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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

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.167685844245\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
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

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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} + ( 2 - 3 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} + ( 2 - 3 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 4 - 2 \zeta_{6} ) q^{12} + ( -1 + 2 \zeta_{6} ) q^{13} -4 \zeta_{6} q^{16} + ( -6 + 3 \zeta_{6} ) q^{19} + ( -5 + 4 \zeta_{6} ) q^{21} + ( 5 - 5 \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 2 + 4 \zeta_{6} ) q^{28} + ( 5 + 5 \zeta_{6} ) q^{31} -6 q^{36} -\zeta_{6} q^{37} + ( 3 - 3 \zeta_{6} ) q^{39} -5 q^{43} + ( -4 + 8 \zeta_{6} ) q^{48} + ( -5 - 3 \zeta_{6} ) q^{49} + ( -2 - 2 \zeta_{6} ) q^{52} + 9 q^{57} + ( 8 - 4 \zeta_{6} ) q^{61} + ( 9 - 3 \zeta_{6} ) q^{63} + 8 q^{64} + ( -11 + 11 \zeta_{6} ) q^{67} + ( -9 - 9 \zeta_{6} ) q^{73} + ( -10 + 5 \zeta_{6} ) q^{75} + ( 6 - 12 \zeta_{6} ) q^{76} + 13 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 2 - 10 \zeta_{6} ) q^{84} + ( 4 + \zeta_{6} ) q^{91} -15 \zeta_{6} q^{93} + ( -8 + 16 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} - 2q^{4} + q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - 3q^{3} - 2q^{4} + q^{7} + 3q^{9} + 6q^{12} - 4q^{16} - 9q^{19} - 6q^{21} + 5q^{25} + 8q^{28} + 15q^{31} - 12q^{36} - q^{37} + 3q^{39} - 10q^{43} - 13q^{49} - 6q^{52} + 18q^{57} + 12q^{61} + 15q^{63} + 16q^{64} - 11q^{67} - 27q^{73} - 15q^{75} + 13q^{79} - 9q^{81} - 6q^{84} + 9q^{91} - 15q^{93} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.

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

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 3 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 27 + 9 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 75 - 15 T + T^{2} \)
$37$ \( 1 + T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 5 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 48 - 12 T + T^{2} \)
$67$ \( 121 + 11 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 243 + 27 T + T^{2} \)
$79$ \( 169 - 13 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 192 + T^{2} \)
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