Properties

Label 1110.2.e.d
Level $1110$
Weight $2$
Character orbit 1110.e
Analytic conductor $8.863$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 2 x^{15} + 2 x^{14} + 8 x^{13} + 138 x^{12} - 220 x^{11} + 196 x^{10} + 744 x^{9} + 4241 x^{8} - 3018 x^{7} + 658 x^{6} - 1584 x^{5} + 16372 x^{4} - 18840 x^{3} + 10952 x^{2} - 1184 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{7} q^{3} + q^{4} + \beta_{11} q^{5} + \beta_{7} q^{6} -\beta_{14} q^{7} + q^{8} - q^{9} +O(q^{10})\) \( q + q^{2} + \beta_{7} q^{3} + q^{4} + \beta_{11} q^{5} + \beta_{7} q^{6} -\beta_{14} q^{7} + q^{8} - q^{9} + \beta_{11} q^{10} + \beta_{2} q^{11} + \beta_{7} q^{12} + \beta_{8} q^{13} -\beta_{14} q^{14} -\beta_{6} q^{15} + q^{16} + ( 3 - \beta_{5} - \beta_{6} - \beta_{9} ) q^{17} - q^{18} + ( \beta_{1} + \beta_{3} + \beta_{7} - \beta_{13} ) q^{19} + \beta_{11} q^{20} + \beta_{10} q^{21} + \beta_{2} q^{22} + ( 1 + \beta_{8} + \beta_{9} + \beta_{10} ) q^{23} + \beta_{7} q^{24} + ( \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{15} ) q^{25} + \beta_{8} q^{26} -\beta_{7} q^{27} -\beta_{14} q^{28} + ( -\beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{14} + \beta_{15} ) q^{29} -\beta_{6} q^{30} + ( -\beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{31} + q^{32} + \beta_{13} q^{33} + ( 3 - \beta_{5} - \beta_{6} - \beta_{9} ) q^{34} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{14} + \beta_{15} ) q^{35} - q^{36} + ( 1 + \beta_{7} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{37} + ( \beta_{1} + \beta_{3} + \beta_{7} - \beta_{13} ) q^{38} + \beta_{1} q^{39} + \beta_{11} q^{40} + ( \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{41} + \beta_{10} q^{42} + ( -\beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{43} + \beta_{2} q^{44} -\beta_{11} q^{45} + ( 1 + \beta_{8} + \beta_{9} + \beta_{10} ) q^{46} + ( 2 \beta_{1} - \beta_{4} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{47} + \beta_{7} q^{48} + ( -\beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{15} ) q^{49} + ( \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{15} ) q^{50} + ( \beta_{3} + 3 \beta_{7} - \beta_{11} + \beta_{12} ) q^{51} + \beta_{8} q^{52} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{7} + 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{53} -\beta_{7} q^{54} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{13} - \beta_{15} ) q^{55} -\beta_{14} q^{56} + ( -1 + \beta_{2} - \beta_{8} + \beta_{9} ) q^{57} + ( -\beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{14} + \beta_{15} ) q^{58} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{11} - \beta_{12} ) q^{59} -\beta_{6} q^{60} + ( \beta_{1} - \beta_{4} + 2 \beta_{7} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{61} + ( -\beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{62} + \beta_{14} q^{63} + q^{64} + ( -\beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{13} + \beta_{14} ) q^{65} + \beta_{13} q^{66} + ( \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{15} ) q^{67} + ( 