# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9408,2,Mod(1,9408)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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 672) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{5} + q^{9}+O(q^{10})$$ q - q^3 - 2 * q^5 + q^9 $$q - q^{3} - 2 q^{5} + q^{9} - 4 q^{11} - 6 q^{13} + 2 q^{15} + 2 q^{17} - 4 q^{19} + 4 q^{23} - q^{25} - q^{27} + 2 q^{29} + 8 q^{31} + 4 q^{33} + 10 q^{37} + 6 q^{39} + 2 q^{41} + 8 q^{43} - 2 q^{45} - 2 q^{51} + 10 q^{53} + 8 q^{55} + 4 q^{57} + 12 q^{59} + 10 q^{61} + 12 q^{65} - 8 q^{67} - 4 q^{69} - 12 q^{71} - 2 q^{73} + q^{75} + q^{81} - 12 q^{83} - 4 q^{85} - 2 q^{87} - 6 q^{89} - 8 q^{93} + 8 q^{95} - 2 q^{97} - 4 q^{99}+O(q^{100})$$ q - q^3 - 2 * q^5 + q^9 - 4 * q^11 - 6 * q^13 + 2 * q^15 + 2 * q^17 - 4 * q^19 + 4 * q^23 - q^25 - q^27 + 2 * q^29 + 8 * q^31 + 4 * q^33 + 10 * q^37 + 6 * q^39 + 2 * q^41 + 8 * q^43 - 2 * q^45 - 2 * q^51 + 10 * q^53 + 8 * q^55 + 4 * q^57 + 12 * q^59 + 10 * q^61 + 12 * q^65 - 8 * q^67 - 4 * q^69 - 12 * q^71 - 2 * q^73 + q^75 + q^81 - 12 * q^83 - 4 * q^85 - 2 * q^87 - 6 * q^89 - 8 * q^93 + 8 * q^95 - 2 * q^97 - 4 * 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
 0
0 −1.00000 0 −2.00000 0 0 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.g 1
4.b odd 2 1 9408.2.a.cb 1
7.b odd 2 1 1344.2.a.r 1
8.b even 2 1 4704.2.a.be 1
8.d odd 2 1 4704.2.a.m 1
21.c even 2 1 4032.2.a.n 1
28.d even 2 1 1344.2.a.h 1
56.e even 2 1 672.2.a.f yes 1
56.h odd 2 1 672.2.a.b 1
84.h odd 2 1 4032.2.a.f 1
112.j even 4 2 5376.2.c.s 2
112.l odd 4 2 5376.2.c.m 2
168.e odd 2 1 2016.2.a.k 1
168.i even 2 1 2016.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.b 1 56.h odd 2 1
672.2.a.f yes 1 56.e even 2 1
1344.2.a.h 1 28.d even 2 1
1344.2.a.r 1 7.b odd 2 1
2016.2.a.k 1 168.e odd 2 1
2016.2.a.l 1 168.i even 2 1
4032.2.a.f 1 84.h odd 2 1
4032.2.a.n 1 21.c even 2 1
4704.2.a.m 1 8.d odd 2 1
4704.2.a.be 1 8.b even 2 1
5376.2.c.m 2 112.l odd 4 2
5376.2.c.s 2 112.j even 4 2
9408.2.a.g 1 1.a even 1 1 trivial
9408.2.a.cb 1 4.b odd 2 1

## Hecke kernels

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

 $$T_{5} + 2$$ T5 + 2 $$T_{11} + 4$$ T11 + 4 $$T_{13} + 6$$ T13 + 6 $$T_{17} - 2$$ T17 - 2 $$T_{19} + 4$$ T19 + 4 $$T_{31} - 8$$ T31 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 2$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T + 6$$
$17$ $$T - 2$$
$19$ $$T + 4$$
$23$ $$T - 4$$
$29$ $$T - 2$$
$31$ $$T - 8$$
$37$ $$T - 10$$
$41$ $$T - 2$$
$43$ $$T - 8$$
$47$ $$T$$
$53$ $$T - 10$$
$59$ $$T - 12$$
$61$ $$T - 10$$
$67$ $$T + 8$$
$71$ $$T + 12$$
$73$ $$T + 2$$
$79$ $$T$$
$83$ $$T + 12$$
$89$ $$T + 6$$
$97$ $$T + 2$$