LMFDB - Embedded newform 3381.2.a.bf.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 15x^{6} + 11x^{5} + 75x^{4} - 35x^{3} - 141x^{2} + 37x + 80 $ x^8 - x^7 - 15x^6 + 11x^5 + 75x^4 - 35x^3 - 141x^2 + 37x + 80
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.818544$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.818544 q^{2} +1.00000 q^{3} -1.32999 q^{4} +3.72041 q^{5} -0.818544 q^{6} +2.72574 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.818544 q^{2} +1.00000 q^{3} -1.32999 q^{4} +3.72041 q^{5} -0.818544 q^{6} +2.72574 q^{8} +1.00000 q^{9} -3.04532 q^{10} +3.70004 q^{11} -1.32999 q^{12} -6.90735 q^{13} +3.72041 q^{15} +0.428834 q^{16} -3.21772 q^{17} -0.818544 q^{18} +6.07700 q^{19} -4.94809 q^{20} -3.02865 q^{22} -1.00000 q^{23} +2.72574 q^{24} +8.84143 q^{25} +5.65397 q^{26} +1.00000 q^{27} -3.33137 q^{29} -3.04532 q^{30} +0.638468 q^{31} -5.80250 q^{32} +3.70004 q^{33} +2.63384 q^{34} -1.32999 q^{36} +7.78032 q^{37} -4.97429 q^{38} -6.90735 q^{39} +10.1409 q^{40} +10.4188 q^{41} +4.34017 q^{43} -4.92100 q^{44} +3.72041 q^{45} +0.818544 q^{46} -6.02770 q^{47} +0.428834 q^{48} -7.23710 q^{50} -3.21772 q^{51} +9.18668 q^{52} +7.27207 q^{53} -0.818544 q^{54} +13.7657 q^{55} +6.07700 q^{57} +2.72687 q^{58} -5.29661 q^{59} -4.94809 q^{60} +8.11466 q^{61} -0.522615 q^{62} +3.89193 q^{64} -25.6981 q^{65} -3.02865 q^{66} -4.01771 q^{67} +4.27952 q^{68} -1.00000 q^{69} -2.92567 q^{71} +2.72574 q^{72} +9.14983 q^{73} -6.36854 q^{74} +8.84143 q^{75} -8.08232 q^{76} +5.65397 q^{78} +8.17142 q^{79} +1.59544 q^{80} +1.00000 q^{81} -8.52824 q^{82} +6.01247 q^{83} -11.9712 q^{85} -3.55262 q^{86} -3.33137 q^{87} +10.0853 q^{88} +14.4303 q^{89} -3.04532 q^{90} +1.32999 q^{92} +0.638468 q^{93} +4.93394 q^{94} +22.6089 q^{95} -5.80250 q^{96} -14.8716 q^{97} +3.70004 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + q^2 + 8 * q^3 + 15 * q^4 + 5 * q^5 + q^6 + 9 * q^8 + 8 * q^9	$ 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9} - 3 q^{10} + 10 q^{11} + 15 q^{12} - 6 q^{13} + 5 q^{15} + 13 q^{16} + 21 q^{17} + q^{18} + 5 q^{19} - q^{20} + 18 q^{22} - 8 q^{23} + 9 q^{24} + 27 q^{25} + 3 q^{26} + 8 q^{27} + 2 q^{29} - 3 q^{30} - 13 q^{31} + 29 q^{32} + 10 q^{33} - 19 q^{34} + 15 q^{36} + 13 q^{37} - 6 q^{38} - 6 q^{39} + 7 q^{40} + 16 q^{41} + 15 q^{43} + 24 q^{44} + 5 q^{45} - q^{46} - q^{47} + 13 q^{48} + 16 q^{50} + 21 q^{51} - 19 q^{52} + 3 q^{53} + q^{54} - 10 q^{55} + 5 q^{57} - 40 q^{58} + 26 q^{59} - q^{60} - 14 q^{61} - 14 q^{62} + 49 q^{64} - 3 q^{65} + 18 q^{66} + 38 q^{67} + 43 q^{68} - 8 q^{69} + 9 q^{71} + 9 q^{72} - 6 q^{73} + 32 q^{74} + 27 q^{75} - 14 q^{76} + 3 q^{78} + 23 q^{79} - 17 q^{80} + 8 q^{81} + 20 q^{82} + 30 q^{83} - 37 q^{85} - 28 q^{86} + 2 q^{87} + 86 q^{88} + 12 q^{89} - 3 q^{90} - 15 q^{92} - 13 q^{93} - 45 q^{94} + 16 q^{95} + 29 q^{96} + 14 q^{97} + 10 q^{99}+O(q^{100}) $ 8 * q + q^2 + 8 * q^3 + 15 * q^4 + 5 * q^5 + q^6 + 9 * q^8 + 8 * q^9 - 3 * q^10 + 10 * q^11 + 15 * q^12 - 6 * q^13 + 5 * q^15 + 13 * q^16 + 21 * q^17 + q^18 + 5 * q^19 - q^20 + 18 * q^22 - 8 * q^23 + 9 * q^24 + 27 * q^25 + 3 * q^26 + 8 * q^27 + 2 * q^29 - 3 * q^30 - 13 * q^31 + 29 * q^32 + 10 * q^33 - 19 * q^34 + 15 * q^36 + 13 * q^37 - 6 * q^38 - 6 * q^39 + 7 * q^40 + 16 * q^41 + 15 * q^43 + 24 * q^44 + 5 * q^45 - q^46 - q^47 + 13 * q^48 + 16 * q^50 + 21 * q^51 - 19 * q^52 + 3 * q^53 + q^54 - 10 * q^55 + 5 * q^57 - 40 * q^58 + 26 * q^59 - q^60 - 14 * q^61 - 14 * q^62 + 49 * q^64 - 3 * q^65 + 18 * q^66 + 38 * q^67 + 43 * q^68 - 8 * q^69 + 9 * q^71 + 9 * q^72 - 6 * q^73 + 32 * q^74 + 27 * q^75 - 14 * q^76 + 3 * q^78 + 23 * q^79 - 17 * q^80 + 8 * q^81 + 20 * q^82 + 30 * q^83 - 37 * q^85 - 28 * q^86 + 2 * q^87 + 86 * q^88 + 12 * q^89 - 3 * q^90 - 15 * q^92 - 13 * q^93 - 45 * q^94 + 16 * q^95 + 29 * q^96 + 14 * q^97 + 10 * q^99

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 $

$2$	−0.818544	−0.578798	−0.289399	−	0.957209i	$-0.593455\pi$
$2$	−0.818544	−0.578798	−0.289399	+	0.957209i	$0.593455\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.32999	−0.664993
$5$	3.72041	1.66382	0.831908	−	0.554913i	$-0.187249\pi$
$5$	3.72041	1.66382	0.831908	+	0.554913i	$0.187249\pi$
$6$	−0.818544	−0.334169
$7$	0	0
$7$	0	0
$8$	2.72574	0.963695
$9$	1.00000	0.333333
$10$	−3.04532	−0.963014
$11$	3.70004	1.11560	0.557802	−	0.829974i	$-0.311645\pi$
$11$	3.70004	1.11560	0.557802	+	0.829974i	$0.311645\pi$
$12$	−1.32999	−0.383934
$13$	−6.90735	−1.91575	−0.957877	−	0.287179i	$-0.907283\pi$
$13$	−6.90735	−1.91575	−0.957877	+	0.287179i	$0.907283\pi$
$14$	0	0
$15$	3.72041	0.960605
$16$	0.428834	0.107208
$17$	−3.21772	−0.780411	−0.390206	−	0.920728i	$-0.627596\pi$
$17$	−3.21772	−0.780411	−0.390206	+	0.920728i	$0.627596\pi$
$18$	−0.818544	−0.192933
$19$	6.07700	1.39416	0.697080	−	0.716994i	$-0.254482\pi$
$19$	6.07700	1.39416	0.697080	+	0.716994i	$0.254482\pi$
$20$	−4.94809	−1.10643
$21$	0	0
$22$	−3.02865	−0.645710
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	2.72574	0.556389
$25$	8.84143	1.76829
$26$	5.65397	1.10883
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.33137	−0.618619	−0.309310	−	0.950961i	$-0.600098\pi$
$29$	−3.33137	−0.618619	−0.309310	+	0.950961i	$0.600098\pi$
$30$	−3.04532	−0.555996
$31$	0.638468	0.114672	0.0573362	−	0.998355i	$-0.481739\pi$
$31$	0.638468	0.114672	0.0573362	+	0.998355i	$0.481739\pi$
$32$	−5.80250	−1.02575
$33$	3.70004	0.644094
$34$	2.63384	0.451700
$35$	0	0
$36$	−1.32999	−0.221664
$37$	7.78032	1.27908	0.639539	−	0.768759i	$-0.279125\pi$
$37$	7.78032	1.27908	0.639539	+	0.768759i	$0.279125\pi$
$38$	−4.97429	−0.806937
$39$	−6.90735	−1.10606
$40$	10.1409	1.60341
$41$	10.4188	1.62714	0.813571	−	0.581466i	$-0.197521\pi$
$41$	10.4188	1.62714	0.813571	+	0.581466i	$0.197521\pi$
