LMFDB - Embedded newform 4009.2.a.c.1.32

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.32
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.784700 q^{2} -2.16047 q^{3} -1.38425 q^{4} +1.59976 q^{5} +1.69532 q^{6} +4.17479 q^{7} +2.65562 q^{8} +1.66763 q^{9} +O(q^{10})$	\(q-0.784700 q^{2} -2.16047 q^{3} -1.38425 q^{4} +1.59976 q^{5} +1.69532 q^{6} +4.17479 q^{7} +2.65562 q^{8} +1.66763 q^{9} -1.25534 q^{10} +3.08275 q^{11} +2.99062 q^{12} -4.21572 q^{13} -3.27596 q^{14} -3.45624 q^{15} +0.684627 q^{16} -0.283116 q^{17} -1.30859 q^{18} +1.00000 q^{19} -2.21447 q^{20} -9.01951 q^{21} -2.41903 q^{22} +7.85371 q^{23} -5.73738 q^{24} -2.44075 q^{25} +3.30808 q^{26} +2.87854 q^{27} -5.77894 q^{28} -6.87222 q^{29} +2.71212 q^{30} -1.52968 q^{31} -5.84846 q^{32} -6.66018 q^{33} +0.222161 q^{34} +6.67868 q^{35} -2.30841 q^{36} -9.42672 q^{37} -0.784700 q^{38} +9.10794 q^{39} +4.24836 q^{40} -11.9841 q^{41} +7.07761 q^{42} -11.3060 q^{43} -4.26728 q^{44} +2.66782 q^{45} -6.16281 q^{46} +0.398003 q^{47} -1.47912 q^{48} +10.4289 q^{49} +1.91526 q^{50} +0.611664 q^{51} +5.83560 q^{52} -10.2451 q^{53} -2.25879 q^{54} +4.93167 q^{55} +11.0867 q^{56} -2.16047 q^{57} +5.39263 q^{58} +8.69174 q^{59} +4.78429 q^{60} +2.70052 q^{61} +1.20034 q^{62} +6.96201 q^{63} +3.22004 q^{64} -6.74416 q^{65} +5.22624 q^{66} +4.04404 q^{67} +0.391902 q^{68} -16.9677 q^{69} -5.24077 q^{70} -6.40653 q^{71} +4.42859 q^{72} +1.15024 q^{73} +7.39715 q^{74} +5.27317 q^{75} -1.38425 q^{76} +12.8698 q^{77} -7.14701 q^{78} -7.79384 q^{79} +1.09524 q^{80} -11.2219 q^{81} +9.40390 q^{82} +16.1133 q^{83} +12.4852 q^{84} -0.452919 q^{85} +8.87184 q^{86} +14.8472 q^{87} +8.18660 q^{88} +10.0068 q^{89} -2.09344 q^{90} -17.5998 q^{91} -10.8715 q^{92} +3.30482 q^{93} -0.312313 q^{94} +1.59976 q^{95} +12.6354 q^{96} -13.7963 q^{97} -8.18355 q^{98} +5.14088 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * 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 $

$2$	−0.784700	−0.554867	−0.277433	−	0.960745i	$-0.589484\pi$
$2$	−0.784700	−0.554867	−0.277433	+	0.960745i	$0.589484\pi$
$3$	−2.16047	−1.24735	−0.623674	−	0.781685i	$-0.714361\pi$
$3$	−2.16047	−1.24735	−0.623674	+	0.781685i	$0.714361\pi$
$4$	−1.38425	−0.692123
$5$	1.59976	0.715436	0.357718	−	0.933830i	$-0.383555\pi$
$5$	1.59976	0.715436	0.357718	+	0.933830i	$0.383555\pi$
$6$	1.69532	0.692112
$7$	4.17479	1.57792	0.788962	−	0.614443i	$-0.210619\pi$
$7$	4.17479	1.57792	0.788962	+	0.614443i	$0.210619\pi$
$8$	2.65562	0.938903
$9$	1.66763	0.555877
$10$	−1.25534	−0.396972
$11$	3.08275	0.929483	0.464741	−	0.885446i	$-0.346147\pi$
$11$	3.08275	0.929483	0.464741	+	0.885446i	$0.346147\pi$
$12$	2.99062	0.863318
$13$	−4.21572	−1.16923	−0.584616	−	0.811310i	$-0.698755\pi$
$13$	−4.21572	−1.16923	−0.584616	+	0.811310i	$0.698755\pi$
$14$	−3.27596	−0.875537
$15$	−3.45624	−0.892398
$16$	0.684627	0.171157
$17$	−0.283116	−0.0686658	−0.0343329	−	0.999410i	$-0.510931\pi$
$17$	−0.283116	−0.0686658	−0.0343329	+	0.999410i	$0.510931\pi$
$18$	−1.30859	−0.308438
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.21447	−0.495170
$21$	−9.01951	−1.96822
$22$	−2.41903	−0.515739
$23$	7.85371	1.63761	0.818806	−	0.574070i	$-0.194636\pi$
$23$	7.85371	1.63761	0.818806	+	0.574070i	$0.194636\pi$
$24$	−5.73738	−1.17114
$25$	−2.44075	−0.488151
$26$	3.30808	0.648768
$27$	2.87854	0.553976
$28$	−5.77894	−1.09212
$29$	−6.87222	−1.27614	−0.638070	−	0.769979i	$-0.720267\pi$
$29$	−6.87222	−1.27614	−0.638070	+	0.769979i	$0.720267\pi$
$30$	2.71212	0.495162
$31$	−1.52968	−0.274738	−0.137369	−	0.990520i	$-0.543865\pi$
$31$	−1.52968	−0.274738	−0.137369	+	0.990520i	$0.543865\pi$
$32$	−5.84846	−1.03387
$33$	−6.66018	−1.15939
$34$	0.222161	0.0381004
$35$	6.67868	1.12890
$36$	−2.30841	−0.384735
$37$	−9.42672	−1.54974	−0.774872	−	0.632119i	$-0.782186\pi$
$37$	−9.42672	−1.54974	−0.774872	+	0.632119i	$0.782186\pi$
$38$	−0.784700	−0.127295
$39$	9.10794	1.45844
$40$	4.24836	0.671725
$41$	−11.9841	−1.87160	−0.935798	−	0.352536i	$-0.885319\pi$
$41$	−11.9841	−1.87160	−0.935798	+	0.352536i	$0.885319\pi$
$42$	7.07761	1.09210
$43$	−11.3060	−1.72415	−0.862076	−	0.506779i	$-0.830836\pi$
$43$	−11.3060	−1.72415	−0.862076	+	0.506779i	$0.830836\pi$
$44$	−4.26728	−0.643316
$45$	2.66782	0.397695
$46$	−6.16281	−0.908657
$47$	0.398003	0.0580547	0.0290273	−	0.999579i	$-0.490759\pi$
$47$	0.398003	0.0580547	0.0290273	+	0.999579i	$0.490759\pi$
$48$	−1.47912	−0.213492
$49$	10.4289	1.48984
$50$	1.91526	0.270859
$51$	0.611664	0.0856501
$52$	5.83560	0.809251
$53$	−10.2451	−1.40727	−0.703635	−	0.710562i	$-0.748441\pi$
$53$	−10.2451	−1.40727	−0.703635	+	0.710562i	$0.748441\pi$
$54$	−2.25879	−0.307383
$55$	4.93167	0.664986
$56$	11.0867	1.48152
$57$	−2.16047	−0.286161
$58$	5.39263	0.708088
$59$	8.69174	1.13157	0.565784	−	0.824554i	$-0.308574\pi$
$59$	8.69174	1.13157	0.565784	+	0.824554i	$0.308574\pi$
$60$	4.78429	0.617649
$61$	2.70052	0.345766	0.172883	−	0.984942i	$-0.444692\pi$
$61$	2.70052	0.345766	0.172883	+	0.984942i	$0.444692\pi$
$62$	1.20034	0.152443
$63$	6.96201	0.877131
$64$	3.22004	0.402505
$65$	−6.74416	−0.836511
$66$	5.22624	0.643306
$67$	4.04404	0.494058	0.247029	−	0.969008i	$-0.420546\pi$
$67$	4.04404	0.494058	0.247029	+	0.969008i	$0.420546\pi$
$68$	0.391902	0.0475251
$69$	−16.9677	−2.04267
$70$	−5.24077	−0.626391
$71$	−6.40653	−0.760315	−0.380158	−	0.924922i	$-0.624130\pi$
$71$	−6.40653	−0.760315	−0.380158	+	0.924922i	$0.624130\pi$
$72$	4.42859	0.521915
$73$	1.15024	0.134625	0.0673125	−	0.997732i	$-0.478558\pi$
