LMFDB - Embedded newform 4029.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4029,2,Mod(1,4029)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4029, 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("4029.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4029 = 3 \cdot 17 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4029.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:	$32.1717269744$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	4029.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17605 q^{2} -1.00000 q^{3} +2.73521 q^{4} +1.58006 q^{5} +2.17605 q^{6} -0.969003 q^{7} -1.59985 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.17605 q^{2} -1.00000 q^{3} +2.73521 q^{4} +1.58006 q^{5} +2.17605 q^{6} -0.969003 q^{7} -1.59985 q^{8} +1.00000 q^{9} -3.43828 q^{10} -1.57718 q^{11} -2.73521 q^{12} +2.04735 q^{13} +2.10860 q^{14} -1.58006 q^{15} -1.98906 q^{16} -1.00000 q^{17} -2.17605 q^{18} +2.91990 q^{19} +4.32178 q^{20} +0.969003 q^{21} +3.43202 q^{22} +1.52439 q^{23} +1.59985 q^{24} -2.50342 q^{25} -4.45515 q^{26} -1.00000 q^{27} -2.65042 q^{28} +1.93820 q^{29} +3.43828 q^{30} -6.19687 q^{31} +7.52799 q^{32} +1.57718 q^{33} +2.17605 q^{34} -1.53108 q^{35} +2.73521 q^{36} +1.87113 q^{37} -6.35386 q^{38} -2.04735 q^{39} -2.52785 q^{40} -0.280574 q^{41} -2.10860 q^{42} +0.309240 q^{43} -4.31391 q^{44} +1.58006 q^{45} -3.31716 q^{46} -1.15268 q^{47} +1.98906 q^{48} -6.06103 q^{49} +5.44758 q^{50} +1.00000 q^{51} +5.59993 q^{52} +1.61591 q^{53} +2.17605 q^{54} -2.49203 q^{55} +1.55026 q^{56} -2.91990 q^{57} -4.21763 q^{58} +8.59327 q^{59} -4.32178 q^{60} +4.56505 q^{61} +13.4847 q^{62} -0.969003 q^{63} -12.4032 q^{64} +3.23493 q^{65} -3.43202 q^{66} -5.25347 q^{67} -2.73521 q^{68} -1.52439 q^{69} +3.33171 q^{70} -15.2180 q^{71} -1.59985 q^{72} -12.9057 q^{73} -4.07168 q^{74} +2.50342 q^{75} +7.98654 q^{76} +1.52829 q^{77} +4.45515 q^{78} -1.00000 q^{79} -3.14283 q^{80} +1.00000 q^{81} +0.610545 q^{82} -8.46462 q^{83} +2.65042 q^{84} -1.58006 q^{85} -0.672922 q^{86} -1.93820 q^{87} +2.52324 q^{88} +4.95946 q^{89} -3.43828 q^{90} -1.98389 q^{91} +4.16953 q^{92} +6.19687 q^{93} +2.50829 q^{94} +4.61361 q^{95} -7.52799 q^{96} +12.4605 q^{97} +13.1891 q^{98} -1.57718 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 2 q^{2} - 22 q^{3} + 16 q^{4} + 5 q^{5} - 2 q^{6} - 4 q^{7} + 6 q^{8} + 22 q^{9}+O(q^{10}) $ 22 * q + 2 * q^2 - 22 * q^3 + 16 * q^4 + 5 * q^5 - 2 * q^6 - 4 * q^7 + 6 * q^8 + 22 * q^9	\( 22 q + 2 q^{2} - 22 q^{3} + 16 q^{4} + 5 q^{5} - 2 q^{6} - 4 q^{7} + 6 q^{8} + 22 q^{9} - 5 q^{10} - 2 q^{11} - 16 q^{12} - 11 q^{13} - 7 q^{14} - 5 q^{15} - 22 q^{17} + 2 q^{18} - 36 q^{19} + 4 q^{21} - 9 q^{22} + 21 q^{23} - 6 q^{24} + 9 q^{25} - 16 q^{26} - 22 q^{27} - 17 q^{28} - q^{29} + 5 q^{30} - 12 q^{31} - 11 q^{32} + 2 q^{33} - 2 q^{34} - 14 q^{35} + 16 q^{36} - 6 q^{37} + q^{38} + 11 q^{39} - 24 q^{40} - 17 q^{41} + 7 q^{42} - 36 q^{43} + 16 q^{44} + 5 q^{45} - 23 q^{46} - 17 q^{47} - 6 q^{49} - 33 q^{50} + 22 q^{51} - 57 q^{52} - 2 q^{53} - 2 q^{54} - 24 q^{55} - 64 q^{56} + 36 q^{57} - 7 q^{58} - 59 q^{59} - 30 q^{61} - 4 q^{62} - 4 q^{63} - 22 q^{64} + 36 q^{65} + 9 q^{66} - 16 q^{67} - 16 q^{68} - 21 q^{69} - 39 q^{70} - 11 q^{71} + 6 q^{72} - 19 q^{73} - 28 q^{74} - 9 q^{75} - 77 q^{76} + 2 q^{77} + 16 q^{78} - 22 q^{79} - 2 q^{80} + 22 q^{81} + 33 q^{82} - 23 q^{83} + 17 q^{84} - 5 q^{85} + 6 q^{86} + q^{87} - 23 q^{88} + 12 q^{89} - 5 q^{90} - 24 q^{91} + 66 q^{92} + 12 q^{93} - 61 q^{94} - 11 q^{95} + 11 q^{96} - 9 q^{97} + 17 q^{98} - 2 q^{99}+O(q^{100}) \) 22 * q + 2 * q^2 - 22 * q^3 + 16 * q^4 + 5 * q^5 - 2 * q^6 - 4 * q^7 + 6 * q^8 + 22 * q^9 - 5 * q^10 - 2 * q^11 - 16 * q^12 - 11 * q^13 - 7 * q^14 - 5 * q^15 - 22 * q^17 + 2 * q^18 - 36 * q^19 + 4 * q^21 - 9 * q^22 + 21 * q^23 - 6 * q^24 + 9 * q^25 - 16 * q^26 - 22 * q^27 - 17 * q^28 - q^29 + 5 * q^30 - 12 * q^31 - 11 * q^32 + 2 * q^33 - 2 * q^34 - 14 * q^35 + 16 * q^36 - 6 * q^37 + q^38 + 11 * q^39 - 24 * q^40 - 17 * q^41 + 7 * q^42 - 36 * q^43 + 16 * q^44 + 5 * q^45 - 23 * q^46 - 17 * q^47 - 6 * q^49 - 33 * q^50 + 22 * q^51 - 57 * q^52 - 2 * q^53 - 2 * q^54 - 24 * q^55 - 64 * q^56 + 36 * q^57 - 7 * q^58 - 59 * q^59 - 30 * q^61 - 4 * q^62 - 4 * q^63 - 22 * q^64 + 36 * q^65 + 9 * q^66 - 16 * q^67 - 16 * q^68 - 21 * q^69 - 39 * q^70 - 11 * q^71 + 6 * q^72 - 19 * q^73 - 28 * q^74 - 9 * q^75 - 77 * q^76 + 2 * q^77 + 16 * q^78 - 22 * q^79 - 2 * q^80 + 22 * q^81 + 33 * q^82 - 23 * q^83 + 17 * q^84 - 5 * q^85 + 6 * q^86 + q^87 - 23 * q^88 + 12 * q^89 - 5 * q^90 - 24 * q^91 + 66 * q^92 + 12 * q^93 - 61 * q^94 - 11 * q^95 + 11 * q^96 - 9 * q^97 + 17 * q^98 - 2 * 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 $

$2$	−2.17605	−1.53870	−0.769351	−	0.638826i	$-0.779420\pi$
$2$	−2.17605	−1.53870	−0.769351	+	0.638826i	$0.779420\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.73521	1.36760
$5$	1.58006	0.706622	0.353311	−	0.935506i	$-0.385056\pi$
$5$	1.58006	0.706622	0.353311	+	0.935506i	$0.385056\pi$
$6$	2.17605	0.888370
$7$	−0.969003	−0.366249	−0.183124	−	0.983090i	$-0.558621\pi$
$7$	−0.969003	−0.366249	−0.183124	+	0.983090i	$0.558621\pi$
$8$	−1.59985	−0.565631
$9$	1.00000	0.333333
$10$	−3.43828	−1.08728
$11$	−1.57718	−0.475537	−0.237769	−	0.971322i	$-0.576416\pi$
$11$	−1.57718	−0.475537	−0.237769	+	0.971322i	$0.576416\pi$
$12$	−2.73521	−0.789586
$13$	2.04735	0.567833	0.283917	−	0.958849i	$-0.408366\pi$
$13$	2.04735	0.567833	0.283917	+	0.958849i	$0.408366\pi$
$14$	2.10860	0.563547
$15$	−1.58006	−0.407969
$16$	−1.98906	−0.497265
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.17605	−0.512901
$19$	2.91990	0.669872	0.334936	−	0.942241i	$-0.391285\pi$
$19$	2.91990	0.669872	0.334936	+	0.942241i	$0.391285\pi$
$20$	4.32178	0.966379
$21$	0.969003	0.211454
$22$	3.43202	0.731710
$23$	1.52439	0.317858	0.158929	−	0.987290i	$-0.449196\pi$
$23$	1.52439	0.317858	0.158929	+	0.987290i	$0.449196\pi$
$24$	1.59985	0.326567
$25$	−2.50342	−0.500685
$26$	−4.45515	−0.873726
$27$	−1.00000	−0.192450
$28$	−2.65042	−0.500883
$29$	1.93820	0.359915	0.179958	−	0.983674i	$-0.442404\pi$
$29$	1.93820	0.359915	0.179958	+	0.983674i	$0.442404\pi$
