Properties

Label 1008.2.df.d
Level $1008$
Weight $2$
Character orbit 1008.df
Analytic conductor $8.049$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.df (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{3} - \beta_{12} ) q^{3} + ( -\beta_{7} + \beta_{9} ) q^{5} + ( -\beta_{5} + \beta_{6} - \beta_{9} + \beta_{14} ) q^{7} + ( -\beta_{5} - \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{3} - \beta_{12} ) q^{3} + ( -\beta_{7} + \beta_{9} ) q^{5} + ( -\beta_{5} + \beta_{6} - \beta_{9} + \beta_{14} ) q^{7} + ( -\beta_{5} - \beta_{11} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{11} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{9} + \beta_{12} - \beta_{13} ) q^{13} + ( 1 - \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{12} - \beta_{13} ) q^{15} + ( -1 + \beta_{3} - \beta_{7} + \beta_{8} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{17} + ( -1 - \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{13} + \beta_{15} ) q^{19} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} + 2 \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{21} + ( 2 - 3 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{23} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{15} ) q^{25} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{27} + ( -1 - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{29} + ( -\beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{12} ) q^{31} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} - 2 \beta_{15} ) q^{33} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} ) q^{35} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{13} ) q^{37} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{15} ) q^{41} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{43} + ( 1 - 2 \beta_{1} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{11} + 3 \beta_{14} ) q^{45} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{49} + ( -2 + 3 \beta_{1} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{51} + ( -1 + \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - \beta_{5} - \beta_{8} + 2 \beta_{9} + \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - \beta_{15} ) q^{53} + ( -1 + 2 \beta_{1} + \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{55} + ( \beta_{2} - 2 \beta_{3} + \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{57} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{12} + \beta_{15} ) q^{59} + ( 1 + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{15} ) q^{61} + ( 4 - 5 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{13} - 2 \beta_{15} ) q^{63} + ( -2 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{15} ) q^{65} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{67} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + \beta_{8} - 5 \beta_{9} - \beta_{11} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{69} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{8} + 