Properties

Label 2394.2.b.b
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: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 2 \beta q^{5} + q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + 2 \beta q^{5} + q^{7} - q^{8} - 2 \beta q^{10} - 4 \beta q^{11} + 3 \beta q^{13} - q^{14} + q^{16} - 4 \beta q^{17} + (3 \beta + 1) q^{19} + 2 \beta q^{20} + 4 \beta q^{22} - 4 \beta q^{23} - 3 q^{25} - 3 \beta q^{26} + q^{28} + 6 q^{29} - 6 \beta q^{31} - q^{32} + 4 \beta q^{34} + 2 \beta q^{35} + ( - 3 \beta - 1) q^{38} - 2 \beta q^{40} - 4 q^{43} - 4 \beta q^{44} + 4 \beta q^{46} - 7 \beta q^{47} + q^{49} + 3 q^{50} + 3 \beta q^{52} + 6 q^{53} + 16 q^{55} - q^{56} - 6 q^{58} - 12 q^{59} + 2 q^{61} + 6 \beta q^{62} + q^{64} - 12 q^{65} - 9 \beta q^{67} - 4 \beta q^{68} - 2 \beta q^{70} - 6 q^{71} + 14 q^{73} + (3 \beta + 1) q^{76} - 4 \beta q^{77} - 9 \beta q^{79} + 2 \beta q^{80} + 11 \beta q^{83} + 16 q^{85} + 4 q^{86} + 4 \beta q^{88} + 6 q^{89} + 3 \beta q^{91} - 4 \beta q^{92} + 7 \beta q^{94} + (2 \beta - 12) q^{95} + 3 \beta q^{97} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 2 q^{14} + 2 q^{16} + 2 q^{19} - 6 q^{25} + 2 q^{28} + 12 q^{29} - 2 q^{32} - 2 q^{38} - 8 q^{43} + 2 q^{49} + 6 q^{50} + 12 q^{53} + 32 q^{55} - 2 q^{56} - 12 q^{58} - 24 q^{59} + 4 q^{61} + 2 q^{64} - 24 q^{65} - 12 q^{71} + 28 q^{73} + 2 q^{76} + 32 q^{85} + 8 q^{86} + 12 q^{89} - 24 q^{95} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 2.82843i 0 1.00000 −1.00000 0 2.82843i
1709.2 −1.00000 0 1.00000 2.82843i 0 1.00000 −1.00000 0 2.82843i
\(n\): 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.b 2
3.b odd 2 1 2394.2.b.e yes 2
19.b odd 2 1 2394.2.b.e yes 2
57.d even 2 1 inner 2394.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.b.b 2 1.a even 1 1 trivial
2394.2.b.b 2 57.d even 2 1 inner
2394.2.b.e yes 2 3.b odd 2 1
2394.2.b.e 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 \(S_{2}^{\mathrm{new}}(2394, [\chi])\):

\( T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display
\( T_{53} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 8 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 32 \) Copy content Toggle raw display
$13$ \( T^{2} + 18 \) Copy content Toggle raw display
$17$ \( T^{2} + 32 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 32 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 72 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 98 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 162 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( (T - 14)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 162 \) Copy content Toggle raw display
$83$ \( T^{2} + 242 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 18 \) Copy content Toggle raw display
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