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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

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(672))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
672.4.a.a 672.a 1.a $1$ $39.649$ \(\Q\) None \(0\) \(-3\) \(-18\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-18q^{5}+7q^{7}+9q^{9}-44q^{11}+\cdots\)
672.4.a.b 672.a 1.a $1$ $39.649$ \(\Q\) None \(0\) \(-3\) \(6\) \(7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+6q^{5}+7q^{7}+9q^{9}-4q^{11}+\cdots\)
672.4.a.c 672.a 1.a $1$ $39.649$ \(\Q\) None \(0\) \(3\) \(-18\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-18q^{5}-7q^{7}+9q^{9}+44q^{11}+\cdots\)
672.4.a.d 672.a 1.a $1$ $39.649$ \(\Q\) None \(0\) \(3\) \(6\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+6q^{5}-7q^{7}+9q^{9}+4q^{11}+\cdots\)
672.4.a.e 672.a 1.a $2$ $39.649$ \(\Q(\sqrt{17}) \) None \(0\) \(-6\) \(-10\) \(-14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-5-\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
672.4.a.f 672.a 1.a $2$ $39.649$ \(\Q(\sqrt{37}) \) None \(0\) \(-6\) \(-4\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-2-\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
672.4.a.g 672.a 1.a $2$ $39.649$ \(\Q(\sqrt{11}) \) None \(0\) \(-6\) \(8\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(4+\beta )q^{5}+7q^{7}+9q^{9}+(22+\cdots)q^{11}+\cdots\)
672.4.a.h 672.a 1.a $2$ $39.649$ \(\Q(\sqrt{137}) \) None \(0\) \(-6\) \(10\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(5-\beta )q^{5}-7q^{7}+9q^{9}+(-11+\cdots)q^{11}+\cdots\)
672.4.a.i 672.a 1.a $2$ $39.649$ \(\Q(\sqrt{43}) \) None \(0\) \(-6\) \(16\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(8+\beta )q^{5}+7q^{7}+9q^{9}+(2+\cdots)q^{11}+\cdots\)
672.4.a.j 672.a 1.a $2$ $39.649$ \(\Q(\sqrt{17}) \) None \(0\) \(6\) \(-10\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-5-\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
672.4.a.k 672.a 1.a $2$ $39.649$ \(\Q(\sqrt{37}) \) None \(0\) \(6\) \(-4\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-2-\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
672.4.a.l 672.a 1.a $2$ $39.649$ \(\Q(\sqrt{11}) \) None \(0\) \(6\) \(8\) \(-14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(4+\beta )q^{5}-7q^{7}+9q^{9}+(-22+\cdots)q^{11}+\cdots\)
672.4.a.m 672.a 1.a $2$ $39.649$ \(\Q(\sqrt{137}) \) None \(0\) \(6\) \(10\) \(14\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(5-\beta )q^{5}+7q^{7}+9q^{9}+(11+\cdots)q^{11}+\cdots\)
672.4.a.n 672.a 1.a $2$ $39.649$ \(\Q(\sqrt{43}) \) None \(0\) \(6\) \(16\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(8+\beta )q^{5}-7q^{7}+9q^{9}+(-2+\cdots)q^{11}+\cdots\)
672.4.a.o 672.a 1.a $3$ $39.649$ 3.3.22700.1 None \(0\) \(-9\) \(-10\) \(21\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-3-\beta _{2})q^{5}+7q^{7}+9q^{9}+\cdots\)
672.4.a.p 672.a 1.a $3$ $39.649$ 3.3.37341.1 None \(0\) \(-9\) \(6\) \(-21\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(2-\beta _{1})q^{5}-7q^{7}+9q^{9}+\cdots\)
672.4.a.q 672.a 1.a $3$ $39.649$ 3.3.22700.1 None \(0\) \(9\) \(-10\) \(-21\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-3-\beta _{2})q^{5}-7q^{7}+9q^{9}+\cdots\)
672.4.a.r 672.a 1.a $3$ $39.649$ 3.3.37341.1 None \(0\) \(9\) \(6\) \(21\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(2-\beta _{1})q^{5}+7q^{7}+9q^{9}+\cdots\)

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

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