LMFDB - Embedded newform 8031.2.a.b.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8031,2,Mod(1,8031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8031, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8031 = 3 \cdot 2677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8031.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:	$64.1278578633$
Analytic rank:	$1$
Dimension:	$102$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17521 q^{2} -1.00000 q^{3} +2.73153 q^{4} -1.76548 q^{5} +2.17521 q^{6} +3.60234 q^{7} -1.59123 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.17521 q^{2} -1.00000 q^{3} +2.73153 q^{4} -1.76548 q^{5} +2.17521 q^{6} +3.60234 q^{7} -1.59123 q^{8} +1.00000 q^{9} +3.84028 q^{10} -6.60487 q^{11} -2.73153 q^{12} -0.678741 q^{13} -7.83584 q^{14} +1.76548 q^{15} -2.00181 q^{16} -1.79266 q^{17} -2.17521 q^{18} +0.792591 q^{19} -4.82245 q^{20} -3.60234 q^{21} +14.3670 q^{22} -7.43018 q^{23} +1.59123 q^{24} -1.88309 q^{25} +1.47640 q^{26} -1.00000 q^{27} +9.83990 q^{28} +3.76746 q^{29} -3.84028 q^{30} +2.72224 q^{31} +7.53680 q^{32} +6.60487 q^{33} +3.89941 q^{34} -6.35985 q^{35} +2.73153 q^{36} +5.44787 q^{37} -1.72405 q^{38} +0.678741 q^{39} +2.80927 q^{40} +0.376054 q^{41} +7.83584 q^{42} +7.25967 q^{43} -18.0414 q^{44} -1.76548 q^{45} +16.1622 q^{46} +4.72079 q^{47} +2.00181 q^{48} +5.97686 q^{49} +4.09612 q^{50} +1.79266 q^{51} -1.85400 q^{52} +2.12637 q^{53} +2.17521 q^{54} +11.6607 q^{55} -5.73214 q^{56} -0.792591 q^{57} -8.19501 q^{58} +10.3184 q^{59} +4.82245 q^{60} +0.962644 q^{61} -5.92144 q^{62} +3.60234 q^{63} -12.3905 q^{64} +1.19830 q^{65} -14.3670 q^{66} -5.61566 q^{67} -4.89671 q^{68} +7.43018 q^{69} +13.8340 q^{70} -10.7259 q^{71} -1.59123 q^{72} -1.84208 q^{73} -11.8502 q^{74} +1.88309 q^{75} +2.16499 q^{76} -23.7930 q^{77} -1.47640 q^{78} +6.58518 q^{79} +3.53414 q^{80} +1.00000 q^{81} -0.817995 q^{82} +7.98035 q^{83} -9.83990 q^{84} +3.16490 q^{85} -15.7913 q^{86} -3.76746 q^{87} +10.5098 q^{88} +11.7906 q^{89} +3.84028 q^{90} -2.44505 q^{91} -20.2957 q^{92} -2.72224 q^{93} -10.2687 q^{94} -1.39930 q^{95} -7.53680 q^{96} -6.25590 q^{97} -13.0009 q^{98} -6.60487 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9}+O(q^{10}) $ 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9} - 16 q^{10} - 28 q^{11} - 96 q^{12} - 2 q^{13} - 41 q^{14} + 20 q^{15} + 88 q^{16} - 77 q^{17} - 6 q^{18} + 10 q^{19} - 50 q^{20} - 12 q^{21} + 24 q^{22} - 29 q^{23} + 21 q^{24} + 74 q^{25} - 45 q^{26} - 102 q^{27} + 19 q^{28} - 68 q^{29} + 16 q^{30} - 29 q^{31} - 48 q^{32} + 28 q^{33} - 19 q^{34} - 49 q^{35} + 96 q^{36} + 4 q^{37} - 44 q^{38} + 2 q^{39} - 41 q^{40} - 122 q^{41} + 41 q^{42} + 85 q^{43} - 86 q^{44} - 20 q^{45} - 28 q^{46} - 39 q^{47} - 88 q^{48} + 24 q^{49} - 37 q^{50} + 77 q^{51} + 8 q^{52} - 37 q^{53} + 6 q^{54} - 13 q^{55} - 130 q^{56} - 10 q^{57} + 17 q^{58} - 58 q^{59} + 50 q^{60} - 114 q^{61} - 64 q^{62} + 12 q^{63} + 47 q^{64} - 92 q^{65} - 24 q^{66} + 121 q^{67} - 138 q^{68} + 29 q^{69} - 2 q^{70} - 67 q^{71} - 21 q^{72} - 72 q^{73} - 111 q^{74} - 74 q^{75} - 17 q^{76} - 57 q^{77} + 45 q^{78} - 24 q^{79} - 97 q^{80} + 102 q^{81} - q^{82} - 78 q^{83} - 19 q^{84} - 24 q^{85} - 80 q^{86} + 68 q^{87} + 54 q^{88} - 176 q^{89} - 16 q^{90} - 3 q^{91} - 82 q^{92} + 29 q^{93} - 41 q^{94} - 90 q^{95} + 48 q^{96} - 77 q^{97} - 48 q^{98} - 28 q^{99}+O(q^{100}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9 - 16 * q^10 - 28 * q^11 - 96 * q^12 - 2 * q^13 - 41 * q^14 + 20 * q^15 + 88 * q^16 - 77 * q^17 - 6 * q^18 + 10 * q^19 - 50 * q^20 - 12 * q^21 + 24 * q^22 - 29 * q^23 + 21 * q^24 + 74 * q^25 - 45 * q^26 - 102 * q^27 + 19 * q^28 - 68 * q^29 + 16 * q^30 - 29 * q^31 - 48 * q^32 + 28 * q^33 - 19 * q^34 - 49 * q^35 + 96 * q^36 + 4 * q^37 - 44 * q^38 + 2 * q^39 - 41 * q^40 - 122 * q^41 + 41 * q^42 + 85 * q^43 - 86 * q^44 - 20 * q^45 - 28 * q^46 - 39 * q^47 - 88 * q^48 + 24 * q^49 - 37 * q^50 + 77 * q^51 + 8 * q^52 - 37 * q^53 + 6 * q^54 - 13 * q^55 - 130 * q^56 - 10 * q^57 + 17 * q^58 - 58 * q^59 + 50 * q^60 - 114 * q^61 - 64 * q^62 + 12 * q^63 + 47 * q^64 - 92 * q^65 - 24 * q^66 + 121 * q^67 - 138 * q^68 + 29 * q^69 - 2 * q^70 - 67 * q^71 - 21 * q^72 - 72 * q^73 - 111 * q^74 - 74 * q^75 - 17 * q^76 - 57 * q^77 + 45 * q^78 - 24 * q^79 - 97 * q^80 + 102 * q^81 - q^82 - 78 * q^83 - 19 * q^84 - 24 * q^85 - 80 * q^86 + 68 * q^87 + 54 * q^88 - 176 * q^89 - 16 * q^90 - 3 * q^91 - 82 * q^92 + 29 * q^93 - 41 * q^94 - 90 * q^95 + 48 * q^96 - 77 * q^97 - 48 * q^98 - 28 * 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 $

$2$	−2.17521	−1.53810	−0.769052	−	0.639186i	$-0.779271\pi$
$2$	−2.17521	−1.53810	−0.769052	+	0.639186i	$0.779271\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.73153	1.36576
$5$	−1.76548	−0.789545	−0.394772	−	0.918779i	$-0.629177\pi$
$5$	−1.76548	−0.789545	−0.394772	+	0.918779i	$0.629177\pi$
$6$	2.17521	0.888025
$7$	3.60234	1.36156	0.680778	−	0.732489i	$-0.261642\pi$
$7$	3.60234	1.36156	0.680778	+	0.732489i	$0.261642\pi$
$8$	−1.59123	−0.562584
$9$	1.00000	0.333333
$10$	3.84028	1.21440
$11$	−6.60487	−1.99144	−0.995722	−	0.0924042i	$-0.970545\pi$
$11$	−6.60487	−1.99144	−0.995722	+	0.0924042i	$0.970545\pi$
$12$	−2.73153	−0.788524
$13$	−0.678741	−0.188249	−0.0941244	−	0.995560i	$-0.530005\pi$
$13$	−0.678741	−0.188249	−0.0941244	+	0.995560i	$0.530005\pi$
$14$	−7.83584	−2.09422
$15$	1.76548	0.455844
$16$	−2.00181	−0.500452
$17$	−1.79266	−0.434785	−0.217392	−	0.976084i	$-0.569755\pi$
$17$	−1.79266	−0.434785	−0.217392	+	0.976084i	$0.569755\pi$
$18$	−2.17521	−0.512701
$19$	0.792591	0.181833	0.0909165	−	0.995859i	$-0.471020\pi$
$19$	0.792591	0.181833	0.0909165	+	0.995859i	$0.471020\pi$
$20$	−4.82245	−1.07833
$21$	−3.60234	−0.786095
$22$	14.3670	3.06305
$23$	−7.43018	−1.54930	−0.774650	−	0.632391i	$-0.782074\pi$
$23$	−7.43018	−1.54930	−0.774650	+	0.632391i	$0.782074\pi$
$24$	1.59123	0.324808
$25$	−1.88309	−0.376619
$26$	1.47640	0.289546
$27$	−1.00000	−0.192450
$28$	9.83990	1.85957
$29$	3.76746	0.699600	0.349800	−	0.936824i	$-0.386250\pi$
$29$	3.76746	0.699600	0.349800	+	0.936824i	$0.386250\pi$
$30$	−3.84028	−0.701135
