Properties

Label 3520.2.a
Level $3520$
Weight $2$
Character orbit 3520.a
Rep. character $\chi_{3520}(1,\cdot)$
Character field $\Q$
Dimension $80$
Newform subspaces $52$
Sturm bound $1152$
Trace bound $13$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3520.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 52 \)
Sturm bound: \(1152\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(7\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3520))\).

Total New Old
Modular forms 600 80 520
Cusp forms 553 80 473
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.

\(2\)\(5\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(10\)
\(+\)\(+\)\(-\)\(-\)\(12\)
\(+\)\(-\)\(+\)\(-\)\(10\)
\(+\)\(-\)\(-\)\(+\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(10\)
\(-\)\(+\)\(-\)\(+\)\(8\)
\(-\)\(-\)\(+\)\(+\)\(10\)
\(-\)\(-\)\(-\)\(-\)\(12\)
Plus space\(+\)\(36\)
Minus space\(-\)\(44\)

Trace form

\( 80 q + 80 q^{9} + O(q^{10}) \) \( 80 q + 80 q^{9} - 32 q^{13} - 32 q^{21} + 80 q^{25} - 32 q^{37} + 80 q^{49} + 32 q^{69} + 112 q^{81} + 32 q^{85} + 32 q^{89} + 64 q^{93} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3520))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 11
3520.2.a.a $1$ $28.107$ \(\Q\) None \(0\) \(-3\) \(-1\) \(1\) $+$ $+$ $-$ \(q-3q^{3}-q^{5}+q^{7}+6q^{9}+q^{11}+6q^{13}+\cdots\)
3520.2.a.b $1$ $28.107$ \(\Q\) None \(0\) \(-2\) \(-1\) \(0\) $-$ $+$ $+$ \(q-2q^{3}-q^{5}+q^{9}-q^{11}-4q^{13}+\cdots\)
3520.2.a.c $1$ $28.107$ \(\Q\) None \(0\) \(-2\) \(-1\) \(0\) $+$ $+$ $+$ \(q-2q^{3}-q^{5}+q^{9}-q^{11}+2q^{15}+\cdots\)
3520.2.a.d $1$ $28.107$ \(\Q\) None \(0\) \(-2\) \(-1\) \(4\) $-$ $+$ $+$ \(q-2q^{3}-q^{5}+4q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
3520.2.a.e $1$ $28.107$ \(\Q\) None \(0\) \(-2\) \(-1\) \(4\) $-$ $+$ $-$ \(q-2q^{3}-q^{5}+4q^{7}+q^{9}+q^{11}-4q^{13}+\cdots\)
3520.2.a.f $1$ $28.107$ \(\Q\) None \(0\) \(-2\) \(1\) \(-2\) $+$ $-$ $-$ \(q-2q^{3}+q^{5}-2q^{7}+q^{9}+q^{11}+2q^{13}+\cdots\)
3520.2.a.g $1$ $28.107$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-3\) $-$ $+$ $-$ \(q-q^{3}-q^{5}-3q^{7}-2q^{9}+q^{11}+2q^{13}+\cdots\)
3520.2.a.h $1$ $28.107$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-3\) $-$ $+$ $-$ \(q-q^{3}-q^{5}-3q^{7}-2q^{9}+q^{11}+6q^{13}+\cdots\)
3520.2.a.i $1$ $28.107$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-1\) $-$ $+$ $+$ \(q-q^{3}-q^{5}-q^{7}-2q^{9}-q^{11}+2q^{13}+\cdots\)
3520.2.a.j $1$ $28.107$ \(\Q\) None \(0\) \(-1\) \(1\) \(-3\) $-$ $-$ $+$ \(q-q^{3}+q^{5}-3q^{7}-2q^{9}-q^{11}+2q^{13}+\cdots\)
3520.2.a.k $1$ $28.107$ \(\Q\) None \(0\) \(-1\) \(1\) \(-1\) $+$ $-$ $-$ \(q-q^{3}+q^{5}-q^{7}-2q^{9}+q^{11}-2q^{13}+\cdots\)
3520.2.a.l $1$ $28.107$ \(\Q\) None \(0\) \(-1\) \(1\) \(5\) $+$ $-$ $+$ \(q-q^{3}+q^{5}+5q^{7}-2q^{9}-q^{11}-2q^{13}+\cdots\)
3520.2.a.m $1$ $28.107$ \(\Q\) None \(0\) \(0\) \(-1\) \(-4\) $-$ $+$ $+$ \(q-q^{5}-4q^{7}-3q^{9}-q^{11}-6q^{13}+\cdots\)
3520.2.a.n $1$ $28.107$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $-$ $+$ $+$ \(q-q^{5}-3q^{9}-q^{11}-2q^{13}+6q^{17}+\cdots\)
