LMFDB - Embedded newform 6046.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6046,2,Mod(1,6046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6046 = 2 \cdot 3023 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6046.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:	$48.2775530621$
Analytic rank:	$0$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.37105 q^{3} +1.00000 q^{4} +3.37148 q^{5} -3.37105 q^{6} +1.10661 q^{7} +1.00000 q^{8} +8.36395 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.37105 q^{3} +1.00000 q^{4} +3.37148 q^{5} -3.37105 q^{6} +1.10661 q^{7} +1.00000 q^{8} +8.36395 q^{9} +3.37148 q^{10} +3.06369 q^{11} -3.37105 q^{12} -2.07859 q^{13} +1.10661 q^{14} -11.3654 q^{15} +1.00000 q^{16} -3.24454 q^{17} +8.36395 q^{18} +4.56616 q^{19} +3.37148 q^{20} -3.73043 q^{21} +3.06369 q^{22} +3.62028 q^{23} -3.37105 q^{24} +6.36689 q^{25} -2.07859 q^{26} -18.0821 q^{27} +1.10661 q^{28} +2.47911 q^{29} -11.3654 q^{30} +4.52613 q^{31} +1.00000 q^{32} -10.3278 q^{33} -3.24454 q^{34} +3.73092 q^{35} +8.36395 q^{36} +7.13547 q^{37} +4.56616 q^{38} +7.00702 q^{39} +3.37148 q^{40} -6.77163 q^{41} -3.73043 q^{42} +10.9333 q^{43} +3.06369 q^{44} +28.1989 q^{45} +3.62028 q^{46} -0.745380 q^{47} -3.37105 q^{48} -5.77541 q^{49} +6.36689 q^{50} +10.9375 q^{51} -2.07859 q^{52} +5.70153 q^{53} -18.0821 q^{54} +10.3292 q^{55} +1.10661 q^{56} -15.3927 q^{57} +2.47911 q^{58} +4.43829 q^{59} -11.3654 q^{60} -9.03837 q^{61} +4.52613 q^{62} +9.25563 q^{63} +1.00000 q^{64} -7.00793 q^{65} -10.3278 q^{66} -10.8540 q^{67} -3.24454 q^{68} -12.2041 q^{69} +3.73092 q^{70} +10.3880 q^{71} +8.36395 q^{72} -13.3698 q^{73} +7.13547 q^{74} -21.4631 q^{75} +4.56616 q^{76} +3.39031 q^{77} +7.00702 q^{78} +12.6342 q^{79} +3.37148 q^{80} +35.8638 q^{81} -6.77163 q^{82} -3.41714 q^{83} -3.73043 q^{84} -10.9389 q^{85} +10.9333 q^{86} -8.35719 q^{87} +3.06369 q^{88} +10.5500 q^{89} +28.1989 q^{90} -2.30019 q^{91} +3.62028 q^{92} -15.2578 q^{93} -0.745380 q^{94} +15.3947 q^{95} -3.37105 q^{96} +0.221101 q^{97} -5.77541 q^{98} +25.6245 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9 + 21 * q^10 + 56 * q^11 + 21 * q^12 + 33 * q^13 + 38 * q^14 + 25 * q^15 + 67 * q^16 + 30 * q^17 + 90 * q^18 + 36 * q^19 + 21 * q^20 + 20 * q^21 + 56 * q^22 + 65 * q^23 + 21 * q^24 + 72 * q^25 + 33 * q^26 + 57 * q^27 + 38 * q^28 + 84 * q^29 + 25 * q^30 + 52 * q^31 + 67 * q^32 - 9 * q^33 + 30 * q^34 + 30 * q^35 + 90 * q^36 + 52 * q^37 + 36 * q^38 + 41 * q^39 + 21 * q^40 + 46 * q^41 + 20 * q^42 + 61 * q^43 + 56 * q^44 + 23 * q^45 + 65 * q^46 + 51 * q^47 + 21 * q^48 + 81 * q^49 + 72 * q^50 + 33 * q^51 + 33 * q^52 + 72 * q^53 + 57 * q^54 + 14 * q^55 + 38 * q^56 - 26 * q^57 + 84 * q^58 + 71 * q^59 + 25 * q^60 + 42 * q^61 + 52 * q^62 + 63 * q^63 + 67 * q^64 - 2 * q^65 - 9 * q^66 + 70 * q^67 + 30 * q^68 + 21 * q^69 + 30 * q^70 + 104 * q^71 + 90 * q^72 - 31 * q^73 + 52 * q^74 + 69 * q^75 + 36 * q^76 + 48 * q^77 + 41 * q^78 + 79 * q^79 + 21 * q^80 + 123 * q^81 + 46 * q^82 + 41 * q^83 + 20 * q^84 + 6 * q^85 + 61 * q^86 + 19 * q^87 + 56 * q^88 + 58 * q^89 + 23 * q^90 + 31 * q^91 + 65 * q^92 + 13 * q^93 + 51 * q^94 + 77 * q^95 + 21 * q^96 - 8 * q^97 + 81 * q^98 + 129 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.37105	−1.94627	−0.973137	−	0.230227i	$-0.926053\pi$
$3$	−3.37105	−1.94627	−0.973137	+	0.230227i	$0.926053\pi$
$4$	1.00000	0.500000
$5$	3.37148	1.50777	0.753886	−	0.657005i	$-0.228177\pi$
$5$	3.37148	1.50777	0.753886	+	0.657005i	$0.228177\pi$
$6$	−3.37105	−1.37622
$7$	1.10661	0.418259	0.209130	−	0.977888i	$-0.432937\pi$
$7$	1.10661	0.418259	0.209130	+	0.977888i	$0.432937\pi$
$8$	1.00000	0.353553
$9$	8.36395	2.78798
$10$	3.37148	1.06616
$11$	3.06369	0.923737	0.461868	−	0.886949i	$-0.347179\pi$
$11$	3.06369	0.923737	0.461868	+	0.886949i	$0.347179\pi$
$12$	−3.37105	−0.973137
$13$	−2.07859	−0.576497	−0.288249	−	0.957556i	$-0.593073\pi$
$13$	−2.07859	−0.576497	−0.288249	+	0.957556i	$0.593073\pi$
$14$	1.10661	0.295754
$15$	−11.3654	−2.93454
$16$	1.00000	0.250000
$17$	−3.24454	−0.786917	−0.393458	−	0.919342i	$-0.628721\pi$
$17$	−3.24454	−0.786917	−0.393458	+	0.919342i	$0.628721\pi$
$18$	8.36395	1.97140
$19$	4.56616	1.04755	0.523774	−	0.851857i	$-0.324524\pi$
$19$	4.56616	1.04755	0.523774	+	0.851857i	$0.324524\pi$
$20$	3.37148	0.753886
$21$	−3.73043	−0.814047
$22$	3.06369	0.653180
$23$	3.62028	0.754880	0.377440	−	0.926034i	$-0.376804\pi$
$23$	3.62028	0.754880	0.377440	+	0.926034i	$0.376804\pi$
$24$	−3.37105	−0.688112
$25$	6.36689	1.27338
$26$	−2.07859	−0.407645
$27$	−18.0821	−3.47990
$28$	1.10661	0.209130
$29$	2.47911	0.460359	0.230179	−	0.973148i	$-0.426069\pi$
$29$	2.47911	0.460359	0.230179	+	0.973148i	$0.426069\pi$
$30$	−11.3654	−2.07503
$31$	4.52613	0.812918	0.406459	−	0.913669i	$-0.366763\pi$
$31$	4.52613	0.812918	0.406459	+	0.913669i	$0.366763\pi$
$32$	1.00000	0.176777
$33$	−10.3278	−1.79784
$34$	−3.24454	−0.556434
$35$	3.73092	0.630640
$36$	8.36395	1.39399
$37$	7.13547	1.17306	0.586532	−	0.809926i	$-0.300493\pi$
$37$	7.13547	1.17306	0.586532	+	0.809926i	$0.300493\pi$
$38$	4.56616	0.740729
$39$	7.00702	1.12202
$40$	3.37148	0.533078
$41$	−6.77163	−1.05755	−0.528776	−	0.848762i	$-0.677349\pi$
$41$	−6.77163	−1.05755	−0.528776	+	0.848762i	$0.677349\pi$
$42$	−3.73043	−0.575618
$43$	10.9333	1.66732	0.833658	−	0.552281i	$-0.186242\pi$
$43$	10.9333	1.66732	0.833658	+	0.552281i	$0.186242\pi$
$44$	3.06369	0.461868
$45$	28.1989	4.20364
$46$	3.62028	0.533781
$47$	−0.745380	−0.108725	−0.0543625	−	0.998521i	$-0.517313\pi$
$47$	−0.745380	−0.108725	−0.0543625	+	0.998521i	$0.517313\pi$
$48$	−3.37105	−0.486568
$49$	−5.77541	−0.825059
$50$	6.36689	0.900414
$51$	10.9375	1.53156
$52$	−2.07859	−0.288249
$53$	5.70153	0.783166	0.391583	−	0.920143i	$-0.371928\pi$
