# Properties

 Label 20.9.f.a Level 20 Weight 9 Character orbit 20.f Analytic conductor 8.148 Analytic rank 0 Dimension 8 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$9$$ Character orbit: $$[\chi]$$ = 20.f (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$8.14757220122$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{13}\cdot 5^{3}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -9 - 9 \beta_{1} + \beta_{2} ) q^{3}$$ $$+ ( 112 - 58 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{5}$$ $$+ ( -255 + 253 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{7}$$ $$+ ( -4 + 2648 \beta_{1} - 31 \beta_{2} - 28 \beta_{3} - 6 \beta_{4} - \beta_{5} - 7 \beta_{6} + 4 \beta_{7} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -9 - 9 \beta_{1} + \beta_{2} ) q^{3}$$ $$+ ( 112 - 58 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{5}$$ $$+ ( -255 + 253 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{7}$$ $$+ ( -4 + 2648 \beta_{1} - 31 \beta_{2} - 28 \beta_{3} - 6 \beta_{4} - \beta_{5} - 7 \beta_{6} + 4 \beta_{7} ) q^{9}$$ $$+ ( -45 - 10 \beta_{1} - 8 \beta_{2} + 16 \beta_{3} + 7 \beta_{4} + \beta_{5} + 19 \beta_{6} + 9 \beta_{7} ) q^{11}$$ $$+ ( 4123 + 4127 \beta_{1} + 94 \beta_{2} + 15 \beta_{3} + 26 \beta_{4} - 4 \beta_{5} - 23 \beta_{6} + 15 \beta_{7} ) q^{13}$$ $$+ ( -6251 + 8475 \beta_{1} + 387 \beta_{2} - 315 \beta_{3} - 14 \beta_{4} + \beta_{6} + 25 \beta_{7} ) q^{15}$$ $$+ ( 5594 - 5639 \beta_{1} + 5 \beta_{2} + 621 \beta_{3} - 45 \beta_{4} - 5 \beta_{5} + 35 \beta_{6} + 25 \beta_{7} ) q^{17}$$ $$+ ( -1 + 28846 \beta_{1} - 1199 \beta_{2} - 1187 \beta_{3} + 21 \beta_{4} + 11 \beta_{5} - 13 \beta_{6} + \beta_{7} ) q^{19}$$ $$+ ( 14226 + 75 \beta_{1} - 1279 \beta_{2} + 1258 \beta_{3} + 6 \beta_{4} - 27 \beta_{5} - 123 \beta_{6} - 48 \beta_{7} ) q^{21}$$ $$+ ( -83350 - 83407 \beta_{1} + 1822 \beta_{2} - 100 \beta_{3} - 143 \beta_{4} + 57 \beta_{5} + 214 \beta_{6} - 100 \beta_{7} ) q^{23}$$ $$+ ( 19591 - 3373 \beta_{1} + 2803 \beta_{2} - 1034 \beta_{3} + 72 \beta_{4} - 25 \beta_{5} - 19 \beta_{6} - 200 \beta_{7} ) q^{25}$$ $$+ ( 198140 - 197664 \beta_{1} - 60 \beta_{2} + 3976 \beta_{3} + 476 \beta_{4} + 60 \beta_{5} - 388 \beta_{6} - 268 \beta_{7} ) q^{27}$$ $$+ ( 233 + 22670 \beta_{1} - 1883 \beta_{2} - 2179 \beta_{3} + 107 \beta_{4} - 63 \beta_{5} + 529 \beta_{6} - 233 \beta_{7} ) q^{29}$$ $$+ ( -397079 - 170 \beta_{1} - 1000 \beta_{2} + 700 \beta_{3} - 535 \beta_{4} + 235 \beta_{5} + 105 \beta_{6} - 65 \beta_{7} ) q^{31}$$ $$+ ( -116983 - 116606 \beta_{1} - 4455 \beta_{2} + 45 \beta_{3} - 287 \beta_{4} - 377 \beta_{5} - 799 \beta_{6} + 45 \beta_{7} ) q^{33}$$ $$+ ( 322640 + 748827 \beta_{1} - 5685 \beta_{2} + 426 \beta_{3} - 117 \beta_{4} + 325 \beta_{5} + 157 \beta_{6} + 375 \beta_{7} ) q^{35}$$ $$+ ( 664893 - 665937 \beta_{1} + 330 \beta_{2} - 11727 \beta_{3} - 1044 \beta_{4} - 330 \beta_{5} + 1347 \beta_{6} + 687 \beta_{7} ) q^{37}$$ $$+ ( -915 + 935772 \beta_{1} + 14550 \beta_{2} + 15750 \beta_{3} - 345 \beta_{4} + 285 \beta_{5} - 2115 \beta_{6} + 915 \beta_{7} ) q^{39}$$ $$+ ( -1280589 + 205 \beta_{1} + 17522 \beta_{2} - 15819 \beta_{3} + 2657 \beta_{4} - 954 \beta_{5} + 544 \beta_{6} + 749 \beta_{7} ) q^{41}$$ $$+ ( -1292779 - 1294253 \beta_{1} + 1235 \beta_{2} + 1020 \beta_{3} + 3514 \beta_{4} + 1474 \beta_{5} + 1928 \beta_{6} + 1020 \beta_{7} ) q^{43}$$ $$+ ( 2547801 + 2850119 \beta_{1} - 23647 \beta_{2} + 24412 \beta_{3} - 305 \beta_{4} - 1750 \beta_{5} - 717 \beta_{6} + 875 \beta_{7} ) q^{45}$$ $$+ ( 2392773 - 2395539 \beta_{1} - 1200 \beta_{2} - 37884 \beta_{3} - 2766 \beta_{4} + 1200 \beta_{5} - 1617 \beta_{6} + 783 \beta_{7} ) q^{47}$$ $$+ ( 182 + 2166976 \beta_{1} + 13153 \beta_{2} + 11914 \beta_{3} - 1932 \beta_{4} - 1057 \beta_{5} + 1421 \beta_{6} - 182 \beta_{7} ) q^{49}$$ $$+ ( -5890818 - 1340 \beta_{1} - 4107 \beta_{2} + 1839 \beta_{3} - 4072 \beta_{4} + 1804 \beta_{5} + 876 \beta_{6} - 464 \beta_{7} ) q^{51}$$ $$+ ( -3029414 - 3026081 \beta_{1} - 45997 \beta_{2} - 1760 \beta_{3} - 6853 \beta_{4} - 3333 \beta_{5} - 4906 \beta_{6} - 1760 \beta_{7} ) q^{53}$$ $$+ ( 2678900 + 6376020 \beta_{1} - 3475 \beta_{2} - 29115 \beta_{3} + 1330 \beta_{4} + 4750 \beta_{5} + 1820 \beta_{6} - 3000 \beta_{7} ) q^{55}$$ $$+ ( 11053602 - 11039268 \beta_{1} + 3390 \beta_{2} + 75272 \beta_{3} + 14334 \beta_{4} - 3390 \beta_{5} + 1308 \beta_{6} - 5472 \beta_{7} ) q^{57}$$ $$+ ( 4453 + 3199282 \beta_{1} + 8687 \beta_{2} + 7131 \beta_{3} + 10247 \beta_{4} + 2897 \beta_{5} + 6009 \beta_{6} - 4453 \beta_{7} ) q^{59}$$ $$+ ( -10313733 + 4465 \beta_{1} - 3934 \beta_{2} + 793 \beta_{3} - 2479 \beta_{4} - 662 \beta_{5} - 8268 \beta_{6} - 3803 \beta_{7} ) q^{61}$$ $$+ ( -9707894 - 9710407 \beta_{1} + 113414 \beta_{2} - 3360 \beta_{3} - 4207 \beta_{4} + 2513 \beta_{5} + 8386 \beta_{6} - 3360 \beta_{7} ) q^{63}$$ $$+ ( 8979436 + 11207790 \beta_{1} + 129168 \beta_{2} + 10285 \beta_{3} - 161 \beta_{4} - 5250 \beta_{5} - 1646 \beta_{6} - 2525 \beta_{7} ) q^{65}$$ $$+ ( 12586883 - 12599567 \beta_{1} - 6720 \beta_{2} + 45393 \beta_{3} - 12684 \beta_{4} + 6720 \beta_{5} - 10458 \beta_{6} + 2982 \beta_{7} ) q^{67}$$ $$+ ( -4271 + 16663315 \beta_{1} - 235854 \beta_{2} - 237327 \beta_{3} - 15759 \beta_{4} - 5744 \beta_{5} - 2798 \beta_{6} + 4271 \beta_{7} ) q^{69}$$ $$+ ( -12305007 - 1810 \beta_{1} - 204584 \beta_{2} + 210468 \beta_{3} + 7921 \beta_{4} - 2037 \beta_{5} + 5657 \beta_{6} + 3847 \beta_{7} ) q^{71}$$ $$+ ( -11689687 - 11697123 \beta_{1} + 1924 \beta_{2} + 6990 \beta_{3} + 21416 \beta_{4} + 7436 \beta_{5} + 7882 \beta_{6} + 6990 \beta_{7} ) q^{73}$$ $$+ ( 9503403 + 24444531 \beta_{1} + 25349 \beta_{2} - 200202 \beta_{3} - 5364 \beta_{4} - 5700 \beta_{5} - 4662 \beta_{6} + 12150 \beta_{7} ) q^{75}$$ $$+ ( 16794783 - 16816720 \beta_{1} + 4585 \beta_{2} + 100745 \beta_{3} - 21937 \beta_{4} - 4585 \beta_{5} + 22431 \beta_{6} + 13261 \beta_{7} ) q^{77}$$ $$+ ( -9112 + 13406084 \beta_{1} + 86032 \beta_{2} + 102916 \beta_{3} + 6432 \beta_{4} + 7772 \beta_{5} - 25996 \beta_{6} + 9112 \beta_{7} ) q^{79}$$ $$+ ( -20390976 - 6175 \beta_{1} + 316424 \beta_{2} - 304673 \beta_{3} + 14539 \beta_{4} - 2788 \beta_{5} + 15138 \beta_{6} + 8963 \beta_{7} ) q^{81}$$ $$+ ( -1345897 - 1326401 \beta_{1} + 138917 \beta_{2} + 10680 \beta_{3} + 1864 \beta_{4} - 19496 \beta_{5} - 49672 \beta_{6} + 10680 \beta_{7} ) q^{83}$$ $$+ ( 6486275 + 11541594 \beta_{1} + 84965 \beta_{2} + 240997 \beta_{3} + 2756 \beta_{4} + 27025 \beta_{5} + 18534 \beta_{6} + 8125 \beta_{7} ) q^{85}$$ $$+ ( 20563576 - 20529648 \beta_{1} + 16380 \beta_{2} - 238338 \beta_{3} + 33928 \beta_{4} - 16380 \beta_{5} + 23986 \beta_{6} - 8774 \beta_{7} ) q^{87}$$ $$+ ( 5654 + 4382972 \beta_{1} + 170566 \beta_{2} + 164958 \beta_{3} + 5746 \beta_{4} + 46 \beta_{5} + 11262 \beta_{6} - 5654 \beta_{7} ) q^{89}$$ $$+ ( -16277796 - 16160 \beta_{1} - 145057 \beta_{2} + 130189 \beta_{3} - 30382 \beta_{4} + 15514 \beta_{5} + 16806 \beta_{6} + 646 \beta_{7} ) q^{91}$$ $$+ ( -6095049 - 6097974 \beta_{1} - 708689 \beta_{2} - 12945 \beta_{3} - 22965 \beta_{4} + 2925 \beta_{5} + 18795 \beta_{6} - 12945 \beta_{7} ) q^{93}$$ $$+ ( 10760501 + 3225234 \beta_{1} - 