Properties

Label 10.16.a
Level $10$
Weight $16$
Character orbit 10.a
Rep. character $\chi_{10}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $4$
Sturm bound $24$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(24\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_0(10))\).

Total New Old
Modular forms 25 5 20
Cusp forms 21 5 16
Eisenstein series 4 0 4

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

\(2\)\(5\)FrickeDim
\(+\)\(+\)$+$\(1\)
\(+\)\(-\)$-$\(1\)
\(-\)\(+\)$-$\(1\)
\(-\)\(-\)$+$\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(2\)

Trace form

\( 5 q + 128 q^{2} - 9632 q^{3} + 81920 q^{4} + 78125 q^{5} + 427520 q^{6} + 535804 q^{7} + 2097152 q^{8} + 11409985 q^{9} + O(q^{10}) \) \( 5 q + 128 q^{2} - 9632 q^{3} + 81920 q^{4} + 78125 q^{5} + 427520 q^{6} + 535804 q^{7} + 2097152 q^{8} + 11409985 q^{9} + 10000000 q^{10} - 21143940 q^{11} - 157810688 q^{12} + 744201478 q^{13} - 343946240 q^{14} - 405625000 q^{15} + 1342177280 q^{16} + 890757354 q^{17} + 654709376 q^{18} - 760960100 q^{19} + 1280000000 q^{20} + 7887208760 q^{21} + 13494391296 q^{22} + 1886355348 q^{23} + 7004487680 q^{24} + 30517578125 q^{25} - 30680218880 q^{26} - 33283167920 q^{27} + 8778612736 q^{28} - 134211118050 q^{29} + 41080000000 q^{30} - 288025204040 q^{31} + 34359738368 q^{32} - 152670425424 q^{33} - 101290755840 q^{34} + 205025312500 q^{35} + 186941194240 q^{36} + 1660154133694 q^{37} + 376608140800 q^{38} - 3672013210480 q^{39} + 163840000000 q^{40} - 349873701990 q^{41} + 4441635721216 q^{42} - 2804034484232 q^{43} - 346422312960 q^{44} + 4978887265625 q^{45} - 4509689748480 q^{46} + 10246116175284 q^{47} - 2585570312192 q^{48} - 7182896417835 q^{49} + 781250000000 q^{50} + 9093956417760 q^{51} + 12192997015552 q^{52} - 25057737033762 q^{53} - 7521119872000 q^{54} + 6415069687500 q^{55} - 5635215196160 q^{56} + 19903369887200 q^{57} - 10077353329920 q^{58} - 60966958907100 q^{59} - 6645760000000 q^{60} + 28196537181910 q^{61} + 24327579406336 q^{62} + 9069525437468 q^{63} + 21990232555520 q^{64} + 41972637343750 q^{65} - 130917115176960 q^{66} - 98148968872856 q^{67} + 14594168487936 q^{68} + 197582993684520 q^{69} - 44127760000000 q^{70} + 111220835423760 q^{71} + 10726758416384 q^{72} + 66038638669618 q^{73} + 179782580565760 q^{74} - 58789062500000 q^{75} - 12467570278400 q^{76} - 516867155383872 q^{77} + 63914271450112 q^{78} + 610332360628000 q^{79} + 20971520000000 q^{80} - 191631237239195 q^{81} - 248954943124224 q^{82} - 169647139364832 q^{83} + 129224028323840 q^{84} - 184603809843750 q^{85} + 18482801144320 q^{86} - 764001995922720 q^{87} + 221092106993664 q^{88} + 74973548668050 q^{89} + 34099570000000 q^{90} + 49650886540760 q^{91} + 30906046021632 q^{92} - 660512111777584 q^{93} + 331947449948160 q^{94} + 223816885937500 q^{95} + 114761526149120 q^{96} + 1420399684126474 q^{97} - 405707494390656 q^{98} + 1515440388237420 q^{99} + O(q^{100}) \)

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_0(10))\) 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 5
10.16.a.a 10.a 1.a $1$ $14.269$ \(\Q\) None \(-128\) \(-5568\) \(78125\) \(2564996\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}-5568q^{3}+2^{14}q^{4}+5^{7}q^{5}+\cdots\)
10.16.a.b 10.a 1.a $1$ $14.269$ \(\Q\) None \(-128\) \(-918\) \(-78125\) \(-953554\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{7}q^{2}-918q^{3}+2^{14}q^{4}-5^{7}q^{5}+\cdots\)
10.16.a.c 10.a 1.a $1$ $14.269$ \(\Q\) None \(128\) \(-1302\) \(-78125\) \(-90706\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}-1302q^{3}+2^{14}q^{4}-5^{7}q^{5}+\cdots\)
10.16.a.d 10.a 1.a $2$ $14.269$ \(\Q(\sqrt{239569}) \) None \(256\) \(-1844\) \(156250\) \(-984932\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{7}q^{2}+(-922-\beta )q^{3}+2^{14}q^{4}+\cdots\)

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_0(10)) \cong \) \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)