Properties

Label 4200.2.a
Level $4200$
Weight $2$
Character orbit 4200.a
Rep. character $\chi_{4200}(1,\cdot)$
Character field $\Q$
Dimension $58$
Newform subspaces $43$
Sturm bound $1920$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4200.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 43 \)
Sturm bound: \(1920\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4200))\).

Total New Old
Modular forms 1008 58 950
Cusp forms 913 58 855
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.

\(2\)\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(3\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(4\)
Plus space\(+\)\(24\)
Minus space\(-\)\(34\)

Trace form

\( 58 q + 58 q^{9} + O(q^{10}) \) \( 58 q + 58 q^{9} - 4 q^{13} - 4 q^{17} + 2 q^{21} - 8 q^{23} - 36 q^{29} - 4 q^{37} + 8 q^{39} - 44 q^{41} + 8 q^{43} + 32 q^{47} + 58 q^{49} - 16 q^{51} - 12 q^{53} + 16 q^{59} - 44 q^{61} - 24 q^{67} + 40 q^{71} - 20 q^{73} + 32 q^{79} + 58 q^{81} + 16 q^{83} - 32 q^{87} + 4 q^{89} - 16 q^{91} - 24 q^{93} + 12 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4200))\) 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 7
4200.2.a.a $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $+$ $+$ $+$ $+$ \(q-q^{3}-q^{7}+q^{9}-3q^{11}-2q^{13}+\cdots\)
4200.2.a.b $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $-$ $+$ $-$ $+$ \(q-q^{3}-q^{7}+q^{9}-q^{11}+4q^{13}+2q^{17}+\cdots\)
4200.2.a.c $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $+$ $+$ $+$ $+$ \(q-q^{3}-q^{7}+q^{9}-2q^{13}-2q^{17}+\cdots\)
4200.2.a.d $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $+$ $+$ $-$ \(q-q^{3}+q^{7}+q^{9}-5q^{11}-4q^{13}+\cdots\)
4200.2.a.e $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $+$ $+$ $-$ \(q-q^{3}+q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
4200.2.a.f $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $+$ $-$ $-$ \(q-q^{3}+q^{7}+q^{9}-3q^{11}-2q^{13}+\cdots\)
4200.2.a.g $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $-$ $-$ \(q-q^{3}+q^{7}+q^{9}-2q^{11}-5q^{13}+\cdots\)
4200.2.a.h $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $-$ \(q-q^{3}+q^{7}+q^{9}-6q^{13}+2q^{17}+\cdots\)
4200.2.a.i $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $-$ \(q-q^{3}+q^{7}+q^{9}+2q^{13}-6q^{17}+\cdots\)
4200.2.a.j $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $+$ $-$ $-$ \(q-q^{3}+q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
4200.2.a.k $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $+$ $-$ $-$ \(q-q^{3}+q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
4200.2.a.l $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $+$ $+$ $-$ \(q-q^{3}+q^{7}+q^{9}+2q^{11}+3q^{13}+\cdots\)
4200.2.a.m $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $-$ \(q-q^{3}+q^{7}+q^{9}+3q^{11}+2q^{17}+\cdots\)
4200.2.a.n $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $+$ $+$ $-$ \(q-q^{3}+q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
4200.2.a.o $1$ $33.537$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $-$ $-$ \(q-q^{3}+q^{7}+q^{9}+5q^{11}+2q^{13}+\cdots\)
4200.2.a.p $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $-$ $+$ \(q+q^{3}-q^{7}+q^{9}-5q^{11}+4q^{13}+\cdots\)
4200.2.a.q $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $+$ $-$ $+$ $+$ \(q+q^{3}-q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
4200.2.a.r $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $+$ $+$ \(q+q^{3}-q^{7}+q^{9}-3q^{11}+2q^{13}+\cdots\)
4200.2.a.s $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $+$ $-$ $+$ $+$ \(q+q^{3}-q^{7}+q^{9}-2q^{11}+5q^{13}+\cdots\)
