LMFDB - Embedded newform 8007.2.a.d.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.85562 q^{2} +1.00000 q^{3} +1.44333 q^{4} -0.427574 q^{5} -1.85562 q^{6} +2.91879 q^{7} +1.03298 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.85562 q^{2} +1.00000 q^{3} +1.44333 q^{4} -0.427574 q^{5} -1.85562 q^{6} +2.91879 q^{7} +1.03298 q^{8} +1.00000 q^{9} +0.793415 q^{10} +3.71762 q^{11} +1.44333 q^{12} -5.72229 q^{13} -5.41616 q^{14} -0.427574 q^{15} -4.80346 q^{16} +1.00000 q^{17} -1.85562 q^{18} +3.37120 q^{19} -0.617129 q^{20} +2.91879 q^{21} -6.89850 q^{22} +1.24607 q^{23} +1.03298 q^{24} -4.81718 q^{25} +10.6184 q^{26} +1.00000 q^{27} +4.21276 q^{28} -2.93984 q^{29} +0.793415 q^{30} +8.39215 q^{31} +6.84745 q^{32} +3.71762 q^{33} -1.85562 q^{34} -1.24800 q^{35} +1.44333 q^{36} -6.95597 q^{37} -6.25566 q^{38} -5.72229 q^{39} -0.441674 q^{40} -8.29470 q^{41} -5.41616 q^{42} -9.44652 q^{43} +5.36574 q^{44} -0.427574 q^{45} -2.31224 q^{46} -2.01875 q^{47} -4.80346 q^{48} +1.51932 q^{49} +8.93886 q^{50} +1.00000 q^{51} -8.25913 q^{52} -10.6712 q^{53} -1.85562 q^{54} -1.58956 q^{55} +3.01504 q^{56} +3.37120 q^{57} +5.45523 q^{58} -0.434172 q^{59} -0.617129 q^{60} +3.44712 q^{61} -15.5726 q^{62} +2.91879 q^{63} -3.09934 q^{64} +2.44671 q^{65} -6.89850 q^{66} -2.83710 q^{67} +1.44333 q^{68} +1.24607 q^{69} +2.31581 q^{70} -8.17504 q^{71} +1.03298 q^{72} -4.55373 q^{73} +12.9076 q^{74} -4.81718 q^{75} +4.86574 q^{76} +10.8509 q^{77} +10.6184 q^{78} +12.8889 q^{79} +2.05384 q^{80} +1.00000 q^{81} +15.3918 q^{82} -8.01878 q^{83} +4.21276 q^{84} -0.427574 q^{85} +17.5291 q^{86} -2.93984 q^{87} +3.84022 q^{88} -4.13744 q^{89} +0.793415 q^{90} -16.7022 q^{91} +1.79849 q^{92} +8.39215 q^{93} +3.74603 q^{94} -1.44144 q^{95} +6.84745 q^{96} -9.39980 q^{97} -2.81927 q^{98} +3.71762 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 7 q^{2} + 40 q^{3} + 29 q^{4} - 15 q^{5} - 7 q^{6} - 13 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) $ 40 * q - 7 * q^2 + 40 * q^3 + 29 * q^4 - 15 * q^5 - 7 * q^6 - 13 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 7 q^{2} + 40 q^{3} + 29 q^{4} - 15 q^{5} - 7 q^{6} - 13 q^{7} - 18 q^{8} + 40 q^{9} - 6 q^{10} - 25 q^{11} + 29 q^{12} - 24 q^{13} - 22 q^{14} - 15 q^{15} + 7 q^{16} + 40 q^{17} - 7 q^{18} - 18 q^{19} - 20 q^{20} - 13 q^{21} - 25 q^{22} - 28 q^{23} - 18 q^{24} - 11 q^{25} - 13 q^{26} + 40 q^{27} - 8 q^{28} - 23 q^{29} - 6 q^{30} - 11 q^{31} - 23 q^{32} - 25 q^{33} - 7 q^{34} - 45 q^{35} + 29 q^{36} - 38 q^{37} - 30 q^{38} - 24 q^{39} - 12 q^{40} - 33 q^{41} - 22 q^{42} - 25 q^{43} - 14 q^{44} - 15 q^{45} + 8 q^{46} - 55 q^{47} + 7 q^{48} - 21 q^{49} + 2 q^{50} + 40 q^{51} - 39 q^{52} - 39 q^{53} - 7 q^{54} - 9 q^{55} - 48 q^{56} - 18 q^{57} - 13 q^{58} - 81 q^{59} - 20 q^{60} - 9 q^{61} - 16 q^{62} - 13 q^{63} - 4 q^{64} - 43 q^{65} - 25 q^{66} - 24 q^{67} + 29 q^{68} - 28 q^{69} + 48 q^{70} - 32 q^{71} - 18 q^{72} - 43 q^{73} - 20 q^{74} - 11 q^{75} - 58 q^{76} - 32 q^{77} - 13 q^{78} - 22 q^{79} - 48 q^{80} + 40 q^{81} - 11 q^{82} - 45 q^{83} - 8 q^{84} - 15 q^{85} - 30 q^{86} - 23 q^{87} - 48 q^{88} - 94 q^{89} - 6 q^{90} - 7 q^{91} - 98 q^{92} - 11 q^{93} + 32 q^{94} - 23 q^{96} - 28 q^{97} - 46 q^{98} - 25 q^{99}+O(q^{100}) \) 40 * q - 7 * q^2 + 40 * q^3 + 29 * q^4 - 15 * q^5 - 7 * q^6 - 13 * q^7 - 18 * q^8 + 40 * q^9 - 6 * q^10 - 25 * q^11 + 29 * q^12 - 24 * q^13 - 22 * q^14 - 15 * q^15 + 7 * q^16 + 40 * q^17 - 7 * q^18 - 18 * q^19 - 20 * q^20 - 13 * q^21 - 25 * q^22 - 28 * q^23 - 18 * q^24 - 11 * q^25 - 13 * q^26 + 40 * q^27 - 8 * q^28 - 23 * q^29 - 6 * q^30 - 11 * q^31 - 23 * q^32 - 25 * q^33 - 7 * q^34 - 45 * q^35 + 29 * q^36 - 38 * q^37 - 30 * q^38 - 24 * q^39 - 12 * q^40 - 33 * q^41 - 22 * q^42 - 25 * q^43 - 14 * q^44 - 15 * q^45 + 8 * q^46 - 55 * q^47 + 7 * q^48 - 21 * q^49 + 2 * q^50 + 40 * q^51 - 39 * q^52 - 39 * q^53 - 7 * q^54 - 9 * q^55 - 48 * q^56 - 18 * q^57 - 13 * q^58 - 81 * q^59 - 20 * q^60 - 9 * q^61 - 16 * q^62 - 13 * q^63 - 4 * q^64 - 43 * q^65 - 25 * q^66 - 24 * q^67 + 29 * q^68 - 28 * q^69 + 48 * q^70 - 32 * q^71 - 18 * q^72 - 43 * q^73 - 20 * q^74 - 11 * q^75 - 58 * q^76 - 32 * q^77 - 13 * q^78 - 22 * q^79 - 48 * q^80 + 40 * q^81 - 11 * q^82 - 45 * q^83 - 8 * q^84 - 15 * q^85 - 30 * q^86 - 23 * q^87 - 48 * q^88 - 94 * q^89 - 6 * q^90 - 7 * q^91 - 98 * q^92 - 11 * q^93 + 32 * q^94 - 23 * q^96 - 28 * q^97 - 46 * q^98 - 25 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.85562	−1.31212	−0.656061	−	0.754708i	$-0.727778\pi$
$2$	−1.85562	−1.31212	−0.656061	+	0.754708i	$0.727778\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.44333	0.721663
$5$	−0.427574	−0.191217	−0.0956085	−	0.995419i	$-0.530480\pi$
$5$	−0.427574	−0.191217	−0.0956085	+	0.995419i	$0.530480\pi$
$6$	−1.85562	−0.757554
$7$	2.91879	1.10320	0.551599	−	0.834110i	$-0.314018\pi$
$7$	2.91879	1.10320	0.551599	+	0.834110i	$0.314018\pi$
$8$	1.03298	0.365212
$9$	1.00000	0.333333
$10$	0.793415	0.250900
$11$	3.71762	1.12091	0.560453	−	0.828186i	$-0.310627\pi$
$11$	3.71762	1.12091	0.560453	+	0.828186i	$0.310627\pi$
$12$	1.44333	0.416652
$13$	−5.72229	−1.58708	−0.793539	−	0.608519i	$-0.791764\pi$
$13$	−5.72229	−1.58708	−0.793539	+	0.608519i	$0.791764\pi$
$14$	−5.41616	−1.44753
$15$	−0.427574	−0.110399
$16$	−4.80346	−1.20087
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.85562	−0.437374
$19$	3.37120	0.773406	0.386703	−	0.922204i	$-0.373614\pi$
$19$	3.37120	0.773406	0.386703	+	0.922204i	$0.373614\pi$
$20$	−0.617129	−0.137994
$21$	2.91879	0.636931
$22$	−6.89850	−1.47076
$23$	1.24607	0.259824	0.129912	−	0.991526i	$-0.458531\pi$
$23$	1.24607	0.259824	0.129912	+	0.991526i	$0.458531\pi$
$24$	1.03298	0.210855
$25$	−4.81718	−0.963436
$26$	10.6184	2.08244
$27$	1.00000	0.192450
$28$	4.21276	0.796137
$29$	−2.93984	−0.545915	−0.272957	−	0.962026i	$-0.588002\pi$
$29$	−2.93984	−0.545915	−0.272957	+	0.962026i	$0.588002\pi$
$30$	0.793415	0.144857
$31$	8.39215	1.50727	0.753637	−	0.657291i	$-0.228298\pi$
$31$	8.39215	1.50727	0.753637	+	0.657291i	$0.228298\pi$
$32$	6.84745	1.21047
$33$	3.71762	0.647155
$34$	−1.85562	−0.318236
$35$	−1.24800	−0.210950
