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.

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

Character Values

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

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

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 \(S_{3}^{\mathrm{new}}(21, [\chi])\).