Properties

Label 7800.2.a
Level $7800$
Weight $2$
Character orbit 7800.a
Rep. character $\chi_{7800}(1,\cdot)$
Character field $\Q$
Dimension $114$
Newform subspaces $53$
Sturm bound $3360$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 53 \)
Sturm bound: \(3360\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7800))\).

Total New Old
Modular forms 1728 114 1614
Cusp forms 1633 114 1519
Eisenstein series 95 0 95

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.
\(+\)\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(7\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(7\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(7\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(7\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(52\)
Minus space\(-\)\(62\)

Trace form

\( 114q + 4q^{7} + 114q^{9} + O(q^{10}) \) \( 114q + 4q^{7} + 114q^{9} - 2q^{13} - 4q^{17} - 12q^{19} + 4q^{21} - 24q^{23} - 4q^{29} + 4q^{31} - 16q^{33} - 4q^{37} - 8q^{41} + 16q^{43} + 16q^{47} + 90q^{49} + 4q^{53} + 4q^{57} + 8q^{59} + 4q^{61} + 4q^{63} + 4q^{67} - 8q^{69} + 56q^{71} - 36q^{73} - 32q^{77} + 16q^{79} + 114q^{81} + 8q^{83} - 12q^{91} - 36q^{93} - 36q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\) 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
7800.2.a.a \(1\) \(62.283\) \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-4q^{7}+q^{9}-q^{13}-2q^{17}+\cdots\)
7800.2.a.b \(1\) \(62.283\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}-q^{7}+q^{9}+3q^{11}+q^{13}+7q^{17}+\cdots\)
7800.2.a.c \(1\) \(62.283\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{9}-4q^{11}+q^{13}+8q^{17}+\cdots\)
7800.2.a.d \(1\) \(62.283\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}+q^{9}-q^{13}-2q^{17}-4q^{19}+\cdots\)
7800.2.a.e \(1\) \(62.283\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{9}+q^{13}-2q^{17}-4q^{23}+\cdots\)
7800.2.a.f \(1\) \(62.283\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{9}+6q^{11}+q^{13}-2q^{17}+\cdots\)
7800.2.a.g \(1\) \(62.283\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+q^{7}+q^{9}-5q^{11}-q^{13}-3q^{17}+\cdots\)
7800.2.a.h \(1\) \(62.283\) \(\Q\) None \(0\) \(-1\) \(0\) \(2\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+2q^{7}+q^{9}-q^{13}+4q^{17}+\cdots\)
7800.2.a.i \(1\) \(62.283\) \(\Q\) None \(0\) \(-1\) \(0\) \(3\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+3q^{7}+q^{9}-3q^{11}+q^{13}+\cdots\)
7800.2.a.j \(1\) \(62.283\) \(\Q\) None \(0\) \(-1\) \(0\) \(4\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+4q^{7}+q^{9}-2q^{11}+q^{13}+\cdots\)
7800.2.a.k \(1\) \(62.283\) \(\Q\) None \(0\) \(-1\) \(0\) \(4\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+4q^{7}+q^{9}+4q^{11}-q^{13}+\cdots\)
7800.2.a.l \(1\) \(62.283\) \(\Q\) None \(0\) \(-1\) \(0\) \(5\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}+5q^{7}+q^{9}+5q^{11}-q^{13}+\cdots\)
7800.2.a.m \(1\) \(62.283\) \(\Q\) None \(0\) \(1\) \(0\) \(-5\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}-5q^{7}+q^{9}+5q^{11}+q^{13}+\cdots\)
7800.2.a.n \(1\) \(62.283\) \(\Q\) None \(0\) \(1\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-4q^{7}+q^{9}-q^{13}-2q^{17}+\cdots\)
7800.2.a.o \(1\) \(62.283\) \(\Q\) None \(0\) \(1\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-4q^{7}+q^{9}+4q^{11}-q^{13}+\cdots\)
