LMFDB - Embedded newform 8007.2.a.f.1.37

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:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.37
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)

$f(q)$	$=$	$q+1.67933 q^{2} -1.00000 q^{3} +0.820135 q^{4} +3.59386 q^{5} -1.67933 q^{6} +1.09506 q^{7} -1.98138 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.67933 q^{2} -1.00000 q^{3} +0.820135 q^{4} +3.59386 q^{5} -1.67933 q^{6} +1.09506 q^{7} -1.98138 q^{8} +1.00000 q^{9} +6.03526 q^{10} -1.52969 q^{11} -0.820135 q^{12} -0.0461842 q^{13} +1.83896 q^{14} -3.59386 q^{15} -4.96765 q^{16} -1.00000 q^{17} +1.67933 q^{18} -1.91613 q^{19} +2.94745 q^{20} -1.09506 q^{21} -2.56885 q^{22} -5.70497 q^{23} +1.98138 q^{24} +7.91581 q^{25} -0.0775583 q^{26} -1.00000 q^{27} +0.898094 q^{28} -8.96107 q^{29} -6.03526 q^{30} +1.93639 q^{31} -4.37954 q^{32} +1.52969 q^{33} -1.67933 q^{34} +3.93548 q^{35} +0.820135 q^{36} -6.02148 q^{37} -3.21781 q^{38} +0.0461842 q^{39} -7.12079 q^{40} -4.24287 q^{41} -1.83896 q^{42} -0.305192 q^{43} -1.25455 q^{44} +3.59386 q^{45} -9.58051 q^{46} +5.72249 q^{47} +4.96765 q^{48} -5.80085 q^{49} +13.2932 q^{50} +1.00000 q^{51} -0.0378772 q^{52} +0.193461 q^{53} -1.67933 q^{54} -5.49749 q^{55} -2.16972 q^{56} +1.91613 q^{57} -15.0486 q^{58} -2.32265 q^{59} -2.94745 q^{60} -14.8783 q^{61} +3.25183 q^{62} +1.09506 q^{63} +2.58062 q^{64} -0.165979 q^{65} +2.56885 q^{66} -4.82522 q^{67} -0.820135 q^{68} +5.70497 q^{69} +6.60895 q^{70} -8.61413 q^{71} -1.98138 q^{72} +14.1239 q^{73} -10.1120 q^{74} -7.91581 q^{75} -1.57149 q^{76} -1.67510 q^{77} +0.0775583 q^{78} +4.00757 q^{79} -17.8530 q^{80} +1.00000 q^{81} -7.12516 q^{82} -3.07445 q^{83} -0.898094 q^{84} -3.59386 q^{85} -0.512517 q^{86} +8.96107 q^{87} +3.03090 q^{88} -2.17593 q^{89} +6.03526 q^{90} -0.0505743 q^{91} -4.67885 q^{92} -1.93639 q^{93} +9.60993 q^{94} -6.88631 q^{95} +4.37954 q^{96} -15.4254 q^{97} -9.74152 q^{98} -1.52969 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * 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 $

$2$	1.67933	1.18746	0.593731	−	0.804663i	$-0.297654\pi$
$2$	1.67933	1.18746	0.593731	+	0.804663i	$0.297654\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.820135	0.410067
$5$	3.59386	1.60722	0.803611	−	0.595155i	$-0.202909\pi$
$5$	3.59386	1.60722	0.803611	+	0.595155i	$0.202909\pi$
$6$	−1.67933	−0.685582
$7$	1.09506	0.413892	0.206946	−	0.978352i	$-0.433647\pi$
$7$	1.09506	0.413892	0.206946	+	0.978352i	$0.433647\pi$
$8$	−1.98138	−0.700523
$9$	1.00000	0.333333
$10$	6.03526	1.90852
$11$	−1.52969	−0.461220	−0.230610	−	0.973046i	$-0.574072\pi$
$11$	−1.52969	−0.461220	−0.230610	+	0.973046i	$0.574072\pi$
$12$	−0.820135	−0.236752
$13$	−0.0461842	−0.0128092	−0.00640459	−	0.999979i	$-0.502039\pi$
$13$	−0.0461842	−0.0128092	−0.00640459	+	0.999979i	$0.502039\pi$
$14$	1.83896	0.491482
$15$	−3.59386	−0.927930
$16$	−4.96765	−1.24191
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.67933	0.395821
$19$	−1.91613	−0.439591	−0.219796	−	0.975546i	$-0.570539\pi$
$19$	−1.91613	−0.439591	−0.219796	+	0.975546i	$0.570539\pi$
$20$	2.94745	0.659069
$21$	−1.09506	−0.238961
$22$	−2.56885	−0.547681
$23$	−5.70497	−1.18957	−0.594785	−	0.803885i	$-0.702763\pi$
$23$	−5.70497	−1.18957	−0.594785	+	0.803885i	$0.702763\pi$
$24$	1.98138	0.404447
$25$	7.91581	1.58316
$26$	−0.0775583	−0.0152104
$27$	−1.00000	−0.192450
$28$	0.898094	0.169724
$29$	−8.96107	−1.66403	−0.832014	−	0.554754i	$-0.812812\pi$
$29$	−8.96107	−1.66403	−0.832014	+	0.554754i	$0.812812\pi$
$30$	−6.03526	−1.10188
$31$	1.93639	0.347786	0.173893	−	0.984765i	$-0.444365\pi$
$31$	1.93639	0.347786	0.173893	+	0.984765i	$0.444365\pi$
$32$	−4.37954	−0.774201
$33$	1.52969	0.266285
$34$	−1.67933	−0.288002
$35$	3.93548	0.665217
$36$	0.820135	0.136689
$37$	−6.02148	−0.989925	−0.494962	−	0.868914i	$-0.664818\pi$
$37$	−6.02148	−0.989925	−0.494962	+	0.868914i	$0.664818\pi$
$38$	−3.21781	−0.521998
$39$	0.0461842	0.00739539
$40$	−7.12079	−1.12590
$41$	−4.24287	−0.662625	−0.331312	−	0.943521i	$-0.607491\pi$
$41$	−4.24287	−0.662625	−0.331312	+	0.943521i	$0.607491\pi$
$42$	−1.83896	−0.283757
$43$	−0.305192	−0.0465414	−0.0232707	−	0.999729i	$-0.507408\pi$
$43$	−0.305192	−0.0465414	−0.0232707	+	0.999729i	$0.507408\pi$
$44$	−1.25455	−0.189131
$45$	3.59386	0.535741
$46$	−9.58051	−1.41257
$47$	5.72249	0.834712	0.417356	−	0.908743i	$-0.362957\pi$
$47$	5.72249	0.834712	0.417356	+	0.908743i	$0.362957\pi$
$48$	4.96765	0.717018
$49$	−5.80085	−0.828693
$50$	13.2932	1.87995
$51$	1.00000	0.140028
$52$	−0.0378772	−0.00525263
$53$	0.193461	0.0265740	0.0132870	−	0.999912i	$-0.495770\pi$
$53$	0.193461	0.0265740	0.0132870	+	0.999912i	$0.495770\pi$
$54$	−1.67933	−0.228527
$55$	−5.49749	−0.741282
$56$	−2.16972	−0.289941
$57$	1.91613	0.253798
$58$	−15.0486	−1.97597
$59$	−2.32265	−0.302383	−0.151191	−	0.988505i	$-0.548311\pi$
$59$	−2.32265	−0.302383	−0.151191	+	0.988505i	$0.548311\pi$
$60$	−2.94745	−0.380514
$61$	−14.8783	−1.90497	−0.952485	−	0.304587i	$-0.901482\pi$
$61$	−14.8783	−1.90497	−0.952485	+	0.304587i	$0.901482\pi$
$62$	3.25183	0.412983
$63$	1.09506	0.137964
$64$	2.58062	0.322577
$65$	−0.165979	−0.0205872
$66$	2.56885	0.316204
$67$	−4.82522	−0.589495	−0.294747	−	0.955575i	$-0.595236\pi$
$67$	−4.82522	−0.589495	−0.294747	+	0.955575i	$0.595236\pi$
$68$	−0.820135	−0.0994559
$69$	5.70497	0.686798
$70$	6.60895	0.789920
$71$	−8.61413	−1.02231	−0.511155	−	0.859489i	$-0.670782\pi$
$71$	−8.61413	−1.02231	−0.511155	+	0.859489i	$0.670782\pi$
$72$	−1.98138	−0.233508
