LMFDB - Embedded newform 4022.2.a.f.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4022,2,Mod(1,4022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4022, 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("4022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4022 = 2 \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4022.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:	$32.1158316930$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	4022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.537568 q^{3} +1.00000 q^{4} +2.09045 q^{5} -0.537568 q^{6} +4.78863 q^{7} +1.00000 q^{8} -2.71102 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.537568 q^{3} +1.00000 q^{4} +2.09045 q^{5} -0.537568 q^{6} +4.78863 q^{7} +1.00000 q^{8} -2.71102 q^{9} +2.09045 q^{10} -0.581956 q^{11} -0.537568 q^{12} -2.31166 q^{13} +4.78863 q^{14} -1.12376 q^{15} +1.00000 q^{16} -3.85853 q^{17} -2.71102 q^{18} +5.75005 q^{19} +2.09045 q^{20} -2.57421 q^{21} -0.581956 q^{22} -6.40957 q^{23} -0.537568 q^{24} -0.630001 q^{25} -2.31166 q^{26} +3.07006 q^{27} +4.78863 q^{28} +10.0252 q^{29} -1.12376 q^{30} -0.952590 q^{31} +1.00000 q^{32} +0.312841 q^{33} -3.85853 q^{34} +10.0104 q^{35} -2.71102 q^{36} +9.58463 q^{37} +5.75005 q^{38} +1.24268 q^{39} +2.09045 q^{40} +6.05842 q^{41} -2.57421 q^{42} -1.50188 q^{43} -0.581956 q^{44} -5.66726 q^{45} -6.40957 q^{46} +7.67306 q^{47} -0.537568 q^{48} +15.9310 q^{49} -0.630001 q^{50} +2.07422 q^{51} -2.31166 q^{52} +11.3320 q^{53} +3.07006 q^{54} -1.21655 q^{55} +4.78863 q^{56} -3.09104 q^{57} +10.0252 q^{58} +3.28283 q^{59} -1.12376 q^{60} +4.19020 q^{61} -0.952590 q^{62} -12.9821 q^{63} +1.00000 q^{64} -4.83243 q^{65} +0.312841 q^{66} -1.67325 q^{67} -3.85853 q^{68} +3.44558 q^{69} +10.0104 q^{70} +4.15060 q^{71} -2.71102 q^{72} -10.8377 q^{73} +9.58463 q^{74} +0.338668 q^{75} +5.75005 q^{76} -2.78677 q^{77} +1.24268 q^{78} -9.18393 q^{79} +2.09045 q^{80} +6.48270 q^{81} +6.05842 q^{82} -1.46287 q^{83} -2.57421 q^{84} -8.06608 q^{85} -1.50188 q^{86} -5.38925 q^{87} -0.581956 q^{88} -2.79907 q^{89} -5.66726 q^{90} -11.0697 q^{91} -6.40957 q^{92} +0.512082 q^{93} +7.67306 q^{94} +12.0202 q^{95} -0.537568 q^{96} +8.47436 q^{97} +15.9310 q^{98} +1.57769 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 50 q^{2} + 18 q^{3} + 50 q^{4} + 11 q^{5} + 18 q^{6} + 30 q^{7} + 50 q^{8} + 66 q^{9}+O(q^{10}) $ 50 * q + 50 * q^2 + 18 * q^3 + 50 * q^4 + 11 * q^5 + 18 * q^6 + 30 * q^7 + 50 * q^8 + 66 * q^9	\( 50 q + 50 q^{2} + 18 q^{3} + 50 q^{4} + 11 q^{5} + 18 q^{6} + 30 q^{7} + 50 q^{8} + 66 q^{9} + 11 q^{10} + 21 q^{11} + 18 q^{12} + 26 q^{13} + 30 q^{14} + 17 q^{15} + 50 q^{16} + 24 q^{17} + 66 q^{18} + 39 q^{19} + 11 q^{20} - q^{21} + 21 q^{22} + 28 q^{23} + 18 q^{24} + 79 q^{25} + 26 q^{26} + 66 q^{27} + 30 q^{28} - 5 q^{29} + 17 q^{30} + 60 q^{31} + 50 q^{32} + 37 q^{33} + 24 q^{34} + 38 q^{35} + 66 q^{36} + 35 q^{37} + 39 q^{38} + 37 q^{39} + 11 q^{40} + 42 q^{41} - q^{42} + 44 q^{43} + 21 q^{44} + 31 q^{45} + 28 q^{46} + 60 q^{47} + 18 q^{48} + 92 q^{49} + 79 q^{50} + 26 q^{51} + 26 q^{52} - 2 q^{53} + 66 q^{54} + 33 q^{55} + 30 q^{56} + 15 q^{57} - 5 q^{58} + 65 q^{59} + 17 q^{60} + 15 q^{61} + 60 q^{62} + 56 q^{63} + 50 q^{64} + 6 q^{65} + 37 q^{66} + 48 q^{67} + 24 q^{68} - 9 q^{69} + 38 q^{70} + 34 q^{71} + 66 q^{72} + 91 q^{73} + 35 q^{74} + 54 q^{75} + 39 q^{76} - 6 q^{77} + 37 q^{78} + 29 q^{79} + 11 q^{80} + 66 q^{81} + 42 q^{82} + 43 q^{83} - q^{84} + 44 q^{86} + 32 q^{87} + 21 q^{88} + 38 q^{89} + 31 q^{90} + 55 q^{91} + 28 q^{92} - 15 q^{93} + 60 q^{94} + 9 q^{95} + 18 q^{96} + 80 q^{97} + 92 q^{98} + 20 q^{99}+O(q^{100}) \) 50 * q + 50 * q^2 + 18 * q^3 + 50 * q^4 + 11 * q^5 + 18 * q^6 + 30 * q^7 + 50 * q^8 + 66 * q^9 + 11 * q^10 + 21 * q^11 + 18 * q^12 + 26 * q^13 + 30 * q^14 + 17 * q^15 + 50 * q^16 + 24 * q^17 + 66 * q^18 + 39 * q^19 + 11 * q^20 - q^21 + 21 * q^22 + 28 * q^23 + 18 * q^24 + 79 * q^25 + 26 * q^26 + 66 * q^27 + 30 * q^28 - 5 * q^29 + 17 * q^30 + 60 * q^31 + 50 * q^32 + 37 * q^33 + 24 * q^34 + 38 * q^35 + 66 * q^36 + 35 * q^37 + 39 * q^38 + 37 * q^39 + 11 * q^40 + 42 * q^41 - q^42 + 44 * q^43 + 21 * q^44 + 31 * q^45 + 28 * q^46 + 60 * q^47 + 18 * q^48 + 92 * q^49 + 79 * q^50 + 26 * q^51 + 26 * q^52 - 2 * q^53 + 66 * q^54 + 33 * q^55 + 30 * q^56 + 15 * q^57 - 5 * q^58 + 65 * q^59 + 17 * q^60 + 15 * q^61 + 60 * q^62 + 56 * q^63 + 50 * q^64 + 6 * q^65 + 37 * q^66 + 48 * q^67 + 24 * q^68 - 9 * q^69 + 38 * q^70 + 34 * q^71 + 66 * q^72 + 91 * q^73 + 35 * q^74 + 54 * q^75 + 39 * q^76 - 6 * q^77 + 37 * q^78 + 29 * q^79 + 11 * q^80 + 66 * q^81 + 42 * q^82 + 43 * q^83 - q^84 + 44 * q^86 + 32 * q^87 + 21 * q^88 + 38 * q^89 + 31 * q^90 + 55 * q^91 + 28 * q^92 - 15 * q^93 + 60 * q^94 + 9 * q^95 + 18 * q^96 + 80 * q^97 + 92 * q^98 + 20 * 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
$3$	−0.537568	−0.310365	−0.155182	−	0.987886i	$-0.549597\pi$
$3$	−0.537568	−0.310365	−0.155182	+	0.987886i	$0.549597\pi$
$4$	1.00000	0.500000
$5$	2.09045	0.934880	0.467440	−	0.884025i	$-0.345176\pi$
$5$	2.09045	0.934880	0.467440	+	0.884025i	$0.345176\pi$
$6$	−0.537568	−0.219461
$7$	4.78863	1.80993	0.904966	−	0.425484i	$-0.139896\pi$
$7$	4.78863	1.80993	0.904966	+	0.425484i	$0.139896\pi$
$8$	1.00000	0.353553
$9$	−2.71102	−0.903674
$10$	2.09045	0.661060
$11$	−0.581956	−0.175466	−0.0877331	−	0.996144i	$-0.527962\pi$
$11$	−0.581956	−0.175466	−0.0877331	+	0.996144i	$0.527962\pi$
$12$	−0.537568	−0.155182
$13$	−2.31166	−0.641140	−0.320570	−	0.947225i	$-0.603875\pi$
$13$	−2.31166	−0.641140	−0.320570	+	0.947225i	$0.603875\pi$
$14$	4.78863	1.27982
$15$	−1.12376	−0.290154
$16$	1.00000	0.250000
$17$	−3.85853	−0.935831	−0.467915	−	0.883773i	$-0.654995\pi$
$17$	−3.85853	−0.935831	−0.467915	+	0.883773i	$0.654995\pi$
$18$	−2.71102	−0.638994
$19$	5.75005	1.31915	0.659576	−	0.751638i	$-0.270736\pi$
$19$	5.75005	1.31915	0.659576	+	0.751638i	$0.270736\pi$
$20$	2.09045	0.467440
$21$	−2.57421	−0.561740
$22$	−0.581956	−0.124073
$23$	−6.40957	−1.33649	−0.668244	−	0.743942i	$-0.732954\pi$
$23$	−6.40957	−1.33649	−0.668244	+	0.743942i	$0.732954\pi$
$24$	−0.537568	−0.109731
$25$	−0.630001	−0.126000
$26$	−2.31166	−0.453355
$27$	3.07006	0.590834
