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Properties

Label 10.12.a
Level $10$
Weight $12$
Character orbit 10.a
Rep. character $\chi_{10}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $4$
Sturm bound $18$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(18\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(10))\).

Total New Old
Modular forms 19 5 14
Cusp forms 15 5 10
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

\( 5 q + 32 q^{2} + 1012 q^{3} + 5120 q^{4} + 3125 q^{5} - 14080 q^{6} - 45224 q^{7} + 32768 q^{8} + 336385 q^{9} + O(q^{10}) \) \( 5 q + 32 q^{2} + 1012 q^{3} + 5120 q^{4} + 3125 q^{5} - 14080 q^{6} - 45224 q^{7} + 32768 q^{8} + 336385 q^{9} + 100000 q^{10} + 672660 q^{11} + 1036288 q^{12} - 2191418 q^{13} - 2176640 q^{14} + 537500 q^{15} + 5242880 q^{16} + 12748506 q^{17} - 1427296 q^{18} - 29636900 q^{19} + 3200000 q^{20} - 23916040 q^{21} + 20705664 q^{22} - 98981928 q^{23} - 14417920 q^{24} + 48828125 q^{25} + 46640320 q^{26} + 423112600 q^{27} - 46309376 q^{28} - 293845650 q^{29} + 167200000 q^{30} - 345535040 q^{31} + 33554432 q^{32} + 134433024 q^{33} - 431509440 q^{34} + 140800000 q^{35} + 344458240 q^{36} + 403327246 q^{37} - 247560320 q^{38} + 1094070920 q^{39} + 102400000 q^{40} - 788322390 q^{41} - 1984111616 q^{42} - 1784459588 q^{43} + 688803840 q^{44} - 770509375 q^{45} + 46024320 q^{46} - 1770523344 q^{47} + 1061158912 q^{48} + 8405476965 q^{49} + 312500000 q^{50} + 1793570760 q^{51} - 2244012032 q^{52} + 219799902 q^{53} + 4277004800 q^{54} - 4194337500 q^{55} - 2228879360 q^{56} - 13068201520 q^{57} + 4214028480 q^{58} + 6798539700 q^{59} + 550400000 q^{60} + 9056775910 q^{61} + 7243427584 q^{62} - 23253311128 q^{63} + 5368709120 q^{64} + 12004693750 q^{65} - 32607605760 q^{66} + 12541795636 q^{67} + 13054470144 q^{68} - 26524371480 q^{69} + 12455600000 q^{70} + 5366257560 q^{71} - 1461551104 q^{72} - 48488064638 q^{73} - 22356607040 q^{74} + 9882812500 q^{75} - 30348185600 q^{76} + 103920657552 q^{77} + 77720217088 q^{78} - 18957078800 q^{79} + 3276800000 q^{80} + 123956164405 q^{81} - 6817242816 q^{82} - 87174300588 q^{83} - 24490024960 q^{84} - 61838043750 q^{85} - 116121121280 q^{86} + 10614977880 q^{87} + 21202599936 q^{88} + 25259764050 q^{89} + 84243700000 q^{90} + 48640223360 q^{91} - 101357494272 q^{92} - 88457439856 q^{93} - 39614240640 q^{94} - 121922562500 q^{95} - 14763950080 q^{96} - 17065411814 q^{97} + 467353825056 q^{98} + 146930040420 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(10))\) 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
10.12.a.a 10.a 1.a $1$ $7.683$ \(\Q\) None \(-32\) \(-12\) \(3125\) \(-14176\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}-12q^{3}+2^{10}q^{4}+5^{5}q^{5}+\cdots\)
10.12.a.b 10.a 1.a $1$ $7.683$ \(\Q\) None \(-32\) \(738\) \(-3125\) \(25574\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+738q^{3}+2^{10}q^{4}-5^{5}q^{5}+\cdots\)
10.12.a.c 10.a 1.a $1$ $7.683$ \(\Q\) None \(32\) \(-318\) \(-3125\) \(-70714\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}-318q^{3}+2^{10}q^{4}-5^{5}q^{5}+\cdots\)
10.12.a.d 10.a 1.a $2$ $7.683$ \(\Q(\sqrt{1969}) \) None \(64\) \(604\) \(6250\) \(14092\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+(302-\beta )q^{3}+2^{10}q^{4}+5^{5}q^{5}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(10)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)