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: \( x^{2} - x + 1 \) 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

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.

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

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 \(S_{6}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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