Properties

Label 10.16.b.a
Level 10
Weight 16
Character orbit 10.b
Analytic conductor 14.269
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 16 \)
Character orbit: \([\chi]\) = 10.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.2693505100\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{38}\cdot 3^{2}\cdot 5^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + \beta_{2} q^{3} -16384 q^{4} + ( 31425 + 34 \beta_{1} - \beta_{4} ) q^{5} + ( -6656 - \beta_{5} ) q^{6} + ( -3769 \beta_{1} + 92 \beta_{2} - 2 \beta_{3} - 7 \beta_{4} - \beta_{6} ) q^{7} + 16384 \beta_{1} q^{8} + ( -5436397 - 14 \beta_{1} - 7 \beta_{3} + 35 \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + \beta_{2} q^{3} -16384 q^{4} + ( 31425 + 34 \beta_{1} - \beta_{4} ) q^{5} + ( -6656 - \beta_{5} ) q^{6} + ( -3769 \beta_{1} + 92 \beta_{2} - 2 \beta_{3} - 7 \beta_{4} - \beta_{6} ) q^{7} + 16384 \beta_{1} q^{8} + ( -5436397 - 14 \beta_{1} - 7 \beta_{3} + 35 \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{9} + ( 550400 - 31347 \beta_{1} + 200 \beta_{2} - 8 \beta_{3} - \beta_{5} + 8 \beta_{6} - 4 \beta_{7} ) q^{10} + ( 11929452 - 76 \beta_{1} - 38 \beta_{3} + 190 \beta_{4} + 41 \beta_{5} + 11 \beta_{7} ) q^{11} -16384 \beta_{2} q^{12} + ( 511638 \beta_{1} + 14712 \beta_{2} + 7 \beta_{3} + 287 \beta_{4} - 84 \beta_{6} ) q^{13} + ( -62418432 + 128 \beta_{1} + 64 \beta_{3} - 320 \beta_{4} - 139 \beta_{5} - 88 \beta_{7} ) q^{14} + ( 681100 - 216903 \beta_{1} + 80050 \beta_{2} + 68 \beta_{3} + 85 \beta_{4} - 379 \beta_{5} - 243 \beta_{6} - 141 \beta_{7} ) q^{15} + 268435456 q^{16} + ( 2355332 \beta_{1} - 255852 \beta_{2} + 574 \beta_{3} + 4814 \beta_{4} - 648 \beta_{6} ) q^{17} + ( 5452281 \beta_{1} + 42448 \beta_{2} + 496 \beta_{3} + 3776 \beta_{4} - 432 \beta_{6} ) q^{18} + ( 809902020 - 3964 \beta_{1} - 1982 \beta_{3} + 9910 \beta_{4} + 3823 \beta_{5} - 611 \beta_{7} ) q^{19} + ( -514867200 - 557056 \beta_{1} + 16384 \beta_{4} ) q^{20} + ( -1845040688 - 8486 \beta_{1} - 4243 \beta_{3} + 21215 \beta_{4} - 5552 \beta_{5} - 2824 \beta_{7} ) q^{21} + ( -11716052 \beta_{1} + 515488 \beta_{2} - 3616 \beta_{3} - 4736 \beta_{4} - 4448 \beta_{6} ) q^{22} + ( 92781241 \beta_{1} - 343494 \beta_{2} - 4322 \beta_{3} - 4357 \beta_{4} - 5751 \beta_{6} ) q^{23} + ( 109051904 + 16384 \beta_{5} ) q^{24} + ( -280231875 - 103464850 \beta_{1} - 1464500 \beta_{2} - 15775 \beta_{3} - 34975 \beta_{4} - 35875 \beta_{5} - 4400 \beta_{6} + 1625 \beta_{7} ) q^{25} + ( 8286065664 + 33152 \beta_{1} + 16576 \beta_{3} - 82880 \beta_{4} - 16910 \beta_{5} - 392 \beta_{7} ) q^{26} + ( -53328978 \beta_{1} + 4743572 \beta_{2} + 13172 \beta_{3} + 2842 \beta_{4} + 21006 \beta_{6} ) q^{27} + ( 61751296 \beta_{1} - 1507328 \beta_{2} + 32768 \beta_{3} + 114688 \beta_{4} + 16384 \beta_{6} ) q^{28} + ( -30568704570 + 11976 \beta_{1} + 5988 \beta_{3} - 29940 \beta_{4} + 225210 \beta_{5} + 7194 \beta_{7} ) q^{29} + ( -4087692800 - 2704604 \beta_{1} - 4984000 \beta_{2} + 97280 \beta_{3} - 38464 \beta_{4} - 86915 \beta_{5} + 25920 \beta_{6} - 3160 \beta_{7} ) q^{30} + ( 65288963152 - 35336 \beta_{1} - 17668 \beta_{3} + 88340 \beta_{4} - 114652 \beta_{5} + 47156 \beta_{7} ) q^{31} -268435456 \beta_{1} q^{32} + ( -629004384 \beta_{1} - 1764676 \beta_{2} - 146636 \beta_{3} - 974236 \beta_{4} + 80352 \beta_{6} ) q^{33} + ( 40320401408 + 322304 \beta_{1} + 161152 \beta_{3} - 805760 \beta_{4} + 244096 \beta_{5} + 17776 \beta_{7} ) q^{34} + ( -228700893300 + 477432534 \beta_{1} + 32802975 \beta_{2} - 199054 \beta_{3} - 270280 \beta_{4} - 523563 \beta_{5} + 18954 \beta_{6} + 1623 \beta_{7} ) q^{35} + ( 89069928448 + 229376 \beta_{1} + 114688 \beta_{3} - 573440 \beta_{4} - 49152 \beta_{5} + 16384 \beta_{7} ) q^{36} + ( 386029922 \beta_{1} - 52867912 \beta_{2} - 282321 \beta_{3} - 1570101 \beta_{4} + 52832 \beta_{6} ) q^{37} + ( -784867596 \beta_{1} + 63129888 \beta_{2} + 266336 \beta_{3} + 1572736 \beta_{4} - 80352 \beta_{6} ) q^{38} + ( -288449478024 - 1155000 \beta_{1} - 577500 \beta_{3} + 2887500 \beta_{4} + 1105218 \beta_{5} + 2550 \beta_{7} ) q^{39} + ( -9017753600 + 513589248 \beta_{1} - 3276800 \beta_{2} + 131072 \beta_{3} + 16384 \beta_{5} - 131072 \beta_{6} + 65536 \beta_{7} ) q^{40} + ( 837389689902 - 836742 \beta_{1} - 418371 \beta_{3} + 2091855 \beta_{4} - 680291 \beta_{5} - 328703 \beta_{7} ) q^{41} + ( 1810650060 \beta_{1} - 78846064 \beta_{2} + 1152304 \beta_{3} + 5695424 \beta_{4} + 22032 \beta_{6} ) q^{42} + ( -135935726 \beta_{1} - 166087517 \beta_{2} - 79132 \beta_{3} + 1284718 \beta_{4} - 560126 \beta_{6} ) q^{43} + ( -195452141568 + 1245184 \beta_{1} + 622592 \beta_{3} - 3112960 \beta_{4} - 671744 \beta_{5} - 180224 \beta_{7} ) q^{44} + ( -1136350934725 + 6488862052 \beta_{1} - 19982000 \beta_{2} + 41865 \beta_{3} + 4759012 \beta_{4} + 1403430 \beta_{5} - 22140 \beta_{6} - 101530 \beta_{7} ) q^{45} + ( 1522341712384 + 1655168 \beta_{1} + 827584 \beta_{3} - 4137920 \beta_{4} + 144997 \beta_{5} - 218888 \beta_{7} ) q^{46} + ( -3850649393 \beta_{1} + 601564752 \beta_{2} - 1743094 \beta_{3} - 3287579 \beta_{4} - 1809297 \beta_{6} ) q^{47} + 268435456 \beta_{2} q^{48} + ( -1976145398193 - 510358 \beta_{1} - 255179 \beta_{3} + 1275895 \beta_{4} - 9152165 \beta_{5} - 451097 \beta_{7} ) q^{49} + ( -1685692800000 + 35031675 \beta_{1} - 605772000 \beta_{2} - 543200 \beta_{3} - 1672000 \beta_{4} + 1218650 \beta_{5} - 453600 \beta_{6} - 543400 \beta_{7} ) q^{50} + ( 5069447456272 - 4762536 \beta_{1} - 2381268 \beta_{3} + 11906340 \beta_{4} + 7972904 \beta_{5} + 1122216 \beta_{7} ) q^{51} + ( -8382676992 \beta_{1} - 241041408 \beta_{2} - 114688 \beta_{3} - 4702208 \beta_{4} + 1376256 \beta_{6} ) q^{52} + ( 24568639362 \beta_{1} - 1278008928 \beta_{2} - 267071 \beta_{3} - 7345231 \beta_{4} + 2003292 \beta_{6} ) q^{53} + ( -905138959360 - 6380288 \beta_{1} - 3190144 \beta_{3} + 15950720 \beta_{4} - 4044690 \beta_{5} + 694928 \beta_{7} ) q^{54} + ( -4968286024900 - 12100693232 \beta_{1} + 1430705500 \beta_{2} + 1411390 \beta_{3} - 17167462 \beta_{4} + 11622105 \beta_{5} + 855460 \beta_{6} + 416295 \beta_{7} ) q^{55} + ( 1022663589888 - 2097152 \beta_{1} - 1048576 \beta_{3} + 5242880 \beta_{4} + 2277376 \beta_{5} + 1441792 \beta_{7} ) q^{56} + ( -76970810088 \beta_{1} - 719092308 \beta_{2} + 6363372 \beta_{3} + 7464372 \beta_{4} + 8117496 \beta_{6} ) q^{57} + ( 32056794666 \beta_{1} + 3649329216 \beta_{2} - 2858304 \beta_{3} - 12966144 \beta_{4} - 441792 \beta_{6} ) q^{58} + ( 473976063660 + 23596012 \beta_{1} + 11798006 \beta_{3} - 58990030 \beta_{4} - 16849755 \beta_{5} + 2819583 \beta_{7} ) q^{59} + ( -11159142400 + 3553738752 \beta_{1} - 1311539200 \beta_{2} - 1114112 \beta_{3} - 1392640 \beta_{4} + 6209536 \beta_{5} + 3981312 \beta_{6} + 2310144 \beta_{7} ) q^{60} + ( 7225078662602 + 45174458 \beta_{1} + 22587229 \beta_{3} - 112936145 \beta_{4} - 9520874 \beta_{5} - 1537538 \beta_{7} ) q^{61} + ( -66175267840 \beta_{1} - 2257844544 \beta_{2} - 17825216 \beta_{3} - 66777856 \beta_{4} - 7449408 \beta_{6} ) q^{62} + ( 39305052207 \beta_{1} - 4060356410 \beta_{2} + 10736218 \beta_{3} + 50117333 \beta_{4} + 1187919 \beta_{6} ) q^{63} -4398046511104 q^{64} + ( 7253549361600 + 156938052302 \beta_{1} + 4418479800 \beta_{2} + 4656573 \beta_{3} + 8493245 \beta_{4} - 14348219 \beta_{5} - 3354048 \beta_{6} + 1279249 \beta_{7} ) q^{65} + ( -10299845773312 - 49624576 \beta_{1} - 24812288 \beta_{3} + 124061440 \beta_{4} + 2467820 \beta_{5} - 5222624 \beta_{7} ) q^{66} + ( -188899662722 \beta_{1} + 4265536171 \beta_{2} + 24753196 \beta_{3} + 150042386 \beta_{4} - 8758802 \beta_{6} ) q^{67} + ( -38589759488 \beta_{1} + 4191879168 \beta_{2} - 9404416 \beta_{3} - 78872576 \beta_{4} + 10616832 \beta_{6} ) q^{68} + ( 7382624541056 - 50717022 \beta_{1} - 25358511 \beta_{3} + 126792555 \beta_{4} + 109548490 \beta_{5} - 5244438 \beta_{7} ) q^{69} + ( 7602152998400 + 225118628812 \beta_{1} - 8746268000 \beta_{2} - 10747040 \beta_{3} + 79400192 \beta_{4} - 33499905 \beta_{5} - 6462560 \beta_{6} - 4683120 \beta_{7} ) q^{70} + ( -30678279427848 - 54005536 \beta_{1} - 27002768 \beta_{3} + 135013840 \beta_{4} - 20555318 \beta_{5} - 363194 \beta_{7} ) q^{71} + ( -89330171904 \beta_{1} - 695468032 \beta_{2} - 8126464 \beta_{3} - 61865984 \beta_{4} + 7077888 \beta_{6} ) q^{72} + ( 82485330952 \beta_{1} - 12059633588 \beta_{2} - 13479536 \beta_{3} + 52357424 \beta_{4} - 39918368 \beta_{6} ) q^{73} + ( 6666386623488 - 56424576 \beta_{1} - 28212288 \beta_{3} + 141061440 \beta_{4} + 51471166 \beta_{5} - 10870184 \beta_{7} ) q^{74} + ( 28306060775000 + 731853000200 \beta_{1} + 992635125 \beta_{2} - 3378800 \beta_{3} + 37798000 \beta_{4} - 133031650 \beta_{5} - 12425400 \beta_{6} - 16062350 \beta_{7} ) q^{75} + ( -13269434695680 + 64946176 \beta_{1} + 32473088 \beta_{3} - 162365440 \beta_{4} - 62636032 \beta_{5} + 10010624 \beta_{7} ) q^{76} + ( -1069495990556 \beta_{1} + 11650434768 \beta_{2} - 39920356 \beta_{3} - 126792176 \beta_{4} - 24269868 \beta_{6} ) q^{77} + ( 295275645432 \beta_{1} + 16934286912 \beta_{2} + 8260800 \beta_{3} + 180883200 \beta_{4} - 46526400 \beta_{6} ) q^{78} + ( 78011825676480 + 152331912 \beta_{1} + 76165956 \beta_{3} - 380829780 \beta_{4} + 88523424 \beta_{5} - 14662152 \beta_{7} ) q^{79} + ( 8435584204800 + 9126805504 \beta_{1} - 268435456 \beta_{4} ) q^{80} + ( -172088950013519 - 59336398 \beta_{1} - 29668199 \beta_{3} + 148340995 \beta_{4} - 71851707 \beta_{5} - 2368487 \beta_{7} ) q^{81} + ( -841550774450 \beta_{1} - 9584412656 \beta_{2} + 132915888 \beta_{3} + 638766528 \beta_{4} + 8604304 \beta_{6} ) q^{82} + ( 1224405036094 \beta_{1} - 11264982525 \beta_{2} + 70686924 \beta_{3} - 56161806 \beta_{4} + 136532142 \beta_{6} ) q^{83} + ( 30229146632192 + 139034624 \beta_{1} + 69517312 \beta_{3} - 347586560 \beta_{4} + 90963968 \beta_{5} + 46268416 \beta_{7} ) q^{84} + ( 136557970305200 + 1515403516424 \beta_{1} + 8610248600 \beta_{2} + 40495066 \beta_{3} + 125060030 \beta_{4} + 121154502 \beta_{5} + 86643284 \beta_{6} + 52057758 \beta_{7} ) q^{85} + ( -1117196837376 + 204959488 \beta_{1} + 102479744 \beta_{3} - 512398720 \beta_{4} + 150172795 \beta_{5} - 7646288 \beta_{7} ) q^{86} + ( -4406798939904 \beta_{1} - 34270751466 \beta_{2} - 203679720 \beta_{3} - 1208346840 \beta_{4} + 63316080 \beta_{6} ) q^{87} + ( 191955795968 \beta_{1} - 8445755392 \beta_{2} + 59244544 \beta_{3} + 77594624 \beta_{4} + 72876032 \beta_{6} ) q^{88} + ( -79406224204710 - 478721320 \beta_{1} - 239360660 \beta_{3} + 1196803300 \beta_{4} - 328821930 \beta_{5} + 81695790 \beta_{7} ) q^{89} + ( 106478054771200 + 1145587006619 \beta_{1} + 22837906600 \beta_{2} + 81783896 \beta_{3} + 121298880 \beta_{4} + 24286137 \beta_{5} - 22870296 \beta_{6} + 19430548 \beta_{7} ) q^{90} + ( -122061463311528 + 57926752 \beta_{1} + 28963376 \beta_{3} - 144816880 \beta_{4} + 232024598 \beta_{5} + 17634578 \beta_{7} ) q^{91} + ( -1520127852544 \beta_{1} + 5627805696 \beta_{2} + 70811648 \beta_{3} + 71385088 \beta_{4} + 94224384 \beta_{6} ) q^{92} + ( 2738778893640 \beta_{1} + 91464330616 \beta_{2} - 528720528 \beta_{3} - 2241849768 \beta_{4} - 133917624 \beta_{6} ) q^{93} + ( -67128798275072 + 471654016 \beta_{1} + 235827008 \beta_{3} - 1179135040 \beta_{4} - 667846711 \beta_{5} - 84198136 \beta_{7} ) q^{94} + ( -232448767075500 + 2898318829480 \beta_{1} - 19095979500 \beta_{2} - 349025810 \beta_{3} - 814129830 \beta_{4} + 618790955 \beta_{5} - 115598340 \beta_{6} - 37099555 \beta_{7} ) q^{95} + ( -1786706395136 - 268435456 \beta_{5} ) q^{96} + ( -6849420087596 \beta_{1} + 10014499660 \beta_{2} + 169150506 \beta_{3} + 879905106 \beta_{4} - 11384192 \beta_{6} ) q^{97} + ( 1916021777805 \beta_{1} - 147168225648 \beta_{2} + 177304112 \beta_{3} + 774542272 \beta_{4} + 37326096 \beta_{6} ) q^{98} + ( 203280853669156 + 117407348 \beta_{1} + 58703674 \beta_{3} - 293518370 \beta_{4} - 1149247977 \beta_{5} - 54225083 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 131072q^{4} + 251400q^{5} - 53248q^{6} - 43491176q^{9} + O(q^{10}) \) \( 8q - 131072q^{4} + 251400q^{5} - 53248q^{6} - 43491176q^{9} + 4403200q^{10} + 95435616q^{11} - 499347456q^{14} + 5448800q^{15} + 2147483648q^{16} + 6479216160q^{19} - 4118937600q^{20} - 14760325504q^{21} + 872415232q^{24} - 2241855000q^{25} + 66288525312q^{26} - 244549636560q^{29} - 32701542400q^{30} + 522311705216q^{31} + 322563211264q^{34} - 1829607146400q^{35} + 712559427584q^{36} - 2307595824192q^{39} - 72142028800q^{40} + 6699117519216q^{41} - 1563617132544q^{44} - 9090807477800q^{45} + 12178733699072q^{46} - 15809163185544q^{49} - 13485542400000q^{50} + 40555579650176q^{51} - 7241111674880q^{54} - 39746288199200q^{55} + 8181308719104q^{56} + 3791808509280q^{59} - 89273139200q^{60} + 57800629300816q^{61} - 35184372088832q^{64} + 58028394892800q^{65} - 82398766186496q^{66} + 59060996328448q^{69} + 60817223987200q^{70} - 245426235422784q^{71} + 53331092987904q^{74} + 226448486200000q^{75} - 106155477565440q^{76} + 