Properties

Label 8024.2.a.v
Level 8024
Weight 2
Character orbit 8024.a
Self dual Yes
Analytic conductor 64.072
Analytic rank 0
Dimension 18
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8024.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0719625819\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 - \beta_{1} ) q^{3} \) \( + \beta_{3} q^{5} \) \( + ( 1 - \beta_{6} ) q^{7} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 - \beta_{1} ) q^{3} \) \( + \beta_{3} q^{5} \) \( + ( 1 - \beta_{6} ) q^{7} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{9} \) \( + \beta_{13} q^{11} \) \( + ( 1 - \beta_{6} - \beta_{12} ) q^{13} \) \( + ( -\beta_{1} - \beta_{5} + 2 \beta_{6} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{15} \) \(+ q^{17}\) \( + ( 2 - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{13} ) q^{19} \) \( + ( -2 \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{17} ) q^{21} \) \( + ( 2 - \beta_{1} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{13} ) q^{23} \) \( + ( 2 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{14} - \beta_{16} + \beta_{17} ) q^{25} \) \( + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} + \beta_{12} + \beta_{14} ) q^{27} \) \( + ( 1 - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{29} \) \( + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{31} \) \( + ( \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{13} + \beta_{15} - \beta_{17} ) q^{33} \) \( + ( \beta_{3} + \beta_{6} - \beta_{13} + \beta_{16} ) q^{35} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{15} + \beta_{17} ) q^{37} \) \( + ( 1 + \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{8} - 3 \beta_{12} - \beta_{14} - \beta_{15} ) q^{39} \) \( + ( -1 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{41} \) \( + ( \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{43} \) \( + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} - 2 \beta_{17} ) q^{45} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{9} + \beta_{11} - 2 \beta_{13} + \beta_{17} ) q^{47} \) \( + ( 1 - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{14} ) q^{49} \) \( + ( 1 - \beta_{1} ) q^{51} \) \( + ( -2 + 3 \beta_{1} + 3 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{17} ) q^{53} \) \( + ( 2 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} ) q^{55} \) \( + ( 1 - 4 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{17} ) q^{57} \) \(- q^{59}\) \( + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{10} - \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{61} \) \( + ( 4 - 5 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + 3 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} ) q^{63} \) \( + ( -3 + 2 \beta_{1} + 5 \beta_{3} + \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{16} ) q^{65} \) \( + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} - \beta_{15} - \beta_{16} ) q^{67} \) \( + ( 4 - 3 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{9} + 2 \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} ) q^{69} \) \( + ( \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{12} + \beta_{14} ) q^{71} \) \( + ( -1 - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{10} + \beta_{12} + \beta_{14} + \beta_{16} ) q^{73} \) \( + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{16} ) q^{75} \) \( + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{16} + \beta_{17} ) q^{77} \) \( + ( -1 + 4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{79} \) \( + ( 4 - 4 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} + \beta_{11} + 3 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} - \beta_{17} ) q^{81} \) \( + ( 3 \beta_{1} + 6 \beta_{3} - \beta_{4} + 3 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - \beta_{9} - 3 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} + \beta_{16} + \beta_{17} ) q^{83} \) \( + \beta_{3} q^{85} \) \( + ( 2 - \beta_{1} + \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{12} - \beta_{13} + 2 \beta_{15} + \beta_{16} - \beta_{17} ) q^{87} \) \( + ( 1 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{12} + \beta_{13} - 3 \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{89} \) \( + ( 5 + \beta_{3} - 2 \beta_{6} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{91} \) \( + ( \beta_{2} + \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} - 3 \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{17} ) q^{93} \) \( + ( -1 + \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{12} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{16} + \beta_{17} ) q^{95} \) \( + ( -3 - 2 \beta_{1} + \beta_{2} - 5 \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{11} + 2 \beta_{12} + \beta_{13} + 3 \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{97} \) \( + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{6} + 3 \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(18q \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 31q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(18q \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 31q^{9} \) \(\mathstrut -\mathstrut q^{11} \) \(\mathstrut +\mathstrut 15q^{13} \) \(\mathstrut +\mathstrut 3q^{15} \) \(\mathstrut +\mathstrut 18q^{17} \) \(\mathstrut +\mathstrut 26q^{19} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 22q^{23} \) \(\mathstrut +\mathstrut 42q^{25} \) \(\mathstrut +\mathstrut 45q^{27} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut +\mathstrut 13q^{31} \) \(\mathstrut -\mathstrut 5q^{33} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 4q^{37} \) \(\mathstrut +\mathstrut 36q^{39} \) \(\mathstrut -\mathstrut 15q^{41} \) \(\mathstrut +\mathstrut 12q^{43} \) \(\mathstrut +\mathstrut 14q^{45} \) \(\mathstrut +\mathstrut 8q^{47} \) \(\mathstrut -\mathstrut 13q^{49} \) \(\mathstrut +\mathstrut 9q^{51} \) \(\mathstrut -\mathstrut 11q^{53} \) \(\mathstrut +\mathstrut 55q^{55} \) \(\mathstrut -\mathstrut 20q^{57} \) \(\mathstrut -\mathstrut 18q^{59} \) \(\mathstrut +\mathstrut 53q^{61} \) \(\mathstrut +\mathstrut 29q^{63} \) \(\mathstrut -\mathstrut 26q^{65} \) \(\mathstrut +\mathstrut 2q^{67} \) \(\mathstrut +\mathstrut 32q^{69} \) \(\mathstrut +\mathstrut 8q^{71} \) \(\mathstrut -\mathstrut 42q^{73} \) \(\mathstrut +\mathstrut 72q^{75} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut -\mathstrut 9q^{79} \) \(\mathstrut +\mathstrut 42q^{81} \) \(\mathstrut -\mathstrut 4q^{83} \) \(\mathstrut +\mathstrut 2q^{85} \) \(\mathstrut +\mathstrut 36q^{87} \) \(\mathstrut +\mathstrut 13q^{89} \) \(\mathstrut +\mathstrut 68q^{91} \) \(\mathstrut +\mathstrut q^{93} \) \(\mathstrut +\mathstrut 3q^{95} \) \(\mathstrut -\mathstrut 56q^{97} \) \(\mathstrut -\mathstrut 13q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18}\mathstrut -\mathstrut \) \(9\) \(x^{17}\mathstrut -\mathstrut \) \(2\) \(x^{16}\mathstrut +\mathstrut \) \(212\) \(x^{15}\mathstrut -\mathstrut \) \(289\) \(x^{14}\mathstrut -\mathstrut \) \(2094\) \(x^{13}\mathstrut +\mathstrut \) \(3933\) \(x^{12}\mathstrut +\mathstrut \) \(11326\) \(x^{11}\mathstrut -\mathstrut \) \(23166\) \(x^{10}\mathstrut -\mathstrut \) \(36429\) \(x^{9}\mathstrut +\mathstrut \) \(72042\) \(x^{8}\mathstrut +\mathstrut \) \(69272\) \(x^{7}\mathstrut -\mathstrut \) \(119982\) \(x^{6}\mathstrut -\mathstrut \) \(69890\) \(x^{5}\mathstrut +\mathstrut \) \(98783\) \(x^{4}\mathstrut +\mathstrut \) \(26872\) \(x^{3}\mathstrut -\mathstrut \) \(30932\) \(x^{2}\mathstrut +\mathstrut \) \(1600\) \(x\mathstrut +\mathstrut \) \(1136\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\(122592670237151\) \(\nu^{17}\mathstrut -\mathstrut \) \(1298419400230158\) \(\nu^{16}\mathstrut +\mathstrut \) \(8917952125020903\) \(\nu^{15}\mathstrut -\mathstrut \) \(45177501288012240\) \(\nu^{14}\mathstrut -\mathstrut \) \(16910838981905815\) \(\nu^{13}\mathstrut +\mathstrut \) \(1068971107829004041\) \(\nu^{12}\mathstrut -\mathstrut \) \(1898824299050997205\) \(\nu^{11}\mathstrut -\mathstrut \) \(8298941431812432865\) \(\nu^{10}\mathstrut +\mathstrut \) \(19958906277888160018\) \(\nu^{9}\mathstrut +\mathstrut \) \(33367043245087548791\) \(\nu^{8}\mathstrut -\mathstrut \) \(84593803297438338739\) \(\nu^{7}\mathstrut -\mathstrut \) \(82251733824418117228\) \(\nu^{6}\mathstrut +\mathstrut \) \(167991325863897118854\) \(\nu^{5}\mathstrut +\mathstrut \) \(126158545696274496020\) \(\nu^{4}\mathstrut -\mathstrut \) \(135744630989828723269\) \(\nu^{3}\mathstrut -\mathstrut \) \(88927587157346062677\) \(\nu^{2}\mathstrut +\mathstrut \) \(22346730059405859106\) \(\nu\mathstrut +\mathstrut \) \(6867367017125660872\)\()/\)\(380020491990339796\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(609319771140861\) \(\nu^{17}\mathstrut +\mathstrut \) \(1683058106282306\) \(\nu^{16}\mathstrut +\mathstrut \) \(36891106721606087\) \(\nu^{15}\mathstrut -\mathstrut \) \(143204048151005224\) \(\nu^{14}\mathstrut -\mathstrut \) \(550071034613862583\) \(\nu^{13}\mathstrut +\mathstrut \) \(2620579382553627489\) \(\nu^{12}\mathstrut +\mathstrut \) \(3559171359085627247\) \(\nu^{11}\mathstrut -\mathstrut \) \(21316422660458068649\) \(\nu^{10}\mathstrut -\mathstrut \) \(11452337014793851938\) \(\nu^{9}\mathstrut +\mathstrut \) \(90376426588817657983\) \(\nu^{8}\mathstrut +\mathstrut \) \(18438131565115721365\) \(\nu^{7}\mathstrut -\mathstrut \) \(201651888562237303008\) \(\nu^{6}\mathstrut -\mathstrut \) \(11440627932045831290\) \(\nu^{5}\mathstrut +\mathstrut \) \(212423816484993120856\) \(\nu^{4}\mathstrut -\mathstrut \) \(9598438403037070149\) \(\nu^{3}\mathstrut -\mathstrut \) \(77691746396577096909\) \(\nu^{2}\mathstrut +\mathstrut \) \(12859716270017882690\) \(\nu\mathstrut +\mathstrut \) \(2004295146728977072\)\()/\)\(380020491990339796\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(1587273663311409\) \(\nu^{17}\mathstrut +\mathstrut \) \(15375801048458298\) \(\nu^{16}\mathstrut -\mathstrut \) \(17918131791014758\) \(\nu^{15}\mathstrut -\mathstrut \) \(226996639618614114\) \(\nu^{14}\mathstrut +\mathstrut \) \(577881124032751143\) \(\nu^{13}\mathstrut +\mathstrut \) \(1041987950611805491\) \(\nu^{12}\mathstrut -\mathstrut \) \(3720724040758125966\) \(\nu^{11}\mathstrut -\mathstrut \) \(381412837173183774\) \(\nu^{10}\mathstrut +\mathstrut \) \(5302356796711099564\) \(\nu^{9}\mathstrut -\mathstrut \) \(11869910731788594483\) \(\nu^{8}\mathstrut +\mathstrut \) \(28237618262508633157\) \(\nu^{7}\mathstrut +\mathstrut \) \(48266235266063981029\) \(\nu^{6}\mathstrut -\mathstrut \) \(111008030768968424501\) \(\nu^{5}\mathstrut -\mathstrut \) \(95683913700501012343\) \(\nu^{4}\mathstrut +\mathstrut \) \(117119701211630831274\) \(\nu^{3}\mathstrut +\mathstrut \) \(78572542590193322132\) \(\nu^{2}\mathstrut -\mathstrut \) \(22074946780103382640\) \(\nu\mathstrut -\mathstrut \) \(6908112440951904184\)\()/\)\(380020491990339796\)
\(\beta_{6}\)\(=\)\((\)\(1645454574373317\) \(\nu^{17}\mathstrut -\mathstrut \) \(14597662963437626\) \(\nu^{16}\mathstrut -\mathstrut \) \(11978113787028513\) \(\nu^{15}\mathstrut +\mathstrut \) \(411983375267988368\) \(\nu^{14}\mathstrut -\mathstrut \) \(460969956879408005\) \(\nu^{13}\mathstrut -\mathstrut \) \(4734740771498861825\) \(\nu^{12}\mathstrut +\mathstrut \) \(8273856112682504103\) \(\nu^{11}\mathstrut +\mathstrut \) \(29177983112459848035\) \(\nu^{10}\mathstrut -\mathstrut \) \(57644230746626669226\) \(\nu^{9}\mathstrut -\mathstrut \) \(106981426925077764031\) \(\nu^{8}\mathstrut +\mathstrut \) \(204070575933799822211\) \(\nu^{7}\mathstrut +\mathstrut \) \(242130459822250769402\) \(\nu^{6}\mathstrut -\mathstrut \) \(369854372345519699080\) \(\nu^{5}\mathstrut -\mathstrut \) \(322343491531637979194\) \(\nu^{4}\mathstrut +\mathstrut \) \(293365062298146487619\) \(\nu^{3}\mathstrut +\mathstrut \) \(194721848009727247951\) \(\nu^{2}\mathstrut -\mathstrut \) \(58008212605629246130\) \(\nu\mathstrut -\mathstrut \) \(12683949158962437572\)\()/\)\(380020491990339796\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(2155746446570540\) \(\nu^{17}\mathstrut +\mathstrut \) \(22723658425676579\) \(\nu^{16}\mathstrut -\mathstrut \) \(36012302274689123\) \(\nu^{15}\mathstrut -\mathstrut \) \(357048777205337504\) \(\nu^{14}\mathstrut +\mathstrut \) \(1196938509560942180\) \(\nu^{13}\mathstrut +\mathstrut \) \(1696060816350981777\) \(\nu^{12}\mathstrut -\mathstrut \) \(10250280830558409326\) \(\nu^{11}\mathstrut +\mathstrut \) \(471666071212712433\) \(\nu^{10}\mathstrut +\mathstrut \) \(39212600148949455436\) \(\nu^{9}\mathstrut -\mathstrut \) \(29533987255600170582\) \(\nu^{8}\mathstrut -\mathstrut \) \(66793628522387237747\) \(\nu^{7}\mathstrut +\mathstrut \) \(103280366913410821826\) \(\nu^{6}\mathstrut +\mathstrut \) \(27531093574330230954\) \(\nu^{5}\mathstrut -\mathstrut \) \(155859517760206007852\) \(\nu^{4}\mathstrut +\mathstrut \) \(33286811006657918320\) \(\nu^{3}\mathstrut +\mathstrut \) \(93301957314807764499\) \(\nu^{2}\mathstrut -\mathstrut \) \(16204549819069239570\) \(\nu\mathstrut -\mathstrut \) \(5503247053984782784\)\()/\)\(380020491990339796\)
