LMFDB - Embedded newform 8013.2.a.c.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8013,2,Mod(1,8013)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8013.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8013, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8013 = 3 \cdot 2671 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8013.a (trivial)

Newform invariants

comment:select newform

sage:traces = [116] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$63.9841271397$
Analytic rank:	$1$
Dimension:	$116$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8013.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.57433 q^{2} -1.00000 q^{3} +4.62715 q^{4} -4.27877 q^{5} +2.57433 q^{6} -4.90523 q^{7} -6.76315 q^{8} +1.00000 q^{9} +11.0149 q^{10} +2.43417 q^{11} -4.62715 q^{12} +3.14146 q^{13} +12.6277 q^{14} +4.27877 q^{15} +8.15625 q^{16} -6.78864 q^{17} -2.57433 q^{18} -5.14142 q^{19} -19.7985 q^{20} +4.90523 q^{21} -6.26636 q^{22} -6.13758 q^{23} +6.76315 q^{24} +13.3079 q^{25} -8.08715 q^{26} -1.00000 q^{27} -22.6972 q^{28} -8.16565 q^{29} -11.0149 q^{30} +3.92961 q^{31} -7.47053 q^{32} -2.43417 q^{33} +17.4762 q^{34} +20.9883 q^{35} +4.62715 q^{36} +0.264720 q^{37} +13.2357 q^{38} -3.14146 q^{39} +28.9379 q^{40} -3.49656 q^{41} -12.6277 q^{42} -9.08989 q^{43} +11.2633 q^{44} -4.27877 q^{45} +15.8001 q^{46} +2.42798 q^{47} -8.15625 q^{48} +17.0612 q^{49} -34.2588 q^{50} +6.78864 q^{51} +14.5360 q^{52} +9.64198 q^{53} +2.57433 q^{54} -10.4153 q^{55} +33.1748 q^{56} +5.14142 q^{57} +21.0211 q^{58} +8.26294 q^{59} +19.7985 q^{60} -11.3254 q^{61} -10.1161 q^{62} -4.90523 q^{63} +2.91910 q^{64} -13.4416 q^{65} +6.26636 q^{66} -8.90650 q^{67} -31.4121 q^{68} +6.13758 q^{69} -54.0308 q^{70} -9.27952 q^{71} -6.76315 q^{72} -0.319119 q^{73} -0.681476 q^{74} -13.3079 q^{75} -23.7901 q^{76} -11.9402 q^{77} +8.08715 q^{78} +12.6079 q^{79} -34.8987 q^{80} +1.00000 q^{81} +9.00128 q^{82} -14.2645 q^{83} +22.6972 q^{84} +29.0470 q^{85} +23.4004 q^{86} +8.16565 q^{87} -16.4627 q^{88} +1.53412 q^{89} +11.0149 q^{90} -15.4096 q^{91} -28.3995 q^{92} -3.92961 q^{93} -6.25042 q^{94} +21.9989 q^{95} +7.47053 q^{96} +12.8594 q^{97} -43.9212 q^{98} +2.43417 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9} + 3 q^{10} - 57 q^{11} - 116 q^{12} + 6 q^{13} - 9 q^{14} + 20 q^{15} + 112 q^{16} - 30 q^{17} - 16 q^{18}+ \cdots - 57 q^{99}+O(q^{100}) $ 116 * q - 16 * q^2 - 116 * q^3 + 116 * q^4 - 20 * q^5 + 16 * q^6 - 33 * q^7 - 45 * q^8 + 116 * q^9 + 3 * q^10 - 57 * q^11 - 116 * q^12 + 6 * q^13 - 9 * q^14 + 20 * q^15 + 112 * q^16 - 30 * q^17 - 16 * q^18 + 3 * q^19 - 54 * q^20 + 33 * q^21 - 22 * q^22 - 58 * q^23 + 45 * q^24 + 126 * q^25 - 21 * q^26 - 116 * q^27 - 77 * q^28 - 38 * q^29 - 3 * q^30 + 17 * q^31 - 106 * q^32 + 57 * q^33 + 35 * q^34 - 72 * q^35 + 116 * q^36 - 41 * q^37 - 45 * q^38 - 6 * q^39 + 5 * q^40 - 39 * q^41 + 9 * q^42 - 118 * q^43 - 103 * q^44 - 20 * q^45 - 8 * q^46 - 65 * q^47 - 112 * q^48 + 165 * q^49 - 72 * q^50 + 30 * q^51 - 10 * q^52 - 58 * q^53 + 16 * q^54 + 14 * q^55 - 23 * q^56 - 3 * q^57 - 27 * q^58 - 75 * q^59 + 54 * q^60 + 45 * q^61 - 73 * q^62 - 33 * q^63 + 111 * q^64 - 86 * q^65 + 22 * q^66 - 127 * q^67 - 94 * q^68 + 58 * q^69 - 7 * q^70 - 61 * q^71 - 45 * q^72 + 15 * q^73 - 51 * q^74 - 126 * q^75 + 96 * q^76 - 57 * q^77 + 21 * q^78 + 7 * q^79 - 144 * q^80 + 116 * q^81 - 37 * q^82 - 194 * q^83 + 77 * q^84 + 3 * q^85 - 57 * q^86 + 38 * q^87 - 42 * q^88 - 56 * q^89 + 3 * q^90 - 39 * q^91 - 138 * q^92 - 17 * q^93 + 51 * q^94 - 127 * q^95 + 106 * q^96 + 57 * q^97 - 105 * q^98 - 57 * q^99