$73$	14.1239	1.65307	0.826537	−	0.562883i	$-0.190308\pi$
$73$	14.1239	1.65307	0.826537	+	0.562883i	$0.190308\pi$
$74$	−10.1120	−1.17550
$75$	−7.91581	−0.914039
$76$	−1.57149	−0.180262
$77$	−1.67510	−0.190895
$78$	0.0775583	0.00878174
$79$	4.00757	0.450887	0.225443	−	0.974256i	$-0.427617\pi$
$79$	4.00757	0.450887	0.225443	+	0.974256i	$0.427617\pi$
$80$	−17.8530	−1.99603
$81$	1.00000	0.111111
$82$	−7.12516	−0.786842
$83$	−3.07445	−0.337465	−0.168732	−	0.985662i	$-0.553967\pi$
$83$	−3.07445	−0.337465	−0.168732	+	0.985662i	$0.553967\pi$
$84$	−0.898094	−0.0979900
$85$	−3.59386	−0.389809
$86$	−0.512517	−0.0552662
$87$	8.96107	0.960727
$88$	3.03090	0.323095
$89$	−2.17593	−0.230648	−0.115324	−	0.993328i	$-0.536791\pi$
$89$	−2.17593	−0.230648	−0.115324	+	0.993328i	$0.536791\pi$
$90$	6.03526	0.636172
$91$	−0.0505743	−0.00530162
$92$	−4.67885	−0.487803
$93$	−1.93639	−0.200794
$94$	9.60993	0.991189
$95$	−6.88631	−0.706521
$96$	4.37954	0.446985
$97$	−15.4254	−1.56621	−0.783105	−	0.621889i	$-0.786365\pi$
$97$	−15.4254	−1.56621	−0.783105	+	0.621889i	$0.786365\pi$
$98$	−9.74152	−0.984042
$99$	−1.52969	−0.153740
$100$	6.49203	0.649203
$101$	11.1376	1.10823	0.554116	−	0.832440i	$-0.313057\pi$
$101$	11.1376	1.10823	0.554116	+	0.832440i	$0.313057\pi$
$102$	1.67933	0.166278
$103$	4.16853	0.410737	0.205369	−	0.978685i	$-0.434161\pi$
$103$	4.16853	0.410737	0.205369	+	0.978685i	$0.434161\pi$
$104$	0.0915083	0.00897313
$105$	−3.93548	−0.384063
$106$	0.324885	0.0315556
$107$	9.95303	0.962196	0.481098	−	0.876667i	$-0.340238\pi$
$107$	9.95303	0.962196	0.481098	+	0.876667i	$0.340238\pi$
$108$	−0.820135	−0.0789175
$109$	−6.25723	−0.599334	−0.299667	−	0.954044i	$-0.596876\pi$
$109$	−6.25723	−0.599334	−0.299667	+	0.954044i	$0.596876\pi$
$110$	−9.23208	−0.880245
$111$	6.02148	0.571533
$112$	−5.43986	−0.514018
$113$	3.19648	0.300700	0.150350	−	0.988633i	$-0.451960\pi$
$113$	3.19648	0.300700	0.150350	+	0.988633i	$0.451960\pi$
$114$	3.21781	0.301376
$115$	−20.5029	−1.91190
$116$	−7.34928	−0.682364
$117$	−0.0461842	−0.00426973
$118$	−3.90048	−0.359068
$119$	−1.09506	−0.100384
$120$	7.12079	0.650036
$121$	−8.66004	−0.787277
$122$	−24.9855	−2.26208
$123$	4.24287	0.382567
$124$	1.58810	0.142616
$125$	10.4790	0.937270
$126$	1.83896	0.163827
$127$	−6.53383	−0.579784	−0.289892	−	0.957059i	$-0.593619\pi$
$127$	−6.53383	−0.579784	−0.289892	+	0.957059i	$0.593619\pi$
$128$	13.0928	1.15725
$129$	0.305192	0.0268707
$130$	−0.278733	−0.0244465
$131$	6.42075	0.560983	0.280492	−	0.959856i	$-0.409502\pi$
$131$	6.42075	0.560983	0.280492	+	0.959856i	$0.409502\pi$
$132$	1.25455	0.109195
$133$	−2.09828	−0.181944
$134$	−8.10312	−0.700003
$135$	−3.59386	−0.309310
$136$	1.98138	0.169902
$137$	8.79125	0.751087	0.375543	−	0.926805i	$-0.377456\pi$
$137$	8.79125	0.751087	0.375543	+	0.926805i	$0.377456\pi$
$138$	9.58051	0.815547
$139$	−0.593136	−0.0503091	−0.0251546	−	0.999684i	$-0.508008\pi$
$139$	−0.593136	−0.0503091	−0.0251546	+	0.999684i	$0.508008\pi$
$140$	3.22762	0.272784
$141$	−5.72249	−0.481921
$142$	−14.4659	−1.21395
$143$	0.0706476	0.00590785
$144$	−4.96765	−0.413971
$145$	−32.2048	−2.67446
$146$	23.7186	1.96296
$147$	5.80085	0.478446
$148$	−4.93842	−0.405936
$149$	9.44340	0.773634	0.386817	−	0.922157i	$-0.373575\pi$
$149$	9.44340	0.773634	0.386817	+	0.922157i	$0.373575\pi$
$150$	−13.2932	−1.08539
$151$	15.8001	1.28580	0.642899	−	0.765951i	$-0.277732\pi$
$151$	15.8001	1.28580	0.642899	+	0.765951i	$0.277732\pi$
$152$	3.79659	0.307944
$153$	−1.00000	−0.0808452
$154$	−2.81304	−0.226681
$155$	6.95911	0.558969
$156$	0.0378772	0.00303261
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	6.73001	0.535411
$159$	−0.193461	−0.0153425
$160$	−15.7395	−1.24431
$161$	−6.24727	−0.492354
$162$	1.67933	0.131940
$163$	8.56463	0.670834	0.335417	−	0.942070i	$-0.391123\pi$
$163$	8.56463	0.670834	0.335417	+	0.942070i	$0.391123\pi$
$164$	−3.47972	−0.271721
$165$	5.49749	0.427979
$166$	−5.16300	−0.400727
$167$	−12.1584	−0.940844	−0.470422	−	0.882442i	$-0.655898\pi$
$167$	−12.1584	−0.940844	−0.470422	+	0.882442i	$0.655898\pi$
$168$	2.16972	0.167398
$169$	−12.9979	−0.999836
$170$	−6.03526	−0.462883
$171$	−1.91613	−0.146530
$172$	−0.250299	−0.0190851
$173$	−13.0161	−0.989598	−0.494799	−	0.869007i	$-0.664758\pi$
$173$	−13.0161	−0.989598	−0.494799	+	0.869007i	$0.664758\pi$
$174$	15.0486	1.14083
$175$	8.66826	0.655259
$176$	7.59897	0.572794
$177$	2.32265	0.174581
$178$	−3.65409	−0.273886
$179$	5.06732	0.378749	0.189375	−	0.981905i	$-0.439354\pi$
$179$	5.06732	0.378749	0.189375	+	0.981905i	$0.439354\pi$
$180$	2.94745	0.219690
$181$	−5.21426	−0.387573	−0.193786	−	0.981044i	$-0.562077\pi$
$181$	−5.21426	−0.387573	−0.193786	+	0.981044i	$0.562077\pi$
$182$	−0.0849307	−0.00629548
$183$	14.8783	1.09983
$184$	11.3037	0.833321
$185$	−21.6403	−1.59103
$186$	−3.25183	−0.238436
$187$	1.52969	0.111862
$188$	4.69322	0.342288
$189$	−1.09506	−0.0796536
$190$	−11.5644	−0.838967
$191$	5.91599	0.428066	0.214033	−	0.976826i	$-0.431340\pi$
$191$	5.91599	0.428066	0.214033	+	0.976826i	$0.431340\pi$
$192$	−2.58062	−0.186240
$193$	22.8297	1.64332	0.821660	−	0.569978i	$-0.193048\pi$
$193$	22.8297	1.64332	0.821660	+	0.569978i	$0.193048\pi$
$194$	−25.9042	−1.85982
$195$	0.165979	0.0118860
$196$	−4.75748	−0.339820
$197$	−0.855946	−0.0609836	−0.0304918	−	0.999535i	$-0.509707\pi$
$197$	−0.855946	−0.0609836	−0.0304918	+	0.999535i	$0.509707\pi$
$198$	−2.56885	−0.182560
$199$	19.4670	1.37998	0.689988	−	0.723820i	$-0.257616\pi$
$199$	19.4670	1.37998	0.689988	+	0.723820i	$0.257616\pi$
$200$	−15.6842	−1.10904
$201$	4.82522	0.340345
$202$	18.7036	1.31598
