Properties

Label 10.12.a
Level 10
Weight 12
Character orbit 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

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

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(10))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
10.12.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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 32 T \))(\( 1 + 32 T \))(\( 1 - 32 T \))(\( ( 1 - 32 T )^{2} \))
$3$ (\( 1 + 12 T + 177147 T^{2} \))(\( 1 - 738 T + 177147 T^{2} \))(\( 1 + 318 T + 177147 T^{2} \))(\( 1 - 604 T + 248598 T^{2} - 106996788 T^{3} + 31381059609 T^{4} \))
$5$ (\( 1 - 3125 T \))(\( 1 + 3125 T \))(\( 1 + 3125 T \))(\( ( 1 - 3125 T )^{2} \))
$7$ (\( 1 + 14176 T + 1977326743 T^{2} \))(\( 1 - 25574 T + 1977326743 T^{2} \))(\( 1 + 70714 T + 1977326743 T^{2} \))(\( 1 - 14092 T - 2164380498 T^{2} - 27864488462356 T^{3} + 3909821048582988049 T^{4} \))
$11$ (\( 1 + 756348 T + 285311670611 T^{2} \))(\( 1 - 769152 T + 285311670611 T^{2} \))(\( 1 - 238272 T + 285311670611 T^{2} \))(\( 1 - 421584 T + 305428208086 T^{2} - 120282835342867824 T^{3} + \)\(81\!\cdots\!21\)\( T^{4} \))
$13$ (\( 1 + 905482 T + 1792160394037 T^{2} \))(\( 1 + 918982 T + 1792160394037 T^{2} \))(\( 1 + 2097478 T + 1792160394037 T^{2} \))(\( 1 - 1730524 T + 3917874059118 T^{2} - 3101376573730485388 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))
$17$ (\( 1 - 2803794 T + 34271896307633 T^{2} \))(\( 1 - 10312794 T + 34271896307633 T^{2} \))(\( 1 - 5955546 T + 34271896307633 T^{2} \))(\( 1 + 6323628 T + 73563074016262 T^{2} + \)\(21\!\cdots\!24\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \))
$19$ (\( 1 + 5428660 T + 116490258898219 T^{2} \))(\( 1 + 5521660 T + 116490258898219 T^{2} \))(\( 1 - 10210820 T + 116490258898219 T^{2} \))(\( 1 + 28897400 T + 430951545856038 T^{2} + \)\(33\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \))
$23$ (\( 1 + 10236672 T + 952809757913927 T^{2} \))(\( 1 + 39973422 T + 952809757913927 T^{2} \))(\( 1 + 3535758 T + 952809757913927 T^{2} \))(\( 1 + 45236076 T + 2116464952736398 T^{2} + \)\(43\!\cdots\!52\)\( T^{3} + \)\(90\!\cdots\!29\)\( T^{4} \))
$29$ (\( 1 + 197498010 T + 12200509765705829 T^{2} \))(\( 1 + 15269010 T + 12200509765705829 T^{2} \))(\( 1 + 139304850 T + 12200509765705829 T^{2} \))(\( 1 - 58226220 T + 22969270731722158 T^{2} - \)\(71\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \))
$31$ (\( 1 + 44362288 T + 25408476896404831 T^{2} \))(\( 1 + 241583788 T + 25408476896404831 T^{2} \))(\( 1 + 101002348 T + 25408476896404831 T^{2} \))(\( 1 - 41413384 T + 48711797584420926 T^{2} - \)\(10\!\cdots\!04\)\( T^{3} + \)\(64\!\cdots\!61\)\( T^{4} \))
$37$ (\( 1 - 576737054 T + 177917621779460413 T^{2} \))(\( 1 + 25751446 T + 177917621779460413 T^{2} \))(\( 1 + 524913814 T + 177917621779460413 T^{2} \))(\( 1 - 377255452 T + 373315553507530302 T^{2} - \)\(67\!\cdots\!76\)\( T^{3} + \)\(31\!\cdots\!69\)\( T^{4} \))
$41$ (\( 1 - 930058362 T + 550329031716248441 T^{2} \))(\( 1 + 1217700138 T + 550329031716248441 T^{2} \))(\( 1 - 284590422 T + 550329031716248441 T^{2} \))(\( 1 + 785271036 T + 985831391781991606 T^{2} + \)\(43\!\cdots\!76\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} \))