$73$	1.15024	0.134625	0.0673125	+	0.997732i	$0.478558\pi$
$74$	7.39715	0.859901
$75$	5.27317	0.608894
$76$	−1.38425	−0.158784
$77$	12.8698	1.46665
$78$	−7.14701	−0.809239
$79$	−7.79384	−0.876876	−0.438438	−	0.898761i	$-0.644468\pi$
$79$	−7.79384	−0.876876	−0.438438	+	0.898761i	$0.644468\pi$
$80$	1.09524	0.122452
$81$	−11.2219	−1.24688
$82$	9.40390	1.03849
$83$	16.1133	1.76866	0.884332	−	0.466859i	$-0.154614\pi$
$83$	16.1133	1.76866	0.884332	+	0.466859i	$0.154614\pi$
$84$	12.4852	1.36225
$85$	−0.452919	−0.0491260
$86$	8.87184	0.956675
$87$	14.8472	1.59179
$88$	8.18660	0.872694
$89$	10.0068	1.06071	0.530357	−	0.847774i	$-0.322058\pi$
$89$	10.0068	1.06071	0.530357	+	0.847774i	$0.322058\pi$
$90$	−2.09344	−0.220668
$91$	−17.5998	−1.84496
$92$	−10.8715	−1.13343
$93$	3.30482	0.342694
$94$	−0.312313	−0.0322126
$95$	1.59976	0.164132
$96$	12.6354	1.28960
$97$	−13.7963	−1.40081	−0.700403	−	0.713747i	$-0.746997\pi$
$97$	−13.7963	−1.40081	−0.700403	+	0.713747i	$0.746997\pi$
$98$	−8.18355	−0.826664
$99$	5.14088	0.516678
$100$	3.37860	0.337860
$101$	−9.58260	−0.953504	−0.476752	−	0.879038i	$-0.658186\pi$
$101$	−9.58260	−0.953504	−0.476752	+	0.879038i	$0.658186\pi$
$102$	−0.479973	−0.0475244
$103$	−18.0829	−1.78177	−0.890883	−	0.454234i	$-0.849913\pi$
$103$	−18.0829	−1.78177	−0.890883	+	0.454234i	$0.849913\pi$
$104$	−11.1954	−1.09779
$105$	−14.4291	−1.40814
$106$	8.03932	0.780847
$107$	17.7815	1.71901	0.859503	−	0.511131i	$-0.170773\pi$
$107$	17.7815	1.71901	0.859503	+	0.511131i	$0.170773\pi$
$108$	−3.98461	−0.383419
$109$	12.2100	1.16950	0.584751	−	0.811213i	$-0.301192\pi$
$109$	12.2100	1.16950	0.584751	+	0.811213i	$0.301192\pi$
$110$	−3.86988	−0.368979
$111$	20.3661	1.93307
$112$	2.85817	0.270072
$113$	10.9832	1.03321	0.516605	−	0.856224i	$-0.327196\pi$
$113$	10.9832	1.03321	0.516605	+	0.856224i	$0.327196\pi$
$114$	1.69532	0.158781
$115$	12.5641	1.17161
$116$	9.51284	0.883245
$117$	−7.03027	−0.649949
$118$	−6.82041	−0.627869
$119$	−1.18195	−0.108349
$120$	−9.17846	−0.837875
$121$	−1.49668	−0.136062
$122$	−2.11910	−0.191854
$123$	25.8912	2.33453
$124$	2.11745	0.190153
$125$	−11.9035	−1.06468
$126$	−5.46309	−0.486691
$127$	−0.639312	−0.0567298	−0.0283649	−	0.999598i	$-0.509030\pi$
$127$	−0.639312	−0.0567298	−0.0283649	+	0.999598i	$0.509030\pi$
$128$	9.17016	0.810536
$129$	24.4263	2.15062
$130$	5.29215	0.464152
$131$	−11.2454	−0.982518	−0.491259	−	0.871014i	$-0.663463\pi$
$131$	−11.2454	−0.982518	−0.491259	+	0.871014i	$0.663463\pi$
$132$	9.21932	0.802439
$133$	4.17479	0.362000
$134$	−3.17336	−0.274136
$135$	4.60499	0.396335
$136$	−0.751849	−0.0644705
$137$	−10.8328	−0.925507	−0.462754	−	0.886487i	$-0.653139\pi$
$137$	−10.8328	−0.925507	−0.462754	+	0.886487i	$0.653139\pi$
$138$	13.3146	1.13341
$139$	−2.80757	−0.238135	−0.119067	−	0.992886i	$-0.537990\pi$
$139$	−2.80757	−0.238135	−0.119067	+	0.992886i	$0.537990\pi$
$140$	−9.24494	−0.781340
$141$	−0.859873	−0.0724144
$142$	5.02721	0.421874
$143$	−12.9960	−1.08678
$144$	1.14170	0.0951421
$145$	−10.9939	−0.912997
$146$	−0.902590	−0.0746989
$147$	−22.5313	−1.85835
$148$	13.0489	1.07261
$149$	22.1012	1.81060	0.905300	−	0.424772i	$-0.139646\pi$
$149$	22.1012	1.81060	0.905300	+	0.424772i	$0.139646\pi$
$150$	−4.13786	−0.337855
$151$	1.32134	0.107529	0.0537646	−	0.998554i	$-0.482878\pi$
$151$	1.32134	0.107529	0.0537646	+	0.998554i	$0.482878\pi$
$152$	2.65562	0.215399
$153$	−0.472133	−0.0381697
$154$	−10.0990	−0.813797
$155$	−2.44712	−0.196558
$156$	−12.6076	−1.00942
$157$	−6.99386	−0.558171	−0.279085	−	0.960266i	$-0.590031\pi$
$157$	−6.99386	−0.558171	−0.279085	+	0.960266i	$0.590031\pi$
$158$	6.11583	0.486549
$159$	22.1342	1.75536
$160$	−9.35616	−0.739670
$161$	32.7876	2.58403
$162$	8.80583	0.691851
$163$	−0.605356	−0.0474151	−0.0237076	−	0.999719i	$-0.507547\pi$
$163$	−0.605356	−0.0474151	−0.0237076	+	0.999719i	$0.507547\pi$
$164$	16.5889	1.29537
$165$	−10.6547	−0.829469
$166$	−12.6441	−0.981373
$167$	−18.3608	−1.42080	−0.710401	−	0.703797i	$-0.751486\pi$
$167$	−18.3608	−1.42080	−0.710401	+	0.703797i	$0.751486\pi$
$168$	−23.9524	−1.84797
$169$	4.77232	0.367101
$170$	0.355406	0.0272584
$171$	1.66763	0.127527
$172$	15.6503	1.19332
$173$	1.09706	0.0834079	0.0417039	−	0.999130i	$-0.486721\pi$
$173$	1.09706	0.0834079	0.0417039	+	0.999130i	$0.486721\pi$
$174$	−11.6506	−0.883232
$175$	−10.1896	−0.770264
$176$	2.11053	0.159087
$177$	−18.7782	−1.41146
$178$	−7.85231	−0.588555
$179$	13.1369	0.981898	0.490949	−	0.871188i	$-0.336650\pi$
$179$	13.1369	0.981898	0.490949	+	0.871188i	$0.336650\pi$
$180$	−3.69291	−0.275254
$181$	−0.384487	−0.0285787	−0.0142893	−	0.999898i	$-0.504549\pi$
$181$	−0.384487	−0.0285787	−0.0142893	+	0.999898i	$0.504549\pi$
$182$	13.8105	1.02371
$183$	−5.83440	−0.431291
$184$	20.8565	1.53756
$185$	−15.0805	−1.10874
$186$	−2.59330	−0.190150
$187$	−0.872775	−0.0638237
$188$	−0.550934	−0.0401810
$189$	12.0173	0.874131
$190$	−1.25534	−0.0910716
$191$	−15.8843	−1.14935	−0.574673	−	0.818383i	$-0.694871\pi$
$191$	−15.8843	−1.14935	−0.574673	+	0.818383i	$0.694871\pi$
$192$	−6.95679	−0.502063
$193$	9.43557	0.679187	0.339594	−	0.940572i	$-0.389710\pi$
$193$	9.43557	0.679187	0.339594	+	0.940572i	$0.389710\pi$
$194$	10.8260	0.777261
$195$	14.5706	1.04342
$196$	−14.4361	−1.03115
$197$	−24.2947	−1.73093	−0.865463	−	0.500973i	$-0.832976\pi$
$197$	−24.2947	−1.73093	−0.865463	+	0.500973i	$0.832976\pi$
$198$	−4.03405	−0.286688
$199$	−9.16300	−0.649548	−0.324774	−	0.945792i	$-0.605288\pi$
$199$	−9.16300	−0.649548	−0.324774	+	0.945792i	$0.605288\pi$