3520.2.a.o $1$ $28.107$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $+$ $+$ \(q-q^{5}-3q^{9}-q^{11}+2q^{13}-2q^{17}+\cdots\)
3520.2.a.p $1$ $28.107$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $+$ $-$ \(q-q^{5}-3q^{9}+q^{11}-2q^{13}+6q^{17}+\cdots\)
3520.2.a.q $1$ $28.107$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $+$ $-$ \(q-q^{5}-3q^{9}+q^{11}+2q^{13}-2q^{17}+\cdots\)
3520.2.a.r $1$ $28.107$ \(\Q\) None \(0\) \(0\) \(-1\) \(4\) $+$ $+$ $-$ \(q-q^{5}+4q^{7}-3q^{9}+q^{11}-6q^{13}+\cdots\)
3520.2.a.s $1$ $28.107$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $+$ $-$ $+$ \(q+q^{5}-2q^{7}-3q^{9}-q^{11}+4q^{13}+\cdots\)
3520.2.a.t $1$ $28.107$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $+$ $-$ $-$ \(q+q^{5}-2q^{7}-3q^{9}+q^{11}+8q^{19}+\cdots\)
3520.2.a.u $1$ $28.107$ \(\Q\) None \(0\) \(0\) \(1\) \(2\) $-$ $-$ $+$ \(q+q^{5}+2q^{7}-3q^{9}-q^{11}-8q^{19}+\cdots\)
3520.2.a.v $1$ $28.107$ \(\Q\) None \(0\) \(0\) \(1\) \(2\) $-$ $-$ $-$ \(q+q^{5}+2q^{7}-3q^{9}+q^{11}+4q^{13}+\cdots\)
3520.2.a.w $1$ $28.107$ \(\Q\) None \(0\) \(1\) \(-1\) \(1\) $-$ $+$ $-$ \(q+q^{3}-q^{5}+q^{7}-2q^{9}+q^{11}+2q^{13}+\cdots\)
3520.2.a.x $1$ $28.107$ \(\Q\) None \(0\) \(1\) \(-1\) \(3\) $-$ $+$ $+$ \(q+q^{3}-q^{5}+3q^{7}-2q^{9}-q^{11}+2q^{13}+\cdots\)
3520.2.a.y $1$ $28.107$ \(\Q\) None \(0\) \(1\) \(-1\) \(3\) $+$ $+$ $+$ \(q+q^{3}-q^{5}+3q^{7}-2q^{9}-q^{11}+6q^{13}+\cdots\)
3520.2.a.z $1$ $28.107$ \(\Q\) None \(0\) \(1\) \(1\) \(-5\) $-$ $-$ $-$ \(q+q^{3}+q^{5}-5q^{7}-2q^{9}+q^{11}-2q^{13}+\cdots\)
3520.2.a.ba $1$ $28.107$ \(\Q\) None \(0\) \(1\) \(1\) \(1\) $-$ $-$ $+$ \(q+q^{3}+q^{5}+q^{7}-2q^{9}-q^{11}-2q^{13}+\cdots\)
3520.2.a.bb $1$ $28.107$ \(\Q\) None \(0\) \(1\) \(1\) \(3\) $-$ $-$ $-$ \(q+q^{3}+q^{5}+3q^{7}-2q^{9}+q^{11}+2q^{13}+\cdots\)
3520.2.a.bc $1$ $28.107$ \(\Q\) None \(0\) \(2\) \(-1\) \(-4\) $-$ $+$ $+$ \(q+2q^{3}-q^{5}-4q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
3520.2.a.bd $1$ $28.107$ \(\Q\) None \(0\) \(2\) \(-1\) \(-4\) $+$ $+$ $-$ \(q+2q^{3}-q^{5}-4q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
3520.2.a.be $1$ $28.107$ \(\Q\) None \(0\) \(2\) \(-1\) \(0\) $-$ $+$ $-$ \(q+2q^{3}-q^{5}+q^{9}+q^{11}-4q^{13}+\cdots\)
3520.2.a.bf $1$ $28.107$ \(\Q\) None \(0\) \(2\) \(-1\) \(0\) $-$ $+$ $-$ \(q+2q^{3}-q^{5}+q^{9}+q^{11}-2q^{15}+\cdots\)
3520.2.a.bg $1$ $28.107$ \(\Q\) None \(0\) \(2\) \(1\) \(2\) $+$ $-$ $+$ \(q+2q^{3}+q^{5}+2q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
3520.2.a.bh $1$ $28.107$ \(\Q\) None \(0\) \(3\) \(-1\) \(-1\) $-$ $+$ $+$ \(q+3q^{3}-q^{5}-q^{7}+6q^{9}-q^{11}+6q^{13}+\cdots\)
3520.2.a.bi $2$ $28.107$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(2\) \(4\) $-$ $-$ $+$ \(q+(-1-\beta )q^{3}+q^{5}+2q^{7}+(3+2\beta )q^{9}+\cdots\)
3520.2.a.bj $2$ $28.107$ \(\Q(\sqrt{33}) \) None \(0\) \(-1\) \(-2\) \(-1\) $-$ $+$ $+$ \(q-\beta q^{3}-q^{5}-\beta q^{7}+(5+\beta )q^{9}-q^{11}+\cdots\)
