Properties

Label 8016.2.a.bf
Level 8016
Weight 2
Character orbit 8016.a
Self dual Yes
Analytic conductor 64.008
Analytic rank 1
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8016.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( + \beta_{1} q^{5} \) \( + ( -1 + \beta_{9} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + \beta_{1} q^{5} \) \( + ( -1 + \beta_{9} ) q^{7} \) \(+ q^{9}\) \( + ( -\beta_{1} + \beta_{10} ) q^{11} \) \( + ( 1 - \beta_{5} ) q^{13} \) \( -\beta_{1} q^{15} \) \( + ( \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} \) \( + ( -1 + \beta_{11} ) q^{19} \) \( + ( 1 - \beta_{9} ) q^{21} \) \( + ( -\beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{23} \) \( + ( 2 + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{25} \) \(- q^{27}\) \( + ( \beta_{1} + \beta_{7} - \beta_{10} ) q^{29} \) \( + ( -3 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{31} \) \( + ( \beta_{1} - \beta_{10} ) q^{33} \) \( + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{35} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{11} ) q^{37} \) \( + ( -1 + \beta_{5} ) q^{39} \) \( + ( -1 + \beta_{2} - 2 \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{41} \) \( + ( -2 - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{43} \) \( + \beta_{1} q^{45} \) \( + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{47} \) \( + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{49} \) \( + ( -\beta_{5} - \beta_{6} + \beta_{7} ) q^{51} \) \( + ( 3 + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{10} ) q^{53} \) \( + ( -4 - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{11} ) q^{55} \) \( + ( 1 - \beta_{11} ) q^{57} \) \( + ( 1 - \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{8} - \beta_{9} ) q^{59} \) \( + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{61} \) \( + ( -1 + \beta_{9} ) q^{63} \) \( + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{65} \) \( + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{67} \) \( + ( \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{69} \) \( + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{9} + \beta_{11} ) q^{71} \) \( + ( 2 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{73} \) \( + ( -2 - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{75} \) \( + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{77} \) \( + ( -5 - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{79} \) \(+ q^{81}\) \( + ( -2 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{83} \) \( + ( 2 - 3 \beta_{1} + 2 \beta_{5} - 2 \beta_{7} + \beta_{10} ) q^{85} \) \( + ( -\beta_{1} - \beta_{7} + \beta_{10} ) q^{87} \) \( + ( -2 - 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + \beta_{9} ) q^{89} \) \( + ( -1 + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{91} \) \( + ( 3 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{93} \) \( + ( 1 - 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{95} \) \( + ( -1 + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{97} \) \( + ( -\beta_{1} + \beta_{10} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 12q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 12q^{9} \) \(\mathstrut -\mathstrut q^{11} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 