LMFDB - Embedded newform 8041.2.a.f.1.46

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, 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("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2077082653$
Analytic rank:	$0$
Dimension:	$66$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.46
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56669 q^{2} +0.629934 q^{3} +0.454508 q^{4} -2.16451 q^{5} +0.986909 q^{6} -5.08174 q^{7} -2.42130 q^{8} -2.60318 q^{9} +O(q^{10})$	\(q+1.56669 q^{2} +0.629934 q^{3} +0.454508 q^{4} -2.16451 q^{5} +0.986909 q^{6} -5.08174 q^{7} -2.42130 q^{8} -2.60318 q^{9} -3.39110 q^{10} -1.00000 q^{11} +0.286310 q^{12} -2.97614 q^{13} -7.96149 q^{14} -1.36350 q^{15} -4.70244 q^{16} +1.00000 q^{17} -4.07837 q^{18} -6.04783 q^{19} -0.983785 q^{20} -3.20116 q^{21} -1.56669 q^{22} +2.54252 q^{23} -1.52526 q^{24} -0.314913 q^{25} -4.66268 q^{26} -3.52963 q^{27} -2.30969 q^{28} +1.74304 q^{29} -2.13617 q^{30} -4.76506 q^{31} -2.52464 q^{32} -0.629934 q^{33} +1.56669 q^{34} +10.9994 q^{35} -1.18317 q^{36} -11.5443 q^{37} -9.47505 q^{38} -1.87477 q^{39} +5.24092 q^{40} +1.39116 q^{41} -5.01521 q^{42} -1.00000 q^{43} -0.454508 q^{44} +5.63461 q^{45} +3.98333 q^{46} -3.18740 q^{47} -2.96223 q^{48} +18.8240 q^{49} -0.493369 q^{50} +0.629934 q^{51} -1.35268 q^{52} -6.48181 q^{53} -5.52983 q^{54} +2.16451 q^{55} +12.3044 q^{56} -3.80973 q^{57} +2.73080 q^{58} -2.15708 q^{59} -0.619720 q^{60} -3.33319 q^{61} -7.46535 q^{62} +13.2287 q^{63} +5.44955 q^{64} +6.44188 q^{65} -0.986909 q^{66} +5.09482 q^{67} +0.454508 q^{68} +1.60162 q^{69} +17.2327 q^{70} +13.6817 q^{71} +6.30309 q^{72} +10.0295 q^{73} -18.0862 q^{74} -0.198374 q^{75} -2.74879 q^{76} +5.08174 q^{77} -2.93718 q^{78} +9.72008 q^{79} +10.1785 q^{80} +5.58611 q^{81} +2.17951 q^{82} -3.17833 q^{83} -1.45495 q^{84} -2.16451 q^{85} -1.56669 q^{86} +1.09800 q^{87} +2.42130 q^{88} -1.09226 q^{89} +8.82766 q^{90} +15.1240 q^{91} +1.15559 q^{92} -3.00167 q^{93} -4.99366 q^{94} +13.0906 q^{95} -1.59036 q^{96} -10.8851 q^{97} +29.4914 q^{98} +2.60318 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 12 q^{2} + 66 q^{4} + 6 q^{5} + 7 q^{6} + 13 q^{7} + 30 q^{8} + 58 q^{9}+O(q^{10}) $ 66 * q + 12 * q^2 + 66 * q^4 + 6 * q^5 + 7 * q^6 + 13 * q^7 + 30 * q^8 + 58 * q^9	\( 66 q + 12 q^{2} + 66 q^{4} + 6 q^{5} + 7 q^{6} + 13 q^{7} + 30 q^{8} + 58 q^{9} + 7 q^{10} - 66 q^{11} + 12 q^{12} + 12 q^{13} + 13 q^{14} + 35 q^{15} + 58 q^{16} + 66 q^{17} + 37 q^{18} + 24 q^{19} + 17 q^{20} + 16 q^{21} - 12 q^{22} + 25 q^{23} + 22 q^{24} + 56 q^{25} + 36 q^{26} + 17 q^{28} + 29 q^{29} + 28 q^{30} + 37 q^{31} + 62 q^{32} + 12 q^{34} + 40 q^{35} + 107 q^{36} - 34 q^{37} + 22 q^{38} + 61 q^{39} + 37 q^{40} + 41 q^{41} + 19 q^{42} - 66 q^{43} - 66 q^{44} + 10 q^{45} + 43 q^{46} + 61 q^{47} + 29 q^{48} + 33 q^{49} + 59 q^{50} + 51 q^{52} - 35 q^{53} - 37 q^{54} - 6 q^{55} + 37 q^{56} - 7 q^{57} + 17 q^{58} + 48 q^{59} - 56 q^{60} + q^{61} + 37 q^{62} + 43 q^{63} + 68 q^{64} + 41 q^{65} - 7 q^{66} + 10 q^{67} + 66 q^{68} + 18 q^{69} + 77 q^{70} + 84 q^{71} + 83 q^{72} + 5 q^{73} + 36 q^{74} + 14 q^{75} + 14 q^{76} - 13 q^{77} + 41 q^{78} + 58 q^{79} + 25 q^{80} + 78 q^{81} - 28 q^{82} + 47 q^{83} + 44 q^{84} + 6 q^{85} - 12 q^{86} + 101 q^{87} - 30 q^{88} + 53 q^{89} + q^{90} + 2 q^{91} + 34 q^{92} - 3 q^{93} + 17 q^{94} + 91 q^{95} + 27 q^{96} - 28 q^{97} + 87 q^{98} - 58 q^{99}+O(q^{100}) \) 66 * q + 12 * q^2 + 66 * q^4 + 6 * q^5 + 7 * q^6 + 13 * q^7 + 30 * q^8 + 58 * q^9 + 7 * q^10 - 66 * q^11 + 12 * q^12 + 12 * q^13 + 13 * q^14 + 35 * q^15 + 58 * q^16 + 66 * q^17 + 37 * q^18 + 24 * q^19 + 17 * q^20 + 16 * q^21 - 12 * q^22 + 25 * q^23 + 22 * q^24 + 56 * q^25 + 36 * q^26 + 17 * q^28 + 29 * q^29 + 28 * q^30 + 37 * q^31 + 62 * q^32 + 12 * q^34 + 40 * q^35 + 107 * q^36 - 34 * q^37 + 22 * q^38 + 61 * q^39 + 37 * q^40 + 41 * q^41 + 19 * q^42 - 66 * q^43 - 66 * q^44 + 10 * q^45 + 43 * q^46 + 61 * q^47 + 29 * q^48 + 33 * q^49 + 59 * q^50 + 51 * q^52 - 35 * q^53 - 37 * q^54 - 6 * q^55 + 37 * q^56 - 7 * q^57 + 17 * q^58 + 48 * q^59 - 56 * q^60 + q^61 + 37 * q^62 + 43 * q^63 + 68 * q^64 + 41 * q^65 - 7 * q^66 + 10 * q^67 + 66 * q^68 + 18 * q^69 + 77 * q^70 + 84 * q^71 + 83 * q^72 + 5 * q^73 + 36 * q^74 + 14 * q^75 + 14 * q^76 - 13 * q^77 + 41 * q^78 + 58 * q^79 + 25 * q^80 + 78 * q^81 - 28 * q^82 + 47 * q^83 + 44 * q^84 + 6 * q^85 - 12 * q^86 + 101 * q^87 - 30 * q^88 + 53 * q^89 + q^90 + 2 * q^91 + 34 * q^92 - 3 * q^93 + 17 * q^94 + 91 * q^95 + 27 * q^96 - 28 * q^97 + 87 * q^98 - 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.56669	1.10781	0.553907	−	0.832578i	$-0.313136\pi$
$2$	1.56669	1.10781	0.553907	+	0.832578i	$0.313136\pi$
$3$	0.629934	0.363692	0.181846	−	0.983327i	$-0.441793\pi$
$3$	0.629934	0.363692	0.181846	+	0.983327i	$0.441793\pi$
$4$	0.454508	0.227254
$5$	−2.16451	−0.967997	−0.483998	−	0.875069i	$-0.660816\pi$
$5$	−2.16451	−0.967997	−0.483998	+	0.875069i	$0.660816\pi$
$6$	0.986909	0.402904
$7$	−5.08174	−1.92072	−0.960358	−	0.278770i	$-0.910073\pi$
$7$	−5.08174	−1.92072	−0.960358	+	0.278770i	$0.910073\pi$
$8$	−2.42130	−0.856060
$9$	−2.60318	−0.867728
$10$	−3.39110	−1.07236
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.286310	0.0826506
$13$	−2.97614	−0.825433	−0.412717	−	0.910859i	$-0.635420\pi$
$13$	−2.97614	−0.825433	−0.412717	+	0.910859i	$0.635420\pi$
$14$	−7.96149	−2.12780
$15$	−1.36350	−0.352053
$16$	−4.70244	−1.17561
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−4.07837	−0.961282
$19$	−6.04783	−1.38747	−0.693733	−	0.720232i	$-0.744035\pi$
$19$	−6.04783	−1.38747	−0.693733	+	0.720232i	$0.744035\pi$
$20$	−0.983785	−0.219981
$21$	−3.20116	−0.698550
$22$	−1.56669	−0.334019
$23$	2.54252	0.530151	0.265076	−	0.964228i	$-0.414603\pi$
$23$	2.54252	0.530151	0.265076	+	0.964228i	$0.414603\pi$
$24$	−1.52526	−0.311342
$25$	−0.314913	−0.0629825
$26$	−4.66268	−0.914428
$27$	−3.52963	−0.679279
$28$	−2.30969	−0.436490
$29$	1.74304	0.323675	0.161838	−	0.986817i	$-0.448258\pi$
$29$	1.74304	0.323675	0.161838	+	0.986817i	$0.448258\pi$
$30$	−2.13617	−0.390010
$31$	−4.76506	−0.855829	−0.427915	−	0.903819i	$-0.640752\pi$
