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Properties

Label 1300.2.r
Level $1300$
Weight $2$
Character orbit 1300.r
Rep. character $\chi_{1300}(957,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $42$
Newform subspaces $6$
Sturm bound $420$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.r (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 65 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 6 \)
Sturm bound: \(420\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 456 42 414
Cusp forms 384 42 342
Eisenstein series 72 0 72

Trace form

\( 42 q + O(q^{10}) \) \( 42 q + 2 q^{13} + 2 q^{17} + 4 q^{19} + 4 q^{21} - 12 q^{23} + 12 q^{27} + 20 q^{31} + 4 q^{37} - 24 q^{39} - 34 q^{41} - 12 q^{43} - 32 q^{47} + 42 q^{49} + 18 q^{53} - 32 q^{59} + 80 q^{69} + 24 q^{71} + 40 q^{77} - 58 q^{81} + 56 q^{83} - 8 q^{87} + 6 q^{89} - 12 q^{91} + 52 q^{93} - 40 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.r.a 1300.r 65.f $2$ $10.381$ \(\Q(\sqrt{-1}) \) None \(0\) \(4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2-2i)q^{3}-4q^{7}-5iq^{9}+(-4+\cdots)q^{11}+\cdots\)
1300.2.r.b 1300.r 65.f $4$ $10.381$ \(\Q(i, \sqrt{5})\) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3})q^{9}+(-2+\cdots)q^{11}+\cdots\)
1300.2.r.c 1300.r 65.f $8$ $10.381$ 8.0.\(\cdots\).2 None \(0\) \(-2\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{3}q^{3}+(1+\beta _{2})q^{7}+(-\beta _{1}-\beta _{3}+\cdots)q^{9}+\cdots\)
1300.2.r.d 1300.r 65.f $8$ $10.381$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\zeta_{24}-\zeta_{24}^{4}-\zeta_{24}^{7})q^{3}+(-\zeta_{24}+\cdots)q^{7}+\cdots\)
1300.2.r.e 1300.r 65.f $8$ $10.381$ 8.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3}+\beta _{7})q^{7}+(1+\cdots)q^{11}+\cdots\)
1300.2.r.f 1300.r 65.f $12$ $10.381$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}-\beta _{8}q^{7}+(2\beta _{3}+\beta _{5})q^{9}-\beta _{7}q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1300, [\chi]) \cong \)