LMFDB - Embedded newform 8047.2.a.e.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, 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("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$0$
Dimension:	$168$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52416 q^{2} +3.25012 q^{3} +4.37136 q^{4} -2.90640 q^{5} -8.20382 q^{6} +1.38708 q^{7} -5.98570 q^{8} +7.56330 q^{9} +O(q^{10})$	\(q-2.52416 q^{2} +3.25012 q^{3} +4.37136 q^{4} -2.90640 q^{5} -8.20382 q^{6} +1.38708 q^{7} -5.98570 q^{8} +7.56330 q^{9} +7.33621 q^{10} +1.12664 q^{11} +14.2075 q^{12} +1.00000 q^{13} -3.50122 q^{14} -9.44616 q^{15} +6.36610 q^{16} +0.122371 q^{17} -19.0910 q^{18} -7.13279 q^{19} -12.7049 q^{20} +4.50820 q^{21} -2.84382 q^{22} +5.25877 q^{23} -19.4543 q^{24} +3.44717 q^{25} -2.52416 q^{26} +14.8313 q^{27} +6.06345 q^{28} +10.5137 q^{29} +23.8436 q^{30} +0.658633 q^{31} -4.09764 q^{32} +3.66173 q^{33} -0.308884 q^{34} -4.03142 q^{35} +33.0620 q^{36} +0.901453 q^{37} +18.0043 q^{38} +3.25012 q^{39} +17.3968 q^{40} +7.27752 q^{41} -11.3794 q^{42} +1.31961 q^{43} +4.92497 q^{44} -21.9820 q^{45} -13.2740 q^{46} +10.5106 q^{47} +20.6906 q^{48} -5.07600 q^{49} -8.70120 q^{50} +0.397722 q^{51} +4.37136 q^{52} -11.3072 q^{53} -37.4365 q^{54} -3.27448 q^{55} -8.30266 q^{56} -23.1824 q^{57} -26.5383 q^{58} +7.28355 q^{59} -41.2926 q^{60} -10.1547 q^{61} -1.66249 q^{62} +10.4909 q^{63} -2.38911 q^{64} -2.90640 q^{65} -9.24277 q^{66} -12.3477 q^{67} +0.534929 q^{68} +17.0917 q^{69} +10.1759 q^{70} +7.47815 q^{71} -45.2716 q^{72} +5.97190 q^{73} -2.27541 q^{74} +11.2037 q^{75} -31.1800 q^{76} +1.56275 q^{77} -8.20382 q^{78} +17.2787 q^{79} -18.5024 q^{80} +25.5137 q^{81} -18.3696 q^{82} -14.8272 q^{83} +19.7070 q^{84} -0.355660 q^{85} -3.33091 q^{86} +34.1709 q^{87} -6.74374 q^{88} -11.5708 q^{89} +55.4860 q^{90} +1.38708 q^{91} +22.9880 q^{92} +2.14064 q^{93} -26.5304 q^{94} +20.7307 q^{95} -13.3178 q^{96} +2.77845 q^{97} +12.8126 q^{98} +8.52114 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) $ 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * 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 $

$2$	−2.52416	−1.78485	−0.892424	−	0.451198i	$-0.850997\pi$
$2$	−2.52416	−1.78485	−0.892424	+	0.451198i	$0.850997\pi$
$3$	3.25012	1.87646	0.938230	−	0.346012i	$-0.112465\pi$
$3$	3.25012	1.87646	0.938230	+	0.346012i	$0.112465\pi$
$4$	4.37136	2.18568
$5$	−2.90640	−1.29978	−0.649891	−	0.760027i	$-0.725186\pi$
$5$	−2.90640	−1.29978	−0.649891	+	0.760027i	$0.725186\pi$
$6$	−8.20382	−3.34920
$7$	1.38708	0.524269	0.262134	−	0.965031i	$-0.415574\pi$
$7$	1.38708	0.524269	0.262134	+	0.965031i	$0.415574\pi$
$8$	−5.98570	−2.11626
$9$	7.56330	2.52110
$10$	7.33621	2.31991
$11$	1.12664	0.339696	0.169848	−	0.985470i	$-0.445672\pi$
$11$	1.12664	0.339696	0.169848	+	0.985470i	$0.445672\pi$
$12$	14.2075	4.10135
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−3.50122	−0.935740
$15$	−9.44616	−2.43899
$16$	6.36610	1.59153
$17$	0.122371	0.0296794	0.0148397	−	0.999890i	$-0.495276\pi$
$17$	0.122371	0.0296794	0.0148397	+	0.999890i	$0.495276\pi$
$18$	−19.0910	−4.49978
$19$	−7.13279	−1.63637	−0.818187	−	0.574952i	$-0.805021\pi$
$19$	−7.13279	−1.63637	−0.818187	+	0.574952i	$0.805021\pi$
$20$	−12.7049	−2.84091
$21$	4.50820	0.983769
$22$	−2.84382	−0.606305
$23$	5.25877	1.09653	0.548265	−	0.836305i	$-0.315289\pi$
$23$	5.25877	1.09653	0.548265	+	0.836305i	$0.315289\pi$
$24$	−19.4543	−3.97108
$25$	3.44717	0.689434
$26$	−2.52416	−0.495028
$27$	14.8313	2.85429
$28$	6.06345	1.14588
$29$	10.5137	1.95235	0.976175	−	0.216983i	$-0.0696216\pi$
$29$	10.5137	1.95235	0.976175	+	0.216983i	$0.0696216\pi$
$30$	23.8436	4.35323
$31$	0.658633	0.118294	0.0591470	−	0.998249i	$-0.481162\pi$
$31$	0.658633	0.118294	0.0591470	+	0.998249i	$0.481162\pi$
$32$	−4.09764	−0.724368
$33$	3.66173	0.637425
$34$	−0.308884	−0.0529732
$35$	−4.03142	−0.681435
$36$	33.0620	5.51033
$37$	0.901453	0.148198	0.0740989	−	0.997251i	$-0.476392\pi$
$37$	0.901453	0.148198	0.0740989	+	0.997251i	$0.476392\pi$
$38$	18.0043	2.92068
$39$	3.25012	0.520436
$40$	17.3968	2.75068
$41$	7.27752	1.13656	0.568279	−	0.822836i	$-0.307609\pi$
$41$	7.27752	1.13656	0.568279	+	0.822836i	$0.307609\pi$
$42$	−11.3794	−1.75588
$43$	1.31961	0.201239	0.100620	−	0.994925i	$-0.467917\pi$
$43$	1.31961	0.201239	0.100620	+	0.994925i	$0.467917\pi$
$44$	4.92497	0.742467
$45$	−21.9820	−3.27688
$46$	−13.2740	−1.95714
$47$	10.5106	1.53313	0.766563	−	0.642169i	$-0.221965\pi$
$47$	10.5106	1.53313	0.766563	+	0.642169i	$0.221965\pi$
$48$	20.6906	2.98643
$49$	−5.07600	−0.725142
$50$	−8.70120	−1.23053
$51$	0.397722	0.0556922
$52$	4.37136	0.606199
$53$	−11.3072	−1.55316	−0.776579	−	0.630020i	$-0.783047\pi$
$53$	−11.3072	−1.55316	−0.776579	+	0.630020i	$0.783047\pi$
$54$	−37.4365	−5.09447
$55$	−3.27448	−0.441530
$56$	−8.30266	−1.10949
$57$	−23.1824	−3.07059
$58$	−26.5383	−3.48465
$59$	7.28355	0.948237	0.474119	−	0.880461i	$-0.342767\pi$
$59$	7.28355	0.948237	0.474119	+	0.880461i	$0.342767\pi$
$60$	−41.2926	−5.33086
$61$	−10.1547	−1.30018	−0.650088	−	0.759859i	$-0.725268\pi$
$61$	−10.1547	−1.30018	−0.650088	+	0.759859i	$0.725268\pi$
$62$	−1.66249	−0.211137
$63$	10.4909	1.32173
$64$	−2.38911	−0.298639
$65$	−2.90640	−0.360495
$66$	−9.24277	−1.13771
$67$	−12.3477	−1.50851	−0.754254	−	0.656583i	$-0.772001\pi$
$67$	−12.3477	−1.50851	−0.754254	+	0.656583i	$0.772001\pi$
$68$	0.534929	0.0648697
$69$	17.0917	2.05759
$70$	10.1759	1.21626
