LMFDB - Embedded newform 6038.2.a.c.1.5

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:	$1$
Dimension:	$57$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.94445 q^{3} +1.00000 q^{4} -1.42608 q^{5} +2.94445 q^{6} +1.57265 q^{7} -1.00000 q^{8} +5.66979 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.94445 q^{3} +1.00000 q^{4} -1.42608 q^{5} +2.94445 q^{6} +1.57265 q^{7} -1.00000 q^{8} +5.66979 q^{9} +1.42608 q^{10} -3.80076 q^{11} -2.94445 q^{12} -2.01762 q^{13} -1.57265 q^{14} +4.19902 q^{15} +1.00000 q^{16} +5.77157 q^{17} -5.66979 q^{18} +4.17143 q^{19} -1.42608 q^{20} -4.63059 q^{21} +3.80076 q^{22} +0.500759 q^{23} +2.94445 q^{24} -2.96630 q^{25} +2.01762 q^{26} -7.86108 q^{27} +1.57265 q^{28} -8.28862 q^{29} -4.19902 q^{30} -6.12765 q^{31} -1.00000 q^{32} +11.1912 q^{33} -5.77157 q^{34} -2.24272 q^{35} +5.66979 q^{36} +1.74510 q^{37} -4.17143 q^{38} +5.94078 q^{39} +1.42608 q^{40} +12.3268 q^{41} +4.63059 q^{42} -0.636286 q^{43} -3.80076 q^{44} -8.08558 q^{45} -0.500759 q^{46} +2.62469 q^{47} -2.94445 q^{48} -4.52677 q^{49} +2.96630 q^{50} -16.9941 q^{51} -2.01762 q^{52} -5.90783 q^{53} +7.86108 q^{54} +5.42019 q^{55} -1.57265 q^{56} -12.2826 q^{57} +8.28862 q^{58} -4.93177 q^{59} +4.19902 q^{60} -3.48312 q^{61} +6.12765 q^{62} +8.91660 q^{63} +1.00000 q^{64} +2.87728 q^{65} -11.1912 q^{66} +11.7861 q^{67} +5.77157 q^{68} -1.47446 q^{69} +2.24272 q^{70} +16.3903 q^{71} -5.66979 q^{72} -16.2235 q^{73} -1.74510 q^{74} +8.73412 q^{75} +4.17143 q^{76} -5.97727 q^{77} -5.94078 q^{78} -6.60841 q^{79} -1.42608 q^{80} +6.13717 q^{81} -12.3268 q^{82} +14.7099 q^{83} -4.63059 q^{84} -8.23072 q^{85} +0.636286 q^{86} +24.4054 q^{87} +3.80076 q^{88} +15.7634 q^{89} +8.08558 q^{90} -3.17301 q^{91} +0.500759 q^{92} +18.0426 q^{93} -2.62469 q^{94} -5.94879 q^{95} +2.94445 q^{96} +0.683844 q^{97} +4.52677 q^{98} -21.5495 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9}+O(q^{10}) $ 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9	\( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9} + 15 q^{10} + 13 q^{11} - 5 q^{12} - 43 q^{13} + 28 q^{14} - 10 q^{15} + 57 q^{16} - 50 q^{18} - 6 q^{19} - 15 q^{20} - 23 q^{21} - 13 q^{22} - q^{23} + 5 q^{24} + 20 q^{25} + 43 q^{26} - 20 q^{27} - 28 q^{28} - 4 q^{29} + 10 q^{30} - 34 q^{31} - 57 q^{32} - 43 q^{33} + 26 q^{35} + 50 q^{36} - 64 q^{37} + 6 q^{38} + 8 q^{39} + 15 q^{40} + 27 q^{41} + 23 q^{42} - 29 q^{43} + 13 q^{44} - 76 q^{45} + q^{46} - 25 q^{47} - 5 q^{48} + 7 q^{49} - 20 q^{50} + 27 q^{51} - 43 q^{52} - 34 q^{53} + 20 q^{54} - 36 q^{55} + 28 q^{56} - 33 q^{57} + 4 q^{58} + 19 q^{59} - 10 q^{60} - 58 q^{61} + 34 q^{62} - 65 q^{63} + 57 q^{64} + 17 q^{65} + 43 q^{66} - 84 q^{67} - 33 q^{69} - 26 q^{70} + 22 q^{71} - 50 q^{72} - 82 q^{73} + 64 q^{74} + 8 q^{75} - 6 q^{76} - 41 q^{77} - 8 q^{78} + 8 q^{79} - 15 q^{80} + 25 q^{81} - 27 q^{82} - 23 q^{83} - 23 q^{84} - 58 q^{85} + 29 q^{86} - 17 q^{87} - 13 q^{88} + 18 q^{89} + 76 q^{90} - 4 q^{91} - q^{92} - 60 q^{93} + 25 q^{94} + 36 q^{95} + 5 q^{96} - 156 q^{97} - 7 q^{98} + 25 q^{99}+O(q^{100}) \) 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9 + 15 * q^10 + 13 * q^11 - 5 * q^12 - 43 * q^13 + 28 * q^14 - 10 * q^15 + 57 * q^16 - 50 * q^18 - 6 * q^19 - 15 * q^20 - 23 * q^21 - 13 * q^22 - q^23 + 5 * q^24 + 20 * q^25 + 43 * q^26 - 20 * q^27 - 28 * q^28 - 4 * q^29 + 10 * q^30 - 34 * q^31 - 57 * q^32 - 43 * q^33 + 26 * q^35 + 50 * q^36 - 64 * q^37 + 6 * q^38 + 8 * q^39 + 15 * q^40 + 27 * q^41 + 23 * q^42 - 29 * q^43 + 13 * q^44 - 76 * q^45 + q^46 - 25 * q^47 - 5 * q^48 + 7 * q^49 - 20 * q^50 + 27 * q^51 - 43 * q^52 - 34 * q^53 + 20 * q^54 - 36 * q^55 + 28 * q^56 - 33 * q^57 + 4 * q^58 + 19 * q^59 - 10 * q^60 - 58 * q^61 + 34 * q^62 - 65 * q^63 + 57 * q^64 + 17 * q^65 + 43 * q^66 - 84 * q^67 - 33 * q^69 - 26 * q^70 + 22 * q^71 - 50 * q^72 - 82 * q^73 + 64 * q^74 + 8 * q^75 - 6 * q^76 - 41 * q^77 - 8 * q^78 + 8 * q^79 - 15 * q^80 + 25 * q^81 - 27 * q^82 - 23 * q^83 - 23 * q^84 - 58 * q^85 + 29 * q^86 - 17 * q^87 - 13 * q^88 + 18 * q^89 + 76 * q^90 - 4 * q^91 - q^92 - 60 * q^93 + 25 * q^94 + 36 * q^95 + 5 * q^96 - 156 * q^97 - 7 * q^98 + 25 * 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
$3$	−2.94445	−1.69998	−0.849990	−	0.526799i	$-0.823392\pi$
$3$	−2.94445	−1.69998	−0.849990	+	0.526799i	$0.823392\pi$
$4$	1.00000	0.500000
$5$	−1.42608	−0.637762	−0.318881	−	0.947795i	$-0.603307\pi$
$5$	−1.42608	−0.637762	−0.318881	+	0.947795i	$0.603307\pi$
$6$	2.94445	1.20207
$7$	1.57265	0.594406	0.297203	−	0.954814i	$-0.403946\pi$
$7$	1.57265	0.594406	0.297203	+	0.954814i	$0.403946\pi$
$8$	−1.00000	−0.353553
$9$	5.66979	1.88993
$10$	1.42608	0.450966
$11$	−3.80076	−1.14597	−0.572986	−	0.819565i	$-0.694215\pi$
$11$	−3.80076	−1.14597	−0.572986	+	0.819565i	$0.694215\pi$
$12$	−2.94445	−0.849990
$13$	−2.01762	−0.559586	−0.279793	−	0.960060i	$-0.590266\pi$
$13$	−2.01762	−0.559586	−0.279793	+	0.960060i	$0.590266\pi$
$14$	−1.57265	−0.420309
$15$	4.19902	1.08418
$16$	1.00000	0.250000
$17$	5.77157	1.39981	0.699906	−	0.714235i	$-0.253225\pi$
$17$	5.77157	1.39981	0.699906	+	0.714235i	$0.253225\pi$
$18$	−5.66979	−1.33638
$19$	4.17143	0.956992	0.478496	−	0.878090i	$-0.341182\pi$
$19$	4.17143	0.956992	0.478496	+	0.878090i	$0.341182\pi$
$20$	−1.42608	−0.318881
$21$	−4.63059	−1.01048
$22$	3.80076	0.810325
$23$	0.500759	0.104415	0.0522077	−	0.998636i	$-0.483374\pi$
$23$	0.500759	0.104415	0.0522077	+	0.998636i	$0.483374\pi$
$24$	2.94445	0.601034
$25$	−2.96630	−0.593259
$26$	2.01762	0.395687
$27$	−7.86108	−1.51286
$28$	1.57265	0.297203
$29$	−8.28862	−1.53916	−0.769579	−	0.638552i	$-0.779534\pi$
$29$	−8.28862	−1.53916	−0.769579	+	0.638552i	$0.779534\pi$
$30$	−4.19902	−0.766633
$31$	−6.12765	−1.10056	−0.550279	−	0.834981i	$-0.685479\pi$
$31$	−6.12765	−1.10056	−0.550279	+	0.834981i	$0.685479\pi$
$32$	−1.00000	−0.176777
$33$	11.1912	1.94813
$34$	−5.77157	−0.989816
$35$	−2.24272	−0.379090
$36$	5.66979	0.944965
$37$	1.74510	0.286893	0.143447	−	0.989658i	$-0.454181\pi$
$37$	1.74510	0.286893	0.143447	+	0.989658i	$0.454181\pi$