$31$	2.72224	0.488929	0.244464	−	0.969658i	$-0.421388\pi$
$31$	2.72224	0.488929	0.244464	+	0.969658i	$0.421388\pi$
$32$	7.53680	1.33233
$33$	6.60487	1.14976
$34$	3.89941	0.668744
$35$	−6.35985	−1.07501
$36$	2.73153	0.455255
$37$	5.44787	0.895624	0.447812	−	0.894128i	$-0.352203\pi$
$37$	5.44787	0.895624	0.447812	+	0.894128i	$0.352203\pi$
$38$	−1.72405	−0.279678
$39$	0.678741	0.108685
$40$	2.80927	0.444185
$41$	0.376054	0.0587297	0.0293649	−	0.999569i	$-0.490652\pi$
$41$	0.376054	0.0587297	0.0293649	+	0.999569i	$0.490652\pi$
$42$	7.83584	1.20910
$43$	7.25967	1.10709	0.553544	−	0.832820i	$-0.313275\pi$
$43$	7.25967	1.10709	0.553544	+	0.832820i	$0.313275\pi$
$44$	−18.0414	−2.71984
$45$	−1.76548	−0.263182
$46$	16.1622	2.38298
$47$	4.72079	0.688598	0.344299	−	0.938860i	$-0.388117\pi$
$47$	4.72079	0.688598	0.344299	+	0.938860i	$0.388117\pi$
$48$	2.00181	0.288936
$49$	5.97686	0.853837
$50$	4.09612	0.579279
$51$	1.79266	0.251023
$52$	−1.85400	−0.257103
$53$	2.12637	0.292079	0.146039	−	0.989279i	$-0.453347\pi$
$53$	2.12637	0.292079	0.146039	+	0.989279i	$0.453347\pi$
$54$	2.17521	0.296008
$55$	11.6607	1.57233
$56$	−5.73214	−0.765990
$57$	−0.792591	−0.104981
$58$	−8.19501	−1.07606
$59$	10.3184	1.34334	0.671668	−	0.740852i	$-0.265578\pi$
$59$	10.3184	1.34334	0.671668	+	0.740852i	$0.265578\pi$
$60$	4.82245	0.622575
$61$	0.962644	0.123254	0.0616270	−	0.998099i	$-0.480371\pi$
$61$	0.962644	0.123254	0.0616270	+	0.998099i	$0.480371\pi$
$62$	−5.92144	−0.752023
$63$	3.60234	0.453852
$64$	−12.3905	−1.54881
$65$	1.19830	0.148631
$66$	−14.3670	−1.76845
$67$	−5.61566	−0.686062	−0.343031	−	0.939324i	$-0.611454\pi$
$67$	−5.61566	−0.686062	−0.343031	+	0.939324i	$0.611454\pi$
$68$	−4.89671	−0.593813
$69$	7.43018	0.894488
$70$	13.8340	1.65348
$71$	−10.7259	−1.27293	−0.636466	−	0.771305i	$-0.719604\pi$
$71$	−10.7259	−1.27293	−0.636466	+	0.771305i	$0.719604\pi$
$72$	−1.59123	−0.187528
$73$	−1.84208	−0.215600	−0.107800	−	0.994173i	$-0.534381\pi$
$73$	−1.84208	−0.215600	−0.107800	+	0.994173i	$0.534381\pi$
$74$	−11.8502	−1.37756
$75$	1.88309	0.217441
$76$	2.16499	0.248341
$77$	−23.7930	−2.71146
$78$	−1.47640	−0.167170
$79$	6.58518	0.740890	0.370445	−	0.928854i	$-0.379205\pi$
$79$	6.58518	0.740890	0.370445	+	0.928854i	$0.379205\pi$
$80$	3.53414	0.395129
$81$	1.00000	0.111111
$82$	−0.817995	−0.0903324
$83$	7.98035	0.875957	0.437979	−	0.898985i	$-0.355695\pi$
$83$	7.98035	0.875957	0.437979	+	0.898985i	$0.355695\pi$
$84$	−9.83990	−1.07362
$85$	3.16490	0.343282
$86$	−15.7913	−1.70282
$87$	−3.76746	−0.403914
$88$	10.5098	1.12035
$89$	11.7906	1.24981	0.624903	−	0.780702i	$-0.285139\pi$
$89$	11.7906	1.24981	0.624903	+	0.780702i	$0.285139\pi$
$90$	3.84028	0.404801
$91$	−2.44505	−0.256311
$92$	−20.2957	−2.11598
$93$	−2.72224	−0.282283
$94$	−10.2687	−1.05914
$95$	−1.39930	−0.143565
$96$	−7.53680	−0.769222
$97$	−6.25590	−0.635190	−0.317595	−	0.948226i	$-0.602875\pi$
$97$	−6.25590	−0.635190	−0.317595	+	0.948226i	$0.602875\pi$
$98$	−13.0009	−1.31329
$99$	−6.60487	−0.663814
$100$	−5.14373	−0.514373
$101$	6.28125	0.625008	0.312504	−	0.949916i	$-0.398832\pi$
$101$	6.28125	0.625008	0.312504	+	0.949916i	$0.398832\pi$
$102$	−3.89941	−0.386100
$103$	5.82981	0.574428	0.287214	−	0.957866i	$-0.407271\pi$
$103$	5.82981	0.574428	0.287214	+	0.957866i	$0.407271\pi$
$104$	1.08003	0.105906
$105$	6.35985	0.620657
$106$	−4.62529	−0.449248
$107$	−13.9693	−1.35047	−0.675234	−	0.737604i	$-0.735957\pi$
$107$	−13.9693	−1.35047	−0.675234	+	0.737604i	$0.735957\pi$
$108$	−2.73153	−0.262841
$109$	−3.69160	−0.353591	−0.176795	−	0.984248i	$-0.556573\pi$
$109$	−3.69160	−0.353591	−0.176795	+	0.984248i	$0.556573\pi$
$110$	−25.3645	−2.41841
$111$	−5.44787	−0.517089
$112$	−7.21119	−0.681394
$113$	−10.6324	−1.00021	−0.500105	−	0.865965i	$-0.666705\pi$
$113$	−10.6324	−1.00021	−0.500105	+	0.865965i	$0.666705\pi$
$114$	1.72405	0.161472
$115$	13.1178	1.22324
$116$	10.2909	0.955488
$117$	−0.678741	−0.0627496
$118$	−22.4446	−2.06619
$119$	−6.45778	−0.591984
$120$	−2.80927	−0.256450
$121$	32.6243	2.96585
$122$	−2.09395	−0.189577
$123$	−0.376054	−0.0339076
$124$	7.43587	0.667761
$125$	12.1519	1.08690
$126$	−7.83584	−0.698072
$127$	−4.22184	−0.374628	−0.187314	−	0.982300i	$-0.559978\pi$
$127$	−4.22184	−0.374628	−0.187314	+	0.982300i	$0.559978\pi$
$128$	11.8783	1.04990
$129$	−7.25967	−0.639178
$130$	−2.60655	−0.228610
$131$	14.4377	1.26142	0.630712	−	0.776017i	$-0.282763\pi$
$131$	14.4377	1.26142	0.630712	+	0.776017i	$0.282763\pi$
$132$	18.0414	1.57030
$133$	2.85518	0.247576
$134$	12.2152	1.05523
$135$	1.76548	0.151948
$136$	2.85253	0.244603
$137$	−8.52152	−0.728043	−0.364021	−	0.931391i	$-0.618596\pi$
$137$	−8.52152	−0.728043	−0.364021	+	0.931391i	$0.618596\pi$
$138$	−16.1622	−1.37582
$139$	−20.1763	−1.71133	−0.855666	−	0.517528i	$-0.826852\pi$
$139$	−20.1763	−1.71133	−0.855666	+	0.517528i	$0.826852\pi$
$140$	−17.3721	−1.46821
$141$	−4.72079	−0.397562
$142$	23.3311	1.95790
$143$	4.48299	0.374887
$144$	−2.00181	−0.166817
$145$	−6.65136	−0.552365
$146$	4.00691	0.331615
$147$	−5.97686	−0.492963
$148$	14.8810	1.22321
$149$	−12.8408	−1.05196	−0.525980	−	0.850497i	$-0.676301\pi$
$149$	−12.8408	−1.05196	−0.525980	+	0.850497i	$0.676301\pi$
$150$	−4.09612	−0.334447
$151$	13.5248	1.10063	0.550317	−	0.834956i	$-0.314507\pi$
$151$	13.5248	1.10063	0.550317	+	0.834956i	$0.314507\pi$
$152$	−1.26119	−0.102296
$153$	−1.79266	−0.144928
$154$	51.7547	4.17051
$155$	−4.80605	−0.386031
$156$	1.85400	0.148439
$157$	23.0195	1.83715	0.918576	−	0.395244i	$-0.129340\pi$
$157$	23.0195	1.83715	0.918576	+	0.395244i	$0.129340\pi$
$158$	−14.3241	−1.13957
$159$	−2.12637	−0.168632
$160$	−13.3060	−1.05194
$161$	−26.7660	−2.10946
$162$	−2.17521	−0.170900
$163$	−6.00591	−0.470419	−0.235210	−	0.971945i	$-0.575578\pi$
$163$	−6.00591	−0.470419	−0.235210	+	0.971945i	$0.575578\pi$
$164$	1.02720	0.0802109
$165$	−11.6607	−0.907787
$166$	−17.3589	−1.34731
$167$	18.4818	1.43016	0.715081	−	0.699041i	$-0.246390\pi$
$167$	18.4818	1.43016	0.715081	+	0.699041i	$0.246390\pi$
$168$	5.73214	0.442244
$169$	−12.5393	−0.964562
$170$	−6.88432	−0.528003
$171$	0.792591	0.0606110
$172$	19.8300	1.51202
