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Properties

Label 42.4.a
Level $42$
Weight $4$
Character orbit 42.a
Rep. character $\chi_{42}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
Sturm bound $32$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 42.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(42))\).

Total New Old
Modular forms 28 2 26
Cusp forms 20 2 18
Eisenstein series 8 0 8

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

\(2\)\(3\)\(7\)FrickeDim
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(+\)$+$\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(0\)

Trace form

\( 2 q + 4 q^{2} + 8 q^{4} + 20 q^{5} + 16 q^{8} + 18 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{2} + 8 q^{4} + 20 q^{5} + 16 q^{8} + 18 q^{9} + 40 q^{10} - 80 q^{11} - 76 q^{13} - 48 q^{15} + 32 q^{16} + 4 q^{17} + 36 q^{18} - 32 q^{19} + 80 q^{20} - 42 q^{21} - 160 q^{22} - 104 q^{23} + 78 q^{25} - 152 q^{26} + 140 q^{29} - 96 q^{30} - 16 q^{31} + 64 q^{32} + 192 q^{33} + 8 q^{34} + 112 q^{35} + 72 q^{36} + 364 q^{37} - 64 q^{38} - 24 q^{39} + 160 q^{40} + 468 q^{41} - 84 q^{42} + 376 q^{43} - 320 q^{44} + 180 q^{45} - 208 q^{46} - 96 q^{47} + 98 q^{49} + 156 q^{50} - 24 q^{51} - 304 q^{52} + 492 q^{53} - 1312 q^{55} - 648 q^{57} + 280 q^{58} - 1264 q^{59} - 192 q^{60} - 220 q^{61} - 32 q^{62} + 128 q^{64} - 696 q^{65} + 384 q^{66} - 888 q^{67} + 16 q^{68} + 768 q^{69} + 224 q^{70} - 200 q^{71} + 144 q^{72} + 1428 q^{73} + 728 q^{74} - 960 q^{75} - 128 q^{76} - 448 q^{77} - 48 q^{78} + 608 q^{79} + 320 q^{80} + 162 q^{81} + 936 q^{82} + 1264 q^{83} - 168 q^{84} + 104 q^{85} + 752 q^{86} + 1104 q^{87} - 640 q^{88} + 420 q^{89} + 360 q^{90} + 56 q^{91} - 416 q^{92} - 384 q^{93} - 192 q^{94} + 1408 q^{95} - 1260 q^{97} + 196 q^{98} - 720 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(42))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
42.4.a.a 42.a 1.a $1$ $2.478$ \(\Q\) None \(2\) \(-3\) \(18\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+18q^{5}-6q^{6}+\cdots\)
42.4.a.b 42.a 1.a $1$ $2.478$ \(\Q\) None \(2\) \(3\) \(2\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+2q^{5}+6q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(42))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(42)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)