Properties

Label 2394.2.a.w
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( 1 - 3 \beta ) q^{5} + q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( 1 - 3 \beta ) q^{5} + q^{7} + q^{8} + ( 1 - 3 \beta ) q^{10} + ( -4 + \beta ) q^{11} + ( 4 - 2 \beta ) q^{13} + q^{14} + q^{16} + 4 \beta q^{17} - q^{19} + ( 1 - 3 \beta ) q^{20} + ( -4 + \beta ) q^{22} + ( 2 - 6 \beta ) q^{23} + ( 5 + 3 \beta ) q^{25} + ( 4 - 2 \beta ) q^{26} + q^{28} + ( 7 - 3 \beta ) q^{29} + ( 2 + 4 \beta ) q^{31} + q^{32} + 4 \beta q^{34} + ( 1 - 3 \beta ) q^{35} + ( 3 - 3 \beta ) q^{37} - q^{38} + ( 1 - 3 \beta ) q^{40} + ( -1 + 7 \beta ) q^{41} + ( -7 + \beta ) q^{43} + ( -4 + \beta ) q^{44} + ( 2 - 6 \beta ) q^{46} + ( 2 - 3 \beta ) q^{47} + q^{49} + ( 5 + 3 \beta ) q^{50} + ( 4 - 2 \beta ) q^{52} + ( 12 + \beta ) q^{53} + ( -7 + 10 \beta ) q^{55} + q^{56} + ( 7 - 3 \beta ) q^{58} + ( 7 - 3 \beta ) q^{59} + ( 2 + 7 \beta ) q^{61} + ( 2 + 4 \beta ) q^{62} + q^{64} + ( 10 - 8 \beta ) q^{65} + ( 6 - 8 \beta ) q^{67} + 4 \beta q^{68} + ( 1 - 3 \beta ) q^{70} + ( -8 + 13 \beta ) q^{71} + ( -6 + 2 \beta ) q^{73} + ( 3 - 3 \beta ) q^{74} - q^{76} + ( -4 + \beta ) q^{77} + ( -9 + 7 \beta ) q^{79} + ( 1 - 3 \beta ) q^{80} + ( -1 + 7 \beta ) q^{82} + ( -4 + 8 \beta ) q^{83} + ( -12 - 8 \beta ) q^{85} + ( -7 + \beta ) q^{86} + ( -4 + \beta ) q^{88} + ( 2 + \beta ) q^{89} + ( 4 - 2 \beta ) q^{91} + ( 2 - 6 \beta ) q^{92} + ( 2 - 3 \beta ) q^{94} + ( -1 + 3 \beta ) q^{95} + ( 2 - 3 \beta ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8} - q^{10} - 7 q^{11} + 6 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{17} - 2 q^{19} - q^{20} - 7 q^{22} - 2 q^{23} + 13 q^{25} + 6 q^{26} + 2 q^{28} + 11 q^{29} + 8 q^{31} + 2 q^{32} + 4 q^{34} - q^{35} + 3 q^{37} - 2 q^{38} - q^{40} + 5 q^{41} - 13 q^{43} - 7 q^{44} - 2 q^{46} + q^{47} + 2 q^{49} + 13 q^{50} + 6 q^{52} + 25 q^{53} - 4 q^{55} + 2 q^{56} + 11 q^{58} + 11 q^{59} + 11 q^{61} + 8 q^{62} + 2 q^{64} + 12 q^{65} + 4 q^{67} + 4 q^{68} - q^{70} - 3 q^{71} - 10 q^{73} + 3 q^{74} - 2 q^{76} - 7 q^{77} - 11 q^{79} - q^{80} + 5 q^{82} - 32 q^{85} - 13 q^{86} - 7 q^{88} + 5 q^{89} + 6 q^{91} - 2 q^{92} + q^{94} + q^{95} + q^{97} + 2 q^{98} + O(q^{100}) \)

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
1.61803
−0.618034
1.00000 0 1.00000 −3.85410 0 1.00000 1.00000 0 −3.85410
1.2 1.00000 0 1.00000 2.85410 0 1.00000 1.00000 0 2.85410
\(n\): 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.w 2
3.b odd 2 1 266.2.a.b 2
12.b even 2 1 2128.2.a.b 2
15.d odd 2 1 6650.2.a.bq 2
21.c even 2 1 1862.2.a.g 2
24.f even 2 1 8512.2.a.bc 2
24.h odd 2 1 8512.2.a.h 2
57.d even 2 1 5054.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.a.b 2 3.b odd 2 1
1862.2.a.g 2 21.c even 2 1
2128.2.a.b 2 12.b even 2 1
2394.2.a.w 2 1.a even 1 1 trivial
5054.2.a.k 2 57.d even 2 1
6650.2.a.bq 2 15.d odd 2 1
8512.2.a.h 2 24.h odd 2 1
8512.2.a.bc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2394))\):

\( T_{5}^{2} + T_{5} - 11 \)
\( T_{11}^{2} + 7 T_{11} + 11 \)
\( T_{13}^{2} - 6 T_{13} + 4 \)
\( T_{17}^{2} - 4 T_{17} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -11 + T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( 11 + 7 T + T^{2} \)
$13$ \( 4 - 6 T + T^{2} \)
$17$ \( -16 - 4 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -44 + 2 T + T^{2} \)
$29$ \( 19 - 11 T + T^{2} \)
$31$ \( -4 - 8 T + T^{2} \)
$37$ \( -9 - 3 T + T^{2} \)
$41$ \( -55 - 5 T + T^{2} \)
$43$ \( 41 + 13 T + T^{2} \)
$47$ \( -11 - T + T^{2} \)
$53$ \( 155 - 25 T + T^{2} \)
$59$ \( 19 - 11 T + T^{2} \)
$61$ \( -31 - 11 T + T^{2} \)
$67$ \( -76 - 4 T + T^{2} \)
$71$ \( -209 + 3 T + T^{2} \)
$73$ \( 20 + 10 T + T^{2} \)
$79$ \( -31 + 11 T + T^{2} \)
$83$ \( -80 + T^{2} \)
$89$ \( 5 - 5 T + T^{2} \)
$97$ \( -11 - T + T^{2} \)
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