Properties

Label 1856.4.a
Level $1856$
Weight $4$
Character orbit 1856.a
Rep. character $\chi_{1856}(1,\cdot)$
Character field $\Q$
Dimension $168$
Newform subspaces $38$
Sturm bound $960$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 38 \)
Sturm bound: \(960\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1856))\).

Total New Old
Modular forms 732 168 564
Cusp forms 708 168 540
Eisenstein series 24 0 24

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

\(2\)\(29\)FrickeDim.
\(+\)\(+\)\(+\)\(45\)
\(+\)\(-\)\(-\)\(39\)
\(-\)\(+\)\(-\)\(39\)
\(-\)\(-\)\(+\)\(45\)
Plus space\(+\)\(90\)
Minus space\(-\)\(78\)

Trace form

\( 168 q + 1512 q^{9} + O(q^{10}) \) \( 168 q + 1512 q^{9} - 144 q^{13} + 240 q^{21} + 4200 q^{25} + 464 q^{33} + 1008 q^{37} + 80 q^{41} - 1968 q^{45} + 8232 q^{49} - 816 q^{53} - 688 q^{57} + 3984 q^{61} + 2544 q^{69} - 3504 q^{77} + 11768 q^{81} - 2832 q^{85} + 8640 q^{93} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 29
1856.4.a.a \(1\) \(109.508\) \(\Q\) None \(0\) \(-7\) \(-5\) \(-2\) \(+\) \(-\) \(q-7q^{3}-5q^{5}-2q^{7}+22q^{9}-37q^{11}+\cdots\)
1856.4.a.b \(1\) \(109.508\) \(\Q\) None \(0\) \(-7\) \(13\) \(-16\) \(-\) \(-\) \(q-7q^{3}+13q^{5}-2^{4}q^{7}+22q^{9}-45q^{11}+\cdots\)
1856.4.a.c \(1\) \(109.508\) \(\Q\) None \(0\) \(-7\) \(15\) \(18\) \(-\) \(-\) \(q-7q^{3}+15q^{5}+18q^{7}+22q^{9}+3^{3}q^{11}+\cdots\)
1856.4.a.d \(1\) \(109.508\) \(\Q\) None \(0\) \(7\) \(-5\) \(2\) \(-\) \(-\) \(q+7q^{3}-5q^{5}+2q^{7}+22q^{9}+37q^{11}+\cdots\)
1856.4.a.e \(1\) \(109.508\) \(\Q\) None \(0\) \(7\) \(13\) \(16\) \(-\) \(-\) \(q+7q^{3}+13q^{5}+2^{4}q^{7}+22q^{9}+45q^{11}+\cdots\)
1856.4.a.f \(1\) \(109.508\) \(\Q\) None \(0\) \(7\) \(15\) \(-18\) \(+\) \(-\) \(q+7q^{3}+15q^{5}-18q^{7}+22q^{9}-3^{3}q^{11}+\cdots\)
1856.4.a.g \(2\) \(109.508\) \(\Q(\sqrt{22}) \) None \(0\) \(-10\) \(-30\) \(0\) \(+\) \(+\) \(q+(-5+\beta )q^{3}-15q^{5}+2\beta q^{7}+(20+\cdots)q^{9}+\cdots\)
1856.4.a.h \(2\) \(109.508\) \(\Q(\sqrt{2}) \) None \(0\) \(-10\) \(10\) \(16\) \(-\) \(+\) \(q+(-5+3\beta )q^{3}+(5+4\beta )q^{5}+(8+10\beta )q^{7}+\cdots\)
1856.4.a.i \(2\) \(109.508\) \(\Q(\sqrt{6}) \) None \(0\) \(-2\) \(10\) \(16\) \(-\) \(+\) \(q+(-1+\beta )q^{3}+(5+6\beta )q^{5}+(8-8\beta )q^{7}+\cdots\)
1856.4.a.j \(2\) \(109.508\) \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(10\) \(-20\) \(+\) \(-\) \(q-\beta q^{3}+(5+2\beta )q^{5}+(-10-4\beta )q^{7}+\cdots\)
1856.4.a.k \(2\) \(109.508\) \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(10\) \(20\) \(-\) \(-\) \(q-\beta q^{3}+(5-2\beta )q^{5}+(10-4\beta )q^{7}+\cdots\)
