LMFDB - Embedded newform 8015.2.a.l.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.78055 q^{2} +0.298033 q^{3} +5.73146 q^{4} -1.00000 q^{5} -0.828697 q^{6} -1.00000 q^{7} -10.3755 q^{8} -2.91118 q^{9} +O(q^{10})$	\(q-2.78055 q^{2} +0.298033 q^{3} +5.73146 q^{4} -1.00000 q^{5} -0.828697 q^{6} -1.00000 q^{7} -10.3755 q^{8} -2.91118 q^{9} +2.78055 q^{10} +1.93021 q^{11} +1.70817 q^{12} +0.800858 q^{13} +2.78055 q^{14} -0.298033 q^{15} +17.3867 q^{16} +0.984021 q^{17} +8.09467 q^{18} -7.86576 q^{19} -5.73146 q^{20} -0.298033 q^{21} -5.36704 q^{22} +3.94812 q^{23} -3.09225 q^{24} +1.00000 q^{25} -2.22683 q^{26} -1.76173 q^{27} -5.73146 q^{28} -9.46879 q^{29} +0.828697 q^{30} -2.41133 q^{31} -27.5937 q^{32} +0.575266 q^{33} -2.73612 q^{34} +1.00000 q^{35} -16.6853 q^{36} +9.35326 q^{37} +21.8711 q^{38} +0.238682 q^{39} +10.3755 q^{40} -4.16333 q^{41} +0.828697 q^{42} -6.96297 q^{43} +11.0629 q^{44} +2.91118 q^{45} -10.9779 q^{46} -8.50129 q^{47} +5.18183 q^{48} +1.00000 q^{49} -2.78055 q^{50} +0.293271 q^{51} +4.59009 q^{52} +5.63325 q^{53} +4.89857 q^{54} -1.93021 q^{55} +10.3755 q^{56} -2.34426 q^{57} +26.3285 q^{58} +1.48487 q^{59} -1.70817 q^{60} +8.07188 q^{61} +6.70483 q^{62} +2.91118 q^{63} +41.9521 q^{64} -0.800858 q^{65} -1.59956 q^{66} -13.2763 q^{67} +5.63988 q^{68} +1.17667 q^{69} -2.78055 q^{70} -3.56191 q^{71} +30.2050 q^{72} +13.7107 q^{73} -26.0072 q^{74} +0.298033 q^{75} -45.0823 q^{76} -1.93021 q^{77} -0.663669 q^{78} -4.47244 q^{79} -17.3867 q^{80} +8.20847 q^{81} +11.5764 q^{82} -7.65176 q^{83} -1.70817 q^{84} -0.984021 q^{85} +19.3609 q^{86} -2.82202 q^{87} -20.0269 q^{88} +5.62676 q^{89} -8.09467 q^{90} -0.800858 q^{91} +22.6285 q^{92} -0.718657 q^{93} +23.6383 q^{94} +7.86576 q^{95} -8.22384 q^{96} +18.7585 q^{97} -2.78055 q^{98} -5.61917 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.78055	−1.96615	−0.983073	−	0.183213i	$-0.941350\pi$
$2$	−2.78055	−1.96615	−0.983073	+	0.183213i	$0.941350\pi$
$3$	0.298033	0.172070	0.0860348	−	0.996292i	$-0.472580\pi$
$3$	0.298033	0.172070	0.0860348	+	0.996292i	$0.472580\pi$
$4$	5.73146	2.86573
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.828697	−0.338314
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−10.3755	−3.66830
$9$	−2.91118	−0.970392
$10$	2.78055	0.879287
$11$	1.93021	0.581979	0.290990	−	0.956726i	$-0.406015\pi$
$11$	1.93021	0.581979	0.290990	+	0.956726i	$0.406015\pi$
$12$	1.70817	0.493105
$13$	0.800858	0.222118	0.111059	−	0.993814i	$-0.464576\pi$
$13$	0.800858	0.222118	0.111059	+	0.993814i	$0.464576\pi$
$14$	2.78055	0.743133
$15$	−0.298033	−0.0769519
$16$	17.3867	4.34668
$17$	0.984021	0.238660	0.119330	−	0.992855i	$-0.461925\pi$
$17$	0.984021	0.238660	0.119330	+	0.992855i	$0.461925\pi$
$18$	8.09467	1.90793
$19$	−7.86576	−1.80453	−0.902264	−	0.431183i	$-0.858096\pi$
$19$	−7.86576	−1.80453	−0.902264	+	0.431183i	$0.858096\pi$
$20$	−5.73146	−1.28159
$21$	−0.298033	−0.0650362
$22$	−5.36704	−1.14426
$23$	3.94812	0.823239	0.411620	−	0.911356i	$-0.364963\pi$
$23$	3.94812	0.823239	0.411620	+	0.911356i	$0.364963\pi$
$24$	−3.09225	−0.631203
$25$	1.00000	0.200000
$26$	−2.22683	−0.436717
$27$	−1.76173	−0.339045
$28$	−5.73146	−1.08314
$29$	−9.46879	−1.75831	−0.879155	−	0.476535i	$-0.841892\pi$
$29$	−9.46879	−1.75831	−0.879155	+	0.476535i	$0.841892\pi$
$30$	0.828697	0.151299
$31$	−2.41133	−0.433088	−0.216544	−	0.976273i	$-0.569478\pi$
$31$	−2.41133	−0.433088	−0.216544	+	0.976273i	$0.569478\pi$
$32$	−27.5937	−4.87792
$33$	0.575266	0.100141
$34$	−2.73612	−0.469241
$35$	1.00000	0.169031
$36$	−16.6853	−2.78088
$37$	9.35326	1.53767	0.768833	−	0.639449i	$-0.220838\pi$
$37$	9.35326	1.53767	0.768833	+	0.639449i	$0.220838\pi$
$38$	21.8711	3.54797
$39$	0.238682	0.0382198
$40$	10.3755	1.64051
$41$	−4.16333	−0.650203	−0.325102	−	0.945679i	$-0.605398\pi$
$41$	−4.16333	−0.650203	−0.325102	+	0.945679i	$0.605398\pi$
$42$	0.828697	0.127871
$43$	−6.96297	−1.06184	−0.530922	−	0.847421i	$-0.678154\pi$
$43$	−6.96297	−1.06184	−0.530922	+	0.847421i	$0.678154\pi$
$44$	11.0629	1.66780
$45$	2.91118	0.433973
$46$	−10.9779	−1.61861
$47$	−8.50129	−1.24004	−0.620020	−	0.784586i	$-0.712876\pi$
$47$	−8.50129	−1.24004	−0.620020	+	0.784586i	$0.712876\pi$
$48$	5.18183	0.747933
$49$	1.00000	0.142857
$50$	−2.78055	−0.393229
$51$	0.293271	0.0410662
$52$	4.59009	0.636531
$53$	5.63325	0.773786	0.386893	−	0.922125i	$-0.373548\pi$
$53$	5.63325	0.773786	0.386893	+	0.922125i	$0.373548\pi$
$54$	4.89857	0.666612
$55$	−1.93021	−0.260269
$56$	10.3755	1.38649
$57$	−2.34426	−0.310505
$58$	26.3285	3.45710
$59$	1.48487	0.193313	0.0966567	−	0.995318i	$-0.469185\pi$
$59$	1.48487	0.193313	0.0966567	+	0.995318i	$0.469185\pi$
$60$	−1.70817	−0.220523
$61$	8.07188	1.03350	0.516749	−	0.856137i	$-0.327142\pi$
$61$	8.07188	1.03350	0.516749	+	0.856137i	$0.327142\pi$
$62$	6.70483	0.851514
$63$	2.91118	0.366774
$64$	41.9521	5.24401
$65$	−0.800858	−0.0993342
$66$	−1.59956	−0.196892
$67$	−13.2763	−1.62196	−0.810982	−	0.585071i	$-0.801066\pi$
$67$	−13.2763	−1.62196	−0.810982	+	0.585071i	$0.801066\pi$
$68$	5.63988	0.683936
$69$	1.17667	0.141655
$70$	−2.78055	−0.332339
$71$	−3.56191	−0.422720	−0.211360	−	0.977408i	$-0.567789\pi$
$71$	−3.56191	−0.422720	−0.211360	+	0.977408i	$0.567789\pi$
$72$	30.2050	3.55969
$73$	13.7107	1.60471	0.802357	−	0.596844i	$-0.203579\pi$
$73$	13.7107	1.60471	0.802357	+	0.596844i	$0.203579\pi$
$74$	−26.0072	−3.02328
$75$	0.298033	0.0344139
$76$	−45.0823	−5.17130
$77$	−1.93021	−0.219968
