Properties

Label 1764.4.f.a
Level $1764$
Weight $4$
Character orbit 1764.f
Analytic conductor $104.079$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + 14344616 x^{9} - 18123280 x^{8} - 273588032 x^{7} + 1239640536 x^{6} - 1407381792 x^{5} - 1961185792 x^{4} + 4297169408 x^{3} + 2991779296 x^{2} - 11217342336 x + 7375227456\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{18}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 252)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{5} +O(q^{10})\) \( q -\beta_{3} q^{5} + \beta_{8} q^{11} -\beta_{4} q^{13} + ( -3 \beta_{3} - \beta_{10} ) q^{17} + ( \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{9} ) q^{19} + ( -\beta_{5} - 3 \beta_{8} ) q^{23} + ( 27 + 2 \beta_{6} - \beta_{12} ) q^{25} + ( \beta_{5} - 2 \beta_{8} - \beta_{15} ) q^{29} + ( -5 \beta_{1} - 7 \beta_{2} + 6 \beta_{4} + 4 \beta_{9} ) q^{31} + ( -10 - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{12} ) q^{37} + ( 2 \beta_{3} + 2 \beta_{10} + \beta_{13} - \beta_{14} ) q^{41} + ( 88 + 3 \beta_{6} + 4 \beta_{7} - 3 \beta_{12} ) q^{43} + ( -14 \beta_{3} + 4 \beta_{10} - \beta_{13} ) q^{47} + ( \beta_{5} + 6 \beta_{8} - \beta_{15} ) q^{53} + ( 7 \beta_{1} - 2 \beta_{2} - 9 \beta_{4} + 2 \beta_{9} ) q^{55} + ( -26 \beta_{3} - 3 \beta_{10} - \beta_{13} + \beta_{14} ) q^{59} + ( 9 \beta_{1} - 3 \beta_{2} - \beta_{9} ) q^{61} + ( 6 \beta_{8} + 3 \beta_{11} + 2 \beta_{15} ) q^{65} + ( 189 + 2 \beta_{6} - \beta_{7} - 6 \beta_{12} ) q^{67} + ( \beta_{5} + 11 \beta_{8} + 8 \beta_{11} - \beta_{15} ) q^{71} + ( -19 \beta_{1} - 9 \beta_{2} - 17 \beta_{4} + 3 \beta_{9} ) q^{73} + ( 44 + \beta_{6} + 8 \beta_{7} - 4 \beta_{12} ) q^{79} + ( -19 \beta_{3} - 4 \beta_{10} - 2 \beta_{13} - 3 \beta_{14} ) q^{83} + ( 461 - \beta_{6} - 9 \beta_{7} - \beta_{12} ) q^{85} + ( 16 \beta_{3} - 10 \beta_{10} - 4 \beta_{14} ) q^{89} + ( -5 \beta_{5} - 9 \beta_{8} + 4 \beta_{11} - 3 \beta_{15} ) q^{95} + ( 48 \beta_{1} + 9 \beta_{2} + 33 \beta_{4} + \beta_{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 424q^{25} - 152q^{37} + 1408q^{43} + 3056q^{67} + 728q^{79} + 7392q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + 14344616 x^{9} - 18123280 x^{8} - 273588032 x^{7} + 1239640536 x^{6} - 1407381792 x^{5} - 1961185792 x^{4} + 4297169408 x^{3} + 2991779296 x^{2} - 11217342336 x + 7375227456\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-8606933517 \nu^{15} + 17491457046 \nu^{14} + 2762073108256 \nu^{13} - 9225620697790 \nu^{12} - 338722846892395 \nu^{11} + 1516388281986214 \nu^{10} + 18962342025989460 \nu^{9} - 105038524605413770 \nu^{8} - 440378181482053668 \nu^{7} + 3029046816285277760 \nu^{6} + 2302641961781731248 \nu^{5} - 25851157990713132280 \nu^{4} - 13012479407317279344 \nu^{3} + 97218732178861288576 \nu^{2} + 22061423532217981184 \nu - 125590689624071796768\)\()/ 6787326577694418304 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(14\!