Properties

Label 21.4.e.a
Level 21
Weight 4
Character orbit 21.e
Analytic conductor 1.239
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 21.e (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( 3 - 3 \zeta_{6} ) q^{2} \) \( -3 \zeta_{6} q^{3} \) \( -\zeta_{6} q^{4} \) \( + ( 3 - 3 \zeta_{6} ) q^{5} \) \( -9 q^{6} \) \( + ( -14 + 21 \zeta_{6} ) q^{7} \) \( + 21 q^{8} \) \( + ( -9 + 9 \zeta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 3 - 3 \zeta_{6} ) q^{2} \) \( -3 \zeta_{6} q^{3} \) \( -\zeta_{6} q^{4} \) \( + ( 3 - 3 \zeta_{6} ) q^{5} \) \( -9 q^{6} \) \( + ( -14 + 21 \zeta_{6} ) q^{7} \) \( + 21 q^{8} \) \( + ( -9 + 9 \zeta_{6} ) q^{9} \) \( -9 \zeta_{6} q^{10} \) \( + 15 \zeta_{6} q^{11} \) \( + ( -3 + 3 \zeta_{6} ) q^{12} \) \( -64 q^{13} \) \( + ( 21 + 42 \zeta_{6} ) q^{14} \) \( -9 q^{15} \) \( + ( 71 - 71 \zeta_{6} ) q^{16} \) \( -84 \zeta_{6} q^{17} \) \( + 27 \zeta_{6} q^{18} \) \( + ( 16 - 16 \zeta_{6} ) q^{19} \) \( -3 q^{20} \) \( + ( 63 - 21 \zeta_{6} ) q^{21} \) \( + 45 q^{22} \) \( + ( 84 - 84 \zeta_{6} ) q^{23} \) \( -63 \zeta_{6} q^{24} \) \( + 116 \zeta_{6} q^{25} \) \( + ( -192 + 192 \zeta_{6} ) q^{26} \) \( + 27 q^{27} \) \( + ( 21 - 7 \zeta_{6} ) q^{28} \) \( -297 q^{29} \) \( + ( -27 + 27 \zeta_{6} ) q^{30} \) \( + 253 \zeta_{6} q^{31} \) \( -45 \zeta_{6} q^{32} \) \( + ( 45 - 45 \zeta_{6} ) q^{33} \) \( -252 q^{34} \) \( + ( 21 + 42 \zeta_{6} ) q^{35} \) \( + 9 q^{36} \) \( + ( 316 - 316 \zeta_{6} ) q^{37} \) \( -48 \zeta_{6} q^{38} \) \( + 192 \zeta_{6} q^{39} \) \( + ( 63 - 63 \zeta_{6} ) q^{40} \) \( + 360 q^{41} \) \( + ( 126 - 189 \zeta_{6} ) q^{42} \) \( + 26 q^{43} \) \( + ( 15 - 15 \zeta_{6} ) q^{44} \) \( + 27 \zeta_{6} q^{45} \) \( -252 \zeta_{6} q^{46} \) \( + ( 30 - 30 \zeta_{6} ) q^{47} \) \( -213 q^{48} \) \( + ( -245 - 147 \zeta_{6} ) q^{49} \) \( + 348 q^{50} \) \( + ( -252 + 252 \zeta_{6} ) q^{51} \) \( + 64 \zeta_{6} q^{52} \) \( -363 \zeta_{6} q^{53} \) \( + ( 81 - 81 \zeta_{6} ) q^{54} \) \( + 45 q^{55} \) \( + ( -294 + 441 \zeta_{6} ) q^{56} \) \( -48 q^{57} \) \( + ( -891 + 891 \zeta_{6} ) q^{58} \) \( + 15 \zeta_{6} q^{59} \) \( + 9 \zeta_{6} q^{60} \) \( + ( 118 - 118 \zeta_{6} ) q^{61} \) \( + 759 q^{62} \) \( + ( -63 - 126 \zeta_{6} ) q^{63} \) \( + 433 q^{64} \) \( + ( -192 + 192 \zeta_{6} ) q^{65} \) \( -135 \zeta_{6} q^{66} \) \( + 370 \zeta_{6} q^{67} \) \( + ( -84 + 84 \zeta_{6} ) q^{68} \) \( -252 q^{69} \) \( + ( 189 - 63 \zeta_{6} ) q^{70} \) \( -342 q^{71} \) \( + ( -189 + 189 \zeta_{6} ) q^{72} \) \( -362 \zeta_{6} q^{73} \) \( -948 \zeta_{6} q^{74} \) \( + ( 348 - 348 \zeta_{6} ) q^{75} \) \( -16 q^{76} \) \( + ( -315 + 105 \zeta_{6} ) q^{77} \) \( + 576 q^{78} \) \( + ( -467 + 467 \zeta_{6} ) q^{79} \) \( -213 \zeta_{6} q^{80} \) \( -81 \zeta_{6} q^{81} \) \( + ( 1080 - 1080 \zeta_{6} ) q^{82} \) \( + 477 q^{83} \) \( + ( -21 - 42 \zeta_{6} ) q^{84} \) \( -252 q^{85} \) \( + ( 78 - 78 \zeta_{6} ) q^{86} \) \( + 891 \zeta_{6} q^{87} \) \( + 315 \zeta_{6} q^{88} \) \( + ( -906 + 906 \zeta_{6} ) q^{89} \) \( + 81 q^{90} \) \( + ( 896 - 1344 \zeta_{6} ) q^{91} \) \( -84 q^{92} \) \( + ( 759 - 759 \zeta_{6} ) q^{93} \) \( -90 \zeta_{6} q^{94} \) \( -48 \zeta_{6} q^{95} \) \( + ( -135 + 135 \zeta_{6} ) q^{96} \) \( + 503 q^{97} \) \( + ( -1176 + 735 \zeta_{6} ) q^{98} \) \( -135 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 18q^{6} \) \(\mathstrut -\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 42q^{8} \) \(\mathstrut -\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 18q^{6} \) \(\mathstrut -\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 42q^{8} \) \(\mathstrut -\mathstrut 9q^{9} \) \(\mathstrut -\mathstrut 9q^{10} \) \(\mathstrut +\mathstrut 15q^{11} \) \(\mathstrut -\mathstrut 3q^{12} \) \(\mathstrut -\mathstrut 128q^{13} \) \(\mathstrut +\mathstrut 84q^{14} \) \(\mathstrut -\mathstrut 18q^{15} \) \(\mathstrut +\mathstrut 71q^{16} \) \(\mathstrut -\mathstrut 84q^{17} \) \(\mathstrut +\mathstrut 27q^{18} \) \(\mathstrut +\mathstrut 16q^{19} \) \(\mathstrut -\mathstrut 6q^{20} \) \(\mathstrut +\mathstrut 105q^{21} \) \(\mathstrut +\mathstrut 90q^{22} \) \(\mathstrut +\mathstrut 84q^{23} \) \(\mathstrut -\mathstrut 63q^{24} \) \(\mathstrut +\mathstrut 116q^{25} \) \(\mathstrut -\mathstrut 192q^{26} \) \(\mathstrut +\mathstrut 54q^{27} \) \(\mathstrut +\mathstrut 35q^{28} \) \(\mathstrut -\mathstrut 594q^{29} \) \(\mathstrut -\mathstrut 27q^{30} \) \(\mathstrut +\mathstrut 253q^{31} \) \(\mathstrut -\mathstrut 45q^{32} \) \(\mathstrut +\mathstrut 45q^{33} \) \(\mathstrut -\mathstrut 504q^{34} \) \(\mathstrut +\mathstrut 84q^{35} \) \(\mathstrut +\mathstrut 18q^{36} \) \(\mathstrut +\mathstrut 316q^{37} \) \(\mathstrut -\mathstrut 48q^{38} \) \(\mathstrut +\mathstrut 192q^{39} \) \(\mathstrut +\mathstrut 63q^{40} \) \(\mathstrut +\mathstrut 720q^{41} \) \(\mathstrut +\mathstrut 63q^{42} \) \(\mathstrut +\mathstrut 52q^{43} \) \(\mathstrut +\mathstrut 15q^{44} \) \(\mathstrut +\mathstrut 27q^{45} \) \(\mathstrut -\mathstrut 252q^{46} \) \(\mathstrut +\mathstrut 30q^{47} \) \(\mathstrut -\mathstrut 426q^{48} \) \(\mathstrut -\mathstrut 637q^{49} \) \(\mathstrut +\mathstrut 696q^{50} \) \(\mathstrut -\mathstrut 252q^{51} \) \(\mathstrut +\mathstrut 64q^{52} \) \(\mathstrut -\mathstrut 363q^{53} \) \(\mathstrut +\mathstrut 81q^{54} \) \(\mathstrut +\mathstrut 90q^{55} \) \(\mathstrut -\mathstrut 147q^{56} \) \(\mathstrut -\mathstrut 96q^{57} \) \(\mathstrut -\mathstrut 891q^{58} \) \(\mathstrut +\mathstrut 15q^{59} \) \(\mathstrut +\mathstrut 9q^{60} \) \(\mathstrut +\mathstrut 118q^{61} \) \(\mathstrut +\mathstrut 1518q^{62} \) \(\mathstrut -\mathstrut 252q^{63} \) \(\mathstrut +\mathstrut 866q^{64} \) \(\mathstrut -\mathstrut 192q^{65} \) \(\mathstrut -\mathstrut 135q^{66} \) \(\mathstrut +\mathstrut 370q^{67} \) \(\mathstrut -\mathstrut 84q^{68} \) \(\mathstrut -\mathstrut 504q^{69} \) \(\mathstrut +\mathstrut 315q^{70} \) \(\mathstrut -\mathstrut 684q^{71} \) \(\mathstrut -\mathstrut 189q^{72} \) \(\mathstrut -\mathstrut 362q^{73} \) \(\mathstrut -\mathstrut 948q^{74} \) \(\mathstrut +\mathstrut 348q^{75} \) \(\mathstrut -\mathstrut 32q^{76} \) \(\mathstrut -\mathstrut 525q^{77} \) \(\mathstrut +\mathstrut 1152q^{78} \) \(\mathstrut -\mathstrut 467q^{79} \) \(\mathstrut -\mathstrut 213q^{80} \) \(\mathstrut -\mathstrut 81q^{81} \) \(\mathstrut +\mathstrut 1080q^{82} \) \(\mathstrut +\mathstrut 954q^{83} \) \(\mathstrut -\mathstrut 84q^{84} \) \(\mathstrut -\mathstrut 504q^{85} \) \(\mathstrut +\mathstrut 78q^{86} \) \(\mathstrut +\mathstrut 891q^{87} \) \(\mathstrut +\mathstrut 315q^{88} \) \(\mathstrut -\mathstrut 906q^{89} \) \(\mathstrut +\mathstrut 162q^{90} \) \(\mathstrut +\mathstrut 448q^{91} \) \(\mathstrut -\mathstrut 168q^{92} \) \(\mathstrut +\mathstrut 759q^{93} \) \(\mathstrut -\mathstrut 90q^{94} \) \(\mathstrut -\mathstrut 48q^{95} \) \(\mathstrut -\mathstrut 135q^{96} \) \(\mathstrut +\mathstrut 1006q^{97} \) \(\mathstrut -\mathstrut 1617q^{98} \) \(\mathstrut -\mathstrut 270q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 0.866025i
0.500000 + 0.866025i
1.50000 + 2.59808i −1.50000 + 2.59808i −0.500000 + 0.866025i 1.50000 + 2.59808i −9.00000 −3.50000 18.1865i 21.0000 −4.50000 7.79423i −4.50000 + 7.79423i
16.1 1.50000 2.59808i −1.50000 2.59808i −0.500000 0.866025i 1.50000 2.59808i −9.00000 −3.50000 + 18.1865i 21.0000 −4.50000 + 7.79423i −4.50000 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.c Even 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 \(S_{4}^{\mathrm{new}}(21, [\chi])\).