Properties

Label 3920.2.a
Level $3920$
Weight $2$
Character orbit 3920.a
Rep. character $\chi_{3920}(1,\cdot)$
Character field $\Q$
Dimension $82$
Newform subspaces $57$
Sturm bound $1344$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 57 \)
Sturm bound: \(1344\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3920))\).

Total New Old
Modular forms 720 82 638
Cusp forms 625 82 543
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.

\(2\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(9\)
\(+\)\(+\)\(-\)\(-\)\(12\)
\(+\)\(-\)\(+\)\(-\)\(11\)
\(+\)\(-\)\(-\)\(+\)\(9\)
\(-\)\(+\)\(+\)\(-\)\(11\)
\(-\)\(+\)\(-\)\(+\)\(9\)
\(-\)\(-\)\(+\)\(+\)\(9\)
\(-\)\(-\)\(-\)\(-\)\(12\)
Plus space\(+\)\(36\)
Minus space\(-\)\(46\)

Trace form

\( 82 q - 2 q^{3} + 78 q^{9} + O(q^{10}) \) \( 82 q - 2 q^{3} + 78 q^{9} + 4 q^{11} - 2 q^{15} - 4 q^{17} - 8 q^{19} - 2 q^{23} + 82 q^{25} - 20 q^{27} - 4 q^{29} - 20 q^{31} - 8 q^{37} - 28 q^{39} + 8 q^{41} - 6 q^{43} + 4 q^{45} + 22 q^{47} + 4 q^{51} + 4 q^{55} - 8 q^{57} + 8 q^{59} + 16 q^{61} + 4 q^{65} + 22 q^{67} + 28 q^{69} + 12 q^{71} - 20 q^{73} - 2 q^{75} + 24 q^{79} + 70 q^{81} + 14 q^{83} + 8 q^{85} - 36 q^{87} - 12 q^{89} + 40 q^{93} - 8 q^{95} - 28 q^{97} - 44 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3920))\) 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 5 7
3920.2.a.a 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-3\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{5}+6q^{9}-q^{11}+3q^{13}+\cdots\)
3920.2.a.b 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-3\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{5}+6q^{9}+2q^{11}+3q^{15}+\cdots\)
3920.2.a.c 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-3\) \(-1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{5}+6q^{9}+5q^{11}+5q^{13}+\cdots\)
3920.2.a.d 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-3\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+q^{5}+6q^{9}+2q^{11}+6q^{13}+\cdots\)
3920.2.a.e 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-2\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{5}+q^{9}-3q^{11}+q^{13}+\cdots\)
3920.2.a.f 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-2\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+q^{9}-4q^{11}+2q^{13}+\cdots\)
3920.2.a.g 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-2\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+q^{9}-3q^{11}-5q^{13}+\cdots\)
3920.2.a.h 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-2\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+q^{9}-2q^{13}-2q^{15}+\cdots\)
3920.2.a.i 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-2\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+q^{9}+q^{11}+3q^{13}+\cdots\)
3920.2.a.j 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-2\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+q^{9}+4q^{11}+2q^{13}+\cdots\)
3920.2.a.k 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{9}-6q^{11}+2q^{13}+\cdots\)
3920.2.a.l 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{9}-3q^{11}-q^{13}+\cdots\)
3920.2.a.m 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{9}-2q^{11}+q^{15}+\cdots\)
3920.2.a.n 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{9}+q^{11}-5q^{13}+\cdots\)
3920.2.a.o 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{9}+5q^{11}+7q^{13}+\cdots\)
3920.2.a.p 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{9}+6q^{11}-4q^{13}+\cdots\)
3920.2.a.q 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-1\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{9}+2q^{11}-4q^{13}+\cdots\)
3920.2.a.r 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(-1\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{9}+5q^{11}-q^{13}+\cdots\)
3920.2.a.s 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{9}-4q^{11}+2q^{13}-2q^{17}+\cdots\)
3920.2.a.t 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{9}-4q^{11}+6q^{13}-2q^{17}+\cdots\)
3920.2.a.u 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(1\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{9}-3q^{11}+q^{13}+\cdots\)
3920.2.a.v 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(1\) \(-1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{9}+2q^{11}+4q^{13}+\cdots\)
3920.2.a.w 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{9}-6q^{11}-2q^{13}+\cdots\)
3920.2.a.x 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{9}-3q^{11}+q^{13}+\cdots\)
3920.2.a.y 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{9}-2q^{11}+q^{15}+\cdots\)
3920.2.a.z 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{9}+q^{11}+5q^{13}+\cdots\)
3920.2.a.ba 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{9}+3q^{11}-5q^{13}+\cdots\)
3920.2.a.bb 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{9}+5q^{11}-7q^{13}+\cdots\)