$78$	−0.663669	−0.0751457
$79$	−4.47244	−0.503189	−0.251594	−	0.967833i	$-0.580955\pi$
$79$	−4.47244	−0.503189	−0.251594	+	0.967833i	$0.580955\pi$
$80$	−17.3867	−1.94390
$81$	8.20847	0.912053
$82$	11.5764	1.27839
$83$	−7.65176	−0.839890	−0.419945	−	0.907549i	$-0.637951\pi$
$83$	−7.65176	−0.839890	−0.419945	+	0.907549i	$0.637951\pi$
$84$	−1.70817	−0.186376
$85$	−0.984021	−0.106732
$86$	19.3609	2.08774
$87$	−2.82202	−0.302552
$88$	−20.0269	−2.13488
$89$	5.62676	0.596436	0.298218	−	0.954498i	$-0.403608\pi$
$89$	5.62676	0.596436	0.298218	+	0.954498i	$0.403608\pi$
$90$	−8.09467	−0.853253
$91$	−0.800858	−0.0839527
$92$	22.6285	2.35918
$93$	−0.718657	−0.0745213
$94$	23.6383	2.43810
$95$	7.86576	0.807010
$96$	−8.22384	−0.839342
$97$	18.7585	1.90464	0.952319	−	0.305104i	$-0.0986913\pi$
$97$	18.7585	1.90464	0.952319	+	0.305104i	$0.0986913\pi$
$98$	−2.78055	−0.280878
$99$	−5.61917	−0.564748
$100$	5.73146	0.573146
$101$	−15.0894	−1.50145	−0.750724	−	0.660616i	$-0.770295\pi$
$101$	−15.0894	−1.50145	−0.750724	+	0.660616i	$0.770295\pi$
$102$	−0.815455	−0.0807421
$103$	3.53153	0.347972	0.173986	−	0.984748i	$-0.444335\pi$
$103$	3.53153	0.347972	0.173986	+	0.984748i	$0.444335\pi$
$104$	−8.30932	−0.814796
$105$	0.298033	0.0290851
$106$	−15.6635	−1.52138
$107$	−19.6309	−1.89779	−0.948894	−	0.315595i	$-0.897796\pi$
$107$	−19.6309	−1.89779	−0.948894	+	0.315595i	$0.897796\pi$
$108$	−10.0973	−0.971611
$109$	1.25448	0.120157	0.0600787	−	0.998194i	$-0.480865\pi$
$109$	1.25448	0.120157	0.0600787	+	0.998194i	$0.480865\pi$
$110$	5.36704	0.511727
$111$	2.78758	0.264586
$112$	−17.3867	−1.64289
$113$	−5.15218	−0.484676	−0.242338	−	0.970192i	$-0.577914\pi$
$113$	−5.15218	−0.484676	−0.242338	+	0.970192i	$0.577914\pi$
$114$	6.51833	0.610498
$115$	−3.94812	−0.368164
$116$	−54.2700	−5.03885
$117$	−2.33144	−0.215542
$118$	−4.12875	−0.380082
$119$	−0.984021	−0.0902050
$120$	3.09225	0.282283
$121$	−7.27430	−0.661300
$122$	−22.4443	−2.03201
$123$	−1.24081	−0.111880
$124$	−13.8204	−1.24111
$125$	−1.00000	−0.0894427
$126$	−8.09467	−0.721131
$127$	14.0867	1.24999	0.624995	−	0.780629i	$-0.285101\pi$
$127$	14.0867	1.24999	0.624995	+	0.780629i	$0.285101\pi$
$128$	−61.4626	−5.43258
$129$	−2.07520	−0.182711
$130$	2.22683	0.195306
$131$	−19.8616	−1.73532	−0.867660	−	0.497158i	$-0.834377\pi$
$131$	−19.8616	−1.73532	−0.867660	+	0.497158i	$0.834377\pi$
$132$	3.29712	0.286977
$133$	7.86576	0.682048
$134$	36.9155	3.18902
$135$	1.76173	0.151625
$136$	−10.2097	−0.875477
$137$	14.7868	1.26332	0.631662	−	0.775244i	$-0.282373\pi$
$137$	14.7868	1.26332	0.631662	+	0.775244i	$0.282373\pi$
$138$	−3.27179	−0.278514
$139$	−21.9510	−1.86186	−0.930931	−	0.365196i	$-0.881002\pi$
$139$	−21.9510	−1.86186	−0.930931	+	0.365196i	$0.881002\pi$
$140$	5.73146	0.484397
$141$	−2.53367	−0.213373
$142$	9.90406	0.831130
$143$	1.54582	0.129268
$144$	−50.6159	−4.21799
$145$	9.46879	0.786341
$146$	−38.1233	−3.15510
$147$	0.298033	0.0245814
$148$	53.6079	4.40654
$149$	13.3356	1.09250	0.546249	−	0.837623i	$-0.316055\pi$
$149$	13.3356	1.09250	0.546249	+	0.837623i	$0.316055\pi$
$150$	−0.828697	−0.0676628
$151$	−1.52962	−0.124479	−0.0622395	−	0.998061i	$-0.519824\pi$
$151$	−1.52962	−0.124479	−0.0622395	+	0.998061i	$0.519824\pi$
$152$	81.6114	6.61955
$153$	−2.86466	−0.231594
$154$	5.36704	0.432488
$155$	2.41133	0.193683
$156$	1.36800	0.109528
$157$	−13.7258	−1.09543	−0.547717	−	0.836663i	$-0.684503\pi$
$157$	−13.7258	−1.09543	−0.547717	+	0.836663i	$0.684503\pi$
$158$	12.4358	0.989343
$159$	1.67890	0.133145
$160$	27.5937	2.18147
$161$	−3.94812	−0.311155
$162$	−22.8241	−1.79323
$163$	13.3747	1.04759	0.523795	−	0.851845i	$-0.324516\pi$
$163$	13.3747	1.04759	0.523795	+	0.851845i	$0.324516\pi$
$164$	−23.8620	−1.86331
$165$	−0.575266	−0.0447844
$166$	21.2761	1.65135
$167$	2.92099	0.226033	0.113017	−	0.993593i	$-0.463949\pi$
$167$	2.92099	0.226033	0.113017	+	0.993593i	$0.463949\pi$
$168$	3.09225	0.238572
$169$	−12.3586	−0.950664
$170$	2.73612	0.209851
$171$	22.8986	1.75110
$172$	−39.9080	−3.04296
$173$	11.8602	0.901711	0.450856	−	0.892597i	$-0.351119\pi$
$173$	11.8602	0.901711	0.450856	+	0.892597i	$0.351119\pi$
$174$	7.84676	0.594861
$175$	−1.00000	−0.0755929
$176$	33.5600	2.52968
$177$	0.442540	0.0332634
$178$	−15.6455	−1.17268
$179$	−6.68431	−0.499609	−0.249805	−	0.968296i	$-0.580366\pi$
$179$	−6.68431	−0.499609	−0.249805	+	0.968296i	$0.580366\pi$
$180$	16.6853	1.24365
$181$	16.3937	1.21854	0.609268	−	0.792965i	$-0.291463\pi$
$181$	16.3937	1.21854	0.609268	+	0.792965i	$0.291463\pi$
$182$	2.22683	0.165063
$183$	2.40569	0.177834
$184$	−40.9638	−3.01989
$185$	−9.35326	−0.687665
$186$	1.99826	0.146520
$187$	1.89936	0.138895
$188$	−48.7248	−3.55362
$189$	1.76173	0.128147
$190$	−21.8711	−1.58670
$191$	10.8073	0.781986	0.390993	−	0.920394i	$-0.372132\pi$
$191$	10.8073	0.781986	0.390993	+	0.920394i	$0.372132\pi$
$192$	12.5031	0.902336
$193$	3.27495	0.235736	0.117868	−	0.993029i	$-0.462394\pi$
$193$	3.27495	0.235736	0.117868	+	0.993029i	$0.462394\pi$
$194$	−52.1590	−3.74480
$195$	−0.238682	−0.0170924
$196$	5.73146	0.409390
$197$	−3.98985	−0.284265	−0.142132	−	0.989848i	$-0.545396\pi$
$197$	−3.98985	−0.284265	−0.142132	+	0.989848i	$0.545396\pi$
$198$	15.6244	1.11038
$199$	9.65928	0.684728	0.342364	−	0.939567i	$-0.388772\pi$
$199$	9.65928	0.684728	0.342364	+	0.939567i	$0.388772\pi$
$200$	−10.3755	−0.733660
$201$	−3.95679	−0.279091
$202$	41.9567	2.95207
$203$	9.46879	0.664579
$204$	1.68087	0.117685
$205$	4.16333	0.290780
