LMFDB - Embedded newform 8002.2.a.d.1.27

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, 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("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.27
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.23475 q^{3} +1.00000 q^{4} -3.13157 q^{5} -1.23475 q^{6} -4.29517 q^{7} +1.00000 q^{8} -1.47540 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.23475 q^{3} +1.00000 q^{4} -3.13157 q^{5} -1.23475 q^{6} -4.29517 q^{7} +1.00000 q^{8} -1.47540 q^{9} -3.13157 q^{10} +2.57262 q^{11} -1.23475 q^{12} +3.94451 q^{13} -4.29517 q^{14} +3.86670 q^{15} +1.00000 q^{16} +2.45228 q^{17} -1.47540 q^{18} -5.68621 q^{19} -3.13157 q^{20} +5.30346 q^{21} +2.57262 q^{22} -2.06750 q^{23} -1.23475 q^{24} +4.80675 q^{25} +3.94451 q^{26} +5.52599 q^{27} -4.29517 q^{28} +8.50176 q^{29} +3.86670 q^{30} -0.553096 q^{31} +1.00000 q^{32} -3.17654 q^{33} +2.45228 q^{34} +13.4506 q^{35} -1.47540 q^{36} -3.11217 q^{37} -5.68621 q^{38} -4.87048 q^{39} -3.13157 q^{40} -6.76637 q^{41} +5.30346 q^{42} +7.21658 q^{43} +2.57262 q^{44} +4.62032 q^{45} -2.06750 q^{46} -4.95110 q^{47} -1.23475 q^{48} +11.4485 q^{49} +4.80675 q^{50} -3.02795 q^{51} +3.94451 q^{52} +13.2235 q^{53} +5.52599 q^{54} -8.05636 q^{55} -4.29517 q^{56} +7.02104 q^{57} +8.50176 q^{58} +2.61991 q^{59} +3.86670 q^{60} -3.35993 q^{61} -0.553096 q^{62} +6.33709 q^{63} +1.00000 q^{64} -12.3525 q^{65} -3.17654 q^{66} +8.67132 q^{67} +2.45228 q^{68} +2.55284 q^{69} +13.4506 q^{70} +1.98721 q^{71} -1.47540 q^{72} -15.0388 q^{73} -3.11217 q^{74} -5.93513 q^{75} -5.68621 q^{76} -11.0499 q^{77} -4.87048 q^{78} +15.9045 q^{79} -3.13157 q^{80} -2.39701 q^{81} -6.76637 q^{82} +6.85849 q^{83} +5.30346 q^{84} -7.67951 q^{85} +7.21658 q^{86} -10.4975 q^{87} +2.57262 q^{88} +0.549439 q^{89} +4.62032 q^{90} -16.9424 q^{91} -2.06750 q^{92} +0.682934 q^{93} -4.95110 q^{94} +17.8068 q^{95} -1.23475 q^{96} -7.92832 q^{97} +11.4485 q^{98} -3.79564 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * 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
$3$	−1.23475	−0.712882	−0.356441	−	0.934318i	$-0.616010\pi$
$3$	−1.23475	−0.712882	−0.356441	+	0.934318i	$0.616010\pi$
$4$	1.00000	0.500000
$5$	−3.13157	−1.40048	−0.700241	−	0.713906i	$-0.746924\pi$
$5$	−3.13157	−1.40048	−0.700241	+	0.713906i	$0.746924\pi$
$6$	−1.23475	−0.504084
$7$	−4.29517	−1.62342	−0.811711	−	0.584059i	$-0.801464\pi$
$7$	−4.29517	−1.62342	−0.811711	+	0.584059i	$0.801464\pi$
$8$	1.00000	0.353553
$9$	−1.47540	−0.491799
$10$	−3.13157	−0.990290
$11$	2.57262	0.775676	0.387838	−	0.921728i	$-0.373222\pi$
$11$	2.57262	0.775676	0.387838	+	0.921728i	$0.373222\pi$
$12$	−1.23475	−0.356441
$13$	3.94451	1.09401	0.547005	−	0.837129i	$-0.315768\pi$
$13$	3.94451	1.09401	0.547005	+	0.837129i	$0.315768\pi$
$14$	−4.29517	−1.14793
$15$	3.86670	0.998378
$16$	1.00000	0.250000
$17$	2.45228	0.594766	0.297383	−	0.954758i	$-0.403886\pi$
$17$	2.45228	0.594766	0.297383	+	0.954758i	$0.403886\pi$
$18$	−1.47540	−0.347755
$19$	−5.68621	−1.30451	−0.652254	−	0.758001i	$-0.726176\pi$
$19$	−5.68621	−1.30451	−0.652254	+	0.758001i	$0.726176\pi$
$20$	−3.13157	−0.700241
$21$	5.30346	1.15731
$22$	2.57262	0.548485
$23$	−2.06750	−0.431103	−0.215551	−	0.976493i	$-0.569155\pi$
$23$	−2.06750	−0.431103	−0.215551	+	0.976493i	$0.569155\pi$
$24$	−1.23475	−0.252042
$25$	4.80675	0.961350
$26$	3.94451	0.773582
$27$	5.52599	1.06348
$28$	−4.29517	−0.811711
$29$	8.50176	1.57874	0.789369	−	0.613919i	$-0.210408\pi$
$29$	8.50176	1.57874	0.789369	+	0.613919i	$0.210408\pi$
$30$	3.86670	0.705960
$31$	−0.553096	−0.0993390	−0.0496695	−	0.998766i	$-0.515817\pi$
$31$	−0.553096	−0.0993390	−0.0496695	+	0.998766i	$0.515817\pi$
$32$	1.00000	0.176777
$33$	−3.17654	−0.552965
$34$	2.45228	0.420563
$35$	13.4506	2.27357
$36$	−1.47540	−0.245900
$37$	−3.11217	−0.511637	−0.255819	−	0.966725i	$-0.582345\pi$
$37$	−3.11217	−0.511637	−0.255819	+	0.966725i	$0.582345\pi$
$38$	−5.68621	−0.922426
$39$	−4.87048	−0.779901
$40$	−3.13157	−0.495145
$41$	−6.76637	−1.05673	−0.528365	−	0.849017i	$-0.677195\pi$
$41$	−6.76637	−1.05673	−0.528365	+	0.849017i	$0.677195\pi$
$42$	5.30346	0.818341
$43$	7.21658	1.10052	0.550259	−	0.834994i	$-0.314529\pi$
$43$	7.21658	1.10052	0.550259	+	0.834994i	$0.314529\pi$
$44$	2.57262	0.387838
$45$	4.62032	0.688756
$46$	−2.06750	−0.304836
$47$	−4.95110	−0.722191	−0.361096	−	0.932529i	$-0.617597\pi$
$47$	−4.95110	−0.722191	−0.361096	+	0.932529i	$0.617597\pi$
$48$	−1.23475	−0.178220
$49$	11.4485	1.63550
$50$	4.80675	0.679777
$51$	−3.02795	−0.423998
$52$	3.94451	0.547005
$53$	13.2235	1.81638	0.908192	−	0.418555i	$-0.137463\pi$
$53$	13.2235	1.81638	0.908192	+	0.418555i	$0.137463\pi$
$54$	5.52599	0.751992
$55$	−8.05636	−1.08632
$56$	−4.29517	−0.573967
$57$	7.02104	0.929960
$58$	8.50176	1.11634
$59$	2.61991	0.341083	0.170542	−	0.985350i	$-0.445448\pi$
$59$	2.61991	0.341083	0.170542	+	0.985350i	$0.445448\pi$
$60$	3.86670	0.499189
$61$	−3.35993	−0.430195	−0.215097	−	0.976593i	$-0.569007\pi$
$61$	−3.35993	−0.430195	−0.215097	+	0.976593i	$0.569007\pi$
$62$	−0.553096	−0.0702433
$63$	6.33709	0.798398
$64$	1.00000	0.125000
$65$	−12.3525	−1.53214
$66$	−3.17654	−0.391005
$67$	8.67132	1.05937	0.529685	−	0.848194i	$-0.322310\pi$
$67$	8.67132	1.05937	0.529685	+	0.848194i	$0.322310\pi$
$68$	2.45228	0.297383
$69$	2.55284	0.307325
$70$	13.4506	1.60766
$71$	1.98721	0.235838	0.117919	−	0.993023i	$-0.462378\pi$
$71$	1.98721	0.235838	0.117919	+	0.993023i	$0.462378\pi$
$72$	−1.47540	−0.173877
