Properties

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

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.x (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 44 x^{14} + 724 x^{12} + 5750 x^{10} + 23344 x^{8} + 47024 x^{6} + 43297 x^{4} + 13976 x^{2} + 676\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 44 x^{14} + 724 x^{12} + 5750 x^{10} + 23344 x^{8} + 47024 x^{6} + 43297 x^{4} + 13976 x^{2} + 676\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -13206 \nu^{15} - 582741 \nu^{13} - 9594515 \nu^{11} - 75642377 \nu^{9} - 298884555 \nu^{7} - 559225583 \nu^{5} - 431800511 \nu^{3} - 88306866 \nu - 9055852 \)\()/18111704\)
\(\beta_{2}\)\(=\)\((\)\(-10324114 \nu^{15} - 231600785 \nu^{14} - 431418313 \nu^{13} - 9788447929 \nu^{12} - 6548710747 \nu^{11} - 150568927483 \nu^{10} - 45846209935 \nu^{9} - 1066046293013 \nu^{8} - 149605457411 \nu^{7} - 3505456229509 \nu^{6} - 183908713051 \nu^{5} - 4574527274083 \nu^{4} + 13243132119 \nu^{3} - 1722644084824 \nu^{2} + 115482059490 \nu - 77890999576\)\()/ 13348325848 \)
\(\beta_{3}\)\(=\)\((\)\(54424395 \nu^{15} - 147256174 \nu^{14} + 2303097077 \nu^{13} - 6219207696 \nu^{12} + 35550593381 \nu^{11} - 95765609498 \nu^{10} + 254037921393 \nu^{9} - 682692894520 \nu^{8} + 856452813701 \nu^{7} - 2297096731332 \nu^{6} + 1209580766261 \nu^{5} - 3219686942498 \nu^{4} + 621706144390 \nu^{3} - 1484648364030 \nu^{2} + 144197912780 \nu - 122757597300\)\()/ 13348325848 \)
\(\beta_{4}\)\(=\)\((\)\( 2649 \nu^{14} + 111621 \nu^{12} + 1710274 \nu^{10} + 12057906 \nu^{8} + 39518805 \nu^{6} + 51587280 \nu^{4} + 19510371 \nu^{2} + 46766 \nu + 980070 \)\()/46766\)
\(\beta_{5}\)\(=\)\((\)\(-71079673 \nu^{15} + 53593657 \nu^{14} - 2995690422 \nu^{13} + 2247594011 \nu^{12} - 45886051749 \nu^{11} + 34160381775 \nu^{10} - 322877995643 \nu^{9} + 237377029149 \nu^{8} - 1051583349562 \nu^{7} + 754965960931 \nu^{6} - 1348129378719 \nu^{5} + 907543313781 \nu^{4} - 493899653872 \nu^{3} + 248883798774 \nu^{2} - 39312620648 \nu - 2258341020\)\()/ 6674162924 \)
\(\beta_{6}\)\(=\)\((\)\(-18809805 \nu^{15} - 44185843 \nu^{14} - 793263294 \nu^{13} - 1863894877 \nu^{12} - 12170127966 \nu^{11} - 28605027321 \nu^{10} - 85967283874 \nu^{9} - 202135749621 \nu^{8} - 282656815476 \nu^{7} - 664771352973 \nu^{6} - 371795294250 \nu^{5} - 873572804013 \nu^{4} - 145114756365 \nu^{3} - 336823640582 \nu^{2} - 9758281326 \nu - 17830174076\)\()/ 1213484168 \)
\(\beta_{7}\)\(=\)\((\)\(18809805 \nu^{15} - 44185843 \nu^{14} + 793263294 \nu^{13} - 1863894877 \nu^{12} + 12170127966 \nu^{11} - 28605027321 \nu^{10} + 85967283874 \nu^{9} - 202135749621 \nu^{8} + 282656815476 \nu^{7} - 664771352973 \nu^{6} + 371795294250 \nu^{5} - 873572804013 \nu^{4} + 145114756365 \nu^{3} - 336823640582 \nu^{2} + 9758281326 \nu - 17830174076\)\()/ 1213484168 \)
