LMFDB - Embedded newform 6038.2.a.e.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, 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("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$0$
Dimension:	$70$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.23514 q^{3} +1.00000 q^{4} +3.00438 q^{5} -2.23514 q^{6} +1.56958 q^{7} +1.00000 q^{8} +1.99587 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.23514 q^{3} +1.00000 q^{4} +3.00438 q^{5} -2.23514 q^{6} +1.56958 q^{7} +1.00000 q^{8} +1.99587 q^{9} +3.00438 q^{10} +2.72533 q^{11} -2.23514 q^{12} -0.479948 q^{13} +1.56958 q^{14} -6.71522 q^{15} +1.00000 q^{16} -3.03644 q^{17} +1.99587 q^{18} -1.72356 q^{19} +3.00438 q^{20} -3.50824 q^{21} +2.72533 q^{22} -2.38948 q^{23} -2.23514 q^{24} +4.02629 q^{25} -0.479948 q^{26} +2.24438 q^{27} +1.56958 q^{28} +8.38595 q^{29} -6.71522 q^{30} +1.50751 q^{31} +1.00000 q^{32} -6.09150 q^{33} -3.03644 q^{34} +4.71562 q^{35} +1.99587 q^{36} +1.01246 q^{37} -1.72356 q^{38} +1.07275 q^{39} +3.00438 q^{40} -6.06494 q^{41} -3.50824 q^{42} +7.69447 q^{43} +2.72533 q^{44} +5.99634 q^{45} -2.38948 q^{46} +5.59201 q^{47} -2.23514 q^{48} -4.53641 q^{49} +4.02629 q^{50} +6.78687 q^{51} -0.479948 q^{52} -3.27922 q^{53} +2.24438 q^{54} +8.18792 q^{55} +1.56958 q^{56} +3.85241 q^{57} +8.38595 q^{58} -2.26795 q^{59} -6.71522 q^{60} +12.5528 q^{61} +1.50751 q^{62} +3.13267 q^{63} +1.00000 q^{64} -1.44195 q^{65} -6.09150 q^{66} +7.36646 q^{67} -3.03644 q^{68} +5.34084 q^{69} +4.71562 q^{70} -5.25083 q^{71} +1.99587 q^{72} +14.5216 q^{73} +1.01246 q^{74} -8.99934 q^{75} -1.72356 q^{76} +4.27762 q^{77} +1.07275 q^{78} +3.24512 q^{79} +3.00438 q^{80} -11.0041 q^{81} -6.06494 q^{82} +14.9675 q^{83} -3.50824 q^{84} -9.12260 q^{85} +7.69447 q^{86} -18.7438 q^{87} +2.72533 q^{88} +1.06264 q^{89} +5.99634 q^{90} -0.753318 q^{91} -2.38948 q^{92} -3.36949 q^{93} +5.59201 q^{94} -5.17823 q^{95} -2.23514 q^{96} -17.4249 q^{97} -4.53641 q^{98} +5.43939 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9}+O(q^{10}) $ 70 * q + 70 * q^2 + 25 * q^3 + 70 * q^4 + 18 * q^5 + 25 * q^6 + 50 * q^7 + 70 * q^8 + 89 * q^9	\( 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9} + 18 q^{10} + 41 q^{11} + 25 q^{12} + 41 q^{13} + 50 q^{14} + 13 q^{15} + 70 q^{16} + 40 q^{17} + 89 q^{18} + 55 q^{19} + 18 q^{20} + 2 q^{21} + 41 q^{22} + 41 q^{23} + 25 q^{24} + 104 q^{25} + 41 q^{26} + 82 q^{27} + 50 q^{28} + 11 q^{29} + 13 q^{30} + 78 q^{31} + 70 q^{32} + 45 q^{33} + 40 q^{34} + 25 q^{35} + 89 q^{36} + 46 q^{37} + 55 q^{38} + 19 q^{39} + 18 q^{40} + 51 q^{41} + 2 q^{42} + 68 q^{43} + 41 q^{44} + 37 q^{45} + 41 q^{46} + 69 q^{47} + 25 q^{48} + 126 q^{49} + 104 q^{50} + 36 q^{51} + 41 q^{52} + 23 q^{53} + 82 q^{54} + 42 q^{55} + 50 q^{56} + 14 q^{57} + 11 q^{58} + 89 q^{59} + 13 q^{60} + 32 q^{61} + 78 q^{62} + 106 q^{63} + 70 q^{64} + 18 q^{65} + 45 q^{66} + 90 q^{67} + 40 q^{68} - 12 q^{69} + 25 q^{70} + 54 q^{71} + 89 q^{72} + 94 q^{73} + 46 q^{74} + 72 q^{75} + 55 q^{76} - 16 q^{77} + 19 q^{78} + 54 q^{79} + 18 q^{80} + 102 q^{81} + 51 q^{82} + 60 q^{83} + 2 q^{84} - 5 q^{85} + 68 q^{86} + 9 q^{87} + 41 q^{88} + 77 q^{89} + 37 q^{90} + 54 q^{91} + 41 q^{92} - 2 q^{93} + 69 q^{94} + 39 q^{95} + 25 q^{96} + 139 q^{97} + 126 q^{98} + 60 q^{99}+O(q^{100}) \) 70 * q + 70 * q^2 + 25 * q^3 + 70 * q^4 + 18 * q^5 + 25 * q^6 + 50 * q^7 + 70 * q^8 + 89 * q^9 + 18 * q^10 + 41 * q^11 + 25 * q^12 + 41 * q^13 + 50 * q^14 + 13 * q^15 + 70 * q^16 + 40 * q^17 + 89 * q^18 + 55 * q^19 + 18 * q^20 + 2 * q^21 + 41 * q^22 + 41 * q^23 + 25 * q^24 + 104 * q^25 + 41 * q^26 + 82 * q^27 + 50 * q^28 + 11 * q^29 + 13 * q^30 + 78 * q^31 + 70 * q^32 + 45 * q^33 + 40 * q^34 + 25 * q^35 + 89 * q^36 + 46 * q^37 + 55 * q^38 + 19 * q^39 + 18 * q^40 + 51 * q^41 + 2 * q^42 + 68 * q^43 + 41 * q^44 + 37 * q^45 + 41 * q^46 + 69 * q^47 + 25 * q^48 + 126 * q^49 + 104 * q^50 + 36 * q^51 + 41 * q^52 + 23 * q^53 + 82 * q^54 + 42 * q^55 + 50 * q^56 + 14 * q^57 + 11 * q^58 + 89 * q^59 + 13 * q^60 + 32 * q^61 + 78 * q^62 + 106 * q^63 + 70 * q^64 + 18 * q^65 + 45 * q^66 + 90 * q^67 + 40 * q^68 - 12 * q^69 + 25 * q^70 + 54 * q^71 + 89 * q^72 + 94 * q^73 + 46 * q^74 + 72 * q^75 + 55 * q^76 - 16 * q^77 + 19 * q^78 + 54 * q^79 + 18 * q^80 + 102 * q^81 + 51 * q^82 + 60 * q^83 + 2 * q^84 - 5 * q^85 + 68 * q^86 + 9 * q^87 + 41 * q^88 + 77 * q^89 + 37 * q^90 + 54 * q^91 + 41 * q^92 - 2 * q^93 + 69 * q^94 + 39 * q^95 + 25 * q^96 + 139 * q^97 + 126 * q^98 + 60 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.23514	−1.29046	−0.645230	−	0.763988i	$-0.723239\pi$
$3$	−2.23514	−1.29046	−0.645230	+	0.763988i	$0.723239\pi$
$4$	1.00000	0.500000
$5$	3.00438	1.34360	0.671800	−	0.740733i	$-0.265522\pi$
$5$	3.00438	1.34360	0.671800	+	0.740733i	$0.265522\pi$
$6$	−2.23514	−0.912493
$7$	1.56958	0.593246	0.296623	−	0.954995i	$-0.404140\pi$
$7$	1.56958	0.593246	0.296623	+	0.954995i	$0.404140\pi$
$8$	1.00000	0.353553
$9$	1.99587	0.665289
$10$	3.00438	0.950068
$11$	2.72533	0.821717	0.410859	−	0.911699i	$-0.365229\pi$
$11$	2.72533	0.821717	0.410859	+	0.911699i	$0.365229\pi$
$12$	−2.23514	−0.645230
$13$	−0.479948	−0.133114	−0.0665568	−	0.997783i	$-0.521201\pi$
$13$	−0.479948	−0.133114	−0.0665568	+	0.997783i	$0.521201\pi$
$14$	1.56958	0.419488
$15$	−6.71522	−1.73386
$16$	1.00000	0.250000
$17$	−3.03644	−0.736444	−0.368222	−	0.929738i	$-0.620033\pi$
$17$	−3.03644	−0.736444	−0.368222	+	0.929738i	$0.620033\pi$
$18$	1.99587	0.470430
$19$	−1.72356	−0.395412	−0.197706	−	0.980261i	$-0.563349\pi$
$19$	−1.72356	−0.395412	−0.197706	+	0.980261i	$0.563349\pi$
$20$	3.00438	0.671800
$21$	−3.50824	−0.765561
$22$	2.72533	0.581042
$23$	−2.38948	−0.498242	−0.249121	−	0.968472i	$-0.580142\pi$
$23$	−2.38948	−0.498242	−0.249121	+	0.968472i	$0.580142\pi$
$24$	−2.23514	−0.456247
$25$	4.02629	0.805258
$26$	−0.479948	−0.0941255
$27$	2.24438	0.431932
$28$	1.56958	0.296623
$29$	8.38595	1.55723	0.778616	−	0.627501i	$-0.215922\pi$
$29$	8.38595	1.55723	0.778616	+	0.627501i	$0.215922\pi$
$30$	−6.71522	−1.22603
$31$	1.50751	0.270756	0.135378	−	0.990794i	$-0.456775\pi$
$31$	1.50751	0.270756	0.135378	+	0.990794i	$0.456775\pi$
