Properties

Label 1110.2.x.d
Level $1110$
Weight $2$
Character orbit 1110.x
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.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: \(x^{16} + 60 x^{14} + 1362 x^{12} + 15028 x^{10} + 86441 x^{8} + 260376 x^{6} + 382684 x^{4} + 224224 x^{2} + 38416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 60 x^{14} + 1362 x^{12} + 15028 x^{10} + 86441 x^{8} + 260376 x^{6} + 382684 x^{4} + 224224 x^{2} + 38416\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 11 \nu^{14} + 613 \nu^{12} + 12389 \nu^{10} + 113739 \nu^{8} + 490228 \nu^{6} + 982004 \nu^{4} + 849840 \nu^{2} - 34944 \nu + 206192 \)\()/69888\)
\(\beta_{3}\)\(=\)\((\)\(-16872 \nu^{15} - 85673 \nu^{14} - 845916 \nu^{13} - 4862599 \nu^{12} - 13686912 \nu^{11} - 100872023 \nu^{10} - 65555292 \nu^{9} - 957858153 \nu^{8} + 258196992 \nu^{7} - 4251636172 \nu^{6} + 2769818064 \nu^{5} - 8143903964 \nu^{4} + 6020259264 \nu^{3} - 5400934224 \nu^{2} + 2496441024 \nu - 1892498384\)\()/ 1061598720 \)
\(\beta_{4}\)\(=\)\((\)\(-1843 \nu^{15} + 87690 \nu^{14} - 55469 \nu^{13} + 5110410 \nu^{12} + 441587 \nu^{11} + 110660190 \nu^{10} + 28000317 \nu^{9} + 1128794190 \nu^{8} + 291477148 \nu^{7} + 5671831800 \nu^{6} + 1033742156 \nu^{5} + 13479437640 \nu^{4} + 1153506576 \nu^{3} + 12745189920 \nu^{2} + 82282256 \nu + 3013123680\)\()/ 303313920 \)
\(\beta_{5}\)\(=\)\((\)\( 1406 \nu^{15} + 70493 \nu^{13} + 1140576 \nu^{11} + 5462941 \nu^{9} - 21516416 \nu^{7} - 230818172 \nu^{5} - 501688272 \nu^{3} - 208036752 \nu \)\()/44233280\)
\(\beta_{6}\)\(=\)\((\)\(-1843 \nu^{15} - 87690 \nu^{14} - 55469 \nu^{13} - 5110410 \nu^{12} + 441587 \nu^{11} - 110660190 \nu^{10} + 28000317 \nu^{9} - 1128794190 \nu^{8} + 291477148 \nu^{7} - 5671831800 \nu^{6} + 1033742156 \nu^{5} - 13479437640 \nu^{4} + 1153506576 \nu^{3} - 12745189920 \nu^{2} + 82282256 \nu - 3013123680\)\()/ 303313920 \)
\(\beta_{7}\)\(=\)\((\)\(-52443 \nu^{15} - 10192 \nu^{14} - 3509229 \nu^{13} - 343196 \nu^{12} - 91328373 \nu^{11} + 666008 \nu^{10} - 1178556723 \nu^{9} + 128790228 \nu^{8} - 7902057732 \nu^{7} + 1556081632 \nu^{6} - 26227642164 \nu^{5} + 7108505264 \nu^{4} - 37538711664 \nu^{3} + 14559519744 \nu^{2} - 15174626544 \nu + 9551468864\)\()/ 1061598720 \)
\(\beta_{8}\)\(=\)\((\)\(12239 \nu^{15} - 23772 \nu^{14} + 694657 \nu^{13} - 1327536 \nu^{12} + 14410289 \nu^{11} - 26856732 \nu^{10} + 136836879 \nu^{9} - 245232792 \nu^{8} + 607376596 \nu^{7} - 1023268848 \nu^{6} + 1163414852 \nu^{5} - 1782414816 \nu^{4} + 771562032 \nu^{3} - 897078336 \nu^{2} + 270356912 \nu - 92593536\)\()/ 151656960 \)
\(\beta_{9}\)\(=\)\((\)\( 263 \nu^{15} + 15241 \nu^{13} + 328169 \nu^{11} + 3345303 \nu^{9} + 17160772 \nu^{7} + 44457716 \nu^{5} + 52527696 \nu^{3} + 17328752 \nu + 1712256 \)\()/3424512\)
\(\beta_{10}\)\(=\)\((\)\(12239 \nu^{15} + 23772 \nu^{14} + 694657 \nu^{13} + 1327536 \nu^{12} + 14410289 \nu^{11} + 26856732 \nu^{10} + 136836879 \nu^{9} + 245232792 \nu^{8} + 607376596 \nu^{7} + 1023268848 \nu^{6} + 1163414852 \nu^{5} + 1782414816 \nu^{4} + 771562032 \nu^{3} + 897078336 \nu^{2} + 270356912 \nu + 92593536\)\()/ 151656960 \)
