Properties

Label 21.8.g.a
Level $21$
Weight $8$
Character orbit 21.g
Analytic conductor $6.560$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.56008553517\)
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.

\(f(q)\) \(=\) \( q + ( 27 + 27 \zeta_{6} ) q^{3} + ( -128 + 128 \zeta_{6} ) q^{4} + ( -1006 + 249 \zeta_{6} ) q^{7} + 2187 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 27 + 27 \zeta_{6} ) q^{3} + ( -128 + 128 \zeta_{6} ) q^{4} + ( -1006 + 249 \zeta_{6} ) q^{7} + 2187 \zeta_{6} q^{9} + ( -6912 + 3456 \zeta_{6} ) q^{12} + ( -9073 + 18146 \zeta_{6} ) q^{13} -16384 \zeta_{6} q^{16} + ( 67026 - 33513 \zeta_{6} ) q^{19} + ( -33885 - 13716 \zeta_{6} ) q^{21} + ( 78125 - 78125 \zeta_{6} ) q^{25} + ( -59049 + 118098 \zeta_{6} ) q^{27} + ( 96896 - 128768 \zeta_{6} ) q^{28} + ( -8815 - 8815 \zeta_{6} ) q^{31} -279936 q^{36} + 335663 \zeta_{6} q^{37} + ( -734913 + 734913 \zeta_{6} ) q^{39} + 409495 q^{43} + ( 442368 - 884736 \zeta_{6} ) q^{48} + ( 950035 - 438987 \zeta_{6} ) q^{49} + ( -1161344 - 1161344 \zeta_{6} ) q^{52} + 2714553 q^{57} + ( -307432 + 153716 \zeta_{6} ) q^{61} + ( -544563 - 1655559 \zeta_{6} ) q^{63} + 2097152 q^{64} + ( -4443527 + 4443527 \zeta_{6} ) q^{67} + ( -3770937 - 3770937 \zeta_{6} ) q^{73} + ( 4218750 - 2109375 \zeta_{6} ) q^{75} + ( -4289664 + 8579328 \zeta_{6} ) q^{76} + 4517617 \zeta_{6} q^{79} + ( -4782969 + 4782969 \zeta_{6} ) q^{81} + ( 6092928 - 4337280 \zeta_{6} ) q^{84} + ( 4609084 - 15995699 \zeta_{6} ) q^{91} -714015 \zeta_{6} q^{93} + ( 7599304 - 15198608 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 81q^{3} - 128q^{4} - 1763q^{7} + 2187q^{9} + O(q^{10}) \) \( 2q + 81q^{3} - 128q^{4} - 1763q^{7} + 2187q^{9} - 10368q^{12} - 16384q^{16} + 100539q^{19} - 81486q^{21} + 78125q^{25} + 65024q^{28} - 26445q^{31} - 559872q^{36} + 335663q^{37} - 734913q^{39} + 818990q^{43} + 1461083q^{49} - 3484032q^{52} + 5429106q^{57} - 461148q^{61} - 2744685q^{63} + 4194304q^{64} - 4443527q^{67} - 11312811q^{73} + 6328125q^{75} + 4517617q^{79} - 4782969q^{81} + 7848576q^{84} - 6777531q^{91} - 714015q^{93} + O(q^{100}) \)

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 40.5000 23.3827i −64.0000 110.851i 0 0 −881.500 215.640i 0 1093.50 1894.00i 0
17.1 0 40.5000 + 23.3827i −64.0000 + 110.851i 0 0 −881.500 + 215.640i 0 1093.50 + 1894.00i 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 21.8.g.a 2
3.b odd 2 1 CM 21.8.g.a 2
7.c even 3 1 147.8.c.a 2
7.d odd 6 1 inner 21.8.g.a 2
7.d odd 6 1 147.8.c.a 2
21.g even 6 1 inner 21.8.g.a 2
21.g even 6 1 147.8.c.a 2
21.h odd 6 1 147.8.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.8.g.a 2 1.a even 1 1 trivial
21.8.g.a 2 3.b odd 2 1 CM
21.8.g.a 2 7.d odd 6 1 inner
21.8.g.a 2 21.g even 6 1 inner
147.8.c.a 2 7.c even 3 1
147.8.c.a 2 7.d odd 6 1
147.8.c.a 2 21.g even 6 1
147.8.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 \(S_{8}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2187 - 81 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 823543 + 1763 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 246957987 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 3369363507 - 100539 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 233112675 + 26445 T + T^{2} \)
$37$ \( 112669649569 - 335663 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -409495 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 70885825968 + 461148 T + T^{2} \)
$67$ \( 19744932199729 + 4443527 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 42659897573907 + 11312811 T + T^{2} \)
$79$ \( 20408863358689 - 4517617 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 173248263853248 + T^{2} \)
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