Properties

Label 2394.2.a.v
Level $2394$
Weight $2$
Character orbit 2394.a
Self dual yes
Analytic conductor $19.116$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( - \beta - 1) q^{5} - q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + ( - \beta - 1) q^{5} - q^{7} + q^{8} + ( - \beta - 1) q^{10} + (\beta - 3) q^{11} + (\beta - 1) q^{13} - q^{14} + q^{16} + (3 \beta + 1) q^{17} + q^{19} + ( - \beta - 1) q^{20} + (\beta - 3) q^{22} + ( - \beta - 3) q^{23} + (2 \beta + 1) q^{25} + (\beta - 1) q^{26} - q^{28} + ( - 2 \beta - 4) q^{29} - 2 q^{31} + q^{32} + (3 \beta + 1) q^{34} + (\beta + 1) q^{35} + ( - 2 \beta - 6) q^{37} + q^{38} + ( - \beta - 1) q^{40} - 6 q^{41} + 4 \beta q^{43} + (\beta - 3) q^{44} + ( - \beta - 3) q^{46} - 4 \beta q^{47} + q^{49} + (2 \beta + 1) q^{50} + (\beta - 1) q^{52} + ( - 2 \beta - 8) q^{53} + (2 \beta - 2) q^{55} - q^{56} + ( - 2 \beta - 4) q^{58} - 8 q^{59} + ( - 2 \beta + 4) q^{61} - 2 q^{62} + q^{64} - 4 q^{65} + ( - 3 \beta + 7) q^{67} + (3 \beta + 1) q^{68} + (\beta + 1) q^{70} + ( - 2 \beta - 6) q^{71} + ( - 2 \beta - 8) q^{73} + ( - 2 \beta - 6) q^{74} + q^{76} + ( - \beta + 3) q^{77} + ( - \beta + 3) q^{79} + ( - \beta - 1) q^{80} - 6 q^{82} + 10 q^{83} + ( - 4 \beta - 16) q^{85} + 4 \beta q^{86} + (\beta - 3) q^{88} - 6 q^{89} + ( - \beta + 1) q^{91} + ( - \beta - 3) q^{92} - 4 \beta q^{94} + ( - \beta - 1) q^{95} + ( - \beta + 7) q^{97} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 6 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{19} - 2 q^{20} - 6 q^{22} - 6 q^{23} + 2 q^{25} - 2 q^{26} - 2 q^{28} - 8 q^{29} - 4 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{35} - 12 q^{37} + 2 q^{38} - 2 q^{40} - 12 q^{41} - 6 q^{44} - 6 q^{46} + 2 q^{49} + 2 q^{50} - 2 q^{52} - 16 q^{53} - 4 q^{55} - 2 q^{56} - 8 q^{58} - 16 q^{59} + 8 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{65} + 14 q^{67} + 2 q^{68} + 2 q^{70} - 12 q^{71} - 16 q^{73} - 12 q^{74} + 2 q^{76} + 6 q^{77} + 6 q^{79} - 2 q^{80} - 12 q^{82} + 20 q^{83} - 32 q^{85} - 6 q^{88} - 12 q^{89} + 2 q^{91} - 6 q^{92} - 2 q^{95} + 14 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
1.61803
−0.618034
1.00000 0 1.00000 −3.23607 0 −1.00000 1.00000 0 −3.23607
1.2 1.00000 0 1.00000 1.23607 0 −1.00000 1.00000 0 1.23607
\(n\): 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.v yes 2
3.b odd 2 1 2394.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.a.s 2 3.b odd 2 1
2394.2.a.v yes 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2394))\):

\( T_{5}^{2} + 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} - 44 \) 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
$5$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 80 \) Copy content Toggle raw display
$47$ \( T^{2} - 80 \) Copy content Toggle raw display
$53$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$59$ \( (T + 8)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$79$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$83$ \( (T - 10)^{2} \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 44 \) Copy content Toggle raw display
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