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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 $

$2$	−2.57433	−1.82032	−0.910162	−	0.414253i	$-0.864043\pi$
$2$	−2.57433	−1.82032	−0.910162	+	0.414253i	$0.864043\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.62715	2.31358
$5$	−4.27877	−1.91352	−0.956762	−	0.290873i	$-0.906054\pi$
$5$	−4.27877	−1.91352	−0.956762	+	0.290873i	$0.906054\pi$
$6$	2.57433	1.05096
$7$	−4.90523	−1.85400	−0.927001	−	0.375060i	$-0.877622\pi$
$7$	−4.90523	−1.85400	−0.927001	+	0.375060i	$0.877622\pi$
$8$	−6.76315	−2.39113
$9$	1.00000	0.333333
$10$	11.0149	3.48323
$11$	2.43417	0.733931	0.366965	−	0.930235i	$-0.380397\pi$
$11$	2.43417	0.733931	0.366965	+	0.930235i	$0.380397\pi$
$12$	−4.62715	−1.33574
$13$	3.14146	0.871285	0.435643	−	0.900120i	$-0.356521\pi$
$13$	3.14146	0.871285	0.435643	+	0.900120i	$0.356521\pi$
$14$	12.6277	3.37488
$15$	4.27877	1.10477
$16$	8.15625	2.03906
$17$	−6.78864	−1.64649	−0.823244	−	0.567688i	$-0.807838\pi$
$17$	−6.78864	−1.64649	−0.823244	+	0.567688i	$0.807838\pi$
$18$	−2.57433	−0.606774
$19$	−5.14142	−1.17952	−0.589761	−	0.807578i	$-0.700778\pi$
$19$	−5.14142	−1.17952	−0.589761	+	0.807578i	$0.700778\pi$
$20$	−19.7985	−4.42708
$21$	4.90523	1.07041
$22$	−6.26636	−1.33599
$23$	−6.13758	−1.27977	−0.639887	−	0.768469i	$-0.721019\pi$
$23$	−6.13758	−1.27977	−0.639887	+	0.768469i	$0.721019\pi$
$24$	6.76315	1.38052
$25$	13.3079	2.66157
$26$	−8.08715	−1.58602
$27$	−1.00000	−0.192450
$28$	−22.6972	−4.28937
$29$	−8.16565	−1.51632	−0.758162	−	0.652066i	$-0.773902\pi$
$29$	−8.16565	−1.51632	−0.758162	+	0.652066i	$0.773902\pi$
$30$	−11.0149	−2.01104
$31$	3.92961	0.705779	0.352889	−	0.935665i	$-0.385199\pi$
$31$	3.92961	0.705779	0.352889	+	0.935665i	$0.385199\pi$
$32$	−7.47053	−1.32062
$33$	−2.43417	−0.423735
$34$	17.4762	2.99714
$35$	20.9883	3.54767
$36$	4.62715	0.771192
$37$	0.264720	0.0435197	0.0217599	−	0.999763i	$-0.493073\pi$
$37$	0.264720	0.0435197	0.0217599	+	0.999763i	$0.493073\pi$
$38$	13.2357	2.14711
$39$	−3.14146	−0.503037
$40$	28.9379	4.57549
$41$	−3.49656	−0.546071	−0.273035	−	0.962004i	$-0.588028\pi$
$41$	−3.49656	−0.546071	−0.273035	+	0.962004i	$0.588028\pi$
$42$	−12.6277	−1.94849
$43$	−9.08989	−1.38620	−0.693098	−	0.720843i	$-0.743755\pi$
$43$	−9.08989	−1.38620	−0.693098	+	0.720843i	$0.743755\pi$