$71$	7.47815	0.887493	0.443747	−	0.896152i	$-0.353649\pi$
$71$	7.47815	0.887493	0.443747	+	0.896152i	$0.353649\pi$
$72$	−45.2716	−5.33531
$73$	5.97190	0.698958	0.349479	−	0.936944i	$-0.386359\pi$
$73$	5.97190	0.698958	0.349479	+	0.936944i	$0.386359\pi$
$74$	−2.27541	−0.264511
$75$	11.2037	1.29370
$76$	−31.1800	−3.57659
$77$	1.56275	0.178092
$78$	−8.20382	−0.928900
$79$	17.2787	1.94401	0.972004	−	0.234964i	$-0.0754971\pi$
$79$	17.2787	1.94401	0.972004	+	0.234964i	$0.0754971\pi$
$80$	−18.5024	−2.06864
$81$	25.5137	2.83485
$82$	−18.3696	−2.02858
$83$	−14.8272	−1.62750	−0.813750	−	0.581215i	$-0.802577\pi$
$83$	−14.8272	−1.62750	−0.813750	+	0.581215i	$0.802577\pi$
$84$	19.7070	2.15021
$85$	−0.355660	−0.0385767
$86$	−3.33091	−0.359181
$87$	34.1709	3.66351
$88$	−6.74374	−0.718885
$89$	−11.5708	−1.22651	−0.613253	−	0.789887i	$-0.710139\pi$
$89$	−11.5708	−1.22651	−0.613253	+	0.789887i	$0.710139\pi$
$90$	55.4860	5.84874
$91$	1.38708	0.145406
$92$	22.9880	2.39667
$93$	2.14064	0.221974
$94$	−26.5304	−2.73640
$95$	20.7307	2.12693
$96$	−13.3178	−1.35925
$97$	2.77845	0.282109	0.141055	−	0.990002i	$-0.454951\pi$
$97$	2.77845	0.282109	0.141055	+	0.990002i	$0.454951\pi$
$98$	12.8126	1.29427
$99$	8.52114	0.856407
$100$	15.0688	1.50688
$101$	−18.3441	−1.82531	−0.912653	−	0.408735i	$-0.865970\pi$
$101$	−18.3441	−1.82531	−0.912653	+	0.408735i	$0.865970\pi$
$102$	−1.00391	−0.0994021
$103$	6.65365	0.655604	0.327802	−	0.944746i	$-0.393692\pi$
$103$	6.65365	0.655604	0.327802	+	0.944746i	$0.393692\pi$
$104$	−5.98570	−0.586946
$105$	−13.1026	−1.27869
$106$	28.5410	2.77215
$107$	15.7021	1.51798	0.758989	−	0.651104i	$-0.225694\pi$
$107$	15.7021	1.51798	0.758989	+	0.651104i	$0.225694\pi$
$108$	64.8330	6.23856
$109$	−3.04666	−0.291817	−0.145908	−	0.989298i	$-0.546610\pi$
$109$	−3.04666	−0.291817	−0.145908	+	0.989298i	$0.546610\pi$
$110$	8.26529	0.788064
$111$	2.92983	0.278087
$112$	8.83032	0.834387
$113$	−1.19601	−0.112511	−0.0562554	−	0.998416i	$-0.517916\pi$
$113$	−1.19601	−0.112511	−0.0562554	+	0.998416i	$0.517916\pi$
$114$	58.5161	5.48054
$115$	−15.2841	−1.42525
$116$	45.9594	4.26722
$117$	7.56330	0.699228
$118$	−18.3848	−1.69246
$119$	0.169739	0.0155600
$120$	56.5419	5.16154
$121$	−9.73068	−0.884607
$122$	25.6321	2.32062
$123$	23.6528	2.13271
$124$	2.87913	0.258553
$125$	4.51315	0.403668
$126$	−26.4808	−2.35909
$127$	−4.58375	−0.406742	−0.203371	−	0.979102i	$-0.565190\pi$
$127$	−4.58375	−0.406742	−0.203371	+	0.979102i	$0.565190\pi$
$128$	14.2258	1.25739
$129$	4.28891	0.377617
$130$	7.33621	0.643428
$131$	13.4516	1.17527	0.587637	−	0.809125i	$-0.300058\pi$
$131$	13.4516	1.17527	0.587637	+	0.809125i	$0.300058\pi$
$132$	16.0068	1.39321
$133$	−9.89378	−0.857899
$134$	31.1675	2.69246
$135$	−43.1057	−3.70995
$136$	−0.732477	−0.0628094
$137$	12.0100	1.02608	0.513041	−	0.858364i	$-0.328519\pi$
$137$	12.0100	1.02608	0.513041	+	0.858364i	$0.328519\pi$
$138$	−43.1420	−3.67249
$139$	0.503495	0.0427059	0.0213529	−	0.999772i	$-0.493203\pi$
$139$	0.503495	0.0427059	0.0213529	+	0.999772i	$0.493203\pi$
$140$	−17.6228	−1.48940
$141$	34.1607	2.87685
$142$	−18.8760	−1.58404
$143$	1.12664	0.0942146
$144$	48.1488	4.01240
$145$	−30.5571	−2.53763
$146$	−15.0740	−1.24753
$147$	−16.4976	−1.36070
$148$	3.94058	0.323914
$149$	−13.5832	−1.11278	−0.556388	−	0.830923i	$-0.687813\pi$
$149$	−13.5832	−1.11278	−0.556388	+	0.830923i	$0.687813\pi$
$150$	−28.2800	−2.30905
$151$	6.55754	0.533645	0.266823	−	0.963746i	$-0.414026\pi$
$151$	6.55754	0.533645	0.266823	+	0.963746i	$0.414026\pi$
$152$	42.6947	3.46300
$153$	0.925531	0.0748248
$154$	−3.94462	−0.317867
$155$	−1.91425	−0.153757
$156$	14.2075	1.13751
$157$	−15.6324	−1.24760	−0.623801	−	0.781583i	$-0.714413\pi$
$157$	−15.6324	−1.24760	−0.623801	+	0.781583i	$0.714413\pi$
$158$	−43.6142	−3.46976
$159$	−36.7497	−2.91444
$160$	11.9094	0.941521
$161$	7.29436	0.574876
$162$	−64.4005	−5.05978
$163$	8.19070	0.641545	0.320773	−	0.947156i	$-0.396057\pi$
$163$	8.19070	0.641545	0.320773	+	0.947156i	$0.396057\pi$
$164$	31.8127	2.48415
$165$	−10.6425	−0.828514
$166$	37.4262	2.90484
$167$	−10.3437	−0.800416	−0.400208	−	0.916424i	$-0.631062\pi$
$167$	−10.3437	−0.800416	−0.400208	+	0.916424i	$0.631062\pi$
$168$	−26.9847	−2.08191
$169$	1.00000	0.0769231
$170$	0.897741	0.0688536
$171$	−53.9475	−4.12546
$172$	5.76851	0.439845
$173$	−5.55574	−0.422395	−0.211198	−	0.977443i	$-0.567736\pi$
$173$	−5.55574	−0.422395	−0.211198	+	0.977443i	$0.567736\pi$
$174$	−86.2528	−6.53881
$175$	4.78152	0.361449
$176$	7.17232	0.540634
$177$	23.6724	1.77933
$178$	29.2066	2.18913
$179$	26.1182	1.95217	0.976084	−	0.217393i	$-0.0697553\pi$
$179$	26.1182	1.95217	0.976084	+	0.217393i	$0.0697553\pi$
$180$	−96.0913	−7.16223
$181$	18.1156	1.34652	0.673262	−	0.739404i	$-0.264893\pi$
$181$	18.1156	1.34652	0.673262	+	0.739404i	$0.264893\pi$
$182$	−3.50122	−0.259528
$183$	−33.0040	−2.43973
$184$	−31.4774	−2.32055
$185$	−2.61998	−0.192625
$186$	−5.40331	−0.396190
$187$	0.137869	0.0100820
$188$	45.9456	3.35093
$189$	20.5723	1.49641
$190$	−52.3276	−3.79625
$191$	1.09002	0.0788712	0.0394356	−	0.999222i	$-0.487444\pi$
$191$	1.09002	0.0788712	0.0394356	+	0.999222i	$0.487444\pi$
$192$	−7.76490	−0.560384
$193$	17.7339	1.27651	0.638255	−	0.769825i	$-0.279656\pi$
$193$	17.7339	1.27651	0.638255	+	0.769825i	$0.279656\pi$
$194$	−7.01325	−0.503522
$195$	−9.44616	−0.676454
$196$	−22.1890	−1.58493
$197$	2.43765	0.173675	0.0868376	−	0.996222i	$-0.472324\pi$
$197$	2.43765	0.173675	0.0868376	+	0.996222i	$0.472324\pi$
$198$	−21.5087	−1.52856
