Properties

Label 9450.2.a.bc
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

Related objects

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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 378)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} + q^{7} - q^{8} - 4q^{11} - 3q^{13} - q^{14} + q^{16} + 7q^{17} + 2q^{19} + 4q^{22} + q^{23} + 3q^{26} + q^{28} + q^{29} - 9q^{31} - q^{32} - 7q^{34} - 2q^{37} - 2q^{38} + 6q^{41} - 11q^{43} - 4q^{44} - q^{46} + 6q^{47} + q^{49} - 3q^{52} + 9q^{53} - q^{56} - q^{58} - 5q^{59} - 6q^{61} + 9q^{62} + q^{64} - 7q^{67} + 7q^{68} - 7q^{71} + 14q^{73} + 2q^{74} + 2q^{76} - 4q^{77} - 6q^{79} - 6q^{82} + 4q^{83} + 11q^{86} + 4q^{88} - 3q^{89} - 3q^{91} + q^{92} - 6q^{94} + 8q^{97} - q^{98} + O(q^{100}) \)

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.bc 1
3.b odd 2 1 9450.2.a.dv 1
5.b even 2 1 378.2.a.h yes 1
15.d odd 2 1 378.2.a.a 1
20.d odd 2 1 3024.2.a.bd 1
35.c odd 2 1 2646.2.a.p 1
45.h odd 6 2 1134.2.f.p 2
45.j even 6 2 1134.2.f.a 2
60.h even 2 1 3024.2.a.a 1
105.g even 2 1 2646.2.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.a 1 15.d odd 2 1
378.2.a.h yes 1 5.b even 2 1
1134.2.f.a 2 45.j even 6 2
1134.2.f.p 2 45.h odd 6 2
2646.2.a.o 1 105.g even 2 1
2646.2.a.p 1 35.c odd 2 1
3024.2.a.a 1 60.h even 2 1
3024.2.a.bd 1 20.d odd 2 1
9450.2.a.bc 1 1.a even 1 1 trivial
9450.2.a.dv 1 3.b odd 2 1

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9450))\):

\( T_{11} + 4 \)
\( T_{13} + 3 \)
\( T_{17} - 7 \)
\( T_{19} - 2 \)