Properties

Label 1764.2.x.a
Level $1764$
Weight $2$
Character orbit 1764.x
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.x (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} - 4374 x + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 252)
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.

\(f(q)\) \(=\) \( q -\beta_{5} q^{3} -\beta_{11} q^{5} + ( \beta_{2} - \beta_{3} + \beta_{14} - \beta_{15} ) q^{9} +O(q^{10})\) \( q -\beta_{5} q^{3} -\beta_{11} q^{5} + ( \beta_{2} - \beta_{3} + \beta_{14} - \beta_{15} ) q^{9} + ( 1 + \beta_{2} + \beta_{5} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{11} + ( -\beta_{4} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{13} - \beta_{14} ) q^{13} + ( -1 + \beta_{1} - 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{15} + ( -1 + \beta_{1} - \beta_{5} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{13} - \beta_{15} ) q^{17} + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{19} + ( \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{10} - 2 \beta_{13} - \beta_{14} ) q^{23} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + 2 \beta_{10} - \beta_{12} - \beta_{14} ) q^{25} + ( -\beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{27} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{29} + ( -2 \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{15} ) q^{31} + ( -2 - \beta_{1} - \beta_{2} + \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{33} + ( -1 + \beta_{1} - \beta_{6} - \beta_{10} - \beta_{11} + 2 \beta_{13} ) q^{37} + ( -1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{39} + ( -\beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{41} + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} - 3 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{15} ) q^{43} + ( 2 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} + 3 \beta_{9} - \beta_{12} - \beta_{14} + 3 \beta_{15} ) q^{45} + ( 1 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - 4 \beta_{15} ) q^{47} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{51} + ( -1 + 2 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{10} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{53} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{55} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{12} + \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{57} + ( 3 + 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{59} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{61} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{65} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{67} + ( -\beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + 3 \beta_{12} + 2 \beta_{14} - 5 \beta_{15} ) q^{69} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{71} + ( -2 + \beta_{2} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 2 \beta_{14} - \beta_{15} ) q^{73} + ( -4 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{75} + ( -1 + \beta_{1} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{10} + \beta_{11} + \beta_{13} ) q^{79} + ( -3 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - 3 \beta_{11} - 4 \beta_{12} + \beta_{13} - 2 \beta_{14} + 6 \beta_{15} ) q^{81} + ( 1 - 3 \beta_{2} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{83} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - \beta_{12} + 2 \beta_{14} - 4 \beta_{15} ) q^{85} + ( -6 - 3 \beta_{1} + 3 \beta_{3} - \beta_{4} - 4 \beta_{5} - 2 \beta_{8} + \beta_{9} - 3 \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{87} + ( -3 - 3 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{89} + ( 5 + 4 \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{7} - \beta_{12} - \beta_{15} ) q^{93} + ( -1 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{8} + 3 \beta_{9} - 6 \beta_{10} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{95} + ( -\beta_{1} + 5 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 6 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} - 4 \beta_{15} ) q^{97} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{9} + O(q^{10}) \) \( 16q - 6q^{9} + 6q^{11} - 3q^{13} - 3q^{15} - 18q^{17} - 21q^{23} - 8q^{25} + 9q^{27} + 6q^{29} + 6q^{31} - 27q^{33} - 2q^{37} + 6q^{39} - 6q^{41} - 2q^{43} + 15q^{45} + 18q^{47} + 18q^{51} + 15q^{57} + 15q^{59} + 3q^{61} + 39q^{65} - 7q^{67} + 21q^{69} + 42q^{75} - q^{79} - 18q^{81} + 6q^{85} - 51q^{87} - 42q^{89} + 48q^{93} - 6q^{95} - 3q^{97} - 9q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} - 4374 x + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-1307 \nu^{15} + 5068 \nu^{14} + 824 \nu^{13} + 49267 \nu^{12} + 2716 \nu^{11} + 77018 \nu^{10} - 113602 \nu^{9} - 7210 \nu^{8} - 181946 \nu^{7} + 84090 \nu^{6} - 1174032 \nu^{5} + 900801 \nu^{4} - 2054484 \nu^{3} + 11094408 \nu^{2} + 4573017 \nu + 19166868\)\()/621108\)
\(\beta_{2}\)\(=\)\((\)\(3695 \nu^{15} - 20725 \nu^{14} + 51544 \nu^{13} - 99223 \nu^{12} + 215537 \nu^{11} - 360098 \nu^{10} + 187876 \nu^{9} - 298928 \nu^{8} + 711356 \nu^{7} - 844320 \nu^{6} + 2586978 \nu^{5} - 9488205 \nu^{4} + 18766647 \nu^{3} - 20620980 \nu^{2} + 33241671 \nu - 46300977\)\()/1242216\)
\(\beta_{3}\)\(=\)\((\)\(-6677 \nu^{15} + 21577 \nu^{14} - 16612 \nu^{13} + 91129 \nu^{12} - 145673 \nu^{11} + 11630 \nu^{10} - 101824 \nu^{9} + 277628 \nu^{8} - 214640 \nu^{7} + 202764 \nu^{6} - 3624714 \nu^{5} + 7713063 \nu^{4} - 3514995 \nu^{3} + 15997176 \nu^{2} - 16161201 \nu - 5872095\)\()/1242216\)
