LMFDB - Embedded newform 6010.2.a.g.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6010,2,Mod(1,6010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6010, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6010 = 2 \cdot 5 \cdot 601 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6010.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:	$47.9900916148$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	6010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.21224 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.21224 q^{6} +2.32820 q^{7} -1.00000 q^{8} -1.53047 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.21224 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.21224 q^{6} +2.32820 q^{7} -1.00000 q^{8} -1.53047 q^{9} -1.00000 q^{10} +2.81381 q^{11} -1.21224 q^{12} -5.17251 q^{13} -2.32820 q^{14} -1.21224 q^{15} +1.00000 q^{16} +2.75752 q^{17} +1.53047 q^{18} -2.93224 q^{19} +1.00000 q^{20} -2.82234 q^{21} -2.81381 q^{22} -2.72164 q^{23} +1.21224 q^{24} +1.00000 q^{25} +5.17251 q^{26} +5.49202 q^{27} +2.32820 q^{28} +1.77083 q^{29} +1.21224 q^{30} -2.88982 q^{31} -1.00000 q^{32} -3.41101 q^{33} -2.75752 q^{34} +2.32820 q^{35} -1.53047 q^{36} +7.74538 q^{37} +2.93224 q^{38} +6.27032 q^{39} -1.00000 q^{40} -9.53383 q^{41} +2.82234 q^{42} -1.99682 q^{43} +2.81381 q^{44} -1.53047 q^{45} +2.72164 q^{46} -0.274397 q^{47} -1.21224 q^{48} -1.57947 q^{49} -1.00000 q^{50} -3.34278 q^{51} -5.17251 q^{52} -8.62015 q^{53} -5.49202 q^{54} +2.81381 q^{55} -2.32820 q^{56} +3.55458 q^{57} -1.77083 q^{58} +4.64325 q^{59} -1.21224 q^{60} +13.5020 q^{61} +2.88982 q^{62} -3.56326 q^{63} +1.00000 q^{64} -5.17251 q^{65} +3.41101 q^{66} -0.401085 q^{67} +2.75752 q^{68} +3.29928 q^{69} -2.32820 q^{70} +5.59843 q^{71} +1.53047 q^{72} +2.36578 q^{73} -7.74538 q^{74} -1.21224 q^{75} -2.93224 q^{76} +6.55112 q^{77} -6.27032 q^{78} +8.10115 q^{79} +1.00000 q^{80} -2.06623 q^{81} +9.53383 q^{82} -3.51170 q^{83} -2.82234 q^{84} +2.75752 q^{85} +1.99682 q^{86} -2.14667 q^{87} -2.81381 q^{88} +15.3172 q^{89} +1.53047 q^{90} -12.0427 q^{91} -2.72164 q^{92} +3.50316 q^{93} +0.274397 q^{94} -2.93224 q^{95} +1.21224 q^{96} +12.6217 q^{97} +1.57947 q^{98} -4.30646 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 27 q^{2} + 6 q^{3} + 27 q^{4} + 27 q^{5} - 6 q^{6} - 27 q^{8} + 37 q^{9}+O(q^{10}) $ 27 * q - 27 * q^2 + 6 * q^3 + 27 * q^4 + 27 * q^5 - 6 * q^6 - 27 * q^8 + 37 * q^9	\( 27 q - 27 q^{2} + 6 q^{3} + 27 q^{4} + 27 q^{5} - 6 q^{6} - 27 q^{8} + 37 q^{9} - 27 q^{10} + 18 q^{11} + 6 q^{12} - 6 q^{13} + 6 q^{15} + 27 q^{16} + 3 q^{17} - 37 q^{18} + 27 q^{19} + 27 q^{20} + 16 q^{21} - 18 q^{22} + 15 q^{23} - 6 q^{24} + 27 q^{25} + 6 q^{26} + 27 q^{27} + 25 q^{29} - 6 q^{30} + 9 q^{31} - 27 q^{32} + 11 q^{33} - 3 q^{34} + 37 q^{36} - 16 q^{37} - 27 q^{38} + 20 q^{39} - 27 q^{40} + 39 q^{41} - 16 q^{42} + 9 q^{43} + 18 q^{44} + 37 q^{45} - 15 q^{46} + 31 q^{47} + 6 q^{48} + 27 q^{49} - 27 q^{50} + 39 q^{51} - 6 q^{52} - 5 q^{53} - 27 q^{54} + 18 q^{55} - 10 q^{57} - 25 q^{58} + 46 q^{59} + 6 q^{60} + 18 q^{61} - 9 q^{62} + 23 q^{63} + 27 q^{64} - 6 q^{65} - 11 q^{66} + 11 q^{67} + 3 q^{68} + 17 q^{69} + 50 q^{71} - 37 q^{72} - 29 q^{73} + 16 q^{74} + 6 q^{75} + 27 q^{76} - 6 q^{77} - 20 q^{78} + 56 q^{79} + 27 q^{80} + 51 q^{81} - 39 q^{82} + 44 q^{83} + 16 q^{84} + 3 q^{85} - 9 q^{86} + 42 q^{87} - 18 q^{88} + 34 q^{89} - 37 q^{90} + 43 q^{91} + 15 q^{92} - 20 q^{93} - 31 q^{94} + 27 q^{95} - 6 q^{96} - 37 q^{97} - 27 q^{98} + 67 q^{99}+O(q^{100}) \) 27 * q - 27 * q^2 + 6 * q^3 + 27 * q^4 + 27 * q^5 - 6 * q^6 - 27 * q^8 + 37 * q^9 - 27 * q^10 + 18 * q^11 + 6 * q^12 - 6 * q^13 + 6 * q^15 + 27 * q^16 + 3 * q^17 - 37 * q^18 + 27 * q^19 + 27 * q^20 + 16 * q^21 - 18 * q^22 + 15 * q^23 - 6 * q^24 + 27 * q^25 + 6 * q^26 + 27 * q^27 + 25 * q^29 - 6 * q^30 + 9 * q^31 - 27 * q^32 + 11 * q^33 - 3 * q^34 + 37 * q^36 - 16 * q^37 - 27 * q^38 + 20 * q^39 - 27 * q^40 + 39 * q^41 - 16 * q^42 + 9 * q^43 + 18 * q^44 + 37 * q^45 - 15 * q^46 + 31 * q^47 + 6 * q^48 + 27 * q^49 - 27 * q^50 + 39 * q^51 - 6 * q^52 - 5 * q^53 - 27 * q^54 + 18 * q^55 - 10 * q^57 - 25 * q^58 + 46 * q^59 + 6 * q^60 + 18 * q^61 - 9 * q^62 + 23 * q^63 + 27 * q^64 - 6 * q^65 - 11 * q^66 + 11 * q^67 + 3 * q^68 + 17 * q^69 + 50 * q^71 - 37 * q^72 - 29 * q^73 + 16 * q^74 + 6 * q^75 + 27 * q^76 - 6 * q^77 - 20 * q^78 + 56 * q^79 + 27 * q^80 + 51 * q^81 - 39 * q^82 + 44 * q^83 + 16 * q^84 + 3 * q^85 - 9 * q^86 + 42 * q^87 - 18 * q^88 + 34 * q^89 - 37 * q^90 + 43 * q^91 + 15 * q^92 - 20 * q^93 - 31 * q^94 + 27 * q^95 - 6 * q^96 - 37 * q^97 - 27 * q^98 + 67 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.21224	−0.699887	−0.349944	−	0.936771i	$-0.613799\pi$
$3$	−1.21224	−0.699887	−0.349944	+	0.936771i	$0.613799\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.21224	0.494895
$7$	2.32820	0.879978	0.439989	−	0.898003i	$-0.354982\pi$
$7$	2.32820	0.879978	0.439989	+	0.898003i	$0.354982\pi$
$8$	−1.00000	−0.353553
$9$	−1.53047	−0.510158
$10$	−1.00000	−0.316228
$11$	2.81381	0.848396	0.424198	−	0.905570i	$-0.360556\pi$
$11$	2.81381	0.848396	0.424198	+	0.905570i	$0.360556\pi$
$12$	−1.21224	−0.349944
$13$	−5.17251	−1.43460	−0.717298	−	0.696767i	$-0.754621\pi$
$13$	−5.17251	−1.43460	−0.717298	+	0.696767i	$0.754621\pi$
$14$	−2.32820	−0.622239
$15$	−1.21224	−0.312999
$16$	1.00000	0.250000
$17$	2.75752	0.668798	0.334399	−	0.942432i	$-0.391467\pi$
$17$	2.75752	0.668798	0.334399	+	0.942432i	$0.391467\pi$
$18$	1.53047	0.360736
$19$	−2.93224	−0.672701	−0.336351	−	0.941737i	$-0.609193\pi$
$19$	−2.93224	−0.672701	−0.336351	+	0.941737i	$0.609193\pi$
$20$	1.00000	0.223607
$21$	−2.82234	−0.615885
$22$	−2.81381	−0.599906
$23$	−2.72164	−0.567501	−0.283751	−	0.958898i	$-0.591579\pi$
$23$	−2.72164	−0.567501	−0.283751	+	0.958898i	$0.591579\pi$
$24$	1.21224	0.247447
$25$	1.00000	0.200000
$26$	5.17251	1.01441
$27$	5.49202	1.05694
$28$	2.32820	0.439989
$29$	1.77083	0.328835	0.164417	−	0.986391i	$-0.447426\pi$
$29$	1.77083	0.328835	0.164417	+	0.986391i	$0.447426\pi$
$30$	1.21224	0.221324
$31$	−2.88982	−0.519028	−0.259514	−	0.965739i	$-0.583562\pi$
