LMFDB - Embedded newform 8002.2.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$0$
Dimension:	$77$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.91842 q^{3} +1.00000 q^{4} +3.04954 q^{5} +2.91842 q^{6} +3.39338 q^{7} -1.00000 q^{8} +5.51716 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.91842 q^{3} +1.00000 q^{4} +3.04954 q^{5} +2.91842 q^{6} +3.39338 q^{7} -1.00000 q^{8} +5.51716 q^{9} -3.04954 q^{10} +3.24275 q^{11} -2.91842 q^{12} -3.02838 q^{13} -3.39338 q^{14} -8.89983 q^{15} +1.00000 q^{16} +2.71304 q^{17} -5.51716 q^{18} +1.92519 q^{19} +3.04954 q^{20} -9.90330 q^{21} -3.24275 q^{22} +1.77570 q^{23} +2.91842 q^{24} +4.29970 q^{25} +3.02838 q^{26} -7.34612 q^{27} +3.39338 q^{28} +1.39033 q^{29} +8.89983 q^{30} -9.18702 q^{31} -1.00000 q^{32} -9.46369 q^{33} -2.71304 q^{34} +10.3482 q^{35} +5.51716 q^{36} +4.71373 q^{37} -1.92519 q^{38} +8.83807 q^{39} -3.04954 q^{40} -11.1910 q^{41} +9.90330 q^{42} -12.0232 q^{43} +3.24275 q^{44} +16.8248 q^{45} -1.77570 q^{46} +6.09693 q^{47} -2.91842 q^{48} +4.51503 q^{49} -4.29970 q^{50} -7.91777 q^{51} -3.02838 q^{52} -8.04949 q^{53} +7.34612 q^{54} +9.88889 q^{55} -3.39338 q^{56} -5.61851 q^{57} -1.39033 q^{58} +6.52281 q^{59} -8.89983 q^{60} +6.23515 q^{61} +9.18702 q^{62} +18.7218 q^{63} +1.00000 q^{64} -9.23516 q^{65} +9.46369 q^{66} +11.9464 q^{67} +2.71304 q^{68} -5.18223 q^{69} -10.3482 q^{70} -1.32813 q^{71} -5.51716 q^{72} +5.31705 q^{73} -4.71373 q^{74} -12.5483 q^{75} +1.92519 q^{76} +11.0039 q^{77} -8.83807 q^{78} +10.2976 q^{79} +3.04954 q^{80} +4.88755 q^{81} +11.1910 q^{82} +11.8365 q^{83} -9.90330 q^{84} +8.27351 q^{85} +12.0232 q^{86} -4.05757 q^{87} -3.24275 q^{88} -1.96091 q^{89} -16.8248 q^{90} -10.2764 q^{91} +1.77570 q^{92} +26.8116 q^{93} -6.09693 q^{94} +5.87095 q^{95} +2.91842 q^{96} +17.6719 q^{97} -4.51503 q^{98} +17.8907 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) $ 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.91842	−1.68495	−0.842474	−	0.538736i	$-0.818902\pi$
$3$	−2.91842	−1.68495	−0.842474	+	0.538736i	$0.818902\pi$
$4$	1.00000	0.500000
$5$	3.04954	1.36380	0.681898	−	0.731447i	$-0.261155\pi$
$5$	3.04954	1.36380	0.681898	+	0.731447i	$0.261155\pi$
$6$	2.91842	1.19144
$7$	3.39338	1.28258	0.641289	−	0.767300i	$-0.278400\pi$
$7$	3.39338	1.28258	0.641289	+	0.767300i	$0.278400\pi$
$8$	−1.00000	−0.353553
$9$	5.51716	1.83905
$10$	−3.04954	−0.964349
$11$	3.24275	0.977725	0.488862	−	0.872361i	$-0.337412\pi$
$11$	3.24275	0.977725	0.488862	+	0.872361i	$0.337412\pi$
$12$	−2.91842	−0.842474
$13$	−3.02838	−0.839921	−0.419960	−	0.907542i	$-0.637956\pi$
$13$	−3.02838	−0.839921	−0.419960	+	0.907542i	$0.637956\pi$
$14$	−3.39338	−0.906919
$15$	−8.89983	−2.29793
$16$	1.00000	0.250000
$17$	2.71304	0.658008	0.329004	−	0.944329i	$-0.393287\pi$
$17$	2.71304	0.658008	0.329004	+	0.944329i	$0.393287\pi$
$18$	−5.51716	−1.30041
$19$	1.92519	0.441669	0.220835	−	0.975311i	$-0.429122\pi$
$19$	1.92519	0.441669	0.220835	+	0.975311i	$0.429122\pi$
$20$	3.04954	0.681898
$21$	−9.90330	−2.16108
$22$	−3.24275	−0.691356
$23$	1.77570	0.370259	0.185129	−	0.982714i	$-0.440730\pi$
$23$	1.77570	0.370259	0.185129	+	0.982714i	$0.440730\pi$
$24$	2.91842	0.595719
$25$	4.29970	0.859939
$26$	3.02838	0.593914
$27$	−7.34612	−1.41376
$28$	3.39338	0.641289
$29$	1.39033	0.258178	0.129089	−	0.991633i	$-0.458795\pi$
$29$	1.39033	0.258178	0.129089	+	0.991633i	$0.458795\pi$
$30$	8.89983	1.62488
$31$	−9.18702	−1.65004	−0.825019	−	0.565105i	$-0.808836\pi$
$31$	−9.18702	−1.65004	−0.825019	+	0.565105i	$0.808836\pi$
$32$	−1.00000	−0.176777
$33$	−9.46369	−1.64742
$34$	−2.71304	−0.465282
$35$	10.3482	1.74917
$36$	5.51716	0.919526
$37$	4.71373	0.774933	0.387467	−	0.921884i	$-0.373350\pi$
$37$	4.71373	0.774933	0.387467	+	0.921884i	$0.373350\pi$
$38$	−1.92519	−0.312307
$39$	8.83807	1.41522
$40$	−3.04954	−0.482175
$41$	−11.1910	−1.74774	−0.873869	−	0.486161i	$-0.838397\pi$
$41$	−11.1910	−1.74774	−0.873869	+	0.486161i	$0.838397\pi$
$42$	9.90330	1.52811
$43$	−12.0232	−1.83353	−0.916764	−	0.399430i	$-0.869208\pi$
$43$	−12.0232	−1.83353	−0.916764	+	0.399430i	$0.869208\pi$
$44$	3.24275	0.488862
$45$	16.8248	2.50809
$46$	−1.77570	−0.261812
$47$	6.09693	0.889328	0.444664	−	0.895697i	$-0.353323\pi$
$47$	6.09693	0.889328	0.444664	+	0.895697i	$0.353323\pi$
$48$	−2.91842	−0.421237
$49$	4.51503	0.645004
$50$	−4.29970	−0.608069
$51$	−7.91777	−1.10871
$52$	−3.02838	−0.419960
$53$	−8.04949	−1.10568	−0.552841	−	0.833287i	$-0.686456\pi$
$53$	−8.04949	−1.10568	−0.552841	+	0.833287i	$0.686456\pi$
$54$	7.34612	0.999680
$55$	9.88889	1.33342
$56$	−3.39338	−0.453460
$57$	−5.61851	−0.744190
$58$	−1.39033	−0.182559
$59$	6.52281	0.849197	0.424599	−	0.905382i	$-0.360415\pi$
$59$	6.52281	0.849197	0.424599	+	0.905382i	$0.360415\pi$
$60$	−8.89983	−1.14896
$61$	6.23515	0.798330	0.399165	−	0.916879i	$-0.369300\pi$
$61$	6.23515	0.798330	0.399165	+	0.916879i	$0.369300\pi$
$62$	9.18702	1.16675
$63$	18.7218	2.35873
$64$	1.00000	0.125000
$65$	−9.23516	−1.14548
$66$	9.46369	1.16490
$67$	11.9464	1.45948	0.729742	−	0.683723i	$-0.239640\pi$
$67$	11.9464	1.45948	0.729742	+	0.683723i	$0.239640\pi$
$68$	2.71304	0.329004
$69$	−5.18223	−0.623867
$70$	−10.3482	−1.23685
$71$	−1.32813	−0.157620	−0.0788098	−	0.996890i	$-0.525112\pi$
$71$	−1.32813	−0.157620	−0.0788098	+	0.996890i	$0.525112\pi$
$72$	−5.51716	−0.650203
$73$	5.31705	0.622314	0.311157	−	0.950359i	$-0.399284\pi$
$73$	5.31705	0.622314	0.311157	+	0.950359i	$0.399284\pi$