$203$	−9.81288	−0.688729
$204$	0.820135	0.0574209
$205$	−15.2483	−1.06499
$206$	7.00031	0.487735
$207$	−5.70497	−0.396523
$208$	0.229427	0.0159079
$209$	2.93110	0.202748
$210$	−6.60895	−0.456061
$211$	−9.18507	−0.632327	−0.316163	−	0.948705i	$-0.602395\pi$
$211$	−9.18507	−0.632327	−0.316163	+	0.948705i	$0.602395\pi$
$212$	0.158664	0.0108971
$213$	8.61413	0.590231
$214$	16.7144	1.14257
$215$	−1.09682	−0.0748023
$216$	1.98138	0.134816
$217$	2.12046	0.143946
$218$	−10.5079	−0.711687
$219$	−14.1239	−0.954402
$220$	−4.50869	−0.303976
$221$	0.0461842	0.00310668
$222$	10.1120	0.678674
$223$	−15.0124	−1.00530	−0.502652	−	0.864489i	$-0.667642\pi$
$223$	−15.0124	−1.00530	−0.502652	+	0.864489i	$0.667642\pi$
$224$	−4.79585	−0.320436
$225$	7.91581	0.527721
$226$	5.36794	0.357070
$227$	19.0896	1.26702	0.633509	−	0.773735i	$-0.281614\pi$
$227$	19.0896	1.26702	0.633509	+	0.773735i	$0.281614\pi$
$228$	1.57149	0.104074
$229$	−14.5725	−0.962978	−0.481489	−	0.876452i	$-0.659904\pi$
$229$	−14.5725	−0.962978	−0.481489	+	0.876452i	$0.659904\pi$
$230$	−34.4310	−2.27031
$231$	1.67510	0.110213
$232$	17.7553	1.16569
$233$	−9.19047	−0.602087	−0.301044	−	0.953610i	$-0.597335\pi$
$233$	−9.19047	−0.602087	−0.301044	+	0.953610i	$0.597335\pi$
$234$	−0.0775583	−0.00507014
$235$	20.5658	1.34157
$236$	−1.90488	−0.123997
$237$	−4.00757	−0.260320
$238$	−1.83896	−0.119202
$239$	−29.9619	−1.93808	−0.969038	−	0.246911i	$-0.920584\pi$
$239$	−29.9619	−1.93808	−0.969038	+	0.246911i	$0.920584\pi$
$240$	17.8530	1.15241
$241$	−0.260683	−0.0167921	−0.00839604	−	0.999965i	$-0.502673\pi$
$241$	−0.260683	−0.0167921	−0.00839604	+	0.999965i	$0.502673\pi$
$242$	−14.5430	−0.934861
$243$	−1.00000	−0.0641500
$244$	−12.2022	−0.781166
$245$	−20.8474	−1.33189
$246$	7.12516	0.454284
$247$	0.0884951	0.00563081
$248$	−3.83672	−0.243632
$249$	3.07445	0.194835
$250$	17.5976	1.11297
$251$	28.8017	1.81795	0.908974	−	0.416853i	$-0.136867\pi$
$251$	28.8017	1.81795	0.908974	+	0.416853i	$0.136867\pi$
$252$	0.898094	0.0565746
$253$	8.72685	0.548653
$254$	−10.9724	−0.688472
$255$	3.59386	0.225056
$256$	16.8258	1.05161
$257$	−13.0022	−0.811057	−0.405529	−	0.914082i	$-0.632913\pi$
$257$	−13.0022	−0.811057	−0.405529	+	0.914082i	$0.632913\pi$
$258$	0.512517	0.0319079
$259$	−6.59386	−0.409722
$260$	−0.136125	−0.00844214
$261$	−8.96107	−0.554676
$262$	10.7825	0.666147
$263$	11.3986	0.702866	0.351433	−	0.936213i	$-0.385695\pi$
$263$	11.3986	0.702866	0.351433	+	0.936213i	$0.385695\pi$
$264$	−3.03090	−0.186539
$265$	0.695273	0.0427103
$266$	−3.52369	−0.216051
$267$	2.17593	0.133165
$268$	−3.95733	−0.241733
$269$	−10.6849	−0.651473	−0.325736	−	0.945461i	$-0.605612\pi$
$269$	−10.6849	−0.651473	−0.325736	+	0.945461i	$0.605612\pi$
$270$	−6.03526	−0.367294
$271$	−10.3987	−0.631677	−0.315839	−	0.948813i	$-0.602286\pi$
$271$	−10.3987	−0.631677	−0.315839	+	0.948813i	$0.602286\pi$
$272$	4.96765	0.301208
$273$	0.0505743	0.00306089
$274$	14.7634	0.891888
$275$	−12.1087	−0.730185
$276$	4.67885	0.281633
$277$	−4.41176	−0.265077	−0.132538	−	0.991178i	$-0.542313\pi$
$277$	−4.41176	−0.265077	−0.132538	+	0.991178i	$0.542313\pi$
$278$	−0.996069	−0.0597402
$279$	1.93639	0.115929
$280$	−7.79767	−0.466000
$281$	−2.71894	−0.162198	−0.0810992	−	0.996706i	$-0.525843\pi$
$281$	−2.71894	−0.162198	−0.0810992	+	0.996706i	$0.525843\pi$
$282$	−9.60993	−0.572263
$283$	−9.80213	−0.582676	−0.291338	−	0.956620i	$-0.594100\pi$
$283$	−9.80213	−0.582676	−0.291338	+	0.956620i	$0.594100\pi$
$284$	−7.06475	−0.419216
$285$	6.88631	0.407910
$286$	0.118640	0.00701535
$287$	−4.64618	−0.274255
$288$	−4.37954	−0.258067
$289$	1.00000	0.0588235
$290$	−54.0823	−3.17582
$291$	15.4254	0.904252
$292$	11.5835	0.677871
$293$	18.5396	1.08309	0.541547	−	0.840670i	$-0.317839\pi$
$293$	18.5396	1.08309	0.541547	+	0.840670i	$0.317839\pi$
$294$	9.74152	0.568137
$295$	−8.34726	−0.485996
$296$	11.9308	0.693465
$297$	1.52969	0.0887617
$298$	15.8585	0.918661
$299$	0.263479	0.0152374
$300$	−6.49203	−0.374817
$301$	−0.334203	−0.0192631
$302$	26.5336	1.52684
$303$	−11.1376	−0.639837
$304$	9.51868	0.545934
$305$	−53.4704	−3.06171
$306$	−1.67933	−0.0960007
$307$	12.4781	0.712161	0.356080	−	0.934455i	$-0.384113\pi$
$307$	12.4781	0.712161	0.356080	+	0.934455i	$0.384113\pi$
$308$	−1.37381	−0.0782799
$309$	−4.16853	−0.237139
$310$	11.6866	0.663755
$311$	−5.40547	−0.306516	−0.153258	−	0.988186i	$-0.548977\pi$
$311$	−5.40547	−0.306516	−0.153258	+	0.988186i	$0.548977\pi$
$312$	−0.0915083	−0.00518064
$313$	22.9439	1.29687	0.648433	−	0.761272i	$-0.275425\pi$
$313$	22.9439	1.29687	0.648433	+	0.761272i	$0.275425\pi$
$314$	−1.67933	−0.0947698
$315$	3.93548	0.221739
$316$	3.28674	0.184894
$317$	32.7680	1.84043	0.920217	−	0.391410i	$-0.128012\pi$
$317$	32.7680	1.84043	0.920217	+	0.391410i	$0.128012\pi$
$318$	−0.324885	−0.0182186
$319$	13.7077	0.767483
$320$	9.27437	0.518453
$321$	−9.95303	−0.555524
$322$	−10.4912	−0.584652
$323$	1.91613	0.106617
$324$	0.820135	0.0455630
$325$	−0.365585	−0.0202790
$326$	14.3828	0.796590
$327$	6.25723	0.346026
$328$	8.40673	0.464184
$329$	6.26645	0.345481
$330$	9.23208	0.508210
$331$	−3.68621	−0.202613	−0.101306	−	0.994855i	$-0.532302\pi$
$331$	−3.68621	−0.202613	−0.101306	+	0.994855i	$0.532302\pi$
$332$	−2.52146	−0.138383
$333$	−6.02148	−0.329975
$334$	−20.4179	−1.11722
$335$	−17.3412	−0.947449
$336$	5.43986	0.296768
$337$	12.2642	0.668074	0.334037	−	0.942560i	$-0.391589\pi$
$337$	12.2642	0.668074	0.334037	+	0.942560i	$0.391589\pi$
$338$	−21.8277	−1.18727
$339$	−3.19648	−0.173609
$340$	−2.94745	−0.159848
$341$	−2.96208	−0.160406
