Properties

Label 363.3.b.n
Level $363$
Weight $3$
Character orbit 363.b
Analytic conductor $9.891$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} + 10 x^{10} - 12 x^{9} + 290 x^{8} + 580 x^{7} + 3656 x^{6} + 5424 x^{5} + 6124 x^{4} + 6920 x^{3} - 32528 x^{2} - 23952 x + 48312\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + \beta_{4} q^{3} + ( -4 + \beta_{2} - \beta_{4} - \beta_{7} ) q^{4} + ( -1 - \beta_{2} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{5} + ( -\beta_{1} + \beta_{5} + \beta_{6} - \beta_{10} ) q^{6} + ( \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{7} + ( 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{9} + \beta_{10} ) q^{8} + ( -3 - 2 \beta_{2} - \beta_{4} - 2 \beta_{7} - \beta_{8} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + \beta_{4} q^{3} + ( -4 + \beta_{2} - \beta_{4} - \beta_{7} ) q^{4} + ( -1 - \beta_{2} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{5} + ( -\beta_{1} + \beta_{5} + \beta_{6} - \beta_{10} ) q^{6} + ( \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{7} + ( 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{9} + \beta_{10} ) q^{8} + ( -3 - 2 \beta_{2} - \beta_{4} - 2 \beta_{7} - \beta_{8} ) q^{9} + ( \beta_{9} + 4 \beta_{10} ) q^{10} + ( -6 + 3 \beta_{2} - 3 \beta_{4} + 3 \beta_{8} - \beta_{11} ) q^{12} + ( \beta_{6} + 6 \beta_{9} - \beta_{10} ) q^{13} + ( -1 - \beta_{2} - 4 \beta_{4} + 4 \beta_{7} - 2 \beta_{8} ) q^{14} + ( 9 + 5 \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} + 4 \beta_{11} ) q^{15} + ( 9 - 8 \beta_{2} + 6 \beta_{4} + 6 \beta_{7} ) q^{16} + ( 3 \beta_{1} - 2 \beta_{3} - 4 \beta_{5} + \beta_{6} + 2 \beta_{9} + 2 \beta_{10} ) q^{17} + ( 4 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} + 4 \beta_{6} + 7 \beta_{9} + 3 \beta_{10} ) q^{18} + ( 3 \beta_{6} - 4 \beta_{10} ) q^{19} + ( \beta_{8} - \beta_{11} ) q^{20} + ( 6 \beta_{1} - 5 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{9} + 2 \beta_{10} ) q^{21} + ( 2 + 2 \beta_{2} - \beta_{4} + \beta_{7} - 2 \beta_{8} + 6 \beta_{11} ) q^{23} + ( 8 \beta_{1} + 4 \beta_{3} + \beta_{5} - 4 \beta_{6} - 18 \beta_{9} - 6 \beta_{10} ) q^{24} + ( -3 - \beta_{2} + 7 \beta_{4} + 7 \beta_{7} ) q^{25} + ( -3 \beta_{4} + 3 \beta_{7} + 7 \beta_{8} - 7 \beta_{11} ) q^{26} + ( -18 + 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{11} ) q^{27} + ( -3 \beta_{6} - 2 \beta_{9} ) q^{28} -5 \beta_{1} q^{29} + ( -4 \beta_{1} + 5 \beta_{3} + \beta_{6} + 8 \beta_{9} - 7 \beta_{10} ) q^{30} + ( -1 - 5 \beta_{2} + 2 \beta_{4} + 2 \beta_{7} ) q^{31} + ( -17 \beta_{1} - 8 \beta_{3} + 4 \beta_{5} - 6 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{32} + ( 17 + 11 \beta_{4} + 11 \beta_{7} ) q^{34} + ( -10 \beta_{1} + 8 \beta_{3} - 2 \beta_{5} + 5 \beta_{6} + \beta_{9} + \beta_{10} ) q^{35} + ( 18 + \beta_{2} - 2 \beta_{4} + 7 \beta_{7} + 11 \beta_{8} - 4 \beta_{11} ) q^{36} + ( -18 + 3 \beta_{2} + 2 \beta_{4} + 2 \beta_{7} ) q^{37} + ( -1 - \beta_{2} - 10 \beta_{4} + 10 \beta_{7} + 2 \beta_{8} - 4 \beta_{11} ) q^{38} + ( 5 \beta_{1} + 4 \beta_{3} - \beta_{5} + 7 \beta_{6} + \beta_{9} + 8 \beta_{10} ) q^{39} + ( -2 \beta_{6} - 4 \beta_{9} + 15 \beta_{10} ) q^{40} + ( -7 \beta_{1} + 4 \beta_{3} + 6 \beta_{5} - \beta_{6} - 3 \beta_{9} - 3 \beta_{10} ) q^{41} + ( 42 + 13 \beta_{2} + 3 \beta_{4} + 4 \beta_{7} + 2 \beta_{8} + 3 \beta_{11} ) q^{42} + ( 12 \beta_{6} - 9 \beta_{9} + \beta_{10} ) q^{43} + ( -9 - 4 \beta_{2} + 4 \beta_{4} - \beta_{7} + \beta_{8} - 14 \beta_{11} ) q^{45} + ( -\beta_{6} + 27 \beta_{9} - \beta_{10} ) q^{46} + ( -1 - \beta_{2} - 4 \beta_{4} + 4 \beta_{7} - 10 \beta_{8} + 8 \beta_{11} ) q^{47} + ( 36 - 20 \beta_{2} + 3 \beta_{4} + 4 \beta_{7} - 22 \beta_{8} + 8 \beta_{11} ) q^{48} + ( 18 + 23 \beta_{2} + 3 \beta_{4} + 3 \beta_{7} ) q^{49} + ( -12 \beta_{1} - 2 \beta_{3} + 14 \beta_{5} - 8 \beta_{6} - 7 \beta_{9} - 7 \beta_{10} ) q^{50} + ( 7 \beta_{1} + 5 \beta_{3} + 8 \beta_{5} - 8 \beta_{6} - 3 \beta_{9} - 3 \beta_{10} ) q^{51} + ( -19 \beta_{6} - 35 \beta_{9} - 8 \beta_{10} ) q^{52} + ( -3 - 3 \beta_{2} + 9 \beta_{4} - 9 \beta_{7} - 3 \beta_{8} - 3 \beta_{11} ) q^{53} + ( 20 \beta_{1} + 6 \beta_{3} + \beta_{5} + \beta_{6} + 21 \beta_{9} + 5 \beta_{10} ) q^{54} + ( -7 - 7 \beta_{2} - 10 \beta_{4} + 10 \beta_{7} - 16 \beta_{8} + 2 \beta_{11} ) q^{56} + ( 16 \beta_{1} - 7 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + \beta_{9} + 8 \beta_{10} ) q^{57} + ( -40 + 5 \beta_{2} - 5 \beta_{4} - 5 \beta_{7} ) q^{58} + ( -3 - 3 \beta_{2} + 9 \beta_{4} - 9 \beta_{7} - 6 \beta_{11} ) q^{59} + ( 3 - 2 \beta_{2} + \beta_{4} + \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{60} + ( 4 \beta_{6} + 24 \beta_{9} - 2 \beta_{10} ) q^{61} + ( -8 \beta_{1} - 10 \beta_{3} + 4 \beta_{5} - 7 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{62} + ( 5 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - 10 \beta_{6} + 30 \beta_{9} + 9 \beta_{10} ) q^{63} + ( -108 + 17 \beta_{2} - 11 \beta_{4} - 11 \beta_{7} ) q^{64} + ( \beta_{1} - 8 \beta_{3} - 14 \beta_{5} + 3 \beta_{6} + 7 \beta_{9} + 7 \beta_{10} ) q^{65} + ( -25 - 15 \beta_{2} - 14 \beta_{4} - 14 \beta_{7} ) q^{67} + ( -27 \beta_{1} - 8 \beta_{3} + 6 \beta_{5} - 7 \beta_{6} - 3 \beta_{9} - 3 \beta_{10} ) q^{68} + ( 6 + 4 \beta_{2} - 3 \beta_{4} + 4 \beta_{7} - 7 \beta_{8} - 12 \beta_{11} ) q^{69} + ( -70 - 20 \beta_{2} + 3 \beta_{4} + 3 \beta_{7} ) q^{70} + ( 19 + 19 \beta_{2} + 13 \beta_{4} - 13 \beta_{7} + 30 \beta_{8} + 8 \beta_{11} ) q^{71} + ( -6 \beta_{1} - 17 \beta_{3} - 7 \beta_{5} - 28 \beta_{6} - 46 \beta_{9} - 4 \beta_{10} ) q^{72} + ( -12 \beta_{6} - 7 \beta_{9} - 23 \beta_{10} ) q^{73} + ( 17 \beta_{1} + 6 \beta_{3} + 4 \beta_{5} + \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{74} + ( 42 - 15 \beta_{2} - 10 \beta_{4} - 12 \beta_{7} - 9 \beta_{8} + \beta_{11} ) q^{75} + ( -21 \beta_{6} - 32 \beta_{9} - 6 \beta_{10} ) q^{76} + ( 57 - 8 \beta_{2} + 10 \beta_{4} + 13 \beta_{7} + 17 \beta_{8} + 7 \beta_{11} ) q^{78} + ( -7 \beta_{6} - 13 \beta_{9} - 17 \beta_{10} ) q^{79} + ( 13 + 13 \beta_{2} + 19 \beta_{4} - 19 \beta_{7} + 11 \beta_{8} + 15 \beta_{11} ) q^{80} + ( 24 + 13 \beta_{2} - 19 \beta_{4} - 5 \beta_{7} + 2 \beta_{8} - 6 \beta_{11} ) q^{81} + ( -44 - \beta_{2} - 18 \beta_{4} - 18 \beta_{7} ) q^{82} + ( 2 \beta_{1} + 14 \beta_{3} - 8 \beta_{5} + 11 \beta_{6} + 4 \beta_{9} + 4 \beta_{10} ) q^{83} + ( -12 \beta_{1} + \beta_{3} + 3 \beta_{5} - 5 \beta_{6} - 9 \beta_{9} - 2 \beta_{10} ) q^{84} + ( 11 \beta_{6} - 44 \beta_{9} + 27 \beta_{10} ) q^{85} + ( 13 + 13 \beta_{2} - 23 \beta_{4} + 23 \beta_{7} + 16 \beta_{8} + 10 \beta_{11} ) q^{86} + ( -5 \beta_{1} + 5 \beta_{5} + 5 \beta_{6} - 5 \beta_{10} ) q^{87} + ( 4 + 4 \beta_{2} + 25 \beta_{4} - 25 \beta_{7} - 3 \beta_{8} + 11 \beta_{11} ) q^{89} + ( 2 \beta_{1} + 5 \beta_{3} + 3 \beta_{5} + 13 \beta_{6} - 46 \beta_{9} + 8 \beta_{10} ) q^{90} + ( 3 - 13 \beta_{2} - 4 \beta_{4} - 4 \beta_{7} ) q^{91} + ( 6 + 6 \beta_{2} - 3 \beta_{4} + 3 \beta_{7} + 16 \beta_{8} - 4 \beta_{11} ) q^{92} + ( 12 - 9 \beta_{2} - 3 \beta_{4} + 6 \beta_{7} - 12 \beta_{8} + 5 \beta_{11} ) q^{93} + ( 9 \beta_{6} + 70 \beta_{9} + 16 \beta_{10} ) q^{94} + ( -18 \beta_{1} + 12 \beta_{3} - 10 \beta_{5} + 11 \beta_{6} + 5 \beta_{9} + 5 \beta_{10} ) q^{95} + ( -31 \beta_{1} - 10 \beta_{3} + 11 \beta_{5} + 17 \beta_{6} + 58 \beta_{9} + 9 \beta_{10} ) q^{96} + ( -16 - 19 \beta_{2} + 5 \beta_{4} + 5 \beta_{7} ) q^{97} + ( -\beta_{1} + 46 \beta_{3} + 6 \beta_{5} + 20 \beta_{6} - 3 \beta_{9} - 3 \beta_{10} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} - 44 q^{4} - 12 q^{9} + O(q^{10}) \) \( 12 q - 4 q^{3} - 44 q^{4} - 12 q^{9} - 80 q^{12} + 68 q^{15} + 92 q^{16} - 88 q^{25} - 232 q^{27} - 8 q^{31} + 116 q^{34} + 164 q^{36} - 244 q^{37} + 404 q^{42} - 52 q^{45} + 540 q^{48} + 100 q^{49} - 460 q^{58} + 24 q^{60} - 1276 q^{64} - 128 q^{67} + 128 q^{69} - 784 q^{70} + 684 q^{75} + 528 q^{78} + 348 q^{81} - 380 q^{82} + 120 q^{91} + 196 q^{93} - 156 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} + 10 x^{10} - 12 x^{9} + 290 x^{8} + 580 x^{7} + 3656 x^{6} + 5424 x^{5} + 6124 x^{4} + 6920 x^{3} - 32528 x^{2} - 23952 x + 