\(\beta_{8}\)\(=\)\((\)\(2899445345842989\) \(\nu^{17}\mathstrut -\mathstrut \) \(32281353023037673\) \(\nu^{16}\mathstrut +\mathstrut \) \(56951967308129513\) \(\nu^{15}\mathstrut +\mathstrut \) \(539441371211316430\) \(\nu^{14}\mathstrut -\mathstrut \) \(1914199856765011043\) \(\nu^{13}\mathstrut -\mathstrut \) \(3129525971552630136\) \(\nu^{12}\mathstrut +\mathstrut \) \(18116305359534657544\) \(\nu^{11}\mathstrut +\mathstrut \) \(6295856115583527717\) \(\nu^{10}\mathstrut -\mathstrut \) \(84465800191630555160\) \(\nu^{9}\mathstrut +\mathstrut \) \(3444653625121760317\) \(\nu^{8}\mathstrut +\mathstrut \) \(216382259239413814262\) \(\nu^{7}\mathstrut -\mathstrut \) \(17630774929745579611\) \(\nu^{6}\mathstrut -\mathstrut \) \(297919314260919361573\) \(\nu^{5}\mathstrut -\mathstrut \) \(18722852965496106793\) \(\nu^{4}\mathstrut +\mathstrut \) \(182750516267868362314\) \(\nu^{3}\mathstrut +\mathstrut \) \(49892398998151622105\) \(\nu^{2}\mathstrut -\mathstrut \) \(22758020131543004038\) \(\nu\mathstrut -\mathstrut \) \(6424015501352120412\)\()/\)\(380020491990339796\)
\(\beta_{9}\)\(=\)\((\)\(4215060968698447\) \(\nu^{17}\mathstrut -\mathstrut \) \(28073965336066934\) \(\nu^{16}\mathstrut -\mathstrut \) \(94307067622634669\) \(\nu^{15}\mathstrut +\mathstrut \) \(871125502935203600\) \(\nu^{14}\mathstrut +\mathstrut \) \(650318221718818285\) \(\nu^{13}\mathstrut -\mathstrut \) \(10968965355282886159\) \(\nu^{12}\mathstrut -\mathstrut \) \(682889959747118289\) \(\nu^{11}\mathstrut +\mathstrut \) \(72800029286840876183\) \(\nu^{10}\mathstrut -\mathstrut \) \(8312888739480360282\) \(\nu^{9}\mathstrut -\mathstrut \) \(272236558820278140229\) \(\nu^{8}\mathstrut +\mathstrut \) \(23909891410593995597\) \(\nu^{7}\mathstrut +\mathstrut \) \(555861345004453236860\) \(\nu^{6}\mathstrut +\mathstrut \) \(769599417620036818\) \(\nu^{5}\mathstrut -\mathstrut \) \(537031629360800327708\) \(\nu^{4}\mathstrut -\mathstrut \) \(18359654476265138349\) \(\nu^{3}\mathstrut +\mathstrut \) \(176842007495950173487\) \(\nu^{2}\mathstrut -\mathstrut \) \(17716168754019672746\) \(\nu\mathstrut -\mathstrut \) \(8189793039128305828\)\()/\)\(380020491990339796\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(4794431721437933\) \(\nu^{17}\mathstrut +\mathstrut \) \(42658716944469609\) \(\nu^{16}\mathstrut +\mathstrut \) \(10497548576101440\) \(\nu^{15}\mathstrut -\mathstrut \) \(979471863622201128\) \(\nu^{14}\mathstrut +\mathstrut \) \(1240606727423931413\) \(\nu^{13}\mathstrut +\mathstrut \) \(9527178478716388790\) \(\nu^{12}\mathstrut -\mathstrut \) \(16088773239713417783\) \(\nu^{11}\mathstrut -\mathstrut \) \(51670645609439856460\) \(\nu^{10}\mathstrut +\mathstrut \) \(88618450697283801706\) \(\nu^{9}\mathstrut +\mathstrut \) \(170695569759297221301\) \(\nu^{8}\mathstrut -\mathstrut \) \(250600346380992213774\) \(\nu^{7}\mathstrut -\mathstrut \) \(341836328407726159962\) \(\nu^{6}\mathstrut +\mathstrut \) \(360477659903277161392\) \(\nu^{5}\mathstrut +\mathstrut \) \(374784608476543199552\) \(\nu^{4}\mathstrut -\mathstrut \) \(233220453496109779213\) \(\nu^{3}\mathstrut -\mathstrut \) \(177529861737967908158\) \(\nu^{2}\mathstrut +\mathstrut \) \(43258835845672450716\) \(\nu\mathstrut +\mathstrut \) \(12471316210180831476\)\()/\)\(380020491990339796\)
\(\beta_{11}\)\(=\)\((\)\(5492949629426422\) \(\nu^{17}\mathstrut -\mathstrut \) \(39735627247266322\) \(\nu^{16}\mathstrut -\mathstrut \) \(87147296867544777\) \(\nu^{15}\mathstrut +\mathstrut \) \(1064532787843635838\) \(\nu^{14}\mathstrut +\mathstrut \) \(284141213144684328\) \(\nu^{13}\mathstrut -\mathstrut \) \(12071523229605049502\) \(\nu^{12}\mathstrut +\mathstrut \) \(1813548826053870941\) \(\nu^{11}\mathstrut +\mathstrut \) \(74521636662952605063\) \(\nu^{10}\mathstrut -\mathstrut \) \(10811738264796031518\) \(\nu^{9}\mathstrut -\mathstrut \) \(263769758386512499916\) \(\nu^{8}\mathstrut -\mathstrut \) \(6523513834488659456\) \(\nu^{7}\mathstrut +\mathstrut \) \(504781186464729878233\) \(\nu^{6}\mathstrut +\mathstrut \) \(112475516922424880929\) \(\nu^{5}\mathstrut -\mathstrut \) \(422566106011048401047\) \(\nu^{4}\mathstrut -\mathstrut \) \(137333516399786632527\) \(\nu^{3}\mathstrut +\mathstrut \) \(82532989290400175625\) \(\nu^{2}\mathstrut +\mathstrut \) \(6804269861444342498\) \(\nu\mathstrut -\mathstrut \) \(360025852626870664\)\()/\)\(380020491990339796\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(5825488733147193\) \(\nu^{17}\mathstrut +\mathstrut \) \(47773817067872047\) \(\nu^{16}\mathstrut +\mathstrut \) \(54177483044544452\) \(\nu^{15}\mathstrut -\mathstrut \) \(1239090767851230792\) \(\nu^{14}\mathstrut +\mathstrut \) \(774190286080761517\) \(\nu^{13}\mathstrut +\mathstrut \) \(13586472879117599116\) \(\nu^{12}\mathstrut -\mathstrut \) \(14739395565968183493\) \(\nu^{11}\mathstrut -\mathstrut \) \(81885441250861356724\) \(\nu^{10}\mathstrut +\mathstrut \) \(93967655297668630406\) \(\nu^{9}\mathstrut +\mathstrut \) \(292353778772984075121\) \(\nu^{8}\mathstrut -\mathstrut \) \(292932132011035079976\) \(\nu^{7}\mathstrut -\mathstrut \) \(610503448481482237612\) \(\nu^{6}\mathstrut +\mathstrut \) \(457751048239555631558\) \(\nu^{5}\mathstrut +\mathstrut \) \(674969615715753532546\) \(\nu^{4}\mathstrut -\mathstrut \) \(324563455593762330599\) \(\nu^{3}\mathstrut -\mathstrut \) \(310523105642083691654\) \(\nu^{2}\mathstrut +\mathstrut \) \(70792891784965523420\) \(\nu\mathstrut +\mathstrut \) \(17975623602172032796\)\()/\)\(380020491990339796\)