1856.4.a.l \(2\) \(109.508\) \(\Q(\sqrt{6}) \) None \(0\) \(2\) \(10\) \(-16\) \(+\) \(+\) \(q+(1+\beta )q^{3}+(5-6\beta )q^{5}+(-8-8\beta )q^{7}+\cdots\)
1856.4.a.m \(2\) \(109.508\) \(\Q(\sqrt{22}) \) None \(0\) \(10\) \(-30\) \(0\) \(-\) \(+\) \(q+(5+\beta )q^{3}-15q^{5}+2\beta q^{7}+(20+10\beta )q^{9}+\cdots\)
1856.4.a.n \(2\) \(109.508\) \(\Q(\sqrt{2}) \) None \(0\) \(10\) \(10\) \(-16\) \(+\) \(+\) \(q+(5+3\beta )q^{3}+(5-4\beta )q^{5}+(-8+10\beta )q^{7}+\cdots\)
1856.4.a.o \(3\) \(109.508\) 3.3.148344.1 None \(0\) \(-10\) \(20\) \(-8\) \(-\) \(+\) \(q+(-3-\beta _{1})q^{3}+(7-\beta _{2})q^{5}+(-4+\cdots)q^{7}+\cdots\)
1856.4.a.p \(3\) \(109.508\) 3.3.229.1 None \(0\) \(-6\) \(-4\) \(16\) \(+\) \(-\) \(q+(-1+3\beta _{2})q^{3}+(-1-3\beta _{1}+4\beta _{2})q^{5}+\cdots\)
1856.4.a.q \(3\) \(109.508\) 3.3.4481.1 None \(0\) \(-3\) \(-11\) \(-38\) \(+\) \(-\) \(q+(-1+\beta _{1})q^{3}+(-4-\beta _{1}-\beta _{2})q^{5}+\cdots\)
1856.4.a.r \(3\) \(109.508\) 3.3.19816.1 None \(0\) \(-2\) \(-20\) \(24\) \(+\) \(+\) \(q+(-1+\beta _{1})q^{3}+(-6-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
1856.4.a.s \(3\) \(109.508\) 3.3.19816.1 None \(0\) \(2\) \(-20\) \(-24\) \(-\) \(+\) \(q+(1-\beta _{1})q^{3}+(-6-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
1856.4.a.t \(3\) \(109.508\) 3.3.4481.1 None \(0\) \(3\) \(-11\) \(38\) \(-\) \(-\) \(q+(1-\beta _{1})q^{3}+(-4-\beta _{1}-\beta _{2})q^{5}+\cdots\)
1856.4.a.u \(3\) \(109.508\) 3.3.229.1 None \(0\) \(6\) \(-4\) \(-16\) \(-\) \(-\) \(q+(1-3\beta _{2})q^{3}+(-1-3\beta _{1}+4\beta _{2})q^{5}+\cdots\)
1856.4.a.v \(3\) \(109.508\) 3.3.148344.1 None \(0\) \(10\) \(20\) \(8\) \(+\) \(+\) \(q+(3+\beta _{1})q^{3}+(7-\beta _{2})q^{5}+(4-2\beta _{1}+\cdots)q^{7}+\cdots\)
1856.4.a.w \(4\) \(109.508\) 4.4.225792.1 None \(0\) \(0\) \(20\) \(-8\) \(+\) \(+\) \(q+\beta _{1}q^{3}+(5+\beta _{2})q^{5}+(-2+2\beta _{1}+\cdots)q^{7}+\cdots\)
1856.4.a.x \(4\) \(109.508\) 4.4.225792.1 None \(0\) \(0\) \(20\) \(8\) \(-\) \(+\) \(q+(\beta _{1}-\beta _{3})q^{3}+(5-\beta _{2})q^{5}+(2+2\beta _{1}+\cdots)q^{7}+\cdots\)
1856.4.a.y \(5\) \(109.508\) 5.5.13458092.1 None \(0\) \(-8\) \(-10\) \(40\) \(+\) \(-\) \(q+(-2+\beta _{3})q^{3}+(-2+2\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
1856.4.a.z \(5\) \(109.508\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-4\) \(-10\) \(32\) \(+\) \(+\) \(q+(-1+\beta _{1})q^{3}+(-2-\beta _{3})q^{5}+(6+\cdots)q^{7}+\cdots\)
