Properties

Label 42.7.g
Level $42$
Weight $7$
Character orbit 42.g
Rep. character $\chi_{42}(19,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $16$
Newform subspaces $2$
Sturm bound $56$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 42.g (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(56\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(42, [\chi])\).

Total New Old
Modular forms 104 16 88
Cusp forms 88 16 72
Eisenstein series 16 0 16

Trace form

\( 16 q - 256 q^{4} + 672 q^{5} - 28 q^{7} + 1944 q^{9} + O(q^{10}) \) \( 16 q - 256 q^{4} + 672 q^{5} - 28 q^{7} + 1944 q^{9} + 2016 q^{10} + 3864 q^{11} + 2208 q^{14} - 4536 q^{15} - 8192 q^{16} - 1680 q^{17} + 44856 q^{19} - 5832 q^{21} - 35328 q^{22} + 17904 q^{23} + 26212 q^{25} - 51072 q^{26} - 9344 q^{28} + 32832 q^{29} + 7776 q^{30} + 97356 q^{31} + 6804 q^{33} + 82872 q^{35} - 124416 q^{36} - 63764 q^{37} - 311136 q^{38} - 176904 q^{39} - 64512 q^{40} - 18144 q^{42} + 480784 q^{43} + 123648 q^{44} + 163296 q^{45} + 112416 q^{46} - 378336 q^{47} - 286424 q^{49} - 275712 q^{50} - 187272 q^{51} - 94080 q^{52} + 21912 q^{53} + 150528 q^{56} - 21384 q^{57} - 135168 q^{58} + 158928 q^{59} + 72576 q^{60} + 1272264 q^{61} - 77760 q^{63} + 524288 q^{64} + 503640 q^{65} - 680872 q^{67} + 53760 q^{68} + 2119296 q^{70} - 317472 q^{71} + 1411452 q^{73} + 1530048 q^{74} - 1061424 q^{75} - 3561600 q^{77} + 228096 q^{78} - 692164 q^{79} - 688128 q^{80} - 472392 q^{81} - 762048 q^{82} + 839808 q^{84} - 3316128 q^{85} + 1080672 q^{86} + 510300 q^{87} + 565248 q^{88} + 4544064 q^{89} - 228192 q^{91} - 1145856 q^{92} - 1795284 q^{93} - 3344544 q^{94} - 2276112 q^{95} + 156672 q^{98} + 1877904 q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(42, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
42.7.g.a $8$ $9.662$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-108\) \(462\) \(580\) \(q+\beta _{3}q^{2}+(-18+9\beta _{1})q^{3}-2^{5}\beta _{1}q^{4}+\cdots\)
42.7.g.b $8$ $9.662$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(108\) \(210\) \(-608\) \(q-\beta _{5}q^{2}+(9+9\beta _{1})q^{3}+(-2^{5}+2^{5}\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(42, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)