LMFDB - Embedded newform 7942.2.a.ca.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, 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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.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:	$63.4171892853$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 33 x^{13} + 101 x^{12} + 408 x^{11} - 1314 x^{10} - 2271 x^{9} + 8292 x^{8} + \cdots - 3592 $ x^15 - 3x^14 - 33x^13 + 101x^12 + 408x^11 - 1314x^10 - 2271x^9 + 8292x^8 + 4944x^7 - 26254x^6 + 1401x^5 + 39318x^4 - 19069x^3 - 20118x^2 + 18180x - 3592
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$3.14157$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +3.14157 q^{3} +1.00000 q^{4} +4.25314 q^{5} +3.14157 q^{6} -0.789032 q^{7} +1.00000 q^{8} +6.86947 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +3.14157 q^{3} +1.00000 q^{4} +4.25314 q^{5} +3.14157 q^{6} -0.789032 q^{7} +1.00000 q^{8} +6.86947 q^{9} +4.25314 q^{10} -1.00000 q^{11} +3.14157 q^{12} -2.86562 q^{13} -0.789032 q^{14} +13.3616 q^{15} +1.00000 q^{16} -4.87274 q^{17} +6.86947 q^{18} +4.25314 q^{20} -2.47880 q^{21} -1.00000 q^{22} -3.00362 q^{23} +3.14157 q^{24} +13.0892 q^{25} -2.86562 q^{26} +12.1562 q^{27} -0.789032 q^{28} +6.79350 q^{29} +13.3616 q^{30} +10.2935 q^{31} +1.00000 q^{32} -3.14157 q^{33} -4.87274 q^{34} -3.35587 q^{35} +6.86947 q^{36} -0.927122 q^{37} -9.00256 q^{39} +4.25314 q^{40} -0.0787280 q^{41} -2.47880 q^{42} -4.27932 q^{43} -1.00000 q^{44} +29.2169 q^{45} -3.00362 q^{46} +1.10688 q^{47} +3.14157 q^{48} -6.37743 q^{49} +13.0892 q^{50} -15.3081 q^{51} -2.86562 q^{52} -1.13808 q^{53} +12.1562 q^{54} -4.25314 q^{55} -0.789032 q^{56} +6.79350 q^{58} +2.50283 q^{59} +13.3616 q^{60} -7.07049 q^{61} +10.2935 q^{62} -5.42023 q^{63} +1.00000 q^{64} -12.1879 q^{65} -3.14157 q^{66} -0.0387536 q^{67} -4.87274 q^{68} -9.43609 q^{69} -3.35587 q^{70} -14.8810 q^{71} +6.86947 q^{72} +9.08659 q^{73} -0.927122 q^{74} +41.1207 q^{75} +0.789032 q^{77} -9.00256 q^{78} -1.40495 q^{79} +4.25314 q^{80} +17.5812 q^{81} -0.0787280 q^{82} +11.0728 q^{83} -2.47880 q^{84} -20.7245 q^{85} -4.27932 q^{86} +21.3423 q^{87} -1.00000 q^{88} -0.532856 q^{89} +29.2169 q^{90} +2.26107 q^{91} -3.00362 q^{92} +32.3376 q^{93} +1.10688 q^{94} +3.14157 q^{96} -11.9267 q^{97} -6.37743 q^{98} -6.86947 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 15 q^{8} + 30 q^{9}+O(q^{10}) $ 15 * q + 15 * q^2 + 3 * q^3 + 15 * q^4 + 9 * q^5 + 3 * q^6 + 15 * q^8 + 30 * q^9	$ 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 15 q^{8} + 30 q^{9} + 9 q^{10} - 15 q^{11} + 3 q^{12} + 21 q^{15} + 15 q^{16} + 21 q^{17} + 30 q^{18} + 9 q^{20} - 9 q^{21} - 15 q^{22} + 21 q^{23} + 3 q^{24} + 24 q^{25} + 3 q^{27} - 9 q^{29} + 21 q^{30} + 18 q^{31} + 15 q^{32} - 3 q^{33} + 21 q^{34} + 18 q^{35} + 30 q^{36} - 9 q^{37} + 9 q^{40} + 15 q^{41} - 9 q^{42} + 3 q^{43} - 15 q^{44} + 54 q^{45} + 21 q^{46} + 39 q^{47} + 3 q^{48} + 33 q^{49} + 24 q^{50} + 30 q^{51} + 18 q^{53} + 3 q^{54} - 9 q^{55} - 9 q^{58} + 6 q^{59} + 21 q^{60} - 30 q^{61} + 18 q^{62} + 24 q^{63} + 15 q^{64} + 6 q^{65} - 3 q^{66} + 9 q^{67} + 21 q^{68} - 42 q^{69} + 18 q^{70} + 9 q^{71} + 30 q^{72} + 12 q^{73} - 9 q^{74} + 21 q^{75} + 12 q^{79} + 9 q^{80} + 63 q^{81} + 15 q^{82} + 30 q^{83} - 9 q^{84} - 3 q^{85} + 3 q^{86} + 9 q^{87} - 15 q^{88} - 6 q^{89} + 54 q^{90} + 96 q^{91} + 21 q^{92} + 102 q^{93} + 39 q^{94} + 3 q^{96} + 33 q^{98} - 30 q^{99}+O(q^{100}) $ 15 * q + 15 * q^2 + 3 * q^3 + 15 * q^4 + 9 * q^5 + 3 * q^6 + 15 * q^8 + 30 * q^9 + 9 * q^10 - 15 * q^11 + 3 * q^12 + 21 * q^15 + 15 * q^16 + 21 * q^17 + 30 * q^18 + 9 * q^20 - 9 * q^21 - 15 * q^22 + 21 * q^23 + 3 * q^24 + 24 * q^25 + 3 * q^27 - 9 * q^29 + 21 * q^30 + 18 * q^31 + 15 * q^32 - 3 * q^33 + 21 * q^34 + 18 * q^35 + 30 * q^36 - 9 * q^37 + 9 * q^40 + 15 * q^41 - 9 * q^42 + 3 * q^43 - 15 * q^44 + 54 * q^45 + 21 * q^46 + 39 * q^47 + 3 * q^48 + 33 * q^49 + 24 * q^50 + 30 * q^51 + 18 * q^53 + 3 * q^54 - 9 * q^55 - 9 * q^58 + 6 * q^59 + 21 * q^60 - 30 * q^61 + 18 * q^62 + 24 * q^63 + 15 * q^64 + 6 * q^65 - 3 * q^66 + 9 * q^67 + 21 * q^68 - 42 * q^69 + 18 * q^70 + 9 * q^71 + 30 * q^72 + 12 * q^73 - 9 * q^74 + 21 * q^75 + 12 * q^79 + 9 * q^80 + 63 * q^81 + 15 * q^82 + 30 * q^83 - 9 * q^84 - 3 * q^85 + 3 * q^86 + 9 * q^87 - 15 * q^88 - 6 * q^89 + 54 * q^90 + 96 * q^91 + 21 * q^92 + 102 * q^93 + 39 * q^94 + 3 * q^96 + 33 * q^98 - 30 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	3.14157	1.81379	0.906894	−	0.421360i	$-0.138447\pi$
$3$	3.14157	1.81379	0.906894	+	0.421360i	$0.138447\pi$
$4$	1.00000	0.500000
$5$	4.25314	1.90206	0.951032	−	0.309093i	$-0.100026\pi$
$5$	4.25314	1.90206	0.951032	+	0.309093i	$0.100026\pi$
$6$	3.14157	1.28254
$7$	−0.789032	−0.298226	−0.149113	−	0.988820i	$-0.547642\pi$
$7$	−0.789032	−0.298226	−0.149113	+	0.988820i	$0.547642\pi$
$8$	1.00000	0.353553
$9$	6.86947	2.28982
$10$	4.25314	1.34496
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	3.14157	0.906894
$13$	−2.86562	−0.794781	−0.397391	−	0.917650i	$-0.630084\pi$
$13$	−2.86562	−0.794781	−0.397391	+	0.917650i	$0.630084\pi$
$14$	−0.789032	−0.210878
$15$	13.3616	3.44994
$16$	1.00000	0.250000
$17$	−4.87274	−1.18181	−0.590906	−	0.806740i	$-0.701230\pi$
$17$	−4.87274	−1.18181	−0.590906	+	0.806740i	$0.701230\pi$
$18$	6.86947	1.61915
$19$	0	0
$19$	0	0
$20$	4.25314	0.951032
$21$	−2.47880	−0.540919
$22$	−1.00000	−0.213201
$23$	−3.00362	−0.626298	−0.313149	−	0.949704i	$-0.601384\pi$
$23$	−3.00362	−0.626298	−0.313149	+	0.949704i	$0.601384\pi$
$24$	3.14157	0.641271
$25$	13.0892	2.61785
$26$	−2.86562	−0.561995
$27$	12.1562	2.33947
$28$	−0.789032	−0.149113
$29$	6.79350	1.26152	0.630760	−	0.775978i	$-0.282743\pi$
$29$	6.79350	1.26152	0.630760	+	0.775978i	$0.282743\pi$
$30$	13.3616	2.43947
$31$	10.2935	1.84876	0.924380	−	0.381473i	$-0.124583\pi$
$31$	10.2935	1.84876	0.924380	+	0.381473i	$0.124583\pi$
$32$	1.00000	0.176777
$33$	−3.14157	−0.546877
$34$	−4.87274	−0.835668
$35$	−3.35587	−0.567245
$36$	6.86947	1.14491
