Newform orbit 9408.2.a.db

• Label: 9408.2.a.db
• Level: $9408$
• Weight: $2$
• Character orbit: 9408.a
• Self dual: yes
• Analytic conductor: $75.123$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

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

Newform invariants

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

$f(q)$

$=$

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 - 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.

Label  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1.1

0

0

1.00000

0

3.00000

0

0

0

1.00000

0

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 $S_{2}^{\mathrm{new}}(\Gamma_0(9408))$:

$T_{5} - 3$

$T_{11} + 3$

$T_{13} + 4$

$T_{17}$

$T_{19} + 4$

$T_{31} - 1$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T - 1$
$5$	$T - 3$
$7$	$T$
$11$	$T + 3$