# Properties

 Label 80.6.a Level 80 Weight 6 Character orbit a Rep. character $$\chi_{80}(1,\cdot)$$ Character field $$\Q$$ Dimension 10 Newform subspaces 9 Sturm bound 72 Trace bound 3

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 80.a (trivial) Character field: $$\Q$$ Newform subspaces: $$9$$ Sturm bound: $$72$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(80))$$.

Total New Old
Modular forms 66 10 56
Cusp forms 54 10 44
Eisenstein series 12 0 12

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

$$2$$$$5$$FrickeDim.
$$+$$$$+$$$$+$$$$2$$
$$+$$$$-$$$$-$$$$3$$
$$-$$$$+$$$$-$$$$3$$
$$-$$$$-$$$$+$$$$2$$
Plus space$$+$$$$4$$
Minus space$$-$$$$6$$

## Trace form

 $$10q + 18q^{3} - 222q^{7} + 854q^{9} + O(q^{10})$$ $$10q + 18q^{3} - 222q^{7} + 854q^{9} + 604q^{11} - 450q^{15} + 1004q^{17} + 4416q^{19} - 820q^{21} - 4010q^{23} + 6250q^{25} + 9564q^{27} + 8052q^{29} - 756q^{31} - 1920q^{33} + 7350q^{35} - 10648q^{37} + 42948q^{39} - 10624q^{41} - 41918q^{43} + 5900q^{45} - 19582q^{47} + 45022q^{49} - 50540q^{51} - 42240q^{53} - 12100q^{55} - 60888q^{57} - 37000q^{59} + 36608q^{61} + 754q^{63} + 21100q^{65} + 70630q^{67} - 84764q^{69} - 64308q^{71} - 106196q^{73} + 11250q^{75} + 67376q^{77} - 22792q^{79} + 88118q^{81} + 229210q^{83} - 66200q^{85} - 34932q^{87} - 138108q^{89} + 24788q^{91} + 329192q^{93} - 72200q^{95} + 40772q^{97} - 32596q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(80))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 5
80.6.a.a $$1$$ $$12.831$$ $$\Q$$ None $$0$$ $$-24$$ $$25$$ $$172$$ $$-$$ $$-$$ $$q-24q^{3}+5^{2}q^{5}+172q^{7}+333q^{9}+\cdots$$
80.6.a.b $$1$$ $$12.831$$ $$\Q$$ None $$0$$ $$-22$$ $$-25$$ $$-218$$ $$-$$ $$+$$ $$q-22q^{3}-5^{2}q^{5}-218q^{7}+241q^{9}+\cdots$$
80.6.a.c $$1$$ $$12.831$$ $$\Q$$ None $$0$$ $$-6$$ $$-25$$ $$118$$ $$-$$ $$+$$ $$q-6q^{3}-5^{2}q^{5}+118q^{7}-207q^{9}+\cdots$$
80.6.a.d $$1$$ $$12.831$$ $$\Q$$ None $$0$$ $$2$$ $$-25$$ $$62$$ $$+$$ $$+$$ $$q+2q^{3}-5^{2}q^{5}+62q^{7}-239q^{9}+\cdots$$
80.6.a.e $$1$$ $$12.831$$ $$\Q$$ None $$0$$ $$4$$ $$25$$ $$-192$$ $$-$$ $$-$$ $$q+4q^{3}+5^{2}q^{5}-192q^{7}-227q^{9}+\cdots$$
80.6.a.f $$1$$ $$12.831$$ $$\Q$$ None $$0$$ $$8$$ $$25$$ $$108$$ $$+$$ $$-$$ $$q+8q^{3}+5^{2}q^{5}+108q^{7}-179q^{9}+\cdots$$
80.6.a.g $$1$$ $$12.831$$ $$\Q$$ None $$0$$ $$18$$ $$-25$$ $$-242$$ $$+$$ $$+$$ $$q+18q^{3}-5^{2}q^{5}-242q^{7}+3^{4}q^{9}+\cdots$$
80.6.a.h $$1$$ $$12.831$$ $$\Q$$ None $$0$$ $$26$$ $$-25$$ $$22$$ $$-$$ $$+$$ $$q+26q^{3}-5^{2}q^{5}+22q^{7}+433q^{9}+\cdots$$