\(\beta_{4}\)\(=\)\((\)\(-818 \nu^{15} - 2453 \nu^{14} - 9016 \nu^{13} - 10490 \nu^{12} - 12167 \nu^{11} + 10856 \nu^{10} + 25058 \nu^{9} + 37994 \nu^{8} + 39934 \nu^{7} + 148242 \nu^{6} + 64386 \nu^{5} - 91530 \nu^{4} - 1898397 \nu^{3} - 3112344 \nu^{2} - 4586868 \nu - 2858409\)\()/103518\)
\(\beta_{5}\)\(=\)\((\)\(4659 \nu^{15} - 5459 \nu^{14} + 29980 \nu^{13} - 59671 \nu^{12} + 62083 \nu^{11} - 182738 \nu^{10} + 119048 \nu^{9} - 112444 \nu^{8} + 265832 \nu^{7} - 648764 \nu^{6} + 2006262 \nu^{5} - 2722977 \nu^{4} + 10424025 \nu^{3} - 11026368 \nu^{2} + 9412119 \nu - 25313067\)\()/414072\)
\(\beta_{6}\)\(=\)\((\)\(-7279 \nu^{15} + 11960 \nu^{14} - 9158 \nu^{13} + 168785 \nu^{12} + 36326 \nu^{11} + 302098 \nu^{10} - 390434 \nu^{9} - 30734 \nu^{8} - 575494 \nu^{7} + 625362 \nu^{6} - 4453344 \nu^{5} + 1286793 \nu^{4} - 9909540 \nu^{3} + 36051966 \nu^{2} + 18445887 \nu + 70277058\)\()/621108\)
\(\beta_{7}\)\(=\)\((\)\(-13862 \nu^{15} + 1333 \nu^{14} - 45760 \nu^{13} + 164782 \nu^{12} + 15775 \nu^{11} + 343040 \nu^{10} - 361210 \nu^{9} + 100070 \nu^{8} - 284726 \nu^{7} + 1370658 \nu^{6} - 5698206 \nu^{5} + 63018 \nu^{4} - 18020475 \nu^{3} + 29851092 \nu^{2} + 7676370 \nu + 61443765\)\()/621108\)
\(\beta_{8}\)\(=\)\((\)\(27959 \nu^{15} + 45263 \nu^{14} + 129088 \nu^{13} - 58111 \nu^{12} - 95755 \nu^{11} - 442826 \nu^{10} + 157996 \nu^{9} - 326936 \nu^{8} - 426532 \nu^{7} - 3257256 \nu^{6} + 5542434 \nu^{5} + 10347075 \nu^{4} + 36196875 \nu^{3} + 5383908 \nu^{2} + 7598367 \nu - 47062053\)\()/1242216\)
\(\beta_{9}\)\(=\)\((\)\(-7241 \nu^{15} + 4357 \nu^{14} - 32353 \nu^{13} + 38470 \nu^{12} - 78698 \nu^{11} + 72614 \nu^{10} - 29722 \nu^{9} + 229040 \nu^{8} - 92132 \nu^{7} + 637584 \nu^{6} - 2307114 \nu^{5} + 2838753 \nu^{4} - 8018433 \nu^{3} + 3449385 \nu^{2} - 14505642 \nu + 1180980\)\()/310554\)
\(\beta_{10}\)\(=\)\((\)\(5228 \nu^{15} + 1525 \nu^{14} + 20418 \nu^{13} - 46974 \nu^{12} - 5255 \nu^{11} - 122904 \nu^{10} + 100970 \nu^{9} - 49402 \nu^{8} + 85518 \nu^{7} - 572506 \nu^{6} + 1811682 \nu^{5} + 307476 \nu^{4} + 7123329 \nu^{3} - 7628742 \nu^{2} - 1004562 \nu - 19887849\)\()/207036\)
\(\beta_{11}\)\(=\)\((\)\(16952 \nu^{15} - 9175 \nu^{14} + 73804 \nu^{13} - 123904 \nu^{12} + 128807 \nu^{11} - 247964 \nu^{10} + 166462 \nu^{9} - 398066 \nu^{8} + 282422 \nu^{7} - 1677798 \nu^{6} + 5939262 \nu^{5} - 5124384 \nu^{4} + 20396205 \nu^{3} - 16524972 \nu^{2} + 21030192 \nu - 23648031\)\()/621108\)
\(\beta_{12}\)\(=\)\((\)\(11445 \nu^{15} - 10925 \nu^{14} + 48460 \nu^{13} - 170497 \nu^{12} + 54037 \nu^{11} - 403262 \nu^{10} + 367448 \nu^{9} - 147196 \nu^{8} + 581144 \nu^{7} - 1306628 \nu^{6} + 5344746 \nu^{5} - 3588687 \nu^{4} + 19817487 \nu^{3} - 33430320 \nu^{2} + 2317977 \nu - 69748533\)\()/414072\)
