LMFDB - Embedded newform 8015.2.a.l.1.57

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			1.57
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)

$f(q)$	$=$	$q+2.42755 q^{2} +1.02702 q^{3} +3.89301 q^{4} -1.00000 q^{5} +2.49316 q^{6} -1.00000 q^{7} +4.59539 q^{8} -1.94522 q^{9} +O(q^{10})$	\(q+2.42755 q^{2} +1.02702 q^{3} +3.89301 q^{4} -1.00000 q^{5} +2.49316 q^{6} -1.00000 q^{7} +4.59539 q^{8} -1.94522 q^{9} -2.42755 q^{10} +0.740098 q^{11} +3.99822 q^{12} -0.862758 q^{13} -2.42755 q^{14} -1.02702 q^{15} +3.36953 q^{16} +2.25071 q^{17} -4.72213 q^{18} +4.76201 q^{19} -3.89301 q^{20} -1.02702 q^{21} +1.79663 q^{22} -0.0894125 q^{23} +4.71958 q^{24} +1.00000 q^{25} -2.09439 q^{26} -5.07886 q^{27} -3.89301 q^{28} +10.3886 q^{29} -2.49316 q^{30} +7.27024 q^{31} -1.01107 q^{32} +0.760098 q^{33} +5.46373 q^{34} +1.00000 q^{35} -7.57277 q^{36} +1.86231 q^{37} +11.5600 q^{38} -0.886074 q^{39} -4.59539 q^{40} +1.00466 q^{41} -2.49316 q^{42} +5.65130 q^{43} +2.88121 q^{44} +1.94522 q^{45} -0.217054 q^{46} +0.888047 q^{47} +3.46059 q^{48} +1.00000 q^{49} +2.42755 q^{50} +2.31154 q^{51} -3.35873 q^{52} +13.1121 q^{53} -12.3292 q^{54} -0.740098 q^{55} -4.59539 q^{56} +4.89070 q^{57} +25.2190 q^{58} +9.57154 q^{59} -3.99822 q^{60} +6.37429 q^{61} +17.6489 q^{62} +1.94522 q^{63} -9.19348 q^{64} +0.862758 q^{65} +1.84518 q^{66} -14.4564 q^{67} +8.76206 q^{68} -0.0918288 q^{69} +2.42755 q^{70} -9.82493 q^{71} -8.93906 q^{72} -14.0816 q^{73} +4.52085 q^{74} +1.02702 q^{75} +18.5386 q^{76} -0.740098 q^{77} -2.15099 q^{78} +8.20700 q^{79} -3.36953 q^{80} +0.619547 q^{81} +2.43887 q^{82} +5.30835 q^{83} -3.99822 q^{84} -2.25071 q^{85} +13.7188 q^{86} +10.6694 q^{87} +3.40104 q^{88} -11.9860 q^{89} +4.72213 q^{90} +0.862758 q^{91} -0.348084 q^{92} +7.46672 q^{93} +2.15578 q^{94} -4.76201 q^{95} -1.03839 q^{96} +16.9852 q^{97} +2.42755 q^{98} -1.43965 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.42755	1.71654	0.858270	−	0.513199i	$-0.171540\pi$
$2$	2.42755	1.71654	0.858270	+	0.513199i	$0.171540\pi$
$3$	1.02702	0.592953	0.296476	−	0.955040i	$-0.404188\pi$
$3$	1.02702	0.592953	0.296476	+	0.955040i	$0.404188\pi$
$4$	3.89301	1.94651
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.49316	1.01783
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	4.59539	1.62472
$9$	−1.94522	−0.648407
$10$	−2.42755	−0.767660
$11$	0.740098	0.223148	0.111574	−	0.993756i	$-0.464411\pi$
$11$	0.740098	0.223148	0.111574	+	0.993756i	$0.464411\pi$
$12$	3.99822	1.15419
$13$	−0.862758	−0.239286	−0.119643	−	0.992817i	$-0.538175\pi$
$13$	−0.862758	−0.239286	−0.119643	+	0.992817i	$0.538175\pi$
$14$	−2.42755	−0.648791
$15$	−1.02702	−0.265177
$16$	3.36953	0.842383
$17$	2.25071	0.545878	0.272939	−	0.962031i	$-0.412004\pi$
$17$	2.25071	0.545878	0.272939	+	0.962031i	$0.412004\pi$
$18$	−4.72213	−1.11302
$19$	4.76201	1.09248	0.546240	−	0.837629i	$-0.316059\pi$
$19$	4.76201	1.09248	0.546240	+	0.837629i	$0.316059\pi$
$20$	−3.89301	−0.870505
$21$	−1.02702	−0.224115
$22$	1.79663	0.383042
$23$	−0.0894125	−0.0186438	−0.00932190	−	0.999957i	$-0.502967\pi$
$23$	−0.0894125	−0.0186438	−0.00932190	+	0.999957i	$0.502967\pi$
$24$	4.71958	0.963380
$25$	1.00000	0.200000
$26$	−2.09439	−0.410744
$27$	−5.07886	−0.977428
$28$	−3.89301	−0.735711
$29$	10.3886	1.92912	0.964562	−	0.263857i	$-0.0849948\pi$
$29$	10.3886	1.92912	0.964562	+	0.263857i	$0.0849948\pi$
$30$	−2.49316	−0.455186
$31$	7.27024	1.30577	0.652887	−	0.757455i	$-0.273558\pi$
$31$	7.27024	1.30577	0.652887	+	0.757455i	$0.273558\pi$
$32$	−1.01107	−0.178733
$33$	0.760098	0.132316
$34$	5.46373	0.937022
$35$	1.00000	0.169031
$36$	−7.57277	−1.26213
$37$	1.86231	0.306161	0.153081	−	0.988214i	$-0.451081\pi$
$37$	1.86231	0.306161	0.153081	+	0.988214i	$0.451081\pi$
$38$	11.5600	1.87529
$39$	−0.886074	−0.141885
$40$	−4.59539	−0.726596
$41$	1.00466	0.156902	0.0784510	−	0.996918i	$-0.475003\pi$
$41$	1.00466	0.156902	0.0784510	+	0.996918i	$0.475003\pi$
$42$	−2.49316	−0.384702
$43$	5.65130	0.861815	0.430908	−	0.902396i	$-0.358193\pi$
$43$	5.65130	0.861815	0.430908	+	0.902396i	$0.358193\pi$
$44$	2.88121	0.434359
$45$	1.94522	0.289976
$46$	−0.217054	−0.0320028
$47$	0.888047	0.129535	0.0647675	−	0.997900i	$-0.479369\pi$
$47$	0.888047	0.129535	0.0647675	+	0.997900i	$0.479369\pi$
$48$	3.46059	0.499494
$49$	1.00000	0.142857
$50$	2.42755	0.343308
$51$	2.31154	0.323680
$52$	−3.35873	−0.465772
$53$	13.1121	1.80109	0.900545	−	0.434764i	$-0.143168\pi$
$53$	13.1121	1.80109	0.900545	+	0.434764i	$0.143168\pi$
$54$	−12.3292	−1.67779
$55$	−0.740098	−0.0997948
$56$	−4.59539	−0.614085
$57$	4.89070	0.647789
$58$	25.2190	3.31142
$59$	9.57154	1.24611	0.623054	−	0.782178i	$-0.285892\pi$
$59$	9.57154	1.24611	0.623054	+	0.782178i	$0.285892\pi$
$60$	−3.99822	−0.516168
$61$	6.37429	0.816145	0.408072	−	0.912950i	$-0.366201\pi$
$61$	6.37429	0.816145	0.408072	+	0.912950i	$0.366201\pi$
$62$	17.6489	2.24141
$63$	1.94522	0.245075
$64$	−9.19348	−1.14919
$65$	0.862758	0.107012
$66$	1.84518	0.227126
$67$	−14.4564	−1.76613	−0.883066	−	0.469248i	$-0.844525\pi$
$67$	−14.4564	−1.76613	−0.883066	+	0.469248i	$0.844525\pi$
$68$	8.76206	1.06256
$69$	−0.0918288	−0.0110549
$70$	2.42755	0.290148
$71$	−9.82493	−1.16601	−0.583003	−	0.812470i	$-0.698122\pi$
$71$	−9.82493	−1.16601	−0.583003	+	0.812470i	$0.698122\pi$
$72$	−8.93906	−1.05348
$73$	−14.0816	−1.64813	−0.824066	−	0.566494i	$-0.808300\pi$
$73$	−14.0816	−1.64813	−0.824066	+	0.566494i	$0.808300\pi$
$74$	4.52085	0.525538
$75$	1.02702	0.118591
$76$	18.5386	2.12652
