# Properties

 Label 12.10.a Level 12 Weight 10 Character orbit a Rep. character $$\chi_{12}(1,\cdot)$$ Character field $$\Q$$ Dimension 1 Newform subspaces 1 Sturm bound 20 Trace bound 0

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## Defining parameters

 Level: $$N$$ $$=$$ $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 12.a (trivial) Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$20$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(\Gamma_0(12))$$.

Total New Old
Modular forms 21 1 20
Cusp forms 15 1 14
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.

$$2$$$$3$$FrickeDim.
$$-$$$$+$$$$-$$$$1$$
Plus space$$+$$$$0$$
Minus space$$-$$$$1$$

## Trace form

 $$q - 81q^{3} + 990q^{5} + 8576q^{7} + 6561q^{9} + O(q^{10})$$ $$q - 81q^{3} + 990q^{5} + 8576q^{7} + 6561q^{9} + 70596q^{11} - 2530q^{13} - 80190q^{15} - 200574q^{17} - 695620q^{19} - 694656q^{21} + 2472696q^{23} - 973025q^{25} - 531441q^{27} + 5474214q^{29} + 3732104q^{31} - 5718276q^{33} + 8490240q^{35} - 21898522q^{37} + 204930q^{39} - 23818950q^{41} + 10612676q^{43} + 6495390q^{45} + 2398464q^{47} + 33194169q^{49} + 16246494q^{51} - 8994978q^{53} + 69890040q^{55} + 56345220q^{57} - 143417916q^{59} - 19804258q^{61} + 56267136q^{63} - 2504700q^{65} - 165625156q^{67} - 200288376q^{69} - 194801400q^{71} + 148729418q^{73} + 78815025q^{75} + 605431296q^{77} - 30134152q^{79} + 43046721q^{81} + 302054076q^{83} - 198568260q^{85} - 443411334q^{87} + 909502650q^{89} - 21697280q^{91} - 302300424q^{93} - 688663800q^{95} - 872463358q^{97} + 463180356q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(\Gamma_0(12))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 3
12.10.a.a $$1$$ $$6.180$$ $$\Q$$ None $$0$$ $$-81$$ $$990$$ $$8576$$ $$-$$ $$+$$ $$q-3^{4}q^{3}+990q^{5}+8576q^{7}+3^{8}q^{9}+\cdots$$

## Decomposition of $$S_{10}^{\mathrm{old}}(\Gamma_0(12))$$ into lower level spaces

$$S_{10}^{\mathrm{old}}(\Gamma_0(12)) \cong$$ $$S_{10}^{\mathrm{new}}(\Gamma_0(2))$$$$^{\oplus 4}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(3))$$$$^{\oplus 3}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 2}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(6))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 81 T$$
$5$ $$1 - 990 T + 1953125 T^{2}$$
$7$ $$1 - 8576 T + 40353607 T^{2}$$
$11$ $$1 - 70596 T + 2357947691 T^{2}$$
$13$ $$1 + 2530 T + 10604499373 T^{2}$$
$17$ $$1 + 200574 T + 118587876497 T^{2}$$
$19$ $$1 + 695620 T + 322687697779 T^{2}$$
$23$ $$1 - 2472696 T + 1801152661463 T^{2}$$
$29$ $$1 - 5474214 T + 14507145975869 T^{2}$$
$31$ $$1 - 3732104 T + 26439622160671 T^{2}$$
$37$ $$1 + 21898522 T + 129961739795077 T^{2}$$
$41$ $$1 + 23818950 T + 327381934393961 T^{2}$$
$43$ $$1 - 10612676 T + 502592611936843 T^{2}$$
$47$ $$1 - 2398464 T + 1119130473102767 T^{2}$$
$53$ $$1 + 8994978 T + 3299763591802133 T^{2}$$
$59$ $$1 + 143417916 T + 8662995818654939 T^{2}$$
$61$ $$1 + 19804258 T + 11694146092834141 T^{2}$$
$67$ $$1 + 165625156 T + 27206534396294947 T^{2}$$
$71$ $$1 + 194801400 T + 45848500718449031 T^{2}$$
$73$ $$1 - 148729418 T + 58871586708267913 T^{2}$$
$79$ $$1 + 30134152 T + 119851595982618319 T^{2}$$
$83$ $$1 - 302054076 T + 186940255267540403 T^{2}$$
$89$ $$1 - 909502650 T + 350356403707485209 T^{2}$$
$97$ $$1 + 872463358 T + 760231058654565217 T^{2}$$
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