# Properties

 Label 7200.2.a.p 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

 $$f(q)$$ $$=$$ $$q - 2 q^{7} + O(q^{10})$$ $$q - 2 q^{7} + 6 q^{11} + 2 q^{13} - 6 q^{17} + 4 q^{19} - 8 q^{23} - 8 q^{31} - 2 q^{37} + 6 q^{41} - 4 q^{43} - 4 q^{47} - 3 q^{49} + 6 q^{53} + 6 q^{59} - 6 q^{61} + 4 q^{71} - 12 q^{73} - 12 q^{77} - 8 q^{79} + 12 q^{83} - 14 q^{89} - 4 q^{91} - 8 q^{97} + O(q^{100})$$

## 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 −2.00000 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.p 1
3.b odd 2 1 2400.2.a.c 1
4.b odd 2 1 7200.2.a.bl 1
5.b even 2 1 7200.2.a.br 1
5.c odd 4 2 1440.2.f.g 2
12.b even 2 1 2400.2.a.bf 1
15.d odd 2 1 2400.2.a.be 1
15.e even 4 2 480.2.f.a 2
20.d odd 2 1 7200.2.a.j 1
20.e even 4 2 1440.2.f.e 2
24.f even 2 1 4800.2.a.z 1
24.h odd 2 1 4800.2.a.bu 1
40.i odd 4 2 2880.2.f.a 2
40.k even 4 2 2880.2.f.g 2
60.h even 2 1 2400.2.a.d 1
60.l odd 4 2 480.2.f.b yes 2
120.i odd 2 1 4800.2.a.ba 1
120.m even 2 1 4800.2.a.bt 1
120.q odd 4 2 960.2.f.g 2
120.w even 4 2 960.2.f.j 2
240.z odd 4 2 3840.2.d.e 2
240.bb even 4 2 3840.2.d.u 2
240.bd odd 4 2 3840.2.d.bc 2
240.bf even 4 2 3840.2.d.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.a 2 15.e even 4 2
480.2.f.b yes 2 60.l odd 4 2
960.2.f.g 2 120.q odd 4 2
960.2.f.j 2 120.w even 4 2
1440.2.f.e 2 20.e even 4 2
1440.2.f.g 2 5.c odd 4 2
2400.2.a.c 1 3.b odd 2 1
2400.2.a.d 1 60.h even 2 1
2400.2.a.be 1 15.d odd 2 1
2400.2.a.bf 1 12.b even 2 1
2880.2.f.a 2 40.i odd 4 2
2880.2.f.g 2 40.k even 4 2
3840.2.d.e 2 240.z odd 4 2
3840.2.d.k 2 240.bf even 4 2
3840.2.d.u 2 240.bb even 4 2
3840.2.d.bc 2 240.bd odd 4 2
4800.2.a.z 1 24.f even 2 1
4800.2.a.ba 1 120.i odd 2 1
4800.2.a.bt 1 120.m even 2 1
4800.2.a.bu 1 24.h odd 2 1
7200.2.a.j 1 20.d odd 2 1
7200.2.a.p 1 1.a even 1 1 trivial
7200.2.a.bl 1 4.b odd 2 1
7200.2.a.br 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7200))$$:

 $$T_{7} + 2$$ $$T_{11} - 6$$ $$T_{13} - 2$$ $$T_{17} + 6$$ $$T_{19} - 4$$ $$T_{23} + 8$$

## Hecke characteristic polynomials

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