$77$	−0.740098	−0.0843420
$78$	−2.15099	−0.243552
$79$	8.20700	0.923360	0.461680	−	0.887047i	$-0.347247\pi$
$79$	8.20700	0.923360	0.461680	+	0.887047i	$0.347247\pi$
$80$	−3.36953	−0.376725
$81$	0.619547	0.0688386
$82$	2.43887	0.269328
$83$	5.30835	0.582667	0.291334	−	0.956621i	$-0.405901\pi$
$83$	5.30835	0.582667	0.291334	+	0.956621i	$0.405901\pi$
$84$	−3.99822	−0.436242
$85$	−2.25071	−0.244124
$86$	13.7188	1.47934
$87$	10.6694	1.14388
$88$	3.40104	0.362552
$89$	−11.9860	−1.27052	−0.635258	−	0.772300i	$-0.719106\pi$
$89$	−11.9860	−1.27052	−0.635258	+	0.772300i	$0.719106\pi$
$90$	4.72213	0.497756
$91$	0.862758	0.0904416
$92$	−0.348084	−0.0362903
$93$	7.46672	0.774262
$94$	2.15578	0.222352
$95$	−4.76201	−0.488572
$96$	−1.03839	−0.105980
$97$	16.9852	1.72459	0.862295	−	0.506406i	$-0.169026\pi$
$97$	16.9852	1.72459	0.862295	+	0.506406i	$0.169026\pi$
$98$	2.42755	0.245220
$99$	−1.43965	−0.144691
$100$	3.89301	0.389301
$101$	8.16261	0.812210	0.406105	−	0.913826i	$-0.366887\pi$
$101$	8.16261	0.812210	0.406105	+	0.913826i	$0.366887\pi$
$102$	5.61138	0.555610
$103$	−8.40818	−0.828483	−0.414241	−	0.910167i	$-0.635953\pi$
$103$	−8.40818	−0.828483	−0.414241	+	0.910167i	$0.635953\pi$
$104$	−3.96471	−0.388772
$105$	1.02702	0.100227
$106$	31.8304	3.09164
$107$	11.4992	1.11167	0.555834	−	0.831294i	$-0.312399\pi$
$107$	11.4992	1.11167	0.555834	+	0.831294i	$0.312399\pi$
$108$	−19.7721	−1.90257
$109$	−1.37478	−0.131680	−0.0658399	−	0.997830i	$-0.520973\pi$
$109$	−1.37478	−0.131680	−0.0658399	+	0.997830i	$0.520973\pi$
$110$	−1.79663	−0.171302
$111$	1.91263	0.181539
$112$	−3.36953	−0.318391
$113$	8.92213	0.839323	0.419662	−	0.907681i	$-0.362149\pi$
$113$	8.92213	0.839323	0.419662	+	0.907681i	$0.362149\pi$
$114$	11.8724	1.11196
$115$	0.0894125	0.00833776
$116$	40.4432	3.75505
$117$	1.67826	0.155155
$118$	23.2354	2.13899
$119$	−2.25071	−0.206323
$120$	−4.71958	−0.430837
$121$	−10.4523	−0.950205
$122$	15.4739	1.40094
$123$	1.03181	0.0930354
$124$	28.3032	2.54170
$125$	−1.00000	−0.0894427
$126$	4.72213	0.420681
$127$	0.785135	0.0696695	0.0348347	−	0.999393i	$-0.488910\pi$
$127$	0.785135	0.0696695	0.0348347	+	0.999393i	$0.488910\pi$
$128$	−20.2955	−1.79389
$129$	5.80402	0.511016
$130$	2.09439	0.183690
$131$	−4.24712	−0.371072	−0.185536	−	0.982637i	$-0.559402\pi$
$131$	−4.24712	−0.371072	−0.185536	+	0.982637i	$0.559402\pi$
$132$	2.95907	0.257554
$133$	−4.76201	−0.412919
$134$	−35.0937	−3.03164
$135$	5.07886	0.437119
$136$	10.3429	0.886898
$137$	21.6224	1.84733	0.923664	−	0.383204i	$-0.125179\pi$
$137$	21.6224	1.84733	0.923664	+	0.383204i	$0.125179\pi$
$138$	−0.222919	−0.0189762
$139$	−5.79999	−0.491949	−0.245974	−	0.969276i	$-0.579108\pi$
$139$	−5.79999	−0.491949	−0.245974	+	0.969276i	$0.579108\pi$
$140$	3.89301	0.329020
$141$	0.912046	0.0768081
$142$	−23.8506	−2.00149
$143$	−0.638525	−0.0533962
$144$	−6.55449	−0.546207
$145$	−10.3886	−0.862730
$146$	−34.1840	−2.82908
$147$	1.02702	0.0847075
$148$	7.24999	0.595945
$149$	−6.37255	−0.522060	−0.261030	−	0.965331i	$-0.584062\pi$
$149$	−6.37255	−0.522060	−0.261030	+	0.965331i	$0.584062\pi$
$150$	2.49316	0.203565
$151$	−22.6558	−1.84370	−0.921852	−	0.387541i	$-0.873325\pi$
$151$	−22.6558	−1.84370	−0.921852	+	0.387541i	$0.873325\pi$
$152$	21.8833	1.77497
$153$	−4.37814	−0.353951
$154$	−1.79663	−0.144776
$155$	−7.27024	−0.583960
$156$	−3.44950	−0.276181
$157$	0.332918	0.0265697	0.0132849	−	0.999912i	$-0.495771\pi$
$157$	0.332918	0.0265697	0.0132849	+	0.999912i	$0.495771\pi$
$158$	19.9229	1.58498
$159$	13.4665	1.06796
$160$	1.01107	0.0799318
$161$	0.0894125	0.00704669
$162$	1.50398	0.118164
$163$	10.2993	0.806700	0.403350	−	0.915046i	$-0.367846\pi$
$163$	10.2993	0.806700	0.403350	+	0.915046i	$0.367846\pi$
$164$	3.91117	0.305411
$165$	−0.760098	−0.0591736
$166$	12.8863	1.00017
$167$	11.7609	0.910085	0.455042	−	0.890470i	$-0.349624\pi$
$167$	11.7609	0.910085	0.455042	+	0.890470i	$0.349624\pi$
$168$	−4.71958	−0.364124
$169$	−12.2556	−0.942742
$170$	−5.46373	−0.419049
$171$	−9.26316	−0.708372
$172$	22.0006	1.67753
$173$	1.63055	0.123969	0.0619843	−	0.998077i	$-0.480257\pi$
$173$	1.63055	0.123969	0.0619843	+	0.998077i	$0.480257\pi$
$174$	25.9005	1.96351
$175$	−1.00000	−0.0755929
$176$	2.49378	0.187976
$177$	9.83021	0.738884
$178$	−29.0967	−2.18089
$179$	−17.9204	−1.33943	−0.669717	−	0.742616i	$-0.733585\pi$
$179$	−17.9204	−1.33943	−0.669717	+	0.742616i	$0.733585\pi$
$180$	7.57277	0.564441
$181$	8.17518	0.607656	0.303828	−	0.952727i	$-0.401735\pi$
$181$	8.17518	0.607656	0.303828	+	0.952727i	$0.401735\pi$
$182$	2.09439	0.155247
$183$	6.54656	0.483935
$184$	−0.410886	−0.0302909
$185$	−1.86231	−0.136919
$186$	18.1259	1.32905
$187$	1.66575	0.121812
$188$	3.45718	0.252141
$189$	5.07886	0.369433
$190$	−11.5600	−0.838653
$191$	−8.00026	−0.578878	−0.289439	−	0.957196i	$-0.593469\pi$
$191$	−8.00026	−0.578878	−0.289439	+	0.957196i	$0.593469\pi$
$192$	−9.44193	−0.681413
$193$	−16.3017	−1.17343	−0.586713	−	0.809795i	$-0.699578\pi$
$193$	−16.3017	−1.17343	−0.586713	+	0.809795i	$0.699578\pi$
$194$	41.2326	2.96033
$195$	0.886074	0.0634531
$196$	3.89301	0.278072
$197$	−6.85481	−0.488385	−0.244193	−	0.969727i	$-0.578523\pi$
$197$	−6.85481	−0.488385	−0.244193	+	0.969727i	$0.578523\pi$
$198$	−3.49484	−0.248367
$199$	1.90350	0.134935	0.0674677	−	0.997721i	$-0.478508\pi$
$199$	1.90350	0.134935	0.0674677	+	0.997721i	$0.478508\pi$
$200$	4.59539	0.324943
$201$	−14.8471	−1.04723
$202$	19.8152	1.39419
$203$	−10.3886	−0.729140
$204$	8.99885	0.630046
$205$	−1.00466	−0.0701687
$206$	−20.4113	−1.42212
