LMFDB - Embedded newform 8041.2.a.f.1.41

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			1.41
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)

$f(q)$	$=$	$q+1.11773 q^{2} +1.08920 q^{3} -0.750671 q^{4} -1.64855 q^{5} +1.21744 q^{6} -2.58128 q^{7} -3.07452 q^{8} -1.81364 q^{9} +O(q^{10})$	\(q+1.11773 q^{2} +1.08920 q^{3} -0.750671 q^{4} -1.64855 q^{5} +1.21744 q^{6} -2.58128 q^{7} -3.07452 q^{8} -1.81364 q^{9} -1.84264 q^{10} -1.00000 q^{11} -0.817633 q^{12} +6.92902 q^{13} -2.88519 q^{14} -1.79560 q^{15} -1.93515 q^{16} +1.00000 q^{17} -2.02716 q^{18} +1.59828 q^{19} +1.23752 q^{20} -2.81154 q^{21} -1.11773 q^{22} -1.30072 q^{23} -3.34878 q^{24} -2.28230 q^{25} +7.74480 q^{26} -5.24303 q^{27} +1.93769 q^{28} -5.52500 q^{29} -2.00701 q^{30} -5.56065 q^{31} +3.98605 q^{32} -1.08920 q^{33} +1.11773 q^{34} +4.25536 q^{35} +1.36144 q^{36} -4.58950 q^{37} +1.78646 q^{38} +7.54711 q^{39} +5.06849 q^{40} -3.57465 q^{41} -3.14255 q^{42} -1.00000 q^{43} +0.750671 q^{44} +2.98986 q^{45} -1.45386 q^{46} +7.65098 q^{47} -2.10777 q^{48} -0.336983 q^{49} -2.55100 q^{50} +1.08920 q^{51} -5.20141 q^{52} -0.234742 q^{53} -5.86031 q^{54} +1.64855 q^{55} +7.93620 q^{56} +1.74086 q^{57} -6.17548 q^{58} -14.9324 q^{59} +1.34791 q^{60} +6.37309 q^{61} -6.21533 q^{62} +4.68151 q^{63} +8.32565 q^{64} -11.4228 q^{65} -1.21744 q^{66} +12.9119 q^{67} -0.750671 q^{68} -1.41675 q^{69} +4.75636 q^{70} +4.42951 q^{71} +5.57606 q^{72} -11.5108 q^{73} -5.12984 q^{74} -2.48588 q^{75} -1.19979 q^{76} +2.58128 q^{77} +8.43566 q^{78} +11.2299 q^{79} +3.19019 q^{80} -0.269814 q^{81} -3.99551 q^{82} +10.3732 q^{83} +2.11054 q^{84} -1.64855 q^{85} -1.11773 q^{86} -6.01785 q^{87} +3.07452 q^{88} +11.2950 q^{89} +3.34187 q^{90} -17.8857 q^{91} +0.976414 q^{92} -6.05668 q^{93} +8.55176 q^{94} -2.63485 q^{95} +4.34162 q^{96} +12.5622 q^{97} -0.376657 q^{98} +1.81364 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.11773	0.790357	0.395179	−	0.918604i	$-0.370683\pi$
$2$	1.11773	0.790357	0.395179	+	0.918604i	$0.370683\pi$
$3$	1.08920	0.628852	0.314426	−	0.949282i	$-0.398188\pi$
$3$	1.08920	0.628852	0.314426	+	0.949282i	$0.398188\pi$
$4$	−0.750671	−0.375335
$5$	−1.64855	−0.737252	−0.368626	−	0.929578i	$-0.620172\pi$
$5$	−1.64855	−0.737252	−0.368626	+	0.929578i	$0.620172\pi$
$6$	1.21744	0.497018
$7$	−2.58128	−0.975633	−0.487816	−	0.872946i	$-0.662206\pi$
$7$	−2.58128	−0.975633	−0.487816	+	0.872946i	$0.662206\pi$
$8$	−3.07452	−1.08701
$9$	−1.81364	−0.604545
$10$	−1.84264	−0.582693
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.817633	−0.236030
$13$	6.92902	1.92176	0.960882	−	0.276958i	$-0.0893264\pi$
$13$	6.92902	1.92176	0.960882	+	0.276958i	$0.0893264\pi$
$14$	−2.88519	−0.771099
$15$	−1.79560	−0.463622
$16$	−1.93515	−0.483788
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.02716	−0.477807
$19$	1.59828	0.366672	0.183336	−	0.983050i	$-0.441310\pi$
$19$	1.59828	0.366672	0.183336	+	0.983050i	$0.441310\pi$
$20$	1.23752	0.276717
$21$	−2.81154	−0.613528
$22$	−1.11773	−0.238302
$23$	−1.30072	−0.271219	−0.135610	−	0.990762i	$-0.543299\pi$
$23$	−1.30072	−0.271219	−0.135610	+	0.990762i	$0.543299\pi$
$24$	−3.34878	−0.683566
$25$	−2.28230	−0.456459
$26$	7.74480	1.51888
$27$	−5.24303	−1.00902
$28$	1.93769	0.366189
$29$	−5.52500	−1.02597	−0.512983	−	0.858399i	$-0.671460\pi$
$29$	−5.52500	−1.02597	−0.512983	+	0.858399i	$0.671460\pi$
$30$	−2.00701	−0.366427
$31$	−5.56065	−0.998722	−0.499361	−	0.866394i	$-0.666432\pi$
$31$	−5.56065	−0.998722	−0.499361	+	0.866394i	$0.666432\pi$
$32$	3.98605	0.704641
$33$	−1.08920	−0.189606
$34$	1.11773	0.191690
$35$	4.25536	0.719288
$36$	1.36144	0.226907
$37$	−4.58950	−0.754509	−0.377255	−	0.926110i	$-0.623132\pi$
$37$	−4.58950	−0.754509	−0.377255	+	0.926110i	$0.623132\pi$
$38$	1.78646	0.289802
$39$	7.54711	1.20850
$40$	5.06849	0.801398
$41$	−3.57465	−0.558266	−0.279133	−	0.960252i	$-0.590047\pi$
$41$	−3.57465	−0.558266	−0.279133	+	0.960252i	$0.590047\pi$
$42$	−3.14255	−0.484907
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0.750671	0.113168
$45$	2.98986	0.445703
$46$	−1.45386	−0.214360
$47$	7.65098	1.11601	0.558005	−	0.829838i	$-0.311567\pi$
$47$	7.65098	1.11601	0.558005	+	0.829838i	$0.311567\pi$
$48$	−2.10777	−0.304231
$49$	−0.336983	−0.0481404
$50$	−2.55100	−0.360766
$51$	1.08920	0.152519
$52$	−5.20141	−0.721306
$53$	−0.234742	−0.0322443	−0.0161222	−	0.999870i	$-0.505132\pi$
$53$	−0.234742	−0.0322443	−0.0161222	+	0.999870i	$0.505132\pi$
$54$	−5.86031	−0.797487
$55$	1.64855	0.222290
$56$	7.93620	1.06052
$57$	1.74086	0.230582
$58$	−6.17548	−0.810880
$59$	−14.9324	−1.94403	−0.972013	−	0.234925i	$-0.924515\pi$
$59$	−14.9324	−1.94403	−0.972013	+	0.234925i	$0.924515\pi$
$60$	1.34791	0.174014
$61$	6.37309	0.815990	0.407995	−	0.912984i	$-0.366228\pi$
$61$	6.37309	0.815990	0.407995	+	0.912984i	$0.366228\pi$
$62$	−6.21533	−0.789347
$63$	4.68151	0.589814
$64$	8.32565	1.04071
$65$	−11.4228	−1.41682
$66$	−1.21744	−0.149856
$67$	12.9119	1.57744	0.788720	−	0.614752i	$-0.210744\pi$
$67$	12.9119	1.57744	0.788720	+	0.614752i	$0.210744\pi$
$68$	−0.750671	−0.0910322
$69$	−1.41675	−0.170557
$70$	4.75636	0.568494
$71$	4.42951	0.525686	0.262843	−	0.964839i	$-0.415340\pi$
$71$	4.42951	0.525686	0.262843	+	0.964839i	$0.415340\pi$
$72$	5.57606	0.657145
$73$	−11.5108	−1.34724	−0.673618	−	0.739080i	$-0.735261\pi$
$73$	−11.5108	−1.34724	−0.673618	+	0.739080i	$0.735261\pi$
$74$	−5.12984	−0.596332
$75$	−2.48588	−0.287045
$76$	−1.19979	−0.137625
$77$	2.58128	0.294164
$78$	8.43566	0.955150