\cdots\!75\)\( \nu^{15} + \)\(87\!\cdots\!44\)\( \nu^{14} + \)\(49\!\cdots\!72\)\( \nu^{13} - \)\(29\!\cdots\!84\)\( \nu^{12} - \)\(62\!\cdots\!73\)\( \nu^{11} + \)\(39\!\cdots\!76\)\( \nu^{10} + \)\(34\!\cdots\!68\)\( \nu^{9} - \)\(24\!\cdots\!36\)\( \nu^{8} - \)\(72\!\cdots\!16\)\( \nu^{7} + \)\(64\!\cdots\!44\)\( \nu^{6} + \)\(14\!\cdots\!48\)\( \nu^{5} - \)\(48\!\cdots\!52\)\( \nu^{4} - \)\(16\!\cdots\!60\)\( \nu^{3} + \)\(18\!\cdots\!64\)\( \nu^{2} + \)\(27\!\cdots\!72\)\( \nu - \)\(22\!\cdots\!68\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(13\!\cdots\!96\)\( \nu^{15} - \)\(28\!\cdots\!71\)\( \nu^{14} - \)\(39\!\cdots\!79\)\( \nu^{13} + \)\(16\!\cdots\!25\)\( \nu^{12} + \)\(39\!\cdots\!36\)\( \nu^{11} - \)\(27\!\cdots\!69\)\( \nu^{10} - \)\(14\!\cdots\!13\)\( \nu^{9} + \)\(16\!\cdots\!47\)\( \nu^{8} - \)\(11\!\cdots\!24\)\( \nu^{7} - \)\(35\!\cdots\!20\)\( \nu^{6} + \)\(13\!\cdots\!36\)\( \nu^{5} - \)\(71\!\cdots\!72\)\( \nu^{4} - \)\(32\!\cdots\!64\)\( \nu^{3} + \)\(23\!\cdots\!56\)\( \nu^{2} + \)\(45\!\cdots\!32\)\( \nu - \)\(88\!\cdots\!72\)\(\)\()/ \)\(35\!\cdots\!44\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(57\!\cdots\!29\)\( \nu^{15} + \)\(24\!\cdots\!09\)\( \nu^{14} + \)\(17\!\cdots\!58\)\( \nu^{13} - \)\(10\!\cdots\!01\)\( \nu^{12} - \)\(18\!\cdots\!23\)\( \nu^{11} + \)\(14\!\cdots\!59\)\( \nu^{10} + \)\(73\!\cdots\!10\)\( \nu^{9} - \)\(85\!\cdots\!11\)\( \nu^{8} + \)\(12\!\cdots\!84\)\( \nu^{7} + \)\(18\!\cdots\!00\)\( \nu^{6} - \)\(54\!\cdots\!64\)\( \nu^{5} + \)\(62\!\cdots\!96\)\( \nu^{4} + \)\(14\!\cdots\!44\)\( \nu^{3} - \)\(52\!\cdots\!92\)\( \nu^{2} - \)\(28\!\cdots\!12\)\( \nu + \)\(29\!\cdots\!84\)\(\)\()/ \)\(35\!\cdots\!44\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(68\!\cdots\!01\)\( \nu^{15} + \)\(66\!\cdots\!94\)\( \nu^{14} + \)\(19\!\cdots\!35\)\( \nu^{13} - \)\(23\!\cdots\!29\)\( \nu^{12} - \)\(17\!\cdots\!35\)\( \nu^{11} + \)\(29\!\cdots\!90\)\( \nu^{10} + \)\(16\!\cdots\!81\)\( \nu^{9} - \)\(15\!\cdots\!43\)\( \nu^{8} + \)\(47\!\cdots\!88\)\( \nu^{7} + \)\(26\!\cdots\!24\)\( \nu^{6} - \)\(17\!\cdots\!12\)\( \nu^{5} + \)\(22\!\cdots\!64\)\( \nu^{4} + \)\(32\!\cdots\!08\)\( \nu^{3} - \)\(60\!\cdots\!08\)\( \nu^{2} - \)\(88\!\cdots\!20\)\( \nu + \)\(14\!\cdots\!24\)\(\)\()/ \)\(35\!