LMFDB - Embedded newform 187.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [187,2,Mod(1,187)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(187, 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("187.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 187 = 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	187.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:	$1.49320251780$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.33844.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 2x + 2 $ x^4 - x^3 - 6x^2 + 2x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.752785$ of defining polynomial
Character	$\chi$	$=$	187.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.752785 q^{2} +2.90402 q^{3} -1.43332 q^{4} +0.247215 q^{5} +2.18610 q^{6} -2.58455 q^{8} +5.43332 q^{9} +O(q^{10})$	\(q+0.752785 q^{2} +2.90402 q^{3} -1.43332 q^{4} +0.247215 q^{5} +2.18610 q^{6} -2.58455 q^{8} +5.43332 q^{9} +0.186100 q^{10} -1.00000 q^{11} -4.16237 q^{12} -0.470702 q^{13} +0.717917 q^{15} +0.921022 q^{16} +1.00000 q^{17} +4.09012 q^{18} -6.84290 q^{19} -0.354337 q^{20} -0.752785 q^{22} -2.40959 q^{23} -7.50557 q^{24} -4.93888 q^{25} -0.354337 q^{26} +7.06639 q^{27} -0.186100 q^{29} +0.540437 q^{30} +7.93302 q^{31} +5.86243 q^{32} -2.90402 q^{33} +0.752785 q^{34} -7.78765 q^{36} +8.40959 q^{37} -5.15123 q^{38} -1.36693 q^{39} -0.638939 q^{40} -3.80803 q^{41} -0.941404 q^{43} +1.43332 q^{44} +1.34320 q^{45} -1.81390 q^{46} -1.15123 q^{47} +2.67466 q^{48} -7.00000 q^{49} -3.71792 q^{50} +2.90402 q^{51} +0.674664 q^{52} +4.84877 q^{53} +5.31947 q^{54} -0.247215 q^{55} -19.8719 q^{57} -0.140093 q^{58} +2.13085 q^{59} -1.02900 q^{60} +5.01114 q^{61} +5.97186 q^{62} +2.57110 q^{64} -0.116365 q^{65} -2.18610 q^{66} -8.10126 q^{67} -1.43332 q^{68} -6.99748 q^{69} +13.4163 q^{71} -14.0427 q^{72} -12.0639 q^{73} +6.33061 q^{74} -14.3426 q^{75} +9.80803 q^{76} -1.02900 q^{78} -1.36693 q^{79} +0.227691 q^{80} +4.22097 q^{81} -2.86663 q^{82} +16.6969 q^{83} +0.247215 q^{85} -0.708675 q^{86} -0.540437 q^{87} +2.58455 q^{88} +16.2815 q^{89} +1.01114 q^{90} +3.45370 q^{92} +23.0376 q^{93} -0.866630 q^{94} -1.69167 q^{95} +17.0246 q^{96} +11.2065 q^{97} -5.26949 q^{98} -5.43332 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{3} + 5 q^{4} + 3 q^{5} - 4 q^{6} + 9 q^{8} + 11 q^{9}+O(q^{10}) $ 4 * q + q^2 + q^3 + 5 * q^4 + 3 * q^5 - 4 * q^6 + 9 * q^8 + 11 * q^9	$ 4 q + q^{2} + q^{3} + 5 q^{4} + 3 q^{5} - 4 q^{6} + 9 q^{8} + 11 q^{9} - 12 q^{10} - 4 q^{11} - 2 q^{13} + 5 q^{15} + 19 q^{16} + 4 q^{17} - 7 q^{18} - 2 q^{19} - 6 q^{20} - q^{22} + 5 q^{23} - 26 q^{24} - 5 q^{25} - 6 q^{26} + q^{27} + 12 q^{29} - 6 q^{30} - 17 q^{31} + 39 q^{32} - q^{33} + q^{34} - 25 q^{36} + 19 q^{37} - 12 q^{38} - 22 q^{39} - 20 q^{40} + 6 q^{41} - 4 q^{43} - 5 q^{44} + 18 q^{45} - 20 q^{46} + 4 q^{47} - 32 q^{48} - 28 q^{49} - 17 q^{50} + q^{51} - 40 q^{52} + 28 q^{53} + 30 q^{54} - 3 q^{55} - 16 q^{57} + 15 q^{59} + 34 q^{60} + 12 q^{61} + 6 q^{62} + 35 q^{64} + 4 q^{65} + 4 q^{66} - q^{67} + 5 q^{68} - 13 q^{69} + 17 q^{71} - 25 q^{72} - 6 q^{73} + 26 q^{74} + 6 q^{75} + 18 q^{76} + 34 q^{78} - 22 q^{79} - 38 q^{80} + 10 q^{82} + 8 q^{83} + 3 q^{85} - 12 q^{86} + 6 q^{87} - 9 q^{88} - 13 q^{89} - 4 q^{90} - 12 q^{92} + 31 q^{93} + 18 q^{94} + 10 q^{95} + 16 q^{96} + 17 q^{97} - 7 q^{98} - 11 q^{99}+O(q^{100}) $ 4 * q + q^2 + q^3 + 5 * q^4 + 3 * q^5 - 4 * q^6 + 9 * q^8 + 11 * q^9 - 12 * q^10 - 4 * q^11 - 2 * q^13 + 5 * q^15 + 19 * q^16 + 4 * q^17 - 7 * q^18 - 2 * q^19 - 6 * q^20 - q^22 + 5 * q^23 - 26 * q^24 - 5 * q^25 - 6 * q^26 + q^27 + 12 * q^29 - 6 * q^30 - 17 * q^31 + 39 * q^32 - q^33 + q^34 - 25 * q^36 + 19 * q^37 - 12 * q^38 - 22 * q^39 - 20 * q^40 + 6 * q^41 - 4 * q^43 - 5 * q^44 + 18 * q^45 - 20 * q^46 + 4 * q^47 - 32 * q^48 - 28 * q^49 - 17 * q^50 + q^51 - 40 * q^52 + 28 * q^53 + 30 * q^54 - 3 * q^55 - 16 * q^57 + 15 * q^59 + 34 * q^60 + 12 * q^61 + 6 * q^62 + 35 * q^64 + 4 * q^65 + 4 * q^66 - q^67 + 5 * q^68 - 13 * q^69 + 17 * q^71 - 25 * q^72 - 6 * q^73 + 26 * q^74 + 6 * q^75 + 18 * q^76 + 34 * q^78 - 22 * q^79 - 38 * q^80 + 10 * q^82 + 8 * q^83 + 3 * q^85 - 12 * q^86 + 6 * q^87 - 9 * q^88 - 13 * q^89 - 4 * q^90 - 12 * q^92 + 31 * q^93 + 18 * q^94 + 10 * q^95 + 16 * q^96 + 17 * q^97 - 7 * q^98 - 11 * 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 $

$2$	0.752785	0.532299	0.266150	−	0.963932i	$-0.414248\pi$
$2$	0.752785	0.532299	0.266150	+	0.963932i	$0.414248\pi$
$3$	2.90402	1.67664	0.838318	−	0.545182i	$-0.183540\pi$
$3$	2.90402	1.67664	0.838318	+	0.545182i	$0.183540\pi$
$4$	−1.43332	−0.716658
$5$	0.247215	0.110558	0.0552790	−	0.998471i	$-0.482395\pi$
$5$	0.247215	0.110558	0.0552790	+	0.998471i	$0.482395\pi$
$6$	2.18610	0.892472
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−2.58455	−0.913775
$9$	5.43332	1.81111
$10$	0.186100	0.0588499
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−4.16237	−1.20157
$13$	−0.470702	−0.130549	−0.0652746	−	0.997867i	$-0.520792\pi$
$13$	−0.470702	−0.130549	−0.0652746	+	0.997867i	$0.520792\pi$
$14$	0	0
$15$	0.717917	0.185365
$16$	0.921022	0.230256
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	4.09012	0.964050
$19$	−6.84290	−1.56987	−0.784935	−	0.619579i	$-0.787304\pi$
$19$	−6.84290	−1.56987	−0.784935	+	0.619579i	$0.787304\pi$
$20$	−0.354337	−0.0792322
$21$	0	0
$22$	−0.752785	−0.160494
$23$	−2.40959	−0.502434	−0.251217	−	0.967931i	$-0.580831\pi$
$23$	−2.40959	−0.502434	−0.251217	+	0.967931i	$0.580831\pi$
$24$	−7.50557	−1.53207
$25$	−4.93888	−0.987777
$26$	−0.354337	−0.0694913
$27$	7.06639	1.35993
$28$	0	0
$29$	−0.186100	−0.0345579	−0.0172789	−	0.999851i	$-0.505500\pi$
$29$	−0.186100	−0.0345579	−0.0172789	+	0.999851i	$0.505500\pi$
$30$	0.540437	0.0986699
$31$	7.93302	1.42481	0.712406	−	0.701767i	$-0.247605\pi$
$31$	7.93302	1.42481	0.712406	+	0.701767i	$0.247605\pi$
$32$	5.86243	1.03634
