Properties

Label 1764.4.a
Level $1764$
Weight $4$
Character orbit 1764.a
Rep. character $\chi_{1764}(1,\cdot)$
Character field $\Q$
Dimension $51$
Newform subspaces $29$
Sturm bound $1344$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 29 \)
Sturm bound: \(1344\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1764))\).

Total New Old
Modular forms 1056 51 1005
Cusp forms 960 51 909
Eisenstein series 96 0 96

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.
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(12\)
\(-\)\(-\)\(+\)\(+\)\(16\)
\(-\)\(-\)\(-\)\(-\)\(15\)
Plus space\(+\)\(28\)
Minus space\(-\)\(23\)

Trace form

\( 51q + O(q^{10}) \) \( 51q + 28q^{11} - 4q^{13} + 30q^{17} + 54q^{19} + 56q^{23} + 1341q^{25} - 214q^{29} + 68q^{31} - 486q^{37} - 162q^{41} - 44q^{43} - 492q^{47} + 298q^{53} + 352q^{55} + 162q^{59} + 512q^{61} + 408q^{65} - 372q^{67} - 480q^{71} - 674q^{73} + 112q^{79} - 798q^{83} + 1948q^{85} - 2862q^{89} + 5224q^{95} + 1258q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1764))\) 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
1764.4.a.a \(1\) \(104.079\) \(\Q\) None \(0\) \(0\) \(-20\) \(0\) \(-\) \(-\) \(-\) \(q-20q^{5}-44q^{11}+44q^{13}+72q^{17}+\cdots\)
1764.4.a.b \(1\) \(104.079\) \(\Q\) None \(0\) \(0\) \(-18\) \(0\) \(-\) \(-\) \(-\) \(q-18q^{5}-6^{2}q^{11}+10q^{13}+18q^{17}+\cdots\)
1764.4.a.c \(1\) \(104.079\) \(\Q\) None \(0\) \(0\) \(-8\) \(0\) \(-\) \(-\) \(-\) \(q-8q^{5}+40q^{11}+12q^{13}-58q^{17}+\cdots\)
1764.4.a.d \(1\) \(104.079\) \(\Q\) None \(0\) \(0\) \(-4\) \(0\) \(-\) \(-\) \(-\) \(q-4q^{5}+20q^{11}-4q^{13}-24q^{17}+\cdots\)
1764.4.a.e \(1\) \(104.079\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-89q^{13}+163q^{19}-5^{3}q^{25}+19q^{31}+\cdots\)
1764.4.a.f \(1\) \(104.079\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-19q^{13}+107q^{19}-5^{3}q^{25}-17^{2}q^{31}+\cdots\)
1764.4.a.g \(1\) \(104.079\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+19q^{13}-107q^{19}-5^{3}q^{25}+17^{2}q^{31}+\cdots\)
1764.4.a.h \(1\) \(104.079\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+89q^{13}-163q^{19}-5^{3}q^{25}-19q^{31}+\cdots\)
1764.4.a.i \(1\) \(104.079\) \(\Q\) None \(0\) \(0\) \(4\) \(0\) \(-\) \(-\) \(-\) \(q+4q^{5}+20q^{11}+4q^{13}+24q^{17}+\cdots\)
1764.4.a.j \(1\) \(104.079\) \(\Q\) None \(0\) \(0\) \(6\) \(0\) \(-\) \(-\) \(-\) \(q+6q^{5}-6^{2}q^{11}-62q^{13}+114q^{17}+\cdots\)
1764.4.a.k \(1\) \(104.079\) \(\Q\) None \(0\) \(0\) \(6\) \(0\) \(-\) \(-\) \(-\) \(q+6q^{5}+12q^{11}+82q^{13}-30q^{17}+\cdots\)
1764.4.a.l \(1\) \(104.079\) \(\Q\) None \(0\) \(0\) \(14\) \(0\) \(-\) \(-\) \(-\) \(q+14q^{5}-4q^{11}-54q^{13}-14q^{17}+\cdots\)
1764.4.a.m \(1\) \(104.079\) \(\Q\) None \(0\) \(0\) \(20\) \(0\) \(-\) \(-\) \(-\) \(q+20q^{5}-44q^{11}-44q^{13}-72q^{17}+\cdots\)