$342$	−3.21781	−0.173999
$343$	−14.0177	−0.756882
$344$	0.604702	0.0326033
$345$	20.5029	1.10384
$346$	−21.8583	−1.17511
$347$	9.28716	0.498561	0.249280	−	0.968431i	$-0.419806\pi$
$347$	9.28716	0.498561	0.249280	+	0.968431i	$0.419806\pi$
$348$	7.34928	0.393963
$349$	−29.4998	−1.57909	−0.789544	−	0.613695i	$-0.789683\pi$
$349$	−29.4998	−1.57909	−0.789544	+	0.613695i	$0.789683\pi$
$350$	14.5568	0.778095
$351$	0.0461842	0.00246513
$352$	6.69935	0.357077
$353$	−15.0848	−0.802882	−0.401441	−	0.915885i	$-0.631491\pi$
$353$	−15.0848	−0.802882	−0.401441	+	0.915885i	$0.631491\pi$
$354$	3.90048	0.207308
$355$	−30.9580	−1.64308
$356$	−1.78455	−0.0945812
$357$	1.09506	0.0579565
$358$	8.50968	0.449750
$359$	3.23478	0.170725	0.0853626	−	0.996350i	$-0.472795\pi$
$359$	3.23478	0.170725	0.0853626	+	0.996350i	$0.472795\pi$
$360$	−7.12079	−0.375299
$361$	−15.3284	−0.806759
$362$	−8.75644	−0.460228
$363$	8.66004	0.454534
$364$	−0.0414777	−0.00217402
$365$	50.7591	2.65686
$366$	24.9855	1.30601
$367$	0.561855	0.0293286	0.0146643	−	0.999892i	$-0.495332\pi$
$367$	0.561855	0.0293286	0.0146643	+	0.999892i	$0.495332\pi$
$368$	28.3403	1.47734
$369$	−4.24287	−0.220875
$370$	−36.3412	−1.88929
$371$	0.211851	0.0109988
$372$	−1.58810	−0.0823392
$373$	27.4552	1.42158	0.710789	−	0.703405i	$-0.248338\pi$
$373$	27.4552	1.42158	0.710789	+	0.703405i	$0.248338\pi$
$374$	2.56885	0.132832
$375$	−10.4790	−0.541133
$376$	−11.3384	−0.584735
$377$	0.413859	0.0213148
$378$	−1.83896	−0.0945857
$379$	9.73443	0.500024	0.250012	−	0.968243i	$-0.419565\pi$
$379$	9.73443	0.500024	0.250012	+	0.968243i	$0.419565\pi$
$380$	−5.64770	−0.289721
$381$	6.53383	0.334738
$382$	9.93488	0.508313
$383$	−16.8676	−0.861892	−0.430946	−	0.902378i	$-0.641820\pi$
$383$	−16.8676	−0.861892	−0.430946	+	0.902378i	$0.641820\pi$
$384$	−13.0928	−0.668138
$385$	−6.02007	−0.306811
$386$	38.3386	1.95138
$387$	−0.305192	−0.0155138
$388$	−12.6509	−0.642252
$389$	11.4566	0.580873	0.290437	−	0.956894i	$-0.406199\pi$
$389$	11.4566	0.580873	0.290437	+	0.956894i	$0.406199\pi$
$390$	0.278733	0.0141142
$391$	5.70497	0.288513
$392$	11.4937	0.580519
$393$	−6.42075	−0.323884
$394$	−1.43741	−0.0724158
$395$	14.4026	0.724675
$396$	−1.25455	−0.0630437
$397$	−32.8386	−1.64812	−0.824061	−	0.566501i	$-0.808297\pi$
$397$	−32.8386	−1.64812	−0.824061	+	0.566501i	$0.808297\pi$
$398$	32.6914	1.63867
$399$	2.09828	0.105045
$400$	−39.3230	−1.96615
$401$	−36.5893	−1.82718	−0.913590	−	0.406636i	$-0.866702\pi$
$401$	−36.5893	−1.82718	−0.913590	+	0.406636i	$0.866702\pi$
$402$	8.10312	0.404147
$403$	−0.0894306	−0.00445486
$404$	9.13432	0.454449
$405$	3.59386	0.178580
$406$	−16.4790	−0.817840
$407$	9.21100	0.456573
$408$	−1.98138	−0.0980928
$409$	−12.4925	−0.617715	−0.308858	−	0.951108i	$-0.599947\pi$
$409$	−12.4925	−0.617715	−0.308858	+	0.951108i	$0.599947\pi$
$410$	−25.6068	−1.26463
$411$	−8.79125	−0.433640
$412$	3.41875	0.168430
$413$	−2.54343	−0.125154
$414$	−9.58051	−0.470856
$415$	−11.0491	−0.542381
$416$	0.202266	0.00991688
$417$	0.593136	0.0290460
$418$	4.92226	0.240756
$419$	−27.6888	−1.35269	−0.676344	−	0.736586i	$-0.736437\pi$
$419$	−27.6888	−1.35269	−0.676344	+	0.736586i	$0.736437\pi$
$420$	−3.22762	−0.157492
$421$	21.9880	1.07163	0.535815	−	0.844335i	$-0.320004\pi$
$421$	21.9880	1.07163	0.535815	+	0.844335i	$0.320004\pi$
$422$	−15.4247	−0.750864
$423$	5.72249	0.278237
$424$	−0.383320	−0.0186157
$425$	−7.91581	−0.383973
$426$	14.4659	0.700877
$427$	−16.2926	−0.788452
$428$	8.16283	0.394565
$429$	−0.0706476	−0.00341090
$430$	−1.84191	−0.0888250
$431$	4.78198	0.230340	0.115170	−	0.993346i	$-0.463259\pi$
$431$	4.78198	0.230340	0.115170	+	0.993346i	$0.463259\pi$
$432$	4.96765	0.239006
$433$	−17.8643	−0.858503	−0.429251	−	0.903185i	$-0.641223\pi$
$433$	−17.8643	−0.858503	−0.429251	+	0.903185i	$0.641223\pi$
$434$	3.56094	0.170931
$435$	32.2048	1.54410
$436$	−5.13177	−0.245767
$437$	10.9315	0.522924
$438$	−23.7186	−1.13332
$439$	12.3881	0.591252	0.295626	−	0.955304i	$-0.404472\pi$
$439$	12.3881	0.591252	0.295626	+	0.955304i	$0.404472\pi$
$440$	10.8926	0.519285
$441$	−5.80085	−0.276231
$442$	0.0775583	0.00368907
$443$	−19.2163	−0.912995	−0.456498	−	0.889725i	$-0.650896\pi$
$443$	−19.2163	−0.912995	−0.456498	+	0.889725i	$0.650896\pi$
$444$	4.93842	0.234367
$445$	−7.81998	−0.370703
$446$	−25.2107	−1.19376
$447$	−9.44340	−0.446658
$448$	2.82592	0.133512
$449$	−8.50966	−0.401596	−0.200798	−	0.979633i	$-0.564353\pi$
$449$	−8.50966	−0.401596	−0.200798	+	0.979633i	$0.564353\pi$
$450$	13.2932	0.626648
$451$	6.49028	0.305616
$452$	2.62155	0.123307
$453$	−15.8001	−0.742356
$454$	32.0576	1.50454
$455$	−0.181757	−0.00852088
$456$	−3.79659	−0.177791
$457$	−30.1291	−1.40938	−0.704689	−	0.709516i	$-0.748914\pi$
$457$	−30.1291	−1.40938	−0.704689	+	0.709516i	$0.748914\pi$
$458$	−24.4720	−1.14350
$459$	1.00000	0.0466760
$460$	−16.8151	−0.784008
$461$	16.4091	0.764249	0.382124	−	0.924111i	$-0.375193\pi$
$461$	16.4091	0.764249	0.382124	+	0.924111i	$0.375193\pi$
$462$	2.81304	0.130874
$463$	−31.4518	−1.46169	−0.730844	−	0.682545i	$-0.760873\pi$
$463$	−31.4518	−1.46169	−0.730844	+	0.682545i	$0.760873\pi$
$464$	44.5154	2.06658
$465$	−6.95911	−0.322721
$466$	−15.4338	−0.714956
$467$	4.39800	0.203515	0.101757	−	0.994809i	$-0.467553\pi$
$467$	4.39800	0.203515	0.101757	+	0.994809i	$0.467553\pi$
$468$	−0.0378772	−0.00175088
$469$	−5.28389	−0.243987
$470$	34.5367	1.59306
$471$	1.00000	0.0460776
$472$	4.60204	0.211826
$473$	0.466850	0.0214658
$474$	−6.73001	−0.309120
$475$	−15.1678	−0.695944
$476$	−0.898094	−0.0411641
$477$	0.193461	0.00885799