$36$	1.44333	0.240554
$37$	−6.95597	−1.14355	−0.571777	−	0.820409i	$-0.693746\pi$
$37$	−6.95597	−1.14355	−0.571777	+	0.820409i	$0.693746\pi$
$38$	−6.25566	−1.01480
$39$	−5.72229	−0.916300
$40$	−0.441674	−0.0698348
$41$	−8.29470	−1.29541	−0.647707	−	0.761889i	$-0.724272\pi$
$41$	−8.29470	−1.29541	−0.647707	+	0.761889i	$0.724272\pi$
$42$	−5.41616	−0.835731
$43$	−9.44652	−1.44058	−0.720290	−	0.693673i	$-0.755991\pi$
$43$	−9.44652	−1.44058	−0.720290	+	0.693673i	$0.755991\pi$
$44$	5.36574	0.808916
$45$	−0.427574	−0.0637390
$46$	−2.31224	−0.340921
$47$	−2.01875	−0.294465	−0.147232	−	0.989102i	$-0.547037\pi$
$47$	−2.01875	−0.294465	−0.147232	+	0.989102i	$0.547037\pi$
$48$	−4.80346	−0.693320
$49$	1.51932	0.217045
$50$	8.93886	1.26415
$51$	1.00000	0.140028
$52$	−8.25913	−1.14534
$53$	−10.6712	−1.46580	−0.732902	−	0.680334i	$-0.761835\pi$
$53$	−10.6712	−1.46580	−0.732902	+	0.680334i	$0.761835\pi$
$54$	−1.85562	−0.252518
$55$	−1.58956	−0.214336
$56$	3.01504	0.402901
$57$	3.37120	0.446526
$58$	5.45523	0.716306
$59$	−0.434172	−0.0565243	−0.0282622	−	0.999601i	$-0.508997\pi$
$59$	−0.434172	−0.0565243	−0.0282622	+	0.999601i	$0.508997\pi$
$60$	−0.617129	−0.0796710
$61$	3.44712	0.441358	0.220679	−	0.975346i	$-0.429173\pi$
$61$	3.44712	0.441358	0.220679	+	0.975346i	$0.429173\pi$
$62$	−15.5726	−1.97773
$63$	2.91879	0.367733
$64$	−3.09934	−0.387417
$65$	2.44671	0.303476
$66$	−6.89850	−0.849146
$67$	−2.83710	−0.346607	−0.173303	−	0.984869i	$-0.555444\pi$
$67$	−2.83710	−0.346607	−0.173303	+	0.984869i	$0.555444\pi$
$68$	1.44333	0.175029
$69$	1.24607	0.150009
$70$	2.31581	0.276792
$71$	−8.17504	−0.970199	−0.485099	−	0.874459i	$-0.661217\pi$
$71$	−8.17504	−0.970199	−0.485099	+	0.874459i	$0.661217\pi$
$72$	1.03298	0.121737
$73$	−4.55373	−0.532974	−0.266487	−	0.963838i	$-0.585863\pi$
$73$	−4.55373	−0.532974	−0.266487	+	0.963838i	$0.585863\pi$
$74$	12.9076	1.50048
$75$	−4.81718	−0.556240
$76$	4.86574	0.558138
$77$	10.8509	1.23658
$78$	10.6184	1.20230
$79$	12.8889	1.45011	0.725057	−	0.688689i	$-0.241813\pi$
$79$	12.8889	1.45011	0.725057	+	0.688689i	$0.241813\pi$
$80$	2.05384	0.229626
$81$	1.00000	0.111111
$82$	15.3918	1.69974
$83$	−8.01878	−0.880176	−0.440088	−	0.897955i	$-0.645053\pi$
$83$	−8.01878	−0.880176	−0.440088	+	0.897955i	$0.645053\pi$
$84$	4.21276	0.459650
$85$	−0.427574	−0.0463769
$86$	17.5291	1.89022
$87$	−2.93984	−0.315184
$88$	3.84022	0.409368
$89$	−4.13744	−0.438568	−0.219284	−	0.975661i	$-0.570372\pi$
$89$	−4.13744	−0.438568	−0.219284	+	0.975661i	$0.570372\pi$
$90$	0.793415	0.0836333
$91$	−16.7022	−1.75086
$92$	1.79849	0.187505
$93$	8.39215	0.870225
$94$	3.74603	0.386374
$95$	−1.44144	−0.147888
$96$	6.84745	0.698865
$97$	−9.39980	−0.954406	−0.477203	−	0.878793i	$-0.658349\pi$
$97$	−9.39980	−0.954406	−0.477203	+	0.878793i	$0.658349\pi$
$98$	−2.81927	−0.284789
$99$	3.71762	0.373635
$100$	−6.95276	−0.695276
$101$	−8.43872	−0.839684	−0.419842	−	0.907597i	$-0.637915\pi$
$101$	−8.43872	−0.839684	−0.419842	+	0.907597i	$0.637915\pi$
$102$	−1.85562	−0.183734
$103$	4.70878	0.463970	0.231985	−	0.972719i	$-0.425478\pi$
$103$	4.70878	0.463970	0.231985	+	0.972719i	$0.425478\pi$
$104$	−5.91099	−0.579621
$105$	−1.24800	−0.121792
$106$	19.8017	1.92331
$107$	−10.0005	−0.966786	−0.483393	−	0.875403i	$-0.660596\pi$
$107$	−10.0005	−0.966786	−0.483393	+	0.875403i	$0.660596\pi$
$108$	1.44333	0.138884
$109$	14.6441	1.40265	0.701327	−	0.712840i	$-0.252591\pi$
$109$	14.6441	1.40265	0.701327	+	0.712840i	$0.252591\pi$
$110$	2.94962	0.281235
$111$	−6.95597	−0.660231
$112$	−14.0203	−1.32479
$113$	−9.43247	−0.887332	−0.443666	−	0.896192i	$-0.646322\pi$
$113$	−9.43247	−0.887332	−0.443666	+	0.896192i	$0.646322\pi$
$114$	−6.25566	−0.585897
$115$	−0.532788	−0.0496828
$116$	−4.24315	−0.393966
$117$	−5.72229	−0.529026
$118$	0.805658	0.0741668
$119$	2.91879	0.267565
$120$	−0.441674	−0.0403191
$121$	2.82072	0.256429
$122$	−6.39654	−0.579116
$123$	−8.29470	−0.747908
$124$	12.1126	1.08774
$125$	4.19757	0.375442
$126$	−5.41616	−0.482510
$127$	−16.7077	−1.48257	−0.741283	−	0.671192i	$-0.765782\pi$
$127$	−16.7077	−1.48257	−0.741283	+	0.671192i	$0.765782\pi$
$128$	−7.94370	−0.702131
$129$	−9.44652	−0.831719
$130$	−4.54016	−0.398198
$131$	−0.154453	−0.0134947	−0.00674733	−	0.999977i	$-0.502148\pi$
$131$	−0.154453	−0.0134947	−0.00674733	+	0.999977i	$0.502148\pi$
$132$	5.36574	0.467028
$133$	9.83981	0.853220
$134$	5.26457	0.454790
$135$	−0.427574	−0.0367997
$136$	1.03298	0.0885770
$137$	−11.8207	−1.00991	−0.504955	−	0.863146i	$-0.668491\pi$
$137$	−11.8207	−1.00991	−0.504955	+	0.863146i	$0.668491\pi$
$138$	−2.31224	−0.196831
$139$	−18.0556	−1.53146	−0.765729	−	0.643163i	$-0.777622\pi$
$139$	−18.0556	−1.53146	−0.765729	+	0.643163i	$0.777622\pi$
$140$	−1.80127	−0.152235
$141$	−2.01875	−0.170009
$142$	15.1698	1.27302
$143$	−21.2733	−1.77897
$144$	−4.80346	−0.400289
$145$	1.25700	0.104388
$146$	8.45000	0.699327
$147$	1.51932	0.125311
$148$	−10.0397	−0.825261
$149$	9.25198	0.757951	0.378976	−	0.925407i	$-0.376276\pi$
$149$	9.25198	0.757951	0.378976	+	0.925407i	$0.376276\pi$
$150$	8.93886	0.729855
$151$	−8.59075	−0.699106	−0.349553	−	0.936917i	$-0.613666\pi$
$151$	−8.59075	−0.699106	−0.349553	+	0.936917i	$0.613666\pi$
$152$	3.48237	0.282457
$153$	1.00000	0.0808452
$154$	−20.1352	−1.62254
$155$	−3.58827	−0.288217
$156$	−8.25913	−0.661260
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−23.9169	−1.90273
$159$	−10.6712	−0.846283
$160$	−2.92779	−0.231462
$161$	3.63702	0.286637
$162$	−1.85562	−0.145791
$163$	20.6647	1.61858	0.809291	−	0.587408i	$-0.199852\pi$
$163$	20.6647	1.61858	0.809291	+	0.587408i	$0.199852\pi$
$164$	−11.9720	−0.934853
$165$	−1.58956	−0.123747
$166$	14.8798	1.15490
$167$	−17.3863	−1.34539	−0.672695	−	0.739919i	$-0.734864\pi$
$167$	−17.3863	−1.34539	−0.672695	+	0.739919i	$0.734864\pi$
$168$	3.01504	0.232615
$169$	19.7446	1.51882
$170$	0.793415	0.0608522
$171$	3.37120	0.257802
$172$	−13.6344	−1.03961
$173$	−13.2493	−1.00732	−0.503662	−	0.863901i	$-0.668014\pi$
$173$	−13.2493	−1.00732	−0.503662	+	0.863901i	$0.668014\pi$
$174$	5.45523	0.413560
$175$	−14.0603	−1.06286
$176$	−17.8575	−1.34606
$177$	−0.434172	−0.0326343
$178$	7.67752	0.575455