$53$	5.70153	0.783166	0.391583	+	0.920143i	$0.371928\pi$
$54$	−18.0821	−2.46066
$55$	10.3292	1.39278
$56$	1.10661	0.147877
$57$	−15.3927	−2.03882
$58$	2.47911	0.325523
$59$	4.43829	0.577816	0.288908	−	0.957357i	$-0.406708\pi$
$59$	4.43829	0.577816	0.288908	+	0.957357i	$0.406708\pi$
$60$	−11.3654	−1.46727
$61$	−9.03837	−1.15725	−0.578623	−	0.815595i	$-0.696410\pi$
$61$	−9.03837	−1.15725	−0.578623	+	0.815595i	$0.696410\pi$
$62$	4.52613	0.574820
$63$	9.25563	1.16610
$64$	1.00000	0.125000
$65$	−7.00793	−0.869227
$66$	−10.3278	−1.27127
$67$	−10.8540	−1.32602	−0.663011	−	0.748610i	$-0.730722\pi$
$67$	−10.8540	−1.32602	−0.663011	+	0.748610i	$0.730722\pi$
$68$	−3.24454	−0.393458
$69$	−12.2041	−1.46920
$70$	3.73092	0.445930
$71$	10.3880	1.23283	0.616414	−	0.787422i	$-0.288585\pi$
$71$	10.3880	1.23283	0.616414	+	0.787422i	$0.288585\pi$
$72$	8.36395	0.985701
$73$	−13.3698	−1.56481	−0.782406	−	0.622769i	$-0.786008\pi$
$73$	−13.3698	−1.56481	−0.782406	+	0.622769i	$0.786008\pi$
$74$	7.13547	0.829482
$75$	−21.4631	−2.47834
$76$	4.56616	0.523774
$77$	3.39031	0.386362
$78$	7.00702	0.793389
$79$	12.6342	1.42145	0.710727	−	0.703468i	$-0.248366\pi$
$79$	12.6342	1.42145	0.710727	+	0.703468i	$0.248366\pi$
$80$	3.37148	0.376943
$81$	35.8638	3.98486
$82$	−6.77163	−0.747802
$83$	−3.41714	−0.375080	−0.187540	−	0.982257i	$-0.560051\pi$
$83$	−3.41714	−0.375080	−0.187540	+	0.982257i	$0.560051\pi$
$84$	−3.73043	−0.407024
$85$	−10.9389	−1.18649
$86$	10.9333	1.17897
$87$	−8.35719	−0.895984
$88$	3.06369	0.326590
$89$	10.5500	1.11830	0.559151	−	0.829066i	$-0.311127\pi$
$89$	10.5500	1.11830	0.559151	+	0.829066i	$0.311127\pi$
$90$	28.1989	2.97242
$91$	−2.30019	−0.241125
$92$	3.62028	0.377440
$93$	−15.2578	−1.58216
$94$	−0.745380	−0.0768801
$95$	15.3947	1.57946
$96$	−3.37105	−0.344056
$97$	0.221101	0.0224495	0.0112247	−	0.999937i	$-0.496427\pi$
$97$	0.221101	0.0224495	0.0112247	+	0.999937i	$0.496427\pi$
$98$	−5.77541	−0.583405
$99$	25.6245	2.57536
$100$	6.36689	0.636689
$101$	−3.47821	−0.346095	−0.173047	−	0.984914i	$-0.555361\pi$
$101$	−3.47821	−0.346095	−0.173047	+	0.984914i	$0.555361\pi$
$102$	10.9375	1.08297
$103$	11.4260	1.12584	0.562918	−	0.826513i	$-0.309679\pi$
$103$	11.4260	1.12584	0.562918	+	0.826513i	$0.309679\pi$
$104$	−2.07859	−0.203823
$105$	−12.5771	−1.22740
$106$	5.70153	0.553782
$107$	−12.1729	−1.17680	−0.588398	−	0.808572i	$-0.700241\pi$
$107$	−12.1729	−1.17680	−0.588398	+	0.808572i	$0.700241\pi$
$108$	−18.0821	−1.73995
$109$	−4.64909	−0.445302	−0.222651	−	0.974898i	$-0.571471\pi$
$109$	−4.64909	−0.445302	−0.222651	+	0.974898i	$0.571471\pi$
$110$	10.3292	0.984848
$111$	−24.0540	−2.28310
$112$	1.10661	0.104565
$113$	−3.63112	−0.341587	−0.170793	−	0.985307i	$-0.554633\pi$
$113$	−3.63112	−0.341587	−0.170793	+	0.985307i	$0.554633\pi$
$114$	−15.3927	−1.44166
$115$	12.2057	1.13819
$116$	2.47911	0.230179
$117$	−17.3852	−1.60726
$118$	4.43829	0.408578
$119$	−3.59044	−0.329135
$120$	−11.3654	−1.03752
$121$	−1.61382	−0.146711
$122$	−9.03837	−0.818296
$123$	22.8275	2.05829
$124$	4.52613	0.406459
$125$	4.60844	0.412192
$126$	9.25563	0.824557
$127$	−8.06975	−0.716075	−0.358037	−	0.933707i	$-0.616554\pi$
$127$	−8.06975	−0.716075	−0.358037	+	0.933707i	$0.616554\pi$
$128$	1.00000	0.0883883
$129$	−36.8567	−3.24505
$130$	−7.00793	−0.614636
$131$	−8.52801	−0.745096	−0.372548	−	0.928013i	$-0.621516\pi$
$131$	−8.52801	−0.745096	−0.372548	+	0.928013i	$0.621516\pi$
$132$	−10.3278	−0.898922
$133$	5.05296	0.438147
$134$	−10.8540	−0.937639
$135$	−60.9635	−5.24690
$136$	−3.24454	−0.278217
$137$	1.56580	0.133775	0.0668874	−	0.997761i	$-0.478693\pi$
$137$	1.56580	0.133775	0.0668874	+	0.997761i	$0.478693\pi$
$138$	−12.2041	−1.03888
$139$	−8.89733	−0.754661	−0.377331	−	0.926079i	$-0.623158\pi$
$139$	−8.89733	−0.754661	−0.377331	+	0.926079i	$0.623158\pi$
$140$	3.73092	0.315320
$141$	2.51271	0.211608
$142$	10.3880	0.871741
$143$	−6.36815	−0.532532
$144$	8.36395	0.696996
$145$	8.35827	0.694116
$146$	−13.3698	−1.10649
$147$	19.4692	1.60579
$148$	7.13547	0.586532
$149$	−11.0430	−0.904679	−0.452339	−	0.891846i	$-0.649410\pi$
$149$	−11.0430	−0.904679	−0.452339	+	0.891846i	$0.649410\pi$
$150$	−21.4631	−1.75245
$151$	6.02353	0.490188	0.245094	−	0.969499i	$-0.421181\pi$
$151$	6.02353	0.490188	0.245094	+	0.969499i	$0.421181\pi$
$152$	4.56616	0.370364
$153$	−27.1372	−2.19391
$154$	3.39031	0.273199
$155$	15.2598	1.22569
$156$	7.00702	0.561011
$157$	−4.08546	−0.326055	−0.163028	−	0.986622i	$-0.552126\pi$
$157$	−4.08546	−0.326055	−0.163028	+	0.986622i	$0.552126\pi$
$158$	12.6342	1.00512
$159$	−19.2201	−1.52426
$160$	3.37148	0.266539
$161$	4.00624	0.315736
$162$	35.8638	2.81772
$163$	−17.0749	−1.33741	−0.668706	−	0.743527i	$-0.733152\pi$
$163$	−17.0749	−1.33741	−0.668706	+	0.743527i	$0.733152\pi$
$164$	−6.77163	−0.528776
$165$	−34.8201	−2.71074
$166$	−3.41714	−0.265221
$167$	14.3537	1.11073	0.555363	−	0.831608i	$-0.312579\pi$
$167$	14.3537	1.11073	0.555363	+	0.831608i	$0.312579\pi$
$168$	−3.73043	−0.287809
$169$	−8.67946	−0.667651
$170$	−10.9389	−0.838976
$171$	38.1911	2.92055
$172$	10.9333	0.833658
$173$	−8.78966	−0.668265	−0.334133	−	0.942526i	$-0.608443\pi$
$173$	−8.78966	−0.668265	−0.334133	+	0.942526i	$0.608443\pi$
$174$	−8.35719	−0.633557
$175$	7.04567	0.532602
$176$	3.06369	0.230934
$177$	−14.9617	−1.12459
$178$	10.5500	0.790758
$179$	17.5751	1.31362	0.656811	−	0.754056i	$-0.271905\pi$
$179$	17.5751	1.31362	0.656811	+	0.754056i	$0.271905\pi$
$180$	28.1989	2.10182
$181$	0.813712	0.0604827	0.0302414	−	0.999543i	$-0.490372\pi$
$181$	0.813712	0.0604827	0.0302414	+	0.999543i	$0.490372\pi$
$182$	−2.30019	−0.170501
$183$	30.4688	2.25232
$184$	3.62028	0.266890
$185$	24.0571	1.76871
$186$	−15.2578	−1.11876
$187$	−9.94026	−0.726904
$188$	−0.745380	−0.0543625
$189$	−20.0098	−1.45550
$190$	15.3947	1.11685