$31$	−4.76506	−0.855829	−0.427915	+	0.903819i	$0.640752\pi$
$32$	−2.52464	−0.446298
$33$	−0.629934	−0.109657
$34$	1.56669	0.268685
$35$	10.9994	1.85925
$36$	−1.18317	−0.197195
$37$	−11.5443	−1.89786	−0.948932	−	0.315481i	$-0.897834\pi$
$37$	−11.5443	−1.89786	−0.948932	+	0.315481i	$0.897834\pi$
$38$	−9.47505	−1.53706
$39$	−1.87477	−0.300204
$40$	5.24092	0.828663
$41$	1.39116	0.217263	0.108631	−	0.994082i	$-0.465353\pi$
$41$	1.39116	0.217263	0.108631	+	0.994082i	$0.465353\pi$
$42$	−5.01521	−0.773864
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	−0.454508	−0.0685196
$45$	5.63461	0.839958
$46$	3.98333	0.587310
$47$	−3.18740	−0.464930	−0.232465	−	0.972605i	$-0.574679\pi$
$47$	−3.18740	−0.464930	−0.232465	+	0.972605i	$0.574679\pi$
$48$	−2.96223	−0.427560
$49$	18.8240	2.68915
$50$	−0.493369	−0.0697730
$51$	0.629934	0.0882084
$52$	−1.35268	−0.187583
$53$	−6.48181	−0.890345	−0.445173	−	0.895445i	$-0.646858\pi$
$53$	−6.48181	−0.890345	−0.445173	+	0.895445i	$0.646858\pi$
$54$	−5.52983	−0.752515
$55$	2.16451	0.291862
$56$	12.3044	1.64425
$57$	−3.80973	−0.504611
$58$	2.73080	0.358572
$59$	−2.15708	−0.280828	−0.140414	−	0.990093i	$-0.544843\pi$
$59$	−2.15708	−0.280828	−0.140414	+	0.990093i	$0.544843\pi$
$60$	−0.619720	−0.0800055
$61$	−3.33319	−0.426771	−0.213385	−	0.976968i	$-0.568449\pi$
$61$	−3.33319	−0.426771	−0.213385	+	0.976968i	$0.568449\pi$
$62$	−7.46535	−0.948101
$63$	13.2287	1.66666
$64$	5.44955	0.681194
$65$	6.44188	0.799017
$66$	−0.986909	−0.121480
$67$	5.09482	0.622431	0.311216	−	0.950339i	$-0.399264\pi$
$67$	5.09482	0.622431	0.311216	+	0.950339i	$0.399264\pi$
$68$	0.454508	0.0551172
$69$	1.60162	0.192812
$70$	17.2327	2.05970
$71$	13.6817	1.62372	0.811858	−	0.583855i	$-0.198456\pi$
$71$	13.6817	1.62372	0.811858	+	0.583855i	$0.198456\pi$
$72$	6.30309	0.742827
$73$	10.0295	1.17387	0.586933	−	0.809636i	$-0.300335\pi$
$73$	10.0295	1.17387	0.586933	+	0.809636i	$0.300335\pi$
$74$	−18.0862	−2.10248
$75$	−0.198374	−0.0229063
$76$	−2.74879	−0.315307
$77$	5.08174	0.579118
$78$	−2.93718	−0.332570
$79$	9.72008	1.09359	0.546797	−	0.837265i	$-0.315847\pi$
$79$	9.72008	1.09359	0.546797	+	0.837265i	$0.315847\pi$
$80$	10.1785	1.13799
$81$	5.58611	0.620679
$82$	2.17951	0.240687
$83$	−3.17833	−0.348868	−0.174434	−	0.984669i	$-0.555809\pi$
$83$	−3.17833	−0.348868	−0.174434	+	0.984669i	$0.555809\pi$
$84$	−1.45495	−0.158748
$85$	−2.16451	−0.234774
$86$	−1.56669	−0.168940
$87$	1.09800	0.117718
$88$	2.42130	0.258112
$89$	−1.09226	−0.115780	−0.0578898	−	0.998323i	$-0.518437\pi$
$89$	−1.09226	−0.115780	−0.0578898	+	0.998323i	$0.518437\pi$
$90$	8.82766	0.930518
$91$	15.1240	1.58542
$92$	1.15559	0.120479
$93$	−3.00167	−0.311259
$94$	−4.99366	−0.515056
$95$	13.0906	1.34306
$96$	−1.59036	−0.162315
$97$	−10.8851	−1.10521	−0.552605	−	0.833443i	$-0.686366\pi$
$97$	−10.8851	−1.10521	−0.552605	+	0.833443i	$0.686366\pi$
$98$	29.4914	2.97908
$99$	2.60318	0.261630
$100$	−0.143130	−0.0143130
$101$	−15.7641	−1.56859	−0.784294	−	0.620390i	$-0.786974\pi$
$101$	−15.7641	−1.56859	−0.784294	+	0.620390i	$0.786974\pi$
$102$	0.986909	0.0977186
$103$	−17.8022	−1.75411	−0.877053	−	0.480393i	$-0.840494\pi$
$103$	−17.8022	−1.75411	−0.877053	+	0.480393i	$0.840494\pi$
$104$	7.20614	0.706620
$105$	6.92892	0.676194
$106$	−10.1550	−0.986338
$107$	−9.99279	−0.966039	−0.483019	−	0.875610i	$-0.660460\pi$
$107$	−9.99279	−0.966039	−0.483019	+	0.875610i	$0.660460\pi$
$108$	−1.60425	−0.154369
$109$	−0.914502	−0.0875934	−0.0437967	−	0.999040i	$-0.513945\pi$
$109$	−0.914502	−0.0875934	−0.0437967	+	0.999040i	$0.513945\pi$
$110$	3.39110	0.323329
$111$	−7.27212	−0.690239
$112$	23.8965	2.25801
$113$	−11.8753	−1.11713	−0.558566	−	0.829460i	$-0.688648\pi$
$113$	−11.8753	−1.11713	−0.558566	+	0.829460i	$0.688648\pi$
$114$	−5.96866	−0.559016
$115$	−5.50329	−0.513185
$116$	0.792227	0.0735564
$117$	7.74744	0.716252
$118$	−3.37947	−0.311106
$119$	−5.08174	−0.465842
$120$	3.30144	0.301378
$121$	1.00000	0.0909091
$122$	−5.22206	−0.472783
$123$	0.876340	0.0790169
$124$	−2.16576	−0.194491
$125$	11.5042	1.02896
$126$	20.7252	1.84635
$127$	−9.91578	−0.879883	−0.439942	−	0.898026i	$-0.645001\pi$
$127$	−9.91578	−0.879883	−0.439942	+	0.898026i	$0.645001\pi$
$128$	13.5870	1.20093
$129$	−0.629934	−0.0554626
$130$	10.0924	0.885163
$131$	−21.0223	−1.83673	−0.918364	−	0.395736i	$-0.870490\pi$
$131$	−21.0223	−1.83673	−0.918364	+	0.395736i	$0.870490\pi$
$132$	−0.286310	−0.0249201
$133$	30.7335	2.66493
$134$	7.98199	0.689538
$135$	7.63992	0.657539
$136$	−2.42130	−0.207625
$137$	−15.2610	−1.30384	−0.651919	−	0.758288i	$-0.726036\pi$
$137$	−15.2610	−1.30384	−0.651919	+	0.758288i	$0.726036\pi$
$138$	2.50923	0.213600
$139$	−17.8581	−1.51470	−0.757351	−	0.653008i	$-0.773507\pi$
$139$	−17.8581	−1.51470	−0.757351	+	0.653008i	$0.773507\pi$
$140$	4.99934	0.422521
$141$	−2.00785	−0.169092
$142$	21.4349	1.79878
$143$	2.97614	0.248878
$144$	12.2413	1.02011
$145$	−3.77283	−0.313316
$146$	15.7131	1.30043
$147$	11.8579	0.978023
$148$	−5.24696	−0.431297
$149$	21.0180	1.72186	0.860931	−	0.508721i	$-0.169882\pi$
$149$	21.0180	1.72186	0.860931	+	0.508721i	$0.169882\pi$
$150$	−0.310790	−0.0253759
$151$	11.8119	0.961242	0.480621	−	0.876928i	$-0.340411\pi$
$151$	11.8119	0.961242	0.480621	+	0.876928i	$0.340411\pi$
$152$	14.6436	1.18775
$153$	−2.60318	−0.210455
$154$	7.96149	0.641555
$155$	10.3140	0.828440
$156$	−0.852099	−0.0682225
$157$	9.98732	0.797075	0.398538	−	0.917152i	$-0.369518\pi$
$157$	9.98732	0.797075	0.398538	+	0.917152i	$0.369518\pi$
$158$	15.2283	1.21150
$159$	−4.08311	−0.323812
$160$	5.46461	0.432015
$161$	−12.9204	−1.01827
$162$	8.75169	0.687598
$163$	−15.2520	−1.19463	−0.597313	−	0.802008i	$-0.703765\pi$
$163$	−15.2520	−1.19463	−0.597313	+	0.802008i	$0.703765\pi$
$164$	0.632294	0.0493739
$165$	1.36350	0.106148
$166$	−4.97946	−0.386481
$167$	8.39677	0.649762	0.324881	−	0.945755i	$-0.394676\pi$
$167$	8.39677	0.649762	0.324881	+	0.945755i	$0.394676\pi$
$168$	7.75097	0.598000
$169$	−4.14258	−0.318660
$170$	−3.39110	−0.260086
$171$	15.7436	1.20394
$172$	−0.454508	−0.0346559
$173$	−15.3916	−1.17020	−0.585101	−	0.810960i	$-0.698945\pi$
$173$	−15.3916	−1.17020	−0.585101	+	0.810960i	$0.698945\pi$
