Properties

Label 63.3.d.c
Level $63$
Weight $3$
Character orbit 63.d
Self dual yes
Analytic conductor $1.717$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,3,Mod(55,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.55");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 63.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.71662566547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 3 q^{4} + 7 q^{7} - \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 3 q^{4} + 7 q^{7} - \beta q^{8} - 8 \beta q^{11} + 7 \beta q^{14} - 19 q^{16} - 56 q^{22} + 16 \beta q^{23} + 25 q^{25} + 21 q^{28} - 8 \beta q^{29} - 15 \beta q^{32} + 38 q^{37} - 58 q^{43} - 24 \beta q^{44} + 112 q^{46} + 49 q^{49} + 25 \beta q^{50} + 40 \beta q^{53} - 7 \beta q^{56} - 56 q^{58} - 29 q^{64} - 118 q^{67} - 32 \beta q^{71} + 38 \beta q^{74} - 56 \beta q^{77} - 94 q^{79} - 58 \beta q^{86} + 56 q^{88} + 48 \beta q^{92} + 49 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{4} + 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{4} + 14 q^{7} - 38 q^{16} - 112 q^{22} + 50 q^{25} + 42 q^{28} + 76 q^{37} - 116 q^{43} + 224 q^{46} + 98 q^{49} - 112 q^{58} - 58 q^{64} - 236 q^{67} - 188 q^{79} + 112 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−2.64575
2.64575
−2.64575 0 3.00000 0 0 7.00000 2.64575 0 0
55.2 2.64575 0 3.00000 0 0 7.00000 −2.64575 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
21.c even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 7 \) acting on \(S_{3}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 7 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 448 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 1792 \) Copy content Toggle raw display
$29$ \( T^{2} - 448 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 38)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 58)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 11200 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T + 118)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 7168 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T + 94)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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