$191$	−9.30218	−0.673082	−0.336541	−	0.941669i	$-0.609257\pi$
$191$	−9.30218	−0.673082	−0.336541	+	0.941669i	$0.609257\pi$
$192$	−3.37105	−0.243284
$193$	13.0787	0.941423	0.470712	−	0.882287i	$-0.343997\pi$
$193$	13.0787	0.941423	0.470712	+	0.882287i	$0.343997\pi$
$194$	0.221101	0.0158742
$195$	23.6241	1.69175
$196$	−5.77541	−0.412530
$197$	−12.2399	−0.872054	−0.436027	−	0.899934i	$-0.643615\pi$
$197$	−12.2399	−0.872054	−0.436027	+	0.899934i	$0.643615\pi$
$198$	25.6245	1.82106
$199$	−4.72811	−0.335167	−0.167583	−	0.985858i	$-0.553596\pi$
$199$	−4.72811	−0.335167	−0.167583	+	0.985858i	$0.553596\pi$
$200$	6.36689	0.450207
$201$	36.5892	2.58080
$202$	−3.47821	−0.244726
$203$	2.74341	0.192549
$204$	10.9375	0.765778
$205$	−22.8304	−1.59455
$206$	11.4260	0.796086
$207$	30.2798	2.10459
$208$	−2.07859	−0.144124
$209$	13.9893	0.967659
$210$	−12.5771	−0.867902
$211$	10.5313	0.725007	0.362503	−	0.931982i	$-0.381922\pi$
$211$	10.5313	0.725007	0.362503	+	0.931982i	$0.381922\pi$
$212$	5.70153	0.391583
$213$	−35.0184	−2.39942
$214$	−12.1729	−0.832120
$215$	36.8615	2.51393
$216$	−18.0821	−1.23033
$217$	5.00867	0.340010
$218$	−4.64909	−0.314876
$219$	45.0701	3.04555
$220$	10.3292	0.696392
$221$	6.74407	0.453655
$222$	−24.0540	−1.61440
$223$	−24.3681	−1.63181	−0.815904	−	0.578188i	$-0.803760\pi$
$223$	−24.3681	−1.63181	−0.815904	+	0.578188i	$0.803760\pi$
$224$	1.10661	0.0739385
$225$	53.2523	3.55016
$226$	−3.63112	−0.241538
$227$	26.2016	1.73906	0.869532	−	0.493876i	$-0.164420\pi$
$227$	26.2016	1.73906	0.869532	+	0.493876i	$0.164420\pi$
$228$	−15.3927	−1.01941
$229$	−10.2702	−0.678672	−0.339336	−	0.940665i	$-0.610202\pi$
$229$	−10.2702	−0.678672	−0.339336	+	0.940665i	$0.610202\pi$
$230$	12.2057	0.804820
$231$	−11.4289	−0.751965
$232$	2.47911	0.162761
$233$	28.9047	1.89361	0.946805	−	0.321808i	$-0.104290\pi$
$233$	28.9047	1.89361	0.946805	+	0.321808i	$0.104290\pi$
$234$	−17.3852	−1.13651
$235$	−2.51304	−0.163932
$236$	4.43829	0.288908
$237$	−42.5903	−2.76654
$238$	−3.59044	−0.232734
$239$	1.40079	0.0906097	0.0453049	−	0.998973i	$-0.485574\pi$
$239$	1.40079	0.0906097	0.0453049	+	0.998973i	$0.485574\pi$
$240$	−11.3654	−0.733635
$241$	28.7808	1.85393	0.926967	−	0.375144i	$-0.122407\pi$
$241$	28.7808	1.85393	0.926967	+	0.375144i	$0.122407\pi$
$242$	−1.61382	−0.103740
$243$	−66.6521	−4.27573
$244$	−9.03837	−0.578623
$245$	−19.4717	−1.24400
$246$	22.8275	1.45543
$247$	−9.49117	−0.603909
$248$	4.52613	0.287410
$249$	11.5193	0.730008
$250$	4.60844	0.291464
$251$	7.85059	0.495525	0.247763	−	0.968821i	$-0.420305\pi$
$251$	7.85059	0.495525	0.247763	+	0.968821i	$0.420305\pi$
$252$	9.25563	0.583050
$253$	11.0914	0.697311
$254$	−8.06975	−0.506341
$255$	36.8756	2.30924
$256$	1.00000	0.0625000
$257$	−29.7942	−1.85851	−0.929254	−	0.369441i	$-0.879549\pi$
$257$	−29.7942	−1.85851	−0.929254	+	0.369441i	$0.879549\pi$
$258$	−36.8567	−2.29460
$259$	7.89619	0.490645
$260$	−7.00793	−0.434613
$261$	20.7351	1.28347
$262$	−8.52801	−0.526862
$263$	10.4898	0.646830	0.323415	−	0.946257i	$-0.395169\pi$
$263$	10.4898	0.646830	0.323415	+	0.946257i	$0.395169\pi$
$264$	−10.3278	−0.635634
$265$	19.2226	1.18084
$266$	5.05296	0.309817
$267$	−35.5646	−2.17652
$268$	−10.8540	−0.663011
$269$	5.45188	0.332407	0.166203	−	0.986091i	$-0.446849\pi$
$269$	5.45188	0.332407	0.166203	+	0.986091i	$0.446849\pi$
$270$	−60.9635	−3.71012
$271$	8.03984	0.488386	0.244193	−	0.969727i	$-0.421477\pi$
$271$	8.03984	0.488386	0.244193	+	0.969727i	$0.421477\pi$
$272$	−3.24454	−0.196729
$273$	7.75405	0.469296
$274$	1.56580	0.0945931
$275$	19.5062	1.17627
$276$	−12.2041	−0.734602
$277$	27.5354	1.65444	0.827221	−	0.561877i	$-0.189920\pi$
$277$	27.5354	1.65444	0.827221	+	0.561877i	$0.189920\pi$
$278$	−8.89733	−0.533626
$279$	37.8563	2.26640
$280$	3.73092	0.222965
$281$	−1.06196	−0.0633510	−0.0316755	−	0.999498i	$-0.510084\pi$
$281$	−1.06196	−0.0633510	−0.0316755	+	0.999498i	$0.510084\pi$
$282$	2.51271	0.149630
$283$	4.48140	0.266391	0.133196	−	0.991090i	$-0.457476\pi$
$283$	4.48140	0.266391	0.133196	+	0.991090i	$0.457476\pi$
$284$	10.3880	0.616414
$285$	−51.8963	−3.07407
$286$	−6.36815	−0.376557
$287$	−7.49356	−0.442331
$288$	8.36395	0.492850
$289$	−6.47296	−0.380762
$290$	8.35827	0.490814
$291$	−0.745343	−0.0436928
$292$	−13.3698	−0.782406
$293$	−22.6152	−1.32119	−0.660597	−	0.750740i	$-0.729697\pi$
$293$	−22.6152	−1.32119	−0.660597	+	0.750740i	$0.729697\pi$
$294$	19.4692	1.13547
$295$	14.9636	0.871215
$296$	7.13547	0.414741
$297$	−55.3979	−3.21451
$298$	−11.0430	−0.639705
$299$	−7.52508	−0.435186
$300$	−21.4631	−1.23917
$301$	12.0989	0.697371
$302$	6.02353	0.346615
$303$	11.7252	0.673595
$304$	4.56616	0.261887
$305$	−30.4727	−1.74486
$306$	−27.1372	−1.55133
$307$	16.2320	0.926410	0.463205	−	0.886251i	$-0.346699\pi$
$307$	16.2320	0.926410	0.463205	+	0.886251i	$0.346699\pi$
$308$	3.39031	0.193181
$309$	−38.5175	−2.19118
$310$	15.2598	0.866697
$311$	13.1060	0.743172	0.371586	−	0.928399i	$-0.378814\pi$
$311$	13.1060	0.743172	0.371586	+	0.928399i	$0.378814\pi$
$312$	7.00702	0.396695
$313$	22.3381	1.26263	0.631313	−	0.775528i	$-0.282516\pi$
$313$	22.3381	1.26263	0.631313	+	0.775528i	$0.282516\pi$
$314$	−4.08546	−0.230556
$315$	31.2052	1.75821
$316$	12.6342	0.710727
$317$	11.9121	0.669048	0.334524	−	0.942387i	$-0.391425\pi$
$317$	11.9121	0.669048	0.334524	+	0.942387i	$0.391425\pi$
$318$	−19.2201	−1.07781
$319$	7.59521	0.425250
$320$	3.37148	0.188472
$321$	41.0353	2.29037
$322$	4.00624	0.223259
$323$	−14.8151	−0.824333
$324$	35.8638	1.99243
$325$	−13.2342	−0.734099
$326$	−17.0749	−0.945693
$327$	15.6723	0.866680
$328$	−6.77163	−0.373901
$329$	−0.824846	−0.0454752
$330$	−34.8201	−1.91678
$331$	30.5717	1.68037	0.840187	−	0.542297i	$-0.182445\pi$
$331$	30.5717	1.68037	0.840187	+	0.542297i	$0.182445\pi$
$332$	−3.41714	−0.187540