\(\beta_{11}\)\(=\)\((\)\( -17481 \nu^{15} - 1169743 \nu^{13} - 30442791 \nu^{11} - 392852241 \nu^{9} - 2634019244 \nu^{7} - 8742547388 \nu^{5} - 12512903888 \nu^{3} - 5058208848 \nu \)\()/ 176933120 \)
\(\beta_{12}\)\(=\)\((\)\(-407712 \nu^{15} + 191303 \nu^{14} - 23678916 \nu^{13} + 10175809 \nu^{12} - 509843112 \nu^{11} + 188699273 \nu^{10} - 5151114132 \nu^{9} + 1453116063 \nu^{8} - 25447889088 \nu^{7} + 4069471252 \nu^{6} - 58519687536 \nu^{5} + 1181806724 \nu^{4} - 50260461696 \nu^{3} - 6855421776 \nu^{2} - 5721258816 \nu - 2614850896\)\()/ 2123197440 \)
\(\beta_{13}\)\(=\)\((\)\(407712 \nu^{15} + 191303 \nu^{14} + 23678916 \nu^{13} + 10175809 \nu^{12} + 509843112 \nu^{11} + 188699273 \nu^{10} + 5151114132 \nu^{9} + 1453116063 \nu^{8} + 25447889088 \nu^{7} + 4069471252 \nu^{6} + 58519687536 \nu^{5} + 1181806724 \nu^{4} + 50260461696 \nu^{3} - 6855421776 \nu^{2} + 5721258816 \nu - 2614850896\)\()/ 2123197440 \)
\(\beta_{14}\)\(=\)\((\)\(-152793 \nu^{15} + 28329 \nu^{14} - 8812379 \nu^{13} + 1614627 \nu^{12} - 188046063 \nu^{11} + 33560079 \nu^{10} - 1883030733 \nu^{9} + 315597429 \nu^{8} - 9316505932 \nu^{7} + 1322651596 \nu^{6} - 22536386924 \nu^{5} + 1901388972 \nu^{4} - 24044200464 \nu^{3} - 586462128 \nu^{2} - 8364998544 \nu - 1258733168\)\()/ 353866240 \)
\(\beta_{15}\)\(=\)\((\)\(-539909 \nu^{15} - 84987 \nu^{14} - 31161847 \nu^{13} - 4843881 \nu^{12} - 665870579 \nu^{11} - 100680237 \nu^{10} - 6686136129 \nu^{9} - 946792287 \nu^{8} - 33269357116 \nu^{7} - 3967954788 \nu^{6} - 81391052732 \nu^{5} - 5704166916 \nu^{4} - 88416187152 \nu^{3} + 1759386384 \nu^{2} - 30466908752 \nu + 4306998864\)\()/ 1061598720 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} - \beta_{14} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - \beta_{1} - 7\)
\(\nu^{3}\)\(=\)\(-\beta_{15} - \beta_{14} - 2 \beta_{11} - 2 \beta_{10} - 13 \beta_{9} - 2 \beta_{8} - 2 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 12 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(-18 \beta_{15} + 18 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \beta_{11} - 28 \beta_{10} - 18 \beta_{9} + 28 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 48 \beta_{2} + 24 \beta_{1} + 95\)
\(\nu^{5}\)\(=\)\(27 \beta_{15} + 27 \beta_{14} + 44 \beta_{11} + 70 \beta_{10} + 303 \beta_{9} + 70 \beta_{8} + 34 \beta_{6} - 139 \beta_{5} + 34 \beta_{4} + 216 \beta_{1} - 165\)
\(\nu^{6}\)\(=\)\(328 \beta_{15} - 328 \beta_{14} + 54 \beta_{13} + 54 \beta_{12} - 63 \beta_{11} + 690 \beta_{10} + 328 \beta_{9} - 690 \beta_{8} + 126 \beta_{7} - 54 \beta_{6} + 243 \beta_{5} + 54 \beta_{4} + 486 \beta_{3} - 1044 \beta_{2} - 522 \beta_{1} - 1631\)
\(\nu^{7}\)\(=\)\(-639 \beta_{15} - 639 \beta_{14} - 72 \beta_{13} + 72 \beta_{12} - 964 \beta_{11} - 1846 \beta_{10} - 6395 \beta_{9} - 1846 \beta_{8} - 458 \beta_{6} + 3619 \beta_{5} - 458 \beta_{4} - 4404 \beta_{1} + 3517\)
