Properties

Label 3900.2.a
Level $3900$
Weight $2$
Character orbit 3900.a
Rep. character $\chi_{3900}(1,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $24$
Sturm bound $1680$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3900.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(1680\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3900))\).

Total New Old
Modular forms 876 38 838
Cusp forms 805 38 767
Eisenstein series 71 0 71

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

\(2\)\(3\)\(5\)\(13\)FrickeDim.
\(-\)\(+\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(6\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(6\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(6\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(4\)
Plus space\(+\)\(19\)
Minus space\(-\)\(19\)

Trace form

\( 38q + 38q^{9} + O(q^{10}) \) \( 38q + 38q^{9} + 4q^{11} + 2q^{13} + 4q^{17} - 8q^{19} - 4q^{21} - 8q^{23} + 12q^{29} - 4q^{33} - 4q^{37} + 12q^{41} - 16q^{43} - 28q^{47} + 54q^{49} + 4q^{53} + 4q^{57} - 12q^{59} + 20q^{61} + 8q^{67} + 8q^{69} - 20q^{71} + 20q^{73} - 16q^{79} + 38q^{81} - 12q^{83} - 8q^{87} - 12q^{89} + 12q^{93} + 20q^{97} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3900))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5 13
3900.2.a.a \(1\) \(31.142\) \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-2q^{7}+q^{9}-q^{13}+6q^{17}+\cdots\)
3900.2.a.b \(1\) \(31.142\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+q^{7}+q^{9}-3q^{11}-q^{13}-3q^{17}+\cdots\)
3900.2.a.c \(1\) \(31.142\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}+q^{7}+q^{9}-3q^{11}-q^{13}+7q^{17}+\cdots\)
3900.2.a.d \(1\) \(31.142\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+q^{7}+q^{9}+3q^{11}-q^{13}-3q^{17}+\cdots\)
3900.2.a.e \(1\) \(31.142\) \(\Q\) None \(0\) \(-1\) \(0\) \(2\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+2q^{7}+q^{9}-6q^{11}+q^{13}+\cdots\)
3900.2.a.f \(1\) \(31.142\) \(\Q\) None \(0\) \(-1\) \(0\) \(3\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+3q^{7}+q^{9}+q^{11}+q^{13}+3q^{17}+\cdots\)
3900.2.a.g \(1\) \(31.142\) \(\Q\) None \(0\) \(-1\) \(0\) \(3\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+3q^{7}+q^{9}+3q^{11}+q^{13}+\cdots\)
3900.2.a.h \(1\) \(31.142\) \(\Q\) None \(0\) \(1\) \(0\) \(-3\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-3q^{7}+q^{9}+q^{11}-q^{13}-3q^{17}+\cdots\)
3900.2.a.i \(1\) \(31.142\) \(\Q\) None \(0\) \(1\) \(0\) \(-3\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-3q^{7}+q^{9}+q^{11}-q^{13}+3q^{17}+\cdots\)
3900.2.a.j \(1\) \(31.142\) \(\Q\) None \(0\) \(1\) \(0\) \(-3\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}-3q^{7}+q^{9}+3q^{11}-q^{13}+\cdots\)
3900.2.a.k \(1\) \(31.142\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}-q^{7}+q^{9}-3q^{11}+q^{13}-7q^{17}+\cdots\)
3900.2.a.l \(1\) \(31.142\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}-q^{7}+q^{9}-3q^{11}+q^{13}+3q^{17}+\cdots\)
3900.2.a.m \(1\) \(31.142\) \(\Q\) None \(0\) \(1\) \(0\) \(2\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}+2q^{7}+q^{9}-4q^{11}-q^{13}+\cdots\)
3900.2.a.n \(1\) \(31.142\) \(\Q\) None \(0\) \(1\) \(0\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+2q^{7}+q^{9}-2q^{11}+q^{13}+\cdots\)
3900.2.a.o \(2\) \(31.142\) \(\Q(\sqrt{6}) \) None \(0\) \(-2\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}-2q^{7}+q^{9}+(2-\beta )q^{11}+q^{13}+\cdots\)
3900.2.a.p \(2\) \(31.142\) \(\Q(\sqrt{17}) \) None \(0\) \(-2\) \(0\) \(-3\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+(-1-\beta )q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)
3900.2.a.q \(2\) \(31.142\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(4\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}+(2+2\beta )q^{7}+q^{9}-\beta q^{11}-q^{13}+\cdots\)
3900.2.a.r \(2\) \(31.142\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(-4\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+(-2+2\beta )q^{7}+q^{9}+\beta q^{11}+\cdots\)
3900.2.a.s \(2\) \(31.142\) \(\Q(\sqrt{73}) \) None \(0\) \(2\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+\beta q^{7}+q^{9}+\beta q^{11}+q^{13}+\cdots\)
3900.2.a.t \(2\) \(31.142\) \(\Q(\sqrt{6}) \) None \(0\) \(2\) \(0\) \(4\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+2q^{7}+q^{9}+(2+\beta )q^{11}-q^{13}+\cdots\)
3900.2.a.u \(3\) \(31.142\) 3.3.3028.1 None \(0\) \(-3\) \(0\) \(-3\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}+(-1-\beta _{1})q^{7}+q^{9}+(2+\beta _{2})q^{11}+\cdots\)
3900.2.a.v \(3\) \(31.142\) 3.3.1940.1 None \(0\) \(-3\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+(\beta _{1}-\beta _{2})q^{7}+q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
3900.2.a.w \(3\) \(31.142\) 3.3.1940.1 None \(0\) \(3\) \(0\) \(1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+(-\beta _{1}+\beta _{2})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3900.2.a.x \(3\) \(31.142\) 3.3.3028.1 None \(0\) \(3\) \(0\) \(3\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+(1+\beta _{1})q^{7}+q^{9}+(2+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3900))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3900)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(650))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(780))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(975))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1950))\)\(^{\oplus 2}\)