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Properties

Label 1300.4.a
Level $1300$
Weight $4$
Character orbit 1300.a
Rep. character $\chi_{1300}(1,\cdot)$
Character field $\Q$
Dimension $57$
Newform subspaces $16$
Sturm bound $840$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1300.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(840\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1300))\).

Total New Old
Modular forms 648 57 591
Cusp forms 612 57 555
Eisenstein series 36 0 36

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
\(-\)\(+\)\(+\)\(-\)\(12\)
\(-\)\(+\)\(-\)\(+\)\(15\)
\(-\)\(-\)\(+\)\(+\)\(16\)
\(-\)\(-\)\(-\)\(-\)\(14\)
Plus space\(+\)\(31\)
Minus space\(-\)\(26\)

Trace form

\( 57 q + 2 q^{3} - 16 q^{7} + 519 q^{9} + O(q^{10}) \) \( 57 q + 2 q^{3} - 16 q^{7} + 519 q^{9} - 100 q^{11} + 13 q^{13} - 84 q^{17} + 32 q^{19} - 124 q^{21} - 180 q^{23} - 226 q^{27} + 118 q^{29} + 280 q^{31} - 108 q^{33} + 426 q^{37} + 322 q^{41} - 98 q^{43} - 280 q^{47} + 2435 q^{49} + 1534 q^{51} - 622 q^{53} - 432 q^{57} - 108 q^{59} + 734 q^{61} - 1032 q^{63} - 1192 q^{67} - 2172 q^{69} - 3080 q^{71} + 1578 q^{73} + 1632 q^{77} - 276 q^{79} + 3273 q^{81} + 1344 q^{83} - 192 q^{87} - 1510 q^{89} - 494 q^{91} + 536 q^{93} - 1570 q^{97} + 516 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1300))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 13
1300.4.a.a 1300.a 1.a $1$ $76.702$ \(\Q\) None 1300.4.a.a \(0\) \(-5\) \(0\) \(-16\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-5q^{3}-2^{4}q^{7}-2q^{9}-3^{3}q^{11}+13q^{13}+\cdots\)
1300.4.a.b 1300.a 1.a $1$ $76.702$ \(\Q\) None 260.4.c.a \(0\) \(-4\) \(0\) \(4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{3}+4q^{7}-11q^{9}-6q^{11}-13q^{13}+\cdots\)
1300.4.a.c 1300.a 1.a $1$ $76.702$ \(\Q\) None 260.4.a.a \(0\) \(2\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+4q^{7}-23q^{9}+18q^{11}-13q^{13}+\cdots\)
1300.4.a.d 1300.a 1.a $1$ $76.702$ \(\Q\) None 52.4.a.a \(0\) \(3\) \(0\) \(11\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+11q^{7}-18q^{9}-2q^{11}+13q^{13}+\cdots\)
1300.4.a.e 1300.a 1.a $1$ $76.702$ \(\Q\) None 260.4.c.a \(0\) \(4\) \(0\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{3}-4q^{7}-11q^{9}-6q^{11}+13q^{13}+\cdots\)
1300.4.a.f 1300.a 1.a $1$ $76.702$ \(\Q\) None 1300.4.a.a \(0\) \(5\) \(0\) \(16\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+5q^{3}+2^{4}q^{7}-2q^{9}-3^{3}q^{11}-13q^{13}+\cdots\)
1300.4.a.g 1300.a 1.a $2$ $76.702$ \(\Q(\sqrt{217}) \) None 52.4.a.b \(0\) \(3\) \(0\) \(-27\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(-13-\beta )q^{7}+(28+3\beta )q^{9}+\cdots\)
1300.4.a.h 1300.a 1.a $3$ $76.702$ 3.3.788.1 None 260.4.a.b \(0\) \(2\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(-2+2\beta _{1}+2\beta _{2})q^{7}+\cdots\)
1300.4.a.i 1300.a 1.a $4$ $76.702$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 260.4.a.d \(0\) \(-4\) \(0\) \(-8\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{2})q^{3}+(-2-\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
1300.4.a.j 1300.a 1.a $4$ $76.702$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 1300.4.a.j \(0\) \(-4\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-1-2\beta _{1}-\beta _{3})q^{7}+\cdots\)
1300.4.a.k 1300.a 1.a $4$ $76.702$ 4.4.2785296.1 None 260.4.a.c \(0\) \(-4\) \(0\) \(8\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{2})q^{3}+(2+\beta _{2}-\beta _{3})q^{7}+\cdots\)
1300.4.a.l 1300.a 1.a $4$ $76.702$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 1300.4.a.j \(0\) \(4\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(1+2\beta _{1}+\beta _{3})q^{7}+(8+\cdots)q^{9}+\cdots\)
1300.4.a.m 1300.a 1.a $7$ $76.702$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 1300.4.a.m \(0\) \(-1\) \(0\) \(12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(2+\beta _{4})q^{7}+(17+\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots\)
1300.4.a.n 1300.a 1.a $7$ $76.702$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 1300.4.a.m \(0\) \(1\) \(0\) \(-12\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-2-\beta _{4})q^{7}+(17+\beta _{2}+\cdots)q^{9}+\cdots\)
1300.4.a.o 1300.a 1.a $8$ $76.702$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 260.4.c.b \(0\) \(-4\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{3}q^{3}+(\beta _{1}-\beta _{2}+\beta _{4}+\beta _{6})q^{7}+\cdots\)
1300.4.a.p 1300.a 1.a $8$ $76.702$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 260.4.c.b \(0\) \(4\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{3}+(-\beta _{1}+\beta _{2}-\beta _{4}-\beta _{6})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1300))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1300)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(650))\)\(^{\oplus 2}\)