LMFDB - Embedded newform 8014.2.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8014,2,Mod(1,8014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8014, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8014 = 2 \cdot 4007 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8014.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:	$63.9921121799$
Analytic rank:	$0$
Dimension:	$88$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.99129 q^{3} +1.00000 q^{4} +3.54646 q^{5} -2.99129 q^{6} +3.26038 q^{7} +1.00000 q^{8} +5.94783 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.99129 q^{3} +1.00000 q^{4} +3.54646 q^{5} -2.99129 q^{6} +3.26038 q^{7} +1.00000 q^{8} +5.94783 q^{9} +3.54646 q^{10} +1.94696 q^{11} -2.99129 q^{12} -5.13670 q^{13} +3.26038 q^{14} -10.6085 q^{15} +1.00000 q^{16} +2.80954 q^{17} +5.94783 q^{18} -3.35046 q^{19} +3.54646 q^{20} -9.75274 q^{21} +1.94696 q^{22} +0.239467 q^{23} -2.99129 q^{24} +7.57737 q^{25} -5.13670 q^{26} -8.81782 q^{27} +3.26038 q^{28} +3.51053 q^{29} -10.6085 q^{30} -5.70064 q^{31} +1.00000 q^{32} -5.82394 q^{33} +2.80954 q^{34} +11.5628 q^{35} +5.94783 q^{36} -5.03384 q^{37} -3.35046 q^{38} +15.3654 q^{39} +3.54646 q^{40} -8.78183 q^{41} -9.75274 q^{42} +8.70345 q^{43} +1.94696 q^{44} +21.0937 q^{45} +0.239467 q^{46} +8.59728 q^{47} -2.99129 q^{48} +3.63007 q^{49} +7.57737 q^{50} -8.40417 q^{51} -5.13670 q^{52} +4.16392 q^{53} -8.81782 q^{54} +6.90483 q^{55} +3.26038 q^{56} +10.0222 q^{57} +3.51053 q^{58} -5.06992 q^{59} -10.6085 q^{60} +11.3864 q^{61} -5.70064 q^{62} +19.3922 q^{63} +1.00000 q^{64} -18.2171 q^{65} -5.82394 q^{66} +14.7225 q^{67} +2.80954 q^{68} -0.716317 q^{69} +11.5628 q^{70} +9.52674 q^{71} +5.94783 q^{72} -12.2384 q^{73} -5.03384 q^{74} -22.6661 q^{75} -3.35046 q^{76} +6.34784 q^{77} +15.3654 q^{78} +9.14994 q^{79} +3.54646 q^{80} +8.53318 q^{81} -8.78183 q^{82} +12.5067 q^{83} -9.75274 q^{84} +9.96393 q^{85} +8.70345 q^{86} -10.5010 q^{87} +1.94696 q^{88} +5.21444 q^{89} +21.0937 q^{90} -16.7476 q^{91} +0.239467 q^{92} +17.0523 q^{93} +8.59728 q^{94} -11.8823 q^{95} -2.99129 q^{96} +5.62501 q^{97} +3.63007 q^{98} +11.5802 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 88 q^{2} + 22 q^{3} + 88 q^{4} + 25 q^{5} + 22 q^{6} + 33 q^{7} + 88 q^{8} + 108 q^{9}+O(q^{10}) $ 88 * q + 88 * q^2 + 22 * q^3 + 88 * q^4 + 25 * q^5 + 22 * q^6 + 33 * q^7 + 88 * q^8 + 108 * q^9	\( 88 q + 88 q^{2} + 22 q^{3} + 88 q^{4} + 25 q^{5} + 22 q^{6} + 33 q^{7} + 88 q^{8} + 108 q^{9} + 25 q^{10} + 70 q^{11} + 22 q^{12} + 31 q^{13} + 33 q^{14} + 47 q^{15} + 88 q^{16} + 19 q^{17} + 108 q^{18} + 33 q^{19} + 25 q^{20} + 48 q^{21} + 70 q^{22} + 77 q^{23} + 22 q^{24} + 109 q^{25} + 31 q^{26} + 88 q^{27} + 33 q^{28} + 83 q^{29} + 47 q^{30} + 51 q^{31} + 88 q^{32} + 30 q^{33} + 19 q^{34} + 40 q^{35} + 108 q^{36} + 45 q^{37} + 33 q^{38} + 82 q^{39} + 25 q^{40} + 35 q^{41} + 48 q^{42} + 78 q^{43} + 70 q^{44} + 37 q^{45} + 77 q^{46} + 59 q^{47} + 22 q^{48} + 103 q^{49} + 109 q^{50} + 21 q^{51} + 31 q^{52} + 58 q^{53} + 88 q^{54} + 35 q^{55} + 33 q^{56} - 16 q^{57} + 83 q^{58} + 54 q^{59} + 47 q^{60} + 18 q^{61} + 51 q^{62} + 47 q^{63} + 88 q^{64} + 34 q^{65} + 30 q^{66} + 88 q^{67} + 19 q^{68} + 62 q^{69} + 40 q^{70} + 139 q^{71} + 108 q^{72} - 6 q^{73} + 45 q^{74} + 45 q^{75} + 33 q^{76} + 37 q^{77} + 82 q^{78} + 94 q^{79} + 25 q^{80} + 112 q^{81} + 35 q^{82} + 58 q^{83} + 48 q^{84} + 83 q^{85} + 78 q^{86} + 21 q^{87} + 70 q^{88} + 99 q^{89} + 37 q^{90} + 53 q^{91} + 77 q^{92} + 57 q^{93} + 59 q^{94} + 92 q^{95} + 22 q^{96} + 16 q^{97} + 103 q^{98} + 150 q^{99}+O(q^{100}) \) 88 * q + 88 * q^2 + 22 * q^3 + 88 * q^4 + 25 * q^5 + 22 * q^6 + 33 * q^7 + 88 * q^8 + 108 * q^9 + 25 * q^10 + 70 * q^11 + 22 * q^12 + 31 * q^13 + 33 * q^14 + 47 * q^15 + 88 * q^16 + 19 * q^17 + 108 * q^18 + 33 * q^19 + 25 * q^20 + 48 * q^21 + 70 * q^22 + 77 * q^23 + 22 * q^24 + 109 * q^25 + 31 * q^26 + 88 * q^27 + 33 * q^28 + 83 * q^29 + 47 * q^30 + 51 * q^31 + 88 * q^32 + 30 * q^33 + 19 * q^34 + 40 * q^35 + 108 * q^36 + 45 * q^37 + 33 * q^38 + 82 * q^39 + 25 * q^40 + 35 * q^41 + 48 * q^42 + 78 * q^43 + 70 * q^44 + 37 * q^45 + 77 * q^46 + 59 * q^47 + 22 * q^48 + 103 * q^49 + 109 * q^50 + 21 * q^51 + 31 * q^52 + 58 * q^53 + 88 * q^54 + 35 * q^55 + 33 * q^56 - 16 * q^57 + 83 * q^58 + 54 * q^59 + 47 * q^60 + 18 * q^61 + 51 * q^62 + 47 * q^63 + 88 * q^64 + 34 * q^65 + 30 * q^66 + 88 * q^67 + 19 * q^68 + 62 * q^69 + 40 * q^70 + 139 * q^71 + 108 * q^72 - 6 * q^73 + 45 * q^74 + 45 * q^75 + 33 * q^76 + 37 * q^77 + 82 * q^78 + 94 * q^79 + 25 * q^80 + 112 * q^81 + 35 * q^82 + 58 * q^83 + 48 * q^84 + 83 * q^85 + 78 * q^86 + 21 * q^87 + 70 * q^88 + 99 * q^89 + 37 * q^90 + 53 * q^91 + 77 * q^92 + 57 * q^93 + 59 * q^94 + 92 * q^95 + 22 * q^96 + 16 * q^97 + 103 * q^98 + 150 * 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
$3$	−2.99129	−1.72702	−0.863512	−	0.504329i	$-0.831740\pi$
$3$	−2.99129	−1.72702	−0.863512	+	0.504329i	$0.831740\pi$
$4$	1.00000	0.500000
$5$	3.54646	1.58602	0.793012	−	0.609205i	$-0.208512\pi$
$5$	3.54646	1.58602	0.793012	+	0.609205i	$0.208512\pi$
$6$	−2.99129	−1.22119
$7$	3.26038	1.23231	0.616154	−	0.787626i	$-0.288690\pi$
$7$	3.26038	1.23231	0.616154	+	0.787626i	$0.288690\pi$
$8$	1.00000	0.353553
$9$	5.94783	1.98261
$10$	3.54646	1.12149
$11$	1.94696	0.587032	0.293516	−	0.955954i	$-0.405175\pi$
$11$	1.94696	0.587032	0.293516	+	0.955954i	$0.405175\pi$
$12$	−2.99129	−0.863512
$13$	−5.13670	−1.42466	−0.712332	−	0.701843i	$-0.752361\pi$
$13$	−5.13670	−1.42466	−0.712332	+	0.701843i	$0.752361\pi$
$14$	3.26038	0.871373
$15$	−10.6085	−2.73910
$16$	1.00000	0.250000
$17$	2.80954	0.681414	0.340707	−	0.940169i	$-0.389334\pi$
$17$	2.80954	0.681414	0.340707	+	0.940169i	$0.389334\pi$
$18$	5.94783	1.40192
$19$	−3.35046	−0.768648	−0.384324	−	0.923198i	$-0.625565\pi$
$19$	−3.35046	−0.768648	−0.384324	+	0.923198i	$0.625565\pi$
$20$	3.54646	0.793012
$21$	−9.75274	−2.12822
$22$	1.94696	0.415094
$23$	0.239467	0.0499324	0.0249662	−	0.999688i	$-0.492052\pi$
$23$	0.239467	0.0499324	0.0249662	+	0.999688i	$0.492052\pi$
$24$	−2.99129	−0.610595
$25$	7.57737	1.51547
$26$	−5.13670	−1.00739
$27$	−8.81782	−1.69699
