Newform orbit 2450.4.a.bz

• Label: 2450.4.a.bz
• Level: $2450$
• Weight: $4$
• Character orbit: 2450.a
• Self dual: yes
• Analytic conductor: $144.555$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(144.554679514\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{46}) \)
Defining polynomial:	\( x^{2} - 46 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{46}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 2 q^{2} + (\beta + 1) q^{3} + 4 q^{4} + (2 \beta + 2) q^{6} + 8 q^{8} + (2 \beta + 20) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} + 4 q^{6} + 16 q^{8} + 40 q^{9}+O(q^{10}) \)

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

−6.78233
6.78233

2.00000

−5.78233

4.00000

0

−11.5647

0

8.00000

6.43534

0

1.2

2.00000

7.78233

4.00000

0

15.5647

0

8.00000

33.5647

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(5\)	\(1\)
\(7\)	\(1\)

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	2450.4.a.bz		2
5.b	even	2	1		490.4.a.r		2
7.b	odd	2	1		2450.4.a.bv		2
7.c	even	3	2		350.4.e.h		4
35.c	odd	2	1		490.4.a.t		2
35.i	odd	6	2		490.4.e.u		4
35.j	even	6	2		70.4.e.d	✓	4
35.l	odd	12	4		350.4.j.g		8
105.o	odd	6	2		630.4.k.l		4
140.p	odd	6	2		560.4.q.j		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.e.d	✓	4	35.j	even	6	2
350.4.e.h		4	7.c	even	3	2
350.4.j.g		8	35.l	odd	12	4
490.4.a.r		2	5.b	even	2	1
490.4.a.t		2	35.c	odd	2	1
490.4.e.u		4	35.i	odd	6	2
560.4.q.j		4	140.p	odd	6	2
630.4.k.l		4	105.o	odd	6	2
2450.4.a.bv		2	7.b	odd	2	1
2450.4.a.bz		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3}^{2} - 2T_{3} - 45 \)

\( T_{11}^{2} + 68T_{11} + 972 \)

\( T_{19}^{2} + 232T_{19} + 12720 \)

\( T_{23}^{2} - 198T_{23} - 549 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( T^{2} - 2T - 45 \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 68T + 972 \)