Properties

Label 33.2.f
Level 33
Weight 2
Character orbit f
Rep. character \(\chi_{33}(2,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 8
Newform subspaces 1
Sturm bound 8
Trace bound 0

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Defining parameters

Level: \( N \) = \( 33 = 3 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 33.f (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 33 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 1 \)
Sturm bound: \(8\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(33, [\chi])\).

Total New Old
Modular forms 24 24 0
Cusp forms 8 8 0
Eisenstein series 16 16 0

Trace form

\( 8q - 6q^{3} - 6q^{4} - 10q^{7} + 10q^{9} + O(q^{10}) \) \( 8q - 6q^{3} - 6q^{4} - 10q^{7} + 10q^{9} + 12q^{12} - 10q^{13} - 6q^{15} + 2q^{16} + 20q^{19} + 20q^{22} - 10q^{24} + 12q^{25} - 12q^{27} - 20q^{30} - 20q^{31} - 4q^{33} - 40q^{34} - 10q^{36} - 6q^{37} + 20q^{39} + 20q^{42} + 24q^{45} + 30q^{46} + 26q^{48} + 16q^{49} + 30q^{51} + 10q^{52} - 32q^{55} - 30q^{57} + 20q^{58} + 2q^{60} - 10q^{61} - 30q^{63} - 34q^{64} - 30q^{66} - 4q^{67} - 16q^{69} + 10q^{70} - 20q^{72} + 6q^{75} - 20q^{78} + 50q^{79} - 2q^{81} - 10q^{82} + 10q^{85} + 50q^{88} + 40q^{90} - 10q^{91} + 10q^{93} - 30q^{94} + 10q^{96} + 6q^{97} + 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(33, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
33.2.f.a \(8\) \(0.264\) \(\Q(\zeta_{20})\) None \(0\) \(-6\) \(0\) \(-10\) \(q+(-\zeta_{20}^{5}-\zeta_{20}^{7})q^{2}+(-1+\zeta_{20}+\cdots)q^{3}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} + 2 T^{4} + 8 T^{6} + 25 T^{8} + 32 T^{10} + 32 T^{12} + 64 T^{14} + 256 T^{16} \)
$3$ \( 1 + 6 T + 13 T^{2} + 10 T^{3} + T^{4} + 30 T^{5} + 117 T^{6} + 162 T^{7} + 81 T^{8} \)
$5$ \( 1 - T^{2} + 51 T^{4} + 109 T^{6} + 1136 T^{8} + 2725 T^{10} + 31875 T^{12} - 15625 T^{14} + 390625 T^{16} \)
$7$ \( ( 1 + 5 T + 12 T^{2} - 5 T^{3} - 51 T^{4} - 35 T^{5} + 588 T^{6} + 1715 T^{7} + 2401 T^{8} )^{2} \)
$11$ \( 1 + 19 T^{2} + 301 T^{4} + 2299 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 + 5 T + 18 T^{2} - 5 T^{3} - 21 T^{4} - 65 T^{5} + 3042 T^{6} + 10985 T^{7} + 28561 T^{8} )^{2} \)
$17$ \( 1 - 34 T^{2} + 267 T^{4} + 8548 T^{6} - 277795 T^{8} + 2470372 T^{10} + 22300107 T^{12} - 820677346 T^{14} + 6975757441 T^{16} \)
$19$ \( ( 1 - 10 T + 69 T^{2} - 410 T^{3} + 1911 T^{4} - 7790 T^{5} + 24909 T^{6} - 68590 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 50 T^{2} + 1363 T^{4} - 26450 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( 1 - 58 T^{2} + 1523 T^{4} - 6056 T^{6} - 679595 T^{8} - 5093096 T^{10} + 1077188963 T^{12} - 34499752618 T^{14} + 500246412961 T^{16} \)
$31$ \( ( 1 + 10 T + 9 T^{2} - 130 T^{3} - 409 T^{4} - 4030 T^{5} + 8649 T^{6} + 297910 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 7215 T^{5} - 38332 T^{6} + 151959 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( 1 - 87 T^{2} + 4308 T^{4} - 137029 T^{6} + 5286975 T^{8} - 230345749 T^{10} + 12173378388 T^{12} - 413259068967 T^{14} + 7984925229121 T^{16} \)
$43$ \( ( 1 - 122 T^{2} + 6919 T^{4} - 225578 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( 1 + 15 T^{2} - 969 T^{4} - 93835 T^{6} + 1517976 T^{8} - 207281515 T^{10} - 4728410889 T^{12} + 161688229935 T^{14} + 23811286661761 T^{16} \)
$53$ \( 1 + 70 T^{2} + 6051 T^{4} + 452660 T^{6} + 19250861 T^{8} + 1271521940 T^{10} + 47745300531 T^{12} + 1551505279030 T^{14} + 62259690411361 T^{16} \)
$59$ \( 1 + 122 T^{2} + 9663 T^{4} + 604444 T^{6} + 40851365 T^{8} + 2104069564 T^{10} + 117090059343 T^{12} + 5146025104202 T^{14} + 146830437604321 T^{16} \)
$61$ \( ( 1 + 5 T + 61 T^{2} - 305 T^{3} + 796 T^{4} - 18605 T^{5} + 226981 T^{6} + 1134905 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + T + 73 T^{2} + 67 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( 1 - 13 T^{2} + 5103 T^{4} - 100031 T^{6} + 31068680 T^{8} - 504256271 T^{10} + 129675808143 T^{12} - 1665303690973 T^{14} + 645753531245761 T^{16} \)
$73$ \( ( 1 + 73 T^{2} - 360 T^{3} + 2449 T^{4} - 26280 T^{5} + 389017 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 25 T + 304 T^{2} - 2435 T^{3} + 19501 T^{4} - 192365 T^{5} + 1897264 T^{6} - 12325975 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( 1 + 149 T^{2} + 16452 T^{4} + 1844407 T^{6} + 200466815 T^{8} + 12706119823 T^{10} + 780784297092 T^{12} + 48714115631981 T^{14} + 2252292232139041 T^{16} \)
$89$ \( ( 1 - 266 T^{2} + 31531 T^{4} - 2106986 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 3 T - 63 T^{2} + 835 T^{3} + 4236 T^{4} + 80995 T^{5} - 592767 T^{6} - 2738019 T^{7} + 88529281 T^{8} )^{2} \)
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