3 - \beta_{5} - \beta_{6} - \beta_{9} ) q^{68} + ( \beta_{1} - \beta_{3} + \beta_{7} + \beta_{14} ) q^{69} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{14} + \beta_{15} ) q^{70} + ( -2 + 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{71} - q^{72} + ( -3 \beta_{1} + 2 \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{73} + ( 1 + \beta_{7} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{74} + ( -\beta_{1} + \beta_{3} + \beta_{4} - \beta_{8} - \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{75} + ( \beta_{1} + \beta_{3} + \beta_{7} - \beta_{13} ) q^{76} + ( -3 \beta_{3} - \beta_{4} - 3 \beta_{7} + 2 \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{77} + \beta_{1} q^{78} + ( \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{14} + \beta_{15} ) q^{79} + \beta_{11} q^{80} + q^{81} + ( \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{82} + ( \beta_{1} - 3 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{83} + \beta_{10} q^{84} + ( -\beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{7} - \beta_{9} + 2 \beta_{11} + \beta_{12} ) q^{85} + ( -\beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{86} + ( \beta_{4} + \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{15} ) q^{87} + \beta_{2} q^{88} + ( -\beta_{1} + 4 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{15} ) q^{89} -\beta_{11} q^{90} + ( -\beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{91} + ( 1 + \beta_{8} + \beta_{9} + \beta_{10} ) q^{92} + ( -2 + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{15} ) q^{93} + ( 2 \beta_{1} - \beta_{4} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{94} + ( -\beta_{1} + 3 \beta_{3} - \beta_{6} + 2 \beta_{7} - 2 \beta_{10} - 2 \beta_{11} - \beta_{14} + \beta_{15} ) q^{95} + \beta_{7} q^{96} + ( -2 + \beta_{2} + 3 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 3 \beta_{15} ) q^{97} + ( -\beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{15} ) q^{98} -\beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{2} + 16q^{4} + 2q^{5} + 16q^{8} - 16q^{9} + O(q^{10}) \) \( 16q + 16q^{2} + 16q^{4} + 2q^{5} + 16q^{8} - 16q^{9} + 2q^{10} + 2q^{11} - 6q^{15} + 16q^{16} + 38q^{17} - 16q^{18} + 2q^{20} + 6q^{21} + 2q^{22} + 20q^{23} - 4q^{25} - 6q^{30} + 16q^{32} + 38q^{34} + 10q^{35} - 16q^{36} + 4q^{37} + 2q^{40} - 6q^{41} + 6q^{42} + 2q^{43} + 2q^{44} - 2q^{45} + 20q^{46} - 18q^{49} - 4q^{50} - 20q^{55} - 16q^{57} - 6q^{60} + 16q^{64} + 12q^{65} + 38q^{68} + 10q^{70} - 24q^{71} - 16q^{72} + 4q^{74} + 2q^{80} + 16q^{81} - 6q^{82} + 6q^{84} + 2q^{86} - 2q^{87} + 2q^{88} - 2q^{90} + 20q^{92} - 22q^{93} - 16q^{95} - 38q^{97} - 18q^{98} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{15} + 2 x^{14} + 8 x^{13} + 138 x^{12} - 220 x^{11} + 196 x^{10} + 744 x^{9} + 4241 x^{8} - 3018 x^{7} + 658 x^{6} - 1584 x^{5} + 16372 x^{4} - 18840 x^{3} + 10952 