$333$	59.6807	3.27048
$334$	14.3537	0.785402
$335$	−36.5939	−1.99934
$336$	−3.73043	−0.203512
$337$	−11.0120	−0.599861	−0.299930	−	0.953961i	$-0.596963\pi$
$337$	−11.0120	−0.599861	−0.299930	+	0.953961i	$0.596963\pi$
$338$	−8.67946	−0.472100
$339$	12.2407	0.664822
$340$	−10.9389	−0.593246
$341$	13.8667	0.750922
$342$	38.1911	2.06514
$343$	−14.1374	−0.763348
$344$	10.9333	0.589485
$345$	−41.1460	−2.21522
$346$	−8.78966	−0.472535
$347$	−3.16400	−0.169852	−0.0849261	−	0.996387i	$-0.527065\pi$
$347$	−3.16400	−0.169852	−0.0849261	+	0.996387i	$0.527065\pi$
$348$	−8.35719	−0.447992
$349$	33.9851	1.81918	0.909591	−	0.415505i	$-0.136395\pi$
$349$	33.9851	1.81918	0.909591	+	0.415505i	$0.136395\pi$
$350$	7.04567	0.376607
$351$	37.5853	2.00616
$352$	3.06369	0.163295
$353$	8.40690	0.447454	0.223727	−	0.974652i	$-0.428178\pi$
$353$	8.40690	0.447454	0.223727	+	0.974652i	$0.428178\pi$
$354$	−14.9617	−0.795204
$355$	35.0229	1.85882
$356$	10.5500	0.559151
$357$	12.1035	0.640587
$358$	17.5751	0.928871
$359$	−9.23629	−0.487473	−0.243736	−	0.969842i	$-0.578373\pi$
$359$	−9.23629	−0.487473	−0.243736	+	0.969842i	$0.578373\pi$
$360$	28.1989	1.48621
$361$	1.84979	0.0973574
$362$	0.813712	0.0427677
$363$	5.44025	0.285539
$364$	−2.30019	−0.120563
$365$	−45.0759	−2.35938
$366$	30.4688	1.59263
$367$	6.23265	0.325341	0.162671	−	0.986680i	$-0.447989\pi$
$367$	6.23265	0.325341	0.162671	+	0.986680i	$0.447989\pi$
$368$	3.62028	0.188720
$369$	−56.6376	−2.94844
$370$	24.0571	1.25067
$371$	6.30938	0.327567
$372$	−15.2578	−0.791080
$373$	19.0716	0.987488	0.493744	−	0.869607i	$-0.335628\pi$
$373$	19.0716	0.987488	0.493744	+	0.869607i	$0.335628\pi$
$374$	−9.94026	−0.513999
$375$	−15.5353	−0.802238
$376$	−0.745380	−0.0384401
$377$	−5.15305	−0.265396
$378$	−20.0098	−1.02920
$379$	5.76840	0.296303	0.148151	−	0.988965i	$-0.452668\pi$
$379$	5.76840	0.296303	0.148151	+	0.988965i	$0.452668\pi$
$380$	15.3947	0.789732
$381$	27.2035	1.39368
$382$	−9.30218	−0.475941
$383$	−16.8383	−0.860396	−0.430198	−	0.902735i	$-0.641556\pi$
$383$	−16.8383	−0.860396	−0.430198	+	0.902735i	$0.641556\pi$
$384$	−3.37105	−0.172028
$385$	11.4304	0.582545
$386$	13.0787	0.665687
$387$	91.4457	4.64845
$388$	0.221101	0.0112247
$389$	−10.2267	−0.518513	−0.259257	−	0.965808i	$-0.583478\pi$
$389$	−10.2267	−0.518513	−0.259257	+	0.965808i	$0.583478\pi$
$390$	23.6241	1.19625
$391$	−11.7461	−0.594028
$392$	−5.77541	−0.291702
$393$	28.7483	1.45016
$394$	−12.2399	−0.616636
$395$	42.5959	2.14323
$396$	25.6245	1.28768
$397$	−6.55362	−0.328917	−0.164458	−	0.986384i	$-0.552588\pi$
$397$	−6.55362	−0.328917	−0.164458	+	0.986384i	$0.552588\pi$
$398$	−4.72811	−0.236999
$399$	−17.0337	−0.852754
$400$	6.36689	0.318345
$401$	−5.36241	−0.267786	−0.133893	−	0.990996i	$-0.542748\pi$
$401$	−5.36241	−0.267786	−0.133893	+	0.990996i	$0.542748\pi$
$402$	36.5892	1.82490
$403$	−9.40798	−0.468645
$404$	−3.47821	−0.173047
$405$	120.914	6.00827
$406$	2.74341	0.136153
$407$	21.8609	1.08360
$408$	10.9375	0.541487
$409$	33.8435	1.67345	0.836727	−	0.547621i	$-0.184466\pi$
$409$	33.8435	1.67345	0.836727	+	0.547621i	$0.184466\pi$
$410$	−22.8304	−1.12752
$411$	−5.27837	−0.260363
$412$	11.4260	0.562918
$413$	4.91146	0.241677
$414$	30.2798	1.48817
$415$	−11.5208	−0.565535
$416$	−2.07859	−0.101911
$417$	29.9933	1.46878
$418$	13.9893	0.684238
$419$	−15.6672	−0.765392	−0.382696	−	0.923874i	$-0.625004\pi$
$419$	−15.6672	−0.765392	−0.382696	+	0.923874i	$0.625004\pi$
$420$	−12.5771	−0.613699
$421$	20.5496	1.00152	0.500762	−	0.865585i	$-0.333053\pi$
$421$	20.5496	1.00152	0.500762	+	0.865585i	$0.333053\pi$
$422$	10.5313	0.512657
$423$	−6.23432	−0.303123
$424$	5.70153	0.276891
$425$	−20.6576	−1.00204
$426$	−35.0184	−1.69665
$427$	−10.0020	−0.484029
$428$	−12.1729	−0.588398
$429$	21.4673	1.03645
$430$	36.8615	1.77762
$431$	−29.9478	−1.44253	−0.721267	−	0.692657i	$-0.756440\pi$
$431$	−29.9478	−1.44253	−0.721267	+	0.692657i	$0.756440\pi$
$432$	−18.0821	−0.869976
$433$	−2.50126	−0.120203	−0.0601014	−	0.998192i	$-0.519142\pi$
$433$	−2.50126	−0.120203	−0.0601014	+	0.998192i	$0.519142\pi$
$434$	5.00867	0.240424
$435$	−28.1761	−1.35094
$436$	−4.64909	−0.222651
$437$	16.5308	0.790773
$438$	45.0701	2.15353
$439$	−2.13092	−0.101703	−0.0508517	−	0.998706i	$-0.516194\pi$
$439$	−2.13092	−0.101703	−0.0508517	+	0.998706i	$0.516194\pi$
$440$	10.3292	0.492424
$441$	−48.3053	−2.30025
$442$	6.74407	0.320783
$443$	−25.4502	−1.20917	−0.604587	−	0.796539i	$-0.706662\pi$
$443$	−25.4502	−1.20917	−0.604587	+	0.796539i	$0.706662\pi$
$444$	−24.0540	−1.14155
$445$	35.5692	1.68614
$446$	−24.3681	−1.15386
$447$	37.2265	1.76075
$448$	1.10661	0.0522824
$449$	21.9124	1.03411	0.517055	−	0.855952i	$-0.327028\pi$
$449$	21.9124	1.03411	0.517055	+	0.855952i	$0.327028\pi$
$450$	53.2523	2.51034
$451$	−20.7462	−0.976899
$452$	−3.63112	−0.170793
$453$	−20.3056	−0.954039
$454$	26.2016	1.22970
$455$	−7.75505	−0.363562
$456$	−15.3927	−0.720830
$457$	−30.3537	−1.41989	−0.709943	−	0.704259i	$-0.751279\pi$
$457$	−30.3537	−1.41989	−0.709943	+	0.704259i	$0.751279\pi$
$458$	−10.2702	−0.479893
$459$	58.6681	2.73839
$460$	12.2057	0.569094
$461$	36.0865	1.68072	0.840358	−	0.542032i	$-0.182345\pi$
$461$	36.0865	1.68072	0.840358	+	0.542032i	$0.182345\pi$
$462$	−11.4289	−0.531720
$463$	6.08864	0.282963	0.141482	−	0.989941i	$-0.454813\pi$
$463$	6.08864	0.282963	0.141482	+	0.989941i	$0.454813\pi$
$464$	2.47911	0.115090
$465$	−51.4414	−2.38554
$466$	28.9047	1.33898
$467$	−21.1367	−0.978090	−0.489045	−	0.872259i	$-0.662655\pi$
$467$	−21.1367	−0.978090	−0.489045	+	0.872259i	$0.662655\pi$
$468$	−17.3852	−0.803632
$469$	−12.0111	−0.554621
$470$	−2.51304	−0.115918
$471$	13.7723	0.634593
$472$	4.43829	0.204289
$473$	33.4963	1.54016
$474$	−42.5903	−1.95624
$475$	29.0722	1.33392
$476$	−3.59044	−0.164568
$477$	47.6873	2.18345