\cdots\!44\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(16\!\cdots\!49\)\( \nu^{15} + \)\(12\!\cdots\!76\)\( \nu^{14} + \)\(50\!\cdots\!60\)\( \nu^{13} - \)\(44\!\cdots\!24\)\( \nu^{12} - \)\(52\!\cdots\!07\)\( \nu^{11} + \)\(57\!\cdots\!36\)\( \nu^{10} + \)\(17\!\cdots\!80\)\( \nu^{9} - \)\(32\!\cdots\!24\)\( \nu^{8} + \)\(34\!\cdots\!96\)\( \nu^{7} + \)\(66\!\cdots\!24\)\( \nu^{6} - \)\(24\!\cdots\!48\)\( \nu^{5} + \)\(69\!\cdots\!92\)\( \nu^{4} + \)\(66\!\cdots\!20\)\( \nu^{3} - \)\(26\!\cdots\!68\)\( \nu^{2} - \)\(12\!\cdots\!48\)\( \nu + \)\(12\!\cdots\!04\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(18\!\cdots\!99\)\( \nu^{15} + \)\(25\!\cdots\!04\)\( \nu^{14} + \)\(53\!\cdots\!68\)\( \nu^{13} - \)\(11\!\cdots\!90\)\( \nu^{12} - \)\(57\!\cdots\!37\)\( \nu^{11} + \)\(24\!\cdots\!80\)\( \nu^{10} + \)\(25\!\cdots\!88\)\( \nu^{9} - \)\(16\!\cdots\!90\)\( \nu^{8} - \)\(20\!\cdots\!00\)\( \nu^{7} + \)\(40\!\cdots\!96\)\( \nu^{6} - \)\(96\!\cdots\!12\)\( \nu^{5} - \)\(15\!\cdots\!88\)\( \nu^{4} + \)\(26\!\cdots\!52\)\( \nu^{3} - \)\(24\!\cdots\!00\)\( \nu^{2} - \)\(52\!\cdots\!96\)\( \nu - \)\(52\!\cdots\!96\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(23\!\cdots\!95\)\( \nu^{15} - \)\(48\!\cdots\!86\)\( \nu^{14} - \)\(68\!\cdots\!98\)\( \nu^{13} + \)\(26\!\cdots\!10\)\( \nu^{12} + \)\(72\!\cdots\!81\)\( \nu^{11} - \)\(43\!\cdots\!46\)\( \nu^{10} - \)\(29\!\cdots\!98\)\( \nu^{9} + \)\(27\!\cdots\!70\)\( \nu^{8} + \)\(70\!\cdots\!24\)\( \nu^{7} - \)\(59\!\cdots\!20\)\( \nu^{6} + \)\(17\!\cdots\!80\)\( \nu^{5} - \)\(49\!\cdots\!96\)\( \nu^{4} - \)\(42\!\cdots\!24\)\( \nu^{3} + \)\(22\!\cdots\!80\)\( \nu^{2} + \)\(94\!\cdots\!36\)\( \nu - \)\(10\!\cdots\!84\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(15\!\cdots\!67\)\( \nu^{15} - \)\(56\!\cdots\!56\)\( \nu^{14} - \)\(45\!\cdots\!36\)\( \nu^{13} + \)\(25\!\cdots\!08\)\( \nu^{12} + \)\(48\!\cdots\!57\)\( \nu^{11} - \)\(36\!\cdots\!64\)\( \nu^{10} - \)\(18\!\cdots\!08\)\( \nu^{9} + \)\(22\!\cdots\!68\)\( \nu^{8} - \)\(94\!\cdots\!40\)\( \nu^{7} - \)\(47\!\cdots\!88\)\( \nu^{6} + \)\(15\!\cdots\!08\)\( \nu^{5} - \)\(47\!\cdots\!52\)\( \nu^{4} - \)\(43\!\cdots\!00\)\( \nu^{3} + \)\(32\!\cdots\!24\)\( \nu^{2} + \)\(84\!\cdots\!92\)\( \nu - \)\(10\!\cdots\!48\)\(\)\()/ \)\(35\!\cdots\!44\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(36\!\cdots\!39\)\( \nu^{15} + \)\(90\!\cdots\!56\)\( \nu^{14} + \)\(10\!\cdots\!74\)\( \nu^{13} - \)\(46\!\cdots\!66\)\( \nu^{12} - \)\(11\!\cdots\!25\)\( \nu^{11} + \)\(73\!\cdots\!36\)\( \nu^{10} + \)\(47\!\cdots\!58\)\( \nu^{9} - \)\(45\!