Properties

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

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.572208555157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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.

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

Character Values

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

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

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

Hecke kernels

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