$42$	0	0
$43$	4.34017	0.661870	0.330935	−	0.943654i	$-0.392636\pi$
$43$	4.34017	0.661870	0.330935	+	0.943654i	$0.392636\pi$
$44$	−4.92100	−0.741869
$45$	3.72041	0.554606
$46$	0.818544	0.120688
$47$	−6.02770	−0.879230	−0.439615	−	0.898186i	$-0.644885\pi$
$47$	−6.02770	−0.879230	−0.439615	+	0.898186i	$0.644885\pi$
$48$	0.428834	0.0618968
$49$	0	0
$50$	−7.23710	−1.02348
$51$	−3.21772	−0.450571
$52$	9.18668	1.27396
$53$	7.27207	0.998896	0.499448	−	0.866344i	$-0.333536\pi$
$53$	7.27207	0.998896	0.499448	+	0.866344i	$0.333536\pi$
$54$	−0.818544	−0.111390
$55$	13.7657	1.85616
$56$	0	0
$57$	6.07700	0.804918
$58$	2.72687	0.358056
$59$	−5.29661	−0.689560	−0.344780	−	0.938683i	$-0.612047\pi$
$59$	−5.29661	−0.689560	−0.344780	+	0.938683i	$0.612047\pi$
$60$	−4.94809	−0.638795
$61$	8.11466	1.03898	0.519488	−	0.854478i	$-0.326123\pi$
$61$	8.11466	1.03898	0.519488	+	0.854478i	$0.326123\pi$
$62$	−0.522615	−0.0663721
$63$	0	0
$64$	3.89193	0.486492
$65$	−25.6981	−3.18746
$66$	−3.02865	−0.372801
$67$	−4.01771	−0.490841	−0.245420	−	0.969417i	$-0.578926\pi$
$67$	−4.01771	−0.490841	−0.245420	+	0.969417i	$0.578926\pi$
$68$	4.27952	0.518968
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−2.92567	−0.347213	−0.173606	−	0.984815i	$-0.555542\pi$
$71$	−2.92567	−0.347213	−0.173606	+	0.984815i	$0.555542\pi$
$72$	2.72574	0.321232
$73$	9.14983	1.07091	0.535453	−	0.844565i	$-0.320141\pi$
$73$	9.14983	1.07091	0.535453	+	0.844565i	$0.320141\pi$
$74$	−6.36854	−0.740327
$75$	8.84143	1.02092
$76$	−8.08232	−0.927106
$77$	0	0
$78$	5.65397	0.640186
$79$	8.17142	0.919357	0.459678	−	0.888085i	$-0.347965\pi$
$79$	8.17142	0.919357	0.459678	+	0.888085i	$0.347965\pi$
$80$	1.59544	0.178375
$81$	1.00000	0.111111
$82$	−8.52824	−0.941786
$83$	6.01247	0.659954	0.329977	−	0.943989i	$-0.392959\pi$
$83$	6.01247	0.659954	0.329977	+	0.943989i	$0.392959\pi$
$84$	0	0
$85$	−11.9712	−1.29846
$86$	−3.55262	−0.383089
$87$	−3.33137	−0.357160
$88$	10.0853	1.07510
$89$	14.4303	1.52961	0.764805	−	0.644262i	$-0.222835\pi$
$89$	14.4303	1.52961	0.764805	+	0.644262i	$0.222835\pi$
$90$	−3.04532	−0.321005
$91$	0	0
$92$	1.32999	0.138661
$93$	0.638468	0.0662061
$94$	4.93394	0.508897
$95$	22.6089	2.31963
$96$	−5.80250	−0.592215
$97$	−14.8716	−1.50998	−0.754991	−	0.655735i	$-0.772359\pi$
$97$	−14.8716	−1.50998	−0.754991	+	0.655735i	$0.772359\pi$
$98$	0	0
$99$	3.70004	0.371868
$100$	−11.7590	−1.17590
$101$	4.20008	0.417924	0.208962	−	0.977924i	$-0.432992\pi$
$101$	4.20008	0.417924	0.208962	+	0.977924i	$0.432992\pi$
$102$	2.63384	0.260789
$103$	−17.4541	−1.71980	−0.859900	−	0.510462i	$-0.829474\pi$
$103$	−17.4541	−1.71980	−0.859900	+	0.510462i	$0.829474\pi$
$104$	−18.8276	−1.84620
$105$	0	0
$106$	−5.95251	−0.578159
$107$	−0.974217	−0.0941811	−0.0470905	−	0.998891i	$-0.514995\pi$
$107$	−0.974217	−0.0941811	−0.0470905	+	0.998891i	$0.514995\pi$
$108$	−1.32999	−0.127978
$109$	0.186541	0.0178674	0.00893371	−	0.999960i	$-0.497156\pi$
$109$	0.186541	0.0178674	0.00893371	+	0.999960i	$0.497156\pi$
$110$	−11.2678	−1.07434
$111$	7.78032	0.738476
$112$	0	0
$113$	2.70857	0.254801	0.127400	−	0.991851i	$-0.459337\pi$
$113$	2.70857	0.254801	0.127400	+	0.991851i	$0.459337\pi$
$114$	−4.97429	−0.465885
$115$	−3.72041	−0.346930
$116$	4.43067	0.411377
$117$	−6.90735	−0.638585
$118$	4.33551	0.399116
$119$	0	0
$120$	10.1409	0.925730
$121$	2.69030	0.244573
$122$	−6.64220	−0.601357
$123$	10.4188	0.939431
$124$	−0.849154	−0.0762563
$125$	14.2917	1.27829
$126$	0	0
$127$	6.67778	0.592557	0.296278	−	0.955102i	$-0.404254\pi$
$127$	6.67778	0.592557	0.296278	+	0.955102i	$0.404254\pi$
$128$	8.41928	0.744166
$129$	4.34017	0.382131
$130$	21.0351	1.84490
$131$	7.70604	0.673280	0.336640	−	0.941633i	$-0.390709\pi$
$131$	7.70604	0.673280	0.336640	+	0.941633i	$0.390709\pi$
$132$	−4.92100	−0.428318
$133$	0	0
$134$	3.28867	0.284098
$135$	3.72041	0.320202
$136$	−8.77066	−0.752078
$137$	−20.3951	−1.74247	−0.871234	−	0.490867i	$-0.836680\pi$
$137$	−20.3951	−1.74247	−0.871234	+	0.490867i	$0.836680\pi$
$138$	0.818544	0.0696791
$139$	−12.1820	−1.03326	−0.516632	−	0.856208i	$-0.672814\pi$
$139$	−12.1820	−1.03326	−0.516632	+	0.856208i	$0.672814\pi$
$140$	0	0
$141$	−6.02770	−0.507624
$142$	2.39479	0.200966
$143$	−25.5575	−2.13722
$144$	0.428834	0.0357361
$145$	−12.3940	−1.02927
$146$	−7.48953	−0.619838
$147$	0	0
$148$	−10.3477	−0.850577
$149$	−1.00394	−0.0822459	−0.0411229	−	0.999154i	$-0.513094\pi$
$149$	−1.00394	−0.0822459	−0.0411229	+	0.999154i	$0.513094\pi$
$150$	−7.23710	−0.590907
$151$	−2.71063	−0.220588	−0.110294	−	0.993899i	$-0.535179\pi$
$151$	−2.71063	−0.220588	−0.110294	+	0.993899i	$0.535179\pi$
$152$	16.5643	1.34354
$153$	−3.21772	−0.260137
$154$	0	0
$155$	2.37536	0.190794
$156$	9.18668	0.735523
$157$	3.02533	0.241448	0.120724	−	0.992686i	$-0.461478\pi$
$157$	3.02533	0.241448	0.120724	+	0.992686i	$0.461478\pi$
$158$	−6.68867	−0.532122
$159$	7.27207	0.576713
$160$	−21.5877	−1.70665
$161$	0	0
$162$	−0.818544	−0.0643109
$163$	7.68199	0.601700	0.300850	−	0.953671i	$-0.402730\pi$
$163$	7.68199	0.601700	0.300850	+	0.953671i	$0.402730\pi$
$164$	−13.8568	−1.08204
$165$	13.7657	1.07166
$166$	−4.92147	−0.381980
$167$	4.58319	0.354658	0.177329	−	0.984152i	$-0.443254\pi$
$167$	4.58319	0.354658	0.177329	+	0.984152i	$0.443254\pi$
$168$	0	0
$169$	34.7115	2.67011
$170$	9.79897	0.751547
$171$	6.07700	0.464720
$172$	−5.77237	−0.440139
$173$	−20.0190	−1.52201	−0.761006	−	0.648744i	$-0.775294\pi$
$173$	−20.0190	−1.52201	−0.761006	+	0.648744i	$0.775294\pi$
$174$	2.72687	0.206723
$175$	0	0
$176$	1.58670	0.119602
$177$	−5.29661	−0.398118
$178$	−11.8118	−0.885336
$179$	−4.71460	−0.352385	−0.176193	−	0.984356i	$-0.556378\pi$
$179$	−4.71460	−0.352385	−0.176193	+	0.984356i	$0.556378\pi$
$180$	−4.94809	−0.368809
