Properties

Label 10.12.a
Level 10
Weight 12
Character orbit a
Rep. character \(\chi_{10}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newforms 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: \(\Q\)
Newforms: \( 4 \)
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

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

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(10))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
10.12.a.a \(1\) \(7.683\) \(\Q\) None \(-32\) \(-12\) \(3125\) \(-14176\) \(+\) \(-\) \(q-2^{5}q^{2}-12q^{3}+2^{10}q^{4}+5^{5}q^{5}+\cdots\)
10.12.a.b \(1\) \(7.683\) \(\Q\) None \(-32\) \(738\) \(-3125\) \(25574\) \(+\) \(+\) \(q-2^{5}q^{2}+738q^{3}+2^{10}q^{4}-5^{5}q^{5}+\cdots\)
10.12.a.c \(1\) \(7.683\) \(\Q\) None \(32\) \(-318\) \(-3125\) \(-70714\) \(-\) \(+\) \(q+2^{5}q^{2}-318q^{3}+2^{10}q^{4}-5^{5}q^{5}+\cdots\)
10.12.a.d \(2\) \(7.683\) \(\Q(\sqrt{1969}) \) None \(64\) \(604\) \(6250\) \(14092\) \(-\) \(-\) \(q+2^{5}q^{2}+(302-\beta )q^{3}+2^{10}q^{4}+5^{5}q^{5}+\cdots\)

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

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