$28$	4.78863	0.904966
$29$	10.0252	1.86164	0.930821	−	0.365477i	$-0.119094\pi$
$29$	10.0252	1.86164	0.930821	+	0.365477i	$0.119094\pi$
$30$	−1.12376	−0.205170
$31$	−0.952590	−0.171090	−0.0855451	−	0.996334i	$-0.527263\pi$
$31$	−0.952590	−0.171090	−0.0855451	+	0.996334i	$0.527263\pi$
$32$	1.00000	0.176777
$33$	0.312841	0.0544586
$34$	−3.85853	−0.661732
$35$	10.0104	1.69207
$36$	−2.71102	−0.451837
$37$	9.58463	1.57570	0.787852	−	0.615865i	$-0.211193\pi$
$37$	9.58463	1.57570	0.787852	+	0.615865i	$0.211193\pi$
$38$	5.75005	0.932781
$39$	1.24268	0.198987
$40$	2.09045	0.330530
$41$	6.05842	0.946167	0.473083	−	0.881018i	$-0.343141\pi$
$41$	6.05842	0.946167	0.473083	+	0.881018i	$0.343141\pi$
$42$	−2.57421	−0.397210
$43$	−1.50188	−0.229034	−0.114517	−	0.993421i	$-0.536532\pi$
$43$	−1.50188	−0.229034	−0.114517	+	0.993421i	$0.536532\pi$
$44$	−0.581956	−0.0877331
$45$	−5.66726	−0.844826
$46$	−6.40957	−0.945040
$47$	7.67306	1.11923	0.559616	−	0.828752i	$-0.310949\pi$
$47$	7.67306	1.11923	0.559616	+	0.828752i	$0.310949\pi$
$48$	−0.537568	−0.0775912
$49$	15.9310	2.27585
$50$	−0.630001	−0.0890956
$51$	2.07422	0.290449
$52$	−2.31166	−0.320570
$53$	11.3320	1.55657	0.778284	−	0.627912i	$-0.216090\pi$
$53$	11.3320	1.55657	0.778284	+	0.627912i	$0.216090\pi$
$54$	3.07006	0.417782
$55$	−1.21655	−0.164040
$56$	4.78863	0.639908
$57$	−3.09104	−0.409418
$58$	10.0252	1.31638
$59$	3.28283	0.427389	0.213694	−	0.976901i	$-0.431450\pi$
$59$	3.28283	0.427389	0.213694	+	0.976901i	$0.431450\pi$
$60$	−1.12376	−0.145077
$61$	4.19020	0.536500	0.268250	−	0.963349i	$-0.413555\pi$
$61$	4.19020	0.536500	0.268250	+	0.963349i	$0.413555\pi$
$62$	−0.952590	−0.120979
$63$	−12.9821	−1.63559
$64$	1.00000	0.125000
$65$	−4.83243	−0.599389
$66$	0.312841	0.0385080
$67$	−1.67325	−0.204420	−0.102210	−	0.994763i	$-0.532591\pi$
$67$	−1.67325	−0.204420	−0.102210	+	0.994763i	$0.532591\pi$
$68$	−3.85853	−0.467915
$69$	3.44558	0.414799
$70$	10.0104	1.19647
$71$	4.15060	0.492586	0.246293	−	0.969195i	$-0.420788\pi$
$71$	4.15060	0.492586	0.246293	+	0.969195i	$0.420788\pi$
$72$	−2.71102	−0.319497
$73$	−10.8377	−1.26846	−0.634229	−	0.773145i	$-0.718682\pi$
$73$	−10.8377	−1.26846	−0.634229	+	0.773145i	$0.718682\pi$
$74$	9.58463	1.11419
$75$	0.338668	0.0391061
$76$	5.75005	0.659576
$77$	−2.78677	−0.317582
$78$	1.24268	0.140705
$79$	−9.18393	−1.03327	−0.516637	−	0.856205i	$-0.672816\pi$
$79$	−9.18393	−1.03327	−0.516637	+	0.856205i	$0.672816\pi$
$80$	2.09045	0.233720
$81$	6.48270	0.720300
$82$	6.05842	0.669041
$83$	−1.46287	−0.160571	−0.0802856	−	0.996772i	$-0.525583\pi$
$83$	−1.46287	−0.160571	−0.0802856	+	0.996772i	$0.525583\pi$
$84$	−2.57421	−0.280870
$85$	−8.06608	−0.874889
$86$	−1.50188	−0.161951
$87$	−5.38925	−0.577788
$88$	−0.581956	−0.0620367
$89$	−2.79907	−0.296701	−0.148351	−	0.988935i	$-0.547396\pi$
$89$	−2.79907	−0.296701	−0.148351	+	0.988935i	$0.547396\pi$
$90$	−5.66726	−0.597382
$91$	−11.0697	−1.16042
$92$	−6.40957	−0.668244
$93$	0.512082	0.0531004
$94$	7.67306	0.791416
$95$	12.0202	1.23325
$96$	−0.537568	−0.0548653
$97$	8.47436	0.860441	0.430221	−	0.902724i	$-0.358436\pi$
$97$	8.47436	0.860441	0.430221	+	0.902724i	$0.358436\pi$
$98$	15.9310	1.60927
$99$	1.57769	0.158564
$100$	−0.630001	−0.0630001
$101$	−17.7498	−1.76617	−0.883087	−	0.469210i	$-0.844539\pi$
$101$	−17.7498	−1.76617	−0.883087	+	0.469210i	$0.844539\pi$
$102$	2.07422	0.205379
$103$	7.12422	0.701970	0.350985	−	0.936381i	$-0.385847\pi$
$103$	7.12422	0.701970	0.350985	+	0.936381i	$0.385847\pi$
$104$	−2.31166	−0.226677
$105$	−5.38128	−0.525159
$106$	11.3320	1.10066
$107$	−7.24404	−0.700307	−0.350154	−	0.936692i	$-0.613871\pi$
$107$	−7.24404	−0.700307	−0.350154	+	0.936692i	$0.613871\pi$
$108$	3.07006	0.295417
$109$	2.02982	0.194421	0.0972107	−	0.995264i	$-0.469008\pi$
$109$	2.02982	0.194421	0.0972107	+	0.995264i	$0.469008\pi$
$110$	−1.21655	−0.115994
$111$	−5.15239	−0.489043
$112$	4.78863	0.452483
$113$	−7.48945	−0.704548	−0.352274	−	0.935897i	$-0.614591\pi$
$113$	−7.48945	−0.704548	−0.352274	+	0.935897i	$0.614591\pi$
$114$	−3.09104	−0.289502
$115$	−13.3989	−1.24946
$116$	10.0252	0.930821
$117$	6.26697	0.579382
$118$	3.28283	0.302209
$119$	−18.4771	−1.69379
$120$	−1.12376	−0.102585
$121$	−10.6613	−0.969212
$122$	4.19020	0.379362
$123$	−3.25681	−0.293657
$124$	−0.952590	−0.0855451
$125$	−11.7693	−1.05267
$126$	−12.9821	−1.15654
$127$	4.97804	0.441730	0.220865	−	0.975304i	$-0.429112\pi$
$127$	4.97804	0.441730	0.220865	+	0.975304i	$0.429112\pi$
$128$	1.00000	0.0883883
$129$	0.807360	0.0710841
$130$	−4.83243	−0.423832
$131$	−3.75157	−0.327777	−0.163888	−	0.986479i	$-0.552404\pi$
$131$	−3.75157	−0.327777	−0.163888	+	0.986479i	$0.552404\pi$
$132$	0.312841	0.0272293
$133$	27.5349	2.38757
$134$	−1.67325	−0.144547
$135$	6.41782	0.552358
$136$	−3.85853	−0.330866
$137$	11.6109	0.991983	0.495992	−	0.868327i	$-0.334805\pi$
$137$	11.6109	0.991983	0.495992	+	0.868327i	$0.334805\pi$
$138$	3.44558	0.293307
$139$	6.38640	0.541688	0.270844	−	0.962623i	$-0.412697\pi$
$139$	6.38640	0.541688	0.270844	+	0.962623i	$0.412697\pi$
$140$	10.0104	0.846034
$141$	−4.12479	−0.347370
$142$	4.15060	0.348311
$143$	1.34529	0.112499
$144$	−2.71102	−0.225918
$145$	20.9573	1.74041
$146$	−10.8377	−0.896935
$147$	−8.56398	−0.706346
$148$	9.58463	0.787852
$149$	−5.56681	−0.456051	−0.228025	−	0.973655i	$-0.573227\pi$
$149$	−5.56681	−0.456051	−0.228025	+	0.973655i	$0.573227\pi$
$150$	0.338668	0.0276522
$151$	1.93643	0.157584	0.0787922	−	0.996891i	$-0.474894\pi$
$151$	1.93643	0.157584	0.0787922	+	0.996891i	$0.474894\pi$
$152$	5.75005	0.466390
$153$	10.4606	0.845686
$154$	−2.78677	−0.224564
$155$	−1.99135	−0.159949
$156$	1.24268	0.0994937
$157$	13.3455	1.06509	0.532543	−	0.846403i	$-0.321236\pi$
$157$	13.3455	1.06509	0.532543	+	0.846403i	$0.321236\pi$
$158$	−9.18393	−0.730634
$159$	−6.09171	−0.483104
$160$	2.09045	0.165265
$161$	−30.6931	−2.41895
$162$	6.48270	0.509329
$163$	4.55341	0.356650	0.178325	−	0.983972i	$-0.442932\pi$
$163$	4.55341	0.356650	0.178325	+	0.983972i	$0.442932\pi$
$164$	6.05842	0.473083
$165$	0.653979	0.0509122
$166$	−1.46287	−0.113541
$167$	−4.04708	−0.313173	−0.156586	−	0.987664i	$-0.550049\pi$
$167$	−4.04708	−0.313173	−0.156586	+	0.987664i	$0.550049\pi$
$168$	−2.57421	−0.198605