2 \beta_{10} + \beta_{12} + \beta_{13} ) q^{71} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{14} ) q^{73} + ( 4 - 5 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{10} - 2 \beta_{11} - \beta_{13} + \beta_{15} ) q^{75} + ( -2 + 2 \beta_{1} + \beta_{2} - 5 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{77} + ( \beta_{1} - \beta_{3} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{13} ) q^{79} + ( 3 - 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{81} + ( 1 - \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{83} + ( 1 - \beta_{1} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{6} + 2 \beta_{10} + 2 \beta_{15} ) q^{85} + ( 1 - 4 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - \beta_{11} - 4 \beta_{12} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{87} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{11} + 2 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{89} + ( -2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{91} + ( -4 - \beta_{1} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} - 3 \beta_{14} ) q^{93} + ( -\beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} + 4 \beta_{10} - \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} ) q^{95} + ( -2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{97} + ( 2 - \beta_{1} + 5 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 4 \beta_{9} - \beta_{11} + 4 \beta_{12} - 3 \beta_{13} - 3 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + q^{7} + O(q^{10}) \) \( 16q + q^{7} + 3q^{13} + 3q^{15} - 9q^{17} + 16q^{25} + 9q^{27} + 6q^{29} - 6q^{31} - 27q^{33} - 15q^{35} + q^{37} + 3q^{39} + 6q^{41} + 2q^{43} - 15q^{45} + 18q^{47} + 13q^{49} - 15q^{51} + 15q^{57} + 15q^{59} + 3q^{61} + 9q^{63} - 39q^{65} + 7q^{67} - 21q^{69} + 15q^{75} - 45q^{77} + q^{79} + 6q^{85} + 3q^{87} - 21q^{89} - 9q^{91} - 69q^{93} - 6q^{95} + 3q^{97} + 9q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-1307 \nu^{15} + 5068 \nu^{14} + 824 \nu^{13} + 49267 \nu^{12} + 2716 \nu^{11} + 77018 \nu^{10} - 113602 \nu^{9} - 7210 \nu^{8} - 181946 \nu^{7} + 84090 \nu^{6} - 1174032 \nu^{5} + 900801 \nu^{4} - 2054484 \nu^{3} + 11094408 \nu^{2} + 4573017 \nu + 19787976\)\()/621108\)
\(\beta_{2}\)\(=\)\((\)\(-3695 \nu^{15} + 20725 \nu^{14} - 51544 \nu^{13} + 99223 \nu^{12} - 215537 \nu^{11} + 360098 \nu^{10} - 187876 \nu^{9} + 298928 \nu^{8} - 711356 \nu^{7} + 844320 \nu^{6} - 2586978 \nu^{5} + 9488205 \nu^{4} - 18766647 \nu^{3} + 20620980 \nu^{2} - 33241671 \nu + 46300977\)\()/1242216\)
\(\beta_{3}\)\(=\)\((\)\(-6613 \nu^{15} - 4165 \nu^{14} - 58592 \nu^{13} + 79853 \nu^{12} - 42655 \nu^{11} + 422782 \nu^{10} - 220796 \nu^{9} + 95200 \nu^{8} - 384796 \nu^{7} + 1316016 \nu^{6} - 2201310 \nu^{5} + 683775 \nu^{4} - 21798153 \nu^{3} + 15850404 \nu^{2} - 11411037 \nu + 63188991\)\()/1242216\)