$79$	11.2299	1.26347	0.631733	−	0.775186i	$-0.282344\pi$
$79$	11.2299	1.26347	0.631733	+	0.775186i	$0.282344\pi$
$80$	3.19019	0.356674
$81$	−0.269814	−0.0299793
$82$	−3.99551	−0.441230
$83$	10.3732	1.13860	0.569302	−	0.822129i	$-0.307214\pi$
$83$	10.3732	1.13860	0.569302	+	0.822129i	$0.307214\pi$
$84$	2.11054	0.230279
$85$	−1.64855	−0.178810
$86$	−1.11773	−0.120528
$87$	−6.01785	−0.645181
$88$	3.07452	0.327745
$89$	11.2950	1.19727	0.598633	−	0.801023i	$-0.295711\pi$
$89$	11.2950	1.19727	0.598633	+	0.801023i	$0.295711\pi$
$90$	3.34187	0.352264
$91$	−17.8857	−1.87494
$92$	0.976414	0.101798
$93$	−6.05668	−0.628048
$94$	8.55176	0.882047
$95$	−2.63485	−0.270330
$96$	4.34162	0.443115
$97$	12.5622	1.27550	0.637751	−	0.770242i	$-0.279865\pi$
$97$	12.5622	1.27550	0.637751	+	0.770242i	$0.279865\pi$
$98$	−0.376657	−0.0380481
$99$	1.81364	0.182277
$100$	1.71325	0.171325
$101$	5.75277	0.572422	0.286211	−	0.958167i	$-0.407604\pi$
$101$	5.75277	0.572422	0.286211	+	0.958167i	$0.407604\pi$
$102$	1.21744	0.120544
$103$	1.58421	0.156097	0.0780486	−	0.996950i	$-0.475131\pi$
$103$	1.58421	0.156097	0.0780486	+	0.996950i	$0.475131\pi$
$104$	−21.3034	−2.08897
$105$	4.63495	0.452325
$106$	−0.262379	−0.0254845
$107$	3.53306	0.341554	0.170777	−	0.985310i	$-0.445372\pi$
$107$	3.53306	0.341554	0.170777	+	0.985310i	$0.445372\pi$
$108$	3.93579	0.378721
$109$	0.801044	0.0767261	0.0383630	−	0.999264i	$-0.487786\pi$
$109$	0.801044	0.0767261	0.0383630	+	0.999264i	$0.487786\pi$
$110$	1.84264	0.175688
$111$	−4.99890	−0.474474
$112$	4.99517	0.472000
$113$	10.2628	0.965440	0.482720	−	0.875775i	$-0.339649\pi$
$113$	10.2628	0.965440	0.482720	+	0.875775i	$0.339649\pi$
$114$	1.94582	0.182242
$115$	2.14430	0.199957
$116$	4.14745	0.385081
$117$	−12.5667	−1.16179
$118$	−16.6904	−1.53648
$119$	−2.58128	−0.236626
$120$	5.52061	0.503960
$121$	1.00000	0.0909091
$122$	7.12342	0.644924
$123$	−3.89352	−0.351067
$124$	4.17422	0.374856
$125$	12.0052	1.07378
$126$	5.23268	0.466164
$127$	−0.493621	−0.0438018	−0.0219009	−	0.999760i	$-0.506972\pi$
$127$	−0.493621	−0.0438018	−0.0219009	+	0.999760i	$0.506972\pi$
$128$	1.33376	0.117889
$129$	−1.08920	−0.0958990
$130$	−12.7677	−1.11980
$131$	11.5337	1.00770	0.503851	−	0.863790i	$-0.331916\pi$
$131$	11.5337	1.00770	0.503851	+	0.863790i	$0.331916\pi$
$132$	0.817633	0.0711658
$133$	−4.12562	−0.357737
$134$	14.4321	1.24674
$135$	8.64337	0.743903
$136$	−3.07452	−0.263638
$137$	17.3166	1.47945	0.739726	−	0.672908i	$-0.234955\pi$
$137$	17.3166	1.47945	0.739726	+	0.672908i	$0.234955\pi$
$138$	−1.58355	−0.134801
$139$	−5.32287	−0.451480	−0.225740	−	0.974188i	$-0.572480\pi$
$139$	−5.32287	−0.451480	−0.225740	+	0.974188i	$0.572480\pi$
$140$	−3.19438	−0.269974
$141$	8.33347	0.701805
$142$	4.95101	0.415480
$143$	−6.92902	−0.579434
$144$	3.50966	0.292472
$145$	9.10822	0.756396
$146$	−12.8660	−1.06480
$147$	−0.367043	−0.0302732
$148$	3.44520	0.283194
$149$	2.78859	0.228450	0.114225	−	0.993455i	$-0.463561\pi$
$149$	2.78859	0.228450	0.114225	+	0.993455i	$0.463561\pi$
$150$	−2.77856	−0.226868
$151$	−0.981609	−0.0798822	−0.0399411	−	0.999202i	$-0.512717\pi$
$151$	−0.981609	−0.0798822	−0.0399411	+	0.999202i	$0.512717\pi$
$152$	−4.91396	−0.398574
$153$	−1.81364	−0.146624
$154$	2.88519	0.232495
$155$	9.16699	0.736310
$156$	−5.66539	−0.453594
$157$	9.67985	0.772536	0.386268	−	0.922387i	$-0.373764\pi$
$157$	9.67985	0.772536	0.386268	+	0.922387i	$0.373764\pi$
$158$	12.5521	0.998590
$159$	−0.255682	−0.0202769
$160$	−6.57119	−0.519498
$161$	3.35753	0.264611
$162$	−0.301580	−0.0236944
$163$	8.00939	0.627344	0.313672	−	0.949531i	$-0.398441\pi$
$163$	8.00939	0.627344	0.313672	+	0.949531i	$0.398441\pi$
$164$	2.68338	0.209537
$165$	1.79560	0.139787
$166$	11.5944	0.899903
$167$	−15.7706	−1.22037	−0.610184	−	0.792260i	$-0.708904\pi$
$167$	−15.7706	−1.22037	−0.610184	+	0.792260i	$0.708904\pi$
$168$	8.64413	0.666909
$169$	35.0113	2.69318
$170$	−1.84264	−0.141324
$171$	−2.89871	−0.221670
$172$	0.750671	0.0572381
$173$	18.0226	1.37023	0.685116	−	0.728434i	$-0.259752\pi$
$173$	18.0226	1.37023	0.685116	+	0.728434i	$0.259752\pi$
$174$	−6.72635	−0.509924
$175$	5.89125	0.445336
$176$	1.93515	0.145868
$177$	−16.2644	−1.22250
$178$	12.6248	0.946268
$179$	11.9768	0.895186	0.447593	−	0.894237i	$-0.352281\pi$
$179$	11.9768	0.895186	0.447593	+	0.894237i	$0.352281\pi$
$180$	−2.24440	−0.167288
$181$	−4.04944	−0.300992	−0.150496	−	0.988611i	$-0.548087\pi$
$181$	−4.04944	−0.300992	−0.150496	+	0.988611i	$0.548087\pi$
$182$	−19.9915	−1.48187
$183$	6.94159	0.513137
$184$	3.99909	0.294817
$185$	7.56600	0.556264
$186$	−6.76975	−0.496382
$187$	−1.00000	−0.0731272
$188$	−5.74336	−0.418878
$189$	13.5337	0.984434
$190$	−2.94506	−0.213657
$191$	−27.1835	−1.96693	−0.983464	−	0.181103i	$-0.942033\pi$
$191$	−27.1835	−1.96693	−0.983464	+	0.181103i	$0.942033\pi$
$192$	9.06832	0.654450
$193$	−8.02490	−0.577645	−0.288822	−	0.957383i	$-0.593264\pi$
$193$	−8.02490	−0.577645	−0.288822	+	0.957383i	$0.593264\pi$
$194$	14.0412	1.00810
$195$	−12.4418	−0.890973
$196$	0.252963	0.0180688
$197$	−25.0160	−1.78232	−0.891159	−	0.453691i	$-0.850107\pi$
$197$	−25.0160	−1.78232	−0.891159	+	0.453691i	$0.850107\pi$
$198$	2.02716	0.144064
$199$	−22.1791	−1.57223	−0.786117	−	0.618078i	$-0.787912\pi$
$199$	−22.1791	−1.57223	−0.786117	+	0.618078i	$0.787912\pi$
$200$	7.01696	0.496174
$201$	14.0637	0.991976
$202$	6.43007	0.452418
$203$	14.2616	1.00097
$204$	−0.817633	−0.0572457
$205$	5.89297	0.411583
$206$	1.77073	0.123373
$207$	2.35904	0.163964
$208$	−13.4087	−0.929727
