LMFDB - Embedded newform 8015.2.a.l.1.61

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8015,2,Mod(1,8015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8015, 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("8015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8015 = 5 \cdot 7 \cdot 229 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8015.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0000972201$
Analytic rank:	$0$
Dimension:	$62$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.61
Character	$\chi$	$=$	8015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.76672 q^{2} +2.46366 q^{3} +5.65472 q^{4} -1.00000 q^{5} +6.81626 q^{6} -1.00000 q^{7} +10.1116 q^{8} +3.06964 q^{9} +O(q^{10})$	\(q+2.76672 q^{2} +2.46366 q^{3} +5.65472 q^{4} -1.00000 q^{5} +6.81626 q^{6} -1.00000 q^{7} +10.1116 q^{8} +3.06964 q^{9} -2.76672 q^{10} -2.78520 q^{11} +13.9313 q^{12} +3.88010 q^{13} -2.76672 q^{14} -2.46366 q^{15} +16.6665 q^{16} -4.28037 q^{17} +8.49282 q^{18} +2.70527 q^{19} -5.65472 q^{20} -2.46366 q^{21} -7.70585 q^{22} +6.92125 q^{23} +24.9116 q^{24} +1.00000 q^{25} +10.7351 q^{26} +0.171568 q^{27} -5.65472 q^{28} -8.72264 q^{29} -6.81626 q^{30} +9.05218 q^{31} +25.8882 q^{32} -6.86178 q^{33} -11.8426 q^{34} +1.00000 q^{35} +17.3580 q^{36} +11.0030 q^{37} +7.48472 q^{38} +9.55925 q^{39} -10.1116 q^{40} -12.7093 q^{41} -6.81626 q^{42} -1.77480 q^{43} -15.7495 q^{44} -3.06964 q^{45} +19.1492 q^{46} +11.5298 q^{47} +41.0606 q^{48} +1.00000 q^{49} +2.76672 q^{50} -10.5454 q^{51} +21.9409 q^{52} -3.40381 q^{53} +0.474680 q^{54} +2.78520 q^{55} -10.1116 q^{56} +6.66488 q^{57} -24.1331 q^{58} +4.06941 q^{59} -13.9313 q^{60} +11.6559 q^{61} +25.0448 q^{62} -3.06964 q^{63} +38.2924 q^{64} -3.88010 q^{65} -18.9846 q^{66} -9.21908 q^{67} -24.2043 q^{68} +17.0516 q^{69} +2.76672 q^{70} -8.15080 q^{71} +31.0389 q^{72} -7.93858 q^{73} +30.4421 q^{74} +2.46366 q^{75} +15.2976 q^{76} +2.78520 q^{77} +26.4477 q^{78} -9.55960 q^{79} -16.6665 q^{80} -8.78623 q^{81} -35.1631 q^{82} -2.97998 q^{83} -13.9313 q^{84} +4.28037 q^{85} -4.91038 q^{86} -21.4897 q^{87} -28.1628 q^{88} +8.83996 q^{89} -8.49282 q^{90} -3.88010 q^{91} +39.1378 q^{92} +22.3015 q^{93} +31.8997 q^{94} -2.70527 q^{95} +63.7798 q^{96} -16.5622 q^{97} +2.76672 q^{98} -8.54954 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 62 q + 2 q^{2} + 11 q^{3} + 64 q^{4} - 62 q^{5} + 3 q^{6} - 62 q^{7} + 15 q^{8} + 69 q^{9}+O(q^{10}) $ 62 * q + 2 * q^2 + 11 * q^3 + 64 * q^4 - 62 * q^5 + 3 * q^6 - 62 * q^7 + 15 * q^8 + 69 * q^9	\( 62 q + 2 q^{2} + 11 q^{3} + 64 q^{4} - 62 q^{5} + 3 q^{6} - 62 q^{7} + 15 q^{8} + 69 q^{9} - 2 q^{10} - 13 q^{11} + 37 q^{12} + 31 q^{13} - 2 q^{14} - 11 q^{15} + 64 q^{16} + 30 q^{17} + 18 q^{18} + 20 q^{19} - 64 q^{20} - 11 q^{21} + 7 q^{22} + 29 q^{24} + 62 q^{25} + 59 q^{27} - 64 q^{28} - 29 q^{29} - 3 q^{30} + 20 q^{31} + 22 q^{32} + 72 q^{33} + 13 q^{34} + 62 q^{35} + 53 q^{36} + 35 q^{37} + 34 q^{38} - 6 q^{39} - 15 q^{40} + 13 q^{41} - 3 q^{42} - 4 q^{43} - 44 q^{44} - 69 q^{45} - 19 q^{46} + 58 q^{47} + 64 q^{48} + 62 q^{49} + 2 q^{50} - 30 q^{51} + 82 q^{52} + 18 q^{53} + 22 q^{54} + 13 q^{55} - 15 q^{56} + 21 q^{57} + 18 q^{58} - 11 q^{59} - 37 q^{60} + 24 q^{61} + 48 q^{62} - 69 q^{63} + 65 q^{64} - 31 q^{65} + 25 q^{66} - 6 q^{67} + 65 q^{68} + 27 q^{69} + 2 q^{70} - 35 q^{71} + 53 q^{72} + 116 q^{73} - 69 q^{74} + 11 q^{75} + 65 q^{76} + 13 q^{77} + 102 q^{78} - 83 q^{79} - 64 q^{80} + 126 q^{81} + 71 q^{82} + 84 q^{83} - 37 q^{84} - 30 q^{85} + 24 q^{86} + 49 q^{87} + 20 q^{88} - 16 q^{89} - 18 q^{90} - 31 q^{91} + 19 q^{92} + 65 q^{93} + 54 q^{94} - 20 q^{95} + 17 q^{96} + 155 q^{97} + 2 q^{98} + 6 q^{99}+O(q^{100}) \) 62 * q + 2 * q^2 + 11 * q^3 + 64 * q^4 - 62 * q^5 + 3 * q^6 - 62 * q^7 + 15 * q^8 + 69 * q^9 - 2 * q^10 - 13 * q^11 + 37 * q^12 + 31 * q^13 - 2 * q^14 - 11 * q^15 + 64 * q^16 + 30 * q^17 + 18 * q^18 + 20 * q^19 - 64 * q^20 - 11 * q^21 + 7 * q^22 + 29 * q^24 + 62 * q^25 + 59 * q^27 - 64 * q^28 - 29 * q^29 - 3 * q^30 + 20 * q^31 + 22 * q^32 + 72 * q^33 + 13 * q^34 + 62 * q^35 + 53 * q^36 + 35 * q^37 + 34 * q^38 - 6 * q^39 - 15 * q^40 + 13 * q^41 - 3 * q^42 - 4 * q^43 - 44 * q^44 - 69 * q^45 - 19 * q^46 + 58 * q^47 + 64 * q^48 + 62 * q^49 + 2 * q^50 - 30 * q^51 + 82 * q^52 + 18 * q^53 + 22 * q^54 + 13 * q^55 - 15 * q^56 + 21 * q^57 + 18 * q^58 - 11 * q^59 - 37 * q^60 + 24 * q^61 + 48 * q^62 - 69 * q^63 + 65 * q^64 - 31 * q^65 + 25 * q^66 - 6 * q^67 + 65 * q^68 + 27 * q^69 + 2 * q^70 - 35 * q^71 + 53 * q^72 + 116 * q^73 - 69 * q^74 + 11 * q^75 + 65 * q^76 + 13 * q^77 + 102 * q^78 - 83 * q^79 - 64 * q^80 + 126 * q^81 + 71 * q^82 + 84 * q^83 - 37 * q^84 - 30 * q^85 + 24 * q^86 + 49 * q^87 + 20 * q^88 - 16 * q^89 - 18 * q^90 - 31 * q^91 + 19 * q^92 + 65 * q^93 + 54 * q^94 - 20 * q^95 + 17 * q^96 + 155 * q^97 + 2 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.76672	1.95636	0.978182	−	0.207748i	$-0.0666136\pi$
$2$	2.76672	1.95636	0.978182	+	0.207748i	$0.0666136\pi$
$3$	2.46366	1.42240	0.711198	−	0.702991i	$-0.248153\pi$
$3$	2.46366	1.42240	0.711198	+	0.702991i	$0.248153\pi$
$4$	5.65472	2.82736
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	6.81626	2.78273
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	10.1116	3.57499
$9$	3.06964	1.02321
$10$	−2.76672	−0.874913
$11$	−2.78520	−0.839768	−0.419884	−	0.907578i	$-0.637929\pi$
$11$	−2.78520	−0.839768	−0.419884	+	0.907578i	$0.637929\pi$
$12$	13.9313	4.02163
$13$	3.88010	1.07614	0.538072	−	0.842899i	$-0.319153\pi$
$13$	3.88010	1.07614	0.538072	+	0.842899i	$0.319153\pi$
$14$	−2.76672	−0.739436
$15$	−2.46366	−0.636115
$16$	16.6665	4.16662
$17$	−4.28037	−1.03814	−0.519071	−	0.854731i	$-0.673722\pi$
$17$	−4.28037	−1.03814	−0.519071	+	0.854731i	$0.673722\pi$
$18$	8.49282	2.00178
$19$	2.70527	0.620632	0.310316	−	0.950634i	$-0.399565\pi$
$19$	2.70527	0.620632	0.310316	+	0.950634i	$0.399565\pi$
$20$	−5.65472	−1.26443
$21$	−2.46366	−0.537616
$22$	−7.70585	−1.64289
$23$	6.92125	1.44318	0.721591	−	0.692320i	$-0.243411\pi$
$23$	6.92125	1.44318	0.721591	+	0.692320i	$0.243411\pi$
$24$	24.9116	5.08505
$25$	1.00000	0.200000
