# Properties

 Label 7488.2.a.bz 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 78) 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 $$q + 2 q^{5} + 4 q^{7} - 4 q^{11} - q^{13} - 2 q^{17} + 8 q^{19} - q^{25} + 6 q^{29} - 4 q^{31} + 8 q^{35} + 2 q^{37} + 10 q^{41} - 4 q^{43} - 8 q^{47} + 9 q^{49} - 10 q^{53} - 8 q^{55} + 4 q^{59} + 2 q^{61} - 2 q^{65} + 16 q^{67} + 8 q^{71} + 2 q^{73} - 16 q^{77} + 8 q^{79} + 12 q^{83} - 4 q^{85} - 14 q^{89} - 4 q^{91} + 16 q^{95} + 10 q^{97}+O(q^{100})$$ q + 2 * q^5 + 4 * q^7 - 4 * q^11 - q^13 - 2 * q^17 + 8 * q^19 - q^25 + 6 * q^29 - 4 * q^31 + 8 * q^35 + 2 * q^37 + 10 * q^41 - 4 * q^43 - 8 * q^47 + 9 * q^49 - 10 * q^53 - 8 * q^55 + 4 * q^59 + 2 * q^61 - 2 * q^65 + 16 * q^67 + 8 * q^71 + 2 * q^73 - 16 * q^77 + 8 * q^79 + 12 * q^83 - 4 * q^85 - 14 * q^89 - 4 * q^91 + 16 * 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.

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.bz 1
3.b odd 2 1 2496.2.a.t 1
4.b odd 2 1 7488.2.a.bk 1
8.b even 2 1 234.2.a.c 1
8.d odd 2 1 1872.2.a.c 1
12.b even 2 1 2496.2.a.b 1
24.f even 2 1 624.2.a.h 1
24.h odd 2 1 78.2.a.a 1
40.f even 2 1 5850.2.a.d 1
40.i odd 4 2 5850.2.e.bb 2
72.j odd 6 2 2106.2.e.q 2
72.n even 6 2 2106.2.e.j 2
104.e even 2 1 3042.2.a.f 1
104.j odd 4 2 3042.2.b.g 2
120.i odd 2 1 1950.2.a.w 1
120.w even 4 2 1950.2.e.i 2
168.i even 2 1 3822.2.a.j 1
264.m even 2 1 9438.2.a.t 1
312.b odd 2 1 1014.2.a.d 1
312.h even 2 1 8112.2.a.v 1
312.y even 4 2 1014.2.b.b 2
312.bg odd 6 2 1014.2.e.c 2
312.bh odd 6 2 1014.2.e.f 2
312.bo even 12 4 1014.2.i.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 24.h odd 2 1
234.2.a.c 1 8.b even 2 1
624.2.a.h 1 24.f even 2 1
1014.2.a.d 1 312.b odd 2 1
1014.2.b.b 2 312.y even 4 2
1014.2.e.c 2 312.bg odd 6 2
1014.2.e.f 2 312.bh odd 6 2
1014.2.i.d 4 312.bo even 12 4
1872.2.a.c 1 8.d odd 2 1
1950.2.a.w 1 120.i odd 2 1
1950.2.e.i 2 120.w even 4 2
2106.2.e.j 2 72.n even 6 2
2106.2.e.q 2 72.j odd 6 2
2496.2.a.b 1 12.b even 2 1
2496.2.a.t 1 3.b odd 2 1
3042.2.a.f 1 104.e even 2 1
3042.2.b.g 2 104.j odd 4 2
3822.2.a.j 1 168.i even 2 1
5850.2.a.d 1 40.f even 2 1
5850.2.e.bb 2 40.i odd 4 2
7488.2.a.bk 1 4.b odd 2 1
7488.2.a.bz 1 1.a even 1 1 trivial
8112.2.a.v 1 312.h even 2 1
9438.2.a.t 1 264.m 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 $$T_{19} - 8$$ T19 - 8 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 2$$
$7$ $$T - 4$$
$11$ $$T + 4$$
$13$ $$T + 1$$
$17$ $$T + 2$$
$19$ $$T - 8$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T + 4$$
$37$ $$T - 2$$
$41$ $$T - 10$$
$43$ $$T + 4$$
$47$ $$T + 8$$
$53$ $$T + 10$$
$59$ $$T - 4$$
$61$ $$T - 2$$
$67$ $$T - 16$$
$71$ $$T - 8$$
$73$ $$T - 2$$
$79$ $$T - 8$$
$83$ $$T - 12$$
$89$ $$T + 14$$
$97$ $$T - 10$$