LMFDB - Embedded newform 8041.2.a.e.1.9

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			1.9
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)

$f(q)$	$=$	$q-1.96819 q^{2} +1.90311 q^{3} +1.87379 q^{4} +3.64418 q^{5} -3.74569 q^{6} +3.36449 q^{7} +0.248405 q^{8} +0.621828 q^{9} +O(q^{10})$	\(q-1.96819 q^{2} +1.90311 q^{3} +1.87379 q^{4} +3.64418 q^{5} -3.74569 q^{6} +3.36449 q^{7} +0.248405 q^{8} +0.621828 q^{9} -7.17246 q^{10} -1.00000 q^{11} +3.56603 q^{12} +2.78123 q^{13} -6.62198 q^{14} +6.93528 q^{15} -4.23649 q^{16} -1.00000 q^{17} -1.22388 q^{18} -2.33626 q^{19} +6.82844 q^{20} +6.40300 q^{21} +1.96819 q^{22} +1.11768 q^{23} +0.472742 q^{24} +8.28008 q^{25} -5.47400 q^{26} -4.52592 q^{27} +6.30436 q^{28} +8.05116 q^{29} -13.6500 q^{30} +6.98619 q^{31} +7.84143 q^{32} -1.90311 q^{33} +1.96819 q^{34} +12.2608 q^{35} +1.16518 q^{36} +1.89369 q^{37} +4.59821 q^{38} +5.29299 q^{39} +0.905234 q^{40} -4.93501 q^{41} -12.6024 q^{42} +1.00000 q^{43} -1.87379 q^{44} +2.26606 q^{45} -2.19981 q^{46} +6.19172 q^{47} -8.06251 q^{48} +4.31981 q^{49} -16.2968 q^{50} -1.90311 q^{51} +5.21144 q^{52} -2.53589 q^{53} +8.90790 q^{54} -3.64418 q^{55} +0.835757 q^{56} -4.44615 q^{57} -15.8462 q^{58} -5.39255 q^{59} +12.9953 q^{60} -8.25877 q^{61} -13.7502 q^{62} +2.09214 q^{63} -6.96048 q^{64} +10.1353 q^{65} +3.74569 q^{66} -2.03405 q^{67} -1.87379 q^{68} +2.12706 q^{69} -24.1317 q^{70} +1.12689 q^{71} +0.154465 q^{72} -0.262475 q^{73} -3.72715 q^{74} +15.7579 q^{75} -4.37765 q^{76} -3.36449 q^{77} -10.4176 q^{78} +5.65063 q^{79} -15.4386 q^{80} -10.4788 q^{81} +9.71306 q^{82} -1.75725 q^{83} +11.9979 q^{84} -3.64418 q^{85} -1.96819 q^{86} +15.3222 q^{87} -0.248405 q^{88} +8.39773 q^{89} -4.46004 q^{90} +9.35743 q^{91} +2.09429 q^{92} +13.2955 q^{93} -12.1865 q^{94} -8.51375 q^{95} +14.9231 q^{96} -10.2247 q^{97} -8.50224 q^{98} -0.621828 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9}+O(q^{10}) $ 66 * q + 7 * q^2 + 3 * q^3 + 61 * q^4 + 4 * q^5 + 10 * q^6 + 14 * q^7 + 21 * q^8 + 65 * q^9	\( 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9} + 13 q^{10} - 66 q^{11} + 9 q^{12} - 12 q^{13} + 25 q^{14} + 13 q^{15} + 47 q^{16} - 66 q^{17} + 37 q^{18} + 19 q^{20} + 26 q^{21} - 7 q^{22} + 47 q^{23} + 15 q^{24} + 52 q^{25} + 16 q^{26} + 9 q^{27} + 3 q^{28} + 57 q^{29} + 2 q^{30} + 31 q^{31} + 39 q^{32} - 3 q^{33} - 7 q^{34} + 36 q^{35} + 39 q^{36} - 14 q^{37} + 18 q^{38} + 71 q^{39} + 29 q^{40} + 62 q^{41} - 3 q^{42} + 66 q^{43} - 61 q^{44} - 2 q^{45} + 19 q^{46} + 32 q^{47} + 26 q^{48} + 42 q^{49} + 10 q^{50} - 3 q^{51} - 7 q^{52} + 33 q^{53} + 100 q^{54} - 4 q^{55} + 61 q^{56} + 35 q^{57} - 16 q^{58} + 59 q^{59} + 50 q^{60} + 26 q^{61} + 29 q^{62} + 62 q^{63} + 29 q^{64} + 55 q^{65} - 10 q^{66} + 5 q^{67} - 61 q^{68} - 36 q^{69} - 35 q^{70} + 128 q^{71} + 87 q^{72} + 23 q^{73} + 64 q^{74} - 11 q^{75} + 74 q^{76} - 14 q^{77} + 45 q^{78} + 39 q^{79} + 95 q^{80} + 54 q^{81} - 6 q^{82} + 48 q^{83} + 38 q^{84} - 4 q^{85} + 7 q^{86} + 14 q^{87} - 21 q^{88} + 28 q^{89} + 135 q^{90} - 18 q^{91} + 108 q^{92} - 9 q^{93} + 37 q^{94} + 149 q^{95} + 104 q^{96} + 19 q^{97} + 30 q^{98} - 65 q^{99}+O(q^{100}) \) 66 * q + 7 * q^2 + 3 * q^3 + 61 * q^4 + 4 * q^5 + 10 * q^6 + 14 * q^7 + 21 * q^8 + 65 * q^9 + 13 * q^10 - 66 * q^11 + 9 * q^12 - 12 * q^13 + 25 * q^14 + 13 * q^15 + 47 * q^16 - 66 * q^17 + 37 * q^18 + 19 * q^20 + 26 * q^21 - 7 * q^22 + 47 * q^23 + 15 * q^24 + 52 * q^25 + 16 * q^26 + 9 * q^27 + 3 * q^28 + 57 * q^29 + 2 * q^30 + 31 * q^31 + 39 * q^32 - 3 * q^33 - 7 * q^34 + 36 * q^35 + 39 * q^36 - 14 * q^37 + 18 * q^38 + 71 * q^39 + 29 * q^40 + 62 * q^41 - 3 * q^42 + 66 * q^43 - 61 * q^44 - 2 * q^45 + 19 * q^46 + 32 * q^47 + 26 * q^48 + 42 * q^49 + 10 * q^50 - 3 * q^51 - 7 * q^52 + 33 * q^53 + 100 * q^54 - 4 * q^55 + 61 * q^56 + 35 * q^57 - 16 * q^58 + 59 * q^59 + 50 * q^60 + 26 * q^61 + 29 * q^62 + 62 * q^63 + 29 * q^64 + 55 * q^65 - 10 * q^66 + 5 * q^67 - 61 * q^68 - 36 * q^69 - 35 * q^70 + 128 * q^71 + 87 * q^72 + 23 * q^73 + 64 * q^74 - 11 * q^75 + 74 * q^76 - 14 * q^77 + 45 * q^78 + 39 * q^79 + 95 * q^80 + 54 * q^81 - 6 * q^82 + 48 * q^83 + 38 * q^84 - 4 * q^85 + 7 * q^86 + 14 * q^87 - 21 * q^88 + 28 * q^89 + 135 * q^90 - 18 * q^91 + 108 * q^92 - 9 * q^93 + 37 * q^94 + 149 * q^95 + 104 * q^96 + 19 * q^97 + 30 * q^98 - 65 * 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 $

$2$	−1.96819	−1.39172	−0.695862	−	0.718176i	$-0.744977\pi$
$2$	−1.96819	−1.39172	−0.695862	+	0.718176i	$0.744977\pi$
$3$	1.90311	1.09876	0.549381	−	0.835572i	$-0.314864\pi$
$3$	1.90311	1.09876	0.549381	+	0.835572i	$0.314864\pi$
$4$	1.87379	0.936895
$5$	3.64418	1.62973	0.814864	−	0.579652i	$-0.196811\pi$
$5$	3.64418	1.62973	0.814864	+	0.579652i	$0.196811\pi$
$6$	−3.74569	−1.52917
$7$	3.36449	1.27166	0.635829	−	0.771830i	$-0.280658\pi$
$7$	3.36449	1.27166	0.635829	+	0.771830i	$0.280658\pi$
$8$	0.248405	0.0878244
$9$	0.621828	0.207276
$10$	−7.17246	−2.26813
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	3.56603	1.02942
$13$	2.78123	0.771374	0.385687	−	0.922630i	$-0.373964\pi$
$13$	2.78123	0.771374	0.385687	+	0.922630i	$0.373964\pi$
$14$	−6.62198	−1.76980
$15$	6.93528	1.79068
$16$	−4.23649	−1.05912
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.22388	−0.288471
$19$	−2.33626	−0.535974	−0.267987	−	0.963423i	$-0.586358\pi$
$19$	−2.33626	−0.535974	−0.267987	+	0.963423i	$0.586358\pi$
$20$	6.82844	1.52689
$21$	6.40300	1.39725
$22$	1.96819	0.419621
$23$	1.11768	0.233052	0.116526	−	0.993188i	$-0.462824\pi$
$23$	1.11768	0.233052	0.116526	+	0.993188i	$0.462824\pi$
$24$	0.472742	0.0964981
$25$	8.28008	1.65602
$26$	−5.47400	−1.07354
$27$	−4.52592	−0.871014
$28$	6.30436	1.19141
$29$	8.05116	1.49506	0.747531	−	0.664226i	$-0.231239\pi$
$29$	8.05116	1.49506	0.747531	+	0.664226i	$0.231239\pi$
$30$	−13.6500	−2.49214
$31$	6.98619	1.25476	0.627378	−	0.778714i	$-0.284128\pi$
$31$	6.98619	1.25476	0.627378	+	0.778714i	$0.284128\pi$
$32$	7.84143	1.38618
$33$	−1.90311	−0.331289
$34$	1.96819	0.337543
$35$	12.2608	2.07246
$36$	1.16518	0.194196