$179$	10.2085	0.763017	0.381509	−	0.924365i	$-0.375405\pi$
$179$	10.2085	0.763017	0.381509	+	0.924365i	$0.375405\pi$
$180$	−0.617129	−0.0459981
$181$	3.06840	0.228072	0.114036	−	0.993477i	$-0.463622\pi$
$181$	3.06840	0.228072	0.114036	+	0.993477i	$0.463622\pi$
$182$	30.9928	2.29734
$183$	3.44712	0.254818
$184$	1.28716	0.0948909
$185$	2.97419	0.218667
$186$	−15.5726	−1.14184
$187$	3.71762	0.271860
$188$	−2.91371	−0.212504
$189$	2.91879	0.212310
$190$	2.67476	0.194048
$191$	−5.98708	−0.433210	−0.216605	−	0.976259i	$-0.569498\pi$
$191$	−5.98708	−0.433210	−0.216605	+	0.976259i	$0.569498\pi$
$192$	−3.09934	−0.223675
$193$	13.1974	0.949971	0.474985	−	0.879994i	$-0.342453\pi$
$193$	13.1974	0.949971	0.474985	+	0.879994i	$0.342453\pi$
$194$	17.4425	1.25230
$195$	2.44671	0.175212
$196$	2.19287	0.156633
$197$	−0.983195	−0.0700497	−0.0350249	−	0.999386i	$-0.511151\pi$
$197$	−0.983195	−0.0700497	−0.0350249	+	0.999386i	$0.511151\pi$
$198$	−6.89850	−0.490255
$199$	−0.250794	−0.0177783	−0.00888915	−	0.999960i	$-0.502830\pi$
$199$	−0.250794	−0.0177783	−0.00888915	+	0.999960i	$0.502830\pi$
$200$	−4.97603	−0.351859
$201$	−2.83710	−0.200113
$202$	15.6591	1.10177
$203$	−8.58077	−0.602252
$204$	1.44333	0.101053
$205$	3.54660	0.247705
$206$	−8.73771	−0.608785
$207$	1.24607	0.0866080
$208$	27.4868	1.90587
$209$	12.5328	0.866915
$210$	2.31581	0.159806
$211$	7.44346	0.512429	0.256215	−	0.966620i	$-0.417525\pi$
$211$	7.44346	0.512429	0.256215	+	0.966620i	$0.417525\pi$
$212$	−15.4020	−1.05782
$213$	−8.17504	−0.560144
$214$	18.5572	1.26854
$215$	4.03909	0.275464
$216$	1.03298	0.0702851
$217$	24.4949	1.66282
$218$	−27.1740	−1.84045
$219$	−4.55373	−0.307713
$220$	−2.29425	−0.154678
$221$	−5.72229	−0.384923
$222$	12.9076	0.866304
$223$	2.86277	0.191705	0.0958525	−	0.995396i	$-0.469442\pi$
$223$	2.86277	0.191705	0.0958525	+	0.995396i	$0.469442\pi$
$224$	19.9862	1.33539
$225$	−4.81718	−0.321145
$226$	17.5031	1.16429
$227$	−2.38216	−0.158109	−0.0790546	−	0.996870i	$-0.525190\pi$
$227$	−2.38216	−0.158109	−0.0790546	+	0.996870i	$0.525190\pi$
$228$	4.86574	0.322241
$229$	−9.43205	−0.623288	−0.311644	−	0.950199i	$-0.600880\pi$
$229$	−9.43205	−0.623288	−0.311644	+	0.950199i	$0.600880\pi$
$230$	0.988653	0.0651898
$231$	10.8509	0.713940
$232$	−3.03679	−0.199375
$233$	3.81786	0.250117	0.125058	−	0.992149i	$-0.460088\pi$
$233$	3.81786	0.250117	0.125058	+	0.992149i	$0.460088\pi$
$234$	10.6184	0.694147
$235$	0.863166	0.0563067
$236$	−0.626651	−0.0407915
$237$	12.8889	0.837224
$238$	−5.41616	−0.351077
$239$	−0.642610	−0.0415670	−0.0207835	−	0.999784i	$-0.506616\pi$
$239$	−0.642610	−0.0415670	−0.0207835	+	0.999784i	$0.506616\pi$
$240$	2.05384	0.132575
$241$	4.81558	0.310199	0.155099	−	0.987899i	$-0.450430\pi$
$241$	4.81558	0.310199	0.155099	+	0.987899i	$0.450430\pi$
$242$	−5.23419	−0.336466
$243$	1.00000	0.0641500
$244$	4.97531	0.318512
$245$	−0.649620	−0.0415027
$246$	15.3918	0.981346
$247$	−19.2910	−1.22746
$248$	8.66889	0.550475
$249$	−8.01878	−0.508170
$250$	−7.78910	−0.492626
$251$	5.87884	0.371069	0.185534	−	0.982638i	$-0.440598\pi$
$251$	5.87884	0.371069	0.185534	+	0.982638i	$0.440598\pi$
$252$	4.21276	0.265379
$253$	4.63243	0.291238
$254$	31.0031	1.94531
$255$	−0.427574	−0.0267757
$256$	20.9392	1.30870
$257$	19.6794	1.22757	0.613783	−	0.789475i	$-0.289647\pi$
$257$	19.6794	1.22757	0.613783	+	0.789475i	$0.289647\pi$
$258$	17.5291	1.09132
$259$	−20.3030	−1.26157
$260$	3.53139	0.219008
$261$	−2.93984	−0.181972
$262$	0.286607	0.0177066
$263$	−13.2604	−0.817669	−0.408835	−	0.912608i	$-0.634065\pi$
$263$	−13.2604	−0.817669	−0.408835	+	0.912608i	$0.634065\pi$
$264$	3.84022	0.236349
$265$	4.56274	0.280287
$266$	−18.2589	−1.11953
$267$	−4.13744	−0.253207
$268$	−4.09485	−0.250133
$269$	10.4806	0.639011	0.319505	−	0.947584i	$-0.396483\pi$
$269$	10.4806	0.639011	0.319505	+	0.947584i	$0.396483\pi$
$270$	0.793415	0.0482857
$271$	10.4553	0.635115	0.317557	−	0.948239i	$-0.397137\pi$
$271$	10.4553	0.635115	0.317557	+	0.948239i	$0.397137\pi$
$272$	−4.80346	−0.291253
$273$	−16.7022	−1.01086
$274$	21.9347	1.32512
$275$	−17.9085	−1.07992
$276$	1.79849	0.108256
$277$	10.3173	0.619904	0.309952	−	0.950752i	$-0.399687\pi$
$277$	10.3173	0.619904	0.309952	+	0.950752i	$0.399687\pi$
$278$	33.5044	2.00946
$279$	8.39215	0.502425
$280$	−1.28915	−0.0770416
$281$	14.5866	0.870161	0.435081	−	0.900392i	$-0.356720\pi$
$281$	14.5866	0.870161	0.435081	+	0.900392i	$0.356720\pi$
$282$	3.74603	0.223073
$283$	27.4171	1.62978	0.814890	−	0.579616i	$-0.196797\pi$
$283$	27.4171	1.62978	0.814890	+	0.579616i	$0.196797\pi$
$284$	−11.7992	−0.700156
$285$	−1.44144	−0.0853834
$286$	39.4752	2.33422
$287$	−24.2105	−1.42910
$288$	6.84745	0.403490
$289$	1.00000	0.0588235
$290$	−2.33251	−0.136970
$291$	−9.39980	−0.551026
$292$	−6.57252	−0.384628
$293$	−18.0735	−1.05586	−0.527932	−	0.849287i	$-0.677032\pi$
$293$	−18.0735	−1.05586	−0.527932	+	0.849287i	$0.677032\pi$
$294$	−2.81927	−0.164423
$295$	0.185641	0.0108084
$296$	−7.18535	−0.417640
$297$	3.71762	0.215718
$298$	−17.1682	−0.994524
$299$	−7.13039	−0.412361
$300$	−6.95276	−0.401418
$301$	−27.5724	−1.58924
$302$	15.9412	0.917311
$303$	−8.43872	−0.484792
$304$	−16.1934	−0.928757
$305$	−1.47390	−0.0843952
$306$	−1.85562	−0.106079
$307$	24.3428	1.38932	0.694658	−	0.719340i	$-0.255556\pi$
$307$	24.3428	1.38932	0.694658	+	0.719340i	$0.255556\pi$
$308$	15.6615	0.892394
$309$	4.70878	0.267873
$310$	6.65846	0.378175
$311$	−16.7261	−0.948451	−0.474225	−	0.880404i	$-0.657272\pi$
$311$	−16.7261	−0.948451	−0.474225	+	0.880404i	$0.657272\pi$
$312$	−5.91099	−0.334644
$313$	−1.66423	−0.0940677	−0.0470339	−	0.998893i	$-0.514977\pi$
$313$	−1.66423	−0.0940677	−0.0470339	+	0.998893i	$0.514977\pi$
$314$	1.85562	0.104719
$315$	−1.24800	−0.0703167
$316$	18.6029	1.04649
$317$	−19.4162	−1.09052	−0.545261	−	0.838266i	$-0.683570\pi$
$317$	−19.4162	−1.09052	−0.545261	+	0.838266i	$0.683570\pi$
$318$	19.8017	1.11043
$319$	−10.9292	−0.611919
$320$	1.32520	0.0740808
$321$	−10.0005	−0.558174
$322$	−6.74892	−0.376103
$323$	3.37120	0.187578
$324$	1.44333	0.0801848
$325$	27.5653	1.52905
$326$	−38.3458	−2.12378
$327$	14.6441	0.809823
$328$	−8.56823	−0.473101