7800.2.a.p \(1\) \(62.283\) \(\Q\) None \(0\) \(1\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-2q^{7}+q^{9}+4q^{11}+q^{13}+\cdots\)
7800.2.a.q \(1\) \(62.283\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}+q^{9}-4q^{11}-q^{13}-8q^{17}+\cdots\)
7800.2.a.r \(1\) \(62.283\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}+q^{9}-4q^{11}-q^{13}-2q^{17}+\cdots\)
7800.2.a.s \(1\) \(62.283\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}+q^{9}-2q^{11}+q^{13}-2q^{17}+\cdots\)
7800.2.a.t \(1\) \(62.283\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}+q^{9}-q^{13}+6q^{17}+4q^{19}+\cdots\)
7800.2.a.u \(1\) \(62.283\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}+q^{9}+4q^{11}-q^{13}+2q^{17}+\cdots\)
7800.2.a.v \(1\) \(62.283\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{7}+q^{9}+3q^{11}-q^{13}-7q^{17}+\cdots\)
7800.2.a.w \(1\) \(62.283\) \(\Q\) None \(0\) \(1\) \(0\) \(4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+4q^{7}+q^{9}-2q^{11}+q^{13}+\cdots\)
7800.2.a.x \(1\) \(62.283\) \(\Q\) None \(0\) \(1\) \(0\) \(5\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}+5q^{7}+q^{9}+q^{11}-q^{13}+3q^{17}+\cdots\)
7800.2.a.y \(2\) \(62.283\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+\beta q^{7}+q^{9}+\beta q^{11}+q^{13}+\cdots\)
7800.2.a.z \(2\) \(62.283\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(2\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}+(1+\beta )q^{7}+q^{9}+(-1+\beta )q^{11}+\cdots\)
7800.2.a.ba \(2\) \(62.283\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}+(-1+\beta )q^{7}+q^{9}+(-1-\beta )q^{11}+\cdots\)
7800.2.a.bb \(2\) \(62.283\) \(\Q(\sqrt{33}) \) None \(0\) \(2\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-\beta q^{7}+q^{9}-\beta q^{11}+q^{13}+\cdots\)
7800.2.a.bc \(2\) \(62.283\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}+\beta q^{7}+q^{9}-\beta q^{11}-q^{13}+\cdots\)
7800.2.a.bd \(2\) \(62.283\) \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(0\) \(1\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}+\beta q^{7}+q^{9}+(-4+\beta )q^{11}+\cdots\)
7800.2.a.be \(2\) \(62.283\) \(\Q(\sqrt{41}) \) None \(0\) \(2\) \(0\) \(1\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}+\beta q^{7}+q^{9}+(-2-\beta )q^{11}+\cdots\)
7800.2.a.bf \(3\) \(62.283\) 3.3.1849.1 None \(0\) \(-3\) \(0\) \(-5\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}+(-2+\beta _{1})q^{7}+q^{9}+(1+\beta _{2})q^{11}+\cdots\)
7800.2.a.bg \(3\) \(62.283\) 3.3.788.1 None \(0\) \(-3\) \(0\) \(-1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}-\beta _{1}q^{7}+q^{9}+(-1-\beta _{1}+\beta _{2})q^{11}+\cdots\)
7800.2.a.bh \(3\) \(62.283\) 3.3.3732.1 None \(0\) \(-3\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}-\beta _{2}q^{7}+q^{9}+(2-\beta _{1})q^{11}+\cdots\)
7800.2.a.bi \(3\) \(62.283\) 3.3.940.1 None \(0\) \(-3\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}-\beta _{2}q^{7}+q^{9}+(2+\beta _{1})q^{11}+\cdots\)
7800.2.a.bj \(3\) \(62.283\) 3.3.568.1 None \(0\) \(-3\) \(0\) \(0\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}+q^{9}+(-2-\beta _{1})q^{11}-q^{13}+\cdots\)
7800.2.a.bk \(3\) \(62.283\) 3.3.2700.1 None \(0\) \(-3\) \(0\) \(0\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}+\beta _{1}q^{7}+q^{9}-\beta _{1}q^{11}-q^{13}+\cdots\)