$44$	11.2633	1.69801
$45$	−4.27877	−0.637841
$46$	15.8001	2.32960
$47$	2.42798	0.354158	0.177079	−	0.984197i	$-0.443335\pi$
$47$	2.42798	0.354158	0.177079	+	0.984197i	$0.443335\pi$
$48$	−8.15625	−1.17725
$49$	17.0612	2.43732
$50$	−34.2588	−4.84492
$51$	6.78864	0.950600
$52$	14.5360	2.01579
$53$	9.64198	1.32443	0.662214	−	0.749315i	$-0.269617\pi$
$53$	9.64198	1.32443	0.662214	+	0.749315i	$0.269617\pi$
$54$	2.57433	0.350321
$55$	−10.4153	−1.40439
$56$	33.1748	4.43317
$57$	5.14142	0.680998
$58$	21.0211	2.76020
$59$	8.26294	1.07574	0.537872	−	0.843027i	$-0.319229\pi$
$59$	8.26294	1.07574	0.537872	+	0.843027i	$0.319229\pi$
$60$	19.7985	2.55598
$61$	−11.3254	−1.45007	−0.725035	−	0.688712i	$-0.758176\pi$
$61$	−11.3254	−1.45007	−0.725035	+	0.688712i	$0.758176\pi$
$62$	−10.1161	−1.28475
$63$	−4.90523	−0.618000
$64$	2.91910	0.364887
$65$	−13.4416	−1.66722
$66$	6.26636	0.771335
$67$	−8.90650	−1.08810	−0.544051	−	0.839052i	$-0.683110\pi$
$67$	−8.90650	−1.08810	−0.544051	+	0.839052i	$0.683110\pi$
$68$	−31.4121	−3.80928
$69$	6.13758	0.738877
$70$	−54.0308	−6.45791
$71$	−9.27952	−1.10128	−0.550638	−	0.834744i	$-0.685616\pi$
$71$	−9.27952	−1.10128	−0.550638	+	0.834744i	$0.685616\pi$
$72$	−6.76315	−0.797045
$73$	−0.319119	−0.0373501	−0.0186751	−	0.999826i	$-0.505945\pi$
$73$	−0.319119	−0.0373501	−0.0186751	+	0.999826i	$0.505945\pi$
$74$	−0.681476	−0.0792200
$75$	−13.3079	−1.53666
$76$	−23.7901	−2.72892
$77$	−11.9402	−1.36071
$78$	8.08715	0.915690
$79$	12.6079	1.41850	0.709252	−	0.704955i	$-0.249033\pi$
$79$	12.6079	1.41850	0.709252	+	0.704955i	$0.249033\pi$
$80$	−34.8987	−3.90179
$81$	1.00000	0.111111
$82$	9.00128	0.994025
$83$	−14.2645	−1.56574	−0.782868	−	0.622188i	$-0.786244\pi$
$83$	−14.2645	−1.56574	−0.782868	+	0.622188i	$0.786244\pi$
$84$	22.6972	2.47647
$85$	29.0470	3.15059
$86$	23.4004	2.52332
$87$	8.16565	0.875450
$88$	−16.4627	−1.75493
$89$	1.53412	0.162617	0.0813084	−	0.996689i	$-0.474090\pi$
$89$	1.53412	0.162617	0.0813084	+	0.996689i	$0.474090\pi$
$90$	11.0149	1.16108
$91$	−15.4096	−1.61536
$92$	−28.3995	−2.96085
$93$	−3.92961	−0.407482
$94$	−6.25042	−0.644681
$95$	21.9989	2.25704
$96$	7.47053	0.762458
$97$	12.8594	1.30568	0.652839	−	0.757496i	$-0.273578\pi$
$97$	12.8594	1.30568	0.652839	+	0.757496i	$0.273578\pi$
$98$	−43.9212	−4.43671
$99$	2.43417	0.244644