$173$	12.6645	0.962864	0.481432	−	0.876483i	$-0.340117\pi$
$173$	12.6645	0.962864	0.481432	+	0.876483i	$0.340117\pi$
$174$	8.19501	0.621262
$175$	−6.78355	−0.512788
$176$	13.2217	0.996622
$177$	−10.3184	−0.775576
$178$	−25.6471	−1.92233
$179$	4.93044	0.368519	0.184259	−	0.982878i	$-0.441011\pi$
$179$	4.93044	0.368519	0.184259	+	0.982878i	$0.441011\pi$
$180$	−4.82245	−0.359444
$181$	−18.9050	−1.40520	−0.702598	−	0.711587i	$-0.747977\pi$
$181$	−18.9050	−1.40520	−0.702598	+	0.711587i	$0.747977\pi$
$182$	5.31850	0.394234
$183$	−0.962644	−0.0711607
$184$	11.8231	0.871611
$185$	−9.61808	−0.707135
$186$	5.92144	0.434181
$187$	11.8403	0.865849
$188$	12.8950	0.940463
$189$	−3.60234	−0.262032
$190$	3.04377	0.220818
$191$	−17.7489	−1.28426	−0.642132	−	0.766594i	$-0.721950\pi$
$191$	−17.7489	−1.28426	−0.642132	+	0.766594i	$0.721950\pi$
$192$	12.3905	0.894207
$193$	−27.4284	−1.97434	−0.987170	−	0.159672i	$-0.948956\pi$
$193$	−27.4284	−1.97434	−0.987170	+	0.159672i	$0.948956\pi$
$194$	13.6079	0.976988
$195$	−1.19830	−0.0858121
$196$	16.3260	1.16614
$197$	−10.1365	−0.722193	−0.361097	−	0.932528i	$-0.617598\pi$
$197$	−10.1365	−0.722193	−0.361097	+	0.932528i	$0.617598\pi$
$198$	14.3670	1.02102
$199$	−8.93611	−0.633464	−0.316732	−	0.948515i	$-0.602586\pi$
$199$	−8.93611	−0.633464	−0.316732	+	0.948515i	$0.602586\pi$
$200$	2.99643	0.211880
$201$	5.61566	0.396098
$202$	−13.6630	−0.961327
$203$	13.5717	0.952545
$204$	4.89671	0.342838
$205$	−0.663914	−0.0463697
$206$	−12.6810	−0.883530
$207$	−7.43018	−0.516433
$208$	1.35871	0.0942095
$209$	−5.23496	−0.362110
$210$	−13.8340	−0.954636
$211$	24.5132	1.68756	0.843780	−	0.536690i	$-0.180325\pi$
$211$	24.5132	1.68756	0.843780	+	0.536690i	$0.180325\pi$
$212$	5.80823	0.398911
$213$	10.7259	0.734927
$214$	30.3862	2.07716
$215$	−12.8168	−0.874096
$216$	1.59123	0.108269
$217$	9.80643	0.665704
$218$	8.02999	0.543860
$219$	1.84208	0.124476
$220$	31.8516	2.14744
$221$	1.21675	0.0818477
$222$	11.8502	0.795336
$223$	−0.583962	−0.0391050	−0.0195525	−	0.999809i	$-0.506224\pi$
$223$	−0.583962	−0.0391050	−0.0195525	+	0.999809i	$0.506224\pi$
$224$	27.1501	1.81404
$225$	−1.88309	−0.125540
$226$	23.1276	1.53843
$227$	3.31400	0.219958	0.109979	−	0.993934i	$-0.464922\pi$
$227$	3.31400	0.219958	0.109979	+	0.993934i	$0.464922\pi$
$228$	−2.16499	−0.143380
$229$	9.86973	0.652210	0.326105	−	0.945334i	$-0.394264\pi$
$229$	9.86973	0.652210	0.326105	+	0.945334i	$0.394264\pi$
$230$	−28.5339	−1.88147
$231$	23.7930	1.56546
$232$	−5.99488	−0.393583
$233$	27.4603	1.79898	0.899492	−	0.436937i	$-0.143937\pi$
$233$	27.4603	1.79898	0.899492	+	0.436937i	$0.143937\pi$
$234$	1.47640	0.0965154
$235$	−8.33445	−0.543679
$236$	28.1849	1.83468
$237$	−6.58518	−0.427753
$238$	14.0470	0.910533
$239$	14.9126	0.964617	0.482308	−	0.876002i	$-0.339798\pi$
$239$	14.9126	0.964617	0.482308	+	0.876002i	$0.339798\pi$
$240$	−3.53414	−0.228128
$241$	−10.2160	−0.658070	−0.329035	−	0.944318i	$-0.606723\pi$
$241$	−10.2160	−0.658070	−0.329035	+	0.944318i	$0.606723\pi$
$242$	−70.9646	−4.56178
$243$	−1.00000	−0.0641500
$244$	2.62949	0.168336
$245$	−10.5520	−0.674142
$246$	0.817995	0.0521534
$247$	−0.537964	−0.0342298
$248$	−4.33170	−0.275063
$249$	−7.98035	−0.505734
$250$	−26.4330	−1.67177
$251$	9.92470	0.626441	0.313221	−	0.949680i	$-0.398592\pi$
$251$	9.92470	0.626441	0.313221	+	0.949680i	$0.398592\pi$
$252$	9.83990	0.619855
$253$	49.0754	3.08534
$254$	9.18339	0.576217
$255$	−3.16490	−0.198194
$256$	−1.05677	−0.0660484
$257$	24.9792	1.55816	0.779080	−	0.626925i	$-0.215687\pi$
$257$	24.9792	1.55816	0.779080	+	0.626925i	$0.215687\pi$
$258$	15.7913	0.983123
$259$	19.6251	1.21944
$260$	3.27319	0.202995
$261$	3.76746	0.233200
$262$	−31.4049	−1.94020
$263$	−15.9064	−0.980829	−0.490414	−	0.871489i	$-0.663155\pi$
$263$	−15.9064	−0.980829	−0.490414	+	0.871489i	$0.663155\pi$
$264$	−10.5098	−0.646836
$265$	−3.75405	−0.230609
$266$	−6.21062	−0.380797
$267$	−11.7906	−0.721576
$268$	−15.3393	−0.936999
$269$	−15.5961	−0.950910	−0.475455	−	0.879740i	$-0.657716\pi$
$269$	−15.5961	−0.950910	−0.475455	+	0.879740i	$0.657716\pi$
$270$	−3.84028	−0.233712
$271$	5.56922	0.338306	0.169153	−	0.985590i	$-0.445897\pi$
$271$	5.56922	0.338306	0.169153	+	0.985590i	$0.445897\pi$
$272$	3.58857	0.217589
$273$	2.44505	0.147981
$274$	18.5361	1.11981
$275$	12.4376	0.750015
$276$	20.2957	1.22166
$277$	−11.6206	−0.698217	−0.349109	−	0.937082i	$-0.613516\pi$
$277$	−11.6206	−0.698217	−0.349109	+	0.937082i	$0.613516\pi$
$278$	43.8877	2.63221
$279$	2.72224	0.162976
$280$	10.1200	0.604783
$281$	4.94493	0.294990	0.147495	−	0.989063i	$-0.452879\pi$
$281$	4.94493	0.294990	0.147495	+	0.989063i	$0.452879\pi$
$282$	10.2687	0.611492
$283$	−16.6779	−0.991396	−0.495698	−	0.868495i	$-0.665088\pi$
$283$	−16.6779	−0.991396	−0.495698	+	0.868495i	$0.665088\pi$
$284$	−29.2981	−1.73852
$285$	1.39930	0.0828874
$286$	−9.75144	−0.576615
$287$	1.35467	0.0799638
$288$	7.53680	0.444110
$289$	−13.7864	−0.810962
$290$	14.4681	0.849595
$291$	6.25590	0.366727
$292$	−5.03170	−0.294458
$293$	−15.5590	−0.908965	−0.454482	−	0.890756i	$-0.650176\pi$
$293$	−15.5590	−0.908965	−0.454482	+	0.890756i	$0.650176\pi$
$294$	13.0009	0.758228
$295$	−18.2168	−1.06062
$296$	−8.66880	−0.503864
$297$	6.60487	0.383253
$298$	27.9314	1.61802
$299$	5.04316	0.291654
$300$	5.14373	0.296973
$301$	26.1518	1.50736
$302$	−29.4193	−1.69289
$303$	−6.28125	−0.360848
$304$	−1.58662	−0.0909987
$305$	−1.69953	−0.0973145
$306$	3.89941	0.222915
$307$	4.96722	0.283494	0.141747	−	0.989903i	$-0.454728\pi$
$307$	4.96722	0.283494	0.141747	+	0.989903i	$0.454728\pi$
$308$	−64.9912	−3.70322
$309$	−5.82981	−0.331646
$310$	10.4542	0.593756
$311$	31.0380	1.76000	0.880001	−	0.474972i	$-0.157542\pi$
$311$	31.0380	1.76000	0.880001	+	0.474972i	$0.157542\pi$
$312$	−1.08003	−0.0611447
$313$	10.7093	0.605324	0.302662	−	0.953098i	$-0.402125\pi$
$313$	10.7093	0.605324	0.302662	+	0.953098i	$0.402125\pi$
$314$	−50.0721	−2.82573
$315$	−6.35985	−0.358337
$316$	17.9876	1.01188
$317$	−5.07979	−0.285309	−0.142655	−	0.989773i	$-0.545564\pi$
$317$	−5.07979	−0.285309	−0.142655	+	0.989773i	$0.545564\pi$
$318$	4.62529	0.259373
$319$	−24.8836	−1.39321
$320$	21.8751	1.22286