\cdots\!14\)\( \nu^{8} - \)\(94\!\cdots\!16\)\( \nu^{7} + \)\(10\!\cdots\!32\)\( \nu^{6} - \)\(29\!\cdots\!28\)\( \nu^{5} - \)\(45\!\cdots\!24\)\( \nu^{4} + \)\(91\!\cdots\!44\)\( \nu^{3} - \)\(21\!\cdots\!16\)\( \nu^{2} - \)\(20\!\cdots\!36\)\( \nu + \)\(12\!\cdots\!28\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{11}\)\(=\)\((\)\(1508211484212094895295 \nu^{15} - 2042933163117031611288 \nu^{14} - 439253308654693863881418 \nu^{13} + 1450495914027627203357166 \nu^{12} + 46994815669119101240110065 \nu^{11} - 247874106788934307120833084 \nu^{10} - 1988938317293089652193634458 \nu^{9} + 16127251053578459040451501386 \nu^{8} + 10266896078569080179037095436 \nu^{7} - 364055953972190375764005961032 \nu^{6} + 981304131032841111857426210616 \nu^{5} - 92670014753316466982364742920 \nu^{4} - 2436280186130887921098606063792 \nu^{3} + 856097779910477052351091216032 \nu^{2} + 5323619205744740232273004797216 \nu - 5489463328251957179851909284576\)\()/ \)\(23\!\cdots\!84\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(52\!\cdots\!07\)\( \nu^{15} - \)\(12\!\cdots\!12\)\( \nu^{14} - \)\(15\!\cdots\!80\)\( \nu^{13} + \)\(64\!\cdots\!54\)\( \nu^{12} + \)\(16\!\cdots\!57\)\( \nu^{11} - \)\(10\!\cdots\!84\)\( \nu^{10} - \)\(66\!\cdots\!68\)\( \nu^{9} + \)\(64\!\cdots\!26\)\( \nu^{8} + \)\(98\!\cdots\!60\)\( \nu^{7} - \)\(14\!\cdots\!48\)\( \nu^{6} + \)\(41\!\cdots\!64\)\( \nu^{5} - \)\(32\!\cdots\!72\)\( \nu^{4} - \)\(11\!\cdots\!52\)\( \nu^{3} + \)\(29\!\cdots\!88\)\( \nu^{2} + \)\(21\!\cdots\!68\)\( \nu - \)\(17\!\cdots\!76\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(30\!\cdots\!29\)\( \nu^{15} - \)\(12\!\cdots\!22\)\( \nu^{14} - \)\(89\!\cdots\!50\)\( \nu^{13} + \)\(52\!\cdots\!54\)\( \nu^{12} + \)\(94\!\cdots\!63\)\( \nu^{11} - \)\(75\!\cdots\!34\)\( \nu^{10} - \)\(35\!\cdots\!58\)\( \nu^{9} + \)\(45\!\cdots\!66\)\( \nu^{8} - \)\(24\!\cdots\!04\)\( \nu^{7} - \)\(97\!\cdots\!28\)\( \nu^{6} + \)\(32\!\cdots\!16\)\( \nu^{5} - \)\(42\!\cdots\!52\)\( \nu^{4} - \)\(95\!\cdots\!20\)\( \nu^{3} + \)\(22\!\cdots\!48\)\( \nu^{2} + \)\(19\!\cdots\!00\)\( \nu - \)\(15\!\cdots\!12\)\(\)\()/ \)\(35\!\cdots\!44\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(65\!\cdots\!01\)\( \nu^{15} - \)\(71\!\cdots\!58\)\( \nu^{14} - \)\(19\!\cdots\!76\)\( \nu^{13} + \)\(59\!\cdots\!96\)\( \nu^{12} + \)\(20\!\cdots\!31\)\( \nu^{11} - \)\(10\!\cdots\!82\)\( \nu^{10} - \)\(86\!\cdots\!00\)\( \nu^{9} + \)\(69\!\cdots\!80\)\( \nu^{8} + \)\(49\!\cdots\!92\)\( \nu^{7} - \)\(16\!\cdots\!48\)\( \nu^{6} + \)\(41\!\cdots\!80\)\( \nu^{5} + \)\(79\!\cdots\!80\)\( \nu^{4} - \)\(12\!