Properties

Label 5.12.a
Level 5
Weight 12
Character orbit a
Rep. character \(\chi_{5}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 6
Trace bound 1

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Defining parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 5.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(6\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(5))\).

Total New Old
Modular forms 7 3 4
Cusp forms 5 3 2
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators.

\(5\)Dim.
\(+\)\(2\)
\(-\)\(1\)

Trace form

\( 3q + 14q^{2} - 1012q^{3} + 6084q^{4} - 3125q^{5} + 33256q^{6} + 40344q^{7} - 346200q^{8} + 429271q^{9} + O(q^{10}) \) \( 3q + 14q^{2} - 1012q^{3} + 6084q^{4} - 3125q^{5} + 33256q^{6} + 40344q^{7} - 346200q^{8} + 429271q^{9} + 168750q^{10} - 1086964q^{11} - 1220576q^{12} + 3040218q^{13} + 737568q^{14} - 1787500q^{15} + 4433808q^{16} - 2406346q^{17} + 2755958q^{18} + 4945860q^{19} - 24587500q^{20} + 17740536q^{21} - 105430632q^{22} + 26485848q^{23} + 201348480q^{24} + 29296875q^{25} - 93479164q^{26} - 242961400q^{27} + 179345712q^{28} + 163837090q^{29} - 272225000q^{30} + 415992096q^{31} - 476025056q^{32} - 70979744q^{33} + 318732348q^{34} - 235800000q^{35} - 219085612q^{36} - 270337326q^{37} + 928850600q^{38} - 327962728q^{39} + 457125000q^{40} - 554400074q^{41} + 1902541008q^{42} - 1129907292q^{43} + 175433008q^{44} + 1471759375q^{45} - 1571769984q^{46} + 221408384q^{47} - 5130637952q^{48} - 3610791021q^{49} + 136718750q^{50} + 5250194296q^{51} + 13174967064q^{52} + 493431938q^{53} - 16555690640q^{54} + 466837500q^{55} - 2235656160q^{56} + 5021442800q^{57} + 9757106100q^{58} + 8243084780q^{59} + 8229700000q^{60} - 12415015014q^{61} - 22735073952q^{62} - 10750886232q^{63} + 9182013504q^{64} - 11838443750q^{65} + 9648059072q^{66} + 19529964204q^{67} - 1251711608q^{68} + 19869613032q^{69} - 6035550000q^{70} + 9971527816q^{71} - 63347763000q^{72} - 20102028402q^{73} + 58862619508q^{74} - 9882812500q^{75} - 986914320q^{76} - 26504706672q^{77} + 120428573776q^{78} - 24071709360q^{79} - 23679550000q^{80} + 47602946083q^{81} - 68124845412q^{82} + 16503964428q^{83} - 28385169792q^{84} - 15756956250q^{85} - 116541245224q^{86} - 21134853400q^{87} - 33934826400q^{88} + 50501550270q^{89} + 87037493750q^{90} + 97204540416q^{91} + 232124189904q^{92} - 301987859184q^{93} - 5091788112q^{94} - 17827437500q^{95} - 14295226624q^{96} + 123413770134q^{97} + 73456940302q^{98} - 92309175748q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(5))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5
5.12.a.a \(1\) \(3.842\) \(\Q\) None \(34\) \(-792\) \(3125\) \(-17556\) \(-\) \(q+34q^{2}-792q^{3}-892q^{4}+5^{5}q^{5}+\cdots\)
5.12.a.b \(2\) \(3.842\) \(\Q(\sqrt{151}) \) None \(-20\) \(-220\) \(-6250\) \(57900\) \(+\) \(q+(-10+3\beta )q^{2}+(-110+2^{4}\beta )q^{3}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(5))\) into lower level spaces

