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Properties

Label 441.7.d
Level $441$
Weight $7$
Character orbit 441.d
Rep. character $\chi_{441}(244,\cdot)$
Character field $\Q$
Dimension $98$
Newform subspaces $8$
Sturm bound $392$
Trace bound $8$

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Defining parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 441.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(392\)
Trace bound: \(8\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(441, [\chi])\).

Total New Old
Modular forms 352 102 250
Cusp forms 320 98 222
Eisenstein series 32 4 28

Trace form

\( 98 q + 8 q^{2} + 3164 q^{4} - 620 q^{8} + O(q^{10}) \) \( 98 q + 8 q^{2} + 3164 q^{4} - 620 q^{8} - 3530 q^{11} + 108060 q^{16} + 3040 q^{22} - 48162 q^{23} - 281176 q^{25} + 81308 q^{29} + 88028 q^{32} + 263706 q^{37} - 268692 q^{43} - 547304 q^{44} - 674648 q^{46} - 49152 q^{50} - 160186 q^{53} + 82432 q^{58} + 4370196 q^{64} - 349384 q^{65} - 286286 q^{67} - 1359980 q^{71} + 903276 q^{74} + 1442002 q^{79} - 628638 q^{85} + 3338236 q^{86} + 324680 q^{88} - 8160632 q^{92} + 4772266 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{7}^{\mathrm{new}}(441, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
441.7.d.a 441.d 7.b $2$ $101.454$ \(\Q(\sqrt{-3}) \) None \(24\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+12q^{2}+80q^{4}-15\zeta_{6}q^{5}+192q^{8}+\cdots\)
441.7.d.b 441.d 7.b $4$ $101.454$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-16\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-4+\beta _{1})q^{2}+(-30-8\beta _{1})q^{4}+\cdots\)
441.7.d.c 441.d 7.b $8$ $101.454$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-10\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta _{3})q^{2}+(42-4\beta _{3}-\beta _{4})q^{4}+\cdots\)
441.7.d.d 441.d 7.b $8$ $101.454$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-10\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta _{3})q^{2}+(43-\beta _{5})q^{4}+(3\beta _{1}+\cdots)q^{5}+\cdots\)
441.7.d.e 441.d 7.b $12$ $101.454$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-20\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-2+\beta _{1})q^{2}+(47-\beta _{1}-\beta _{5})q^{4}+\cdots\)
441.7.d.f 441.d 7.b $16$ $101.454$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}+(21-\beta _{1})q^{4}-\beta _{10}q^{5}+(31\beta _{3}+\cdots)q^{8}+\cdots\)
441.7.d.g 441.d 7.b $24$ $101.454$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
441.7.d.h 441.d 7.b $24$ $101.454$ None \(40\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{7}^{\mathrm{old}}(441, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(441, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)