Properties

Label 8112.2.a.bv
Level $8112$
Weight $2$
Character orbit 8112.a
Self dual yes
Analytic conductor $64.775$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} -\beta q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} -\beta q^{7} + q^{9} -\beta q^{11} + 6 q^{17} -\beta q^{19} -\beta q^{21} -5 q^{25} + q^{27} + 6 q^{29} + \beta q^{31} -\beta q^{33} -2 \beta q^{37} -2 \beta q^{41} -4 q^{43} + \beta q^{47} + 5 q^{49} + 6 q^{51} + 6 q^{53} -\beta q^{57} + 3 \beta q^{59} -2 q^{61} -\beta q^{63} + 3 \beta q^{67} -\beta q^{71} -5 q^{75} + 12 q^{77} + 8 q^{79} + q^{81} + \beta q^{83} + 6 q^{87} + 2 \beta q^{89} + \beta q^{93} -4 \beta q^{97} -\beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{9} + 12q^{17} - 10q^{25} + 2q^{27} + 12q^{29} - 8q^{43} + 10q^{49} + 12q^{51} + 12q^{53} - 4q^{61} - 10q^{75} + 24q^{77} + 16q^{79} + 2q^{81} + 12q^{87} + O(q^{100}) \)

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
1.73205
−1.73205
0 1.00000 0 0 0 −3.46410 0 1.00000 0
1.2 0 1.00000 0 0 0 3.46410 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

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8112.2.a.bv 2
4.b odd 2 1 507.2.a.f 2
12.b even 2 1 1521.2.a.l 2
13.b even 2 1 inner 8112.2.a.bv 2
13.d odd 4 2 624.2.c.e 2
39.f even 4 2 1872.2.c.e 2
52.b odd 2 1 507.2.a.f 2
52.f even 4 2 39.2.b.a 2
52.i odd 6 2 507.2.e.e 4
52.j odd 6 2 507.2.e.e 4
52.l even 12 2 507.2.j.a 2
52.l even 12 2 507.2.j.c 2
104.j odd 4 2 2496.2.c.d 2
104.m even 4 2 2496.2.c.k 2
156.h even 2 1 1521.2.a.l 2
156.l odd 4 2 117.2.b.a 2
260.l odd 4 2 975.2.h.f 4
260.s odd 4 2 975.2.h.f 4
260.u even 4 2 975.2.b.d 2
364.p odd 4 2 1911.2.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 52.f even 4 2
117.2.b.a 2 156.l odd 4 2
507.2.a.f 2 4.b odd 2 1
507.2.a.f 2 52.b odd 2 1
507.2.e.e 4 52.i odd 6 2
507.2.e.e 4 52.j odd 6 2
507.2.j.a 2 52.l even 12 2
507.2.j.c 2 52.l even 12 2
624.2.c.e 2 13.d odd 4 2
975.2.b.d 2 260.u even 4 2
975.2.h.f 4 260.l odd 4 2
975.2.h.f 4 260.s odd 4 2
1521.2.a.l 2 12.b even 2 1
1521.2.a.l 2 156.h even 2 1
1872.2.c.e 2 39.f even 4 2
1911.2.c.d 2 364.p odd 4 2
2496.2.c.d 2 104.j odd 4 2
2496.2.c.k 2 104.m even 4 2
8112.2.a.bv 2 1.a even 1 1 trivial
8112.2.a.bv 2 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8112))\):

\( T_{5} \)
\( T_{7}^{2} - 12 \)
\( T_{11}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -12 + T^{2} \)
$11$ \( -12 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( -12 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( -12 + T^{2} \)
$37$ \( -48 + T^{2} \)
$41$ \( -48 + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( -12 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( -108 + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( -108 + T^{2} \)
$71$ \( -12 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( -12 + T^{2} \)
$89$ \( -48 + T^{2} \)
$97$ \( -192 + T^{2} \)
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