$169$	−7.65621	−0.588939
$170$	−8.06608	−0.618640
$171$	−15.5885	−1.19208
$172$	−1.50188	−0.114517
$173$	11.0711	0.841719	0.420860	−	0.907126i	$-0.361729\pi$
$173$	11.0711	0.841719	0.420860	+	0.907126i	$0.361729\pi$
$174$	−5.38925	−0.408558
$175$	−3.01684	−0.228052
$176$	−0.581956	−0.0438666
$177$	−1.76475	−0.132646
$178$	−2.79907	−0.209799
$179$	4.93855	0.369125	0.184562	−	0.982821i	$-0.440913\pi$
$179$	4.93855	0.369125	0.184562	+	0.982821i	$0.440913\pi$
$180$	−5.66726	−0.422413
$181$	−9.74308	−0.724198	−0.362099	−	0.932140i	$-0.617940\pi$
$181$	−9.74308	−0.724198	−0.362099	+	0.932140i	$0.617940\pi$
$182$	−11.0697	−0.820541
$183$	−2.25251	−0.166511
$184$	−6.40957	−0.472520
$185$	20.0362	1.47309
$186$	0.512082	0.0375476
$187$	2.24549	0.164207
$188$	7.67306	0.559616
$189$	14.7014	1.06937
$190$	12.0202	0.872038
$191$	−20.2009	−1.46169	−0.730845	−	0.682544i	$-0.760874\pi$
$191$	−20.2009	−1.46169	−0.730845	+	0.682544i	$0.760874\pi$
$192$	−0.537568	−0.0387956
$193$	16.1642	1.16353	0.581764	−	0.813358i	$-0.302363\pi$
$193$	16.1642	1.16353	0.581764	+	0.813358i	$0.302363\pi$
$194$	8.47436	0.608424
$195$	2.59776	0.186029
$196$	15.9310	1.13793
$197$	−6.54669	−0.466432	−0.233216	−	0.972425i	$-0.574925\pi$
$197$	−6.54669	−0.466432	−0.233216	+	0.972425i	$0.574925\pi$
$198$	1.57769	0.112122
$199$	5.81975	0.412551	0.206276	−	0.978494i	$-0.433866\pi$
$199$	5.81975	0.412551	0.206276	+	0.978494i	$0.433866\pi$
$200$	−0.630001	−0.0445478
$201$	0.899486	0.0634449
$202$	−17.7498	−1.24887
$203$	48.0072	3.36944
$204$	2.07422	0.145225
$205$	12.6649	0.884552
$206$	7.12422	0.496368
$207$	17.3765	1.20775
$208$	−2.31166	−0.160285
$209$	−3.34627	−0.231467
$210$	−5.38128	−0.371343
$211$	0.737638	0.0507811	0.0253906	−	0.999678i	$-0.491917\pi$
$211$	0.737638	0.0507811	0.0253906	+	0.999678i	$0.491917\pi$
$212$	11.3320	0.778284
$213$	−2.23123	−0.152881
$214$	−7.24404	−0.495192
$215$	−3.13960	−0.214119
$216$	3.07006	0.208891
$217$	−4.56160	−0.309662
$218$	2.02982	0.137477
$219$	5.82600	0.393685
$220$	−1.21655	−0.0820199
$221$	8.91963	0.599999
$222$	−5.15239	−0.345806
$223$	−18.4618	−1.23629	−0.618146	−	0.786063i	$-0.712116\pi$
$223$	−18.4618	−1.23629	−0.618146	+	0.786063i	$0.712116\pi$
$224$	4.78863	0.319954
$225$	1.70795	0.113863
$226$	−7.48945	−0.498191
$227$	10.9028	0.723647	0.361824	−	0.932247i	$-0.382154\pi$
$227$	10.9028	0.723647	0.361824	+	0.932247i	$0.382154\pi$
$228$	−3.09104	−0.204709
$229$	6.82912	0.451281	0.225641	−	0.974211i	$-0.427552\pi$
$229$	6.82912	0.451281	0.225641	+	0.974211i	$0.427552\pi$
$230$	−13.3989	−0.883499
$231$	1.49808	0.0985663
$232$	10.0252	0.658190
$233$	−21.2998	−1.39539	−0.697697	−	0.716393i	$-0.745792\pi$
$233$	−21.2998	−1.39539	−0.697697	+	0.716393i	$0.745792\pi$
$234$	6.26697	0.409685
$235$	16.0402	1.04635
$236$	3.28283	0.213694
$237$	4.93699	0.320692
$238$	−18.4771	−1.19769
$239$	−26.9090	−1.74060	−0.870301	−	0.492521i	$-0.836076\pi$
$239$	−26.9090	−1.74060	−0.870301	+	0.492521i	$0.836076\pi$
$240$	−1.12376	−0.0725385
$241$	−8.28611	−0.533756	−0.266878	−	0.963730i	$-0.585992\pi$
$241$	−8.28611	−0.533756	−0.266878	+	0.963730i	$0.585992\pi$
$242$	−10.6613	−0.685336
$243$	−12.6951	−0.814389
$244$	4.19020	0.268250
$245$	33.3030	2.12765
$246$	−3.25681	−0.207647
$247$	−13.2922	−0.845761
$248$	−0.952590	−0.0604895
$249$	0.786393	0.0498356
$250$	−11.7693	−0.744353
$251$	−25.2201	−1.59188	−0.795941	−	0.605374i	$-0.793023\pi$
$251$	−25.2201	−1.59188	−0.795941	+	0.605374i	$0.793023\pi$
$252$	−12.9821	−0.817794
$253$	3.73009	0.234509
$254$	4.97804	0.312350
$255$	4.33607	0.271535
$256$	1.00000	0.0625000
$257$	−29.0465	−1.81187	−0.905934	−	0.423418i	$-0.860830\pi$
$257$	−29.0465	−1.81187	−0.905934	+	0.423418i	$0.860830\pi$
$258$	0.807360	0.0502640
$259$	45.8973	2.85192
$260$	−4.83243	−0.299695
$261$	−27.1786	−1.68232
$262$	−3.75157	−0.231773
$263$	−11.4824	−0.708035	−0.354018	−	0.935239i	$-0.615185\pi$
$263$	−11.4824	−0.708035	−0.354018	+	0.935239i	$0.615185\pi$
$264$	0.312841	0.0192540
$265$	23.6890	1.45520
$266$	27.5349	1.68827
$267$	1.50469	0.0920856
$268$	−1.67325	−0.102210
$269$	7.73768	0.471775	0.235887	−	0.971780i	$-0.424200\pi$
$269$	7.73768	0.471775	0.235887	+	0.971780i	$0.424200\pi$
$270$	6.41782	0.390576
$271$	−4.33187	−0.263143	−0.131571	−	0.991307i	$-0.542002\pi$
$271$	−4.33187	−0.263143	−0.131571	+	0.991307i	$0.542002\pi$
$272$	−3.85853	−0.233958
$273$	5.95072	0.360154
$274$	11.6109	0.701438
$275$	0.366633	0.0221088
$276$	3.44558	0.207400
$277$	1.34011	0.0805193	0.0402597	−	0.999189i	$-0.487181\pi$
$277$	1.34011	0.0805193	0.0402597	+	0.999189i	$0.487181\pi$
$278$	6.38640	0.383031
$279$	2.58249	0.154610
$280$	10.0104	0.598237
$281$	−21.8099	−1.30107	−0.650535	−	0.759476i	$-0.725455\pi$
$281$	−21.8099	−1.30107	−0.650535	+	0.759476i	$0.725455\pi$
$282$	−4.12479	−0.245628
$283$	−1.66731	−0.0991115	−0.0495558	−	0.998771i	$-0.515781\pi$
$283$	−1.66731	−0.0991115	−0.0495558	+	0.998771i	$0.515781\pi$
$284$	4.15060	0.246293
$285$	−6.46168	−0.382757
$286$	1.34529	0.0795485
$287$	29.0116	1.71250
$288$	−2.71102	−0.159748
$289$	−2.11175	−0.124220
$290$	20.9573	1.23066
$291$	−4.55554	−0.267051
$292$	−10.8377	−0.634229
$293$	−23.2412	−1.35777	−0.678883	−	0.734246i	$-0.737536\pi$
$293$	−23.2412	−1.35777	−0.678883	+	0.734246i	$0.737536\pi$
$294$	−8.56398	−0.499462
$295$	6.86261	0.399557
$296$	9.58463	0.557095
$297$	−1.78664	−0.103671
$298$	−5.56681	−0.322477
$299$	14.8168	0.856877
$300$	0.338668	0.0195530
$301$	−7.19193	−0.414536
$302$	1.93643	0.111429
$303$	9.54173	0.548158
$304$	5.75005	0.329788
$305$	8.75941	0.501562
$306$	10.4606	0.597990
$307$	20.3231	1.15990	0.579949	−	0.814653i	$-0.303072\pi$
$307$	20.3231	1.15990	0.579949	+	0.814653i	$0.303072\pi$
$308$	−2.78677	−0.158791
$309$	−3.82975	−0.217867
$310$	−1.99135	−0.113101
$311$	11.1883	0.634432	0.317216	−	0.948353i	$-0.397252\pi$
$311$	11.1883	0.634432	0.317216	+	0.948353i	$0.397252\pi$
$312$	1.24268	0.0703527
$313$	−24.9258	−1.40889	−0.704446	−	0.709758i	$-0.748804\pi$
$313$	−24.9258	−1.40889	−0.704446	+	0.709758i	$0.748804\pi$
$314$	13.3455	0.753130
$315$	−27.1384	−1.52908
$316$	−9.18393	−0.516637
$317$	31.5113	1.76985	0.884927	−	0.465730i	$-0.154208\pi$
$317$	31.5113	1.76985	0.884927	+	0.465730i	$0.154208\pi$