$206$	−9.81960	−0.684164
$207$	−11.4937	−0.798865
$208$	13.9243	0.965477
$209$	−15.1825	−1.05020
$210$	−0.828697	−0.0571855
$211$	−20.8322	−1.43415	−0.717075	−	0.696996i	$-0.754519\pi$
$211$	−20.8322	−1.43415	−0.717075	+	0.696996i	$0.754519\pi$
$212$	32.2867	2.21746
$213$	−1.06157	−0.0727374
$214$	54.5846	3.73133
$215$	6.96297	0.474871
$216$	18.2788	1.24372
$217$	2.41133	0.163692
$218$	−3.48815	−0.236247
$219$	4.08624	0.276123
$220$	−11.0629	−0.745861
$221$	0.788061	0.0530107
$222$	−7.75102	−0.520214
$223$	5.42935	0.363576	0.181788	−	0.983338i	$-0.441812\pi$
$223$	5.42935	0.363576	0.181788	+	0.983338i	$0.441812\pi$
$224$	27.5937	1.84368
$225$	−2.91118	−0.194078
$226$	14.3259	0.952945
$227$	7.57562	0.502812	0.251406	−	0.967882i	$-0.419107\pi$
$227$	7.57562	0.502812	0.251406	+	0.967882i	$0.419107\pi$
$228$	−13.4360	−0.889823
$229$	1.00000	0.0660819
$229$	1.00000	0.0660819
$230$	10.9779	0.723864
$231$	−0.575266	−0.0378497
$232$	98.2437	6.45001
$233$	−22.7454	−1.49010	−0.745050	−	0.667008i	$-0.767575\pi$
$233$	−22.7454	−1.49010	−0.745050	+	0.667008i	$0.767575\pi$
$234$	6.48268	0.423786
$235$	8.50129	0.554563
$236$	8.51046	0.553984
$237$	−1.33294	−0.0865835
$238$	2.73612	0.177356
$239$	9.30016	0.601578	0.300789	−	0.953691i	$-0.402750\pi$
$239$	9.30016	0.601578	0.300789	+	0.953691i	$0.402750\pi$
$240$	−5.18183	−0.334486
$241$	0.572380	0.0368703	0.0184351	−	0.999830i	$-0.494132\pi$
$241$	0.572380	0.0368703	0.0184351	+	0.999830i	$0.494132\pi$
$242$	20.2266	1.30021
$243$	7.73158	0.495981
$244$	46.2637	2.96173
$245$	−1.00000	−0.0638877
$246$	3.45014	0.219973
$247$	−6.29936	−0.400818
$248$	25.0188	1.58870
$249$	−2.28048	−0.144520
$250$	2.78055	0.175857
$251$	−14.2096	−0.896903	−0.448451	−	0.893807i	$-0.648024\pi$
$251$	−14.2096	−0.896903	−0.448451	+	0.893807i	$0.648024\pi$
$252$	16.6853	1.05107
$253$	7.62069	0.479108
$254$	−39.1687	−2.45766
$255$	−0.293271	−0.0183653
$256$	86.9958	5.43724
$257$	6.65911	0.415384	0.207692	−	0.978194i	$-0.433405\pi$
$257$	6.65911	0.415384	0.207692	+	0.978194i	$0.433405\pi$
$258$	5.77020	0.359237
$259$	−9.35326	−0.581183
$260$	−4.59009	−0.284665
$261$	27.5653	1.70625
$262$	55.2263	3.41189
$263$	26.6547	1.64360	0.821799	−	0.569777i	$-0.192970\pi$
$263$	26.6547	1.64360	0.821799	+	0.569777i	$0.192970\pi$
$264$	−5.96869	−0.367347
$265$	−5.63325	−0.346048
$266$	−21.8711	−1.34101
$267$	1.67696	0.102629
$268$	−76.0929	−4.64811
$269$	−5.50963	−0.335928	−0.167964	−	0.985793i	$-0.553719\pi$
$269$	−5.50963	−0.335928	−0.167964	+	0.985793i	$0.553719\pi$
$270$	−4.89857	−0.298118
$271$	19.3282	1.17411	0.587053	−	0.809549i	$-0.300288\pi$
$271$	19.3282	1.17411	0.587053	+	0.809549i	$0.300288\pi$
$272$	17.1089	1.03738
$273$	−0.238682	−0.0144457
$274$	−41.1155	−2.48388
$275$	1.93021	0.116396
$276$	6.74405	0.405944
$277$	9.36540	0.562712	0.281356	−	0.959603i	$-0.409216\pi$
$277$	9.36540	0.562712	0.281356	+	0.959603i	$0.409216\pi$
$278$	61.0359	3.66069
$279$	7.01981	0.420265
$280$	−10.3755	−0.620056
$281$	27.6185	1.64758	0.823792	−	0.566892i	$-0.191854\pi$
$281$	27.6185	1.64758	0.823792	+	0.566892i	$0.191854\pi$
$282$	7.04500	0.419523
$283$	26.9369	1.60123	0.800617	−	0.599176i	$-0.204505\pi$
$283$	26.9369	1.60123	0.800617	+	0.599176i	$0.204505\pi$
$284$	−20.4149	−1.21140
$285$	2.34426	0.138862
$286$	−4.29824	−0.254160
$287$	4.16333	0.245754
$288$	80.3300	4.73349
$289$	−16.0317	−0.943041
$290$	−26.3285	−1.54606
$291$	5.59066	0.327730
$292$	78.5823	4.59868
$293$	−14.5291	−0.848797	−0.424398	−	0.905476i	$-0.639514\pi$
$293$	−14.5291	−0.848797	−0.424398	+	0.905476i	$0.639514\pi$
$294$	−0.828697	−0.0483306
$295$	−1.48487	−0.0864523
$296$	−97.0449	−5.64062
$297$	−3.40050	−0.197317
$298$	−37.0804	−2.14801
$299$	3.16188	0.182856
$300$	1.70817	0.0986211
$301$	6.96297	0.401339
$302$	4.25319	0.244744
$303$	−4.49714	−0.258354
$304$	−136.760	−7.84372
$305$	−8.07188	−0.462195
$306$	7.96533	0.455347
$307$	32.6927	1.86587	0.932935	−	0.360044i	$-0.117238\pi$
$307$	32.6927	1.86587	0.932935	+	0.360044i	$0.117238\pi$
$308$	−11.0629	−0.630368
$309$	1.05251	0.0598754
$310$	−6.70483	−0.380808
$311$	−26.4579	−1.50029	−0.750145	−	0.661274i	$-0.770016\pi$
$311$	−26.4579	−1.50029	−0.750145	+	0.661274i	$0.770016\pi$
$312$	−2.47646	−0.140202
$313$	31.1423	1.76027	0.880135	−	0.474724i	$-0.157452\pi$
$313$	31.1423	1.76027	0.880135	+	0.474724i	$0.157452\pi$
$314$	38.1652	2.15378
$315$	−2.91118	−0.164026
$316$	−25.6336	−1.44200
$317$	−14.3876	−0.808085	−0.404043	−	0.914740i	$-0.632395\pi$
$317$	−14.3876	−0.808085	−0.404043	+	0.914740i	$0.632395\pi$
$318$	−4.66825	−0.261783
$319$	−18.2767	−1.02330
$320$	−41.9521	−2.34519
$321$	−5.85066	−0.326552
$322$	10.9779	0.611777
$323$	−7.74007	−0.430669
$324$	47.0466	2.61370
$325$	0.800858	0.0444236
$326$	−37.1891	−2.05971
$327$	0.373877	0.0206754
$328$	43.1967	2.38514
$329$	8.50129	0.468691
$330$	1.59956	0.0880527
$331$	2.76125	0.151772	0.0758859	−	0.997117i	$-0.475822\pi$
$331$	2.76125	0.151772	0.0758859	+	0.997117i	$0.475822\pi$
$332$	−43.8558	−2.40690
$333$	−27.2290	−1.49214
$334$	−8.12197	−0.444414
$335$	13.2763	0.725364
$336$	−5.18183	−0.282692
$337$	−31.0714	−1.69257	−0.846284	−	0.532732i	$-0.821165\pi$
$337$	−31.0714	−1.69257	−0.846284	+	0.532732i	$0.821165\pi$
$338$	34.3638	1.86914
$339$	−1.53552	−0.0833981
$340$	−5.63988	−0.305865
$341$	−4.65437	−0.252048
$342$	−63.6707	−3.44292
$343$	−1.00000	−0.0539949
$344$	72.2445	3.89516
$345$	−1.17667	−0.0633498
$346$	−32.9778	−1.77290
$347$	−29.2759	−1.57161	−0.785806	−	0.618473i	$-0.787752\pi$