$26$	10.7351	2.10533
$27$	0.171568	0.0330183
$28$	−5.65472	−1.06864
$29$	−8.72264	−1.61975	−0.809877	−	0.586599i	$-0.800466\pi$
$29$	−8.72264	−1.61975	−0.809877	+	0.586599i	$0.800466\pi$
$30$	−6.81626	−1.24447
$31$	9.05218	1.62582	0.812910	−	0.582390i	$-0.197882\pi$
$31$	9.05218	1.62582	0.812910	+	0.582390i	$0.197882\pi$
$32$	25.8882	4.57643
$33$	−6.86178	−1.19448
$34$	−11.8426	−2.03098
$35$	1.00000	0.169031
$36$	17.3580	2.89299
$37$	11.0030	1.80888	0.904440	−	0.426601i	$-0.140289\pi$
$37$	11.0030	1.80888	0.904440	+	0.426601i	$0.140289\pi$
$38$	7.48472	1.21418
$39$	9.55925	1.53071
$40$	−10.1116	−1.59878
$41$	−12.7093	−1.98486	−0.992431	−	0.122805i	$-0.960811\pi$
$41$	−12.7093	−1.98486	−0.992431	+	0.122805i	$0.960811\pi$
$42$	−6.81626	−1.05177
$43$	−1.77480	−0.270655	−0.135327	−	0.990801i	$-0.543209\pi$
$43$	−1.77480	−0.270655	−0.135327	+	0.990801i	$0.543209\pi$
$44$	−15.7495	−2.37433
$45$	−3.06964	−0.457595
$46$	19.1492	2.82339
$47$	11.5298	1.68180	0.840898	−	0.541193i	$-0.182027\pi$
$47$	11.5298	1.68180	0.840898	+	0.541193i	$0.182027\pi$
$48$	41.0606	5.92658
$49$	1.00000	0.142857
$50$	2.76672	0.391273
$51$	−10.5454	−1.47665
$52$	21.9409	3.04265
$53$	−3.40381	−0.467550	−0.233775	−	0.972291i	$-0.575108\pi$
$53$	−3.40381	−0.467550	−0.233775	+	0.972291i	$0.575108\pi$
$54$	0.474680	0.0645958
$55$	2.78520	0.375556
$56$	−10.1116	−1.35122
$57$	6.66488	0.882784
$58$	−24.1331	−3.16883
$59$	4.06941	0.529791	0.264896	−	0.964277i	$-0.414662\pi$
$59$	4.06941	0.529791	0.264896	+	0.964277i	$0.414662\pi$
$60$	−13.9313	−1.79853
$61$	11.6559	1.49238	0.746190	−	0.665734i	$-0.231881\pi$
$61$	11.6559	1.49238	0.746190	+	0.665734i	$0.231881\pi$
$62$	25.0448	3.18070
$63$	−3.06964	−0.386738
$64$	38.2924	4.78655
$65$	−3.88010	−0.481267
$66$	−18.9846	−2.33684
$67$	−9.21908	−1.12629	−0.563145	−	0.826358i	$-0.690409\pi$
$67$	−9.21908	−1.12629	−0.563145	+	0.826358i	$0.690409\pi$
$68$	−24.2043	−2.93520
$69$	17.0516	2.05278
$70$	2.76672	0.330686
$71$	−8.15080	−0.967322	−0.483661	−	0.875255i	$-0.660693\pi$
$71$	−8.15080	−0.967322	−0.483661	+	0.875255i	$0.660693\pi$
$72$	31.0389	3.65797
$73$	−7.93858	−0.929141	−0.464570	−	0.885536i	$-0.653791\pi$
$73$	−7.93858	−0.929141	−0.464570	+	0.885536i	$0.653791\pi$
$74$	30.4421	3.53883
$75$	2.46366	0.284479
$76$	15.2976	1.75475
$77$	2.78520	0.317402
$78$	26.4477	2.99462
$79$	−9.55960	−1.07554	−0.537769	−	0.843092i	$-0.680733\pi$
$79$	−9.55960	−1.07554	−0.537769	+	0.843092i	$0.680733\pi$
$80$	−16.6665	−1.86337
$81$	−8.78623	−0.976248
$82$	−35.1631	−3.88311
$83$	−2.97998	−0.327096	−0.163548	−	0.986535i	$-0.552294\pi$
$83$	−2.97998	−0.327096	−0.163548	+	0.986535i	$0.552294\pi$
$84$	−13.9313	−1.52003
$85$	4.28037	0.464271
$86$	−4.91038	−0.529499
$87$	−21.4897	−2.30393
$88$	−28.1628	−3.00216
$89$	8.83996	0.937034	0.468517	−	0.883455i	$-0.344789\pi$
$89$	8.83996	0.937034	0.468517	+	0.883455i	$0.344789\pi$
$90$	−8.49282	−0.895222
$91$	−3.88010	−0.406745
$92$	39.1378	4.08040
$93$	22.3015	2.31256
$94$	31.8997	3.29021
$95$	−2.70527	−0.277555
$96$	63.7798	6.50950
$97$	−16.5622	−1.68164	−0.840818	−	0.541318i	$-0.817926\pi$
$97$	−16.5622	−1.68164	−0.840818	+	0.541318i	$0.817926\pi$
$98$	2.76672	0.279481
$99$	−8.54954	−0.859262
$100$	5.65472	0.565472
$101$	−3.94900	−0.392940	−0.196470	−	0.980510i	$-0.562948\pi$
$101$	−3.94900	−0.392940	−0.196470	+	0.980510i	$0.562948\pi$
$102$	−29.1761	−2.88887
$103$	17.0324	1.67825	0.839126	−	0.543937i	$-0.183067\pi$
$103$	17.0324	1.67825	0.839126	+	0.543937i	$0.183067\pi$
$104$	39.2339	3.84720
$105$	2.46366	0.240429
$106$	−9.41739	−0.914697
$107$	5.11239	0.494233	0.247117	−	0.968986i	$-0.420517\pi$
$107$	5.11239	0.494233	0.247117	+	0.968986i	$0.420517\pi$
$108$	0.970169	0.0933546
$109$	−3.60978	−0.345754	−0.172877	−	0.984943i	$-0.555306\pi$
$109$	−3.60978	−0.345754	−0.172877	+	0.984943i	$0.555306\pi$
$110$	7.70585	0.734724
$111$	27.1077	2.57294
$112$	−16.6665	−1.57483
$113$	−4.59166	−0.431947	−0.215973	−	0.976399i	$-0.569292\pi$
$113$	−4.59166	−0.431947	−0.215973	+	0.976399i	$0.569292\pi$
$114$	18.4398	1.72705
$115$	−6.92125	−0.645410
$116$	−49.3241	−4.57963
$117$	11.9105	1.10113
$118$	11.2589	1.03647
$119$	4.28037	0.392381
$120$	−24.9116	−2.27410
$121$	−3.24269	−0.294790
$122$	32.2484	2.91964
$123$	−31.3115	−2.82326
$124$	51.1876	4.59678
$125$	−1.00000	−0.0894427
$126$	−8.49282	−0.756601
$127$	3.89538	0.345659	0.172829	−	0.984952i	$-0.444709\pi$
$127$	3.89538	0.345659	0.172829	+	0.984952i	$0.444709\pi$
$128$	54.1679	4.78781
$129$	−4.37252	−0.384979
$130$	−10.7351	−0.941533
$131$	−6.57928	−0.574834	−0.287417	−	0.957806i	$-0.592797\pi$
$131$	−6.57928	−0.574834	−0.287417	+	0.957806i	$0.592797\pi$
$132$	−38.8015	−3.37724
$133$	−2.70527	−0.234577
$134$	−25.5066	−2.20343
$135$	−0.171568	−0.0147662
$136$	−43.2813	−3.71134
$137$	−15.2407	−1.30210	−0.651048	−	0.759036i	$-0.725671\pi$
$137$	−15.2407	−1.30210	−0.651048	+	0.759036i	$0.725671\pi$
$138$	47.1771	4.01598
$139$	6.80209	0.576946	0.288473	−	0.957488i	$-0.406852\pi$
$139$	6.80209	0.576946	0.288473	+	0.957488i	$0.406852\pi$
$140$	5.65472	0.477911
$141$	28.4056	2.39218
$142$	−22.5510	−1.89243
$143$	−10.8068	−0.903712
$144$	51.1600	4.26334
$145$	8.72264	0.724376
$146$	−21.9638	−1.81774
$147$	2.46366	0.203200
$148$	62.2188	5.11436
$149$	−10.9390	−0.896155	−0.448077	−	0.893995i	$-0.647891\pi$
$149$	−10.9390	−0.896155	−0.448077	+	0.893995i	$0.647891\pi$
$150$	6.81626	0.556545
$151$	−9.05777	−0.737111	−0.368556	−	0.929606i	$-0.620148\pi$
$151$	−9.05777	−0.737111	−0.368556	+	0.929606i	$0.620148\pi$
$152$	27.3546	2.21875
$153$	−13.1392	−1.06224
$154$	7.70585	0.620955
$155$	−9.05218	−0.727088
$156$	54.0549	4.32786
$157$	−21.7165	−1.73317	−0.866584	−	0.499031i	$-0.833689\pi$
$157$	−21.7165	−1.73317	−0.866584	+	0.499031i	$0.833689\pi$
$158$	−26.4487	−2.10415
$159$	−8.38585	−0.665041
$160$	−25.8882	−2.04664
$161$	−6.92125	−0.545471
$162$	−24.3090	−1.90990
$163$	−3.33544	−0.261252	−0.130626	−	0.991432i	$-0.541699\pi$
$163$	−3.33544	−0.261252	−0.130626	+	0.991432i	$0.541699\pi$
$164$	−71.8677	−5.61192
$165$	6.86178	0.534189
$166$	−8.24477	−0.639918
$167$	−4.89072	−0.378456	−0.189228	−	0.981933i	$-0.560598\pi$