$28$	3.26038	0.616154
$29$	3.51053	0.651890	0.325945	−	0.945389i	$-0.394318\pi$
$29$	3.51053	0.651890	0.325945	+	0.945389i	$0.394318\pi$
$30$	−10.6085	−1.93684
$31$	−5.70064	−1.02386	−0.511932	−	0.859026i	$-0.671070\pi$
$31$	−5.70064	−1.02386	−0.511932	+	0.859026i	$0.671070\pi$
$32$	1.00000	0.176777
$33$	−5.82394	−1.01382
$34$	2.80954	0.481833
$35$	11.5628	1.95447
$36$	5.94783	0.991305
$37$	−5.03384	−0.827559	−0.413779	−	0.910377i	$-0.635791\pi$
$37$	−5.03384	−0.827559	−0.413779	+	0.910377i	$0.635791\pi$
$38$	−3.35046	−0.543516
$39$	15.3654	2.46043
$40$	3.54646	0.560744
$41$	−8.78183	−1.37149	−0.685746	−	0.727841i	$-0.740524\pi$
$41$	−8.78183	−1.37149	−0.685746	+	0.727841i	$0.740524\pi$
$42$	−9.75274	−1.50488
$43$	8.70345	1.32726	0.663632	−	0.748060i	$-0.269014\pi$
$43$	8.70345	1.32726	0.663632	+	0.748060i	$0.269014\pi$
$44$	1.94696	0.293516
$45$	21.0937	3.14447
$46$	0.239467	0.0353075
$47$	8.59728	1.25404	0.627021	−	0.779002i	$-0.284274\pi$
$47$	8.59728	1.25404	0.627021	+	0.779002i	$0.284274\pi$
$48$	−2.99129	−0.431756
$49$	3.63007	0.518581
$50$	7.57737	1.07160
$51$	−8.40417	−1.17682
$52$	−5.13670	−0.712332
$53$	4.16392	0.571959	0.285979	−	0.958236i	$-0.407681\pi$
$53$	4.16392	0.571959	0.285979	+	0.958236i	$0.407681\pi$
$54$	−8.81782	−1.19995
$55$	6.90483	0.931047
$56$	3.26038	0.435686
$57$	10.0222	1.32747
$58$	3.51053	0.460956
$59$	−5.06992	−0.660048	−0.330024	−	0.943973i	$-0.607057\pi$
$59$	−5.06992	−0.660048	−0.330024	+	0.943973i	$0.607057\pi$
$60$	−10.6085	−1.36955
$61$	11.3864	1.45788	0.728939	−	0.684579i	$-0.240014\pi$
$61$	11.3864	1.45788	0.728939	+	0.684579i	$0.240014\pi$
$62$	−5.70064	−0.723982
$63$	19.3922	2.44318
$64$	1.00000	0.125000
$65$	−18.2171	−2.25955
$66$	−5.82394	−0.716877
$67$	14.7225	1.79864	0.899318	−	0.437294i	$-0.144063\pi$
$67$	14.7225	1.79864	0.899318	+	0.437294i	$0.144063\pi$
$68$	2.80954	0.340707
$69$	−0.716317	−0.0862344
$70$	11.5628	1.38202
$71$	9.52674	1.13062	0.565308	−	0.824880i	$-0.308757\pi$
$71$	9.52674	1.13062	0.565308	+	0.824880i	$0.308757\pi$
$72$	5.94783	0.700958
$73$	−12.2384	−1.43240	−0.716199	−	0.697897i	$-0.754120\pi$
$73$	−12.2384	−1.43240	−0.716199	+	0.697897i	$0.754120\pi$
$74$	−5.03384	−0.585172
$75$	−22.6661	−2.61726
$76$	−3.35046	−0.384324
$77$	6.34784	0.723403
$78$	15.3654	1.73978
$79$	9.14994	1.02945	0.514724	−	0.857356i	$-0.327894\pi$
$79$	9.14994	1.02945	0.514724	+	0.857356i	$0.327894\pi$
$80$	3.54646	0.396506
$81$	8.53318	0.948131
$82$	−8.78183	−0.969791
$83$	12.5067	1.37279	0.686397	−	0.727227i	$-0.259191\pi$
$83$	12.5067	1.37279	0.686397	+	0.727227i	$0.259191\pi$
$84$	−9.75274	−1.06411
$85$	9.96393	1.08074
$86$	8.70345	0.938517
$87$	−10.5010	−1.12583
$88$	1.94696	0.207547
$89$	5.21444	0.552729	0.276365	−	0.961053i	$-0.410870\pi$
$89$	5.21444	0.552729	0.276365	+	0.961053i	$0.410870\pi$
$90$	21.0937	2.22347
$91$	−16.7476	−1.75562
$92$	0.239467	0.0249662
$93$	17.0523	1.76824
$94$	8.59728	0.886742
$95$	−11.8823	−1.21909
$96$	−2.99129	−0.305297
$97$	5.62501	0.571133	0.285567	−	0.958359i	$-0.407818\pi$
$97$	5.62501	0.571133	0.285567	+	0.958359i	$0.407818\pi$
$98$	3.63007	0.366692
$99$	11.5802	1.16385
$100$	7.57737	0.757737
$101$	−13.6203	−1.35527	−0.677634	−	0.735399i	$-0.736995\pi$
$101$	−13.6203	−1.35527	−0.677634	+	0.735399i	$0.736995\pi$
$102$	−8.40417	−0.832136
$103$	−1.74578	−0.172017	−0.0860083	−	0.996294i	$-0.527411\pi$
$103$	−1.74578	−0.172017	−0.0860083	+	0.996294i	$0.527411\pi$
$104$	−5.13670	−0.503695
$105$	−34.5877	−3.37541
$106$	4.16392	0.404436
$107$	9.76030	0.943564	0.471782	−	0.881715i	$-0.343611\pi$
$107$	9.76030	0.943564	0.471782	+	0.881715i	$0.343611\pi$
$108$	−8.81782	−0.848495
$109$	12.7535	1.22157	0.610784	−	0.791797i	$-0.290854\pi$
$109$	12.7535	1.22157	0.610784	+	0.791797i	$0.290854\pi$
$110$	6.90483	0.658349
$111$	15.0577	1.42921
$112$	3.26038	0.308077
$113$	−2.35238	−0.221293	−0.110647	−	0.993860i	$-0.535292\pi$
$113$	−2.35238	−0.221293	−0.110647	+	0.993860i	$0.535292\pi$
$114$	10.0222	0.938665
$115$	0.849261	0.0791940
$116$	3.51053	0.325945
$117$	−30.5522	−2.82455
$118$	−5.06992	−0.466724
$119$	9.16017	0.839712
$120$	−10.6085	−0.968419
$121$	−7.20933	−0.655394
$122$	11.3864	1.03087
$123$	26.2690	2.36860
$124$	−5.70064	−0.511932
$125$	9.14055	0.817556
$126$	19.3922	1.72759
$127$	16.1594	1.43392	0.716959	−	0.697115i	$-0.245534\pi$
$127$	16.1594	1.43392	0.716959	+	0.697115i	$0.245534\pi$
$128$	1.00000	0.0883883
$129$	−26.0346	−2.29221
$130$	−18.2171	−1.59774
$131$	10.3052	0.900369	0.450184	−	0.892936i	$-0.351358\pi$
$131$	10.3052	0.900369	0.450184	+	0.892936i	$0.351358\pi$
$132$	−5.82394	−0.506909
$133$	−10.9238	−0.947210
$134$	14.7225	1.27183
$135$	−31.2720	−2.69147
$136$	2.80954	0.240916
$137$	11.7010	0.999682	0.499841	−	0.866117i	$-0.333392\pi$
$137$	11.7010	0.999682	0.499841	+	0.866117i	$0.333392\pi$
$138$	−0.716317	−0.0609770
$139$	10.5054	0.891058	0.445529	−	0.895267i	$-0.353016\pi$
$139$	10.5054	0.891058	0.445529	+	0.895267i	$0.353016\pi$
$140$	11.5628	0.977235
$141$	−25.7170	−2.16576
$142$	9.52674	0.799466
$143$	−10.0010	−0.836322
$144$	5.94783	0.495652
$145$	12.4500	1.03391
$146$	−12.2384	−1.01286
$147$	−10.8586	−0.895601
$148$	−5.03384	−0.413779
$149$	−1.96542	−0.161013	−0.0805066	−	0.996754i	$-0.525654\pi$
$149$	−1.96542	−0.161013	−0.0805066	+	0.996754i	$0.525654\pi$
$150$	−22.6661	−1.85068
$151$	−18.9517	−1.54227	−0.771135	−	0.636671i	$-0.780311\pi$
$151$	−18.9517	−1.54227	−0.771135	+	0.636671i	$0.780311\pi$
$152$	−3.35046	−0.271758
$153$	16.7107	1.35098
$154$	6.34784	0.511523
$155$	−20.2171	−1.62388
$156$	15.3654	1.23021
$157$	−19.2834	−1.53899	−0.769493	−	0.638655i	$-0.779491\pi$
$157$	−19.2834	−1.53899	−0.769493	+	0.638655i	$0.779491\pi$
$158$	9.14994	0.727930
$159$	−12.4555	−0.987786
$160$	3.54646	0.280372
$161$	0.780754	0.0615321
$162$	8.53318	0.670430
$163$	0.560467	0.0438992	0.0219496	−	0.999759i	$-0.493013\pi$
$163$	0.560467	0.0438992	0.0219496	+	0.999759i	$0.493013\pi$
$164$	−8.78183	−0.685746
$165$	−20.6544	−1.60794
$166$	12.5067	0.970712
$167$	−9.13219	−0.706670	−0.353335	−	0.935497i	$-0.614952\pi$
$167$	−9.13219	−0.706670	−0.353335	+	0.935497i	$0.614952\pi$
$168$	−9.75274	−0.752440
$169$	13.3856	1.02967