$37$	1.89369	0.311321	0.155660	−	0.987811i	$-0.450249\pi$
$37$	1.89369	0.311321	0.155660	+	0.987811i	$0.450249\pi$
$38$	4.59821	0.745927
$39$	5.29299	0.847556
$40$	0.905234	0.143130
$41$	−4.93501	−0.770719	−0.385359	−	0.922767i	$-0.625923\pi$
$41$	−4.93501	−0.770719	−0.385359	+	0.922767i	$0.625923\pi$
$42$	−12.6024	−1.94459
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−1.87379	−0.282485
$45$	2.26606	0.337804
$46$	−2.19981	−0.324344
$47$	6.19172	0.903155	0.451578	−	0.892232i	$-0.350861\pi$
$47$	6.19172	0.903155	0.451578	+	0.892232i	$0.350861\pi$
$48$	−8.06251	−1.16372
$49$	4.31981	0.617116
$50$	−16.2968	−2.30472
$51$	−1.90311	−0.266489
$52$	5.21144	0.722697
$53$	−2.53589	−0.348331	−0.174165	−	0.984716i	$-0.555723\pi$
$53$	−2.53589	−0.348331	−0.174165	+	0.984716i	$0.555723\pi$
$54$	8.90790	1.21221
$55$	−3.64418	−0.491382
$56$	0.835757	0.111683
$57$	−4.44615	−0.588907
$58$	−15.8462	−2.08071
$59$	−5.39255	−0.702051	−0.351025	−	0.936366i	$-0.614167\pi$
$59$	−5.39255	−0.702051	−0.351025	+	0.936366i	$0.614167\pi$
$60$	12.9953	1.67768
$61$	−8.25877	−1.05743	−0.528714	−	0.848800i	$-0.677325\pi$
$61$	−8.25877	−1.05743	−0.528714	+	0.848800i	$0.677325\pi$
$62$	−13.7502	−1.74628
$63$	2.09214	0.263584
$64$	−6.96048	−0.870060
$65$	10.1353	1.25713
$66$	3.74569	0.461063
$67$	−2.03405	−0.248499	−0.124249	−	0.992251i	$-0.539652\pi$
$67$	−2.03405	−0.248499	−0.124249	+	0.992251i	$0.539652\pi$
$68$	−1.87379	−0.227230
$69$	2.12706	0.256068
$70$	−24.1317	−2.88429
$71$	1.12689	0.133738	0.0668688	−	0.997762i	$-0.478699\pi$
$71$	1.12689	0.133738	0.0668688	+	0.997762i	$0.478699\pi$
$72$	0.154465	0.0182039
$73$	−0.262475	−0.0307203	−0.0153602	−	0.999882i	$-0.504889\pi$
$73$	−0.262475	−0.0307203	−0.0153602	+	0.999882i	$0.504889\pi$
$74$	−3.72715	−0.433272
$75$	15.7579	1.81957
$76$	−4.37765	−0.502151
$77$	−3.36449	−0.383420
$78$	−10.4176	−1.17956
$79$	5.65063	0.635746	0.317873	−	0.948133i	$-0.397031\pi$
$79$	5.65063	0.635746	0.317873	+	0.948133i	$0.397031\pi$
$80$	−15.4386	−1.72608
$81$	−10.4788	−1.16431
$82$	9.71306	1.07263
$83$	−1.75725	−0.192883	−0.0964415	−	0.995339i	$-0.530746\pi$
$83$	−1.75725	−0.192883	−0.0964415	+	0.995339i	$0.530746\pi$
$84$	11.9979	1.30908
$85$	−3.64418	−0.395267
$86$	−1.96819	−0.212236
$87$	15.3222	1.64272
$88$	−0.248405	−0.0264801
$89$	8.39773	0.890157	0.445079	−	0.895492i	$-0.353176\pi$
$89$	8.39773	0.890157	0.445079	+	0.895492i	$0.353176\pi$
$90$	−4.46004	−0.470129
$91$	9.35743	0.980925
$92$	2.09429	0.218345
$93$	13.2955	1.37868
$94$	−12.1865	−1.25694
$95$	−8.51375	−0.873492
$96$	14.9231	1.52308
$97$	−10.2247	−1.03816	−0.519079	−	0.854726i	$-0.673725\pi$
$97$	−10.2247	−1.03816	−0.519079	+	0.854726i	$0.673725\pi$
$98$	−8.50224	−0.858856
$99$	−0.621828	−0.0624961
$100$	15.5151	1.55151
$101$	9.86117	0.981224	0.490612	−	0.871378i	$-0.336773\pi$
$101$	9.86117	0.981224	0.490612	+	0.871378i	$0.336773\pi$
$102$	3.74569	0.370879
$103$	3.58729	0.353466	0.176733	−	0.984259i	$-0.443447\pi$
$103$	3.58729	0.353466	0.176733	+	0.984259i	$0.443447\pi$
$104$	0.690871	0.0677455
$105$	23.3337	2.27714
$106$	4.99112	0.484781
$107$	8.03940	0.777198	0.388599	−	0.921407i	$-0.372959\pi$
$107$	8.03940	0.777198	0.388599	+	0.921407i	$0.372959\pi$
$108$	−8.48063	−0.816049
$109$	12.5532	1.20238	0.601189	−	0.799107i	$-0.294694\pi$
$109$	12.5532	1.20238	0.601189	+	0.799107i	$0.294694\pi$
$110$	7.17246	0.683868
$111$	3.60390	0.342067
$112$	−14.2536	−1.34684
$113$	−11.2256	−1.05602	−0.528010	−	0.849238i	$-0.677062\pi$
$113$	−11.2256	−1.05602	−0.528010	+	0.849238i	$0.677062\pi$
$114$	8.75089	0.819596
$115$	4.07302	0.379811
$116$	15.0862	1.40072
$117$	1.72945	0.159887
$118$	10.6136	0.977061
$119$	−3.36449	−0.308423
$120$	1.72276	0.157266
$121$	1.00000	0.0909091
$122$	16.2549	1.47165
$123$	−9.39187	−0.846836
$124$	13.0907	1.17558
$125$	11.9532	1.06913
$126$	−4.11773	−0.366837
$127$	−4.91659	−0.436277	−0.218139	−	0.975918i	$-0.569998\pi$
$127$	−4.91659	−0.436277	−0.218139	+	0.975918i	$0.569998\pi$
$128$	−1.98328	−0.175299
$129$	1.90311	0.167560
$130$	−19.9483	−1.74958
$131$	−3.05168	−0.266627	−0.133313	−	0.991074i	$-0.542562\pi$
$131$	−3.05168	−0.266627	−0.133313	+	0.991074i	$0.542562\pi$
$132$	−3.56603	−0.310383
$133$	−7.86032	−0.681576
$134$	4.00340	0.345841
$135$	−16.4933	−1.41952
$136$	−0.248405	−0.0213006
$137$	9.80444	0.837650	0.418825	−	0.908067i	$-0.362442\pi$
$137$	9.80444	0.837650	0.418825	+	0.908067i	$0.362442\pi$
$138$	−4.18647	−0.356376
$139$	16.2539	1.37864	0.689318	−	0.724459i	$-0.257910\pi$
$139$	16.2539	1.37864	0.689318	+	0.724459i	$0.257910\pi$
$140$	22.9742	1.94168
$141$	11.7835	0.992352
$142$	−2.21794	−0.186126
$143$	−2.78123	−0.232578
$144$	−2.63437	−0.219531
$145$	29.3399	2.43655
$146$	0.516601	0.0427542
$147$	8.22108	0.678064
$148$	3.54838	0.291675
$149$	0.192010	0.0157301	0.00786506	−	0.999969i	$-0.497496\pi$
$149$	0.192010	0.0157301	0.00786506	+	0.999969i	$0.497496\pi$
$150$	−31.0146	−2.53233
$151$	−9.30307	−0.757073	−0.378536	−	0.925586i	$-0.623573\pi$
$151$	−9.30307	−0.757073	−0.378536	+	0.925586i	$0.623573\pi$
$152$	−0.580338	−0.0470716
$153$	−0.621828	−0.0502718
$154$	6.62198	0.533614
$155$	25.4590	2.04491
$156$	9.91795	0.794071
$157$	11.6390	0.928896	0.464448	−	0.885600i	$-0.346253\pi$
$157$	11.6390	0.928896	0.464448	+	0.885600i	$0.346253\pi$
$158$	−11.1215	−0.884783
$159$	−4.82607	−0.382733
$160$	28.5756	2.25910
$161$	3.76042	0.296362
$162$	20.6243	1.62040
$163$	−10.6649	−0.835338	−0.417669	−	0.908599i	$-0.637153\pi$
$163$	−10.6649	−0.835338	−0.417669	+	0.908599i	$0.637153\pi$
$164$	−9.24717	−0.722083
$165$	−6.93528	−0.539911
$166$	3.45860	0.268440
$167$	13.1989	1.02136	0.510680	−	0.859771i	$-0.329394\pi$
$167$	13.1989	1.02136	0.510680	+	0.859771i	$0.329394\pi$
$168$	1.59054	0.122713
$169$	−5.26476	−0.404982
$170$	7.17246	0.550103
$171$	−1.45275	−0.111095
$172$	1.87379	0.142875
$173$	18.2926	1.39076	0.695379	−	0.718643i	$-0.255236\pi$
$173$	18.2926	1.39076	0.695379	+	0.718643i	$0.255236\pi$
$174$	−30.1572	−2.28621
$175$	27.8583	2.10589
$176$	4.23649	0.319337
$177$	−10.2626	−0.771386
$178$	−16.5284	−1.23885
$179$	−20.0032	−1.49511	−0.747553	−	0.664202i	$-0.768771\pi$