3520.2.a.bk $2$ $28.107$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(-2\) \(1\) $-$ $+$ $-$ \(q-\beta q^{3}-q^{5}+\beta q^{7}+(1+\beta )q^{9}+q^{11}+\cdots\)
3520.2.a.bl $2$ $28.107$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(2\) \(3\) $+$ $-$ $+$ \(q-\beta q^{3}+q^{5}+(2-\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
3520.2.a.bm $2$ $28.107$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(2\) \(5\) $+$ $-$ $-$ \(q-\beta q^{3}+q^{5}+(2+\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
3520.2.a.bn $2$ $28.107$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(2\) \(-4\) $+$ $-$ $+$ \(q+\beta q^{3}+q^{5}-2q^{7}+5q^{9}-q^{11}+\cdots\)
3520.2.a.bo $2$ $28.107$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(2\) \(4\) $-$ $-$ $-$ \(q+\beta q^{3}+q^{5}+2q^{7}+5q^{9}+q^{11}+\cdots\)
3520.2.a.bp $2$ $28.107$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(-2\) \(-1\) $+$ $+$ $+$ \(q+\beta q^{3}-q^{5}-\beta q^{7}+(1+\beta )q^{9}-q^{11}+\cdots\)
3520.2.a.bq $2$ $28.107$ \(\Q(\sqrt{33}) \) None \(0\) \(1\) \(-2\) \(1\) $+$ $+$ $-$ \(q+\beta q^{3}-q^{5}+\beta q^{7}+(5+\beta )q^{9}+q^{11}+\cdots\)
3520.2.a.br $2$ $28.107$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(2\) \(-5\) $-$ $-$ $+$ \(q+\beta q^{3}+q^{5}+(-2-\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
3520.2.a.bs $2$ $28.107$ \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(2\) \(-3\) $-$ $-$ $-$ \(q+\beta q^{3}+q^{5}+(-2+\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
3520.2.a.bt $2$ $28.107$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(2\) \(-4\) $-$ $-$ $-$ \(q+(1+\beta )q^{3}+q^{5}-2q^{7}+(3+2\beta )q^{9}+\cdots\)
3520.2.a.bu $3$ $28.107$ 3.3.229.1 None \(0\) \(-2\) \(3\) \(-4\) $+$ $-$ $-$ \(q+(-1-\beta _{2})q^{3}+q^{5}+(-1+\beta _{1})q^{7}+\cdots\)
3520.2.a.bv $3$ $28.107$ 3.3.229.1 None \(0\) \(-1\) \(3\) \(-7\) $-$ $-$ $+$ \(q+\beta _{2}q^{3}+q^{5}+(-2+\beta _{1})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3520.2.a.bw $3$ $28.107$ 3.3.229.1 None \(0\) \(1\) \(3\) \(7\) $-$ $-$ $-$ \(q-\beta _{2}q^{3}+q^{5}+(2-\beta _{1})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
3520.2.a.bx $3$ $28.107$ 3.3.229.1 None \(0\) \(2\) \(3\) \(4\) $+$ $-$ $+$ \(q+(1+\beta _{2})q^{3}+q^{5}+(1-\beta _{1})q^{7}+(2+\cdots)q^{9}+\cdots\)
3520.2.a.by $5$ $28.107$ 5.5.792644.1 None \(0\) \(0\) \(-5\) \(-2\) $+$ $+$ $-$ \(q+\beta _{3}q^{3}-q^{5}-\beta _{4}q^{7}+(2-\beta _{1})q^{9}+\cdots\)
3520.2.a.bz $5$ $28.107$ 5.5.792644.1 None \(0\) \(0\) \(-5\) \(2\) $+$ $+$ $+$ \(q-\beta _{3}q^{3}-q^{5}+\beta _{4}q^{7}+(2-\beta _{1})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3520))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3520)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 14}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(352))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(440))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(704))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(880))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1760))\)\(^{\oplus 2}\)