Properties

Label 1225.2.b
Level $1225$
Weight $2$
Character orbit 1225.b
Rep. character $\chi_{1225}(99,\cdot)$
Character field $\Q$
Dimension $56$
Newform subspaces $14$
Sturm bound $280$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(280\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\), \(19\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1225, [\chi])\).

Total New Old
Modular forms 164 66 98
Cusp forms 116 56 60
Eisenstein series 48 10 38

Trace form

\( 56 q - 56 q^{4} + 8 q^{6} - 36 q^{9} + O(q^{10}) \) \( 56 q - 56 q^{4} + 8 q^{6} - 36 q^{9} - 8 q^{11} + 40 q^{16} - 8 q^{19} - 28 q^{24} + 40 q^{29} + 20 q^{31} - 36 q^{34} + 48 q^{36} - 16 q^{39} - 8 q^{41} + 64 q^{44} + 24 q^{46} - 52 q^{51} + 24 q^{54} + 4 q^{59} - 16 q^{61} - 36 q^{64} - 8 q^{66} - 12 q^{69} - 68 q^{71} + 60 q^{74} + 52 q^{76} - 4 q^{79} + 8 q^{81} - 24 q^{86} + 48 q^{89} + 52 q^{94} - 20 q^{96} + 44 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1225, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1225.2.b.a 1225.b 5.b $2$ $9.782$ \(\Q(\sqrt{-1}) \) None 245.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+3iq^{3}-2q^{4}-6q^{6}-6q^{9}+\cdots\)
1225.2.b.b 1225.b 5.b $2$ $9.782$ \(\Q(\sqrt{-1}) \) None 245.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-3iq^{3}-2q^{4}+6q^{6}-6q^{9}+\cdots\)
1225.2.b.c 1225.b 5.b $2$ $9.782$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-7}) \) 49.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+iq^{2}+q^{4}+3iq^{8}+3q^{9}+4q^{11}+\cdots\)
1225.2.b.d 1225.b 5.b $2$ $9.782$ \(\Q(\sqrt{-1}) \) None 35.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+2q^{4}+2q^{9}-3q^{11}-2iq^{12}+\cdots\)
1225.2.b.e 1225.b 5.b $4$ $9.782$ \(\Q(i, \sqrt{21})\) \(\Q(\sqrt{-7}) \) 1225.2.a.o \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{1}q^{2}+(-4+\beta _{2})q^{4}+(-2\beta _{1}+\beta _{3})q^{8}+\cdots\)
1225.2.b.f 1225.b 5.b $4$ $9.782$ \(\Q(i, \sqrt{17})\) None 35.2.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{2})q^{3}+(-3+\beta _{3})q^{4}+\cdots\)
1225.2.b.g 1225.b 5.b $4$ $9.782$ \(\Q(\zeta_{8})\) None 35.2.e.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}q^{2}-\zeta_{8}^{2}q^{3}+(-1-\zeta_{8}^{3})q^{4}+\cdots\)
1225.2.b.h 1225.b 5.b $4$ $9.782$ \(\Q(\zeta_{8})\) None 35.2.e.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{3}+(-1-\zeta_{8}^{3})q^{4}+\cdots\)
1225.2.b.i 1225.b 5.b $4$ $9.782$ \(\Q(\zeta_{8})\) None 245.2.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{2}q^{2}+(\zeta_{8}+\zeta_{8}^{2})q^{3}+(-2-\zeta_{8}^{3})q^{6}+\cdots\)
1225.2.b.j 1225.b 5.b $4$ $9.782$ \(\Q(\zeta_{8})\) None 245.2.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{2}q^{2}+(-\zeta_{8}-\zeta_{8}^{2})q^{3}+(2+\zeta_{8}^{3})q^{6}+\cdots\)
1225.2.b.k 1225.b 5.b $4$ $9.782$ \(\Q(i, \sqrt{5})\) None 175.2.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(2\beta _{1}+2\beta _{3})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.2.b.l 1225.b 5.b $6$ $9.782$ 6.0.4227136.2 None 175.2.e.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-\beta _{3}+\beta _{4})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.2.b.m 1225.b 5.b $6$ $9.782$ 6.0.4227136.2 None 175.2.e.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{3}-\beta _{4})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.2.b.n 1225.b 5.b $8$ $9.782$ 8.0.40960000.1 None 1225.2.a.ba \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{2}-\beta _{3})q^{2}+(-\beta _{5}+\beta _{7})q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1225, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1225, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 2}\)