\(\beta_{8}\)\(=\)\((\)\(-109394411 \nu^{15} - 394487353 \nu^{14} - 4631516922 \nu^{13} - 16655667834 \nu^{12} - 71545576365 \nu^{11} - 255907946385 \nu^{10} - 511794695604 \nu^{9} - 1810543749131 \nu^{8} - 1727111569368 \nu^{7} - 5959533316776 \nu^{6} - 2427112866829 \nu^{5} - 7827014551031 \nu^{4} - 1147108653689 \nu^{3} - 3000525724378 \nu^{2} - 116168789070 \nu - 152403200300\)\()/ 6674162924 \)
\(\beta_{9}\)\(=\)\((\)\(-152483460 \nu^{15} - 6422799157 \nu^{13} - 98320814245 \nu^{11} - 691602201221 \nu^{9} - 2252772156535 \nu^{7} - 2880167470489 \nu^{5} - 974556175625 \nu^{3} + 36856818194 \nu\)\()/ 6674162924 \)
\(\beta_{10}\)\(=\)\((\)\(-180474084 \nu^{15} - 826130396 \nu^{14} - 7627207344 \nu^{13} - 34833141239 \nu^{12} - 117431628114 \nu^{11} - 534148371796 \nu^{10} - 834672691247 \nu^{9} - 3768752775164 \nu^{8} - 2778694918930 \nu^{7} - 12354386014477 \nu^{6} - 3775242245548 \nu^{5} - 16096784942732 \nu^{4} - 1641008307561 \nu^{3} - 6033812610046 \nu^{2} - 148807246794 \nu - 283340406336\)\()/ 6674162924 \)
\(\beta_{11}\)\(=\)\((\)\(180474084 \nu^{15} - 826130396 \nu^{14} + 7627207344 \nu^{13} - 34833141239 \nu^{12} + 117431628114 \nu^{11} - 534148371796 \nu^{10} + 834672691247 \nu^{9} - 3768752775164 \nu^{8} + 2778694918930 \nu^{7} - 12354386014477 \nu^{6} + 3775242245548 \nu^{5} - 16096784942732 \nu^{4} + 1641008307561 \nu^{3} - 6033812610046 \nu^{2} + 148807246794 \nu - 283340406336\)\()/ 6674162924 \)
\(\beta_{12}\)\(=\)\((\)\(-664665858 \nu^{15} - 716837485 \nu^{14} - 28023362596 \nu^{13} - 30213515345 \nu^{12} - 429677117692 \nu^{11} - 462969734669 \nu^{10} - 3031013989376 \nu^{9} - 3261632348195 \nu^{8} - 9931123320356 \nu^{7} - 10655274888141 \nu^{6} - 12918094576644 \nu^{5} - 13751840979565 \nu^{4} - 4803947370306 \nu^{3} - 5011488932190 \nu^{2} - 228113543072 \nu - 246614842136\)\()/ 13348325848 \)
\(\beta_{13}\)\(=\)\((\)\(817149318 \nu^{15} - 841250956 \nu^{14} + 34446161753 \nu^{13} - 35506775252 \nu^{12} + 527997931937 \nu^{11} - 545217898602 \nu^{10} + 3722616190597 \nu^{9} - 3852924582910 \nu^{8} + 12183895476891 \nu^{7} - 12650799195788 \nu^{6} + 15798262047133 \nu^{5} - 16511281626086 \nu^{4} + 5778503545931 \nu^{3} - 6236365419466 \nu^{2} + 191256724878 \nu - 315674197904\)\()/ 13348325848 \)
\(\beta_{14}\)\(=\)\((\)\(-1240593227 \nu^{15} - 431724501 \nu^{14} - 52322147503 \nu^{13} - 18205033405 \nu^{12} - 802702515889 \nu^{11} - 279214096005 \nu^{10} - 5668426141845 \nu^{9} - 1970674050267 \nu^{8} - 18615042339947 \nu^{7} - 6463613063453 \nu^{6} - 24366401654201 \nu^{5} - 8423226198985 \nu^{4} - 9254818590818 \nu^{3} - 3117800473762 \nu^{2} - 411912946812 \nu - 111042129536\)\()/ 13348325848 \)
