LMFDB - Embedded newform 8038.2.a.a.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.10614 q^{3} +1.00000 q^{4} -0.0733427 q^{5} -2.10614 q^{6} +4.17239 q^{7} +1.00000 q^{8} +1.43582 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.10614 q^{3} +1.00000 q^{4} -0.0733427 q^{5} -2.10614 q^{6} +4.17239 q^{7} +1.00000 q^{8} +1.43582 q^{9} -0.0733427 q^{10} +3.04833 q^{11} -2.10614 q^{12} -5.28486 q^{13} +4.17239 q^{14} +0.154470 q^{15} +1.00000 q^{16} +1.26189 q^{17} +1.43582 q^{18} +0.617075 q^{19} -0.0733427 q^{20} -8.78763 q^{21} +3.04833 q^{22} -4.17171 q^{23} -2.10614 q^{24} -4.99462 q^{25} -5.28486 q^{26} +3.29438 q^{27} +4.17239 q^{28} -6.94128 q^{29} +0.154470 q^{30} +6.71719 q^{31} +1.00000 q^{32} -6.42021 q^{33} +1.26189 q^{34} -0.306014 q^{35} +1.43582 q^{36} -2.70741 q^{37} +0.617075 q^{38} +11.1306 q^{39} -0.0733427 q^{40} +0.105799 q^{41} -8.78763 q^{42} -9.93029 q^{43} +3.04833 q^{44} -0.105307 q^{45} -4.17171 q^{46} -7.42797 q^{47} -2.10614 q^{48} +10.4088 q^{49} -4.99462 q^{50} -2.65771 q^{51} -5.28486 q^{52} -10.1293 q^{53} +3.29438 q^{54} -0.223573 q^{55} +4.17239 q^{56} -1.29964 q^{57} -6.94128 q^{58} +11.6365 q^{59} +0.154470 q^{60} -6.65436 q^{61} +6.71719 q^{62} +5.99080 q^{63} +1.00000 q^{64} +0.387606 q^{65} -6.42021 q^{66} +11.1446 q^{67} +1.26189 q^{68} +8.78619 q^{69} -0.306014 q^{70} -12.8784 q^{71} +1.43582 q^{72} -15.6780 q^{73} -2.70741 q^{74} +10.5194 q^{75} +0.617075 q^{76} +12.7188 q^{77} +11.1306 q^{78} -14.3847 q^{79} -0.0733427 q^{80} -11.2459 q^{81} +0.105799 q^{82} -0.991837 q^{83} -8.78763 q^{84} -0.0925501 q^{85} -9.93029 q^{86} +14.6193 q^{87} +3.04833 q^{88} -3.60769 q^{89} -0.105307 q^{90} -22.0505 q^{91} -4.17171 q^{92} -14.1473 q^{93} -7.42797 q^{94} -0.0452579 q^{95} -2.10614 q^{96} -15.2101 q^{97} +10.4088 q^{98} +4.37686 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) $ 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * 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
$3$	−2.10614	−1.21598	−0.607990	−	0.793945i	$-0.708024\pi$
$3$	−2.10614	−1.21598	−0.607990	+	0.793945i	$0.708024\pi$
$4$	1.00000	0.500000
$5$	−0.0733427	−0.0327998	−0.0163999	−	0.999866i	$-0.505220\pi$
$5$	−0.0733427	−0.0327998	−0.0163999	+	0.999866i	$0.505220\pi$
$6$	−2.10614	−0.859827
$7$	4.17239	1.57701	0.788507	−	0.615025i	$-0.210854\pi$
$7$	4.17239	1.57701	0.788507	+	0.615025i	$0.210854\pi$
$8$	1.00000	0.353553
$9$	1.43582	0.478607
$10$	−0.0733427	−0.0231930
$11$	3.04833	0.919107	0.459554	−	0.888150i	$-0.348009\pi$
$11$	3.04833	0.919107	0.459554	+	0.888150i	$0.348009\pi$
$12$	−2.10614	−0.607990
$13$	−5.28486	−1.46576	−0.732878	−	0.680360i	$-0.761823\pi$
$13$	−5.28486	−1.46576	−0.732878	+	0.680360i	$0.761823\pi$
$14$	4.17239	1.11512
$15$	0.154470	0.0398839
$16$	1.00000	0.250000
$17$	1.26189	0.306052	0.153026	−	0.988222i	$-0.451098\pi$
$17$	1.26189	0.306052	0.153026	+	0.988222i	$0.451098\pi$
$18$	1.43582	0.338426
$19$	0.617075	0.141567	0.0707833	−	0.997492i	$-0.477450\pi$
$19$	0.617075	0.141567	0.0707833	+	0.997492i	$0.477450\pi$
$20$	−0.0733427	−0.0163999
$21$	−8.78763	−1.91762
$22$	3.04833	0.649907
$23$	−4.17171	−0.869861	−0.434930	−	0.900464i	$-0.643227\pi$
$23$	−4.17171	−0.869861	−0.434930	+	0.900464i	$0.643227\pi$
$24$	−2.10614	−0.429914
$25$	−4.99462	−0.998924
$26$	−5.28486	−1.03645
$27$	3.29438	0.634004
$28$	4.17239	0.788507
$29$	−6.94128	−1.28896	−0.644482	−	0.764620i	$-0.722927\pi$
$29$	−6.94128	−1.28896	−0.644482	+	0.764620i	$0.722927\pi$
$30$	0.154470	0.0282022
$31$	6.71719	1.20644	0.603221	−	0.797574i	$-0.293884\pi$
$31$	6.71719	1.20644	0.603221	+	0.797574i	$0.293884\pi$
$32$	1.00000	0.176777
$33$	−6.42021	−1.11762
$34$	1.26189	0.216412
$35$	−0.306014	−0.0517258
$36$	1.43582	0.239303
$37$	−2.70741	−0.445095	−0.222548	−	0.974922i	$-0.571437\pi$
$37$	−2.70741	−0.445095	−0.222548	+	0.974922i	$0.571437\pi$
$38$	0.617075	0.100103
$39$	11.1306	1.78233
$40$	−0.0733427	−0.0115965
$41$	0.105799	0.0165230	0.00826152	−	0.999966i	$-0.497370\pi$
$41$	0.105799	0.0165230	0.00826152	+	0.999966i	$0.497370\pi$
$42$	−8.78763	−1.35596
$43$	−9.93029	−1.51435	−0.757177	−	0.653209i	$-0.773422\pi$
$43$	−9.93029	−1.51435	−0.757177	+	0.653209i	$0.773422\pi$
$44$	3.04833	0.459554
$45$	−0.105307	−0.0156982
$46$	−4.17171	−0.615085
$47$	−7.42797	−1.08348	−0.541741	−	0.840546i	$-0.682234\pi$
$47$	−7.42797	−1.08348	−0.541741	+	0.840546i	$0.682234\pi$
$48$	−2.10614	−0.303995
$49$	10.4088	1.48697
$50$	−4.99462	−0.706346
$51$	−2.65771	−0.372153
$52$	−5.28486	−0.732878
$53$	−10.1293	−1.39136	−0.695682	−	0.718350i	$-0.744898\pi$
$53$	−10.1293	−1.39136	−0.695682	+	0.718350i	$0.744898\pi$
$54$	3.29438	0.448308
$55$	−0.223573	−0.0301466
$56$	4.17239	0.557559
$57$	−1.29964	−0.172142
$58$	−6.94128	−0.911435
$59$	11.6365	1.51495	0.757474	−	0.652866i	$-0.226433\pi$
$59$	11.6365	1.51495	0.757474	+	0.652866i	$0.226433\pi$
$60$	0.154470	0.0199420
$61$	−6.65436	−0.852003	−0.426002	−	0.904722i	$-0.640078\pi$
$61$	−6.65436	−0.852003	−0.426002	+	0.904722i	$0.640078\pi$
$62$	6.71719	0.853084
$63$	5.99080	0.754770
$64$	1.00000	0.125000
$65$	0.387606	0.0480766
$66$	−6.42021	−0.790274
$67$	11.1446	1.36153	0.680765	−	0.732502i	$-0.261648\pi$
$67$	11.1446	1.36153	0.680765	+	0.732502i	$0.261648\pi$
$68$	1.26189	0.153026
$69$	8.78619	1.05773
$70$	−0.306014	−0.0365757
$71$	−12.8784	−1.52838	−0.764192	−	0.644989i	$-0.776862\pi$
$71$	−12.8784	−1.52838	−0.764192	+	0.644989i	$0.776862\pi$
$72$	1.43582	0.169213
$73$	−15.6780	−1.83497	−0.917485	−	0.397772i	$-0.869784\pi$
$73$	−15.6780	−1.83497	−0.917485	+	0.397772i	$0.869784\pi$
