Properties

Label 2475.2.a
Level $2475$
Weight $2$
Character orbit 2475.a
Rep. character $\chi_{2475}(1,\cdot)$
Character field $\Q$
Dimension $78$
Newform subspaces $36$
Sturm bound $720$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 36 \)
Sturm bound: \(720\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(7\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2475))\).

Total New Old
Modular forms 384 78 306
Cusp forms 337 78 259
Eisenstein series 47 0 47

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

\(3\)\(5\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(8\)
\(+\)\(+\)\(-\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(7\)
\(+\)\(-\)\(-\)\(+\)\(7\)
\(-\)\(+\)\(+\)\(-\)\(12\)
\(-\)\(+\)\(-\)\(+\)\(10\)
\(-\)\(-\)\(+\)\(+\)\(11\)
\(-\)\(-\)\(-\)\(-\)\(15\)
Plus space\(+\)\(36\)
Minus space\(-\)\(42\)

Trace form

\( 78 q - q^{2} + 79 q^{4} - 2 q^{7} - 15 q^{8} + O(q^{10}) \) \( 78 q - q^{2} + 79 q^{4} - 2 q^{7} - 15 q^{8} + 2 q^{11} - 10 q^{13} - 12 q^{14} + 77 q^{16} + q^{22} + 3 q^{23} + 22 q^{26} + 16 q^{28} + 18 q^{29} - 7 q^{31} - 11 q^{32} + 14 q^{34} - 17 q^{37} + 22 q^{41} - 18 q^{43} + q^{44} + 10 q^{46} - 12 q^{47} + 30 q^{49} - 2 q^{52} - 12 q^{53} - 24 q^{56} + 42 q^{58} - 7 q^{59} + 6 q^{61} + 10 q^{62} + 101 q^{64} - 9 q^{67} + 66 q^{68} - 15 q^{71} + 2 q^{73} - 28 q^{74} - 24 q^{76} - 2 q^{77} - 50 q^{79} + 30 q^{82} - 6 q^{83} + 52 q^{86} + 15 q^{88} + 5 q^{89} - 36 q^{91} - 22 q^{92} - 40 q^{94} - 35 q^{97} + 63 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\) 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 11
2475.2.a.a 2475.a 1.a $1$ $19.763$ \(\Q\) None \(-2\) \(0\) \(0\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+2q^{7}-q^{11}-4q^{13}+\cdots\)
2475.2.a.b 2475.a 1.a $1$ $19.763$ \(\Q\) None \(-1\) \(0\) \(0\) \(-3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-3q^{7}+3q^{8}+q^{11}+2q^{13}+\cdots\)
2475.2.a.c 2475.a 1.a $1$ $19.763$ \(\Q\) None \(-1\) \(0\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+2q^{7}+3q^{8}+q^{11}+2q^{13}+\cdots\)
2475.2.a.d 2475.a 1.a $1$ $19.763$ \(\Q\) None \(-1\) \(0\) \(0\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+3q^{7}+3q^{8}-q^{11}-2q^{13}+\cdots\)
2475.2.a.e 2475.a 1.a $1$ $19.763$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}-q^{7}+q^{11}-q^{13}+4q^{16}+\cdots\)
2475.2.a.f 2475.a 1.a $1$ $19.763$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}+q^{7}+q^{11}+q^{13}+4q^{16}+\cdots\)
2475.2.a.g 2475.a 1.a $1$ $19.763$ \(\Q\) None \(1\) \(0\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-4q^{7}-3q^{8}-q^{11}+2q^{13}+\cdots\)
2475.2.a.h 2475.a 1.a $1$ $19.763$ \(\Q\) None \(1\) \(0\) \(0\) \(-3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-3q^{7}-3q^{8}-q^{11}+2q^{13}+\cdots\)
2475.2.a.i 2475.a 1.a $1$ $19.763$ \(\Q\) None \(1\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-3q^{8}+q^{11}-2q^{13}+\cdots\)
2475.2.a.j 2475.a 1.a $1$ $19.763$ \(\Q\) None \(1\) \(0\) \(0\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+2q^{7}-3q^{8}-q^{11}+2q^{13}+\cdots\)
2475.2.a.k 2475.a 1.a $1$ $19.763$ \(\Q\) None \(1\) \(0\) \(0\) \(3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+3q^{7}-3q^{8}+q^{11}-2q^{13}+\cdots\)
2475.2.a.l 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(-1-\beta )q^{7}+\cdots\)
2475.2.a.m 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(2+2\beta )q^{7}+\cdots\)
2475.2.a.n 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{5}) \) None \(-1\) \(0\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+(2-3\beta )q^{7}+\cdots\)
