Newform orbit 7225.2.a.by

• Label: 7225.2.a.by
• Level: $7225$
• Weight: $2$
• Character orbit: 7225.a
• Self dual: yes
• Analytic conductor: $57.692$
• Analytic rank: $1$
• Dimension: $24$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$7225 = 5^{2} \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$57.6919154604$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) =$ $24 q + 8 q^{4} - 8 q^{9} - 8 q^{16} - 16 q^{19} - 32 q^{21} - 48 q^{26} + 8 q^{36} - 56 q^{49} - 64 q^{59} - 104 q^{64} - 16 q^{66} - 32 q^{69} - 48 q^{76} - 88 q^{81} - 96 q^{84} - 112 q^{89} - 128 q^{94}+O(q^{100})$

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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
1.1	−2.30578			−1.87767			3.31660				0		4.32949			1.54574			−3.03579			0.525655				0
1.2	−2.30578			1.87767			3.31660				0		−4.32949			−1.54574			−3.03579			0.525655				0
1.3	−2.10495			−1.80557			2.43082				0		3.80063			2.68338			−0.906851			0.260076				0
1.4	−2.10495			1.80557			2.43082				0		−3.80063			−2.68338			−0.906851			0.260076				0
1.5	−1.74231			−0.598698			1.03565				0		1.04312			1.03798			1.68020			−2.64156				0
1.6	−1.74231			0.598698			1.03565				0		−1.04312			−1.03798			1.68020			−2.64156				0
1.7	−0.951739			−2.47483			−1.09419				0		2.35540			0.533377			2.94486			3.12480				0
1.8	−0.951739			2.47483			−1.09419				0		−2.35540			−0.533377			2.94486			3.12480				0
1.9	−0.499161			−0.171281			−1.75084				0		0.0854968			3.87301			1.87227			−2.97066				0
1.10	−0.499161			0.171281			−1.75084				0		−0.0854968			−3.87301			1.87227			−2.97066				0
1.11	−0.248918			−1.64368			−1.93804				0		0.409143			−1.43109			0.980251			−0.298307				0
1.12	−0.248918			1.64368			−1.93804				0		−0.409143			1.43109			0.980251			−0.298307				0
1.13	0.248918			−1.64368			−1.93804				0		−0.409143			−1.43109			−0.980251			−0.298307				0
1.14	0.248918			1.64368			−1.93804				0		0.409143			1.43109			−0.980251			−0.298307				0
1.15	0.499161			−0.171281			−1.75084				0		−0.0854968			3.87301			−1.87227			−2.97066				0
1.16	0.499161			0.171281			−1.75084				0		0.0854968			−3.87301			−1.87227			−2.97066				0
1.17	0.951739			−2.47483			−1.09419				0		−2.35540			0.533377			−2.94486			3.12480				0
1.18	0.951739			2.47483			−1.09419				0		2.35540			−0.533377			−2.94486			3.12480				0
1.19	1.74231			−0.598698			1.03565				0		−1.04312			1.03798			−1.68020			−2.64156				0
1.20	1.74231			0.598698			1.03565				0		1.04312			−1.03798			−1.68020			−2.64156				0

Atkin-Lehner signs

$p$	Sign
$5$	$-1$
$17$	$-1$

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7225.2.a.by		24
5.b	even	2	1	inner	7225.2.a.by		24
5.c	odd	4	2		1445.2.b.i		24
17.b	even	2	1	inner	7225.2.a.by		24
17.e	odd	16	2		425.2.m.e		24
85.c	even	2	1	inner	7225.2.a.by		24
85.g	odd	4	2		1445.2.b.i		24
85.o	even	16	2		85.2.m.a	✓	24
85.p	odd	16	2		425.2.m.e		24
85.r	even	16	2		85.2.m.a	✓	24
255.bc	odd	16	2		765.2.bh.b		24
255.bj	odd	16	2		765.2.bh.b		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
85.2.m.a	✓	24	85.o	even	16	2
85.2.m.a	✓	24	85.r	even	16	2
425.2.m.e		24	17.e	odd	16	2
425.2.m.e		24	85.p	odd	16	2
765.2.bh.b		24	255.bc	odd	16	2
765.2.bh.b		24	255.bj	odd	16	2
1445.2.b.i		24	5.c	odd	4	2
1445.2.b.i		24	85.g	odd	4	2
7225.2.a.by		24	1.a	even	1	1	trivial
7225.2.a.by		24	5.b	even	2	1	inner
7225.2.a.by		24	17.b	even	2	1	inner
7225.2.a.by		24	85.c	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 $S_{2}^{\mathrm{new}}(\Gamma_0(7225))$:

$T_{2}^{12} - 14T_{2}^{10} + 69T_{2}^{8} - 140T_{2}^{6} + 103T_{2}^{4} - 22T_{2}^{2} + 1$

$T_{3}^{12} - 16T_{3}^{10} + 94T_{3}^{8} - 248T_{3}^{6} + 274T_{3}^{4} - 76T_{3}^{2} + 2$

$T_{7}^{12} - 28T_{7}^{10} + 248T_{7}^{8} - 884T_{7}^{6} + 1394T_{7}^{4} - 900T_{7}^{2} + 162$

$T_{11}^{12} - 60T_{11}^{10} + 1270T_{11}^{8} - 11252T_{11}^{6} + 38776T_{11}^{4} - 27936T_{11}^{2} + 162$