LMFDB - Embedded newform 4010.2.a.o.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4010,2,Mod(1,4010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4010, 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("4010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4010 = 2 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4010.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:	$32.0200112105$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	4010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.31697 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.31697 q^{6} -3.01403 q^{7} +1.00000 q^{8} -1.26559 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.31697 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.31697 q^{6} -3.01403 q^{7} +1.00000 q^{8} -1.26559 q^{9} -1.00000 q^{10} -4.14944 q^{11} -1.31697 q^{12} +4.55376 q^{13} -3.01403 q^{14} +1.31697 q^{15} +1.00000 q^{16} -1.18544 q^{17} -1.26559 q^{18} +2.76979 q^{19} -1.00000 q^{20} +3.96939 q^{21} -4.14944 q^{22} -3.56973 q^{23} -1.31697 q^{24} +1.00000 q^{25} +4.55376 q^{26} +5.61766 q^{27} -3.01403 q^{28} -6.21799 q^{29} +1.31697 q^{30} +2.36560 q^{31} +1.00000 q^{32} +5.46470 q^{33} -1.18544 q^{34} +3.01403 q^{35} -1.26559 q^{36} -9.95108 q^{37} +2.76979 q^{38} -5.99717 q^{39} -1.00000 q^{40} +3.57940 q^{41} +3.96939 q^{42} -7.80349 q^{43} -4.14944 q^{44} +1.26559 q^{45} -3.56973 q^{46} +4.65151 q^{47} -1.31697 q^{48} +2.08439 q^{49} +1.00000 q^{50} +1.56119 q^{51} +4.55376 q^{52} -1.53610 q^{53} +5.61766 q^{54} +4.14944 q^{55} -3.01403 q^{56} -3.64774 q^{57} -6.21799 q^{58} -7.45163 q^{59} +1.31697 q^{60} +13.2771 q^{61} +2.36560 q^{62} +3.81452 q^{63} +1.00000 q^{64} -4.55376 q^{65} +5.46470 q^{66} +10.3042 q^{67} -1.18544 q^{68} +4.70123 q^{69} +3.01403 q^{70} -5.86976 q^{71} -1.26559 q^{72} +9.19353 q^{73} -9.95108 q^{74} -1.31697 q^{75} +2.76979 q^{76} +12.5066 q^{77} -5.99717 q^{78} +5.83414 q^{79} -1.00000 q^{80} -3.60153 q^{81} +3.57940 q^{82} +14.6425 q^{83} +3.96939 q^{84} +1.18544 q^{85} -7.80349 q^{86} +8.18892 q^{87} -4.14944 q^{88} +10.7976 q^{89} +1.26559 q^{90} -13.7252 q^{91} -3.56973 q^{92} -3.11543 q^{93} +4.65151 q^{94} -2.76979 q^{95} -1.31697 q^{96} +3.67055 q^{97} +2.08439 q^{98} +5.25148 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 22 q^{2} + 2 q^{3} + 22 q^{4} - 22 q^{5} + 2 q^{6} + 13 q^{7} + 22 q^{8} + 32 q^{9}+O(q^{10}) $ 22 * q + 22 * q^2 + 2 * q^3 + 22 * q^4 - 22 * q^5 + 2 * q^6 + 13 * q^7 + 22 * q^8 + 32 * q^9	\( 22 q + 22 q^{2} + 2 q^{3} + 22 q^{4} - 22 q^{5} + 2 q^{6} + 13 q^{7} + 22 q^{8} + 32 q^{9} - 22 q^{10} - 3 q^{11} + 2 q^{12} + 6 q^{13} + 13 q^{14} - 2 q^{15} + 22 q^{16} + 17 q^{17} + 32 q^{18} + 13 q^{19} - 22 q^{20} + 16 q^{21} - 3 q^{22} + 19 q^{23} + 2 q^{24} + 22 q^{25} + 6 q^{26} + 14 q^{27} + 13 q^{28} + 14 q^{29} - 2 q^{30} + 13 q^{31} + 22 q^{32} + 12 q^{33} + 17 q^{34} - 13 q^{35} + 32 q^{36} + 35 q^{37} + 13 q^{38} + 30 q^{39} - 22 q^{40} - 5 q^{41} + 16 q^{42} + 19 q^{43} - 3 q^{44} - 32 q^{45} + 19 q^{46} + 29 q^{47} + 2 q^{48} + 61 q^{49} + 22 q^{50} + q^{51} + 6 q^{52} + 29 q^{53} + 14 q^{54} + 3 q^{55} + 13 q^{56} + 33 q^{57} + 14 q^{58} - 4 q^{59} - 2 q^{60} + 20 q^{61} + 13 q^{62} + 50 q^{63} + 22 q^{64} - 6 q^{65} + 12 q^{66} + 48 q^{67} + 17 q^{68} + 19 q^{69} - 13 q^{70} + 2 q^{71} + 32 q^{72} + 16 q^{73} + 35 q^{74} + 2 q^{75} + 13 q^{76} + 53 q^{77} + 30 q^{78} + 29 q^{79} - 22 q^{80} + 54 q^{81} - 5 q^{82} + 13 q^{83} + 16 q^{84} - 17 q^{85} + 19 q^{86} + 56 q^{87} - 3 q^{88} + 20 q^{89} - 32 q^{90} + 42 q^{91} + 19 q^{92} + 50 q^{93} + 29 q^{94} - 13 q^{95} + 2 q^{96} + 36 q^{97} + 61 q^{98} - q^{99}+O(q^{100}) \) 22 * q + 22 * q^2 + 2 * q^3 + 22 * q^4 - 22 * q^5 + 2 * q^6 + 13 * q^7 + 22 * q^8 + 32 * q^9 - 22 * q^10 - 3 * q^11 + 2 * q^12 + 6 * q^13 + 13 * q^14 - 2 * q^15 + 22 * q^16 + 17 * q^17 + 32 * q^18 + 13 * q^19 - 22 * q^20 + 16 * q^21 - 3 * q^22 + 19 * q^23 + 2 * q^24 + 22 * q^25 + 6 * q^26 + 14 * q^27 + 13 * q^28 + 14 * q^29 - 2 * q^30 + 13 * q^31 + 22 * q^32 + 12 * q^33 + 17 * q^34 - 13 * q^35 + 32 * q^36 + 35 * q^37 + 13 * q^38 + 30 * q^39 - 22 * q^40 - 5 * q^41 + 16 * q^42 + 19 * q^43 - 3 * q^44 - 32 * q^45 + 19 * q^46 + 29 * q^47 + 2 * q^48 + 61 * q^49 + 22 * q^50 + q^51 + 6 * q^52 + 29 * q^53 + 14 * q^54 + 3 * q^55 + 13 * q^56 + 33 * q^57 + 14 * q^58 - 4 * q^59 - 2 * q^60 + 20 * q^61 + 13 * q^62 + 50 * q^63 + 22 * q^64 - 6 * q^65 + 12 * q^66 + 48 * q^67 + 17 * q^68 + 19 * q^69 - 13 * q^70 + 2 * q^71 + 32 * q^72 + 16 * q^73 + 35 * q^74 + 2 * q^75 + 13 * q^76 + 53 * q^77 + 30 * q^78 + 29 * q^79 - 22 * q^80 + 54 * q^81 - 5 * q^82 + 13 * q^83 + 16 * q^84 - 17 * q^85 + 19 * q^86 + 56 * q^87 - 3 * q^88 + 20 * q^89 - 32 * q^90 + 42 * q^91 + 19 * q^92 + 50 * q^93 + 29 * q^94 - 13 * q^95 + 2 * q^96 + 36 * q^97 + 61 * q^98 - 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
$3$	−1.31697	−0.760354	−0.380177	−	0.924914i	$-0.624137\pi$
$3$	−1.31697	−0.760354	−0.380177	+	0.924914i	$0.624137\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.31697	−0.537651
$7$	−3.01403	−1.13920	−0.569599	−	0.821923i	$-0.692901\pi$
$7$	−3.01403	−1.13920	−0.569599	+	0.821923i	$0.692901\pi$
$8$	1.00000	0.353553
$9$	−1.26559	−0.421862
$10$	−1.00000	−0.316228
$11$	−4.14944	−1.25110	−0.625552	−	0.780182i	$-0.715126\pi$
$11$	−4.14944	−1.25110	−0.625552	+	0.780182i	$0.715126\pi$
$12$	−1.31697	−0.380177
$13$	4.55376	1.26299	0.631493	−	0.775381i	$-0.282442\pi$
$13$	4.55376	1.26299	0.631493	+	0.775381i	$0.282442\pi$
$14$	−3.01403	−0.805534
$15$	1.31697	0.340040
$16$	1.00000	0.250000
$17$	−1.18544	−0.287511	−0.143756	−	0.989613i	$-0.545918\pi$
$17$	−1.18544	−0.287511	−0.143756	+	0.989613i	$0.545918\pi$
$18$	−1.26559	−0.298302
$19$	2.76979	0.635434	0.317717	−	0.948186i	$-0.397084\pi$
$19$	2.76979	0.635434	0.317717	+	0.948186i	$0.397084\pi$
$20$	−1.00000	−0.223607
$21$	3.96939	0.866193
$22$	−4.14944	−0.884664
$23$	−3.56973	−0.744340	−0.372170	−	0.928165i	$-0.621386\pi$
$23$	−3.56973	−0.744340	−0.372170	+	0.928165i	$0.621386\pi$
$24$	−1.31697	−0.268826
$25$	1.00000	0.200000
$26$	4.55376	0.893066
$27$	5.61766	1.08112
$28$	−3.01403	−0.569599
$29$	−6.21799	−1.15465	−0.577326	−	0.816514i	$-0.695904\pi$
