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Properties

Label 63.2.a.a
Level 63
Weight 2
Character orbit 63.a
Self dual yes
Analytic conductor 0.503
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.503057532734\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + 2q^{5} - q^{7} - 3q^{8} + O(q^{10}) \) \( q + q^{2} - q^{4} + 2q^{5} - q^{7} - 3q^{8} + 2q^{10} - 4q^{11} - 2q^{13} - q^{14} - q^{16} + 6q^{17} + 4q^{19} - 2q^{20} - 4q^{22} - q^{25} - 2q^{26} + q^{28} + 2q^{29} + 5q^{32} + 6q^{34} - 2q^{35} + 6q^{37} + 4q^{38} - 6q^{40} - 2q^{41} - 4q^{43} + 4q^{44} + q^{49} - q^{50} + 2q^{52} - 6q^{53} - 8q^{55} + 3q^{56} + 2q^{58} - 12q^{59} - 2q^{61} + 7q^{64} - 4q^{65} + 4q^{67} - 6q^{68} - 2q^{70} - 6q^{73} + 6q^{74} - 4q^{76} + 4q^{77} - 16q^{79} - 2q^{80} - 2q^{82} + 12q^{83} + 12q^{85} - 4q^{86} + 12q^{88} + 14q^{89} + 2q^{91} + 8q^{95} + 18q^{97} + q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 2.00000 0 −1.00000 −3.00000 0 2.00000
\(n\): 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 63.2.a.a 1
3.b odd 2 1 21.2.a.a ✓ 1
4.b odd 2 1 1008.2.a.l 1
5.b even 2 1 1575.2.a.c 1
5.c odd 4 2 1575.2.d.a 2
7.b odd 2 1 441.2.a.f 1
7.c even 3 2 441.2.e.a 2
7.d odd 6 2 441.2.e.b 2
8.b even 2 1 4032.2.a.h 1
8.d odd 2 1 4032.2.a.k 1
9.c even 3 2 567.2.f.b 2
9.d odd 6 2 567.2.f.g 2
11.b odd 2 1 7623.2.a.g 1
12.b even 2 1 336.2.a.a 1
15.d odd 2 1 525.2.a.d 1
15.e even 4 2 525.2.d.a 2
21.c even 2 1 147.2.a.a 1
21.g even 6 2 147.2.e.c 2
21.h odd 6 2 147.2.e.b 2
24.f even 2 1 1344.2.a.s 1
24.h odd 2 1 1344.2.a.g 1
28.d even 2 1 7056.2.a.p 1
33.d even 2 1 2541.2.a.j 1
39.d odd 2 1 3549.2.a.c 1
48.i odd 4 2 5376.2.c.r 2
48.k even 4 2 5376.2.c.l 2
51.c odd 2 1 6069.2.a.b 1
57.d even 2 1 7581.2.a.d 1
60.h even 2 1 8400.2.a.bn 1
84.h odd 2 1 2352.2.a.v 1
84.j odd 6 2 2352.2.q.e 2
84.n even 6 2 2352.2.q.x 2
105.g even 2 1 3675.2.a.n 1
168.e odd 2 1 9408.2.a.m 1
168.i even 2 1 9408.2.a.bv 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a ✓ 1 3.b odd 2 1
63.2.a.a 1 1.a even 1 1 trivial
147.2.a.a 1 21.c even 2 1
147.2.e.b 2 21.h odd 6 2
147.2.e.c 2 21.g even 6 2
336.2.a.a 1 12.b even 2 1
441.2.a.f 1 7.b odd 2 1
441.2.e.a 2 7.c even 3 2
441.2.e.b 2 7.d odd 6 2
525.2.a.d 1 15.d odd 2 1
525.2.d.a 2 15.e even 4 2
567.2.f.b 2 9.c even 3 2
567.2.f.g 2 9.d odd 6 2
1008.2.a.l 1 4.b odd 2 1
1344.2.a.g 1 24.h odd 2 1
1344.2.a.s 1 24.f even 2 1
1575.2.a.c 1 5.b even 2 1
1575.2.d.a 2 5.c odd 4 2
2352.2.a.v 1 84.h odd 2 1
2352.2.q.e 2 84.j odd 6 2
2352.2.q.x 2 84.n even 6 2
2541.2.a.j 1 33.d even 2 1
3549.2.a.c 1 39.d odd 2 1
3675.2.a.n 1 105.g even 2 1
4032.2.a.h 1 8.b even 2 1
4032.2.a.k 1 8.d odd 2 1
5376.2.c.l 2 48.k even 4 2
5376.2.c.r 2 48.i odd 4 2
6069.2.a.b 1 51.c odd 2 1
7056.2.a.p 1 28.d even 2 1
7581.2.a.d 1 57.d even 2 1
7623.2.a.g 1 11.b odd 2 1
8400.2.a.bn 1 60.h even 2 1
9408.2.a.m 1 168.e odd 2 1
9408.2.a.bv 1 168.i even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 2 T^{2} \)
$3$ 1
$5$ \( 1 - 2 T + 5 T^{2} \)
$7$ \( 1 + T \)
$11$ \( 1 + 4 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 - 6 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 + 2 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 + 6 T + 53 T^{2} \)
$59$ \( 1 + 12 T + 59 T^{2} \)
$61$ \( 1 + 2 T + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 6 T + 73 T^{2} \)
$79$ \( 1 + 16 T + 79 T^{2} \)
$83$ \( 1 - 12 T + 83 T^{2} \)
$89$ \( 1 - 14 T + 89 T^{2} \)
$97$ \( 1 - 18 T + 97 T^{2} \)
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