$199$	26.9978	1.91382	0.956912	−	0.290379i	$-0.0937815\pi$
$199$	26.9978	1.91382	0.956912	+	0.290379i	$0.0937815\pi$
$200$	−20.6337	−1.45902
$201$	−40.1315	−2.83066
$202$	46.3034	3.25789
$203$	14.5834	1.02356
$204$	1.73859	0.121725
$205$	−21.1514	−1.47728
$206$	−16.7949	−1.17015
$207$	39.7737	2.76446
$208$	6.36610	0.441410
$209$	−8.03610	−0.555869
$210$	33.0731	2.28226
$211$	−5.39625	−0.371493	−0.185747	−	0.982598i	$-0.559470\pi$
$211$	−5.39625	−0.371493	−0.185747	+	0.982598i	$0.559470\pi$
$212$	−49.4277	−3.39471
$213$	24.3049	1.66535
$214$	−39.6345	−2.70936
$215$	−3.83533	−0.261567
$216$	−88.7757	−6.04042
$217$	0.913580	0.0620179
$218$	7.69023	0.520848
$219$	19.4094	1.31157
$220$	−14.3139	−0.965045
$221$	0.122371	0.00823158
$222$	−7.39535	−0.496344
$223$	15.0643	1.00878	0.504389	−	0.863477i	$-0.331718\pi$
$223$	15.0643	1.00878	0.504389	+	0.863477i	$0.331718\pi$
$224$	−5.68378	−0.379763
$225$	26.0720	1.73813
$226$	3.01891	0.200815
$227$	20.3093	1.34798	0.673989	−	0.738741i	$-0.264579\pi$
$227$	20.3093	1.34798	0.673989	+	0.738741i	$0.264579\pi$
$228$	−101.339	−6.71133
$229$	−19.8740	−1.31331	−0.656655	−	0.754191i	$-0.728029\pi$
$229$	−19.8740	−1.31331	−0.656655	+	0.754191i	$0.728029\pi$
$230$	38.5795	2.54385
$231$	5.07913	0.334182
$232$	−62.9320	−4.13169
$233$	15.4757	1.01384	0.506922	−	0.861992i	$-0.330783\pi$
$233$	15.4757	1.01384	0.506922	+	0.861992i	$0.330783\pi$
$234$	−19.0910	−1.24802
$235$	−30.5480	−1.99273
$236$	31.8390	2.07255
$237$	56.1580	3.64785
$238$	−0.428448	−0.0277722
$239$	−12.1800	−0.787856	−0.393928	−	0.919141i	$-0.628884\pi$
$239$	−12.1800	−0.787856	−0.393928	+	0.919141i	$0.628884\pi$
$240$	−60.1352	−3.88171
$241$	0.602245	0.0387940	0.0193970	−	0.999812i	$-0.493825\pi$
$241$	0.602245	0.0387940	0.0193970	+	0.999812i	$0.493825\pi$
$242$	24.5617	1.57889
$243$	38.4287	2.46520
$244$	−44.3899	−2.84177
$245$	14.7529	0.942527
$246$	−59.7035	−3.80656
$247$	−7.13279	−0.453848
$248$	−3.94238	−0.250341
$249$	−48.1903	−3.05394
$250$	−11.3919	−0.720486
$251$	6.53550	0.412517	0.206259	−	0.978498i	$-0.433871\pi$
$251$	6.53550	0.412517	0.206259	+	0.978498i	$0.433871\pi$
$252$	45.8597	2.88889
$253$	5.92476	0.372486
$254$	11.5701	0.725973
$255$	−1.15594	−0.0723877
$256$	−31.1299	−1.94562
$257$	−11.5205	−0.718629	−0.359314	−	0.933217i	$-0.616989\pi$
$257$	−11.5205	−0.718629	−0.359314	+	0.933217i	$0.616989\pi$
$258$	−10.8259	−0.673990
$259$	1.25039	0.0776955
$260$	−12.7049	−0.787927
$261$	79.5186	4.92207
$262$	−33.9540	−2.09769
$263$	−22.3998	−1.38123	−0.690616	−	0.723221i	$-0.742661\pi$
$263$	−22.3998	−1.38123	−0.690616	+	0.723221i	$0.742661\pi$
$264$	−21.9180	−1.34896
$265$	32.8631	2.01877
$266$	24.9734	1.53122
$267$	−37.6066	−2.30149
$268$	−53.9762	−3.29712
$269$	9.82189	0.598851	0.299426	−	0.954120i	$-0.403205\pi$
$269$	9.82189	0.598851	0.299426	+	0.954120i	$0.403205\pi$
$270$	108.806	6.62170
$271$	15.6826	0.952653	0.476326	−	0.879269i	$-0.341968\pi$
$271$	15.6826	0.952653	0.476326	+	0.879269i	$0.341968\pi$
$272$	0.779028	0.0472355
$273$	4.50820	0.272848
$274$	−30.3151	−1.83140
$275$	3.88373	0.234198
$276$	74.7139	4.49725
$277$	−32.4662	−1.95071	−0.975354	−	0.220648i	$-0.929183\pi$
$277$	−32.4662	−1.95071	−0.975354	+	0.220648i	$0.929183\pi$
$278$	−1.27090	−0.0762235
$279$	4.98145	0.298231
$280$	24.1309	1.44210
$281$	12.6778	0.756295	0.378147	−	0.925745i	$-0.376561\pi$
$281$	12.6778	0.756295	0.378147	+	0.925745i	$0.376561\pi$
$282$	−86.2269	−5.13474
$283$	21.6337	1.28599	0.642994	−	0.765871i	$-0.277692\pi$
$283$	21.6337	1.28599	0.642994	+	0.765871i	$0.277692\pi$
$284$	32.6897	1.93978
$285$	67.3775	3.99110
$286$	−2.84382	−0.168159
$287$	10.0945	0.595862
$288$	−30.9917	−1.82621
$289$	−16.9850	−0.999119
$290$	77.1310	4.52929
$291$	9.03032	0.529367
$292$	26.1054	1.52770
$293$	9.87993	0.577192	0.288596	−	0.957451i	$-0.406812\pi$
$293$	9.87993	0.577192	0.288596	+	0.957451i	$0.406812\pi$
$294$	41.6426	2.42864
$295$	−21.1689	−1.23250
$296$	−5.39582	−0.313626
$297$	16.7096	0.969588
$298$	34.2860	1.98613
$299$	5.25877	0.304123
$300$	48.9756	2.82761
$301$	1.83042	0.105503
$302$	−16.5523	−0.952475
$303$	−59.6206	−3.42511
$304$	−45.4081	−2.60433
$305$	29.5136	1.68995
$306$	−2.33619	−0.133551
$307$	−14.7751	−0.843258	−0.421629	−	0.906768i	$-0.638542\pi$
$307$	−14.7751	−0.843258	−0.421629	+	0.906768i	$0.638542\pi$
$308$	6.83134	0.389252
$309$	21.6252	1.23021
$310$	4.83187	0.274432
$311$	8.27663	0.469325	0.234662	−	0.972077i	$-0.424602\pi$
$311$	8.27663	0.469325	0.234662	+	0.972077i	$0.424602\pi$
$312$	−19.4543	−1.10138
$313$	2.42489	0.137063	0.0685314	−	0.997649i	$-0.478169\pi$
$313$	2.42489	0.137063	0.0685314	+	0.997649i	$0.478169\pi$
$314$	39.4587	2.22678
$315$	−30.4909	−1.71797
$316$	75.5316	4.24899
$317$	−5.59285	−0.314126	−0.157063	−	0.987589i	$-0.550202\pi$
$317$	−5.59285	−0.314126	−0.157063	+	0.987589i	$0.550202\pi$
$318$	92.7619	5.20183
$319$	11.8452	0.663205
$320$	6.94371	0.388165
$321$	51.0337	2.84842
$322$	−18.4121	−1.02607
$323$	−0.872848	−0.0485666
$324$	111.530	6.19609
$325$	3.44717	0.191215
$326$	−20.6746	−1.14506
$327$	−9.90201	−0.547582
$328$	−43.5610	−2.40526
$329$	14.5791	0.803770
$330$	26.8632	1.47877
$331$	26.0614	1.43247	0.716233	−	0.697862i	$-0.245865\pi$
$331$	26.0614	1.43247	0.716233	+	0.697862i	$0.245865\pi$
$332$	−64.8152	−3.55720
$333$	6.81796	0.373622
$334$	26.1090	1.42862
$335$	35.8873	1.96073
$336$	28.6996	1.56569
$337$	−22.6970	−1.23638	−0.618192	−	0.786027i	$-0.712134\pi$
