86766	−0.188335
$669$	49.8517	1.92738
$670$	−64.4224	−2.48885
$671$	0	0
$672$	−5.75254	−0.221909
$673$	33.4248	1.28843	0.644216	−	0.764844i	$-0.277184\pi$
$673$	33.4248	1.28843	0.644216	+	0.764844i	$0.277184\pi$
$674$	−31.9855	−1.23203
$675$	55.2689	2.12730
$676$	−16.1069	−0.619495
$677$	33.9964	1.30659	0.653294	−	0.757104i	$-0.273387\pi$
$677$	33.9964	1.30659	0.653294	+	0.757104i	$0.273387\pi$
$678$	−66.4789	−2.55311
$679$	1.06023	0.0406881
$680$	−2.41881	−0.0927570
$681$	−27.5276	−1.05486
$682$	0	0
$683$	14.0398	0.537218	0.268609	−	0.963249i	$-0.413436\pi$
$683$	14.0398	0.537218	0.268609	+	0.963249i	$0.413436\pi$
$684$	−59.5297	−2.27617
$685$	45.0615	1.72171
$686$	−7.29274	−0.278438
$687$	−34.8118	−1.32815
$688$	7.96495	0.303661
$689$	−1.95732	−0.0745679
$690$	127.584	4.85705
$691$	−8.49435	−0.323140	−0.161570	−	0.986861i	$-0.551656\pi$
$691$	−8.49435	−0.323140	−0.161570	+	0.986861i	$0.551656\pi$
$692$	17.1523	0.652033
$693$	0	0
$694$	−36.0580	−1.36874
$695$	−1.88274	−0.0714162
$696$	27.7883	1.05331
$697$	4.33289	0.164120
$698$	34.6769	1.31254
$699$	−48.6916	−1.84168
$700$	1.88172	0.0711224
$701$	−42.8308	−1.61770	−0.808849	−	0.588016i	$-0.799909\pi$
$701$	−42.8308	−1.61770	−0.808849	+	0.588016i	$0.799909\pi$
$702$	−18.5482	−0.700057
$703$	−64.8018	−2.44405
$704$	0	0
$705$	111.736	4.20821
$706$	−51.5700	−1.94086
$707$	2.90150	0.109122
$708$	16.7591	0.629845
$709$	43.5929	1.63716	0.818582	−	0.574389i	$-0.194760\pi$
$709$	43.5929	1.63716	0.818582	+	0.574389i	$0.194760\pi$
$710$	−28.1760	−1.05743
$711$	32.2952	1.21116
$712$	16.6342	0.623392
$713$	−41.0650	−1.53790
$714$	−1.00978	−0.0377901
$715$	0	0
$716$	−13.3585	−0.499229
$717$	−19.4788	−0.727448
$718$	34.6654	1.29370
$719$	29.8724	1.11405	0.557026	−	0.830495i	$-0.311942\pi$
$719$	29.8724	1.11405	0.557026	+	0.830495i	$0.311942\pi$
$720$	101.978	3.80049
$721$	−3.02857	−0.112790
$722$	49.9133	1.85758
$723$	−11.4728	−0.426677
$724$	−21.8347	−0.811482
$725$	−35.9596	−1.33551
$726$	0	0
$727$	9.05098	0.335682	0.167841	−	0.985814i	$-0.446320\pi$
$727$	9.05098	0.335682	0.167841	+	0.985814i	$0.446320\pi$
$728$	0.322854	0.0119658
$729$	−6.04913	−0.224042
$730$	55.8443	2.06689
$731$	1.01167	0.0374179
$732$	4.10152	0.151596
$733$	−34.7099	−1.28204	−0.641019	−	0.767525i	$-0.721488\pi$
$733$	−34.7099	−1.28204	−0.641019	+	0.767525i	$0.721488\pi$
$734$	3.25748	0.120236
$735$	67.6133	2.49395
$736$	46.2363	1.70429
$737$	0	0
$738$	−83.9078	−3.08869
$739$	5.55188	0.204229	0.102115	−	0.994773i	$-0.467439\pi$
$739$	5.55188	0.204229	0.102115	+	0.994773i	$0.467439\pi$
$740$	−39.7155	−1.45997
$741$	19.2203	0.706077
$742$	1.12628	0.0413470
$743$	2.08882	0.0766315	0.0383158	−	0.999266i	$-0.487801\pi$