$170$	9.96393	0.764199
$171$	−19.9280	−1.52393
$172$	8.70345	0.663632
$173$	−12.8343	−0.975773	−0.487887	−	0.872907i	$-0.662232\pi$
$173$	−12.8343	−0.975773	−0.487887	+	0.872907i	$0.662232\pi$
$174$	−10.5010	−0.796081
$175$	24.7051	1.86753
$176$	1.94696	0.146758
$177$	15.1656	1.13992
$178$	5.21444	0.390838
$179$	−0.119520	−0.00893336	−0.00446668	−	0.999990i	$-0.501422\pi$
$179$	−0.119520	−0.00893336	−0.00446668	+	0.999990i	$0.501422\pi$
$180$	21.0937	1.57223
$181$	−11.8463	−0.880525	−0.440262	−	0.897869i	$-0.645115\pi$
$181$	−11.8463	−0.880525	−0.440262	+	0.897869i	$0.645115\pi$
$182$	−16.7476	−1.24141
$183$	−34.0600	−2.51779
$184$	0.239467	0.0176538
$185$	−17.8523	−1.31253
$186$	17.0523	1.25033
$187$	5.47008	0.400012
$188$	8.59728	0.627021
$189$	−28.7494	−2.09121
$190$	−11.8823	−0.862030
$191$	−14.4002	−1.04196	−0.520980	−	0.853569i	$-0.674434\pi$
$191$	−14.4002	−1.04196	−0.520980	+	0.853569i	$0.674434\pi$
$192$	−2.99129	−0.215878
$193$	14.5424	1.04678	0.523392	−	0.852092i	$-0.324666\pi$
$193$	14.5424	1.04678	0.523392	+	0.852092i	$0.324666\pi$
$194$	5.62501	0.403852
$195$	54.4926	3.90230
$196$	3.63007	0.259290
$197$	2.41084	0.171765	0.0858826	−	0.996305i	$-0.472629\pi$
$197$	2.41084	0.171765	0.0858826	+	0.996305i	$0.472629\pi$
$198$	11.5802	0.822969
$199$	−1.82763	−0.129558	−0.0647788	−	0.997900i	$-0.520634\pi$
$199$	−1.82763	−0.129558	−0.0647788	+	0.997900i	$0.520634\pi$
$200$	7.57737	0.535801
$201$	−44.0392	−3.10629
$202$	−13.6203	−0.958320
$203$	11.4457	0.803328
$204$	−8.40417	−0.588409
$205$	−31.1444	−2.17522
$206$	−1.74578	−0.121634
$207$	1.42431	0.0989965
$208$	−5.13670	−0.356166
$209$	−6.52322	−0.451221
$210$	−34.5877	−2.38678
$211$	−8.76677	−0.603530	−0.301765	−	0.953382i	$-0.597576\pi$
$211$	−8.76677	−0.603530	−0.301765	+	0.953382i	$0.597576\pi$
$212$	4.16392	0.285979
$213$	−28.4973	−1.95260
$214$	9.76030	0.667200
$215$	30.8664	2.10507
$216$	−8.81782	−0.599977
$217$	−18.5862	−1.26172
$218$	12.7535	0.863779
$219$	36.6087	2.47378
$220$	6.90483	0.465523
$221$	−14.4318	−0.970786
$222$	15.0577	1.01061
$223$	−12.1631	−0.814502	−0.407251	−	0.913316i	$-0.633513\pi$
$223$	−12.1631	−0.814502	−0.407251	+	0.913316i	$0.633513\pi$
$224$	3.26038	0.217843
$225$	45.0689	3.00460
$226$	−2.35238	−0.156478
$227$	−27.5228	−1.82675	−0.913377	−	0.407115i	$-0.866535\pi$
$227$	−27.5228	−1.82675	−0.913377	+	0.407115i	$0.866535\pi$
$228$	10.0222	0.663736
$229$	29.3803	1.94151	0.970754	−	0.240077i	$-0.0771728\pi$
$229$	29.3803	1.94151	0.970754	+	0.240077i	$0.0771728\pi$
$230$	0.849261	0.0559986
$231$	−18.9882	−1.24933
$232$	3.51053	0.230478
$233$	−2.51176	−0.164551	−0.0822753	−	0.996610i	$-0.526219\pi$
$233$	−2.51176	−0.164551	−0.0822753	+	0.996610i	$0.526219\pi$
$234$	−30.5522	−1.99726
$235$	30.4899	1.98894
$236$	−5.06992	−0.330024
$237$	−27.3701	−1.77788
$238$	9.16017	0.593766
$239$	7.51308	0.485981	0.242990	−	0.970029i	$-0.421872\pi$
$239$	7.51308	0.485981	0.242990	+	0.970029i	$0.421872\pi$
$240$	−10.6085	−0.684775
$241$	8.13479	0.524008	0.262004	−	0.965067i	$-0.415617\pi$
$241$	8.13479	0.524008	0.262004	+	0.965067i	$0.415617\pi$
$242$	−7.20933	−0.463434
$243$	0.928210	0.0595447
$244$	11.3864	0.728939
$245$	12.8739	0.822482
$246$	26.2690	1.67485
$247$	17.2103	1.09506
$248$	−5.70064	−0.361991
$249$	−37.4113	−2.37085
$250$	9.14055	0.578099
$251$	−7.35558	−0.464280	−0.232140	−	0.972682i	$-0.574573\pi$
$251$	−7.35558	−0.464280	−0.232140	+	0.972682i	$0.574573\pi$
$252$	19.3922	1.22159
$253$	0.466234	0.0293119
$254$	16.1594	1.01393
$255$	−29.8050	−1.86646
$256$	1.00000	0.0625000
$257$	1.26902	0.0791592	0.0395796	−	0.999216i	$-0.487398\pi$
$257$	1.26902	0.0791592	0.0395796	+	0.999216i	$0.487398\pi$
$258$	−26.0346	−1.62084
$259$	−16.4122	−1.01981
$260$	−18.2171	−1.12978
$261$	20.8801	1.29244
$262$	10.3052	0.636657
$263$	−22.3280	−1.37680	−0.688402	−	0.725330i	$-0.741687\pi$
$263$	−22.3280	−1.37680	−0.688402	+	0.725330i	$0.741687\pi$
$264$	−5.82394	−0.358439
$265$	14.7672	0.907141
$266$	−10.9238	−0.669779
$267$	−15.5979	−0.954576
$268$	14.7225	0.899318
$269$	29.6471	1.80762	0.903809	−	0.427937i	$-0.140759\pi$
$269$	29.6471	1.80762	0.903809	+	0.427937i	$0.140759\pi$
$270$	−31.2720	−1.90316
$271$	31.0403	1.88556	0.942781	−	0.333413i	$-0.108200\pi$
$271$	31.0403	1.88556	0.942781	+	0.333413i	$0.108200\pi$
$272$	2.80954	0.170354
$273$	50.0969	3.03200
$274$	11.7010	0.706882
$275$	14.7529	0.889632
$276$	−0.716317	−0.0431172
$277$	15.4516	0.928395	0.464198	−	0.885732i	$-0.346343\pi$
$277$	15.4516	0.928395	0.464198	+	0.885732i	$0.346343\pi$
$278$	10.5054	0.630073
$279$	−33.9064	−2.02992
$280$	11.5628	0.691009
$281$	−4.07378	−0.243021	−0.121511	−	0.992590i	$-0.538774\pi$
$281$	−4.07378	−0.243021	−0.121511	+	0.992590i	$0.538774\pi$
$282$	−25.7170	−1.53142
$283$	15.6107	0.927961	0.463980	−	0.885845i	$-0.346421\pi$
$283$	15.6107	0.927961	0.463980	+	0.885845i	$0.346421\pi$
$284$	9.52674	0.565308
$285$	35.5433	2.10540
$286$	−10.0010	−0.591369
$287$	−28.6321	−1.69010
$288$	5.94783	0.350479
$289$	−9.10647	−0.535674
$290$	12.4500	0.731087
$291$	−16.8260	−0.986360
$292$	−12.2384	−0.716199
$293$	−22.6870	−1.32539	−0.662693	−	0.748891i	$-0.730587\pi$
$293$	−22.6870	−1.32539	−0.662693	+	0.748891i	$0.730587\pi$
$294$	−10.8586	−0.633286
$295$	−17.9803	−1.04685
$296$	−5.03384	−0.292586
$297$	−17.1680	−0.996187
$298$	−1.96542	−0.113854
$299$	−1.23007	−0.0711369
$300$	−22.6661	−1.30863
$301$	28.3765	1.63560
$302$	−18.9517	−1.09055
$303$	40.7422	2.34058
$304$	−3.35046	−0.192162
$305$	40.3814	2.31223
$306$	16.7107	0.955286
$307$	16.4903	0.941154	0.470577	−	0.882359i	$-0.344046\pi$
$307$	16.4903	0.941154	0.470577	+	0.882359i	$0.344046\pi$
$308$	6.34784	0.361702
$309$	5.22213	0.297077
$310$	−20.2171	−1.14825
$311$	4.32705	0.245364	0.122682	−	0.992446i	$-0.460850\pi$
$311$	4.32705	0.245364	0.122682	+	0.992446i	$0.460850\pi$
$312$	15.3654	0.869892
$313$	16.5654	0.936334	0.468167	−	0.883640i	$-0.344915\pi$
$313$	16.5654	0.936334	0.468167	+	0.883640i	$0.344915\pi$
$314$	−19.2834	−1.08823
$315$	68.7735	3.87495
$316$	9.14994	0.514724
$317$	1.99336	0.111958	0.0559792	−	0.998432i	$-0.482172\pi$
$317$	1.99336	0.111958	0.0559792	+	0.998432i	$0.482172\pi$
$318$	−12.4555	−0.698470