48312\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(26448881484039 \nu^{11} - 417791761524872 \nu^{10} + 1310082049276159 \nu^{9} - 3024676745468031 \nu^{8} + 13577919523074024 \nu^{7} - 92142433681723612 \nu^{6} - 93463398566058652 \nu^{5} - 1276036664666480508 \nu^{4} - 1938775534922449992 \nu^{3} - 3625740947249225360 \nu^{2} - 1071774178392867524 \nu + 6692372878650359808\)\()/ 3323537421696563124 \)
\(\beta_{2}\)\(=\)\((\)\(-11927197578380 \nu^{11} + 5134113130431 \nu^{10} + 150881956031212 \nu^{9} - 372094129814679 \nu^{8} - 3829018100161522 \nu^{7} - 17613565372016056 \nu^{6} - 35937786948450608 \nu^{5} - 78652747314959200 \nu^{4} - 11957094161013092 \nu^{3} + 258071482856074256 \nu^{2} + 78960387780624720 \nu - 2020820004543909752\)\()/ 1107845807232187708 \)
\(\beta_{3}\)\(=\)\((\)\(8822693138623285 \nu^{11} - 57882282931575184 \nu^{10} + 31447134194477824 \nu^{9} + 300866100101886591 \nu^{8} + 2611326289453704128 \nu^{7} - 6807677848883533112 \nu^{6} - 11819129858771009236 \nu^{5} - 82437644860272245874 \nu^{4} - 257375018632556899976 \nu^{3} - 381555055669710098800 \nu^{2} + 575633285070152337904 \nu + 670893485451951566856\)\()/ \)\(80\!\cdots\!84\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-48975271419376623 \nu^{11} + 363007005409259499 \nu^{10} - 1277766703331862934 \nu^{9} + 2287828660717298340 \nu^{8} - 14915674233345665226 \nu^{7} + 13924240122380881806 \nu^{6} - 106267167179216788776 \nu^{5} + 128969872433151188586 \nu^{4} + 161815700072224958888 \nu^{3} - 1427027257539369392052 \nu^{2} - 534982761765271818720 \nu - 7036509238495228040384\)\()/ \)\(29\!\cdots\!08\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-177706076056576919 \nu^{11} - 2804781998624474028 \nu^{10} + 14858367510231974573 \nu^{9} - 47806983107408448309 \nu^{8} + 28179863062791853082 \nu^{7} - 1187454388064646162504 \nu^{6} - 1800541346662496588552 \nu^{5} - 14161350320107014514722 \nu^{4} - 14044419534094934229728 \nu^{3} - 33370340310114586174464 \nu^{2} - 33103757365332729475324 \nu + 64861604038464242056476\)\()/ \)\(88\!\cdots\!24\)\( \)
\(\beta_{6}\)\(=\)\((\)\(18668369032428349 \nu^{11} - 125391624092071578 \nu^{10} + 394160792924366246 \nu^{9} - 475655019654191499 \nu^{8} + 4608352588862406692 \nu^{7} - 830326065821126658 \nu^{6} + 41372736583160355424 \nu^{5} - 33229403435334680076 \nu^{4} - 27297923529737395292 \nu^{3} + 432306287007330727032 \nu^{2} - 875528847962700872848 \nu - 787224451537901179308\)\()/ \)\(80\!\cdots\!84\)\( \)
