Properties

Label 1710.2.t.c
Level $1710$
Weight $2$
Character orbit 1710.t
Analytic conductor $13.654$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 49 x^{16} - 8 x^{15} + 72 x^{13} + 2145 x^{12} - 648 x^{11} + 32 x^{10} - 7056 x^{9} - 11968 x^{8} + 10368 x^{7} + 9344 x^{6} + 18176 x^{5} + 56320 x^{4} + 28160 x^{3} + 8192 x^{2} + 4096 x + 1024\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 570)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{5} q^{2} + \beta_{12} q^{4} + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{7} + \beta_{9} ) q^{5} + ( \beta_{1} - \beta_{5} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{17} ) q^{7} + ( -\beta_{5} - \beta_{11} ) q^{8} +O(q^{10})\) \( q -\beta_{5} q^{2} + \beta_{12} q^{4} + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{7} + \beta_{9} ) q^{5} + ( \beta_{1} - \beta_{5} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{17} ) q^{7} + ( -\beta_{5} - \beta_{11} ) q^{8} + ( -1 + \beta_{1} + \beta_{3} - \beta_{9} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{10} + ( \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{11} + ( -\beta_{5} + \beta_{7} + \beta_{8} - \beta_{11} - \beta_{17} - 2 \beta_{18} ) q^{13} + ( -2 + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{16} - \beta_{17} ) q^{14} + ( -1 + \beta_{12} ) q^{16} + ( \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{14} - \beta_{16} ) q^{17} + ( -1 + 2 \beta_{1} - \beta_{4} - \beta_{6} + \beta_{8} + 2 \beta_{13} + \beta_{15} - \beta_{19} ) q^{19} + ( -\beta_{2} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{16} ) q^{20} + ( \beta_{5} - \beta_{14} + \beta_{16} ) q^{22} + ( -\beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{16} + \beta_{17} ) q^{23} + ( -1 - \beta_{2} + \beta_{3} - \beta_{8} - \beta_{11} + 2 \beta_{12} - \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{25} + ( -1 + 2 \beta_{4} - \beta_{6} ) q^{26} + ( -\beta_{2} - \beta_{8} - 2 \beta_{11} - \beta_{16} + \beta_{17} ) q^{28} + ( -2 \beta_{1} - \beta_{3} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} - 2 \beta_{19} ) q^{29} + ( 1 + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{31} -\beta_{11} q^{32} + ( 2 - \beta_{3} - \beta_{8} - \beta_{15} + 2 \beta_{19} ) q^{34} + ( -2 + \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{9} + 2 \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} + \beta_{19} ) q^{35} + ( \beta_{1} - 2 \beta_{5} - \beta_{8} - \beta_{9} - 3 \beta_{11} - \beta_{14} + 2 \beta_{17} + \beta_{18} ) q^{37} + ( \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{38} + ( -1 + \beta_{1} + \beta_{3} - \beta_{6} - 2 \beta_{9} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{40} + ( 4 - 2 \beta_{3} + 2 \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} - 5 \beta_{12} + 4 \beta_{13} - 3 \beta_{15} + \beta_{16} - \beta_{17} ) q^{41} + ( \beta_{1} + 2 \beta_{2} - 4 \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{14} - \beta_{16} ) q^{43} + ( -1 + \beta_{3} - \beta_{19} ) q^{44} + ( 2 - 2 \beta_{1} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{16} - 2 \beta_{17} ) q^{46} + ( -\beta_{2} + \beta_{5} + \beta_{7} - 2 \beta_{10} - 2 \beta_{11} + \beta_{16} + \beta_{18} ) q^{47} + ( -2 + \beta_{1} + \beta_{4} - 2 \beta_{6} + \beta_{8} + \beta_{13} + \beta_{15} ) q^{49} + ( -1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} + \beta_{18} ) q^{50} + ( -\beta_{1} - \beta_{2} + \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{14} ) q^{52} + ( 3 \beta_{2} + \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} + 2 \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{53} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{11} + \beta_{13} + 2 \beta_{14} - \beta_{17} ) q^{55} + ( -2 - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{56} + ( 2 \beta_{5} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{14} + \beta_{16} + 2 \beta_{17} + 2 \beta_{18} ) q^{58} + ( 2 + \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - 2 \beta_{15} - \beta_{16} + \beta_{17} + \beta_{19} ) q^{59} + ( 3 - 3 \beta_{8} + 3 \beta_{9} - 