$30$	3.43828	0.627742
$31$	−6.19687	−1.11299	−0.556495	−	0.830851i	$-0.687854\pi$
$31$	−6.19687	−1.11299	−0.556495	+	0.830851i	$0.687854\pi$
$32$	7.52799	1.33077
$33$	1.57718	0.274551
$34$	2.17605	0.373190
$35$	−1.53108	−0.258800
$36$	2.73521	0.455868
$37$	1.87113	0.307612	0.153806	−	0.988101i	$-0.450847\pi$
$37$	1.87113	0.307612	0.153806	+	0.988101i	$0.450847\pi$
$38$	−6.35386	−1.03073
$39$	−2.04735	−0.327839
$40$	−2.52785	−0.399688
$41$	−0.280574	−0.0438184	−0.0219092	−	0.999760i	$-0.506974\pi$
$41$	−0.280574	−0.0438184	−0.0219092	+	0.999760i	$0.506974\pi$
$42$	−2.10860	−0.325364
$43$	0.309240	0.0471586	0.0235793	−	0.999722i	$-0.492494\pi$
$43$	0.309240	0.0471586	0.0235793	+	0.999722i	$0.492494\pi$
$44$	−4.31391	−0.650346
$45$	1.58006	0.235541
$46$	−3.31716	−0.489089
$47$	−1.15268	−0.168136	−0.0840678	−	0.996460i	$-0.526791\pi$
$47$	−1.15268	−0.168136	−0.0840678	+	0.996460i	$0.526791\pi$
$48$	1.98906	0.287096
$49$	−6.06103	−0.865862
$50$	5.44758	0.770405
$51$	1.00000	0.140028
$52$	5.59993	0.776570
$53$	1.61591	0.221962	0.110981	−	0.993823i	$-0.464601\pi$
$53$	1.61591	0.221962	0.110981	+	0.993823i	$0.464601\pi$
$54$	2.17605	0.296123
$55$	−2.49203	−0.336025
$56$	1.55026	0.207162
$57$	−2.91990	−0.386751
$58$	−4.21763	−0.553802
$59$	8.59327	1.11875	0.559374	−	0.828915i	$-0.311041\pi$
$59$	8.59327	1.11875	0.559374	+	0.828915i	$0.311041\pi$
$60$	−4.32178	−0.557939
$61$	4.56505	0.584495	0.292247	−	0.956343i	$-0.405597\pi$
$61$	4.56505	0.584495	0.292247	+	0.956343i	$0.405597\pi$
$62$	13.4847	1.71256
$63$	−0.969003	−0.122083
$64$	−12.4032	−1.55040
$65$	3.23493	0.401244
$66$	−3.43202	−0.422453
$67$	−5.25347	−0.641814	−0.320907	−	0.947111i	$-0.603988\pi$
$67$	−5.25347	−0.641814	−0.320907	+	0.947111i	$0.603988\pi$
$68$	−2.73521	−0.331692
$69$	−1.52439	−0.183515
$70$	3.33171	0.398215
$71$	−15.2180	−1.80604	−0.903020	−	0.429598i	$-0.858655\pi$
$71$	−15.2180	−1.80604	−0.903020	+	0.429598i	$0.858655\pi$
$72$	−1.59985	−0.188544
$73$	−12.9057	−1.51050	−0.755251	−	0.655435i	$-0.772485\pi$
$73$	−12.9057	−1.51050	−0.755251	+	0.655435i	$0.772485\pi$
$74$	−4.07168	−0.473323
$75$	2.50342	0.289071
$76$	7.98654	0.916119
$77$	1.52829	0.174165
$78$	4.45515	0.504446
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	−3.14283	−0.351379
$81$	1.00000	0.111111
$82$	0.610545	0.0674234
$83$	−8.46462	−0.929113	−0.464556	−	0.885544i	$-0.653786\pi$
$83$	−8.46462	−0.929113	−0.464556	+	0.885544i	$0.653786\pi$
$84$	2.65042	0.289185
$85$	−1.58006	−0.171381
$86$	−0.672922	−0.0725630
$87$	−1.93820	−0.207797
$88$	2.52324	0.268979
$89$	4.95946	0.525702	0.262851	−	0.964836i	$-0.415337\pi$
$89$	4.95946	0.525702	0.262851	+	0.964836i	$0.415337\pi$
$90$	−3.43828	−0.362427
$91$	−1.98389	−0.207968
$92$	4.16953	0.434703
$93$	6.19687	0.642585
$94$	2.50829	0.258711
$95$	4.61361	0.473346
$96$	−7.52799	−0.768323
$97$	12.4605	1.26517	0.632584	−	0.774492i	$-0.281994\pi$
$97$	12.4605	1.26517	0.632584	+	0.774492i	$0.281994\pi$
$98$	13.1891	1.33230
$99$	−1.57718	−0.158512
$100$	−6.84738	−0.684738
$101$	−0.163302	−0.0162491	−0.00812456	−	0.999967i	$-0.502586\pi$
$101$	−0.163302	−0.0162491	−0.00812456	+	0.999967i	$0.502586\pi$
$102$	−2.17605	−0.215461
$103$	−12.8213	−1.26332	−0.631659	−	0.775247i	$-0.717626\pi$
$103$	−12.8213	−1.26332	−0.631659	+	0.775247i	$0.717626\pi$
$104$	−3.27545	−0.321184
$105$	1.53108	0.149418
$106$	−3.51630	−0.341533
$107$	−0.450237	−0.0435261	−0.0217630	−	0.999763i	$-0.506928\pi$
$107$	−0.450237	−0.0435261	−0.0217630	+	0.999763i	$0.506928\pi$
$108$	−2.73521	−0.263195
$109$	2.36713	0.226730	0.113365	−	0.993553i	$-0.463837\pi$
$109$	2.36713	0.226730	0.113365	+	0.993553i	$0.463837\pi$
$110$	5.42279	0.517042
$111$	−1.87113	−0.177600
$112$	1.92741	0.182123
$113$	9.28405	0.873370	0.436685	−	0.899615i	$-0.356153\pi$
$113$	9.28405	0.873370	0.436685	+	0.899615i	$0.356153\pi$
$114$	6.35386	0.595094
$115$	2.40863	0.224606
$116$	5.30139	0.492221
$117$	2.04735	0.189278
$118$	−18.6994	−1.72142
$119$	0.969003	0.0888284
$120$	2.52785	0.230760
$121$	−8.51251	−0.773865
$122$	−9.93379	−0.899363
$123$	0.280574	0.0252985
$124$	−16.9497	−1.52213
$125$	−11.8558	−1.06042
$126$	2.10860	0.187849
$127$	−11.4743	−1.01818	−0.509090	−	0.860713i	$-0.670018\pi$
$127$	−11.4743	−1.01818	−0.509090	+	0.860713i	$0.670018\pi$
$128$	11.9340	1.05483
$129$	−0.309240	−0.0272270
$130$	−7.03938	−0.617394
$131$	9.98745	0.872607	0.436304	−	0.899800i	$-0.356287\pi$
$131$	9.98745	0.872607	0.436304	+	0.899800i	$0.356287\pi$
$132$	4.31391	0.375477
$133$	−2.82940	−0.245340
$134$	11.4318	0.987560
$135$	−1.58006	−0.135990
$136$	1.59985	0.137186
$137$	−10.8766	−0.929249	−0.464625	−	0.885508i	$-0.653811\pi$
$137$	−10.8766	−0.929249	−0.464625	+	0.885508i	$0.653811\pi$
$138$	3.31716	0.282375
$139$	−2.72332	−0.230989	−0.115495	−	0.993308i	$-0.536845\pi$
$139$	−2.72332	−0.230989	−0.115495	+	0.993308i	$0.536845\pi$
$140$	−4.18782	−0.353935
$141$	1.15268	0.0970731
$142$	33.1151	2.77896
$143$	−3.22904	−0.270026
$144$	−1.98906	−0.165755
$145$	3.06247	0.254324
$146$	28.0836	2.32421
$147$	6.06103	0.499906
$148$	5.11793	0.420691
$149$	9.25598	0.758280	0.379140	−	0.925339i	$-0.376220\pi$
$149$	9.25598	0.758280	0.379140	+	0.925339i	$0.376220\pi$
$150$	−5.44758	−0.444793
$151$	12.2607	0.997765	0.498882	−	0.866670i	$-0.333744\pi$
$151$	12.2607	0.997765	0.498882	+	0.866670i	$0.333744\pi$
$152$	−4.67140	−0.378900
$153$	−1.00000	−0.0808452
$154$	−3.32564	−0.267988
$155$	−9.79140	−0.786464
$156$	−5.59993	−0.448353
$157$	5.31369	0.424079	0.212039	−	0.977261i	$-0.431990\pi$
$157$	5.31369	0.424079	0.212039	+	0.977261i	$0.431990\pi$
$158$	2.17605	0.173117
$159$	−1.61591	−0.128150
$160$	11.8947	0.940355
$161$	−1.47714	−0.116415
$162$	−2.17605	−0.170967
$163$	12.6998	0.994722	0.497361	−	0.867544i	$-0.334303\pi$
$163$	12.6998	0.994722	0.497361	+	0.867544i	$0.334303\pi$
$164$	−0.767429	−0.0599261
$165$	2.49203	0.194004
$166$	18.4195	1.42963
$167$	11.9065	0.921353	0.460677	−	0.887568i	$-0.347607\pi$
$167$	11.9065	0.921353	0.460677	+	0.887568i	$0.347607\pi$
$168$	−1.55026	−0.119605
$169$	−8.80835	−0.677565
$170$	3.43828	0.263704
$171$	2.91990	0.223291
$172$	0.845834	0.0644942
$173$	21.2163	1.61304	0.806522	−	0.591204i	$-0.201347\pi$
