Properties

Label 21.3.f.c
Level 21
Weight 3
Character orbit 21.f
Analytic conductor 0.572
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 21.f (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + 2 \zeta_{6} q^{2} \) \( + ( -1 - \zeta_{6} ) q^{3} \) \( + ( -4 + 2 \zeta_{6} ) q^{5} \) \( + ( 2 - 4 \zeta_{6} ) q^{6} \) \( -7 \zeta_{6} q^{7} \) \( + 8 q^{8} \) \( + 3 \zeta_{6} q^{9} \) \(+O(q^{10})\) \( q\) \( + 2 \zeta_{6} q^{2} \) \( + ( -1 - \zeta_{6} ) q^{3} \) \( + ( -4 + 2 \zeta_{6} ) q^{5} \) \( + ( 2 - 4 \zeta_{6} ) q^{6} \) \( -7 \zeta_{6} q^{7} \) \( + 8 q^{8} \) \( + 3 \zeta_{6} q^{9} \) \( + ( -4 - 4 \zeta_{6} ) q^{10} \) \( + ( -10 + 10 \zeta_{6} ) q^{11} \) \( + ( -7 + 14 \zeta_{6} ) q^{13} \) \( + ( 14 - 14 \zeta_{6} ) q^{14} \) \( + 6 q^{15} \) \( + 16 \zeta_{6} q^{16} \) \( + ( -4 - 4 \zeta_{6} ) q^{17} \) \( + ( -6 + 6 \zeta_{6} ) q^{18} \) \( + ( 38 - 19 \zeta_{6} ) q^{19} \) \( + ( -7 + 14 \zeta_{6} ) q^{21} \) \( -20 q^{22} \) \( -40 \zeta_{6} q^{23} \) \( + ( -8 - 8 \zeta_{6} ) q^{24} \) \( + ( -13 + 13 \zeta_{6} ) q^{25} \) \( + ( -28 + 14 \zeta_{6} ) q^{26} \) \( + ( 3 - 6 \zeta_{6} ) q^{27} \) \( + 16 q^{29} \) \( + 12 \zeta_{6} q^{30} \) \( + ( 3 + 3 \zeta_{6} ) q^{31} \) \( + ( 20 - 10 \zeta_{6} ) q^{33} \) \( + ( 8 - 16 \zeta_{6} ) q^{34} \) \( + ( 14 + 14 \zeta_{6} ) q^{35} \) \( -5 \zeta_{6} q^{37} \) \( + ( 38 + 38 \zeta_{6} ) q^{38} \) \( + ( 21 - 21 \zeta_{6} ) q^{39} \) \( + ( -32 + 16 \zeta_{6} ) q^{40} \) \( + ( -14 + 28 \zeta_{6} ) q^{41} \) \( + ( -28 + 14 \zeta_{6} ) q^{42} \) \( -19 q^{43} \) \( + ( -6 - 6 \zeta_{6} ) q^{45} \) \( + ( 80 - 80 \zeta_{6} ) q^{46} \) \( + ( -60 + 30 \zeta_{6} ) q^{47} \) \( + ( 16 - 32 \zeta_{6} ) q^{48} \) \( + ( -49 + 49 \zeta_{6} ) q^{49} \) \( -26 q^{50} \) \( + 12 \zeta_{6} q^{51} \) \( + ( 32 - 32 \zeta_{6} ) q^{53} \) \( + ( 12 - 6 \zeta_{6} ) q^{54} \) \( + ( 20 - 40 \zeta_{6} ) q^{55} \) \( -56 \zeta_{6} q^{56} \) \( -57 q^{57} \) \( + 32 \zeta_{6} q^{58} \) \( + ( 24 + 24 \zeta_{6} ) q^{59} \) \( + ( 24 - 12 \zeta_{6} ) q^{61} \) \( + ( -6 + 12 \zeta_{6} ) q^{62} \) \( + ( 21 - 21 \zeta_{6} ) q^{63} \) \( + 64 q^{64} \) \( -42 \zeta_{6} q^{65} \) \( + ( 20 + 20 \zeta_{6} ) q^{66} \) \( + ( -59 + 59 \zeta_{6} ) q^{67} \) \( + ( -40 + 80 \zeta_{6} ) q^{69} \) \( + ( -28 + 56 \zeta_{6} ) q^{70} \) \( -26 q^{71} \) \( + 24 \zeta_{6} q^{72} \) \( + ( -11 - 11 \zeta_{6} ) q^{73} \) \( + ( 10 - 10 \zeta_{6} ) q^{74} \) \( + ( 26 - 13 \zeta_{6} ) q^{75} \) \( + 70 q^{77} \) \( + 42 q^{78} \) \( -47 \zeta_{6} q^{79} \) \( + ( -32 - 32 \zeta_{6} ) q^{80} \) \( + ( -9 + 9 \zeta_{6} ) q^{81} \) \( + ( -56 + 28 \zeta_{6} ) q^{82} \) \( + ( 14 - 28 \zeta_{6} ) q^{83} \) \( + 24 q^{85} \) \( -38 \zeta_{6} q^{86} \) \( + ( -16 - 16 \zeta_{6} ) q^{87} \) \( + ( -80 + 80 \zeta_{6} ) q^{88} \) \( + ( 136 - 68 \zeta_{6} ) q^{89} \) \( + ( 12 - 24 \zeta_{6} ) q^{90} \) \( + ( 98 - 49 \zeta_{6} ) q^{91} \) \( -9 \zeta_{6} q^{93} \) \( + ( -60 - 60 \zeta_{6} ) q^{94} \) \( + ( -114 + 114 \zeta_{6} ) q^{95} \) \( + ( 28 - 56 \zeta_{6} ) q^{97} \) \( -98 q^{98} \) \( -30 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 16q^{8} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 16q^{8} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut -\mathstrut 12q^{10} \) \(\mathstrut -\mathstrut 10q^{11} \) \(\mathstrut +\mathstrut 14q^{14} \) \(\mathstrut +\mathstrut 12q^{15} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut -\mathstrut 6q^{18} \) \(\mathstrut +\mathstrut 57q^{19} \) \(\mathstrut -\mathstrut 40q^{22} \) \(\mathstrut -\mathstrut 40q^{23} \) \(\mathstrut -\mathstrut 24q^{24} \) \(\mathstrut -\mathstrut 13q^{25} \) \(\mathstrut -\mathstrut 42q^{26} \) \(\mathstrut +\mathstrut 32q^{29} \) \(\mathstrut +\mathstrut 12q^{30} \) \(\mathstrut +\mathstrut 9q^{31} \) \(\mathstrut +\mathstrut 30q^{33} \) \(\mathstrut +\mathstrut 42q^{35} \) \(\mathstrut -\mathstrut 5q^{37} \) \(\mathstrut +\mathstrut 114q^{38} \) \(\mathstrut +\mathstrut 21q^{39} \) \(\mathstrut -\mathstrut 48q^{40} \) \(\mathstrut -\mathstrut 42q^{42} \) \(\mathstrut -\mathstrut 38q^{43} \) \(\mathstrut -\mathstrut 18q^{45} \) \(\mathstrut +\mathstrut 80q^{46} \) \(\mathstrut -\mathstrut 90q^{47} \) \(\mathstrut -\mathstrut 49q^{49} \) \(\mathstrut -\mathstrut 52q^{50} \) \(\mathstrut +\mathstrut 12q^{51} \) \(\mathstrut +\mathstrut 32q^{53} \) \(\mathstrut +\mathstrut 18q^{54} \) \(\mathstrut -\mathstrut 56q^{56} \) \(\mathstrut -\mathstrut 114q^{57} \) \(\mathstrut +\mathstrut 32q^{58} \) \(\mathstrut +\mathstrut 72q^{59} \) \(\mathstrut +\mathstrut 36q^{61} \) \(\mathstrut +\mathstrut 21q^{63} \) \(\mathstrut +\mathstrut 128q^{64} \) \(\mathstrut -\mathstrut 42q^{65} \) \(\mathstrut +\mathstrut 60q^{66} \) \(\mathstrut -\mathstrut 59q^{67} \) \(\mathstrut -\mathstrut 52q^{71} \) \(\mathstrut +\mathstrut 24q^{72} \) \(\mathstrut -\mathstrut 33q^{73} \) \(\mathstrut +\mathstrut 10q^{74} \) \(\mathstrut +\mathstrut 39q^{75} \) \(\mathstrut +\mathstrut 140q^{77} \) \(\mathstrut +\mathstrut 84q^{78} \) \(\mathstrut -\mathstrut 47q^{79} \) \(\mathstrut -\mathstrut 96q^{80} \) \(\mathstrut -\mathstrut 9q^{81} \) \(\mathstrut -\mathstrut 84q^{82} \) \(\mathstrut +\mathstrut 48q^{85} \) \(\mathstrut -\mathstrut 38q^{86} \) \(\mathstrut -\mathstrut 48q^{87} \) \(\mathstrut -\mathstrut 80q^{88} \) \(\mathstrut +\mathstrut 204q^{89} \) \(\mathstrut +\mathstrut 147q^{91} \) \(\mathstrut -\mathstrut 9q^{93} \) \(\mathstrut -\mathstrut 180q^{94} \) \(\mathstrut -\mathstrut 114q^{95} \) \(\mathstrut -\mathstrut 196q^{98} \) \(\mathstrut -\mathstrut 60q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 + 1.73205i −1.50000 0.866025i 0 −3.00000 + 1.73205i 3.46410i −3.50000 6.06218i 8.00000 1.50000 + 2.59808i −6.00000 3.46410i
19.1 1.00000 1.73205i −1.50000 + 0.866025i 0 −3.00000 1.73205i 3.46410i −3.50000 + 6.06218i 8.00000 1.50000 2.59808i −6.00000 + 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.d Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 2 T_{2} \) \(\mathstrut +\mathstrut 4 \) acting on \(S_{3}^{\mathrm{new}}(21, [\chi])\).