$318$	−6.09171	−0.341606
$319$	−5.83425	−0.326655
$320$	2.09045	0.116860
$321$	3.89416	0.217351
$322$	−30.6931	−1.71046
$323$	−22.1867	−1.23450
$324$	6.48270	0.360150
$325$	1.45635	0.0807839
$326$	4.55341	0.252190
$327$	−1.09117	−0.0603416
$328$	6.05842	0.334521
$329$	36.7435	2.02573
$330$	0.653979	0.0360004
$331$	21.5314	1.18347	0.591737	−	0.806131i	$-0.298442\pi$
$331$	21.5314	1.18347	0.591737	+	0.806131i	$0.298442\pi$
$332$	−1.46287	−0.0802856
$333$	−25.9841	−1.42392
$334$	−4.04708	−0.221446
$335$	−3.49786	−0.191108
$336$	−2.57421	−0.140435
$337$	−26.5485	−1.44619	−0.723094	−	0.690749i	$-0.757281\pi$
$337$	−26.5485	−1.44619	−0.723094	+	0.690749i	$0.757281\pi$
$338$	−7.65621	−0.416443
$339$	4.02609	0.218667
$340$	−8.06608	−0.437445
$341$	0.554365	0.0300206
$342$	−15.5885	−0.842929
$343$	42.7672	2.30921
$344$	−1.50188	−0.0809757
$345$	7.20283	0.387787
$346$	11.0711	0.595185
$347$	−6.57360	−0.352889	−0.176445	−	0.984311i	$-0.556460\pi$
$347$	−6.57360	−0.352889	−0.176445	+	0.984311i	$0.556460\pi$
$348$	−5.38925	−0.288894
$349$	7.74933	0.414812	0.207406	−	0.978255i	$-0.433498\pi$
$349$	7.74933	0.414812	0.207406	+	0.978255i	$0.433498\pi$
$350$	−3.01684	−0.161257
$351$	−7.09695	−0.378807
$352$	−0.581956	−0.0310183
$353$	15.2890	0.813753	0.406876	−	0.913483i	$-0.366618\pi$
$353$	15.2890	0.813753	0.406876	+	0.913483i	$0.366618\pi$
$354$	−1.76475	−0.0937952
$355$	8.67664	0.460508
$356$	−2.79907	−0.148351
$357$	9.93268	0.525693
$358$	4.93855	0.261011
$359$	22.1511	1.16909	0.584546	−	0.811360i	$-0.301273\pi$
$359$	22.1511	1.16909	0.584546	+	0.811360i	$0.301273\pi$
$360$	−5.66726	−0.298691
$361$	14.0630	0.740160
$362$	−9.74308	−0.512085
$363$	5.73119	0.300809
$364$	−11.0697	−0.580210
$365$	−22.6557	−1.18585
$366$	−2.25251	−0.117741
$367$	−9.81128	−0.512145	−0.256072	−	0.966658i	$-0.582429\pi$
$367$	−9.81128	−0.512145	−0.256072	+	0.966658i	$0.582429\pi$
$368$	−6.40957	−0.334122
$369$	−16.4245	−0.855026
$370$	20.0362	1.04163
$371$	54.2647	2.81728
$372$	0.512082	0.0265502
$373$	7.53525	0.390161	0.195080	−	0.980787i	$-0.437503\pi$
$373$	7.53525	0.390161	0.195080	+	0.980787i	$0.437503\pi$
$374$	2.24549	0.116112
$375$	6.32678	0.326713
$376$	7.67306	0.395708
$377$	−23.1750	−1.19357
$378$	14.7014	0.756158
$379$	23.0683	1.18494	0.592468	−	0.805594i	$-0.298153\pi$
$379$	23.0683	1.18494	0.592468	+	0.805594i	$0.298153\pi$
$380$	12.0202	0.616624
$381$	−2.67604	−0.137097
$382$	−20.2009	−1.03357
$383$	7.46431	0.381408	0.190704	−	0.981648i	$-0.438923\pi$
$383$	7.46431	0.381408	0.190704	+	0.981648i	$0.438923\pi$
$384$	−0.537568	−0.0274326
$385$	−5.82562	−0.296901
$386$	16.1642	0.822738
$387$	4.07162	0.206972
$388$	8.47436	0.430221
$389$	−6.13264	−0.310937	−0.155469	−	0.987841i	$-0.549689\pi$
$389$	−6.13264	−0.310937	−0.155469	+	0.987841i	$0.549689\pi$
$390$	2.59776	0.131543
$391$	24.7315	1.25073
$392$	15.9310	0.804636
$393$	2.01673	0.101730
$394$	−6.54669	−0.329817
$395$	−19.1986	−0.965986
$396$	1.57769	0.0792821
$397$	2.65026	0.133013	0.0665064	−	0.997786i	$-0.478815\pi$
$397$	2.65026	0.133013	0.0665064	+	0.997786i	$0.478815\pi$
$398$	5.81975	0.291718
$399$	−14.8019	−0.741019
$400$	−0.630001	−0.0315001
$401$	−3.44324	−0.171947	−0.0859736	−	0.996297i	$-0.527400\pi$
$401$	−3.44324	−0.171947	−0.0859736	+	0.996297i	$0.527400\pi$
$402$	0.899486	0.0448623
$403$	2.20207	0.109693
$404$	−17.7498	−0.883087
$405$	13.5518	0.673393
$406$	48.0072	2.38256
$407$	−5.57783	−0.276483
$408$	2.07422	0.102689
$409$	−21.1312	−1.04487	−0.522434	−	0.852680i	$-0.674976\pi$
$409$	−21.1312	−1.04487	−0.522434	+	0.852680i	$0.674976\pi$
$410$	12.6649	0.625473
$411$	−6.24163	−0.307877
$412$	7.12422	0.350985
$413$	15.7203	0.773544
$414$	17.3765	0.854008
$415$	−3.05807	−0.150115
$416$	−2.31166	−0.113339
$417$	−3.43313	−0.168121
$418$	−3.34627	−0.163672
$419$	−21.6150	−1.05596	−0.527980	−	0.849257i	$-0.677051\pi$
$419$	−21.6150	−1.05596	−0.527980	+	0.849257i	$0.677051\pi$
$420$	−5.38128	−0.262579
$421$	24.0618	1.17270	0.586350	−	0.810058i	$-0.300564\pi$
$421$	24.0618	1.17270	0.586350	+	0.810058i	$0.300564\pi$
$422$	0.737638	0.0359077
$423$	−20.8018	−1.01142
$424$	11.3320	0.550330
$425$	2.43088	0.117915
$426$	−2.23123	−0.108103
$427$	20.0653	0.971028
$428$	−7.24404	−0.350154
$429$	−0.723183	−0.0349156
$430$	−3.13960	−0.151405
$431$	15.5569	0.749349	0.374675	−	0.927156i	$-0.377754\pi$
$431$	15.5569	0.749349	0.374675	+	0.927156i	$0.377754\pi$
$432$	3.07006	0.147708
$433$	−28.2854	−1.35931	−0.679656	−	0.733531i	$-0.737871\pi$
$433$	−28.2854	−1.35931	−0.679656	+	0.733531i	$0.737871\pi$
$434$	−4.56160	−0.218964
$435$	−11.2660	−0.540162
$436$	2.02982	0.0972107
$437$	−36.8554	−1.76303
$438$	5.82600	0.278377
$439$	−4.87276	−0.232564	−0.116282	−	0.993216i	$-0.537098\pi$
$439$	−4.87276	−0.232564	−0.116282	+	0.993216i	$0.537098\pi$
$440$	−1.21655	−0.0579968
$441$	−43.1892	−2.05663
$442$	8.91963	0.424263
$443$	−4.14981	−0.197163	−0.0985816	−	0.995129i	$-0.531431\pi$
$443$	−4.14981	−0.197163	−0.0985816	+	0.995129i	$0.531431\pi$
$444$	−5.15239	−0.244522
$445$	−5.85133	−0.277380
$446$	−18.4618	−0.874190
$447$	2.99254	0.141542
$448$	4.78863	0.226242
$449$	31.9899	1.50970	0.754849	−	0.655899i	$-0.227710\pi$
$449$	31.9899	1.50970	0.754849	+	0.655899i	$0.227710\pi$
$450$	1.70795	0.0805134
$451$	−3.52573	−0.166020
$452$	−7.48945	−0.352274
$453$	−1.04096	−0.0489086
$454$	10.9028	0.511696
$455$	−23.1407	−1.08485
$456$	−3.09104	−0.144751
$457$	−18.9655	−0.887166	−0.443583	−	0.896233i	$-0.646293\pi$
$457$	−18.9655	−0.887166	−0.443583	+	0.896233i	$0.646293\pi$
$458$	6.82912	0.319104
$459$	−11.8459	−0.552920
$460$	−13.3989	−0.624728
$461$	−3.79039	−0.176536	−0.0882679	−	0.996097i	$-0.528133\pi$
$461$	−3.79039	−0.176536	−0.0882679	+	0.996097i	$0.528133\pi$
$462$	1.49808	0.0696969
$463$	18.4978	0.859666	0.429833	−	0.902908i	$-0.358572\pi$
$463$	18.4978	0.859666	0.429833	+	0.902908i	$0.358572\pi$
$464$	10.0252	0.465410
$465$	1.07048	0.0496425
$466$	−21.2998	−0.986692
$467$	−35.2104	−1.62934	−0.814671	−	0.579923i	$-0.803083\pi$
$467$	−35.2104	−1.62934	−0.814671	+	0.579923i	$0.803083\pi$
$468$	6.26697	0.289691
$469$	−8.01258	−0.369987
$470$	16.0402	0.739878
$471$	−7.17411	−0.330566
$472$	3.28283	0.151105
$473$	0.874025	0.0401877