$31$	−2.88982	−0.519028	−0.259514	+	0.965739i	$0.583562\pi$
$32$	−1.00000	−0.176777
$33$	−3.41101	−0.593781
$34$	−2.75752	−0.472911
$35$	2.32820	0.393538
$36$	−1.53047	−0.255079
$37$	7.74538	1.27333	0.636666	−	0.771139i	$-0.280313\pi$
$37$	7.74538	1.27333	0.636666	+	0.771139i	$0.280313\pi$
$38$	2.93224	0.475672
$39$	6.27032	1.00405
$40$	−1.00000	−0.158114
$41$	−9.53383	−1.48893	−0.744467	−	0.667659i	$-0.767296\pi$
$41$	−9.53383	−1.48893	−0.744467	+	0.667659i	$0.767296\pi$
$42$	2.82234	0.435497
$43$	−1.99682	−0.304512	−0.152256	−	0.988341i	$-0.548654\pi$
$43$	−1.99682	−0.304512	−0.152256	+	0.988341i	$0.548654\pi$
$44$	2.81381	0.424198
$45$	−1.53047	−0.228150
$46$	2.72164	0.401284
$47$	−0.274397	−0.0400250	−0.0200125	−	0.999800i	$-0.506371\pi$
$47$	−0.274397	−0.0400250	−0.0200125	+	0.999800i	$0.506371\pi$
$48$	−1.21224	−0.174972
$49$	−1.57947	−0.225638
$50$	−1.00000	−0.141421
$51$	−3.34278	−0.468083
$52$	−5.17251	−0.717298
$53$	−8.62015	−1.18407	−0.592034	−	0.805913i	$-0.701675\pi$
$53$	−8.62015	−1.18407	−0.592034	+	0.805913i	$0.701675\pi$
$54$	−5.49202	−0.747370
$55$	2.81381	0.379414
$56$	−2.32820	−0.311119
$57$	3.55458	0.470815
$58$	−1.77083	−0.232521
$59$	4.64325	0.604500	0.302250	−	0.953229i	$-0.402262\pi$
$59$	4.64325	0.604500	0.302250	+	0.953229i	$0.402262\pi$
$60$	−1.21224	−0.156500
$61$	13.5020	1.72875	0.864374	−	0.502850i	$-0.167715\pi$
$61$	13.5020	1.72875	0.864374	+	0.502850i	$0.167715\pi$
$62$	2.88982	0.367008
$63$	−3.56326	−0.448928
$64$	1.00000	0.125000
$65$	−5.17251	−0.641571
$66$	3.41101	0.419867
$67$	−0.401085	−0.0490004	−0.0245002	−	0.999700i	$-0.507799\pi$
$67$	−0.401085	−0.0490004	−0.0245002	+	0.999700i	$0.507799\pi$
$68$	2.75752	0.334399
$69$	3.29928	0.397187
$70$	−2.32820	−0.278274
$71$	5.59843	0.664411	0.332206	−	0.943207i	$-0.392207\pi$
$71$	5.59843	0.664411	0.332206	+	0.943207i	$0.392207\pi$
$72$	1.53047	0.180368
$73$	2.36578	0.276894	0.138447	−	0.990370i	$-0.455789\pi$
$73$	2.36578	0.276894	0.138447	+	0.990370i	$0.455789\pi$
$74$	−7.74538	−0.900382
$75$	−1.21224	−0.139977
$76$	−2.93224	−0.336351
$77$	6.55112	0.746570
$78$	−6.27032	−0.709974
$79$	8.10115	0.911451	0.455725	−	0.890120i	$-0.349380\pi$
$79$	8.10115	0.911451	0.455725	+	0.890120i	$0.349380\pi$
$80$	1.00000	0.111803
$81$	−2.06623	−0.229581
$82$	9.53383	1.05284
$83$	−3.51170	−0.385459	−0.192730	−	0.981252i	$-0.561734\pi$
$83$	−3.51170	−0.385459	−0.192730	+	0.981252i	$0.561734\pi$
$84$	−2.82234	−0.307943
$85$	2.75752	0.299095
$86$	1.99682	0.215323
$87$	−2.14667	−0.230147
$88$	−2.81381	−0.299953
$89$	15.3172	1.62362	0.811812	−	0.583919i	$-0.198482\pi$
$89$	15.3172	1.62362	0.811812	+	0.583919i	$0.198482\pi$
$90$	1.53047	0.161326
$91$	−12.0427	−1.26241
$92$	−2.72164	−0.283751
$93$	3.50316	0.363261
$94$	0.274397	0.0283019
$95$	−2.93224	−0.300841
$96$	1.21224	0.123724
$97$	12.6217	1.28154	0.640769	−	0.767733i	$-0.278616\pi$
$97$	12.6217	1.28154	0.640769	+	0.767733i	$0.278616\pi$
$98$	1.57947	0.159550
$99$	−4.30646	−0.432816
$100$	1.00000	0.100000
$101$	14.0437	1.39740	0.698699	−	0.715416i	$-0.253763\pi$
$101$	14.0437	1.39740	0.698699	+	0.715416i	$0.253763\pi$
$102$	3.34278	0.330985
$103$	3.09383	0.304844	0.152422	−	0.988315i	$-0.451293\pi$
$103$	3.09383	0.304844	0.152422	+	0.988315i	$0.451293\pi$
$104$	5.17251	0.507206
$105$	−2.82234	−0.275432
$106$	8.62015	0.837263
$107$	−9.31993	−0.900992	−0.450496	−	0.892778i	$-0.648753\pi$
$107$	−9.31993	−0.900992	−0.450496	+	0.892778i	$0.648753\pi$
$108$	5.49202	0.528470
$109$	20.6332	1.97630	0.988150	−	0.153490i	$-0.0490514\pi$
$109$	20.6332	1.97630	0.988150	+	0.153490i	$0.0490514\pi$
$110$	−2.81381	−0.268286
$111$	−9.38926	−0.891189
$112$	2.32820	0.219995
$113$	1.76347	0.165893	0.0829467	−	0.996554i	$-0.473567\pi$
$113$	1.76347	0.165893	0.0829467	+	0.996554i	$0.473567\pi$
$114$	−3.55458	−0.332916
$115$	−2.72164	−0.253794
$116$	1.77083	0.164417
$117$	7.91639	0.731870
$118$	−4.64325	−0.427446
$119$	6.42008	0.588527
$120$	1.21224	0.110662
$121$	−3.08247	−0.280225
$122$	−13.5020	−1.22241
$123$	11.5573	1.04209
$124$	−2.88982	−0.259514
$125$	1.00000	0.0894427
$126$	3.56326	0.317440
$127$	−5.40703	−0.479797	−0.239898	−	0.970798i	$-0.577114\pi$
$127$	−5.40703	−0.479797	−0.239898	+	0.970798i	$0.577114\pi$
$128$	−1.00000	−0.0883883
$129$	2.42063	0.213124
$130$	5.17251	0.453659
$131$	−16.8462	−1.47186	−0.735930	−	0.677058i	$-0.763255\pi$
$131$	−16.8462	−1.47186	−0.735930	+	0.677058i	$0.763255\pi$
$132$	−3.41101	−0.296891
$133$	−6.82684	−0.591963
$134$	0.401085	0.0346485
$135$	5.49202	0.472678
$136$	−2.75752	−0.236456
$137$	−2.72810	−0.233077	−0.116538	−	0.993186i	$-0.537180\pi$
$137$	−2.72810	−0.233077	−0.116538	+	0.993186i	$0.537180\pi$
$138$	−3.29928	−0.280853
$139$	1.01592	0.0861691	0.0430846	−	0.999071i	$-0.486282\pi$
$139$	1.01592	0.0861691	0.0430846	+	0.999071i	$0.486282\pi$
$140$	2.32820	0.196769
$141$	0.332635	0.0280130
$142$	−5.59843	−0.469810
$143$	−14.5545	−1.21710
$144$	−1.53047	−0.127540
$145$	1.77083	0.147059
$146$	−2.36578	−0.195794
$147$	1.91470	0.157921
$148$	7.74538	0.636666
$149$	17.0533	1.39706	0.698529	−	0.715582i	$-0.253838\pi$
$149$	17.0533	1.39706	0.698529	+	0.715582i	$0.253838\pi$
$150$	1.21224	0.0989790
$151$	13.4725	1.09638	0.548190	−	0.836354i	$-0.315317\pi$
$151$	13.4725	1.09638	0.548190	+	0.836354i	$0.315317\pi$
$152$	2.93224	0.237836
$153$	−4.22032	−0.341193
$154$	−6.55112	−0.527904
$155$	−2.88982	−0.232116
$156$	6.27032	0.502027
$157$	−16.7960	−1.34047	−0.670234	−	0.742150i	$-0.733806\pi$
$157$	−16.7960	−1.34047	−0.670234	+	0.742150i	$0.733806\pi$
$158$	−8.10115	−0.644493
$159$	10.4497	0.828714
$160$	−1.00000	−0.0790569
$161$	−6.33653	−0.499389
$162$	2.06623	0.162338
$163$	−7.47065	−0.585147	−0.292573	−	0.956243i	$-0.594512\pi$
$163$	−7.47065	−0.585147	−0.292573	+	0.956243i	$0.594512\pi$
$164$	−9.53383	−0.744467
$165$	−3.41101	−0.265547
$166$	3.51170	0.272561
$167$	5.86097	0.453536	0.226768	−	0.973949i	$-0.427184\pi$
$167$	5.86097	0.453536	0.226768	+	0.973949i	$0.427184\pi$
$168$	2.82234	0.217748
$169$	13.7548	1.05806
$170$	−2.75752	−0.211492
$171$	4.48771	0.343184
$172$	−1.99682	−0.152256
$173$	−8.62481	−0.655732	−0.327866	−	0.944724i	$-0.606329\pi$