$207$	0.173927	0.0120888
$208$	−2.90709	−0.201571
$209$	3.52435	0.243785
$210$	2.49316	0.172044
$211$	−10.3078	−0.709619	−0.354810	−	0.934939i	$-0.615454\pi$
$211$	−10.3078	−0.709619	−0.354810	+	0.934939i	$0.615454\pi$
$212$	51.0457	3.50583
$213$	−10.0904	−0.691386
$214$	27.9149	1.90822
$215$	−5.65130	−0.385416
$216$	−23.3394	−1.58804
$217$	−7.27024	−0.493536
$218$	−3.33734	−0.226033
$219$	−14.4622	−0.977265
$220$	−2.88121	−0.194251
$221$	−1.94182	−0.130621
$222$	4.64302	0.311619
$223$	15.4523	1.03476	0.517381	−	0.855755i	$-0.326907\pi$
$223$	15.4523	1.03476	0.517381	+	0.855755i	$0.326907\pi$
$224$	1.01107	0.0675547
$225$	−1.94522	−0.129681
$226$	21.6589	1.44073
$227$	−6.62030	−0.439405	−0.219702	−	0.975567i	$-0.570509\pi$
$227$	−6.62030	−0.439405	−0.219702	+	0.975567i	$0.570509\pi$
$228$	19.0396	1.26093
$229$	1.00000	0.0660819
$229$	1.00000	0.0660819
$230$	0.217054	0.0143121
$231$	−0.760098	−0.0500108
$232$	47.7399	3.13428
$233$	−5.37938	−0.352415	−0.176208	−	0.984353i	$-0.556383\pi$
$233$	−5.37938	−0.352415	−0.176208	+	0.984353i	$0.556383\pi$
$234$	4.07405	0.266329
$235$	−0.888047	−0.0579298
$236$	37.2622	2.42556
$237$	8.42879	0.547509
$238$	−5.46373	−0.354161
$239$	−19.9990	−1.29363	−0.646815	−	0.762647i	$-0.723899\pi$
$239$	−19.9990	−1.29363	−0.646815	+	0.762647i	$0.723899\pi$
$240$	−3.46059	−0.223380
$241$	−7.25801	−0.467530	−0.233765	−	0.972293i	$-0.575105\pi$
$241$	−7.25801	−0.467530	−0.233765	+	0.972293i	$0.575105\pi$
$242$	−25.3734	−1.63106
$243$	15.8729	1.01825
$244$	24.8152	1.58863
$245$	−1.00000	−0.0638877
$246$	2.50478	0.159699
$247$	−4.10846	−0.261415
$248$	33.4096	2.12151
$249$	5.45181	0.345494
$250$	−2.42755	−0.153532
$251$	12.3043	0.776644	0.388322	−	0.921524i	$-0.373055\pi$
$251$	12.3043	0.776644	0.388322	+	0.921524i	$0.373055\pi$
$252$	7.57277	0.477040
$253$	−0.0661740	−0.00416032
$254$	1.90596	0.119590
$255$	−2.31154	−0.144754
$256$	−30.8815	−1.93010
$257$	−6.09684	−0.380310	−0.190155	−	0.981754i	$-0.560899\pi$
$257$	−6.09684	−0.380310	−0.190155	+	0.981754i	$0.560899\pi$
$258$	14.0896	0.877179
$259$	−1.86231	−0.115718
$260$	3.35873	0.208300
$261$	−20.2082	−1.25086
$262$	−10.3101	−0.636960
$263$	−9.85239	−0.607525	−0.303762	−	0.952748i	$-0.598243\pi$
$263$	−9.85239	−0.607525	−0.303762	+	0.952748i	$0.598243\pi$
$264$	3.49295	0.214976
$265$	−13.1121	−0.805472
$266$	−11.5600	−0.708791
$267$	−12.3099	−0.753356
$268$	−56.2790	−3.43779
$269$	−2.05818	−0.125489	−0.0627446	−	0.998030i	$-0.519985\pi$
$269$	−2.05818	−0.125489	−0.0627446	+	0.998030i	$0.519985\pi$
$270$	12.3292	0.750332
$271$	−3.42212	−0.207879	−0.103939	−	0.994584i	$-0.533145\pi$
$271$	−3.42212	−0.207879	−0.103939	+	0.994584i	$0.533145\pi$
$272$	7.58386	0.459839
$273$	0.886074	0.0536276
$274$	52.4896	3.17101
$275$	0.740098	0.0446296
$276$	−0.357491	−0.0215184
$277$	32.3702	1.94494	0.972469	−	0.233034i	$-0.0748653\pi$
$277$	32.3702	1.94494	0.972469	+	0.233034i	$0.0748653\pi$
$278$	−14.0798	−0.844449
$279$	−14.1422	−0.846673
$280$	4.59539	0.274627
$281$	1.93110	0.115200	0.0576000	−	0.998340i	$-0.481655\pi$
$281$	1.93110	0.115200	0.0576000	+	0.998340i	$0.481655\pi$
$282$	2.21404	0.131844
$283$	19.5305	1.16097	0.580483	−	0.814273i	$-0.302864\pi$
$283$	19.5305	1.16097	0.580483	+	0.814273i	$0.302864\pi$
$284$	−38.2486	−2.26964
$285$	−4.89070	−0.289700
$286$	−1.55005	−0.0916566
$287$	−1.00466	−0.0593034
$288$	1.96675	0.115892
$289$	−11.9343	−0.702017
$290$	−25.2190	−1.48091
$291$	17.4443	1.02260
$292$	−54.8201	−3.20810
$293$	−25.2603	−1.47572	−0.737862	−	0.674952i	$-0.764164\pi$
$293$	−25.2603	−1.47572	−0.737862	+	0.674952i	$0.764164\pi$
$294$	2.49316	0.145404
$295$	−9.57154	−0.557277
$296$	8.55803	0.497425
$297$	−3.75885	−0.218111
$298$	−15.4697	−0.896136
$299$	0.0771414	0.00446120
$300$	3.99822	0.230837
$301$	−5.65130	−0.325736
$302$	−54.9982	−3.16479
$303$	8.38320	0.481602
$304$	16.0458	0.920287
$305$	−6.37429	−0.364991
$306$	−10.6282	−0.607572
$307$	11.1238	0.634867	0.317434	−	0.948280i	$-0.397179\pi$
$307$	11.1238	0.634867	0.317434	+	0.948280i	$0.397179\pi$
$308$	−2.88121	−0.164172
$309$	−8.63541	−0.491251
$310$	−17.6489	−1.00239
$311$	27.6045	1.56531	0.782654	−	0.622457i	$-0.213866\pi$
$311$	27.6045	1.56531	0.782654	+	0.622457i	$0.213866\pi$
$312$	−4.07186	−0.230524
$313$	−18.7243	−1.05836	−0.529181	−	0.848509i	$-0.677501\pi$
$313$	−18.7243	−1.05836	−0.529181	+	0.848509i	$0.677501\pi$
$314$	0.808175	0.0456080
$315$	−1.94522	−0.109601
$316$	31.9500	1.79733
$317$	−1.46196	−0.0821117	−0.0410558	−	0.999157i	$-0.513072\pi$
$317$	−1.46196	−0.0821117	−0.0410558	+	0.999157i	$0.513072\pi$
$318$	32.6906	1.83320
$319$	7.68861	0.430480
$320$	9.19348	0.513931
$321$	11.8099	0.659166
$322$	0.217054	0.0120959
$323$	10.7179	0.596361
$324$	2.41191	0.133995
$325$	−0.862758	−0.0478572
$326$	25.0020	1.38473
$327$	−1.41193	−0.0780799
$328$	4.61682	0.254921
$329$	−0.888047	−0.0489596
$330$	−1.84518	−0.101574
$331$	10.0826	0.554190	0.277095	−	0.960843i	$-0.410628\pi$
$331$	10.0826	0.554190	0.277095	+	0.960843i	$0.410628\pi$
$332$	20.6655	1.13417
$333$	−3.62260	−0.198517
$334$	28.5502	1.56220
$335$	14.4564	0.789838
$336$	−3.46059	−0.188791
$337$	−16.4132	−0.894086	−0.447043	−	0.894512i	$-0.647523\pi$
$337$	−16.4132	−0.894086	−0.447043	+	0.894512i	$0.647523\pi$
$338$	−29.7512	−1.61825
$339$	9.16324	0.497679
$340$	−8.76206	−0.475190
$341$	5.38069	0.291381
$342$	−22.4868	−1.21595
$343$	−1.00000	−0.0539949
$344$	25.9700	1.40021
$345$	0.0918288	0.00494390
$346$	3.95825	0.212797
$347$	−31.8614	−1.71041	−0.855204	−	0.518291i	$-0.826568\pi$