$32$	1.00000	0.176777
$33$	−6.09150	−1.06039
$34$	−3.03644	−0.520745
$35$	4.71562	0.797085
$36$	1.99587	0.332644
$37$	1.01246	0.166448	0.0832238	−	0.996531i	$-0.473478\pi$
$37$	1.01246	0.166448	0.0832238	+	0.996531i	$0.473478\pi$
$38$	−1.72356	−0.279599
$39$	1.07275	0.171778
$40$	3.00438	0.475034
$41$	−6.06494	−0.947185	−0.473593	−	0.880744i	$-0.657043\pi$
$41$	−6.06494	−0.947185	−0.473593	+	0.880744i	$0.657043\pi$
$42$	−3.50824	−0.541333
$43$	7.69447	1.17340	0.586698	−	0.809806i	$-0.300428\pi$
$43$	7.69447	1.17340	0.586698	+	0.809806i	$0.300428\pi$
$44$	2.72533	0.410859
$45$	5.99634	0.893881
$46$	−2.38948	−0.352310
$47$	5.59201	0.815678	0.407839	−	0.913054i	$-0.366282\pi$
$47$	5.59201	0.815678	0.407839	+	0.913054i	$0.366282\pi$
$48$	−2.23514	−0.322615
$49$	−4.53641	−0.648059
$50$	4.02629	0.569404
$51$	6.78687	0.950352
$52$	−0.479948	−0.0665568
$53$	−3.27922	−0.450436	−0.225218	−	0.974308i	$-0.572309\pi$
$53$	−3.27922	−0.450436	−0.225218	+	0.974308i	$0.572309\pi$
$54$	2.24438	0.305422
$55$	8.18792	1.10406
$56$	1.56958	0.209744
$57$	3.85241	0.510264
$58$	8.38595	1.10113
$59$	−2.26795	−0.295262	−0.147631	−	0.989043i	$-0.547165\pi$
$59$	−2.26795	−0.295262	−0.147631	+	0.989043i	$0.547165\pi$
$60$	−6.71522	−0.866931
$61$	12.5528	1.60722	0.803610	−	0.595157i	$-0.202910\pi$
$61$	12.5528	1.60722	0.803610	+	0.595157i	$0.202910\pi$
$62$	1.50751	0.191454
$63$	3.13267	0.394680
$64$	1.00000	0.125000
$65$	−1.44195	−0.178851
$66$	−6.09150	−0.749812
$67$	7.36646	0.899957	0.449978	−	0.893039i	$-0.351432\pi$
$67$	7.36646	0.899957	0.449978	+	0.893039i	$0.351432\pi$
$68$	−3.03644	−0.368222
$69$	5.34084	0.642962
$70$	4.71562	0.563624
$71$	−5.25083	−0.623159	−0.311579	−	0.950220i	$-0.600858\pi$
$71$	−5.25083	−0.623159	−0.311579	+	0.950220i	$0.600858\pi$
$72$	1.99587	0.235215
$73$	14.5216	1.69962	0.849811	−	0.527088i	$-0.176716\pi$
$73$	14.5216	1.69962	0.849811	+	0.527088i	$0.176716\pi$
$74$	1.01246	0.117696
$75$	−8.99934	−1.03915
$76$	−1.72356	−0.197706
$77$	4.27762	0.487481
$78$	1.07275	0.121465
$79$	3.24512	0.365104	0.182552	−	0.983196i	$-0.441564\pi$
$79$	3.24512	0.365104	0.182552	+	0.983196i	$0.441564\pi$
$80$	3.00438	0.335900
$81$	−11.0041	−1.22268
$82$	−6.06494	−0.669761
$83$	14.9675	1.64290	0.821448	−	0.570284i	$-0.193167\pi$
$83$	14.9675	1.64290	0.821448	+	0.570284i	$0.193167\pi$
$84$	−3.50824	−0.382780
$85$	−9.12260	−0.989485
$86$	7.69447	0.829716
$87$	−18.7438	−2.00955
$88$	2.72533	0.290521
$89$	1.06264	0.112640	0.0563198	−	0.998413i	$-0.482063\pi$
$89$	1.06264	0.112640	0.0563198	+	0.998413i	$0.482063\pi$
$90$	5.99634	0.632069
$91$	−0.753318	−0.0789691
$92$	−2.38948	−0.249121
$93$	−3.36949	−0.349400
$94$	5.59201	0.576771
$95$	−5.17823	−0.531275
$96$	−2.23514	−0.228123
$97$	−17.4249	−1.76923	−0.884616	−	0.466319i	$-0.845580\pi$
$97$	−17.4249	−1.76923	−0.884616	+	0.466319i	$0.845580\pi$
$98$	−4.53641	−0.458247
$99$	5.43939	0.546679
$100$	4.02629	0.402629
$101$	12.7242	1.26610	0.633050	−	0.774111i	$-0.281803\pi$
$101$	12.7242	1.26610	0.633050	+	0.774111i	$0.281803\pi$
$102$	6.78687	0.672000
$103$	−8.55858	−0.843302	−0.421651	−	0.906758i	$-0.638549\pi$
$103$	−8.55858	−0.843302	−0.421651	+	0.906758i	$0.638549\pi$
$104$	−0.479948	−0.0470628
$105$	−10.5401	−1.02861
$106$	−3.27922	−0.318506
$107$	−8.45008	−0.816900	−0.408450	−	0.912781i	$-0.633931\pi$
$107$	−8.45008	−0.816900	−0.408450	+	0.912781i	$0.633931\pi$
$108$	2.24438	0.215966
$109$	11.2302	1.07566	0.537830	−	0.843053i	$-0.319244\pi$
$109$	11.2302	1.07566	0.537830	+	0.843053i	$0.319244\pi$
$110$	8.18792	0.780687
$111$	−2.26300	−0.214794
$112$	1.56958	0.148312
$113$	−5.00697	−0.471016	−0.235508	−	0.971872i	$-0.575675\pi$
$113$	−5.00697	−0.471016	−0.235508	+	0.971872i	$0.575675\pi$
$114$	3.85241	0.360811
$115$	−7.17892	−0.669437
$116$	8.38595	0.778616
$117$	−0.957912	−0.0885590
$118$	−2.26795	−0.208782
$119$	−4.76594	−0.436893
$120$	−6.71522	−0.613013
$121$	−3.57259	−0.324781
$122$	12.5528	1.13648
$123$	13.5560	1.22231
$124$	1.50751	0.135378
$125$	−2.92539	−0.261655
$126$	3.13267	0.279081
$127$	−1.37445	−0.121963	−0.0609815	−	0.998139i	$-0.519423\pi$
$127$	−1.37445	−0.121963	−0.0609815	+	0.998139i	$0.519423\pi$
$128$	1.00000	0.0883883
$129$	−17.1982	−1.51422
$130$	−1.44195	−0.126467
$131$	10.1333	0.885349	0.442675	−	0.896682i	$-0.354030\pi$
$131$	10.1333	0.885349	0.442675	+	0.896682i	$0.354030\pi$
$132$	−6.09150	−0.530197
$133$	−2.70527	−0.234577
$134$	7.36646	0.636365
$135$	6.74298	0.580343
$136$	−3.03644	−0.260372
$137$	−6.02264	−0.514549	−0.257274	−	0.966338i	$-0.582824\pi$
$137$	−6.02264	−0.514549	−0.257274	+	0.966338i	$0.582824\pi$
$138$	5.34084	0.454643
$139$	3.25915	0.276437	0.138219	−	0.990402i	$-0.455862\pi$
$139$	3.25915	0.276437	0.138219	+	0.990402i	$0.455862\pi$
$140$	4.71562	0.398542
$141$	−12.4989	−1.05260
$142$	−5.25083	−0.440640
$143$	−1.30802	−0.109382
$144$	1.99587	0.166322
$145$	25.1946	2.09230
$146$	14.5216	1.20181
$147$	10.1395	0.836295
$148$	1.01246	0.0832238
$149$	−16.9095	−1.38528	−0.692640	−	0.721284i	$-0.743552\pi$
$149$	−16.9095	−1.38528	−0.692640	+	0.721284i	$0.743552\pi$
$150$	−8.99934	−0.734793
$151$	1.19977	0.0976358	0.0488179	−	0.998808i	$-0.484455\pi$
$151$	1.19977	0.0976358	0.0488179	+	0.998808i	$0.484455\pi$
$152$	−1.72356	−0.139799
$153$	−6.06032	−0.489948
$154$	4.27762	0.344701
$155$	4.52912	0.363788
$156$	1.07275	0.0858889
$157$	14.5963	1.16491	0.582457	−	0.812862i	$-0.302092\pi$
$157$	14.5963	1.16491	0.582457	+	0.812862i	$0.302092\pi$
$158$	3.24512	0.258168
$159$	7.32953	0.581270
$160$	3.00438	0.237517
$161$	−3.75049	−0.295580
$162$	−11.0041	−0.864565
$163$	−2.41927	−0.189492	−0.0947459	−	0.995501i	$-0.530204\pi$
$163$	−2.41927	−0.189492	−0.0947459	+	0.995501i	$0.530204\pi$
$164$	−6.06494	−0.473593
$165$	−18.3012	−1.42474
$166$	14.9675	1.16170
$167$	8.45064	0.653930	0.326965	−	0.945036i	$-0.393974\pi$
$167$	8.45064	0.653930	0.326965	+	0.945036i	$0.393974\pi$
$168$	−3.50824	−0.270667
$169$	−12.7696	−0.982281
$170$	−9.12260	−0.699672
$171$	−3.44000	−0.263063
$172$	7.69447	0.586698
$173$	0.189483	0.0144062	0.00720308	−	0.999974i	$-0.497707\pi$
$173$	0.189483	0.0144062	0.00720308	+	0.999974i	$0.497707\pi$