$209$	−1.59828	−0.110556
$210$	5.18065	0.357499
$211$	−20.9681	−1.44350	−0.721752	−	0.692151i	$-0.756663\pi$
$211$	−20.9681	−1.44350	−0.721752	+	0.692151i	$0.756663\pi$
$212$	0.176214	0.0121024
$213$	4.82463	0.330578
$214$	3.94902	0.269949
$215$	1.64855	0.112430
$216$	16.1198	1.09681
$217$	14.3536	0.974386
$218$	0.895354	0.0606410
$219$	−12.5376	−0.847212
$220$	−1.23752	−0.0834333
$221$	6.92902	0.466096
$222$	−5.58744	−0.375004
$223$	7.08870	0.474694	0.237347	−	0.971425i	$-0.423722\pi$
$223$	7.08870	0.474694	0.237347	+	0.971425i	$0.423722\pi$
$224$	−10.2891	−0.687471
$225$	4.13925	0.275950
$226$	11.4710	0.763042
$227$	−1.79998	−0.119469	−0.0597344	−	0.998214i	$-0.519025\pi$
$227$	−1.79998	−0.119469	−0.0597344	+	0.998214i	$0.519025\pi$
$228$	−1.30681	−0.0865456
$229$	3.40906	0.225277	0.112638	−	0.993636i	$-0.464070\pi$
$229$	3.40906	0.225277	0.112638	+	0.993636i	$0.464070\pi$
$230$	2.39676	0.158038
$231$	2.81154	0.184986
$232$	16.9867	1.11523
$233$	2.45451	0.160800	0.0804001	−	0.996763i	$-0.474380\pi$
$233$	2.45451	0.160800	0.0804001	+	0.996763i	$0.474380\pi$
$234$	−14.0462	−0.918232
$235$	−12.6130	−0.822781
$236$	11.2093	0.729662
$237$	12.2317	0.794533
$238$	−2.88519	−0.187019
$239$	21.1336	1.36702	0.683510	−	0.729941i	$-0.260452\pi$
$239$	21.1336	1.36702	0.683510	+	0.729941i	$0.260452\pi$
$240$	3.47476	0.224295
$241$	−12.3860	−0.797851	−0.398926	−	0.916983i	$-0.630617\pi$
$241$	−12.3860	−0.797851	−0.398926	+	0.916983i	$0.630617\pi$
$242$	1.11773	0.0718507
$243$	15.4352	0.990169
$244$	−4.78409	−0.306270
$245$	0.555532	0.0354916
$246$	−4.35192	−0.277468
$247$	11.0745	0.704656
$248$	17.0963	1.08562
$249$	11.2985	0.716013
$250$	13.4186	0.848668
$251$	21.9754	1.38708	0.693539	−	0.720419i	$-0.256051\pi$
$251$	21.9754	1.38708	0.693539	+	0.720419i	$0.256051\pi$
$252$	−3.51427	−0.221378
$253$	1.30072	0.0817757
$254$	−0.551737	−0.0346191
$255$	−1.79560	−0.112445
$256$	−15.1605	−0.947532
$257$	0.100544	0.00627174	0.00313587	−	0.999995i	$-0.499002\pi$
$257$	0.100544	0.00627174	0.00313587	+	0.999995i	$0.499002\pi$
$258$	−1.21744	−0.0757945
$259$	11.8468	0.736124
$260$	8.57476	0.531784
$261$	10.0203	0.620243
$262$	12.8916	0.796445
$263$	14.0568	0.866777	0.433388	−	0.901207i	$-0.357318\pi$
$263$	14.0568	0.866777	0.433388	+	0.901207i	$0.357318\pi$
$264$	3.34878	0.206103
$265$	0.386983	0.0237722
$266$	−4.61135	−0.282740
$267$	12.3025	0.752903
$268$	−9.69259	−0.592069
$269$	−14.3835	−0.876976	−0.438488	−	0.898737i	$-0.644486\pi$
$269$	−14.3835	−0.876976	−0.438488	+	0.898737i	$0.644486\pi$
$270$	9.66099	0.587949
$271$	3.88582	0.236047	0.118023	−	0.993011i	$-0.462344\pi$
$271$	3.88582	0.236047	0.118023	+	0.993011i	$0.462344\pi$
$272$	−1.93515	−0.117336
$273$	−19.4812	−1.17906
$274$	19.3553	1.16930
$275$	2.28230	0.137628
$276$	1.06351	0.0640160
$277$	−1.70431	−0.102402	−0.0512011	−	0.998688i	$-0.516305\pi$
$277$	−1.70431	−0.102402	−0.0512011	+	0.998688i	$0.516305\pi$
$278$	−5.94955	−0.356830
$279$	10.0850	0.603773
$280$	−13.0832	−0.781870
$281$	−20.2133	−1.20583	−0.602913	−	0.797807i	$-0.705993\pi$
$281$	−20.2133	−1.20583	−0.602913	+	0.797807i	$0.705993\pi$
$282$	9.31460	0.554677
$283$	−3.37052	−0.200357	−0.100178	−	0.994970i	$-0.531941\pi$
$283$	−3.37052	−0.200357	−0.100178	+	0.994970i	$0.531941\pi$
$284$	−3.32510	−0.197308
$285$	−2.86988	−0.169997
$286$	−7.74480	−0.457960
$287$	9.22718	0.544663
$288$	−7.22925	−0.425987
$289$	1.00000	0.0588235
$290$	10.1806	0.597823
$291$	13.6828	0.802102
$292$	8.64081	0.505665
$293$	22.7312	1.32797	0.663985	−	0.747746i	$-0.268864\pi$
$293$	22.7312	1.32797	0.663985	+	0.747746i	$0.268864\pi$
$294$	−0.410256	−0.0239266
$295$	24.6167	1.43324
$296$	14.1105	0.820156
$297$	5.24303	0.304231
$298$	3.11690	0.180557
$299$	−9.01273	−0.521220
$300$	1.86608	0.107738
$301$	2.58128	0.148783
$302$	−1.09718	−0.0631355
$303$	6.26594	0.359969
$304$	−3.09293	−0.177391
$305$	−10.5063	−0.601591
$306$	−2.02716	−0.115885
$307$	−1.88975	−0.107854	−0.0539270	−	0.998545i	$-0.517174\pi$
$307$	−1.88975	−0.107854	−0.0539270	+	0.998545i	$0.517174\pi$
$308$	−1.93769	−0.110410
$309$	1.72553	0.0981620
$310$	10.2463	0.581948
$311$	33.2899	1.88770	0.943848	−	0.330380i	$-0.107177\pi$
$311$	33.2899	1.88770	0.943848	+	0.330380i	$0.107177\pi$
$312$	−23.2037	−1.31365
$313$	31.2417	1.76588	0.882941	−	0.469483i	$-0.155560\pi$
$313$	31.2417	1.76588	0.882941	+	0.469483i	$0.155560\pi$
$314$	10.8195	0.610580
$315$	−7.71768	−0.434842
$316$	−8.42998	−0.474223
$317$	−32.7096	−1.83715	−0.918576	−	0.395244i	$-0.870660\pi$
$317$	−32.7096	−1.83715	−0.918576	+	0.395244i	$0.870660\pi$
$318$	−0.285784	−0.0160260
$319$	5.52500	0.309341
$320$	−13.7252	−0.767263
$321$	3.84822	0.214787
$322$	3.75283	0.209137
$323$	1.59828	0.0889310
$324$	0.202541	0.0112523
$325$	−15.8141	−0.877206
$326$	8.95237	0.495826
$327$	0.872499	0.0482493
$328$	10.9903	0.606839
$329$	−19.7493	−1.08882
$330$	2.00701	0.110482
$331$	29.8090	1.63845	0.819224	−	0.573473i	$-0.194404\pi$
$331$	29.8090	1.63845	0.819224	+	0.573473i	$0.194404\pi$
$332$	−7.78683	−0.427358
$333$	8.32368	0.456135
$334$	−17.6274	−0.964526
$335$	−21.2859	−1.16297
$336$	5.44076	0.296818
$337$	5.23260	0.285038	0.142519	−	0.989792i	$-0.454480\pi$
$337$	5.23260	0.285038	0.142519	+	0.989792i	$0.454480\pi$
$338$	39.1333	2.12857
$339$	11.1782	0.607118
$340$	1.23752	0.0671137
$341$	5.56065	0.301126
$342$	−3.23998	−0.175198
$343$	18.9388	1.02260
$344$	3.07452	0.165767
$345$	2.33558	0.125743
$346$	20.1445	1.08297
$347$	20.5245	1.10181	0.550907	−	0.834567i	$-0.314282\pi$