624094605411840q^{79} + 67484673638400q^{80} - 1376711600108152q^{81} + 241833173057536q^{84} + 1092463762441600q^{85} - 8937574699008q^{86} - 635249793637680q^{89} + 851824438169600q^{90} - 976491706492224q^{91} - 537030386200576q^{94} - 1859590136604000q^{95} - 14293651161088q^{96} + 1626246829353248q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 8 x^{6} + 6172534 x^{5} + 23752924445 x^{4} + 1095295465934 x^{3} + 14478882731282 x^{2} - 41384481648202964 x + 59143904715457383364\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-64631454965757861480288 \nu^{7} - 190210628213944223576783584 \nu^{6} + 3352317624629619746282965728 \nu^{5} - 446222755875853786209452541216 \nu^{4} - 2695340257204035207363257830796352 \nu^{3} - 6050803925287062975900266285093564736 \nu^{2} - 125266809290974978921494641595461945088 \nu + 1649429145461498232805098793733374695936\)\()/ \)\(22\!\cdots\!75\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-14849096797892152834494293 \nu^{7} + 354073482124671874636786426 \nu^{6} - 5317755617417827568909290542 \nu^{5} - 90420906995745184998648633140676 \nu^{4} - 465882990321225458024151453955165497 \nu^{3} - 6782502632577644273726423038346708946 \nu^{2} - 22759233124909892127626956035916547576168 \nu + 297838293235491922687608089510161970469846\)\()/ \)\(22\!\cdots\!75\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-1769903380506502482086107286897 \nu^{7} + 238182762673673343382098694213871254 \nu^{6} - 53529602259395671192261513750731300118 \nu^{5} - 8818841841975084646175858020529421283704 \nu^{4} + 3292653572766033699957736997017248396862987 \nu^{3} + 3389304839722019432663863577307267831110211166 \nu^{2} - 635920359442498099345410614196408743527079352472 \nu - 136439319949381249750328781743597133595299295799616\)\()/ \)\(22\!\cdots\!25\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-137441371795353580927749672367061 \nu^{7} + 49967997736650698682803306710426102 \nu^{6} - 3419834934250262012610859159784264134 \nu^{5} + 1144296532317881669184519650031886419048 \nu^{4} - 2197919560728568685327124438654241273240769 \nu^{3} + 608397161877261829514559146784331473259764358 \nu^{2} + 25269916056081197447145737826710696919810807464 \nu + 28579328697541907910618001537611874964055827921292\)\()/ \)\(22\!\cdots\!25\)\( \)
\(\beta_{5}\)\(=\)\((\)\(99067145033162432 \nu^{7} - 1743882156830683584 \nu^{6} + 24592864898365207488 \nu^{5} + 603421399371408949197504 \nu^{4} + 3110334880139552113052771328 \nu^{3} + 64667263621522485826250561664 \nu^{2} - 150858299102349160942460343707648 \nu - 1836803156436590728497470781251584\)\()/ \)\(11\!\cdots\!95\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-32142075671203191908712778485386 \nu^{7} - 3074285292066163118245427802710248 \nu^{6} - 1433026832913019747466545016771064684 \nu^{5} - 64378050826990905672452383605591787852 \nu^{4} - 645368816970190513111189234733849658402194 \nu^{3} - 98958636908456696248421627537584892343207292 \nu^{2} - 26354886517579894646357039566052932902119504736 \nu + 1977616627995112051007771137562731804544875769942\)\()/ \)\(43\!\cdots\!25\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-188766630054076381120864 \nu^{7} + 3186918370360379295519968 \nu^{6} + 13400377645397961613889038624 \nu^{5} - 6376996500022107985945798677408 \nu^{4} - 3267226742586191239272606543014656 \nu^{3} - 85994218297372530146701426096952128 \nu^{2} + 184039733026365880005498727523289565696 \nu - 42433743872592732005318000988208635990432\)\()/ \)\(11\!\cdots\!85\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} - 128 \beta_{2} - 62 \beta_{1} + 1280\)\()/2560\)