$173$	−8.62481	−0.655732	−0.327866	+	0.944724i	$0.606329\pi$
$174$	2.14667	0.162739
$175$	2.32820	0.175996
$176$	2.81381	0.212099
$177$	−5.62873	−0.423082
$178$	−15.3172	−1.14808
$179$	−15.3420	−1.14671	−0.573357	−	0.819306i	$-0.694359\pi$
$179$	−15.3420	−1.14671	−0.573357	+	0.819306i	$0.694359\pi$
$180$	−1.53047	−0.114075
$181$	−21.0159	−1.56210	−0.781051	−	0.624467i	$-0.785316\pi$
$181$	−21.0159	−1.56210	−0.781051	+	0.624467i	$0.785316\pi$
$182$	12.0427	0.892661
$183$	−16.3676	−1.20993
$184$	2.72164	0.200642
$185$	7.74538	0.569452
$186$	−3.50316	−0.256864
$187$	7.75915	0.567405
$188$	−0.274397	−0.0200125
$189$	12.7865	0.930084
$190$	2.93224	0.212727
$191$	16.1874	1.17128	0.585638	−	0.810573i	$-0.300844\pi$
$191$	16.1874	1.17128	0.585638	+	0.810573i	$0.300844\pi$
$192$	−1.21224	−0.0874859
$193$	2.09542	0.150832	0.0754158	−	0.997152i	$-0.475972\pi$
$193$	2.09542	0.150832	0.0754158	+	0.997152i	$0.475972\pi$
$194$	−12.6217	−0.906185
$195$	6.27032	0.449027
$196$	−1.57947	−0.112819
$197$	−5.02438	−0.357972	−0.178986	−	0.983852i	$-0.557282\pi$
$197$	−5.02438	−0.357972	−0.178986	+	0.983852i	$0.557282\pi$
$198$	4.30646	0.306047
$199$	−12.3769	−0.877375	−0.438687	−	0.898640i	$-0.644556\pi$
$199$	−12.3769	−0.877375	−0.438687	+	0.898640i	$0.644556\pi$
$200$	−1.00000	−0.0707107
$201$	0.486212	0.0342947
$202$	−14.0437	−0.988109
$203$	4.12285	0.289367
$204$	−3.34278	−0.234041
$205$	−9.53383	−0.665872
$206$	−3.09383	−0.215557
$207$	4.16540	0.289515
$208$	−5.17251	−0.358649
$209$	−8.25076	−0.570717
$210$	2.82234	0.194760
$211$	4.69560	0.323259	0.161629	−	0.986852i	$-0.448325\pi$
$211$	4.69560	0.323259	0.161629	+	0.986852i	$0.448325\pi$
$212$	−8.62015	−0.592034
$213$	−6.78664	−0.465013
$214$	9.31993	0.637097
$215$	−1.99682	−0.136182
$216$	−5.49202	−0.373685
$217$	−6.72810	−0.456733
$218$	−20.6332	−1.39746
$219$	−2.86790	−0.193794
$220$	2.81381	0.189707
$221$	−14.2633	−0.959454
$222$	9.38926	0.630166
$223$	23.9237	1.60205	0.801024	−	0.598632i	$-0.204289\pi$
$223$	23.9237	1.60205	0.801024	+	0.598632i	$0.204289\pi$
$224$	−2.32820	−0.155560
$225$	−1.53047	−0.102032
$226$	−1.76347	−0.117304
$227$	23.3241	1.54808	0.774039	−	0.633138i	$-0.218234\pi$
$227$	23.3241	1.54808	0.774039	+	0.633138i	$0.218234\pi$
$228$	3.55458	0.235408
$229$	−1.37040	−0.0905589	−0.0452794	−	0.998974i	$-0.514418\pi$
$229$	−1.37040	−0.0905589	−0.0452794	+	0.998974i	$0.514418\pi$
$230$	2.72164	0.179460
$231$	−7.94153	−0.522514
$232$	−1.77083	−0.116261
$233$	−25.6872	−1.68283	−0.841414	−	0.540392i	$-0.818276\pi$
$233$	−25.6872	−1.68283	−0.841414	+	0.540392i	$0.818276\pi$
$234$	−7.91639	−0.517511
$235$	−0.274397	−0.0178997
$236$	4.64325	0.302250
$237$	−9.82054	−0.637912
$238$	−6.42008	−0.416152
$239$	20.3712	1.31770	0.658850	−	0.752274i	$-0.271043\pi$
$239$	20.3712	1.31770	0.658850	+	0.752274i	$0.271043\pi$
$240$	−1.21224	−0.0782498
$241$	−15.7238	−1.01286	−0.506428	−	0.862282i	$-0.669034\pi$
$241$	−15.7238	−1.01286	−0.506428	+	0.862282i	$0.669034\pi$
$242$	3.08247	0.198149
$243$	−13.9713	−0.896260
$244$	13.5020	0.864374
$245$	−1.57947	−0.100909
$246$	−11.5573	−0.736866
$247$	15.1670	0.965054
$248$	2.88982	0.183504
$249$	4.25702	0.269778
$250$	−1.00000	−0.0632456
$251$	16.6129	1.04860	0.524300	−	0.851534i	$-0.324327\pi$
$251$	16.6129	1.04860	0.524300	+	0.851534i	$0.324327\pi$
$252$	−3.56326	−0.224464
$253$	−7.65818	−0.481465
$254$	5.40703	0.339267
$255$	−3.34278	−0.209333
$256$	1.00000	0.0625000
$257$	17.1622	1.07055	0.535273	−	0.844679i	$-0.320209\pi$
$257$	17.1622	1.07055	0.535273	+	0.844679i	$0.320209\pi$
$258$	−2.42063	−0.150702
$259$	18.0328	1.12050
$260$	−5.17251	−0.320785
$261$	−2.71021	−0.167758
$262$	16.8462	1.04076
$263$	−3.67758	−0.226769	−0.113385	−	0.993551i	$-0.536169\pi$
$263$	−3.67758	−0.226769	−0.113385	+	0.993551i	$0.536169\pi$
$264$	3.41101	0.209933
$265$	−8.62015	−0.529532
$266$	6.82684	0.418581
$267$	−18.5682	−1.13635
$268$	−0.401085	−0.0245002
$269$	23.3824	1.42565	0.712825	−	0.701342i	$-0.247415\pi$
$269$	23.3824	1.42565	0.712825	+	0.701342i	$0.247415\pi$
$270$	−5.49202	−0.334234
$271$	19.0068	1.15458	0.577290	−	0.816539i	$-0.304110\pi$
$271$	19.0068	1.15458	0.577290	+	0.816539i	$0.304110\pi$
$272$	2.75752	0.167199
$273$	14.5986	0.883546
$274$	2.72810	0.164810
$275$	2.81381	0.169679
$276$	3.29928	0.198593
$277$	−19.5903	−1.17707	−0.588535	−	0.808472i	$-0.700295\pi$
$277$	−19.5903	−1.17707	−0.588535	+	0.808472i	$0.700295\pi$
$278$	−1.01592	−0.0609308
$279$	4.42280	0.264786
$280$	−2.32820	−0.139137
$281$	3.68199	0.219649	0.109824	−	0.993951i	$-0.464971\pi$
$281$	3.68199	0.219649	0.109824	+	0.993951i	$0.464971\pi$
$282$	−0.332635	−0.0198081
$283$	−2.22974	−0.132544	−0.0662720	−	0.997802i	$-0.521111\pi$
$283$	−2.22974	−0.132544	−0.0662720	+	0.997802i	$0.521111\pi$
$284$	5.59843	0.332206
$285$	3.55458	0.210555
$286$	14.5545	0.860623
$287$	−22.1967	−1.31023
$288$	1.53047	0.0901841
$289$	−9.39607	−0.552710
$290$	−1.77083	−0.103987
$291$	−15.3005	−0.896932
$292$	2.36578	0.138447
$293$	14.3486	0.838251	0.419126	−	0.907928i	$-0.362337\pi$
$293$	14.3486	0.838251	0.419126	+	0.907928i	$0.362337\pi$
$294$	−1.91470	−0.111667
$295$	4.64325	0.270340
$296$	−7.74538	−0.450191
$297$	15.4535	0.896703
$298$	−17.0533	−0.987869
$299$	14.0777	0.814135
$300$	−1.21224	−0.0699887
$301$	−4.64900	−0.267964
$302$	−13.4725	−0.775258
$303$	−17.0243	−0.978020
$304$	−2.93224	−0.168175
$305$	13.5020	0.773120
$306$	4.22032	0.241260
$307$	−1.66944	−0.0952801	−0.0476400	−	0.998865i	$-0.515170\pi$
$307$	−1.66944	−0.0952801	−0.0476400	+	0.998865i	$0.515170\pi$
$308$	6.55112	0.373285
$309$	−3.75047	−0.213357
$310$	2.88982	0.164131
$311$	34.3210	1.94617	0.973083	−	0.230454i	$-0.0740211\pi$
$311$	34.3210	1.94617	0.973083	+	0.230454i	$0.0740211\pi$
$312$	−6.27032	−0.354987
$313$	16.2482	0.918405	0.459202	−	0.888332i	$-0.348135\pi$
$313$	16.2482	0.918405	0.459202	+	0.888332i	$0.348135\pi$
$314$	16.7960	0.947854
$315$	−3.56326	−0.200767
$316$	8.10115	0.455725
$317$	−1.24915	−0.0701594	−0.0350797	−	0.999385i	$-0.511169\pi$
$317$	−1.24915	−0.0701594	−0.0350797	+	0.999385i	$0.511169\pi$
$318$	−10.4497	−0.585990
$319$	4.98278	0.278982
$320$	1.00000	0.0559017
$321$	11.2980	0.630593