$74$	−4.71373	−0.547961
$75$	−12.5483	−1.44895
$76$	1.92519	0.220835
$77$	11.0039	1.25401
$78$	−8.83807	−1.00071
$79$	10.2976	1.15857	0.579285	−	0.815125i	$-0.303332\pi$
$79$	10.2976	1.15857	0.579285	+	0.815125i	$0.303332\pi$
$80$	3.04954	0.340949
$81$	4.88755	0.543062
$82$	11.1910	1.23584
$83$	11.8365	1.29922	0.649610	−	0.760267i	$-0.274932\pi$
$83$	11.8365	1.29922	0.649610	+	0.760267i	$0.274932\pi$
$84$	−9.90330	−1.08054
$85$	8.27351	0.897389
$86$	12.0232	1.29650
$87$	−4.05757	−0.435017
$88$	−3.24275	−0.345678
$89$	−1.96091	−0.207856	−0.103928	−	0.994585i	$-0.533141\pi$
$89$	−1.96091	−0.207856	−0.103928	+	0.994585i	$0.533141\pi$
$90$	−16.8248	−1.77349
$91$	−10.2764	−1.07726
$92$	1.77570	0.185129
$93$	26.8116	2.78023
$94$	−6.09693	−0.628850
$95$	5.87095	0.602347
$96$	2.91842	0.297860
$97$	17.6719	1.79431	0.897153	−	0.441721i	$-0.145632\pi$
$97$	17.6719	1.79431	0.897153	+	0.441721i	$0.145632\pi$
$98$	−4.51503	−0.456087
$99$	17.8907	1.79809
$100$	4.29970	0.429970
$101$	−12.1834	−1.21229	−0.606144	−	0.795355i	$-0.707285\pi$
$101$	−12.1834	−1.21229	−0.606144	+	0.795355i	$0.707285\pi$
$102$	7.91777	0.783976
$103$	9.85463	0.971006	0.485503	−	0.874235i	$-0.338637\pi$
$103$	9.85463	0.971006	0.485503	+	0.874235i	$0.338637\pi$
$104$	3.02838	0.296957
$105$	−30.2005	−2.94727
$106$	8.04949	0.781836
$107$	−16.3987	−1.58532	−0.792660	−	0.609664i	$-0.791305\pi$
$107$	−16.3987	−1.58532	−0.792660	+	0.609664i	$0.791305\pi$
$108$	−7.34612	−0.706880
$109$	20.1882	1.93368	0.966838	−	0.255389i	$-0.0822036\pi$
$109$	20.1882	1.93368	0.966838	+	0.255389i	$0.0822036\pi$
$110$	−9.88889	−0.942868
$111$	−13.7566	−1.30572
$112$	3.39338	0.320644
$113$	2.38781	0.224626	0.112313	−	0.993673i	$-0.464174\pi$
$113$	2.38781	0.224626	0.112313	+	0.993673i	$0.464174\pi$
$114$	5.61851	0.526222
$115$	5.41506	0.504957
$116$	1.39033	0.129089
$117$	−16.7080	−1.54466
$118$	−6.52281	−0.600473
$119$	9.20637	0.843946
$120$	8.89983	0.812440
$121$	−0.484593	−0.0440539
$122$	−6.23515	−0.564504
$123$	32.6600	2.94485
$124$	−9.18702	−0.825019
$125$	−2.13561	−0.191015
$126$	−18.7218	−1.66787
$127$	1.82067	0.161558	0.0807791	−	0.996732i	$-0.474259\pi$
$127$	1.82067	0.161558	0.0807791	+	0.996732i	$0.474259\pi$
$128$	−1.00000	−0.0883883
$129$	35.0888	3.08940
$130$	9.23516	0.809977
$131$	10.1233	0.884480	0.442240	−	0.896897i	$-0.354184\pi$
$131$	10.1233	0.884480	0.442240	+	0.896897i	$0.354184\pi$
$132$	−9.46369	−0.823708
$133$	6.53291	0.566475
$134$	−11.9464	−1.03201
$135$	−22.4023	−1.92808
$136$	−2.71304	−0.232641
$137$	5.59503	0.478016	0.239008	−	0.971018i	$-0.423178\pi$
$137$	5.59503	0.478016	0.239008	+	0.971018i	$0.423178\pi$
$138$	5.18223	0.441140
$139$	11.3239	0.960481	0.480241	−	0.877137i	$-0.340549\pi$
$139$	11.3239	0.960481	0.480241	+	0.877137i	$0.340549\pi$
$140$	10.3482	0.874587
$141$	−17.7934	−1.49847
$142$	1.32813	0.111454
$143$	−9.82026	−0.821211
$144$	5.51716	0.459763
$145$	4.23987	0.352102
$146$	−5.31705	−0.440042
$147$	−13.1767	−1.08680
$148$	4.71373	0.387467
$149$	−3.07453	−0.251876	−0.125938	−	0.992038i	$-0.540194\pi$
$149$	−3.07453	−0.251876	−0.125938	+	0.992038i	$0.540194\pi$
$150$	12.5483	1.02456
$151$	−0.609059	−0.0495646	−0.0247823	−	0.999693i	$-0.507889\pi$
$151$	−0.609059	−0.0495646	−0.0247823	+	0.999693i	$0.507889\pi$
$152$	−1.92519	−0.156154
$153$	14.9683	1.21011
$154$	−11.0039	−0.886717
$155$	−28.0162	−2.25032
$156$	8.83807	0.707612
$157$	11.0608	0.882747	0.441374	−	0.897323i	$-0.354491\pi$
$157$	11.0608	0.882747	0.441374	+	0.897323i	$0.354491\pi$
$158$	−10.2976	−0.819233
$159$	23.4918	1.86302
$160$	−3.04954	−0.241087
$161$	6.02562	0.474885
$162$	−4.88755	−0.384003
$163$	15.4803	1.21251	0.606255	−	0.795270i	$-0.292671\pi$
$163$	15.4803	1.21251	0.606255	+	0.795270i	$0.292671\pi$
$164$	−11.1910	−0.873869
$165$	−28.8599	−2.24674
$166$	−11.8365	−0.918688
$167$	−0.310574	−0.0240329	−0.0120165	−	0.999928i	$-0.503825\pi$
$167$	−0.310574	−0.0240329	−0.0120165	+	0.999928i	$0.503825\pi$
$168$	9.90330	0.764056
$169$	−3.82893	−0.294533
$170$	−8.27351	−0.634550
$171$	10.6216	0.812253
$172$	−12.0232	−0.916764
$173$	−2.50587	−0.190518	−0.0952588	−	0.995453i	$-0.530368\pi$
$173$	−2.50587	−0.190518	−0.0952588	+	0.995453i	$0.530368\pi$
$174$	4.05757	0.307603
$175$	14.5905	1.10294
$176$	3.24275	0.244431
$177$	−19.0363	−1.43085
$178$	1.96091	0.146976
$179$	−3.41102	−0.254951	−0.127476	−	0.991842i	$-0.540687\pi$
$179$	−3.41102	−0.254951	−0.127476	+	0.991842i	$0.540687\pi$
$180$	16.8248	1.25405
$181$	−19.0067	−1.41276	−0.706380	−	0.707833i	$-0.749673\pi$
$181$	−19.0067	−1.41276	−0.706380	+	0.707833i	$0.749673\pi$
$182$	10.2764	0.761740
$183$	−18.1968	−1.34514
$184$	−1.77570	−0.130906
$185$	14.3747	1.05685
$186$	−26.8116	−1.96592
$187$	8.79769	0.643351
$188$	6.09693	0.444664
$189$	−24.9282	−1.81326
$190$	−5.87095	−0.425924
$191$	5.35334	0.387354	0.193677	−	0.981065i	$-0.437959\pi$
$191$	5.35334	0.387354	0.193677	+	0.981065i	$0.437959\pi$
$192$	−2.91842	−0.210619
$193$	9.31689	0.670644	0.335322	−	0.942103i	$-0.391155\pi$
$193$	9.31689	0.670644	0.335322	+	0.942103i	$0.391155\pi$
$194$	−17.6719	−1.26877
$195$	26.9520	1.93008
$196$	4.51503	0.322502
$197$	16.2169	1.15541	0.577703	−	0.816247i	$-0.303949\pi$
$197$	16.2169	1.15541	0.577703	+	0.816247i	$0.303949\pi$
$198$	−17.8907	−1.27144
$199$	3.69204	0.261722	0.130861	−	0.991401i	$-0.458226\pi$
$199$	3.69204	0.261722	0.130861	+	0.991401i	$0.458226\pi$
$200$	−4.29970	−0.304034
$201$	−34.8645	−2.45915
$202$	12.1834	0.857218