$33$	−2.90402	−0.505524
$34$	0.752785	0.129102
$35$	0	0
$36$	−7.78765	−1.29794
$37$	8.40959	1.38253	0.691264	−	0.722603i	$-0.257054\pi$
$37$	8.40959	1.38253	0.691264	+	0.722603i	$0.257054\pi$
$38$	−5.15123	−0.835640
$39$	−1.36693	−0.218883
$40$	−0.638939	−0.101025
$41$	−3.80803	−0.594715	−0.297358	−	0.954766i	$-0.596105\pi$
$41$	−3.80803	−0.594715	−0.297358	+	0.954766i	$0.596105\pi$
$42$	0	0
$43$	−0.941404	−0.143563	−0.0717814	−	0.997420i	$-0.522868\pi$
$43$	−0.941404	−0.143563	−0.0717814	+	0.997420i	$0.522868\pi$
$44$	1.43332	0.216080
$45$	1.34320	0.200232
$46$	−1.81390	−0.267445
$47$	−1.15123	−0.167924	−0.0839622	−	0.996469i	$-0.526758\pi$
$47$	−1.15123	−0.167924	−0.0839622	+	0.996469i	$0.526758\pi$
$48$	2.67466	0.386055
$49$	−7.00000	−1.00000
$50$	−3.71792	−0.525793
$51$	2.90402	0.406644
$52$	0.674664	0.0935591
$53$	4.84877	0.666029	0.333015	−	0.942922i	$-0.391934\pi$
$53$	4.84877	0.666029	0.333015	+	0.942922i	$0.391934\pi$
$54$	5.31947	0.723888
$55$	−0.247215	−0.0333345
$56$	0	0
$57$	−19.8719	−2.63210
$58$	−0.140093	−0.0183951
$59$	2.13085	0.277413	0.138707	−	0.990334i	$-0.455706\pi$
$59$	2.13085	0.277413	0.138707	+	0.990334i	$0.455706\pi$
$60$	−1.02900	−0.132844
$61$	5.01114	0.641611	0.320805	−	0.947145i	$-0.396046\pi$
$61$	5.01114	0.641611	0.320805	+	0.947145i	$0.396046\pi$
$62$	5.97186	0.758426
$63$	0	0
$64$	2.57110	0.321388
$65$	−0.116365	−0.0144333
$66$	−2.18610	−0.269090
$67$	−8.10126	−0.989726	−0.494863	−	0.868971i	$-0.664782\pi$
$67$	−8.10126	−0.989726	−0.494863	+	0.868971i	$0.664782\pi$
$68$	−1.43332	−0.173815
$69$	−6.99748	−0.842398
$70$	0	0
$71$	13.4163	1.59222	0.796112	−	0.605150i	$-0.206887\pi$
$71$	13.4163	1.59222	0.796112	+	0.605150i	$0.206887\pi$
$72$	−14.0427	−1.65494
$73$	−12.0639	−1.41197	−0.705985	−	0.708227i	$-0.749495\pi$
$73$	−12.0639	−1.41197	−0.705985	+	0.708227i	$0.749495\pi$
$74$	6.33061	0.735918
$75$	−14.3426	−1.65614
$76$	9.80803	1.12506
$77$	0	0
$78$	−1.02900	−0.116511
$79$	−1.36693	−0.153791	−0.0768956	−	0.997039i	$-0.524501\pi$
$79$	−1.36693	−0.153791	−0.0768956	+	0.997039i	$0.524501\pi$
$80$	0.227691	0.0254566
$81$	4.22097	0.468996
$82$	−2.86663	−0.316566
$83$	16.6969	1.83273	0.916364	−	0.400347i	$-0.131110\pi$
$83$	16.6969	1.83273	0.916364	+	0.400347i	$0.131110\pi$
$84$	0	0
$85$	0.247215	0.0268143
$86$	−0.708675	−0.0764183
$87$	−0.540437	−0.0579409
$88$	2.58455	0.275514
$89$	16.2815	1.72583	0.862917	−	0.505345i	$-0.168635\pi$
$89$	16.2815	1.72583	0.862917	+	0.505345i	$0.168635\pi$
$90$	1.01114	0.106583
$91$	0	0
$92$	3.45370	0.360073
$93$	23.0376	2.38889
$94$	−0.866630	−0.0893861
$95$	−1.69167	−0.173562
$96$	17.0246	1.73756
$97$	11.2065	1.13785	0.568923	−	0.822391i	$-0.307360\pi$
$97$	11.2065	1.13785	0.568923	+	0.822391i	$0.307360\pi$
$98$	−5.26949	−0.532299
$99$	−5.43332	−0.546069
$100$	7.07898	0.707898
$101$	15.5176	1.54406	0.772028	−	0.635589i	$-0.219243\pi$
$101$	15.5176	1.54406	0.772028	+	0.635589i	$0.219243\pi$
$102$	2.18610	0.216456
$103$	−6.09264	−0.600325	−0.300163	−	0.953888i	$-0.597041\pi$
$103$	−6.09264	−0.600325	−0.300163	+	0.953888i	$0.597041\pi$
$104$	1.21655	0.119293
$105$	0	0
$106$	3.65008	0.354527
$107$	−10.1861	−0.984727	−0.492364	−	0.870390i	$-0.663867\pi$
$107$	−10.1861	−0.984727	−0.492364	+	0.870390i	$0.663867\pi$
$108$	−10.1284	−0.974602
$109$	−9.49970	−0.909907	−0.454953	−	0.890515i	$-0.650344\pi$
$109$	−9.49970	−0.909907	−0.454953	+	0.890515i	$0.650344\pi$
$110$	−0.186100	−0.0177439
$111$	24.4216	2.31799
$112$	0	0
$113$	2.93361	0.275971	0.137985	−	0.990434i	$-0.455937\pi$
$113$	2.93361	0.275971	0.137985	+	0.990434i	$0.455937\pi$
$114$	−14.9593	−1.40106
$115$	−0.595686	−0.0555481
$116$	0.266740	0.0247662
$117$	−2.55747	−0.236438
$118$	1.60407	0.147667
$119$	0	0
$120$	−1.85549	−0.169382
$121$	1.00000	0.0909091
$122$	3.77231	0.341529
$123$	−11.0586	−0.997120
$124$	−11.3705	−1.02110
$125$	−2.45704	−0.219765
$126$	0	0
$127$	−20.2500	−1.79689	−0.898447	−	0.439082i	$-0.855304\pi$
$127$	−20.2500	−1.79689	−0.898447	+	0.439082i	$0.855304\pi$
$128$	−9.78937	−0.865266
$129$	−2.73385	−0.240702
$130$	−0.0875976	−0.00768282
$131$	6.18023	0.539970	0.269985	−	0.962865i	$-0.412981\pi$
$131$	6.18023	0.539970	0.269985	+	0.962865i	$0.412981\pi$
$132$	4.16237	0.362288
$133$	0	0
$134$	−6.09850	−0.526830
$135$	1.74692	0.150351
$136$	−2.58455	−0.221623
$137$	5.77651	0.493521	0.246760	−	0.969077i	$-0.420634\pi$
$137$	5.77651	0.493521	0.246760	+	0.969077i	$0.420634\pi$
$138$	−5.26760	−0.448408
$139$	−2.30833	−0.195790	−0.0978950	−	0.995197i	$-0.531211\pi$
$139$	−2.30833	−0.195790	−0.0978950	+	0.995197i	$0.531211\pi$
$140$	0	0
$141$	−3.34320	−0.281548
$142$	10.0996	0.847539
$143$	0.470702	0.0393621
$144$	5.00420	0.417017
$145$	−0.0460067	−0.00382065
$146$	−9.08150	−0.751590
$147$	−20.3281	−1.67664
$148$	−12.0536	−0.990799
$149$	6.77317	0.554879	0.277440	−	0.960743i	$-0.410514\pi$
$149$	6.77317	0.554879	0.277440	+	0.960743i	$0.410514\pi$
$150$	−10.7969	−0.881563
$151$	20.5762	1.67446	0.837232	−	0.546848i	$-0.184172\pi$
$151$	20.5762	1.67446	0.837232	+	0.546848i	$0.184172\pi$
$152$	17.6858	1.43451
$153$	5.43332	0.439257
$154$	0	0
$155$	1.96116	0.157524
$156$	1.95924	0.156864
$157$	12.4037	0.989925	0.494962	−	0.868914i	$-0.335182\pi$
$157$	12.4037	0.989925	0.494962	+	0.868914i	$0.335182\pi$
$158$	−1.02900	−0.0818630
$159$	14.0809	1.11669
$160$	1.44928	0.114576
$161$	0	0
$162$	3.17748	0.249646
$163$	−15.6161	−1.22314	−0.611572	−	0.791188i	$-0.709463\pi$
$163$	−15.6161	−1.22314	−0.611572	+	0.791188i	$0.709463\pi$
$164$	5.45811	0.426207
$165$	−0.717917	−0.0558898
$166$	12.5692	0.975559
$167$	−22.9305	−1.77442	−0.887208	−	0.461370i	$-0.847358\pi$
$167$	−22.9305	−1.77442	−0.887208	+	0.461370i	$0.847358\pi$
$168$	0	0
$169$	−12.7784	−0.982957
$170$	0.186100	0.0142732
$171$	−37.1796	−2.84320
$172$	1.34933	0.102885
$173$	−9.87777	−0.750993	−0.375496	−	0.926824i	$-0.622528\pi$