7800.2.a.bl \(3\) \(62.283\) 3.3.148.1 None \(0\) \(-3\) \(0\) \(1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}+(\beta _{1}+2\beta _{2})q^{7}+q^{9}+(-1+\beta _{2})q^{11}+\cdots\)
7800.2.a.bm \(3\) \(62.283\) 3.3.568.1 None \(0\) \(-3\) \(0\) \(2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+(1-\beta _{1})q^{7}+q^{9}+(1+\beta _{1}+2\beta _{2})q^{11}+\cdots\)
7800.2.a.bn \(3\) \(62.283\) 3.3.568.1 None \(0\) \(3\) \(0\) \(-2\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+(-1+\beta _{1})q^{7}+q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
7800.2.a.bo \(3\) \(62.283\) 3.3.148.1 None \(0\) \(3\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+(-\beta _{1}-2\beta _{2})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
7800.2.a.bp \(3\) \(62.283\) 3.3.568.1 None \(0\) \(3\) \(0\) \(0\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{9}+(-2-\beta _{1})q^{11}+q^{13}+\cdots\)
7800.2.a.bq \(3\) \(62.283\) 3.3.2700.1 None \(0\) \(3\) \(0\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-\beta _{1}q^{7}+q^{9}-\beta _{1}q^{11}+q^{13}+\cdots\)
7800.2.a.br \(3\) \(62.283\) 3.3.788.1 None \(0\) \(3\) \(0\) \(1\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}+\beta _{1}q^{7}+q^{9}+(-1-\beta _{1}+\beta _{2})q^{11}+\cdots\)
7800.2.a.bs \(3\) \(62.283\) 3.3.3732.1 None \(0\) \(3\) \(0\) \(1\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}+\beta _{2}q^{7}+q^{9}+(2-\beta _{1})q^{11}+\cdots\)
7800.2.a.bt \(4\) \(62.283\) 4.4.30056.1 None \(0\) \(-4\) \(0\) \(-7\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+(-2-\beta _{1})q^{7}+q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
7800.2.a.bu \(4\) \(62.283\) 4.4.461844.1 None \(0\) \(-4\) \(0\) \(-3\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+(-1+\beta _{1})q^{7}+q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
7800.2.a.bv \(4\) \(62.283\) 4.4.10304.1 None \(0\) \(-4\) \(0\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}-\beta _{3}q^{7}+q^{9}+(-2-\beta _{1})q^{11}+\cdots\)
7800.2.a.bw \(4\) \(62.283\) 4.4.10304.1 None \(0\) \(4\) \(0\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}+\beta _{3}q^{7}+q^{9}+(-2-\beta _{1})q^{11}+\cdots\)
7800.2.a.bx \(4\) \(62.283\) 4.4.461844.1 None \(0\) \(4\) \(0\) \(3\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+(1-\beta _{1})q^{7}+q^{9}+(1+\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
7800.2.a.by \(4\) \(62.283\) 4.4.30056.1 None \(0\) \(4\) \(0\) \(7\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+(2+\beta _{1})q^{7}+q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
7800.2.a.bz \(5\) \(62.283\) 5.5.504568.1 None \(0\) \(-5\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}-\beta _{1}q^{7}+q^{9}+(1-\beta _{3})q^{11}+\cdots\)
7800.2.a.ca \(5\) \(62.283\) 5.5.504568.1 None \(0\) \(5\) \(0\) \(1\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+\beta _{1}q^{7}+q^{9}+(1-\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7800)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(650))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(780))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(975))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1560))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1950))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2600))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3900))\)\(^{\oplus 2}\)