Properties

Label 21.3.f.a
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.

\(f(q)\) \(=\) \( q\) \( -3 \zeta_{6} q^{2} \) \( + ( -1 - \zeta_{6} ) q^{3} \) \( + ( -5 + 5 \zeta_{6} ) q^{4} \) \( + ( 6 - 3 \zeta_{6} ) q^{5} \) \( + ( -3 + 6 \zeta_{6} ) q^{6} \) \( + ( 5 + 3 \zeta_{6} ) q^{7} \) \( + 3 q^{8} \) \( + 3 \zeta_{6} q^{9} \) \(+O(q^{10})\) \( q\) \( -3 \zeta_{6} q^{2} \) \( + ( -1 - \zeta_{6} ) q^{3} \) \( + ( -5 + 5 \zeta_{6} ) q^{4} \) \( + ( 6 - 3 \zeta_{6} ) q^{5} \) \( + ( -3 + 6 \zeta_{6} ) q^{6} \) \( + ( 5 + 3 \zeta_{6} ) q^{7} \) \( + 3 q^{8} \) \( + 3 \zeta_{6} q^{9} \) \( + ( -9 - 9 \zeta_{6} ) q^{10} \) \( + ( -15 + 15 \zeta_{6} ) q^{11} \) \( + ( 10 - 5 \zeta_{6} ) q^{12} \) \( + ( 8 - 16 \zeta_{6} ) q^{13} \) \( + ( 9 - 24 \zeta_{6} ) q^{14} \) \( -9 q^{15} \) \( + 11 \zeta_{6} q^{16} \) \( + ( 6 + 6 \zeta_{6} ) q^{17} \) \( + ( 9 - 9 \zeta_{6} ) q^{18} \) \( + ( -12 + 6 \zeta_{6} ) q^{19} \) \( + ( -15 + 30 \zeta_{6} ) q^{20} \) \( + ( -2 - 11 \zeta_{6} ) q^{21} \) \( + 45 q^{22} \) \( + ( -3 - 3 \zeta_{6} ) q^{24} \) \( + ( 2 - 2 \zeta_{6} ) q^{25} \) \( + ( -48 + 24 \zeta_{6} ) q^{26} \) \( + ( 3 - 6 \zeta_{6} ) q^{27} \) \( + ( -40 + 25 \zeta_{6} ) q^{28} \) \( -9 q^{29} \) \( + 27 \zeta_{6} q^{30} \) \( + ( -7 - 7 \zeta_{6} ) q^{31} \) \( + ( 45 - 45 \zeta_{6} ) q^{32} \) \( + ( 30 - 15 \zeta_{6} ) q^{33} \) \( + ( 18 - 36 \zeta_{6} ) q^{34} \) \( + ( 39 - 6 \zeta_{6} ) q^{35} \) \( -15 q^{36} \) \( -10 \zeta_{6} q^{37} \) \( + ( 18 + 18 \zeta_{6} ) q^{38} \) \( + ( -24 + 24 \zeta_{6} ) q^{39} \) \( + ( 18 - 9 \zeta_{6} ) q^{40} \) \( + ( 6 - 12 \zeta_{6} ) q^{41} \) \( + ( -33 + 39 \zeta_{6} ) q^{42} \) \( -74 q^{43} \) \( -75 \zeta_{6} q^{44} \) \( + ( 9 + 9 \zeta_{6} ) q^{45} \) \( + ( 11 - 22 \zeta_{6} ) q^{48} \) \( + ( 16 + 39 \zeta_{6} ) q^{49} \) \( -6 q^{50} \) \( -18 \zeta_{6} q^{51} \) \( + ( 40 + 40 \zeta_{6} ) q^{52} \) \( + ( -33 + 33 \zeta_{6} ) q^{53} \) \( + ( -18 + 9 \zeta_{6} ) q^{54} \) \( + ( -45 + 90 \zeta_{6} ) q^{55} \) \( + ( 15 + 9 \zeta_{6} ) q^{56} \) \( + 18 q^{57} \) \( + 27 \zeta_{6} q^{58} \) \( + ( 9 + 9 \zeta_{6} ) q^{59} \) \( + ( 45 - 45 \zeta_{6} ) q^{60} \) \( + ( 104 - 52 \zeta_{6} ) q^{61} \) \( + ( -21 + 42 \zeta_{6} ) q^{62} \) \( + ( -9 + 24 \zeta_{6} ) q^{63} \) \( -91 q^{64} \) \( -72 \zeta_{6} q^{65} \) \( + ( -45 - 45 \zeta_{6} ) q^{66} \) \( + ( 76 - 76 \zeta_{6} ) q^{67} \) \( + ( -60 + 30 \zeta_{6} ) q^{68} \) \( + ( -18 - 99 \zeta_{6} ) q^{70} \) \( + 84 q^{71} \) \( + 9 \zeta_{6} q^{72} \) \( + ( -36 - 36 \zeta_{6} ) q^{73} \) \( + ( -30 + 30 \zeta_{6} ) q^{74} \) \( + ( -4 + 2 \zeta_{6} ) q^{75} \) \( + ( 30 - 60 \zeta_{6} ) q^{76} \) \( + ( -120 + 75 \zeta_{6} ) q^{77} \) \( + 72 q^{78} \) \( + 43 \zeta_{6} q^{79} \) \( + ( 33 + 33 \zeta_{6} ) q^{80} \) \( + ( -9 + 9 \zeta_{6} ) q^{81} \) \( + ( -36 + 18 \zeta_{6} ) q^{82} \) \( + ( 69 - 138 \zeta_{6} ) q^{83} \) \( + ( 65 - 10 \zeta_{6} ) q^{84} \) \( + 54 q^{85} \) \( + 222 \zeta_{6} q^{86} \) \( + ( 9 + 9 \zeta_{6} ) q^{87} \) \( + ( -45 + 45 \zeta_{6} ) q^{88} \) \( + ( -84 + 42 \zeta_{6} ) q^{89} \) \( + ( 27 - 54 \zeta_{6} ) q^{90} \) \( + ( 88 - 104 \zeta_{6} ) q^{91} \) \( + 21 \zeta_{6} q^{93} \) \( + ( -54 + 54 \zeta_{6} ) q^{95} \) \( + ( -90 + 45 \zeta_{6} ) q^{96} \) \( + ( -107 + 214 \zeta_{6} ) q^{97} \) \( + ( 117 - 165 \zeta_{6} ) q^{98} \) \( -45 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 