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Properties

Label 21.2.g.a
Level $21$
Weight $2$
Character orbit 21.g
Analytic conductor $0.168$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [21,2,Mod(5,21)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("21.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.167685844245\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + ( - 3 \zeta_{6} + 2) q^{7} + 3 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 4) q^{12} + (2 \zeta_{6} - 1) q^{13} - 4 \zeta_{6} q^{16} + (3 \zeta_{6} - 6) q^{19} + \cdots + (16 \zeta_{6} - 8) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 2 q^{4} + q^{7} + 3 q^{9} + 6 q^{12} - 4 q^{16} - 9 q^{19} - 6 q^{21} + 5 q^{25} + 8 q^{28} + 15 q^{31} - 12 q^{36} - q^{37} + 3 q^{39} - 10 q^{43} - 13 q^{49} - 6 q^{52} + 18 q^{57} + 12 q^{61}+ \cdots - 15 q^{93}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 −1.50000 + 0.866025i −1.00000 − 1.73205i 0 0 0.500000 + 2.59808i 0 1.50000 − 2.59808i 0
17.1 0 −1.50000 − 0.866025i −1.00000 + 1.73205i 0 0 0.500000 − 2.59808i 0 1.50000 + 2.59808i 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 21.2.g.a ✓ 2
3.b odd 2 1 CM 21.2.g.a ✓ 2
4.b odd 2 1 336.2.bc.c 2
5.b even 2 1 525.2.t.c 2
5.c odd 4 2 525.2.q.d 4
7.b odd 2 1 147.2.g.a 2
7.c even 3 1 147.2.c.a 2
7.c even 3 1 147.2.g.a 2
7.d odd 6 1 inner 21.2.g.a ✓ 2
7.d odd 6 1 147.2.c.a 2
9.c even 3 1 567.2.i.b 2
9.c even 3 1 567.2.s.a 2
9.d odd 6 1 567.2.i.b 2
9.d odd 6 1 567.2.s.a 2
12.b even 2 1 336.2.bc.c 2
15.d odd 2 1 525.2.t.c 2
15.e even 4 2 525.2.q.d 4
21.c even 2 1 147.2.g.a 2
21.g even 6 1 inner 21.2.g.a ✓ 2
21.g even 6 1 147.2.c.a 2
21.h odd 6 1 147.2.c.a 2
21.h odd 6 1 147.2.g.a 2
28.f even 6 1 336.2.bc.c 2
28.f even 6 1 2352.2.k.c 2
28.g odd 6 1 2352.2.k.c 2
35.i odd 6 1 525.2.t.c 2
35.k even 12 2 525.2.q.d 4
63.i even 6 1 567.2.s.a 2
63.k odd 6 1 567.2.i.b 2
63.s even 6 1 567.2.i.b 2
63.t odd 6 1 567.2.s.a 2
84.j odd 6 1 336.2.bc.c 2
84.j odd 6 1 2352.2.k.c 2
84.n even 6 1 2352.2.k.c 2
105.p even 6 1 525.2.t.c 2
105.w odd 12 2 525.2.q.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.g.a ✓ 2 1.a even 1 1 trivial
21.2.g.a ✓ 2 3.b odd 2 1 CM
21.2.g.a ✓ 2 7.d odd 6 1 inner
21.2.g.a ✓ 2 21.g even 6 1 inner
147.2.c.a 2 7.c even 3 1
147.2.c.a 2 7.d odd 6 1
147.2.c.a 2 21.g even 6 1
147.2.c.a 2 21.h odd 6 1
147.2.g.a 2 7.b odd 2 1
147.2.g.a 2 7.c even 3 1
147.2.g.a 2 21.c even 2 1
147.2.g.a 2 21.h odd 6 1
336.2.bc.c 2 4.b odd 2 1
336.2.bc.c 2 12.b even 2 1
336.2.bc.c 2 28.f even 6 1
336.2.bc.c 2 84.j odd 6 1
525.2.q.d 4 5.c odd 4 2
525.2.q.d 4 15.e even 4 2
525.2.q.d 4 35.k even 12 2
525.2.q.d 4 105.w odd 12 2
525.2.t.c 2 5.b even 2 1
525.2.t.c 2 15.d odd 2 1
525.2.t.c 2 35.i odd 6 1
525.2.t.c 2 105.p even 6 1
567.2.i.b 2 9.c even 3 1
567.2.i.b 2 9.d odd 6 1
567.2.i.b 2 63.k odd 6 1
567.2.i.b 2 63.s even 6 1
567.2.s.a 2 9.c even 3 1
567.2.s.a 2 9.d odd 6 1
567.2.s.a 2 63.i even 6 1
567.2.s.a 2 63.t odd 6 1
2352.2.k.c 2 28.f even 6 1
2352.2.k.c 2 28.g odd 6 1
2352.2.k.c 2 84.j odd 6 1
2352.2.k.c 2 84.n even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$37$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 5)^{2} \) 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
$61$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$67$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 27T + 243 \) Copy content Toggle raw display
$79$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 192 \) Copy content Toggle raw display
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