\cdots\!28\)\( \nu^{3} + \)\(29\!\cdots\!56\)\( \nu^{2} + \)\(28\!\cdots\!36\)\( \nu - \)\(18\!\cdots\!56\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(17\!\cdots\!73\)\( \nu^{15} - \)\(50\!\cdots\!70\)\( \nu^{14} - \)\(52\!\cdots\!52\)\( \nu^{13} + \)\(11\!\cdots\!52\)\( \nu^{12} + \)\(57\!\cdots\!39\)\( \nu^{11} - \)\(23\!\cdots\!22\)\( \nu^{10} - \)\(26\!\cdots\!56\)\( \nu^{9} + \)\(16\!\cdots\!88\)\( \nu^{8} + \)\(30\!\cdots\!72\)\( \nu^{7} - \)\(39\!\cdots\!12\)\( \nu^{6} + \)\(71\!\cdots\!32\)\( \nu^{5} + \)\(69\!\cdots\!40\)\( \nu^{4} - \)\(21\!\cdots\!84\)\( \nu^{3} - \)\(99\!\cdots\!32\)\( \nu^{2} + \)\(40\!\cdots\!28\)\( \nu - \)\(22\!\cdots\!64\)\(\)\()/ \)\(70\!\cdots\!88\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{14} + \beta_{13} - 21 \beta_{12} + 7 \beta_{11} - \beta_{10} - 15 \beta_{7} - 12 \beta_{6} + 7 \beta_{3} + 78\)\()/378\)
\(\nu^{2}\)\(=\)\((\)\(15 \beta_{15} + 3 \beta_{14} - 20 \beta_{13} + 21 \beta_{12} + \beta_{11} - 43 \beta_{10} - 57 \beta_{8} + 195 \beta_{7} + 30 \beta_{6} - 21 \beta_{5} + 301 \beta_{3} + 54 \beta_{1} + 14106\)\()/378\)
\(\nu^{3}\)\(=\)\((\)\(-207 \beta_{15} + 801 \beta_{14} + 204 \beta_{13} - 2520 \beta_{12} + 1655 \beta_{11} + 1119 \beta_{10} - 378 \beta_{9} - 423 \beta_{8} - 2952 \beta_{7} - 2412 \beta_{6} + 441 \beta_{5} - 756 \beta_{4} - 1029 \beta_{3} + 756 \beta_{2} + 162 \beta_{1} - 79956\)\()/756\)
\(\nu^{4}\)\(=\)\((\)\(1890 \beta_{15} - 966 \beta_{14} - 3304 \beta_{13} + 4851 \beta_{12} - 3458 \beta_{11} - 7154 \beta_{10} + 378 \beta_{9} - 7686 \beta_{8} + 16929 \beta_{7} + 3690 \beta_{6} - 3654 \beta_{5} + 2268 \beta_{4} + 30422 \beta_{3} - 6048 \beta_{2} + 10746 \beta_{1} + 1002852\)\()/378\)
\(\nu^{5}\)\(=\)\((\)\(-38925 \beta_{15} + 89691 \beta_{14} + 36932 \beta_{13} - 194082 \beta_{12} + 201581 \beta_{11} + 161833 \beta_{10} - 73080 \beta_{9} - 14625 \beta_{8} - 283926 \beta_{7} - 196548 \beta_{6} + 72135 \beta_{5} - 196560 \beta_{4} - 365491 \beta_{3} + 194040 \beta_{2} - 88920 \beta_{1} - 10523976\)\()/756\)
\(\nu^{6}\)\(=\)\((\)\(247251 \beta_{15} - 218997 \beta_{14} - 375588 \beta_{13} + 575946 \beta_{12} - 671387 \beta_{11} - 906399 \beta_{10} + 214515 \beta_{9} - 837897 \beta_{8} + 1532430 \beta_{7} + 462636 \beta_{6} - 497553 \beta_{5} + 670950 \beta_{4} + 3253509 \beta_{3} - 1372140 \beta_{2} + 1793205 \beta_{1} + 82934064\)\()/378\)
\(\nu^{7}\)\(=\)\((\)\(-5457375 \beta_{15} + 9167565 \beta_{14} + 5256004 \beta_{13} - 16598400 \beta_{12} + 24293255 \beta_{11} + 19349591 \beta_{10} - 11248146 \beta_{9} + 2556477 \beta_{8} - 27504192 \beta_{7} - 16546848 \beta_{6} + 10081701 \beta_{5} - 32036004 \beta_{4} - 51500141 \beta_{3} + 35283528 \beta_{2} - 24950142 \beta_{1} - 1143777696\)\()/756\)