$347$	20.5245	1.10181	0.550907	+	0.834567i	$0.314282\pi$
$348$	4.51742	0.242159
$349$	23.7964	1.27379	0.636897	−	0.770949i	$-0.280218\pi$
$349$	23.7964	1.27379	0.636897	+	0.770949i	$0.280218\pi$
$350$	6.58485	0.351975
$351$	−36.3290	−1.93910
$352$	−3.98605	−0.212457
$353$	26.4217	1.40629	0.703143	−	0.711049i	$-0.251780\pi$
$353$	26.4217	1.40629	0.703143	+	0.711049i	$0.251780\pi$
$354$	−18.1792	−0.966216
$355$	−7.30225	−0.387563
$356$	−8.47881	−0.449376
$357$	−2.81154	−0.148803
$358$	13.3868	0.707517
$359$	−22.8008	−1.20338	−0.601690	−	0.798729i	$-0.705506\pi$
$359$	−22.8008	−1.20338	−0.601690	+	0.798729i	$0.705506\pi$
$360$	−9.19239	−0.484481
$361$	−16.4455	−0.865552
$362$	−4.52620	−0.237892
$363$	1.08920	0.0571683
$364$	13.4263	0.703730
$365$	18.9761	0.993253
$366$	7.75885	0.405562
$367$	−4.83845	−0.252565	−0.126283	−	0.991994i	$-0.540305\pi$
$367$	−4.83845	−0.252565	−0.126283	+	0.991994i	$0.540305\pi$
$368$	2.51710	0.131213
$369$	6.48311	0.337497
$370$	8.45678	0.439647
$371$	0.605935	0.0314586
$372$	4.54657	0.235729
$373$	0.782459	0.0405142	0.0202571	−	0.999795i	$-0.493552\pi$
$373$	0.782459	0.0405142	0.0202571	+	0.999795i	$0.493552\pi$
$374$	−1.11773	−0.0577967
$375$	13.0761	0.675247
$376$	−23.5231	−1.21311
$377$	−38.2828	−1.97167
$378$	15.1271	0.778055
$379$	2.34598	0.120505	0.0602524	−	0.998183i	$-0.480809\pi$
$379$	2.34598	0.120505	0.0602524	+	0.998183i	$0.480809\pi$
$380$	1.97790	0.101464
$381$	−0.537654	−0.0275448
$382$	−30.3839	−1.55458
$383$	37.8256	1.93280	0.966398	−	0.257051i	$-0.0827509\pi$
$383$	37.8256	1.93280	0.966398	+	0.257051i	$0.0827509\pi$
$384$	1.45274	0.0741347
$385$	−4.25536	−0.216873
$386$	−8.96970	−0.456546
$387$	1.81364	0.0921923
$388$	−9.43011	−0.478741
$389$	−8.55593	−0.433803	−0.216901	−	0.976194i	$-0.569595\pi$
$389$	−8.55593	−0.433803	−0.216901	+	0.976194i	$0.569595\pi$
$390$	−13.9066	−0.704187
$391$	−1.30072	−0.0657804
$392$	1.03606	0.0523289
$393$	12.5625	0.633696
$394$	−27.9613	−1.40867
$395$	−18.5131	−0.931493
$396$	−1.36144	−0.0684151
$397$	7.38176	0.370480	0.185240	−	0.982693i	$-0.440694\pi$
$397$	7.38176	0.370480	0.185240	+	0.982693i	$0.440694\pi$
$398$	−24.7903	−1.24263
$399$	−4.49364	−0.224964
$400$	4.41659	0.220829
$401$	−12.9039	−0.644388	−0.322194	−	0.946674i	$-0.604420\pi$
$401$	−12.9039	−0.644388	−0.322194	+	0.946674i	$0.604420\pi$
$402$	15.7195	0.784016
$403$	−38.5298	−1.91931
$404$	−4.31844	−0.214850
$405$	0.444801	0.0221023
$406$	15.9407	0.791122
$407$	4.58950	0.227493
$408$	−3.34878	−0.165789
$409$	−37.8239	−1.87027	−0.935135	−	0.354290i	$-0.884722\pi$
$409$	−37.8239	−1.87027	−0.935135	+	0.354290i	$0.884722\pi$
$410$	6.58678	0.325298
$411$	18.8612	0.930357
$412$	−1.18922	−0.0585888
$413$	38.5446	1.89666
$414$	2.63678	0.129591
$415$	−17.1006	−0.839438
$416$	27.6194	1.35415
$417$	−5.79769	−0.283914
$418$	−1.78646	−0.0873785
$419$	27.2869	1.33305	0.666527	−	0.745481i	$-0.267780\pi$
$419$	27.2869	1.33305	0.666527	+	0.745481i	$0.267780\pi$
$420$	−3.47932	−0.169774
$421$	−14.1026	−0.687321	−0.343661	−	0.939094i	$-0.611667\pi$
$421$	−14.1026	−0.687321	−0.343661	+	0.939094i	$0.611667\pi$
$422$	−23.4368	−1.14088
$423$	−13.8761	−0.674679
$424$	0.721719	0.0350498
$425$	−2.28230	−0.110708
$426$	5.39266	0.261275
$427$	−16.4507	−0.796107
$428$	−2.65216	−0.128197
$429$	−7.54711	−0.364378
$430$	1.84264	0.0888598
$431$	33.7551	1.62593	0.812963	−	0.582316i	$-0.197853\pi$
$431$	33.7551	1.62593	0.812963	+	0.582316i	$0.197853\pi$
$432$	10.1461	0.488153
$433$	−13.7853	−0.662481	−0.331241	−	0.943546i	$-0.607467\pi$
$433$	−13.7853	−0.662481	−0.331241	+	0.943546i	$0.607467\pi$
$434$	16.0435	0.770113
$435$	9.92070	0.475661
$436$	−0.601320	−0.0287980
$437$	−2.07893	−0.0994485
$438$	−14.0137	−0.669600
$439$	−24.3806	−1.16362	−0.581811	−	0.813324i	$-0.697656\pi$
$439$	−24.3806	−1.16362	−0.581811	+	0.813324i	$0.697656\pi$
$440$	−5.06849	−0.241631
$441$	0.611164	0.0291031
$442$	7.74480	0.368383
$443$	−33.3994	−1.58685	−0.793426	−	0.608667i	$-0.791705\pi$
$443$	−33.3994	−1.58685	−0.793426	+	0.608667i	$0.791705\pi$
$444$	3.75253	0.178087
$445$	−18.6203	−0.882687
$446$	7.92328	0.375178
$447$	3.03734	0.143661
$448$	−21.4908	−1.01535
$449$	14.4086	0.679984	0.339992	−	0.940428i	$-0.389576\pi$
$449$	14.4086	0.679984	0.339992	+	0.940428i	$0.389576\pi$
$450$	4.62658	0.218099
$451$	3.57465	0.168324
$452$	−7.70395	−0.362364
$453$	−1.06917	−0.0502340
$454$	−2.01190	−0.0944230
$455$	29.4855	1.38230
$456$	−5.35230	−0.250644
$457$	−21.2753	−0.995214	−0.497607	−	0.867403i	$-0.665788\pi$
$457$	−21.2753	−0.995214	−0.497607	+	0.867403i	$0.665788\pi$
$458$	3.81042	0.178049
$459$	−5.24303	−0.244724
$460$	−1.60966	−0.0750510
$461$	−40.2226	−1.87335	−0.936676	−	0.350197i	$-0.886115\pi$
$461$	−40.2226	−1.87335	−0.936676	+	0.350197i	$0.886115\pi$
$462$	3.14255	0.146205
$463$	−14.6128	−0.679116	−0.339558	−	0.940585i	$-0.610277\pi$
$463$	−14.6128	−0.679116	−0.339558	+	0.940585i	$0.610277\pi$
$464$	10.6917	0.496351
$465$	9.98471	0.463030
$466$	2.74349	0.127090
$467$	−1.23342	−0.0570757	−0.0285378	−	0.999593i	$-0.509085\pi$
$467$	−1.23342	−0.0570757	−0.0285378	+	0.999593i	$0.509085\pi$
$468$	9.43347	0.436062
$469$	−33.3293	−1.53900
$470$	−14.0980	−0.650291
$471$	10.5433	0.485811
$472$	45.9098	2.11317
$473$	1.00000	0.0459800
$474$	13.6718	0.627965
$475$	−3.64776	−0.167371
$476$	1.93769	0.0888140
$477$	0.425737	0.0194931
$478$	23.6218	1.08043
$479$	8.42270	0.384843	0.192421	−	0.981312i	$-0.438366\pi$
$479$	8.42270	0.384843	0.192421	+	0.981312i	$0.438366\pi$
$480$	−7.15736	−0.326687
$481$	−31.8007	−1.44999
