Properties

Label 7168.2.a.k
Level $7168$
Weight $2$
Character orbit 7168.a
Self dual yes
Analytic conductor $57.237$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7168 = 2^{10} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7168.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.2367681689\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \beta q^{5} + q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{5} + q^{7} - 3 q^{9} + \beta q^{11} + 2 q^{17} - 2 \beta q^{19} - 6 q^{23} + 3 q^{25} - 7 \beta q^{29} + 8 q^{31} + 2 \beta q^{35} - 5 \beta q^{37} - 10 q^{41} - \beta q^{43} - 6 \beta q^{45} - 12 q^{47} + q^{49} + \beta q^{53} + 4 q^{55} - 8 \beta q^{59} - 6 \beta q^{61} - 3 q^{63} + 3 \beta q^{67} + 6 q^{73} + \beta q^{77} + 10 q^{79} + 9 q^{81} + 10 \beta q^{83} + 4 \beta q^{85} - 14 q^{89} - 8 q^{95} - 2 q^{97} - 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} - 6 q^{9} + 4 q^{17} - 12 q^{23} + 6 q^{25} + 16 q^{31} - 20 q^{41} - 24 q^{47} + 2 q^{49} + 8 q^{55} - 6 q^{63} + 12 q^{73} + 20 q^{79} + 18 q^{81} - 28 q^{89} - 16 q^{95} - 4 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
0 0 0 −2.82843 0 1.00000 0 −3.00000 0
1.2 0 0 0 2.82843 0 1.00000 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7168.2.a.k 2
4.b odd 2 1 7168.2.a.b 2
8.b even 2 1 inner 7168.2.a.k 2
8.d odd 2 1 7168.2.a.b 2
32.g even 8 2 448.2.m.a 2
32.g even 8 2 896.2.m.c 2
32.h odd 8 2 112.2.m.b 2
32.h odd 8 2 896.2.m.b 2
224.x even 8 2 784.2.m.a 2
224.be even 24 4 784.2.x.e 4
224.bf odd 24 4 784.2.x.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.b 2 32.h odd 8 2
448.2.m.a 2 32.g even 8 2
784.2.m.a 2 224.x even 8 2
784.2.x.d 4 224.bf odd 24 4
784.2.x.e 4 224.be even 24 4
896.2.m.b 2 32.h odd 8 2
896.2.m.c 2 32.g even 8 2
7168.2.a.b 2 4.b odd 2 1
7168.2.a.b 2 8.d odd 2 1
7168.2.a.k 2 1.a even 1 1 trivial
7168.2.a.k 2 8.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7168))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 2 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display
\( T_{23} + 6 \) Copy content Toggle raw display
\( T_{31} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 8 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 8 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 98 \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 50 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 2 \) Copy content Toggle raw display
$47$ \( (T + 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 2 \) Copy content Toggle raw display
$59$ \( T^{2} - 128 \) Copy content Toggle raw display
$61$ \( T^{2} - 72 \) Copy content Toggle raw display
$67$ \( T^{2} - 18 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( (T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 200 \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
show more
show less