Properties

Label 4704.2.a
Level $4704$
Weight $2$
Character orbit 4704.a
Rep. character $\chi_{4704}(1,\cdot)$
Character field $\Q$
Dimension $82$
Newform subspaces $52$
Sturm bound $1792$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4704.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 52 \)
Sturm bound: \(1792\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\), \(19\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4704))\).

Total New Old
Modular forms 960 82 878
Cusp forms 833 82 751
Eisenstein series 127 0 127

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

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(9\)
\(+\)\(+\)\(-\)\(-\)\(12\)
\(+\)\(-\)\(+\)\(-\)\(11\)
\(+\)\(-\)\(-\)\(+\)\(9\)
\(-\)\(+\)\(+\)\(-\)\(11\)
\(-\)\(+\)\(-\)\(+\)\(9\)
\(-\)\(-\)\(+\)\(+\)\(9\)
\(-\)\(-\)\(-\)\(-\)\(12\)
Plus space\(+\)\(36\)
Minus space\(-\)\(46\)

Trace form

\( 82q + 4q^{5} + 82q^{9} + O(q^{10}) \) \( 82q + 4q^{5} + 82q^{9} + 12q^{13} - 12q^{17} + 62q^{25} + 4q^{29} - 8q^{33} + 12q^{37} - 28q^{41} + 4q^{45} - 12q^{53} + 8q^{57} - 4q^{61} - 40q^{65} - 12q^{73} + 82q^{81} + 40q^{85} - 12q^{89} - 8q^{93} + 36q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4704))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
4704.2.a.a \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(-4\) \(0\) \(-\) \(+\) \(+\) \(q-q^{3}-4q^{5}+q^{9}+6q^{11}+5q^{13}+\cdots\)
4704.2.a.b \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(-3\) \(0\) \(+\) \(+\) \(-\) \(q-q^{3}-3q^{5}+q^{9}-q^{11}+4q^{13}+\cdots\)
4704.2.a.c \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(-2\) \(0\) \(-\) \(+\) \(-\) \(q-q^{3}-2q^{5}+q^{9}-2q^{13}+2q^{15}+\cdots\)
4704.2.a.d \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(-2\) \(0\) \(-\) \(+\) \(-\) \(q-q^{3}-2q^{5}+q^{9}+2q^{11}+2q^{15}+\cdots\)
4704.2.a.e \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(-2\) \(0\) \(+\) \(+\) \(-\) \(q-q^{3}-2q^{5}+q^{9}+4q^{11}+2q^{13}+\cdots\)
4704.2.a.f \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) \(-\) \(+\) \(-\) \(q-q^{3}-q^{5}+q^{9}-q^{11}+q^{15}+8q^{17}+\cdots\)
4704.2.a.g \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{3}+q^{9}-2q^{11}+q^{13}-2q^{17}+\cdots\)
4704.2.a.h \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{3}+q^{9}-2q^{11}+5q^{13}-2q^{17}+\cdots\)
4704.2.a.i \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{9}+2q^{11}-5q^{13}+2q^{17}+\cdots\)
4704.2.a.j \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-q^{3}+q^{9}+2q^{11}-q^{13}+2q^{17}+\cdots\)
4704.2.a.k \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-q^{3}+q^{9}+2q^{11}+2q^{13}-4q^{17}+\cdots\)
4704.2.a.l \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(1\) \(0\) \(-\) \(+\) \(+\) \(q-q^{3}+q^{5}+q^{9}+q^{11}-q^{15}-8q^{17}+\cdots\)
4704.2.a.m \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(2\) \(0\) \(-\) \(+\) \(-\) \(q-q^{3}+2q^{5}+q^{9}-4q^{11}+6q^{13}+\cdots\)
4704.2.a.n \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(2\) \(0\) \(+\) \(+\) \(-\) \(q-q^{3}+2q^{5}+q^{9}-2q^{11}-2q^{15}+\cdots\)
4704.2.a.o \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(3\) \(0\) \(+\) \(+\) \(+\) \(q-q^{3}+3q^{5}+q^{9}+q^{11}-4q^{13}+\cdots\)
4704.2.a.p \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(4\) \(0\) \(-\) \(+\) \(-\) \(q-q^{3}+4q^{5}+q^{9}-6q^{11}-5q^{13}+\cdots\)
4704.2.a.q \(1\) \(37.562\) \(\Q\) None \(0\) \(-1\) \(4\) \(0\) \(+\) \(+\) \(-\) \(q-q^{3}+4q^{5}+q^{9}-2q^{11}+2q^{13}+\cdots\)
4704.2.a.r \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(-4\) \(0\) \(-\) \(-\) \(+\) \(q+q^{3}-4q^{5}+q^{9}-6q^{11}+5q^{13}+\cdots\)
4704.2.a.s \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(-3\) \(0\) \(+\) \(-\) \(-\) \(q+q^{3}-3q^{5}+q^{9}+q^{11}+4q^{13}+\cdots\)
4704.2.a.t \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(-2\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}-2q^{5}+q^{9}-4q^{11}+2q^{13}+\cdots\)
4704.2.a.u \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(-2\) \(0\) \(+\) \(-\) \(-\) \(q+q^{3}-2q^{5}+q^{9}-2q^{11}-2q^{15}+\cdots\)
4704.2.a.v \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(-2\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}-2q^{5}+q^{9}-2q^{13}-2q^{15}+\cdots\)
4704.2.a.w \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(-1\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}-q^{5}+q^{9}+q^{11}-q^{15}+8q^{17}+\cdots\)