$74$	−2.70741	−0.314730
$75$	10.5194	1.21467
$76$	0.617075	0.0707833
$77$	12.7188	1.44945
$78$	11.1306	1.26030
$79$	−14.3847	−1.61841	−0.809204	−	0.587528i	$-0.800101\pi$
$79$	−14.3847	−1.61841	−0.809204	+	0.587528i	$0.800101\pi$
$80$	−0.0733427	−0.00819996
$81$	−11.2459	−1.24954
$82$	0.105799	0.0116836
$83$	−0.991837	−0.108868	−0.0544341	−	0.998517i	$-0.517335\pi$
$83$	−0.991837	−0.108868	−0.0544341	+	0.998517i	$0.517335\pi$
$84$	−8.78763	−0.958809
$85$	−0.0925501	−0.0100385
$86$	−9.93029	−1.07081
$87$	14.6193	1.56735
$88$	3.04833	0.324954
$89$	−3.60769	−0.382414	−0.191207	−	0.981550i	$-0.561240\pi$
$89$	−3.60769	−0.382414	−0.191207	+	0.981550i	$0.561240\pi$
$90$	−0.105307	−0.0111003
$91$	−22.0505	−2.31152
$92$	−4.17171	−0.434930
$93$	−14.1473	−1.46701
$94$	−7.42797	−0.766137
$95$	−0.0452579	−0.00464336
$96$	−2.10614	−0.214957
$97$	−15.2101	−1.54435	−0.772175	−	0.635410i	$-0.780831\pi$
$97$	−15.2101	−1.54435	−0.772175	+	0.635410i	$0.780831\pi$
$98$	10.4088	1.05145
$99$	4.37686	0.439891
$100$	−4.99462	−0.499462
$101$	1.59550	0.158758	0.0793792	−	0.996844i	$-0.474706\pi$
$101$	1.59550	0.158758	0.0793792	+	0.996844i	$0.474706\pi$
$102$	−2.65771	−0.263152
$103$	−1.23434	−0.121623	−0.0608116	−	0.998149i	$-0.519369\pi$
$103$	−1.23434	−0.121623	−0.0608116	+	0.998149i	$0.519369\pi$
$104$	−5.28486	−0.518223
$105$	0.644508	0.0628975
$106$	−10.1293	−0.983842
$107$	19.0974	1.84621	0.923106	−	0.384546i	$-0.125642\pi$
$107$	19.0974	1.84621	0.923106	+	0.384546i	$0.125642\pi$
$108$	3.29438	0.317002
$109$	7.69123	0.736687	0.368343	−	0.929690i	$-0.379925\pi$
$109$	7.69123	0.736687	0.368343	+	0.929690i	$0.379925\pi$
$110$	−0.223573	−0.0213168
$111$	5.70218	0.541227
$112$	4.17239	0.394254
$113$	9.58577	0.901754	0.450877	−	0.892586i	$-0.351111\pi$
$113$	9.58577	0.901754	0.450877	+	0.892586i	$0.351111\pi$
$114$	−1.29964	−0.121723
$115$	0.305964	0.0285313
$116$	−6.94128	−0.644482
$117$	−7.58810	−0.701521
$118$	11.6365	1.07123
$119$	5.26508	0.482649
$120$	0.154470	0.0141011
$121$	−1.70766	−0.155242
$122$	−6.65436	−0.602457
$123$	−0.222828	−0.0200917
$124$	6.71719	0.603221
$125$	0.733032	0.0655644
$126$	5.99080	0.533703
$127$	18.6436	1.65435	0.827176	−	0.561943i	$-0.189946\pi$
$127$	18.6436	1.65435	0.827176	+	0.561943i	$0.189946\pi$
$128$	1.00000	0.0883883
$129$	20.9146	1.84142
$130$	0.387606	0.0339953
$131$	−2.56426	−0.224041	−0.112020	−	0.993706i	$-0.535732\pi$
$131$	−2.56426	−0.224041	−0.112020	+	0.993706i	$0.535732\pi$
$132$	−6.42021	−0.558808
$133$	2.57467	0.223253
$134$	11.1446	0.962747
$135$	−0.241619	−0.0207952
$136$	1.26189	0.108206
$137$	−10.7491	−0.918359	−0.459180	−	0.888343i	$-0.651857\pi$
$137$	−10.7491	−0.918359	−0.459180	+	0.888343i	$0.651857\pi$
$138$	8.78619	0.747930
$139$	16.3594	1.38758	0.693792	−	0.720176i	$-0.255939\pi$
$139$	16.3594	1.38758	0.693792	+	0.720176i	$0.255939\pi$
$140$	−0.306014	−0.0258629
$141$	15.6443	1.31749
$142$	−12.8784	−1.08073
$143$	−16.1100	−1.34719
$144$	1.43582	0.119652
$145$	0.509092	0.0422778
$146$	−15.6780	−1.29752
$147$	−21.9224	−1.80813
$148$	−2.70741	−0.222548
$149$	7.18589	0.588691	0.294345	−	0.955699i	$-0.404898\pi$
$149$	7.18589	0.588691	0.294345	+	0.955699i	$0.404898\pi$
$150$	10.5194	0.858902
$151$	2.78738	0.226834	0.113417	−	0.993547i	$-0.463820\pi$
$151$	2.78738	0.226834	0.113417	+	0.993547i	$0.463820\pi$
$152$	0.617075	0.0500514
$153$	1.81184	0.146479
$154$	12.7188	1.02491
$155$	−0.492656	−0.0395711
$156$	11.1306	0.891165
$157$	−11.6284	−0.928044	−0.464022	−	0.885824i	$-0.653594\pi$
$157$	−11.6284	−0.928044	−0.464022	+	0.885824i	$0.653594\pi$
$158$	−14.3847	−1.14439
$159$	21.3337	1.69187
$160$	−0.0733427	−0.00579825
$161$	−17.4060	−1.37178
$162$	−11.2459	−0.883560
$163$	20.1718	1.57998	0.789989	−	0.613121i	$-0.210086\pi$
$163$	20.1718	1.57998	0.789989	+	0.613121i	$0.210086\pi$
$164$	0.105799	0.00826152
$165$	0.470876	0.0366576
$166$	−0.991837	−0.0769815
$167$	7.06157	0.546441	0.273220	−	0.961951i	$-0.411911\pi$
$167$	7.06157	0.546441	0.273220	+	0.961951i	$0.411911\pi$
$168$	−8.78763	−0.677980
$169$	14.9297	1.14844
$170$	−0.0925501	−0.00709827
$171$	0.886008	0.0677547
$172$	−9.93029	−0.757177
$173$	−7.63698	−0.580629	−0.290314	−	0.956931i	$-0.593760\pi$
$173$	−7.63698	−0.580629	−0.290314	+	0.956931i	$0.593760\pi$
$174$	14.6193	1.10829
$175$	−20.8395	−1.57532
$176$	3.04833	0.229777
$177$	−24.5081	−1.84215
$178$	−3.60769	−0.270408
$179$	−7.23413	−0.540704	−0.270352	−	0.962762i	$-0.587140\pi$
$179$	−7.23413	−0.540704	−0.270352	+	0.962762i	$0.587140\pi$
$180$	−0.105307	−0.00784911
$181$	5.22122	0.388090	0.194045	−	0.980993i	$-0.437839\pi$
$181$	5.22122	0.388090	0.194045	+	0.980993i	$0.437839\pi$
$182$	−22.0505	−1.63449
$183$	14.0150	1.03602
$184$	−4.17171	−0.307542
$185$	0.198569	0.0145991
$186$	−14.1473	−1.03733
$187$	3.84665	0.281295
$188$	−7.42797	−0.541741
$189$	13.7454	0.999833
$190$	−0.0452579	−0.00328335
$191$	23.9496	1.73293	0.866467	−	0.499235i	$-0.166385\pi$
$191$	23.9496	1.73293	0.866467	+	0.499235i	$0.166385\pi$
$192$	−2.10614	−0.151997
$193$	−8.37461	−0.602818	−0.301409	−	0.953495i	$-0.597457\pi$
$193$	−8.37461	−0.602818	−0.301409	+	0.953495i	$0.597457\pi$
$194$	−15.2101	−1.09202
$195$	−0.816351	−0.0584601
$196$	10.4088	0.743487
$197$	−17.4420	−1.24269	−0.621346	−	0.783537i	$-0.713414\pi$
$197$	−17.4420	−1.24269	−0.621346	+	0.783537i	$0.713414\pi$
$198$	4.37686	0.311050
$199$	−14.6853	−1.04101	−0.520506	−	0.853858i	$-0.674257\pi$
$199$	−14.6853	−1.04101	−0.520506	+	0.853858i	$0.674257\pi$
$200$	−4.99462	−0.353173
$201$	−23.4721	−1.65559
$202$	1.59550	0.112259
