Properties

Label 21.3.f.b
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q\) \( -\zeta_{6} q^{2} \) \( + ( 1 + \zeta_{6} ) q^{3} \) \( + ( 3 - 3 \zeta_{6} ) q^{4} \) \( + ( -6 + 3 \zeta_{6} ) q^{5} \) \( + ( 1 - 2 \zeta_{6} ) q^{6} \) \( + ( -7 + 7 \zeta_{6} ) q^{7} \) \( -7 q^{8} \) \( + 3 \zeta_{6} q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{6} q^{2} \) \( + ( 1 + \zeta_{6} ) q^{3} \) \( + ( 3 - 3 \zeta_{6} ) q^{4} \) \( + ( -6 + 3 \zeta_{6} ) q^{5} \) \( + ( 1 - 2 \zeta_{6} ) q^{6} \) \( + ( -7 + 7 \zeta_{6} ) q^{7} \) \( -7 q^{8} \) \( + 3 \zeta_{6} q^{9} \) \( + ( 3 + 3 \zeta_{6} ) q^{10} \) \( + ( 11 - 11 \zeta_{6} ) q^{11} \) \( + ( 6 - 3 \zeta_{6} ) q^{12} \) \( + ( 4 - 8 \zeta_{6} ) q^{13} \) \( + 7 q^{14} \) \( -9 q^{15} \) \( -5 \zeta_{6} q^{16} \) \( + ( 14 + 14 \zeta_{6} ) q^{17} \) \( + ( 3 - 3 \zeta_{6} ) q^{18} \) \( + ( -4 + 2 \zeta_{6} ) q^{19} \) \( + ( -9 + 18 \zeta_{6} ) q^{20} \) \( + ( -14 + 7 \zeta_{6} ) q^{21} \) \( -11 q^{22} \) \( -28 \zeta_{6} q^{23} \) \( + ( -7 - 7 \zeta_{6} ) q^{24} \) \( + ( 2 - 2 \zeta_{6} ) q^{25} \) \( + ( -8 + 4 \zeta_{6} ) q^{26} \) \( + ( -3 + 6 \zeta_{6} ) q^{27} \) \( + 21 \zeta_{6} q^{28} \) \( + 25 q^{29} \) \( + 9 \zeta_{6} q^{30} \) \( + ( -19 - 19 \zeta_{6} ) q^{31} \) \( + ( -33 + 33 \zeta_{6} ) q^{32} \) \( + ( 22 - 11 \zeta_{6} ) q^{33} \) \( + ( 14 - 28 \zeta_{6} ) q^{34} \) \( + ( 21 - 42 \zeta_{6} ) q^{35} \) \( + 9 q^{36} \) \( + 58 \zeta_{6} q^{37} \) \( + ( 2 + 2 \zeta_{6} ) q^{38} \) \( + ( 12 - 12 \zeta_{6} ) q^{39} \) \( + ( 42 - 21 \zeta_{6} ) q^{40} \) \( + ( 2 - 4 \zeta_{6} ) q^{41} \) \( + ( 7 + 7 \zeta_{6} ) q^{42} \) \( + 26 q^{43} \) \( -33 \zeta_{6} q^{44} \) \( + ( -9 - 9 \zeta_{6} ) q^{45} \) \( + ( -28 + 28 \zeta_{6} ) q^{46} \) \( + ( -88 + 44 \zeta_{6} ) q^{47} \) \( + ( 5 - 10 \zeta_{6} ) q^{48} \) \( -49 \zeta_{6} q^{49} \) \( -2 q^{50} \) \( + 42 \zeta_{6} q^{51} \) \( + ( -12 - 12 \zeta_{6} ) q^{52} \) \( + ( -31 + 31 \zeta_{6} ) q^{53} \) \( + ( 6 - 3 \zeta_{6} ) q^{54} \) \( + ( -33 + 66 \zeta_{6} ) q^{55} \) \( + ( 49 - 49 \zeta_{6} ) q^{56} \) \( -6 q^{57} \) \( -25 \zeta_{6} q^{58} \) \( + ( -5 - 5 \zeta_{6} ) q^{59} \) \( + ( -27 + 27 \zeta_{6} ) q^{60} \) \( + ( 16 - 8 \zeta_{6} ) q^{61} \) \( + ( -19 + 38 \zeta_{6} ) q^{62} \) \( -21 q^{63} \) \( + 13 q^{64} \) \( + 