$174$	1.72023	0.130410
$175$	1.60030	0.120971
$176$	4.70244	0.354460
$177$	−1.35882	−0.102135
$178$	−1.71123	−0.128262
$179$	−17.8419	−1.33356	−0.666781	−	0.745253i	$-0.732329\pi$
$179$	−17.8419	−1.33356	−0.666781	+	0.745253i	$0.732329\pi$
$180$	2.56097	0.190884
$181$	−5.42245	−0.403047	−0.201524	−	0.979484i	$-0.564589\pi$
$181$	−5.42245	−0.403047	−0.201524	+	0.979484i	$0.564589\pi$
$182$	23.6945	1.75636
$183$	−2.09969	−0.155213
$184$	−6.15620	−0.453841
$185$	24.9876	1.83713
$186$	−4.70268	−0.344817
$187$	−1.00000	−0.0731272
$188$	−1.44870	−0.105657
$189$	17.9367	1.30470
$190$	20.5088	1.48787
$191$	22.3555	1.61759	0.808794	−	0.588092i	$-0.200121\pi$
$191$	22.3555	1.61759	0.808794	+	0.588092i	$0.200121\pi$
$192$	3.43286	0.247745
$193$	17.7094	1.27475	0.637376	−	0.770553i	$-0.280020\pi$
$193$	17.7094	1.27475	0.637376	+	0.770553i	$0.280020\pi$
$194$	−17.0535	−1.22437
$195$	4.05796	0.290596
$196$	8.55567	0.611120
$197$	−15.5660	−1.10903	−0.554517	−	0.832173i	$-0.687097\pi$
$197$	−15.5660	−1.10903	−0.554517	+	0.832173i	$0.687097\pi$
$198$	4.07837	0.289837
$199$	11.6103	0.823031	0.411515	−	0.911403i	$-0.365000\pi$
$199$	11.6103	0.823031	0.411515	+	0.911403i	$0.365000\pi$
$200$	0.762498	0.0539168
$201$	3.20940	0.226374
$202$	−24.6974	−1.73770
$203$	−8.85769	−0.621688
$204$	0.286310	0.0200457
$205$	−3.01118	−0.210310
$206$	−27.8905	−1.94323
$207$	−6.61864	−0.460027
$208$	13.9951	0.970387
$209$	6.04783	0.418337
$210$	10.8555	0.749098
$211$	7.07451	0.487029	0.243515	−	0.969897i	$-0.421700\pi$
$211$	7.07451	0.487029	0.243515	+	0.969897i	$0.421700\pi$
$212$	−2.94604	−0.202335
$213$	8.61855	0.590533
$214$	−15.6556	−1.07019
$215$	2.16451	0.147618
$216$	8.54631	0.581503
$217$	24.2148	1.64380
$218$	−1.43274	−0.0970373
$219$	6.31793	0.426926
$220$	0.983785	0.0663268
$221$	−2.97614	−0.200197
$222$	−11.3931	−0.764657
$223$	10.9540	0.733531	0.366766	−	0.930313i	$-0.380465\pi$
$223$	10.9540	0.733531	0.366766	+	0.930313i	$0.380465\pi$
$224$	12.8296	0.857212
$225$	0.819775	0.0546517
$226$	−18.6048	−1.23757
$227$	−28.4848	−1.89060	−0.945301	−	0.326198i	$-0.894232\pi$
$227$	−28.4848	−1.89060	−0.945301	+	0.326198i	$0.894232\pi$
$228$	−1.73155	−0.114675
$229$	19.3709	1.28007	0.640034	−	0.768347i	$-0.278920\pi$
$229$	19.3709	1.28007	0.640034	+	0.768347i	$0.278920\pi$
$230$	−8.62194	−0.568514
$231$	3.20116	0.210621
$232$	−4.22044	−0.277085
$233$	−0.0784137	−0.00513705	−0.00256852	−	0.999997i	$-0.500818\pi$
$233$	−0.0784137	−0.00513705	−0.00256852	+	0.999997i	$0.500818\pi$
$234$	12.1378	0.793474
$235$	6.89915	0.450051
$236$	−0.980411	−0.0638193
$237$	6.12301	0.397732
$238$	−7.96149	−0.516067
$239$	−29.6117	−1.91542	−0.957712	−	0.287727i	$-0.907100\pi$
$239$	−29.6117	−1.91542	−0.957712	+	0.287727i	$0.907100\pi$
$240$	6.41175	0.413877
$241$	−10.9636	−0.706230	−0.353115	−	0.935580i	$-0.614878\pi$
$241$	−10.9636	−0.706230	−0.353115	+	0.935580i	$0.614878\pi$
$242$	1.56669	0.100710
$243$	14.1078	0.905015
$244$	−1.51496	−0.0969854
$245$	−40.7447	−2.60309
$246$	1.37295	0.0875361
$247$	17.9992	1.14526
$248$	11.5376	0.732641
$249$	−2.00214	−0.126881
$250$	18.0234	1.13990
$251$	−8.71374	−0.550007	−0.275003	−	0.961443i	$-0.588679\pi$
$251$	−8.71374	−0.550007	−0.275003	+	0.961443i	$0.588679\pi$
$252$	6.01254	0.378755
$253$	−2.54252	−0.159847
$254$	−15.5349	−0.974748
$255$	−1.36350	−0.0853854
$256$	10.3875	0.649220
$257$	−18.9474	−1.18190	−0.590952	−	0.806707i	$-0.701248\pi$
$257$	−18.9474	−1.18190	−0.590952	+	0.806707i	$0.701248\pi$
$258$	−0.986909	−0.0614423
$259$	58.6649	3.64526
$260$	2.92789	0.181580
$261$	−4.53746	−0.280862
$262$	−32.9354	−2.03476
$263$	−8.58628	−0.529453	−0.264726	−	0.964324i	$-0.585282\pi$
$263$	−8.58628	−0.529453	−0.264726	+	0.964324i	$0.585282\pi$
$264$	1.52526	0.0938733
$265$	14.0299	0.861851
$266$	48.1497	2.95225
$267$	−0.688054	−0.0421082
$268$	2.31564	0.141450
$269$	13.1918	0.804319	0.402159	−	0.915570i	$-0.368260\pi$
$269$	13.1918	0.804319	0.402159	+	0.915570i	$0.368260\pi$
$270$	11.9694	0.728432
$271$	21.2422	1.29037	0.645187	−	0.764025i	$-0.276779\pi$
$271$	21.2422	1.29037	0.645187	+	0.764025i	$0.276779\pi$
$272$	−4.70244	−0.285127
$273$	9.52710	0.576606
$274$	−23.9093	−1.44441
$275$	0.314913	0.0189899
$276$	0.727948	0.0438173
$277$	−21.0921	−1.26730	−0.633651	−	0.773619i	$-0.718444\pi$
$277$	−21.0921	−1.26730	−0.633651	+	0.773619i	$0.718444\pi$
$278$	−27.9780	−1.67801
$279$	12.4043	0.742627
$280$	−26.6330	−1.59163
$281$	15.6359	0.932757	0.466379	−	0.884585i	$-0.345558\pi$
$281$	15.6359	0.932757	0.466379	+	0.884585i	$0.345558\pi$
$282$	−3.14567	−0.187322
$283$	13.8618	0.823997	0.411999	−	0.911184i	$-0.364831\pi$
$283$	13.8618	0.823997	0.411999	+	0.911184i	$0.364831\pi$
$284$	6.21843	0.368996
$285$	8.24619	0.488462
$286$	4.66268	0.275710
$287$	−7.06952	−0.417300
$288$	6.57211	0.387265
$289$	1.00000	0.0588235
$290$	−5.91084	−0.347097
$291$	−6.85687	−0.401957
$292$	4.55849	0.266766
$293$	−26.8652	−1.56948	−0.784740	−	0.619825i	$-0.787203\pi$
$293$	−26.8652	−1.56948	−0.784740	+	0.619825i	$0.787203\pi$
$294$	18.5776	1.08347
$295$	4.66902	0.271841
$296$	27.9521	1.62468
$297$	3.52963	0.204810
$298$	32.9286	1.90751
$299$	−7.56689	−0.437605
$300$	−0.0901626	−0.00520554
$301$	5.08174	0.292906
$302$	18.5056	1.06488
$303$	−9.93034	−0.570483
$304$	28.4395	1.63112
$305$	7.21471	0.413113
$306$	−4.07837	−0.233145
$307$	−2.71255	−0.154813	−0.0774067	−	0.997000i	$-0.524664\pi$
$307$	−2.71255	−0.154813	−0.0774067	+	0.997000i	$0.524664\pi$
$308$	2.30969	0.131607
$309$	−11.2142	−0.637955
$310$	16.1588	0.917758
$311$	3.03068	0.171854	0.0859270	−	0.996301i	$-0.472615\pi$
$311$	3.03068	0.171854	0.0859270	+	0.996301i	$0.472615\pi$
$312$	4.53939	0.256992
$313$	−9.30409	−0.525898	−0.262949	−	0.964810i	$-0.584695\pi$
$313$	−9.30409	−0.525898	−0.262949	+	0.964810i	$0.584695\pi$
$314$	15.6470	0.883012
$315$	−28.6336	−1.61332
$316$	4.41786	0.248524
$317$	0.111814	0.00628009	0.00314004	−	0.999995i	$-0.499000\pi$
$317$	0.111814	0.00628009	0.00314004	+	0.999995i	$0.499000\pi$
$318$	−6.39696	−0.358724
$319$	−1.74304	−0.0975917
$320$	−11.7956	−0.659393
$321$	−6.29479	−0.351341
$322$	−20.2422	−1.12805