1856.4.a.ba \(5\) \(109.508\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(4\) \(-10\) \(-32\) \(-\) \(+\) \(q+(1-\beta _{1})q^{3}+(-2-\beta _{3})q^{5}+(-6+\cdots)q^{7}+\cdots\)
1856.4.a.bb \(5\) \(109.508\) 5.5.13458092.1 None \(0\) \(8\) \(-10\) \(-40\) \(-\) \(-\) \(q+(2-\beta _{3})q^{3}+(-2+2\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
1856.4.a.bc \(6\) \(109.508\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-5\) \(5\) \(38\) \(-\) \(-\) \(q+(-1+\beta _{3})q^{3}+(1+\beta _{3}+\beta _{4})q^{5}+\cdots\)
1856.4.a.bd \(6\) \(109.508\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(5\) \(5\) \(-38\) \(+\) \(-\) \(q+(1-\beta _{3})q^{3}+(1+\beta _{3}+\beta _{4})q^{5}+(-6+\cdots)q^{7}+\cdots\)
1856.4.a.be \(8\) \(109.508\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(20\) \(0\) \(-\) \(+\) \(q-\beta _{1}q^{3}+(3+\beta _{2})q^{5}-\beta _{4}q^{7}+(5+\beta _{3}+\cdots)q^{9}+\cdots\)
1856.4.a.bf \(9\) \(109.508\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-4\) \(-10\) \(12\) \(+\) \(-\) \(q-\beta _{1}q^{3}+(-1-\beta _{1}+\beta _{4})q^{5}+(1+\beta _{6}+\cdots)q^{7}+\cdots\)
1856.4.a.bg \(9\) \(109.508\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(4\) \(-10\) \(-12\) \(+\) \(-\) \(q+\beta _{1}q^{3}+(-1-\beta _{1}+\beta _{4})q^{5}+(-1+\cdots)q^{7}+\cdots\)
1856.4.a.bh \(10\) \(109.508\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(-40\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{3}+(-4-\beta _{3})q^{5}+\beta _{8}q^{7}+(6+\cdots)q^{9}+\cdots\)
1856.4.a.bi \(10\) \(109.508\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(4\) \(0\) \(-\) \(-\) \(q-\beta _{1}q^{3}+\beta _{2}q^{5}-\beta _{6}q^{7}+(5-\beta _{5}+\cdots)q^{9}+\cdots\)
1856.4.a.bj \(12\) \(109.508\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-14\) \(10\) \(44\) \(+\) \(+\) \(q+(-1-\beta _{1})q^{3}+(1-\beta _{4})q^{5}+(4-\beta _{6}+\cdots)q^{7}+\cdots\)
1856.4.a.bk \(12\) \(109.508\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(-10\) \(0\) \(-\) \(-\) \(q+\beta _{1}q^{3}+(-1+\beta _{2})q^{5}-\beta _{7}q^{7}+(14+\cdots)q^{9}+\cdots\)
1856.4.a.bl \(12\) \(109.508\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(14\) \(10\) \(-44\) \(+\) \(+\) \(q+(1+\beta _{1})q^{3}+(1-\beta _{4})q^{5}+(-4+\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1856))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1856)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(232))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(464))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(928))\)\(^{\oplus 2}\)