Properties

Label 4032.2.a
Level $4032$
Weight $2$
Character orbit 4032.a
Rep. character $\chi_{4032}(1,\cdot)$
Character field $\Q$
Dimension $60$
Newform subspaces $50$
Sturm bound $1536$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 50 \)
Sturm bound: \(1536\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4032))\).

Total New Old
Modular forms 816 60 756
Cusp forms 721 60 661
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.

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(6\)
\(+\)\(+\)\(-\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(9\)
\(+\)\(-\)\(-\)\(+\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(9\)
\(-\)\(-\)\(-\)\(-\)\(10\)
Plus space\(+\)\(27\)
Minus space\(-\)\(33\)

Trace form

\( 60 q + O(q^{10}) \) \( 60 q - 8 q^{17} + 52 q^{25} - 8 q^{29} - 8 q^{37} + 8 q^{41} + 60 q^{49} - 56 q^{53} + 32 q^{61} + 8 q^{73} - 8 q^{77} + 32 q^{85} + 8 q^{89} - 8 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4032))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
4032.2.a.a $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-4\) \(-1\) $-$ $-$ $+$ \(q-4q^{5}-q^{7}+2q^{17}-2q^{19}+8q^{23}+\cdots\)
4032.2.a.b $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-4\) \(-1\) $+$ $-$ $+$ \(q-4q^{5}-q^{7}+2q^{11}+2q^{13}-4q^{19}+\cdots\)
4032.2.a.c $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-4\) \(1\) $+$ $-$ $-$ \(q-4q^{5}+q^{7}-2q^{11}+2q^{13}+4q^{19}+\cdots\)
4032.2.a.d $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-4\) \(1\) $+$ $-$ $-$ \(q-4q^{5}+q^{7}+2q^{17}+2q^{19}-8q^{23}+\cdots\)
4032.2.a.e $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) $+$ $-$ $+$ \(q-2q^{5}-q^{7}-4q^{11}-6q^{13}-2q^{17}+\cdots\)
4032.2.a.f $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) $-$ $-$ $+$ \(q-2q^{5}-q^{7}-4q^{11}+6q^{13}+2q^{17}+\cdots\)
4032.2.a.g $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) $+$ $+$ $+$ \(q-2q^{5}-q^{7}-2q^{11}-2q^{13}+6q^{17}+\cdots\)
4032.2.a.h $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) $+$ $-$ $+$ \(q-2q^{5}-q^{7}+4q^{11}+2q^{13}+6q^{17}+\cdots\)
4032.2.a.i $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) $-$ $+$ $+$ \(q-2q^{5}-q^{7}+6q^{11}+6q^{13}-2q^{17}+\cdots\)
4032.2.a.j $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $+$ $+$ $-$ \(q-2q^{5}+q^{7}-6q^{11}+6q^{13}-2q^{17}+\cdots\)
4032.2.a.k $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $-$ $-$ $-$ \(q-2q^{5}+q^{7}-4q^{11}+2q^{13}+6q^{17}+\cdots\)
4032.2.a.l $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $-$ $+$ $-$ \(q-2q^{5}+q^{7}+2q^{11}-2q^{13}+6q^{17}+\cdots\)
4032.2.a.m $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $-$ $-$ $-$ \(q-2q^{5}+q^{7}+4q^{11}-6q^{13}-2q^{17}+\cdots\)
4032.2.a.n $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $-$ $-$ $-$ \(q-2q^{5}+q^{7}+4q^{11}+6q^{13}+2q^{17}+\cdots\)
4032.2.a.o $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $+$ $+$ \(q-q^{7}-4q^{11}-2q^{13}-4q^{17}+4q^{23}+\cdots\)
4032.2.a.p $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ \(q-q^{7}-4q^{11}+4q^{13}+2q^{17}+6q^{19}+\cdots\)
4032.2.a.q $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $+$ $-$ $+$ \(q-q^{7}-2q^{11}+2q^{13}-4q^{17}+4q^{19}+\cdots\)
4032.2.a.r $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ \(q-q^{7}+4q^{13}-6q^{17}+2q^{19}-5q^{25}+\cdots\)
4032.2.a.s $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $+$ $+$ \(q-q^{7}+4q^{11}-2q^{13}+4q^{17}-4q^{23}+\cdots\)
4032.2.a.t $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ \(q-q^{7}+6q^{11}-2q^{13}-4q^{19}-6q^{23}+\cdots\)
4032.2.a.u $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $+$ $-$ $-$ \(q+q^{7}-6q^{11}-2q^{13}+4q^{19}+6q^{23}+\cdots\)
4032.2.a.v $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $-$ $+$ $-$ \(q+q^{7}-4q^{11}-2q^{13}+4q^{17}+4q^{23}+\cdots\)
4032.2.a.w $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $+$ $-$ $-$ \(q+q^{7}+4q^{13}-6q^{17}-2q^{19}-5q^{25}+\cdots\)
4032.2.a.x $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $+$ $-$ $-$ \(q+q^{7}+2q^{11}+2q^{13}-4q^{17}-4q^{19}+\cdots\)
4032.2.a.y $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $-$ $+$ $-$ \(q+q^{7}+4q^{11}-2q^{13}-4q^{17}-4q^{23}+\cdots\)