$321$	13.9693	0.779693
$322$	58.2217	3.24457
$323$	−1.42085	−0.0790582
$324$	2.73153	0.151752
$325$	1.27813	0.0708980
$326$	13.0641	0.723554
$327$	3.69160	0.204146
$328$	−0.598387	−0.0330404
$329$	17.0059	0.937566
$330$	25.3645	1.39627
$331$	10.6718	0.586575	0.293287	−	0.956024i	$-0.405251\pi$
$331$	10.6718	0.586575	0.293287	+	0.956024i	$0.405251\pi$
$332$	21.7986	1.19635
$333$	5.44787	0.298541
$334$	−40.2017	−2.19974
$335$	9.91431	0.541677
$336$	7.21119	0.393403
$337$	24.3936	1.32880	0.664401	−	0.747376i	$-0.268687\pi$
$337$	24.3936	1.32880	0.664401	+	0.747376i	$0.268687\pi$
$338$	27.2756	1.48360
$339$	10.6324	0.577472
$340$	8.64502	0.468842
$341$	−17.9800	−0.973673
$342$	−1.72405	−0.0932260
$343$	−3.68571	−0.199009
$344$	−11.5518	−0.622830
$345$	−13.1178	−0.706239
$346$	−27.5479	−1.48098
$347$	−4.44323	−0.238525	−0.119263	−	0.992863i	$-0.538053\pi$
$347$	−4.44323	−0.238525	−0.119263	+	0.992863i	$0.538053\pi$
$348$	−10.2909	−0.551651
$349$	11.3294	0.606449	0.303224	−	0.952919i	$-0.401937\pi$
$349$	11.3294	0.606449	0.303224	+	0.952919i	$0.401937\pi$
$350$	14.7556	0.788721
$351$	0.678741	0.0362285
$352$	−49.7796	−2.65326
$353$	−23.6794	−1.26033	−0.630164	−	0.776462i	$-0.717012\pi$
$353$	−23.6794	−1.26033	−0.630164	+	0.776462i	$0.717012\pi$
$354$	22.4446	1.19292
$355$	18.9363	1.00504
$356$	32.2065	1.70694
$357$	6.45778	0.341782
$358$	−10.7247	−0.566820
$359$	−16.3526	−0.863056	−0.431528	−	0.902099i	$-0.642025\pi$
$359$	−16.3526	−0.863056	−0.431528	+	0.902099i	$0.642025\pi$
$360$	2.80927	0.148062
$361$	−18.3718	−0.966937
$362$	41.1223	2.16134
$363$	−32.6243	−1.71233
$364$	−6.67874	−0.350061
$365$	3.25215	0.170226
$366$	2.09395	0.109453
$367$	−2.23834	−0.116840	−0.0584202	−	0.998292i	$-0.518606\pi$
$367$	−2.23834	−0.116840	−0.0584202	+	0.998292i	$0.518606\pi$
$368$	14.8738	0.775350
$369$	0.376054	0.0195766
$370$	20.9213	1.08765
$371$	7.65989	0.397682
$372$	−7.43587	−0.385532
$373$	−12.6619	−0.655607	−0.327803	−	0.944746i	$-0.606308\pi$
$373$	−12.6619	−0.655607	−0.327803	+	0.944746i	$0.606308\pi$
$374$	−25.7551	−1.33177
$375$	−12.1519	−0.627523
$376$	−7.51185	−0.387394
$377$	−2.55713	−0.131699
$378$	7.83584	0.403032
$379$	−20.7674	−1.06675	−0.533375	−	0.845879i	$-0.679076\pi$
$379$	−20.7674	−1.06675	−0.533375	+	0.845879i	$0.679076\pi$
$380$	−3.82223	−0.196076
$381$	4.22184	0.216292
$382$	38.6075	1.97533
$383$	−37.8089	−1.93194	−0.965972	−	0.258648i	$-0.916723\pi$
$383$	−37.8089	−1.93194	−0.965972	+	0.258648i	$0.916723\pi$
$384$	−11.8783	−0.606162
$385$	42.0060	2.14082
$386$	59.6625	3.03674
$387$	7.25967	0.369030
$388$	−17.0882	−0.867520
$389$	−18.0050	−0.912892	−0.456446	−	0.889751i	$-0.650878\pi$
$389$	−18.0050	−0.912892	−0.456446	+	0.889751i	$0.650878\pi$
$390$	2.60655	0.131988
$391$	13.3198	0.673611
$392$	−9.51054	−0.480355
$393$	−14.4377	−0.728284
$394$	22.0489	1.11081
$395$	−11.6260	−0.584966
$396$	−18.0414	−0.906614
$397$	−7.67388	−0.385141	−0.192570	−	0.981283i	$-0.561682\pi$
$397$	−7.67388	−0.385141	−0.192570	+	0.981283i	$0.561682\pi$
$398$	19.4379	0.974333
$399$	−2.85518	−0.142938
$400$	3.76959	0.188480
$401$	−34.5312	−1.72441	−0.862203	−	0.506562i	$-0.830916\pi$
$401$	−34.5312	−1.72441	−0.862203	+	0.506562i	$0.830916\pi$
$402$	−12.2152	−0.609240
$403$	−1.84769	−0.0920402
$404$	17.1574	0.853614
$405$	−1.76548	−0.0877272
$406$	−29.5212	−1.46511
$407$	−35.9825	−1.78358
$408$	−2.85253	−0.141221
$409$	20.6727	1.02220	0.511100	−	0.859521i	$-0.329238\pi$
$409$	20.6727	1.02220	0.511100	+	0.859521i	$0.329238\pi$
$410$	1.44415	0.0713215
$411$	8.52152	0.420336
$412$	15.9243	0.784533
$413$	37.1703	1.82903
$414$	16.1622	0.794328
$415$	−14.0891	−0.691607
$416$	−5.11553	−0.250810
$417$	20.1763	0.988038
$418$	11.3871	0.556963
$419$	−1.71942	−0.0839990	−0.0419995	−	0.999118i	$-0.513373\pi$
$419$	−1.71942	−0.0839990	−0.0419995	+	0.999118i	$0.513373\pi$
$420$	17.3721	0.847672
$421$	29.8912	1.45681	0.728404	−	0.685148i	$-0.240263\pi$
$421$	29.8912	1.45681	0.728404	+	0.685148i	$0.240263\pi$
$422$	−53.3213	−2.59564
$423$	4.72079	0.229533
$424$	−3.38353	−0.164319
$425$	3.37575	0.163748
$426$	−23.3311	−1.13039
$427$	3.46777	0.167817
$428$	−38.1577	−1.84442
$429$	−4.48299	−0.216441
$430$	27.8791	1.34445
$431$	−11.8552	−0.571045	−0.285523	−	0.958372i	$-0.592167\pi$
$431$	−11.8552	−0.571045	−0.285523	+	0.958372i	$0.592167\pi$
$432$	2.00181	0.0963120
$433$	−26.1646	−1.25739	−0.628696	−	0.777651i	$-0.716411\pi$
$433$	−26.1646	−1.25739	−0.628696	+	0.777651i	$0.716411\pi$
$434$	−21.3310	−1.02392
$435$	6.65136	0.318908
$436$	−10.0837	−0.482922
$437$	−5.88910	−0.281714
$438$	−4.00691	−0.191458
$439$	−21.7930	−1.04012	−0.520062	−	0.854129i	$-0.674091\pi$
$439$	−21.7930	−1.04012	−0.520062	+	0.854129i	$0.674091\pi$
$440$	−18.5549	−0.884569
$441$	5.97686	0.284612
$442$	−2.64669	−0.125890
$443$	−3.26698	−0.155219	−0.0776096	−	0.996984i	$-0.524729\pi$
$443$	−3.26698	−0.155219	−0.0776096	+	0.996984i	$0.524729\pi$
$444$	−14.8810	−0.706221
$445$	−20.8161	−0.986778
$446$	1.27024	0.0601475
$447$	12.8408	0.607349
$448$	−44.6348	−2.10880
$449$	−5.13278	−0.242231	−0.121115	−	0.992638i	$-0.538647\pi$
$449$	−5.13278	−0.242231	−0.121115	+	0.992638i	$0.538647\pi$
$450$	4.09612	0.193093
$451$	−2.48378	−0.116957
$452$	−29.0427	−1.36605
$453$	−13.5248	−0.635451
$454$	−7.20865	−0.338319
$455$	4.31669	0.202369
$456$	1.26119	0.0590608
$457$	−12.8442	−0.600825	−0.300413	−	0.953809i	$-0.597124\pi$
$457$	−12.8442	−0.600825	−0.300413	+	0.953809i	$0.597124\pi$
$458$	−21.4687	−1.00317
$459$	1.79266	0.0836743
$460$	35.8317	1.67066
$461$	36.3331	1.69220	0.846100	−	0.533025i	$-0.178945\pi$
$461$	36.3331	1.69220	0.846100	+	0.533025i	$0.178945\pi$
$462$	−51.7547	−2.40785
$463$	22.5755	1.04917	0.524586	−	0.851358i	$-0.324220\pi$
$463$	22.5755	1.04917	0.524586	+	0.851358i	$0.324220\pi$
$464$	−7.54173	−0.350116
$465$	4.80605	0.222875
$466$	−59.7319	−2.76702
$467$	36.3329	1.68129	0.840643	−	0.541590i	$-0.182177\pi$
$467$	36.3329	1.68129	0.840643	+	0.541590i	$0.182177\pi$
$468$	−1.85400	−0.0857012
$469$	−20.2295	−0.934112
$470$	18.1292	0.836235
$471$	−23.0195	−1.06068
$472$	−16.4189	−0.755739
$473$	−47.9492	−2.20470
$474$	14.3241	0.657929
