Properties

Label 6900.2.a
Level $6900$
Weight $2$
Character orbit 6900.a
Rep. character $\chi_{6900}(1,\cdot)$
Character field $\Q$
Dimension $68$
Newform subspaces $30$
Sturm bound $2880$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6900.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 30 \)
Sturm bound: \(2880\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6900))\).

Total New Old
Modular forms 1476 68 1408
Cusp forms 1405 68 1337
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.

\(2\)\(3\)\(5\)\(23\)FrickeDim.
\(-\)\(+\)\(+\)\(+\)\(-\)\(9\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(7\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(8\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(10\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(10\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(6\)
Plus space\(+\)\(28\)
Minus space\(-\)\(40\)

Trace form

\( 68 q - 4 q^{7} + 68 q^{9} + O(q^{10}) \) \( 68 q - 4 q^{7} + 68 q^{9} + 4 q^{17} - 8 q^{19} - 8 q^{21} + 40 q^{29} - 12 q^{31} - 8 q^{33} - 12 q^{39} + 40 q^{41} - 12 q^{43} - 8 q^{47} + 48 q^{49} - 4 q^{51} - 12 q^{53} + 4 q^{57} + 16 q^{59} + 28 q^{61} - 4 q^{63} + 20 q^{67} - 4 q^{69} - 8 q^{71} + 40 q^{77} - 12 q^{79} + 68 q^{81} + 32 q^{83} + 20 q^{89} + 28 q^{91} - 8 q^{93} - 16 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6900))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 23
6900.2.a.a 6900.a 1.a $1$ $55.097$ \(\Q\) None \(0\) \(-1\) \(0\) \(-3\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{7}+q^{9}-5q^{11}-q^{13}+\cdots\)
6900.2.a.b 6900.a 1.a $1$ $55.097$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}+q^{9}+4q^{13}+3q^{17}+\cdots\)
6900.2.a.c 6900.a 1.a $1$ $55.097$ \(\Q\) None \(0\) \(-1\) \(0\) \(3\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
6900.2.a.d 6900.a 1.a $1$ $55.097$ \(\Q\) None \(0\) \(-1\) \(0\) \(3\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{7}+q^{9}+q^{11}-q^{13}-5q^{19}+\cdots\)
6900.2.a.e 6900.a 1.a $1$ $55.097$ \(\Q\) None \(0\) \(-1\) \(0\) \(4\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{7}+q^{9}-2q^{13}-6q^{17}+\cdots\)
6900.2.a.f 6900.a 1.a $1$ $55.097$ \(\Q\) None \(0\) \(1\) \(0\) \(-3\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{7}+q^{9}+q^{11}+q^{13}-5q^{19}+\cdots\)
6900.2.a.g 6900.a 1.a $1$ $55.097$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{9}+6q^{13}-2q^{17}+6q^{19}+\cdots\)
6900.2.a.h 6900.a 1.a $1$ $55.097$ \(\Q\) None \(0\) \(1\) \(0\) \(3\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{7}+q^{9}-5q^{11}+q^{13}+\cdots\)
6900.2.a.i 6900.a 1.a $1$ $55.097$ \(\Q\) None \(0\) \(1\) \(0\) \(5\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+5q^{7}+q^{9}-4q^{13}+3q^{17}+\cdots\)
6900.2.a.j 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{15}) \) None \(0\) \(-2\) \(0\) \(-6\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{7}+q^{9}+(1-\beta )q^{11}+(-1+\cdots)q^{13}+\cdots\)
6900.2.a.k 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{13}) \) None \(0\) \(-2\) \(0\) \(-5\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(-2-\beta )q^{7}+q^{9}+\beta q^{11}+\cdots\)
6900.2.a.l 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{13}) \) None \(0\) \(-2\) \(0\) \(-5\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-2-\beta )q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)
6900.2.a.m 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{7}+q^{9}+4\beta q^{11}-4\beta q^{13}+\cdots\)
6900.2.a.n 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{73}) \) None \(0\) \(-2\) \(0\) \(2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}+q^{9}+(-1+\beta )q^{11}+q^{13}+\cdots\)
6900.2.a.o 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{21}) \) None \(0\) \(-2\) \(0\) \(3\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(1+\beta )q^{7}+q^{9}+(1-\beta )q^{11}+\cdots\)
6900.2.a.p 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{10}) \) None \(0\) \(2\) \(0\) \(-4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-2+\beta )q^{7}+q^{9}-4q^{13}+\cdots\)
6900.2.a.q 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{21}) \) None \(0\) \(2\) \(0\) \(-3\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1-\beta )q^{7}+q^{9}+(1-\beta )q^{11}+\cdots\)
6900.2.a.r 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{6}) \) None \(0\) \(2\) \(0\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}+q^{9}+(-2-\beta )q^{11}-\beta q^{13}+\cdots\)
6900.2.a.s 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(-2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1+2\beta )q^{7}+q^{9}+(-1+3\beta )q^{11}+\cdots\)
6900.2.a.t 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{73}) \) None \(0\) \(2\) \(0\) \(-2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}+q^{9}+(-1+\beta )q^{11}-q^{13}+\cdots\)
6900.2.a.u 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(0\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-\beta q^{7}+q^{9}+2\beta q^{11}-2q^{13}+\cdots\)
6900.2.a.v 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{13}) \) None \(0\) \(2\) \(0\) \(5\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(3-\beta )q^{7}+q^{9}+(1-\beta )q^{11}+\cdots\)
6900.2.a.w 6900.a 1.a $2$ $55.097$ \(\Q(\sqrt{13}) \) None \(0\) \(2\) \(0\) \(5\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(3-\beta )q^{7}+q^{9}+(3-\beta )q^{11}+\cdots\)
6900.2.a.x 6900.a 1.a $3$ $55.097$ 3.3.3144.1 None \(0\) \(-3\) \(0\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1+\beta _{1})q^{7}+q^{9}+(1+\beta _{2})q^{11}+\cdots\)
6900.2.a.y 6900.a 1.a $3$ $55.097$ 3.3.148.1 None \(0\) \(-3\) \(0\) \(3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}+q^{9}-\beta _{2}q^{11}+(1-\beta _{1}+\cdots)q^{13}+\cdots\)
6900.2.a.z 6900.a 1.a $3$ $55.097$ 3.3.148.1 None \(0\) \(3\) \(0\) \(-3\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}+q^{9}-\beta _{2}q^{11}+(-1+\beta _{1}+\cdots)q^{13}+\cdots\)
6900.2.a.ba 6900.a 1.a $4$ $55.097$ 4.4.175557.1 None \(0\) \(-4\) \(0\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(1-\beta _{1}-\beta _{3})q^{7}+q^{9}-\beta _{1}q^{11}+\cdots\)
6900.2.a.bb 6900.a 1.a $4$ $55.097$ 4.4.175557.1 None \(0\) \(4\) \(0\) \(-3\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1+\beta _{1}+\beta _{3})q^{7}+q^{9}-\beta _{1}q^{11}+\cdots\)
6900.2.a.bc 6900.a 1.a $7$ $55.097$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-7\) \(0\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta _{1}q^{7}+q^{9}+(\beta _{1}-\beta _{2}-\beta _{4}+\cdots)q^{11}+\cdots\)
6900.2.a.bd 6900.a 1.a $7$ $55.097$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(7\) \(0\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{1}q^{7}+q^{9}+(\beta _{1}-\beta _{2}-\beta _{4}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6900))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6900)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(460))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(575))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(690))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1380))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1725))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3450))\)\(^{\oplus 2}\)