# Properties

 Label 9360.2.a.d Level $9360$ Weight $2$ Character orbit 9360.a Self dual yes Analytic conductor $74.740$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9360.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: $$74.7399762919$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) 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)

 $$f(q)$$ $$=$$ $$q - q^{5} - 3 q^{7}+O(q^{10})$$ q - q^5 - 3 * q^7 $$q - q^{5} - 3 q^{7} - q^{11} - q^{13} + q^{17} + 2 q^{19} - 3 q^{23} + q^{25} + 2 q^{29} + 6 q^{31} + 3 q^{35} + 11 q^{37} + 5 q^{41} - 4 q^{43} - 10 q^{47} + 2 q^{49} - 11 q^{53} + q^{55} + 8 q^{59} + 13 q^{61} + q^{65} - 12 q^{67} - 5 q^{71} + 10 q^{73} + 3 q^{77} + 3 q^{79} - 12 q^{83} - q^{85} + 15 q^{89} + 3 q^{91} - 2 q^{95} + 17 q^{97}+O(q^{100})$$ q - q^5 - 3 * q^7 - q^11 - q^13 + q^17 + 2 * q^19 - 3 * q^23 + q^25 + 2 * q^29 + 6 * q^31 + 3 * q^35 + 11 * q^37 + 5 * q^41 - 4 * q^43 - 10 * q^47 + 2 * q^49 - 11 * q^53 + q^55 + 8 * q^59 + 13 * q^61 + q^65 - 12 * q^67 - 5 * q^71 + 10 * q^73 + 3 * q^77 + 3 * q^79 - 12 * q^83 - q^85 + 15 * q^89 + 3 * q^91 - 2 * q^95 + 17 * q^97

## 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 0 0 −1.00000 0 −3.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 9360.2.a.d 1
3.b odd 2 1 3120.2.a.u 1
4.b odd 2 1 585.2.a.b 1
12.b even 2 1 195.2.a.b 1
20.d odd 2 1 2925.2.a.q 1
20.e even 4 2 2925.2.c.c 2
52.b odd 2 1 7605.2.a.u 1
60.h even 2 1 975.2.a.c 1
60.l odd 4 2 975.2.c.a 2
84.h odd 2 1 9555.2.a.v 1
156.h even 2 1 2535.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.b 1 12.b even 2 1
585.2.a.b 1 4.b odd 2 1
975.2.a.c 1 60.h even 2 1
975.2.c.a 2 60.l odd 4 2
2535.2.a.a 1 156.h even 2 1
2925.2.a.q 1 20.d odd 2 1
2925.2.c.c 2 20.e even 4 2
3120.2.a.u 1 3.b odd 2 1
7605.2.a.u 1 52.b odd 2 1
9360.2.a.d 1 1.a even 1 1 trivial
9555.2.a.v 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9360))$$:

 $$T_{7} + 3$$ T7 + 3 $$T_{11} + 1$$ T11 + 1 $$T_{17} - 1$$ T17 - 1 $$T_{19} - 2$$ T19 - 2 $$T_{31} - 6$$ T31 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 3$$
$11$ $$T + 1$$
$13$ $$T + 1$$
$17$ $$T - 1$$
$19$ $$T - 2$$
$23$ $$T + 3$$
$29$ $$T - 2$$
$31$ $$T - 6$$
$37$ $$T - 11$$
$41$ $$T - 5$$
$43$ $$T + 4$$
$47$ $$T + 10$$
$53$ $$T + 11$$
$59$ $$T - 8$$
$61$ $$T - 13$$
$67$ $$T + 12$$
$71$ $$T + 5$$
$73$ $$T - 10$$
$79$ $$T - 3$$
$83$ $$T + 12$$
$89$ $$T - 15$$
$97$ $$T - 17$$
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