$29$	−6.21799	−1.15465	−0.577326	+	0.816514i	$0.695904\pi$
$30$	1.31697	0.240445
$31$	2.36560	0.424874	0.212437	−	0.977175i	$-0.431860\pi$
$31$	2.36560	0.424874	0.212437	+	0.977175i	$0.431860\pi$
$32$	1.00000	0.176777
$33$	5.46470	0.951282
$34$	−1.18544	−0.203301
$35$	3.01403	0.509464
$36$	−1.26559	−0.210931
$37$	−9.95108	−1.63595	−0.817973	−	0.575256i	$-0.804902\pi$
$37$	−9.95108	−1.63595	−0.817973	+	0.575256i	$0.804902\pi$
$38$	2.76979	0.449319
$39$	−5.99717	−0.960316
$40$	−1.00000	−0.158114
$41$	3.57940	0.559008	0.279504	−	0.960144i	$-0.409830\pi$
$41$	3.57940	0.559008	0.279504	+	0.960144i	$0.409830\pi$
$42$	3.96939	0.612491
$43$	−7.80349	−1.19002	−0.595011	−	0.803718i	$-0.702852\pi$
$43$	−7.80349	−1.19002	−0.595011	+	0.803718i	$0.702852\pi$
$44$	−4.14944	−0.625552
$45$	1.26559	0.188663
$46$	−3.56973	−0.526328
$47$	4.65151	0.678493	0.339246	−	0.940698i	$-0.389828\pi$
$47$	4.65151	0.678493	0.339246	+	0.940698i	$0.389828\pi$
$48$	−1.31697	−0.190088
$49$	2.08439	0.297770
$50$	1.00000	0.141421
$51$	1.56119	0.218610
$52$	4.55376	0.631493
$53$	−1.53610	−0.211000	−0.105500	−	0.994419i	$-0.533644\pi$
$53$	−1.53610	−0.211000	−0.105500	+	0.994419i	$0.533644\pi$
$54$	5.61766	0.764466
$55$	4.14944	0.559511
$56$	−3.01403	−0.402767
$57$	−3.64774	−0.483154
$58$	−6.21799	−0.816462
$59$	−7.45163	−0.970119	−0.485060	−	0.874481i	$-0.661202\pi$
$59$	−7.45163	−0.970119	−0.485060	+	0.874481i	$0.661202\pi$
$60$	1.31697	0.170020
$61$	13.2771	1.69996	0.849980	−	0.526816i	$-0.176614\pi$
$61$	13.2771	1.69996	0.849980	+	0.526816i	$0.176614\pi$
$62$	2.36560	0.300431
$63$	3.81452	0.480584
$64$	1.00000	0.125000
$65$	−4.55376	−0.564825
$66$	5.46470	0.672658
$67$	10.3042	1.25885	0.629427	−	0.777060i	$-0.283290\pi$
$67$	10.3042	1.25885	0.629427	+	0.777060i	$0.283290\pi$
$68$	−1.18544	−0.143756
$69$	4.70123	0.565962
$70$	3.01403	0.360246
$71$	−5.86976	−0.696612	−0.348306	−	0.937381i	$-0.613243\pi$
$71$	−5.86976	−0.696612	−0.348306	+	0.937381i	$0.613243\pi$
$72$	−1.26559	−0.149151
$73$	9.19353	1.07602	0.538010	−	0.842938i	$-0.319176\pi$
$73$	9.19353	1.07602	0.538010	+	0.842938i	$0.319176\pi$
$74$	−9.95108	−1.15679
$75$	−1.31697	−0.152071
$76$	2.76979	0.317717
$77$	12.5066	1.42525
$78$	−5.99717	−0.679046
$79$	5.83414	0.656392	0.328196	−	0.944610i	$-0.393559\pi$
$79$	5.83414	0.656392	0.328196	+	0.944610i	$0.393559\pi$
$80$	−1.00000	−0.111803
$81$	−3.60153	−0.400170
$82$	3.57940	0.395279
$83$	14.6425	1.60722	0.803612	−	0.595154i	$-0.202909\pi$
$83$	14.6425	1.60722	0.803612	+	0.595154i	$0.202909\pi$
$84$	3.96939	0.433096
$85$	1.18544	0.128579
$86$	−7.80349	−0.841472
$87$	8.18892	0.877944
$88$	−4.14944	−0.442332
$89$	10.7976	1.14454	0.572272	−	0.820064i	$-0.306062\pi$
$89$	10.7976	1.14454	0.572272	+	0.820064i	$0.306062\pi$
$90$	1.26559	0.133405
$91$	−13.7252	−1.43879
$92$	−3.56973	−0.372170
$93$	−3.11543	−0.323055
$94$	4.65151	0.479767
$95$	−2.76979	−0.284175
$96$	−1.31697	−0.134413
$97$	3.67055	0.372688	0.186344	−	0.982485i	$-0.440336\pi$
$97$	3.67055	0.372688	0.186344	+	0.982485i	$0.440336\pi$
$98$	2.08439	0.210555
$99$	5.25148	0.527794
$100$	1.00000	0.100000
$101$	16.3971	1.63158	0.815788	−	0.578352i	$-0.196304\pi$
$101$	16.3971	1.63158	0.815788	+	0.578352i	$0.196304\pi$
$102$	1.56119	0.154581
$103$	3.40271	0.335279	0.167639	−	0.985848i	$-0.446386\pi$
$103$	3.40271	0.335279	0.167639	+	0.985848i	$0.446386\pi$
$104$	4.55376	0.446533
$105$	−3.96939	−0.387373
$106$	−1.53610	−0.149200
$107$	1.75418	0.169583	0.0847914	−	0.996399i	$-0.472978\pi$
$107$	1.75418	0.169583	0.0847914	+	0.996399i	$0.472978\pi$
$108$	5.61766	0.540559
$109$	17.2945	1.65652	0.828258	−	0.560347i	$-0.189332\pi$
$109$	17.2945	1.65652	0.828258	+	0.560347i	$0.189332\pi$
$110$	4.14944	0.395634
$111$	13.1053	1.24390
$112$	−3.01403	−0.284799
$113$	5.09591	0.479383	0.239692	−	0.970849i	$-0.422954\pi$
$113$	5.09591	0.479383	0.239692	+	0.970849i	$0.422954\pi$
$114$	−3.64774	−0.341642
$115$	3.56973	0.332879
$116$	−6.21799	−0.577326
$117$	−5.76318	−0.532806
$118$	−7.45163	−0.685978
$119$	3.57296	0.327532
$120$	1.31697	0.120222
$121$	6.21788	0.565262
$122$	13.2771	1.20205
$123$	−4.71397	−0.425044
$124$	2.36560	0.212437
$125$	−1.00000	−0.0894427
$126$	3.81452	0.339824
$127$	12.1206	1.07553	0.537767	−	0.843094i	$-0.319268\pi$
$127$	12.1206	1.07553	0.537767	+	0.843094i	$0.319268\pi$
$128$	1.00000	0.0883883
$129$	10.2770	0.904837
$130$	−4.55376	−0.399391
$131$	−17.6493	−1.54202	−0.771012	−	0.636821i	$-0.780249\pi$
$131$	−17.6493	−1.54202	−0.771012	+	0.636821i	$0.780249\pi$
$132$	5.46470	0.475641
$133$	−8.34824	−0.723884
$134$	10.3042	0.890144
$135$	−5.61766	−0.483491
$136$	−1.18544	−0.101651
$137$	−1.44857	−0.123760	−0.0618800	−	0.998084i	$-0.519710\pi$
$137$	−1.44857	−0.123760	−0.0618800	+	0.998084i	$0.519710\pi$
$138$	4.70123	0.400195
$139$	−13.4126	−1.13764	−0.568820	−	0.822462i	$-0.692600\pi$
$139$	−13.4126	−1.13764	−0.568820	+	0.822462i	$0.692600\pi$
$140$	3.01403	0.254732
$141$	−6.12591	−0.515895
$142$	−5.86976	−0.492579
$143$	−18.8956	−1.58013
$144$	−1.26559	−0.105466
$145$	6.21799	0.516376
$146$	9.19353	0.760862
$147$	−2.74508	−0.226411
$148$	−9.95108	−0.817973
$149$	−9.55353	−0.782656	−0.391328	−	0.920251i	$-0.627984\pi$
$149$	−9.55353	−0.782656	−0.391328	+	0.920251i	$0.627984\pi$
$150$	−1.31697	−0.107530
$151$	0.796880	0.0648492	0.0324246	−	0.999474i	$-0.489677\pi$
$151$	0.796880	0.0648492	0.0324246	+	0.999474i	$0.489677\pi$
$152$	2.76979	0.224660
$153$	1.50028	0.121290
$154$	12.5066	1.00781
$155$	−2.36560	−0.190009
$156$	−5.99717	−0.480158
$157$	10.7678	0.859366	0.429683	−	0.902980i	$-0.358625\pi$
$157$	10.7678	0.859366	0.429683	+	0.902980i	$0.358625\pi$
$158$	5.83414	0.464139
$159$	2.02301	0.160435
$160$	−1.00000	−0.0790569
$161$	10.7593	0.847950
$162$	−3.60153	−0.282963
$163$	17.6654	1.38366	0.691830	−	0.722060i	$-0.256805\pi$
$163$	17.6654	1.38366	0.691830	+	0.722060i	$0.256805\pi$
$164$	3.57940	0.279504
$165$	−5.46470	−0.425426
$166$	14.6425	1.13648
$167$	−7.06177	−0.546457	−0.273228	−	0.961949i	$-0.588091\pi$
$167$	−7.06177	−0.546457	−0.273228	+	0.961949i	$0.588091\pi$
$168$	3.96939	0.306245
$169$	7.73675	0.595135
$170$	1.18544	0.0909191
$171$	−3.50541	−0.268066