$482$	−13.8442	−0.630588
$483$	3.65703	0.166401
$484$	−0.750671	−0.0341214
$485$	−20.7094	−0.940367
$486$	17.2525	0.782587
$487$	−17.4380	−0.790192	−0.395096	−	0.918640i	$-0.629289\pi$
$487$	−17.4380	−0.790192	−0.395096	+	0.918640i	$0.629289\pi$
$488$	−19.5942	−0.886987
$489$	8.72386	0.394506
$490$	0.620937	0.0280511
$491$	−22.7570	−1.02701	−0.513504	−	0.858087i	$-0.671653\pi$
$491$	−22.7570	−1.02701	−0.513504	+	0.858087i	$0.671653\pi$
$492$	2.92275	0.131768
$493$	−5.52500	−0.248833
$494$	12.3784	0.556930
$495$	−2.98986	−0.134384
$496$	10.7607	0.483170
$497$	−11.4338	−0.512876
$498$	12.6287	0.565906
$499$	−2.83178	−0.126768	−0.0633840	−	0.997989i	$-0.520189\pi$
$499$	−2.83178	−0.126768	−0.0633840	+	0.997989i	$0.520189\pi$
$500$	−9.01195	−0.403027
$501$	−17.1774	−0.767430
$502$	24.5627	1.09629
$503$	22.5124	1.00378	0.501890	−	0.864932i	$-0.332638\pi$
$503$	22.5124	1.00378	0.501890	+	0.864932i	$0.332638\pi$
$504$	−14.3934	−0.641132
$505$	−9.48371	−0.422020
$506$	1.45386	0.0646320
$507$	38.1344	1.69361
$508$	0.370547	0.0164404
$509$	−32.1324	−1.42424	−0.712121	−	0.702057i	$-0.752265\pi$
$509$	−32.1324	−1.42424	−0.712121	+	0.702057i	$0.752265\pi$
$510$	−2.00701	−0.0888717
$511$	29.7126	1.31441
$512$	−19.6129	−0.866778
$513$	−8.37985	−0.369980
$514$	0.112381	0.00495692
$515$	−2.61165	−0.115083
$516$	0.817633	0.0359943
$517$	−7.65098	−0.336490
$518$	13.2416	0.581801
$519$	19.6303	0.861673
$520$	35.1196	1.54010
$521$	0.704174	0.0308504	0.0154252	−	0.999881i	$-0.495090\pi$
$521$	0.704174	0.0308504	0.0154252	+	0.999881i	$0.495090\pi$
$522$	11.2001	0.490214
$523$	20.9663	0.916792	0.458396	−	0.888748i	$-0.348424\pi$
$523$	20.9663	0.916792	0.458396	+	0.888748i	$0.348424\pi$
$524$	−8.65800	−0.378226
$525$	6.41677	0.280051
$526$	15.7117	0.685063
$527$	−5.56065	−0.242226
$528$	2.10777	0.0917291
$529$	−21.3081	−0.926440
$530$	0.432544	0.0187885
$531$	27.0819	1.17525
$532$	3.09698	0.134271
$533$	−24.7688	−1.07286
$534$	13.7510	0.595062
$535$	−5.82441	−0.251811
$536$	−39.6979	−1.71469
$537$	13.0451	0.562939
$538$	−16.0769	−0.693124
$539$	0.336983	0.0145149
$540$	−6.48833	−0.279213
$541$	22.1689	0.953114	0.476557	−	0.879144i	$-0.341885\pi$
$541$	22.1689	0.953114	0.476557	+	0.879144i	$0.341885\pi$
$542$	4.34331	0.186561
$543$	−4.41066	−0.189280
$544$	3.98605	0.170901
$545$	−1.32056	−0.0565665
$546$	−21.7748	−0.931876
$547$	−38.1235	−1.63004	−0.815022	−	0.579430i	$-0.803275\pi$
$547$	−38.1235	−1.63004	−0.815022	+	0.579430i	$0.803275\pi$
$548$	−12.9990	−0.555291
$549$	−11.5585	−0.493303
$550$	2.55100	0.108775
$551$	−8.83052	−0.376193
$552$	4.35583	0.185396
$553$	−28.9876	−1.23268
$554$	−1.90497	−0.0809344
$555$	8.24091	0.349807
$556$	3.99572	0.169456
$557$	−14.0221	−0.594134	−0.297067	−	0.954857i	$-0.596009\pi$
$557$	−14.0221	−0.594134	−0.297067	+	0.954857i	$0.596009\pi$
$558$	11.2723	0.477196
$559$	−6.92902	−0.293066
$560$	−8.23478	−0.347983
$561$	−1.08920	−0.0459862
$562$	−22.5931	−0.953033
$563$	19.4658	0.820386	0.410193	−	0.911999i	$-0.365461\pi$
$563$	19.4658	0.820386	0.410193	+	0.911999i	$0.365461\pi$
$564$	−6.25569	−0.263412
$565$	−16.9186	−0.711773
$566$	−3.76734	−0.158353
$567$	0.696466	0.0292488
$568$	−13.6186	−0.571424
$569$	13.1610	0.551736	0.275868	−	0.961195i	$-0.411035\pi$
$569$	13.1610	0.551736	0.275868	+	0.961195i	$0.411035\pi$
$570$	−3.20777	−0.134359
$571$	−39.8266	−1.66669	−0.833345	−	0.552753i	$-0.813577\pi$
$571$	−39.8266	−1.66669	−0.833345	+	0.552753i	$0.813577\pi$
$572$	5.20141	0.217482
$573$	−29.6083	−1.23691
$574$	10.3135	0.430478
$575$	2.96863	0.123801
$576$	−15.0997	−0.629154
$577$	13.7426	0.572113	0.286057	−	0.958213i	$-0.407655\pi$
$577$	13.7426	0.572113	0.286057	+	0.958213i	$0.407655\pi$
$578$	1.11773	0.0464916
$579$	−8.74074	−0.363253
$580$	−6.83727	−0.283902
$581$	−26.7761	−1.11086
$582$	15.2938	0.633947
$583$	0.234742	0.00972202
$584$	35.3901	1.46445
$585$	20.7168	0.856535
$586$	25.4074	1.04957
$587$	26.6741	1.10096	0.550479	−	0.834849i	$-0.314445\pi$
$587$	26.6741	1.10096	0.550479	+	0.834849i	$0.314445\pi$
$588$	0.275528	0.0113626
$589$	−8.88750	−0.366203
$590$	27.5149	1.13277
$591$	−27.2475	−1.12081
$592$	8.88138	0.365023
$593$	41.5676	1.70698	0.853488	−	0.521112i	$-0.174483\pi$
$593$	41.5676	1.70698	0.853488	+	0.521112i	$0.174483\pi$
$594$	5.86031	0.240451
$595$	4.25536	0.174453
$596$	−2.09331	−0.0857454
$597$	−24.1575	−0.988702
$598$	−10.0738	−0.411950
$599$	4.40382	0.179935	0.0899676	−	0.995945i	$-0.471324\pi$
$599$	4.40382	0.179935	0.0899676	+	0.995945i	$0.471324\pi$
$600$	7.64289	0.312020
$601$	−40.5501	−1.65407	−0.827037	−	0.562148i	$-0.809975\pi$
$601$	−40.5501	−1.65407	−0.827037	+	0.562148i	$0.809975\pi$
$602$	2.88519	0.117591
$603$	−23.4175	−0.953634
$604$	0.736865	0.0299826
$605$	−1.64855	−0.0670229
$606$	7.00365	0.284504
$607$	0.420105	0.0170515	0.00852576	−	0.999964i	$-0.497286\pi$
$607$	0.420105	0.0170515	0.00852576	+	0.999964i	$0.497286\pi$
$608$	6.37084	0.258372
$609$	15.5338	0.629460
$610$	−11.7433	−0.475472
$611$	53.0138	2.14471
$612$	1.36144	0.0550331
$613$	13.4915	0.544918	0.272459	−	0.962167i	$-0.412163\pi$
$613$	13.4915	0.544918	0.272459	+	0.962167i	$0.412163\pi$
$614$	−2.11224	−0.0852432
$615$	6.41865	0.258825
$616$	−7.93620	−0.319759
$617$	17.1805	0.691662	0.345831	−	0.938297i	$-0.387597\pi$
$617$	17.1805	0.691662	0.345831	+	0.938297i	$0.387597\pi$
$618$	1.92868	0.0775831
$619$	20.8121	0.836509	0.418254	−	0.908330i	$-0.362642\pi$
$619$	20.8121	0.836509	0.418254	+	0.908330i	$0.362642\pi$
$620$	−6.88139	−0.276363