$337$	−22.6970	−1.23638	−0.618192	+	0.786027i	$0.712134\pi$
$338$	−2.52416	−0.137296
$339$	−3.88717	−0.211122
$340$	−1.55472	−0.0843165
$341$	0.742045	0.0401840
$342$	136.172	7.36333
$343$	−16.7504	−0.904438
$344$	−7.89881	−0.425875
$345$	−49.6752	−2.67442
$346$	14.0236	0.753911
$347$	−8.71133	−0.467648	−0.233824	−	0.972279i	$-0.575124\pi$
$347$	−8.71133	−0.467648	−0.233824	+	0.972279i	$0.575124\pi$
$348$	149.374	8.00727
$349$	−25.6005	−1.37036	−0.685182	−	0.728372i	$-0.740277\pi$
$349$	−25.6005	−1.37036	−0.685182	+	0.728372i	$0.740277\pi$
$350$	−12.0693	−0.645131
$351$	14.8313	0.791637
$352$	−4.61658	−0.246065
$353$	35.4457	1.88659	0.943293	−	0.331961i	$-0.107710\pi$
$353$	35.4457	1.88659	0.943293	+	0.331961i	$0.107710\pi$
$354$	−59.7529	−3.17583
$355$	−21.7345	−1.15355
$356$	−50.5803	−2.68075
$357$	0.551674	0.0291977
$358$	−65.9265	−3.48432
$359$	15.7547	0.831501	0.415751	−	0.909479i	$-0.363519\pi$
$359$	15.7547	0.831501	0.415751	+	0.909479i	$0.363519\pi$
$360$	131.578	6.93475
$361$	31.8767	1.67772
$362$	−45.7267	−2.40334
$363$	−31.6259	−1.65993
$364$	6.06345	0.317811
$365$	−17.3567	−0.908493
$366$	83.3074	4.35455
$367$	14.0771	0.734820	0.367410	−	0.930059i	$-0.380245\pi$
$367$	14.0771	0.734820	0.367410	+	0.930059i	$0.380245\pi$
$368$	33.4779	1.74515
$369$	55.0421	2.86538
$370$	6.61325	0.343806
$371$	−15.6840	−0.814272
$372$	9.35752	0.485165
$373$	9.97694	0.516586	0.258293	−	0.966067i	$-0.416840\pi$
$373$	9.97694	0.516586	0.258293	+	0.966067i	$0.416840\pi$
$374$	−0.348002	−0.0179948
$375$	14.6683	0.757467
$376$	−62.9132	−3.24450
$377$	10.5137	0.541485
$378$	−51.9276	−2.67087
$379$	6.50084	0.333926	0.166963	−	0.985963i	$-0.446604\pi$
$379$	6.50084	0.333926	0.166963	+	0.985963i	$0.446604\pi$
$380$	90.6217	4.64879
$381$	−14.8978	−0.763235
$382$	−2.75138	−0.140773
$383$	−22.6493	−1.15733	−0.578663	−	0.815567i	$-0.696425\pi$
$383$	−22.6493	−1.15733	−0.578663	+	0.815567i	$0.696425\pi$
$384$	46.2355	2.35945
$385$	−4.54197	−0.231480
$386$	−44.7630	−2.27838
$387$	9.98064	0.507345
$388$	12.1456	0.616601
$389$	−12.5525	−0.636438	−0.318219	−	0.948017i	$-0.603085\pi$
$389$	−12.5525	−0.636438	−0.318219	+	0.948017i	$0.603085\pi$
$390$	23.8436	1.20737
$391$	0.643523	0.0325443
$392$	30.3834	1.53459
$393$	43.7194	2.20535
$394$	−6.15300	−0.309984
$395$	−50.2189	−2.52679
$396$	37.2490	1.87183
$397$	−10.2732	−0.515598	−0.257799	−	0.966199i	$-0.582997\pi$
$397$	−10.2732	−0.515598	−0.257799	+	0.966199i	$0.582997\pi$
$398$	−68.1467	−3.41588
$399$	−32.1560	−1.60981
$400$	21.9450	1.09725
$401$	25.0249	1.24968	0.624842	−	0.780751i	$-0.285163\pi$
$401$	25.0249	1.24968	0.624842	+	0.780751i	$0.285163\pi$
$402$	101.298	5.05229
$403$	0.658633	0.0328089
$404$	−80.1888	−3.98954
$405$	−74.1530	−3.68469
$406$	−36.8109	−1.82689
$407$	1.01562	0.0503422
$408$	−2.38064	−0.117859
$409$	27.9903	1.38403	0.692016	−	0.721882i	$-0.256723\pi$
$409$	27.9903	1.38403	0.692016	+	0.721882i	$0.256723\pi$
$410$	53.3894	2.63672
$411$	39.0339	1.92540
$412$	29.0855	1.43294
$413$	10.1029	0.497131
$414$	−100.395	−4.93415
$415$	43.0939	2.11540
$416$	−4.09764	−0.200904
$417$	1.63642	0.0801359
$418$	20.2844	0.992142
$419$	−9.47399	−0.462835	−0.231417	−	0.972855i	$-0.574336\pi$
$419$	−9.47399	−0.462835	−0.231417	+	0.972855i	$0.574336\pi$
$420$	−57.2764	−2.79480
$421$	−26.6819	−1.30040	−0.650198	−	0.759765i	$-0.725314\pi$
$421$	−26.6819	−1.30040	−0.650198	+	0.759765i	$0.725314\pi$
$422$	13.6210	0.663059
$423$	79.4947	3.86517
$424$	67.6812	3.28689
$425$	0.421835	0.0204620
$426$	−61.3494	−2.97239
$427$	−14.0854	−0.681642
$428$	68.6395	3.31782
$429$	3.66173	0.176790
$430$	9.68097	0.466858
$431$	37.7622	1.81894	0.909471	−	0.415768i	$-0.136487\pi$
$431$	37.7622	1.81894	0.909471	+	0.415768i	$0.136487\pi$
$432$	94.4176	4.54267
$433$	30.0938	1.44622	0.723108	−	0.690735i	$-0.242713\pi$
$433$	30.0938	1.44622	0.723108	+	0.690735i	$0.242713\pi$
$434$	−2.30602	−0.110692
$435$	−99.3144	−4.76176
$436$	−13.3180	−0.637819
$437$	−37.5097	−1.79433
$438$	−48.9924	−2.34095
$439$	−23.6057	−1.12664	−0.563320	−	0.826239i	$-0.690476\pi$
$439$	−23.6057	−1.12664	−0.563320	+	0.826239i	$0.690476\pi$
$440$	19.6000	0.934394
$441$	−38.3913	−1.82816
$442$	−0.308884	−0.0146921
$443$	7.03031	0.334020	0.167010	−	0.985955i	$-0.446589\pi$
$443$	7.03031	0.334020	0.167010	+	0.985955i	$0.446589\pi$
$444$	12.8074	0.607811
$445$	33.6295	1.59419
$446$	−38.0246	−1.80051
$447$	−44.1469	−2.08808
$448$	−3.31390	−0.156567
$449$	−16.3122	−0.769821	−0.384911	−	0.922954i	$-0.625768\pi$
$449$	−16.3122	−0.769821	−0.384911	+	0.922954i	$0.625768\pi$
$450$	−65.8098	−3.10230
$451$	8.19917	0.386084
$452$	−5.22818	−0.245913
$453$	21.3128	1.00136
$454$	−51.2640	−2.40594
$455$	−4.03142	−0.188996
$456$	138.763	6.49818
$457$	−15.0140	−0.702325	−0.351162	−	0.936315i	$-0.614213\pi$
$457$	−15.0140	−0.702325	−0.351162	+	0.936315i	$0.614213\pi$
$458$	50.1650	2.34406
$459$	1.81493	0.0847135
$460$	−66.8124	−3.11514
$461$	1.40565	0.0654677	0.0327339	−	0.999464i	$-0.489579\pi$
$461$	1.40565	0.0654677	0.0327339	+	0.999464i	$0.489579\pi$
$462$	−12.8205	−0.596464
$463$	5.13023	0.238422	0.119211	−	0.992869i	$-0.461963\pi$
$463$	5.13023	0.238422	0.119211	+	0.992869i	$0.461963\pi$
$464$	66.9315	3.10722
$465$	−6.22156	−0.288518
$466$	−39.0630	−1.80956
$467$	−25.4023	−1.17548	−0.587739	−	0.809051i	$-0.699982\pi$
$467$	−25.4023	−1.17548	−0.587739	+	0.809051i	$0.699982\pi$
$468$	33.0620	1.52829
$469$	−17.1273	−0.790863
$470$	77.1079	3.55672
$471$	−50.8073	−2.34108
$472$	−43.5971	−2.00672