$203$	−28.9617	−2.03271
$204$	−2.65771	−0.186077
$205$	−0.00775959	−0.000541953 0
$206$	−1.23434	−0.0860005
$207$	−5.98982	−0.416321
$208$	−5.28486	−0.366439
$209$	1.88105	0.130115
$210$	0.644508	0.0444753
$211$	21.8638	1.50517	0.752583	−	0.658497i	$-0.228808\pi$
$211$	21.8638	1.50517	0.752583	+	0.658497i	$0.228808\pi$
$212$	−10.1293	−0.695682
$213$	27.1237	1.85848
$214$	19.0974	1.30547
$215$	0.728314	0.0496706
$216$	3.29438	0.224154
$217$	28.0267	1.90258
$218$	7.69123	0.520916
$219$	33.0200	2.23129
$220$	−0.223573	−0.0150733
$221$	−6.66889	−0.448598
$222$	5.70218	0.382705
$223$	13.7880	0.923311	0.461655	−	0.887059i	$-0.347256\pi$
$223$	13.7880	0.923311	0.461655	+	0.887059i	$0.347256\pi$
$224$	4.17239	0.278779
$225$	−7.17138	−0.478092
$226$	9.58577	0.637636
$227$	−16.7539	−1.11200	−0.555998	−	0.831184i	$-0.687664\pi$
$227$	−16.7539	−1.11200	−0.555998	+	0.831184i	$0.687664\pi$
$228$	−1.29964	−0.0860711
$229$	21.0604	1.39171	0.695854	−	0.718183i	$-0.255026\pi$
$229$	21.0604	1.39171	0.695854	+	0.718183i	$0.255026\pi$
$230$	0.305964	0.0201747
$231$	−26.7876	−1.76250
$232$	−6.94128	−0.455717
$233$	8.10612	0.531049	0.265525	−	0.964104i	$-0.414455\pi$
$233$	8.10612	0.531049	0.265525	+	0.964104i	$0.414455\pi$
$234$	−7.58810	−0.496050
$235$	0.544787	0.0355380
$236$	11.6365	0.757474
$237$	30.2962	1.96795
$238$	5.26508	0.341284
$239$	−10.7037	−0.692366	−0.346183	−	0.938167i	$-0.612522\pi$
$239$	−10.7037	−0.692366	−0.346183	+	0.938167i	$0.612522\pi$
$240$	0.154470	0.00997098
$241$	−17.4088	−1.12140	−0.560700	−	0.828019i	$-0.689468\pi$
$241$	−17.4088	−1.12140	−0.560700	+	0.828019i	$0.689468\pi$
$242$	−1.70766	−0.109772
$243$	13.8022	0.885414
$244$	−6.65436	−0.426002
$245$	−0.763411	−0.0487725
$246$	−0.222828	−0.0142070
$247$	−3.26115	−0.207502
$248$	6.71719	0.426542
$249$	2.08895	0.132382
$250$	0.733032	0.0463610
$251$	−17.7499	−1.12037	−0.560183	−	0.828369i	$-0.689269\pi$
$251$	−17.7499	−1.12037	−0.560183	+	0.828369i	$0.689269\pi$
$252$	5.99080	0.377385
$253$	−12.7168	−0.799496
$254$	18.6436	1.16980
$255$	0.194923	0.0122066
$256$	1.00000	0.0625000
$257$	−15.8953	−0.991520	−0.495760	−	0.868460i	$-0.665110\pi$
$257$	−15.8953	−0.991520	−0.495760	+	0.868460i	$0.665110\pi$
$258$	20.9146	1.30208
$259$	−11.2964	−0.701922
$260$	0.387606	0.0240383
$261$	−9.96643	−0.616906
$262$	−2.56426	−0.158421
$263$	−5.70267	−0.351641	−0.175821	−	0.984422i	$-0.556258\pi$
$263$	−5.70267	−0.351641	−0.175821	+	0.984422i	$0.556258\pi$
$264$	−6.42021	−0.395137
$265$	0.742908	0.0456365
$266$	2.57467	0.157863
$267$	7.59830	0.465008
$268$	11.1446	0.680765
$269$	−23.7108	−1.44567	−0.722837	−	0.691018i	$-0.757162\pi$
$269$	−23.7108	−1.44567	−0.722837	+	0.691018i	$0.757162\pi$
$270$	−0.241619	−0.0147044
$271$	2.27972	0.138483	0.0692416	−	0.997600i	$-0.477942\pi$
$271$	2.27972	0.138483	0.0692416	+	0.997600i	$0.477942\pi$
$272$	1.26189	0.0765131
$273$	46.4414	2.81076
$274$	−10.7491	−0.649378
$275$	−15.2253	−0.918119
$276$	8.78619	0.528867
$277$	−6.81698	−0.409593	−0.204796	−	0.978805i	$-0.565653\pi$
$277$	−6.81698	−0.409593	−0.204796	+	0.978805i	$0.565653\pi$
$278$	16.3594	0.981170
$279$	9.64467	0.577411
$280$	−0.306014	−0.0182878
$281$	5.01289	0.299044	0.149522	−	0.988758i	$-0.452227\pi$
$281$	5.01289	0.299044	0.149522	+	0.988758i	$0.452227\pi$
$282$	15.6443	0.931607
$283$	−21.7618	−1.29361	−0.646803	−	0.762657i	$-0.723895\pi$
$283$	−21.7618	−1.29361	−0.646803	+	0.762657i	$0.723895\pi$
$284$	−12.8784	−0.764192
$285$	0.0953194	0.00564623
$286$	−16.1100	−0.952605
$287$	0.441435	0.0260571
$288$	1.43582	0.0846065
$289$	−15.4076	−0.906332
$290$	0.509092	0.0298949
$291$	32.0345	1.87790
$292$	−15.6780	−0.917485
$293$	−11.6073	−0.678106	−0.339053	−	0.940767i	$-0.610107\pi$
$293$	−11.6073	−0.678106	−0.339053	+	0.940767i	$0.610107\pi$
$294$	−21.9224	−1.27854
$295$	−0.853454	−0.0496900
$296$	−2.70741	−0.157365
$297$	10.0424	0.582718
$298$	7.18589	0.416267
$299$	22.0469	1.27500
$300$	10.5194	0.607336
$301$	−41.4330	−2.38816
$302$	2.78738	0.160396
$303$	−3.36035	−0.193047
$304$	0.617075	0.0353917
$305$	0.488048	0.0279456
$306$	1.81184	0.103576
$307$	−13.0468	−0.744620	−0.372310	−	0.928108i	$-0.621434\pi$
$307$	−13.0468	−0.744620	−0.372310	+	0.928108i	$0.621434\pi$
$308$	12.7188	0.724723
$309$	2.59969	0.147891
$310$	−0.492656	−0.0279810
$311$	0.294032	0.0166730	0.00833652	−	0.999965i	$-0.497346\pi$
$311$	0.294032	0.0166730	0.00833652	+	0.999965i	$0.497346\pi$
$312$	11.1306	0.630149
$313$	−13.5178	−0.764071	−0.382036	−	0.924148i	$-0.624777\pi$
$313$	−13.5178	−0.764071	−0.382036	+	0.924148i	$0.624777\pi$
$314$	−11.6284	−0.656226
$315$	−0.439381	−0.0247563
$316$	−14.3847	−0.809204
$317$	−3.99120	−0.224168	−0.112084	−	0.993699i	$-0.535753\pi$
$317$	−3.99120	−0.224168	−0.112084	+	0.993699i	$0.535753\pi$
$318$	21.3337	1.19633
$319$	−21.1593	−1.18470
$320$	−0.0733427	−0.00409998
$321$	−40.2217	−2.24496
$322$	−17.4060	−0.969997
$323$	0.778678	0.0433268
$324$	−11.2459	−0.624771
$325$	26.3959	1.46418
$326$	20.1718	1.11721
$327$	−16.1988	−0.895796
$328$	0.105799	0.00584178
$329$	−30.9924	−1.70867
$330$	0.470876	0.0259208
$331$	−13.0516	−0.717379	−0.358690	−	0.933457i	$-0.616776\pi$
$331$	−13.0516	−0.717379	−0.358690	+	0.933457i	$0.616776\pi$
$332$	−0.991837	−0.0544341
$333$	−3.88735	−0.213026
$334$	7.06157	0.386392
$335$	−0.817374	−0.0446579
$336$	−8.78763	−0.479404
$337$	−30.0862	−1.63890	−0.819449	−	0.573152i	$-0.805721\pi$
$337$	−30.0862	−1.63890	−0.819449	+	0.573152i	$0.805721\pi$
$338$	14.9297	0.812070
$339$	−20.1890	−1.09651
$340$	−0.0925501	−0.00501923
$341$	20.4762	1.10885
