Properties

Label 2450.2.a.x
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 350)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} - 2q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} - 2q^{9} + 3q^{11} - q^{12} - 2q^{13} + q^{16} - 3q^{17} - 2q^{18} + 7q^{19} + 3q^{22} - q^{24} - 2q^{26} + 5q^{27} - 6q^{29} + 4q^{31} + q^{32} - 3q^{33} - 3q^{34} - 2q^{36} + 8q^{37} + 7q^{38} + 2q^{39} + 9q^{41} + 8q^{43} + 3q^{44} + 6q^{47} - q^{48} + 3q^{51} - 2q^{52} - 12q^{53} + 5q^{54} - 7q^{57} - 6q^{58} - 12q^{59} + 10q^{61} + 4q^{62} + q^{64} - 3q^{66} - 7q^{67} - 3q^{68} + 6q^{71} - 2q^{72} - 5q^{73} + 8q^{74} + 7q^{76} + 2q^{78} + 14q^{79} + q^{81} + 9q^{82} + 9q^{83} + 8q^{86} + 6q^{87} + 3q^{88} + 15q^{89} - 4q^{93} + 6q^{94} - q^{96} + 10q^{97} - 6q^{99} + O(q^{100}) \)

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 −1.00000 1.00000 0 −1.00000 0 1.00000 −2.00000 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.x 1
5.b even 2 1 2450.2.a.m 1
5.c odd 4 2 2450.2.c.h 2
7.b odd 2 1 350.2.a.e yes 1
21.c even 2 1 3150.2.a.m 1
28.d even 2 1 2800.2.a.h 1
35.c odd 2 1 350.2.a.a 1
35.f even 4 2 350.2.c.c 2
105.g even 2 1 3150.2.a.x 1
105.k odd 4 2 3150.2.g.f 2
140.c even 2 1 2800.2.a.x 1
140.j odd 4 2 2800.2.g.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.a 1 35.c odd 2 1
350.2.a.e yes 1 7.b odd 2 1
350.2.c.c 2 35.f even 4 2
2450.2.a.m 1 5.b even 2 1
2450.2.a.x 1 1.a even 1 1 trivial
2450.2.c.h 2 5.c odd 4 2
2800.2.a.h 1 28.d even 2 1
2800.2.a.x 1 140.c even 2 1
2800.2.g.i 2 140.j odd 4 2
3150.2.a.m 1 21.c even 2 1
3150.2.a.x 1 105.g even 2 1
3150.2.g.f 2 105.k odd 4 2

Hecke kernels

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

\( T_{3} + 1 \)
\( T_{11} - 3 \)
\( T_{13} + 2 \)
\( T_{17} + 3 \)
\( T_{19} - 7 \)
\( T_{23} \)
\( T_{37} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -3 + T \)
$13$ \( 2 + T \)
$17$ \( 3 + T \)
$19$ \( -7 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( -4 + T \)
$37$ \( -8 + T \)
$41$ \( -9 + T \)
$43$ \( -8 + T \)
$47$ \( -6 + T \)
$53$ \( 12 + T \)
$59$ \( 12 + T \)
$61$ \( -10 + T \)
$67$ \( 7 + T \)
$71$ \( -6 + T \)
$73$ \( 5 + T \)
$79$ \( -14 + T \)
$83$ \( -9 + T \)
$89$ \( -15 + T \)
$97$ \( -10 + T \)
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