$319$	6.83488	0.382680
$320$	3.54646	0.198253
$321$	−29.1959	−1.62956
$322$	0.780754	0.0435097
$323$	−9.41326	−0.523768
$324$	8.53318	0.474066
$325$	−38.9227	−2.15904
$326$	0.560467	0.0310414
$327$	−38.1496	−2.10968
$328$	−8.78183	−0.484895
$329$	28.0304	1.54537
$330$	−20.6544	−1.13698
$331$	35.3432	1.94264	0.971319	−	0.237782i	$-0.0764204\pi$
$331$	35.3432	1.94264	0.971319	+	0.237782i	$0.0764204\pi$
$332$	12.5067	0.686397
$333$	−29.9404	−1.64073
$334$	−9.13219	−0.499691
$335$	52.2127	2.85268
$336$	−9.75274	−0.532056
$337$	−9.05172	−0.493078	−0.246539	−	0.969133i	$-0.579293\pi$
$337$	−9.05172	−0.493078	−0.246539	+	0.969133i	$0.579293\pi$
$338$	13.3856	0.728083
$339$	7.03665	0.382178
$340$	9.96393	0.540370
$341$	−11.0989	−0.601041
$342$	−19.9280	−1.07758
$343$	−10.9873	−0.593256
$344$	8.70345	0.469258
$345$	−2.54039	−0.136770
$346$	−12.8343	−0.689976
$347$	−0.348197	−0.0186922	−0.00934609	−	0.999956i	$-0.502975\pi$
$347$	−0.348197	−0.0186922	−0.00934609	+	0.999956i	$0.502975\pi$
$348$	−10.5010	−0.562914
$349$	15.9182	0.852082	0.426041	−	0.904704i	$-0.359908\pi$
$349$	15.9182	0.852082	0.426041	+	0.904704i	$0.359908\pi$
$350$	24.7051	1.32054
$351$	45.2945	2.41764
$352$	1.94696	0.103773
$353$	−0.0241859	−0.00128729	−0.000643644	−	1.00000i	$-0.500205\pi$
$353$	−0.0241859	−0.00128729	−0.000643644		1.00000i	$0.500205\pi$
$354$	15.1656	0.806044
$355$	33.7862	1.79319
$356$	5.21444	0.276365
$357$	−27.4008	−1.45020
$358$	−0.119520	−0.00631684
$359$	11.6967	0.617326	0.308663	−	0.951172i	$-0.400119\pi$
$359$	11.6967	0.617326	0.308663	+	0.951172i	$0.400119\pi$
$360$	21.0937	1.11174
$361$	−7.77443	−0.409180
$362$	−11.8463	−0.622625
$363$	21.5652	1.13188
$364$	−16.7476	−0.877811
$365$	−43.4030	−2.27182
$366$	−34.0600	−1.78035
$367$	25.8291	1.34827	0.674134	−	0.738609i	$-0.264517\pi$
$367$	25.8291	1.34827	0.674134	+	0.738609i	$0.264517\pi$
$368$	0.239467	0.0124831
$369$	−52.2328	−2.71913
$370$	−17.8523	−0.928098
$371$	13.5760	0.704829
$372$	17.0523	0.884119
$373$	8.57692	0.444096	0.222048	−	0.975036i	$-0.428726\pi$
$373$	8.57692	0.444096	0.222048	+	0.975036i	$0.428726\pi$
$374$	5.47008	0.282851
$375$	−27.3421	−1.41194
$376$	8.59728	0.443371
$377$	−18.0325	−0.928723
$378$	−28.7494	−1.47871
$379$	−6.93582	−0.356269	−0.178135	−	0.984006i	$-0.557006\pi$
$379$	−6.93582	−0.356269	−0.178135	+	0.984006i	$0.557006\pi$
$380$	−11.8823	−0.609547
$381$	−48.3376	−2.47641
$382$	−14.4002	−0.736777
$383$	−23.7773	−1.21496	−0.607482	−	0.794334i	$-0.707820\pi$
$383$	−23.7773	−1.21496	−0.607482	+	0.794334i	$0.707820\pi$
$384$	−2.99129	−0.152649
$385$	22.5123	1.14734
$386$	14.5424	0.740188
$387$	51.7666	2.63144
$388$	5.62501	0.285567
$389$	3.90711	0.198098	0.0990491	−	0.995083i	$-0.468420\pi$
$389$	3.90711	0.198098	0.0990491	+	0.995083i	$0.468420\pi$
$390$	54.4926	2.75934
$391$	0.672794	0.0340247
$392$	3.63007	0.183346
$393$	−30.8258	−1.55496
$394$	2.41084	0.121456
$395$	32.4499	1.63273
$396$	11.5802	0.581927
$397$	−18.1664	−0.911747	−0.455873	−	0.890045i	$-0.650673\pi$
$397$	−18.1664	−0.911747	−0.455873	+	0.890045i	$0.650673\pi$
$398$	−1.82763	−0.0916111
$399$	32.6762	1.63585
$400$	7.57737	0.378869
$401$	−11.1573	−0.557171	−0.278586	−	0.960411i	$-0.589866\pi$
$401$	−11.1573	−0.557171	−0.278586	+	0.960411i	$0.589866\pi$
$402$	−44.0392	−2.19648
$403$	29.2824	1.45866
$404$	−13.6203	−0.677634
$405$	30.2626	1.50376
$406$	11.4457	0.568039
$407$	−9.80071	−0.485803
$408$	−8.40417	−0.416068
$409$	−18.6293	−0.921161	−0.460580	−	0.887618i	$-0.652359\pi$
$409$	−18.6293	−0.921161	−0.460580	+	0.887618i	$0.652359\pi$
$410$	−31.1444	−1.53811
$411$	−35.0010	−1.72647
$412$	−1.74578	−0.0860083
$413$	−16.5299	−0.813382
$414$	1.42431	0.0700011
$415$	44.3547	2.17728
$416$	−5.13670	−0.251847
$417$	−31.4248	−1.53888
$418$	−6.52322	−0.319061
$419$	3.38193	0.165218	0.0826091	−	0.996582i	$-0.473675\pi$
$419$	3.38193	0.165218	0.0826091	+	0.996582i	$0.473675\pi$
$420$	−34.5877	−1.68771
$421$	0.848449	0.0413509	0.0206754	−	0.999786i	$-0.493418\pi$
$421$	0.848449	0.0413509	0.0206754	+	0.999786i	$0.493418\pi$
$422$	−8.76677	−0.426760
$423$	51.1352	2.48628
$424$	4.16392	0.202218
$425$	21.2890	1.03267
$426$	−28.4973	−1.38070
$427$	37.1239	1.79655
$428$	9.76030	0.471782
$429$	29.9158	1.44435
$430$	30.8664	1.48851
$431$	4.79599	0.231015	0.115507	−	0.993307i	$-0.463151\pi$
$431$	4.79599	0.231015	0.115507	+	0.993307i	$0.463151\pi$
$432$	−8.81782	−0.424247
$433$	−28.7837	−1.38326	−0.691628	−	0.722254i	$-0.743106\pi$
$433$	−28.7837	−1.38326	−0.691628	+	0.722254i	$0.743106\pi$
$434$	−18.5862	−0.892168
$435$	−37.2415	−1.78559
$436$	12.7535	0.610784
$437$	−0.802326	−0.0383804
$438$	36.6087	1.74923
$439$	6.84453	0.326672	0.163336	−	0.986571i	$-0.447775\pi$
$439$	6.84453	0.326672	0.163336	+	0.986571i	$0.447775\pi$
$440$	6.90483	0.329175
$441$	21.5910	1.02814
$442$	−14.4318	−0.686449
$443$	−22.0743	−1.04878	−0.524390	−	0.851478i	$-0.675707\pi$
$443$	−22.0743	−1.04878	−0.524390	+	0.851478i	$0.675707\pi$
$444$	15.0577	0.714607
$445$	18.4928	0.876642
$446$	−12.1631	−0.575940
$447$	5.87914	0.278074
$448$	3.26038	0.154038
$449$	−2.49312	−0.117658	−0.0588288	−	0.998268i	$-0.518737\pi$
$449$	−2.49312	−0.117658	−0.0588288	+	0.998268i	$0.518737\pi$
$450$	45.0689	2.12457
$451$	−17.0979	−0.805109
$452$	−2.35238	−0.110647
$453$	56.6902	2.66354
$454$	−27.5228	−1.29171
$455$	−59.3946	−2.78446
$456$	10.0222	0.469333
$457$	−1.07535	−0.0503027	−0.0251513	−	0.999684i	$-0.508007\pi$
$457$	−1.07535	−0.0503027	−0.0251513	+	0.999684i	$0.508007\pi$
$458$	29.3803	1.37285
$459$	−24.7740	−1.15635
$460$	0.849261	0.0395970
$461$	0.163451	0.00761267	0.00380634	−	0.999993i	$-0.498788\pi$
$461$	0.163451	0.00761267	0.00380634	+	0.999993i	$0.498788\pi$
$462$	−18.9882	−0.883413
$463$	−30.7175	−1.42756	−0.713781	−	0.700369i	$-0.753019\pi$
$463$	−30.7175	−1.42756	−0.713781	+	0.700369i	$0.753019\pi$
$464$	3.51053	0.162972
$465$	60.4752	2.80447
$466$	−2.51176	−0.116355
$467$	19.3530	0.895549	0.447774	−	0.894147i	$-0.352217\pi$
$467$	19.3530	0.895549	0.447774	+	0.894147i	$0.352217\pi$
$468$	−30.5522	−1.41228
$469$	48.0008	2.21647
$470$	30.4899	1.40639
$471$	57.6824	2.65787
$472$	−5.06992	−0.233362
$473$	16.9453	0.779145
$474$	−27.3701	−1.25715