$172$	−7.80349	−0.595011
$173$	11.7879	0.896220	0.448110	−	0.893978i	$-0.352097\pi$
$173$	11.7879	0.896220	0.448110	+	0.893978i	$0.352097\pi$
$174$	8.18892	0.620800
$175$	−3.01403	−0.227839
$176$	−4.14944	−0.312776
$177$	9.81358	0.737634
$178$	10.7976	0.809315
$179$	10.5110	0.785632	0.392816	−	0.919617i	$-0.371501\pi$
$179$	10.5110	0.785632	0.392816	+	0.919617i	$0.371501\pi$
$180$	1.26559	0.0943313
$181$	25.1278	1.86774	0.933869	−	0.357616i	$-0.116410\pi$
$181$	25.1278	1.86774	0.933869	+	0.357616i	$0.116410\pi$
$182$	−13.7252	−1.01738
$183$	−17.4856	−1.29257
$184$	−3.56973	−0.263164
$185$	9.95108	0.731618
$186$	−3.11543	−0.228434
$187$	4.91892	0.359707
$188$	4.65151	0.339246
$189$	−16.9318	−1.23161
$190$	−2.76979	−0.200942
$191$	−17.2311	−1.24680	−0.623401	−	0.781903i	$-0.714249\pi$
$191$	−17.2311	−1.24680	−0.623401	+	0.781903i	$0.714249\pi$
$192$	−1.31697	−0.0950442
$193$	2.14138	0.154140	0.0770700	−	0.997026i	$-0.475444\pi$
$193$	2.14138	0.154140	0.0770700	+	0.997026i	$0.475444\pi$
$194$	3.67055	0.263530
$195$	5.99717	0.429467
$196$	2.08439	0.148885
$197$	−18.1721	−1.29471	−0.647353	−	0.762190i	$-0.724124\pi$
$197$	−18.1721	−1.29471	−0.647353	+	0.762190i	$0.724124\pi$
$198$	5.25148	0.373207
$199$	−0.996004	−0.0706048	−0.0353024	−	0.999377i	$-0.511239\pi$
$199$	−0.996004	−0.0706048	−0.0353024	+	0.999377i	$0.511239\pi$
$200$	1.00000	0.0707107
$201$	−13.5703	−0.957174
$202$	16.3971	1.15370
$203$	18.7412	1.31538
$204$	1.56119	0.109305
$205$	−3.57940	−0.249996
$206$	3.40271	0.237078
$207$	4.51780	0.314009
$208$	4.55376	0.315747
$209$	−11.4931	−0.794994
$210$	−3.96939	−0.273914
$211$	−4.32853	−0.297988	−0.148994	−	0.988838i	$-0.547604\pi$
$211$	−4.32853	−0.297988	−0.148994	+	0.988838i	$0.547604\pi$
$212$	−1.53610	−0.105500
$213$	7.73030	0.529671
$214$	1.75418	0.119913
$215$	7.80349	0.532194
$216$	5.61766	0.382233
$217$	−7.12999	−0.484015
$218$	17.2945	1.17133
$219$	−12.1076	−0.818156
$220$	4.14944	0.279755
$221$	−5.39821	−0.363123
$222$	13.1053	0.879569
$223$	−25.3890	−1.70017	−0.850086	−	0.526644i	$-0.823450\pi$
$223$	−25.3890	−1.70017	−0.850086	+	0.526644i	$0.823450\pi$
$224$	−3.01403	−0.201384
$225$	−1.26559	−0.0843725
$226$	5.09591	0.338975
$227$	−15.3789	−1.02073	−0.510367	−	0.859957i	$-0.670490\pi$
$227$	−15.3789	−1.02073	−0.510367	+	0.859957i	$0.670490\pi$
$228$	−3.64774	−0.241577
$229$	13.1714	0.870392	0.435196	−	0.900336i	$-0.356679\pi$
$229$	13.1714	0.870392	0.435196	+	0.900336i	$0.356679\pi$
$230$	3.56973	0.235381
$231$	−16.4708	−1.08370
$232$	−6.21799	−0.408231
$233$	19.8450	1.30009	0.650046	−	0.759895i	$-0.274750\pi$
$233$	19.8450	1.30009	0.650046	+	0.759895i	$0.274750\pi$
$234$	−5.76318	−0.376751
$235$	−4.65151	−0.303431
$236$	−7.45163	−0.485060
$237$	−7.68339	−0.499090
$238$	3.57296	0.231600
$239$	−9.30931	−0.602169	−0.301085	−	0.953597i	$-0.597349\pi$
$239$	−9.30931	−0.602169	−0.301085	+	0.953597i	$0.597349\pi$
$240$	1.31697	0.0850101
$241$	−14.2415	−0.917376	−0.458688	−	0.888597i	$-0.651680\pi$
$241$	−14.2415	−0.917376	−0.458688	+	0.888597i	$0.651680\pi$
$242$	6.21788	0.399701
$243$	−12.1099	−0.776848
$244$	13.2771	0.849980
$245$	−2.08439	−0.133167
$246$	−4.71397	−0.300552
$247$	12.6130	0.802544
$248$	2.36560	0.150216
$249$	−19.2837	−1.22206
$250$	−1.00000	−0.0632456
$251$	8.66403	0.546869	0.273434	−	0.961891i	$-0.411840\pi$
$251$	8.66403	0.546869	0.273434	+	0.961891i	$0.411840\pi$
$252$	3.81452	0.240292
$253$	14.8124	0.931247
$254$	12.1206	0.760517
$255$	−1.56119	−0.0977655
$256$	1.00000	0.0625000
$257$	2.56785	0.160178	0.0800890	−	0.996788i	$-0.474480\pi$
$257$	2.56785	0.160178	0.0800890	+	0.996788i	$0.474480\pi$
$258$	10.2770	0.639816
$259$	29.9929	1.86367
$260$	−4.55376	−0.282412
$261$	7.86941	0.487104
$262$	−17.6493	−1.09038
$263$	6.35687	0.391982	0.195991	−	0.980606i	$-0.437208\pi$
$263$	6.35687	0.391982	0.195991	+	0.980606i	$0.437208\pi$
$264$	5.46470	0.336329
$265$	1.53610	0.0943621
$266$	−8.34824	−0.511864
$267$	−14.2201	−0.870258
$268$	10.3042	0.629427
$269$	−16.0267	−0.977163	−0.488582	−	0.872518i	$-0.662486\pi$
$269$	−16.0267	−0.977163	−0.488582	+	0.872518i	$0.662486\pi$
$270$	−5.61766	−0.341880
$271$	22.3196	1.35582	0.677910	−	0.735145i	$-0.262886\pi$
$271$	22.3196	1.35582	0.677910	+	0.735145i	$0.262886\pi$
$272$	−1.18544	−0.0718779
$273$	18.0757	1.09399
$274$	−1.44857	−0.0875115
$275$	−4.14944	−0.250221
$276$	4.70123	0.282981
$277$	16.3501	0.982384	0.491192	−	0.871051i	$-0.336561\pi$
$277$	16.3501	0.982384	0.491192	+	0.871051i	$0.336561\pi$
$278$	−13.4126	−0.804433
$279$	−2.99387	−0.179238
$280$	3.01403	0.180123
$281$	−4.05141	−0.241687	−0.120843	−	0.992672i	$-0.538560\pi$
$281$	−4.05141	−0.241687	−0.120843	+	0.992672i	$0.538560\pi$
$282$	−6.12591	−0.364793
$283$	−30.7268	−1.82652	−0.913261	−	0.407376i	$-0.866444\pi$
$283$	−30.7268	−1.82652	−0.913261	+	0.407376i	$0.866444\pi$
$284$	−5.86976	−0.348306
$285$	3.64774	0.216073
$286$	−18.8956	−1.11732
$287$	−10.7884	−0.636821
$288$	−1.26559	−0.0745754
$289$	−15.5947	−0.917337
$290$	6.21799	0.365133
$291$	−4.83401	−0.283375
$292$	9.19353	0.538010
$293$	22.7251	1.32761	0.663807	−	0.747904i	$-0.268940\pi$
$293$	22.7251	1.32761	0.663807	+	0.747904i	$0.268940\pi$
$294$	−2.74508	−0.160097
$295$	7.45163	0.433850
$296$	−9.95108	−0.578395
$297$	−23.3101	−1.35259
$298$	−9.55353	−0.553421
$299$	−16.2557	−0.940092
$300$	−1.31697	−0.0760354
$301$	23.5200	1.35567
$302$	0.796880	0.0458553
$303$	−21.5945	−1.24057
$304$	2.76979	0.158858
$305$	−13.2771	−0.760245
$306$	1.50028	0.0857652
$307$	2.93454	0.167483	0.0837415	−	0.996488i	$-0.473313\pi$
$307$	2.93454	0.167483	0.0837415	+	0.996488i	$0.473313\pi$
$308$	12.5066	0.712627
$309$	−4.48127	−0.254930
$310$	−2.36560	−0.134357
$311$	−17.3237	−0.982335	−0.491168	−	0.871065i	$-0.663430\pi$
$311$	−17.3237	−0.982335	−0.491168	+	0.871065i	$0.663430\pi$
$312$	−5.99717	−0.339523
$313$	−6.11995	−0.345920	−0.172960	−	0.984929i	$-0.555333\pi$
$313$	−6.11995	−0.345920	−0.172960	+	0.984929i	$0.555333\pi$
$314$	10.7678	0.607664
$315$	−3.81452	−0.214924
$316$	5.83414	0.328196
$317$	7.31559	0.410884	0.205442	−	0.978669i	$-0.434137\pi$
$317$	7.31559	0.410884	0.205442	+	0.978669i	$0.434137\pi$
$318$	2.02301	0.113445