$743$	2.08882	0.0766315	0.0383158	+	0.999266i	$0.487801\pi$
$744$	−21.9242	−0.803780
$745$	48.7674	1.78670
$746$	−34.9319	−1.27895
$747$	58.9864	2.15820
$748$	0	0
$749$	−4.68681	−0.171252
$750$	0.948866	0.0346477
$751$	10.8303	0.395203	0.197601	−	0.980282i	$-0.436685\pi$
$751$	10.8303	0.395203	0.197601	+	0.980282i	$0.436685\pi$
$752$	55.9616	2.04071
$753$	21.8691	0.796953
$754$	12.0680	0.439491
$755$	23.2662	0.846742
$756$	4.25008	0.154574
$757$	−23.2966	−0.846730	−0.423365	−	0.905959i	$-0.639151\pi$
$757$	−23.2966	−0.846730	−0.423365	+	0.905959i	$0.639151\pi$
$758$	−41.6622	−1.51324
$759$	0	0
$760$	−26.4923	−0.960978
$761$	53.5346	1.94063	0.970314	−	0.241849i	$-0.0777538\pi$
$761$	53.5346	1.94063	0.970314	+	0.241849i	$0.0777538\pi$
$762$	99.8248	3.61627
$763$	−1.57894	−0.0571615
$764$	9.93956	0.359601
$765$	12.9527	0.468306
$766$	13.2775	0.479737
$767$	−3.72096	−0.134356
$768$	64.8423	2.33979
$769$	−26.4098	−0.952363	−0.476182	−	0.879347i	$-0.657980\pi$
$769$	−26.4098	−0.952363	−0.476182	+	0.879347i	$0.657980\pi$
$770$	0	0
$771$	−14.2267	−0.512363
$772$	26.9257	0.969076
$773$	23.6677	0.851269	0.425635	−	0.904895i	$-0.360051\pi$
$773$	23.6677	0.851269	0.425635	+	0.904895i	$0.360051\pi$
$774$	−19.5913	−0.704193
$775$	28.3711	1.01912
$776$	−4.54965	−0.163323
$777$	8.47648	0.304092
$778$	37.2698	1.33619
$779$	47.4566	1.70031
$780$	11.7797	0.421781
$781$	0	0
$782$	8.11616	0.290233
$783$	−81.2188	−2.90252
$784$	33.8634	1.20941
$785$	−24.5907	−0.877679
$786$	36.8868	1.31571
$787$	45.9007	1.63618	0.818092	−	0.575088i	$-0.195032\pi$
$787$	45.9007	1.63618	0.818092	+	0.575088i	$0.195032\pi$
$788$	13.5129	0.481378
$789$	−2.97858	−0.106040
$790$	−28.1122	−1.00019
$791$	−3.38207	−0.120253
$792$	0	0
$793$	−0.910644	−0.0323379
$794$	29.0534	1.03107
$795$	−21.0089	−0.745108
$796$	12.4772	0.442243
$797$	−32.7460	−1.15992	−0.579962	−	0.814644i	$-0.696933\pi$
$797$	−32.7460	−1.15992	−0.579962	+	0.814644i	$0.696933\pi$
$798$	−11.0598	−0.391511
$799$	7.10796	0.251462
$800$	−31.9439	−1.12939
$801$	−89.0759	−3.14734
$802$	67.8057	2.39430
$803$	0	0
$804$	45.9567	1.62077
$805$	6.49078	0.228770
$806$	−9.52133	−0.335375
$807$	42.2287	1.48652
$808$	−12.4509	−0.438020
$809$	48.2215	1.69538	0.847688	−	0.530495i	$-0.177994\pi$
$809$	48.2215	1.69538	0.847688	+	0.530495i	$0.177994\pi$
$810$	−85.1594	−2.99220
$811$	28.8635	1.01353	0.506767	−	0.862083i	$-0.330840\pi$
$811$	28.8635	1.01353	0.506767	+	0.862083i	$0.330840\pi$
$812$	−2.76523	−0.0970405
$813$	89.9293	3.15396
$814$	0	0
$815$	1.86573	0.0653536
$816$	9.43370	0.330246
$817$	11.0804	0.387655
$818$	−61.0647	−2.13508
$819$	−1.72888	−0.0604121
$820$	29.0850	1.01569
$821$	−50.5520	−1.76428	−0.882139	−	0.470989i	$-0.843897\pi$