$475$	−1.49252	−0.0684817
$476$	−17.6396	−0.808511
$477$	2.12637	0.0973596
$478$	−32.4380	−1.48368
$479$	−14.6134	−0.667703	−0.333851	−	0.942626i	$-0.608348\pi$
$479$	−14.6134	−0.667703	−0.333851	+	0.942626i	$0.608348\pi$
$480$	13.3060	0.607335
$481$	−3.69769	−0.168600
$482$	22.2219	1.01218
$483$	26.7660	1.21790
$484$	89.1142	4.05065
$485$	11.0446	0.501511
$486$	2.17521	0.0986694
$487$	−32.4082	−1.46855	−0.734277	−	0.678850i	$-0.762479\pi$
$487$	−32.4082	−1.46855	−0.734277	+	0.678850i	$0.762479\pi$
$488$	−1.53179	−0.0693407
$489$	6.00591	0.271597
$490$	22.9528	1.03690
$491$	20.2054	0.911855	0.455927	−	0.890017i	$-0.349308\pi$
$491$	20.2054	0.911855	0.455927	+	0.890017i	$0.349308\pi$
$492$	−1.02720	−0.0463098
$493$	−6.75378	−0.304175
$494$	1.17018	0.0526490
$495$	11.6607	0.524111
$496$	−5.44940	−0.244685
$497$	−38.6384	−1.73317
$498$	17.3589	0.777872
$499$	−15.6954	−0.702623	−0.351311	−	0.936259i	$-0.614264\pi$
$499$	−15.6954	−0.702623	−0.351311	+	0.936259i	$0.614264\pi$
$500$	33.1934	1.48445
$501$	−18.4818	−0.825705
$502$	−21.5883	−0.963532
$503$	33.1424	1.47775	0.738873	−	0.673845i	$-0.235358\pi$
$503$	33.1424	1.47775	0.738873	+	0.673845i	$0.235358\pi$
$504$	−5.73214	−0.255330
$505$	−11.0894	−0.493472
$506$	−106.749	−4.74558
$507$	12.5393	0.556890
$508$	−11.5321	−0.511654
$509$	−31.6086	−1.40103	−0.700513	−	0.713640i	$-0.747045\pi$
$509$	−31.6086	−1.40103	−0.700513	+	0.713640i	$0.747045\pi$
$510$	6.88432	0.304843
$511$	−6.63581	−0.293551
$512$	−21.4579	−0.948314
$513$	−0.792591	−0.0349938
$514$	−54.3349	−2.39661
$515$	−10.2924	−0.453537
$516$	−19.8300	−0.872967
$517$	−31.1802	−1.37130
$518$	−42.6886	−1.87563
$519$	−12.6645	−0.555910
$520$	−1.90677	−0.0836173
$521$	−32.4410	−1.42126	−0.710632	−	0.703564i	$-0.751591\pi$
$521$	−32.4410	−1.42126	−0.710632	+	0.703564i	$0.751591\pi$
$522$	−8.19501	−0.358686
$523$	24.0926	1.05350	0.526748	−	0.850021i	$-0.323411\pi$
$523$	24.0926	1.05350	0.526748	+	0.850021i	$0.323411\pi$
$524$	39.4369	1.72281
$525$	6.78355	0.296058
$526$	34.5997	1.50862
$527$	−4.88006	−0.212579
$528$	−13.2217	−0.575400
$529$	32.2076	1.40033
$530$	8.16583	0.354701
$531$	10.3184	0.447779
$532$	7.79902	0.338130
$533$	−0.255243	−0.0110558
$534$	25.6471	1.10986
$535$	24.6625	1.06625
$536$	8.93579	0.385967
$537$	−4.93044	−0.212764
$538$	33.9247	1.46260
$539$	−39.4764	−1.70037
$540$	4.82245	0.207525
$541$	−4.00927	−0.172372	−0.0861860	−	0.996279i	$-0.527468\pi$
$541$	−4.00927	−0.172372	−0.0861860	+	0.996279i	$0.527468\pi$
$542$	−12.1142	−0.520350
$543$	18.9050	0.811291
$544$	−13.5109	−0.579277
$545$	6.51743	0.279176
$546$	−5.31850	−0.227611
$547$	−32.9124	−1.40723	−0.703616	−	0.710580i	$-0.748433\pi$
$547$	−32.9124	−1.40723	−0.703616	+	0.710580i	$0.748433\pi$
$548$	−23.2768	−0.994335
$549$	0.962644	0.0410847
$550$	−27.0544	−1.15360
$551$	2.98606	0.127210
$552$	−11.8231	−0.503225
$553$	23.7220	1.00876
$554$	25.2773	1.07393
$555$	9.61808	0.408265
$556$	−55.1122	−2.33728
$557$	−13.1380	−0.556673	−0.278336	−	0.960484i	$-0.589783\pi$
$557$	−13.1380	−0.556673	−0.278336	+	0.960484i	$0.589783\pi$
$558$	−5.92144	−0.250674
$559$	−4.92743	−0.208408
$560$	12.7312	0.537991
$561$	−11.8403	−0.499898
$562$	−10.7562	−0.453725
$563$	10.3074	0.434407	0.217204	−	0.976126i	$-0.430306\pi$
$563$	10.3074	0.434407	0.217204	+	0.976126i	$0.430306\pi$
$564$	−12.8950	−0.542977
$565$	18.7712	0.789711
$566$	36.2778	1.52487
$567$	3.60234	0.151284
$568$	17.0674	0.716130
$569$	−10.2052	−0.427826	−0.213913	−	0.976853i	$-0.568621\pi$
$569$	−10.2052	−0.427826	−0.213913	+	0.976853i	$0.568621\pi$
$570$	−3.04377	−0.127490
$571$	19.8669	0.831404	0.415702	−	0.909501i	$-0.363536\pi$
$571$	19.8669	0.831404	0.415702	+	0.909501i	$0.363536\pi$
$572$	12.2454	0.512007
$573$	17.7489	0.741471
$574$	−2.94670	−0.122993
$575$	13.9917	0.583495
$576$	−12.3905	−0.516271
$577$	−22.9393	−0.954975	−0.477488	−	0.878638i	$-0.658452\pi$
$577$	−22.9393	−0.954975	−0.477488	+	0.878638i	$0.658452\pi$
$578$	29.9882	1.24734
$579$	27.4284	1.13989
$580$	−18.1684	−0.754401
$581$	28.7479	1.19267
$582$	−13.6079	−0.564064
$583$	−14.0444	−0.581658
$584$	2.93117	0.121293
$585$	1.19830	0.0495436
$586$	33.8440	1.39808
$587$	20.3591	0.840311	0.420155	−	0.907452i	$-0.361976\pi$
$587$	20.3591	0.840311	0.420155	+	0.907452i	$0.361976\pi$
$588$	−16.3260	−0.673271
$589$	2.15762	0.0889033
$590$	39.6254	1.63135
$591$	10.1365	0.416958
$592$	−10.9056	−0.448217
$593$	3.40418	0.139793	0.0698966	−	0.997554i	$-0.477733\pi$
$593$	3.40418	0.139793	0.0698966	+	0.997554i	$0.477733\pi$
$594$	−14.3670	−0.589484
$595$	11.4011	0.467398
$596$	−35.0750	−1.43673
$597$	8.93611	0.365731
$598$	−10.9699	−0.448594
$599$	16.3208	0.666849	0.333425	−	0.942777i	$-0.391796\pi$
$599$	16.3208	0.666849	0.333425	+	0.942777i	$0.391796\pi$
$600$	−2.99643	−0.122329
$601$	−13.1339	−0.535745	−0.267872	−	0.963454i	$-0.586321\pi$
$601$	−13.1339	−0.535745	−0.267872	+	0.963454i	$0.586321\pi$
$602$	−56.8856	−2.31848
$603$	−5.61566	−0.228687
$604$	36.9434	1.50321
$605$	−57.5974	−2.34167
$606$	13.6630	0.555023
$607$	30.6338	1.24339	0.621693	−	0.783261i	$-0.286445\pi$
$607$	30.6338	1.24339	0.621693	+	0.783261i	$0.286445\pi$
$608$	5.97361	0.242262
$609$	−13.5717	−0.549952
$610$	3.69682	0.149680
$611$	−3.20419	−0.129628
$612$	−4.89671	−0.197938
$613$	29.8474	1.20552	0.602762	−	0.797921i	$-0.294067\pi$
$613$	29.8474	1.20552	0.602762	+	0.797921i	$0.294067\pi$
$614$	−10.8047	−0.436044
$615$	0.663914	0.0267716
$616$	37.8601	1.52543
$617$	35.8990	1.44524	0.722620	−	0.691245i	$-0.242937\pi$
$617$	35.8990	1.44524	0.722620	+	0.691245i	$0.242937\pi$
$618$	12.6810	0.510106
$619$	−45.6643	−1.83540	−0.917702	−	0.397269i	$-0.869958\pi$
$619$	−45.6643	−1.83540	−0.917702	+	0.397269i	$0.869958\pi$
$620$	−13.1279	−0.527227
$621$	7.43018	0.298163
$622$	−67.5140	−2.70707
$623$	42.4739	1.70168
$624$	−1.35871	−0.0543919
$625$	−12.0385	−0.481539
$626$	−23.2949	−0.931051
$627$	5.23496	0.209064
$628$	62.8783	2.50912
$629$	−9.76619	−0.389404
$630$	13.8340	0.551159
$631$	−2.82997	−0.112659	−0.0563297	−	0.998412i	$-0.517940\pi$
$631$	−2.82997	−0.112659	−0.0563297	+	0.998412i	$0.517940\pi$
$632$	−10.4785	−0.416813
$633$	−24.5132	−0.974313