\(\beta_{15}\)\(=\)\((\)\(1240593227 \nu^{15} - 431724501 \nu^{14} + 52322147503 \nu^{13} - 18205033405 \nu^{12} + 802702515889 \nu^{11} - 279214096005 \nu^{10} + 5668426141845 \nu^{9} - 1970674050267 \nu^{8} + 18615042339947 \nu^{7} - 6463613063453 \nu^{6} + 24366401654201 \nu^{5} - 8423226198985 \nu^{4} + 9254818590818 \nu^{3} - 3117800473762 \nu^{2} + 411912946812 \nu - 111042129536\)\()/ 13348325848 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{13} - 2 \beta_{12} + \beta_{10} - \beta_{9} - \beta_{8} + 3 \beta_{7} + 2 \beta_{5} - \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 9\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} + 3 \beta_{11} - 5 \beta_{10} + 6 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + 6 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 7 \beta_{2} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\((\)\(-8 \beta_{15} - 8 \beta_{14} + 14 \beta_{13} + 14 \beta_{12} + 20 \beta_{11} - 11 \beta_{10} + 21 \beta_{9} + 31 \beta_{8} - 43 \beta_{7} - 8 \beta_{6} - 32 \beta_{5} + 31 \beta_{4} + 35 \beta_{3} - 63 \beta_{2} + 91\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(82 \beta_{15} - 82 \beta_{14} - 178 \beta_{13} + 178 \beta_{12} - 152 \beta_{11} + 137 \beta_{10} - 171 \beta_{9} + 15 \beta_{8} + 299 \beta_{7} - 334 \beta_{6} + 128 \beta_{5} - 15 \beta_{4} + 35 \beta_{3} + 213 \beta_{2} - 336 \beta_{1} + 25\)\()/2\)
\(\nu^{6}\)\(=\)\(119 \beta_{15} + 119 \beta_{14} - 31 \beta_{13} - 31 \beta_{12} - 256 \beta_{11} + 70 \beta_{10} - 206 \beta_{9} - 326 \beta_{8} + 299 \beta_{7} + 84 \beta_{6} + 270 \beta_{5} - 326 \beta_{4} - 215 \beta_{3} + 596 \beta_{2} - 615\)
\(\nu^{7}\)\(=\)\((\)\(-1456 \beta_{15} + 1456 \beta_{14} + 3334 \beta_{13} - 3334 \beta_{12} + 2998 \beta_{11} - 2137 \beta_{10} + 2793 \beta_{9} - 861 \beta_{8} - 6735 \beta_{7} + 6860 \beta_{6} - 2348 \beta_{5} + 861 \beta_{4} - 125 \beta_{3} - 3459 \beta_{2} + 6948 \beta_{1} - 721\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-5016 \beta_{15} - 5016 \beta_{14} - 406 \beta_{13} - 406 \beta_{12} + 10228 \beta_{11} - 2165 \beta_{10} + 7837 \beta_{9} + 12393 \beta_{8} - 8631 \beta_{7} - 2828 \beta_{6} - 9490 \beta_{5} + 12393 \beta_{4} + 5803 \beta_{3} - 21883 \beta_{2} + 19109\)\()/2\)
\(\nu^{9}\)\(=\)\(12624 \beta_{15} - 12624 \beta_{14} - 29928 \beta_{13} + 29928 \beta_{12} - 27525 \beta_{11} + 17694 \beta_{10} - 24117 \beta_{9} + 9831 \beta_{8} + 65116 \beta_{7} - 64420 \beta_{6} + 20793 \beta_{5} - 9831 \beta_{4} - 696 \beta_{3} + 29232 \beta_{2} - 65406 \beta_{1} + 7056\)
\(\nu^{10}\)\(=\)\((\)\(95208 \beta_{15} + 95208 \beta_{14} + 19626 \beta_{13} + 19626 \beta_{12} - 189672 \beta_{11} + 36865 \beta_{10} - 144919 \beta_{9} - 226537 \beta_{8} + 131633 \beta_{7} + 45948 \beta_{6} + 168612 \beta_{5} - 226537 \beta_{4} - 85685 \beta_{3} + 395149 \beta_{2} - 317701\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-438354 \beta_{15} + 438354 \beta_{14} + 1061474 \beta_{13} - 1061474 \beta_{12} + 986636 \beta_{11} - 604297 \beta_{10} + 848187 \beta_{9} - 382339 \beta_{8} - 2390635 \beta_{7} + 2338710 \beta_{6} - 731060 \beta_{5} + 382339 \beta_{4} + 51925 \beta_{3} - 1009549 \beta_{2} + 2376928 \beta_{1} - 255349\)\()/2\)
\(\nu^{12}\)\(=\)\(-868373 \beta_{15} - 868373 \beta_{14} - 226657 \beta_{13} - 226657 \beta_{12} + 1710670 \beta_{11} - 323997 \beta_{10} + 1314315 \beta_{9} + 2034667 \beta_{8} - 1056308 \beta_{7} - 378236 \beta_{6} - 1498692 \beta_{5} + 2034667 \beta_{4} + 678072 \beta_{3} - 3533359 \beta_{2} + 2726774\)