2475.2.a.o 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{13}) \) None \(-1\) \(0\) \(0\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(1+\beta )q^{4}+(2+\beta )q^{7}-3q^{8}+\cdots\)
2475.2.a.p 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{17}) \) None \(-1\) \(0\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(2+\beta )q^{4}-q^{7}+(-4-\beta )q^{8}+\cdots\)
2475.2.a.q 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{17}) \) None \(-1\) \(0\) \(0\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(2+\beta )q^{4}+q^{7}+(-4-\beta )q^{8}+\cdots\)
2475.2.a.r 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}-2q^{7}-\beta q^{8}+q^{11}+\cdots\)
2475.2.a.s 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{5}) \) None \(1\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+(-2+3\beta )q^{7}+\cdots\)
2475.2.a.t 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{13}) \) None \(1\) \(0\) \(0\) \(-5\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1+\beta )q^{4}+(-2-\beta )q^{7}+3q^{8}+\cdots\)
2475.2.a.u 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{17}) \) None \(1\) \(0\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(2+\beta )q^{4}-q^{7}+(4+\beta )q^{8}+\cdots\)
2475.2.a.v 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{17}) \) None \(1\) \(0\) \(0\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(2+\beta )q^{4}+q^{7}+(4+\beta )q^{8}+\cdots\)
2475.2.a.w 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(1-\beta )q^{7}+\cdots\)
2475.2.a.x 2475.a 1.a $2$ $19.763$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+2q^{7}+(3+\cdots)q^{8}+\cdots\)
2475.2.a.y 2475.a 1.a $3$ $19.763$ 3.3.148.1 None \(-3\) \(0\) \(0\) \(8\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2475.2.a.z 2475.a 1.a $3$ $19.763$ 3.3.568.1 None \(-2\) \(0\) \(0\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(3+\beta _{2})q^{4}+(-1+\cdots)q^{7}+\cdots\)
2475.2.a.ba 2475.a 1.a $3$ $19.763$ 3.3.148.1 None \(-1\) \(0\) \(0\) \(4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
2475.2.a.bb 2475.a 1.a $3$ $19.763$ 3.3.148.1 None \(-1\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
2475.2.a.bc 2475.a 1.a $3$ $19.763$ 3.3.148.1 None \(1\) \(0\) \(0\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
2475.2.a.bd 2475.a 1.a $3$ $19.763$ 3.3.568.1 None \(2\) \(0\) \(0\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(3+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
2475.2.a.be 2475.a 1.a $3$ $19.763$ 3.3.148.1 None \(3\) \(0\) \(0\) \(-8\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(-3+\cdots)q^{7}+\cdots\)
2475.2.a.bf 2475.a 1.a $4$ $19.763$ 4.4.48704.2 None \(-2\) \(0\) \(0\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(2+\beta _{2})q^{4}+(-1+\cdots)q^{7}+\cdots\)
2475.2.a.bg 2475.a 1.a $4$ $19.763$ \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}-\beta _{3})q^{7}+\beta _{3}q^{8}+\cdots\)
2475.2.a.bh 2475.a 1.a $4$ $19.763$ \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{1}+\beta _{3})q^{7}+\cdots\)
2475.2.a.bi 2475.a 1.a $4$ $19.763$ \(\Q(\sqrt{3}, \sqrt{11})\) None \(0\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{3})q^{4}+2\beta _{2}q^{7}+(\beta _{1}+\cdots)q^{8}+\cdots\)
2475.2.a.bj 2475.a 1.a $4$ $19.763$ 4.4.48704.2 None \(2\) \(0\) \(0\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(2+\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2475))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2475)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(495))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(825))\)\(^{\oplus 2}\)