Properties

Label 1176.4.a
Level $1176$
Weight $4$
Character orbit 1176.a
Rep. character $\chi_{1176}(1,\cdot)$
Character field $\Q$
Dimension $61$
Newform subspaces $31$
Sturm bound $896$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1176.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 31 \)
Sturm bound: \(896\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1176))\).

Total New Old
Modular forms 704 61 643
Cusp forms 640 61 579
Eisenstein series 64 0 64

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.
\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(-\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(7\)
\(+\)\(-\)\(-\)\(+\)\(8\)
\(-\)\(+\)\(+\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(10\)
\(-\)\(-\)\(+\)\(+\)\(8\)
\(-\)\(-\)\(-\)\(-\)\(7\)
Plus space\(+\)\(33\)
Minus space\(-\)\(28\)

Trace form

\( 61q - 3q^{3} + 14q^{5} + 549q^{9} + O(q^{10}) \) \( 61q - 3q^{3} + 14q^{5} + 549q^{9} + 28q^{11} - 58q^{13} - 42q^{15} + 90q^{17} - 92q^{19} - 160q^{23} + 1391q^{25} - 27q^{27} - 418q^{29} - 256q^{31} + 84q^{33} + 234q^{37} + 198q^{39} - 606q^{41} + 512q^{43} + 126q^{45} + 240q^{47} + 366q^{51} - 2194q^{53} + 1384q^{55} - 504q^{57} - 1156q^{59} + 454q^{61} + 956q^{65} + 1064q^{67} - 24q^{69} + 1928q^{71} + 2082q^{73} - 213q^{75} - 1192q^{79} + 4941q^{81} + 2100q^{83} - 1588q^{85} - 1362q^{87} + 2722q^{89} - 204q^{93} + 328q^{95} - 1478q^{97} + 252q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1176))\) 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
1176.4.a.a \(1\) \(69.386\) \(\Q\) None \(0\) \(-3\) \(-14\) \(0\) \(-\) \(+\) \(-\) \(q-3q^{3}-14q^{5}+9q^{9}-28q^{11}+74q^{13}+\cdots\)
1176.4.a.b \(1\) \(69.386\) \(\Q\) None \(0\) \(-3\) \(-12\) \(0\) \(-\) \(+\) \(-\) \(q-3q^{3}-12q^{5}+9q^{9}-60q^{11}-44q^{13}+\cdots\)
1176.4.a.c \(1\) \(69.386\) \(\Q\) None \(0\) \(-3\) \(-11\) \(0\) \(+\) \(+\) \(+\) \(q-3q^{3}-11q^{5}+9q^{9}+39q^{11}-2^{5}q^{13}+\cdots\)
1176.4.a.d \(1\) \(69.386\) \(\Q\) None \(0\) \(-3\) \(2\) \(0\) \(-\) \(+\) \(+\) \(q-3q^{3}+2q^{5}+9q^{9}-18q^{11}+33q^{13}+\cdots\)
1176.4.a.e \(1\) \(69.386\) \(\Q\) None \(0\) \(-3\) \(2\) \(0\) \(-\) \(+\) \(-\) \(q-3q^{3}+2q^{5}+9q^{9}+52q^{11}-86q^{13}+\cdots\)
1176.4.a.f \(1\) \(69.386\) \(\Q\) None \(0\) \(-3\) \(7\) \(0\) \(-\) \(+\) \(+\) \(q-3q^{3}+7q^{5}+9q^{9}+7q^{11}-52q^{13}+\cdots\)
1176.4.a.g \(1\) \(69.386\) \(\Q\) None \(0\) \(-3\) \(10\) \(0\) \(+\) \(+\) \(-\) \(q-3q^{3}+10q^{5}+9q^{9}-52q^{11}+10q^{13}+\cdots\)
1176.4.a.h \(1\) \(69.386\) \(\Q\) None \(0\) \(-3\) \(16\) \(0\) \(-\) \(+\) \(-\) \(q-3q^{3}+2^{4}q^{5}+9q^{9}-18q^{11}+54q^{13}+\cdots\)
1176.4.a.i \(1\) \(69.386\) \(\Q\) None \(0\) \(3\) \(-7\) \(0\) \(-\) \(-\) \(-\) \(q+3q^{3}-7q^{5}+9q^{9}+7q^{11}+52q^{13}+\cdots\)
1176.4.a.j \(1\) \(69.386\) \(\Q\) None \(0\) \(3\) \(-4\) \(0\) \(-\) \(-\) \(-\) \(q+3q^{3}-4q^{5}+9q^{9}-26q^{11}-2q^{13}+\cdots\)
1176.4.a.k \(1\) \(69.386\) \(\Q\) None \(0\) \(3\) \(-2\) \(0\) \(-\) \(-\) \(-\) \(q+3q^{3}-2q^{5}+9q^{9}-18q^{11}-33q^{13}+\cdots\)
1176.4.a.l \(1\) \(69.386\) \(\Q\) None \(0\) \(3\) \(2\) \(0\) \(+\) \(-\) \(-\) \(q+3q^{3}+2q^{5}+9q^{9}+12q^{11}+66q^{13}+\cdots\)
