Properties

Label 672.4.a
Level $672$
Weight $4$
Character orbit 672.a
Rep. character $\chi_{672}(1,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $18$
Sturm bound $512$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(512\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(672))\).

Total New Old
Modular forms 400 36 364
Cusp forms 368 36 332
Eisenstein series 32 0 32

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.
\(+\)\(+\)\(+\)\(+\)\(5\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(4\)
Plus space\(+\)\(20\)
Minus space\(-\)\(16\)

Trace form

\( 36q + 8q^{5} + 324q^{9} + O(q^{10}) \) \( 36q + 8q^{5} + 324q^{9} - 184q^{13} - 312q^{17} + 1164q^{25} - 568q^{29} + 264q^{37} + 1416q^{41} + 72q^{45} + 1764q^{49} - 1144q^{53} - 312q^{61} + 1680q^{65} - 888q^{73} + 2916q^{81} - 1504q^{85} + 2984q^{89} + 912q^{93} + 7400q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(672))\) 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
672.4.a.a \(1\) \(39.649\) \(\Q\) None \(0\) \(-3\) \(-18\) \(7\) \(-\) \(+\) \(-\) \(q-3q^{3}-18q^{5}+7q^{7}+9q^{9}-44q^{11}+\cdots\)
672.4.a.b \(1\) \(39.649\) \(\Q\) None \(0\) \(-3\) \(6\) \(7\) \(+\) \(+\) \(-\) \(q-3q^{3}+6q^{5}+7q^{7}+9q^{9}-4q^{11}+\cdots\)
672.4.a.c \(1\) \(39.649\) \(\Q\) None \(0\) \(3\) \(-18\) \(-7\) \(+\) \(-\) \(+\) \(q+3q^{3}-18q^{5}-7q^{7}+9q^{9}+44q^{11}+\cdots\)
672.4.a.d \(1\) \(39.649\) \(\Q\) None \(0\) \(3\) \(6\) \(-7\) \(+\) \(-\) \(+\) \(q+3q^{3}+6q^{5}-7q^{7}+9q^{9}+4q^{11}+\cdots\)
672.4.a.e \(2\) \(39.649\) \(\Q(\sqrt{17}) \) None \(0\) \(-6\) \(-10\) \(-14\) \(+\) \(+\) \(+\) \(q-3q^{3}+(-5-\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
672.4.a.f \(2\) \(39.649\) \(\Q(\sqrt{37}) \) None \(0\) \(-6\) \(-4\) \(-14\) \(-\) \(+\) \(+\) \(q-3q^{3}+(-2-\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
672.4.a.g \(2\) \(39.649\) \(\Q(\sqrt{11}) \) None \(0\) \(-6\) \(8\) \(14\) \(-\) \(+\) \(-\) \(q-3q^{3}+(4+\beta )q^{5}+7q^{7}+9q^{9}+(22+\cdots)q^{11}+\cdots\)
672.4.a.h \(2\) \(39.649\) \(\Q(\sqrt{137}) \) None \(0\) \(-6\) \(10\) \(-14\) \(-\) \(+\) \(+\) \(q-3q^{3}+(5-\beta )q^{5}-7q^{7}+9q^{9}+(-11+\cdots)q^{11}+\cdots\)
672.4.a.i \(2\) \(39.649\) \(\Q(\sqrt{43}) \) None \(0\) \(-6\) \(16\) \(14\) \(-\) \(+\) \(-\) \(q-3q^{3}+(8+\beta )q^{5}+7q^{7}+9q^{9}+(2+\cdots)q^{11}+\cdots\)
672.4.a.j \(2\) \(39.649\) \(\Q(\sqrt{17}) \) None \(0\) \(6\) \(-10\) \(14\) \(-\) \(-\) \(-\) \(q+3q^{3}+(-5-\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
672.4.a.k \(2\) \(39.649\) \(\Q(\sqrt{37}) \) None \(0\) \(6\) \(-4\) \(14\) \(-\) \(-\) \(-\) \(q+3q^{3}+(-2-\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
672.4.a.l \(2\) \(39.649\) \(\Q(\sqrt{11}) \) None \(0\) \(6\) \(8\) \(-14\) \(+\) \(-\) \(+\) \(q+3q^{3}+(4+\beta )q^{5}-7q^{7}+9q^{9}+(-22+\cdots)q^{11}+\cdots\)
672.4.a.m \(2\) \(39.649\) \(\Q(\sqrt{137}) \) None \(0\) \(6\) \(10\) \(14\) \(+\) \(-\) \(-\) \(q+3q^{3}+(5-\beta )q^{5}+7q^{7}+9q^{9}+(11+\cdots)q^{11}+\cdots\)
672.4.a.n \(2\) \(39.649\) \(\Q(\sqrt{43}) \) None \(0\) \(6\) \(16\) \(-14\) \(-\) \(-\) \(+\) \(q+3q^{3}+(8+\beta )q^{5}-7q^{7}+9q^{9}+(-2+\cdots)q^{11}+\cdots\)
672.4.a.o \(3\) \(39.649\) 3.3.22700.1 None \(0\) \(-9\) \(-10\) \(21\) \(+\) \(+\) \(-\) \(q-3q^{3}+(-3-\beta _{2})q^{5}+7q^{7}+9q^{9}+\cdots\)
672.4.a.p \(3\) \(39.649\) 3.3.37341.1 None \(0\) \(-9\) \(6\) \(-21\) \(+\) \(+\) \(+\) \(q-3q^{3}+(2-\beta _{1})q^{5}-7q^{7}+9q^{9}+\cdots\)
672.4.a.q \(3\) \(39.649\) 3.3.22700.1 None \(0\) \(9\) \(-10\) \(-21\) \(-\) \(-\) \(+\) \(q+3q^{3}+(-3-\beta _{2})q^{5}-7q^{7}+9q^{9}+\cdots\)
672.4.a.r \(3\) \(39.649\) 3.3.37341.1 None \(0\) \(9\) \(6\) \(21\) \(+\) \(-\) \(-\) \(q+3q^{3}+(2-\beta _{1})q^{5}+7q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(672))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(672)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)