Properties

Label 8112.2.a.bo
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: \(x^{2} - x - 4\)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 + \beta ) q^{5} + ( -2 + \beta ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 + \beta ) q^{5} + ( -2 + \beta ) q^{7} + q^{9} -2 q^{11} + ( -1 - \beta ) q^{15} + ( 1 - \beta ) q^{17} + ( 2 + 2 \beta ) q^{19} + ( 2 - \beta ) q^{21} -2 q^{23} + 3 \beta q^{25} - q^{27} + ( -1 + 3 \beta ) q^{29} + \beta q^{31} + 2 q^{33} + 2 q^{35} + ( -5 - \beta ) q^{37} + ( -1 + \beta ) q^{41} + ( -2 - \beta ) q^{43} + ( 1 + \beta ) q^{45} + ( -2 + 4 \beta ) q^{47} + ( 1 - 3 \beta ) q^{49} + ( -1 + \beta ) q^{51} + ( 7 - 3 \beta ) q^{53} + ( -2 - 2 \beta ) q^{55} + ( -2 - 2 \beta ) q^{57} + ( -8 + 2 \beta ) q^{59} + ( 9 - 2 \beta ) q^{61} + ( -2 + \beta ) q^{63} + ( 2 + \beta ) q^{67} + 2 q^{69} + 14 q^{71} + ( 5 + 2 \beta ) q^{73} -3 \beta q^{75} + ( 4 - 2 \beta ) q^{77} + ( -8 + \beta ) q^{79} + q^{81} + ( -6 + 2 \beta ) q^{83} + ( -3 - \beta ) q^{85} + ( 1 - 3 \beta ) q^{87} + ( -10 + 2 \beta ) q^{89} -\beta q^{93} + ( 10 + 6 \beta ) q^{95} + ( 6 + \beta ) q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 3q^{5} - 3q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 3q^{5} - 3q^{7} + 2q^{9} - 4q^{11} - 3q^{15} + q^{17} + 6q^{19} + 3q^{21} - 4q^{23} + 3q^{25} - 2q^{27} + q^{29} + q^{31} + 4q^{33} + 4q^{35} - 11q^{37} - q^{41} - 5q^{43} + 3q^{45} - q^{49} - q^{51} + 11q^{53} - 6q^{55} - 6q^{57} - 14q^{59} + 16q^{61} - 3q^{63} + 5q^{67} + 4q^{69} + 28q^{71} + 12q^{73} - 3q^{75} + 6q^{77} - 15q^{79} + 2q^{81} - 10q^{83} - 7q^{85} - q^{87} - 18q^{89} - q^{93} + 26q^{95} + 13q^{97} - 4q^{99} + O(q^{100}) \)

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
−1.56155
2.56155
0 −1.00000 0 −0.561553 0 −3.56155 0 1.00000 0
1.2 0 −1.00000 0 3.56155 0 0.561553 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.bo 2
4.b odd 2 1 507.2.a.d 2
12.b even 2 1 1521.2.a.m 2
13.b even 2 1 8112.2.a.bk 2
13.e even 6 2 624.2.q.h 4
39.h odd 6 2 1872.2.t.r 4
52.b odd 2 1 507.2.a.g 2
52.f even 4 2 507.2.b.d 4
52.i odd 6 2 39.2.e.b 4
52.j odd 6 2 507.2.e.g 4
52.l even 12 4 507.2.j.g 8
156.h even 2 1 1521.2.a.g 2
156.l odd 4 2 1521.2.b.h 4
156.r even 6 2 117.2.g.c 4
260.w odd 6 2 975.2.i.k 4
260.bg 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.i odd 6 2
117.2.g.c 4 156.r even 6 2
507.2.a.d 2 4.b odd 2 1
507.2.a.g 2 52.b odd 2 1
507.2.b.d 4 52.f even 4 2
507.2.e.g 4 52.j odd 6 2
507.2.j.g 8 52.l even 12 4
624.2.q.h 4 13.e even 6 2
975.2.i.k 4 260.w odd 6 2
975.2.bb.i 8 260.bg even 12 4
1521.2.a.g 2 156.h even 2 1
1521.2.a.m 2 12.b even 2 1
1521.2.b.h 4 156.l odd 4 2
1872.2.t.r 4 39.h odd 6 2
8112.2.a.bk 2 13.b even 2 1
8112.2.a.bo 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8112))\):

\( T_{5}^{2} - 3 T_{5} - 2 \)
\( T_{7}^{2} + 3 T_{7} - 2 \)
\( T_{11} + 2 \)

Hecke characteristic polynomials

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