Properties

Label 9450.2.a.cf
Level 9450
Weight 2
Character orbit 9450.a
Self dual yes
Analytic conductor 75.459
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 9450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.4586299101\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1890)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} - q^{7} + q^{8} - 3q^{11} + q^{13} - q^{14} + q^{16} - q^{19} - 3q^{22} + 6q^{23} + q^{26} - q^{28} - 6q^{29} - 4q^{31} + q^{32} - 2q^{37} - q^{38} - 3q^{41} + q^{43} - 3q^{44} + 6q^{46} + 9q^{47} + q^{49} + q^{52} - 3q^{53} - q^{56} - 6q^{58} - 6q^{59} - 4q^{61} - 4q^{62} + q^{64} - 11q^{67} + q^{73} - 2q^{74} - q^{76} + 3q^{77} - 4q^{79} - 3q^{82} - 3q^{83} + q^{86} - 3q^{88} + 9q^{89} - q^{91} + 6q^{92} + 9q^{94} - 2q^{97} + q^{98} + O(q^{100}) \)

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

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 9450.2.a.cf 1
3.b odd 2 1 9450.2.a.v 1
5.b even 2 1 1890.2.a.j 1
15.d odd 2 1 1890.2.a.s yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.a.j 1 5.b even 2 1
1890.2.a.s yes 1 15.d odd 2 1
9450.2.a.v 1 3.b odd 2 1
9450.2.a.cf 1 1.a even 1 1 trivial

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9450))\):

\( T_{11} + 3 \)
\( T_{13} - 1 \)
\( T_{17} \)
\( T_{19} + 1 \)