$181$	−1.17108	−0.0870460	−0.0435230	−	0.999052i	$-0.513858\pi$
$181$	−1.17108	−0.0870460	−0.0435230	+	0.999052i	$0.513858\pi$
$182$	0	0
$183$	8.11466	0.599853
$184$	−2.72574	−0.200944
$185$	28.9460	2.12815
$186$	−0.522615	−0.0383200
$187$	−11.9057	−0.870630
$188$	8.01676	0.584682
$189$	0	0
$190$	−18.5064	−1.34259
$191$	24.1866	1.75008	0.875039	−	0.484052i	$-0.160836\pi$
$191$	24.1866	1.75008	0.875039	+	0.484052i	$0.160836\pi$
$192$	3.89193	0.280876
$193$	12.3253	0.887194	0.443597	−	0.896226i	$-0.353702\pi$
$193$	12.3253	0.887194	0.443597	+	0.896226i	$0.353702\pi$
$194$	12.1731	0.873975
$195$	−25.6981	−1.84028
$196$	0	0
$197$	−2.76210	−0.196792	−0.0983958	−	0.995147i	$-0.531371\pi$
$197$	−2.76210	−0.196792	−0.0983958	+	0.995147i	$0.531371\pi$
$198$	−3.02865	−0.215237
$199$	16.5812	1.17541	0.587704	−	0.809076i	$-0.300032\pi$
$199$	16.5812	1.17541	0.587704	+	0.809076i	$0.300032\pi$
$200$	24.0994	1.70409
$201$	−4.01771	−0.283387
$202$	−3.43795	−0.241893
$203$	0	0
$204$	4.27952	0.299626
$205$	38.7621	2.70727
$206$	14.2869	0.995417
$207$	−1.00000	−0.0695048
$208$	−2.96210	−0.205385
$209$	22.4851	1.55533
$210$	0	0
$211$	2.75812	0.189877	0.0949386	−	0.995483i	$-0.469735\pi$
$211$	2.75812	0.189877	0.0949386	+	0.995483i	$0.469735\pi$
$212$	−9.67175	−0.664259
$213$	−2.92567	−0.200463
$214$	0.797439	0.0545118
$215$	16.1472	1.10123
$216$	2.72574	0.185463
$217$	0	0
$218$	−0.152692	−0.0103416
$219$	9.14983	0.618288
$220$	−18.3081	−1.23433
$221$	22.2259	1.49508
$222$	−6.36854	−0.427428
$223$	−25.8192	−1.72898	−0.864490	−	0.502650i	$-0.832358\pi$
$223$	−25.8192	−1.72898	−0.864490	+	0.502650i	$0.832358\pi$
$224$	0	0
$225$	8.84143	0.589429
$226$	−2.21708	−0.147478
$227$	8.37096	0.555600	0.277800	−	0.960639i	$-0.410395\pi$
$227$	8.37096	0.555600	0.277800	+	0.960639i	$0.410395\pi$
$228$	−8.08232	−0.535265
$229$	19.5187	1.28983	0.644915	−	0.764254i	$-0.276893\pi$
$229$	19.5187	1.28983	0.644915	+	0.764254i	$0.276893\pi$
$230$	3.04532	0.200802
$231$	0	0
$232$	−9.08044	−0.596160
$233$	6.25868	0.410020	0.205010	−	0.978760i	$-0.434277\pi$
$233$	6.25868	0.410020	0.205010	+	0.978760i	$0.434277\pi$
$234$	5.65397	0.369612
$235$	−22.4255	−1.46288
$236$	7.04442	0.458553
$237$	8.17142	0.530791
$238$	0	0
$239$	−4.53876	−0.293588	−0.146794	−	0.989167i	$-0.546895\pi$
$239$	−4.53876	−0.293588	−0.146794	+	0.989167i	$0.546895\pi$
$240$	1.59544	0.102985
$241$	−19.6771	−1.26751	−0.633756	−	0.773533i	$-0.718488\pi$
$241$	−19.6771	−1.26751	−0.633756	+	0.773533i	$0.718488\pi$
$242$	−2.20213	−0.141558
$243$	1.00000	0.0641500
$244$	−10.7924	−0.690911
$245$	0	0
$246$	−8.52824	−0.543741
$247$	−41.9760	−2.67087
$248$	1.74030	0.110509
$249$	6.01247	0.381025
$250$	−11.6984	−0.739869
$251$	−10.3015	−0.650226	−0.325113	−	0.945675i	$-0.605402\pi$
$251$	−10.3015	−0.650226	−0.325113	+	0.945675i	$0.605402\pi$
$252$	0	0
$253$	−3.70004	−0.232620
$254$	−5.46606	−0.342971
$255$	−11.9712	−0.749667
$256$	−14.6754	−0.917214
$257$	15.3148	0.955312	0.477656	−	0.878547i	$-0.341487\pi$
$257$	15.3148	0.955312	0.477656	+	0.878547i	$0.341487\pi$
$258$	−3.55262	−0.221177
$259$	0	0
$260$	34.1782	2.11964
$261$	−3.33137	−0.206206
$262$	−6.30773	−0.389693
$263$	−5.31749	−0.327891	−0.163945	−	0.986469i	$-0.552422\pi$
$263$	−5.31749	−0.327891	−0.163945	+	0.986469i	$0.552422\pi$
$264$	10.0853	0.620710
$265$	27.0551	1.66198
$266$	0	0
$267$	14.4303	0.883121
$268$	5.34349	0.326406
$269$	−1.76465	−0.107592	−0.0537962	−	0.998552i	$-0.517132\pi$
$269$	−1.76465	−0.107592	−0.0537962	+	0.998552i	$0.517132\pi$
$270$	−3.04532	−0.185332
$271$	16.6166	1.00938	0.504692	−	0.863300i	$-0.331606\pi$
$271$	16.6166	1.00938	0.504692	+	0.863300i	$0.331606\pi$
$272$	−1.37987	−0.0836666
$273$	0	0
$274$	16.6943	1.00854
$275$	32.7136	1.97271
$276$	1.32999	0.0800557
$277$	1.95295	0.117342	0.0586708	−	0.998277i	$-0.481314\pi$
$277$	1.95295	0.117342	0.0586708	+	0.998277i	$0.481314\pi$
$278$	9.97150	0.598051
$279$	0.638468	0.0382241
$280$	0	0
$281$	13.1183	0.782575	0.391287	−	0.920268i	$-0.372030\pi$
$281$	13.1183	0.782575	0.391287	+	0.920268i	$0.372030\pi$
$282$	4.93394	0.293812
$283$	0.979872	0.0582473	0.0291237	−	0.999576i	$-0.490728\pi$
$283$	0.979872	0.0582473	0.0291237	+	0.999576i	$0.490728\pi$
$284$	3.89110	0.230894
$285$	22.6089	1.33924
$286$	20.9199	1.23702
$287$	0	0
$288$	−5.80250	−0.341916
$289$	−6.64630	−0.390959
$290$	10.1451	0.595739
$291$	−14.8716	−0.871789
$292$	−12.1691	−0.712145
$293$	−22.4505	−1.31157	−0.655786	−	0.754947i	$-0.727663\pi$
$293$	−22.4505	−1.31157	−0.655786	+	0.754947i	$0.727663\pi$
$294$	0	0
$295$	−19.7056	−1.14730
$296$	21.2071	1.23264
$297$	3.70004	0.214698
$298$	0.821768	0.0476038
$299$	6.90735	0.399462
$300$	−11.7590	−0.678905
$301$	0	0
$302$	2.21877	0.127676
$303$	4.20008	0.241288
$304$	2.60602	0.149466
$305$	30.1898	1.72866
$306$	2.63384	0.150567
$307$	29.2019	1.66664	0.833322	−	0.552789i	$-0.186436\pi$
$307$	29.2019	1.66664	0.833322	+	0.552789i	$0.186436\pi$
$308$	0	0
$309$	−17.4541	−0.992927
$310$	−1.94434	−0.110431
$311$	1.79193	0.101611	0.0508055	−	0.998709i	$-0.483821\pi$
$311$	1.79193	0.101611	0.0508055	+	0.998709i	$0.483821\pi$
$312$	−18.8276	−1.06591
$313$	−8.83165	−0.499194	−0.249597	−	0.968350i	$-0.580298\pi$
$313$	−8.83165	−0.499194	−0.249597	+	0.968350i	$0.580298\pi$
$314$	−2.47637	−0.139750
$315$	0	0
$316$	−10.8679	−0.611366
$317$	−21.5495	−1.21034	−0.605170	−	0.796096i	$-0.706895\pi$
$317$	−21.5495	−1.21034	−0.605170	+	0.796096i	$0.706895\pi$
$318$	−5.95251	−0.333800
$319$	−12.3262	−0.690134
$320$	14.4796	0.809433
$321$	−0.974217	−0.0543755
$322$	0	0
$323$	−19.5541	−1.08802
$324$	−1.32999	−0.0738881
$325$	−61.0708	−3.38760
$326$	−6.28805	−0.348263
$327$	0.186541	0.0103158
$328$	28.3989	1.56807
$329$	0	0
$330$	−11.2678	−0.620272
