Properties

Label 84.7.m
Level $84$
Weight $7$
Character orbit 84.m
Rep. character $\chi_{84}(61,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $16$
Newform subspaces $2$
Sturm bound $112$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 84.m (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(112\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(84, [\chi])\).

Total New Old
Modular forms 204 16 188
Cusp forms 180 16 164
Eisenstein series 24 0 24

Trace form

\( 16 q - 336 q^{5} + 140 q^{7} + 1944 q^{9} + O(q^{10}) \) \( 16 q - 336 q^{5} + 140 q^{7} + 1944 q^{9} + 504 q^{11} - 4536 q^{15} - 1680 q^{17} + 23688 q^{19} - 1944 q^{21} - 48 q^{23} + 46084 q^{25} + 60864 q^{29} - 19068 q^{31} + 6804 q^{33} + 167352 q^{35} + 25780 q^{37} + 54432 q^{39} - 241808 q^{43} - 81648 q^{45} + 189168 q^{47} + 471928 q^{49} + 30456 q^{51} + 588120 q^{53} - 293544 q^{57} - 1215984 q^{59} - 1033032 q^{61} + 231336 q^{63} + 604344 q^{65} + 1004600 q^{67} - 513408 q^{71} - 2468340 q^{73} - 1061424 q^{75} + 2594304 q^{77} + 199940 q^{79} - 472392 q^{81} - 2632416 q^{85} - 2673972 q^{87} - 810432 q^{89} + 3065280 q^{91} + 586116 q^{93} + 3496512 q^{95} + 244944 q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(84, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
84.7.m.a $8$ $19.325$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-108\) \(-42\) \(-92\) \(q+(-9+9\beta _{1})q^{3}+(-7-3\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
84.7.m.b $8$ $19.325$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(108\) \(-294\) \(232\) \(q+(9+9\beta _{1})q^{3}+(-7^{2}+5^{2}\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(84, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(84, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)