# Properties

 Label 106.2.a Level 106 Weight 2 Character orbit a Rep. character $$\chi_{106}(1,\cdot)$$ Character field $$\Q$$ Dimension 4 Newform subspaces 4 Sturm bound 27 Trace bound 3

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## Defining parameters

 Level: $$N$$ $$=$$ $$106 = 2 \cdot 53$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 106.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$27$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(106))$$.

Total New Old
Modular forms 15 4 11
Cusp forms 12 4 8
Eisenstein series 3 0 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$$$53$$FrickeDim.
$$+$$$$+$$$$+$$$$1$$
$$+$$$$-$$$$-$$$$1$$
$$-$$$$+$$$$-$$$$2$$
Plus space$$+$$$$1$$
Minus space$$-$$$$3$$

## Trace form

 $$4q + 4q^{4} - 2q^{6} - 4q^{7} - 2q^{9} + O(q^{10})$$ $$4q + 4q^{4} - 2q^{6} - 4q^{7} - 2q^{9} + 6q^{10} - 2q^{11} - 2q^{13} + 4q^{16} + 8q^{17} - 16q^{19} - 12q^{21} - 4q^{22} - 8q^{23} - 2q^{24} + 6q^{25} + 4q^{26} - 4q^{28} + 14q^{29} - 12q^{30} + 4q^{31} + 20q^{33} - 8q^{34} + 4q^{35} - 2q^{36} - 10q^{37} + 6q^{38} + 4q^{39} + 6q^{40} + 4q^{41} - 4q^{42} - 14q^{43} - 2q^{44} + 12q^{45} - 4q^{46} + 4q^{47} - 4q^{49} - 8q^{50} - 8q^{51} - 2q^{52} - 2q^{53} - 2q^{54} + 12q^{55} + 6q^{57} + 16q^{58} + 22q^{59} + 4q^{61} - 2q^{62} + 8q^{63} + 4q^{64} - 20q^{65} - 8q^{66} + 12q^{67} + 8q^{68} + 14q^{69} + 8q^{70} + 36q^{71} - 16q^{73} - 32q^{75} - 16q^{76} - 16q^{77} + 22q^{78} - 8q^{79} - 20q^{81} + 20q^{82} - 20q^{83} - 12q^{84} - 8q^{85} - 8q^{86} - 20q^{87} - 4q^{88} + 46q^{89} - 6q^{90} - 20q^{91} - 8q^{92} + 4q^{93} + 8q^{94} + 12q^{95} - 2q^{96} + 16q^{98} + 10q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(106))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 53
106.2.a.a $$1$$ $$0.846$$ $$\Q$$ None $$-1$$ $$-1$$ $$-4$$ $$0$$ $$+$$ $$+$$ $$q-q^{2}-q^{3}+q^{4}-4q^{5}+q^{6}-q^{8}+\cdots$$
106.2.a.b $$1$$ $$0.846$$ $$\Q$$ None $$-1$$ $$2$$ $$1$$ $$-2$$ $$+$$ $$-$$ $$q-q^{2}+2q^{3}+q^{4}+q^{5}-2q^{6}-2q^{7}+\cdots$$
106.2.a.c $$1$$ $$0.846$$ $$\Q$$ None $$1$$ $$-2$$ $$3$$ $$2$$ $$-$$ $$+$$ $$q+q^{2}-2q^{3}+q^{4}+3q^{5}-2q^{6}+2q^{7}+\cdots$$
106.2.a.d $$1$$ $$0.846$$ $$\Q$$ None $$1$$ $$1$$ $$0$$ $$-4$$ $$-$$ $$+$$ $$q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(106))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(106)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(53))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + T$$)($$1 + T$$)($$1 - T$$)($$1 - T$$)