\(\beta_{13}\)\(=\)\((\)\(19793 \nu^{15} + 1430 \nu^{14} + 108688 \nu^{13} - 147133 \nu^{12} + 121322 \nu^{11} - 529214 \nu^{10} + 286834 \nu^{9} - 407870 \nu^{8} + 442766 \nu^{7} - 2453682 \nu^{6} + 6290388 \nu^{5} - 3336147 \nu^{4} + 34358094 \nu^{3} - 21512304 \nu^{2} + 26451765 \nu - 66410442\)\()/621108\)
\(\beta_{14}\)\(=\)\((\)\(59447 \nu^{15} + 8969 \nu^{14} + 284176 \nu^{13} - 415519 \nu^{12} + 220139 \nu^{11} - 1262618 \nu^{10} + 758536 \nu^{9} - 1010876 \nu^{8} + 904208 \nu^{7} - 6809388 \nu^{6} + 18944982 \nu^{5} - 4839237 \nu^{4} + 86642541 \nu^{3} - 56752164 \nu^{2} + 46993527 \nu - 158830875\)\()/1242216\)
\(\beta_{15}\)\(=\)\((\)\(24909 \nu^{15} - 6205 \nu^{14} + 135476 \nu^{13} - 180977 \nu^{12} + 218117 \nu^{11} - 566494 \nu^{10} + 299296 \nu^{9} - 605132 \nu^{8} + 530200 \nu^{7} - 2915308 \nu^{6} + 8256786 \nu^{5} - 7163775 \nu^{4} + 40104855 \nu^{3} - 24898752 \nu^{2} + 41926977 \nu - 60703101\)\()/414072\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{15} - 2 \beta_{14} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{14} - \beta_{13} - 2 \beta_{12} + 2 \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - \beta_{12} - 3 \beta_{10} - \beta_{9} - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} - 5 \beta_{1} + 4\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(6 \beta_{15} - \beta_{13} + 2 \beta_{12} - 12 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - \beta_{6} - 7 \beta_{5} + 5 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + \beta_{1} + 7\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-13 \beta_{15} + 5 \beta_{14} - 7 \beta_{13} + 9 \beta_{12} - 3 \beta_{11} + 10 \beta_{10} - 16 \beta_{9} - 9 \beta_{8} - 9 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} - 10 \beta_{4} + 10 \beta_{3} + 8 \beta_{2} + 12 \beta_{1} - 4\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{15} + 7 \beta_{14} + 24 \beta_{13} - 25 \beta_{12} - 12 \beta_{11} - 25 \beta_{10} - 3 \beta_{9} - 18 \beta_{8} - 23 \beta_{7} - 30 \beta_{6} + \beta_{5} - 6 \beta_{4} - \beta_{3} - 13 \beta_{2} + 31 \beta_{1} - 13\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(35 \beta_{15} + 4 \beta_{14} - 29 \beta_{13} - 8 \beta_{12} - 27 \beta_{11} + 3 \beta_{10} - 2 \beta_{9} - 3 \beta_{8} + 20 \beta_{7} + \beta_{6} + 4 \beta_{5} + 5 \beta_{4} + 17 \beta_{3} - 33 \beta_{2} - 82 \beta_{1} + 20\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(73 \beta_{15} - 51 \beta_{14} - 30 \beta_{13} + 4 \beta_{12} - 63 \beta_{11} + 52 \beta_{10} - 24 \beta_{9} - 6 \beta_{8} + 23 \beta_{7} + 21 \beta_{6} + 65 \beta_{5} + 61 \beta_{4} + 39 \beta_{3} - 94 \beta_{2} + 59 \beta_{1} - 26\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-132 \beta_{15} + 4 \beta_{14} - 100 \beta_{13} + 15 \beta_{12} + 48 \beta_{11} + 61 \beta_{10} - 49 \beta_{9} + 