Properties

Label 1305.2.a
Level $1305$
Weight $2$
Character orbit 1305.a
Rep. character $\chi_{1305}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $20$
Sturm bound $360$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 20 \)
Sturm bound: \(360\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1305))\).

Total New Old
Modular forms 188 48 140
Cusp forms 173 48 125
Eisenstein series 15 0 15

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

\(3\)\(5\)\(29\)FrickeDim
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(7\)
\(+\)\(-\)\(+\)$-$\(7\)
\(+\)\(-\)\(-\)$+$\(3\)
\(-\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(-\)$+$\(6\)
\(-\)\(-\)\(+\)$+$\(6\)
\(-\)\(-\)\(-\)$-$\(8\)
Plus space\(+\)\(18\)
Minus space\(-\)\(30\)

Trace form

\( 48 q + 2 q^{2} + 56 q^{4} + 4 q^{7} + 6 q^{8} + O(q^{10}) \) \( 48 q + 2 q^{2} + 56 q^{4} + 4 q^{7} + 6 q^{8} + 2 q^{10} - 4 q^{11} + 8 q^{13} + 8 q^{14} + 64 q^{16} + 16 q^{17} - 4 q^{19} - 8 q^{20} - 4 q^{23} + 48 q^{25} + 20 q^{26} + 20 q^{28} - 4 q^{31} + 14 q^{32} - 32 q^{34} - 4 q^{35} + 16 q^{37} + 16 q^{38} - 6 q^{40} + 8 q^{41} - 24 q^{43} - 28 q^{44} - 16 q^{46} + 80 q^{49} + 2 q^{50} + 8 q^{53} + 72 q^{56} - 2 q^{58} - 32 q^{59} + 16 q^{61} - 56 q^{62} + 64 q^{64} - 16 q^{65} - 20 q^{67} + 84 q^{68} - 12 q^{70} - 24 q^{71} - 36 q^{73} + 8 q^{74} - 28 q^{76} + 36 q^{77} - 12 q^{79} - 4 q^{82} + 20 q^{83} + 12 q^{85} + 56 q^{86} + 16 q^{88} - 40 q^{89} + 8 q^{91} + 20 q^{92} + 16 q^{94} + 16 q^{95} - 20 q^{97} - 34 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1305))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 29
1305.2.a.a 1305.a 1.a $1$ $10.420$ \(\Q\) None \(-2\) \(0\) \(1\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+q^{5}-2q^{7}-2q^{10}+\cdots\)
1305.2.a.b 1305.a 1.a $1$ $10.420$ \(\Q\) None \(-1\) \(0\) \(-1\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-q^{5}+4q^{7}+3q^{8}+q^{10}+\cdots\)
1305.2.a.c 1305.a 1.a $1$ $10.420$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{4}+q^{5}-2q^{7}-q^{11}+6q^{13}+\cdots\)
1305.2.a.d 1305.a 1.a $1$ $10.420$ \(\Q\) None \(0\) \(0\) \(1\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}+q^{5}+2q^{7}-3q^{11}+2q^{13}+\cdots\)
1305.2.a.e 1305.a 1.a $1$ $10.420$ \(\Q\) None \(1\) \(0\) \(-1\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{5}-4q^{7}-3q^{8}-q^{10}+\cdots\)
1305.2.a.f 1305.a 1.a $1$ $10.420$ \(\Q\) None \(1\) \(0\) \(1\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+q^{5}-2q^{7}-3q^{8}+q^{10}+\cdots\)
1305.2.a.g 1305.a 1.a $1$ $10.420$ \(\Q\) None \(2\) \(0\) \(-1\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-q^{5}-2q^{7}-2q^{10}+\cdots\)
1305.2.a.h 1305.a 1.a $2$ $10.420$ \(\Q(\sqrt{5}) \) None \(-1\) \(0\) \(-2\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}-q^{5}+(-1-2\beta )q^{7}+\cdots\)
1305.2.a.i 1305.a 1.a $2$ $10.420$ \(\Q(\sqrt{17}) \) None \(-1\) \(0\) \(-2\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(2+\beta )q^{4}-q^{5}+(2-2\beta )q^{7}+\cdots\)
1305.2.a.j 1305.a 1.a $2$ $10.420$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+3q^{4}+q^{5}+2q^{7}-\beta q^{8}+\cdots\)
1305.2.a.k 1305.a 1.a $2$ $10.420$ \(\Q(\sqrt{5}) \) None \(1\) \(0\) \(2\) \(-6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+q^{5}-3q^{7}+\cdots\)
1305.2.a.l 1305.a 1.a $2$ $10.420$ \(\Q(\sqrt{5}) \) None \(1\) \(0\) \(2\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+q^{5}+(-1-2\beta )q^{7}+\cdots\)
1305.2.a.m 1305.a 1.a $2$ $10.420$ \(\Q(\sqrt{21}) \) None \(1\) \(0\) \(-2\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(3+\beta )q^{4}-q^{5}+q^{7}+(5+2\beta )q^{8}+\cdots\)
1305.2.a.n 1305.a 1.a $2$ $10.420$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-2\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}-q^{5}+(-2+\cdots)q^{7}+\cdots\)
1305.2.a.o 1305.a 1.a $3$ $10.420$ 3.3.148.1 None \(-3\) \(0\) \(3\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1305.2.a.p 1305.a 1.a $3$ $10.420$ 3.3.148.1 None \(-1\) \(0\) \(-3\) \(4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}-q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
1305.2.a.q 1305.a 1.a $3$ $10.420$ 3.3.469.1 None \(-1\) \(0\) \(-3\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}-q^{5}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)
1305.2.a.r 1305.a 1.a $4$ $10.420$ 4.4.2225.1 None \(3\) \(0\) \(4\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)
1305.2.a.s 1305.a 1.a $7$ $10.420$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-1\) \(0\) \(-7\) \(10\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}-q^{5}+(1-\beta _{5}+\cdots)q^{7}+\cdots\)
1305.2.a.t 1305.a 1.a $7$ $10.420$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(1\) \(0\) \(7\) \(10\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+q^{5}+(1-\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1305))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1305)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(435))\)\(^{\oplus 2}\)