2 \beta_{12} + 3 \beta_{13} - 6 \beta_{15} ) q^{61} + ( -4 \beta_{1} - 4 \beta_{2} - 2 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{14} - \beta_{16} ) q^{62} - q^{64} + ( -4 + 5 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} + 4 \beta_{8} - 3 \beta_{9} + \beta_{11} + 3 \beta_{13} + \beta_{14} + 3 \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{65} + ( 2 \beta_{2} - 3 \beta_{7} - \beta_{8} + 3 \beta_{10} + 2 \beta_{11} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{67} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{14} - \beta_{17} - 2 \beta_{18} ) q^{68} + ( 2 - \beta_{1} - 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} + \beta_{16} - 3 \beta_{17} - \beta_{18} + \beta_{19} ) q^{70} + ( -7 + 2 \beta_{1} + 5 \beta_{3} - 3 \beta_{4} - 5 \beta_{6} + 2 \beta_{7} + \beta_{8} - 7 \beta_{9} - 2 \beta_{10} + 5 \beta_{12} - \beta_{13} + 6 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} - 3 \beta_{19} ) q^{71} + ( 3 \beta_{1} - 2 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 2 \beta_{14} + \beta_{16} ) q^{73} + ( -3 - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{10} + 2 \beta_{12} + 2 \beta_{13} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{74} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{9} + \beta_{13} + \beta_{15} ) q^{76} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{5} + 2 \beta_{7} - \beta_{8} + 3 \beta_{9} - 3 \beta_{10} - 4 \beta_{11} - \beta_{14} + 2 \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{77} + ( -2 - 2 \beta_{1} + 3 \beta_{3} + \beta_{4} - 3 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{15} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{79} + ( \beta_{1} - \beta_{5} + \beta_{10} - \beta_{16} ) q^{80} + ( -\beta_{2} + 3 \beta_{5} - 3 \beta_{7} - 4 \beta_{8} + \beta_{11} - \beta_{16} + 4 \beta_{17} ) q^{82} + ( -\beta_{1} - 4 \beta_{2} - \beta_{7} - 4 \beta_{8} + \beta_{9} - \beta_{10} - 4 \beta_{11} - 4 \beta_{14} - \beta_{16} + 3 \beta_{17} + 4 \beta_{18} ) q^{83} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + 3 \beta_{7} + 5 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} + 3 \beta_{16} - 4 \beta_{17} + \beta_{18} - 3 \beta_{19} ) q^{85} + ( 3 - 2 \beta_{1} - 3 \beta_{3} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{12} + 2 \beta_{13} - 2 \beta_{15} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{86} + ( -\beta_{2} + \beta_{5} - \beta_{8} - \beta_{14} + \beta_{17} + \beta_{18} ) q^{88} + ( -3 + 2 \beta_{1} + 2 \beta_{3} - \beta_{7} + \beta_{10} + 2 \beta_{12} + \beta_{15} - \beta_{16} + \beta_{17} - 3 \beta_{19} ) q^{89} + ( 1 - 4 \beta_{1} - 5 \beta_{3} + 2 \beta_{7} + 6 \beta_{8} + 5 \beta_{9} - 2 \beta_{10} - 8 \beta_{12} + 5 \beta_{13} - \beta_{15} + 2 \beta_{16} - 2 \beta_{17} - 4 \beta_{19} ) q^{91} + ( -\beta_{1} + \beta_{2} - 3 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{92} + ( -2 + 2 \beta_{1} - \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{94} + ( 5 - 4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - 4 \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} - 4 \beta_{17} - \beta_{18} ) q^{95} + ( \beta_{1} + 4 \beta_{2} - 4 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + \beta_{14} + 2 \beta_{16} ) q^{97} + ( -\beta_{2} + \beta_{7} - 2 \beta_{10} - 2 \beta_{11} - \beta_{14} - \beta_{16} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 10q^{4} + O(q^{10}) \) \( 20q + 10q^{4} - 2q^{10} - 12q^{11} - 10q^{14} - 10q^{16} + 6q^{19} + 14q^{25} - 8q^{29} + 40q^{31} + 12q^{34} - 2q^{35} + 2q^{40} + 14q^{41} - 6q^{44} + 44q^{46} - 8q^{49} + 8q^{50} - 20q^{56} - 8q^{59} + 16q^{61} - 20q^{64} - 40q^{65} + 8q^{70} + 4q^{71} - 26q^{74} + 8q^{79} - 16q^{85} + 20q^{86} + 2q^{89} - 44q^{91} - 32q^{94} + 80q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 49 x^{16} - 8 x^{15} + 72 x^{13} + 2145 x^{12} - 648 x^{11} + 32 x^{10} - 7056 x^{9} - 11968 x^{8} + 10368 x^{7} + 9344 x^{6} + 18176 x^{5} + 56320 x^{4} + 28160 x^{3} + 8192 x^{2} + 4096 x + 1024\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-1247415232500299 \nu^{19} + 14121847713683436 \nu^{18} - 2452242241905736 \nu^{17} - 1345766534397896 \nu^{16} + 65703114624924227 \nu^{15} - 691819782689747060 \nu^{14} + 19186881322460808 \nu^{13} - 7078576395524816 \nu^{12} - 1843195025438242611 \nu^{11} + 31306801912862989668 \nu^{10} - 15100640410954392168 \nu^{9} + 8364294482985111976 \nu^{8} - 76266479634225026904 \nu^{7} - 185075074826781213152 \nu^{6} + 207634332638339783200 \nu^{5} + 54202215092729018944 \nu^{4} + 226095576498151131648 \nu^{3} + 646913736064602024448 \nu^{2} + 74315506995497216 \nu + 7896690386388411392\)\()/ \)\(12\!