$475$	−25.3877	−1.16487
$476$	9.16017	0.419856
$477$	24.7663	1.13397
$478$	7.51308	0.343640
$479$	34.5486	1.57857	0.789283	−	0.614030i	$-0.210453\pi$
$479$	34.5486	1.57857	0.789283	+	0.614030i	$0.210453\pi$
$480$	−10.6085	−0.484209
$481$	25.8573	1.17899
$482$	8.13479	0.370530
$483$	−2.33546	−0.106267
$484$	−7.20933	−0.327697
$485$	19.9489	0.905831
$486$	0.928210	0.0421045
$487$	1.92363	0.0871678	0.0435839	−	0.999050i	$-0.486122\pi$
$487$	1.92363	0.0871678	0.0435839	+	0.999050i	$0.486122\pi$
$488$	11.3864	0.515437
$489$	−1.67652	−0.0758149
$490$	12.8739	0.581583
$491$	12.2167	0.551331	0.275665	−	0.961254i	$-0.411102\pi$
$491$	12.2167	0.551331	0.275665	+	0.961254i	$0.411102\pi$
$492$	26.2690	1.18430
$493$	9.86300	0.444207
$494$	17.2103	0.774327
$495$	41.0687	1.84590
$496$	−5.70064	−0.255966
$497$	31.0608	1.39327
$498$	−37.4113	−1.67644
$499$	13.0656	0.584899	0.292449	−	0.956281i	$-0.405530\pi$
$499$	13.0656	0.584899	0.292449	+	0.956281i	$0.405530\pi$
$500$	9.14055	0.408778
$501$	27.3171	1.22044
$502$	−7.35558	−0.328296
$503$	−7.04500	−0.314121	−0.157060	−	0.987589i	$-0.550202\pi$
$503$	−7.04500	−0.314121	−0.157060	+	0.987589i	$0.550202\pi$
$504$	19.3922	0.863796
$505$	−48.3038	−2.14949
$506$	0.466234	0.0207266
$507$	−40.0404	−1.77826
$508$	16.1594	0.716959
$509$	15.3957	0.682400	0.341200	−	0.939991i	$-0.389167\pi$
$509$	15.3957	0.682400	0.341200	+	0.939991i	$0.389167\pi$
$510$	−29.8050	−1.31979
$511$	−39.9018	−1.76515
$512$	1.00000	0.0441942
$513$	29.5437	1.30439
$514$	1.26902	0.0559740
$515$	−6.19133	−0.272823
$516$	−26.0346	−1.14611
$517$	16.7386	0.736163
$518$	−16.4122	−0.721112
$519$	38.3911	1.68518
$520$	−18.2171	−0.798872
$521$	4.83456	0.211806	0.105903	−	0.994376i	$-0.466227\pi$
$521$	4.83456	0.211806	0.105903	+	0.994376i	$0.466227\pi$
$522$	20.8801	0.913895
$523$	−8.05734	−0.352323	−0.176161	−	0.984361i	$-0.556368\pi$
$523$	−8.05734	−0.352323	−0.176161	+	0.984361i	$0.556368\pi$
$524$	10.3052	0.450184
$525$	−73.9002	−3.22527
$526$	−22.3280	−0.973547
$527$	−16.0162	−0.697676
$528$	−5.82394	−0.253454
$529$	−22.9427	−0.997507
$530$	14.7672	0.641445
$531$	−30.1550	−1.30862
$532$	−10.9238	−0.473605
$533$	45.1096	1.95391
$534$	−15.5979	−0.674987
$535$	34.6145	1.49652
$536$	14.7225	0.635914
$537$	0.357520	0.0154281
$538$	29.6471	1.27818
$539$	7.06760	0.304423
$540$	−31.2720	−1.34573
$541$	−5.70905	−0.245451	−0.122726	−	0.992441i	$-0.539164\pi$
$541$	−5.70905	−0.245451	−0.122726	+	0.992441i	$0.539164\pi$
$542$	31.0403	1.33329
$543$	35.4356	1.52069
$544$	2.80954	0.120458
$545$	45.2299	1.93744
$546$	50.0969	2.14395
$547$	−40.2517	−1.72104	−0.860520	−	0.509416i	$-0.829861\pi$
$547$	−40.2517	−1.72104	−0.860520	+	0.509416i	$0.829861\pi$
$548$	11.7010	0.499841
$549$	67.7243	2.89040
$550$	14.7529	0.629064
$551$	−11.7619	−0.501074
$552$	−0.716317	−0.0304885
$553$	29.8322	1.26860
$554$	15.4516	0.656475
$555$	53.4015	2.26677
$556$	10.5054	0.445529
$557$	28.3752	1.20229	0.601147	−	0.799138i	$-0.294711\pi$
$557$	28.3752	1.20229	0.601147	+	0.799138i	$0.294711\pi$
$558$	−33.9064	−1.43537
$559$	−44.7070	−1.89090
$560$	11.5628	0.488617
$561$	−16.3626	−0.690830
$562$	−4.07378	−0.171842
$563$	−4.88754	−0.205985	−0.102993	−	0.994682i	$-0.532842\pi$
$563$	−4.88754	−0.205985	−0.102993	+	0.994682i	$0.532842\pi$
$564$	−25.7170	−1.08288
$565$	−8.34261	−0.350976
$566$	15.6107	0.656167
$567$	27.8214	1.16839
$568$	9.52674	0.399733
$569$	−29.4545	−1.23480	−0.617399	−	0.786650i	$-0.711814\pi$
$569$	−29.4545	−1.23480	−0.617399	+	0.786650i	$0.711814\pi$
$570$	35.5433	1.48875
$571$	3.42642	0.143391	0.0716955	−	0.997427i	$-0.477159\pi$
$571$	3.42642	0.143391	0.0716955	+	0.997427i	$0.477159\pi$
$572$	−10.0010	−0.418161
$573$	43.0752	1.79949
$574$	−28.6321	−1.19508
$575$	1.81453	0.0756713
$576$	5.94783	0.247826
$577$	−37.0759	−1.54349	−0.771745	−	0.635932i	$-0.780616\pi$
$577$	−37.0759	−1.54349	−0.771745	+	0.635932i	$0.780616\pi$
$578$	−9.10647	−0.378779
$579$	−43.5005	−1.80782
$580$	12.4500	0.516957
$581$	40.7767	1.69170
$582$	−16.8260	−0.697462
$583$	8.10700	0.335758
$584$	−12.2384	−0.506429
$585$	−108.352	−4.47981
$586$	−22.6870	−0.937190
$587$	−1.82969	−0.0755195	−0.0377597	−	0.999287i	$-0.512022\pi$
$587$	−1.82969	−0.0755195	−0.0377597	+	0.999287i	$0.512022\pi$
$588$	−10.8586	−0.447801
$589$	19.0997	0.786991
$590$	−17.9803	−0.740237
$591$	−7.21153	−0.296643
$592$	−5.03384	−0.206890
$593$	−26.6640	−1.09496	−0.547480	−	0.836819i	$-0.684413\pi$
$593$	−26.6640	−1.09496	−0.547480	+	0.836819i	$0.684413\pi$
$594$	−17.1680	−0.704410
$595$	32.4862	1.33180
$596$	−1.96542	−0.0805066
$597$	5.46699	0.223749
$598$	−1.23007	−0.0503014
$599$	24.2107	0.989224	0.494612	−	0.869114i	$-0.335310\pi$
$599$	24.2107	0.989224	0.494612	+	0.869114i	$0.335310\pi$
$600$	−22.6661	−0.925341
$601$	33.4755	1.36549	0.682747	−	0.730655i	$-0.260785\pi$
$601$	33.4755	1.36549	0.682747	+	0.730655i	$0.260785\pi$
$602$	28.3765	1.15654
$603$	87.5668	3.56599
$604$	−18.9517	−0.771135
$605$	−25.5676	−1.03947
$606$	40.7422	1.65504
$607$	−11.9219	−0.483894	−0.241947	−	0.970289i	$-0.577786\pi$
$607$	−11.9219	−0.483894	−0.241947	+	0.970289i	$0.577786\pi$
$608$	−3.35046	−0.135879
$609$	−34.2373	−1.38737
$610$	40.3814	1.63499
$611$	−44.1616	−1.78659
$612$	16.7107	0.675489
$613$	5.57361	0.225116	0.112558	−	0.993645i	$-0.464096\pi$
$613$	5.57361	0.225116	0.112558	+	0.993645i	$0.464096\pi$
$614$	16.4903	0.665496
$615$	93.1620	3.75665
$616$	6.34784	0.255762
$617$	−34.9679	−1.40775	−0.703877	−	0.710322i	$-0.748549\pi$
$617$	−34.9679	−1.40775	−0.703877	+	0.710322i	$0.748549\pi$
$618$	5.22213	0.210065
$619$	31.0613	1.24846	0.624229	−	0.781242i	$-0.285413\pi$
$619$	31.0613	1.24846	0.624229	+	0.781242i	$0.285413\pi$
$620$	−20.2171	−0.811938
$621$	−2.11158	−0.0847348
$622$	4.32705	0.173499
$623$	17.0010	0.681132
$624$	15.3654	0.615107
$625$	−5.47027	−0.218811
$626$	16.5654	0.662088
$627$	19.5129	0.779268
$628$	−19.2834	−0.769493
$629$	−14.1428	−0.563910
$630$	68.7735	2.74000
$631$	−14.5439	−0.578984	−0.289492	−	0.957180i	$-0.593486\pi$
$631$	−14.5439	−0.578984	−0.289492	+	0.957180i	$0.593486\pi$
$632$	9.14994	0.363965
$633$	26.2240	1.04231
$634$	1.99336	0.0791666
$635$	57.3088	2.27423
$636$	−12.4555	−0.493893
