LMFDB - Embedded newform 7168.2.a.bh.1.9

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 = [12,0,-8,0,0,0,12,0,12,0,-16,0,0,0,0,0,0] 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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 14 x^{10} + 60 x^{9} + 71 x^{8} - 312 x^{7} - 164 x^{6} + 648 x^{5} + 167 x^{4} + \cdots + 1 $ x^12 - 4x^11 - 14x^10 + 60x^9 + 71x^8 - 312x^7 - 164x^6 + 648x^5 + 167x^4 - 484x^3 - 30x^2 + 92*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 3584)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.68305$ 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)

$f(q)$	$=$	$q+0.683046 q^{3} -3.22341 q^{5} +1.00000 q^{7} -2.53345 q^{9} -2.93389 q^{11} +2.02031 q^{13} -2.20174 q^{15} +2.77550 q^{17} +3.99867 q^{19} +0.683046 q^{21} +5.18972 q^{23} +5.39039 q^{25} -3.77960 q^{27} +0.990901 q^{29} +0.128216 q^{31} -2.00398 q^{33} -3.22341 q^{35} -2.57962 q^{37} +1.37997 q^{39} -11.7075 q^{41} -1.84101 q^{43} +8.16635 q^{45} +3.30830 q^{47} +1.00000 q^{49} +1.89579 q^{51} -2.87777 q^{53} +9.45713 q^{55} +2.73127 q^{57} -7.17589 q^{59} -9.33838 q^{61} -2.53345 q^{63} -6.51231 q^{65} +6.75189 q^{67} +3.54482 q^{69} +15.3517 q^{71} +2.03560 q^{73} +3.68189 q^{75} -2.93389 q^{77} +5.37833 q^{79} +5.01870 q^{81} +8.04251 q^{83} -8.94658 q^{85} +0.676831 q^{87} +10.9096 q^{89} +2.02031 q^{91} +0.0875774 q^{93} -12.8894 q^{95} +2.78580 q^{97} +7.43285 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{3} + 12 q^{7} + 12 q^{9} - 16 q^{11} - 8 q^{19} - 8 q^{21} + 12 q^{25} - 32 q^{27} - 24 q^{29} + 16 q^{33} - 8 q^{37} - 32 q^{39} + 16 q^{41} - 16 q^{43} + 12 q^{49} - 16 q^{51} - 24 q^{53}+ \cdots - 48 q^{99}+O(q^{100}) $ 12 * q - 8 * q^3 + 12 * q^7 + 12 * q^9 - 16 * q^11 - 8 * q^19 - 8 * q^21 + 12 * q^25 - 32 * q^27 - 24 * q^29 + 16 * q^33 - 8 * q^37 - 32 * q^39 + 16 * q^41 - 16 * q^43 + 12 * q^49 - 16 * q^51 - 24 * q^53 + 16 * q^57 - 40 * q^59 + 12 * q^63 - 8 * q^65 - 32 * q^67 - 8 * q^73 - 40 * q^75 - 16 * q^77 + 28 * q^81 - 56 * q^83 - 8 * q^89 + 32 * q^93 - 32 * q^95 - 48 * 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
$3$	0.683046	0.394357	0.197178	−	0.980368i	$-0.436822\pi$
$3$	0.683046	0.394357	0.197178	+	0.980368i	$0.436822\pi$
$4$	0	0
$5$	−3.22341	−1.44155	−0.720777	−	0.693167i	$-0.756215\pi$
$5$	−3.22341	−1.44155	−0.720777	+	0.693167i	$0.756215\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.53345	−0.844483
$10$	0	0
$11$	−2.93389	−0.884601	−0.442300	−	0.896867i	$-0.645837\pi$
$11$	−2.93389	−0.884601	−0.442300	+	0.896867i	$0.645837\pi$
$12$	0	0
$13$	2.02031	0.560334	0.280167	−	0.959951i	$-0.409610\pi$
$13$	2.02031	0.560334	0.280167	+	0.959951i	$0.409610\pi$
$14$	0	0
$15$	−2.20174	−0.568487
$16$	0	0
$17$	2.77550	0.673157	0.336579	−	0.941655i	$-0.390730\pi$