$342$	0.886008	0.0479098
$343$	14.2229	0.767967
$344$	−9.93029	−0.535405
$345$	−0.644403	−0.0346935
$346$	−7.63698	−0.410566
$347$	13.6591	0.733260	0.366630	−	0.930367i	$-0.380511\pi$
$347$	13.6591	0.733260	0.366630	+	0.930367i	$0.380511\pi$
$348$	14.6193	0.783677
$349$	−0.522464	−0.0279668	−0.0139834	−	0.999902i	$-0.504451\pi$
$349$	−0.522464	−0.0279668	−0.0139834	+	0.999902i	$0.504451\pi$
$350$	−20.8395	−1.11392
$351$	−17.4103	−0.929295
$352$	3.04833	0.162477
$353$	−32.2401	−1.71597	−0.857985	−	0.513675i	$-0.828284\pi$
$353$	−32.2401	−1.71597	−0.857985	+	0.513675i	$0.828284\pi$
$354$	−24.5081	−1.30259
$355$	0.944535	0.0501307
$356$	−3.60769	−0.191207
$357$	−11.0890	−0.586891
$358$	−7.23413	−0.382335
$359$	−3.90150	−0.205913	−0.102957	−	0.994686i	$-0.532830\pi$
$359$	−3.90150	−0.205913	−0.102957	+	0.994686i	$0.532830\pi$
$360$	−0.105307	−0.00555016
$361$	−18.6192	−0.979959
$362$	5.22122	0.274421
$363$	3.59657	0.188771
$364$	−22.0505	−1.15576
$365$	1.14986	0.0601867
$366$	14.0150	0.732576
$367$	25.7272	1.34295	0.671476	−	0.741027i	$-0.265661\pi$
$367$	25.7272	1.34295	0.671476	+	0.741027i	$0.265661\pi$
$368$	−4.17171	−0.217465
$369$	0.151908	0.00790804
$370$	0.198569	0.0103231
$371$	−42.2633	−2.19420
$372$	−14.1473	−0.733505
$373$	−18.9998	−0.983771	−0.491886	−	0.870660i	$-0.663692\pi$
$373$	−18.9998	−0.983771	−0.491886	+	0.870660i	$0.663692\pi$
$374$	3.84665	0.198906
$375$	−1.54387	−0.0797250
$376$	−7.42797	−0.383068
$377$	36.6837	1.88931
$378$	13.7454	0.706989
$379$	9.44808	0.485315	0.242658	−	0.970112i	$-0.421981\pi$
$379$	9.44808	0.485315	0.242658	+	0.970112i	$0.421981\pi$
$380$	−0.0452579	−0.00232168
$381$	−39.2660	−2.01166
$382$	23.9496	1.22537
$383$	−21.6669	−1.10713	−0.553565	−	0.832806i	$-0.686733\pi$
$383$	−21.6669	−1.10713	−0.553565	+	0.832806i	$0.686733\pi$
$384$	−2.10614	−0.107478
$385$	−0.932833	−0.0475416
$386$	−8.37461	−0.426256
$387$	−14.2581	−0.724780
$388$	−15.2101	−0.772175
$389$	14.9040	0.755662	0.377831	−	0.925875i	$-0.376670\pi$
$389$	14.9040	0.755662	0.377831	+	0.925875i	$0.376670\pi$
$390$	−0.816351	−0.0413375
$391$	−5.26422	−0.266223
$392$	10.4088	0.525725
$393$	5.40069	0.272429
$394$	−17.4420	−0.878715
$395$	1.05501	0.0530835
$396$	4.37686	0.219945
$397$	20.3285	1.02026	0.510128	−	0.860098i	$-0.329598\pi$
$397$	20.3285	1.02026	0.510128	+	0.860098i	$0.329598\pi$
$398$	−14.6853	−0.736106
$399$	−5.42262	−0.271471
$400$	−4.99462	−0.249731
$401$	22.9664	1.14689	0.573444	−	0.819245i	$-0.305607\pi$
$401$	22.9664	1.14689	0.573444	+	0.819245i	$0.305607\pi$
$402$	−23.4721	−1.17068
$403$	−35.4994	−1.76835
$404$	1.59550	0.0793792
$405$	0.824803	0.0409848
$406$	−28.9617	−1.43735
$407$	−8.25309	−0.409090
$408$	−2.65771	−0.131576
$409$	2.03075	0.100414	0.0502070	−	0.998739i	$-0.484012\pi$
$409$	2.03075	0.100414	0.0502070	+	0.998739i	$0.484012\pi$
$410$	−0.00775959	−0.000383219 0
$411$	22.6391	1.11671
$412$	−1.23434	−0.0608116
$413$	48.5521	2.38909
$414$	−5.98982	−0.294384
$415$	0.0727440	0.00357086
$416$	−5.28486	−0.259111
$417$	−34.4551	−1.68727
$418$	1.88105	0.0920051
$419$	4.68884	0.229065	0.114532	−	0.993420i	$-0.463463\pi$
$419$	4.68884	0.229065	0.114532	+	0.993420i	$0.463463\pi$
$420$	0.644508	0.0314488
$421$	40.8317	1.99002	0.995008	−	0.0997915i	$-0.0318176\pi$
$421$	40.8317	1.99002	0.995008	+	0.0997915i	$0.0318176\pi$
$422$	21.8638	1.06431
$423$	−10.6652	−0.518561
$424$	−10.1293	−0.491921
$425$	−6.30264	−0.305723
$426$	27.1237	1.31415
$427$	−27.7646	−1.34362
$428$	19.0974	0.923106
$429$	33.9299	1.63815
$430$	0.728314	0.0351224
$431$	−6.51364	−0.313751	−0.156875	−	0.987618i	$-0.550142\pi$
$431$	−6.51364	−0.313751	−0.156875	+	0.987618i	$0.550142\pi$
$432$	3.29438	0.158501
$433$	−4.78063	−0.229742	−0.114871	−	0.993380i	$-0.536646\pi$
$433$	−4.78063	−0.229742	−0.114871	+	0.993380i	$0.536646\pi$
$434$	28.0267	1.34533
$435$	−1.07222	−0.0514089
$436$	7.69123	0.368343
$437$	−2.57425	−0.123143
$438$	33.0200	1.57776
$439$	−20.3115	−0.969414	−0.484707	−	0.874677i	$-0.661074\pi$
$439$	−20.3115	−0.969414	−0.484707	+	0.874677i	$0.661074\pi$
$440$	−0.223573	−0.0106584
$441$	14.9452	0.711676
$442$	−6.66889	−0.317207
$443$	−13.9885	−0.664614	−0.332307	−	0.943171i	$-0.607827\pi$
$443$	−13.9885	−0.664614	−0.332307	+	0.943171i	$0.607827\pi$
$444$	5.70218	0.270613
$445$	0.264598	0.0125431
$446$	13.7880	0.652879
$447$	−15.1345	−0.715836
$448$	4.17239	0.197127
$449$	40.3841	1.90584	0.952921	−	0.303218i	$-0.0980612\pi$
$449$	40.3841	1.90584	0.952921	+	0.303218i	$0.0980612\pi$
$450$	−7.17138	−0.338062
$451$	0.322511	0.0151865
$452$	9.58577	0.450877
$453$	−5.87061	−0.275826
$454$	−16.7539	−0.786300
$455$	1.61724	0.0758174
$456$	−1.29964	−0.0608614
$457$	0.949312	0.0444070	0.0222035	−	0.999753i	$-0.492932\pi$
$457$	0.949312	0.0444070	0.0222035	+	0.999753i	$0.492932\pi$
$458$	21.0604	0.984087
$459$	4.15713	0.194038
$460$	0.305964	0.0142656
$461$	21.5510	1.00373	0.501865	−	0.864946i	$-0.332648\pi$
$461$	21.5510	1.00373	0.501865	+	0.864946i	$0.332648\pi$
$462$	−26.7876	−1.24627
$463$	−10.6555	−0.495201	−0.247601	−	0.968862i	$-0.579642\pi$
$463$	−10.6555	−0.495201	−0.247601	+	0.968862i	$0.579642\pi$
$464$	−6.94128	−0.322241
$465$	1.03760	0.0481177
$466$	8.10612	0.375509
$467$	−21.9152	−1.01411	−0.507056	−	0.861913i	$-0.669266\pi$
$467$	−21.9152	−1.01411	−0.507056	+	0.861913i	$0.669266\pi$
$468$	−7.58810	−0.350760
$469$	46.4996	2.14715
$470$	0.544787	0.0251292
$471$	24.4909	1.12848
$472$	11.6365	0.535615
$473$	−30.2708	−1.39185
$474$	30.2962	1.39155
$475$	−3.08205	−0.141414
$476$	5.26508	0.241324
$477$	−14.5438	−0.665916
