Properties

Label 2450.2.a.d
Level $2450$
Weight $2$
Character orbit 2450.a
Self dual yes
Analytic conductor $19.563$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 490)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - 2q^{3} + q^{4} + 2q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - 2q^{3} + q^{4} + 2q^{6} - q^{8} + q^{9} - 4q^{11} - 2q^{12} - 2q^{13} + q^{16} - 8q^{17} - q^{18} - 6q^{19} + 4q^{22} + 4q^{23} + 2q^{24} + 2q^{26} + 4q^{27} - 6q^{29} + 4q^{31} - q^{32} + 8q^{33} + 8q^{34} + q^{36} + 10q^{37} + 6q^{38} + 4q^{39} + 4q^{41} - 4q^{43} - 4q^{44} - 4q^{46} - 4q^{47} - 2q^{48} + 16q^{51} - 2q^{52} - 10q^{53} - 4q^{54} + 12q^{57} + 6q^{58} - 14q^{59} - 10q^{61} - 4q^{62} + q^{64} - 8q^{66} + 4q^{67} - 8q^{68} - 8q^{69} + 12q^{71} - q^{72} - 4q^{73} - 10q^{74} - 6q^{76} - 4q^{78} + 4q^{79} - 11q^{81} - 4q^{82} + 2q^{83} + 4q^{86} + 12q^{87} + 4q^{88} + 8q^{89} + 4q^{92} - 8q^{93} + 4q^{94} + 2q^{96} - 4q^{99} + O(q^{100}) \)

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 −2.00000 1.00000 0 2.00000 0 −1.00000 1.00000 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.d 1
5.b even 2 1 490.2.a.i yes 1
5.c odd 4 2 2450.2.c.n 2
7.b odd 2 1 2450.2.a.n 1
15.d odd 2 1 4410.2.a.i 1
20.d odd 2 1 3920.2.a.j 1
35.c odd 2 1 490.2.a.f 1
35.f even 4 2 2450.2.c.b 2
35.i odd 6 2 490.2.e.e 2
35.j even 6 2 490.2.e.b 2
105.g even 2 1 4410.2.a.s 1
140.c even 2 1 3920.2.a.bg 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.f 1 35.c odd 2 1
490.2.a.i yes 1 5.b even 2 1
490.2.e.b 2 35.j even 6 2
490.2.e.e 2 35.i odd 6 2
2450.2.a.d 1 1.a even 1 1 trivial
2450.2.a.n 1 7.b odd 2 1
2450.2.c.b 2 35.f even 4 2
2450.2.c.n 2 5.c odd 4 2
3920.2.a.j 1 20.d odd 2 1
3920.2.a.bg 1 140.c even 2 1
4410.2.a.i 1 15.d odd 2 1
4410.2.a.s 1 105.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3} + 2 \)
\( T_{11} + 4 \)
\( T_{13} + 2 \)
\( T_{17} + 8 \)
\( T_{19} + 6 \)
\( T_{23} - 4 \)
\( T_{37} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 2 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 4 + T \)
$13$ \( 2 + T \)
$17$ \( 8 + T \)
$19$ \( 6 + T \)
$23$ \( -4 + T \)
$29$ \( 6 + T \)
$31$ \( -4 + T \)
$37$ \( -10 + T \)
$41$ \( -4 + T \)
$43$ \( 4 + T \)
$47$ \( 4 + T \)
$53$ \( 10 + T \)
$59$ \( 14 + T \)
$61$ \( 10 + T \)
$67$ \( -4 + T \)
$71$ \( -12 + T \)
$73$ \( 4 + T \)
$79$ \( -4 + T \)
$83$ \( -2 + T \)
$89$ \( -8 + T \)
$97$ \( T \)
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