$174$	−18.7438	−1.42096
$175$	6.31959	0.477716
$176$	2.72533	0.205429
$177$	5.06919	0.381024
$178$	1.06264	0.0796482
$179$	−21.0683	−1.57472	−0.787361	−	0.616492i	$-0.788553\pi$
$179$	−21.0683	−1.57472	−0.787361	+	0.616492i	$0.788553\pi$
$180$	5.99634	0.446941
$181$	4.54442	0.337784	0.168892	−	0.985635i	$-0.445981\pi$
$181$	4.54442	0.337784	0.168892	+	0.985635i	$0.445981\pi$
$182$	−0.753318	−0.0558396
$183$	−28.0573	−2.07405
$184$	−2.38948	−0.176155
$185$	3.04182	0.223639
$186$	−3.36949	−0.247063
$187$	−8.27528	−0.605149
$188$	5.59201	0.407839
$189$	3.52274	0.256242
$190$	−5.17823	−0.375668
$191$	22.8068	1.65025	0.825123	−	0.564954i	$-0.191106\pi$
$191$	22.8068	1.65025	0.825123	+	0.564954i	$0.191106\pi$
$192$	−2.23514	−0.161308
$193$	3.48820	0.251086	0.125543	−	0.992088i	$-0.459933\pi$
$193$	3.48820	0.251086	0.125543	+	0.992088i	$0.459933\pi$
$194$	−17.4249	−1.25104
$195$	3.22295	0.230801
$196$	−4.53641	−0.324029
$197$	20.7955	1.48162	0.740808	−	0.671717i	$-0.234443\pi$
$197$	20.7955	1.48162	0.740808	+	0.671717i	$0.234443\pi$
$198$	5.43939	0.386560
$199$	1.14345	0.0810569	0.0405284	−	0.999178i	$-0.487096\pi$
$199$	1.14345	0.0810569	0.0405284	+	0.999178i	$0.487096\pi$
$200$	4.02629	0.284702
$201$	−16.4651	−1.16136
$202$	12.7242	0.895269
$203$	13.1624	0.923822
$204$	6.78687	0.475176
$205$	−18.2214	−1.27264
$206$	−8.55858	−0.596304
$207$	−4.76909	−0.331475
$208$	−0.479948	−0.0332784
$209$	−4.69727	−0.324917
$210$	−10.5401	−0.727335
$211$	1.29049	0.0888408	0.0444204	−	0.999013i	$-0.485856\pi$
$211$	1.29049	0.0888408	0.0444204	+	0.999013i	$0.485856\pi$
$212$	−3.27922	−0.225218
$213$	11.7364	0.804162
$214$	−8.45008	−0.577636
$215$	23.1171	1.57657
$216$	2.24438	0.152711
$217$	2.36616	0.160625
$218$	11.2302	0.760606
$219$	−32.4578	−2.19329
$220$	8.18792	0.552029
$221$	1.45733	0.0980307
$222$	−2.26300	−0.151882
$223$	−1.73875	−0.116435	−0.0582176	−	0.998304i	$-0.518542\pi$
$223$	−1.73875	−0.116435	−0.0582176	+	0.998304i	$0.518542\pi$
$224$	1.56958	0.104872
$225$	8.03594	0.535729
$226$	−5.00697	−0.333059
$227$	6.56393	0.435663	0.217832	−	0.975986i	$-0.430102\pi$
$227$	6.56393	0.435663	0.217832	+	0.975986i	$0.430102\pi$
$228$	3.85241	0.255132
$229$	−8.08281	−0.534127	−0.267064	−	0.963679i	$-0.586053\pi$
$229$	−8.08281	−0.534127	−0.267064	+	0.963679i	$0.586053\pi$
$230$	−7.17892	−0.473364
$231$	−9.56110	−0.629074
$232$	8.38595	0.550565
$233$	3.38366	0.221671	0.110835	−	0.993839i	$-0.464647\pi$
$233$	3.38366	0.221671	0.110835	+	0.993839i	$0.464647\pi$
$234$	−0.957912	−0.0626206
$235$	16.8005	1.09594
$236$	−2.26795	−0.147631
$237$	−7.25330	−0.471153
$238$	−4.76594	−0.308930
$239$	10.5331	0.681332	0.340666	−	0.940184i	$-0.389347\pi$
$239$	10.5331	0.681332	0.340666	+	0.940184i	$0.389347\pi$
$240$	−6.71522	−0.433465
$241$	−25.8009	−1.66198	−0.830991	−	0.556285i	$-0.812226\pi$
$241$	−25.8009	−1.66198	−0.830991	+	0.556285i	$0.812226\pi$
$242$	−3.57259	−0.229655
$243$	17.8626	1.14589
$244$	12.5528	0.803610
$245$	−13.6291	−0.870731
$246$	13.5560	0.864300
$247$	0.827220	0.0526347
$248$	1.50751	0.0957268
$249$	−33.4545	−2.12009
$250$	−2.92539	−0.185018
$251$	4.94944	0.312406	0.156203	−	0.987725i	$-0.450075\pi$
$251$	4.94944	0.312406	0.156203	+	0.987725i	$0.450075\pi$
$252$	3.13267	0.197340
$253$	−6.51213	−0.409414
$254$	−1.37445	−0.0862409
$255$	20.3903	1.27689
$256$	1.00000	0.0625000
$257$	17.0354	1.06264	0.531319	−	0.847172i	$-0.321697\pi$
$257$	17.0354	1.06264	0.531319	+	0.847172i	$0.321697\pi$
$258$	−17.1982	−1.07072
$259$	1.58914	0.0987444
$260$	−1.44195	−0.0894257
$261$	16.7372	1.03601
$262$	10.1333	0.626037
$263$	14.6775	0.905054	0.452527	−	0.891751i	$-0.350523\pi$
$263$	14.6775	0.905054	0.452527	+	0.891751i	$0.350523\pi$
$264$	−6.09150	−0.374906
$265$	−9.85203	−0.605205
$266$	−2.70527	−0.165871
$267$	−2.37515	−0.145357
$268$	7.36646	0.449978
$269$	8.74269	0.533051	0.266526	−	0.963828i	$-0.414124\pi$
$269$	8.74269	0.533051	0.266526	+	0.963828i	$0.414124\pi$
$270$	6.74298	0.410365
$271$	−2.60752	−0.158396	−0.0791978	−	0.996859i	$-0.525236\pi$
$271$	−2.60752	−0.158396	−0.0791978	+	0.996859i	$0.525236\pi$
$272$	−3.03644	−0.184111
$273$	1.68377	0.101907
$274$	−6.02264	−0.363841
$275$	10.9730	0.661695
$276$	5.34084	0.321481
$277$	−23.3939	−1.40560	−0.702802	−	0.711385i	$-0.748068\pi$
$277$	−23.3939	−1.40560	−0.702802	+	0.711385i	$0.748068\pi$
$278$	3.25915	0.195471
$279$	3.00878	0.180131
$280$	4.71562	0.281812
$281$	−4.95604	−0.295653	−0.147826	−	0.989013i	$-0.547228\pi$
$281$	−4.95604	−0.295653	−0.147826	+	0.989013i	$0.547228\pi$
$282$	−12.4989	−0.744301
$283$	27.8635	1.65631	0.828156	−	0.560498i	$-0.189390\pi$
$283$	27.8635	1.65631	0.828156	+	0.560498i	$0.189390\pi$
$284$	−5.25083	−0.311579
$285$	11.5741	0.685590
$286$	−1.30802	−0.0773446
$287$	−9.51943	−0.561914
$288$	1.99587	0.117608
$289$	−7.78005	−0.457650
$290$	25.1946	1.47948
$291$	38.9472	2.28313
$292$	14.5216	0.849811
$293$	−26.9512	−1.57451	−0.787254	−	0.616629i	$-0.788498\pi$
$293$	−26.9512	−1.57451	−0.787254	+	0.616629i	$0.788498\pi$
$294$	10.1395	0.591350
$295$	−6.81378	−0.396714
$296$	1.01246	0.0588481
$297$	6.11668	0.354926
$298$	−16.9095	−0.979540
$299$	1.14683	0.0663228
$300$	−8.99934	−0.519577
$301$	12.0771	0.696112
$302$	1.19977	0.0690390
$303$	−28.4403	−1.63385
$304$	−1.72356	−0.0988531
$305$	37.7133	2.15946
$306$	−6.06032	−0.346445
$307$	25.5082	1.45583	0.727914	−	0.685668i	$-0.240490\pi$
$307$	25.5082	1.45583	0.727914	+	0.685668i	$0.240490\pi$
$308$	4.27762	0.243740
$309$	19.1296	1.08825
$310$	4.52912	0.257237
$311$	−1.89803	−0.107627	−0.0538137	−	0.998551i	$-0.517138\pi$
$311$	−1.89803	−0.107627	−0.0538137	+	0.998551i	$0.517138\pi$
$312$	1.07275	0.0607327
$313$	1.82320	0.103053	0.0515266	−	0.998672i	$-0.483591\pi$
$313$	1.82320	0.103053	0.0515266	+	0.998672i	$0.483591\pi$
$314$	14.5963	0.823718
$315$	9.41174	0.530292
$316$	3.24512	0.182552
$317$	−13.8918	−0.780242	−0.390121	−	0.920764i	$-0.627567\pi$
$317$	−13.8918	−0.780242	−0.390121	+	0.920764i	$0.627567\pi$
$318$	7.32953	0.411020
$319$	22.8545	1.27960
$320$	3.00438	0.167950
$321$	18.8871	1.05418
$322$	−3.75049	−0.209007
$323$	5.23349	0.291199
