Properties

Label 2394.2.a.t
Level $2394$
Weight $2$
Character orbit 2394.a
Self dual yes
Analytic conductor $19.116$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 2 q^{5} + q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + 2 q^{5} + q^{7} - q^{8} -2 q^{10} + ( 2 + \beta ) q^{11} + 2 \beta q^{13} - q^{14} + q^{16} + ( 4 + \beta ) q^{17} - q^{19} + 2 q^{20} + ( -2 - \beta ) q^{22} + 4 q^{23} - q^{25} -2 \beta q^{26} + q^{28} + 2 q^{29} + ( -2 - \beta ) q^{31} - q^{32} + ( -4 - \beta ) q^{34} + 2 q^{35} -3 \beta q^{37} + q^{38} -2 q^{40} + ( 2 - 2 \beta ) q^{41} + ( 4 + 2 \beta ) q^{43} + ( 2 + \beta ) q^{44} -4 q^{46} + ( 4 - 3 \beta ) q^{47} + q^{49} + q^{50} + 2 \beta q^{52} + ( -2 - 2 \beta ) q^{53} + ( 4 + 2 \beta ) q^{55} - q^{56} -2 q^{58} + 8 q^{59} + ( -2 - 4 \beta ) q^{61} + ( 2 + \beta ) q^{62} + q^{64} + 4 \beta q^{65} + ( -4 - \beta ) q^{67} + ( 4 + \beta ) q^{68} -2 q^{70} -2 \beta q^{71} + ( -2 - 2 \beta ) q^{73} + 3 \beta q^{74} - q^{76} + ( 2 + \beta ) q^{77} + ( 2 + 2 \beta ) q^{79} + 2 q^{80} + ( -2 + 2 \beta ) q^{82} + ( 6 + 2 \beta ) q^{83} + ( 8 + 2 \beta ) q^{85} + ( -4 - 2 \beta ) q^{86} + ( -2 - \beta ) q^{88} + ( 2 + 4 \beta ) q^{89} + 2 \beta q^{91} + 4 q^{92} + ( -4 + 3 \beta ) q^{94} -2 q^{95} + ( -10 - 3 \beta ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} - 2 q^{8} - 4 q^{10} + 4 q^{11} - 2 q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 2 q^{25} + 2 q^{28} + 4 q^{29} - 4 q^{31} - 2 q^{32} - 8 q^{34} + 4 q^{35} + 2 q^{38} - 4 q^{40} + 4 q^{41} + 8 q^{43} + 4 q^{44} - 8 q^{46} + 8 q^{47} + 2 q^{49} + 2 q^{50} - 4 q^{53} + 8 q^{55} - 2 q^{56} - 4 q^{58} + 16 q^{59} - 4 q^{61} + 4 q^{62} + 2 q^{64} - 8 q^{67} + 8 q^{68} - 4 q^{70} - 4 q^{73} - 2 q^{76} + 4 q^{77} + 4 q^{79} + 4 q^{80} - 4 q^{82} + 12 q^{83} + 16 q^{85} - 8 q^{86} - 4 q^{88} + 4 q^{89} + 8 q^{92} - 8 q^{94} - 4 q^{95} - 20 q^{97} - 2 q^{98} + O(q^{100}) \)

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
−1.41421
1.41421
−1.00000 0 1.00000 2.00000 0 1.00000 −1.00000 0 −2.00000
1.2 −1.00000 0 1.00000 2.00000 0 1.00000 −1.00000 0 −2.00000
\(n\): 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.t 2
3.b odd 2 1 2394.2.a.u yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.a.t 2 1.a even 1 1 trivial
2394.2.a.u yes 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2394))\):

\( T_{5} - 2 \)
\( T_{11}^{2} - 4 T_{11} - 4 \)
\( T_{13}^{2} - 32 \)
\( T_{17}^{2} - 8 T_{17} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -2 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -4 - 4 T + T^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( 8 - 8 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( -4 + 4 T + T^{2} \)
$37$ \( -72 + T^{2} \)
$41$ \( -28 - 4 T + T^{2} \)
$43$ \( -16 - 8 T + T^{2} \)
$47$ \( -56 - 8 T + T^{2} \)
$53$ \( -28 + 4 T + T^{2} \)
$59$ \( ( -8 + T )^{2} \)
$61$ \( -124 + 4 T + T^{2} \)
$67$ \( 8 + 8 T + T^{2} \)
$71$ \( -32 + T^{2} \)
$73$ \( -28 + 4 T + T^{2} \)
$79$ \( -28 - 4 T + T^{2} \)
$83$ \( 4 - 12 T + T^{2} \)
$89$ \( -124 - 4 T + T^{2} \)
$97$ \( 28 + 20 T + T^{2} \)
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