\(\nu^{2}\)\(=\)\((\)\(54 \beta_{6} - 472 \beta_{4} - 62 \beta_{3} - 7290 \beta_{2} - 2475629 \beta_{1}\)\()/3200\)
\(\nu^{3}\)\(=\)\((\)\(8389 \beta_{7} + 34614 \beta_{6} + 309201 \beta_{5} + 205568 \beta_{4} - 22038 \beta_{3} - 38443930 \beta_{2} - 131451026 \beta_{1} - 14814107200\)\()/6400\)
\(\nu^{4}\)\(=\)\((\)\(-3148390 \beta_{7} + 19906715 \beta_{5} + 111200480 \beta_{4} - 22240096 \beta_{3} - 44480192 \beta_{1} - 38044183380800\)\()/3200\)
\(\nu^{5}\)\(=\)\((\)\(1560843715 \beta_{7} - 7181039070 \beta_{6} + 42362136545 \beta_{5} + 48712219680 \beta_{4} - 12550804642 \beta_{3} + 5192880502130 \beta_{2} + 39291994283146 \beta_{1} - 4761426176148800\)\()/6400\)
\(\nu^{6}\)\(=\)\((\)\(-227391930747 \beta_{6} + 1857732780786 \beta_{4} + 235111397709 \beta_{3} + 66072745219545 \beta_{2} + 5046855719970688 \beta_{1}\)\()/400\)
\(\nu^{7}\)\(=\)\((\)\(-225416540685531 \beta_{7} - 1219509347305266 \beta_{6} - 6126880500350879 \beta_{5} - 6681369714705312 \beta_{4} + 1113117429858930 \beta_{3} + 746539641060343870 \beta_{2} + 8535254599303236730 \beta_{1} + 1053281256096782084800\)\()/6400\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−249.000 + 249.000i
−182.037 + 182.037i
149.707 149.707i
283.331 283.331i
283.331 + 283.331i
149.707 + 149.707i
−182.037 182.037i
−249.000 249.000i
128.000i 5042.00i −16384.0 79524.6 155542.i −645376. 3.81841e6i 2.09715e6i −1.10729e7 −1.99094e7 1.01791e7i
9.2 128.000i 3702.75i −16384.0 44593.5 + 168905.i −473951. 2.91723e6i 2.09715e6i 638585. 2.16199e7 5.70797e6i
9.3 128.000i 2932.14i −16384.0 −160703. + 68498.7i 375313. 1.80018e6i 2.09715e6i 5.75148e6 8.76783e6 + 2.05700e7i
9.4 128.000i 5604.61i −16384.0 162285. 64661.7i 717390. 750784.i 2.09715e6i −1.70628e7 −8.27669e6 2.07725e7i
9.5 128.000i 5604.61i −16384.0 162285. + 64661.7i 717390. 750784.i 2.09715e6i −1.70628e7 −8.27669e6 + 2.07725e7i
9.6 128.000i 2932.14i −16384.0 −160703. 68498.7i 375313. 1.80018e6i 2.09715e6i 5.75148e6 8.76783e6 2.05700e7i
9.7 128.000i 3702.75i −16384.0 44593.5 168905.i −473951. 2.91723e6i 2.09715e6i 638585. 2.16199e7 + 5.70797e6i
9.8 128.000i 5042.00i −16384.0 79524.6 + 155542.i −645376. 3.81841e6i 2.09715e6i −1.10729e7 −1.99094e7 + 1.01791e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.16.b.a 8
3.b odd 2 1 90.16.c.c 8
4.b odd 2 1 80.16.c.c 8
5.b even 2 1 inner 10.16.b.a 8
5.c odd 4 1 50.16.a.j 4
5.c odd 4 1 50.16.a.k 4
15.d odd 2 1 90.16.c.c 8
20.d odd 2 1 80.16.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.16.b.a 8 1.a even 1 1 trivial
10.16.b.a 8 5.b even 2 1 inner
50.16.a.j 4 5.c odd 4 1
50.16.a.k 4 5.c odd 4 1
80.16.c.c 8 4.b odd 2 1
80.16.c.c 8 20.d odd 2 1
90.16.c.c 8 3.b odd 2 1
90.16.c.c 8 15.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(10, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 16384 T^{2} )^{4} \)
