Newform orbit 1225.4.a.bf

• Label: 1225.4.a.bf
• Level: $1225$
• Weight: $4$
• Character orbit: 1225.a
• Self dual: yes
• Analytic conductor: $72.277$
• Analytic rank: $1$
• Dimension: $5$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(72.2773397570\)
Analytic rank:	\(1\)
Dimension:	\(5\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial:	\( x^{5} - x^{4} - 37x^{3} + 21x^{2} + 288x + 64 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b1 * q^2 + (b2 - 2) * q^3 + (b3 + 7) * q^4 + (-b4 - 2*b3 - b2 + 3*b1 - 3) * q^6 + (-4*b2 - 5*b1 - 4) * q^8 + (-b4 + b3 - 3*b2 - b1 + 18) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} - 8 q^{3} + 35 q^{4} - 16 q^{6} - 33 q^{8} + 81 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 37x^{3} + 21x^{2} + 288x + 64 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} - 21\nu - 4 ) / 4 \)
\(\beta_{3}\)	\(=\)	\( \nu^{2} - 15 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{4} - \nu^{3} - 29\nu^{2} + 21\nu + 112 ) / 4 \)

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

5.20362
3.84623
−0.227497
−2.79706
−5.02529

−5.20362

4.90652

19.0777

0

−25.5317

0

−57.6442

−2.92609

0

1.2

−3.84623

−8.96795

6.79345

0

34.4927

0

4.64067

53.4241

0

1.3

0.227497

−1.80858

−7.94824

0

−0.411448

0

−3.62818

−23.7290

0

1.4

2.79706

6.21383

−0.176445

0

17.3805

0

−22.8700

11.6117

0

1.5

5.02529

−8.34382

17.2535

0

−41.9301

0

46.5017

42.6193

0

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\(1\)
\(7\)	\(-1\)

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	1225.4.a.bf		5
5.b	even	2	1		245.4.a.n		5
7.b	odd	2	1		1225.4.a.bg		5
7.d	odd	6	2		175.4.e.d		10
15.d	odd	2	1		2205.4.a.bt		5
35.c	odd	2	1		245.4.a.m		5
35.i	odd	6	2		35.4.e.c	✓	10
35.j	even	6	2		245.4.e.o		10
35.k	even	12	4		175.4.k.d		20
105.g	even	2	1		2205.4.a.bu		5
105.p	even	6	2		315.4.j.g		10
140.s	even	6	2		560.4.q.n		10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.4.e.c	✓	10	35.i	odd	6	2
175.4.e.d		10	7.d	odd	6	2
175.4.k.d		20	35.k	even	12	4
245.4.a.m		5	35.c	odd	2	1
245.4.a.n		5	5.b	even	2	1
245.4.e.o		10	35.j	even	6	2
315.4.j.g		10	105.p	even	6	2
560.4.q.n		10	140.s	even	6	2
1225.4.a.bf		5	1.a	even	1	1	trivial
1225.4.a.bg		5	7.b	odd	2	1
2205.4.a.bt		5	15.d	odd	2	1
2205.4.a.bu		5	105.g	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2}^{5} + T_{2}^{4} - 37T_{2}^{3} - 21T_{2}^{2} + 288T_{2} - 64 \)

\( T_{3}^{5} + 8T_{3}^{4} - 76T_{3}^{3} - 462T_{3}^{2} + 1731T_{3} + 4126 \)

\( T_{19}^{5} - 21T_{19}^{4} - 13760T_{19}^{3} - 109220T_{19}^{2} + 52536096T_{19} + 1626396544 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^5 + T^4 - 37*T^3 - 21*T^2 + 288*T - 64
$3$	T^5 + 8*T^4 - 76*T^3 - 462*T^2 + 1731*T + 4126
$5$	\( T^{5} \)
$7$	\( T^{5} \)
$11$	T^5 - 47*T^4 - 2156*T^3 + 109968*T^2 - 1050784*T - 1022160