$322$	6.33653	0.353121
$323$	−8.08571	−0.449901
$324$	−2.06623	−0.114790
$325$	−5.17251	−0.286919
$326$	7.47065	0.413761
$327$	−25.0124	−1.38319
$328$	9.53383	0.526418
$329$	−0.638853	−0.0352211
$330$	3.41101	0.187770
$331$	34.5850	1.90096	0.950481	−	0.310782i	$-0.100591\pi$
$331$	34.5850	1.90096	0.950481	+	0.310782i	$0.100591\pi$
$332$	−3.51170	−0.192730
$333$	−11.8541	−0.649601
$334$	−5.86097	−0.320698
$335$	−0.401085	−0.0219136
$336$	−2.82234	−0.153971
$337$	5.83893	0.318067	0.159034	−	0.987273i	$-0.449162\pi$
$337$	5.83893	0.318067	0.159034	+	0.987273i	$0.449162\pi$
$338$	−13.7548	−0.748165
$339$	−2.13775	−0.116107
$340$	2.75752	0.149548
$341$	−8.13141	−0.440341
$342$	−4.48771	−0.242668
$343$	−19.9747	−1.07854
$344$	1.99682	0.107661
$345$	3.29928	0.177627
$346$	8.62481	0.463673
$347$	11.0791	0.594756	0.297378	−	0.954760i	$-0.403888\pi$
$347$	11.0791	0.594756	0.297378	+	0.954760i	$0.403888\pi$
$348$	−2.14667	−0.115074
$349$	−20.1301	−1.07754	−0.538771	−	0.842452i	$-0.681111\pi$
$349$	−20.1301	−1.07754	−0.538771	+	0.842452i	$0.681111\pi$
$350$	−2.32820	−0.124448
$351$	−28.4075	−1.51628
$352$	−2.81381	−0.149977
$353$	13.0234	0.693164	0.346582	−	0.938020i	$-0.387342\pi$
$353$	13.0234	0.693164	0.346582	+	0.938020i	$0.387342\pi$
$354$	5.62873	0.299164
$355$	5.59843	0.297134
$356$	15.3172	0.811812
$357$	−7.78267	−0.411903
$358$	15.3420	0.810849
$359$	−20.8587	−1.10088	−0.550441	−	0.834874i	$-0.685541\pi$
$359$	−20.8587	−1.10088	−0.550441	+	0.834874i	$0.685541\pi$
$360$	1.53047	0.0806631
$361$	−10.4020	−0.547473
$362$	21.0159	1.10457
$363$	3.73670	0.196126
$364$	−12.0427	−0.631206
$365$	2.36578	0.123831
$366$	16.3676	0.855548
$367$	−26.0064	−1.35752	−0.678761	−	0.734359i	$-0.737483\pi$
$367$	−26.0064	−1.35752	−0.678761	+	0.734359i	$0.737483\pi$
$368$	−2.72164	−0.141875
$369$	14.5913	0.759592
$370$	−7.74538	−0.402663
$371$	−20.0695	−1.04195
$372$	3.50316	0.181630
$373$	32.2760	1.67119	0.835593	−	0.549349i	$-0.185124\pi$
$373$	32.2760	1.67119	0.835593	+	0.549349i	$0.185124\pi$
$374$	−7.75915	−0.401216
$375$	−1.21224	−0.0625998
$376$	0.274397	0.0141510
$377$	−9.15963	−0.471745
$378$	−12.7865	−0.657669
$379$	0.0405680	0.00208384	0.00104192	−	0.999999i	$-0.499668\pi$
$379$	0.0405680	0.00208384	0.00104192	+	0.999999i	$0.499668\pi$
$380$	−2.93224	−0.150421
$381$	6.55462	0.335803
$382$	−16.1874	−0.828217
$383$	30.0361	1.53477	0.767387	−	0.641184i	$-0.221556\pi$
$383$	30.0361	1.53477	0.767387	+	0.641184i	$0.221556\pi$
$384$	1.21224	0.0618619
$385$	6.55112	0.333876
$386$	−2.09542	−0.106654
$387$	3.05608	0.155349
$388$	12.6217	0.640769
$389$	−16.0849	−0.815539	−0.407770	−	0.913085i	$-0.633693\pi$
$389$	−16.0849	−0.815539	−0.407770	+	0.913085i	$0.633693\pi$
$390$	−6.27032	−0.317510
$391$	−7.50499	−0.379543
$392$	1.57947	0.0797752
$393$	20.4216	1.03014
$394$	5.02438	0.253125
$395$	8.10115	0.407613
$396$	−4.30646	−0.216408
$397$	−12.3909	−0.621883	−0.310941	−	0.950429i	$-0.600644\pi$
$397$	−12.3909	−0.621883	−0.310941	+	0.950429i	$0.600644\pi$
$398$	12.3769	0.620398
$399$	8.27577	0.414307
$400$	1.00000	0.0500000
$401$	14.4029	0.719245	0.359622	−	0.933098i	$-0.382905\pi$
$401$	14.4029	0.719245	0.359622	+	0.933098i	$0.382905\pi$
$402$	−0.486212	−0.0242500
$403$	14.9476	0.744595
$404$	14.0437	0.698699
$405$	−2.06623	−0.102672
$406$	−4.12285	−0.204614
$407$	21.7940	1.08029
$408$	3.34278	0.165492
$409$	−26.5854	−1.31456	−0.657282	−	0.753645i	$-0.728294\pi$
$409$	−26.5854	−1.31456	−0.657282	+	0.753645i	$0.728294\pi$
$410$	9.53383	0.470842
$411$	3.30711	0.163128
$412$	3.09383	0.152422
$413$	10.8104	0.531947
$414$	−4.16540	−0.204718
$415$	−3.51170	−0.172383
$416$	5.17251	0.253603
$417$	−1.23154	−0.0603087
$418$	8.25076	0.403558
$419$	13.7919	0.673778	0.336889	−	0.941544i	$-0.390625\pi$
$419$	13.7919	0.673778	0.336889	+	0.941544i	$0.390625\pi$
$420$	−2.82234	−0.137716
$421$	23.3922	1.14007	0.570033	−	0.821622i	$-0.306930\pi$
$421$	23.3922	1.14007	0.570033	+	0.821622i	$0.306930\pi$
$422$	−4.69560	−0.228578
$423$	0.419958	0.0204191
$424$	8.62015	0.418632
$425$	2.75752	0.133760
$426$	6.78664	0.328814
$427$	31.4353	1.52126
$428$	−9.31993	−0.450496
$429$	17.6435	0.851836
$430$	1.99682	0.0962952
$431$	4.77496	0.230002	0.115001	−	0.993365i	$-0.463313\pi$
$431$	4.77496	0.230002	0.115001	+	0.993365i	$0.463313\pi$
$432$	5.49202	0.264235
$433$	−6.30177	−0.302844	−0.151422	−	0.988469i	$-0.548385\pi$
$433$	−6.30177	−0.302844	−0.151422	+	0.988469i	$0.548385\pi$
$434$	6.72810	0.322959
$435$	−2.14667	−0.102925
$436$	20.6332	0.988150
$437$	7.98049	0.381759
$438$	2.86790	0.137033
$439$	−2.35656	−0.112473	−0.0562363	−	0.998417i	$-0.517910\pi$
$439$	−2.35656	−0.112473	−0.0562363	+	0.998417i	$0.517910\pi$
$440$	−2.81381	−0.134143
$441$	2.41734	0.115111
$442$	14.2633	0.678437
$443$	37.9229	1.80177	0.900885	−	0.434059i	$-0.142919\pi$
$443$	37.9229	1.80177	0.900885	+	0.434059i	$0.142919\pi$
$444$	−9.38926	−0.445594
$445$	15.3172	0.726106
$446$	−23.9237	−1.13282
$447$	−20.6726	−0.977783
$448$	2.32820	0.109997
$449$	14.3889	0.679052	0.339526	−	0.940597i	$-0.389733\pi$
$449$	14.3889	0.679052	0.339526	+	0.940597i	$0.389733\pi$
$450$	1.53047	0.0721472
$451$	−26.8264	−1.26321
$452$	1.76347	0.0829467
$453$	−16.3320	−0.767342
$454$	−23.3241	−1.09466
$455$	−12.0427	−0.564568
$456$	−3.55458	−0.166458
$457$	−11.4413	−0.535204	−0.267602	−	0.963530i	$-0.586231\pi$
$457$	−11.4413	−0.535204	−0.267602	+	0.963530i	$0.586231\pi$
$458$	1.37040	0.0640348
$459$	15.1444	0.706879
$460$	−2.72164	−0.126897
$461$	26.4040	1.22976	0.614878	−	0.788622i	$-0.289205\pi$
$461$	26.4040	1.22976	0.614878	+	0.788622i	$0.289205\pi$
$462$	7.94153	0.369473
$463$	−17.1908	−0.798926	−0.399463	−	0.916749i	$-0.630803\pi$
$463$	−17.1908	−0.798926	−0.399463	+	0.916749i	$0.630803\pi$
$464$	1.77083	0.0822087
$465$	3.50316	0.162455
$466$	25.6872	1.18994
$467$	−7.20861	−0.333575	−0.166787	−	0.985993i	$-0.553339\pi$
$467$	−7.20861	−0.333575	−0.166787	+	0.985993i	$0.553339\pi$
$468$	7.91639	0.365935
$469$	−0.933808	−0.0431193
$470$	0.274397	0.0126570
$471$	20.3608	0.938176
$472$	−4.64325	−0.213723
$473$	−5.61867	−0.258347
$474$	9.82054	0.451072
$475$	−2.93224	−0.134540
$476$	6.42008	0.294264