$37$	−0.927122	−0.152418	−0.0762089	−	0.997092i	$-0.524282\pi$
$37$	−0.927122	−0.152418	−0.0762089	+	0.997092i	$0.524282\pi$
$38$	0	0
$39$	−9.00256	−1.44156
$40$	4.25314	0.672481
$41$	−0.0787280	−0.0122952	−0.00614762	−	0.999981i	$-0.501957\pi$
$41$	−0.0787280	−0.0122952	−0.00614762	+	0.999981i	$0.501957\pi$
$42$	−2.47880	−0.382487
$43$	−4.27932	−0.652590	−0.326295	−	0.945268i	$-0.605800\pi$
$43$	−4.27932	−0.652590	−0.326295	+	0.945268i	$0.605800\pi$
$44$	−1.00000	−0.150756
$45$	29.2169	4.35539
$46$	−3.00362	−0.442860
$47$	1.10688	0.161455	0.0807275	−	0.996736i	$-0.474276\pi$
$47$	1.10688	0.161455	0.0807275	+	0.996736i	$0.474276\pi$
$48$	3.14157	0.453447
$49$	−6.37743	−0.911061
$50$	13.0892	1.85110
$51$	−15.3081	−2.14356
$52$	−2.86562	−0.397391
$53$	−1.13808	−0.156327	−0.0781636	−	0.996941i	$-0.524906\pi$
$53$	−1.13808	−0.156327	−0.0781636	+	0.996941i	$0.524906\pi$
$54$	12.1562	1.65425
$55$	−4.25314	−0.573494
$56$	−0.789032	−0.105439
$57$	0	0
$58$	6.79350	0.892030
$59$	2.50283	0.325840	0.162920	−	0.986639i	$-0.447909\pi$
$59$	2.50283	0.325840	0.162920	+	0.986639i	$0.447909\pi$
$60$	13.3616	1.72497
$61$	−7.07049	−0.905283	−0.452642	−	0.891693i	$-0.649518\pi$
$61$	−7.07049	−0.905283	−0.452642	+	0.891693i	$0.649518\pi$
$62$	10.2935	1.30727
$63$	−5.42023	−0.682885
$64$	1.00000	0.125000
$65$	−12.1879	−1.51172
$66$	−3.14157	−0.386701
$67$	−0.0387536	−0.00473451	−0.00236726	−	0.999997i	$-0.500754\pi$
$67$	−0.0387536	−0.00473451	−0.00236726	+	0.999997i	$0.500754\pi$
$68$	−4.87274	−0.590906
$69$	−9.43609	−1.13597
$70$	−3.35587	−0.401103
$71$	−14.8810	−1.76605	−0.883027	−	0.469323i	$-0.844498\pi$
$71$	−14.8810	−1.76605	−0.883027	+	0.469323i	$0.844498\pi$
$72$	6.86947	0.809575
$73$	9.08659	1.06350	0.531752	−	0.846900i	$-0.321534\pi$
$73$	9.08659	1.06350	0.531752	+	0.846900i	$0.321534\pi$
$74$	−0.927122	−0.107776
$75$	41.1207	4.74821
$76$	0	0
$77$	0.789032	0.0899185
$78$	−9.00256	−1.01934
$79$	−1.40495	−0.158069	−0.0790344	−	0.996872i	$-0.525184\pi$
$79$	−1.40495	−0.158069	−0.0790344	+	0.996872i	$0.525184\pi$
$80$	4.25314	0.475516
$81$	17.5812	1.95347
$82$	−0.0787280	−0.00869405
$83$	11.0728	1.21539	0.607696	−	0.794169i	$-0.292094\pi$
$83$	11.0728	1.21539	0.607696	+	0.794169i	$0.292094\pi$
$84$	−2.47880	−0.270459
$85$	−20.7245	−2.24788
$86$	−4.27932	−0.461451
$87$	21.3423	2.28813
$88$	−1.00000	−0.106600
$89$	−0.532856	−0.0564826	−0.0282413	−	0.999601i	$-0.508991\pi$
$89$	−0.532856	−0.0564826	−0.0282413	+	0.999601i	$0.508991\pi$
$90$	29.2169	3.07973
$91$	2.26107	0.237024
$92$	−3.00362	−0.313149
$93$	32.3376	3.35326
$94$	1.10688	0.114166
$95$	0	0
$96$	3.14157	0.320635
$97$	−11.9267	−1.21097	−0.605487	−	0.795855i	$-0.707022\pi$
$97$	−11.9267	−1.21097	−0.605487	+	0.795855i	$0.707022\pi$
$98$	−6.37743	−0.644218
$99$	−6.86947	−0.690408
$100$	13.0892	1.30892
$101$	−16.0343	−1.59547	−0.797736	−	0.603007i	$-0.793969\pi$
$101$	−16.0343	−1.59547	−0.797736	+	0.603007i	$0.793969\pi$
$102$	−15.3081	−1.51572
$103$	−7.04010	−0.693682	−0.346841	−	0.937924i	$-0.612746\pi$
$103$	−7.04010	−0.693682	−0.346841	+	0.937924i	$0.612746\pi$
$104$	−2.86562	−0.280998
$105$	−10.5427	−1.02886
$106$	−1.13808	−0.110540
$107$	−14.5660	−1.40815	−0.704076	−	0.710124i	$-0.748639\pi$
$107$	−14.5660	−1.40815	−0.704076	+	0.710124i	$0.748639\pi$
$108$	12.1562	1.16973
$109$	−1.32900	−0.127295	−0.0636474	−	0.997972i	$-0.520273\pi$
$109$	−1.32900	−0.127295	−0.0636474	+	0.997972i	$0.520273\pi$
$110$	−4.25314	−0.405521
$111$	−2.91262	−0.276454
$112$	−0.789032	−0.0745565
$113$	−5.13343	−0.482913	−0.241456	−	0.970412i	$-0.577625\pi$
$113$	−5.13343	−0.482913	−0.241456	+	0.970412i	$0.577625\pi$
$114$	0	0
$115$	−12.7748	−1.19126
$116$	6.79350	0.630760
$117$	−19.6853	−1.81991
$118$	2.50283	0.230404
$119$	3.84475	0.352447
$120$	13.3616	1.21974
$121$	1.00000	0.0909091
$122$	−7.07049	−0.640132
$123$	−0.247330	−0.0223010
$124$	10.2935	0.924380
$125$	34.4046	3.07724
$126$	−5.42023	−0.482873
$127$	0.480932	0.0426758	0.0213379	−	0.999772i	$-0.493207\pi$
$127$	0.480932	0.0426758	0.0213379	+	0.999772i	$0.493207\pi$
$128$	1.00000	0.0883883
$129$	−13.4438	−1.18366
$130$	−12.1879	−1.06895
$131$	19.3980	1.69482	0.847408	−	0.530943i	$-0.178162\pi$
$131$	19.3980	1.69482	0.847408	+	0.530943i	$0.178162\pi$
$132$	−3.14157	−0.273439
$133$	0	0
$134$	−0.0387536	−0.00334781
$135$	51.7022	4.44982
$136$	−4.87274	−0.417834
$137$	13.3184	1.13786	0.568932	−	0.822384i	$-0.307357\pi$
$137$	13.3184	1.13786	0.568932	+	0.822384i	$0.307357\pi$
$138$	−9.43609	−0.803253
$139$	−1.65530	−0.140400	−0.0702002	−	0.997533i	$-0.522364\pi$
$139$	−1.65530	−0.140400	−0.0702002	+	0.997533i	$0.522364\pi$
$140$	−3.35587	−0.283622
$141$	3.47734	0.292845
$142$	−14.8810	−1.24879
$143$	2.86562	0.239635
$144$	6.86947	0.572456
$145$	28.8937	2.39949
$146$	9.08659	0.752011
$147$	−20.0352	−1.65247
$148$	−0.927122	−0.0762089
$149$	11.9696	0.980588	0.490294	−	0.871557i	$-0.336889\pi$
$149$	11.9696	0.980588	0.490294	+	0.871557i	$0.336889\pi$
$150$	41.1207	3.35749
$151$	9.55950	0.777941	0.388970	−	0.921250i	$-0.372831\pi$
$151$	9.55950	0.777941	0.388970	+	0.921250i	$0.372831\pi$
$152$	0	0
$153$	−33.4732	−2.70614
$154$	0.789032	0.0635820
$155$	43.7796	3.51646
$156$	−9.00256	−0.720782
$157$	15.0603	1.20194	0.600970	−	0.799272i	$-0.294781\pi$
$157$	15.0603	1.20194	0.600970	+	0.799272i	$0.294781\pi$
$158$	−1.40495	−0.111771
$159$	−3.57536	−0.283544
$160$	4.25314	0.336240
$161$	2.36995	0.186778
$162$	17.5812	1.38131
$163$	9.68699	0.758744	0.379372	−	0.925244i	$-0.376140\pi$
$163$	9.68699	0.758744	0.379372	+	0.925244i	$0.376140\pi$
$164$	−0.0787280	−0.00614762
$165$	−13.3616	−1.04020
$166$	11.0728	0.859413
$167$	10.1358	0.784335	0.392168	−	0.919894i	$-0.371725\pi$
$167$	10.1358	0.784335	0.392168	+	0.919894i	$0.371725\pi$
$168$	−2.47880	−0.191244
$169$	−4.78820	−0.368323
$170$	−20.7245	−1.58949
$171$	0	0
$172$	−4.27932	−0.326295
$173$	−19.3926	−1.47439	−0.737197	−	0.675677i	$-0.763851\pi$
$173$	−19.3926	−1.47439	−0.737197	+	0.675677i	$0.763851\pi$
$174$	21.3423	1.61795
$175$	−10.3278	−0.780710