\cdots\!52\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-3386282915786266757 \nu^{19} + 1405722860355455696 \nu^{18} + 1691818199538348336 \nu^{17} - 3166847588073419904 \nu^{16} + 159444937358634070533 \nu^{15} - 37371725192928414376 \nu^{14} - 106497758856045419280 \nu^{13} - 90380106099253612008 \nu^{12} - 6831828743877138610549 \nu^{11} + 5163070527522030233720 \nu^{10} + 2884616161494605905520 \nu^{9} + 15222516788927210798288 \nu^{8} + 19762356858230169519952 \nu^{7} - 51018372230969002051328 \nu^{6} - 40605238149096354346208 \nu^{5} + 77784244236226642424704 \nu^{4} - 174407667837172815736320 \nu^{3} - 48014712778533209974528 \nu^{2} - 25197546629123584491008 \nu - 93198922890801102759936\)\()/ \)\(10\!\cdots\!76\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-4232101711420771201 \nu^{19} + 2181486017059807400 \nu^{18} - 4124378885648077616 \nu^{17} + 7897757488520220768 \nu^{16} + 203874060607586606945 \nu^{15} - 73137442077379054496 \nu^{14} + 177447029744981331984 \nu^{13} - 646457938616487397544 \nu^{12} - 8781772937239702318673 \nu^{11} + 7146456142840655417264 \nu^{10} - 9377744037444471122096 \nu^{9} + 49144256794412031271536 \nu^{8} + 21103295458928210307824 \nu^{7} - 38635350105783020220544 \nu^{6} - 38809047547892025412960 \nu^{5} - 153076350746732603062912 \nu^{4} - 81038318609659313118720 \nu^{3} - 22795926509410679341824 \nu^{2} - 11496553397312437555712 \nu + 275149990140827868475392\)\()/ \)\(10\!\cdots\!76\)\( \)
\(\beta_{5}\)\(=\)\((\)\(3133260286595853563 \nu^{19} - 8917971619032791856 \nu^{18} + 4476196673936002298 \nu^{17} - 1384964632967199976 \nu^{16} - 153147956971322109391 \nu^{15} + 411566901687289627400 \nu^{14} - 147180711438880372858 \nu^{13} + 256929796626116377360 \nu^{12} + 6070954064014948604319 \nu^{11} - 20831081326414491767240 \nu^{10} + 15345591852660473358458 \nu^{9} - 28211644710196789844200 \nu^{8} + 27257710650423835000892 \nu^{7} + 107001943669920263298480 \nu^{6} - 105079993696826594907168 \nu^{5} + 31913630923549524706080 \nu^{4} + 39990879319161082027072 \nu^{3} - 366215923858014787519488 \nu^{2} + 4751904279189369378688 \nu + 2229839295497973177344\)\()/ \)\(54\!\cdots\!88\)\( \)
\(\beta_{6}\)\(=\)\((\)\(1648552185841481858 \nu^{19} - 868072745834387710 \nu^{18} + 1407550487586609139 \nu^{17} - 3367177978530011772 \nu^{16} - 79066071146916754258 \nu^{15} + 29043521716966332070 \nu^{14} - 60745525849893456627 \nu^{13} + 273164235167645503348 \nu^{12} + 3416635977348824906626 \nu^{11} - 2825268522512545867918 \nu^{10} + 3345480027873051449347 \nu^{9} - 19652502825124211122740 \nu^{8} - 7717350402685912928368 \nu^{7} + 15688567928872081773416 \nu^{6} + 16310030431118385830240 \nu^{5} + 62558402718936493084736 \nu^{4} + 33010525034365549456992 \nu^{3} + 9306741062666398339968 \nu^{2} + 4686268323500600734720 \nu - 52286566420136855591424\)\()/ \)\(27\!\cdots\!44\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-10889321835906579181 \nu^{19} + 20480885828464279044 \nu^{18} - 14508099400000947584 \nu^{17} + 6115691526109346312 \nu^{16} + 526434800852081918149 \nu^{15} - 908100348750920697724 \nu^{14} + 535220462230478409792 \nu^{13} - 965811755004771615856 \nu^{12} - 21616254071927221899477 \nu^{11} + 49545471162119392308300 \nu^{10} - 43767528870466275198432 \nu^{9} + 99449244681413964232280 \nu^{8} - 31860523066686255721560 \nu^{7} - 231871919707911102984928 \nu^{6} + 209477736315213849314528 \nu^{5} - 168539132167513140495040 \nu^{4} - 356775643605785589354240 \nu^{3} + 498778830461939754037760 \nu^{2} - 17947123472360399154944 \nu - 15616434662069267182592\)\()/ \)\(10\!