$634$	11.0496	0.438835
$635$	7.45356	0.295786
$636$	−5.80823	−0.230311
$637$	−4.05674	−0.160734
$638$	54.1270	2.14291
$639$	−10.7259	−0.424310
$640$	−20.9708	−0.828946
$641$	31.9970	1.26380	0.631902	−	0.775048i	$-0.282274\pi$
$641$	31.9970	1.26380	0.631902	+	0.775048i	$0.282274\pi$
$642$	−30.3862	−1.19925
$643$	8.67047	0.341930	0.170965	−	0.985277i	$-0.445311\pi$
$643$	8.67047	0.341930	0.170965	+	0.985277i	$0.445311\pi$
$644$	−73.1122	−2.88102
$645$	12.8168	0.504660
$646$	3.09064	0.121600
$647$	−27.4345	−1.07856	−0.539282	−	0.842125i	$-0.681304\pi$
$647$	−27.4345	−1.07856	−0.539282	+	0.842125i	$0.681304\pi$
$648$	−1.59123	−0.0625093
$649$	−68.1514	−2.67518
$650$	−2.78020	−0.109049
$651$	−9.80643	−0.384344
$652$	−16.4053	−0.642482
$653$	28.0943	1.09942	0.549708	−	0.835357i	$-0.314739\pi$
$653$	28.0943	1.09942	0.549708	+	0.835357i	$0.314739\pi$
$654$	−8.02999	−0.313997
$655$	−25.4894	−0.995951
$656$	−0.752787	−0.0293914
$657$	−1.84208	−0.0718665
$658$	−36.9914	−1.44207
$659$	7.16603	0.279149	0.139574	−	0.990212i	$-0.455427\pi$
$659$	7.16603	0.279149	0.139574	+	0.990212i	$0.455427\pi$
$660$	−31.8516	−1.23982
$661$	15.4606	0.601346	0.300673	−	0.953727i	$-0.402789\pi$
$661$	15.4606	0.601346	0.300673	+	0.953727i	$0.402789\pi$
$662$	−23.2134	−0.902213
$663$	−1.21675	−0.0472548
$664$	−12.6985	−0.492799
$665$	−5.04076	−0.195472
$666$	−11.8502	−0.459188
$667$	−27.9929	−1.08389
$668$	50.4835	1.95326
$669$	0.583962	0.0225773
$670$	−21.5657	−0.833155
$671$	−6.35814	−0.245453
$672$	−27.1501	−1.04734
$673$	−9.25178	−0.356630	−0.178315	−	0.983973i	$-0.557065\pi$
$673$	−9.25178	−0.356630	−0.178315	+	0.983973i	$0.557065\pi$
$674$	−53.0611	−2.04384
$675$	1.88309	0.0724803
$676$	−34.2515	−1.31737
$677$	−27.3045	−1.04940	−0.524699	−	0.851288i	$-0.675822\pi$
$677$	−27.3045	−1.04940	−0.524699	+	0.851288i	$0.675822\pi$
$678$	−23.1276	−0.888212
$679$	−22.5359	−0.864847
$680$	−5.03608	−0.193125
$681$	−3.31400	−0.126993
$682$	39.1103	1.49761
$683$	−25.9670	−0.993599	−0.496799	−	0.867865i	$-0.665492\pi$
$683$	−25.9670	−0.993599	−0.496799	+	0.867865i	$0.665492\pi$
$684$	2.16499	0.0827803
$685$	15.0445	0.574822
$686$	8.01718	0.306097
$687$	−9.86973	−0.376554
$688$	−14.5325	−0.554045
$689$	−1.44325	−0.0549835
$690$	28.5339	1.08627
$691$	9.98700	0.379923	0.189962	−	0.981792i	$-0.439164\pi$
$691$	9.98700	0.379923	0.189962	+	0.981792i	$0.439164\pi$
$692$	34.5934	1.31505
$693$	−23.7930	−0.903821
$694$	9.66495	0.366876
$695$	35.6208	1.35117
$696$	5.99488	0.227236
$697$	−0.674137	−0.0255348
$698$	−24.6438	−0.932781
$699$	−27.4603	−1.03864
$700$	−18.5295	−0.700348
$701$	−2.96507	−0.111989	−0.0559945	−	0.998431i	$-0.517833\pi$
$701$	−2.96507	−0.111989	−0.0559945	+	0.998431i	$0.517833\pi$
$702$	−1.47640	−0.0557232
$703$	4.31793	0.162854
$704$	81.8376	3.08437
$705$	8.33445	0.313893
$706$	51.5076	1.93852
$707$	22.6272	0.850984
$708$	−28.1849	−1.05925
$709$	8.15222	0.306163	0.153081	−	0.988214i	$-0.451080\pi$
$709$	8.15222	0.306163	0.153081	+	0.988214i	$0.451080\pi$
$710$	−41.1905	−1.54585
$711$	6.58518	0.246963
$712$	−18.7616	−0.703121
$713$	−20.2267	−0.757497
$714$	−14.0470	−0.525696
$715$	−7.91462	−0.295990
$716$	13.4676	0.503310
$717$	−14.9126	−0.556922
$718$	35.5703	1.32747
$719$	1.55495	0.0579898	0.0289949	−	0.999580i	$-0.490769\pi$
$719$	1.55495	0.0579898	0.0289949	+	0.999580i	$0.490769\pi$
$720$	3.53414	0.131710
$721$	21.0009	0.782116
$722$	39.9625	1.48725
$723$	10.2160	0.379937
$724$	−51.6395	−1.91917
$725$	−7.09448	−0.263482
$726$	70.9646	2.63374
$727$	−0.365796	−0.0135666	−0.00678332	−	0.999977i	$-0.502159\pi$
$727$	−0.365796	−0.0135666	−0.00678332	+	0.999977i	$0.502159\pi$
$728$	3.89064	0.144197
$729$	1.00000	0.0370370
$730$	−7.07411	−0.261825
$731$	−13.0141	−0.481345
$732$	−2.62949	−0.0971888
$733$	44.3099	1.63662	0.818311	−	0.574776i	$-0.194911\pi$
$733$	44.3099	1.63662	0.818311	+	0.574776i	$0.194911\pi$
$734$	4.86886	0.179713
$735$	10.5520	0.389216
$736$	−55.9998	−2.06418
$737$	37.0907	1.36625
$738$	−0.817995	−0.0301108
$739$	−21.2556	−0.781900	−0.390950	−	0.920412i	$-0.627853\pi$
$739$	−21.2556	−0.781900	−0.390950	+	0.920412i	$0.627853\pi$
$740$	−26.2721	−0.965780
$741$	0.537964	0.0197626
$742$	−16.6619	−0.611676
$743$	−37.1455	−1.36274	−0.681368	−	0.731941i	$-0.738614\pi$
$743$	−37.1455	−1.36274	−0.681368	+	0.731941i	$0.738614\pi$
$744$	4.33170	0.158808
$745$	22.6701	0.830569
$746$	27.5422	1.00839
$747$	7.98035	0.291986
$748$	32.3421	1.18255
$749$	−50.3223	−1.83874
$750$	26.4330	0.965196
$751$	−15.0751	−0.550099	−0.275050	−	0.961430i	$-0.588694\pi$
$751$	−15.0751	−0.550099	−0.275050	+	0.961430i	$0.588694\pi$
$752$	−9.45012	−0.344610
$753$	−9.92470	−0.361676
$754$	5.56228	0.202566
$755$	−23.8777	−0.868999
$756$	−9.83990	−0.357874
$757$	−36.9279	−1.34217	−0.671084	−	0.741382i	$-0.734171\pi$
$757$	−36.9279	−1.34217	−0.671084	+	0.741382i	$0.734171\pi$
$758$	45.1734	1.64077
$759$	−49.0754	−1.78132
$760$	2.22661	0.0807675
$761$	−21.4779	−0.778575	−0.389287	−	0.921116i	$-0.627279\pi$
$761$	−21.4779	−0.778575	−0.389287	+	0.921116i	$0.627279\pi$
$762$	−9.18339	−0.332679
$763$	−13.2984	−0.481434
$764$	−48.4816	−1.75400
$765$	3.16490	0.114427
$766$	82.2422	2.97153
$767$	−7.00349	−0.252881
$768$	1.05677	0.0381331
$769$	−21.0915	−0.760580	−0.380290	−	0.924867i	$-0.624176\pi$
$769$	−21.0915	−0.760580	−0.380290	+	0.924867i	$0.624176\pi$
$770$	−91.3717	−3.29281
$771$	−24.9792	−0.899604
$772$	−74.9215	−2.69648
$773$	−32.0344	−1.15220	−0.576100	−	0.817380i	$-0.695426\pi$
$773$	−32.0344	−1.15220	−0.576100	+	0.817380i	$0.695426\pi$
$774$	−15.7913	−0.567606
$775$	−5.12623	−0.184140
$776$	9.95455	0.357348
$777$	−19.6251	−0.704046
$778$	39.1647	1.40412
$779$	0.298057	0.0106790
$780$	−3.27319	−0.117199
$781$	70.8432	2.53497
$782$	−28.9733	−1.03608
$783$	−3.76746	−0.134638
$784$	−11.9645	−0.427304
$785$	−40.6403	−1.45051
$786$	31.4049	1.12018
$787$	39.6158	1.41215	0.706076	−	0.708136i	$-0.250464\pi$
$787$	39.6158	1.41215	0.706076	+	0.708136i	$0.250464\pi$
$788$	−27.6880	−0.986346
$789$	15.9064	0.566282
$790$	25.2889	0.899739
$791$	−38.3015	−1.36184
$792$	10.5098	0.373451
$793$	−0.653386	−0.0232024
$794$	16.6923	0.592387
