# Properties

 Label 7488.2.a.g Level $7488$ Weight $2$ Character orbit 7488.a Self dual yes Analytic conductor $59.792$ Analytic rank $0$ 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] = [7488,2,Mod(1,7488)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7488, 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("7488.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7488 = 2^{6} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7488.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: $$59.7919810335$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{5} - q^{7}+O(q^{10})$$ q - 3 * q^5 - q^7 $$q - 3 q^{5} - q^{7} + 6 q^{11} - q^{13} + 3 q^{17} - 2 q^{19} + 4 q^{25} + 6 q^{29} - 4 q^{31} + 3 q^{35} + 7 q^{37} + q^{43} - 3 q^{47} - 6 q^{49} - 18 q^{55} - 6 q^{59} - 8 q^{61} + 3 q^{65} - 14 q^{67} + 3 q^{71} + 2 q^{73} - 6 q^{77} + 8 q^{79} + 12 q^{83} - 9 q^{85} + 6 q^{89} + q^{91} + 6 q^{95} - 10 q^{97}+O(q^{100})$$ q - 3 * q^5 - q^7 + 6 * q^11 - q^13 + 3 * q^17 - 2 * q^19 + 4 * q^25 + 6 * q^29 - 4 * q^31 + 3 * q^35 + 7 * q^37 + q^43 - 3 * q^47 - 6 * q^49 - 18 * q^55 - 6 * q^59 - 8 * q^61 + 3 * q^65 - 14 * q^67 + 3 * q^71 + 2 * q^73 - 6 * q^77 + 8 * q^79 + 12 * q^83 - 9 * q^85 + 6 * q^89 + q^91 + 6 * q^95 - 10 * 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 −3.00000 0 −1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$13$$ $$1$$

## 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 7488.2.a.g 1
3.b odd 2 1 832.2.a.d 1
4.b odd 2 1 7488.2.a.h 1
8.b even 2 1 234.2.a.e 1
8.d odd 2 1 1872.2.a.q 1
12.b even 2 1 832.2.a.i 1
24.f even 2 1 208.2.a.a 1
24.h odd 2 1 26.2.a.a 1
40.f even 2 1 5850.2.a.p 1
40.i odd 4 2 5850.2.e.a 2
48.i odd 4 2 3328.2.b.m 2
48.k even 4 2 3328.2.b.j 2
72.j odd 6 2 2106.2.e.ba 2
72.n even 6 2 2106.2.e.b 2
104.e even 2 1 3042.2.a.a 1
104.j odd 4 2 3042.2.b.a 2
120.i odd 2 1 650.2.a.j 1
120.m even 2 1 5200.2.a.x 1
120.w even 4 2 650.2.b.d 2
168.i even 2 1 1274.2.a.d 1
168.s odd 6 2 1274.2.f.p 2
168.ba even 6 2 1274.2.f.r 2
264.m even 2 1 3146.2.a.n 1
312.b odd 2 1 338.2.a.f 1
312.h even 2 1 2704.2.a.f 1
312.w odd 4 2 2704.2.f.d 2
312.y even 4 2 338.2.b.c 2
312.bg odd 6 2 338.2.c.a 2
312.bh odd 6 2 338.2.c.d 2
312.bo even 12 4 338.2.e.a 4
408.b odd 2 1 7514.2.a.c 1
456.p even 2 1 9386.2.a.j 1
1560.y odd 2 1 8450.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 24.h odd 2 1
208.2.a.a 1 24.f even 2 1
234.2.a.e 1 8.b even 2 1
338.2.a.f 1 312.b odd 2 1
338.2.b.c 2 312.y even 4 2
338.2.c.a 2 312.bg odd 6 2
338.2.c.d 2 312.bh odd 6 2
338.2.e.a 4 312.bo even 12 4
650.2.a.j 1 120.i odd 2 1
650.2.b.d 2 120.w even 4 2
832.2.a.d 1 3.b odd 2 1
832.2.a.i 1 12.b even 2 1
1274.2.a.d 1 168.i even 2 1
1274.2.f.p 2 168.s odd 6 2
1274.2.f.r 2 168.ba even 6 2
1872.2.a.q 1 8.d odd 2 1
2106.2.e.b 2 72.n even 6 2
2106.2.e.ba 2 72.j odd 6 2
2704.2.a.f 1 312.h even 2 1
2704.2.f.d 2 312.w odd 4 2
3042.2.a.a 1 104.e even 2 1
3042.2.b.a 2 104.j odd 4 2
3146.2.a.n 1 264.m even 2 1
3328.2.b.j 2 48.k even 4 2
3328.2.b.m 2 48.i odd 4 2
5200.2.a.x 1 120.m even 2 1
5850.2.a.p 1 40.f even 2 1
5850.2.e.a 2 40.i odd 4 2
7488.2.a.g 1 1.a even 1 1 trivial
7488.2.a.h 1 4.b odd 2 1
7514.2.a.c 1 408.b odd 2 1
8450.2.a.c 1 1560.y odd 2 1
9386.2.a.j 1 456.p even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7488))$$:

 $$T_{5} + 3$$ T5 + 3 $$T_{7} + 1$$ T7 + 1 $$T_{11} - 6$$ T11 - 6 $$T_{17} - 3$$ T17 - 3 $$T_{19} + 2$$ T19 + 2 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 3$$
$7$ $$T + 1$$
$11$ $$T - 6$$
$13$ $$T + 1$$
$17$ $$T - 3$$
$19$ $$T + 2$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T + 4$$
$37$ $$T - 7$$
$41$ $$T$$
$43$ $$T - 1$$
$47$ $$T + 3$$
$53$ $$T$$
$59$ $$T + 6$$
$61$ $$T + 8$$
$67$ $$T + 14$$
$71$ $$T - 3$$
$73$ $$T - 2$$
$79$ $$T - 8$$
$83$ $$T - 12$$
$89$ $$T - 6$$
$97$ $$T + 10$$