$637$	−18.6465	−0.738803
$638$	6.83488	0.270595
$639$	56.6634	2.24157
$640$	3.54646	0.140186
$641$	6.26905	0.247613	0.123806	−	0.992306i	$-0.460490\pi$
$641$	6.26905	0.247613	0.123806	+	0.992306i	$0.460490\pi$
$642$	−29.1959	−1.15227
$643$	14.2221	0.560863	0.280432	−	0.959874i	$-0.409522\pi$
$643$	14.2221	0.560863	0.280432	+	0.959874i	$0.409522\pi$
$644$	0.780754	0.0307660
$645$	−92.3305	−3.63551
$646$	−9.41326	−0.370360
$647$	−32.2386	−1.26743	−0.633715	−	0.773567i	$-0.718471\pi$
$647$	−32.2386	−1.26743	−0.633715	+	0.773567i	$0.718471\pi$
$648$	8.53318	0.335215
$649$	−9.87096	−0.387469
$650$	−38.9227	−1.52667
$651$	55.5969	2.17901
$652$	0.560467	0.0219496
$653$	11.9277	0.466768	0.233384	−	0.972385i	$-0.425020\pi$
$653$	11.9277	0.466768	0.233384	+	0.972385i	$0.425020\pi$
$654$	−38.1496	−1.49177
$655$	36.5469	1.42801
$656$	−8.78183	−0.342873
$657$	−72.7920	−2.83988
$658$	28.0304	1.09274
$659$	46.3824	1.80680	0.903401	−	0.428796i	$-0.141062\pi$
$659$	46.3824	1.80680	0.903401	+	0.428796i	$0.141062\pi$
$660$	−20.6544	−0.803970
$661$	49.7538	1.93520	0.967599	−	0.252491i	$-0.0812499\pi$
$661$	49.7538	1.93520	0.967599	+	0.252491i	$0.0812499\pi$
$662$	35.3432	1.37365
$663$	43.1696	1.67657
$664$	12.5067	0.485356
$665$	−38.7407	−1.50230
$666$	−29.9404	−1.16017
$667$	0.840658	0.0325504
$668$	−9.13219	−0.353335
$669$	36.3834	1.40666
$670$	52.2127	2.01715
$671$	22.1689	0.855820
$672$	−9.75274	−0.376220
$673$	−8.34569	−0.321703	−0.160851	−	0.986979i	$-0.551424\pi$
$673$	−8.34569	−0.321703	−0.160851	+	0.986979i	$0.551424\pi$
$674$	−9.05172	−0.348659
$675$	−66.8159	−2.57175
$676$	13.3856	0.514833
$677$	−32.5469	−1.25088	−0.625439	−	0.780273i	$-0.715080\pi$
$677$	−32.5469	−1.25088	−0.625439	+	0.780273i	$0.715080\pi$
$678$	7.03665	0.270241
$679$	18.3397	0.703811
$680$	9.96393	0.382099
$681$	82.3288	3.15485
$682$	−11.0989	−0.425000
$683$	−29.0736	−1.11247	−0.556236	−	0.831024i	$-0.687755\pi$
$683$	−29.0736	−1.11247	−0.556236	+	0.831024i	$0.687755\pi$
$684$	−19.9280	−0.761964
$685$	41.4970	1.58552
$686$	−10.9873	−0.419496
$687$	−87.8852	−3.35303
$688$	8.70345	0.331816
$689$	−21.3888	−0.814849
$690$	−2.54039	−0.0967110
$691$	13.6855	0.520620	0.260310	−	0.965525i	$-0.416175\pi$
$691$	13.6855	0.520620	0.260310	+	0.965525i	$0.416175\pi$
$692$	−12.8343	−0.487887
$693$	37.7558	1.43423
$694$	−0.348197	−0.0132174
$695$	37.2570	1.41324
$696$	−10.5010	−0.398041
$697$	−24.6729	−0.934554
$698$	15.9182	0.602513
$699$	7.51340	0.284183
$700$	24.7051	0.933765
$701$	−5.61990	−0.212261	−0.106130	−	0.994352i	$-0.533846\pi$
$701$	−5.61990	−0.212261	−0.106130	+	0.994352i	$0.533846\pi$
$702$	45.2945	1.70953
$703$	16.8657	0.636101
$704$	1.94696	0.0733789
$705$	−91.2042	−3.43495
$706$	−0.0241859	−0.000910250 0
$707$	−44.4073	−1.67011
$708$	15.1656	0.569959
$709$	−41.9598	−1.57583	−0.787916	−	0.615782i	$-0.788840\pi$
$709$	−41.9598	−1.57583	−0.787916	+	0.615782i	$0.788840\pi$
$710$	33.7862	1.26797
$711$	54.4223	2.04099
$712$	5.21444	0.195419
$713$	−1.36512	−0.0511240
$714$	−27.4008	−1.02545
$715$	−35.4680	−1.32643
$716$	−0.119520	−0.00446668
$717$	−22.4738	−0.839300
$718$	11.6967	0.436515
$719$	−50.2970	−1.87576	−0.937881	−	0.346958i	$-0.887215\pi$
$719$	−50.2970	−1.87576	−0.937881	+	0.346958i	$0.887215\pi$
$720$	21.0937	0.786117
$721$	−5.69190	−0.211977
$722$	−7.77443	−0.289334
$723$	−24.3335	−0.904974
$724$	−11.8463	−0.440262
$725$	26.6006	0.987922
$726$	21.5652	0.800361
$727$	−29.3195	−1.08740	−0.543701	−	0.839279i	$-0.682977\pi$
$727$	−29.3195	−1.08740	−0.543701	+	0.839279i	$0.682977\pi$
$728$	−16.7476	−0.620706
$729$	−28.3761	−1.05097
$730$	−43.4030	−1.60642
$731$	24.4527	0.904416
$732$	−34.0600	−1.25889
$733$	−47.4413	−1.75228	−0.876142	−	0.482053i	$-0.839891\pi$
$733$	−47.4413	−1.75228	−0.876142	+	0.482053i	$0.839891\pi$
$734$	25.8291	0.953370
$735$	−38.5095	−1.42045
$736$	0.239467	0.00882689
$737$	28.6641	1.05586
$738$	−52.2328	−1.92272
$739$	−16.8602	−0.620214	−0.310107	−	0.950702i	$-0.600365\pi$
$739$	−16.8602	−0.620214	−0.310107	+	0.950702i	$0.600365\pi$
$740$	−17.8523	−0.656264
$741$	−51.4810	−1.89120
$742$	13.5760	0.498389
$743$	7.31872	0.268498	0.134249	−	0.990948i	$-0.457138\pi$
$743$	7.31872	0.268498	0.134249	+	0.990948i	$0.457138\pi$
$744$	17.0523	0.625167
$745$	−6.97027	−0.255371
$746$	8.57692	0.314023
$747$	74.3880	2.72171
$748$	5.47008	0.200006
$749$	31.8223	1.16276
$750$	−27.3421	−0.998391
$751$	−29.7007	−1.08379	−0.541897	−	0.840445i	$-0.682294\pi$
$751$	−29.7007	−1.08379	−0.541897	+	0.840445i	$0.682294\pi$
$752$	8.59728	0.313511
$753$	22.0027	0.801823
$754$	−18.0325	−0.656706
$755$	−67.2116	−2.44608
$756$	−28.7494	−1.04561
$757$	43.4574	1.57949	0.789743	−	0.613438i	$-0.210214\pi$
$757$	43.4574	1.57949	0.789743	+	0.613438i	$0.210214\pi$
$758$	−6.93582	−0.251920
$759$	−1.39464	−0.0506223
$760$	−11.8823	−0.431015
$761$	46.3591	1.68052	0.840259	−	0.542186i	$-0.182403\pi$
$761$	46.3591	1.68052	0.840259	+	0.542186i	$0.182403\pi$
$762$	−48.3376	−1.75109
$763$	41.5814	1.50535
$764$	−14.4002	−0.520980
$765$	59.2638	2.14269
$766$	−23.7773	−0.859109
$767$	26.0427	0.940346
$768$	−2.99129	−0.107939
$769$	−35.0543	−1.26409	−0.632045	−	0.774932i	$-0.717784\pi$
$769$	−35.0543	−1.26409	−0.632045	+	0.774932i	$0.717784\pi$
$770$	22.5123	0.811289
$771$	−3.79600	−0.136710
$772$	14.5424	0.523392
$773$	−3.68746	−0.132629	−0.0663144	−	0.997799i	$-0.521124\pi$
$773$	−3.68746	−0.132629	−0.0663144	+	0.997799i	$0.521124\pi$
$774$	51.7666	1.86071
$775$	−43.1959	−1.55164
$776$	5.62501	0.201926
$777$	49.0938	1.76123
$778$	3.90711	0.140077
$779$	29.4232	1.05419
$780$	54.4926	1.95115
$781$	18.5482	0.663707
$782$	0.672794	0.0240591
$783$	−30.9552	−1.10625
$784$	3.63007	0.129645
$785$	−68.3880	−2.44087
$786$	−30.8258	−1.09952
$787$	−32.1588	−1.14634	−0.573169	−	0.819437i	$-0.694286\pi$
$787$	−32.1588	−1.14634	−0.573169	+	0.819437i	$0.694286\pi$
$788$	2.41084	0.0858826
$789$	66.7896	2.37777
$790$	32.4499	1.15451
$791$	−7.66964	−0.272701
$792$	11.5802	0.411485
$793$	−58.4884	−2.07698
$794$	−18.1664	−0.644702
$795$	−44.1730	−1.56665
$796$	−1.82763	−0.0647788
$797$	−1.50337	−0.0532522	−0.0266261	−	0.999645i	$-0.508476\pi$
$797$	−1.50337	−0.0532522	−0.0266261	+	0.999645i	$0.508476\pi$