$478$	−50.3158	−2.30139
$479$	−15.6060	−0.713056	−0.356528	−	0.934285i	$-0.616040\pi$
$479$	−15.6060	−0.713056	−0.356528	+	0.934285i	$0.616040\pi$
$480$	15.7395	0.718404
$481$	0.278097	0.0126801
$482$	−0.437772	−0.0199400
$483$	6.24727	0.284261
$484$	−7.10240	−0.322836
$485$	−55.4366	−2.51725
$486$	−1.67933	−0.0761758
$487$	40.2640	1.82454	0.912268	−	0.409594i	$-0.134330\pi$
$487$	40.2640	1.82454	0.912268	+	0.409594i	$0.134330\pi$
$488$	29.4795	1.33447
$489$	−8.56463	−0.387306
$490$	−35.0096	−1.58157
$491$	−30.9388	−1.39625	−0.698124	−	0.715976i	$-0.745982\pi$
$491$	−30.9388	−1.39625	−0.698124	+	0.715976i	$0.745982\pi$
$492$	3.47972	0.156878
$493$	8.96107	0.403586
$494$	0.148612	0.00668637
$495$	−5.49749	−0.247094
$496$	−9.61931	−0.431920
$497$	−9.43296	−0.423126
$498$	5.16300	0.231360
$499$	−5.81475	−0.260304	−0.130152	−	0.991494i	$-0.541547\pi$
$499$	−5.81475	−0.260304	−0.130152	+	0.991494i	$0.541547\pi$
$500$	8.59419	0.384344
$501$	12.1584	0.543197
$502$	48.3674	2.15874
$503$	−26.0840	−1.16303	−0.581514	−	0.813536i	$-0.697539\pi$
$503$	−26.0840	−1.16303	−0.581514	+	0.813536i	$0.697539\pi$
$504$	−2.16972	−0.0966471
$505$	40.0269	1.78117
$506$	14.6552	0.651504
$507$	12.9979	0.577256
$508$	−5.35862	−0.237750
$509$	−4.63729	−0.205544	−0.102772	−	0.994705i	$-0.532771\pi$
$509$	−4.63729	−0.205544	−0.102772	+	0.994705i	$0.532771\pi$
$510$	6.03526	0.267246
$511$	15.4664	0.684195
$512$	2.07045	0.0915017
$513$	1.91613	0.0845994
$514$	−21.8350	−0.963100
$515$	14.9811	0.660146
$516$	0.250299	0.0110188
$517$	−8.75366	−0.384985
$518$	−11.0732	−0.486530
$519$	13.0161	0.571345
$520$	0.328868	0.0144218
$521$	−14.0071	−0.613662	−0.306831	−	0.951764i	$-0.599269\pi$
$521$	−14.0071	−0.613662	−0.306831	+	0.951764i	$0.599269\pi$
$522$	−15.0486	−0.658657
$523$	19.8939	0.869901	0.434951	−	0.900454i	$-0.356766\pi$
$523$	19.8939	0.869901	0.434951	+	0.900454i	$0.356766\pi$
$524$	5.26588	0.230041
$525$	−8.66826	−0.378314
$526$	19.1419	0.834627
$527$	−1.93639	−0.0843505
$528$	−7.59897	−0.330703
$529$	9.54673	0.415075
$530$	1.16759	0.0507168
$531$	−2.32265	−0.100794
$532$	−1.72087	−0.0746091
$533$	0.195953	0.00848768
$534$	3.65409	0.158128
$535$	35.7698	1.54646
$536$	9.56059	0.412955
$537$	−5.06732	−0.218671
$538$	−17.9435	−0.773599
$539$	8.87352	0.382209
$540$	−2.94745	−0.126838
$541$	−28.7557	−1.23631	−0.618153	−	0.786058i	$-0.712119\pi$
$541$	−28.7557	−1.23631	−0.618153	+	0.786058i	$0.712119\pi$
$542$	−17.4628	−0.750093
$543$	5.21426	0.223765
$544$	4.37954	0.187771
$545$	−22.4876	−0.963263
$546$	0.0849307	0.00363470
$547$	−4.15252	−0.177549	−0.0887745	−	0.996052i	$-0.528295\pi$
$547$	−4.15252	−0.177549	−0.0887745	+	0.996052i	$0.528295\pi$
$548$	7.21001	0.307996
$549$	−14.8783	−0.634990
$550$	−20.3345	−0.867067
$551$	17.1706	0.731493
$552$	−11.3037	−0.481118
$553$	4.38851	0.186619
$554$	−7.40878	−0.314769
$555$	21.6403	0.918581
$556$	−0.486451	−0.0206301
$557$	−33.7307	−1.42922	−0.714609	−	0.699524i	$-0.753395\pi$
$557$	−33.7307	−1.42922	−0.714609	+	0.699524i	$0.753395\pi$
$558$	3.25183	0.137661
$559$	0.0140951	0.000596157 0
$560$	−19.5501	−0.826141
$561$	−1.52969	−0.0645837
$562$	−4.56599	−0.192605
$563$	33.3227	1.40439	0.702193	−	0.711987i	$-0.252204\pi$
$563$	33.3227	1.40439	0.702193	+	0.711987i	$0.252204\pi$
$564$	−4.69322	−0.197620
$565$	11.4877	0.483291
$566$	−16.4610	−0.691906
$567$	1.09506	0.0459880
$568$	17.0679	0.716151
$569$	−37.3867	−1.56733	−0.783665	−	0.621183i	$-0.786652\pi$
$569$	−37.3867	−1.56733	−0.783665	+	0.621183i	$0.786652\pi$
$570$	11.5644	0.484378
$571$	0.500722	0.0209546	0.0104773	−	0.999945i	$-0.496665\pi$
$571$	0.500722	0.0209546	0.0104773	+	0.999945i	$0.496665\pi$
$572$	0.0579405	0.00242261
$573$	−5.91599	−0.247144
$574$	−7.80245	−0.325668
$575$	−45.1595	−1.88328
$576$	2.58062	0.107526
$577$	−8.80915	−0.366730	−0.183365	−	0.983045i	$-0.558699\pi$
$577$	−8.80915	−0.366730	−0.183365	+	0.983045i	$0.558699\pi$
$578$	1.67933	0.0698507
$579$	−22.8297	−0.948772
$580$	−26.4123	−1.09671
$581$	−3.36670	−0.139674
$582$	25.9042	1.07377
$583$	−0.295936	−0.0122564
$584$	−27.9847	−1.15802
$585$	−0.165979	−0.00686240
$586$	31.1340	1.28613
$587$	47.2663	1.95089	0.975444	−	0.220246i	$-0.0706860\pi$
$587$	47.2663	1.95089	0.975444	+	0.220246i	$0.0706860\pi$
$588$	4.75748	0.196195
$589$	−3.71039	−0.152884
$590$	−14.0178	−0.577102
$591$	0.855946	0.0352089
$592$	29.9126	1.22940
$593$	−0.274025	−0.0112528	−0.00562642	−	0.999984i	$-0.501791\pi$
$593$	−0.274025	−0.0112528	−0.00562642	+	0.999984i	$0.501791\pi$
$594$	2.56885	0.105401
$595$	−3.93548	−0.161339
$596$	7.74486	0.317242
$597$	−19.4670	−0.796730
$598$	0.442468	0.0180939
$599$	36.0529	1.47308	0.736542	−	0.676392i	$-0.236458\pi$
$599$	36.0529	1.47308	0.736542	+	0.676392i	$0.236458\pi$
$600$	15.6842	0.640305
$601$	−14.6526	−0.597693	−0.298846	−	0.954301i	$-0.596602\pi$
$601$	−14.6526	−0.597693	−0.298846	+	0.954301i	$0.596602\pi$
$602$	−0.561235	−0.0228742
$603$	−4.82522	−0.196498
$604$	12.9582	0.527264
$605$	−31.1230	−1.26533
$606$	−18.7036	−0.759783
$607$	−15.6217	−0.634064	−0.317032	−	0.948415i	$-0.602686\pi$
$607$	−15.6217	−0.634064	−0.317032	+	0.948415i	$0.602686\pi$
$608$	8.39179	0.340332
$609$	9.81288	0.397638
$610$	−89.7943	−3.63566
$611$	−0.264289	−0.0106920
$612$	−0.820135	−0.0331520
$613$	12.0078	0.484991	0.242496	−	0.970152i	$-0.422034\pi$
$613$	12.0078	0.484991	0.242496	+	0.970152i	$0.422034\pi$
$614$	20.9547	0.845664
$615$	15.2483	0.614870
$616$	3.31901	0.133727
$617$	14.9841	0.603237	0.301619	−	0.953429i	$-0.402473\pi$