$621$	6.81972	0.273666
$622$	37.2092	1.49195
$623$	−29.1555	−1.16809
$624$	−14.6048	−0.584660
$625$	−8.37965	−0.335186
$626$	34.9199	1.39568
$627$	−1.74086	−0.0695231
$628$	−7.26638	−0.289960
$629$	−4.58950	−0.182995
$630$	−8.62631	−0.343681
$631$	−6.89938	−0.274660	−0.137330	−	0.990525i	$-0.543852\pi$
$631$	−6.89938	−0.274660	−0.137330	+	0.990525i	$0.543852\pi$
$632$	−34.5266	−1.37340
$633$	−22.8385	−0.907751
$634$	−36.5606	−1.45201
$635$	0.813757	0.0322930
$636$	0.191933	0.00761063
$637$	−2.33496	−0.0925145
$638$	6.17548	0.244490
$639$	−8.03352	−0.317801
$640$	−2.19877	−0.0869139
$641$	31.4310	1.24145	0.620724	−	0.784029i	$-0.286839\pi$
$641$	31.4310	1.24145	0.620724	+	0.784029i	$0.286839\pi$
$642$	4.30128	0.169758
$643$	−11.5758	−0.456503	−0.228252	−	0.973602i	$-0.573301\pi$
$643$	−11.5758	−0.456503	−0.228252	+	0.973602i	$0.573301\pi$
$644$	−2.52040	−0.0993177
$645$	1.79560	0.0707018
$646$	1.78646	0.0702872
$647$	0.550608	0.0216466	0.0108233	−	0.999941i	$-0.496555\pi$
$647$	0.550608	0.0216466	0.0108233	+	0.999941i	$0.496555\pi$
$648$	0.829548	0.0325877
$649$	14.9324	0.586146
$650$	−17.6759	−0.693307
$651$	15.6340	0.612744
$652$	−6.01242	−0.235464
$653$	31.2202	1.22174	0.610871	−	0.791730i	$-0.290819\pi$
$653$	31.2202	1.22174	0.610871	+	0.791730i	$0.290819\pi$
$654$	0.975222	0.0381342
$655$	−19.0138	−0.742931
$656$	6.91749	0.270083
$657$	20.8764	0.814465
$658$	−22.0745	−0.860554
$659$	−22.5824	−0.879685	−0.439842	−	0.898075i	$-0.644966\pi$
$659$	−22.5824	−0.879685	−0.439842	+	0.898075i	$0.644966\pi$
$660$	−1.34791	−0.0524671
$661$	−18.8904	−0.734750	−0.367375	−	0.930073i	$-0.619743\pi$
$661$	−18.8904	−0.734750	−0.367375	+	0.930073i	$0.619743\pi$
$662$	33.3185	1.29496
$663$	7.54711	0.293105
$664$	−31.8925	−1.23767
$665$	6.80128	0.263742
$666$	9.30366	0.360510
$667$	7.18649	0.278262
$668$	11.8385	0.458047
$669$	7.72103	0.298512
$670$	−23.7919	−0.919163
$671$	−6.37309	−0.246030
$672$	−11.2069	−0.432317
$673$	−17.6329	−0.679697	−0.339848	−	0.940480i	$-0.610376\pi$
$673$	−17.6329	−0.679697	−0.339848	+	0.940480i	$0.610376\pi$
$674$	5.84865	0.225282
$675$	11.9661	0.460577
$676$	−26.2819	−1.01084
$677$	13.2901	0.510779	0.255390	−	0.966838i	$-0.417796\pi$
$677$	13.2901	0.510779	0.255390	+	0.966838i	$0.417796\pi$
$678$	12.4943	0.479840
$679$	−32.4267	−1.24442
$680$	5.06849	0.194368
$681$	−1.96054	−0.0751281
$682$	6.21533	0.237997
$683$	−37.8104	−1.44678	−0.723388	−	0.690442i	$-0.757416\pi$
$683$	−37.8104	−1.44678	−0.723388	+	0.690442i	$0.757416\pi$
$684$	2.17597	0.0832005
$685$	−28.5471	−1.09073
$686$	21.1686	0.808220
$687$	3.71316	0.141666
$688$	1.93515	0.0737770
$689$	−1.62653	−0.0619659
$690$	2.61056	0.0993822
$691$	21.0044	0.799045	0.399523	−	0.916723i	$-0.369176\pi$
$691$	21.0044	0.799045	0.399523	+	0.916723i	$0.369176\pi$
$692$	−13.5290	−0.514296
$693$	−4.68151	−0.177836
$694$	22.9409	0.870826
$695$	8.77500	0.332855
$696$	18.5020	0.701316
$697$	−3.57465	−0.135399
$698$	26.5981	1.00675
$699$	2.67346	0.101119
$700$	−4.42239	−0.167150
$701$	39.6228	1.49653	0.748265	−	0.663400i	$-0.230887\pi$
$701$	39.6228	1.49653	0.748265	+	0.663400i	$0.230887\pi$
$702$	−40.6062	−1.53258
$703$	−7.33533	−0.276657
$704$	−8.32565	−0.313785
$705$	−13.7381	−0.517407
$706$	29.5324	1.11147
$707$	−14.8495	−0.558474
$708$	12.2092	0.458849
$709$	−21.8163	−0.819327	−0.409664	−	0.912237i	$-0.634354\pi$
$709$	−21.8163	−0.819327	−0.409664	+	0.912237i	$0.634354\pi$
$710$	−8.16197	−0.306313
$711$	−20.3670	−0.763823
$712$	−34.7266	−1.30144
$713$	7.23286	0.270873
$714$	−3.14255	−0.117607
$715$	11.4228	0.427189
$716$	−8.99061	−0.335995
$717$	23.0188	0.859653
$718$	−25.4852	−0.951101
$719$	−29.4614	−1.09872	−0.549362	−	0.835585i	$-0.685129\pi$
$719$	−29.4614	−1.09872	−0.549362	+	0.835585i	$0.685129\pi$
$720$	−5.78584	−0.215626
$721$	−4.08930	−0.152294
$722$	−18.3817	−0.684095
$723$	−13.4909	−0.501730
$724$	3.03979	0.112973
$725$	12.6097	0.468312
$726$	1.21744	0.0451834
$727$	−34.4548	−1.27786	−0.638929	−	0.769265i	$-0.720622\pi$
$727$	−34.4548	−1.27786	−0.638929	+	0.769265i	$0.720622\pi$
$728$	54.9901	2.03807
$729$	17.6215	0.652649
$730$	21.2102	0.785024
$731$	−1.00000	−0.0369863
$732$	−5.21085	−0.192598
$733$	41.3641	1.52782	0.763909	−	0.645324i	$-0.223278\pi$
$733$	41.3641	1.52782	0.763909	+	0.645324i	$0.223278\pi$
$734$	−5.40810	−0.199617
$735$	0.605087	0.0223190
$736$	−5.18475	−0.191112
$737$	−12.9119	−0.475616
$738$	7.24640	0.266744
$739$	−16.5857	−0.610114	−0.305057	−	0.952334i	$-0.598675\pi$
$739$	−16.5857	−0.610114	−0.305057	+	0.952334i	$0.598675\pi$
$740$	−5.67958	−0.208785
$741$	12.0624	0.443124
$742$	0.677275	0.0248635
$743$	31.5760	1.15841	0.579205	−	0.815182i	$-0.303363\pi$
$743$	31.5760	1.15841	0.579205	+	0.815182i	$0.303363\pi$
$744$	18.6214	0.682692
$745$	−4.59712	−0.168425
$746$	0.874582	0.0320207
$747$	−18.8132	−0.688337
$748$	0.750671	0.0274472
$749$	−9.11982	−0.333231
$750$	14.6156	0.533686
$751$	−16.8088	−0.613364	−0.306682	−	0.951812i	$-0.599219\pi$
$751$	−16.8088	−0.613364	−0.306682	+	0.951812i	$0.599219\pi$
$752$	−14.8058	−0.539912
$753$	23.9357	0.872266
$754$	−42.7900	−1.55832
$755$	1.61823	0.0588933
$756$	−10.1594	−0.369493
$757$	−26.0385	−0.946385	−0.473192	−	0.880959i	$-0.656898\pi$
$757$	−26.0385	−0.946385	−0.473192	+	0.880959i	$0.656898\pi$
$758$	2.62218	0.0952419
$759$	1.41675	0.0514248
$760$	8.10088	0.293850
$761$	−25.8992	−0.938846	−0.469423	−	0.882973i	$-0.655538\pi$
$761$	−25.8992	−0.938846	−0.469423	+	0.882973i	$0.655538\pi$
$762$	−0.600954	−0.0217703
$763$	−2.06772	−0.0748565