$203$	4.71792	0.331133
$204$	−7.91777	−0.554355
$205$	−34.1274	−2.38356
$206$	−9.85463	−0.686605
$207$	9.79681	0.680925
$208$	−3.02838	−0.209980
$209$	6.24291	0.431831
$210$	30.2005	2.08403
$211$	−15.4489	−1.06355	−0.531773	−	0.846887i	$-0.678474\pi$
$211$	−15.4489	−1.06355	−0.531773	+	0.846887i	$0.678474\pi$
$212$	−8.04949	−0.552841
$213$	3.87603	0.265581
$214$	16.3987	1.12099
$215$	−36.6654	−2.50056
$216$	7.34612	0.499840
$217$	−31.1751	−2.11630
$218$	−20.1882	−1.36732
$219$	−15.5174	−1.04857
$220$	9.88889	0.666709
$221$	−8.21610	−0.552675
$222$	13.7566	0.923285
$223$	−13.5607	−0.908094	−0.454047	−	0.890978i	$-0.650020\pi$
$223$	−13.5607	−0.908094	−0.454047	+	0.890978i	$0.650020\pi$
$224$	−3.39338	−0.226730
$225$	23.7221	1.58147
$226$	−2.38781	−0.158835
$227$	8.04828	0.534183	0.267091	−	0.963671i	$-0.413937\pi$
$227$	8.04828	0.534183	0.267091	+	0.963671i	$0.413937\pi$
$228$	−5.61851	−0.372095
$229$	−14.2222	−0.939829	−0.469914	−	0.882712i	$-0.655715\pi$
$229$	−14.2222	−0.939829	−0.469914	+	0.882712i	$0.655715\pi$
$230$	−5.41506	−0.357059
$231$	−32.1139	−2.11294
$232$	−1.39033	−0.0912797
$233$	2.83488	0.185719	0.0928597	−	0.995679i	$-0.470399\pi$
$233$	2.83488	0.185719	0.0928597	+	0.995679i	$0.470399\pi$
$234$	16.7080	1.09224
$235$	18.5928	1.21286
$236$	6.52281	0.424599
$237$	−30.0527	−1.95213
$238$	−9.20637	−0.596760
$239$	0.659329	0.0426485	0.0213242	−	0.999773i	$-0.493212\pi$
$239$	0.659329	0.0426485	0.0213242	+	0.999773i	$0.493212\pi$
$240$	−8.89983	−0.574482
$241$	−5.93920	−0.382577	−0.191289	−	0.981534i	$-0.561267\pi$
$241$	−5.93920	−0.382577	−0.191289	+	0.981534i	$0.561267\pi$
$242$	0.484593	0.0311508
$243$	7.77442	0.498729
$244$	6.23515	0.399165
$245$	13.7688	0.879654
$246$	−32.6600	−2.08232
$247$	−5.83021	−0.370967
$248$	9.18702	0.583377
$249$	−34.5437	−2.18912
$250$	2.13561	0.135068
$251$	16.9407	1.06929	0.534644	−	0.845077i	$-0.320446\pi$
$251$	16.9407	1.06929	0.534644	+	0.845077i	$0.320446\pi$
$252$	18.7218	1.17936
$253$	5.75814	0.362011
$254$	−1.82067	−0.114239
$255$	−24.1456	−1.51205
$256$	1.00000	0.0625000
$257$	11.0742	0.690792	0.345396	−	0.938457i	$-0.387745\pi$
$257$	11.0742	0.690792	0.345396	+	0.938457i	$0.387745\pi$
$258$	−35.0888	−2.18454
$259$	15.9955	0.993912
$260$	−9.23516	−0.572740
$261$	7.67068	0.474803
$262$	−10.1233	−0.625422
$263$	17.3168	1.06780	0.533899	−	0.845548i	$-0.320726\pi$
$263$	17.3168	1.06780	0.533899	+	0.845548i	$0.320726\pi$
$264$	9.46369	0.582450
$265$	−24.5472	−1.50793
$266$	−6.53291	−0.400558
$267$	5.72274	0.350226
$268$	11.9464	0.729742
$269$	−0.794480	−0.0484403	−0.0242201	−	0.999707i	$-0.507710\pi$
$269$	−0.794480	−0.0484403	−0.0242201	+	0.999707i	$0.507710\pi$
$270$	22.4023	1.36336
$271$	8.63822	0.524735	0.262367	−	0.964968i	$-0.415497\pi$
$271$	8.63822	0.524735	0.262367	+	0.964968i	$0.415497\pi$
$272$	2.71304	0.164502
$273$	29.9909	1.81513
$274$	−5.59503	−0.338008
$275$	13.9428	0.840784
$276$	−5.18223	−0.311933
$277$	−16.8967	−1.01522	−0.507612	−	0.861586i	$-0.669472\pi$
$277$	−16.8967	−1.01522	−0.507612	+	0.861586i	$0.669472\pi$
$278$	−11.3239	−0.679163
$279$	−50.6863	−3.03451
$280$	−10.3482	−0.618426
$281$	12.0704	0.720058	0.360029	−	0.932941i	$-0.382767\pi$
$281$	12.0704	0.720058	0.360029	+	0.932941i	$0.382767\pi$
$282$	17.7934	1.05958
$283$	−7.86758	−0.467679	−0.233839	−	0.972275i	$-0.575129\pi$
$283$	−7.86758	−0.467679	−0.233839	+	0.972275i	$0.575129\pi$
$284$	−1.32813	−0.0788098
$285$	−17.1339	−1.01492
$286$	9.82026	0.580684
$287$	−37.9753	−2.24161
$288$	−5.51716	−0.325102
$289$	−9.63943	−0.567025
$290$	−4.23987	−0.248974
$291$	−51.5738	−3.02331
$292$	5.31705	0.311157
$293$	8.28476	0.484001	0.242000	−	0.970276i	$-0.422196\pi$
$293$	8.28476	0.484001	0.242000	+	0.970276i	$0.422196\pi$
$294$	13.1767	0.768483
$295$	19.8916	1.15813
$296$	−4.71373	−0.273980
$297$	−23.8216	−1.38227
$298$	3.07453	0.178103
$299$	−5.37748	−0.310988
$300$	−12.5483	−0.724477
$301$	−40.7994	−2.35164
$302$	0.609059	0.0350474
$303$	35.5561	2.04264
$304$	1.92519	0.110417
$305$	19.0143	1.08876
$306$	−14.9683	−0.855678
$307$	−9.29988	−0.530772	−0.265386	−	0.964142i	$-0.585499\pi$
$307$	−9.29988	−0.530772	−0.265386	+	0.964142i	$0.585499\pi$
$308$	11.0039	0.627004
$309$	−28.7599	−1.63609
$310$	28.0162	1.59121
$311$	18.1058	1.02668	0.513342	−	0.858184i	$-0.328407\pi$
$311$	18.1058	1.02668	0.513342	+	0.858184i	$0.328407\pi$
$312$	−8.83807	−0.500357
$313$	−19.7732	−1.11765	−0.558825	−	0.829286i	$-0.688748\pi$
$313$	−19.7732	−1.11765	−0.558825	+	0.829286i	$0.688748\pi$
$314$	−11.0608	−0.624196
$315$	57.0929	3.21682
$316$	10.2976	0.579285
$317$	−25.3471	−1.42363	−0.711817	−	0.702365i	$-0.752127\pi$
$317$	−25.3471	−1.42363	−0.711817	+	0.702365i	$0.752127\pi$
$318$	−23.4918	−1.31735
$319$	4.50849	0.252427
$320$	3.04954	0.170474
$321$	47.8582	2.67118
$322$	−6.02562	−0.335795
$323$	5.22312	0.290622
$324$	4.88755	0.271531
$325$	−13.0211	−0.722280
$326$	−15.4803	−0.857374
$327$	−58.9175	−3.25815
$328$	11.1910	0.617919
$329$	20.6892	1.14063
$330$	28.8599	1.58868
$331$	12.8359	0.705522	0.352761	−	0.935713i	$-0.385243\pi$
$331$	12.8359	0.705522	0.352761	+	0.935713i	$0.385243\pi$
$332$	11.8365	0.649610
$333$	26.0064	1.42514
$334$	0.310574	0.0169938
$335$	36.4310	1.99044
$336$	−9.90330	−0.540269
$337$	12.4219	0.676663	0.338332	−	0.941027i	$-0.390137\pi$
$337$	12.4219	0.676663	0.338332	+	0.941027i	$0.390137\pi$
$338$	3.82893	0.208267
$339$	−6.96862	−0.378483
$340$	8.27351	0.448694
$341$	−29.7912	−1.61328