$347$	−31.8614	−1.71041	−0.855204	+	0.518291i	$0.826568\pi$
$348$	41.5361	2.22657
$349$	14.7371	0.788858	0.394429	−	0.918926i	$-0.370942\pi$
$349$	14.7371	0.788858	0.394429	+	0.918926i	$0.370942\pi$
$350$	−2.42755	−0.129758
$351$	4.38183	0.233885
$352$	−0.748287	−0.0398839
$353$	32.6134	1.73584	0.867919	−	0.496706i	$-0.165457\pi$
$353$	32.6134	1.73584	0.867919	+	0.496706i	$0.165457\pi$
$354$	23.8634	1.26832
$355$	9.82493	0.521453
$356$	−46.6618	−2.47307
$357$	−2.31154	−0.122340
$358$	−43.5027	−2.29919
$359$	−33.7478	−1.78114	−0.890569	−	0.454848i	$-0.849694\pi$
$359$	−33.7478	−1.78114	−0.890569	+	0.454848i	$0.849694\pi$
$360$	8.93906	0.471130
$361$	3.67674	0.193513
$362$	19.8457	1.04307
$363$	−10.7347	−0.563427
$364$	3.35873	0.176045
$365$	14.0816	0.737067
$366$	15.8921	0.830694
$367$	27.9198	1.45740	0.728701	−	0.684831i	$-0.240124\pi$
$367$	27.9198	1.45740	0.728701	+	0.684831i	$0.240124\pi$
$368$	−0.301278	−0.0157052
$369$	−1.95429	−0.101736
$370$	−4.52085	−0.235028
$371$	−13.1121	−0.680748
$372$	29.0680	1.50711
$373$	−0.956503	−0.0495258	−0.0247629	−	0.999693i	$-0.507883\pi$
$373$	−0.956503	−0.0495258	−0.0247629	+	0.999693i	$0.507883\pi$
$374$	4.04369	0.209094
$375$	−1.02702	−0.0530353
$376$	4.08093	0.210458
$377$	−8.96289	−0.461612
$378$	12.3292	0.634146
$379$	25.1473	1.29173	0.645864	−	0.763452i	$-0.276497\pi$
$379$	25.1473	1.29173	0.645864	+	0.763452i	$0.276497\pi$
$380$	−18.5386	−0.951009
$381$	0.806353	0.0413107
$382$	−19.4211	−0.993667
$383$	−27.6762	−1.41419	−0.707095	−	0.707118i	$-0.749995\pi$
$383$	−27.6762	−1.41419	−0.707095	+	0.707118i	$0.749995\pi$
$384$	−20.8440	−1.06369
$385$	0.740098	0.0377189
$386$	−39.5734	−2.01423
$387$	−10.9930	−0.558807
$388$	66.1238	3.35693
$389$	7.36030	0.373182	0.186591	−	0.982438i	$-0.440256\pi$
$389$	7.36030	0.373182	0.186591	+	0.982438i	$0.440256\pi$
$390$	2.15099	0.108920
$391$	−0.201242	−0.0101772
$392$	4.59539	0.232102
$393$	−4.36189	−0.220028
$394$	−16.6404	−0.838333
$395$	−8.20700	−0.412939
$396$	−5.60459	−0.281641
$397$	−27.2854	−1.36941	−0.684707	−	0.728819i	$-0.740070\pi$
$397$	−27.2854	−1.36941	−0.684707	+	0.728819i	$0.740070\pi$
$398$	4.62084	0.231622
$399$	−4.89070	−0.244841
$400$	3.36953	0.168477
$401$	17.8903	0.893401	0.446700	−	0.894684i	$-0.352599\pi$
$401$	17.8903	0.893401	0.446700	+	0.894684i	$0.352599\pi$
$402$	−36.0421	−1.79762
$403$	−6.27246	−0.312454
$404$	31.7772	1.58097
$405$	−0.619547	−0.0307855
$406$	−25.2190	−1.25160
$407$	1.37829	0.0683192
$408$	10.6224	0.525889
$409$	9.77211	0.483200	0.241600	−	0.970376i	$-0.422328\pi$
$409$	9.77211	0.483200	0.241600	+	0.970376i	$0.422328\pi$
$410$	−2.43887	−0.120447
$411$	22.2067	1.09538
$412$	−32.7332	−1.61265
$413$	−9.57154	−0.470985
$414$	0.422217	0.0207508
$415$	−5.30835	−0.260577
$416$	0.872305	0.0427683
$417$	−5.95673	−0.291702
$418$	8.55555	0.418466
$419$	−5.52138	−0.269737	−0.134868	−	0.990864i	$-0.543061\pi$
$419$	−5.52138	−0.269737	−0.134868	+	0.990864i	$0.543061\pi$
$420$	3.99822	0.195093
$421$	2.37531	0.115766	0.0578828	−	0.998323i	$-0.481565\pi$
$421$	2.37531	0.115766	0.0578828	+	0.998323i	$0.481565\pi$
$422$	−25.0228	−1.21809
$423$	−1.72745	−0.0839914
$424$	60.2554	2.92626
$425$	2.25071	0.109176
$426$	−24.4951	−1.18679
$427$	−6.37429	−0.308474
$428$	44.7664	2.16387
$429$	−0.655781	−0.0316614
$430$	−13.7188	−0.661581
$431$	−15.0116	−0.723085	−0.361542	−	0.932356i	$-0.617750\pi$
$431$	−15.0116	−0.723085	−0.361542	+	0.932356i	$0.617750\pi$
$432$	−17.1134	−0.823369
$433$	−5.97353	−0.287069	−0.143535	−	0.989645i	$-0.545847\pi$
$433$	−5.97353	−0.287069	−0.143535	+	0.989645i	$0.545847\pi$
$434$	−17.6489	−0.847174
$435$	−10.6694	−0.511558
$436$	−5.35203	−0.256316
$437$	−0.425783	−0.0203680
$438$	−35.1078	−1.67751
$439$	−35.1370	−1.67700	−0.838498	−	0.544905i	$-0.816566\pi$
$439$	−35.1370	−1.67700	−0.838498	+	0.544905i	$0.816566\pi$
$440$	−3.40104	−0.162138
$441$	−1.94522	−0.0926296
$442$	−4.71388	−0.224216
$443$	−14.8550	−0.705783	−0.352892	−	0.935664i	$-0.614802\pi$
$443$	−14.8550	−0.705783	−0.352892	+	0.935664i	$0.614802\pi$
$444$	7.44591	0.353367
$445$	11.9860	0.568192
$446$	37.5112	1.77621
$447$	−6.54477	−0.309557
$448$	9.19348	0.434351
$449$	13.3464	0.629855	0.314927	−	0.949116i	$-0.398020\pi$
$449$	13.3464	0.629855	0.314927	+	0.949116i	$0.398020\pi$
$450$	−4.72213	−0.222603
$451$	0.743548	0.0350123
$452$	34.7340	1.63375
$453$	−23.2681	−1.09323
$454$	−16.0711	−0.754256
$455$	−0.862758	−0.0404467
$456$	22.4747	1.05247
$457$	15.9288	0.745118	0.372559	−	0.928009i	$-0.378480\pi$
$457$	15.9288	0.745118	0.372559	+	0.928009i	$0.378480\pi$
$458$	2.42755	0.113432
$459$	−11.4311	−0.533557
$460$	0.348084	0.0162295
$461$	28.4199	1.32365	0.661824	−	0.749659i	$-0.269783\pi$
$461$	28.4199	1.32365	0.661824	+	0.749659i	$0.269783\pi$
$462$	−1.84518	−0.0858455
$463$	9.35157	0.434605	0.217302	−	0.976104i	$-0.430274\pi$
$463$	9.35157	0.434605	0.217302	+	0.976104i	$0.430274\pi$
$464$	35.0049	1.62506
$465$	−7.46672	−0.346261
$466$	−13.0587	−0.604934
$467$	−3.27005	−0.151320	−0.0756598	−	0.997134i	$-0.524106\pi$
$467$	−3.27005	−0.151320	−0.0756598	+	0.997134i	$0.524106\pi$
$468$	6.53347	0.302010
$469$	14.4564	0.667535
$470$	−2.15578	−0.0994388
$471$	0.341915	0.0157546
$472$	43.9850	2.02457
$473$	4.18252	0.192312
$474$	20.4613	0.939820
$475$	4.76201	0.218496
$476$	−8.76206	−0.401609
$477$	−25.5060	−1.16784
$478$	−48.5487	−2.22057
$479$	18.5245	0.846408	0.423204	−	0.906034i	$-0.360905\pi$
$479$	18.5245	0.846408	0.423204	+	0.906034i	$0.360905\pi$
$480$	1.03839	0.0473958
$481$	−1.60672	−0.0732601