$200$	−6.48171	−0.458326
$201$	−8.73703	−0.616262
$202$	7.51947	0.529068
$203$	−28.6901	−2.01365
$204$	−0.846693	−0.0592804
$205$	−19.1717	−1.33901
$206$	14.1897	0.988643
$207$	13.0971	0.910311
$208$	−2.88620	−0.200122
$209$	3.08275	0.213238
$210$	11.3225	0.781328
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	14.1817	0.974003
$213$	13.8411	0.948378
$214$	−13.9532	−0.953819
$215$	−18.0870	−1.23352
$216$	7.64431	0.520130
$217$	−6.38609	−0.433516
$218$	−9.58116	−0.648918
$219$	−2.48505	−0.167924
$220$	−6.82664	−0.460252
$221$	1.19354	0.0802862
$222$	−15.9813	−1.07260
$223$	−19.1803	−1.28441	−0.642204	−	0.766534i	$-0.721980\pi$
$223$	−19.1803	−1.28441	−0.642204	+	0.766534i	$0.721980\pi$
$224$	−24.4161	−1.63137
$225$	−4.07028	−0.271352
$226$	−8.61850	−0.573294
$227$	−3.49656	−0.232075	−0.116037	−	0.993245i	$-0.537019\pi$
$227$	−3.49656	−0.232075	−0.116037	+	0.993245i	$0.537019\pi$
$228$	2.99062	0.198059
$229$	7.37149	0.487122	0.243561	−	0.969886i	$-0.421684\pi$
$229$	7.37149	0.487122	0.243561	+	0.969886i	$0.421684\pi$
$230$	−9.85905	−0.650086
$231$	−27.8049	−1.82943
$232$	−18.2500	−1.19817
$233$	−22.2866	−1.46004	−0.730020	−	0.683426i	$-0.760489\pi$
$233$	−22.2866	−1.46004	−0.730020	+	0.683426i	$0.760489\pi$
$234$	5.51666	0.360635
$235$	0.636711	0.0415344
$236$	−12.0315	−0.783184
$237$	16.8384	1.09377
$238$	0.927478	0.0601194
$239$	−12.6201	−0.816329	−0.408164	−	0.912909i	$-0.633831\pi$
$239$	−12.6201	−0.816329	−0.408164	+	0.912909i	$0.633831\pi$
$240$	−2.36624	−0.152740
$241$	−5.24954	−0.338153	−0.169076	−	0.985603i	$-0.554078\pi$
$241$	−5.24954	−0.338153	−0.169076	+	0.985603i	$0.554078\pi$
$242$	1.17445	0.0754962
$243$	15.6090	1.00131
$244$	−3.73818	−0.239313
$245$	16.6838	1.06589
$246$	−20.3168	−1.29535
$247$	−4.21572	−0.268240
$248$	−4.06224	−0.257953
$249$	−34.8123	−2.20614
$250$	9.34064	0.590754
$251$	−4.06438	−0.256542	−0.128271	−	0.991739i	$-0.540943\pi$
$251$	−4.06438	−0.256542	−0.128271	+	0.991739i	$0.540943\pi$
$252$	−9.63714	−0.607083
$253$	24.2110	1.52213
$254$	0.501668	0.0314775
$255$	0.978519	0.0612772
$256$	−13.6359	−0.852244
$257$	12.3073	0.767707	0.383854	−	0.923394i	$-0.374597\pi$
$257$	12.3073	0.767707	0.383854	+	0.923394i	$0.374597\pi$
$258$	−19.1673	−1.19331
$259$	−39.3546	−2.44538
$260$	9.33558	0.578968
$261$	−11.4603	−0.709377
$262$	8.82429	0.545167
$263$	20.0077	1.23373	0.616863	−	0.787070i	$-0.288403\pi$
$263$	20.0077	1.23373	0.616863	+	0.787070i	$0.288403\pi$
$264$	−17.6869	−1.08855
$265$	−16.3897	−1.00681
$266$	−3.27596	−0.200862
$267$	−21.6193	−1.32308
$268$	−5.59794	−0.341949
$269$	−17.8626	−1.08910	−0.544551	−	0.838728i	$-0.683300\pi$
$269$	−17.8626	−1.08910	−0.544551	+	0.838728i	$0.683300\pi$
$270$	−3.61354	−0.219913
$271$	22.3578	1.35814	0.679068	−	0.734075i	$-0.262384\pi$
$271$	22.3578	1.35814	0.679068	+	0.734075i	$0.262384\pi$
$272$	−0.193829	−0.0117526
$273$	38.0238	2.30130
$274$	8.50049	0.513533
$275$	−7.52422	−0.453728
$276$	23.4875	1.41378
$277$	−26.0701	−1.56640	−0.783202	−	0.621768i	$-0.786415\pi$
$277$	−26.0701	−1.56640	−0.783202	+	0.621768i	$0.786415\pi$
$278$	2.20310	0.132133
$279$	−2.55094	−0.152721
$280$	17.7360	1.05993
$281$	−4.13328	−0.246571	−0.123285	−	0.992371i	$-0.539343\pi$
$281$	−4.13328	−0.246571	−0.123285	+	0.992371i	$0.539343\pi$
$282$	0.674743	0.0401803
$283$	9.64851	0.573544	0.286772	−	0.957999i	$-0.407418\pi$
$283$	9.64851	0.573544	0.286772	+	0.957999i	$0.407418\pi$
$284$	8.86821	0.526231
$285$	−3.45624	−0.204730
$286$	10.1980	0.603018
$287$	−50.0310	−2.95324
$288$	−9.75308	−0.574706
$289$	−16.9198	−0.995285
$290$	8.62694	0.506592
$291$	29.8066	1.74729
$292$	−1.59221	−0.0931770
$293$	20.8739	1.21947	0.609734	−	0.792606i	$-0.291276\pi$
$293$	20.8739	1.21947	0.609734	+	0.792606i	$0.291276\pi$
$294$	17.6803	1.03114
$295$	13.9047	0.809565
$296$	−25.0338	−1.45506
$297$	8.87382	0.514911
$298$	−17.3428	−1.00464
$299$	−33.1091	−1.91475
$300$	−7.29937	−0.421429
$301$	−47.2003	−2.72058
$302$	−1.03686	−0.0596644
$303$	20.7029	1.18935
$304$	0.684627	0.0392660
$305$	4.32020	0.247374
$306$	0.370483	0.0211791
$307$	31.3103	1.78697	0.893487	−	0.449089i	$-0.148251\pi$
$307$	31.3103	1.78697	0.893487	+	0.449089i	$0.148251\pi$
$308$	−17.8150	−1.01510
$309$	39.0677	2.22248
$310$	1.92026	0.109063
$311$	−18.0209	−1.02187	−0.510936	−	0.859619i	$-0.670701\pi$
$311$	−18.0209	−1.02187	−0.510936	+	0.859619i	$0.670701\pi$
$312$	24.1872	1.36933
$313$	−5.87918	−0.332311	−0.166156	−	0.986100i	$-0.553135\pi$
$313$	−5.87918	−0.332311	−0.166156	+	0.986100i	$0.553135\pi$
$314$	5.48808	0.309710
$315$	11.1376	0.627532
$316$	10.7886	0.606906
$317$	23.1078	1.29786	0.648932	−	0.760846i	$-0.275216\pi$
$317$	23.1078	1.29786	0.648932	+	0.760846i	$0.275216\pi$
$318$	−17.3687	−0.973988
$319$	−21.1853	−1.18615
$320$	5.15130	0.287967
$321$	−38.4165	−2.14420
$322$	−25.7285	−1.43379
$323$	−0.283116	−0.0157530
$324$	15.5339	0.862992
$325$	10.2895	0.570761
$326$	0.475023	0.0263091
$327$	−26.3793	−1.45878
$328$	−31.8251	−1.75725
$329$	1.66158	0.0916058
$330$	8.36076	0.460245
$331$	29.7840	1.63708	0.818539	−	0.574451i	$-0.194784\pi$
$331$	29.7840	1.63708	0.818539	+	0.574451i	$0.194784\pi$
$332$	−22.3048	−1.22413
$333$	−15.7203	−0.861467
$334$	14.4077	0.788356
$335$	6.46951	0.353467
$336$	−6.17500	−0.336874
$337$	−5.92920	−0.322984	−0.161492	−	0.986874i	$-0.551631\pi$
$337$	−5.92920	−0.322984	−0.161492	+	0.986874i	$0.551631\pi$
$338$	−3.74484	−0.203692
$339$	−23.7288	−1.28877
$340$	0.626952	0.0340012
$341$	−4.71561	−0.255364