$38$	−4.17143	−0.676696
$39$	5.94078	0.951285
$40$	1.42608	0.225483
$41$	12.3268	1.92512	0.962558	−	0.271075i	$-0.0873791\pi$
$41$	12.3268	1.92512	0.962558	+	0.271075i	$0.0873791\pi$
$42$	4.63059	0.714516
$43$	−0.636286	−0.0970326	−0.0485163	−	0.998822i	$-0.515449\pi$
$43$	−0.636286	−0.0970326	−0.0485163	+	0.998822i	$0.515449\pi$
$44$	−3.80076	−0.572986
$45$	−8.08558	−1.20533
$46$	−0.500759	−0.0738329
$47$	2.62469	0.382851	0.191425	−	0.981507i	$-0.438689\pi$
$47$	2.62469	0.382851	0.191425	+	0.981507i	$0.438689\pi$
$48$	−2.94445	−0.424995
$49$	−4.52677	−0.646682
$50$	2.96630	0.419498
$51$	−16.9941	−2.37965
$52$	−2.01762	−0.279793
$53$	−5.90783	−0.811502	−0.405751	−	0.913984i	$-0.632990\pi$
$53$	−5.90783	−0.811502	−0.405751	+	0.913984i	$0.632990\pi$
$54$	7.86108	1.06976
$55$	5.42019	0.730858
$56$	−1.57265	−0.210154
$57$	−12.2826	−1.62687
$58$	8.28862	1.08835
$59$	−4.93177	−0.642062	−0.321031	−	0.947069i	$-0.604029\pi$
$59$	−4.93177	−0.642062	−0.321031	+	0.947069i	$0.604029\pi$
$60$	4.19902	0.542091
$61$	−3.48312	−0.445967	−0.222984	−	0.974822i	$-0.571580\pi$
$61$	−3.48312	−0.445967	−0.222984	+	0.974822i	$0.571580\pi$
$62$	6.12765	0.778212
$63$	8.91660	1.12339
$64$	1.00000	0.125000
$65$	2.87728	0.356883
$66$	−11.1912	−1.37754
$67$	11.7861	1.43990	0.719949	−	0.694027i	$-0.244165\pi$
$67$	11.7861	1.43990	0.719949	+	0.694027i	$0.244165\pi$
$68$	5.77157	0.699906
$69$	−1.47446	−0.177504
$70$	2.24272	0.268057
$71$	16.3903	1.94517	0.972586	−	0.232544i	$-0.0747049\pi$
$71$	16.3903	1.94517	0.972586	+	0.232544i	$0.0747049\pi$
$72$	−5.66979	−0.668192
$73$	−16.2235	−1.89882	−0.949410	−	0.314039i	$-0.898318\pi$
$73$	−16.2235	−1.89882	−0.949410	+	0.314039i	$0.898318\pi$
$74$	−1.74510	−0.202864
$75$	8.73412	1.00853
$76$	4.17143	0.478496
$77$	−5.97727	−0.681173
$78$	−5.94078	−0.672660
$79$	−6.60841	−0.743504	−0.371752	−	0.928332i	$-0.621243\pi$
$79$	−6.60841	−0.743504	−0.371752	+	0.928332i	$0.621243\pi$
$80$	−1.42608	−0.159441
$81$	6.13717	0.681908
$82$	−12.3268	−1.36126
$83$	14.7099	1.61463	0.807313	−	0.590123i	$-0.200921\pi$
$83$	14.7099	1.61463	0.807313	+	0.590123i	$0.200921\pi$
$84$	−4.63059	−0.505239
$85$	−8.23072	−0.892746
$86$	0.636286	0.0686124
$87$	24.4054	2.61654
$88$	3.80076	0.405162
$89$	15.7634	1.67092	0.835459	−	0.549552i	$-0.185202\pi$
$89$	15.7634	1.67092	0.835459	+	0.549552i	$0.185202\pi$
$90$	8.08558	0.852294
$91$	−3.17301	−0.332621
$92$	0.500759	0.0522077
$93$	18.0426	1.87093
$94$	−2.62469	−0.270716
$95$	−5.94879	−0.610333
$96$	2.94445	0.300517
$97$	0.683844	0.0694339	0.0347169	−	0.999397i	$-0.488947\pi$
$97$	0.683844	0.0694339	0.0347169	+	0.999397i	$0.488947\pi$
$98$	4.52677	0.457273
$99$	−21.5495	−2.16581
$100$	−2.96630	−0.296630
$101$	1.47094	0.146364	0.0731818	−	0.997319i	$-0.476685\pi$
$101$	1.47094	0.146364	0.0731818	+	0.997319i	$0.476685\pi$
$102$	16.9941	1.68267
$103$	−8.00953	−0.789202	−0.394601	−	0.918852i	$-0.629117\pi$
$103$	−8.00953	−0.789202	−0.394601	+	0.918852i	$0.629117\pi$
$104$	2.01762	0.197844
$105$	6.60359	0.644445
$106$	5.90783	0.573819
$107$	−9.64412	−0.932332	−0.466166	−	0.884697i	$-0.654365\pi$
$107$	−9.64412	−0.932332	−0.466166	+	0.884697i	$0.654365\pi$
$108$	−7.86108	−0.756432
$109$	−12.3471	−1.18264	−0.591320	−	0.806437i	$-0.701393\pi$
$109$	−12.3471	−1.18264	−0.591320	+	0.806437i	$0.701393\pi$
$110$	−5.42019	−0.516794
$111$	−5.13837	−0.487713
$112$	1.57265	0.148601
$113$	6.14153	0.577746	0.288873	−	0.957367i	$-0.406719\pi$
$113$	6.14153	0.577746	0.288873	+	0.957367i	$0.406719\pi$
$114$	12.2826	1.15037
$115$	−0.714122	−0.0665922
$116$	−8.28862	−0.769579
$117$	−11.4395	−1.05758
$118$	4.93177	0.454006
$119$	9.07666	0.832056
$120$	−4.19902	−0.383316
$121$	3.44578	0.313253
$122$	3.48312	0.315347
$123$	−36.2955	−3.27266
$124$	−6.12765	−0.550279
$125$	11.3606	1.01612
$126$	−8.91660	−0.794354
$127$	−1.58730	−0.140850	−0.0704252	−	0.997517i	$-0.522436\pi$
$127$	−1.58730	−0.140850	−0.0704252	+	0.997517i	$0.522436\pi$
$128$	−1.00000	−0.0883883
$129$	1.87351	0.164954
$130$	−2.87728	−0.252354
$131$	10.8882	0.951310	0.475655	−	0.879632i	$-0.342211\pi$
$131$	10.8882	0.951310	0.475655	+	0.879632i	$0.342211\pi$
$132$	11.1912	0.974065
$133$	6.56020	0.568842
$134$	−11.7861	−1.01816
$135$	11.2105	0.964848
$136$	−5.77157	−0.494908
$137$	1.29616	0.110739	0.0553694	−	0.998466i	$-0.482366\pi$
$137$	1.29616	0.110739	0.0553694	+	0.998466i	$0.482366\pi$
$138$	1.47446	0.125514
$139$	15.0543	1.27689	0.638445	−	0.769667i	$-0.279578\pi$
$139$	15.0543	1.27689	0.638445	+	0.769667i	$0.279578\pi$
$140$	−2.24272	−0.189545
$141$	−7.72828	−0.650839
$142$	−16.3903	−1.37544
$143$	7.66848	0.641270
$144$	5.66979	0.472483
$145$	11.8202	0.981616
$146$	16.2235	1.34267
$147$	13.3289	1.09935
$148$	1.74510	0.143447
$149$	12.6069	1.03280	0.516398	−	0.856349i	$-0.327272\pi$
$149$	12.6069	1.03280	0.516398	+	0.856349i	$0.327272\pi$
$150$	−8.73412	−0.713138
$151$	9.03906	0.735588	0.367794	−	0.929907i	$-0.380113\pi$
$151$	9.03906	0.735588	0.367794	+	0.929907i	$0.380113\pi$
$152$	−4.17143	−0.338348
$153$	32.7236	2.64555
$154$	5.97727	0.481662
$155$	8.73851	0.701894
$156$	5.94078	0.475643
$157$	−19.7409	−1.57550	−0.787749	−	0.615996i	$-0.788754\pi$
$157$	−19.7409	−1.57550	−0.787749	+	0.615996i	$0.788754\pi$
$158$	6.60841	0.525737
$159$	17.3953	1.37954
$160$	1.42608	0.112741
$161$	0.787519	0.0620652
$162$	−6.13717	−0.482182
$163$	7.45604	0.584002	0.292001	−	0.956418i	$-0.405679\pi$
$163$	7.45604	0.584002	0.292001	+	0.956418i	$0.405679\pi$
$164$	12.3268	0.962558
$165$	−15.9595	−1.24244
$166$	−14.7099	−1.14171
$167$	5.78119	0.447362	0.223681	−	0.974662i	$-0.428193\pi$
$167$	5.78119	0.447362	0.223681	+	0.974662i	$0.428193\pi$
$168$	4.63059	0.357258
$169$	−8.92922	−0.686863
$170$	8.23072	0.631267
$171$	23.6512	1.80865
$172$	−0.636286	−0.0485163
$173$	−11.0620	−0.841026	−0.420513	−	0.907287i	$-0.638150\pi$
$173$	−11.0620	−0.841026	−0.420513	+	0.907287i	$0.638150\pi$
$174$	−24.4054	−1.85017
$175$	−4.66495	−0.352637
$176$	−3.80076	−0.286493
$177$	14.5214	1.09149
$178$	−15.7634	−1.18152
$179$	8.12025	0.606936	0.303468	−	0.952842i	$-0.401856\pi$