$323$	−6.04783	−0.336510
$324$	2.53893	0.141052
$325$	0.937225	0.0519879
$326$	−23.8950	−1.32342
$327$	−0.576076	−0.0318571
$328$	−3.36842	−0.185990
$329$	16.1975	0.892998
$330$	2.13617	0.117592
$331$	10.3632	0.569612	0.284806	−	0.958585i	$-0.408071\pi$
$331$	10.3632	0.569612	0.284806	+	0.958585i	$0.408071\pi$
$332$	−1.44458	−0.0792815
$333$	30.0518	1.64683
$334$	13.1551	0.719816
$335$	−11.0278	−0.602511
$336$	15.0532	0.821222
$337$	8.39881	0.457513	0.228756	−	0.973484i	$-0.426534\pi$
$337$	8.39881	0.457513	0.228756	+	0.973484i	$0.426534\pi$
$338$	−6.49012	−0.353016
$339$	−7.48063	−0.406292
$340$	−0.983785	−0.0533532
$341$	4.76506	0.258042
$342$	24.6653	1.33375
$343$	−60.0866	−3.24437
$344$	2.42130	0.130548
$345$	−3.46671	−0.186641
$346$	−24.1138	−1.29637
$347$	−18.6269	−0.999945	−0.499973	−	0.866041i	$-0.666657\pi$
$347$	−18.6269	−0.999945	−0.499973	+	0.866041i	$0.666657\pi$
$348$	0.499051	0.0267519
$349$	−26.3264	−1.40922	−0.704611	−	0.709594i	$-0.748879\pi$
$349$	−26.3264	−1.40922	−0.704611	+	0.709594i	$0.748879\pi$
$350$	2.50717	0.134014
$351$	10.5047	0.560699
$352$	2.52464	0.134564
$353$	2.66815	0.142011	0.0710057	−	0.997476i	$-0.477379\pi$
$353$	2.66815	0.142011	0.0710057	+	0.997476i	$0.477379\pi$
$354$	−2.12884	−0.113147
$355$	−29.6141	−1.57175
$356$	−0.496442	−0.0263114
$357$	−3.20116	−0.169423
$358$	−27.9526	−1.47734
$359$	16.1491	0.852318	0.426159	−	0.904648i	$-0.359866\pi$
$359$	16.1491	0.852318	0.426159	+	0.904648i	$0.359866\pi$
$360$	−13.6431	−0.719054
$361$	17.5762	0.925064
$362$	−8.49528	−0.446502
$363$	0.629934	0.0330630
$364$	6.87396	0.360294
$365$	−21.7089	−1.13630
$366$	−3.28955	−0.171948
$367$	11.5868	0.604826	0.302413	−	0.953177i	$-0.402208\pi$
$367$	11.5868	0.604826	0.302413	+	0.953177i	$0.402208\pi$
$368$	−11.9560	−0.623251
$369$	−3.62145	−0.188525
$370$	39.1478	2.03520
$371$	32.9389	1.71010
$372$	−1.36428	−0.0707348
$373$	−26.9663	−1.39626	−0.698132	−	0.715969i	$-0.745985\pi$
$373$	−26.9663	−1.39626	−0.698132	+	0.715969i	$0.745985\pi$
$374$	−1.56669	−0.0810115
$375$	7.24686	0.374226
$376$	7.71766	0.398008
$377$	−5.18755	−0.267172
$378$	28.1011	1.44537
$379$	18.6871	0.959889	0.479945	−	0.877299i	$-0.340657\pi$
$379$	18.6871	0.959889	0.479945	+	0.877299i	$0.340657\pi$
$380$	5.94976	0.305216
$381$	−6.24629	−0.320007
$382$	35.0241	1.79199
$383$	23.9666	1.22463	0.612317	−	0.790612i	$-0.290238\pi$
$383$	23.9666	1.22463	0.612317	+	0.790612i	$0.290238\pi$
$384$	8.55893	0.436771
$385$	−10.9994	−0.560584
$386$	27.7451	1.41219
$387$	2.60318	0.132327
$388$	−4.94734	−0.251163
$389$	−18.6189	−0.944014	−0.472007	−	0.881595i	$-0.656470\pi$
$389$	−18.6189	−0.944014	−0.472007	+	0.881595i	$0.656470\pi$
$390$	6.35755	0.321927
$391$	2.54252	0.128581
$392$	−45.5787	−2.30207
$393$	−13.2427	−0.668004
$394$	−24.3871	−1.22860
$395$	−21.0392	−1.05860
$396$	1.18317	0.0594564
$397$	−10.3442	−0.519162	−0.259581	−	0.965721i	$-0.583584\pi$
$397$	−10.3442	−0.519162	−0.259581	+	0.965721i	$0.583584\pi$
$398$	18.1897	0.911766
$399$	19.3600	0.969215
$400$	1.48086	0.0740428
$401$	−30.8948	−1.54281	−0.771407	−	0.636342i	$-0.780447\pi$
$401$	−30.8948	−1.54281	−0.771407	+	0.636342i	$0.780447\pi$
$402$	5.02812	0.250780
$403$	14.1815	0.706430
$404$	−7.16491	−0.356468
$405$	−12.0912	−0.600815
$406$	−13.8772	−0.688715
$407$	11.5443	0.572227
$408$	−1.52526	−0.0755116
$409$	12.3358	0.609964	0.304982	−	0.952358i	$-0.401350\pi$
$409$	12.3358	0.609964	0.304982	+	0.952358i	$0.401350\pi$
$410$	−4.71757	−0.232984
$411$	−9.61345	−0.474196
$412$	−8.09126	−0.398628
$413$	10.9617	0.539391
$414$	−10.3693	−0.509625
$415$	6.87953	0.337703
$416$	7.51370	0.368390
$417$	−11.2494	−0.550886
$418$	9.47505	0.463440
$419$	−17.5947	−0.859558	−0.429779	−	0.902934i	$-0.641408\pi$
$419$	−17.5947	−0.859558	−0.429779	+	0.902934i	$0.641408\pi$
$420$	3.14925	0.153668
$421$	13.2859	0.647513	0.323756	−	0.946140i	$-0.395054\pi$
$421$	13.2859	0.647513	0.323756	+	0.946140i	$0.395054\pi$
$422$	11.0835	0.539538
$423$	8.29739	0.403433
$424$	15.6944	0.762189
$425$	−0.314913	−0.0152755
$426$	13.5026	0.654202
$427$	16.9384	0.819706
$428$	−4.54180	−0.219536
$429$	1.87477	0.0905149
$430$	3.39110	0.163534
$431$	−9.85066	−0.474490	−0.237245	−	0.971450i	$-0.576244\pi$
$431$	−9.85066	−0.474490	−0.237245	+	0.971450i	$0.576244\pi$
$432$	16.5979	0.798566
$433$	−31.2540	−1.50197	−0.750985	−	0.660319i	$-0.770421\pi$
$433$	−31.2540	−1.50197	−0.750985	+	0.660319i	$0.770421\pi$
$434$	37.9369	1.82103
$435$	−2.37663	−0.113951
$436$	−0.415648	−0.0199059
$437$	−15.3767	−0.735567
$438$	9.89821	0.472955
$439$	31.5349	1.50508	0.752539	−	0.658547i	$-0.228829\pi$
$439$	31.5349	1.50508	0.752539	+	0.658547i	$0.228829\pi$
$440$	−5.24092	−0.249851
$441$	−49.0024	−2.33345
$442$	−4.66268	−0.221781
$443$	8.55422	0.406423	0.203212	−	0.979135i	$-0.434862\pi$
$443$	8.55422	0.406423	0.203212	+	0.979135i	$0.434862\pi$
$444$	−3.30523	−0.156859
$445$	2.36421	0.112074
$446$	17.1614	0.812617
$447$	13.2400	0.626228
$448$	−27.6932	−1.30838
$449$	12.0337	0.567903	0.283952	−	0.958839i	$-0.408354\pi$
$449$	12.0337	0.567903	0.283952	+	0.958839i	$0.408354\pi$
$450$	1.28433	0.0605439
$451$	−1.39116	−0.0655072
$452$	−5.39740	−0.253872
$453$	7.44074	0.349597
$454$	−44.6268	−2.09444
$455$	−32.7359	−1.53468
$456$	9.22451	0.431977
$457$	32.4115	1.51615	0.758074	−	0.652169i	$-0.226141\pi$
$457$	32.4115	1.51615	0.758074	+	0.652169i	$0.226141\pi$
$458$	30.3482	1.41808
$459$	−3.52963	−0.164749
$460$	−2.50129	−0.116623
$461$	−6.70361	−0.312218	−0.156109	−	0.987740i	$-0.549895\pi$
$461$	−6.70361	−0.312218	−0.156109	+	0.987740i	$0.549895\pi$
$462$	5.01521	0.233329
$463$	−8.38243	−0.389565	−0.194782	−	0.980846i	$-0.562400\pi$
$463$	−8.38243	−0.389565	−0.194782	+	0.980846i	$0.562400\pi$
$464$	−8.19656	−0.380516
$465$	6.49713	0.301297
$466$	−0.122850	−0.00569090
$467$	7.20317	0.333323	0.166661	−	0.986014i	$-0.446701\pi$
$467$	7.20317	0.333323	0.166661	+	0.986014i	$0.446701\pi$
$468$	3.52128	0.162771
$469$	−25.8905	−1.19551
$470$	10.8088	0.498573
$471$	6.29135	0.289890
$472$	5.22295	0.240406
$473$	1.00000	0.0459800
$474$	9.59284	0.440614
$475$	1.90454	0.0873861
$476$	−2.30969	−0.105864
$477$	16.8733	0.772577