$3$ \( 1 - 35650040 T^{2} + 1135653285963996 T^{4} - \)\(21\!\cdots\!80\)\( T^{6} + \)\(38\!\cdots\!06\)\( T^{8} - \)\(45\!\cdots\!20\)\( T^{10} + \)\(48\!\cdots\!96\)\( T^{12} - \)\(31\!\cdots\!60\)\( T^{14} + \)\(17\!\cdots\!01\)\( T^{16} \)
$5$ \( 1 - 251400 T + 32721907500 T^{2} + 2834486711625000 T^{3} - \)\(13\!\cdots\!50\)\( T^{4} + \)\(86\!\cdots\!00\)\( T^{5} + \)\(30\!\cdots\!00\)\( T^{6} - \)\(71\!\cdots\!00\)\( T^{7} + \)\(86\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - 11085664447000 T^{2} + \)\(78\!\cdots\!96\)\( T^{4} - \)\(44\!\cdots\!00\)\( T^{6} + \)\(22\!\cdots\!06\)\( T^{8} - \)\(10\!\cdots\!00\)\( T^{10} + \)\(40\!\cdots\!96\)\( T^{12} - \)\(12\!\cdots\!00\)\( T^{14} + \)\(25\!\cdots\!01\)\( T^{16} \)
$11$ \( ( 1 - 47717808 T + 9659899185285228 T^{2} - \)\(39\!\cdots\!56\)\( T^{3} + \)\(58\!\cdots\!70\)\( T^{4} - \)\(16\!\cdots\!56\)\( T^{5} + \)\(16\!\cdots\!28\)\( T^{6} - \)\(34\!\cdots\!08\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} )^{2} \)
$13$ \( 1 - 280229833918042280 T^{2} + \)\(35\!\cdots\!96\)\( T^{4} - \)\(29\!\cdots\!60\)\( T^{6} + \)\(16\!\cdots\!06\)\( T^{8} - \)\(75\!\cdots\!40\)\( T^{10} + \)\(24\!\cdots\!96\)\( T^{12} - \)\(50\!\cdots\!20\)\( T^{14} + \)\(47\!\cdots\!01\)\( T^{16} \)
$17$ \( 1 - 7591262165361338760 T^{2} + \)\(30\!\cdots\!96\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{6} + \)\(43\!\cdots\!06\)\( T^{8} - \)\(10\!\cdots\!80\)\( T^{10} + \)\(20\!\cdots\!96\)\( T^{12} - \)\(41\!\cdots\!40\)\( T^{14} + \)\(45\!\cdots\!01\)\( T^{16} \)
$19$ \( ( 1 - 3239608080 T + 35126038636215448396 T^{2} - \)\(86\!\cdots\!60\)\( T^{3} + \)\(62\!\cdots\!06\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{5} + \)\(80\!\cdots\!96\)\( T^{6} - \)\(11\!\cdots\!20\)\( T^{7} + \)\(53\!\cdots\!01\)\( T^{8} )^{2} \)
$23$ \( 1 - \)\(11\!\cdots\!20\)\( T^{2} + \)\(70\!\cdots\!96\)\( T^{4} - \)\(29\!\cdots\!40\)\( T^{6} + \)\(90\!\cdots\!06\)\( T^{8} - \)\(20\!\cdots\!60\)\( T^{10} + \)\(35\!\cdots\!96\)\( T^{12} - \)\(41\!\cdots\!80\)\( T^{14} + \)\(25\!\cdots\!01\)\( T^{16} \)
$29$ \( ( 1 + 122274818280 T + \)\(65\!\cdots\!96\)\( T^{2} + \)\(72\!\cdots\!60\)\( T^{3} + \)\(78\!\cdots\!06\)\( T^{4} + \)\(62\!\cdots\!40\)\( T^{5} + \)\(49\!\cdots\!96\)\( T^{6} + \)\(78\!\cdots\!20\)\( T^{7} + \)\(55\!\cdots\!01\)\( T^{8} )^{2} \)
$31$ \( ( 1 - 261155852608 T + \)\(50\!\cdots\!28\)\( T^{2} - \)\(97\!\cdots\!56\)\( T^{3} + \)\(12\!\cdots\!70\)\( T^{4} - \)\(22\!\cdots\!56\)\( T^{5} + \)\(27\!\cdots\!28\)\( T^{6} - \)\(33\!\cdots\!08\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} )^{2} \)
$37$ \( 1 - \)\(18\!\cdots\!80\)\( T^{2} + \)\(16\!\cdots\!96\)\( T^{4} - \)\(91\!\cdots\!60\)\( T^{6} + \)\(36\!\cdots\!06\)\( T^{8} - \)\(10\!\cdots\!40\)\( T^{10} + \)\(19\!\cdots\!96\)\( T^{12} - \)\(24\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!01\)\( T^{16} \)
$41$ \( ( 1 - 3349558759608 T + \)\(70\!\cdots\!28\)\( T^{2} - \)\(10\!\cdots\!56\)\( T^{3} + \)\(14\!\cdots\!70\)\( T^{4} - \)\(16\!\cdots\!56\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} - \)\(12\!\cdots\!08\)\( T^{7} + \)\(58\!\cdots\!01\)\( T^{8} )^{2} \)