80.6.a.i $$2$$ $$12.831$$ $$\Q(\sqrt{129})$$ None $$0$$ $$12$$ $$50$$ $$-52$$ $$+$$ $$-$$ $$q+(6+\beta )q^{3}+5^{2}q^{5}+(-26+3\beta )q^{7}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(80))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(80)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(5))$$$$^{\oplus 5}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(20))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(40))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 24 T + 243 T^{2}$$)($$1 + 22 T + 243 T^{2}$$)($$1 + 6 T + 243 T^{2}$$)($$1 - 2 T + 243 T^{2}$$)($$1 - 4 T + 243 T^{2}$$)($$1 - 8 T + 243 T^{2}$$)($$1 - 18 T + 243 T^{2}$$)($$1 - 26 T + 243 T^{2}$$)($$1 - 12 T + 6 T^{2} - 2916 T^{3} + 59049 T^{4}$$)
$5$ ($$1 - 25 T$$)($$1 + 25 T$$)($$1 + 25 T$$)($$1 + 25 T$$)($$1 - 25 T$$)($$1 - 25 T$$)($$1 + 25 T$$)($$1 + 25 T$$)($$( 1 - 25 T )^{2}$$)
$7$ ($$1 - 172 T + 16807 T^{2}$$)($$1 + 218 T + 16807 T^{2}$$)($$1 - 118 T + 16807 T^{2}$$)($$1 - 62 T + 16807 T^{2}$$)($$1 + 192 T + 16807 T^{2}$$)($$1 - 108 T + 16807 T^{2}$$)($$1 + 242 T + 16807 T^{2}$$)($$1 - 22 T + 16807 T^{2}$$)($$1 + 52 T + 29646 T^{2} + 873964 T^{3} + 282475249 T^{4}$$)
$11$ ($$1 + 132 T + 161051 T^{2}$$)($$1 - 480 T + 161051 T^{2}$$)($$1 + 192 T + 161051 T^{2}$$)($$1 - 144 T + 161051 T^{2}$$)($$1 - 148 T + 161051 T^{2}$$)($$1 - 604 T + 161051 T^{2}$$)($$1 + 656 T + 161051 T^{2}$$)($$1 - 768 T + 161051 T^{2}$$)($$1 + 560 T + 381926 T^{2} + 90188560 T^{3} + 25937424601 T^{4}$$)
$13$ ($$1 + 946 T + 371293 T^{2}$$)($$1 + 622 T + 371293 T^{2}$$)($$1 - 1106 T + 371293 T^{2}$$)($$1 + 654 T + 371293 T^{2}$$)($$1 - 286 T + 371293 T^{2}$$)($$1 + 306 T + 371293 T^{2}$$)($$1 + 206 T + 371293 T^{2}$$)($$1 + 46 T + 371293 T^{2}$$)($$1 - 1388 T + 1149918 T^{2} - 515354684 T^{3} + 137858491849 T^{4}$$)
$17$ ($$1 + 222 T + 1419857 T^{2}$$)($$1 - 186 T + 1419857 T^{2}$$)($$1 - 762 T + 1419857 T^{2}$$)($$1 + 1190 T + 1419857 T^{2}$$)($$1 + 1678 T + 1419857 T^{2}$$)($$1 - 930 T + 1419857 T^{2}$$)($$1 - 1690 T + 1419857 T^{2}$$)($$1 - 378 T + 1419857 T^{2}$$)($$1 - 148 T - 795706 T^{2} - 210138836 T^{3} + 2015993900449 T^{4}$$)
$19$ ($$1 + 500 T + 2476099 T^{2}$$)($$1 - 1204 T + 2476099 T^{2}$$)($$1 - 2740 T + 2476099 T^{2}$$)($$1 + 556 T + 2476099 T^{2}$$)($$1 + 1060 T + 2476099 T^{2}$$)($$1 - 1324 T + 2476099 T^{2}$$)($$1 - 1364 T + 2476099 T^{2}$$)($$1 + 1100 T + 2476099 T^{2}$$)($$1 - 1000 T + 4533462 T^{2} - 2476099000 T^{3} + 6131066257801 T^{4}$$)
$23$ ($$1 + 3564 T + 6436343 T^{2}$$)($$1 - 3186 T + 6436343 T^{2}$$)($$1 + 1566 T + 6436343 T^{2}$$)($$1 + 2182 T + 6436343 T^{2}$$)($$1 + 2976 T + 6436343 T^{2}$$)($$1 - 852 T + 6436343 T^{2}$$)($$1 + 2198 T + 6436343 T^{2}$$)($$1 - 1986 T + 6436343 T^{2}$$)($$1 - 2452 T + 6569198 T^{2} - 15781913036 T^{3} + 41426511213649 T^{4}$$)
$29$ ($$1 - 2190 T + 20511149 T^{2}$$)($$1 - 5526 T + 20511149 T^{2}$$)($$1 - 5910 T + 20511149 T^{2}$$)($$1 + 1578 T + 20511149 T^{2}$$)($$1 + 3410 T + 20511149 T^{2}$$)($$1 - 5902 T + 20511149 T^{2}$$)($$1 + 2218 T + 20511149 T^{2}$$)($$1 + 5610 T + 20511149 T^{2}$$)($$1 - 1340 T - 8758306 T^{2} - 27484939660 T^{3} + 420707233300201 T^{4}$$)
