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Properties

Label 16.48.a
Level $16$
Weight $48$
Character orbit 16.a
Rep. character $\chi_{16}(1,\cdot)$
Character field $\Q$
Dimension $23$
Newform subspaces $6$
Sturm bound $96$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 48 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{48}(\Gamma_0(16))\).

Total New Old
Modular forms 97 24 73
Cusp forms 91 23 68
Eisenstein series 6 1 5

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

\(2\)Dim
\(+\)\(12\)
\(-\)\(11\)

Trace form

\( 23 q + 94143178828 q^{3} + 13063095838313538 q^{5} - 98284592784472971016 q^{7} + 213132560493416036629459 q^{9} + O(q^{10}) \) \( 23 q + 94143178828 q^{3} + 13063095838313538 q^{5} - 98284592784472971016 q^{7} + 213132560493416036629459 q^{9} + 1461151866018050437432452 q^{11} + 44844593209717838242931722 q^{13} + 5082864689309761311347331112 q^{15} - 37921963394608935092945611074 q^{17} + 1383900104945454276673627033148 q^{19} + 6442770801261087379322216949600 q^{21} + 19800399242294138908048440650088 q^{23} + 3633073046879264936987822819848049 q^{25} - 8937729443551685440342552433986760 q^{27} - 5376259732907861563059402947806182 q^{29} - 176784696304024185801368049298913056 q^{31} - 642122070944104119265645688803024816 q^{33} + 1278082237611466678695142528731174288 q^{35} + 2835998750506926165721522361233230946 q^{37} - 88316723824979138206915224517803853880 q^{39} - 46076242037062523744145631439018681130 q^{41} + 668988354161550122133826696535935218788 q^{43} + 742052204486566494046562428023938735242 q^{45} + 3609522635447635335059835946518684035664 q^{47} + 14899347136267501797682140016783786339791 q^{49} - 14982046869177524271489310315271850120232 q^{51} - 22902647515953494969296988935502957774478 q^{53} + 196387249649643558529822949433395265385272 q^{55} - 245358434724275222404236175177680516122960 q^{57} + 495867317165030717761019097437640737310036 q^{59} + 181534683347947768800560476494017501492794 q^{61} - 2659360929324967703463943150023770808922152 q^{63} - 2161209037737044215470115497898902854157684 q^{65} + 3243649800883396742290538710769592094118924 q^{67} + 3115210349365092234175493529812817021720608 q^{69} + 64033041808239277153413814508372701603421880 q^{71} - 83230607270438025139551735959185457153402378 q^{73} - 25871928611174089828448728480747162246317484 q^{75} + 413361502483828366498981686314176300203354912 q^{77} + 13336587451931898683609731332192541389784880 q^{79} + 2634255080278205642123733363101120569063220911 q^{81} + 4314773962022408918463387381623361306374205948 q^{83} - 359836030026899479382680183169870106283432860 q^{85} - 3405549717053152967619093582511389904739733240 q^{87} + 3058139401280398148252178171787377597205071750 q^{89} - 5018305109188514644851698590372727454052372016 q^{91} - 1160333151758765235906881873097915942861209216 q^{93} - 19760123712391969409612542553811109297399098552 q^{95} - 9363777488742546430993561438680108381592885874 q^{97} + 435786508572965749286360049709213137883661505812 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{48}^{\mathrm{new}}(\Gamma_0(16))\) 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
16.48.a.a 16.a 1.a $1$ $223.852$ \(\Q\) None \(0\) \(196634580372\) \(20\!\cdots\!50\) \(51\!\cdots\!96\) $-$ $\mathrm{SU}(2)$ \(q+196634580372q^{3}+20669962168980750q^{5}+\cdots\)
16.48.a.b 16.a 1.a $2$ $223.852$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(0\) \(-122289844824\) \(18\!\cdots\!40\) \(-16\!\cdots\!32\) $-$ $\mathrm{SU}(2)$ \(q+(-61144922412-5\beta )q^{3}+(9089248919597070+\cdots)q^{5}+\cdots\)
16.48.a.c 16.a 1.a $4$ $223.852$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-127448165040\) \(-23\!\cdots\!00\) \(90\!\cdots\!20\) $-$ $\mathrm{SU}(2)$ \(q+(-31862041260+\beta _{1})q^{3}+\cdots\)
16.48.a.d 16.a 1.a $4$ $223.852$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-38461494960\) \(-31\!\cdots\!00\) \(39\!\cdots\!00\) $-$ $\mathrm{SU}(2)$ \(q+(-9615373740-\beta _{1})q^{3}+\cdots\)
16.48.a.e 16.a 1.a $6$ $223.852$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(59074429624\) \(49\!\cdots\!80\) \(-20\!\cdots\!20\) $+$ $\mathrm{SU}(2)$ \(q+(9845738271+\beta _{1})q^{3}+(8297943011013387+\cdots)q^{5}+\cdots\)
16.48.a.f 16.a 1.a $6$ $223.852$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(126633673656\) \(-20\!\cdots\!32\) \(-97\!\cdots\!80\) $+$ $\mathrm{SU}(2)$ \(q+(21105612276-\beta _{1})q^{3}+(-3486029198212722+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{48}^{\mathrm{old}}(\Gamma_0(16))\) into lower level spaces

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