$173$	−9.87777	−0.750993	−0.375496	+	0.926824i	$0.622528\pi$
$174$	−0.406833	−0.0308419
$175$	0	0
$176$	−0.921022	−0.0694247
$177$	6.18803	0.465121
$178$	12.2565	0.918660
$179$	−5.04266	−0.376906	−0.188453	−	0.982082i	$-0.560347\pi$
$179$	−5.04266	−0.376906	−0.188453	+	0.982082i	$0.560347\pi$
$180$	−1.92523	−0.143498
$181$	−23.3460	−1.73529	−0.867645	−	0.497183i	$-0.834368\pi$
$181$	−23.3460	−1.73529	−0.867645	+	0.497183i	$0.834368\pi$
$182$	0	0
$183$	14.5524	1.07575
$184$	6.22769	0.459111
$185$	2.07898	0.152849
$186$	17.3424	1.27160
$187$	−1.00000	−0.0731272
$188$	1.65008	0.120344
$189$	0	0
$190$	−1.27346	−0.0923867
$191$	−19.7648	−1.43013	−0.715065	−	0.699058i	$-0.753603\pi$
$191$	−19.7648	−1.43013	−0.715065	+	0.699058i	$0.753603\pi$
$192$	7.46652	0.538850
$193$	17.7324	1.27641	0.638204	−	0.769867i	$-0.279678\pi$
$193$	17.7324	1.27641	0.638204	+	0.769867i	$0.279678\pi$
$194$	8.43607	0.605674
$195$	−0.337925	−0.0241993
$196$	10.0332	0.716658
$197$	5.51144	0.392674	0.196337	−	0.980537i	$-0.437095\pi$
$197$	5.51144	0.392674	0.196337	+	0.980537i	$0.437095\pi$
$198$	−4.09012	−0.290672
$199$	−11.8030	−0.836692	−0.418346	−	0.908288i	$-0.637390\pi$
$199$	−11.8030	−0.836692	−0.418346	+	0.908288i	$0.637390\pi$
$200$	12.7648	0.902606
$201$	−23.5262	−1.65941
$202$	11.6814	0.821900
$203$	0	0
$204$	−4.16237	−0.291424
$205$	−0.941404	−0.0657505
$206$	−4.58644	−0.319553
$207$	−13.0920	−0.909960
$208$	−0.433527	−0.0300597
$209$	6.84290	0.473333
$210$	0	0
$211$	−11.6908	−0.804831	−0.402415	−	0.915457i	$-0.631829\pi$
$211$	−11.6908	−0.804831	−0.402415	+	0.915457i	$0.631829\pi$
$212$	−6.94981	−0.477315
$213$	38.9612	2.66958
$214$	−7.66794	−0.524170
$215$	−0.232729	−0.0158720
$216$	−18.2634	−1.24267
$217$	0	0
$218$	−7.15123	−0.484343
$219$	−35.0337	−2.36736
$220$	0.354337	0.0238894
$221$	−0.470702	−0.0316628
$222$	18.3842	1.23387
$223$	−4.80551	−0.321801	−0.160901	−	0.986971i	$-0.551440\pi$
$223$	−4.80551	−0.321801	−0.160901	+	0.986971i	$0.551440\pi$
$224$	0	0
$225$	−26.8345	−1.78897
$226$	2.20838	0.146899
$227$	0.658252	0.0436897	0.0218449	−	0.999761i	$-0.493046\pi$
$227$	0.658252	0.0436897	0.0218449	+	0.999761i	$0.493046\pi$
$228$	28.4827	1.88631
$229$	26.0494	1.72139	0.860696	−	0.509119i	$-0.170029\pi$
$229$	26.0494	1.72139	0.860696	+	0.509119i	$0.170029\pi$
$230$	−0.448424	−0.0295682
$231$	0	0
$232$	0.480984	0.0315781
$233$	24.2433	1.58823	0.794115	−	0.607768i	$-0.207935\pi$
$233$	24.2433	1.58823	0.794115	+	0.607768i	$0.207935\pi$
$234$	−1.92523	−0.125856
$235$	−0.284602	−0.0185654
$236$	−3.05418	−0.198810
$237$	−3.96958	−0.257852
$238$	0	0
$239$	−24.9757	−1.61554	−0.807771	−	0.589496i	$-0.799326\pi$
$239$	−24.9757	−1.61554	−0.807771	+	0.589496i	$0.799326\pi$
$240$	0.661218	0.0426814
$241$	−15.7606	−1.01523	−0.507614	−	0.861585i	$-0.669472\pi$
$241$	−15.7606	−1.01523	−0.507614	+	0.861585i	$0.669472\pi$
$242$	0.752785	0.0483908
$243$	−8.94140	−0.573591
$244$	−7.18254	−0.459815
$245$	−1.73051	−0.110558
$246$	−8.32474	−0.530766
$247$	3.22097	0.204945
$248$	−20.5033	−1.30196
$249$	48.4882	3.07282
$250$	−1.84962	−0.116981
$251$	−17.8345	−1.12570	−0.562852	−	0.826557i	$-0.690296\pi$
$251$	−17.8345	−1.12570	−0.562852	+	0.826557i	$0.690296\pi$
$252$	0	0
$253$	2.40959	0.151489
$254$	−15.2439	−0.956485
$255$	0.717917	0.0449577
$256$	−12.5115	−0.781968
$257$	−7.13950	−0.445350	−0.222675	−	0.974893i	$-0.571479\pi$
$257$	−7.13950	−0.445350	−0.222675	+	0.974893i	$0.571479\pi$
$258$	−2.05800	−0.128126
$259$	0	0
$260$	0.166787	0.0103437
$261$	−1.01114	−0.0625879
$262$	4.65239	0.287425
$263$	10.7207	0.661065	0.330532	−	0.943795i	$-0.392772\pi$
$263$	10.7207	0.661065	0.330532	+	0.943795i	$0.392772\pi$
$264$	7.50557	0.461936
$265$	1.19869	0.0736349
$266$	0	0
$267$	47.2817	2.89359
$268$	11.6117	0.709295
$269$	−19.3019	−1.17686	−0.588428	−	0.808550i	$-0.700253\pi$
$269$	−19.3019	−1.17686	−0.588428	+	0.808550i	$0.700253\pi$
$270$	1.31505	0.0800316
$271$	0.679703	0.0412890	0.0206445	−	0.999787i	$-0.493428\pi$
$271$	0.679703	0.0412890	0.0206445	+	0.999787i	$0.493428\pi$
$272$	0.921022	0.0558452
$273$	0	0
$274$	4.34847	0.262701
$275$	4.93888	0.297826
$276$	10.0296	0.603711
$277$	−4.96368	−0.298239	−0.149119	−	0.988819i	$-0.547644\pi$
$277$	−4.96368	−0.298239	−0.149119	+	0.988819i	$0.547644\pi$
$278$	−1.73768	−0.104219
$279$	43.1026	2.58048
$280$	0	0
$281$	19.8423	1.18369	0.591846	−	0.806051i	$-0.298399\pi$
$281$	19.8423	1.18369	0.591846	+	0.806051i	$0.298399\pi$
$282$	−2.51671	−0.149868
$283$	−3.85466	−0.229136	−0.114568	−	0.993415i	$-0.536548\pi$
$283$	−3.85466	−0.229136	−0.114568	+	0.993415i	$0.536548\pi$
$284$	−19.2298	−1.14108
$285$	−4.91264	−0.290999
$286$	0.354337	0.0209524
$287$	0	0
$288$	31.8524	1.87692
$289$	1.00000	0.0588235
$290$	−0.0346332	−0.00203373
$291$	32.5438	1.90775
$292$	17.2913	1.01190
$293$	16.7494	0.978513	0.489256	−	0.872140i	$-0.337268\pi$
$293$	16.7494	0.978513	0.489256	+	0.872140i	$0.337268\pi$
$294$	−15.3027	−0.892472
$295$	0.526779	0.0306702
$296$	−21.7350	−1.26332
$297$	−7.06639	−0.410033
$298$	5.09874	0.295362
$299$	1.13420	0.0655923
$300$	20.5575	1.18689
$301$	0	0
$302$	15.4894	0.891316
$303$	45.0633	2.58882
$304$	−6.30246	−0.361471
$305$	1.23883	0.0709352
$306$	4.09012	0.233816
$307$	−9.57675	−0.546574	−0.273287	−	0.961933i	$-0.588111\pi$
$307$	−9.57675	−0.546574	−0.273287	+	0.961933i	$0.588111\pi$
$308$	0	0
$309$	−17.6931	−1.00653
$310$	1.47633	0.0838501
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	3.53289	0.200010
$313$	−2.22494	−0.125761	−0.0628804	−	0.998021i	$-0.520029\pi$
$313$	−2.22494	−0.125761	−0.0628804	+	0.998021i	$0.520029\pi$
$314$	9.33733	0.526936
$315$	0	0
$316$	1.95924	0.110216
$317$	14.5022	0.814526	0.407263	−	0.913311i	$-0.366483\pi$
$317$	14.5022	0.814526	0.407263	+	0.913311i	$0.366483\pi$
$318$	10.5999	0.594412
$319$	0.186100	0.0104196
$320$	0.635615	0.0355320
$321$	−29.5806	−1.65103
$322$	0	0
$323$	−6.84290	−0.380749
$324$	−6.04998	−0.336110