\(\nu^{13}\)\(=\)\((\)\(7652012 \beta_{15} - 7652012 \beta_{14} - 18758706 \beta_{13} + 18758706 \beta_{12} - 17527426 \beta_{11} + 10486519 \beta_{10} - 14987307 \beta_{9} + 7040907 \beta_{8} + 42973229 \beta_{7} - 41840472 \beta_{6} + 12850556 \beta_{5} - 7040907 \beta_{4} - 1132757 \beta_{3} + 17625949 \beta_{2} - 42548332 \beta_{1} + 4525447\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(31149580 \beta_{15} + 31149580 \beta_{14} + 8945262 \beta_{13} + 8945262 \beta_{12} - 61014980 \beta_{11} + 11476453 \beta_{10} - 47126821 \beta_{9} - 72491433 \beta_{8} + 35214467 \beta_{7} + 12732272 \beta_{6} + 53189666 \beta_{5} - 72491433 \beta_{4} - 22482195 \beta_{3} + 125681099 \beta_{2} - 95017697\)\()/2\)
\(\nu^{15}\)\(=\)\(-67095230 \beta_{15} + 67095230 \beta_{14} + 165621522 \beta_{13} - 165621522 \beta_{12} + 155161453 \beta_{11} - 91757091 \beta_{10} + 132517776 \beta_{9} - 63404362 \beta_{8} - 382700249 \beta_{7} + 371815306 \beta_{6} - 113102103 \beta_{5} + 63404362 \beta_{4} + 10884943 \beta_{3} - 154736579 \beta_{2} + 378242802 \beta_{1} - 39904483\)

Character values

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

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
3.06293i
4.20281i
1.35087i
0.241301i
2.47236i
1.35400i
2.57049i
0.720074i
3.06293i
4.20281i
1.35087i
0.241301i
2.47236i
1.35400i
2.57049i
0.720074i
−0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i −1.55211 + 2.68834i 1.00000i −0.500000 0.866025i −1.00000
751.2 −0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i −0.839478 + 1.45402i 1.00000i −0.500000 0.866025i −1.00000
751.3 −0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i 1.60020 2.77163i 1.00000i −0.500000 0.866025i −1.00000
751.4 −0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i 2.38947 4.13868i 1.00000i −0.500000 0.866025i −1.00000
751.5 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i −2.53766 + 4.39536i 1.00000i −0.500000 0.866025i −1.00000
751.6 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i −2.34998 + 4.07029i 1.00000i −0.500000 0.866025i −1.00000
751.7 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i 0.243282 0.421377i 1.00000i −0.500000 0.866025i −1.00000
751.8 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i 1.04628 1.81222i 1.00000i −0.500000 0.866025i −1.00000
841.1 −0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i −1.55211 2.68834i 1.00000i −0.500000 + 0.866025i −1.00000
841.2 −0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i −0.839478 1.45402i 1.00000i −0.500000 + 0.866025i −1.00000
841.3 −0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i 1.60020 + 2.77163i 1.00000i −0.500000 + 0.866025i −1.00000
841.4 −0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i 2.38947 + 4.13868i 1.00000i −0.500000 + 0.866025i −1.00000
