Properties

Label 21.42.g.a
Level $21$
Weight $42$
Character orbit 21.g
Analytic conductor $223.591$
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] = [21,42,Mod(5,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 42, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.5");
 
S:= CuspForms(chi, 42);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 21.g (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: \(223.590507958\)
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: \(\Z[a_1, \ldots, a_{4}]\)
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.

\(f(q)\) \(=\) \( q + (3486784401 \zeta_{6} + 3486784401) q^{3} + (2199023255552 \zeta_{6} - 2199023255552) q^{4} + (14\!\cdots\!23 \zeta_{6} + 20\!\cdots\!78) q^{7}+ \cdots + 36\!\cdots\!03 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3486784401 \zeta_{6} + 3486784401) q^{3} + (2199023255552 \zeta_{6} - 2199023255552) q^{4} + (14\!\cdots\!23 \zeta_{6} + 20\!\cdots\!78) q^{7}+ \cdots + ( - 67\!\cdots\!56 \zeta_{6} + 33\!\cdots\!28) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10460353203 q^{3} - 2199023255552 q^{4} + 42\!\cdots\!79 q^{7}+ \cdots + 36\!\cdots\!03 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10460353203 q^{3} - 2199023255552 q^{4} + 42\!\cdots\!79 q^{7}+ \cdots + 41\!\cdots\!85 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 0.866025i
0.500000 + 0.866025i
0 5.23018e9 3.01964e9i −1.09951e12 1.90441e12i 0 0 2.10745e17 1.24129e16i 0 1.82365e19 3.15865e19i 0
17.1 0 5.23018e9 + 3.01964e9i −1.09951e12 + 1.90441e12i 0 0 2.10745e17 + 1.24129e16i 0 1.82365e19 + 3.15865e19i 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 21.42.g.a 2
3.b odd 2 1 CM 21.42.g.a 2
7.d odd 6 1 inner 21.42.g.a 2
21.g even 6 1 inner 21.42.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.42.g.a 2 1.a even 1 1 trivial
21.42.g.a 2 3.b odd 2 1 CM
21.42.g.a 2 7.d odd 6 1 inner
21.42.g.a 2 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{42}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 10460353203 T + 36\!\cdots\!03 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots + 44\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 46\!\cdots\!23 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 80\!\cdots\!87 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 47\!\cdots\!75 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 41\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 19\!\cdots\!45)^{2} \) 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
$61$ \( T^{2} + \cdots + 46\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 25\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 66\!\cdots\!03 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 16\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 33\!\cdots\!52 \) Copy content Toggle raw display
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