$324$	−11.0041	−0.611340
$325$	−1.93241	−0.107191
$326$	−2.41927	−0.133991
$327$	−25.1011	−1.38810
$328$	−6.06494	−0.334880
$329$	8.77711	0.483898
$330$	−18.3012	−1.00745
$331$	4.83471	0.265740	0.132870	−	0.991133i	$-0.457581\pi$
$331$	4.83471	0.265740	0.132870	+	0.991133i	$0.457581\pi$
$332$	14.9675	0.821448
$333$	2.02074	0.110736
$334$	8.45064	0.462399
$335$	22.1316	1.20918
$336$	−3.50824	−0.191390
$337$	−11.1178	−0.605624	−0.302812	−	0.953050i	$-0.597925\pi$
$337$	−11.1178	−0.605624	−0.302812	+	0.953050i	$0.597925\pi$
$338$	−12.7696	−0.694577
$339$	11.1913	0.607827
$340$	−9.12260	−0.494743
$341$	4.10845	0.222485
$342$	−3.44000	−0.186014
$343$	−18.1073	−0.977705
$344$	7.69447	0.414858
$345$	16.0459	0.863883
$346$	0.189483	0.0101867
$347$	5.25984	0.282363	0.141181	−	0.989984i	$-0.454910\pi$
$347$	5.25984	0.282363	0.141181	+	0.989984i	$0.454910\pi$
$348$	−18.7438	−1.00477
$349$	23.9226	1.28055	0.640274	−	0.768147i	$-0.278821\pi$
$349$	23.9226	1.28055	0.640274	+	0.768147i	$0.278821\pi$
$350$	6.31959	0.337797
$351$	−1.07719	−0.0574960
$352$	2.72533	0.145260
$353$	−19.4603	−1.03576	−0.517882	−	0.855452i	$-0.673279\pi$
$353$	−19.4603	−1.03576	−0.517882	+	0.855452i	$0.673279\pi$
$354$	5.06919	0.269425
$355$	−15.7755	−0.837275
$356$	1.06264	0.0563198
$357$	10.6525	0.563793
$358$	−21.0683	−1.11350
$359$	32.7558	1.72879	0.864394	−	0.502816i	$-0.167703\pi$
$359$	32.7558	1.72879	0.864394	+	0.502816i	$0.167703\pi$
$360$	5.99634	0.316035
$361$	−16.0293	−0.843649
$362$	4.54442	0.238850
$363$	7.98525	0.419117
$364$	−0.753318	−0.0394846
$365$	43.6283	2.28361
$366$	−28.0573	−1.46658
$367$	33.7725	1.76291	0.881456	−	0.472267i	$-0.156564\pi$
$367$	33.7725	1.76291	0.881456	+	0.472267i	$0.156564\pi$
$368$	−2.38948	−0.124560
$369$	−12.1048	−0.630151
$370$	3.04182	0.158136
$371$	−5.14701	−0.267219
$372$	−3.36949	−0.174700
$373$	12.0188	0.622310	0.311155	−	0.950359i	$-0.399284\pi$
$373$	12.0188	0.622310	0.311155	+	0.950359i	$0.399284\pi$
$374$	−8.27528	−0.427905
$375$	6.53866	0.337655
$376$	5.59201	0.288386
$377$	−4.02482	−0.207289
$378$	3.52274	0.181190
$379$	19.3835	0.995663	0.497832	−	0.867274i	$-0.334130\pi$
$379$	19.3835	0.995663	0.497832	+	0.867274i	$0.334130\pi$
$380$	−5.17823	−0.265638
$381$	3.07210	0.157389
$382$	22.8068	1.16690
$383$	7.64169	0.390472	0.195236	−	0.980756i	$-0.437453\pi$
$383$	7.64169	0.390472	0.195236	+	0.980756i	$0.437453\pi$
$384$	−2.23514	−0.114062
$385$	12.8516	0.654978
$386$	3.48820	0.177545
$387$	15.3571	0.780646
$388$	−17.4249	−0.884616
$389$	−27.2735	−1.38282	−0.691409	−	0.722463i	$-0.743010\pi$
$389$	−27.2735	−1.38282	−0.691409	+	0.722463i	$0.743010\pi$
$390$	3.22295	0.163201
$391$	7.25552	0.366927
$392$	−4.53641	−0.229123
$393$	−22.6493	−1.14251
$394$	20.7955	1.04766
$395$	9.74956	0.490554
$396$	5.43939	0.273340
$397$	−6.17351	−0.309840	−0.154920	−	0.987927i	$-0.549512\pi$
$397$	−6.17351	−0.309840	−0.154920	+	0.987927i	$0.549512\pi$
$398$	1.14345	0.0573159
$399$	6.04667	0.302712
$400$	4.02629	0.201315
$401$	4.72314	0.235863	0.117931	−	0.993022i	$-0.462374\pi$
$401$	4.72314	0.235863	0.117931	+	0.993022i	$0.462374\pi$
$402$	−16.4651	−0.821205
$403$	−0.723525	−0.0360413
$404$	12.7242	0.633050
$405$	−33.0605	−1.64279
$406$	13.1624	0.653241
$407$	2.75929	0.136773
$408$	6.78687	0.336000
$409$	9.25036	0.457401	0.228700	−	0.973497i	$-0.426552\pi$
$409$	9.25036	0.457401	0.228700	+	0.973497i	$0.426552\pi$
$410$	−18.2214	−0.899890
$411$	13.4615	0.664005
$412$	−8.55858	−0.421651
$413$	−3.55973	−0.175163
$414$	−4.76909	−0.234388
$415$	44.9680	2.20739
$416$	−0.479948	−0.0235314
$417$	−7.28466	−0.356731
$418$	−4.69727	−0.229751
$419$	1.73934	0.0849725	0.0424863	−	0.999097i	$-0.486472\pi$
$419$	1.73934	0.0849725	0.0424863	+	0.999097i	$0.486472\pi$
$420$	−10.5401	−0.514303
$421$	19.5696	0.953765	0.476882	−	0.878967i	$-0.341767\pi$
$421$	19.5696	0.953765	0.476882	+	0.878967i	$0.341767\pi$
$422$	1.29049	0.0628199
$423$	11.1609	0.542661
$424$	−3.27922	−0.159253
$425$	−12.2256	−0.593028
$426$	11.7364	0.568628
$427$	19.7026	0.953477
$428$	−8.45008	−0.408450
$429$	2.92360	0.141153
$430$	23.1171	1.11481
$431$	−1.45623	−0.0701440	−0.0350720	−	0.999385i	$-0.511166\pi$
$431$	−1.45623	−0.0701440	−0.0350720	+	0.999385i	$0.511166\pi$
$432$	2.24438	0.107983
$433$	26.0883	1.25372	0.626861	−	0.779131i	$-0.284339\pi$
$433$	26.0883	1.25372	0.626861	+	0.779131i	$0.284339\pi$
$434$	2.36616	0.113579
$435$	−56.3135	−2.70003
$436$	11.2302	0.537830
$437$	4.11842	0.197011
$438$	−32.4578	−1.55089
$439$	1.81765	0.0867519	0.0433759	−	0.999059i	$-0.486189\pi$
$439$	1.81765	0.0867519	0.0433759	+	0.999059i	$0.486189\pi$
$440$	8.18792	0.390344
$441$	−9.05407	−0.431146
$442$	1.45733	0.0693182
$443$	34.0095	1.61584	0.807919	−	0.589293i	$-0.200594\pi$
$443$	34.0095	1.61584	0.807919	+	0.589293i	$0.200594\pi$
$444$	−2.26300	−0.107397
$445$	3.19257	0.151342
$446$	−1.73875	−0.0823321
$447$	37.7951	1.78765
$448$	1.56958	0.0741558
$449$	3.76947	0.177892	0.0889461	−	0.996036i	$-0.471650\pi$
$449$	3.76947	0.177892	0.0889461	+	0.996036i	$0.471650\pi$
$450$	8.03594	0.378818
$451$	−16.5290	−0.778318
$452$	−5.00697	−0.235508
$453$	−2.68166	−0.125995
$454$	6.56393	0.308060
$455$	−2.26325	−0.106103
$456$	3.85241	0.180406
$457$	−0.944645	−0.0441886	−0.0220943	−	0.999756i	$-0.507033\pi$
$457$	−0.944645	−0.0441886	−0.0220943	+	0.999756i	$0.507033\pi$
$458$	−8.08281	−0.377685
$459$	−6.81493	−0.318094
$460$	−7.17892	−0.334719
$461$	5.90574	0.275058	0.137529	−	0.990498i	$-0.456084\pi$
$461$	5.90574	0.275058	0.137529	+	0.990498i	$0.456084\pi$
$462$	−9.56110	−0.444823
$463$	−16.2473	−0.755077	−0.377538	−	0.925994i	$-0.623229\pi$
$463$	−16.2473	−0.755077	−0.377538	+	0.925994i	$0.623229\pi$
$464$	8.38595	0.389308
$465$	−10.1232	−0.469454
$466$	3.38366	0.156745
$467$	−3.66122	−0.169421	−0.0847106	−	0.996406i	$-0.526997\pi$
$467$	−3.66122	−0.169421	−0.0847106	+	0.996406i	$0.526997\pi$
$468$	−0.957912	−0.0442795
$469$	11.5623	0.533896
$470$	16.8005	0.774949
$471$	−32.6249	−1.50328
$472$	−2.26795	−0.104391
$473$	20.9699	0.964199
$474$	−7.25330	−0.333155
$475$	−6.93956	−0.318409
$476$	−4.76594	−0.218446
$477$	−6.54489	−0.299670