36 \zeta_{6} q^{65} \) \( + ( -11 - 11 \zeta_{6} ) q^{66} \) \( + ( 52 - 52 \zeta_{6} ) q^{67} \) \( + ( 84 - 42 \zeta_{6} ) q^{68} \) \( + ( 28 - 56 \zeta_{6} ) q^{69} \) \( + ( -42 + 21 \zeta_{6} ) q^{70} \) \( + 64 q^{71} \) \( -21 \zeta_{6} q^{72} \) \( + ( 4 + 4 \zeta_{6} ) q^{73} \) \( + ( 58 - 58 \zeta_{6} ) q^{74} \) \( + ( 4 - 2 \zeta_{6} ) q^{75} \) \( + ( -6 + 12 \zeta_{6} ) q^{76} \) \( + 77 \zeta_{6} q^{77} \) \( -12 q^{78} \) \( -17 \zeta_{6} q^{79} \) \( + ( 15 + 15 \zeta_{6} ) q^{80} \) \( + ( -9 + 9 \zeta_{6} ) q^{81} \) \( + ( -4 + 2 \zeta_{6} ) q^{82} \) \( + ( 31 - 62 \zeta_{6} ) q^{83} \) \( + ( -21 + 42 \zeta_{6} ) q^{84} \) \( -126 q^{85} \) \( -26 \zeta_{6} q^{86} \) \( + ( 25 + 25 \zeta_{6} ) q^{87} \) \( + ( -77 + 77 \zeta_{6} ) q^{88} \) \( + ( -92 + 46 \zeta_{6} ) q^{89} \) \( + ( -9 + 18 \zeta_{6} ) q^{90} \) \( + ( 28 + 28 \zeta_{6} ) q^{91} \) \( -84 q^{92} \) \( -57 \zeta_{6} q^{93} \) \( + ( 44 + 44 \zeta_{6} ) q^{94} \) \( + ( 18 - 18 \zeta_{6} ) q^{95} \) \( + ( -66 + 33 \zeta_{6} ) q^{96} \) \( + ( 53 - 106 \zeta_{6} ) q^{97} \) \( + ( -49 + 49 \zeta_{6} ) q^{98} \) \( + 33 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 7q^{7} \) \(\mathstrut -\mathstrut 14q^{8} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 7q^{7} \) \(\mathstrut -\mathstrut 14q^{8} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut 9q^{10} \) \(\mathstrut +\mathstrut 11q^{11} \) \(\mathstrut +\mathstrut 9q^{12} \) \(\mathstrut +\mathstrut 14q^{14} \) \(\mathstrut -\mathstrut 18q^{15} \) \(\mathstrut -\mathstrut 5q^{16} \) \(\mathstrut +\mathstrut 42q^{17} \) \(\mathstrut +\mathstrut 3q^{18} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut 21q^{21} \) \(\mathstrut -\mathstrut 22q^{22} \) \(\mathstrut -\mathstrut 28q^{23} \) \(\mathstrut -\mathstrut 21q^{24} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 12q^{26} \) \(\mathstrut +\mathstrut 21q^{28} \) \(\mathstrut +\mathstrut 50q^{29} \) \(\mathstrut +\mathstrut 9q^{30} \) \(\mathstrut -\mathstrut 57q^{31} \) \(\mathstrut -\mathstrut 33q^{32} \) \(\mathstrut +\mathstrut 33q^{33} \) \(\mathstrut +\mathstrut 18q^{36} \) \(\mathstrut +\mathstrut 58q^{37} \) \(\mathstrut +\mathstrut 6q^{38} \) \(\mathstrut +\mathstrut 12q^{39} \) \(\mathstrut +\mathstrut 63q^{40} \) \(\mathstrut +\mathstrut 