$478$	−10.7037	−0.489577
$479$	4.39329	0.200734	0.100367	−	0.994950i	$-0.467998\pi$
$479$	4.39329	0.200734	0.100367	+	0.994950i	$0.467998\pi$
$480$	0.154470	0.00705055
$481$	14.3083	0.652401
$482$	−17.4088	−0.792950
$483$	36.6594	1.66806
$484$	−1.70766	−0.0776208
$485$	1.11555	0.0506544
$486$	13.8022	0.626082
$487$	−12.3108	−0.557854	−0.278927	−	0.960312i	$-0.589979\pi$
$487$	−12.3108	−0.557854	−0.278927	+	0.960312i	$0.589979\pi$
$488$	−6.65436	−0.301229
$489$	−42.4846	−1.92122
$490$	−0.763411	−0.0344874
$491$	−21.0570	−0.950291	−0.475145	−	0.879907i	$-0.657605\pi$
$491$	−21.0570	−0.950291	−0.475145	+	0.879907i	$0.657605\pi$
$492$	−0.222828	−0.0100458
$493$	−8.75911	−0.394490
$494$	−3.26115	−0.146726
$495$	−0.321010	−0.0144283
$496$	6.71719	0.301611
$497$	−53.7336	−2.41028
$498$	2.08895	0.0936079
$499$	27.6080	1.23590	0.617951	−	0.786216i	$-0.287963\pi$
$499$	27.6080	1.23590	0.617951	+	0.786216i	$0.287963\pi$
$500$	0.733032	0.0327822
$501$	−14.8726	−0.664461
$502$	−17.7499	−0.792219
$503$	−31.9815	−1.42598	−0.712992	−	0.701172i	$-0.752660\pi$
$503$	−31.9815	−1.42598	−0.712992	+	0.701172i	$0.752660\pi$
$504$	5.99080	0.266851
$505$	−0.117018	−0.00520725
$506$	−12.7168	−0.565329
$507$	−31.4441	−1.39648
$508$	18.6436	0.827176
$509$	6.76846	0.300007	0.150003	−	0.988685i	$-0.452072\pi$
$509$	6.76846	0.300007	0.150003	+	0.988685i	$0.452072\pi$
$510$	0.194923	0.00863135
$511$	−65.4146	−2.89377
$512$	1.00000	0.0441942
$513$	2.03288	0.0897538
$514$	−15.8953	−0.701111
$515$	0.0905298	0.00398922
$516$	20.9146	0.920712
$517$	−22.6429	−0.995835
$518$	−11.2964	−0.496334
$519$	16.0845	0.706033
$520$	0.387606	0.0169976
$521$	15.1936	0.665642	0.332821	−	0.942990i	$-0.391999\pi$
$521$	15.1936	0.665642	0.332821	+	0.942990i	$0.391999\pi$
$522$	−9.96643	−0.436219
$523$	−4.83981	−0.211630	−0.105815	−	0.994386i	$-0.533745\pi$
$523$	−4.83981	−0.211630	−0.105815	+	0.994386i	$0.533745\pi$
$524$	−2.56426	−0.112020
$525$	43.8909	1.91555
$526$	−5.70267	−0.248648
$527$	8.47632	0.369234
$528$	−6.42021	−0.279404
$529$	−5.59687	−0.243342
$530$	0.742908	0.0322699
$531$	16.7080	0.725064
$532$	2.57467	0.111626
$533$	−0.559133	−0.0242188
$534$	7.59830	0.328810
$535$	−1.40065	−0.0605554
$536$	11.1446	0.481373
$537$	15.2361	0.657485
$538$	−23.7108	−1.02225
$539$	31.7296	1.36669
$540$	−0.241619	−0.0103976
$541$	40.5887	1.74505	0.872523	−	0.488573i	$-0.162483\pi$
$541$	40.5887	1.74505	0.872523	+	0.488573i	$0.162483\pi$
$542$	2.27972	0.0979224
$543$	−10.9966	−0.471910
$544$	1.26189	0.0541029
$545$	−0.564096	−0.0241632
$546$	46.4414	1.98751
$547$	3.44667	0.147369	0.0736845	−	0.997282i	$-0.476524\pi$
$547$	3.44667	0.147369	0.0736845	+	0.997282i	$0.476524\pi$
$548$	−10.7491	−0.459180
$549$	−9.55446	−0.407774
$550$	−15.2253	−0.649208
$551$	−4.28329	−0.182474
$552$	8.78619	0.373965
$553$	−60.0187	−2.55225
$554$	−6.81698	−0.289626
$555$	−0.418213	−0.0177521
$556$	16.3594	0.693792
$557$	37.0019	1.56782	0.783911	−	0.620873i	$-0.213222\pi$
$557$	37.0019	1.56782	0.783911	+	0.620873i	$0.213222\pi$
$558$	9.64467	0.408291
$559$	52.4802	2.21967
$560$	−0.306014	−0.0129315
$561$	−8.10158	−0.342049
$562$	5.01289	0.211456
$563$	8.94686	0.377065	0.188533	−	0.982067i	$-0.439627\pi$
$563$	8.94686	0.377065	0.188533	+	0.982067i	$0.439627\pi$
$564$	15.6443	0.658746
$565$	−0.703046	−0.0295774
$566$	−21.7618	−0.914718
$567$	−46.9222	−1.97055
$568$	−12.8784	−0.540365
$569$	−29.3880	−1.23201	−0.616005	−	0.787743i	$-0.711250\pi$
$569$	−29.3880	−1.23201	−0.616005	+	0.787743i	$0.711250\pi$
$570$	0.0953194	0.00399249
$571$	−23.2670	−0.973694	−0.486847	−	0.873487i	$-0.661853\pi$
$571$	−23.2670	−0.973694	−0.486847	+	0.873487i	$0.661853\pi$
$572$	−16.1100	−0.673594
$573$	−50.4412	−2.10721
$574$	0.441435	0.0184251
$575$	20.8361	0.868925
$576$	1.43582	0.0598258
$577$	16.4162	0.683415	0.341707	−	0.939806i	$-0.388995\pi$
$577$	16.4162	0.683415	0.341707	+	0.939806i	$0.388995\pi$
$578$	−15.4076	−0.640873
$579$	17.6381	0.733014
$580$	0.509092	0.0211389
$581$	−4.13833	−0.171687
$582$	32.0345	1.32787
$583$	−30.8774	−1.27881
$584$	−15.6780	−0.648760
$585$	0.556532	0.0230098
$586$	−11.6073	−0.479493
$587$	−6.63778	−0.273971	−0.136985	−	0.990573i	$-0.543741\pi$
$587$	−6.63778	−0.273971	−0.136985	+	0.990573i	$0.543741\pi$
$588$	−21.9224	−0.904066
$589$	4.14500	0.170792
$590$	−0.853454	−0.0351361
$591$	36.7353	1.51109
$592$	−2.70741	−0.111274
$593$	−2.17072	−0.0891408	−0.0445704	−	0.999006i	$-0.514192\pi$
$593$	−2.17072	−0.0891408	−0.0445704	+	0.999006i	$0.514192\pi$
$594$	10.0424	0.412044
$595$	−0.386155	−0.0158308
$596$	7.18589	0.294345
$597$	30.9292	1.26585
$598$	22.0469	0.901564
$599$	−11.1860	−0.457048	−0.228524	−	0.973538i	$-0.573390\pi$
$599$	−11.1860	−0.457048	−0.228524	+	0.973538i	$0.573390\pi$
$600$	10.5194	0.429451
$601$	−28.8680	−1.17755	−0.588776	−	0.808296i	$-0.700390\pi$
$601$	−28.8680	−1.17755	−0.588776	+	0.808296i	$0.700390\pi$
$602$	−41.4330	−1.68868
$603$	16.0016	0.651637
$604$	2.78738	0.113417
$605$	0.125244	0.00509190
$606$	−3.36035	−0.136505
$607$	0.654624	0.0265704	0.0132852	−	0.999912i	$-0.495771\pi$
$607$	0.654624	0.0265704	0.0132852	+	0.999912i	$0.495771\pi$
$608$	0.617075	0.0250257
$609$	60.9974	2.47174
$610$	0.488048	0.0197605
$611$	39.2558	1.58812
$612$	1.81184	0.0732393
$613$	−5.17520	−0.209025	−0.104512	−	0.994524i	$-0.533328\pi$
$613$	−5.17520	−0.209025	−0.104512	+	0.994524i	$0.533328\pi$
$614$	−13.0468	−0.526526
$615$	0.0163428	0.000659004 0
$616$	12.7188	0.512456
$617$	−24.1533	−0.972376	−0.486188	−	0.873854i	$-0.661613\pi$
$617$	−24.1533	−0.972376	−0.486188	+	0.873854i	$0.661613\pi$