$325$	2.32474	0.128954
$326$	−11.7555	−0.651079
$327$	−27.5873	−1.52558
$328$	9.84204	0.543436
$329$	0	0
$330$	−0.540437	−0.0297501
$331$	3.49633	0.192176	0.0960878	−	0.995373i	$-0.469367\pi$
$331$	3.49633	0.192176	0.0960878	+	0.995373i	$0.469367\pi$
$332$	−23.9320	−1.31344
$333$	45.6919	2.50390
$334$	−17.2617	−0.944520
$335$	−2.00275	−0.109422
$336$	0	0
$337$	−7.09936	−0.386727	−0.193363	−	0.981127i	$-0.561940\pi$
$337$	−7.09936	−0.386727	−0.193363	+	0.981127i	$0.561940\pi$
$338$	−9.61941	−0.523227
$339$	8.51926	0.462703
$340$	−0.354337	−0.0192166
$341$	−7.93302	−0.429597
$342$	−27.9883	−1.51343
$343$	0	0
$344$	2.43310	0.131184
$345$	−1.72988	−0.0931338
$346$	−7.43583	−0.399753
$347$	20.1105	1.07959	0.539794	−	0.841797i	$-0.318502\pi$
$347$	20.1105	1.07959	0.539794	+	0.841797i	$0.318502\pi$
$348$	0.774617	0.0415238
$349$	−30.2382	−1.61862	−0.809308	−	0.587385i	$-0.800158\pi$
$349$	−30.2382	−1.61862	−0.809308	+	0.587385i	$0.800158\pi$
$350$	0	0
$351$	−3.32616	−0.177537
$352$	−5.86243	−0.312468
$353$	−23.0555	−1.22712	−0.613560	−	0.789648i	$-0.710263\pi$
$353$	−23.0555	−1.22712	−0.613560	+	0.789648i	$0.710263\pi$
$354$	4.65825	0.247583
$355$	3.31672	0.176033
$356$	−23.3365	−1.23683
$357$	0	0
$358$	−3.79604	−0.200627
$359$	19.1676	1.01163	0.505815	−	0.862642i	$-0.331192\pi$
$359$	19.1676	1.01163	0.505815	+	0.862642i	$0.331192\pi$
$360$	−3.47156	−0.182967
$361$	27.8253	1.46449
$362$	−17.5745	−0.923694
$363$	2.90402	0.152421
$364$	0	0
$365$	−2.98237	−0.156104
$366$	10.9549	0.572619
$367$	28.6417	1.49508	0.747542	−	0.664214i	$-0.231234\pi$
$367$	28.6417	1.49508	0.747542	+	0.664214i	$0.231234\pi$
$368$	−2.21928	−0.115688
$369$	−20.6902	−1.07709
$370$	1.56502	0.0813617
$371$	0	0
$372$	−33.0202	−1.71202
$373$	−3.40707	−0.176411	−0.0882056	−	0.996102i	$-0.528113\pi$
$373$	−3.40707	−0.176411	−0.0882056	+	0.996102i	$0.528113\pi$
$374$	−0.752785	−0.0389256
$375$	−7.13530	−0.368465
$376$	2.97541	0.153445
$377$	0.0875976	0.00451150
$378$	0	0
$379$	−6.54464	−0.336176	−0.168088	−	0.985772i	$-0.553759\pi$
$379$	−6.54464	−0.336176	−0.168088	+	0.985772i	$0.553759\pi$
$380$	2.42470	0.124384
$381$	−58.8063	−3.01274
$382$	−14.8786	−0.761257
$383$	−15.2704	−0.780278	−0.390139	−	0.920756i	$-0.627573\pi$
$383$	−15.2704	−0.780278	−0.390139	+	0.920756i	$0.627573\pi$
$384$	−28.4285	−1.45074
$385$	0	0
$386$	13.3487	0.679431
$387$	−5.11494	−0.260007
$388$	−16.0624	−0.815446
$389$	26.0767	1.32214	0.661071	−	0.750324i	$-0.270102\pi$
$389$	26.0767	1.32214	0.661071	+	0.750324i	$0.270102\pi$
$390$	−0.254385	−0.0128813
$391$	−2.40959	−0.121858
$392$	18.0918	0.913775
$393$	17.9475	0.905332
$394$	4.14892	0.209020
$395$	−0.337925	−0.0170029
$396$	7.78765	0.391344
$397$	5.05128	0.253516	0.126758	−	0.991934i	$-0.459543\pi$
$397$	5.05128	0.253516	0.126758	+	0.991934i	$0.459543\pi$
$398$	−8.88512	−0.445371
$399$	0	0
$400$	−4.54882	−0.227441
$401$	−18.3203	−0.914873	−0.457437	−	0.889242i	$-0.651232\pi$
$401$	−18.3203	−0.914873	−0.457437	+	0.889242i	$0.651232\pi$
$402$	−17.7102	−0.883302
$403$	−3.73409	−0.186008
$404$	−22.2416	−1.10656
$405$	1.04349	0.0518513
$406$	0	0
$407$	−8.40959	−0.416848
$408$	−7.50557	−0.371581
$409$	27.8608	1.37763	0.688814	−	0.724938i	$-0.258132\pi$
$409$	27.8608	1.37763	0.688814	+	0.724938i	$0.258132\pi$
$410$	−0.708675	−0.0349989
$411$	16.7751	0.827454
$412$	8.73267	0.430228
$413$	0	0
$414$	−9.85549	−0.484371
$415$	4.12774	0.202623
$416$	−2.75946	−0.135293
$417$	−6.70343	−0.328268
$418$	5.15123	0.251955
$419$	−28.8822	−1.41099	−0.705494	−	0.708716i	$-0.749275\pi$
$419$	−28.8822	−1.41099	−0.705494	+	0.708716i	$0.749275\pi$
$420$	0	0
$421$	8.55975	0.417177	0.208588	−	0.978004i	$-0.433113\pi$
$421$	8.55975	0.417177	0.208588	+	0.978004i	$0.433113\pi$
$422$	−8.80069	−0.428411
$423$	−6.25501	−0.304129
$424$	−12.5319	−0.608601
$425$	−4.93888	−0.239571
$426$	29.3294	1.42101
$427$	0	0
$428$	14.5999	0.705712
$429$	1.36693	0.0659958
$430$	−0.175195	−0.00844866
$431$	−13.8769	−0.668429	−0.334214	−	0.942497i	$-0.608471\pi$
$431$	−13.8769	−0.668429	−0.334214	+	0.942497i	$0.608471\pi$
$432$	6.50830	0.313131
$433$	9.32222	0.447997	0.223999	−	0.974589i	$-0.428089\pi$
$433$	9.32222	0.447997	0.223999	+	0.974589i	$0.428089\pi$
$434$	0	0
$435$	−0.133604	−0.00640584
$436$	13.6161	0.652091
$437$	16.4886	0.788755
$438$	−26.3728	−1.26014
$439$	24.1521	1.15272	0.576358	−	0.817197i	$-0.304473\pi$
$439$	24.1521	1.15272	0.576358	+	0.817197i	$0.304473\pi$
$440$	0.638939	0.0304602
$441$	−38.0332	−1.81111
$442$	−0.354337	−0.0168541
$443$	32.3741	1.53814	0.769071	−	0.639164i	$-0.220719\pi$
$443$	32.3741	1.53814	0.769071	+	0.639164i	$0.220719\pi$
$444$	−35.0038	−1.66121
$445$	4.02503	0.190805
$446$	−3.61752	−0.171294
$447$	19.6694	0.930330
$448$	0	0
$449$	−35.6590	−1.68285	−0.841425	−	0.540374i	$-0.818283\pi$
$449$	−35.6590	−1.68285	−0.841425	+	0.540374i	$0.818283\pi$
$450$	−20.2006	−0.952266
$451$	3.80803	0.179313
$452$	−4.20479	−0.197777
$453$	59.7535	2.80747
$454$	0.495522	0.0232560
$455$	0	0
$456$	51.3599	2.40515
$457$	15.7028	0.734548	0.367274	−	0.930113i	$-0.380291\pi$
$457$	15.7028	0.734548	0.367274	+	0.930113i	$0.380291\pi$
$458$	19.6096	0.916295
$459$	7.06639	0.329831
$460$	0.853806	0.0398089
$461$	29.2137	1.36062	0.680308	−	0.732927i	$-0.261846\pi$
$461$	29.2137	1.36062	0.680308	+	0.732927i	$0.261846\pi$
$462$	0	0
$463$	−19.8322	−0.921682	−0.460841	−	0.887483i	$-0.652452\pi$
$463$	−19.8322	−0.921682	−0.460841	+	0.887483i	$0.652452\pi$
$464$	−0.171402	−0.00795714
$465$	5.69525	0.264111
$466$	18.2500	0.845413
$467$	−15.6141	−0.722536	−0.361268	−	0.932462i	$-0.617656\pi$
$467$	−15.6141	−0.722536	−0.361268	+	0.932462i	$0.617656\pi$
$468$	3.66566	0.169445
$469$	0	0
$470$	−0.214244	−0.00988235
$471$	36.0206	1.65974
$472$	−5.50728	−0.253493
$473$	0.941404	0.0432858
$474$	−2.98824	−0.137254
$475$	33.7963	1.55068
$476$	0	0
$477$	26.3449	1.20625
$478$	−18.8013	−0.859952