$478$	10.5331	0.481775
$479$	−23.8220	−1.08845	−0.544227	−	0.838938i	$-0.683177\pi$
$479$	−23.8220	−1.08845	−0.544227	+	0.838938i	$0.683177\pi$
$480$	−6.71522	−0.306506
$481$	−0.485929	−0.0221564
$482$	−25.8009	−1.17520
$483$	8.38289	0.381435
$484$	−3.57259	−0.162390
$485$	−52.3511	−2.37714
$486$	17.8626	0.810265
$487$	−8.67073	−0.392908	−0.196454	−	0.980513i	$-0.562943\pi$
$487$	−8.67073	−0.392908	−0.196454	+	0.980513i	$0.562943\pi$
$488$	12.5528	0.568238
$489$	5.40742	0.244532
$490$	−13.6291	−0.615700
$491$	−19.2200	−0.867385	−0.433693	−	0.901061i	$-0.642790\pi$
$491$	−19.2200	−0.867385	−0.433693	+	0.901061i	$0.642790\pi$
$492$	13.5560	0.611153
$493$	−25.4634	−1.14681
$494$	0.827220	0.0372184
$495$	16.3420	0.734517
$496$	1.50751	0.0676891
$497$	−8.24160	−0.369686
$498$	−33.4545	−1.49913
$499$	−36.8229	−1.64842	−0.824210	−	0.566285i	$-0.808380\pi$
$499$	−36.8229	−1.64842	−0.824210	+	0.566285i	$0.808380\pi$
$500$	−2.92539	−0.130827
$501$	−18.8884	−0.843871
$502$	4.94944	0.220904
$503$	19.6351	0.875485	0.437743	−	0.899100i	$-0.355778\pi$
$503$	19.6351	0.875485	0.437743	+	0.899100i	$0.355778\pi$
$504$	3.13267	0.139540
$505$	38.2282	1.70113
$506$	−6.51213	−0.289499
$507$	28.5420	1.26759
$508$	−1.37445	−0.0609815
$509$	−1.44762	−0.0641645	−0.0320823	−	0.999485i	$-0.510214\pi$
$509$	−1.44762	−0.0641645	−0.0320823	+	0.999485i	$0.510214\pi$
$510$	20.3903	0.902899
$511$	22.7928	1.00829
$512$	1.00000	0.0441942
$513$	−3.86833	−0.170791
$514$	17.0354	0.751399
$515$	−25.7132	−1.13306
$516$	−17.1982	−0.757110
$517$	15.2400	0.670257
$518$	1.58914	0.0698228
$519$	−0.423522	−0.0185906
$520$	−1.44195	−0.0632335
$521$	−2.81633	−0.123386	−0.0616929	−	0.998095i	$-0.519650\pi$
$521$	−2.81633	−0.123386	−0.0616929	+	0.998095i	$0.519650\pi$
$522$	16.7372	0.732569
$523$	5.77352	0.252458	0.126229	−	0.992001i	$-0.459713\pi$
$523$	5.77352	0.252458	0.126229	+	0.992001i	$0.459713\pi$
$524$	10.1333	0.442675
$525$	−14.1252	−0.616474
$526$	14.6775	0.639970
$527$	−4.57745	−0.199397
$528$	−6.09150	−0.265098
$529$	−17.2904	−0.751755
$530$	−9.85203	−0.427945
$531$	−4.52652	−0.196434
$532$	−2.70527	−0.117288
$533$	2.91086	0.126083
$534$	−2.37515	−0.102783
$535$	−25.3872	−1.09759
$536$	7.36646	0.318183
$537$	47.0908	2.03212
$538$	8.74269	0.376924
$539$	−12.3632	−0.532521
$540$	6.74298	0.290172
$541$	−17.2773	−0.742810	−0.371405	−	0.928471i	$-0.621124\pi$
$541$	−17.2773	−0.742810	−0.371405	+	0.928471i	$0.621124\pi$
$542$	−2.60752	−0.112003
$543$	−10.1574	−0.435897
$544$	−3.03644	−0.130186
$545$	33.7398	1.44526
$546$	1.68377	0.0720588
$547$	−35.4302	−1.51488	−0.757442	−	0.652902i	$-0.773551\pi$
$547$	−35.4302	−1.51488	−0.757442	+	0.652902i	$0.773551\pi$
$548$	−6.02264	−0.257274
$549$	25.0537	1.06926
$550$	10.9730	0.467889
$551$	−14.4537	−0.615749
$552$	5.34084	0.227321
$553$	5.09348	0.216597
$554$	−23.3939	−0.993913
$555$	−6.79889	−0.288597
$556$	3.25915	0.138219
$557$	25.8944	1.09718	0.548590	−	0.836091i	$-0.315165\pi$
$557$	25.8944	1.09718	0.548590	+	0.836091i	$0.315165\pi$
$558$	3.00878	0.127372
$559$	−3.69294	−0.156195
$560$	4.71562	0.199271
$561$	18.4964	0.780920
$562$	−4.95604	−0.209058
$563$	16.6541	0.701887	0.350943	−	0.936397i	$-0.385861\pi$
$563$	16.6541	0.701887	0.350943	+	0.936397i	$0.385861\pi$
$564$	−12.4989	−0.526300
$565$	−15.0428	−0.632856
$566$	27.8635	1.17119
$567$	−17.2719	−0.725350
$568$	−5.25083	−0.220320
$569$	−45.1686	−1.89357	−0.946784	−	0.321870i	$-0.895689\pi$
$569$	−45.1686	−1.89357	−0.946784	+	0.321870i	$0.895689\pi$
$570$	11.5741	0.484785
$571$	8.70363	0.364235	0.182118	−	0.983277i	$-0.441705\pi$
$571$	8.70363	0.364235	0.182118	+	0.983277i	$0.441705\pi$
$572$	−1.30802	−0.0546909
$573$	−50.9766	−2.12958
$574$	−9.51943	−0.397333
$575$	−9.62076	−0.401214
$576$	1.99587	0.0831611
$577$	14.2242	0.592162	0.296081	−	0.955163i	$-0.404320\pi$
$577$	14.2242	0.592162	0.296081	+	0.955163i	$0.404320\pi$
$578$	−7.78005	−0.323608
$579$	−7.79663	−0.324017
$580$	25.1946	1.04615
$581$	23.4927	0.974641
$582$	38.9472	1.61441
$583$	−8.93696	−0.370131
$584$	14.5216	0.600907
$585$	−2.87793	−0.118988
$586$	−26.9512	−1.11334
$587$	−45.3845	−1.87322	−0.936610	−	0.350373i	$-0.886055\pi$
$587$	−45.3845	−1.87322	−0.936610	+	0.350373i	$0.886055\pi$
$588$	10.1395	0.418147
$589$	−2.59828	−0.107060
$590$	−6.81378	−0.280519
$591$	−46.4809	−1.91197
$592$	1.01246	0.0416119
$593$	33.4306	1.37283	0.686415	−	0.727210i	$-0.259183\pi$
$593$	33.4306	1.37283	0.686415	+	0.727210i	$0.259183\pi$
$594$	6.11668	0.250970
$595$	−14.3187	−0.587008
$596$	−16.9095	−0.692640
$597$	−2.55577	−0.104601
$598$	1.14683	0.0468973
$599$	−31.0187	−1.26739	−0.633695	−	0.773583i	$-0.718463\pi$
$599$	−31.0187	−1.26739	−0.633695	+	0.773583i	$0.718463\pi$
$600$	−8.99934	−0.367396
$601$	19.6955	0.803398	0.401699	−	0.915772i	$-0.368420\pi$
$601$	19.6955	0.803398	0.401699	+	0.915772i	$0.368420\pi$
$602$	12.0771	0.492226
$603$	14.7025	0.598731
$604$	1.19977	0.0488179
$605$	−10.7334	−0.436375
$606$	−28.4403	−1.15531
$607$	47.1364	1.91321	0.956604	−	0.291392i	$-0.0941185\pi$
$607$	47.1364	1.91321	0.956604	+	0.291392i	$0.0941185\pi$
$608$	−1.72356	−0.0698997
$609$	−29.4199	−1.19216
$610$	37.7133	1.52697
$611$	−2.68387	−0.108578
$612$	−6.06032	−0.244974
$613$	9.58673	0.387205	0.193602	−	0.981080i	$-0.437983\pi$
$613$	9.58673	0.387205	0.193602	+	0.981080i	$0.437983\pi$
$614$	25.5082	1.02943
$615$	40.7274	1.64229
$616$	4.27762	0.172350
$617$	−33.5102	−1.34907	−0.674535	−	0.738243i	$-0.735656\pi$
$617$	−33.5102	−1.34907	−0.674535	+	0.738243i	$0.735656\pi$
$618$	19.1296	0.769507
$619$	−42.1908	−1.69579	−0.847896	−	0.530163i	$-0.822131\pi$
$619$	−42.1908	−1.69579	−0.847896	+	0.530163i	$0.822131\pi$
$620$	4.52912	0.181894
$621$	−5.36292	−0.215207
$622$	−1.89803	−0.0761041
$623$	1.66790	0.0668230
$624$	1.07275	0.0429445
$625$	−28.9204	−1.15682
$626$	1.82320	0.0728696
$627$	10.4991	0.419293
$628$	14.5963	0.582457
$629$	−3.07427	−0.122579
$630$	9.41174	0.374973
$631$	15.2308	0.606327	0.303164	−	0.952938i	$-0.401957\pi$
$631$	15.2308	0.606327	0.303164	+	0.952938i	$0.401957\pi$
$632$	3.24512	0.129084
$633$	−2.88442	−0.114646
$634$	−13.8918	−0.551714
$635$	−4.12938	−0.163869
