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Properties

Label 63.6.e.c
Level $63$
Weight $6$
Character orbit 63.e
Analytic conductor $10.104$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,6,Mod(37,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.37");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), degree \(2\), 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: no
Analytic conductor: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-83})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 20x^{2} - 21x + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 2 \beta_1 - 2) q^{2} + (3 \beta_{3} - 3 \beta_{2} + 34 \beta_1) q^{4} + ( - 7 \beta_{3} - 20 \beta_1 - 20) q^{5} + ( - 14 \beta_{3} + 7 \beta_{2} + \cdots - 91) q^{7} + (5 \beta_{2} + 190) q^{8}+ \cdots + ( - 343 \beta_{3} + 3675 \beta_{2} + \cdots + 80164) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 65 q^{4} - 33 q^{5} - 350 q^{7} + 750 q^{8} - 921 q^{10} - 1137 q^{11} + 1850 q^{13} - 2352 q^{14} + 895 q^{16} - 324 q^{17} - 2311 q^{19} + 7374 q^{20} + 3162 q^{22} + 1596 q^{23} - 395 q^{25}+ \cdots + 462021 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 20x^{2} - 21x + 441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 20\nu^{2} - 20\nu - 441 ) / 420 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 41\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 20\nu - 41 ) / 20 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 61\beta _1 + 62 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 40\beta_{3} - 20\beta_{2} + 20\beta _1 + 103 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1 - \beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
4.19493 − 1.84460i
−3.69493 + 2.71062i
4.19493 + 1.84460i
−3.69493 − 2.71062i
−4.69493 + 8.13186i 0 −28.0848 − 48.6443i −35.8645 + 62.1192i 0 −87.5000 + 95.6596i 226.949 0 −336.763 − 583.291i
37.2 3.19493 − 5.53379i 0 −4.41520 − 7.64735i 19.3645 − 33.5404i 0 −87.5000 − 95.6596i 148.051 0 −123.737 − 214.318i
46.1 −4.69493 − 8.13186i 0 −28.0848 + 48.6443i −35.8645 − 62.1192i 0 −87.5000 − 95.6596i 226.949 0 −336.763 + 583.291i
46.2 3.19493 + 5.53379i 0 −4.41520 + 7.64735i 19.3645 + 33.5404i 0 −87.5000 + 95.6596i 148.051 0 −123.737 + 214.318i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.e.c 4
3.b odd 2 1 21.6.e.b ✓ 4
7.c even 3 1 inner 63.6.e.c 4
7.c even 3 1 441.6.a.t 2
7.d odd 6 1 441.6.a.s 2
12.b even 2 1 336.6.q.e 4
21.c even 2 1 147.6.e.l 4
21.g even 6 1 147.6.a.k 2
21.g even 6 1 147.6.e.l 4
21.h odd 6 1 21.6.e.b ✓ 4
21.h odd 6 1 147.6.a.i 2
84.n even 6 1 336.6.q.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.b ✓ 4 3.b odd 2 1
21.6.e.b ✓ 4 21.h odd 6 1
63.6.e.c 4 1.a even 1 1 trivial
63.6.e.c 4 7.c even 3 1 inner
147.6.a.i 2 21.h odd 6 1
147.6.a.k 2 21.g even 6 1
147.6.e.l 4 21.c even 2 1
147.6.e.l 4 21.g even 6 1
336.6.q.e 4 12.b even 2 1
336.6.q.e 4 84.n even 6 1
441.6.a.s 2 7.d odd 6 1
441.6.a.t 2 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{3} + 69T_{2}^{2} - 180T_{2} + 3600 \) acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} + \cdots + 3600 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 33 T^{3} + \cdots + 7717284 \) Copy content Toggle raw display
$7$ \( (T^{2} + 175 T + 16807)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 104412996900 \) Copy content Toggle raw display
$13$ \( (T^{2} - 925 T + 208864)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 1788317798400 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 1663584040000 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 27756881510400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2217 T + 1102716)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{2} - 12858 T - 3221280)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2771 T - 257902490)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{2} + 102096 T + 2483190108)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 79\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{2} + 61179 T + 711231498)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{2} + 9791 T - 40418570)^{2} \) Copy content Toggle raw display
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