Properties

Label 21.6.g.a
Level $21$
Weight $6$
Character orbit 21.g
Analytic conductor $3.368$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.36806021607\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
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

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.

\(f(q)\) \(=\) \( q + ( 9 + 9 \zeta_{6} ) q^{3} + ( -32 + 32 \zeta_{6} ) q^{4} + ( 62 + 87 \zeta_{6} ) q^{7} + 243 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 9 + 9 \zeta_{6} ) q^{3} + ( -32 + 32 \zeta_{6} ) q^{4} + ( 62 + 87 \zeta_{6} ) q^{7} + 243 \zeta_{6} q^{9} + ( -576 + 288 \zeta_{6} ) q^{12} + ( 659 - 1318 \zeta_{6} ) q^{13} -1024 \zeta_{6} q^{16} + ( -186 + 93 \zeta_{6} ) q^{19} + ( -225 + 2124 \zeta_{6} ) q^{21} + ( 3125 - 3125 \zeta_{6} ) q^{25} + ( -2187 + 4374 \zeta_{6} ) q^{27} + ( -4768 + 1984 \zeta_{6} ) q^{28} + ( 5975 + 5975 \zeta_{6} ) q^{31} -7776 q^{36} -6661 \zeta_{6} q^{37} + ( 17793 - 17793 \zeta_{6} ) q^{39} -22475 q^{43} + ( 9216 - 18432 \zeta_{6} ) q^{48} + ( -3725 + 18357 \zeta_{6} ) q^{49} + ( 21088 + 21088 \zeta_{6} ) q^{52} -2511 q^{57} + ( -50152 + 25076 \zeta_{6} ) q^{61} + ( -21141 + 36207 \zeta_{6} ) q^{63} + 32768 q^{64} + ( 37939 - 37939 \zeta_{6} ) q^{67} + ( -27009 - 27009 \zeta_{6} ) q^{73} + ( 56250 - 28125 \zeta_{6} ) q^{75} + ( 2976 - 5952 \zeta_{6} ) q^{76} -90857 \zeta_{6} q^{79} + ( -59049 + 59049 \zeta_{6} ) q^{81} + ( -60768 - 7200 \zeta_{6} ) q^{84} + ( 155524 - 139049 \zeta_{6} ) q^{91} + 161325 \zeta_{6} q^{93} + ( -73688 + 147376 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 27q^{3} - 32q^{4} + 211q^{7} + 243q^{9} + O(q^{10}) \) \( 2q + 27q^{3} - 32q^{4} + 211q^{7} + 243q^{9} - 864q^{12} - 1024q^{16} - 279q^{19} + 1674q^{21} + 3125q^{25} - 7552q^{28} + 17925q^{31} - 15552q^{36} - 6661q^{37} + 17793q^{39} - 44950q^{43} + 10907q^{49} + 63264q^{52} - 5022q^{57} - 75228q^{61} - 6075q^{63} + 65536q^{64} + 37939q^{67} - 81027q^{73} + 84375q^{75} - 90857q^{79} - 59049q^{81} - 128736q^{84} + 171999q^{91} + 161325q^{93} + O(q^{100}) \)

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 13.5000 7.79423i −16.0000 27.7128i 0 0 105.500 75.3442i 0 121.500 210.444i 0
17.1 0 13.5000 + 7.79423i −16.0000 + 27.7128i 0 0 105.500 + 75.3442i 0 121.500 + 210.444i 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 21.6.g.a 2
3.b odd 2 1 CM 21.6.g.a 2
7.c even 3 1 147.6.c.a 2
7.d odd 6 1 inner 21.6.g.a 2
7.d odd 6 1 147.6.c.a 2
21.g even 6 1 inner 21.6.g.a 2
21.g even 6 1 147.6.c.a 2
21.h odd 6 1 147.6.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.g.a 2 1.a even 1 1 trivial
21.6.g.a 2 3.b odd 2 1 CM
21.6.g.a 2 7.d odd 6 1 inner
21.6.g.a 2 21.g even 6 1 inner
147.6.c.a 2 7.c even 3 1
147.6.c.a 2 7.d odd 6 1
147.6.c.a 2 21.g even 6 1
147.6.c.a 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{6}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 243 - 27 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 16807 - 211 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 1302843 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 25947 + 279 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 107101875 - 17925 T + T^{2} \)
$37$ \( 44368921 + 6661 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 22475 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 1886417328 + 75228 T + T^{2} \)
$67$ \( 1439367721 - 37939 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 2188458243 + 81027 T + T^{2} \)
$79$ \( 8254994449 + 90857 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 16289764032 + T^{2} \)
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