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8013.2.a.c.1.10	✓	116

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8013.2.a.c.1.10	✓	116	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8013 = 3 \cdot 2671 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.57433 q^{2} -1.00000 q^{3} +4.62715 q^{4} -4.27877 q^{5} +2.57433 q^{6} -4.90523 q^{7} -6.76315 q^{8} +1.00000 q^{9} +11.0149 q^{10} +2.43417 q^{11} -4.62715 q^{12} +3.14146 q^{13} +12.6277 q^{14} +4.27877 q^{15} +8.15625 q^{16} -6.78864 q^{17} -2.57433 q^{18} -5.14142 q^{19} -19.7985 q^{20} +4.90523 q^{21} -6.26636 q^{22} -6.13758 q^{23} +6.76315 q^{24} +13.3079 q^{25} -8.08715 q^{26} -1.00000 q^{27} -22.6972 q^{28} -8.16565 q^{29} -11.0149 q^{30} +3.92961 q^{31} -7.47053 q^{32} -2.43417 q^{33} +17.4762 q^{34} +20.9883 q^{35} +4.62715 q^{36} +0.264720 q^{37} +13.2357 q^{38} -3.14146 q^{39} +28.9379 q^{40} -3.49656 q^{41} -12.6277 q^{42} -9.08989 q^{43} +11.2633 q^{44} -4.27877 q^{45} +15.8001 q^{46} +2.42798 q^{47} -8.15625 q^{48} +17.0612 q^{49} -34.2588 q^{50} +6.78864 q^{51} +14.5360 q^{52} +9.64198 q^{53} +2.57433 q^{54} -10.4153 q^{55} +33.1748 q^{56} +5.14142 q^{57} +21.0211 q^{58} +8.26294 q^{59} +19.7985 q^{60} -11.3254 q^{61} -10.1161 q^{62} -4.90523 q^{63} +2.91910 q^{64} -13.4416 q^{65} +6.26636 q^{66} -8.90650 q^{67} -31.4121 q^{68} +6.13758 q^{69} -54.0308 q^{70} -9.27952 q^{71} -6.76315 q^{72} -0.319119 q^{73} -0.681476 q^{74} -13.3079 q^{75} -23.7901 q^{76} -11.9402 q^{77} +8.08715 q^{78} +12.6079 q^{79} -34.8987 q^{80} +1.00000 q^{81} +9.00128 q^{82} -14.2645 q^{83} +22.6972 q^{84} +29.0470 q^{85} +23.4004 q^{86} +8.16565 q^{87} -16.4627 q^{88} +1.53412 q^{89} +11.0149 q^{90} -15.4096 q^{91} -28.3995 q^{92} -3.92961 q^{93} -6.25042 q^{94} +21.9989 q^{95} +7.47053 q^{96} +12.8594 q^{97} -43.9212 q^{98} +2.43417 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9} + 3 q^{10} - 57 q^{11} - 116 q^{12} + 6 q^{13} - 9 q^{14} + 20 q^{15} + 112 q^{16} - 30 q^{17} - 16 q^{18}+ \cdots - 57 q^{99}+O(q^{100}) \) 116 * q - 16 * q^2 - 116 * q^3 + 116 * q^4 - 20 * q^5 + 16 * q^6 - 33 * q^7 - 45 * q^8 + 116 * q^9 + 3 * q^10 - 57 * q^11 - 116 * q^12 + 6 * q^13 - 9 * q^14 + 20 * q^15 + 112 * q^16 - 30 * q^17 - 16 * q^18 + 3 * q^19 - 54 * q^20 + 33 * q^21 - 22 * q^22 - 58 * q^23 + 45 * q^24 + 126 * q^25 - 21 * q^26 - 116 * q^27 - 77 * q^28 - 38 * q^29 - 3 * q^30 + 17 * q^31 - 106 * q^32 + 57 * q^33 + 35 * q^34 - 72 * q^35 + 116 * q^36 - 41 * q^37 - 45 * q^38 - 6 * q^39 + 5 * q^40 - 39 * q^41 + 9 * q^42 - 118 * q^43 - 103 * q^44 - 20 * q^45 - 8 * q^46 - 65 * q^47 - 112 * q^48 + 165 * q^49 - 72 * q^50 + 30 * q^51 - 10 * q^52 - 58 * q^53 + 16 * q^54 + 14 * q^55 - 23 * q^56 - 3 * q^57 - 27 * q^58 - 75 * q^59 + 54 * q^60 + 45 * q^61 - 73 * q^62 - 33 * q^63 + 111 * q^64 - 86 * q^65 + 22 * q^66 - 127 * q^67 - 94 * q^68 + 58 * q^69 - 7 * q^70 - 61 * q^71 - 45 * q^72 + 15 * q^73 - 51 * q^74 - 126 * q^75 + 96 * q^76 - 57 * q^77 + 21 * q^78 + 7 * q^79 - 144 * q^80 + 116 * q^81 - 37 * q^82 - 194 * q^83 + 77 * q^84 + 3 * q^85 - 57 * q^86 + 38 * q^87 - 42 * q^88 - 56 * q^89 + 3 * q^90 - 39 * q^91 - 138 * q^92 - 17 * q^93 + 51 * q^94 - 127 * q^95 + 106 * q^96 + 57 * q^97 - 105 * q^98 - 57 * q^99

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