$473$	1.48673	0.0683601
$474$	−141.752	−6.51086
$475$	−24.5879	−1.12817
$476$	0.741992	0.0340092
$477$	−85.5195	−3.91567
$478$	30.7441	1.40620
$479$	19.4456	0.888490	0.444245	−	0.895905i	$-0.353472\pi$
$479$	19.4456	0.888490	0.444245	+	0.895905i	$0.353472\pi$
$480$	38.7070	1.76673
$481$	0.901453	0.0411027
$482$	−1.52016	−0.0692414
$483$	23.7076	1.07873
$484$	−42.5363	−1.93347
$485$	−8.07530	−0.366681
$486$	−96.9999	−4.40001
$487$	1.35883	0.0615744	0.0307872	−	0.999526i	$-0.490199\pi$
$487$	1.35883	0.0615744	0.0307872	+	0.999526i	$0.490199\pi$
$488$	60.7830	2.75152
$489$	26.6208	1.20383
$490$	−37.2386	−1.68227
$491$	−43.7923	−1.97632	−0.988160	−	0.153426i	$-0.950969\pi$
$491$	−43.7923	−1.97632	−0.988160	+	0.153426i	$0.950969\pi$
$492$	103.395	4.66142
$493$	1.28658	0.0579446
$494$	18.0043	0.810051
$495$	−24.7659	−1.11314
$496$	4.19293	0.188268
$497$	10.3728	0.465285
$498$	121.640	5.45082
$499$	7.94505	0.355669	0.177835	−	0.984060i	$-0.443091\pi$
$499$	7.94505	0.355669	0.177835	+	0.984060i	$0.443091\pi$
$500$	19.7286	0.882291
$501$	−33.6182	−1.50195
$502$	−16.4966	−0.736280
$503$	−18.9801	−0.846280	−0.423140	−	0.906064i	$-0.639072\pi$
$503$	−18.9801	−0.846280	−0.423140	+	0.906064i	$0.639072\pi$
$504$	−62.7956	−2.79714
$505$	53.3153	2.37250
$506$	−14.9550	−0.664831
$507$	3.25012	0.144343
$508$	−20.0372	−0.889009
$509$	29.5819	1.31119	0.655597	−	0.755111i	$-0.272417\pi$
$509$	29.5819	1.31119	0.655597	+	0.755111i	$0.272417\pi$
$510$	2.91777	0.129201
$511$	8.28353	0.366442
$512$	50.1251	2.21524
$513$	−105.789	−4.67068
$514$	29.0795	1.28264
$515$	−19.3382	−0.852142
$516$	18.7484	0.825352
$517$	11.8417	0.520796
$518$	−3.15618	−0.138675
$519$	−18.0568	−0.792607
$520$	17.3968	0.762902
$521$	40.7491	1.78525	0.892626	−	0.450798i	$-0.148860\pi$
$521$	40.7491	1.78525	0.892626	+	0.450798i	$0.148860\pi$
$522$	−200.717	−8.78516
$523$	19.9266	0.871331	0.435665	−	0.900109i	$-0.356513\pi$
$523$	19.9266	0.871331	0.435665	+	0.900109i	$0.356513\pi$
$524$	58.8020	2.56878
$525$	15.5405	0.678244
$526$	56.5407	2.46529
$527$	0.0805978	0.00351090
$528$	23.3109	1.01448
$529$	4.65468	0.202378
$530$	−82.9517	−3.60319
$531$	55.0877	2.39060
$532$	−43.2493	−1.87510
$533$	7.27752	0.315224
$534$	94.9250	4.10781
$535$	−45.6366	−1.97304
$536$	73.9094	3.19240
$537$	84.8875	3.66317
$538$	−24.7920	−1.06886
$539$	−5.71884	−0.246328
$540$	−188.431	−8.10877
$541$	22.6725	0.974769	0.487384	−	0.873188i	$-0.337951\pi$
$541$	22.6725	0.974769	0.487384	+	0.873188i	$0.337951\pi$
$542$	−39.5854	−1.70034
$543$	58.8780	2.52670
$544$	−0.501434	−0.0214988
$545$	8.85480	0.379298
$546$	−11.3794	−0.486993
$547$	11.7228	0.501232	0.250616	−	0.968087i	$-0.419367\pi$
$547$	11.7228	0.501232	0.250616	+	0.968087i	$0.419367\pi$
$548$	52.5000	2.24269
$549$	−76.8031	−3.27788
$550$	−9.80314	−0.418007
$551$	−74.9922	−3.19478
$552$	−102.305	−4.35441
$553$	23.9670	1.01918
$554$	81.9499	3.48172
$555$	−8.51527	−0.361453
$556$	2.20096	0.0933415
$557$	−16.4818	−0.698354	−0.349177	−	0.937057i	$-0.613539\pi$
$557$	−16.4818	−0.698354	−0.349177	+	0.937057i	$0.613539\pi$
$558$	−12.5739	−0.532298
$559$	1.31961	0.0558137
$560$	−25.6645	−1.08452
$561$	0.448090	0.0189184
$562$	−32.0008	−1.34987
$563$	44.8278	1.88926	0.944632	−	0.328130i	$-0.106419\pi$
$563$	44.8278	1.88926	0.944632	+	0.328130i	$0.106419\pi$
$564$	149.329	6.28788
$565$	3.47608	0.146240
$566$	−54.6068	−2.29529
$567$	35.3896	1.48622
$568$	−44.7619	−1.87817
$569$	−4.23889	−0.177703	−0.0888517	−	0.996045i	$-0.528320\pi$
$569$	−4.23889	−0.177703	−0.0888517	+	0.996045i	$0.528320\pi$
$570$	−170.071	−7.12350
$571$	−22.6739	−0.948873	−0.474436	−	0.880290i	$-0.657348\pi$
$571$	−22.6739	−0.948873	−0.474436	+	0.880290i	$0.657348\pi$
$572$	4.92497	0.205923
$573$	3.54271	0.147999
$574$	−25.4802	−1.06352
$575$	18.1279	0.755985
$576$	−18.0696	−0.752899
$577$	17.8773	0.744240	0.372120	−	0.928185i	$-0.378631\pi$
$577$	17.8773	0.744240	0.372120	+	0.928185i	$0.378631\pi$
$578$	42.8729	1.78328
$579$	57.6372	2.39532
$580$	−133.576	−5.54646
$581$	−20.5666	−0.853247
$582$	−22.7939	−0.944839
$583$	−12.7391	−0.527601
$584$	−35.7460	−1.47918
$585$	−21.9820	−0.908844
$586$	−24.9385	−1.03020
$587$	1.46996	0.0606719	0.0303359	−	0.999540i	$-0.490342\pi$
$587$	1.46996	0.0606719	0.0303359	+	0.999540i	$0.490342\pi$
$588$	−72.1171	−2.97406
$589$	−4.69789	−0.193573
$590$	53.4336	2.19983
$591$	7.92265	0.325894
$592$	5.73874	0.235861
$593$	−8.31249	−0.341353	−0.170676	−	0.985327i	$-0.554595\pi$
$593$	−8.31249	−0.341353	−0.170676	+	0.985327i	$0.554595\pi$
$594$	−42.1776	−1.73057
$595$	−0.493330	−0.0202246
$596$	−59.3769	−2.43217
$597$	87.7462	3.59121
$598$	−13.2740	−0.542813
$599$	−16.0348	−0.655165	−0.327583	−	0.944823i	$-0.606234\pi$
$599$	−16.0348	−0.655165	−0.327583	+	0.944823i	$0.606234\pi$
$600$	−67.0621	−2.73780
$601$	−19.9680	−0.814511	−0.407255	−	0.913314i	$-0.633514\pi$
$601$	−19.9680	−0.814511	−0.407255	+	0.913314i	$0.633514\pi$
$602$	−4.62025	−0.188308
$603$	−93.3892	−3.80310
$604$	28.6654	1.16638
$605$	28.2813	1.14980
$606$	150.492	6.11331
$607$	8.24107	0.334495	0.167247	−	0.985915i	$-0.446512\pi$
$607$	8.24107	0.334495	0.167247	+	0.985915i	$0.446512\pi$
$608$	29.2276	1.18534
$609$	47.3980	1.92066
$610$	−74.4971	−3.01630
$611$	10.5106	0.425213
$612$	4.04583	0.163543
$613$	−19.7577	−0.798007	−0.399004	−	0.916949i	$-0.630644\pi$
$613$	−19.7577	−0.798007	−0.399004	+	0.916949i	$0.630644\pi$
$614$	37.2946	1.50509
$615$	−68.7447	−2.77205
$616$	−9.35414	−0.376889