$179$	8.12025	0.606936	0.303468	+	0.952842i	$0.401856\pi$
$180$	−8.08558	−0.602663
$181$	4.12095	0.306308	0.153154	−	0.988202i	$-0.451057\pi$
$181$	4.12095	0.306308	0.153154	+	0.988202i	$0.451057\pi$
$182$	3.17301	0.235199
$183$	10.2559	0.758136
$184$	−0.500759	−0.0369164
$185$	−2.48866	−0.182970
$186$	−18.0426	−1.32294
$187$	−21.9363	−1.60414
$188$	2.62469	0.191425
$189$	−12.3627	−0.899256
$190$	5.94879	0.431571
$191$	3.44893	0.249556	0.124778	−	0.992185i	$-0.460178\pi$
$191$	3.44893	0.249556	0.124778	+	0.992185i	$0.460178\pi$
$192$	−2.94445	−0.212497
$193$	21.6532	1.55863	0.779314	−	0.626634i	$-0.215568\pi$
$193$	21.6532	1.55863	0.779314	+	0.626634i	$0.215568\pi$
$194$	−0.683844	−0.0490972
$195$	−8.47202	−0.606694
$196$	−4.52677	−0.323341
$197$	−3.75818	−0.267759	−0.133880	−	0.990998i	$-0.542744\pi$
$197$	−3.75818	−0.267759	−0.133880	+	0.990998i	$0.542744\pi$
$198$	21.5495	1.53146
$199$	0.218863	0.0155148	0.00775741	−	0.999970i	$-0.497531\pi$
$199$	0.218863	0.0155148	0.00775741	+	0.999970i	$0.497531\pi$
$200$	2.96630	0.209749
$201$	−34.7035	−2.44780
$202$	−1.47094	−0.103495
$203$	−13.0351	−0.914885
$204$	−16.9941	−1.18983
$205$	−17.5789	−1.22777
$206$	8.00953	0.558050
$207$	2.83920	0.197338
$208$	−2.01762	−0.139897
$209$	−15.8546	−1.09669
$210$	−6.60359	−0.455691
$211$	−8.44463	−0.581352	−0.290676	−	0.956822i	$-0.593880\pi$
$211$	−8.44463	−0.581352	−0.290676	+	0.956822i	$0.593880\pi$
$212$	−5.90783	−0.405751
$213$	−48.2605	−3.30675
$214$	9.64412	0.659258
$215$	0.907394	0.0618837
$216$	7.86108	0.534878
$217$	−9.63665	−0.654178
$218$	12.3471	0.836252
$219$	47.7694	3.22796
$220$	5.42019	0.365429
$221$	−11.6448	−0.783315
$222$	5.13837	0.344865
$223$	8.10682	0.542873	0.271437	−	0.962456i	$-0.412501\pi$
$223$	8.10682	0.542873	0.271437	+	0.962456i	$0.412501\pi$
$224$	−1.57265	−0.105077
$225$	−16.8183	−1.12122
$226$	−6.14153	−0.408528
$227$	1.98719	0.131894	0.0659472	−	0.997823i	$-0.478993\pi$
$227$	1.98719	0.131894	0.0659472	+	0.997823i	$0.478993\pi$
$228$	−12.2826	−0.813434
$229$	20.4835	1.35359	0.676793	−	0.736174i	$-0.263369\pi$
$229$	20.4835	1.35359	0.676793	+	0.736174i	$0.263369\pi$
$230$	0.714122	0.0470878
$231$	17.5998	1.15798
$232$	8.28862	0.544174
$233$	2.13471	0.139850	0.0699249	−	0.997552i	$-0.477724\pi$
$233$	2.13471	0.139850	0.0699249	+	0.997552i	$0.477724\pi$
$234$	11.4395	0.747822
$235$	−3.74302	−0.244168
$236$	−4.93177	−0.321031
$237$	19.4581	1.26394
$238$	−9.07666	−0.588353
$239$	27.7722	1.79643	0.898217	−	0.439551i	$-0.144863\pi$
$239$	27.7722	1.79643	0.898217	+	0.439551i	$0.144863\pi$
$240$	4.19902	0.271046
$241$	21.7423	1.40055	0.700273	−	0.713875i	$-0.253061\pi$
$241$	21.7423	1.40055	0.700273	+	0.713875i	$0.253061\pi$
$242$	−3.44578	−0.221503
$243$	5.51262	0.353635
$244$	−3.48312	−0.222984
$245$	6.45553	0.412429
$246$	36.2955	2.31412
$247$	−8.41635	−0.535520
$248$	6.12765	0.389106
$249$	−43.3127	−2.74483
$250$	−11.3606	−0.718506
$251$	−15.7343	−0.993141	−0.496571	−	0.867996i	$-0.665408\pi$
$251$	−15.7343	−0.993141	−0.496571	+	0.867996i	$0.665408\pi$
$252$	8.91660	0.561693
$253$	−1.90326	−0.119657
$254$	1.58730	0.0995963
$255$	24.2349	1.51765
$256$	1.00000	0.0625000
$257$	−19.9895	−1.24691	−0.623455	−	0.781859i	$-0.714271\pi$
$257$	−19.9895	−1.24691	−0.623455	+	0.781859i	$0.714271\pi$
$258$	−1.87351	−0.116640
$259$	2.74444	0.170531
$260$	2.87728	0.178441
$261$	−46.9947	−2.90890
$262$	−10.8882	−0.672678
$263$	−16.3272	−1.00678	−0.503390	−	0.864059i	$-0.667914\pi$
$263$	−16.3272	−1.00678	−0.503390	+	0.864059i	$0.667914\pi$
$264$	−11.1912	−0.688768
$265$	8.42503	0.517545
$266$	−6.56020	−0.402232
$267$	−46.4146	−2.84053
$268$	11.7861	0.719949
$269$	19.2157	1.17160	0.585802	−	0.810454i	$-0.300780\pi$
$269$	19.2157	1.17160	0.585802	+	0.810454i	$0.300780\pi$
$270$	−11.2105	−0.682250
$271$	15.6227	0.949011	0.474505	−	0.880253i	$-0.342627\pi$
$271$	15.6227	0.949011	0.474505	+	0.880253i	$0.342627\pi$
$272$	5.77157	0.349953
$273$	9.34276	0.565450
$274$	−1.29616	−0.0783042
$275$	11.2742	0.679859
$276$	−1.47446	−0.0887521
$277$	−12.8887	−0.774406	−0.387203	−	0.921995i	$-0.626559\pi$
$277$	−12.8887	−0.774406	−0.387203	+	0.921995i	$0.626559\pi$
$278$	−15.0543	−0.902898
$279$	−34.7425	−2.07998
$280$	2.24272	0.134028
$281$	23.9721	1.43006	0.715028	−	0.699096i	$-0.246414\pi$
$281$	23.9721	1.43006	0.715028	+	0.699096i	$0.246414\pi$
$282$	7.72828	0.460212
$283$	−4.70509	−0.279688	−0.139844	−	0.990174i	$-0.544660\pi$
$283$	−4.70509	−0.279688	−0.139844	+	0.990174i	$0.544660\pi$
$284$	16.3903	0.972586
$285$	17.5159	1.03755
$286$	−7.66848	−0.453447
$287$	19.3857	1.14430
$288$	−5.66979	−0.334096
$289$	16.3110	0.959471
$290$	−11.8202	−0.694108
$291$	−2.01355	−0.118036
$292$	−16.2235	−0.949410
$293$	−11.9292	−0.696912	−0.348456	−	0.937325i	$-0.613294\pi$
$293$	−11.9292	−0.696912	−0.348456	+	0.937325i	$0.613294\pi$
$294$	−13.3289	−0.777355
$295$	7.03310	0.409483
$296$	−1.74510	−0.101432
$297$	29.8781	1.73370
$298$	−12.6069	−0.730297
$299$	−1.01034	−0.0584295
$300$	8.73412	0.504265
$301$	−1.00065	−0.0576768
$302$	−9.03906	−0.520139
$303$	−4.33110	−0.248815
$304$	4.17143	0.239248
$305$	4.96720	0.284421
$306$	−32.7236	−1.87068
$307$	1.59515	0.0910400	0.0455200	−	0.998963i	$-0.485506\pi$
$307$	1.59515	0.0910400	0.0455200	+	0.998963i	$0.485506\pi$
$308$	−5.97727	−0.340586
$309$	23.5837	1.34163
$310$	−8.73851	−0.496314
$311$	−30.1563	−1.71001	−0.855004	−	0.518621i	$-0.826445\pi$
$311$	−30.1563	−1.71001	−0.855004	+	0.518621i	$0.826445\pi$
$312$	−5.94078	−0.336330
$313$	2.03499	0.115024	0.0575122	−	0.998345i	$-0.481683\pi$
$313$	2.03499	0.115024	0.0575122	+	0.998345i	$0.481683\pi$
$314$	19.7409	1.11405
$315$	−12.7158	−0.716453
$316$	−6.60841	−0.371752
$317$	−7.41987	−0.416741	−0.208371	−	0.978050i	$-0.566816\pi$
$317$	−7.41987	−0.416741	−0.208371	+	0.978050i	$0.566816\pi$
$318$	−17.3953	−0.975480
$319$	31.5030	1.76383
$320$	−1.42608	−0.0797203
$321$	28.3966	1.58495
$322$	−0.787519	−0.0438867
$323$	24.0757	1.33961
$324$	6.13717	0.340954
$325$	5.98485	0.331980
$326$	−7.45604	−0.412952
$327$	36.3555	2.01046