\(\beta_{13}\)\(=\)\((\)\(6496459109723310\) \(\nu^{17}\mathstrut -\mathstrut \) \(66319409537457946\) \(\nu^{16}\mathstrut +\mathstrut \) \(69660048779369049\) \(\nu^{15}\mathstrut +\mathstrut \) \(1255670681930968686\) \(\nu^{14}\mathstrut -\mathstrut \) \(3251993606658771360\) \(\nu^{13}\mathstrut -\mathstrut \) \(9286471323264579526\) \(\nu^{12}\mathstrut +\mathstrut \) \(33362552134951427659\) \(\nu^{11}\mathstrut +\mathstrut \) \(34519689627591335493\) \(\nu^{10}\mathstrut -\mathstrut \) \(163414236778153685302\) \(\nu^{9}\mathstrut -\mathstrut \) \(72134029343582697496\) \(\nu^{8}\mathstrut +\mathstrut \) \(430208023989172542408\) \(\nu^{7}\mathstrut +\mathstrut \) \(102705512323436338943\) \(\nu^{6}\mathstrut -\mathstrut \) \(593960656049984420265\) \(\nu^{5}\mathstrut -\mathstrut \) \(124933329741605122625\) \(\nu^{4}\mathstrut +\mathstrut \) \(361913163695762400867\) \(\nu^{3}\mathstrut +\mathstrut \) \(91928976025600392911\) \(\nu^{2}\mathstrut -\mathstrut \) \(49248032508852844662\) \(\nu\mathstrut -\mathstrut \) \(7905991124351893836\)\()/\)\(380020491990339796\)
\(\beta_{14}\)\(=\)\((\)\(7301100073797654\) \(\nu^{17}\mathstrut -\mathstrut \) \(60312989171745605\) \(\nu^{16}\mathstrut -\mathstrut \) \(61210972607689246\) \(\nu^{15}\mathstrut +\mathstrut \) \(1519509908720202702\) \(\nu^{14}\mathstrut -\mathstrut \) \(958291437652873612\) \(\nu^{13}\mathstrut -\mathstrut \) \(16571546309043870923\) \(\nu^{12}\mathstrut +\mathstrut \) \(17147424436807807259\) \(\nu^{11}\mathstrut +\mathstrut \) \(101670444666303232658\) \(\nu^{10}\mathstrut -\mathstrut \) \(107809612047667718174\) \(\nu^{9}\mathstrut -\mathstrut \) \(375452495747192594578\) \(\nu^{8}\mathstrut +\mathstrut \) \(339488827865507596387\) \(\nu^{7}\mathstrut +\mathstrut \) \(815553195202966665607\) \(\nu^{6}\mathstrut -\mathstrut \) \(548707315130201812033\) \(\nu^{5}\mathstrut -\mathstrut \) \(935817257230174272027\) \(\nu^{4}\mathstrut +\mathstrut \) \(414074569789506618317\) \(\nu^{3}\mathstrut +\mathstrut \) \(443241662178254792082\) \(\nu^{2}\mathstrut -\mathstrut \) \(97988324994247751836\) \(\nu\mathstrut -\mathstrut \) \(26559896255073611376\)\()/\)\(380020491990339796\)
\(\beta_{15}\)\(=\)\((\)\(9315252936800621\) \(\nu^{17}\mathstrut -\mathstrut \) \(75661731821851047\) \(\nu^{16}\mathstrut -\mathstrut \) \(74247095725639959\) \(\nu^{15}\mathstrut +\mathstrut \) \(1806791571566250226\) \(\nu^{14}\mathstrut -\mathstrut \) \(1047929132857691311\) \(\nu^{13}\mathstrut -\mathstrut \) \(18411368758700884454\) \(\nu^{12}\mathstrut +\mathstrut \) \(16460663200163554658\) \(\nu^{11}\mathstrut +\mathstrut \) \(104066684569316829853\) \(\nu^{10}\mathstrut -\mathstrut \) \(83372649103324939320\) \(\nu^{9}\mathstrut -\mathstrut \) \(346617696868159503227\) \(\nu^{8}\mathstrut +\mathstrut \) \(178112661977340319868\) \(\nu^{7}\mathstrut +\mathstrut \) \(644739305731437767939\) \(\nu^{6}\mathstrut -\mathstrut \) \(118928011200846828791\) \(\nu^{5}\mathstrut -\mathstrut \) \(548009895438680011751\) \(\nu^{4}\mathstrut -\mathstrut \) \(16857947279464779708\) \(\nu^{3}\mathstrut +\mathstrut \) \(127896352186272514209\) \(\nu^{2}\mathstrut -\mathstrut \) \(9808157146173717262\) \(\nu\mathstrut -\mathstrut \) \(3057609649456819704\)\()/\)\(380020491990339796\)
\(\beta_{16}\)\(=\)\((\)\(12859036804817362\) \(\nu^{17}\mathstrut -\mathstrut \) \(113419748259092336\) \(\nu^{16}\mathstrut -\mathstrut \) \(34209208371415515\) \(\nu^{15}\mathstrut +\mathstrut \) \(2604235231211404686\) \(\nu^{14}\mathstrut -\mathstrut \) \(3157705148777585220\) \(\nu^{13}\mathstrut -\mathstrut \) \(25318827620210899992\) \(\nu^{12}\mathstrut +\mathstrut \) \(41149242651925490525\) \(\nu^{11}\mathstrut +\mathstrut \) \(137012676033537426397\) \(\nu^{10}\mathstrut -\mathstrut \) \(225093759486933631810\) \(\nu^{9}\mathstrut -\mathstrut \) \(449800041483918874196\) \(\nu^{8}\mathstrut +\mathstrut \) \(628208179058093848846\) \(\nu^{7}\mathstrut +\mathstrut \) \(888915705750737641777\) \(\nu^{6}\mathstrut -\mathstrut \) \(886435847496065251703\) \(\nu^{5}\mathstrut -\mathstrut \) \(952221190096461101411\) \(\nu^{4}\mathstrut +\mathstrut \) \(561366881202777187469\) \(\nu^{3}\mathstrut +\mathstrut \) \(434180255920490206419\) \(\nu^{2}\mathstrut -\mathstrut \) \(105905159471142343622\) \(\nu\mathstrut -\mathstrut \) \(27863832008111642444\)\()/\)\(380020491990339796\)