\(\nu^{8}\)\(=\)\((\)\(31761492 \beta_{15} - 30732828 \beta_{14} - 38908016 \beta_{13} + 61087341 \beta_{12} - 99578612 \beta_{11} - 102373012 \beta_{10} + 44463384 \beta_{9} - 89225724 \beta_{8} + 143167695 \beta_{7} + 52162518 \beta_{6} - 63447132 \beta_{5} + 131107536 \beta_{4} + 341214748 \beta_{3} - 225763776 \beta_{2} + 258792984 \beta_{1} + 7342936392\)\()/378\)
\(\nu^{9}\)\(=\)\((\)\(-705685329 \beta_{15} + 905768487 \beta_{14} + 630291996 \beta_{13} - 1491982674 \beta_{12} + 2914570457 \beta_{11} + 2109655557 \beta_{10} - 1553081040 \beta_{9} + 690659955 \beta_{8} - 2667928134 \beta_{7} - 1451788764 \beta_{6} + 1320955083 \beta_{5} - 4478646816 \beta_{4} - 5995144071 \beta_{3} + 5381818848 \beta_{2} - 4415367024 \beta_{1} - 117465566928\)\()/756\)
\(\nu^{10}\)\(=\)\((\)\(3983508633 \beta_{15} - 3593813727 \beta_{14} - 3873978556 \beta_{13} + 6187643826 \beta_{12} - 13391555297 \beta_{11} - 10779333005 \beta_{10} + 7062339375 \beta_{9} - 9641062059 \beta_{8} + 13519561638 \beta_{7} + 5484433020 \beta_{6} - 7865474883 \beta_{5} + 20610765630 \beta_{4} + 34535195807 \beta_{3} - 32288423580 \beta_{2} + 34454087721 \beta_{1} + 670379739216\)\()/378\)
\(\nu^{11}\)\(=\)\((\)\(-88019325471 \beta_{15} + 87423864621 \beta_{14} + 68068170740 \beta_{13} - 136607395848 \beta_{12} + 348057970967 \beta_{11} + 216338895511 \beta_{10} - 201306893634 \beta_{9} + 114069661629 \beta_{8} - 256189015128 \beta_{7} - 130393288752 \beta_{6} + 166608558501 \beta_{5} - 582527009988 \beta_{4} - 635602694125 \beta_{3} + 740695680648 \beta_{2} - 652161282894 \beta_{1} - 11644698741504\)\()/756\)
\(\nu^{12}\)\(=\)\((\)\(69826854858 \beta_{15} - 54148201326 \beta_{14} - 53369153208 \beta_{13} + 86322951231 \beta_{12} - 244021801370 \beta_{11} - 153640517418 \beta_{10} + 140859532440 \beta_{9} - 151770071838 \beta_{8} + 181047726741 \beta_{7} + 78162369810 \beta_{6} - 136670744334 \beta_{5} + 409904987184 \beta_{4} + 481197260142 \beta_{3} - 609337507680 \beta_{2} + 623361422376 \beta_{1} + 8788001773704\)\()/54\)
\(\nu^{13}\)\(=\)\((\)\(-10715369975505 \beta_{15} + 8205381038199 \beta_{14} + 6819476356444 \beta_{13} - 12444995066322 \beta_{12} + 41256721282841 \beta_{11} + 21060178184021 \beta_{10} - 25034840334504 \beta_{9} + 15991071712083 \beta_{8} - 24038443432518 \beta_{7} - 11724814079292 \beta_{6} + 20434048828011 \beta_{5} - 72543618293328 \beta_{4} - 63021735353591 \beta_{3} + 95498829657504 \beta_{2} - 87324431058072 \beta_{1} - 1113034475400816\)\()/756\)