$319$	25.8012	1.44459
$320$	−1.00000	−0.0559017
$321$	−2.31020	−0.128943
$322$	10.7593	0.599591
$323$	−3.28342	−0.182694
$324$	−3.60153	−0.200085
$325$	4.55376	0.252597
$326$	17.6654	0.978396
$327$	−22.7764	−1.25954
$328$	3.57940	0.197639
$329$	−14.0198	−0.772937
$330$	−5.46470	−0.300822
$331$	−33.9008	−1.86336	−0.931679	−	0.363283i	$-0.881656\pi$
$331$	−33.9008	−1.86336	−0.931679	+	0.363283i	$0.881656\pi$
$332$	14.6425	0.803612
$333$	12.5940	0.690144
$334$	−7.06177	−0.386403
$335$	−10.3042	−0.562976
$336$	3.96939	0.216548
$337$	24.2677	1.32195	0.660974	−	0.750409i	$-0.270143\pi$
$337$	24.2677	1.32195	0.660974	+	0.750409i	$0.270143\pi$
$338$	7.73675	0.420824
$339$	−6.71117	−0.364501
$340$	1.18544	0.0642895
$341$	−9.81592	−0.531562
$342$	−3.50541	−0.189551
$343$	14.8158	0.799978
$344$	−7.80349	−0.420736
$345$	−4.70123	−0.253106
$346$	11.7879	0.633723
$347$	0.865500	0.0464625	0.0232312	−	0.999730i	$-0.492605\pi$
$347$	0.865500	0.0464625	0.0232312	+	0.999730i	$0.492605\pi$
$348$	8.18892	0.438972
$349$	21.1529	1.13229	0.566146	−	0.824305i	$-0.308434\pi$
$349$	21.1529	1.13229	0.566146	+	0.824305i	$0.308434\pi$
$350$	−3.01403	−0.161107
$351$	25.5815	1.36544
$352$	−4.14944	−0.221166
$353$	30.6441	1.63102	0.815509	−	0.578744i	$-0.196457\pi$
$353$	30.6441	1.63102	0.815509	+	0.578744i	$0.196457\pi$
$354$	9.81358	0.521586
$355$	5.86976	0.311534
$356$	10.7976	0.572272
$357$	−4.70548	−0.249040
$358$	10.5110	0.555525
$359$	−31.9399	−1.68572	−0.842862	−	0.538129i	$-0.819131\pi$
$359$	−31.9399	−1.68572	−0.842862	+	0.538129i	$0.819131\pi$
$360$	1.26559	0.0667023
$361$	−11.3283	−0.596224
$362$	25.1278	1.32069
$363$	−8.18877	−0.429799
$364$	−13.7252	−0.719395
$365$	−9.19353	−0.481211
$366$	−17.4856	−0.913985
$367$	−7.18943	−0.375285	−0.187643	−	0.982237i	$-0.560085\pi$
$367$	−7.18943	−0.375285	−0.187643	+	0.982237i	$0.560085\pi$
$368$	−3.56973	−0.186085
$369$	−4.53004	−0.235825
$370$	9.95108	0.517332
$371$	4.62987	0.240371
$372$	−3.11543	−0.161527
$373$	12.2078	0.632097	0.316048	−	0.948743i	$-0.397644\pi$
$373$	12.2078	0.632097	0.316048	+	0.948743i	$0.397644\pi$
$374$	4.91892	0.254351
$375$	1.31697	0.0680081
$376$	4.65151	0.239883
$377$	−28.3153	−1.45831
$378$	−16.9318	−0.870878
$379$	13.5105	0.693987	0.346993	−	0.937868i	$-0.387203\pi$
$379$	13.5105	0.693987	0.346993	+	0.937868i	$0.387203\pi$
$380$	−2.76979	−0.142087
$381$	−15.9625	−0.817786
$382$	−17.2311	−0.881622
$383$	23.1583	1.18333	0.591666	−	0.806183i	$-0.298471\pi$
$383$	23.1583	1.18333	0.591666	+	0.806183i	$0.298471\pi$
$384$	−1.31697	−0.0672064
$385$	−12.5066	−0.637393
$386$	2.14138	0.108993
$387$	9.87600	0.502025
$388$	3.67055	0.186344
$389$	−1.84393	−0.0934908	−0.0467454	−	0.998907i	$-0.514885\pi$
$389$	−1.84393	−0.0934908	−0.0467454	+	0.998907i	$0.514885\pi$
$390$	5.99717	0.303679
$391$	4.23170	0.214006
$392$	2.08439	0.105278
$393$	23.2436	1.17248
$394$	−18.1721	−0.915496
$395$	−5.83414	−0.293547
$396$	5.25148	0.263897
$397$	−32.8056	−1.64647	−0.823233	−	0.567704i	$-0.807832\pi$
$397$	−32.8056	−1.64647	−0.823233	+	0.567704i	$0.807832\pi$
$398$	−0.996004	−0.0499252
$399$	10.9944	0.550408
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	−13.5703	−0.676824
$403$	10.7724	0.536610
$404$	16.3971	0.815788
$405$	3.60153	0.178961
$406$	18.7412	0.930112
$407$	41.2914	2.04674
$408$	1.56119	0.0772905
$409$	−26.9793	−1.33404	−0.667019	−	0.745041i	$-0.732430\pi$
$409$	−26.9793	−1.33404	−0.667019	+	0.745041i	$0.732430\pi$
$410$	−3.57940	−0.176774
$411$	1.90773	0.0941013
$412$	3.40271	0.167639
$413$	22.4594	1.10516
$414$	4.51780	0.222038
$415$	−14.6425	−0.718772
$416$	4.55376	0.223267
$417$	17.6640	0.865009
$418$	−11.4931	−0.562146
$419$	26.2672	1.28324	0.641619	−	0.767023i	$-0.278263\pi$
$419$	26.2672	1.28324	0.641619	+	0.767023i	$0.278263\pi$
$420$	−3.96939	−0.193687
$421$	−19.2845	−0.939868	−0.469934	−	0.882701i	$-0.655722\pi$
$421$	−19.2845	−0.939868	−0.469934	+	0.882701i	$0.655722\pi$
$422$	−4.32853	−0.210709
$423$	−5.88689	−0.286231
$424$	−1.53610	−0.0745998
$425$	−1.18544	−0.0575023
$426$	7.73030	0.374534
$427$	−40.0176	−1.93659
$428$	1.75418	0.0847914
$429$	24.8849	1.20146
$430$	7.80349	0.376318
$431$	18.4546	0.888927	0.444463	−	0.895797i	$-0.353394\pi$
$431$	18.4546	0.888927	0.444463	+	0.895797i	$0.353394\pi$
$432$	5.61766	0.270280
$433$	35.2647	1.69471	0.847357	−	0.531024i	$-0.178192\pi$
$433$	35.2647	1.69471	0.847357	+	0.531024i	$0.178192\pi$
$434$	−7.12999	−0.342251
$435$	−8.18892	−0.392628
$436$	17.2945	0.828258
$437$	−9.88741	−0.472979
$438$	−12.1076	−0.578524
$439$	−28.1107	−1.34165	−0.670825	−	0.741616i	$-0.734060\pi$
$439$	−28.1107	−1.34165	−0.670825	+	0.741616i	$0.734060\pi$
$440$	4.14944	0.197817
$441$	−2.63798	−0.125618
$442$	−5.39821	−0.256767
$443$	15.3803	0.730741	0.365371	−	0.930862i	$-0.380942\pi$
$443$	15.3803	0.730741	0.365371	+	0.930862i	$0.380942\pi$
$444$	13.1053	0.621949
$445$	−10.7976	−0.511856
$446$	−25.3890	−1.20220
$447$	12.5817	0.595095
$448$	−3.01403	−0.142400
$449$	7.62498	0.359845	0.179923	−	0.983681i	$-0.442415\pi$
$449$	7.62498	0.359845	0.179923	+	0.983681i	$0.442415\pi$
$450$	−1.26559	−0.0596603
$451$	−14.8525	−0.699378
$452$	5.09591	0.239692
$453$	−1.04947	−0.0493083
$454$	−15.3789	−0.721767
$455$	13.7252	0.643447
$456$	−3.64774	−0.170821
$457$	−22.6036	−1.05735	−0.528675	−	0.848824i	$-0.677311\pi$
$457$	−22.6036	−1.05735	−0.528675	+	0.848824i	$0.677311\pi$
$458$	13.1714	0.615460
$459$	−6.65939	−0.310834
$460$	3.56973	0.166440
$461$	2.84937	0.132708	0.0663542	−	0.997796i	$-0.478863\pi$
$461$	2.84937	0.132708	0.0663542	+	0.997796i	$0.478863\pi$
$462$	−16.4708	−0.766290
$463$	30.9923	1.44033	0.720167	−	0.693801i	$-0.244065\pi$
$463$	30.9923	1.44033	0.720167	+	0.693801i	$0.244065\pi$
$464$	−6.21799	−0.288663
$465$	3.11543	0.144474
$466$	19.8450	0.919303
$467$	−33.4093	−1.54600	−0.772999	−	0.634407i	$-0.781244\pi$
$467$	−33.4093	−1.54600	−0.772999	+	0.634407i	$0.781244\pi$
$468$	−5.76318	−0.266403
$469$	−31.0571	−1.43408
$470$	−4.65151	−0.214558
$471$	−14.1809	−0.653422
$472$	−7.45163	−0.342989
$473$	32.3801	1.48884
$474$	−7.68339	−0.352910
$475$	2.76979	0.127087
$476$	3.57296	0.163766
$477$	1.94407	0.0890130
$478$	−9.30931	−0.425798