$477$	13.1929	0.604062
$478$	−20.3712	−0.931755
$479$	26.2192	1.19798	0.598992	−	0.800755i	$-0.295568\pi$
$479$	26.2192	1.19798	0.598992	+	0.800755i	$0.295568\pi$
$480$	1.21224	0.0553309
$481$	−40.0630	−1.82672
$482$	15.7238	0.716197
$483$	7.68140	0.349516
$484$	−3.08247	−0.140112
$485$	12.6217	0.573121
$486$	13.9713	0.633751
$487$	−1.74683	−0.0791563	−0.0395782	−	0.999216i	$-0.512601\pi$
$487$	−1.74683	−0.0791563	−0.0395782	+	0.999216i	$0.512601\pi$
$488$	−13.5020	−0.611205
$489$	9.05623	0.409537
$490$	1.57947	0.0713531
$491$	16.7032	0.753806	0.376903	−	0.926253i	$-0.376989\pi$
$491$	16.7032	0.753806	0.376903	+	0.926253i	$0.376989\pi$
$492$	11.5573	0.521043
$493$	4.88310	0.219924
$494$	−15.1670	−0.682397
$495$	−4.30646	−0.193561
$496$	−2.88982	−0.129757
$497$	13.0343	0.584667
$498$	−4.25702	−0.190762
$499$	19.9902	0.894884	0.447442	−	0.894313i	$-0.352335\pi$
$499$	19.9902	0.894884	0.447442	+	0.894313i	$0.352335\pi$
$500$	1.00000	0.0447214
$501$	−7.10490	−0.317424
$502$	−16.6129	−0.741472
$503$	−14.9181	−0.665167	−0.332583	−	0.943074i	$-0.607920\pi$
$503$	−14.9181	−0.665167	−0.332583	+	0.943074i	$0.607920\pi$
$504$	3.56326	0.158720
$505$	14.0437	0.624935
$506$	7.65818	0.340447
$507$	−16.6742	−0.740526
$508$	−5.40703	−0.239898
$509$	5.56364	0.246604	0.123302	−	0.992369i	$-0.460652\pi$
$509$	5.56364	0.246604	0.123302	+	0.992369i	$0.460652\pi$
$510$	3.34278	0.148021
$511$	5.50802	0.243661
$512$	−1.00000	−0.0441942
$513$	−16.1039	−0.711005
$514$	−17.1622	−0.756991
$515$	3.09383	0.136330
$516$	2.42063	0.106562
$517$	−0.772102	−0.0339570
$518$	−18.0328	−0.792316
$519$	10.4553	0.458938
$520$	5.17251	0.226829
$521$	25.1583	1.10221	0.551103	−	0.834437i	$-0.314207\pi$
$521$	25.1583	1.10221	0.551103	+	0.834437i	$0.314207\pi$
$522$	2.71021	0.118623
$523$	37.7927	1.65256	0.826279	−	0.563262i	$-0.190454\pi$
$523$	37.7927	1.65256	0.826279	+	0.563262i	$0.190454\pi$
$524$	−16.8462	−0.735930
$525$	−2.82234	−0.123177
$526$	3.67758	0.160350
$527$	−7.96875	−0.347124
$528$	−3.41101	−0.148445
$529$	−15.5927	−0.677942
$530$	8.62015	0.374435
$531$	−7.10638	−0.308390
$532$	−6.82684	−0.295981
$533$	49.3138	2.13602
$534$	18.5682	0.803523
$535$	−9.31993	−0.402936
$536$	0.401085	0.0173242
$537$	18.5982	0.802570
$538$	−23.3824	−1.00809
$539$	−4.44433	−0.191431
$540$	5.49202	0.236339
$541$	17.9259	0.770697	0.385348	−	0.922771i	$-0.374081\pi$
$541$	17.9259	0.770697	0.385348	+	0.922771i	$0.374081\pi$
$542$	−19.0068	−0.816411
$543$	25.4764	1.09330
$544$	−2.75752	−0.118228
$545$	20.6332	0.883828
$546$	−14.5986	−0.624762
$547$	−2.76215	−0.118101	−0.0590505	−	0.998255i	$-0.518807\pi$
$547$	−2.76215	−0.118101	−0.0590505	+	0.998255i	$0.518807\pi$
$548$	−2.72810	−0.116538
$549$	−20.6644	−0.881935
$550$	−2.81381	−0.119981
$551$	−5.19249	−0.221208
$552$	−3.29928	−0.140427
$553$	18.8611	0.802057
$554$	19.5903	0.832314
$555$	−9.38926	−0.398552
$556$	1.01592	0.0430846
$557$	4.91498	0.208254	0.104127	−	0.994564i	$-0.466795\pi$
$557$	4.91498	0.208254	0.104127	+	0.994564i	$0.466795\pi$
$558$	−4.42280	−0.187232
$559$	10.3286	0.436852
$560$	2.32820	0.0983845
$561$	−9.40595	−0.397119
$562$	−3.68199	−0.155315
$563$	25.5871	1.07837	0.539184	−	0.842188i	$-0.318733\pi$
$563$	25.5871	1.07837	0.539184	+	0.842188i	$0.318733\pi$
$564$	0.332635	0.0140065
$565$	1.76347	0.0741898
$566$	2.22974	0.0937228
$567$	−4.81059	−0.202026
$568$	−5.59843	−0.234905
$569$	−7.84544	−0.328898	−0.164449	−	0.986386i	$-0.552585\pi$
$569$	−7.84544	−0.328898	−0.164449	+	0.986386i	$0.552585\pi$
$570$	−3.55458	−0.148885
$571$	15.5426	0.650439	0.325220	−	0.945639i	$-0.394562\pi$
$571$	15.5426	0.650439	0.325220	+	0.945639i	$0.394562\pi$
$572$	−14.5545	−0.608552
$573$	−19.6230	−0.819761
$574$	22.1967	0.926472
$575$	−2.72164	−0.113500
$576$	−1.53047	−0.0637698
$577$	−22.2381	−0.925783	−0.462892	−	0.886415i	$-0.653188\pi$
$577$	−22.2381	−0.925783	−0.462892	+	0.886415i	$0.653188\pi$
$578$	9.39607	0.390825
$579$	−2.54015	−0.105565
$580$	1.77083	0.0735297
$581$	−8.17595	−0.339196
$582$	15.3005	0.634227
$583$	−24.2555	−1.00456
$584$	−2.36578	−0.0978968
$585$	7.91639	0.327302
$586$	−14.3486	−0.592733
$587$	−21.5746	−0.890481	−0.445240	−	0.895411i	$-0.646882\pi$
$587$	−21.5746	−0.890481	−0.445240	+	0.895411i	$0.646882\pi$
$588$	1.91470	0.0789607
$589$	8.47365	0.349151
$590$	−4.64325	−0.191160
$591$	6.09076	0.250540
$592$	7.74538	0.318333
$593$	18.3814	0.754835	0.377418	−	0.926043i	$-0.376812\pi$
$593$	18.3814	0.754835	0.377418	+	0.926043i	$0.376812\pi$
$594$	−15.4535	−0.634065
$595$	6.42008	0.263197
$596$	17.0533	0.698529
$597$	15.0038	0.614063
$598$	−14.0777	−0.575680
$599$	8.70074	0.355503	0.177751	−	0.984075i	$-0.443118\pi$
$599$	8.70074	0.355503	0.177751	+	0.984075i	$0.443118\pi$
$600$	1.21224	0.0494895
$601$	−1.00000	−0.0407909
$601$	−1.00000	−0.0407909
$602$	4.64900	0.189479
$603$	0.613851	0.0249979
$604$	13.4725	0.548190
$605$	−3.08247	−0.125320
$606$	17.0243	0.691565
$607$	36.8531	1.49582	0.747910	−	0.663800i	$-0.231057\pi$
$607$	36.8531	1.49582	0.747910	+	0.663800i	$0.231057\pi$
$608$	2.93224	0.118918
$609$	−4.99788	−0.202524
$610$	−13.5020	−0.546678
$611$	1.41932	0.0574196
$612$	−4.22032	−0.170596
$613$	−32.5481	−1.31461	−0.657304	−	0.753626i	$-0.728303\pi$
$613$	−32.5481	−1.31461	−0.657304	+	0.753626i	$0.728303\pi$
$614$	1.66944	0.0673732
$615$	11.5573	0.466035
$616$	−6.55112	−0.263952
$617$	21.9410	0.883312	0.441656	−	0.897184i	$-0.354391\pi$
$617$	21.9410	0.883312	0.441656	+	0.897184i	$0.354391\pi$
$618$	3.75047	0.150866
$619$	1.58368	0.0636535	0.0318267	−	0.999493i	$-0.489868\pi$
$619$	1.58368	0.0636535	0.0318267	+	0.999493i	$0.489868\pi$
$620$	−2.88982	−0.116058
$621$	−14.9473	−0.599815
$622$	−34.3210	−1.37615
$623$	35.6616	1.42875
$624$	6.27032	0.251014
$625$	1.00000	0.0400000
$626$	−16.2482	−0.649410
$627$	10.0019	0.399437
$628$	−16.7960	−0.670234
$629$	21.3581	0.851602
$630$	3.56326	0.141963
$631$	−33.1249	−1.31868	−0.659341	−	0.751844i	$-0.729165\pi$
$631$	−33.1249	−1.31868	−0.659341	+	0.751844i	$0.729165\pi$
$632$	−8.10115	−0.322246
$633$	−5.69220	−0.226244
$634$	1.24915	0.0496102
$635$	−5.40703	−0.214572
$636$	10.4497	0.414357
$637$	8.16982	0.323700
$638$	−4.98278	−0.197270