$176$	−1.00000	−0.0753778
$177$	7.86281	0.591005
$178$	−0.532856	−0.0399392
$179$	3.17158	0.237055	0.118527	−	0.992951i	$-0.462183\pi$
$179$	3.17158	0.237055	0.118527	+	0.992951i	$0.462183\pi$
$180$	29.2169	2.17770
$181$	−16.1966	−1.20388	−0.601942	−	0.798540i	$-0.705606\pi$
$181$	−16.1966	−1.20388	−0.601942	+	0.798540i	$0.705606\pi$
$182$	2.26107	0.167602
$183$	−22.2124	−1.64199
$184$	−3.00362	−0.221430
$185$	−3.94318	−0.289908
$186$	32.3376	2.37111
$187$	4.87274	0.356330
$188$	1.10688	0.0807275
$189$	−9.59165	−0.697690
$190$	0	0
$191$	−0.789906	−0.0571556	−0.0285778	−	0.999592i	$-0.509098\pi$
$191$	−0.789906	−0.0571556	−0.0285778	+	0.999592i	$0.509098\pi$
$192$	3.14157	0.226723
$193$	−6.02776	−0.433888	−0.216944	−	0.976184i	$-0.569609\pi$
$193$	−6.02776	−0.433888	−0.216944	+	0.976184i	$0.569609\pi$
$194$	−11.9267	−0.856288
$195$	−38.2892	−2.74195
$196$	−6.37743	−0.455531
$197$	10.7225	0.763950	0.381975	−	0.924173i	$-0.375244\pi$
$197$	10.7225	0.763950	0.381975	+	0.924173i	$0.375244\pi$
$198$	−6.86947	−0.488192
$199$	−8.73696	−0.619347	−0.309673	−	0.950843i	$-0.600220\pi$
$199$	−8.73696	−0.619347	−0.309673	+	0.950843i	$0.600220\pi$
$200$	13.0892	0.925548
$201$	−0.121747	−0.00858740
$202$	−16.0343	−1.12817
$203$	−5.36029	−0.376218
$204$	−15.3081	−1.07178
$205$	−0.334841	−0.0233863
$206$	−7.04010	−0.490507
$207$	−20.6333	−1.43411
$208$	−2.86562	−0.198695
$209$	0	0
$210$	−10.5427	−0.727515
$211$	1.55294	0.106909	0.0534544	−	0.998570i	$-0.482977\pi$
$211$	1.55294	0.106909	0.0534544	+	0.998570i	$0.482977\pi$
$212$	−1.13808	−0.0781636
$213$	−46.7498	−3.20325
$214$	−14.5660	−0.995714
$215$	−18.2005	−1.24127
$216$	12.1562	0.827127
$217$	−8.12187	−0.551348
$218$	−1.32900	−0.0900110
$219$	28.5462	1.92897
$220$	−4.25314	−0.286747
$221$	13.9634	0.939282
$222$	−2.91262	−0.195482
$223$	−10.1442	−0.679306	−0.339653	−	0.940551i	$-0.610310\pi$
$223$	−10.1442	−0.679306	−0.339653	+	0.940551i	$0.610310\pi$
$224$	−0.789032	−0.0527194
$225$	89.9161	5.99441
$226$	−5.13343	−0.341471
$227$	−18.0026	−1.19487	−0.597436	−	0.801917i	$-0.703814\pi$
$227$	−18.0026	−1.19487	−0.597436	+	0.801917i	$0.703814\pi$
$228$	0	0
$229$	21.0845	1.39330	0.696652	−	0.717409i	$-0.254672\pi$
$229$	21.0845	1.39330	0.696652	+	0.717409i	$0.254672\pi$
$230$	−12.7748	−0.842347
$231$	2.47880	0.163093
$232$	6.79350	0.446015
$233$	−0.852182	−0.0558283	−0.0279142	−	0.999610i	$-0.508887\pi$
$233$	−0.852182	−0.0558283	−0.0279142	+	0.999610i	$0.508887\pi$
$234$	−19.6853	−1.28687
$235$	4.70772	0.307098
$236$	2.50283	0.162920
$237$	−4.41374	−0.286703
$238$	3.84475	0.249218
$239$	−27.7837	−1.79718	−0.898588	−	0.438793i	$-0.855406\pi$
$239$	−27.7837	−1.79718	−0.898588	+	0.438793i	$0.855406\pi$
$240$	13.3616	0.862485
$241$	−14.7320	−0.948972	−0.474486	−	0.880263i	$-0.657366\pi$
$241$	−14.7320	−0.948972	−0.474486	+	0.880263i	$0.657366\pi$
$242$	1.00000	0.0642824
$243$	18.7641	1.20371
$244$	−7.07049	−0.452642
$245$	−27.1241	−1.73290
$246$	−0.247330	−0.0157692
$247$	0	0
$248$	10.2935	0.653635
$249$	34.7859	2.20446
$250$	34.4046	2.17594
$251$	−14.1660	−0.894151	−0.447076	−	0.894496i	$-0.647534\pi$
$251$	−14.1660	−0.894151	−0.447076	+	0.894496i	$0.647534\pi$
$252$	−5.42023	−0.341443
$253$	3.00362	0.188836
$254$	0.480932	0.0301764
$255$	−65.1074	−4.07718
$256$	1.00000	0.0625000
$257$	−25.8309	−1.61128	−0.805642	−	0.592402i	$-0.798180\pi$
$257$	−25.8309	−1.61128	−0.805642	+	0.592402i	$0.798180\pi$
$258$	−13.4438	−0.836973
$259$	0.731529	0.0454550
$260$	−12.1879	−0.755862
$261$	46.6677	2.88866
$262$	19.3980	1.19842
$263$	2.52559	0.155735	0.0778673	−	0.996964i	$-0.475189\pi$
$263$	2.52559	0.155735	0.0778673	+	0.996964i	$0.475189\pi$
$264$	−3.14157	−0.193350
$265$	−4.84041	−0.297344
$266$	0	0
$267$	−1.67400	−0.102447
$268$	−0.0387536	−0.00236726
$269$	6.20436	0.378287	0.189143	−	0.981949i	$-0.439429\pi$
$269$	6.20436	0.378287	0.189143	+	0.981949i	$0.439429\pi$
$270$	51.7022	3.14649
$271$	−21.4449	−1.30268	−0.651342	−	0.758785i	$-0.725793\pi$
$271$	−21.4449	−1.30268	−0.651342	+	0.758785i	$0.725793\pi$
$272$	−4.87274	−0.295453
$273$	7.10331	0.429912
$274$	13.3184	0.804592
$275$	−13.0892	−0.789310
$276$	−9.43609	−0.567986
$277$	−16.5444	−0.994055	−0.497027	−	0.867735i	$-0.665575\pi$
$277$	−16.5444	−0.994055	−0.497027	+	0.867735i	$0.665575\pi$
$278$	−1.65530	−0.0992781
$279$	70.7106	4.23334
$280$	−3.35587	−0.200551
$281$	−7.17752	−0.428175	−0.214087	−	0.976815i	$-0.568678\pi$
$281$	−7.17752	−0.428175	−0.214087	+	0.976815i	$0.568678\pi$
$282$	3.47734	0.207073
$283$	−18.0992	−1.07588	−0.537942	−	0.842982i	$-0.680798\pi$
$283$	−18.0992	−1.07588	−0.537942	+	0.842982i	$0.680798\pi$
$284$	−14.8810	−0.883027
$285$	0	0
$286$	2.86562	0.169448
$287$	0.0621189	0.00366676
$288$	6.86947	0.404788
$289$	6.74359	0.396681
$290$	28.8937	1.69670
$291$	−37.4686	−2.19645
$292$	9.08659	0.531752
$293$	28.4730	1.66341	0.831706	−	0.555217i	$-0.187365\pi$
$293$	28.4730	1.66341	0.831706	+	0.555217i	$0.187365\pi$
$294$	−20.0352	−1.16847
$295$	10.6449	0.619769
$296$	−0.927122	−0.0538879
$297$	−12.1562	−0.705376
$298$	11.9696	0.693380
$299$	8.60725	0.497770
$300$	41.1207	2.37411
$301$	3.37652	0.194619
$302$	9.55950	0.550087
$303$	−50.3729	−2.89385
$304$	0	0
$305$	−30.0718	−1.72191
$306$	−33.4732	−1.91353
$307$	−0.660887	−0.0377188	−0.0188594	−	0.999822i	$-0.506003\pi$
$307$	−0.660887	−0.0377188	−0.0188594	+	0.999822i	$0.506003\pi$
$308$	0.789032	0.0449593
$309$	−22.1170	−1.25819
$310$	43.7796	2.48651
$311$	26.1008	1.48004	0.740019	−	0.672586i	$-0.234816\pi$
$311$	26.1008	1.48004	0.740019	+	0.672586i	$0.234816\pi$
$312$	−9.00256	−0.509670
$313$	−6.18994	−0.349876	−0.174938	−	0.984579i	$-0.555972\pi$
$313$	−6.18994	−0.349876	−0.174938	+	0.984579i	$0.555972\pi$
$314$	15.0603	0.849900
$315$	−23.0530	−1.29889
$316$	−1.40495	−0.0790344
$317$	30.5155	1.71392	0.856961	−	0.515381i	$-0.172350\pi$
$317$	30.5155	1.71392	0.856961	+	0.515381i	$0.172350\pi$
$318$	−3.57536	−0.200496
$319$	−6.79350	−0.380363
$320$	4.25314	0.237758
$321$	−45.7603	−2.55409
