Properties

Label 21.3.h.b
Level 21
Weight 3
Character orbit 21.h
Analytic conductor 0.572
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 21.h (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.572208555157\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{3} \) \( + \beta_{2} q^{4} \) \( -\beta_{1} q^{5} \) \( + ( -5 - 2 \beta_{3} ) q^{6} \) \( + 7 \beta_{2} q^{7} \) \( -3 \beta_{3} q^{8} \) \( + ( 1 + 4 \beta_{1} - \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{3} \) \( + \beta_{2} q^{4} \) \( -\beta_{1} q^{5} \) \( + ( -5 - 2 \beta_{3} ) q^{6} \) \( + 7 \beta_{2} q^{7} \) \( -3 \beta_{3} q^{8} \) \( + ( 1 + 4 \beta_{1} - \beta_{2} ) q^{9} \) \( -5 \beta_{2} q^{10} \) \( + ( -5 \beta_{1} + 5 \beta_{3} ) q^{11} \) \( + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{12} \) \( -2 q^{13} \) \( + 7 \beta_{3} q^{14} \) \( + ( 5 + 2 \beta_{3} ) q^{15} \) \( + ( 19 - 19 \beta_{2} ) q^{16} \) \( + ( 12 \beta_{1} - 12 \beta_{3} ) q^{17} \) \( + ( \beta_{1} + 20 \beta_{2} - \beta_{3} ) q^{18} \) \( + ( -16 + 16 \beta_{2} ) q^{19} \) \( -\beta_{3} q^{20} \) \( + ( 14 - 7 \beta_{1} - 14 \beta_{2} ) q^{21} \) \( -25 q^{22} \) \( -6 \beta_{1} q^{23} \) \( + ( -6 \beta_{1} + 15 \beta_{2} + 6 \beta_{3} ) q^{24} \) \( -20 \beta_{2} q^{25} \) \( -2 \beta_{1} q^{26} \) \( + ( -22 - 7 \beta_{3} ) q^{27} \) \( + ( -7 + 7 \beta_{2} ) q^{28} \) \( + 7 \beta_{3} q^{29} \) \( + ( -10 + 5 \beta_{1} + 10 \beta_{2} ) q^{30} \) \( + 3 \beta_{2} q^{31} \) \( + ( 7 \beta_{1} - 7 \beta_{3} ) q^{32} \) \( + ( 25 + 10 \beta_{1} - 25 \beta_{2} ) q^{33} \) \( + 60 q^{34} \) \( -7 \beta_{3} q^{35} \) \( + ( 1 + 4 \beta_{3} ) q^{36} \) \( + ( -12 + 12 \beta_{2} ) q^{37} \) \( + ( -16 \beta_{1} + 16 \beta_{3} ) q^{38} \) \( + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{39} \) \( + ( -15 + 15 \beta_{2} ) q^{40} \) \( + 14 \beta_{3} q^{41} \) \( + ( 14 \beta_{1} - 35 \beta_{2} - 14 \beta_{3} ) q^{42} \) \( + 44 q^{43} \) \( -5 \beta_{1} q^{44} \) \( + ( -\beta_{1} - 20 \beta_{2} + \beta_{3} ) q^{45} \) \( -30 \beta_{2} q^{46} \) \( + 6 \beta_{1} q^{47} \) \( + ( -38 + 19 \beta_{3} ) q^{48} \) \( + ( -49 + 49 \beta_{2} ) q^{49} \) \( -20 \beta_{3} q^{50} \) \( + ( -60 - 24 \beta_{1} + 60 \beta_{2} ) q^{51} \) \( -2 \beta_{2} q^{52} \) \( + ( 9 \beta_{1} - 9 \beta_{3} ) q^{53} \) \( + ( 35 - 22 \beta_{1} - 35 \beta_{2} ) q^{54} \) \( + 25 q^{55} \) \( + ( 21 \beta_{1} - 21 \beta_{3} ) q^{56} \) \( + ( 32 - 16 \beta_{3} ) q^{57} \) \( + ( -35 + 35 \beta_{2} ) q^{58} \) \( + ( -9 \beta_{1} + 9 \beta_{3} ) q^{59} \) \( + ( -2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{60} \) \( + ( 26 - 26 \beta_{2} ) q^{61} \) \( + 3 \beta_{3} q^{62} \) \( + ( 7 + 28 \beta_{3} ) q^{63} \) \( -41 q^{64} \) \( + 2 \beta_{1} q^{65} \) \( + ( 25 \beta_{1} + 50 \beta_{2} - 25 \beta_{3} ) q^{66} \) \( -52 \beta_{2} q^{67} \) \( + 12 \beta_{1} q^{68} \) \( + ( 30 + 12 \beta_{3} ) q^{69} \) \( + ( 35 - 35 \beta_{2} ) q^{70} \) \( + 42 \beta_{3} q^{71} \) \( + ( 60 - 3 \beta_{1} - 60 \beta_{2} ) q^{72} \) \( -18 \beta_{2} q^{73} \) \( + ( -12 \beta_{1} + 12 \beta_{3} ) q^{74} \) \( + ( -40 + 20 \beta_{1} + 40 \beta_{2} ) q^{75} \) \( -16 q^{76} \) \( -35 \beta_{1} q^{77} \) \( + ( 10 + 4 \beta_{3} ) q^{78} \) \( + ( 79 - 79 \beta_{2} ) q^{79} \) \( + ( -19 \beta_{1} + 19 \beta_{3} ) q^{80} \) \( + ( 8 \beta_{1} + 79 \beta_{2} - 8 \beta_{3} ) q^{81} \) \( + ( -70 + 70 \beta_{2} ) q^{82} \) \( -63 \beta_{3} q^{83} \) \( + ( 14 - 7 \beta_{3} ) q^{84} \) \( -60 q^{85} \) \( + 44 \beta_{1} q^{86} \) \( + ( 14 \beta_{1} - 35 \beta_{2} - 14 \beta_{3} ) q^{87} \) \( + 75 \beta_{2} q^{88} \) \( -22 \beta_{1} q^{89} \) \( + ( -5 - 20 \beta_{3} ) q^{90} \) \( -14 \beta_{2} q^{91} \) \( -6 \beta_{3} q^{92} \) \( + ( 6 - 3 \beta_{1} - 6 \beta_{2} ) q^{93} \) \( + 30 \beta_{2} q^{94} \) \( + ( 16 \beta_{1} - 16 \beta_{3} ) q^{95} \) \( + ( -35 - 14 \beta_{1} + 35 \beta_{2} ) q^{96} \) \( -93 q^{97} \) \( + ( -49 \beta_{1} + 49 \beta_{3} ) q^{98} \) \( + ( -100 + 5 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 20q^{6} \) \(\mathstrut +\mathstrut 14q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 20q^{6} \) \(\mathstrut +\mathstrut 14q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 10q^{10} \) \(\mathstrut +\mathstrut 4q^{12} \) \(\mathstrut -\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut 20q^{15} \) \(\mathstrut +\mathstrut 38q^{16} \) \(\mathstrut +\mathstrut 40q^{18} \) \(\mathstrut -\mathstrut 32q^{19} \) \(\mathstrut +\mathstrut 28q^{21} \) \(\mathstrut -\mathstrut 100q^{22} \) \(\mathstrut +\mathstrut 30q^{24} \) \(\mathstrut -\mathstrut 40q^{25} \) \(\mathstrut -\mathstrut 88q^{27} \) \(\mathstrut -\mathstrut 14q^{28} \) \(\mathstrut -\mathstrut 20q^{30} \) \(\mathstrut +\mathstrut 6q^{31} \) \(\mathstrut +\mathstrut 50q^{33} \) \(\mathstrut +\mathstrut 240q^{34} \) \(\mathstrut +\mathstrut 4q^{36} \) \(\mathstrut -\mathstrut 24q^{37} \) \(\mathstrut +\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 30q^{40} \) \(\mathstrut -\mathstrut 70q^{42} \) \(\mathstrut +\mathstrut 176q^{43} \) \(\mathstrut -\mathstrut 40q^{45} \) \(\mathstrut -\mathstrut 