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Properties

Label 48.26.a
Level $48$
Weight $26$
Character orbit 48.a
Rep. character $\chi_{48}(1,\cdot)$
Character field $\Q$
Dimension $25$
Newform subspaces $11$
Sturm bound $208$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(208\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{26}(\Gamma_0(48))\).

Total New Old
Modular forms 206 25 181
Cusp forms 194 25 169
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
\(+\)\(+\)$+$\(6\)
\(+\)\(-\)$-$\(7\)
\(-\)\(+\)$-$\(6\)
\(-\)\(-\)$+$\(6\)
Plus space\(+\)\(12\)
Minus space\(-\)\(13\)

Trace form

\( 25 q + 531441 q^{3} - 306268562 q^{5} - 63411416640 q^{7} + 7060738412025 q^{9} + O(q^{10}) \) \( 25 q + 531441 q^{3} - 306268562 q^{5} - 63411416640 q^{7} + 7060738412025 q^{9} + 14000777409596 q^{11} - 71041880304722 q^{13} - 259492675781250 q^{15} + 1186009327105202 q^{17} - 2578604427051452 q^{19} - 210628999613059192 q^{23} + 1274805191542732103 q^{25} + 150094635296999121 q^{27} + 43681834572009846 q^{29} - 8054758362722135624 q^{31} - 3049957046009763684 q^{33} + 82695663023527612800 q^{35} + 64673679591921712566 q^{37} + 10854313217460675198 q^{39} + 63829338646905784362 q^{41} + 450874508564631051836 q^{43} - 86499288004362410322 q^{45} + 2339689137142659822720 q^{47} + 3405064238448957852305 q^{49} - 383572863745541182494 q^{51} + 6944426420025337294318 q^{53} + 2819919639420398251016 q^{55} - 3604454647232349033756 q^{57} - 17410891012837930430788 q^{59} - 1087710830380529600210 q^{61} - 17909257009238770443840 q^{63} - 17555063407355653294748 q^{65} + 84198024120459583657924 q^{67} - 98171499459672143695800 q^{69} - 305471954924433131163272 q^{71} - 400647608020398642315590 q^{73} - 115481336846373088045713 q^{75} + 97098828148653745927680 q^{77} + 1373253171665092982781128 q^{79} + 1994161076921812746584025 q^{81} - 1341605711179509990632636 q^{83} + 2135095860447835007773148 q^{85} - 571038370718366720120634 q^{87} + 3618649100407410002267562 q^{89} + 3790565764657602294859392 q^{91} + 191566278774054478404792 q^{93} + 5166708997028552715559672 q^{95} - 3500882872836870493052878 q^{97} + 3954233074165854161471676 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{26}^{\mathrm{new}}(\Gamma_0(48))\) 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
48.26.a.a 48.a 1.a $1$ $190.078$ \(\Q\) None \(0\) \(-531441\) \(590425734\) \(-57857417576\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3^{12}q^{3}+590425734q^{5}-57857417576q^{7}+\cdots\)
48.26.a.b 48.a 1.a $1$ $190.078$ \(\Q\) None \(0\) \(531441\) \(-799327650\) \(-7962409664\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{12}q^{3}-799327650q^{5}-7962409664q^{7}+\cdots\)
48.26.a.c 48.a 1.a $1$ $190.078$ \(\Q\) None \(0\) \(531441\) \(-292754850\) \(-3580644032\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{12}q^{3}-292754850q^{5}-3580644032q^{7}+\cdots\)
48.26.a.d 48.a 1.a $2$ $190.078$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(0\) \(-1062882\) \(500431500\) \(24013399472\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3^{12}q^{3}+(250215750-5\beta )q^{5}+\cdots\)
48.26.a.e 48.a 1.a $2$ $190.078$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(0\) \(1062882\) \(198560700\) \(-50461213312\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{12}q^{3}+(99280350-\beta )q^{5}+(-25230606656+\cdots)q^{7}+\cdots\)
48.26.a.f 48.a 1.a $2$ $190.078$ \(\Q(\sqrt{1287001}) \) None \(0\) \(1062882\) \(570861756\) \(29687385728\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{12}q^{3}+(285430878-179\beta )q^{5}+\cdots\)
48.26.a.g 48.a 1.a $3$ $190.078$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(-1594323\) \(-560960334\) \(-27161710296\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3^{12}q^{3}+(-186986778-\beta _{2})q^{5}+\cdots\)
48.26.a.h 48.a 1.a $3$ $190.078$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(-1594323\) \(-275737806\) \(19677447336\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3^{12}q^{3}+(-91912602-\beta _{1})q^{5}+\cdots\)
48.26.a.i 48.a 1.a $3$ $190.078$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(-1594323\) \(-163152750\) \(9622572744\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3^{12}q^{3}+(-54384250+3\beta _{1}+\beta _{2})q^{5}+\cdots\)
48.26.a.j 48.a 1.a $3$ $190.078$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(1594323\) \(-341050950\) \(1095757536\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3^{12}q^{3}+(-113683650+\beta _{1})q^{5}+\cdots\)
48.26.a.k 48.a 1.a $4$ $190.078$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(2125764\) \(266436088\) \(-484584576\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3^{12}q^{3}+(66609022+\beta _{1})q^{5}+(-121146144+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{26}^{\mathrm{old}}(\Gamma_0(48))\) into lower level spaces

\( S_{26}^{\mathrm{old}}(\Gamma_0(48)) \cong \) \(S_{26}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 10}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 5}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)