$617$	14.9841	0.603237	0.301619	+	0.953429i	$0.402473\pi$
$618$	−7.00031	−0.281594
$619$	−1.87009	−0.0751654	−0.0375827	−	0.999294i	$-0.511966\pi$
$619$	−1.87009	−0.0751654	−0.0375827	+	0.999294i	$0.511966\pi$
$620$	5.70741	0.229215
$621$	5.70497	0.228933
$622$	−9.07755	−0.363977
$623$	−2.38277	−0.0954635
$624$	−0.229427	−0.00918442
$625$	−1.91902	−0.0767608
$626$	38.5303	1.53998
$627$	−2.93110	−0.117057
$628$	−0.820135	−0.0327269
$629$	6.02148	0.240092
$630$	6.60895	0.263307
$631$	45.2024	1.79948	0.899739	−	0.436428i	$-0.143756\pi$
$631$	45.2024	1.79948	0.899739	+	0.436428i	$0.143756\pi$
$632$	−7.94051	−0.315856
$633$	9.18507	0.365074
$634$	55.0281	2.18545
$635$	−23.4817	−0.931841
$636$	−0.158664	−0.00629145
$637$	0.267907	0.0106149
$638$	23.0197	0.911357
$639$	−8.61413	−0.340770
$640$	47.0536	1.85996
$641$	−21.0744	−0.832388	−0.416194	−	0.909276i	$-0.636636\pi$
$641$	−21.0744	−0.832388	−0.416194	+	0.909276i	$0.636636\pi$
$642$	−16.7144	−0.659664
$643$	−10.7588	−0.424284	−0.212142	−	0.977239i	$-0.568044\pi$
$643$	−10.7588	−0.424284	−0.212142	+	0.977239i	$0.568044\pi$
$644$	−5.12360	−0.201898
$645$	1.09682	0.0431872
$646$	3.21781	0.126603
$647$	−43.4902	−1.70978	−0.854888	−	0.518812i	$-0.826374\pi$
$647$	−43.4902	−1.70978	−0.854888	+	0.518812i	$0.826374\pi$
$648$	−1.98138	−0.0778359
$649$	3.55293	0.139465
$650$	−0.613936	−0.0240806
$651$	−2.12046	−0.0831073
$652$	7.02415	0.275087
$653$	18.7651	0.734334	0.367167	−	0.930155i	$-0.380328\pi$
$653$	18.7651	0.734334	0.367167	+	0.930155i	$0.380328\pi$
$654$	10.5079	0.410893
$655$	23.0752	0.901625
$656$	21.0771	0.822922
$657$	14.1239	0.551024
$658$	10.5234	0.410246
$659$	−9.87754	−0.384774	−0.192387	−	0.981319i	$-0.561623\pi$
$659$	−9.87754	−0.384774	−0.192387	+	0.981319i	$0.561623\pi$
$660$	4.50869	0.175500
$661$	44.2858	1.72252	0.861259	−	0.508167i	$-0.169677\pi$
$661$	44.2858	1.72252	0.861259	+	0.508167i	$0.169677\pi$
$662$	−6.19035	−0.240595
$663$	−0.0461842	−0.00179364
$664$	6.09165	0.236402
$665$	−7.54090	−0.292424
$666$	−10.1120	−0.391833
$667$	51.1227	1.97948
$668$	−9.97151	−0.385809
$669$	15.0124	0.580413
$670$	−29.1215	−1.12506
$671$	22.7592	0.878609
$672$	4.79585	0.185004
$673$	33.3030	1.28374	0.641868	−	0.766815i	$-0.278160\pi$
$673$	33.3030	1.28374	0.641868	+	0.766815i	$0.278160\pi$
$674$	20.5956	0.793312
$675$	−7.91581	−0.304680
$676$	−10.6600	−0.410000
$677$	−37.4141	−1.43794	−0.718970	−	0.695041i	$-0.755386\pi$
$677$	−37.4141	−1.43794	−0.718970	+	0.695041i	$0.755386\pi$
$678$	−5.36794	−0.206154
$679$	−16.8917	−0.648243
$680$	7.12079	0.273070
$681$	−19.0896	−0.731513
$682$	−4.97430	−0.190476
$683$	−18.1151	−0.693156	−0.346578	−	0.938021i	$-0.612656\pi$
$683$	−18.1151	−0.693156	−0.346578	+	0.938021i	$0.612656\pi$
$684$	−1.57149	−0.0600873
$685$	31.5945	1.20716
$686$	−23.5402	−0.898769
$687$	14.5725	0.555976
$688$	1.51609	0.0578003
$689$	−0.00893485	−0.000340391 0
$690$	34.4310	1.31076
$691$	39.2338	1.49252	0.746261	−	0.665653i	$-0.231847\pi$
$691$	39.2338	1.49252	0.746261	+	0.665653i	$0.231847\pi$
$692$	−10.6750	−0.405802
$693$	−1.67510	−0.0636318
$694$	15.5962	0.592022
$695$	−2.13165	−0.0808579
$696$	−17.7553	−0.673012
$697$	4.24287	0.160710
$698$	−49.5397	−1.87511
$699$	9.19047	0.347615
$700$	7.10914	0.268700
$701$	45.6812	1.72535	0.862677	−	0.505756i	$-0.168786\pi$
$701$	45.6812	1.72535	0.862677	+	0.505756i	$0.168786\pi$
$702$	0.0775583	0.00292725
$703$	11.5380	0.435162
$704$	−3.94755	−0.148779
$705$	−20.5658	−0.774554
$706$	−25.3323	−0.953392
$707$	12.1963	0.458688
$708$	1.90488	0.0715899
$709$	−41.7290	−1.56717	−0.783583	−	0.621287i	$-0.786610\pi$
$709$	−41.7290	−1.56717	−0.783583	+	0.621287i	$0.786610\pi$
$710$	−51.9885	−1.95109
$711$	4.00757	0.150296
$712$	4.31134	0.161574
$713$	−11.0471	−0.413716
$714$	1.83896	0.0688212
$715$	0.253897	0.00949522
$716$	4.15588	0.155313
$717$	29.9619	1.11895
$718$	5.43225	0.202730
$719$	35.2895	1.31608	0.658038	−	0.752985i	$-0.271387\pi$
$719$	35.2895	1.31608	0.658038	+	0.752985i	$0.271387\pi$
$720$	−17.8530	−0.665343
$721$	4.56477	0.170001
$722$	−25.7414	−0.957997
$723$	0.260683	0.00969491
$724$	−4.27639	−0.158931
$725$	−70.9341	−2.63443
$726$	14.5430	0.539742
$727$	−15.0621	−0.558621	−0.279311	−	0.960201i	$-0.590106\pi$
$727$	−15.0621	−0.558621	−0.279311	+	0.960201i	$0.590106\pi$
$728$	0.100207	0.00371391
$729$	1.00000	0.0370370
$730$	85.2411	3.15492
$731$	0.305192	0.0112879
$732$	12.2022	0.451006
$733$	−5.35920	−0.197947	−0.0989733	−	0.995090i	$-0.531556\pi$
$733$	−5.35920	−0.197947	−0.0989733	+	0.995090i	$0.531556\pi$
$734$	0.943537	0.0348266
$735$	20.8474	0.768969
$736$	24.9852	0.920966
$737$	7.38111	0.271887
$738$	−7.12516	−0.262281
$739$	21.0427	0.774067	0.387033	−	0.922066i	$-0.373500\pi$
$739$	21.0427	0.774067	0.387033	+	0.922066i	$0.373500\pi$
$740$	−17.7480	−0.652429
$741$	−0.0884951	−0.00325095
$742$	0.355767	0.0130606
$743$	38.4467	1.41047	0.705236	−	0.708973i	$-0.250841\pi$
$743$	38.4467	1.41047	0.705236	+	0.708973i	$0.250841\pi$
$744$	3.83672	0.140661
$745$	33.9382	1.24340
$746$	46.1063	1.68807
$747$	−3.07445	−0.112488
$748$	1.25455	0.0458710
$749$	10.8991	0.398246
$750$	−17.5976	−0.642575
$751$	−14.4525	−0.527377	−0.263689	−	0.964608i	$-0.584939\pi$
$751$	−14.4525	−0.527377	−0.263689	+	0.964608i	$0.584939\pi$
$752$	−28.4273	−1.03664
$753$	−28.8017	−1.04959
$754$	0.695005	0.0253106
$755$	56.7835	2.06656
$756$	−0.898094	−0.0326633
$757$	16.9998	0.617867	0.308934	−	0.951084i	$-0.400028\pi$
$757$	16.9998	0.617867	0.308934	+	0.951084i	$0.400028\pi$
$758$	16.3473	0.593760
$759$	−8.72685	−0.316765
$760$	13.6444	0.494934