\(\beta_{17}\)\(=\)\((\)\(13783223077520056\) \(\nu^{17}\mathstrut -\mathstrut \) \(117868852258168011\) \(\nu^{16}\mathstrut -\mathstrut \) \(74032072000453052\) \(\nu^{15}\mathstrut +\mathstrut \) \(2832777360936221178\) \(\nu^{14}\mathstrut -\mathstrut \) \(2724155509238984062\) \(\nu^{13}\mathstrut -\mathstrut \) \(28852686242929469855\) \(\nu^{12}\mathstrut +\mathstrut \) \(39650609343998259627\) \(\nu^{11}\mathstrut +\mathstrut \) \(162864555758759784136\) \(\nu^{10}\mathstrut -\mathstrut \) \(226870286780523757750\) \(\nu^{9}\mathstrut -\mathstrut \) \(551631582271446720736\) \(\nu^{8}\mathstrut +\mathstrut \) \(651022912769327052319\) \(\nu^{7}\mathstrut +\mathstrut \) \(1106361845260493698201\) \(\nu^{6}\mathstrut -\mathstrut \) \(939852238915508452323\) \(\nu^{5}\mathstrut -\mathstrut \) \(1179370602867702624565\) \(\nu^{4}\mathstrut +\mathstrut \) \(619553059887906583505\) \(\nu^{3}\mathstrut +\mathstrut \) \(521830997687310359572\) \(\nu^{2}\mathstrut -\mathstrut \) \(133602309476643986684\) \(\nu\mathstrut -\mathstrut \) \(29621775796267516520\)\()/\)\(380020491990339796\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(3\) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(31\)
\(\nu^{5}\)\(=\)\(-\)\(3\) \(\beta_{17}\mathstrut +\mathstrut \) \(4\) \(\beta_{16}\mathstrut -\mathstrut \) \(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(21\) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(16\) \(\beta_{12}\mathstrut +\mathstrut \) \(4\) \(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(20\) \(\beta_{8}\mathstrut +\mathstrut \) \(23\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(26\) \(\beta_{4}\mathstrut +\mathstrut \) \(20\) \(\beta_{3}\mathstrut +\mathstrut \) \(32\) \(\beta_{2}\mathstrut +\mathstrut \) \(87\) \(\beta_{1}\mathstrut +\mathstrut \) \(66\)
\(\nu^{6}\)\(=\)\(-\)\(21\) \(\beta_{17}\mathstrut +\mathstrut \) \(21\) \(\beta_{16}\mathstrut -\mathstrut \) \(7\) \(\beta_{15}\mathstrut -\mathstrut \) \(74\) \(\beta_{14}\mathstrut +\mathstrut \) \(21\) \(\beta_{13}\mathstrut -\mathstrut \) \(35\) \(\beta_{12}\mathstrut +\mathstrut \) \(24\) \(\beta_{11}\mathstrut -\mathstrut \) \(13\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(70\) \(\beta_{8}\mathstrut +\mathstrut \) \(94\) \(\beta_{7}\mathstrut +\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(6\) \(\beta_{5}\mathstrut -\mathstrut \) \(109\) \(\beta_{4}\mathstrut +\mathstrut \) \(62\) \(\beta_{3}\mathstrut +\mathstrut \) \(134\) \(\beta_{2}\mathstrut +\mathstrut \) \(256\) \(\beta_{1}\mathstrut +\mathstrut \) \(316\)
\(\nu^{7}\)\(=\)\(-\)\(77\) \(\beta_{17}\mathstrut +\mathstrut \) \(78\) \(\beta_{16}\mathstrut -\mathstrut \) \(49\) \(\beta_{15}\mathstrut -\mathstrut \) \(353\) \(\beta_{14}\mathstrut +\mathstrut \) \(70\) \(\beta_{13}\mathstrut -\mathstrut \) \(227\) \(\beta_{12}\mathstrut +\mathstrut \) \(96\) \(\beta_{11}\mathstrut -\mathstrut \) \(87\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(324\) \(\beta_{8}\mathstrut +\mathstrut \) \(409\) \(\beta_{7}\mathstrut +\mathstrut \) \(22\) \(\beta_{6}\mathstrut +\mathstrut \) \(35\) \(\beta_{5}\mathstrut -\mathstrut \) \(465\) \(\beta_{4}\mathstrut +\mathstrut \) \(329\) \(\beta_{3}\mathstrut +\mathstrut \) \(451\) \(\beta_{2}\mathstrut +\mathstrut \) \(1092\) \(\beta_{1}\mathstrut +\mathstrut \) \(916\)
\(\nu^{8}\)\(=\)\(-\)\(384\) \(\beta_{17}\mathstrut +\mathstrut \) \(320\) \(\beta_{16}\mathstrut -\mathstrut \) \(187\) \(\beta_{15}\mathstrut -\mathstrut \) \(1317\) \(\beta_{14}\mathstrut +\mathstrut \) \(408\) \(\beta_{13}\mathstrut -\mathstrut \) \(683\) \(\beta_{12}\mathstrut +\mathstrut \) \(437\) \(\beta_{11}\mathstrut -\mathstrut \) \(364\) \(\beta_{10}\mathstrut +\mathstrut \) \(88\) \(\beta_{9}\mathstrut +\mathstrut \) \(1198\) \(\beta_{8}\mathstrut +\mathstrut \) \(1637\) \(\beta_{7}\mathstrut +\mathstrut \) \(212\) \(\beta_{6}\mathstrut +\mathstrut \) \(111\) \(\beta_{5}\mathstrut -\mathstrut \) \(1853\) \(\beta_{4}\mathstrut +\mathstrut \) \(1168\) \(\beta_{3}\mathstrut +\mathstrut \) \(1744\) \(\beta_{2}\mathstrut +\mathstrut \) \(3705\) \(\beta_{1}\mathstrut +\mathstrut \) \(3725\)