$173$	21.2163	1.61304	0.806522	+	0.591204i	$0.201347\pi$
$174$	4.21763	0.319738
$175$	2.42583	0.183375
$176$	3.13710	0.236468
$177$	−8.59327	−0.645910
$178$	−10.7921	−0.808898
$179$	12.5897	0.941001	0.470501	−	0.882400i	$-0.344073\pi$
$179$	12.5897	0.941001	0.470501	+	0.882400i	$0.344073\pi$
$180$	4.32178	0.322126
$181$	3.74482	0.278350	0.139175	−	0.990268i	$-0.455555\pi$
$181$	3.74482	0.278350	0.139175	+	0.990268i	$0.455555\pi$
$182$	4.31705	0.320001
$183$	−4.56505	−0.337458
$184$	−2.43880	−0.179790
$185$	2.95649	0.217366
$186$	−13.4847	−0.988747
$187$	1.57718	0.115335
$188$	−3.15282	−0.229943
$189$	0.969003	0.0704846
$190$	−10.0395	−0.728339
$191$	9.58988	0.693899	0.346950	−	0.937884i	$-0.387217\pi$
$191$	9.58988	0.693899	0.346950	+	0.937884i	$0.387217\pi$
$192$	12.4032	0.895123
$193$	−6.91063	−0.497438	−0.248719	−	0.968576i	$-0.580010\pi$
$193$	−6.91063	−0.497438	−0.248719	+	0.968576i	$0.580010\pi$
$194$	−27.1146	−1.94672
$195$	−3.23493	−0.231658
$196$	−16.5782	−1.18416
$197$	17.6251	1.25573	0.627867	−	0.778321i	$-0.283928\pi$
$197$	17.6251	1.25573	0.627867	+	0.778321i	$0.283928\pi$
$198$	3.43202	0.243903
$199$	−20.1977	−1.43178	−0.715889	−	0.698214i	$-0.753978\pi$
$199$	−20.1977	−1.43178	−0.715889	+	0.698214i	$0.753978\pi$
$200$	4.00509	0.283203
$201$	5.25347	0.370551
$202$	0.355353	0.0250025
$203$	−1.87812	−0.131819
$204$	2.73521	0.191503
$205$	−0.443323	−0.0309630
$206$	27.8998	1.94387
$207$	1.52439	0.105953
$208$	−4.07231	−0.282364
$209$	−4.60521	−0.318549
$210$	−3.33171	−0.229910
$211$	−24.4918	−1.68609	−0.843044	−	0.537844i	$-0.819239\pi$
$211$	−24.4918	−1.68609	−0.843044	+	0.537844i	$0.819239\pi$
$212$	4.41984	0.303556
$213$	15.2180	1.04272
$214$	0.979740	0.0669737
$215$	0.488616	0.0333233
$216$	1.59985	0.108856
$217$	6.00478	0.407631
$218$	−5.15101	−0.348870
$219$	12.9057	0.872089
$220$	−6.81621	−0.459549
$221$	−2.04735	−0.137720
$222$	4.07168	0.273273
$223$	−21.1179	−1.41416	−0.707080	−	0.707133i	$-0.749988\pi$
$223$	−21.1179	−1.41416	−0.707080	+	0.707133i	$0.749988\pi$
$224$	−7.29465	−0.487394
$225$	−2.50342	−0.166895
$226$	−20.2026	−1.34386
$227$	20.7853	1.37957	0.689786	−	0.724014i	$-0.257705\pi$
$227$	20.7853	1.37957	0.689786	+	0.724014i	$0.257705\pi$
$228$	−7.98654	−0.528921
$229$	−3.52244	−0.232769	−0.116385	−	0.993204i	$-0.537131\pi$
$229$	−3.52244	−0.232769	−0.116385	+	0.993204i	$0.537131\pi$
$230$	−5.24130	−0.345601
$231$	−1.52829	−0.100554
$232$	−3.10083	−0.203579
$233$	−25.6310	−1.67914	−0.839570	−	0.543251i	$-0.817193\pi$
$233$	−25.6310	−1.67914	−0.839570	+	0.543251i	$0.817193\pi$
$234$	−4.45515	−0.291242
$235$	−1.82130	−0.118808
$236$	23.5044	1.53000
$237$	1.00000	0.0649570
$238$	−2.10860	−0.136680
$239$	−10.4297	−0.674640	−0.337320	−	0.941390i	$-0.609520\pi$
$239$	−10.4297	−0.674640	−0.337320	+	0.941390i	$0.609520\pi$
$240$	3.14283	0.202869
$241$	−13.5955	−0.875761	−0.437881	−	0.899033i	$-0.644271\pi$
$241$	−13.5955	−0.875761	−0.437881	+	0.899033i	$0.644271\pi$
$242$	18.5237	1.19075
$243$	−1.00000	−0.0641500
$244$	12.4863	0.799357
$245$	−9.57677	−0.611837
$246$	−0.610545	−0.0389269
$247$	5.97807	0.380375
$248$	9.91404	0.629542
$249$	8.46462	0.536423
$250$	25.7989	1.63167
$251$	−6.91049	−0.436186	−0.218093	−	0.975928i	$-0.569984\pi$
$251$	−6.91049	−0.436186	−0.218093	+	0.975928i	$0.569984\pi$
$252$	−2.65042	−0.166961
$253$	−2.40424	−0.151153
$254$	24.9687	1.56668
$255$	1.58006	0.0989469
$256$	−1.16266	−0.0726660
$257$	16.0883	1.00356	0.501782	−	0.864994i	$-0.332678\pi$
$257$	16.0883	1.00356	0.501782	+	0.864994i	$0.332678\pi$
$258$	0.672922	0.0418943
$259$	−1.81313	−0.112662
$260$	8.84820	0.548742
$261$	1.93820	0.119972
$262$	−21.7332	−1.34268
$263$	4.14906	0.255842	0.127921	−	0.991784i	$-0.459170\pi$
$263$	4.14906	0.255842	0.127921	+	0.991784i	$0.459170\pi$
$264$	−2.52324	−0.155295
$265$	2.55322	0.156843
$266$	6.15691	0.377505
$267$	−4.95946	−0.303514
$268$	−14.3693	−0.877746
$269$	−24.1572	−1.47289	−0.736445	−	0.676497i	$-0.763497\pi$
$269$	−24.1572	−1.47289	−0.736445	+	0.676497i	$0.763497\pi$
$270$	3.43828	0.209247
$271$	−19.3444	−1.17509	−0.587543	−	0.809193i	$-0.699905\pi$
$271$	−19.3444	−1.17509	−0.587543	+	0.809193i	$0.699905\pi$
$272$	1.98906	0.120605
$273$	1.98389	0.120070
$274$	23.6680	1.42984
$275$	3.94835	0.238094
$276$	−4.16953	−0.250976
$277$	3.15210	0.189391	0.0946955	−	0.995506i	$-0.469812\pi$
$277$	3.15210	0.189391	0.0946955	+	0.995506i	$0.469812\pi$
$278$	5.92609	0.355423
$279$	−6.19687	−0.370997
$280$	2.44949	0.146385
$281$	−12.2909	−0.733217	−0.366608	−	0.930375i	$-0.619481\pi$
$281$	−12.2909	−0.733217	−0.366608	+	0.930375i	$0.619481\pi$
$282$	−2.50829	−0.149367
$283$	3.71871	0.221054	0.110527	−	0.993873i	$-0.464746\pi$
$283$	3.71871	0.221054	0.110527	+	0.993873i	$0.464746\pi$
$284$	−41.6243	−2.46995
$285$	−4.61361	−0.273287
$286$	7.02656	0.415489
$287$	0.271877	0.0160484
$288$	7.52799	0.443591
$289$	1.00000	0.0588235
$290$	−6.66410	−0.391329
$291$	−12.4605	−0.730445
$292$	−35.2999	−2.06577
$293$	13.1619	0.768924	0.384462	−	0.923141i	$-0.374387\pi$
$293$	13.1619	0.768924	0.384462	+	0.923141i	$0.374387\pi$
$294$	−13.1891	−0.769206
$295$	13.5779	0.790533
$296$	−2.99352	−0.173995
$297$	1.57718	0.0915171
$298$	−20.1415	−1.16677
$299$	3.12097	0.180490
$300$	6.84738	0.395334
$301$	−0.299654	−0.0172718
$302$	−26.6800	−1.53526
$303$	0.163302	0.00938143
$304$	−5.80787	−0.333104
$305$	7.21303	0.413017
$306$	2.17605	0.124397
$307$	−6.16109	−0.351632	−0.175816	−	0.984423i	$-0.556256\pi$
$307$	−6.16109	−0.351632	−0.175816	+	0.984423i	$0.556256\pi$
$308$	4.18019	0.238188
$309$	12.8213	0.729376
$310$	21.3066	1.21013
$311$	−23.4767	−1.33124	−0.665622	−	0.746289i	$-0.731834\pi$
$311$	−23.4767	−1.33124	−0.665622	+	0.746289i	$0.731834\pi$
$312$	3.27545	0.185436
$313$	30.6469	1.73226	0.866132	−	0.499815i	$-0.166599\pi$
$313$	30.6469	1.73226	0.866132	+	0.499815i	$0.166599\pi$
$314$	−11.5629	−0.652531
$315$	−1.53108	−0.0862665
$316$	−2.73521	−0.153867
$317$	26.3354	1.47915	0.739573	−	0.673077i	$-0.235028\pi$
$317$	26.3354	1.47915	0.739573	+	0.673077i	$0.235028\pi$
$318$	3.51630	0.197184
$319$	−3.05689	−0.171153
$320$	−19.5977	−1.09555
$321$	0.450237	0.0251298
$322$	3.21434	0.179128
