Properties

Label 10.10.a
Level 10
Weight 10
Character orbit a
Rep. character \(\chi_{10}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 3
Sturm bound 15
Trace bound 3

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Defining parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 10.a (trivial)
Character field: \(\Q\)
Newforms: \( 3 \)
Sturm bound: \(15\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 15 3 12
Cusp forms 11 3 8
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.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(2\)

Trace form

\(3q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 768q^{4} \) \(\mathstrut -\mathstrut 625q^{5} \) \(\mathstrut +\mathstrut 5312q^{6} \) \(\mathstrut -\mathstrut 228q^{7} \) \(\mathstrut -\mathstrut 4096q^{8} \) \(\mathstrut +\mathstrut 14959q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 768q^{4} \) \(\mathstrut -\mathstrut 625q^{5} \) \(\mathstrut +\mathstrut 5312q^{6} \) \(\mathstrut -\mathstrut 228q^{7} \) \(\mathstrut -\mathstrut 4096q^{8} \) \(\mathstrut +\mathstrut 14959q^{9} \) \(\mathstrut -\mathstrut 10000q^{10} \) \(\mathstrut +\mathstrut 97356q^{11} \) \(\mathstrut +\mathstrut 4096q^{12} \) \(\mathstrut -\mathstrut 232974q^{13} \) \(\mathstrut +\mathstrut 152704q^{14} \) \(\mathstrut -\mathstrut 265000q^{15} \) \(\mathstrut +\mathstrut 196608q^{16} \) \(\mathstrut -\mathstrut 934458q^{17} \) \(\mathstrut +\mathstrut 99632q^{18} \) \(\mathstrut +\mathstrut 1372140q^{19} \) \(\mathstrut -\mathstrut 160000q^{20} \) \(\mathstrut -\mathstrut 772264q^{21} \) \(\mathstrut -\mathstrut 629952q^{22} \) \(\mathstrut +\mathstrut 2695956q^{23} \) \(\mathstrut +\mathstrut 1359872q^{24} \) \(\mathstrut +\mathstrut 1171875q^{25} \) \(\mathstrut -\mathstrut 1534688q^{26} \) \(\mathstrut -\mathstrut 3754160q^{27} \) \(\mathstrut -\mathstrut 58368q^{28} \) \(\mathstrut +\mathstrut 3847050q^{29} \) \(\mathstrut +\mathstrut 760000q^{30} \) \(\mathstrut -\mathstrut 4230024q^{31} \) \(\mathstrut -\mathstrut 1048576q^{32} \) \(\mathstrut -\mathstrut 10293648q^{33} \) \(\mathstrut -\mathstrut 4082976q^{34} \) \(\mathstrut +\mathstrut 6932500q^{35} \) \(\mathstrut +\mathstrut 3829504q^{36} \) \(\mathstrut +\mathstrut 28529802q^{37} \) \(\mathstrut -\mathstrut 31419200q^{38} \) \(\mathstrut -\mathstrut 3137392q^{39} \) \(\mathstrut -\mathstrut 2560000q^{40} \) \(\mathstrut -\mathstrut 2660874q^{41} \) \(\mathstrut +\mathstrut 38291968q^{42} \) \(\mathstrut -\mathstrut 39518424q^{43} \) \(\mathstrut +\mathstrut 24923136q^{44} \) \(\mathstrut +\mathstrut 18066875q^{45} \) \(\mathstrut +\mathstrut 38289792q^{46} \) \(\mathstrut -\mathstrut 27549708q^{47} \) \(\mathstrut +\mathstrut 1048576q^{48} \) \(\mathstrut +\mathstrut 36603891q^{49} \) \(\mathstrut -\mathstrut 