$347$	−29.2759	−1.57161	−0.785806	+	0.618473i	$0.787752\pi$
$348$	−16.1743	−0.867033
$349$	10.6770	0.571528	0.285764	−	0.958300i	$-0.407753\pi$
$349$	10.6770	0.571528	0.285764	+	0.958300i	$0.407753\pi$
$350$	2.78055	0.148627
$351$	−1.41089	−0.0753080
$352$	−53.2615	−2.83885
$353$	−9.63726	−0.512940	−0.256470	−	0.966552i	$-0.582559\pi$
$353$	−9.63726	−0.512940	−0.256470	+	0.966552i	$0.582559\pi$
$354$	−1.23051	−0.0654006
$355$	3.56191	0.189046
$356$	32.2496	1.70922
$357$	−0.293271	−0.0155216
$358$	18.5861	0.982304
$359$	19.4085	1.02434	0.512171	−	0.858883i	$-0.328841\pi$
$359$	19.4085	1.02434	0.512171	+	0.858883i	$0.328841\pi$
$360$	−30.2050	−1.59194
$361$	42.8702	2.25632
$362$	−45.5836	−2.39582
$363$	−2.16798	−0.113790
$364$	−4.59009	−0.240586
$365$	−13.7107	−0.717650
$366$	−6.68915	−0.349647
$367$	12.5807	0.656709	0.328354	−	0.944555i	$-0.393506\pi$
$367$	12.5807	0.656709	0.328354	+	0.944555i	$0.393506\pi$
$368$	68.6449	3.57836
$369$	12.1202	0.630952
$370$	26.0072	1.35205
$371$	−5.63325	−0.292464
$372$	−4.11896	−0.213558
$373$	−27.3935	−1.41838	−0.709192	−	0.705016i	$-0.750940\pi$
$373$	−27.3935	−1.41838	−0.709192	+	0.705016i	$0.750940\pi$
$374$	−5.28128	−0.273088
$375$	−0.298033	−0.0153904
$376$	88.2053	4.54884
$377$	−7.58316	−0.390553
$378$	−4.89857	−0.251955
$379$	−19.5606	−1.00476	−0.502381	−	0.864646i	$-0.667542\pi$
$379$	−19.5606	−1.00476	−0.502381	+	0.864646i	$0.667542\pi$
$380$	45.0823	2.31267
$381$	4.19830	0.215085
$382$	−30.0501	−1.53750
$383$	22.0264	1.12549	0.562747	−	0.826629i	$-0.309744\pi$
$383$	22.0264	1.12549	0.562747	+	0.826629i	$0.309744\pi$
$384$	−18.3179	−0.934783
$385$	1.93021	0.0983725
$386$	−9.10616	−0.463491
$387$	20.2704	1.03040
$388$	107.514	5.45818
$389$	−39.1240	−1.98367	−0.991833	−	0.127546i	$-0.959290\pi$
$389$	−39.1240	−1.98367	−0.991833	+	0.127546i	$0.959290\pi$
$390$	0.663669	0.0336062
$391$	3.88503	0.196474
$392$	−10.3755	−0.524043
$393$	−5.91943	−0.298596
$394$	11.0940	0.558906
$395$	4.47244	0.225033
$396$	−32.2061	−1.61842
$397$	2.21011	0.110922	0.0554611	−	0.998461i	$-0.482337\pi$
$397$	2.21011	0.110922	0.0554611	+	0.998461i	$0.482337\pi$
$398$	−26.8581	−1.34628
$399$	2.34426	0.117360
$400$	17.3867	0.869337
$401$	−7.74623	−0.386828	−0.193414	−	0.981117i	$-0.561956\pi$
$401$	−7.74623	−0.386828	−0.193414	+	0.981117i	$0.561956\pi$
$402$	11.0021	0.548733
$403$	−1.93113	−0.0961966
$404$	−86.4841	−4.30275
$405$	−8.20847	−0.407882
$406$	−26.3285	−1.30666
$407$	18.0537	0.894890
$408$	−3.04284	−0.150643
$409$	0.843997	0.0417329	0.0208665	−	0.999782i	$-0.493358\pi$
$409$	0.843997	0.0417329	0.0208665	+	0.999782i	$0.493358\pi$
$410$	−11.5764	−0.571715
$411$	4.40697	0.217380
$412$	20.2408	0.997194
$413$	−1.48487	−0.0730656
$414$	31.9587	1.57069
$415$	7.65176	0.375610
$416$	−22.0986	−1.08347
$417$	−6.54214	−0.320370
$418$	42.2158	2.06484
$419$	13.5588	0.662391	0.331196	−	0.943562i	$-0.392548\pi$
$419$	13.5588	0.662391	0.331196	+	0.943562i	$0.392548\pi$
$420$	1.70817	0.0833500
$421$	34.5469	1.68371	0.841855	−	0.539703i	$-0.181464\pi$
$421$	34.5469	1.68371	0.841855	+	0.539703i	$0.181464\pi$
$422$	57.9251	2.81975
$423$	24.7488	1.20333
$424$	−58.4479	−2.83848
$425$	0.984021	0.0477320
$426$	2.95174	0.143012
$427$	−8.07188	−0.390626
$428$	−112.514	−5.43855
$429$	0.460707	0.0222431
$430$	−19.3609	−0.933666
$431$	−16.4036	−0.790134	−0.395067	−	0.918652i	$-0.629279\pi$
$431$	−16.4036	−0.790134	−0.395067	+	0.918652i	$0.629279\pi$
$432$	−30.6307	−1.47372
$433$	7.60672	0.365555	0.182778	−	0.983154i	$-0.441491\pi$
$433$	7.60672	0.365555	0.182778	+	0.983154i	$0.441491\pi$
$434$	−6.70483	−0.321842
$435$	2.82202	0.135305
$436$	7.19001	0.344339
$437$	−31.0549	−1.48556
$438$	−11.3620	−0.542898
$439$	−18.1859	−0.867964	−0.433982	−	0.900921i	$-0.642892\pi$
$439$	−18.1859	−0.867964	−0.433982	+	0.900921i	$0.642892\pi$
$440$	20.0269	0.954745
$441$	−2.91118	−0.138627
$442$	−2.19124	−0.104227
$443$	−11.7409	−0.557826	−0.278913	−	0.960316i	$-0.589974\pi$
$443$	−11.7409	−0.557826	−0.278913	+	0.960316i	$0.589974\pi$
$444$	15.9769	0.758232
$445$	−5.62676	−0.266734
$446$	−15.0966	−0.714843
$447$	3.97447	0.187986
$448$	−41.9521	−1.98205
$449$	−7.45839	−0.351983	−0.175992	−	0.984392i	$-0.556313\pi$
$449$	−7.45839	−0.351983	−0.175992	+	0.984392i	$0.556313\pi$
$450$	8.09467	0.381587
$451$	−8.03609	−0.378405
$452$	−29.5295	−1.38895
$453$	−0.455879	−0.0214191
$454$	−21.0644	−0.988602
$455$	0.800858	0.0375448
$456$	24.3229	1.13902
$457$	−21.8356	−1.02143	−0.510714	−	0.859751i	$-0.670619\pi$
$457$	−21.8356	−1.02143	−0.510714	+	0.859751i	$0.670619\pi$
$458$	−2.78055	−0.129927
$459$	−1.73358	−0.0809164
$460$	−22.6285	−1.05506
$461$	15.6388	0.728371	0.364185	−	0.931327i	$-0.381347\pi$
$461$	15.6388	0.728371	0.364185	+	0.931327i	$0.381347\pi$
$462$	1.59956	0.0744181
$463$	29.1462	1.35454	0.677269	−	0.735736i	$-0.263164\pi$
$463$	29.1462	1.35454	0.677269	+	0.735736i	$0.263164\pi$
$464$	−164.631	−7.64282
$465$	0.718657	0.0333269
$466$	63.2447	2.92976
$467$	−3.70976	−0.171667	−0.0858337	−	0.996309i	$-0.527355\pi$
$467$	−3.70976	−0.171667	−0.0858337	+	0.996309i	$0.527355\pi$
$468$	−13.3626	−0.617684
$469$	13.2763	0.613045
$470$	−23.6383	−1.09035
$471$	−4.09073	−0.188491
$472$	−15.4063	−0.709131
$473$	−13.4400	−0.617971
$474$	3.70630	0.170236
$475$	−7.86576	−0.360906
$476$	−5.63988	−0.258503
$477$	−16.3994	−0.750876
$478$	−25.8596	−1.18279
$479$	−28.0449	−1.28140	−0.640701	−	0.767790i	$-0.721356\pi$