$479$	−0.479632	−0.0219149	−0.0109575	−	0.999940i	$-0.503488\pi$
$479$	−0.479632	−0.0219149	−0.0109575	+	0.999940i	$0.503488\pi$
$480$	1.31697	0.0601112
$481$	−45.3148	−2.06618
$482$	−14.2415	−0.648683
$483$	−14.1697	−0.644742
$484$	6.21788	0.282631
$485$	−3.67055	−0.166671
$486$	−12.1099	−0.549314
$487$	−8.98378	−0.407094	−0.203547	−	0.979065i	$-0.565247\pi$
$487$	−8.98378	−0.407094	−0.203547	+	0.979065i	$0.565247\pi$
$488$	13.2771	0.601026
$489$	−23.2648	−1.05207
$490$	−2.08439	−0.0941632
$491$	−23.9670	−1.08162	−0.540809	−	0.841146i	$-0.681882\pi$
$491$	−23.9670	−1.08162	−0.540809	+	0.841146i	$0.681882\pi$
$492$	−4.71397	−0.212522
$493$	7.37106	0.331976
$494$	12.6130	0.567484
$495$	−5.25148	−0.236037
$496$	2.36560	0.106219
$497$	17.6916	0.793578
$498$	−19.2837	−0.864125
$499$	−9.90452	−0.443387	−0.221694	−	0.975116i	$-0.571158\pi$
$499$	−9.90452	−0.443387	−0.221694	+	0.975116i	$0.571158\pi$
$500$	−1.00000	−0.0447214
$501$	9.30015	0.415500
$502$	8.66403	0.386695
$503$	23.1749	1.03332	0.516660	−	0.856191i	$-0.327175\pi$
$503$	23.1749	1.03332	0.516660	+	0.856191i	$0.327175\pi$
$504$	3.81452	0.169912
$505$	−16.3971	−0.729663
$506$	14.8124	0.658491
$507$	−10.1891	−0.452513
$508$	12.1206	0.537767
$509$	−4.69259	−0.207995	−0.103998	−	0.994578i	$-0.533163\pi$
$509$	−4.69259	−0.207995	−0.103998	+	0.994578i	$0.533163\pi$
$510$	−1.56119	−0.0691307
$511$	−27.7096	−1.22580
$512$	1.00000	0.0441942
$513$	15.5597	0.686979
$514$	2.56785	0.113263
$515$	−3.40271	−0.149941
$516$	10.2770	0.452419
$517$	−19.3012	−0.848865
$518$	29.9929	1.31781
$519$	−15.5244	−0.681444
$520$	−4.55376	−0.199696
$521$	30.5415	1.33805	0.669024	−	0.743241i	$-0.266712\pi$
$521$	30.5415	1.33805	0.669024	+	0.743241i	$0.266712\pi$
$522$	7.86941	0.344435
$523$	36.3034	1.58744	0.793719	−	0.608284i	$-0.208142\pi$
$523$	36.3034	1.58744	0.793719	+	0.608284i	$0.208142\pi$
$524$	−17.6493	−0.771012
$525$	3.96939	0.173239
$526$	6.35687	0.277173
$527$	−2.80428	−0.122156
$528$	5.46470	0.237820
$529$	−10.2570	−0.445958
$530$	1.53610	0.0667241
$531$	9.43068	0.409257
$532$	−8.34824	−0.361942
$533$	16.2997	0.706020
$534$	−14.2201	−0.615366
$535$	−1.75418	−0.0758397
$536$	10.3042	0.445072
$537$	−13.8427	−0.597358
$538$	−16.0267	−0.690959
$539$	−8.64907	−0.372542
$540$	−5.61766	−0.241745
$541$	−24.2298	−1.04172	−0.520861	−	0.853642i	$-0.674389\pi$
$541$	−24.2298	−1.04172	−0.520861	+	0.853642i	$0.674389\pi$
$542$	22.3196	0.958710
$543$	−33.0926	−1.42014
$544$	−1.18544	−0.0508253
$545$	−17.2945	−0.740817
$546$	18.0757	0.773568
$547$	45.5726	1.94854	0.974271	−	0.225378i	$-0.0723619\pi$
$547$	45.5726	1.94854	0.974271	+	0.225378i	$0.0723619\pi$
$548$	−1.44857	−0.0618800
$549$	−16.8033	−0.717149
$550$	−4.14944	−0.176933
$551$	−17.2225	−0.733705
$552$	4.70123	0.200098
$553$	−17.5843	−0.747760
$554$	16.3501	0.694651
$555$	−13.1053	−0.556288
$556$	−13.4126	−0.568820
$557$	−6.32102	−0.267830	−0.133915	−	0.990993i	$-0.542755\pi$
$557$	−6.32102	−0.267830	−0.133915	+	0.990993i	$0.542755\pi$
$558$	−2.99387	−0.126741
$559$	−35.5352	−1.50298
$560$	3.01403	0.127366
$561$	−6.47807	−0.273504
$562$	−4.05141	−0.170898
$563$	19.1945	0.808950	0.404475	−	0.914549i	$-0.367454\pi$
$563$	19.1945	0.808950	0.404475	+	0.914549i	$0.367454\pi$
$564$	−6.12591	−0.257947
$565$	−5.09591	−0.214387
$566$	−30.7268	−1.29155
$567$	10.8551	0.455872
$568$	−5.86976	−0.246289
$569$	17.7866	0.745652	0.372826	−	0.927901i	$-0.378389\pi$
$569$	17.7866	0.745652	0.372826	+	0.927901i	$0.378389\pi$
$570$	3.64774	0.152787
$571$	−4.80376	−0.201031	−0.100516	−	0.994935i	$-0.532049\pi$
$571$	−4.80376	−0.201031	−0.100516	+	0.994935i	$0.532049\pi$
$572$	−18.8956	−0.790064
$573$	22.6929	0.948010
$574$	−10.7884	−0.450300
$575$	−3.56973	−0.148868
$576$	−1.26559	−0.0527328
$577$	32.7924	1.36517	0.682583	−	0.730808i	$-0.260857\pi$
$577$	32.7924	1.36517	0.682583	+	0.730808i	$0.260857\pi$
$578$	−15.5947	−0.648655
$579$	−2.82014	−0.117201
$580$	6.21799	0.258188
$581$	−44.1330	−1.83094
$582$	−4.83401	−0.200376
$583$	6.37398	0.263983
$584$	9.19353	0.380431
$585$	5.76318	0.238278
$586$	22.7251	0.938765
$587$	22.2207	0.917145	0.458572	−	0.888657i	$-0.348361\pi$
$587$	22.2207	0.917145	0.458572	+	0.888657i	$0.348361\pi$
$588$	−2.74508	−0.113205
$589$	6.55222	0.269979
$590$	7.45163	0.306779
$591$	23.9321	0.984435
$592$	−9.95108	−0.408987
$593$	42.8834	1.76101	0.880505	−	0.474036i	$-0.157203\pi$
$593$	42.8834	1.76101	0.880505	+	0.474036i	$0.157203\pi$
$594$	−23.3101	−0.956427
$595$	−3.57296	−0.146477
$596$	−9.55353	−0.391328
$597$	1.31171	0.0536846
$598$	−16.2557	−0.664745
$599$	29.3506	1.19923	0.599617	−	0.800287i	$-0.295319\pi$
$599$	29.3506	1.19923	0.599617	+	0.800287i	$0.295319\pi$
$600$	−1.31697	−0.0537651
$601$	21.5853	0.880484	0.440242	−	0.897879i	$-0.354893\pi$
$601$	21.5853	0.880484	0.440242	+	0.897879i	$0.354893\pi$
$602$	23.5200	0.958603
$603$	−13.0408	−0.531063
$604$	0.796880	0.0324246
$605$	−6.21788	−0.252793
$606$	−21.5945	−0.877218
$607$	−15.3343	−0.622401	−0.311201	−	0.950344i	$-0.600731\pi$
$607$	−15.3343	−0.622401	−0.311201	+	0.950344i	$0.600731\pi$
$608$	2.76979	0.112330
$609$	−24.6817	−1.00015
$610$	−13.2771	−0.537574
$611$	21.1819	0.856927
$612$	1.50028	0.0606451
$613$	−12.1200	−0.489522	−0.244761	−	0.969583i	$-0.578709\pi$
$613$	−12.1200	−0.489522	−0.244761	+	0.969583i	$0.578709\pi$
$614$	2.93454	0.118428
$615$	4.71397	0.190086
$616$	12.5066	0.503904
$617$	−31.1063	−1.25229	−0.626147	−	0.779705i	$-0.715369\pi$
$617$	−31.1063	−1.25229	−0.626147	+	0.779705i	$0.715369\pi$
$618$	−4.48127	−0.180263
$619$	−37.0185	−1.48790	−0.743950	−	0.668235i	$-0.767050\pi$
$619$	−37.0185	−1.48790	−0.743950	+	0.668235i	$0.767050\pi$
$620$	−2.36560	−0.0950047
$621$	−20.0535	−0.804720
$622$	−17.3237	−0.694616
$623$	−32.5443	−1.30386
$624$	−5.99717	−0.240079
$625$	1.00000	0.0400000
$626$	−6.11995	−0.244603
$627$	15.1361	0.604477
$628$	10.7678	0.429683
$629$	11.7964	0.470354
$630$	−3.81452	−0.151974
$631$	24.6165	0.979967	0.489984	−	0.871732i	$-0.337003\pi$
$631$	24.6165	0.979967	0.489984	+	0.871732i	$0.337003\pi$
$632$	5.83414	0.232069
$633$	5.70055	0.226576
$634$	7.31559	0.290539
$635$	−12.1206	−0.480993
$636$	2.02301	0.0802174
$637$	9.49183	0.376080
$638$	25.8012	1.02148