\cdots\!76\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-4348683772245218088 \nu^{19} + 450224447508307223 \nu^{18} - 250293380222287660 \nu^{17} + 252414601602031042 \nu^{16} + 211936627997478730656 \nu^{15} + 13307367314496253097 \nu^{14} + 8336154124993271236 \nu^{13} - 324187926250202751138 \nu^{12} - 9242242633044145305656 \nu^{11} + 3747665020505397420871 \nu^{10} - 940060277648747221732 \nu^{9} + 31353272552620295570722 \nu^{8} + 46262382297897708509688 \nu^{7} - 46795392525082251159280 \nu^{6} - 35851029569796743766544 \nu^{5} - 73660308958475322375328 \nu^{4} - 225492176012277395225344 \nu^{3} - 112287673008321933193152 \nu^{2} - 32741613241344859875840 \nu - 16365664137352620048896\)\()/ \)\(27\!\cdots\!44\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-6169005809303 \nu^{19} + 3083823703934 \nu^{18} - 1600056848304 \nu^{17} + 905444482148 \nu^{16} + 301739777308423 \nu^{15} - 101624356612614 \nu^{14} + 50753283745472 \nu^{13} - 474300220611292 \nu^{12} - 12992549378199047 \nu^{11} + 10490522679024534 \nu^{10} - 5431574304698976 \nu^{9} + 46523908980338900 \nu^{8} + 50324957261568352 \nu^{7} - 88946154925346464 \nu^{6} - 19365916040629696 \nu^{5} - 102172307974395328 \nu^{4} - 295267134460650752 \nu^{3} - 15600348856665216 \nu^{2} - 7910772435618304 \nu - 21232184143012864\)\()/ 34667475668677632 \)
\(\beta_{10}\)\(=\)\((\)\(11647748895677625177 \nu^{19} - 2678420044235495016 \nu^{18} - 1006896450659993441 \nu^{17} + 876149593765785408 \nu^{16} - 570537544771009026203 \nu^{15} + 40308075782636361480 \nu^{14} + 65847772884033547121 \nu^{13} + 806794749290181775616 \nu^{12} + 24771131063838911503947 \nu^{11} - 13481413914046234941392 \nu^{10} + 221704128229926374015 \nu^{9} - 79832070942964433951880 \nu^{8} - 120402255969919478060242 \nu^{7} + 164331049679298755322760 \nu^{6} + 75811190849997163996192 \nu^{5} + 173733884004395553256912 \nu^{4} + 581090240131191728140768 \nu^{3} + 172547640415924211795968 \nu^{2} + 12619980319796627765184 \nu + 22984806829298006831104\)\()/ \)\(27\!\cdots\!44\)\( \)
\(\beta_{11}\)\(=\)\((\)\(31964187768266836033 \nu^{19} - 8697367544490436176 \nu^{18} + 900448895016614446 \nu^{17} - 500586760444575320 \nu^{16} - 1565740371441870903533 \nu^{15} + 168159753848822773048 \nu^{14} + 26614734628992506194 \nu^{13} + 2318093827565198736848 \nu^{12} + 67914806910431957788509 \nu^{11} - 39197278939925200360696 \nu^{10} + 8518184049595333594798 \nu^{9} - 227419429448188289492312 \nu^{8} - 319840854105376902501500 \nu^{7} + 423929463377185973009520 \nu^{6} + 205082585456520813573792 \nu^{5} + 509279017736424524202720 \nu^{4} + 1652902437191837560627904 \nu^{3} + 449127175529839312238592 \nu^{2} + 37275280180998054396032 \nu + 65442086616131240639488\)\()/ \)\(54\!\cdots\!88\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-20734554827161 \nu^{19} + 6169005809303 \nu^{18} - 3083823703934 \nu^{17} + 1600056848304 \nu^{16} + 1015087742048741 \nu^{15} - 135863338691135 \nu^{14} + 101624356612614 \nu^{13} - 1543641231301064 \nu^{12} - 44001319883649053 \nu^{11} + 26428540906199375 \nu^{10} - 11154028433493686 \nu^{9} + 151734593165146992 \nu^{8} + 201627243191123948 \nu^{7} - 265300821709573600 \nu^{6} - 104797525379645920 \nu^{5} - 357505352497848640 \nu^{4} - 1065597819891312192 \nu^{3} - 288617929472203008 \nu^{2} - 154257124287437696 \nu - 42350488467755520\)\()/ 34667475668677632 \)
