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Properties

Label 33.8.a
Level $33$
Weight $8$
Character orbit 33.a
Rep. character $\chi_{33}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $5$
Sturm bound $32$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(32\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(33))\).

Total New Old
Modular forms 30 12 18
Cusp forms 26 12 14
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(11\)FrickeDim
\(+\)\(+\)$+$\(3\)
\(+\)\(-\)$-$\(2\)
\(-\)\(+\)$-$\(3\)
\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(7\)
Minus space\(-\)\(5\)

Trace form

\( 12 q + 16 q^{2} + 54 q^{3} + 716 q^{4} - 776 q^{5} - 108 q^{6} + 892 q^{7} + 5076 q^{8} + 8748 q^{9} + O(q^{10}) \) \( 12 q + 16 q^{2} + 54 q^{3} + 716 q^{4} - 776 q^{5} - 108 q^{6} + 892 q^{7} + 5076 q^{8} + 8748 q^{9} - 12688 q^{10} + 10368 q^{12} + 5992 q^{13} + 30724 q^{14} + 13500 q^{15} + 23732 q^{16} - 37220 q^{17} + 11664 q^{18} - 66244 q^{19} - 196 q^{20} - 40500 q^{21} + 21296 q^{22} + 34112 q^{23} - 54756 q^{24} + 112196 q^{25} + 164572 q^{26} + 39366 q^{27} + 518640 q^{28} - 442164 q^{29} - 162864 q^{30} + 272200 q^{31} - 13876 q^{32} + 71874 q^{33} - 308800 q^{34} - 1052120 q^{35} + 521964 q^{36} + 606536 q^{37} - 797280 q^{38} - 244620 q^{39} - 1899528 q^{40} + 1031716 q^{41} - 127116 q^{42} - 1354492 q^{43} + 1032856 q^{44} - 565704 q^{45} - 3767312 q^{46} + 112040 q^{47} + 2093040 q^{48} + 3599196 q^{49} + 2649728 q^{50} + 333936 q^{51} - 2113272 q^{52} - 4281984 q^{53} - 78732 q^{54} + 1331000 q^{55} + 9320004 q^{56} - 3877740 q^{57} + 3793752 q^{58} + 4634144 q^{59} - 3053268 q^{60} + 7975664 q^{61} - 9043688 q^{62} + 650268 q^{63} - 5223924 q^{64} + 2806480 q^{65} + 1149984 q^{66} + 7746456 q^{67} - 9632392 q^{68} - 236088 q^{69} + 13542824 q^{70} + 1080360 q^{71} + 3700404 q^{72} - 10042064 q^{73} + 732120 q^{74} + 4258170 q^{75} - 22267024 q^{76} + 69212 q^{77} - 7101540 q^{78} - 3758484 q^{79} - 15843652 q^{80} + 6377292 q^{81} + 10266464 q^{82} + 16095864 q^{83} + 2022192 q^{84} - 13327720 q^{85} - 34146408 q^{86} - 3052728 q^{87} + 846516 q^{88} + 5262936 q^{89} - 9249552 q^{90} - 4197240 q^{91} + 28775068 q^{92} - 5656392 q^{93} - 27713696 q^{94} + 25753992 q^{95} + 30246804 q^{96} + 9482328 q^{97} + 24587664 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(33))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 11
33.8.a.a 33.a 1.a $1$ $10.309$ \(\Q\) None \(10\) \(27\) \(-410\) \(-1028\) $-$ $+$ $\mathrm{SU}(2)$ \(q+10q^{2}+3^{3}q^{3}-28q^{4}-410q^{5}+\cdots\)
33.8.a.b 33.a 1.a $2$ $10.309$ \(\Q(\sqrt{177}) \) None \(-19\) \(54\) \(-34\) \(-166\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-9-\beta )q^{2}+3^{3}q^{3}+(-3+19\beta )q^{4}+\cdots\)
33.8.a.c 33.a 1.a $2$ $10.309$ \(\Q(\sqrt{97}) \) None \(1\) \(-54\) \(-194\) \(-418\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-3^{3}q^{3}+(90+\beta )q^{4}+(-90+\cdots)q^{5}+\cdots\)
33.8.a.d 33.a 1.a $3$ $10.309$ 3.3.115512.1 None \(9\) \(-81\) \(-444\) \(1614\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{2}-3^{3}q^{3}+(-5+13\beta _{1}+\cdots)q^{4}+\cdots\)
33.8.a.e 33.a 1.a $4$ $10.309$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(15\) \(108\) \(306\) \(890\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(4-\beta _{1})q^{2}+3^{3}q^{3}+(142-2\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(33))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(33)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)