LMFDB - Embedded newform 7168.2.a.ba.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7168,2,Mod(1,7168)] 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("7168.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-8,0,-8,0,12,0,12,0,-20,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$57.2367681689$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.9433055232.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 6x^{6} + 32x^{5} + 9x^{4} - 76x^{3} - 4x^{2} + 48x - 8 $ x^8 - 4x^7 - 6x^6 + 32x^5 + 9x^4 - 76x^3 - 4x^2 + 48*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1792)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.177920$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67251 q^{3} +2.65618 q^{5} -1.00000 q^{7} -0.202696 q^{9} +0.826484 q^{11} -5.57281 q^{13} -4.44250 q^{15} -1.74896 q^{17} +5.93216 q^{19} +1.67251 q^{21} -3.04150 q^{23} +2.05530 q^{25} +5.35655 q^{27} -6.26943 q^{29} +7.90794 q^{31} -1.38231 q^{33} -2.65618 q^{35} +8.30514 q^{37} +9.32061 q^{39} +1.38922 q^{41} +2.45934 q^{43} -0.538396 q^{45} +1.80017 q^{47} +1.00000 q^{49} +2.92516 q^{51} -13.7698 q^{53} +2.19529 q^{55} -9.92162 q^{57} +6.70340 q^{59} +4.38770 q^{61} +0.202696 q^{63} -14.8024 q^{65} -6.80390 q^{67} +5.08696 q^{69} -1.11625 q^{71} +11.2521 q^{73} -3.43752 q^{75} -0.826484 q^{77} +7.61158 q^{79} -8.35083 q^{81} -15.8207 q^{83} -4.64556 q^{85} +10.4857 q^{87} -0.428825 q^{89} +5.57281 q^{91} -13.2261 q^{93} +15.7569 q^{95} -19.2163 q^{97} -0.167525 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} - 8 q^{7} + 12 q^{9} + 12 q^{11} - 20 q^{13} + 4 q^{17} + 4 q^{19} - 8 q^{23} + 12 q^{25} - 12 q^{27} - 8 q^{29} + 4 q^{31} + 8 q^{33} + 8 q^{35} - 8 q^{37} + 16 q^{39} - 12 q^{41} - 4 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) $ 8 * q - 8 * q^5 - 8 * q^7 + 12 * q^9 + 12 * q^11 - 20 * q^13 + 4 * q^17 + 4 * q^19 - 8 * q^23 + 12 * q^25 - 12 * q^27 - 8 * q^29 + 4 * q^31 + 8 * q^33 + 8 * q^35 - 8 * q^37 + 16 * q^39 - 12 * q^41 - 4 * q^43 - 52 * q^45 - 20 * q^47 + 8 * q^49 + 32 * q^51 - 40 * q^53 - 24 * q^55 - 4 * q^57 + 4 * q^59 + 8 * q^61 - 12 * q^63 + 36 * q^65 + 28 * q^67 - 4 * q^69 + 16 * q^71 + 16 * q^73 - 28 * q^75 - 12 * q^77 + 20 * q^81 - 8 * q^83 - 16 * q^85 - 20 * q^87 + 16 * q^89 + 20 * q^91 - 16 * q^93 + 40 * q^95 - 36 * q^97 - 4 * 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 $

$2$	0	0
$2$	0	0
$3$	−1.67251	−0.965627	−0.482813	−	0.875723i	$-0.660385\pi$
$3$	−1.67251	−0.965627	−0.482813	+	0.875723i	$0.660385\pi$
$4$	0	0
$5$	2.65618	1.18788	0.593940	−	0.804509i	$-0.297572\pi$
$5$	2.65618	1.18788	0.593940	+	0.804509i	$0.297572\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.202696	−0.0675652
$10$	0	0
$11$	0.826484	0.249194	0.124597	−	0.992207i	$-0.460236\pi$
$11$	0.826484	0.249194	0.124597	+	0.992207i	$0.460236\pi$
$12$	0	0
$13$	−5.57281	−1.54562	−0.772810	−	0.634637i	$-0.781150\pi$
