# Properties

 Label 75.2.b Level 75 Weight 2 Character orbit b Rep. character $$\chi_{75}(49,\cdot)$$ Character field $$\Q$$ Dimension 4 Newform subspaces 2 Sturm bound 20 Trace bound 4

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$20$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(75, [\chi])$$.

Total New Old
Modular forms 16 4 12
Cusp forms 4 4 0
Eisenstein series 12 0 12

## Trace form

 $$4q - 2q^{4} - 2q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 2q^{4} - 2q^{6} - 4q^{9} - 4q^{11} + 12q^{14} - 10q^{16} + 2q^{19} + 6q^{21} + 6q^{24} + 8q^{26} - 16q^{29} - 6q^{31} - 4q^{34} + 2q^{36} - 2q^{39} + 4q^{41} - 16q^{44} + 24q^{46} + 10q^{49} - 8q^{51} + 2q^{54} + 28q^{59} + 10q^{61} + 2q^{64} - 16q^{66} + 12q^{69} - 32q^{71} - 28q^{74} - 28q^{76} + 4q^{81} - 12q^{84} - 4q^{86} + 12q^{89} - 6q^{91} + 8q^{94} + 26q^{96} + 4q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(75, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
75.2.b.a $$2$$ $$0.599$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{2}+iq^{3}-2q^{4}-2q^{6}-3iq^{7}+\cdots$$
75.2.b.b $$2$$ $$0.599$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}+q^{4}+q^{6}+3iq^{8}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$)($$1 - 3 T^{2} + 4 T^{4}$$)
$3$ ($$1 + T^{2}$$)($$1 + T^{2}$$)
$5$ 1
$7$ ($$1 - 5 T^{2} + 49 T^{4}$$)($$( 1 - 7 T^{2} )^{2}$$)
$11$ ($$( 1 - 2 T + 11 T^{2} )^{2}$$)($$( 1 + 4 T + 11 T^{2} )^{2}$$)
$13$ ($$1 - 25 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)
$17$ ($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)
$19$ ($$( 1 - 5 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)
$23$ ($$1 - 10 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)
$29$ ($$( 1 + 10 T + 29 T^{2} )^{2}$$)($$( 1 - 2 T + 29 T^{2} )^{2}$$)
$31$ ($$( 1 + 3 T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)
$37$ ($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$1 + 26 T^{2} + 1369 T^{4}$$)
$41$ ($$( 1 + 8 T + 41 T^{2} )^{2}$$)($$( 1 - 10 T + 41 T^{2} )^{2}$$)
$43$ ($$1 - 85 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)
$47$ ($$1 - 90 T^{2} + 2209 T^{4}$$)($$1 - 30 T^{2} + 2209 T^{4}$$)
$53$ ($$( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} )$$)($$1 - 6 T^{2} + 2809 T^{4}$$)
$59$ ($$( 1 - 10 T + 59 T^{2} )^{2}$$)($$( 1 - 4 T + 59 T^{2} )^{2}$$)
$61$ ($$( 1 - 7 T + 61 T^{2} )^{2}$$)($$( 1 + 2 T + 61 T^{2} )^{2}$$)
$67$ ($$1 - 125 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)
$71$ ($$( 1 + 8 T + 71 T^{2} )^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{2}$$)
$73$ ($$1 + 50 T^{2} + 5329 T^{4}$$)($$1 - 46 T^{2} + 5329 T^{4}$$)
$79$ ($$( 1 + 79 T^{2} )^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)
$83$ ($$1 - 130 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)
$89$ ($$( 1 + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)
$97$ ($$1 + 95 T^{2} + 9409 T^{4}$$)($$1 - 190 T^{2} + 9409 T^{4}$$)