$331$	23.1086	1.27016	0.635082	−	0.772444i	$-0.280966\pi$
$331$	23.1086	1.27016	0.635082	+	0.772444i	$0.280966\pi$
$332$	−7.99650	−0.438865
$333$	7.78032	0.426359
$334$	−3.75154	−0.205275
$335$	−14.9475	−0.816669
$336$	0	0
$337$	4.83315	0.263279	0.131639	−	0.991298i	$-0.457976\pi$
$337$	4.83315	0.263279	0.131639	+	0.991298i	$0.457976\pi$
$338$	−28.4129	−1.54546
$339$	2.70857	0.147109
$340$	15.9215	0.863467
$341$	2.36236	0.127929
$342$	−4.97429	−0.268979
$343$	0	0
$344$	11.8302	0.637841
$345$	−3.72041	−0.200300
$346$	16.3864	0.880938
$347$	2.31584	0.124321	0.0621605	−	0.998066i	$-0.480201\pi$
$347$	2.31584	0.124321	0.0621605	+	0.998066i	$0.480201\pi$
$348$	4.43067	0.237509
$349$	2.05541	0.110024	0.0550119	−	0.998486i	$-0.482480\pi$
$349$	2.05541	0.110024	0.0550119	+	0.998486i	$0.482480\pi$
$350$	0	0
$351$	−6.90735	−0.368687
$352$	−21.4695	−1.14433
$353$	18.5639	0.988058	0.494029	−	0.869445i	$-0.335524\pi$
$353$	18.5639	0.988058	0.494029	+	0.869445i	$0.335524\pi$
$354$	4.33551	0.230430
$355$	−10.8847	−0.577698
$356$	−19.1921	−1.01718
$357$	0	0
$358$	3.85910	0.203960
$359$	−7.27408	−0.383911	−0.191956	−	0.981404i	$-0.561483\pi$
$359$	−7.27408	−0.383911	−0.191956	+	0.981404i	$0.561483\pi$
$360$	10.1409	0.534470
$361$	17.9299	0.943681
$362$	0.958584	0.0503821
$363$	2.69030	0.141204
$364$	0	0
$365$	34.0411	1.78179
$366$	−6.64220	−0.347194
$367$	−20.2397	−1.05651	−0.528253	−	0.849087i	$-0.677153\pi$
$367$	−20.2397	−1.05651	−0.528253	+	0.849087i	$0.677153\pi$
$368$	−0.428834	−0.0223545
$369$	10.4188	0.542381
$370$	−23.6935	−1.23177
$371$	0	0
$372$	−0.849154	−0.0440266
$373$	13.3237	0.689876	0.344938	−	0.938626i	$-0.387900\pi$
$373$	13.3237	0.689876	0.344938	+	0.938626i	$0.387900\pi$
$374$	9.74533	0.503919
$375$	14.2917	0.738019
$376$	−16.4299	−0.847310
$377$	23.0109	1.18512
$378$	0	0
$379$	9.15684	0.470355	0.235178	−	0.971952i	$-0.424433\pi$
$379$	9.15684	0.470355	0.235178	+	0.971952i	$0.424433\pi$
$380$	−30.0695	−1.54253
$381$	6.67778	0.342113
$382$	−19.7978	−1.01294
$383$	8.98554	0.459140	0.229570	−	0.973292i	$-0.426268\pi$
$383$	8.98554	0.459140	0.229570	+	0.973292i	$0.426268\pi$
$384$	8.41928	0.429645
$385$	0	0
$386$	−10.0888	−0.513506
$387$	4.34017	0.220623
$388$	19.7790	1.00413
$389$	−16.3724	−0.830115	−0.415057	−	0.909795i	$-0.636238\pi$
$389$	−16.3724	−0.830115	−0.415057	+	0.909795i	$0.636238\pi$
$390$	21.0351	1.06515
$391$	3.21772	0.162727
$392$	0	0
$393$	7.70604	0.388718
$394$	2.26090	0.113903
$395$	30.4010	1.52964
$396$	−4.92100	−0.247290
$397$	9.68491	0.486072	0.243036	−	0.970017i	$-0.421857\pi$
$397$	9.68491	0.486072	0.243036	+	0.970017i	$0.421857\pi$
$398$	−13.5724	−0.680323
$399$	0	0
$400$	3.79150	0.189575
$401$	11.4662	0.572593	0.286297	−	0.958141i	$-0.407576\pi$
$401$	11.4662	0.572593	0.286297	+	0.958141i	$0.407576\pi$
$402$	3.28867	0.164024
$403$	−4.41012	−0.219684
$404$	−5.58605	−0.277916
$405$	3.72041	0.184869
$406$	0	0
$407$	28.7875	1.42694
$408$	−8.77066	−0.434212
$409$	−38.6532	−1.91128	−0.955639	−	0.294541i	$-0.904833\pi$
$409$	−38.6532	−1.91128	−0.955639	+	0.294541i	$0.904833\pi$
$410$	−31.7285	−1.56696
$411$	−20.3951	−1.00601
$412$	23.2137	1.14365
$413$	0	0
$414$	0.818544	0.0402292
$415$	22.3688	1.09804
$416$	40.0799	1.96508
$417$	−12.1820	−0.596555
$418$	−18.4051	−0.900222
$419$	−6.70860	−0.327736	−0.163868	−	0.986482i	$-0.552397\pi$
$419$	−6.70860	−0.327736	−0.163868	+	0.986482i	$0.552397\pi$
$420$	0	0
$421$	−34.0515	−1.65957	−0.829783	−	0.558086i	$-0.811536\pi$
$421$	−34.0515	−1.65957	−0.829783	+	0.558086i	$0.811536\pi$
$422$	−2.25765	−0.109900
$423$	−6.02770	−0.293077
$424$	19.8218	0.962630
$425$	−28.4492	−1.37999
$426$	2.39479	0.116028
$427$	0	0
$428$	1.29569	0.0626298
$429$	−25.5575	−1.23393
$430$	−13.2172	−0.637390
$431$	7.46897	0.359768	0.179884	−	0.983688i	$-0.442428\pi$
$431$	7.46897	0.359768	0.179884	+	0.983688i	$0.442428\pi$
$432$	0.428834	0.0206323
$433$	−25.8698	−1.24322	−0.621611	−	0.783326i	$-0.713522\pi$
$433$	−25.8698	−1.24322	−0.621611	+	0.783326i	$0.713522\pi$
$434$	0	0
$435$	−12.3940	−0.594249
$436$	−0.248097	−0.0118817
$437$	−6.07700	−0.290702
$438$	−7.48953	−0.357864
$439$	−15.9723	−0.762318	−0.381159	−	0.924509i	$-0.624475\pi$
$439$	−15.9723	−0.762318	−0.381159	+	0.924509i	$0.624475\pi$
$440$	37.5216	1.78877
$441$	0	0
$442$	−18.1929	−0.865347
$443$	30.6283	1.45520	0.727598	−	0.686004i	$-0.240637\pi$
$443$	30.6283	1.45520	0.727598	+	0.686004i	$0.240637\pi$
$444$	−10.3477	−0.491081
$445$	53.6867	2.54499
$446$	21.1341	1.00073
$447$	−1.00394	−0.0474847
$448$	0	0
$449$	−7.12584	−0.336289	−0.168144	−	0.985762i	$-0.553778\pi$
$449$	−7.12584	−0.336289	−0.168144	+	0.985762i	$0.553778\pi$
$450$	−7.23710	−0.341160
$451$	38.5500	1.81525
$452$	−3.60236	−0.169441
$453$	−2.71063	−0.127357
$454$	−6.85200	−0.321580
$455$	0	0
$456$	16.5643	0.775695
$457$	20.6851	0.967609	0.483805	−	0.875176i	$-0.339254\pi$
$457$	20.6851	0.967609	0.483805	+	0.875176i	$0.339254\pi$
$458$	−15.9769	−0.746551
$459$	−3.21772	−0.150190
$460$	4.94809	0.230706
$461$	12.1611	0.566401	0.283200	−	0.959061i	$-0.408604\pi$
$461$	12.1611	0.566401	0.283200	+	0.959061i	$0.408604\pi$
$462$	0	0
$463$	−1.96022	−0.0910992	−0.0455496	−	0.998962i	$-0.514504\pi$
$463$	−1.96022	−0.0910992	−0.0455496	+	0.998962i	$0.514504\pi$
$464$	−1.42860	−0.0663212
$465$	2.37536	0.110155
$466$	−5.12301	−0.237319
$467$	−39.3922	−1.82285	−0.911427	−	0.411463i	$-0.865018\pi$
$467$	−39.3922	−1.82285	−0.911427	+	0.411463i	$0.865018\pi$
$468$	9.18668	0.424654
$469$	0	0
$470$	18.3563	0.846711
$471$	3.02533	0.139400
$472$	−14.4372	−0.664525
$473$	16.0588	0.738385
$474$	−6.68867	−0.307221
$475$	53.7294	2.46527
$476$	0	0
$477$	7.27207	0.332965
$478$	3.71517	0.169928
$479$	−2.82732	−0.129183	−0.0645917	−	0.997912i	$-0.520575\pi$