$167$	−4.89072	−0.378456	−0.189228	+	0.981933i	$0.560598\pi$
$168$	−24.9116	−1.92197
$169$	2.05514	0.158088
$170$	11.8426	0.908284
$171$	8.30420	0.635038
$172$	−10.0360	−0.765239
$173$	−6.13092	−0.466125	−0.233063	−	0.972462i	$-0.574875\pi$
$173$	−6.13092	−0.466125	−0.233063	+	0.972462i	$0.574875\pi$
$174$	−59.4558	−4.50733
$175$	−1.00000	−0.0755929
$176$	−46.4193	−3.49899
$177$	10.0256	0.753574
$178$	24.4577	1.83318
$179$	18.0566	1.34962	0.674808	−	0.737994i	$-0.264227\pi$
$179$	18.0566	1.34962	0.674808	+	0.737994i	$0.264227\pi$
$180$	−17.3580	−1.29379
$181$	−6.85899	−0.509825	−0.254912	−	0.966964i	$-0.582047\pi$
$181$	−6.85899	−0.509825	−0.254912	+	0.966964i	$0.582047\pi$
$182$	−10.7351	−0.795741
$183$	28.7161	2.12276
$184$	69.9849	5.15935
$185$	−11.0030	−0.808956
$186$	61.7020	4.52421
$187$	11.9217	0.871799
$188$	65.1979	4.75505
$189$	−0.171568	−0.0124797
$190$	−7.48472	−0.542998
$191$	−12.0769	−0.873857	−0.436928	−	0.899496i	$-0.643934\pi$
$191$	−12.0769	−0.873857	−0.436928	+	0.899496i	$0.643934\pi$
$192$	94.3397	6.80838
$193$	5.91986	0.426121	0.213061	−	0.977039i	$-0.431657\pi$
$193$	5.91986	0.426121	0.213061	+	0.977039i	$0.431657\pi$
$194$	−45.8229	−3.28989
$195$	−9.55925	−0.684552
$196$	5.65472	0.403909
$197$	−11.6607	−0.830789	−0.415394	−	0.909641i	$-0.636356\pi$
$197$	−11.6607	−0.830789	−0.415394	+	0.909641i	$0.636356\pi$
$198$	−23.6542	−1.68103
$199$	17.4871	1.23963	0.619813	−	0.784750i	$-0.287209\pi$
$199$	17.4871	1.23963	0.619813	+	0.784750i	$0.287209\pi$
$200$	10.1116	0.714997
$201$	−22.7127	−1.60203
$202$	−10.9258	−0.768735
$203$	8.72264	0.612210
$204$	−59.6313	−4.17503
$205$	12.7093	0.887657
$206$	47.1238	3.28327
$207$	21.2458	1.47668
$208$	64.6675	4.48388
$209$	−7.53471	−0.521186
$210$	6.81626	0.470367
$211$	−6.26379	−0.431217	−0.215608	−	0.976480i	$-0.569173\pi$
$211$	−6.26379	−0.431217	−0.215608	+	0.976480i	$0.569173\pi$
$212$	−19.2476	−1.32193
$213$	−20.0808	−1.37592
$214$	14.1445	0.966900
$215$	1.77480	0.121041
$216$	1.73482	0.118040
$217$	−9.05218	−0.614502
$218$	−9.98724	−0.676421
$219$	−19.5580	−1.32161
$220$	15.7495	1.06183
$221$	−16.6082	−1.11719
$222$	74.9992	5.03362
$223$	3.39706	0.227484	0.113742	−	0.993510i	$-0.463716\pi$
$223$	3.39706	0.227484	0.113742	+	0.993510i	$0.463716\pi$
$224$	−25.8882	−1.72973
$225$	3.06964	0.204643
$226$	−12.7038	−0.845046
$227$	−2.15517	−0.143044	−0.0715218	−	0.997439i	$-0.522786\pi$
$227$	−2.15517	−0.143044	−0.0715218	+	0.997439i	$0.522786\pi$
$228$	37.6880	2.49595
$229$	1.00000	0.0660819
$229$	1.00000	0.0660819
$230$	−19.1492	−1.26266
$231$	6.86178	0.451472
$232$	−88.1998	−5.79060
$233$	−10.3177	−0.675938	−0.337969	−	0.941157i	$-0.609740\pi$
$233$	−10.3177	−0.675938	−0.337969	+	0.941157i	$0.609740\pi$
$234$	32.9530	2.15420
$235$	−11.5298	−0.752122
$236$	23.0114	1.49791
$237$	−23.5516	−1.52984
$238$	11.8426	0.767640
$239$	13.7291	0.888064	0.444032	−	0.896011i	$-0.353548\pi$
$239$	13.7291	0.888064	0.444032	+	0.896011i	$0.353548\pi$
$240$	−41.0606	−2.65045
$241$	−17.0265	−1.09677	−0.548387	−	0.836225i	$-0.684758\pi$
$241$	−17.0265	−1.09677	−0.548387	+	0.836225i	$0.684758\pi$
$242$	−8.97160	−0.576716
$243$	−22.1610	−1.42163
$244$	65.9106	4.21950
$245$	−1.00000	−0.0638877
$246$	−86.6300	−5.52333
$247$	10.4967	0.667889
$248$	91.5319	5.81228
$249$	−7.34168	−0.465260
$250$	−2.76672	−0.174983
$251$	−15.4408	−0.974613	−0.487307	−	0.873231i	$-0.662021\pi$
$251$	−15.4408	−0.974613	−0.487307	+	0.873231i	$0.662021\pi$
$252$	−17.3580	−1.09345
$253$	−19.2770	−1.21194
$254$	10.7774	0.676235
$255$	10.5454	0.660378
$256$	73.2824	4.58015
$257$	0.0234149	0.00146058	0.000730290	−	1.00000i	$-0.499768\pi$
$257$	0.0234149	0.00146058	0.000730290		1.00000i	$0.499768\pi$
$258$	−12.0975	−0.753158
$259$	−11.0030	−0.683692
$260$	−21.9409	−1.36072
$261$	−26.7754	−1.65735
$262$	−18.2030	−1.12459
$263$	−15.2066	−0.937681	−0.468841	−	0.883283i	$-0.655328\pi$
$263$	−15.2066	−0.937681	−0.468841	+	0.883283i	$0.655328\pi$
$264$	−69.3836	−4.27026
$265$	3.40381	0.209095
$266$	−7.48472	−0.458917
$267$	21.7787	1.33283
$268$	−52.1313	−3.18443
$269$	1.93604	0.118042	0.0590211	−	0.998257i	$-0.481202\pi$
$269$	1.93604	0.118042	0.0590211	+	0.998257i	$0.481202\pi$
$270$	−0.474680	−0.0288881
$271$	23.0187	1.39828	0.699142	−	0.714983i	$-0.253565\pi$
$271$	23.0187	1.39828	0.699142	+	0.714983i	$0.253565\pi$
$272$	−71.3386	−4.32554
$273$	−9.55925	−0.578552
$274$	−42.1666	−2.54738
$275$	−2.78520	−0.167954
$276$	96.4224	5.80394
$277$	27.2538	1.63752	0.818760	−	0.574136i	$-0.194662\pi$
$277$	27.2538	1.63752	0.818760	+	0.574136i	$0.194662\pi$
$278$	18.8195	1.12872
$279$	27.7869	1.66356
$280$	10.1116	0.604283
$281$	26.3117	1.56962	0.784812	−	0.619734i	$-0.212760\pi$
$281$	26.3117	1.56962	0.784812	+	0.619734i	$0.212760\pi$
$282$	78.5902	4.67998
$283$	−16.5230	−0.982192	−0.491096	−	0.871105i	$-0.663404\pi$
$283$	−16.5230	−0.982192	−0.491096	+	0.871105i	$0.663404\pi$
$284$	−46.0905	−2.73497
$285$	−6.66488	−0.394793
$286$	−29.8994	−1.76799
$287$	12.7093	0.750207
$288$	79.4675	4.68266
$289$	1.32156	0.0777391
$290$	24.1331	1.41714
$291$	−40.8037	−2.39195
$292$	−44.8905	−2.62702
$293$	−13.8816	−0.810973	−0.405487	−	0.914101i	$-0.632898\pi$
$293$	−13.8816	−0.810973	−0.405487	+	0.914101i	$0.632898\pi$
$294$	6.81626	0.397532
$295$	−4.06941	−0.236930
$296$	111.258	6.46672
$297$	−0.477850	−0.0277277
$298$	−30.2650	−1.75321
$299$	26.8551	1.55307
$300$	13.9313	0.804326
$301$	1.77480	0.102298
$302$	−25.0603	−1.44206
$303$	−9.72902	−0.558917
$304$	45.0873	2.58593
$305$	−11.6559	−0.667412
$306$	−36.3524	−2.07813
$307$	13.1992	0.753319	0.376659	−	0.926352i	$-0.377073\pi$
$307$	13.1992	0.753319	0.376659	+	0.926352i	$0.377073\pi$
$308$	15.7495	0.897412
$309$	41.9621	2.38714
$310$	−25.0448	−1.42245
$311$	28.8564	1.63630	0.818148	−	0.575008i	$-0.195001\pi$
$311$	28.8564	1.63630	0.818148	+	0.575008i	$0.195001\pi$
$312$	96.6592	5.47225
$313$	2.52541	0.142744	0.0713722	−	0.997450i	$-0.477262\pi$
$313$	2.52541	0.142744	0.0713722	+	0.997450i	$0.477262\pi$
$314$	−60.0835	−3.39071
$315$	3.06964	0.172955
$316$	−54.0569	−3.04094
$317$	7.30718	0.410412	0.205206	−	0.978719i	$-0.434214\pi$
$317$	7.30718	0.410412	0.205206	+	0.978719i	$0.434214\pi$