$618$	2.59969	0.104575
$619$	−34.2577	−1.37693	−0.688466	−	0.725268i	$-0.741716\pi$
$619$	−34.2577	−1.37693	−0.688466	+	0.725268i	$0.741716\pi$
$620$	−0.492656	−0.0197856
$621$	−13.7432	−0.551495
$622$	0.294032	0.0117896
$623$	−15.0527	−0.603073
$624$	11.1306	0.445582
$625$	24.9193	0.996774
$626$	−13.5178	−0.540280
$627$	−3.96175	−0.158217
$628$	−11.6284	−0.464022
$629$	−3.41644	−0.136222
$630$	−0.439381	−0.0175054
$631$	48.2199	1.91961	0.959803	−	0.280676i	$-0.0905587\pi$
$631$	48.2199	1.91961	0.959803	+	0.280676i	$0.0905587\pi$
$632$	−14.3847	−0.572194
$633$	−46.0482	−1.83025
$634$	−3.99120	−0.158511
$635$	−1.36737	−0.0542625
$636$	21.3337	0.845935
$637$	−55.0092	−2.17954
$638$	−21.1593	−0.837706
$639$	−18.4910	−0.731494
$640$	−0.0733427	−0.00289912
$641$	−16.1727	−0.638782	−0.319391	−	0.947623i	$-0.603478\pi$
$641$	−16.1727	−0.638782	−0.319391	+	0.947623i	$0.603478\pi$
$642$	−40.2217	−1.58742
$643$	21.6812	0.855025	0.427512	−	0.904010i	$-0.359390\pi$
$643$	21.6812	0.855025	0.427512	+	0.904010i	$0.359390\pi$
$644$	−17.4060	−0.685892
$645$	−1.53393	−0.0603984
$646$	0.778678	0.0306367
$647$	17.4499	0.686025	0.343012	−	0.939331i	$-0.388553\pi$
$647$	17.4499	0.686025	0.343012	+	0.939331i	$0.388553\pi$
$648$	−11.2459	−0.441780
$649$	35.4720	1.39240
$650$	26.3959	1.03533
$651$	−59.0281	−2.31350
$652$	20.1718	0.789989
$653$	31.0886	1.21659	0.608296	−	0.793711i	$-0.291853\pi$
$653$	31.0886	1.21659	0.608296	+	0.793711i	$0.291853\pi$
$654$	−16.1988	−0.633423
$655$	0.188070	0.00734850
$656$	0.105799	0.00413076
$657$	−22.5108	−0.878228
$658$	−30.9924	−1.20821
$659$	−45.1264	−1.75788	−0.878938	−	0.476936i	$-0.841747\pi$
$659$	−45.1264	−1.75788	−0.878938	+	0.476936i	$0.841747\pi$
$660$	0.470876	0.0183288
$661$	13.4807	0.524337	0.262169	−	0.965022i	$-0.415562\pi$
$661$	13.4807	0.524337	0.262169	+	0.965022i	$0.415562\pi$
$662$	−13.0516	−0.507264
$663$	14.0456	0.545486
$664$	−0.991837	−0.0384907
$665$	−0.188833	−0.00732265
$666$	−3.88735	−0.150632
$667$	28.9570	1.12122
$668$	7.06157	0.273220
$669$	−29.0394	−1.12273
$670$	−0.817374	−0.0315779
$671$	−20.2847	−0.783083
$672$	−8.78763	−0.338990
$673$	13.0475	0.502946	0.251473	−	0.967864i	$-0.419085\pi$
$673$	13.0475	0.502946	0.251473	+	0.967864i	$0.419085\pi$
$674$	−30.0862	−1.15888
$675$	−16.4542	−0.633322
$676$	14.9297	0.574220
$677$	38.9483	1.49690	0.748452	−	0.663189i	$-0.230798\pi$
$677$	38.9483	1.49690	0.748452	+	0.663189i	$0.230798\pi$
$678$	−20.1890	−0.775353
$679$	−63.4624	−2.43546
$680$	−0.0925501	−0.00354913
$681$	35.2860	1.35216
$682$	20.4762	0.784075
$683$	−31.8651	−1.21928	−0.609642	−	0.792677i	$-0.708687\pi$
$683$	−31.8651	−1.21928	−0.609642	+	0.792677i	$0.708687\pi$
$684$	0.886008	0.0338774
$685$	0.788369	0.0301220
$686$	14.2229	0.543034
$687$	−44.3561	−1.69229
$688$	−9.93029	−0.378589
$689$	53.5318	2.03940
$690$	−0.644403	−0.0245320
$691$	−4.66361	−0.177412	−0.0887061	−	0.996058i	$-0.528273\pi$
$691$	−4.66361	−0.177412	−0.0887061	+	0.996058i	$0.528273\pi$
$692$	−7.63698	−0.290314
$693$	18.2620	0.693714
$694$	13.6591	0.518493
$695$	−1.19984	−0.0455125
$696$	14.6193	0.554143
$697$	0.133506	0.00505692
$698$	−0.522464	−0.0197755
$699$	−17.0726	−0.645745
$700$	−20.8395	−0.787659
$701$	25.5680	0.965689	0.482845	−	0.875706i	$-0.339604\pi$
$701$	25.5680	0.965689	0.482845	+	0.875706i	$0.339604\pi$
$702$	−17.4103	−0.657111
$703$	−1.67067	−0.0630106
$704$	3.04833	0.114888
$705$	−1.14740	−0.0432135
$706$	−32.2401	−1.21337
$707$	6.65705	0.250364
$708$	−24.5081	−0.921073
$709$	29.9543	1.12496	0.562478	−	0.826812i	$-0.309848\pi$
$709$	29.9543	1.12496	0.562478	+	0.826812i	$0.309848\pi$
$710$	0.944535	0.0354478
$711$	−20.6539	−0.774581
$712$	−3.60769	−0.135204
$713$	−28.0221	−1.04944
$714$	−11.0890	−0.414995
$715$	1.18155	0.0441875
$716$	−7.23413	−0.270352
$717$	22.5435	0.841903
$718$	−3.90150	−0.145603
$719$	−4.91004	−0.183114	−0.0915568	−	0.995800i	$-0.529184\pi$
$719$	−4.91004	−0.183114	−0.0915568	+	0.995800i	$0.529184\pi$
$720$	−0.105307	−0.00392455
$721$	−5.15014	−0.191801
$722$	−18.6192	−0.692936
$723$	36.6654	1.36360
$724$	5.22122	0.194045
$725$	34.6691	1.28758
$726$	3.59657	0.133481
$727$	7.22164	0.267836	0.133918	−	0.990992i	$-0.457244\pi$
$727$	7.22164	0.267836	0.133918	+	0.990992i	$0.457244\pi$
$728$	−22.0505	−0.817245
$729$	4.66820	0.172896
$730$	1.14986	0.0425584
$731$	−12.5309	−0.463472
$732$	14.0150	0.518009
$733$	22.6979	0.838367	0.419184	−	0.907902i	$-0.362316\pi$
$733$	22.6979	0.838367	0.419184	+	0.907902i	$0.362316\pi$
$734$	25.7272	0.949610
$735$	1.60785	0.0593064
$736$	−4.17171	−0.153771
$737$	33.9725	1.25139
$738$	0.151908	0.00559183
$739$	22.9445	0.844028	0.422014	−	0.906589i	$-0.361323\pi$
$739$	22.9445	0.844028	0.422014	+	0.906589i	$0.361323\pi$
$740$	0.198569	0.00729953
$741$	6.86844	0.252318
$742$	−42.2633	−1.55153
$743$	−7.60162	−0.278876	−0.139438	−	0.990231i	$-0.544530\pi$
$743$	−7.60162	−0.278876	−0.139438	+	0.990231i	$0.544530\pi$
$744$	−14.1473	−0.518666
$745$	−0.527032	−0.0193090
$746$	−18.9998	−0.695631
$747$	−1.42410	−0.0521051
$748$	3.84665	0.140647
$749$	79.6816	2.91150
$750$	−1.54387	−0.0563741
$751$	−29.2343	−1.06677	−0.533387	−	0.845871i	$-0.679081\pi$
$751$	−29.2343	−1.06677	−0.533387	+	0.845871i	$0.679081\pi$
$752$	−7.42797	−0.270870
$753$	37.3838	1.36234
$754$	36.6837	1.33594
$755$	−0.204434	−0.00744012
$756$	13.7454	0.499917
$757$	−5.25363	−0.190946	−0.0954732	−	0.995432i	$-0.530436\pi$
$757$	−5.25363	−0.190946	−0.0954732	+	0.995432i	$0.530436\pi$
$758$	9.44808	0.343170
$759$	26.7832	0.972170
$760$	−0.0452579	−0.00164168