$482$	−17.6192	−0.802533
$483$	0.0918288	0.00417836
$484$	−40.6908	−1.84958
$485$	−16.9852	−0.771260
$486$	38.5323	1.74786
$487$	−9.59755	−0.434907	−0.217453	−	0.976071i	$-0.569775\pi$
$487$	−9.59755	−0.434907	−0.217453	+	0.976071i	$0.569775\pi$
$488$	29.2924	1.32600
$489$	10.5776	0.478335
$490$	−2.42755	−0.109666
$491$	−20.5181	−0.925968	−0.462984	−	0.886367i	$-0.653221\pi$
$491$	−20.5181	−0.925968	−0.462984	+	0.886367i	$0.653221\pi$
$492$	4.01686	0.181094
$493$	23.3819	1.05307
$494$	−9.97352	−0.448730
$495$	1.43965	0.0647076
$496$	24.4973	1.09996
$497$	9.82493	0.440709
$498$	13.2346	0.593055
$499$	10.6166	0.475264	0.237632	−	0.971355i	$-0.423629\pi$
$499$	10.6166	0.475264	0.237632	+	0.971355i	$0.423629\pi$
$500$	−3.89301	−0.174101
$501$	12.0787	0.539637
$502$	29.8695	1.33314
$503$	−19.1511	−0.853906	−0.426953	−	0.904274i	$-0.640413\pi$
$503$	−19.1511	−0.853906	−0.426953	+	0.904274i	$0.640413\pi$
$504$	8.93906	0.398177
$505$	−8.16261	−0.363231
$506$	−0.160641	−0.00714136
$507$	−12.5868	−0.559002
$508$	3.05654	0.135612
$509$	−7.65209	−0.339173	−0.169586	−	0.985515i	$-0.554243\pi$
$509$	−7.65209	−0.339173	−0.169586	+	0.985515i	$0.554243\pi$
$510$	−5.61138	−0.248476
$511$	14.0816	0.622935
$512$	−34.3755	−1.51920
$513$	−24.1856	−1.06782
$514$	−14.8004	−0.652818
$515$	8.40818	0.370509
$516$	22.5952	0.994696
$517$	0.657242	0.0289055
$518$	−4.52085	−0.198635
$519$	1.67462	0.0735076
$520$	3.96471	0.173864
$521$	−26.2816	−1.15142	−0.575708	−	0.817655i	$-0.695274\pi$
$521$	−26.2816	−1.15142	−0.575708	+	0.817655i	$0.695274\pi$
$522$	−49.0565	−2.14715
$523$	12.2118	0.533986	0.266993	−	0.963698i	$-0.413970\pi$
$523$	12.2118	0.533986	0.266993	+	0.963698i	$0.413970\pi$
$524$	−16.5341	−0.722295
$525$	−1.02702	−0.0448230
$526$	−23.9172	−1.04284
$527$	16.3632	0.712794
$528$	2.56118	0.111461
$529$	−22.9920	−0.999652
$530$	−31.8304	−1.38262
$531$	−18.6188	−0.807986
$532$	−18.5386	−0.803749
$533$	−0.866781	−0.0375445
$534$	−29.8830	−1.29317
$535$	−11.4992	−0.497153
$536$	−66.4329	−2.86947
$537$	−18.4047	−0.794221
$538$	−4.99633	−0.215407
$539$	0.740098	0.0318783
$540$	19.7721	0.850855
$541$	23.8332	1.02467	0.512334	−	0.858786i	$-0.328781\pi$
$541$	23.8332	1.02467	0.512334	+	0.858786i	$0.328781\pi$
$542$	−8.30737	−0.356832
$543$	8.39611	0.360312
$544$	−2.27562	−0.0975664
$545$	1.37478	0.0588890
$546$	2.15099	0.0920539
$547$	31.5100	1.34727	0.673636	−	0.739063i	$-0.264732\pi$
$547$	31.5100	1.34727	0.673636	+	0.739063i	$0.264732\pi$
$548$	84.1764	3.59584
$549$	−12.3994	−0.529194
$550$	1.79663	0.0766084
$551$	49.4708	2.10753
$552$	−0.421990	−0.0179611
$553$	−8.20700	−0.348997
$554$	78.5804	3.33856
$555$	−1.91263	−0.0811868
$556$	−22.5794	−0.957582
$557$	15.8148	0.670096	0.335048	−	0.942201i	$-0.391248\pi$
$557$	15.8148	0.670096	0.335048	+	0.942201i	$0.391248\pi$
$558$	−34.3310	−1.45335
$559$	−4.87571	−0.206220
$560$	3.36953	0.142389
$561$	1.71076	0.0722285
$562$	4.68785	0.197745
$563$	31.3699	1.32208	0.661041	−	0.750350i	$-0.270115\pi$
$563$	31.3699	1.32208	0.661041	+	0.750350i	$0.270115\pi$
$564$	3.55061	0.149508
$565$	−8.92213	−0.375357
$566$	47.4112	1.99284
$567$	−0.619547	−0.0260185
$568$	−45.1494	−1.89443
$569$	−43.9884	−1.84409	−0.922044	−	0.387085i	$-0.873482\pi$
$569$	−43.9884	−1.84409	−0.922044	+	0.387085i	$0.873482\pi$
$570$	−11.8724	−0.497282
$571$	−31.3586	−1.31231	−0.656157	−	0.754624i	$-0.727819\pi$
$571$	−31.3586	−1.31231	−0.656157	+	0.754624i	$0.727819\pi$
$572$	−2.48579	−0.103936
$573$	−8.21646	−0.343248
$574$	−2.43887	−0.101797
$575$	−0.0894125	−0.00372876
$576$	17.8834	0.745140
$577$	−8.86781	−0.369172	−0.184586	−	0.982816i	$-0.559094\pi$
$577$	−8.86781	−0.369172	−0.184586	+	0.982816i	$0.559094\pi$
$578$	−28.9711	−1.20504
$579$	−16.7423	−0.695786
$580$	−40.4432	−1.67931
$581$	−5.30835	−0.220228
$582$	42.3469	1.75533
$583$	9.70426	0.401909
$584$	−64.7107	−2.67775
$585$	−1.67826	−0.0693873
$586$	−61.3208	−2.53314
$587$	36.7954	1.51871	0.759354	−	0.650678i	$-0.225515\pi$
$587$	36.7954	1.51871	0.759354	+	0.650678i	$0.225515\pi$
$588$	3.99822	0.164884
$589$	34.6210	1.42653
$590$	−23.2354	−0.956588
$591$	−7.04006	−0.289589
$592$	6.27510	0.257905
$593$	−10.1339	−0.416151	−0.208075	−	0.978113i	$-0.566720\pi$
$593$	−10.1339	−0.416151	−0.208075	+	0.978113i	$0.566720\pi$
$594$	−9.12482	−0.374396
$595$	2.25071	0.0922703
$596$	−24.8084	−1.01619
$597$	1.95494	0.0800103
$598$	0.187265	0.00765783
$599$	−40.0591	−1.63677	−0.818385	−	0.574670i	$-0.805130\pi$
$599$	−40.0591	−1.63677	−0.818385	+	0.574670i	$0.805130\pi$
$600$	4.71958	0.192676
$601$	−20.7435	−0.846145	−0.423073	−	0.906096i	$-0.639048\pi$
$601$	−20.7435	−0.846145	−0.423073	+	0.906096i	$0.639048\pi$
$602$	−13.7188	−0.559138
$603$	28.1209	1.14517
$604$	−88.1995	−3.58878
$605$	10.4523	0.424945
$606$	20.3507	0.826689
$607$	−26.5072	−1.07589	−0.537947	−	0.842978i	$-0.680800\pi$
$607$	−26.5072	−1.07589	−0.537947	+	0.842978i	$0.680800\pi$
$608$	−4.81471	−0.195262
$609$	−10.6694	−0.432346
$610$	−15.4739	−0.626522
$611$	−0.766170	−0.0309959
$612$	−17.0442	−0.688969
$613$	9.87842	0.398986	0.199493	−	0.979899i	$-0.436071\pi$
$613$	9.87842	0.398986	0.199493	+	0.979899i	$0.436071\pi$
$614$	27.0036	1.08977
$615$	−1.03181	−0.0416067
$616$	−3.40104	−0.137032
$617$	12.0395	0.484692	0.242346	−	0.970190i	$-0.422083\pi$
$617$	12.0395	0.484692	0.242346	+	0.970190i	$0.422083\pi$
$618$	−20.9629	−0.843252
$619$	−13.2428	−0.532275	−0.266137	−	0.963935i	$-0.585747\pi$
$619$	−13.2428	−0.532275	−0.266137	+	0.963935i	$0.585747\pi$
$620$	−28.3032	−1.13668