$73$	−15.0388	−1.76016	−0.880081	−	0.474823i	$-0.842512\pi$
$73$	−15.0388	−1.76016	−0.880081	+	0.474823i	$0.842512\pi$
$74$	−3.11217	−0.361782
$75$	−5.93513	−0.685329
$76$	−5.68621	−0.652254
$77$	−11.0499	−1.25925
$78$	−4.87048	−0.551473
$79$	15.9045	1.78940	0.894698	−	0.446671i	$-0.147390\pi$
$79$	15.9045	1.78940	0.894698	+	0.446671i	$0.147390\pi$
$80$	−3.13157	−0.350121
$81$	−2.39701	−0.266334
$82$	−6.76637	−0.747221
$83$	6.85849	0.752817	0.376408	−	0.926454i	$-0.377159\pi$
$83$	6.85849	0.752817	0.376408	+	0.926454i	$0.377159\pi$
$84$	5.30346	0.578654
$85$	−7.67951	−0.832960
$86$	7.21658	0.778184
$87$	−10.4975	−1.12545
$88$	2.57262	0.274243
$89$	0.549439	0.0582404	0.0291202	−	0.999576i	$-0.490729\pi$
$89$	0.549439	0.0582404	0.0291202	+	0.999576i	$0.490729\pi$
$90$	4.62032	0.487024
$91$	−16.9424	−1.77604
$92$	−2.06750	−0.215551
$93$	0.682934	0.0708170
$94$	−4.95110	−0.510666
$95$	17.8068	1.82694
$96$	−1.23475	−0.126021
$97$	−7.92832	−0.804999	−0.402500	−	0.915420i	$-0.631858\pi$
$97$	−7.92832	−0.804999	−0.402500	+	0.915420i	$0.631858\pi$
$98$	11.4485	1.15647
$99$	−3.79564	−0.381477
$100$	4.80675	0.480675
$101$	−1.24407	−0.123790	−0.0618950	−	0.998083i	$-0.519714\pi$
$101$	−1.24407	−0.123790	−0.0618950	+	0.998083i	$0.519714\pi$
$102$	−3.02795	−0.299812
$103$	−3.63007	−0.357682	−0.178841	−	0.983878i	$-0.557235\pi$
$103$	−3.63007	−0.357682	−0.178841	+	0.983878i	$0.557235\pi$
$104$	3.94451	0.386791
$105$	−16.6082	−1.62079
$106$	13.2235	1.28438
$107$	−1.97517	−0.190947	−0.0954735	−	0.995432i	$-0.530437\pi$
$107$	−1.97517	−0.190947	−0.0954735	+	0.995432i	$0.530437\pi$
$108$	5.52599	0.531738
$109$	−2.06769	−0.198049	−0.0990243	−	0.995085i	$-0.531572\pi$
$109$	−2.06769	−0.198049	−0.0990243	+	0.995085i	$0.531572\pi$
$110$	−8.05636	−0.768144
$111$	3.84274	0.364737
$112$	−4.29517	−0.405856
$113$	−20.0070	−1.88210	−0.941051	−	0.338266i	$-0.890160\pi$
$113$	−20.0070	−1.88210	−0.941051	+	0.338266i	$0.890160\pi$
$114$	7.02104	0.657581
$115$	6.47452	0.603752
$116$	8.50176	0.789369
$117$	−5.81972	−0.538034
$118$	2.61991	0.241182
$119$	−10.5330	−0.965557
$120$	3.86670	0.352980
$121$	−4.38160	−0.398327
$122$	−3.35993	−0.304194
$123$	8.35476	0.753324
$124$	−0.553096	−0.0496695
$125$	0.605174	0.0541284
$126$	6.33709	0.564553
$127$	2.65776	0.235838	0.117919	−	0.993023i	$-0.462378\pi$
$127$	2.65776	0.235838	0.117919	+	0.993023i	$0.462378\pi$
$128$	1.00000	0.0883883
$129$	−8.91066	−0.784540
$130$	−12.3525	−1.08339
$131$	11.7648	1.02789	0.513946	−	0.857822i	$-0.328183\pi$
$131$	11.7648	1.02789	0.513946	+	0.857822i	$0.328183\pi$
$132$	−3.17654	−0.276483
$133$	24.4233	2.11777
$134$	8.67132	0.749088
$135$	−17.3050	−1.48938
$136$	2.45228	0.210282
$137$	−6.47065	−0.552825	−0.276413	−	0.961039i	$-0.589146\pi$
$137$	−6.47065	−0.552825	−0.276413	+	0.961039i	$0.589146\pi$
$138$	2.55284	0.217312
$139$	9.93333	0.842534	0.421267	−	0.906937i	$-0.361585\pi$
$139$	9.93333	0.842534	0.421267	+	0.906937i	$0.361585\pi$
$140$	13.4506	1.13679
$141$	6.11335	0.514837
$142$	1.98721	0.166763
$143$	10.1477	0.848597
$144$	−1.47540	−0.122950
$145$	−26.6239	−2.21099
$146$	−15.0388	−1.24462
$147$	−14.1360	−1.16592
$148$	−3.11217	−0.255819
$149$	2.33998	0.191699	0.0958493	−	0.995396i	$-0.469443\pi$
$149$	2.33998	0.191699	0.0958493	+	0.995396i	$0.469443\pi$
$150$	−5.93513	−0.484601
$151$	0.877446	0.0714055	0.0357028	−	0.999362i	$-0.488633\pi$
$151$	0.877446	0.0714055	0.0357028	+	0.999362i	$0.488633\pi$
$152$	−5.68621	−0.461213
$153$	−3.61809	−0.292506
$154$	−11.0499	−0.890424
$155$	1.73206	0.139122
$156$	−4.87048	−0.389950
$157$	−15.0337	−1.19982	−0.599912	−	0.800066i	$-0.704798\pi$
$157$	−15.0337	−1.19982	−0.599912	+	0.800066i	$0.704798\pi$
$158$	15.9045	1.26529
$159$	−16.3277	−1.29487
$160$	−3.13157	−0.247573
$161$	8.88025	0.699862
$162$	−2.39701	−0.188327
$163$	6.36762	0.498751	0.249375	−	0.968407i	$-0.419775\pi$
$163$	6.36762	0.498751	0.249375	+	0.968407i	$0.419775\pi$
$164$	−6.76637	−0.528365
$165$	9.94758	0.774418
$166$	6.85849	0.532322
$167$	2.64472	0.204655	0.102327	−	0.994751i	$-0.467371\pi$
$167$	2.64472	0.204655	0.102327	+	0.994751i	$0.467371\pi$
$168$	5.30346	0.409170
$169$	2.55918	0.196860
$170$	−7.67951	−0.588991
$171$	8.38943	0.641556
$172$	7.21658	0.550259
$173$	−10.3320	−0.785528	−0.392764	−	0.919639i	$-0.628481\pi$
$173$	−10.3320	−0.785528	−0.392764	+	0.919639i	$0.628481\pi$
$174$	−10.4975	−0.795816
$175$	−20.6458	−1.56068
$176$	2.57262	0.193919
$177$	−3.23493	−0.243152
$178$	0.549439	0.0411822
$179$	−0.288516	−0.0215647	−0.0107823	−	0.999942i	$-0.503432\pi$
$179$	−0.288516	−0.0215647	−0.0107823	+	0.999942i	$0.503432\pi$
$180$	4.62032	0.344378
$181$	−13.5320	−1.00583	−0.502913	−	0.864337i	$-0.667739\pi$
$181$	−13.5320	−1.00583	−0.502913	+	0.864337i	$0.667739\pi$
$182$	−16.9424	−1.25585
$183$	4.14866	0.306678
$184$	−2.06750	−0.152418
$185$	9.74598	0.716539
$186$	0.682934	0.0500752
$187$	6.30881	0.461346
$188$	−4.95110	−0.361096
$189$	−23.7351	−1.72647
$190$	17.8068	1.29184
$191$	−22.6088	−1.63591	−0.817956	−	0.575281i	$-0.804893\pi$
$191$	−22.6088	−1.63591	−0.817956	+	0.575281i	$0.804893\pi$
$192$	−1.23475	−0.0891102
$193$	1.51015	0.108703	0.0543514	−	0.998522i	$-0.482691\pi$
$193$	1.51015	0.108703	0.0543514	+	0.998522i	$0.482691\pi$
$194$	−7.92832	−0.569220
$195$	15.2523	1.09224
$196$	11.4485	0.817751
$197$	−11.6428	−0.829514	−0.414757	−	0.909932i	$-0.636133\pi$
$197$	−11.6428	−0.829514	−0.414757	+	0.909932i	$0.636133\pi$
$198$	−3.79564	−0.269745
$199$	−27.3739	−1.94048	−0.970242	−	0.242139i	$-0.922151\pi$
$199$	−27.3739	−1.94048	−0.970242	+	0.242139i	$0.922151\pi$