$342$	−1.30859	−0.0707605
$343$	14.3149	0.772932
$344$	−30.0245	−1.61881
$345$	−27.1443	−1.46140
$346$	−0.860863	−0.0462803
$347$	−7.97597	−0.428173	−0.214086	−	0.976815i	$-0.568677\pi$
$347$	−7.97597	−0.428173	−0.214086	+	0.976815i	$0.568677\pi$
$348$	−20.5522	−1.10171
$349$	−26.3415	−1.41003	−0.705013	−	0.709194i	$-0.749059\pi$
$349$	−26.3415	−1.41003	−0.705013	+	0.709194i	$0.749059\pi$
$350$	7.99581	0.427394
$351$	−12.1351	−0.647726
$352$	−18.0293	−0.960966
$353$	−2.68512	−0.142914	−0.0714572	−	0.997444i	$-0.522765\pi$
$353$	−2.68512	−0.142914	−0.0714572	+	0.997444i	$0.522765\pi$
$354$	14.7353	0.783172
$355$	−10.2489	−0.543957
$356$	−13.8518	−0.734145
$357$	2.55357	0.135149
$358$	−10.3085	−0.544822
$359$	−20.5067	−1.08230	−0.541152	−	0.840925i	$-0.682011\pi$
$359$	−20.5067	−1.08230	−0.541152	+	0.840925i	$0.682011\pi$
$360$	7.08470	0.373397
$361$	1.00000	0.0526316
$362$	0.301707	0.0158574
$363$	3.23353	0.169716
$364$	24.3624	1.27694
$365$	1.84011	0.0963156
$366$	4.57825	0.239309
$367$	14.7830	0.771666	0.385833	−	0.922569i	$-0.373914\pi$
$367$	14.7830	0.771666	0.385833	+	0.922569i	$0.373914\pi$
$368$	5.37686	0.280288
$369$	−19.9850	−1.04038
$370$	11.8337	0.615205
$371$	−42.7711	−2.22056
$372$	−4.57469	−0.237186
$373$	−0.242901	−0.0125770	−0.00628848	−	0.999980i	$-0.502002\pi$
$373$	−0.242901	−0.0125770	−0.00628848	+	0.999980i	$0.502002\pi$
$374$	0.684867	0.0354136
$375$	25.7171	1.32802
$376$	1.05694	0.0545077
$377$	28.9714	1.49210
$378$	−9.42999	−0.485026
$379$	29.8260	1.53206	0.766030	−	0.642804i	$-0.222229\pi$
$379$	29.8260	1.53206	0.766030	+	0.642804i	$0.222229\pi$
$380$	−2.21447	−0.113600
$381$	1.38121	0.0707618
$382$	12.4644	0.637734
$383$	−30.4660	−1.55674	−0.778370	−	0.627806i	$-0.783953\pi$
$383$	−30.4660	−1.55674	−0.778370	+	0.627806i	$0.783953\pi$
$384$	−19.8119	−1.01102
$385$	20.5887	1.04930
$386$	−7.40410	−0.376859
$387$	−18.8543	−0.958417
$388$	19.0975	0.969530
$389$	30.2035	1.53138	0.765688	−	0.643212i	$-0.222399\pi$
$389$	30.2035	1.53138	0.765688	+	0.643212i	$0.222399\pi$
$390$	−11.4335	−0.578959
$391$	−2.22351	−0.112448
$392$	27.6951	1.39882
$393$	24.2954	1.22554
$394$	19.0641	0.960433
$395$	−12.4683	−0.627349
$396$	−7.11624	−0.357605
$397$	−29.0058	−1.45576	−0.727881	−	0.685704i	$-0.759495\pi$
$397$	−29.0058	−1.45576	−0.727881	+	0.685704i	$0.759495\pi$
$398$	7.19021	0.360412
$399$	−9.01951	−0.451540
$400$	−1.67100	−0.0835502
$401$	−7.01949	−0.350537	−0.175268	−	0.984521i	$-0.556079\pi$
$401$	−7.01949	−0.350537	−0.175268	+	0.984521i	$0.556079\pi$
$402$	6.85595	0.341944
$403$	6.44870	0.321233
$404$	13.2647	0.659942
$405$	−17.9524	−0.892062
$406$	22.5131	1.11731
$407$	−29.0602	−1.44046
$408$	1.62435	0.0804171
$409$	18.8247	0.930821	0.465410	−	0.885095i	$-0.345907\pi$
$409$	18.8247	0.930821	0.465410	+	0.885095i	$0.345907\pi$
$410$	15.0440	0.742971
$411$	23.4039	1.15443
$412$	25.0312	1.23320
$413$	36.2862	1.78553
$414$	−10.2773	−0.505101
$415$	25.7775	1.26537
$416$	24.6555	1.20884
$417$	6.06567	0.297037
$418$	−2.41903	−0.118319
$419$	5.06813	0.247595	0.123797	−	0.992308i	$-0.460493\pi$
$419$	5.06813	0.247595	0.123797	+	0.992308i	$0.460493\pi$
$420$	19.9734	0.974603
$421$	−15.0934	−0.735605	−0.367803	−	0.929904i	$-0.619890\pi$
$421$	−15.0934	−0.735605	−0.367803	+	0.929904i	$0.619890\pi$
$422$	−0.784700	−0.0381986
$423$	0.663722	0.0322713
$424$	−27.2070	−1.32129
$425$	0.691017	0.0335192
$426$	−10.8611	−0.526223
$427$	11.2741	0.545593
$428$	−24.6140	−1.18976
$429$	28.0775	1.35559
$430$	14.1929	0.684440
$431$	−16.6358	−0.801318	−0.400659	−	0.916227i	$-0.631219\pi$
$431$	−16.6358	−0.801318	−0.400659	+	0.916227i	$0.631219\pi$
$432$	1.97073	0.0948167
$433$	25.7494	1.23744	0.618719	−	0.785613i	$-0.287652\pi$
$433$	25.7494	1.23744	0.618719	+	0.785613i	$0.287652\pi$
$434$	5.01116	0.240544
$435$	23.7521	1.13882
$436$	−16.9016	−0.809439
$437$	7.85371	0.375694
$438$	1.95002	0.0931755
$439$	−19.4237	−0.927045	−0.463522	−	0.886085i	$-0.653415\pi$
$439$	−19.4237	−0.927045	−0.463522	+	0.886085i	$0.653415\pi$
$440$	13.0966	0.624357
$441$	17.3915	0.828169
$442$	−0.936571	−0.0445481
$443$	36.6021	1.73902	0.869510	−	0.493916i	$-0.164435\pi$
$443$	36.6021	1.73902	0.869510	+	0.493916i	$0.164435\pi$
$444$	−28.1917	−1.33792
$445$	16.0085	0.758874
$446$	15.0508	0.712675
$447$	−47.7490	−2.25845
$448$	13.4430	0.635121
$449$	11.0473	0.521355	0.260677	−	0.965426i	$-0.416054\pi$
$449$	11.0473	0.521355	0.260677	+	0.965426i	$0.416054\pi$
$450$	3.19395	0.150564
$451$	−36.9438	−1.73962
$452$	−15.2034	−0.715108
$453$	−2.85472	−0.134126
$454$	2.74375	0.128771
$455$	−28.1555	−1.31995
$456$	−5.73738	−0.268678
$457$	8.67842	0.405960	0.202980	−	0.979183i	$-0.434937\pi$
$457$	8.67842	0.405960	0.202980	+	0.979183i	$0.434937\pi$
$458$	−5.78441	−0.270288
$459$	−0.814962	−0.0380392
$460$	−17.3918	−0.810896
$461$	32.9103	1.53278	0.766392	−	0.642373i	$-0.222050\pi$
$461$	32.9103	1.53278	0.766392	+	0.642373i	$0.222050\pi$
$462$	21.8185	1.01509
$463$	13.7592	0.639443	0.319721	−	0.947512i	$-0.396411\pi$
$463$	13.7592	0.639443	0.319721	+	0.947512i	$0.396411\pi$
$464$	−4.70491	−0.218420
$465$	5.28694	0.245176
$466$	17.4883	0.810128
$467$	−9.25902	−0.428457	−0.214228	−	0.976784i	$-0.568724\pi$
$467$	−9.25902	−0.428457	−0.214228	+	0.976784i	$0.568724\pi$
$468$	9.73162	0.449844
$469$	16.8830	0.779586
$470$	−0.499627	−0.0230461
$471$	15.1100	0.696233
$472$	23.0819	1.06243
$473$	−34.8536	−1.60257
$474$	−13.2131	−0.606896
$475$	−2.44075	−0.111989
$476$	1.63611	0.0749910
$477$	−17.0850	−0.782269