$329$	−5.89230	−0.324853
$330$	2.94962	0.162371
$331$	−12.7576	−0.701219	−0.350610	−	0.936522i	$-0.614026\pi$
$331$	−12.7576	−0.701219	−0.350610	+	0.936522i	$0.614026\pi$
$332$	−11.5737	−0.635190
$333$	−6.95597	−0.381185
$334$	32.2623	1.76532
$335$	1.21307	0.0662771
$336$	−14.0203	−0.764869
$337$	33.1491	1.80575	0.902874	−	0.429905i	$-0.141453\pi$
$337$	33.1491	1.80575	0.902874	+	0.429905i	$0.141453\pi$
$338$	−36.6385	−1.99287
$339$	−9.43247	−0.512301
$340$	−0.617129	−0.0334685
$341$	31.1988	1.68951
$342$	−6.25566	−0.338268
$343$	−15.9969	−0.863754
$344$	−9.75803	−0.526118
$345$	−0.532788	−0.0286844
$346$	24.5856	1.32173
$347$	−13.0993	−0.703205	−0.351603	−	0.936149i	$-0.614363\pi$
$347$	−13.0993	−0.703205	−0.351603	+	0.936149i	$0.614363\pi$
$348$	−4.24315	−0.227457
$349$	3.61839	0.193688	0.0968440	−	0.995300i	$-0.469125\pi$
$349$	3.61839	0.193688	0.0968440	+	0.995300i	$0.469125\pi$
$350$	26.0906	1.39460
$351$	−5.72229	−0.305433
$352$	25.4562	1.35682
$353$	31.1246	1.65660	0.828298	−	0.560288i	$-0.189309\pi$
$353$	31.1246	1.65660	0.828298	+	0.560288i	$0.189309\pi$
$354$	0.805658	0.0428202
$355$	3.49544	0.185519
$356$	−5.97168	−0.316498
$357$	2.91879	0.154479
$358$	−18.9430	−1.00117
$359$	23.8266	1.25752	0.628761	−	0.777599i	$-0.283562\pi$
$359$	23.8266	1.25752	0.628761	+	0.777599i	$0.283562\pi$
$360$	−0.441674	−0.0232783
$361$	−7.63502	−0.401843
$362$	−5.69378	−0.299258
$363$	2.82072	0.148049
$364$	−24.1066	−1.26353
$365$	1.94706	0.101914
$366$	−6.39654	−0.334353
$367$	−17.2677	−0.901364	−0.450682	−	0.892685i	$-0.648819\pi$
$367$	−17.2677	−0.901364	−0.450682	+	0.892685i	$0.648819\pi$
$368$	−5.98546	−0.312014
$369$	−8.29470	−0.431805
$370$	−5.51897	−0.286918
$371$	−31.1470	−1.61707
$372$	12.1126	0.628009
$373$	36.5371	1.89182	0.945909	−	0.324433i	$-0.105173\pi$
$373$	36.5371	1.89182	0.945909	+	0.324433i	$0.105173\pi$
$374$	−6.89850	−0.356713
$375$	4.19757	0.216762
$376$	−2.08532	−0.107542
$377$	16.8226	0.866409
$378$	−5.41616	−0.278577
$379$	7.33334	0.376688	0.188344	−	0.982103i	$-0.439688\pi$
$379$	7.33334	0.376688	0.188344	+	0.982103i	$0.439688\pi$
$380$	−2.08046	−0.106726
$381$	−16.7077	−0.855960
$382$	11.1097	0.568424
$383$	8.31090	0.424667	0.212334	−	0.977197i	$-0.431894\pi$
$383$	8.31090	0.424667	0.212334	+	0.977197i	$0.431894\pi$
$384$	−7.94370	−0.405375
$385$	−4.63959	−0.236455
$386$	−24.4894	−1.24648
$387$	−9.44652	−0.480193
$388$	−13.5670	−0.688759
$389$	5.87528	0.297888	0.148944	−	0.988846i	$-0.452413\pi$
$389$	5.87528	0.297888	0.148944	+	0.988846i	$0.452413\pi$
$390$	−4.54016	−0.229900
$391$	1.24607	0.0630166
$392$	1.56942	0.0792675
$393$	−0.154453	−0.00779114
$394$	1.82444	0.0919138
$395$	−5.51096	−0.277287
$396$	5.36574	0.269639
$397$	23.0778	1.15824	0.579122	−	0.815241i	$-0.303396\pi$
$397$	23.0778	1.15824	0.579122	+	0.815241i	$0.303396\pi$
$398$	0.465378	0.0233273
$399$	9.83981	0.492607
$400$	23.1391	1.15696
$401$	−5.08570	−0.253968	−0.126984	−	0.991905i	$-0.540530\pi$
$401$	−5.08570	−0.253968	−0.126984	+	0.991905i	$0.540530\pi$
$402$	5.26457	0.262573
$403$	−48.0223	−2.39216
$404$	−12.1798	−0.605969
$405$	−0.427574	−0.0212463
$406$	15.9226	0.790227
$407$	−25.8597	−1.28182
$408$	1.03298	0.0511399
$409$	−35.0299	−1.73212	−0.866059	−	0.499942i	$-0.833355\pi$
$409$	−35.0299	−1.73212	−0.866059	+	0.499942i	$0.833355\pi$
$410$	−6.58115	−0.325020
$411$	−11.8207	−0.583072
$412$	6.79631	0.334830
$413$	−1.26725	−0.0623575
$414$	−2.31224	−0.113640
$415$	3.42863	0.168305
$416$	−39.1831	−1.92111
$417$	−18.0556	−0.884188
$418$	−23.2562	−1.13750
$419$	−8.90107	−0.434846	−0.217423	−	0.976077i	$-0.569765\pi$
$419$	−8.90107	−0.434846	−0.217423	+	0.976077i	$0.569765\pi$
$420$	−1.80127	−0.0878929
$421$	4.15828	0.202662	0.101331	−	0.994853i	$-0.467690\pi$
$421$	4.15828	0.202662	0.101331	+	0.994853i	$0.467690\pi$
$422$	−13.8122	−0.672369
$423$	−2.01875	−0.0981550
$424$	−11.0231	−0.535330
$425$	−4.81718	−0.233668
$426$	15.1698	0.734978
$427$	10.0614	0.486905
$428$	−14.4340	−0.697694
$429$	−21.2733	−1.02709
$430$	−7.49501	−0.361442
$431$	23.9612	1.15417	0.577086	−	0.816683i	$-0.304190\pi$
$431$	23.9612	1.15417	0.577086	+	0.816683i	$0.304190\pi$
$432$	−4.80346	−0.231107
$433$	−18.0279	−0.866367	−0.433183	−	0.901306i	$-0.642610\pi$
$433$	−18.0279	−0.866367	−0.433183	+	0.901306i	$0.642610\pi$
$434$	−45.4532	−2.18182
$435$	1.25700	0.0602686
$436$	21.1363	1.01224
$437$	4.20076	0.200949
$438$	8.45000	0.403757
$439$	31.7430	1.51501	0.757506	−	0.652828i	$-0.226418\pi$
$439$	31.7430	1.51501	0.757506	+	0.652828i	$0.226418\pi$
$440$	−1.64198	−0.0782782
$441$	1.51932	0.0723483
$442$	10.6184	0.505066
$443$	−8.46094	−0.401992	−0.200996	−	0.979592i	$-0.564418\pi$
$443$	−8.46094	−0.401992	−0.200996	+	0.979592i	$0.564418\pi$
$444$	−10.0397	−0.476464
$445$	1.76906	0.0838617
$446$	−5.31221	−0.251540
$447$	9.25198	0.437603
$448$	−9.04630	−0.427398
$449$	13.7390	0.648382	0.324191	−	0.945992i	$-0.394908\pi$
$449$	13.7390	0.648382	0.324191	+	0.945992i	$0.394908\pi$
$450$	8.93886	0.421382
$451$	−30.8366	−1.45204
$452$	−13.6141	−0.640355
$453$	−8.59075	−0.403629
$454$	4.42037	0.207458
$455$	7.14141	0.334795
$456$	3.48237	0.163077
$457$	−23.7654	−1.11170	−0.555849	−	0.831283i	$-0.687607\pi$
$457$	−23.7654	−1.11170	−0.555849	+	0.831283i	$0.687607\pi$
$458$	17.5023	0.817829
$459$	1.00000	0.0466760
$460$	−0.768987	−0.0358542
$461$	−36.2353	−1.68765	−0.843823	−	0.536621i	$-0.819700\pi$
$461$	−36.2353	−1.68765	−0.843823	+	0.536621i	$0.819700\pi$
$462$	−20.1352	−0.936776
$463$	−22.6740	−1.05375	−0.526876	−	0.849942i	$-0.676637\pi$
$463$	−22.6740	−1.05375	−0.526876	+	0.849942i	$0.676637\pi$
$464$	14.1214	0.655570
$465$	−3.58827	−0.166402
$466$	−7.08450	−0.328183
$467$	20.7558	0.960466	0.480233	−	0.877141i	$-0.340552\pi$
$467$	20.7558	0.960466	0.480233	+	0.877141i	$0.340552\pi$
$468$	−8.25913	−0.381779
$469$	−8.28088	−0.382375
$470$	−1.60171	−0.0738813
$471$	−1.00000	−0.0460776
$472$	−0.448489	−0.0206434
$473$	−35.1186	−1.61475
$474$	−23.9169	−1.09854
$475$	−16.2397	−0.745127
$476$	4.21276	0.193092
$477$	−10.6712	−0.488602
$478$	1.19244	0.0545409
$479$	−8.04393	−0.367536	−0.183768	−	0.982970i	$-0.558830\pi$