$323$	−2.91990	−0.162468
$324$	2.73521	0.151956
$325$	−5.12539	−0.284305
$326$	−27.6353	−1.53058
$327$	−2.36713	−0.130903
$328$	0.448876	0.0247850
$329$	1.11695	0.0615794
$330$	−5.42279	−0.298515
$331$	−28.5510	−1.56930	−0.784652	−	0.619936i	$-0.787159\pi$
$331$	−28.5510	−1.56930	−0.784652	+	0.619936i	$0.787159\pi$
$332$	−23.1525	−1.27066
$333$	1.87113	0.102537
$334$	−25.9092	−1.41769
$335$	−8.30078	−0.453520
$336$	−1.92741	−0.105149
$337$	9.28256	0.505653	0.252827	−	0.967512i	$-0.418640\pi$
$337$	9.28256	0.505653	0.252827	+	0.967512i	$0.418640\pi$
$338$	19.1674	1.04257
$339$	−9.28405	−0.504240
$340$	−4.32178	−0.234381
$341$	9.77356	0.529268
$342$	−6.35386	−0.343578
$343$	12.6562	0.683369
$344$	−0.494736	−0.0266744
$345$	−2.40863	−0.129676
$346$	−46.1677	−2.48199
$347$	−29.2453	−1.56997	−0.784984	−	0.619516i	$-0.787329\pi$
$347$	−29.2453	−1.56997	−0.784984	+	0.619516i	$0.787329\pi$
$348$	−5.30139	−0.284184
$349$	−28.2610	−1.51278	−0.756390	−	0.654121i	$-0.773039\pi$
$349$	−28.2610	−1.51278	−0.756390	+	0.654121i	$0.773039\pi$
$350$	−5.27872	−0.282160
$351$	−2.04735	−0.109280
$352$	−11.8730	−0.632832
$353$	17.7348	0.943929	0.471964	−	0.881618i	$-0.343545\pi$
$353$	17.7348	0.943929	0.471964	+	0.881618i	$0.343545\pi$
$354$	18.6994	0.993863
$355$	−24.0452	−1.27619
$356$	13.5651	0.718951
$357$	−0.969003	−0.0512851
$358$	−27.3959	−1.44792
$359$	−19.2352	−1.01520	−0.507598	−	0.861594i	$-0.669467\pi$
$359$	−19.2352	−1.01520	−0.507598	+	0.861594i	$0.669467\pi$
$360$	−2.52785	−0.133229
$361$	−10.4742	−0.551272
$362$	−8.14892	−0.428298
$363$	8.51251	0.446791
$364$	−5.42635	−0.284418
$365$	−20.3918	−1.06735
$366$	9.93379	0.519247
$367$	20.2717	1.05818	0.529088	−	0.848567i	$-0.322534\pi$
$367$	20.2717	1.05818	0.529088	+	0.848567i	$0.322534\pi$
$368$	−3.03211	−0.158060
$369$	−0.280574	−0.0146061
$370$	−6.43348	−0.334461
$371$	−1.56582	−0.0812932
$372$	16.9497	0.878801
$373$	−21.6686	−1.12196	−0.560980	−	0.827830i	$-0.689575\pi$
$373$	−21.6686	−1.12196	−0.560980	+	0.827830i	$0.689575\pi$
$374$	−3.43202	−0.177466
$375$	11.8558	0.612232
$376$	1.84411	0.0951027
$377$	3.96818	0.204372
$378$	−2.10860	−0.108455
$379$	−24.4667	−1.25677	−0.628384	−	0.777903i	$-0.716283\pi$
$379$	−24.4667	−1.25677	−0.628384	+	0.777903i	$0.716283\pi$
$380$	12.6192	0.647350
$381$	11.4743	0.587846
$382$	−20.8681	−1.06770
$383$	−17.9296	−0.916159	−0.458080	−	0.888911i	$-0.651463\pi$
$383$	−17.9296	−0.916159	−0.458080	+	0.888911i	$0.651463\pi$
$384$	−11.9340	−0.609005
$385$	2.41478	0.123069
$386$	15.0379	0.765409
$387$	0.309240	0.0157195
$388$	34.0819	1.73025
$389$	−12.0856	−0.612763	−0.306382	−	0.951909i	$-0.599118\pi$
$389$	−12.0856	−0.612763	−0.306382	+	0.951909i	$0.599118\pi$
$390$	7.03938	0.356453
$391$	−1.52439	−0.0770919
$392$	9.69672	0.489759
$393$	−9.98745	−0.503800
$394$	−38.3531	−1.93220
$395$	−1.58006	−0.0795012
$396$	−4.31391	−0.216782
$397$	−23.8061	−1.19479	−0.597396	−	0.801946i	$-0.703798\pi$
$397$	−23.8061	−1.19479	−0.597396	+	0.801946i	$0.703798\pi$
$398$	43.9513	2.20308
$399$	2.82940	0.141647
$400$	4.97946	0.248973
$401$	−15.5181	−0.774938	−0.387469	−	0.921883i	$-0.626651\pi$
$401$	−15.5181	−0.774938	−0.387469	+	0.921883i	$0.626651\pi$
$402$	−11.4318	−0.570168
$403$	−12.6872	−0.631993
$404$	−0.446664	−0.0222223
$405$	1.58006	0.0785136
$406$	4.08690	0.202829
$407$	−2.95111	−0.146281
$408$	−1.59985	−0.0792042
$409$	6.40439	0.316677	0.158338	−	0.987385i	$-0.449386\pi$
$409$	6.40439	0.316677	0.158338	+	0.987385i	$0.449386\pi$
$410$	0.964694	0.0476429
$411$	10.8766	0.536502
$412$	−35.0688	−1.72772
$413$	−8.32691	−0.409740
$414$	−3.31716	−0.163030
$415$	−13.3746	−0.656532
$416$	15.4125	0.755658
$417$	2.72332	0.133362
$418$	10.0212	0.490152
$419$	−11.8529	−0.579053	−0.289526	−	0.957170i	$-0.593498\pi$
$419$	−11.8529	−0.579053	−0.289526	+	0.957170i	$0.593498\pi$
$420$	4.18782	0.204344
$421$	−12.8437	−0.625963	−0.312981	−	0.949759i	$-0.601328\pi$
$421$	−12.8437	−0.625963	−0.312981	+	0.949759i	$0.601328\pi$
$422$	53.2956	2.59439
$423$	−1.15268	−0.0560452
$424$	−2.58520	−0.125549
$425$	2.50342	0.121434
$426$	−33.1151	−1.60443
$427$	−4.42355	−0.214070
$428$	−1.23149	−0.0595264
$429$	3.22904	0.155899
$430$	−1.06325	−0.0512747
$431$	−2.91979	−0.140641	−0.0703206	−	0.997524i	$-0.522402\pi$
$431$	−2.91979	−0.140641	−0.0703206	+	0.997524i	$0.522402\pi$
$432$	1.98906	0.0956987
$433$	−17.8669	−0.858631	−0.429315	−	0.903155i	$-0.641245\pi$
$433$	−17.8669	−0.858631	−0.429315	+	0.903155i	$0.641245\pi$
$434$	−13.0667	−0.627223
$435$	−3.06247	−0.146834
$436$	6.47460	0.310077
$437$	4.45108	0.212924
$438$	−28.0836	−1.34188
$439$	−7.58405	−0.361967	−0.180983	−	0.983486i	$-0.557928\pi$
$439$	−7.58405	−0.361967	−0.180983	+	0.983486i	$0.557928\pi$
$440$	3.98686	0.190066
$441$	−6.06103	−0.288621
$442$	4.45515	0.211910
$443$	18.9483	0.900260	0.450130	−	0.892963i	$-0.351378\pi$
$443$	18.9483	0.900260	0.450130	+	0.892963i	$0.351378\pi$
$444$	−5.11793	−0.242886
$445$	7.83623	0.371473
$446$	45.9537	2.17597
$447$	−9.25598	−0.437793
$448$	12.0187	0.567832
$449$	−6.76031	−0.319039	−0.159519	−	0.987195i	$-0.550994\pi$
$449$	−6.76031	−0.319039	−0.159519	+	0.987195i	$0.550994\pi$
$450$	5.44758	0.256802
$451$	0.442516	0.0208372
$452$	25.3938	1.19442
$453$	−12.2607	−0.576060
$454$	−45.2300	−2.12275
$455$	−3.13466	−0.146955
$456$	4.67140	0.218758
$457$	28.7610	1.34538	0.672691	−	0.739924i	$-0.265139\pi$
$457$	28.7610	1.34538	0.672691	+	0.739924i	$0.265139\pi$
$458$	7.66501	0.358163
$459$	1.00000	0.0466760
$460$	6.58809	0.307171
$461$	1.28235	0.0597250	0.0298625	−	0.999554i	$-0.490493\pi$
$461$	1.28235	0.0597250	0.0298625	+	0.999554i	$0.490493\pi$
$462$	3.32564	0.154723
$463$	0.545758	0.0253635	0.0126818	−	0.999920i	$-0.495963\pi$
$463$	0.545758	0.0253635	0.0126818	+	0.999920i	$0.495963\pi$
$464$	−3.85520	−0.178973
$465$	9.79140	0.454065
$466$	55.7743	2.58370
$467$	−20.1829	−0.933955	−0.466977	−	0.884269i	$-0.654657\pi$
$467$	−20.1829	−0.933955	−0.466977	+	0.884269i	$0.654657\pi$
$468$	5.59993	0.258857
$469$	5.09063	0.235063
$470$	3.96324	0.182811
$471$	−5.31369	−0.244842
$472$	−13.7479	−0.632799
$473$	−0.487726	−0.0224257
$474$	−2.17605	−0.0999494
$475$	−7.30976	−0.335395