$479$	−2.82732	−0.129183	−0.0645917	+	0.997912i	$0.520575\pi$
$480$	−21.5877	−0.985337
$481$	−53.7414	−2.45040
$482$	16.1066	0.733634
$483$	0	0
$484$	−3.57806	−0.162639
$485$	−55.3284	−2.51233
$486$	−0.818544	−0.0371299
$487$	−24.7848	−1.12311	−0.561554	−	0.827440i	$-0.689796\pi$
$487$	−24.7848	−1.12311	−0.561554	+	0.827440i	$0.689796\pi$
$488$	22.1184	1.00125
$489$	7.68199	0.347392
$490$	0	0
$491$	13.1356	0.592802	0.296401	−	0.955064i	$-0.404214\pi$
$491$	13.1356	0.592802	0.296401	+	0.955064i	$0.404214\pi$
$492$	−13.8568	−0.624715
$493$	10.7194	0.482777
$494$	34.3592	1.54589
$495$	13.7657	0.618720
$496$	0.273797	0.0122938
$497$	0	0
$498$	−4.92147	−0.220536
$499$	16.4810	0.737790	0.368895	−	0.929471i	$-0.379736\pi$
$499$	16.4810	0.737790	0.368895	+	0.929471i	$0.379736\pi$
$500$	−19.0077	−0.850051
$501$	4.58319	0.204762
$502$	8.43224	0.376349
$503$	17.3522	0.773696	0.386848	−	0.922144i	$-0.373564\pi$
$503$	17.3522	0.773696	0.386848	+	0.922144i	$0.373564\pi$
$504$	0	0
$505$	15.6260	0.695348
$506$	3.02865	0.134640
$507$	34.7115	1.54159
$508$	−8.88135	−0.394046
$509$	−34.8099	−1.54292	−0.771461	−	0.636277i	$-0.780473\pi$
$509$	−34.8099	−1.54292	−0.771461	+	0.636277i	$0.780473\pi$
$510$	9.79897	0.433906
$511$	0	0
$512$	−4.82609	−0.213285
$513$	6.07700	0.268306
$514$	−12.5358	−0.552933
$515$	−64.9362	−2.86143
$516$	−5.77237	−0.254114
$517$	−22.3027	−0.980873
$518$	0	0
$519$	−20.0190	−0.878735
$520$	−70.0465	−3.07174
$521$	16.9148	0.741052	0.370526	−	0.928822i	$-0.379177\pi$
$521$	16.9148	0.741052	0.370526	+	0.928822i	$0.379177\pi$
$522$	2.72687	0.119352
$523$	−29.0471	−1.27014	−0.635071	−	0.772454i	$-0.719029\pi$
$523$	−29.0471	−1.27014	−0.635071	+	0.772454i	$0.719029\pi$
$524$	−10.2489	−0.447726
$525$	0	0
$526$	4.35260	0.189783
$527$	−2.05441	−0.0894915
$528$	1.58670	0.0690524
$529$	1.00000	0.0434783
$530$	−22.1458	−0.961950
$531$	−5.29661	−0.229853
$532$	0	0
$533$	−71.9662	−3.11720
$534$	−11.8118	−0.511149
$535$	−3.62448	−0.156700
$536$	−10.9512	−0.473021
$537$	−4.71460	−0.203450
$538$	1.44444	0.0622743
$539$	0	0
$540$	−4.94809	−0.212932
$541$	−3.79565	−0.163188	−0.0815938	−	0.996666i	$-0.526001\pi$
$541$	−3.79565	−0.163188	−0.0815938	+	0.996666i	$0.526001\pi$
$542$	−13.6014	−0.584229
$543$	−1.17108	−0.0502560
$544$	18.6708	0.800504
$545$	0.694010	0.0297281
$546$	0	0
$547$	3.31002	0.141526	0.0707630	−	0.997493i	$-0.477457\pi$
$547$	3.31002	0.141526	0.0707630	+	0.997493i	$0.477457\pi$
$548$	27.1252	1.15873
$549$	8.11466	0.346325
$550$	−26.7776	−1.14180
$551$	−20.2447	−0.862454
$552$	−2.72574	−0.116015
$553$	0	0
$554$	−1.59858	−0.0679171
$555$	28.9460	1.22869
$556$	16.2019	0.687113
$557$	−7.75267	−0.328491	−0.164246	−	0.986419i	$-0.552519\pi$
$557$	−7.75267	−0.328491	−0.164246	+	0.986419i	$0.552519\pi$
$558$	−0.522615	−0.0221240
$559$	−29.9791	−1.26798
$560$	0	0
$561$	−11.9057	−0.502658
$562$	−10.7379	−0.452953
$563$	−16.2180	−0.683508	−0.341754	−	0.939789i	$-0.611021\pi$
$563$	−16.2180	−0.683508	−0.341754	+	0.939789i	$0.611021\pi$
$564$	8.01676	0.337566
$565$	10.0770	0.423942
$566$	−0.802068	−0.0337134
$567$	0	0
$568$	−7.97461	−0.334607
$569$	−24.0539	−1.00839	−0.504196	−	0.863589i	$-0.668211\pi$
$569$	−24.0539	−1.00839	−0.504196	+	0.863589i	$0.668211\pi$
$570$	−18.5064	−0.775147
$571$	43.9333	1.83855	0.919276	−	0.393613i	$-0.128775\pi$
$571$	43.9333	1.83855	0.919276	+	0.393613i	$0.128775\pi$
$572$	33.9911	1.42124
$573$	24.1866	1.01041
$574$	0	0
$575$	−8.84143	−0.368713
$576$	3.89193	0.162164
$577$	−18.0483	−0.751359	−0.375679	−	0.926750i	$-0.622591\pi$
$577$	−18.0483	−0.751359	−0.375679	+	0.926750i	$0.622591\pi$
$578$	5.44029	0.226286
$579$	12.3253	0.512222
$580$	16.4839	0.684456
$581$	0	0
$582$	12.1731	0.504590
$583$	26.9070	1.11437
$584$	24.9400	1.03203
$585$	−25.6981	−1.06249
$586$	18.3767	0.759136
$587$	−0.826642	−0.0341192	−0.0170596	−	0.999854i	$-0.505431\pi$
$587$	−0.826642	−0.0341192	−0.0170596	+	0.999854i	$0.505431\pi$
$588$	0	0
$589$	3.87997	0.159871
$590$	16.1299	0.664056
$591$	−2.76210	−0.113618
$592$	3.33646	0.137128
$593$	−0.685463	−0.0281486	−0.0140743	−	0.999901i	$-0.504480\pi$
$593$	−0.685463	−0.0281486	−0.0140743	+	0.999901i	$0.504480\pi$
$594$	−3.02865	−0.124267
$595$	0	0
$596$	1.33522	0.0546929
$597$	16.5812	0.678622
$598$	−5.65397	−0.231208
$599$	−27.3692	−1.11827	−0.559137	−	0.829075i	$-0.688867\pi$
$599$	−27.3692	−1.11827	−0.559137	+	0.829075i	$0.688867\pi$
$600$	24.0994	0.983855
$601$	−10.3969	−0.424096	−0.212048	−	0.977259i	$-0.568013\pi$
$601$	−10.3969	−0.424096	−0.212048	+	0.977259i	$0.568013\pi$
$602$	0	0
$603$	−4.01771	−0.163614
$604$	3.60510	0.146689
$605$	10.0090	0.406925
$606$	−3.43795	−0.139657
$607$	15.8582	0.643663	0.321832	−	0.946797i	$-0.395701\pi$
$607$	15.8582	0.643663	0.321832	+	0.946797i	$0.395701\pi$
$608$	−35.2618	−1.43005
$609$	0	0
$610$	−24.7117	−1.00055
$611$	41.6354	1.68439
$612$	4.27952	0.172989
$613$	8.49341	0.343046	0.171523	−	0.985180i	$-0.445131\pi$
$613$	8.49341	0.343046	0.171523	+	0.985180i	$0.445131\pi$
$614$	−23.9031	−0.964650
$615$	38.7621	1.56304
$616$	0	0
$617$	−20.0404	−0.806797	−0.403399	−	0.915024i	$-0.632171\pi$
$617$	−20.0404	−0.806797	−0.403399	+	0.915024i	$0.632171\pi$
$618$	14.2869	0.574704
$619$	−27.3390	−1.09885	−0.549424	−	0.835544i	$-0.685153\pi$
$619$	−27.3390	−1.09885	−0.549424	+	0.835544i	$0.685153\pi$
$620$	−3.15920	−0.126876
$621$	−1.00000	−0.0401286
$622$	−1.46677	−0.0588123
$623$	0	0
$624$	−2.96210	−0.118579
$625$	8.96370	0.358548
$626$	7.22909	0.288933
$627$	22.4851	0.897970
$628$	−4.02365	−0.160561
$629$	−25.0349	−0.998206
$630$	0	0
$631$	−12.7286	−0.506717	−0.253358	−	0.967373i	$-0.581535\pi$
$631$	−12.7286	−0.506717	−0.253358	+	0.967373i	$0.581535\pi$
$632$	22.2732	0.885979
$633$	2.75812	0.109626
$634$	17.6392	0.700543