$200$	4.80675	0.339889
$201$	−10.7069	−0.755206
$202$	−1.24407	−0.0875327
$203$	−36.5166	−2.56296
$204$	−3.02795	−0.211999
$205$	21.1894	1.47993
$206$	−3.63007	−0.252919
$207$	3.05038	0.212016
$208$	3.94451	0.273503
$209$	−14.6285	−1.01187
$210$	−16.6082	−1.14607
$211$	−2.01112	−0.138451	−0.0692256	−	0.997601i	$-0.522053\pi$
$211$	−2.01112	−0.138451	−0.0692256	+	0.997601i	$0.522053\pi$
$212$	13.2235	0.908192
$213$	−2.45370	−0.168125
$214$	−1.97517	−0.135020
$215$	−22.5993	−1.54126
$216$	5.52599	0.375996
$217$	2.37564	0.161269
$218$	−2.06769	−0.140041
$219$	18.5692	1.25479
$220$	−8.05636	−0.543160
$221$	9.67306	0.650681
$222$	3.84274	0.257908
$223$	−21.9644	−1.47085	−0.735424	−	0.677607i	$-0.763017\pi$
$223$	−21.9644	−1.47085	−0.735424	+	0.677607i	$0.763017\pi$
$224$	−4.29517	−0.286983
$225$	−7.09187	−0.472791
$226$	−20.0070	−1.33085
$227$	22.2000	1.47346	0.736732	−	0.676185i	$-0.236368\pi$
$227$	22.2000	1.47346	0.736732	+	0.676185i	$0.236368\pi$
$228$	7.02104	0.464980
$229$	−5.68021	−0.375359	−0.187679	−	0.982230i	$-0.560097\pi$
$229$	−5.68021	−0.375359	−0.187679	+	0.982230i	$0.560097\pi$
$230$	6.47452	0.426917
$231$	13.6438	0.897696
$232$	8.50176	0.558168
$233$	−14.4748	−0.948277	−0.474138	−	0.880450i	$-0.657240\pi$
$233$	−14.4748	−0.948277	−0.474138	+	0.880450i	$0.657240\pi$
$234$	−5.81972	−0.380447
$235$	15.5047	1.01142
$236$	2.61991	0.170542
$237$	−19.6381	−1.27563
$238$	−10.5330	−0.682752
$239$	−8.95159	−0.579030	−0.289515	−	0.957173i	$-0.593494\pi$
$239$	−8.95159	−0.579030	−0.289515	+	0.957173i	$0.593494\pi$
$240$	3.86670	0.249595
$241$	−14.0989	−0.908193	−0.454097	−	0.890952i	$-0.650038\pi$
$241$	−14.0989	−0.908193	−0.454097	+	0.890952i	$0.650038\pi$
$242$	−4.38160	−0.281660
$243$	−13.6183	−0.873612
$244$	−3.35993	−0.215097
$245$	−35.8519	−2.29049
$246$	8.35476	0.532680
$247$	−22.4293	−1.42714
$248$	−0.553096	−0.0351216
$249$	−8.46850	−0.536670
$250$	0.605174	0.0382746
$251$	−0.462283	−0.0291790	−0.0145895	−	0.999894i	$-0.504644\pi$
$251$	−0.462283	−0.0291790	−0.0145895	+	0.999894i	$0.504644\pi$
$252$	6.33709	0.399199
$253$	−5.31889	−0.334396
$254$	2.65776	0.166763
$255$	9.48226	0.593802
$256$	1.00000	0.0625000
$257$	27.3141	1.70381	0.851904	−	0.523699i	$-0.175448\pi$
$257$	27.3141	1.70381	0.851904	+	0.523699i	$0.175448\pi$
$258$	−8.91066	−0.554754
$259$	13.3673	0.830604
$260$	−12.3525	−0.766071
$261$	−12.5435	−0.776422
$262$	11.7648	0.726830
$263$	1.24817	0.0769657	0.0384829	−	0.999259i	$-0.487747\pi$
$263$	1.24817	0.0769657	0.0384829	+	0.999259i	$0.487747\pi$
$264$	−3.17654	−0.195503
$265$	−41.4103	−2.54381
$266$	24.4233	1.49749
$267$	−0.678419	−0.0415186
$268$	8.67132	0.529685
$269$	22.9201	1.39746	0.698732	−	0.715383i	$-0.253748\pi$
$269$	22.9201	1.39746	0.698732	+	0.715383i	$0.253748\pi$
$270$	−17.3050	−1.05315
$271$	27.9139	1.69565	0.847823	−	0.530279i	$-0.177913\pi$
$271$	27.9139	1.69565	0.847823	+	0.530279i	$0.177913\pi$
$272$	2.45228	0.148692
$273$	20.9195	1.26611
$274$	−6.47065	−0.390906
$275$	12.3660	0.745696
$276$	2.55284	0.153663
$277$	−11.1532	−0.670131	−0.335065	−	0.942195i	$-0.608758\pi$
$277$	−11.1532	−0.670131	−0.335065	+	0.942195i	$0.608758\pi$
$278$	9.93333	0.595762
$279$	0.816037	0.0488548
$280$	13.4506	0.803830
$281$	16.8808	1.00702	0.503511	−	0.863989i	$-0.332041\pi$
$281$	16.8808	1.00702	0.503511	+	0.863989i	$0.332041\pi$
$282$	6.11335	0.364045
$283$	−7.23109	−0.429843	−0.214922	−	0.976631i	$-0.568950\pi$
$283$	−7.23109	−0.429843	−0.214922	+	0.976631i	$0.568950\pi$
$284$	1.98721	0.117919
$285$	−21.9869	−1.30239
$286$	10.1477	0.600049
$287$	29.0627	1.71552
$288$	−1.47540	−0.0869387
$289$	−10.9863	−0.646253
$290$	−26.6239	−1.56341
$291$	9.78948	0.573869
$292$	−15.0388	−0.880081
$293$	−1.88031	−0.109849	−0.0549246	−	0.998491i	$-0.517492\pi$
$293$	−1.88031	−0.109849	−0.0549246	+	0.998491i	$0.517492\pi$
$294$	−14.1360	−0.824430
$295$	−8.20444	−0.477681
$296$	−3.11217	−0.180891
$297$	14.2163	0.824913
$298$	2.33998	0.135551
$299$	−8.15526	−0.471631
$300$	−5.93513	−0.342665
$301$	−30.9965	−1.78661
$302$	0.877446	0.0504913
$303$	1.53612	0.0882476
$304$	−5.68621	−0.326127
$305$	10.5219	0.602480
$306$	−3.61809	−0.206833
$307$	−8.14649	−0.464945	−0.232472	−	0.972603i	$-0.574682\pi$
$307$	−8.14649	−0.464945	−0.232472	+	0.972603i	$0.574682\pi$
$308$	−11.0499	−0.629625
$309$	4.48222	0.254985
$310$	1.73206	0.0983744
$311$	−28.9275	−1.64033	−0.820165	−	0.572128i	$-0.806118\pi$
$311$	−28.9275	−1.64033	−0.820165	+	0.572128i	$0.806118\pi$
$312$	−4.87048	−0.275737
$313$	−8.00595	−0.452523	−0.226261	−	0.974067i	$-0.572650\pi$
$313$	−8.00595	−0.452523	−0.226261	+	0.974067i	$0.572650\pi$
$314$	−15.0337	−0.848403
$315$	−19.8451	−1.11814
$316$	15.9045	0.894698
$317$	12.9073	0.724945	0.362473	−	0.931994i	$-0.381933\pi$
$317$	12.9073	0.724945	0.362473	+	0.931994i	$0.381933\pi$
$318$	−16.3277	−0.915609
$319$	21.8719	1.22459
$320$	−3.13157	−0.175060
$321$	2.43884	0.136123
$322$	8.88025	0.494877
$323$	−13.9442	−0.775877
$324$	−2.39701	−0.133167
$325$	18.9603	1.05173
$326$	6.36762	0.352670
$327$	2.55307	0.141185
$328$	−6.76637	−0.373610
$329$	21.2658	1.17242
$330$	9.94758	0.547596
$331$	−6.63949	−0.364939	−0.182470	−	0.983211i	$-0.558409\pi$
$331$	−6.63949	−0.364939	−0.182470	+	0.983211i	$0.558409\pi$
$332$	6.85849	0.376408
$333$	4.59169	0.251623
$334$	2.64472	0.144713
$335$	−27.1549	−1.48363
$336$	5.30346	0.289327
$337$	−17.1772	−0.935701	−0.467851	−	0.883808i	$-0.654972\pi$
$337$	−17.1772	−0.935701	−0.467851	+	0.883808i	$0.654972\pi$