\(\nu^{9}\)\(=\)\(-\)\(1523\) \(\beta_{17}\mathstrut +\mathstrut \) \(1165\) \(\beta_{16}\mathstrut -\mathstrut \) \(905\) \(\beta_{15}\mathstrut -\mathstrut \) \(5522\) \(\beta_{14}\mathstrut +\mathstrut \) \(1579\) \(\beta_{13}\mathstrut -\mathstrut \) \(3238\) \(\beta_{12}\mathstrut +\mathstrut \) \(1747\) \(\beta_{11}\mathstrut -\mathstrut \) \(1756\) \(\beta_{10}\mathstrut +\mathstrut \) \(429\) \(\beta_{9}\mathstrut +\mathstrut \) \(4942\) \(\beta_{8}\mathstrut +\mathstrut \) \(6572\) \(\beta_{7}\mathstrut +\mathstrut \) \(750\) \(\beta_{6}\mathstrut +\mathstrut \) \(465\) \(\beta_{5}\mathstrut -\mathstrut \) \(7438\) \(\beta_{4}\mathstrut +\mathstrut \) \(5091\) \(\beta_{3}\mathstrut +\mathstrut \) \(6225\) \(\beta_{2}\mathstrut +\mathstrut \) \(14703\) \(\beta_{1}\mathstrut +\mathstrut \) \(12357\)
\(\nu^{10}\)\(=\)\(-\)\(6744\) \(\beta_{17}\mathstrut +\mathstrut \) \(4428\) \(\beta_{16}\mathstrut -\mathstrut \) \(3514\) \(\beta_{15}\mathstrut -\mathstrut \) \(21002\) \(\beta_{14}\mathstrut +\mathstrut \) \(7417\) \(\beta_{13}\mathstrut -\mathstrut \) \(11166\) \(\beta_{12}\mathstrut +\mathstrut \) \(7309\) \(\beta_{11}\mathstrut -\mathstrut \) \(7154\) \(\beta_{10}\mathstrut +\mathstrut \) \(2642\) \(\beta_{9}\mathstrut +\mathstrut \) \(18701\) \(\beta_{8}\mathstrut +\mathstrut \) \(25757\) \(\beta_{7}\mathstrut +\mathstrut \) \(4252\) \(\beta_{6}\mathstrut +\mathstrut \) \(1557\) \(\beta_{5}\mathstrut -\mathstrut \) \(29081\) \(\beta_{4}\mathstrut +\mathstrut \) \(18997\) \(\beta_{3}\mathstrut +\mathstrut \) \(23480\) \(\beta_{2}\mathstrut +\mathstrut \) \(53119\) \(\beta_{1}\mathstrut +\mathstrut \) \(47338\)
\(\nu^{11}\)\(=\)\(-\)\(27344\) \(\beta_{17}\mathstrut +\mathstrut \) \(16049\) \(\beta_{16}\mathstrut -\mathstrut \) \(14889\) \(\beta_{15}\mathstrut -\mathstrut \) \(83662\) \(\beta_{14}\mathstrut +\mathstrut \) \(29711\) \(\beta_{13}\mathstrut -\mathstrut \) \(46630\) \(\beta_{12}\mathstrut +\mathstrut \) \(29110\) \(\beta_{11}\mathstrut -\mathstrut \) \(30672\) \(\beta_{10}\mathstrut +\mathstrut \) \(12123\) \(\beta_{9}\mathstrut +\mathstrut \) \(74002\) \(\beta_{8}\mathstrut +\mathstrut \) \(100353\) \(\beta_{7}\mathstrut +\mathstrut \) \(18024\) \(\beta_{6}\mathstrut +\mathstrut \) \(5973\) \(\beta_{5}\mathstrut -\mathstrut \) \(114034\) \(\beta_{4}\mathstrut +\mathstrut \) \(76933\) \(\beta_{3}\mathstrut +\mathstrut \) \(85932\) \(\beta_{2}\mathstrut +\mathstrut \) \(205099\) \(\beta_{1}\mathstrut +\mathstrut \) \(166841\)
\(\nu^{12}\)\(=\)\(-\)\(115062\) \(\beta_{17}\mathstrut +\mathstrut \) \(59365\) \(\beta_{16}\mathstrut -\mathstrut \) \(57635\) \(\beta_{15}\mathstrut -\mathstrut \) \(320138\) \(\beta_{14}\mathstrut +\mathstrut \) \(127306\) \(\beta_{13}\mathstrut -\mathstrut \) \(170294\) \(\beta_{12}\mathstrut +\mathstrut \) \(117748\) \(\beta_{11}\mathstrut -\mathstrut \) \(122547\) \(\beta_{10}\mathstrut +\mathstrut \) \(58242\) \(\beta_{9}\mathstrut +\mathstrut \) \(283099\) \(\beta_{8}\mathstrut +\mathstrut \) \(387839\) \(\beta_{7}\mathstrut +\mathstrut \) \(85964\) \(\beta_{6}\mathstrut +\mathstrut \) \(21263\) \(\beta_{5}\mathstrut -\mathstrut \) \(442386\) \(\beta_{4}\mathstrut +\mathstrut \) \(292728\) \(\beta_{3}\mathstrut +\mathstrut \) \(321754\) \(\beta_{2}\mathstrut +\mathstrut \) \(762399\) \(\beta_{1}\mathstrut +\mathstrut \) \(625793\)
\(\nu^{13}\)\(=\)\(-\)\(466947\) \(\beta_{17}\mathstrut +\mathstrut \) \(215469\) \(\beta_{16}\mathstrut -\mathstrut \) \(230415\) \(\beta_{15}\mathstrut -\mathstrut \) \(1248025\) \(\beta_{14}\mathstrut +\mathstrut \) \(510804\) \(\beta_{13}\mathstrut -\mathstrut \) \(674114\) \(\beta_{12}\mathstrut +\mathstrut \) \(466502\) \(\beta_{11}\mathstrut -\mathstrut \) \(498405\) \(\beta_{10}\mathstrut +\mathstrut \) \(254217\) \(\beta_{9}\mathstrut +\mathstrut \) \(1103493\) \(\beta_{8}\mathstrut +\mathstrut \) \(1491457\) \(\beta_{7}\mathstrut +\mathstrut \) \(371670\) \(\beta_{6}\mathstrut +\mathstrut \) \(81559\) \(\beta_{5}\mathstrut -\mathstrut \) \(1718307\) \(\beta_{4}\mathstrut +\mathstrut \) \(1151586\) \(\beta_{3}\mathstrut +\mathstrut \) \(1191042\) \(\beta_{2}\mathstrut +\mathstrut \) \(2916159\) \(\beta_{1}\mathstrut +\mathstrut \) \(2269038\)
