Properties

Label 20.5.d
Level 20
Weight 5
Character orbit d
Rep. character \(\chi_{20}(19,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 3
Sturm bound 15
Trace bound 2

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Defining parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 20 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(15\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(20, [\chi])\).

Total New Old
Modular forms 14 14 0
Cusp forms 10 10 0
Eisenstein series 4 4 0

Trace form

\( 10q - 16q^{4} + 10q^{5} - 16q^{6} + 158q^{9} + O(q^{10}) \) \( 10q - 16q^{4} + 10q^{5} - 16q^{6} + 158q^{9} - 160q^{10} - 16q^{14} - 320q^{16} + 400q^{20} - 632q^{21} + 1664q^{24} - 790q^{25} + 2496q^{26} + 308q^{29} - 2320q^{30} - 2176q^{34} - 8176q^{36} + 6400q^{40} + 2420q^{41} + 1920q^{44} + 2270q^{45} + 14064q^{46} + 638q^{49} - 12480q^{50} - 8992q^{54} - 25216q^{56} + 21120q^{60} - 1580q^{61} + 15104q^{64} - 1920q^{65} + 46080q^{66} - 15992q^{69} - 22800q^{70} - 23616q^{74} - 48000q^{76} + 32320q^{80} - 8302q^{81} + 30208q^{84} + 8960q^{85} + 53744q^{86} + 36308q^{89} - 48480q^{90} - 24336q^{94} - 80896q^{96} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(20, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
20.5.d.a \(1\) \(2.067\) \(\Q\) \(\Q(\sqrt{-5}) \) \(-4\) \(2\) \(25\) \(82\) \(q-4q^{2}+2q^{3}+2^{4}q^{4}+5^{2}q^{5}-8q^{6}+\cdots\)
20.5.d.b \(1\) \(2.067\) \(\Q\) \(\Q(\sqrt{-5}) \) \(4\) \(-2\) \(25\) \(-82\) \(q+4q^{2}-2q^{3}+2^{4}q^{4}+5^{2}q^{5}-8q^{6}+\cdots\)
20.5.d.c \(8\) \(2.067\) 8.0.\(\cdots\).2 None \(0\) \(0\) \(-40\) \(0\) \(q+\beta _{1}q^{2}-\beta _{2}q^{3}+(-6+\beta _{3})q^{4}+(-5+\cdots)q^{5}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T \))(\( 1 - 4 T \))(\( 1 + 24 T^{2} + 496 T^{4} + 6144 T^{6} + 65536 T^{8} \))
$3$ (\( 1 - 2 T + 81 T^{2} \))(\( 1 + 2 T + 81 T^{2} \))(\( ( 1 + 84 T^{2} + 9126 T^{4} + 551124 T^{6} + 43046721 T^{8} )^{2} \))
$5$ (\( 1 - 25 T \))(\( 1 - 25 T \))(\( ( 1 + 20 T + 710 T^{2} + 12500 T^{3} + 390625 T^{4} )^{2} \))
$7$ (\( 1 - 82 T + 2401 T^{2} \))(\( 1 + 82 T + 2401 T^{2} \))(\( ( 1 + 6804 T^{2} + 23087206 T^{4} + 39223706004 T^{6} + 33232930569601 T^{8} )^{2} \))
$11$ (\( ( 1 - 121 T )( 1 + 121 T ) \))(\( ( 1 - 121 T )( 1 + 121 T ) \))(\( ( 1 - 10564 T^{2} + 433024326 T^{4} - 2264487218884 T^{6} + 45949729863572161 T^{8} )^{2} \))
$13$ (\( ( 1 - 169 T )( 1 + 169 T ) \))(\( ( 1 - 169 T )( 1 + 169 T ) \))(\( ( 1 - 93316 T^{2} + 3781796166 T^{4} - 76120727960836 T^{6} + 665416609183179841 T^{8} )^{2} \))
$17$ (\( ( 1 - 289 T )( 1 + 289 T ) \))(\( ( 1 - 289 T )( 1 + 289 T ) \))(\( ( 1 - 307716 T^{2} + 37466348806 T^{4} - 2146552176714756 T^{6} + 48661191875666868481 T^{8} )^{2} \))
$19$ (\( ( 1 - 361 T )( 1 + 361 T ) \))(\( ( 1 - 361 T )( 1 + 361 T ) \))(\( ( 1 - 73924 T^{2} - 4326450234 T^{4} - 1255492914242884 T^{6} + \)\(28\!\cdots\!81\)\( T^{8} )^{2} \))
$23$ (\( 1 + 878 T + 279841 T^{2} \))(\( 1 - 878 T + 279841 T^{2} \))(\( ( 1 + 769044 T^{2} + 304476266086 T^{4} + 60224593364441364 T^{6} + \)\(61\!\cdots\!61\)\( T^{8} )^{2} \))
