# Properties

 Label 7200.2.f.m Level $7200$ Weight $2$ Character orbit 7200.f Analytic conductor $57.492$ Analytic rank $1$ Dimension $2$ CM discriminant -4 Inner twists $4$

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q +O(q^{10})$$ $$q + 6 i q^{13} + 2 i q^{17} -10 q^{29} + 2 i q^{37} -10 q^{41} + 7 q^{49} -14 i q^{53} -10 q^{61} -6 i q^{73} + 10 q^{89} -18 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 20q^{29} - 20q^{41} + 14q^{49} - 20q^{61} + 20q^{89} + O(q^{100})$$

## 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 0 0 0 0
6049.2 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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.b even 2 1 inner
20.d odd 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.m 2
3.b odd 2 1 800.2.c.e 2
4.b odd 2 1 CM 7200.2.f.m 2
5.b even 2 1 inner 7200.2.f.m 2
5.c odd 4 1 288.2.a.d 1
5.c odd 4 1 7200.2.a.v 1
12.b even 2 1 800.2.c.e 2
15.d odd 2 1 800.2.c.e 2
15.e even 4 1 32.2.a.a 1
15.e even 4 1 800.2.a.d 1
20.d odd 2 1 inner 7200.2.f.m 2
20.e even 4 1 288.2.a.d 1
20.e even 4 1 7200.2.a.v 1
24.f even 2 1 1600.2.c.l 2
24.h odd 2 1 1600.2.c.l 2
40.i odd 4 1 576.2.a.c 1
40.k even 4 1 576.2.a.c 1
45.k odd 12 2 2592.2.i.e 2
45.l even 12 2 2592.2.i.t 2
60.h even 2 1 800.2.c.e 2
60.l odd 4 1 32.2.a.a 1
60.l odd 4 1 800.2.a.d 1
80.i odd 4 1 2304.2.d.j 2
80.j even 4 1 2304.2.d.j 2
80.s even 4 1 2304.2.d.j 2
80.t odd 4 1 2304.2.d.j 2
105.k odd 4 1 1568.2.a.e 1
105.w odd 12 2 1568.2.i.f 2
105.x even 12 2 1568.2.i.g 2
120.i odd 2 1 1600.2.c.l 2
120.m even 2 1 1600.2.c.l 2
120.q odd 4 1 64.2.a.a 1
120.q odd 4 1 1600.2.a.n 1
120.w even 4 1 64.2.a.a 1
120.w even 4 1 1600.2.a.n 1
165.l odd 4 1 3872.2.a.f 1
180.v odd 12 2 2592.2.i.t 2
180.x even 12 2 2592.2.i.e 2
195.s even 4 1 5408.2.a.g 1
240.z odd 4 1 256.2.b.b 2
240.bb even 4 1 256.2.b.b 2
240.bd odd 4 1 256.2.b.b 2
240.bf even 4 1 256.2.b.b 2
255.o even 4 1 9248.2.a.f 1
420.w even 4 1 1568.2.a.e 1
420.bp odd 12 2 1568.2.i.g 2
420.br even 12 2 1568.2.i.f 2
480.bq odd 8 2 1024.2.e.j 4
480.br even 8 2 1024.2.e.j 4
480.ca odd 8 2 1024.2.e.j 4
480.cb even 8 2 1024.2.e.j 4
660.q even 4 1 3872.2.a.f 1
780.w odd 4 1 5408.2.a.g 1
840.bm even 4 1 3136.2.a.m 1
840.bp odd 4 1 3136.2.a.m 1
1020.x odd 4 1 9248.2.a.f 1
1320.bn odd 4 1 7744.2.a.v 1
1320.bt even 4 1 7744.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 15.e even 4 1
32.2.a.a 1 60.l odd 4 1
64.2.a.a 1 120.q odd 4 1
64.2.a.a 1 120.w even 4 1
256.2.b.b 2 240.z odd 4 1
256.2.b.b 2 240.bb even 4 1
256.2.b.b 2 240.bd odd 4 1
256.2.b.b 2 240.bf even 4 1
288.2.a.d 1 5.c odd 4 1
288.2.a.d 1 20.e even 4 1
576.2.a.c 1 40.i odd 4 1
576.2.a.c 1 40.k even 4 1
800.2.a.d 1 15.e even 4 1
800.2.a.d 1 60.l odd 4 1
800.2.c.e 2 3.b odd 2 1
800.2.c.e 2 12.b even 2 1
800.2.c.e 2 15.d odd 2 1
800.2.c.e 2 60.h even 2 1
1024.2.e.j 4 480.bq odd 8 2
1024.2.e.j 4 480.br even 8 2
1024.2.e.j 4 480.ca odd 8 2
1024.2.e.j 4 480.cb even 8 2
1568.2.a.e 1 105.k odd 4 1
1568.2.a.e 1 420.w even 4 1
1568.2.i.f 2 105.w odd 12 2
1568.2.i.f 2 420.br even 12 2
1568.2.i.g 2 105.x even 12 2
1568.2.i.g 2 420.bp odd 12 2
1600.2.a.n 1 120.q odd 4 1
1600.2.a.n 1 120.w even 4 1
1600.2.c.l 2 24.f even 2 1
1600.2.c.l 2 24.h odd 2 1
1600.2.c.l 2 120.i odd 2 1
1600.2.c.l 2 120.m even 2 1
2304.2.d.j 2 80.i odd 4 1
2304.2.d.j 2 80.j even 4 1
2304.2.d.j 2 80.s even 4 1
2304.2.d.j 2 80.t odd 4 1
2592.2.i.e 2 45.k odd 12 2
2592.2.i.e 2 180.x even 12 2
2592.2.i.t 2 45.l even 12 2
2592.2.i.t 2 180.v odd 12 2
3136.2.a.m 1 840.bm even 4 1
3136.2.a.m 1 840.bp odd 4 1
3872.2.a.f 1 165.l odd 4 1
3872.2.a.f 1 660.q even 4 1
5408.2.a.g 1 195.s even 4 1
5408.2.a.g 1 780.w odd 4 1
7200.2.a.v 1 5.c odd 4 1
7200.2.a.v 1 20.e even 4 1
7200.2.f.m 2 1.a even 1 1 trivial
7200.2.f.m 2 4.b odd 2 1 CM
7200.2.f.m 2 5.b even 2 1 inner
7200.2.f.m 2 20.d odd 2 1 inner
7744.2.a.v 1 1320.bn odd 4 1
7744.2.a.v 1 1320.bt even 4 1
9248.2.a.f 1 255.o even 4 1
9248.2.a.f 1 1020.x odd 4 1

## Hecke kernels

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

 $$T_{7}$$ $$T_{11}$$ $$T_{13}^{2} + 36$$ $$T_{17}^{2} + 4$$ $$T_{19}$$ $$T_{29} + 10$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 10 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$196 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$324 + T^{2}$$