$479$	−8.04393	−0.367536	−0.183768	+	0.982970i	$0.558830\pi$
$480$	−2.92779	−0.133635
$481$	39.8041	1.81491
$482$	−8.93589	−0.407019
$483$	3.63702	0.165490
$484$	4.07122	0.185055
$485$	4.01912	0.182499
$486$	−1.85562	−0.0841726
$487$	−27.0537	−1.22592	−0.612960	−	0.790114i	$-0.710021\pi$
$487$	−27.0537	−1.22592	−0.612960	+	0.790114i	$0.710021\pi$
$488$	3.56079	0.161189
$489$	20.6647	0.934488
$490$	1.20545	0.0544566
$491$	4.05362	0.182937	0.0914687	−	0.995808i	$-0.470844\pi$
$491$	4.05362	0.182937	0.0914687	+	0.995808i	$0.470844\pi$
$492$	−11.9720	−0.539737
$493$	−2.93984	−0.132404
$494$	35.7967	1.61057
$495$	−1.58956	−0.0714454
$496$	−40.3114	−1.81003
$497$	−23.8612	−1.07032
$498$	14.8798	0.666780
$499$	−7.78092	−0.348322	−0.174161	−	0.984717i	$-0.555721\pi$
$499$	−7.78092	−0.348322	−0.174161	+	0.984717i	$0.555721\pi$
$500$	6.05847	0.270943
$501$	−17.3863	−0.776762
$502$	−10.9089	−0.486887
$503$	−40.1455	−1.79000	−0.895001	−	0.446065i	$-0.852825\pi$
$503$	−40.1455	−1.79000	−0.895001	+	0.446065i	$0.852825\pi$
$504$	3.01504	0.134300
$505$	3.60818	0.160562
$506$	−8.59602	−0.382140
$507$	19.7446	0.876890
$508$	−24.1146	−1.06991
$509$	24.7446	1.09678	0.548392	−	0.836221i	$-0.315240\pi$
$509$	24.7446	1.09678	0.548392	+	0.836221i	$0.315240\pi$
$510$	0.793415	0.0351330
$511$	−13.2914	−0.587976
$512$	−22.9677	−1.01504
$513$	3.37120	0.148842
$514$	−36.5174	−1.61071
$515$	−2.01336	−0.0887190
$516$	−13.6344	−0.600221
$517$	−7.50495	−0.330067
$518$	37.6746	1.65533
$519$	−13.2493	−0.581579
$520$	2.52739	0.110833
$521$	17.7095	0.775868	0.387934	−	0.921687i	$-0.373189\pi$
$521$	17.7095	0.775868	0.387934	+	0.921687i	$0.373189\pi$
$522$	5.45523	0.238769
$523$	13.7538	0.601412	0.300706	−	0.953717i	$-0.402778\pi$
$523$	13.7538	0.601412	0.300706	+	0.953717i	$0.402778\pi$
$524$	−0.222927	−0.00973859
$525$	−14.0603	−0.613643
$526$	24.6062	1.07288
$527$	8.39215	0.365568
$528$	−17.8575	−0.777146
$529$	−21.4473	−0.932492
$530$	−8.46671	−0.367770
$531$	−0.434172	−0.0188414
$532$	14.2020	0.615737
$533$	47.4647	2.05593
$534$	7.67752	0.332239
$535$	4.27596	0.184866
$536$	−2.93065	−0.126585
$537$	10.2085	0.440528
$538$	−19.4479	−0.838460
$539$	5.64824	0.243287
$540$	−0.617129	−0.0265570
$541$	34.7253	1.49296	0.746479	−	0.665409i	$-0.231743\pi$
$541$	34.7253	1.49296	0.746479	+	0.665409i	$0.231743\pi$
$542$	−19.4011	−0.833348
$543$	3.06840	0.131678
$544$	6.84745	0.293582
$545$	−6.26146	−0.268211
$546$	30.9928	1.32637
$547$	24.0341	1.02762	0.513812	−	0.857903i	$-0.328233\pi$
$547$	24.0341	1.02762	0.513812	+	0.857903i	$0.328233\pi$
$548$	−17.0611	−0.728814
$549$	3.44712	0.147119
$550$	33.2313	1.41699
$551$	−9.91079	−0.422214
$552$	1.28716	0.0547853
$553$	37.6199	1.59976
$554$	−19.1449	−0.813389
$555$	2.97419	0.126248
$556$	−26.0601	−1.10520
$557$	−32.4315	−1.37417	−0.687084	−	0.726578i	$-0.741109\pi$
$557$	−32.4315	−1.37417	−0.687084	+	0.726578i	$0.741109\pi$
$558$	−15.5726	−0.659242
$559$	54.0557	2.28631
$560$	5.99471	0.253323
$561$	3.71762	0.156958
$562$	−27.0671	−1.14176
$563$	11.1632	0.470472	0.235236	−	0.971938i	$-0.424414\pi$
$563$	11.1632	0.470472	0.235236	+	0.971938i	$0.424414\pi$
$564$	−2.91371	−0.122689
$565$	4.03308	0.169673
$566$	−50.8758	−2.13847
$567$	2.91879	0.122578
$568$	−8.44462	−0.354328
$569$	−26.5734	−1.11401	−0.557007	−	0.830508i	$-0.688050\pi$
$569$	−26.5734	−1.11401	−0.557007	+	0.830508i	$0.688050\pi$
$570$	2.67476	0.112033
$571$	26.9808	1.12911	0.564557	−	0.825394i	$-0.309047\pi$
$571$	26.9808	1.12911	0.564557	+	0.825394i	$0.309047\pi$
$572$	−30.7043	−1.28381
$573$	−5.98708	−0.250114
$574$	44.9254	1.87515
$575$	−6.00255	−0.250324
$576$	−3.09934	−0.129139
$577$	0.149905	0.00624061	0.00312030	−	0.999995i	$-0.499007\pi$
$577$	0.149905	0.00624061	0.00312030	+	0.999995i	$0.499007\pi$
$578$	−1.85562	−0.0771836
$579$	13.1974	0.548466
$580$	1.81426	0.0753331
$581$	−23.4051	−0.971008
$582$	17.4425	0.723013
$583$	−39.6716	−1.64303
$584$	−4.70390	−0.194649
$585$	2.44671	0.101159
$586$	33.5375	1.38542
$587$	36.4593	1.50484	0.752418	−	0.658686i	$-0.228887\pi$
$587$	36.4593	1.50484	0.752418	+	0.658686i	$0.228887\pi$
$588$	2.19287	0.0904323
$589$	28.2916	1.16574
$590$	−0.344479	−0.0141820
$591$	−0.983195	−0.0404432
$592$	33.4127	1.37326
$593$	−3.03483	−0.124626	−0.0623128	−	0.998057i	$-0.519848\pi$
$593$	−3.03483	−0.124626	−0.0623128	+	0.998057i	$0.519848\pi$
$594$	−6.89850	−0.283049
$595$	−1.24800	−0.0511629
$596$	13.3536	0.546985
$597$	−0.250794	−0.0102643
$598$	13.2313	0.541068
$599$	14.7169	0.601318	0.300659	−	0.953732i	$-0.402793\pi$
$599$	14.7169	0.601318	0.300659	+	0.953732i	$0.402793\pi$
$600$	−4.97603	−0.203146
$601$	−1.02679	−0.0418836	−0.0209418	−	0.999781i	$-0.506666\pi$
$601$	−1.02679	−0.0418836	−0.0209418	+	0.999781i	$0.506666\pi$
$602$	51.1638	2.08528
$603$	−2.83710	−0.115536
$604$	−12.3993	−0.504518
$605$	−1.20607	−0.0490336
$606$	15.6591	0.636106
$607$	−34.0810	−1.38330	−0.691652	−	0.722231i	$-0.743117\pi$
$607$	−34.0810	−1.38330	−0.691652	+	0.722231i	$0.743117\pi$
$608$	23.0841	0.936184
$609$	−8.58077	−0.347710
$610$	2.73500	0.110737
$611$	11.5519	0.467339
$612$	1.44333	0.0583430
$613$	4.13484	0.167005	0.0835023	−	0.996508i	$-0.473389\pi$
$613$	4.13484	0.167005	0.0835023	+	0.996508i	$0.473389\pi$
$614$	−45.1710	−1.82295
$615$	3.54660	0.143013
$616$	11.2088	0.451614
$617$	−9.92687	−0.399641	−0.199820	−	0.979833i	$-0.564036\pi$
$617$	−9.92687	−0.399641	−0.199820	+	0.979833i	$0.564036\pi$
$618$	−8.73771	−0.351482
$619$	−3.88666	−0.156218	−0.0781090	−	0.996945i	$-0.524888\pi$
$619$	−3.88666	−0.156218	−0.0781090	+	0.996945i	$0.524888\pi$
$620$	−5.17904	−0.207995
$621$	1.24607	0.0500031
$622$	31.0373	1.24448
$623$	−12.0763	−0.483827
$624$	27.4868	1.10035
$625$	22.2911	0.891645
$626$	3.08818	0.123428
$627$	12.5328	0.500514
$628$	−1.44333	−0.0575950
$629$	−6.95597	−0.277353
$630$	2.31581	0.0922641
$631$	−38.6394	−1.53821	−0.769105	−	0.639122i	$-0.779298\pi$
$631$	−38.6394	−1.53821	−0.769105	+	0.639122i	$0.779298\pi$
$632$	13.3139	0.529599
$633$	7.44346	0.295851
$634$	36.0291	1.43090
$635$	7.14377	0.283492
$636$	−15.4020	−0.610731
$637$	−8.69397	−0.344468
$638$	20.2805	0.802912
$639$	−8.17504	−0.323400
