Properties

Label 8664.2.a.j
Level $8664$
Weight $2$
Character orbit 8664.a
Self dual yes
Analytic conductor $69.182$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8664.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.1823883112\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + q^{3} - 2q^{5} + q^{9} + O(q^{10}) \) \( q + q^{3} - 2q^{5} + q^{9} + 4q^{11} + 2q^{13} - 2q^{15} + 2q^{17} - 8q^{23} - q^{25} + q^{27} - 6q^{29} - 8q^{31} + 4q^{33} - 6q^{37} + 2q^{39} + 6q^{41} + 4q^{43} - 2q^{45} - 7q^{49} + 2q^{51} + 2q^{53} - 8q^{55} - 4q^{59} - 2q^{61} - 4q^{65} + 4q^{67} - 8q^{69} - 8q^{71} + 10q^{73} - q^{75} + 8q^{79} + q^{81} - 4q^{83} - 4q^{85} - 6q^{87} + 6q^{89} - 8q^{93} - 2q^{97} + 4q^{99} + O(q^{100}) \)

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
\(2\) \(1\)
\(3\) \(-1\)
\(19\) \(-1\)

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 8664.2.a.j 1
19.b odd 2 1 24.2.a.a 1
57.d even 2 1 72.2.a.a 1
76.d even 2 1 48.2.a.a 1
95.d odd 2 1 600.2.a.h 1
95.g even 4 2 600.2.f.e 2
133.c even 2 1 1176.2.a.i 1
133.o even 6 2 1176.2.q.a 2
133.r odd 6 2 1176.2.q.i 2
152.b even 2 1 192.2.a.b 1
152.g odd 2 1 192.2.a.d 1
171.l even 6 2 648.2.i.b 2
171.o odd 6 2 648.2.i.g 2
209.d even 2 1 2904.2.a.c 1
228.b odd 2 1 144.2.a.b 1
247.d odd 2 1 4056.2.a.i 1
247.i even 4 2 4056.2.c.e 2
285.b even 2 1 1800.2.a.m 1
285.j odd 4 2 1800.2.f.c 2
304.j odd 4 2 768.2.d.e 2
304.m even 4 2 768.2.d.d 2
323.c odd 2 1 6936.2.a.p 1
380.d even 2 1 1200.2.a.d 1
380.j odd 4 2 1200.2.f.b 2
399.h odd 2 1 3528.2.a.d 1
399.s odd 6 2 3528.2.s.y 2
399.w even 6 2 3528.2.s.j 2
456.l odd 2 1 576.2.a.b 1
456.p even 2 1 576.2.a.d 1
532.b odd 2 1 2352.2.a.i 1
532.t even 6 2 2352.2.q.l 2
532.bh odd 6 2 2352.2.q.r 2
627.b odd 2 1 8712.2.a.u 1
684.w even 6 2 1296.2.i.m 2
684.bh odd 6 2 1296.2.i.e 2
760.b odd 2 1 4800.2.a.q 1
760.p even 2 1 4800.2.a.cc 1
760.t even 4 2 4800.2.f.d 2
760.y odd 4 2 4800.2.f.bg 2
836.h odd 2 1 5808.2.a.s 1
912.r even 4 2 2304.2.d.i 2
912.w odd 4 2 2304.2.d.k 2
988.g even 2 1 8112.2.a.be 1
1064.f even 2 1 9408.2.a.h 1
1064.p odd 2 1 9408.2.a.cc 1
1140.p odd 2 1 3600.2.a.v 1
1140.w even 4 2 3600.2.f.r 2
1596.p even 2 1 7056.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 19.b odd 2 1
48.2.a.a 1 76.d even 2 1
72.2.a.a 1 57.d even 2 1
144.2.a.b 1 228.b odd 2 1
192.2.a.b 1 152.b even 2 1
192.2.a.d 1 152.g odd 2 1
576.2.a.b 1 456.l odd 2 1
576.2.a.d 1 456.p even 2 1
600.2.a.h 1 95.d odd 2 1
600.2.f.e 2 95.g even 4 2
648.2.i.b 2 171.l even 6 2
648.2.i.g 2 171.o odd 6 2
768.2.d.d 2 304.m even 4 2
768.2.d.e 2 304.j odd 4 2
1176.2.a.i 1 133.c even 2 1
1176.2.q.a 2 133.o even 6 2
1176.2.q.i 2 133.r odd 6 2
1200.2.a.d 1 380.d even 2 1
1200.2.f.b 2 380.j odd 4 2
1296.2.i.e 2 684.bh odd 6 2
1296.2.i.m 2 684.w even 6 2
1800.2.a.m 1 285.b even 2 1
1800.2.f.c 2 285.j odd 4 2
2304.2.d.i 2 912.r even 4 2
2304.2.d.k 2 912.w odd 4 2
2352.2.a.i 1 532.b odd 2 1
2352.2.q.l 2 532.t even 6 2
2352.2.q.r 2 532.bh odd 6 2
2904.2.a.c 1 209.d even 2 1
3528.2.a.d 1 399.h odd 2 1
3528.2.s.j 2 399.w even 6 2
3528.2.s.y 2 399.s odd 6 2
3600.2.a.v 1 1140.p odd 2 1
3600.2.f.r 2 1140.w even 4 2
4056.2.a.i 1 247.d odd 2 1
4056.2.c.e 2 247.i even 4 2
4800.2.a.q 1 760.b odd 2 1
4800.2.a.cc 1 760.p even 2 1
4800.2.f.d 2 760.t even 4 2
4800.2.f.bg 2 760.y odd 4 2
5808.2.a.s 1 836.h odd 2 1
6936.2.a.p 1 323.c odd 2 1
7056.2.a.q 1 1596.p even 2 1
8112.2.a.be 1 988.g even 2 1
8664.2.a.j 1 1.a even 1 1 trivial
8712.2.a.u 1 627.b odd 2 1
9408.2.a.h 1 1064.f even 2 1
9408.2.a.cc 1 1064.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8664))\):

\( T_{5} + 2 \)
\( T_{7} \)
\( T_{13} - 2 \)
\( T_{29} + 6 \)

Hecke characteristic polynomials

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