Properties

Label 63.6.e.b
Level $63$
Weight $6$
Character orbit 63.e
Analytic conductor $10.104$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.1041806482\)
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: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + 32 \zeta_{6} q^{4} + (87 \zeta_{6} - 149) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 32 \zeta_{6} q^{4} + (87 \zeta_{6} - 149) q^{7} - 427 q^{13} + (1024 \zeta_{6} - 1024) q^{16} + (3143 \zeta_{6} - 3143) q^{19} + 3125 \zeta_{6} q^{25} + ( - 1984 \zeta_{6} - 2784) q^{28} - 2723 \zeta_{6} q^{31} + ( - 6661 \zeta_{6} + 6661) q^{37} + 22475 q^{43} + ( - 18357 \zeta_{6} + 14632) q^{49} - 13664 \zeta_{6} q^{52} + ( - 38626 \zeta_{6} + 38626) q^{61} - 32768 q^{64} + 37939 \zeta_{6} q^{67} + 78127 \zeta_{6} q^{73} - 100576 q^{76} + (90857 \zeta_{6} - 90857) q^{79} + ( - 37149 \zeta_{6} + 63623) q^{91} - 134386 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32 q^{4} - 211 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 32 q^{4} - 211 q^{7} - 854 q^{13} - 1024 q^{16} - 3143 q^{19} + 3125 q^{25} - 7552 q^{28} - 2723 q^{31} + 6661 q^{37} + 44950 q^{43} + 10907 q^{49} - 13664 q^{52} + 38626 q^{61} - 65536 q^{64} + 37939 q^{67} + 78127 q^{73} - 201152 q^{76} - 90857 q^{79} + 90097 q^{91} - 268772 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 16.0000 + 27.7128i 0 0 −105.500 + 75.3442i 0 0 0
46.1 0 0 16.0000 27.7128i 0 0 −105.500 75.3442i 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 63.6.e.b 2
3.b odd 2 1 CM 63.6.e.b 2
7.c even 3 1 inner 63.6.e.b 2
7.c even 3 1 441.6.a.e 1
7.d odd 6 1 441.6.a.f 1
21.g even 6 1 441.6.a.f 1
21.h odd 6 1 inner 63.6.e.b 2
21.h odd 6 1 441.6.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.e.b 2 1.a even 1 1 trivial
63.6.e.b 2 3.b odd 2 1 CM
63.6.e.b 2 7.c even 3 1 inner
63.6.e.b 2 21.h odd 6 1 inner
441.6.a.e 1 7.c even 3 1
441.6.a.e 1 21.h odd 6 1
441.6.a.f 1 7.d odd 6 1
441.6.a.f 1 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\). 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
$7$ \( T^{2} + 211T + 16807 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 427)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 3143 T + 9878449 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2723 T + 7414729 \) Copy content Toggle raw display
$37$ \( T^{2} - 6661 T + 44368921 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 22475)^{2} \) 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
$61$ \( T^{2} - 38626 T + 1491967876 \) Copy content Toggle raw display
$67$ \( T^{2} - 37939 T + 1439367721 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 78127 T + 6103828129 \) Copy content Toggle raw display
$79$ \( T^{2} + 90857 T + 8254994449 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 134386)^{2} \) Copy content Toggle raw display
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