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Properties

Label 1300.2.a
Level $1300$
Weight $2$
Character orbit 1300.a
Rep. character $\chi_{1300}(1,\cdot)$
Character field $\Q$
Dimension $19$
Newform subspaces $11$
Sturm bound $420$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(420\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1300))\).

Total New Old
Modular forms 228 19 209
Cusp forms 193 19 174
Eisenstein series 35 0 35

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(5\)\(13\)FrickeDim
\(-\)\(+\)\(+\)$-$\(6\)
\(-\)\(+\)\(-\)$+$\(3\)
\(-\)\(-\)\(+\)$+$\(4\)
\(-\)\(-\)\(-\)$-$\(6\)
Plus space\(+\)\(7\)
Minus space\(-\)\(12\)

Trace form

\( 19 q - 4 q^{3} + 2 q^{7} + 19 q^{9} + O(q^{10}) \) \( 19 q - 4 q^{3} + 2 q^{7} + 19 q^{9} - 2 q^{11} - q^{13} - 10 q^{17} - 2 q^{19} + 12 q^{21} + 8 q^{23} - 4 q^{27} - 2 q^{29} - 2 q^{31} + 4 q^{33} - 2 q^{37} + 10 q^{41} - 4 q^{43} + 18 q^{47} + 11 q^{49} - 8 q^{51} + 10 q^{53} + 20 q^{57} - 6 q^{59} - 6 q^{61} + 42 q^{63} - 18 q^{67} + 52 q^{69} + 42 q^{71} - 26 q^{73} + 24 q^{77} + 16 q^{79} + 67 q^{81} + 6 q^{83} + 20 q^{87} + 50 q^{89} - 2 q^{91} + 20 q^{93} - 30 q^{97} - 58 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1300))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 13
1300.2.a.a 1300.a 1.a $1$ $10.381$ \(\Q\) None \(0\) \(-2\) \(0\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-2q^{7}+q^{9}+4q^{11}+q^{13}+\cdots\)
1300.2.a.b 1300.a 1.a $1$ $10.381$ \(\Q\) None \(0\) \(-1\) \(0\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{7}-2q^{9}-2q^{11}-q^{13}+\cdots\)
1300.2.a.c 1300.a 1.a $1$ $10.381$ \(\Q\) None \(0\) \(0\) \(0\) \(-3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{7}-3q^{9}+3q^{11}+q^{13}-q^{17}+\cdots\)
1300.2.a.d 1300.a 1.a $1$ $10.381$ \(\Q\) None \(0\) \(0\) \(0\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{7}-3q^{9}-2q^{11}+q^{13}-6q^{17}+\cdots\)
1300.2.a.e 1300.a 1.a $1$ $10.381$ \(\Q\) None \(0\) \(0\) \(0\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{7}-3q^{9}+3q^{11}-q^{13}+q^{17}+\cdots\)
1300.2.a.f 1300.a 1.a $1$ $10.381$ \(\Q\) None \(0\) \(1\) \(0\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{7}-2q^{9}-2q^{11}+q^{13}+\cdots\)
1300.2.a.g 1300.a 1.a $2$ $10.381$ \(\Q(\sqrt{33}) \) None \(0\) \(-1\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(1-\beta )q^{7}+(5+\beta )q^{9}+(-1+\cdots)q^{11}+\cdots\)
1300.2.a.h 1300.a 1.a $2$ $10.381$ \(\Q(\sqrt{33}) \) None \(0\) \(1\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(-1+\beta )q^{7}+(5+\beta )q^{9}+(-1+\cdots)q^{11}+\cdots\)
1300.2.a.i 1300.a 1.a $3$ $10.381$ 3.3.564.1 None \(0\) \(-2\) \(0\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}+(1+\beta _{1})q^{7}+(4+\beta _{1}+\cdots)q^{9}+\cdots\)
1300.2.a.j 1300.a 1.a $3$ $10.381$ 3.3.148.1 None \(0\) \(0\) \(0\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(-1-\beta _{1}-\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
1300.2.a.k 1300.a 1.a $3$ $10.381$ 3.3.148.1 None \(0\) \(0\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(1+\beta _{1}+\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1300))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1300)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(650))\)\(^{\oplus 2}\)