\(\nu^{8}\)\(=\)\(-6356 \beta_{15} + 6356 \beta_{14} - 1086 \beta_{13} - 1086 \beta_{12} + 2027 \beta_{11} - 16442 \beta_{10} - 6356 \beta_{9} + 16442 \beta_{8} - 4054 \beta_{7} + 1206 \beta_{6} - 8747 \beta_{5} - 1206 \beta_{4} - 17494 \beta_{3} + 22740 \beta_{2} + 11370 \beta_{1} + 31135\)
\(\nu^{9}\)\(=\)\(14483 \beta_{15} + 14483 \beta_{14} + 2848 \beta_{13} - 2848 \beta_{12} + 21832 \beta_{11} + 44942 \beta_{10} + 135095 \beta_{9} + 44942 \beta_{8} + 5690 \beta_{6} - 90523 \beta_{5} + 5690 \beta_{4} + 94124 \beta_{1} - 74789\)
\(\nu^{10}\)\(=\)\(130184 \beta_{15} - 130184 \beta_{14} + 19958 \beta_{13} + 19958 \beta_{12} - 53735 \beta_{11} + 384754 \beta_{10} + 130184 \beta_{9} - 384754 \beta_{8} + 107470 \beta_{7} - 26118 \beta_{6} + 239715 \beta_{5} + 26118 \beta_{4} + 479430 \beta_{3} - 500340 \beta_{2} - 250170 \beta_{1} - 632007\)
\(\nu^{11}\)\(=\)\(-323863 \beta_{15} - 323863 \beta_{14} - 81352 \beta_{13} + 81352 \beta_{12} - 501140 \beta_{11} - 1058558 \beta_{10} - 2896595 \beta_{9} - 1058558 \beta_{8} - 65386 \beta_{6} + 2192763 \beta_{5} - 65386 \beta_{4} - 2058660 \beta_{1} + 1610229\)
\(\nu^{12}\)\(=\)\(-2771924 \beta_{15} + 2771924 \beta_{14} - 358558 \beta_{13} - 358558 \beta_{12} + 1317035 \beta_{11} - 8899826 \beta_{10} - 2771924 \beta_{9} + 8899826 \beta_{8} - 2634070 \beta_{7} + 566374 \beta_{6} - 5980091 \beta_{5} - 566374 \beta_{4} - 11960182 \beta_{3} + 11104900 \beta_{2} + 5552450 \beta_{1} + 13344047\)
\(\nu^{13}\)\(=\)\(7228043 \beta_{15} + 7228043 \beta_{14} + 2067696 \beta_{13} - 2067696 \beta_{12} + 11521008 \beta_{11} + 24521326 \beta_{10} + 63035871 \beta_{9} + 24521326 \beta_{8} + 634266 \beta_{6} - 51912235 \beta_{5} + 634266 \beta_{4} + 45624252 \beta_{1} - 35131957\)
\(\nu^{14}\)\(=\)\(60481488 \beta_{15} - 60481488 \beta_{14} + 6504422 \beta_{13} + 6504422 \beta_{12} - 31115103 \beta_{11} + 204305602 \beta_{10} + 60481488 \beta_{9} - 204305602 \beta_{8} + 62230206 \beta_{7} - 12388390 \beta_{6} + 142871283 \beta_{5} + 12388390 \beta_{4} + 285742566 \beta_{3} - 248053172 \beta_{2} - 124026586 \beta_{1} - 289019607\)
\(\nu^{15}\)\(=\)\(-161646111 \beta_{15} - 161646111 \beta_{14} - 49841816 \beta_{13} + 49841816 \beta_{12} - 264166636 \beta_{11} - 562800062 \beta_{10} - 1388334603 \beta_{9} - 562800062 \beta_{8} - 3006986 \beta_{6} + 1210024891 \beta_{5} - 3006986 \beta_{4} - 1019231764 \beta_{1} + 774990357\)

Character values

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

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(\beta_{9}\) \(1\)

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
4.75759i
0.535537i
0.871333i
1.68974i
3.25684i
1.95985i
2.16125i
3.78749i
4.75759i
0.535537i
0.871333i
1.68974i
3.25684i
1.95985i
2.16125i
3.78749i
