Properties

Label 9450.2.a.eb
Level 9450
Weight 2
Character orbit 9450.a
Self dual yes
Analytic conductor 75.459
Analytic rank 1
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1890)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} - q^{7} - q^{8} + \beta q^{11} + ( 2 - 2 \beta ) q^{13} + q^{14} + q^{16} -4 q^{17} + ( 4 - \beta ) q^{19} -\beta q^{22} + ( 1 + 2 \beta ) q^{23} + ( -2 + 2 \beta ) q^{26} - q^{28} + ( 2 + 2 \beta ) q^{29} + ( -2 - \beta ) q^{31} - q^{32} + 4 q^{34} + ( 2 + 3 \beta ) q^{37} + ( -4 + \beta ) q^{38} + ( -3 - 4 \beta ) q^{41} + ( -8 - 2 \beta ) q^{43} + \beta q^{44} + ( -1 - 2 \beta ) q^{46} + ( -8 + 2 \beta ) q^{47} + q^{49} + ( 2 - 2 \beta ) q^{52} -8 \beta q^{53} + q^{56} + ( -2 - 2 \beta ) q^{58} + ( 2 + 4 \beta ) q^{59} + 8 \beta q^{61} + ( 2 + \beta ) q^{62} + q^{64} + ( -2 + 6 \beta ) q^{67} -4 q^{68} + ( -6 - \beta ) q^{71} + ( 10 - 4 \beta ) q^{73} + ( -2 - 3 \beta ) q^{74} + ( 4 - \beta ) q^{76} -\beta q^{77} + ( 2 - 4 \beta ) q^{79} + ( 3 + 4 \beta ) q^{82} + ( -6 + 2 \beta ) q^{83} + ( 8 + 2 \beta ) q^{86} -\beta q^{88} - q^{89} + ( -2 + 2 \beta ) q^{91} + ( 1 + 2 \beta ) q^{92} + ( 8 - 2 \beta ) q^{94} + 4 \beta q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{7} - 2q^{8} + 4q^{13} + 2q^{14} + 2q^{16} - 8q^{17} + 8q^{19} + 2q^{23} - 4q^{26} - 2q^{28} + 4q^{29} - 4q^{31} - 2q^{32} + 8q^{34} + 4q^{37} - 8q^{38} - 6q^{41} - 16q^{43} - 2q^{46} - 16q^{47} + 2q^{49} + 4q^{52} + 2q^{56} - 4q^{58} + 4q^{59} + 4q^{62} + 2q^{64} - 4q^{67} - 8q^{68} - 12q^{71} + 20q^{73} - 4q^{74} + 8q^{76} + 4q^{79} + 6q^{82} - 12q^{83} + 16q^{86} - 2q^{89} - 4q^{91} + 2q^{92} + 16q^{94} - 2q^{98} + O(q^{100}) \)

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
−1.73205
1.73205
−1.00000 0 1.00000 0 0 −1.00000 −1.00000 0 0
1.2 −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.eb 2
3.b odd 2 1 9450.2.a.er 2
5.b even 2 1 9450.2.a.eu 2
5.c odd 4 2 1890.2.g.n 4
15.d odd 2 1 9450.2.a.ei 2
15.e even 4 2 1890.2.g.q yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.g.n 4 5.c odd 4 2
1890.2.g.q yes 4 15.e even 4 2
9450.2.a.eb 2 1.a even 1 1 trivial
9450.2.a.ei 2 15.d odd 2 1
9450.2.a.er 2 3.b odd 2 1
9450.2.a.eu 2 5.b even 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9450))\):

\( T_{11}^{2} - 3 \)
\( T_{13}^{2} - 4 T_{13} - 8 \)
\( T_{17} + 4 \)
\( T_{19}^{2} - 8 T_{19} + 13 \)