$179$	−20.0032	−1.49511	−0.747553	+	0.664202i	$0.768771\pi$
$180$	4.24611	0.316487
$181$	−15.0871	−1.12142	−0.560708	−	0.828014i	$-0.689471\pi$
$181$	−15.0871	−1.12142	−0.560708	+	0.828014i	$0.689471\pi$
$182$	−18.4172	−1.36518
$183$	−15.7173	−1.16186
$184$	0.277637	0.0204676
$185$	6.90096	0.507368
$186$	−26.1681	−1.91874
$187$	1.00000	0.0731272
$188$	11.6020	0.846162
$189$	−15.2274	−1.10763
$190$	16.7567	1.21566
$191$	−15.6883	−1.13517	−0.567584	−	0.823316i	$-0.692122\pi$
$191$	−15.6883	−1.13517	−0.567584	+	0.823316i	$0.692122\pi$
$192$	−13.2466	−0.955988
$193$	3.95637	0.284785	0.142393	−	0.989810i	$-0.454520\pi$
$193$	3.95637	0.284785	0.142393	+	0.989810i	$0.454520\pi$
$194$	20.1241	1.44483
$195$	19.2886	1.38129
$196$	8.09443	0.578173
$197$	24.1469	1.72040	0.860198	−	0.509961i	$-0.170340\pi$
$197$	24.1469	1.72040	0.860198	+	0.509961i	$0.170340\pi$
$198$	1.22388	0.0869773
$199$	−15.7903	−1.11935	−0.559673	−	0.828713i	$-0.689073\pi$
$199$	−15.7903	−1.11935	−0.559673	+	0.828713i	$0.689073\pi$
$200$	2.05681	0.145439
$201$	−3.87102	−0.273041
$202$	−19.4087	−1.36559
$203$	27.0881	1.90121
$204$	−3.56603	−0.249672
$205$	−17.9841	−1.25606
$206$	−7.06048	−0.491927
$207$	0.695003	0.0483061
$208$	−11.7827	−0.816980
$209$	2.33626	0.161602
$210$	−45.9253	−3.16915
$211$	10.3734	0.714134	0.357067	−	0.934079i	$-0.383777\pi$
$211$	10.3734	0.714134	0.357067	+	0.934079i	$0.383777\pi$
$212$	−4.75172	−0.326350
$213$	2.14460	0.146946
$214$	−15.8231	−1.08165
$215$	3.64418	0.248531
$216$	−1.12426	−0.0764963
$217$	23.5050	1.59562
$218$	−24.7071	−1.67338
$219$	−0.499518	−0.0337543
$220$	−6.82844	−0.460373
$221$	−2.78123	−0.187086
$222$	−7.09318	−0.476063
$223$	20.4049	1.36642	0.683208	−	0.730224i	$-0.260584\pi$
$223$	20.4049	1.36642	0.683208	+	0.730224i	$0.260584\pi$
$224$	26.3824	1.76275
$225$	5.14879	0.343252
$226$	22.0943	1.46969
$227$	−11.3912	−0.756063	−0.378032	−	0.925793i	$-0.623399\pi$
$227$	−11.3912	−0.756063	−0.378032	+	0.925793i	$0.623399\pi$
$228$	−8.33116	−0.551744
$229$	21.8994	1.44715	0.723577	−	0.690244i	$-0.242497\pi$
$229$	21.8994	1.44715	0.723577	+	0.690244i	$0.242497\pi$
$230$	−8.01650	−0.528592
$231$	−6.40300	−0.421287
$232$	1.99995	0.131303
$233$	5.12214	0.335562	0.167781	−	0.985824i	$-0.446340\pi$
$233$	5.12214	0.335562	0.167781	+	0.985824i	$0.446340\pi$
$234$	−3.40389	−0.222519
$235$	22.5638	1.47190
$236$	−10.1045	−0.657748
$237$	10.7538	0.698533
$238$	6.62198	0.429239
$239$	19.0055	1.22937	0.614683	−	0.788774i	$-0.289284\pi$
$239$	19.0055	1.22937	0.614683	+	0.788774i	$0.289284\pi$
$240$	−29.3813	−1.89655
$241$	−14.0790	−0.906909	−0.453454	−	0.891279i	$-0.649809\pi$
$241$	−14.0790	−0.906909	−0.453454	+	0.891279i	$0.649809\pi$
$242$	−1.96819	−0.126520
$243$	−6.36457	−0.408287
$244$	−15.4752	−0.990699
$245$	15.7422	1.00573
$246$	18.4850	1.17856
$247$	−6.49766	−0.413436
$248$	1.73540	0.110198
$249$	−3.34423	−0.211932
$250$	−23.5263	−1.48793
$251$	−1.96938	−0.124306	−0.0621530	−	0.998067i	$-0.519797\pi$
$251$	−1.96938	−0.124306	−0.0621530	+	0.998067i	$0.519797\pi$
$252$	3.92022	0.246951
$253$	−1.11768	−0.0702678
$254$	9.67681	0.607177
$255$	−6.93528	−0.434304
$256$	17.8244	1.11403
$257$	−11.8182	−0.737197	−0.368599	−	0.929589i	$-0.620162\pi$
$257$	−11.8182	−0.737197	−0.368599	+	0.929589i	$0.620162\pi$
$258$	−3.74569	−0.233197
$259$	6.37131	0.395894
$260$	18.9915	1.17780
$261$	5.00644	0.309891
$262$	6.00630	0.371071
$263$	−11.3527	−0.700040	−0.350020	−	0.936742i	$-0.613825\pi$
$263$	−11.3527	−0.700040	−0.350020	+	0.936742i	$0.613825\pi$
$264$	−0.472742	−0.0290953
$265$	−9.24124	−0.567685
$266$	15.4706	0.948565
$267$	15.9818	0.978070
$268$	−3.81138	−0.232817
$269$	−8.84332	−0.539186	−0.269593	−	0.962974i	$-0.586889\pi$
$269$	−8.84332	−0.539186	−0.269593	+	0.962974i	$0.586889\pi$
$270$	32.4620	1.97558
$271$	−8.75114	−0.531594	−0.265797	−	0.964029i	$-0.585635\pi$
$271$	−8.75114	−0.531594	−0.265797	+	0.964029i	$0.585635\pi$
$272$	4.23649	0.256875
$273$	17.8082	1.07780
$274$	−19.2971	−1.16578
$275$	−8.28008	−0.499308
$276$	3.98567	0.239909
$277$	−17.3922	−1.04500	−0.522499	−	0.852640i	$-0.675000\pi$
$277$	−17.3922	−1.04500	−0.522499	+	0.852640i	$0.675000\pi$
$278$	−31.9908	−1.91868
$279$	4.34421	0.260081
$280$	3.04565	0.182013
$281$	−4.98862	−0.297596	−0.148798	−	0.988868i	$-0.547540\pi$
$281$	−4.98862	−0.297596	−0.148798	+	0.988868i	$0.547540\pi$
$282$	−23.1923	−1.38108
$283$	21.4046	1.27237	0.636187	−	0.771535i	$-0.280511\pi$
$283$	21.4046	1.27237	0.636187	+	0.771535i	$0.280511\pi$
$284$	2.11156	0.125298
$285$	−16.2026	−0.959759
$286$	5.47400	0.323684
$287$	−16.6038	−0.980092
$288$	4.87602	0.287322
$289$	1.00000	0.0588235
$290$	−57.7467	−3.39100
$291$	−19.4587	−1.14069
$292$	−0.491822	−0.0287817
$293$	19.5884	1.14437	0.572183	−	0.820126i	$-0.306097\pi$
$293$	19.5884	1.14437	0.572183	+	0.820126i	$0.306097\pi$
$294$	−16.1807	−0.943677
$295$	−19.6515	−1.14415
$296$	0.470402	0.0273416
$297$	4.52592	0.262621
$298$	−0.377914	−0.0218920
$299$	3.10852	0.179770
$300$	29.5270	1.70474
$301$	3.36449	0.193926
$302$	18.3102	1.05364
$303$	18.7669	1.07813
$304$	9.89752	0.567662
$305$	−30.0965	−1.72332
$306$	1.22388	0.0699645
$307$	16.5983	0.947317	0.473658	−	0.880709i	$-0.342933\pi$
$307$	16.5983	0.947317	0.473658	+	0.880709i	$0.342933\pi$
$308$	−6.30436	−0.359224
$309$	6.82700	0.388375
$310$	−50.1082	−2.84596
$311$	16.6386	0.943489	0.471744	−	0.881735i	$-0.343624\pi$
$311$	16.6386	0.943489	0.471744	+	0.881735i	$0.343624\pi$
$312$	1.31480	0.0744361
$313$	−15.5561	−0.879284	−0.439642	−	0.898173i	$-0.644895\pi$
$313$	−15.5561	−0.879284	−0.439642	+	0.898173i	$0.644895\pi$
$314$	−22.9079	−1.29277
$315$	7.62413	0.429571
$316$	10.5881	0.595627
$317$	−31.2428	−1.75477	−0.877384	−	0.479789i	$-0.840713\pi$
$317$	−31.2428	−1.75477	−0.877384	+	0.479789i	$0.840713\pi$
$318$	9.49865	0.532658
$319$	−8.05116	−0.450778
$320$	−25.3653	−1.41796
$321$	15.2999	0.853955
$322$	−7.40123	−0.412455
$323$	2.33626	0.129993
$324$	−19.6351	−1.09084
$325$	23.0288	1.27741
$326$	20.9906	1.16256
$327$	23.8901	1.32113
$328$	−1.22588	−0.0676879
$329$	20.8320	1.14851