$479$	24.5457	1.12152	0.560762	−	0.827977i	$-0.310508\pi$
$479$	24.5457	1.12152	0.560762	+	0.827977i	$0.310508\pi$
$480$	4.20874	0.192102
$481$	−3.95841	−0.180488
$482$	−11.8643	−0.540405
$483$	0	0
$484$	−1.43332	−0.0651507
$485$	2.77041	0.125798
$486$	−6.73095	−0.305322
$487$	−25.5039	−1.15569	−0.577846	−	0.816146i	$-0.696107\pi$
$487$	−25.5039	−1.15569	−0.577846	+	0.816146i	$0.696107\pi$
$488$	−12.9515	−0.586288
$489$	−45.3493	−2.05077
$490$	−1.30270	−0.0588499
$491$	8.38923	0.378601	0.189300	−	0.981919i	$-0.439378\pi$
$491$	8.38923	0.378601	0.189300	+	0.981919i	$0.439378\pi$
$492$	15.8505	0.714594
$493$	−0.186100	−0.00838152
$494$	2.42470	0.109092
$495$	−1.34320	−0.0603723
$496$	7.30649	0.328071
$497$	0	0
$498$	36.5012	1.63566
$499$	30.9950	1.38753	0.693763	−	0.720203i	$-0.255951\pi$
$499$	30.9950	1.38753	0.693763	+	0.720203i	$0.255951\pi$
$500$	3.52172	0.157496
$501$	−66.5906	−2.97505
$502$	−13.4256	−0.599212
$503$	−25.2330	−1.12508	−0.562541	−	0.826769i	$-0.690176\pi$
$503$	−25.2330	−1.12508	−0.562541	+	0.826769i	$0.690176\pi$
$504$	0	0
$505$	3.83618	0.170708
$506$	1.81390	0.0806377
$507$	−37.1088	−1.64806
$508$	29.0246	1.28776
$509$	−18.1370	−0.803908	−0.401954	−	0.915660i	$-0.631669\pi$
$509$	−18.1370	−0.803908	−0.401954	+	0.915660i	$0.631669\pi$
$510$	0.540437	0.0239310
$511$	0	0
$512$	10.1603	0.449025
$513$	−48.3546	−2.13491
$514$	−5.37451	−0.237059
$515$	−1.50619	−0.0663708
$516$	3.91847	0.172501
$517$	1.15123	0.0506311
$518$	0	0
$519$	−28.6852	−1.25914
$520$	0.300750	0.0131888
$521$	−1.03066	−0.0451542	−0.0225771	−	0.999745i	$-0.507187\pi$
$521$	−1.03066	−0.0451542	−0.0225771	+	0.999745i	$0.507187\pi$
$522$	−0.761170	−0.0333155
$523$	24.1277	1.05503	0.527516	−	0.849545i	$-0.323123\pi$
$523$	24.1277	1.05503	0.527516	+	0.849545i	$0.323123\pi$
$524$	−8.85822	−0.386973
$525$	0	0
$526$	8.07036	0.351884
$527$	7.93302	0.345568
$528$	−2.67466	−0.116400
$529$	−17.1939	−0.747561
$530$	0.902355	0.0391958
$531$	11.5776	0.502424
$532$	0	0
$533$	1.79245	0.0776396
$534$	35.5930	1.54026
$535$	−2.51816	−0.108870
$536$	20.9381	0.904387
$537$	−14.6440	−0.631934
$538$	−14.5302	−0.626439
$539$	7.00000	0.301511
$540$	−2.50388	−0.107750
$541$	16.3256	0.701891	0.350946	−	0.936396i	$-0.385860\pi$
$541$	16.3256	0.701891	0.350946	+	0.936396i	$0.385860\pi$
$542$	0.511670	0.0219781
$543$	−67.7970	−2.90945
$544$	5.86243	0.251349
$545$	−2.34847	−0.100597
$546$	0	0
$547$	46.1152	1.97174	0.985871	−	0.167504i	$-0.0535708\pi$
$547$	46.1152	1.97174	0.985871	+	0.167504i	$0.0535708\pi$
$548$	−8.27956	−0.353685
$549$	27.2271	1.16202
$550$	3.71792	0.158533
$551$	1.27346	0.0542513
$552$	18.0853	0.769762
$553$	0	0
$554$	−3.73658	−0.158752
$555$	6.03739	0.256273
$556$	3.30857	0.140314
$557$	8.61162	0.364886	0.182443	−	0.983216i	$-0.441599\pi$
$557$	8.61162	0.364886	0.182443	+	0.983216i	$0.441599\pi$
$558$	32.4470	1.37359
$559$	0.443121	0.0187420
$560$	0	0
$561$	−2.90402	−0.122608
$562$	14.9370	0.630079
$563$	22.9417	0.966876	0.483438	−	0.875379i	$-0.339388\pi$
$563$	22.9417	0.966876	0.483438	+	0.875379i	$0.339388\pi$
$564$	4.79186	0.201774
$565$	0.725233	0.0305108
$566$	−2.90173	−0.121969
$567$	0	0
$568$	−34.6751	−1.45493
$569$	28.9246	1.21258	0.606292	−	0.795242i	$-0.292656\pi$
$569$	28.9246	1.21258	0.606292	+	0.795242i	$0.292656\pi$
$570$	−3.69816	−0.154899
$571$	−4.28315	−0.179244	−0.0896222	−	0.995976i	$-0.528566\pi$
$571$	−4.28315	−0.179244	−0.0896222	+	0.995976i	$0.528566\pi$
$572$	−0.674664	−0.0282091
$573$	−57.3973	−2.39781
$574$	0	0
$575$	11.9007	0.496292
$576$	13.9696	0.582067
$577$	−29.0834	−1.21076	−0.605379	−	0.795937i	$-0.706979\pi$
$577$	−29.0834	−1.21076	−0.605379	+	0.795937i	$0.706979\pi$
$578$	0.752785	0.0313117
$579$	51.4953	2.14007
$580$	0.0659421	0.00273810
$581$	0	0
$582$	24.4985	1.01549
$583$	−4.84877	−0.200815
$584$	31.1796	1.29022
$585$	−0.632246	−0.0261402
$586$	12.6087	0.520862
$587$	−25.6154	−1.05726	−0.528631	−	0.848852i	$-0.677295\pi$
$587$	−25.6154	−1.05726	−0.528631	+	0.848852i	$0.677295\pi$
$588$	29.1366	1.20157
$589$	−54.2849	−2.23677
$590$	0.396551	0.0163257
$591$	16.0053	0.658370
$592$	7.74542	0.318335
$593$	−7.32679	−0.300875	−0.150438	−	0.988620i	$-0.548068\pi$
$593$	−7.32679	−0.300875	−0.150438	+	0.988620i	$0.548068\pi$
$594$	−5.31947	−0.218260
$595$	0	0
$596$	−9.70808	−0.397659
$597$	−34.2761	−1.40283
$598$	0.853806	0.0349147
$599$	3.97710	0.162500	0.0812499	−	0.996694i	$-0.474109\pi$
$599$	3.97710	0.162500	0.0812499	+	0.996694i	$0.474109\pi$
$600$	37.0691	1.51334
$601$	−43.0129	−1.75453	−0.877265	−	0.480006i	$-0.840635\pi$
$601$	−43.0129	−1.75453	−0.877265	+	0.480006i	$0.840635\pi$
$602$	0	0
$603$	−44.0167	−1.79250
$604$	−29.4921	−1.20002
$605$	0.247215	0.0100507
$606$	33.9229	1.37803
$607$	26.0853	1.05877	0.529385	−	0.848382i	$-0.322423\pi$
$607$	26.0853	1.05877	0.529385	+	0.848382i	$0.322423\pi$
$608$	−40.1160	−1.62692
$609$	0	0
$610$	0.932572	0.0377587
$611$	0.541887	0.0219224
$612$	−7.78765	−0.314797
$613$	39.6266	1.60050	0.800252	−	0.599664i	$-0.204699\pi$
$613$	39.6266	1.60050	0.800252	+	0.599664i	$0.204699\pi$
$614$	−7.20924	−0.290941
$615$	−2.73385	−0.110240
$616$	0	0
$617$	39.4296	1.58738	0.793688	−	0.608325i	$-0.208158\pi$
$617$	39.4296	1.58738	0.793688	+	0.608325i	$0.208158\pi$
$618$	−13.3191	−0.535773
$619$	−16.1830	−0.650449	−0.325225	−	0.945637i	$-0.605440\pi$
$619$	−16.1830	−0.650449	−0.325225	+	0.945637i	$0.605440\pi$
$620$	−2.81096	−0.112891
$621$	−17.0271	−0.683273
$622$	0	0
$623$	0	0
$624$	−1.25897	−0.0503991
$625$	24.0870	0.963480
$626$	−1.67490	−0.0669424
$627$	19.8719	0.793607
$628$	−17.7784	−0.709437
$629$	8.40959	0.335312
$630$	0	0
$631$	0.717917	0.0285798	0.0142899	−	0.999898i	$-0.495451\pi$
$631$	0.717917	0.0285798	0.0142899	+	0.999898i	$0.495451\pi$
$632$	3.53289	0.140531
$633$	−33.9504	−1.34941
$634$	10.9171	0.433572
$635$	−5.00610	−0.198661
$636$	−20.1824	−0.800283
$637$	3.29491	0.130549
$638$	0.140093	0.00554634
