Properties

Label 21.41.h.a
Level $21$
Weight $41$
Character orbit 21.h
Analytic conductor $212.819$
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] = [21,41,Mod(2,21)]
 
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, 2]))
 
N = Newforms(chi, 41, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.2");
 
S:= CuspForms(chi, 41);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 41 \)
Character orbit: \([\chi]\) \(=\) 21.h (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: \(212.818798913\)
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.

\(f(q)\) \(=\) \( q - 3486784401 \zeta_{6} q^{3} - 1099511627776 \zeta_{6} q^{4} + ( - 89\!\cdots\!51 \zeta_{6} + 64\!\cdots\!75) q^{7} + \cdots + (12\!\cdots\!01 \zeta_{6} - 12\!\cdots\!01) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3486784401 \zeta_{6} q^{3} - 1099511627776 \zeta_{6} q^{4} + ( - 89\!\cdots\!51 \zeta_{6} + 64\!\cdots\!75) q^{7} + \cdots + 10\!\cdots\!74 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3486784401 q^{3} - 1099511627776 q^{4} + 39\!\cdots\!99 q^{7}+ \cdots - 12\!\cdots\!01 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3486784401 q^{3} - 1099511627776 q^{4} + 39\!\cdots\!99 q^{7}+ \cdots + 21\!\cdots\!48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.74339e9 3.01964e9i −5.49756e11 9.52205e11i 0 0 1.96603e16 7.73323e16i 0 −6.07883e18 + 1.05288e19i 0
11.1 0 −1.74339e9 + 3.01964e9i −5.49756e11 + 9.52205e11i 0 0 1.96603e16 + 7.73323e16i 0 −6.07883e18 1.05288e19i 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 21.41.h.a 2
3.b odd 2 1 CM 21.41.h.a 2
7.c even 3 1 inner 21.41.h.a 2
21.h odd 6 1 inner 21.41.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.41.h.a 2 1.a even 1 1 trivial
21.41.h.a 2 3.b odd 2 1 CM
21.41.h.a 2 7.c even 3 1 inner
21.41.h.a 2 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{41}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3486784401 T + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 36\!\cdots\!01)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 47\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 39\!\cdots\!29 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 20\!\cdots\!01 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 93\!\cdots\!73)^{2} \) 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
$61$ \( T^{2} + \cdots + 82\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 58\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 20\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 99\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 10\!\cdots\!74)^{2} \) Copy content Toggle raw display
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