90 \beta_{8} + 3 \beta_{7} - 37 \beta_{6} + 235 \beta_{5} - 110 \beta_{4} - 97 \beta_{3} + 85 \beta_{2} + 72 \beta_{1} - 16\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(79 \beta_{15} + 41 \beta_{14} - 17 \beta_{13} - 71 \beta_{12} + 15 \beta_{11} - 174 \beta_{10} + 82 \beta_{9} + 63 \beta_{8} + 17 \beta_{7} + 19 \beta_{6} + 49 \beta_{5} + 36 \beta_{4} + 10 \beta_{3} - 113 \beta_{2} - 472 \beta_{1} - 463\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-5 \beta_{15} + 275 \beta_{14} - 241 \beta_{13} + 185 \beta_{12} - 225 \beta_{11} - 63 \beta_{10} - 28 \beta_{9} + 120 \beta_{8} + 118 \beta_{7} + 197 \beta_{6} - 142 \beta_{5} + 121 \beta_{4} - 152 \beta_{3} + 177 \beta_{2} - 113 \beta_{1} - 80\)\()/3\)
\(\nu^{12}\)\(=\)\((\)\(-995 \beta_{15} + 438 \beta_{14} - 26 \beta_{13} + 764 \beta_{12} + 51 \beta_{11} + 139 \beta_{10} - 452 \beta_{9} - 39 \beta_{8} + 4 \beta_{7} + 331 \beta_{6} + 174 \beta_{5} - 547 \beta_{4} - 387 \beta_{3} + 397 \beta_{2} + 727 \beta_{1} - 198\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(376 \beta_{15} - 4 \beta_{14} + 276 \beta_{13} - 699 \beta_{12} + 408 \beta_{11} - 924 \beta_{10} + 825 \beta_{9} - 129 \beta_{8} - 189 \beta_{7} - 1467 \beta_{6} - 393 \beta_{5} - 625 \beta_{4} - 476 \beta_{3} - 295 \beta_{2} + 558 \beta_{1} - 1935\)\()/3\)
\(\nu^{14}\)\(=\)\((\)\(453 \beta_{15} + 679 \beta_{14} - 1199 \beta_{13} - 40 \beta_{12} - 1635 \beta_{11} + 2518 \beta_{10} - 287 \beta_{9} - 732 \beta_{8} + 1228 \beta_{7} + 1309 \beta_{6} - 150 \beta_{5} - 336 \beta_{4} - 781 \beta_{3} - 253 \beta_{2} - 4148 \beta_{1} - 1881\)\()/3\)
\(\nu^{15}\)\(=\)\((\)\(1349 \beta_{15} - 1166 \beta_{14} + 1447 \beta_{13} + 472 \beta_{12} - 1887 \beta_{11} + 861 \beta_{10} + 454 \beta_{9} + 1194 \beta_{8} + 2750 \beta_{7} - 320 \beta_{6} + 949 \beta_{5} + 2555 \beta_{4} - 2791 \beta_{3} - 2679 \beta_{2} + 3770 \beta_{1} + 611\)\()/3\)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1 + \beta_{1}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
−0.213160 1.71888i
−0.811340 + 1.53027i
−0.544978 + 1.64408i
1.68042 + 0.419752i
1.08696 + 1.34852i
1.68124 0.416458i
−1.61108 + 0.635951i
−0.268067 1.71118i
−0.213160 + 1.71888i
−0.811340 1.53027i
−0.544978 1.64408i
1.68042 0.419752i
1.08696 1.34852i
1.68124 + 0.416458i
−1.61108 0.635951i
−0.268067 + 1.71118i
0 −1.55054 0.771901i 0 1.43402 2.48379i 0 0 0 1.80834 + 2.39372i 0
293.2 0 −1.20245 + 1.24664i 0 1.37166 2.37578i 0 0 0 −0.108243 2.99805i 0
293.3 0 −1.09400 + 1.34282i 0 −1.95741 + 3.39033i 0 0 0 −0.606348 2.93808i 0
293.4 0 −0.240682 1.71525i 0 −1.48494 + 2.57199i 0 0 0 −2.88414 + 0.825658i 0
293.5 0 0.317569 1.70269i 0 0.0382122 0.0661855i 0 0 0 −2.79830 1.08144i 0
293.6 0 0.647613 + 1.60642i 0 0.349828 0.605920i 0 0 0 −2.16119 + 2.08068i 0