\(\nu^{14}\)\(=\)\((\)\(58781025379719 \beta_{15} - 37030892703297 \beta_{14} - 34684564933124 \beta_{13} + 56607602007642 \beta_{12} - 210142914779519 \beta_{11} - 101893537055827 \beta_{10} + 127569023016435 \beta_{9} - 118772375499189 \beta_{8} + 115884521667102 \beta_{7} + 51884296051884 \beta_{6} - 114394442978781 \beta_{5} + 370792334073798 \beta_{4} + 314924159092417 \beta_{3} - 535058753448492 \beta_{2} + 533742420199077 \beta_{1} + 5550942388854192\)\()/378\)
\(\nu^{15}\)\(=\)\((\)\(-1277826382488615 \beta_{15} + 739525552505109 \beta_{14} + 638478540273492 \beta_{13} - 1102930179866256 \beta_{12} + 4839594059223839 \beta_{11} + 1939900022482959 \beta_{10} - 3020067066896082 \beta_{9} + 2060641280479125 \beta_{8} - 2169287020890672 \beta_{7} - 1030495300713696 \beta_{6} + 2448113612899773 \beta_{5} - 8757454706027748 \beta_{4} - 5866179196536981 \beta_{3} + 11772210945075336 \beta_{2} - 10996020069922494 \beta_{1} - 101543997959861376\)\()/756\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
4.65022 0.707107i
4.65022 + 0.707107i
−10.4548 0.707107i
−10.4548 + 0.707107i
−1.35249 0.707107i
−1.35249 + 0.707107i
8.15703 + 0.707107i
8.15703 0.707107i
5.70754 + 0.707107i
5.70754 0.707107i
1.09700 0.707107i
1.09700 + 0.707107i
−8.00527 0.707107i
−8.00527 + 0.707107i
2.20073 0.707107i
2.20073 + 0.707107i
0 0 0 −20.2518 0 0 0 0 0
881.2 0 0 0 −20.2518 0 0 0 0 0
881.3 0 0 0 −8.73625 0 0 0 0 0
881.4 0 0 0 −8.73625 0 0 0 0 0
881.5 0 0 0 −8.54212 0 0 0 0 0
881.6 0 0 0 −8.54212 0 0 0 0 0
881.7 0 0 0 −6.82452 0 0 0 0 0
881.8 0 0 0 −6.82452 0 0 0 0 0
881.9 0 0 0 6.82452 0 0 0 0 0
881.10 0 0 0 6.82452 0 0 0 0 0
881.11 0 0 0 8.54212 0 0 0 0 0
881.12 0 0 0 8.54212 0 0 0 0 0
881.13 0 0 0 8.73625 0 0 0 0 0
881.14 0 0 0 8.73625 0 0 0 0 0
881.15 0 0 0 20.2518 0 0 0 0 0
881.16 0 0 0 20.2518 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.f.a 16
3.b odd 2 1 inner 1764.4.f.a 16
7.b odd 2 1 inner 1764.4.f.a 16
7.c even 3 1 252.4.t.a 16
7.c even 3 1 1764.4.t.b 16
7.d odd 6 1 252.4.t.a 16
7.d odd 6 1 1764.4.t.b 16
21.c even 2 1 inner 1764.4.f.a 16
21.g even 6 1 252.4.t.a 16
21.g even 6 1 1764.4.t.b 16
21.h odd 6 1 252.4.t.a 16
21.h odd 6 1 1764.4.t.b 16
28.f even 6 1 1008.4.bt.b 16
28.g odd 6 1 1008.4.bt.b 16
84.j odd 6 1 1008.4.bt.b 16
84.n even 6 1 1008.4.bt.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.t.a 16 7.c even 3 1
252.4.t.a 16 7.d odd 6 1
252.4.t.a 16 21.g even 6 1
252.4.t.a 16 21.h odd 6 1
1008.4.bt.b 16 28.f even 6 1
1008.4.bt.b 16 28.g odd 6 1
1008.4.bt.b 16 84.j odd 6 1
1008.4.bt.b 16 84.n even 6 1
1764.4.f.a 16 1.a even 1 1 trivial