$330$	13.6500	0.751407
$331$	5.81348	0.319538	0.159769	−	0.987154i	$-0.448925\pi$
$331$	5.81348	0.319538	0.159769	+	0.987154i	$0.448925\pi$
$332$	−3.29271	−0.180711
$333$	1.17755	0.0645293
$334$	−25.9779	−1.42145
$335$	−7.41245	−0.404985
$336$	−27.1263	−1.47986
$337$	−12.0094	−0.654195	−0.327097	−	0.944991i	$-0.606071\pi$
$337$	−12.0094	−0.654195	−0.327097	+	0.944991i	$0.606071\pi$
$338$	10.3621	0.563623
$339$	−21.3636	−1.16031
$340$	−6.82844	−0.370324
$341$	−6.98619	−0.378323
$342$	2.85929	0.154613
$343$	−9.01747	−0.486897
$344$	0.248405	0.0133931
$345$	7.75141	0.417322
$346$	−36.0033	−1.93555
$347$	2.78906	0.149725	0.0748624	−	0.997194i	$-0.476148\pi$
$347$	2.78906	0.149725	0.0748624	+	0.997194i	$0.476148\pi$
$348$	28.7107	1.53905
$349$	−32.9632	−1.76448	−0.882240	−	0.470800i	$-0.843966\pi$
$349$	−32.9632	−1.76448	−0.882240	+	0.470800i	$0.843966\pi$
$350$	−54.8305	−2.93081
$351$	−12.5876	−0.671878
$352$	−7.84143	−0.417950
$353$	23.7085	1.26188	0.630938	−	0.775833i	$-0.282670\pi$
$353$	23.7085	1.26188	0.630938	+	0.775833i	$0.282670\pi$
$354$	20.1988	1.07356
$355$	4.10661	0.217956
$356$	15.7356	0.833984
$357$	−6.40300	−0.338883
$358$	39.3701	2.08077
$359$	−26.4946	−1.39833	−0.699166	−	0.714959i	$-0.746445\pi$
$359$	−26.4946	−1.39833	−0.699166	+	0.714959i	$0.746445\pi$
$360$	0.562900	0.0296674
$361$	−13.5419	−0.712732
$362$	29.6944	1.56070
$363$	1.90311	0.0998874
$364$	17.5339	0.919024
$365$	−0.956506	−0.0500658
$366$	30.9348	1.61699
$367$	26.7109	1.39430	0.697149	−	0.716927i	$-0.254452\pi$
$367$	26.7109	1.39430	0.697149	+	0.716927i	$0.254452\pi$
$368$	−4.73503	−0.246830
$369$	−3.06873	−0.159752
$370$	−13.5824	−0.706117
$371$	−8.53198	−0.442958
$372$	24.9130	1.29168
$373$	15.6399	0.809803	0.404902	−	0.914360i	$-0.367306\pi$
$373$	15.6399	0.809803	0.404902	+	0.914360i	$0.367306\pi$
$374$	−1.96819	−0.101773
$375$	22.7483	1.17472
$376$	1.53805	0.0793191
$377$	22.3921	1.15325
$378$	29.9706	1.54152
$379$	−13.0215	−0.668869	−0.334435	−	0.942419i	$-0.608545\pi$
$379$	−13.0215	−0.668869	−0.334435	+	0.942419i	$0.608545\pi$
$380$	−15.9530	−0.818370
$381$	−9.35682	−0.479364
$382$	30.8777	1.57984
$383$	19.0675	0.974303	0.487151	−	0.873318i	$-0.338036\pi$
$383$	19.0675	0.974303	0.487151	+	0.873318i	$0.338036\pi$
$384$	−3.77441	−0.192612
$385$	−12.2608	−0.624870
$386$	−7.78690	−0.396343
$387$	0.621828	0.0316093
$388$	−19.1589	−0.972645
$389$	0.833128	0.0422413	0.0211206	−	0.999777i	$-0.493277\pi$
$389$	0.833128	0.0422413	0.0211206	+	0.999777i	$0.493277\pi$
$390$	−37.9638	−1.92237
$391$	−1.11768	−0.0565234
$392$	1.07306	0.0541979
$393$	−5.80768	−0.292959
$394$	−47.5258	−2.39432
$395$	20.5920	1.03609
$396$	−1.16518	−0.0585523
$397$	−6.85972	−0.344279	−0.172140	−	0.985073i	$-0.555068\pi$
$397$	−6.85972	−0.344279	−0.172140	+	0.985073i	$0.555068\pi$
$398$	31.0784	1.55782
$399$	−14.9590	−0.748889
$400$	−35.0785	−1.75392
$401$	12.3090	0.614680	0.307340	−	0.951600i	$-0.400561\pi$
$401$	12.3090	0.614680	0.307340	+	0.951600i	$0.400561\pi$
$402$	7.61892	0.379997
$403$	19.4302	0.967887
$404$	18.4778	0.919304
$405$	−38.1867	−1.89751
$406$	−53.3146	−2.64596
$407$	−1.89369	−0.0938667
$408$	−0.472742	−0.0234042
$409$	−5.28067	−0.261112	−0.130556	−	0.991441i	$-0.541676\pi$
$409$	−5.28067	−0.261112	−0.130556	+	0.991441i	$0.541676\pi$
$410$	35.3962	1.74809
$411$	18.6589	0.920378
$412$	6.72183	0.331161
$413$	−18.1432	−0.892769
$414$	−1.36790	−0.0672287
$415$	−6.40373	−0.314347
$416$	21.8088	1.06926
$417$	30.9329	1.51479
$418$	−4.59821	−0.224906
$419$	−24.7416	−1.20871	−0.604353	−	0.796716i	$-0.706568\pi$
$419$	−24.7416	−1.20871	−0.604353	+	0.796716i	$0.706568\pi$
$420$	43.7225	2.13344
$421$	−28.2748	−1.37803	−0.689016	−	0.724747i	$-0.741957\pi$
$421$	−28.2748	−1.37803	−0.689016	+	0.724747i	$0.741957\pi$
$422$	−20.4169	−0.993877
$423$	3.85019	0.187202
$424$	−0.629927	−0.0305920
$425$	−8.28008	−0.401643
$426$	−4.22099	−0.204508
$427$	−27.7866	−1.34469
$428$	15.0641	0.728153
$429$	−5.29299	−0.255548
$430$	−7.17246	−0.345887
$431$	−10.9189	−0.525947	−0.262973	−	0.964803i	$-0.584703\pi$
$431$	−10.9189	−0.525947	−0.262973	+	0.964803i	$0.584703\pi$
$432$	19.1740	0.922511
$433$	−0.224869	−0.0108065	−0.00540326	−	0.999985i	$-0.501720\pi$
$433$	−0.224869	−0.0108065	−0.00540326	+	0.999985i	$0.501720\pi$
$434$	−46.2624	−2.22067
$435$	55.8371	2.67718
$436$	23.5220	1.12650
$437$	−2.61118	−0.124910
$438$	0.983149	0.0469767
$439$	16.9203	0.807560	0.403780	−	0.914856i	$-0.367696\pi$
$439$	16.9203	0.807560	0.403780	+	0.914856i	$0.367696\pi$
$440$	−0.905234	−0.0431553
$441$	2.68618	0.127913
$442$	5.47400	0.260372
$443$	−14.7798	−0.702208	−0.351104	−	0.936336i	$-0.614194\pi$
$443$	−14.7798	−0.702208	−0.351104	+	0.936336i	$0.614194\pi$
$444$	6.75295	0.320481
$445$	30.6029	1.45071
$446$	−40.1609	−1.90167
$447$	0.365417	0.0172836
$448$	−23.4185	−1.10642
$449$	−25.5680	−1.20663	−0.603313	−	0.797504i	$-0.706153\pi$
$449$	−25.5680	−1.20663	−0.603313	+	0.797504i	$0.706153\pi$
$450$	−10.1338	−0.477713
$451$	4.93501	0.232380
$452$	−21.0345	−0.989380
$453$	−17.7048	−0.831842
$454$	22.4202	1.05223
$455$	34.1002	1.59864
$456$	−1.10445	−0.0517204
$457$	−27.8026	−1.30055	−0.650275	−	0.759699i	$-0.725346\pi$
$457$	−27.8026	−1.30055	−0.650275	+	0.759699i	$0.725346\pi$
$458$	−43.1023	−2.01404
$459$	4.52592	0.211252
$460$	7.63199	0.355843
$461$	1.85943	0.0866022	0.0433011	−	0.999062i	$-0.486213\pi$
$461$	1.85943	0.0866022	0.0433011	+	0.999062i	$0.486213\pi$
$462$	12.6024	0.586315
$463$	−22.1654	−1.03012	−0.515058	−	0.857156i	$-0.672229\pi$
$463$	−22.1654	−1.03012	−0.515058	+	0.857156i	$0.672229\pi$
$464$	−34.1087	−1.58345
$465$	48.4512	2.24687
$466$	−10.0814	−0.467010
$467$	0.852281	0.0394389	0.0197194	−	0.999806i	$-0.493723\pi$
$467$	0.852281	0.0394389	0.0197194	+	0.999806i	$0.493723\pi$
$468$	3.24062	0.149798
$469$	−6.84354	−0.316006
$470$	−44.4099	−2.04848
$471$	22.1504	1.02063
$472$	−1.33954	−0.0616572
$473$	−1.00000	−0.0459800
$474$	−21.1655	−0.972165
$475$	−19.3444	−0.887581
$476$	−6.30436	−0.288960
$477$	−1.57689	−0.0722007
$478$	−37.4066	−1.71094
$479$	25.1947	1.15117	0.575587	−	0.817741i	$-0.304774\pi$
