Properties

Label 1764.4.k
Level $1764$
Weight $4$
Character orbit 1764.k
Rep. character $\chi_{1764}(361,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $100$
Newform subspaces $30$
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.k (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 30 \)
Sturm bound: \(1344\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1764, [\chi])\).

Total New Old
Modular forms 2112 100 2012
Cusp forms 1920 100 1820
Eisenstein series 192 0 192

Trace form

\( 100q + 6q^{5} + O(q^{10}) \) \( 100q + 6q^{5} + 20q^{11} + 30q^{17} - 40q^{19} + 40q^{23} - 1100q^{25} - 224q^{29} + 168q^{31} + 50q^{37} - 816q^{41} - 488q^{43} - 48q^{47} + 86q^{53} + 656q^{55} + 1152q^{59} - 918q^{61} + 1764q^{65} - 808q^{67} + 2856q^{71} - 1262q^{73} - 1448q^{79} - 2832q^{83} - 244q^{85} + 858q^{89} - 3664q^{95} + 976q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1764, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1764.4.k.a \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-20\) \(0\) \(q-20\zeta_{6}q^{5}+(44-44\zeta_{6})q^{11}-44q^{13}+\cdots\)
1764.4.k.b \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-18\) \(0\) \(q-18\zeta_{6}q^{5}+(6^{2}-6^{2}\zeta_{6})q^{11}-10q^{13}+\cdots\)
1764.4.k.c \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-14\) \(0\) \(q-14\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}-54q^{13}+\cdots\)
1764.4.k.d \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-8\) \(0\) \(q-8\zeta_{6}q^{5}+(-40+40\zeta_{6})q^{11}-12q^{13}+\cdots\)
1764.4.k.e \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-6\) \(0\) \(q-6\zeta_{6}q^{5}+(-12+12\zeta_{6})q^{11}+82q^{13}+\cdots\)
1764.4.k.f \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-6\) \(0\) \(q-6\zeta_{6}q^{5}+(6^{2}-6^{2}\zeta_{6})q^{11}-62q^{13}+\cdots\)
1764.4.k.g \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-4\) \(0\) \(q-4\zeta_{6}q^{5}+(-20+20\zeta_{6})q^{11}+4q^{13}+\cdots\)
1764.4.k.h \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q-89q^{13}-163\zeta_{6}q^{19}+(5^{3}-5^{3}\zeta_{6})q^{25}+\cdots\)
1764.4.k.i \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+19q^{13}+107\zeta_{6}q^{19}+(5^{3}-5^{3}\zeta_{6})q^{25}+\cdots\)
1764.4.k.j \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(4\) \(0\) \(q+4\zeta_{6}q^{5}+(-20+20\zeta_{6})q^{11}-4q^{13}+\cdots\)
1764.4.k.k \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(6\) \(0\) \(q+6\zeta_{6}q^{5}+(-12+12\zeta_{6})q^{11}-82q^{13}+\cdots\)
1764.4.k.l \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(6\) \(0\) \(q+6\zeta_{6}q^{5}+(6^{2}-6^{2}\zeta_{6})q^{11}+62q^{13}+\cdots\)
1764.4.k.m \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(8\) \(0\) \(q+8\zeta_{6}q^{5}+(-40+40\zeta_{6})q^{11}+12q^{13}+\cdots\)
1764.4.k.n \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(14\) \(0\) \(q+14\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+54q^{13}+\cdots\)
1764.4.k.o \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(18\) \(0\) \(q+18\zeta_{6}q^{5}+(6^{2}-6^{2}\zeta_{6})q^{11}+10q^{13}+\cdots\)
1764.4.k.p \(2\) \(104.079\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(20\) \(0\) \(q+20\zeta_{6}q^{5}+(44-44\zeta_{6})q^{11}+44q^{13}+\cdots\)
1764.4.k.q \(4\) \(104.079\) \(\Q(\sqrt{-3}, \sqrt{193})\) None \(0\) \(0\) \(-11\) \(0\) \(q+(\beta _{1}-6\beta _{2})q^{5}+(1-7\beta _{1}+6\beta _{2}-7\beta _{3})q^{11}+\cdots\)
1764.4.k.r \(4\) \(104.079\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+7\beta _{1}q^{5}+28\beta _{2}q^{11}+3\beta _{3}q^{13}+\cdots\)
1764.4.k.s \(4\) \(104.079\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+2\beta _{1}q^{5}+26\beta _{2}q^{11}-24\beta _{3}q^{13}+\cdots\)
1764.4.k.t \(4\) \(104.079\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}+(13\beta _{1}+13\beta _{3})q^{11}-30q^{13}+\cdots\)
1764.4.k.u \(4\) \(104.079\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}+(\beta _{1}+\beta _{3})q^{11}-26q^{13}+\cdots\)
1764.4.k.v \(4\) \(104.079\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}+(-\beta _{1}-\beta _{3})q^{11}+26q^{13}+\cdots\)
1764.4.k.w \(4\) \(104.079\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}+(-13\beta _{1}-13\beta _{3})q^{11}+30q^{13}+\cdots\)
1764.4.k.x \(4\) \(104.079\) \(\Q(\sqrt{-3}, \sqrt{385})\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{5}+(\beta _{2}+\beta _{3})q^{11}+54q^{13}+\cdots\)
1764.4.k.y \(4\) \(104.079\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+7\beta _{1}q^{5}-28\beta _{2}q^{11}-3\beta _{3}q^{13}+\cdots\)
1764.4.k.z \(4\) \(104.079\) \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(0\) \(0\) \(3\) \(0\) \(q+(1+\beta _{1}+\beta _{3})q^{5}+(-1+5^{2}\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
1764.4.k.ba \(4\) \(104.079\) \(\Q(\sqrt{-3}, \sqrt{37})\) None \(0\) \(0\) \(14\) \(0\) \(q+(7\beta _{1}+2\beta _{2})q^{5}+(2^{4}-2^{4}\beta _{1}+7\beta _{2}+\cdots)q^{11}+\cdots\)
1764.4.k.bb \(8\) \(104.079\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{6}q^{5}+(-3\beta _{2}-\beta _{4})q^{11}+(3\beta _{4}+\cdots)q^{13}+\cdots\)
1764.4.k.bc \(8\) \(104.079\) 8.0.\(\cdots\).42 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{5}q^{5}+(\beta _{2}+\beta _{3})q^{11}+\beta _{6}q^{13}+\cdots\)
1764.4.k.bd \(8\) \(104.079\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{6}q^{5}+(-3\beta _{2}-\beta _{4})q^{11}+(-3\beta _{4}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1764, [\chi])\) into lower level spaces

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