$639$	−8.56825	−0.338955
$640$	−1.00000	−0.0395285
$641$	47.0186	1.85712	0.928561	−	0.371179i	$-0.121047\pi$
$641$	47.0186	1.85712	0.928561	+	0.371179i	$0.121047\pi$
$642$	−11.2980	−0.445896
$643$	20.8773	0.823320	0.411660	−	0.911338i	$-0.364949\pi$
$643$	20.8773	0.823320	0.411660	+	0.911338i	$0.364949\pi$
$644$	−6.33653	−0.249694
$645$	2.42063	0.0953120
$646$	8.08571	0.318128
$647$	11.9246	0.468806	0.234403	−	0.972140i	$-0.424687\pi$
$647$	11.9246	0.468806	0.234403	+	0.972140i	$0.424687\pi$
$648$	2.06623	0.0811690
$649$	13.0652	0.512855
$650$	5.17251	0.202882
$651$	8.15607	0.319661
$652$	−7.47065	−0.292573
$653$	−36.6127	−1.43277	−0.716383	−	0.697707i	$-0.754204\pi$
$653$	−36.6127	−1.43277	−0.716383	+	0.697707i	$0.754204\pi$
$654$	25.0124	0.978061
$655$	−16.8462	−0.658235
$656$	−9.53383	−0.372233
$657$	−3.62077	−0.141260
$658$	0.638853	0.0249051
$659$	6.94324	0.270470	0.135235	−	0.990814i	$-0.456821\pi$
$659$	6.94324	0.270470	0.135235	+	0.990814i	$0.456821\pi$
$660$	−3.41101	−0.132773
$661$	8.97482	0.349080	0.174540	−	0.984650i	$-0.444156\pi$
$661$	8.97482	0.349080	0.174540	+	0.984650i	$0.444156\pi$
$662$	−34.5850	−1.34418
$663$	17.2906	0.671510
$664$	3.51170	0.136280
$665$	−6.82684	−0.264734
$666$	11.8541	0.459337
$667$	−4.81956	−0.186614
$668$	5.86097	0.226768
$669$	−29.0012	−1.12125
$670$	0.401085	0.0154953
$671$	37.9919	1.46666
$672$	2.82234	0.108874
$673$	−20.7173	−0.798594	−0.399297	−	0.916822i	$-0.630746\pi$
$673$	−20.7173	−0.798594	−0.399297	+	0.916822i	$0.630746\pi$
$674$	−5.83893	−0.224907
$675$	5.49202	0.211388
$676$	13.7548	0.529032
$677$	27.8621	1.07083	0.535414	−	0.844589i	$-0.320155\pi$
$677$	27.8621	1.07083	0.535414	+	0.844589i	$0.320155\pi$
$678$	2.13775	0.0820998
$679$	29.3859	1.12773
$680$	−2.75752	−0.105746
$681$	−28.2744	−1.08348
$682$	8.13141	0.311368
$683$	−29.2973	−1.12103	−0.560516	−	0.828144i	$-0.689397\pi$
$683$	−29.2973	−1.12103	−0.560516	+	0.828144i	$0.689397\pi$
$684$	4.48771	0.171592
$685$	−2.72810	−0.104235
$686$	19.9747	0.762639
$687$	1.66126	0.0633810
$688$	−1.99682	−0.0761281
$689$	44.5878	1.69866
$690$	−3.29928	−0.125601
$691$	−41.6866	−1.58583	−0.792917	−	0.609330i	$-0.791439\pi$
$691$	−41.6866	−1.58583	−0.792917	+	0.609330i	$0.791439\pi$
$692$	−8.62481	−0.327866
$693$	−10.0263	−0.380868
$694$	−11.0791	−0.420556
$695$	1.01592	0.0385360
$696$	2.14667	0.0813693
$697$	−26.2898	−0.995796
$698$	20.1301	0.761937
$699$	31.1391	1.17779
$700$	2.32820	0.0879978
$701$	−11.0054	−0.415667	−0.207833	−	0.978164i	$-0.566641\pi$
$701$	−11.0054	−0.415667	−0.207833	+	0.978164i	$0.566641\pi$
$702$	28.4075	1.07217
$703$	−22.7113	−0.856573
$704$	2.81381	0.106049
$705$	0.332635	0.0125278
$706$	−13.0234	−0.490141
$707$	32.6965	1.22968
$708$	−5.62873	−0.211541
$709$	16.1813	0.607701	0.303851	−	0.952720i	$-0.401728\pi$
$709$	16.1813	0.607701	0.303851	+	0.952720i	$0.401728\pi$
$710$	−5.59843	−0.210105
$711$	−12.3986	−0.464984
$712$	−15.3172	−0.574038
$713$	7.86506	0.294549
$714$	7.78267	0.291259
$715$	−14.5545	−0.544306
$716$	−15.3420	−0.573357
$717$	−24.6947	−0.922242
$718$	20.8587	0.778442
$719$	19.8163	0.739022	0.369511	−	0.929226i	$-0.379525\pi$
$719$	19.8163	0.739022	0.369511	+	0.929226i	$0.379525\pi$
$720$	−1.53047	−0.0570374
$721$	7.20307	0.268256
$722$	10.4020	0.387122
$723$	19.0610	0.708885
$724$	−21.0159	−0.781051
$725$	1.77083	0.0657669
$726$	−3.73670	−0.138682
$727$	−4.45995	−0.165410	−0.0827052	−	0.996574i	$-0.526356\pi$
$727$	−4.45995	−0.165410	−0.0827052	+	0.996574i	$0.526356\pi$
$728$	12.0427	0.446330
$729$	23.1353	0.856861
$730$	−2.36578	−0.0875615
$731$	−5.50628	−0.203657
$732$	−16.3676	−0.604964
$733$	4.11177	0.151872	0.0759359	−	0.997113i	$-0.475806\pi$
$733$	4.11177	0.151872	0.0759359	+	0.997113i	$0.475806\pi$
$734$	26.0064	0.959913
$735$	1.91470	0.0706246
$736$	2.72164	0.100321
$737$	−1.12858	−0.0415717
$738$	−14.5913	−0.537112
$739$	41.5997	1.53027	0.765135	−	0.643870i	$-0.222672\pi$
$739$	41.5997	1.53027	0.765135	+	0.643870i	$0.222672\pi$
$740$	7.74538	0.284726
$741$	−18.3861	−0.675429
$742$	20.0695	0.736773
$743$	54.1954	1.98824	0.994118	−	0.108299i	$-0.0345405\pi$
$743$	54.1954	1.98824	0.994118	+	0.108299i	$0.0345405\pi$
$744$	−3.50316	−0.128432
$745$	17.0533	0.624783
$746$	−32.2760	−1.18171
$747$	5.37457	0.196645
$748$	7.75915	0.283702
$749$	−21.6987	−0.792853
$750$	1.21224	0.0442647
$751$	4.05712	0.148046	0.0740232	−	0.997257i	$-0.476416\pi$
$751$	4.05712	0.148046	0.0740232	+	0.997257i	$0.476416\pi$
$752$	−0.274397	−0.0100062
$753$	−20.1389	−0.733901
$754$	9.15963	0.333574
$755$	13.4725	0.490316
$756$	12.7865	0.465042
$757$	−33.6909	−1.22452	−0.612259	−	0.790658i	$-0.709739\pi$
$757$	−33.6909	−1.22452	−0.612259	+	0.790658i	$0.709739\pi$
$758$	−0.0405680	−0.00147350
$759$	9.28355	0.336971
$760$	2.93224	0.106363
$761$	37.5564	1.36142	0.680710	−	0.732553i	$-0.261671\pi$
$761$	37.5564	1.36142	0.680710	+	0.732553i	$0.261671\pi$
$762$	−6.55462	−0.237449
$763$	48.0382	1.73910
$764$	16.1874	0.585638
$765$	−4.22032	−0.152586
$766$	−30.0361	−1.08525
$767$	−24.0173	−0.867213
$768$	−1.21224	−0.0437429
$769$	−45.3288	−1.63460	−0.817299	−	0.576214i	$-0.804529\pi$
$769$	−45.3288	−1.63460	−0.817299	+	0.576214i	$0.804529\pi$
$770$	−6.55112	−0.236086
$771$	−20.8047	−0.749262
$772$	2.09542	0.0754158
$773$	−19.9792	−0.718601	−0.359300	−	0.933222i	$-0.616985\pi$
$773$	−19.9792	−0.718601	−0.359300	+	0.933222i	$0.616985\pi$
$774$	−3.05608	−0.109849
$775$	−2.88982	−0.103806
$776$	−12.6217	−0.453092
$777$	−21.8601	−0.784227
$778$	16.0849	0.576673
$779$	27.9555	1.00161
$780$	6.27032	0.224514
$781$	15.7529	0.563684
$782$	7.50499	0.268378
$783$	9.72543	0.347559
$784$	−1.57947	−0.0564096
$785$	−16.7960	−0.599476
$786$	−20.4216	−0.728416
$787$	−17.7292	−0.631977	−0.315989	−	0.948763i	$-0.602336\pi$
$787$	−17.7292	−0.631977	−0.315989	+	0.948763i	$0.602336\pi$
$788$	−5.02438	−0.178986
$789$	4.45811	0.158713
$790$	−8.10115	−0.288226
$791$	4.10572	0.145983
$792$	4.30646	0.153024
$793$	−69.8390	−2.48005
$794$	12.3909	0.439737
$795$	10.4497	0.370612
$796$	−12.3769	−0.438687
$797$	−49.5283	−1.75438	−0.877192	−	0.480140i	$-0.840586\pi$
$797$	−49.5283	−1.75438	−0.877192	+	0.480140i	$0.840586\pi$
