# Properties

 Label 20.10.e Level 20 Weight 10 Character orbit e Rep. character $$\chi_{20}(3,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 50 Newform subspaces 2 Sturm bound 30 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 20.e (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$20$$ Character field: $$\Q(i)$$ Newform subspaces: $$2$$ Sturm bound: $$30$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(20, [\chi])$$.

Total New Old
Modular forms 58 58 0
Cusp forms 50 50 0
Eisenstein series 8 8 0

## Trace form

 $$50q - 2q^{2} - 4q^{5} + 6152q^{6} - 716q^{8} + O(q^{10})$$ $$50q - 2q^{2} - 4q^{5} + 6152q^{6} - 716q^{8} + 11446q^{10} - 155360q^{12} - 86162q^{13} + 3000q^{16} - 509994q^{17} - 679086q^{18} - 334324q^{20} - 634176q^{21} + 3176920q^{22} + 4304406q^{25} - 11222732q^{26} + 12731840q^{28} + 15498680q^{30} - 29092792q^{32} - 2488000q^{33} + 44773372q^{36} - 1602814q^{37} - 26115120q^{38} - 55995604q^{40} - 3634000q^{41} + 82830200q^{42} + 31366486q^{45} - 92534488q^{46} + 46448320q^{48} - 54429014q^{50} + 88926156q^{52} - 195634782q^{53} - 177356448q^{56} + 62365440q^{57} + 82723048q^{58} + 166923520q^{60} + 180242200q^{61} - 292810200q^{62} + 64634162q^{65} + 614341200q^{66} - 289868412q^{68} - 203227600q^{70} + 930217668q^{72} + 504789018q^{73} - 1061841600q^{76} - 277316160q^{77} + 49362600q^{78} + 210150736q^{80} - 1707948546q^{81} - 591442904q^{82} + 1383913502q^{85} - 874588728q^{86} + 865939360q^{88} + 1779829206q^{90} - 1106673600q^{92} - 2554477120q^{93} + 1870685312q^{96} + 3440322826q^{97} - 3885590026q^{98} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(20, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
20.10.e.a $$2$$ $$10.301$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$-32$$ $$0$$ $$1436$$ $$0$$ $$q+(-2^{4}-2^{4}i)q^{2}+2^{9}iq^{4}+(718+\cdots)q^{5}+\cdots$$
20.10.e.b $$48$$ $$10.301$$ None $$30$$ $$0$$ $$-1440$$ $$0$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 32 T + 512 T^{2}$$)
$3$ ($$1 + 387420489 T^{4}$$)
$5$ ($$1 - 1436 T + 1953125 T^{2}$$)
$7$ ($$1 + 1628413597910449 T^{4}$$)
$11$ ($$( 1 - 2357947691 T^{2} )^{2}$$)
$13$ ($$( 1 + 112806 T + 10604499373 T^{2} )( 1 + 172316 T + 10604499373 T^{2} )$$)
$17$ ($$( 1 + 407992 T + 118587876497 T^{2} )( 1 + 554882 T + 118587876497 T^{2} )$$)
$19$ ($$( 1 + 322687697779 T^{2} )^{2}$$)
$23$ ($$1 +$$$$32\!\cdots\!69$$$$T^{4}$$)
$29$ ($$( 1 - 7314710 T + 14507145975869 T^{2} )( 1 + 7314710 T + 14507145975869 T^{2} )$$)
$31$ ($$( 1 - 26439622160671 T^{2} )^{2}$$)
$37$ ($$( 1 + 1923372 T + 129961739795077 T^{2} )( 1 + 22718882 T + 129961739795077 T^{2} )$$)
$41$ ($$( 1 - 7561912 T + 327381934393961 T^{2} )^{2}$$)
$43$ ($$1 +$$$$25\!\cdots\!49$$$$T^{4}$$)
$47$ ($$1 +$$$$12\!\cdots\!89$$$$T^{4}$$)
$53$ ($$( 1 - 68323684 T + 3299763591802133 T^{2} )( 1 + 92363026 T + 3299763591802133 T^{2} )$$)
$59$ ($$( 1 + 8662995818654939 T^{2} )^{2}$$)
$61$ ($$( 1 - 216178092 T + 11694146092834141 T^{2} )^{2}$$)
$67$ ($$1 +$$$$74\!\cdots\!09$$$$T^{4}$$)
$71$ ($$( 1 - 45848500718449031 T^{2} )^{2}$$)
$73$ ($$( 1 - 483419504 T + 58871586708267913 T^{2} )( 1 - 42331194 T + 58871586708267913 T^{2} )$$)
$79$ ($$( 1 + 119851595982618319 T^{2} )^{2}$$)
$83$ ($$1 +$$$$34\!\cdots\!09$$$$T^{4}$$)
$89$ ($$( 1 - 1125568310 T + 350356403707485209 T^{2} )( 1 + 1125568310 T + 350356403707485209 T^{2} )$$)
$97$ ($$( 1 + 1016663992 T + 760231058654565217 T^{2} )( 1 + 1416798702 T + 760231058654565217 T^{2} )$$)