$478$	1.40079	0.0640708
$479$	−38.5116	−1.75964	−0.879820	−	0.475308i	$-0.842337\pi$
$479$	−38.5116	−1.75964	−0.879820	+	0.475308i	$0.842337\pi$
$480$	−11.3654	−0.518758
$481$	−14.8317	−0.676268
$482$	28.7808	1.31093
$483$	−13.5052	−0.614508
$484$	−1.61382	−0.0733553
$485$	0.745440	0.0338487
$486$	−66.6521	−3.02340
$487$	−26.7519	−1.21224	−0.606122	−	0.795371i	$-0.707276\pi$
$487$	−26.7519	−1.21224	−0.606122	+	0.795371i	$0.707276\pi$
$488$	−9.03837	−0.409148
$489$	57.5604	2.60297
$490$	−19.4717	−0.879642
$491$	10.3011	0.464882	0.232441	−	0.972610i	$-0.425329\pi$
$491$	10.3011	0.464882	0.232441	+	0.972610i	$0.425329\pi$
$492$	22.8275	1.02914
$493$	−8.04357	−0.362264
$494$	−9.49117	−0.427028
$495$	86.3926	3.88306
$496$	4.52613	0.203229
$497$	11.4955	0.515642
$498$	11.5193	0.516194
$499$	31.5724	1.41337	0.706687	−	0.707526i	$-0.250189\pi$
$499$	31.5724	1.41337	0.706687	+	0.707526i	$0.250189\pi$
$500$	4.60844	0.206096
$501$	−48.3871	−2.16178
$502$	7.85059	0.350389
$503$	20.1882	0.900146	0.450073	−	0.892992i	$-0.351398\pi$
$503$	20.1882	0.900146	0.450073	+	0.892992i	$0.351398\pi$
$504$	9.25563	0.412279
$505$	−11.7267	−0.521832
$506$	11.0914	0.493073
$507$	29.2589	1.29943
$508$	−8.06975	−0.358037
$509$	−2.58631	−0.114636	−0.0573181	−	0.998356i	$-0.518255\pi$
$509$	−2.58631	−0.114636	−0.0573181	+	0.998356i	$0.518255\pi$
$510$	36.8756	1.63288
$511$	−14.7951	−0.654497
$512$	1.00000	0.0441942
$513$	−82.5658	−3.64537
$514$	−29.7942	−1.31416
$515$	38.5225	1.69750
$516$	−36.8567	−1.62253
$517$	−2.28361	−0.100433
$518$	7.89619	0.346939
$519$	29.6303	1.30063
$520$	−7.00793	−0.307318
$521$	3.73686	0.163715	0.0818573	−	0.996644i	$-0.473915\pi$
$521$	3.73686	0.163715	0.0818573	+	0.996644i	$0.473915\pi$
$522$	20.7351	0.907552
$523$	−13.7746	−0.602322	−0.301161	−	0.953573i	$-0.597374\pi$
$523$	−13.7746	−0.602322	−0.301161	+	0.953573i	$0.597374\pi$
$524$	−8.52801	−0.372548
$525$	−23.7513	−1.03659
$526$	10.4898	0.457378
$527$	−14.6852	−0.639698
$528$	−10.3278	−0.449461
$529$	−9.89358	−0.430156
$530$	19.2226	0.834977
$531$	37.1216	1.61094
$532$	5.05296	0.219073
$533$	14.0755	0.609676
$534$	−35.5646	−1.53903
$535$	−41.0406	−1.77434
$536$	−10.8540	−0.468819
$537$	−59.2463	−2.55667
$538$	5.45188	0.235047
$539$	−17.6941	−0.762137
$540$	−60.9635	−2.62345
$541$	37.2818	1.60287	0.801435	−	0.598082i	$-0.204070\pi$
$541$	37.2818	1.60287	0.801435	+	0.598082i	$0.204070\pi$
$542$	8.03984	0.345341
$543$	−2.74306	−0.117716
$544$	−3.24454	−0.139109
$545$	−15.6743	−0.671415
$546$	7.75405	0.331843
$547$	12.3836	0.529484	0.264742	−	0.964319i	$-0.414713\pi$
$547$	12.3836	0.529484	0.264742	+	0.964319i	$0.414713\pi$
$548$	1.56580	0.0668874
$549$	−75.5965	−3.22638
$550$	19.5062	0.831746
$551$	11.3200	0.482248
$552$	−12.2041	−0.519442
$553$	13.9811	0.594537
$554$	27.5354	1.16987
$555$	−81.0976	−3.44240
$556$	−8.89733	−0.377331
$557$	−25.0366	−1.06083	−0.530416	−	0.847737i	$-0.677964\pi$
$557$	−25.0366	−1.06083	−0.530416	+	0.847737i	$0.677964\pi$
$558$	37.8563	1.60259
$559$	−22.7259	−0.961203
$560$	3.73092	0.157660
$561$	33.5091	1.41475
$562$	−1.06196	−0.0447959
$563$	−10.7027	−0.451064	−0.225532	−	0.974236i	$-0.572412\pi$
$563$	−10.7027	−0.451064	−0.225532	+	0.974236i	$0.572412\pi$
$564$	2.51271	0.105804
$565$	−12.2422	−0.515035
$566$	4.48140	0.188367
$567$	39.6872	1.66671
$568$	10.3880	0.435871
$569$	11.8831	0.498164	0.249082	−	0.968482i	$-0.419871\pi$
$569$	11.8831	0.498164	0.249082	+	0.968482i	$0.419871\pi$
$570$	−51.8963	−2.17370
$571$	−39.7121	−1.66190	−0.830949	−	0.556348i	$-0.812202\pi$
$571$	−39.7121	−1.66190	−0.830949	+	0.556348i	$0.812202\pi$
$572$	−6.36815	−0.266266
$573$	31.3581	1.31000
$574$	−7.49356	−0.312775
$575$	23.0499	0.961248
$576$	8.36395	0.348498
$577$	21.7520	0.905549	0.452774	−	0.891625i	$-0.350434\pi$
$577$	21.7520	0.905549	0.452774	+	0.891625i	$0.350434\pi$
$578$	−6.47296	−0.269240
$579$	−44.0888	−1.83227
$580$	8.35827	0.347058
$581$	−3.78144	−0.156881
$582$	−0.745343	−0.0308955
$583$	17.4677	0.723439
$584$	−13.3698	−0.553245
$585$	−58.6140	−2.42339
$586$	−22.6152	−0.934226
$587$	35.7581	1.47590	0.737948	−	0.674857i	$-0.235795\pi$
$587$	35.7581	1.47590	0.737948	+	0.674857i	$0.235795\pi$
$588$	19.4692	0.802895
$589$	20.6670	0.851571
$590$	14.9636	0.616042
$591$	41.2611	1.69726
$592$	7.13547	0.293266
$593$	−40.4228	−1.65997	−0.829984	−	0.557788i	$-0.811650\pi$
$593$	−40.4228	−1.65997	−0.829984	+	0.557788i	$0.811650\pi$
$594$	−55.3979	−2.27300
$595$	−12.1051	−0.496261
$596$	−11.0430	−0.452339
$597$	15.9387	0.652326
$598$	−7.52508	−0.307723
$599$	−13.7108	−0.560209	−0.280104	−	0.959970i	$-0.590369\pi$
$599$	−13.7108	−0.560209	−0.280104	+	0.959970i	$0.590369\pi$
$600$	−21.4631	−0.876226
$601$	−6.95424	−0.283669	−0.141835	−	0.989890i	$-0.545300\pi$
$601$	−6.95424	−0.283669	−0.141835	+	0.989890i	$0.545300\pi$
$602$	12.0989	0.493116
$603$	−90.7819	−3.69692
$604$	6.02353	0.245094
$605$	−5.44095	−0.221206
$606$	11.7252	0.476304
$607$	−37.1881	−1.50942	−0.754709	−	0.656060i	$-0.772222\pi$
$607$	−37.1881	−1.50942	−0.754709	+	0.656060i	$0.772222\pi$
$608$	4.56616	0.185182
$609$	−9.24815	−0.374754
$610$	−30.4727	−1.23380
$611$	1.54934	0.0626796
$612$	−27.1372	−1.09695
$613$	−33.4111	−1.34946	−0.674730	−	0.738064i	$-0.735740\pi$
$613$	−33.4111	−1.34946	−0.674730	+	0.738064i	$0.735740\pi$
$614$	16.2320	0.655071
$615$	76.9625	3.10343
$616$	3.39031	0.136599
$617$	−17.4136	−0.701045	−0.350523	−	0.936554i	$-0.613996\pi$
$617$	−17.4136	−0.701045	−0.350523	+	0.936554i	$0.613996\pi$
$618$	−38.5175	−1.54940
$619$	−13.8571	−0.556963	−0.278482	−	0.960442i	$-0.589831\pi$
$619$	−13.8571	−0.556963	−0.278482	+	0.960442i	$0.589831\pi$
$620$	15.2598	0.612847
$621$	−65.4623	−2.62691
$622$	13.1060	0.525502
$623$	11.6748	0.467740
$624$	7.00702	0.280505
$625$	−16.2972	−0.651886