$476$	2.65042	0.121482
$477$	1.61591	0.0739873
$478$	22.6955	1.03807
$479$	−36.2551	−1.65654	−0.828269	−	0.560331i	$-0.810674\pi$
$479$	−36.2551	−1.65654	−0.828269	+	0.560331i	$0.810674\pi$
$480$	−11.8947	−0.542914
$481$	3.83086	0.174672
$482$	29.5845	1.34754
$483$	1.47714	0.0672123
$484$	−23.2835	−1.05834
$485$	19.6882	0.893996
$486$	2.17605	0.0987078
$487$	−8.90219	−0.403397	−0.201698	−	0.979448i	$-0.564646\pi$
$487$	−8.90219	−0.403397	−0.201698	+	0.979448i	$0.564646\pi$
$488$	−7.30338	−0.330608
$489$	−12.6998	−0.574303
$490$	20.8396	0.941435
$491$	−20.1526	−0.909474	−0.454737	−	0.890626i	$-0.650267\pi$
$491$	−20.1526	−0.909474	−0.454737	+	0.890626i	$0.650267\pi$
$492$	0.767429	0.0345984
$493$	−1.93820	−0.0872923
$494$	−13.0086	−0.585284
$495$	−2.49203	−0.112008
$496$	12.3259	0.553451
$497$	14.7463	0.661460
$498$	−18.4195	−0.825396
$499$	−7.12128	−0.318792	−0.159396	−	0.987215i	$-0.550955\pi$
$499$	−7.12128	−0.318792	−0.159396	+	0.987215i	$0.550955\pi$
$500$	−32.4281	−1.45023
$501$	−11.9065	−0.531944
$502$	15.0376	0.671161
$503$	−19.1551	−0.854085	−0.427042	−	0.904232i	$-0.640444\pi$
$503$	−19.1551	−0.854085	−0.427042	+	0.904232i	$0.640444\pi$
$504$	1.55026	0.0690539
$505$	−0.258026	−0.0114820
$506$	5.23175	0.232580
$507$	8.80835	0.391193
$508$	−31.3846	−1.39247
$509$	2.51026	0.111265	0.0556327	−	0.998451i	$-0.482282\pi$
$509$	2.51026	0.111265	0.0556327	+	0.998451i	$0.482282\pi$
$510$	−3.43828	−0.152250
$511$	12.5057	0.553220
$512$	−21.3380	−0.943016
$513$	−2.91990	−0.128917
$514$	−35.0091	−1.54418
$515$	−20.2583	−0.892688
$516$	−0.845834	−0.0372358
$517$	1.81798	0.0799547
$518$	3.94547	0.173354
$519$	−21.2163	−0.931291
$520$	−5.17539	−0.226956
$521$	−29.4290	−1.28931	−0.644654	−	0.764475i	$-0.722999\pi$
$521$	−29.4290	−1.28931	−0.644654	+	0.764475i	$0.722999\pi$
$522$	−4.21763	−0.184601
$523$	−0.146565	−0.00640884	−0.00320442	−	0.999995i	$-0.501020\pi$
$523$	−0.146565	−0.00640884	−0.00320442	+	0.999995i	$0.501020\pi$
$524$	27.3177	1.19338
$525$	−2.42583	−0.105872
$526$	−9.02858	−0.393665
$527$	6.19687	0.269940
$528$	−3.13710	−0.136525
$529$	−20.6762	−0.898966
$530$	−5.55595	−0.241335
$531$	8.59327	0.372916
$532$	−7.73898	−0.335527
$533$	−0.574434	−0.0248815
$534$	10.7921	0.467018
$535$	−0.711400	−0.0307565
$536$	8.40475	0.363030
$537$	−12.5897	−0.543287
$538$	52.5674	2.26634
$539$	9.55933	0.411749
$540$	−4.32178	−0.185980
$541$	−11.1455	−0.479183	−0.239591	−	0.970874i	$-0.577013\pi$
$541$	−11.1455	−0.479183	−0.239591	+	0.970874i	$0.577013\pi$
$542$	42.0944	1.80811
$543$	−3.74482	−0.160706
$544$	−7.52799	−0.322760
$545$	3.74020	0.160213
$546$	−4.31705	−0.184753
$547$	−40.3041	−1.72328	−0.861639	−	0.507521i	$-0.830562\pi$
$547$	−40.3041	−1.72328	−0.861639	+	0.507521i	$0.830562\pi$
$548$	−29.7497	−1.27084
$549$	4.56505	0.194832
$550$	−8.59181	−0.366356
$551$	5.65937	0.241097
$552$	2.43880	0.103802
$553$	0.969003	0.0412062
$554$	−6.85913	−0.291416
$555$	−2.95649	−0.125496
$556$	−7.44884	−0.315901
$557$	−1.45233	−0.0615372	−0.0307686	−	0.999527i	$-0.509795\pi$
$557$	−1.45233	−0.0615372	−0.0307686	+	0.999527i	$0.509795\pi$
$558$	13.4847	0.570853
$559$	0.633122	0.0267782
$560$	3.04541	0.128692
$561$	−1.57718	−0.0665885
$562$	26.7458	1.12820
$563$	23.6944	0.998599	0.499299	−	0.866429i	$-0.333591\pi$
$563$	23.6944	0.998599	0.499299	+	0.866429i	$0.333591\pi$
$564$	3.15282	0.132757
$565$	14.6693	0.617142
$566$	−8.09210	−0.340136
$567$	−0.969003	−0.0406943
$568$	24.3464	1.02155
$569$	−15.2774	−0.640461	−0.320231	−	0.947340i	$-0.603760\pi$
$569$	−15.2774	−0.640461	−0.320231	+	0.947340i	$0.603760\pi$
$570$	10.0395	0.420507
$571$	43.8526	1.83517	0.917587	−	0.397535i	$-0.130134\pi$
$571$	43.8526	1.83517	0.917587	+	0.397535i	$0.130134\pi$
$572$	−8.83208	−0.369288
$573$	−9.58988	−0.400623
$574$	−0.591620	−0.0246937
$575$	−3.81620	−0.159147
$576$	−12.4032	−0.516800
$577$	−2.62032	−0.109085	−0.0545426	−	0.998511i	$-0.517370\pi$
$577$	−2.62032	−0.109085	−0.0545426	+	0.998511i	$0.517370\pi$
$578$	−2.17605	−0.0905119
$579$	6.91063	0.287196
$580$	8.37648	0.347815
$581$	8.20224	0.340286
$582$	27.1146	1.12394
$583$	−2.54857	−0.105551
$584$	20.6472	0.854387
$585$	3.23493	0.133748
$586$	−28.6409	−1.18314
$587$	0.116292	0.00479987	0.00239993	−	0.999997i	$-0.499236\pi$
$587$	0.116292	0.00479987	0.00239993	+	0.999997i	$0.499236\pi$
$588$	16.5782	0.683672
$589$	−18.0943	−0.745561
$590$	−29.5461	−1.21639
$591$	−17.6251	−0.724998
$592$	−3.72179	−0.152965
$593$	−35.3342	−1.45100	−0.725500	−	0.688222i	$-0.758391\pi$
$593$	−35.3342	−1.45100	−0.725500	+	0.688222i	$0.758391\pi$
$594$	−3.43202	−0.140818
$595$	1.53108	0.0627681
$596$	25.3170	1.03703
$597$	20.1977	0.826637
$598$	−6.79139	−0.277721
$599$	7.13981	0.291725	0.145862	−	0.989305i	$-0.453404\pi$
$599$	7.13981	0.291725	0.145862	+	0.989305i	$0.453404\pi$
$600$	−4.00509	−0.163507
$601$	8.18347	0.333811	0.166905	−	0.985973i	$-0.446623\pi$
$601$	8.18347	0.333811	0.166905	+	0.985973i	$0.446623\pi$
$602$	0.652063	0.0265761
$603$	−5.25347	−0.213938
$604$	33.5356	1.36455
$605$	−13.4502	−0.546830
$606$	−0.355353	−0.0144352
$607$	−18.1520	−0.736769	−0.368384	−	0.929674i	$-0.620089\pi$
$607$	−18.1520	−0.736769	−0.368384	+	0.929674i	$0.620089\pi$
$608$	21.9810	0.891448
$609$	1.87812	0.0761055
$610$	−15.6959	−0.635510
$611$	−2.35994	−0.0954730
$612$	−2.73521	−0.110564
$613$	17.7318	0.716180	0.358090	−	0.933687i	$-0.383428\pi$
$613$	17.7318	0.716180	0.358090	+	0.933687i	$0.383428\pi$
$614$	13.4069	0.541057
$615$	0.443323	0.0178765
$616$	−2.44503	−0.0985131
$617$	30.7917	1.23963	0.619813	−	0.784749i	$-0.287208\pi$
$617$	30.7917	1.23963	0.619813	+	0.784749i	$0.287208\pi$
$618$	−27.8998	−1.12229
$619$	20.6159	0.828622	0.414311	−	0.910135i	$-0.364023\pi$
$619$	20.6159	0.828622	0.414311	+	0.910135i	$0.364023\pi$
$620$	−26.7815	−1.07557
$621$	−1.52439	−0.0611718
$622$	51.0866	2.04839
$623$	−4.80573	−0.192538
$624$	4.07231	0.163023
$625$	−6.21575	−0.248630
$626$	−66.6892	−2.66544
$627$	4.60521	0.183914
$628$	14.5340	0.579971
$629$	−1.87113	−0.0746069
$630$	3.33171	0.132738
$631$	−30.8790	−1.22927	−0.614636	−	0.788811i	$-0.710697\pi$
$631$	−30.8790	−1.22927	−0.614636	+	0.788811i	$0.710697\pi$
$632$	1.59985	0.0636385
