# Properties

 Label 7488.2.a.bl 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

## Newspace parameters

 Level: $$N$$ $$=$$ $$7488 = 2^{6} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7488.a (trivial)

## Newform invariants

 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 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{5} - 4 q^{7}+O(q^{10})$$ q + 2 * q^5 - 4 * q^7 $$q + 2 q^{5} - 4 q^{7} + 4 q^{11} - q^{13} - 2 q^{17} - q^{25} - 10 q^{29} + 4 q^{31} - 8 q^{35} + 2 q^{37} - 6 q^{41} + 12 q^{43} + 9 q^{49} + 6 q^{53} + 8 q^{55} + 12 q^{59} + 2 q^{61} - 2 q^{65} + 8 q^{67} + 2 q^{73} - 16 q^{77} + 8 q^{79} + 4 q^{83} - 4 q^{85} + 2 q^{89} + 4 q^{91} + 10 q^{97}+O(q^{100})$$ q + 2 * q^5 - 4 * q^7 + 4 * q^11 - q^13 - 2 * q^17 - q^25 - 10 * q^29 + 4 * q^31 - 8 * q^35 + 2 * q^37 - 6 * q^41 + 12 * q^43 + 9 * q^49 + 6 * q^53 + 8 * q^55 + 12 * q^59 + 2 * q^61 - 2 * q^65 + 8 * q^67 + 2 * q^73 - 16 * q^77 + 8 * q^79 + 4 * q^83 - 4 * q^85 + 2 * q^89 + 4 * q^91 + 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.

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 2.00000 0 −4.00000 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.bl 1
3.b odd 2 1 2496.2.a.q 1
4.b odd 2 1 7488.2.a.by 1
8.b even 2 1 117.2.a.a 1
8.d odd 2 1 1872.2.a.h 1
12.b even 2 1 2496.2.a.e 1
24.f even 2 1 624.2.a.i 1
24.h odd 2 1 39.2.a.a 1
40.f even 2 1 2925.2.a.p 1
40.i odd 4 2 2925.2.c.e 2
56.h odd 2 1 5733.2.a.e 1
72.j odd 6 2 1053.2.e.b 2
72.n even 6 2 1053.2.e.d 2
104.e even 2 1 1521.2.a.e 1
104.j odd 4 2 1521.2.b.b 2
120.i odd 2 1 975.2.a.f 1
120.w even 4 2 975.2.c.f 2
168.i even 2 1 1911.2.a.f 1
264.m even 2 1 4719.2.a.c 1
312.b odd 2 1 507.2.a.a 1
312.h even 2 1 8112.2.a.s 1
312.y even 4 2 507.2.b.a 2
312.bg odd 6 2 507.2.e.b 2
312.bh odd 6 2 507.2.e.a 2
312.bo even 12 4 507.2.j.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 24.h odd 2 1
117.2.a.a 1 8.b even 2 1
507.2.a.a 1 312.b odd 2 1
507.2.b.a 2 312.y even 4 2
507.2.e.a 2 312.bh odd 6 2
507.2.e.b 2 312.bg odd 6 2
507.2.j.e 4 312.bo even 12 4
624.2.a.i 1 24.f even 2 1
975.2.a.f 1 120.i odd 2 1
975.2.c.f 2 120.w even 4 2
1053.2.e.b 2 72.j odd 6 2
1053.2.e.d 2 72.n even 6 2
1521.2.a.e 1 104.e even 2 1
1521.2.b.b 2 104.j odd 4 2
1872.2.a.h 1 8.d odd 2 1
1911.2.a.f 1 168.i even 2 1
2496.2.a.e 1 12.b even 2 1
2496.2.a.q 1 3.b odd 2 1
2925.2.a.p 1 40.f even 2 1
2925.2.c.e 2 40.i odd 4 2
4719.2.a.c 1 264.m even 2 1
5733.2.a.e 1 56.h odd 2 1
7488.2.a.bl 1 1.a even 1 1 trivial
7488.2.a.by 1 4.b odd 2 1
8112.2.a.s 1 312.h even 2 1

## Hecke kernels

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

 $$T_{5} - 2$$ T5 - 2 $$T_{7} + 4$$ T7 + 4 $$T_{11} - 4$$ T11 - 4 $$T_{17} + 2$$ T17 + 2 $$T_{19}$$ T19 $$T_{29} + 10$$ T29 + 10

## Hecke characteristic polynomials

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