$478$	9.90302	0.452954
$479$	−26.5380	−1.21255	−0.606275	−	0.795255i	$-0.707337\pi$
$479$	−26.5380	−1.21255	−0.606275	+	0.795255i	$0.707337\pi$
$480$	20.2137	0.922626
$481$	39.7404	1.81201
$482$	4.11932	0.187630
$483$	−70.8367	−3.22318
$484$	2.07177	0.0941715
$485$	−22.0709	−1.00219
$486$	−12.2483	−0.555596
$487$	5.65525	0.256264	0.128132	−	0.991757i	$-0.459102\pi$
$487$	5.65525	0.256264	0.128132	+	0.991757i	$0.459102\pi$
$488$	7.17155	0.324641
$489$	1.30785	0.0591431
$490$	−13.0918	−0.591425
$491$	−5.97748	−0.269760	−0.134880	−	0.990862i	$-0.543065\pi$
$491$	−5.97748	−0.269760	−0.134880	+	0.990862i	$0.543065\pi$
$492$	−35.8398	−1.61578
$493$	1.94564	0.0876271
$494$	3.30808	0.148838
$495$	8.22420	0.369650
$496$	−1.04726	−0.0470233
$497$	−26.7459	−1.19972
$498$	27.3172	1.22411
$499$	13.1794	0.589991	0.294995	−	0.955499i	$-0.404682\pi$
$499$	13.1794	0.589991	0.294995	+	0.955499i	$0.404682\pi$
$500$	16.4773	0.736887
$501$	39.6680	1.77223
$502$	3.18932	0.142346
$503$	−3.96736	−0.176896	−0.0884480	−	0.996081i	$-0.528191\pi$
$503$	−3.96736	−0.176896	−0.0884480	+	0.996081i	$0.528191\pi$
$504$	18.4885	0.823541
$505$	−15.3299	−0.682172
$506$	−18.9984	−0.844581
$507$	−10.3105	−0.457903
$508$	0.884965	0.0392640
$509$	−23.5958	−1.04587	−0.522933	−	0.852374i	$-0.675162\pi$
$509$	−23.5958	−1.04587	−0.522933	+	0.852374i	$0.675162\pi$
$510$	−0.767844	−0.0340007
$511$	4.80200	0.212428
$512$	−7.64023	−0.337654
$513$	2.87854	0.127091
$514$	−9.65753	−0.425975
$515$	−28.9285	−1.27474
$516$	−33.8120	−1.48849
$517$	1.22694	0.0539608
$518$	30.8816	1.35686
$519$	−2.37016	−0.104039
$520$	−17.9099	−0.785402
$521$	−5.55998	−0.243587	−0.121793	−	0.992555i	$-0.538865\pi$
$521$	−5.55998	−0.243587	−0.121793	+	0.992555i	$0.538865\pi$
$522$	8.99292	0.393610
$523$	−13.5329	−0.591754	−0.295877	−	0.955226i	$-0.595612\pi$
$523$	−13.5329	−0.591754	−0.295877	+	0.955226i	$0.595612\pi$
$524$	15.5664	0.680023
$525$	22.0144	0.960788
$526$	−15.7000	−0.684554
$527$	0.433077	0.0188651
$528$	−4.55974	−0.198437
$529$	38.6808	1.68177
$530$	12.8610	0.558647
$531$	14.4946	0.629013
$532$	−5.77894	−0.250549
$533$	50.5215	2.18833
$534$	16.9647	0.734134
$535$	28.4463	1.22984
$536$	10.7394	0.463872
$537$	−28.3819	−1.22477
$538$	14.0168	0.604307
$539$	32.1496	1.38478
$540$	−6.37444	−0.274312
$541$	−41.3546	−1.77797	−0.888987	−	0.457932i	$-0.848590\pi$
$541$	−41.3546	−1.77797	−0.888987	+	0.457932i	$0.848590\pi$
$542$	−17.5441	−0.753585
$543$	0.830673	0.0356476
$544$	1.65579	0.0709916
$545$	19.5331	0.836705
$546$	−29.8373	−1.27692
$547$	44.2848	1.89348	0.946741	−	0.321996i	$-0.104354\pi$
$547$	44.2848	1.89348	0.946741	+	0.321996i	$0.104354\pi$
$548$	14.9952	0.640565
$549$	4.50347	0.192204
$550$	5.90426	0.251758
$551$	−6.87222	−0.292766
$552$	−45.0598	−1.91787
$553$	−32.5377	−1.38364
$554$	20.4573	0.869145
$555$	32.5810	1.38299
$556$	3.88637	0.164819
$557$	1.71125	0.0725081	0.0362540	−	0.999343i	$-0.488457\pi$
$557$	1.71125	0.0725081	0.0362540	+	0.999343i	$0.488457\pi$
$558$	2.00172	0.0847397
$559$	47.6631	2.01593
$560$	4.57241	0.193219
$561$	1.88561	0.0796103
$562$	3.24338	0.136814
$563$	0.420186	0.0177087	0.00885436	−	0.999961i	$-0.497182\pi$
$563$	0.420186	0.0177087	0.00885436	+	0.999961i	$0.497182\pi$
$564$	1.19028	0.0501196
$565$	17.5705	0.739196
$566$	−7.57118	−0.318241
$567$	−46.8491	−1.96748
$568$	−17.0133	−0.713862
$569$	−20.7853	−0.871366	−0.435683	−	0.900100i	$-0.643493\pi$
$569$	−20.7853	−0.871366	−0.435683	+	0.900100i	$0.643493\pi$
$570$	2.71212	0.113598
$571$	−17.7308	−0.742013	−0.371006	−	0.928630i	$-0.620987\pi$
$571$	−17.7308	−0.742013	−0.371006	+	0.928630i	$0.620987\pi$
$572$	17.9897	0.752185
$573$	34.3175	1.43363
$574$	39.2593	1.63865
$575$	−19.1690	−0.799402
$576$	5.36983	0.223743
$577$	−8.41486	−0.350315	−0.175158	−	0.984540i	$-0.556044\pi$
$577$	−8.41486	−0.350315	−0.175158	+	0.984540i	$0.556044\pi$
$578$	13.2770	0.552251
$579$	−20.3853	−0.847183
$580$	15.2183	0.631906
$581$	67.2697	2.79082
$582$	−23.3892	−0.969515
$583$	−31.5830	−1.30803
$584$	3.05459	0.126400
$585$	−11.2468	−0.464997
$586$	−16.3798	−0.676642
$587$	30.8121	1.27175	0.635876	−	0.771791i	$-0.280639\pi$
$587$	30.8121	1.27175	0.635876	+	0.771791i	$0.280639\pi$
$588$	31.1889	1.28621
$589$	−1.52968	−0.0630293
$590$	−10.9110	−0.449201
$591$	52.4880	2.15907
$592$	−6.45378	−0.265249
$593$	29.3866	1.20676	0.603381	−	0.797453i	$-0.293820\pi$
$593$	29.3866	1.20676	0.603381	+	0.797453i	$0.293820\pi$
$594$	−6.96329	−0.285707
$595$	−1.89084	−0.0775170
$596$	−30.5935	−1.25316
$597$	19.7964	0.810212
$598$	25.9807	1.06243
$599$	32.8491	1.34218	0.671090	−	0.741376i	$-0.265827\pi$
$599$	32.8491	1.34218	0.671090	+	0.741376i	$0.265827\pi$
$600$	14.0035	0.571692
$601$	21.8660	0.891933	0.445966	−	0.895050i	$-0.352860\pi$
$601$	21.8660	0.891933	0.445966	+	0.895050i	$0.352860\pi$
$602$	37.0381	1.50956
$603$	6.74397	0.274636
$604$	−1.82906	−0.0744234
$605$	−2.39434	−0.0973436
$606$	−16.2456	−0.659932
$607$	−9.51514	−0.386207	−0.193104	−	0.981178i	$-0.561855\pi$
$607$	−9.51514	−0.386207	−0.193104	+	0.981178i	$0.561855\pi$
$608$	−5.84846	−0.237187
$609$	61.9841	2.51172
$610$	−3.39006	−0.137260
$611$	−1.67787	−0.0678793
$612$	0.653549	0.0264181
$613$	−4.13152	−0.166871	−0.0834354	−	0.996513i	$-0.526589\pi$
$613$	−4.13152	−0.166871	−0.0834354	+	0.996513i	$0.526589\pi$
$614$	−24.5692	−0.991533
$615$	41.4198	1.67021
$616$	34.1773	1.37704
$617$	−41.6291	−1.67593	−0.837963	−	0.545727i	$-0.816253\pi$
$617$	−41.6291	−1.67593	−0.837963	+	0.545727i	$0.816253\pi$