4032.2.a.z $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $-$ $-$ $-$ \(q+q^{7}+4q^{11}+4q^{13}+2q^{17}-6q^{19}+\cdots\)
4032.2.a.ba $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(2\) \(-1\) $-$ $+$ $+$ \(q+2q^{5}-q^{7}-6q^{11}+6q^{13}+2q^{17}+\cdots\)
4032.2.a.bb $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(2\) \(-1\) $+$ $-$ $+$ \(q+2q^{5}-q^{7}-4q^{11}-2q^{13}+6q^{17}+\cdots\)
4032.2.a.bc $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(2\) \(-1\) $-$ $-$ $+$ \(q+2q^{5}-q^{7}-6q^{13}+2q^{17}+4q^{19}+\cdots\)
4032.2.a.bd $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(2\) \(-1\) $-$ $-$ $+$ \(q+2q^{5}-q^{7}-2q^{13}-2q^{17}-4q^{19}+\cdots\)
4032.2.a.be $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(2\) \(-1\) $+$ $-$ $+$ \(q+2q^{5}-q^{7}+2q^{13}-6q^{17}+4q^{19}+\cdots\)
4032.2.a.bf $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(2\) \(-1\) $+$ $+$ $+$ \(q+2q^{5}-q^{7}+2q^{11}-2q^{13}-6q^{17}+\cdots\)
4032.2.a.bg $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(2\) \(1\) $-$ $+$ $-$ \(q+2q^{5}+q^{7}-2q^{11}-2q^{13}-6q^{17}+\cdots\)
4032.2.a.bh $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(2\) \(1\) $+$ $-$ $-$ \(q+2q^{5}+q^{7}-6q^{13}+2q^{17}-4q^{19}+\cdots\)
4032.2.a.bi $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(2\) \(1\) $-$ $-$ $-$ \(q+2q^{5}+q^{7}-2q^{13}-2q^{17}+4q^{19}+\cdots\)
4032.2.a.bj $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(2\) \(1\) $-$ $-$ $-$ \(q+2q^{5}+q^{7}+2q^{13}-6q^{17}-4q^{19}+\cdots\)
4032.2.a.bk $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(2\) \(1\) $-$ $-$ $-$ \(q+2q^{5}+q^{7}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
4032.2.a.bl $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(2\) \(1\) $+$ $+$ $-$ \(q+2q^{5}+q^{7}+6q^{11}+6q^{13}+2q^{17}+\cdots\)
4032.2.a.bm $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(4\) \(-1\) $+$ $-$ $+$ \(q+4q^{5}-q^{7}+2q^{11}+6q^{13}+4q^{17}+\cdots\)
4032.2.a.bn $1$ $32.196$ \(\Q\) None \(0\) \(0\) \(4\) \(1\) $-$ $-$ $-$ \(q+4q^{5}+q^{7}-2q^{11}+6q^{13}+4q^{17}+\cdots\)
4032.2.a.bo $2$ $32.196$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-2\) \(-2\) $+$ $+$ $+$ \(q+(-1-\beta )q^{5}-q^{7}+(1+\beta )q^{11}-2\beta q^{13}+\cdots\)
4032.2.a.bp $2$ $32.196$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-2\) \(2\) $+$ $+$ $-$ \(q+(-1-\beta )q^{5}+q^{7}+(-1-\beta )q^{11}+\cdots\)
4032.2.a.bq $2$ $32.196$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-2\) $-$ $+$ $+$ \(q+\beta q^{5}-q^{7}+\beta q^{11}-2q^{13}+\beta q^{17}+\cdots\)
4032.2.a.br $2$ $32.196$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-2\) $-$ $-$ $+$ \(q+\beta q^{5}-q^{7}+(2-\beta )q^{11}-2q^{13}+\cdots\)
4032.2.a.bs $2$ $32.196$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(2\) $-$ $-$ $-$ \(q+\beta q^{5}+q^{7}+(-2+\beta )q^{11}-2q^{13}+\cdots\)
4032.2.a.bt $2$ $32.196$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(2\) $+$ $+$ $-$ \(q+\beta q^{5}+q^{7}-\beta q^{11}-2q^{13}+\beta q^{17}+\cdots\)
4032.2.a.bu $2$ $32.196$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(-2\) $+$ $+$ $+$ \(q+(1+\beta )q^{5}-q^{7}+(-1-\beta )q^{11}-2\beta q^{13}+\cdots\)
4032.2.a.bv $2$ $32.196$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(-2\) $+$ $-$ $+$ \(q+(1+\beta )q^{5}-q^{7}+(2+2\beta )q^{11}+(-3+\cdots)q^{13}+\cdots\)
4032.2.a.bw $2$ $32.196$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(2\) $+$ $-$ $-$ \(q+(1+\beta )q^{5}+q^{7}+(-2-2\beta )q^{11}+\cdots\)
4032.2.a.bx $2$ $32.196$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(2\) $+$ $+$ $-$ \(q+(1+\beta )q^{5}+q^{7}+(1+\beta )q^{11}-2\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4032))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4032)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 14}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(448))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(504))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(672))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1008))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1344))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2016))\)\(^{\oplus 2}\)