Properties

Label 1575.2.a
Level 1575
Weight 2
Character orbit a
Rep. character \(\chi_{1575}(1,\cdot)\)
Character field \(\Q\)
Dimension 47
Newforms 26
Sturm bound 480
Trace bound 13

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Defining parameters

Level: \( N \) = \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1575.a (trivial)
Character field: \(\Q\)
Newforms: \( 26 \)
Sturm bound: \(480\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1575))\).

Total New Old
Modular forms 264 47 217
Cusp forms 217 47 170
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.

\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(6\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(9\)
Plus space\(+\)\(16\)
Minus space\(-\)\(31\)

Trace form

\(47q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 51q^{4} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(47q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut 51q^{4} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 8q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 3q^{14} \) \(\mathstrut +\mathstrut 67q^{16} \) \(\mathstrut -\mathstrut 10q^{17} \) \(\mathstrut +\mathstrut 20q^{19} \) \(\mathstrut +\mathstrut 12q^{22} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 18q^{26} \) \(\mathstrut +\mathstrut 3q^{28} \) \(\mathstrut +\mathstrut 18q^{29} \) \(\mathstrut +\mathstrut 12q^{31} \) \(\mathstrut -\mathstrut 9q^{32} \) \(\mathstrut +\mathstrut 34q^{34} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut +\mathstrut 28q^{38} \) \(\mathstrut +\mathstrut 10q^{41} \) \(\mathstrut +\mathstrut 24q^{43} \) \(\mathstrut +\mathstrut 50q^{44} \) \(\mathstrut +\mathstrut 2q^{46} \) \(\mathstrut +\mathstrut 20q^{47} \) \(\mathstrut +\mathstrut 47q^{49} \) \(\mathstrut +\mathstrut 34q^{52} \) \(\mathstrut +\mathstrut 10q^{53} \) \(\mathstrut +\mathstrut 21q^{56} \) \(\mathstrut -\mathstrut 6q^{58} \) \(\mathstrut -\mathstrut 24q^{59} \) \(\mathstrut +\mathstrut 22q^{61} \) \(\mathstrut +\mathstrut 24q^{62} \) \(\mathstrut +\mathstrut 133q^{64} \) \(\mathstrut +\mathstrut 24q^{67} \) \(\mathstrut -\mathstrut 18q^{68} \) \(\mathstrut -\mathstrut 16q^{71} \) \(\mathstrut -\mathstrut 6q^{73} \) \(\mathstrut +\mathstrut 60q^{74} \) \(\mathstrut -\mathstrut 8q^{76} \) \(\mathstrut -\mathstrut 4q^{77} \) \(\mathstrut -\mathstrut 4q^{79} \) \(\mathstrut -\mathstrut 34q^{82} \) \(\mathstrut -\mathstrut 12q^{83} \) \(\mathstrut -\mathstrut 66q^{86} \) \(\mathstrut +\mathstrut 8q^{88} \) \(\mathstrut -\mathstrut 62q^{89} \) \(\mathstrut -\mathstrut 14q^{91} \) \(\mathstrut +\mathstrut 40q^{92} \) \(\mathstrut -\mathstrut 92q^{94} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut -\mathstrut q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
1575.2.a.a \(1\) \(12.576\) \(\Q\) None \(-2\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-2q^{2}+2q^{4}-q^{7}+3q^{11}+q^{13}+\cdots\)
1575.2.a.b \(1\) \(12.576\) \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{4}-q^{7}+3q^{8}+4q^{13}+q^{14}+\cdots\)
1575.2.a.c \(1\) \(12.576\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-q^{2}-q^{4}+q^{7}+3q^{8}-4q^{11}+2q^{13}+\cdots\)
1575.2.a.d \(1\) \(12.576\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{4}+q^{7}+3q^{8}-4q^{13}-q^{14}+\cdots\)
