Properties

Label 48.28.a
Level $48$
Weight $28$
Character orbit 48.a
Rep. character $\chi_{48}(1,\cdot)$
Character field $\Q$
Dimension $27$
Newform subspaces $12$
Sturm bound $224$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(224\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{28}(\Gamma_0(48))\).

Total New Old
Modular forms 222 27 195
Cusp forms 210 27 183
Eisenstein series 12 0 12

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

\(2\)\(3\)FrickeDim
\(+\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(7\)
\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(14\)
Minus space\(-\)\(13\)

Trace form

\( 27 q - 1594323 q^{3} + 2726446322 q^{5} + 314746286432 q^{7} + 68630377364883 q^{9} + O(q^{10}) \) \( 27 q - 1594323 q^{3} + 2726446322 q^{5} + 314746286432 q^{7} + 68630377364883 q^{9} + 34039821810220 q^{11} + 182714776311554 q^{13} + 3892390136718750 q^{15} - 19308849445239146 q^{17} - 283193242114551756 q^{19} - 3605769732956008568 q^{23} + 43907705843941389181 q^{25} - 4052555153018976267 q^{27} + 14350007589432084378 q^{29} + 60631848978988210008 q^{31} - 166006103446804034412 q^{33} - 2200593807907526101824 q^{35} - 594282631082432506262 q^{37} + 5291124453625179187566 q^{39} - 5197757424407616932658 q^{41} - 2603643070315236988244 q^{43} + 6930260738665085455938 q^{45} - 26204361341315254001856 q^{47} + 303169414744699057590035 q^{49} - 121466095992424097802054 q^{51} - 348758202622664087167918 q^{53} - 1103196419396213625191992 q^{55} - 10999854261479163444084 q^{57} + 2070270006997691323192492 q^{59} + 1175145341542792571930978 q^{61} + 800042830074952373932128 q^{63} + 7779450420595273317336812 q^{65} - 15823692246889887899185740 q^{67} + 5870296284113552631371064 q^{69} - 9874520673380575154228872 q^{71} + 28652629739754323779058366 q^{73} - 23029472354760297274542597 q^{75} - 69207851784751579601532672 q^{77} - 12711573738864585050097560 q^{79} + 174449211009120179071170507 q^{81} + 234738996402292760681908148 q^{83} + 47705931444117881168882468 q^{85} - 158203708017368228601121674 q^{87} - 179257875026818440382974354 q^{89} + 213436952254568248915641664 q^{91} - 207847131476128799145978744 q^{93} + 202959728274321026886720632 q^{95} - 34267114580160548819753994 q^{97} + 86524659861806420537722380 q^{99} + O(q^{100}) \)

Decomposition of \(S_{28}^{\mathrm{new}}(\Gamma_0(48))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3
48.28.a.a 48.a 1.a $1$ $221.691$ \(\Q\) None 6.28.a.c \(0\) \(-1594323\) \(1220703150\) \(-96889207016\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3^{13}q^{3}+1220703150q^{5}-96889207016q^{7}+\cdots\)
48.28.a.b 48.a 1.a $1$ $221.691$ \(\Q\) None 6.28.a.a \(0\) \(-1594323\) \(1992850350\) \(321751224088\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3^{13}q^{3}+1992850350q^{5}+321751224088q^{7}+\cdots\)
48.28.a.c 48.a 1.a $1$ $221.691$ \(\Q\) None 6.28.a.b \(0\) \(1594323\) \(2904255750\) \(493494294832\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{13}q^{3}+2904255750q^{5}+493494294832q^{7}+\cdots\)
48.28.a.d 48.a 1.a $2$ $221.691$ \(\Q(\sqrt{6469}) \) None 3.28.a.a \(0\) \(-3188646\) \(-4906065060\) \(151657089584\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3^{13}q^{3}+(-2453032530-12265\beta )q^{5}+\cdots\)
48.28.a.e 48.a 1.a $2$ $221.691$ \(\Q(\sqrt{30001}) \) None 3.28.a.b \(0\) \(3188646\) \(-1771946100\) \(-369665199904\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{13}q^{3}+(-885973050-275\beta )q^{5}+\cdots\)
48.28.a.f 48.a 1.a $2$ $221.691$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None 12.28.a.a \(0\) \(3188646\) \(-1503115380\) \(127562115296\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{13}q^{3}+(-751557690-11\beta )q^{5}+\cdots\)
48.28.a.g 48.a 1.a $2$ $221.691$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None 6.28.a.d \(0\) \(3188646\) \(291441036\) \(-121646295328\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{13}q^{3}+(145720518-7\beta )q^{5}+\cdots\)
48.28.a.h 48.a 1.a $3$ $221.691$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 24.28.a.c \(0\) \(-4782969\) \(-3653328150\) \(-100803674328\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3^{13}q^{3}+(-1217776050-5^{2}\beta _{1}+\cdots)q^{5}+\cdots\)
48.28.a.i 48.a 1.a $3$ $221.691$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 12.28.a.b \(0\) \(-4782969\) \(539309322\) \(-206592020664\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3^{13}q^{3}+(179769774-\beta _{1})q^{5}+\cdots\)
48.28.a.j 48.a 1.a $3$ $221.691$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 24.28.a.a \(0\) \(4782969\) \(1164343410\) \(-199831078608\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3^{13}q^{3}+(388114470+\beta _{1})q^{5}+\cdots\)
48.28.a.k 48.a 1.a $3$ $221.691$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 24.28.a.b \(0\) \(4782969\) \(1498947570\) \(227459306928\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3^{13}q^{3}+(499649190-\beta _{1})q^{5}+\cdots\)
48.28.a.l 48.a 1.a $4$ $221.691$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 24.28.a.d \(0\) \(-6377292\) \(4949050424\) \(88249731552\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3^{13}q^{3}+(1237262606-\beta _{1})q^{5}+\cdots\)

Decomposition of \(S_{28}^{\mathrm{old}}(\Gamma_0(48))\) into lower level spaces

\( S_{28}^{\mathrm{old}}(\Gamma_0(48)) \simeq \) \(S_{28}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 10}\)\(\oplus\)\(S_{28}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{28}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 5}\)\(\oplus\)\(S_{28}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{28}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{28}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{28}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{28}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{28}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)