$761$	−43.8294	−1.58882	−0.794408	−	0.607385i	$-0.792219\pi$
$761$	−43.8294	−1.58882	−0.794408	+	0.607385i	$0.792219\pi$
$762$	−39.2660	−1.42246
$763$	32.0908	1.16177
$764$	23.9496	0.866467
$765$	−0.132885	−0.00480448
$766$	−21.6669	−0.782858
$767$	−61.4974	−2.22054
$768$	−2.10614	−0.0759987
$769$	−26.8740	−0.969103	−0.484551	−	0.874763i	$-0.661017\pi$
$769$	−26.8740	−0.969103	−0.484551	+	0.874763i	$0.661017\pi$
$770$	−0.932833	−0.0336170
$771$	33.4777	1.20567
$772$	−8.37461	−0.301409
$773$	15.9741	0.574549	0.287274	−	0.957848i	$-0.407251\pi$
$773$	15.9741	0.574549	0.287274	+	0.957848i	$0.407251\pi$
$774$	−14.2581	−0.512497
$775$	−33.5498	−1.20514
$776$	−15.2101	−0.546010
$777$	23.7917	0.853523
$778$	14.9040	0.534334
$779$	0.0652859	0.00233911
$780$	−0.816351	−0.0292301
$781$	−39.2576	−1.40475
$782$	−5.26422	−0.188248
$783$	−22.8672	−0.817208
$784$	10.4088	0.371744
$785$	0.852855	0.0304397
$786$	5.40069	0.192636
$787$	9.05133	0.322645	0.161322	−	0.986902i	$-0.448424\pi$
$787$	9.05133	0.322645	0.161322	+	0.986902i	$0.448424\pi$
$788$	−17.4420	−0.621346
$789$	12.0106	0.427589
$790$	1.05501	0.0375357
$791$	39.9956	1.42208
$792$	4.37686	0.155525
$793$	35.1673	1.24883
$794$	20.3285	0.721431
$795$	−1.56467	−0.0554930
$796$	−14.6853	−0.520506
$797$	−29.2502	−1.03610	−0.518048	−	0.855351i	$-0.673341\pi$
$797$	−29.2502	−1.03610	−0.518048	+	0.855351i	$0.673341\pi$
$798$	−5.42262	−0.191959
$799$	−9.37325	−0.331602
$800$	−4.99462	−0.176587
$801$	−5.17999	−0.183026
$802$	22.9664	0.810972
$803$	−47.7917	−1.68653
$804$	−23.4721	−0.827796
$805$	1.27660	0.0449943
$806$	−35.4994	−1.25041
$807$	49.9383	1.75791
$808$	1.59550	0.0561295
$809$	3.73538	0.131329	0.0656644	−	0.997842i	$-0.479083\pi$
$809$	3.73538	0.131329	0.0656644	+	0.997842i	$0.479083\pi$
$810$	0.824803	0.0289806
$811$	14.7717	0.518704	0.259352	−	0.965783i	$-0.416491\pi$
$811$	14.7717	0.518704	0.259352	+	0.965783i	$0.416491\pi$
$812$	−28.9617	−1.01636
$813$	−4.80141	−0.168393
$814$	−8.25309	−0.289271
$815$	−1.47945	−0.0518230
$816$	−2.65771	−0.0930384
$817$	−6.12773	−0.214382
$818$	2.03075	0.0710034
$819$	−31.6605	−1.10631
$820$	−0.00775959	−0.000270977 0
$821$	−3.40083	−0.118690	−0.0593449	−	0.998238i	$-0.518901\pi$
$821$	−3.40083	−0.118690	−0.0593449	+	0.998238i	$0.518901\pi$
$822$	22.6391	0.789630
$823$	18.2752	0.637035	0.318517	−	0.947917i	$-0.396815\pi$
$823$	18.2752	0.637035	0.318517	+	0.947917i	$0.396815\pi$
$824$	−1.23434	−0.0430003
$825$	32.0665	1.11641
$826$	48.5521	1.68934
$827$	−7.55358	−0.262664	−0.131332	−	0.991338i	$-0.541925\pi$
$827$	−7.55358	−0.262664	−0.131332	+	0.991338i	$0.541925\pi$
$828$	−5.98982	−0.208161
$829$	−31.8264	−1.10538	−0.552688	−	0.833389i	$-0.686398\pi$
$829$	−31.8264	−1.10538	−0.552688	+	0.833389i	$0.686398\pi$
$830$	0.0727440	0.00252498
$831$	14.3575	0.498056
$832$	−5.28486	−0.183219
$833$	13.1348	0.455092
$834$	−34.4551	−1.19308
$835$	−0.517914	−0.0179232
$836$	1.88105	0.0650575
$837$	22.1290	0.764889
$838$	4.68884	0.161973
$839$	26.5846	0.917801	0.458901	−	0.888488i	$-0.348243\pi$
$839$	26.5846	0.917801	0.458901	+	0.888488i	$0.348243\pi$
$840$	0.644508	0.0222376
$841$	19.1814	0.661427
$842$	40.8317	1.40715
$843$	−10.5578	−0.363631
$844$	21.8638	0.752583
$845$	−1.09499	−0.0376687
$846$	−10.6652	−0.366678
$847$	−7.12501	−0.244818
$848$	−10.1293	−0.347841
$849$	45.8334	1.57300
$850$	−6.30264	−0.216179
$851$	11.2945	0.387171
$852$	27.1237	0.929242
$853$	49.8588	1.70713	0.853567	−	0.520984i	$-0.174435\pi$
$853$	49.8588	1.70713	0.853567	+	0.520984i	$0.174435\pi$
$854$	−27.7646	−0.950084
$855$	−0.0649822	−0.00222234
$856$	19.0974	0.652734
$857$	16.0488	0.548217	0.274108	−	0.961699i	$-0.411617\pi$
$857$	16.0488	0.548217	0.274108	+	0.961699i	$0.411617\pi$
$858$	33.9299	1.15835
$859$	−37.8341	−1.29088	−0.645442	−	0.763809i	$-0.723327\pi$
$859$	−37.8341	−1.29088	−0.645442	+	0.763809i	$0.723327\pi$
$860$	0.728314	0.0248353
$861$	−0.929723	−0.0316849
$862$	−6.51364	−0.221855
$863$	−27.7462	−0.944490	−0.472245	−	0.881467i	$-0.656556\pi$
$863$	−27.7462	−0.944490	−0.472245	+	0.881467i	$0.656556\pi$
$864$	3.29438	0.112077
$865$	0.560116	0.0190445
$866$	−4.78063	−0.162452
$867$	32.4506	1.10208
$868$	28.0267	0.951289
$869$	−43.8494	−1.48749
$870$	−1.07222	−0.0363516
$871$	−58.8976	−1.99567
$872$	7.69123	0.260458
$873$	−21.8389	−0.739136
$874$	−2.57425	−0.0870754
$875$	3.05849	0.103396
$876$	33.0200	1.11564
$877$	−5.15402	−0.174039	−0.0870195	−	0.996207i	$-0.527734\pi$
$877$	−5.15402	−0.174039	−0.0870195	+	0.996207i	$0.527734\pi$
$878$	−20.3115	−0.685479
$879$	24.4466	0.824563
$880$	−0.223573	−0.00753664
$881$	−19.9149	−0.670950	−0.335475	−	0.942049i	$-0.608897\pi$
$881$	−19.9149	−0.670950	−0.335475	+	0.942049i	$0.608897\pi$
$882$	14.9452	0.503231
$883$	35.8322	1.20585	0.602925	−	0.797798i	$-0.294002\pi$
$883$	35.8322	1.20585	0.602925	+	0.797798i	$0.294002\pi$
$884$	−6.66889	−0.224299
$885$	1.79749	0.0604221
$886$	−13.9885	−0.469953
$887$	31.6797	1.06370	0.531850	−	0.846838i	$-0.321497\pi$
$887$	31.6797	1.06370	0.531850	+	0.846838i	$0.321497\pi$
$888$	5.70218	0.191353
$889$	77.7883	2.60894
$890$	0.264598	0.00886933
$891$	−34.2812	−1.14846
$892$	13.7880	0.461655
$893$	−4.58361	−0.153385
$894$	−15.1345	−0.506173
$895$	0.530570	0.0177350
$896$	4.17239	0.139390
$897$	−46.4338	−1.55038
$898$	40.3841	1.34763
$899$	−46.6259	−1.55506
$900$	−7.17138	−0.239046
$901$	−12.7820	−0.425830
$902$	0.322511	0.0107384
$903$	87.2637	2.90395
$904$	9.58577	0.318818
$905$	−0.382938	−0.0127293
$906$	−5.87061	−0.195038
$907$	49.0325	1.62810	0.814049	−	0.580796i	$-0.197258\pi$