x^{2} - 1184 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(3598583486206356 \nu^{15} + 12442080899330691 \nu^{14} - 14569191096743604 \nu^{13} + 39468243127992567 \nu^{12} + 700061195842025382 \nu^{11} + 1963789626693085265 \nu^{10} - 974073933365141220 \nu^{9} + 3679942840789531641 \nu^{8} + 34385434998275232414 \nu^{7} + 74659043196870143072 \nu^{6} + 39504937564477019472 \nu^{5} - 6917313879099567612 \nu^{4} + 37944345263941329576 \nu^{3} + 57688770756908863728 \nu^{2} + 32004315856384857120 \nu - 1913670067113595968\)\()/ 8776577717871405136 \)
\(\beta_{2}\)\(=\)\((\)\(-5019160714525244 \nu^{15} + 7302267297319127 \nu^{14} - 7397401618349153 \nu^{13} - 27789629224556003 \nu^{12} - 739353168840258249 \nu^{11} + 714784098392565685 \nu^{10} - 629714596255782179 \nu^{9} - 2097418788500457013 \nu^{8} - 25695585297125062851 \nu^{7} + 2402052437683258296 \nu^{6} + 7074323891414175984 \nu^{5} + 51547587830387511516 \nu^{4} - 78239049472925927388 \nu^{3} + 48290456870435422360 \nu^{2} - 5254999736306159424 \nu + 23585035147057026960\)\()/ 8776577717871405136 \)
\(\beta_{3}\)\(=\)\((\)\(9567165448408746 \nu^{15} - 2762615888851235 \nu^{14} + 7897444240156323 \nu^{13} + 80748754982251899 \nu^{12} + 1460496131630607439 \nu^{11} + 369699396297893537 \nu^{10} + 1338037921924729557 \nu^{9} + 7563565552882221785 \nu^{8} + 53127278044494191983 \nu^{7} + 60306332089804664822 \nu^{6} + 56448524114568934548 \nu^{5} - 10785012944336174024 \nu^{4} + 82276993584777983684 \nu^{3} + 20812138400247811624 \nu^{2} + 64382242804295173056 \nu - 3846979372012104640\)\()/ 8776577717871405136 \)
\(\beta_{4}\)\(=\)\((\)\(-10747369620592049 \nu^{15} + 49507716150097919 \nu^{14} - 59292368394327099 \nu^{13} - 69436016039014749 \nu^{12} - 1234651710034151587 \nu^{11} + 6372699016733347429 \nu^{10} - 5776516973434541039 \nu^{9} - 7026367954906604219 \nu^{8} - 23166341077983495856 \nu^{7} + 163792457990670368680 \nu^{6} - 16498400553027426082 \nu^{5} - 39842557895772371100 \nu^{4} - 290703731837693963136 \nu^{3} + 568979970447300755648 \nu^{2} - 350212003392840468312 \nu + 38099452666799307040\)\()/ 8776577717871405136 \)
\(\beta_{5}\)\(=\)\((\)\(-23656976378661003 \nu^{15} + 30447616523487557 \nu^{14} - 5668307481205641 \nu^{13} - 227788358439582645 \nu^{12} - 3371744582497375019 \nu^{11} + 2928984459775409985 \nu^{10} + 183920614188561673 \nu^{9} - 20557827614532110923 \nu^{8} - 109211225452771384886 \nu^{7} + 4858276358302257694 \nu^{6} + 68517725209056647836 \nu^{5} + 104967433823144605564 \nu^{4} - 263375550046521833464 \nu^{3} + 174184766214316358360 \nu^{2} - 1536500573724590784 \nu + 39218675595124970912\)\()/ 17553155435742810272 \)
\(\beta_{6}\)\(=\)\((\)\(-27270263108412109 \nu^{15} + 35845002295782009 \nu^{14} - 10266165292669687 \nu^{13} - 258385448345775489 \nu^{12} - 3888009054007888081 \nu^{11} + 3464120130945415681 \nu^{10} - 265227763229672233 \nu^{9} - 23400222459835818047 \nu^{8} - 126253736340414947654 \nu^{7} + 7307757322320005194 \nu^{6} + 67175719164418371636 \nu^{5} + 111309066839156150716 \nu^{4} - 328339387849681228408 \nu^{3} + 212479679502370074056 \nu^{2} - 40789217922827096384 \nu + 41557711249901783200\)\()/ 17553155435742810272 \)
