Properties

Label 8112.2.a.be
Level $8112$
Weight $2$
Character orbit 8112.a
Self dual yes
Analytic conductor $64.775$
Analytic rank $0$
Dimension $1$
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
0 1.00000 0 2.00000 0 0 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.be 1
4.b odd 2 1 4056.2.a.i 1
13.b even 2 1 48.2.a.a 1
39.d odd 2 1 144.2.a.b 1
52.b odd 2 1 24.2.a.a 1
52.f even 4 2 4056.2.c.e 2
65.d even 2 1 1200.2.a.d 1
65.h odd 4 2 1200.2.f.b 2
91.b odd 2 1 2352.2.a.i 1
91.r even 6 2 2352.2.q.l 2
91.s odd 6 2 2352.2.q.r 2
104.e even 2 1 192.2.a.b 1
104.h odd 2 1 192.2.a.d 1
117.n odd 6 2 1296.2.i.e 2
117.t even 6 2 1296.2.i.m 2
143.d odd 2 1 5808.2.a.s 1
156.h even 2 1 72.2.a.a 1
195.e odd 2 1 3600.2.a.v 1
195.s even 4 2 3600.2.f.r 2
208.o odd 4 2 768.2.d.e 2
208.p even 4 2 768.2.d.d 2
260.g odd 2 1 600.2.a.h 1
260.p even 4 2 600.2.f.e 2
273.g even 2 1 7056.2.a.q 1
312.b odd 2 1 576.2.a.b 1
312.h even 2 1 576.2.a.d 1
364.h even 2 1 1176.2.a.i 1
364.x even 6 2 1176.2.q.a 2
364.bl odd 6 2 1176.2.q.i 2
468.x even 6 2 648.2.i.b 2
468.bg odd 6 2 648.2.i.g 2
520.b odd 2 1 4800.2.a.q 1
520.p even 2 1 4800.2.a.cc 1
520.bc even 4 2 4800.2.f.d 2
520.bg odd 4 2 4800.2.f.bg 2
572.b even 2 1 2904.2.a.c 1
624.v even 4 2 2304.2.d.i 2
624.bi odd 4 2 2304.2.d.k 2
728.b even 2 1 9408.2.a.h 1
728.l odd 2 1 9408.2.a.cc 1
780.d even 2 1 1800.2.a.m 1
780.w odd 4 2 1800.2.f.c 2
884.h odd 2 1 6936.2.a.p 1
988.g even 2 1 8664.2.a.j 1
1092.d odd 2 1 3528.2.a.d 1
1092.by even 6 2 3528.2.s.j 2
1092.ct odd 6 2 3528.2.s.y 2
1716.m odd 2 1 8712.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 52.b odd 2 1
48.2.a.a 1 13.b even 2 1
72.2.a.a 1 156.h even 2 1
144.2.a.b 1 39.d odd 2 1
192.2.a.b 1 104.e even 2 1
192.2.a.d 1 104.h odd 2 1
576.2.a.b 1 312.b odd 2 1
576.2.a.d 1 312.h even 2 1
600.2.a.h 1 260.g odd 2 1
600.2.f.e 2 260.p even 4 2
648.2.i.b 2 468.x even 6 2
648.2.i.g 2 468.bg odd 6 2
768.2.d.d 2 208.p even 4 2
768.2.d.e 2 208.o odd 4 2
1176.2.a.i 1 364.h even 2 1
1176.2.q.a 2 364.x even 6 2
1176.2.q.i 2 364.bl odd 6 2
1200.2.a.d 1 65.d even 2 1
1200.2.f.b 2 65.h odd 4 2
1296.2.i.e 2 117.n odd 6 2
1296.2.i.m 2 117.t even 6 2
1800.2.a.m 1 780.d even 2 1
1800.2.f.c 2 780.w odd 4 2
2304.2.d.i 2 624.v even 4 2
2304.2.d.k 2 624.bi odd 4 2
2352.2.a.i 1 91.b odd 2 1
2352.2.q.l 2 91.r even 6 2
2352.2.q.r 2 91.s odd 6 2
2904.2.a.c 1 572.b even 2 1
3528.2.a.d 1 1092.d odd 2 1
3528.2.s.j 2 1092.by even 6 2
3528.2.s.y 2 1092.ct odd 6 2
3600.2.a.v 1 195.e odd 2 1
3600.2.f.r 2 195.s even 4 2
4056.2.a.i 1 4.b odd 2 1
4056.2.c.e 2 52.f even 4 2
4800.2.a.q 1 520.b odd 2 1
4800.2.a.cc 1 520.p even 2 1
4800.2.f.d 2 520.bc even 4 2
4800.2.f.bg 2 520.bg odd 4 2
5808.2.a.s 1 143.d odd 2 1
6936.2.a.p 1 884.h odd 2 1
7056.2.a.q 1 273.g even 2 1
8112.2.a.be 1 1.a even 1 1 trivial
8664.2.a.j 1 988.g even 2 1
8712.2.a.u 1 1716.m odd 2 1
9408.2.a.h 1 728.b even 2 1
9408.2.a.cc 1 728.l odd 2 1

Hecke kernels

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

\( T_{5} - 2 \)
\( T_{7} \)
\( T_{11} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T \)
$5$ \( 1 - 2 T + 5 T^{2} \)
$7$ \( 1 + 7 T^{2} \)
$11$ \( 1 - 4 T + 11 T^{2} \)
$13$ 1
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 - 8 T + 23 T^{2} \)
$29$ \( 1 - 6 T + 29 T^{2} \)
$31$ \( 1 - 8 T + 31 T^{2} \)
$37$ \( 1 + 6 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 + 2 T + 53 T^{2} \)
$59$ \( 1 - 4 T + 59 T^{2} \)
$61$ \( 1 + 2 T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 - 8 T + 71 T^{2} \)
$73$ \( 1 + 10 T + 73 T^{2} \)
$79$ \( 1 - 8 T + 79 T^{2} \)
$83$ \( 1 + 4 T + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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