1764.4.a.n \(2\) \(104.079\) \(\Q(\sqrt{37}) \) None \(0\) \(0\) \(-14\) \(0\) \(-\) \(-\) \(-\) \(q+(-7-2\beta )q^{5}+(-2^{4}+7\beta )q^{11}+\cdots\)
1764.4.a.o \(2\) \(104.079\) \(\Q(\sqrt{193}) \) None \(0\) \(0\) \(-11\) \(0\) \(-\) \(-\) \(+\) \(q+(-5-\beta )q^{5}+(-1+7\beta )q^{11}+5\beta q^{13}+\cdots\)
1764.4.a.p \(2\) \(104.079\) \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(-3\) \(0\) \(-\) \(-\) \(-\) \(q+(-2-\beta )q^{5}+(26+\beta )q^{11}+(-33+\cdots)q^{13}+\cdots\)
1764.4.a.q \(2\) \(104.079\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+7\beta q^{5}-28q^{11}-3\beta q^{13}-35\beta q^{17}+\cdots\)
1764.4.a.r \(2\) \(104.079\) \(\Q(\sqrt{385}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q-\beta q^{5}+\beta q^{11}-54q^{13}-2\beta q^{17}+\cdots\)
1764.4.a.s \(2\) \(104.079\) \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+\beta q^{5}-\beta q^{11}-26q^{13}-5\beta q^{17}+\cdots\)
1764.4.a.t \(2\) \(104.079\) \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+\beta q^{5}+13\beta q^{11}+30q^{13}+23\beta q^{17}+\cdots\)
1764.4.a.u \(2\) \(104.079\) \(\Q(\sqrt{385}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-\beta q^{5}-\beta q^{11}+54q^{13}-2\beta q^{17}+\cdots\)
1764.4.a.v \(2\) \(104.079\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+2\beta q^{5}+26q^{11}-24\beta q^{13}-73\beta q^{17}+\cdots\)
1764.4.a.w \(2\) \(104.079\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+7\beta q^{5}+28q^{11}+3\beta q^{13}-35\beta q^{17}+\cdots\)
1764.4.a.x \(2\) \(104.079\) \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(3\) \(0\) \(-\) \(-\) \(+\) \(q+(2+\beta )q^{5}+(26+\beta )q^{11}+(33+5\beta )q^{13}+\cdots\)
1764.4.a.y \(2\) \(104.079\) \(\Q(\sqrt{193}) \) None \(0\) \(0\) \(11\) \(0\) \(-\) \(-\) \(-\) \(q+(6-\beta )q^{5}+(6-7\beta )q^{11}+(-5+5\beta )q^{13}+\cdots\)
1764.4.a.z \(2\) \(104.079\) \(\Q(\sqrt{37}) \) None \(0\) \(0\) \(14\) \(0\) \(-\) \(-\) \(+\) \(q+(7+2\beta )q^{5}+(-2^{4}+7\beta )q^{11}+(-14+\cdots)q^{13}+\cdots\)
1764.4.a.ba \(4\) \(104.079\) 4.4.136768.1 None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q-\beta _{1}q^{5}+(-3\beta _{1}+\beta _{3})q^{11}-3\beta _{3}q^{13}+\cdots\)
1764.4.a.bb \(4\) \(104.079\) \(\Q(\sqrt{7}, \sqrt{109})\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q+\beta _{1}q^{5}+\beta _{3}q^{11}-\beta _{2}q^{13}+9\beta _{1}q^{17}+\cdots\)
1764.4.a.bc \(4\) \(104.079\) 4.4.136768.1 None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+\beta _{1}q^{5}+(-3\beta _{1}+\beta _{3})q^{11}+3\beta _{3}q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1764))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1764)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(882))\)\(^{\oplus 2}\)