−0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i −2.37879 + 4.12019i 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 −0.267768 + 0.463788i 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 0.435667 0.754597i 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 0.844871 1.46336i 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 −1.62842 + 2.82051i 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 −0.979923 + 1.69728i 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 1.08063 1.87170i 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 1.89374 3.28006i 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 −2.37879 4.12019i 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 −0.267768 0.463788i 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 0.435667 + 0.754597i 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 0.844871 + 1.46336i 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 −1.62842 2.82051i 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 −0.979923 1.69728i 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 1.08063 + 1.87170i 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 1.89374 + 3.28006i 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.d 16
37.e even 6 1 inner 1110.2.x.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.x.d 16 1.a even 1 1 trivial
1110.2.x.d 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

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$3$ \( ( 1 + T + T^{2} )^{8} \)
$5$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$7$ \( 38416 + 16464 T + 115640 T^{2} - 75544 T^{3} + 256600 T^{4} - 85068 T^{5} + 121838 T^{6} - 20766 T^{7} + 37621 T^{8} - 4080 T^{9} + 6068 T^{10} + 70 T^{11} + 623 T^{12} + 8 T^{13} + 32 T^{14} + 2 T^{15} + T^{16} \)
$11$ \( ( 24336 - 18096 T - 8696 T^{2} + 3972 T^{3} + 1225 T^{4} - 232 T^{5} - 62 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$13$ \( 37417689 - 96049134 T + 59698176 T^{2} + 57720552 T^{3} - 29643110 T^{4} - 22780614 T^{5} + 10287144 T^{6} + 6809838 T^{7} + 139387 T^{8} - 450438 T^{9} - 23896 T^{10} + 22062 T^{11} + 2626 T^{12} - 408 T^{13} - 56 T^{14} + 6 T^{15} + T^{16} \)
$17$ \( 82944 + 539136 T + 1480320 T^{2} + 2029248 T^{3} + 1344688 T^{4} + 182016 T^{5} - 252172 T^{6} - 90288 T^{7} + 37025 T^{8} + 20298 T^{9} - 2622 T^{10} - 2196 T^{11} + 219 T^{12} + 156 T^{13} - 14 T^{14} - 6 T^{15} + T^{16} \)
$19$ \( 26873856 - 223948800 T + 607564800 T^{2} + 120960000 T^{3} - 184735232 T^{4} - 32244480 T^{5} + 42256192 T^{6} + 11835072 T^{7} - 1904480 T^{8} - 818640 T^{9} + 77348 T^{10} + 40860 T^{11} + 493 T^{12} - 828 T^{13} - 21 T^{14} + 12 T^{15} + T^{16} \)
$23$ \( 5308416 + 130351104 T^{2} + 92787712 T^{4} + 25499136 T^{6} + 3454144 T^{8} + 246944 T^{10} + 9172 T^{12} + 160 T^{14} + T^{16} \)
$29$ \( 10585940544 + 8081097696 T^{2} + 2286456385 T^{4} + 307823166 T^{6} + 22051311 T^{8} + 880484 T^{10} + 19503 T^{12} + 222 T^{14} + T^{16} \)
$31$ \( 201867264 + 341458944 T^{2} + 214373632 T^{4} + 61980544 T^{6} + 8458480 T^{8} + 510584 T^{10} + 14689 T^{12} + 198 T^{14} + T^{16} \)
$37$ \( 3512479453921 - 1139182525596 T + 130852046859 T^{2} + 13868791400 T^{3} - 7554742991 T^{4} + 825238676 T^{5} + 2395750 T^{6} - 10020932 T^{7} + 2319866 T^{8} - 270836 T^{9} + 1750 T^{10} + 16292 T^{11} - 4031 T^{12} + 200 T^{13} + 51 T^{14} - 12 T^{15} + T^{16} \)
$41$ \( 215838879399936 - 118141872316416 T + 53445202209792 T^{2} - 12022810071552 T^{3} + 2626522960960 T^{4} - 345353064320 T^{5} + 58702638208 T^{6} - 5856113472 T^{7} + 909029056 T^{8} - 63476960 T^{9} + 9193152 T^{10} - 449168 T^{11} + 69280 T^{12} - 1976 T^{13} + 320 T^{14} - 4 T^{15} + T^{16} \)