797397 \beta_{2} - 67593 \beta_{3} + 17255 \beta_{4} - 33875 \beta_{5} - 26027 \beta_{6} - 34625 \beta_{7} ) q^{95}$$ $$+ ( -22540049 + 22533123 \beta_{1} - 49530 \beta_{2} - 240602 \beta_{3} - 6926 \beta_{4} + 49530 \beta_{5} - 120362 \beta_{6} - 21302 \beta_{7} ) q^{97}$$ $$+ ( 16577 - 34340848 \beta_{1} + 473888 \beta_{2} + 420644 \beta_{3} - 56757 \beta_{4} - 36667 \beta_{5} + 69821 \beta_{6} - 16577 \beta_{7} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut -\mathstrut 70q^{3}$$ $$\mathstrut +\mathstrut 894q^{5}$$ $$\mathstrut -\mathstrut 2030q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut -\mathstrut 70q^{3}$$ $$\mathstrut +\mathstrut 894q^{5}$$ $$\mathstrut -\mathstrut 2030q^{7}$$ $$\mathstrut -\mathstrut 420q^{11}$$ $$\mathstrut +\mathstrut 33180q^{13}$$ $$\mathstrut -\mathstrut 48478q^{15}$$ $$\mathstrut +\mathstrut 43620q^{17}$$ $$\mathstrut +\mathstrut 108668q^{21}$$ $$\mathstrut -\mathstrut 663270q^{23}$$ $$\mathstrut +\mathstrut 163396q^{25}$$ $$\mathstrut +\mathstrut 1576040q^{27}$$ $$\mathstrut -\mathstrut 3178492q^{31}$$ $$\mathstrut -\mathstrut 944020q^{33}$$ $$\mathstrut +\mathstrut 2571618q^{35}$$ $$\mathstrut +\mathstrut 5344080q^{37}$$ $$\mathstrut -\mathstrut 10185252q^{41}$$ $$\mathstrut -\mathstrut 10342710q^{43}$$ $$\mathstrut +\mathstrut 20284834q^{45}$$ $$\mathstrut +\mathstrut 19232250q^{47}$$ $$\mathstrut -\mathstrut 47126684q^{51}$$ $$\mathstrut -\mathstrut 24320640q^{53}$$ $$\mathstrut +\mathstrut 21483180q^{55}$$ $$\mathstrut +\mathstrut 88218320q^{57}$$ $$\mathstrut -\mathstrut 82515684q^{61}$$ $$\mathstrut -\mathstrut 77441350q^{63}$$ $$\mathstrut +\mathstrut 72045768q^{65}$$ $$\mathstrut +\mathstrut 100675930q^{67}$$ $$\mathstrut -\mathstrut 99290076q^{71}$$ $$\mathstrut -\mathstrut 93528520q^{73}$$ $$\mathstrut +\mathstrut 76524178q^{75}$$ $$\mathstrut +\mathstrut 134199660q^{77}$$ $$\mathstrut -\mathstrut 161920268q^{81}$$ $$\mathstrut -\mathstrut 10450350q^{83}$$ $$\mathstrut +\mathstrut 51676156q^{85}$$ $$\mathstrut +\mathstrut 164801600q^{87}$$ $$\mathstrut -\mathstrut 130681068q^{91}$$ $$\mathstrut -\mathstrut 50183620q^{93}$$ $$\mathstrut +\mathstrut 84367944q^{95}$$ $$\mathstrut -\mathstrut 179570760q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8}\mathstrut -\mathstrut$$ $$4$$ $$x^{7}\mathstrut +\mathstrut$$ $$8$$ $$x^{6}\mathstrut +\mathstrut$$ $$22254$$ $$x^{5}\mathstrut +\mathstrut$$ $$4820745$$ $$x^{4}\mathstrut +\mathstrut$$ $$50131374$$ $$x^{3}\mathstrut +\mathstrut$$ $$307615702$$ $$x^{2}\mathstrut -\mathstrut$$ $$1770757924$$ $$x\mathstrut +\mathstrut$$ $$2405464244$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$17543298991401$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$16868436332617$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$231983879936421$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$392578007792287377$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$85750328228947671494$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$1137909215057435119602$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$7291639495351253703576$$ $$\nu\mathstrut -\mathstrut$$ $$17201080872629180854312$$$$)/$$$$11\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$8397852473178398$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$9230087829528791$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$246098412514904642$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$183559907930723391146$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$41049105751594389826687$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$544847636925257658459946$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$5023198320683908632074548$$ $$\nu\mathstrut -\mathstrut$$ $$4411956177038996224861776$$$$)/$$$$75\!