$322$	2.36995	0.132072
$323$	0	0
$324$	17.5812	0.976736
$325$	−37.5088	−2.08061
$326$	9.68699	0.536513
$327$	−4.17514	−0.230886
$328$	−0.0787280	−0.00434703
$329$	−0.873363	−0.0481501
$330$	−13.3616	−0.735529
$331$	9.69672	0.532980	0.266490	−	0.963838i	$-0.414136\pi$
$331$	9.69672	0.532980	0.266490	+	0.963838i	$0.414136\pi$
$332$	11.0728	0.607696
$333$	−6.36884	−0.349010
$334$	10.1358	0.554609
$335$	−0.164825	−0.00900534
$336$	−2.47880	−0.135230
$337$	−20.9048	−1.13876	−0.569379	−	0.822075i	$-0.692816\pi$
$337$	−20.9048	−1.13876	−0.569379	+	0.822075i	$0.692816\pi$
$338$	−4.78820	−0.260444
$339$	−16.1270	−0.875901
$340$	−20.7245	−1.12394
$341$	−10.2935	−0.557422
$342$	0	0
$343$	10.5552	0.569928
$344$	−4.27932	−0.230725
$345$	−40.1330	−2.16069
$346$	−19.3926	−1.04255
$347$	−13.5329	−0.726485	−0.363243	−	0.931695i	$-0.618330\pi$
$347$	−13.5329	−0.726485	−0.363243	+	0.931695i	$0.618330\pi$
$348$	21.3423	1.14406
$349$	0.617963	0.0330788	0.0165394	−	0.999863i	$-0.494735\pi$
$349$	0.617963	0.0330788	0.0165394	+	0.999863i	$0.494735\pi$
$350$	−10.3278	−0.552045
$351$	−34.8352	−1.85936
$352$	−1.00000	−0.0533002
$353$	−10.7603	−0.572714	−0.286357	−	0.958123i	$-0.592444\pi$
$353$	−10.7603	−0.572714	−0.286357	+	0.958123i	$0.592444\pi$
$354$	7.86281	0.417904
$355$	−63.2912	−3.35915
$356$	−0.532856	−0.0282413
$357$	12.0785	0.639265
$358$	3.17158	0.167623
$359$	21.1931	1.11853	0.559265	−	0.828989i	$-0.311084\pi$
$359$	21.1931	1.11853	0.559265	+	0.828989i	$0.311084\pi$
$360$	29.2169	1.53986
$361$	0	0
$362$	−16.1966	−0.851274
$363$	3.14157	0.164890
$364$	2.26107	0.118512
$365$	38.6466	2.02285
$366$	−22.2124	−1.16106
$367$	10.1420	0.529408	0.264704	−	0.964330i	$-0.414726\pi$
$367$	10.1420	0.529408	0.264704	+	0.964330i	$0.414726\pi$
$368$	−3.00362	−0.156575
$369$	−0.540820	−0.0281540
$370$	−3.94318	−0.204996
$371$	0.897981	0.0466209
$372$	32.3376	1.67663
$373$	7.87532	0.407769	0.203884	−	0.978995i	$-0.434643\pi$
$373$	7.87532	0.407769	0.203884	+	0.978995i	$0.434643\pi$
$374$	4.87274	0.251963
$375$	108.085	5.58147
$376$	1.10688	0.0570830
$377$	−19.4676	−1.00263
$378$	−9.59165	−0.493341
$379$	14.8933	0.765016	0.382508	−	0.923952i	$-0.375060\pi$
$379$	14.8933	0.765016	0.382508	+	0.923952i	$0.375060\pi$
$380$	0	0
$381$	1.51088	0.0774049
$382$	−0.789906	−0.0404151
$383$	−9.65125	−0.493156	−0.246578	−	0.969123i	$-0.579306\pi$
$383$	−9.65125	−0.493156	−0.246578	+	0.969123i	$0.579306\pi$
$384$	3.14157	0.160318
$385$	3.35587	0.171031
$386$	−6.02776	−0.306805
$387$	−29.3967	−1.49432
$388$	−11.9267	−0.605487
$389$	−9.10826	−0.461807	−0.230904	−	0.972977i	$-0.574168\pi$
$389$	−9.10826	−0.461807	−0.230904	+	0.972977i	$0.574168\pi$
$390$	−38.2892	−1.93885
$391$	14.6359	0.740167
$392$	−6.37743	−0.322109
$393$	60.9404	3.07403
$394$	10.7225	0.540194
$395$	−5.97544	−0.300657
$396$	−6.86947	−0.345204
$397$	1.57061	0.0788266	0.0394133	−	0.999223i	$-0.487451\pi$
$397$	1.57061	0.0788266	0.0394133	+	0.999223i	$0.487451\pi$
$398$	−8.73696	−0.437944
$399$	0	0
$400$	13.0892	0.654461
$401$	−7.96088	−0.397547	−0.198774	−	0.980045i	$-0.563696\pi$
$401$	−7.96088	−0.397547	−0.198774	+	0.980045i	$0.563696\pi$
$402$	−0.121747	−0.00607221
$403$	−29.4972	−1.46936
$404$	−16.0343	−0.797736
$405$	74.7755	3.71563
$406$	−5.36029	−0.266026
$407$	0.927122	0.0459557
$408$	−15.3081	−0.757862
$409$	−4.38108	−0.216630	−0.108315	−	0.994117i	$-0.534546\pi$
$409$	−4.38108	−0.216630	−0.108315	+	0.994117i	$0.534546\pi$
$410$	−0.334841	−0.0165366
$411$	41.8406	2.06384
$412$	−7.04010	−0.346841
$413$	−1.97481	−0.0971740
$414$	−20.6333	−1.01407
$415$	47.0940	2.31175
$416$	−2.86562	−0.140499
$417$	−5.20023	−0.254656
$418$	0	0
$419$	−13.1059	−0.640266	−0.320133	−	0.947373i	$-0.603728\pi$
$419$	−13.1059	−0.640266	−0.320133	+	0.947373i	$0.603728\pi$
$420$	−10.5427	−0.514431
$421$	−35.0448	−1.70798	−0.853990	−	0.520289i	$-0.825824\pi$
$421$	−35.0448	−1.70798	−0.853990	+	0.520289i	$0.825824\pi$
$422$	1.55294	0.0755960
$423$	7.60368	0.369704
$424$	−1.13808	−0.0552700
$425$	−63.7804	−3.09380
$426$	−46.7498	−2.26504
$427$	5.57884	0.269979
$428$	−14.5660	−0.704076
$429$	9.00256	0.434648
$430$	−18.2005	−0.877708
$431$	−15.0634	−0.725581	−0.362790	−	0.931871i	$-0.618176\pi$
$431$	−15.0634	−0.725581	−0.362790	+	0.931871i	$0.618176\pi$
$432$	12.1562	0.584867
$433$	20.4220	0.981420	0.490710	−	0.871323i	$-0.336737\pi$
$433$	20.4220	0.981420	0.490710	+	0.871323i	$0.336737\pi$
$434$	−8.12187	−0.389862
$435$	90.7717	4.35217
$436$	−1.32900	−0.0636474
$437$	0	0
$438$	28.5462	1.36399
$439$	−32.9051	−1.57048	−0.785238	−	0.619194i	$-0.787459\pi$
$439$	−32.9051	−1.57048	−0.785238	+	0.619194i	$0.787459\pi$
$440$	−4.25314	−0.202761
$441$	−43.8096	−2.08617
$442$	13.9634	0.664173
$443$	34.9998	1.66289	0.831446	−	0.555606i	$-0.187514\pi$
$443$	34.9998	1.66289	0.831446	+	0.555606i	$0.187514\pi$
$444$	−2.91262	−0.138227
$445$	−2.26631	−0.107433
$446$	−10.1442	−0.480342
$447$	37.6034	1.77858
$448$	−0.789032	−0.0372783
$449$	12.2917	0.580080	0.290040	−	0.957015i	$-0.406331\pi$
$449$	12.2917	0.580080	0.290040	+	0.957015i	$0.406331\pi$
$450$	89.9161	4.23869
$451$	0.0787280	0.00370716
$452$	−5.13343	−0.241456
$453$	30.0318	1.41102
$454$	−18.0026	−0.844902
$455$	9.61665	0.450835
$456$	0	0
$457$	40.8163	1.90931	0.954653	−	0.297721i	$-0.0962264\pi$
$457$	40.8163	1.90931	0.954653	+	0.297721i	$0.0962264\pi$
$458$	21.0845	0.985215
$459$	−59.2341	−2.76481
$460$	−12.7748	−0.595629
$461$	11.6324	0.541776	0.270888	−	0.962611i	$-0.412683\pi$
$461$	11.6324	0.541776	0.270888	+	0.962611i	$0.412683\pi$
$462$	2.47880	0.115324
$463$	19.2052	0.892540	0.446270	−	0.894898i	$-0.352752\pi$
$463$	19.2052	0.892540	0.446270	+	0.894898i	$0.352752\pi$
$464$	6.79350	0.315380
$465$	137.537	6.37811
$466$	−0.852182	−0.0394766
$467$	9.52417	0.440726	0.220363	−	0.975418i	$-0.429276\pi$
$467$	9.52417	0.440726	0.220363	+	0.975418i	$0.429276\pi$
$468$	−19.6853	−0.909954
$469$	0.0305779	0.00141195
$470$	4.70772	0.217151
$471$	47.3129	2.18006
$472$	2.50283	0.115202
$473$	4.27932	0.196763
$474$	−4.41374	−0.202730