$639$	7.42869	0.293874
$640$	−1.00000	−0.0395285
$641$	−19.0100	−0.750848	−0.375424	−	0.926853i	$-0.622503\pi$
$641$	−19.0100	−0.750848	−0.375424	+	0.926853i	$0.622503\pi$
$642$	−2.31020	−0.0911764
$643$	37.9879	1.49810	0.749048	−	0.662516i	$-0.230511\pi$
$643$	37.9879	1.49810	0.749048	+	0.662516i	$0.230511\pi$
$644$	10.7593	0.423975
$645$	−10.2770	−0.404655
$646$	−3.28342	−0.129185
$647$	−50.4526	−1.98350	−0.991748	−	0.128202i	$-0.959079\pi$
$647$	−50.4526	−1.98350	−0.991748	+	0.128202i	$0.959079\pi$
$648$	−3.60153	−0.141481
$649$	30.9201	1.21372
$650$	4.55376	0.178613
$651$	9.38999	0.368023
$652$	17.6654	0.691830
$653$	−21.1393	−0.827245	−0.413623	−	0.910448i	$-0.635737\pi$
$653$	−21.1393	−0.827245	−0.413623	+	0.910448i	$0.635737\pi$
$654$	−22.7764	−0.890628
$655$	17.6493	0.689614
$656$	3.57940	0.139752
$657$	−11.6352	−0.453933
$658$	−14.0198	−0.546549
$659$	32.7672	1.27643	0.638214	−	0.769859i	$-0.279673\pi$
$659$	32.7672	1.27643	0.638214	+	0.769859i	$0.279673\pi$
$660$	−5.46470	−0.212713
$661$	−42.1849	−1.64080	−0.820401	−	0.571789i	$-0.806250\pi$
$661$	−42.1849	−1.64080	−0.820401	+	0.571789i	$0.806250\pi$
$662$	−33.9008	−1.31759
$663$	7.10929	0.276102
$664$	14.6425	0.568239
$665$	8.34824	0.323731
$666$	12.5940	0.488006
$667$	22.1966	0.859454
$668$	−7.06177	−0.273228
$669$	33.4366	1.29273
$670$	−10.3042	−0.398084
$671$	−55.0926	−2.12683
$672$	3.96939	0.153123
$673$	17.9719	0.692767	0.346384	−	0.938093i	$-0.387410\pi$
$673$	17.9719	0.692767	0.346384	+	0.938093i	$0.387410\pi$
$674$	24.2677	0.934758
$675$	5.61766	0.216224
$676$	7.73675	0.297567
$677$	35.8226	1.37678	0.688388	−	0.725343i	$-0.258319\pi$
$677$	35.8226	1.37678	0.688388	+	0.725343i	$0.258319\pi$
$678$	−6.71117	−0.257741
$679$	−11.0632	−0.424565
$680$	1.18544	0.0454596
$681$	20.2536	0.776118
$682$	−9.81592	−0.375871
$683$	0.459535	0.0175836	0.00879181	−	0.999961i	$-0.497201\pi$
$683$	0.459535	0.0175836	0.00879181	+	0.999961i	$0.497201\pi$
$684$	−3.50541	−0.134033
$685$	1.44857	0.0553471
$686$	14.8158	0.565670
$687$	−17.3464	−0.661806
$688$	−7.80349	−0.297505
$689$	−6.99505	−0.266490
$690$	−4.70123	−0.178973
$691$	−9.50125	−0.361445	−0.180722	−	0.983534i	$-0.557844\pi$
$691$	−9.50125	−0.361445	−0.180722	+	0.983534i	$0.557844\pi$
$692$	11.7879	0.448110
$693$	−15.8281	−0.601261
$694$	0.865500	0.0328539
$695$	13.4126	0.508768
$696$	8.18892	0.310400
$697$	−4.24317	−0.160721
$698$	21.1529	0.800651
$699$	−26.1353	−0.988529
$700$	−3.01403	−0.113920
$701$	12.6815	0.478975	0.239488	−	0.970899i	$-0.423021\pi$
$701$	12.6815	0.478975	0.239488	+	0.970899i	$0.423021\pi$
$702$	25.5815	0.965510
$703$	−27.5624	−1.03954
$704$	−4.14944	−0.156388
$705$	6.12591	0.230715
$706$	30.6441	1.15330
$707$	−49.4215	−1.85869
$708$	9.81358	0.368817
$709$	−13.6433	−0.512386	−0.256193	−	0.966626i	$-0.582468\pi$
$709$	−13.6433	−0.512386	−0.256193	+	0.966626i	$0.582468\pi$
$710$	5.86976	0.220288
$711$	−7.38361	−0.276907
$712$	10.7976	0.404657
$713$	−8.44455	−0.316251
$714$	−4.70548	−0.176098
$715$	18.8956	0.706655
$716$	10.5110	0.392816
$717$	12.2601	0.457862
$718$	−31.9399	−1.19199
$719$	−19.8022	−0.738499	−0.369249	−	0.929330i	$-0.620385\pi$
$719$	−19.8022	−0.738499	−0.369249	+	0.929330i	$0.620385\pi$
$720$	1.26559	0.0471656
$721$	−10.2559	−0.381949
$722$	−11.3283	−0.421594
$723$	18.7557	0.697530
$724$	25.1278	0.933869
$725$	−6.21799	−0.230930
$726$	−8.18877	−0.303914
$727$	−32.1613	−1.19280	−0.596398	−	0.802689i	$-0.703402\pi$
$727$	−32.1613	−1.19280	−0.596398	+	0.802689i	$0.703402\pi$
$728$	−13.7252	−0.508689
$729$	26.7529	0.990849
$730$	−9.19353	−0.340268
$731$	9.25057	0.342145
$732$	−17.4856	−0.646285
$733$	−44.8373	−1.65611	−0.828053	−	0.560650i	$-0.810551\pi$
$733$	−44.8373	−1.65611	−0.828053	+	0.560650i	$0.810551\pi$
$734$	−7.18943	−0.265367
$735$	2.74508	0.101254
$736$	−3.56973	−0.131582
$737$	−42.7565	−1.57496
$738$	−4.53004	−0.166753
$739$	26.6661	0.980930	0.490465	−	0.871461i	$-0.336827\pi$
$739$	26.6661	0.980930	0.490465	+	0.871461i	$0.336827\pi$
$740$	9.95108	0.365809
$741$	−16.6109	−0.610217
$742$	4.62987	0.169968
$743$	27.7259	1.01716	0.508582	−	0.861014i	$-0.330170\pi$
$743$	27.7259	1.01716	0.508582	+	0.861014i	$0.330170\pi$
$744$	−3.11543	−0.114217
$745$	9.55353	0.350014
$746$	12.2078	0.446960
$747$	−18.5314	−0.678027
$748$	4.91892	0.179853
$749$	−5.28715	−0.193188
$750$	1.31697	0.0480890
$751$	31.1507	1.13671	0.568353	−	0.822785i	$-0.307581\pi$
$751$	31.1507	1.13671	0.568353	+	0.822785i	$0.307581\pi$
$752$	4.65151	0.169623
$753$	−11.4103	−0.415814
$754$	−28.3153	−1.03118
$755$	−0.796880	−0.0290014
$756$	−16.9318	−0.615803
$757$	16.1171	0.585785	0.292893	−	0.956145i	$-0.405382\pi$
$757$	16.1171	0.585785	0.292893	+	0.956145i	$0.405382\pi$
$758$	13.5105	0.490723
$759$	−19.5075	−0.708077
$760$	−2.76979	−0.100471
$761$	10.3572	0.375449	0.187725	−	0.982222i	$-0.439889\pi$
$761$	10.3572	0.375449	0.187725	+	0.982222i	$0.439889\pi$
$762$	−15.9625	−0.578262
$763$	−52.1263	−1.88710
$764$	−17.2311	−0.623401
$765$	−1.50028	−0.0542427
$766$	23.1583	0.836742
$767$	−33.9329	−1.22525
$768$	−1.31697	−0.0475221
$769$	8.14321	0.293652	0.146826	−	0.989162i	$-0.453094\pi$
$769$	8.14321	0.293652	0.146826	+	0.989162i	$0.453094\pi$
$770$	−12.5066	−0.450705
$771$	−3.38178	−0.121792
$772$	2.14138	0.0770700
$773$	−32.6673	−1.17496	−0.587481	−	0.809238i	$-0.699880\pi$
$773$	−32.6673	−1.17496	−0.587481	+	0.809238i	$0.699880\pi$
$774$	9.87600	0.354985
$775$	2.36560	0.0849748
$776$	3.67055	0.131765
$777$	−39.4997	−1.41705
$778$	−1.84393	−0.0661080
$779$	9.91419	0.355213
$780$	5.99717	0.214733
$781$	24.3562	0.871534
$782$	4.23170	0.151325
$783$	−34.9305	−1.24832
$784$	2.08439	0.0744426
$785$	−10.7678	−0.384320
$786$	23.2436	0.829071
$787$	42.8221	1.52644	0.763221	−	0.646137i	$-0.223617\pi$
$787$	42.8221	1.52644	0.763221	+	0.646137i	$0.223617\pi$
$788$	−18.1721	−0.647353
$789$	−8.37182	−0.298045
$790$	−5.83414	−0.207569
$791$	−15.3592	−0.546112
$792$	5.25148	0.186603
$793$	60.4608	2.14703
$794$	−32.8056	−1.16423
$795$	−2.02301	−0.0717486
$796$	−0.996004	−0.0353024
$797$	−9.50592	−0.336717	−0.168359	−	0.985726i	$-0.553847\pi$
$797$	−9.50592	−0.336717	−0.168359	+	0.985726i	$0.553847\pi$
$798$	10.9944	0.389197