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 34 T + 2048 T^{2} \))(\( 1 + 20 T - 1240 T^{2} + 40960 T^{3} + 4194304 T^{4} \))
$3$ (\( 1 + 792 T + 177147 T^{2} \))(\( 1 + 220 T + 211770 T^{2} + 38972340 T^{3} + 31381059609 T^{4} \))
$5$ (\( 1 - 3125 T \))(\( ( 1 + 3125 T )^{2} \))
$7$ (\( 1 + 17556 T + 1977326743 T^{2} \))(\( 1 - 57900 T + 4624370450 T^{2} - 114487218419700 T^{3} + 3909821048582988049 T^{4} \))
$11$ (\( 1 + 468788 T + 285311670611 T^{2} \))(\( 1 + 618176 T + 245194892966 T^{2} + 176372827291625536 T^{3} + \)\(81\!\cdots\!21\)\( T^{4} \))
$13$ (\( 1 + 374042 T + 1792160394037 T^{2} \))(\( 1 - 3414260 T + 6398662197390 T^{2} - 6118901546944767620 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))
$17$ (\( 1 + 3724286 T + 34271896307633 T^{2} \))(\( 1 - 1317940 T + 59308395866630 T^{2} - 45168303019681836020 T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \))
$19$ (\( 1 + 379460 T + 116490258898219 T^{2} \))(\( 1 - 5325320 T + 194538827137638 T^{2} - \)\(62\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \))
$23$ (\( 1 + 32458092 T + 952809757913927 T^{2} \))(\( 1 - 58943940 T + 2773540471931410 T^{2} - \)\(56\!\cdots\!80\)\( T^{3} + \)\(90\!\cdots\!29\)\( T^{4} \))
$29$ (\( 1 - 69696710 T + 12200509765705829 T^{2} \))(\( 1 - 94140380 T + 23426350431097358 T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \))
$31$ (\( 1 - 171448632 T + 25408476896404831 T^{2} \))(\( 1 - 244543464 T + 34393316207729486 T^{2} - \)\(62\!\cdots\!84\)\( T^{3} + \)\(64\!\cdots\!61\)\( T^{4} \))
$37$ (\( 1 + 291340546 T + 177917621779460413 T^{2} \))(\( 1 - 21003220 T + 137126715218410590 T^{2} - \)\(37\!\cdots\!60\)\( T^{3} + \)\(31\!\cdots\!69\)\( T^{4} \))
$41$ (\( 1 - 191343242 T + 550329031716248441 T^{2} \))(\( 1 + 745743316 T + 929792912462405846 T^{2} + \)\(41\!\cdots\!56\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} \))
$43$ (\( 1 + 1759857392 T + 929293739471222707 T^{2} \))(\( 1 - 629950100 T + 1840945003918927050 T^{2} - \)\(58\!\cdots\!00\)\( T^{3} + \)\(86\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 - 1623469924 T + 2472159215084012303 T^{2} \))(\( 1 + 1402061540 T + 5181805952108806370 T^{2} + \)\(34\!\cdots\!20\)\( T^{3} + \)\(61\!\cdots\!09\)\( T^{4} \))
$53$ (\( 1 + 644888642 T + 9269035929372191597 T^{2} \))(\( 1 - 1138320580 T - 2203723231625575330 T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(85\!\cdots\!09\)\( T^{4} \))
$59$ (\( 1 - 925569220 T + 30155888444737842659 T^{2} \))(\( 1 - 7317515560 T + 55027608950440780118 T^{2} - \)\(22\!\cdots\!40\)\( T^{3} + \)\(90\!\cdots\!81\)\( T^{4} \))
$61$ (\( 1 + 10898589338 T + 43513917611435838661 T^{2} \))(\( 1 + 1516425676 T + 85869525433683691566 T^{2} + \)\(65\!\cdots\!36\)\( T^{3} + \)\(18\!\cdots\!21\)\( T^{4} \))
$67$ (\( 1 - 3795674064 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 - 15734290140 T + \)\(30\!\cdots\!30\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!89\)\( T^{4} \))
$71$ (\( 1 + 22966943728 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 - 32938471544 T + \)\(73\!\cdots\!26\)\( T^{2} - \)\(76\!\cdots\!24\)\( T^{3} + \)\(53\!\cdots\!41\)\( T^{4} \))
$73$ (\( 1 - 9880820458 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 + 29982848860 T + \)\(78\!\cdots\!10\)\( T^{2} + \)\(94\!\cdots\!20\)\( T^{3} + \)\(98\!\cdots\!29\)\( T^{4} \))
$79$ (\( 1 + 20768886240 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 + 3302823120 T + \)\(12\!\cdots\!58\)\( T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \))
$83$ (\( 1 - 3204862008 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 - 13299102420 T + \)\(18\!\cdots\!30\)\( T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \))
$89$ (\( 1 - 63176321130 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 + 12674770860 T + \)\(43\!\cdots\!78\)\( T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + \)\(77\!\cdots\!21\)\( T^{4} \))
$97$ (\( 1 - 126494473874 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 + 3080703740 T + \)\(18\!\cdots\!70\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(51\!\cdots\!09\)\( T^{4} \))
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