$475$	0	0
$476$	3.84475	0.176224
$477$	−7.81801	−0.357962
$478$	−27.7837	−1.27080
$479$	25.8981	1.18331	0.591656	−	0.806190i	$-0.298474\pi$
$479$	25.8981	1.18331	0.591656	+	0.806190i	$0.298474\pi$
$480$	13.3616	0.609869
$481$	2.65678	0.121139
$482$	−14.7320	−0.671025
$483$	7.44538	0.338776
$484$	1.00000	0.0454545
$485$	−50.7260	−2.30335
$486$	18.7641	0.851155
$487$	−1.13302	−0.0513418	−0.0256709	−	0.999670i	$-0.508172\pi$
$487$	−1.13302	−0.0513418	−0.0256709	+	0.999670i	$0.508172\pi$
$488$	−7.07049	−0.320066
$489$	30.4324	1.37620
$490$	−27.1241	−1.22534
$491$	−2.02215	−0.0912582	−0.0456291	−	0.998958i	$-0.514529\pi$
$491$	−2.02215	−0.0912582	−0.0456291	+	0.998958i	$0.514529\pi$
$492$	−0.247330	−0.0111505
$493$	−33.1029	−1.49088
$494$	0	0
$495$	−29.2169	−1.31320
$496$	10.2935	0.462190
$497$	11.7416	0.526683
$498$	34.7859	1.55879
$499$	39.8484	1.78386	0.891930	−	0.452173i	$-0.149351\pi$
$499$	39.8484	1.78386	0.891930	+	0.452173i	$0.149351\pi$
$500$	34.4046	1.53862
$501$	31.8425	1.42262
$502$	−14.1660	−0.632260
$503$	43.6692	1.94711	0.973556	−	0.228447i	$-0.0733648\pi$
$503$	43.6692	1.94711	0.973556	+	0.228447i	$0.0733648\pi$
$504$	−5.42023	−0.241436
$505$	−68.1961	−3.03469
$506$	3.00362	0.133527
$507$	−15.0425	−0.668060
$508$	0.480932	0.0213379
$509$	−24.4491	−1.08369	−0.541844	−	0.840479i	$-0.682274\pi$
$509$	−24.4491	−1.08369	−0.541844	+	0.840479i	$0.682274\pi$
$510$	−65.1074	−2.88300
$511$	−7.16961	−0.317165
$512$	1.00000	0.0441942
$513$	0	0
$514$	−25.8309	−1.13935
$515$	−29.9425	−1.31943
$516$	−13.4438	−0.591829
$517$	−1.10688	−0.0486805
$518$	0.731529	0.0321415
$519$	−60.9234	−2.67424
$520$	−12.1879	−0.534475
$521$	−22.6422	−0.991972	−0.495986	−	0.868330i	$-0.665193\pi$
$521$	−22.6422	−0.991972	−0.495986	+	0.868330i	$0.665193\pi$
$522$	46.6677	2.04259
$523$	24.7889	1.08394	0.541972	−	0.840397i	$-0.317678\pi$
$523$	24.7889	1.08394	0.541972	+	0.840397i	$0.317678\pi$
$524$	19.3980	0.847408
$525$	−32.4456	−1.41604
$526$	2.52559	0.110121
$527$	−50.1573	−2.18489
$528$	−3.14157	−0.136719
$529$	−13.9783	−0.607751
$530$	−4.84041	−0.210254
$531$	17.1931	0.746117
$532$	0	0
$533$	0.225605	0.00977203
$534$	−1.67400	−0.0724412
$535$	−61.9515	−2.67840
$536$	−0.0387536	−0.00167390
$537$	9.96374	0.429967
$538$	6.20436	0.267489
$539$	6.37743	0.274695
$540$	51.7022	2.22491
$541$	−1.68305	−0.0723598	−0.0361799	−	0.999345i	$-0.511519\pi$
$541$	−1.68305	−0.0723598	−0.0361799	+	0.999345i	$0.511519\pi$
$542$	−21.4449	−0.921136
$543$	−50.8828	−2.18359
$544$	−4.87274	−0.208917
$545$	−5.65241	−0.242123
$546$	7.10331	0.303994
$547$	1.39005	0.0594341	0.0297170	−	0.999558i	$-0.490539\pi$
$547$	1.39005	0.0594341	0.0297170	+	0.999558i	$0.490539\pi$
$548$	13.3184	0.568932
$549$	−48.5705	−2.07294
$550$	−13.0892	−0.558126
$551$	0	0
$552$	−9.43609	−0.401627
$553$	1.10855	0.0471402
$554$	−16.5444	−0.702903
$555$	−12.3878	−0.525832
$556$	−1.65530	−0.0702002
$557$	10.1719	0.430995	0.215498	−	0.976504i	$-0.430863\pi$
$557$	10.1719	0.430995	0.215498	+	0.976504i	$0.430863\pi$
$558$	70.7106	2.99342
$559$	12.2629	0.518666
$560$	−3.35587	−0.141811
$561$	15.3081	0.646307
$562$	−7.17752	−0.302765
$563$	11.4443	0.482319	0.241160	−	0.970485i	$-0.422472\pi$
$563$	11.4443	0.482319	0.241160	+	0.970485i	$0.422472\pi$
$564$	3.47734	0.146423
$565$	−21.8332	−0.918530
$566$	−18.0992	−0.760764
$567$	−13.8722	−0.582576
$568$	−14.8810	−0.624394
$569$	−25.3528	−1.06285	−0.531423	−	0.847107i	$-0.678343\pi$
$569$	−25.3528	−1.06285	−0.531423	+	0.847107i	$0.678343\pi$
$570$	0	0
$571$	−31.6040	−1.32259	−0.661293	−	0.750128i	$-0.729992\pi$
$571$	−31.6040	−1.32259	−0.661293	+	0.750128i	$0.729992\pi$
$572$	2.86562	0.119818
$573$	−2.48155	−0.103668
$574$	0.0621189	0.00259279
$575$	−39.3151	−1.63955
$576$	6.86947	0.286228
$577$	20.2670	0.843726	0.421863	−	0.906660i	$-0.361376\pi$
$577$	20.2670	0.843726	0.421863	+	0.906660i	$0.361376\pi$
$578$	6.74359	0.280496
$579$	−18.9366	−0.786980
$580$	28.8937	1.19975
$581$	−8.73676	−0.362462
$582$	−37.4686	−1.55312
$583$	1.13808	0.0471344
$584$	9.08659	0.376006
$585$	−83.7245	−3.46158
$586$	28.4730	1.17621
$587$	13.3771	0.552132	0.276066	−	0.961139i	$-0.410969\pi$
$587$	13.3771	0.552132	0.276066	+	0.961139i	$0.410969\pi$
$588$	−20.0352	−0.826236
$589$	0	0
$590$	10.6449	0.438243
$591$	33.6856	1.38564
$592$	−0.927122	−0.0381045
$593$	−21.7005	−0.891134	−0.445567	−	0.895249i	$-0.646998\pi$
$593$	−21.7005	−0.891134	−0.445567	+	0.895249i	$0.646998\pi$
$594$	−12.1562	−0.498776
$595$	16.3523	0.670377
$596$	11.9696	0.490294
$597$	−27.4478	−1.12336
$598$	8.60725	0.351976
$599$	−43.5240	−1.77834	−0.889171	−	0.457574i	$-0.848718\pi$
$599$	−43.5240	−1.77834	−0.889171	+	0.457574i	$0.848718\pi$
$600$	41.1207	1.67875
$601$	37.3107	1.52193	0.760967	−	0.648790i	$-0.224725\pi$
$601$	37.3107	1.52193	0.760967	+	0.648790i	$0.224725\pi$
$602$	3.37652	0.137617
$603$	−0.266217	−0.0108412
$604$	9.55950	0.388970
$605$	4.25314	0.172915
$606$	−50.3729	−2.04626
$607$	−1.12996	−0.0458635	−0.0229318	−	0.999737i	$-0.507300\pi$
$607$	−1.12996	−0.0458635	−0.0229318	+	0.999737i	$0.507300\pi$
$608$	0	0
$609$	−16.8397	−0.682380
$610$	−30.0718	−1.21757
$611$	−3.17190	−0.128321
$612$	−33.4732	−1.35307
$613$	−10.0991	−0.407899	−0.203950	−	0.978981i	$-0.565378\pi$
$613$	−10.0991	−0.407899	−0.203950	+	0.978981i	$0.565378\pi$
$614$	−0.660887	−0.0266712
$615$	−1.05193	−0.0424178
$616$	0.789032	0.0317910
$617$	−36.7553	−1.47971	−0.739857	−	0.672764i	$-0.765107\pi$
$617$	−36.7553	−1.47971	−0.739857	+	0.672764i	$0.765107\pi$
$618$	−22.1170	−0.889675
$619$	23.7732	0.955526	0.477763	−	0.878489i	$-0.341448\pi$
$619$	23.7732	0.955526	0.477763	+	0.878489i	$0.341448\pi$
$620$	43.7796	1.75823
$621$	−36.5127	−1.46520
$622$	26.1008	1.04655
$623$	0.420440	0.0168446
$624$	−9.00256	−0.360391
$625$	80.8817	3.23527
$626$	−6.18994	−0.247400
$627$	0	0
$628$	15.0603	0.600970
$629$	4.51762	0.180129
$630$	−23.0530	−0.918455
$631$	25.9645	1.03363	0.516815	−	0.856097i	$-0.327118\pi$
$631$	25.9645	1.03363	0.516815	+	0.856097i	$0.327118\pi$