$618$	−30.6564	−1.23318
$619$	−21.7546	−0.874390	−0.437195	−	0.899367i	$-0.644028\pi$
$619$	−21.7546	−0.874390	−0.437195	+	0.899367i	$0.644028\pi$
$620$	3.38742	0.136042
$621$	22.6072	0.907198
$622$	14.1410	0.567003
$623$	41.7762	1.67373
$624$	6.23554	0.249621
$625$	−6.83896	−0.273558
$626$	4.61340	0.184388
$627$	−6.66018	−0.265982
$628$	9.68122	0.386323
$629$	2.66886	0.106414
$630$	−8.73966	−0.348197
$631$	−29.8491	−1.18827	−0.594137	−	0.804364i	$-0.702506\pi$
$631$	−29.8491	−1.18827	−0.594137	+	0.804364i	$0.702506\pi$
$632$	−20.6975	−0.823301
$633$	−2.16047	−0.0858710
$634$	−18.1327	−0.720142
$635$	−1.02275	−0.0405865
$636$	−30.6391	−1.21492
$637$	−43.9653	−1.74197
$638$	16.6241	0.658155
$639$	−10.6837	−0.422642
$640$	14.6701	0.579887
$641$	27.9733	1.10488	0.552440	−	0.833553i	$-0.313697\pi$
$641$	27.9733	1.10488	0.552440	+	0.833553i	$0.313697\pi$
$642$	30.1454	1.18974
$643$	−38.4074	−1.51464	−0.757321	−	0.653043i	$-0.773492\pi$
$643$	−38.4074	−1.51464	−0.757321	+	0.653043i	$0.773492\pi$
$644$	−45.3861	−1.78846
$645$	39.0764	1.53863
$646$	0.222161	0.00874082
$647$	−5.92907	−0.233096	−0.116548	−	0.993185i	$-0.537183\pi$
$647$	−5.92907	−0.233096	−0.116548	+	0.993185i	$0.537183\pi$
$648$	−29.8011	−1.17070
$649$	26.7944	1.05177
$650$	−8.07420	−0.316696
$651$	13.7970	0.540745
$652$	0.837961	0.0328171
$653$	−8.08334	−0.316325	−0.158163	−	0.987413i	$-0.550557\pi$
$653$	−8.08334	−0.316325	−0.158163	+	0.987413i	$0.550557\pi$
$654$	20.6998	0.809427
$655$	−17.9900	−0.702929
$656$	−8.20461	−0.320336
$657$	1.91817	0.0748349
$658$	−1.30384	−0.0508290
$659$	26.2596	1.02293	0.511463	−	0.859305i	$-0.329104\pi$
$659$	26.2596	1.02293	0.511463	+	0.859305i	$0.329104\pi$
$660$	14.7487	0.574094
$661$	−14.9417	−0.581164	−0.290582	−	0.956850i	$-0.593849\pi$
$661$	−14.9417	−0.581164	−0.290582	+	0.956850i	$0.593849\pi$
$662$	−23.3715	−0.908360
$663$	−2.57861	−0.100145
$664$	42.7908	1.66060
$665$	6.67868	0.258988
$666$	12.3357	0.477999
$667$	−53.9725	−2.08982
$668$	25.4159	0.983369
$669$	41.4385	1.60210
$670$	−5.07663	−0.196127
$671$	8.32502	0.321384
$672$	52.7503	2.03489
$673$	−45.6147	−1.75832	−0.879158	−	0.476530i	$-0.841894\pi$
$673$	−45.6147	−1.75832	−0.879158	+	0.476530i	$0.841894\pi$
$674$	4.65264	0.179213
$675$	−7.02581	−0.270424
$676$	−6.60606	−0.254079
$677$	−13.8283	−0.531465	−0.265732	−	0.964047i	$-0.585614\pi$
$677$	−13.8283	−0.531465	−0.265732	+	0.964047i	$0.585614\pi$
$678$	18.6200	0.715097
$679$	−57.5969	−2.21037
$680$	−1.20278	−0.0461245
$681$	7.55422	0.289478
$682$	3.70034	0.141693
$683$	−15.5367	−0.594496	−0.297248	−	0.954800i	$-0.596069\pi$
$683$	−15.5367	−0.594496	−0.297248	+	0.954800i	$0.596069\pi$
$684$	−2.30841	−0.0882643
$685$	−17.3299	−0.662142
$686$	−11.2329	−0.428874
$687$	−15.9259	−0.607610
$688$	−7.74040	−0.295100
$689$	43.1904	1.64542
$690$	21.3002	0.810884
$691$	9.29464	0.353585	0.176792	−	0.984248i	$-0.443428\pi$
$691$	9.29464	0.353585	0.176792	+	0.984248i	$0.443428\pi$
$692$	−1.51860	−0.0577285
$693$	21.4621	0.815278
$694$	6.25875	0.237579
$695$	−4.49145	−0.170370
$696$	39.4286	1.49454
$697$	3.39288	0.128515
$698$	20.6701	0.782377
$699$	48.1494	1.82118
$700$	14.1050	0.533117
$701$	−36.1617	−1.36581	−0.682905	−	0.730508i	$-0.739284\pi$
$701$	−36.1617	−1.36581	−0.682905	+	0.730508i	$0.739284\pi$
$702$	9.52245	0.359402
$703$	−9.42672	−0.355535
$704$	9.92656	0.374121
$705$	−1.37559	−0.0518079
$706$	2.10701	0.0792985
$707$	−40.0054	−1.50456
$708$	25.9937	0.976903
$709$	23.7161	0.890676	0.445338	−	0.895362i	$-0.353083\pi$
$709$	23.7161	0.890676	0.445338	+	0.895362i	$0.353083\pi$
$710$	8.04235	0.301824
$711$	−12.9973	−0.487435
$712$	26.5741	0.995908
$713$	−12.0137	−0.449915
$714$	−2.00379	−0.0749899
$715$	−20.7905	−0.777522
$716$	−18.1847	−0.679594
$717$	27.2654	1.01825
$718$	16.0916	0.600534
$719$	−23.0454	−0.859448	−0.429724	−	0.902960i	$-0.641389\pi$
$719$	−23.0454	−0.859448	−0.429724	+	0.902960i	$0.641389\pi$
$720$	1.82646	0.0680681
$721$	−75.4925	−2.81149
$722$	−0.784700	−0.0292035
$723$	11.3415	0.421794
$724$	0.532224	0.0197800
$725$	16.7734	0.622948
$726$	−2.53735	−0.0941701
$727$	39.8595	1.47831	0.739153	−	0.673537i	$-0.235226\pi$
$727$	39.8595	1.47831	0.739153	+	0.673537i	$0.235226\pi$
$728$	−46.7383	−1.73224
$729$	−0.0569724	−0.00211009
$730$	−1.44393	−0.0534423
$731$	3.20092	0.118390
$732$	8.07624	0.298506
$733$	−11.0601	−0.408514	−0.204257	−	0.978917i	$-0.565478\pi$
$733$	−11.0601	−0.408514	−0.204257	+	0.978917i	$0.565478\pi$
$734$	−11.6002	−0.428172
$735$	−36.0448	−1.32953
$736$	−45.9322	−1.69308
$737$	12.4667	0.459218
$738$	15.6822	0.577271
$739$	40.3621	1.48474	0.742372	−	0.669988i	$-0.233701\pi$
$739$	40.3621	1.48474	0.742372	+	0.669988i	$0.233701\pi$
$740$	20.8752	0.767386
$741$	9.10794	0.334589
$742$	33.5625	1.23212
$743$	−2.61929	−0.0960925	−0.0480462	−	0.998845i	$-0.515299\pi$
$743$	−2.61929	−0.0960925	−0.0480462	+	0.998845i	$0.515299\pi$
$744$	8.77635	0.321757
$745$	35.3567	1.29537
$746$	0.190605	0.00697854
$747$	26.8710	0.983160
$748$	1.20814	0.0441738
$749$	74.2342	2.71246
$750$	−20.1802	−0.736876
$751$	−49.3830	−1.80201	−0.901007	−	0.433806i	$-0.857170\pi$
$751$	−49.3830	−1.80201	−0.901007	+	0.433806i	$0.857170\pi$
$752$	0.272483	0.00993644
$753$	8.78098	0.319997
$754$	−22.7339	−0.827918
$755$	2.11383	0.0769303
$756$	−16.6349	−0.605006
$757$	27.5371	1.00085	0.500427	−	0.865779i	$-0.333176\pi$
$757$	27.5371	1.00085	0.500427	+	0.865779i	$0.333176\pi$
$758$	−23.4045	−0.850090
$759$	−52.3071	−1.89863
$760$	4.24836	0.154104