293.7 0 1.55979 0.753039i 0 1.09150 1.89054i 0 0 0 1.86586 2.34916i 0
293.8 0 1.56269 + 0.746985i 0 −0.842869 + 1.45989i 0 0 0 1.88403 + 2.33462i 0
1469.1 0 −1.55054 + 0.771901i 0 1.43402 + 2.48379i 0 0 0 1.80834 2.39372i 0
1469.2 0 −1.20245 1.24664i 0 1.37166 + 2.37578i 0 0 0 −0.108243 + 2.99805i 0
1469.3 0 −1.09400 1.34282i 0 −1.95741 3.39033i 0 0 0 −0.606348 + 2.93808i 0
1469.4 0 −0.240682 + 1.71525i 0 −1.48494 2.57199i 0 0 0 −2.88414 0.825658i 0
1469.5 0 0.317569 + 1.70269i 0 0.0382122 + 0.0661855i 0 0 0 −2.79830 + 1.08144i 0
1469.6 0 0.647613 1.60642i 0 0.349828 + 0.605920i 0 0 0 −2.16119 2.08068i 0
1469.7 0 1.55979 + 0.753039i 0 1.09150 + 1.89054i 0 0 0 1.86586 + 2.34916i 0
1469.8 0 1.56269 0.746985i 0 −0.842869 1.45989i 0 0 0 1.88403 2.33462i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1469.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.x.a 16
3.b odd 2 1 5292.2.x.a 16
7.b odd 2 1 1764.2.x.b 16
7.c even 3 1 252.2.w.a 16
7.c even 3 1 1764.2.bm.a 16
7.d odd 6 1 252.2.bm.a yes 16
7.d odd 6 1 1764.2.w.b 16
9.c even 3 1 5292.2.x.b 16
9.d odd 6 1 1764.2.x.b 16
21.c even 2 1 5292.2.x.b 16
21.g even 6 1 756.2.bm.a 16
21.g even 6 1 5292.2.w.b 16
21.h odd 6 1 756.2.w.a 16
21.h odd 6 1 5292.2.bm.a 16
28.f even 6 1 1008.2.df.d 16
28.g odd 6 1 1008.2.ca.d 16
63.g even 3 1 2268.2.t.b 16
63.g even 3 1 5292.2.w.b 16
63.h even 3 1 756.2.bm.a 16
63.i even 6 1 1764.2.bm.a 16
63.i even 6 1 2268.2.t.b 16
63.j odd 6 1 252.2.bm.a yes 16
63.k odd 6 1 756.2.w.a 16
63.l odd 6 1 5292.2.x.a 16
63.n odd 6 1 1764.2.w.b 16
63.n odd 6 1 2268.2.t.a 16
63.o even 6 1 inner 1764.2.x.a 16
63.s even 6 1 252.2.w.a 16
63.t odd 6 1 2268.2.t.a 16
63.t odd 6 1 5292.2.bm.a 16
84.j odd 6 1 3024.2.df.d 16
84.n even 6 1 3024.2.ca.d 16
252.n even 6 1 3024.2.ca.d 16
252.u odd 6 1 3024.2.df.d 16
252.bb even 6 1 1008.2.df.d 16
252.bn odd 6 1 1008.2.ca.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.w.a 16 7.c even 3 1
252.2.w.a 16 63.s even 6 1
252.2.bm.a yes 16 7.d odd 6 1
252.2.bm.a yes 16 63.j odd 6 1
756.2.w.a 16 21.h odd 6 1
756.2.w.a 16 63.k odd 6 1
756.2.bm.a 16 21.g even 6 1
756.2.bm.a 16 63.h even 3 1
1008.2.ca.d 16 28.g odd 6 1
1008.2.ca.d 16 252.bn odd 6 1
1008.2.df.d 16 28.f even 6 1
1008.2.df.d 16 252.bb even 6 1
1764.2.w.b 16 7.d odd 6 1
1764.2.w.b 16 63.n odd 6 1
1764.2.x.a 16 1.a even 1 1 trivial
1764.2.x.a 16 63.o even 6 1 inner
1764.2.x.b 16 7.b odd 2 1
1764.2.x.b 16 9.d odd 6 1
1764.2.bm.a 16 7.c even 3 1
1764.2.bm.a 16 63.i even 6 1
2268.2.t.a 16 63.n odd 6 1
2268.2.t.a 16 63.t odd 6 1
2268.2.t.b 16 63.g even 3 1
2268.2.t.b 16 63.i even 6 1
3024.2.ca.d 16 84.n even 6 1
3024.2.ca.d 16 252.n even 6 1
3024.2.df.d 16 84.j odd 6 1
3024.2.df.d 16 252.u odd 6 1
5292.2.w.b 16 21.g even 6 1
