Properties

Label 21.3.d.a
Level 21
Weight 3
Character orbit 21.d
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.d (of order \(2\) and degree \(1\))

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: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} -3 q^{4} -4 \beta q^{5} + \beta q^{6} + ( 1 + 4 \beta ) q^{7} -7 q^{8} -3 q^{9} +O(q^{10})\) \( q + q^{2} + \beta q^{3} -3 q^{4} -4 \beta q^{5} + \beta q^{6} + ( 1 + 4 \beta ) q^{7} -7 q^{8} -3 q^{9} -4 \beta q^{10} + 10 q^{11} -3 \beta q^{12} + 4 \beta q^{13} + ( 1 + 4 \beta ) q^{14} + 12 q^{15} + 5 q^{16} -3 q^{18} -12 \beta q^{19} + 12 \beta q^{20} + ( -12 + \beta ) q^{21} + 10 q^{22} -14 q^{23} -7 \beta q^{24} -23 q^{25} + 4 \beta q^{26} -3 \beta q^{27} + ( -3 - 12 \beta ) q^{28} -38 q^{29} + 12 q^{30} + 16 \beta q^{31} + 33 q^{32} + 10 \beta q^{33} + ( 48 - 4 \beta ) q^{35} + 9 q^{36} + 26 q^{37} -12 \beta q^{38} -12 q^{39} + 28 \beta q^{40} -40 \beta q^{41} + ( -12 + \beta ) q^{42} + 26 q^{43} -30 q^{44} + 12 \beta q^{45} -14 q^{46} + 16 \beta q^{47} + 5 \beta q^{48} + ( -47 + 8 \beta ) q^{49} -23 q^{50} -12 \beta q^{52} + 10 q^{53} -3 \beta q^{54} -40 \beta q^{55} + ( -7 - 28 \beta ) q^{56} + 36 q^{57} -38 q^{58} + 44 \beta q^{59} -36 q^{60} + 20 \beta q^{61} + 16 \beta q^{62} + ( -3 - 12 \beta ) q^{63} + 13 q^{64} + 48 q^{65} + 10 \beta q^{66} + 74 q^{67} -14 \beta q^{69} + ( 48 - 4 \beta ) q^{70} -62 q^{71} + 21 q^{72} -24 \beta q^{73} + 26 q^{74} -23 \beta q^{75} + 36 \beta q^{76} + ( 10 + 40 \beta ) q^{77} -12 q^{78} -46 q^{79} -20 \beta q^{80} + 9 q^{81} -40 \beta q^{82} + 52 \beta q^{83} + ( 36 - 3 \beta ) q^{84} + 26 q^{86} -38 \beta q^{87} -70 q^{88} -24 \beta q^{89} + 12 \beta q^{90} + ( -48 + 4 \beta ) q^{91} + 42 q^{92} -48 q^{93} + 16 \beta q^{94} -144 q^{95} + 33 \beta q^{96} + 32 \beta q^{97} + ( -47 + 8 \beta ) q^{98} -30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 6q^{4} + 2q^{7} - 14q^{8} - 6q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 6q^{4} + 2q^{7} - 14q^{8} - 6q^{9} + 20q^{11} + 2q^{14} + 24q^{15} + 10q^{16} - 6q^{18} - 24q^{21} + 20q^{22} - 28q^{23} - 46q^{25} - 6q^{28} - 76q^{29} + 24q^{30} + 66q^{32} + 96q^{35} + 18q^{36} + 52q^{37} - 24q^{39} - 24q^{42} + 52q^{43} - 60q^{44} - 28q^{46} - 94q^{49} - 46q^{50} + 20q^{53} - 14q^{56} + 72q^{57} - 76q^{58} - 72q^{60} - 6q^{63} + 26q^{64} + 96q^{65} + 148q^{67} + 96q^{70} - 124q^{71} + 42q^{72} + 52q^{74} + 20q^{77} - 24q^{78} - 92q^{79} + 18q^{81} + 72q^{84} + 52q^{86} - 140q^{88} - 96q^{91} + 84q^{92} - 96q^{93} - 288q^{95} - 94q^{98} - 60q^{99} + O(q^{100}) \)

Character Values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 1.73205i −3.00000 6.92820i 1.73205i 1.00000 6.92820i −7.00000 −3.00000 6.92820i
13.2 1.00000 1.73205i −3.00000 6.92820i 1.73205i 1.00000 + 6.92820i −7.00000 −3.00000 6.92820i
\(n\): 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.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(21, [\chi])\).