$328$	−12.3268	−0.680631
$329$	4.12772	0.227569
$330$	15.9595	0.878540
$331$	−25.9405	−1.42582	−0.712910	−	0.701255i	$-0.752623\pi$
$331$	−25.9405	−1.42582	−0.712910	+	0.701255i	$0.752623\pi$
$332$	14.7099	0.807313
$333$	9.89437	0.542208
$334$	−5.78119	−0.316333
$335$	−16.8079	−0.918312
$336$	−4.63059	−0.252620
$337$	−33.3933	−1.81905	−0.909525	−	0.415649i	$-0.863554\pi$
$337$	−33.3933	−1.81905	−0.909525	+	0.415649i	$0.863554\pi$
$338$	8.92922	0.485686
$339$	−18.0834	−0.982157
$340$	−8.23072	−0.446373
$341$	23.2897	1.26121
$342$	−23.6512	−1.27891
$343$	−18.1276	−0.978797
$344$	0.636286	0.0343062
$345$	2.10270	0.113205
$346$	11.0620	0.594695
$347$	3.43680	0.184497	0.0922485	−	0.995736i	$-0.470595\pi$
$347$	3.43680	0.184497	0.0922485	+	0.995736i	$0.470595\pi$
$348$	24.4054	1.30827
$349$	−30.4297	−1.62886	−0.814432	−	0.580259i	$-0.802951\pi$
$349$	−30.4297	−1.62886	−0.814432	+	0.580259i	$0.802951\pi$
$350$	4.66495	0.249352
$351$	15.8606	0.846578
$352$	3.80076	0.202581
$353$	18.6107	0.990550	0.495275	−	0.868736i	$-0.335067\pi$
$353$	18.6107	0.990550	0.495275	+	0.868736i	$0.335067\pi$
$354$	−14.5214	−0.771802
$355$	−23.3739	−1.24056
$356$	15.7634	0.835459
$357$	−26.7258	−1.41448
$358$	−8.12025	−0.429168
$359$	−16.5362	−0.872745	−0.436373	−	0.899766i	$-0.643737\pi$
$359$	−16.5362	−0.872745	−0.436373	+	0.899766i	$0.643737\pi$
$360$	8.08558	0.426147
$361$	−1.59916	−0.0841661
$362$	−4.12095	−0.216592
$363$	−10.1459	−0.532523
$364$	−3.17301	−0.166311
$365$	23.1360	1.21100
$366$	−10.2559	−0.536083
$367$	−20.2979	−1.05954	−0.529772	−	0.848140i	$-0.677722\pi$
$367$	−20.2979	−1.05954	−0.529772	+	0.848140i	$0.677722\pi$
$368$	0.500759	0.0261039
$369$	69.8902	3.63834
$370$	2.48866	0.129379
$371$	−9.29095	−0.482362
$372$	18.0426	0.935463
$373$	−17.1977	−0.890462	−0.445231	−	0.895416i	$-0.646878\pi$
$373$	−17.1977	−0.890462	−0.445231	+	0.895416i	$0.646878\pi$
$374$	21.9363	1.13430
$375$	−33.4507	−1.72738
$376$	−2.62469	−0.135358
$377$	16.7233	0.861292
$378$	12.3627	0.635870
$379$	31.4188	1.61388	0.806938	−	0.590637i	$-0.201123\pi$
$379$	31.4188	1.61388	0.806938	+	0.590637i	$0.201123\pi$
$380$	−5.94879	−0.305167
$381$	4.67374	0.239443
$382$	−3.44893	−0.176463
$383$	−33.7737	−1.72575	−0.862877	−	0.505414i	$-0.831340\pi$
$383$	−33.7737	−1.72575	−0.862877	+	0.505414i	$0.831340\pi$
$384$	2.94445	0.150258
$385$	8.52406	0.434426
$386$	−21.6532	−1.10212
$387$	−3.60761	−0.183385
$388$	0.683844	0.0347169
$389$	−23.6373	−1.19846	−0.599229	−	0.800578i	$-0.704526\pi$
$389$	−23.6373	−1.19846	−0.599229	+	0.800578i	$0.704526\pi$
$390$	8.47202	0.428997
$391$	2.89017	0.146162
$392$	4.52677	0.228636
$393$	−32.0599	−1.61721
$394$	3.75818	0.189335
$395$	9.42411	0.474179
$396$	−21.5495	−1.08290
$397$	−34.7974	−1.74643	−0.873216	−	0.487333i	$-0.837970\pi$
$397$	−34.7974	−1.74643	−0.873216	+	0.487333i	$0.837970\pi$
$398$	−0.218863	−0.0109706
$399$	−19.3162	−0.967020
$400$	−2.96630	−0.148315
$401$	−6.95757	−0.347444	−0.173722	−	0.984795i	$-0.555579\pi$
$401$	−6.95757	−0.347444	−0.173722	+	0.984795i	$0.555579\pi$
$402$	34.7035	1.73085
$403$	12.3632	0.615857
$404$	1.47094	0.0731818
$405$	−8.75210	−0.434895
$406$	13.0351	0.646921
$407$	−6.63272	−0.328772
$408$	16.9941	0.841333
$409$	−8.00132	−0.395640	−0.197820	−	0.980238i	$-0.563386\pi$
$409$	−8.00132	−0.395640	−0.197820	+	0.980238i	$0.563386\pi$
$410$	17.5789	0.868162
$411$	−3.81649	−0.188254
$412$	−8.00953	−0.394601
$413$	−7.75595	−0.381646
$414$	−2.83920	−0.139539
$415$	−20.9776	−1.02975
$416$	2.01762	0.0989218
$417$	−44.3267	−2.17069
$418$	15.8546	0.775474
$419$	−32.0309	−1.56481	−0.782407	−	0.622768i	$-0.786008\pi$
$419$	−32.0309	−1.56481	−0.782407	+	0.622768i	$0.786008\pi$
$420$	6.60359	0.322222
$421$	−11.0033	−0.536266	−0.268133	−	0.963382i	$-0.586407\pi$
$421$	−11.0033	−0.536266	−0.268133	+	0.963382i	$0.586407\pi$
$422$	8.44463	0.411078
$423$	14.8815	0.723562
$424$	5.90783	0.286909
$425$	−17.1202	−0.830451
$426$	48.2605	2.33823
$427$	−5.47773	−0.265086
$428$	−9.64412	−0.466166
$429$	−22.5795	−1.09015
$430$	−0.907394	−0.0437584
$431$	−35.7072	−1.71995	−0.859977	−	0.510332i	$-0.829522\pi$
$431$	−35.7072	−1.71995	−0.859977	+	0.510332i	$0.829522\pi$
$432$	−7.86108	−0.378216
$433$	26.1368	1.25605	0.628027	−	0.778191i	$-0.283863\pi$
$433$	26.1368	1.25605	0.628027	+	0.778191i	$0.283863\pi$
$434$	9.63665	0.462574
$435$	−34.8041	−1.66873
$436$	−12.3471	−0.591320
$437$	2.08888	0.0999248
$438$	−47.7694	−2.28251
$439$	29.1259	1.39010	0.695051	−	0.718961i	$-0.255382\pi$
$439$	29.1259	1.39010	0.695051	+	0.718961i	$0.255382\pi$
$440$	−5.42019	−0.258397
$441$	−25.6659	−1.22218
$442$	11.6448	0.553887
$443$	−16.8012	−0.798251	−0.399126	−	0.916896i	$-0.630686\pi$
$443$	−16.8012	−0.798251	−0.399126	+	0.916896i	$0.630686\pi$
$444$	−5.13837	−0.243856
$445$	−22.4799	−1.06565
$446$	−8.10682	−0.383869
$447$	−37.1204	−1.75573
$448$	1.57265	0.0743007
$449$	4.42888	0.209012	0.104506	−	0.994524i	$-0.466674\pi$
$449$	4.42888	0.209012	0.104506	+	0.994524i	$0.466674\pi$
$450$	16.8183	0.792822
$451$	−46.8511	−2.20613
$452$	6.14153	0.288873
$453$	−26.6151	−1.25049
$454$	−1.98719	−0.0932634
$455$	4.52496	0.212133
$456$	12.2826	0.575184
$457$	−41.5020	−1.94138	−0.970691	−	0.240333i	$-0.922743\pi$
$457$	−41.5020	−1.94138	−0.970691	+	0.240333i	$0.922743\pi$
$458$	−20.4835	−0.957129
$459$	−45.3707	−2.11772
$460$	−0.714122	−0.0332961
$461$	−20.0817	−0.935297	−0.467649	−	0.883914i	$-0.654899\pi$
$461$	−20.0817	−0.935297	−0.467649	+	0.883914i	$0.654899\pi$
$462$	−17.5998	−0.818815
$463$	−17.1012	−0.794760	−0.397380	−	0.917654i	$-0.630081\pi$
$463$	−17.1012	−0.794760	−0.397380	+	0.917654i	$0.630081\pi$
$464$	−8.28862	−0.384789
$465$	−25.7301	−1.19321
$466$	−2.13471	−0.0988887
$467$	−39.5943	−1.83221	−0.916104	−	0.400942i	$-0.868683\pi$
$467$	−39.5943	−1.83221	−0.916104	+	0.400942i	$0.868683\pi$
$468$	−11.4395	−0.528790
$469$	18.5354	0.855884
$470$	3.74302	0.172653
$471$	58.1262	2.67832
$472$	4.93177	0.227003
$473$	2.41837	0.111197
$474$	−19.4581	−0.893742
$475$	−12.3737	−0.567745
$476$	9.07666	0.416028
$477$	−33.4962	−1.53368
$478$	−27.7722	−1.27027
