Properties

Label 1300.2.i
Level $1300$
Weight $2$
Character orbit 1300.i
Rep. character $\chi_{1300}(601,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $44$
Newform subspaces $9$
Sturm bound $420$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 9 \)
Sturm bound: \(420\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(3\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1300, [\chi])\).

Total New Old
Modular forms 456 44 412
Cusp forms 384 44 340
Eisenstein series 72 0 72

Trace form

\( 44 q - q^{3} + 3 q^{7} - 19 q^{9} + O(q^{10}) \) \( 44 q - q^{3} + 3 q^{7} - 19 q^{9} - 7 q^{11} + 5 q^{13} - 8 q^{17} - 13 q^{19} + 18 q^{21} - 5 q^{23} + 2 q^{27} + 8 q^{29} - 4 q^{31} - 13 q^{33} - 14 q^{37} + 33 q^{39} + 20 q^{41} + 3 q^{43} - 36 q^{47} - 29 q^{49} - 22 q^{51} + 2 q^{53} + 30 q^{57} + 9 q^{59} + 12 q^{61} + 14 q^{63} + 23 q^{67} + 41 q^{69} - 3 q^{71} - 2 q^{73} + 54 q^{77} - 16 q^{79} - 10 q^{81} - 72 q^{83} - 11 q^{87} + 13 q^{89} + 35 q^{91} - 12 q^{93} + 23 q^{97} + 76 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1300, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1300.2.i.a 1300.i 13.c $2$ $10.381$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(0\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}+3\zeta_{6}q^{7}-6\zeta_{6}q^{9}+\cdots\)
1300.2.i.b 1300.i 13.c $2$ $10.381$ \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}-4\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
1300.2.i.c 1300.i 13.c $2$ $10.381$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(0\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+5\zeta_{6}q^{7}+2\zeta_{6}q^{9}+\cdots\)
1300.2.i.d 1300.i 13.c $2$ $10.381$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{7}+2\zeta_{6}q^{9}+(-3+\cdots)q^{11}+\cdots\)
1300.2.i.e 1300.i 13.c $2$ $10.381$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{7}+2\zeta_{6}q^{9}+(-3+\cdots)q^{11}+\cdots\)
1300.2.i.f 1300.i 13.c $2$ $10.381$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(0\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}+\zeta_{6}q^{7}-6\zeta_{6}q^{9}+(5+\cdots)q^{11}+\cdots\)
1300.2.i.g 1300.i 13.c $10$ $10.381$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-1\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{3}+(\beta _{3}+\beta _{5}-\beta _{7})q^{7}+(\beta _{3}+\beta _{5}+\cdots)q^{9}+\cdots\)
1300.2.i.h 1300.i 13.c $10$ $10.381$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(1\) \(0\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(-\beta _{3}-\beta _{5}+\beta _{7})q^{7}+(\beta _{3}+\cdots)q^{9}+\cdots\)
1300.2.i.i 1300.i 13.c $12$ $10.381$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{3}q^{3}+\beta _{9}q^{7}+(-\beta _{4}+\beta _{5}-\beta _{6}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1300, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1300, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(650, [\chi])\)\(^{\oplus 2}\)