6250000q^{50} \) \(\mathstrut -\mathstrut 129542784q^{51} \) \(\mathstrut -\mathstrut 59641344q^{52} \) \(\mathstrut +\mathstrut 181541706q^{53} \) \(\mathstrut +\mathstrut 9453440q^{54} \) \(\mathstrut +\mathstrut 31567500q^{55} \) \(\mathstrut +\mathstrut 39092224q^{56} \) \(\mathstrut -\mathstrut 239106400q^{57} \) \(\mathstrut -\mathstrut 180687840q^{58} \) \(\mathstrut +\mathstrut 32450100q^{59} \) \(\mathstrut -\mathstrut 67840000q^{60} \) \(\mathstrut -\mathstrut 108436254q^{61} \) \(\mathstrut +\mathstrut 142425088q^{62} \) \(\mathstrut +\mathstrut 349738556q^{63} \) \(\mathstrut +\mathstrut 50331648q^{64} \) \(\mathstrut +\mathstrut 2466250q^{65} \) \(\mathstrut +\mathstrut 326125824q^{66} \) \(\mathstrut -\mathstrut 56692488q^{67} \) \(\mathstrut -\mathstrut 239221248q^{68} \) \(\mathstrut +\mathstrut 49880328q^{69} \) \(\mathstrut -\mathstrut 204080000q^{70} \) \(\mathstrut +\mathstrut 26724816q^{71} \) \(\mathstrut +\mathstrut 25505792q^{72} \) \(\mathstrut +\mathstrut 283392846q^{73} \) \(\mathstrut -\mathstrut 109583456q^{74} \) \(\mathstrut +\mathstrut 6250000q^{75} \) \(\mathstrut +\mathstrut 351267840q^{76} \) \(\mathstrut +\mathstrut 594093984q^{77} \) \(\mathstrut -\mathstrut 865437056q^{78} \) \(\mathstrut -\mathstrut 1186742880q^{79} \) \(\mathstrut -\mathstrut 40960000q^{80} \) \(\mathstrut -\mathstrut 554831837q^{81} \) \(\mathstrut +\mathstrut 733577568q^{82} \) \(\mathstrut +\mathstrut 560234736q^{83} \) \(\mathstrut -\mathstrut 197699584q^{84} \) \(\mathstrut +\mathstrut 636138750q^{85} \) \(\mathstrut +\mathstrut 978928192q^{86} \) \(\mathstrut -\mathstrut 845703360q^{87} \) \(\mathstrut -\mathstrut 161267712q^{88} \) \(\mathstrut -\mathstrut 950284530q^{89} \) \(\mathstrut -\mathstrut 500930000q^{90} \) \(\mathstrut -\mathstrut 1862513064q^{91} \) \(\mathstrut +\mathstrut 690164736q^{92} \) \(\mathstrut +\mathstrut 2509309712q^{93} \) \(\mathstrut +\mathstrut 606303744q^{94} \) \(\mathstrut +\mathstrut 464237500q^{95} \) \(\mathstrut +\mathstrut 348127232q^{96} \) \(\mathstrut -\mathstrut 961141338q^{97} \) \(\mathstrut -\mathstrut 1182674832q^{98} \) \(\mathstrut +\mathstrut 2026475868q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(10))\) 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
10.10.a.a \(1\) \(5.150\) \(\Q\) None \(-16\) \(-204\) \(625\) \(5432\) \(+\) \(-\) \(q-2^{4}q^{2}-204q^{3}+2^{8}q^{4}+5^{4}q^{5}+\cdots\)
10.10.a.b \(1\) \(5.150\) \(\Q\) None \(-16\) \(46\) \(-625\) \(-10318\) \(+\) \(+\) \(q-2^{4}q^{2}+46q^{3}+2^{8}q^{4}-5^{4}q^{5}+\cdots\)
10.10.a.c \(1\) \(5.150\) \(\Q\) None \(16\) \(174\) \(-625\) \(4658\) \(-\) \(+\) \(q+2^{4}q^{2}+174q^{3}+2^{8}q^{4}-5^{4}q^{5}+\cdots\)

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

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