# Properties

 Label 403.2.l Level 403 Weight 2 Character orbit l Rep. character $$\chi_{403}(25,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 72 Newform subspaces 3 Sturm bound 74 Trace bound 5

# Related objects

## Defining parameters

 Level: $$N$$ = $$403 = 13 \cdot 31$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 403.l (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$403$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$74$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(403, [\chi])$$.

Total New Old
Modular forms 80 80 0
Cusp forms 72 72 0
Eisenstein series 8 8 0

## Trace form

 $$72q - 4q^{3} - 80q^{4} - 36q^{9} + O(q^{10})$$ $$72q - 4q^{3} - 80q^{4} - 36q^{9} + 2q^{10} - 12q^{12} + 2q^{13} + 16q^{14} + 64q^{16} - 14q^{22} - 44q^{23} + 26q^{25} + 6q^{26} + 8q^{27} + 24q^{29} - 16q^{30} - 36q^{35} + 36q^{36} + 42q^{38} - 4q^{39} + 14q^{40} - 16q^{42} - 10q^{43} + 22q^{48} + 50q^{49} + 26q^{51} - 32q^{52} + 2q^{53} - 18q^{55} - 44q^{56} + 28q^{61} - 38q^{62} - 4q^{64} - 24q^{65} + 128q^{66} + 10q^{68} - 34q^{69} - 22q^{74} + 24q^{75} - 164q^{77} - 40q^{78} - 70q^{79} - 84q^{81} - 48q^{82} + 40q^{87} - 2q^{88} - 2q^{90} - 88q^{91} + 108q^{92} + 20q^{94} + 40q^{95} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(403, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
403.2.l.a $$2$$ $$3.218$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$-3$$ $$-3$$ $$q+(1-2\zeta_{6})q^{2}+\zeta_{6}q^{3}-q^{4}+(-1+\cdots)q^{5}+\cdots$$
403.2.l.b $$2$$ $$3.218$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$3$$ $$3$$ $$q+(-1+2\zeta_{6})q^{2}+\zeta_{6}q^{3}-q^{4}+(1+\cdots)q^{5}+\cdots$$
403.2.l.c $$68$$ $$3.218$$ None $$0$$ $$-6$$ $$0$$ $$0$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - T^{2} + 4 T^{4}$$)($$1 - T^{2} + 4 T^{4}$$)
$3$ ($$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$)($$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$)
$5$ ($$1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4}$$)($$1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4}$$)
$7$ ($$( 1 - T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$)($$( 1 - 4 T + 7 T^{2} )( 1 + T + 7 T^{2} )$$)
$11$ ($$1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4}$$)($$1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4}$$)
$13$ ($$1 - 7 T + 13 T^{2}$$)($$1 + 2 T + 13 T^{2}$$)
$17$ ($$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$)
$19$ ($$( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} )$$)($$( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)
$23$ ($$( 1 + 23 T^{2} )^{2}$$)($$( 1 + 23 T^{2} )^{2}$$)
$29$ ($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)
$31$ ($$1 - 4 T + 31 T^{2}$$)($$1 + 4 T + 31 T^{2}$$)
$37$ ($$( 1 - 10 T + 37 T^{2} )( 1 + T + 37 T^{2} )$$)($$( 1 - T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$)
$41$ ($$1 - 9 T + 68 T^{2} - 369 T^{3} + 1681 T^{4}$$)($$1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4}$$)
$43$ ($$1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4}$$)($$1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4}$$)
$47$ ($$1 - 82 T^{2} + 2209 T^{4}$$)($$1 - 82 T^{2} + 2209 T^{4}$$)
$53$ ($$1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4}$$)($$1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4}$$)
$59$ ($$1 + 3 T + 62 T^{2} + 177 T^{3} + 3481 T^{4}$$)($$1 - 3 T + 62 T^{2} - 177 T^{3} + 3481 T^{4}$$)
$61$ ($$( 1 - 10 T + 61 T^{2} )^{2}$$)($$( 1 - 10 T + 61 T^{2} )^{2}$$)
$67$ ($$1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4}$$)($$1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4}$$)
$71$ ($$1 - 15 T + 146 T^{2} - 1065 T^{3} + 5041 T^{4}$$)($$1 + 15 T + 146 T^{2} + 1065 T^{3} + 5041 T^{4}$$)
$73$ ($$1 + 15 T + 148 T^{2} + 1095 T^{3} + 5329 T^{4}$$)($$1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4}$$)
$79$ ($$1 - 5 T - 54 T^{2} - 395 T^{3} + 6241 T^{4}$$)($$1 - 5 T - 54 T^{2} - 395 T^{3} + 6241 T^{4}$$)
$83$ ($$1 + 21 T + 230 T^{2} + 1743 T^{3} + 6889 T^{4}$$)($$1 - 21 T + 230 T^{2} - 1743 T^{3} + 6889 T^{4}$$)
$89$ ($$1 - 130 T^{2} + 7921 T^{4}$$)($$1 - 130 T^{2} + 7921 T^{4}$$)
$97$ ($$1 - 146 T^{2} + 9409 T^{4}$$)($$1 - 146 T^{2} + 9409 T^{4}$$)