$633$	24.4918	0.973464
$634$	−57.3073	−2.27596
$635$	−18.1300	−0.719469
$636$	−4.41984	−0.175258
$637$	−12.4091	−0.491665
$638$	6.65196	0.263354
$639$	−15.2180	−0.602013
$640$	18.8564	0.745365
$641$	−10.3285	−0.407950	−0.203975	−	0.978976i	$-0.565386\pi$
$641$	−10.3285	−0.407950	−0.203975	+	0.978976i	$0.565386\pi$
$642$	−0.979740	−0.0386673
$643$	37.5151	1.47945	0.739726	−	0.672908i	$-0.234955\pi$
$643$	37.5151	1.47945	0.739726	+	0.672908i	$0.234955\pi$
$644$	−4.04029	−0.159210
$645$	−0.488616	−0.0192392
$646$	6.35386	0.249989
$647$	−43.9718	−1.72871	−0.864356	−	0.502881i	$-0.832273\pi$
$647$	−43.9718	−1.72871	−0.864356	+	0.502881i	$0.832273\pi$
$648$	−1.59985	−0.0628479
$649$	−13.5531	−0.532007
$650$	11.1531	0.437461
$651$	−6.00478	−0.235346
$652$	34.7364	1.36038
$653$	−4.24654	−0.166180	−0.0830901	−	0.996542i	$-0.526479\pi$
$653$	−4.24654	−0.166180	−0.0830901	+	0.996542i	$0.526479\pi$
$654$	5.15101	0.201420
$655$	15.7807	0.616604
$656$	0.558079	0.0217893
$657$	−12.9057	−0.503501
$658$	−2.43054	−0.0947524
$659$	4.13053	0.160903	0.0804513	−	0.996759i	$-0.474364\pi$
$659$	4.13053	0.160903	0.0804513	+	0.996759i	$0.474364\pi$
$660$	6.81621	0.265321
$661$	36.3118	1.41236	0.706182	−	0.708030i	$-0.250416\pi$
$661$	36.3118	1.41236	0.706182	+	0.708030i	$0.250416\pi$
$662$	62.1285	2.41469
$663$	2.04735	0.0795125
$664$	13.5421	0.525535
$665$	−4.47060	−0.173363
$666$	−4.07168	−0.157774
$667$	2.95458	0.114402
$668$	32.5668	1.26005
$669$	21.1179	0.816466
$670$	18.0629	0.697832
$671$	−7.19990	−0.277949
$672$	7.29465	0.281397
$673$	37.7476	1.45506	0.727531	−	0.686075i	$-0.240668\pi$
$673$	37.7476	1.45506	0.727531	+	0.686075i	$0.240668\pi$
$674$	−20.1993	−0.778050
$675$	2.50342	0.0963568
$676$	−24.0927	−0.926641
$677$	11.6431	0.447480	0.223740	−	0.974649i	$-0.428173\pi$
$677$	11.6431	0.447480	0.223740	+	0.974649i	$0.428173\pi$
$678$	20.2026	0.775875
$679$	−12.0742	−0.463366
$680$	2.52785	0.0969385
$681$	−20.7853	−0.796496
$682$	−21.2678	−0.814386
$683$	−15.2031	−0.581732	−0.290866	−	0.956764i	$-0.593943\pi$
$683$	−15.2031	−0.581732	−0.290866	+	0.956764i	$0.593943\pi$
$684$	7.98654	0.305373
$685$	−17.1856	−0.656628
$686$	−27.5405	−1.05150
$687$	3.52244	0.134389
$688$	−0.615096	−0.0234503
$689$	3.30833	0.126037
$690$	5.24130	0.199533
$691$	32.9376	1.25301	0.626503	−	0.779419i	$-0.284486\pi$
$691$	32.9376	1.25301	0.626503	+	0.779419i	$0.284486\pi$
$692$	58.0309	2.20600
$693$	1.52829	0.0580549
$694$	63.6393	2.41571
$695$	−4.30300	−0.163222
$696$	3.10083	0.117537
$697$	0.280574	0.0106275
$698$	61.4975	2.32772
$699$	25.6310	0.969452
$700$	6.63513	0.250784
$701$	24.0922	0.909952	0.454976	−	0.890504i	$-0.349648\pi$
$701$	24.0922	0.909952	0.454976	+	0.890504i	$0.349648\pi$
$702$	4.45515	0.168149
$703$	5.46352	0.206061
$704$	19.5620	0.737272
$705$	1.82130	0.0685940
$706$	−38.5919	−1.45242
$707$	0.158240	0.00595122
$708$	−23.5044	−0.883348
$709$	33.7935	1.26914	0.634571	−	0.772864i	$-0.281177\pi$
$709$	33.7935	1.26914	0.634571	+	0.772864i	$0.281177\pi$
$710$	52.3237	1.96367
$711$	−1.00000	−0.0375029
$712$	−7.93438	−0.297353
$713$	−9.44646	−0.353773
$714$	2.10860	0.0789124
$715$	−5.10206	−0.190806
$716$	34.4355	1.28692
$717$	10.4297	0.389504
$718$	41.8569	1.56208
$719$	−9.56744	−0.356805	−0.178403	−	0.983958i	$-0.557093\pi$
$719$	−9.56744	−0.356805	−0.178403	+	0.983958i	$0.557093\pi$
$720$	−3.14283	−0.117126
$721$	12.4238	0.462688
$722$	22.7923	0.848243
$723$	13.5955	0.505621
$724$	10.2428	0.380673
$725$	−4.85215	−0.180204
$726$	−18.5237	−0.687478
$727$	−14.7260	−0.546157	−0.273078	−	0.961992i	$-0.588042\pi$
$727$	−14.7260	−0.546157	−0.273078	+	0.961992i	$0.588042\pi$
$728$	3.17392	0.117633
$729$	1.00000	0.0370370
$730$	44.3736	1.64234
$731$	−0.309240	−0.0114376
$732$	−12.4863	−0.461509
$733$	−43.7528	−1.61605	−0.808024	−	0.589150i	$-0.799463\pi$
$733$	−43.7528	−1.61605	−0.808024	+	0.589150i	$0.799463\pi$
$734$	−44.1124	−1.62822
$735$	9.57677	0.353244
$736$	11.4756	0.422997
$737$	8.28566	0.305206
$738$	0.610545	0.0224745
$739$	−13.2707	−0.488171	−0.244086	−	0.969754i	$-0.578488\pi$
$739$	−13.2707	−0.488171	−0.244086	+	0.969754i	$0.578488\pi$
$740$	8.08661	0.297270
$741$	−5.97807	−0.219610
$742$	3.40730	0.125086
$743$	−17.6038	−0.645822	−0.322911	−	0.946429i	$-0.604661\pi$
$743$	−17.6038	−0.645822	−0.322911	+	0.946429i	$0.604661\pi$
$744$	−9.91404	−0.363466
$745$	14.6250	0.535817
$746$	47.1521	1.72636
$747$	−8.46462	−0.309704
$748$	4.31391	0.157732
$749$	0.436281	0.0159414
$750$	−25.7989	−0.942043
$751$	48.3837	1.76555	0.882773	−	0.469799i	$-0.155674\pi$
$751$	48.3837	1.76555	0.882773	+	0.469799i	$0.155674\pi$
$752$	2.29275	0.0836080
$753$	6.91049	0.251832
$754$	−8.63498	−0.314467
$755$	19.3727	0.705043
$756$	2.65042	0.0963949
$757$	9.69125	0.352234	0.176117	−	0.984369i	$-0.443646\pi$
$757$	9.69125	0.352234	0.176117	+	0.984369i	$0.443646\pi$
$758$	53.2407	1.93379
$759$	2.40424	0.0872683
$760$	−7.38107	−0.267740
$761$	−38.2083	−1.38505	−0.692525	−	0.721394i	$-0.743502\pi$
$761$	−38.2083	−1.38505	−0.692525	+	0.721394i	$0.743502\pi$
$762$	−24.9687	−0.904520
$763$	−2.29376	−0.0830397
$764$	26.2303	0.948978
$765$	−1.58006	−0.0571270
$766$	39.0157	1.40970
$767$	17.5935	0.635263
$768$	1.16266	0.0419537
$769$	3.48053	0.125511	0.0627556	−	0.998029i	$-0.480011\pi$
$769$	3.48053	0.125511	0.0627556	+	0.998029i	$0.480011\pi$
$770$	−5.25470	−0.189366
$771$	−16.0883	−0.579408
$772$	−18.9020	−0.680298
$773$	−49.0224	−1.76321	−0.881606	−	0.471986i	$-0.843537\pi$
$773$	−49.0224	−1.76321	−0.881606	+	0.471986i	$0.843537\pi$
$774$	−0.672922	−0.0241877
$775$	15.5134	0.557257
$776$	−19.9348	−0.715618
$777$	1.81313	0.0650457
$778$	26.2989	0.942860
$779$	−0.819250	−0.0293527
$780$	−8.84820	−0.316816
$781$	24.0014	0.858839
$782$	3.31716	0.118621
$783$	−1.93820	−0.0692657
$784$	12.0558	0.430563
$785$	8.39593	0.299663
$786$	21.7332	0.775198
$787$	−26.3678	−0.939912	−0.469956	−	0.882690i	$-0.655730\pi$
$787$	−26.3678	−0.939912	−0.469956	+	0.882690i	$0.655730\pi$
$788$	48.2082	1.71735
$789$	−4.14906	−0.147711
$790$	3.43828	0.122329
$791$	−8.99627	−0.319870
$792$	2.52324	0.0896595
$793$	9.34626	0.331895
$794$	51.8032	1.83843
$795$	−2.55322	−0.0905535
$796$	−55.2449	−1.95810
