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

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

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(10))\) into newform subspaces

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 16 T \))(\( 1 + 16 T \))(\( 1 - 16 T \))
$3$ (\( 1 + 204 T + 19683 T^{2} \))(\( 1 - 46 T + 19683 T^{2} \))(\( 1 - 174 T + 19683 T^{2} \))
$5$ (\( 1 - 625 T \))(\( 1 + 625 T \))(\( 1 + 625 T \))
$7$ (\( 1 - 5432 T + 40353607 T^{2} \))(\( 1 + 10318 T + 40353607 T^{2} \))(\( 1 - 4658 T + 40353607 T^{2} \))
$11$ (\( 1 - 73932 T + 2357947691 T^{2} \))(\( 1 + 5568 T + 2357947691 T^{2} \))(\( 1 - 28992 T + 2357947691 T^{2} \))
$13$ (\( 1 + 114514 T + 10604499373 T^{2} \))(\( 1 - 45986 T + 10604499373 T^{2} \))(\( 1 + 164446 T + 10604499373 T^{2} \))
$17$ (\( 1 - 41682 T + 118587876497 T^{2} \))(\( 1 + 381318 T + 118587876497 T^{2} \))(\( 1 + 594822 T + 118587876497 T^{2} \))
$19$ (\( 1 - 1057460 T + 322687697779 T^{2} \))(\( 1 - 610460 T + 322687697779 T^{2} \))(\( 1 + 295780 T + 322687697779 T^{2} \))
$23$ (\( 1 - 1599336 T + 1801152661463 T^{2} \))(\( 1 + 1447914 T + 1801152661463 T^{2} \))(\( 1 - 2544534 T + 1801152661463 T^{2} \))
$29$ (\( 1 - 2184510 T + 14507145975869 T^{2} \))(\( 1 - 5385510 T + 14507145975869 T^{2} \))(\( 1 + 3722970 T + 14507145975869 T^{2} \))
$31$ (\( 1 + 9619648 T + 26439622160671 T^{2} \))(\( 1 - 3053852 T + 26439622160671 T^{2} \))(\( 1 - 2335772 T + 26439622160671 T^{2} \))
$37$ (\( 1 - 4799942 T + 129961739795077 T^{2} \))(\( 1 - 12889442 T + 129961739795077 T^{2} \))(\( 1 - 10840418 T + 129961739795077 T^{2} \))
$41$ (\( 1 - 9531882 T + 327381934393961 T^{2} \))(\( 1 + 33786618 T + 327381934393961 T^{2} \))(\( 1 - 21593862 T + 327381934393961 T^{2} \))
$43$ (\( 1 + 13464484 T + 502592611936843 T^{2} \))(\( 1 + 36886234 T + 502592611936843 T^{2} \))(\( 1 - 10832294 T + 502592611936843 T^{2} \))
$47$ (\( 1 - 11441952 T + 1119130473102767 T^{2} \))(\( 1 + 44163798 T + 1119130473102767 T^{2} \))(\( 1 - 5172138 T + 1119130473102767 T^{2} \))
$53$ (\( 1 - 53615766 T + 3299763591802133 T^{2} \))(\( 1 - 29746266 T + 3299763591802133 T^{2} \))(\( 1 - 98179674 T + 3299763591802133 T^{2} \))
$59$ (\( 1 - 81862620 T + 8662995818654939 T^{2} \))(\( 1 + 65575380 T + 8662995818654939 T^{2} \))(\( 1 - 16162860 T + 8662995818654939 T^{2} \))
$61$ (\( 1 + 104691298 T + 11694146092834141 T^{2} \))(\( 1 - 40183202 T + 11694146092834141 T^{2} \))(\( 1 + 43928158 T + 11694146092834141 T^{2} \))
$67$ (\( 1 - 140571092 T + 27206534396294947 T^{2} \))(\( 1 + 115706158 T + 27206534396294947 T^{2} \))(\( 1 + 81557422 T + 27206534396294947 T^{2} \))
$71$ (\( 1 - 97098792 T + 45848500718449031 T^{2} \))(\( 1 + 231681708 T + 45848500718449031 T^{2} \))(\( 1 - 161307732 T + 45848500718449031 T^{2} \))
$73$ (\( 1 - 171848906 T + 58871586708267913 T^{2} \))(\( 1 - 358691906 T + 58871586708267913 T^{2} \))(\( 1 + 247147966 T + 58871586708267913 T^{2} \))
$79$ (\( 1 + 117380080 T + 119851595982618319 T^{2} \))(\( 1 + 486017080 T + 119851595982618319 T^{2} \))(\( 1 + 583345720 T + 119851595982618319 T^{2} \))
$83$ (\( 1 - 323637636 T + 186940255267540403 T^{2} \))(\( 1 - 251168886 T + 186940255267540403 T^{2} \))(\( 1 + 14571786 T + 186940255267540403 T^{2} \))
$89$ (\( 1 + 894379110 T + 350356403707485209 T^{2} \))(\( 1 + 526039110 T + 350356403707485209 T^{2} \))(\( 1 - 470133690 T + 350356403707485209 T^{2} \))
$97$ (\( 1 - 232678562 T + 760231058654565217 T^{2} \))(\( 1 + 1075981438 T + 760231058654565217 T^{2} \))(\( 1 + 117838462 T + 760231058654565217 T^{2} \))
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