$821$	−50.5520	−1.76428	−0.882139	+	0.470989i	$0.843897\pi$
$822$	−80.7249	−2.81560
$823$	−29.2399	−1.01924	−0.509619	−	0.860400i	$-0.670214\pi$
$823$	−29.2399	−1.01924	−0.509619	+	0.860400i	$0.670214\pi$
$824$	12.9961	0.452742
$825$	0	0
$826$	2.14111	0.0744988
$827$	−17.1084	−0.594918	−0.297459	−	0.954735i	$-0.596139\pi$
$827$	−17.1084	−0.594918	−0.297459	+	0.954735i	$0.596139\pi$
$828$	−62.5872	−2.17506
$829$	−35.4183	−1.23013	−0.615064	−	0.788477i	$-0.710870\pi$
$829$	−35.4183	−1.23013	−0.615064	+	0.788477i	$0.710870\pi$
$830$	−51.3462	−1.78225
$831$	51.4333	1.78420
$832$	1.80441	0.0625567
$833$	4.30116	0.149026
$834$	3.37280	0.116791
$835$	11.6002	0.401442
$836$	0	0
$837$	64.0793	2.21491
$838$	−19.4345	−0.671353
$839$	−32.8599	−1.13445	−0.567224	−	0.823564i	$-0.691983\pi$
$839$	−32.8599	−1.13445	−0.567224	+	0.823564i	$0.691983\pi$
$840$	3.46536	0.119566
$841$	23.8434	0.822185
$842$	62.7677	2.16312
$843$	−21.9003	−0.754286
$844$	15.9318	0.548396
$845$	38.3846	1.32047
$846$	−137.648	−4.73243
$847$	0	0
$848$	−10.5221	−0.361329
$849$	44.6373	1.53195
$850$	−5.60733	−0.192330
$851$	−68.1302	−2.33547
$852$	20.0998	0.688607
$853$	−1.85750	−0.0635996	−0.0317998	−	0.999494i	$-0.510124\pi$
$853$	−1.85750	−0.0635996	−0.0317998	+	0.999494i	$0.510124\pi$
$854$	0.524002	0.0179310
$855$	141.866	4.85173
$856$	20.1119	0.687411
$857$	−43.9149	−1.50010	−0.750052	−	0.661379i	$-0.769971\pi$
$857$	−43.9149	−1.50010	−0.750052	+	0.661379i	$0.769971\pi$
$858$	0	0
$859$	14.8800	0.507698	0.253849	−	0.967244i	$-0.418303\pi$
$859$	14.8800	0.507698	0.253849	+	0.967244i	$0.418303\pi$
$860$	6.79094	0.231569
$861$	−6.20762	−0.211555
$862$	19.7684	0.673313
$863$	−44.5369	−1.51605	−0.758027	−	0.652224i	$-0.773836\pi$
$863$	−44.5369	−1.51605	−0.758027	+	0.652224i	$0.773836\pi$
$864$	−72.1489	−2.45455
$865$	−40.8760	−1.38983
$866$	−2.08825	−0.0709615
$867$	−51.4882	−1.74863
$868$	2.18169	0.0740513
$869$	0	0
$870$	129.532	4.39155
$871$	−10.2036	−0.345735
$872$	6.77550	0.229447
$873$	24.3634	0.824575
$874$	88.8934	3.00686
$875$	0.0482730	0.00163192
$876$	−39.8374	−1.34598
$877$	20.2794	0.684787	0.342393	−	0.939557i	$-0.388762\pi$
$877$	20.2794	0.684787	0.342393	+	0.939557i	$0.388762\pi$
$878$	−55.7723	−1.88222
$879$	88.2910	2.97798
$880$	0	0
$881$	−40.5359	−1.36569	−0.682844	−	0.730564i	$-0.739257\pi$
$881$	−40.5359	−1.36569	−0.682844	+	0.730564i	$0.739257\pi$
$882$	−83.2932	−2.80463
$883$	8.55778	0.287992	0.143996	−	0.989578i	$-0.454005\pi$
$883$	8.55778	0.287992	0.143996	+	0.989578i	$0.454005\pi$
$884$	0.749354	0.0252035
$885$	−39.9389	−1.34253
$886$	−17.3907	−0.584251
$887$	−10.5486	−0.354188	−0.177094	−	0.984194i	$-0.556670\pi$
