Properties

Label 8112.2.a.bc
Level $8112$
Weight $2$
Character orbit 8112.a
Self dual yes
Analytic conductor $64.775$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.7746461197\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} + 2 q^{7} + q^{9} - 2 q^{11} + q^{15} - 7 q^{17} - 6 q^{19} + 2 q^{21} + 6 q^{23} - 4 q^{25} + q^{27} - q^{29} + 4 q^{31} - 2 q^{33} + 2 q^{35} - q^{37} - 9 q^{41} - 6 q^{43} + q^{45} + 6 q^{47} - 3 q^{49} - 7 q^{51} - 9 q^{53} - 2 q^{55} - 6 q^{57} + q^{61} + 2 q^{63} - 2 q^{67} + 6 q^{69} + 6 q^{71} - 11 q^{73} - 4 q^{75} - 4 q^{77} + 4 q^{79} + q^{81} - 14 q^{83} - 7 q^{85} - q^{87} + 14 q^{89} + 4 q^{93} - 6 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100}) \) 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
0
0 1.00000 0 1.00000 0 2.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(1\)

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 8112.2.a.bc 1
4.b odd 2 1 507.2.a.b 1
12.b even 2 1 1521.2.a.d 1
13.b even 2 1 8112.2.a.w 1
13.e even 6 2 624.2.q.c 2
39.h odd 6 2 1872.2.t.j 2
52.b odd 2 1 507.2.a.c 1
52.f even 4 2 507.2.b.b 2
52.i odd 6 2 39.2.e.a 2
52.j odd 6 2 507.2.e.c 2
52.l even 12 4 507.2.j.d 4
156.h even 2 1 1521.2.a.a 1
156.l odd 4 2 1521.2.b.c 2
156.r even 6 2 117.2.g.b 2
260.w odd 6 2 975.2.i.f 2
260.bg even 12 4 975.2.bb.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 52.i odd 6 2
117.2.g.b 2 156.r even 6 2
507.2.a.b 1 4.b odd 2 1
507.2.a.c 1 52.b odd 2 1
507.2.b.b 2 52.f even 4 2
507.2.e.c 2 52.j odd 6 2
507.2.j.d 4 52.l even 12 4
624.2.q.c 2 13.e even 6 2
975.2.i.f 2 260.w odd 6 2
975.2.bb.d 4 260.bg even 12 4
1521.2.a.a 1 156.h even 2 1
1521.2.a.d 1 12.b even 2 1
1521.2.b.c 2 156.l odd 4 2
1872.2.t.j 2 39.h odd 6 2
8112.2.a.w 1 13.b even 2 1
8112.2.a.bc 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8112))\):

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 7 \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T + 1 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 1 \) Copy content Toggle raw display
$41$ \( T + 9 \) Copy content Toggle raw display
$43$ \( T + 6 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T + 9 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 1 \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T - 6 \) Copy content Toggle raw display
$73$ \( T + 11 \) Copy content Toggle raw display
$79$ \( T - 4 \) Copy content Toggle raw display
$83$ \( T + 14 \) Copy content Toggle raw display
$89$ \( T - 14 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
show more
show less