Properties

Label 1521.2.a
Level $1521$
Weight $2$
Character orbit 1521.a
Rep. character $\chi_{1521}(1,\cdot)$
Character field $\Q$
Dimension $59$
Newform subspaces $23$
Sturm bound $364$
Trace bound $23$

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Defining parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 23 \)
Sturm bound: \(364\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1521))\).

Total New Old
Modular forms 210 70 140
Cusp forms 155 59 96
Eisenstein series 55 11 44

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

\(3\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(10\)
\(+\)\(-\)\(-\)\(16\)
\(-\)\(+\)\(-\)\(18\)
\(-\)\(-\)\(+\)\(15\)
Plus space\(+\)\(25\)
Minus space\(-\)\(34\)

Trace form

\( 59q - q^{2} + 55q^{4} + 2q^{5} - 9q^{8} + O(q^{10}) \) \( 59q - q^{2} + 55q^{4} + 2q^{5} - 9q^{8} + 2q^{10} + 18q^{14} + 47q^{16} + 8q^{17} - 4q^{19} + 14q^{20} + 2q^{22} - 6q^{23} + 47q^{25} + 8q^{28} - 2q^{29} + 11q^{32} + 10q^{34} - 28q^{35} + 2q^{37} - 16q^{38} - 10q^{40} + 22q^{41} - 18q^{43} - 8q^{44} - 32q^{46} - 12q^{47} + 21q^{49} - 7q^{50} + 22q^{53} - 20q^{55} + 40q^{56} - 10q^{58} + 16q^{59} + 20q^{61} + 22q^{62} + 25q^{64} - 28q^{67} + 30q^{68} - 8q^{70} + 4q^{71} + 6q^{73} - 16q^{74} - 20q^{76} - 16q^{77} - 18q^{79} - 2q^{80} - 32q^{82} + 28q^{85} - 36q^{86} - 6q^{88} + 22q^{89} + 8q^{92} + 46q^{94} + 18q^{95} + 14q^{97} + 7q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1521))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 13
1521.2.a.a \(1\) \(12.145\) \(\Q\) None \(-1\) \(0\) \(1\) \(2\) \(-\) \(+\) \(q-q^{2}-q^{4}+q^{5}+2q^{7}+3q^{8}-q^{10}+\cdots\)
1521.2.a.b \(1\) \(12.145\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-1\) \(+\) \(+\) \(q-2q^{4}-q^{7}+4q^{16}+8q^{19}-5q^{25}+\cdots\)
1521.2.a.c \(1\) \(12.145\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(1\) \(+\) \(+\) \(q-2q^{4}+q^{7}+4q^{16}-8q^{19}-5q^{25}+\cdots\)
1521.2.a.d \(1\) \(12.145\) \(\Q\) None \(1\) \(0\) \(-1\) \(-2\) \(-\) \(+\) \(q+q^{2}-q^{4}-q^{5}-2q^{7}-3q^{8}-q^{10}+\cdots\)
1521.2.a.e \(1\) \(12.145\) \(\Q\) None \(1\) \(0\) \(2\) \(4\) \(-\) \(+\) \(q+q^{2}-q^{4}+2q^{5}+4q^{7}-3q^{8}+2q^{10}+\cdots\)
1521.2.a.f \(2\) \(12.145\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}-2\beta q^{5}+\cdots\)
1521.2.a.g \(2\) \(12.145\) \(\Q(\sqrt{17}) \) None \(-1\) \(0\) \(3\) \(-3\) \(-\) \(+\) \(q-\beta q^{2}+(2+\beta )q^{4}+(2-\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
1521.2.a.h \(2\) \(12.145\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-2q^{4}+2\beta q^{5}-\beta q^{7}-2\beta q^{11}+4q^{16}+\cdots\)
1521.2.a.i \(2\) \(12.145\) \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-2q^{4}-3\beta q^{7}+4q^{16}-2\beta q^{19}+\cdots\)
1521.2.a.j \(2\) \(12.145\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-4\) \(+\) \(+\) \(q+\beta q^{2}+q^{4}-2q^{7}-\beta q^{8}-2\beta q^{11}+\cdots\)
1521.2.a.k \(2\) \(12.145\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta q^{2}+q^{4}-\beta q^{5}-\beta q^{8}-3q^{10}+\cdots\)
1521.2.a.l \(2\) \(12.145\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta q^{2}+q^{4}-2\beta q^{7}-\beta q^{8}+2\beta q^{11}+\cdots\)
1521.2.a.m \(2\) \(12.145\) \(\Q(\sqrt{17}) \) None \(1\) \(0\) \(-3\) \(3\) \(-\) \(+\) \(q+\beta q^{2}+(2+\beta )q^{4}+(-2+\beta )q^{5}+(1+\cdots)q^{7}+\cdots\)
1521.2.a.n \(3\) \(12.145\) \(\Q(\zeta_{14})^+\) None \(-3\) \(0\) \(-6\) \(2\) \(-\) \(-\) \(q+(-1+\beta _{1}+\beta _{2})q^{2}+(4-\beta _{1})q^{4}+\cdots\)
1521.2.a.o \(3\) \(12.145\) \(\Q(\zeta_{14})^+\) None \(-2\) \(0\) \(-4\) \(3\) \(-\) \(-\) \(q+(-1-\beta _{2})q^{2}+(\beta _{1}+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
1521.2.a.p \(3\) \(12.145\) \(\Q(\zeta_{14})^+\) None \(-1\) \(0\) \(-4\) \(10\) \(-\) \(+\) \(q-\beta _{1}q^{2}+\beta _{2}q^{4}+(-2+\beta _{1}-\beta _{2})q^{5}+\cdots\)
1521.2.a.q \(3\) \(12.145\) \(\Q(\zeta_{14})^+\) None \(1\) \(0\) \(4\) \(-10\) \(-\) \(-\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(2-\beta _{1}+\beta _{2})q^{5}+\cdots\)
1521.2.a.r \(3\) \(12.145\) \(\Q(\zeta_{14})^+\) None \(2\) \(0\) \(4\) \(-3\) \(-\) \(+\) \(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
1521.2.a.s \(3\) \(12.145\) \(\Q(\zeta_{14})^+\) None \(3\) \(0\) \(6\) \(-2\) \(-\) \(+\) \(q+(1-\beta _{1}-\beta _{2})q^{2}+(4-\beta _{1})q^{4}+(1+\cdots)q^{5}+\cdots\)
1521.2.a.t \(4\) \(12.145\) 4.4.8112.1 \(\Q(\sqrt{-39}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-\beta _{2}q^{2}+(2-\beta _{3})q^{4}+\beta _{1}q^{5}+(\beta _{1}+\cdots)q^{8}+\cdots\)
1521.2.a.u \(4\) \(12.145\) \(\Q(\sqrt{3}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta _{3}q^{2}+3q^{4}+\beta _{3}q^{5}+2\beta _{1}q^{7}+\cdots\)
1521.2.a.v \(6\) \(12.145\) 6.6.1997632.1 None \(0\) \(0\) \(0\) \(-12\) \(+\) \(+\) \(q+\beta _{5}q^{2}+(-2\beta _{2}+\beta _{4})q^{4}+(\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
1521.2.a.w \(6\) \(12.145\) 6.6.1997632.1 None \(0\) \(0\) \(0\) \(12\) \(+\) \(-\) \(q+\beta _{5}q^{2}+(-2\beta _{2}+\beta _{4})q^{4}+(\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1521))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1521)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(507))\)\(^{\oplus 2}\)