841.5 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i −2.53766 4.39536i 1.00000i −0.500000 + 0.866025i −1.00000
841.6 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i −2.34998 4.07029i 1.00000i −0.500000 + 0.866025i −1.00000
841.7 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 0.243282 + 0.421377i 1.00000i −0.500000 + 0.866025i −1.00000
841.8 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 1.04628 + 1.81222i 1.00000i −0.500000 + 0.866025i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 841.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.x.e 16
37.e even 6 1 inner 1110.2.x.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.x.e 16 1.a even 1 1 trivial
1110.2.x.e 16 37.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$3$ \( ( 1 - T + T^{2} )^{8} \)
$5$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$7$ \( 3748096 - 6473984 T + 15216960 T^{2} + 1563584 T^{3} + 8039408 T^{4} + 165456 T^{5} + 2348008 T^{6} + 237112 T^{7} + 341688 T^{8} + 29500 T^{9} + 32146 T^{10} + 3366 T^{11} + 1763 T^{12} + 128 T^{13} + 57 T^{14} + 4 T^{15} + T^{16} \)
$11$ \( ( -16316 - 21364 T - 2579 T^{2} + 4370 T^{3} + 1159 T^{4} - 184 T^{5} - 65 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$13$ \( 33721249 + 108428304 T + 133310336 T^{2} + 54970368 T^{3} - 13427918 T^{4} - 14402160 T^{5} + 2099840 T^{6} + 4030464 T^{7} + 882163 T^{8} - 144624 T^{9} - 59776 T^{10} + 4608 T^{11} + 2962 T^{12} - 64 T^{14} + T^{16} \)
$17$ \( 33962066944 - 8376995328 T - 5915395456 T^{2} + 1628961216 T^{3} + 701880304 T^{4} - 191859888 T^{5} - 49788508 T^{6} + 13727220 T^{7} + 2755417 T^{8} - 678318 T^{9} - 108478 T^{10} + 22776 T^{11} + 3427 T^{12} - 492 T^{13} - 70 T^{14} + 6 T^{15} + T^{16} \)
$19$ \( 1234872117504 - 314918793216 T - 130929084288 T^{2} + 40216725504 T^{3} + 10247463792 T^{4} - 2806303968 T^{5} - 505049688 T^{6} + 124886232 T^{7} + 20127825 T^{8} - 3605742 T^{9} - 534366 T^{10} + 71604 T^{11} + 10887 T^{12} - 828 T^{13} - 126 T^{14} + 6 T^{15} + T^{16} \)
$23$ \( 301925376 + 1012391424 T^{2} + 639566784 T^{4} + 143806752 T^{6} + 14068884 T^{8} + 681864 T^{10} + 17061 T^{12} + 210 T^{14} + T^{16} \)
$29$ \( 290225296 + 334673336 T^{2} + 139947985 T^{4} + 28698518 T^{6} + 3210367 T^{8} + 203276 T^{10} + 7231 T^{12} + 134 T^{14} + T^{16} \)
$31$ \( 31719424 + 128319488 T^{2} + 161812480 T^{4} + 65527808 T^{6} + 9687040 T^{8} + 611456 T^{10} + 17728 T^{12} + 224 T^{14} + T^{16} \)
$37$ \( 3512479453921 + 1139182525596 T + 82103245088 T^{2} + 4992764904 T^{3} + 6585801754 T^{4} + 1455767220 T^{5} + 123253808 T^{6} + 27728244 T^{7} + 7382059 T^{8} + 749412 T^{9} + 90032 T^{10} + 28740 T^{11} + 3514 T^{12} + 72 T^{13} + 32 T^{14} + 12 T^{15} + T^{16} \)
$41$ \( 33398293504 + 86884686848 T + 263872263936 T^{2} - 86286828800 T^{3} + 56394251840 T^{4} - 4991859840 T^{5} + 3295748992 T^{6} - 215701696 T^{7} + 136509696 T^{8} - 3453376 T^{9} + 2680000 T^{10} - 89040 T^{11} + 35264 T^{12} - 632 T^{13} + 228 T^{14} - 4 T^{15} + T^{16} \)