\(\beta_{7}\)\(=\)\((\)\(18273716052943846 \nu^{15} - 34740788741012139 \nu^{14} + 33848739219740466 \nu^{13} + 148488657329282791 \nu^{12} + 2537071360259347170 \nu^{11} - 3762085295892389589 \nu^{10} + 3314080510791990968 \nu^{9} + 13820218932099338377 \nu^{8} + 78920027203186704448 \nu^{7} - 46628819603962745844 \nu^{6} + 10799364680828176918 \nu^{5} - 28274563205543913964 \nu^{4} + 296006462710790874136 \nu^{3} - 311794891535882361168 \nu^{2} + 180986281567814143544 \nu - 10786298850005666000\)\()/ 8776577717871405136 \)
\(\beta_{8}\)\(=\)\((\)\(-21379513563644685 \nu^{15} + 30517922200804757 \nu^{14} - 24808754347643015 \nu^{13} - 155365536772579389 \nu^{12} - 3105505244059654869 \nu^{11} + 2986426268519524431 \nu^{10} - 2060018014627625601 \nu^{9} - 13421212405044163915 \nu^{8} - 105151622543957426978 \nu^{7} + 9942740781114420512 \nu^{6} + 28298939092911453964 \nu^{5} + 127950227372178531508 \nu^{4} - 324346718373826828408 \nu^{3} + 200025933426027654000 \nu^{2} - 21765048210218762368 \nu + 3543341730699935552\)\()/ 8776577717871405136 \)
\(\beta_{9}\)\(=\)\((\)\(24216523190739559 \nu^{15} - 32826769491708205 \nu^{14} + 15191250088530019 \nu^{13} + 218066683956470309 \nu^{12} + 3466704866406129737 \nu^{11} - 3190075392666935965 \nu^{10} + 946193107947504485 \nu^{9} + 19727658351803595975 \nu^{8} + 113783901684109617420 \nu^{7} - 8456590044398138202 \nu^{6} - 47181359656513821644 \nu^{5} - 95486908052697491976 \nu^{4} + 321463007301980811984 \nu^{3} - 203241633505665399864 \nu^{2} + 22172893866218873632 \nu - 5271315577848317504\)\()/ 8776577717871405136 \)
\(\beta_{10}\)\(=\)\((\)\(-12817278110967634 \nu^{15} + 17727172949013492 \nu^{14} - 9114568564984668 \nu^{13} - 118516004781748970 \nu^{12} - 1827553739206846274 \nu^{11} + 1731816050083731475 \nu^{10} - 682650127547049224 \nu^{9} - 10968243851454832160 \nu^{8} - 59671526276950665476 \nu^{7} + 5472559110178869761 \nu^{6} + 17389998241110704844 \nu^{5} + 28357264878402748114 \nu^{4} - 183727634589725420008 \nu^{3} + 113728649431467197100 \nu^{2} - 12379865386235900624 \nu - 13297933955641522072\)\()/ 4388288858935702568 \)
\(\beta_{11}\)\(=\)\((\)\(83459165975960877 \nu^{15} - 142854828745674593 \nu^{14} + 150156848474466771 \nu^{13} + 677067493024494429 \nu^{12} + 11703383233180742765 \nu^{11} - 14685343779497331501 \nu^{10} + 15464887800268384161 \nu^{9} + 63102508808192934739 \nu^{8} + 370192197216499213802 \nu^{7} - 116570414566402812598 \nu^{6} + 120150201939706955400 \nu^{5} - 120684227016313567036 \nu^{4} + 1228622370375316235368 \nu^{3} - 1224969327699563325368 \nu^{2} + 789420777992913348560 \nu - 46015836365558307488\)\()/ 17553155435742810272 \)
