Properties

Label 2704.2.a
Level $2704$
Weight $2$
Character orbit 2704.a
Rep. character $\chi_{2704}(1,\cdot)$
Character field $\Q$
Dimension $72$
Newform subspaces $33$
Sturm bound $728$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 33 \)
Sturm bound: \(728\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2704))\).

Total New Old
Modular forms 406 83 323
Cusp forms 323 72 251
Eisenstein series 83 11 72

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

\(2\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(18\)
\(+\)\(-\)\(-\)\(21\)
\(-\)\(+\)\(-\)\(18\)
\(-\)\(-\)\(+\)\(15\)
Plus space\(+\)\(33\)
Minus space\(-\)\(39\)

Trace form

\( 72 q + 2 q^{7} + 64 q^{9} + O(q^{10}) \) \( 72 q + 2 q^{7} + 64 q^{9} - 2 q^{11} - 8 q^{15} + 4 q^{17} + 2 q^{19} + 8 q^{21} - 8 q^{23} + 56 q^{25} + 14 q^{31} + 8 q^{33} - 8 q^{37} - 4 q^{41} + 4 q^{43} + 8 q^{45} + 10 q^{47} + 44 q^{49} + 28 q^{51} - 20 q^{53} - 12 q^{55} - 18 q^{59} - 16 q^{61} - 6 q^{63} + 26 q^{67} - 8 q^{69} + 6 q^{71} + 20 q^{73} - 20 q^{75} - 8 q^{77} + 40 q^{79} + 24 q^{81} - 22 q^{83} + 28 q^{87} - 4 q^{89} + 16 q^{93} - 24 q^{95} + 4 q^{97} - 34 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2704))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 13
2704.2.a.a 2704.a 1.a $1$ $21.592$ \(\Q\) None 52.2.e.a \(0\) \(-3\) \(-2\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-2q^{5}+q^{7}+6q^{9}-5q^{11}+\cdots\)
2704.2.a.b 2704.a 1.a $1$ $21.592$ \(\Q\) None 52.2.e.a \(0\) \(-3\) \(2\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+2q^{5}-q^{7}+6q^{9}+5q^{11}+\cdots\)
2704.2.a.c 2704.a 1.a $1$ $21.592$ \(\Q\) None 104.2.i.a \(0\) \(-1\) \(-2\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}-q^{7}-2q^{9}+q^{11}+2q^{15}+\cdots\)
2704.2.a.d 2704.a 1.a $1$ $21.592$ \(\Q\) None 104.2.a.a \(0\) \(-1\) \(1\) \(5\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+5q^{7}-2q^{9}-2q^{11}+\cdots\)
2704.2.a.e 2704.a 1.a $1$ $21.592$ \(\Q\) None 104.2.i.a \(0\) \(-1\) \(2\) \(1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}+q^{7}-2q^{9}-q^{11}-2q^{15}+\cdots\)
2704.2.a.f 2704.a 1.a $1$ $21.592$ \(\Q\) None 26.2.a.a \(0\) \(-1\) \(3\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{5}-q^{7}-2q^{9}+6q^{11}+\cdots\)
2704.2.a.g 2704.a 1.a $1$ $21.592$ \(\Q\) None 52.2.a.a \(0\) \(0\) \(-2\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}-2q^{7}-3q^{9}-2q^{11}+6q^{17}+\cdots\)
2704.2.a.h 2704.a 1.a $1$ $21.592$ \(\Q\) None 26.2.c.a \(0\) \(0\) \(-1\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-4q^{7}-3q^{9}-4q^{11}+3q^{17}+\cdots\)
2704.2.a.i 2704.a 1.a $1$ $21.592$ \(\Q\) None 26.2.c.a \(0\) \(0\) \(1\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+4q^{7}-3q^{9}+4q^{11}+3q^{17}+\cdots\)
2704.2.a.j 2704.a 1.a $1$ $21.592$ \(\Q\) None 26.2.b.a \(0\) \(1\) \(-3\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{5}+3q^{7}-2q^{9}-3q^{15}+\cdots\)
2704.2.a.k 2704.a 1.a $1$ $21.592$ \(\Q\) None 26.2.b.a \(0\) \(1\) \(3\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{5}-3q^{7}-2q^{9}+3q^{15}+\cdots\)
2704.2.a.l 2704.a 1.a $1$ $21.592$ \(\Q\) None 52.2.e.b \(0\) \(2\) \(-3\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-3q^{5}+4q^{7}+q^{9}-6q^{15}+\cdots\)
2704.2.a.m 2704.a 1.a $1$ $21.592$ \(\Q\) None 52.2.e.b \(0\) \(2\) \(3\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+3q^{5}-4q^{7}+q^{9}+6q^{15}+\cdots\)
2704.2.a.n 2704.a 1.a $1$ $21.592$ \(\Q\) None 26.2.a.b \(0\) \(3\) \(1\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{5}+q^{7}+6q^{9}-2q^{11}+\cdots\)
2704.2.a.o 2704.a 1.a $2$ $21.592$ \(\Q(\sqrt{3}) \) None 13.2.e.a \(0\) \(-4\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-\beta q^{5}+q^{9}+2\beta q^{15}+3q^{17}+\cdots\)