$17$	2.77550	0.673157	0.336579	+	0.941655i	$0.390730\pi$
$18$	0	0
$19$	3.99867	0.917357	0.458678	−	0.888602i	$-0.348323\pi$
$19$	3.99867	0.917357	0.458678	+	0.888602i	$0.348323\pi$
$20$	0	0
$21$	0.683046	0.149053
$22$	0	0
$23$	5.18972	1.08213	0.541066	−	0.840980i	$-0.318021\pi$
$23$	5.18972	1.08213	0.541066	+	0.840980i	$0.318021\pi$
$24$	0	0
$25$	5.39039	1.07808
$26$	0	0
$27$	−3.77960	−0.727384
$28$	0	0
$29$	0.990901	0.184006	0.0920029	−	0.995759i	$-0.470673\pi$
$29$	0.990901	0.184006	0.0920029	+	0.995759i	$0.470673\pi$
$30$	0	0
$31$	0.128216	0.0230283	0.0115141	−	0.999934i	$-0.496335\pi$
$31$	0.128216	0.0230283	0.0115141	+	0.999934i	$0.496335\pi$
$32$	0	0
$33$	−2.00398	−0.348848
$34$	0	0
$35$	−3.22341	−0.544856
$36$	0	0
$37$	−2.57962	−0.424087	−0.212043	−	0.977260i	$-0.568012\pi$
$37$	−2.57962	−0.424087	−0.212043	+	0.977260i	$0.568012\pi$
$38$	0	0
$39$	1.37997	0.220972
$40$	0	0
$41$	−11.7075	−1.82840	−0.914199	−	0.405265i	$-0.867179\pi$
$41$	−11.7075	−1.82840	−0.914199	+	0.405265i	$0.867179\pi$
$42$	0	0
$43$	−1.84101	−0.280751	−0.140375	−	0.990098i	$-0.544831\pi$
$43$	−1.84101	−0.280751	−0.140375	+	0.990098i	$0.544831\pi$
$44$	0	0
$45$	8.16635	1.21737
$46$	0	0
$47$	3.30830	0.482565	0.241283	−	0.970455i	$-0.422432\pi$
$47$	3.30830	0.482565	0.241283	+	0.970455i	$0.422432\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.89579	0.265464
$52$	0	0
$53$	−2.87777	−0.395293	−0.197646	−	0.980273i	$-0.563330\pi$
$53$	−2.87777	−0.395293	−0.197646	+	0.980273i	$0.563330\pi$
$54$	0	0
$55$	9.45713	1.27520
$56$	0	0
$57$	2.73127	0.361766
$58$	0	0
$59$	−7.17589	−0.934222	−0.467111	−	0.884199i	$-0.654705\pi$
$59$	−7.17589	−0.934222	−0.467111	+	0.884199i	$0.654705\pi$
$60$	0	0
$61$	−9.33838	−1.19566	−0.597828	−	0.801624i	$-0.703970\pi$
$61$	−9.33838	−1.19566	−0.597828	+	0.801624i	$0.703970\pi$
$62$	0	0
$63$	−2.53345	−0.319184
$64$	0	0
$65$	−6.51231	−0.807752
$66$	0	0
$67$	6.75189	0.824875	0.412437	−	0.910986i	$-0.364678\pi$
$67$	6.75189	0.824875	0.412437	+	0.910986i	$0.364678\pi$
$68$	0	0
$69$	3.54482	0.426746
$70$	0	0
$71$	15.3517	1.82192	0.910958	−	0.412498i	$-0.135344\pi$
$71$	15.3517	1.82192	0.910958	+	0.412498i	$0.135344\pi$
$72$	0	0
$73$	2.03560	0.238249	0.119124	−	0.992879i	$-0.461991\pi$
$73$	2.03560	0.238249	0.119124	+	0.992879i	$0.461991\pi$
$74$	0	0
$75$	3.68189	0.425148
$76$	0	0
$77$	−2.93389	−0.334348
$78$	0	0
$79$	5.37833	0.605110	0.302555	−	0.953132i	$-0.402160\pi$
$79$	5.37833	0.605110	0.302555	+	0.953132i	$0.402160\pi$
$80$	0	0
$81$	5.01870	0.557634
$82$	0	0
$83$	8.04251	0.882780	0.441390	−	0.897315i	$-0.354485\pi$
$83$	8.04251	0.882780	0.441390	+	0.897315i	$0.354485\pi$
$84$	0	0
$85$	−8.94658	−0.970393
$86$	0	0
$87$	0.676831	0.0725639
$88$	0	0
$89$	10.9096	1.15642	0.578209	−	0.815889i	$-0.303752\pi$
