Properties

Label 20.10.a
Level 20
Weight 10
Character orbit a
Rep. character \(\chi_{20}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 30
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 20.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(30\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(20))\).

Total New Old
Modular forms 30 3 27
Cusp forms 24 3 21
Eisenstein series 6 0 6

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.
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(2\)

Trace form

\( 3q - 308q^{3} - 625q^{5} - 912q^{7} + 17503q^{9} + O(q^{10}) \) \( 3q - 308q^{3} - 625q^{5} - 912q^{7} + 17503q^{9} + 69540q^{11} + 79458q^{13} + 132500q^{15} - 127434q^{17} - 219972q^{19} + 2865848q^{21} - 808416q^{23} + 1171875q^{25} - 8154584q^{27} - 5365782q^{29} + 7615248q^{31} - 10547520q^{33} - 95000q^{35} + 8594538q^{37} + 4309208q^{39} + 21834438q^{41} - 32882700q^{43} - 32663125q^{45} + 78552936q^{47} + 71867331q^{49} - 5720904q^{51} - 78404454q^{53} - 84937500q^{55} + 208014032q^{57} + 27694356q^{59} + 289250466q^{61} - 723019072q^{63} - 174263750q^{65} - 26145732q^{67} - 165212376q^{69} + 728970504q^{71} - 549596802q^{73} - 120312500q^{75} + 81862320q^{77} + 450836304q^{79} + 2137918387q^{81} - 1909553076q^{83} - 474671250q^{85} + 585010632q^{87} - 632351058q^{89} + 1593068592q^{91} - 1451222128q^{93} - 309072500q^{95} + 1660769718q^{97} + 2052680340q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(20))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
20.10.a.a \(1\) \(10.301\) \(\Q\) None \(0\) \(-48\) \(625\) \(-532\) \(-\) \(-\) \(q-48q^{3}+5^{4}q^{5}-532q^{7}-17379q^{9}+\cdots\)
20.10.a.b \(2\) \(10.301\) \(\Q(\sqrt{79}) \) None \(0\) \(-260\) \(-1250\) \(-380\) \(-\) \(+\) \(q+(-130+\beta )q^{3}-5^{4}q^{5}+(-190+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(20))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(20)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 48 T + 19683 T^{2} \))(\( 1 + 260 T + 36042 T^{2} + 5117580 T^{3} + 387420489 T^{4} \))
$5$ (\( 1 - 625 T \))(\( ( 1 + 625 T )^{2} \))
$7$ (\( 1 + 532 T + 40353607 T^{2} \))(\( 1 + 380 T - 15543150 T^{2} + 15334370660 T^{3} + 1628413597910449 T^{4} \))
$11$ (\( 1 + 33180 T + 2357947691 T^{2} \))(\( 1 - 102720 T + 7335543382 T^{2} - 242208386819520 T^{3} + 5559917313492231481 T^{4} \))
$13$ (\( 1 + 99682 T + 10604499373 T^{2} \))(\( 1 - 179140 T + 22798610142 T^{2} - 1899690017679220 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))
$17$ (\( 1 + 443454 T + 118587876497 T^{2} \))(\( 1 - 316020 T + 259693705798 T^{2} - 37476140730581940 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \))
$19$ (\( 1 + 357244 T + 322687697779 T^{2} \))(\( 1 - 137272 T + 111610161654 T^{2} - 44295985649518888 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \))
$23$ (\( 1 + 142956 T + 1801152661463 T^{2} \))(\( 1 + 665460 T + 2886450615250 T^{2} + 1198595050097167980 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))
$29$ (\( 1 - 1527966 T + 14507145975869 T^{2} \))(\( 1 + 6893748 T + 40195999658014 T^{2} + \)\(10\!\cdots\!12\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))
$31$ (\( 1 - 7323416 T + 26439622160671 T^{2} \))(\( 1 - 291832 T + 38964935800398 T^{2} - 7715927814392939272 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \))
$37$ (\( 1 + 2666842 T + 129961739795077 T^{2} \))(\( 1 - 11261380 T + 218879982937230 T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \))
$41$ (\( 1 + 7939014 T + 327381934393961 T^{2} \))(\( 1 - 29773452 T + 771012402449398 T^{2} - \)\(97\!\cdots\!72\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \))
$43$ (\( 1 + 21174520 T + 502592611936843 T^{2} \))(\( 1 + 11708180 T + 838769843899386 T^{2} + \)\(58\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 - 16059636 T + 1119130473102767 T^{2} \))(\( 1 - 62493300 T + 3177958884734338 T^{2} - \)\(69\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \))
$53$ (\( 1 + 87822234 T + 3299763591802133 T^{2} \))(\( 1 - 9417780 T + 5708185761526990 T^{2} - \)\(31\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \))
$59$ (\( 1 - 120625212 T + 8662995818654939 T^{2} \))(\( 1 + 92930856 T + 16656477955483462 T^{2} + \)\(80\!\cdots\!84\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))
$61$ (\( 1 - 93576542 T + 11694146092834141 T^{2} \))(\( 1 - 195673924 T + 22434263296171326 T^{2} - \)\(22\!\cdots\!84\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \))
$67$ (\( 1 - 193621688 T + 27206534396294947 T^{2} \))(\( 1 + 219767420 T + 65652945987990090 T^{2} + \)\(59\!\cdots\!40\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \))
$71$ (\( 1 - 417763488 T + 45848500718449031 T^{2} \))(\( 1 - 311207016 T + 76405636625293726 T^{2} - \)\(14\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))
$73$ (\( 1 + 450372742 T + 58871586708267913 T^{2} \))(\( 1 + 99224060 T + 35402447061205782 T^{2} + \)\(58\!\cdots\!80\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \))
$79$ (\( 1 + 91425472 T + 119851595982618319 T^{2} \))(\( 1 - 542261776 T + 313115996157615582 T^{2} - \)\(64\!\cdots\!44\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \))
$83$ (\( 1 + 652637376 T + 186940255267540403 T^{2} \))(\( 1 + 1256915700 T + 768086791626261130 T^{2} + \)\(23\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \))
$89$ (\( 1 + 170059206 T + 350356403707485209 T^{2} \))(\( 1 + 462291852 T + 159603168035249494 T^{2} + \)\(16\!\cdots\!68\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \))
$97$ (\( 1 + 10947022 T + 760231058654565217 T^{2} \))(\( 1 - 1671716740 T + 2048690578856969670 T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \))
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