4200.2.a.t $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $+$ $+$ \(q+q^{3}-q^{7}+q^{9}-6q^{13}+2q^{17}+\cdots\)
4200.2.a.u $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $+$ $+$ \(q+q^{3}-q^{7}+q^{9}+2q^{13}-6q^{17}+\cdots\)
4200.2.a.v $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $-$ $+$ \(q+q^{3}-q^{7}+q^{9}+2q^{11}-3q^{13}+\cdots\)
4200.2.a.w $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $-$ $+$ \(q+q^{3}-q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
4200.2.a.x $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $-$ $+$ \(q+q^{3}-q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
4200.2.a.y $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $+$ $-$ $-$ $+$ \(q+q^{3}-q^{7}+q^{9}+3q^{11}-2q^{17}+\cdots\)
4200.2.a.z $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $+$ $-$ $+$ $+$ \(q+q^{3}-q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
4200.2.a.ba $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $+$ $-$ $+$ $+$ \(q+q^{3}-q^{7}+q^{9}+5q^{11}-2q^{13}+\cdots\)
4200.2.a.bb $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $+$ $-$ $+$ $-$ \(q+q^{3}+q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
4200.2.a.bc $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $-$ $-$ $-$ \(q+q^{3}+q^{7}+q^{9}-3q^{11}+2q^{13}+\cdots\)
4200.2.a.bd $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $+$ $-$ $+$ $-$ \(q+q^{3}+q^{7}+q^{9}-q^{11}-4q^{13}-2q^{17}+\cdots\)
4200.2.a.be $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $-$ $+$ $-$ \(q+q^{3}+q^{7}+q^{9}-2q^{13}-2q^{17}+\cdots\)
4200.2.a.bf $1$ $33.537$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $-$ $+$ $-$ \(q+q^{3}+q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
4200.2.a.bg $2$ $33.537$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(-2\) $-$ $+$ $+$ $+$ \(q-q^{3}-q^{7}+q^{9}+\beta q^{11}-2q^{13}+\cdots\)
4200.2.a.bh $2$ $33.537$ \(\Q(\sqrt{73}) \) None \(0\) \(-2\) \(0\) \(-2\) $+$ $+$ $-$ $+$ \(q-q^{3}-q^{7}+q^{9}+\beta q^{11}+(-1-\beta )q^{13}+\cdots\)
4200.2.a.bi $2$ $33.537$ \(\Q(\sqrt{17}) \) None \(0\) \(-2\) \(0\) \(-2\) $-$ $+$ $+$ $+$ \(q-q^{3}-q^{7}+q^{9}+(1+\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
4200.2.a.bj $2$ $33.537$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(0\) \(2\) $-$ $+$ $-$ $-$ \(q-q^{3}+q^{7}+q^{9}-2q^{11}-\beta q^{13}+\cdots\)
4200.2.a.bk $2$ $33.537$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(0\) \(-2\) $+$ $-$ $-$ $+$ \(q+q^{3}-q^{7}+q^{9}-2q^{11}+\beta q^{13}+\cdots\)
4200.2.a.bl $2$ $33.537$ \(\Q(\sqrt{73}) \) None \(0\) \(2\) \(0\) \(2\) $-$ $-$ $+$ $-$ \(q+q^{3}+q^{7}+q^{9}+\beta q^{11}+(1+\beta )q^{13}+\cdots\)
4200.2.a.bm $2$ $33.537$ \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(0\) \(2\) $+$ $-$ $-$ $-$ \(q+q^{3}+q^{7}+q^{9}+(1+\beta )q^{11}+(2-3\beta )q^{13}+\cdots\)
4200.2.a.bn $3$ $33.537$ 3.3.148.1 None \(0\) \(-3\) \(0\) \(-3\) $+$ $+$ $-$ $+$ \(q-q^{3}-q^{7}+q^{9}+(-1-\beta _{1}+\beta _{2})q^{11}+\cdots\)
4200.2.a.bo $3$ $33.537$ 3.3.148.1 None \(0\) \(-3\) \(0\) \(-3\) $-$ $+$ $-$ $+$ \(q-q^{3}-q^{7}+q^{9}+(1+\beta _{1}+\beta _{2})q^{11}+\cdots\)
4200.2.a.bp $3$ $33.537$ 3.3.148.1 None \(0\) \(3\) \(0\) \(3\) $-$ $-$ $-$ $-$ \(q+q^{3}+q^{7}+q^{9}+(-1-\beta _{1}+\beta _{2})q^{11}+\cdots\)
4200.2.a.bq $3$ $33.537$ 3.3.148.1 None \(0\) \(3\) \(0\) \(3\) $+$ $-$ $-$ $-$ \(q+q^{3}+q^{7}+q^{9}+(1+\beta _{1}+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4200))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4200)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(840))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1050))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2100))\)\(^{\oplus 2}\)