$635$	24.8441	0.985906
$636$	−9.67175	−0.383510
$637$	0	0
$638$	10.0895	0.399448
$639$	−2.92567	−0.115738
$640$	31.3231	1.23816
$641$	14.0195	0.553738	0.276869	−	0.960908i	$-0.410703\pi$
$641$	14.0195	0.553738	0.276869	+	0.960908i	$0.410703\pi$
$642$	0.797439	0.0314724
$643$	10.2819	0.405477	0.202738	−	0.979233i	$-0.435016\pi$
$643$	10.2819	0.405477	0.202738	+	0.979233i	$0.435016\pi$
$644$	0	0
$645$	16.1472	0.635796
$646$	16.0059	0.629742
$647$	35.5195	1.39641	0.698207	−	0.715896i	$-0.253982\pi$
$647$	35.5195	1.39641	0.698207	+	0.715896i	$0.253982\pi$
$648$	2.72574	0.107077
$649$	−19.5977	−0.769276
$650$	49.9892	1.96074
$651$	0	0
$652$	−10.2169	−0.400126
$653$	27.4136	1.07278	0.536389	−	0.843971i	$-0.319788\pi$
$653$	27.4136	1.07278	0.536389	+	0.843971i	$0.319788\pi$
$654$	−0.152692	−0.00597074
$655$	28.6696	1.12021
$656$	4.46793	0.174443
$657$	9.14983	0.356969
$658$	0	0
$659$	32.7484	1.27570	0.637848	−	0.770162i	$-0.279825\pi$
$659$	32.7484	1.27570	0.637848	+	0.770162i	$0.279825\pi$
$660$	−18.3081	−0.712643
$661$	−38.3045	−1.48987	−0.744936	−	0.667136i	$-0.767520\pi$
$661$	−38.3045	−1.48987	−0.744936	+	0.667136i	$0.767520\pi$
$662$	−18.9154	−0.735169
$663$	22.2259	0.863182
$664$	16.3884	0.635994
$665$	0	0
$666$	−6.36854	−0.246776
$667$	3.33137	0.128991
$668$	−6.09557	−0.235845
$669$	−25.8192	−0.998227
$670$	12.2352	0.472686
$671$	30.0246	1.15909
$672$	0	0
$673$	−0.487689	−0.0187990	−0.00939951	−	0.999956i	$-0.502992\pi$
$673$	−0.487689	−0.0187990	−0.00939951	+	0.999956i	$0.502992\pi$
$674$	−3.95615	−0.152385
$675$	8.84143	0.340307
$676$	−46.1658	−1.77561
$677$	−47.3426	−1.81953	−0.909763	−	0.415129i	$-0.863736\pi$
$677$	−47.3426	−1.81953	−0.909763	+	0.415129i	$0.863736\pi$
$678$	−2.21708	−0.0851466
$679$	0	0
$680$	−32.6304	−1.25132
$681$	8.37096	0.320776
$682$	−1.93370	−0.0740450
$683$	38.7091	1.48116	0.740582	−	0.671966i	$-0.234550\pi$
$683$	38.7091	1.48116	0.740582	+	0.671966i	$0.234550\pi$
$684$	−8.08232	−0.309035
$685$	−75.8780	−2.89915
$686$	0	0
$687$	19.5187	0.744684
$688$	1.86121	0.0709581
$689$	−50.2307	−1.91364
$690$	3.04532	0.115933
$691$	−21.9568	−0.835277	−0.417639	−	0.908613i	$-0.637142\pi$
$691$	−21.9568	−0.835277	−0.417639	+	0.908613i	$0.637142\pi$
$692$	26.6249	1.01213
$693$	0	0
$694$	−1.89562	−0.0719567
$695$	−45.3220	−1.71916
$696$	−9.08044	−0.344193
$697$	−33.5247	−1.26984
$698$	−1.68245	−0.0636815
$699$	6.25868	0.236725
$700$	0	0
$701$	7.52091	0.284061	0.142031	−	0.989862i	$-0.454637\pi$
$701$	7.52091	0.284061	0.142031	+	0.989862i	$0.454637\pi$
$702$	5.65397	0.213395
$703$	47.2810	1.78324
$704$	14.4003	0.542732
$705$	−22.4255	−0.844593
$706$	−15.1954	−0.571886
$707$	0	0
$708$	7.04442	0.264745
$709$	0.693914	0.0260605	0.0130302	−	0.999915i	$-0.495852\pi$
$709$	0.693914	0.0260605	0.0130302	+	0.999915i	$0.495852\pi$
$710$	8.90958	0.334371
$711$	8.17142	0.306452
$712$	39.3333	1.47408
$713$	−0.638468	−0.0239108
$714$	0	0
$715$	−95.0842	−3.55595
$716$	6.27034	0.234334
$717$	−4.53876	−0.169503
$718$	5.95415	0.222207
$719$	−9.37630	−0.349677	−0.174838	−	0.984597i	$-0.555940\pi$
$719$	−9.37630	−0.349677	−0.174838	+	0.984597i	$0.555940\pi$
$720$	1.59544	0.0594584
$721$	0	0
$722$	−14.6764	−0.546200
$723$	−19.6771	−0.731799
$724$	1.55753	0.0578850
$725$	−29.4540	−1.09390
$726$	−2.20213	−0.0817288
$727$	−18.9568	−0.703069	−0.351535	−	0.936175i	$-0.614340\pi$
$727$	−18.9568	−0.703069	−0.351535	+	0.936175i	$0.614340\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−27.8641	−1.03130
$731$	−13.9654	−0.516531
$732$	−10.7924	−0.398898
$733$	−6.39615	−0.236247	−0.118124	−	0.992999i	$-0.537688\pi$
$733$	−6.39615	−0.236247	−0.118124	+	0.992999i	$0.537688\pi$
$734$	16.5671	0.611504
$735$	0	0
$736$	5.80250	0.213883
$737$	−14.8657	−0.547584
$738$	−8.52824	−0.313929
$739$	29.8857	1.09936	0.549682	−	0.835374i	$-0.314749\pi$
$739$	29.8857	1.09936	0.549682	+	0.835374i	$0.314749\pi$
$740$	−38.4977	−1.41520
$741$	−41.9760	−1.54203
$742$	0	0
$743$	16.7564	0.614731	0.307366	−	0.951591i	$-0.400553\pi$
$743$	16.7564	0.614731	0.307366	+	0.951591i	$0.400553\pi$
$744$	1.74030	0.0638025
$745$	−3.73506	−0.136842
$746$	−10.9060	−0.399299
$747$	6.01247	0.219985
$748$	15.8344	0.578963
$749$	0	0
$750$	−11.6984	−0.427164
$751$	−19.1378	−0.698348	−0.349174	−	0.937058i	$-0.613538\pi$
$751$	−19.1378	−0.698348	−0.349174	+	0.937058i	$0.613538\pi$
$752$	−2.58488	−0.0942609
$753$	−10.3015	−0.375408
$754$	−18.8354	−0.685946
$755$	−10.0846	−0.367018
$756$	0	0
$757$	−43.3744	−1.57647	−0.788234	−	0.615376i	$-0.789004\pi$
$757$	−43.3744	−1.57647	−0.788234	+	0.615376i	$0.789004\pi$
$758$	−7.49527	−0.272241
$759$	−3.70004	−0.134303
$760$	61.6260	2.23541
$761$	6.16432	0.223456	0.111728	−	0.993739i	$-0.464361\pi$
$761$	6.16432	0.223456	0.111728	+	0.993739i	$0.464361\pi$
$762$	−5.46606	−0.198014
$763$	0	0
$764$	−32.1678	−1.16379
$765$	−11.9712	−0.432820
$766$	−7.35506	−0.265749
$767$	36.5855	1.32103
$768$	−14.6754	−0.529553
$769$	2.61523	0.0943076	0.0471538	−	0.998888i	$-0.484985\pi$
$769$	2.61523	0.0943076	0.0471538	+	0.998888i	$0.484985\pi$
$770$	0	0
$771$	15.3148	0.551550
$772$	−16.3925	−0.589977
$773$	−52.4555	−1.88669	−0.943346	−	0.331810i	$-0.892341\pi$
$773$	−52.4555	−1.88669	−0.943346	+	0.331810i	$0.892341\pi$
$774$	−3.55262	−0.127696
$775$	5.64497	0.202773
$776$	−40.5361	−1.45516
$777$	0	0
$778$	13.4015	0.480469
$779$	63.3150	2.26850
$780$	34.1782	1.22377
$781$	−10.8251	−0.387352
$782$	−2.63384	−0.0941860
$783$	−3.33137	−0.119053
$784$	0	0
$785$	11.2555	0.401725
$786$	−6.30773	−0.224989
$787$	−33.2853	−1.18649	−0.593246	−	0.805021i	$-0.702154\pi$
$787$	−33.2853	−1.18649	−0.593246	+	0.805021i	$0.702154\pi$
$788$	3.67356	0.130865
$789$	−5.31749	−0.189308
$790$	−24.8846	−0.885353
$791$	0	0
$792$	10.0853	0.358367