$13$	−5.57281	−1.54562	−0.772810	+	0.634637i	$0.781150\pi$
$14$	0	0
$15$	−4.44250	−1.14705
$16$	0	0
$17$	−1.74896	−0.424185	−0.212093	−	0.977250i	$-0.568028\pi$
$17$	−1.74896	−0.424185	−0.212093	+	0.977250i	$0.568028\pi$
$18$	0	0
$19$	5.93216	1.36093	0.680465	−	0.732781i	$-0.261778\pi$
$19$	5.93216	1.36093	0.680465	+	0.732781i	$0.261778\pi$
$20$	0	0
$21$	1.67251	0.364973
$22$	0	0
$23$	−3.04150	−0.634198	−0.317099	−	0.948393i	$-0.602709\pi$
$23$	−3.04150	−0.634198	−0.317099	+	0.948393i	$0.602709\pi$
$24$	0	0
$25$	2.05530	0.411060
$26$	0	0
$27$	5.35655	1.03087
$28$	0	0
$29$	−6.26943	−1.16420	−0.582102	−	0.813116i	$-0.697770\pi$
$29$	−6.26943	−1.16420	−0.582102	+	0.813116i	$0.697770\pi$
$30$	0	0
$31$	7.90794	1.42031	0.710154	−	0.704046i	$-0.248625\pi$
$31$	7.90794	1.42031	0.710154	+	0.704046i	$0.248625\pi$
$32$	0	0
$33$	−1.38231	−0.240629
$34$	0	0
$35$	−2.65618	−0.448977
$36$	0	0
$37$	8.30514	1.36536	0.682678	−	0.730719i	$-0.260815\pi$
$37$	8.30514	1.36536	0.682678	+	0.730719i	$0.260815\pi$
$38$	0	0
$39$	9.32061	1.49249
$40$	0	0
$41$	1.38922	0.216960	0.108480	−	0.994099i	$-0.465402\pi$
$41$	1.38922	0.216960	0.108480	+	0.994099i	$0.465402\pi$
$42$	0	0
$43$	2.45934	0.375046	0.187523	−	0.982260i	$-0.439954\pi$
$43$	2.45934	0.375046	0.187523	+	0.982260i	$0.439954\pi$
$44$	0	0
$45$	−0.538396	−0.0802594
$46$	0	0
$47$	1.80017	0.262582	0.131291	−	0.991344i	$-0.458088\pi$
$47$	1.80017	0.262582	0.131291	+	0.991344i	$0.458088\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.92516	0.409604
$52$	0	0
$53$	−13.7698	−1.89143	−0.945717	−	0.324992i	$-0.894639\pi$
$53$	−13.7698	−1.89143	−0.945717	+	0.324992i	$0.894639\pi$
$54$	0	0
$55$	2.19529	0.296013
$56$	0	0
$57$	−9.92162	−1.31415
$58$	0	0
$59$	6.70340	0.872708	0.436354	−	0.899775i	$-0.356269\pi$
$59$	6.70340	0.872708	0.436354	+	0.899775i	$0.356269\pi$
$60$	0	0
$61$	4.38770	0.561787	0.280893	−	0.959739i	$-0.409369\pi$
$61$	4.38770	0.561787	0.280893	+	0.959739i	$0.409369\pi$
$62$	0	0
$63$	0.202696	0.0255372
$64$	0	0
$65$	−14.8024	−1.83601
$66$	0	0
$67$	−6.80390	−0.831228	−0.415614	−	0.909541i	$-0.636433\pi$
$67$	−6.80390	−0.831228	−0.415614	+	0.909541i	$0.636433\pi$
$68$	0	0
$69$	5.08696	0.612398
$70$	0	0
$71$	−1.11625	−0.132475	−0.0662375	−	0.997804i	$-0.521100\pi$
$71$	−1.11625	−0.132475	−0.0662375	+	0.997804i	$0.521100\pi$
$72$	0	0
$73$	11.2521	1.31696	0.658478	−	0.752600i	$-0.271201\pi$
$73$	11.2521	1.31696	0.658478	+	0.752600i	$0.271201\pi$
$74$	0	0
$75$	−3.43752	−0.396931
$76$	0	0
$77$	−0.826484	−0.0941866
$78$	0	0
$79$	7.61158	0.856370	0.428185	−	0.903691i	$-0.359153\pi$
$79$	7.61158	0.856370	0.428185	+	0.903691i	$0.359153\pi$
$80$	0	0
$81$	−8.35083	−0.927870
$82$	0	0
$83$	−15.8207	−1.73655	−0.868273	−	0.496086i	$-0.834770\pi$
$83$	−15.8207	−1.73655	−0.868273	+	0.496086i	$0.834770\pi$
$84$	0	0
$85$	−4.64556	−0.503881
$86$	0	0
$87$	10.4857	1.12419
$88$	0	0
$89$	−0.428825	−0.0454554	−0.0227277	−	0.999742i	$-0.507235\pi$