\(\beta_{12}\)\(=\)\((\)\(-87072452705711983 \nu^{15} + 148252214517969045 \nu^{14} - 154754706285930817 \nu^{13} - 707664582930687273 \nu^{12} - 12219647704691255827 \nu^{11} + 15220479450667337197 \nu^{10} - 15914036177686618067 \nu^{9} - 65944903653496641863 \nu^{8} - 387234708104142776570 \nu^{7} + 119019895530420560098 \nu^{6} - 121492207984345231600 \nu^{5} + 127025860032325112188 \nu^{4} - 1293586208178475630312 \nu^{3} + 1263264240987617041064 \nu^{2} - 793567184470530233616 \nu + 48354872020335119776\)\()/ 17553155435742810272 \)
\(\beta_{13}\)\(=\)\((\)\(48118279587099894 \nu^{15} - 84633788878151129 \nu^{14} + 81011792960581768 \nu^{13} + 395068305903552419 \nu^{12} + 6753590184184493108 \nu^{11} - 8963801050572878679 \nu^{10} + 8071383029047362522 \nu^{9} + 36768398939132960325 \nu^{8} + 214738955638592641550 \nu^{7} - 94977690341340840884 \nu^{6} + 39676510248119000838 \nu^{5} - 75392677385581876676 \nu^{4} + 777438211441026403136 \nu^{3} - 805058818164724298664 \nu^{2} + 477800959874748124440 \nu - 28477425678718074000\)\()/ 8776577717871405136 \)
\(\beta_{14}\)\(=\)\((\)\(-58599146537079738 \nu^{15} + 123266479279552438 \nu^{14} - 115989458543362729 \nu^{13} - 473747488729663366 \nu^{12} - 8038705530926452605 \nu^{11} + 13879856120878845218 \nu^{10} - 10956815022075974549 \nu^{9} - 44060072494787372566 \nu^{8} - 244375119257663357073 \nu^{7} + 216915316012027345288 \nu^{6} + 1175795409551302370 \nu^{5} + 93341958315418191324 \nu^{4} - 999232944627119782372 \nu^{3} + 1203959456301324504768 \nu^{2} - 601092945511793476232 \nu + 35816544221695449712\)\()/ 8776577717871405136 \)
\(\beta_{15}\)\(=\)\((\)\(-59923129802572879 \nu^{15} + 117113881990875321 \nu^{14} - 95951644204439093 \nu^{13} - 495669233213731053 \nu^{12} - 8288330089041619629 \nu^{11} + 12958762198920303113 \nu^{10} - 8373350468913758013 \nu^{9} - 45494715681625362075 \nu^{8} - 255837304581511347840 \nu^{7} + 182824512099824497730 \nu^{6} + 67554990970372022286 \nu^{5} + 154569987779827399044 \nu^{4} - 971789885387320911184 \nu^{3} + 995490369264377828984 \nu^{2} - 396696295674212250488 \nu + 6396540916876267712\)\()/ 8776577717871405136 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{12} + \beta_{11} - \beta_{6} + \beta_{5}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} + 5 \beta_{7} + \beta_{4} + \beta_{3} - \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{14} - 2 \beta_{13} - 7 \beta_{12} - 7 \beta_{11} + 2 \beta_{10} + 2 \beta_{7} - 7 \beta_{6} + 7 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 2\)\()/2\)
\(\nu^{4}\)\(=\)\(-10 \beta_{15} - 2 \beta_{12} - 2 \beta_{11} + 11 \beta_{10} + 11 \beta_{9} + 11 \beta_{8} + 10 \beta_{6} + 10 \beta_{5} + 10 \beta_{4} - 2 \beta_{2} - 41\)
\(\nu^{5}\)\(=\)\((\)\(-22 \beta_{15} + 22 \beta_{14} + 26 \beta_{13} - 61 \beta_{12} - 61 \beta_{11} + 22 \beta_{10} - 2 \beta_{8} - 34 \beta_{7} + 61 \beta_{6} - 61 \beta_{5} + 26 \beta_{2} - 2 \beta_{1} - 34\)\()/2\)
