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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 11
2475.2.a.a \(1\) \(19.763\) \(\Q\) None \(-2\) \(0\) \(0\) \(2\) \(-\) \(+\) \(+\) \(q-2q^{2}+2q^{4}+2q^{7}-q^{11}-4q^{13}+\cdots\)
2475.2.a.b \(1\) \(19.763\) \(\Q\) None \(-1\) \(0\) \(0\) \(-3\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{4}-3q^{7}+3q^{8}+q^{11}+2q^{13}+\cdots\)
2475.2.a.c \(1\) \(19.763\) \(\Q\) None \(-1\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{4}+2q^{7}+3q^{8}+q^{11}+2q^{13}+\cdots\)
2475.2.a.d \(1\) \(19.763\) \(\Q\) None \(-1\) \(0\) \(0\) \(3\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{4}+3q^{7}+3q^{8}-q^{11}-2q^{13}+\cdots\)
2475.2.a.e \(1\) \(19.763\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(q-2q^{4}-q^{7}+q^{11}-q^{13}+4q^{16}+\cdots\)
2475.2.a.f \(1\) \(19.763\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-2q^{4}+q^{7}+q^{11}+q^{13}+4q^{16}+\cdots\)
2475.2.a.g \(1\) \(19.763\) \(\Q\) None \(1\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{4}-4q^{7}-3q^{8}-q^{11}+2q^{13}+\cdots\)
2475.2.a.h \(1\) \(19.763\) \(\Q\) None \(1\) \(0\) \(0\) \(-3\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{4}-3q^{7}-3q^{8}-q^{11}+2q^{13}+\cdots\)
2475.2.a.i \(1\) \(19.763\) \(\Q\) None \(1\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+q^{2}-q^{4}-3q^{8}+q^{11}-2q^{13}+\cdots\)
2475.2.a.j \(1\) \(19.763\) \(\Q\) None \(1\) \(0\) \(0\) \(2\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{4}+2q^{7}-3q^{8}-q^{11}+2q^{13}+\cdots\)
2475.2.a.k \(1\) \(19.763\) \(\Q\) None \(1\) \(0\) \(0\) \(3\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{4}+3q^{7}-3q^{8}+q^{11}-2q^{13}+\cdots\)
2475.2.a.l \(2\) \(19.763\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(-1-\beta )q^{7}+\cdots\)
2475.2.a.m \(2\) \(19.763\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(4\) \(-\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(2+2\beta )q^{7}+\cdots\)
2475.2.a.n \(2\) \(19.763\) \(\Q(\sqrt{5}) \) None \(-1\) \(0\) \(0\) \(1\) \(-\) \(+\) \(+\) \(q-\beta q^{2}+(-1+\beta )q^{4}+(2-3\beta )q^{7}+\cdots\)
2475.2.a.o \(2\) \(19.763\) \(\Q(\sqrt{13}) \) None \(-1\) \(0\) \(0\) \(5\) \(-\) \(-\) \(-\) \(q-\beta q^{2}+(1+\beta )q^{4}+(2+\beta )q^{7}-3q^{8}+\cdots\)
2475.2.a.p \(2\) \(19.763\) \(\Q(\sqrt{17}) \) None \(-1\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q-\beta q^{2}+(2+\beta )q^{4}-q^{7}+(-4-\beta )q^{8}+\cdots\)
2475.2.a.q \(2\) \(19.763\) \(\Q(\sqrt{17}) \) None \(-1\) \(0\) \(0\) \(2\) \(+\) \(-\) \(-\) \(q-\beta q^{2}+(2+\beta )q^{4}+q^{7}+(-4-\beta )q^{8}+\cdots\)
2475.2.a.r \(2\) \(19.763\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(q+\beta q^{2}+q^{4}-2q^{7}-\beta q^{8}+q^{11}+\cdots\)
2475.2.a.s \(2\) \(19.763\) \(\Q(\sqrt{5}) \) None \(1\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q+\beta q^{2}+(-1+\beta )q^{4}+(-2+3\beta )q^{7}+\cdots\)
2475.2.a.t \(2\) \(19.763\) \(\Q(\sqrt{13}) \) None \(1\) \(0\) \(0\) \(-5\) \(-\) \(+\) \(-\) \(q+\beta q^{2}+(1+\beta )q^{4}+(-2-\beta )q^{7}+3q^{8}+\cdots\)
2475.2.a.u \(2\) \(19.763\) \(\Q(\sqrt{17}) \) None \(1\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(-\) \(q+\beta q^{2}+(2+\beta )q^{4}-q^{7}+(4+\beta )q^{8}+\cdots\)
2475.2.a.v \(2\) \(19.763\) \(\Q(\sqrt{17}) \) None \(1\) \(0\) \(0\) \(2\) \(+\) \(-\) \(+\) \(q+\beta q^{2}+(2+\beta )q^{4}+q^{7}+(4+\beta )q^{8}+\cdots\)
2475.2.a.w \(2\) \(19.763\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(1-\beta )q^{7}+\cdots\)
2475.2.a.x \(2\) \(19.763\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(4\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+2q^{7}+(3+\cdots)q^{8}+\cdots\)
2475.2.a.y \(3\) \(19.763\) 3.3.148.1 None \(-3\) \(0\) \(0\) \(8\) \(-\) \(-\) \(-\) \(q+(-1-\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2475.2.a.z \(3\) \(19.763\) 3.3.568.1 None \(-2\) \(0\) \(0\) \(-3\) \(-\) \(-\) \(+\) \(q+(-1+\beta _{1})q^{2}+(3+\beta _{2})q^{4}+(-1+\cdots)q^{7}+\cdots\)
2475.2.a.ba \(3\) \(19.763\) 3.3.148.1 None \(-1\) \(0\) \(0\) \(4\) \(-\) \(-\) \(+\) \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
2475.2.a.bb \(3\) \(19.763\) 3.3.148.1 None \(-1\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
2475.2.a.bc \(3\) \(19.763\) 3.3.148.1 None \(1\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
2475.2.a.bd \(3\) \(19.763\) 3.3.568.1 None \(2\) \(0\) \(0\) \(3\) \(-\) \(+\) \(+\) \(q+(1-\beta _{1})q^{2}+(3+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
2475.2.a.be \(3\) \(19.763\) 3.3.148.1 None \(3\) \(0\) \(0\) \(-8\) \(-\) \(-\) \(-\) \(q+(1+\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(-3+\cdots)q^{7}+\cdots\)
2475.2.a.bf \(4\) \(19.763\) 4.4.48704.2 None \(-2\) \(0\) \(0\) \(-4\) \(+\) \(+\) \(+\) \(q+(-1+\beta _{1})q^{2}+(2+\beta _{2})q^{4}+(-1+\cdots)q^{7}+\cdots\)
2475.2.a.bg \(4\) \(19.763\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}-\beta _{3})q^{7}+\beta _{3}q^{8}+\cdots\)
2475.2.a.bh \(4\) \(19.763\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{1}+\beta _{3})q^{7}+\cdots\)
2475.2.a.bi \(4\) \(19.763\) \(\Q(\sqrt{3}, \sqrt{11})\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(2+\beta _{3})q^{4}+2\beta _{2}q^{7}+(\beta _{1}+\cdots)q^{8}+\cdots\)
2475.2.a.bj \(4\) \(19.763\) 4.4.48704.2 None \(2\) \(0\) \(0\) \(-4\) \(+\) \(+\) \(-\) \(q+(1-\beta _{1})q^{2}+(2+\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)

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

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