$342$	−10.6216	−0.574350
$343$	−8.43245	−0.455309
$344$	12.0232	0.648250
$345$	−15.8034	−0.850827
$346$	2.50587	0.134716
$347$	15.7866	0.847468	0.423734	−	0.905787i	$-0.360719\pi$
$347$	15.7866	0.847468	0.423734	+	0.905787i	$0.360719\pi$
$348$	−4.05757	−0.217508
$349$	−30.6239	−1.63926	−0.819630	−	0.572893i	$-0.805821\pi$
$349$	−30.6239	−1.63926	−0.819630	+	0.572893i	$0.805821\pi$
$350$	−14.5905	−0.779895
$351$	22.2468	1.18745
$352$	−3.24275	−0.172839
$353$	−1.54201	−0.0820731	−0.0410366	−	0.999158i	$-0.513066\pi$
$353$	−1.54201	−0.0820731	−0.0410366	+	0.999158i	$0.513066\pi$
$354$	19.0363	1.01177
$355$	−4.05017	−0.214961
$356$	−1.96091	−0.103928
$357$	−26.8680	−1.42201
$358$	3.41102	0.180278
$359$	5.82488	0.307425	0.153713	−	0.988116i	$-0.450877\pi$
$359$	5.82488	0.307425	0.153713	+	0.988116i	$0.450877\pi$
$360$	−16.8248	−0.886744
$361$	−15.2936	−0.804928
$362$	19.0067	0.998972
$363$	1.41424	0.0742286
$364$	−10.2764	−0.538632
$365$	16.2146	0.848709
$366$	18.1968	0.951161
$367$	30.0128	1.56665	0.783327	−	0.621610i	$-0.213521\pi$
$367$	30.0128	1.56665	0.783327	+	0.621610i	$0.213521\pi$
$368$	1.77570	0.0925647
$369$	−61.7425	−3.21418
$370$	−14.3747	−0.747306
$371$	−27.3150	−1.41812
$372$	26.8116	1.39011
$373$	−22.8789	−1.18463	−0.592313	−	0.805708i	$-0.701785\pi$
$373$	−22.8789	−1.18463	−0.592313	+	0.805708i	$0.701785\pi$
$374$	−8.79769	−0.454918
$375$	6.23259	0.321850
$376$	−6.09693	−0.314425
$377$	−4.21045	−0.216849
$378$	24.9282	1.28217
$379$	5.67831	0.291675	0.145838	−	0.989309i	$-0.453412\pi$
$379$	5.67831	0.291675	0.145838	+	0.989309i	$0.453412\pi$
$380$	5.87095	0.301173
$381$	−5.31347	−0.272217
$382$	−5.35334	−0.273901
$383$	17.1966	0.878703	0.439352	−	0.898315i	$-0.355208\pi$
$383$	17.1966	0.878703	0.439352	+	0.898315i	$0.355208\pi$
$384$	2.91842	0.148930
$385$	33.5568	1.71021
$386$	−9.31689	−0.474217
$387$	−66.3341	−3.37195
$388$	17.6719	0.897153
$389$	37.2248	1.88737	0.943686	−	0.330843i	$-0.107333\pi$
$389$	37.2248	1.88737	0.943686	+	0.330843i	$0.107333\pi$
$390$	−26.9520	−1.36477
$391$	4.81753	0.243633
$392$	−4.51503	−0.228043
$393$	−29.5441	−1.49030
$394$	−16.2169	−0.816996
$395$	31.4029	1.58005
$396$	17.8907	0.899044
$397$	−14.7102	−0.738286	−0.369143	−	0.929373i	$-0.620349\pi$
$397$	−14.7102	−0.738286	−0.369143	+	0.929373i	$0.620349\pi$
$398$	−3.69204	−0.185065
$399$	−19.0658	−0.954482
$400$	4.29970	0.214985
$401$	−29.6325	−1.47978	−0.739889	−	0.672729i	$-0.765122\pi$
$401$	−29.6325	−1.47978	−0.739889	+	0.672729i	$0.765122\pi$
$402$	34.8645	1.73888
$403$	27.8218	1.38590
$404$	−12.1834	−0.606144
$405$	14.9048	0.740625
$406$	−4.71792	−0.234147
$407$	15.2854	0.757672
$408$	7.91777	0.391988
$409$	31.4794	1.55656	0.778278	−	0.627919i	$-0.216093\pi$
$409$	31.4794	1.55656	0.778278	+	0.627919i	$0.216093\pi$
$410$	34.1274	1.68543
$411$	−16.3286	−0.805432
$412$	9.85463	0.485503
$413$	22.1344	1.08916
$414$	−9.79681	−0.481487
$415$	36.0958	1.77187
$416$	3.02838	0.148478
$417$	−33.0479	−1.61836
$418$	−6.24291	−0.305351
$419$	−6.87683	−0.335955	−0.167978	−	0.985791i	$-0.553724\pi$
$419$	−6.87683	−0.335955	−0.167978	+	0.985791i	$0.553724\pi$
$420$	−30.2005	−1.47363
$421$	−14.7388	−0.718325	−0.359163	−	0.933275i	$-0.616938\pi$
$421$	−14.7388	−0.718325	−0.359163	+	0.933275i	$0.616938\pi$
$422$	15.4489	0.752041
$423$	33.6377	1.63552
$424$	8.04949	0.390918
$425$	11.6652	0.565847
$426$	−3.87603	−0.187794
$427$	21.1582	1.02392
$428$	−16.3987	−0.792660
$429$	28.6596	1.38370
$430$	36.6654	1.76816
$431$	−0.510707	−0.0245999	−0.0122999	−	0.999924i	$-0.503915\pi$
$431$	−0.510707	−0.0245999	−0.0122999	+	0.999924i	$0.503915\pi$
$432$	−7.34612	−0.353440
$433$	13.0265	0.626013	0.313006	−	0.949751i	$-0.398664\pi$
$433$	13.0265	0.626013	0.313006	+	0.949751i	$0.398664\pi$
$434$	31.1751	1.49645
$435$	−12.3737	−0.593274
$436$	20.1882	0.966838
$437$	3.41856	0.163532
$438$	15.5174	0.741449
$439$	15.0265	0.717175	0.358587	−	0.933496i	$-0.383259\pi$
$439$	15.0265	0.717175	0.358587	+	0.933496i	$0.383259\pi$
$440$	−9.88889	−0.471434
$441$	24.9101	1.18620
$442$	8.21610	0.390800
$443$	24.2179	1.15063	0.575313	−	0.817934i	$-0.304880\pi$
$443$	24.2179	1.15063	0.575313	+	0.817934i	$0.304880\pi$
$444$	−13.7566	−0.652861
$445$	−5.97986	−0.283473
$446$	13.5607	0.642119
$447$	8.97277	0.424397
$448$	3.39338	0.160322
$449$	3.31168	0.156288	0.0781439	−	0.996942i	$-0.475101\pi$
$449$	3.31168	0.156288	0.0781439	+	0.996942i	$0.475101\pi$
$450$	−23.7221	−1.11827
$451$	−36.2895	−1.70881
$452$	2.38781	0.112313
$453$	1.77749	0.0835137
$454$	−8.04828	−0.377724
$455$	−31.3384	−1.46917
$456$	5.61851	0.263111
$457$	7.98224	0.373393	0.186697	−	0.982418i	$-0.440222\pi$
$457$	7.98224	0.373393	0.186697	+	0.982418i	$0.440222\pi$
$458$	14.2222	0.664559
$459$	−19.9303	−0.930266
$460$	5.41506	0.252479
$461$	32.5697	1.51692	0.758461	−	0.651719i	$-0.225952\pi$
$461$	32.5697	1.51692	0.758461	+	0.651719i	$0.225952\pi$
$462$	32.1139	1.49407
$463$	−30.9159	−1.43678	−0.718391	−	0.695640i	$-0.755121\pi$
$463$	−30.9159	−1.43678	−0.718391	+	0.695640i	$0.755121\pi$
$464$	1.39033	0.0645445
$465$	81.7629	3.79167
$466$	−2.83488	−0.131323
$467$	26.2417	1.21432	0.607160	−	0.794580i	$-0.292309\pi$
$467$	26.2417	1.21432	0.607160	+	0.794580i	$0.292309\pi$
$468$	−16.7080	−0.772329
$469$	40.5386	1.87190
$470$	−18.5928	−0.857623
$471$	−32.2800	−1.48738
$472$	−6.52281	−0.300237
$473$	−38.9883	−1.79269
$474$	30.0527	1.38037
$475$	8.27774	0.379809
$476$	9.20637	0.421973
$477$	−44.4103	−2.03341
$478$	−0.659329	−0.0301570
