Properties

Label 21.5.h.a
Level $21$
Weight $5$
Character orbit 21.h
Analytic conductor $2.171$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.2");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(2.17076922476\)
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.

\(f(q)\) \(=\) \( q - 9 \zeta_{6} q^{3} - 16 \zeta_{6} q^{4} + ( - 39 \zeta_{6} + 55) q^{7} + (81 \zeta_{6} - 81) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 9 \zeta_{6} q^{3} - 16 \zeta_{6} q^{4} + ( - 39 \zeta_{6} + 55) q^{7} + (81 \zeta_{6} - 81) q^{9} + (144 \zeta_{6} - 144) q^{12} + 191 q^{13} + (256 \zeta_{6} - 256) q^{16} + ( - 601 \zeta_{6} + 601) q^{19} + ( - 144 \zeta_{6} - 351) q^{21} - 625 \zeta_{6} q^{25} + 729 q^{27} + ( - 256 \zeta_{6} - 624) q^{28} + 1753 \zeta_{6} q^{31} + 1296 q^{36} + (2591 \zeta_{6} - 2591) q^{37} - 1719 \zeta_{6} q^{39} + 23 q^{43} + 2304 q^{48} + ( - 2769 \zeta_{6} + 1504) q^{49} - 3056 \zeta_{6} q^{52} - 5409 q^{57} + ( - 1966 \zeta_{6} + 1966) q^{61} + (4455 \zeta_{6} - 1296) q^{63} + 4096 q^{64} + 8809 \zeta_{6} q^{67} + 1249 \zeta_{6} q^{73} + (5625 \zeta_{6} - 5625) q^{75} - 9616 q^{76} + ( - 12361 \zeta_{6} + 12361) q^{79} - 6561 \zeta_{6} q^{81} + (7920 \zeta_{6} - 2304) q^{84} + ( - 7449 \zeta_{6} + 10505) q^{91} + ( - 15777 \zeta_{6} + 15777) q^{93} - 18814 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{3} - 16 q^{4} + 71 q^{7} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{3} - 16 q^{4} + 71 q^{7} - 81 q^{9} - 144 q^{12} + 382 q^{13} - 256 q^{16} + 601 q^{19} - 846 q^{21} - 625 q^{25} + 1458 q^{27} - 1504 q^{28} + 1753 q^{31} + 2592 q^{36} - 2591 q^{37} - 1719 q^{39} + 46 q^{43} + 4608 q^{48} + 239 q^{49} - 3056 q^{52} - 10818 q^{57} + 1966 q^{61} + 1863 q^{63} + 8192 q^{64} + 8809 q^{67} + 1249 q^{73} - 5625 q^{75} - 19232 q^{76} + 12361 q^{79} - 6561 q^{81} + 3312 q^{84} + 13561 q^{91} + 15777 q^{93} - 37628 q^{97}+O(q^{100}) \) 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 −4.50000 7.79423i −8.00000 13.8564i 0 0 35.5000 33.7750i 0 −40.5000 + 70.1481i 0
11.1 0 −4.50000 + 7.79423i −8.00000 + 13.8564i 0 0 35.5000 + 33.7750i 0 −40.5000 70.1481i 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 21.5.h.a 2
3.b odd 2 1 CM 21.5.h.a 2
7.b odd 2 1 147.5.h.a 2
7.c even 3 1 inner 21.5.h.a 2
7.c even 3 1 147.5.b.b 1
7.d odd 6 1 147.5.b.a 1
7.d odd 6 1 147.5.h.a 2
21.c even 2 1 147.5.h.a 2
21.g even 6 1 147.5.b.a 1
21.g even 6 1 147.5.h.a 2
21.h odd 6 1 inner 21.5.h.a 2
21.h odd 6 1 147.5.b.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.5.h.a 2 1.a even 1 1 trivial
21.5.h.a 2 3.b odd 2 1 CM
21.5.h.a 2 7.c even 3 1 inner
21.5.h.a 2 21.h odd 6 1 inner
147.5.b.a 1 7.d odd 6 1
147.5.b.a 1 21.g even 6 1
147.5.b.b 1 7.c even 3 1
147.5.b.b 1 21.h odd 6 1
147.5.h.a 2 7.b odd 2 1
147.5.h.a 2 7.d odd 6 1
147.5.h.a 2 21.c even 2 1
147.5.h.a 2 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{5}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9T + 81 \) 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
$13$ \( (T - 191)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 601T + 361201 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 1753 T + 3073009 \) 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
$61$ \( T^{2} - 1966 T + 3865156 \) Copy content Toggle raw display
$67$ \( T^{2} - 8809 T + 77598481 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 1249 T + 1560001 \) 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
$97$ \( (T + 18814)^{2} \) Copy content Toggle raw display
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