$31$ ($$1 + 2312 T + 28629151 T^{2}$$)($$1 + 9356 T + 28629151 T^{2}$$)($$1 - 6868 T + 28629151 T^{2}$$)($$1 + 9660 T + 28629151 T^{2}$$)($$1 - 2448 T + 28629151 T^{2}$$)($$1 - 3320 T + 28629151 T^{2}$$)($$1 - 1700 T + 28629151 T^{2}$$)($$1 - 3988 T + 28629151 T^{2}$$)($$1 - 2248 T + 57017022 T^{2} - 64358331448 T^{3} + 819628286980801 T^{4}$$)
$37$ ($$1 + 11242 T + 69343957 T^{2}$$)($$1 - 5618 T + 69343957 T^{2}$$)($$1 + 5518 T + 69343957 T^{2}$$)($$1 + 3534 T + 69343957 T^{2}$$)($$1 - 182 T + 69343957 T^{2}$$)($$1 - 10774 T + 69343957 T^{2}$$)($$1 + 846 T + 69343957 T^{2}$$)($$1 + 142 T + 69343957 T^{2}$$)($$1 + 5940 T + 123434318 T^{2} + 411903104580 T^{3} + 4808584372417849 T^{4}$$)
$41$ ($$1 - 1242 T + 115856201 T^{2}$$)($$1 + 14394 T + 115856201 T^{2}$$)($$1 + 378 T + 115856201 T^{2}$$)($$1 - 7462 T + 115856201 T^{2}$$)($$1 + 9398 T + 115856201 T^{2}$$)($$1 + 17958 T + 115856201 T^{2}$$)($$1 + 1818 T + 115856201 T^{2}$$)($$1 - 1542 T + 115856201 T^{2}$$)($$1 - 23076 T + 352280470 T^{2} - 2673497694276 T^{3} + 13422659310152401 T^{4}$$)
$43$ ($$1 + 20624 T + 147008443 T^{2}$$)($$1 - 370 T + 147008443 T^{2}$$)($$1 - 2434 T + 147008443 T^{2}$$)($$1 - 7114 T + 147008443 T^{2}$$)($$1 - 1244 T + 147008443 T^{2}$$)($$1 + 9264 T + 147008443 T^{2}$$)($$1 + 10534 T + 147008443 T^{2}$$)($$1 - 5026 T + 147008443 T^{2}$$)($$1 + 17684 T + 312898614 T^{2} + 2599697306012 T^{3} + 21611482313284249 T^{4}$$)
$47$ ($$1 + 6588 T + 229345007 T^{2}$$)($$1 + 16146 T + 229345007 T^{2}$$)($$1 + 13122 T + 229345007 T^{2}$$)($$1 - 28294 T + 229345007 T^{2}$$)($$1 - 12088 T + 229345007 T^{2}$$)($$1 - 9796 T + 229345007 T^{2}$$)($$1 + 12074 T + 229345007 T^{2}$$)($$1 + 24738 T + 229345007 T^{2}$$)($$1 - 2908 T + 56660030 T^{2} - 666935280356 T^{3} + 52599132235830049 T^{4}$$)
$53$ ($$1 + 21066 T + 418195493 T^{2}$$)($$1 + 4374 T + 418195493 T^{2}$$)($$1 + 9174 T + 418195493 T^{2}$$)($$1 + 13046 T + 418195493 T^{2}$$)($$1 - 23846 T + 418195493 T^{2}$$)($$1 + 31434 T + 418195493 T^{2}$$)($$1 - 32586 T + 418195493 T^{2}$$)($$1 + 14166 T + 418195493 T^{2}$$)($$1 + 5412 T + 693247822 T^{2} + 2263274008116 T^{3} + 174887470365513049 T^{4}$$)
$59$ ($$1 + 7980 T + 714924299 T^{2}$$)($$1 - 11748 T + 714924299 T^{2}$$)($$1 - 34980 T + 714924299 T^{2}$$)($$1 - 37092 T + 714924299 T^{2}$$)($$1 - 20020 T + 714924299 T^{2}$$)($$1 + 33228 T + 714924299 T^{2}$$)($$1 + 8668 T + 714924299 T^{2}$$)($$1 + 28380 T + 714924299 T^{2}$$)($$1 + 62584 T + 2277965606 T^{2} + 44742822328616 T^{3} + 511116753300641401 T^{4}$$)
$61$ ($$1 - 16622 T + 844596301 T^{2}$$)($$1 - 13202 T + 844596301 T^{2}$$)($$1 + 9838 T + 844596301 T^{2}$$)($$1 - 39570 T + 844596301 T^{2}$$)($$1 - 32302 T + 844596301 T^{2}$$)($$1 + 40210 T + 844596301 T^{2}$$)($$1 + 34670 T + 844596301 T^{2}$$)($$1 - 5522 T + 844596301 T^{2}$$)($$1 - 14108 T + 1110042462 T^{2} - 11915564614508 T^{3} + 713342911662882601 T^{4}$$)