$479$	25.1947	1.15117	0.575587	+	0.817741i	$0.304774\pi$
$480$	54.3825	2.48221
$481$	5.26679	0.240145
$482$	27.7102	1.26217
$483$	7.15649	0.325632
$484$	1.87379	0.0851723
$485$	−37.2606	−1.69192
$486$	12.5267	0.568223
$487$	−2.50131	−0.113345	−0.0566726	−	0.998393i	$-0.518049\pi$
$487$	−2.50131	−0.113345	−0.0566726	+	0.998393i	$0.518049\pi$
$488$	−2.05152	−0.0928679
$489$	−20.2964	−0.917837
$490$	−30.9837	−1.39970
$491$	17.1685	0.774804	0.387402	−	0.921911i	$-0.373373\pi$
$491$	17.1685	0.774804	0.387402	+	0.921911i	$0.373373\pi$
$492$	−17.5984	−0.793397
$493$	−8.05116	−0.362606
$494$	12.7887	0.575389
$495$	−2.26606	−0.101852
$496$	−29.5969	−1.32894
$497$	3.79142	0.170069
$498$	6.58210	0.294951
$499$	−27.9513	−1.25127	−0.625635	−	0.780116i	$-0.715160\pi$
$499$	−27.9513	−1.25127	−0.625635	+	0.780116i	$0.715160\pi$
$500$	22.3978	1.00166
$501$	25.1189	1.12223
$502$	3.87612	0.173000
$503$	10.7062	0.477365	0.238682	−	0.971098i	$-0.423285\pi$
$503$	10.7062	0.477365	0.238682	+	0.971098i	$0.423285\pi$
$504$	0.519697	0.0231491
$505$	35.9359	1.59913
$506$	2.19981	0.0977933
$507$	−10.0194	−0.444978
$508$	−9.21266	−0.408746
$509$	−3.92988	−0.174189	−0.0870944	−	0.996200i	$-0.527758\pi$
$509$	−3.92988	−0.174189	−0.0870944	+	0.996200i	$0.527758\pi$
$510$	13.6500	0.604432
$511$	−0.883094	−0.0390658
$512$	−31.1154	−1.37512
$513$	10.5737	0.466841
$514$	23.2605	1.02598
$515$	13.0727	0.576054
$516$	3.56603	0.156986
$517$	−6.19172	−0.272312
$518$	−12.5400	−0.550975
$519$	34.8128	1.52811
$520$	2.51766	0.110407
$521$	−0.262435	−0.0114975	−0.00574874	−	0.999983i	$-0.501830\pi$
$521$	−0.262435	−0.0114975	−0.00574874	+	0.999983i	$0.501830\pi$
$522$	−9.85364	−0.431282
$523$	−22.3487	−0.977242	−0.488621	−	0.872496i	$-0.662500\pi$
$523$	−22.3487	−0.977242	−0.488621	+	0.872496i	$0.662500\pi$
$524$	−5.71821	−0.249801
$525$	53.0174	2.31387
$526$	22.3444	0.974263
$527$	−6.98619	−0.304323
$528$	8.06251	0.350876
$529$	−21.7508	−0.945687
$530$	18.1886	0.790061
$531$	−3.35324	−0.145518
$532$	−14.7286	−0.638565
$533$	−13.7254	−0.594513
$534$	−31.4553	−1.36120
$535$	29.2971	1.26662
$536$	−0.505268	−0.0218243
$537$	−38.0682	−1.64276
$538$	17.4054	0.750399
$539$	−4.31981	−0.186068
$540$	−30.9050	−1.32994
$541$	−0.571498	−0.0245706	−0.0122853	−	0.999925i	$-0.503911\pi$
$541$	−0.571498	−0.0245706	−0.0122853	+	0.999925i	$0.503911\pi$
$542$	17.2240	0.739832
$543$	−28.7124	−1.23217
$544$	−7.84143	−0.336198
$545$	45.7461	1.95955
$546$	−35.0500	−1.50000
$547$	28.5662	1.22140	0.610702	−	0.791861i	$-0.290887\pi$
$547$	28.5662	1.22140	0.610702	+	0.791861i	$0.290887\pi$
$548$	18.3715	0.784790
$549$	−5.13554	−0.219179
$550$	16.2968	0.694898
$551$	−18.8096	−0.801314
$552$	0.528373	0.0224891
$553$	19.0115	0.808452
$554$	34.2313	1.45435
$555$	13.1333	0.557477
$556$	30.4564	1.29164
$557$	−9.74116	−0.412746	−0.206373	−	0.978473i	$-0.566166\pi$
$557$	−9.74116	−0.412746	−0.206373	+	0.978473i	$0.566166\pi$
$558$	−8.55025	−0.361961
$559$	2.78123	0.117633
$560$	−51.9429	−2.19499
$561$	1.90311	0.0803494
$562$	9.81857	0.414172
$563$	−7.81263	−0.329263	−0.164631	−	0.986355i	$-0.552643\pi$
$563$	−7.81263	−0.329263	−0.164631	+	0.986355i	$0.552643\pi$
$564$	22.0799	0.929730
$565$	−40.9083	−1.72103
$566$	−42.1285	−1.77079
$567$	−35.2559	−1.48061
$568$	0.279926	0.0117454
$569$	−10.8061	−0.453014	−0.226507	−	0.974010i	$-0.572731\pi$
$569$	−10.8061	−0.453014	−0.226507	+	0.974010i	$0.572731\pi$
$570$	31.8899	1.33572
$571$	38.5344	1.61262	0.806308	−	0.591497i	$-0.201463\pi$
$571$	38.5344	1.61262	0.806308	+	0.591497i	$0.201463\pi$
$572$	−5.21144	−0.217901
$573$	−29.8566	−1.24728
$574$	32.6795	1.36402
$575$	9.25446	0.385938
$576$	−4.32822	−0.180342
$577$	33.2026	1.38224	0.691121	−	0.722739i	$-0.257117\pi$
$577$	33.2026	1.38224	0.691121	+	0.722739i	$0.257117\pi$
$578$	−1.96819	−0.0818661
$579$	7.52940	0.312911
$580$	54.9768	2.28279
$581$	−5.91225	−0.245281
$582$	38.2984	1.58752
$583$	2.53589	0.105026
$584$	−0.0652000	−0.00269800
$585$	6.30242	0.260573
$586$	−38.5538	−1.59264
$587$	−7.95955	−0.328526	−0.164263	−	0.986417i	$-0.552525\pi$
$587$	−7.95955	−0.328526	−0.164263	+	0.986417i	$0.552525\pi$
$588$	15.4046	0.635274
$589$	−16.3215	−0.672517
$590$	38.6779	1.59234
$591$	45.9542	1.89030
$592$	−8.02260	−0.329727
$593$	39.6527	1.62834	0.814172	−	0.580624i	$-0.197191\pi$
$593$	39.6527	1.62834	0.814172	+	0.580624i	$0.197191\pi$
$594$	−8.90790	−0.365495
$595$	−12.2608	−0.502645
$596$	0.359787	0.0147375
$597$	−30.0507	−1.22989
$598$	−6.11817	−0.250190
$599$	8.64178	0.353094	0.176547	−	0.984292i	$-0.443507\pi$
$599$	8.64178	0.353094	0.176547	+	0.984292i	$0.443507\pi$
$600$	3.91434	0.159802
$601$	−16.6141	−0.677704	−0.338852	−	0.940840i	$-0.610039\pi$
$601$	−16.6141	−0.677704	−0.338852	+	0.940840i	$0.610039\pi$
$602$	−6.62198	−0.269892
$603$	−1.26483	−0.0515078
$604$	−17.4320	−0.709298
$605$	3.64418	0.148157
$606$	−36.9369	−1.50046
$607$	−8.45775	−0.343290	−0.171645	−	0.985159i	$-0.554908\pi$
$607$	−8.45775	−0.343290	−0.171645	+	0.985159i	$0.554908\pi$
$608$	−18.3196	−0.742957
$609$	51.5516	2.08898
$610$	59.2357	2.39839
$611$	17.2206	0.696671
$612$	−1.16518	−0.0470994
$613$	−30.3629	−1.22635	−0.613173	−	0.789949i	$-0.710107\pi$
$613$	−30.3629	−1.22635	−0.613173	+	0.789949i	$0.710107\pi$
$614$	−32.6687	−1.31840
$615$	−34.2257	−1.38011
$616$	−0.835757	−0.0336736
$617$	30.4166	1.22452	0.612262	−	0.790655i	$-0.290260\pi$
$617$	30.4166	1.22452	0.612262	+	0.790655i	$0.290260\pi$
$618$	−13.4369	−0.540510
$619$	8.06109	0.324003	0.162001	−	0.986791i	$-0.448205\pi$
$619$	8.06109	0.324003	0.162001	+	0.986791i	$0.448205\pi$
$620$	47.7048	1.91587
$621$	−5.05852	−0.202991
$622$	−32.7480	−1.31308
$623$	28.2541	1.13198
$624$	−22.4237	−0.897666
$625$	2.15934	0.0863738
$626$	30.6175	1.22372
$627$	4.44615	0.177562
$628$	21.8091	0.870278
$629$	−1.89369	−0.0755064
$630$	−15.0058	−0.597844
$631$	−14.2622	−0.567769	−0.283885	−	0.958858i	$-0.591623\pi$
$631$	−14.2622	−0.567769	−0.283885	+	0.958858i	$0.591623\pi$
$632$	1.40365	0.0558340
$633$	19.7417	0.784662
$634$	61.4918	2.44215
$635$	−17.9170	−0.711013
$636$	−9.04305	−0.358580