$43$ \( 1 - \)\(19\!\cdots\!00\)\( T^{2} + \)\(17\!\cdots\!96\)\( T^{4} - \)\(98\!\cdots\!00\)\( T^{6} + \)\(37\!\cdots\!06\)\( T^{8} - \)\(99\!\cdots\!00\)\( T^{10} + \)\(17\!\cdots\!96\)\( T^{12} - \)\(19\!\cdots\!00\)\( T^{14} + \)\(10\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - \)\(25\!\cdots\!40\)\( T^{2} + \)\(43\!\cdots\!96\)\( T^{4} - \)\(74\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!06\)\( T^{8} - \)\(10\!\cdots\!20\)\( T^{10} + \)\(93\!\cdots\!96\)\( T^{12} - \)\(77\!\cdots\!60\)\( T^{14} + \)\(44\!\cdots\!01\)\( T^{16} \)
$53$ \( 1 - \)\(35\!\cdots\!40\)\( T^{2} + \)\(69\!\cdots\!96\)\( T^{4} - \)\(85\!\cdots\!80\)\( T^{6} + \)\(74\!\cdots\!06\)\( T^{8} - \)\(45\!\cdots\!20\)\( T^{10} + \)\(19\!\cdots\!96\)\( T^{12} - \)\(54\!\cdots\!60\)\( T^{14} + \)\(81\!\cdots\!01\)\( T^{16} \)
$59$ \( ( 1 - 1895904254640 T + \)\(70\!\cdots\!96\)\( T^{2} - \)\(48\!\cdots\!80\)\( T^{3} + \)\(30\!\cdots\!06\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{5} + \)\(94\!\cdots\!96\)\( T^{6} - \)\(92\!\cdots\!60\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} )^{2} \)
$61$ \( ( 1 - 28900314650408 T + \)\(11\!\cdots\!28\)\( T^{2} - \)\(23\!\cdots\!56\)\( T^{3} + \)\(58\!\cdots\!70\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{5} + \)\(43\!\cdots\!28\)\( T^{6} - \)\(63\!\cdots\!08\)\( T^{7} + \)\(13\!\cdots\!01\)\( T^{8} )^{2} \)
$67$ \( 1 - \)\(97\!\cdots\!60\)\( T^{2} + \)\(50\!\cdots\!96\)\( T^{4} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(49\!\cdots\!06\)\( T^{8} - \)\(10\!\cdots\!80\)\( T^{10} + \)\(18\!\cdots\!96\)\( T^{12} - \)\(21\!\cdots\!40\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} \)
$71$ \( ( 1 + 122713117711392 T + \)\(26\!\cdots\!28\)\( T^{2} + \)\(20\!\cdots\!44\)\( T^{3} + \)\(24\!\cdots\!70\)\( T^{4} + \)\(12\!\cdots\!44\)\( T^{5} + \)\(92\!\cdots\!28\)\( T^{6} + \)\(24\!\cdots\!92\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} )^{2} \)
$73$ \( 1 - \)\(40\!\cdots\!20\)\( T^{2} + \)\(86\!\cdots\!96\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!06\)\( T^{8} - \)\(97\!\cdots\!60\)\( T^{10} + \)\(54\!\cdots\!96\)\( T^{12} - \)\(20\!\cdots\!80\)\( T^{14} + \)\(39\!\cdots\!01\)\( T^{16} \)
$79$ \( ( 1 - 312047302705920 T + \)\(12\!\cdots\!96\)\( T^{2} - \)\(22\!\cdots\!40\)\( T^{3} + \)\(52\!\cdots\!06\)\( T^{4} - \)\(66\!\cdots\!60\)\( T^{5} + \)\(10\!\cdots\!96\)\( T^{6} - \)\(77\!\cdots\!80\)\( T^{7} + \)\(72\!\cdots\!01\)\( T^{8} )^{2} \)
$83$ \( 1 - \)\(16\!\cdots\!60\)\( T^{2} + \)\(10\!\cdots\!96\)\( T^{4} - \)\(23\!\cdots\!20\)\( T^{6} - \)\(15\!\cdots\!94\)\( T^{8} - \)\(87\!\cdots\!80\)\( T^{10} + \)\(14\!\cdots\!96\)\( T^{12} - \)\(87\!\cdots\!40\)\( T^{14} + \)\(19\!\cdots\!01\)\( T^{16} \)
$89$ \( ( 1 + 317624896818840 T + \)\(29\!\cdots\!96\)\( T^{2} + \)\(83\!\cdots\!80\)\( T^{3} + \)\(39\!\cdots\!06\)\( T^{4} + \)\(14\!\cdots\!20\)\( T^{5} + \)\(90\!\cdots\!96\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{7} + \)\(91\!\cdots\!01\)\( T^{8} )^{2} \)
$97$ \( 1 - \)\(17\!\cdots\!40\)\( T^{2} + \)\(24\!\cdots\!96\)\( T^{4} - \)\(21\!\cdots\!80\)\( T^{6} + \)\(16\!\cdots\!06\)\( T^{8} - \)\(87\!\cdots\!20\)\( T^{10} + \)\(39\!\cdots\!96\)\( T^{12} - \)\(11\!\cdots\!60\)\( T^{14} + \)\(25\!\cdots\!01\)\( T^{16} \)
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