Properties

Label 7350.2.a.bh
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + 2q^{11} + q^{12} - 2q^{13} + q^{16} + 4q^{17} - q^{18} - 2q^{22} - 8q^{23} - q^{24} + 2q^{26} + q^{27} - 2q^{31} - q^{32} + 2q^{33} - 4q^{34} + q^{36} - 8q^{37} - 2q^{39} - 2q^{41} + 2q^{43} + 2q^{44} + 8q^{46} - 10q^{47} + q^{48} + 4q^{51} - 2q^{52} + 2q^{53} - q^{54} + 4q^{59} - 10q^{61} + 2q^{62} + q^{64} - 2q^{66} - 2q^{67} + 4q^{68} - 8q^{69} - 12q^{71} - q^{72} - 10q^{73} + 8q^{74} + 2q^{78} + 16q^{79} + q^{81} + 2q^{82} - 16q^{83} - 2q^{86} - 2q^{88} + 14q^{89} - 8q^{92} - 2q^{93} + 10q^{94} - q^{96} - 6q^{97} + 2q^{99} + O(q^{100}) \)

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.bh 1
5.b even 2 1 1470.2.a.n 1
7.b odd 2 1 7350.2.a.o 1
15.d odd 2 1 4410.2.a.e 1
35.c odd 2 1 1470.2.a.p yes 1
35.i odd 6 2 1470.2.i.c 2
35.j even 6 2 1470.2.i.g 2
105.g even 2 1 4410.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.n 1 5.b even 2 1
1470.2.a.p yes 1 35.c odd 2 1
1470.2.i.c 2 35.i odd 6 2
1470.2.i.g 2 35.j even 6 2
4410.2.a.e 1 15.d odd 2 1
4410.2.a.n 1 105.g even 2 1
7350.2.a.o 1 7.b odd 2 1
7350.2.a.bh 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):

\( T_{11} - 2 \)
\( T_{13} + 2 \)
\( T_{17} - 4 \)
\( T_{19} \)
\( T_{23} + 8 \)
\( T_{31} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -2 + T \)
$13$ \( 2 + T \)
$17$ \( -4 + T \)
$19$ \( T \)
$23$ \( 8 + T \)
$29$ \( T \)
$31$ \( 2 + T \)
$37$ \( 8 + T \)
$41$ \( 2 + T \)
$43$ \( -2 + T \)
$47$ \( 10 + T \)
$53$ \( -2 + T \)
$59$ \( -4 + T \)
$61$ \( 10 + T \)
$67$ \( 2 + T \)
$71$ \( 12 + T \)
$73$ \( 10 + T \)
$79$ \( -16 + T \)
$83$ \( 16 + T \)
$89$ \( -14 + T \)
$97$ \( 6 + T \)
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