60q^{46} \) \(\mathstrut -\mathstrut 152q^{48} \) \(\mathstrut -\mathstrut 98q^{49} \) \(\mathstrut -\mathstrut 120q^{51} \) \(\mathstrut -\mathstrut 4q^{52} \) \(\mathstrut +\mathstrut 70q^{54} \) \(\mathstrut +\mathstrut 100q^{55} \) \(\mathstrut +\mathstrut 128q^{57} \) \(\mathstrut -\mathstrut 70q^{58} \) \(\mathstrut +\mathstrut 10q^{60} \) \(\mathstrut +\mathstrut 52q^{61} \) \(\mathstrut +\mathstrut 28q^{63} \) \(\mathstrut -\mathstrut 164q^{64} \) \(\mathstrut +\mathstrut 100q^{66} \) \(\mathstrut -\mathstrut 104q^{67} \) \(\mathstrut +\mathstrut 120q^{69} \) \(\mathstrut +\mathstrut 70q^{70} \) \(\mathstrut +\mathstrut 120q^{72} \) \(\mathstrut -\mathstrut 36q^{73} \) \(\mathstrut -\mathstrut 80q^{75} \) \(\mathstrut -\mathstrut 64q^{76} \) \(\mathstrut +\mathstrut 40q^{78} \) \(\mathstrut +\mathstrut 158q^{79} \) \(\mathstrut +\mathstrut 158q^{81} \) \(\mathstrut -\mathstrut 140q^{82} \) \(\mathstrut +\mathstrut 56q^{84} \) \(\mathstrut -\mathstrut 240q^{85} \) \(\mathstrut -\mathstrut 70q^{87} \) \(\mathstrut +\mathstrut 150q^{88} \) \(\mathstrut -\mathstrut 20q^{90} \) \(\mathstrut -\mathstrut 28q^{91} \) \(\mathstrut +\mathstrut 12q^{93} \) \(\mathstrut +\mathstrut 60q^{94} \) \(\mathstrut -\mathstrut 70q^{96} \) \(\mathstrut -\mathstrut 372q^{97} \) \(\mathstrut -\mathstrut 400q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(5\) \(x^{2}\mathstrut +\mathstrut \) \(25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5\) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(5\) \(\beta_{3}\)

Character Values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−1.93649 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 1.11803i
−1.93649 1.11803i 0.936492 2.85008i 0.500000 + 0.866025i 1.93649 + 1.11803i −5.00000 + 4.47214i 3.50000 + 6.06218i 6.70820i −7.24597 5.33816i −2.50000 4.33013i
2.2 1.93649 + 1.11803i −2.93649 0.614017i 0.500000 + 0.866025i −1.93649 1.11803i −5.00000 4.47214i 3.50000 + 6.06218i 6.70820i 8.24597 + 3.60611i −2.50000 4.33013i
11.1 −1.93649 + 1.11803i 0.936492 + 2.85008i 0.500000 0.866025i 1.93649 1.11803i −5.00000 4.47214i 3.50000 6.06218i 6.70820i −7.24597 + 5.33816i −2.50000 + 4.33013i
11.2 1.93649 1.11803i −2.93649 + 0.614017i 0.500000 0.866025i −1.93649 + 1.11803i −5.00000 + 4.47214i 3.50000 6.06218i 6.70820i 8.24597 3.60611i −2.50000 + 4.33013i
\(n\): 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 yes
7.c Even 1 yes
21.h Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut -\mathstrut 5 T_{2}^{2} \) \(\mathstrut +\mathstrut 25 \) acting on \(S_{3}^{\mathrm{new}}(21, [\chi])\).