$3$ ($$1 + T + 3 T^{2}$$)($$1 - 2 T + 3 T^{2}$$)($$1 + 2 T + 3 T^{2}$$)($$1 - T + 3 T^{2}$$)
$5$ ($$1 + 4 T + 5 T^{2}$$)($$1 - T + 5 T^{2}$$)($$1 - 3 T + 5 T^{2}$$)($$1 + 5 T^{2}$$)
$7$ ($$1 + 7 T^{2}$$)($$1 + 2 T + 7 T^{2}$$)($$1 - 2 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)
$11$ ($$1 + 4 T + 11 T^{2}$$)($$1 - 5 T + 11 T^{2}$$)($$1 + 3 T + 11 T^{2}$$)($$1 + 11 T^{2}$$)
$13$ ($$1 - T + 13 T^{2}$$)($$1 + 4 T + 13 T^{2}$$)($$1 + 4 T + 13 T^{2}$$)($$1 - 5 T + 13 T^{2}$$)
$17$ ($$1 - 5 T + 17 T^{2}$$)($$1 - 3 T + 17 T^{2}$$)($$1 - 3 T + 17 T^{2}$$)($$1 + 3 T + 17 T^{2}$$)
$19$ ($$1 + 7 T + 19 T^{2}$$)($$1 + 4 T + 19 T^{2}$$)($$1 + 4 T + 19 T^{2}$$)($$1 + T + 19 T^{2}$$)
$23$ ($$1 - T + 23 T^{2}$$)($$1 + 3 T + 23 T^{2}$$)($$1 + 9 T + 23 T^{2}$$)($$1 - 3 T + 23 T^{2}$$)
$29$ ($$1 - 5 T + 29 T^{2}$$)($$1 + 6 T + 29 T^{2}$$)($$1 - 6 T + 29 T^{2}$$)($$1 - 9 T + 29 T^{2}$$)
$31$ ($$1 + 4 T + 31 T^{2}$$)($$1 - 7 T + 31 T^{2}$$)($$1 - 5 T + 31 T^{2}$$)($$1 + 4 T + 31 T^{2}$$)
$37$ ($$1 - T + 37 T^{2}$$)($$1 + 6 T + 37 T^{2}$$)($$1 + 10 T + 37 T^{2}$$)($$1 - 5 T + 37 T^{2}$$)
$41$ ($$1 + 10 T + 41 T^{2}$$)($$1 - 2 T + 41 T^{2}$$)($$1 - 6 T + 41 T^{2}$$)($$1 - 6 T + 41 T^{2}$$)
$43$ ($$1 + 10 T + 43 T^{2}$$)($$1 - 7 T + 43 T^{2}$$)($$1 + T + 43 T^{2}$$)($$1 + 10 T + 43 T^{2}$$)
$47$ ($$1 + 6 T + 47 T^{2}$$)($$1 - 4 T + 47 T^{2}$$)($$1 + 47 T^{2}$$)($$1 - 6 T + 47 T^{2}$$)
$53$ ($$1 + T$$)($$1 - T$$)($$1 + T$$)($$1 + T$$)
$59$ ($$1 + 6 T + 59 T^{2}$$)($$1 - 7 T + 59 T^{2}$$)($$1 - 15 T + 59 T^{2}$$)($$1 - 6 T + 59 T^{2}$$)
$61$ ($$1 - 4 T + 61 T^{2}$$)($$1 - 2 T + 61 T^{2}$$)($$1 + 10 T + 61 T^{2}$$)($$1 - 8 T + 61 T^{2}$$)
$67$ ($$1 - 4 T + 67 T^{2}$$)($$1 - 16 T + 67 T^{2}$$)($$1 + 4 T + 67 T^{2}$$)($$1 + 4 T + 67 T^{2}$$)
$71$ ($$1 - 15 T + 71 T^{2}$$)($$1 - 12 T + 71 T^{2}$$)($$1 - 12 T + 71 T^{2}$$)($$1 + 3 T + 71 T^{2}$$)
$73$ ($$1 + 8 T + 73 T^{2}$$)($$1 + 12 T + 73 T^{2}$$)($$1 - 8 T + 73 T^{2}$$)($$1 + 4 T + 73 T^{2}$$)
$79$ ($$1 - T + 79 T^{2}$$)($$1 + 7 T + 79 T^{2}$$)($$1 - 11 T + 79 T^{2}$$)($$1 + 13 T + 79 T^{2}$$)
$83$ ($$1 + 3 T + 83 T^{2}$$)($$1 + 14 T + 83 T^{2}$$)($$1 + 6 T + 83 T^{2}$$)($$1 - 3 T + 83 T^{2}$$)
$89$ ($$1 - 2 T + 89 T^{2}$$)($$1 - 17 T + 89 T^{2}$$)($$1 - 9 T + 89 T^{2}$$)($$1 - 18 T + 89 T^{2}$$)
$97$ ($$1 - 17 T + 97 T^{2}$$)($$1 - 3 T + 97 T^{2}$$)($$1 + 13 T + 97 T^{2}$$)($$1 + 7 T + 97 T^{2}$$)
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