Properties

Label 21.4.c.a
Level 21
Weight 4
Character orbit 21.c
Analytic conductor 1.239
Analytic rank 0
Dimension 2
CM disc. -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 21.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta q^{3} \) \( + 8 q^{4} \) \( + ( -10 + 3 \beta ) q^{7} \) \( -27 q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta q^{3} \) \( + 8 q^{4} \) \( + ( -10 + 3 \beta ) q^{7} \) \( -27 q^{9} \) \( -8 \beta q^{12} \) \( + 12 \beta q^{13} \) \( + 64 q^{16} \) \( -30 \beta q^{19} \) \( + ( 81 + 10 \beta ) q^{21} \) \( -125 q^{25} \) \( + 27 \beta q^{27} \) \( + ( -80 + 24 \beta ) q^{28} \) \( -30 \beta q^{31} \) \( -216 q^{36} \) \( -110 q^{37} \) \( + 324 q^{39} \) \( + 520 q^{43} \) \( -64 \beta q^{48} \) \( + ( -143 - 60 \beta ) q^{49} \) \( + 96 \beta q^{52} \) \( -810 q^{57} \) \( + 180 \beta q^{61} \) \( + ( 270 - 81 \beta ) q^{63} \) \( + 512 q^{64} \) \( -880 q^{67} \) \( -72 \beta q^{73} \) \( + 125 \beta q^{75} \) \( -240 \beta q^{76} \) \( + 884 q^{79} \) \( + 729 q^{81} \) \( + ( 648 + 80 \beta ) q^{84} \) \( + ( -972 - 120 \beta ) q^{91} \) \( -810 q^{93} \) \( + 264 \beta q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 16q^{4} \) \(\mathstrut -\mathstrut 20q^{7} \) \(\mathstrut -\mathstrut 54q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 16q^{4} \) \(\mathstrut -\mathstrut 20q^{7} \) \(\mathstrut -\mathstrut 54q^{9} \) \(\mathstrut +\mathstrut 128q^{16} \) \(\mathstrut +\mathstrut 162q^{21} \) \(\mathstrut -\mathstrut 250q^{25} \) \(\mathstrut -\mathstrut 160q^{28} \) \(\mathstrut -\mathstrut 432q^{36} \) \(\mathstrut -\mathstrut 220q^{37} \) \(\mathstrut +\mathstrut 648q^{39} \) \(\mathstrut +\mathstrut 1040q^{43} \) \(\mathstrut -\mathstrut 286q^{49} \) \(\mathstrut -\mathstrut 1620q^{57} \) \(\mathstrut +\mathstrut 540q^{63} \) \(\mathstrut +\mathstrut 1024q^{64} \) \(\mathstrut -\mathstrut 1760q^{67} \) \(\mathstrut +\mathstrut 1768q^{79} \) \(\mathstrut +\mathstrut 1458q^{81} \) \(\mathstrut +\mathstrut 1296q^{84} \) \(\mathstrut -\mathstrut 1944q^{91} \) \(\mathstrut -\mathstrut 1620q^{93} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1
0.500000 + 0.866025i
0.500000 0.866025i
0 5.19615i 8.00000 0 0 −10.0000 + 15.5885i 0 −27.0000 0
20.2 0 5.19615i 8.00000 0 0 −10.0000 15.5885i 0 −27.0000 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.b Odd 1 yes
21.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) acting on \(S_{4}^{\mathrm{new}}(21, [\chi])\).