$478$	−46.3923	−2.12194
$479$	18.1635	0.829914	0.414957	−	0.909841i	$-0.363797\pi$
$479$	18.1635	0.829914	0.414957	+	0.909841i	$0.363797\pi$
$480$	3.44234	0.157121
$481$	34.3573	1.56656
$482$	−17.1766	−0.782372
$483$	−8.13899	−0.370337
$484$	0.454508	0.0206595
$485$	23.5608	1.06984
$486$	22.1025	1.00259
$487$	−5.11109	−0.231606	−0.115803	−	0.993272i	$-0.536944\pi$
$487$	−5.11109	−0.231606	−0.115803	+	0.993272i	$0.536944\pi$
$488$	8.07065	0.365341
$489$	−9.60773	−0.434476
$490$	−63.8343	−2.88374
$491$	−11.2012	−0.505504	−0.252752	−	0.967531i	$-0.581336\pi$
$491$	−11.2012	−0.505504	−0.252752	+	0.967531i	$0.581336\pi$
$492$	0.398303	0.0179569
$493$	1.74304	0.0785027
$494$	28.1991	1.26874
$495$	−5.63461	−0.253257
$496$	22.4074	1.00612
$497$	−69.5267	−3.11870
$498$	−3.13673	−0.140560
$499$	−18.0883	−0.809744	−0.404872	−	0.914373i	$-0.632684\pi$
$499$	−18.0883	−0.809744	−0.404872	+	0.914373i	$0.632684\pi$
$500$	5.22873	0.233836
$501$	5.28941	0.236314
$502$	−13.6517	−0.609306
$503$	15.7378	0.701712	0.350856	−	0.936429i	$-0.385891\pi$
$503$	15.7378	0.701712	0.350856	+	0.936429i	$0.385891\pi$
$504$	−32.0307	−1.42676
$505$	34.1215	1.51839
$506$	−3.98333	−0.177080
$507$	−2.60955	−0.115894
$508$	−4.50680	−0.199957
$509$	22.5955	1.00153	0.500763	−	0.865584i	$-0.333053\pi$
$509$	22.5955	1.00153	0.500763	+	0.865584i	$0.333053\pi$
$510$	−2.13617	−0.0945912
$511$	−50.9673	−2.25466
$512$	−10.9001	−0.481719
$513$	21.3466	0.942476
$514$	−29.6846	−1.30933
$515$	38.5331	1.69797
$516$	−0.286310	−0.0126041
$517$	3.18740	0.140182
$518$	91.9095	4.03827
$519$	−9.69570	−0.425594
$520$	−15.5977	−0.684006
$521$	−37.7269	−1.65284	−0.826422	−	0.563051i	$-0.809628\pi$
$521$	−37.7269	−1.65284	−0.826422	+	0.563051i	$0.809628\pi$
$522$	−7.10878	−0.311143
$523$	−21.6502	−0.946697	−0.473348	−	0.880875i	$-0.656955\pi$
$523$	−21.6502	−0.946697	−0.473348	+	0.880875i	$0.656955\pi$
$524$	−9.55481	−0.417404
$525$	1.00808	0.0439964
$526$	−13.4520	−0.586536
$527$	−4.76506	−0.207569
$528$	2.96223	0.128914
$529$	−16.5356	−0.718940
$530$	21.9805	0.954772
$531$	5.61528	0.243682
$532$	13.9686	0.605616
$533$	−4.14030	−0.179336
$534$	−1.07796	−0.0466481
$535$	21.6294	0.935123
$536$	−12.3361	−0.532838
$537$	−11.2392	−0.485007
$538$	20.6674	0.891036
$539$	−18.8240	−0.810809
$540$	3.47240	0.149428
$541$	12.1670	0.523102	0.261551	−	0.965190i	$-0.415766\pi$
$541$	12.1670	0.523102	0.261551	+	0.965190i	$0.415766\pi$
$542$	33.2799	1.42950
$543$	−3.41578	−0.146585
$544$	−2.52464	−0.108243
$545$	1.97944	0.0847901
$546$	14.9260	0.638773
$547$	−6.39262	−0.273329	−0.136664	−	0.990617i	$-0.543638\pi$
$547$	−6.39262	−0.273329	−0.136664	+	0.990617i	$0.543638\pi$
$548$	−6.93626	−0.296303
$549$	8.67690	0.370321
$550$	0.493369	0.0210373
$551$	−10.5416	−0.449088
$552$	−3.87800	−0.165059
$553$	−49.3949	−2.10048
$554$	−33.0447	−1.40394
$555$	15.7405	0.668149
$556$	−8.11663	−0.344222
$557$	12.1065	0.512968	0.256484	−	0.966548i	$-0.417436\pi$
$557$	12.1065	0.512968	0.256484	+	0.966548i	$0.417436\pi$
$558$	19.4337	0.822693
$559$	2.97614	0.125877
$560$	−51.7242	−2.18575
$561$	−0.629934	−0.0265958
$562$	24.4965	1.03332
$563$	38.5236	1.62357	0.811787	−	0.583953i	$-0.198495\pi$
$563$	38.5236	1.62357	0.811787	+	0.583953i	$0.198495\pi$
$564$	−0.912584	−0.0384267
$565$	25.7041	1.08138
$566$	21.7171	0.912837
$567$	−28.3872	−1.19215
$568$	−33.1275	−1.39000
$569$	13.7109	0.574792	0.287396	−	0.957812i	$-0.407210\pi$
$569$	13.7109	0.574792	0.287396	+	0.957812i	$0.407210\pi$
$570$	12.9192	0.541125
$571$	−31.7878	−1.33028	−0.665138	−	0.746720i	$-0.731627\pi$
$571$	−31.7878	−1.33028	−0.665138	+	0.746720i	$0.731627\pi$
$572$	1.35268	0.0565584
$573$	14.0825	0.588305
$574$	−11.0757	−0.462292
$575$	−0.800670	−0.0333903
$576$	−14.1862	−0.591091
$577$	17.9768	0.748383	0.374192	−	0.927351i	$-0.377920\pi$
$577$	17.9768	0.748383	0.374192	+	0.927351i	$0.377920\pi$
$578$	1.56669	0.0651656
$579$	11.1558	0.463617
$580$	−1.71478	−0.0712024
$581$	16.1515	0.670075
$582$	−10.7426	−0.445293
$583$	6.48181	0.268449
$584$	−24.2845	−1.00490
$585$	−16.7694	−0.693329
$586$	−42.0893	−1.73869
$587$	−12.2140	−0.504127	−0.252063	−	0.967711i	$-0.581109\pi$
$587$	−12.2140	−0.504127	−0.252063	+	0.967711i	$0.581109\pi$
$588$	5.38951	0.222260
$589$	28.8182	1.18743
$590$	7.31489	0.301149
$591$	−9.80556	−0.403347
$592$	54.2861	2.23115
$593$	18.4969	0.759577	0.379788	−	0.925073i	$-0.375997\pi$
$593$	18.4969	0.759577	0.379788	+	0.925073i	$0.375997\pi$
$594$	5.52983	0.226892
$595$	10.9994	0.450933
$596$	9.55285	0.391300
$597$	7.31371	0.299330
$598$	−11.8549	−0.484785
$599$	14.1644	0.578744	0.289372	−	0.957217i	$-0.406554\pi$
$599$	14.1644	0.578744	0.289372	+	0.957217i	$0.406554\pi$
$600$	0.480324	0.0196091
$601$	1.71394	0.0699132	0.0349566	−	0.999389i	$-0.488871\pi$
$601$	1.71394	0.0699132	0.0349566	+	0.999389i	$0.488871\pi$
$602$	7.96149	0.324486
$603$	−13.2627	−0.540101
$604$	5.36862	0.218446
$605$	−2.16451	−0.0879997
$606$	−15.5577	−0.631990
$607$	5.75555	0.233610	0.116805	−	0.993155i	$-0.462735\pi$
$607$	5.75555	0.233610	0.116805	+	0.993155i	$0.462735\pi$
$608$	15.2686	0.619224
$609$	−5.57976	−0.226103
$610$	11.3032	0.457653
$611$	9.48616	0.383769
$612$	−1.18317	−0.0478267
$613$	42.8158	1.72931	0.864656	−	0.502364i	$-0.167536\pi$
$613$	42.8158	1.72931	0.864656	+	0.502364i	$0.167536\pi$
$614$	−4.24971	−0.171505
$615$	−1.89684	−0.0764881
$616$	−12.3044	−0.495759
$617$	4.80588	0.193478	0.0967388	−	0.995310i	$-0.469159\pi$
$617$	4.80588	0.193478	0.0967388	+	0.995310i	$0.469159\pi$
$618$	−17.5692	−0.706736
$619$	−21.8256	−0.877246	−0.438623	−	0.898671i	$-0.644534\pi$
$619$	−21.8256	−0.877246	−0.438623	+	0.898671i	$0.644534\pi$
$620$	4.68779	0.188266
$621$	−8.97415	−0.360120
$622$	4.74812	0.190382
$623$	5.55059	0.222380
$624$	8.81600	0.352923
$625$	−23.3263	−0.933051
$626$	−14.5766	−0.582598
$627$	3.80973	0.152146
$628$	4.53932	0.181139
$629$	−11.5443	−0.460300
$630$	−44.8599	−1.78726
$631$	13.3284	0.530597	0.265298	−	0.964166i	$-0.414530\pi$
$631$	13.3284	0.530597	0.265298	+	0.964166i	$0.414530\pi$
$632$	−23.5353	−0.936182
$633$	4.45647	0.177129
$634$	0.175177	0.00695718
$635$	21.4628	0.851724