5292.2.w.b 16 63.g even 3 1
5292.2.x.a 16 3.b odd 2 1
5292.2.x.a 16 63.l odd 6 1
5292.2.x.b 16 9.c even 3 1
5292.2.x.b 16 21.c even 2 1
5292.2.bm.a 16 21.h odd 6 1
5292.2.bm.a 16 63.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 6561 + 2187 T^{2} - 729 T^{3} + 729 T^{4} - 486 T^{5} + 405 T^{6} - 27 T^{7} + 171 T^{8} - 9 T^{9} + 45 T^{10} - 18 T^{11} + 9 T^{12} - 3 T^{13} + 3 T^{14} + T^{16} \)
$5$ \( 324 - 4698 T + 62289 T^{2} - 89424 T^{3} + 143289 T^{4} - 45738 T^{5} + 71361 T^{6} - 21573 T^{7} + 23103 T^{8} - 4167 T^{9} + 3600 T^{10} - 423 T^{11} + 405 T^{12} - 24 T^{13} + 24 T^{14} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( 26244 + 13122 T - 269001 T^{2} - 135594 T^{3} + 2920374 T^{4} - 4745790 T^{5} + 3147093 T^{6} - 725355 T^{7} - 137538 T^{8} + 80433 T^{9} + 7047 T^{10} - 8019 T^{11} + 711 T^{12} + 234 T^{13} - 27 T^{14} - 6 T^{15} + T^{16} \)
$13$ \( 3337929 - 17462466 T + 38097783 T^{2} - 40000230 T^{3} + 18330786 T^{4} + 831303 T^{5} - 3499173 T^{6} + 321543 T^{7} + 494064 T^{8} - 66258 T^{9} - 35379 T^{10} + 4608 T^{11} + 1980 T^{12} - 162 T^{13} - 51 T^{14} + 3 T^{15} + T^{16} \)
$17$ \( ( 3681 - 9711 T + 45 T^{2} + 3384 T^{3} + 117 T^{4} - 318 T^{5} - 24 T^{6} + 9 T^{7} + T^{8} )^{2} \)
$19$ \( 2099601 + 85577148 T^{2} + 403026354 T^{4} + 101546541 T^{6} + 10092969 T^{8} + 507069 T^{10} + 13635 T^{12} + 186 T^{14} + T^{16} \)
$23$ \( 15198451524 - 14629258530 T + 2044217331 T^{2} + 2550348180 T^{3} - 604378908 T^{4} - 340227216 T^{5} + 112333068 T^{6} + 20560716 T^{7} - 7612947 T^{8} - 1006263 T^{9} + 350730 T^{10} + 37422 T^{11} - 7767 T^{12} - 1008 T^{13} + 99 T^{14} + 21 T^{15} + T^{16} \)
$29$ \( 15752961 - 9001692 T - 182284263 T^{2} + 105142212 T^{3} + 2135730888 T^{4} + 464916834 T^{5} - 210471048 T^{6} - 53675541 T^{7} + 20028303 T^{8} + 2348109 T^{9} - 577287 T^{10} - 58239 T^{11} + 12753 T^{12} + 828 T^{13} - 126 T^{14} - 6 T^{15} + T^{16} \)
$31$ \( 3910251024 - 4355979120 T - 2625353532 T^{2} + 4726500660 T^{3} + 4449942117 T^{4} + 478336266 T^{5} - 309231837 T^{6} - 48203910 T^{7} + 16995015 T^{8} + 2259819 T^{9} - 490239 T^{10} - 57834 T^{11} + 11187 T^{12} + 792 T^{13} - 120 T^{14} - 6 T^{15} + T^{16} \)
$37$ \( ( 7264 - 512 T - 15848 T^{2} + 916 T^{3} + 2590 T^{4} - 137 T^{5} - 107 T^{6} + T^{7} + T^{8} )^{2} \)
$41$ \( 91647269289 - 3013404282 T + 36627754095 T^{2} - 8471239848 T^{3} + 12508926552 T^{4} - 1968463728 T^{5} + 1038978252 T^{6} - 28504467 T^{7} + 44578395 T^{8} - 430317 T^{9} + 1135899 T^{10} + 20637 T^{11} + 18891 T^{12} + 330 T^{13} + 186 T^{14} + 6 T^{15} + T^{16} \)
$43$ \( 28009034881 + 1271593682 T + 16892706132 T^{2} + 953140042 T^{3} + 8330008163 T^{4} + 324910827 T^{5} + 1058091937 T^{6} - 87622111 T^{7} + 97936317 T^{8} - 3925843 T^{9} + 2048842 T^{10} - 18162 T^{11} + 30590 T^{12} - 104 T^{13} + 207 T^{14} + 2 T^{15} + T^{16} \)
