Properties

Label 6498.2.a.f
Level $6498$
Weight $2$
Character orbit 6498.a
Self dual yes
Analytic conductor $51.887$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6498.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.8867912334\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{7} - q^{8} + 6q^{11} - 5q^{13} + q^{14} + q^{16} - 3q^{17} - 6q^{22} - 3q^{23} - 5q^{25} + 5q^{26} - q^{28} + 9q^{29} + 4q^{31} - q^{32} + 3q^{34} - 2q^{37} + 8q^{43} + 6q^{44} + 3q^{46} - 6q^{49} + 5q^{50} - 5q^{52} - 3q^{53} + q^{56} - 9q^{58} + 9q^{59} - 10q^{61} - 4q^{62} + q^{64} - 5q^{67} - 3q^{68} - 6q^{71} - 7q^{73} + 2q^{74} - 6q^{77} + 10q^{79} + 6q^{83} - 8q^{86} - 6q^{88} - 12q^{89} + 5q^{91} - 3q^{92} + 10q^{97} + 6q^{98} + O(q^{100}) \)

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
−1.00000 0 1.00000 0 0 −1.00000 −1.00000 0 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 6498.2.a.f 1
3.b odd 2 1 722.2.a.e 1
12.b even 2 1 5776.2.a.m 1
19.b odd 2 1 342.2.a.e 1
57.d even 2 1 38.2.a.a 1
57.f even 6 2 722.2.c.e 2
57.h odd 6 2 722.2.c.c 2
57.j even 18 6 722.2.e.f 6
57.l odd 18 6 722.2.e.e 6
76.d even 2 1 2736.2.a.n 1
95.d odd 2 1 8550.2.a.m 1
228.b odd 2 1 304.2.a.c 1
285.b even 2 1 950.2.a.d 1
285.j odd 4 2 950.2.b.b 2
399.h odd 2 1 1862.2.a.b 1
456.l odd 2 1 1216.2.a.m 1
456.p even 2 1 1216.2.a.e 1
627.b odd 2 1 4598.2.a.p 1
741.d even 2 1 6422.2.a.h 1
1140.p odd 2 1 7600.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.a 1 57.d even 2 1
304.2.a.c 1 228.b odd 2 1
342.2.a.e 1 19.b odd 2 1
722.2.a.e 1 3.b odd 2 1
722.2.c.c 2 57.h odd 6 2
722.2.c.e 2 57.f even 6 2
722.2.e.e 6 57.l odd 18 6
722.2.e.f 6 57.j even 18 6
950.2.a.d 1 285.b even 2 1
950.2.b.b 2 285.j odd 4 2
1216.2.a.e 1 456.p even 2 1
1216.2.a.m 1 456.l odd 2 1
1862.2.a.b 1 399.h odd 2 1
2736.2.a.n 1 76.d even 2 1
4598.2.a.p 1 627.b odd 2 1
5776.2.a.m 1 12.b even 2 1
6422.2.a.h 1 741.d even 2 1
6498.2.a.f 1 1.a even 1 1 trivial
7600.2.a.n 1 1140.p odd 2 1
8550.2.a.m 1 95.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6498))\):

\( T_{5} \)
\( T_{7} + 1 \)
\( T_{11} - 6 \)
\( T_{13} + 5 \)
\( T_{29} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 1 + T \)
$11$ \( -6 + T \)
$13$ \( 5 + T \)
$17$ \( 3 + T \)
$19$ \( T \)
$23$ \( 3 + T \)
$29$ \( -9 + T \)
$31$ \( -4 + T \)
$37$ \( 2 + T \)
$41$ \( T \)
$43$ \( -8 + T \)
$47$ \( T \)
$53$ \( 3 + T \)
$59$ \( -9 + T \)
$61$ \( 10 + T \)
$67$ \( 5 + T \)
$71$ \( 6 + T \)
$73$ \( 7 + T \)
$79$ \( -10 + T \)
$83$ \( -6 + T \)
$89$ \( 12 + T \)
$97$ \( -10 + T \)
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