$479$	11.5139	0.526081	0.263041	−	0.964785i	$-0.415275\pi$
$479$	11.5139	0.526081	0.263041	+	0.964785i	$0.415275\pi$
$480$	−4.19902	−0.191658
$481$	−3.52095	−0.160541
$482$	−21.7423	−0.990336
$483$	−2.31881	−0.105510
$484$	3.44578	0.156626
$485$	−0.975217	−0.0442823
$486$	−5.51262	−0.250057
$487$	0.150171	0.00680488	0.00340244	−	0.999994i	$-0.498917\pi$
$487$	0.150171	0.00680488	0.00340244	+	0.999994i	$0.498917\pi$
$488$	3.48312	0.157673
$489$	−21.9539	−0.992792
$490$	−6.45553	−0.291631
$491$	4.56227	0.205892	0.102946	−	0.994687i	$-0.467173\pi$
$491$	4.56227	0.205892	0.102946	+	0.994687i	$0.467173\pi$
$492$	−36.2955	−1.63633
$493$	−47.8383	−2.15453
$494$	8.41635	0.378670
$495$	30.7313	1.38127
$496$	−6.12765	−0.275139
$497$	25.7762	1.15622
$498$	43.3127	1.94089
$499$	23.7195	1.06183	0.530915	−	0.847425i	$-0.321848\pi$
$499$	23.7195	1.06183	0.530915	+	0.847425i	$0.321848\pi$
$500$	11.3606	0.508060
$501$	−17.0224	−0.760506
$502$	15.7343	0.702257
$503$	−23.7208	−1.05766	−0.528830	−	0.848728i	$-0.677369\pi$
$503$	−23.7208	−1.05766	−0.528830	+	0.848728i	$0.677369\pi$
$504$	−8.91660	−0.397177
$505$	−2.09767	−0.0933452
$506$	1.90326	0.0846104
$507$	26.2917	1.16765
$508$	−1.58730	−0.0704252
$509$	−12.8265	−0.568524	−0.284262	−	0.958747i	$-0.591749\pi$
$509$	−12.8265	−0.568524	−0.284262	+	0.958747i	$0.591749\pi$
$510$	−24.2349	−1.07314
$511$	−25.5139	−1.12867
$512$	−1.00000	−0.0441942
$513$	−32.7919	−1.44780
$514$	19.9895	0.881698
$515$	11.4222	0.503323
$516$	1.87351	0.0824768
$517$	−9.97583	−0.438736
$518$	−2.74444	−0.120584
$519$	32.5714	1.42973
$520$	−2.87728	−0.126177
$521$	0.748727	0.0328023	0.0164012	−	0.999865i	$-0.494779\pi$
$521$	0.748727	0.0328023	0.0164012	+	0.999865i	$0.494779\pi$
$522$	46.9947	2.05690
$523$	−5.13934	−0.224728	−0.112364	−	0.993667i	$-0.535842\pi$
$523$	−5.13934	−0.224728	−0.112364	+	0.993667i	$0.535842\pi$
$524$	10.8882	0.475655
$525$	13.7357	0.599476
$526$	16.3272	0.711901
$527$	−35.3661	−1.54057
$528$	11.1912	0.487032
$529$	−22.7492	−0.989097
$530$	−8.42503	−0.365960
$531$	−27.9621	−1.21345
$532$	6.56020	0.284421
$533$	−24.8707	−1.07727
$534$	46.4146	2.00856
$535$	13.7533	0.594606
$536$	−11.7861	−0.509081
$537$	−23.9097	−1.03178
$538$	−19.2157	−0.828449
$539$	17.2052	0.741079
$540$	11.2105	0.482424
$541$	−7.13919	−0.306938	−0.153469	−	0.988153i	$-0.549044\pi$
$541$	−7.13919	−0.306938	−0.153469	+	0.988153i	$0.549044\pi$
$542$	−15.6227	−0.671052
$543$	−12.1339	−0.520717
$544$	−5.77157	−0.247454
$545$	17.6080	0.754243
$546$	−9.34276	−0.399833
$547$	33.5402	1.43408	0.717038	−	0.697034i	$-0.245497\pi$
$547$	33.5402	1.43408	0.717038	+	0.697034i	$0.245497\pi$
$548$	1.29616	0.0553694
$549$	−19.7486	−0.842848
$550$	−11.2742	−0.480733
$551$	−34.5754	−1.47296
$552$	1.47446	0.0627572
$553$	−10.3927	−0.441943
$554$	12.8887	0.547587
$555$	7.32772	0.311045
$556$	15.0543	0.638445
$557$	−8.20474	−0.347646	−0.173823	−	0.984777i	$-0.555612\pi$
$557$	−8.20474	−0.347646	−0.173823	+	0.984777i	$0.555612\pi$
$558$	34.7425	1.47077
$559$	1.28378	0.0542981
$560$	−2.24272	−0.0947724
$561$	64.5905	2.72701
$562$	−23.9721	−1.01120
$563$	25.8919	1.09121	0.545606	−	0.838042i	$-0.316299\pi$
$563$	25.8919	1.09121	0.545606	+	0.838042i	$0.316299\pi$
$564$	−7.72828	−0.325419
$565$	−8.75831	−0.368465
$566$	4.70509	0.197769
$567$	9.65163	0.405330
$568$	−16.3903	−0.687722
$569$	39.2733	1.64642	0.823212	−	0.567734i	$-0.192180\pi$
$569$	39.2733	1.64642	0.823212	+	0.567734i	$0.192180\pi$
$570$	−17.5159	−0.733662
$571$	8.51594	0.356381	0.178190	−	0.983996i	$-0.442976\pi$
$571$	8.51594	0.356381	0.178190	+	0.983996i	$0.442976\pi$
$572$	7.66848	0.320635
$573$	−10.1552	−0.424240
$574$	−19.3857	−0.809143
$575$	−1.48540	−0.0619455
$576$	5.66979	0.236241
$577$	−30.9830	−1.28984	−0.644920	−	0.764250i	$-0.723109\pi$
$577$	−30.9830	−1.28984	−0.644920	+	0.764250i	$0.723109\pi$
$578$	−16.3110	−0.678448
$579$	−63.7567	−2.64964
$580$	11.8202	0.490808
$581$	23.1336	0.959744
$582$	2.01355	0.0834642
$583$	22.4542	0.929959
$584$	16.2235	0.671334
$585$	16.3136	0.674484
$586$	11.9292	0.492791
$587$	21.6949	0.895444	0.447722	−	0.894173i	$-0.352235\pi$
$587$	21.6949	0.895444	0.447722	+	0.894173i	$0.352235\pi$
$588$	13.3289	0.549673
$589$	−25.5611	−1.05323
$590$	−7.03310	−0.289548
$591$	11.0658	0.455186
$592$	1.74510	0.0717233
$593$	22.9502	0.942450	0.471225	−	0.882013i	$-0.343812\pi$
$593$	22.9502	0.942450	0.471225	+	0.882013i	$0.343812\pi$
$594$	−29.8781	−1.22591
$595$	−12.9440	−0.530654
$596$	12.6069	0.516398
$597$	−0.644432	−0.0263749
$598$	1.01034	0.0413159
$599$	39.0034	1.59364	0.796818	−	0.604219i	$-0.206515\pi$
$599$	39.0034	1.59364	0.796818	+	0.604219i	$0.206515\pi$
$600$	−8.73412	−0.356569
$601$	−33.2861	−1.35777	−0.678884	−	0.734245i	$-0.737536\pi$
$601$	−33.2861	−1.35777	−0.678884	+	0.734245i	$0.737536\pi$
$602$	1.00065	0.0407836
$603$	66.8246	2.72131
$604$	9.03906	0.367794
$605$	−4.91395	−0.199781
$606$	4.33110	0.175939
$607$	−23.8840	−0.969423	−0.484712	−	0.874674i	$-0.661075\pi$
$607$	−23.8840	−0.969423	−0.484712	+	0.874674i	$0.661075\pi$
$608$	−4.17143	−0.169174
$609$	38.3812	1.55529
$610$	−4.96720	−0.201116
$611$	−5.29563	−0.214238
$612$	32.7236	1.32277
$613$	0.103564	0.00418291	0.00209145	−	0.999998i	$-0.499334\pi$
$613$	0.103564	0.00418291	0.00209145	+	0.999998i	$0.499334\pi$
$614$	−1.59515	−0.0643750
$615$	51.7603	2.08718
$616$	5.97727	0.240831
$617$	−23.0228	−0.926864	−0.463432	−	0.886132i	$-0.653382\pi$
$617$	−23.0228	−0.926864	−0.463432	+	0.886132i	$0.653382\pi$
$618$	−23.5837	−0.948674
$619$	−13.4243	−0.539568	−0.269784	−	0.962921i	$-0.586952\pi$
$619$	−13.4243	−0.539568	−0.269784	+	0.962921i	$0.586952\pi$
$620$	8.73851	0.350947
$621$	−3.93650	−0.157966
$622$	30.1563	1.20916
$623$	24.7903	0.993204
$624$	5.94078	0.237821
$625$	−1.36959	−0.0547837
$626$	−2.03499	−0.0813346
$627$	46.6831	1.86434
$628$	−19.7409	−0.787749
$629$	10.0720	0.401596
$630$	12.7158	0.506609
$631$	34.4028	1.36956	0.684778	−	0.728752i	$-0.259899\pi$
$631$	34.4028	1.36956	0.684778	+	0.728752i	$0.259899\pi$
$632$	6.60841	0.262868
$633$	24.8648	0.988287