$479$	−5.21508	−0.238283	−0.119142	−	0.992877i	$-0.538014\pi$
$479$	−5.21508	−0.238283	−0.119142	+	0.992877i	$0.538014\pi$
$480$	8.89983	0.406220
$481$	−14.2750	−0.650882
$482$	5.93920	0.270523
$483$	−17.5853	−0.800157
$484$	−0.484593	−0.0220270
$485$	53.8910	2.44707
$486$	−7.77442	−0.352655
$487$	41.3148	1.87215	0.936077	−	0.351796i	$-0.114429\pi$
$487$	41.3148	1.87215	0.936077	+	0.351796i	$0.114429\pi$
$488$	−6.23515	−0.282252
$489$	−45.1779	−2.04302
$490$	−13.7688	−0.622010
$491$	−43.2059	−1.94985	−0.974927	−	0.222525i	$-0.928570\pi$
$491$	−43.2059	−1.94985	−0.974927	+	0.222525i	$0.928570\pi$
$492$	32.6600	1.47243
$493$	3.77202	0.169883
$494$	5.83021	0.262313
$495$	54.5585	2.45222
$496$	−9.18702	−0.412510
$497$	−4.50684	−0.202159
$498$	34.5437	1.54794
$499$	2.55810	0.114516	0.0572582	−	0.998359i	$-0.481764\pi$
$499$	2.55810	0.114516	0.0572582	+	0.998359i	$0.481764\pi$
$500$	−2.13561	−0.0955073
$501$	0.906384	0.0404942
$502$	−16.9407	−0.756101
$503$	29.5012	1.31540	0.657698	−	0.753282i	$-0.271530\pi$
$503$	29.5012	1.31540	0.657698	+	0.753282i	$0.271530\pi$
$504$	−18.7218	−0.833936
$505$	−37.1536	−1.65331
$506$	−5.75814	−0.255981
$507$	11.1744	0.496274
$508$	1.82067	0.0807791
$509$	7.95135	0.352438	0.176219	−	0.984351i	$-0.443613\pi$
$509$	7.95135	0.352438	0.176219	+	0.984351i	$0.443613\pi$
$510$	24.1456	1.06918
$511$	18.0428	0.798166
$512$	−1.00000	−0.0441942
$513$	−14.1427	−0.624415
$514$	−11.0742	−0.488464
$515$	30.0521	1.32425
$516$	35.0888	1.54470
$517$	19.7708	0.869519
$518$	−15.9955	−0.702802
$519$	7.31317	0.321013
$520$	9.23516	0.404988
$521$	−6.42823	−0.281626	−0.140813	−	0.990036i	$-0.544972\pi$
$521$	−6.42823	−0.281626	−0.140813	+	0.990036i	$0.544972\pi$
$522$	−7.67068	−0.335736
$523$	30.1877	1.32001	0.660007	−	0.751259i	$-0.270553\pi$
$523$	30.1877	1.32001	0.660007	+	0.751259i	$0.270553\pi$
$524$	10.1233	0.442240
$525$	−42.5812	−1.85839
$526$	−17.3168	−0.755047
$527$	−24.9247	−1.08574
$528$	−9.46369	−0.411854
$529$	−19.8469	−0.862909
$530$	24.5472	1.06626
$531$	35.9874	1.56172
$532$	6.53291	0.283238
$533$	33.8905	1.46796
$534$	−5.72274	−0.247647
$535$	−50.0084	−2.16205
$536$	−11.9464	−0.516005
$537$	9.95477	0.429580
$538$	0.794480	0.0342524
$539$	14.6411	0.630637
$540$	−22.4023	−0.964040
$541$	27.8226	1.19619	0.598093	−	0.801427i	$-0.295925\pi$
$541$	27.8226	1.19619	0.598093	+	0.801427i	$0.295925\pi$
$542$	−8.63822	−0.371044
$543$	55.4696	2.38043
$544$	−2.71304	−0.116320
$545$	61.5647	2.63714
$546$	−29.9909	−1.28349
$547$	−22.2276	−0.950382	−0.475191	−	0.879883i	$-0.657621\pi$
$547$	−22.2276	−0.950382	−0.475191	+	0.879883i	$0.657621\pi$
$548$	5.59503	0.239008
$549$	34.4003	1.46817
$550$	−13.9428	−0.594524
$551$	2.67665	0.114029
$552$	5.18223	0.220570
$553$	34.9437	1.48596
$554$	16.8967	0.717872
$555$	−41.9514	−1.78074
$556$	11.3239	0.480241
$557$	28.1460	1.19258	0.596291	−	0.802768i	$-0.296640\pi$
$557$	28.1460	1.19258	0.596291	+	0.802768i	$0.296640\pi$
$558$	50.6863	2.14572
$559$	36.4109	1.54002
$560$	10.3482	0.437293
$561$	−25.6753	−1.08401
$562$	−12.0704	−0.509158
$563$	−17.9140	−0.754986	−0.377493	−	0.926013i	$-0.623214\pi$
$563$	−17.9140	−0.754986	−0.377493	+	0.926013i	$0.623214\pi$
$564$	−17.7934	−0.749236
$565$	7.28171	0.306344
$566$	7.86758	0.330699
$567$	16.5853	0.696519
$568$	1.32813	0.0557269
$569$	10.7515	0.450726	0.225363	−	0.974275i	$-0.427643\pi$
$569$	10.7515	0.450726	0.225363	+	0.974275i	$0.427643\pi$
$570$	17.1339	0.717659
$571$	5.07165	0.212242	0.106121	−	0.994353i	$-0.466157\pi$
$571$	5.07165	0.212242	0.106121	+	0.994353i	$0.466157\pi$
$572$	−9.82026	−0.410606
$573$	−15.6233	−0.652672
$574$	37.9753	1.58506
$575$	7.63496	0.318400
$576$	5.51716	0.229882
$577$	18.2492	0.759724	0.379862	−	0.925043i	$-0.375971\pi$
$577$	18.2492	0.759724	0.379862	+	0.925043i	$0.375971\pi$
$578$	9.63943	0.400947
$579$	−27.1906	−1.13000
$580$	4.23987	0.176051
$581$	40.1656	1.66635
$582$	51.5738	2.13780
$583$	−26.1025	−1.08105
$584$	−5.31705	−0.220021
$585$	−50.9518	−2.10660
$586$	−8.28476	−0.342240
$587$	−10.1541	−0.419105	−0.209552	−	0.977797i	$-0.567201\pi$
$587$	−10.1541	−0.419105	−0.209552	+	0.977797i	$0.567201\pi$
$588$	−13.1767	−0.543400
$589$	−17.6868	−0.728771
$590$	−19.8916	−0.818923
$591$	−47.3277	−1.94680
$592$	4.71373	0.193733
$593$	4.77348	0.196023	0.0980117	−	0.995185i	$-0.468752\pi$
$593$	4.77348	0.196023	0.0980117	+	0.995185i	$0.468752\pi$
$594$	23.8216	0.977412
$595$	28.0752	1.15097
$596$	−3.07453	−0.125938
$597$	−10.7749	−0.440988
$598$	5.37748	0.219902
$599$	−27.4626	−1.12209	−0.561047	−	0.827784i	$-0.689601\pi$
$599$	−27.4626	−1.12209	−0.561047	+	0.827784i	$0.689601\pi$
$600$	12.5483	0.512282
$601$	32.7509	1.33594	0.667969	−	0.744189i	$-0.267164\pi$
$601$	32.7509	1.33594	0.667969	+	0.744189i	$0.267164\pi$
$602$	40.7994	1.66286
$603$	65.9101	2.68407
$604$	−0.609059	−0.0247823
$605$	−1.47779	−0.0600805
$606$	−35.5561	−1.44437
$607$	41.0293	1.66533	0.832665	−	0.553777i	$-0.186814\pi$
$607$	41.0293	1.66533	0.832665	+	0.553777i	$0.186814\pi$
$608$	−1.92519	−0.0780769
$609$	−13.7689	−0.557943
$610$	−19.0143	−0.769869
$611$	−18.4638	−0.746965
$612$	14.9683	0.605056
$613$	23.1268	0.934083	0.467041	−	0.884235i	$-0.345320\pi$
$613$	23.1268	0.934083	0.467041	+	0.884235i	$0.345320\pi$
$614$	9.29988	0.375313
$615$	99.5979	4.01618
$616$	−11.0039	−0.443359
$617$	−9.09422	−0.366119	−0.183060	−	0.983102i	$-0.558600\pi$
$617$	−9.09422	−0.366119	−0.183060	+	0.983102i	$0.558600\pi$
$618$	28.7599	1.15689
