Properties

Label 608.4.a
Level $608$
Weight $4$
Character orbit 608.a
Rep. character $\chi_{608}(1,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $12$
Sturm bound $320$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 608.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(320\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(608))\).

Total New Old
Modular forms 248 54 194
Cusp forms 232 54 178
Eisenstein series 16 0 16

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(19\)FrickeDim
\(+\)\(+\)$+$\(15\)
\(+\)\(-\)$-$\(12\)
\(-\)\(+\)$-$\(12\)
\(-\)\(-\)$+$\(15\)
Plus space\(+\)\(30\)
Minus space\(-\)\(24\)

Trace form

\( 54 q - 4 q^{5} + 486 q^{9} + O(q^{10}) \) \( 54 q - 4 q^{5} + 486 q^{9} + 92 q^{13} + 156 q^{17} - 272 q^{21} + 882 q^{25} + 396 q^{29} + 48 q^{33} - 660 q^{37} - 708 q^{41} + 2668 q^{45} + 3462 q^{49} + 1388 q^{53} + 780 q^{61} - 840 q^{65} - 1536 q^{69} + 2412 q^{73} + 440 q^{77} + 8070 q^{81} + 4352 q^{85} + 2412 q^{89} - 4720 q^{93} + 1740 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(608))\) 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 19
608.4.a.a 608.a 1.a $1$ $35.873$ \(\Q\) None \(0\) \(-1\) \(-8\) \(-17\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-8q^{5}-17q^{7}-26q^{9}-70q^{11}+\cdots\)
608.4.a.b 608.a 1.a $1$ $35.873$ \(\Q\) None \(0\) \(1\) \(-8\) \(17\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-8q^{5}+17q^{7}-26q^{9}+70q^{11}+\cdots\)
608.4.a.c 608.a 1.a $2$ $35.873$ \(\Q(\sqrt{93}) \) None \(0\) \(-2\) \(-4\) \(44\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(-2-2\beta )q^{5}+(22+\beta )q^{7}+\cdots\)
608.4.a.d 608.a 1.a $2$ $35.873$ \(\Q(\sqrt{93}) \) None \(0\) \(2\) \(-4\) \(-44\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-2-2\beta )q^{5}+(-22-\beta )q^{7}+\cdots\)
608.4.a.e 608.a 1.a $5$ $35.873$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-6\) \(5\) \(7\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}+(1-\beta _{2}+\beta _{3})q^{5}+\cdots\)
608.4.a.f 608.a 1.a $5$ $35.873$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-3\) \(27\) \(20\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{3})q^{3}+(5-\beta _{1})q^{5}+(5-\beta _{1}+\cdots)q^{7}+\cdots\)
608.4.a.g 608.a 1.a $5$ $35.873$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(3\) \(27\) \(-20\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{3})q^{3}+(5-\beta _{1})q^{5}+(-5+\beta _{1}+\cdots)q^{7}+\cdots\)
608.4.a.h 608.a 1.a $5$ $35.873$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(6\) \(5\) \(-7\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}+(1-\beta _{2}+\beta _{3})q^{5}+(-1+\cdots)q^{7}+\cdots\)
608.4.a.i 608.a 1.a $7$ $35.873$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-9\) \(-5\) \(-28\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{3}+(-1+\beta _{5})q^{5}+(-4+\cdots)q^{7}+\cdots\)
608.4.a.j 608.a 1.a $7$ $35.873$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-3\) \(-17\) \(-42\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-2-\beta _{3})q^{5}+(-6+\beta _{2}+\cdots)q^{7}+\cdots\)
608.4.a.k 608.a 1.a $7$ $35.873$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(3\) \(-17\) \(42\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-2-\beta _{3})q^{5}+(6-\beta _{2}+\beta _{5}+\cdots)q^{7}+\cdots\)
608.4.a.l 608.a 1.a $7$ $35.873$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(9\) \(-5\) \(28\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+(-1+\beta _{5})q^{5}+(4+\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(608))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(608)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 2}\)