$626$	22.3381	0.892811
$627$	−47.1585	−1.88333
$628$	−4.08546	−0.163028
$629$	−23.1513	−0.923104
$630$	31.2052	1.24324
$631$	16.9403	0.674384	0.337192	−	0.941436i	$-0.390523\pi$
$631$	16.9403	0.674384	0.337192	+	0.941436i	$0.390523\pi$
$632$	12.6342	0.502560
$633$	−35.5016	−1.41106
$634$	11.9121	0.473088
$635$	−27.2070	−1.07968
$636$	−19.2201	−0.762128
$637$	12.0047	0.475644
$638$	7.59521	0.300697
$639$	86.8846	3.43710
$640$	3.37148	0.133270
$641$	−22.3355	−0.882198	−0.441099	−	0.897459i	$-0.645411\pi$
$641$	−22.3355	−0.882198	−0.441099	+	0.897459i	$0.645411\pi$
$642$	41.0353	1.61953
$643$	41.0495	1.61883	0.809417	−	0.587234i	$-0.199783\pi$
$643$	41.0495	1.61883	0.809417	+	0.587234i	$0.199783\pi$
$644$	4.00624	0.157868
$645$	−124.262	−4.89280
$646$	−14.8151	−0.582892
$647$	3.97781	0.156384	0.0781919	−	0.996938i	$-0.475085\pi$
$647$	3.97781	0.156384	0.0781919	+	0.996938i	$0.475085\pi$
$648$	35.8638	1.40886
$649$	13.5975	0.533750
$650$	−13.2342	−0.519086
$651$	−16.8844	−0.661754
$652$	−17.0749	−0.668706
$653$	−27.2465	−1.06624	−0.533120	−	0.846040i	$-0.678980\pi$
$653$	−27.2465	−1.06624	−0.533120	+	0.846040i	$0.678980\pi$
$654$	15.6723	0.612836
$655$	−28.7520	−1.12344
$656$	−6.77163	−0.264388
$657$	−111.824	−4.36267
$658$	−0.824846	−0.0321558
$659$	40.3947	1.57356	0.786778	−	0.617236i	$-0.211748\pi$
$659$	40.3947	1.57356	0.786778	+	0.617236i	$0.211748\pi$
$660$	−34.8201	−1.35537
$661$	−29.8681	−1.16174	−0.580868	−	0.813998i	$-0.697287\pi$
$661$	−29.8681	−1.16174	−0.580868	+	0.813998i	$0.697287\pi$
$662$	30.5717	1.18820
$663$	−22.7346	−0.882938
$664$	−3.41714	−0.132611
$665$	17.0360	0.660626
$666$	59.6807	2.31258
$667$	8.97506	0.347516
$668$	14.3537	0.555363
$669$	82.1459	3.17594
$670$	−36.5939	−1.41375
$671$	−27.6908	−1.06899
$672$	−3.73043	−0.143905
$673$	−35.1399	−1.35454	−0.677272	−	0.735732i	$-0.736838\pi$
$673$	−35.1399	−1.35454	−0.677272	+	0.735732i	$0.736838\pi$
$674$	−11.0120	−0.424165
$675$	−115.127	−4.43123
$676$	−8.67946	−0.333825
$677$	44.0772	1.69402	0.847012	−	0.531574i	$-0.178399\pi$
$677$	44.0772	1.69402	0.847012	+	0.531574i	$0.178399\pi$
$678$	12.2407	0.470100
$679$	0.244673	0.00938969
$680$	−10.9389	−0.419488
$681$	−88.3269	−3.38470
$682$	13.8667	0.530982
$683$	−30.5199	−1.16781	−0.583906	−	0.811822i	$-0.698476\pi$
$683$	−30.5199	−1.16781	−0.583906	+	0.811822i	$0.698476\pi$
$684$	38.1911	1.46027
$685$	5.27905	0.201702
$686$	−14.1374	−0.539769
$687$	34.6212	1.32088
$688$	10.9333	0.416829
$689$	−11.8512	−0.451493
$690$	−41.1460	−1.56640
$691$	15.8712	0.603770	0.301885	−	0.953344i	$-0.402384\pi$
$691$	15.8712	0.603770	0.301885	+	0.953344i	$0.402384\pi$
$692$	−8.78966	−0.334133
$693$	28.3564	1.07717
$694$	−3.16400	−0.120104
$695$	−29.9972	−1.13786
$696$	−8.35719	−0.316778
$697$	21.9708	0.832205
$698$	33.9851	1.28636
$699$	−97.4391	−3.68548
$700$	7.04567	0.266301
$701$	−6.16455	−0.232832	−0.116416	−	0.993201i	$-0.537141\pi$
$701$	−6.16455	−0.232832	−0.116416	+	0.993201i	$0.537141\pi$
$702$	37.5853	1.41857
$703$	32.5817	1.22884
$704$	3.06369	0.115467
$705$	8.47156	0.319057
$706$	8.40690	0.316398
$707$	−3.84902	−0.144757
$708$	−14.9617	−0.562294
$709$	−6.36927	−0.239203	−0.119601	−	0.992822i	$-0.538162\pi$
$709$	−6.36927	−0.239203	−0.119601	+	0.992822i	$0.538162\pi$
$710$	35.0229	1.31439
$711$	105.672	3.96299
$712$	10.5500	0.395379
$713$	16.3859	0.613655
$714$	12.1035	0.452964
$715$	−21.4701	−0.802937
$716$	17.5751	0.656811
$717$	−4.72214	−0.176351
$718$	−9.23629	−0.344695
$719$	1.25770	0.0469041	0.0234521	−	0.999725i	$-0.492534\pi$
$719$	1.25770	0.0469041	0.0234521	+	0.999725i	$0.492534\pi$
$720$	28.1989	1.05091
$721$	12.6441	0.470891
$722$	1.84979	0.0688421
$723$	−97.0213	−3.60826
$724$	0.813712	0.0302414
$725$	15.7842	0.586211
$726$	5.44025	0.201907
$727$	−38.7312	−1.43646	−0.718231	−	0.695805i	$-0.755048\pi$
$727$	−38.7312	−1.43646	−0.718231	+	0.695805i	$0.755048\pi$
$728$	−2.30019	−0.0852507
$729$	117.096	4.33688
$730$	−45.0759	−1.66833
$731$	−35.4736	−1.31204
$732$	30.4688	1.12616
$733$	18.8258	0.695348	0.347674	−	0.937615i	$-0.386972\pi$
$733$	18.8258	0.695348	0.347674	+	0.937615i	$0.386972\pi$
$734$	6.23265	0.230051
$735$	65.6400	2.42117
$736$	3.62028	0.133445
$737$	−33.2531	−1.22489
$738$	−56.6376	−2.08486
$739$	−26.9735	−0.992238	−0.496119	−	0.868255i	$-0.665242\pi$
$739$	−26.9735	−0.992238	−0.496119	+	0.868255i	$0.665242\pi$
$740$	24.0571	0.884357
$741$	31.9952	1.17537
$742$	6.30938	0.231625
$743$	−15.0399	−0.551759	−0.275880	−	0.961192i	$-0.588969\pi$
$743$	−15.0399	−0.551759	−0.275880	+	0.961192i	$0.588969\pi$
$744$	−15.2578	−0.559378
$745$	−37.2313	−1.36405
$746$	19.0716	0.698259
$747$	−28.5808	−1.04572
$748$	−9.94026	−0.363452
$749$	−13.4706	−0.492206
$750$	−15.5353	−0.567268
$751$	29.3462	1.07086	0.535428	−	0.844581i	$-0.320150\pi$
$751$	29.3462	1.07086	0.535428	+	0.844581i	$0.320150\pi$
$752$	−0.745380	−0.0271812
$753$	−26.4647	−0.964427
$754$	−5.15305	−0.187663
$755$	20.3082	0.739091
$756$	−20.0098	−0.727751
$757$	−18.5016	−0.672453	−0.336226	−	0.941781i	$-0.609151\pi$
$757$	−18.5016	−0.672453	−0.336226	+	0.941781i	$0.609151\pi$
$758$	5.76840	0.209518
$759$	−37.3896	−1.35716
$760$	15.3947	0.558425
$761$	−2.56436	−0.0929580	−0.0464790	−	0.998919i	$-0.514800\pi$
$761$	−2.56436	−0.0929580	−0.0464790	+	0.998919i	$0.514800\pi$
$762$	27.2035	0.985479
$763$	−5.14473	−0.186252
$764$	−9.30218	−0.336541
$765$	−91.4925	−3.30792
$766$	−16.8383	−0.608392
$767$	−9.22538	−0.333109
$768$	−3.37105	−0.121642
$769$	−17.3721	−0.626455	−0.313228	−	0.949678i	$-0.601410\pi$
$769$	−17.3721	−0.626455	−0.313228	+	0.949678i	$0.601410\pi$
$770$	11.4304	0.411922
$771$	100.437	3.61717
$772$	13.0787	0.470712
$773$	39.4380	1.41849	0.709243	−	0.704964i	$-0.249037\pi$
$773$	39.4380	1.41849	0.709243	+	0.704964i	$0.249037\pi$
$774$	91.4457	3.28695