3q^{17} \) \(\mathstrut -\mathstrut 12q^{19} \) \(\mathstrut +\mathstrut 11q^{21} \) \(\mathstrut -\mathstrut 7q^{23} \) \(\mathstrut +\mathstrut 18q^{25} \) \(\mathstrut -\mathstrut 12q^{27} \) \(\mathstrut +\mathstrut 5q^{29} \) \(\mathstrut -\mathstrut 33q^{31} \) \(\mathstrut +\mathstrut q^{33} \) \(\mathstrut -\mathstrut 15q^{35} \) \(\mathstrut +\mathstrut 8q^{37} \) \(\mathstrut -\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 16q^{43} \) \(\mathstrut +\mathstrut 4q^{45} \) \(\mathstrut -\mathstrut 18q^{47} \) \(\mathstrut +\mathstrut 25q^{49} \) \(\mathstrut -\mathstrut 3q^{51} \) \(\mathstrut +\mathstrut 20q^{53} \) \(\mathstrut -\mathstrut 39q^{55} \) \(\mathstrut +\mathstrut 12q^{57} \) \(\mathstrut -\mathstrut 4q^{59} \) \(\mathstrut +\mathstrut 10q^{61} \) \(\mathstrut -\mathstrut 11q^{63} \) \(\mathstrut -\mathstrut 9q^{67} \) \(\mathstrut +\mathstrut 7q^{69} \) \(\mathstrut -\mathstrut 11q^{71} \) \(\mathstrut +\mathstrut 22q^{73} \) \(\mathstrut -\mathstrut 18q^{75} \) \(\mathstrut +\mathstrut 24q^{77} \) \(\mathstrut -\mathstrut 56q^{79} \) \(\mathstrut +\mathstrut 12q^{81} \) \(\mathstrut -\mathstrut 26q^{83} \) \(\mathstrut +\mathstrut 15q^{85} \) \(\mathstrut -\mathstrut 5q^{87} \) \(\mathstrut -\mathstrut 15q^{89} \) \(\mathstrut -\mathstrut 11q^{91} \) \(\mathstrut +\mathstrut 33q^{93} \) \(\mathstrut -\mathstrut 3q^{95} \) \(\mathstrut +\mathstrut 8q^{97} \) \(\mathstrut -\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(4\) \(x^{11}\mathstrut -\mathstrut \) \(31\) \(x^{10}\mathstrut +\mathstrut \) \(131\) \(x^{9}\mathstrut +\mathstrut \) \(309\) \(x^{8}\mathstrut -\mathstrut \) \(1453\) \(x^{7}\mathstrut -\mathstrut \) \(1072\) \(x^{6}\mathstrut +\mathstrut \) \(6350\) \(x^{5}\mathstrut +\mathstrut \) \(1411\) \(x^{4}\mathstrut -\mathstrut \) \(11022\) \(x^{3}\mathstrut -\mathstrut \) \(2450\) \(x^{2}\mathstrut +\mathstrut \) \(6960\) \(x\mathstrut +\mathstrut \) \(3008\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(2612041\) \(\nu^{11}\mathstrut +\mathstrut \) \(172266458\) \(\nu^{10}\mathstrut -\mathstrut \) \(431149861\) \(\nu^{9}\mathstrut -\mathstrut \) \(5604811029\) \(\nu^{8}\mathstrut +\mathstrut \) \(15598783254\) \(\nu^{7}\mathstrut +\mathstrut \) \(61532588360\) \(\nu^{6}\mathstrut -\mathstrut \) \(170251220041\) \(\nu^{5}\mathstrut -\mathstrut \) \(269226343929\) \(\nu^{4}\mathstrut +\mathstrut \) \(657292058507\) \(\nu^{3}\mathstrut +\mathstrut \) \(528815192804\) \(\nu^{2}\mathstrut -\mathstrut \) \(748850317708\) \(\nu\mathstrut -\mathstrut \) \(506734859797\)\()/\)\(6887737669\)
\(\beta_{3}\)\(=\)\((\)\(48270257\) \(\nu^{11}\mathstrut -\mathstrut \) \(393971708\) \(\nu^{10}\mathstrut -\mathstrut \) \(803884765\) \(\nu^{9}\mathstrut +\mathstrut \) \(12731441603\) \(\nu^{8}\mathstrut -\mathstrut \) \(7245605191\) \(\nu^{7}\mathstrut -\mathstrut \) \(136791943933\) \(\nu^{6}\mathstrut +\mathstrut \) \(180010099322\) \(\nu^{5}\mathstrut +\mathstrut \) \(557383751450\) \(\nu^{4}\mathstrut -\mathstrut \) \(796092987441\) \(\nu^{3}\mathstrut -\mathstrut \) \(886580239346\) \(\nu^{2}\mathstrut +\mathstrut \) \(843664925302\) \(\nu\mathstrut +\mathstrut \) \(623931970190\)\()/\)\(13775475338\)