$338$	2.55918	0.139201
$339$	24.7036	1.34172
$340$	−7.67951	−0.416480
$341$	−1.42291	−0.0770548
$342$	8.38943	0.453648
$343$	−19.1071	−1.03169
$344$	7.21658	0.389092
$345$	−7.99439	−0.430404
$346$	−10.3320	−0.555452
$347$	−25.5023	−1.36904	−0.684519	−	0.728995i	$-0.739988\pi$
$347$	−25.5023	−1.36904	−0.684519	+	0.728995i	$0.739988\pi$
$348$	−10.4975	−0.562727
$349$	9.07639	0.485848	0.242924	−	0.970045i	$-0.421893\pi$
$349$	9.07639	0.485848	0.242924	+	0.970045i	$0.421893\pi$
$350$	−20.6458	−1.10357
$351$	21.7973	1.16346
$352$	2.57262	0.137121
$353$	4.04348	0.215213	0.107606	−	0.994194i	$-0.465681\pi$
$353$	4.04348	0.215213	0.107606	+	0.994194i	$0.465681\pi$
$354$	−3.23493	−0.171934
$355$	−6.22309	−0.330287
$356$	0.549439	0.0291202
$357$	13.0056	0.688328
$358$	−0.288516	−0.0152485
$359$	−9.33350	−0.492603	−0.246302	−	0.969193i	$-0.579215\pi$
$359$	−9.33350	−0.492603	−0.246302	+	0.969193i	$0.579215\pi$
$360$	4.62032	0.243512
$361$	13.3330	0.701739
$362$	−13.5320	−0.711227
$363$	5.41017	0.283960
$364$	−16.9424	−0.888021
$365$	47.0952	2.46508
$366$	4.14866	0.216854
$367$	33.1304	1.72939	0.864697	−	0.502294i	$-0.167510\pi$
$367$	33.1304	1.72939	0.864697	+	0.502294i	$0.167510\pi$
$368$	−2.06750	−0.107776
$369$	9.98309	0.519699
$370$	9.74598	0.506670
$371$	−56.7971	−2.94876
$372$	0.682934	0.0354085
$373$	2.24084	0.116026	0.0580132	−	0.998316i	$-0.481523\pi$
$373$	2.24084	0.116026	0.0580132	+	0.998316i	$0.481523\pi$
$374$	6.30881	0.326221
$375$	−0.747237	−0.0385872
$376$	−4.95110	−0.255333
$377$	33.5353	1.72716
$378$	−23.7351	−1.22080
$379$	−22.8624	−1.17436	−0.587182	−	0.809455i	$-0.699763\pi$
$379$	−22.8624	−1.17436	−0.587182	+	0.809455i	$0.699763\pi$
$380$	17.8068	0.913469
$381$	−3.28167	−0.168125
$382$	−22.6088	−1.15676
$383$	−23.0013	−1.17531	−0.587657	−	0.809110i	$-0.699950\pi$
$383$	−23.0013	−1.17531	−0.587657	+	0.809110i	$0.699950\pi$
$384$	−1.23475	−0.0630105
$385$	34.6035	1.76356
$386$	1.51015	0.0768645
$387$	−10.6473	−0.541234
$388$	−7.92832	−0.402500
$389$	9.03734	0.458211	0.229106	−	0.973402i	$-0.426420\pi$
$389$	9.03734	0.458211	0.229106	+	0.973402i	$0.426420\pi$
$390$	15.2523	0.772328
$391$	−5.07009	−0.256405
$392$	11.4485	0.578237
$393$	−14.5265	−0.732766
$394$	−11.6428	−0.586555
$395$	−49.8061	−2.50602
$396$	−3.79564	−0.190738
$397$	−14.5733	−0.731413	−0.365706	−	0.930730i	$-0.619173\pi$
$397$	−14.5733	−0.731413	−0.365706	+	0.930730i	$0.619173\pi$
$398$	−27.3739	−1.37213
$399$	−30.1566	−1.50972
$400$	4.80675	0.240338
$401$	−27.4179	−1.36918	−0.684592	−	0.728927i	$-0.740020\pi$
$401$	−27.4179	−1.36918	−0.684592	+	0.728927i	$0.740020\pi$
$402$	−10.7069	−0.534011
$403$	−2.18169	−0.108678
$404$	−1.24407	−0.0618950
$405$	7.50641	0.372996
$406$	−36.5166	−1.81229
$407$	−8.00644	−0.396865
$408$	−3.02795	−0.149906
$409$	−0.426619	−0.0210950	−0.0105475	−	0.999944i	$-0.503357\pi$
$409$	−0.426619	−0.0210950	−0.0105475	+	0.999944i	$0.503357\pi$
$410$	21.1894	1.04647
$411$	7.98963	0.394099
$412$	−3.63007	−0.178841
$413$	−11.2530	−0.553722
$414$	3.05038	0.149918
$415$	−21.4779	−1.05431
$416$	3.94451	0.193396
$417$	−12.2652	−0.600628
$418$	−14.6285	−0.715503
$419$	−19.7661	−0.965638	−0.482819	−	0.875720i	$-0.660387\pi$
$419$	−19.7661	−0.965638	−0.482819	+	0.875720i	$0.660387\pi$
$420$	−16.6082	−0.810395
$421$	−10.1763	−0.495960	−0.247980	−	0.968765i	$-0.579767\pi$
$421$	−10.1763	−0.495960	−0.247980	+	0.968765i	$0.579767\pi$
$422$	−2.01112	−0.0978998
$423$	7.30484	0.355173
$424$	13.2235	0.642188
$425$	11.7875	0.571779
$426$	−2.45370	−0.118882
$427$	14.4315	0.698388
$428$	−1.97517	−0.0954735
$429$	−12.5299	−0.604950
$430$	−22.5993	−1.08983
$431$	41.0293	1.97631	0.988156	−	0.153450i	$-0.0490384\pi$
$431$	41.0293	1.97631	0.988156	+	0.153450i	$0.0490384\pi$
$432$	5.52599	0.265869
$433$	3.01772	0.145022	0.0725111	−	0.997368i	$-0.476899\pi$
$433$	3.01772	0.145022	0.0725111	+	0.997368i	$0.476899\pi$
$434$	2.37564	0.114035
$435$	32.8738	1.57618
$436$	−2.06769	−0.0990243
$437$	11.7562	0.562377
$438$	18.5692	0.887269
$439$	3.17838	0.151696	0.0758479	−	0.997119i	$-0.475834\pi$
$439$	3.17838	0.151696	0.0758479	+	0.997119i	$0.475834\pi$
$440$	−8.05636	−0.384072
$441$	−16.8911	−0.804339
$442$	9.67306	0.460101
$443$	−13.1379	−0.624199	−0.312100	−	0.950049i	$-0.601032\pi$
$443$	−13.1379	−0.624199	−0.312100	+	0.950049i	$0.601032\pi$
$444$	3.84274	0.182369
$445$	−1.72061	−0.0815647
$446$	−21.9644	−1.04005
$447$	−2.88928	−0.136658
$448$	−4.29517	−0.202928
$449$	−29.9802	−1.41485	−0.707426	−	0.706788i	$-0.750144\pi$
$449$	−29.9802	−1.41485	−0.707426	+	0.706788i	$0.750144\pi$
$450$	−7.09187	−0.334314
$451$	−17.4073	−0.819679
$452$	−20.0070	−0.941051
$453$	−1.08342	−0.0509037
$454$	22.2000	1.04190
$455$	53.0562	2.48732
$456$	7.02104	0.328790
$457$	29.5186	1.38082	0.690411	−	0.723417i	$-0.257430\pi$
$457$	29.5186	1.38082	0.690411	+	0.723417i	$0.257430\pi$
$458$	−5.68021	−0.265419
$459$	13.5513	0.632520
$460$	6.47452	0.301876
$461$	−11.0242	−0.513450	−0.256725	−	0.966484i	$-0.582644\pi$
$461$	−11.0242	−0.513450	−0.256725	+	0.966484i	$0.582644\pi$
$462$	13.6438	0.634767
$463$	−17.4398	−0.810498	−0.405249	−	0.914206i	$-0.632815\pi$
$463$	−17.4398	−0.810498	−0.405249	+	0.914206i	$0.632815\pi$
$464$	8.50176	0.394685
$465$	−2.13866	−0.0991779
$466$	−14.4748	−0.670533
$467$	32.5273	1.50519	0.752593	−	0.658486i	$-0.228803\pi$
$467$	32.5273	1.50519	0.752593	+	0.658486i	$0.228803\pi$
$468$	−5.81972	−0.269017
$469$	−37.2448	−1.71981
$470$	15.5047	0.715179
$471$	18.5629	0.855332
$472$	2.61991	0.120591
$473$	18.5656	0.853646
