Properties

Label 7350.2.a.cf
Level $7350$
Weight $2$
Character orbit 7350.a
Self dual yes
Analytic conductor $58.690$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.6900454856\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1470)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

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 7350.2.a.cf 1
5.b even 2 1 1470.2.a.i yes 1
7.b odd 2 1 7350.2.a.da 1
15.d odd 2 1 4410.2.a.v 1
35.c odd 2 1 1470.2.a.c 1
35.i odd 6 2 1470.2.i.r 2
35.j even 6 2 1470.2.i.k 2
105.g even 2 1 4410.2.a.be 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.c 1 35.c odd 2 1
1470.2.a.i yes 1 5.b even 2 1
1470.2.i.k 2 35.j even 6 2
1470.2.i.r 2 35.i odd 6 2
4410.2.a.v 1 15.d odd 2 1
4410.2.a.be 1 105.g even 2 1
7350.2.a.cf 1 1.a even 1 1 trivial
7350.2.a.da 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):

\( T_{11} - 6 \)
\( T_{13} + 6 \)
\( T_{17} \)
\( T_{19} + 4 \)
\( T_{23} \)
\( T_{31} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -6 + T \)
$13$ \( 6 + T \)
$17$ \( T \)
$19$ \( 4 + T \)
$23$ \( T \)
$29$ \( 8 + T \)
$31$ \( -2 + T \)
$37$ \( 4 + T \)
$41$ \( 10 + T \)
$43$ \( -6 + T \)
$47$ \( -2 + T \)
$53$ \( 10 + T \)
$59$ \( 4 + T \)
$61$ \( -14 + T \)
$67$ \( 14 + T \)
$71$ \( -8 + T \)
$73$ \( 6 + T \)
$79$ \( 8 + T \)
$83$ \( -8 + T \)
$89$ \( 18 + T \)
$97$ \( 2 + T \)
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