$639$	72.8950	2.88368
$640$	−2.42008	−0.0956621
$641$	−25.8337	−1.02037	−0.510185	−	0.860065i	$-0.670423\pi$
$641$	−25.8337	−1.02037	−0.510185	+	0.860065i	$0.670423\pi$
$642$	−22.2678	−0.878841
$643$	21.5039	0.848031	0.424016	−	0.905655i	$-0.360620\pi$
$643$	21.5039	0.848031	0.424016	+	0.905655i	$0.360620\pi$
$644$	0	0
$645$	−0.675850	−0.0266116
$646$	−5.15123	−0.202673
$647$	21.3630	0.839866	0.419933	−	0.907555i	$-0.362054\pi$
$647$	21.3630	0.839866	0.419933	+	0.907555i	$0.362054\pi$
$648$	−10.9093	−0.428557
$649$	−2.13085	−0.0836432
$650$	1.75003	0.0686419
$651$	0	0
$652$	22.3827	0.876576
$653$	35.7293	1.39820	0.699098	−	0.715026i	$-0.253585\pi$
$653$	35.7293	1.39820	0.699098	+	0.715026i	$0.253585\pi$
$654$	−20.7673	−0.812066
$655$	1.52785	0.0596980
$656$	−3.50728	−0.136936
$657$	−65.5468	−2.55722
$658$	0	0
$659$	−29.9372	−1.16619	−0.583094	−	0.812404i	$-0.698158\pi$
$659$	−29.9372	−1.16619	−0.583094	+	0.812404i	$0.698158\pi$
$660$	1.02900	0.0400538
$661$	37.5315	1.45981	0.729903	−	0.683551i	$-0.239565\pi$
$661$	37.5315	1.45981	0.729903	+	0.683551i	$0.239565\pi$
$662$	2.63198	0.102295
$663$	−1.36693	−0.0530870
$664$	−43.1540	−1.67470
$665$	0	0
$666$	34.3962	1.33283
$667$	0.448424	0.0173630
$668$	32.8666	1.27165
$669$	−13.9553	−0.539543
$670$	−1.50764	−0.0582453
$671$	−5.01114	−0.193453
$672$	0	0
$673$	9.49970	0.366187	0.183093	−	0.983096i	$-0.441389\pi$
$673$	9.49970	0.366187	0.183093	+	0.983096i	$0.441389\pi$
$674$	−5.34429	−0.205854
$675$	−34.9001	−1.34330
$676$	18.3155	0.704443
$677$	16.8557	0.647818	0.323909	−	0.946088i	$-0.395003\pi$
$677$	16.8557	0.647818	0.323909	+	0.946088i	$0.395003\pi$
$678$	6.41317	0.246296
$679$	0	0
$680$	−0.638939	−0.0245022
$681$	1.91158	0.0732517
$682$	−5.97186	−0.228674
$683$	−38.0463	−1.45580	−0.727900	−	0.685683i	$-0.759504\pi$
$683$	−38.0463	−1.45580	−0.727900	+	0.685683i	$0.759504\pi$
$684$	53.2901	2.03760
$685$	1.42804	0.0545627
$686$	0	0
$687$	75.6479	2.88615
$688$	−0.867054	−0.0330561
$689$	−2.28232	−0.0869496
$690$	−1.30223	−0.0495751
$691$	6.68246	0.254213	0.127106	−	0.991889i	$-0.459431\pi$
$691$	6.68246	0.254213	0.127106	+	0.991889i	$0.459431\pi$
$692$	14.1580	0.538205
$693$	0	0
$694$	15.1389	0.574664
$695$	−0.570654	−0.0216462
$696$	1.39679	0.0529450
$697$	−3.80803	−0.144240
$698$	−22.7629	−0.861588
$699$	70.4029	2.66288
$700$	0	0
$701$	25.4821	0.962446	0.481223	−	0.876598i	$-0.340193\pi$
$701$	25.4821	0.962446	0.481223	+	0.876598i	$0.340193\pi$
$702$	−2.50388	−0.0945030
$703$	−57.5460	−2.17039
$704$	−2.57110	−0.0969020
$705$	−0.826489	−0.0311274
$706$	−17.3558	−0.653195
$707$	0	0
$708$	−8.86939	−0.333332
$709$	25.1423	0.944237	0.472119	−	0.881535i	$-0.343489\pi$
$709$	25.1423	0.944237	0.472119	+	0.881535i	$0.343489\pi$
$710$	2.49677	0.0937022
$711$	−7.42694	−0.278532
$712$	−42.0803	−1.57703
$713$	−19.1153	−0.715873
$714$	0	0
$715$	0.116365	0.00435179
$716$	7.22772	0.270113
$717$	−72.5298	−2.70867
$718$	14.4291	0.538490
$719$	−2.99052	−0.111528	−0.0557638	−	0.998444i	$-0.517759\pi$
$719$	−2.99052	−0.111528	−0.0557638	+	0.998444i	$0.517759\pi$
$720$	1.23712	0.0461046
$721$	0	0
$722$	20.9465	0.779547
$723$	−45.7690	−1.70217
$724$	33.4621	1.24361
$725$	0.919126	0.0341355
$726$	2.18610	0.0811338
$727$	12.1130	0.449246	0.224623	−	0.974446i	$-0.427885\pi$
$727$	12.1130	0.449246	0.224623	+	0.974446i	$0.427885\pi$
$728$	0	0
$729$	−38.6289	−1.43070
$730$	−2.24508	−0.0830943
$731$	−0.941404	−0.0348191
$732$	−20.8582	−0.770942
$733$	−38.2567	−1.41304	−0.706522	−	0.707691i	$-0.749737\pi$
$733$	−38.2567	−1.41304	−0.706522	+	0.707691i	$0.749737\pi$
$734$	21.5610	0.795832
$735$	−5.02542	−0.185365
$736$	−14.1260	−0.520692
$737$	8.10126	0.298414
$738$	−15.5753	−0.573335
$739$	−20.8954	−0.768650	−0.384325	−	0.923198i	$-0.625566\pi$
$739$	−20.8954	−0.768650	−0.384325	+	0.923198i	$0.625566\pi$
$740$	−2.97983	−0.109541
$741$	9.35374	0.343618
$742$	0	0
$743$	−17.6266	−0.646658	−0.323329	−	0.946287i	$-0.604802\pi$
$743$	−17.6266	−0.646658	−0.323329	+	0.946287i	$0.604802\pi$
$744$	−59.5418	−2.18291
$745$	1.67443	0.0613464
$746$	−2.56479	−0.0939036
$747$	90.7197	3.31926
$748$	1.43332	0.0524072
$749$	0	0
$750$	−5.37134	−0.196134
$751$	−10.2355	−0.373498	−0.186749	−	0.982408i	$-0.559795\pi$
$751$	−10.2355	−0.373498	−0.186749	+	0.982408i	$0.559795\pi$
$752$	−1.06031	−0.0386655
$753$	−51.7917	−1.88740
$754$	0.0659421	0.00240147
$755$	5.08674	0.185125
$756$	0	0
$757$	−28.3856	−1.03169	−0.515847	−	0.856681i	$-0.672523\pi$
$757$	−28.3856	−1.03169	−0.515847	+	0.856681i	$0.672523\pi$
$758$	−4.92671	−0.178946
$759$	6.99748	0.253992
$760$	4.37220	0.158596
$761$	−33.7095	−1.22197	−0.610985	−	0.791642i	$-0.709226\pi$
$761$	−33.7095	−1.22197	−0.610985	+	0.791642i	$0.709226\pi$
$762$	−44.2685	−1.60368
$763$	0	0
$764$	28.3292	1.02491
$765$	1.34320	0.0485634
$766$	−11.4953	−0.415342
$767$	−1.00300	−0.0362161
$768$	−36.3336	−1.31107
$769$	−12.9994	−0.468771	−0.234385	−	0.972144i	$-0.575308\pi$
$769$	−12.9994	−0.468771	−0.234385	+	0.972144i	$0.575308\pi$
$770$	0	0
$771$	−20.7332	−0.746689
$772$	−25.4162	−0.914748
$773$	45.0580	1.62062	0.810312	−	0.585998i	$-0.199297\pi$
$773$	45.0580	1.62062	0.810312	+	0.585998i	$0.199297\pi$
$774$	−3.85045	−0.138402
$775$	−39.1803	−1.40740
$776$	−28.9637	−1.03974
$777$	0	0
$778$	19.6301	0.703775
$779$	26.0580	0.933625
$780$	0.484353	0.0173426
$781$	−13.4163	−0.480073
$782$	−1.81390	−0.0648649
$783$	−1.31505	−0.0469962
$784$	−6.44716	−0.230256
$785$	3.06639	0.109444
$786$	13.5106	0.481907
$787$	−1.99917	−0.0712628	−0.0356314	−	0.999365i	$-0.511344\pi$
$787$	−1.99917	−0.0712628	−0.0356314	+	0.999365i	$0.511344\pi$
$788$	−7.89962	−0.281412
$789$	31.1330	1.10836
$790$	−0.254385	−0.00905061
$791$	0	0
$792$	14.0427	0.498984
$793$	−2.35875	−0.0837618
$794$	3.80253	0.134947
$795$	3.48101	0.123459
$796$	16.9174	0.599622
$797$	18.4406	0.653201	0.326600	−	0.945163i	$-0.394097\pi$