\(\beta_{4}\)\(=\)\((\)\(-1730 \nu^{15} + 4365 \nu^{14} - 8834 \nu^{13} + 21044 \nu^{12} - 39051 \nu^{11} + 29320 \nu^{10} - 27966 \nu^{9} + 78114 \nu^{8} - 73522 \nu^{7} + 146582 \nu^{6} - 864450 \nu^{5} + 1801458 \nu^{4} - 2415663 \nu^{3} + 3505194 \nu^{2} - 5662872 \nu + 1850931\)\()/207036\)
\(\beta_{5}\)\(=\)\((\)\(-8480 \nu^{15} + 13039 \nu^{14} - 27196 \nu^{13} + 146632 \nu^{12} - 38759 \nu^{11} + 271028 \nu^{10} - 294706 \nu^{9} + 100154 \nu^{8} - 462590 \nu^{7} + 777042 \nu^{6} - 4479570 \nu^{5} + 3575664 \nu^{4} - 12408957 \nu^{3} + 28867428 \nu^{2} + 2373624 \nu + 50810571\)\()/621108\)
\(\beta_{6}\)\(=\)\((\)\(-9197 \nu^{15} + 929 \nu^{14} - 62076 \nu^{13} + 49233 \nu^{12} - 112105 \nu^{11} + 233886 \nu^{10} - 68192 \nu^{9} + 273436 \nu^{8} - 205752 \nu^{7} + 1227580 \nu^{6} - 2641242 \nu^{5} + 3259071 \nu^{4} - 17595171 \nu^{3} + 4704480 \nu^{2} - 23810841 \nu + 20080305\)\()/414072\)
\(\beta_{7}\)\(=\)\((\)\(13862 \nu^{15} - 1333 \nu^{14} + 45760 \nu^{13} - 164782 \nu^{12} - 15775 \nu^{11} - 343040 \nu^{10} + 361210 \nu^{9} - 100070 \nu^{8} + 284726 \nu^{7} - 1370658 \nu^{6} + 5698206 \nu^{5} - 63018 \nu^{4} + 18020475 \nu^{3} - 29851092 \nu^{2} - 7676370 \nu - 61443765\)\()/621108\)
\(\beta_{8}\)\(=\)\((\)\(-10991 \nu^{15} + 435 \nu^{14} - 52928 \nu^{13} + 71471 \nu^{12} - 66663 \nu^{11} + 211882 \nu^{10} - 117624 \nu^{9} + 240180 \nu^{8} - 166912 \nu^{7} + 1197116 \nu^{6} - 3459822 \nu^{5} + 1996173 \nu^{4} - 15639345 \nu^{3} + 9058068 \nu^{2} - 13830831 \nu + 23210631\)\()/414072\)
\(\beta_{9}\)\(=\)\((\)\(-16952 \nu^{15} + 9175 \nu^{14} - 73804 \nu^{13} + 123904 \nu^{12} - 128807 \nu^{11} + 247964 \nu^{10} - 166462 \nu^{9} + 398066 \nu^{8} - 282422 \nu^{7} + 1677798 \nu^{6} - 5939262 \nu^{5} + 5124384 \nu^{4} - 20396205 \nu^{3} + 16524972 \nu^{2} - 21030192 \nu + 23648031\)\()/621108\)
\(\beta_{10}\)\(=\)\((\)\(17318 \nu^{15} + 11843 \nu^{14} + 72850 \nu^{13} - 74620 \nu^{12} + 24959 \nu^{11} - 227228 \nu^{10} + 129910 \nu^{9} - 302282 \nu^{8} - 105946 \nu^{7} - 1730010 \nu^{6} + 4583574 \nu^{5} + 1317006 \nu^{4} + 19855935 \nu^{3} - 5306634 \nu^{2} + 11879784 \nu - 20594979\)\()/621108\)
\(\beta_{11}\)\(=\)\((\)\(8906 \nu^{15} - 13891 \nu^{14} + 29326 \nu^{13} - 153874 \nu^{12} + 48131 \nu^{11} - 284234 \nu^{10} + 321118 \nu^{9} - 122306 \nu^{8} + 484742 \nu^{7} - 843498 \nu^{6} + 4717278 \nu^{5} - 3932226 \nu^{4} + 13168089 \nu^{3} - 30627234 \nu^{2} - 820854 \nu - 52673895\)\()/310554\)
\(\beta_{12}\)\(=\)\((\)\(-20668 \nu^{15} + 11675 \nu^{14} - 91538 \nu^{13} + 200162 \nu^{12} - 134761 \nu^{11} + 470224 \nu^{10} - 364490 \nu^{9} + 414682 \nu^{8} - 589198 \nu^{7} + 2170158 \nu^{6} - 7868934 \nu^{5} + 5845284 \nu^{4} - 29583873 \nu^{3} + 32737446 \nu^{2} - 19369530 \nu + 65179161\)\()/621108\)
\(\beta_{13}\)\(=\)\((\)\(-7981 \nu^{15} + 3790 \nu^{14} - 34392 \nu^{13} + 96837 \nu^{12} - 29234 \nu^{11} + 242874 \nu^{10} - 207322 \nu^{9} + 112070 \nu^{8} - 288162 \nu^{7} + 882698 \nu^{6} - 3277620 \nu^{5} + 1569195 \nu^{4} - 12954546 \nu^{3} + 17811576 \nu^{2} - 2638737 \nu + 39766950\)\()/207036\)