3920.2.a.bc 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-2q^{9}+6q^{11}+4q^{13}+\cdots\)
3920.2.a.bd 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(2\) \(-1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
3920.2.a.be 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(2\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}+q^{9}-3q^{11}+5q^{13}+\cdots\)
3920.2.a.bf 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(2\) \(-1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}+q^{9}+q^{11}-3q^{13}+\cdots\)
3920.2.a.bg 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(2\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}+q^{9}+4q^{11}-2q^{13}+\cdots\)
3920.2.a.bh 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(2\) \(1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}+q^{9}-3q^{11}-q^{13}+\cdots\)
3920.2.a.bi 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(3\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-q^{5}+6q^{9}+2q^{11}-6q^{13}+\cdots\)
3920.2.a.bj 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(3\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{5}+6q^{9}-q^{11}-3q^{13}+\cdots\)
3920.2.a.bk 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(3\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{5}+6q^{9}+2q^{11}+3q^{15}+\cdots\)
3920.2.a.bl 3920.a 1.a $1$ $31.301$ \(\Q\) None \(0\) \(3\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{5}+6q^{9}+5q^{11}+3q^{13}+\cdots\)
3920.2.a.bm 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{2}) \) None \(0\) \(-4\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{3}+q^{5}+(3-4\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
3920.2.a.bn 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-2\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-q^{5}-2\beta q^{9}+q^{11}+\cdots\)
3920.2.a.bo 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-q^{5}-2\beta q^{9}+(1+2\beta )q^{11}+\cdots\)
3920.2.a.bp 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-q^{5}-2\beta q^{9}+(2-2\beta )q^{11}+\cdots\)
3920.2.a.bq 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+q^{5}-2\beta q^{9}+(-2+\cdots)q^{11}+\cdots\)
3920.2.a.br 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+q^{5}-2\beta q^{9}+(3-2\beta )q^{11}+\cdots\)
3920.2.a.bs 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-q^{5}+(1+\beta )q^{9}-\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
3920.2.a.bt 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{33}) \) None \(0\) \(-1\) \(2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+q^{5}+(5+\beta )q^{9}+(-4+\beta )q^{11}+\cdots\)
3920.2.a.bu 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(-2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}+(1+\beta )q^{9}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
3920.2.a.bv 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-q^{5}+2\beta q^{9}+(-2-2\beta )q^{11}+\cdots\)
3920.2.a.bw 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(-2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-q^{5}+2\beta q^{9}+(3+2\beta )q^{11}+\cdots\)
3920.2.a.bx 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+q^{5}+2\beta q^{9}+(1-2\beta )q^{11}+\cdots\)
3920.2.a.by 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+q^{5}+2\beta q^{9}+q^{11}+(-1+\cdots)q^{13}+\cdots\)
3920.2.a.bz 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+q^{5}+2\beta q^{9}+(2+2\beta )q^{11}+\cdots\)
3920.2.a.ca 3920.a 1.a $2$ $31.301$ \(\Q(\sqrt{2}) \) None \(0\) \(4\) \(-2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{3}-q^{5}+(3+4\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
3920.2.a.cb 3920.a 1.a $3$ $31.301$ 3.3.1944.1 None \(0\) \(0\) \(-3\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-q^{5}+(3+\beta _{1}+\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)
3920.2.a.cc 3920.a 1.a $3$ $31.301$ 3.3.1944.1 None \(0\) \(0\) \(3\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+q^{5}+(3+\beta _{1}+\beta _{2})q^{9}+(-1+\cdots)q^{11}+\cdots\)
3920.2.a.cd 3920.a 1.a $4$ $31.301$ 4.4.16448.2 None \(0\) \(-2\) \(4\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+q^{5}+(1+\beta _{1}+\beta _{2})q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
3920.2.a.ce 3920.a 1.a $4$ $31.301$ 4.4.16448.2 None \(0\) \(2\) \(-4\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-q^{5}+(1+\beta _{1}+\beta _{2})q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3920))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3920)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(560))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(784))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(980))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1960))\)\(^{\oplus 2}\)