Properties

Label 9450.2.a.l
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 378)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{7} - q^{8} - 5q^{13} + q^{14} + q^{16} - 3q^{17} + 2q^{19} + 9q^{23} + 5q^{26} - q^{28} - 3q^{29} + 5q^{31} - q^{32} + 3q^{34} - 2q^{37} - 2q^{38} - 6q^{41} + q^{43} - 9q^{46} + 6q^{47} + q^{49} - 5q^{52} - 3q^{53} + q^{56} + 3q^{58} - 3q^{59} - 10q^{61} - 5q^{62} + q^{64} + 13q^{67} - 3q^{68} + 9q^{71} - 2q^{73} + 2q^{74} + 2q^{76} - 10q^{79} + 6q^{82} + 12q^{83} - q^{86} + 15q^{89} + 5q^{91} + 9q^{92} - 6q^{94} - 8q^{97} - q^{98} + O(q^{100}) \)

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.l 1
3.b odd 2 1 9450.2.a.cl 1
5.b even 2 1 378.2.a.e yes 1
15.d odd 2 1 378.2.a.d 1
20.d odd 2 1 3024.2.a.p 1
35.c odd 2 1 2646.2.a.y 1
45.h odd 6 2 1134.2.f.k 2
45.j even 6 2 1134.2.f.e 2
60.h even 2 1 3024.2.a.o 1
105.g even 2 1 2646.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.d 1 15.d odd 2 1
378.2.a.e yes 1 5.b even 2 1
1134.2.f.e 2 45.j even 6 2
1134.2.f.k 2 45.h odd 6 2
2646.2.a.f 1 105.g even 2 1
2646.2.a.y 1 35.c odd 2 1
3024.2.a.o 1 60.h even 2 1
3024.2.a.p 1 20.d odd 2 1
9450.2.a.l 1 1.a even 1 1 trivial
9450.2.a.cl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9450))\):

\( T_{11} \)
\( T_{13} + 5 \)
\( T_{17} + 3 \)
\( T_{19} - 2 \)

Hecke Characteristic Polynomials

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