$621$	0.454114	0.0182230
$622$	67.0114	2.68691
$623$	11.9860	0.480210
$624$	−2.98566	−0.119522
$625$	1.00000	0.0400000
$626$	−45.4543	−1.81672
$627$	3.61960	0.144553
$628$	1.29605	0.0517182
$629$	4.19152	0.167127
$630$	−4.72213	−0.188134
$631$	−1.71942	−0.0684492	−0.0342246	−	0.999414i	$-0.510896\pi$
$631$	−1.71942	−0.0684492	−0.0342246	+	0.999414i	$0.510896\pi$
$632$	37.7144	1.50020
$633$	−10.5864	−0.420771
$634$	−3.54898	−0.140948
$635$	−0.785135	−0.0311571
$636$	52.4252	2.07879
$637$	−0.862758	−0.0341837
$638$	18.6645	0.738935
$639$	19.1117	0.756046
$640$	20.2955	0.802252
$641$	−12.9327	−0.510811	−0.255405	−	0.966834i	$-0.582209\pi$
$641$	−12.9327	−0.510811	−0.255405	+	0.966834i	$0.582209\pi$
$642$	28.6692	1.13148
$643$	22.2330	0.876782	0.438391	−	0.898784i	$-0.355549\pi$
$643$	22.2330	0.876782	0.438391	+	0.898784i	$0.355549\pi$
$644$	0.348084	0.0137164
$645$	−5.80402	−0.228533
$646$	26.0183	1.02368
$647$	4.19045	0.164744	0.0823718	−	0.996602i	$-0.473751\pi$
$647$	4.19045	0.164744	0.0823718	+	0.996602i	$0.473751\pi$
$648$	2.84706	0.111843
$649$	7.08388	0.278067
$650$	−2.09439	−0.0821488
$651$	−7.46672	−0.292644
$652$	40.0952	1.57025
$653$	−22.2995	−0.872646	−0.436323	−	0.899790i	$-0.643720\pi$
$653$	−22.2995	−0.872646	−0.436323	+	0.899790i	$0.643720\pi$
$654$	−3.42753	−0.134027
$655$	4.24712	0.165949
$656$	3.38524	0.132172
$657$	27.3919	1.06866
$658$	−2.15578	−0.0840411
$659$	23.4989	0.915389	0.457694	−	0.889110i	$-0.348675\pi$
$659$	23.4989	0.915389	0.457694	+	0.889110i	$0.348675\pi$
$660$	−2.95907	−0.115182
$661$	28.0632	1.09153	0.545767	−	0.837937i	$-0.316239\pi$
$661$	28.0632	1.09153	0.545767	+	0.837937i	$0.316239\pi$
$662$	24.4760	0.951288
$663$	−1.99430	−0.0774522
$664$	24.3940	0.946670
$665$	4.76201	0.184663
$666$	−8.79405	−0.340762
$667$	−0.928875	−0.0359662
$668$	45.7853	1.77149
$669$	15.8699	0.613565
$670$	35.0937	1.35579
$671$	4.71760	0.182121
$672$	1.03839	0.0400567
$673$	−32.3781	−1.24809	−0.624043	−	0.781390i	$-0.714511\pi$
$673$	−32.3781	−1.24809	−0.624043	+	0.781390i	$0.714511\pi$
$674$	−39.8440	−1.53473
$675$	−5.07886	−0.195486
$676$	−47.7114	−1.83505
$677$	41.4384	1.59261	0.796303	−	0.604898i	$-0.206786\pi$
$677$	41.4384	1.59261	0.796303	+	0.604898i	$0.206786\pi$
$678$	22.2443	0.854286
$679$	−16.9852	−0.651834
$680$	−10.3429	−0.396633
$681$	−6.79921	−0.260546
$682$	13.0619	0.500166
$683$	−9.20148	−0.352085	−0.176042	−	0.984383i	$-0.556330\pi$
$683$	−9.20148	−0.352085	−0.176042	+	0.984383i	$0.556330\pi$
$684$	−36.0616	−1.37885
$685$	−21.6224	−0.826150
$686$	−2.42755	−0.0926844
$687$	1.02702	0.0391834
$688$	19.0422	0.725979
$689$	−11.3126	−0.430976
$690$	0.222919	0.00848639
$691$	45.6389	1.73619	0.868093	−	0.496402i	$-0.165346\pi$
$691$	45.6389	1.73619	0.868093	+	0.496402i	$0.165346\pi$
$692$	6.34777	0.241306
$693$	1.43965	0.0546879
$694$	−77.3452	−2.93598
$695$	5.79999	0.220006
$696$	49.0301	1.85848
$697$	2.26121	0.0856494
$698$	35.7751	1.35411
$699$	−5.52476	−0.208966
$700$	−3.89301	−0.147142
$701$	−38.9258	−1.47021	−0.735104	−	0.677954i	$-0.762867\pi$
$701$	−38.9258	−1.47021	−0.735104	+	0.677954i	$0.762867\pi$
$702$	10.6371	0.401473
$703$	8.86832	0.334475
$704$	−6.80408	−0.256438
$705$	−0.912046	−0.0343496
$706$	79.1709	2.97963
$707$	−8.16261	−0.306987
$708$	38.2691	1.43824
$709$	−41.2626	−1.54965	−0.774825	−	0.632176i	$-0.782162\pi$
$709$	−41.2626	−1.54965	−0.774825	+	0.632176i	$0.782162\pi$
$710$	23.8506	0.895095
$711$	−15.9644	−0.598713
$712$	−55.0805	−2.06423
$713$	−0.650051	−0.0243446
$714$	−5.61138	−0.210001
$715$	0.638525	0.0238795
$716$	−69.7644	−2.60722
$717$	−20.5395	−0.767061
$718$	−81.9245	−3.05739
$719$	12.8627	0.479699	0.239849	−	0.970810i	$-0.422902\pi$
$719$	12.8627	0.479699	0.239849	+	0.970810i	$0.422902\pi$
$720$	6.55449	0.244271
$721$	8.40818	0.313137
$722$	8.92549	0.332172
$723$	−7.45416	−0.277223
$724$	31.8261	1.18281
$725$	10.3886	0.385825
$726$	−26.0591	−0.967144
$727$	7.79574	0.289128	0.144564	−	0.989495i	$-0.453822\pi$
$727$	7.79574	0.289128	0.144564	+	0.989495i	$0.453822\pi$
$728$	3.96471	0.146942
$729$	14.4432	0.534933
$730$	34.1840	1.26520
$731$	12.7195	0.470446
$732$	25.4858	0.941984
$733$	−9.51550	−0.351463	−0.175732	−	0.984438i	$-0.556229\pi$
$733$	−9.51550	−0.351463	−0.175732	+	0.984438i	$0.556229\pi$
$734$	67.7769	2.50169
$735$	−1.02702	−0.0378824
$736$	0.0904019	0.00333226
$737$	−10.6992	−0.394109
$738$	−4.74414	−0.174634
$739$	18.8977	0.695165	0.347582	−	0.937649i	$-0.387003\pi$
$739$	18.8977	0.695165	0.347582	+	0.937649i	$0.387003\pi$
$740$	−7.24999	−0.266515
$741$	−4.21949	−0.155007
$742$	−31.8304	−1.16853
$743$	45.6012	1.67294	0.836472	−	0.548010i	$-0.184614\pi$
$743$	45.6012	1.67294	0.836472	+	0.548010i	$0.184614\pi$
$744$	34.3125	1.25796
$745$	6.37255	0.233472
$746$	−2.32196	−0.0850130
$747$	−10.3259	−0.377806
$748$	6.48478	0.237107
$749$	−11.4992	−0.420171
$750$	−2.49316	−0.0910372
$751$	13.4090	0.489302	0.244651	−	0.969611i	$-0.421327\pi$
$751$	13.4090	0.489302	0.244651	+	0.969611i	$0.421327\pi$
$752$	2.99230	0.109118
$753$	12.6369	0.460513
$754$	−21.7579	−0.792376
$755$	22.6558	0.824530
$756$	19.7721	0.719104
$757$	−36.5920	−1.32996	−0.664979	−	0.746862i	$-0.731560\pi$
$757$	−36.5920	−1.32996	−0.664979	+	0.746862i	$0.731560\pi$
$758$	61.0463	2.21730
$759$	−0.0679623	−0.00246687
$760$	−21.8833	−0.793791
$761$	51.2608	1.85820	0.929101	−	0.369827i	$-0.120583\pi$
$761$	51.2608	1.85820	0.929101	+	0.369827i	$0.120583\pi$
$762$	1.95746	0.0709115
$763$	1.37478	0.0497703
$764$	−31.1451	−1.12679
$765$	4.37814	0.158292
