Properties

Label 3312.2.a
Level $3312$
Weight $2$
Character orbit 3312.a
Rep. character $\chi_{3312}(1,\cdot)$
Character field $\Q$
Dimension $55$
Newform subspaces $34$
Sturm bound $1152$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3312.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 34 \)
Sturm bound: \(1152\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3312))\).

Total New Old
Modular forms 600 55 545
Cusp forms 553 55 498
Eisenstein series 47 0 47

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

\(2\)\(3\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(6\)
\(+\)\(+\)\(-\)\(-\)\(6\)
\(+\)\(-\)\(+\)\(-\)\(8\)
\(+\)\(-\)\(-\)\(+\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(10\)
Plus space\(+\)\(26\)
Minus space\(-\)\(29\)

Trace form

\( 55 q - 2 q^{5} - 4 q^{7} + O(q^{10}) \) \( 55 q - 2 q^{5} - 4 q^{7} - 2 q^{13} + 6 q^{17} - 4 q^{19} + 3 q^{23} + 65 q^{25} - 2 q^{29} + 2 q^{31} - 24 q^{35} - 2 q^{37} - 2 q^{41} + 24 q^{43} - 2 q^{47} + 55 q^{49} - 10 q^{53} + 8 q^{55} - 28 q^{59} - 10 q^{61} + 12 q^{65} - 16 q^{67} - 26 q^{71} - 2 q^{73} - 8 q^{77} - 28 q^{79} + 20 q^{83} - 4 q^{85} + 14 q^{89} - 12 q^{91} - 32 q^{95} - 26 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3312))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 23
3312.2.a.a $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(-4\) \(-2\) $+$ $-$ $-$ \(q-4q^{5}-2q^{7}+2q^{13}+4q^{17}+6q^{19}+\cdots\)
3312.2.a.b $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(-4\) \(4\) $-$ $-$ $-$ \(q-4q^{5}+4q^{7}+2q^{11}-2q^{13}+2q^{17}+\cdots\)
3312.2.a.c $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(-2\) \(-2\) $-$ $+$ $+$ \(q-2q^{5}-2q^{7}+4q^{11}-2q^{13}-2q^{17}+\cdots\)
3312.2.a.d $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(-2\) \(0\) $-$ $-$ $+$ \(q-2q^{5}-2q^{13}-2q^{17}+8q^{19}-q^{23}+\cdots\)
3312.2.a.e $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(-2\) \(4\) $+$ $-$ $-$ \(q-2q^{5}+4q^{7}-4q^{11}-2q^{13}+2q^{17}+\cdots\)
3312.2.a.f $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(0\) \(-4\) $+$ $-$ $-$ \(q-4q^{7}+6q^{11}-2q^{13}-6q^{17}+6q^{19}+\cdots\)
3312.2.a.g $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $-$ $-$ $+$ \(q-2q^{7}-q^{13}+6q^{17}-2q^{19}-q^{23}+\cdots\)
3312.2.a.h $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $-$ $-$ $+$ \(q-2q^{7}+2q^{13}-2q^{19}-q^{23}-5q^{25}+\cdots\)
3312.2.a.i $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(0\) \(2\) $+$ $-$ $-$ \(q+2q^{7}-5q^{13}+6q^{17}-6q^{19}+q^{23}+\cdots\)
3312.2.a.j $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(0\) \(2\) $+$ $-$ $-$ \(q+2q^{7}+2q^{13}-8q^{17}-6q^{19}+q^{23}+\cdots\)
3312.2.a.k $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(0\) \(2\) $-$ $-$ $+$ \(q+2q^{7}+4q^{11}-6q^{13}-4q^{17}-2q^{19}+\cdots\)
3312.2.a.l $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(2\) \(-2\) $-$ $+$ $-$ \(q+2q^{5}-2q^{7}-4q^{11}-2q^{13}+2q^{17}+\cdots\)
3312.2.a.m $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(2\) \(-2\) $+$ $-$ $+$ \(q+2q^{5}-2q^{7}-2q^{11}-2q^{13}+4q^{17}+\cdots\)
3312.2.a.n $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(2\) \(2\) $-$ $-$ $+$ \(q+2q^{5}+2q^{7}-6q^{11}-2q^{13}-q^{23}+\cdots\)
3312.2.a.o $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(2\) \(4\) $+$ $-$ $+$ \(q+2q^{5}+4q^{7}-2q^{11}+7q^{13}+4q^{17}+\cdots\)