$474$	4.93699	0.226763
$475$	−3.62254	−0.166213
$476$	−18.4771	−0.846895
$477$	−30.7213	−1.40663
$478$	−26.9090	−1.23079
$479$	−18.2487	−0.833806	−0.416903	−	0.908951i	$-0.636884\pi$
$479$	−18.2487	−0.833806	−0.416903	+	0.908951i	$0.636884\pi$
$480$	−1.12376	−0.0512924
$481$	−22.1565	−1.01025
$482$	−8.28611	−0.377422
$483$	16.4996	0.750758
$484$	−10.6613	−0.484606
$485$	17.7153	0.804409
$486$	−12.6951	−0.575860
$487$	39.9157	1.80875	0.904377	−	0.426735i	$-0.140336\pi$
$487$	39.9157	1.80875	0.904377	+	0.426735i	$0.140336\pi$
$488$	4.19020	0.189681
$489$	−2.44776	−0.110692
$490$	33.3030	1.50448
$491$	−29.3738	−1.32562	−0.662810	−	0.748787i	$-0.730636\pi$
$491$	−29.3738	−1.32562	−0.662810	+	0.748787i	$0.730636\pi$
$492$	−3.25681	−0.146829
$493$	−38.6827	−1.74218
$494$	−13.2922	−0.598043
$495$	3.29810	0.148238
$496$	−0.952590	−0.0427725
$497$	19.8757	0.891547
$498$	0.786393	0.0352391
$499$	13.5242	0.605426	0.302713	−	0.953082i	$-0.402108\pi$
$499$	13.5242	0.605426	0.302713	+	0.953082i	$0.402108\pi$
$500$	−11.7693	−0.526337
$501$	2.17558	0.0971978
$502$	−25.2201	−1.12563
$503$	41.0282	1.82936	0.914678	−	0.404182i	$-0.132444\pi$
$503$	41.0282	1.82936	0.914678	+	0.404182i	$0.132444\pi$
$504$	−12.9821	−0.578268
$505$	−37.1052	−1.65116
$506$	3.73009	0.165823
$507$	4.11573	0.182786
$508$	4.97804	0.220865
$509$	18.7424	0.830744	0.415372	−	0.909652i	$-0.363651\pi$
$509$	18.7424	0.830744	0.415372	+	0.909652i	$0.363651\pi$
$510$	4.33607	0.192004
$511$	−51.8978	−2.29582
$512$	1.00000	0.0441942
$513$	17.6530	0.779399
$514$	−29.0465	−1.28118
$515$	14.8928	0.656257
$516$	0.807360	0.0355420
$517$	−4.46538	−0.196387
$518$	45.8973	2.01661
$519$	−5.95146	−0.261240
$520$	−4.83243	−0.211916
$521$	7.15293	0.313376	0.156688	−	0.987648i	$-0.449918\pi$
$521$	7.15293	0.313376	0.156688	+	0.987648i	$0.449918\pi$
$522$	−27.1786	−1.18958
$523$	5.25777	0.229906	0.114953	−	0.993371i	$-0.463328\pi$
$523$	5.25777	0.229906	0.114953	+	0.993371i	$0.463328\pi$
$524$	−3.75157	−0.163888
$525$	1.62176	0.0707793
$526$	−11.4824	−0.500657
$527$	3.67560	0.160111
$528$	0.312841	0.0136146
$529$	18.0826	0.786202
$530$	23.6890	1.02898
$531$	−8.89983	−0.386220
$532$	27.5349	1.19379
$533$	−14.0050	−0.606626
$534$	1.50469	0.0651144
$535$	−15.1433	−0.654703
$536$	−1.67325	−0.0722734
$537$	−2.65481	−0.114563
$538$	7.73768	0.333595
$539$	−9.27113	−0.399336
$540$	6.41782	0.276179
$541$	3.72040	0.159952	0.0799762	−	0.996797i	$-0.474516\pi$
$541$	3.72040	0.159952	0.0799762	+	0.996797i	$0.474516\pi$
$542$	−4.33187	−0.186070
$543$	5.23757	0.224766
$544$	−3.85853	−0.165433
$545$	4.24325	0.181761
$546$	5.95072	0.254667
$547$	−23.3325	−0.997626	−0.498813	−	0.866710i	$-0.666231\pi$
$547$	−23.3325	−0.997626	−0.498813	+	0.866710i	$0.666231\pi$
$548$	11.6109	0.495992
$549$	−11.3597	−0.484820
$550$	0.366633	0.0156333
$551$	57.6456	2.45579
$552$	3.44558	0.146654
$553$	−43.9785	−1.87015
$554$	1.34011	0.0569358
$555$	−10.7708	−0.457197
$556$	6.38640	0.270844
$557$	−3.20266	−0.135701	−0.0678504	−	0.997696i	$-0.521614\pi$
$557$	−3.20266	−0.135701	−0.0678504	+	0.997696i	$0.521614\pi$
$558$	2.58249	0.109326
$559$	3.47183	0.146843
$560$	10.0104	0.423017
$561$	−1.20711	−0.0509640
$562$	−21.8099	−0.919996
$563$	−9.39431	−0.395923	−0.197961	−	0.980210i	$-0.563432\pi$
$563$	−9.39431	−0.395923	−0.197961	+	0.980210i	$0.563432\pi$
$564$	−4.12479	−0.173685
$565$	−15.6563	−0.658668
$566$	−1.66731	−0.0700824
$567$	31.0432	1.30369
$568$	4.15060	0.174155
$569$	3.66622	0.153696	0.0768479	−	0.997043i	$-0.475514\pi$
$569$	3.66622	0.153696	0.0768479	+	0.997043i	$0.475514\pi$
$570$	−6.46168	−0.270650
$571$	1.28854	0.0539238	0.0269619	−	0.999636i	$-0.491417\pi$
$571$	1.28854	0.0539238	0.0269619	+	0.999636i	$0.491417\pi$
$572$	1.34529	0.0562493
$573$	10.8594	0.453657
$574$	29.0116	1.21092
$575$	4.03804	0.168398
$576$	−2.71102	−0.112959
$577$	30.1383	1.25467	0.627337	−	0.778748i	$-0.284145\pi$
$577$	30.1383	1.25467	0.627337	+	0.778748i	$0.284145\pi$
$578$	−2.11175	−0.0878371
$579$	−8.68938	−0.361118
$580$	20.9573	0.870205
$581$	−7.00516	−0.290623
$582$	−4.55554	−0.188833
$583$	−6.59472	−0.273125
$584$	−10.8377	−0.448467
$585$	13.1008	0.541652
$586$	−23.2412	−0.960086
$587$	−41.0613	−1.69478	−0.847390	−	0.530971i	$-0.821827\pi$
$587$	−41.0613	−1.69478	−0.847390	+	0.530971i	$0.821827\pi$
$588$	−8.56398	−0.353173
$589$	−5.47744	−0.225694
$590$	6.86261	0.282529
$591$	3.51929	0.144764
$592$	9.58463	0.393926
$593$	8.76682	0.360010	0.180005	−	0.983666i	$-0.442389\pi$
$593$	8.76682	0.360010	0.180005	+	0.983666i	$0.442389\pi$
$594$	−1.78664	−0.0733067
$595$	−38.6255	−1.58349
$596$	−5.56681	−0.228025
$597$	−3.12851	−0.128041
$598$	14.8168	0.605903
$599$	25.4963	1.04175	0.520876	−	0.853633i	$-0.325606\pi$
$599$	25.4963	1.04175	0.520876	+	0.853633i	$0.325606\pi$
$600$	0.338668	0.0138261
$601$	−19.1742	−0.782131	−0.391065	−	0.920363i	$-0.627893\pi$
$601$	−19.1742	−0.782131	−0.391065	+	0.920363i	$0.627893\pi$
$602$	−7.19193	−0.293121
$603$	4.53622	0.184729
$604$	1.93643	0.0787922
$605$	−22.2870	−0.906096
$606$	9.54173	0.387606
$607$	7.35411	0.298494	0.149247	−	0.988800i	$-0.452315\pi$
$607$	7.35411	0.298494	0.149247	+	0.988800i	$0.452315\pi$
$608$	5.75005	0.233195
$609$	−25.8071	−1.04576
$610$	8.75941	0.354658
$611$	−17.7375	−0.717584
$612$	10.4606	0.422843
$613$	−19.1464	−0.773317	−0.386659	−	0.922223i	$-0.626371\pi$
$613$	−19.1464	−0.773317	−0.386659	+	0.922223i	$0.626371\pi$
$614$	20.3231	0.820172
$615$	−6.80822	−0.274534
$616$	−2.78677	−0.112282
$617$	−48.3822	−1.94779	−0.973896	−	0.226993i	$-0.927110\pi$
$617$	−48.3822	−1.94779	−0.973896	+	0.226993i	$0.927110\pi$
$618$	−3.82975	−0.154055
$619$	−6.48599	−0.260694	−0.130347	−	0.991468i	$-0.541609\pi$
$619$	−6.48599	−0.260694	−0.130347	+	0.991468i	$0.541609\pi$
$620$	−1.99135	−0.0799743
$621$	−19.6778	−0.789642
$622$	11.1883	0.448611
$623$	−13.4037	−0.537009
$624$	1.24268	0.0497469
$625$	−21.4531	−0.858124
$626$	−24.9258	−0.996236
$627$	1.79885	0.0718391
$628$	13.3455	0.532543
$629$	−36.9826	−1.47459
$630$	−27.1384	−1.08122
$631$	23.5351	0.936919	0.468459	−	0.883485i	$-0.344809\pi$
$631$	23.5351	0.936919	0.468459	+	0.883485i	$0.344809\pi$
$632$	−9.18393	−0.365317
$633$	−0.396531	−0.0157607
$634$	31.5113	1.25148
