Properties

Label 2394.2.a.y
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: \(\Q(\sqrt{29}) \)
Defining polynomial: \(x^{2} - x - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 266)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{29})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta q^{5} - q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + \beta q^{5} - q^{7} + q^{8} + \beta q^{10} + ( -1 - \beta ) q^{11} + ( -2 + 2 \beta ) q^{13} - q^{14} + q^{16} + 4 q^{17} + q^{19} + \beta q^{20} + ( -1 - \beta ) q^{22} + ( 4 - 2 \beta ) q^{23} + ( 2 + \beta ) q^{25} + ( -2 + 2 \beta ) q^{26} - q^{28} + ( -2 - \beta ) q^{29} + 10 q^{31} + q^{32} + 4 q^{34} -\beta q^{35} + ( 2 - \beta ) q^{37} + q^{38} + \beta q^{40} + 3 \beta q^{41} + ( 8 - \beta ) q^{43} + ( -1 - \beta ) q^{44} + ( 4 - 2 \beta ) q^{46} + ( 7 + \beta ) q^{47} + q^{49} + ( 2 + \beta ) q^{50} + ( -2 + 2 \beta ) q^{52} + ( 3 - \beta ) q^{53} + ( -7 - 2 \beta ) q^{55} - q^{56} + ( -2 - \beta ) q^{58} + ( -2 - 3 \beta ) q^{59} + ( -3 - \beta ) q^{61} + 10 q^{62} + q^{64} + 14 q^{65} -2 q^{67} + 4 q^{68} -\beta q^{70} + ( -1 - \beta ) q^{71} + 2 \beta q^{73} + ( 2 - \beta ) q^{74} + q^{76} + ( 1 + \beta ) q^{77} + ( 4 + \beta ) q^{79} + \beta q^{80} + 3 \beta q^{82} -8 q^{83} + 4 \beta q^{85} + ( 8 - \beta ) q^{86} + ( -1 - \beta ) q^{88} + ( -9 - 3 \beta ) q^{89} + ( 2 - 2 \beta ) q^{91} + ( 4 - 2 \beta ) q^{92} + ( 7 + \beta ) q^{94} + \beta q^{95} + ( 11 + \beta ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8} + q^{10} - 3 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 8 q^{17} + 2 q^{19} + q^{20} - 3 q^{22} + 6 q^{23} + 5 q^{25} - 2 q^{26} - 2 q^{28} - 5 q^{29} + 20 q^{31} + 2 q^{32} + 8 q^{34} - q^{35} + 3 q^{37} + 2 q^{38} + q^{40} + 3 q^{41} + 15 q^{43} - 3 q^{44} + 6 q^{46} + 15 q^{47} + 2 q^{49} + 5 q^{50} - 2 q^{52} + 5 q^{53} - 16 q^{55} - 2 q^{56} - 5 q^{58} - 7 q^{59} - 7 q^{61} + 20 q^{62} + 2 q^{64} + 28 q^{65} - 4 q^{67} + 8 q^{68} - q^{70} - 3 q^{71} + 2 q^{73} + 3 q^{74} + 2 q^{76} + 3 q^{77} + 9 q^{79} + q^{80} + 3 q^{82} - 16 q^{83} + 4 q^{85} + 15 q^{86} - 3 q^{88} - 21 q^{89} + 2 q^{91} + 6 q^{92} + 15 q^{94} + q^{95} + 23 q^{97} + 2 q^{98} + O(q^{100}) \)

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
−2.19258
3.19258
1.00000 0 1.00000 −2.19258 0 −1.00000 1.00000 0 −2.19258
1.2 1.00000 0 1.00000 3.19258 0 −1.00000 1.00000 0 3.19258
\(n\): 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.y 2
3.b odd 2 1 266.2.a.a 2
12.b even 2 1 2128.2.a.g 2
15.d odd 2 1 6650.2.a.bu 2
21.c even 2 1 1862.2.a.h 2
24.f even 2 1 8512.2.a.w 2
24.h odd 2 1 8512.2.a.p 2
57.d even 2 1 5054.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.a.a 2 3.b odd 2 1
1862.2.a.h 2 21.c even 2 1
2128.2.a.g 2 12.b even 2 1
2394.2.a.y 2 1.a even 1 1 trivial
5054.2.a.n 2 57.d even 2 1
6650.2.a.bu 2 15.d odd 2 1
8512.2.a.p 2 24.h odd 2 1
8512.2.a.w 2 24.f even 2 1

Hecke kernels

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

\( T_{5}^{2} - T_{5} - 7 \)
\( T_{11}^{2} + 3 T_{11} - 5 \)
\( T_{13}^{2} + 2 T_{13} - 28 \)
\( T_{17} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -7 - T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -5 + 3 T + T^{2} \)
$13$ \( -28 + 2 T + T^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -20 - 6 T + T^{2} \)
$29$ \( -1 + 5 T + T^{2} \)
$31$ \( ( -10 + T )^{2} \)
$37$ \( -5 - 3 T + T^{2} \)
$41$ \( -63 - 3 T + T^{2} \)
$43$ \( 49 - 15 T + T^{2} \)
$47$ \( 49 - 15 T + T^{2} \)
$53$ \( -1 - 5 T + T^{2} \)
$59$ \( -53 + 7 T + T^{2} \)
$61$ \( 5 + 7 T + T^{2} \)
$67$ \( ( 2 + T )^{2} \)
$71$ \( -5 + 3 T + T^{2} \)
$73$ \( -28 - 2 T + T^{2} \)
$79$ \( 13 - 9 T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( 45 + 21 T + T^{2} \)
$97$ \( 125 - 23 T + T^{2} \)
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