Properties

Label 8112.2.a.by
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.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 - \beta_{1} ) q^{5} + ( 3 + \beta_{1} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 - \beta_{1} ) q^{5} + ( 3 + \beta_{1} ) q^{7} + q^{9} + ( 2 - 4 \beta_{1} + 3 \beta_{2} ) q^{11} + ( 1 + \beta_{1} ) q^{15} + ( -3 + \beta_{1} - \beta_{2} ) q^{17} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{19} + ( -3 - \beta_{1} ) q^{21} + ( -3 + 5 \beta_{1} - 2 \beta_{2} ) q^{23} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{25} - q^{27} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{29} + ( 5 - 2 \beta_{1} + 5 \beta_{2} ) q^{31} + ( -2 + 4 \beta_{1} - 3 \beta_{2} ) q^{33} + ( -5 - 4 \beta_{1} - \beta_{2} ) q^{35} + ( -4 - \beta_{1} + \beta_{2} ) q^{37} + \beta_{2} q^{41} + ( -1 + 2 \beta_{1} - 4 \beta_{2} ) q^{43} + ( -1 - \beta_{1} ) q^{45} + ( -7 + 3 \beta_{1} - 9 \beta_{2} ) q^{47} + ( 4 + 6 \beta_{1} + \beta_{2} ) q^{49} + ( 3 - \beta_{1} + \beta_{2} ) q^{51} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{53} + ( 3 + 2 \beta_{1} - 2 \beta_{2} ) q^{55} + ( -3 + \beta_{1} + 3 \beta_{2} ) q^{57} + ( -8 + 6 \beta_{1} - 4 \beta_{2} ) q^{59} + ( -7 + 3 \beta_{1} - 5 \beta_{2} ) q^{61} + ( 3 + \beta_{1} ) q^{63} + ( 4 - 3 \beta_{1} + 4 \beta_{2} ) q^{67} + ( 3 - 5 \beta_{1} + 2 \beta_{2} ) q^{69} + ( -1 + 2 \beta_{1} + 5 \beta_{2} ) q^{71} + ( -7 + \beta_{1} - 2 \beta_{2} ) q^{73} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{75} + ( 1 - 10 \beta_{1} + 8 \beta_{2} ) q^{77} + ( 5 \beta_{1} - 4 \beta_{2} ) q^{79} + q^{81} + ( -6 + 3 \beta_{1} + \beta_{2} ) q^{83} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{85} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{87} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{89} + ( -5 + 2 \beta_{1} - 5 \beta_{2} ) q^{93} + ( 2 - 2 \beta_{1} + 7 \beta_{2} ) q^{95} + ( 3 - 10 \beta_{1} + 4 \beta_{2} ) q^{97} + ( 2 - 4 \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 4q^{5} + 10q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 4q^{5} + 10q^{7} + 3q^{9} - q^{11} + 4q^{15} - 7q^{17} + 11q^{19} - 10q^{21} - 2q^{23} - 5q^{25} - 3q^{27} - 8q^{29} + 8q^{31} + q^{33} - 18q^{35} - 14q^{37} - q^{41} + 3q^{43} - 4q^{45} - 9q^{47} + 17q^{49} + 7q^{51} - 13q^{53} + 13q^{55} - 11q^{57} - 14q^{59} - 13q^{61} + 10q^{63} + 5q^{67} + 2q^{69} - 6q^{71} - 18q^{73} + 5q^{75} - 15q^{77} + 9q^{79} + 3q^{81} - 16q^{83} + 7q^{85} + 8q^{87} + 5q^{89} - 8q^{93} - 3q^{95} - 5q^{97} - q^{99} + O(q^{100}) \)

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
1.80194
0.445042
−1.24698
0 −1.00000 0 −2.80194 0 4.80194 0 1.00000 0
1.2 0 −1.00000 0 −1.44504 0 3.44504 0 1.00000 0
1.3 0 −1.00000 0 0.246980 0 1.75302 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.by 3
4.b odd 2 1 507.2.a.j 3
12.b even 2 1 1521.2.a.q 3
13.b even 2 1 8112.2.a.cf 3
52.b odd 2 1 507.2.a.k yes 3
52.f even 4 2 507.2.b.g 6
52.i odd 6 2 507.2.e.j 6
52.j odd 6 2 507.2.e.k 6
52.l even 12 4 507.2.j.h 12
156.h even 2 1 1521.2.a.p 3
156.l odd 4 2 1521.2.b.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 4.b odd 2 1
507.2.a.k yes 3 52.b odd 2 1
507.2.b.g 6 52.f even 4 2
507.2.e.j 6 52.i odd 6 2
507.2.e.k 6 52.j odd 6 2
507.2.j.h 12 52.l even 12 4
1521.2.a.p 3 156.h even 2 1
1521.2.a.q 3 12.b even 2 1
1521.2.b.m 6 156.l odd 4 2
8112.2.a.by 3 1.a even 1 1 trivial
8112.2.a.cf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8112))\):

\( T_{5}^{3} + 4 T_{5}^{2} + 3 T_{5} - 1 \)
\( T_{7}^{3} - 10 T_{7}^{2} + 31 T_{7} - 29 \)
\( T_{11}^{3} + T_{11}^{2} - 30 T_{11} - 43 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -1 + 3 T + 4 T^{2} + T^{3} \)
$7$ \( -29 + 31 T - 10 T^{2} + T^{3} \)
$11$ \( -43 - 30 T + T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 7 + 14 T + 7 T^{2} + T^{3} \)
$19$ \( 113 + 10 T - 11 T^{2} + T^{3} \)
$23$ \( 83 - 43 T + 2 T^{2} + T^{3} \)
$29$ \( -43 + 5 T + 8 T^{2} + T^{3} \)
$31$ \( 197 - 23 T - 8 T^{2} + T^{3} \)
$37$ \( 91 + 63 T + 14 T^{2} + T^{3} \)
$41$ \( -1 - 2 T + T^{2} + T^{3} \)
$43$ \( -29 - 25 T - 3 T^{2} + T^{3} \)
$47$ \( -911 - 120 T + 9 T^{2} + T^{3} \)
$53$ \( 29 + 40 T + 13 T^{2} + T^{3} \)
$59$ \( -56 + 14 T^{2} + T^{3} \)
$61$ \( -223 + 12 T + 13 T^{2} + T^{3} \)
$67$ \( 97 - 22 T - 5 T^{2} + T^{3} \)
$71$ \( -461 - 79 T + 6 T^{2} + T^{3} \)
$73$ \( 167 + 101 T + 18 T^{2} + T^{3} \)
$79$ \( 169 - 22 T - 9 T^{2} + T^{3} \)
$83$ \( -43 + 55 T + 16 T^{2} + T^{3} \)
$89$ \( -1 - 8 T - 5 T^{2} + T^{3} \)
$97$ \( -1189 - 169 T + 5 T^{2} + T^{3} \)
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