$766$	−67.1856	−2.42751
$767$	−8.25793	−0.298177
$768$	−31.7161	−1.14446
$769$	−29.3683	−1.05905	−0.529524	−	0.848295i	$-0.677629\pi$
$769$	−29.3683	−1.05905	−0.529524	+	0.848295i	$0.677629\pi$
$770$	1.79663	0.0647459
$771$	−6.26160	−0.225506
$772$	−63.4629	−2.28408
$773$	−13.6105	−0.489537	−0.244769	−	0.969582i	$-0.578712\pi$
$773$	−13.6105	−0.489537	−0.244769	+	0.969582i	$0.578712\pi$
$774$	−26.6862	−0.959214
$775$	7.27024	0.261155
$776$	78.0539	2.80197
$777$	−1.91263	−0.0686154
$778$	17.8675	0.640581
$779$	4.78421	0.171412
$780$	3.44950	0.123512
$781$	−7.27141	−0.260192
$782$	−0.488526	−0.0174696
$783$	−52.7625	−1.88558
$784$	3.36953	0.120340
$785$	−0.332918	−0.0118823
$786$	−10.5887	−0.377687
$787$	6.09675	0.217326	0.108663	−	0.994079i	$-0.465343\pi$
$787$	6.09675	0.217326	0.108663	+	0.994079i	$0.465343\pi$
$788$	−26.6859	−0.950646
$789$	−10.1186	−0.360233
$790$	−19.9229	−0.708826
$791$	−8.92213	−0.317234
$792$	−6.61577	−0.235081
$793$	−5.49948	−0.195292
$794$	−66.2367	−2.35065
$795$	−13.4665	−0.477607
$796$	7.41035	0.262653
$797$	1.10018	0.0389702	0.0194851	−	0.999810i	$-0.493797\pi$
$797$	1.10018	0.0389702	0.0194851	+	0.999810i	$0.493797\pi$
$798$	−11.8724	−0.420280
$799$	1.99874	0.0707104
$800$	−1.01107	−0.0357466
$801$	23.3155	0.823812
$802$	43.4297	1.53356
$803$	−10.4218	−0.367777
$804$	−57.7999	−2.03845
$805$	−0.0894125	−0.00315138
$806$	−15.2267	−0.536339
$807$	−2.11380	−0.0744092
$808$	37.5104	1.31961
$809$	−6.31862	−0.222151	−0.111076	−	0.993812i	$-0.535430\pi$
$809$	−6.31862	−0.222151	−0.111076	+	0.993812i	$0.535430\pi$
$810$	−1.50398	−0.0528446
$811$	−19.3061	−0.677929	−0.338964	−	0.940799i	$-0.610077\pi$
$811$	−19.3061	−0.677929	−0.338964	+	0.940799i	$0.610077\pi$
$812$	−40.4432	−1.41928
$813$	−3.51460	−0.123262
$814$	3.34587	0.117273
$815$	−10.2993	−0.360767
$816$	7.78881	0.272663
$817$	26.9116	0.941516
$818$	23.7223	0.829431
$819$	−1.67826	−0.0586430
$820$	−3.91117	−0.136584
$821$	−42.5637	−1.48548	−0.742741	−	0.669579i	$-0.766475\pi$
$821$	−42.5637	−1.48548	−0.742741	+	0.669579i	$0.766475\pi$
$822$	53.9081	1.88026
$823$	−47.9332	−1.67084	−0.835422	−	0.549608i	$-0.814777\pi$
$823$	−47.9332	−1.67084	−0.835422	+	0.549608i	$0.814777\pi$
$824$	−38.6389	−1.34605
$825$	0.760098	0.0264632
$826$	−23.2354	−0.808464
$827$	2.82311	0.0981690	0.0490845	−	0.998795i	$-0.484370\pi$
$827$	2.82311	0.0981690	0.0490845	+	0.998795i	$0.484370\pi$
$828$	0.677101	0.0235309
$829$	54.3745	1.88851	0.944253	−	0.329221i	$-0.106786\pi$
$829$	54.3745	1.88851	0.944253	+	0.329221i	$0.106786\pi$
$830$	−12.8863	−0.447290
$831$	33.2450	1.15326
$832$	7.93175	0.274984
$833$	2.25071	0.0779826
$834$	−14.4603	−0.500719
$835$	−11.7609	−0.407002
$836$	13.7204	0.474528
$837$	−36.9246	−1.27630
$838$	−13.4034	−0.463014
$839$	−10.8966	−0.376193	−0.188097	−	0.982151i	$-0.560232\pi$
$839$	−10.8966	−0.376193	−0.188097	+	0.982151i	$0.560232\pi$
$840$	4.71958	0.162841
$841$	78.9240	2.72152
$842$	5.76619	0.198716
$843$	1.98329	0.0683081
$844$	−40.1285	−1.38128
$845$	12.2556	0.421607
$846$	−4.19347	−0.144175
$847$	10.4523	0.359144
$848$	44.1818	1.51721
$849$	20.0583	0.688397
$850$	5.46373	0.187404
$851$	−0.166513	−0.00570801
$852$	−39.2823	−1.34579
$853$	8.16287	0.279491	0.139746	−	0.990187i	$-0.455372\pi$
$853$	8.16287	0.279491	0.139746	+	0.990187i	$0.455372\pi$
$854$	−15.4739	−0.529507
$855$	9.26316	0.316793
$856$	52.8432	1.80614
$857$	1.47516	0.0503906	0.0251953	−	0.999683i	$-0.491979\pi$
$857$	1.47516	0.0503906	0.0251953	+	0.999683i	$0.491979\pi$
$858$	−1.59194	−0.0543481
$859$	11.2257	0.383016	0.191508	−	0.981491i	$-0.438662\pi$
$859$	11.2257	0.383016	0.191508	+	0.981491i	$0.438662\pi$
$860$	−22.0006	−0.750214
$861$	−1.03181	−0.0351641
$862$	−36.4415	−1.24120
$863$	5.66106	0.192705	0.0963524	−	0.995347i	$-0.469282\pi$
$863$	5.66106	0.192705	0.0963524	+	0.995347i	$0.469282\pi$
$864$	5.13506	0.174698
$865$	−1.63055	−0.0554405
$866$	−14.5011	−0.492766
$867$	−12.2568	−0.416263
$868$	−28.3032	−0.960672
$869$	6.07398	0.206046
$870$	−25.9005	−0.878110
$871$	12.4724	0.422611
$872$	−6.31764	−0.213942
$873$	−33.0400	−1.11824
$874$	−1.03361	−0.0349624
$875$	1.00000	0.0338062
$876$	−56.3015	−1.90225
$877$	2.87821	0.0971901	0.0485951	−	0.998819i	$-0.484526\pi$
$877$	2.87821	0.0971901	0.0485951	+	0.998819i	$0.484526\pi$
$878$	−85.2968	−2.87863
$879$	−25.9430	−0.875034
$880$	−2.49378	−0.0840654
$881$	−26.6780	−0.898805	−0.449403	−	0.893329i	$-0.648363\pi$
$881$	−26.6780	−0.898805	−0.449403	+	0.893329i	$0.648363\pi$
$882$	−4.72213	−0.159002
$883$	−26.6351	−0.896341	−0.448171	−	0.893948i	$-0.647924\pi$
$883$	−26.6351	−0.896341	−0.448171	+	0.893948i	$0.647924\pi$
$884$	−7.55954	−0.254255
$885$	−9.83021	−0.330439
$886$	−36.0613	−1.21150
$887$	−2.75874	−0.0926293	−0.0463146	−	0.998927i	$-0.514748\pi$
$887$	−2.75874	−0.0926293	−0.0463146	+	0.998927i	$0.514748\pi$
$888$	8.78931	0.294950
$889$	−0.785135	−0.0263326
$890$	29.0967	0.975324
$891$	0.458525	0.0153612
$892$	60.1560	2.01417
$893$	4.22889	0.141514
$894$	−15.8878	−0.531366
$895$	17.9204	0.599013
$896$	20.2955	0.678026
$897$	0.0792261	0.00264528
$898$	32.3990	1.08117
$899$	75.5280	2.51900
$900$	−7.57277	−0.252426
$901$	29.5117	0.983176
$902$	1.80500	0.0601000
$903$	−5.80402	−0.193146
$904$	41.0007	1.36366
$905$	−8.17518	−0.271752
$906$	−56.4845	−1.87657
$907$	32.6785	1.08507	0.542536	−	0.840032i	$-0.317464\pi$
$907$	32.6785	1.08507	0.542536	+	0.840032i	$0.317464\pi$
$908$	−25.7729	−0.855305
$909$	−15.8781	−0.526643
$910$	−2.09439	−0.0694284
$911$	5.72577	0.189703	0.0948516	−	0.995491i	$-0.469762\pi$