21q^{42} \) \(\mathstrut +\mathstrut 52q^{43} \) \(\mathstrut -\mathstrut 33q^{44} \) \(\mathstrut -\mathstrut 27q^{45} \) \(\mathstrut -\mathstrut 28q^{46} \) \(\mathstrut -\mathstrut 132q^{47} \) \(\mathstrut -\mathstrut 49q^{49} \) \(\mathstrut -\mathstrut 4q^{50} \) \(\mathstrut +\mathstrut 42q^{51} \) \(\mathstrut -\mathstrut 36q^{52} \) \(\mathstrut -\mathstrut 31q^{53} \) \(\mathstrut +\mathstrut 9q^{54} \) \(\mathstrut +\mathstrut 49q^{56} \) \(\mathstrut -\mathstrut 12q^{57} \) \(\mathstrut -\mathstrut 25q^{58} \) \(\mathstrut -\mathstrut 15q^{59} \) \(\mathstrut -\mathstrut 27q^{60} \) \(\mathstrut +\mathstrut 24q^{61} \) \(\mathstrut -\mathstrut 42q^{63} \) \(\mathstrut +\mathstrut 26q^{64} \) \(\mathstrut +\mathstrut 36q^{65} \) \(\mathstrut -\mathstrut 33q^{66} \) \(\mathstrut +\mathstrut 52q^{67} \) \(\mathstrut +\mathstrut 126q^{68} \) \(\mathstrut -\mathstrut 63q^{70} \) \(\mathstrut +\mathstrut 128q^{71} \) \(\mathstrut -\mathstrut 21q^{72} \) \(\mathstrut +\mathstrut 12q^{73} \) \(\mathstrut +\mathstrut 58q^{74} \) \(\mathstrut +\mathstrut 6q^{75} \) \(\mathstrut +\mathstrut 77q^{77} \) \(\mathstrut -\mathstrut 24q^{78} \) \(\mathstrut -\mathstrut 17q^{79} \) \(\mathstrut +\mathstrut 45q^{80} \) \(\mathstrut -\mathstrut 9q^{81} \) \(\mathstrut -\mathstrut 6q^{82} \) \(\mathstrut -\mathstrut 252q^{85} \) \(\mathstrut -\mathstrut 26q^{86} \) \(\mathstrut +\mathstrut 75q^{87} \) \(\mathstrut -\mathstrut 77q^{88} \) \(\mathstrut -\mathstrut 138q^{89} \) \(\mathstrut +\mathstrut 84q^{91} \) \(\mathstrut -\mathstrut 168q^{92} \) \(\mathstrut -\mathstrut 57q^{93} \) \(\mathstrut +\mathstrut 132q^{94} \) \(\mathstrut +\mathstrut 18q^{95} \) \(\mathstrut -\mathstrut 99q^{96} \) \(\mathstrut -\mathstrut 49q^{98} \) \(\mathstrut +\mathstrut 66q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−0.500000 0.866025i 1.50000 + 0.866025i 1.50000 2.59808i −4.50000 + 2.59808i 1.73205i −3.50000 + 6.06218i −7.00000 1.50000 + 2.59808i 4.50000 + 2.59808i
19.1 −0.500000 + 0.866025i 1.50000 0.866025i 1.50000 + 2.59808i −4.50000 2.59808i 1.73205i −3.50000 6.06218i −7.00000 1.50000 2.59808i 4.50000 2.59808i
\(n\): 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 \(T_{2}^{2} \) \(\mathstrut +\mathstrut T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{3}^{\mathrm{new}}(21, [\chi])\).