\(\beta_{14}\)\(=\)\((\)\(-25616 \nu^{15} - 1811 \nu^{14} - 72430 \nu^{13} + 225742 \nu^{12} + 48469 \nu^{11} + 389576 \nu^{10} - 437878 \nu^{9} + 251534 \nu^{8} - 223118 \nu^{7} + 2242098 \nu^{6} - 9228186 \nu^{5} - 1362096 \nu^{4} - 24510843 \nu^{3} + 35643726 \nu^{2} + 13260510 \nu + 67869171\)\()/621108\)
\(\beta_{15}\)\(=\)\((\)\(-19601 \nu^{15} + 19537 \nu^{14} - 71044 \nu^{13} + 245365 \nu^{12} - 112385 \nu^{11} + 469334 \nu^{10} - 472072 \nu^{9} + 355172 \nu^{8} - 709136 \nu^{7} + 1851660 \nu^{6} - 8655978 \nu^{5} + 6647859 \nu^{4} - 26067339 \nu^{3} + 44697096 \nu^{2} - 8267589 \nu + 77064777\)\()/414072\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{15} - \beta_{13} - \beta_{12} - \beta_{9} - \beta_{8} + 2 \beta_{6} - \beta_{4} - \beta_{2} - \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{15} - 2 \beta_{14} + \beta_{11} - \beta_{10} + 2 \beta_{9} - 2 \beta_{6} + \beta_{5} + 3 \beta_{3} - 2 \beta_{2} + \beta_{1} - 2\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{15} + \beta_{14} - 3 \beta_{13} + 7 \beta_{12} - 2 \beta_{11} - \beta_{10} - 2 \beta_{9} + 2 \beta_{7} - 4 \beta_{6} + \beta_{5} + \beta_{4} + 5 \beta_{3} + \beta_{2} - 6 \beta_{1} + 9\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{13} + 2 \beta_{12} - 6 \beta_{11} - 5 \beta_{9} + 9 \beta_{8} + \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} + 7 \beta_{3} - 3 \beta_{2} - 5 \beta_{1} + 13\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{15} - 3 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} - 3 \beta_{11} - 12 \beta_{10} + 16 \beta_{9} - 11 \beta_{8} + 6 \beta_{7} + \beta_{6} + 3 \beta_{5} - 8 \beta_{4} - 6 \beta_{3} + 7 \beta_{2} + 19 \beta_{1} - 21\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-7 \beta_{15} + 7 \beta_{14} - 11 \beta_{13} + 2 \beta_{12} - 11 \beta_{11} + 5 \beta_{10} + 11 \beta_{9} + 7 \beta_{8} - 10 \beta_{7} - 9 \beta_{6} - 23 \beta_{5} - 10 \beta_{4} + 5 \beta_{3} + 26 \beta_{2} + 4 \beta_{1} - 3\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-4 \beta_{15} + 6 \beta_{14} + 11 \beta_{13} - 39 \beta_{12} - 15 \beta_{11} - 6 \beta_{10} + 16 \beta_{9} + 11 \beta_{8} + 5 \beta_{7} - 15 \beta_{6} + 3 \beta_{5} - 6 \beta_{4} - 4 \beta_{3} + 32 \beta_{2} - 62 \beta_{1} + 116\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(51 \beta_{15} + 43 \beta_{14} - 43 \beta_{13} - 37 \beta_{12} + 10 \beta_{11} + 2 \beta_{10} - 41 \beta_{9} - 37 \beta_{8} + 30 \beta_{7} + 63 \beta_{6} - 8 \beta_{5} + 20 \beta_{4} + 57 \beta_{3} - 18 \beta_{2} + 51 \beta_{1} - 116\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-4 \beta_{15} - 48 \beta_{14} - 4 \beta_{13} + 35 \beta_{12} + 126 \beta_{11} - 33 \beta_{10} + 110 \beta_{9} - 16 \beta_{8} - 42 \beta_{7} - 112 \beta_{6} + 123 \beta_{5} + 20 \beta_{4} + 111 \beta_{3} + 29 \beta_{2} + 143 \beta_{1} - 