$636$	7.32953	0.290635
$637$	2.17724	0.0862655
$638$	22.8545	0.904817
$639$	−10.4799	−0.414580
$640$	3.00438	0.118758
$641$	−26.8361	−1.05996	−0.529980	−	0.848010i	$-0.677801\pi$
$641$	−26.8361	−1.05996	−0.529980	+	0.848010i	$0.677801\pi$
$642$	18.8871	0.745416
$643$	−11.2722	−0.444531	−0.222265	−	0.974986i	$-0.571345\pi$
$643$	−11.2722	−0.444531	−0.222265	+	0.974986i	$0.571345\pi$
$644$	−3.75049	−0.147790
$645$	−51.6700	−2.03450
$646$	5.23349	0.205909
$647$	−16.6603	−0.654982	−0.327491	−	0.944854i	$-0.606203\pi$
$647$	−16.6603	−0.654982	−0.327491	+	0.944854i	$0.606203\pi$
$648$	−11.0041	−0.432283
$649$	−6.18091	−0.242622
$650$	−1.93241	−0.0757954
$651$	−5.28870	−0.207280
$652$	−2.41927	−0.0947459
$653$	3.26060	0.127597	0.0637986	−	0.997963i	$-0.479678\pi$
$653$	3.26060	0.127597	0.0637986	+	0.997963i	$0.479678\pi$
$654$	−25.1011	−0.981533
$655$	30.4442	1.18955
$656$	−6.06494	−0.236796
$657$	28.9831	1.13074
$658$	8.77711	0.342167
$659$	−11.9465	−0.465369	−0.232685	−	0.972552i	$-0.574751\pi$
$659$	−11.9465	−0.465369	−0.232685	+	0.972552i	$0.574751\pi$
$660$	−18.3012	−0.712372
$661$	11.3439	0.441228	0.220614	−	0.975361i	$-0.429194\pi$
$661$	11.3439	0.441228	0.220614	+	0.975361i	$0.429194\pi$
$662$	4.83471	0.187906
$663$	−3.25734	−0.126505
$664$	14.9675	0.580851
$665$	−8.12766	−0.315177
$666$	2.02074	0.0783019
$667$	−20.0381	−0.775878
$668$	8.45064	0.326965
$669$	3.88635	0.150255
$670$	22.1316	0.855020
$671$	34.2104	1.32068
$672$	−3.50824	−0.135333
$673$	−32.6430	−1.25829	−0.629147	−	0.777286i	$-0.716596\pi$
$673$	−32.6430	−1.25829	−0.629147	+	0.777286i	$0.716596\pi$
$674$	−11.1178	−0.428241
$675$	9.03654	0.347817
$676$	−12.7696	−0.491140
$677$	13.8396	0.531901	0.265950	−	0.963987i	$-0.414314\pi$
$677$	13.8396	0.531901	0.265950	+	0.963987i	$0.414314\pi$
$678$	11.1913	0.429799
$679$	−27.3498	−1.04959
$680$	−9.12260	−0.349836
$681$	−14.6713	−0.562206
$682$	4.10845	0.157321
$683$	14.8849	0.569555	0.284777	−	0.958594i	$-0.408080\pi$
$683$	14.8849	0.569555	0.284777	+	0.958594i	$0.408080\pi$
$684$	−3.44000	−0.131532
$685$	−18.0943	−0.691347
$686$	−18.1073	−0.691342
$687$	18.0662	0.689270
$688$	7.69447	0.293349
$689$	1.57386	0.0599591
$690$	16.0459	0.610857
$691$	−21.8616	−0.831654	−0.415827	−	0.909444i	$-0.636508\pi$
$691$	−21.8616	−0.831654	−0.415827	+	0.909444i	$0.636508\pi$
$692$	0.189483	0.00720308
$693$	8.53756	0.324315
$694$	5.25984	0.199661
$695$	9.79171	0.371421
$696$	−18.7438	−0.710482
$697$	18.4158	0.697549
$698$	23.9226	0.905484
$699$	−7.56295	−0.286057
$700$	6.31959	0.238858
$701$	40.7947	1.54080	0.770398	−	0.637563i	$-0.220058\pi$
$701$	40.7947	1.54080	0.770398	+	0.637563i	$0.220058\pi$
$702$	−1.07719	−0.0406558
$703$	−1.74504	−0.0658154
$704$	2.72533	0.102715
$705$	−37.5515	−1.41427
$706$	−19.4603	−0.732396
$707$	19.9716	0.751109
$708$	5.06919	0.190512
$709$	−45.9439	−1.72546	−0.862731	−	0.505664i	$-0.831248\pi$
$709$	−45.9439	−1.72546	−0.862731	+	0.505664i	$0.831248\pi$
$710$	−15.7755	−0.592043
$711$	6.47682	0.242900
$712$	1.06264	0.0398241
$713$	−3.60217	−0.134902
$714$	10.6525	0.398662
$715$	−3.92977	−0.146965
$716$	−21.0683	−0.787361
$717$	−23.5431	−0.879232
$718$	32.7558	1.22244
$719$	−7.08374	−0.264179	−0.132089	−	0.991238i	$-0.542169\pi$
$719$	−7.08374	−0.264179	−0.132089	+	0.991238i	$0.542169\pi$
$720$	5.99634	0.223470
$721$	−13.4334	−0.500286
$722$	−16.0293	−0.596550
$723$	57.6687	2.14472
$724$	4.54442	0.168892
$725$	33.7643	1.25397
$726$	7.98525	0.296360
$727$	−48.4057	−1.79527	−0.897634	−	0.440741i	$-0.854716\pi$
$727$	−48.4057	−1.79527	−0.897634	+	0.440741i	$0.854716\pi$
$728$	−0.753318	−0.0279198
$729$	−6.91318	−0.256044
$730$	43.6283	1.61476
$731$	−23.3638	−0.864140
$732$	−28.0573	−1.03703
$733$	−30.8534	−1.13960	−0.569799	−	0.821784i	$-0.692979\pi$
$733$	−30.8534	−1.13960	−0.569799	+	0.821784i	$0.692979\pi$
$734$	33.7725	1.24657
$735$	30.4630	1.12364
$736$	−2.38948	−0.0880776
$737$	20.0760	0.739510
$738$	−12.1048	−0.445584
$739$	22.6406	0.832848	0.416424	−	0.909171i	$-0.363283\pi$
$739$	22.6406	0.832848	0.416424	+	0.909171i	$0.363283\pi$
$740$	3.04182	0.111819
$741$	−1.84896	−0.0679231
$742$	−5.14701	−0.188953
$743$	1.91113	0.0701126	0.0350563	−	0.999385i	$-0.488839\pi$
$743$	1.91113	0.0701126	0.0350563	+	0.999385i	$0.488839\pi$
$744$	−3.36949	−0.123532
$745$	−50.8025	−1.86126
$746$	12.0188	0.440040
$747$	29.8731	1.09300
$748$	−8.27528	−0.302574
$749$	−13.2631	−0.484623
$750$	6.53866	0.238758
$751$	19.4792	0.710804	0.355402	−	0.934713i	$-0.384344\pi$
$751$	19.4792	0.710804	0.355402	+	0.934713i	$0.384344\pi$
$752$	5.59201	0.203919
$753$	−11.0627	−0.403148
$754$	−4.02482	−0.146575
$755$	3.60456	0.131183
$756$	3.52274	0.128121
$757$	−28.8056	−1.04696	−0.523479	−	0.852039i	$-0.675366\pi$
$757$	−28.8056	−1.04696	−0.523479	+	0.852039i	$0.675366\pi$
$758$	19.3835	0.704040
$759$	14.5555	0.528333
$760$	−5.17823	−0.187834
$761$	18.5913	0.673934	0.336967	−	0.941516i	$-0.390599\pi$
$761$	18.5913	0.673934	0.336967	+	0.941516i	$0.390599\pi$
$762$	3.07210	0.111291
$763$	17.6267	0.638131
$764$	22.8068	0.825123
$765$	−18.2075	−0.658293
$766$	7.64169	0.276105
$767$	1.08850	0.0393034
$768$	−2.23514	−0.0806538
$769$	−46.3017	−1.66968	−0.834841	−	0.550491i	$-0.814441\pi$
$769$	−46.3017	−1.66968	−0.834841	+	0.550491i	$0.814441\pi$
$770$	12.8516	0.463140
$771$	−38.0765	−1.37129
$772$	3.48820	0.125543
$773$	−9.35781	−0.336577	−0.168288	−	0.985738i	$-0.553824\pi$
$773$	−9.35781	−0.336577	−0.168288	+	0.985738i	$0.553824\pi$
$774$	15.3571	0.552000
$775$	6.06966	0.218029
$776$	−17.4249	−0.625518
$777$	−3.55196	−0.127426
$778$	−27.2735	−0.977801
$779$	10.4533	0.374529
$780$	3.22295	0.115400
$781$	−14.3102	−0.512060
$782$	7.25552	0.259457
$783$	18.8213	0.672618
$784$	−4.53641	−0.162015
$785$	43.8529	1.56518
$786$	−22.6493	−0.807875
$787$	−37.6965	−1.34374	−0.671868	−	0.740671i	$-0.734508\pi$
$787$	−37.6965	−1.34374	−0.671868	+	0.740671i	$0.734508\pi$
$788$	20.7955	0.740808
$789$	−32.8063	−1.16794
$790$	9.74956	0.346874
$791$	−7.85884	−0.279428
$792$	5.43939	0.193280
$793$	−6.02468	−0.213943
$794$	−6.17351	−0.219090
$795$	22.0207	0.780993
$796$	1.14345	0.0405284