$793$	−56.0508	−1.99042
$794$	−7.92752	−0.281337
$795$	27.0551	0.959544
$796$	−22.0527	−0.781638
$797$	−26.0207	−0.921702	−0.460851	−	0.887477i	$-0.652456\pi$
$797$	−26.0207	−0.921702	−0.460851	+	0.887477i	$0.652456\pi$
$798$	0	0
$799$	19.3954	0.686161
$800$	−51.3024	−1.81381
$801$	14.4303	0.509870
$802$	−9.38557	−0.331416
$803$	33.8547	1.19471
$804$	5.34349	0.188450
$805$	0	0
$806$	3.60988	0.127153
$807$	−1.76465	−0.0621185
$808$	11.4483	0.402751
$809$	−41.8504	−1.47138	−0.735691	−	0.677317i	$-0.763142\pi$
$809$	−41.8504	−1.47138	−0.735691	+	0.677317i	$0.763142\pi$
$810$	−3.04532	−0.107002
$811$	−17.1370	−0.601760	−0.300880	−	0.953662i	$-0.597280\pi$
$811$	−17.1370	−0.601760	−0.300880	+	0.953662i	$0.597280\pi$
$812$	0	0
$813$	16.6166	0.582768
$814$	−23.5638	−0.825912
$815$	28.5801	1.00112
$816$	−1.37987	−0.0483050
$817$	26.3752	0.922752
$818$	31.6393	1.10624
$819$	0	0
$820$	−51.5531	−1.80031
$821$	−4.76075	−0.166151	−0.0830757	−	0.996543i	$-0.526474\pi$
$821$	−4.76075	−0.166151	−0.0830757	+	0.996543i	$0.526474\pi$
$822$	16.6943	0.582279
$823$	−46.9815	−1.63767	−0.818835	−	0.574029i	$-0.805380\pi$
$823$	−46.9815	−1.63767	−0.818835	+	0.574029i	$0.805380\pi$
$824$	−47.5752	−1.65736
$825$	32.7136	1.13894
$826$	0	0
$827$	−13.3820	−0.465336	−0.232668	−	0.972556i	$-0.574746\pi$
$827$	−13.3820	−0.465336	−0.232668	+	0.972556i	$0.574746\pi$
$828$	1.32999	0.0462202
$829$	−0.371697	−0.0129096	−0.00645478	−	0.999979i	$-0.502055\pi$
$829$	−0.371697	−0.0129096	−0.00645478	+	0.999979i	$0.502055\pi$
$830$	−18.3099	−0.635545
$831$	1.95295	0.0677472
$832$	−26.8829	−0.931998
$833$	0	0
$834$	9.97150	0.345285
$835$	17.0513	0.590085
$836$	−29.9049	−1.03428
$837$	0.638468	0.0220687
$838$	5.49128	0.189693
$839$	45.6631	1.57647	0.788233	−	0.615377i	$-0.210996\pi$
$839$	45.6631	1.57647	0.788233	+	0.615377i	$0.210996\pi$
$840$	0	0
$841$	−17.9020	−0.617310
$842$	27.8726	0.960554
$843$	13.1183	0.451820
$844$	−3.66827	−0.126267
$845$	129.141	4.44258
$846$	4.93394	0.169632
$847$	0	0
$848$	3.11851	0.107090
$849$	0.979872	0.0336291
$850$	23.2869	0.798735
$851$	−7.78032	−0.266706
$852$	3.89110	0.133307
$853$	42.5768	1.45780	0.728901	−	0.684619i	$-0.240031\pi$
$853$	42.5768	1.45780	0.728901	+	0.684619i	$0.240031\pi$
$854$	0	0
$855$	22.6089	0.773209
$856$	−2.65546	−0.0907618
$857$	41.3922	1.41393	0.706965	−	0.707248i	$-0.250064\pi$
$857$	41.3922	1.41393	0.706965	+	0.707248i	$0.250064\pi$
$858$	20.9199	0.714194
$859$	−52.7547	−1.79997	−0.899984	−	0.435923i	$-0.856422\pi$
$859$	−52.7547	−1.79997	−0.899984	+	0.435923i	$0.856422\pi$
$860$	−21.4756	−0.732310
$861$	0	0
$862$	−6.11368	−0.208233
$863$	22.7441	0.774217	0.387109	−	0.922034i	$-0.373474\pi$
$863$	22.7441	0.774217	0.387109	+	0.922034i	$0.373474\pi$
$864$	−5.80250	−0.197405
$865$	−74.4787	−2.53235
$866$	21.1756	0.719575
$867$	−6.64630	−0.225720
$868$	0	0
$869$	30.2346	1.02564
$870$	10.1451	0.343950
$871$	27.7517	0.940330
$872$	0.508463	0.0172187
$873$	−14.8716	−0.503328
$874$	4.97429	0.168258
$875$	0	0
$876$	−12.1691	−0.411157
$877$	5.28592	0.178493	0.0892464	−	0.996010i	$-0.471554\pi$
$877$	5.28592	0.178493	0.0892464	+	0.996010i	$0.471554\pi$
$878$	13.0741	0.441228
$879$	−22.4505	−0.757237
$880$	5.90318	0.198996
$881$	−51.9769	−1.75115	−0.875573	−	0.483086i	$-0.839516\pi$
$881$	−51.9769	−1.75115	−0.875573	+	0.483086i	$0.839516\pi$
$882$	0	0
$883$	13.0607	0.439526	0.219763	−	0.975553i	$-0.429472\pi$
$883$	13.0607	0.439526	0.219763	+	0.975553i	$0.429472\pi$
$884$	−29.5601	−0.994215
$885$	−19.7056	−0.662395
$886$	−25.0706	−0.842264
$887$	−49.1688	−1.65093	−0.825464	−	0.564455i	$-0.809087\pi$
$887$	−49.1688	−1.65093	−0.825464	+	0.564455i	$0.809087\pi$
$888$	21.2071	0.711665
$889$	0	0
$890$	−43.9449	−1.47304
$891$	3.70004	0.123956
$892$	34.3391	1.14976
$893$	−36.6303	−1.22579
$894$	0.821768	0.0274840
$895$	−17.5402	−0.586305
$896$	0	0
$897$	6.90735	0.230630
$898$	5.83281	0.194643
$899$	−2.12697	−0.0709385
$900$	−11.7590	−0.391966
$901$	−23.3995	−0.779549
$902$	−31.5548	−1.05066
$903$	0	0
$904$	7.38286	0.245550
$905$	−4.35691	−0.144829
$906$	2.21877	0.0737137
$907$	−51.1121	−1.69715	−0.848575	−	0.529074i	$-0.822539\pi$
$907$	−51.1121	−1.69715	−0.848575	+	0.529074i	$0.822539\pi$
$908$	−11.1333	−0.369470
$909$	4.20008	0.139308
$910$	0	0
$911$	10.9073	0.361376	0.180688	−	0.983540i	$-0.442168\pi$
$911$	10.9073	0.361376	0.180688	+	0.983540i	$0.442168\pi$
$912$	2.60602	0.0862940
$913$	22.2464	0.736248
$914$	−16.9317	−0.560050
$915$	30.1898	0.998045
$916$	−25.9596	−0.857728
$917$	0	0
$918$	2.63384	0.0869298
$919$	−34.8033	−1.14805	−0.574027	−	0.818836i	$-0.694619\pi$
$919$	−34.8033	−1.14805	−0.574027	+	0.818836i	$0.694619\pi$
$920$	−10.1409	−0.334334
$921$	29.2019	0.962237
$922$	−9.95443	−0.327832
$923$	20.2086	0.665174
$924$	0	0
$925$	68.7892	2.26177
$926$	1.60453	0.0527280
$927$	−17.4541	−0.573267
$928$	19.3302	0.634547
$929$	−7.12498	−0.233763	−0.116882	−	0.993146i	$-0.537290\pi$
$929$	−7.12498	−0.233763	−0.116882	+	0.993146i	$0.537290\pi$
$930$	−1.94434	−0.0637574
$931$	0	0
$932$	−8.32396	−0.272660
$933$	1.79193	0.0586652
$934$	32.2442	1.05506
$935$	−44.2940	−1.44857
$936$	−18.8276	−0.615401
$937$	−2.12786	−0.0695142	−0.0347571	−	0.999396i	$-0.511066\pi$
$937$	−2.12786	−0.0695142	−0.0347571	+	0.999396i	$0.511066\pi$
$938$	0	0
$939$	−8.83165	−0.288210
$940$	29.8256	0.972804
$941$	−46.7565	−1.52422	−0.762109	−	0.647449i	$-0.775836\pi$
$941$	−46.7565	−1.52422	−0.762109	+	0.647449i	$0.775836\pi$
$942$	−2.47637	−0.0806844
$943$	−10.4188	−0.339283
$944$	−2.27137	−0.0739267
$945$	0	0
$946$	−13.1448	−0.427376
$947$	29.9575	0.973487	0.486744	−	0.873545i	$-0.338185\pi$
$947$	29.9575	0.973487	0.486744	+	0.873545i	$0.338185\pi$
$948$	−10.8679	−0.352972
$949$	−63.2010	−2.05159
$950$	−43.9798	−1.42689
