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Properties

Label 30.7.f
Level $30$
Weight $7$
Character orbit 30.f
Rep. character $\chi_{30}(7,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $12$
Newform subspaces $2$
Sturm bound $42$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 30.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(42\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(30, [\chi])\).

Total New Old
Modular forms 80 12 68
Cusp forms 64 12 52
Eisenstein series 16 0 16

Trace form

\( 12 q - 16 q^{2} - 360 q^{5} + 984 q^{7} + 512 q^{8} + O(q^{10}) \) \( 12 q - 16 q^{2} - 360 q^{5} + 984 q^{7} + 512 q^{8} - 3216 q^{10} - 3248 q^{11} + 3372 q^{13} + 4536 q^{15} - 12288 q^{16} - 10916 q^{17} - 3888 q^{18} + 1408 q^{20} + 31104 q^{21} + 20544 q^{22} + 12064 q^{23} + 46332 q^{25} - 99168 q^{26} - 31488 q^{28} + 15552 q^{30} + 138960 q^{31} + 16384 q^{32} + 57672 q^{33} - 205952 q^{35} + 93312 q^{36} - 25548 q^{37} - 137984 q^{38} + 19968 q^{40} - 221264 q^{41} + 88128 q^{42} + 313248 q^{43} - 10692 q^{45} - 141696 q^{46} - 197664 q^{47} + 479632 q^{50} + 290304 q^{51} + 107904 q^{52} - 786700 q^{53} - 418008 q^{55} - 53248 q^{56} - 598752 q^{57} + 136704 q^{58} - 165888 q^{60} + 716592 q^{61} + 537920 q^{62} + 239112 q^{63} + 1169556 q^{65} + 404352 q^{66} - 391776 q^{67} + 349312 q^{68} - 926592 q^{70} - 1749088 q^{71} - 124416 q^{72} - 208548 q^{73} - 832032 q^{75} + 32256 q^{76} + 4441712 q^{77} + 798336 q^{78} + 368640 q^{80} - 708588 q^{81} - 2237568 q^{82} + 700512 q^{83} - 51012 q^{85} - 120192 q^{86} - 3130488 q^{87} - 657408 q^{88} - 151632 q^{90} + 1588224 q^{91} + 386048 q^{92} + 322704 q^{93} + 3279584 q^{95} - 173988 q^{97} - 2034704 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{7}^{\mathrm{new}}(30, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
30.7.f.a 30.f 5.c $4$ $6.902$ \(\Q(i, \sqrt{6})\) None \(16\) \(0\) \(-420\) \(596\) $\mathrm{SU}(2)[C_{4}]$ \(q+(4-4\beta _{2})q^{2}+3\beta _{1}q^{3}-2^{5}\beta _{2}q^{4}+\cdots\)
30.7.f.b 30.f 5.c $8$ $6.902$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(-32\) \(0\) \(60\) \(388\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-4-4\beta _{1})q^{2}-\beta _{3}q^{3}+2^{5}\beta _{1}q^{4}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(30, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(30, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)