\(\beta_{13}\)\(=\)\((\)\(-3646259655903443123 \nu^{19} + 1049158889148839496 \nu^{18} - 561240697228913042 \nu^{17} + 22333734971175960 \nu^{16} + 178756408928343520931 \nu^{15} - 22334906367696174976 \nu^{14} + 18774799517307073746 \nu^{13} - 258873072705287923424 \nu^{12} - 7748987885156851181299 \nu^{11} + 4576100104703020725408 \nu^{10} - 1981019837387064789970 \nu^{9} + 26180658128977366937560 \nu^{8} + 36346923085589964549856 \nu^{7} - 46661932408703203780016 \nu^{6} - 17174198126279741012192 \nu^{5} - 62665768111016438619008 \nu^{4} - 190137108819849924869184 \nu^{3} - 51452042331087350676992 \nu^{2} - 32528346917521801635840 \nu - 17696182533796593435648\)\()/ \)\(49\!\cdots\!08\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-46527052292546473825 \nu^{19} - 13575408670476075672 \nu^{18} + 12754703821981264468 \nu^{17} - 3220161156818882080 \nu^{16} + 2281387719715237522501 \nu^{15} + 1036205686456293995312 \nu^{14} - 521292289525887764148 \nu^{13} - 3292602276348860657320 \nu^{12} - 100844332868165628305397 \nu^{11} + 1979670180196761370848 \nu^{10} + 34682937247625982645972 \nu^{9} + 312741418200420979610192 \nu^{8} + 658987268537276688945012 \nu^{7} - 412980240669076675040848 \nu^{6} - 714625482892316831898208 \nu^{5} - 806608960105343683192864 \nu^{4} - 2821712631754951968739584 \nu^{3} - 1815147994886654017121536 \nu^{2} - 51040974120829335778688 \nu - 109576270124468940804608\)\()/ \)\(54\!\cdots\!88\)\( \)
\(\beta_{15}\)\(=\)\((\)\(100048750214915279139 \nu^{19} - 26347596329974818268 \nu^{18} + 13559910524286036752 \nu^{17} - 9559785219930873160 \nu^{16} - 4895119672668795944515 \nu^{15} + 486618328506871659076 \nu^{14} - 457981518467565561040 \nu^{13} + 7576611074740505420320 \nu^{12} + 212450896844449840710707 \nu^{11} - 120247059670561064548404 \nu^{10} + 49005250239774782740912 \nu^{9} - 735964318230292003394168 \nu^{8} - 990727124139759613489872 \nu^{7} + 1243650893808756763834432 \nu^{6} + 557854431525641700600928 \nu^{5} + 1797096405692255675904256 \nu^{4} + 5149244069846542601718784 \nu^{3} + 1599418858853573468076032 \nu^{2} + 745812124127402779890176 \nu + 147877792913541010915328\)\()/ \)\(10\!\cdots\!76\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-59060093438929888077 \nu^{19} + 22096477805655091752 \nu^{18} - 5150082873762744724 \nu^{17} + 2319697375049917824 \nu^{16} + 2893979547600525960065 \nu^{15} - 610061920292864514288 \nu^{14} + 67430556229633727284 \nu^{13} - 4320321462853326166760 \nu^{12} - 125128149124225422722673 \nu^{11} + 85303995485854728439808 \nu^{10} - 26699430163015910787860 \nu^{9} + 425587997041208138986992 \nu^{8} + 549956425935581348941444 \nu^{7} - 840988015348757728234768 \nu^{6} - 294305508105010452269536 \nu^{5} - 934263483799541782017184 \nu^{4} - 2981676149031596296847872 \nu^{3} - 405251627508347704559872 \nu^{2} - 70048591237586813293440 \nu - 118495627306460833513984\)\()/ \)\(54\!\cdots\!88\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-157764561058411272331 \nu^{19} + 66435128268844438016 \nu^{18} - 18010750078952063316 \nu^{17} + 6486893757396609936 \nu^{16} + 7723278107177057031283 \nu^{15} - 1995095069589434426504 \nu^{14} + 344020371171181834228 \nu^{13} - 11533855176405836475208 \nu^{12} - 333355619254332956043779 \nu^{11} + 243588462743034976864600 \nu^{10} - 85846485453464908062868 \nu^{9} + 1140440383014881957128032 \nu^{8} + 1400628750647165447287704 \nu^{7} - 2302457793493065079545824 \nu^{6} - 632305027203562291899296 \nu^{5} - 2459052955740344136274880 \nu^{4} - 7735591835136637047905920 \nu^{3} - 960083458241244776247040 \nu^{2} - 297250893657428925801728 \nu - 338129999525234653437952\)\()/ \)\(10\!\cdots\!76\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-40223802192232846787 \nu^{19} + 9890552072338621158 \nu^{18} + 484772892229500262 \nu^{17} - 195047145168744320 \nu^{16} + 1972023170728282320859 \nu^{15} - 162861158695652904014 \nu^{14} - 103384232028996576086 \nu^{13} - 2889399513322311849544 \nu^{12} - 85612809236527723004667 \nu^{11} + 47294703425592575999630 \nu^{10} - 6677456366575321201994 \nu^{9} + 283440629233831139367616 \nu^{8} + 413628553340261730937576 \nu^{7} - 539467094386649320965120 \nu^{6} - 277266319277329450639776 \nu^{5} - 636215367432115864474176 \nu^{4} - 2084826245599925660715712 \nu^{3} - 567132525750879492500736 \nu^{2} - 46149434678891103030016 \nu - 82433651039597839952384\)\()/ \)\(27\!\cdots\!44\)\( \)