$764$	20.4058	0.738258
$765$	2.98986	0.108099
$766$	42.2789	1.52760
$767$	−103.467	−3.73596
$768$	−16.5129	−0.595857
$769$	5.40654	0.194965	0.0974825	−	0.995237i	$-0.468921\pi$
$769$	5.40654	0.194965	0.0974825	+	0.995237i	$0.468921\pi$
$770$	−4.75636	−0.171407
$771$	0.109512	0.00394400
$772$	6.02405	0.216810
$773$	−35.0717	−1.26144	−0.630721	−	0.776010i	$-0.717241\pi$
$773$	−35.0717	−1.26144	−0.630721	+	0.776010i	$0.717241\pi$
$774$	2.02716	0.0728649
$775$	12.6910	0.455876
$776$	−38.6229	−1.38648
$777$	12.9036	0.462913
$778$	−9.56325	−0.342859
$779$	−5.71331	−0.204700
$780$	9.33966	0.334414
$781$	−4.42951	−0.158500
$782$	−1.45386	−0.0519900
$783$	28.9677	1.03522
$784$	0.652113	0.0232897
$785$	−15.9577	−0.569554
$786$	14.0416	0.500846
$787$	29.0669	1.03612	0.518062	−	0.855343i	$-0.326654\pi$
$787$	29.0669	1.03612	0.518062	+	0.855343i	$0.326654\pi$
$788$	18.7788	0.668967
$789$	15.3107	0.545074
$790$	−20.6927	−0.736213
$791$	−26.4911	−0.941915
$792$	−5.57606	−0.198137
$793$	44.1592	1.56814
$794$	8.25085	0.292812
$795$	0.421503	0.0149492
$796$	16.6492	0.590115
$797$	16.7351	0.592788	0.296394	−	0.955066i	$-0.404216\pi$
$797$	16.7351	0.592788	0.296394	+	0.955066i	$0.404216\pi$
$798$	−5.02270	−0.177802
$799$	7.65098	0.270672
$800$	−9.09734	−0.321640
$801$	−20.4850	−0.723802
$802$	−14.4231	−0.509297
$803$	11.5108	0.406207
$804$	−10.5572	−0.372324
$805$	−5.53505	−0.195085
$806$	−43.0661	−1.51694
$807$	−15.6665	−0.551488
$808$	−17.6870	−0.622227
$809$	21.4673	0.754749	0.377374	−	0.926061i	$-0.376827\pi$
$809$	21.4673	0.754749	0.377374	+	0.926061i	$0.376827\pi$
$810$	0.497169	0.0174687
$811$	28.4078	0.997532	0.498766	−	0.866737i	$-0.333787\pi$
$811$	28.4078	0.997532	0.498766	+	0.866737i	$0.333787\pi$
$812$	−10.7058	−0.375698
$813$	4.23245	0.148438
$814$	5.12984	0.179801
$815$	−13.2039	−0.462511
$816$	−2.10777	−0.0737869
$817$	−1.59828	−0.0559169
$818$	−42.2770	−1.47818
$819$	32.4382	1.13348
$820$	−4.42368	−0.154482
$821$	23.9540	0.836000	0.418000	−	0.908447i	$-0.362731\pi$
$821$	23.9540	0.836000	0.418000	+	0.908447i	$0.362731\pi$
$822$	21.0819	0.735314
$823$	16.8631	0.587811	0.293905	−	0.955835i	$-0.405045\pi$
$823$	16.8631	0.587811	0.293905	+	0.955835i	$0.405045\pi$
$824$	−4.87069	−0.169679
$825$	2.48588	0.0865474
$826$	43.0826	1.49904
$827$	−20.6037	−0.716463	−0.358231	−	0.933633i	$-0.616620\pi$
$827$	−20.6037	−0.716463	−0.358231	+	0.933633i	$0.616620\pi$
$828$	−1.77086	−0.0615416
$829$	−8.25250	−0.286621	−0.143311	−	0.989678i	$-0.545775\pi$
$829$	−8.25250	−0.286621	−0.143311	+	0.989678i	$0.545775\pi$
$830$	−19.1140	−0.663456
$831$	−1.85634	−0.0643958
$832$	57.6886	1.99999
$833$	−0.336983	−0.0116758
$834$	−6.48027	−0.224393
$835$	25.9986	0.899718
$836$	1.19979	0.0414954
$837$	29.1546	1.00773
$838$	30.4995	1.05359
$839$	−21.4878	−0.741841	−0.370921	−	0.928665i	$-0.620958\pi$
$839$	−21.4878	−0.741841	−0.370921	+	0.928665i	$0.620958\pi$
$840$	−14.2503	−0.491680
$841$	1.52562	0.0526076
$842$	−15.7630	−0.543229
$843$	−22.0164	−0.758286
$844$	15.7401	0.541798
$845$	−57.7177	−1.98555
$846$	−15.5098	−0.533237
$847$	−2.58128	−0.0886939
$848$	0.454262	0.0155994
$849$	−3.67118	−0.125995
$850$	−2.55100	−0.0874986
$851$	5.96967	0.204637
$852$	−3.62171	−0.124078
$853$	6.83319	0.233964	0.116982	−	0.993134i	$-0.462678\pi$
$853$	6.83319	0.233964	0.116982	+	0.993134i	$0.462678\pi$
$854$	−18.3875	−0.629209
$855$	4.77865	0.163427
$856$	−10.8624	−0.371271
$857$	38.8981	1.32873	0.664367	−	0.747406i	$-0.268701\pi$
$857$	38.8981	1.32873	0.664367	+	0.747406i	$0.268701\pi$
$858$	−8.43566	−0.287989
$859$	22.1012	0.754083	0.377042	−	0.926196i	$-0.376941\pi$
$859$	22.1012	0.754083	0.377042	+	0.926196i	$0.376941\pi$
$860$	−1.23752	−0.0421989
$861$	10.0503	0.342512
$862$	37.7292	1.28506
$863$	19.0404	0.648144	0.324072	−	0.946032i	$-0.394948\pi$
$863$	19.0404	0.648144	0.324072	+	0.946032i	$0.394948\pi$
$864$	−20.8990	−0.710998
$865$	−29.7111	−1.01021
$866$	−15.4083	−0.523597
$867$	1.08920	0.0369913
$868$	−10.7748	−0.365722
$869$	−11.2299	−0.380949
$870$	11.0887	0.375942
$871$	89.4668	3.03147
$872$	−2.46282	−0.0834017
$873$	−22.7833	−0.771099
$874$	−2.32369	−0.0785998
$875$	−30.9888	−1.04761
$876$	9.41160	0.317988
$877$	19.7142	0.665702	0.332851	−	0.942980i	$-0.391989\pi$
$877$	19.7142	0.665702	0.332851	+	0.942980i	$0.391989\pi$
$878$	−27.2510	−0.919677
$879$	24.7589	0.835097
$880$	−3.19019	−0.107541
$881$	−1.84594	−0.0621912	−0.0310956	−	0.999516i	$-0.509900\pi$
$881$	−1.84594	−0.0621912	−0.0310956	+	0.999516i	$0.509900\pi$
$882$	0.683119	0.0230018
$883$	59.1862	1.99177	0.995887	−	0.0906095i	$-0.0288815\pi$
$883$	59.1862	1.99177	0.995887	+	0.0906095i	$0.0288815\pi$
$884$	−5.20141	−0.174942
$885$	26.8126	0.901294
$886$	−37.3316	−1.25418
$887$	41.7443	1.40164	0.700818	−	0.713340i	$-0.252818\pi$
$887$	41.7443	1.40164	0.700818	+	0.713340i	$0.252818\pi$
$888$	15.3692	0.515757
$889$	1.27418	0.0427345
$890$	−20.8126	−0.697638
$891$	0.269814	0.00903911
$892$	−5.32128	−0.178170
$893$	12.2284	0.409209
$894$	3.39494	0.113544
$895$	−19.7443	−0.659978
$896$	−3.44281	−0.115016
$897$	−9.81669	−0.327770
$898$	16.1050	0.537430
$899$	30.7226	1.02466
$900$	−3.10722	−0.103574
$901$	−0.234742	−0.00782039
$902$	3.99551	0.133036
$903$	2.81154	0.0935622
$904$	−31.5531	−1.04944
$905$	6.67569	0.221907
$906$	−1.19505	−0.0397028
$907$	20.8066	0.690872	0.345436	−	0.938442i	$-0.387731\pi$
$907$	20.8066	0.690872	0.345436	+	0.938442i	$0.387731\pi$
$908$	1.35119	0.0448408
$909$	−10.4334	−0.346055
$910$	32.9569	1.09251
