Properties

Label 3600.2.f
Level $3600$
Weight $2$
Character orbit 3600.f
Rep. character $\chi_{3600}(2449,\cdot)$
Character field $\Q$
Dimension $44$
Newform subspaces $22$
Sturm bound $1440$
Trace bound $29$

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Defining parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(1440\)
Trace bound: \(29\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3600, [\chi])\).

Total New Old
Modular forms 792 46 746
Cusp forms 648 44 604
Eisenstein series 144 2 142

Trace form

\( 44 q + O(q^{10}) \) \( 44 q + 4 q^{11} - 4 q^{19} - 12 q^{29} + 16 q^{31} - 8 q^{41} - 20 q^{49} - 16 q^{61} - 56 q^{71} - 32 q^{79} + 24 q^{89} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3600, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
3600.2.f.a 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 90.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{7}-6q^{11}-2iq^{13}+3iq^{17}+\cdots\)
3600.2.f.b 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 600.2.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+5iq^{7}-6q^{11}+3iq^{13}+2iq^{17}+\cdots\)
3600.2.f.c 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 120.2.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4q^{11}-3iq^{13}+3iq^{17}-4q^{19}+\cdots\)
3600.2.f.d 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 1800.2.a.k \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{7}-4q^{11}-iq^{13}-4iq^{17}+\cdots\)
3600.2.f.e 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 15.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4q^{11}+iq^{13}-iq^{17}+4q^{19}+\cdots\)
3600.2.f.f 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 50.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{7}-3q^{11}-4iq^{13}-3iq^{17}+\cdots\)
3600.2.f.g 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 360.2.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{7}-2q^{11}+2iq^{13}+iq^{17}+\cdots\)
3600.2.f.h 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) 900.2.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+iq^{7}+7iq^{13}-7q^{19}-11q^{31}+\cdots\)
3600.2.f.i 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 30.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{7}-iq^{13}-3iq^{17}-4q^{19}+\cdots\)
3600.2.f.j 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 20.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{7}+iq^{13}-3iq^{17}-4q^{19}+\cdots\)
3600.2.f.k 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) 225.2.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+iq^{7}+iq^{13}-q^{19}+7q^{31}+2iq^{37}+\cdots\)
3600.2.f.l 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 120.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{7}-3iq^{13}-iq^{17}+4q^{19}+\cdots\)
3600.2.f.m 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) 36.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2iq^{7}-iq^{13}+8q^{19}+4q^{31}+\cdots\)
3600.2.f.n 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 200.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{7}+q^{11}+4iq^{13}+5iq^{17}+\cdots\)
3600.2.f.o 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 600.2.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{7}+2q^{11}-3iq^{13}+6iq^{17}+\cdots\)
3600.2.f.p 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 75.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{7}+2q^{11}-iq^{13}-2iq^{17}+\cdots\)
3600.2.f.q 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 360.2.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{7}+2q^{11}+2iq^{13}-iq^{17}+\cdots\)
3600.2.f.r 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 24.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4q^{11}-iq^{13}+iq^{17}-4q^{19}+\cdots\)
3600.2.f.s 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 1800.2.a.k \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{7}+4q^{11}-iq^{13}+4iq^{17}+\cdots\)
3600.2.f.t 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 40.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{7}+4q^{11}+iq^{13}-iq^{17}+\cdots\)
3600.2.f.u 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 90.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{7}+6q^{11}-2iq^{13}-3iq^{17}+\cdots\)
3600.2.f.v 3600.f 5.b $2$ $28.746$ \(\Q(\sqrt{-1}) \) None 300.2.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{7}+6q^{11}-5iq^{13}+6iq^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3600, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3600, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(720, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1200, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1800, [\chi])\)\(^{\oplus 2}\)