$636$	−1.85581	−0.0735875
$637$	−56.0230	−2.21971
$638$	−2.73080	−0.108114
$639$	−35.6159	−1.40894
$640$	−29.4092	−1.16250
$641$	−38.8411	−1.53413	−0.767065	−	0.641569i	$-0.778284\pi$
$641$	−38.8411	−1.53413	−0.767065	+	0.641569i	$0.778284\pi$
$642$	−9.86197	−0.389221
$643$	−47.1691	−1.86017	−0.930084	−	0.367347i	$-0.880266\pi$
$643$	−47.1691	−1.86017	−0.930084	+	0.367347i	$0.880266\pi$
$644$	−5.87242	−0.231406
$645$	1.36350	0.0536876
$646$	−9.47505	−0.372791
$647$	33.0048	1.29755	0.648775	−	0.760980i	$-0.275281\pi$
$647$	33.0048	1.29755	0.648775	+	0.760980i	$0.275281\pi$
$648$	−13.5257	−0.531338
$649$	2.15708	0.0846729
$650$	1.46834	0.0575929
$651$	15.2537	0.597839
$652$	−6.93214	−0.271483
$653$	23.4943	0.919404	0.459702	−	0.888073i	$-0.347956\pi$
$653$	23.4943	0.919404	0.459702	+	0.888073i	$0.347956\pi$
$654$	−0.902530	−0.0352917
$655$	45.5029	1.77795
$656$	−6.54185	−0.255416
$657$	−26.1087	−1.01860
$658$	25.3764	0.989277
$659$	−43.4618	−1.69303	−0.846517	−	0.532362i	$-0.821304\pi$
$659$	−43.4618	−1.69303	−0.846517	+	0.532362i	$0.821304\pi$
$660$	0.619720	0.0241226
$661$	−47.8325	−1.86047	−0.930234	−	0.366968i	$-0.880396\pi$
$661$	−47.8325	−1.86047	−0.930234	+	0.366968i	$0.880396\pi$
$662$	16.2359	0.631025
$663$	−1.87477	−0.0728101
$664$	7.69571	0.298651
$665$	−66.5228	−2.57964
$666$	47.0818	1.82438
$667$	4.43172	0.171597
$668$	3.81640	0.147661
$669$	6.90027	0.266780
$670$	−17.2771	−0.667471
$671$	3.33319	0.128676
$672$	8.08178	0.311762
$673$	29.5896	1.14060	0.570298	−	0.821438i	$-0.306828\pi$
$673$	29.5896	1.14060	0.570298	+	0.821438i	$0.306828\pi$
$674$	13.1583	0.506839
$675$	1.11153	0.0427827
$676$	−1.88283	−0.0724167
$677$	2.28690	0.0878926	0.0439463	−	0.999034i	$-0.486007\pi$
$677$	2.28690	0.0878926	0.0439463	+	0.999034i	$0.486007\pi$
$678$	−11.7198	−0.450097
$679$	55.3150	2.12279
$680$	5.24092	0.200980
$681$	−17.9435	−0.687598
$682$	7.46535	0.285863
$683$	−32.2247	−1.23304	−0.616522	−	0.787338i	$-0.711459\pi$
$683$	−32.2247	−1.23304	−0.616522	+	0.787338i	$0.711459\pi$
$684$	7.15559	0.273601
$685$	33.0326	1.26211
$686$	−94.1369	−3.59416
$687$	12.2024	0.465551
$688$	4.70244	0.179279
$689$	19.2908	0.734921
$690$	−5.43125	−0.206764
$691$	−36.0679	−1.37209	−0.686044	−	0.727560i	$-0.740654\pi$
$691$	−36.0679	−1.37209	−0.686044	+	0.727560i	$0.740654\pi$
$692$	−6.99561	−0.265933
$693$	−13.2287	−0.502516
$694$	−29.1825	−1.10775
$695$	38.6539	1.46623
$696$	−2.65860	−0.100774
$697$	1.39116	0.0526940
$698$	−41.2453	−1.56116
$699$	−0.0493954	−0.00186831
$700$	0.727350	0.0274913
$701$	−27.6313	−1.04362	−0.521810	−	0.853062i	$-0.674743\pi$
$701$	−27.6313	−1.04362	−0.521810	+	0.853062i	$0.674743\pi$
$702$	16.4576	0.621151
$703$	69.8177	2.63322
$704$	−5.44955	−0.205388
$705$	4.34601	0.163680
$706$	4.18016	0.157322
$707$	80.1090	3.01281
$708$	−0.617594	−0.0232106
$709$	−31.9970	−1.20167	−0.600836	−	0.799373i	$-0.705165\pi$
$709$	−31.9970	−1.20167	−0.600836	+	0.799373i	$0.705165\pi$
$710$	−46.3960	−1.74121
$711$	−25.3032	−0.948943
$712$	2.64470	0.0991143
$713$	−12.1152	−0.453719
$714$	−5.01521	−0.187690
$715$	−6.44188	−0.240913
$716$	−8.10926	−0.303057
$717$	−18.6534	−0.696626
$718$	25.3006	0.944211
$719$	28.1625	1.05028	0.525142	−	0.851015i	$-0.324012\pi$
$719$	28.1625	1.05028	0.525142	+	0.851015i	$0.324012\pi$
$720$	−26.4964	−0.987462
$721$	90.4663	3.36914
$722$	27.5364	1.02480
$723$	−6.90637	−0.256851
$724$	−2.46455	−0.0915941
$725$	−0.548906	−0.0203859
$726$	0.986909	0.0366276
$727$	−17.0335	−0.631737	−0.315868	−	0.948803i	$-0.602296\pi$
$727$	−17.0335	−0.631737	−0.315868	+	0.948803i	$0.602296\pi$
$728$	−36.6197	−1.35722
$729$	−7.87137	−0.291532
$730$	−34.0111	−1.25881
$731$	−1.00000	−0.0369863
$732$	−0.954325	−0.0352729
$733$	−12.9720	−0.479133	−0.239567	−	0.970880i	$-0.577005\pi$
$733$	−12.9720	−0.479133	−0.239567	+	0.970880i	$0.577005\pi$
$734$	18.1529	0.670035
$735$	−25.6665	−0.946723
$736$	−6.41895	−0.236606
$737$	−5.09482	−0.187670
$738$	−5.67368	−0.208851
$739$	−27.5274	−1.01261	−0.506307	−	0.862353i	$-0.668990\pi$
$739$	−27.5274	−1.01261	−0.506307	+	0.862353i	$0.668990\pi$
$740$	11.3571	0.417494
$741$	11.3383	0.416523
$742$	51.6049	1.89447
$743$	−28.4767	−1.04471	−0.522354	−	0.852728i	$-0.674946\pi$
$743$	−28.4767	−1.04471	−0.522354	+	0.852728i	$0.674946\pi$
$744$	7.26795	0.266456
$745$	−45.4936	−1.66676
$746$	−42.2478	−1.54680
$747$	8.27379	0.302722
$748$	−0.454508	−0.0166185
$749$	50.7807	1.85549
$750$	11.3536	0.414573
$751$	−47.3137	−1.72650	−0.863250	−	0.504776i	$-0.831575\pi$
$751$	−47.3137	−1.72650	−0.863250	+	0.504776i	$0.831575\pi$
$752$	14.9886	0.546576
$753$	−5.48908	−0.200033
$754$	−8.12726	−0.295977
$755$	−25.5670	−0.930479
$756$	8.15236	0.296498
$757$	−27.7097	−1.00712	−0.503562	−	0.863959i	$-0.667978\pi$
$757$	−27.7097	−1.00712	−0.503562	+	0.863959i	$0.667978\pi$
$758$	29.2768	1.06338
$759$	−1.60162	−0.0581350
$760$	−31.6962	−1.14974
$761$	15.2571	0.553068	0.276534	−	0.961004i	$-0.410814\pi$
$761$	15.2571	0.553068	0.276534	+	0.961004i	$0.410814\pi$
$762$	−9.78598	−0.354509
$763$	4.64726	0.168242
$764$	10.1608	0.367603
$765$	5.63461	0.203720
$766$	37.5481	1.35667
$767$	6.41978	0.231805
$768$	6.54345	0.236116
$769$	25.5780	0.922366	0.461183	−	0.887305i	$-0.347425\pi$
$769$	25.5780	0.922366	0.461183	+	0.887305i	$0.347425\pi$
$770$	−17.2327	−0.621023
$771$	−11.9356	−0.429850
$772$	8.04907	0.289692
$773$	−26.5470	−0.954829	−0.477415	−	0.878678i	$-0.658426\pi$
$773$	−26.5470	−0.954829	−0.477415	+	0.878678i	$0.658426\pi$
$774$	4.07837	0.146594
$775$	1.50058	0.0539023
$776$	26.3560	0.946126
$777$	36.9550	1.32575
$778$	−29.1699	−1.04579
$779$	−8.41351	−0.301445
$780$	1.84437	0.0660392
$781$	−13.6817	−0.489569
$782$	3.98333	0.142443
$783$	−6.15231	−0.219866
$784$	−88.5189	−3.16139
$785$	−21.6176	−0.771566
$786$	−20.7471	−0.740025
$787$	−6.66587	−0.237613	−0.118806	−	0.992917i	$-0.537907\pi$
$787$	−6.66587	−0.237613	−0.118806	+	0.992917i	$0.537907\pi$
$788$	−7.07488	−0.252032
$789$	−5.40879	−0.192558
$790$	−32.9618	−1.17273
$791$	60.3470	2.14569
$792$	−6.30309	−0.223971
$793$	9.92004	0.352271
$794$	−16.2062	−0.575135
$795$	8.83792	0.313449
$796$	5.27696	0.187037