$761$	1.18758	0.0430498	0.0215249	−	0.999768i	$-0.493148\pi$
$761$	1.18758	0.0430498	0.0215249	+	0.999768i	$0.493148\pi$
$762$	−1.08384	−0.0392634
$763$	50.9741	1.84538
$764$	21.9877	0.795488
$765$	−0.755302	−0.0273080
$766$	23.9067	0.863784
$767$	−36.6420	−1.32306
$768$	29.4600	1.06304
$769$	24.5647	0.885824	0.442912	−	0.896565i	$-0.353945\pi$
$769$	24.5647	0.885824	0.442912	+	0.896565i	$0.353945\pi$
$770$	−16.1559	−0.582220
$771$	−26.5895	−0.957598
$772$	−13.0612	−0.470081
$773$	8.50473	0.305894	0.152947	−	0.988234i	$-0.451124\pi$
$773$	8.50473	0.305894	0.152947	+	0.988234i	$0.451124\pi$
$774$	14.7950	0.531794
$775$	3.73357	0.134114
$776$	−36.6378	−1.31522
$777$	85.0244	3.05023
$778$	−23.7007	−0.849710
$779$	−11.9841	−0.429374
$780$	−20.1692	−0.722175
$781$	−19.7497	−0.706700
$782$	1.74479	0.0623936
$783$	−19.7820	−0.706950
$784$	7.13990	0.254996
$785$	−11.1885	−0.399336
$786$	−19.0646	−0.680013
$787$	10.6525	0.379720	0.189860	−	0.981811i	$-0.439197\pi$
$787$	10.6525	0.379720	0.189860	+	0.981811i	$0.439197\pi$
$788$	33.6298	1.19801
$789$	−43.2260	−1.53889
$790$	9.78389	0.348095
$791$	45.8525	1.63033
$792$	13.6522	0.485111
$793$	−11.3846	−0.404281
$794$	22.7609	0.807754
$795$	35.4095	1.25585
$796$	12.6838	0.449567
$797$	−47.3615	−1.67763	−0.838815	−	0.544416i	$-0.816751\pi$
$797$	−47.3615	−1.67763	−0.838815	+	0.544416i	$0.816751\pi$
$798$	7.07761	0.250545
$799$	−0.112681	−0.00398637
$800$	14.2747	0.504685
$801$	16.6876	0.589627
$802$	5.50820	0.194501
$803$	3.54588	0.125132
$804$	12.0942	0.426529
$805$	52.4525	1.84871
$806$	−5.06030	−0.178241
$807$	38.5916	1.35849
$808$	−25.4477	−0.895248
$809$	−56.1648	−1.97465	−0.987325	−	0.158712i	$-0.949266\pi$
$809$	−56.1648	−1.97465	−0.987325	+	0.158712i	$0.949266\pi$
$810$	14.0873	0.494976
$811$	25.6752	0.901579	0.450789	−	0.892630i	$-0.351143\pi$
$811$	25.6752	0.901579	0.450789	+	0.892630i	$0.351143\pi$
$812$	39.7141	1.39369
$813$	−48.3033	−1.69407
$814$	22.8035	0.799263
$815$	−0.968426	−0.0339225
$816$	0.418762	0.0146596
$817$	−11.3060	−0.395548
$818$	−14.7717	−0.516482
$819$	−29.3499	−1.02557
$820$	26.5383	0.926758
$821$	−50.1658	−1.75080	−0.875399	−	0.483402i	$-0.839401\pi$
$821$	−50.1658	−1.75080	−0.875399	+	0.483402i	$0.839401\pi$
$822$	−18.3651	−0.640555
$823$	−20.0795	−0.699926	−0.349963	−	0.936764i	$-0.613806\pi$
$823$	−20.0795	−0.699926	−0.349963	+	0.936764i	$0.613806\pi$
$824$	−48.0214	−1.67290
$825$	16.2559	0.565956
$826$	−28.4738	−0.990730
$827$	−39.2122	−1.36354	−0.681771	−	0.731565i	$-0.738790\pi$
$827$	−39.2122	−1.36354	−0.681771	+	0.731565i	$0.738790\pi$
$828$	−18.1296	−0.630047
$829$	4.35677	0.151317	0.0756584	−	0.997134i	$-0.475894\pi$
$829$	4.35677	0.151317	0.0756584	+	0.997134i	$0.475894\pi$
$830$	−20.2276	−0.702110
$831$	56.3238	1.95385
$832$	−13.5748	−0.470621
$833$	−2.95259	−0.102301
$834$	−4.75973	−0.164816
$835$	−29.3730	−1.01649
$836$	−4.26728	−0.147587
$837$	−4.40324	−0.152198
$838$	−3.97697	−0.137382
$839$	13.1809	0.455055	0.227528	−	0.973772i	$-0.426936\pi$
$839$	13.1809	0.455055	0.227528	+	0.973772i	$0.426936\pi$
$840$	−38.3182	−1.32210
$841$	18.2274	0.628532
$842$	11.8438	0.408163
$843$	8.92982	0.307559
$844$	−1.38425	−0.0476477
$845$	7.63459	0.262638
$846$	−0.520823	−0.0179063
$847$	−6.24833	−0.214695
$848$	−7.01405	−0.240864
$849$	−20.8453	−0.715409
$850$	−0.542241	−0.0185987
$851$	−74.0348	−2.53788
$852$	−19.1595	−0.656394
$853$	−3.19736	−0.109476	−0.0547378	−	0.998501i	$-0.517432\pi$
$853$	−3.19736	−0.109476	−0.0547378	+	0.998501i	$0.517432\pi$
$854$	−8.84680	−0.302731
$855$	2.66782	0.0912374
$856$	47.2210	1.61398
$857$	17.7806	0.607373	0.303687	−	0.952772i	$-0.401782\pi$
$857$	17.7806	0.607373	0.303687	+	0.952772i	$0.401782\pi$
$858$	−22.0324	−0.752174
$859$	21.6562	0.738899	0.369450	−	0.929251i	$-0.379546\pi$
$859$	21.6562	0.738899	0.369450	+	0.929251i	$0.379546\pi$
$860$	25.0368	0.853748
$861$	108.090	3.68371
$862$	13.0541	0.444625
$863$	44.0499	1.49948	0.749738	−	0.661735i	$-0.230180\pi$
$863$	44.0499	1.49948	0.749738	+	0.661735i	$0.230180\pi$
$864$	−16.8351	−0.572740
$865$	1.75504	0.0596730
$866$	−20.2056	−0.686613
$867$	36.5548	1.24147
$868$	8.83991	0.300046
$869$	−24.0264	−0.815041
$870$	−18.6383	−0.631896
$871$	−17.0485	−0.577668
$872$	32.4250	1.09805
$873$	−23.0072	−0.778676
$874$	−6.16281	−0.208460
$875$	−49.6944	−1.67998
$876$	3.43992	0.116224
$877$	−28.1790	−0.951536	−0.475768	−	0.879571i	$-0.657830\pi$
$877$	−28.1790	−0.951536	−0.475768	+	0.879571i	$0.657830\pi$
$878$	15.2418	0.514386
$879$	−45.0975	−1.52110
$880$	3.37635	0.113817
$881$	−31.8622	−1.07346	−0.536732	−	0.843753i	$-0.680341\pi$
$881$	−31.8622	−1.07346	−0.536732	+	0.843753i	$0.680341\pi$
$882$	−13.6471	−0.459523
$883$	3.32464	0.111883	0.0559415	−	0.998434i	$-0.482184\pi$
$883$	3.32464	0.111883	0.0559415	+	0.998434i	$0.482184\pi$
$884$	−1.65215	−0.0555679
$885$	−30.0408	−1.00981
$886$	−28.7217	−0.964924
$887$	53.0496	1.78123	0.890616	−	0.454757i	$-0.150274\pi$
$887$	53.0496	1.78123	0.890616	+	0.454757i	$0.150274\pi$
$888$	54.0847	1.81496
$889$	−2.66899	−0.0895152
$890$	−12.5618	−0.421074
$891$	−34.5943	−1.15895
$892$	26.5502	0.888968
$893$	0.398003	0.0133187
$894$	37.4686	1.25314
$895$	21.0159	0.702485
$896$	38.2835	1.27896
$897$	71.5312	2.38836
$898$	−8.66883	−0.289283
$899$	10.5123	0.350604
$900$	5.63426	0.187809
$901$	2.90055	0.0966313
$902$	28.9898	0.965256
$903$	101.975	3.39351
$904$	29.1671	0.970084
$905$	−0.615089	−0.0204462
$906$	2.24010	0.0744222
$907$	52.1833	1.73272	0.866360	−	0.499420i	$-0.166454\pi$