$795$	3.75405	0.133142
$796$	−24.4092	−0.865162
$797$	−4.49531	−0.159232	−0.0796160	−	0.996826i	$-0.525369\pi$
$797$	−4.49531	−0.159232	−0.0796160	+	0.996826i	$0.525369\pi$
$798$	6.21062	0.219854
$799$	−8.46279	−0.299392
$800$	−14.1925	−0.501781
$801$	11.7906	0.416602
$802$	75.1126	2.65232
$803$	12.1667	0.429354
$804$	15.3393	0.540976
$805$	47.2548	1.66551
$806$	4.01912	0.141567
$807$	15.5961	0.549008
$808$	−9.99490	−0.351619
$809$	1.78740	0.0628416	0.0314208	−	0.999506i	$-0.489997\pi$
$809$	1.78740	0.0628416	0.0314208	+	0.999506i	$0.489997\pi$
$810$	3.84028	0.134934
$811$	35.1646	1.23480	0.617398	−	0.786651i	$-0.288187\pi$
$811$	35.1646	1.23480	0.617398	+	0.786651i	$0.288187\pi$
$812$	37.0714	1.30095
$813$	−5.56922	−0.195321
$814$	78.2693	2.74334
$815$	10.6033	0.371417
$816$	−3.58857	−0.125625
$817$	5.75395	0.201305
$818$	−44.9675	−1.57225
$819$	−2.44505	−0.0854371
$820$	−1.81350	−0.0633301
$821$	14.9696	0.522442	0.261221	−	0.965279i	$-0.415875\pi$
$821$	14.9696	0.522442	0.261221	+	0.965279i	$0.415875\pi$
$822$	−18.5361	−0.646520
$823$	−40.1776	−1.40050	−0.700252	−	0.713896i	$-0.746929\pi$
$823$	−40.1776	−1.40050	−0.700252	+	0.713896i	$0.746929\pi$
$824$	−9.27655	−0.323164
$825$	−12.4376	−0.433021
$826$	−80.8530	−2.81324
$827$	−22.0472	−0.766656	−0.383328	−	0.923612i	$-0.625222\pi$
$827$	−22.0472	−0.766656	−0.383328	+	0.923612i	$0.625222\pi$
$828$	−20.2957	−0.705326
$829$	12.3162	0.427758	0.213879	−	0.976860i	$-0.431390\pi$
$829$	12.3162	0.427758	0.213879	+	0.976860i	$0.431390\pi$
$830$	30.6468	1.06376
$831$	11.6206	0.403116
$832$	8.40993	0.291562
$833$	−10.7145	−0.371235
$834$	−43.8877	−1.51971
$835$	−32.6291	−1.12918
$836$	−14.2995	−0.494557
$837$	−2.72224	−0.0940943
$838$	3.74009	0.129199
$839$	−32.0564	−1.10671	−0.553355	−	0.832945i	$-0.686653\pi$
$839$	−32.0564	−1.10671	−0.553355	+	0.832945i	$0.686653\pi$
$840$	−10.1200	−0.349172
$841$	−14.8062	−0.510560
$842$	−65.0196	−2.24072
$843$	−4.94493	−0.170312
$844$	66.9586	2.30481
$845$	22.1379	0.761565
$846$	−10.2687	−0.353045
$847$	117.524	4.03817
$848$	−4.25658	−0.146171
$849$	16.6779	0.572383
$850$	−7.34297	−0.251862
$851$	−40.4786	−1.38759
$852$	29.2981	1.00374
$853$	34.6238	1.18550	0.592749	−	0.805388i	$-0.298043\pi$
$853$	34.6238	1.18550	0.592749	+	0.805388i	$0.298043\pi$
$854$	−7.54313	−0.258120
$855$	−1.39930	−0.0478551
$856$	22.2284	0.759751
$857$	−37.3002	−1.27415	−0.637075	−	0.770802i	$-0.719856\pi$
$857$	−37.3002	−1.27415	−0.637075	+	0.770802i	$0.719856\pi$
$858$	9.75144	0.332909
$859$	−39.8649	−1.36017	−0.680086	−	0.733132i	$-0.738058\pi$
$859$	−39.8649	−1.36017	−0.680086	+	0.733132i	$0.738058\pi$
$860$	−35.0094	−1.19381
$861$	−1.35467	−0.0461671
$862$	25.7875	0.878327
$863$	−8.41699	−0.286518	−0.143259	−	0.989685i	$-0.545758\pi$
$863$	−8.41699	−0.286518	−0.143259	+	0.989685i	$0.545758\pi$
$864$	−7.53680	−0.256407
$865$	−22.3589	−0.760224
$866$	56.9135	1.93400
$867$	13.7864	0.468209
$868$	26.7866	0.909195
$869$	−43.4942	−1.47544
$870$	−14.4681	−0.490514
$871$	3.81157	0.129150
$872$	5.87417	0.198924
$873$	−6.25590	−0.211730
$874$	12.8100	0.433305
$875$	43.7754	1.47988
$876$	5.03170	0.170006
$877$	−26.0562	−0.879856	−0.439928	−	0.898033i	$-0.644996\pi$
$877$	−26.0562	−0.879856	−0.439928	+	0.898033i	$0.644996\pi$
$878$	47.4043	1.59982
$879$	15.5590	0.524791
$880$	−23.3426	−0.786877
$881$	−37.2321	−1.25438	−0.627190	−	0.778866i	$-0.715795\pi$
$881$	−37.2321	−1.25438	−0.627190	+	0.778866i	$0.715795\pi$
$882$	−13.0009	−0.437763
$883$	29.0964	0.979173	0.489586	−	0.871955i	$-0.337148\pi$
$883$	29.0964	0.979173	0.489586	+	0.871955i	$0.337148\pi$
$884$	3.32360	0.111785
$885$	18.2168	0.612352
$886$	7.10637	0.238743
$887$	16.7397	0.562064	0.281032	−	0.959698i	$-0.409323\pi$
$887$	16.7397	0.562064	0.281032	+	0.959698i	$0.409323\pi$
$888$	8.66880	0.290906
$889$	−15.2085	−0.510077
$890$	45.2793	1.51777
$891$	−6.60487	−0.221271
$892$	−1.59511	−0.0534082
$893$	3.74166	0.125210
$894$	−27.9314	−0.934166
$895$	−8.70458	−0.290962
$896$	42.7897	1.42950
$897$	−5.04316	−0.168386
$898$	11.1649	0.372576
$899$	10.2559	0.342054
$900$	−5.14373	−0.171458
$901$	−3.81186	−0.126991
$902$	5.40275	0.179892
$903$	−26.1518	−0.870277
$904$	16.9185	0.562702
$905$	33.3763	1.10947
$906$	29.4193	0.977390
$907$	−38.5712	−1.28074	−0.640368	−	0.768068i	$-0.721218\pi$
$907$	−38.5712	−1.28074	−0.640368	+	0.768068i	$0.721218\pi$
$908$	9.05230	0.300411
$909$	6.28125	0.208336
$910$	−9.38969	−0.311265
$911$	−33.9433	−1.12459	−0.562296	−	0.826936i	$-0.690082\pi$
$911$	−33.9433	−1.12459	−0.562296	+	0.826936i	$0.690082\pi$
$912$	1.58662	0.0525381
$913$	−52.7092	−1.74442
$914$	27.9388	0.924132
$915$	1.69953	0.0561846
$916$	26.9594	0.890765
$917$	52.0094	1.71750
$918$	−3.89941	−0.128700
$919$	−31.7669	−1.04789	−0.523947	−	0.851751i	$-0.675541\pi$
$919$	−31.7669	−1.04789	−0.523947	+	0.851751i	$0.675541\pi$
$920$	−20.8734	−0.688176
$921$	−4.96722	−0.163675
$922$	−79.0319	−2.60278
$923$	7.28011	0.239628
$924$	64.9912	2.13805
$925$	−10.2589	−0.337309
$926$	−49.1064	−1.61373
$927$	5.82981	0.191476
$928$	28.3946	0.932098
$929$	−7.07572	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$929$	−7.07572	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$930$	−10.4542	−0.342805
$931$	4.73721	0.155256
$932$	75.0086	2.45699
$933$	−31.0380	−1.01614
$934$	−79.0316	−2.58599
$935$	−20.9038	−0.683626
$936$	1.08003	0.0353019
$937$	24.0142	0.784511	0.392255	−	0.919856i	$-0.371695\pi$
$937$	24.0142	0.784511	0.392255	+	0.919856i	$0.371695\pi$
$938$	44.0034	1.43676
$939$	−10.7093	−0.349484
$940$	−22.7658	−0.742538
$941$	−34.1165	−1.11217	−0.556084	−	0.831126i	$-0.687697\pi$
$941$	−34.1165	−1.11217	−0.556084	+	0.831126i	$0.687697\pi$
$942$	50.0721	1.63144
$943$	−2.79415	−0.0909899
$944$	−20.6554	−0.672275
$945$	6.35985	0.206886
$946$	104.299	3.39107
$947$	46.4196	1.50844	0.754218	−	0.656624i	$-0.228016\pi$
$947$	46.4196	1.50844	0.754218	+	0.656624i	$0.228016\pi$
$948$	−17.9876	−0.584210
$949$	1.25030	0.0405864
$950$	3.24655	0.105332
$951$	5.07979	0.164723
$952$	10.2758	0.333041
$953$	15.5779	0.504617	0.252309	−	0.967647i	$-0.418810\pi$
$953$	15.5779	0.504617	0.252309	+	0.967647i	$0.418810\pi$