$479$	−28.0449	−1.28140	−0.640701	+	0.767790i	$0.721356\pi$
$480$	8.22384	0.375365
$481$	7.49063	0.341543
$482$	−1.59153	−0.0724923
$483$	−1.17667	−0.0535404
$484$	−41.6924	−1.89511
$485$	−18.7585	−0.851780
$486$	−21.4981	−0.975172
$487$	−21.3870	−0.969139	−0.484569	−	0.874753i	$-0.661024\pi$
$487$	−21.3870	−0.969139	−0.484569	+	0.874753i	$0.661024\pi$
$488$	−83.7500	−3.79118
$489$	3.98612	0.180258
$490$	2.78055	0.125612
$491$	13.6251	0.614892	0.307446	−	0.951566i	$-0.400526\pi$
$491$	13.6251	0.614892	0.307446	+	0.951566i	$0.400526\pi$
$492$	−7.11167	−0.320619
$493$	−9.31749	−0.419639
$494$	17.5157	0.788068
$495$	5.61917	0.252563
$496$	−41.9252	−1.88250
$497$	3.56191	0.159773
$498$	6.34100	0.284147
$499$	9.29854	0.416260	0.208130	−	0.978101i	$-0.433262\pi$
$499$	9.29854	0.416260	0.208130	+	0.978101i	$0.433262\pi$
$500$	−5.73146	−0.256319
$501$	0.870553	0.0388935
$502$	39.5105	1.76344
$503$	−11.1887	−0.498879	−0.249439	−	0.968390i	$-0.580246\pi$
$503$	−11.1887	−0.498879	−0.249439	+	0.968390i	$0.580246\pi$
$504$	−30.2050	−1.34544
$505$	15.0894	0.671468
$506$	−21.1897	−0.941997
$507$	−3.68328	−0.163580
$508$	80.7372	3.58213
$509$	42.8827	1.90074	0.950371	−	0.311118i	$-0.100704\pi$
$509$	42.8827	1.90074	0.950371	+	0.311118i	$0.100704\pi$
$510$	0.815455	0.0361090
$511$	−13.7107	−0.606525
$512$	−118.971	−5.25782
$513$	13.8573	0.611816
$514$	−18.5160	−0.816705
$515$	−3.53153	−0.155618
$516$	−11.8939	−0.523601
$517$	−16.4093	−0.721678
$518$	26.0072	1.14269
$519$	3.53472	0.155157
$520$	8.30932	0.364388
$521$	19.0920	0.836434	0.418217	−	0.908347i	$-0.362655\pi$
$521$	19.0920	0.836434	0.418217	+	0.908347i	$0.362655\pi$
$522$	−76.6468	−3.35474
$523$	0.946269	0.0413775	0.0206887	−	0.999786i	$-0.493414\pi$
$523$	0.946269	0.0413775	0.0206887	+	0.999786i	$0.493414\pi$
$524$	−113.836	−4.97296
$525$	−0.298033	−0.0130072
$526$	−74.1147	−3.23156
$527$	−2.37280	−0.103361
$528$	10.0020	0.435281
$529$	−7.41237	−0.322277
$530$	15.6635	0.680380
$531$	−4.32271	−0.187590
$532$	45.0823	1.95457
$533$	−3.33424	−0.144422
$534$	−4.66288	−0.201783
$535$	19.6309	0.848717
$536$	137.749	5.94985
$537$	−1.99215	−0.0859676
$538$	15.3198	0.660484
$539$	1.93021	0.0831399
$540$	10.0973	0.434518
$541$	6.05878	0.260487	0.130244	−	0.991482i	$-0.458424\pi$
$541$	6.05878	0.260487	0.130244	+	0.991482i	$0.458424\pi$
$542$	−53.7431	−2.30846
$543$	4.88588	0.209673
$544$	−27.1527	−1.16416
$545$	−1.25448	−0.0537360
$546$	0.663669	0.0284024
$547$	22.3395	0.955167	0.477584	−	0.878586i	$-0.341513\pi$
$547$	22.3395	0.955167	0.477584	+	0.878586i	$0.341513\pi$
$548$	84.7501	3.62035
$549$	−23.4987	−1.00290
$550$	−5.36704	−0.228851
$551$	74.4793	3.17292
$552$	−12.2086	−0.519632
$553$	4.47244	0.190187
$554$	−26.0410	−1.10637
$555$	−2.78758	−0.118326
$556$	−125.811	−5.33560
$557$	−28.3183	−1.19988	−0.599942	−	0.800044i	$-0.704810\pi$
$557$	−28.3183	−1.19988	−0.599942	+	0.800044i	$0.704810\pi$
$558$	−19.5189	−0.826302
$559$	−5.57635	−0.235855
$560$	17.3867	0.734724
$561$	0.566074	0.0238997
$562$	−76.7948	−3.23939
$563$	34.7133	1.46299	0.731495	−	0.681847i	$-0.238823\pi$
$563$	34.7133	1.46299	0.731495	+	0.681847i	$0.238823\pi$
$564$	−14.5216	−0.611471
$565$	5.15218	0.216754
$566$	−74.8995	−3.14826
$567$	−8.20847	−0.344724
$568$	36.9566	1.55067
$569$	32.8004	1.37506	0.687532	−	0.726154i	$-0.258694\pi$
$569$	32.8004	1.37506	0.687532	+	0.726154i	$0.258694\pi$
$570$	−6.51833	−0.273023
$571$	28.7894	1.20480	0.602398	−	0.798196i	$-0.294212\pi$
$571$	28.7894	1.20480	0.602398	+	0.798196i	$0.294212\pi$
$572$	8.85982	0.370448
$573$	3.22092	0.134556
$574$	−11.5764	−0.483188
$575$	3.94812	0.164648
$576$	−122.130	−5.08875
$577$	18.8693	0.785540	0.392770	−	0.919637i	$-0.371517\pi$
$577$	18.8693	0.785540	0.392770	+	0.919637i	$0.371517\pi$
$578$	44.5770	1.85416
$579$	0.976044	0.0405630
$580$	54.2700	2.25344
$581$	7.65176	0.317449
$582$	−15.5451	−0.644366
$583$	10.8733	0.450327
$584$	−142.256	−5.88658
$585$	2.33144	0.0963931
$586$	40.3988	1.66886
$587$	−20.2809	−0.837081	−0.418540	−	0.908198i	$-0.637458\pi$
$587$	−20.2809	−0.837081	−0.418540	+	0.908198i	$0.637458\pi$
$588$	1.70817	0.0704436
$589$	18.9669	0.781519
$590$	4.12875	0.169978
$591$	−1.18911	−0.0489134
$592$	162.623	6.68375
$593$	41.7801	1.71570	0.857852	−	0.513897i	$-0.171799\pi$
$593$	41.7801	1.71570	0.857852	+	0.513897i	$0.171799\pi$
$594$	9.45526	0.387954
$595$	0.984021	0.0403409
$596$	76.4328	3.13081
$597$	2.87879	0.117821
$598$	−8.79177	−0.359522
$599$	12.7717	0.521836	0.260918	−	0.965361i	$-0.415975\pi$
$599$	12.7717	0.521836	0.260918	+	0.965361i	$0.415975\pi$
$600$	−3.09225	−0.126241
$601$	20.7937	0.848192	0.424096	−	0.905617i	$-0.360592\pi$
$601$	20.7937	0.848192	0.424096	+	0.905617i	$0.360592\pi$
$602$	−19.3609	−0.789091
$603$	38.6498	1.57394
$604$	−8.76698	−0.356723
$605$	7.27430	0.295742
$606$	12.5045	0.507961
$607$	−30.7120	−1.24656	−0.623280	−	0.781999i	$-0.714200\pi$
$607$	−30.7120	−1.24656	−0.623280	+	0.781999i	$0.714200\pi$
$608$	217.045	8.80234
$609$	2.82202	0.114354
$610$	22.4443	0.908742
$611$	−6.80833	−0.275435
$612$	−16.4187	−0.663686
$613$	−31.8057	−1.28462	−0.642310	−	0.766445i	$-0.722024\pi$
$613$	−31.8057	−1.28462	−0.642310	+	0.766445i	$0.722024\pi$
$614$	−90.9037	−3.66857
$615$	1.24081	0.0500344
$616$	20.0269	0.806907
$617$	13.5320	0.544778	0.272389	−	0.962187i	$-0.412186\pi$
$617$	13.5320	0.544778	0.272389	+	0.962187i	$0.412186\pi$
$618$	−2.92657	−0.117724
$619$	6.94027	0.278953	0.139477	−	0.990225i	$-0.455458\pi$
