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Properties

Label 450.4.c
Level $450$
Weight $4$
Character orbit 450.c
Rep. character $\chi_{450}(199,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $11$
Sturm bound $360$
Trace bound $14$

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Defining parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(360\)
Trace bound: \(14\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(450, [\chi])\).

Total New Old
Modular forms 294 22 272
Cusp forms 246 22 224
Eisenstein series 48 0 48

Trace form

\( 22 q - 88 q^{4} + O(q^{10}) \) \( 22 q - 88 q^{4} + 90 q^{11} - 136 q^{14} + 352 q^{16} + 82 q^{19} - 32 q^{26} - 816 q^{29} - 460 q^{31} + 204 q^{34} + 294 q^{41} - 360 q^{44} - 360 q^{46} + 282 q^{49} + 544 q^{56} + 480 q^{59} + 1076 q^{61} - 1408 q^{64} - 600 q^{71} - 712 q^{74} - 328 q^{76} - 6260 q^{79} + 2464 q^{86} + 318 q^{89} + 4984 q^{91} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
450.4.c.a 450.c 5.b $2$ $26.551$ \(\Q(\sqrt{-1}) \) None 150.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2 i q^{2}-4 q^{4}+i q^{7}+8 i q^{8}+\cdots\)
450.4.c.b 450.c 5.b $2$ $26.551$ \(\Q(\sqrt{-1}) \) None 450.4.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2 i q^{2}-4 q^{4}+11 i q^{7}+8 i q^{8}+\cdots\)
450.4.c.c 450.c 5.b $2$ $26.551$ \(\Q(\sqrt{-1}) \) None 50.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2 i q^{2}-4 q^{4}+34 i q^{7}-8 i q^{8}+\cdots\)
450.4.c.d 450.c 5.b $2$ $26.551$ \(\Q(\sqrt{-1}) \) None 10.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}-4 q^{4}+2\beta q^{7}-4\beta q^{8}+\cdots\)
450.4.c.e 450.c 5.b $2$ $26.551$ \(\Q(\sqrt{-1}) \) None 6.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta q^{2}-4 q^{4}+8\beta q^{7}+4\beta q^{8}+\cdots\)
450.4.c.f 450.c 5.b $2$ $26.551$ \(\Q(\sqrt{-1}) \) None 90.4.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}-4 q^{4}+7\beta q^{7}-4\beta q^{8}+\cdots\)
450.4.c.g 450.c 5.b $2$ $26.551$ \(\Q(\sqrt{-1}) \) None 90.4.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta q^{2}-4 q^{4}+7\beta q^{7}+4\beta q^{8}+\cdots\)
450.4.c.h 450.c 5.b $2$ $26.551$ \(\Q(\sqrt{-1}) \) None 150.4.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2 i q^{2}-4 q^{4}+23 i q^{7}+8 i q^{8}+\cdots\)
450.4.c.i 450.c 5.b $2$ $26.551$ \(\Q(\sqrt{-1}) \) None 450.4.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2 i q^{2}-4 q^{4}+11 i q^{7}-8 i q^{8}+\cdots\)
450.4.c.j 450.c 5.b $2$ $26.551$ \(\Q(\sqrt{-1}) \) None 30.4.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}-4 q^{4}+2\beta q^{7}-4\beta q^{8}+\cdots\)
450.4.c.k 450.c 5.b $2$ $26.551$ \(\Q(\sqrt{-1}) \) None 30.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}-4 q^{4}+16\beta q^{7}-4\beta q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(450, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)