$954$	−4.62529	−0.149749
$955$	31.3352	1.01398
$956$	40.7342	1.31744
$957$	24.8836	0.804372
$958$	31.7871	1.02700
$959$	−30.6974	−0.991271
$960$	−21.8751	−0.706017
$961$	−23.5894	−0.760949
$962$	8.04324	0.259325
$963$	−13.9693	−0.450156
$964$	−27.9053	−0.898769
$965$	48.4242	1.55883
$966$	−58.2217	−1.87325
$967$	−24.0111	−0.772144	−0.386072	−	0.922469i	$-0.626168\pi$
$967$	−24.0111	−0.772144	−0.386072	+	0.922469i	$0.626168\pi$
$968$	−51.9127	−1.66854
$969$	1.42085	0.0456443
$970$	−24.0244	−0.771376
$971$	−15.0727	−0.483705	−0.241853	−	0.970313i	$-0.577755\pi$
$971$	−15.0727	−0.483705	−0.241853	+	0.970313i	$0.577755\pi$
$972$	−2.73153	−0.0876138
$973$	−72.6819	−2.33008
$974$	70.4945	2.25879
$975$	−1.27813	−0.0409330
$976$	−1.92703	−0.0616827
$977$	15.3765	0.491939	0.245969	−	0.969278i	$-0.420894\pi$
$977$	15.3765	0.491939	0.245969	+	0.969278i	$0.420894\pi$
$978$	−13.0641	−0.417744
$979$	−77.8757	−2.48892
$980$	−28.8231	−0.920720
$981$	−3.69160	−0.117864
$982$	−43.9508	−1.40253
$983$	−21.6719	−0.691228	−0.345614	−	0.938377i	$-0.612329\pi$
$983$	−21.6719	−0.691228	−0.345614	+	0.938377i	$0.612329\pi$
$984$	0.598387	0.0190759
$985$	17.8957	0.570204
$986$	14.6909	0.467853
$987$	−17.0059	−0.541304
$988$	−1.46946	−0.0467499
$989$	−53.9406	−1.71521
$990$	−25.3645	−0.806138
$991$	1.34153	0.0426152	0.0213076	−	0.999773i	$-0.493217\pi$
$991$	1.34153	0.0426152	0.0213076	+	0.999773i	$0.493217\pi$
$992$	20.5170	0.651415
$993$	−10.6718	−0.338659
$994$	84.0465	2.66579
$995$	15.7765	0.500148
$996$	−21.7986	−0.690714
$997$	−26.7227	−0.846317	−0.423159	−	0.906056i	$-0.639079\pi$
$997$	−26.7227	−0.846317	−0.423159	+	0.906056i	$0.639079\pi$
$998$	34.1408	1.08071
$999$	−5.44787	−0.172363

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8031.2.a.b.1.15	✓	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8031.2.a.b.1.15	✓	102	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:	\( N \)	\(=\)	\( 8031 = 3 \cdot 2677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.17521 q^{2} -1.00000 q^{3} +2.73153 q^{4} -1.76548 q^{5} +2.17521 q^{6} +3.60234 q^{7} -1.59123 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.17521 q^{2} -1.00000 q^{3} +2.73153 q^{4} -1.76548 q^{5} +2.17521 q^{6} +3.60234 q^{7} -1.59123 q^{8} +1.00000 q^{9} +3.84028 q^{10} -6.60487 q^{11} -2.73153 q^{12} -0.678741 q^{13} -7.83584 q^{14} +1.76548 q^{15} -2.00181 q^{16} -1.79266 q^{17} -2.17521 q^{18} +0.792591 q^{19} -4.82245 q^{20} -3.60234 q^{21} +14.3670 q^{22} -7.43018 q^{23} +1.59123 q^{24} -1.88309 q^{25} +1.47640 q^{26} -1.00000 q^{27} +9.83990 q^{28} +3.76746 q^{29} -3.84028 q^{30} +2.72224 q^{31} +7.53680 q^{32} +6.60487 q^{33} +3.89941 q^{34} -6.35985 q^{35} +2.73153 q^{36} +5.44787 q^{37} -1.72405 q^{38} +0.678741 q^{39} +2.80927 q^{40} +0.376054 q^{41} +7.83584 q^{42} +7.25967 q^{43} -18.0414 q^{44} -1.76548 q^{45} +16.1622 q^{46} +4.72079 q^{47} +2.00181 q^{48} +5.97686 q^{49} +4.09612 q^{50} +1.79266 q^{51} -1.85400 q^{52} +2.12637 q^{53} +2.17521 q^{54} +11.6607 q^{55} -5.73214 q^{56} -0.792591 q^{57} -8.19501 q^{58} +10.3184 q^{59} +4.82245 q^{60} +0.962644 q^{61} -5.92144 q^{62} +3.60234 q^{63} -12.3905 q^{64} +1.19830 q^{65} -14.3670 q^{66} -5.61566 q^{67} -4.89671 q^{68} +7.43018 q^{69} +13.8340 q^{70} -10.7259 q^{71} -1.59123 q^{72} -1.84208 q^{73} -11.8502 q^{74} +1.88309 q^{75} +2.16499 q^{76} -23.7930 q^{77} -1.47640 q^{78} +6.58518 q^{79} +3.53414 q^{80} +1.00000 q^{81} -0.817995 q^{82} +7.98035 q^{83} -9.83990 q^{84} +3.16490 q^{85} -15.7913 q^{86} -3.76746 q^{87} +10.5098 q^{88} +11.7906 q^{89} +3.84028 q^{90} -2.44505 q^{91} -20.2957 q^{92} -2.72224 q^{93} -10.2687 q^{94} -1.39930 q^{95} -7.53680 q^{96} -6.25590 q^{97} -13.0009 q^{98} -6.60487 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9}+O(q^{10}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9} - 16 q^{10} - 28 q^{11} - 96 q^{12} - 2 q^{13} - 41 q^{14} + 20 q^{15} + 88 q^{16} - 77 q^{17} - 6 q^{18} + 10 q^{19} - 50 q^{20} - 12 q^{21} + 24 q^{22} - 29 q^{23} + 21 q^{24} + 74 q^{25} - 45 q^{26} - 102 q^{27} + 19 q^{28} - 68 q^{29} + 16 q^{30} - 29 q^{31} - 48 q^{32} + 28 q^{33} - 19 q^{34} - 49 q^{35} + 96 q^{36} + 4 q^{37} - 44 q^{38} + 2 q^{39} - 41 q^{40} - 122 q^{41} + 41 q^{42} + 85 q^{43} - 86 q^{44} - 20 q^{45} - 28 q^{46} - 39 q^{47} - 88 q^{48} + 24 q^{49} - 37 q^{50} + 77 q^{51} + 8 q^{52} - 37 q^{53} + 6 q^{54} - 13 q^{55} - 130 q^{56} - 10 q^{57} + 17 q^{58} - 58 q^{59} + 50 q^{60} - 114 q^{61} - 64 q^{62} + 12 q^{63} + 47 q^{64} - 92 q^{65} - 24 q^{66} + 121 q^{67} - 138 q^{68} + 29 q^{69} - 2 q^{70} - 67 q^{71} - 21 q^{72} - 72 q^{73} - 111 q^{74} - 74 q^{75} - 17 q^{76} - 57 q^{77} + 45 q^{78} - 24 q^{79} - 97 q^{80} + 102 q^{81} - q^{82} - 78 q^{83} - 19 q^{84} - 24 q^{85} - 80 q^{86} + 68 q^{87} + 54 q^{88} - 176 q^{89} - 16 q^{90} - 3 q^{91} - 82 q^{92} + 29 q^{93} - 41 q^{94} - 90 q^{95} + 48 q^{96} - 77 q^{97} - 48 q^{98} - 28 q^{99}+O(q^{100}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9 - 16 * q^10 - 28 * q^11 - 96 * q^12 - 2 * q^13 - 41 * q^14 + 20 * q^15 + 88 * q^16 - 77 * q^17 - 6 * q^18 + 10 * q^19 - 50 * q^20 - 12 * q^21 + 24 * q^22 - 29 * q^23 + 21 * q^24 + 74 * q^25 - 45 * q^26 - 102 * q^27 + 19 * q^28 - 68 * q^29 + 16 * q^30 - 29 * q^31 - 48 * q^32 + 28 * q^33 - 19 * q^34 - 49 * q^35 + 96 * q^36 + 4 * q^37 - 44 * q^38 + 2 * q^39 - 41 * q^40 - 122 * q^41 + 41 * q^42 + 85 * q^43 - 86 * q^44 - 20 * q^45 - 28 * q^46 - 39 * q^47 - 88 * q^48 + 24 * q^49 - 37 * q^50 + 77 * q^51 + 8 * q^52 - 37 * q^53 + 6 * q^54 - 13 * q^55 - 130 * q^56 - 10 * q^57 + 17 * q^58 - 58 * q^59 + 50 * q^60 - 114 * q^61 - 64 * q^62 + 12 * q^63 + 47 * q^64 - 92 * q^65 - 24 * q^66 + 121 * q^67 - 138 * q^68 + 29 * q^69 - 2 * q^70 - 67 * q^71 - 21 * q^72 - 72 * q^73 - 111 * q^74 - 74 * q^75 - 17 * q^76 - 57 * q^77 + 45 * q^78 - 24 * q^79 - 97 * q^80 + 102 * q^81 - q^82 - 78 * q^83 - 19 * q^84 - 24 * q^85 - 80 * q^86 + 68 * q^87 + 54 * q^88 - 176 * q^89 - 16 * q^90 - 3 * q^91 - 82 * q^92 + 29 * q^93 - 41 * q^94 - 90 * q^95 + 48 * q^96 - 77 * q^97 - 48 * q^98 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more