$29$ (\( 1 + 1198 T + 707281 T^{2} \))(\( 1 + 1198 T + 707281 T^{2} \))(\( ( 1 - 676 T + 1518566 T^{2} - 478121956 T^{3} + 500246412961 T^{4} )^{4} \))
$31$ (\( ( 1 - 961 T )( 1 + 961 T ) \))(\( ( 1 - 961 T )( 1 + 961 T ) \))(\( ( 1 - 2680324 T^{2} + 3246634805766 T^{4} - 2286024317038010884 T^{6} + \)\(72\!\cdots\!81\)\( T^{8} )^{2} \))
$37$ (\( ( 1 - 1369 T )( 1 + 1369 T ) \))(\( ( 1 - 1369 T )( 1 + 1369 T ) \))(\( ( 1 - 4488196 T^{2} + 9924276565446 T^{4} - 15764696235170416516 T^{6} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \))
$41$ (\( 1 - 482 T + 2825761 T^{2} \))(\( 1 - 482 T + 2825761 T^{2} \))(\( ( 1 - 364 T + 1690406 T^{2} - 1028577004 T^{3} + 7984925229121 T^{4} )^{4} \))
$43$ (\( 1 + 2078 T + 3418801 T^{2} \))(\( 1 - 2078 T + 3418801 T^{2} \))(\( ( 1 + 3640404 T^{2} + 1840427380006 T^{4} + 42549771043379790804 T^{6} + \)\(13\!\cdots\!01\)\( T^{8} )^{2} \))
$47$ (\( 1 - 4402 T + 4879681 T^{2} \))(\( 1 + 4402 T + 4879681 T^{2} \))(\( ( 1 + 18767124 T^{2} + 135582308428006 T^{4} + \)\(44\!\cdots\!64\)\( T^{6} + \)\(56\!\cdots\!21\)\( T^{8} )^{2} \))
$53$ (\( ( 1 - 2809 T )( 1 + 2809 T ) \))(\( ( 1 - 2809 T )( 1 + 2809 T ) \))(\( ( 1 - 7023876 T^{2} + 6572518202566 T^{4} - \)\(43\!\cdots\!36\)\( T^{6} + \)\(38\!\cdots\!21\)\( T^{8} )^{2} \))
$59$ (\( ( 1 - 3481 T )( 1 + 3481 T ) \))(\( ( 1 - 3481 T )( 1 + 3481 T ) \))(\( ( 1 - 18192964 T^{2} + 372799944667206 T^{4} - \)\(26\!\cdots\!44\)\( T^{6} + \)\(21\!\cdots\!41\)\( T^{8} )^{2} \))
$61$ (\( 1 + 4078 T + 13845841 T^{2} \))(\( 1 + 4078 T + 13845841 T^{2} \))(\( ( 1 - 1644 T + 17384326 T^{2} - 22762562604 T^{3} + 191707312997281 T^{4} )^{4} \))
$67$ (\( 1 + 4478 T + 20151121 T^{2} \))(\( 1 - 4478 T + 20151121 T^{2} \))(\( ( 1 + 69567444 T^{2} + 1999220647242406 T^{4} + \)\(28\!\cdots\!04\)\( T^{6} + \)\(16\!\cdots\!81\)\( T^{8} )^{2} \))
$71$ (\( ( 1 - 5041 T )( 1 + 5041 T ) \))(\( ( 1 - 5041 T )( 1 + 5041 T ) \))(\( ( 1 - 36881284 T^{2} + 596367162990726 T^{4} - \)\(23\!\cdots\!24\)\( T^{6} + \)\(41\!\cdots\!21\)\( T^{8} )^{2} \))
$73$ (\( ( 1 - 5329 T )( 1 + 5329 T ) \))(\( ( 1 - 5329 T )( 1 + 5329 T ) \))(\( ( 1 - 45412996 T^{2} + 2127886061569926 T^{4} - \)\(36\!\cdots\!76\)\( T^{6} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \))
$79$ (\( ( 1 - 6241 T )( 1 + 6241 T ) \))(\( ( 1 - 6241 T )( 1 + 6241 T ) \))(\( ( 1 - 128520964 T^{2} + 7130553234151686 T^{4} - \)\(19\!\cdots\!04\)\( T^{6} + \)\(23\!\cdots\!21\)\( T^{8} )^{2} \))
$83$ (\( 1 - 8002 T + 47458321 T^{2} \))(\( 1 + 8002 T + 47458321 T^{2} \))(\( ( 1 + 71614164 T^{2} + 3304864070081446 T^{4} + \)\(16\!\cdots\!24\)\( T^{6} + \)\(50\!\cdots\!81\)\( T^{8} )^{2} \))
$89$ (\( 1 - 4322 T + 62742241 T^{2} \))(\( 1 - 4322 T + 62742241 T^{2} \))(\( ( 1 - 6916 T + 63458246 T^{2} - 433925338756 T^{3} + 3936588805702081 T^{4} )^{4} \))
$97$ (\( ( 1 - 9409 T )( 1 + 9409 T ) \))(\( ( 1 - 9409 T )( 1 + 9409 T ) \))(\( ( 1 - 267347716 T^{2} + 33011083228247046 T^{4} - \)\(20\!\cdots\!76\)\( T^{6} + \)\(61\!\cdots\!21\)\( T^{8} )^{2} \))
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