Properties

Label 9450.2.a.dc
Level $9450$
Weight $2$
Character orbit 9450.a
Self dual yes
Analytic conductor $75.459$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{7} + q^{8} - 5 q^{11} + q^{14} + q^{16} + 2 q^{17} - q^{19} - 5 q^{22} - q^{23} + q^{28} - 4 q^{29} - 9 q^{31} + q^{32} + 2 q^{34} - 5 q^{37} - q^{38} + 9 q^{41} + 10 q^{43} - 5 q^{44} - q^{46} + 6 q^{47} + q^{49} + 12 q^{53} + q^{56} - 4 q^{58} + 14 q^{59} - 9 q^{62} + q^{64} + 8 q^{67} + 2 q^{68} + 13 q^{71} + 2 q^{73} - 5 q^{74} - q^{76} - 5 q^{77} + 6 q^{79} + 9 q^{82} - 4 q^{83} + 10 q^{86} - 5 q^{88} + 9 q^{89} - q^{92} + 6 q^{94} - 16 q^{97} + q^{98} + O(q^{100}) \)

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:

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 9450.2.a.dc 1
3.b odd 2 1 9450.2.a.bx 1
5.b even 2 1 378.2.a.c 1
15.d odd 2 1 378.2.a.f yes 1
20.d odd 2 1 3024.2.a.m 1
35.c odd 2 1 2646.2.a.i 1
45.h odd 6 2 1134.2.f.c 2
45.j even 6 2 1134.2.f.n 2
60.h even 2 1 3024.2.a.t 1
105.g even 2 1 2646.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.c 1 5.b even 2 1
378.2.a.f yes 1 15.d odd 2 1
1134.2.f.c 2 45.h odd 6 2
1134.2.f.n 2 45.j even 6 2
2646.2.a.i 1 35.c odd 2 1
2646.2.a.v 1 105.g even 2 1
3024.2.a.m 1 20.d odd 2 1
3024.2.a.t 1 60.h even 2 1
9450.2.a.bx 1 3.b odd 2 1
9450.2.a.dc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9450))\):

\( T_{11} + 5 \)
\( T_{13} \)
\( T_{17} - 2 \)
\( T_{19} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -1 + T \)
$11$ \( 5 + T \)
$13$ \( T \)
$17$ \( -2 + T \)
$19$ \( 1 + T \)
$23$ \( 1 + T \)
$29$ \( 4 + T \)
$31$ \( 9 + T \)
$37$ \( 5 + T \)
$41$ \( -9 + T \)
$43$ \( -10 + T \)
$47$ \( -6 + T \)
$53$ \( -12 + T \)
$59$ \( -14 + T \)
$61$ \( T \)
$67$ \( -8 + T \)
$71$ \( -13 + T \)
$73$ \( -2 + T \)
$79$ \( -6 + T \)
$83$ \( 4 + T \)
$89$ \( -9 + T \)
$97$ \( 16 + T \)
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