5q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut +\mathstrut 13q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 5q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut +\mathstrut 13q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut -\mathstrut 27q^{10} \) \(\mathstrut -\mathstrut 15q^{11} \) \(\mathstrut +\mathstrut 15q^{12} \) \(\mathstrut -\mathstrut 6q^{14} \) \(\mathstrut -\mathstrut 18q^{15} \) \(\mathstrut +\mathstrut 11q^{16} \) \(\mathstrut +\mathstrut 18q^{17} \) \(\mathstrut +\mathstrut 9q^{18} \) \(\mathstrut -\mathstrut 18q^{19} \) \(\mathstrut -\mathstrut 15q^{21} \) \(\mathstrut +\mathstrut 90q^{22} \) \(\mathstrut -\mathstrut 9q^{24} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 72q^{26} \) \(\mathstrut -\mathstrut 55q^{28} \) \(\mathstrut -\mathstrut 18q^{29} \) \(\mathstrut +\mathstrut 27q^{30} \) \(\mathstrut -\mathstrut 21q^{31} \) \(\mathstrut +\mathstrut 45q^{32} \) \(\mathstrut +\mathstrut 45q^{33} \) \(\mathstrut +\mathstrut 72q^{35} \) \(\mathstrut -\mathstrut 30q^{36} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut +\mathstrut 54q^{38} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 27q^{40} \) \(\mathstrut -\mathstrut 27q^{42} \) \(\mathstrut -\mathstrut 148q^{43} \) \(\mathstrut -\mathstrut 75q^{44} \) \(\mathstrut +\mathstrut 27q^{45} \) \(\mathstrut +\mathstrut 71q^{49} \) \(\mathstrut -\mathstrut 12q^{50} \) \(\mathstrut -\mathstrut 18q^{51} \) \(\mathstrut +\mathstrut 120q^{52} \) \(\mathstrut -\mathstrut 33q^{53} \) \(\mathstrut -\mathstrut 27q^{54} \) \(\mathstrut +\mathstrut 39q^{56} \) \(\mathstrut +\mathstrut 36q^{57} \) \(\mathstrut +\mathstrut 27q^{58} \) \(\mathstrut +\mathstrut 27q^{59} \) \(\mathstrut +\mathstrut 45q^{60} \) \(\mathstrut +\mathstrut 156q^{61} \) \(\mathstrut +\mathstrut 6q^{63} \) \(\mathstrut -\mathstrut 182q^{64} \) \(\mathstrut -\mathstrut 72q^{65} \) \(\mathstrut -\mathstrut 135q^{66} \) \(\mathstrut +\mathstrut 76q^{67} \) \(\mathstrut -\mathstrut 90q^{68} \) \(\mathstrut -\mathstrut 135q^{70} \) \(\mathstrut +\mathstrut 168q^{71} \) \(\mathstrut +\mathstrut 9q^{72} \) \(\mathstrut -\mathstrut 108q^{73} \) \(\mathstrut -\mathstrut 30q^{74} \) \(\mathstrut -\mathstrut 6q^{75} \) \(\mathstrut -\mathstrut 165q^{77} \) \(\mathstrut +\mathstrut 144q^{78} \) \(\mathstrut +\mathstrut 43q^{79} \) \(\mathstrut +\mathstrut 99q^{80} \) \(\mathstrut -\mathstrut 9q^{81} \) \(\mathstrut -\mathstrut 54q^{82} \) \(\mathstrut +\mathstrut 120q^{84} \) \(\mathstrut +\mathstrut 108q^{85} \) \(\mathstrut +\mathstrut 222q^{86} \) \(\mathstrut +\mathstrut 27q^{87} \) \(\mathstrut -\mathstrut 45q^{88} \) \(\mathstrut -\mathstrut 126q^{89} \) \(\mathstrut +\mathstrut 72q^{91} \) \(\mathstrut +\mathstrut 21q^{93} \) \(\mathstrut -\mathstrut 54q^{95} \) \(\mathstrut -\mathstrut 135q^{96} \) \(\mathstrut +\mathstrut 69q^{98} \) \(\mathstrut -\mathstrut 90q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−1.50000 2.59808i −1.50000 0.866025i −2.50000 + 4.33013i 4.50000 2.59808i 5.19615i 6.50000 + 2.59808i 3.00000 1.50000 + 2.59808i −13.5000 7.79423i
19.1 −1.50000 + 2.59808i −1.50000 + 0.866025i −2.50000 4.33013i 4.50000 + 2.59808i 5.19615i 6.50000 2.59808i 3.00000 1.50000 2.59808i −13.5000 + 7.79423i
\(n\): 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 \(T_{2}^{2} \) \(\mathstrut +\mathstrut 3 T_{2} \) \(\mathstrut +\mathstrut 9 \) acting on \(S_{3}^{\mathrm{new}}(21, [\chi])\).