2704.2.a.p 2704.a 1.a $2$ $21.592$ \(\Q(\sqrt{17}) \) None 104.2.a.b \(0\) \(-1\) \(-3\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(-2+\beta )q^{5}-\beta q^{7}+(1+\beta )q^{9}+\cdots\)
2704.2.a.q 2704.a 1.a $2$ $21.592$ \(\Q(\sqrt{17}) \) None 104.2.i.b \(0\) \(-1\) \(-3\) \(1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(-1-\beta )q^{5}+\beta q^{7}+(1+\beta )q^{9}+\cdots\)
2704.2.a.r 2704.a 1.a $2$ $21.592$ \(\Q(\sqrt{17}) \) None 104.2.i.b \(0\) \(-1\) \(3\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(1+\beta )q^{5}-\beta q^{7}+(1+\beta )q^{9}+\cdots\)
2704.2.a.s 2704.a 1.a $2$ $21.592$ \(\Q(\sqrt{17}) \) None 104.2.f.a \(0\) \(1\) \(-1\) \(-5\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-\beta q^{5}+(-2-\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
2704.2.a.t 2704.a 1.a $2$ $21.592$ \(\Q(\sqrt{17}) \) None 104.2.f.a \(0\) \(1\) \(1\) \(5\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+\beta q^{5}+(2+\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
2704.2.a.u 2704.a 1.a $2$ $21.592$ \(\Q(\sqrt{3}) \) None 52.2.h.a \(0\) \(2\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-\beta q^{7}-2q^{9}+3\beta q^{11}-3q^{17}+\cdots\)
2704.2.a.v 2704.a 1.a $3$ $21.592$ \(\Q(\zeta_{14})^+\) None 338.2.a.g \(0\) \(-3\) \(-2\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1}+\beta _{2})q^{3}-2\beta _{1}q^{5}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
2704.2.a.w 2704.a 1.a $3$ $21.592$ \(\Q(\zeta_{14})^+\) None 338.2.a.g \(0\) \(-3\) \(2\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1}+\beta _{2})q^{3}+2\beta _{1}q^{5}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
2704.2.a.x 2704.a 1.a $3$ $21.592$ \(\Q(\zeta_{14})^+\) None 676.2.a.g \(0\) \(0\) \(-8\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-2\beta _{1}+\beta _{2})q^{3}+(-3-\beta _{2})q^{5}+\cdots\)
2704.2.a.y 2704.a 1.a $3$ $21.592$ \(\Q(\zeta_{14})^+\) None 676.2.a.g \(0\) \(0\) \(8\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-2\beta _{1}+\beta _{2})q^{3}+(3+\beta _{2})q^{5}+(2+\cdots)q^{7}+\cdots\)
2704.2.a.z 2704.a 1.a $3$ $21.592$ \(\Q(\zeta_{14})^+\) None 169.2.a.b \(0\) \(2\) \(-4\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(-1-\beta _{1})q^{5}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)
2704.2.a.ba 2704.a 1.a $3$ $21.592$ \(\Q(\zeta_{14})^+\) None 169.2.a.b \(0\) \(2\) \(4\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(1+\beta _{1})q^{5}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)
2704.2.a.bb 2704.a 1.a $3$ $21.592$ \(\Q(\zeta_{14})^+\) None 1352.2.a.i \(0\) \(4\) \(-4\) \(1\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+(-1-\beta _{1})q^{5}+\beta _{1}q^{7}+\cdots\)
2704.2.a.bc 2704.a 1.a $3$ $21.592$ \(\Q(\zeta_{14})^+\) None 1352.2.a.i \(0\) \(4\) \(4\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+(1+\beta _{1})q^{5}-\beta _{1}q^{7}+\cdots\)
2704.2.a.bd 2704.a 1.a $4$ $21.592$ 4.4.13968.1 None 104.2.o.a \(0\) \(2\) \(-4\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-1-\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)
2704.2.a.be 2704.a 1.a $4$ $21.592$ 4.4.13968.1 None 104.2.o.a \(0\) \(2\) \(4\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+\beta _{1}+\beta _{2}-\beta _{3})q^{5}+(-1+\cdots)q^{7}+\cdots\)
2704.2.a.bf 2704.a 1.a $6$ $21.592$ 6.6.3728753.1 None 1352.2.a.m \(0\) \(-3\) \(-2\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1}-\beta _{2}-\beta _{5})q^{3}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
2704.2.a.bg 2704.a 1.a $6$ $21.592$ 6.6.3728753.1 None 1352.2.a.m \(0\) \(-3\) \(2\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1}-\beta _{2}-\beta _{5})q^{3}+(\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2704))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2704)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1352))\)\(^{\oplus 2}\)