$887$	−10.5486	−0.354188	−0.177094	+	0.984194i	$0.556670\pi$
$888$	−36.3740	−1.22063
$889$	5.07852	0.170328
$890$	77.5384	2.59909
$891$	0	0
$892$	21.2875	0.712759
$893$	77.8509	2.60518
$894$	−87.3636	−2.92188
$895$	31.8348	1.06412
$896$	2.67398	0.0893313
$897$	20.2075	0.674710
$898$	30.3617	1.01318
$899$	−41.6920	−1.39050
$900$	43.2405	1.44135
$901$	−1.33646	−0.0445240
$902$	0	0
$903$	−1.44939	−0.0482327
$904$	14.5130	0.482697
$905$	52.0348	1.72970
$906$	−41.6798	−1.38472
$907$	3.88830	0.129109	0.0645544	−	0.997914i	$-0.479437\pi$
$907$	3.88830	0.129109	0.0645544	+	0.997914i	$0.479437\pi$
$908$	−11.7548	−0.390095
$909$	66.6743	2.21145
$910$	1.50495	0.0498886
$911$	36.2379	1.20061	0.600307	−	0.799769i	$-0.295045\pi$
$911$	36.2379	1.20061	0.600307	+	0.799769i	$0.295045\pi$
$912$	103.324	3.42140
$913$	0	0
$914$	−12.0431	−0.398352
$915$	−9.77441	−0.323132
$916$	−14.8652	−0.491160
$917$	1.87659	0.0619706
$918$	−12.6648	−0.417999
$919$	43.8784	1.44741	0.723707	−	0.690107i	$-0.242437\pi$
$919$	43.8784	1.44741	0.723707	+	0.690107i	$0.242437\pi$
$920$	−27.8530	−0.918287
$921$	−32.6519	−1.07592
$922$	−18.0821	−0.595503
$923$	−4.46268	−0.146891
$924$	0	0
$925$	47.0700	1.54765
$926$	−35.9691	−1.18202
$927$	−69.5942	−2.28577
$928$	46.9422	1.54095
$929$	29.9320	0.982036	0.491018	−	0.871149i	$-0.336625\pi$
$929$	29.9320	0.982036	0.491018	+	0.871149i	$0.336625\pi$
$930$	−102.197	−3.35118
$931$	47.1090	1.54394
$932$	−20.7921	−0.681068
$933$	51.4351	1.68391
$934$	44.2638	1.44836
$935$	0	0
$936$	7.41894	0.242496
$937$	0.358168	0.0117008	0.00585041	−	0.999983i	$-0.498138\pi$
$937$	0.358168	0.0117008	0.00585041	+	0.999983i	$0.498138\pi$
$938$	5.87134	0.191706
$939$	30.5353	0.996482
$940$	47.7130	1.55623
$941$	46.6762	1.52160	0.760801	−	0.648985i	$-0.224806\pi$
$941$	46.6762	1.52160	0.760801	+	0.648985i	$0.224806\pi$
$942$	44.0526	1.43531
$943$	49.8941	1.62478
$944$	−20.0030	−0.651041
$945$	−10.1284	−0.329478
$946$	0	0
$947$	26.4645	0.859981	0.429990	−	0.902833i	$-0.358517\pi$
$947$	26.4645	0.859981	0.429990	+	0.902833i	$0.358517\pi$
$948$	20.0543	0.651332
$949$	8.84494	0.287119
$950$	−61.4150	−1.99257
$951$	−92.3669	−2.99520
$952$	0.220446	0.00714469
$953$	41.6677	1.34975	0.674874	−	0.737933i	$-0.264198\pi$
$953$	41.6677	1.34975	0.674874	+	0.737933i	$0.264198\pi$
$954$	25.8810	0.837928
$955$	−23.6872	−0.766498
$956$	−8.31776	−0.269016
$957$	0	0
$958$	−5.75228	−0.185848
$959$	−4.10683	−0.132616
$960$	19.3677	0.625088
$961$	1.89379	0.0610901
$962$	−15.7967	−0.509305
$963$	−107.699	−3.47056
$964$	−4.89907	−0.157788
$965$	−64.1671	−2.06561
$966$	−11.6278	−0.374119
$967$	26.2577	0.844390	0.422195	−	0.906505i	$-0.361260\pi$
$967$	26.2577	0.844390	0.422195	+	0.906505i	$0.361260\pi$