$318$	−23.2013	−1.30106
$319$	24.2943	1.36022
$320$	−38.2924	−2.14061
$321$	12.5952	0.702996
$322$	−19.1492	−1.06714
$323$	−11.5796	−0.644304
$324$	−49.6837	−2.76021
$325$	3.88010	0.215229
$326$	−9.22821	−0.511103
$327$	−8.89328	−0.491800
$328$	−128.511	−7.09585
$329$	−11.5298	−0.635659
$330$	18.9846	1.04507
$331$	23.2382	1.27729	0.638643	−	0.769503i	$-0.279496\pi$
$331$	23.2382	1.27729	0.638643	+	0.769503i	$0.279496\pi$
$332$	−16.8510	−0.924818
$333$	33.7752	1.85087
$334$	−13.5312	−0.740397
$335$	9.21908	0.503692
$336$	−41.0606	−2.24004
$337$	−13.8003	−0.751752	−0.375876	−	0.926670i	$-0.622658\pi$
$337$	−13.8003	−0.751752	−0.375876	+	0.926670i	$0.622658\pi$
$338$	5.68599	0.309277
$339$	−11.3123	−0.614400
$340$	24.2043	1.31266
$341$	−25.2121	−1.36531
$342$	22.9754	1.24237
$343$	−1.00000	−0.0539949
$344$	−17.9461	−0.967587
$345$	−17.0516	−0.918030
$346$	−16.9625	−0.911911
$347$	−0.938045	−0.0503569	−0.0251784	−	0.999683i	$-0.508015\pi$
$347$	−0.938045	−0.0503569	−0.0251784	+	0.999683i	$0.508015\pi$
$348$	−121.518	−6.51406
$349$	14.7934	0.791874	0.395937	−	0.918278i	$-0.370420\pi$
$349$	14.7934	0.791874	0.395937	+	0.918278i	$0.370420\pi$
$350$	−2.76672	−0.147887
$351$	0.665700	0.0355324
$352$	−72.1037	−3.84314
$353$	−34.8870	−1.85685	−0.928423	−	0.371524i	$-0.878835\pi$
$353$	−34.8870	−1.85685	−0.928423	+	0.371524i	$0.878835\pi$
$354$	27.7381	1.47427
$355$	8.15080	0.432600
$356$	49.9875	2.64933
$357$	10.5454	0.558121
$358$	49.9576	2.64034
$359$	−16.7591	−0.884509	−0.442255	−	0.896890i	$-0.645821\pi$
$359$	−16.7591	−0.884509	−0.442255	+	0.896890i	$0.645821\pi$
$360$	−31.0389	−1.63590
$361$	−11.6815	−0.614817
$362$	−18.9769	−0.997403
$363$	−7.98889	−0.419308
$364$	−21.9409	−1.15001
$365$	7.93858	0.415524
$366$	79.4493	4.15288
$367$	−4.39917	−0.229635	−0.114817	−	0.993387i	$-0.536628\pi$
$367$	−4.39917	−0.229635	−0.114817	+	0.993387i	$0.536628\pi$
$368$	115.353	6.01318
$369$	−39.0130	−2.03094
$370$	−30.4421	−1.58261
$371$	3.40381	0.176717
$372$	126.109	6.53845
$373$	16.4329	0.850861	0.425431	−	0.904991i	$-0.360123\pi$
$373$	16.4329	0.850861	0.425431	+	0.904991i	$0.360123\pi$
$374$	32.9839	1.70556
$375$	−2.46366	−0.127223
$376$	116.585	6.01240
$377$	−33.8447	−1.74309
$378$	−0.474680	−0.0244149
$379$	16.9307	0.869673	0.434837	−	0.900509i	$-0.356806\pi$
$379$	16.9307	0.869673	0.434837	+	0.900509i	$0.356806\pi$
$380$	−15.2976	−0.784748
$381$	9.59690	0.491664
$382$	−33.4135	−1.70958
$383$	−18.1103	−0.925393	−0.462697	−	0.886517i	$-0.653118\pi$
$383$	−18.1103	−0.925393	−0.462697	+	0.886517i	$0.653118\pi$
$384$	133.452	6.81017
$385$	−2.78520	−0.141947
$386$	16.3786	0.833648
$387$	−5.44800	−0.276938
$388$	−93.6547	−4.75459
$389$	−11.6034	−0.588316	−0.294158	−	0.955757i	$-0.595039\pi$
$389$	−11.6034	−0.588316	−0.294158	+	0.955757i	$0.595039\pi$
$390$	−26.4477	−1.33923
$391$	−29.6255	−1.49823
$392$	10.1116	0.510712
$393$	−16.2091	−0.817642
$394$	−32.2618	−1.62533
$395$	9.55960	0.480996
$396$	−48.3453	−2.42944
$397$	25.7939	1.29456	0.647281	−	0.762252i	$-0.275906\pi$
$397$	25.7939	1.29456	0.647281	+	0.762252i	$0.275906\pi$
$398$	48.3818	2.42516
$399$	−6.66488	−0.333661
$400$	16.6665	0.833323
$401$	−19.4018	−0.968879	−0.484439	−	0.874825i	$-0.660976\pi$
$401$	−19.4018	−0.968879	−0.484439	+	0.874825i	$0.660976\pi$
$402$	−62.8396	−3.13416
$403$	35.1233	1.74962
$404$	−22.3305	−1.11099
$405$	8.78623	0.436591
$406$	24.1331	1.19771
$407$	−30.6455	−1.51904
$408$	−106.631	−5.27901
$409$	34.5242	1.70711	0.853557	−	0.521000i	$-0.174441\pi$
$409$	34.5242	1.70711	0.853557	+	0.521000i	$0.174441\pi$
$410$	35.1631	1.73658
$411$	−37.5478	−1.85210
$412$	96.3135	4.74503
$413$	−4.06941	−0.200242
$414$	58.7810	2.88893
$415$	2.97998	0.146282
$416$	100.449	4.92490
$417$	16.7581	0.820646
$418$	−20.8464	−1.01963
$419$	14.6639	0.716376	0.358188	−	0.933649i	$-0.383395\pi$
$419$	14.6639	0.716376	0.358188	+	0.933649i	$0.383395\pi$
$420$	13.9313	0.679780
$421$	−26.3449	−1.28397	−0.641986	−	0.766716i	$-0.721889\pi$
$421$	−26.3449	−1.28397	−0.641986	+	0.766716i	$0.721889\pi$
$422$	−17.3301	−0.843617
$423$	35.3924	1.72084
$424$	−34.4180	−1.67148
$425$	−4.28037	−0.207628
$426$	−55.5580	−2.69179
$427$	−11.6559	−0.564066
$428$	28.9092	1.39738
$429$	−26.6244	−1.28544
$430$	4.91038	0.236799
$431$	−27.7047	−1.33449	−0.667244	−	0.744839i	$-0.732526\pi$
$431$	−27.7047	−1.33449	−0.667244	+	0.744839i	$0.732526\pi$
$432$	2.85943	0.137574
$433$	−16.9803	−0.816022	−0.408011	−	0.912977i	$-0.633777\pi$
$433$	−16.9803	−0.816022	−0.408011	+	0.912977i	$0.633777\pi$
$434$	−25.0448	−1.20219
$435$	21.4897	1.03035
$436$	−20.4123	−0.977573
$437$	18.7239	0.895684
$438$	−54.1114	−2.58554
$439$	34.0794	1.62652	0.813260	−	0.581900i	$-0.197691\pi$
$439$	34.0794	1.62652	0.813260	+	0.581900i	$0.197691\pi$
$440$	28.1628	1.34261
$441$	3.06964	0.146173
$442$	−45.9503	−2.18563
$443$	0.461355	0.0219196	0.0109598	−	0.999940i	$-0.496511\pi$
$443$	0.461355	0.0219196	0.0109598	+	0.999940i	$0.496511\pi$
$444$	153.286	7.27465
$445$	−8.83996	−0.419054
$446$	9.39871	0.445042
$447$	−26.9499	−1.27469
$448$	−38.2924	−1.80915
$449$	−8.12986	−0.383672	−0.191836	−	0.981427i	$-0.561444\pi$
$449$	−8.12986	−0.383672	−0.191836	+	0.981427i	$0.561444\pi$
$450$	8.49282	0.400356
$451$	35.3979	1.66682
$452$	−25.9646	−1.22127
$453$	−22.3153	−1.04846
$454$	−5.96274	−0.279845
$455$	3.88010	0.181902
$456$	67.3925	3.15594
$457$	−25.6777	−1.20115	−0.600576	−	0.799567i	$-0.705062\pi$
$457$	−25.6777	−1.20115	−0.600576	+	0.799567i	$0.705062\pi$
$458$	2.76672	0.129280
$459$	−0.734374	−0.0342776
$460$	−39.1378	−1.82481
$461$	−18.4427	−0.858960	−0.429480	−	0.903076i	$-0.641303\pi$
$461$	−18.4427	−0.858960	−0.429480	+	0.903076i	$0.641303\pi$
$462$	18.9846	0.883244
$463$	35.5395	1.65166	0.825831	−	0.563918i	$-0.190707\pi$
$463$	35.5395	1.65166	0.825831	+	0.563918i	$0.190707\pi$
$464$	−145.376	−6.74889
$465$	−22.3015	−1.03421
$466$	−28.5463	−1.32238
$467$	38.3440	1.77435	0.887175	−	0.461433i	$-0.152665\pi$
$467$	38.3440	1.77435	0.887175	+	0.461433i	$0.152665\pi$
$468$	67.3506	3.11328
$469$	9.21908	0.425697
$470$	−31.8997	−1.47143
$471$	−53.5023	−2.46525
$472$	41.1482	1.89400