$761$	−7.92681	−0.287346	−0.143673	−	0.989625i	$-0.545891\pi$
$761$	−7.92681	−0.287346	−0.143673	+	0.989625i	$0.545891\pi$
$762$	10.9724	0.397489
$763$	−6.85202	−0.248060
$764$	4.85191	0.175536
$765$	−3.59386	−0.129936
$766$	−28.3261	−1.02346
$767$	0.107270	0.00387328
$768$	−16.8258	−0.607149
$769$	−3.34933	−0.120780	−0.0603900	−	0.998175i	$-0.519234\pi$
$769$	−3.34933	−0.120780	−0.0603900	+	0.998175i	$0.519234\pi$
$770$	−10.1097	−0.364327
$771$	13.0022	0.468264
$772$	18.7235	0.673872
$773$	30.4161	1.09399	0.546996	−	0.837135i	$-0.315771\pi$
$773$	30.4161	1.09399	0.546996	+	0.837135i	$0.315771\pi$
$774$	−0.512517	−0.0184221
$775$	15.3281	0.550602
$776$	30.5635	1.09717
$777$	6.59386	0.236553
$778$	19.2394	0.689765
$779$	8.12991	0.291284
$780$	0.136125	0.00487407
$781$	13.1770	0.471509
$782$	9.58051	0.342598
$783$	8.96107	0.320242
$784$	28.8166	1.02916
$785$	−3.59386	−0.128270
$786$	−10.7825	−0.384600
$787$	−8.20172	−0.292360	−0.146180	−	0.989258i	$-0.546698\pi$
$787$	−8.20172	−0.292360	−0.146180	+	0.989258i	$0.546698\pi$
$788$	−0.701991	−0.0250074
$789$	−11.3986	−0.405800
$790$	24.1867	0.860524
$791$	3.50033	0.124457
$792$	3.03090	0.107698
$793$	0.687141	0.0244011
$794$	−55.1467	−1.95708
$795$	−0.695273	−0.0246588
$796$	15.9655	0.565883
$797$	−21.7121	−0.769083	−0.384541	−	0.923108i	$-0.625640\pi$
$797$	−21.7121	−0.769083	−0.384541	+	0.923108i	$0.625640\pi$
$798$	3.52369	0.124737
$799$	−5.72249	−0.202447
$800$	−34.6676	−1.22569
$801$	−2.17593	−0.0768827
$802$	−61.4453	−2.16971
$803$	−21.6052	−0.762430
$804$	3.95733	0.139564
$805$	−22.4518	−0.791322
$806$	−0.150183	−0.00528998
$807$	10.6849	0.376128
$808$	−22.0678	−0.776341
$809$	−43.3879	−1.52544	−0.762718	−	0.646731i	$-0.776136\pi$
$809$	−43.3879	−1.52544	−0.762718	+	0.646731i	$0.776136\pi$
$810$	6.03526	0.212057
$811$	−46.3570	−1.62781	−0.813907	−	0.580995i	$-0.802664\pi$
$811$	−46.3570	−1.62781	−0.813907	+	0.580995i	$0.802664\pi$
$812$	−8.04788	−0.282425
$813$	10.3987	0.364699
$814$	15.4683	0.542163
$815$	30.7801	1.07818
$816$	−4.96765	−0.173902
$817$	0.584790	0.0204592
$818$	−20.9790	−0.733514
$819$	−0.0505743	−0.00176721
$820$	−12.5056	−0.436716
$821$	−0.510003	−0.0177992	−0.00889961	−	0.999960i	$-0.502833\pi$
$821$	−0.510003	−0.0177992	−0.00889961	+	0.999960i	$0.502833\pi$
$822$	−14.7634	−0.514932
$823$	−33.4611	−1.16638	−0.583190	−	0.812336i	$-0.698196\pi$
$823$	−33.4611	−1.16638	−0.583190	+	0.812336i	$0.698196\pi$
$824$	−8.25943	−0.287731
$825$	12.1087	0.421573
$826$	−4.27125	−0.148616
$827$	−40.2051	−1.39807	−0.699035	−	0.715088i	$-0.746387\pi$
$827$	−40.2051	−1.39807	−0.699035	+	0.715088i	$0.746387\pi$
$828$	−4.67885	−0.162601
$829$	16.1426	0.560655	0.280327	−	0.959904i	$-0.409557\pi$
$829$	16.1426	0.560655	0.280327	+	0.959904i	$0.409557\pi$
$830$	−18.5551	−0.644057
$831$	4.41176	0.153042
$832$	−0.119184	−0.00413195
$833$	5.80085	0.200988
$834$	0.996069	0.0344910
$835$	−43.6955	−1.51214
$836$	2.40389	0.0831404
$837$	−1.93639	−0.0669315
$838$	−46.4985	−1.60627
$839$	49.4308	1.70654	0.853270	−	0.521469i	$-0.174616\pi$
$839$	49.4308	1.70654	0.853270	+	0.521469i	$0.174616\pi$
$840$	7.79767	0.269045
$841$	51.3008	1.76899
$842$	36.9251	1.27252
$843$	2.71894	0.0936453
$844$	−7.53300	−0.259296
$845$	−46.7125	−1.60696
$846$	9.60993	0.330396
$847$	−9.48323	−0.325848
$848$	−0.961048	−0.0330025
$849$	9.80213	0.336408
$850$	−13.2932	−0.455954
$851$	34.3524	1.17758
$852$	7.06475	0.242034
$853$	17.4755	0.598351	0.299176	−	0.954198i	$-0.403288\pi$
$853$	17.4755	0.598351	0.299176	+	0.954198i	$0.403288\pi$
$854$	−27.3605	−0.936258
$855$	−6.88631	−0.235507
$856$	−19.7207	−0.674040
$857$	−15.6295	−0.533892	−0.266946	−	0.963711i	$-0.586015\pi$
$857$	−15.6295	−0.533892	−0.266946	+	0.963711i	$0.586015\pi$
$858$	−0.118640	−0.00405031
$859$	7.67136	0.261743	0.130872	−	0.991399i	$-0.458222\pi$
$859$	7.67136	0.261743	0.130872	+	0.991399i	$0.458222\pi$
$860$	−0.899538	−0.0306740
$861$	4.64618	0.158341
$862$	8.03050	0.273520
$863$	22.4197	0.763176	0.381588	−	0.924333i	$-0.375377\pi$
$863$	22.4197	0.763176	0.381588	+	0.924333i	$0.375377\pi$
$864$	4.37954	0.148995
$865$	−46.7781	−1.59050
$866$	−29.9999	−1.01944
$867$	−1.00000	−0.0339618
$868$	1.73906	0.0590276
$869$	−6.13035	−0.207958
$870$	54.0823	1.83356
$871$	0.222849	0.00755095
$872$	12.3979	0.419847
$873$	−15.4254	−0.522070
$874$	18.3575	0.620953
$875$	11.4751	0.387929
$876$	−11.5835	−0.391369
$877$	6.51262	0.219915	0.109958	−	0.993936i	$-0.464928\pi$
$877$	6.51262	0.219915	0.109958	+	0.993936i	$0.464928\pi$
$878$	20.8037	0.702089
$879$	−18.5396	−0.625325
$880$	27.3096	0.920607
$881$	−8.28718	−0.279202	−0.139601	−	0.990208i	$-0.544582\pi$
$881$	−8.28718	−0.279202	−0.139601	+	0.990208i	$0.544582\pi$
$882$	−9.74152	−0.328014
$883$	13.4417	0.452348	0.226174	−	0.974087i	$-0.427378\pi$
$883$	13.4417	0.452348	0.226174	+	0.974087i	$0.427378\pi$
$884$	0.0378772	0.00127395
$885$	8.34726	0.280590
$886$	−32.2705	−1.08415
$887$	15.7191	0.527794	0.263897	−	0.964551i	$-0.414992\pi$
$887$	15.7191	0.527794	0.263897	+	0.964551i	$0.414992\pi$
$888$	−11.9308	−0.400372
$889$	−7.15491	−0.239968
$890$	−13.1323	−0.440195
$891$	−1.52969	−0.0512466
$892$	−12.3122	−0.412243
$893$	−10.9651	−0.366932
$894$	−15.8585	−0.530389
$895$	18.2112	0.608734
$896$	14.3373	0.478977
$897$	−0.263479	−0.00879732
$898$	−14.2905	−0.476880
$899$	−17.3521	−0.578726
$900$	6.49203	0.216401
$901$	−0.193461	−0.00644513
$902$	10.8993	0.362907
$903$	0.334203	0.0111216
$904$	−6.33344	−0.210647
$905$	−18.7393	−0.622915
$906$	−26.5336	−0.881520
$907$	45.1858	1.50037	0.750185	−	0.661228i	$-0.229965\pi$