1575.2.a.e \(1\) \(12.576\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q-q^{2}-q^{4}+q^{7}+3q^{8}+6q^{11}+2q^{13}+\cdots\)
1575.2.a.f \(1\) \(12.576\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q-2q^{4}-q^{7}+3q^{11}-5q^{13}+4q^{16}+\cdots\)
1575.2.a.g \(1\) \(12.576\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{4}-q^{7}-3q^{8}+4q^{13}-q^{14}+\cdots\)
1575.2.a.h \(1\) \(12.576\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{4}-q^{7}-3q^{8}+6q^{13}-q^{14}+\cdots\)
1575.2.a.i \(1\) \(12.576\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q+q^{2}-q^{4}-q^{7}-3q^{8}+6q^{11}-2q^{13}+\cdots\)
1575.2.a.j \(1\) \(12.576\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{4}+q^{7}-3q^{8}-4q^{13}+q^{14}+\cdots\)
1575.2.a.k \(1\) \(12.576\) \(\Q\) None \(2\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+2q^{2}+2q^{4}+q^{7}+3q^{11}-q^{13}+\cdots\)
1575.2.a.l \(2\) \(12.576\) \(\Q(\sqrt{5}) \) None \(-3\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+(-1-\beta )q^{2}+3\beta q^{4}-q^{7}+(-1+\cdots)q^{8}+\cdots\)
1575.2.a.m \(2\) \(12.576\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+q^{7}+(-3+\cdots)q^{8}+\cdots\)
1575.2.a.n \(2\) \(12.576\) \(\Q(\sqrt{5}) \) None \(-1\) \(0\) \(0\) \(2\) \(-\) \(+\) \(-\) \(q-\beta q^{2}+(-1+\beta )q^{4}+q^{7}+(-1+2\beta )q^{8}+\cdots\)
1575.2.a.o \(2\) \(12.576\) \(\Q(\sqrt{13}) \) None \(-1\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q-\beta q^{2}+(1+\beta )q^{4}+q^{7}-3q^{8}+3q^{11}+\cdots\)
1575.2.a.p \(2\) \(12.576\) \(\Q(\sqrt{17}) \) None \(-1\) \(0\) \(0\) \(2\) \(-\) \(+\) \(-\) \(q-\beta q^{2}+(2+\beta )q^{4}+q^{7}+(-4-\beta )q^{8}+\cdots\)
1575.2.a.q \(2\) \(12.576\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q+\beta q^{2}+q^{4}-q^{7}-\beta q^{8}-2\beta q^{11}+\cdots\)
1575.2.a.r \(2\) \(12.576\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q-\beta q^{2}+3q^{4}-q^{7}-\beta q^{8}+(-2-2\beta )q^{11}+\cdots\)
1575.2.a.s \(2\) \(12.576\) \(\Q(\sqrt{5}) \) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(q+\beta q^{2}+(-1+\beta )q^{4}-q^{7}+(1-2\beta )q^{8}+\cdots\)
1575.2.a.t \(2\) \(12.576\) \(\Q(\sqrt{13}) \) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+\beta q^{2}+(1+\beta )q^{4}-q^{7}+3q^{8}+3q^{11}+\cdots\)
1575.2.a.u \(2\) \(12.576\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{7}+(3+\beta )q^{8}+\cdots\)
1575.2.a.v \(2\) \(12.576\) \(\Q(\sqrt{5}) \) None \(3\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+(1+\beta )q^{2}+3\beta q^{4}+q^{7}+(1+4\beta )q^{8}+\cdots\)
1575.2.a.w \(3\) \(12.576\) 3.3.148.1 None \(-1\) \(0\) \(0\) \(-3\) \(-\) \(-\) \(+\) \(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}-q^{7}+(-2+\cdots)q^{8}+\cdots\)
1575.2.a.x \(3\) \(12.576\) 3.3.148.1 None \(1\) \(0\) \(0\) \(3\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+q^{7}+(2+\cdots)q^{8}+\cdots\)
1575.2.a.y \(4\) \(12.576\) 4.4.174928.1 None \(0\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}-q^{7}+(2\beta _{1}+\beta _{3})q^{8}+\cdots\)
1575.2.a.z \(4\) \(12.576\) 4.4.174928.1 None \(0\) \(0\) \(0\) \(4\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+q^{7}+(2\beta _{1}+\beta _{3})q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1575))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1575)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 2}\)