Properties

Label 63.6.a.c
Level $63$
Weight $6$
Character orbit 63.a
Self dual yes
Analytic conductor $10.104$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - 31q^{4} + 34q^{5} - 49q^{7} + 63q^{8} + O(q^{10}) \) \( q - q^{2} - 31q^{4} + 34q^{5} - 49q^{7} + 63q^{8} - 34q^{10} + 340q^{11} + 454q^{13} + 49q^{14} + 929q^{16} + 798q^{17} + 892q^{19} - 1054q^{20} - 340q^{22} + 3192q^{23} - 1969q^{25} - 454q^{26} + 1519q^{28} + 8242q^{29} - 2496q^{31} - 2945q^{32} - 798q^{34} - 1666q^{35} + 9798q^{37} - 892q^{38} + 2142q^{40} - 19834q^{41} - 17236q^{43} - 10540q^{44} - 3192q^{46} - 8928q^{47} + 2401q^{49} + 1969q^{50} - 14074q^{52} - 150q^{53} + 11560q^{55} - 3087q^{56} - 8242q^{58} + 42396q^{59} + 14758q^{61} + 2496q^{62} - 26783q^{64} + 15436q^{65} - 1676q^{67} - 24738q^{68} + 1666q^{70} - 14568q^{71} + 78378q^{73} - 9798q^{74} - 27652q^{76} - 16660q^{77} - 2272q^{79} + 31586q^{80} + 19834q^{82} + 37764q^{83} + 27132q^{85} + 17236q^{86} + 21420q^{88} + 117286q^{89} - 22246q^{91} - 98952q^{92} + 8928q^{94} + 30328q^{95} + 10002q^{97} - 2401q^{98} + O(q^{100}) \)

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 −31.0000 34.0000 0 −49.0000 63.0000 0 −34.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

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 63.6.a.c 1
3.b odd 2 1 21.6.a.b 1
4.b odd 2 1 1008.6.a.t 1
7.b odd 2 1 441.6.a.d 1
12.b even 2 1 336.6.a.l 1
15.d odd 2 1 525.6.a.c 1
15.e even 4 2 525.6.d.d 2
21.c even 2 1 147.6.a.e 1
21.g even 6 2 147.6.e.e 2
21.h odd 6 2 147.6.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.b 1 3.b odd 2 1
63.6.a.c 1 1.a even 1 1 trivial
147.6.a.e 1 21.c even 2 1
147.6.e.e 2 21.g even 6 2
147.6.e.f 2 21.h odd 6 2
336.6.a.l 1 12.b even 2 1
441.6.a.d 1 7.b odd 2 1
525.6.a.c 1 15.d odd 2 1
525.6.d.d 2 15.e even 4 2
1008.6.a.t 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(63))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( -34 + T \)
$7$ \( 49 + T \)
$11$ \( -340 + T \)
$13$ \( -454 + T \)
$17$ \( -798 + T \)
$19$ \( -892 + T \)
$23$ \( -3192 + T \)
$29$ \( -8242 + T \)
$31$ \( 2496 + T \)
$37$ \( -9798 + T \)
$41$ \( 19834 + T \)
$43$ \( 17236 + T \)
$47$ \( 8928 + T \)
$53$ \( 150 + T \)
$59$ \( -42396 + T \)
$61$ \( -14758 + T \)
$67$ \( 1676 + T \)
$71$ \( 14568 + T \)
$73$ \( -78378 + T \)
$79$ \( 2272 + T \)
$83$ \( -37764 + T \)
$89$ \( -117286 + T \)
$97$ \( -10002 + T \)
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