$798$	−8.27577	−0.292959
$799$	−0.756657	−0.0267686
$800$	−1.00000	−0.0353553
$801$	−23.4426	−0.828305
$802$	−14.4029	−0.508583
$803$	6.65686	0.234916
$804$	0.486212	0.0171474
$805$	−6.33653	−0.223333
$806$	−14.9476	−0.526508
$807$	−28.3451	−0.997794
$808$	−14.0437	−0.494055
$809$	−31.3877	−1.10353	−0.551766	−	0.833999i	$-0.686046\pi$
$809$	−31.3877	−1.10353	−0.551766	+	0.833999i	$0.686046\pi$
$810$	2.06623	0.0725998
$811$	17.2391	0.605348	0.302674	−	0.953094i	$-0.402121\pi$
$811$	17.2391	0.605348	0.302674	+	0.953094i	$0.402121\pi$
$812$	4.12285	0.144684
$813$	−23.0408	−0.808076
$814$	−21.7940	−0.763880
$815$	−7.47065	−0.261686
$816$	−3.34278	−0.117021
$817$	5.85515	0.204846
$818$	26.5854	0.929537
$819$	18.4310	0.644030
$820$	−9.53383	−0.332936
$821$	−2.54462	−0.0888080	−0.0444040	−	0.999014i	$-0.514139\pi$
$821$	−2.54462	−0.0888080	−0.0444040	+	0.999014i	$0.514139\pi$
$822$	−3.30711	−0.115349
$823$	−34.1394	−1.19003	−0.595013	−	0.803716i	$-0.702853\pi$
$823$	−34.1394	−1.19003	−0.595013	+	0.803716i	$0.702853\pi$
$824$	−3.09383	−0.107779
$825$	−3.41101	−0.118756
$826$	−10.8104	−0.376143
$827$	47.4803	1.65105	0.825527	−	0.564363i	$-0.190878\pi$
$827$	47.4803	1.65105	0.825527	+	0.564363i	$0.190878\pi$
$828$	4.16540	0.144758
$829$	56.7793	1.97203	0.986013	−	0.166669i	$-0.0533012\pi$
$829$	56.7793	1.97203	0.986013	+	0.166669i	$0.0533012\pi$
$830$	3.51170	0.121893
$831$	23.7482	0.823816
$832$	−5.17251	−0.179324
$833$	−4.35542	−0.150906
$834$	1.23154	0.0426447
$835$	5.86097	0.202827
$836$	−8.25076	−0.285358
$837$	−15.8710	−0.548581
$838$	−13.7919	−0.476433
$839$	−21.1236	−0.729267	−0.364633	−	0.931151i	$-0.618806\pi$
$839$	−21.1236	−0.729267	−0.364633	+	0.931151i	$0.618806\pi$
$840$	2.82234	0.0973800
$841$	−25.8642	−0.891868
$842$	−23.3922	−0.806149
$843$	−4.46345	−0.153729
$844$	4.69560	0.161629
$845$	13.7548	0.473181
$846$	−0.419958	−0.0144385
$847$	−7.17663	−0.246592
$848$	−8.62015	−0.296017
$849$	2.70298	0.0927659
$850$	−2.75752	−0.0945823
$851$	−21.0801	−0.722618
$852$	−6.78664	−0.232506
$853$	57.1452	1.95661	0.978307	−	0.207158i	$-0.0664214\pi$
$853$	57.1452	1.95661	0.978307	+	0.207158i	$0.0664214\pi$
$854$	−31.4353	−1.07569
$855$	4.48771	0.153477
$856$	9.31993	0.318549
$857$	−3.23980	−0.110669	−0.0553347	−	0.998468i	$-0.517623\pi$
$857$	−3.23980	−0.110669	−0.0553347	+	0.998468i	$0.517623\pi$
$858$	−17.6435	−0.602339
$859$	55.6599	1.89909	0.949545	−	0.313631i	$-0.101546\pi$
$859$	55.6599	1.89909	0.949545	+	0.313631i	$0.101546\pi$
$860$	−1.99682	−0.0680910
$861$	26.9077	0.917013
$862$	−4.77496	−0.162636
$863$	−42.2410	−1.43790	−0.718950	−	0.695061i	$-0.755377\pi$
$863$	−42.2410	−1.43790	−0.718950	+	0.695061i	$0.755377\pi$
$864$	−5.49202	−0.186842
$865$	−8.62481	−0.293252
$866$	6.30177	0.214143
$867$	11.3903	0.386834
$868$	−6.72810	−0.228366
$869$	22.7951	0.773271
$870$	2.14667	0.0727789
$871$	2.07462	0.0702957
$872$	−20.6332	−0.698728
$873$	−19.3172	−0.653787
$874$	−7.98049	−0.269944
$875$	2.32820	0.0787076
$876$	−2.86790	−0.0968972
$877$	−29.0672	−0.981528	−0.490764	−	0.871292i	$-0.663282\pi$
$877$	−29.0672	−0.981528	−0.490764	+	0.871292i	$0.663282\pi$
$878$	2.35656	0.0795301
$879$	−17.3939	−0.586681
$880$	2.81381	0.0948535
$881$	32.9381	1.10971	0.554857	−	0.831946i	$-0.312773\pi$
$881$	32.9381	1.10971	0.554857	+	0.831946i	$0.312773\pi$
$882$	−2.41734	−0.0813960
$883$	−38.4275	−1.29319	−0.646594	−	0.762835i	$-0.723807\pi$
$883$	−38.4275	−1.29319	−0.646594	+	0.762835i	$0.723807\pi$
$884$	−14.2633	−0.479727
$885$	−5.62873	−0.189208
$886$	−37.9229	−1.27404
$887$	7.98579	0.268136	0.134068	−	0.990972i	$-0.457196\pi$
$887$	7.98579	0.268136	0.134068	+	0.990972i	$0.457196\pi$
$888$	9.38926	0.315083
$889$	−12.5887	−0.422211
$890$	−15.3172	−0.513435
$891$	−5.81397	−0.194775
$892$	23.9237	0.801024
$893$	0.804598	0.0269248
$894$	20.6726	0.691397
$895$	−15.3420	−0.512826
$896$	−2.32820	−0.0777798
$897$	−17.0656	−0.569802
$898$	−14.3889	−0.480162
$899$	−5.11738	−0.170674
$900$	−1.53047	−0.0510158
$901$	−23.7703	−0.791902
$902$	26.8264	0.893221
$903$	5.63571	0.187545
$904$	−1.76347	−0.0586522
$905$	−21.0159	−0.698593
$906$	16.3320	0.542593
$907$	26.1402	0.867970	0.433985	−	0.900920i	$-0.357107\pi$
$907$	26.1402	0.867970	0.433985	+	0.900920i	$0.357107\pi$
$908$	23.3241	0.774039
$909$	−21.4935	−0.712894
$910$	12.0427	0.399210
$911$	−18.7090	−0.619857	−0.309929	−	0.950760i	$-0.600305\pi$
$911$	−18.7090	−0.619857	−0.309929	+	0.950760i	$0.600305\pi$
$912$	3.55458	0.117704
$913$	−9.88126	−0.327022
$914$	11.4413	0.378446
$915$	−16.3676	−0.541096
$916$	−1.37040	−0.0452794
$917$	−39.2214	−1.29520
$918$	−15.1444	−0.499839
$919$	−46.1488	−1.52231	−0.761154	−	0.648571i	$-0.775367\pi$
$919$	−46.1488	−1.52231	−0.761154	+	0.648571i	$0.775367\pi$
$920$	2.72164	0.0897298
$921$	2.02376	0.0666853
$922$	−26.4040	−0.869568
$923$	−28.9579	−0.953161
$924$	−7.94153	−0.261257
$925$	7.74538	0.254666
$926$	17.1908	0.564926
$927$	−4.73503	−0.155519
$928$	−1.77083	−0.0581303
$929$	−30.5224	−1.00141	−0.500703	−	0.865619i	$-0.666925\pi$
$929$	−30.5224	−1.00141	−0.500703	+	0.865619i	$0.666925\pi$
$930$	−3.50316	−0.114873
$931$	4.63138	0.151787
$932$	−25.6872	−0.841414
$933$	−41.6053	−1.36210
$934$	7.20861	0.235873
$935$	7.75915	0.253751
$936$	−7.91639	−0.258755
$937$	52.6935	1.72142	0.860711	−	0.509093i	$-0.170019\pi$
$937$	52.6935	1.72142	0.860711	+	0.509093i	$0.170019\pi$
$938$	0.933808	0.0304899
$939$	−19.6968	−0.642780
$940$	−0.274397	−0.00894985
$941$	32.7698	1.06827	0.534133	−	0.845400i	$-0.320638\pi$
$941$	32.7698	1.06827	0.534133	+	0.845400i	$0.320638\pi$
$942$	−20.3608	−0.663391
$943$	25.9476	0.844972
$944$	4.64325	0.151125
$945$	12.7865	0.415946
$946$	5.61867	0.182679
$947$	12.4140	0.403399	0.201700	−	0.979447i	$-0.435354\pi$
$947$	12.4140	0.403399	0.201700	+	0.979447i	$0.435354\pi$
$948$	−9.82054	−0.318956
$949$	−12.2370	−0.397231
$950$	2.93224	0.0951343
$951$	1.51427	0.0491037
$952$	−6.42008	−0.208076
$953$	−3.83353	−0.124180	−0.0620901	−	0.998071i	$-0.519777\pi$
$953$	−3.83353	−0.124180	−0.0620901	+	0.998071i	$0.519777\pi$
$954$	−13.1929	−0.427137
$955$	16.1874	0.523810
$956$	20.3712	0.658850
$957$	−6.04032	−0.195256
