# Properties

 Label 7200.2.a.bw 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

 $$f(q)$$ $$=$$ $$q + 4 q^{7} + O(q^{10})$$ $$q + 4 q^{7} - 4 q^{11} - 6 q^{13} + 2 q^{17} + 4 q^{19} - 10 q^{29} - 4 q^{31} + 10 q^{37} - 2 q^{41} + 4 q^{43} + 8 q^{47} + 9 q^{49} + 2 q^{53} - 12 q^{59} - 10 q^{61} - 12 q^{67} - 10 q^{73} - 16 q^{77} - 4 q^{79} + 4 q^{83} + 6 q^{89} - 24 q^{91} + 14 q^{97} + O(q^{100})$$

## 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 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.bw 1
3.b odd 2 1 2400.2.a.bh 1
4.b odd 2 1 7200.2.a.d 1
5.b even 2 1 1440.2.a.g 1
5.c odd 4 2 7200.2.f.c 2
12.b even 2 1 2400.2.a.a 1
15.d odd 2 1 480.2.a.a 1
15.e even 4 2 2400.2.f.o 2
20.d odd 2 1 1440.2.a.n 1
20.e even 4 2 7200.2.f.ba 2
24.f even 2 1 4800.2.a.bo 1
24.h odd 2 1 4800.2.a.bg 1
40.e odd 2 1 2880.2.a.p 1
40.f even 2 1 2880.2.a.c 1
60.h even 2 1 480.2.a.f yes 1
60.l odd 4 2 2400.2.f.d 2
120.i odd 2 1 960.2.a.m 1
120.m even 2 1 960.2.a.h 1
120.q odd 4 2 4800.2.f.bb 2
120.w even 4 2 4800.2.f.h 2
240.t even 4 2 3840.2.k.c 2
240.bm odd 4 2 3840.2.k.bb 2

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

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