Properties

Label 8112.2.a.bk
Level $8112$
Weight $2$
Character orbit 8112.a
Self dual yes
Analytic conductor $64.775$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta - 1) q^{5} + ( - \beta + 2) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + ( - \beta - 1) q^{5} + ( - \beta + 2) q^{7} + q^{9} + 2 q^{11} + (\beta + 1) q^{15} + ( - \beta + 1) q^{17} + ( - 2 \beta - 2) q^{19} + (\beta - 2) q^{21} - 2 q^{23} + 3 \beta q^{25} - q^{27} + (3 \beta - 1) q^{29} - \beta q^{31} - 2 q^{33} + 2 q^{35} + (\beta + 5) q^{37} + ( - \beta + 1) q^{41} + ( - \beta - 2) q^{43} + ( - \beta - 1) q^{45} + ( - 4 \beta + 2) q^{47} + ( - 3 \beta + 1) q^{49} + (\beta - 1) q^{51} + ( - 3 \beta + 7) q^{53} + ( - 2 \beta - 2) q^{55} + (2 \beta + 2) q^{57} + ( - 2 \beta + 8) q^{59} + ( - 2 \beta + 9) q^{61} + ( - \beta + 2) q^{63} + ( - \beta - 2) q^{67} + 2 q^{69} - 14 q^{71} + ( - 2 \beta - 5) q^{73} - 3 \beta q^{75} + ( - 2 \beta + 4) q^{77} + (\beta - 8) q^{79} + q^{81} + ( - 2 \beta + 6) q^{83} + (\beta + 3) q^{85} + ( - 3 \beta + 1) q^{87} + ( - 2 \beta + 10) q^{89} + \beta q^{93} + (6 \beta + 10) q^{95} + ( - \beta - 6) q^{97} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 2 q^{9} + 4 q^{11} + 3 q^{15} + q^{17} - 6 q^{19} - 3 q^{21} - 4 q^{23} + 3 q^{25} - 2 q^{27} + q^{29} - q^{31} - 4 q^{33} + 4 q^{35} + 11 q^{37} + q^{41} - 5 q^{43} - 3 q^{45} - q^{49} - q^{51} + 11 q^{53} - 6 q^{55} + 6 q^{57} + 14 q^{59} + 16 q^{61} + 3 q^{63} - 5 q^{67} + 4 q^{69} - 28 q^{71} - 12 q^{73} - 3 q^{75} + 6 q^{77} - 15 q^{79} + 2 q^{81} + 10 q^{83} + 7 q^{85} - q^{87} + 18 q^{89} + q^{93} + 26 q^{95} - 13 q^{97} + 4 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
0 −1.00000 0 −3.56155 0 −0.561553 0 1.00000 0
1.2 0 −1.00000 0 0.561553 0 3.56155 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.bk 2
4.b odd 2 1 507.2.a.g 2
12.b even 2 1 1521.2.a.g 2
13.b even 2 1 8112.2.a.bo 2
13.c even 3 2 624.2.q.h 4
39.i odd 6 2 1872.2.t.r 4
52.b odd 2 1 507.2.a.d 2
52.f even 4 2 507.2.b.d 4
52.i odd 6 2 507.2.e.g 4
52.j odd 6 2 39.2.e.b 4
52.l even 12 4 507.2.j.g 8
156.h even 2 1 1521.2.a.m 2
156.l odd 4 2 1521.2.b.h 4
156.p even 6 2 117.2.g.c 4
260.v odd 6 2 975.2.i.k 4
260.bj even 12 4 975.2.bb.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 52.j odd 6 2
117.2.g.c 4 156.p even 6 2
507.2.a.d 2 52.b odd 2 1
507.2.a.g 2 4.b odd 2 1
507.2.b.d 4 52.f even 4 2
507.2.e.g 4 52.i odd 6 2
507.2.j.g 8 52.l even 12 4
624.2.q.h 4 13.c even 3 2
975.2.i.k 4 260.v odd 6 2
975.2.bb.i 8 260.bj even 12 4
1521.2.a.g 2 12.b even 2 1
1521.2.a.m 2 156.h even 2 1
1521.2.b.h 4 156.l odd 4 2
1872.2.t.r 4 39.i odd 6 2
8112.2.a.bk 2 1.a even 1 1 trivial
8112.2.a.bo 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8112))\):

\( T_{5}^{2} + 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$23$ \( (T + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 26 \) Copy content Toggle raw display
$41$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$47$ \( T^{2} - 68 \) Copy content Toggle raw display
$53$ \( T^{2} - 11T - 8 \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$61$ \( T^{2} - 16T + 47 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$71$ \( (T + 14)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 19 \) Copy content Toggle raw display
$79$ \( T^{2} + 15T + 52 \) Copy content Toggle raw display
$83$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 64 \) Copy content Toggle raw display
$97$ \( T^{2} + 13T + 38 \) Copy content Toggle raw display
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