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Properties

Label 42.6.a
Level $42$
Weight $6$
Character orbit 42.a
Rep. character $\chi_{42}(1,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $6$
Sturm bound $48$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 42.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(48\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(42))\).

Total New Old
Modular forms 44 6 38
Cusp forms 36 6 30
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
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(4\)

Trace form

\( 6 q - 8 q^{2} + 96 q^{4} + 44 q^{5} - 128 q^{8} + 486 q^{9} + O(q^{10}) \) \( 6 q - 8 q^{2} + 96 q^{4} + 44 q^{5} - 128 q^{8} + 486 q^{9} + 624 q^{10} + 712 q^{11} - 36 q^{13} - 792 q^{15} + 1536 q^{16} + 1468 q^{17} - 648 q^{18} - 4080 q^{19} + 704 q^{20} + 882 q^{21} + 2880 q^{22} - 2336 q^{23} - 1686 q^{25} + 7024 q^{26} - 12508 q^{29} - 576 q^{30} + 2976 q^{31} - 2048 q^{32} - 720 q^{33} - 12048 q^{34} - 12152 q^{35} + 7776 q^{36} - 1692 q^{37} - 6016 q^{38} - 11160 q^{39} + 9984 q^{40} + 1548 q^{41} + 3528 q^{42} - 6888 q^{43} + 11392 q^{44} + 3564 q^{45} - 22560 q^{46} + 5904 q^{47} + 14406 q^{49} + 7560 q^{50} - 21888 q^{51} - 576 q^{52} + 59652 q^{53} + 65712 q^{55} + 39096 q^{57} + 24720 q^{58} + 54560 q^{59} - 12672 q^{60} - 55476 q^{61} - 92480 q^{62} + 24576 q^{64} + 39576 q^{65} - 39168 q^{66} + 59928 q^{67} + 23488 q^{68} + 94320 q^{69} + 28224 q^{70} - 128912 q^{71} - 10368 q^{72} - 79716 q^{73} - 29296 q^{74} - 37728 q^{75} - 65280 q^{76} - 47824 q^{77} - 3168 q^{78} - 119760 q^{79} + 11264 q^{80} + 39366 q^{81} - 90576 q^{82} - 56144 q^{83} + 14112 q^{84} + 109416 q^{85} + 78944 q^{86} - 138816 q^{87} + 46080 q^{88} + 154140 q^{89} + 50544 q^{90} + 5880 q^{91} - 37376 q^{92} - 170352 q^{93} - 50304 q^{94} - 369440 q^{95} - 2436 q^{97} - 19208 q^{98} + 57672 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(42))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
42.6.a.a 42.a 1.a $1$ $6.736$ \(\Q\) None 42.6.a.a \(-4\) \(-9\) \(-54\) \(49\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}-54q^{5}+6^{2}q^{6}+\cdots\)
42.6.a.b 42.a 1.a $1$ $6.736$ \(\Q\) None 42.6.a.b \(-4\) \(-9\) \(44\) \(-49\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+44q^{5}+6^{2}q^{6}+\cdots\)
42.6.a.c 42.a 1.a $1$ $6.736$ \(\Q\) None 42.6.a.c \(-4\) \(9\) \(-72\) \(49\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}-72q^{5}-6^{2}q^{6}+\cdots\)
42.6.a.d 42.a 1.a $1$ $6.736$ \(\Q\) None 42.6.a.d \(-4\) \(9\) \(26\) \(-49\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+26q^{5}-6^{2}q^{6}+\cdots\)
42.6.a.e 42.a 1.a $1$ $6.736$ \(\Q\) None 42.6.a.e \(4\) \(-9\) \(76\) \(-49\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+76q^{5}-6^{2}q^{6}+\cdots\)
42.6.a.f 42.a 1.a $1$ $6.736$ \(\Q\) None 42.6.a.f \(4\) \(9\) \(24\) \(49\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+24q^{5}+6^{2}q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(42))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(42)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)