$907$	52.1833	1.73272	0.866360	+	0.499420i	$0.166454\pi$
$908$	4.84010	0.160624
$909$	−15.9802	−0.530031
$910$	22.0936	0.732396
$911$	−42.9378	−1.42259	−0.711297	−	0.702891i	$-0.751892\pi$
$911$	−42.9378	−1.42259	−0.711297	+	0.702891i	$0.751892\pi$
$912$	−1.47912	−0.0489784
$913$	49.6732	1.64394
$914$	−6.80996	−0.225254
$915$	−9.33366	−0.308561
$916$	−10.2039	−0.337148
$917$	−46.9473	−1.55034
$918$	0.639501	0.0211067
$919$	27.8787	0.919634	0.459817	−	0.888014i	$-0.347915\pi$
$919$	27.8787	0.919634	0.459817	+	0.888014i	$0.347915\pi$
$920$	33.3654	1.10003
$921$	−67.6450	−2.22898
$922$	−25.8247	−0.850491
$923$	27.0082	0.888984
$924$	38.4888	1.26619
$925$	23.0083	0.756508
$926$	−10.7968	−0.354806
$927$	−30.1557	−0.990442
$928$	40.1919	1.31936
$929$	11.0760	0.363392	0.181696	−	0.983355i	$-0.441841\pi$
$929$	11.0760	0.363392	0.181696	+	0.983355i	$0.441841\pi$
$930$	−4.14866	−0.136040
$931$	10.4289	0.341793
$932$	30.8501	1.01053
$933$	38.9336	1.27463
$934$	7.26556	0.237736
$935$	−1.39624	−0.0456618
$936$	−18.6697	−0.610239
$937$	31.7493	1.03720	0.518602	−	0.855016i	$-0.326453\pi$
$937$	31.7493	1.03720	0.518602	+	0.855016i	$0.326453\pi$
$938$	−13.2481	−0.432566
$939$	12.7018	0.414508
$940$	−0.881364	−0.0287469
$941$	−0.429173	−0.0139906	−0.00699531	−	0.999976i	$-0.502227\pi$
$941$	−0.429173	−0.0139906	−0.00699531	+	0.999976i	$0.502227\pi$
$942$	−11.8568	−0.386317
$943$	−94.1194	−3.06495
$944$	5.95059	0.193675
$945$	19.2249	0.625385
$946$	27.3496	0.889213
$947$	36.0823	1.17252	0.586259	−	0.810124i	$-0.300600\pi$
$947$	36.0823	1.17252	0.586259	+	0.810124i	$0.300600\pi$
$948$	−23.3084	−0.757023
$949$	−4.84908	−0.157408
$950$	1.91526	0.0621392
$951$	−49.9237	−1.61889
$952$	−3.13881	−0.101729
$953$	−39.7044	−1.28615	−0.643076	−	0.765802i	$-0.722342\pi$
$953$	−39.7044	−1.28615	−0.643076	+	0.765802i	$0.722342\pi$
$954$	13.4066	0.434055
$955$	−25.4111	−0.822283
$956$	17.4694	0.565000
$957$	45.7702	1.47954
$958$	20.8244	0.672804
$959$	−45.2246	−1.46038
$960$	−11.1292	−0.359194
$961$	−28.6601	−0.924519
$962$	−31.1843	−1.00542
$963$	29.6530	0.955556
$964$	7.26665	0.234043
$965$	15.0947	0.485915
$966$	55.5856	1.78844
$967$	−1.28839	−0.0414317	−0.0207158	−	0.999785i	$-0.506595\pi$
$967$	−1.28839	−0.0414317	−0.0207158	+	0.999785i	$0.506595\pi$
$968$	−3.97461	−0.127749
$969$	0.611664	0.0196495
$970$	17.3190	0.556081
$971$	5.74020	0.184212	0.0921060	−	0.995749i	$-0.470640\pi$
$971$	5.74020	0.184212	0.0921060	+	0.995749i	$0.470640\pi$
$972$	−21.6066	−0.693033
$973$	−11.7210	−0.375759
$974$	−4.43768	−0.142192
$975$	−22.2302	−0.711938
$976$	1.84885	0.0591802
$977$	−35.6439	−1.14035	−0.570175	−	0.821523i	$-0.693125\pi$
$977$	−35.6439	−1.14035	−0.570175	+	0.821523i	$0.693125\pi$
$978$	−1.02627	−0.0328166
$979$	30.8483	0.985916
$980$	−23.0944	−0.737725
$981$	20.3617	0.650100
$982$	4.69053	0.149681
$983$	−18.6443	−0.594661	−0.297331	−	0.954775i	$-0.596096\pi$
$983$	−18.6443	−0.594661	−0.297331	+	0.954775i	$0.596096\pi$
$984$	68.7572	2.19190
$985$	−38.8658	−1.23837
$986$	−1.52674	−0.0486214
$987$	−3.58979	−0.114264
$988$	5.83560	0.185655
$989$	−88.7942	−2.82349
$990$	−6.45353	−0.205107
$991$	−38.4592	−1.22170	−0.610849	−	0.791747i	$-0.709172\pi$
$991$	−38.4592	−1.22170	−0.610849	+	0.791747i	$0.709172\pi$
$992$	8.94627	0.284044
$993$	−64.3475	−2.04201
$994$	20.9875	0.665684
$995$	−14.6586	−0.464710
$996$	48.1888	1.52692
$997$	27.1995	0.861416	0.430708	−	0.902491i	$-0.358264\pi$
$997$	27.1995	0.861416	0.430708	+	0.902491i	$0.358264\pi$
$998$	−10.3419	−0.327366
$999$	−27.1352	−0.858520

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.32	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.32	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-0.784700 q^{2} -2.16047 q^{3} -1.38425 q^{4} +1.59976 q^{5} +1.69532 q^{6} +4.17479 q^{7} +2.65562 q^{8} +1.66763 q^{9} +O(q^{10})\)	\(q-0.784700 q^{2} -2.16047 q^{3} -1.38425 q^{4} +1.59976 q^{5} +1.69532 q^{6} +4.17479 q^{7} +2.65562 q^{8} +1.66763 q^{9} -1.25534 q^{10} +3.08275 q^{11} +2.99062 q^{12} -4.21572 q^{13} -3.27596 q^{14} -3.45624 q^{15} +0.684627 q^{16} -0.283116 q^{17} -1.30859 q^{18} +1.00000 q^{19} -2.21447 q^{20} -9.01951 q^{21} -2.41903 q^{22} +7.85371 q^{23} -5.73738 q^{24} -2.44075 q^{25} +3.30808 q^{26} +2.87854 q^{27} -5.77894 q^{28} -6.87222 q^{29} +2.71212 q^{30} -1.52968 q^{31} -5.84846 q^{32} -6.66018 q^{33} +0.222161 q^{34} +6.67868 q^{35} -2.30841 q^{36} -9.42672 q^{37} -0.784700 q^{38} +9.10794 q^{39} +4.24836 q^{40} -11.9841 q^{41} +7.07761 q^{42} -11.3060 q^{43} -4.26728 q^{44} +2.66782 q^{45} -6.16281 q^{46} +0.398003 q^{47} -1.47912 q^{48} +10.4289 q^{49} +1.91526 q^{50} +0.611664 q^{51} +5.83560 q^{52} -10.2451 q^{53} -2.25879 q^{54} +4.93167 q^{55} +11.0867 q^{56} -2.16047 q^{57} +5.39263 q^{58} +8.69174 q^{59} +4.78429 q^{60} +2.70052 q^{61} +1.20034 q^{62} +6.96201 q^{63} +3.22004 q^{64} -6.74416 q^{65} +5.22624 q^{66} +4.04404 q^{67} +0.391902 q^{68} -16.9677 q^{69} -5.24077 q^{70} -6.40653 q^{71} +4.42859 q^{72} +1.15024 q^{73} +7.39715 q^{74} +5.27317 q^{75} -1.38425 q^{76} +12.8698 q^{77} -7.14701 q^{78} -7.79384 q^{79} +1.09524 q^{80} -11.2219 q^{81} +9.40390 q^{82} +16.1133 q^{83} +12.4852 q^{84} -0.452919 q^{85} +8.87184 q^{86} +14.8472 q^{87} +8.18660 q^{88} +10.0068 q^{89} -2.09344 q^{90} -17.5998 q^{91} -10.8715 q^{92} +3.30482 q^{93} -0.312313 q^{94} +1.59976 q^{95} +12.6354 q^{96} -13.7963 q^{97} -8.18355 q^{98} +5.14088 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more