Properties

Label 2394.2.a.bc
Level $2394$
Weight $2$
Character orbit 2394.a
Self dual yes
Analytic conductor $19.116$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( 1 + \beta_{1} ) q^{5} - q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( 1 + \beta_{1} ) q^{5} - q^{7} + q^{8} + ( 1 + \beta_{1} ) q^{10} + ( 2 + \beta_{2} ) q^{11} + ( 3 + \beta_{1} ) q^{13} - q^{14} + q^{16} -\beta_{2} q^{17} - q^{19} + ( 1 + \beta_{1} ) q^{20} + ( 2 + \beta_{2} ) q^{22} + ( 1 - \beta_{1} ) q^{23} + ( 5 + 2 \beta_{1} + 2 \beta_{2} ) q^{25} + ( 3 + \beta_{1} ) q^{26} - q^{28} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -1 - \beta_{1} - \beta_{2} ) q^{31} + q^{32} -\beta_{2} q^{34} + ( -1 - \beta_{1} ) q^{35} + ( 1 - \beta_{1} - \beta_{2} ) q^{37} - q^{38} + ( 1 + \beta_{1} ) q^{40} -2 \beta_{1} q^{41} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 2 + \beta_{2} ) q^{44} + ( 1 - \beta_{1} ) q^{46} + ( 5 - \beta_{1} + \beta_{2} ) q^{47} + q^{49} + ( 5 + 2 \beta_{1} + 2 \beta_{2} ) q^{50} + ( 3 + \beta_{1} ) q^{52} + 2 q^{53} + 4 \beta_{1} q^{55} - q^{56} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{58} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -1 - \beta_{1} - \beta_{2} ) q^{62} + q^{64} + ( 12 + 4 \beta_{1} + 2 \beta_{2} ) q^{65} + \beta_{2} q^{67} -\beta_{2} q^{68} + ( -1 - \beta_{1} ) q^{70} + ( 8 + 2 \beta_{2} ) q^{71} + ( -2 - 2 \beta_{2} ) q^{73} + ( 1 - \beta_{1} - \beta_{2} ) q^{74} - q^{76} + ( -2 - \beta_{2} ) q^{77} + ( -7 - 3 \beta_{1} + 2 \beta_{2} ) q^{79} + ( 1 + \beta_{1} ) q^{80} -2 \beta_{1} q^{82} + ( 4 - 2 \beta_{1} ) q^{83} + ( 2 - 2 \beta_{1} ) q^{85} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{86} + ( 2 + \beta_{2} ) q^{88} + ( 6 - 2 \beta_{2} ) q^{89} + ( -3 - \beta_{1} ) q^{91} + ( 1 - \beta_{1} ) q^{92} + ( 5 - \beta_{1} + \beta_{2} ) q^{94} + ( -1 - \beta_{1} ) q^{95} + ( 2 \beta_{1} - \beta_{2} ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8} + O(q^{10}) \) \( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8} + 2 q^{10} + 6 q^{11} + 8 q^{13} - 3 q^{14} + 3 q^{16} - 3 q^{19} + 2 q^{20} + 6 q^{22} + 4 q^{23} + 13 q^{25} + 8 q^{26} - 3 q^{28} + 2 q^{29} - 2 q^{31} + 3 q^{32} - 2 q^{35} + 4 q^{37} - 3 q^{38} + 2 q^{40} + 2 q^{41} - 4 q^{43} + 6 q^{44} + 4 q^{46} + 16 q^{47} + 3 q^{49} + 13 q^{50} + 8 q^{52} + 6 q^{53} - 4 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{61} - 2 q^{62} + 3 q^{64} + 32 q^{65} - 2 q^{70} + 24 q^{71} - 6 q^{73} + 4 q^{74} - 3 q^{76} - 6 q^{77} - 18 q^{79} + 2 q^{80} + 2 q^{82} + 14 q^{83} + 8 q^{85} - 4 q^{86} + 6 q^{88} + 18 q^{89} - 8 q^{91} + 4 q^{92} + 16 q^{94} - 2 q^{95} - 2 q^{97} + 3 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 5\)\()/2\)

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
−1.48119
0.311108
2.17009
1.00000 0 1.00000 −2.96239 0 −1.00000 1.00000 0 −2.96239
1.2 1.00000 0 1.00000 0.622216 0 −1.00000 1.00000 0 0.622216
1.3 1.00000 0 1.00000 4.34017 0 −1.00000 1.00000 0 4.34017
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)
\(19\) \(1\)

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 2394.2.a.bc yes 3
3.b odd 2 1 2394.2.a.bb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.a.bb 3 3.b odd 2 1
2394.2.a.bc yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2394))\):

\( T_{5}^{3} - 2 T_{5}^{2} - 12 T_{5} + 8 \)
\( T_{11}^{3} - 6 T_{11}^{2} - 4 T_{11} + 40 \)
\( T_{13}^{3} - 8 T_{13}^{2} + 8 T_{13} + 16 \)
\( T_{17}^{3} - 16 T_{17} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( 8 - 12 T - 2 T^{2} + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 40 - 4 T - 6 T^{2} + T^{3} \)
$13$ \( 16 + 8 T - 8 T^{2} + T^{3} \)
$17$ \( -16 - 16 T + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( 16 - 8 T - 4 T^{2} + T^{3} \)
$29$ \( 104 - 84 T - 2 T^{2} + T^{3} \)
$31$ \( -8 - 20 T + 2 T^{2} + T^{3} \)
$37$ \( 32 - 16 T - 4 T^{2} + T^{3} \)
$41$ \( 40 - 52 T - 2 T^{2} + T^{3} \)
$43$ \( -64 - 80 T + 4 T^{2} + T^{3} \)
$47$ \( -32 + 48 T - 16 T^{2} + T^{3} \)
$53$ \( ( -2 + T )^{3} \)
$59$ \( T^{3} \)
$61$ \( 200 - 52 T - 10 T^{2} + T^{3} \)
$67$ \( 16 - 16 T + T^{3} \)
$71$ \( 128 + 128 T - 24 T^{2} + T^{3} \)
$73$ \( -248 - 52 T + 6 T^{2} + T^{3} \)
$79$ \( -2536 - 124 T + 18 T^{2} + T^{3} \)
$83$ \( 152 + 12 T - 14 T^{2} + T^{3} \)
$89$ \( 40 + 44 T - 18 T^{2} + T^{3} \)
$97$ \( 232 - 84 T + 2 T^{2} + T^{3} \)
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