$632$	−1.40495	−0.0558857
$633$	4.87867	0.193910
$634$	30.5155	1.21193
$635$	2.04547	0.0811721
$636$	−3.57536	−0.141772
$637$	18.2753	0.724094
$638$	−6.79350	−0.268957
$639$	−102.225	−4.04395
$640$	4.25314	0.168120
$641$	−0.942753	−0.0372365	−0.0186183	−	0.999827i	$-0.505927\pi$
$641$	−0.942753	−0.0372365	−0.0186183	+	0.999827i	$0.505927\pi$
$642$	−45.7603	−1.80601
$643$	24.9667	0.984591	0.492296	−	0.870428i	$-0.336158\pi$
$643$	24.9667	0.984591	0.492296	+	0.870428i	$0.336158\pi$
$644$	2.36995	0.0933892
$645$	−57.1783	−2.25139
$646$	0	0
$647$	6.15151	0.241841	0.120920	−	0.992662i	$-0.461415\pi$
$647$	6.15151	0.241841	0.120920	+	0.992662i	$0.461415\pi$
$648$	17.5812	0.690657
$649$	−2.50283	−0.0982445
$650$	−37.5088	−1.47122
$651$	−25.5154	−1.00003
$652$	9.68699	0.379372
$653$	−40.1091	−1.56959	−0.784795	−	0.619756i	$-0.787232\pi$
$653$	−40.1091	−1.56959	−0.784795	+	0.619756i	$0.787232\pi$
$654$	−4.17514	−0.163261
$655$	82.5027	3.22365
$656$	−0.0787280	−0.00307381
$657$	62.4201	2.43524
$658$	−0.873363	−0.0340473
$659$	−2.57003	−0.100114	−0.0500570	−	0.998746i	$-0.515940\pi$
$659$	−2.57003	−0.100114	−0.0500570	+	0.998746i	$0.515940\pi$
$660$	−13.3616	−0.520098
$661$	−14.3037	−0.556348	−0.278174	−	0.960531i	$-0.589729\pi$
$661$	−14.3037	−0.556348	−0.278174	+	0.960531i	$0.589729\pi$
$662$	9.69672	0.376874
$663$	43.8671	1.70366
$664$	11.0728	0.429706
$665$	0	0
$666$	−6.36884	−0.246787
$667$	−20.4051	−0.790088
$668$	10.1358	0.392168
$669$	−31.8687	−1.23212
$670$	−0.164825	−0.00636774
$671$	7.07049	0.272953
$672$	−2.47880	−0.0956218
$673$	30.6801	1.18263	0.591315	−	0.806441i	$-0.298609\pi$
$673$	30.6801	1.18263	0.591315	+	0.806441i	$0.298609\pi$
$674$	−20.9048	−0.805224
$675$	159.116	6.12436
$676$	−4.78820	−0.184162
$677$	23.3072	0.895768	0.447884	−	0.894092i	$-0.352178\pi$
$677$	23.3072	0.895768	0.447884	+	0.894092i	$0.352178\pi$
$678$	−16.1270	−0.619355
$679$	9.41056	0.361144
$680$	−20.7245	−0.794747
$681$	−56.5564	−2.16724
$682$	−10.2935	−0.394157
$683$	−16.6896	−0.638609	−0.319304	−	0.947652i	$-0.603449\pi$
$683$	−16.6896	−0.638609	−0.319304	+	0.947652i	$0.603449\pi$
$684$	0	0
$685$	56.6449	2.16429
$686$	10.5552	0.403000
$687$	66.2385	2.52716
$688$	−4.27932	−0.163147
$689$	3.26131	0.124246
$690$	−40.1330	−1.52784
$691$	−2.39287	−0.0910289	−0.0455145	−	0.998964i	$-0.514493\pi$
$691$	−2.39287	−0.0910289	−0.0455145	+	0.998964i	$0.514493\pi$
$692$	−19.3926	−0.737197
$693$	5.42023	0.205898
$694$	−13.5329	−0.513703
$695$	−7.04021	−0.267050
$696$	21.3423	0.808976
$697$	0.383621	0.0145307
$698$	0.617963	0.0233903
$699$	−2.67719	−0.101261
$700$	−10.3278	−0.390355
$701$	−29.2046	−1.10304	−0.551522	−	0.834161i	$-0.685953\pi$
$701$	−29.2046	−1.10304	−0.551522	+	0.834161i	$0.685953\pi$
$702$	−34.8352	−1.31477
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	14.7896	0.557010
$706$	−10.7603	−0.404970
$707$	12.6516	0.475811
$708$	7.86281	0.295502
$709$	−10.3287	−0.387902	−0.193951	−	0.981011i	$-0.562130\pi$
$709$	−10.3287	−0.387902	−0.193951	+	0.981011i	$0.562130\pi$
$710$	−63.2912	−2.37527
$711$	−9.65124	−0.361950
$712$	−0.532856	−0.0199696
$713$	−30.9176	−1.15787
$714$	12.0785	0.452028
$715$	12.1879	0.455802
$716$	3.17158	0.118527
$717$	−87.2844	−3.25970
$718$	21.1931	0.790920
$719$	10.5567	0.393698	0.196849	−	0.980434i	$-0.436929\pi$
$719$	10.5567	0.393698	0.196849	+	0.980434i	$0.436929\pi$
$720$	29.2169	1.08885
$721$	5.55486	0.206874
$722$	0	0
$723$	−46.2817	−1.72123
$724$	−16.1966	−0.601942
$725$	88.9216	3.30247
$726$	3.14157	0.116595
$727$	10.0662	0.373337	0.186668	−	0.982423i	$-0.440231\pi$
$727$	10.0662	0.373337	0.186668	+	0.982423i	$0.440231\pi$
$728$	2.26107	0.0838008
$729$	6.20490	0.229811
$730$	38.6466	1.43037
$731$	20.8520	0.771239
$732$	−22.2124	−0.820995
$733$	−7.05178	−0.260464	−0.130232	−	0.991484i	$-0.541572\pi$
$733$	−7.05178	−0.260464	−0.130232	+	0.991484i	$0.541572\pi$
$734$	10.1420	0.374348
$735$	−85.2124	−3.14311
$736$	−3.00362	−0.110715
$737$	0.0387536	0.00142751
$738$	−0.540820	−0.0199079
$739$	5.21930	0.191995	0.0959975	−	0.995382i	$-0.469396\pi$
$739$	5.21930	0.191995	0.0959975	+	0.995382i	$0.469396\pi$
$740$	−3.94318	−0.144954
$741$	0	0
$742$	0.897981	0.0329659
$743$	7.24624	0.265839	0.132919	−	0.991127i	$-0.457565\pi$
$743$	7.24624	0.265839	0.132919	+	0.991127i	$0.457565\pi$
$744$	32.3376	1.18556
$745$	50.9084	1.86514
$746$	7.87532	0.288336
$747$	76.0640	2.78304
$748$	4.87274	0.178165
$749$	11.4931	0.419948
$750$	108.085	3.94669
$751$	−9.15541	−0.334086	−0.167043	−	0.985950i	$-0.553422\pi$
$751$	−9.15541	−0.334086	−0.167043	+	0.985950i	$0.553422\pi$
$752$	1.10688	0.0403638
$753$	−44.5036	−1.62180
$754$	−19.4676	−0.708968
$755$	40.6579	1.47969
$756$	−9.59165	−0.348845
$757$	8.64501	0.314208	0.157104	−	0.987582i	$-0.449784\pi$
$757$	8.64501	0.314208	0.157104	+	0.987582i	$0.449784\pi$
$758$	14.8933	0.540948
$759$	9.43609	0.342508
$760$	0	0
$761$	−45.2300	−1.63959	−0.819793	−	0.572659i	$-0.805912\pi$
$761$	−45.2300	−1.63959	−0.819793	+	0.572659i	$0.805912\pi$
$762$	1.51088	0.0547335
$763$	1.04862	0.0379626
$764$	−0.789906	−0.0285778
$765$	−142.366	−5.14726
$766$	−9.65125	−0.348714
$767$	−7.17216	−0.258972
$768$	3.14157	0.113362
$769$	0.255007	0.00919579	0.00459789	−	0.999989i	$-0.498536\pi$
$769$	0.255007	0.00919579	0.00459789	+	0.999989i	$0.498536\pi$
$770$	3.35587	0.120937
$771$	−81.1495	−2.92253
$772$	−6.02776	−0.216944
$773$	−30.5492	−1.09878	−0.549389	−	0.835567i	$-0.685139\pi$
$773$	−30.5492	−1.09878	−0.549389	+	0.835567i	$0.685139\pi$
$774$	−29.3967	−1.05664
$775$	134.733	4.83977
$776$	−11.9267	−0.428144
$777$	2.29815	0.0824457
$778$	−9.10826	−0.326547
$779$	0	0
$780$	−38.2892	−1.37097
$781$	14.8810	0.532485
$782$	14.6359	0.523377
$783$	82.5833	2.95129
$784$	−6.37743	−0.227765
$785$	64.0534	2.28617
$786$	60.9404	2.17367
$787$	15.3750	0.548058	0.274029	−	0.961721i	$-0.411644\pi$
$787$	15.3750	0.548058	0.274029	+	0.961721i	$0.411644\pi$
$788$	10.7225	0.381975
$789$	7.93433	0.282470
$790$	−5.97544	−0.212596
$791$	4.05044	0.144017
$792$	−6.86947	−0.244096