$619$	6.94027	0.278953	0.139477	+	0.990225i	$0.455458\pi$
$620$	13.8204	0.555043
$621$	−6.95551	−0.279115
$622$	73.5675	2.94979
$623$	−5.62676	−0.225432
$624$	4.14991	0.166129
$625$	1.00000	0.0400000
$626$	−86.5929	−3.46095
$627$	−4.52491	−0.180707
$628$	−78.6687	−3.13922
$629$	9.20380	0.366980
$630$	8.09467	0.322499
$631$	16.5062	0.657102	0.328551	−	0.944486i	$-0.393440\pi$
$631$	16.5062	0.657102	0.328551	+	0.944486i	$0.393440\pi$
$632$	46.4039	1.84585
$633$	−6.20870	−0.246774
$634$	40.0053	1.58881
$635$	−14.0867	−0.559012
$636$	9.62253	0.381558
$637$	0.800858	0.0317312
$638$	50.8194	2.01196
$639$	10.3693	0.410205
$640$	61.4626	2.42952
$641$	2.48703	0.0982320	0.0491160	−	0.998793i	$-0.484360\pi$
$641$	2.48703	0.0982320	0.0491160	+	0.998793i	$0.484360\pi$
$642$	16.2680	0.642049
$643$	−40.9647	−1.61549	−0.807745	−	0.589532i	$-0.799312\pi$
$643$	−40.9647	−1.61549	−0.807745	+	0.589532i	$0.799312\pi$
$644$	−22.6285	−0.891687
$645$	2.07520	0.0817109
$646$	21.5217	0.846758
$647$	−15.3833	−0.604780	−0.302390	−	0.953184i	$-0.597785\pi$
$647$	−15.3833	−0.604780	−0.302390	+	0.953184i	$0.597785\pi$
$648$	−85.1672	−3.34568
$649$	2.86610	0.112504
$650$	−2.22683	−0.0873433
$651$	0.718657	0.0281664
$652$	76.6567	3.00211
$653$	31.0434	1.21482	0.607411	−	0.794388i	$-0.292208\pi$
$653$	31.0434	1.21482	0.607411	+	0.794388i	$0.292208\pi$
$654$	−1.03958	−0.0406510
$655$	19.8616	0.776059
$656$	−72.3868	−2.82623
$657$	−39.9142	−1.55720
$658$	−23.6383	−0.921516
$659$	45.5061	1.77266	0.886332	−	0.463049i	$-0.153245\pi$
$659$	45.5061	1.77266	0.886332	+	0.463049i	$0.153245\pi$
$660$	−3.29712	−0.128340
$661$	34.7740	1.35255	0.676276	−	0.736648i	$-0.263592\pi$
$661$	34.7740	1.35255	0.676276	+	0.736648i	$0.263592\pi$
$662$	−7.67779	−0.298406
$663$	0.234868	0.00912154
$664$	79.3910	3.08097
$665$	−7.86576	−0.305021
$666$	75.7116	2.93376
$667$	−37.3839	−1.44751
$668$	16.7416	0.647750
$669$	1.61813	0.0625604
$670$	−36.9155	−1.42617
$671$	15.5804	0.601475
$672$	8.22384	0.317241
$673$	33.1060	1.27614	0.638072	−	0.769977i	$-0.279732\pi$
$673$	33.1060	1.27614	0.638072	+	0.769977i	$0.279732\pi$
$674$	86.3957	3.32784
$675$	−1.76173	−0.0678089
$676$	−70.8330	−2.72435
$677$	2.59983	0.0999196	0.0499598	−	0.998751i	$-0.484091\pi$
$677$	2.59983	0.0999196	0.0499598	+	0.998751i	$0.484091\pi$
$678$	4.26960	0.163973
$679$	−18.7585	−0.719886
$680$	10.2097	0.391525
$681$	2.25779	0.0865187
$682$	12.9417	0.495563
$683$	−6.29909	−0.241028	−0.120514	−	0.992712i	$-0.538454\pi$
$683$	−6.29909	−0.241028	−0.120514	+	0.992712i	$0.538454\pi$
$684$	131.243	5.01818
$685$	−14.7868	−0.564975
$686$	2.78055	0.106162
$687$	0.298033	0.0113707
$688$	−121.063	−4.61550
$689$	4.51143	0.171872
$690$	3.27179	0.124555
$691$	6.69528	0.254701	0.127350	−	0.991858i	$-0.459353\pi$
$691$	6.69528	0.254701	0.127350	+	0.991858i	$0.459353\pi$
$692$	67.9761	2.58406
$693$	5.61917	0.213455
$694$	81.4031	3.09002
$695$	21.9510	0.832650
$696$	29.2799	1.10985
$697$	−4.09680	−0.155178
$698$	−29.6880	−1.12371
$699$	−6.77889	−0.256401
$700$	−5.73146	−0.216629
$701$	19.6673	0.742823	0.371411	−	0.928468i	$-0.378874\pi$
$701$	19.6673	0.742823	0.371411	+	0.928468i	$0.378874\pi$
$702$	3.92306	0.148066
$703$	−73.5705	−2.77476
$704$	80.9763	3.05191
$705$	2.53367	0.0954235
$706$	26.7969	1.00851
$707$	15.0894	0.567494
$708$	2.53640	0.0953239
$709$	−15.2843	−0.574015	−0.287007	−	0.957928i	$-0.592660\pi$
$709$	−15.2843	−0.574015	−0.287007	+	0.957928i	$0.592660\pi$
$710$	−9.90406	−0.371693
$711$	13.0201	0.488290
$712$	−58.3806	−2.18791
$713$	−9.52022	−0.356535
$714$	0.815455	0.0305176
$715$	−1.54582	−0.0578105
$716$	−38.3109	−1.43175
$717$	2.77176	0.103513
$718$	−53.9664	−2.01401
$719$	−35.5503	−1.32580	−0.662902	−	0.748706i	$-0.730675\pi$
$719$	−35.5503	−1.32580	−0.662902	+	0.748706i	$0.730675\pi$
$720$	50.6159	1.88634
$721$	−3.53153	−0.131521
$722$	−119.203	−4.43626
$723$	0.170589	0.00634425
$724$	93.9600	3.49200
$725$	−9.46879	−0.351662
$726$	6.02819	0.223727
$727$	−13.6884	−0.507675	−0.253837	−	0.967247i	$-0.581693\pi$
$727$	−13.6884	−0.507675	−0.253837	+	0.967247i	$0.581693\pi$
$728$	8.30932	0.307964
$729$	−22.3212	−0.826709
$730$	38.1233	1.41101
$731$	−6.85171	−0.253420
$732$	13.7881	0.509624
$733$	48.7271	1.79978	0.899889	−	0.436120i	$-0.143648\pi$
$733$	48.7271	1.79978	0.899889	+	0.436120i	$0.143648\pi$
$734$	−34.9814	−1.29119
$735$	−0.298033	−0.0109931
$736$	−108.943	−4.01569
$737$	−25.6261	−0.943949
$738$	−33.7008	−1.24054
$739$	20.4498	0.752258	0.376129	−	0.926567i	$-0.377255\pi$
$739$	20.4498	0.752258	0.376129	+	0.926567i	$0.377255\pi$
$740$	−53.6079	−1.97066
$741$	−1.87742	−0.0689687
$742$	15.6635	0.575026
$743$	28.5780	1.04842	0.524212	−	0.851588i	$-0.324360\pi$
$743$	28.5780	1.04842	0.524212	+	0.851588i	$0.324360\pi$
$744$	7.45644	0.273366
$745$	−13.3356	−0.488580
$746$	76.1691	2.78875
$747$	22.2756	0.815023
$748$	10.8861	0.398036
$749$	19.6309	0.717296
$750$	0.828697	0.0302597
$751$	−3.69614	−0.134874	−0.0674370	−	0.997724i	$-0.521482\pi$
$751$	−3.69614	−0.134874	−0.0674370	+	0.997724i	$0.521482\pi$
$752$	−147.810	−5.39007
$753$	−4.23494	−0.154330
$754$	21.0854	0.767884
$755$	1.52962	0.0556687
$756$	10.0973	0.367234
$757$	−35.7776	−1.30036	−0.650179	−	0.759781i	$-0.725306\pi$
$757$	−35.7776	−1.30036	−0.650179	+	0.759781i	$0.725306\pi$
$758$	54.3893	1.97551
$759$	2.27122	0.0824400
$760$	−81.6114	−2.96035
$761$	13.1374	0.476231	0.238116	−	0.971237i	$-0.423470\pi$
$761$	13.1374	0.476231	0.238116	+	0.971237i	$0.423470\pi$