$473$	4.94317	0.227287
$474$	−65.1607	−2.99293
$475$	2.70527	0.124126
$476$	24.2043	1.10940
$477$	−10.4485	−0.478403
$478$	37.9846	1.73738
$479$	17.0528	0.779162	0.389581	−	0.920992i	$-0.372620\pi$
$479$	17.0528	0.779162	0.389581	+	0.920992i	$0.372620\pi$
$480$	−63.7798	−2.91114
$481$	42.6926	1.94662
$482$	−47.1075	−2.14569
$483$	−17.0516	−0.775877
$484$	−18.3365	−0.833477
$485$	16.5622	0.752051
$486$	−61.3133	−2.78123
$487$	9.20575	0.417152	0.208576	−	0.978006i	$-0.433117\pi$
$487$	9.20575	0.417152	0.208576	+	0.978006i	$0.433117\pi$
$488$	117.859	5.33524
$489$	−8.21740	−0.371603
$490$	−2.76672	−0.124988
$491$	39.8874	1.80009	0.900046	−	0.435794i	$-0.143533\pi$
$491$	39.8874	1.80009	0.900046	+	0.435794i	$0.143533\pi$
$492$	−177.058	−7.98238
$493$	37.3361	1.68154
$494$	29.0414	1.30664
$495$	8.54954	0.384273
$496$	150.868	6.77416
$497$	8.15080	0.365613
$498$	−20.3123	−0.910218
$499$	−38.3664	−1.71752	−0.858758	−	0.512382i	$-0.828763\pi$
$499$	−38.3664	−1.71752	−0.858758	+	0.512382i	$0.828763\pi$
$500$	−5.65472	−0.252887
$501$	−12.0491	−0.538314
$502$	−42.7203	−1.90670
$503$	−0.943900	−0.0420864	−0.0210432	−	0.999779i	$-0.506699\pi$
$503$	−0.943900	−0.0420864	−0.0210432	+	0.999779i	$0.506699\pi$
$504$	−31.0389	−1.38258
$505$	3.94900	0.175728
$506$	−53.3341	−2.37099
$507$	5.06317	0.224863
$508$	22.0273	0.977303
$509$	19.5367	0.865950	0.432975	−	0.901406i	$-0.357464\pi$
$509$	19.5367	0.865950	0.432975	+	0.901406i	$0.357464\pi$
$510$	29.1761	1.29194
$511$	7.93858	0.351182
$512$	94.4159	4.17263
$513$	0.464138	0.0204922
$514$	0.0647823	0.00285743
$515$	−17.0324	−0.750537
$516$	−24.7254	−1.08847
$517$	−32.1128	−1.41232
$518$	−30.4421	−1.33755
$519$	−15.1045	−0.663015
$520$	−39.2339	−1.72052
$521$	15.8024	0.692317	0.346159	−	0.938176i	$-0.387486\pi$
$521$	15.8024	0.692317	0.346159	+	0.938176i	$0.387486\pi$
$522$	−74.0799	−3.24239
$523$	16.3958	0.716939	0.358470	−	0.933541i	$-0.383299\pi$
$523$	16.3958	0.716939	0.358470	+	0.933541i	$0.383299\pi$
$524$	−37.2040	−1.62526
$525$	−2.46366	−0.107523
$526$	−42.0725	−1.83445
$527$	−38.7467	−1.68783
$528$	−114.362	−4.97695
$529$	24.9038	1.08277
$530$	9.41739	0.409065
$531$	12.4916	0.542090
$532$	−15.2976	−0.663233
$533$	−49.3134	−2.13600
$534$	60.2555	2.60751
$535$	−5.11239	−0.221028
$536$	−93.2195	−4.02647
$537$	44.4854	1.91969
$538$	5.35646	0.230934
$539$	−2.78520	−0.119967
$540$	−0.970169	−0.0417494
$541$	40.4855	1.74061	0.870304	−	0.492514i	$-0.163922\pi$
$541$	40.4855	1.74061	0.870304	+	0.492514i	$0.163922\pi$
$542$	63.6861	2.73555
$543$	−16.8982	−0.725173
$544$	−110.811	−4.75099
$545$	3.60978	0.154626
$546$	−26.4477	−1.13186
$547$	−11.5683	−0.494626	−0.247313	−	0.968936i	$-0.579548\pi$
$547$	−11.5683	−0.494626	−0.247313	+	0.968936i	$0.579548\pi$
$548$	−86.1817	−3.68150
$549$	35.7793	1.52702
$550$	−7.70585	−0.328578
$551$	−23.5971	−1.00527
$552$	172.419	7.33865
$553$	9.55960	0.406515
$554$	75.4034	3.20359
$555$	−27.1077	−1.15066
$556$	38.4640	1.63124
$557$	5.09219	0.215763	0.107881	−	0.994164i	$-0.465593\pi$
$557$	5.09219	0.215763	0.107881	+	0.994164i	$0.465593\pi$
$558$	76.8786	3.25453
$559$	−6.88640	−0.291264
$560$	16.6665	0.704287
$561$	29.3710	1.24004
$562$	72.7970	3.07076
$563$	−8.79425	−0.370634	−0.185317	−	0.982679i	$-0.559331\pi$
$563$	−8.79425	−0.370634	−0.185317	+	0.982679i	$0.559331\pi$
$564$	160.626	6.76356
$565$	4.59166	0.193173
$566$	−45.7146	−1.92153
$567$	8.78623	0.368987
$568$	−82.4176	−3.45816
$569$	11.9491	0.500930	0.250465	−	0.968126i	$-0.419416\pi$
$569$	11.9491	0.500930	0.250465	+	0.968126i	$0.419416\pi$
$570$	−18.4398	−0.772359
$571$	1.20752	0.0505333	0.0252666	−	0.999681i	$-0.491957\pi$
$571$	1.20752	0.0505333	0.0252666	+	0.999681i	$0.491957\pi$
$572$	−61.1096	−2.55512
$573$	−29.7535	−1.24297
$574$	35.1631	1.46768
$575$	6.92125	0.288636
$576$	117.544	4.89766
$577$	35.5590	1.48034	0.740171	−	0.672419i	$-0.234745\pi$
$577$	35.5590	1.48034	0.740171	+	0.672419i	$0.234745\pi$
$578$	3.65640	0.152086
$579$	14.5846	0.606113
$580$	49.3241	2.04807
$581$	2.97998	0.123631
$582$	−112.892	−4.67953
$583$	9.48028	0.392633
$584$	−80.2717	−3.32167
$585$	−11.9105	−0.492438
$586$	−38.4065	−1.58656
$587$	11.5101	0.475074	0.237537	−	0.971378i	$-0.423660\pi$
$587$	11.5101	0.475074	0.237537	+	0.971378i	$0.423660\pi$
$588$	13.9313	0.574519
$589$	24.4886	1.00903
$590$	−11.2589	−0.463521
$591$	−28.7280	−1.18171
$592$	183.381	7.53690
$593$	11.1301	0.457057	0.228529	−	0.973537i	$-0.426609\pi$
$593$	11.1301	0.457057	0.228529	+	0.973537i	$0.426609\pi$
$594$	−1.32208	−0.0542454
$595$	−4.28037	−0.175478
$596$	−61.8568	−2.53375
$597$	43.0823	1.76324
$598$	74.3005	3.03838
$599$	−4.49294	−0.183576	−0.0917882	−	0.995779i	$-0.529258\pi$
$599$	−4.49294	−0.183576	−0.0917882	+	0.995779i	$0.529258\pi$
$600$	24.9116	1.01701
$601$	1.20991	0.0493533	0.0246766	−	0.999695i	$-0.492144\pi$
$601$	1.20991	0.0493533	0.0246766	+	0.999695i	$0.492144\pi$
$602$	4.91038	0.200132
$603$	−28.2992	−1.15243
$604$	−51.2192	−2.08408
$605$	3.24269	0.131834
$606$	−26.9174	−1.09345
$607$	−19.6635	−0.798116	−0.399058	−	0.916926i	$-0.630663\pi$
$607$	−19.6635	−0.798116	−0.399058	+	0.916926i	$0.630663\pi$
$608$	70.0346	2.84028
$609$	21.4897	0.870805
$610$	−32.2484	−1.30570
$611$	44.7368	1.80986
$612$	−74.2985	−3.00334
$613$	12.1251	0.489727	0.244863	−	0.969558i	$-0.421257\pi$
$613$	12.1251	0.489727	0.244863	+	0.969558i	$0.421257\pi$
$614$	36.5185	1.47377
$615$	31.3115	1.26260
$616$	28.1628	1.13471
$617$	9.86111	0.396993	0.198497	−	0.980102i	$-0.436394\pi$
$617$	9.86111	0.396993	0.198497	+	0.980102i	$0.436394\pi$
$618$	116.097	4.67012
$619$	−13.2257	−0.531584	−0.265792	−	0.964030i	$-0.585633\pi$
$619$	−13.2257	−0.531584	−0.265792	+	0.964030i	$0.585633\pi$
$620$	−51.1876	−2.05574
$621$	1.18747	0.0476513
$622$	79.8374	3.20119
$623$	−8.83996	−0.354165
$624$	159.319	6.37786
$625$	1.00000	0.0400000
$626$	6.98708	0.279260
$627$	−18.5630	−0.741334
$628$	−122.801	−4.90030
$629$	−47.0968	−1.87787
$630$	8.49282	0.338362
$631$	−36.3077	−1.44539	−0.722694	−	0.691168i	$-0.757096\pi$
$631$	−36.3077	−1.44539	−0.722694	+	0.691168i	$0.757096\pi$
$632$	−96.6627	−3.84504
$633$	−15.4319	−0.613362
$634$	20.2169	0.802916