$793$	20.2614	0.719502
$794$	1.57061	0.0557389
$795$	−15.2065	−0.539319
$796$	−8.73696	−0.309673
$797$	−12.5306	−0.443858	−0.221929	−	0.975063i	$-0.571235\pi$
$797$	−12.5306	−0.443858	−0.221929	+	0.975063i	$0.571235\pi$
$798$	0	0
$799$	−5.39354	−0.190810
$800$	13.0892	0.462774
$801$	−3.66044	−0.129335
$802$	−7.96088	−0.281108
$803$	−9.08659	−0.320659
$804$	−0.121747	−0.00429370
$805$	10.0797	0.355264
$806$	−29.4972	−1.03899
$807$	19.4915	0.686132
$808$	−16.0343	−0.564084
$809$	20.8700	0.733751	0.366876	−	0.930270i	$-0.380427\pi$
$809$	20.8700	0.733751	0.366876	+	0.930270i	$0.380427\pi$
$810$	74.7755	2.62735
$811$	43.5142	1.52799	0.763995	−	0.645222i	$-0.223235\pi$
$811$	43.5142	1.52799	0.763995	+	0.645222i	$0.223235\pi$
$812$	−5.36029	−0.188109
$813$	−67.3706	−2.36279
$814$	0.927122	0.0324956
$815$	41.2002	1.44318
$816$	−15.3081	−0.535889
$817$	0	0
$818$	−4.38108	−0.153181
$819$	15.5324	0.542744
$820$	−0.334841	−0.0116932
$821$	−31.3062	−1.09259	−0.546297	−	0.837592i	$-0.683963\pi$
$821$	−31.3062	−1.09259	−0.546297	+	0.837592i	$0.683963\pi$
$822$	41.8406	1.45936
$823$	5.93510	0.206884	0.103442	−	0.994635i	$-0.467014\pi$
$823$	5.93510	0.206884	0.103442	+	0.994635i	$0.467014\pi$
$824$	−7.04010	−0.245253
$825$	−41.1207	−1.43164
$826$	−1.97481	−0.0687124
$827$	2.38933	0.0830853	0.0415426	−	0.999137i	$-0.486773\pi$
$827$	2.38933	0.0830853	0.0415426	+	0.999137i	$0.486773\pi$
$828$	−20.6333	−0.717056
$829$	−7.02302	−0.243920	−0.121960	−	0.992535i	$-0.538918\pi$
$829$	−7.02302	−0.243920	−0.121960	+	0.992535i	$0.538918\pi$
$830$	47.0940	1.63466
$831$	−51.9753	−1.80300
$832$	−2.86562	−0.0993476
$833$	31.0755	1.07670
$834$	−5.20023	−0.180069
$835$	43.1092	1.49185
$836$	0	0
$837$	125.130	4.32511
$838$	−13.1059	−0.452737
$839$	12.5878	0.434579	0.217289	−	0.976107i	$-0.430278\pi$
$839$	12.5878	0.434579	0.217289	+	0.976107i	$0.430278\pi$
$840$	−10.5427	−0.363757
$841$	17.1516	0.591434
$842$	−35.0448	−1.20772
$843$	−22.5487	−0.776618
$844$	1.55294	0.0534544
$845$	−20.3649	−0.700574
$846$	7.60368	0.261420
$847$	−0.789032	−0.0271115
$848$	−1.13808	−0.0390818
$849$	−56.8598	−1.95142
$850$	−63.7804	−2.18765
$851$	2.78472	0.0954591
$852$	−46.7498	−1.60162
$853$	−46.4032	−1.58882	−0.794408	−	0.607385i	$-0.792219\pi$
$853$	−46.4032	−1.58882	−0.794408	+	0.607385i	$0.792219\pi$
$854$	5.57884	0.190904
$855$	0	0
$856$	−14.5660	−0.497857
$857$	53.0866	1.81340	0.906702	−	0.421772i	$-0.138592\pi$
$857$	53.0866	1.81340	0.906702	+	0.421772i	$0.138592\pi$
$858$	9.00256	0.307342
$859$	−27.8557	−0.950424	−0.475212	−	0.879871i	$-0.657629\pi$
$859$	−27.8557	−0.950424	−0.475212	+	0.879871i	$0.657629\pi$
$860$	−18.2005	−0.620633
$861$	0.195151	0.00665073
$862$	−15.0634	−0.513063
$863$	4.59189	0.156310	0.0781549	−	0.996941i	$-0.475097\pi$
$863$	4.59189	0.156310	0.0781549	+	0.996941i	$0.475097\pi$
$864$	12.1562	0.413563
$865$	−82.4797	−2.80439
$866$	20.4220	0.693969
$867$	21.1855	0.719496
$868$	−8.12187	−0.275674
$869$	1.40495	0.0476595
$870$	90.7717	3.07745
$871$	0.111053	0.00376290
$872$	−1.32900	−0.0450055
$873$	−81.9303	−2.77292
$874$	0	0
$875$	−27.1464	−0.917714
$876$	28.5462	0.964486
$877$	24.7295	0.835057	0.417529	−	0.908664i	$-0.362896\pi$
$877$	24.7295	0.835057	0.417529	+	0.908664i	$0.362896\pi$
$878$	−32.9051	−1.11049
$879$	89.4500	3.01707
$880$	−4.25314	−0.143373
$881$	−35.6142	−1.19987	−0.599937	−	0.800047i	$-0.704808\pi$
$881$	−35.6142	−1.19987	−0.599937	+	0.800047i	$0.704808\pi$
$882$	−43.8096	−1.47515
$883$	−30.3291	−1.02065	−0.510327	−	0.859980i	$-0.670476\pi$
$883$	−30.3291	−1.02065	−0.510327	+	0.859980i	$0.670476\pi$
$884$	13.9634	0.469641
$885$	33.4416	1.12413
$886$	34.9998	1.17584
$887$	24.6424	0.827412	0.413706	−	0.910410i	$-0.364234\pi$
$887$	24.6424	0.827412	0.413706	+	0.910410i	$0.364234\pi$
$888$	−2.91262	−0.0977411
$889$	−0.379471	−0.0127270
$890$	−2.26631	−0.0759669
$891$	−17.5812	−0.588994
$892$	−10.1442	−0.339653
$893$	0	0
$894$	37.6034	1.25764
$895$	13.4892	0.450893
$896$	−0.789032	−0.0263597
$897$	27.0403	0.902849
$898$	12.2917	0.410178
$899$	69.9286	2.33225
$900$	89.9161	2.99720
$901$	5.54556	0.184750
$902$	0.0787280	0.00262136
$903$	10.6076	0.352998
$904$	−5.13343	−0.170735
$905$	−68.8864	−2.28986
$906$	30.0318	0.997741
$907$	32.5163	1.07969	0.539843	−	0.841766i	$-0.318484\pi$
$907$	32.5163	1.07969	0.539843	+	0.841766i	$0.318484\pi$
$908$	−18.0026	−0.597436
$909$	−110.147	−3.65335
$910$	9.61665	0.318789
$911$	25.5544	0.846654	0.423327	−	0.905977i	$-0.360862\pi$
$911$	25.5544	0.846654	0.423327	+	0.905977i	$0.360862\pi$
$912$	0	0
$913$	−11.0728	−0.366455
$914$	40.8163	1.35008
$915$	−94.4727	−3.12317
$916$	21.0845	0.696652
$917$	−15.3057	−0.505438
$918$	−59.2341	−1.95502
$919$	−16.7964	−0.554061	−0.277031	−	0.960861i	$-0.589350\pi$
$919$	−16.7964	−0.554061	−0.277031	+	0.960861i	$0.589350\pi$
$920$	−12.7748	−0.421174
$921$	−2.07622	−0.0684139
$922$	11.6324	0.383094
$923$	42.6434	1.40363
$924$	2.47880	0.0815465
$925$	−12.1353	−0.399006
$926$	19.2052	0.631121
$927$	−48.3618	−1.58841
$928$	6.79350	0.223007
$929$	48.8769	1.60360	0.801799	−	0.597594i	$-0.203876\pi$
$929$	48.8769	1.60360	0.801799	+	0.597594i	$0.203876\pi$
$930$	137.537	4.51000
$931$	0	0
$932$	−0.852182	−0.0279142
$933$	81.9974	2.68448
$934$	9.52417	0.311640
$935$	20.7245	0.677762
$936$	−19.6853	−0.643435
$937$	−9.21994	−0.301202	−0.150601	−	0.988595i	$-0.548121\pi$
$937$	−9.21994	−0.301202	−0.150601	+	0.988595i	$0.548121\pi$
$938$	0.0305779	0.000998403 0
$939$	−19.4461	−0.634601
$940$	4.70772	0.153549
$941$	−46.9982	−1.53210	−0.766049	−	0.642782i	$-0.777780\pi$
$941$	−46.9982	−1.53210	−0.766049	+	0.642782i	$0.777780\pi$
$942$	47.3129	1.54154
$943$	0.236469	0.00770049
$944$	2.50283	0.0814601
$945$	−40.7947	−1.32705
$946$	4.27932	0.139133
$947$	45.8260	1.48914	0.744572	−	0.667542i	$-0.232654\pi$
$947$	45.8260	1.48914	0.744572	+	0.667542i	$0.232654\pi$
$948$	−4.41374	−0.143352
$949$	−26.0387	−0.845253
$950$	0	0
$951$	95.8667	3.10869
$952$	3.84475	0.124609
$953$	2.80221	0.0907726	0.0453863	−	0.998970i	$-0.485548\pi$