$635$	10.4064	0.412964
$636$	−6.09171	−0.241552
$637$	−36.8271	−1.45914
$638$	−5.83425	−0.230980
$639$	−11.2524	−0.445137
$640$	2.09045	0.0826325
$641$	−24.3693	−0.962531	−0.481266	−	0.876575i	$-0.659823\pi$
$641$	−24.3693	−0.962531	−0.481266	+	0.876575i	$0.659823\pi$
$642$	3.89416	0.153690
$643$	−12.0369	−0.474688	−0.237344	−	0.971426i	$-0.576277\pi$
$643$	−12.0369	−0.474688	−0.237344	+	0.971426i	$0.576277\pi$
$644$	−30.6931	−1.20948
$645$	1.68775	0.0664550
$646$	−22.1867	−0.872925
$647$	40.1450	1.57826	0.789132	−	0.614224i	$-0.210531\pi$
$647$	40.1450	1.57826	0.789132	+	0.614224i	$0.210531\pi$
$648$	6.48270	0.254664
$649$	−1.91046	−0.0749923
$650$	1.45635	0.0571228
$651$	2.45217	0.0961081
$652$	4.55341	0.178325
$653$	−2.90967	−0.113864	−0.0569321	−	0.998378i	$-0.518132\pi$
$653$	−2.90967	−0.113864	−0.0569321	+	0.998378i	$0.518132\pi$
$654$	−1.09117	−0.0426680
$655$	−7.84249	−0.306432
$656$	6.05842	0.236542
$657$	29.3812	1.14627
$658$	36.7435	1.43241
$659$	−12.0940	−0.471115	−0.235558	−	0.971860i	$-0.575692\pi$
$659$	−12.0940	−0.471115	−0.235558	+	0.971860i	$0.575692\pi$
$660$	0.653979	0.0254561
$661$	4.83656	0.188120	0.0940602	−	0.995567i	$-0.470015\pi$
$661$	4.83656	0.188120	0.0940602	+	0.995567i	$0.470015\pi$
$662$	21.5314	0.836843
$663$	−4.79490	−0.186219
$664$	−1.46287	−0.0567705
$665$	57.5604	2.23209
$666$	−25.9841	−1.00686
$667$	−64.2575	−2.48806
$668$	−4.04708	−0.156586
$669$	9.92445	0.383702
$670$	−3.49786	−0.135134
$671$	−2.43851	−0.0941376
$672$	−2.57421	−0.0993025
$673$	−30.7943	−1.18703	−0.593517	−	0.804821i	$-0.702261\pi$
$673$	−30.7943	−1.18703	−0.593517	+	0.804821i	$0.702261\pi$
$674$	−26.5485	−1.02261
$675$	−1.93414	−0.0744452
$676$	−7.65621	−0.294469
$677$	−40.7819	−1.56738	−0.783688	−	0.621154i	$-0.786664\pi$
$677$	−40.7819	−1.56738	−0.783688	+	0.621154i	$0.786664\pi$
$678$	4.02609	0.154621
$679$	40.5806	1.55734
$680$	−8.06608	−0.309320
$681$	−5.86102	−0.224595
$682$	0.554365	0.0212277
$683$	7.30609	0.279560	0.139780	−	0.990183i	$-0.455361\pi$
$683$	7.30609	0.279560	0.139780	+	0.990183i	$0.455361\pi$
$684$	−15.5885	−0.596041
$685$	24.2720	0.927385
$686$	42.7672	1.63286
$687$	−3.67112	−0.140062
$688$	−1.50188	−0.0572585
$689$	−26.1958	−0.997979
$690$	7.20283	0.274207
$691$	19.4948	0.741617	0.370808	−	0.928709i	$-0.379081\pi$
$691$	19.4948	0.741617	0.370808	+	0.928709i	$0.379081\pi$
$692$	11.0711	0.420860
$693$	7.55499	0.286991
$694$	−6.57360	−0.249530
$695$	13.3505	0.506413
$696$	−5.38925	−0.204279
$697$	−23.3766	−0.885452
$698$	7.74933	0.293316
$699$	11.4501	0.433081
$700$	−3.01684	−0.114026
$701$	−18.8736	−0.712845	−0.356423	−	0.934325i	$-0.616004\pi$
$701$	−18.8736	−0.712845	−0.356423	+	0.934325i	$0.616004\pi$
$702$	−7.09695	−0.267857
$703$	55.1121	2.07859
$704$	−0.581956	−0.0219333
$705$	−8.62269	−0.324749
$706$	15.2890	0.575410
$707$	−84.9973	−3.19665
$708$	−1.76475	−0.0663232
$709$	−37.5882	−1.41165	−0.705827	−	0.708384i	$-0.749424\pi$
$709$	−37.5882	−1.41165	−0.705827	+	0.708384i	$0.749424\pi$
$710$	8.67664	0.325628
$711$	24.8978	0.933742
$712$	−2.79907	−0.104900
$713$	6.10569	0.228660
$714$	9.93268	0.371721
$715$	2.81226	0.105173
$716$	4.93855	0.184562
$717$	14.4654	0.540222
$718$	22.1511	0.826673
$719$	−43.8644	−1.63587	−0.817933	−	0.575313i	$-0.804880\pi$
$719$	−43.8644	−1.63587	−0.817933	+	0.575313i	$0.804880\pi$
$720$	−5.66726	−0.211206
$721$	34.1152	1.27052
$722$	14.0630	0.523372
$723$	4.45435	0.165659
$724$	−9.74308	−0.362099
$725$	−6.31592	−0.234567
$726$	5.73119	0.212704
$727$	14.4206	0.534830	0.267415	−	0.963581i	$-0.413831\pi$
$727$	14.4206	0.534830	0.267415	+	0.963581i	$0.413831\pi$
$728$	−11.0697	−0.410271
$729$	−12.6236	−0.467542
$730$	−22.6557	−0.838526
$731$	5.79503	0.214337
$732$	−2.25251	−0.0832553
$733$	8.00926	0.295829	0.147914	−	0.989000i	$-0.452744\pi$
$733$	8.00926	0.295829	0.147914	+	0.989000i	$0.452744\pi$
$734$	−9.81128	−0.362141
$735$	−17.9026	−0.660348
$736$	−6.40957	−0.236260
$737$	0.973758	0.0358688
$738$	−16.4245	−0.604595
$739$	−1.54153	−0.0567060	−0.0283530	−	0.999598i	$-0.509026\pi$
$739$	−1.54153	−0.0567060	−0.0283530	+	0.999598i	$0.509026\pi$
$740$	20.0362	0.736547
$741$	7.14545	0.262495
$742$	54.2647	1.99212
$743$	−45.5314	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$743$	−45.5314	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$744$	0.512082	0.0187738
$745$	−11.6372	−0.426353
$746$	7.53525	0.275885
$747$	3.96588	0.145104
$748$	2.24549	0.0821034
$749$	−34.6890	−1.26751
$750$	6.32678	0.231021
$751$	51.0034	1.86114	0.930570	−	0.366113i	$-0.119312\pi$
$751$	51.0034	1.86114	0.930570	+	0.366113i	$0.119312\pi$
$752$	7.67306	0.279808
$753$	13.5575	0.494064
$754$	−23.1750	−0.843984
$755$	4.04802	0.147322
$756$	14.7014	0.534684
$757$	2.46429	0.0895660	0.0447830	−	0.998997i	$-0.485740\pi$
$757$	2.46429	0.0895660	0.0447830	+	0.998997i	$0.485740\pi$
$758$	23.0683	0.837877
$759$	−2.00518	−0.0727833
$760$	12.0202	0.436019
$761$	−35.8073	−1.29801	−0.649007	−	0.760782i	$-0.724816\pi$
$761$	−35.8073	−1.29801	−0.649007	+	0.760782i	$0.724816\pi$
$762$	−2.67604	−0.0969426
$763$	9.72006	0.351890
$764$	−20.2009	−0.730845
$765$	21.8673	0.790614
$766$	7.46431	0.269697
$767$	−7.58881	−0.274016
$768$	−0.537568	−0.0193978
$769$	−50.7229	−1.82911	−0.914557	−	0.404457i	$-0.867461\pi$
$769$	−50.7229	−1.82911	−0.914557	+	0.404457i	$0.867461\pi$
$770$	−5.82562	−0.209941
$771$	15.6144	0.562340
$772$	16.1642	0.581764
$773$	41.1977	1.48178	0.740889	−	0.671627i	$-0.234404\pi$
$773$	41.1977	1.48178	0.740889	+	0.671627i	$0.234404\pi$
$774$	4.07162	0.146351
$775$	0.600133	0.0215574
$776$	8.47436	0.304212
$777$	−24.6729	−0.885135
$778$	−6.13264	−0.219866
$779$	34.8362	1.24814
$780$	2.59776	0.0930147
$781$	−2.41547	−0.0864321
$782$	24.7315	0.884398
$783$	30.7781	1.09992
$784$	15.9310	0.568964
$785$	27.8982	0.995728
$786$	2.01673	0.0719342
$787$	31.5906	1.12608	0.563041	−	0.826429i	$-0.309631\pi$
$787$	31.5906	1.12608	0.563041	+	0.826429i	$0.309631\pi$
$788$	−6.54669	−0.233216
$789$	6.17257	0.219749
$790$	−19.1986	−0.683055
$791$	−35.8642	−1.27518
$792$	1.57769	0.0560609
$793$	−9.68633	−0.343972
$794$	2.65026	0.0940542
$795$	−12.7345	−0.451644
$796$	5.81975	0.206276
$797$	35.0158	1.24032	0.620162	−	0.784474i	$-0.287067\pi$