$634$	7.41987	0.294681
$635$	2.26362	0.0898291
$636$	17.3953	0.689769
$637$	9.13329	0.361874
$638$	−31.5030	−1.24722
$639$	92.9297	3.67624
$640$	1.42608	0.0563707
$641$	−14.5476	−0.574596	−0.287298	−	0.957841i	$-0.592757\pi$
$641$	−14.5476	−0.574596	−0.287298	+	0.957841i	$0.592757\pi$
$642$	−28.3966	−1.12073
$643$	−4.59064	−0.181037	−0.0905185	−	0.995895i	$-0.528852\pi$
$643$	−4.59064	−0.181037	−0.0905185	+	0.995895i	$0.528852\pi$
$644$	0.787519	0.0310326
$645$	−2.67178	−0.105201
$646$	−24.0757	−0.947246
$647$	15.6435	0.615010	0.307505	−	0.951546i	$-0.400506\pi$
$647$	15.6435	0.615010	0.307505	+	0.951546i	$0.400506\pi$
$648$	−6.13717	−0.241091
$649$	18.7445	0.735785
$650$	−5.98485	−0.234745
$651$	28.3746	1.11209
$652$	7.45604	0.292001
$653$	14.3697	0.562329	0.281165	−	0.959660i	$-0.409279\pi$
$653$	14.3697	0.562329	0.281165	+	0.959660i	$0.409279\pi$
$654$	−36.3555	−1.42161
$655$	−15.5275	−0.606710
$656$	12.3268	0.481279
$657$	−91.9840	−3.58864
$658$	−4.12772	−0.160915
$659$	32.3221	1.25909	0.629545	−	0.776964i	$-0.283241\pi$
$659$	32.3221	1.25909	0.629545	+	0.776964i	$0.283241\pi$
$660$	−15.9595	−0.621222
$661$	−47.4941	−1.84731	−0.923653	−	0.383230i	$-0.874812\pi$
$661$	−47.4941	−1.84731	−0.923653	+	0.383230i	$0.874812\pi$
$662$	25.9405	1.00821
$663$	34.2876	1.33162
$664$	−14.7099	−0.570857
$665$	−9.35537	−0.362786
$666$	−9.89437	−0.383399
$667$	−4.15060	−0.160712
$668$	5.78119	0.223681
$669$	−23.8701	−0.922873
$670$	16.8079	0.649345
$671$	13.2385	0.511066
$672$	4.63059	0.178629
$673$	20.2700	0.781352	0.390676	−	0.920528i	$-0.372241\pi$
$673$	20.2700	0.781352	0.390676	+	0.920528i	$0.372241\pi$
$674$	33.3933	1.28626
$675$	23.3183	0.897521
$676$	−8.92922	−0.343432
$677$	−22.3110	−0.857480	−0.428740	−	0.903428i	$-0.641042\pi$
$677$	−22.3110	−0.857480	−0.428740	+	0.903428i	$0.641042\pi$
$678$	18.0834	0.694490
$679$	1.07545	0.0412719
$680$	8.23072	0.315634
$681$	−5.85118	−0.224218
$682$	−23.2897	−0.891809
$683$	−15.0461	−0.575722	−0.287861	−	0.957672i	$-0.592944\pi$
$683$	−15.0461	−0.575722	−0.287861	+	0.957672i	$0.592944\pi$
$684$	23.6512	0.904325
$685$	−1.84843	−0.0706250
$686$	18.1276	0.692114
$687$	−60.3125	−2.30107
$688$	−0.636286	−0.0242582
$689$	11.9197	0.454106
$690$	−2.10270	−0.0800483
$691$	−1.91685	−0.0729205	−0.0364602	−	0.999335i	$-0.511608\pi$
$691$	−1.91685	−0.0729205	−0.0364602	+	0.999335i	$0.511608\pi$
$692$	−11.0620	−0.420513
$693$	−33.8899	−1.28737
$694$	−3.43680	−0.130459
$695$	−21.4687	−0.814353
$696$	−24.4054	−0.925085
$697$	71.1447	2.69480
$698$	30.4297	1.15178
$699$	−6.28556	−0.237742
$700$	−4.66495	−0.176318
$701$	−11.4224	−0.431419	−0.215710	−	0.976458i	$-0.569206\pi$
$701$	−11.4224	−0.431419	−0.215710	+	0.976458i	$0.569206\pi$
$702$	−15.8606	−0.598621
$703$	7.27958	0.274554
$704$	−3.80076	−0.143247
$705$	11.0211	0.415080
$706$	−18.6107	−0.700425
$707$	2.31327	0.0869994
$708$	14.5214	0.545746
$709$	−35.3013	−1.32577	−0.662884	−	0.748723i	$-0.730668\pi$
$709$	−35.3013	−1.32577	−0.662884	+	0.748723i	$0.730668\pi$
$710$	23.3739	0.877206
$711$	−37.4683	−1.40517
$712$	−15.7634	−0.590759
$713$	−3.06847	−0.114915
$714$	26.7258	1.00019
$715$	−10.9359	−0.408978
$716$	8.12025	0.303468
$717$	−81.7739	−3.05390
$718$	16.5362	0.617124
$719$	0.987241	0.0368179	0.0184089	−	0.999831i	$-0.494140\pi$
$719$	0.987241	0.0368179	0.0184089	+	0.999831i	$0.494140\pi$
$720$	−8.08558	−0.301332
$721$	−12.5962	−0.469107
$722$	1.59916	0.0595144
$723$	−64.0192	−2.38090
$724$	4.12095	0.153154
$725$	24.5865	0.913120
$726$	10.1459	0.376551
$727$	−16.1917	−0.600517	−0.300258	−	0.953858i	$-0.597073\pi$
$727$	−16.1917	−0.600517	−0.300258	+	0.953858i	$0.597073\pi$
$728$	3.17301	0.117599
$729$	−34.6432	−1.28308
$730$	−23.1360	−0.856303
$731$	−3.67237	−0.135827
$732$	10.2559	0.379068
$733$	−4.18082	−0.154422	−0.0772111	−	0.997015i	$-0.524602\pi$
$733$	−4.18082	−0.154422	−0.0772111	+	0.997015i	$0.524602\pi$
$734$	20.2979	0.749211
$735$	−19.0080	−0.701121
$736$	−0.500759	−0.0184582
$737$	−44.7960	−1.65008
$738$	−69.8902	−2.57269
$739$	45.5269	1.67473	0.837367	−	0.546642i	$-0.184094\pi$
$739$	45.5269	1.67473	0.837367	+	0.546642i	$0.184094\pi$
$740$	−2.48866	−0.0914848
$741$	24.7815	0.910373
$742$	9.29095	0.341081
$743$	−13.6297	−0.500024	−0.250012	−	0.968243i	$-0.580435\pi$
$743$	−13.6297	−0.500024	−0.250012	+	0.968243i	$0.580435\pi$
$744$	−18.0426	−0.661472
$745$	−17.9784	−0.658678
$746$	17.1977	0.629652
$747$	83.4023	3.05153
$748$	−21.9363	−0.802072
$749$	−15.1668	−0.554184
$750$	33.4507	1.22145
$751$	18.2892	0.667383	0.333691	−	0.942682i	$-0.391706\pi$
$751$	18.2892	0.667383	0.333691	+	0.942682i	$0.391706\pi$
$752$	2.62469	0.0957127
$753$	46.3289	1.68832
$754$	−16.7233	−0.609025
$755$	−12.8904	−0.469130
$756$	−12.3627	−0.449628
$757$	37.8747	1.37658	0.688289	−	0.725436i	$-0.258362\pi$
$757$	37.8747	1.37658	0.688289	+	0.725436i	$0.258362\pi$
$758$	−31.4188	−1.14118
$759$	5.60407	0.203415
$760$	5.94879	0.215785
$761$	13.1383	0.476264	0.238132	−	0.971233i	$-0.423465\pi$
$761$	13.1383	0.476264	0.238132	+	0.971233i	$0.423465\pi$
$762$	−4.67374	−0.169312
$763$	−19.4177	−0.702968
$764$	3.44893	0.124778
$765$	−46.6665	−1.68723
$766$	33.7737	1.22029
$767$	9.95043	0.359289
$768$	−2.94445	−0.106249
$769$	34.3352	1.23816	0.619080	−	0.785328i	$-0.287506\pi$
$769$	34.3352	1.23816	0.619080	+	0.785328i	$0.287506\pi$
$770$	−8.52406	−0.307186
$771$	58.8581	2.11972
$772$	21.6532	0.779314
$773$	17.8270	0.641194	0.320597	−	0.947216i	$-0.396116\pi$
$773$	17.8270	0.641194	0.320597	+	0.947216i	$0.396116\pi$
$774$	3.60761	0.129673
$775$	18.1764	0.652916
$776$	−0.683844	−0.0245486
$777$	−8.08086	−0.289899
$778$	23.6373	0.847438
$779$	51.4202	1.84232
$780$	−8.47202	−0.303347
$781$	−62.2956	−2.22911
$782$	−2.89017	−0.103352
$783$	65.1574	2.32854
$784$	−4.52677	−0.161670
$785$	28.1522	1.00479
$786$	32.0599	1.14354
$787$	0.302620	0.0107872	0.00539362	−	0.999985i	$-0.498283\pi$
$787$	0.302620	0.0107872	0.00539362	+	0.999985i	$0.498283\pi$
$788$	−3.75818	−0.133880
$789$	48.0747	1.71150
$790$	−9.42411	−0.335295
$791$	9.65848	0.343416
$792$	21.5495	0.765729