$953$	2.80221	0.0907726	0.0453863	+	0.998970i	$0.485548\pi$
$954$	−7.81801	−0.253117
$955$	−3.35958	−0.108714
$956$	−27.7837	−0.898588
$957$	−21.3423	−0.689897
$958$	25.8981	0.836728
$959$	−10.5086	−0.339341
$960$	13.3616	0.431242
$961$	74.9553	2.41791
$962$	2.65678	0.0856581
$963$	−100.061	−3.22442
$964$	−14.7320	−0.474486
$965$	−25.6369	−0.825282
$966$	7.44538	0.239551
$967$	28.8507	0.927776	0.463888	−	0.885894i	$-0.346454\pi$
$967$	28.8507	0.927776	0.463888	+	0.885894i	$0.346454\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−50.7260	−1.62871
$971$	−18.5452	−0.595145	−0.297572	−	0.954699i	$-0.596177\pi$
$971$	−18.5452	−0.595145	−0.297572	+	0.954699i	$0.596177\pi$
$972$	18.7641	0.601857
$973$	1.30608	0.0418711
$974$	−1.13302	−0.0363042
$975$	−117.837	−3.77379
$976$	−7.07049	−0.226321
$977$	39.6590	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$977$	39.6590	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$978$	30.4324	0.973120
$979$	0.532856	0.0170301
$980$	−27.1241	−0.866448
$981$	−9.12950	−0.291483
$982$	−2.02215	−0.0645293
$983$	−13.0948	−0.417660	−0.208830	−	0.977952i	$-0.566966\pi$
$983$	−13.0948	−0.417660	−0.208830	+	0.977952i	$0.566966\pi$
$984$	−0.247330	−0.00788458
$985$	45.6045	1.45308
$986$	−33.1029	−1.05421
$987$	−2.74373	−0.0873340
$988$	0	0
$989$	12.8534	0.408716
$990$	−29.2169	−0.928573
$991$	7.87788	0.250249	0.125125	−	0.992141i	$-0.460067\pi$
$991$	7.87788	0.250249	0.125125	+	0.992141i	$0.460067\pi$
$992$	10.2935	0.326818
$993$	30.4629	0.966712
$994$	11.7416	0.372421
$995$	−37.1595	−1.17804
$996$	34.7859	1.10223
$997$	46.9150	1.48581	0.742906	−	0.669396i	$-0.233447\pi$
$997$	46.9150	1.48581	0.742906	+	0.669396i	$0.233447\pi$
$998$	39.8484	1.26138
$999$	−11.2703	−0.356577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.ca.1.14		15
19.6	even	9		418.2.j.d.397.5	yes	30
19.16	even	9		418.2.j.d.199.5	✓	30
19.18	odd	2		7942.2.a.by.1.2		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.j.d.199.5	✓	30	19.16	even	9
418.2.j.d.397.5	yes	30	19.6	even	9
7942.2.a.by.1.2		15	19.18	odd	2
7942.2.a.ca.1.14		15	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:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.14157 q^{3} +1.00000 q^{4} +4.25314 q^{5} +3.14157 q^{6} -0.789032 q^{7} +1.00000 q^{8} +6.86947 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +3.14157 q^{3} +1.00000 q^{4} +4.25314 q^{5} +3.14157 q^{6} -0.789032 q^{7} +1.00000 q^{8} +6.86947 q^{9} +4.25314 q^{10} -1.00000 q^{11} +3.14157 q^{12} -2.86562 q^{13} -0.789032 q^{14} +13.3616 q^{15} +1.00000 q^{16} -4.87274 q^{17} +6.86947 q^{18} +4.25314 q^{20} -2.47880 q^{21} -1.00000 q^{22} -3.00362 q^{23} +3.14157 q^{24} +13.0892 q^{25} -2.86562 q^{26} +12.1562 q^{27} -0.789032 q^{28} +6.79350 q^{29} +13.3616 q^{30} +10.2935 q^{31} +1.00000 q^{32} -3.14157 q^{33} -4.87274 q^{34} -3.35587 q^{35} +6.86947 q^{36} -0.927122 q^{37} -9.00256 q^{39} +4.25314 q^{40} -0.0787280 q^{41} -2.47880 q^{42} -4.27932 q^{43} -1.00000 q^{44} +29.2169 q^{45} -3.00362 q^{46} +1.10688 q^{47} +3.14157 q^{48} -6.37743 q^{49} +13.0892 q^{50} -15.3081 q^{51} -2.86562 q^{52} -1.13808 q^{53} +12.1562 q^{54} -4.25314 q^{55} -0.789032 q^{56} +6.79350 q^{58} +2.50283 q^{59} +13.3616 q^{60} -7.07049 q^{61} +10.2935 q^{62} -5.42023 q^{63} +1.00000 q^{64} -12.1879 q^{65} -3.14157 q^{66} -0.0387536 q^{67} -4.87274 q^{68} -9.43609 q^{69} -3.35587 q^{70} -14.8810 q^{71} +6.86947 q^{72} +9.08659 q^{73} -0.927122 q^{74} +41.1207 q^{75} +0.789032 q^{77} -9.00256 q^{78} -1.40495 q^{79} +4.25314 q^{80} +17.5812 q^{81} -0.0787280 q^{82} +11.0728 q^{83} -2.47880 q^{84} -20.7245 q^{85} -4.27932 q^{86} +21.3423 q^{87} -1.00000 q^{88} -0.532856 q^{89} +29.2169 q^{90} +2.26107 q^{91} -3.00362 q^{92} +32.3376 q^{93} +1.10688 q^{94} +3.14157 q^{96} -11.9267 q^{97} -6.37743 q^{98} -6.86947 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 15 q^{8} + 30 q^{9}+O(q^{10}) \) 15 * q + 15 * q^2 + 3 * q^3 + 15 * q^4 + 9 * q^5 + 3 * q^6 + 15 * q^8 + 30 * q^9	\( 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 15 q^{8} + 30 q^{9} + 9 q^{10} - 15 q^{11} + 3 q^{12} + 21 q^{15} + 15 q^{16} + 21 q^{17} + 30 q^{18} + 9 q^{20} - 9 q^{21} - 15 q^{22} + 21 q^{23} + 3 q^{24} + 24 q^{25} + 3 q^{27} - 9 q^{29} + 21 q^{30} + 18 q^{31} + 15 q^{32} - 3 q^{33} + 21 q^{34} + 18 q^{35} + 30 q^{36} - 9 q^{37} + 9 q^{40} + 15 q^{41} - 9 q^{42} + 3 q^{43} - 15 q^{44} + 54 q^{45} + 21 q^{46} + 39 q^{47} + 3 q^{48} + 33 q^{49} + 24 q^{50} + 30 q^{51} + 18 q^{53} + 3 q^{54} - 9 q^{55} - 9 q^{58} + 6 q^{59} + 21 q^{60} - 30 q^{61} + 18 q^{62} + 24 q^{63} + 15 q^{64} + 6 q^{65} - 3 q^{66} + 9 q^{67} + 21 q^{68} - 42 q^{69} + 18 q^{70} + 9 q^{71} + 30 q^{72} + 12 q^{73} - 9 q^{74} + 21 q^{75} + 12 q^{79} + 9 q^{80} + 63 q^{81} + 15 q^{82} + 30 q^{83} - 9 q^{84} - 3 q^{85} + 3 q^{86} + 9 q^{87} - 15 q^{88} - 6 q^{89} + 54 q^{90} + 96 q^{91} + 21 q^{92} + 102 q^{93} + 39 q^{94} + 3 q^{96} + 33 q^{98} - 30 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 + 3 * q^3 + 15 * q^4 + 9 * q^5 + 3 * q^6 + 15 * q^8 + 30 * q^9 + 9 * q^10 - 15 * q^11 + 3 * q^12 + 21 * q^15 + 15 * q^16 + 21 * q^17 + 30 * q^18 + 9 * q^20 - 9 * q^21 - 15 * q^22 + 21 * q^23 + 3 * q^24 + 24 * q^25 + 3 * q^27 - 9 * q^29 + 21 * q^30 + 18 * q^31 + 15 * q^32 - 3 * q^33 + 21 * q^34 + 18 * q^35 + 30 * q^36 - 9 * q^37 + 9 * q^40 + 15 * q^41 - 9 * q^42 + 3 * q^43 - 15 * q^44 + 54 * q^45 + 21 * q^46 + 39 * q^47 + 3 * q^48 + 33 * q^49 + 24 * q^50 + 30 * q^51 + 18 * q^53 + 3 * q^54 - 9 * q^55 - 9 * q^58 + 6 * q^59 + 21 * q^60 - 30 * q^61 + 18 * q^62 + 24 * q^63 + 15 * q^64 + 6 * q^65 - 3 * q^66 + 9 * q^67 + 21 * q^68 - 42 * q^69 + 18 * q^70 + 9 * q^71 + 30 * q^72 + 12 * q^73 - 9 * q^74 + 21 * q^75 + 12 * q^79 + 9 * q^80 + 63 * q^81 + 15 * q^82 + 30 * q^83 - 9 * q^84 - 3 * q^85 + 3 * q^86 + 9 * q^87 - 15 * q^88 - 6 * q^89 + 54 * q^90 + 96 * q^91 + 21 * q^92 + 102 * q^93 + 39 * q^94 + 3 * q^96 + 33 * q^98 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more