$619$	−22.7072	−0.912678	−0.456339	−	0.889806i	$-0.650840\pi$
$619$	−22.7072	−0.912678	−0.456339	+	0.889806i	$0.650840\pi$
$620$	−28.0162	−1.12516
$621$	−13.0445	−0.523457
$622$	−18.1058	−0.725975
$623$	−6.65410	−0.266591
$624$	8.83807	0.353806
$625$	−28.0111	−1.12044
$626$	19.7732	0.790298
$627$	−18.2194	−0.727613
$628$	11.0608	0.441374
$629$	12.7885	0.509912
$630$	−57.0929	−2.27464
$631$	29.3641	1.16897	0.584484	−	0.811405i	$-0.301297\pi$
$631$	29.3641	1.16897	0.584484	+	0.811405i	$0.301297\pi$
$632$	−10.2976	−0.409616
$633$	45.0863	1.79202
$634$	25.3471	1.00666
$635$	5.55220	0.220332
$636$	23.4918	0.931509
$637$	−13.6732	−0.541752
$638$	−4.50849	−0.178493
$639$	−7.32748	−0.289871
$640$	−3.04954	−0.120544
$641$	40.8880	1.61498	0.807489	−	0.589883i	$-0.200826\pi$
$641$	40.8880	1.61498	0.807489	+	0.589883i	$0.200826\pi$
$642$	−47.8582	−1.88881
$643$	−30.6968	−1.21056	−0.605282	−	0.796011i	$-0.706939\pi$
$643$	−30.6968	−1.21056	−0.605282	+	0.796011i	$0.706939\pi$
$644$	6.02562	0.237443
$645$	107.005	4.21331
$646$	−5.22312	−0.205501
$647$	17.6111	0.692363	0.346181	−	0.938168i	$-0.387478\pi$
$647$	17.6111	0.692363	0.346181	+	0.938168i	$0.387478\pi$
$648$	−4.88755	−0.192001
$649$	21.1518	0.830281
$650$	13.0211	0.510729
$651$	90.9818	3.56586
$652$	15.4803	0.606255
$653$	−34.1366	−1.33587	−0.667934	−	0.744221i	$-0.732821\pi$
$653$	−34.1366	−1.33587	−0.667934	+	0.744221i	$0.732821\pi$
$654$	58.9175	2.30386
$655$	30.8715	1.20625
$656$	−11.1910	−0.436935
$657$	29.3350	1.14447
$658$	−20.6892	−0.806549
$659$	−15.5750	−0.606717	−0.303359	−	0.952876i	$-0.598108\pi$
$659$	−15.5750	−0.606717	−0.303359	+	0.952876i	$0.598108\pi$
$660$	−28.8599	−1.12337
$661$	−47.6998	−1.85531	−0.927653	−	0.373444i	$-0.878177\pi$
$661$	−47.6998	−1.85531	−0.927653	+	0.373444i	$0.878177\pi$
$662$	−12.8359	−0.498880
$663$	23.9780	0.931228
$664$	−11.8365	−0.459344
$665$	19.9224	0.772556
$666$	−26.0064	−1.00773
$667$	2.46881	0.0955926
$668$	−0.310574	−0.0120165
$669$	39.5759	1.53009
$670$	−36.4310	−1.40745
$671$	20.2190	0.780547
$672$	9.90330	0.382028
$673$	−43.1214	−1.66221	−0.831104	−	0.556116i	$-0.812291\pi$
$673$	−43.1214	−1.66221	−0.831104	+	0.556116i	$0.812291\pi$
$674$	−12.4219	−0.478473
$675$	−31.5861	−1.21575
$676$	−3.82893	−0.147267
$677$	−15.3548	−0.590134	−0.295067	−	0.955477i	$-0.595342\pi$
$677$	−15.3548	−0.590134	−0.295067	+	0.955477i	$0.595342\pi$
$678$	6.96862	0.267628
$679$	59.9673	2.30134
$680$	−8.27351	−0.317275
$681$	−23.4882	−0.900071
$682$	29.7912	1.14076
$683$	16.1837	0.619252	0.309626	−	0.950858i	$-0.399796\pi$
$683$	16.1837	0.619252	0.309626	+	0.950858i	$0.399796\pi$
$684$	10.6216	0.406127
$685$	17.0623	0.651916
$686$	8.43245	0.321952
$687$	41.5063	1.58356
$688$	−12.0232	−0.458382
$689$	24.3769	0.928685
$690$	15.8034	0.601626
$691$	−33.0997	−1.25917	−0.629586	−	0.776931i	$-0.716775\pi$
$691$	−33.0997	−1.25917	−0.629586	+	0.776931i	$0.716775\pi$
$692$	−2.50587	−0.0952588
$693$	60.7101	2.30619
$694$	−15.7866	−0.599250
$695$	34.5327	1.30990
$696$	4.05757	0.153802
$697$	−30.3616	−1.15003
$698$	30.6239	1.15913
$699$	−8.27337	−0.312928
$700$	14.5905	0.551469
$701$	−22.7439	−0.859026	−0.429513	−	0.903061i	$-0.641315\pi$
$701$	−22.7439	−0.859026	−0.429513	+	0.903061i	$0.641315\pi$
$702$	−22.2468	−0.839652
$703$	9.07485	0.342264
$704$	3.24275	0.122216
$705$	−54.2616	−2.04361
$706$	1.54201	0.0580345
$707$	−41.3427	−1.55485
$708$	−19.0363	−0.715427
$709$	−40.2044	−1.50991	−0.754955	−	0.655777i	$-0.772341\pi$
$709$	−40.2044	−1.50991	−0.754955	+	0.655777i	$0.772341\pi$
$710$	4.05017	0.152000
$711$	56.8135	2.13067
$712$	1.96091	0.0734881
$713$	−16.3134	−0.610941
$714$	26.8680	1.00551
$715$	−29.9473	−1.11996
$716$	−3.41102	−0.127476
$717$	−1.92420	−0.0718605
$718$	−5.82488	−0.217383
$719$	−4.50222	−0.167904	−0.0839521	−	0.996470i	$-0.526754\pi$
$719$	−4.50222	−0.167904	−0.0839521	+	0.996470i	$0.526754\pi$
$720$	16.8248	0.627023
$721$	33.4405	1.24539
$722$	15.2936	0.569170
$723$	17.3331	0.644623
$724$	−19.0067	−0.706380
$725$	5.97800	0.222017
$726$	−1.41424	−0.0524875
$727$	48.7315	1.80735	0.903675	−	0.428219i	$-0.140859\pi$
$727$	48.7315	1.80735	0.903675	+	0.428219i	$0.140859\pi$
$728$	10.2764	0.380870
$729$	−37.3517	−1.38340
$730$	−16.2146	−0.600128
$731$	−32.6195	−1.20648
$732$	−18.1968	−0.672572
$733$	−52.0038	−1.92081	−0.960403	−	0.278616i	$-0.910124\pi$
$733$	−52.0038	−1.92081	−0.960403	+	0.278616i	$0.910124\pi$
$734$	−30.0128	−1.10779
$735$	−40.1830	−1.48217
$736$	−1.77570	−0.0654531
$737$	38.7391	1.42697
$738$	61.7425	2.27277
$739$	1.30695	0.0480768	0.0240384	−	0.999711i	$-0.492348\pi$
$739$	1.30695	0.0480768	0.0240384	+	0.999711i	$0.492348\pi$
$740$	14.3747	0.528425
$741$	17.0150	0.625061
$742$	27.3150	1.00276
$743$	−25.6873	−0.942376	−0.471188	−	0.882033i	$-0.656175\pi$
$743$	−25.6873	−0.942376	−0.471188	+	0.882033i	$0.656175\pi$
$744$	−26.8116	−0.982960
$745$	−9.37591	−0.343507
$746$	22.8789	0.837657
$747$	65.3036	2.38934
$748$	8.79769	0.321675
$749$	−55.6470	−2.03330
$750$	−6.23259	−0.227582
$751$	−41.3433	−1.50864	−0.754319	−	0.656508i	$-0.772033\pi$
$751$	−41.3433	−1.50864	−0.754319	+	0.656508i	$0.772033\pi$
$752$	6.09693	0.222332
$753$	−49.4401	−1.80170
$754$	4.21045	0.153335
$755$	−1.85735	−0.0675959
$756$	−24.9282	−0.906629
$757$	−25.9620	−0.943603	−0.471802	−	0.881705i	$-0.656396\pi$
$757$	−25.9620	−0.943603	−0.471802	+	0.881705i	$0.656396\pi$
$758$	−5.67831	−0.206246
$759$	−16.8047	−0.609970
$760$	−5.87095	−0.212962