$907$	45.1858	1.50037	0.750185	+	0.661228i	$0.229965\pi$
$908$	15.6560	0.519563
$909$	11.1376	0.369410
$910$	−0.305229	−0.0101182
$911$	52.5562	1.74127	0.870633	−	0.491933i	$-0.163710\pi$
$911$	52.5562	1.74127	0.870633	+	0.491933i	$0.163710\pi$
$912$	−9.51868	−0.315195
$913$	4.70296	0.155645
$914$	−50.5965	−1.67358
$915$	53.4704	1.76768
$916$	−11.9514	−0.394886
$917$	7.03108	0.232187
$918$	1.67933	0.0554260
$919$	59.2062	1.95303	0.976517	−	0.215441i	$-0.0691188\pi$
$919$	59.2062	1.95303	0.976517	+	0.215441i	$0.0691188\pi$
$920$	40.6239	1.33933
$921$	−12.4781	−0.411166
$922$	27.5563	0.907517
$923$	0.397837	0.0130950
$924$	1.37381	0.0451949
$925$	−47.6649	−1.56721
$926$	−52.8178	−1.73570
$927$	4.16853	0.136912
$928$	39.2454	1.28829
$929$	54.0557	1.77351	0.886755	−	0.462239i	$-0.152954\pi$
$929$	54.0557	1.77351	0.886755	+	0.462239i	$0.152954\pi$
$930$	−11.6866	−0.383219
$931$	11.1152	0.364286
$932$	−7.53742	−0.246896
$933$	5.40547	0.176967
$934$	7.38567	0.241666
$935$	5.49749	0.179787
$936$	0.0915083	0.00299104
$937$	−56.3169	−1.83979	−0.919897	−	0.392160i	$-0.871728\pi$
$937$	−56.3169	−1.83979	−0.919897	+	0.392160i	$0.871728\pi$
$938$	−8.87338	−0.289726
$939$	−22.9439	−0.748746
$940$	16.8667	0.550133
$941$	32.9031	1.07261	0.536305	−	0.844024i	$-0.319820\pi$
$941$	32.9031	1.07261	0.536305	+	0.844024i	$0.319820\pi$
$942$	1.67933	0.0547154
$943$	24.2055	0.788238
$944$	11.5381	0.375533
$945$	−3.93548	−0.128021
$946$	0.783994	0.0254898
$947$	−24.3724	−0.791995	−0.395998	−	0.918252i	$-0.629601\pi$
$947$	−24.3724	−0.791995	−0.395998	+	0.918252i	$0.629601\pi$
$948$	−3.28674	−0.106749
$949$	−0.652299	−0.0211745
$950$	−25.4716	−0.826408
$951$	−32.7680	−1.06257
$952$	2.16972	0.0703211
$953$	19.9179	0.645204	0.322602	−	0.946535i	$-0.395442\pi$
$953$	19.9179	0.645204	0.322602	+	0.946535i	$0.395442\pi$
$954$	0.324885	0.0105185
$955$	21.2612	0.687997
$956$	−24.5728	−0.794742
$957$	−13.7077	−0.443106
$958$	−26.2075	−0.846727
$959$	9.62691	0.310869
$960$	−9.27437	−0.299329
$961$	−27.2504	−0.879045
$962$	0.467015	0.0150572
$963$	9.95303	0.320732
$964$	−0.213795	−0.00688588
$965$	82.0468	2.64118
$966$	10.4912	0.337549
$967$	−23.6814	−0.761542	−0.380771	−	0.924669i	$-0.624341\pi$
$967$	−23.6814	−0.761542	−0.380771	+	0.924669i	$0.624341\pi$
$968$	17.1588	0.551505
$969$	−1.91613	−0.0615551
$970$	−93.0962	−2.98914
$971$	−7.95210	−0.255195	−0.127598	−	0.991826i	$-0.540727\pi$
$971$	−7.95210	−0.255195	−0.127598	+	0.991826i	$0.540727\pi$
$972$	−0.820135	−0.0263058
$973$	−0.649517	−0.0208226
$974$	67.6164	2.16657
$975$	0.365585	0.0117081
$976$	73.9101	2.36580
$977$	58.5191	1.87219	0.936095	−	0.351747i	$-0.114412\pi$
$977$	58.5191	1.87219	0.936095	+	0.351747i	$0.114412\pi$
$978$	−14.3828	−0.459911
$979$	3.32850	0.106379
$980$	−17.0977	−0.546166
$981$	−6.25723	−0.199778
$982$	−51.9563	−1.65799
$983$	15.3438	0.489390	0.244695	−	0.969600i	$-0.421312\pi$
$983$	15.3438	0.489390	0.244695	+	0.969600i	$0.421312\pi$
$984$	−8.40673	−0.267997
$985$	−3.07615	−0.0980142
$986$	15.0486	0.479244
$987$	−6.26645	−0.199463
$988$	0.0725779	0.00230901
$989$	1.74111	0.0553642
$990$	−9.23208	−0.293415
$991$	15.1365	0.480825	0.240413	−	0.970671i	$-0.422717\pi$
$991$	15.1365	0.480825	0.240413	+	0.970671i	$0.422717\pi$
$992$	−8.48051	−0.269256
$993$	3.68621	0.116978
$994$	−15.8410	−0.502446
$995$	69.9615	2.21793
$996$	2.52146	0.0798956
$997$	−37.3031	−1.18140	−0.590701	−	0.806890i	$-0.701149\pi$
$997$	−37.3031	−1.18140	−0.590701	+	0.806890i	$0.701149\pi$
$998$	−9.76486	−0.309101
$999$	6.02148	0.190511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.37	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.37	✓	48	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)

\(f(q)\)	\(=\)	\(q+1.67933 q^{2} -1.00000 q^{3} +0.820135 q^{4} +3.59386 q^{5} -1.67933 q^{6} +1.09506 q^{7} -1.98138 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.67933 q^{2} -1.00000 q^{3} +0.820135 q^{4} +3.59386 q^{5} -1.67933 q^{6} +1.09506 q^{7} -1.98138 q^{8} +1.00000 q^{9} +6.03526 q^{10} -1.52969 q^{11} -0.820135 q^{12} -0.0461842 q^{13} +1.83896 q^{14} -3.59386 q^{15} -4.96765 q^{16} -1.00000 q^{17} +1.67933 q^{18} -1.91613 q^{19} +2.94745 q^{20} -1.09506 q^{21} -2.56885 q^{22} -5.70497 q^{23} +1.98138 q^{24} +7.91581 q^{25} -0.0775583 q^{26} -1.00000 q^{27} +0.898094 q^{28} -8.96107 q^{29} -6.03526 q^{30} +1.93639 q^{31} -4.37954 q^{32} +1.52969 q^{33} -1.67933 q^{34} +3.93548 q^{35} +0.820135 q^{36} -6.02148 q^{37} -3.21781 q^{38} +0.0461842 q^{39} -7.12079 q^{40} -4.24287 q^{41} -1.83896 q^{42} -0.305192 q^{43} -1.25455 q^{44} +3.59386 q^{45} -9.58051 q^{46} +5.72249 q^{47} +4.96765 q^{48} -5.80085 q^{49} +13.2932 q^{50} +1.00000 q^{51} -0.0378772 q^{52} +0.193461 q^{53} -1.67933 q^{54} -5.49749 q^{55} -2.16972 q^{56} +1.91613 q^{57} -15.0486 q^{58} -2.32265 q^{59} -2.94745 q^{60} -14.8783 q^{61} +3.25183 q^{62} +1.09506 q^{63} +2.58062 q^{64} -0.165979 q^{65} +2.56885 q^{66} -4.82522 q^{67} -0.820135 q^{68} +5.70497 q^{69} +6.60895 q^{70} -8.61413 q^{71} -1.98138 q^{72} +14.1239 q^{73} -10.1120 q^{74} -7.91581 q^{75} -1.57149 q^{76} -1.67510 q^{77} +0.0775583 q^{78} +4.00757 q^{79} -17.8530 q^{80} +1.00000 q^{81} -7.12516 q^{82} -3.07445 q^{83} -0.898094 q^{84} -3.59386 q^{85} -0.512517 q^{86} +8.96107 q^{87} +3.03090 q^{88} -2.17593 q^{89} +6.03526 q^{90} -0.0505743 q^{91} -4.67885 q^{92} -1.93639 q^{93} +9.60993 q^{94} -6.88631 q^{95} +4.37954 q^{96} -15.4254 q^{97} -9.74152 q^{98} -1.52969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more