$793$	7.02760	0.249557
$794$	34.7974	1.23491
$795$	−24.8071	−0.879817
$796$	0.218863	0.00775741
$797$	−10.3651	−0.367152	−0.183576	−	0.983005i	$-0.558767\pi$
$797$	−10.3651	−0.367152	−0.183576	+	0.983005i	$0.558767\pi$
$798$	19.3162	0.683786
$799$	15.1486	0.535919
$800$	2.96630	0.104874
$801$	89.3753	3.15792
$802$	6.95757	0.245680
$803$	61.6617	2.17600
$804$	−34.7035	−1.22390
$805$	−1.12306	−0.0395828
$806$	−12.3632	−0.435477
$807$	−56.5798	−1.99170
$808$	−1.47094	−0.0517474
$809$	−40.7175	−1.43155	−0.715776	−	0.698330i	$-0.753927\pi$
$809$	−40.7175	−1.43155	−0.715776	+	0.698330i	$0.753927\pi$
$810$	8.75210	0.307517
$811$	11.2408	0.394718	0.197359	−	0.980331i	$-0.436764\pi$
$811$	11.2408	0.394718	0.197359	+	0.980331i	$0.436764\pi$
$812$	−13.0351	−0.457442
$813$	−46.0002	−1.61330
$814$	6.63272	0.232477
$815$	−10.6329	−0.372454
$816$	−16.9941	−0.594913
$817$	−2.65422	−0.0928595
$818$	8.00132	0.279760
$819$	−17.9903	−0.628632
$820$	−17.5789	−0.613883
$821$	−9.17982	−0.320378	−0.160189	−	0.987086i	$-0.551210\pi$
$821$	−9.17982	−0.320378	−0.160189	+	0.987086i	$0.551210\pi$
$822$	3.81649	0.133116
$823$	45.5263	1.58695	0.793473	−	0.608605i	$-0.208271\pi$
$823$	45.5263	1.58695	0.793473	+	0.608605i	$0.208271\pi$
$824$	8.00953	0.279025
$825$	−33.1963	−1.15575
$826$	7.75595	0.269864
$827$	−29.3490	−1.02056	−0.510282	−	0.860007i	$-0.670459\pi$
$827$	−29.3490	−1.02056	−0.510282	+	0.860007i	$0.670459\pi$
$828$	2.83920	0.0986690
$829$	−36.2463	−1.25889	−0.629443	−	0.777047i	$-0.716717\pi$
$829$	−36.2463	−1.25889	−0.629443	+	0.777047i	$0.716717\pi$
$830$	20.9776	0.728141
$831$	37.9501	1.31647
$832$	−2.01762	−0.0699483
$833$	−26.1266	−0.905232
$834$	44.3267	1.53491
$835$	−8.24444	−0.285311
$836$	−15.8546	−0.548343
$837$	48.1699	1.66500
$838$	32.0309	1.10649
$839$	−12.4538	−0.429951	−0.214976	−	0.976619i	$-0.568967\pi$
$839$	−12.4538	−0.429951	−0.214976	+	0.976619i	$0.568967\pi$
$840$	−6.60359	−0.227846
$841$	39.7012	1.36901
$842$	11.0033	0.379198
$843$	−70.5847	−2.43107
$844$	−8.44463	−0.290676
$845$	12.7338	0.438055
$846$	−14.8815	−0.511635
$847$	5.41900	0.186199
$848$	−5.90783	−0.202876
$849$	13.8539	0.475464
$850$	17.1202	0.587218
$851$	0.873876	0.0299561
$852$	−48.2605	−1.65338
$853$	42.1967	1.44479	0.722394	−	0.691481i	$-0.243042\pi$
$853$	42.1967	1.44479	0.722394	+	0.691481i	$0.243042\pi$
$854$	5.47773	0.187444
$855$	−33.7284	−1.15349
$856$	9.64412	0.329629
$857$	−49.5638	−1.69307	−0.846534	−	0.532335i	$-0.821315\pi$
$857$	−49.5638	−1.69307	−0.846534	+	0.532335i	$0.821315\pi$
$858$	22.5795	0.770850
$859$	18.9712	0.647289	0.323645	−	0.946179i	$-0.395092\pi$
$859$	18.9712	0.647289	0.323645	+	0.946179i	$0.395092\pi$
$860$	0.907394	0.0309419
$861$	−57.0802	−1.94529
$862$	35.7072	1.21619
$863$	48.9287	1.66555	0.832776	−	0.553610i	$-0.186750\pi$
$863$	48.9287	1.66555	0.832776	+	0.553610i	$0.186750\pi$
$864$	7.86108	0.267439
$865$	15.7752	0.536374
$866$	−26.1368	−0.888165
$867$	−48.0270	−1.63108
$868$	−9.63665	−0.327089
$869$	25.1170	0.852035
$870$	34.8041	1.17997
$871$	−23.7798	−0.805747
$872$	12.3471	0.418126
$873$	3.87726	0.131225
$874$	−2.08888	−0.0706575
$875$	17.8662	0.603988
$876$	47.7694	1.61398
$877$	26.4213	0.892184	0.446092	−	0.894987i	$-0.352815\pi$
$877$	26.4213	0.892184	0.446092	+	0.894987i	$0.352815\pi$
$878$	−29.1259	−0.982950
$879$	35.1250	1.18474
$880$	5.42019	0.182714
$881$	−9.61048	−0.323785	−0.161893	−	0.986808i	$-0.551760\pi$
$881$	−9.61048	−0.323785	−0.161893	+	0.986808i	$0.551760\pi$
$882$	25.6659	0.864214
$883$	20.3276	0.684078	0.342039	−	0.939686i	$-0.388882\pi$
$883$	20.3276	0.684078	0.342039	+	0.939686i	$0.388882\pi$
$884$	−11.6448	−0.391658
$885$	−20.7086	−0.696113
$886$	16.8012	0.564449
$887$	−37.2458	−1.25059	−0.625296	−	0.780388i	$-0.715022\pi$
$887$	−37.2458	−1.25059	−0.625296	+	0.780388i	$0.715022\pi$
$888$	5.13837	0.172432
$889$	−2.49627	−0.0837223
$890$	22.4799	0.753527
$891$	−23.3259	−0.781448
$892$	8.10682	0.271437
$893$	10.9487	0.366385
$894$	37.1204	1.24149
$895$	−11.5801	−0.387081
$896$	−1.57265	−0.0525386
$897$	2.97490	0.0993289
$898$	−4.42888	−0.147794
$899$	50.7897	1.69393
$900$	−16.8183	−0.560610
$901$	−34.0974	−1.13595
$902$	46.8511	1.55997
$903$	2.94638	0.0980494
$904$	−6.14153	−0.204264
$905$	−5.87681	−0.195352
$906$	26.6151	0.884227
$907$	−24.5269	−0.814401	−0.407201	−	0.913339i	$-0.633495\pi$
$907$	−24.5269	−0.814401	−0.407201	+	0.913339i	$0.633495\pi$
$908$	1.98719	0.0659472
$909$	8.33990	0.276617
$910$	−4.52496	−0.150001
$911$	−26.7677	−0.886853	−0.443426	−	0.896311i	$-0.646237\pi$
$911$	−26.7677	−0.886853	−0.443426	+	0.896311i	$0.646237\pi$
$912$	−12.2826	−0.406717
$913$	−55.9090	−1.85032
$914$	41.5020	1.37276
$915$	−14.6257	−0.483510
$916$	20.4835	0.676793
$917$	17.1234	0.565465
$918$	45.3707	1.49746
$919$	−12.7513	−0.420627	−0.210313	−	0.977634i	$-0.567448\pi$
$919$	−12.7513	−0.420627	−0.210313	+	0.977634i	$0.567448\pi$
$920$	0.714122	0.0235439
$921$	−4.69684	−0.154766
$922$	20.0817	0.661355
$923$	−33.0694	−1.08849
$924$	17.5998	0.578990
$925$	−5.17649	−0.170202
$926$	17.1012	0.561980
$927$	−45.4124	−1.49154
$928$	8.28862	0.272087
$929$	0.345927	0.0113495	0.00567474	−	0.999984i	$-0.498194\pi$
$929$	0.345927	0.0113495	0.00567474	+	0.999984i	$0.498194\pi$
$930$	25.7301	0.843724
$931$	−18.8831	−0.618869
$932$	2.13471	0.0699249
$933$	88.7938	2.90698
$934$	39.5943	1.29557
$935$	31.2830	1.02306
$936$	11.4395	0.373911
$937$	−12.8315	−0.419188	−0.209594	−	0.977788i	$-0.567214\pi$
$937$	−12.8315	−0.419188	−0.209594	+	0.977788i	$0.567214\pi$
$938$	−18.5354	−0.605201
$939$	−5.99193	−0.195539
$940$	−3.74302	−0.122084
$941$	−8.19829	−0.267257	−0.133628	−	0.991032i	$-0.542663\pi$
$941$	−8.19829	−0.267257	−0.133628	+	0.991032i	$0.542663\pi$
$942$	−58.1262	−1.89386
$943$	6.17274	0.201012
$944$	−4.93177	−0.160516
$945$	17.6302	0.573511
$946$	−2.41837	−0.0786280
$947$	−24.8072	−0.806127	−0.403063	−	0.915172i	$-0.632055\pi$
$947$	−24.8072	−0.806127	−0.403063	+	0.915172i	$0.632055\pi$
$948$	19.4581	0.631971
$949$	32.7329	1.06255
$950$	12.3737	0.401456
$951$	21.8474	0.708452
