Properties

Label 1225.2.a
Level $1225$
Weight $2$
Character orbit 1225.a
Rep. character $\chi_{1225}(1,\cdot)$
Character field $\Q$
Dimension $58$
Newform subspaces $29$
Sturm bound $280$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 29 \)
Sturm bound: \(280\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1225))\).

Total New Old
Modular forms 164 73 91
Cusp forms 117 58 59
Eisenstein series 47 15 32

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

\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(13\)
\(+\)\(-\)\(-\)\(15\)
\(-\)\(+\)\(-\)\(17\)
\(-\)\(-\)\(+\)\(13\)
Plus space\(+\)\(26\)
Minus space\(-\)\(32\)

Trace form

\( 58q + 54q^{4} - 4q^{6} + 38q^{9} + O(q^{10}) \) \( 58q + 54q^{4} - 4q^{6} + 38q^{9} + 4q^{12} + 10q^{13} + 54q^{16} - 2q^{17} + 20q^{18} + 4q^{19} + 16q^{22} - 12q^{23} + 24q^{24} + 2q^{26} - 12q^{27} - 8q^{29} + 12q^{31} - 12q^{33} + 2q^{34} + 22q^{36} + 20q^{38} - 8q^{39} - 22q^{41} - 4q^{43} + 40q^{44} + 24q^{46} + 4q^{47} + 28q^{48} + 20q^{51} - 6q^{52} + 36q^{53} + 8q^{54} - 32q^{57} - 8q^{58} - 32q^{59} + 14q^{61} + 34q^{64} - 4q^{67} - 10q^{68} + 12q^{69} - 36q^{71} + 44q^{72} - 6q^{73} + 16q^{76} + 32q^{78} + 4q^{79} - 46q^{81} - 18q^{82} + 20q^{83} - 48q^{86} + 28q^{87} - 32q^{88} - 54q^{89} - 12q^{92} - 8q^{93} - 12q^{94} - 8q^{96} - 10q^{97} - 60q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 7
1225.2.a.a \(1\) \(9.782\) \(\Q\) None \(-2\) \(1\) \(0\) \(0\) \(-\) \(-\) \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}-2q^{9}+\cdots\)
1225.2.a.b \(1\) \(9.782\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(-\) \(-\) \(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{8}-2q^{9}+\cdots\)
1225.2.a.c \(1\) \(9.782\) \(\Q\) \(\Q(\sqrt{-7}) \) \(-1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-q^{2}-q^{4}+3q^{8}-3q^{9}+4q^{11}+\cdots\)
1225.2.a.d \(1\) \(9.782\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(-\) \(+\) \(q-q^{2}+q^{3}-q^{4}-q^{6}+3q^{8}-2q^{9}+\cdots\)
1225.2.a.e \(1\) \(9.782\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(q+q^{3}-2q^{4}-2q^{9}-3q^{11}-2q^{12}+\cdots\)
1225.2.a.f \(1\) \(9.782\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}-2q^{9}+\cdots\)
1225.2.a.g \(1\) \(9.782\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}-2q^{9}+\cdots\)
1225.2.a.h \(1\) \(9.782\) \(\Q\) None \(2\) \(-3\) \(0\) \(0\) \(+\) \(-\) \(q+2q^{2}-3q^{3}+2q^{4}-6q^{6}+6q^{9}+\cdots\)
1225.2.a.i \(1\) \(9.782\) \(\Q\) None \(2\) \(-1\) \(0\) \(0\) \(-\) \(-\) \(q+2q^{2}-q^{3}+2q^{4}-2q^{6}-2q^{9}+\cdots\)
1225.2.a.j \(1\) \(9.782\) \(\Q\) None \(2\) \(3\) \(0\) \(0\) \(+\) \(-\) \(q+2q^{2}+3q^{3}+2q^{4}+6q^{6}+6q^{9}+\cdots\)
1225.2.a.k \(2\) \(9.782\) \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(q+(-1+\beta )q^{2}+(-1-\beta )q^{3}+(1-2\beta )q^{4}+\cdots\)
1225.2.a.l \(2\) \(9.782\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-q^{2}+2\beta q^{3}-q^{4}-2\beta q^{6}+3q^{8}+\cdots\)
1225.2.a.m \(2\) \(9.782\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(0\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+(1+\beta )q^{3}+(1-2\beta )q^{4}+\cdots\)
1225.2.a.n \(2\) \(9.782\) \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(0\) \(0\) \(-\) \(-\) \(q-\beta q^{2}+(-2+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
1225.2.a.o \(2\) \(9.782\) \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(-1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-\beta q^{2}+(3+\beta )q^{4}+(-5-2\beta )q^{8}+\cdots\)
1225.2.a.p \(2\) \(9.782\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(q+\beta q^{2}+(-1+\beta )q^{3}+(2-\beta )q^{6}-2\beta q^{8}+\cdots\)
1225.2.a.q \(2\) \(9.782\) \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-35}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-\beta q^{3}-2q^{4}+2q^{9}-3q^{11}+2\beta q^{12}+\cdots\)
1225.2.a.r \(2\) \(9.782\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) \(+\) \(+\) \(q+\beta q^{2}+(1-\beta )q^{3}+(-2+\beta )q^{6}-2\beta q^{8}+\cdots\)
1225.2.a.s \(2\) \(9.782\) \(\Q(\sqrt{17}) \) None \(1\) \(-1\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}+(-1+\beta )q^{3}+(2+\beta )q^{4}+4q^{6}+\cdots\)
1225.2.a.t \(2\) \(9.782\) \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}+(3+\beta )q^{4}+(5+2\beta )q^{8}-3q^{9}+\cdots\)
1225.2.a.u \(2\) \(9.782\) \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}+(2-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
1225.2.a.v \(2\) \(9.782\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+q^{2}-2\beta q^{3}-q^{4}-2\beta q^{6}-3q^{8}+\cdots\)
1225.2.a.w \(3\) \(9.782\) 3.3.257.1 None \(-1\) \(-3\) \(0\) \(0\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1225.2.a.x \(3\) \(9.782\) 3.3.257.1 None \(-1\) \(3\) \(0\) \(0\) \(-\) \(+\) \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1225.2.a.y \(3\) \(9.782\) 3.3.257.1 None \(1\) \(-3\) \(0\) \(0\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1225.2.a.z \(3\) \(9.782\) 3.3.257.1 None \(1\) \(3\) \(0\) \(0\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1225.2.a.ba \(4\) \(9.782\) \(\Q(\sqrt{2}, \sqrt{5})\) None \(-6\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+(-1-\beta _{2})q^{2}+\beta _{1}q^{3}+3\beta _{2}q^{4}+\cdots\)
1225.2.a.bb \(4\) \(9.782\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-\beta _{3}q^{2}-\beta _{1}q^{3}+4q^{4}+3\beta _{2}q^{6}+\cdots\)
1225.2.a.bc \(4\) \(9.782\) \(\Q(\sqrt{2}, \sqrt{5})\) None \(6\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+(1+\beta _{2})q^{2}+\beta _{1}q^{3}+3\beta _{2}q^{4}+(2\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1225))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1225)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)