$907$	49.0325	1.62810	0.814049	+	0.580796i	$0.197258\pi$
$908$	−16.7539	−0.555998
$909$	2.29085	0.0759828
$910$	1.61724	0.0536110
$911$	10.1620	0.336682	0.168341	−	0.985729i	$-0.446159\pi$
$911$	10.1620	0.336682	0.168341	+	0.985729i	$0.446159\pi$
$912$	−1.29964	−0.0430355
$913$	−3.02345	−0.100062
$914$	0.949312	0.0314005
$915$	−1.02790	−0.0339812
$916$	21.0604	0.695854
$917$	−10.6991	−0.353316
$918$	4.15713	0.137206
$919$	−37.1982	−1.22705	−0.613527	−	0.789674i	$-0.710250\pi$
$919$	−37.1982	−1.22705	−0.613527	+	0.789674i	$0.710250\pi$
$920$	0.305964	0.0100873
$921$	27.4784	0.905443
$922$	21.5510	0.709744
$923$	68.0605	2.24024
$924$	−26.7876	−0.881248
$925$	13.5225	0.444616
$926$	−10.6555	−0.350160
$927$	−1.77229	−0.0582096
$928$	−6.94128	−0.227859
$929$	−3.87721	−0.127207	−0.0636035	−	0.997975i	$-0.520259\pi$
$929$	−3.87721	−0.127207	−0.0636035	+	0.997975i	$0.520259\pi$
$930$	1.03760	0.0340243
$931$	6.42302	0.210506
$932$	8.10612	0.265525
$933$	−0.619273	−0.0202741
$934$	−21.9152	−0.717086
$935$	−0.282124	−0.00922643
$936$	−7.58810	−0.248025
$937$	−40.9338	−1.33725	−0.668625	−	0.743600i	$-0.733117\pi$
$937$	−40.9338	−1.33725	−0.668625	+	0.743600i	$0.733117\pi$
$938$	46.4996	1.51827
$939$	28.4704	0.929095
$940$	0.544787	0.0177690
$941$	−0.0682344	−0.00222438	−0.00111219	−	0.999999i	$-0.500354\pi$
$941$	−0.0682344	−0.00222438	−0.00111219	+	0.999999i	$0.500354\pi$
$942$	24.4909	0.797958
$943$	−0.441363	−0.0143727
$944$	11.6365	0.378737
$945$	−1.00813	−0.0327944
$946$	−30.2708	−0.984190
$947$	17.5042	0.568810	0.284405	−	0.958704i	$-0.408204\pi$
$947$	17.5042	0.568810	0.284405	+	0.958704i	$0.408204\pi$
$948$	30.2962	0.983976
$949$	82.8559	2.68962
$950$	−3.08205	−0.0999950
$951$	8.40602	0.272584
$952$	5.26508	0.170642
$953$	25.3180	0.820130	0.410065	−	0.912056i	$-0.365506\pi$
$953$	25.3180	0.820130	0.410065	+	0.912056i	$0.365506\pi$
$954$	−14.5438	−0.470873
$955$	−1.75653	−0.0568399
$956$	−10.7037	−0.346183
$957$	44.5645	1.44057
$958$	4.39329	0.141941
$959$	−44.8495	−1.44827
$960$	0.154470	0.00498549
$961$	14.1206	0.455503
$962$	14.3083	0.461317
$963$	27.4204	0.883609
$964$	−17.4088	−0.560700
$965$	0.614216	0.0197723
$966$	36.6594	1.17950
$967$	41.3344	1.32922	0.664612	−	0.747189i	$-0.268597\pi$
$967$	41.3344	1.32922	0.664612	+	0.747189i	$0.268597\pi$
$968$	−1.70766	−0.0548862
$969$	−1.64000	−0.0526845
$970$	1.11555	0.0358181
$971$	−49.6235	−1.59249	−0.796247	−	0.604971i	$-0.793185\pi$
$971$	−49.6235	−1.59249	−0.796247	+	0.604971i	$0.793185\pi$
$972$	13.8022	0.442707
$973$	68.2576	2.18824
$974$	−12.3108	−0.394462
$975$	−55.5933	−1.78041
$976$	−6.65436	−0.213001
$977$	9.14544	0.292589	0.146294	−	0.989241i	$-0.453265\pi$
$977$	9.14544	0.292589	0.146294	+	0.989241i	$0.453265\pi$
$978$	−42.4846	−1.35851
$979$	−10.9974	−0.351480
$980$	−0.763411	−0.0243863
$981$	11.0432	0.352583
$982$	−21.0570	−0.671957
$983$	−31.8551	−1.01602	−0.508010	−	0.861351i	$-0.669619\pi$
$983$	−31.8551	−1.01602	−0.508010	+	0.861351i	$0.669619\pi$
$984$	−0.222828	−0.00710348
$985$	1.27924	0.0407601
$986$	−8.75911	−0.278947
$987$	65.2743	2.07770
$988$	−3.26115	−0.103751
$989$	41.4262	1.31728
$990$	−0.321010	−0.0102024
$991$	14.7506	0.468569	0.234284	−	0.972168i	$-0.424725\pi$
$991$	14.7506	0.468569	0.234284	+	0.972168i	$0.424725\pi$
$992$	6.71719	0.213271
$993$	27.4884	0.872318
$994$	−53.7336	−1.70433
$995$	1.07706	0.0341450
$996$	2.08895	0.0661908
$997$	−46.7080	−1.47926	−0.739628	−	0.673015i	$-0.764999\pi$
$997$	−46.7080	−1.47926	−0.739628	+	0.673015i	$0.764999\pi$
$998$	27.6080	0.873915
$999$	−8.91923	−0.282192

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.a.1.19	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.a.1.19	✓	75	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.10614 q^{3} +1.00000 q^{4} -0.0733427 q^{5} -2.10614 q^{6} +4.17239 q^{7} +1.00000 q^{8} +1.43582 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.10614 q^{3} +1.00000 q^{4} -0.0733427 q^{5} -2.10614 q^{6} +4.17239 q^{7} +1.00000 q^{8} +1.43582 q^{9} -0.0733427 q^{10} +3.04833 q^{11} -2.10614 q^{12} -5.28486 q^{13} +4.17239 q^{14} +0.154470 q^{15} +1.00000 q^{16} +1.26189 q^{17} +1.43582 q^{18} +0.617075 q^{19} -0.0733427 q^{20} -8.78763 q^{21} +3.04833 q^{22} -4.17171 q^{23} -2.10614 q^{24} -4.99462 q^{25} -5.28486 q^{26} +3.29438 q^{27} +4.17239 q^{28} -6.94128 q^{29} +0.154470 q^{30} +6.71719 q^{31} +1.00000 q^{32} -6.42021 q^{33} +1.26189 q^{34} -0.306014 q^{35} +1.43582 q^{36} -2.70741 q^{37} +0.617075 q^{38} +11.1306 q^{39} -0.0733427 q^{40} +0.105799 q^{41} -8.78763 q^{42} -9.93029 q^{43} +3.04833 q^{44} -0.105307 q^{45} -4.17171 q^{46} -7.42797 q^{47} -2.10614 q^{48} +10.4088 q^{49} -4.99462 q^{50} -2.65771 q^{51} -5.28486 q^{52} -10.1293 q^{53} +3.29438 q^{54} -0.223573 q^{55} +4.17239 q^{56} -1.29964 q^{57} -6.94128 q^{58} +11.6365 q^{59} +0.154470 q^{60} -6.65436 q^{61} +6.71719 q^{62} +5.99080 q^{63} +1.00000 q^{64} +0.387606 q^{65} -6.42021 q^{66} +11.1446 q^{67} +1.26189 q^{68} +8.78619 q^{69} -0.306014 q^{70} -12.8784 q^{71} +1.43582 q^{72} -15.6780 q^{73} -2.70741 q^{74} +10.5194 q^{75} +0.617075 q^{76} +12.7188 q^{77} +11.1306 q^{78} -14.3847 q^{79} -0.0733427 q^{80} -11.2459 q^{81} +0.105799 q^{82} -0.991837 q^{83} -8.78763 q^{84} -0.0925501 q^{85} -9.93029 q^{86} +14.6193 q^{87} +3.04833 q^{88} -3.60769 q^{89} -0.105307 q^{90} -22.0505 q^{91} -4.17171 q^{92} -14.1473 q^{93} -7.42797 q^{94} -0.0452579 q^{95} -2.10614 q^{96} -15.2101 q^{97} +10.4088 q^{98} +4.37686 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more