$952$	−9.07666	−0.294176
$953$	9.56936	0.309982	0.154991	−	0.987916i	$-0.450465\pi$
$953$	9.56936	0.309982	0.154991	+	0.987916i	$0.450465\pi$
$954$	33.4962	1.08448
$955$	−4.91845	−0.159157
$956$	27.7722	0.898217
$957$	−92.7592	−2.99848
$958$	−11.5139	−0.371996
$959$	2.03841	0.0658238
$960$	4.19902	0.135523
$961$	6.54807	0.211228
$962$	3.52095	0.113520
$963$	−54.6801	−1.76204
$964$	21.7423	0.700273
$965$	−30.8791	−0.994034
$966$	2.31881	0.0746065
$967$	23.7801	0.764718	0.382359	−	0.924014i	$-0.375112\pi$
$967$	23.7801	0.764718	0.382359	+	0.924014i	$0.375112\pi$
$968$	−3.44578	−0.110752
$969$	−70.8897	−2.27731
$970$	0.975217	0.0313123
$971$	7.02573	0.225466	0.112733	−	0.993625i	$-0.464039\pi$
$971$	7.02573	0.225466	0.112733	+	0.993625i	$0.464039\pi$
$972$	5.51262	0.176817
$973$	23.6752	0.758992
$974$	−0.150171	−0.00481178
$975$	−17.6221	−0.564359
$976$	−3.48312	−0.111492
$977$	46.4225	1.48519	0.742594	−	0.669742i	$-0.233595\pi$
$977$	46.4225	1.48519	0.742594	+	0.669742i	$0.233595\pi$
$978$	21.9539	0.702010
$979$	−59.9130	−1.91483
$980$	6.45553	0.206214
$981$	−70.0056	−2.23511
$982$	−4.56227	−0.145588
$983$	−18.1567	−0.579109	−0.289554	−	0.957162i	$-0.593507\pi$
$983$	−18.1567	−0.579109	−0.289554	+	0.957162i	$0.593507\pi$
$984$	36.2955	1.15706
$985$	5.35947	0.170767
$986$	47.8383	1.52348
$987$	−12.1539	−0.386862
$988$	−8.41635	−0.267760
$989$	−0.318626	−0.0101317
$990$	−30.7313	−0.976706
$991$	−9.44484	−0.300025	−0.150013	−	0.988684i	$-0.547931\pi$
$991$	−9.44484	−0.300025	−0.150013	+	0.988684i	$0.547931\pi$
$992$	6.12765	0.194553
$993$	76.3806	2.42387
$994$	−25.7762	−0.817572
$995$	−0.312116	−0.00989476
$996$	−43.3127	−1.37242
$997$	−41.0771	−1.30092	−0.650462	−	0.759539i	$-0.725425\pi$
$997$	−41.0771	−1.30092	−0.650462	+	0.759539i	$0.725425\pi$
$998$	−23.7195	−0.750827
$999$	−13.7184	−0.434031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.c.1.5	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.c.1.5	✓	57	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.94445 q^{3} +1.00000 q^{4} -1.42608 q^{5} +2.94445 q^{6} +1.57265 q^{7} -1.00000 q^{8} +5.66979 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.94445 q^{3} +1.00000 q^{4} -1.42608 q^{5} +2.94445 q^{6} +1.57265 q^{7} -1.00000 q^{8} +5.66979 q^{9} +1.42608 q^{10} -3.80076 q^{11} -2.94445 q^{12} -2.01762 q^{13} -1.57265 q^{14} +4.19902 q^{15} +1.00000 q^{16} +5.77157 q^{17} -5.66979 q^{18} +4.17143 q^{19} -1.42608 q^{20} -4.63059 q^{21} +3.80076 q^{22} +0.500759 q^{23} +2.94445 q^{24} -2.96630 q^{25} +2.01762 q^{26} -7.86108 q^{27} +1.57265 q^{28} -8.28862 q^{29} -4.19902 q^{30} -6.12765 q^{31} -1.00000 q^{32} +11.1912 q^{33} -5.77157 q^{34} -2.24272 q^{35} +5.66979 q^{36} +1.74510 q^{37} -4.17143 q^{38} +5.94078 q^{39} +1.42608 q^{40} +12.3268 q^{41} +4.63059 q^{42} -0.636286 q^{43} -3.80076 q^{44} -8.08558 q^{45} -0.500759 q^{46} +2.62469 q^{47} -2.94445 q^{48} -4.52677 q^{49} +2.96630 q^{50} -16.9941 q^{51} -2.01762 q^{52} -5.90783 q^{53} +7.86108 q^{54} +5.42019 q^{55} -1.57265 q^{56} -12.2826 q^{57} +8.28862 q^{58} -4.93177 q^{59} +4.19902 q^{60} -3.48312 q^{61} +6.12765 q^{62} +8.91660 q^{63} +1.00000 q^{64} +2.87728 q^{65} -11.1912 q^{66} +11.7861 q^{67} +5.77157 q^{68} -1.47446 q^{69} +2.24272 q^{70} +16.3903 q^{71} -5.66979 q^{72} -16.2235 q^{73} -1.74510 q^{74} +8.73412 q^{75} +4.17143 q^{76} -5.97727 q^{77} -5.94078 q^{78} -6.60841 q^{79} -1.42608 q^{80} +6.13717 q^{81} -12.3268 q^{82} +14.7099 q^{83} -4.63059 q^{84} -8.23072 q^{85} +0.636286 q^{86} +24.4054 q^{87} +3.80076 q^{88} +15.7634 q^{89} +8.08558 q^{90} -3.17301 q^{91} +0.500759 q^{92} +18.0426 q^{93} -2.62469 q^{94} -5.94879 q^{95} +2.94445 q^{96} +0.683844 q^{97} +4.52677 q^{98} -21.5495 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9}+O(q^{10}) \) 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9	\( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9} + 15 q^{10} + 13 q^{11} - 5 q^{12} - 43 q^{13} + 28 q^{14} - 10 q^{15} + 57 q^{16} - 50 q^{18} - 6 q^{19} - 15 q^{20} - 23 q^{21} - 13 q^{22} - q^{23} + 5 q^{24} + 20 q^{25} + 43 q^{26} - 20 q^{27} - 28 q^{28} - 4 q^{29} + 10 q^{30} - 34 q^{31} - 57 q^{32} - 43 q^{33} + 26 q^{35} + 50 q^{36} - 64 q^{37} + 6 q^{38} + 8 q^{39} + 15 q^{40} + 27 q^{41} + 23 q^{42} - 29 q^{43} + 13 q^{44} - 76 q^{45} + q^{46} - 25 q^{47} - 5 q^{48} + 7 q^{49} - 20 q^{50} + 27 q^{51} - 43 q^{52} - 34 q^{53} + 20 q^{54} - 36 q^{55} + 28 q^{56} - 33 q^{57} + 4 q^{58} + 19 q^{59} - 10 q^{60} - 58 q^{61} + 34 q^{62} - 65 q^{63} + 57 q^{64} + 17 q^{65} + 43 q^{66} - 84 q^{67} - 33 q^{69} - 26 q^{70} + 22 q^{71} - 50 q^{72} - 82 q^{73} + 64 q^{74} + 8 q^{75} - 6 q^{76} - 41 q^{77} - 8 q^{78} + 8 q^{79} - 15 q^{80} + 25 q^{81} - 27 q^{82} - 23 q^{83} - 23 q^{84} - 58 q^{85} + 29 q^{86} - 17 q^{87} - 13 q^{88} + 18 q^{89} + 76 q^{90} - 4 q^{91} - q^{92} - 60 q^{93} + 25 q^{94} + 36 q^{95} + 5 q^{96} - 156 q^{97} - 7 q^{98} + 25 q^{99}+O(q^{100}) \) 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9 + 15 * q^10 + 13 * q^11 - 5 * q^12 - 43 * q^13 + 28 * q^14 - 10 * q^15 + 57 * q^16 - 50 * q^18 - 6 * q^19 - 15 * q^20 - 23 * q^21 - 13 * q^22 - q^23 + 5 * q^24 + 20 * q^25 + 43 * q^26 - 20 * q^27 - 28 * q^28 - 4 * q^29 + 10 * q^30 - 34 * q^31 - 57 * q^32 - 43 * q^33 + 26 * q^35 + 50 * q^36 - 64 * q^37 + 6 * q^38 + 8 * q^39 + 15 * q^40 + 27 * q^41 + 23 * q^42 - 29 * q^43 + 13 * q^44 - 76 * q^45 + q^46 - 25 * q^47 - 5 * q^48 + 7 * q^49 - 20 * q^50 + 27 * q^51 - 43 * q^52 - 34 * q^53 + 20 * q^54 - 36 * q^55 + 28 * q^56 - 33 * q^57 + 4 * q^58 + 19 * q^59 - 10 * q^60 - 58 * q^61 + 34 * q^62 - 65 * q^63 + 57 * q^64 + 17 * q^65 + 43 * q^66 - 84 * q^67 - 33 * q^69 - 26 * q^70 + 22 * q^71 - 50 * q^72 - 82 * q^73 + 64 * q^74 + 8 * q^75 - 6 * q^76 - 41 * q^77 - 8 * q^78 + 8 * q^79 - 15 * q^80 + 25 * q^81 - 27 * q^82 - 23 * q^83 - 23 * q^84 - 58 * q^85 + 29 * q^86 - 17 * q^87 - 13 * q^88 + 18 * q^89 + 76 * q^90 - 4 * q^91 - q^92 - 60 * q^93 + 25 * q^94 + 36 * q^95 + 5 * q^96 - 156 * q^97 - 7 * q^98 + 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more