$474$	−19.6381	−0.902006
$475$	−27.3322	−1.25409
$476$	−10.5330	−0.482779
$477$	−19.5099	−0.893296
$478$	−8.95159	−0.409436
$479$	7.70350	0.351982	0.175991	−	0.984392i	$-0.443687\pi$
$479$	7.70350	0.351982	0.175991	+	0.984392i	$0.443687\pi$
$480$	3.86670	0.176490
$481$	−12.2760	−0.559737
$482$	−14.0989	−0.642190
$483$	−10.9649	−0.498919
$484$	−4.38160	−0.199164
$485$	24.8281	1.12739
$486$	−13.6183	−0.617737
$487$	0.0434217	0.00196762	0.000983812	−	1.00000i	$-0.499687\pi$
$487$	0.0434217	0.00196762	0.000983812		1.00000i	$0.499687\pi$
$488$	−3.35993	−0.152097
$489$	−7.86241	−0.355551
$490$	−35.8519	−1.61962
$491$	−34.8772	−1.57398	−0.786992	−	0.616963i	$-0.788363\pi$
$491$	−34.8772	−1.57398	−0.786992	+	0.616963i	$0.788363\pi$
$492$	8.35476	0.376662
$493$	20.8487	0.938980
$494$	−22.4293	−1.00914
$495$	11.8863	0.534251
$496$	−0.553096	−0.0248347
$497$	−8.53541	−0.382865
$498$	−8.46850	−0.379483
$499$	25.3724	1.13582	0.567912	−	0.823090i	$-0.307752\pi$
$499$	25.3724	1.13582	0.567912	+	0.823090i	$0.307752\pi$
$500$	0.605174	0.0270642
$501$	−3.26556	−0.145895
$502$	−0.462283	−0.0206327
$503$	23.8755	1.06456	0.532279	−	0.846569i	$-0.321336\pi$
$503$	23.8755	1.06456	0.532279	+	0.846569i	$0.321336\pi$
$504$	6.33709	0.282276
$505$	3.89591	0.173366
$506$	−5.31889	−0.236454
$507$	−3.15994	−0.140338
$508$	2.65776	0.117919
$509$	4.91600	0.217898	0.108949	−	0.994047i	$-0.465252\pi$
$509$	4.91600	0.217898	0.108949	+	0.994047i	$0.465252\pi$
$510$	9.48226	0.419881
$511$	64.5944	2.85749
$512$	1.00000	0.0441942
$513$	−31.4220	−1.38731
$514$	27.3141	1.20477
$515$	11.3678	0.500927
$516$	−8.91066	−0.392270
$517$	−12.7373	−0.560186
$518$	13.3673	0.587326
$519$	12.7574	0.559989
$520$	−12.3525	−0.541694
$521$	−9.94953	−0.435897	−0.217948	−	0.975960i	$-0.569936\pi$
$521$	−9.94953	−0.435897	−0.217948	+	0.975960i	$0.569936\pi$
$522$	−12.5435	−0.549013
$523$	15.7169	0.687253	0.343627	−	0.939106i	$-0.388345\pi$
$523$	15.7169	0.687253	0.343627	+	0.939106i	$0.388345\pi$
$524$	11.7648	0.513946
$525$	25.4924	1.11258
$526$	1.24817	0.0544230
$527$	−1.35635	−0.0590835
$528$	−3.17654	−0.138241
$529$	−18.7255	−0.814150
$530$	−41.4103	−1.79875
$531$	−3.86541	−0.167744
$532$	24.4233	1.05888
$533$	−26.6900	−1.15607
$534$	−0.678419	−0.0293581
$535$	6.18539	0.267418
$536$	8.67132	0.374544
$537$	0.356244	0.0153731
$538$	22.9201	0.988157
$539$	29.4527	1.26862
$540$	−17.3050	−0.744690
$541$	45.0158	1.93538	0.967691	−	0.252138i	$-0.0811338\pi$
$541$	45.0158	1.93538	0.967691	+	0.252138i	$0.0811338\pi$
$542$	27.9139	1.19900
$543$	16.7086	0.717036
$544$	2.45228	0.105141
$545$	6.47512	0.277363
$546$	20.9195	0.895274
$547$	38.2547	1.63565	0.817827	−	0.575464i	$-0.195178\pi$
$547$	38.2547	1.63565	0.817827	+	0.575464i	$0.195178\pi$
$548$	−6.47065	−0.276413
$549$	4.95723	0.211569
$550$	12.3660	0.527287
$551$	−48.3429	−2.05948
$552$	2.55284	0.108656
$553$	−68.3126	−2.90495
$554$	−11.1532	−0.473854
$555$	−12.0338	−0.510808
$556$	9.93333	0.421267
$557$	15.2193	0.644864	0.322432	−	0.946593i	$-0.395500\pi$
$557$	15.2193	0.644864	0.322432	+	0.946593i	$0.395500\pi$
$558$	0.816037	0.0345456
$559$	28.4659	1.20398
$560$	13.4506	0.568394
$561$	−7.78979	−0.328885
$562$	16.8808	0.712072
$563$	33.7670	1.42311	0.711555	−	0.702630i	$-0.247991\pi$
$563$	33.7670	1.42311	0.711555	+	0.702630i	$0.247991\pi$
$564$	6.11335	0.257419
$565$	62.6534	2.63585
$566$	−7.23109	−0.303945
$567$	10.2956	0.432373
$568$	1.98721	0.0833815
$569$	33.5199	1.40523	0.702614	−	0.711571i	$-0.252016\pi$
$569$	33.5199	1.40523	0.702614	+	0.711571i	$0.252016\pi$
$570$	−21.9869	−0.920930
$571$	38.1487	1.59647	0.798236	−	0.602345i	$-0.205767\pi$
$571$	38.1487	1.59647	0.798236	+	0.602345i	$0.205767\pi$
$572$	10.1477	0.424299
$573$	27.9161	1.16621
$574$	29.0627	1.21306
$575$	−9.93794	−0.414441
$576$	−1.47540	−0.0614749
$577$	2.07201	0.0862590	0.0431295	−	0.999069i	$-0.486267\pi$
$577$	2.07201	0.0862590	0.0431295	+	0.999069i	$0.486267\pi$
$578$	−10.9863	−0.456970
$579$	−1.86465	−0.0774923
$580$	−26.6239	−1.10550
$581$	−29.4584	−1.22214
$582$	9.78948	0.405787
$583$	34.0190	1.40892
$584$	−15.0388	−0.622312
$585$	18.2249	0.753507
$586$	−1.88031	−0.0776751
$587$	−7.58277	−0.312974	−0.156487	−	0.987680i	$-0.550017\pi$
$587$	−7.58277	−0.312974	−0.156487	+	0.987680i	$0.550017\pi$
$588$	−14.1360	−0.582960
$589$	3.14502	0.129588
$590$	−8.20444	−0.337771
$591$	14.3759	0.591346
$592$	−3.11217	−0.127909
$593$	−33.9420	−1.39383	−0.696915	−	0.717154i	$-0.745444\pi$
$593$	−33.9420	−1.39383	−0.696915	+	0.717154i	$0.745444\pi$
$594$	14.2163	0.583302
$595$	32.9848	1.35225
$596$	2.33998	0.0958493
$597$	33.7998	1.38334
$598$	−8.15526	−0.333494
$599$	36.5794	1.49459	0.747297	−	0.664491i	$-0.231351\pi$
$599$	36.5794	1.49459	0.747297	+	0.664491i	$0.231351\pi$
$600$	−5.93513	−0.242300
$601$	−1.36063	−0.0555012	−0.0277506	−	0.999615i	$-0.508834\pi$
$601$	−1.36063	−0.0555012	−0.0277506	+	0.999615i	$0.508834\pi$
$602$	−30.9965	−1.26332
$603$	−12.7936	−0.520998
$604$	0.877446	0.0357028
$605$	13.7213	0.557850
$606$	1.53612	0.0624005
$607$	−0.297025	−0.0120559	−0.00602793	−	0.999982i	$-0.501919\pi$
$607$	−0.297025	−0.0120559	−0.00602793	+	0.999982i	$0.501919\pi$
$608$	−5.68621	−0.230606
$609$	45.0887	1.82709
$610$	10.5219	0.426018
$611$	−19.5297	−0.790085
$612$	−3.61809	−0.146253
$613$	24.7961	1.00150	0.500752	−	0.865591i	$-0.333057\pi$
$613$	24.7961	1.00150	0.500752	+	0.865591i	$0.333057\pi$
$614$	−8.14649	−0.328766
$615$	−26.1636	−1.05502
$616$	−11.0499	−0.445212
$617$	14.1575	0.569960	0.284980	−	0.958533i	$-0.408013\pi$