\(\nu^{6}\)\(=\)\(-97 \beta_{15} + 109 \beta_{14} - 26 \beta_{13} - 99 \beta_{12} + 99 \beta_{11} - 379 \beta_{7} + 28 \beta_{6} - 28 \beta_{5} - 97 \beta_{4} - 113 \beta_{3} + 111 \beta_{1}\)
\(\nu^{7}\)\(=\)\((\)\(222 \beta_{14} + 278 \beta_{13} + 569 \beta_{12} + 577 \beta_{11} - 222 \beta_{10} - 8 \beta_{9} + 28 \beta_{8} - 394 \beta_{7} + 569 \beta_{6} - 577 \beta_{5} - 218 \beta_{4} - 8 \beta_{3} - 278 \beta_{2} - 28 \beta_{1} + 394\)\()/2\)
\(\nu^{8}\)\(=\)\(942 \beta_{15} + 306 \beta_{12} + 306 \beta_{11} - 1073 \beta_{10} - 1141 \beta_{9} - 1097 \beta_{8} - 978 \beta_{6} - 978 \beta_{5} - 942 \beta_{4} + 270 \beta_{2} + 3643\)
\(\nu^{9}\)\(=\)\((\)\(2122 \beta_{15} - 2218 \beta_{14} - 2822 \beta_{13} + 5615 \beta_{12} + 5439 \beta_{11} - 2218 \beta_{10} - 192 \beta_{9} + 302 \beta_{8} + 4126 \beta_{7} - 5615 \beta_{6} + 5439 \beta_{5} + 192 \beta_{3} - 2822 \beta_{2} + 302 \beta_{1} + 4126\)\()/2\)
\(\nu^{10}\)\(=\)\(9163 \beta_{15} - 10567 \beta_{14} + 2630 \beta_{13} + 9657 \beta_{12} - 9657 \beta_{11} + 35497 \beta_{7} - 3124 \beta_{6} + 3124 \beta_{5} + 9163 \beta_{4} + 11459 \beta_{3} - 10765 \beta_{1}\)
\(\nu^{11}\)\(=\)\((\)\(-22122 \beta_{14} - 28178 \beta_{13} - 52459 \beta_{12} - 55187 \beta_{11} + 22122 \beta_{10} + 3128 \beta_{9} - 3012 \beta_{8} + 41854 \beta_{7} - 52459 \beta_{6} + 55187 \beta_{5} + 20574 \beta_{4} + 3128 \beta_{3} + 28178 \beta_{2} + 3012 \beta_{1} - 41854\)\()/2\)
\(\nu^{12}\)\(=\)\(-89202 \beta_{15} - 31214 \beta_{12} - 31214 \beta_{11} + 104123 \beta_{10} + 114855 \beta_{9} + 105371 \beta_{8} + 95358 \beta_{6} + 95358 \beta_{5} + 89202 \beta_{4} - 25050 \beta_{2} - 347593\)
\(\nu^{13}\)\(=\)\((\)\(-199358 \beta_{15} + 220574 \beta_{14} + 279810 \beta_{13} - 544365 \beta_{12} - 507629 \beta_{11} + 220574 \beta_{10} + 43376 \beta_{9} - 29162 \beta_{8} - 420186 \beta_{7} + 544365 \beta_{6} - 507629 \beta_{5} - 43376 \beta_{3} + 279810 \beta_{2} - 29162 \beta_{1} - 420186\)\()/2\)
\(\nu^{14}\)\(=\)\(-868761 \beta_{15} + 1026381 \beta_{14} - 236434 \beta_{13} - 941739 \beta_{12} + 941739 \beta_{11} - 3410179 \beta_{7} + 309772 \beta_{6} - 309772 \beta_{5} - 868761 \beta_{4} - 1150209 \beta_{3} + 1030535 \beta_{1}\)
\(\nu^{15}\)\(=\)\((\)\(2199438 \beta_{14} + 2773286 \beta_{13} + 4918065 \beta_{12} + 5377337 \beta_{11} - 2199438 \beta_{10} - 551944 \beta_{9} + 278572 \beta_{8} - 4204746 \beta_{7} + 4918065 \beta_{6} - 5377337 \beta_{5} - 1932538 \beta_{4} - 551944 \beta_{3} - 2773286 \beta_{2} - 278572 \beta_{1} + 4204746\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
739.1
−2.20888 + 2.20888i
−1.32920 + 1.32920i
−1.28817 + 1.28817i
0.0603067 0.0603067i
0.623809 0.623809i
0.718347 0.718347i
2.18974 2.18974i
2.23405 2.23405i
−2.20888 2.20888i
−1.32920 1.32920i
−1.28817 1.28817i
0.0603067 + 0.0603067i
0.623809 + 0.623809i
0.718347 + 0.718347i
2.18974 + 2.18974i
2.23405 + 2.23405i
1.00000 1.00000i 1.00000 −2.20888 0.347650i 1.00000i 2.08112i 1.00000 −1.00000 −2.20888 0.347650i