177\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-41 \beta_{15} + 60 \beta_{14} - 71 \beta_{13} + 164 \beta_{12} - 57 \beta_{11} + 6 \beta_{10} - 111 \beta_{9} + 94 \beta_{8} + 65 \beta_{7} - 169 \beta_{6} + 57 \beta_{5} + 20 \beta_{4} + 116 \beta_{3} + 118 \beta_{2} - 426 \beta_{1} + 47\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-275 \beta_{15} + 134 \beta_{14} - 31 \beta_{13} + 20 \beta_{12} - 181 \beta_{11} + 58 \beta_{10} - 30 \beta_{9} + 323 \beta_{8} + 168 \beta_{7} - 152 \beta_{6} + 62 \beta_{5} + 215 \beta_{4} + 213 \beta_{3} - 23 \beta_{2} + 52 \beta_{1} + 17\)\()/3\)
\(\nu^{12}\)\(=\)\((\)\(-438 \beta_{15} + 208 \beta_{14} + 101 \beta_{13} + 9 \beta_{12} + 133 \beta_{11} - 184 \beta_{10} + 288 \beta_{9} - 442 \beta_{8} - 97 \beta_{7} + 181 \beta_{6} + 145 \beta_{5} + 24 \beta_{4} - 391 \beta_{3} + 588 \beta_{2} + 1181 \beta_{1} - 1389\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(4 \beta_{15} - 144 \beta_{14} + 283 \beta_{13} - 602 \beta_{12} + 243 \beta_{11} + 237 \beta_{10} + 361 \beta_{9} + 772 \beta_{8} - 663 \beta_{7} - 329 \beta_{6} - 366 \beta_{5} - 386 \beta_{4} - 861 \beta_{3} + 916 \beta_{2} + 331 \beta_{1} - 1269\)\()/3\)
\(\nu^{14}\)\(=\)\((\)\(-679 \beta_{15} + 815 \beta_{14} + 2796 \beta_{13} - 2187 \beta_{12} - 229 \beta_{11} + 331 \beta_{10} + 1156 \beta_{9} - 957 \beta_{8} + 984 \beta_{7} + 344 \beta_{6} - 1063 \beta_{5} + 114 \beta_{4} - 2346 \beta_{3} + 1268 \beta_{2} - 1600 \beta_{1} + 1187\)\()/3\)
\(\nu^{15}\)\(=\)\((\)\(1166 \beta_{15} + 2420 \beta_{14} - 2349 \beta_{13} + 662 \beta_{12} + 1109 \beta_{11} + 2848 \beta_{10} - 3088 \beta_{9} - 576 \beta_{8} - 1778 \beta_{7} + 3292 \beta_{6} - 1654 \beta_{5} + 2498 \beta_{4} + 1636 \beta_{3} - 1042 \beta_{2} + 2187 \beta_{1} - 3948\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(\beta_{1}\) \(1\) \(1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
689.1
1.68124 + 0.416458i
−0.268067 + 1.71118i
−0.544978 1.64408i
−0.811340 1.53027i
−1.61108 0.635951i
1.08696 1.34852i
−0.213160 + 1.71888i
1.68042 0.419752i
1.68124 0.416458i
−0.268067 1.71118i
−0.544978 + 1.64408i
−0.811340 + 1.53027i
−1.61108 + 0.635951i
1.08696 + 1.34852i
−0.213160 1.71888i
1.68042 + 0.419752i
0 −1.71501 0.242362i 0 0.699656 0 0.461278 2.60523i 0 2.88252 + 0.831308i 0
689.2 0 −1.42826 + 0.979841i 0 −1.68574 0 0.0236360 + 2.64565i 0 1.07983 2.79892i 0
689.3 0 −0.615921 1.61884i 0 −3.91482 0 2.51757 + 0.813537i 0 −2.24128 + 1.99416i 0
689.4 0 −0.478402 1.66467i 0 2.74332 0 −1.70417 + 2.02381i 0 −2.54226 + 1.59276i 0
689.5 0 −0.127742 + 1.72733i 0 2.18300 0 −2.64473 0.0736382i 0 −2.96736 0.441307i 0
689.6 0 1.31579 + 1.12637i 0 0.0764245 0 2.39886 + 1.11601i 0 0.462593 + 2.96412i 0
689.7 0 1.44376 0.956855i 0 2.86804 0 1.83240 1.90848i 0 1.16886 2.76293i 0
689.8 0 1.60579 + 0.649187i 0 −2.96988 0 −2.38485 1.14563i 0 2.15711 + 2.08491i 0
929.1 0 −1.71501 + 0.242362i 0 0.699656 0 0.461278 + 2.60523i 0 2.88252 0.831308i 0