$797$	−33.7256	−1.19462	−0.597311	−	0.802010i	$-0.703764\pi$
$797$	−33.7256	−1.19462	−0.597311	+	0.802010i	$0.703764\pi$
$798$	30.3311	1.07371
$799$	−3.18740	−0.112762
$800$	0.795042	0.0281090
$801$	2.84336	0.100465
$802$	−48.4025	−1.70915
$803$	−10.0295	−0.353934
$804$	1.45870	0.0514443
$805$	27.9663	0.985682
$806$	22.2180	0.782594
$807$	8.30996	0.292525
$808$	38.1697	1.34280
$809$	48.1316	1.69222	0.846108	−	0.533012i	$-0.178940\pi$
$809$	48.1316	1.69222	0.846108	+	0.533012i	$0.178940\pi$
$810$	−18.9431	−0.665592
$811$	−22.6159	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$811$	−22.6159	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$812$	−4.02589	−0.141281
$813$	13.3812	0.469299
$814$	18.0862	0.633922
$815$	33.0130	1.15639
$816$	−2.96223	−0.103699
$817$	6.04783	0.211587
$818$	19.3263	0.675727
$819$	−39.3705	−1.37572
$820$	−1.36860	−0.0477937
$821$	38.9210	1.35835	0.679177	−	0.733975i	$-0.262337\pi$
$821$	38.9210	1.35835	0.679177	+	0.733975i	$0.262337\pi$
$822$	−15.0613	−0.525322
$823$	−33.9889	−1.18478	−0.592389	−	0.805652i	$-0.701815\pi$
$823$	−33.9889	−1.18478	−0.592389	+	0.805652i	$0.701815\pi$
$824$	43.1046	1.50162
$825$	0.198374	0.00690650
$826$	17.1736	0.597546
$827$	25.1509	0.874582	0.437291	−	0.899320i	$-0.355938\pi$
$827$	25.1509	0.874582	0.437291	+	0.899320i	$0.355938\pi$
$828$	−3.00822	−0.104543
$829$	3.29669	0.114499	0.0572495	−	0.998360i	$-0.481767\pi$
$829$	3.29669	0.114499	0.0572495	+	0.998360i	$0.481767\pi$
$830$	10.7781	0.374112
$831$	−13.2866	−0.460908
$832$	−16.2186	−0.562280
$833$	18.8240	0.652214
$834$	−17.6243	−0.610279
$835$	−18.1749	−0.628967
$836$	2.74879	0.0950687
$837$	16.8189	0.581347
$838$	−27.5654	−0.952231
$839$	5.04326	0.174113	0.0870564	−	0.996203i	$-0.472254\pi$
$839$	5.04326	0.174113	0.0870564	+	0.996203i	$0.472254\pi$
$840$	−16.7770	−0.578862
$841$	−25.9618	−0.895234
$842$	20.8148	0.717324
$843$	9.84956	0.339237
$844$	3.21542	0.110679
$845$	8.96663	0.308461
$846$	12.9994	0.446929
$847$	−5.08174	−0.174611
$848$	30.4803	1.04670
$849$	8.73200	0.299682
$850$	−0.493369	−0.0169224
$851$	−29.3515	−1.00615
$852$	3.91720	0.134201
$853$	6.25291	0.214095	0.107048	−	0.994254i	$-0.465860\pi$
$853$	6.25291	0.214095	0.107048	+	0.994254i	$0.465860\pi$
$854$	26.5371	0.908082
$855$	−34.0771	−1.16541
$856$	24.1956	0.826987
$857$	−43.8554	−1.49807	−0.749037	−	0.662529i	$-0.769483\pi$
$857$	−43.8554	−1.49807	−0.749037	+	0.662529i	$0.769483\pi$
$858$	2.93718	0.100274
$859$	23.5808	0.804567	0.402283	−	0.915515i	$-0.368217\pi$
$859$	23.5808	0.804567	0.402283	+	0.915515i	$0.368217\pi$
$860$	0.983785	0.0335468
$861$	−4.45333	−0.151769
$862$	−15.4329	−0.525647
$863$	31.7604	1.08114	0.540568	−	0.841301i	$-0.318210\pi$
$863$	31.7604	1.08114	0.540568	+	0.841301i	$0.318210\pi$
$864$	8.91107	0.303161
$865$	33.3152	1.13275
$866$	−48.9652	−1.66390
$867$	0.629934	0.0213937
$868$	11.0058	0.373561
$869$	−9.72008	−0.329731
$870$	−3.72344	−0.126236
$871$	−15.1629	−0.513775
$872$	2.21428	0.0749852
$873$	28.3358	0.959021
$874$	−24.0905	−0.814872
$875$	−58.4611	−1.97635
$876$	2.87155	0.0970206
$877$	−27.9723	−0.944558	−0.472279	−	0.881449i	$-0.656569\pi$
$877$	−27.9723	−0.944558	−0.472279	+	0.881449i	$0.656569\pi$
$878$	49.4053	1.66735
$879$	−16.9233	−0.570808
$880$	−10.1785	−0.343116
$881$	−31.4662	−1.06012	−0.530062	−	0.847959i	$-0.677831\pi$
$881$	−31.4662	−1.06012	−0.530062	+	0.847959i	$0.677831\pi$
$882$	−76.7715	−2.58503
$883$	27.7318	0.933249	0.466625	−	0.884455i	$-0.345470\pi$
$883$	27.7318	0.933249	0.466625	+	0.884455i	$0.345470\pi$
$884$	−1.35268	−0.0454956
$885$	2.94117	0.0988664
$886$	13.4018	0.450242
$887$	14.3566	0.482046	0.241023	−	0.970519i	$-0.422517\pi$
$887$	14.3566	0.482046	0.241023	+	0.970519i	$0.422517\pi$
$888$	17.6080	0.590886
$889$	50.3894	1.69001
$890$	3.70398	0.124158
$891$	−5.58611	−0.187142
$892$	4.97866	0.166698
$893$	19.2768	0.645075
$894$	20.7429	0.693745
$895$	38.6188	1.29088
$896$	−69.0457	−2.30665
$897$	−4.76664	−0.159153
$898$	18.8530	0.629132
$899$	−8.30570	−0.277011
$900$	0.372594	0.0124198
$901$	−6.48181	−0.215940
$902$	−2.17951	−0.0725699
$903$	3.20116	0.106528
$904$	28.7536	0.956331
$905$	11.7369	0.390148
$906$	11.6573	0.387288
$907$	−1.95467	−0.0649037	−0.0324519	−	0.999473i	$-0.510332\pi$
$907$	−1.95467	−0.0649037	−0.0324519	+	0.999473i	$0.510332\pi$
$908$	−12.9466	−0.429647
$909$	41.0369	1.36111
$910$	−51.2869	−1.70015
$911$	−2.21168	−0.0732764	−0.0366382	−	0.999329i	$-0.511665\pi$
$911$	−2.21168	−0.0732764	−0.0366382	+	0.999329i	$0.511665\pi$
$912$	17.9150	0.593226
$913$	3.17833	0.105188
$914$	50.7787	1.67961
$915$	4.54479	0.150246
$916$	8.80424	0.290900
$917$	106.830	3.52783
$918$	−5.52983	−0.182512
$919$	−8.08206	−0.266602	−0.133301	−	0.991076i	$-0.542558\pi$
$919$	−8.08206	−0.266602	−0.133301	+	0.991076i	$0.542558\pi$
$920$	13.3251	0.439317
$921$	−1.70873	−0.0563044
$922$	−10.5025	−0.345880
$923$	−40.7186	−1.34027
$924$	1.45495	0.0478644
$925$	3.63543	0.119532
$926$	−13.1327	−0.431566
$927$	46.3425	1.52209
$928$	−4.40057	−0.144456
$929$	8.28327	0.271765	0.135883	−	0.990725i	$-0.456613\pi$
$929$	8.28327	0.271765	0.135883	+	0.990725i	$0.456613\pi$
$930$	10.1790	0.333782
$931$	−113.845	−3.73110
$932$	−0.0356396	−0.00116741
$933$	1.90913	0.0625020
$934$	11.2851	0.369260
$935$	2.16451	0.0707869
$936$	−18.7589	−0.613154
$937$	42.0003	1.37209	0.686045	−	0.727559i	$-0.259345\pi$
$937$	42.0003	1.37209	0.686045	+	0.727559i	$0.259345\pi$
$938$	−40.5623	−1.32441
$939$	−5.86096	−0.191265
$940$	3.13572	0.102276
$941$	−21.1720	−0.690186	−0.345093	−	0.938568i	$-0.612153\pi$
$941$	−21.1720	−0.690186	−0.345093	+	0.938568i	$0.612153\pi$
$942$	9.85658	0.321145
$943$	3.53705	0.115182
$944$	10.1435	0.330144
$945$	−38.8240	−1.26295
$946$	1.56669	0.0509374
$947$	−30.4354	−0.989019	−0.494509	−	0.869172i	$-0.664652\pi$
$947$	−30.4354	−0.989019	−0.494509	+	0.869172i	$0.664652\pi$
$948$	2.78296	0.0903862
$949$	−29.8492	−0.968948
$950$	2.98381	0.0968077
$951$	0.0704353	0.00228402
$952$	12.3044	0.398788
$953$	−24.5406	−0.794947	−0.397473	−	0.917614i	$-0.630113\pi$
$953$	−24.5406	−0.794947	−0.397473	+	0.917614i	$0.630113\pi$
$954$	26.4353	0.855873
$955$	−48.3887	−1.56582