$911$	21.7244	0.719761	0.359880	−	0.932998i	$-0.382818\pi$
$911$	21.7244	0.719761	0.359880	+	0.932998i	$0.382818\pi$
$912$	−3.36882	−0.111553
$913$	−10.3732	−0.343302
$914$	−23.7801	−0.786575
$915$	−11.4435	−0.378311
$916$	−2.55908	−0.0845544
$917$	−29.7717	−0.983148
$918$	−5.86031	−0.193419
$919$	−22.5355	−0.743378	−0.371689	−	0.928357i	$-0.621221\pi$
$919$	−22.5355	−0.743378	−0.371689	+	0.928357i	$0.621221\pi$
$920$	−6.59269	−0.217355
$921$	−2.05833	−0.0678242
$922$	−44.9581	−1.48062
$923$	30.6921	1.01024
$924$	−2.11054	−0.0694317
$925$	10.4746	0.344403
$926$	−16.3333	−0.536744
$927$	−2.87319	−0.0943679
$928$	−22.0229	−0.722938
$929$	−15.5635	−0.510621	−0.255311	−	0.966859i	$-0.582178\pi$
$929$	−15.5635	−0.510621	−0.255311	+	0.966859i	$0.582178\pi$
$930$	11.1603	0.365959
$931$	−0.538594	−0.0176517
$932$	−1.84253	−0.0603540
$933$	36.2595	1.18708
$934$	−1.37863	−0.0451102
$935$	1.64855	0.0539132
$936$	38.6366	1.26288
$937$	23.1036	0.754762	0.377381	−	0.926058i	$-0.376825\pi$
$937$	23.1036	0.754762	0.377381	+	0.926058i	$0.376825\pi$
$938$	−37.2533	−1.21636
$939$	34.0285	1.11048
$940$	9.46820	0.308819
$941$	−1.85866	−0.0605905	−0.0302952	−	0.999541i	$-0.509645\pi$
$941$	−1.85866	−0.0605905	−0.0302952	+	0.999541i	$0.509645\pi$
$942$	11.7846	0.383964
$943$	4.64963	0.151413
$944$	28.8964	0.940497
$945$	−22.3110	−0.725776
$946$	1.11773	0.0363407
$947$	7.43751	0.241687	0.120843	−	0.992672i	$-0.461440\pi$
$947$	7.43751	0.241687	0.120843	+	0.992672i	$0.461440\pi$
$948$	−9.18196	−0.298216
$949$	−79.7584	−2.58907
$950$	−4.07722	−0.132283
$951$	−35.6274	−1.15530
$952$	7.93620	0.257214
$953$	−46.3536	−1.50154	−0.750770	−	0.660564i	$-0.770317\pi$
$953$	−46.3536	−1.50154	−0.750770	+	0.660564i	$0.770317\pi$
$954$	0.475860	0.0154066
$955$	44.8132	1.45012
$956$	−15.8644	−0.513091
$957$	6.01785	0.194529
$958$	9.41434	0.304163
$959$	−44.6989	−1.44340
$960$	−14.9496	−0.482495
$961$	−0.0791760	−0.00255407
$962$	−35.5448	−1.14601
$963$	−6.40768	−0.206485
$964$	9.29779	0.299462
$965$	13.2294	0.425870
$966$	4.08759	0.131516
$967$	16.1120	0.518127	0.259063	−	0.965860i	$-0.416586\pi$
$967$	16.1120	0.518127	0.259063	+	0.965860i	$0.416586\pi$
$968$	−3.07452	−0.0988188
$969$	1.74086	0.0559244
$970$	−23.1476	−0.743226
$971$	−33.3216	−1.06934	−0.534670	−	0.845061i	$-0.679564\pi$
$971$	−33.3216	−1.06934	−0.534670	+	0.845061i	$0.679564\pi$
$972$	−11.5868	−0.371645
$973$	13.7398	0.440479
$974$	−19.4911	−0.624534
$975$	−17.2247	−0.551633
$976$	−12.3329	−0.394766
$977$	−21.9509	−0.702270	−0.351135	−	0.936325i	$-0.614204\pi$
$977$	−21.9509	−0.702270	−0.351135	+	0.936325i	$0.614204\pi$
$978$	9.75095	0.311801
$979$	−11.2950	−0.360989
$980$	−0.417021	−0.0133213
$981$	−1.45280	−0.0463844
$982$	−25.4362	−0.811703
$983$	37.4724	1.19518	0.597592	−	0.801800i	$-0.296124\pi$
$983$	37.4724	1.19518	0.597592	+	0.801800i	$0.296124\pi$
$984$	11.9707	0.381612
$985$	41.2401	1.31402
$986$	−6.17548	−0.196667
$987$	−21.5110	−0.684704
$988$	−8.31334	−0.264482
$989$	1.30072	0.0413606
$990$	−3.34187	−0.106212
$991$	11.2614	0.357730	0.178865	−	0.983874i	$-0.442758\pi$
$991$	11.2614	0.357730	0.178865	+	0.983874i	$0.442758\pi$
$992$	−22.1650	−0.703740
$993$	32.4680	1.03034
$994$	−12.7800	−0.405356
$995$	36.5633	1.15913
$996$	−8.48144	−0.268745
$997$	30.8162	0.975959	0.487980	−	0.872855i	$-0.337734\pi$
$997$	30.8162	0.975959	0.487980	+	0.872855i	$0.337734\pi$
$998$	−3.16518	−0.100192
$999$	24.0629	0.761316

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.f.1.41	✓	66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.f.1.41	✓	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)

\(f(q)\)	\(=\)	\(q+1.11773 q^{2} +1.08920 q^{3} -0.750671 q^{4} -1.64855 q^{5} +1.21744 q^{6} -2.58128 q^{7} -3.07452 q^{8} -1.81364 q^{9} +O(q^{10})\)	\(q+1.11773 q^{2} +1.08920 q^{3} -0.750671 q^{4} -1.64855 q^{5} +1.21744 q^{6} -2.58128 q^{7} -3.07452 q^{8} -1.81364 q^{9} -1.84264 q^{10} -1.00000 q^{11} -0.817633 q^{12} +6.92902 q^{13} -2.88519 q^{14} -1.79560 q^{15} -1.93515 q^{16} +1.00000 q^{17} -2.02716 q^{18} +1.59828 q^{19} +1.23752 q^{20} -2.81154 q^{21} -1.11773 q^{22} -1.30072 q^{23} -3.34878 q^{24} -2.28230 q^{25} +7.74480 q^{26} -5.24303 q^{27} +1.93769 q^{28} -5.52500 q^{29} -2.00701 q^{30} -5.56065 q^{31} +3.98605 q^{32} -1.08920 q^{33} +1.11773 q^{34} +4.25536 q^{35} +1.36144 q^{36} -4.58950 q^{37} +1.78646 q^{38} +7.54711 q^{39} +5.06849 q^{40} -3.57465 q^{41} -3.14255 q^{42} -1.00000 q^{43} +0.750671 q^{44} +2.98986 q^{45} -1.45386 q^{46} +7.65098 q^{47} -2.10777 q^{48} -0.336983 q^{49} -2.55100 q^{50} +1.08920 q^{51} -5.20141 q^{52} -0.234742 q^{53} -5.86031 q^{54} +1.64855 q^{55} +7.93620 q^{56} +1.74086 q^{57} -6.17548 q^{58} -14.9324 q^{59} +1.34791 q^{60} +6.37309 q^{61} -6.21533 q^{62} +4.68151 q^{63} +8.32565 q^{64} -11.4228 q^{65} -1.21744 q^{66} +12.9119 q^{67} -0.750671 q^{68} -1.41675 q^{69} +4.75636 q^{70} +4.42951 q^{71} +5.57606 q^{72} -11.5108 q^{73} -5.12984 q^{74} -2.48588 q^{75} -1.19979 q^{76} +2.58128 q^{77} +8.43566 q^{78} +11.2299 q^{79} +3.19019 q^{80} -0.269814 q^{81} -3.99551 q^{82} +10.3732 q^{83} +2.11054 q^{84} -1.64855 q^{85} -1.11773 q^{86} -6.01785 q^{87} +3.07452 q^{88} +11.2950 q^{89} +3.34187 q^{90} -17.8857 q^{91} +0.976414 q^{92} -6.05668 q^{93} +8.55176 q^{94} -2.63485 q^{95} +4.34162 q^{96} +12.5622 q^{97} -0.376657 q^{98} +1.81364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 12 q^{2} + 66 q^{4} + 6 q^{5} + 7 q^{6} + 13 q^{7} + 30 q^{8} + 58 q^{9}+O(q^{10}) \) 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