Properties

Label 63.3.m.b
Level $63$
Weight $3$
Character orbit 63.m
Analytic conductor $1.717$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 63.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.71662566547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 4 \zeta_{6} + 4) q^{4} + ( - 3 \zeta_{6} + 8) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 \zeta_{6} + 4) q^{4} + ( - 3 \zeta_{6} + 8) q^{7} + (14 \zeta_{6} - 7) q^{13} - 16 \zeta_{6} q^{16} + (21 \zeta_{6} - 42) q^{19} + (25 \zeta_{6} - 25) q^{25} + ( - 32 \zeta_{6} + 20) q^{28} + (35 \zeta_{6} + 35) q^{31} - 73 \zeta_{6} q^{37} + 61 q^{43} + ( - 39 \zeta_{6} + 55) q^{49} + (28 \zeta_{6} + 28) q^{52} + (56 \zeta_{6} - 112) q^{61} - 64 q^{64} + ( - 13 \zeta_{6} + 13) q^{67} + ( - 63 \zeta_{6} - 63) q^{73} + (168 \zeta_{6} - 84) q^{76} - 11 \zeta_{6} q^{79} + (91 \zeta_{6} - 14) q^{91} + ( - 224 \zeta_{6} + 112) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} + 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} + 13 q^{7} - 16 q^{16} - 63 q^{19} - 25 q^{25} + 8 q^{28} + 105 q^{31} - 73 q^{37} + 122 q^{43} + 71 q^{49} + 84 q^{52} - 168 q^{61} - 128 q^{64} + 13 q^{67} - 189 q^{73} - 11 q^{79} + 63 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 2.00000 3.46410i 0 0 6.50000 2.59808i 0 0 0
19.1 0 0 2.00000 + 3.46410i 0 0 6.50000 + 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.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.3.m.b 2
3.b odd 2 1 CM 63.3.m.b 2
4.b odd 2 1 1008.3.cg.d 2
7.b odd 2 1 441.3.m.c 2
7.c even 3 1 441.3.d.c 2
7.c even 3 1 441.3.m.c 2
7.d odd 6 1 inner 63.3.m.b 2
7.d odd 6 1 441.3.d.c 2
12.b even 2 1 1008.3.cg.d 2
21.c even 2 1 441.3.m.c 2
21.g even 6 1 inner 63.3.m.b 2
21.g even 6 1 441.3.d.c 2
21.h odd 6 1 441.3.d.c 2
21.h odd 6 1 441.3.m.c 2
28.f even 6 1 1008.3.cg.d 2
84.j odd 6 1 1008.3.cg.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.m.b 2 1.a even 1 1 trivial
63.3.m.b 2 3.b odd 2 1 CM
63.3.m.b 2 7.d odd 6 1 inner
63.3.m.b 2 21.g even 6 1 inner
441.3.d.c 2 7.c even 3 1
441.3.d.c 2 7.d odd 6 1
441.3.d.c 2 21.g even 6 1
441.3.d.c 2 21.h odd 6 1
441.3.m.c 2 7.b odd 2 1
441.3.m.c 2 7.c even 3 1
441.3.m.c 2 21.c even 2 1
441.3.m.c 2 21.h odd 6 1
1008.3.cg.d 2 4.b odd 2 1
1008.3.cg.d 2 12.b even 2 1
1008.3.cg.d 2 28.f even 6 1
1008.3.cg.d 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 13T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 147 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 63T + 1323 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 105T + 3675 \) Copy content Toggle raw display
$37$ \( T^{2} + 73T + 5329 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 61)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 168T + 9408 \) Copy content Toggle raw display
$67$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 189T + 11907 \) Copy content Toggle raw display
$79$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 37632 \) Copy content Toggle raw display
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