\(\beta_{19}\)\(=\)\((\)\(2227909443849796827 \nu^{19} - 656774505610434658 \nu^{18} + 364647356926557260 \nu^{17} - 213174748204975680 \nu^{16} - 109161387048547456643 \nu^{15} + 14367159029100068202 \nu^{14} - 12743321084439647196 \nu^{13} + 167929933156791916280 \nu^{12} + 4732955902985575909635 \nu^{11} - 2826807527039742448090 \nu^{10} + 1269085700510484763548 \nu^{9} - 16431980908444778161232 \nu^{8} - 21862622741273844294072 \nu^{7} + 28389111128407692155520 \nu^{6} + 10905220420002583240992 \nu^{5} + 40761879076292763448320 \nu^{4} + 114488831094101234698368 \nu^{3} + 30997659803712423966976 \nu^{2} + 16574288883169887321856 \nu + 2186718120392649538560\)\()/ \)\(13\!\cdots\!72\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{16} + \beta_{14} - 4 \beta_{5}\)
\(\nu^{3}\)\(=\)\(7 \beta_{17} - 6 \beta_{16} - \beta_{15} - \beta_{13} + 7 \beta_{10} - \beta_{8} - 6 \beta_{7} + \beta_{6} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(7 \beta_{19} - 2 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{13} + 13 \beta_{12} - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 9 \beta_{3} - 4 \beta_{1} + 9\)
\(\nu^{5}\)\(=\)\(-9 \beta_{17} + 18 \beta_{16} + 9 \beta_{13} - 4 \beta_{12} + 9 \beta_{11} - 47 \beta_{9} + 9 \beta_{8} + 9 \beta_{7} - 9 \beta_{6} + 4 \beta_{5} + 9 \beta_{3} + 9 \beta_{2} + 38 \beta_{1} - 5\)
\(\nu^{6}\)\(=\)\(-47 \beta_{18} - 51 \beta_{17} + 22 \beta_{16} + 47 \beta_{14} - 87 \beta_{11} - 22 \beta_{10} + 4 \beta_{9} + 47 \beta_{8} + 22 \beta_{7} - 134 \beta_{5} + 65 \beta_{2} - 4 \beta_{1}\)
\(\nu^{7}\)\(=\)\(-4 \beta_{19} + 4 \beta_{18} + 250 \beta_{17} - 250 \beta_{16} - 65 \beta_{15} - 48 \beta_{12} - 17 \beta_{11} + 181 \beta_{10} - 315 \beta_{8} - 181 \beta_{7} + 69 \beta_{3} - 69 \beta_{2} + 181 \beta_{1} - 4\)
\(\nu^{8}\)\(=\)\(311 \beta_{19} + 319 \beta_{17} - 319 \beta_{16} - 441 \beta_{15} - 189 \beta_{13} + 413 \beta_{12} + 319 \beta_{10} + 635 \beta_{9} - 125 \beta_{8} - 319 \beta_{7} + 449 \beta_{6} + 311 \beta_{4} - 449 \beta_{3} - 186 \beta_{1} - 94\)
\(\nu^{9}\)\(=\)\(64 \beta_{18} - 8 \beta_{17} + 505 \beta_{16} + 505 \beta_{15} - 64 \beta_{14} + 505 \beta_{13} + 428 \beta_{11} - 64 \beta_{10} - 2119 \beta_{9} + 441 \beta_{8} + 505 \beta_{7} - 513 \beta_{6} + 492 \beta_{5} - 64 \beta_{4} - 72 \beta_{2} + 433 \beta_{1} - 13\)
\(\nu^{10}\)\(=\)\(-2047 \beta_{18} - 2747 \beta_{17} + 3629 \beta_{16} - 3383 \beta_{11} - 1438 \beta_{10} + 2747 \beta_{8} + 556 \beta_{7} + 882 \beta_{5} + 2191 \beta_{2}\)
\(\nu^{11}\)\(=\)\(-700 \beta_{19} - 3629 \beta_{17} + 844 \beta_{15} + 700 \beta_{14} + 3485 \beta_{13} - 3288 \beta_{12} - 2929 \beta_{11} - 6558 \beta_{10} - 988 \beta_{9} - 9994 \beta_{8} + 3773 \beta_{7} - 3773 \beta_{6} - 4276 \beta_{5} - 700 \beta_{4} + 3773 \beta_{3} - 2785 \beta_{2} - 2785 \beta_{1} - 341\)
\(\nu^{12}\)\(=\)\(25857 \beta_{17} - 25857 \beta_{16} - 15167 \beta_{15} - 15167 \beta_{13} + 25857 \beta_{10} + 25857 \beta_{9} - 15167 \beta_{8} - 25857 \beta_{7} + 21025 \beta_{6} + 13479 \beta_{4} + 10690 \beta_{1} - 21751\)
\(\nu^{13}\)\(=\)\(6520 \beta_{19} + 6520 \beta_{18} + 24169 \beta_{17} - 25857 \beta_{16} + 17649 \beta_{15} + 1688 \beta_{13} + 24524 \beta_{12} + 6875 \beta_{11} + 1688 \beta_{9} - 4832 \beta_{8} + 1688 \beta_{7} - 1688 \beta_{5} - 27545 \beta_{3} - 25857 \beta_{2} - 53773 \beta_{1} + 8208\)
\(\nu^{14}\)\(=\)\(105487 \beta_{16} - 89071 \beta_{14} + 16416 \beta_{11} + 16416 \beta_{10} - 55668 \beta_{9} + 55668 \beta_{8} - 38674 \beta_{7} + 285688 \beta_{5} - 22258 \beta_{2} + 55668 \beta_{1}\)
\(\nu^{15}\)\(=\)\(-55668 \beta_{18} - 719327 \beta_{17} + 552330 \beta_{16} + 183413 \beta_{15} + 55668 \beta_{14} + 183413 \beta_{13} - 195104 \beta_{11} - 680075 \beta_{10} - 72084 \beta_{9} + 239081 \beta_{8} + 552330 \beta_{7} - 199829 \beta_{6} - 250772 \beta_{5} - 55668 \beta_{4} + 72084 \beta_{2} - 480246 \beta_{1} + 67359\)
\(\nu^{16}\)\(=\)\(-591575 \beta_{19} + 561930 \beta_{17} - 561930 \beta_{16} + 111322 \beta_{15} - 255490 \beta_{13} - 818429 \beta_{12} + 561930 \beta_{10} - 255490 \beta_{9} - 706098 \beta_{8} - 561930 \beta_{7} + 991233 \beta_{3} + 1123860 \beta_{1} - 847065\)
