Properties

Label 2450.2.a.j
Level $2450$
Weight $2$
Character orbit 2450.a
Self dual yes
Analytic conductor $19.563$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(19.5633484952\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{8} - 3q^{9} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{8} - 3q^{9} + 3q^{11} - 5q^{13} + q^{16} - 2q^{17} + 3q^{18} + 5q^{19} - 3q^{22} + 7q^{23} + 5q^{26} - 4q^{29} + 2q^{31} - q^{32} + 2q^{34} - 3q^{36} - q^{37} - 5q^{38} - 3q^{41} - 2q^{43} + 3q^{44} - 7q^{46} - 7q^{47} - 5q^{52} - 9q^{53} + 4q^{58} + 4q^{59} - 6q^{61} - 2q^{62} + q^{64} - 2q^{67} - 2q^{68} - 6q^{71} + 3q^{72} - 16q^{73} + q^{74} + 5q^{76} + 14q^{79} + 9q^{81} + 3q^{82} - 6q^{83} + 2q^{86} - 3q^{88} - 2q^{89} + 7q^{92} + 7q^{94} - 12q^{97} - 9q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 0 1.00000 0 0 0 −1.00000 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 2450.2.a.j 1
5.b even 2 1 2450.2.a.bb 1
5.c odd 4 2 490.2.c.d 2
7.b odd 2 1 2450.2.a.k 1
7.d odd 6 2 350.2.e.j 2
35.c odd 2 1 2450.2.a.ba 1
35.f even 4 2 490.2.c.a 2
35.i odd 6 2 350.2.e.c 2
35.k even 12 4 70.2.i.b 4
35.l odd 12 4 490.2.i.a 4
105.w odd 12 4 630.2.u.a 4
140.x odd 12 4 560.2.bw.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.b 4 35.k even 12 4
350.2.e.c 2 35.i odd 6 2
350.2.e.j 2 7.d odd 6 2
490.2.c.a 2 35.f even 4 2
490.2.c.d 2 5.c odd 4 2
490.2.i.a 4 35.l odd 12 4
560.2.bw.d 4 140.x odd 12 4
630.2.u.a 4 105.w odd 12 4
2450.2.a.j 1 1.a even 1 1 trivial
2450.2.a.k 1 7.b odd 2 1
2450.2.a.ba 1 35.c odd 2 1
2450.2.a.bb 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3} \)
\( T_{11} - 3 \)
\( T_{13} + 5 \)
\( T_{17} + 2 \)
\( T_{19} - 5 \)
\( T_{23} - 7 \)
\( T_{37} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -3 + T \)
$13$ \( 5 + T \)
$17$ \( 2 + T \)
$19$ \( -5 + T \)
$23$ \( -7 + T \)
$29$ \( 4 + T \)
$31$ \( -2 + T \)
$37$ \( 1 + T \)
$41$ \( 3 + T \)
$43$ \( 2 + T \)
$47$ \( 7 + T \)
$53$ \( 9 + T \)
$59$ \( -4 + T \)
$61$ \( 6 + T \)
$67$ \( 2 + T \)
$71$ \( 6 + T \)
$73$ \( 16 + T \)
$79$ \( -14 + T \)
$83$ \( 6 + T \)
$89$ \( 2 + T \)
$97$ \( 12 + T \)
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