Properties

Label 21.2.e.a
Level 21
Weight 2
Character orbit 21.e
Analytic conductor 0.168
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 21.e (of order \(3\) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( -2 + 2 \zeta_{6} ) q^{2} \) \( -\zeta_{6} q^{3} \) \( -2 \zeta_{6} q^{4} \) \( + ( 2 - 2 \zeta_{6} ) q^{5} \) \( + 2 q^{6} \) \( + ( -3 + \zeta_{6} ) q^{7} \) \( + ( -1 + \zeta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 + 2 \zeta_{6} ) q^{2} \) \( -\zeta_{6} q^{3} \) \( -2 \zeta_{6} q^{4} \) \( + ( 2 - 2 \zeta_{6} ) q^{5} \) \( + 2 q^{6} \) \( + ( -3 + \zeta_{6} ) q^{7} \) \( + ( -1 + \zeta_{6} ) q^{9} \) \( + 4 \zeta_{6} q^{10} \) \( + 2 \zeta_{6} q^{11} \) \( + ( -2 + 2 \zeta_{6} ) q^{12} \) \(+ q^{13}\) \( + ( 4 - 6 \zeta_{6} ) q^{14} \) \( -2 q^{15} \) \( + ( 4 - 4 \zeta_{6} ) q^{16} \) \( -2 \zeta_{6} q^{18} \) \( + ( -1 + \zeta_{6} ) q^{19} \) \( -4 q^{20} \) \( + ( 1 + 2 \zeta_{6} ) q^{21} \) \( -4 q^{22} \) \( + \zeta_{6} q^{25} \) \( + ( -2 + 2 \zeta_{6} ) q^{26} \) \(+ q^{27}\) \( + ( 2 + 4 \zeta_{6} ) q^{28} \) \( + 4 q^{29} \) \( + ( 4 - 4 \zeta_{6} ) q^{30} \) \( -9 \zeta_{6} q^{31} \) \( + 8 \zeta_{6} q^{32} \) \( + ( 2 - 2 \zeta_{6} ) q^{33} \) \( + ( -4 + 6 \zeta_{6} ) q^{35} \) \( + 2 q^{36} \) \( + ( -3 + 3 \zeta_{6} ) q^{37} \) \( -2 \zeta_{6} q^{38} \) \( -\zeta_{6} q^{39} \) \( -10 q^{41} \) \( + ( -6 + 2 \zeta_{6} ) q^{42} \) \( + 5 q^{43} \) \( + ( 4 - 4 \zeta_{6} ) q^{44} \) \( + 2 \zeta_{6} q^{45} \) \( + ( 6 - 6 \zeta_{6} ) q^{47} \) \( -4 q^{48} \) \( + ( 8 - 5 \zeta_{6} ) q^{49} \) \( -2 q^{50} \) \( -2 \zeta_{6} q^{52} \) \( -12 \zeta_{6} q^{53} \) \( + ( -2 + 2 \zeta_{6} ) q^{54} \) \( + 4 q^{55} \) \(+ q^{57}\) \( + ( -8 + 8 \zeta_{6} ) q^{58} \) \( + 12 \zeta_{6} q^{59} \) \( + 4 \zeta_{6} q^{60} \) \( + ( -10 + 10 \zeta_{6} ) q^{61} \) \( + 18 q^{62} \) \( + ( 2 - 3 \zeta_{6} ) q^{63} \) \( -8 q^{64} \) \( + ( 2 - 2 \zeta_{6} ) q^{65} \) \( + 4 \zeta_{6} q^{66} \) \( + 5 \zeta_{6} q^{67} \) \( + ( -4 - 8 \zeta_{6} ) q^{70} \) \( -6 q^{71} \) \( + 3 \zeta_{6} q^{73} \) \( -6 \zeta_{6} q^{74} \) \( + ( 1 - \zeta_{6} ) q^{75} \) \( + 2 q^{76} \) \( + ( -2 - 4 \zeta_{6} ) q^{77} \) \( + 2 q^{78} \) \( + ( 1 - \zeta_{6} ) q^{79} \) \( -8 \zeta_{6} q^{80} \) \( -\zeta_{6} q^{81} \) \( + ( 20 - 20 \zeta_{6} ) q^{82} \) \( + 6 q^{83} \) \( + ( 4 - 6 \zeta_{6} ) q^{84} \) \( + ( -10 + 10 \zeta_{6} ) q^{86} \) \( -4 \zeta_{6} q^{87} \) \( + ( -16 + 16 \zeta_{6} ) q^{89} \) \( -4 q^{90} \) \( + ( -3 + \zeta_{6} ) q^{91} \) \( + ( -9 + 9 \zeta_{6} ) q^{93} \) \( + 12 \zeta_{6} q^{94} \) \( + 2 \zeta_{6} q^{95} \) \( + ( 8 - 8 \zeta_{6} ) q^{96} \) \( -6 q^{97} \) \( + ( -6 + 16 \zeta_{6} ) q^{98} \) \( -2 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 5q^{7} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 5q^{7} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut 4q^{10} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut 2q^{12} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 