LMFDB - Newform orbit 9408.2.a.db

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9408,2,Mod(1,9408)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9408.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$9408 = 2^{6} \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9408.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,1,0,3,0,0,0,1,0,-3,0,-4,0,3,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$75.1232582216$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

f(q)

=

q + q^{3} + 3 q^{5} + q^{9} - 3 q^{11} - 4 q^{13} + 3 q^{15} - 4 q^{19} + 4 q^{25} + q^{27} - 9 q^{29} + q^{31} - 3 q^{33} - 8 q^{37} - 4 q^{39} + 10 q^{43} + 3 q^{45} + 6 q^{47} + 3 q^{53} - 9 q^{55}+ \cdots - 3 q^{99}+O(q^{100})

q + q^3 + 3 * q^5 + q^9 - 3 * q^11 - 4 * q^13 + 3 * q^15 - 4 * q^19 + 4 * q^25 + q^27 - 9 * q^29 + q^31 - 3 * q^33 - 8 * q^37 - 4 * q^39 + 10 * q^43 + 3 * q^45 + 6 * q^47 + 3 * q^53 - 9 * q^55 - 4 * q^57 + 3 * q^59 - 10 * q^61 - 12 * q^65 + 10 * q^67 - 6 * q^71 - 2 * q^73 + 4 * q^75 - q^79 + q^81 - 9 * q^83 - 9 * q^87 - 6 * q^89 + q^93 - 12 * q^95 + q^97 - 3 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

1.00000

3.00000

1.00000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$
$7$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9408.2.a.db		1
4.b	odd	2	1		9408.2.a.bm		1
7.b	odd	2	1		9408.2.a.d		1
7.d	odd	6	2		1344.2.q.v		2
8.b	even	2	1		294.2.a.a		1
8.d	odd	2	1		2352.2.a.n		1
24.f	even	2	1		7056.2.a.bz		1
24.h	odd	2	1		882.2.a.k		1
28.d	even	2	1		9408.2.a.bu		1
28.f	even	6	2		1344.2.q.j		2
40.f	even	2	1		7350.2.a.cw		1
56.e	even	2	1		2352.2.a.m		1
56.h	odd	2	1		294.2.a.d		1
56.j	odd	6	2		42.2.e.b	✓	2
56.k	odd	6	2		2352.2.q.m		2
56.m	even	6	2		336.2.q.d		2
56.p	even	6	2		294.2.e.f		2
168.e	odd	2	1		7056.2.a.g		1
168.i	even	2	1		882.2.a.g		1
168.s	odd	6	2		882.2.g.b		2
168.ba	even	6	2		126.2.g.b		2
168.be	odd	6	2		1008.2.s.n		2
280.c	odd	2	1		7350.2.a.ce		1
280.bk	odd	6	2		1050.2.i.e		2
280.bv	even	12	4		1050.2.o.b		4
504.y	even	6	2		1134.2.h.a		2
504.bp	odd	6	2		1134.2.e.a		2
504.ca	even	6	2		1134.2.e.p		2
504.cw	odd	6	2		1134.2.h.p		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.2.e.b	✓	2	56.j	odd	6	2
126.2.g.b		2	168.ba	even	6	2
294.2.a.a		1	8.b	even	2	1
294.2.a.d		1	56.h	odd	2	1
294.2.e.f		2	56.p	even	6	2
336.2.q.d		2	56.m	even	6	2
882.2.a.g		1	168.i	even	2	1
882.2.a.k		1	24.h	odd	2	1
882.2.g.b		2	168.s	odd	6	2
1008.2.s.n		2	168.be	odd	6	2
1050.2.i.e		2	280.bk	odd	6	2
1050.2.o.b		4	280.bv	even	12	4
1134.2.e.a		2	504.bp	odd	6	2
1134.2.e.p		2	504.ca	even	6	2
1134.2.h.a		2	504.y	even	6	2
1134.2.h.p		2	504.cw	odd	6	2
1344.2.q.j		2	28.f	even	6	2
1344.2.q.v		2	7.d	odd	6	2
2352.2.a.m		1	56.e	even	2	1
2352.2.a.n		1	8.d	odd	2	1
2352.2.q.m		2	56.k	odd	6	2
7056.2.a.g		1	168.e	odd	2	1
7056.2.a.bz		1	24.f	even	2	1
7350.2.a.ce		1	280.c	odd	2	1
7350.2.a.cw		1	40.f	even	2	1
9408.2.a.d		1	7.b	odd	2	1
9408.2.a.bm		1	4.b	odd	2	1
9408.2.a.bu		1	28.d	even	2	1
9408.2.a.db		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9408))$ :

T_{5} - 3

T5 - 3

T_{11} + 3

T11 + 3

T_{13} + 4

T13 + 4

T_{17}

T17

T_{19} + 4

T19 + 4

T_{31} - 1

T31 - 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 1$ T - 1
$5$	$T - 3$ T - 3
$7$	$T$ T
$11$	$T + 3$ T + 3
$13$	$T + 4$ T + 4
$17$	$T$ T
$19$	$T + 4$ T + 4
$23$	$T$ T
$29$	$T + 9$ T + 9
$31$	$T - 1$ T - 1
$37$	$T + 8$ T + 8
$41$	$T$ T
$43$	$T - 10$ T - 10
$47$	$T - 6$ T - 6
$53$	$T - 3$ T - 3
$59$	$T - 3$ T - 3
$61$	$T + 10$ T + 10
$67$	$T - 10$ T - 10
$71$	$T + 6$ T + 6
$73$	$T + 2$ T + 2
$79$	$T + 1$ T + 1
$83$	$T + 9$ T + 9
$89$	$T + 6$ T + 6
$97$	$T - 1$ T - 1
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