$640$	3.39652	0.134259
$641$	0.809694	0.0319810	0.0159905	−	0.999872i	$-0.494910\pi$
$641$	0.809694	0.0319810	0.0159905	+	0.999872i	$0.494910\pi$
$642$	18.5572	0.732392
$643$	−44.6352	−1.76024	−0.880120	−	0.474751i	$-0.842538\pi$
$643$	−44.6352	−1.76024	−0.880120	+	0.474751i	$0.842538\pi$
$644$	5.24940	0.206855
$645$	4.03909	0.159039
$646$	−6.25566	−0.246126
$647$	−2.59508	−0.102023	−0.0510115	−	0.998698i	$-0.516245\pi$
$647$	−2.59508	−0.102023	−0.0510115	+	0.998698i	$0.516245\pi$
$648$	1.03298	0.0405791
$649$	−1.61409	−0.0633584
$650$	−51.1508	−2.00630
$651$	24.4949	0.960031
$652$	29.8258	1.16807
$653$	−21.9742	−0.859917	−0.429958	−	0.902849i	$-0.641472\pi$
$653$	−21.9742	−0.859917	−0.429958	+	0.902849i	$0.641472\pi$
$654$	−27.1740	−1.06259
$655$	0.0660403	0.00258041
$656$	39.8433	1.55562
$657$	−4.55373	−0.177658
$658$	10.9339	0.426247
$659$	23.8289	0.928240	0.464120	−	0.885772i	$-0.346371\pi$
$659$	23.8289	0.928240	0.464120	+	0.885772i	$0.346371\pi$
$660$	−2.29425	−0.0893037
$661$	6.39294	0.248657	0.124328	−	0.992241i	$-0.460322\pi$
$661$	6.39294	0.248657	0.124328	+	0.992241i	$0.460322\pi$
$662$	23.6732	0.920085
$663$	−5.72229	−0.222235
$664$	−8.28321	−0.321451
$665$	−4.20725	−0.163150
$666$	12.9076	0.500161
$667$	−3.66325	−0.141842
$668$	−25.0941	−0.970919
$669$	2.86277	0.110681
$670$	−2.25100	−0.0869636
$671$	12.8151	0.494721
$672$	19.9862	0.770986
$673$	−27.7245	−1.06870	−0.534351	−	0.845263i	$-0.679444\pi$
$673$	−27.7245	−1.06870	−0.534351	+	0.845263i	$0.679444\pi$
$674$	−61.5122	−2.36936
$675$	−4.81718	−0.185413
$676$	28.4979	1.09607
$677$	7.61930	0.292834	0.146417	−	0.989223i	$-0.453226\pi$
$677$	7.61930	0.292834	0.146417	+	0.989223i	$0.453226\pi$
$678$	17.5031	0.672202
$679$	−27.4360	−1.05290
$680$	−0.441674	−0.0169374
$681$	−2.38216	−0.0912844
$682$	−57.8932	−2.21685
$683$	32.6609	1.24974	0.624868	−	0.780730i	$-0.285153\pi$
$683$	32.6609	1.24974	0.624868	+	0.780730i	$0.285153\pi$
$684$	4.86574	0.186046
$685$	5.05422	0.193112
$686$	29.6843	1.13335
$687$	−9.43205	−0.359855
$688$	45.3760	1.72994
$689$	61.0638	2.32635
$690$	0.988653	0.0376374
$691$	−32.8681	−1.25036	−0.625180	−	0.780481i	$-0.714974\pi$
$691$	−32.8681	−1.25036	−0.625180	+	0.780481i	$0.714974\pi$
$692$	−19.1230	−0.726949
$693$	10.8509	0.412193
$694$	24.3073	0.922691
$695$	7.72012	0.292841
$696$	−3.03679	−0.115109
$697$	−8.29470	−0.314184
$698$	−6.71436	−0.254142
$699$	3.81786	0.144405
$700$	−20.2936	−0.767027
$701$	−48.4506	−1.82995	−0.914977	−	0.403507i	$-0.867791\pi$
$701$	−48.4506	−1.82995	−0.914977	+	0.403507i	$0.867791\pi$
$702$	10.6184	0.400766
$703$	−23.4500	−0.884432
$704$	−11.5222	−0.434258
$705$	0.863166	0.0325087
$706$	−57.7555	−2.17366
$707$	−24.6308	−0.926337
$708$	−0.626651	−0.0235510
$709$	18.5949	0.698348	0.349174	−	0.937058i	$-0.386462\pi$
$709$	18.5949	0.698348	0.349174	+	0.937058i	$0.386462\pi$
$710$	−6.48620	−0.243423
$711$	12.8889	0.483371
$712$	−4.27388	−0.160170
$713$	10.4572	0.391626
$714$	−5.41616	−0.202695
$715$	9.09593	0.340168
$716$	14.7341	0.550641
$717$	−0.642610	−0.0239987
$718$	−44.2132	−1.65002
$719$	−1.78344	−0.0665110	−0.0332555	−	0.999447i	$-0.510588\pi$
$719$	−1.78344	−0.0665110	−0.0332555	+	0.999447i	$0.510588\pi$
$720$	2.05384	0.0765420
$721$	13.7439	0.511851
$722$	14.1677	0.527267
$723$	4.81558	0.179093
$724$	4.42870	0.164591
$725$	14.1617	0.525954
$726$	−5.23419	−0.194259
$727$	−27.2743	−1.01155	−0.505774	−	0.862666i	$-0.668793\pi$
$727$	−27.2743	−1.01155	−0.505774	+	0.862666i	$0.668793\pi$
$728$	−17.2529	−0.639436
$729$	1.00000	0.0370370
$730$	−3.61300	−0.133723
$731$	−9.44652	−0.349392
$732$	4.97531	0.183893
$733$	−27.3904	−1.01169	−0.505843	−	0.862626i	$-0.668818\pi$
$733$	−27.3904	−1.01169	−0.505843	+	0.862626i	$0.668818\pi$
$734$	32.0422	1.18270
$735$	−0.649620	−0.0239616
$736$	8.53241	0.314509
$737$	−10.5473	−0.388513
$738$	15.3918	0.566581
$739$	30.4595	1.12047	0.560236	−	0.828333i	$-0.310710\pi$
$739$	30.4595	1.12047	0.560236	+	0.828333i	$0.310710\pi$
$740$	4.29273	0.157804
$741$	−19.2910	−0.708672
$742$	57.7970	2.12180
$743$	−39.2635	−1.44044	−0.720219	−	0.693746i	$-0.755959\pi$
$743$	−39.2635	−1.44044	−0.720219	+	0.693746i	$0.755959\pi$
$744$	8.66889	0.317817
$745$	−3.95591	−0.144933
$746$	−67.7989	−2.48229
$747$	−8.01878	−0.293392
$748$	5.36574	0.196191
$749$	−29.1894	−1.06656
$750$	−7.78910	−0.284418
$751$	−3.85273	−0.140588	−0.0702940	−	0.997526i	$-0.522394\pi$
$751$	−3.85273	−0.140588	−0.0702940	+	0.997526i	$0.522394\pi$
$752$	9.69699	0.353613
$753$	5.87884	0.214237
$754$	−31.2164	−1.13683
$755$	3.67319	0.133681
$756$	4.21276	0.153217
$757$	17.0003	0.617888	0.308944	−	0.951080i	$-0.400024\pi$
$757$	17.0003	0.617888	0.308944	+	0.951080i	$0.400024\pi$
$758$	−13.6079	−0.494261
$759$	4.63243	0.168146
$760$	−1.48897	−0.0540107
$761$	−33.4041	−1.21090	−0.605448	−	0.795885i	$-0.707006\pi$
$761$	−33.4041	−1.21090	−0.605448	+	0.795885i	$0.707006\pi$
$762$	31.0031	1.12312
$763$	42.7431	1.54740
$764$	−8.64130	−0.312631
$765$	−0.427574	−0.0154590
$766$	−15.4219	−0.557215
$767$	2.48446	0.0897086
$768$	20.9392	0.755577
$769$	−39.5218	−1.42519	−0.712597	−	0.701574i	$-0.752481\pi$
$769$	−39.5218	−1.42519	−0.712597	+	0.701574i	$0.752481\pi$
$770$	8.60931	0.310258
$771$	19.6794	0.708735
$772$	19.0482	0.685558
$773$	35.0460	1.26052	0.630259	−	0.776385i	$-0.282949\pi$
$773$	35.0460	1.26052	0.630259	+	0.776385i	$0.282949\pi$
$774$	17.5291	0.630072
$775$	−40.4265	−1.45216
$776$	−9.70977	−0.348561
$777$	−20.3030	−0.728366
$778$	−10.9023	−0.390866
$779$	−27.9631	−1.00188
$780$	3.53139	0.126444
$781$	−30.3917	−1.08750
$782$	−2.31224	−0.0826854
$783$	−2.93984	−0.105061
$784$	−7.29797	−0.260642
$785$	0.427574	0.0152608
$786$	0.286607	0.0102229
$787$	−42.7226	−1.52290	−0.761448	−	0.648226i	$-0.775511\pi$
$787$	−42.7226	−1.52290	−0.761448	+	0.648226i	$0.775511\pi$
$788$	−1.41907	−0.0505523
$789$	−13.2604	−0.472082
$790$	10.2263	0.363834
$791$	−27.5314	−0.978903
$792$	3.84022	0.136456
$793$	−19.7254	−0.700470
$794$	−42.8237	−1.51976
$795$	4.56274	0.161824
$796$	−0.361977	−0.0128299
$797$	19.1437	0.678103	0.339052	−	0.940768i	$-0.389894\pi$
$797$	19.1437	0.678103	0.339052	+	0.940768i	$0.389894\pi$