\(\beta_{4}\)\(=\)\((\)\(433554181\) \(\nu^{11}\mathstrut -\mathstrut \) \(1027661208\) \(\nu^{10}\mathstrut -\mathstrut \) \(15151502491\) \(\nu^{9}\mathstrut +\mathstrut \) \(32370295155\) \(\nu^{8}\mathstrut +\mathstrut \) \(187572944809\) \(\nu^{7}\mathstrut -\mathstrut \) \(331978409981\) \(\nu^{6}\mathstrut -\mathstrut \) \(1009649126248\) \(\nu^{5}\mathstrut +\mathstrut \) \(1181123521726\) \(\nu^{4}\mathstrut +\mathstrut \) \(2516794688379\) \(\nu^{3}\mathstrut -\mathstrut \) \(933458223074\) \(\nu^{2}\mathstrut -\mathstrut \) \(2492427824318\) \(\nu\mathstrut -\mathstrut \) \(810005934996\)\()/\)\(27550950676\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(924523625\) \(\nu^{11}\mathstrut +\mathstrut \) \(3335040372\) \(\nu^{10}\mathstrut +\mathstrut \) \(28727151823\) \(\nu^{9}\mathstrut -\mathstrut \) \(105506939067\) \(\nu^{8}\mathstrut -\mathstrut \) \(286439821885\) \(\nu^{7}\mathstrut +\mathstrut \) \(1096386897181\) \(\nu^{6}\mathstrut +\mathstrut \) \(971661553200\) \(\nu^{5}\mathstrut -\mathstrut \) \(4121786159918\) \(\nu^{4}\mathstrut -\mathstrut \) \(965164697859\) \(\nu^{3}\mathstrut +\mathstrut \) \(4869923492678\) \(\nu^{2}\mathstrut +\mathstrut \) \(619274447522\) \(\nu\mathstrut -\mathstrut \) \(1194286516536\)\()/\)\(55101901352\)
\(\beta_{6}\)\(=\)\((\)\(497566513\) \(\nu^{11}\mathstrut -\mathstrut \) \(583717328\) \(\nu^{10}\mathstrut -\mathstrut \) \(19421435387\) \(\nu^{9}\mathstrut +\mathstrut \) \(18043299379\) \(\nu^{8}\mathstrut +\mathstrut \) \(279234229797\) \(\nu^{7}\mathstrut -\mathstrut \) \(177472297861\) \(\nu^{6}\mathstrut -\mathstrut \) \(1817022049764\) \(\nu^{5}\mathstrut +\mathstrut \) \(523553323290\) \(\nu^{4}\mathstrut +\mathstrut \) \(5297394080111\) \(\nu^{3}\mathstrut +\mathstrut \) \(403005890338\) \(\nu^{2}\mathstrut -\mathstrut \) \(5364621239478\) \(\nu\mathstrut -\mathstrut \) \(2291097916892\)\()/\)\(27550950676\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(608147925\) \(\nu^{11}\mathstrut +\mathstrut \) \(2540267068\) \(\nu^{10}\mathstrut +\mathstrut \) \(17866911943\) \(\nu^{9}\mathstrut -\mathstrut \) \(80927081243\) \(\nu^{8}\mathstrut -\mathstrut \) \(156734313797\) \(\nu^{7}\mathstrut +\mathstrut \) \(851051501325\) \(\nu^{6}\mathstrut +\mathstrut \) \(322943033664\) \(\nu^{5}\mathstrut -\mathstrut \) \(3288682206998\) \(\nu^{4}\mathstrut +\mathstrut \) \(476137596409\) \(\nu^{3}\mathstrut +\mathstrut \) \(4250845840470\) \(\nu^{2}\mathstrut -\mathstrut \) \(751582083438\) \(\nu\mathstrut -\mathstrut \) \(1570311572784\)\()/\)\(27550950676\)
\(\beta_{8}\)\(=\)\((\)\(1276718751\) \(\nu^{11}\mathstrut -\mathstrut \) \(4447291652\) \(\nu^{10}\mathstrut -\mathstrut \) \(40378997177\) \(\nu^{9}\mathstrut +\mathstrut \) \(141144188077\) \(\nu^{8}\mathstrut +\mathstrut \) \(420342586963\) \(\nu^{7}\mathstrut -\mathstrut \) \(1472464513643\) \(\nu^{6}\mathstrut -\mathstrut \) \(1640879715920\) \(\nu^{5}\mathstrut +\mathstrut \) \(5556472146210\) \(\nu^{4}\mathstrut +\mathstrut \) \(2723364140149\) \(\nu^{3}\mathstrut -\mathstrut \) \(6468711211946\) \(\nu^{2}\mathstrut -\mathstrut \) \(2707550657158\) \(\nu\mathstrut +\mathstrut \) \(1003995641592\)\()/\)\(55101901352\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(672160257\) \(\nu^{11}\mathstrut +\mathstrut \) \(2096323188\) \(\nu^{10}\mathstrut +\mathstrut \) \(22136844839\) \(\nu^{9}\mathstrut -\mathstrut \) \(66600085467\) \(\nu^{8}\mathstrut -\mathstrut \) \(248395598785\) \(\nu^{7}\mathstrut +\mathstrut \) \(696545389205\) \(\nu^{6}\mathstrut +\mathstrut \) \(1130315957180\) \(\nu^{5}\mathstrut -\mathstrut \) \(2631112008562\) \(\nu^{4}\mathstrut -\mathstrut \) \(2304461795323\) \(\nu^{3}\mathstrut +\mathstrut \) \(2886830776382\) \(\nu^{2}\mathstrut +\mathstrut \) \(2120611331722\) \(\nu\mathstrut +\mathstrut \) \(103637063844\)\()/\)\(27550950676\)
\(\beta_{10}\)\(=\)\((\)\(825574067\) \(\nu^{11}\mathstrut -\mathstrut \) \(1232246264\) \(\nu^{10}\mathstrut -\mathstrut \) \(31068710789\) \(\nu^{9}\mathstrut +\mathstrut \) \(37574547953\) \(\nu^{8}\mathstrut +\mathstrut \) \(427577508547\) \(\nu^{7}\mathstrut -\mathstrut \) \(358636986823\) \(\nu^{6}\mathstrut -\mathstrut \) \(2660357207468\) \(\nu^{5}\mathstrut +\mathstrut \) \(975184226114\) \(\nu^{4}\mathstrut +\mathstrut \) \(7596696525145\) \(\nu^{3}\mathstrut +\mathstrut \) \(904825188458\) \(\nu^{2}\mathstrut -\mathstrut \) \(7836145638226\) \(\nu\mathstrut -\mathstrut \) \(3936944027888\)\()/\)\(27550950676\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(1494430261\) \(\nu^{11}\mathstrut +\mathstrut \) \(4167534940\) \(\nu^{10}\mathstrut +\mathstrut \) \(50046579515\) \(\nu^{9}\mathstrut -\mathstrut \) \(131337919583\) \(\nu^{8}\mathstrut -\mathstrut \) \(578035323361\) \(\nu^{7}\mathstrut +\mathstrut \) \(1352380306305\) \(\nu^{6}\mathstrut +\mathstrut \) \(2776874943512\) \(\nu^{5}\mathstrut -\mathstrut \) \(4915467010946\) \(\nu^{4}\mathstrut -\mathstrut \) \(6107654784727\) \(\nu^{3}\mathstrut +\mathstrut \) \(4699818876386\) \(\nu^{2}\mathstrut +\mathstrut \) \(5874998351882\) \(\nu\mathstrut +\mathstrut \) \(1214478641340\)\()/\)\(27550950676\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(\beta_{11}\mathstrut -\mathstrut \) \(13\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(15\) \(\beta_{7}\mathstrut -\mathstrut \) \(15\) \(\beta_{6}\mathstrut +\mathstrut \) \(19\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(78\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(15\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(31\) \(\beta_{5}\mathstrut +\mathstrut \) \(23\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(30\) \(\beta_{2}\mathstrut +\mathstrut \) \(144\) \(\beta_{1}\mathstrut +\mathstrut \) \(31\)
\(\nu^{6}\)\(=\)\(26\) \(\beta_{11}\mathstrut +\mathstrut \) \(6\) \(\beta_{10}\mathstrut -\mathstrut \) \(157\) \(\beta_{9}\mathstrut +\mathstrut \) \(34\) \(\beta_{8}\mathstrut +\mathstrut \) \(188\) \(\beta_{7}\mathstrut -\mathstrut \) \(202\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(292\) \(\beta_{4}\mathstrut -\mathstrut \) \(38\) \(\beta_{3}\mathstrut -\mathstrut \) \(41\) \(\beta_{2}\mathstrut -\mathstrut \) \(46\) \(\beta_{1}\mathstrut +\mathstrut \) \(976\)
\(\nu^{7}\)\(=\)\(49\) \(\beta_{11}\mathstrut -\mathstrut \) \(199\) \(\beta_{10}\mathstrut +\mathstrut \) \(29\) \(\beta_{9}\mathstrut -\mathstrut \) \(57\) \(\beta_{8}\mathstrut +\mathstrut \) \(96\) \(\beta_{7}\mathstrut -\mathstrut \) \(32\) \(\beta_{6}\mathstrut -\mathstrut \) \(401\) \(\beta_{5}\mathstrut +\mathstrut \) \(406\) \(\beta_{4}\mathstrut +\mathstrut \) \(108\) \(\beta_{3}\mathstrut +\mathstrut \) \(394\) \(\beta_{2}\mathstrut +\mathstrut \) \(1764\) \(\beta_{1}\mathstrut +\mathstrut \) \(484\)