$635$	−3.89538	−0.154583
$636$	−47.4197	−1.88031
$637$	3.88010	0.153735
$638$	67.2154	2.66108
$639$	−25.0200	−0.989777
$640$	−54.1679	−2.14117
$641$	−36.0976	−1.42577	−0.712884	−	0.701282i	$-0.752611\pi$
$641$	−36.0976	−1.42577	−0.712884	+	0.701282i	$0.752611\pi$
$642$	34.8474	1.37532
$643$	−17.4409	−0.687802	−0.343901	−	0.939006i	$-0.611748\pi$
$643$	−17.4409	−0.687802	−0.343901	+	0.939006i	$0.611748\pi$
$644$	−39.1378	−1.54224
$645$	4.37252	0.172168
$646$	−32.0374	−1.26049
$647$	−29.4329	−1.15713	−0.578563	−	0.815638i	$-0.696386\pi$
$647$	−29.4329	−1.15713	−0.578563	+	0.815638i	$0.696386\pi$
$648$	−88.8428	−3.49007
$649$	−11.3341	−0.444902
$650$	10.7351	0.421066
$651$	−22.3015	−0.874066
$652$	−18.8610	−0.738653
$653$	−30.2109	−1.18225	−0.591123	−	0.806582i	$-0.701315\pi$
$653$	−30.2109	−1.18225	−0.591123	+	0.806582i	$0.701315\pi$
$654$	−24.6052	−0.962140
$655$	6.57928	0.257074
$656$	−211.819	−8.27015
$657$	−24.3686	−0.950709
$658$	−31.8997	−1.24358
$659$	−37.2550	−1.45125	−0.725625	−	0.688090i	$-0.758449\pi$
$659$	−37.2550	−1.45125	−0.725625	+	0.688090i	$0.758449\pi$
$660$	38.8015	1.51035
$661$	48.2225	1.87564	0.937818	−	0.347126i	$-0.112843\pi$
$661$	48.2225	1.87564	0.937818	+	0.347126i	$0.112843\pi$
$662$	64.2935	2.49884
$663$	−40.9171	−1.58909
$664$	−30.1324	−1.16936
$665$	2.70527	0.104906
$666$	93.4464	3.62098
$667$	−60.3716	−2.33760
$668$	−27.6557	−1.07003
$669$	8.36922	0.323573
$670$	25.5066	0.985405
$671$	−32.4638	−1.25325
$672$	−63.7798	−2.46036
$673$	6.49244	0.250265	0.125133	−	0.992140i	$-0.460064\pi$
$673$	6.49244	0.250265	0.125133	+	0.992140i	$0.460064\pi$
$674$	−38.1816	−1.47070
$675$	0.171568	0.00660365
$676$	11.6212	0.446971
$677$	22.5654	0.867260	0.433630	−	0.901091i	$-0.357232\pi$
$677$	22.5654	0.867260	0.433630	+	0.901091i	$0.357232\pi$
$678$	−31.2979	−1.20199
$679$	16.5622	0.635599
$680$	43.2813	1.65976
$681$	−5.30961	−0.203465
$682$	−69.7547	−2.67105
$683$	8.32378	0.318501	0.159250	−	0.987238i	$-0.449092\pi$
$683$	8.32378	0.318501	0.159250	+	0.987238i	$0.449092\pi$
$684$	46.9580	1.79548
$685$	15.2407	0.582315
$686$	−2.76672	−0.105634
$687$	2.46366	0.0939946
$688$	−29.5797	−1.12771
$689$	−13.2071	−0.503151
$690$	−47.1771	−1.79600
$691$	3.78947	0.144158	0.0720790	−	0.997399i	$-0.477037\pi$
$691$	3.78947	0.144158	0.0720790	+	0.997399i	$0.477037\pi$
$692$	−34.6687	−1.31791
$693$	8.54954	0.324770
$694$	−2.59530	−0.0985164
$695$	−6.80209	−0.258018
$696$	−217.295	−8.23653
$697$	54.4006	2.06057
$698$	40.9292	1.54919
$699$	−25.4194	−0.961452
$700$	−5.65472	−0.213728
$701$	18.1619	0.685966	0.342983	−	0.939342i	$-0.388563\pi$
$701$	18.1619	0.685966	0.342983	+	0.939342i	$0.388563\pi$
$702$	1.84180	0.0695144
$703$	29.7660	1.12265
$704$	−106.652	−4.01959
$705$	−28.4056	−1.06982
$706$	−96.5224	−3.63267
$707$	3.94900	0.148518
$708$	56.6923	2.13063
$709$	−30.7243	−1.15388	−0.576938	−	0.816788i	$-0.695752\pi$
$709$	−30.7243	−1.15388	−0.576938	+	0.816788i	$0.695752\pi$
$710$	22.5510	0.846323
$711$	−29.3445	−1.10051
$712$	89.3860	3.34988
$713$	62.6524	2.34635
$714$	29.1761	1.09189
$715$	10.8068	0.404152
$716$	102.105	3.81585
$717$	33.8240	1.26318
$718$	−46.3676	−1.73042
$719$	37.1924	1.38704	0.693520	−	0.720437i	$-0.256059\pi$
$719$	37.1924	1.38704	0.693520	+	0.720437i	$0.256059\pi$
$720$	−51.1600	−1.90662
$721$	−17.0324	−0.634320
$722$	−32.3194	−1.20281
$723$	−41.9476	−1.56005
$724$	−38.7857	−1.44146
$725$	−8.72264	−0.323951
$726$	−22.1030	−0.820319
$727$	52.6578	1.95297	0.976485	−	0.215585i	$-0.0691659\pi$
$727$	52.6578	1.95297	0.976485	+	0.215585i	$0.0691659\pi$
$728$	−39.2339	−1.45411
$729$	−28.2386	−1.04587
$730$	21.9638	0.812917
$731$	7.59681	0.280978
$732$	162.382	6.00180
$733$	−10.0745	−0.372110	−0.186055	−	0.982539i	$-0.559570\pi$
$733$	−10.0745	−0.372110	−0.186055	+	0.982539i	$0.559570\pi$
$734$	−12.1713	−0.449249
$735$	−2.46366	−0.0908736
$736$	179.179	6.60462
$737$	25.6769	0.945822
$738$	−107.938	−3.97325
$739$	−15.2860	−0.562305	−0.281152	−	0.959663i	$-0.590717\pi$
$739$	−15.2860	−0.562305	−0.281152	+	0.959663i	$0.590717\pi$
$740$	−62.2188	−2.28721
$741$	25.8604	0.950004
$742$	9.41739	0.345723
$743$	23.6187	0.866485	0.433243	−	0.901277i	$-0.357369\pi$
$743$	23.6187	0.866485	0.433243	+	0.901277i	$0.357369\pi$
$744$	225.504	8.26737
$745$	10.9390	0.400773
$746$	45.4651	1.66459
$747$	−9.14747	−0.334689
$748$	67.4137	2.46489
$749$	−5.11239	−0.186803
$750$	−6.81626	−0.248895
$751$	−20.5806	−0.750998	−0.375499	−	0.926823i	$-0.622529\pi$
$751$	−20.5806	−0.750998	−0.375499	+	0.926823i	$0.622529\pi$
$752$	192.161	7.00740
$753$	−38.0409	−1.38629
$754$	−93.6387	−3.41012
$755$	9.05777	0.329646
$756$	−0.970169	−0.0352847
$757$	30.0680	1.09284	0.546420	−	0.837511i	$-0.315990\pi$
$757$	30.0680	1.09284	0.546420	+	0.837511i	$0.315990\pi$
$758$	46.8426	1.70140
$759$	−47.4922	−1.72386
$760$	−27.3546	−0.992255
$761$	34.7817	1.26084	0.630419	−	0.776255i	$-0.282883\pi$
$761$	34.7817	1.26084	0.630419	+	0.776255i	$0.282883\pi$
$762$	26.5519	0.961874
$763$	3.60978	0.130683
$764$	−68.2918	−2.47071
$765$	13.1392	0.475048
$766$	−50.1061	−1.81041
$767$	15.7897	0.570132
$768$	180.543	6.51479
$769$	−23.7711	−0.857209	−0.428605	−	0.903492i	$-0.640995\pi$
$769$	−23.7711	−0.857209	−0.428605	+	0.903492i	$0.640995\pi$
$770$	−7.70585	−0.277699
$771$	0.0576864	0.00207752
$772$	33.4752	1.20480
$773$	15.3469	0.551991	0.275995	−	0.961159i	$-0.410993\pi$
$773$	15.3469	0.551991	0.275995	+	0.961159i	$0.410993\pi$
$774$	−15.0731	−0.541791
$775$	9.05218	0.325164
$776$	−167.470	−6.01183
$777$	−27.1077	−0.972482
$778$	−32.1033	−1.15096
$779$	−34.3821	−1.23187
$780$	−54.0549	−1.93548
$781$	22.7016	0.812326
$782$	−81.9655	−2.93108
$783$	−1.49653	−0.0534815
$784$	16.6665	0.595231
$785$	21.7165	0.775097
$786$	−44.8461	−1.59961
$787$	27.4856	0.979755	0.489878	−	0.871791i	$-0.337041\pi$
$787$	27.4856	0.979755	0.489878	+	0.871791i	$0.337041\pi$
$788$	−65.9379	−2.34894
$789$	−37.4640	−1.33375
$790$	26.4487	0.941003
$791$	4.59166	0.163261
$792$	−86.4495	−3.07185
$793$	45.2258	1.60602
$794$	71.3646	2.53263
$795$	8.38585	0.297415
$796$	98.8846	3.50487
$797$	47.8270	1.69412	0.847060	−	0.531497i	$-0.178370\pi$
