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Properties

Label 450.7.d
Level $450$
Weight $7$
Character orbit 450.d
Rep. character $\chi_{450}(251,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $9$
Sturm bound $630$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 450.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(630\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(450, [\chi])\).

Total New Old
Modular forms 564 38 526
Cusp forms 516 38 478
Eisenstein series 48 0 48

Trace form

\( 38 q - 1216 q^{4} - 152 q^{7} + O(q^{10}) \) \( 38 q - 1216 q^{4} - 152 q^{7} + 544 q^{13} + 38912 q^{16} + 4192 q^{19} + 15168 q^{22} + 4864 q^{28} - 85688 q^{31} - 93936 q^{34} - 29132 q^{37} - 188240 q^{43} + 196032 q^{46} + 1204746 q^{49} - 17408 q^{52} - 374448 q^{58} + 548620 q^{61} - 1245184 q^{64} + 571984 q^{67} - 3089168 q^{73} - 134144 q^{76} - 352328 q^{79} + 1408752 q^{82} - 485376 q^{88} + 4497824 q^{91} - 768960 q^{94} - 224912 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{7}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
450.7.d.a 450.d 3.b $2$ $103.524$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(968\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta q^{2}-2^{5}q^{4}+22^{2}q^{7}+2^{7}\beta q^{8}+\cdots\)
450.7.d.b 450.d 3.b $4$ $103.524$ \(\Q(\sqrt{-2}, \sqrt{-5})\) None \(0\) \(0\) \(0\) \(-1424\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{1}q^{2}-2^{5}q^{4}+(-356+3\beta _{2}+\cdots)q^{7}+\cdots\)
450.7.d.c 450.d 3.b $4$ $103.524$ \(\Q(\sqrt{-2}, \sqrt{-17})\) None \(0\) \(0\) \(0\) \(-716\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{1}q^{2}-2^{5}q^{4}+(-179+\beta _{2})q^{7}+\cdots\)
450.7.d.d 450.d 3.b $4$ $103.524$ \(\Q(\sqrt{-2}, \sqrt{109})\) None \(0\) \(0\) \(0\) \(-164\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}-2^{5}q^{4}+(-41+\beta _{2})q^{7}+\cdots\)
450.7.d.e 450.d 3.b $4$ $103.524$ \(\Q(\sqrt{-2}, \sqrt{109})\) None \(0\) \(0\) \(0\) \(164\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}-2^{5}q^{4}+(41+\beta _{2})q^{7}+2^{7}\beta _{1}q^{8}+\cdots\)
450.7.d.f 450.d 3.b $4$ $103.524$ \(\Q(\sqrt{-2}, \sqrt{-5})\) None \(0\) \(0\) \(0\) \(304\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{1}q^{2}-2^{5}q^{4}+(76+3\beta _{2})q^{7}+\cdots\)
450.7.d.g 450.d 3.b $4$ $103.524$ \(\Q(\sqrt{-2}, \sqrt{-17})\) None \(0\) \(0\) \(0\) \(716\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}-2^{5}q^{4}+(179+\beta _{2})q^{7}+\cdots\)
450.7.d.h 450.d 3.b $6$ $103.524$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(-104\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{1}q^{2}-2^{5}q^{4}+(-19+3\beta _{2}-2\beta _{5})q^{7}+\cdots\)
450.7.d.i 450.d 3.b $6$ $103.524$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(104\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{1}q^{2}-2^{5}q^{4}+(19-3\beta _{2}+2\beta _{5})q^{7}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(6, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)