$762$	−11.6736	−0.422889
$763$	−1.25448	−0.0454152
$764$	61.9414	2.24096
$765$	2.86466	0.103572
$766$	−61.2454	−2.21289
$767$	1.18917	0.0429384
$768$	25.9277	0.935584
$769$	46.7354	1.68532	0.842661	−	0.538444i	$-0.180988\pi$
$769$	46.7354	1.68532	0.842661	+	0.538444i	$0.180988\pi$
$770$	−5.36704	−0.193415
$771$	1.98464	0.0714749
$772$	18.7702	0.675556
$773$	−36.9751	−1.32990	−0.664950	−	0.746888i	$-0.731547\pi$
$773$	−36.9751	−1.32990	−0.664950	+	0.746888i	$0.731547\pi$
$774$	−56.3630	−2.02593
$775$	−2.41133	−0.0866175
$776$	−194.629	−6.98679
$777$	−2.78758	−0.100004
$778$	108.786	3.90018
$779$	32.7478	1.17331
$780$	−1.36800	−0.0489822
$781$	−6.87522	−0.246015
$782$	−10.8025	−0.386297
$783$	16.6814	0.596146
$784$	17.3867	0.620955
$785$	13.7258	0.489893
$786$	16.4593	0.587083
$787$	31.8596	1.13567	0.567836	−	0.823142i	$-0.307781\pi$
$787$	31.8596	1.13567	0.567836	+	0.823142i	$0.307781\pi$
$788$	−22.8677	−0.814627
$789$	7.94399	0.282814
$790$	−12.4358	−0.442447
$791$	5.15218	0.183190
$792$	58.3019	2.07167
$793$	6.46443	0.229559
$794$	−6.14532	−0.218089
$795$	−1.67890	−0.0595443
$796$	55.3618	1.96225
$797$	0.294649	0.0104370	0.00521850	−	0.999986i	$-0.498339\pi$
$797$	0.294649	0.0104370	0.00521850	+	0.999986i	$0.498339\pi$
$798$	−6.51833	−0.230746
$799$	−8.36545	−0.295948
$800$	−27.5937	−0.975583
$801$	−16.3805	−0.578777
$802$	21.5388	0.760561
$803$	26.4645	0.933911
$804$	−22.6782	−0.799799
$805$	3.94812	0.139153
$806$	5.36961	0.189137
$807$	−1.64205	−0.0578031
$808$	156.560	5.50776
$809$	−47.6044	−1.67368	−0.836841	−	0.547446i	$-0.815600\pi$
$809$	−47.6044	−1.67368	−0.836841	+	0.547446i	$0.815600\pi$
$810$	22.8241	0.801956
$811$	−23.3322	−0.819305	−0.409652	−	0.912242i	$-0.634350\pi$
$811$	−23.3322	−0.819305	−0.409652	+	0.912242i	$0.634350\pi$
$812$	54.2700	1.90450
$813$	5.76045	0.202028
$814$	−50.1993	−1.75948
$815$	−13.3747	−0.468496
$816$	5.09903	0.178502
$817$	54.7691	1.91613
$818$	−2.34678	−0.0820531
$819$	2.33144	0.0814671
$820$	23.8620	0.833296
$821$	28.6336	0.999320	0.499660	−	0.866222i	$-0.333458\pi$
$821$	28.6336	0.999320	0.499660	+	0.866222i	$0.333458\pi$
$822$	−12.2538	−0.427400
$823$	47.1786	1.64454	0.822271	−	0.569096i	$-0.192707\pi$
$823$	47.1786	1.64454	0.822271	+	0.569096i	$0.192707\pi$
$824$	−36.6415	−1.27647
$825$	0.575266	0.0200282
$826$	4.12875	0.143658
$827$	−33.4694	−1.16384	−0.581922	−	0.813244i	$-0.697699\pi$
$827$	−33.4694	−1.16384	−0.581922	+	0.813244i	$0.697699\pi$
$828$	−65.8755	−2.28933
$829$	11.2550	0.390902	0.195451	−	0.980713i	$-0.437383\pi$
$829$	11.2550	0.390902	0.195451	+	0.980713i	$0.437383\pi$
$830$	−21.2761	−0.738505
$831$	2.79120	0.0968257
$832$	33.5977	1.16479
$833$	0.984021	0.0340943
$834$	18.1908	0.629894
$835$	−2.92099	−0.101085
$836$	−87.0182	−3.00959
$837$	4.24811	0.146836
$838$	−37.7010	−1.30236
$839$	22.1711	0.765431	0.382715	−	0.923866i	$-0.374989\pi$
$839$	22.1711	0.765431	0.382715	+	0.923866i	$0.374989\pi$
$840$	−3.09225	−0.106693
$841$	60.6581	2.09166
$842$	−96.0593	−3.31042
$843$	8.23125	0.283499
$844$	−119.399	−4.10989
$845$	12.3586	0.425150
$846$	−68.8152	−2.36591
$847$	7.27430	0.249948
$848$	97.9438	3.36340
$849$	8.02811	0.275524
$850$	−2.73612	−0.0938481
$851$	36.9278	1.26587
$852$	−6.08433	−0.208446
$853$	−13.6775	−0.468309	−0.234155	−	0.972199i	$-0.575232\pi$
$853$	−13.6775	−0.468309	−0.234155	+	0.972199i	$0.575232\pi$
$854$	22.4443	0.768027
$855$	−22.8986	−0.783116
$856$	203.681	6.96166
$857$	42.9241	1.46626	0.733130	−	0.680088i	$-0.238059\pi$
$857$	42.9241	1.46626	0.733130	+	0.680088i	$0.238059\pi$
$858$	−1.28102	−0.0437332
$859$	−37.0842	−1.26530	−0.632649	−	0.774439i	$-0.718032\pi$
$859$	−37.0842	−1.26530	−0.632649	+	0.774439i	$0.718032\pi$
$860$	39.9080	1.36085
$861$	1.24081	0.0422868
$862$	45.6111	1.55352
$863$	19.8413	0.675406	0.337703	−	0.941253i	$-0.390350\pi$
$863$	19.8413	0.675406	0.337703	+	0.941253i	$0.390350\pi$
$864$	48.6125	1.65383
$865$	−11.8602	−0.403258
$866$	−21.1509	−0.718735
$867$	−4.77798	−0.162269
$868$	13.8204	0.469097
$869$	−8.63273	−0.292845
$870$	−7.84676	−0.266030
$871$	−10.6325	−0.360267
$872$	−13.0159	−0.440774
$873$	−54.6093	−1.84825
$874$	86.3499	2.92083
$875$	1.00000	0.0338062
$876$	23.4201	0.791294
$877$	30.8011	1.04008	0.520040	−	0.854142i	$-0.325917\pi$
$877$	30.8011	1.04008	0.520040	+	0.854142i	$0.325917\pi$
$878$	50.5667	1.70654
$879$	−4.33015	−0.146052
$880$	−33.5600	−1.13131
$881$	−3.35327	−0.112974	−0.0564872	−	0.998403i	$-0.517990\pi$
$881$	−3.35327	−0.112974	−0.0564872	+	0.998403i	$0.517990\pi$
$882$	8.09467	0.272562
$883$	−19.9101	−0.670027	−0.335014	−	0.942213i	$-0.608741\pi$
$883$	−19.9101	−0.670027	−0.335014	+	0.942213i	$0.608741\pi$
$884$	4.51674	0.151914
$885$	−0.442540	−0.0148758
$886$	32.6461	1.09677
$887$	38.0937	1.27906	0.639530	−	0.768766i	$-0.279129\pi$
$887$	38.0937	1.27906	0.639530	+	0.768766i	$0.279129\pi$
$888$	−28.9226	−0.970580
$889$	−14.0867	−0.472452
$890$	15.6455	0.524438
$891$	15.8441	0.530796
$892$	31.1181	1.04191
$893$	66.8691	2.23769
$894$	−11.0512	−0.369608
$895$	6.68431	0.223432
$896$	61.4626	2.05332
$897$	0.942347	0.0314640
$898$	20.7384	0.692051
$899$	22.8324	0.761503
$900$	−16.6853	−0.556177
$901$	5.54323	0.184672
$902$	22.3448	0.743999
$903$	2.07520	0.0690583
$904$	53.4566	1.77794
$905$	−16.3937	−0.544946
$906$	1.26759	0.0421130
$907$	21.5946	0.717037	0.358518	−	0.933523i	$-0.383282\pi$
$907$	21.5946	0.717037	0.358518	+	0.933523i	$0.383282\pi$
$908$	43.4194	1.44092
$909$	43.9278	1.45699