$43$ \( 9216 + 5839872 T^{2} + 9376384 T^{4} + 4691136 T^{6} + 1017988 T^{8} + 109136 T^{10} + 5869 T^{12} + 142 T^{14} + T^{16} \)
$47$ \( ( -178848 + 499824 T - 473498 T^{2} + 179494 T^{3} - 26883 T^{4} + 276 T^{5} + 344 T^{6} - 34 T^{7} + T^{8} )^{2} \)
$53$ \( 20517855952896 - 13690746372096 T + 13913866727424 T^{2} + 1215940125696 T^{3} + 1637041543744 T^{4} - 64700852864 T^{5} + 66716028288 T^{6} + 951316800 T^{7} + 1150433152 T^{8} + 22932192 T^{9} + 13348736 T^{10} + 411120 T^{11} + 94656 T^{12} + 2552 T^{13} + 440 T^{14} + 12 T^{15} + T^{16} \)
$59$ \( 14648745024 - 45018094464 T + 50131696464 T^{2} - 12340623456 T^{3} - 7865362388 T^{4} + 2914125996 T^{5} + 1716226410 T^{6} - 827699526 T^{7} + 76915537 T^{8} + 13970424 T^{9} - 1822588 T^{10} - 199410 T^{11} + 31363 T^{12} + 1272 T^{13} - 200 T^{14} - 6 T^{15} + T^{16} \)
$61$ \( 940868960256 + 882204327936 T - 12368259072 T^{2} - 270137240064 T^{3} + 44475681856 T^{4} + 24641235456 T^{5} - 2547287936 T^{6} - 1300823040 T^{7} + 165145216 T^{8} + 30558048 T^{9} - 2458336 T^{10} - 402816 T^{11} + 32320 T^{12} + 2880 T^{13} - 192 T^{14} - 12 T^{15} + T^{16} \)
$67$ \( 1874890000 + 372380000 T + 12584672300 T^{2} + 12796283000 T^{3} + 86449237661 T^{4} + 51519123736 T^{5} + 21496069505 T^{6} + 5372873976 T^{7} + 1059594766 T^{8} + 146381784 T^{9} + 18551009 T^{10} + 1901916 T^{11} + 193366 T^{12} + 14660 T^{13} + 941 T^{14} + 36 T^{15} + T^{16} \)
$71$ \( 2190240000 + 5717088000 T + 13985006400 T^{2} + 4958178240 T^{3} + 3721014256 T^{4} - 219046768 T^{5} + 486408456 T^{6} - 75604392 T^{7} + 54419464 T^{8} - 12349860 T^{9} + 3327266 T^{10} - 413550 T^{11} + 51183 T^{12} - 2630 T^{13} + 233 T^{14} - 6 T^{15} + T^{16} \)
$73$ \( ( -914048 - 771840 T - 22336 T^{2} + 82576 T^{3} + 8222 T^{4} - 2330 T^{5} - 253 T^{6} + 8 T^{7} + T^{8} )^{2} \)
$79$ \( 239802172416 - 163221553152 T - 3193291776 T^{2} + 27379580928 T^{3} - 1184115008 T^{4} - 3097499328 T^{5} + 347682496 T^{6} + 188828208 T^{7} - 21704492 T^{8} - 8096532 T^{9} + 1045082 T^{10} + 174978 T^{11} - 13037 T^{12} - 2376 T^{13} + 93 T^{14} + 24 T^{15} + T^{16} \)
$83$ \( 180864626782464 + 44258993505792 T + 22659527995200 T^{2} + 2224959660288 T^{3} + 1086833567344 T^{4} + 57563024560 T^{5} + 38203820088 T^{6} + 123377448 T^{7} + 807075688 T^{8} - 22466700 T^{9} + 13213646 T^{10} - 577770 T^{11} + 101319 T^{12} - 2944 T^{13} + 437 T^{14} - 12 T^{15} + T^{16} \)
$89$ \( 7020380160000 - 18222465024000 T + 21015230054400 T^{2} - 13624153620480 T^{3} + 5372879908864 T^{4} - 1288823685120 T^{5} + 169986216960 T^{6} - 6197713920 T^{7} - 1250760704 T^{8} + 108853248 T^{9} + 12447296 T^{10} - 1947840 T^{11} + 21184 T^{12} + 7392 T^{13} - 116 T^{14} - 24 T^{15} + T^{16} \)
$97$ \( 5760000 + 251289600 T^{2} + 231569536 T^{4} + 75288064 T^{6} + 10775044 T^{8} + 694496 T^{10} + 20893 T^{12} + 270 T^{14} + T^{16} \)
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