Properties

Label 2394.2.b.a
Level $2394$
Weight $2$
Character orbit 2394.b
Analytic conductor $19.116$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} - q^{7} - q^{8} -\beta q^{13} + q^{14} + q^{16} + ( -1 + \beta ) q^{19} + 2 \beta q^{23} + 5 q^{25} + \beta q^{26} - q^{28} -6 q^{29} - q^{32} -2 \beta q^{37} + ( 1 - \beta ) q^{38} + 8 q^{43} -2 \beta q^{46} + \beta q^{47} + q^{49} -5 q^{50} -\beta q^{52} + 6 q^{53} + q^{56} + 6 q^{58} + 10 q^{61} + q^{64} -\beta q^{67} + 6 q^{71} + 2 q^{73} + 2 \beta q^{74} + ( -1 + \beta ) q^{76} + \beta q^{79} + \beta q^{83} -8 q^{86} -6 q^{89} + \beta q^{91} + 2 \beta q^{92} -\beta q^{94} + \beta q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 2 q^{14} + 2 q^{16} - 2 q^{19} + 10 q^{25} - 2 q^{28} - 12 q^{29} - 2 q^{32} + 2 q^{38} + 16 q^{43} + 2 q^{49} - 10 q^{50} + 12 q^{53} + 2 q^{56} + 12 q^{58} + 20 q^{61} + 2 q^{64} + 12 q^{71} + 4 q^{73} - 2 q^{76} - 16 q^{86} - 12 q^{89} - 2 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1709.1
1.41421i
1.41421i
−1.00000 0 1.00000 0 0 −1.00000 −1.00000 0 0
1709.2 −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

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2394.2.b.a 2
3.b odd 2 1 2394.2.b.d yes 2
19.b odd 2 1 2394.2.b.d yes 2
57.d even 2 1 inner 2394.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.b.a 2 1.a even 1 1 trivial
2394.2.b.a 2 57.d even 2 1 inner
2394.2.b.d yes 2 3.b odd 2 1
2394.2.b.d yes 2 19.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 \(S_{2}^{\mathrm{new}}(2394, [\chi])\):

\( T_{5} \)
\( T_{29} + 6 \)
\( T_{53} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 18 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 19 + 2 T + T^{2} \)
$23$ \( 72 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 72 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( 18 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( 18 + T^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( ( -2 + T )^{2} \)
$79$ \( 18 + T^{2} \)
$83$ \( 18 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 18 + T^{2} \)
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