$637$	12.0144	0.476028
$638$	15.8462	0.627359
$639$	0.700734	0.0277206
$640$	−7.22745	−0.285690
$641$	−8.05565	−0.318179	−0.159089	−	0.987264i	$-0.550856\pi$
$641$	−8.05565	−0.318179	−0.159089	+	0.987264i	$0.550856\pi$
$642$	−30.1131	−1.18847
$643$	−0.107736	−0.00424869	−0.00212434	−	0.999998i	$-0.500676\pi$
$643$	−0.107736	−0.00424869	−0.00212434	+	0.999998i	$0.500676\pi$
$644$	7.04624	0.277661
$645$	6.93528	0.273077
$646$	−4.59821	−0.180914
$647$	−17.2577	−0.678471	−0.339236	−	0.940701i	$-0.610168\pi$
$647$	−17.2577	−0.678471	−0.339236	+	0.940701i	$0.610168\pi$
$648$	−2.60299	−0.102255
$649$	5.39255	0.211676
$650$	−45.3252	−1.77780
$651$	44.7326	1.75321
$652$	−19.9838	−0.782624
$653$	−21.6962	−0.849038	−0.424519	−	0.905419i	$-0.639557\pi$
$653$	−21.6962	−0.849038	−0.424519	+	0.905419i	$0.639557\pi$
$654$	−47.0204	−1.83864
$655$	−11.1209	−0.434529
$656$	20.9071	0.816286
$657$	−0.163214	−0.00636759
$658$	−41.0014	−1.59840
$659$	17.2661	0.672592	0.336296	−	0.941756i	$-0.390826\pi$
$659$	17.2661	0.672592	0.336296	+	0.941756i	$0.390826\pi$
$660$	−12.9953	−0.505840
$661$	−46.1119	−1.79355	−0.896773	−	0.442491i	$-0.854095\pi$
$661$	−46.1119	−1.79355	−0.896773	+	0.442491i	$0.854095\pi$
$662$	−11.4421	−0.444708
$663$	−5.29299	−0.205563
$664$	−0.436509	−0.0169398
$665$	−28.6444	−1.11078
$666$	−2.31765	−0.0898070
$667$	8.99860	0.348427
$668$	24.7319	0.956907
$669$	38.8328	1.50136
$670$	14.5891	0.563628
$671$	8.25877	0.318826
$672$	50.2087	1.93684
$673$	5.74721	0.221539	0.110769	−	0.993846i	$-0.464668\pi$
$673$	5.74721	0.221539	0.110769	+	0.993846i	$0.464668\pi$
$674$	23.6369	0.910458
$675$	−37.4750	−1.44241
$676$	−9.86506	−0.379426
$677$	−5.56198	−0.213764	−0.106882	−	0.994272i	$-0.534087\pi$
$677$	−5.56198	−0.213764	−0.106882	+	0.994272i	$0.534087\pi$
$678$	42.0478	1.61484
$679$	−34.4008	−1.32018
$680$	−0.905234	−0.0347141
$681$	−21.6788	−0.830733
$682$	13.7502	0.526522
$683$	4.62977	0.177153	0.0885767	−	0.996069i	$-0.471768\pi$
$683$	4.62977	0.177153	0.0885767	+	0.996069i	$0.471768\pi$
$684$	−2.72215	−0.104084
$685$	35.7292	1.36514
$686$	17.7481	0.677627
$687$	41.6770	1.59008
$688$	−4.23649	−0.161515
$689$	−7.05289	−0.268694
$690$	−15.2563	−0.580797
$691$	30.3585	1.15489	0.577446	−	0.816429i	$-0.304049\pi$
$691$	30.3585	1.15489	0.577446	+	0.816429i	$0.304049\pi$
$692$	34.2765	1.30300
$693$	−2.09214	−0.0794737
$694$	−5.48942	−0.208376
$695$	59.2321	2.24680
$696$	3.80612	0.144271
$697$	4.93501	0.186927
$698$	64.8780	2.45567
$699$	9.74799	0.368703
$700$	52.2006	1.97300
$701$	−40.7165	−1.53784	−0.768920	−	0.639345i	$-0.779205\pi$
$701$	−40.7165	−1.53784	−0.768920	+	0.639345i	$0.779205\pi$
$702$	24.7749	0.935069
$703$	−4.42414	−0.166860
$704$	6.96048	0.262333
$705$	42.9413	1.61726
$706$	−46.6630	−1.75618
$707$	33.1779	1.24778
$708$	−19.2300	−0.722708
$709$	−40.7976	−1.53219	−0.766093	−	0.642730i	$-0.777802\pi$
$709$	−40.7976	−1.53219	−0.766093	+	0.642730i	$0.777802\pi$
$710$	−8.08260	−0.303335
$711$	3.51372	0.131775
$712$	2.08604	0.0781775
$713$	7.80831	0.292423
$714$	12.6024	0.471631
$715$	−10.1353	−0.379039
$716$	−37.4817	−1.40076
$717$	36.1696	1.35078
$718$	52.1466	1.94609
$719$	−11.3019	−0.421490	−0.210745	−	0.977541i	$-0.567589\pi$
$719$	−11.3019	−0.421490	−0.210745	+	0.977541i	$0.567589\pi$
$720$	−9.60012	−0.357776
$721$	12.0694	0.449488
$722$	26.6531	0.991926
$723$	−26.7939	−0.996476
$724$	−28.2701	−1.05065
$725$	66.6643	2.47585
$726$	−3.74569	−0.139016
$727$	1.80028	0.0667688	0.0333844	−	0.999443i	$-0.489371\pi$
$727$	1.80028	0.0667688	0.0333844	+	0.999443i	$0.489371\pi$
$728$	2.32443	0.0861492
$729$	19.3240	0.715703
$730$	1.88259	0.0696778
$731$	−1.00000	−0.0369863
$732$	−29.4510	−1.08854
$733$	29.5082	1.08991	0.544955	−	0.838465i	$-0.316547\pi$
$733$	29.5082	1.08991	0.544955	+	0.838465i	$0.316547\pi$
$734$	−52.5722	−1.94048
$735$	29.9591	1.10506
$736$	8.76419	0.323052
$737$	2.03405	0.0749252
$738$	6.03985	0.222330
$739$	15.9719	0.587535	0.293768	−	0.955877i	$-0.405091\pi$
$739$	15.9719	0.587535	0.293768	+	0.955877i	$0.405091\pi$
$740$	12.9309	0.475351
$741$	−12.3658	−0.454268
$742$	16.7926	0.616475
$743$	28.6834	1.05229	0.526146	−	0.850394i	$-0.323637\pi$
$743$	28.6834	1.05229	0.526146	+	0.850394i	$0.323637\pi$
$744$	3.30267	0.121082
$745$	0.699722	0.0256358
$746$	−30.7824	−1.12702
$747$	−1.09271	−0.0399800
$748$	1.87379	0.0685126
$749$	27.0485	0.988331
$750$	−44.7731	−1.63488
$751$	18.5827	0.678093	0.339047	−	0.940770i	$-0.389896\pi$
$751$	18.5827	0.678093	0.339047	+	0.940770i	$0.389896\pi$
$752$	−26.2312	−0.956552
$753$	−3.74794	−0.136583
$754$	−44.0721	−1.60501
$755$	−33.9021	−1.23382
$756$	−28.5330	−1.03774
$757$	43.7661	1.59071	0.795353	−	0.606146i	$-0.207285\pi$
$757$	43.7661	1.59071	0.795353	+	0.606146i	$0.207285\pi$
$758$	25.6288	0.930881
$759$	−2.12706	−0.0772075
$760$	−2.11486	−0.0767139
$761$	−33.8257	−1.22618	−0.613090	−	0.790013i	$-0.710074\pi$
$761$	−33.8257	−1.22618	−0.613090	+	0.790013i	$0.710074\pi$
$762$	18.4160	0.667143
$763$	42.2351	1.52901
$764$	−29.3966	−1.06353
$765$	−2.26606	−0.0819294
$766$	−37.5285	−1.35596
$767$	−14.9979	−0.541544
$768$	33.9219	1.22405
$769$	−39.0182	−1.40703	−0.703516	−	0.710680i	$-0.748388\pi$
$769$	−39.0182	−1.40703	−0.703516	+	0.710680i	$0.748388\pi$
$770$	24.1317	0.869646
$771$	−22.4913	−0.810004
$772$	7.41340	0.266814
$773$	−4.62433	−0.166326	−0.0831628	−	0.996536i	$-0.526502\pi$
$773$	−4.62433	−0.166326	−0.0831628	+	0.996536i	$0.526502\pi$
$774$	−1.22388	−0.0439914
$775$	57.8462	2.07790
$776$	−2.53986	−0.0911756
$777$	12.1253	0.434993
$778$	−1.63976	−0.0587882
$779$	11.5294	0.413085
$780$	36.1428	1.29412
$781$	−1.12689	−0.0403234
$782$	2.19981	0.0786649
$783$	−36.4389	−1.30222
$784$	−18.3009	−0.653602
$785$	42.4148	1.51385
$786$	11.4307	0.407718
$787$	−39.2452	−1.39894	−0.699470	−	0.714662i	$-0.746581\pi$
$787$	−39.2452	−1.39894	−0.699470	+	0.714662i	$0.746581\pi$
$788$	45.2462	1.61183
$789$	−21.6055	−0.769177
$790$	−40.5290	−1.44196
$791$	−37.7686	−1.34290
$792$	−0.154465	−0.00548868
$793$	−22.9695	−0.815672
$794$	13.5013	0.479142
$795$	−17.5871	−0.623750
$796$	−29.5878	−1.04871