$761$	−12.6899	−0.460008	−0.230004	−	0.973190i	$-0.573874\pi$
$761$	−12.6899	−0.460008	−0.230004	+	0.973190i	$0.573874\pi$
$762$	5.31347	0.192487
$763$	68.5062	2.48009
$764$	5.35334	0.193677
$765$	45.6463	1.65034
$766$	−17.1966	−0.621337
$767$	−19.7535	−0.713258
$768$	−2.91842	−0.105309
$769$	−42.6967	−1.53968	−0.769842	−	0.638235i	$-0.779665\pi$
$769$	−42.6967	−1.53968	−0.769842	+	0.638235i	$0.779665\pi$
$770$	−33.5568	−1.20930
$771$	−32.3193	−1.16395
$772$	9.31689	0.335322
$773$	−6.04844	−0.217547	−0.108774	−	0.994067i	$-0.534692\pi$
$773$	−6.04844	−0.217547	−0.108774	+	0.994067i	$0.534692\pi$
$774$	66.3341	2.38433
$775$	−39.5014	−1.41893
$776$	−17.6719	−0.634383
$777$	−46.6815	−1.67469
$778$	−37.2248	−1.33457
$779$	−21.5448	−0.771923
$780$	26.9520	0.965038
$781$	−4.30678	−0.154109
$782$	−4.81753	−0.172275
$783$	−10.2135	−0.365002
$784$	4.51503	0.161251
$785$	33.7303	1.20389
$786$	29.5441	1.05380
$787$	49.7478	1.77332	0.886659	−	0.462424i	$-0.153020\pi$
$787$	49.7478	1.77332	0.886659	+	0.462424i	$0.153020\pi$
$788$	16.2169	0.577703
$789$	−50.5376	−1.79918
$790$	−31.4029	−1.11727
$791$	8.10274	0.288100
$792$	−17.8907	−0.635720
$793$	−18.8824	−0.670533
$794$	14.7102	0.522047
$795$	71.6391	2.54078
$796$	3.69204	0.130861
$797$	28.0085	0.992112	0.496056	−	0.868291i	$-0.334781\pi$
$797$	28.0085	0.992112	0.496056	+	0.868291i	$0.334781\pi$
$798$	19.0658	0.674920
$799$	16.5412	0.585185
$800$	−4.29970	−0.152017
$801$	−10.8186	−0.382257
$802$	29.6325	1.04636
$803$	17.2419	0.608452
$804$	−34.8645	−1.22958
$805$	18.3754	0.647647
$806$	−27.8218	−0.979980
$807$	2.31862	0.0816194
$808$	12.1834	0.428609
$809$	13.1264	0.461498	0.230749	−	0.973013i	$-0.425882\pi$
$809$	13.1264	0.461498	0.230749	+	0.973013i	$0.425882\pi$
$810$	−14.9048	−0.523701
$811$	−17.1909	−0.603655	−0.301828	−	0.953363i	$-0.597597\pi$
$811$	−17.1909	−0.603655	−0.301828	+	0.953363i	$0.597597\pi$
$812$	4.71792	0.165567
$813$	−25.2099	−0.884151
$814$	−15.2854	−0.535755
$815$	47.2078	1.65362
$816$	−7.91777	−0.277177
$817$	−23.1471	−0.809813
$818$	−31.4794	−1.10065
$819$	−56.6967	−1.98114
$820$	−34.1274	−1.19178
$821$	14.6297	0.510582	0.255291	−	0.966864i	$-0.417829\pi$
$821$	14.6297	0.510582	0.255291	+	0.966864i	$0.417829\pi$
$822$	16.3286	0.569526
$823$	−17.4531	−0.608378	−0.304189	−	0.952612i	$-0.598385\pi$
$823$	−17.4531	−0.608378	−0.304189	+	0.952612i	$0.598385\pi$
$824$	−9.85463	−0.343302
$825$	−40.6910	−1.41668
$826$	−22.1344	−0.770153
$827$	14.1441	0.491840	0.245920	−	0.969290i	$-0.420910\pi$
$827$	14.1441	0.491840	0.245920	+	0.969290i	$0.420910\pi$
$828$	9.79681	0.340463
$829$	3.27491	0.113742	0.0568711	−	0.998382i	$-0.481888\pi$
$829$	3.27491	0.113742	0.0568711	+	0.998382i	$0.481888\pi$
$830$	−36.0958	−1.25290
$831$	49.3116	1.71060
$832$	−3.02838	−0.104990
$833$	12.2494	0.424418
$834$	33.0479	1.14435
$835$	−0.947107	−0.0327760
$836$	6.24291	0.215916
$837$	67.4889	2.33276
$838$	6.87683	0.237556
$839$	15.8008	0.545504	0.272752	−	0.962084i	$-0.412066\pi$
$839$	15.8008	0.545504	0.272752	+	0.962084i	$0.412066\pi$
$840$	30.2005	1.04202
$841$	−27.0670	−0.933344
$842$	14.7388	0.507933
$843$	−35.2264	−1.21326
$844$	−15.4489	−0.531773
$845$	−11.6765	−0.401683
$846$	−33.6377	−1.15649
$847$	−1.64441	−0.0565025
$848$	−8.04949	−0.276421
$849$	22.9609	0.788015
$850$	−11.6652	−0.400114
$851$	8.37017	0.286926
$852$	3.87603	0.132790
$853$	−2.67157	−0.0914727	−0.0457364	−	0.998954i	$-0.514563\pi$
$853$	−2.67157	−0.0914727	−0.0457364	+	0.998954i	$0.514563\pi$
$854$	−21.1582	−0.724020
$855$	32.3910	1.10775
$856$	16.3987	0.560495
$857$	17.3020	0.591026	0.295513	−	0.955339i	$-0.404509\pi$
$857$	17.3020	0.591026	0.295513	+	0.955339i	$0.404509\pi$
$858$	−28.6596	−0.978423
$859$	35.3105	1.20478	0.602389	−	0.798203i	$-0.294216\pi$
$859$	35.3105	1.20478	0.602389	+	0.798203i	$0.294216\pi$
$860$	−36.6654	−1.25028
$861$	110.828	3.77700
$862$	0.510707	0.0173947
$863$	28.3673	0.965633	0.482817	−	0.875721i	$-0.339614\pi$
$863$	28.3673	0.965633	0.482817	+	0.875721i	$0.339614\pi$
$864$	7.34612	0.249920
$865$	−7.64175	−0.259827
$866$	−13.0265	−0.442658
$867$	28.1319	0.955409
$868$	−31.1751	−1.05815
$869$	33.3925	1.13276
$870$	12.3737	0.419508
$871$	−36.1782	−1.22585
$872$	−20.1882	−0.683658
$873$	97.4984	3.29982
$874$	−3.41856	−0.115635
$875$	−7.24693	−0.244991
$876$	−15.5174	−0.524284
$877$	13.3551	0.450968	0.225484	−	0.974247i	$-0.427604\pi$
$877$	13.3551	0.450968	0.225484	+	0.974247i	$0.427604\pi$
$878$	−15.0265	−0.507119
$879$	−24.1784	−0.815517
$880$	9.88889	0.333354
$881$	33.5661	1.13087	0.565435	−	0.824793i	$-0.308708\pi$
$881$	33.5661	1.13087	0.565435	+	0.824793i	$0.308708\pi$
$882$	−24.9101	−0.838768
$883$	−41.6195	−1.40061	−0.700303	−	0.713846i	$-0.746952\pi$
$883$	−41.6195	−1.40061	−0.700303	+	0.713846i	$0.746952\pi$
$884$	−8.21610	−0.276337
$885$	−58.0519	−1.95139
$886$	−24.2179	−0.813615
$887$	−29.2027	−0.980531	−0.490265	−	0.871573i	$-0.663100\pi$
$887$	−29.2027	−0.980531	−0.490265	+	0.871573i	$0.663100\pi$
$888$	13.7566	0.461643
$889$	6.17822	0.207211
$890$	5.97986	0.200445
$891$	15.8491	0.530965
$892$	−13.5607	−0.454047
$893$	11.7378	0.392789
$894$	−8.97277	−0.300094
$895$	−10.4020	−0.347702
$896$	−3.39338	−0.113365
$897$	15.6937	0.523999
$898$	−3.31168	−0.110512
$899$	−12.7730	−0.426004
$900$	23.7221	0.790737
$901$	−21.8386	−0.727548
$902$	36.2895	1.20831
$903$	119.070	3.96239
$904$	−2.38781	−0.0794173
$905$	−57.9618	−1.92672
$906$	−1.77749	−0.0590531
$907$	18.8095	0.624559	0.312279	−	0.949990i	$-0.398907\pi$