739.2 1.00000 1.00000i 1.00000 −1.32920 + 1.79812i 1.00000i 1.87266i 1.00000 −1.00000 −1.32920 + 1.79812i
739.3 1.00000 1.00000i 1.00000 −1.28817 1.82773i 1.00000i 1.25548i 1.00000 −1.00000 −1.28817 1.82773i
739.4 1.00000 1.00000i 1.00000 0.0603067 2.23525i 1.00000i 3.18244i 1.00000 −1.00000 0.0603067 2.23525i
739.5 1.00000 1.00000i 1.00000 0.623809 2.14729i 1.00000i 5.25373i 1.00000 −1.00000 0.623809 2.14729i
739.6 1.00000 1.00000i 1.00000 0.718347 + 2.11754i 1.00000i 2.74990i 1.00000 −1.00000 0.718347 + 2.11754i
739.7 1.00000 1.00000i 1.00000 2.18974 0.452819i 1.00000i 0.0776078i 1.00000 −1.00000 2.18974 0.452819i
739.8 1.00000 1.00000i 1.00000 2.23405 + 0.0950871i 1.00000i 3.20752i 1.00000 −1.00000 2.23405 + 0.0950871i
739.9 1.00000 1.00000i 1.00000 −2.20888 + 0.347650i 1.00000i 2.08112i 1.00000 −1.00000 −2.20888 + 0.347650i
739.10 1.00000 1.00000i 1.00000 −1.32920 1.79812i 1.00000i 1.87266i 1.00000 −1.00000 −1.32920 1.79812i
739.11 1.00000 1.00000i 1.00000 −1.28817 + 1.82773i 1.00000i 1.25548i 1.00000 −1.00000 −1.28817 + 1.82773i
739.12 1.00000 1.00000i 1.00000 0.0603067 + 2.23525i 1.00000i 3.18244i 1.00000 −1.00000 0.0603067 + 2.23525i
739.13 1.00000 1.00000i 1.00000 0.623809 + 2.14729i 1.00000i 5.25373i 1.00000 −1.00000 0.623809 + 2.14729i
739.14 1.00000 1.00000i 1.00000 0.718347 2.11754i 1.00000i 2.74990i 1.00000 −1.00000 0.718347 2.11754i
739.15 1.00000 1.00000i 1.00000 2.18974 + 0.452819i 1.00000i 0.0776078i 1.00000 −1.00000 2.18974 + 0.452819i
739.16 1.00000 1.00000i 1.00000 2.23405 0.0950871i 1.00000i 3.20752i 1.00000 −1.00000 2.23405 0.0950871i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 739.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.e.d yes 16
3.b odd 2 1 3330.2.e.e 16
5.b even 2 1 1110.2.e.c 16
15.d odd 2 1 3330.2.e.f 16
37.b even 2 1 1110.2.e.c 16
111.d odd 2 1 3330.2.e.f 16
185.d even 2 1 inner 1110.2.e.d yes 16
555.b odd 2 1 3330.2.e.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.e.c 16 5.b even 2 1
1110.2.e.c 16 37.b even 2 1
1110.2.e.d yes 16 1.a even 1 1 trivial
1110.2.e.d yes 16 185.d even 2 1 inner
3330.2.e.e 16 3.b odd 2 1
3330.2.e.e 16 555.b odd 2 1
3330.2.e.f 16 15.d odd 2 1
3330.2.e.f 16 111.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\(T_{7}^{16} + \cdots\)
\( T_{13}^{8} - 34 T_{13}^{6} + 285 T_{13}^{4} - 16 T_{13}^{3} - 600 T_{13}^{2} - 80 T_{13} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{16} \)
$3$ \( ( 1 + T^{2} )^{8} \)
$5$ \( 390625 - 156250 T + 62500 T^{2} - 43750 T^{3} - 5000 T^{4} - 5250 T^{5} - 900 T^{6} + 2770 T^{7} - 338 T^{8} + 554 T^{9} - 36 T^{10} - 42 T^{11} - 8 T^{12} - 14 T^{13} + 4 T^{14} - 2 T^{15} + T^{16} \)
$7$ \( 3136 + 525424 T^{2} + 791452 T^{4} + 448469 T^{6} + 126197 T^{8} + 19194 T^{10} + 1582 T^{12} + 65 T^{14} + T^{16} \)
$11$ \( ( 716 + 716 T - 1312 T^{2} - 373 T^{3} + 537 T^{4} + 22 T^{5} - 50 T^{6} - T^{7} + T^{8} )^{2} \)
$13$ \( ( 64 - 80 T - 600 T^{2} - 16 T^{3} + 285 T^{4} - 34 T^{6} + T^{8} )^{2} \)