4704.2.a.x \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+q^{3}+q^{9}-2q^{11}-5q^{13}+2q^{17}+\cdots\)
4704.2.a.y \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+q^{3}+q^{9}-2q^{11}-q^{13}+2q^{17}+\cdots\)
4704.2.a.z \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+q^{3}+q^{9}-2q^{11}+2q^{13}-4q^{17}+\cdots\)
4704.2.a.ba \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{9}+2q^{11}+q^{13}-2q^{17}+\cdots\)
4704.2.a.bb \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+q^{3}+q^{9}+2q^{11}+5q^{13}-2q^{17}+\cdots\)
4704.2.a.bc \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(1\) \(0\) \(-\) \(-\) \(+\) \(q+q^{3}+q^{5}+q^{9}-q^{11}+q^{15}-8q^{17}+\cdots\)
4704.2.a.bd \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}+2q^{5}+q^{9}+2q^{11}+2q^{15}+\cdots\)
4704.2.a.be \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}+2q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
4704.2.a.bf \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(3\) \(0\) \(+\) \(-\) \(+\) \(q+q^{3}+3q^{5}+q^{9}-q^{11}-4q^{13}+\cdots\)
4704.2.a.bg \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(4\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}+4q^{5}+q^{9}+2q^{11}+2q^{13}+\cdots\)
4704.2.a.bh \(1\) \(37.562\) \(\Q\) None \(0\) \(1\) \(4\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}+4q^{5}+q^{9}+6q^{11}-5q^{13}+\cdots\)
4704.2.a.bi \(2\) \(37.562\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-q^{3}+\beta q^{5}+q^{9}+\beta q^{11}-\beta q^{15}+\cdots\)
4704.2.a.bj \(2\) \(37.562\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q-q^{3}+\beta q^{5}+q^{9}-2\beta q^{11}+\beta q^{13}+\cdots\)
4704.2.a.bk \(2\) \(37.562\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{9}+\beta q^{11}+\beta q^{13}+\beta q^{17}+\cdots\)
4704.2.a.bl \(2\) \(37.562\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-q^{3}+\beta q^{5}+q^{9}+2\beta q^{11}-3\beta q^{13}+\cdots\)
4704.2.a.bm \(2\) \(37.562\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{3}+\beta q^{5}+q^{9}+(2+\beta )q^{11}-2q^{13}+\cdots\)
4704.2.a.bn \(2\) \(37.562\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+q^{3}+\beta q^{5}+q^{9}+(-2-\beta )q^{11}+\cdots\)
4704.2.a.bo \(2\) \(37.562\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+q^{3}+\beta q^{5}+q^{9}-2\beta q^{11}-3\beta q^{13}+\cdots\)
4704.2.a.bp \(2\) \(37.562\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+q^{3}+q^{9}-\beta q^{11}+\beta q^{13}+\beta q^{17}+\cdots\)
4704.2.a.bq \(2\) \(37.562\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+q^{3}+\beta q^{5}+q^{9}+2\beta q^{11}+\beta q^{13}+\cdots\)
4704.2.a.br \(2\) \(37.562\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}+\beta q^{5}+q^{9}-\beta q^{11}+\beta q^{15}+\cdots\)
4704.2.a.bs \(3\) \(37.562\) 3.3.621.1 None \(0\) \(-3\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-q^{3}+\beta _{2}q^{5}+q^{9}+(\beta _{1}+\beta _{2})q^{11}+\cdots\)
4704.2.a.bt \(3\) \(37.562\) 3.3.621.1 None \(0\) \(-3\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{3}-\beta _{2}q^{5}+q^{9}+(-\beta _{1}-\beta _{2})q^{11}+\cdots\)
4704.2.a.bu \(3\) \(37.562\) 3.3.621.1 None \(0\) \(3\) \(0\) \(0\) \(+\) \(-\) \(+\) \(q+q^{3}+\beta _{2}q^{5}+q^{9}+(-\beta _{1}-\beta _{2})q^{11}+\cdots\)
4704.2.a.bv \(3\) \(37.562\) 3.3.621.1 None \(0\) \(3\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}-\beta _{2}q^{5}+q^{9}+(\beta _{1}+\beta _{2})q^{11}+\cdots\)
4704.2.a.bw \(4\) \(37.562\) 4.4.2624.1 None \(0\) \(-4\) \(-4\) \(0\) \(+\) \(+\) \(+\) \(q-q^{3}+(-1+\beta _{3})q^{5}+q^{9}+(1-\beta _{2}+\cdots)q^{11}+\cdots\)
4704.2.a.bx \(4\) \(37.562\) 4.4.2624.1 None \(0\) \(-4\) \(4\) \(0\) \(-\) \(+\) \(+\) \(q-q^{3}+(1+\beta _{3})q^{5}+q^{9}+(-1-\beta _{2}+\cdots)q^{11}+\cdots\)
4704.2.a.by \(4\) \(37.562\) 4.4.2624.1 None \(0\) \(4\) \(-4\) \(0\) \(-\) \(-\) \(+\) \(q+q^{3}+(-1+\beta _{3})q^{5}+q^{9}+(-1+\beta _{2}+\cdots)q^{11}+\cdots\)
4704.2.a.bz \(4\) \(37.562\) 4.4.2624.1 None \(0\) \(4\) \(4\) \(0\) \(+\) \(-\) \(+\) \(q+q^{3}+(1+\beta _{3})q^{5}+q^{9}+(1+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4704))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4704)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(672))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(784))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1176))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1568))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2352))\)\(^{\oplus 2}\)