$89$	−0.428825	−0.0454554	−0.0227277	+	0.999742i	$0.507235\pi$
$90$	0	0
$91$	5.57281	0.584190
$92$	0	0
$93$	−13.2261	−1.37149
$94$	0	0
$95$	15.7569	1.61662
$96$	0	0
$97$	−19.2163	−1.95111	−0.975557	−	0.219745i	$-0.929478\pi$
$97$	−19.2163	−1.95111	−0.975557	+	0.219745i	$0.929478\pi$
$98$	0	0
$99$	−0.167525	−0.0168369

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	7168.2.a.ba.1.3		8
4.3	odd	2		7168.2.a.bb.1.6		8
8.3	odd	2		7168.2.a.bf.1.3		8
8.5	even	2		7168.2.a.be.1.6		8
32.3	odd	8		1792.2.m.e.1345.6	yes	16
32.5	even	8		1792.2.m.f.449.6	yes	16
32.11	odd	8		1792.2.m.e.449.6	✓	16
32.13	even	8		1792.2.m.f.1345.6	yes	16
32.19	odd	8		1792.2.m.h.1345.3	yes	16
32.21	even	8		1792.2.m.g.449.3	yes	16
32.27	odd	8		1792.2.m.h.449.3	yes	16
32.29	even	8		1792.2.m.g.1345.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1792.2.m.e.449.6	✓	16	32.11	odd	8
1792.2.m.e.1345.6	yes	16	32.3	odd	8
1792.2.m.f.449.6	yes	16	32.5	even	8
1792.2.m.f.1345.6	yes	16	32.13	even	8
1792.2.m.g.449.3	yes	16	32.21	even	8
1792.2.m.g.1345.3	yes	16	32.29	even	8
1792.2.m.h.449.3	yes	16	32.27	odd	8
1792.2.m.h.1345.3	yes	16	32.19	odd	8
7168.2.a.ba.1.3		8	1.1	even	1	trivial
7168.2.a.bb.1.6		8	4.3	odd	2
7168.2.a.be.1.6		8	8.5	even	2
7168.2.a.bf.1.3		8	8.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67251 q^{3} +2.65618 q^{5} -1.00000 q^{7} -0.202696 q^{9} +0.826484 q^{11} -5.57281 q^{13} -4.44250 q^{15} -1.74896 q^{17} +5.93216 q^{19} +1.67251 q^{21} -3.04150 q^{23} +2.05530 q^{25} +5.35655 q^{27} -6.26943 q^{29} +7.90794 q^{31} -1.38231 q^{33} -2.65618 q^{35} +8.30514 q^{37} +9.32061 q^{39} +1.38922 q^{41} +2.45934 q^{43} -0.538396 q^{45} +1.80017 q^{47} +1.00000 q^{49} +2.92516 q^{51} -13.7698 q^{53} +2.19529 q^{55} -9.92162 q^{57} +6.70340 q^{59} +4.38770 q^{61} +0.202696 q^{63} -14.8024 q^{65} -6.80390 q^{67} +5.08696 q^{69} -1.11625 q^{71} +11.2521 q^{73} -3.43752 q^{75} -0.826484 q^{77} +7.61158 q^{79} -8.35083 q^{81} -15.8207 q^{83} -4.64556 q^{85} +10.4857 q^{87} -0.428825 q^{89} +5.57281 q^{91} -13.2261 q^{93} +15.7569 q^{95} -19.2163 q^{97} -0.167525 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} - 8 q^{7} + 12 q^{9} + 12 q^{11} - 20 q^{13} + 4 q^{17} + 4 q^{19} - 8 q^{23} + 12 q^{25} - 12 q^{27} - 8 q^{29} + 4 q^{31} + 8 q^{33} + 8 q^{35} - 8 q^{37} + 16 q^{39} - 12 q^{41} - 4 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^5 - 8 * q^7 + 12 * q^9 + 12 * q^11 - 20 * q^13 + 4 * q^17 + 4 * q^19 - 8 * q^23 + 12 * q^25 - 12 * q^27 - 8 * q^29 + 4 * q^31 + 8 * q^33 + 8 * q^35 - 8 * q^37 + 16 * q^39 - 12 * q^41 - 4 * q^43 - 52 * q^45 - 20 * q^47 + 8 * q^49 + 32 * q^51 - 40 * q^53 - 24 * q^55 - 4 * q^57 + 4 * q^59 + 8 * q^61 - 12 * q^63 + 36 * q^65 + 28 * q^67 - 4 * q^69 + 16 * q^71 + 16 * q^73 - 28 * q^75 - 12 * q^77 + 20 * q^81 - 8 * q^83 - 16 * q^85 - 20 * q^87 + 16 * q^89 + 20 * q^91 - 16 * q^93 + 40 * q^95 - 36 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more