$798$	32.6762	1.15672
$799$	24.1544	0.854523
$800$	7.57737	0.267901
$801$	31.0146	1.09585
$802$	−11.1573	−0.393980
$803$	−23.8277	−0.840862
$804$	−44.0392	−1.55314
$805$	2.76891	0.0975914
$806$	29.2824	1.03143
$807$	−88.6832	−3.12180
$808$	−13.6203	−0.479160
$809$	31.6261	1.11191	0.555957	−	0.831211i	$-0.312352\pi$
$809$	31.6261	1.11191	0.555957	+	0.831211i	$0.312352\pi$
$810$	30.2626	1.06332
$811$	−11.9986	−0.421327	−0.210664	−	0.977559i	$-0.567562\pi$
$811$	−11.9986	−0.421327	−0.210664	+	0.977559i	$0.567562\pi$
$812$	11.4457	0.401664
$813$	−92.8505	−3.25641
$814$	−9.80071	−0.343515
$815$	1.98767	0.0696251
$816$	−8.40417	−0.294205
$817$	−29.1605	−1.02020
$818$	−18.6293	−0.651359
$819$	−99.6117	−3.48071
$820$	−31.1444	−1.08761
$821$	−16.0598	−0.560492	−0.280246	−	0.959928i	$-0.590416\pi$
$821$	−16.0598	−0.560492	−0.280246	+	0.959928i	$0.590416\pi$
$822$	−35.0010	−1.22080
$823$	40.8928	1.42543	0.712717	−	0.701452i	$-0.247464\pi$
$823$	40.8928	1.42543	0.712717	+	0.701452i	$0.247464\pi$
$824$	−1.74578	−0.0608171
$825$	−44.1301	−1.53641
$826$	−16.5299	−0.575148
$827$	−15.7939	−0.549207	−0.274604	−	0.961558i	$-0.588547\pi$
$827$	−15.7939	−0.549207	−0.274604	+	0.961558i	$0.588547\pi$
$828$	1.42431	0.0494982
$829$	28.8890	1.00336	0.501678	−	0.865054i	$-0.332716\pi$
$829$	28.8890	1.00336	0.501678	+	0.865054i	$0.332716\pi$
$830$	44.3547	1.53957
$831$	−46.2202	−1.60336
$832$	−5.13670	−0.178083
$833$	10.1988	0.353368
$834$	−31.4248	−1.08815
$835$	−32.3869	−1.12080
$836$	−6.52322	−0.225610
$837$	50.2672	1.73749
$838$	3.38193	0.116827
$839$	−48.2691	−1.66643	−0.833217	−	0.552945i	$-0.813504\pi$
$839$	−48.2691	−1.66643	−0.833217	+	0.552945i	$0.813504\pi$
$840$	−34.5877	−1.19339
$841$	−16.6762	−0.575040
$842$	0.848449	0.0292395
$843$	12.1859	0.419704
$844$	−8.76677	−0.301765
$845$	47.4717	1.63307
$846$	51.1352	1.75806
$847$	−23.5052	−0.807647
$848$	4.16392	0.142990
$849$	−46.6962	−1.60261
$850$	21.2890	0.730205
$851$	−1.20544	−0.0413220
$852$	−28.4973	−0.976300
$853$	9.04863	0.309819	0.154910	−	0.987929i	$-0.450491\pi$
$853$	9.04863	0.309819	0.154910	+	0.987929i	$0.450491\pi$
$854$	37.1239	1.27035
$855$	−70.6737	−2.41699
$856$	9.76030	0.333600
$857$	−48.7387	−1.66488	−0.832441	−	0.554113i	$-0.813057\pi$
$857$	−48.7387	−1.66488	−0.832441	+	0.554113i	$0.813057\pi$
$858$	29.9158	1.02131
$859$	34.1451	1.16501	0.582507	−	0.812826i	$-0.302072\pi$
$859$	34.1451	1.16501	0.582507	+	0.812826i	$0.302072\pi$
$860$	30.8664	1.05254
$861$	85.6469	2.91884
$862$	4.79599	0.163352
$863$	6.69887	0.228032	0.114016	−	0.993479i	$-0.463628\pi$
$863$	6.69887	0.228032	0.114016	+	0.993479i	$0.463628\pi$
$864$	−8.81782	−0.299988
$865$	−45.5163	−1.54760
$866$	−28.7837	−0.978109
$867$	27.2401	0.925122
$868$	−18.5862	−0.630858
$869$	17.8146	0.604319
$870$	−37.2415	−1.26260
$871$	−75.6249	−2.56245
$872$	12.7535	0.431890
$873$	33.4566	1.13233
$874$	−0.802326	−0.0271391
$875$	29.8017	1.00748
$876$	36.6087	1.23689
$877$	14.2242	0.480317	0.240158	−	0.970734i	$-0.422801\pi$
$877$	14.2242	0.480317	0.240158	+	0.970734i	$0.422801\pi$
$878$	6.84453	0.230992
$879$	67.8633	2.28897
$880$	6.90483	0.232762
$881$	8.27131	0.278668	0.139334	−	0.990245i	$-0.455504\pi$
$881$	8.27131	0.278668	0.139334	+	0.990245i	$0.455504\pi$
$882$	21.5910	0.727007
$883$	−25.1185	−0.845304	−0.422652	−	0.906292i	$-0.638901\pi$
$883$	−25.1185	−0.845304	−0.422652	+	0.906292i	$0.638901\pi$
$884$	−14.4318	−0.485393
$885$	53.7843	1.80794
$886$	−22.0743	−0.741600
$887$	−6.34877	−0.213171	−0.106585	−	0.994304i	$-0.533992\pi$
$887$	−6.34877	−0.213171	−0.106585	+	0.994304i	$0.533992\pi$
$888$	15.0577	0.505303
$889$	52.6859	1.76703
$890$	18.4928	0.619880
$891$	16.6138	0.556583
$892$	−12.1631	−0.407251
$893$	−28.8048	−0.963917
$894$	5.87914	0.196628
$895$	−0.423874	−0.0141685
$896$	3.26038	0.108922
$897$	3.67950	0.122855
$898$	−2.49312	−0.0831965
$899$	−20.0123	−0.667447
$900$	45.0689	1.50230
$901$	11.6987	0.389741
$902$	−17.0979	−0.569298
$903$	−84.8825	−2.82471
$904$	−2.35238	−0.0782389
$905$	−42.0122	−1.39653
$906$	56.6902	1.88341
$907$	4.37548	0.145285	0.0726427	−	0.997358i	$-0.476857\pi$
$907$	4.37548	0.145285	0.0726427	+	0.997358i	$0.476857\pi$
$908$	−27.5228	−0.913377
$909$	−81.0111	−2.68697
$910$	−59.3946	−1.96891
$911$	43.6404	1.44587	0.722935	−	0.690916i	$-0.242792\pi$
$911$	43.6404	1.44587	0.722935	+	0.690916i	$0.242792\pi$
$912$	10.0222	0.331868
$913$	24.3502	0.805873
$914$	−1.07535	−0.0355694
$915$	−120.792	−3.99327
$916$	29.3803	0.970754
$917$	33.5988	1.10953
$918$	−24.7740	−0.817665
$919$	−18.5174	−0.610834	−0.305417	−	0.952219i	$-0.598796\pi$
$919$	−18.5174	−0.610834	−0.305417	+	0.952219i	$0.598796\pi$
$920$	0.849261	0.0279993
$921$	−49.3274	−1.62539
$922$	0.163451	0.00538297
$923$	−48.9360	−1.61075
$924$	−18.9882	−0.624667
$925$	−38.1433	−1.25414
$926$	−30.7175	−1.00944
$927$	−10.3836	−0.341042
$928$	3.51053	0.115239
$929$	−15.1237	−0.496193	−0.248097	−	0.968735i	$-0.579805\pi$
$929$	−15.1237	−0.496193	−0.248097	+	0.968735i	$0.579805\pi$
$930$	60.4752	1.98306
$931$	−12.1624	−0.398606
$932$	−2.51176	−0.0822753
$933$	−12.9435	−0.423750
$934$	19.3530	0.633248
$935$	19.3994	0.634429
$936$	−30.5522	−0.998630
$937$	35.9306	1.17380	0.586901	−	0.809659i	$-0.300348\pi$
$937$	35.9306	1.17380	0.586901	+	0.809659i	$0.300348\pi$
$938$	48.0008	1.56728
$939$	−49.5521	−1.61707
$940$	30.4899	0.994471
$941$	−28.8562	−0.940685	−0.470343	−	0.882484i	$-0.655870\pi$
$941$	−28.8562	−0.940685	−0.470343	+	0.882484i	$0.655870\pi$
$942$	57.6824	1.87940
$943$	−2.10296	−0.0684819
$944$	−5.06992	−0.165012
$945$	−101.959	−3.31672
$946$	16.9453	0.550939
$947$	−12.0048	−0.390103	−0.195052	−	0.980793i	$-0.562487\pi$
$947$	−12.0048	−0.390103	−0.195052	+	0.980793i	$0.562487\pi$
$948$	−27.3701	−0.888941
$949$	62.8650	2.04068
$950$	−25.3877	−0.823685
$951$	−5.96273	−0.193355
$952$	9.16017	0.296883
$953$	−36.0919	−1.16913	−0.584566	−	0.811346i	$-0.698735\pi$
$953$	−36.0919	−1.16913	−0.584566	+	0.811346i	$0.698735\pi$
$954$	24.7663	0.801838
$955$	−51.0697	−1.65258
$956$	7.51308	0.242990
$957$	−20.4451	−0.660897
$958$	34.5486	1.11621
$959$	38.1496	1.23192
$960$	−10.6085	−0.342388
$961$	1.49727	0.0482991