$47$ \( 1971620372736 - 2703471458688 T + 2461461067440 T^{2} - 1330775271864 T^{3} + 540676969353 T^{4} - 148161520953 T^{5} + 32201307486 T^{6} - 4775169699 T^{7} + 712315611 T^{8} - 79062246 T^{9} + 10379754 T^{10} - 846819 T^{11} + 84231 T^{12} - 4884 T^{13} + 438 T^{14} - 18 T^{15} + T^{16} \)
$53$ \( 531441 + 231708276 T^{2} + 438734070 T^{4} + 151959321 T^{6} + 20447721 T^{8} + 1206981 T^{10} + 30699 T^{12} + 306 T^{14} + T^{16} \)
$59$ \( 165574120464 - 266986987488 T + 484571771388 T^{2} + 11513862288 T^{3} + 77431736565 T^{4} - 9743030109 T^{5} + 7073934444 T^{6} - 677695707 T^{7} + 309484062 T^{8} - 42107274 T^{9} + 8617212 T^{10} - 720099 T^{11} + 79218 T^{12} - 3882 T^{13} + 393 T^{14} - 15 T^{15} + T^{16} \)
$61$ \( 1475481744 + 5807894400 T + 9308572164 T^{2} + 6644786400 T^{3} + 1891535733 T^{4} - 166075839 T^{5} - 176063652 T^{6} - 1463967 T^{7} + 11523348 T^{8} + 569754 T^{9} - 366912 T^{10} - 19287 T^{11} + 8784 T^{12} + 342 T^{13} - 111 T^{14} - 3 T^{15} + T^{16} \)
$67$ \( 2114953586944 + 2140118586496 T + 2010327590448 T^{2} + 427777738280 T^{3} + 163978476161 T^{4} + 24180967287 T^{5} + 7944661591 T^{6} + 951833224 T^{7} + 231304086 T^{8} + 20606821 T^{9} + 4411018 T^{10} + 318924 T^{11} + 50012 T^{12} + 1859 T^{13} + 270 T^{14} + 7 T^{15} + T^{16} \)
$71$ \( 780959242139904 + 134744717006208 T^{2} + 7956570857364 T^{4} + 234856231407 T^{6} + 3949834995 T^{8} + 39561129 T^{10} + 233316 T^{12} + 747 T^{14} + T^{16} \)
$73$ \( 7523023152969 + 3389821273674 T^{2} + 452072281275 T^{4} + 27475295445 T^{6} + 870857073 T^{8} + 15042177 T^{10} + 138438 T^{12} + 612 T^{14} + T^{16} \)
$79$ \( 10549504 + 14992768 T + 30606480 T^{2} + 29157800 T^{3} + 44789945 T^{4} + 38599749 T^{5} + 35635933 T^{6} + 16316434 T^{7} + 7394544 T^{8} + 943669 T^{9} + 497152 T^{10} + 68454 T^{11} + 18914 T^{12} + 779 T^{13} + 144 T^{14} + T^{15} + T^{16} \)
$83$ \( 669184533369 - 2061188196012 T + 5773577517018 T^{2} - 1903485074616 T^{3} + 720461504880 T^{4} - 85919425596 T^{5} + 25248534300 T^{6} - 2208157416 T^{7} + 614588409 T^{8} - 32727636 T^{9} + 8300268 T^{10} - 290448 T^{11} + 80712 T^{12} - 1272 T^{13} + 330 T^{14} + T^{16} \)
$89$ \( ( -2676159 - 392931 T + 425979 T^{2} + 75492 T^{3} - 11583 T^{4} - 2754 T^{5} - 18 T^{6} + 21 T^{7} + T^{8} )^{2} \)
$97$ \( 22864161681 - 104576900445 T - 8224102287 T^{2} + 766864072890 T^{3} + 1208668941423 T^{4} - 71751605004 T^{5} - 41675283648 T^{6} + 2243025486 T^{7} + 1044816804 T^{8} - 41968233 T^{9} - 12357171 T^{10} + 394884 T^{11} + 109872 T^{12} - 1161 T^{13} - 384 T^{14} + 3 T^{15} + T^{16} \)
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