Properties

Label 3360.2.a
Level $3360$
Weight $2$
Character orbit 3360.a
Rep. character $\chi_{3360}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $36$
Sturm bound $1536$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 36 \)
Sturm bound: \(1536\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3360))\).

Total New Old
Modular forms 800 48 752
Cusp forms 737 48 689
Eisenstein series 63 0 63

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
\(+\)\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(+\)\(-\)\(+\)$-$\(3\)
\(+\)\(+\)\(-\)\(-\)$+$\(2\)
\(+\)\(-\)\(+\)\(+\)$-$\(3\)
\(+\)\(-\)\(+\)\(-\)$+$\(2\)
\(+\)\(-\)\(-\)\(+\)$+$\(2\)
\(+\)\(-\)\(-\)\(-\)$-$\(5\)
\(-\)\(+\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(+\)\(-\)$+$\(3\)
\(-\)\(+\)\(-\)\(+\)$+$\(3\)
\(-\)\(+\)\(-\)\(-\)$-$\(4\)
\(-\)\(-\)\(+\)\(+\)$+$\(3\)
\(-\)\(-\)\(+\)\(-\)$-$\(4\)
\(-\)\(-\)\(-\)\(+\)$-$\(4\)
\(-\)\(-\)\(-\)\(-\)$+$\(1\)
Plus space\(+\)\(20\)
Minus space\(-\)\(28\)

Trace form

\( 48 q + 48 q^{9} + O(q^{10}) \) \( 48 q + 48 q^{9} + 48 q^{25} + 48 q^{49} + 48 q^{81} + 64 q^{89} + 64 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3360))\) 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 7
3360.2.a.a 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
3360.2.a.b 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-q^{7}+q^{9}-2q^{11}+q^{15}+\cdots\)
3360.2.a.c 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
3360.2.a.d 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
3360.2.a.e 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
3360.2.a.f 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
3360.2.a.g 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(1\) \(-1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
3360.2.a.h 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
3360.2.a.i 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
3360.2.a.j 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(1\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{7}+q^{9}-2q^{11}-4q^{13}+\cdots\)
3360.2.a.k 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(1\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{7}+q^{9}+2q^{13}-q^{15}+\cdots\)
3360.2.a.l 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
3360.2.a.m 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(-1\) \(1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{7}+q^{9}+6q^{11}+4q^{13}+\cdots\)
3360.2.a.n 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
3360.2.a.o 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-q^{7}+q^{9}+2q^{13}-q^{15}+\cdots\)
3360.2.a.p 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(-1\) \(1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
3360.2.a.q 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(-1\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
3360.2.a.r 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(-1\) \(1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{7}+q^{9}+2q^{11}-q^{15}+\cdots\)
3360.2.a.s 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(-1\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
3360.2.a.t 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(1\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-q^{7}+q^{9}-6q^{11}+4q^{13}+\cdots\)
3360.2.a.u 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(1\) \(-1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
3360.2.a.v 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(1\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
3360.2.a.w 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(1\) \(-1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-q^{7}+q^{9}+2q^{11}-4q^{13}+\cdots\)
3360.2.a.x 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(1\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
3360.2.a.y 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(1\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
3360.2.a.z 3360.a 1.a $1$ $26.830$ \(\Q\) None \(0\) \(1\) \(1\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
3360.2.a.ba 3360.a 1.a $2$ $26.830$ \(\Q(\sqrt{17}) \) None \(0\) \(-2\) \(-2\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-q^{7}+q^{9}+(-1-\beta )q^{11}+\cdots\)
3360.2.a.bb 3360.a 1.a $2$ $26.830$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(-2\) \(2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots\)
3360.2.a.bc 3360.a 1.a $2$ $26.830$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-2\) \(2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{7}+q^{9}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
3360.2.a.bd 3360.a 1.a $2$ $26.830$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(2\) \(-2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-q^{7}+q^{9}+\beta q^{13}-q^{15}+\cdots\)
3360.2.a.be 3360.a 1.a $2$ $26.830$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(-2\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-q^{7}+q^{9}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
3360.2.a.bf 3360.a 1.a $2$ $26.830$ \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(-2\) \(-2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)
3360.2.a.bg 3360.a 1.a $2$ $26.830$ \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(-2\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)
3360.2.a.bh 3360.a 1.a $2$ $26.830$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(2\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+q^{7}+q^{9}+\beta q^{13}+q^{15}+\cdots\)
3360.2.a.bi 3360.a 1.a $3$ $26.830$ 3.3.148.1 None \(0\) \(-3\) \(3\) \(-3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-q^{7}+q^{9}+(-1+\beta _{2})q^{11}+\cdots\)
3360.2.a.bj 3360.a 1.a $3$ $26.830$ 3.3.148.1 None \(0\) \(3\) \(3\) \(3\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+q^{7}+q^{9}+(1-\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3360))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3360)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(480))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(560))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(672))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(840))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1680))\)\(^{\oplus 2}\)