3312.2.a.p $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(2\) \(4\) $+$ $-$ $+$ \(q+2q^{5}+4q^{7}-2q^{13}+2q^{17}+4q^{19}+\cdots\)
3312.2.a.q $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(2\) \(4\) $-$ $-$ $-$ \(q+2q^{5}+4q^{7}+2q^{11}-5q^{13}-4q^{17}+\cdots\)
3312.2.a.r $1$ $26.446$ \(\Q\) None \(0\) \(0\) \(4\) \(-2\) $+$ $-$ $-$ \(q+4q^{5}-2q^{7}-4q^{11}-5q^{13}+2q^{17}+\cdots\)
3312.2.a.s $2$ $26.446$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-4\) \(0\) $-$ $-$ $-$ \(q+(-2+\beta )q^{5}+\beta q^{7}+4\beta q^{11}+4\beta q^{13}+\cdots\)
3312.2.a.t $2$ $26.446$ \(\Q(\sqrt{17}) \) None \(0\) \(0\) \(-4\) \(0\) $+$ $-$ $+$ \(q-2q^{5}+(2-2\beta )q^{11}+(3-\beta )q^{13}+\cdots\)
3312.2.a.u $2$ $26.446$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-4\) \(4\) $-$ $+$ $-$ \(q+(-2+\beta )q^{5}+(2+\beta )q^{7}-2\beta q^{11}+\cdots\)
3312.2.a.v $2$ $26.446$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(-2\) \(-4\) $-$ $+$ $+$ \(q+(-1+\beta )q^{5}-2q^{7}+(-1+\beta )q^{11}+\cdots\)
3312.2.a.w $2$ $26.446$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-2\) \(-4\) $+$ $-$ $-$ \(q+(-1-\beta )q^{5}-2q^{7}+(1+\beta )q^{11}+\cdots\)
3312.2.a.x $2$ $26.446$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-2\) \(2\) $+$ $+$ $+$ \(q+(-1+\beta )q^{5}+(1-\beta )q^{7}+(-2+2\beta )q^{11}+\cdots\)
3312.2.a.y $2$ $26.446$ \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(0\) \(-4\) $-$ $-$ $+$ \(q+\beta q^{5}+(-2-\beta )q^{7}+4q^{13}+(-4+\cdots)q^{17}+\cdots\)
3312.2.a.z $2$ $26.446$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(2\) \(-4\) $-$ $+$ $-$ \(q+(1+\beta )q^{5}-2q^{7}+(1+\beta )q^{11}-2\beta q^{13}+\cdots\)
3312.2.a.ba $2$ $26.446$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(-2\) $-$ $-$ $-$ \(q+(1+\beta )q^{5}+(-1+\beta )q^{7}+(-3+\beta )q^{11}+\cdots\)
3312.2.a.bb $2$ $26.446$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(-2\) $-$ $-$ $-$ \(q+(1+\beta )q^{5}+(-1+\beta )q^{7}+4q^{11}+\cdots\)
3312.2.a.bc $2$ $26.446$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(0\) $-$ $-$ $-$ \(q+(1+\beta )q^{5}-2\beta q^{7}+(-3+\beta )q^{11}+\cdots\)
3312.2.a.bd $2$ $26.446$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(2\) \(2\) $+$ $+$ $-$ \(q+(1+\beta )q^{5}+(1+\beta )q^{7}+(2+2\beta )q^{11}+\cdots\)
3312.2.a.be $2$ $26.446$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(4\) \(4\) $-$ $+$ $+$ \(q+(2+\beta )q^{5}+(2-\beta )q^{7}-2\beta q^{11}+(6+\cdots)q^{17}+\cdots\)
3312.2.a.bf $3$ $26.446$ 3.3.148.1 None \(0\) \(0\) \(0\) \(-2\) $+$ $-$ $+$ \(q-\beta _{2}q^{5}+(-1-\beta _{1})q^{7}+(1-\beta _{1}-\beta _{2})q^{11}+\cdots\)
3312.2.a.bg $4$ $26.446$ 4.4.44688.2 None \(0\) \(0\) \(-4\) \(-2\) $+$ $+$ $+$ \(q+(-1-\beta _{2})q^{5}+\beta _{1}q^{7}+(\beta _{2}-\beta _{3})q^{11}+\cdots\)
3312.2.a.bh $4$ $26.446$ 4.4.44688.2 None \(0\) \(0\) \(4\) \(-2\) $+$ $+$ $-$ \(q+(1+\beta _{2})q^{5}+(-1-\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3312))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3312)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(368))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(414))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(552))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(828))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1656))\)\(^{\oplus 2}\)