$775$	28.8174	1.03515
$776$	0.221101	0.00793708
$777$	−26.6184	−0.954930
$778$	−10.2267	−0.366644
$779$	−30.9203	−1.10784
$780$	23.6241	0.845877
$781$	31.8256	1.13881
$782$	−11.7461	−0.420041
$783$	−44.8275	−1.60200
$784$	−5.77541	−0.206265
$785$	−13.7741	−0.491617
$786$	28.7483	1.02542
$787$	−0.103359	−0.00368436	−0.00184218	−	0.999998i	$-0.500586\pi$
$787$	−0.103359	−0.00368436	−0.00184218	+	0.999998i	$0.500586\pi$
$788$	−12.2399	−0.436027
$789$	−35.3617	−1.25891
$790$	42.5959	1.51549
$791$	−4.01823	−0.142872
$792$	25.6245	0.910528
$793$	18.7871	0.667149
$794$	−6.55362	−0.232579
$795$	−64.8003	−2.29823
$796$	−4.72811	−0.167583
$797$	−11.2030	−0.396829	−0.198415	−	0.980118i	$-0.563579\pi$
$797$	−11.2030	−0.396829	−0.198415	+	0.980118i	$0.563579\pi$
$798$	−17.0337	−0.602988
$799$	2.41842	0.0855574
$800$	6.36689	0.225104
$801$	88.2399	3.11780
$802$	−5.36241	−0.189353
$803$	−40.9608	−1.44547
$804$	36.5892	1.29040
$805$	13.5070	0.476058
$806$	−9.40798	−0.331382
$807$	−18.3785	−0.646955
$808$	−3.47821	−0.122363
$809$	4.03596	0.141897	0.0709484	−	0.997480i	$-0.477397\pi$
$809$	4.03596	0.141897	0.0709484	+	0.997480i	$0.477397\pi$
$810$	120.914	4.24849
$811$	39.7196	1.39474	0.697372	−	0.716709i	$-0.254352\pi$
$811$	39.7196	1.39474	0.697372	+	0.716709i	$0.254352\pi$
$812$	2.74341	0.0962747
$813$	−27.1027	−0.950533
$814$	21.8609	0.766223
$815$	−57.5678	−2.01651
$816$	10.9375	0.382889
$817$	49.9233	1.74659
$818$	33.8435	1.18331
$819$	−19.2387	−0.672253
$820$	−22.8304	−0.797274
$821$	−16.4620	−0.574528	−0.287264	−	0.957852i	$-0.592746\pi$
$821$	−16.4620	−0.574528	−0.287264	+	0.957852i	$0.592746\pi$
$822$	−5.27837	−0.184104
$823$	−48.5356	−1.69184	−0.845922	−	0.533307i	$-0.820949\pi$
$823$	−48.5356	−1.69184	−0.845922	+	0.533307i	$0.820949\pi$
$824$	11.4260	0.398043
$825$	−65.7562	−2.28934
$826$	4.91146	0.170891
$827$	21.5667	0.749948	0.374974	−	0.927035i	$-0.377652\pi$
$827$	21.5667	0.749948	0.374974	+	0.927035i	$0.377652\pi$
$828$	30.2798	1.05230
$829$	15.5298	0.539372	0.269686	−	0.962948i	$-0.413080\pi$
$829$	15.5298	0.539372	0.269686	+	0.962948i	$0.413080\pi$
$830$	−11.5208	−0.399894
$831$	−92.8231	−3.22000
$832$	−2.07859	−0.0720622
$833$	18.7386	0.649253
$834$	29.9933	1.03858
$835$	48.3934	1.67472
$836$	13.9893	0.483829
$837$	−81.8420	−2.82888
$838$	−15.6672	−0.541214
$839$	10.6536	0.367804	0.183902	−	0.982945i	$-0.441127\pi$
$839$	10.6536	0.367804	0.183902	+	0.982945i	$0.441127\pi$
$840$	−12.5771	−0.433951
$841$	−22.8540	−0.788070
$842$	20.5496	0.708185
$843$	3.57990	0.123298
$844$	10.5313	0.362503
$845$	−29.2626	−1.00667
$846$	−6.23432	−0.214340
$847$	−1.78587	−0.0613631
$848$	5.70153	0.195792
$849$	−15.1070	−0.518471
$850$	−20.6576	−0.708551
$851$	25.8324	0.885523
$852$	−35.0184	−1.19971
$853$	−29.2910	−1.00290	−0.501452	−	0.865186i	$-0.667201\pi$
$853$	−29.2910	−1.00290	−0.501452	+	0.865186i	$0.667201\pi$
$854$	−10.0020	−0.342260
$855$	128.761	4.40352
$856$	−12.1729	−0.416060
$857$	23.4278	0.800279	0.400140	−	0.916454i	$-0.368962\pi$
$857$	23.4278	0.800279	0.400140	+	0.916454i	$0.368962\pi$
$858$	21.4673	0.732883
$859$	43.4381	1.48209	0.741044	−	0.671457i	$-0.234331\pi$
$859$	43.4381	1.48209	0.741044	+	0.671457i	$0.234331\pi$
$860$	36.8615	1.25697
$861$	25.2611	0.860897
$862$	−29.9478	−1.02003
$863$	−49.4540	−1.68343	−0.841716	−	0.539920i	$-0.818454\pi$
$863$	−49.4540	−1.68343	−0.841716	+	0.539920i	$0.818454\pi$
$864$	−18.0821	−0.615166
$865$	−29.6342	−1.00759
$866$	−2.50126	−0.0849963
$867$	21.8206	0.741068
$868$	5.00867	0.170005
$869$	38.7071	1.31305
$870$	−28.1761	−0.955259
$871$	22.5609	0.764448
$872$	−4.64909	−0.157438
$873$	1.84928	0.0625887
$874$	16.5308	0.559161
$875$	5.09975	0.172403
$876$	45.0701	1.52278
$877$	4.15161	0.140190	0.0700950	−	0.997540i	$-0.477670\pi$
$877$	4.15161	0.140190	0.0700950	+	0.997540i	$0.477670\pi$
$878$	−2.13092	−0.0719151
$879$	76.2369	2.57141
$880$	10.3292	0.348196
$881$	48.6911	1.64044	0.820222	−	0.572045i	$-0.193850\pi$
$881$	48.6911	1.64044	0.820222	+	0.572045i	$0.193850\pi$
$882$	−48.3053	−1.62652
$883$	7.21676	0.242863	0.121432	−	0.992600i	$-0.461251\pi$
$883$	7.21676	0.242863	0.121432	+	0.992600i	$0.461251\pi$
$884$	6.74407	0.226828
$885$	−50.4430	−1.69562
$886$	−25.4502	−0.855016
$887$	59.0790	1.98368	0.991839	−	0.127499i	$-0.0406949\pi$
$887$	59.0790	1.98368	0.991839	+	0.127499i	$0.0406949\pi$
$888$	−24.0540	−0.807199
$889$	−8.93007	−0.299505
$890$	35.5692	1.19228
$891$	109.875	3.68096
$892$	−24.3681	−0.815904
$893$	−3.40352	−0.113895
$894$	37.2265	1.24504
$895$	59.2540	1.98064
$896$	1.10661	0.0369693
$897$	25.3674	0.846992
$898$	21.9124	0.731226
$899$	11.2208	0.374234
$900$	53.2523	1.77508
$901$	−18.4989	−0.616286
$902$	−20.7462	−0.690772
$903$	−40.7860	−1.35727
$904$	−3.63112	−0.120769
$905$	2.74341	0.0911942
$906$	−20.3056	−0.674608
$907$	34.0548	1.13077	0.565385	−	0.824827i	$-0.308728\pi$
$907$	34.0548	1.13077	0.565385	+	0.824827i	$0.308728\pi$
$908$	26.2016	0.869532
$909$	−29.0916	−0.964906
$910$	−7.75505	−0.257077
$911$	14.1254	0.467994	0.233997	−	0.972237i	$-0.424819\pi$
$911$	14.1254	0.467994	0.233997	+	0.972237i	$0.424819\pi$
$912$	−15.3927	−0.509704
$913$	−10.4690	−0.346475
$914$	−30.3537	−1.00401
$915$	102.725	3.39598
$916$	−10.2702	−0.339336
$917$	−9.43719	−0.311643
$918$	58.6681	1.93634
$919$	−7.55790	−0.249312	−0.124656	−	0.992200i	$-0.539783\pi$
$919$	−7.55790	−0.249312	−0.124656	+	0.992200i	$0.539783\pi$
$920$	12.2057	0.402410
$921$	−54.7189	−1.80305
$922$	36.0865	1.18845
$923$	−21.5924	−0.710722
$924$	−11.4289	−0.375983
$925$	45.4308	1.49375
$926$	6.08864	0.200085
$927$	95.5663	3.13881
$928$	2.47911	0.0813807
$929$	5.81448	0.190767	0.0953834	−	0.995441i	$-0.469592\pi$
$929$	5.81448	0.190767	0.0953834	+	0.995441i	$0.469592\pi$
$930$	−51.4414	−1.68683
$931$	−26.3714	−0.864289