$797$	35.0158	1.24032	0.620162	+	0.784474i	$0.287067\pi$
$798$	−14.8019	−0.523980
$799$	−29.6067	−1.04741
$800$	−0.630001	−0.0222739
$801$	7.58835	0.268121
$802$	−3.44324	−0.121585
$803$	6.30707	0.222571
$804$	0.899486	0.0317224
$805$	−64.1625	−2.26143
$806$	2.20207	0.0775645
$807$	−4.15953	−0.146422
$808$	−17.7498	−0.624437
$809$	6.08795	0.214041	0.107020	−	0.994257i	$-0.465869\pi$
$809$	6.08795	0.214041	0.107020	+	0.994257i	$0.465869\pi$
$810$	13.5518	0.476161
$811$	7.17922	0.252097	0.126048	−	0.992024i	$-0.459771\pi$
$811$	7.17922	0.252097	0.126048	+	0.992024i	$0.459771\pi$
$812$	48.0072	1.68472
$813$	2.32868	0.0816702
$814$	−5.57783	−0.195503
$815$	9.51869	0.333425
$816$	2.07422	0.0726123
$817$	−8.63585	−0.302130
$818$	−21.1312	−0.738833
$819$	30.0102	1.04864
$820$	12.6649	0.442276
$821$	−29.3610	−1.02471	−0.512353	−	0.858775i	$-0.671226\pi$
$821$	−29.3610	−1.02471	−0.512353	+	0.858775i	$0.671226\pi$
$822$	−6.24163	−0.217702
$823$	−16.7656	−0.584413	−0.292206	−	0.956355i	$-0.594389\pi$
$823$	−16.7656	−0.584413	−0.292206	+	0.956355i	$0.594389\pi$
$824$	7.12422	0.248184
$825$	−0.197090	−0.00686179
$826$	15.7203	0.546979
$827$	34.0707	1.18475	0.592377	−	0.805661i	$-0.298190\pi$
$827$	34.0707	1.18475	0.592377	+	0.805661i	$0.298190\pi$
$828$	17.3765	0.603875
$829$	−42.0349	−1.45993	−0.729966	−	0.683484i	$-0.760464\pi$
$829$	−42.0349	−1.45993	−0.729966	+	0.683484i	$0.760464\pi$
$830$	−3.05807	−0.106147
$831$	−0.720399	−0.0249904
$832$	−2.31166	−0.0801425
$833$	−61.4702	−2.12982
$834$	−3.43313	−0.118879
$835$	−8.46024	−0.292779
$836$	−3.34627	−0.115733
$837$	−2.92451	−0.101086
$838$	−21.6150	−0.746677
$839$	−54.7803	−1.89123	−0.945614	−	0.325292i	$-0.894537\pi$
$839$	−54.7803	−1.89123	−0.945614	+	0.325292i	$0.894537\pi$
$840$	−5.38128	−0.185672
$841$	71.5055	2.46571
$842$	24.0618	0.829225
$843$	11.7243	0.403807
$844$	0.737638	0.0253906
$845$	−16.0049	−0.550587
$846$	−20.8018	−0.715182
$847$	−51.0532	−1.75421
$848$	11.3320	0.389142
$849$	0.896294	0.0307607
$850$	2.43088	0.0833785
$851$	−61.4334	−2.10591
$852$	−2.23123	−0.0764406
$853$	39.4154	1.34956	0.674778	−	0.738020i	$-0.264239\pi$
$853$	39.4154	1.34956	0.674778	+	0.738020i	$0.264239\pi$
$854$	20.0653	0.686620
$855$	−32.5870	−1.11445
$856$	−7.24404	−0.247596
$857$	−45.3610	−1.54950	−0.774752	−	0.632265i	$-0.782125\pi$
$857$	−45.3610	−1.54950	−0.774752	+	0.632265i	$0.782125\pi$
$858$	−0.723183	−0.0246891
$859$	27.0796	0.923945	0.461973	−	0.886894i	$-0.347142\pi$
$859$	27.0796	0.923945	0.461973	+	0.886894i	$0.347142\pi$
$860$	−3.13960	−0.107060
$861$	−15.5957	−0.531499
$862$	15.5569	0.529870
$863$	46.5591	1.58489	0.792446	−	0.609943i	$-0.208807\pi$
$863$	46.5591	1.58489	0.792446	+	0.609943i	$0.208807\pi$
$864$	3.07006	0.104446
$865$	23.1436	0.786906
$866$	−28.2854	−0.961178
$867$	1.13521	0.0385536
$868$	−4.56160	−0.154831
$869$	5.34464	0.181305
$870$	−11.2660	−0.381952
$871$	3.86800	0.131062
$872$	2.02982	0.0687384
$873$	−22.9742	−0.777558
$874$	−36.8554	−1.24665
$875$	−56.3586	−1.90527
$876$	5.82600	0.196842
$877$	23.8160	0.804209	0.402105	−	0.915594i	$-0.368279\pi$
$877$	23.8160	0.804209	0.402105	+	0.915594i	$0.368279\pi$
$878$	−4.87276	−0.164448
$879$	12.4937	0.421403
$880$	−1.21655	−0.0410100
$881$	−7.06206	−0.237927	−0.118963	−	0.992899i	$-0.537957\pi$
$881$	−7.06206	−0.237927	−0.118963	+	0.992899i	$0.537957\pi$
$882$	−43.1892	−1.45426
$883$	−12.1888	−0.410187	−0.205093	−	0.978742i	$-0.565750\pi$
$883$	−12.1888	−0.410187	−0.205093	+	0.978742i	$0.565750\pi$
$884$	8.91963	0.300000
$885$	−3.68912	−0.124008
$886$	−4.14981	−0.139415
$887$	7.32498	0.245949	0.122974	−	0.992410i	$-0.460757\pi$
$887$	7.32498	0.245949	0.122974	+	0.992410i	$0.460757\pi$
$888$	−5.15239	−0.172903
$889$	23.8380	0.799501
$890$	−5.85133	−0.196137
$891$	−3.77264	−0.126388
$892$	−18.4618	−0.618146
$893$	44.1205	1.47644
$894$	2.99254	0.100085
$895$	10.3238	0.345087
$896$	4.78863	0.159977
$897$	−7.96503	−0.265945
$898$	31.9899	1.06752
$899$	−9.54994	−0.318508
$900$	1.70795	0.0569316
$901$	−43.7248	−1.45669
$902$	−3.52573	−0.117394
$903$	3.86615	0.128657
$904$	−7.48945	−0.249095
$905$	−20.3675	−0.677038
$906$	−1.04096	−0.0345836
$907$	−0.821907	−0.0272910	−0.0136455	−	0.999907i	$-0.504344\pi$
$907$	−0.821907	−0.0272910	−0.0136455	+	0.999907i	$0.504344\pi$
$908$	10.9028	0.361824
$909$	48.1201	1.59604
$910$	−23.1407	−0.767107
$911$	50.8377	1.68433	0.842165	−	0.539220i	$-0.181281\pi$
$911$	50.8377	1.68433	0.842165	+	0.539220i	$0.181281\pi$
$912$	−3.09104	−0.102355
$913$	0.851327	0.0281748
$914$	−18.9655	−0.627321
$915$	−4.70878	−0.155667
$916$	6.82912	0.225641
$917$	−17.9649	−0.593253
$918$	−11.8459	−0.390974
$919$	19.7712	0.652193	0.326096	−	0.945337i	$-0.394267\pi$
$919$	19.7712	0.652193	0.326096	+	0.945337i	$0.394267\pi$
$920$	−13.3989	−0.441749
$921$	−10.9250	−0.359992
$922$	−3.79039	−0.124830
$923$	−9.59479	−0.315817
$924$	1.49808	0.0492832
$925$	−6.03833	−0.198539
$926$	18.4978	0.607876
$927$	−19.3139	−0.634352
$928$	10.0252	0.329095
$929$	39.9875	1.31195	0.655974	−	0.754784i	$-0.272258\pi$
$929$	39.9875	1.31195	0.655974	+	0.754784i	$0.272258\pi$
$930$	1.07048	0.0351025
$931$	91.6039	3.00220
$932$	−21.2998	−0.697697
$933$	−6.01448	−0.196905
$934$	−35.2104	−1.15212
$935$	4.69410	0.153514
$936$	6.26697	0.204842
$937$	−10.3140	−0.336942	−0.168471	−	0.985707i	$-0.553883\pi$
$937$	−10.3140	−0.336942	−0.168471	+	0.985707i	$0.553883\pi$
$938$	−8.01258	−0.261620
$939$	13.3993	0.437270
$940$	16.0402	0.523173
$941$	−12.1315	−0.395475	−0.197738	−	0.980255i	$-0.563359\pi$
$941$	−12.1315	−0.395475	−0.197738	+	0.980255i	$0.563359\pi$
$942$	−7.17411	−0.233745
$943$	−38.8319	−1.26454
$944$	3.28283	0.106847
$945$	30.7326	0.999731
$946$	0.874025	0.0284170
$947$	−51.1272	−1.66141	−0.830705	−	0.556713i	$-0.812062\pi$
$947$	−51.1272	−1.66141	−0.830705	+	0.556713i	$0.812062\pi$
$948$	4.93699	0.160346
$949$	25.0531	0.813259
$950$	−3.62254	−0.117531
$951$	−16.9395	−0.549300
$952$	−18.4771	−0.598845
$953$	−11.4739	−0.371677	−0.185838	−	0.982580i	$-0.559500\pi$
$953$	−11.4739	−0.371677	−0.185838	+	0.982580i	$0.559500\pi$
$954$	−30.7213	−0.994638
$955$	−42.2292	−1.36650
$956$	−26.9090	−0.870301
$957$	3.13630	0.101382
$958$	−18.2487	−0.589590
$959$	55.6002	1.79542
