Properties

Label 6272.2.a.a
Level $6272$
Weight $2$
Character orbit 6272.a
Self dual yes
Analytic conductor $50.082$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6272 = 2^{7} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6272.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.0821721477\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{3} - 2q^{5} + q^{9} + O(q^{10}) \) \( q - 2q^{3} - 2q^{5} + q^{9} - 2q^{11} - 2q^{13} + 4q^{15} + 2q^{17} - 2q^{19} + 4q^{23} - q^{25} + 4q^{27} - 6q^{29} + 4q^{33} + 10q^{37} + 4q^{39} + 6q^{41} + 6q^{43} - 2q^{45} + 8q^{47} - 4q^{51} - 6q^{53} + 4q^{55} + 4q^{57} - 14q^{59} - 2q^{61} + 4q^{65} + 10q^{67} - 8q^{69} + 12q^{71} - 14q^{73} + 2q^{75} - 8q^{79} - 11q^{81} + 6q^{83} - 4q^{85} + 12q^{87} + 2q^{89} + 4q^{95} + 2q^{97} - 2q^{99} + O(q^{100}) \)

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 −2.00000 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
\(2\) \(-1\)
\(7\) \(-1\)

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 6272.2.a.a 1
4.b odd 2 1 6272.2.a.g 1
7.b odd 2 1 128.2.a.d yes 1
8.b even 2 1 6272.2.a.h 1
8.d odd 2 1 6272.2.a.b 1
21.c even 2 1 1152.2.a.c 1
28.d even 2 1 128.2.a.b yes 1
35.c odd 2 1 3200.2.a.h 1
35.f even 4 2 3200.2.c.e 2
56.e even 2 1 128.2.a.c yes 1
56.h odd 2 1 128.2.a.a 1
84.h odd 2 1 1152.2.a.h 1
112.j even 4 2 256.2.b.a 2
112.l odd 4 2 256.2.b.c 2
140.c even 2 1 3200.2.a.u 1
140.j odd 4 2 3200.2.c.k 2
168.e odd 2 1 1152.2.a.r 1
168.i even 2 1 1152.2.a.m 1
224.v odd 8 4 1024.2.e.i 4
224.x even 8 4 1024.2.e.m 4
280.c odd 2 1 3200.2.a.x 1
280.n even 2 1 3200.2.a.e 1
280.s even 4 2 3200.2.c.l 2
280.y odd 4 2 3200.2.c.f 2
336.v odd 4 2 2304.2.d.b 2
336.y even 4 2 2304.2.d.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 56.h odd 2 1
128.2.a.b yes 1 28.d even 2 1
128.2.a.c yes 1 56.e even 2 1
128.2.a.d yes 1 7.b odd 2 1
256.2.b.a 2 112.j even 4 2
256.2.b.c 2 112.l odd 4 2
1024.2.e.i 4 224.v odd 8 4
1024.2.e.m 4 224.x even 8 4
1152.2.a.c 1 21.c even 2 1
1152.2.a.h 1 84.h odd 2 1
1152.2.a.m 1 168.i even 2 1
1152.2.a.r 1 168.e odd 2 1
2304.2.d.b 2 336.v odd 4 2
2304.2.d.r 2 336.y even 4 2
3200.2.a.e 1 280.n even 2 1
3200.2.a.h 1 35.c odd 2 1
3200.2.a.u 1 140.c even 2 1
3200.2.a.x 1 280.c odd 2 1
3200.2.c.e 2 35.f even 4 2
3200.2.c.f 2 280.y odd 4 2
3200.2.c.k 2 140.j odd 4 2
3200.2.c.l 2 280.s even 4 2
6272.2.a.a 1 1.a even 1 1 trivial
6272.2.a.b 1 8.d odd 2 1
6272.2.a.g 1 4.b odd 2 1
6272.2.a.h 1 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6272))\):

\( T_{3} + 2 \)
\( T_{5} + 2 \)
\( T_{11} + 2 \)
\( T_{13} + 2 \)
\( T_{23} - 4 \)
\( T_{29} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 2 + T \)
$5$ \( 2 + T \)
$7$ \( T \)
$11$ \( 2 + T \)
$13$ \( 2 + T \)
$17$ \( -2 + T \)
$19$ \( 2 + T \)
$23$ \( -4 + T \)
$29$ \( 6 + T \)
$31$ \( T \)
$37$ \( -10 + T \)
$41$ \( -6 + T \)
$43$ \( -6 + T \)
$47$ \( -8 + T \)
$53$ \( 6 + T \)
$59$ \( 14 + T \)
$61$ \( 2 + T \)
$67$ \( -10 + T \)
$71$ \( -12 + T \)
$73$ \( 14 + T \)
$79$ \( 8 + T \)
$83$ \( -6 + T \)
$89$ \( -2 + T \)
$97$ \( -2 + T \)
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