929.2 0 −1.42826 0.979841i 0 −1.68574 0 0.0236360 2.64565i 0 1.07983 + 2.79892i 0
929.3 0 −0.615921 + 1.61884i 0 −3.91482 0 2.51757 0.813537i 0 −2.24128 1.99416i 0
929.4 0 −0.478402 + 1.66467i 0 2.74332 0 −1.70417 2.02381i 0 −2.54226 1.59276i 0
929.5 0 −0.127742 1.72733i 0 2.18300 0 −2.64473 + 0.0736382i 0 −2.96736 + 0.441307i 0
929.6 0 1.31579 1.12637i 0 0.0764245 0 2.39886 1.11601i 0 0.462593 2.96412i 0
929.7 0 1.44376 + 0.956855i 0 2.86804 0 1.83240 + 1.90848i 0 1.16886 + 2.76293i 0
929.8 0 1.60579 0.649187i 0 −2.96988 0 −2.38485 + 1.14563i 0 2.15711 2.08491i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 929.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.df.d 16
3.b odd 2 1 3024.2.df.d 16
4.b odd 2 1 252.2.bm.a yes 16
7.d odd 6 1 1008.2.ca.d 16
9.c even 3 1 3024.2.ca.d 16
9.d odd 6 1 1008.2.ca.d 16
12.b even 2 1 756.2.bm.a 16
21.g even 6 1 3024.2.ca.d 16
28.d even 2 1 1764.2.bm.a 16
28.f even 6 1 252.2.w.a 16
28.f even 6 1 1764.2.x.a 16
28.g odd 6 1 1764.2.w.b 16
28.g odd 6 1 1764.2.x.b 16
36.f odd 6 1 756.2.w.a 16
36.f odd 6 1 2268.2.t.a 16
36.h even 6 1 252.2.w.a 16
36.h even 6 1 2268.2.t.b 16
63.k odd 6 1 3024.2.df.d 16
63.s even 6 1 inner 1008.2.df.d 16
84.h odd 2 1 5292.2.bm.a 16
84.j odd 6 1 756.2.w.a 16
84.j odd 6 1 5292.2.x.a 16
84.n even 6 1 5292.2.w.b 16
84.n even 6 1 5292.2.x.b 16
252.n even 6 1 756.2.bm.a 16
252.o even 6 1 1764.2.bm.a 16
252.r odd 6 1 1764.2.x.b 16
252.r odd 6 1 2268.2.t.a 16
252.s odd 6 1 1764.2.w.b 16
252.u odd 6 1 5292.2.x.a 16
252.bb even 6 1 1764.2.x.a 16
252.bi even 6 1 5292.2.w.b 16
252.bj even 6 1 2268.2.t.b 16
252.bj even 6 1 5292.2.x.b 16
252.bl odd 6 1 5292.2.bm.a 16
252.bn odd 6 1 252.2.bm.a yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.w.a 16 28.f even 6 1
252.2.w.a 16 36.h even 6 1
252.2.bm.a yes 16 4.b odd 2 1
252.2.bm.a yes 16 252.bn odd 6 1
756.2.w.a 16 36.f odd 6 1
756.2.w.a 16 84.j odd 6 1
756.2.bm.a 16 12.b even 2 1
756.2.bm.a 16 252.n even 6 1
1008.2.ca.d 16 7.d odd 6 1
1008.2.ca.d 16 9.d odd 6 1
1008.2.df.d 16 1.a even 1 1 trivial
1008.2.df.d 16 63.s even 6 1 inner
1764.2.w.b 16 28.g odd 6 1
1764.2.w.b 16 252.s odd 6 1
1764.2.x.a 16 28.f even 6 1
1764.2.x.a 16 252.bb even 6 1
1764.2.x.b 16 28.g odd 6 1
1764.2.x.b 16 252.r odd 6 1
1764.2.bm.a 16 28.d even 2 1
1764.2.bm.a 16 252.o even 6 1
2268.2.t.a 16 36.f odd 6 1
2268.2.t.a 16 252.r odd 6 1
2268.2.t.b 16 36.h even 6 1
2268.2.t.b 16 252.bj even 6 1
3024.2.ca.d 16 9.c even 3 1
3024.2.ca.d 16 21.g even 6 1
3024.2.df.d 16 3.b odd 2 1
3024.2.df.d 16 63.k odd 6 1
5292.2.w.b 16 84.n even 6 1
5292.2.w.b 16 252.bi even 6 1
5292.2.x.a 16 84.j odd 6 1
5292.2.x.a 16 252.u odd 6 1
5292.2.x.b 16 84.n even 6 1
5292.2.x.b 16 252.bj even 6 1
5292.2.bm.a 16 84.h odd 2 1
5292.2.bm.a 16 252.bl odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 24 T_{5}^{6} + 12 T_{5}^{5} + 171 T_{5}^{4} - 135 T_{5}^{3} - 324 T_{5}^{2} + 261 T_{5} - 18 \) acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 6561 - 729 T^{3} + 729 T^{5} + 243 T^{6} - 27 T^{7} - 72 T^{8} - 9 T^{9} + 27 T^{10} + 27 T^{11} - 3 T^{13} + T^{16} \)