\(\beta_{7}\)\(=\)\((\)\(77507244184705304 \nu^{11} - 175490175978103846 \nu^{10} + 30121810996129108 \nu^{9} + 852465132936363965 \nu^{8} + 20760523931491058260 \nu^{7} + 81553012761474194692 \nu^{6} + 312148172520011476812 \nu^{5} + 642682255961817256390 \nu^{4} + 498555667778332357296 \nu^{3} - 1022199286691062364728 \nu^{2} - 893844699712890328176 \nu - 7580546103022258396140\)\()/ \)\(29\!\cdots\!08\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-91632993383928363 \nu^{11} + 310074343020662741 \nu^{10} - 1786541204802508138 \nu^{9} + 5790920913626171749 \nu^{8} - 37809316858156274082 \nu^{7} - 56432339444806846210 \nu^{6} - 702548115947121672664 \nu^{5} - 1012830454341432561618 \nu^{4} - 3593966389810815693144 \nu^{3} - 2999855282911066338732 \nu^{2} + 4210442663611165231104 \nu + 2731580660469197743316\)\()/ \)\(29\!\cdots\!08\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-5455461050 \nu^{11} + 18803987505 \nu^{10} - 36561278785 \nu^{9} + 48997956888 \nu^{8} - 1677987011788 \nu^{7} - 3837116959578 \nu^{6} - 19125164847332 \nu^{5} - 29712476995860 \nu^{4} - 16222688639972 \nu^{3} + 13607393606556 \nu^{2} + 249716816535836 \nu + 54834547813980\)\()/ 171019613016516 \)
\(\beta_{10}\)\(=\)\((\)\(565648472884900591 \nu^{11} - 2543415786466377987 \nu^{10} + 7007217062402978777 \nu^{9} - 9789663250409380644 \nu^{8} + 163024020650920302524 \nu^{7} + 260488675954912819632 \nu^{6} + 1954735263081844532488 \nu^{5} + 2446358227335643520352 \nu^{4} + 1750895972907903512656 \nu^{3} + 2565892016752213042656 \nu^{2} - 30935079782997690260860 \nu - 11717995658357891256420\)\()/ \)\(88\!\cdots\!24\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-535775362755005232 \nu^{11} + 2294504838275490491 \nu^{10} - 7042172449860218080 \nu^{9} + 12333664360019392218 \nu^{8} - 170446860857946380528 \nu^{7} - 236939516904455160008 \nu^{6} - 2234385565980594203824 \nu^{5} - 2894814306606005203966 \nu^{4} - 6962035354557271757960 \nu^{3} - 7148135067883614154012 \nu^{2} + 7761927936870282685008 \nu + 6651844396592211954236\)\()/ \)\(29\!\cdots\!08\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{10} - \beta_{9} + \beta_{7} - \beta_{4} + \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{11} - \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(3 \beta_{11} + 4 \beta_{9} - 6 \beta_{8} + 9 \beta_{7} + 5 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} + 12 \beta_{3} - 6 \beta_{2} - 15 \beta_{1} - 4\)
\(\nu^{4}\)\(=\)\(14 \beta_{11} + 34 \beta_{10} + 60 \beta_{9} - 24 \beta_{8} + 26 \beta_{7} - 14 \beta_{6} + 8 \beta_{5} - 46 \beta_{4} + 44 \beta_{3} - 42 \beta_{2} - 46 \beta_{1} - 120\)
\(\nu^{5}\)\(=\)\(33 \beta_{11} + 277 \beta_{10} + 546 \beta_{9} - 67 \beta_{8} + 10 \beta_{7} - 131 \beta_{6} - 118 \beta_{4} + 60 \beta_{3} - 357 \beta_{2} - 15 \beta_{1} - 876\)
\(\nu^{6}\)\(=\)\(-246 \beta_{11} + 1390 \beta_{10} + 2686 \beta_{9} + 470 \beta_{8} - 826 \beta_{7} - 1084 \beta_{6} - 98 \beta_{5} + 230 \beta_{4} - 718 \beta_{3} - 1706 \beta_{2} + 590 \beta_{1} - 4516\)