$797$	18.4406	0.653201	0.326600	+	0.945163i	$0.394097\pi$
$798$	0	0
$799$	−1.15123	−0.0407277
$800$	−28.9538	−1.02367
$801$	88.4625	3.12567
$802$	−13.7913	−0.486986
$803$	12.0639	0.425725
$804$	33.7204	1.18923
$805$	0	0
$806$	−2.81096	−0.0990120
$807$	−56.0530	−1.97316
$808$	−40.1059	−1.41092
$809$	0.204585	0.00719283	0.00359641	−	0.999994i	$-0.498855\pi$
$809$	0.204585	0.00719283	0.00359641	+	0.999994i	$0.498855\pi$
$810$	0.785521	0.0276004
$811$	−37.0990	−1.30272	−0.651361	−	0.758768i	$-0.725802\pi$
$811$	−37.0990	−1.30272	−0.651361	+	0.758768i	$0.725802\pi$
$812$	0	0
$813$	1.97387	0.0692266
$814$	−6.33061	−0.221888
$815$	−3.86053	−0.135228
$816$	2.67466	0.0936320
$817$	6.44193	0.225375
$818$	20.9732	0.733310
$819$	0	0
$820$	1.34933	0.0471206
$821$	2.34234	0.0817483	0.0408741	−	0.999164i	$-0.486986\pi$
$821$	2.34234	0.0817483	0.0408741	+	0.999164i	$0.486986\pi$
$822$	12.6280	0.440453
$823$	−20.7522	−0.723376	−0.361688	−	0.932299i	$-0.617799\pi$
$823$	−20.7522	−0.723376	−0.361688	+	0.932299i	$0.617799\pi$
$824$	15.7467	0.548563
$825$	14.3426	0.499345
$826$	0	0
$827$	45.8022	1.59270	0.796349	−	0.604838i	$-0.206762\pi$
$827$	45.8022	1.59270	0.796349	+	0.604838i	$0.206762\pi$
$828$	18.7650	0.652130
$829$	31.3586	1.08913	0.544564	−	0.838719i	$-0.316695\pi$
$829$	31.3586	1.08913	0.544564	+	0.838719i	$0.316695\pi$
$830$	3.10730	0.107856
$831$	−14.4146	−0.500038
$832$	−1.21022	−0.0419569
$833$	−7.00000	−0.242536
$834$	−5.04624	−0.174737
$835$	−5.66877	−0.196176
$836$	−9.80803	−0.339218
$837$	56.0578	1.93764
$838$	−21.7421	−0.751068
$839$	−24.0983	−0.831965	−0.415983	−	0.909373i	$-0.636562\pi$
$839$	−24.0983	−0.831965	−0.415983	+	0.909373i	$0.636562\pi$
$840$	0	0
$841$	−28.9654	−0.998806
$842$	6.44365	0.222063
$843$	57.6224	1.98462
$844$	16.7567	0.576788
$845$	−3.15902	−0.108674
$846$	−4.70867	−0.161888
$847$	0	0
$848$	4.46582	0.153357
$849$	−11.1940	−0.384177
$850$	−3.71792	−0.127524
$851$	−20.2636	−0.694628
$852$	−55.8437	−1.91317
$853$	−34.6911	−1.18780	−0.593900	−	0.804539i	$-0.702412\pi$
$853$	−34.6911	−1.18780	−0.593900	+	0.804539i	$0.702412\pi$
$854$	0	0
$855$	−9.19137	−0.314338
$856$	26.3265	0.899820
$857$	44.2218	1.51059	0.755294	−	0.655386i	$-0.227494\pi$
$857$	44.2218	1.51059	0.755294	+	0.655386i	$0.227494\pi$
$858$	1.02900	0.0351295
$859$	10.7921	0.368221	0.184110	−	0.982906i	$-0.441060\pi$
$859$	10.7921	0.368221	0.184110	+	0.982906i	$0.441060\pi$
$860$	0.333575	0.0113748
$861$	0	0
$862$	−10.4464	−0.355804
$863$	−9.70870	−0.330488	−0.165244	−	0.986253i	$-0.552841\pi$
$863$	−9.70870	−0.330488	−0.165244	+	0.986253i	$0.552841\pi$
$864$	41.4262	1.40935
$865$	−2.44193	−0.0830283
$866$	7.01763	0.238469
$867$	2.90402	0.0986256
$868$	0	0
$869$	1.36693	0.0463698
$870$	−0.100575	−0.00340982
$871$	3.81328	0.129208
$872$	24.5524	0.831450
$873$	60.8883	2.06076
$874$	12.4123	0.419854
$875$	0	0
$876$	50.2143	1.69658
$877$	32.3596	1.09271	0.546354	−	0.837555i	$-0.316015\pi$
$877$	32.3596	1.09271	0.546354	+	0.837555i	$0.316015\pi$
$878$	18.1813	0.613590
$879$	48.6407	1.64061
$880$	−0.227691	−0.00767545
$881$	−56.5212	−1.90425	−0.952123	−	0.305714i	$-0.901105\pi$
$881$	−56.5212	−1.90425	−0.952123	+	0.305714i	$0.901105\pi$
$882$	−28.6308	−0.964050
$883$	20.2210	0.680491	0.340245	−	0.940337i	$-0.389490\pi$
$883$	20.2210	0.680491	0.340245	+	0.940337i	$0.389490\pi$
$884$	0.674664	0.0226914
$885$	1.52977	0.0514228
$886$	24.3707	0.818751
$887$	−19.1964	−0.644552	−0.322276	−	0.946646i	$-0.604448\pi$
$887$	−19.1964	−0.644552	−0.322276	+	0.946646i	$0.604448\pi$
$888$	−63.1187	−2.11813
$889$	0	0
$890$	3.02998	0.101565
$891$	−4.22097	−0.141408
$892$	6.88782	0.230621
$893$	7.87777	0.263619
$894$	14.8068	0.495214
$895$	−1.24662	−0.0416700
$896$	0	0
$897$	3.29373	0.109974
$898$	−26.8435	−0.895780
$899$	−1.47633	−0.0492385
$900$	38.4623	1.28208
$901$	4.84877	0.161536
$902$	2.86663	0.0954483
$903$	0	0
$904$	−7.58206	−0.252175
$905$	−5.77147	−0.191850
$906$	44.9815	1.49441
$907$	12.1495	0.403419	0.201710	−	0.979445i	$-0.435350\pi$
$907$	12.1495	0.403419	0.201710	+	0.979445i	$0.435350\pi$
$908$	−0.943483	−0.0313106
$909$	84.3118	2.79645
$910$	0	0
$911$	−55.2605	−1.83086	−0.915432	−	0.402473i	$-0.868151\pi$
$911$	−55.2605	−1.83086	−0.915432	+	0.402473i	$0.868151\pi$
$912$	−18.3025	−0.606055
$913$	−16.6969	−0.552588
$914$	11.8209	0.390999
$915$	3.59758	0.118932
$916$	−37.3370	−1.23365
$917$	0	0
$918$	5.31947	0.175569
$919$	−29.0100	−0.956950	−0.478475	−	0.878101i	$-0.658810\pi$
$919$	−29.0100	−0.956950	−0.478475	+	0.878101i	$0.658810\pi$
$920$	1.53958	0.0507584
$921$	−27.8111	−0.916406
$922$	21.9916	0.724255
$923$	−6.31508	−0.207864
$924$	0	0
$925$	−41.5340	−1.36563
$926$	−14.9294	−0.490611
$927$	−33.1032	−1.08725
$928$	−1.09100	−0.0358137
$929$	30.2109	0.991188	0.495594	−	0.868554i	$-0.334950\pi$
$929$	30.2109	0.991188	0.495594	+	0.868554i	$0.334950\pi$
$930$	4.28730	0.140586
$931$	47.9003	1.56987
$932$	−34.7483	−1.13822
$933$	0	0
$934$	−11.7541	−0.384606
$935$	−0.247215	−0.00808480
$936$	6.60991	0.216052
$937$	−7.52930	−0.245971	−0.122986	−	0.992408i	$-0.539247\pi$
$937$	−7.52930	−0.245971	−0.122986	+	0.992408i	$0.539247\pi$
$938$	0	0
$939$	−6.46125	−0.210855
$940$	0.407925	0.0133050
$941$	−2.97713	−0.0970516	−0.0485258	−	0.998822i	$-0.515452\pi$
$941$	−2.97713	−0.0970516	−0.0485258	+	0.998822i	$0.515452\pi$
$942$	27.1158	0.883480
$943$	9.17579	0.298805
$944$	1.96256	0.0638759
$945$	0	0
$946$	0.708675	0.0230410
$947$	18.0775	0.587441	0.293720	−	0.955891i	$-0.405107\pi$
$947$	18.0775	0.587441	0.293720	+	0.955891i	$0.405107\pi$
$948$	5.68966	0.184791
$949$	5.67849	0.184331
$950$	25.4413	0.825426
$951$	42.1147	1.36566
$952$	0	0
$953$	43.0603	1.39486	0.697430	−	0.716653i	$-0.254327\pi$
$953$	43.0603	1.39486	0.697430	+	0.716653i	$0.254327\pi$
$954$	19.8320	0.642086
$955$	−4.88615	−0.158112
$956$	35.7980	1.15779
$957$	0.540437	0.0174699
$958$	18.4777	0.596986
$959$	0	0
$960$	1.84584	0.0595741