$797$	−19.5715	−0.693257	−0.346629	−	0.938002i	$-0.612674\pi$
$797$	−19.5715	−0.693257	−0.346629	+	0.938002i	$0.612674\pi$
$798$	−6.15691	−0.217952
$799$	1.15268	0.0407789
$800$	−18.8458	−0.666298
$801$	4.95946	0.175234
$802$	33.7682	1.19240
$803$	20.3546	0.718300
$804$	14.3693	0.506767
$805$	−2.33397	−0.0822615
$806$	27.6079	0.972448
$807$	24.1572	0.850374
$808$	0.261258	0.00919101
$809$	38.4263	1.35100	0.675499	−	0.737361i	$-0.263928\pi$
$809$	38.4263	1.35100	0.675499	+	0.737361i	$0.263928\pi$
$810$	−3.43828	−0.120809
$811$	−4.39730	−0.154410	−0.0772051	−	0.997015i	$-0.524600\pi$
$811$	−4.39730	−0.154410	−0.0772051	+	0.997015i	$0.524600\pi$
$812$	−5.13706	−0.180275
$813$	19.3444	0.678437
$814$	6.42176	0.225083
$815$	20.0663	0.702893
$816$	−1.98906	−0.0696311
$817$	0.902950	0.0315902
$818$	−13.9363	−0.487271
$819$	−1.98389	−0.0693227
$820$	−1.21258	−0.0423451
$821$	19.4992	0.680528	0.340264	−	0.940330i	$-0.389483\pi$
$821$	19.4992	0.680528	0.340264	+	0.940330i	$0.389483\pi$
$822$	−23.6680	−0.825517
$823$	13.0831	0.456049	0.228025	−	0.973655i	$-0.426773\pi$
$823$	13.0831	0.456049	0.228025	+	0.973655i	$0.426773\pi$
$824$	20.5121	0.714572
$825$	−3.94835	−0.137464
$826$	18.1198	0.630468
$827$	10.5378	0.366436	0.183218	−	0.983072i	$-0.441349\pi$
$827$	10.5378	0.366436	0.183218	+	0.983072i	$0.441349\pi$
$828$	4.16953	0.144901
$829$	−13.3966	−0.465282	−0.232641	−	0.972563i	$-0.574737\pi$
$829$	−13.3966	−0.465282	−0.232641	+	0.972563i	$0.574737\pi$
$830$	29.1038	1.01021
$831$	−3.15210	−0.109345
$832$	−25.3937	−0.880368
$833$	6.06103	0.210002
$834$	−5.92609	−0.205204
$835$	18.8129	0.651049
$836$	−12.5962	−0.435648
$837$	6.19687	0.214195
$838$	25.7926	0.890989
$839$	8.06286	0.278361	0.139180	−	0.990267i	$-0.455553\pi$
$839$	8.06286	0.278361	0.139180	+	0.990267i	$0.455553\pi$
$840$	−2.44949	−0.0845155
$841$	−25.2434	−0.870461
$842$	27.9485	0.963170
$843$	12.2909	0.423323
$844$	−66.9902	−2.30590
$845$	−13.9177	−0.478783
$846$	2.50829	0.0862368
$847$	8.24865	0.283427
$848$	−3.21414	−0.110374
$849$	−3.71871	−0.127626
$850$	−5.44758	−0.186851
$851$	2.85234	0.0977769
$852$	41.6243	1.42602
$853$	11.1648	0.382274	0.191137	−	0.981563i	$-0.438782\pi$
$853$	11.1648	0.382274	0.191137	+	0.981563i	$0.438782\pi$
$854$	9.62587	0.329390
$855$	4.61361	0.157782
$856$	0.720311	0.0246197
$857$	−47.8579	−1.63480	−0.817398	−	0.576074i	$-0.804584\pi$
$857$	−47.8579	−1.63480	−0.817398	+	0.576074i	$0.804584\pi$
$858$	−7.02656	−0.239883
$859$	−16.3454	−0.557696	−0.278848	−	0.960335i	$-0.589953\pi$
$859$	−16.3454	−0.557696	−0.278848	+	0.960335i	$0.589953\pi$
$860$	1.33646	0.0455731
$861$	−0.271877	−0.00926556
$862$	6.35361	0.216405
$863$	−32.7055	−1.11331	−0.556654	−	0.830744i	$-0.687915\pi$
$863$	−32.7055	−1.11331	−0.556654	+	0.830744i	$0.687915\pi$
$864$	−7.52799	−0.256108
$865$	33.5229	1.13981
$866$	38.8794	1.32118
$867$	−1.00000	−0.0339618
$868$	16.4243	0.557478
$869$	1.57718	0.0535021
$870$	6.66410	0.225934
$871$	−10.7557	−0.364443
$872$	−3.78705	−0.128246
$873$	12.4605	0.421722
$874$	−9.68579	−0.327627
$875$	11.4883	0.388376
$876$	35.2999	1.19267
$877$	38.3901	1.29634	0.648171	−	0.761495i	$-0.275534\pi$
$877$	38.3901	1.29634	0.648171	+	0.761495i	$0.275534\pi$
$878$	16.5033	0.556959
$879$	−13.1619	−0.443938
$880$	4.95680	0.167094
$881$	−10.9521	−0.368985	−0.184493	−	0.982834i	$-0.559064\pi$
$881$	−10.9521	−0.368985	−0.184493	+	0.982834i	$0.559064\pi$
$882$	13.1891	0.444101
$883$	−18.8905	−0.635717	−0.317859	−	0.948138i	$-0.602964\pi$
$883$	−18.8905	−0.635717	−0.317859	+	0.948138i	$0.602964\pi$
$884$	−5.59993	−0.188346
$885$	−13.5779	−0.456414
$886$	−41.2325	−1.38523
$887$	−6.67657	−0.224177	−0.112089	−	0.993698i	$-0.535754\pi$
$887$	−6.67657	−0.224177	−0.112089	+	0.993698i	$0.535754\pi$
$888$	2.99352	0.100456
$889$	11.1186	0.372907
$890$	−17.0520	−0.571586
$891$	−1.57718	−0.0528374
$892$	−57.7619	−1.93401
$893$	−3.36571	−0.112629
$894$	20.1415	0.673633
$895$	19.8925	0.664933
$896$	−11.5641	−0.386329
$897$	−3.12097	−0.104206
$898$	14.7108	0.490905
$899$	−12.0108	−0.400582
$900$	−6.84738	−0.228246
$901$	−1.61591	−0.0538337
$902$	−0.962937	−0.0320623
$903$	0.299654	0.00997186
$904$	−14.8530	−0.494005
$905$	5.91702	0.196688
$906$	26.6800	0.886384
$907$	36.9248	1.22607	0.613034	−	0.790057i	$-0.289949\pi$
$907$	36.9248	1.22607	0.613034	+	0.790057i	$0.289949\pi$
$908$	56.8522	1.88671
$909$	−0.163302	−0.00541637
$910$	6.82118	0.226120
$911$	−19.3077	−0.639693	−0.319847	−	0.947469i	$-0.603631\pi$
$911$	−19.3077	−0.639693	−0.319847	+	0.947469i	$0.603631\pi$
$912$	5.80787	0.192318
$913$	13.3502	0.441828
$914$	−62.5854	−2.07014
$915$	−7.21303	−0.238455
$916$	−9.63460	−0.318336
$917$	−9.67786	−0.319591
$918$	−2.17605	−0.0718204
$919$	−35.6430	−1.17576	−0.587878	−	0.808950i	$-0.700036\pi$
$919$	−35.6430	−1.17576	−0.587878	+	0.808950i	$0.700036\pi$
$920$	−3.85343	−0.127044
$921$	6.16109	0.203015
$922$	−2.79046	−0.0918990
$923$	−31.1565	−1.02553
$924$	−4.18019	−0.137518
$925$	−4.68423	−0.154017
$926$	−1.18760	−0.0390269
$927$	−12.8213	−0.421106
$928$	14.5908	0.478966
$929$	42.2161	1.38506	0.692532	−	0.721387i	$-0.256495\pi$
$929$	42.2161	1.38506	0.692532	+	0.721387i	$0.256495\pi$
$930$	−21.3066	−0.698671
$931$	−17.6976	−0.580017
$932$	−70.1060	−2.29640
$933$	23.4767	0.768594
$934$	43.9191	1.43708
$935$	2.49203	0.0814981
$936$	−3.27545	−0.107061
$937$	44.5212	1.45444	0.727222	−	0.686402i	$-0.240811\pi$
$937$	44.5212	1.45444	0.727222	+	0.686402i	$0.240811\pi$
$938$	−11.0775	−0.361692
$939$	−30.6469	−1.00012
$940$	−4.98162	−0.162483
$941$	23.6022	0.769408	0.384704	−	0.923040i	$-0.374303\pi$
$941$	23.6022	0.769408	0.384704	+	0.923040i	$0.374303\pi$
$942$	11.5629	0.376739
$943$	−0.427706	−0.0139280
$944$	−17.0925	−0.556315
$945$	1.53108	0.0498060
$946$	1.06132	0.0345064
$947$	−14.7973	−0.480849	−0.240425	−	0.970668i	$-0.577287\pi$
$947$	−14.7973	−0.480849	−0.240425	+	0.970668i	$0.577287\pi$
$948$	2.73521	0.0888354
$949$	−26.4226	−0.857713
$950$	15.9064	0.516072
$951$	−26.3354	−0.853985
$952$	−1.55026	−0.0502441
$953$	3.03159	0.0982027	0.0491014	−	0.998794i	$-0.484364\pi$
$953$	3.03159	0.0982027	0.0491014	+	0.998794i	$0.484364\pi$
$954$	−3.51630	−0.113844