$798$	−18.2589	−0.646360
$799$	−2.01875	−0.0714182
$800$	−32.9854	−1.16621
$801$	−4.13744	−0.146189
$802$	9.43713	0.333237
$803$	−16.9291	−0.597414
$804$	−4.09485	−0.144414
$805$	−1.55510	−0.0548099
$806$	89.1112	3.13881
$807$	10.4806	0.368933
$808$	−8.71700	−0.306663
$809$	−39.4115	−1.38563	−0.692817	−	0.721114i	$-0.743631\pi$
$809$	−39.4115	−1.38563	−0.692817	+	0.721114i	$0.743631\pi$
$810$	0.793415	0.0278778
$811$	−29.0804	−1.02115	−0.510575	−	0.859833i	$-0.670567\pi$
$811$	−29.0804	−1.02115	−0.510575	+	0.859833i	$0.670567\pi$
$812$	−12.3848	−0.434623
$813$	10.4553	0.366684
$814$	47.9857	1.68190
$815$	−8.83568	−0.309500
$816$	−4.80346	−0.168155
$817$	−31.8461	−1.11415
$818$	65.0022	2.27275
$819$	−16.7022	−0.583620
$820$	5.11890	0.178760
$821$	−42.1353	−1.47053	−0.735267	−	0.677778i	$-0.762943\pi$
$821$	−42.1353	−1.47053	−0.735267	+	0.677778i	$0.762943\pi$
$822$	21.9347	0.765061
$823$	−43.5826	−1.51919	−0.759596	−	0.650395i	$-0.774603\pi$
$823$	−43.5826	−1.51919	−0.759596	+	0.650395i	$0.774603\pi$
$824$	4.86406	0.169448
$825$	−17.9085	−0.623493
$826$	2.35154	0.0818206
$827$	−20.7407	−0.721224	−0.360612	−	0.932716i	$-0.617432\pi$
$827$	−20.7407	−0.721224	−0.360612	+	0.932716i	$0.617432\pi$
$828$	1.79849	0.0625018
$829$	45.3062	1.57355	0.786774	−	0.617241i	$-0.211750\pi$
$829$	45.3062	1.57355	0.786774	+	0.617241i	$0.211750\pi$
$830$	−6.36223	−0.220836
$831$	10.3173	0.357902
$832$	17.7353	0.614862
$833$	1.51932	0.0526412
$834$	33.5044	1.16016
$835$	7.43393	0.257262
$836$	18.0890	0.625620
$837$	8.39215	0.290075
$838$	16.5170	0.570571
$839$	−36.5098	−1.26046	−0.630229	−	0.776409i	$-0.717039\pi$
$839$	−36.5098	−1.26046	−0.630229	+	0.776409i	$0.717039\pi$
$840$	−1.28915	−0.0444800
$841$	−20.3573	−0.701977
$842$	−7.71619	−0.265917
$843$	14.5866	0.502388
$844$	10.7433	0.369801
$845$	−8.44230	−0.290424
$846$	3.74603	0.128791
$847$	8.23308	0.282892
$848$	51.2588	1.76023
$849$	27.4171	0.940954
$850$	8.93886	0.306600
$851$	−8.66764	−0.297123
$852$	−11.7992	−0.404235
$853$	43.9990	1.50650	0.753248	−	0.657736i	$-0.228486\pi$
$853$	43.9990	1.50650	0.753248	+	0.657736i	$0.228486\pi$
$854$	−18.6701	−0.638879
$855$	−1.44144	−0.0492961
$856$	−10.3303	−0.353082
$857$	−0.476595	−0.0162802	−0.00814009	−	0.999967i	$-0.502591\pi$
$857$	−0.476595	−0.0162802	−0.00814009	+	0.999967i	$0.502591\pi$
$858$	39.4752	1.34766
$859$	−9.32646	−0.318215	−0.159107	−	0.987261i	$-0.550862\pi$
$859$	−9.32646	−0.318215	−0.159107	+	0.987261i	$0.550862\pi$
$860$	5.82972	0.198792
$861$	−24.2105	−0.825090
$862$	−44.4629	−1.51441
$863$	56.6647	1.92889	0.964445	−	0.264285i	$-0.0851360\pi$
$863$	56.6647	1.92889	0.964445	+	0.264285i	$0.0851360\pi$
$864$	6.84745	0.232955
$865$	5.66505	0.192618
$866$	33.4530	1.13678
$867$	1.00000	0.0339618
$868$	35.3541	1.20000
$869$	47.9161	1.62544
$870$	−2.33251	−0.0790797
$871$	16.2347	0.550092
$872$	15.1270	0.512267
$873$	−9.39980	−0.318135
$874$	−7.79501	−0.263670
$875$	12.2518	0.414187
$876$	−6.57252	−0.222065
$877$	35.6003	1.20214	0.601068	−	0.799198i	$-0.294742\pi$
$877$	35.6003	1.20214	0.601068	+	0.799198i	$0.294742\pi$
$878$	−58.9030	−1.98788
$879$	−18.0735	−0.609603
$880$	7.63539	0.257389
$881$	52.5686	1.77108	0.885540	−	0.464563i	$-0.153789\pi$
$881$	52.5686	1.77108	0.885540	+	0.464563i	$0.153789\pi$
$882$	−2.81927	−0.0949298
$883$	37.1871	1.25144	0.625722	−	0.780046i	$-0.284804\pi$
$883$	37.1871	1.25144	0.625722	+	0.780046i	$0.284804\pi$
$884$	−8.25913	−0.277785
$885$	0.185641	0.00624024
$886$	15.7003	0.527462
$887$	7.38797	0.248064	0.124032	−	0.992278i	$-0.460417\pi$
$887$	7.38797	0.248064	0.124032	+	0.992278i	$0.460417\pi$
$888$	−7.18535	−0.241125
$889$	−48.7661	−1.63556
$890$	−3.28271	−0.110037
$891$	3.71762	0.124545
$892$	4.13191	0.138346
$893$	−6.80561	−0.227741
$894$	−17.1682	−0.574189
$895$	−4.36488	−0.145902
$896$	−23.1860	−0.774589
$897$	−7.13039	−0.238077
$898$	−25.4943	−0.850756
$899$	−24.6716	−0.822843
$900$	−6.95276	−0.231759
$901$	−10.6712	−0.355510
$902$	57.2210	1.90525
$903$	−27.5724	−0.917551
$904$	−9.74352	−0.324065
$905$	−1.31197	−0.0436113
$906$	15.9412	0.529610
$907$	10.3892	0.344969	0.172485	−	0.985012i	$-0.444821\pi$
$907$	10.3892	0.344969	0.172485	+	0.985012i	$0.444821\pi$
$908$	−3.43823	−0.114102
$909$	−8.43872	−0.279895
$910$	−13.2517	−0.439291
$911$	6.53421	0.216488	0.108244	−	0.994124i	$-0.465477\pi$
$911$	6.53421	0.216488	0.108244	+	0.994124i	$0.465477\pi$
$912$	−16.1934	−0.536218
$913$	−29.8108	−0.986594
$914$	44.0996	1.45868
$915$	−1.47390	−0.0487256
$916$	−13.6135	−0.449804
$917$	−0.450816	−0.0148873
$918$	−1.85562	−0.0612446
$919$	20.7273	0.683731	0.341865	−	0.939749i	$-0.388941\pi$
$919$	20.7273	0.683731	0.341865	+	0.939749i	$0.388941\pi$
$920$	−0.550358	−0.0181448
$921$	24.3428	0.802122
$922$	67.2390	2.21440
$923$	46.7800	1.53978
$924$	15.6615	0.515224
$925$	33.5082	1.10174
$926$	42.0744	1.38265
$927$	4.70878	0.154657
$928$	−20.1304	−0.660813
$929$	−51.2603	−1.68180	−0.840898	−	0.541193i	$-0.817973\pi$
$929$	−51.2603	−1.68180	−0.840898	+	0.541193i	$0.817973\pi$
$930$	6.65846	0.218340
$931$	5.12191	0.167864
$932$	5.51042	0.180500
$933$	−16.7261	−0.547588
$934$	−38.5150	−1.26025
$935$	−1.58956	−0.0519842
$936$	−5.91099	−0.193207
$937$	−2.83270	−0.0925404	−0.0462702	−	0.998929i	$-0.514734\pi$
$937$	−2.83270	−0.0925404	−0.0462702	+	0.998929i	$0.514734\pi$
$938$	15.3662	0.501723
$939$	−1.66423	−0.0543100
$940$	1.24583	0.0406345
$941$	60.5525	1.97396	0.986978	−	0.160857i	$-0.0514259\pi$
$941$	60.5525	1.97396	0.986978	+	0.160857i	$0.0514259\pi$
$942$	1.85562	0.0604594
$943$	−10.3358	−0.336580
$944$	2.08553	0.0678781
$945$	−1.24800	−0.0405974
$946$	65.1667	2.11875
$947$	−48.1629	−1.56508	−0.782542	−	0.622598i	$-0.786077\pi$
$947$	−48.1629	−1.56508	−0.782542	+	0.622598i	$0.786077\pi$
$948$	18.6029	0.604193
$949$	26.0578	0.845872
$950$	30.1347	0.977697
$951$	−19.4162	−0.629614
$952$	3.01504	0.0977179
$953$	5.31333	0.172116	0.0860578	−	0.996290i	$-0.472573\pi$
$953$	5.31333	0.172116	0.0860578	+	0.996290i	$0.472573\pi$
$954$	19.8017	0.641105
$955$	2.55992	0.0828371
$956$	−0.927496	−0.0299974
$957$	−10.9292	−0.353291
$958$	14.9265	0.482252