\(\nu^{17}\)\(=\)\(450608 \beta_{19} + 1297673 \beta_{17} - 2595346 \beta_{16} - 594776 \beta_{15} + 450608 \beta_{14} - 1153505 \beta_{13} + 1286132 \beta_{12} - 847065 \beta_{11} + 450608 \beta_{10} + 4690143 \beta_{9} - 1153505 \beta_{8} - 1153505 \beta_{7} + 1441841 \beta_{6} - 2025076 \beta_{5} + 450608 \beta_{4} - 1441841 \beta_{3} - 702897 \beta_{2} - 3248302 \beta_{1} + 11541\)
\(\nu^{18}\)\(=\)\(3951199 \beta_{18} + 7472267 \beta_{17} - 4025646 \beta_{16} - 3951199 \beta_{14} + 7347879 \beta_{11} + 5215198 \beta_{10} - 3521068 \beta_{9} - 3951199 \beta_{8} - 4025646 \beta_{7} + 11299078 \beta_{5} - 5645329 \beta_{2} + 3521068 \beta_{1}\)
\(\nu^{19}\)\(=\)\(3521068 \beta_{19} - 3521068 \beta_{18} - 25606226 \beta_{17} + 26795778 \beta_{16} + 4455777 \beta_{15} + 1189552 \beta_{13} + 9193256 \beta_{12} - 4737479 \beta_{11} - 17629381 \beta_{10} + 1189552 \beta_{9} + 31251555 \beta_{8} + 16439829 \beta_{7} + 1189552 \beta_{5} - 10355949 \beta_{3} + 9166397 \beta_{2} - 17629381 \beta_{1} + 4710620\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
919.1
0.320085 + 1.19457i
−0.477979 1.78384i
0.686074 + 2.56046i
0.0996880 + 0.372041i
−0.627868 2.34324i
−0.372041 + 0.0996880i
−2.56046 + 0.686074i
2.34324 0.627868i
−1.19457 + 0.320085i
1.78384 0.477979i
0.320085 1.19457i
−0.477979 + 1.78384i
0.686074 2.56046i
0.0996880 0.372041i
−0.627868 + 2.34324i
−0.372041 0.0996880i
−2.56046 0.686074i
2.34324 + 0.627868i
−1.19457 0.320085i
1.78384 + 0.477979i
−0.866025 + 0.500000i 0 0.500000 0.866025i −2.21950 + 0.271659i 0 4.03495i 1.00000i 0 1.78632 1.34502i
919.2 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.919023 + 2.03848i 0 1.07560i 1.00000i 0 −0.223342 2.22489i
919.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.118742 2.23291i 0 2.79875i 1.00000i 0 1.01362 + 1.99313i
919.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.78590 1.34557i 0 2.32136i 1.00000i 0 −0.873846 + 2.05825i
919.5 −0.866025 + 0.500000i 0 0.500000 0.866025i 2.09991 + 0.768349i 0 2.51805i 1.00000i 0 −2.20275 + 0.384547i
919.6 0.866025 0.500000i 0 0.500000 0.866025i −2.05825 + 0.873846i 0 2.32136i 1.00000i 0 −1.34557 + 1.78590i
919.7 0.866025 0.500000i 0 0.500000 0.866025i −1.99313 1.01362i 0 2.79875i 1.00000i 0 −2.23291 + 0.118742i
919.8 0.866025 0.500000i 0 0.500000 0.866025i −0.384547 + 2.20275i 0 2.51805i 1.00000i 0 0.768349 + 2.09991i
919.9 0.866025 0.500000i 0 0.500000 0.866025i 1.34502 1.78632i 0 4.03495i 1.00000i 0 0.271659 2.21950i
919.10 0.866025 0.500000i 0 0.500000 0.866025i 2.22489 + 0.223342i 0 1.07560i 1.00000i 0 2.03848 0.919023i
1189.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −2.21950 0.271659i 0 4.03495i 1.00000i 0 1.78632 + 1.34502i
1189.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.919023 2.03848i 0 1.07560i 1.00000i 0 −0.223342 + 2.22489i
1189.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.118742 + 2.23291i 0 2.79875i 1.00000i 0 1.01362 1.99313i
1189.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.78590 + 1.34557i 0 2.32136i 1.00000i 0 −0.873846 2.05825i
1189.5 −0.866025 0.500000i 0 0.500000 + 0.866025i 2.09991 0.768349i 0 2.51805i 1.00000i 0 −2.20275 0.384547i
1189.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −2.05825 0.873846i 0 2.32136i 1.00000i 0 −1.34557 1.78590i
1189.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.99313 + 1.01362i 0 2.79875i 1.00000i 0 −2.23291 0.118742i
1189.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.384547 2.20275i 0 2.51805i 1.00000i 0 0.768349 2.09991i
1189.9 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.34502 + 1.78632i 0 4.03495i 1.00000i 0 0.271659 + 2.21950i
1189.10 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.22489 0.223342i 0 1.07560i 1.00000i 0 2.03848 + 0.919023i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1189.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.t.c 20
3.b odd 2 1 570.2.q.c 20
5.b even 2 1 inner 1710.2.t.c 20
15.d odd 2 1 570.2.q.c 20
19.c even 3 1 inner 1710.2.t.c 20
57.h odd 6 1 570.2.q.c 20
95.i even 6 1 inner 1710.2.t.c 20
285.n odd 6 1 570.2.q.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.q.c 20 3.b odd 2 1
570.2.q.c 20 15.d odd 2 1
570.2.q.c 20 57.h odd 6 1
570.2.q.c 20 285.n odd 6 1
1710.2.t.c 20 1.a even 1 1 trivial