$617$	27.3037	1.09920	0.549602	−	0.835427i	$-0.314779\pi$
$617$	27.3037	1.09920	0.549602	+	0.835427i	$0.314779\pi$
$618$	−54.5854	−2.19575
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−8.36790	−0.336063
$621$	77.9945	3.12981
$622$	−20.8915	−0.837673
$623$	−16.0497	−0.643018
$624$	20.6906	0.828288
$625$	−30.3529	−1.21411
$626$	−6.12080	−0.244636
$627$	−26.1183	−1.04307
$628$	−68.3350	−2.72686
$629$	0.110312	0.00439842
$630$	76.9638	3.06631
$631$	2.21475	0.0881679	0.0440840	−	0.999028i	$-0.485963\pi$
$631$	2.21475	0.0881679	0.0440840	+	0.999028i	$0.485963\pi$
$632$	−103.425	−4.11403
$633$	−17.5385	−0.697092
$634$	14.1172	0.560666
$635$	13.3222	0.528676
$636$	−160.646	−6.37004
$637$	−5.07600	−0.201118
$638$	−29.8992	−1.18372
$639$	56.5595	2.23746
$640$	−41.3458	−1.63434
$641$	−4.26615	−0.168503	−0.0842513	−	0.996445i	$-0.526850\pi$
$641$	−4.26615	−0.168503	−0.0842513	+	0.996445i	$0.526850\pi$
$642$	−128.817	−5.08400
$643$	2.43295	0.0959460	0.0479730	−	0.998849i	$-0.484724\pi$
$643$	2.43295	0.0959460	0.0479730	+	0.998849i	$0.484724\pi$
$644$	31.8863	1.25650
$645$	−12.4653	−0.490820
$646$	2.20321	0.0866840
$647$	25.3429	0.996333	0.498166	−	0.867081i	$-0.334007\pi$
$647$	25.3429	0.996333	0.498166	+	0.867081i	$0.334007\pi$
$648$	−152.717	−5.99929
$649$	8.20596	0.322112
$650$	−8.70120	−0.341289
$651$	2.96925	0.116374
$652$	35.8045	1.40221
$653$	20.1332	0.787874	0.393937	−	0.919137i	$-0.371113\pi$
$653$	20.1332	0.787874	0.393937	+	0.919137i	$0.371113\pi$
$654$	24.9942	0.977351
$655$	−39.0958	−1.52760
$656$	46.3294	1.80886
$657$	45.1673	1.76214
$658$	−36.7998	−1.43461
$659$	5.49551	0.214075	0.107037	−	0.994255i	$-0.465864\pi$
$659$	5.49551	0.214075	0.107037	+	0.994255i	$0.465864\pi$
$660$	−46.5220	−1.81087
$661$	−36.8687	−1.43403	−0.717013	−	0.697060i	$-0.754491\pi$
$661$	−36.8687	−1.43403	−0.717013	+	0.697060i	$0.754491\pi$
$662$	−65.7831	−2.55673
$663$	0.397722	0.0154462
$664$	88.7513	3.44422
$665$	28.7553	1.11508
$666$	−17.2096	−0.666858
$667$	55.2893	2.14081
$668$	−45.2159	−1.74946
$669$	48.9607	1.89293
$670$	−90.5851	−3.49961
$671$	−11.4407	−0.441664
$672$	−18.4730	−0.712611
$673$	−31.8686	−1.22844	−0.614222	−	0.789133i	$-0.710530\pi$
$673$	−31.8686	−1.22844	−0.614222	+	0.789133i	$0.710530\pi$
$674$	57.2907	2.20676
$675$	51.1260	1.96784
$676$	4.37136	0.168129
$677$	39.5991	1.52192	0.760959	−	0.648799i	$-0.224729\pi$
$677$	39.5991	1.52192	0.760959	+	0.648799i	$0.224729\pi$
$678$	9.81182	0.376821
$679$	3.85395	0.147901
$680$	2.12887	0.0816385
$681$	66.0079	2.52943
$682$	−1.87304	−0.0717223
$683$	−9.90598	−0.379042	−0.189521	−	0.981877i	$-0.560693\pi$
$683$	−9.90598	−0.379042	−0.189521	+	0.981877i	$0.560693\pi$
$684$	−235.824	−9.01696
$685$	−34.9058	−1.33368
$686$	42.2807	1.61428
$687$	−64.5929	−2.46437
$688$	8.40080	0.320277
$689$	−11.3072	−0.430768
$690$	125.388	4.77344
$691$	23.8222	0.906240	0.453120	−	0.891449i	$-0.350311\pi$
$691$	23.8222	0.906240	0.453120	+	0.891449i	$0.350311\pi$
$692$	−24.2862	−0.923222
$693$	11.8195	0.448987
$694$	21.9887	0.834681
$695$	−1.46336	−0.0555084
$696$	−204.537	−7.75295
$697$	0.890559	0.0337323
$698$	64.6197	2.44589
$699$	50.2978	1.90244
$700$	20.9017	0.790012
$701$	−45.9235	−1.73451	−0.867254	−	0.497866i	$-0.834117\pi$
$701$	−45.9235	−1.73451	−0.867254	+	0.497866i	$0.834117\pi$
$702$	−37.4365	−1.41295
$703$	−6.42987	−0.242507
$704$	−2.69167	−0.101446
$705$	−99.2847	−3.73928
$706$	−89.4706	−3.36727
$707$	−25.4448	−0.956951
$708$	103.481	3.88905
$709$	−7.51065	−0.282068	−0.141034	−	0.990005i	$-0.545043\pi$
$709$	−7.51065	−0.282068	−0.141034	+	0.990005i	$0.545043\pi$
$710$	54.8613	2.05891
$711$	130.684	4.90104
$712$	69.2594	2.59561
$713$	3.46360	0.129713
$714$	−1.39251	−0.0521134
$715$	−3.27448	−0.122458
$716$	114.172	4.26682
$717$	−39.5864	−1.47838
$718$	−39.7673	−1.48410
$719$	9.69615	0.361605	0.180803	−	0.983519i	$-0.442130\pi$
$719$	9.69615	0.361605	0.180803	+	0.983519i	$0.442130\pi$
$720$	−139.940	−5.21524
$721$	9.22918	0.343713
$722$	−80.4617	−2.99447
$723$	1.95737	0.0727954
$724$	79.1900	2.94307
$725$	36.2426	1.34602
$726$	79.8287	2.96272
$727$	44.7735	1.66056	0.830279	−	0.557348i	$-0.188181\pi$
$727$	44.7735	1.66056	0.830279	+	0.557348i	$0.188181\pi$
$728$	−8.30266	−0.307717
$729$	48.3569	1.79100
$730$	43.8111	1.62152
$731$	0.161483	0.00597266
$732$	−144.273	−5.33247
$733$	9.27231	0.342481	0.171240	−	0.985229i	$-0.445223\pi$
$733$	9.27231	0.342481	0.171240	+	0.985229i	$0.445223\pi$
$734$	−35.5328	−1.31154
$735$	47.9487	1.76861
$736$	−21.5486	−0.794291
$737$	−13.9114	−0.512434
$738$	−138.935	−5.11426
$739$	11.8021	0.434149	0.217074	−	0.976155i	$-0.430349\pi$
$739$	11.8021	0.434149	0.217074	+	0.976155i	$0.430349\pi$
$740$	−11.4529	−0.421017
$741$	−23.1824	−0.851628
$742$	39.5888	1.45335
$743$	−21.0245	−0.771313	−0.385657	−	0.922642i	$-0.626025\pi$
$743$	−21.0245	−0.771313	−0.385657	+	0.922642i	$0.626025\pi$
$744$	−12.8132	−0.469755
$745$	39.4781	1.44637
$746$	−25.1834	−0.922028
$747$	−112.143	−4.10309
$748$	0.602674	0.0220360
$749$	21.7801	0.795828
$750$	−37.0251	−1.35196
$751$	−8.85213	−0.323019	−0.161509	−	0.986871i	$-0.551636\pi$
$751$	−8.85213	−0.323019	−0.161509	+	0.986871i	$0.551636\pi$
$752$	66.9114	2.44001
$753$	21.2412	0.774072
$754$	−26.5383	−0.966468
$755$	−19.0588	−0.693622
$756$	89.9289	3.27068
$757$	−40.1345	−1.45871	−0.729357	−	0.684134i	$-0.760180\pi$
$757$	−40.1345	−1.45871	−0.729357	+	0.684134i	$0.760180\pi$
$758$	−16.4091	−0.596007
$759$	19.2562	0.698956
$760$	−124.088	−4.50114