$17$ \( ( -4504 - 4756 T + 2522 T^{2} + 2117 T^{3} - 939 T^{4} - 138 T^{5} + 116 T^{6} - 19 T^{7} + T^{8} )^{2} \)
$19$ \( 86415616 + 191618064 T^{2} + 131703136 T^{4} + 33815600 T^{6} + 4128521 T^{8} + 265804 T^{10} + 9150 T^{12} + 156 T^{14} + T^{16} \)
$23$ \( ( 31168 + 30624 T - 2872 T^{2} - 7398 T^{3} + 121 T^{4} + 568 T^{5} - 42 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$29$ \( 25563136 + 666106880 T^{2} + 682409472 T^{4} + 141650064 T^{6} + 12695440 T^{8} + 591144 T^{10} + 14920 T^{12} + 193 T^{14} + T^{16} \)
$31$ \( 1349533696 + 2328440832 T^{2} + 1261970432 T^{4} + 232163072 T^{6} + 19589056 T^{8} + 844944 T^{10} + 19276 T^{12} + 221 T^{14} + T^{16} \)
$37$ \( 3512479453921 - 379727508532 T - 164206490176 T^{2} + 46876514932 T^{3} + 52476508 T^{4} - 1747528500 T^{5} + 248216128 T^{6} + 21167700 T^{7} - 10388826 T^{8} + 572100 T^{9} + 181312 T^{10} - 34500 T^{11} + 28 T^{12} + 676 T^{13} - 64 T^{14} - 4 T^{15} + T^{16} \)
$41$ \( ( -1327648 + 924944 T - 109768 T^{2} - 51364 T^{3} + 12404 T^{4} + 188 T^{5} - 198 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$43$ \( ( 1418624 + 701632 T - 328544 T^{2} - 30192 T^{3} + 14008 T^{4} + 372 T^{5} - 210 T^{6} - T^{7} + T^{8} )^{2} \)
$47$ \( 1124663296 + 2649827584 T^{2} + 1492471808 T^{4} + 336668224 T^{6} + 33366928 T^{8} + 1471856 T^{10} + 30120 T^{12} + 284 T^{14} + T^{16} \)
$53$ \( 24049806400 + 29886136816 T^{2} + 10162555740 T^{4} + 1454019757 T^{6} + 101632285 T^{8} + 3563794 T^{10} + 59542 T^{12} + 417 T^{14} + T^{16} \)
$59$ \( 50462269542400 + 19267322691584 T^{2} + 2261293218304 T^{4} + 107067830528 T^{6} + 2471674256 T^{8} + 30616704 T^{10} + 207304 T^{12} + 720 T^{14} + T^{16} \)
$61$ \( 229356703744 + 173643603968 T^{2} + 37408965120 T^{4} + 3500166656 T^{6} + 165744656 T^{8} + 4193168 T^{10} + 56572 T^{12} + 381 T^{14} + T^{16} \)
$67$ \( 505031950336 + 281635979264 T^{2} + 54447439872 T^{4} + 4935598080 T^{6} + 231633920 T^{8} + 5718272 T^{10} + 72128 T^{12} + 436 T^{14} + T^{16} \)
$71$ \( ( -6032896 - 2772096 T + 60096 T^{2} + 149760 T^{3} + 8388 T^{4} - 2488 T^{5} - 192 T^{6} + 12 T^{7} + T^{8} )^{2} \)
$73$ \( 6742191359279104 + 1155730613146624 T^{2} + 53868268316928 T^{4} + 1187286889776 T^{6} + 14590829401 T^{8} + 105748588 T^{10} + 449422 T^{12} + 1036 T^{14} + T^{16} \)
$79$ \( 19573129216 + 61879579648 T^{2} + 50912610048 T^{4} + 9389416896 T^{6} + 598045328 T^{8} + 15399904 T^{10} + 161144 T^{12} + 684 T^{14} + T^{16} \)
$83$ \( 502941945856 + 1080407259136 T^{2} + 366873553792 T^{4} + 37621399908 T^{6} + 1371955505 T^{8} + 22746676 T^{10} + 184822 T^{12} + 704 T^{14} + T^{16} \)
$89$ \( 6083069027617024 + 672700101772544 T^{2} + 30280182703584 T^{4} + 723121793424 T^{6} + 9991901601 T^{8} + 81686876 T^{10} + 386566 T^{12} + 972 T^{14} + T^{16} \)
$97$ \( ( 26655952 - 11690080 T - 2282912 T^{2} + 710636 T^{3} + 50176 T^{4} - 7384 T^{5} - 412 T^{6} + 19 T^{7} + T^{8} )^{2} \)
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