Properties

Label 2394.2.b.e
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: \( 1 \)
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.

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

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.e yes 2
3.b odd 2 1 2394.2.b.b 2
19.b odd 2 1 2394.2.b.b 2
57.d even 2 1 inner 2394.2.b.e yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.b.b 2 3.b odd 2 1
2394.2.b.b 2 19.b odd 2 1
2394.2.b.e yes 2 1.a even 1 1 trivial
2394.2.b.e yes 2 57.d even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} + 8 \)
\( T_{29} + 6 \)
\( T_{53} + 6 \)

Hecke characteristic polynomials

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