$797$	−40.5862	−1.43764	−0.718818	−	0.695198i	$-0.755317\pi$
$797$	−40.5862	−1.43764	−0.718818	+	0.695198i	$0.755317\pi$
$798$	6.04667	0.214050
$799$	−16.9798	−0.600701
$800$	4.02629	0.142351
$801$	2.12089	0.0749378
$802$	4.72314	0.166780
$803$	39.5760	1.39661
$804$	−16.4651	−0.580679
$805$	−11.2679	−0.397141
$806$	−0.723525	−0.0254851
$807$	−19.5412	−0.687882
$808$	12.7242	0.447634
$809$	18.7516	0.659271	0.329635	−	0.944108i	$-0.393074\pi$
$809$	18.7516	0.659271	0.329635	+	0.944108i	$0.393074\pi$
$810$	−33.0605	−1.16163
$811$	16.8291	0.590950	0.295475	−	0.955350i	$-0.404522\pi$
$811$	16.8291	0.590950	0.295475	+	0.955350i	$0.404522\pi$
$812$	13.1624	0.461911
$813$	5.82818	0.204403
$814$	2.75929	0.0967130
$815$	−7.26840	−0.254601
$816$	6.78687	0.237588
$817$	−13.2619	−0.463975
$818$	9.25036	0.323431
$819$	−1.50352	−0.0525373
$820$	−18.2214	−0.636318
$821$	−39.0744	−1.36371	−0.681854	−	0.731489i	$-0.738826\pi$
$821$	−39.0744	−1.36371	−0.681854	+	0.731489i	$0.738826\pi$
$822$	13.4615	0.469522
$823$	−34.8656	−1.21534	−0.607669	−	0.794191i	$-0.707895\pi$
$823$	−34.8656	−1.21534	−0.607669	+	0.794191i	$0.707895\pi$
$824$	−8.55858	−0.298152
$825$	−24.5261	−0.853891
$826$	−3.55973	−0.123859
$827$	17.9526	0.624274	0.312137	−	0.950037i	$-0.398955\pi$
$827$	17.9526	0.624274	0.312137	+	0.950037i	$0.398955\pi$
$828$	−4.76909	−0.165737
$829$	13.2054	0.458643	0.229322	−	0.973351i	$-0.426349\pi$
$829$	13.2054	0.458643	0.229322	+	0.973351i	$0.426349\pi$
$830$	44.9680	1.56086
$831$	52.2888	1.81388
$832$	−0.479948	−0.0166392
$833$	13.7745	0.477259
$834$	−7.28466	−0.252247
$835$	25.3889	0.878620
$836$	−4.69727	−0.162458
$837$	3.38342	0.116948
$838$	1.73934	0.0600847
$839$	−45.2608	−1.56258	−0.781288	−	0.624171i	$-0.785437\pi$
$839$	−45.2608	−1.56258	−0.781288	+	0.624171i	$0.785437\pi$
$840$	−10.5401	−0.363667
$841$	41.3242	1.42497
$842$	19.5696	0.674414
$843$	11.0775	0.381528
$844$	1.29049	0.0444204
$845$	−38.3649	−1.31979
$846$	11.1609	0.383719
$847$	−5.60747	−0.192675
$848$	−3.27922	−0.112609
$849$	−62.2789	−2.13741
$850$	−12.2256	−0.419334
$851$	−2.41926	−0.0829312
$852$	11.7364	0.402081
$853$	12.1187	0.414937	0.207468	−	0.978242i	$-0.433478\pi$
$853$	12.1187	0.414937	0.207468	+	0.978242i	$0.433478\pi$
$854$	19.7026	0.674210
$855$	−10.3351	−0.353451
$856$	−8.45008	−0.288818
$857$	2.39930	0.0819586	0.0409793	−	0.999160i	$-0.486952\pi$
$857$	2.39930	0.0819586	0.0409793	+	0.999160i	$0.486952\pi$
$858$	2.92360	0.0998101
$859$	−13.9438	−0.475756	−0.237878	−	0.971295i	$-0.576452\pi$
$859$	−13.9438	−0.475756	−0.237878	+	0.971295i	$0.576452\pi$
$860$	23.1171	0.788286
$861$	21.2773	0.725128
$862$	−1.45623	−0.0495993
$863$	−18.6895	−0.636197	−0.318099	−	0.948058i	$-0.603044\pi$
$863$	−18.6895	−0.636197	−0.318099	+	0.948058i	$0.603044\pi$
$864$	2.24438	0.0763555
$865$	0.569280	0.0193561
$866$	26.0883	0.886516
$867$	17.3895	0.590580
$868$	2.36616	0.0803126
$869$	8.84401	0.300012
$870$	−56.3135	−1.90921
$871$	−3.53552	−0.119796
$872$	11.2302	0.380303
$873$	−34.7778	−1.17705
$874$	4.11842	0.139308
$875$	−4.59164	−0.155226
$876$	−32.4578	−1.09665
$877$	35.6431	1.20358	0.601790	−	0.798654i	$-0.294454\pi$
$877$	35.6431	1.20358	0.601790	+	0.798654i	$0.294454\pi$
$878$	1.81765	0.0613428
$879$	60.2398	2.03184
$880$	8.18792	0.276015
$881$	−13.1282	−0.442299	−0.221150	−	0.975240i	$-0.570981\pi$
$881$	−13.1282	−0.442299	−0.221150	+	0.975240i	$0.570981\pi$
$882$	−9.05407	−0.304866
$883$	5.58082	0.187810	0.0939048	−	0.995581i	$-0.470065\pi$
$883$	5.58082	0.187810	0.0939048	+	0.995581i	$0.470065\pi$
$884$	1.45733	0.0490154
$885$	15.2298	0.511943
$886$	34.0095	1.14257
$887$	−35.1931	−1.18167	−0.590835	−	0.806793i	$-0.701202\pi$
$887$	−35.1931	−1.18167	−0.590835	+	0.806793i	$0.701202\pi$
$888$	−2.26300	−0.0759411
$889$	−2.15732	−0.0723541
$890$	3.19257	0.107015
$891$	−29.9898	−1.00470
$892$	−1.73875	−0.0582176
$893$	−9.63817	−0.322529
$894$	37.7951	1.26406
$895$	−63.2973	−2.11580
$896$	1.56958	0.0524360
$897$	−2.56333	−0.0855870
$898$	3.76947	0.125789
$899$	12.6419	0.421630
$900$	8.03594	0.267865
$901$	9.95715	0.331721
$902$	−16.5290	−0.550354
$903$	−26.9940	−0.898305
$904$	−5.00697	−0.166529
$905$	13.6532	0.453847
$906$	−2.68166	−0.0890921
$907$	22.0272	0.731401	0.365701	−	0.930733i	$-0.380829\pi$
$907$	22.0272	0.731401	0.365701	+	0.930733i	$0.380829\pi$
$908$	6.56393	0.217832
$909$	25.3957	0.842322
$910$	−2.26325	−0.0750261
$911$	9.68551	0.320895	0.160448	−	0.987044i	$-0.448706\pi$
$911$	9.68551	0.320895	0.160448	+	0.987044i	$0.448706\pi$
$912$	3.85241	0.127566
$913$	40.7913	1.35000
$914$	−0.944645	−0.0312461
$915$	−84.2947	−2.78670
$916$	−8.08281	−0.267064
$917$	15.9050	0.525230
$918$	−6.81493	−0.224926
$919$	57.3600	1.89213	0.946066	−	0.323973i	$-0.105019\pi$
$919$	57.3600	1.89213	0.946066	+	0.323973i	$0.105019\pi$
$920$	−7.17892	−0.236682
$921$	−57.0144	−1.87869
$922$	5.90574	0.194495
$923$	2.52012	0.0829509
$924$	−9.56110	−0.314537
$925$	4.07646	0.134033
$926$	−16.2473	−0.533920
$927$	−17.0818	−0.561039
$928$	8.38595	0.275282
$929$	6.27720	0.205948	0.102974	−	0.994684i	$-0.467164\pi$
$929$	6.27720	0.205948	0.102974	+	0.994684i	$0.467164\pi$
$930$	−10.1232	−0.331954
$931$	7.81879	0.256250
$932$	3.38366	0.110835
$933$	4.24237	0.138889
$934$	−3.66122	−0.119799
$935$	−24.8621	−0.813077
$936$	−0.957912	−0.0313103
$937$	−28.0377	−0.915953	−0.457977	−	0.888964i	$-0.651426\pi$
$937$	−28.0377	−0.915953	−0.457977	+	0.888964i	$0.651426\pi$
$938$	11.5623	0.377521
$939$	−4.07511	−0.132986
$940$	16.8005	0.547972
$941$	−31.3268	−1.02122	−0.510612	−	0.859811i	$-0.670581\pi$
$941$	−31.3268	−1.02122	−0.510612	+	0.859811i	$0.670581\pi$
$942$	−32.6249	−1.06298
$943$	14.4921	0.471927
$944$	−2.26795	−0.0738155
$945$	10.5837	0.344286
$946$	20.9699	0.681792
$947$	9.81084	0.318809	0.159405	−	0.987213i	$-0.449043\pi$
$947$	9.81084	0.318809	0.159405	+	0.987213i	$0.449043\pi$
$948$	−7.25330	−0.235576
$949$	−6.96960	−0.226243
$950$	−6.93956	−0.225149
$951$	31.0502	1.00687
$952$	−4.76594	−0.154465
$953$	−8.05385	−0.260890	−0.130445	−	0.991456i	$-0.541641\pi$
$953$	−8.05385	−0.260890	−0.130445	+	0.991456i	$0.541641\pi$
$954$	−6.54489	−0.211899