$911$	5.72577	0.189703	0.0948516	+	0.995491i	$0.469762\pi$
$912$	16.4794	0.545687
$913$	3.92870	0.130021
$914$	38.6680	1.27902
$915$	−6.54656	−0.216422
$916$	3.89301	0.128629
$917$	4.24712	0.140252
$918$	−27.7495	−0.915871
$919$	−50.5430	−1.66726	−0.833630	−	0.552323i	$-0.813742\pi$
$919$	−50.5430	−1.66726	−0.833630	+	0.552323i	$0.813742\pi$
$920$	0.410886	0.0135465
$921$	11.4244	0.376446
$922$	68.9909	2.27209
$923$	8.47654	0.279009
$924$	−2.95907	−0.0973464
$925$	1.86231	0.0612323
$926$	22.7014	0.746016
$927$	16.3558	0.537194
$928$	−10.5036	−0.344798
$929$	−36.4764	−1.19675	−0.598377	−	0.801215i	$-0.704187\pi$
$929$	−36.4764	−1.19675	−0.598377	+	0.801215i	$0.704187\pi$
$930$	−18.1259	−0.594370
$931$	4.76201	0.156069
$932$	−20.9420	−0.685979
$933$	28.3505	0.928154
$934$	−7.93821	−0.259746
$935$	−1.66575	−0.0544758
$936$	7.71224	0.252083
$937$	26.1983	0.855860	0.427930	−	0.903812i	$-0.359243\pi$
$937$	26.1983	0.855860	0.427930	+	0.903812i	$0.359243\pi$
$938$	35.0937	1.14585
$939$	−19.2303	−0.627559
$940$	−3.45718	−0.112761
$941$	1.11635	0.0363918	0.0181959	−	0.999834i	$-0.494208\pi$
$941$	1.11635	0.0363918	0.0181959	+	0.999834i	$0.494208\pi$
$942$	0.830016	0.0270434
$943$	−0.0898294	−0.00292525
$944$	32.2516	1.04970
$945$	−5.07886	−0.165215
$946$	10.1533	0.330112
$947$	46.3600	1.50650	0.753249	−	0.657735i	$-0.228485\pi$
$947$	46.3600	1.50650	0.753249	+	0.657735i	$0.228485\pi$
$948$	32.8134	1.06573
$949$	12.1491	0.394375
$950$	11.5600	0.375057
$951$	−1.50147	−0.0486884
$952$	−10.3429	−0.335216
$953$	48.8790	1.58335	0.791673	−	0.610945i	$-0.209210\pi$
$953$	48.8790	1.58335	0.791673	+	0.610945i	$0.209210\pi$
$954$	−61.9171	−2.00464
$955$	8.00026	0.258882
$956$	−77.8565	−2.51806
$957$	7.89639	0.255254
$958$	44.9693	1.45289
$959$	−21.6224	−0.698224
$960$	9.44193	0.304737
$961$	21.8564	0.705046
$962$	−3.90040	−0.125754
$963$	−22.3684	−0.720813
$964$	−28.2556	−0.910050
$965$	16.3017	0.524772
$966$	0.222919	0.00717231
$967$	5.25610	0.169025	0.0845124	−	0.996422i	$-0.473067\pi$
$967$	5.25610	0.169025	0.0845124	+	0.996422i	$0.473067\pi$
$968$	−48.0322	−1.54381
$969$	11.0076	0.353614
$970$	−41.2326	−1.32390
$971$	−2.41366	−0.0774581	−0.0387290	−	0.999250i	$-0.512331\pi$
$971$	−2.41366	−0.0774581	−0.0387290	+	0.999250i	$0.512331\pi$
$972$	61.7933	1.98202
$973$	5.79999	0.185939
$974$	−23.2986	−0.746534
$975$	−0.886074	−0.0283771
$976$	21.4784	0.687507
$977$	−8.57934	−0.274477	−0.137239	−	0.990538i	$-0.543823\pi$
$977$	−8.57934	−0.274477	−0.137239	+	0.990538i	$0.543823\pi$
$978$	25.6777	0.821081
$979$	−8.87083	−0.283513
$980$	−3.89301	−0.124358
$981$	2.67425	0.0853821
$982$	−49.8087	−1.58946
$983$	−31.5729	−1.00702	−0.503510	−	0.863989i	$-0.667958\pi$
$983$	−31.5729	−1.00702	−0.503510	+	0.863989i	$0.667958\pi$
$984$	4.74159	0.151156
$985$	6.85481	0.218413
$986$	56.7608	1.80763
$987$	−0.912046	−0.0290307
$988$	−15.9943	−0.508847
$989$	−0.505297	−0.0160675
$990$	3.49484	0.111073
$991$	10.7656	0.341980	0.170990	−	0.985273i	$-0.445303\pi$
$991$	10.7656	0.341980	0.170990	+	0.985273i	$0.445303\pi$
$992$	−7.35069	−0.233385
$993$	10.3551	0.328608
$994$	23.8506	0.756494
$995$	−1.90350	−0.0603450
$996$	21.2240	0.672507
$997$	−14.5568	−0.461017	−0.230509	−	0.973070i	$-0.574039\pi$
$997$	−14.5568	−0.461017	−0.230509	+	0.973070i	$0.574039\pi$
$998$	25.7723	0.815809
$999$	−9.45840	−0.299250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8015.2.a.l.1.57	✓	62

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8015.2.a.l.1.57	✓	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)

\(f(q)\)	\(=\)	\(q+2.42755 q^{2} +1.02702 q^{3} +3.89301 q^{4} -1.00000 q^{5} +2.49316 q^{6} -1.00000 q^{7} +4.59539 q^{8} -1.94522 q^{9} +O(q^{10})\)	\(q+2.42755 q^{2} +1.02702 q^{3} +3.89301 q^{4} -1.00000 q^{5} +2.49316 q^{6} -1.00000 q^{7} +4.59539 q^{8} -1.94522 q^{9} -2.42755 q^{10} +0.740098 q^{11} +3.99822 q^{12} -0.862758 q^{13} -2.42755 q^{14} -1.02702 q^{15} +3.36953 q^{16} +2.25071 q^{17} -4.72213 q^{18} +4.76201 q^{19} -3.89301 q^{20} -1.02702 q^{21} +1.79663 q^{22} -0.0894125 q^{23} +4.71958 q^{24} +1.00000 q^{25} -2.09439 q^{26} -5.07886 q^{27} -3.89301 q^{28} +10.3886 q^{29} -2.49316 q^{30} +7.27024 q^{31} -1.01107 q^{32} +0.760098 q^{33} +5.46373 q^{34} +1.00000 q^{35} -7.57277 q^{36} +1.86231 q^{37} +11.5600 q^{38} -0.886074 q^{39} -4.59539 q^{40} +1.00466 q^{41} -2.49316 q^{42} +5.65130 q^{43} +2.88121 q^{44} +1.94522 q^{45} -0.217054 q^{46} +0.888047 q^{47} +3.46059 q^{48} +1.00000 q^{49} +2.42755 q^{50} +2.31154 q^{51} -3.35873 q^{52} +13.1121 q^{53} -12.3292 q^{54} -0.740098 q^{55} -4.59539 q^{56} +4.89070 q^{57} +25.2190 q^{58} +9.57154 q^{59} -3.99822 q^{60} +6.37429 q^{61} +17.6489 q^{62} +1.94522 q^{63} -9.19348 q^{64} +0.862758 q^{65} +1.84518 q^{66} -14.4564 q^{67} +8.76206 q^{68} -0.0918288 q^{69} +2.42755 q^{70} -9.82493 q^{71} -8.93906 q^{72} -14.0816 q^{73} +4.52085 q^{74} +1.02702 q^{75} +18.5386 q^{76} -0.740098 q^{77} -2.15099 q^{78} +8.20700 q^{79} -3.36953 q^{80} +0.619547 q^{81} +2.43887 q^{82} +5.30835 q^{83} -3.99822 q^{84} -2.25071 q^{85} +13.7188 q^{86} +10.6694 q^{87} +3.40104 q^{88} -11.9860 q^{89} +4.72213 q^{90} +0.862758 q^{91} -0.348084 q^{92} +7.46672 q^{93} +2.15578 q^{94} -4.76201 q^{95} -1.03839 q^{96} +16.9852 q^{97} +2.42755 q^{98} -1.43965 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 62 q + 2 q^{2} + 11 q^{3} + 64 q^{4} - 62 q^{5} + 3 q^{6} - 62 q^{7} + 15 q^{8} + 69 q^{9}+O(q^{10}) \) 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