$958$	−26.2192	−0.847103
$959$	−6.35156	−0.205103
$960$	−1.21224	−0.0391249
$961$	−22.6489	−0.730610
$962$	40.0630	1.29168
$963$	14.2639	0.459648
$964$	−15.7238	−0.506428
$965$	2.09542	0.0674539
$966$	−7.68140	−0.247145
$967$	13.5858	0.436889	0.218445	−	0.975849i	$-0.429902\pi$
$967$	13.5858	0.436889	0.218445	+	0.975849i	$0.429902\pi$
$968$	3.08247	0.0990745
$969$	9.80182	0.314880
$970$	−12.6217	−0.405258
$971$	16.5935	0.532510	0.266255	−	0.963903i	$-0.414214\pi$
$971$	16.5935	0.532510	0.266255	+	0.963903i	$0.414214\pi$
$972$	−13.9713	−0.448130
$973$	2.36527	0.0758270
$974$	1.74683	0.0559720
$975$	6.27032	0.200811
$976$	13.5020	0.432187
$977$	−45.1349	−1.44399	−0.721997	−	0.691897i	$-0.756775\pi$
$977$	−45.1349	−1.44399	−0.721997	+	0.691897i	$0.756775\pi$
$978$	−9.05623	−0.289586
$979$	43.0998	1.37747
$980$	−1.57947	−0.0504543
$981$	−31.5785	−1.00823
$982$	−16.7032	−0.533022
$983$	−8.41747	−0.268476	−0.134238	−	0.990949i	$-0.542859\pi$
$983$	−8.41747	−0.268476	−0.134238	+	0.990949i	$0.542859\pi$
$984$	−11.5573	−0.368433
$985$	−5.02438	−0.160090
$986$	−4.88310	−0.155510
$987$	0.774443	0.0246508
$988$	15.1670	0.482527
$989$	5.43463	0.172811
$990$	4.30646	0.136868
$991$	10.3690	0.329383	0.164691	−	0.986345i	$-0.447337\pi$
$991$	10.3690	0.329383	0.164691	+	0.986345i	$0.447337\pi$
$992$	2.88982	0.0917520
$993$	−41.9253	−1.33046
$994$	−13.0343	−0.413422
$995$	−12.3769	−0.392374
$996$	4.25702	0.134889
$997$	−8.43355	−0.267093	−0.133547	−	0.991043i	$-0.542637\pi$
$997$	−8.43355	−0.267093	−0.133547	+	0.991043i	$0.542637\pi$
$998$	−19.9902	−0.632779
$999$	42.5378	1.34584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6010.2.a.g.1.8	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6010.2.a.g.1.8	✓	27	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:	\( N \)	\(=\)	\( 6010 = 2 \cdot 5 \cdot 601 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6010.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.21224 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.21224 q^{6} +2.32820 q^{7} -1.00000 q^{8} -1.53047 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.21224 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.21224 q^{6} +2.32820 q^{7} -1.00000 q^{8} -1.53047 q^{9} -1.00000 q^{10} +2.81381 q^{11} -1.21224 q^{12} -5.17251 q^{13} -2.32820 q^{14} -1.21224 q^{15} +1.00000 q^{16} +2.75752 q^{17} +1.53047 q^{18} -2.93224 q^{19} +1.00000 q^{20} -2.82234 q^{21} -2.81381 q^{22} -2.72164 q^{23} +1.21224 q^{24} +1.00000 q^{25} +5.17251 q^{26} +5.49202 q^{27} +2.32820 q^{28} +1.77083 q^{29} +1.21224 q^{30} -2.88982 q^{31} -1.00000 q^{32} -3.41101 q^{33} -2.75752 q^{34} +2.32820 q^{35} -1.53047 q^{36} +7.74538 q^{37} +2.93224 q^{38} +6.27032 q^{39} -1.00000 q^{40} -9.53383 q^{41} +2.82234 q^{42} -1.99682 q^{43} +2.81381 q^{44} -1.53047 q^{45} +2.72164 q^{46} -0.274397 q^{47} -1.21224 q^{48} -1.57947 q^{49} -1.00000 q^{50} -3.34278 q^{51} -5.17251 q^{52} -8.62015 q^{53} -5.49202 q^{54} +2.81381 q^{55} -2.32820 q^{56} +3.55458 q^{57} -1.77083 q^{58} +4.64325 q^{59} -1.21224 q^{60} +13.5020 q^{61} +2.88982 q^{62} -3.56326 q^{63} +1.00000 q^{64} -5.17251 q^{65} +3.41101 q^{66} -0.401085 q^{67} +2.75752 q^{68} +3.29928 q^{69} -2.32820 q^{70} +5.59843 q^{71} +1.53047 q^{72} +2.36578 q^{73} -7.74538 q^{74} -1.21224 q^{75} -2.93224 q^{76} +6.55112 q^{77} -6.27032 q^{78} +8.10115 q^{79} +1.00000 q^{80} -2.06623 q^{81} +9.53383 q^{82} -3.51170 q^{83} -2.82234 q^{84} +2.75752 q^{85} +1.99682 q^{86} -2.14667 q^{87} -2.81381 q^{88} +15.3172 q^{89} +1.53047 q^{90} -12.0427 q^{91} -2.72164 q^{92} +3.50316 q^{93} +0.274397 q^{94} -2.93224 q^{95} +1.21224 q^{96} +12.6217 q^{97} +1.57947 q^{98} -4.30646 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 27 q^{2} + 6 q^{3} + 27 q^{4} + 27 q^{5} - 6 q^{6} - 27 q^{8} + 37 q^{9}+O(q^{10}) \) 27 * q - 27 * q^2 + 6 * q^3 + 27 * q^4 + 27 * q^5 - 6 * q^6 - 27 * q^8 + 37 * q^9	\( 27 q - 27 q^{2} + 6 q^{3} + 27 q^{4} + 27 q^{5} - 6 q^{6} - 27 q^{8} + 37 q^{9} - 27 q^{10} + 18 q^{11} + 6 q^{12} - 6 q^{13} + 6 q^{15} + 27 q^{16} + 3 q^{17} - 37 q^{18} + 27 q^{19} + 27 q^{20} + 16 q^{21} - 18 q^{22} + 15 q^{23} - 6 q^{24} + 27 q^{25} + 6 q^{26} + 27 q^{27} + 25 q^{29} - 6 q^{30} + 9 q^{31} - 27 q^{32} + 11 q^{33} - 3 q^{34} + 37 q^{36} - 16 q^{37} - 27 q^{38} + 20 q^{39} - 27 q^{40} + 39 q^{41} - 16 q^{42} + 9 q^{43} + 18 q^{44} + 37 q^{45} - 15 q^{46} + 31 q^{47} + 6 q^{48} + 27 q^{49} - 27 q^{50} + 39 q^{51} - 6 q^{52} - 5 q^{53} - 27 q^{54} + 18 q^{55} - 10 q^{57} - 25 q^{58} + 46 q^{59} + 6 q^{60} + 18 q^{61} - 9 q^{62} + 23 q^{63} + 27 q^{64} - 6 q^{65} - 11 q^{66} + 11 q^{67} + 3 q^{68} + 17 q^{69} + 50 q^{71} - 37 q^{72} - 29 q^{73} + 16 q^{74} + 6 q^{75} + 27 q^{76} - 6 q^{77} - 20 q^{78} + 56 q^{79} + 27 q^{80} + 51 q^{81} - 39 q^{82} + 44 q^{83} + 16 q^{84} + 3 q^{85} - 9 q^{86} + 42 q^{87} - 18 q^{88} + 34 q^{89} - 37 q^{90} + 43 q^{91} + 15 q^{92} - 20 q^{93} - 31 q^{94} + 27 q^{95} - 6 q^{96} - 37 q^{97} - 27 q^{98} + 67 q^{99}+O(q^{100}) \) 27 * q - 27 * q^2 + 6 * q^3 + 27 * q^4 + 27 * q^5 - 6 * q^6 - 27 * q^8 + 37 * q^9 - 27 * q^10 + 18 * q^11 + 6 * q^12 - 6 * q^13 + 6 * q^15 + 27 * q^16 + 3 * q^17 - 37 * q^18 + 27 * q^19 + 27 * q^20 + 16 * q^21 - 18 * q^22 + 15 * q^23 - 6 * q^24 + 27 * q^25 + 6 * q^26 + 27 * q^27 + 25 * q^29 - 6 * q^30 + 9 * q^31 - 27 * q^32 + 11 * q^33 - 3 * q^34 + 37 * q^36 - 16 * q^37 - 27 * q^38 + 20 * q^39 - 27 * q^40 + 39 * q^41 - 16 * q^42 + 9 * q^43 + 18 * q^44 + 37 * q^45 - 15 * q^46 + 31 * q^47 + 6 * q^48 + 27 * q^49 - 27 * q^50 + 39 * q^51 - 6 * q^52 - 5 * q^53 - 27 * q^54 + 18 * q^55 - 10 * q^57 - 25 * q^58 + 46 * q^59 + 6 * q^60 + 18 * q^61 - 9 * q^62 + 23 * q^63 + 27 * q^64 - 6 * q^65 - 11 * q^66 + 11 * q^67 + 3 * q^68 + 17 * q^69 + 50 * q^71 - 37 * q^72 - 29 * q^73 + 16 * q^74 + 6 * q^75 + 27 * q^76 - 6 * q^77 - 20 * q^78 + 56 * q^79 + 27 * q^80 + 51 * q^81 - 39 * q^82 + 44 * q^83 + 16 * q^84 + 3 * q^85 - 9 * q^86 + 42 * q^87 - 18 * q^88 + 34 * q^89 - 37 * q^90 + 43 * q^91 + 15 * q^92 - 20 * q^93 - 31 * q^94 + 27 * q^95 - 6 * q^96 - 37 * q^97 - 27 * q^98 + 67 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more