Properties

Label 8112.2.a.cg
Level $8112$
Weight $2$
Character orbit 8112.a
Self dual yes
Analytic conductor $64.775$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -2 + \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} - \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -2 + \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} - \beta_{2} ) q^{7} + q^{9} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{11} + ( -2 + \beta_{1} + \beta_{2} ) q^{15} + ( -2 \beta_{1} - \beta_{2} ) q^{17} + ( -2 + \beta_{2} ) q^{19} + ( \beta_{1} - \beta_{2} ) q^{21} + ( 2 - 5 \beta_{1} + \beta_{2} ) q^{23} + ( 4 - 3 \beta_{1} - 2 \beta_{2} ) q^{25} + q^{27} + ( 1 - 3 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -3 - 5 \beta_{1} + 2 \beta_{2} ) q^{31} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{33} + ( 1 - 3 \beta_{1} + 4 \beta_{2} ) q^{35} + ( -7 + \beta_{2} ) q^{37} + ( -2 - 3 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 5 + 2 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -2 + \beta_{1} + \beta_{2} ) q^{45} + ( 2 - \beta_{2} ) q^{47} + ( -6 + \beta_{1} - 2 \beta_{2} ) q^{49} + ( -2 \beta_{1} - \beta_{2} ) q^{51} + ( -8 + 3 \beta_{1} - 4 \beta_{2} ) q^{53} + ( 5 - 8 \beta_{1} + 10 \beta_{2} ) q^{55} + ( -2 + \beta_{2} ) q^{57} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -8 + 6 \beta_{1} - 5 \beta_{2} ) q^{61} + ( \beta_{1} - \beta_{2} ) q^{63} + ( 7 - 7 \beta_{1} + 3 \beta_{2} ) q^{67} + ( 2 - 5 \beta_{1} + \beta_{2} ) q^{69} + ( 1 - 5 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 2 - 9 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 4 - 3 \beta_{1} - 2 \beta_{2} ) q^{75} + ( 3 + 2 \beta_{1} - 4 \beta_{2} ) q^{77} + ( -3 + 3 \beta_{1} - 3 \beta_{2} ) q^{79} + q^{81} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{83} + ( -8 + 3 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 1 - 3 \beta_{1} + 2 \beta_{2} ) q^{87} + ( -3 + \beta_{1} - 7 \beta_{2} ) q^{89} + ( -3 - 5 \beta_{1} + 2 \beta_{2} ) q^{93} + ( 6 - \beta_{1} - 4 \beta_{2} ) q^{95} + ( -1 - 2 \beta_{1} - 10 \beta_{2} ) q^{97} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} - 6q^{5} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} - 6q^{5} + 2q^{7} + 3q^{9} + 5q^{11} - 6q^{15} - q^{17} - 7q^{19} + 2q^{21} + 11q^{25} + 3q^{27} - 2q^{29} - 16q^{31} + 5q^{33} - 4q^{35} - 22q^{37} - 11q^{41} + 15q^{43} - 6q^{45} + 7q^{47} - 15q^{49} - q^{51} - 17q^{53} - 3q^{55} - 7q^{57} - 6q^{59} - 13q^{61} + 2q^{63} + 11q^{67} - 6q^{73} + 11q^{75} + 15q^{77} - 3q^{79} + 3q^{81} + 12q^{83} - 19q^{85} - 2q^{87} - q^{89} - 16q^{93} + 21q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

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
−1.24698
0.445042
1.80194
0 1.00000 0 −3.69202 0 −0.801938 0 1.00000 0
1.2 0 1.00000 0 −3.35690 0 2.24698 0 1.00000 0
1.3 0 1.00000 0 1.04892 0 0.554958 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.cg 3
4.b odd 2 1 507.2.a.i 3
12.b even 2 1 1521.2.a.s 3
13.b even 2 1 8112.2.a.cp 3
52.b odd 2 1 507.2.a.l yes 3
52.f even 4 2 507.2.b.f 6
52.i odd 6 2 507.2.e.i 6
52.j odd 6 2 507.2.e.l 6
52.l even 12 4 507.2.j.i 12
156.h even 2 1 1521.2.a.n 3
156.l odd 4 2 1521.2.b.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 4.b odd 2 1
507.2.a.l yes 3 52.b odd 2 1
507.2.b.f 6 52.f even 4 2
507.2.e.i 6 52.i odd 6 2
507.2.e.l 6 52.j odd 6 2
507.2.j.i 12 52.l even 12 4
1521.2.a.n 3 156.h even 2 1
1521.2.a.s 3 12.b even 2 1
1521.2.b.k 6 156.l odd 4 2
8112.2.a.cg 3 1.a even 1 1 trivial
8112.2.a.cp 3 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8112))\):

\( T_{5}^{3} + 6 T_{5}^{2} + 5 T_{5} - 13 \)
\( T_{7}^{3} - 2 T_{7}^{2} - T_{7} + 1 \)
\( T_{11}^{3} - 5 T_{11}^{2} - 8 T_{11} + 41 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( -13 + 5 T + 6 T^{2} + T^{3} \)
$7$ \( 1 - T - 2 T^{2} + T^{3} \)
$11$ \( 41 - 8 T - 5 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 13 - 16 T + T^{2} + T^{3} \)
$19$ \( 7 + 14 T + 7 T^{2} + T^{3} \)
$23$ \( -91 - 49 T + T^{3} \)
$29$ \( -29 - 15 T + 2 T^{2} + T^{3} \)
$31$ \( -197 + 41 T + 16 T^{2} + T^{3} \)
$37$ \( 377 + 159 T + 22 T^{2} + T^{3} \)
$41$ \( -29 + 24 T + 11 T^{2} + T^{3} \)
$43$ \( -41 + 47 T - 15 T^{2} + T^{3} \)
$47$ \( -7 + 14 T - 7 T^{2} + T^{3} \)
$53$ \( -41 + 66 T + 17 T^{2} + T^{3} \)
$59$ \( -104 - 16 T + 6 T^{2} + T^{3} \)
$61$ \( -167 - 16 T + 13 T^{2} + T^{3} \)
$67$ \( -41 - 46 T - 11 T^{2} + T^{3} \)
$71$ \( 203 - 91 T + T^{3} \)
$73$ \( -923 - 135 T + 6 T^{2} + T^{3} \)
$79$ \( -27 - 18 T + 3 T^{2} + T^{3} \)
$83$ \( -43 + 41 T - 12 T^{2} + T^{3} \)
$89$ \( -113 - 100 T + T^{2} + T^{3} \)
$97$ \( 1637 - 281 T - 5 T^{2} + T^{3} \)
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