$67$ ($$1 + 1808 T + 1350125107 T^{2}$$)($$1 - 11542 T + 1350125107 T^{2}$$)($$1 + 33722 T + 1350125107 T^{2}$$)($$1 - 56734 T + 1350125107 T^{2}$$)($$1 + 60972 T + 1350125107 T^{2}$$)($$1 + 58864 T + 1350125107 T^{2}$$)($$1 - 47566 T + 1350125107 T^{2}$$)($$1 - 24742 T + 1350125107 T^{2}$$)($$1 - 85412 T + 4371910566 T^{2} - 115316885639084 T^{3} + 1822837804551761449 T^{4}$$)
$71$ ($$1 - 24528 T + 1804229351 T^{2}$$)($$1 - 29532 T + 1804229351 T^{2}$$)($$1 + 70212 T + 1804229351 T^{2}$$)($$1 + 45588 T + 1804229351 T^{2}$$)($$1 - 32648 T + 1804229351 T^{2}$$)($$1 - 55312 T + 1804229351 T^{2}$$)($$1 + 948 T + 1804229351 T^{2}$$)($$1 + 42372 T + 1804229351 T^{2}$$)($$1 + 47208 T + 4011779662 T^{2} + 85174059202008 T^{3} + 3255243551009881201 T^{4}$$)
$73$ ($$1 - 20474 T + 2073071593 T^{2}$$)($$1 - 33698 T + 2073071593 T^{2}$$)($$1 - 21986 T + 2073071593 T^{2}$$)($$1 - 11842 T + 2073071593 T^{2}$$)($$1 + 38774 T + 2073071593 T^{2}$$)($$1 - 27258 T + 2073071593 T^{2}$$)($$1 + 63102 T + 2073071593 T^{2}$$)($$1 + 52126 T + 2073071593 T^{2}$$)($$1 + 67452 T + 4400780438 T^{2} + 139832825091036 T^{3} + 4297625829703557649 T^{4}$$)
$79$ ($$1 - 46240 T + 3077056399 T^{2}$$)($$1 + 31208 T + 3077056399 T^{2}$$)($$1 + 4520 T + 3077056399 T^{2}$$)($$1 + 94216 T + 3077056399 T^{2}$$)($$1 - 33360 T + 3077056399 T^{2}$$)($$1 + 31456 T + 3077056399 T^{2}$$)($$1 + 46536 T + 3077056399 T^{2}$$)($$1 - 39640 T + 3077056399 T^{2}$$)($$1 - 65904 T + 3994274078 T^{2} - 202790324919696 T^{3} + 9468276082626847201 T^{4}$$)
$83$ ($$1 - 51576 T + 3939040643 T^{2}$$)($$1 - 38466 T + 3939040643 T^{2}$$)($$1 - 109074 T + 3939040643 T^{2}$$)($$1 - 31482 T + 3939040643 T^{2}$$)($$1 + 16716 T + 3939040643 T^{2}$$)($$1 + 24552 T + 3939040643 T^{2}$$)($$1 - 88778 T + 3939040643 T^{2}$$)($$1 - 59826 T + 3939040643 T^{2}$$)($$1 + 108724 T + 10572459494 T^{2} + 428268254869532 T^{3} + 15516041187205853449 T^{4}$$)
$89$ ($$1 + 110310 T + 5584059449 T^{2}$$)($$1 - 119514 T + 5584059449 T^{2}$$)($$1 - 38490 T + 5584059449 T^{2}$$)($$1 + 94054 T + 5584059449 T^{2}$$)($$1 - 101370 T + 5584059449 T^{2}$$)($$1 + 90854 T + 5584059449 T^{2}$$)($$1 + 104934 T + 5584059449 T^{2}$$)($$1 - 57690 T + 5584059449 T^{2}$$)($$1 + 55020 T + 10818978262 T^{2} + 307234950883980 T^{3} + 31181719929966183601 T^{4}$$)
$97$ ($$1 + 78382 T + 8587340257 T^{2}$$)($$1 - 94658 T + 8587340257 T^{2}$$)($$1 + 1918 T + 8587340257 T^{2}$$)($$1 - 23714 T + 8587340257 T^{2}$$)($$1 + 119038 T + 8587340257 T^{2}$$)($$1 - 154706 T + 8587340257 T^{2}$$)($$1 + 36254 T + 8587340257 T^{2}$$)($$1 + 144382 T + 8587340257 T^{2}$$)($$1 - 147668 T + 11612429670 T^{2} - 1268075361070676 T^{3} + 73742412689492826049 T^{4}$$)