\(\nu^{7}\)\(=\)\(-2852 \beta_{11} + 4952 \beta_{10} + 8728 \beta_{9} + 5182 \beta_{8} - 8026 \beta_{7} - 6044 \beta_{6} - 1330 \beta_{5} + 5742 \beta_{4} - 7672 \beta_{3} - 4496 \beta_{2} + 7490 \beta_{1} - 14538\)
\(\nu^{8}\)\(=\)\(-18376 \beta_{11} + 5128 \beta_{10} + 6216 \beta_{9} + 33444 \beta_{8} - 43944 \beta_{7} - 20576 \beta_{6} - 8896 \beta_{5} + 44772 \beta_{4} - 50176 \beta_{3} + 7080 \beta_{2} + 49712 \beta_{1} + 9292\)
\(\nu^{9}\)\(=\)\(-83244 \beta_{11} - 106266 \beta_{10} - 223510 \beta_{9} + 152856 \beta_{8} - 167646 \beta_{7} - 18188 \beta_{6} - 38952 \beta_{5} + 224342 \beta_{4} - 230004 \beta_{3} + 199098 \beta_{2} + 220680 \beta_{1} + 461822\)
\(\nu^{10}\)\(=\)\(-205132 \beta_{11} - 1096120 \beta_{10} - 2171256 \beta_{9} + 373152 \beta_{8} - 245160 \beta_{7} + 462180 \beta_{6} - 101432 \beta_{5} + 745812 \beta_{4} - 575376 \beta_{3} + 1556996 \beta_{2} + 567864 \beta_{1} + 3889544\)
\(\nu^{11}\)\(=\)\(537664 \beta_{11} - 6702328 \beta_{10} - 12815952 \beta_{9} - 983848 \beta_{8} + 2755380 \beta_{7} + 4677200 \beta_{6} + 257048 \beta_{5} + 198560 \beta_{4} + 1491644 \beta_{3} + 8199768 \beta_{2} - 1448656 \beta_{1} + 21809804\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1
−0.156151 + 2.44140i
−2.05018 2.44140i
0.293152 + 2.99036i
4.48261 2.99036i
−1.59440 + 0.311973i
1.02498 0.311973i
1.02498 + 0.311973i
−1.59440 0.311973i
4.48261 + 2.99036i
0.293152 2.99036i
−2.05018 + 2.44140i
−0.156151 2.44140i
3.83317i 1.74344 2.44140i −10.6932 0.620885i −9.35829 6.68291i −1.92745 25.6562i −2.92083 8.51286i −2.37996
122.2 3.83317i 1.74344 + 2.44140i −10.6932 0.620885i 9.35829 6.68291i 1.92745 25.6562i −2.92083 + 8.51286i 2.37996
122.3 2.16906i 0.240294 2.99036i −0.704830 5.33026i −6.48628 0.521212i −12.4613 7.14743i −8.88452 1.43713i 11.5617
122.4 2.16906i 0.240294 + 2.99036i −0.704830 5.33026i 6.48628 0.521212i 12.4613 7.14743i −8.88452 + 1.43713i −11.5617
122.5 1.89788i −2.98373 0.311973i 0.398042 8.25850i −0.592088 + 5.66278i −3.60566 8.34697i 8.80535 + 1.86169i 15.6737
122.6 1.89788i −2.98373 + 0.311973i 0.398042 8.25850i 0.592088 + 5.66278i 3.60566 8.34697i 8.80535 1.86169i −15.6737
122.7 1.89788i −2.98373 0.311973i 0.398042 8.25850i 0.592088 5.66278i 3.60566 8.34697i 8.80535 + 1.86169i −15.6737
122.8 1.89788i −2.98373 + 0.311973i 0.398042 8.25850i −0.592088 5.66278i −3.60566 8.34697i 8.80535 1.86169i 15.6737
122.9 2.16906i 0.240294 2.99036i −0.704830 5.33026i 6.48628 + 0.521212i 12.4613 7.14743i −8.88452 1.43713i −11.5617
122.10 2.16906i 0.240294 + 2.99036i −0.704830 5.33026i −6.48628 + 0.521212i −12.4613 7.14743i −8.88452 + 1.43713i 11.5617