$797$	−9.04572	−0.320416	−0.160208	−	0.987083i	$-0.551216\pi$
$797$	−9.04572	−0.320416	−0.160208	+	0.987083i	$0.551216\pi$
$798$	29.4423	1.04225
$799$	−6.19172	−0.219047
$800$	64.9277	2.29554
$801$	5.22194	0.184508
$802$	−24.2264	−0.855465
$803$	0.262475	0.00926253
$804$	−7.25348	−0.255810
$805$	13.7037	0.482990
$806$	−38.2424	−1.34703
$807$	−16.8298	−0.592437
$808$	2.44956	0.0861754
$809$	−13.4678	−0.473504	−0.236752	−	0.971570i	$-0.576083\pi$
$809$	−13.4678	−0.473504	−0.236752	+	0.971570i	$0.576083\pi$
$810$	75.1589	2.64082
$811$	−0.930373	−0.0326698	−0.0163349	−	0.999867i	$-0.505200\pi$
$811$	−0.930373	−0.0326698	−0.0163349	+	0.999867i	$0.505200\pi$
$812$	50.7574	1.78123
$813$	−16.6544	−0.584095
$814$	3.72715	0.130637
$815$	−38.8648	−1.36137
$816$	8.06251	0.282244
$817$	−2.33626	−0.0817352
$818$	10.3934	0.363396
$819$	5.81871	0.203322
$820$	−33.6984	−1.17680
$821$	11.3437	0.395898	0.197949	−	0.980212i	$-0.436572\pi$
$821$	11.3437	0.395898	0.197949	+	0.980212i	$0.436572\pi$
$822$	−36.7244	−1.28091
$823$	−43.6572	−1.52179	−0.760897	−	0.648873i	$-0.775241\pi$
$823$	−43.6572	−1.52179	−0.760897	+	0.648873i	$0.775241\pi$
$824$	0.891100	0.0310429
$825$	−15.7579	−0.548620
$826$	35.7094	1.24249
$827$	9.00874	0.313265	0.156632	−	0.987657i	$-0.449936\pi$
$827$	9.00874	0.313265	0.156632	+	0.987657i	$0.449936\pi$
$828$	1.30229	0.0452577
$829$	17.6970	0.614643	0.307321	−	0.951606i	$-0.400567\pi$
$829$	17.6970	0.614643	0.307321	+	0.951606i	$0.400567\pi$
$830$	12.6038	0.437484
$831$	−33.0994	−1.14820
$832$	−19.3587	−0.671141
$833$	−4.31981	−0.149673
$834$	−60.8820	−2.10817
$835$	48.0991	1.66454
$836$	4.37765	0.151404
$837$	−31.6190	−1.09291
$838$	48.6963	1.68219
$839$	−22.3481	−0.771543	−0.385771	−	0.922594i	$-0.626065\pi$
$839$	−22.3481	−0.771543	−0.385771	+	0.922594i	$0.626065\pi$
$840$	5.79621	0.199988
$841$	35.8212	1.23521
$842$	55.6504	1.91784
$843$	−9.49389	−0.326987
$844$	19.4376	0.669068
$845$	−19.1858	−0.660011
$846$	−7.57791	−0.260534
$847$	3.36449	0.115605
$848$	10.7433	0.368925
$849$	40.7354	1.39803
$850$	16.2968	0.558976
$851$	2.11653	0.0725539
$852$	4.01853	0.137673
$853$	−26.1667	−0.895932	−0.447966	−	0.894051i	$-0.647851\pi$
$853$	−26.1667	−0.895932	−0.447966	+	0.894051i	$0.647851\pi$
$854$	54.6894	1.87143
$855$	−5.29409	−0.181054
$856$	1.99703	0.0682570
$857$	46.6460	1.59340	0.796699	−	0.604376i	$-0.206577\pi$
$857$	46.6460	1.59340	0.796699	+	0.604376i	$0.206577\pi$
$858$	10.4176	0.355652
$859$	−3.80756	−0.129912	−0.0649561	−	0.997888i	$-0.520691\pi$
$859$	−3.80756	−0.129912	−0.0649561	+	0.997888i	$0.520691\pi$
$860$	6.82844	0.232848
$861$	−31.5989	−1.07689
$862$	21.4906	0.731973
$863$	−36.9929	−1.25925	−0.629627	−	0.776898i	$-0.716792\pi$
$863$	−36.9929	−1.25925	−0.629627	+	0.776898i	$0.716792\pi$
$864$	−35.4897	−1.20738
$865$	66.6615	2.26656
$866$	0.442587	0.0150397
$867$	1.90311	0.0646330
$868$	44.0434	1.49493
$869$	−5.65063	−0.191685
$870$	−109.898	−3.72590
$871$	−5.65716	−0.191685
$872$	3.11828	0.105598
$873$	−6.35798	−0.215185
$874$	5.13931	0.173840
$875$	40.2165	1.35957
$876$	−0.935992	−0.0316242
$877$	−27.6133	−0.932435	−0.466217	−	0.884670i	$-0.654384\pi$
$877$	−27.6133	−0.932435	−0.466217	+	0.884670i	$0.654384\pi$
$878$	−33.3024	−1.12390
$879$	37.2789	1.25739
$880$	15.4386	0.520434
$881$	−34.7886	−1.17206	−0.586028	−	0.810291i	$-0.699309\pi$
$881$	−34.7886	−1.17206	−0.586028	+	0.810291i	$0.699309\pi$
$882$	−5.28693	−0.178020
$883$	−34.9233	−1.17526	−0.587632	−	0.809129i	$-0.699940\pi$
$883$	−34.9233	−1.17526	−0.587632	+	0.809129i	$0.699940\pi$
$884$	−5.21144	−0.175280
$885$	−37.3989	−1.25715
$886$	29.0895	0.977280
$887$	−8.59419	−0.288565	−0.144282	−	0.989537i	$-0.546087\pi$
$887$	−8.59419	−0.288565	−0.144282	+	0.989537i	$0.546087\pi$
$888$	0.895227	0.0300418
$889$	−16.5418	−0.554796
$890$	−60.2324	−2.01899
$891$	10.4788	0.351053
$892$	38.2346	1.28019
$893$	−14.4654	−0.484068
$894$	−0.719212	−0.0240540
$895$	−72.8952	−2.43662
$896$	−6.67274	−0.222921
$897$	5.91585	0.197524
$898$	50.3227	1.67929
$899$	56.2469	1.87594
$900$	9.64775	0.321592
$901$	2.53589	0.0844827
$902$	−9.71306	−0.323409
$903$	6.40300	0.213079
$904$	−2.78851	−0.0927443
$905$	−54.9802	−1.82760
$906$	34.8464	1.15769
$907$	13.5482	0.449862	0.224931	−	0.974375i	$-0.427784\pi$
$907$	13.5482	0.449862	0.224931	+	0.974375i	$0.427784\pi$
$908$	−21.3448	−0.708352
$909$	6.13195	0.203384
$910$	−67.1158	−2.22487
$911$	23.9144	0.792320	0.396160	−	0.918182i	$-0.370343\pi$
$911$	23.9144	0.792320	0.396160	+	0.918182i	$0.370343\pi$
$912$	18.8361	0.623725
$913$	1.75725	0.0581564
$914$	54.7209	1.81001
$915$	−57.2769	−1.89352
$916$	41.0349	1.35583
$917$	−10.2674	−0.339058
$918$	−8.90790	−0.294004
$919$	−49.3834	−1.62901	−0.814504	−	0.580158i	$-0.802991\pi$
$919$	−49.3834	−1.62901	−0.814504	+	0.580158i	$0.802991\pi$
$920$	1.01176	0.0333567
$921$	31.5884	1.04087
$922$	−3.65972	−0.120526
$923$	3.13415	0.103162
$924$	−11.9979	−0.394701
$925$	15.6799	0.515552
$926$	43.6259	1.43364
$927$	2.23068	0.0732650
$928$	63.1326	2.07243
$929$	38.0489	1.24834	0.624172	−	0.781287i	$-0.285436\pi$
$929$	38.0489	1.24834	0.624172	+	0.781287i	$0.285436\pi$
$930$	−95.3614	−3.12702
$931$	−10.0922	−0.330758
$932$	9.59781	0.314387
$933$	31.6651	1.03667
$934$	−1.67745	−0.0548880
$935$	3.64418	0.119178
$936$	0.429603	0.0140420
$937$	20.6367	0.674171	0.337086	−	0.941474i	$-0.390559\pi$
$937$	20.6367	0.674171	0.337086	+	0.941474i	$0.390559\pi$
$938$	13.4694	0.439792
$939$	−29.6050	−0.966123
$940$	42.2798	1.37901
$941$	18.8069	0.613088	0.306544	−	0.951856i	$-0.400827\pi$
$941$	18.8069	0.613088	0.306544	+	0.951856i	$0.400827\pi$
$942$	−43.5962	−1.42044
$943$	−5.51575	−0.179617
$944$	22.8455	0.743558
$945$	−55.4916	−1.80514
$946$	1.96819	0.0639915
$947$	30.0540	0.976625	0.488312	−	0.872669i	$-0.337613\pi$
$947$	30.0540	0.976625	0.488312	+	0.872669i	$0.337613\pi$
$948$	20.1503	0.654452
$949$	−0.730002	−0.0236969
$950$	38.0735	1.23527
$951$	−59.4584	−1.92807
$952$	−0.835757	−0.0270870
$953$	−10.1350	−0.328306	−0.164153	−	0.986435i	$-0.552489\pi$
$953$	−10.1350	−0.328306	−0.164153	+	0.986435i	$0.552489\pi$
