Properties

Label 63.5.m.b
Level $63$
Weight $5$
Character orbit 63.m
Analytic conductor $6.512$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,5,Mod(10,63)]
 
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, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.10");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 63.m (of order \(6\), 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: \(6.51230767428\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
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

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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 16 \zeta_{6} + 16) q^{4} + ( - 39 \zeta_{6} - 16) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 16 \zeta_{6} + 16) q^{4} + ( - 39 \zeta_{6} - 16) q^{7} + ( - 322 \zeta_{6} + 161) q^{13} - 256 \zeta_{6} q^{16} + ( - 231 \zeta_{6} + 462) q^{19} + (625 \zeta_{6} - 625) q^{25} + (256 \zeta_{6} - 880) q^{28} + (455 \zeta_{6} + 455) q^{31} + 2591 \zeta_{6} q^{37} - 23 q^{43} + (2769 \zeta_{6} - 1265) q^{49} + ( - 2576 \zeta_{6} - 2576) q^{52} + ( - 4144 \zeta_{6} + 8288) q^{61} - 4096 q^{64} + ( - 8809 \zeta_{6} + 8809) q^{67} + ( - 6111 \zeta_{6} - 6111) q^{73} + ( - 7392 \zeta_{6} + 3696) q^{76} + 12361 \zeta_{6} q^{79} + (11431 \zeta_{6} - 15134) q^{91} + (448 \zeta_{6} - 224) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{4} - 71 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{4} - 71 q^{7} - 256 q^{16} + 693 q^{19} - 625 q^{25} - 1504 q^{28} + 1365 q^{31} + 2591 q^{37} - 46 q^{43} + 239 q^{49} - 7728 q^{52} + 12432 q^{61} - 8192 q^{64} + 8809 q^{67} - 18333 q^{73} + 12361 q^{79} - 18837 q^{91}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 8.00000 13.8564i 0 0 −35.5000 33.7750i 0 0 0
19.1 0 0 8.00000 + 13.8564i 0 0 −35.5000 + 33.7750i 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 63.5.m.b 2
3.b odd 2 1 CM 63.5.m.b 2
7.c even 3 1 441.5.d.b 2
7.d odd 6 1 inner 63.5.m.b 2
7.d odd 6 1 441.5.d.b 2
21.g even 6 1 inner 63.5.m.b 2
21.g even 6 1 441.5.d.b 2
21.h odd 6 1 441.5.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.5.m.b 2 1.a even 1 1 trivial
63.5.m.b 2 3.b odd 2 1 CM
63.5.m.b 2 7.d odd 6 1 inner
63.5.m.b 2 21.g even 6 1 inner
441.5.d.b 2 7.c even 3 1
441.5.d.b 2 7.d odd 6 1
441.5.d.b 2 21.g even 6 1
441.5.d.b 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{5}^{\mathrm{new}}(63, [\chi])\). 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
$7$ \( T^{2} + 71T + 2401 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 77763 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 693T + 160083 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 1365 T + 621075 \) Copy content Toggle raw display
$37$ \( T^{2} - 2591 T + 6713281 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 23)^{2} \) 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
$61$ \( T^{2} - 12432 T + 51518208 \) Copy content Toggle raw display
$67$ \( T^{2} - 8809 T + 77598481 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 18333 T + 112032963 \) Copy content Toggle raw display
$79$ \( T^{2} - 12361 T + 152794321 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 150528 \) Copy content Toggle raw display
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