\cdots\!25$$ $$\beta_{3}$$ $$=$$ $$($$$$13457794229710593$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$59832656428792231$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$242583434153172547$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$299911930994464745036$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$64687778407869036729017$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$648976954800072987776086$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$4389568545832306052594368$$ $$\nu\mathstrut -\mathstrut$$ $$17013291057978802265646116$$$$)/$$$$75\!\cdots\!25$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$1028024730304656$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$1833256378746647$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$12425146347198256$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$22540552675418216062$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$5011260047344589826739$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$61939253049816035577182$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$330544148248550425211036$$ $$\nu\mathstrut +\mathstrut$$ $$2351099624835978245751547$$$$)/$$$$30\!\cdots\!25$$ $$\beta_{5}$$ $$=$$ $$($$$$33439120500318801$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$163005900253367417$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$998791185526011929$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$707846830264245732427$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$161762277349949492884594$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$1541673738389898660923402$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$12232929887608677736990976$$ $$\nu\mathstrut -\mathstrut$$ $$111948976722265098250898362$$$$)/$$$$75\!\cdots\!25$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$49991586418956737$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$85892049784663529$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$238364561878400677$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$1121418502899574600349$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$243829259109382716241378$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$3038353980754440303475474$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$20355655902990107229901912$$ $$\nu\mathstrut +\mathstrut$$ $$36593076596292260533747944$$$$)/$$$$60\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$98099736970983353$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$43412380202449001$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$1436445366484139413$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$2192985283134569394281$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$480384742045082514781582$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$6555611328264449485284706$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$45353464038745268052047128$$ $$\nu\mathstrut -\mathstrut$$ $$91270018197703135167207336$$$$)/$$$$60\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$40$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$41$$$$)/80$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$10$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$17$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$21$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$22373$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$11$$$$)/20$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$1186$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$3069$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1051$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$1348$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$7005$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$1000$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$471403$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$335834$$$$)/40$$ $$\nu^{4}$$ $$=$$ $$($$$$-$$$$7210$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$24413$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$8517$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$13436$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$4613$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1264$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$339043$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$19635522$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$2281412$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$4085197$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$3032697$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$6712166$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2158117$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$15529226$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1249726535$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1744275052$$$$)/40$$ $$\nu^{6}$$ $$=$$ $$($$$$90808300$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$137079741$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$111157765$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$332476258$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$76823883$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$172268926$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$228089847159$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$9103682528$$$$)/40$$ $$\nu^{7}$$ $$=$$ $$($$$$7750380034$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$15133586801$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$5187688299$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$11848395692$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$34723292945$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$5351768600$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$5446495667247$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3835969981826$$$$)/40$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$17$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 −7.21849 − 8.21849i 36.4975 + 35.4975i −29.1758 − 30.1758i 1.89682 + 0.896816i −7.21849 + 8.21849i 36.4975 − 35.4975i −29.1758 + 30.1758i 1.89682 − 0.896816i
0 −103.804 103.804i 0 304.194 545.977i 0 −1625.71 + 1625.71i 0 14989.6i 0
13.2 0 −29.4437 29.4437i 0 98.2101 + 617.236i 0 1846.83 1846.83i 0 4827.14i 0
13.3 0 18.1203 + 18.1203i 0 −577.254 239.589i 0 −190.873 + 190.873i 0 5904.31i 0
13.4 0 80.1275 + 80.1275i 0 621.850 62.6699i 0 −1045.24 + 1045.24i 0 6279.84i 0
17.1 0 −103.804 + 103.804i 0 304.194 + 545.977i 0 −1625.71 1625.71i 0 14989.6i 0
17.2 0 −29.4437 + 29.4437i 0 98.2101 617.236i 0 1846.83 + 1846.83i 0 4827.14i 0
17.3 0 18.1203 18.1203i 0 −577.254 + 239.589i 0 −190.873 190.873i 0 5904.31i 0
17.4 0 80.1275 80.1275i 0 621.850 + 62.6699i 0 −1045.24 1045.24i 0 6279.84i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.c Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{9}^{\mathrm{new}}(20, [\chi])$$.