1176.4.a.m \(1\) \(69.386\) \(\Q\) None \(0\) \(3\) \(10\) \(0\) \(-\) \(-\) \(-\) \(q+3q^{3}+10q^{5}+9q^{9}-12q^{11}-30q^{13}+\cdots\)
1176.4.a.n \(1\) \(69.386\) \(\Q\) None \(0\) \(3\) \(11\) \(0\) \(+\) \(-\) \(-\) \(q+3q^{3}+11q^{5}+9q^{9}+39q^{11}+2^{5}q^{13}+\cdots\)
1176.4.a.o \(1\) \(69.386\) \(\Q\) None \(0\) \(3\) \(12\) \(0\) \(-\) \(-\) \(-\) \(q+3q^{3}+12q^{5}+9q^{9}-60q^{11}+44q^{13}+\cdots\)
1176.4.a.p \(2\) \(69.386\) \(\Q(\sqrt{177}) \) None \(0\) \(-6\) \(-14\) \(0\) \(+\) \(+\) \(-\) \(q-3q^{3}+(-7-\beta )q^{5}+9q^{9}+(9-3\beta )q^{11}+\cdots\)
1176.4.a.q \(2\) \(69.386\) \(\Q(\sqrt{113}) \) None \(0\) \(-6\) \(-6\) \(0\) \(+\) \(+\) \(-\) \(q-3q^{3}+(-3-\beta )q^{5}+9q^{9}+(1+3\beta )q^{11}+\cdots\)
1176.4.a.r \(2\) \(69.386\) \(\Q(\sqrt{505}) \) None \(0\) \(-6\) \(9\) \(0\) \(+\) \(+\) \(+\) \(q-3q^{3}+(5-\beta )q^{5}+9q^{9}+(5-5\beta )q^{11}+\cdots\)
1176.4.a.s \(2\) \(69.386\) \(\Q(\sqrt{137}) \) None \(0\) \(-6\) \(18\) \(0\) \(-\) \(+\) \(-\) \(q-3q^{3}+(9-\beta )q^{5}+9q^{9}+(29+3\beta )q^{11}+\cdots\)
1176.4.a.t \(2\) \(69.386\) \(\Q(\sqrt{137}) \) None \(0\) \(6\) \(-18\) \(0\) \(-\) \(-\) \(-\) \(q+3q^{3}+(-9-\beta )q^{5}+9q^{9}+(29-3\beta )q^{11}+\cdots\)
1176.4.a.u \(2\) \(69.386\) \(\Q(\sqrt{505}) \) None \(0\) \(6\) \(-9\) \(0\) \(+\) \(-\) \(-\) \(q+3q^{3}+(-4-\beta )q^{5}+9q^{9}+5\beta q^{11}+\cdots\)
1176.4.a.v \(2\) \(69.386\) \(\Q(\sqrt{113}) \) None \(0\) \(6\) \(6\) \(0\) \(+\) \(-\) \(-\) \(q+3q^{3}+(3+\beta )q^{5}+9q^{9}+(1+3\beta )q^{11}+\cdots\)
1176.4.a.w \(2\) \(69.386\) \(\Q(\sqrt{337}) \) None \(0\) \(6\) \(6\) \(0\) \(+\) \(-\) \(-\) \(q+3q^{3}+(3+\beta )q^{5}+9q^{9}+(13+\beta )q^{11}+\cdots\)
1176.4.a.x \(3\) \(69.386\) 3.3.58461.1 None \(0\) \(-9\) \(11\) \(0\) \(+\) \(+\) \(-\) \(q-3q^{3}+(4+\beta _{1})q^{5}+9q^{9}+(-6+\beta _{1}+\cdots)q^{11}+\cdots\)
1176.4.a.y \(3\) \(69.386\) 3.3.58461.1 None \(0\) \(9\) \(-11\) \(0\) \(+\) \(-\) \(+\) \(q+3q^{3}+(-4-\beta _{1})q^{5}+9q^{9}+(-6+\cdots)q^{11}+\cdots\)
1176.4.a.z \(4\) \(69.386\) 4.4.391168.1 None \(0\) \(-12\) \(-8\) \(0\) \(-\) \(+\) \(+\) \(q-3q^{3}+(-2-\beta _{1})q^{5}+9q^{9}+(10+\cdots)q^{11}+\cdots\)
1176.4.a.ba \(4\) \(69.386\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-12\) \(-4\) \(0\) \(-\) \(+\) \(-\) \(q-3q^{3}+(-1-\beta _{1})q^{5}+9q^{9}+(3+2\beta _{1}+\cdots)q^{11}+\cdots\)
1176.4.a.bb \(4\) \(69.386\) 4.4.145408.2 None \(0\) \(-12\) \(8\) \(0\) \(+\) \(+\) \(+\) \(q-3q^{3}+(2+\beta _{2}+3\beta _{3})q^{5}+9q^{9}+\cdots\)
1176.4.a.bc \(4\) \(69.386\) 4.4.145408.2 None \(0\) \(12\) \(-8\) \(0\) \(+\) \(-\) \(+\) \(q+3q^{3}+(-2+\beta _{2}-3\beta _{3})q^{5}+9q^{9}+\cdots\)
1176.4.a.bd \(4\) \(69.386\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(12\) \(4\) \(0\) \(-\) \(-\) \(+\) \(q+3q^{3}+(1+\beta _{1})q^{5}+9q^{9}+(3+2\beta _{1}+\cdots)q^{11}+\cdots\)
1176.4.a.be \(4\) \(69.386\) 4.4.391168.1 None \(0\) \(12\) \(8\) \(0\) \(-\) \(-\) \(+\) \(q+3q^{3}+(2+\beta _{1})q^{5}+9q^{9}+(10-\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1176))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1176)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 2}\)