Properties

Label 21.2.g.a
Level 21
Weight 2
Character orbit 21.g
Analytic conductor 0.168
Analytic rank 0
Dimension 2
CM disc. -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 \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.167685844245\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

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

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 −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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes
7.d Odd 1 yes
21.g Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(21, [\chi])\).