\(\nu^{8}\)\(=\)\(465\) \(\beta_{11}\mathstrut +\mathstrut \) \(136\) \(\beta_{10}\mathstrut -\mathstrut \) \(1897\) \(\beta_{9}\mathstrut +\mathstrut \) \(446\) \(\beta_{8}\mathstrut +\mathstrut \) \(2264\) \(\beta_{7}\mathstrut -\mathstrut \) \(2645\) \(\beta_{6}\mathstrut +\mathstrut \) \(150\) \(\beta_{5}\mathstrut +\mathstrut \) \(4221\) \(\beta_{4}\mathstrut -\mathstrut \) \(533\) \(\beta_{3}\mathstrut -\mathstrut \) \(623\) \(\beta_{2}\mathstrut -\mathstrut \) \(730\) \(\beta_{1}\mathstrut +\mathstrut \) \(12688\)
\(\nu^{9}\)\(=\)\(810\) \(\beta_{11}\mathstrut -\mathstrut \) \(2629\) \(\beta_{10}\mathstrut +\mathstrut \) \(581\) \(\beta_{9}\mathstrut -\mathstrut \) \(879\) \(\beta_{8}\mathstrut +\mathstrut \) \(663\) \(\beta_{7}\mathstrut -\mathstrut \) \(715\) \(\beta_{6}\mathstrut -\mathstrut \) \(4895\) \(\beta_{5}\mathstrut +\mathstrut \) \(6479\) \(\beta_{4}\mathstrut +\mathstrut \) \(759\) \(\beta_{3}\mathstrut +\mathstrut \) \(5122\) \(\beta_{2}\mathstrut +\mathstrut \) \(22012\) \(\beta_{1}\mathstrut +\mathstrut \) \(7620\)
\(\nu^{10}\)\(=\)\(7141\) \(\beta_{11}\mathstrut +\mathstrut \) \(2177\) \(\beta_{10}\mathstrut -\mathstrut \) \(22913\) \(\beta_{9}\mathstrut +\mathstrut \) \(5288\) \(\beta_{8}\mathstrut +\mathstrut \) \(26926\) \(\beta_{7}\mathstrut -\mathstrut \) \(34360\) \(\beta_{6}\mathstrut +\mathstrut \) \(3630\) \(\beta_{5}\mathstrut +\mathstrut \) \(59518\) \(\beta_{4}\mathstrut -\mathstrut \) \(6696\) \(\beta_{3}\mathstrut -\mathstrut \) \(8532\) \(\beta_{2}\mathstrut -\mathstrut \) \(9985\) \(\beta_{1}\mathstrut +\mathstrut \) \(167606\)
\(\nu^{11}\)\(=\)\(11158\) \(\beta_{11}\mathstrut -\mathstrut \) \(35307\) \(\beta_{10}\mathstrut +\mathstrut \) \(10456\) \(\beta_{9}\mathstrut -\mathstrut \) \(13224\) \(\beta_{8}\mathstrut +\mathstrut \) \(1478\) \(\beta_{7}\mathstrut -\mathstrut \) \(13396\) \(\beta_{6}\mathstrut -\mathstrut \) \(57692\) \(\beta_{5}\mathstrut +\mathstrut \) \(98497\) \(\beta_{4}\mathstrut +\mathstrut \) \(2215\) \(\beta_{3}\mathstrut +\mathstrut \) \(67138\) \(\beta_{2}\mathstrut +\mathstrut \) \(278574\) \(\beta_{1}\mathstrut +\mathstrut \) \(119224\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.64003
−3.49351
−2.09997
−1.14709
−0.716151
−0.619387
1.53322
1.56937
2.31305
2.93317
3.59055
3.77676
0 −1.00000 0 −3.64003 0 1.45249 0 1.00000 0
1.2 0 −1.00000 0 −3.49351 0 −4.58171 0 1.00000 0
1.3 0 −1.00000 0 −2.09997 0 −2.97416 0 1.00000 0
1.4 0 −1.00000 0 −1.14709 0 3.32442 0 1.00000 0
1.5 0 −1.00000 0 −0.716151 0 3.33002 0 1.00000 0
1.6 0 −1.00000 0 −0.619387 0 −0.954126 0 1.00000 0
1.7 0 −1.00000 0 1.53322 0 0.915565 0 1.00000 0
1.8 0 −1.00000 0 1.56937 0 −3.21458 0 1.00000 0
1.9 0 −1.00000 0 2.31305 0 −4.58648 0 1.00000 0
1.10 0 −1.00000 0 2.93317 0 −1.35675 0 1.00000 0
1.11 0 −1.00000 0 3.59055 0 −4.10828 0 1.00000 0
1.12 0 −1.00000 0 3.77676 0 1.75359 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(167\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{12} - \cdots\)
\(T_{7}^{12} + \cdots\)
\(T_{11}^{12} + \cdots\)
\(T_{13}^{12} - \cdots\)