$968$	0	0
$969$	13.1237	0.421594
$970$	−21.2077	−0.680938
$971$	−35.2185	−1.13021	−0.565107	−	0.825017i	$-0.691165\pi$
$971$	−35.2185	−1.13021	−0.565107	+	0.825017i	$0.691165\pi$
$972$	16.3912	0.525749
$973$	0.171589	0.00550090
$974$	−68.9577	−2.20955
$975$	−13.9611	−0.447112
$976$	−4.89541	−0.156698
$977$	61.8806	1.97974	0.989868	−	0.141987i	$-0.0453493\pi$
$977$	61.8806	1.97974	0.989868	+	0.141987i	$0.0453493\pi$
$978$	−3.34233	−0.106876
$979$	0	0
$980$	28.8720	0.922282
$981$	−36.2828	−1.15842
$982$	−60.0825	−1.91731
$983$	28.0245	0.893842	0.446921	−	0.894573i	$-0.352521\pi$
$983$	28.0245	0.893842	0.446921	+	0.894573i	$0.352521\pi$
$984$	26.6380	0.849187
$985$	−32.2029	−1.02607
$986$	8.24007	0.262417
$987$	−10.1834	−0.324141
$988$	8.20741	0.261112
$989$	11.6496	0.370434
$990$	0	0
$991$	2.64549	0.0840367	0.0420184	−	0.999117i	$-0.486621\pi$
$991$	2.64549	0.0840367	0.0420184	+	0.999117i	$0.486621\pi$
$992$	−37.0361	−1.17590
$993$	−103.177	−3.27422
$994$	2.56791	0.0814492
$995$	−29.7347	−0.942654
$996$	36.6286	1.16062
$997$	−57.5224	−1.82175	−0.910876	−	0.412680i	$-0.864593\pi$
$997$	−57.5224	−1.82175	−0.910876	+	0.412680i	$0.864593\pi$
$998$	−30.0678	−0.951780
$999$	106.313	3.36359

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7381.2.a.i.1.16		19
11.10	odd	2		671.2.a.c.1.4	✓	19
33.32	even	2		6039.2.a.k.1.16		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.4	✓	19	11.10	odd	2
6039.2.a.k.1.16		19	33.32	even	2
7381.2.a.i.1.16		19	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)

\(f(q)\)	\(=\)	\(q+1.82302 q^{2} +3.09920 q^{3} +1.32341 q^{4} -3.15385 q^{5} +5.64991 q^{6} +0.287436 q^{7} -1.23344 q^{8} +6.60505 q^{9} +O(q^{10})\)	\(q+1.82302 q^{2} +3.09920 q^{3} +1.32341 q^{4} -3.15385 q^{5} +5.64991 q^{6} +0.287436 q^{7} -1.23344 q^{8} +6.60505 q^{9} -5.74953 q^{10} +4.10152 q^{12} -0.910644 q^{13} +0.524002 q^{14} -9.77441 q^{15} -4.89541 q^{16} -0.621790 q^{17} +12.0412 q^{18} -6.81024 q^{19} -4.17384 q^{20} +0.890822 q^{21} -7.16003 q^{23} -3.82267 q^{24} +4.94675 q^{25} -1.66012 q^{26} +11.1728 q^{27} +0.380396 q^{28} -7.26934 q^{29} -17.8190 q^{30} +5.73531 q^{31} -6.45756 q^{32} -1.13354 q^{34} -0.906529 q^{35} +8.74120 q^{36} +9.51534 q^{37} -12.4152 q^{38} -2.82227 q^{39} +3.89007 q^{40} -6.96842 q^{41} +1.62399 q^{42} -1.62703 q^{43} -20.8313 q^{45} -13.0529 q^{46} -11.4314 q^{47} -15.1718 q^{48} -6.91738 q^{49} +9.01804 q^{50} -1.92705 q^{51} -1.20516 q^{52} +2.14938 q^{53} +20.3682 q^{54} -0.354534 q^{56} -21.1063 q^{57} -13.2522 q^{58} +4.08607 q^{59} -12.9356 q^{60} +1.00000 q^{61} +10.4556 q^{62} +1.89853 q^{63} -1.98147 q^{64} +2.87203 q^{65} +11.2048 q^{67} -0.822884 q^{68} -22.1904 