$959$	−34.5021	−1.11413
$960$	1.32520	0.0427706
$961$	39.4282	1.27188
$962$	−73.8613	−2.38138
$963$	−10.0005	−0.322262
$964$	6.95045	0.223859
$965$	−5.64287	−0.181651
$966$	−6.74892	−0.217143
$967$	34.2603	1.10174	0.550868	−	0.834592i	$-0.314297\pi$
$967$	34.2603	1.10174	0.550868	+	0.834592i	$0.314297\pi$
$968$	2.91374	0.0936511
$969$	3.37120	0.108298
$970$	−7.45795	−0.239460
$971$	−25.6579	−0.823402	−0.411701	−	0.911319i	$-0.635065\pi$
$971$	−25.6579	−0.823402	−0.411701	+	0.911319i	$0.635065\pi$
$972$	1.44333	0.0462947
$973$	−52.7005	−1.68950
$974$	50.2014	1.60856
$975$	27.5653	0.882797
$976$	−16.5581	−0.530012
$977$	−31.1670	−0.997120	−0.498560	−	0.866855i	$-0.666138\pi$
$977$	−31.1670	−0.997120	−0.498560	+	0.866855i	$0.666138\pi$
$978$	−38.3458	−1.22616
$979$	−15.3815	−0.491593
$980$	−0.937613	−0.0299510
$981$	14.6441	0.467551
$982$	−7.52199	−0.240036
$983$	−5.68313	−0.181264	−0.0906319	−	0.995884i	$-0.528889\pi$
$983$	−5.68313	−0.181264	−0.0906319	+	0.995884i	$0.528889\pi$
$984$	−8.56823	−0.273145
$985$	0.420389	0.0133947
$986$	5.45523	0.173730
$987$	−5.89230	−0.187554
$988$	−27.8432	−0.885809
$989$	−11.7710	−0.374297
$990$	2.94962	0.0937451
$991$	7.84091	0.249075	0.124537	−	0.992215i	$-0.460255\pi$
$991$	7.84091	0.249075	0.124537	+	0.992215i	$0.460255\pi$
$992$	57.4648	1.82451
$993$	−12.7576	−0.404849
$994$	44.2773	1.40439
$995$	0.107233	0.00339951
$996$	−11.5737	−0.366727
$997$	37.6229	1.19153	0.595764	−	0.803160i	$-0.296849\pi$
$997$	37.6229	1.19153	0.595764	+	0.803160i	$0.296849\pi$
$998$	14.4384	0.457041
$999$	−6.95597	−0.220077

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.d.1.9	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.d.1.9	✓	40	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-1.85562 q^{2} +1.00000 q^{3} +1.44333 q^{4} -0.427574 q^{5} -1.85562 q^{6} +2.91879 q^{7} +1.03298 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.85562 q^{2} +1.00000 q^{3} +1.44333 q^{4} -0.427574 q^{5} -1.85562 q^{6} +2.91879 q^{7} +1.03298 q^{8} +1.00000 q^{9} +0.793415 q^{10} +3.71762 q^{11} +1.44333 q^{12} -5.72229 q^{13} -5.41616 q^{14} -0.427574 q^{15} -4.80346 q^{16} +1.00000 q^{17} -1.85562 q^{18} +3.37120 q^{19} -0.617129 q^{20} +2.91879 q^{21} -6.89850 q^{22} +1.24607 q^{23} +1.03298 q^{24} -4.81718 q^{25} +10.6184 q^{26} +1.00000 q^{27} +4.21276 q^{28} -2.93984 q^{29} +0.793415 q^{30} +8.39215 q^{31} +6.84745 q^{32} +3.71762 q^{33} -1.85562 q^{34} -1.24800 q^{35} +1.44333 q^{36} -6.95597 q^{37} -6.25566 q^{38} -5.72229 q^{39} -0.441674 q^{40} -8.29470 q^{41} -5.41616 q^{42} -9.44652 q^{43} +5.36574 q^{44} -0.427574 q^{45} -2.31224 q^{46} -2.01875 q^{47} -4.80346 q^{48} +1.51932 q^{49} +8.93886 q^{50} +1.00000 q^{51} -8.25913 q^{52} -10.6712 q^{53} -1.85562 q^{54} -1.58956 q^{55} +3.01504 q^{56} +3.37120 q^{57} +5.45523 q^{58} -0.434172 q^{59} -0.617129 q^{60} +3.44712 q^{61} -15.5726 q^{62} +2.91879 q^{63} -3.09934 q^{64} +2.44671 q^{65} -6.89850 q^{66} -2.83710 q^{67} +1.44333 q^{68} +1.24607 q^{69} +2.31581 q^{70} -8.17504 q^{71} +1.03298 q^{72} -4.55373 q^{73} +12.9076 q^{74} -4.81718 q^{75} +4.86574 q^{76} +10.8509 q^{77} +10.6184 q^{78} +12.8889 q^{79} +2.05384 q^{80} +1.00000 q^{81} +15.3918 q^{82} -8.01878 q^{83} +4.21276 q^{84} -0.427574 q^{85} +17.5291 q^{86} -2.93984 q^{87} +3.84022 q^{88} -4.13744 q^{89} +0.793415 q^{90} -16.7022 q^{91} +1.79849 q^{92} +8.39215 q^{93} +3.74603 q^{94} -1.44144 q^{95} +6.84745 q^{96} -9.39980 q^{97} -2.81927 q^{98} +3.71762 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 7 q^{2} + 40 q^{3} + 29 q^{4} - 15 q^{5} - 7 q^{6} - 13 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) \) 40 * q - 7 * q^2 + 40 * q^3 + 29 * q^4 - 15 * q^5 - 7 * q^6 - 13 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 7 q^{2} + 40 q^{3} + 29 q^{4} - 15 q^{5} - 7 q^{6} - 13 q^{7} - 18 q^{8} + 40 q^{9} - 6 q^{10} - 25 q^{11} + 29 q^{12} - 24 q^{13} - 22 q^{14} - 15 q^{15} + 7 q^{16} + 40 q^{17} - 7 q^{18} - 18 q^{19} - 20 q^{20} - 13 q^{21} - 25 q^{22} - 28 q^{23} - 18 q^{24} - 11 q^{25} - 13 q^{26} + 40 q^{27} - 8 q^{28} - 23 q^{29} - 6 q^{30} - 11 q^{31} - 23 q^{32} - 25 q^{33} - 7 q^{34} - 45 q^{35} + 29 q^{36} - 38 q^{37} - 30 q^{38} - 24 q^{39} - 12 q^{40} - 33 q^{41} - 22 q^{42} - 25 q^{43} - 14 q^{44} - 15 q^{45} + 8 q^{46} - 55 q^{47} + 7 q^{48} - 21 q^{49} + 2 q^{50} + 40 q^{51} - 39 q^{52} - 39 q^{53} - 7 q^{54} - 9 q^{55} - 48 q^{56} - 18 q^{57} - 13 q^{58} - 81 q^{59} - 20 q^{60} - 9 q^{61} - 16 q^{62} - 13 q^{63} - 4 q^{64} - 43 q^{65} - 25 q^{66} - 24 q^{67} + 29 q^{68} - 28 q^{69} + 48 q^{70} - 32 q^{71} - 18 q^{72} - 43 q^{73} - 20 q^{74} - 11 q^{75} - 58 q^{76} - 32 q^{77} - 13 q^{78} - 22 q^{79} - 48 q^{80} + 40 q^{81} - 11 q^{82} - 45 q^{83} - 8 q^{84} - 15 q^{85} - 30 q^{86} - 23 q^{87} - 48 q^{88} - 94 q^{89} - 6 q^{90} - 7 q^{91} - 98 q^{92} - 11 q^{93} + 32 q^{94} - 23 q^{96} - 28 q^{97} - 46 q^{98} - 25 q^{99}+O(q^{100}) \) 40 * q - 7 * q^2 + 40 * q^3 + 29 * q^4 - 15 * q^5 - 7 * q^6 - 13 * q^7 - 18 * q^8 + 40 * q^9 - 6 * q^10 - 25 * q^11 + 29 * q^12 - 24 * q^13 - 22 * q^14 - 15 * q^15 + 7 * q^16 + 40 * q^17 - 7 * q^18 - 18 * q^19 - 20 * q^20 - 13 * q^21 - 25 * q^22 - 28 * q^23 - 18 * q^24 - 11 * q^25 - 13 * q^26 + 40 * q^27 - 8 * q^28 - 23 * q^29 - 6 * q^30 - 11 * q^31 - 23 * q^32 - 25 * q^33 - 7 * q^34 - 45 * q^35 + 29 * q^36 - 38 * q^37 - 30 * q^38 - 24 * q^39 - 12 * q^40 - 33 * q^41 - 22 * q^42 - 25 * q^43 - 14 * q^44 - 15 * q^45 + 8 * q^46 - 55 * q^47 + 7 * q^48 - 21 * q^49 + 2 * q^50 + 40 * q^51 - 39 * q^52 - 39 * q^53 - 7 * q^54 - 9 * q^55 - 48 * q^56 - 18 * q^57 - 13 * q^58 - 81 * q^59 - 20 * q^60 - 9 * q^61 - 16 * q^62 - 13 * q^63 - 4 * q^64 - 43 * q^65 - 25 * q^66 - 24 * q^67 + 29 * q^68 - 28 * q^69 + 48 * q^70 - 32 * q^71 - 18 * q^72 - 43 * q^73 - 20 * q^74 - 11 * q^75 - 58 * q^76 - 32 * q^77 - 13 * q^78 - 22 * q^79 - 48 * q^80 + 40 * q^81 - 11 * q^82 - 45 * q^83 - 8 * q^84 - 15 * q^85 - 30 * q^86 - 23 * q^87 - 48 * q^88 - 94 * q^89 - 6 * q^90 - 7 * q^91 - 98 * q^92 - 11 * q^93 + 32 * q^94 - 23 * q^96 - 28 * q^97 - 46 * q^98 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more