$932$	28.9047	0.946805
$933$	−44.1808	−1.44642
$934$	−21.1367	−0.691614
$935$	−33.5134	−1.09601
$936$	−17.3852	−0.568254
$937$	−25.4053	−0.829953	−0.414977	−	0.909832i	$-0.636210\pi$
$937$	−25.4053	−0.829953	−0.414977	+	0.909832i	$0.636210\pi$
$938$	−12.0111	−0.392176
$939$	−75.3028	−2.45742
$940$	−2.51304	−0.0819662
$941$	−10.3249	−0.336581	−0.168291	−	0.985737i	$-0.553825\pi$
$941$	−10.3249	−0.336581	−0.168291	+	0.985737i	$0.553825\pi$
$942$	13.7723	0.448725
$943$	−24.5152	−0.798325
$944$	4.43829	0.144454
$945$	−67.4628	−2.19457
$946$	33.4963	1.08906
$947$	−3.47287	−0.112853	−0.0564265	−	0.998407i	$-0.517971\pi$
$947$	−3.47287	−0.112853	−0.0564265	+	0.998407i	$0.517971\pi$
$948$	−42.5903	−1.38327
$949$	27.7903	0.902110
$950$	29.0722	0.943227
$951$	−40.1561	−1.30215
$952$	−3.59044	−0.116367
$953$	28.4527	0.921675	0.460837	−	0.887485i	$-0.347549\pi$
$953$	28.4527	0.921675	0.460837	+	0.887485i	$0.347549\pi$
$954$	47.6873	1.54393
$955$	−31.3621	−1.01485
$956$	1.40079	0.0453049
$957$	−25.6038	−0.827654
$958$	−38.5116	−1.24425
$959$	1.73273	0.0559526
$960$	−11.3654	−0.366817
$961$	−10.5141	−0.339165
$962$	−14.8317	−0.478194
$963$	−101.813	−3.28088
$964$	28.7808	0.926967
$965$	44.0945	1.41945
$966$	−13.5052	−0.434523
$967$	19.9483	0.641496	0.320748	−	0.947165i	$-0.396066\pi$
$967$	19.9483	0.641496	0.320748	+	0.947165i	$0.396066\pi$
$968$	−1.61382	−0.0518700
$969$	49.9423	1.60438
$970$	0.745440	0.0239346
$971$	46.9515	1.50675	0.753373	−	0.657594i	$-0.228426\pi$
$971$	46.9515	1.50675	0.753373	+	0.657594i	$0.228426\pi$
$972$	−66.6521	−2.13787
$973$	−9.84587	−0.315644
$974$	−26.7519	−0.857186
$975$	44.6130	1.42876
$976$	−9.03837	−0.289311
$977$	9.48098	0.303323	0.151662	−	0.988432i	$-0.451538\pi$
$977$	9.48098	0.303323	0.151662	+	0.988432i	$0.451538\pi$
$978$	57.5604	1.84058
$979$	32.3220	1.03302
$980$	−19.4717	−0.622001
$981$	−38.8848	−1.24149
$982$	10.3011	0.328721
$983$	−22.7990	−0.727174	−0.363587	−	0.931560i	$-0.618448\pi$
$983$	−22.7990	−0.727174	−0.363587	+	0.931560i	$0.618448\pi$
$984$	22.8275	0.727714
$985$	−41.2665	−1.31486
$986$	−8.04357	−0.256159
$987$	2.78059	0.0885072
$988$	−9.49117	−0.301954
$989$	39.5817	1.25862
$990$	86.3926	2.74574
$991$	−51.4971	−1.63586	−0.817929	−	0.575319i	$-0.804878\pi$
$991$	−51.4971	−1.63586	−0.817929	+	0.575319i	$0.804878\pi$
$992$	4.52613	0.143705
$993$	−103.059	−3.27047
$994$	11.4955	0.364614
$995$	−15.9407	−0.505355
$996$	11.5193	0.365004
$997$	−8.53730	−0.270379	−0.135189	−	0.990820i	$-0.543164\pi$
$997$	−8.53730	−0.270379	−0.135189	+	0.990820i	$0.543164\pi$
$998$	31.5724	0.999406
$999$	−129.024	−4.08215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.f.1.1	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.f.1.1	✓	67	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:	\( N \)	\(=\)	\( 6046 = 2 \cdot 3023 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.37105 q^{3} +1.00000 q^{4} +3.37148 q^{5} -3.37105 q^{6} +1.10661 q^{7} +1.00000 q^{8} +8.36395 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.37105 q^{3} +1.00000 q^{4} +3.37148 q^{5} -3.37105 q^{6} +1.10661 q^{7} +1.00000 q^{8} +8.36395 q^{9} +3.37148 q^{10} +3.06369 q^{11} -3.37105 q^{12} -2.07859 q^{13} +1.10661 q^{14} -11.3654 q^{15} +1.00000 q^{16} -3.24454 q^{17} +8.36395 q^{18} +4.56616 q^{19} +3.37148 q^{20} -3.73043 q^{21} +3.06369 q^{22} +3.62028 q^{23} -3.37105 q^{24} +6.36689 q^{25} -2.07859 q^{26} -18.0821 q^{27} +1.10661 q^{28} +2.47911 q^{29} -11.3654 q^{30} +4.52613 q^{31} +1.00000 q^{32} -10.3278 q^{33} -3.24454 q^{34} +3.73092 q^{35} +8.36395 q^{36} +7.13547 q^{37} +4.56616 q^{38} +7.00702 q^{39} +3.37148 q^{40} -6.77163 q^{41} -3.73043 q^{42} +10.9333 q^{43} +3.06369 q^{44} +28.1989 q^{45} +3.62028 q^{46} -0.745380 q^{47} -3.37105 q^{48} -5.77541 q^{49} +6.36689 q^{50} +10.9375 q^{51} -2.07859 q^{52} +5.70153 q^{53} -18.0821 q^{54} +10.3292 q^{55} +1.10661 q^{56} -15.3927 q^{57} +2.47911 q^{58} +4.43829 q^{59} -11.3654 q^{60} -9.03837 q^{61} +4.52613 q^{62} +9.25563 q^{63} +1.00000 q^{64} -7.00793 q^{65} -10.3278 q^{66} -10.8540 q^{67} -3.24454 q^{68} -12.2041 q^{69} +3.73092 q^{70} +10.3880 q^{71} +8.36395 q^{72} -13.3698 q^{73} +7.13547 q^{74} -21.4631 q^{75} +4.56616 q^{76} +3.39031 q^{77} +7.00702 q^{78} +12.6342 q^{79} +3.37148 q^{80} +35.8638 q^{81} -6.77163 q^{82} -3.41714 q^{83} -3.73043 q^{84} -10.9389 q^{85} +10.9333 q^{86} -8.35719 q^{87} +3.06369 q^{88} +10.5500 q^{89} +28.1989 q^{90} -2.30019 q^{91} +3.62028 q^{92} -15.2578 q^{93} -0.745380 q^{94} +15.3947 q^{95} -3.37105 q^{96} +0.221101 q^{97} -5.77541 q^{98} +25.6245 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9 + 21 * q^10 + 56 * q^11 + 21 * q^12 + 33 * q^13 + 38 * q^14 + 25 * q^15 + 67 * q^16 + 30 * q^17 + 90 * q^18 + 36 * q^19 + 21 * q^20 + 20 * q^21 + 56 * q^22 + 65 * q^23 + 21 * q^24 + 72 * q^25 + 33 * q^26 + 57 * q^27 + 38 * q^28 + 84 * q^29 + 25 * q^30 + 52 * q^31 + 67 * q^32 - 9 * q^33 + 30 * q^34 + 30 * q^35 + 90 * q^36 + 52 * q^37 + 36 * q^38 + 41 * q^39 + 21 * q^40 + 46 * q^41 + 20 * q^42 + 61 * q^43 + 56 * q^44 + 23 * q^45 + 65 * q^46 + 51 * q^47 + 21 * q^48 + 81 * q^49 + 72 * q^50 + 33 * q^51 + 33 * q^52 + 72 * q^53 + 57 * q^54 + 14 * q^55 + 38 * q^56 - 26 * q^57 + 84 * q^58 + 71 * q^59 + 25 * q^60 + 42 * q^61 + 52 * q^62 + 63 * q^63 + 67 * q^64 - 2 * q^65 - 9 * q^66 + 70 * q^67 + 30 * q^68 + 21 * q^69 + 30 * q^70 + 104 * q^71 + 90 * q^72 - 31 * q^73 + 52 * q^74 + 69 * q^75 + 36 * q^76 + 48 * q^77 + 41 * q^78 + 79 * q^79 + 21 * q^80 + 123 * q^81 + 46 * q^82 + 41 * q^83 + 20 * q^84 + 6 * q^85 + 61 * q^86 + 19 * q^87 + 56 * q^88 + 58 * q^89 + 23 * q^90 + 31 * q^91 + 65 * q^92 + 13 * q^93 + 51 * q^94 + 77 * q^95 + 21 * q^96 - 8 * q^97 + 81 * q^98 + 129 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more