122.11 3.83317i 1.74344 2.44140i −10.6932 0.620885i 9.35829 + 6.68291i 1.92745 25.6562i −2.92083 8.51286i 2.37996
122.12 3.83317i 1.74344 + 2.44140i −10.6932 0.620885i −9.35829 + 6.68291i −1.92745 25.6562i −2.92083 + 8.51286i −2.37996
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 122.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.n 12
3.b odd 2 1 inner 363.3.b.n 12
11.b odd 2 1 inner 363.3.b.n 12
11.c even 5 4 363.3.h.r 48
11.d odd 10 4 363.3.h.r 48
33.d even 2 1 inner 363.3.b.n 12
33.f even 10 4 363.3.h.r 48
33.h odd 10 4 363.3.h.r 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.n 12 1.a even 1 1 trivial
363.3.b.n 12 3.b odd 2 1 inner
363.3.b.n 12 11.b odd 2 1 inner
363.3.b.n 12 33.d even 2 1 inner
363.3.h.r 48 11.c even 5 4
363.3.h.r 48 11.d odd 10 4
363.3.h.r 48 33.f even 10 4
363.3.h.r 48 33.h odd 10 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):

\( T_{2}^{6} + 23 T_{2}^{4} + 139 T_{2}^{2} + 249 \)
\( T_{5}^{6} + 97 T_{5}^{4} + 1975 T_{5}^{2} + 747 \)
\( T_{7}^{6} - 172 T_{7}^{4} + 2644 T_{7}^{2} - 7500 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 249 + 139 T^{2} + 23 T^{4} + T^{6} )^{2} \)
$3$ \( ( 729 + 162 T + 45 T^{2} + 46 T^{3} + 5 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$5$ \( ( 747 + 1975 T^{2} + 97 T^{4} + T^{6} )^{2} \)
$7$ \( ( -7500 + 2644 T^{2} - 172 T^{4} + T^{6} )^{2} \)
$11$ \( T^{12} \)
$13$ \( ( -648675 + 50031 T^{2} - 441 T^{4} + T^{6} )^{2} \)
$17$ \( ( 14462169 + 228355 T^{2} + 971 T^{4} + T^{6} )^{2} \)
$19$ \( ( -20750700 + 251476 T^{2} - 916 T^{4} + T^{6} )^{2} \)
$23$ \( ( 43747308 + 529012 T^{2} + 1384 T^{4} + T^{6} )^{2} \)
$29$ \( ( 3890625 + 86875 T^{2} + 575 T^{4} + T^{6} )^{2} \)
$31$ \( ( -2118 - 432 T + 2 T^{2} + T^{3} )^{4} \)
$37$ \( ( 4085 + 1031 T + 61 T^{2} + T^{3} )^{4} \)
$41$ \( ( 579080625 + 2143027 T^{2} + 2579 T^{4} + T^{6} )^{2} \)
$43$ \( ( -17046433200 + 21354544 T^{2} - 8248 T^{4} + T^{6} )^{2} \)
$47$ \( ( 2578766508 + 8838088 T^{2} + 5728 T^{4} + T^{6} )^{2} \)
$53$ \( ( 3398617683 + 13393431 T^{2} + 7065 T^{4} + T^{6} )^{2} \)
$59$ \( ( 5445630000 + 11912832 T^{2} + 6876 T^{4} + T^{6} )^{2} \)
$61$ \( ( -3248019648 + 10350640 T^{2} - 6436 T^{4} + T^{6} )^{2} \)
$67$ \( ( -186570 - 7092 T + 32 T^{2} + T^{3} )^{4} \)
$71$ \( ( 2030778961200 + 482567872 T^{2} + 38104 T^{4} + T^{6} )^{2} \)
$73$ \( ( -31864090800 + 89698576 T^{2} - 19084 T^{4} + T^{6} )^{2} \)
$79$ \( ( -21498899148 + 27334728 T^{2} - 9492 T^{4} + T^{6} )^{2} \)
$83$ \( ( 128420280900 + 88590148 T^{2} + 17072 T^{4} + T^{6} )^{2} \)
$89$ \( ( 1493575125075 + 484596907 T^{2} + 40801 T^{4} + T^{6} )^{2} \)
$97$ \( ( -85235 - 4953 T + 39 T^{2} + T^{3} )^{4} \)
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