$797$	47.8270	1.69412	0.847060	+	0.531497i	$0.178370\pi$
$798$	−18.4398	−0.652763
$799$	−49.3519	−1.74594
$800$	25.8882	0.915286
$801$	27.1355	0.958785
$802$	−53.6793	−1.89548
$803$	22.1105	0.780262
$804$	−128.434	−4.52952
$805$	6.92125	0.243942
$806$	97.1763	3.42289
$807$	4.76974	0.167903
$808$	−39.9307	−1.40476
$809$	−13.7959	−0.485039	−0.242519	−	0.970147i	$-0.577974\pi$
$809$	−13.7959	−0.485039	−0.242519	+	0.970147i	$0.577974\pi$
$810$	24.3090	0.854132
$811$	−26.7332	−0.938728	−0.469364	−	0.883005i	$-0.655517\pi$
$811$	−26.7332	−0.938728	−0.469364	+	0.883005i	$0.655517\pi$
$812$	49.3241	1.73094
$813$	56.7102	1.98892
$814$	−84.7873	−2.97179
$815$	3.33544	0.116835
$816$	−175.754	−6.15263
$817$	−4.80132	−0.167977
$818$	95.5188	3.33974
$819$	−11.9105	−0.416186
$820$	71.8677	2.50973
$821$	−6.92473	−0.241675	−0.120837	−	0.992672i	$-0.538558\pi$
$821$	−6.92473	−0.241675	−0.120837	+	0.992672i	$0.538558\pi$
$822$	−103.884	−3.62338
$823$	50.6906	1.76696	0.883482	−	0.468466i	$-0.155193\pi$
$823$	50.6906	1.76696	0.883482	+	0.468466i	$0.155193\pi$
$824$	172.225	5.99973
$825$	−6.86178	−0.238897
$826$	−11.2589	−0.391747
$827$	−8.29416	−0.288416	−0.144208	−	0.989547i	$-0.546063\pi$
$827$	−8.29416	−0.288416	−0.144208	+	0.989547i	$0.546063\pi$
$828$	120.139	4.17512
$829$	−9.57303	−0.332485	−0.166243	−	0.986085i	$-0.553163\pi$
$829$	−9.57303	−0.332485	−0.166243	+	0.986085i	$0.553163\pi$
$830$	8.24477	0.286180
$831$	67.1441	2.32920
$832$	148.578	5.15102
$833$	−4.28037	−0.148306
$834$	46.3648	1.60548
$835$	4.89072	0.169250
$836$	−42.6067	−1.47358
$837$	1.55306	0.0536817
$838$	40.5707	1.40149
$839$	46.0697	1.59050	0.795251	−	0.606281i	$-0.207339\pi$
$839$	46.0697	1.59050	0.795251	+	0.606281i	$0.207339\pi$
$840$	24.9116	0.859530
$841$	47.0845	1.62360
$842$	−72.8889	−2.51192
$843$	64.8231	2.23263
$844$	−35.4200	−1.21921
$845$	−2.05514	−0.0706990
$846$	97.9207	3.36658
$847$	3.24269	0.111420
$848$	−56.7295	−1.94810
$849$	−40.7072	−1.39707
$850$	−11.8426	−0.406197
$851$	76.1545	2.61054
$852$	−113.552	−3.89021
$853$	−55.2158	−1.89055	−0.945276	−	0.326272i	$-0.894208\pi$
$853$	−55.2158	−1.89055	−0.945276	+	0.326272i	$0.894208\pi$
$854$	−32.2484	−1.10352
$855$	−8.30420	−0.283998
$856$	51.6944	1.76688
$857$	−54.8924	−1.87509	−0.937545	−	0.347865i	$-0.886907\pi$
$857$	−54.8924	−1.87509	−0.937545	+	0.347865i	$0.886907\pi$
$858$	−73.6621	−2.51478
$859$	−44.1250	−1.50553	−0.752763	−	0.658291i	$-0.771279\pi$
$859$	−44.1250	−1.50553	−0.752763	+	0.658291i	$0.771279\pi$
$860$	10.0360	0.342225
$861$	31.3115	1.06709
$862$	−76.6510	−2.61074
$863$	−1.04695	−0.0356387	−0.0178193	−	0.999841i	$-0.505672\pi$
$863$	−1.04695	−0.0356387	−0.0178193	+	0.999841i	$0.505672\pi$
$864$	4.44159	0.151106
$865$	6.13092	0.208458
$866$	−46.9797	−1.59644
$867$	3.25589	0.110576
$868$	−51.1876	−1.73742
$869$	26.6253	0.903203
$870$	59.4558	2.01574
$871$	−35.7709	−1.21205
$872$	−36.5006	−1.23607
$873$	−50.8400	−1.72067
$874$	51.8036	1.75228
$875$	1.00000	0.0338062
$876$	−110.595	−3.73666
$877$	−41.5066	−1.40158	−0.700789	−	0.713368i	$-0.747169\pi$
$877$	−41.5066	−1.40158	−0.700789	+	0.713368i	$0.747169\pi$
$878$	94.2880	3.18207
$879$	−34.1997	−1.15353
$880$	46.4193	1.56480
$881$	−30.6281	−1.03189	−0.515944	−	0.856622i	$-0.672559\pi$
$881$	−30.6281	−1.03189	−0.515944	+	0.856622i	$0.672559\pi$
$882$	8.49282	0.285968
$883$	37.8262	1.27295	0.636476	−	0.771296i	$-0.280391\pi$
$883$	37.8262	1.27295	0.636476	+	0.771296i	$0.280391\pi$
$884$	−93.9150	−3.15870
$885$	−10.0256	−0.337008
$886$	1.27644	0.0428828
$887$	1.49090	0.0500596	0.0250298	−	0.999687i	$-0.492032\pi$
$887$	1.49090	0.0500596	0.0250298	+	0.999687i	$0.492032\pi$
$888$	274.101	9.19824
$889$	−3.89538	−0.130647
$890$	−24.4577	−0.819823
$891$	24.4714	0.819822
$892$	19.2095	0.643180
$893$	31.1913	1.04378
$894$	−74.5628	−2.49375
$895$	−18.0566	−0.603566
$896$	−54.1679	−1.80962
$897$	66.1620	2.20909
$898$	−22.4930	−0.750602
$899$	−78.9589	−2.63343
$900$	17.3580	0.578599
$901$	14.5696	0.485383
$902$	97.9360	3.26091
$903$	4.37252	0.145508
$904$	−46.4290	−1.54420
$905$	6.85899	0.228001
$906$	−61.7401	−2.05118
$907$	−45.0913	−1.49723	−0.748615	−	0.663005i	$-0.769281\pi$
$907$	−45.0913	−1.49723	−0.748615	+	0.663005i	$0.769281\pi$
$908$	−12.1869	−0.404436
$909$	−12.1220	−0.402062
$910$	10.7351	0.355866
$911$	29.4533	0.975830	0.487915	−	0.872891i	$-0.337758\pi$
$911$	29.4533	0.975830	0.487915	+	0.872891i	$0.337758\pi$
$912$	111.080	3.67822
$913$	8.29983	0.274684
$914$	−71.0430	−2.34989
$915$	−28.7161	−0.949325
$916$	5.65472	0.186837
$917$	6.57928	0.217267
$918$	−2.03181	−0.0670596
$919$	32.0222	1.05632	0.528158	−	0.849146i	$-0.322883\pi$
$919$	32.0222	1.05632	0.528158	+	0.849146i	$0.322883\pi$
$920$	−69.9849	−2.30733
$921$	32.5184	1.07152
$922$	−51.0256	−1.68044
$923$	−31.6259	−1.04098
$924$	38.8015	1.27648
$925$	11.0030	0.361776
$926$	98.3278	3.23125
$927$	52.2833	1.71721
$928$	−225.814	−7.41269
$929$	39.6002	1.29924	0.649620	−	0.760259i	$-0.274928\pi$
$929$	39.6002	1.29924	0.649620	+	0.760259i	$0.274928\pi$
$930$	−61.7020	−2.02329
$931$	2.70527	0.0886616
$932$	−58.3440	−1.91112
$933$	71.0924	2.32746
$934$	106.087	3.47128
$935$	−11.9217	−0.389880
$936$	120.434	3.93651
$937$	−15.3496	−0.501450	−0.250725	−	0.968058i	$-0.580669\pi$
$937$	−15.3496	−0.501450	−0.250725	+	0.968058i	$0.580669\pi$
$938$	25.5066	0.832819
$939$	6.22175	0.203039
$940$	−65.1979	−2.12652
$941$	−14.3675	−0.468368	−0.234184	−	0.972192i	$-0.575242\pi$
$941$	−14.3675	−0.468368	−0.234184	+	0.972192i	$0.575242\pi$
$942$	−148.026	−4.82293
$943$	−87.9644	−2.86452
$944$	67.8226	2.20744
$945$	0.171568	0.00558110
$946$	13.6764	0.444657
$947$	18.5760	0.603638	0.301819	−	0.953365i	$-0.402406\pi$
$947$	18.5760	0.603638	0.301819	+	0.953365i	$0.402406\pi$
$948$	−133.178	−4.32542
$949$	−30.8024	−0.999890
$950$	7.48472	0.242836
$951$	18.0024	0.583769
$952$	43.2813	1.40276
$953$	−42.2489	−1.36858	−0.684288	−	0.729212i	$-0.739887\pi$
$953$	−42.2489	−1.36858	−0.684288	+	0.729212i	$0.739887\pi$
$954$	−28.9080	−0.935930
$955$	12.0769	0.390801
$956$	77.6345	2.51088
$957$	59.8529	1.93477
$958$	47.1803	1.52433
$959$	15.2407	0.492146
$960$	−94.3397	−3.04480
$961$	50.9419	1.64329