$617$	14.1575	0.569960	0.284980	+	0.958533i	$0.408013\pi$
$618$	4.48222	0.180301
$619$	8.87753	0.356818	0.178409	−	0.983956i	$-0.442905\pi$
$619$	8.87753	0.356818	0.178409	+	0.983956i	$0.442905\pi$
$620$	1.73206	0.0695612
$621$	−11.4250	−0.458468
$622$	−28.9275	−1.15989
$623$	−2.35994	−0.0945488
$624$	−4.87048	−0.194975
$625$	−25.9289	−1.03716
$626$	−8.00595	−0.319982
$627$	18.0625	0.721347
$628$	−15.0337	−0.599912
$629$	−7.63192	−0.304305
$630$	−19.8451	−0.790646
$631$	−7.47786	−0.297689	−0.148844	−	0.988861i	$-0.547555\pi$
$631$	−7.47786	−0.297689	−0.148844	+	0.988861i	$0.547555\pi$
$632$	15.9045	0.632647
$633$	2.48323	0.0986993
$634$	12.9073	0.512614
$635$	−8.32298	−0.330288
$636$	−16.3277	−0.647433
$637$	45.1588	1.78926
$638$	21.8719	0.865915
$639$	−2.93193	−0.115985
$640$	−3.13157	−0.123786
$641$	−16.5597	−0.654067	−0.327034	−	0.945013i	$-0.606049\pi$
$641$	−16.5597	−0.654067	−0.327034	+	0.945013i	$0.606049\pi$
$642$	2.43884	0.0962533
$643$	−17.7719	−0.700854	−0.350427	−	0.936590i	$-0.613964\pi$
$643$	−17.7719	−0.700854	−0.350427	+	0.936590i	$0.613964\pi$
$644$	8.88025	0.349931
$645$	27.9044	1.09873
$646$	−13.9442	−0.548628
$647$	−10.5752	−0.415753	−0.207876	−	0.978155i	$-0.566655\pi$
$647$	−10.5752	−0.415753	−0.207876	+	0.978155i	$0.566655\pi$
$648$	−2.39701	−0.0941634
$649$	6.74004	0.264570
$650$	18.9603	0.743684
$651$	−2.93332	−0.114966
$652$	6.36762	0.249375
$653$	24.4877	0.958277	0.479138	−	0.877739i	$-0.340949\pi$
$653$	24.4877	0.958277	0.479138	+	0.877739i	$0.340949\pi$
$654$	2.55307	0.0998331
$655$	−36.8422	−1.43954
$656$	−6.76637	−0.264182
$657$	22.1883	0.865647
$658$	21.2658	0.829028
$659$	30.6406	1.19359	0.596795	−	0.802394i	$-0.296441\pi$
$659$	30.6406	1.19359	0.596795	+	0.802394i	$0.296441\pi$
$660$	9.94758	0.387209
$661$	−33.4574	−1.30134	−0.650671	−	0.759360i	$-0.725512\pi$
$661$	−33.4574	−1.30134	−0.650671	+	0.759360i	$0.725512\pi$
$662$	−6.63949	−0.258051
$663$	−11.9438	−0.463859
$664$	6.85849	0.266161
$665$	−76.4833	−2.96589
$666$	4.59169	0.177924
$667$	−17.5774	−0.680598
$668$	2.64472	0.102327
$669$	27.1206	1.04854
$670$	−27.1549	−1.04908
$671$	−8.64384	−0.333692
$672$	5.30346	0.204585
$673$	1.45764	0.0561879	0.0280940	−	0.999605i	$-0.491056\pi$
$673$	1.45764	0.0561879	0.0280940	+	0.999605i	$0.491056\pi$
$674$	−17.1772	−0.661641
$675$	26.5620	1.02237
$676$	2.55918	0.0984298
$677$	13.6475	0.524515	0.262258	−	0.964998i	$-0.415533\pi$
$677$	13.6475	0.524515	0.262258	+	0.964998i	$0.415533\pi$
$678$	24.7036	0.948737
$679$	34.0535	1.30685
$680$	−7.67951	−0.294496
$681$	−27.4114	−1.05041
$682$	−1.42291	−0.0544860
$683$	−11.3819	−0.435515	−0.217758	−	0.976003i	$-0.569874\pi$
$683$	−11.3819	−0.435515	−0.217758	+	0.976003i	$0.569874\pi$
$684$	8.38943	0.320778
$685$	20.2633	0.774222
$686$	−19.1071	−0.729514
$687$	7.01363	0.267587
$688$	7.21658	0.275130
$689$	52.1601	1.98714
$690$	−7.99439	−0.304341
$691$	11.8486	0.450740	0.225370	−	0.974273i	$-0.427641\pi$
$691$	11.8486	0.450740	0.225370	+	0.974273i	$0.427641\pi$
$692$	−10.3320	−0.392764
$693$	16.3030	0.619298
$694$	−25.5023	−0.968056
$695$	−31.1070	−1.17995
$696$	−10.4975	−0.397908
$697$	−16.5931	−0.628507
$698$	9.07639	0.343546
$699$	17.8728	0.676010
$700$	−20.6458	−0.780339
$701$	−18.7265	−0.707292	−0.353646	−	0.935379i	$-0.615058\pi$
$701$	−18.7265	−0.707292	−0.353646	+	0.935379i	$0.615058\pi$
$702$	21.7973	0.822687
$703$	17.6965	0.667435
$704$	2.57262	0.0969594
$705$	−19.1444	−0.721020
$706$	4.04348	0.152178
$707$	5.34351	0.200963
$708$	−3.23493	−0.121576
$709$	36.9767	1.38869	0.694345	−	0.719643i	$-0.255694\pi$
$709$	36.9767	1.38869	0.694345	+	0.719643i	$0.255694\pi$
$710$	−6.22309	−0.233549
$711$	−23.4655	−0.880024
$712$	0.549439	0.0205911
$713$	1.14352	0.0428253
$714$	13.0056	0.486722
$715$	−31.7784	−1.18845
$716$	−0.288516	−0.0107823
$717$	11.0530	0.412780
$718$	−9.33350	−0.348323
$719$	−49.0469	−1.82914	−0.914571	−	0.404425i	$-0.867472\pi$
$719$	−49.0469	−1.82914	−0.914571	+	0.404425i	$0.867472\pi$
$720$	4.62032	0.172189
$721$	15.5918	0.580668
$722$	13.3330	0.496204
$723$	17.4086	0.647435
$724$	−13.5320	−0.502913
$725$	40.8659	1.51772
$726$	5.41017	0.200790
$727$	8.28721	0.307355	0.153678	−	0.988121i	$-0.450888\pi$
$727$	8.28721	0.307355	0.153678	+	0.988121i	$0.450888\pi$
$728$	−16.9424	−0.627926
$729$	24.0061	0.889116
$730$	47.0952	1.74307
$731$	17.6971	0.654551
$732$	4.14866	0.153339
$733$	28.8559	1.06582	0.532908	−	0.846173i	$-0.321099\pi$
$733$	28.8559	1.06582	0.532908	+	0.846173i	$0.321099\pi$
$734$	33.1304	1.22287
$735$	44.2680	1.63285
$736$	−2.06750	−0.0762089
$737$	22.3081	0.821728
$738$	9.98309	0.367483
$739$	13.5410	0.498113	0.249057	−	0.968489i	$-0.419880\pi$
$739$	13.5410	0.498113	0.249057	+	0.968489i	$0.419880\pi$
$740$	9.74598	0.358269
$741$	27.6946	1.01739
$742$	−56.7971	−2.08509
$743$	−13.1250	−0.481508	−0.240754	−	0.970586i	$-0.577395\pi$
$743$	−13.1250	−0.481508	−0.240754	+	0.970586i	$0.577395\pi$
$744$	0.682934	0.0250376
$745$	−7.32782	−0.268470
$746$	2.24084	0.0820431
$747$	−10.1190	−0.370235
$748$	6.30881	0.230673
$749$	8.48370	0.309988
$750$	−0.747237	−0.0272852
$751$	−8.51611	−0.310757	−0.155379	−	0.987855i	$-0.549660\pi$
$751$	−8.51611	−0.310757	−0.155379	+	0.987855i	$0.549660\pi$
$752$	−4.95110	−0.180548
$753$	0.570803	0.0208012
$754$	33.5353	1.22128
$755$	−2.74779	−0.100002
$756$	−23.7351	−0.863236
$757$	−52.0017	−1.89003	−0.945017	−	0.327022i	$-0.893955\pi$
$757$	−52.0017	−1.89003	−0.945017	+	0.327022i	$0.893955\pi$
$758$	−22.8624	−0.830400
$759$	6.56749	0.238385