2q^{14} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 4q^{16} \) \(\mathstrut -\mathstrut 2q^{18} \) \(\mathstrut -\mathstrut q^{19} \) \(\mathstrut -\mathstrut 8q^{20} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 8q^{22} \) \(\mathstrut +\mathstrut q^{25} \) \(\mathstrut -\mathstrut 2q^{26} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut +\mathstrut 8q^{28} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut +\mathstrut 4q^{30} \) \(\mathstrut -\mathstrut 9q^{31} \) \(\mathstrut +\mathstrut 8q^{32} \) \(\mathstrut +\mathstrut 2q^{33} \) \(\mathstrut -\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 4q^{36} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut -\mathstrut 2q^{38} \) \(\mathstrut -\mathstrut q^{39} \) \(\mathstrut -\mathstrut 20q^{41} \) \(\mathstrut -\mathstrut 10q^{42} \) \(\mathstrut +\mathstrut 10q^{43} \) \(\mathstrut +\mathstrut 4q^{44} \) \(\mathstrut +\mathstrut 2q^{45} \) \(\mathstrut +\mathstrut 6q^{47} \) \(\mathstrut -\mathstrut 8q^{48} \) \(\mathstrut +\mathstrut 11q^{49} \) \(\mathstrut -\mathstrut 4q^{50} \) \(\mathstrut -\mathstrut 2q^{52} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut -\mathstrut 2q^{54} \) \(\mathstrut +\mathstrut 8q^{55} \) \(\mathstrut +\mathstrut 2q^{57} \) \(\mathstrut -\mathstrut 8q^{58} \) \(\mathstrut +\mathstrut 12q^{59} \) \(\mathstrut +\mathstrut 4q^{60} \) \(\mathstrut -\mathstrut 10q^{61} \) \(\mathstrut +\mathstrut 36q^{62} \) \(\mathstrut +\mathstrut q^{63} \) \(\mathstrut -\mathstrut 16q^{64} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 4q^{66} \) \(\mathstrut +\mathstrut 5q^{67} \) \(\mathstrut -\mathstrut 16q^{70} \) \(\mathstrut -\mathstrut 12q^{71} \) \(\mathstrut +\mathstrut 3q^{73} \) \(\mathstrut -\mathstrut 6q^{74} \) \(\mathstrut +\mathstrut q^{75} \) \(\mathstrut +\mathstrut 4q^{76} \) \(\mathstrut -\mathstrut 8q^{77} \) \(\mathstrut +\mathstrut 4q^{78} \) \(\mathstrut +\mathstrut q^{79} \) \(\mathstrut -\mathstrut 8q^{80} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut +\mathstrut 20q^{82} \) \(\mathstrut +\mathstrut 12q^{83} \) \(\mathstrut +\mathstrut 2q^{84} \) \(\mathstrut -\mathstrut 10q^{86} \) \(\mathstrut -\mathstrut 4q^{87} \) \(\mathstrut -\mathstrut 16q^{89} \) \(\mathstrut -\mathstrut 8q^{90} \) \(\mathstrut -\mathstrut 5q^{91} \) \(\mathstrut -\mathstrut 9q^{93} \) \(\mathstrut +\mathstrut 12q^{94} \) \(\mathstrut +\mathstrut 2q^{95} \) \(\mathstrut +\mathstrut 8q^{96} \) \(\mathstrut -\mathstrut 12q^{97} \) \(\mathstrut +\mathstrut 4q^{98} \) \(\mathstrut -\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 1.73205i −0.500000 + 0.866025i −1.00000 + 1.73205i 1.00000 + 1.73205i 2.00000 −2.50000 0.866025i 0 −0.500000 0.866025i 2.00000 3.46410i
16.1 −1.00000 + 1.73205i −0.500000 0.866025i −1.00000 1.73205i 1.00000 1.73205i 2.00000 −2.50000 + 0.866025i 0 −0.500000 + 0.866025i 2.00000 + 3.46410i
\(n\): 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
7.c Even 1 yes

Hecke kernels

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