Properties

Label 3120.2.a
Level $3120$
Weight $2$
Character orbit 3120.a
Rep. character $\chi_{3120}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $36$
Sturm bound $1344$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 36 \)
Sturm bound: \(1344\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\), \(19\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3120))\).

Total New Old
Modular forms 696 48 648
Cusp forms 649 48 601
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.

\(2\)\(3\)\(5\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(18\)
Minus space\(-\)\(30\)

Trace form

\( 48 q - 8 q^{7} + 48 q^{9} + O(q^{10}) \) \( 48 q - 8 q^{7} + 48 q^{9} - 16 q^{11} - 4 q^{15} - 8 q^{19} - 16 q^{23} + 48 q^{25} + 32 q^{29} - 8 q^{31} + 32 q^{37} - 16 q^{43} + 48 q^{49} - 8 q^{51} + 16 q^{53} - 8 q^{63} + 8 q^{67} + 16 q^{69} + 32 q^{71} + 16 q^{77} - 24 q^{79} + 48 q^{81} + 48 q^{83} + 16 q^{85} - 16 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3120))\) 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 13
3120.2.a.a 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-4q^{7}+q^{9}-q^{13}+q^{15}+\cdots\)
3120.2.a.b 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{7}+q^{9}+q^{13}+q^{15}+\cdots\)
3120.2.a.c 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{9}-q^{13}+q^{15}+2q^{17}+\cdots\)
3120.2.a.d 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{7}+q^{9}-5q^{11}-q^{13}+\cdots\)
3120.2.a.e 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{7}+q^{9}-3q^{11}+q^{13}+\cdots\)
3120.2.a.f 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{7}+q^{9}+5q^{11}+q^{13}+\cdots\)
3120.2.a.g 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(-1\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+2q^{7}+q^{9}+6q^{11}-q^{13}+\cdots\)
3120.2.a.h 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(-1\) \(4\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+4q^{7}+q^{9}-4q^{11}+q^{13}+\cdots\)
3120.2.a.i 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(1\) \(-4\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-4q^{7}+q^{9}+q^{13}-q^{15}+\cdots\)
3120.2.a.j 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(1\) \(-2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{7}+q^{9}+q^{13}-q^{15}+\cdots\)
3120.2.a.k 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{9}-4q^{11}+q^{13}-q^{15}+\cdots\)
3120.2.a.l 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(1\) \(2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+2q^{7}+q^{9}+q^{13}-q^{15}+\cdots\)
3120.2.a.m 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(1\) \(3\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+3q^{7}+q^{9}+3q^{11}-q^{13}+\cdots\)
3120.2.a.n 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(-1\) \(1\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+3q^{7}+q^{9}+5q^{11}+q^{13}+\cdots\)
3120.2.a.o 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(-1\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{7}+q^{9}-4q^{11}-q^{13}+\cdots\)
3120.2.a.p 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(-1\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{9}-4q^{11}+q^{13}-q^{15}+\cdots\)
3120.2.a.q 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(-1\) \(0\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{9}-q^{13}-q^{15}-6q^{17}+\cdots\)
3120.2.a.r 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(-1\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{9}+q^{13}-q^{15}-6q^{17}+\cdots\)
3120.2.a.s 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(1\) \(-4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-4q^{7}+q^{9}-4q^{11}+q^{13}+\cdots\)
3120.2.a.t 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(1\) \(-3\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-3q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
3120.2.a.u 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(1\) \(-3\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-3q^{7}+q^{9}+q^{11}-q^{13}+\cdots\)
3120.2.a.v 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(1\) \(-2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{7}+q^{9}-4q^{11}-q^{13}+\cdots\)
3120.2.a.w 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+q^{9}-4q^{11}+q^{13}+q^{15}+\cdots\)
3120.2.a.x 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+q^{9}+4q^{11}+q^{13}+q^{15}+\cdots\)
3120.2.a.y 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+2q^{7}+q^{9}-4q^{11}-q^{13}+\cdots\)
3120.2.a.z 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+2q^{7}+q^{9}+2q^{11}-q^{13}+\cdots\)
3120.2.a.ba 3120.a 1.a $1$ $24.913$ \(\Q\) None \(0\) \(1\) \(1\) \(5\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+5q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
3120.2.a.bb 3120.a 1.a $2$ $24.913$ \(\Q(\sqrt{17}) \) None \(0\) \(-2\) \(2\) \(-3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+(-1-\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3120.2.a.bc 3120.a 1.a $2$ $24.913$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(2\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+\beta q^{7}+q^{9}-2\beta q^{11}+\cdots\)
3120.2.a.bd 3120.a 1.a $2$ $24.913$ \(\Q(\sqrt{33}) \) None \(0\) \(2\) \(-2\) \(-1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-\beta q^{7}+q^{9}+\beta q^{11}-q^{13}+\cdots\)
3120.2.a.be 3120.a 1.a $2$ $24.913$ \(\Q(\sqrt{73}) \) None \(0\) \(2\) \(-2\) \(1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+\beta q^{7}+q^{9}-\beta q^{11}-q^{13}+\cdots\)
3120.2.a.bf 3120.a 1.a $2$ $24.913$ \(\Q(\sqrt{41}) \) None \(0\) \(2\) \(-2\) \(1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+\beta q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)
3120.2.a.bg 3120.a 1.a $2$ $24.913$ \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(2\) \(1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+\beta q^{7}+q^{9}+(4-\beta )q^{11}+\cdots\)
3120.2.a.bh 3120.a 1.a $3$ $24.913$ 3.3.1849.1 None \(0\) \(-3\) \(-3\) \(-5\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+(-2+\beta _{1})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3120.2.a.bi 3120.a 1.a $3$ $24.913$ 3.3.940.1 None \(0\) \(-3\) \(3\) \(-1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-\beta _{2}q^{7}+q^{9}+(-2-\beta _{1}+\cdots)q^{11}+\cdots\)
3120.2.a.bj 3120.a 1.a $3$ $24.913$ 3.3.316.1 None \(0\) \(3\) \(-3\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+\beta _{1}q^{7}+q^{9}+\beta _{1}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3120))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3120)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(624))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(780))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1040))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1560))\)\(^{\oplus 2}\)