$760$	17.8068	0.645920
$761$	−25.1045	−0.910036	−0.455018	−	0.890482i	$-0.650367\pi$
$761$	−25.1045	−0.910036	−0.455018	+	0.890482i	$0.650367\pi$
$762$	−3.28167	−0.118882
$763$	8.88108	0.321517
$764$	−22.6088	−0.817956
$765$	11.3303	0.409649
$766$	−23.0013	−0.831072
$767$	10.3343	0.373149
$768$	−1.23475	−0.0445551
$769$	−20.5327	−0.740429	−0.370214	−	0.928946i	$-0.620716\pi$
$769$	−20.5327	−0.740429	−0.370214	+	0.928946i	$0.620716\pi$
$770$	34.6035	1.24702
$771$	−33.7260	−1.21461
$772$	1.51015	0.0543514
$773$	9.02250	0.324517	0.162258	−	0.986748i	$-0.448122\pi$
$773$	9.02250	0.324517	0.162258	+	0.986748i	$0.448122\pi$
$774$	−10.6473	−0.382710
$775$	−2.65859	−0.0954995
$776$	−7.92832	−0.284610
$777$	−16.5052	−0.592122
$778$	9.03734	0.324004
$779$	38.4750	1.37851
$780$	15.2523	0.546118
$781$	5.11235	0.182934
$782$	−5.07009	−0.181306
$783$	46.9806	1.67895
$784$	11.4485	0.408875
$785$	47.0793	1.68033
$786$	−14.5265	−0.518144
$787$	11.4549	0.408324	0.204162	−	0.978937i	$-0.434553\pi$
$787$	11.4549	0.408324	0.204162	+	0.978937i	$0.434553\pi$
$788$	−11.6428	−0.414757
$789$	−1.54118	−0.0548675
$790$	−49.8061	−1.77202
$791$	85.9336	3.05545
$792$	−3.79564	−0.134872
$793$	−13.2533	−0.470638
$794$	−14.5733	−0.517187
$795$	51.1312	1.81344
$796$	−27.3739	−0.970242
$797$	−18.3151	−0.648754	−0.324377	−	0.945928i	$-0.605155\pi$
$797$	−18.3151	−0.648754	−0.324377	+	0.945928i	$0.605155\pi$
$798$	−30.1566	−1.06753
$799$	−12.1415	−0.429535
$800$	4.80675	0.169944
$801$	−0.810641	−0.0286426
$802$	−27.4179	−0.968159
$803$	−38.6893	−1.36532
$804$	−10.7069	−0.377603
$805$	−27.8092	−0.980144
$806$	−2.18169	−0.0768469
$807$	−28.3006	−0.996227
$808$	−1.24407	−0.0437663
$809$	−39.6990	−1.39574	−0.697872	−	0.716223i	$-0.745870\pi$
$809$	−39.6990	−1.39574	−0.697872	+	0.716223i	$0.745870\pi$
$810$	7.50641	0.263748
$811$	−20.4514	−0.718145	−0.359073	−	0.933310i	$-0.616907\pi$
$811$	−20.4514	−0.718145	−0.359073	+	0.933310i	$0.616907\pi$
$812$	−36.5166	−1.28148
$813$	−34.4666	−1.20880
$814$	−8.00644	−0.280626
$815$	−19.9407	−0.698492
$816$	−3.02795	−0.106000
$817$	−41.0350	−1.43563
$818$	−0.426619	−0.0149164
$819$	24.9967	0.873456
$820$	21.1894	0.739966
$821$	−43.2811	−1.51052	−0.755261	−	0.655425i	$-0.772490\pi$
$821$	−43.2811	−1.51052	−0.755261	+	0.655425i	$0.772490\pi$
$822$	7.98963	0.278670
$823$	−38.4809	−1.34136	−0.670680	−	0.741746i	$-0.733998\pi$
$823$	−38.4809	−1.34136	−0.670680	+	0.741746i	$0.733998\pi$
$824$	−3.63007	−0.126460
$825$	−15.2688	−0.531593
$826$	−11.2530	−0.391541
$827$	−5.60942	−0.195059	−0.0975294	−	0.995233i	$-0.531094\pi$
$827$	−5.60942	−0.195059	−0.0975294	+	0.995233i	$0.531094\pi$
$828$	3.05038	0.106008
$829$	28.3499	0.984632	0.492316	−	0.870416i	$-0.336150\pi$
$829$	28.3499	0.984632	0.492316	+	0.870416i	$0.336150\pi$
$830$	−21.4779	−0.745507
$831$	13.7714	0.477724
$832$	3.94451	0.136751
$833$	28.0750	0.972741
$834$	−12.2652	−0.424708
$835$	−8.28213	−0.286615
$836$	−14.6285	−0.505937
$837$	−3.05640	−0.105645
$838$	−19.7661	−0.682809
$839$	−9.45677	−0.326484	−0.163242	−	0.986586i	$-0.552195\pi$
$839$	−9.45677	−0.326484	−0.163242	+	0.986586i	$0.552195\pi$
$840$	−16.6082	−0.573036
$841$	43.2800	1.49241
$842$	−10.1763	−0.350697
$843$	−20.8435	−0.717888
$844$	−2.01112	−0.0692256
$845$	−8.01425	−0.275698
$846$	7.30484	0.251145
$847$	18.8197	0.646654
$848$	13.2235	0.454096
$849$	8.92857	0.306428
$850$	11.7875	0.404309
$851$	6.43440	0.220568
$852$	−2.45370	−0.0840625
$853$	−48.4542	−1.65904	−0.829521	−	0.558476i	$-0.811386\pi$
$853$	−48.4542	−1.65904	−0.829521	+	0.558476i	$0.811386\pi$
$854$	14.4315	0.493835
$855$	−26.2721	−0.898487
$856$	−1.97517	−0.0675100
$857$	−22.3955	−0.765015	−0.382508	−	0.923952i	$-0.624939\pi$
$857$	−22.3955	−0.765015	−0.382508	+	0.923952i	$0.624939\pi$
$858$	−12.5299	−0.427764
$859$	−21.3105	−0.727106	−0.363553	−	0.931573i	$-0.618437\pi$
$859$	−21.3105	−0.727106	−0.363553	+	0.931573i	$0.618437\pi$
$860$	−22.5993	−0.770628
$861$	−35.8852	−1.22296
$862$	41.0293	1.39746
$863$	33.8218	1.15131	0.575653	−	0.817694i	$-0.304748\pi$
$863$	33.8218	1.15131	0.575653	+	0.817694i	$0.304748\pi$
$864$	5.52599	0.187998
$865$	32.3554	1.10012
$866$	3.01772	0.102546
$867$	13.5653	0.460702
$868$	2.37564	0.0806346
$869$	40.9163	1.38799
$870$	32.8738	1.11453
$871$	34.2041	1.15896
$872$	−2.06769	−0.0700207
$873$	11.6974	0.395898
$874$	11.7562	0.397660
$875$	−2.59933	−0.0878733
$876$	18.5692	0.627394
$877$	−16.3915	−0.553503	−0.276751	−	0.960942i	$-0.589258\pi$
$877$	−16.3915	−0.553503	−0.276751	+	0.960942i	$0.589258\pi$
$878$	3.17838	0.107265
$879$	2.32171	0.0783095
$880$	−8.05636	−0.271580
$881$	52.6926	1.77526	0.887629	−	0.460558i	$-0.152351\pi$
$881$	52.6926	1.77526	0.887629	+	0.460558i	$0.152351\pi$
$882$	−16.8911	−0.568753
$883$	−41.2533	−1.38828	−0.694142	−	0.719838i	$-0.744216\pi$
$883$	−41.2533	−1.38828	−0.694142	+	0.719838i	$0.744216\pi$
$884$	9.67306	0.325340
$885$	10.1304	0.340530
$886$	−13.1379	−0.441375
$887$	−6.37033	−0.213895	−0.106947	−	0.994265i	$-0.534108\pi$
$887$	−6.37033	−0.213895	−0.106947	+	0.994265i	$0.534108\pi$
$888$	3.84274	0.128954
$889$	−11.4156	−0.382866
$890$	−1.72061	−0.0576749
$891$	−6.16660	−0.206589
$892$	−21.9644	−0.735424
$893$	28.1530	0.942104
$894$	−2.88928	−0.0966321
$895$	0.903508	0.0302009
$896$	−4.29517	−0.143492
$897$	10.0697	0.336217
$898$	−29.9802	−1.00045
$899$	−4.70229	−0.156830
$900$	−7.09187	−0.236396
$901$	32.4277	1.08032
$902$	−17.4073	−0.579601
$903$	38.2728	1.27364
$904$	−20.0070	−0.665423
$905$	42.3765	1.40864
$906$	−1.08342	−0.0359944
