# Properties

 Label 1340.2.d Level 1340 Weight 2 Character orbit d Rep. character $$\chi_{1340}(269,\cdot)$$ Character field $$\Q$$ Dimension 34 Newform subspaces 3 Sturm bound 408 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$1340 = 2^{2} \cdot 5 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1340.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$5$$ Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$408$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1340, [\chi])$$.

Total New Old
Modular forms 210 34 176
Cusp forms 198 34 164
Eisenstein series 12 0 12

## Trace form

 $$34q + 2q^{5} - 42q^{9} + O(q^{10})$$ $$34q + 2q^{5} - 42q^{9} - 4q^{11} + 10q^{15} - 2q^{25} + 8q^{31} - 14q^{35} - 4q^{41} - 2q^{45} - 26q^{49} + 28q^{51} - 4q^{59} - 4q^{61} + 2q^{65} + 8q^{69} - 20q^{71} - 18q^{75} - 20q^{79} + 74q^{81} - 22q^{85} + 8q^{89} - 28q^{91} - 32q^{95} + 48q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1340, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1340.2.d.a $$2$$ $$10.700$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+iq^{3}+(-2+i)q^{5}+3iq^{7}+2q^{9}+\cdots$$
1340.2.d.b $$8$$ $$10.700$$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$16$$ $$0$$ $$q+\beta _{6}q^{3}+(2-\beta _{5})q^{5}+(-\beta _{1}-\beta _{5}+\cdots)q^{7}+\cdots$$
1340.2.d.c $$24$$ $$10.700$$ None $$0$$ $$0$$ $$-10$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1340, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1340, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(335, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(670, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ()()
$3$ ($$1 - 5 T^{2} + 9 T^{4}$$)($$1 - 9 T^{2} + 33 T^{4} - 85 T^{6} + 241 T^{8} - 765 T^{10} + 2673 T^{12} - 6561 T^{14} + 6561 T^{16}$$)
$5$ ($$1 + 4 T + 5 T^{2}$$)($$( 1 - 4 T + 5 T^{2} )^{4}$$)
$7$ ($$1 - 5 T^{2} + 49 T^{4}$$)($$1 - 13 T^{2} + 141 T^{4} - 1233 T^{6} + 10809 T^{8} - 60417 T^{10} + 338541 T^{12} - 1529437 T^{14} + 5764801 T^{16}$$)
$11$ ($$( 1 - 6 T + 11 T^{2} )^{2}$$)($$( 1 + 2 T + 4 T^{2} + 26 T^{3} + 230 T^{4} + 286 T^{5} + 484 T^{6} + 2662 T^{7} + 14641 T^{8} )^{2}$$)
$13$ ($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 49 T^{2} + 1257 T^{4} - 23686 T^{6} + 349158 T^{8} - 4002934 T^{10} + 35901177 T^{12} - 236513641 T^{14} + 815730721 T^{16}$$)
$17$ ($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$( 1 - 18 T^{2} + 289 T^{4} )^{4}$$)
$19$ ($$( 1 + 19 T^{2} )^{2}$$)($$( 1 + 3 T + 39 T^{2} + 128 T^{3} + 748 T^{4} + 2432 T^{5} + 14079 T^{6} + 20577 T^{7} + 130321 T^{8} )^{2}$$)
$23$ ($$1 - 30 T^{2} + 529 T^{4}$$)($$1 - 92 T^{2} + 4852 T^{4} - 173476 T^{6} + 4617942 T^{8} - 91768804 T^{10} + 1357788532 T^{12} - 13619301788 T^{14} + 78310985281 T^{16}$$)