$962$	118.118	3.80829
$963$	15.6932	0.505706
$964$	−96.2802	−3.10098
$965$	−5.91986	−0.190567
$966$	−47.1771	−1.51790
$967$	−53.2374	−1.71200	−0.856000	−	0.516975i	$-0.827058\pi$
$967$	−53.2374	−1.71200	−0.856000	+	0.516975i	$0.827058\pi$
$968$	−32.7887	−1.05387
$969$	−28.5281	−0.916456
$970$	45.8229	1.47129
$971$	10.9588	0.351683	0.175842	−	0.984418i	$-0.443735\pi$
$971$	10.9588	0.351683	0.175842	+	0.984418i	$0.443735\pi$
$972$	−125.314	−4.01946
$973$	−6.80209	−0.218065
$974$	25.4697	0.816102
$975$	9.55925	0.306141
$976$	194.262	6.21817
$977$	34.4869	1.10333	0.551667	−	0.834064i	$-0.313992\pi$
$977$	34.4869	1.10333	0.551667	+	0.834064i	$0.313992\pi$
$978$	−22.7352	−0.726992
$979$	−24.6210	−0.786891
$980$	−5.65472	−0.180634
$981$	−11.0807	−0.353780
$982$	110.357	3.52164
$983$	17.3874	0.554572	0.277286	−	0.960787i	$-0.410565\pi$
$983$	17.3874	0.554572	0.277286	+	0.960787i	$0.410565\pi$
$984$	−316.609	−10.0931
$985$	11.6607	0.371540
$986$	103.299	3.28970
$987$	−28.4056	−0.904160
$988$	59.3560	1.88837
$989$	−12.2839	−0.390604
$990$	23.6542	0.751779
$991$	−6.01968	−0.191221	−0.0956107	−	0.995419i	$-0.530480\pi$
$991$	−6.01968	−0.191221	−0.0956107	+	0.995419i	$0.530480\pi$
$992$	234.345	7.44045
$993$	57.2511	1.81681
$994$	22.5510	0.715273
$995$	−17.4871	−0.554377
$996$	−41.5152	−1.31546
$997$	−28.1454	−0.891375	−0.445687	−	0.895189i	$-0.647041\pi$
$997$	−28.1454	−0.891375	−0.445687	+	0.895189i	$0.647041\pi$
$998$	−106.149	−3.36009
$999$	1.88776	0.0597261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8015.2.a.l.1.61	✓	62

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8015.2.a.l.1.61	✓	62	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:	\( N \)	\(=\)	\( 8015 = 5 \cdot 7 \cdot 229 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8015.a (trivial)

\(f(q)\)	\(=\)	\(q+2.76672 q^{2} +2.46366 q^{3} +5.65472 q^{4} -1.00000 q^{5} +6.81626 q^{6} -1.00000 q^{7} +10.1116 q^{8} +3.06964 q^{9} +O(q^{10})\)	\(q+2.76672 q^{2} +2.46366 q^{3} +5.65472 q^{4} -1.00000 q^{5} +6.81626 q^{6} -1.00000 q^{7} +10.1116 q^{8} +3.06964 q^{9} -2.76672 q^{10} -2.78520 q^{11} +13.9313 q^{12} +3.88010 q^{13} -2.76672 q^{14} -2.46366 q^{15} +16.6665 q^{16} -4.28037 q^{17} +8.49282 q^{18} +2.70527 q^{19} -5.65472 q^{20} -2.46366 q^{21} -7.70585 q^{22} +6.92125 q^{23} +24.9116 q^{24} +1.00000 q^{25} +10.7351 q^{26} +0.171568 q^{27} -5.65472 q^{28} -8.72264 q^{29} -6.81626 q^{30} +9.05218 q^{31} +25.8882 q^{32} -6.86178 q^{33} -11.8426 q^{34} +1.00000 q^{35} +17.3580 q^{36} +11.0030 q^{37} +7.48472 q^{38} +9.55925 q^{39} -10.1116 q^{40} -12.7093 q^{41} -6.81626 q^{42} -1.77480 q^{43} -15.7495 q^{44} -3.06964 q^{45} +19.1492 q^{46} +11.5298 q^{47} +41.0606 q^{48} +1.00000 q^{49} +2.76672 q^{50} -10.5454 q^{51} +21.9409 q^{52} -3.40381 q^{53} +0.474680 q^{54} +2.78520 q^{55} -10.1116 q^{56} +6.66488 q^{57} -24.1331 q^{58} +4.06941 q^{59} -13.9313 q^{60} +11.6559 q^{61} +25.0448 q^{62} -3.06964 q^{63} +38.2924 q^{64} -3.88010 q^{65} -18.9846 q^{66} -9.21908 q^{67} -24.2043 q^{68} +17.0516 q^{69} +2.76672 q^{70} -8.15080 q^{71} +31.0389 q^{72} -7.93858 q^{73} +30.4421 q^{74} +2.46366 q^{75} +15.2976 q^{76} +2.78520 q^{77} +26.4477 q^{78} -9.55960 q^{79} -16.6665 q^{80} -8.78623 q^{81} -35.1631 q^{82} -2.97998 q^{83} -13.9313 q^{84} +4.28037 q^{85} -4.91038 q^{86} -21.4897 q^{87} -28.1628 q^{88} +8.83996 q^{89} -8.49282 q^{90} -3.88010 q^{91} +39.1378 q^{92} +22.3015 q^{93} +31.8997 q^{94} -2.70527 q^{95} +63.7798 q^{96} -16.5622 q^{97} +2.76672 q^{98} -8.54954 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 62 q + 2 q^{2} + 11 q^{3} + 64 q^{4} - 62 q^{5} + 3 q^{6} - 62 q^{7} + 15 q^{8} + 69 q^{9}+O(q^{10}) \) 62 * q + 2 * q^2 + 11 * q^3 + 64 * q^4 - 62 * q^5 + 3 * q^6 - 62 * q^7 + 15 * q^8 + 69 * q^9	\( 62 q + 2 q^{2} + 11 q^{3} + 64 q^{4} - 62 q^{5} + 3 q^{6} - 62 q^{7} + 15 q^{8} + 69 q^{9} - 2 q^{10} - 13 q^{11} + 37 q^{12} + 31 q^{13} - 2 q^{14} - 11 q^{15} + 64 q^{16} + 30 q^{17} + 18 q^{18} + 20 q^{19} - 64 q^{20} - 11 q^{21} + 7 q^{22} + 29 q^{24} + 62 q^{25} + 59 q^{27} - 64 q^{28} - 29 q^{29} - 3 q^{30} + 20 q^{31} + 22 q^{32} + 72 q^{33} + 13 q^{34} + 62 q^{35} + 53 q^{36} + 35 q^{37} + 34 q^{38} - 6 q^{39} - 15 q^{40} + 13 q^{41} - 3 q^{42} - 4 q^{43} - 44 q^{44} - 69 q^{45} - 19 q^{46} + 58 q^{47} + 64 q^{48} + 62 q^{49} + 2 q^{50} - 30 q^{51} + 82 q^{52} + 18 q^{53} + 22 q^{54} + 13 q^{55} - 15 q^{56} + 21 q^{57} + 18 q^{58} - 11 q^{59} - 37 q^{60} + 24 q^{61} + 48 q^{62} - 69 q^{63} + 65 q^{64} - 31 q^{65} + 25 q^{66} - 6 q^{67} + 65 q^{68} + 27 q^{69} + 2 q^{70} - 35 q^{71} + 53 q^{72} + 116 q^{73} - 69 q^{74} + 11 q^{75} + 65 q^{76} + 13 q^{77} + 102 q^{78} - 83 q^{79} - 64 q^{80} + 126 q^{81} + 71 q^{82} + 84 q^{83} - 37 q^{84} - 30 q^{85} + 24 q^{86} + 49 q^{87} + 20 q^{88} - 16 q^{89} - 18 q^{90} - 31 q^{91} + 19 q^{92} + 65 q^{93} + 54 q^{94} - 20 q^{95} + 17 q^{96} + 155 q^{97} + 2 q^{98} + 6 q^{99}+O(q^{100}) \) 62 * q + 2 * q^2 + 11 * q^3 + 64 * q^4 - 62 * q^5 + 3 * q^6 - 62 * q^7 + 15 * q^8 + 69 * q^9 - 2 * q^10 - 13 * q^11 + 37 * q^12 + 31 * q^13 - 2 * q^14 - 11 * q^15 + 64 * q^16 + 30 * q^17 + 18 * q^18 + 20 * q^19 - 64 * q^20 - 11 * q^21 + 7 * q^22 + 29 * q^24 + 62 * q^25 + 59 * q^27 - 64 * q^28 - 29 * q^29 - 3 * q^30 + 20 * q^31 + 22 * q^32 + 72 * q^33 + 13 * q^34 + 62 * q^35 + 53 * q^36 + 35 * q^37 + 34 * q^38 - 6 * q^39 - 15 * q^40 + 13 * q^41 - 3 * q^42 - 4 * q^43 - 44 * q^44 - 69 * q^45 - 19 * q^46 + 58 * q^47 + 64 * q^48 + 62 * q^49 + 2 * q^50 - 30 * q^51 + 82 * q^52 + 18 * q^53 + 22 * q^54 + 13 * q^55 - 15 * q^56 + 21 * q^57 + 18 * q^58 - 11 * q^59 - 37 * q^60 + 24 * q^61 + 48 * q^62 - 69 * q^63 + 65 * q^64 - 31 * q^65 + 25 * q^66 - 6 * q^67 + 65 * q^68 + 27 * q^69 + 2 * q^70 - 35 * q^71 + 53 * q^72 + 116 * q^73 - 69 * q^74 + 11 * q^75 + 65 * q^76 + 13 * q^77 + 102 * q^78 - 83 * q^79 - 64 * q^80 + 126 * q^81 + 71 * q^82 + 84 * q^83 - 37 * q^84 - 30 * q^85 + 24 * q^86 + 49 * q^87 + 20 * q^88 - 16 * q^89 - 18 * q^90 - 31 * q^91 + 19 * q^92 + 65 * q^93 + 54 * q^94 - 20 * q^95 + 17 * q^96 + 155 * q^97 + 2 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more