$907$	−49.3964	−1.64018	−0.820090	−	0.572235i	$-0.806076\pi$
$907$	−49.3964	−1.64018	−0.820090	+	0.572235i	$0.806076\pi$
$908$	22.2000	0.736732
$909$	1.83550	0.0608798
$910$	53.0562	1.75880
$911$	28.1014	0.931040	0.465520	−	0.885037i	$-0.345867\pi$
$911$	28.1014	0.931040	0.465520	+	0.885037i	$0.345867\pi$
$912$	7.02104	0.232490
$913$	17.6443	0.583942
$914$	29.5186	0.976388
$915$	−12.9918	−0.429497
$916$	−5.68021	−0.187679
$917$	−50.5317	−1.66870
$918$	13.5513	0.447259
$919$	30.5010	1.00614	0.503068	−	0.864247i	$-0.332205\pi$
$919$	30.5010	1.00614	0.503068	+	0.864247i	$0.332205\pi$
$920$	6.47452	0.213458
$921$	10.0589	0.331451
$922$	−11.0242	−0.363064
$923$	7.83857	0.258010
$924$	13.6438	0.448848
$925$	−14.9594	−0.491863
$926$	−17.4398	−0.573109
$927$	5.35580	0.175908
$928$	8.50176	0.279084
$929$	−45.8142	−1.50311	−0.751557	−	0.659668i	$-0.770697\pi$
$929$	−45.8142	−1.50311	−0.751557	+	0.659668i	$0.770697\pi$
$930$	−2.13866	−0.0701294
$931$	−65.0987	−2.13352
$932$	−14.4748	−0.474138
$933$	35.7182	1.16936
$934$	32.5273	1.06433
$935$	−19.7565	−0.646106
$936$	−5.81972	−0.190224
$937$	26.2775	0.858448	0.429224	−	0.903198i	$-0.358787\pi$
$937$	26.2775	0.858448	0.429224	+	0.903198i	$0.358787\pi$
$938$	−37.2448	−1.21609
$939$	9.88532	0.322595
$940$	15.5047	0.505708
$941$	−57.9504	−1.88913	−0.944565	−	0.328324i	$-0.893516\pi$
$941$	−57.9504	−1.88913	−0.944565	+	0.328324i	$0.893516\pi$
$942$	18.5629	0.604811
$943$	13.9894	0.455559
$944$	2.61991	0.0852708
$945$	74.3281	2.41789
$946$	18.5656	0.603619
$947$	17.6495	0.573533	0.286766	−	0.958001i	$-0.407420\pi$
$947$	17.6495	0.573533	0.286766	+	0.958001i	$0.407420\pi$
$948$	−19.6381	−0.637814
$949$	−59.3209	−1.92564
$950$	−27.3322	−0.886774
$951$	−15.9372	−0.516800
$952$	−10.5330	−0.341376
$953$	−23.0273	−0.745926	−0.372963	−	0.927846i	$-0.621658\pi$
$953$	−23.0273	−0.745926	−0.372963	+	0.927846i	$0.621658\pi$
$954$	−19.5099	−0.631656
$955$	70.8010	2.29107
$956$	−8.95159	−0.289515
$957$	−27.0062	−0.872987
$958$	7.70350	0.248889
$959$	27.7926	0.897469
$960$	3.86670	0.124797
$961$	−30.6941	−0.990132
$962$	−12.2760	−0.395794
$963$	2.91416	0.0939076
$964$	−14.0989	−0.454097
$965$	−4.72914	−0.152236
$966$	−10.9649	−0.352789
$967$	28.4643	0.915351	0.457675	−	0.889119i	$-0.348682\pi$
$967$	28.4643	0.915351	0.457675	+	0.889119i	$0.348682\pi$
$968$	−4.38160	−0.140830
$969$	17.2176	0.553109
$970$	24.8281	0.797183
$971$	10.2194	0.327955	0.163977	−	0.986464i	$-0.447568\pi$
$971$	10.2194	0.327955	0.163977	+	0.986464i	$0.447568\pi$
$972$	−13.6183	−0.436806
$973$	−42.6654	−1.36779
$974$	0.0434217	0.00139132
$975$	−23.4112	−0.749758
$976$	−3.35993	−0.107549
$977$	21.0489	0.673413	0.336707	−	0.941610i	$-0.390687\pi$
$977$	21.0489	0.673413	0.336707	+	0.941610i	$0.390687\pi$
$978$	−7.86241	−0.251412
$979$	1.41350	0.0451757
$980$	−35.8519	−1.14525
$981$	3.05066	0.0974001
$982$	−34.8772	−1.11298
$983$	−37.8242	−1.20640	−0.603202	−	0.797589i	$-0.706109\pi$
$983$	−37.8242	−1.20640	−0.603202	+	0.797589i	$0.706109\pi$
$984$	8.35476	0.266340
$985$	36.4602	1.16172
$986$	20.8487	0.663959
$987$	−26.2579	−0.835799
$988$	−22.4293	−0.713572
$989$	−14.9203	−0.474437
$990$	11.8863	0.377773
$991$	55.6173	1.76674	0.883371	−	0.468675i	$-0.155268\pi$
$991$	55.6173	1.76674	0.883371	+	0.468675i	$0.155268\pi$
$992$	−0.553096	−0.0175608
$993$	8.19810	0.260159
$994$	−8.53541	−0.270727
$995$	85.7233	2.71761
$996$	−8.46850	−0.268335
$997$	31.4467	0.995927	0.497964	−	0.867198i	$-0.334081\pi$
$997$	31.4467	0.995927	0.497964	+	0.867198i	$0.334081\pi$
$998$	25.3724	0.803148
$999$	−17.1978	−0.544114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.27	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.27	✓	69	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.23475 q^{3} +1.00000 q^{4} -3.13157 q^{5} -1.23475 q^{6} -4.29517 q^{7} +1.00000 q^{8} -1.47540 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.23475 q^{3} +1.00000 q^{4} -3.13157 q^{5} -1.23475 q^{6} -4.29517 q^{7} +1.00000 q^{8} -1.47540 q^{9} -3.13157 q^{10} +2.57262 q^{11} -1.23475 q^{12} +3.94451 q^{13} -4.29517 q^{14} +3.86670 q^{15} +1.00000 q^{16} +2.45228 q^{17} -1.47540 q^{18} -5.68621 q^{19} -3.13157 q^{20} +5.30346 q^{21} +2.57262 q^{22} -2.06750 q^{23} -1.23475 q^{24} +4.80675 q^{25} +3.94451 q^{26} +5.52599 q^{27} -4.29517 q^{28} +8.50176 q^{29} +3.86670 q^{30} -0.553096 q^{31} +1.00000 q^{32} -3.17654 q^{33} +2.45228 q^{34} +13.4506 q^{35} -1.47540 q^{36} -3.11217 q^{37} -5.68621 q^{38} -4.87048 q^{39} -3.13157 q^{40} -6.76637 q^{41} +5.30346 q^{42} +7.21658 q^{43} +2.57262 q^{44} +4.62032 q^{45} -2.06750 q^{46} -4.95110 q^{47} -1.23475 q^{48} +11.4485 q^{49} +4.80675 q^{50} -3.02795 q^{51} +3.94451 q^{52} +13.2235 q^{53} +5.52599 q^{54} -8.05636 q^{55} -4.29517 q^{56} +7.02104 q^{57} +8.50176 q^{58} +2.61991 q^{59} +3.86670 q^{60} -3.35993 q^{61} -0.553096 q^{62} +6.33709 q^{63} +1.00000 q^{64} -12.3525 q^{65} -3.17654 q^{66} +8.67132 q^{67} +2.45228 q^{68} +2.55284 q^{69} +13.4506 q^{70} +1.98721 q^{71} -1.47540 q^{72} -15.0388 q^{73} -3.11217 q^{74} -5.93513 q^{75} -5.68621 q^{76} -11.0499 q^{77} -4.87048 q^{78} +15.9045 q^{79} -3.13157 q^{80} -2.39701 q^{81} -6.76637 q^{82} +6.85849 q^{83} +5.30346 q^{84} -7.67951 q^{85} +7.21658 q^{86} -10.4975 q^{87} +2.57262 q^{88} +0.549439 q^{89} +4.62032 q^{90} -16.9424 q^{91} -2.06750 q^{92} +0.682934 q^{93} -4.95110 q^{94} +17.8068 q^{95} -1.23475 q^{96} -7.92832 q^{97} +11.4485 q^{98} -3.79564 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more