Newform orbit 7200.2.f.t

• Label: 7200.2.f.t
• Level: $7200$
• Weight: $2$
• Character orbit: 7200.f
• Analytic conductor: $57.492$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

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:	$1$
Twist minimal:	no (minimal twist has level 2400)
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 + 3*i * q^7 + 5*i * q^13 + 5 * q^19 - 4*i * q^23 + 4 * q^29 + 5 * q^31 - 10*i * q^37 + 10 * q^41 + i * q^43 - 2*i * q^47 - 2 * q^49 - 10*i * q^53 + 10 * q^59 - 5 * q^61 + 3*i * q^67 - 10 * q^71 + 10*i * q^73 + 14*i * q^83 + 16 * q^89 - 15 * q^91 - 5*i * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 10 q^{19} + 8 q^{29} + 10 q^{31} + 20 q^{41} - 4 q^{49} + 20 q^{59} - 10 q^{61} - 20 q^{71} + 32 q^{89} - 30 q^{91}+O(q^{100})$

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

−

3.00000i

0

0

0

6049.2

0

0

0

0

0

3.00000i

0

0

0

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.t		2
3.b	odd	2	1		2400.2.f.m		2
4.b	odd	2	1		7200.2.f.j		2
5.b	even	2	1	inner	7200.2.f.t		2
5.c	odd	4	1		7200.2.a.h		1
5.c	odd	4	1		7200.2.a.bt		1
12.b	even	2	1		2400.2.f.f		2
15.d	odd	2	1		2400.2.f.m		2
15.e	even	4	1		2400.2.a.l	✓	1
15.e	even	4	1		2400.2.a.w	yes	1
20.d	odd	2	1		7200.2.f.j		2
20.e	even	4	1		7200.2.a.g		1
20.e	even	4	1		7200.2.a.bu		1
24.f	even	2	1		4800.2.f.x		2
24.h	odd	2	1		4800.2.f.m		2
60.h	even	2	1		2400.2.f.f		2
60.l	odd	4	1		2400.2.a.m	yes	1
60.l	odd	4	1		2400.2.a.v	yes	1
120.i	odd	2	1		4800.2.f.m		2
120.m	even	2	1		4800.2.f.x		2
120.q	odd	4	1		4800.2.a.i		1
120.q	odd	4	1		4800.2.a.cl		1
120.w	even	4	1		4800.2.a.h		1
120.w	even	4	1		4800.2.a.cm		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2400.2.a.l	✓	1	15.e	even	4	1
2400.2.a.m	yes	1	60.l	odd	4	1
2400.2.a.v	yes	1	60.l	odd	4	1
2400.2.a.w	yes	1	15.e	even	4	1
2400.2.f.f		2	12.b	even	2	1
2400.2.f.f		2	60.h	even	2	1
2400.2.f.m		2	3.b	odd	2	1
2400.2.f.m		2	15.d	odd	2	1
4800.2.a.h		1	120.w	even	4	1
4800.2.a.i		1	120.q	odd	4	1
4800.2.a.cl		1	120.q	odd	4	1
4800.2.a.cm		1	120.w	even	4	1
4800.2.f.m		2	24.h	odd	2	1
4800.2.f.m		2	120.i	odd	2	1
4800.2.f.x		2	24.f	even	2	1
4800.2.f.x		2	120.m	even	2	1
7200.2.a.g		1	20.e	even	4	1
7200.2.a.h		1	5.c	odd	4	1
7200.2.a.bt		1	5.c	odd	4	1
7200.2.a.bu		1	20.e	even	4	1
7200.2.f.j		2	4.b	odd	2	1
7200.2.f.j		2	20.d	odd	2	1
7200.2.f.t		2	1.a	even	1	1	trivial
7200.2.f.t		2	5.b	even	2	1	inner

Hecke kernels

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

$T_{7}^{2} + 9$

$T_{11}$

$T_{13}^{2} + 25$

$T_{17}$

$T_{19} - 5$

$T_{29} - 4$

$T_{31} - 5$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2}$
$7$	$T^{2} + 9$
$11$	$T^{2}$