$5$ \( ( -18 + 261 T - 324 T^{2} - 135 T^{3} + 171 T^{4} + 12 T^{5} - 24 T^{6} + T^{8} )^{2} \)
$7$ \( 5764801 - 823543 T - 705894 T^{2} + 268912 T^{3} - 24010 T^{4} - 13377 T^{5} + 18865 T^{6} + 1673 T^{7} - 2880 T^{8} + 239 T^{9} + 385 T^{10} - 39 T^{11} - 10 T^{12} + 16 T^{13} - 6 T^{14} - T^{15} + T^{16} \)
$11$ \( 26244 + 544563 T^{2} + 2566080 T^{4} + 1858950 T^{6} + 464616 T^{8} + 53703 T^{10} + 3141 T^{12} + 90 T^{14} + T^{16} \)
$13$ \( 3337929 + 17462466 T + 38097783 T^{2} + 40000230 T^{3} + 18330786 T^{4} - 831303 T^{5} - 3499173 T^{6} - 321543 T^{7} + 494064 T^{8} + 66258 T^{9} - 35379 T^{10} - 4608 T^{11} + 1980 T^{12} + 162 T^{13} - 51 T^{14} - 3 T^{15} + T^{16} \)
$17$ \( 13549761 - 35746191 T + 94137876 T^{2} - 25350003 T^{3} + 32433372 T^{4} + 1254096 T^{5} + 8181405 T^{6} + 690741 T^{7} + 912402 T^{8} + 115920 T^{9} + 73566 T^{10} + 8910 T^{11} + 3321 T^{12} + 420 T^{13} + 105 T^{14} + 9 T^{15} + T^{16} \)
$19$ \( 2099601 - 29655234 T + 166640004 T^{2} - 381649968 T^{3} + 320108679 T^{4} + 85622103 T^{5} - 40684761 T^{6} - 9517095 T^{7} + 4276089 T^{8} + 316089 T^{9} - 188478 T^{10} - 7974 T^{11} + 6156 T^{12} - 93 T^{14} + T^{16} \)
$23$ \( 15198451524 + 9992947563 T^{2} + 2594476008 T^{4} + 342510444 T^{6} + 24904827 T^{8} + 1015092 T^{10} + 22446 T^{12} + 243 T^{14} + T^{16} \)
$29$ \( 15752961 - 9001692 T - 182284263 T^{2} + 105142212 T^{3} + 2135730888 T^{4} + 464916834 T^{5} - 210471048 T^{6} - 53675541 T^{7} + 20028303 T^{8} + 2348109 T^{9} - 577287 T^{10} - 58239 T^{11} + 12753 T^{12} + 828 T^{13} - 126 T^{14} - 6 T^{15} + T^{16} \)
$31$ \( 3910251024 + 4355979120 T - 2625353532 T^{2} - 4726500660 T^{3} + 4449942117 T^{4} - 478336266 T^{5} - 309231837 T^{6} + 48203910 T^{7} + 16995015 T^{8} - 2259819 T^{9} - 490239 T^{10} + 57834 T^{11} + 11187 T^{12} - 792 T^{13} - 120 T^{14} + 6 T^{15} + T^{16} \)
$37$ \( 52765696 + 3719168 T + 115382016 T^{2} + 5193472 T^{3} + 232814336 T^{4} + 12859776 T^{5} + 40260736 T^{6} + 1907864 T^{7} + 5129568 T^{8} + 175166 T^{9} + 263287 T^{10} - 10395 T^{11} + 8996 T^{12} - 167 T^{13} + 108 T^{14} - T^{15} + T^{16} \)
$41$ \( 91647269289 + 3013404282 T + 36627754095 T^{2} + 8471239848 T^{3} + 12508926552 T^{4} + 1968463728 T^{5} + 1038978252 T^{6} + 28504467 T^{7} + 44578395 T^{8} + 430317 T^{9} + 1135899 T^{10} - 20637 T^{11} + 18891 T^{12} - 330 T^{13} + 186 T^{14} - 6 T^{15} + T^{16} \)
$43$ \( 28009034881 - 1271593682 T + 16892706132 T^{2} - 953140042 T^{3} + 8330008163 T^{4} - 324910827 T^{5} + 1058091937 T^{6} + 87622111 T^{7} + 97936317 T^{8} + 3925843 T^{9} + 2048842 T^{10} + 18162 T^{11} + 30590 T^{12} + 104 T^{13} + 207 T^{14} - 2 T^{15} + T^{16} \)