$960$	−1.12376	−0.0362692
$961$	−30.0926	−0.970728
$962$	−22.1565	−0.714353
$963$	19.6387	0.632849
$964$	−8.28611	−0.266878
$965$	33.7906	1.08776
$966$	16.4996	0.530866
$967$	25.4615	0.818785	0.409393	−	0.912358i	$-0.365741\pi$
$967$	25.4615	0.818785	0.409393	+	0.912358i	$0.365741\pi$
$968$	−10.6613	−0.342668
$969$	11.9269	0.383146
$970$	17.7153	0.568803
$971$	44.1783	1.41775	0.708875	−	0.705334i	$-0.249203\pi$
$971$	44.1783	1.41775	0.708875	+	0.705334i	$0.249203\pi$
$972$	−12.6951	−0.407195
$973$	30.5821	0.980418
$974$	39.9157	1.27898
$975$	−0.782888	−0.0250725
$976$	4.19020	0.134125
$977$	16.6804	0.533654	0.266827	−	0.963744i	$-0.414025\pi$
$977$	16.6804	0.533654	0.266827	+	0.963744i	$0.414025\pi$
$978$	−2.44776	−0.0782709
$979$	1.62894	0.0520610
$980$	33.3030	1.06383
$981$	−5.50288	−0.175694
$982$	−29.3738	−0.937355
$983$	−41.8041	−1.33334	−0.666672	−	0.745351i	$-0.732282\pi$
$983$	−41.8041	−1.33334	−0.666672	+	0.745351i	$0.732282\pi$
$984$	−3.25681	−0.103823
$985$	−13.6855	−0.436058
$986$	−38.6827	−1.23191
$987$	−19.7521	−0.628716
$988$	−13.2922	−0.422881
$989$	9.62638	0.306101
$990$	3.29810	0.104820
$991$	19.5169	0.619975	0.309988	−	0.950741i	$-0.399675\pi$
$991$	19.5169	0.619975	0.309988	+	0.950741i	$0.399675\pi$
$992$	−0.952590	−0.0302448
$993$	−11.5746	−0.367309
$994$	19.8757	0.630419
$995$	12.1659	0.385686
$996$	0.786393	0.0249178
$997$	−12.7564	−0.404000	−0.202000	−	0.979386i	$-0.564744\pi$
$997$	−12.7564	−0.404000	−0.202000	+	0.979386i	$0.564744\pi$
$998$	13.5242	0.428101
$999$	29.4254	0.930979

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4022.2.a.f.1.17	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4022.2.a.f.1.17	✓	50	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:	\( N \)	\(=\)	\( 4022 = 2 \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4022.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.537568 q^{3} +1.00000 q^{4} +2.09045 q^{5} -0.537568 q^{6} +4.78863 q^{7} +1.00000 q^{8} -2.71102 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.537568 q^{3} +1.00000 q^{4} +2.09045 q^{5} -0.537568 q^{6} +4.78863 q^{7} +1.00000 q^{8} -2.71102 q^{9} +2.09045 q^{10} -0.581956 q^{11} -0.537568 q^{12} -2.31166 q^{13} +4.78863 q^{14} -1.12376 q^{15} +1.00000 q^{16} -3.85853 q^{17} -2.71102 q^{18} +5.75005 q^{19} +2.09045 q^{20} -2.57421 q^{21} -0.581956 q^{22} -6.40957 q^{23} -0.537568 q^{24} -0.630001 q^{25} -2.31166 q^{26} +3.07006 q^{27} +4.78863 q^{28} +10.0252 q^{29} -1.12376 q^{30} -0.952590 q^{31} +1.00000 q^{32} +0.312841 q^{33} -3.85853 q^{34} +10.0104 q^{35} -2.71102 q^{36} +9.58463 q^{37} +5.75005 q^{38} +1.24268 q^{39} +2.09045 q^{40} +6.05842 q^{41} -2.57421 q^{42} -1.50188 q^{43} -0.581956 q^{44} -5.66726 q^{45} -6.40957 q^{46} +7.67306 q^{47} -0.537568 q^{48} +15.9310 q^{49} -0.630001 q^{50} +2.07422 q^{51} -2.31166 q^{52} +11.3320 q^{53} +3.07006 q^{54} -1.21655 q^{55} +4.78863 q^{56} -3.09104 q^{57} +10.0252 q^{58} +3.28283 q^{59} -1.12376 q^{60} +4.19020 q^{61} -0.952590 q^{62} -12.9821 q^{63} +1.00000 q^{64} -4.83243 q^{65} +0.312841 q^{66} -1.67325 q^{67} -3.85853 q^{68} +3.44558 q^{69} +10.0104 q^{70} +4.15060 q^{71} -2.71102 q^{72} -10.8377 q^{73} +9.58463 q^{74} +0.338668 q^{75} +5.75005 q^{76} -2.78677 q^{77} +1.24268 q^{78} -9.18393 q^{79} +2.09045 q^{80} +6.48270 q^{81} +6.05842 q^{82} -1.46287 q^{83} -2.57421 q^{84} -8.06608 q^{85} -1.50188 q^{86} -5.38925 q^{87} -0.581956 q^{88} -2.79907 q^{89} -5.66726 q^{90} -11.0697 q^{91} -6.40957 q^{92} +0.512082 q^{93} +7.67306 q^{94} +12.0202 q^{95} -0.537568 q^{96} +8.47436 q^{97} +15.9310 q^{98} +1.57769 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 50 q^{2} + 18 q^{3} + 50 q^{4} + 11 q^{5} + 18 q^{6} + 30 q^{7} + 50 q^{8} + 66 q^{9}+O(q^{10}) \) 50 * q + 50 * q^2 + 18 * q^3 + 50 * q^4 + 11 * q^5 + 18 * q^6 + 30 * q^7 + 50 * q^8 + 66 * q^9	\( 50 q + 50 q^{2} + 18 q^{3} + 50 q^{4} + 11 q^{5} + 18 q^{6} + 30 q^{7} + 50 q^{8} + 66 q^{9} + 11 q^{10} + 21 q^{11} + 18 q^{12} + 26 q^{13} + 30 q^{14} + 17 q^{15} + 50 q^{16} + 24 q^{17} + 66 q^{18} + 39 q^{19} + 11 q^{20} - q^{21} + 21 q^{22} + 28 q^{23} + 18 q^{24} + 79 q^{25} + 26 q^{26} + 66 q^{27} + 30 q^{28} - 5 q^{29} + 17 q^{30} + 60 q^{31} + 50 q^{32} + 37 q^{33} + 24 q^{34} + 38 q^{35} + 66 q^{36} + 35 q^{37} + 39 q^{38} + 37 q^{39} + 11 q^{40} + 42 q^{41} - q^{42} + 44 q^{43} + 21 q^{44} + 31 q^{45} + 28 q^{46} + 60 q^{47} + 18 q^{48} + 92 q^{49} + 79 q^{50} + 26 q^{51} + 26 q^{52} - 2 q^{53} + 66 q^{54} + 33 q^{55} + 30 q^{56} + 15 q^{57} - 5 q^{58} + 65 q^{59} + 17 q^{60} + 15 q^{61} + 60 q^{62} + 56 q^{63} + 50 q^{64} + 6 q^{65} + 37 q^{66} + 48 q^{67} + 24 q^{68} - 9 q^{69} + 38 q^{70} + 34 q^{71} + 66 q^{72} + 91 q^{73} + 35 q^{74} + 54 q^{75} + 39 q^{76} - 6 q^{77} + 37 q^{78} + 29 q^{79} + 11 q^{80} + 66 q^{81} + 42 q^{82} + 43 q^{83} - q^{84} + 44 q^{86} + 32 q^{87} + 21 q^{88} + 38 q^{89} + 31 q^{90} + 55 q^{91} + 28 q^{92} - 15 q^{93} + 60 q^{94} + 9 q^{95} + 18 q^{96} + 80 q^{97} + 92 q^{98} + 20 q^{99}+O(q^{100}) \) 50 * q + 50 * q^2 + 18 * q^3 + 50 * q^4 + 11 * q^5 + 18 * q^6 + 30 * q^7 + 50 * q^8 + 66 * q^9 + 11 * q^10 + 21 * q^11 + 18 * q^12 + 26 * q^13 + 30 * q^14 + 17 * q^15 + 50 * q^16 + 24 * q^17 + 66 * q^18 + 39 * q^19 + 11 * q^20 - q^21 + 21 * q^22 + 28 * q^23 + 18 * q^24 + 79 * q^25 + 26 * q^26 + 66 * q^27 + 30 * q^28 - 5 * q^29 + 17 * q^30 + 60 * q^31 + 50 * q^32 + 37 * q^33 + 24 * q^34 + 38 * q^35 + 66 * q^36 + 35 * q^37 + 39 * q^38 + 37 * q^39 + 11 * q^40 + 42 * q^41 - q^42 + 44 * q^43 + 21 * q^44 + 31 * q^45 + 28 * q^46 + 60 * q^47 + 18 * q^48 + 92 * q^49 + 79 * q^50 + 26 * q^51 + 26 * q^52 - 2 * q^53 + 66 * q^54 + 33 * q^55 + 30 * q^56 + 15 * q^57 - 5 * q^58 + 65 * q^59 + 17 * q^60 + 15 * q^61 + 60 * q^62 + 56 * q^63 + 50 * q^64 + 6 * q^65 + 37 * q^66 + 48 * q^67 + 24 * q^68 - 9 * q^69 + 38 * q^70 + 34 * q^71 + 66 * q^72 + 91 * q^73 + 35 * q^74 + 54 * q^75 + 39 * q^76 - 6 * q^77 + 37 * q^78 + 29 * q^79 + 11 * q^80 + 66 * q^81 + 42 * q^82 + 43 * q^83 - q^84 + 44 * q^86 + 32 * q^87 + 21 * q^88 + 38 * q^89 + 31 * q^90 + 55 * q^91 + 28 * q^92 - 15 * q^93 + 60 * q^94 + 9 * q^95 + 18 * q^96 + 80 * q^97 + 92 * q^98 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more