$955$	68.5204	2.21727
$956$	10.5331	0.340666
$957$	−51.0830	−1.65128
$958$	−23.8220	−0.769653
$959$	−9.45302	−0.305254
$960$	−6.71522	−0.216733
$961$	−28.7274	−0.926691
$962$	−0.485929	−0.0156670
$963$	−16.8652	−0.543474
$964$	−25.8009	−0.830991
$965$	10.4799	0.337359
$966$	8.38289	0.269715
$967$	−1.37560	−0.0442364	−0.0221182	−	0.999755i	$-0.507041\pi$
$967$	−1.37560	−0.0442364	−0.0221182	+	0.999755i	$0.507041\pi$
$968$	−3.57259	−0.114827
$969$	−11.6976	−0.375781
$970$	−52.3511	−1.68089
$971$	29.5770	0.949170	0.474585	−	0.880210i	$-0.342598\pi$
$971$	29.5770	0.949170	0.474585	+	0.880210i	$0.342598\pi$
$972$	17.8626	0.572944
$973$	5.11549	0.163995
$974$	−8.67073	−0.277828
$975$	4.31921	0.138326
$976$	12.5528	0.401805
$977$	2.37554	0.0760002	0.0380001	−	0.999278i	$-0.487901\pi$
$977$	2.37554	0.0760002	0.0380001	+	0.999278i	$0.487901\pi$
$978$	5.40742	0.172910
$979$	2.89604	0.0925579
$980$	−13.6291	−0.435366
$981$	22.4140	0.715624
$982$	−19.2200	−0.613334
$983$	−10.3800	−0.331070	−0.165535	−	0.986204i	$-0.552935\pi$
$983$	−10.3800	−0.331070	−0.165535	+	0.986204i	$0.552935\pi$
$984$	13.5560	0.432150
$985$	62.4775	1.99070
$986$	−25.4634	−0.810920
$987$	−19.6181	−0.624451
$988$	0.827220	0.0263174
$989$	−18.3858	−0.584635
$990$	16.3420	0.519382
$991$	−12.1862	−0.387108	−0.193554	−	0.981090i	$-0.562002\pi$
$991$	−12.1862	−0.387108	−0.193554	+	0.981090i	$0.562002\pi$
$992$	1.50751	0.0478634
$993$	−10.8063	−0.342927
$994$	−8.24160	−0.261408
$995$	3.43535	0.108908
$996$	−33.4545	−1.06005
$997$	18.4559	0.584505	0.292253	−	0.956341i	$-0.405595\pi$
$997$	18.4559	0.584505	0.292253	+	0.956341i	$0.405595\pi$
$998$	−36.8229	−1.16561
$999$	2.27235	0.0718940

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.e.1.11	✓	70

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.e.1.11	✓	70	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.23514 q^{3} +1.00000 q^{4} +3.00438 q^{5} -2.23514 q^{6} +1.56958 q^{7} +1.00000 q^{8} +1.99587 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.23514 q^{3} +1.00000 q^{4} +3.00438 q^{5} -2.23514 q^{6} +1.56958 q^{7} +1.00000 q^{8} +1.99587 q^{9} +3.00438 q^{10} +2.72533 q^{11} -2.23514 q^{12} -0.479948 q^{13} +1.56958 q^{14} -6.71522 q^{15} +1.00000 q^{16} -3.03644 q^{17} +1.99587 q^{18} -1.72356 q^{19} +3.00438 q^{20} -3.50824 q^{21} +2.72533 q^{22} -2.38948 q^{23} -2.23514 q^{24} +4.02629 q^{25} -0.479948 q^{26} +2.24438 q^{27} +1.56958 q^{28} +8.38595 q^{29} -6.71522 q^{30} +1.50751 q^{31} +1.00000 q^{32} -6.09150 q^{33} -3.03644 q^{34} +4.71562 q^{35} +1.99587 q^{36} +1.01246 q^{37} -1.72356 q^{38} +1.07275 q^{39} +3.00438 q^{40} -6.06494 q^{41} -3.50824 q^{42} +7.69447 q^{43} +2.72533 q^{44} +5.99634 q^{45} -2.38948 q^{46} +5.59201 q^{47} -2.23514 q^{48} -4.53641 q^{49} +4.02629 q^{50} +6.78687 q^{51} -0.479948 q^{52} -3.27922 q^{53} +2.24438 q^{54} +8.18792 q^{55} +1.56958 q^{56} +3.85241 q^{57} +8.38595 q^{58} -2.26795 q^{59} -6.71522 q^{60} +12.5528 q^{61} +1.50751 q^{62} +3.13267 q^{63} +1.00000 q^{64} -1.44195 q^{65} -6.09150 q^{66} +7.36646 q^{67} -3.03644 q^{68} +5.34084 q^{69} +4.71562 q^{70} -5.25083 q^{71} +1.99587 q^{72} +14.5216 q^{73} +1.01246 q^{74} -8.99934 q^{75} -1.72356 q^{76} +4.27762 q^{77} +1.07275 q^{78} +3.24512 q^{79} +3.00438 q^{80} -11.0041 q^{81} -6.06494 q^{82} +14.9675 q^{83} -3.50824 q^{84} -9.12260 q^{85} +7.69447 q^{86} -18.7438 q^{87} +2.72533 q^{88} +1.06264 q^{89} +5.99634 q^{90} -0.753318 q^{91} -2.38948 q^{92} -3.36949 q^{93} +5.59201 q^{94} -5.17823 q^{95} -2.23514 q^{96} -17.4249 q^{97} -4.53641 q^{98} +5.43939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9}+O(q^{10}) \) 70 * q + 70 * q^2 + 25 * q^3 + 70 * q^4 + 18 * q^5 + 25 * q^6 + 50 * q^7 + 70 * q^8 + 89 * q^9	\( 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9} + 18 q^{10} + 41 q^{11} + 25 q^{12} + 41 q^{13} + 50 q^{14} + 13 q^{15} + 70 q^{16} + 40 q^{17} + 89 q^{18} + 55 q^{19} + 18 q^{20} + 2 q^{21} + 41 q^{22} + 41 q^{23} + 25 q^{24} + 104 q^{25} + 41 q^{26} + 82 q^{27} + 50 q^{28} + 11 q^{29} + 13 q^{30} + 78 q^{31} + 70 q^{32} + 45 q^{33} + 40 q^{34} + 25 q^{35} + 89 q^{36} + 46 q^{37} + 55 q^{38} + 19 q^{39} + 18 q^{40} + 51 q^{41} + 2 q^{42} + 68 q^{43} + 41 q^{44} + 37 q^{45} + 41 q^{46} + 69 q^{47} + 25 q^{48} + 126 q^{49} + 104 q^{50} + 36 q^{51} + 41 q^{52} + 23 q^{53} + 82 q^{54} + 42 q^{55} + 50 q^{56} + 14 q^{57} + 11 q^{58} + 89 q^{59} + 13 q^{60} + 32 q^{61} + 78 q^{62} + 106 q^{63} + 70 q^{64} + 18 q^{65} + 45 q^{66} + 90 q^{67} + 40 q^{68} - 12 q^{69} + 25 q^{70} + 54 q^{71} + 89 q^{72} + 94 q^{73} + 46 q^{74} + 72 q^{75} + 55 q^{76} - 16 q^{77} + 19 q^{78} + 54 q^{79} + 18 q^{80} + 102 q^{81} + 51 q^{82} + 60 q^{83} + 2 q^{84} - 5 q^{85} + 68 q^{86} + 9 q^{87} + 41 q^{88} + 77 q^{89} + 37 q^{90} + 54 q^{91} + 41 q^{92} - 2 q^{93} + 69 q^{94} + 39 q^{95} + 25 q^{96} + 139 q^{97} + 126 q^{98} + 60 q^{99}+O(q^{100}) \) 70 * q + 70 * q^2 + 25 * q^3 + 70 * q^4 + 18 * q^5 + 25 * q^6 + 50 * q^7 + 70 * q^8 + 89 * q^9 + 18 * q^10 + 41 * q^11 + 25 * q^12 + 41 * q^13 + 50 * q^14 + 13 * q^15 + 70 * q^16 + 40 * q^17 + 89 * q^18 + 55 * q^19 + 18 * q^20 + 2 * q^21 + 41 * q^22 + 41 * q^23 + 25 * q^24 + 104 * q^25 + 41 * q^26 + 82 * q^27 + 50 * q^28 + 11 * q^29 + 13 * q^30 + 78 * q^31 + 70 * q^32 + 45 * q^33 + 40 * q^34 + 25 * q^35 + 89 * q^36 + 46 * q^37 + 55 * q^38 + 19 * q^39 + 18 * q^40 + 51 * q^41 + 2 * q^42 + 68 * q^43 + 41 * q^44 + 37 * q^45 + 41 * q^46 + 69 * q^47 + 25 * q^48 + 126 * q^49 + 104 * q^50 + 36 * q^51 + 41 * q^52 + 23 * q^53 + 82 * q^54 + 42 * q^55 + 50 * q^56 + 14 * q^57 + 11 * q^58 + 89 * q^59 + 13 * q^60 + 32 * q^61 + 78 * q^62 + 106 * q^63 + 70 * q^64 + 18 * q^65 + 45 * q^66 + 90 * q^67 + 40 * q^68 - 12 * q^69 + 25 * q^70 + 54 * q^71 + 89 * q^72 + 94 * q^73 + 46 * q^74 + 72 * q^75 + 55 * q^76 - 16 * q^77 + 19 * q^78 + 54 * q^79 + 18 * q^80 + 102 * q^81 + 51 * q^82 + 60 * q^83 + 2 * q^84 - 5 * q^85 + 68 * q^86 + 9 * q^87 + 41 * q^88 + 77 * q^89 + 37 * q^90 + 54 * q^91 + 41 * q^92 - 2 * q^93 + 69 * q^94 + 39 * q^95 + 25 * q^96 + 139 * q^97 + 126 * q^98 + 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more