$955$	15.1525	0.490325
$956$	−28.5273	−0.922640
$957$	3.05689	0.0988153
$958$	78.8930	2.54892
$959$	10.5394	0.340336
$960$	19.5977	0.632514
$961$	7.40117	0.238747
$962$	−8.33616	−0.268769
$963$	−0.450237	−0.0145087
$964$	−37.1864	−1.19769
$965$	−10.9192	−0.351501
$966$	−3.21434	−0.103420
$967$	−54.3526	−1.74786	−0.873931	−	0.486051i	$-0.838437\pi$
$967$	−54.3526	−1.74786	−0.873931	+	0.486051i	$0.838437\pi$
$968$	13.6187	0.437722
$969$	2.91990	0.0938008
$970$	−42.8426	−1.37559
$971$	11.6090	0.372552	0.186276	−	0.982497i	$-0.440358\pi$
$971$	11.6090	0.372552	0.186276	+	0.982497i	$0.440358\pi$
$972$	−2.73521	−0.0877318
$973$	2.63891	0.0845994
$974$	19.3716	0.620708
$975$	5.12539	0.164144
$976$	−9.08016	−0.290649
$977$	−3.05826	−0.0978425	−0.0489212	−	0.998803i	$-0.515578\pi$
$977$	−3.05826	−0.0978425	−0.0489212	+	0.998803i	$0.515578\pi$
$978$	27.6353	0.883681
$979$	−7.82195	−0.249991
$980$	−26.1944	−0.836751
$981$	2.36713	0.0755768
$982$	43.8531	1.39941
$983$	−31.3946	−1.00133	−0.500666	−	0.865640i	$-0.666912\pi$
$983$	−31.3946	−1.00133	−0.500666	+	0.865640i	$0.666912\pi$
$984$	−0.448876	−0.0143096
$985$	27.8486	0.887330
$986$	4.21763	0.134317
$987$	−1.11695	−0.0355529
$988$	16.3513	0.520203
$989$	0.471403	0.0149897
$990$	5.42279	0.172347
$991$	−54.7325	−1.73864	−0.869318	−	0.494253i	$-0.835442\pi$
$991$	−54.7325	−1.73864	−0.869318	+	0.494253i	$0.835442\pi$
$992$	−46.6500	−1.48114
$993$	28.5510	0.906038
$994$	−32.0886	−1.01779
$995$	−31.9135	−1.01173
$996$	23.1525	0.733614
$997$	20.8664	0.660844	0.330422	−	0.943833i	$-0.392809\pi$
$997$	20.8664	0.660844	0.330422	+	0.943833i	$0.392809\pi$
$998$	15.4963	0.490526
$999$	−1.87113	−0.0592000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.g.1.2	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.g.1.2	✓	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:	\( N \)	\(=\)	\( 4029 = 3 \cdot 17 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4029.a (trivial)

\(f(q)\)	\(=\)	\(q-2.17605 q^{2} -1.00000 q^{3} +2.73521 q^{4} +1.58006 q^{5} +2.17605 q^{6} -0.969003 q^{7} -1.59985 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.17605 q^{2} -1.00000 q^{3} +2.73521 q^{4} +1.58006 q^{5} +2.17605 q^{6} -0.969003 q^{7} -1.59985 q^{8} +1.00000 q^{9} -3.43828 q^{10} -1.57718 q^{11} -2.73521 q^{12} +2.04735 q^{13} +2.10860 q^{14} -1.58006 q^{15} -1.98906 q^{16} -1.00000 q^{17} -2.17605 q^{18} +2.91990 q^{19} +4.32178 q^{20} +0.969003 q^{21} +3.43202 q^{22} +1.52439 q^{23} +1.59985 q^{24} -2.50342 q^{25} -4.45515 q^{26} -1.00000 q^{27} -2.65042 q^{28} +1.93820 q^{29} +3.43828 q^{30} -6.19687 q^{31} +7.52799 q^{32} +1.57718 q^{33} +2.17605 q^{34} -1.53108 q^{35} +2.73521 q^{36} +1.87113 q^{37} -6.35386 q^{38} -2.04735 q^{39} -2.52785 q^{40} -0.280574 q^{41} -2.10860 q^{42} +0.309240 q^{43} -4.31391 q^{44} +1.58006 q^{45} -3.31716 q^{46} -1.15268 q^{47} +1.98906 q^{48} -6.06103 q^{49} +5.44758 q^{50} +1.00000 q^{51} +5.59993 q^{52} +1.61591 q^{53} +2.17605 q^{54} -2.49203 q^{55} +1.55026 q^{56} -2.91990 q^{57} -4.21763 q^{58} +8.59327 q^{59} -4.32178 q^{60} +4.56505 q^{61} +13.4847 q^{62} -0.969003 q^{63} -12.4032 q^{64} +3.23493 q^{65} -3.43202 q^{66} -5.25347 q^{67} -2.73521 q^{68} -1.52439 q^{69} +3.33171 q^{70} -15.2180 q^{71} -1.59985 q^{72} -12.9057 q^{73} -4.07168 q^{74} +2.50342 q^{75} +7.98654 q^{76} +1.52829 q^{77} +4.45515 q^{78} -1.00000 q^{79} -3.14283 q^{80} +1.00000 q^{81} +0.610545 q^{82} -8.46462 q^{83} +2.65042 q^{84} -1.58006 q^{85} -0.672922 q^{86} -1.93820 q^{87} +2.52324 q^{88} +4.95946 q^{89} -3.43828 q^{90} -1.98389 q^{91} +4.16953 q^{92} +6.19687 q^{93} +2.50829 q^{94} +4.61361 q^{95} -7.52799 q^{96} +12.4605 q^{97} +13.1891 q^{98} -1.57718 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 2 q^{2} - 22 q^{3} + 16 q^{4} + 5 q^{5} - 2 q^{6} - 4 q^{7} + 6 q^{8} + 22 q^{9}+O(q^{10}) \) 22 * q + 2 * q^2 - 22 * q^3 + 16 * q^4 + 5 * q^5 - 2 * q^6 - 4 * q^7 + 6 * q^8 + 22 * q^9	\( 22 q + 2 q^{2} - 22 q^{3} + 16 q^{4} + 5 q^{5} - 2 q^{6} - 4 q^{7} + 6 q^{8} + 22 q^{9} - 5 q^{10} - 2 q^{11} - 16 q^{12} - 11 q^{13} - 7 q^{14} - 5 q^{15} - 22 q^{17} + 2 q^{18} - 36 q^{19} + 4 q^{21} - 9 q^{22} + 21 q^{23} - 6 q^{24} + 9 q^{25} - 16 q^{26} - 22 q^{27} - 17 q^{28} - q^{29} + 5 q^{30} - 12 q^{31} - 11 q^{32} + 2 q^{33} - 2 q^{34} - 14 q^{35} + 16 q^{36} - 6 q^{37} + q^{38} + 11 q^{39} - 24 q^{40} - 17 q^{41} + 7 q^{42} - 36 q^{43} + 16 q^{44} + 5 q^{45} - 23 q^{46} - 17 q^{47} - 6 q^{49} - 33 q^{50} + 22 q^{51} - 57 q^{52} - 2 q^{53} - 2 q^{54} - 24 q^{55} - 64 q^{56} + 36 q^{57} - 7 q^{58} - 59 q^{59} - 30 q^{61} - 4 q^{62} - 4 q^{63} - 22 q^{64} + 36 q^{65} + 9 q^{66} - 16 q^{67} - 16 q^{68} - 21 q^{69} - 39 q^{70} - 11 q^{71} + 6 q^{72} - 19 q^{73} - 28 q^{74} - 9 q^{75} - 77 q^{76} + 2 q^{77} + 16 q^{78} - 22 q^{79} - 2 q^{80} + 22 q^{81} + 33 q^{82} - 23 q^{83} + 17 q^{84} - 5 q^{85} + 6 q^{86} + q^{87} - 23 q^{88} + 12 q^{89} - 5 q^{90} - 24 q^{91} + 66 q^{92} + 12 q^{93} - 61 q^{94} - 11 q^{95} + 11 q^{96} - 9 q^{97} + 17 q^{98} - 2 q^{99}+O(q^{100}) \) 22 * q + 2 * q^2 - 22 * q^3 + 16 * q^4 + 5 * q^5 - 2 * q^6 - 4 * q^7 + 6 * q^8 + 22 * q^9 - 5 * q^10 - 2 * q^11 - 16 * q^12 - 11 * q^13 - 7 * q^14 - 5 * q^15 - 22 * q^17 + 2 * q^18 - 36 * q^19 + 4 * q^21 - 9 * q^22 + 21 * q^23 - 6 * q^24 + 9 * q^25 - 16 * q^26 - 22 * q^27 - 17 * q^28 - q^29 + 5 * q^30 - 12 * q^31 - 11 * q^32 + 2 * q^33 - 2 * q^34 - 14 * q^35 + 16 * q^36 - 6 * q^37 + q^38 + 11 * q^39 - 24 * q^40 - 17 * q^41 + 7 * q^42 - 36 * q^43 + 16 * q^44 + 5 * q^45 - 23 * q^46 - 17 * q^47 - 6 * q^49 - 33 * q^50 + 22 * q^51 - 57 * q^52 - 2 * q^53 - 2 * q^54 - 24 * q^55 - 64 * q^56 + 36 * q^57 - 7 * q^58 - 59 * q^59 - 30 * q^61 - 4 * q^62 - 4 * q^63 - 22 * q^64 + 36 * q^65 + 9 * q^66 - 16 * q^67 - 16 * q^68 - 21 * q^69 - 39 * q^70 - 11 * q^71 + 6 * q^72 - 19 * q^73 - 28 * q^74 - 9 * q^75 - 77 * q^76 + 2 * q^77 + 16 * q^78 - 22 * q^79 - 2 * q^80 + 22 * q^81 + 33 * q^82 - 23 * q^83 + 17 * q^84 - 5 * q^85 + 6 * q^86 + q^87 - 23 * q^88 + 12 * q^89 - 5 * q^90 - 24 * q^91 + 66 * q^92 + 12 * q^93 - 61 * q^94 - 11 * q^95 + 11 * q^96 - 9 * q^97 + 17 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more