$954$	3.10362	0.100483
$955$	−57.1711	−1.85001
$956$	35.6124	1.15179
$957$	−15.3222	−0.495298
$958$	−49.5880	−1.60212
$959$	32.9870	1.06521
$960$	−48.2729	−1.55800
$961$	17.8069	0.574415
$962$	−10.3661	−0.334215
$963$	4.99912	0.161095
$964$	−26.3811	−0.849679
$965$	14.4177	0.464123
$966$	−14.0854	−0.453189
$967$	−32.6070	−1.04857	−0.524285	−	0.851543i	$-0.675668\pi$
$967$	−32.6070	−1.04857	−0.524285	+	0.851543i	$0.675668\pi$
$968$	0.248405	0.00798404
$969$	4.44615	0.142831
$970$	73.3361	2.35468
$971$	57.6574	1.85031	0.925157	−	0.379584i	$-0.123933\pi$
$971$	57.6574	1.85031	0.925157	+	0.379584i	$0.123933\pi$
$972$	−11.9259	−0.382522
$973$	54.6861	1.75315
$974$	4.92306	0.157745
$975$	43.8264	1.40357
$976$	34.9882	1.11995
$977$	38.8705	1.24358	0.621789	−	0.783185i	$-0.286406\pi$
$977$	38.8705	1.24358	0.621789	+	0.783185i	$0.286406\pi$
$978$	39.9474	1.27738
$979$	−8.39773	−0.268392
$980$	29.4976	0.942266
$981$	7.80593	0.249224
$982$	−33.7910	−1.07831
$983$	−24.3406	−0.776345	−0.388172	−	0.921587i	$-0.626894\pi$
$983$	−24.3406	−0.776345	−0.388172	+	0.921587i	$0.626894\pi$
$984$	−2.33299	−0.0743729
$985$	87.9958	2.80378
$986$	15.8462	0.504647
$987$	39.6456	1.26193
$988$	−12.1753	−0.387347
$989$	1.11768	0.0355401
$990$	4.46004	0.141749
$991$	35.5478	1.12921	0.564606	−	0.825361i	$-0.309028\pi$
$991$	35.5478	1.12921	0.564606	+	0.825361i	$0.309028\pi$
$992$	54.7817	1.73932
$993$	11.0637	0.351095
$994$	−7.46226	−0.236689
$995$	−57.5429	−1.82423
$996$	−6.26639	−0.198558
$997$	47.2803	1.49738	0.748691	−	0.662919i	$-0.230683\pi$
$997$	47.2803	1.49738	0.748691	+	0.662919i	$0.230683\pi$
$998$	55.0135	1.74142
$999$	−8.57069	−0.271165

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.e.1.9	✓	66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.e.1.9	✓	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)

\(f(q)\)	\(=\)	\(q-1.96819 q^{2} +1.90311 q^{3} +1.87379 q^{4} +3.64418 q^{5} -3.74569 q^{6} +3.36449 q^{7} +0.248405 q^{8} +0.621828 q^{9} +O(q^{10})\)	\(q-1.96819 q^{2} +1.90311 q^{3} +1.87379 q^{4} +3.64418 q^{5} -3.74569 q^{6} +3.36449 q^{7} +0.248405 q^{8} +0.621828 q^{9} -7.17246 q^{10} -1.00000 q^{11} +3.56603 q^{12} +2.78123 q^{13} -6.62198 q^{14} +6.93528 q^{15} -4.23649 q^{16} -1.00000 q^{17} -1.22388 q^{18} -2.33626 q^{19} +6.82844 q^{20} +6.40300 q^{21} +1.96819 q^{22} +1.11768 q^{23} +0.472742 q^{24} +8.28008 q^{25} -5.47400 q^{26} -4.52592 q^{27} +6.30436 q^{28} +8.05116 q^{29} -13.6500 q^{30} +6.98619 q^{31} +7.84143 q^{32} -1.90311 q^{33} +1.96819 q^{34} +12.2608 q^{35} +1.16518 q^{36} +1.89369 q^{37} +4.59821 q^{38} +5.29299 q^{39} +0.905234 q^{40} -4.93501 q^{41} -12.6024 q^{42} +1.00000 q^{43} -1.87379 q^{44} +2.26606 q^{45} -2.19981 q^{46} +6.19172 q^{47} -8.06251 q^{48} +4.31981 q^{49} -16.2968 q^{50} -1.90311 q^{51} +5.21144 q^{52} -2.53589 q^{53} +8.90790 q^{54} -3.64418 q^{55} +0.835757 q^{56} -4.44615 q^{57} -15.8462 q^{58} -5.39255 q^{59} +12.9953 q^{60} -8.25877 q^{61} -13.7502 q^{62} +2.09214 q^{63} -6.96048 q^{64} +10.1353 q^{65} +3.74569 q^{66} -2.03405 q^{67} -1.87379 q^{68} +2.12706 q^{69} -24.1317 q^{70} +1.12689 q^{71} +0.154465 q^{72} -0.262475 q^{73} -3.72715 q^{74} +15.7579 q^{75} -4.37765 q^{76} -3.36449 q^{77} -10.4176 q^{78} +5.65063 q^{79} -15.4386 q^{80} -10.4788 q^{81} +9.71306 q^{82} -1.75725 q^{83} +11.9979 q^{84} -3.64418 q^{85} -1.96819 q^{86} +15.3222 q^{87} -0.248405 q^{88} +8.39773 q^{89} -4.46004 q^{90} +9.35743 q^{91} +2.09429 q^{92} +13.2955 q^{93} -12.1865 q^{94} -8.51375 q^{95} +14.9231 q^{96} -10.2247 q^{97} -8.50224 q^{98} -0.621828 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9}+O(q^{10}) \) 66 * q + 7 * q^2 + 3 * q^3 + 61 * q^4 + 4 * q^5 + 10 * q^6 + 14 * q^7 + 21 * q^8 + 65 * q^9	\( 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9} + 13 q^{10} - 66 q^{11} + 9 q^{12} - 12 q^{13} + 25 q^{14} + 13 q^{15} + 47 q^{16} - 66 q^{17} + 37 q^{18} + 19 q^{20} + 26 q^{21} - 7 q^{22} + 47 q^{23} + 15 q^{24} + 52 q^{25} + 16 q^{26} + 9 q^{27} + 3 q^{28} + 57 q^{29} + 2 q^{30} + 31 q^{31} + 39 q^{32} - 3 q^{33} - 7 q^{34} + 36 q^{35} + 39 q^{36} - 14 q^{37} + 18 q^{38} + 71 q^{39} + 29 q^{40} + 62 q^{41} - 3 q^{42} + 66 q^{43} - 61 q^{44} - 2 q^{45} + 19 q^{46} + 32 q^{47} + 26 q^{48} + 42 q^{49} + 10 q^{50} - 3 q^{51} - 7 q^{52} + 33 q^{53} + 100 q^{54} - 4 q^{55} + 61 q^{56} + 35 q^{57} - 16 q^{58} + 59 q^{59} + 50 q^{60} + 26 q^{61} + 29 q^{62} + 62 q^{63} + 29 q^{64} + 55 q^{65} - 10 q^{66} + 5 q^{67} - 61 q^{68} - 36 q^{69} - 35 q^{70} + 128 q^{71} + 87 q^{72} + 23 q^{73} + 64 q^{74} - 11 q^{75} + 74 q^{76} - 14 q^{77} + 45 q^{78} + 39 q^{79} + 95 q^{80} + 54 q^{81} - 6 q^{82} + 48 q^{83} + 38 q^{84} - 4 q^{85} + 7 q^{86} + 14 q^{87} - 21 q^{88} + 28 q^{89} + 135 q^{90} - 18 q^{91} + 108 q^{92} - 9 q^{93} + 37 q^{94} + 149 q^{95} + 104 q^{96} + 19 q^{97} + 30 q^{98} - 65 q^{99}+O(q^{100}) \) 66 * q + 7 * q^2 + 3 * q^3 + 61 * q^4 + 4 * q^5 + 10 * q^6 + 14 * q^7 + 21 * q^8 + 65 * q^9 + 13 * q^10 - 66 * q^11 + 9 * q^12 - 12 * q^13 + 25 * q^14 + 13 * q^15 + 47 * q^16 - 66 * q^17 + 37 * q^18 + 19 * q^20 + 26 * q^21 - 7 * q^22 + 47 * q^23 + 15 * q^24 + 52 * q^25 + 16 * q^26 + 9 * q^27 + 3 * q^28 + 57 * q^29 + 2 * q^30 + 31 * q^31 + 39 * q^32 - 3 * q^33 - 7 * q^34 + 36 * q^35 + 39 * q^36 - 14 * q^37 + 18 * q^38 + 71 * q^39 + 29 * q^40 + 62 * q^41 - 3 * q^42 + 66 * q^43 - 61 * q^44 - 2 * q^45 + 19 * q^46 + 32 * q^47 + 26 * q^48 + 42 * q^49 + 10 * q^50 - 3 * q^51 - 7 * q^52 + 33 * q^53 + 100 * q^54 - 4 * q^55 + 61 * q^56 + 35 * q^57 - 16 * q^58 + 59 * q^59 + 50 * q^60 + 26 * q^61 + 29 * q^62 + 62 * q^63 + 29 * q^64 + 55 * q^65 - 10 * q^66 + 5 * q^67 - 61 * q^68 - 36 * q^69 - 35 * q^70 + 128 * q^71 + 87 * q^72 + 23 * q^73 + 64 * q^74 - 11 * q^75 + 74 * q^76 - 14 * q^77 + 45 * q^78 + 39 * q^79 + 95 * q^80 + 54 * q^81 - 6 * q^82 + 48 * q^83 + 38 * q^84 - 4 * q^85 + 7 * q^86 + 14 * q^87 - 21 * q^88 + 28 * q^89 + 135 * q^90 - 18 * q^91 + 108 * q^92 - 9 * q^93 + 37 * q^94 + 149 * q^95 + 104 * q^96 + 19 * q^97 + 30 * q^98 - 65 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more