Properties

Label 63.2.e.a
Level $63$
Weight $2$
Character orbit 63.e
Analytic conductor $0.503$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{4} + ( 1 - 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 2 \zeta_{6} q^{4} + ( 1 - 3 \zeta_{6} ) q^{7} -7 q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} + ( 7 - 7 \zeta_{6} ) q^{19} + 5 \zeta_{6} q^{25} + ( 6 - 4 \zeta_{6} ) q^{28} + 7 \zeta_{6} q^{31} + ( 1 - \zeta_{6} ) q^{37} + 5 q^{43} + ( -8 + 3 \zeta_{6} ) q^{49} -14 \zeta_{6} q^{52} + ( -14 + 14 \zeta_{6} ) q^{61} -8 q^{64} -11 \zeta_{6} q^{67} + 7 \zeta_{6} q^{73} + 14 q^{76} + ( 13 - 13 \zeta_{6} ) q^{79} + ( -7 + 21 \zeta_{6} ) q^{91} + 14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} - q^{7} + O(q^{10}) \) \( 2q + 2q^{4} - q^{7} - 14q^{13} - 4q^{16} + 7q^{19} + 5q^{25} + 8q^{28} + 7q^{31} + q^{37} + 10q^{43} - 13q^{49} - 14q^{52} - 14q^{61} - 16q^{64} - 11q^{67} + 7q^{73} + 28q^{76} + 13q^{79} + 7q^{91} + 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 1.00000 + 1.73205i 0 0 −0.500000 2.59808i 0 0 0
46.1 0 0 1.00000 1.73205i 0 0 −0.500000 + 2.59808i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.2.e.a 2
3.b odd 2 1 CM 63.2.e.a 2
4.b odd 2 1 1008.2.s.j 2
7.b odd 2 1 441.2.e.c 2
7.c even 3 1 inner 63.2.e.a 2
7.c even 3 1 441.2.a.d 1
7.d odd 6 1 441.2.a.e 1
7.d odd 6 1 441.2.e.c 2
9.c even 3 1 567.2.g.d 2
9.c even 3 1 567.2.h.c 2
9.d odd 6 1 567.2.g.d 2
9.d odd 6 1 567.2.h.c 2
12.b even 2 1 1008.2.s.j 2
21.c even 2 1 441.2.e.c 2
21.g even 6 1 441.2.a.e 1
21.g even 6 1 441.2.e.c 2
21.h odd 6 1 inner 63.2.e.a 2
21.h odd 6 1 441.2.a.d 1
28.f even 6 1 7056.2.a.bf 1
28.g odd 6 1 1008.2.s.j 2
28.g odd 6 1 7056.2.a.y 1
63.g even 3 1 567.2.h.c 2
63.h even 3 1 567.2.g.d 2
63.j odd 6 1 567.2.g.d 2
63.n odd 6 1 567.2.h.c 2
84.j odd 6 1 7056.2.a.bf 1
84.n even 6 1 1008.2.s.j 2
84.n even 6 1 7056.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.e.a 2 1.a even 1 1 trivial
63.2.e.a 2 3.b odd 2 1 CM
63.2.e.a 2 7.c even 3 1 inner
63.2.e.a 2 21.h odd 6 1 inner
441.2.a.d 1 7.c even 3 1
441.2.a.d 1 21.h odd 6 1
441.2.a.e 1 7.d odd 6 1
441.2.a.e 1 21.g even 6 1
441.2.e.c 2 7.b odd 2 1
441.2.e.c 2 7.d odd 6 1
441.2.e.c 2 21.c even 2 1
441.2.e.c 2 21.g even 6 1
567.2.g.d 2 9.c even 3 1
567.2.g.d 2 9.d odd 6 1
567.2.g.d 2 63.h even 3 1
567.2.g.d 2 63.j odd 6 1
567.2.h.c 2 9.c even 3 1
567.2.h.c 2 9.d odd 6 1
567.2.h.c 2 63.g even 3 1
567.2.h.c 2 63.n odd 6 1
1008.2.s.j 2 4.b odd 2 1
1008.2.s.j 2 12.b even 2 1
1008.2.s.j 2 28.g odd 6 1
1008.2.s.j 2 84.n even 6 1
7056.2.a.y 1 28.g odd 6 1
7056.2.a.y 1 84.n even 6 1
7056.2.a.bf 1 28.f even 6 1
7056.2.a.bf 1 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(63, [\chi])\).