$47$ \( 1971620372736 - 2703471458688 T + 2461461067440 T^{2} - 1330775271864 T^{3} + 540676969353 T^{4} - 148161520953 T^{5} + 32201307486 T^{6} - 4775169699 T^{7} + 712315611 T^{8} - 79062246 T^{9} + 10379754 T^{10} - 846819 T^{11} + 84231 T^{12} - 4884 T^{13} + 438 T^{14} - 18 T^{15} + T^{16} \)
$53$ \( 531441 - 18068994 T + 191318760 T^{2} + 457747848 T^{3} + 292233501 T^{4} - 102922407 T^{5} - 60026589 T^{6} + 16579647 T^{7} + 9705177 T^{8} - 1451439 T^{9} - 466074 T^{10} + 56376 T^{11} + 19764 T^{12} - 153 T^{14} + T^{16} \)
$59$ \( 165574120464 - 266986987488 T + 484571771388 T^{2} + 11513862288 T^{3} + 77431736565 T^{4} - 9743030109 T^{5} + 7073934444 T^{6} - 677695707 T^{7} + 309484062 T^{8} - 42107274 T^{9} + 8617212 T^{10} - 720099 T^{11} + 79218 T^{12} - 3882 T^{13} + 393 T^{14} - 15 T^{15} + T^{16} \)
$61$ \( 1475481744 + 5807894400 T + 9308572164 T^{2} + 6644786400 T^{3} + 1891535733 T^{4} - 166075839 T^{5} - 176063652 T^{6} - 1463967 T^{7} + 11523348 T^{8} + 569754 T^{9} - 366912 T^{10} - 19287 T^{11} + 8784 T^{12} + 342 T^{13} - 111 T^{14} - 3 T^{15} + T^{16} \)
$67$ \( 2114953586944 - 2140118586496 T + 2010327590448 T^{2} - 427777738280 T^{3} + 163978476161 T^{4} - 24180967287 T^{5} + 7944661591 T^{6} - 951833224 T^{7} + 231304086 T^{8} - 20606821 T^{9} + 4411018 T^{10} - 318924 T^{11} + 50012 T^{12} - 1859 T^{13} + 270 T^{14} - 7 T^{15} + T^{16} \)
$71$ \( 780959242139904 + 134744717006208 T^{2} + 7956570857364 T^{4} + 234856231407 T^{6} + 3949834995 T^{8} + 39561129 T^{10} + 233316 T^{12} + 747 T^{14} + T^{16} \)
$73$ \( 7523023152969 + 2752955922474 T - 1191205799235 T^{2} - 558789791238 T^{3} + 238887968904 T^{4} + 213060248478 T^{5} + 62170270956 T^{6} + 8564672457 T^{7} + 317278665 T^{8} - 59680611 T^{9} - 5521941 T^{10} + 432909 T^{11} + 71235 T^{12} - 306 T^{14} + T^{16} \)
$79$ \( 10549504 - 14992768 T + 30606480 T^{2} - 29157800 T^{3} + 44789945 T^{4} - 38599749 T^{5} + 35635933 T^{6} - 16316434 T^{7} + 7394544 T^{8} - 943669 T^{9} + 497152 T^{10} - 68454 T^{11} + 18914 T^{12} - 779 T^{13} + 144 T^{14} - T^{15} + T^{16} \)
$83$ \( 669184533369 - 2061188196012 T + 5773577517018 T^{2} - 1903485074616 T^{3} + 720461504880 T^{4} - 85919425596 T^{5} + 25248534300 T^{6} - 2208157416 T^{7} + 614588409 T^{8} - 32727636 T^{9} + 8300268 T^{10} - 290448 T^{11} + 80712 T^{12} - 1272 T^{13} + 330 T^{14} + T^{16} \)
$89$ \( 7161826993281 + 1051545832029 T + 1294382305422 T^{2} + 236676836007 T^{3} + 180123305796 T^{4} + 30425509008 T^{5} + 9647366571 T^{6} + 1422741915 T^{7} + 335911536 T^{8} + 43169922 T^{9} + 6642648 T^{10} + 611550 T^{11} + 69741 T^{12} + 5130 T^{13} + 459 T^{14} + 21 T^{15} + T^{16} \)
$97$ \( 22864161681 + 104576900445 T - 8224102287 T^{2} - 766864072890 T^{3} + 1208668941423 T^{4} + 71751605004 T^{5} - 41675283648 T^{6} - 2243025486 T^{7} + 1044816804 T^{8} + 41968233 T^{9} - 12357171 T^{10} - 394884 T^{11} + 109872 T^{12} + 1161 T^{13} - 384 T^{14} - 3 T^{15} + T^{16} \)
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