# Properties

 Label 7200.2.f.x Level $7200$ Weight $2$ Character orbit 7200.f Analytic conductor $57.492$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7200 = 2^{5} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7200.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$57.4922894553$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 96) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 i q^{7} +O(q^{10})$$ $$q + 4 i q^{7} + 4 q^{11} + 2 i q^{13} + 6 i q^{17} -4 q^{19} + 2 q^{29} -4 q^{31} -2 i q^{37} -2 q^{41} + 4 i q^{43} + 8 i q^{47} -9 q^{49} + 10 i q^{53} + 4 q^{59} + 6 q^{61} -4 i q^{67} -16 q^{71} + 6 i q^{73} + 16 i q^{77} + 4 q^{79} -12 i q^{83} + 10 q^{89} -8 q^{91} -14 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 8q^{11} - 8q^{19} + 4q^{29} - 8q^{31} - 4q^{41} - 18q^{49} + 8q^{59} + 12q^{61} - 32q^{71} + 8q^{79} + 20q^{89} - 16q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$6401$$ $$6751$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6049.1
 − 1.00000i 1.00000i
0 0 0 0 0 4.00000i 0 0 0
6049.2 0 0 0 0 0 4.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.f.x 2
3.b odd 2 1 2400.2.f.a 2
4.b odd 2 1 7200.2.f.f 2
5.b even 2 1 inner 7200.2.f.x 2
5.c odd 4 1 288.2.a.c 1
5.c odd 4 1 7200.2.a.e 1
12.b even 2 1 2400.2.f.r 2
15.d odd 2 1 2400.2.f.a 2
15.e even 4 1 96.2.a.a 1
15.e even 4 1 2400.2.a.r 1
20.d odd 2 1 7200.2.f.f 2
20.e even 4 1 288.2.a.b 1
20.e even 4 1 7200.2.a.bx 1
24.f even 2 1 4800.2.f.e 2
24.h odd 2 1 4800.2.f.bh 2
40.i odd 4 1 576.2.a.h 1
40.k even 4 1 576.2.a.g 1
45.k odd 12 2 2592.2.i.q 2
45.l even 12 2 2592.2.i.b 2
60.h even 2 1 2400.2.f.r 2
60.l odd 4 1 96.2.a.b yes 1
60.l odd 4 1 2400.2.a.q 1
80.i odd 4 1 2304.2.d.c 2
80.j even 4 1 2304.2.d.s 2
80.s even 4 1 2304.2.d.s 2
80.t odd 4 1 2304.2.d.c 2
105.k odd 4 1 4704.2.a.t 1
120.i odd 2 1 4800.2.f.bh 2
120.m even 2 1 4800.2.f.e 2
120.q odd 4 1 192.2.a.a 1
120.q odd 4 1 4800.2.a.co 1
120.w even 4 1 192.2.a.c 1
120.w even 4 1 4800.2.a.f 1
180.v odd 12 2 2592.2.i.h 2
180.x even 12 2 2592.2.i.w 2
240.z odd 4 1 768.2.d.h 2
240.bb even 4 1 768.2.d.a 2
240.bd odd 4 1 768.2.d.h 2
240.bf even 4 1 768.2.d.a 2
420.w even 4 1 4704.2.a.e 1
840.bm even 4 1 9408.2.a.ct 1
840.bp odd 4 1 9408.2.a.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.a.a 1 15.e even 4 1
96.2.a.b yes 1 60.l odd 4 1
192.2.a.a 1 120.q odd 4 1
192.2.a.c 1 120.w even 4 1
288.2.a.b 1 20.e even 4 1
288.2.a.c 1 5.c odd 4 1
576.2.a.g 1 40.k even 4 1
576.2.a.h 1 40.i odd 4 1
768.2.d.a 2 240.bb even 4 1
768.2.d.a 2 240.bf even 4 1
768.2.d.h 2 240.z odd 4 1
768.2.d.h 2 240.bd odd 4 1
2304.2.d.c 2 80.i odd 4 1
2304.2.d.c 2 80.t odd 4 1
2304.2.d.s 2 80.j even 4 1
2304.2.d.s 2 80.s even 4 1
2400.2.a.q 1 60.l odd 4 1
2400.2.a.r 1 15.e even 4 1
2400.2.f.a 2 3.b odd 2 1
2400.2.f.a 2 15.d odd 2 1
2400.2.f.r 2 12.b even 2 1
2400.2.f.r 2 60.h even 2 1
2592.2.i.b 2 45.l even 12 2
2592.2.i.h 2 180.v odd 12 2
2592.2.i.q 2 45.k odd 12 2
2592.2.i.w 2 180.x even 12 2
4704.2.a.e 1 420.w even 4 1
4704.2.a.t 1 105.k odd 4 1
4800.2.a.f 1 120.w even 4 1
4800.2.a.co 1 120.q odd 4 1
4800.2.f.e 2 24.f even 2 1
4800.2.f.e 2 120.m even 2 1
4800.2.f.bh 2 24.h odd 2 1
4800.2.f.bh 2 120.i odd 2 1
7200.2.a.e 1 5.c odd 4 1
7200.2.a.bx 1 20.e even 4 1
7200.2.f.f 2 4.b odd 2 1
7200.2.f.f 2 20.d odd 2 1
7200.2.f.x 2 1.a even 1 1 trivial
7200.2.f.x 2 5.b even 2 1 inner
9408.2.a.bj 1 840.bp odd 4 1
9408.2.a.ct 1 840.bm even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 16$$ $$T_{11} - 4$$ $$T_{13}^{2} + 4$$ $$T_{17}^{2} + 36$$ $$T_{19} + 4$$ $$T_{29} - 2$$ $$T_{31} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( 16 + T )^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$196 + T^{2}$$