$956$	−13.4588	−0.435288
$957$	−1.09800	−0.0354934
$958$	28.4566	0.919391
$959$	77.5526	2.50430
$960$	−7.43044	−0.239816
$961$	−8.29423	−0.267556
$962$	53.8272	1.73546
$963$	26.0131	0.838259
$964$	−4.98306	−0.160494
$965$	−38.3321	−1.23396
$966$	−12.7513	−0.410265
$967$	1.11302	0.0357922	0.0178961	−	0.999840i	$-0.494303\pi$
$967$	1.11302	0.0357922	0.0178961	+	0.999840i	$0.494303\pi$
$968$	−2.42130	−0.0778236
$969$	−3.80973	−0.122386
$970$	36.9124	1.18518
$971$	4.40124	0.141243	0.0706213	−	0.997503i	$-0.477502\pi$
$971$	4.40124	0.141243	0.0706213	+	0.997503i	$0.477502\pi$
$972$	6.41210	0.205668
$973$	90.7500	2.90931
$974$	−8.00749	−0.256576
$975$	0.590390	0.0189076
$976$	15.6741	0.501716
$977$	−3.61620	−0.115692	−0.0578462	−	0.998326i	$-0.518423\pi$
$977$	−3.61620	−0.115692	−0.0578462	+	0.998326i	$0.518423\pi$
$978$	−15.0523	−0.481319
$979$	1.09226	0.0349089
$980$	−18.5188	−0.591562
$981$	2.38062	0.0760072
$982$	−17.5488	−0.560005
$983$	−46.1466	−1.47185	−0.735925	−	0.677063i	$-0.763252\pi$
$983$	−46.1466	−1.47185	−0.735925	+	0.677063i	$0.763252\pi$
$984$	−2.12188	−0.0676432
$985$	33.6928	1.07354
$986$	2.73080	0.0869665
$987$	10.2034	0.324777
$988$	8.18078	0.260265
$989$	−2.54252	−0.0808473
$990$	−8.82766	−0.280562
$991$	11.3089	0.359240	0.179620	−	0.983736i	$-0.442513\pi$
$991$	11.3089	0.359240	0.179620	+	0.983736i	$0.442513\pi$
$992$	12.0301	0.381955
$993$	6.52812	0.207164
$994$	−108.927	−3.45494
$995$	−25.1305	−0.796691
$996$	−0.909989	−0.0288341
$997$	1.28073	0.0405611	0.0202806	−	0.999794i	$-0.493544\pi$
$997$	1.28073	0.0405611	0.0202806	+	0.999794i	$0.493544\pi$
$998$	−28.3387	−0.897046
$999$	40.7470	1.28918

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.f.1.46	✓	66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.f.1.46	✓	66	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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56669 q^{2} +0.629934 q^{3} +0.454508 q^{4} -2.16451 q^{5} +0.986909 q^{6} -5.08174 q^{7} -2.42130 q^{8} -2.60318 q^{9} +O(q^{10})\)	\(q+1.56669 q^{2} +0.629934 q^{3} +0.454508 q^{4} -2.16451 q^{5} +0.986909 q^{6} -5.08174 q^{7} -2.42130 q^{8} -2.60318 q^{9} -3.39110 q^{10} -1.00000 q^{11} +0.286310 q^{12} -2.97614 q^{13} -7.96149 q^{14} -1.36350 q^{15} -4.70244 q^{16} +1.00000 q^{17} -4.07837 q^{18} -6.04783 q^{19} -0.983785 q^{20} -3.20116 q^{21} -1.56669 q^{22} +2.54252 q^{23} -1.52526 q^{24} -0.314913 q^{25} -4.66268 q^{26} -3.52963 q^{27} -2.30969 q^{28} +1.74304 q^{29} -2.13617 q^{30} -4.76506 q^{31} -2.52464 q^{32} -0.629934 q^{33} +1.56669 q^{34} +10.9994 q^{35} -1.18317 q^{36} -11.5443 q^{37} -9.47505 q^{38} -1.87477 q^{39} +5.24092 q^{40} +1.39116 q^{41} -5.01521 q^{42} -1.00000 q^{43} -0.454508 q^{44} +5.63461 q^{45} +3.98333 q^{46} -3.18740 q^{47} -2.96223 q^{48} +18.8240 q^{49} -0.493369 q^{50} +0.629934 q^{51} -1.35268 q^{52} -6.48181 q^{53} -5.52983 q^{54} +2.16451 q^{55} +12.3044 q^{56} -3.80973 q^{57} +2.73080 q^{58} -2.15708 q^{59} -0.619720 q^{60} -3.33319 q^{61} -7.46535 q^{62} +13.2287 q^{63} +5.44955 q^{64} +6.44188 q^{65} -0.986909 q^{66} +5.09482 q^{67} +0.454508 q^{68} +1.60162 q^{69} +17.2327 q^{70} +13.6817 q^{71} +6.30309 q^{72} +10.0295 q^{73} -18.0862 q^{74} -0.198374 q^{75} -2.74879 q^{76} +5.08174 q^{77} -2.93718 q^{78} +9.72008 q^{79} +10.1785 q^{80} +5.58611 q^{81} +2.17951 q^{82} -3.17833 q^{83} -1.45495 q^{84} -2.16451 q^{85} -1.56669 q^{86} +1.09800 q^{87} +2.42130 q^{88} -1.09226 q^{89} +8.82766 q^{90} +15.1240 q^{91} +1.15559 q^{92} -3.00167 q^{93} -4.99366 q^{94} +13.0906 q^{95} -1.59036 q^{96} -10.8851 q^{97} +29.4914 q^{98} +2.60318 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 12 q^{2} + 66 q^{4} + 6 q^{5} + 7 q^{6} + 13 q^{7} + 30 q^{8} + 58 q^{9}+O(q^{10}) \) 66 * q + 12 * q^2 + 66 * q^4 + 6 * q^5 + 7 * q^6 + 13 * q^7 + 30 * q^8 + 58 * q^9	\( 66 q + 12 q^{2} + 66 q^{4} + 6 q^{5} + 7 q^{6} + 13 q^{7} + 30 q^{8} + 58 q^{9} + 7 q^{10} - 66 q^{11} + 12 q^{12} + 12 q^{13} + 13 q^{14} + 35 q^{15} + 58 q^{16} + 66 q^{17} + 37 q^{18} + 24 q^{19} + 17 q^{20} + 16 q^{21} - 12 q^{22} + 25 q^{23} + 22 q^{24} + 56 q^{25} + 36 q^{26} + 17 q^{28} + 29 q^{29} + 28 q^{30} + 37 q^{31} + 62 q^{32} + 12 q^{34} + 40 q^{35} + 107 q^{36} - 34 q^{37} + 22 q^{38} + 61 q^{39} + 37 q^{40} + 41 q^{41} + 19 q^{42} - 66 q^{43} - 66 q^{44} + 10 q^{45} + 43 q^{46} + 61 q^{47} + 29 q^{48} + 33 q^{49} + 59 q^{50} + 51 q^{52} - 35 q^{53} - 37 q^{54} - 6 q^{55} + 37 q^{56} - 7 q^{57} + 17 q^{58} + 48 q^{59} - 56 q^{60} + q^{61} + 37 q^{62} + 43 q^{63} + 68 q^{64} + 41 q^{65} - 7 q^{66} + 10 q^{67} + 66 q^{68} + 18 q^{69} + 77 q^{70} + 84 q^{71} + 83 q^{72} + 5 q^{73} + 36 q^{74} + 14 q^{75} + 14 q^{76} - 13 q^{77} + 41 q^{78} + 58 q^{79} + 25 q^{80} + 78 q^{81} - 28 q^{82} + 47 q^{83} + 44 q^{84} + 6 q^{85} - 12 q^{86} + 101 q^{87} - 30 q^{88} + 53 q^{89} + q^{90} + 2 q^{91} + 34 q^{92} - 3 q^{93} + 17 q^{94} + 91 q^{95} + 27 q^{96} - 28 q^{97} + 87 q^{98} - 58 q^{99}+O(q^{100}) \) 66 * q + 12 * q^2 + 66 * q^4 + 6 * q^5 + 7 * q^6 + 13 * q^7 + 30 * q^8 + 58 * q^9 + 7 * q^10 - 66 * q^11 + 12 * q^12 + 12 * q^13 + 13 * q^14 + 35 * q^15 + 58 * q^16 + 66 * q^17 + 37 * q^18 + 24 * q^19 + 17 * q^20 + 16 * q^21 - 12 * q^22 + 25 * q^23 + 22 * q^24 + 56 * q^25 + 36 * q^26 + 17 * q^28 + 29 * q^29 + 28 * q^30 + 37 * q^31 + 62 * q^32 + 12 * q^34 + 40 * q^35 + 107 * q^36 - 34 * q^37 + 22 * q^38 + 61 * q^39 + 37 * q^40 + 41 * q^41 + 19 * q^42 - 66 * q^43 - 66 * q^44 + 10 * q^45 + 43 * q^46 + 61 * q^47 + 29 * q^48 + 33 * q^49 + 59 * q^50 + 51 * q^52 - 35 * q^53 - 37 * q^54 - 6 * q^55 + 37 * q^56 - 7 * q^57 + 17 * q^58 + 48 * q^59 - 56 * q^60 + q^61 + 37 * q^62 + 43 * q^63 + 68 * q^64 + 41 * q^65 - 7 * q^66 + 10 * q^67 + 66 * q^68 + 18 * q^69 + 77 * q^70 + 84 * q^71 + 83 * q^72 + 5 * q^73 + 36 * q^74 + 14 * q^75 + 14 * q^76 - 13 * q^77 + 41 * q^78 + 58 * q^79 + 25 * q^80 + 78 * q^81 - 28 * q^82 + 47 * q^83 + 44 * q^84 + 6 * q^85 - 12 * q^86 + 101 * q^87 - 30 * q^88 + 53 * q^89 + q^90 + 2 * q^91 + 34 * q^92 - 3 * q^93 + 17 * q^94 + 91 * q^95 + 27 * q^96 - 28 * q^97 + 87 * q^98 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more