Properties

Label 21.3.b
Level $21$
Weight $3$
Character orbit 21.b
Rep. character $\chi_{21}(8,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $1$
Sturm bound $8$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 21.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(8\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(21, [\chi])\).

Total New Old
Modular forms 8 4 4
Cusp forms 4 4 0
Eisenstein series 4 0 4

Trace form

\( 4q - 2q^{3} - 12q^{4} + 14q^{6} - 20q^{9} + O(q^{10}) \) \( 4q - 2q^{3} - 12q^{4} + 14q^{6} - 20q^{9} + 28q^{10} - 22q^{12} - 36q^{13} + 28q^{15} + 36q^{16} + 56q^{18} + 12q^{19} + 14q^{21} - 56q^{22} - 126q^{24} - 12q^{25} + 10q^{27} - 56q^{28} - 28q^{30} + 136q^{31} + 28q^{33} + 116q^{36} + 16q^{37} + 4q^{39} + 84q^{40} + 70q^{42} - 160q^{43} - 140q^{45} - 168q^{46} + 38q^{48} + 28q^{49} - 84q^{51} + 164q^{52} - 154q^{54} + 56q^{55} + 64q^{57} + 112q^{58} + 140q^{60} - 156q^{61} - 28q^{63} + 4q^{64} - 28q^{66} - 24q^{67} + 168q^{69} - 28q^{70} - 32q^{73} + 146q^{75} - 316q^{76} - 196q^{78} + 128q^{79} - 68q^{81} + 392q^{82} - 14q^{84} + 168q^{85} + 28q^{87} + 168q^{88} - 112q^{90} - 28q^{91} - 96q^{93} - 336q^{94} - 98q^{96} - 8q^{97} + 112q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(21, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
21.3.b.a \(4\) \(0.572\) 4.0.65856.1 None \(0\) \(-2\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{2}+\beta _{3})q^{3}+(-3+\cdots)q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 5 T^{4} - 32 T^{6} + 256 T^{8} \)
$3$ \( 1 + 2 T + 12 T^{2} + 18 T^{3} + 81 T^{4} \)
$5$ \( 1 - 44 T^{2} + 1034 T^{4} - 27500 T^{6} + 390625 T^{8} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( 1 - 428 T^{2} + 74630 T^{4} - 6266348 T^{6} + 214358881 T^{8} \)
$13$ \( ( 1 + 18 T + 412 T^{2} + 3042 T^{3} + 28561 T^{4} )^{2} \)
$17$ \( 1 - 988 T^{2} + 407046 T^{4} - 82518748 T^{6} + 6975757441 T^{8} \)
$19$ \( ( 1 - 6 T + 556 T^{2} - 2166 T^{3} + 130321 T^{4} )^{2} \)
$23$ \( 1 - 1444 T^{2} + 980166 T^{4} - 404090404 T^{6} + 78310985281 T^{8} \)
$29$ \( 1 - 2972 T^{2} + 3611558 T^{4} - 2102039132 T^{6} + 500246412961 T^{8} \)
$31$ \( ( 1 - 68 T + 3050 T^{2} - 65348 T^{3} + 923521 T^{4} )^{2} \)
$37$ \( ( 1 - 8 T + 1382 T^{2} - 10952 T^{3} + 1874161 T^{4} )^{2} \)
$41$ \( 1 - 1292 T^{2} + 2832038 T^{4} - 3650883212 T^{6} + 7984925229121 T^{8} \)
$43$ \( ( 1 + 80 T + 5046 T^{2} + 147920 T^{3} + 3418801 T^{4} )^{2} \)
$47$ \( 1 - 6148 T^{2} + 19144326 T^{4} - 30000278788 T^{6} + 23811286661761 T^{8} \)
$53$ \( 1 + 20 T^{2} - 13350138 T^{4} + 157809620 T^{6} + 62259690411361 T^{8} \)
$59$ \( 1 - 3676 T^{2} + 15964266 T^{4} - 44543419036 T^{6} + 146830437604321 T^{8} \)
$61$ \( ( 1 + 78 T + 8620 T^{2} + 290238 T^{3} + 13845841 T^{4} )^{2} \)
$67$ \( ( 1 + 12 T + 8566 T^{2} + 53868 T^{3} + 20151121 T^{4} )^{2} \)
$71$ \( 1 - 10588 T^{2} + 78813510 T^{4} - 269058878428 T^{6} + 645753531245761 T^{8} \)
$73$ \( ( 1 + 16 T + 5990 T^{2} + 85264 T^{3} + 28398241 T^{4} )^{2} \)
$79$ \( ( 1 - 64 T + 4434 T^{2} - 399424 T^{3} + 38950081 T^{4} )^{2} \)
$83$ \( 1 - 13948 T^{2} + 141899946 T^{4} - 661948661308 T^{6} + 2252292232139041 T^{8} \)
$89$ \( 1 - 11468 T^{2} + 120945830 T^{4} - 719528019788 T^{6} + 3936588805702081 T^{8} \)
$97$ \( ( 1 + 4 T + 18374 T^{2} + 37636 T^{3} + 88529281 T^{4} )^{2} \)
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