$910$	−2.22683	−0.0738186
$911$	43.0329	1.42574	0.712872	−	0.701294i	$-0.247394\pi$
$911$	43.0329	1.42574	0.712872	+	0.701294i	$0.247394\pi$
$912$	−40.7590	−1.34967
$913$	−14.7695	−0.488799
$914$	60.7151	2.00828
$915$	−2.40569	−0.0795297
$916$	5.73146	0.189373
$917$	19.8616	0.655889
$918$	4.82030	0.159094
$919$	−3.59329	−0.118532	−0.0592658	−	0.998242i	$-0.518876\pi$
$919$	−3.59329	−0.118532	−0.0592658	+	0.998242i	$0.518876\pi$
$920$	40.9638	1.35054
$921$	9.74352	0.321060
$922$	−43.4844	−1.43208
$923$	−2.85258	−0.0938938
$924$	−3.29712	−0.108467
$925$	9.35326	0.307533
$926$	−81.0424	−2.66322
$927$	−10.2809	−0.337669
$928$	261.279	8.57689
$929$	6.21078	0.203769	0.101885	−	0.994796i	$-0.467513\pi$
$929$	6.21078	0.203769	0.101885	+	0.994796i	$0.467513\pi$
$930$	−1.99826	−0.0655256
$931$	−7.86576	−0.257790
$932$	−130.364	−4.27023
$933$	−7.88534	−0.258154
$934$	10.3152	0.337523
$935$	−1.89936	−0.0621158
$936$	24.1899	0.790671
$937$	−32.9998	−1.07806	−0.539028	−	0.842288i	$-0.681208\pi$
$937$	−32.9998	−1.07806	−0.539028	+	0.842288i	$0.681208\pi$
$938$	−36.9155	−1.20534
$939$	9.28146	0.302889
$940$	48.7248	1.58923
$941$	−22.3146	−0.727436	−0.363718	−	0.931509i	$-0.618493\pi$
$941$	−22.3146	−0.727436	−0.363718	+	0.931509i	$0.618493\pi$
$942$	11.3745	0.370601
$943$	−16.4373	−0.535273
$944$	25.8170	0.840272
$945$	−1.76173	−0.0573090
$946$	37.3706	1.21502
$947$	35.2445	1.14529	0.572646	−	0.819802i	$-0.305917\pi$
$947$	35.2445	1.14529	0.572646	+	0.819802i	$0.305917\pi$
$948$	−7.63967	−0.248125
$949$	10.9803	0.356436
$950$	21.8711	0.709594
$951$	−4.28797	−0.139047
$952$	10.2097	0.330899
$953$	−25.9978	−0.842150	−0.421075	−	0.907026i	$-0.638347\pi$
$953$	−25.9978	−0.842150	−0.421075	+	0.907026i	$0.638347\pi$
$954$	45.5993	1.47633
$955$	−10.8073	−0.349715
$956$	53.3035	1.72396
$957$	−5.44708	−0.176079
$958$	77.9802	2.51943
$959$	−14.7868	−0.477491
$960$	−12.5031	−0.403537
$961$	−25.1855	−0.812435
$962$	−20.8281	−0.671524
$963$	57.1489	1.84160
$964$	3.28058	0.105660
$965$	−3.27495	−0.105424
$966$	3.27179	0.105268
$967$	−27.8976	−0.897127	−0.448564	−	0.893751i	$-0.648064\pi$
$967$	−27.8976	−0.897127	−0.448564	+	0.893751i	$0.648064\pi$
$968$	75.4747	2.42585
$969$	−2.30680	−0.0741051
$970$	52.1590	1.67472
$971$	8.17443	0.262330	0.131165	−	0.991361i	$-0.458128\pi$
$971$	8.17443	0.262330	0.131165	+	0.991361i	$0.458128\pi$
$972$	44.3133	1.42135
$973$	21.9510	0.703718
$974$	59.4677	1.90547
$975$	0.238682	0.00764396
$976$	140.344	4.49229
$977$	54.1255	1.73163	0.865814	−	0.500365i	$-0.166801\pi$
$977$	54.1255	1.73163	0.865814	+	0.500365i	$0.166801\pi$
$978$	−11.0836	−0.354414
$979$	10.8608	0.347113
$980$	−5.73146	−0.183085
$981$	−3.65201	−0.116600
$982$	−37.8853	−1.20897
$983$	−27.7172	−0.884043	−0.442021	−	0.897005i	$-0.645738\pi$
$983$	−27.7172	−0.884043	−0.442021	+	0.897005i	$0.645738\pi$
$984$	12.8741	0.410410
$985$	3.98985	0.127127
$986$	25.9078	0.825071
$987$	2.53367	0.0806476
$988$	−36.1045	−1.14864
$989$	−27.4906	−0.874152
$990$	−15.6244	−0.496576
$991$	24.2773	0.771193	0.385596	−	0.922668i	$-0.373996\pi$
$991$	24.2773	0.771193	0.385596	+	0.922668i	$0.373996\pi$
$992$	66.5374	2.11257
$993$	0.822944	0.0261153
$994$	−9.90406	−0.314138
$995$	−9.65928	−0.306220
$996$	−13.0705	−0.414155
$997$	32.3716	1.02522	0.512610	−	0.858622i	$-0.328679\pi$
$997$	32.3716	1.02522	0.512610	+	0.858622i	$0.328679\pi$
$998$	−25.8551	−0.818428
$999$	−16.4779	−0.521338

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8015.2.a.l.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-2.78055 q^{2} +0.298033 q^{3} +5.73146 q^{4} -1.00000 q^{5} -0.828697 q^{6} -1.00000 q^{7} -10.3755 q^{8} -2.91118 q^{9} +O(q^{10})\)	\(q-2.78055 q^{2} +0.298033 q^{3} +5.73146 q^{4} -1.00000 q^{5} -0.828697 q^{6} -1.00000 q^{7} -10.3755 q^{8} -2.91118 q^{9} +2.78055 q^{10} +1.93021 q^{11} +1.70817 q^{12} +0.800858 q^{13} +2.78055 q^{14} -0.298033 q^{15} +17.3867 q^{16} +0.984021 q^{17} +8.09467 q^{18} -7.86576 q^{19} -5.73146 q^{20} -0.298033 q^{21} -5.36704 q^{22} +3.94812 q^{23} -3.09225 q^{24} +1.00000 q^{25} -2.22683 q^{26} -1.76173 q^{27} -5.73146 q^{28} -9.46879 q^{29} +0.828697 q^{30} -2.41133 q^{31} -27.5937 q^{32} +0.575266 q^{33} -2.73612 q^{34} +1.00000 q^{35} -16.6853 q^{36} +9.35326 q^{37} +21.8711 q^{38} +0.238682 q^{39} +10.3755 q^{40} -4.16333 q^{41} +0.828697 q^{42} -6.96297 q^{43} +11.0629 q^{44} +2.91118 q^{45} -10.9779 q^{46} -8.50129 q^{47} +5.18183 q^{48} +1.00000 q^{49} -2.78055 q^{50} +0.293271 q^{51} +4.59009 q^{52} +5.63325 q^{53} +4.89857 q^{54} -1.93021 q^{55} +10.3755 q^{56} -2.34426 q^{57} +26.3285 q^{58} +1.48487 q^{59} -1.70817 q^{60} +8.07188 q^{61} +6.70483 q^{62} +2.91118 q^{63} +41.9521 q^{64} -0.800858 q^{65} -1.59956 q^{66} -13.2763 q^{67} +5.63988 q^{68} +1.17667 q^{69} -2.78055 q^{70} -3.56191 q^{71} +30.2050 q^{72} +13.7107 q^{73} -26.0072 q^{74} +0.298033 q^{75} -45.0823 q^{76} -1.93021 q^{77} -0.663669 q^{78} -4.47244 q^{79} -17.3867 q^{80} +8.20847 q^{81} +11.5764 q^{82} -7.65176 q^{83} -1.70817 q^{84} -0.984021 q^{85} +19.3609 q^{86} -2.82202 q^{87} -20.0269 q^{88} +5.62676 q^{89} -8.09467 q^{90} -0.800858 q^{91} +22.6285 q^{92} -0.718657 q^{93} +23.6383 q^{94} +7.86576 q^{95} -8.22384 q^{96} +18.7585 q^{97} -2.78055 q^{98} -5.61917 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 62 q + 2 q^{2} + 11 q^{3} + 64 q^{4} - 62 q^{5} + 3 q^{6} - 62 q^{7} + 15 q^{8} + 69 q^{9}+O(q^{10}) \) 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