1710.2.t.c 20 5.b even 2 1 inner
1710.2.t.c 20 19.c even 3 1 inner
1710.2.t.c 20 95.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 37 T_{7}^{8} + 486 T_{7}^{6} + 2834 T_{7}^{4} + 7041 T_{7}^{2} + 5041 \) acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{5} \)
$3$ \( T^{20} \)
$5$ \( 9765625 - 2734375 T^{2} + 78125 T^{4} + 100000 T^{5} + 82500 T^{6} - 56000 T^{7} - 11975 T^{8} + 7040 T^{9} + 469 T^{10} + 1408 T^{11} - 479 T^{12} - 448 T^{13} + 132 T^{14} + 32 T^{15} + 5 T^{16} - 7 T^{18} + T^{20} \)
$7$ \( ( 5041 + 7041 T^{2} + 2834 T^{4} + 486 T^{6} + 37 T^{8} + T^{10} )^{2} \)
$11$ \( ( 20 + 40 T - 27 T^{2} - 13 T^{3} + 3 T^{4} + T^{5} )^{4} \)
$13$ \( 12960000 - 1055808000 T^{2} + 85821940800 T^{4} - 15552797280 T^{6} + 1800048976 T^{8} - 127417516 T^{10} + 6618713 T^{12} - 234906 T^{14} + 6123 T^{16} - 98 T^{18} + T^{20} \)
$17$ \( 65536 - 7487488 T^{2} + 849775616 T^{4} - 646519040 T^{6} + 418209296 T^{8} - 49368480 T^{10} + 4043920 T^{12} - 183128 T^{14} + 5992 T^{16} - 92 T^{18} + T^{20} \)
$19$ \( ( 2476099 - 390963 T - 68590 T^{2} - 8303 T^{3} + 4579 T^{4} + 416 T^{5} + 241 T^{6} - 23 T^{7} - 10 T^{8} - 3 T^{9} + T^{10} )^{2} \)
$23$ \( 283982410000 - 175281468000 T^{2} + 74887978300 T^{4} - 15690636480 T^{6} + 2336334861 T^{8} - 202124527 T^{10} + 12555946 T^{12} - 454523 T^{14} + 11566 T^{16} - 127 T^{18} + T^{20} \)
$29$ \( ( 501264 + 662688 T + 588648 T^{2} + 303552 T^{3} + 117124 T^{4} + 28704 T^{5} + 5476 T^{6} + 596 T^{7} + 70 T^{8} + 4 T^{9} + T^{10} )^{2} \)
$31$ \( ( -1648 - 1776 T + 880 T^{2} - 63 T^{3} - 10 T^{4} + T^{5} )^{4} \)
$37$ \( ( 18769 + 20977 T^{2} + 8286 T^{4} + 1358 T^{6} + 81 T^{8} + T^{10} )^{2} \)
$41$ \( ( 244421956 - 31987164 T + 16146126 T^{2} - 1717950 T^{3} + 690617 T^{4} - 67315 T^{5} + 14334 T^{6} - 795 T^{7} + 154 T^{8} - 7 T^{9} + T^{10} )^{2} \)
$43$ \( 3110228525056 - 1367751098368 T^{2} + 404269887488 T^{4} - 65107314176 T^{6} + 7514928512 T^{8} - 479284944 T^{10} + 21804673 T^{12} - 597638 T^{14} + 11827 T^{16} - 134 T^{18} + T^{20} \)
$47$ \( 72699496960000 - 16441627648000 T^{2} + 2277933900800 T^{4} - 204839636480 T^{6} + 13672733696 T^{8} - 666747648 T^{10} + 24715984 T^{12} - 654992 T^{14} + 12508 T^{16} - 140 T^{18} + T^{20} \)
$53$ \( 341163726137266176 - 36504913561721856 T^{2} + 2477905857494208 T^{4} - 102420693536496 T^{6} + 3082031016961 T^{8} - 63189130523 T^{10} + 959806506 T^{12} - 9906483 T^{14} + 74510 T^{16} - 343 T^{18} + T^{20} \)
$59$ \( ( 6400 - 19200 T + 53440 T^{2} - 22080 T^{3} + 17424 T^{4} + 1120 T^{5} + 3568 T^{6} - 136 T^{7} + 76 T^{8} + 4 T^{9} + T^{10} )^{2} \)
$61$ \( ( 313219204 - 217720796 T + 173780268 T^{2} + 7139292 T^{3} + 4406418 T^{4} - 123918 T^{5} + 54963 T^{6} - 624 T^{7} + 303 T^{8} - 8 T^{9} + T^{10} )^{2} \)
$67$ \( 1068375472506250000 - 108660805659500000 T^{2} + 7325106223930000 T^{4} - 265746538410200 T^{6} + 6843255536416 T^{8} - 118227868560 T^{10} + 1519085153 T^{12} - 13717478 T^{14} + 91139 T^{16} - 382 T^{18} + T^{20} \)
$71$ \( ( 593994384 + 282812688 T + 135189000 T^{2} + 10078440 T^{3} + 2509276 T^{4} + 75452 T^{5} + 33296 T^{6} + 468 T^{7} + 216 T^{8} - 2 T^{9} + T^{10} )^{2} \)
$73$ \( 31556334337506250000 - 2162167162495750000 T^{2} + 94220789523830000 T^{4} - 2487464205894200 T^{6} + 47743459183676 T^{8} - 620049736956 T^{10} + 5961691453 T^{12} - 39048950 T^{14} + 186295 T^{16} - 542 T^{18} + T^{20} \)
$79$ \( ( 263997504 - 44649504 T + 23702016 T^{2} - 2695320 T^{3} + 1381960 T^{4} - 160262 T^{5} + 29117 T^{6} - 1320 T^{7} + 183 T^{8} - 4 T^{9} + T^{10} )^{2} \)
$83$ \( ( 184090624 + 52994048 T^{2} + 2476288 T^{4} + 45124 T^{6} + 356 T^{8} + T^{10} )^{2} \)
$89$ \( ( 13468900 - 8110700 T + 5665810 T^{2} - 373370 T^{3} + 295849 T^{4} - 23745 T^{5} + 11228 T^{6} - 311 T^{7} + 116 T^{8} - T^{9} + T^{10} )^{2} \)
$97$ \( 18509302102818816 - 4855835428208640 T^{2} + 987520707136512 T^{4} - 62834590764288 T^{6} + 2767948486416 T^{8} - 69005700144 T^{10} + 1232336272 T^{12} - 12422072 T^{14} + 90228 T^{16} - 368 T^{18} + T^{20} \)
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