Properties

Label 21.3.h.a
Level $21$
Weight $3$
Character orbit 21.h
Analytic conductor $0.572$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.572208555157\)
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.

\(f(q)\) \(=\) \( q + 3 \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + ( - 3 \zeta_{6} - 5) q^{7} + (9 \zeta_{6} - 9) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + ( - 3 \zeta_{6} - 5) q^{7} + (9 \zeta_{6} - 9) q^{9} + ( - 12 \zeta_{6} + 12) q^{12} + 23 q^{13} + (16 \zeta_{6} - 16) q^{16} + (11 \zeta_{6} - 11) q^{19} + ( - 24 \zeta_{6} + 9) q^{21} - 25 \zeta_{6} q^{25} - 27 q^{27} + (32 \zeta_{6} - 12) q^{28} + 13 \zeta_{6} q^{31} + 36 q^{36} + ( - 73 \zeta_{6} + 73) q^{37} + 69 \zeta_{6} q^{39} - 61 q^{43} - 48 q^{48} + (39 \zeta_{6} + 16) q^{49} - 92 \zeta_{6} q^{52} - 33 q^{57} + (74 \zeta_{6} - 74) q^{61} + ( - 45 \zeta_{6} + 72) q^{63} + 64 q^{64} + 13 \zeta_{6} q^{67} + 97 \zeta_{6} q^{73} + ( - 75 \zeta_{6} + 75) q^{75} + 44 q^{76} + (11 \zeta_{6} - 11) q^{79} - 81 \zeta_{6} q^{81} + (60 \zeta_{6} - 96) q^{84} + ( - 69 \zeta_{6} - 115) q^{91} + (39 \zeta_{6} - 39) q^{93} + 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 4 q^{4} - 13 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 4 q^{4} - 13 q^{7} - 9 q^{9} + 12 q^{12} + 46 q^{13} - 16 q^{16} - 11 q^{19} - 6 q^{21} - 25 q^{25} - 54 q^{27} + 8 q^{28} + 13 q^{31} + 72 q^{36} + 73 q^{37} + 69 q^{39} - 122 q^{43} - 96 q^{48} + 71 q^{49} - 92 q^{52} - 66 q^{57} - 74 q^{61} + 99 q^{63} + 128 q^{64} + 13 q^{67} + 97 q^{73} + 75 q^{75} + 88 q^{76} - 11 q^{79} - 81 q^{81} - 132 q^{84} - 299 q^{91} - 39 q^{93} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 + 2.59808i −2.00000 3.46410i 0 0 −6.50000 2.59808i 0 −4.50000 + 7.79423i 0
11.1 0 1.50000 2.59808i −2.00000 + 3.46410i 0 0 −6.50000 + 2.59808i 0 −4.50000 7.79423i 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.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.3.h.a 2
3.b odd 2 1 CM 21.3.h.a 2
4.b odd 2 1 336.3.bn.b 2
7.b odd 2 1 147.3.h.a 2
7.c even 3 1 inner 21.3.h.a 2
7.c even 3 1 147.3.b.a 1
7.d odd 6 1 147.3.b.b 1
7.d odd 6 1 147.3.h.a 2
12.b even 2 1 336.3.bn.b 2
21.c even 2 1 147.3.h.a 2
21.g even 6 1 147.3.b.b 1
21.g even 6 1 147.3.h.a 2
21.h odd 6 1 inner 21.3.h.a 2
21.h odd 6 1 147.3.b.a 1
28.g odd 6 1 336.3.bn.b 2
84.n even 6 1 336.3.bn.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.h.a 2 1.a even 1 1 trivial
21.3.h.a 2 3.b odd 2 1 CM
21.3.h.a 2 7.c even 3 1 inner
21.3.h.a 2 21.h odd 6 1 inner
147.3.b.a 1 7.c even 3 1
147.3.b.a 1 21.h odd 6 1
147.3.b.b 1 7.d odd 6 1
147.3.b.b 1 21.g even 6 1
147.3.h.a 2 7.b odd 2 1
147.3.h.a 2 7.d odd 6 1
147.3.h.a 2 21.c even 2 1
147.3.h.a 2 21.g even 6 1
336.3.bn.b 2 4.b odd 2 1
336.3.bn.b 2 12.b even 2 1
336.3.bn.b 2 28.g odd 6 1
336.3.bn.b 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 13T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 23)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$37$ \( T^{2} - 73T + 5329 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 61)^{2} \) 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
$61$ \( T^{2} + 74T + 5476 \) Copy content Toggle raw display
$67$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 97T + 9409 \) Copy content Toggle raw display
$79$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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