$907$	18.8095	0.624559	0.312279	+	0.949990i	$0.398907\pi$
$908$	8.04828	0.267091
$909$	−67.2175	−2.22946
$910$	31.3384	1.03886
$911$	−46.1255	−1.52821	−0.764103	−	0.645094i	$-0.776818\pi$
$911$	−46.1255	−1.52821	−0.764103	+	0.645094i	$0.776818\pi$
$912$	−5.61851	−0.186048
$913$	38.3827	1.27028
$914$	−7.98224	−0.264029
$915$	−55.4918	−1.83450
$916$	−14.2222	−0.469914
$917$	34.3523	1.13441
$918$	19.9303	0.657797
$919$	1.42934	0.0471497	0.0235748	−	0.999722i	$-0.492495\pi$
$919$	1.42934	0.0471497	0.0235748	+	0.999722i	$0.492495\pi$
$920$	−5.41506	−0.178529
$921$	27.1409	0.894324
$922$	−32.5697	−1.07263
$923$	4.02207	0.132388
$924$	−32.1139	−1.05647
$925$	20.2676	0.666395
$926$	30.9159	1.01596
$927$	54.3696	1.78573
$928$	−1.39033	−0.0456399
$929$	8.45362	0.277354	0.138677	−	0.990338i	$-0.455715\pi$
$929$	8.45362	0.277354	0.138677	+	0.990338i	$0.455715\pi$
$930$	−81.7629	−2.68111
$931$	8.69230	0.284879
$932$	2.83488	0.0928597
$933$	−52.8402	−1.72991
$934$	−26.2417	−0.858654
$935$	26.8289	0.877399
$936$	16.7080	0.546119
$937$	−2.89589	−0.0946048	−0.0473024	−	0.998881i	$-0.515062\pi$
$937$	−2.89589	−0.0946048	−0.0473024	+	0.998881i	$0.515062\pi$
$938$	−40.5386	−1.32363
$939$	57.7066	1.88318
$940$	18.5928	0.606431
$941$	44.8877	1.46330	0.731649	−	0.681682i	$-0.238751\pi$
$941$	44.8877	1.46330	0.731649	+	0.681682i	$0.238751\pi$
$942$	32.2800	1.05174
$943$	−19.8718	−0.647115
$944$	6.52281	0.212299
$945$	−76.0194	−2.47291
$946$	38.9883	1.26762
$947$	−1.36305	−0.0442930	−0.0221465	−	0.999755i	$-0.507050\pi$
$947$	−1.36305	−0.0442930	−0.0221465	+	0.999755i	$0.507050\pi$
$948$	−30.0527	−0.976066
$949$	−16.1020	−0.522694
$950$	−8.27774	−0.268565
$951$	73.9733	2.39875
$952$	−9.20637	−0.298380
$953$	40.5528	1.31363	0.656817	−	0.754050i	$-0.271902\pi$
$953$	40.5528	1.31363	0.656817	+	0.754050i	$0.271902\pi$
$954$	44.4103	1.43784
$955$	16.3252	0.528272
$956$	0.659329	0.0213242
$957$	−13.1577	−0.425327
$958$	5.21508	0.168492
$959$	18.9861	0.613092
$960$	−8.89983	−0.287241
$961$	53.4014	1.72263
$962$	14.2750	0.460243
$963$	−90.4741	−2.91549
$964$	−5.93920	−0.191289
$965$	28.4122	0.914622
$966$	17.5853	0.565797
$967$	12.1999	0.392322	0.196161	−	0.980572i	$-0.437153\pi$
$967$	12.1999	0.392322	0.196161	+	0.980572i	$0.437153\pi$
$968$	0.484593	0.0155754
$969$	−15.2432	−0.489683
$970$	−53.8910	−1.73034
$971$	−34.4508	−1.10558	−0.552789	−	0.833321i	$-0.686436\pi$
$971$	−34.4508	−1.10558	−0.552789	+	0.833321i	$0.686436\pi$
$972$	7.77442	0.249365
$973$	38.4263	1.23189
$974$	−41.3148	−1.32381
$975$	38.0010	1.21701
$976$	6.23515	0.199582
$977$	3.84417	0.122986	0.0614930	−	0.998108i	$-0.480414\pi$
$977$	3.84417	0.122986	0.0614930	+	0.998108i	$0.480414\pi$
$978$	45.1779	1.44463
$979$	−6.35872	−0.203226
$980$	13.7688	0.439827
$981$	111.381	3.55613
$982$	43.2059	1.37875
$983$	62.0659	1.97959	0.989797	−	0.142486i	$-0.0455095\pi$
$983$	62.0659	1.97959	0.989797	+	0.142486i	$0.0455095\pi$
$984$	−32.6600	−1.04116
$985$	49.4541	1.57574
$986$	−3.77202	−0.120126
$987$	−60.3797	−1.92191
$988$	−5.83021	−0.185484
$989$	−21.3497	−0.678880
$990$	−54.5585	−1.73398
$991$	−10.0605	−0.319584	−0.159792	−	0.987151i	$-0.551082\pi$
$991$	−10.0605	−0.319584	−0.159792	+	0.987151i	$0.551082\pi$
$992$	9.18702	0.291688
$993$	−37.4604	−1.18877
$994$	4.50684	0.142948
$995$	11.2590	0.356935
$996$	−34.5437	−1.09456
$997$	7.89603	0.250070	0.125035	−	0.992152i	$-0.460096\pi$
$997$	7.89603	0.250070	0.125035	+	0.992152i	$0.460096\pi$
$998$	−2.55810	−0.0809754
$999$	−34.6276	−1.09557

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.5	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.5	✓	77	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.91842 q^{3} +1.00000 q^{4} +3.04954 q^{5} +2.91842 q^{6} +3.39338 q^{7} -1.00000 q^{8} +5.51716 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.91842 q^{3} +1.00000 q^{4} +3.04954 q^{5} +2.91842 q^{6} +3.39338 q^{7} -1.00000 q^{8} +5.51716 q^{9} -3.04954 q^{10} +3.24275 q^{11} -2.91842 q^{12} -3.02838 q^{13} -3.39338 q^{14} -8.89983 q^{15} +1.00000 q^{16} +2.71304 q^{17} -5.51716 q^{18} +1.92519 q^{19} +3.04954 q^{20} -9.90330 q^{21} -3.24275 q^{22} +1.77570 q^{23} +2.91842 q^{24} +4.29970 q^{25} +3.02838 q^{26} -7.34612 q^{27} +3.39338 q^{28} +1.39033 q^{29} +8.89983 q^{30} -9.18702 q^{31} -1.00000 q^{32} -9.46369 q^{33} -2.71304 q^{34} +10.3482 q^{35} +5.51716 q^{36} +4.71373 q^{37} -1.92519 q^{38} +8.83807 q^{39} -3.04954 q^{40} -11.1910 q^{41} +9.90330 q^{42} -12.0232 q^{43} +3.24275 q^{44} +16.8248 q^{45} -1.77570 q^{46} +6.09693 q^{47} -2.91842 q^{48} +4.51503 q^{49} -4.29970 q^{50} -7.91777 q^{51} -3.02838 q^{52} -8.04949 q^{53} +7.34612 q^{54} +9.88889 q^{55} -3.39338 q^{56} -5.61851 q^{57} -1.39033 q^{58} +6.52281 q^{59} -8.89983 q^{60} +6.23515 q^{61} +9.18702 q^{62} +18.7218 q^{63} +1.00000 q^{64} -9.23516 q^{65} +9.46369 q^{66} +11.9464 q^{67} +2.71304 q^{68} -5.18223 q^{69} -10.3482 q^{70} -1.32813 q^{71} -5.51716 q^{72} +5.31705 q^{73} -4.71373 q^{74} -12.5483 q^{75} +1.92519 q^{76} +11.0039 q^{77} -8.83807 q^{78} +10.2976 q^{79} +3.04954 q^{80} +4.88755 q^{81} +11.1910 q^{82} +11.8365 q^{83} -9.90330 q^{84} +8.27351 q^{85} +12.0232 q^{86} -4.05757 q^{87} -3.24275 q^{88} -1.96091 q^{89} -16.8248 q^{90} -10.2764 q^{91} +1.77570 q^{92} +26.8116 q^{93} -6.09693 q^{94} +5.87095 q^{95} +2.91842 q^{96} +17.6719 q^{97} -4.51503 q^{98} +17.8907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more