$799$	−5.51409	−0.195075
$800$	1.00000	0.0353553
$801$	−13.6653	−0.482840
$802$	−1.00000	−0.0353112
$803$	−38.1480	−1.34621
$804$	−13.5703	−0.478587
$805$	−10.7593	−0.379215
$806$	10.7724	0.379441
$807$	21.1067	0.742990
$808$	16.3971	0.576849
$809$	0.0893244	0.00314048	0.00157024	−	0.999999i	$-0.499500\pi$
$809$	0.0893244	0.00314048	0.00157024	+	0.999999i	$0.499500\pi$
$810$	3.60153	0.126545
$811$	46.7301	1.64091	0.820457	−	0.571708i	$-0.193719\pi$
$811$	46.7301	1.64091	0.820457	+	0.571708i	$0.193719\pi$
$812$	18.7412	0.657688
$813$	−29.3943	−1.03090
$814$	41.2914	1.44726
$815$	−17.6654	−0.618792
$816$	1.56119	0.0546526
$817$	−21.6140	−0.756180
$818$	−26.9793	−0.943307
$819$	17.3704	0.606972
$820$	−3.57940	−0.124998
$821$	−17.7105	−0.618099	−0.309050	−	0.951046i	$-0.600011\pi$
$821$	−17.7105	−0.618099	−0.309050	+	0.951046i	$0.600011\pi$
$822$	1.90773	0.0665397
$823$	−40.7814	−1.42155	−0.710775	−	0.703419i	$-0.751656\pi$
$823$	−40.7814	−1.42155	−0.710775	+	0.703419i	$0.751656\pi$
$824$	3.40271	0.118539
$825$	5.46470	0.190256
$826$	22.4594	0.781464
$827$	−2.94042	−0.102248	−0.0511242	−	0.998692i	$-0.516280\pi$
$827$	−2.94042	−0.102248	−0.0511242	+	0.998692i	$0.516280\pi$
$828$	4.51780	0.157005
$829$	17.0507	0.592195	0.296098	−	0.955158i	$-0.404315\pi$
$829$	17.0507	0.592195	0.296098	+	0.955158i	$0.404315\pi$
$830$	−14.6425	−0.508249
$831$	−21.5327	−0.746960
$832$	4.55376	0.157873
$833$	−2.47092	−0.0856124
$834$	17.6640	0.611654
$835$	7.06177	0.244383
$836$	−11.4931	−0.397497
$837$	13.2891	0.459339
$838$	26.2672	0.907386
$839$	14.8663	0.513241	0.256620	−	0.966512i	$-0.417391\pi$
$839$	14.8663	0.513241	0.256620	+	0.966512i	$0.417391\pi$
$840$	−3.96939	−0.136957
$841$	9.66342	0.333221
$842$	−19.2845	−0.664587
$843$	5.33559	0.183767
$844$	−4.32853	−0.148994
$845$	−7.73675	−0.266152
$846$	−5.88689	−0.202396
$847$	−18.7409	−0.643945
$848$	−1.53610	−0.0527500
$849$	40.4664	1.38880
$850$	−1.18544	−0.0406603
$851$	35.5227	1.21770
$852$	7.73030	0.264836
$853$	30.9528	1.05980	0.529902	−	0.848059i	$-0.322229\pi$
$853$	30.9528	1.05980	0.529902	+	0.848059i	$0.322229\pi$
$854$	−40.0176	−1.36938
$855$	3.50541	0.119883
$856$	1.75418	0.0599566
$857$	6.42423	0.219447	0.109724	−	0.993962i	$-0.465003\pi$
$857$	6.42423	0.219447	0.109724	+	0.993962i	$0.465003\pi$
$858$	24.8849	0.849558
$859$	−38.4042	−1.31033	−0.655167	−	0.755484i	$-0.727402\pi$
$859$	−38.4042	−1.31033	−0.655167	+	0.755484i	$0.727402\pi$
$860$	7.80349	0.266097
$861$	14.2081	0.484209
$862$	18.4546	0.628566
$863$	46.4007	1.57950	0.789748	−	0.613431i	$-0.210211\pi$
$863$	46.4007	1.57950	0.789748	+	0.613431i	$0.210211\pi$
$864$	5.61766	0.191117
$865$	−11.7879	−0.400802
$866$	35.2647	1.19834
$867$	20.5378	0.697501
$868$	−7.12999	−0.242008
$869$	−24.2084	−0.821214
$870$	−8.18892	−0.277630
$871$	46.9227	1.58991
$872$	17.2945	0.585667
$873$	−4.64540	−0.157223
$874$	−9.88741	−0.334447
$875$	3.01403	0.101893
$876$	−12.1076	−0.409078
$877$	−6.96580	−0.235218	−0.117609	−	0.993060i	$-0.537523\pi$
$877$	−6.96580	−0.235218	−0.117609	+	0.993060i	$0.537523\pi$
$878$	−28.1107	−0.948690
$879$	−29.9283	−1.00946
$880$	4.14944	0.139878
$881$	38.2362	1.28821	0.644104	−	0.764938i	$-0.277230\pi$
$881$	38.2362	1.28821	0.644104	+	0.764938i	$0.277230\pi$
$882$	−2.63798	−0.0888254
$883$	15.9949	0.538271	0.269135	−	0.963102i	$-0.413262\pi$
$883$	15.9949	0.538271	0.269135	+	0.963102i	$0.413262\pi$
$884$	−5.39821	−0.181562
$885$	−9.81358	−0.329880
$886$	15.3803	0.516712
$887$	36.6979	1.23219	0.616097	−	0.787670i	$-0.288713\pi$
$887$	36.6979	1.23219	0.616097	+	0.787670i	$0.288713\pi$
$888$	13.1053	0.439784
$889$	−36.5320	−1.22524
$890$	−10.7976	−0.361937
$891$	14.9443	0.500654
$892$	−25.3890	−0.850086
$893$	12.8837	0.431137
$894$	12.5817	0.420796
$895$	−10.5110	−0.351345
$896$	−3.01403	−0.100692
$897$	21.4083	0.714802
$898$	7.62498	0.254449
$899$	−14.7093	−0.490582
$900$	−1.26559	−0.0421862
$901$	1.82096	0.0606650
$902$	−14.8525	−0.494535
$903$	−30.9751	−1.03079
$904$	5.09591	0.169487
$905$	−25.1278	−0.835278
$906$	−1.04947	−0.0348662
$907$	−46.8825	−1.55671	−0.778354	−	0.627826i	$-0.783945\pi$
$907$	−46.8825	−1.55671	−0.778354	+	0.627826i	$0.783945\pi$
$908$	−15.3789	−0.510367
$909$	−20.7520	−0.688300
$910$	13.7252	0.454986
$911$	−24.8133	−0.822101	−0.411051	−	0.911612i	$-0.634838\pi$
$911$	−24.8133	−0.822101	−0.411051	+	0.911612i	$0.634838\pi$
$912$	−3.64774	−0.120789
$913$	−60.7582	−2.01080
$914$	−22.6036	−0.747660
$915$	17.4856	0.578055
$916$	13.1714	0.435196
$917$	53.1955	1.75667
$918$	−6.65939	−0.219793
$919$	−0.525957	−0.0173497	−0.00867487	−	0.999962i	$-0.502761\pi$
$919$	−0.525957	−0.0173497	−0.00867487	+	0.999962i	$0.502761\pi$
$920$	3.56973	0.117691
$921$	−3.86470	−0.127346
$922$	2.84937	0.0938391
$923$	−26.7295	−0.879811
$924$	−16.4708	−0.541849
$925$	−9.95108	−0.327189
$926$	30.9923	1.01847
$927$	−4.30642	−0.141441
$928$	−6.21799	−0.204116
$929$	−36.0982	−1.18434	−0.592172	−	0.805812i	$-0.701729\pi$
$929$	−36.0982	−1.18434	−0.592172	+	0.805812i	$0.701729\pi$
$930$	3.11543	0.102159
$931$	5.77333	0.189213
$932$	19.8450	0.650046
$933$	22.8148	0.746922
$934$	−33.4093	−1.09319
$935$	−4.91892	−0.160866
$936$	−5.76318	−0.188375
$937$	36.6279	1.19658	0.598290	−	0.801279i	$-0.295847\pi$
$937$	36.6279	1.19658	0.598290	+	0.801279i	$0.295847\pi$
$938$	−31.0571	−1.01405
$939$	8.05980	0.263022
$940$	−4.65151	−0.151716
$941$	−26.5791	−0.866454	−0.433227	−	0.901285i	$-0.642625\pi$
$941$	−26.5791	−0.866454	−0.433227	+	0.901285i	$0.642625\pi$
$942$	−14.1809	−0.462039
$943$	−12.7775	−0.416092
$944$	−7.45163	−0.242530
$945$	16.9318	0.550791
$946$	32.3801	1.05277
$947$	−47.2404	−1.53511	−0.767553	−	0.640986i	$-0.778526\pi$
$947$	−47.2404	−1.53511	−0.767553	+	0.640986i	$0.778526\pi$
$948$	−7.68339	−0.249545
$949$	41.8651	1.35900
$950$	2.76979	0.0898639
$951$	−9.63442	−0.312417
$952$	3.57296	0.115800
$953$	−5.79227	−0.187630	−0.0938151	−	0.995590i	$-0.529906\pi$
$953$	−5.79227	−0.187630	−0.0938151	+	0.995590i	$0.529906\pi$
$954$	1.94407	0.0629417
$955$	17.2311	0.557587
$956$	−9.30931	−0.301085
$957$	−33.9794	−1.09840
$958$	−0.479632	−0.0154962
$959$	4.36605	0.140987
$960$	1.31697	0.0425051
$961$	−25.4039	−0.819482