\(\nu^{14}\)\(=\)\(-\)\(1915590\) \(\beta_{17}\mathstrut +\mathstrut \) \(790426\) \(\beta_{16}\mathstrut -\mathstrut \) \(885654\) \(\beta_{15}\mathstrut -\mathstrut \) \(4784387\) \(\beta_{14}\mathstrut +\mathstrut \) \(2095904\) \(\beta_{13}\mathstrut -\mathstrut \) \(2521552\) \(\beta_{12}\mathstrut +\mathstrut \) \(1856393\) \(\beta_{11}\mathstrut -\mathstrut \) \(1964242\) \(\beta_{10}\mathstrut +\mathstrut \) \(1116217\) \(\beta_{9}\mathstrut +\mathstrut \) \(4245461\) \(\beta_{8}\mathstrut +\mathstrut \) \(5717124\) \(\beta_{7}\mathstrut +\mathstrut \) \(1647138\) \(\beta_{6}\mathstrut +\mathstrut \) \(308757\) \(\beta_{5}\mathstrut -\mathstrut \) \(6645160\) \(\beta_{4}\mathstrut +\mathstrut \) \(4417641\) \(\beta_{3}\mathstrut +\mathstrut \) \(4452503\) \(\beta_{2}\mathstrut +\mathstrut \) \(10988820\) \(\beta_{1}\mathstrut +\mathstrut \) \(8457084\)
\(\nu^{15}\)\(=\)\(-\)\(7731717\) \(\beta_{17}\mathstrut +\mathstrut \) \(2880251\) \(\beta_{16}\mathstrut -\mathstrut \) \(3443726\) \(\beta_{15}\mathstrut -\mathstrut \) \(18480803\) \(\beta_{14}\mathstrut +\mathstrut \) \(8357266\) \(\beta_{13}\mathstrut -\mathstrut \) \(9757789\) \(\beta_{12}\mathstrut +\mathstrut \) \(7316432\) \(\beta_{11}\mathstrut -\mathstrut \) \(7783058\) \(\beta_{10}\mathstrut +\mathstrut \) \(4714637\) \(\beta_{9}\mathstrut +\mathstrut \) \(16465931\) \(\beta_{8}\mathstrut +\mathstrut \) \(21849172\) \(\beta_{7}\mathstrut +\mathstrut \) \(7017426\) \(\beta_{6}\mathstrut +\mathstrut \) \(1219232\) \(\beta_{5}\mathstrut -\mathstrut \) \(25714064\) \(\beta_{4}\mathstrut +\mathstrut \) \(17182717\) \(\beta_{3}\mathstrut +\mathstrut \) \(16574820\) \(\beta_{2}\mathstrut +\mathstrut \) \(41933601\) \(\beta_{1}\mathstrut +\mathstrut \) \(31101868\)
\(\nu^{16}\)\(=\)\(-\)\(31261528\) \(\beta_{17}\mathstrut +\mathstrut \) \(10562222\) \(\beta_{16}\mathstrut -\mathstrut \) \(13155222\) \(\beta_{15}\mathstrut -\mathstrut \) \(70893352\) \(\beta_{14}\mathstrut +\mathstrut \) \(33537363\) \(\beta_{13}\mathstrut -\mathstrut \) \(36872896\) \(\beta_{12}\mathstrut +\mathstrut \) \(28849339\) \(\beta_{11}\mathstrut -\mathstrut \) \(30390186\) \(\beta_{10}\mathstrut +\mathstrut \) \(19846429\) \(\beta_{9}\mathstrut +\mathstrut \) \(63553479\) \(\beta_{8}\mathstrut +\mathstrut \) \(83383131\) \(\beta_{7}\mathstrut +\mathstrut \) \(29919398\) \(\beta_{6}\mathstrut +\mathstrut \) \(4838767\) \(\beta_{5}\mathstrut -\mathstrut \) \(99335384\) \(\beta_{4}\mathstrut +\mathstrut \) \(66166029\) \(\beta_{3}\mathstrut +\mathstrut \) \(61977708\) \(\beta_{2}\mathstrut +\mathstrut \) \(159137086\) \(\beta_{1}\mathstrut +\mathstrut \) \(115834959\)
\(\nu^{17}\)\(=\)\(-\)\(125371317\) \(\beta_{17}\mathstrut +\mathstrut \) \(38679802\) \(\beta_{16}\mathstrut -\mathstrut \) \(50447645\) \(\beta_{15}\mathstrut -\mathstrut \) \(272821665\) \(\beta_{14}\mathstrut +\mathstrut \) \(132816430\) \(\beta_{13}\mathstrut -\mathstrut \) \(141360258\) \(\beta_{12}\mathstrut +\mathstrut \) \(113195450\) \(\beta_{11}\mathstrut -\mathstrut \) \(118783125\) \(\beta_{10}\mathstrut +\mathstrut \) \(82044009\) \(\beta_{9}\mathstrut +\mathstrut \) \(246184941\) \(\beta_{8}\mathstrut +\mathstrut \) \(317714685\) \(\beta_{7}\mathstrut +\mathstrut \) \(125173462\) \(\beta_{6}\mathstrut +\mathstrut \) \(19618396\) \(\beta_{5}\mathstrut -\mathstrut \) \(383884189\) \(\beta_{4}\mathstrut +\mathstrut \) \(256294219\) \(\beta_{3}\mathstrut +\mathstrut \) \(231428554\) \(\beta_{2}\mathstrut +\mathstrut \) \(607439823\) \(\beta_{1}\mathstrut +\mathstrut \) \(429304259\)

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
3.87400
3.69527
3.16635
2.83830
2.44623
2.26378
1.28813
1.27925
0.379802
0.365311
−0.164030
−0.947778
−1.45822
−1.57328
−1.86517
−2.05229
−2.08137
−2.45430
0 −2.87400 0 −0.461304 0 −2.48202 0 5.25986 0
1.2 0 −2.69527 0 −0.822149 0 4.27633 0 4.26450 0
1.3 0 −2.16635 0 3.18635 0 1.65768 0 1.69308 0
1.4 0 −1.83830 0 −1.17026 0 −0.774148 0 0.379331 0
1.5 0 −1.44623 0 3.72024 0 0.836543 0 −0.908408 0
1.6 0 −1.26378 0 −4.20424 0 4.08685 0 −1.40287 0
1.7 0 −0.288129 0 2.13990 0 1.97111 0 −2.91698 0
1.8 0 −0.279252 0 −0.263460 0 −1.45782 0 −2.92202 0
1.9 0 0.620198 0 −3.37733 0 −0.896557 0 −2.61535 0
1.10 0 0.634689 0 −0.976475 0 3.68579 0 −2.59717 0
1.11 0 1.16403 0 3.89012 0 −0.552876 0 −1.64503 0
1.12 0 1.94778 0 −1.72233 0 −3.84469 0 0.793839 0
1.13 0 2.45822 0 −0.323529 0 −2.56007 0 3.04282 0
1.14 0 2.57328 0 3.36856 0 0.305606 0 3.62177 0
1.15 0 2.86517 0 −3.00205 0 3.89738 0 5.20917 0
1.16 0 3.05229 0 −3.77561 0 −1.23812 0 6.31647 0
1.17 0 3.08137 0 3.75209 0 1.50295 0 6.49484 0
1.18 0 3.45430 0 2.04147 0 2.58609 0 8.93216 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(17\) \(-1\)
\(59\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8024))\):

\(T_{3}^{18} - \cdots\)
\(T_{5}^{18} - \cdots\)
\(T_{7}^{18} - \cdots\)