$43$ (\( 1 - 1605598988 T + 929293739471222707 T^{2} \))(\( 1 + 683436262 T + 929293739471222707 T^{2} \))(\( 1 + 1253635078 T + 929293739471222707 T^{2} \))(\( 1 + 1452987236 T + 2194695988642931238 T^{2} + \)\(13\!\cdots\!52\)\( T^{3} + \)\(86\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 + 1803684456 T + 2472159215084012303 T^{2} \))(\( 1 - 1537395294 T + 2472159215084012303 T^{2} \))(\( 1 + 216106434 T + 2472159215084012303 T^{2} \))(\( 1 + 1288127748 T + 4501637759066496382 T^{2} + \)\(31\!\cdots\!44\)\( T^{3} + \)\(61\!\cdots\!09\)\( T^{4} \))
$53$ (\( 1 - 1558674798 T + 9269035929372191597 T^{2} \))(\( 1 - 3572891298 T + 9269035929372191597 T^{2} \))(\( 1 + 4881275358 T + 9269035929372191597 T^{2} \))(\( 1 + 30490836 T + 1934178302201224318 T^{2} + \)\(28\!\cdots\!92\)\( T^{3} + \)\(85\!\cdots\!09\)\( T^{4} \))
$59$ (\( 1 + 9501997020 T + 30155888444737842659 T^{2} \))(\( 1 + 1069039020 T + 30155888444737842659 T^{2} \))(\( 1 - 8692473300 T + 30155888444737842659 T^{2} \))(\( 1 - 8677102440 T + 78821502141035647318 T^{2} - \)\(26\!\cdots\!60\)\( T^{3} + \)\(90\!\cdots\!81\)\( T^{4} \))
$61$ (\( 1 - 6736320422 T + 43513917611435838661 T^{2} \))(\( 1 + 2091535078 T + 43513917611435838661 T^{2} \))(\( 1 - 3296491802 T + 43513917611435838661 T^{2} \))(\( 1 - 1115498764 T - 16676167592870147154 T^{2} - \)\(48\!\cdots\!04\)\( T^{3} + \)\(18\!\cdots\!21\)\( T^{4} \))
$67$ (\( 1 - 8402906564 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 + 1462369186 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 - 18275027966 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 + 12673769708 T + \)\(18\!\cdots\!82\)\( T^{2} + \)\(15\!\cdots\!64\)\( T^{3} + \)\(14\!\cdots\!89\)\( T^{4} \))
$71$ (\( 1 + 4806306168 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 - 9660178332 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 + 13287447588 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 - 13799832984 T + \)\(49\!\cdots\!06\)\( T^{2} - \)\(31\!\cdots\!64\)\( T^{3} + \)\(53\!\cdots\!41\)\( T^{4} \))
$73$ (\( 1 - 7462713338 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 + 5603447662 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 + 32505250798 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 + 17842079516 T + \)\(53\!\cdots\!18\)\( T^{2} + \)\(55\!\cdots\!32\)\( T^{3} + \)\(98\!\cdots\!29\)\( T^{4} \))
$79$ (\( 1 + 20644540720 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 - 5026936280 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 - 9297455960 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 + 12636930320 T + \)\(13\!\cdots\!58\)\( T^{2} + \)\(94\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \))
$83$ (\( 1 + 68013349212 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 + 38405955462 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 + 22741484838 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 - 41986488924 T + \)\(18\!\cdots\!78\)\( T^{2} - \)\(54\!\cdots\!08\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \))
$89$ (\( 1 - 69871323210 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 - 35558583210 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 + 93378882390 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 - 13208740020 T + \)\(53\!\cdots\!78\)\( T^{2} - \)\(36\!\cdots\!80\)\( T^{3} + \)\(77\!\cdots\!21\)\( T^{4} \))
$97$ (\( 1 - 39960952514 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 - 10572232514 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 + 5811134014 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 + 61787462828 T + \)\(91\!\cdots\!02\)\( T^{2} + \)\(44\!\cdots\!84\)\( T^{3} + \)\(51\!\cdots\!09\)\( T^{4} \))
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