# Properties

 Label 7200.2.a.z Level $7200$ Weight $2$ Character orbit 7200.a Self dual yes Analytic conductor $57.492$ Analytic rank $0$ 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q - 2 q^{13} + 6 q^{17} - 4 q^{19} + 8 q^{23} + 2 q^{29} + 4 q^{31} - 10 q^{37} - 2 q^{41} + 4 q^{43} + 8 q^{47} - 7 q^{49} - 2 q^{53} - 8 q^{59} - 2 q^{61} + 12 q^{67} - 8 q^{71} + 14 q^{73} - 12 q^{79} - 4 q^{83} + 14 q^{89} - 2 q^{97}+O(q^{100})$$ q - 2 * q^13 + 6 * q^17 - 4 * q^19 + 8 * q^23 + 2 * q^29 + 4 * q^31 - 10 * q^37 - 2 * q^41 + 4 * q^43 + 8 * q^47 - 7 * q^49 - 2 * q^53 - 8 * q^59 - 2 * q^61 + 12 * q^67 - 8 * q^71 + 14 * q^73 - 12 * q^79 - 4 * q^83 + 14 * q^89 - 2 * 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 0 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.z 1
3.b odd 2 1 2400.2.a.z 1
4.b odd 2 1 7200.2.a.ba 1
5.b even 2 1 1440.2.a.c 1
5.c odd 4 2 7200.2.f.s 2
12.b even 2 1 2400.2.a.i 1
15.d odd 2 1 480.2.a.d 1
15.e even 4 2 2400.2.f.l 2
20.d odd 2 1 1440.2.a.d 1
20.e even 4 2 7200.2.f.k 2
24.f even 2 1 4800.2.a.cb 1
24.h odd 2 1 4800.2.a.s 1
40.e odd 2 1 2880.2.a.z 1
40.f even 2 1 2880.2.a.ba 1
60.h even 2 1 480.2.a.g yes 1
60.l odd 4 2 2400.2.f.g 2
120.i odd 2 1 960.2.a.k 1
120.m even 2 1 960.2.a.b 1
120.q odd 4 2 4800.2.f.v 2
120.w even 4 2 4800.2.f.o 2
240.t even 4 2 3840.2.k.s 2
240.bm odd 4 2 3840.2.k.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.d 1 15.d odd 2 1
480.2.a.g yes 1 60.h even 2 1
960.2.a.b 1 120.m even 2 1
960.2.a.k 1 120.i odd 2 1
1440.2.a.c 1 5.b even 2 1
1440.2.a.d 1 20.d odd 2 1
2400.2.a.i 1 12.b even 2 1
2400.2.a.z 1 3.b odd 2 1
2400.2.f.g 2 60.l odd 4 2
2400.2.f.l 2 15.e even 4 2
2880.2.a.z 1 40.e odd 2 1
2880.2.a.ba 1 40.f even 2 1
3840.2.k.n 2 240.bm odd 4 2
3840.2.k.s 2 240.t even 4 2
4800.2.a.s 1 24.h odd 2 1
4800.2.a.cb 1 24.f even 2 1
4800.2.f.o 2 120.w even 4 2
4800.2.f.v 2 120.q odd 4 2
7200.2.a.z 1 1.a even 1 1 trivial
7200.2.a.ba 1 4.b odd 2 1
7200.2.f.k 2 20.e even 4 2
7200.2.f.s 2 5.c odd 4 2

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{13} + 2$$ T13 + 2 $$T_{17} - 6$$ T17 - 6 $$T_{19} + 4$$ T19 + 4 $$T_{23} - 8$$ T23 - 8

## Hecke characteristic polynomials

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