$951$	−21.5495	−0.698791
$952$	0	0
$953$	60.6214	1.96372	0.981860	−	0.189605i	$-0.0607207\pi$
$953$	60.6214	1.96372	0.981860	+	0.189605i	$0.0607207\pi$
$954$	−5.95251	−0.192720
$955$	89.9838	2.91181
$956$	6.03648	0.195234
$957$	−12.3262	−0.398449
$958$	2.31428	0.0747711
$959$	0	0
$960$	14.4796	0.467326
$961$	−30.5924	−0.986850
$962$	43.9897	1.41828
$963$	−0.974217	−0.0313937
$964$	26.1703	0.842887
$965$	45.8551	1.47613
$966$	0	0
$967$	6.34634	0.204084	0.102042	−	0.994780i	$-0.467462\pi$
$967$	6.34634	0.204084	0.102042	+	0.994780i	$0.467462\pi$
$968$	7.33307	0.235694
$969$	−19.5541	−0.628167
$970$	45.2888	1.45413
$971$	−31.3344	−1.00557	−0.502784	−	0.864412i	$-0.667691\pi$
$971$	−31.3344	−1.00557	−0.502784	+	0.864412i	$0.667691\pi$
$972$	−1.32999	−0.0426593
$973$	0	0
$974$	20.2875	0.650052
$975$	−61.0708	−1.95583
$976$	3.47984	0.111387
$977$	−17.6550	−0.564833	−0.282417	−	0.959292i	$-0.591136\pi$
$977$	−17.6550	−0.564833	−0.282417	+	0.959292i	$0.591136\pi$
$978$	−6.28805	−0.201070
$979$	53.3928	1.70644
$980$	0	0
$981$	0.186541	0.00595581
$982$	−10.7521	−0.343113
$983$	3.40755	0.108684	0.0543420	−	0.998522i	$-0.482694\pi$
$983$	3.40755	0.108684	0.0543420	+	0.998522i	$0.482694\pi$
$984$	28.3989	0.905324
$985$	−10.2761	−0.327425
$986$	−8.77430	−0.279431
$987$	0	0
$988$	55.8274	1.77611
$989$	−4.34017	−0.138009
$990$	−11.2678	−0.358114
$991$	1.05296	0.0334485	0.0167242	−	0.999860i	$-0.494676\pi$
$991$	1.05296	0.0334485	0.0167242	+	0.999860i	$0.494676\pi$
$992$	−3.70471	−0.117625
$993$	23.1086	0.733330
$994$	0	0
$995$	61.6887	1.95566
$996$	−7.99650	−0.253379
$997$	−54.9471	−1.74019	−0.870095	−	0.492883i	$-0.835943\pi$
$997$	−54.9471	−1.74019	−0.870095	+	0.492883i	$0.835943\pi$
$998$	−13.4904	−0.427032
$999$	7.78032	0.246159

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bf.1.4		8
7.2	even	3		483.2.i.g.277.5	✓	16
7.4	even	3		483.2.i.g.415.5	yes	16
7.6	odd	2		3381.2.a.be.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.i.g.277.5	✓	16	7.2	even	3
483.2.i.g.415.5	yes	16	7.4	even	3
3381.2.a.be.1.4		8	7.6	odd	2
3381.2.a.bf.1.4		8	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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q-0.818544 q^{2} +1.00000 q^{3} -1.32999 q^{4} +3.72041 q^{5} -0.818544 q^{6} +2.72574 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.818544 q^{2} +1.00000 q^{3} -1.32999 q^{4} +3.72041 q^{5} -0.818544 q^{6} +2.72574 q^{8} +1.00000 q^{9} -3.04532 q^{10} +3.70004 q^{11} -1.32999 q^{12} -6.90735 q^{13} +3.72041 q^{15} +0.428834 q^{16} -3.21772 q^{17} -0.818544 q^{18} +6.07700 q^{19} -4.94809 q^{20} -3.02865 q^{22} -1.00000 q^{23} +2.72574 q^{24} +8.84143 q^{25} +5.65397 q^{26} +1.00000 q^{27} -3.33137 q^{29} -3.04532 q^{30} +0.638468 q^{31} -5.80250 q^{32} +3.70004 q^{33} +2.63384 q^{34} -1.32999 q^{36} +7.78032 q^{37} -4.97429 q^{38} -6.90735 q^{39} +10.1409 q^{40} +10.4188 q^{41} +4.34017 q^{43} -4.92100 q^{44} +3.72041 q^{45} +0.818544 q^{46} -6.02770 q^{47} +0.428834 q^{48} -7.23710 q^{50} -3.21772 q^{51} +9.18668 q^{52} +7.27207 q^{53} -0.818544 q^{54} +13.7657 q^{55} +6.07700 q^{57} +2.72687 q^{58} -5.29661 q^{59} -4.94809 q^{60} +8.11466 q^{61} -0.522615 q^{62} +3.89193 q^{64} -25.6981 q^{65} -3.02865 q^{66} -4.01771 q^{67} +4.27952 q^{68} -1.00000 q^{69} -2.92567 q^{71} +2.72574 q^{72} +9.14983 q^{73} -6.36854 q^{74} +8.84143 q^{75} -8.08232 q^{76} +5.65397 q^{78} +8.17142 q^{79} +1.59544 q^{80} +1.00000 q^{81} -8.52824 q^{82} +6.01247 q^{83} -11.9712 q^{85} -3.55262 q^{86} -3.33137 q^{87} +10.0853 q^{88} +14.4303 q^{89} -3.04532 q^{90} +1.32999 q^{92} +0.638468 q^{93} +4.93394 q^{94} +22.6089 q^{95} -5.80250 q^{96} -14.8716 q^{97} +3.70004 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + q^2 + 8 * q^3 + 15 * q^4 + 5 * q^5 + q^6 + 9 * q^8 + 8 * q^9	\( 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9} - 3 q^{10} + 10 q^{11} + 15 q^{12} - 6 q^{13} + 5 q^{15} + 13 q^{16} + 21 q^{17} + q^{18} + 5 q^{19} - q^{20} + 18 q^{22} - 8 q^{23} + 9 q^{24} + 27 q^{25} + 3 q^{26} + 8 q^{27} + 2 q^{29} - 3 q^{30} - 13 q^{31} + 29 q^{32} + 10 q^{33} - 19 q^{34} + 15 q^{36} + 13 q^{37} - 6 q^{38} - 6 q^{39} + 7 q^{40} + 16 q^{41} + 15 q^{43} + 24 q^{44} + 5 q^{45} - q^{46} - q^{47} + 13 q^{48} + 16 q^{50} + 21 q^{51} - 19 q^{52} + 3 q^{53} + q^{54} - 10 q^{55} + 5 q^{57} - 40 q^{58} + 26 q^{59} - q^{60} - 14 q^{61} - 14 q^{62} + 49 q^{64} - 3 q^{65} + 18 q^{66} + 38 q^{67} + 43 q^{68} - 8 q^{69} + 9 q^{71} + 9 q^{72} - 6 q^{73} + 32 q^{74} + 27 q^{75} - 14 q^{76} + 3 q^{78} + 23 q^{79} - 17 q^{80} + 8 q^{81} + 20 q^{82} + 30 q^{83} - 37 q^{85} - 28 q^{86} + 2 q^{87} + 86 q^{88} + 12 q^{89} - 3 q^{90} - 15 q^{92} - 13 q^{93} - 45 q^{94} + 16 q^{95} + 29 q^{96} + 14 q^{97} + 10 q^{99}+O(q^{100}) \) 8 * q + q^2 + 8 * q^3 + 15 * q^4 + 5 * q^5 + q^6 + 9 * q^8 + 8 * q^9 - 3 * q^10 + 10 * q^11 + 15 * q^12 - 6 * q^13 + 5 * q^15 + 13 * q^16 + 21 * q^17 + q^18 + 5 * q^19 - q^20 + 18 * q^22 - 8 * q^23 + 9 * q^24 + 27 * q^25 + 3 * q^26 + 8 * q^27 + 2 * q^29 - 3 * q^30 - 13 * q^31 + 29 * q^32 + 10 * q^33 - 19 * q^34 + 15 * q^36 + 13 * q^37 - 6 * q^38 - 6 * q^39 + 7 * q^40 + 16 * q^41 + 15 * q^43 + 24 * q^44 + 5 * q^45 - q^46 - q^47 + 13 * q^48 + 16 * q^50 + 21 * q^51 - 19 * q^52 + 3 * q^53 + q^54 - 10 * q^55 + 5 * q^57 - 40 * q^58 + 26 * q^59 - q^60 - 14 * q^61 - 14 * q^62 + 49 * q^64 - 3 * q^65 + 18 * q^66 + 38 * q^67 + 43 * q^68 - 8 * q^69 + 9 * q^71 + 9 * q^72 - 6 * q^73 + 32 * q^74 + 27 * q^75 - 14 * q^76 + 3 * q^78 + 23 * q^79 - 17 * q^80 + 8 * q^81 + 20 * q^82 + 30 * q^83 - 37 * q^85 - 28 * q^86 + 2 * q^87 + 86 * q^88 + 12 * q^89 - 3 * q^90 - 15 * q^92 - 13 * q^93 - 45 * q^94 + 16 * q^95 + 29 * q^96 + 14 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more