$89$	10.9096	1.15642	0.578209	+	0.815889i	$0.303752\pi$
$90$	0	0
$91$	2.02031	0.211787
$92$	0	0
$93$	0.0875774	0.00908135
$94$	0	0
$95$	−12.8894	−1.32242
$96$	0	0
$97$	2.78580	0.282855	0.141427	−	0.989949i	$-0.454831\pi$
$97$	2.78580	0.282855	0.141427	+	0.989949i	$0.454831\pi$
$98$	0	0
$99$	7.43285	0.747030

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.bh.1.9		12
4.3	odd	2		7168.2.a.bk.1.4		12
8.3	odd	2		7168.2.a.bg.1.9		12
8.5	even	2		7168.2.a.bl.1.4		12
32.3	odd	8		3584.2.m.bn.2689.5	yes	24
32.5	even	8		3584.2.m.bl.897.5	yes	24
32.11	odd	8		3584.2.m.bn.897.5	yes	24
32.13	even	8		3584.2.m.bl.2689.5	yes	24
32.19	odd	8		3584.2.m.bk.2689.8	yes	24
32.21	even	8		3584.2.m.bm.897.8	yes	24
32.27	odd	8		3584.2.m.bk.897.8	✓	24
32.29	even	8		3584.2.m.bm.2689.8	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.m.bk.897.8	✓	24	32.27	odd	8
3584.2.m.bk.2689.8	yes	24	32.19	odd	8
3584.2.m.bl.897.5	yes	24	32.5	even	8
3584.2.m.bl.2689.5	yes	24	32.13	even	8
3584.2.m.bm.897.8	yes	24	32.21	even	8
3584.2.m.bm.2689.8	yes	24	32.29	even	8
3584.2.m.bn.897.5	yes	24	32.11	odd	8
3584.2.m.bn.2689.5	yes	24	32.3	odd	8
7168.2.a.bg.1.9		12	8.3	odd	2
7168.2.a.bh.1.9		12	1.1	even	1	trivial
7168.2.a.bk.1.4		12	4.3	odd	2
7168.2.a.bl.1.4		12	8.5	even	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)

\(f(q)\)	\(=\)	\(q+0.683046 q^{3} -3.22341 q^{5} +1.00000 q^{7} -2.53345 q^{9} -2.93389 q^{11} +2.02031 q^{13} -2.20174 q^{15} +2.77550 q^{17} +3.99867 q^{19} +0.683046 q^{21} +5.18972 q^{23} +5.39039 q^{25} -3.77960 q^{27} +0.990901 q^{29} +0.128216 q^{31} -2.00398 q^{33} -3.22341 q^{35} -2.57962 q^{37} +1.37997 q^{39} -11.7075 q^{41} -1.84101 q^{43} +8.16635 q^{45} +3.30830 q^{47} +1.00000 q^{49} +1.89579 q^{51} -2.87777 q^{53} +9.45713 q^{55} +2.73127 q^{57} -7.17589 q^{59} -9.33838 q^{61} -2.53345 q^{63} -6.51231 q^{65} +6.75189 q^{67} +3.54482 q^{69} +15.3517 q^{71} +2.03560 q^{73} +3.68189 q^{75} -2.93389 q^{77} +5.37833 q^{79} +5.01870 q^{81} +8.04251 q^{83} -8.94658 q^{85} +0.676831 q^{87} +10.9096 q^{89} +2.02031 q^{91} +0.0875774 q^{93} -12.8894 q^{95} +2.78580 q^{97} +7.43285 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{3} + 12 q^{7} + 12 q^{9} - 16 q^{11} - 8 q^{19} - 8 q^{21} + 12 q^{25} - 32 q^{27} - 24 q^{29} + 16 q^{33} - 8 q^{37} - 32 q^{39} + 16 q^{41} - 16 q^{43} + 12 q^{49} - 16 q^{51} - 24 q^{53}+ \cdots - 48 q^{99}+O(q^{100}) \) 12 * q - 8 * q^3 + 12 * q^7 + 12 * q^9 - 16 * q^11 - 8 * q^19 - 8 * q^21 + 12 * q^25 - 32 * q^27 - 24 * q^29 + 16 * q^33 - 8 * q^37 - 32 * q^39 + 16 * q^41 - 16 * q^43 + 12 * q^49 - 16 * q^51 - 24 * q^53 + 16 * q^57 - 40 * q^59 + 12 * q^63 - 8 * q^65 - 32 * q^67 - 8 * q^73 - 40 * q^75 - 16 * q^77 + 28 * q^81 - 56 * q^83 - 8 * q^89 + 32 * q^93 - 32 * q^95 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more