Properties

Label 1368.2.a.k
Level $1368$
Weight $2$
Character orbit 1368.a
Self dual yes
Analytic conductor $10.924$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.9235349965\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 456)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{5} + \beta q^{7} +O(q^{10})\) \( q -\beta q^{5} + \beta q^{7} + ( -4 + \beta ) q^{11} -2 \beta q^{13} + ( -2 + 3 \beta ) q^{17} + q^{19} + ( -6 + 2 \beta ) q^{23} + ( -1 + \beta ) q^{25} + ( 2 - 4 \beta ) q^{29} -2 q^{31} + ( -4 - \beta ) q^{35} -8 q^{37} + ( 2 - 2 \beta ) q^{41} + \beta q^{43} + ( 2 - 3 \beta ) q^{47} + ( -3 + \beta ) q^{49} + ( 2 + 4 \beta ) q^{53} + ( -4 + 3 \beta ) q^{55} -12 q^{59} + ( 2 - 3 \beta ) q^{61} + ( 8 + 2 \beta ) q^{65} -4 \beta q^{67} + ( -6 + 7 \beta ) q^{73} + ( 4 - 3 \beta ) q^{77} + ( 2 - 6 \beta ) q^{79} -4 q^{83} + ( -12 - \beta ) q^{85} + 6 q^{89} + ( -8 - 2 \beta ) q^{91} -\beta q^{95} + ( -2 - 4 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + q^{7} + O(q^{10}) \) \( 2 q - q^{5} + q^{7} - 7 q^{11} - 2 q^{13} - q^{17} + 2 q^{19} - 10 q^{23} - q^{25} - 4 q^{31} - 9 q^{35} - 16 q^{37} + 2 q^{41} + q^{43} + q^{47} - 5 q^{49} + 8 q^{53} - 5 q^{55} - 24 q^{59} + q^{61} + 18 q^{65} - 4 q^{67} - 5 q^{73} + 5 q^{77} - 2 q^{79} - 8 q^{83} - 25 q^{85} + 12 q^{89} - 18 q^{91} - q^{95} - 8 q^{97} + O(q^{100}) \)

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
2.56155
−1.56155
0 0 0 −2.56155 0 2.56155 0 0 0
1.2 0 0 0 1.56155 0 −1.56155 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(19\) \(-1\)

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 1368.2.a.k 2
3.b odd 2 1 456.2.a.f 2
4.b odd 2 1 2736.2.a.z 2
12.b even 2 1 912.2.a.m 2
24.f even 2 1 3648.2.a.br 2
24.h odd 2 1 3648.2.a.bl 2
57.d even 2 1 8664.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.a.f 2 3.b odd 2 1
912.2.a.m 2 12.b even 2 1
1368.2.a.k 2 1.a even 1 1 trivial
2736.2.a.z 2 4.b odd 2 1
3648.2.a.bl 2 24.h odd 2 1
3648.2.a.br 2 24.f even 2 1
8664.2.a.r 2 57.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1368))\):

\( T_{5}^{2} + T_{5} - 4 \)
\( T_{7}^{2} - T_{7} - 4 \)
\( T_{11}^{2} + 7 T_{11} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -4 + T + T^{2} \)
$7$ \( -4 - T + T^{2} \)
$11$ \( 8 + 7 T + T^{2} \)
$13$ \( -16 + 2 T + T^{2} \)
$17$ \( -38 + T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 8 + 10 T + T^{2} \)
$29$ \( -68 + T^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( ( 8 + T )^{2} \)
$41$ \( -16 - 2 T + T^{2} \)
$43$ \( -4 - T + T^{2} \)
$47$ \( -38 - T + T^{2} \)
$53$ \( -52 - 8 T + T^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( -38 - T + T^{2} \)
$67$ \( -64 + 4 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( -202 + 5 T + T^{2} \)
$79$ \( -152 + 2 T + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( -52 + 8 T + T^{2} \)
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