$962$	25.8573	0.833674
$963$	58.0526	1.87072
$964$	8.13479	0.262004
$965$	51.5740	1.66023
$966$	−2.33546	−0.0751423
$967$	−12.9600	−0.416766	−0.208383	−	0.978047i	$-0.566820\pi$
$967$	−12.9600	−0.416766	−0.208383	+	0.978047i	$0.566820\pi$
$968$	−7.20933	−0.231717
$969$	28.1578	0.904559
$970$	19.9489	0.640520
$971$	−24.2329	−0.777672	−0.388836	−	0.921307i	$-0.627123\pi$
$971$	−24.2329	−0.777672	−0.388836	+	0.921307i	$0.627123\pi$
$972$	0.928210	0.0297724
$973$	34.2516	1.09806
$974$	1.92363	0.0616369
$975$	116.429	3.72871
$976$	11.3864	0.364469
$977$	59.0321	1.88860	0.944302	−	0.329081i	$-0.106739\pi$
$977$	59.0321	1.88860	0.944302	+	0.329081i	$0.106739\pi$
$978$	−1.67652	−0.0536092
$979$	10.1523	0.324469
$980$	12.8739	0.411241
$981$	75.8559	2.42189
$982$	12.2167	0.389850
$983$	−19.9101	−0.635033	−0.317516	−	0.948253i	$-0.602849\pi$
$983$	−19.9101	−0.635033	−0.317516	+	0.948253i	$0.602849\pi$
$984$	26.2690	0.837426
$985$	8.54994	0.272424
$986$	9.86300	0.314102
$987$	−83.8471	−2.66888
$988$	17.2103	0.547532
$989$	2.08419	0.0662734
$990$	41.0687	1.30525
$991$	−23.9945	−0.762209	−0.381105	−	0.924532i	$-0.624456\pi$
$991$	−23.9945	−0.762209	−0.381105	+	0.924532i	$0.624456\pi$
$992$	−5.70064	−0.180995
$993$	−105.722	−3.35498
$994$	31.0608	0.985188
$995$	−6.48163	−0.205482
$996$	−37.4113	−1.18542
$997$	30.2370	0.957614	0.478807	−	0.877920i	$-0.341069\pi$
$997$	30.2370	0.957614	0.478807	+	0.877920i	$0.341069\pi$
$998$	13.0656	0.413586
$999$	44.3875	1.40436

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8014.2.a.d.1.5	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8014.2.a.d.1.5	✓	88	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:	\( N \)	\(=\)	\( 8014 = 2 \cdot 4007 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.99129 q^{3} +1.00000 q^{4} +3.54646 q^{5} -2.99129 q^{6} +3.26038 q^{7} +1.00000 q^{8} +5.94783 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.99129 q^{3} +1.00000 q^{4} +3.54646 q^{5} -2.99129 q^{6} +3.26038 q^{7} +1.00000 q^{8} +5.94783 q^{9} +3.54646 q^{10} +1.94696 q^{11} -2.99129 q^{12} -5.13670 q^{13} +3.26038 q^{14} -10.6085 q^{15} +1.00000 q^{16} +2.80954 q^{17} +5.94783 q^{18} -3.35046 q^{19} +3.54646 q^{20} -9.75274 q^{21} +1.94696 q^{22} +0.239467 q^{23} -2.99129 q^{24} +7.57737 q^{25} -5.13670 q^{26} -8.81782 q^{27} +3.26038 q^{28} +3.51053 q^{29} -10.6085 q^{30} -5.70064 q^{31} +1.00000 q^{32} -5.82394 q^{33} +2.80954 q^{34} +11.5628 q^{35} +5.94783 q^{36} -5.03384 q^{37} -3.35046 q^{38} +15.3654 q^{39} +3.54646 q^{40} -8.78183 q^{41} -9.75274 q^{42} +8.70345 q^{43} +1.94696 q^{44} +21.0937 q^{45} +0.239467 q^{46} +8.59728 q^{47} -2.99129 q^{48} +3.63007 q^{49} +7.57737 q^{50} -8.40417 q^{51} -5.13670 q^{52} +4.16392 q^{53} -8.81782 q^{54} +6.90483 q^{55} +3.26038 q^{56} +10.0222 q^{57} +3.51053 q^{58} -5.06992 q^{59} -10.6085 q^{60} +11.3864 q^{61} -5.70064 q^{62} +19.3922 q^{63} +1.00000 q^{64} -18.2171 q^{65} -5.82394 q^{66} +14.7225 q^{67} +2.80954 q^{68} -0.716317 q^{69} +11.5628 q^{70} +9.52674 q^{71} +5.94783 q^{72} -12.2384 q^{73} -5.03384 q^{74} -22.6661 q^{75} -3.35046 q^{76} +6.34784 q^{77} +15.3654 q^{78} +9.14994 q^{79} +3.54646 q^{80} +8.53318 q^{81} -8.78183 q^{82} +12.5067 q^{83} -9.75274 q^{84} +9.96393 q^{85} +8.70345 q^{86} -10.5010 q^{87} +1.94696 q^{88} +5.21444 q^{89} +21.0937 q^{90} -16.7476 q^{91} +0.239467 q^{92} +17.0523 q^{93} +8.59728 q^{94} -11.8823 q^{95} -2.99129 q^{96} +5.62501 q^{97} +3.63007 q^{98} +11.5802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 88 q^{2} + 22 q^{3} + 88 q^{4} + 25 q^{5} + 22 q^{6} + 33 q^{7} + 88 q^{8} + 108 q^{9}+O(q^{10}) \) 88 * q + 88 * q^2 + 22 * q^3 + 88 * q^4 + 25 * q^5 + 22 * q^6 + 33 * q^7 + 88 * q^8 + 108 * q^9	\( 88 q + 88 q^{2} + 22 q^{3} + 88 q^{4} + 25 q^{5} + 22 q^{6} + 33 q^{7} + 88 q^{8} + 108 q^{9} + 25 q^{10} + 70 q^{11} + 22 q^{12} + 31 q^{13} + 33 q^{14} + 47 q^{15} + 88 q^{16} + 19 q^{17} + 108 q^{18} + 33 q^{19} + 25 q^{20} + 48 q^{21} + 70 q^{22} + 77 q^{23} + 22 q^{24} + 109 q^{25} + 31 q^{26} + 88 q^{27} + 33 q^{28} + 83 q^{29} + 47 q^{30} + 51 q^{31} + 88 q^{32} + 30 q^{33} + 19 q^{34} + 40 q^{35} + 108 q^{36} + 45 q^{37} + 33 q^{38} + 82 q^{39} + 25 q^{40} + 35 q^{41} + 48 q^{42} + 78 q^{43} + 70 q^{44} + 37 q^{45} + 77 q^{46} + 59 q^{47} + 22 q^{48} + 103 q^{49} + 109 q^{50} + 21 q^{51} + 31 q^{52} + 58 q^{53} + 88 q^{54} + 35 q^{55} + 33 q^{56} - 16 q^{57} + 83 q^{58} + 54 q^{59} + 47 q^{60} + 18 q^{61} + 51 q^{62} + 47 q^{63} + 88 q^{64} + 34 q^{65} + 30 q^{66} + 88 q^{67} + 19 q^{68} + 62 q^{69} + 40 q^{70} + 139 q^{71} + 108 q^{72} - 6 q^{73} + 45 q^{74} + 45 q^{75} + 33 q^{76} + 37 q^{77} + 82 q^{78} + 94 q^{79} + 25 q^{80} + 112 q^{81} + 35 q^{82} + 58 q^{83} + 48 q^{84} + 83 q^{85} + 78 q^{86} + 21 q^{87} + 70 q^{88} + 99 q^{89} + 37 q^{90} + 53 q^{91} + 77 q^{92} + 57 q^{93} + 59 q^{94} + 92 q^{95} + 22 q^{96} + 16 q^{97} + 103 q^{98} + 150 q^{99}+O(q^{100}) \) 88 * q + 88 * q^2 + 22 * q^3 + 88 * q^4 + 25 * q^5 + 22 * q^6 + 33 * q^7 + 88 * q^8 + 108 * q^9 + 25 * q^10 + 70 * q^11 + 22 * q^12 + 31 * q^13 + 33 * q^14 + 47 * q^15 + 88 * q^16 + 19 * q^17 + 108 * q^18 + 33 * q^19 + 25 * q^20 + 48 * q^21 + 70 * q^22 + 77 * q^23 + 22 * q^24 + 109 * q^25 + 31 * q^26 + 88 * q^27 + 33 * q^28 + 83 * q^29 + 47 * q^30 + 51 * q^31 + 88 * q^32 + 30 * q^33 + 19 * q^34 + 40 * q^35 + 108 * q^36 + 45 * q^37 + 33 * q^38 + 82 * q^39 + 25 * q^40 + 35 * q^41 + 48 * q^42 + 78 * q^43 + 70 * q^44 + 37 * q^45 + 77 * q^46 + 59 * q^47 + 22 * q^48 + 103 * q^49 + 109 * q^50 + 21 * q^51 + 31 * q^52 + 58 * q^53 + 88 * q^54 + 35 * q^55 + 33 * q^56 - 16 * q^57 + 83 * q^58 + 54 * q^59 + 47 * q^60 + 18 * q^61 + 51 * q^62 + 47 * q^63 + 88 * q^64 + 34 * q^65 + 30 * q^66 + 88 * q^67 + 19 * q^68 + 62 * q^69 + 40 * q^70 + 139 * q^71 + 108 * q^72 - 6 * q^73 + 45 * q^74 + 45 * q^75 + 33 * q^76 + 37 * q^77 + 82 * q^78 + 94 * q^79 + 25 * q^80 + 112 * q^81 + 35 * q^82 + 58 * q^83 + 48 * q^84 + 83 * q^85 + 78 * q^86 + 21 * q^87 + 70 * q^88 + 99 * q^89 + 37 * q^90 + 53 * q^91 + 77 * q^92 + 57 * q^93 + 59 * q^94 + 92 * q^95 + 22 * q^96 + 16 * q^97 + 103 * q^98 + 150 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more