$961$	31.9328	1.03009
$962$	−2.97983	−0.0960736
$963$	−55.3443	−1.78344
$964$	22.5899	0.727571
$965$	4.38373	0.141117
$966$	0	0
$967$	−13.8368	−0.444962	−0.222481	−	0.974937i	$-0.571415\pi$
$967$	−13.8368	−0.444962	−0.222481	+	0.974937i	$0.571415\pi$
$968$	−2.58455	−0.0830705
$969$	−19.8719	−0.638378
$970$	2.08552	0.0669622
$971$	−10.6425	−0.341535	−0.170767	−	0.985311i	$-0.554625\pi$
$971$	−10.6425	−0.341535	−0.170767	+	0.985311i	$0.554625\pi$
$972$	12.8158	0.411069
$973$	0	0
$974$	−19.1990	−0.615174
$975$	6.75109	0.216208
$976$	4.61537	0.147734
$977$	14.0884	0.450729	0.225364	−	0.974275i	$-0.427643\pi$
$977$	14.0884	0.450729	0.225364	+	0.974275i	$0.427643\pi$
$978$	−34.1383	−1.09162
$979$	−16.2815	−0.520359
$980$	2.48036	0.0792322
$981$	−51.6149	−1.64794
$982$	6.31529	0.201529
$983$	5.94356	0.189570	0.0947851	−	0.995498i	$-0.469784\pi$
$983$	5.94356	0.189570	0.0947851	+	0.995498i	$0.469784\pi$
$984$	28.5815	0.911144
$985$	1.36251	0.0434132
$986$	−0.140093	−0.00446147
$987$	0	0
$988$	−4.61666	−0.146876
$989$	2.26839	0.0721307
$990$	−1.01114	−0.0321361
$991$	23.9425	0.760557	0.380279	−	0.924872i	$-0.375828\pi$
$991$	23.9425	0.760557	0.380279	+	0.924872i	$0.375828\pi$
$992$	46.5067	1.47659
$993$	10.1534	0.322208
$994$	0	0
$995$	−2.91788	−0.0925030
$996$	−69.4989	−2.20216
$997$	−59.4657	−1.88330	−0.941649	−	0.336597i	$-0.890724\pi$
$997$	−59.4657	−1.88330	−0.941649	+	0.336597i	$0.890724\pi$
$998$	23.3326	0.738579
$999$	59.4254	1.88014

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.a.f.1.3	✓	4
3.2	odd	2		1683.2.a.y.1.2		4
4.3	odd	2		2992.2.a.v.1.1		4
5.4	even	2		4675.2.a.bd.1.2		4
7.6	odd	2		9163.2.a.l.1.3		4
11.10	odd	2		2057.2.a.s.1.2		4
17.16	even	2		3179.2.a.w.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.a.f.1.3	✓	4	1.1	even	1	trivial
1683.2.a.y.1.2		4	3.2	odd	2
2057.2.a.s.1.2		4	11.10	odd	2
2992.2.a.v.1.1		4	4.3	odd	2
3179.2.a.w.1.3		4	17.16	even	2
4675.2.a.bd.1.2		4	5.4	even	2
9163.2.a.l.1.3		4	7.6	odd	2

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:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.a (trivial)

\(f(q)\)	\(=\)	\(q+0.752785 q^{2} +2.90402 q^{3} -1.43332 q^{4} +0.247215 q^{5} +2.18610 q^{6} -2.58455 q^{8} +5.43332 q^{9} +O(q^{10})\)	\(q+0.752785 q^{2} +2.90402 q^{3} -1.43332 q^{4} +0.247215 q^{5} +2.18610 q^{6} -2.58455 q^{8} +5.43332 q^{9} +0.186100 q^{10} -1.00000 q^{11} -4.16237 q^{12} -0.470702 q^{13} +0.717917 q^{15} +0.921022 q^{16} +1.00000 q^{17} +4.09012 q^{18} -6.84290 q^{19} -0.354337 q^{20} -0.752785 q^{22} -2.40959 q^{23} -7.50557 q^{24} -4.93888 q^{25} -0.354337 q^{26} +7.06639 q^{27} -0.186100 q^{29} +0.540437 q^{30} +7.93302 q^{31} +5.86243 q^{32} -2.90402 q^{33} +0.752785 q^{34} -7.78765 q^{36} +8.40959 q^{37} -5.15123 q^{38} -1.36693 q^{39} -0.638939 q^{40} -3.80803 q^{41} -0.941404 q^{43} +1.43332 q^{44} +1.34320 q^{45} -1.81390 q^{46} -1.15123 q^{47} +2.67466 q^{48} -7.00000 q^{49} -3.71792 q^{50} +2.90402 q^{51} +0.674664 q^{52} +4.84877 q^{53} +5.31947 q^{54} -0.247215 q^{55} -19.8719 q^{57} -0.140093 q^{58} +2.13085 q^{59} -1.02900 q^{60} +5.01114 q^{61} +5.97186 q^{62} +2.57110 q^{64} -0.116365 q^{65} -2.18610 q^{66} -8.10126 q^{67} -1.43332 q^{68} -6.99748 q^{69} +13.4163 q^{71} -14.0427 q^{72} -12.0639 q^{73} +6.33061 q^{74} -14.3426 q^{75} +9.80803 q^{76} -1.02900 q^{78} -1.36693 q^{79} +0.227691 q^{80} +4.22097 q^{81} -2.86663 q^{82} +16.6969 q^{83} +0.247215 q^{85} -0.708675 q^{86} -0.540437 q^{87} +2.58455 q^{88} +16.2815 q^{89} +1.01114 q^{90} +3.45370 q^{92} +23.0376 q^{93} -0.866630 q^{94} -1.69167 q^{95} +17.0246 q^{96} +11.2065 q^{97} -5.26949 q^{98} -5.43332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{3} + 5 q^{4} + 3 q^{5} - 4 q^{6} + 9 q^{8} + 11 q^{9}+O(q^{10}) \) 4 * q + q^2 + q^3 + 5 * q^4 + 3 * q^5 - 4 * q^6 + 9 * q^8 + 11 * q^9	\( 4 q + q^{2} + q^{3} + 5 q^{4} + 3 q^{5} - 4 q^{6} + 9 q^{8} + 11 q^{9} - 12 q^{10} - 4 q^{11} - 2 q^{13} + 5 q^{15} + 19 q^{16} + 4 q^{17} - 7 q^{18} - 2 q^{19} - 6 q^{20} - q^{22} + 5 q^{23} - 26 q^{24} - 5 q^{25} - 6 q^{26} + q^{27} + 12 q^{29} - 6 q^{30} - 17 q^{31} + 39 q^{32} - q^{33} + q^{34} - 25 q^{36} + 19 q^{37} - 12 q^{38} - 22 q^{39} - 20 q^{40} + 6 q^{41} - 4 q^{43} - 5 q^{44} + 18 q^{45} - 20 q^{46} + 4 q^{47} - 32 q^{48} - 28 q^{49} - 17 q^{50} + q^{51} - 40 q^{52} + 28 q^{53} + 30 q^{54} - 3 q^{55} - 16 q^{57} + 15 q^{59} + 34 q^{60} + 12 q^{61} + 6 q^{62} + 35 q^{64} + 4 q^{65} + 4 q^{66} - q^{67} + 5 q^{68} - 13 q^{69} + 17 q^{71} - 25 q^{72} - 6 q^{73} + 26 q^{74} + 6 q^{75} + 18 q^{76} + 34 q^{78} - 22 q^{79} - 38 q^{80} + 10 q^{82} + 8 q^{83} + 3 q^{85} - 12 q^{86} + 6 q^{87} - 9 q^{88} - 13 q^{89} - 4 q^{90} - 12 q^{92} + 31 q^{93} + 18 q^{94} + 10 q^{95} + 16 q^{96} + 17 q^{97} - 7 q^{98} - 11 q^{99}+O(q^{100}) \) 4 * q + q^2 + q^3 + 5 * q^4 + 3 * q^5 - 4 * q^6 + 9 * q^8 + 11 * q^9 - 12 * q^10 - 4 * q^11 - 2 * q^13 + 5 * q^15 + 19 * q^16 + 4 * q^17 - 7 * q^18 - 2 * q^19 - 6 * q^20 - q^22 + 5 * q^23 - 26 * q^24 - 5 * q^25 - 6 * q^26 + q^27 + 12 * q^29 - 6 * q^30 - 17 * q^31 + 39 * q^32 - q^33 + q^34 - 25 * q^36 + 19 * q^37 - 12 * q^38 - 22 * q^39 - 20 * q^40 + 6 * q^41 - 4 * q^43 - 5 * q^44 + 18 * q^45 - 20 * q^46 + 4 * q^47 - 32 * q^48 - 28 * q^49 - 17 * q^50 + q^51 - 40 * q^52 + 28 * q^53 + 30 * q^54 - 3 * q^55 - 16 * q^57 + 15 * q^59 + 34 * q^60 + 12 * q^61 + 6 * q^62 + 35 * q^64 + 4 * q^65 + 4 * q^66 - q^67 + 5 * q^68 - 13 * q^69 + 17 * q^71 - 25 * q^72 - 6 * q^73 + 26 * q^74 + 6 * q^75 + 18 * q^76 + 34 * q^78 - 22 * q^79 - 38 * q^80 + 10 * q^82 + 8 * q^83 + 3 * q^85 - 12 * q^86 + 6 * q^87 - 9 * q^88 - 13 * q^89 - 4 * q^90 - 12 * q^92 + 31 * q^93 + 18 * q^94 + 10 * q^95 + 16 * q^96 + 17 * q^97 - 7 * q^98 - 11 * q^99

Introduction

L-functions

Modular forms

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Database

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