Properties

Label 4.18
Level 4
Weight 18
Dimension 2
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 18
Trace bound 0

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Defining parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 18 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(18\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_1(4))\).

Total New Old
Modular forms 10 2 8
Cusp forms 7 2 5
Eisenstein series 3 0 3

Trace form

\( 2q - 5880q^{3} + 604044q^{5} + 25350160q^{7} + 449174682q^{9} + O(q^{10}) \) \( 2q - 5880q^{3} + 604044q^{5} + 25350160q^{7} + 449174682q^{9} + 1259648280q^{11} - 1320052580q^{13} - 26621930448q^{15} - 27498226140q^{17} + 101133633832q^{19} + 335430207552q^{21} + 134767491120q^{23} - 448986850114q^{25} - 4619416117680q^{27} + 2337155582652q^{29} + 278836113472q^{31} + 22602380058720q^{33} - 7102242382752q^{35} + 20929802888140q^{37} - 108447997173648q^{39} + 4166592315732q^{41} - 111143148534440q^{43} + 281755357404444q^{45} + 196772651157600q^{47} + 99570326741874q^{49} - 39956666918256q^{51} - 487965122736660q^{53} - 566565363246960q^{55} - 2272313631999840q^{57} + 2835904197813624q^{59} - 67544034994436q^{61} + 3282762121966800q^{63} + 3645157343001768q^{65} - 995171321546360q^{67} - 10188302100105024q^{69} + 3882245493215376q^{71} - 12746425881769580q^{73} - 13688080703624712q^{75} + 31591755846002880q^{77} - 14984271534065504q^{79} + 33380230287529458q^{81} + 43899417809893800q^{83} - 3956216991540264q^{85} - 100892756486768400q^{87} + 51909007958846388q^{89} - 83455169400462112q^{91} - 89807116632334080q^{93} + 101643889304423664q^{95} - 49281007789848380q^{97} + 128223271309133880q^{99} + O(q^{100}) \)

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_1(4))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
4.18.a \(\chi_{4}(1, \cdot)\) 4.18.a.a 2 1

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_1(4))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_1(4)) \cong \) \(S_{18}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 5880 T - 78159978 T^{2} + 759344158440 T^{3} + 16677181699666569 T^{4} \)
$5$ \( 1 - 604044 T + 1169867455150 T^{2} - 460848999023437500 T^{3} + \)\(58\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 - 25350160 T + 504160656629070 T^{2} - \)\(58\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 1259648280 T + 906250749893736742 T^{2} - \)\(63\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!41\)\( T^{4} \)
$13$ \( 1 + 1320052580 T + 8595361106022307422 T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(74\!\cdots\!89\)\( T^{4} \)
$17$ \( 1 + 27498226140 T + \)\(18\!\cdots\!58\)\( T^{2} + \)\(22\!\cdots\!80\)\( T^{3} + \)\(68\!\cdots\!29\)\( T^{4} \)
$19$ \( 1 - 101133633832 T + \)\(10\!\cdots\!34\)\( T^{2} - \)\(55\!\cdots\!48\)\( T^{3} + \)\(30\!\cdots\!21\)\( T^{4} \)
$23$ \( 1 - 134767491120 T + \)\(21\!\cdots\!70\)\( T^{2} - \)\(19\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!09\)\( T^{4} \)
$29$ \( 1 - 2337155582652 T + \)\(94\!\cdots\!94\)\( T^{2} - \)\(16\!\cdots\!68\)\( T^{3} + \)\(52\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 - 278836113472 T + \)\(39\!\cdots\!18\)\( T^{2} - \)\(62\!\cdots\!92\)\( T^{3} + \)\(50\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 - 20929802888140 T + \)\(93\!\cdots\!10\)\( T^{2} - \)\(95\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 - 4166592315732 T + \)\(39\!\cdots\!18\)\( T^{2} - \)\(10\!\cdots\!92\)\( T^{3} + \)\(68\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 + 111143148534440 T + \)\(13\!\cdots\!86\)\( T^{2} + \)\(65\!\cdots\!20\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 196772651157600 T + \)\(53\!\cdots\!18\)\( T^{2} - \)\(52\!\cdots\!00\)\( T^{3} + \)\(71\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 + 487965122736660 T + \)\(41\!\cdots\!50\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(42\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 - 2835904197813624 T + \)\(45\!\cdots\!82\)\( T^{2} - \)\(36\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 + 67544034994436 T - \)\(59\!\cdots\!34\)\( T^{2} + \)\(15\!\cdots\!56\)\( T^{3} + \)\(50\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 + 995171321546360 T + \)\(18\!\cdots\!70\)\( T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 - 3882245493215376 T + \)\(21\!\cdots\!26\)\( T^{2} - \)\(11\!\cdots\!16\)\( T^{3} + \)\(87\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 + 12746425881769580 T + \)\(13\!\cdots\!82\)\( T^{2} + \)\(60\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 + 14984271534065504 T + \)\(37\!\cdots\!22\)\( T^{2} + \)\(27\!\cdots\!36\)\( T^{3} + \)\(33\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 - 43899417809893800 T + \)\(13\!\cdots\!30\)\( T^{2} - \)\(18\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 - 51909007958846388 T + \)\(34\!\cdots\!94\)\( T^{2} - \)\(71\!\cdots\!52\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 + 49281007789848380 T + \)\(91\!\cdots\!10\)\( T^{2} + \)\(29\!\cdots\!60\)\( T^{3} + \)\(35\!\cdots\!69\)\( T^{4} \)
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