q^{69} -1.65262 q^{70} +4.90057 q^{71} -8.14691 q^{72} -9.71284 q^{73} +17.3467 q^{74} +15.3310 q^{75} -9.01275 q^{76} -5.14506 q^{78} +4.88947 q^{79} +15.4394 q^{80} +14.8115 q^{81} -12.7036 q^{82} +8.93050 q^{83} +1.17892 q^{84} +1.96103 q^{85} -2.96610 q^{86} -22.5292 q^{87} -13.4860 q^{89} -37.9760 q^{90} -0.261752 q^{91} -9.47567 q^{92} +17.7749 q^{93} -20.8398 q^{94} +21.4785 q^{95} -20.0133 q^{96} +3.68860 q^{97} -12.6105 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9}+O(q^{10}) \) 19 * q - 5 * q^2 + 23 * q^4 - q^6 - 9 * q^7 - 9 * q^8 + 29 * q^9	\( 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9} - 7 q^{10} - 4 q^{12} - 8 q^{13} - 11 q^{14} - 5 q^{15} + 31 q^{16} - 9 q^{17} - 10 q^{18} - 17 q^{19} - 6 q^{20} - 18 q^{21} - 10 q^{23} - 26 q^{24} + 45 q^{25} + 5 q^{26} - 33 q^{27} - 36 q^{28} - 27 q^{29} + 30 q^{30} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 38 q^{36} + 20 q^{37} - 37 q^{38} - 24 q^{39} - 10 q^{40} - 19 q^{41} + 21 q^{42} - 20 q^{43} - 32 q^{45} - 41 q^{46} - 19 q^{47} + 5 q^{48} + 42 q^{49} - 36 q^{50} - 47 q^{51} + 28 q^{52} + 3 q^{53} + 33 q^{54} - 44 q^{56} - 11 q^{57} + 23 q^{58} - 28 q^{59} - 96 q^{60} + 19 q^{61} + 11 q^{62} + 32 q^{63} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} - 19 q^{71} - 34 q^{72} - 20 q^{73} + 22 q^{74} - 50 q^{75} + 25 q^{76} - 94 q^{78} - 69 q^{79} - 36 q^{80} + 47 q^{81} - 61 q^{82} - q^{83} + 28 q^{84} - 24 q^{85} - 27 q^{86} + 58 q^{87} - 36 q^{90} + 24 q^{91} - 67 q^{92} - 14 q^{93} - 64 q^{94} + 3 q^{95} + 26 q^{96} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 19 * q - 5 * q^2 + 23 * q^4 - q^6 - 9 * q^7 - 9 * q^8 + 29 * q^9 - 7 * q^10 - 4 * q^12 - 8 * q^13 - 11 * q^14 - 5 * q^15 + 31 * q^16 - 9 * q^17 - 10 * q^18 - 17 * q^19 - 6 * q^20 - 18 * q^21 - 10 * q^23 - 26 * q^24 + 45 * q^25 + 5 * q^26 - 33 * q^27 - 36 * q^28 - 27 * q^29 + 30 * q^30 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 38 * q^36 + 20 * q^37 - 37 * q^38 - 24 * q^39 - 10 * q^40 - 19 * q^41 + 21 * q^42 - 20 * q^43 - 32 * q^45 - 41 * q^46 - 19 * q^47 + 5 * q^48 + 42 * q^49 - 36 * q^50 - 47 * q^51 + 28 * q^52 + 3 * q^53 + 33 * q^54 - 44 * q^56 - 11 * q^57 + 23 * q^58 - 28 * q^59 - 96 * q^60 + 19 * q^61 + 11 * q^62 + 32 * q^63 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 - 19 * q^71 - 34 * q^72 - 20 * q^73 + 22 * q^74 - 50 * q^75 + 25 * q^76 - 94 * q^78 - 69 * q^79 - 36 * q^80 + 47 * q^81 - 61 * q^82 - q^83 + 28 * q^84 - 24 * q^85 - 27 * q^86 + 58 * q^87 - 36 * q^90 + 24 * q^91 - 67 * q^92 - 14 * q^93 - 64 * q^94 + 3 * q^95 + 26 * q^96 + 21 * q^97 + 87 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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