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Properties

Label 10.10.a
Level $10$
Weight $10$
Character orbit 10.a
Rep. character $\chi_{10}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
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: \(\Q\)
Newform subspaces: \( 3 \)
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

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

Display 100 coefficients 10 coefficients

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

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

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