$43$ \( 99680256 + 2038136832 T^{2} + 982361088 T^{4} + 174081024 T^{6} + 14977920 T^{8} + 691776 T^{10} + 17316 T^{12} + 216 T^{14} + T^{16} \)
$47$ \( ( -924282 + 723762 T + 46791 T^{2} - 84258 T^{3} + 8445 T^{4} + 1446 T^{5} - 195 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$53$ \( 2390818816 - 1448886272 T + 3609581568 T^{2} + 2576953856 T^{3} + 2560675904 T^{4} + 951171456 T^{5} + 466256512 T^{6} + 136045888 T^{7} + 53366016 T^{8} + 11881600 T^{9} + 2611456 T^{10} + 289776 T^{11} + 36464 T^{12} + 2552 T^{13} + 312 T^{14} + 16 T^{15} + T^{16} \)
$59$ \( 8625839772484 - 6146877617628 T + 14752782242 T^{2} + 1029981207750 T^{3} + 83766171835 T^{4} - 64067798232 T^{5} - 2355729199 T^{6} + 2234335290 T^{7} + 31189408 T^{8} - 56342544 T^{9} + 2633789 T^{10} + 615438 T^{11} - 51668 T^{12} - 4560 T^{13} + 863 T^{14} - 48 T^{15} + T^{16} \)
$61$ \( 219503494144 - 957833428992 T + 875746378496 T^{2} + 2258032939008 T^{3} + 1569523633216 T^{4} + 551543059584 T^{5} + 105557767040 T^{6} + 9006307200 T^{7} - 270255872 T^{8} - 103042080 T^{9} - 228064 T^{10} + 1163088 T^{11} + 62128 T^{12} - 3792 T^{13} - 268 T^{14} + 12 T^{15} + T^{16} \)
$67$ \( 389376 + 1198080 T + 2683008 T^{2} + 3237120 T^{3} + 3173616 T^{4} + 1943712 T^{5} + 1184040 T^{6} + 481176 T^{7} + 272217 T^{8} + 76254 T^{9} + 31974 T^{10} + 2712 T^{11} + 1803 T^{12} + 48 T^{13} + 78 T^{14} - 6 T^{15} + T^{16} \)
$71$ \( 39359261500416 + 19103253751296 T + 12416882645376 T^{2} + 3710784762432 T^{3} + 1795657616784 T^{4} + 494541254160 T^{5} + 148091846760 T^{6} + 22640187816 T^{7} + 3408648672 T^{8} + 219772020 T^{9} + 27267462 T^{10} + 1256106 T^{11} + 159507 T^{12} + 3774 T^{13} + 465 T^{14} + 6 T^{15} + T^{16} \)
$73$ \( ( 467968 - 610816 T + 41344 T^{2} + 105344 T^{3} - 10544 T^{4} - 3304 T^{5} + 22 T^{6} + 26 T^{7} + T^{8} )^{2} \)
$79$ \( 815702845702144 + 1515540492386304 T + 733679094247424 T^{2} - 380739398221824 T^{3} + 40486045413376 T^{4} + 3189591948288 T^{5} - 708827666176 T^{6} - 33735906816 T^{7} + 14897061376 T^{8} - 1026136128 T^{9} - 19439152 T^{10} + 4868832 T^{11} + 30700 T^{12} - 34788 T^{13} + 2474 T^{14} - 78 T^{15} + T^{16} \)
$83$ \( 54711081216 + 302948718336 T + 1773476147520 T^{2} - 497163104064 T^{3} + 255007704240 T^{4} - 61694979984 T^{5} + 22656095928 T^{6} - 4834067976 T^{7} + 1045759512 T^{8} - 139625652 T^{9} + 18986070 T^{10} - 1793430 T^{11} + 202623 T^{12} - 15072 T^{13} + 1005 T^{14} - 36 T^{15} + T^{16} \)
$89$ \( 300268929024 - 8521998336 T - 197108951040 T^{2} + 5596480512 T^{3} + 129314131200 T^{4} - 78820994304 T^{5} + 17660685312 T^{6} - 421588800 T^{7} - 346978320 T^{8} + 17143200 T^{9} + 8867424 T^{10} - 1300464 T^{11} + 31161 T^{12} + 4698 T^{13} - 153 T^{14} - 18 T^{15} + T^{16} \)
$97$ \( 14030626134491136 + 1798814020829184 T^{2} + 80436097336320 T^{4} + 1741566375936 T^{6} + 20775540096 T^{8} + 143116416 T^{10} + 564228 T^{12} + 1176 T^{14} + T^{16} \)
show more
show less