# Properties

 Label 7200.2.a.a Level $7200$ Weight $2$ Character orbit 7200.a Self dual yes Analytic conductor $57.492$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7200 = 2^{5} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$57.4922894553$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 4 q^{7}+O(q^{10})$$ q - 4 * q^7 $$q - 4 q^{7} - 4 q^{13} + 8 q^{19} - 4 q^{23} + 6 q^{29} + 8 q^{31} + 4 q^{37} - 6 q^{41} + 4 q^{43} + 4 q^{47} + 9 q^{49} - 12 q^{53} - 6 q^{61} + 12 q^{67} - 16 q^{71} + 8 q^{79} + 12 q^{83} + 10 q^{89} + 16 q^{91} - 8 q^{97}+O(q^{100})$$ q - 4 * q^7 - 4 * q^13 + 8 * q^19 - 4 * q^23 + 6 * q^29 + 8 * q^31 + 4 * q^37 - 6 * q^41 + 4 * q^43 + 4 * q^47 + 9 * q^49 - 12 * q^53 - 6 * q^61 + 12 * q^67 - 16 * q^71 + 8 * q^79 + 12 * q^83 + 10 * q^89 + 16 * q^91 - 8 * 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 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## 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 7200.2.a.a 1
3.b odd 2 1 2400.2.a.s 1
4.b odd 2 1 7200.2.a.by 1
5.b even 2 1 7200.2.a.ca 1
5.c odd 4 2 1440.2.f.b 2
12.b even 2 1 2400.2.a.o 1
15.d odd 2 1 2400.2.a.p 1
15.e even 4 2 480.2.f.c 2
20.d odd 2 1 7200.2.a.c 1
20.e even 4 2 1440.2.f.d 2
24.f even 2 1 4800.2.a.cr 1
24.h odd 2 1 4800.2.a.e 1
40.i odd 4 2 2880.2.f.o 2
40.k even 4 2 2880.2.f.m 2
60.h even 2 1 2400.2.a.t 1
60.l odd 4 2 480.2.f.d yes 2
120.i odd 2 1 4800.2.a.cp 1
120.m even 2 1 4800.2.a.c 1
120.q odd 4 2 960.2.f.e 2
120.w even 4 2 960.2.f.d 2
240.z odd 4 2 3840.2.d.bf 2
240.bb even 4 2 3840.2.d.p 2
240.bd odd 4 2 3840.2.d.a 2
240.bf even 4 2 3840.2.d.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.c 2 15.e even 4 2
480.2.f.d yes 2 60.l odd 4 2
960.2.f.d 2 120.w even 4 2
960.2.f.e 2 120.q odd 4 2
1440.2.f.b 2 5.c odd 4 2
1440.2.f.d 2 20.e even 4 2
2400.2.a.o 1 12.b even 2 1
2400.2.a.p 1 15.d odd 2 1
2400.2.a.s 1 3.b odd 2 1
2400.2.a.t 1 60.h even 2 1
2880.2.f.m 2 40.k even 4 2
2880.2.f.o 2 40.i odd 4 2
3840.2.d.a 2 240.bd odd 4 2
3840.2.d.p 2 240.bb even 4 2
3840.2.d.q 2 240.bf even 4 2
3840.2.d.bf 2 240.z odd 4 2
4800.2.a.c 1 120.m even 2 1
4800.2.a.e 1 24.h odd 2 1
4800.2.a.cp 1 120.i odd 2 1
4800.2.a.cr 1 24.f even 2 1
7200.2.a.a 1 1.a even 1 1 trivial
7200.2.a.c 1 20.d odd 2 1
7200.2.a.by 1 4.b odd 2 1
7200.2.a.ca 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7200))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11}$$ T11 $$T_{13} + 4$$ T13 + 4 $$T_{17}$$ T17 $$T_{19} - 8$$ T19 - 8 $$T_{23} + 4$$ T23 + 4

## Hecke characteristic polynomials

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