$761$	13.3232	0.482966	0.241483	−	0.970405i	$-0.422366\pi$
$761$	13.3232	0.482966	0.241483	+	0.970405i	$0.422366\pi$
$762$	37.6043	1.36226
$763$	−4.22597	−0.152990
$764$	4.76488	0.172387
$765$	−2.68996	−0.0972559
$766$	57.1704	2.06565
$767$	7.28355	0.262994
$768$	−101.176	−3.65087
$769$	−29.1423	−1.05090	−0.525450	−	0.850825i	$-0.676103\pi$
$769$	−29.1423	−1.05090	−0.525450	+	0.850825i	$0.676103\pi$
$770$	11.4647	0.413157
$771$	−37.4430	−1.34848
$772$	77.5212	2.79005
$773$	2.75657	0.0991471	0.0495735	−	0.998770i	$-0.484214\pi$
$773$	2.75657	0.0991471	0.0495735	+	0.998770i	$0.484214\pi$
$774$	−25.1927	−0.905533
$775$	2.27042	0.0815559
$776$	−16.6310	−0.597017
$777$	4.06392	0.145792
$778$	31.6845	1.13595
$779$	−51.9090	−1.85983
$780$	−41.2926	−1.47851
$781$	8.42520	0.301477
$782$	−1.62435	−0.0580867
$783$	155.932	5.57257
$784$	−32.3143	−1.15408
$785$	45.4341	1.62161
$786$	−110.355	−3.93622
$787$	28.1499	1.00343	0.501717	−	0.865032i	$-0.332702\pi$
$787$	28.1499	1.00343	0.501717	+	0.865032i	$0.332702\pi$
$788$	10.6558	0.379599
$789$	−72.8022	−2.59183
$790$	126.760	4.50993
$791$	−1.65896	−0.0589859
$792$	−51.0050	−1.81238
$793$	−10.1547	−0.360604
$794$	25.9312	0.920264
$795$	106.809	3.78814
$796$	118.017	4.18301
$797$	−36.3891	−1.28897	−0.644484	−	0.764618i	$-0.722928\pi$
$797$	−36.3891	−1.28897	−0.644484	+	0.764618i	$0.722928\pi$
$798$	81.1668	2.87327
$799$	1.28619	0.0455022
$800$	−14.1253	−0.499404
$801$	−87.5137	−3.09214
$802$	−63.1668	−2.23050
$803$	6.72820	0.237433
$804$	−175.429	−6.18691
$805$	−21.2003	−0.747214
$806$	−1.66249	−0.0585588
$807$	31.9224	1.12372
$808$	109.802	3.86283
$809$	−24.7857	−0.871418	−0.435709	−	0.900088i	$-0.643502\pi$
$809$	−24.7857	−0.871418	−0.435709	+	0.900088i	$0.643502\pi$
$810$	187.174	6.57661
$811$	−11.9360	−0.419130	−0.209565	−	0.977795i	$-0.567205\pi$
$811$	−11.9360	−0.419130	−0.209565	+	0.977795i	$0.567205\pi$
$812$	63.7495	2.23717
$813$	50.9705	1.78761
$814$	−2.56357	−0.0898531
$815$	−23.8055	−0.833869
$816$	2.53194	0.0886355
$817$	−9.41253	−0.329303
$818$	−70.6520	−2.47029
$819$	10.4909	0.366583
$820$	−92.4605	−3.22886
$821$	−43.9054	−1.53231	−0.766155	−	0.642656i	$-0.777832\pi$
$821$	−43.9054	−1.53231	−0.766155	+	0.642656i	$0.777832\pi$
$822$	−98.5277	−3.43655
$823$	34.2598	1.19422	0.597111	−	0.802158i	$-0.296315\pi$
$823$	34.2598	1.19422	0.597111	+	0.802158i	$0.296315\pi$
$824$	−39.8267	−1.38743
$825$	12.6226	0.439463
$826$	−25.5013	−0.887303
$827$	−57.1888	−1.98865	−0.994325	−	0.106388i	$-0.966071\pi$
$827$	−57.1888	−1.98865	−0.994325	+	0.106388i	$0.966071\pi$
$828$	173.865	6.04224
$829$	−3.12127	−0.108406	−0.0542031	−	0.998530i	$-0.517262\pi$
$829$	−3.12127	−0.108406	−0.0542031	+	0.998530i	$0.517262\pi$
$830$	−108.776	−3.77566
$831$	−105.519	−3.66042
$832$	−2.38911	−0.0828275
$833$	−0.621156	−0.0215218
$834$	−4.13058	−0.143030
$835$	30.0628	1.04037
$836$	−35.1287	−1.21495
$837$	9.76839	0.337645
$838$	23.9138	0.826089
$839$	−33.5773	−1.15922	−0.579608	−	0.814895i	$-0.696794\pi$
$839$	−33.5773	−1.15922	−0.579608	+	0.814895i	$0.696794\pi$
$840$	78.4283	2.70603
$841$	81.5385	2.81167
$842$	67.3492	2.32101
$843$	41.2045	1.41916
$844$	−23.5890	−0.811966
$845$	−2.90640	−0.0999833
$846$	−200.657	−6.89874
$847$	−13.4973	−0.463772
$848$	−71.9825	−2.47189
$849$	70.3121	2.41311
$850$	−1.06478	−0.0365215
$851$	4.74053	0.162503
$852$	106.246	3.63992
$853$	24.6827	0.845119	0.422559	−	0.906335i	$-0.361132\pi$
$853$	24.6827	0.845119	0.422559	+	0.906335i	$0.361132\pi$
$854$	35.5538	1.21663
$855$	156.793	5.36221
$856$	−93.9879	−3.21244
$857$	−2.33120	−0.0796322	−0.0398161	−	0.999207i	$-0.512677\pi$
$857$	−2.33120	−0.0796322	−0.0398161	+	0.999207i	$0.512677\pi$
$858$	−9.24277	−0.315543
$859$	20.3033	0.692740	0.346370	−	0.938098i	$-0.387414\pi$
$859$	20.3033	0.692740	0.346370	+	0.938098i	$0.387414\pi$
$860$	−16.7656	−0.571703
$861$	32.8085	1.11811
$862$	−95.3177	−3.24653
$863$	3.71086	0.126319	0.0631595	−	0.998003i	$-0.479882\pi$
$863$	3.71086	0.126319	0.0631595	+	0.998003i	$0.479882\pi$
$864$	−60.7734	−2.06755
$865$	16.1472	0.549022
$866$	−75.9615	−2.58128
$867$	−55.2034	−1.87481
$868$	3.99359	0.135551
$869$	19.4669	0.660371
$870$	250.685	8.49902
$871$	−12.3477	−0.418385
$872$	18.2364	0.617561
$873$	21.0143	0.711226
$874$	94.6804	3.20261
$875$	6.26012	0.211631
$876$	84.8456	2.86667
$877$	35.4927	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$877$	35.4927	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$878$	59.5846	2.01088
$879$	32.1110	1.08308
$880$	−20.8456	−0.702707
$881$	20.2531	0.682345	0.341172	−	0.940001i	$-0.389176\pi$
$881$	20.2531	0.682345	0.341172	+	0.940001i	$0.389176\pi$
$882$	96.9057	3.26298
$883$	21.0121	0.707114	0.353557	−	0.935413i	$-0.384972\pi$
$883$	21.0121	0.707114	0.353557	+	0.935413i	$0.384972\pi$
$884$	0.534929	0.0179916
$885$	−68.8016	−2.31274
$886$	−17.7456	−0.596175
$887$	−2.22715	−0.0747805	−0.0373902	−	0.999301i	$-0.511904\pi$
$887$	−2.22715	−0.0747805	−0.0373902	+	0.999301i	$0.511904\pi$
$888$	−17.5371	−0.588506
$889$	−6.35805	−0.213242
$890$	−84.8860	−2.84539
$891$	28.7448	0.962987
$892$	65.8514	2.20487
$893$	−74.9698	−2.50877
$894$	111.434	3.72690
$895$	−75.9101	−2.53739
$896$	19.7323	0.659211
$897$	17.0917	0.570674
$898$	41.1746	1.37401
$899$	6.92469	0.230952
$900$	113.970	3.79901
$901$	−1.38367	−0.0460968
$902$	−20.6960	−0.689101
$903$	5.94908	0.197973
$904$	7.15893	0.238103
$905$	−52.6513	−1.75019
$906$	−53.7969	−1.78728
$907$	54.7404	1.81762	0.908812	−	0.417205i	$-0.136990\pi$
