Properties

Label 1350.4.c
Level $1350$
Weight $4$
Character orbit 1350.c
Rep. character $\chi_{1350}(649,\cdot)$
Character field $\Q$
Dimension $72$
Newform subspaces $28$
Sturm bound $1080$
Trace bound $14$

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Defining parameters

Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 28 \)
Sturm bound: \(1080\)
Trace bound: \(14\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1350, [\chi])\).

Total New Old
Modular forms 846 72 774
Cusp forms 774 72 702
Eisenstein series 72 0 72

Trace form

\( 72 q - 288 q^{4} + O(q^{10}) \) \( 72 q - 288 q^{4} + 1152 q^{16} + 24 q^{19} + 564 q^{31} + 912 q^{34} + 288 q^{46} - 6324 q^{49} - 576 q^{61} - 4608 q^{64} - 96 q^{76} + 3396 q^{79} - 7572 q^{91} + 6192 q^{94} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1350, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1350.4.c.a 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-4q^{4}+7iq^{7}-8iq^{8}-60q^{11}+\cdots\)
1350.4.c.b 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-4q^{4}+29iq^{7}-8iq^{8}+\cdots\)
1350.4.c.c 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}-4q^{4}+34iq^{7}+8iq^{8}+\cdots\)
1350.4.c.d 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-4q^{4}+4iq^{7}-8iq^{8}-42q^{11}+\cdots\)
1350.4.c.e 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-4q^{4}+23iq^{7}-8iq^{8}+\cdots\)
1350.4.c.f 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}-4q^{4}+13iq^{7}+8iq^{8}+\cdots\)
1350.4.c.g 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}-4q^{4}+8iq^{7}+8iq^{8}-18q^{11}+\cdots\)
1350.4.c.h 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-4q^{4}+19iq^{7}-8iq^{8}+\cdots\)
1350.4.c.i 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}-4q^{4}+22iq^{7}+8iq^{8}+\cdots\)
1350.4.c.j 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-4q^{4}+14iq^{7}-8iq^{8}+\cdots\)
1350.4.c.k 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}-4q^{4}+14iq^{7}+8iq^{8}+\cdots\)
1350.4.c.l 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-4q^{4}+22iq^{7}-8iq^{8}+\cdots\)
1350.4.c.m 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}-4q^{4}+19iq^{7}+8iq^{8}+\cdots\)
1350.4.c.n 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-4q^{4}+8iq^{7}-8iq^{8}+18q^{11}+\cdots\)
1350.4.c.o 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-4q^{4}+13iq^{7}-8iq^{8}+\cdots\)
1350.4.c.p 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}-4q^{4}+23iq^{7}+8iq^{8}+\cdots\)
1350.4.c.q 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}-4q^{4}+4iq^{7}+8iq^{8}+42q^{11}+\cdots\)
1350.4.c.r 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-4q^{4}+34iq^{7}-8iq^{8}+\cdots\)
1350.4.c.s 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}-4q^{4}+29iq^{7}+8iq^{8}+\cdots\)
1350.4.c.t 1350.c 5.b $2$ $79.653$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}-4q^{4}+7iq^{7}+8iq^{8}+60q^{11}+\cdots\)
1350.4.c.u 1350.c 5.b $4$ $79.653$ \(\Q(i, \sqrt{401})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2\beta _{1}q^{2}-4q^{4}-\beta _{2}q^{7}+8\beta _{1}q^{8}+\cdots\)
1350.4.c.v 1350.c 5.b $4$ $79.653$ \(\Q(i, \sqrt{21})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2\beta _{1}q^{2}-4q^{4}+(-5\beta _{1}+\beta _{2})q^{7}+\cdots\)
1350.4.c.w 1350.c 5.b $4$ $79.653$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2\beta _{1}q^{2}-4q^{4}+(-10\beta _{1}-\beta _{2})q^{7}+\cdots\)
1350.4.c.x 1350.c 5.b $4$ $79.653$ \(\Q(i, \sqrt{209})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2\beta _{1}q^{2}-4q^{4}+(-7\beta _{1}-\beta _{2})q^{7}+\cdots\)
1350.4.c.y 1350.c 5.b $4$ $79.653$ \(\Q(i, \sqrt{209})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2\beta _{1}q^{2}-4q^{4}+(-7\beta _{1}-\beta _{2})q^{7}+\cdots\)
1350.4.c.z 1350.c 5.b $4$ $79.653$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2\beta _{1}q^{2}-4q^{4}+(-10\beta _{1}-\beta _{2})q^{7}+\cdots\)
1350.4.c.ba 1350.c 5.b $4$ $79.653$ \(\Q(i, \sqrt{21})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2\beta _{1}q^{2}-4q^{4}+(-5\beta _{1}+\beta _{2})q^{7}+\cdots\)
1350.4.c.bb 1350.c 5.b $4$ $79.653$ \(\Q(i, \sqrt{401})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2\beta _{1}q^{2}-4q^{4}-\beta _{2}q^{7}-8\beta _{1}q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1350, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1350, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(675, [\chi])\)\(^{\oplus 2}\)