1764.4.f.a 16 3.b odd 2 1 inner
1764.4.f.a 16 7.b odd 2 1 inner
1764.4.f.a 16 21.c even 2 1 inner
1764.4.t.b 16 7.c even 3 1
1764.4.t.b 16 7.d odd 6 1
1764.4.t.b 16 21.g even 6 1
1764.4.t.b 16 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 606 T_{5}^{6} + 92853 T_{5}^{4} - 5395140 T_{5}^{2} + 106378596 \) acting on \(S_{4}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 106378596 - 5395140 T^{2} + 92853 T^{4} - 606 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 84182380164 + 1463257548 T^{2} + 5194125 T^{4} + 4806 T^{6} + T^{8} )^{2} \)
$13$ \( ( 22573860516 + 426062052 T^{2} + 2267613 T^{4} + 2826 T^{6} + T^{8} )^{2} \)
$17$ \( ( 70607585620224 - 141379508160 T^{2} + 91834020 T^{4} - 21396 T^{6} + T^{8} )^{2} \)
$19$ \( ( 46677071171844 + 379172228220 T^{2} + 262304325 T^{4} + 30858 T^{6} + T^{8} )^{2} \)
$23$ \( ( 22211376380153856 + 10406803131648 T^{2} + 1497771972 T^{4} + 71460 T^{6} + T^{8} )^{2} \)
$29$ \( ( 416548977121475136 + 86086124811216 T^{2} + 5723596161 T^{4} + 133110 T^{6} + T^{8} )^{2} \)
$31$ \( ( 17938763430477561 + 336040273495584 T^{2} + 14869021014 T^{4} + 213456 T^{6} + T^{8} )^{2} \)
$37$ \( ( -424087754 - 14086516 T - 109317 T^{2} + 38 T^{3} + T^{4} )^{4} \)
$41$ \( ( 3423608728179335424 - 403281542442816 T^{2} + 16046933220 T^{4} - 240060 T^{6} + T^{8} )^{2} \)
$43$ \( ( -6254809742 + 68765576 T - 146451 T^{2} - 352 T^{3} + T^{4} )^{4} \)
$47$ \( ( 68567622309159789456 - 4436251230240000 T^{2} + 83116153944 T^{4} - 522912 T^{6} + T^{8} )^{2} \)
$53$ \( ( 1998740432850076224 + 378603077947344 T^{2} + 18621022689 T^{4} + 252054 T^{6} + T^{8} )^{2} \)
$59$ \( ( 236957268508674624 - 4728735561416688 T^{2} + 134996565465 T^{4} - 705414 T^{6} + T^{8} )^{2} \)
$61$ \( ( 10077652565550144 + 15350015516832 T^{2} + 2414692692 T^{4} + 93852 T^{6} + T^{8} )^{2} \)
$67$ \( ( 30097773916 + 189716044 T - 345693 T^{2} - 764 T^{3} + T^{4} )^{4} \)
$71$ \( ( \)\(17\!\cdots\!44\)\( + 42297868577973888 T^{2} + 337035071448 T^{4} + 1006848 T^{6} + T^{8} )^{2} \)
$73$ \( ( 11798164103488007184 + 5973381293368680 T^{2} + 408979621881 T^{4} + 1634910 T^{6} + T^{8} )^{2} \)
$79$ \( ( 1237158301 + 173275438 T - 635574 T^{2} - 182 T^{3} + T^{4} )^{4} \)
$83$ \( ( \)\(10\!\cdots\!84\)\( - 79735840708215204 T^{2} + 734790325365 T^{4} - 1589970 T^{6} + T^{8} )^{2} \)
$89$ \( ( \)\(31\!\cdots\!64\)\( - 534309537696420096 T^{2} + 2392343285328 T^{4} - 3076488 T^{6} + T^{8} )^{2} \)
$97$ \( ( \)\(47\!\cdots\!36\)\( + 3276863917344744120 T^{2} + 6899032044873 T^{4} + 5135454 T^{6} + T^{8} )^{2} \)
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