$907$	54.7404	1.81762	0.908812	+	0.417205i	$0.136990\pi$
$908$	88.7796	2.94625
$909$	−138.742	−4.60178
$910$	10.1759	0.337329
$911$	54.0907	1.79211	0.896053	−	0.443948i	$-0.146422\pi$
$911$	54.0907	1.79211	0.896053	+	0.443948i	$0.146422\pi$
$912$	−147.582	−4.88692
$913$	−16.7050	−0.552855
$914$	37.8977	1.25354
$915$	95.9230	3.17112
$916$	−86.8764	−2.87048
$917$	18.6585	0.616159
$918$	−4.58116	−0.151201
$919$	−27.8106	−0.917387	−0.458693	−	0.888595i	$-0.651682\pi$
$919$	−27.8106	−0.917387	−0.458693	+	0.888595i	$0.651682\pi$
$920$	91.4860	3.01620
$921$	−48.0208	−1.58234
$922$	−3.54808	−0.116850
$923$	7.47815	0.246146
$924$	22.2027	0.730416
$925$	3.10746	0.102173
$926$	−12.9495	−0.425547
$927$	50.3236	1.65284
$928$	−43.0815	−1.41422
$929$	24.0882	0.790308	0.395154	−	0.918615i	$-0.370691\pi$
$929$	24.0882	0.790308	0.395154	+	0.918615i	$0.370691\pi$
$930$	15.7042	0.514961
$931$	36.2060	1.18660
$932$	67.6498	2.21594
$933$	26.9001	0.880669
$934$	64.1193	2.09805
$935$	−0.400702	−0.0131043
$936$	−45.2716	−1.47975
$937$	2.04266	0.0667307	0.0333654	−	0.999443i	$-0.489378\pi$
$937$	2.04266	0.0667307	0.0333654	+	0.999443i	$0.489378\pi$
$938$	43.2319	1.41157
$939$	7.88119	0.257193
$940$	−133.536	−4.35548
$941$	10.7170	0.349363	0.174682	−	0.984625i	$-0.444110\pi$
$941$	10.7170	0.349363	0.174682	+	0.984625i	$0.444110\pi$
$942$	128.246	4.17846
$943$	38.2708	1.24627
$944$	46.3678	1.50914
$945$	−59.7913	−1.94501
$946$	−3.75275	−0.122012
$947$	13.6900	0.444865	0.222433	−	0.974948i	$-0.428600\pi$
$947$	13.6900	0.444865	0.222433	+	0.974948i	$0.428600\pi$
$948$	245.487	7.97305
$949$	5.97190	0.193856
$950$	62.0638	2.01362
$951$	−18.1774	−0.589444
$952$	−1.01601	−0.0329290
$953$	9.21942	0.298646	0.149323	−	0.988788i	$-0.452291\pi$
$953$	9.21942	0.298646	0.149323	+	0.988788i	$0.452291\pi$
$954$	215.865	6.98887
$955$	−3.16804	−0.102515
$956$	−53.2431	−1.72200
$957$	38.4984	1.24448
$958$	−49.0836	−1.58582
$959$	16.6589	0.537943
$960$	22.5679	0.728377
$961$	−30.5662	−0.986007
$962$	−2.27541	−0.0733621
$963$	118.760	3.82698
$964$	2.63263	0.0847914
$965$	−51.5417	−1.65919
$966$	−59.8416	−1.92537
$967$	−53.8013	−1.73013	−0.865067	−	0.501657i	$-0.832724\pi$
$967$	−53.8013	−1.73013	−0.865067	+	0.501657i	$0.832724\pi$
$968$	58.2449	1.87206
$969$	−2.83686	−0.0911332
$970$	20.3833	0.654469
$971$	−7.34760	−0.235796	−0.117898	−	0.993026i	$-0.537616\pi$
$971$	−7.34760	−0.235796	−0.117898	+	0.993026i	$0.537616\pi$
$972$	167.986	5.38814
$973$	0.698390	0.0223894
$974$	−3.42989	−0.109901
$975$	11.2037	0.358806
$976$	−64.6459	−2.06926
$977$	12.8181	0.410089	0.205044	−	0.978753i	$-0.434266\pi$
$977$	12.8181	0.410089	0.205044	+	0.978753i	$0.434266\pi$
$978$	−67.1950	−2.14866
$979$	−13.0362	−0.416638
$980$	64.4902	2.06007
$981$	−23.0428	−0.735699
$982$	110.539	3.52743
$983$	0.0207258	0.000661052 0	0.000330526	−	1.00000i	$-0.499895\pi$
$983$	0.0207258	0.000661052 0	0.000330526		1.00000i	$0.499895\pi$
$984$	−141.579	−4.51337
$985$	−7.08478	−0.225740
$986$	−3.24753	−0.103422
$987$	47.3838	1.50824
$988$	−31.1800	−0.991969
$989$	6.93955	0.220665
$990$	62.5129	1.98679
$991$	27.3979	0.870323	0.435162	−	0.900352i	$-0.356691\pi$
$991$	27.3979	0.870323	0.435162	+	0.900352i	$0.356691\pi$
$992$	−2.69885	−0.0856884
$993$	84.7028	2.68796
$994$	−26.1826	−0.830463
$995$	−78.4664	−2.48755
$996$	−210.658	−6.67494
$997$	35.1312	1.11262	0.556309	−	0.830976i	$-0.312217\pi$
$997$	35.1312	1.11262	0.556309	+	0.830976i	$0.312217\pi$
$998$	−20.0545	−0.634815
$999$	13.3697	0.422999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.e.1.11	✓	168

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.e.1.11	✓	168	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.52416 q^{2} +3.25012 q^{3} +4.37136 q^{4} -2.90640 q^{5} -8.20382 q^{6} +1.38708 q^{7} -5.98570 q^{8} +7.56330 q^{9} +O(q^{10})\)	\(q-2.52416 q^{2} +3.25012 q^{3} +4.37136 q^{4} -2.90640 q^{5} -8.20382 q^{6} +1.38708 q^{7} -5.98570 q^{8} +7.56330 q^{9} +7.33621 q^{10} +1.12664 q^{11} +14.2075 q^{12} +1.00000 q^{13} -3.50122 q^{14} -9.44616 q^{15} +6.36610 q^{16} +0.122371 q^{17} -19.0910 q^{18} -7.13279 q^{19} -12.7049 q^{20} +4.50820 q^{21} -2.84382 q^{22} +5.25877 q^{23} -19.4543 q^{24} +3.44717 q^{25} -2.52416 q^{26} +14.8313 q^{27} +6.06345 q^{28} +10.5137 q^{29} +23.8436 q^{30} +0.658633 q^{31} -4.09764 q^{32} +3.66173 q^{33} -0.308884 q^{34} -4.03142 q^{35} +33.0620 q^{36} +0.901453 q^{37} +18.0043 q^{38} +3.25012 q^{39} +17.3968 q^{40} +7.27752 q^{41} -11.3794 q^{42} +1.31961 q^{43} +4.92497 q^{44} -21.9820 q^{45} -13.2740 q^{46} +10.5106 q^{47} +20.6906 q^{48} -5.07600 q^{49} -8.70120 q^{50} +0.397722 q^{51} +4.37136 q^{52} -11.3072 q^{53} -37.4365 q^{54} -3.27448 q^{55} -8.30266 q^{56} -23.1824 q^{57} -26.5383 q^{58} +7.28355 q^{59} -41.2926 q^{60} -10.1547 q^{61} -1.66249 q^{62} +10.4909 q^{63} -2.38911 q^{64} -2.90640 q^{65} -9.24277 q^{66} -12.3477 q^{67} +0.534929 q^{68} +17.0917 q^{69} +10.1759 q^{70} +7.47815 q^{71} -45.2716 q^{72} +5.97190 q^{73} -2.27541 q^{74} +11.2037 q^{75} -31.1800 q^{76} +1.56275 q^{77} -8.20382 q^{78} +17.2787 q^{79} -18.5024 q^{80} +25.5137 q^{81} -18.3696 q^{82} -14.8272 q^{83} +19.7070 q^{84} -0.355660 q^{85} -3.33091 q^{86} +34.1709 q^{87} -6.74374 q^{88} -11.5708 q^{89} +55.4860 q^{90} +1.38708 q^{91} +22.9880 q^{92} +2.14064 q^{93} -26.5304 q^{94} +20.7307 q^{95} -13.3178 q^{96} +2.77845 q^{97} +12.8126 q^{98} +8.52114 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more