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

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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:

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 \(S_{9}^{\mathrm{new}}(20, [\chi])\).