$962$	−45.3148	−1.46101
$963$	−2.22007	−0.0715406
$964$	−14.2415	−0.458688
$965$	−2.14138	−0.0689335
$966$	−14.1697	−0.455902
$967$	16.2824	0.523606	0.261803	−	0.965121i	$-0.415683\pi$
$967$	16.2824	0.523606	0.261803	+	0.965121i	$0.415683\pi$
$968$	6.21788	0.199850
$969$	4.32417	0.138912
$970$	−3.67055	−0.117854
$971$	−22.6831	−0.727936	−0.363968	−	0.931412i	$-0.618578\pi$
$971$	−22.6831	−0.727936	−0.363968	+	0.931412i	$0.618578\pi$
$972$	−12.1099	−0.388424
$973$	40.4260	1.29600
$974$	−8.98378	−0.287859
$975$	−5.99717	−0.192063
$976$	13.2771	0.424990
$977$	−30.4171	−0.973130	−0.486565	−	0.873644i	$-0.661750\pi$
$977$	−30.4171	−0.973130	−0.486565	+	0.873644i	$0.661750\pi$
$978$	−23.2648	−0.743927
$979$	−44.8041	−1.43194
$980$	−2.08439	−0.0665835
$981$	−21.8877	−0.698822
$982$	−23.9670	−0.764819
$983$	−46.3018	−1.47680	−0.738399	−	0.674364i	$-0.764418\pi$
$983$	−46.3018	−1.47680	−0.738399	+	0.674364i	$0.764418\pi$
$984$	−4.71397	−0.150276
$985$	18.1721	0.579010
$986$	7.37106	0.234742
$987$	18.4637	0.587706
$988$	12.6130	0.401272
$989$	27.8564	0.885781
$990$	−5.25148	−0.166903
$991$	42.4122	1.34727	0.673635	−	0.739065i	$-0.264732\pi$
$991$	42.4122	1.34727	0.673635	+	0.739065i	$0.264732\pi$
$992$	2.36560	0.0751078
$993$	44.6464	1.41681
$994$	17.6916	0.561145
$995$	0.996004	0.0315754
$996$	−19.2837	−0.611029
$997$	−8.26464	−0.261744	−0.130872	−	0.991399i	$-0.541778\pi$
$997$	−8.26464	−0.261744	−0.130872	+	0.991399i	$0.541778\pi$
$998$	−9.90452	−0.313522
$999$	−55.9017	−1.76865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.o.1.8	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.o.1.8	✓	22	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:	\( N \)	\(=\)	\( 4010 = 2 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4010.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.31697 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.31697 q^{6} -3.01403 q^{7} +1.00000 q^{8} -1.26559 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.31697 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.31697 q^{6} -3.01403 q^{7} +1.00000 q^{8} -1.26559 q^{9} -1.00000 q^{10} -4.14944 q^{11} -1.31697 q^{12} +4.55376 q^{13} -3.01403 q^{14} +1.31697 q^{15} +1.00000 q^{16} -1.18544 q^{17} -1.26559 q^{18} +2.76979 q^{19} -1.00000 q^{20} +3.96939 q^{21} -4.14944 q^{22} -3.56973 q^{23} -1.31697 q^{24} +1.00000 q^{25} +4.55376 q^{26} +5.61766 q^{27} -3.01403 q^{28} -6.21799 q^{29} +1.31697 q^{30} +2.36560 q^{31} +1.00000 q^{32} +5.46470 q^{33} -1.18544 q^{34} +3.01403 q^{35} -1.26559 q^{36} -9.95108 q^{37} +2.76979 q^{38} -5.99717 q^{39} -1.00000 q^{40} +3.57940 q^{41} +3.96939 q^{42} -7.80349 q^{43} -4.14944 q^{44} +1.26559 q^{45} -3.56973 q^{46} +4.65151 q^{47} -1.31697 q^{48} +2.08439 q^{49} +1.00000 q^{50} +1.56119 q^{51} +4.55376 q^{52} -1.53610 q^{53} +5.61766 q^{54} +4.14944 q^{55} -3.01403 q^{56} -3.64774 q^{57} -6.21799 q^{58} -7.45163 q^{59} +1.31697 q^{60} +13.2771 q^{61} +2.36560 q^{62} +3.81452 q^{63} +1.00000 q^{64} -4.55376 q^{65} +5.46470 q^{66} +10.3042 q^{67} -1.18544 q^{68} +4.70123 q^{69} +3.01403 q^{70} -5.86976 q^{71} -1.26559 q^{72} +9.19353 q^{73} -9.95108 q^{74} -1.31697 q^{75} +2.76979 q^{76} +12.5066 q^{77} -5.99717 q^{78} +5.83414 q^{79} -1.00000 q^{80} -3.60153 q^{81} +3.57940 q^{82} +14.6425 q^{83} +3.96939 q^{84} +1.18544 q^{85} -7.80349 q^{86} +8.18892 q^{87} -4.14944 q^{88} +10.7976 q^{89} +1.26559 q^{90} -13.7252 q^{91} -3.56973 q^{92} -3.11543 q^{93} +4.65151 q^{94} -2.76979 q^{95} -1.31697 q^{96} +3.67055 q^{97} +2.08439 q^{98} +5.25148 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 22 q^{2} + 2 q^{3} + 22 q^{4} - 22 q^{5} + 2 q^{6} + 13 q^{7} + 22 q^{8} + 32 q^{9}+O(q^{10}) \) 22 * q + 22 * q^2 + 2 * q^3 + 22 * q^4 - 22 * q^5 + 2 * q^6 + 13 * q^7 + 22 * q^8 + 32 * q^9	\( 22 q + 22 q^{2} + 2 q^{3} + 22 q^{4} - 22 q^{5} + 2 q^{6} + 13 q^{7} + 22 q^{8} + 32 q^{9} - 22 q^{10} - 3 q^{11} + 2 q^{12} + 6 q^{13} + 13 q^{14} - 2 q^{15} + 22 q^{16} + 17 q^{17} + 32 q^{18} + 13 q^{19} - 22 q^{20} + 16 q^{21} - 3 q^{22} + 19 q^{23} + 2 q^{24} + 22 q^{25} + 6 q^{26} + 14 q^{27} + 13 q^{28} + 14 q^{29} - 2 q^{30} + 13 q^{31} + 22 q^{32} + 12 q^{33} + 17 q^{34} - 13 q^{35} + 32 q^{36} + 35 q^{37} + 13 q^{38} + 30 q^{39} - 22 q^{40} - 5 q^{41} + 16 q^{42} + 19 q^{43} - 3 q^{44} - 32 q^{45} + 19 q^{46} + 29 q^{47} + 2 q^{48} + 61 q^{49} + 22 q^{50} + q^{51} + 6 q^{52} + 29 q^{53} + 14 q^{54} + 3 q^{55} + 13 q^{56} + 33 q^{57} + 14 q^{58} - 4 q^{59} - 2 q^{60} + 20 q^{61} + 13 q^{62} + 50 q^{63} + 22 q^{64} - 6 q^{65} + 12 q^{66} + 48 q^{67} + 17 q^{68} + 19 q^{69} - 13 q^{70} + 2 q^{71} + 32 q^{72} + 16 q^{73} + 35 q^{74} + 2 q^{75} + 13 q^{76} + 53 q^{77} + 30 q^{78} + 29 q^{79} - 22 q^{80} + 54 q^{81} - 5 q^{82} + 13 q^{83} + 16 q^{84} - 17 q^{85} + 19 q^{86} + 56 q^{87} - 3 q^{88} + 20 q^{89} - 32 q^{90} + 42 q^{91} + 19 q^{92} + 50 q^{93} + 29 q^{94} - 13 q^{95} + 2 q^{96} + 36 q^{97} + 61 q^{98} - q^{99}+O(q^{100}) \) 22 * q + 22 * q^2 + 2 * q^3 + 22 * q^4 - 22 * q^5 + 2 * q^6 + 13 * q^7 + 22 * q^8 + 32 * q^9 - 22 * q^10 - 3 * q^11 + 2 * q^12 + 6 * q^13 + 13 * q^14 - 2 * q^15 + 22 * q^16 + 17 * q^17 + 32 * q^18 + 13 * q^19 - 22 * q^20 + 16 * q^21 - 3 * q^22 + 19 * q^23 + 2 * q^24 + 22 * q^25 + 6 * q^26 + 14 * q^27 + 13 * q^28 + 14 * q^29 - 2 * q^30 + 13 * q^31 + 22 * q^32 + 12 * q^33 + 17 * q^34 - 13 * q^35 + 32 * q^36 + 35 * q^37 + 13 * q^38 + 30 * q^39 - 22 * q^40 - 5 * q^41 + 16 * q^42 + 19 * q^43 - 3 * q^44 - 32 * q^45 + 19 * q^46 + 29 * q^47 + 2 * q^48 + 61 * q^49 + 22 * q^50 + q^51 + 6 * q^52 + 29 * q^53 + 14 * q^54 + 3 * q^55 + 13 * q^56 + 33 * q^57 + 14 * q^58 - 4 * q^59 - 2 * q^60 + 20 * q^61 + 13 * q^62 + 50 * q^63 + 22 * q^64 - 6 * q^65 + 12 * q^66 + 48 * q^67 + 17 * q^68 + 19 * q^69 - 13 * q^70 + 2 * q^71 + 32 * q^72 + 16 * q^73 + 35 * q^74 + 2 * q^75 + 13 * q^76 + 53 * q^77 + 30 * q^78 + 29 * q^79 - 22 * q^80 + 54 * q^81 - 5 * q^82 + 13 * q^83 + 16 * q^84 - 17 * q^85 + 19 * q^86 + 56 * q^87 - 3 * q^88 + 20 * q^89 - 32 * q^90 + 42 * q^91 + 19 * q^92 + 50 * q^93 + 29 * q^94 - 13 * q^95 + 2 * q^96 + 36 * q^97 + 61 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more