$29$ ($$( 1 - 3 T + 29 T^{2} )^{2}$$)($$( 1 + T + 73 T^{2} + 61 T^{3} + 2921 T^{4} + 1769 T^{5} + 61393 T^{6} + 24389 T^{7} + 707281 T^{8} )^{2}$$)
$31$ ($$( 1 + 2 T + 31 T^{2} )^{2}$$)($$( 1 + 2 T + 96 T^{2} + 186 T^{3} + 4046 T^{4} + 5766 T^{5} + 92256 T^{6} + 59582 T^{7} + 923521 T^{8} )^{2}$$)
$37$ ($$1 - 38 T^{2} + 1369 T^{4}$$)($$1 - 212 T^{2} + 22212 T^{4} - 1453452 T^{6} + 64614998 T^{8} - 1989775788 T^{10} + 41628864132 T^{12} - 543933998708 T^{14} + 3512479453921 T^{16}$$)
$41$ ($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 + 4 T + 56 T^{2} + 68 T^{3} + 2094 T^{4} + 2788 T^{5} + 94136 T^{6} + 275684 T^{7} + 2825761 T^{8} )^{2}$$)
$43$ ($$1 - 85 T^{2} + 1849 T^{4}$$)($$1 - 185 T^{2} + 15117 T^{4} - 763065 T^{6} + 32303121 T^{8} - 1410907185 T^{10} + 51682014717 T^{12} - 1169452164065 T^{14} + 11688200277601 T^{16}$$)
$47$ ($$1 - 58 T^{2} + 2209 T^{4}$$)($$1 - 52 T^{2} - 1484 T^{4} + 58164 T^{6} + 2274582 T^{8} + 128484276 T^{10} - 7241446604 T^{12} - 560519197108 T^{14} + 23811286661761 T^{16}$$)
$53$ ($$1 + 15 T^{2} + 2809 T^{4}$$)($$1 - 385 T^{2} + 66777 T^{4} - 6802705 T^{6} + 444793201 T^{8} - 19108798345 T^{10} + 526902649737 T^{12} - 8533279034665 T^{14} + 62259690411361 T^{16}$$)
$59$ ($$( 1 + 5 T + 59 T^{2} )^{2}$$)($$( 1 + T + 37 T^{2} + 151 T^{3} + 5701 T^{4} + 8909 T^{5} + 128797 T^{6} + 205379 T^{7} + 12117361 T^{8} )^{2}$$)
$61$ ($$( 1 + 10 T + 61 T^{2} )^{2}$$)($$( 1 + 216 T^{2} - 8 T^{3} + 18926 T^{4} - 488 T^{5} + 803736 T^{6} + 13845841 T^{8} )^{2}$$)
$67$ ($$1 + T^{2}$$)($$( 1 + T^{2} )^{4}$$)
$71$ ($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 - T + 187 T^{2} - 628 T^{3} + 16000 T^{4} - 44588 T^{5} + 942667 T^{6} - 357911 T^{7} + 25411681 T^{8} )^{2}$$)
$73$ ($$( 1 - 6 T + 73 T^{2} )( 1 + 6 T + 73 T^{2} )$$)($$1 - 492 T^{2} + 111652 T^{4} - 15200276 T^{6} + 1355341942 T^{8} - 81002270804 T^{10} + 3170720404132 T^{12} - 74456439334188 T^{14} + 806460091894081 T^{16}$$)
$79$ ($$( 1 - 2 T + 79 T^{2} )^{2}$$)($$( 1 + 6 T + 164 T^{2} + 1102 T^{3} + 13558 T^{4} + 87058 T^{5} + 1023524 T^{6} + 2958234 T^{7} + 38950081 T^{8} )^{2}$$)
$83$ ($$( 1 - 83 T^{2} )^{2}$$)($$1 - 424 T^{2} + 86652 T^{4} - 11513176 T^{6} + 1107256998 T^{8} - 79314269464 T^{10} + 4112358431292 T^{12} - 138622718308456 T^{14} + 2252292232139041 T^{16}$$)
$89$ ($$( 1 + T + 89 T^{2} )^{2}$$)($$( 1 - 7 T + 209 T^{2} - 1315 T^{3} + 27601 T^{4} - 117035 T^{5} + 1655489 T^{6} - 4934783 T^{7} + 62742241 T^{8} )^{2}$$)
$97$ ($$1 - 25 T^{2} + 9409 T^{4}$$)($$1 - 481 T^{2} + 118921 T^{4} - 19065337 T^{6} + 2172063385 T^{8} - 179385755833 T^{10} + 10527990625801 T^{12} - 400659534370849 T^{14} + 7837433594376961 T^{16}$$)