Properties

Label 8004.2.a.g
Level 8004
Weight 2
Character orbit 8004.a
Self dual Yes
Analytic conductor 63.912
Analytic rank 1
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8004.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( -\beta_{1} q^{5} \) \( + \beta_{3} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( -\beta_{1} q^{5} \) \( + \beta_{3} q^{7} \) \(+ q^{9}\) \( -\beta_{2} q^{11} \) \( + \beta_{8} q^{13} \) \( + \beta_{1} q^{15} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{17} \) \( -\beta_{4} q^{19} \) \( -\beta_{3} q^{21} \) \(+ q^{23}\) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{25} \) \(- q^{27}\) \(- q^{29}\) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} ) q^{31} \) \( + \beta_{2} q^{33} \) \( + ( -\beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} ) q^{35} \) \( + ( -3 + \beta_{2} - \beta_{5} - \beta_{8} + \beta_{11} ) q^{37} \) \( -\beta_{8} q^{39} \) \( + ( -1 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{41} \) \( + ( \beta_{2} - \beta_{3} - \beta_{9} + \beta_{11} ) q^{43} \) \( -\beta_{1} q^{45} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} ) q^{47} \) \( + ( -1 + 2 \beta_{1} + \beta_{4} + \beta_{6} + \beta_{11} ) q^{49} \) \( + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{51} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{11} ) q^{53} \) \( + ( 2 + \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{55} \) \( + \beta_{4} q^{57} \) \( + ( -1 - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{10} - 2 \beta_{11} ) q^{59} \) \( + ( -2 + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} ) q^{61} \) \( + \beta_{3} q^{63} \) \( + ( -1 + 2 \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{65} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{67} \) \(- q^{69}\) \( + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{7} + \beta_{11} ) q^{71} \) \( + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{7} + \beta_{10} ) q^{73} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{75} \) \( + ( -2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{10} - \beta_{11} ) q^{77} \) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{79} \) \(+ q^{81}\) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{83} \) \( + ( -1 + 2 \beta_{1} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{85} \) \(+ q^{87}\) \( + ( -3 + \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{10} + \beta_{11} ) q^{89} \) \( + ( 1 - 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{91} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{93} \) \( + ( -3 + \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{95} \) \( + ( 2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{97} \) \( -\beta_{2} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 12q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 12q^{9} \) \(\mathstrut -\mathstrut 5q^{11} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 3q^{15} \) \(\mathstrut -\mathstrut 7q^{17} \) \(\mathstrut -\mathstrut 3q^{19} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 12q^{23} \) \(\mathstrut +\mathstrut 11q^{25} \) \(\mathstrut -\mathstrut 12q^{27} \) \(\mathstrut -\mathstrut 12q^{29} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut +\mathstrut 5q^{33} \) \(\mathstrut -\mathstrut 9q^{35} \) \(\mathstrut -\mathstrut 20q^{37} \) \(\mathstrut +\mathstrut 6q^{39} \) \(\mathstrut -\mathstrut 3q^{41} \) \(\mathstrut +\mathstrut 5q^{43} \) \(\mathstrut -\mathstrut 3q^{45} \) \(\mathstrut -\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 7q^{51} \) \(\mathstrut -\mathstrut 3q^{53} \) \(\mathstrut +\mathstrut 19q^{55} \) \(\mathstrut +\mathstrut 3q^{57} \) \(\mathstrut -\mathstrut 20q^{59} \) \(\mathstrut -\mathstrut 17q^{61} \) \(\mathstrut +\mathstrut 4q^{63} \) \(\mathstrut -\mathstrut 4q^{65} \) \(\mathstrut -\mathstrut 9q^{67} \) \(\mathstrut -\mathstrut 12q^{69} \) \(\mathstrut +\mathstrut 7q^{71} \) \(\mathstrut -\mathstrut 9q^{73} \) \(\mathstrut -\mathstrut 11q^{75} \) \(\mathstrut -\mathstrut 34q^{77} \) \(\mathstrut +\mathstrut 14q^{79} \) \(\mathstrut +\mathstrut 12q^{81} \) \(\mathstrut +\mathstrut 5q^{83} \) \(\mathstrut -\mathstrut 12q^{85} \) \(\mathstrut +\mathstrut 12q^{87} \) \(\mathstrut -\mathstrut 22q^{89} \) \(\mathstrut -\mathstrut 3q^{91} \) \(\mathstrut -\mathstrut 2q^{93} \) \(\mathstrut -\mathstrut 27q^{95} \) \(\mathstrut +\mathstrut 17q^{97} \) \(\mathstrut -\mathstrut 5q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(3\) \(x^{11}\mathstrut -\mathstrut \) \(31\) \(x^{10}\mathstrut +\mathstrut \) \(80\) \(x^{9}\mathstrut +\mathstrut \) \(347\) \(x^{8}\mathstrut -\mathstrut \) \(697\) \(x^{7}\mathstrut -\mathstrut \) \(1714\) \(x^{6}\mathstrut +\mathstrut \) \(2146\) \(x^{5}\mathstrut +\mathstrut \) \(3304\) \(x^{4}\mathstrut -\mathstrut \) \(1156\) \(x^{3}\mathstrut -\mathstrut \) \(616\) \(x^{2}\mathstrut +\mathstrut \) \(136\) \(x\mathstrut +\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(8071\) \(\nu^{11}\mathstrut +\mathstrut \) \(19357\) \(\nu^{10}\mathstrut +\mathstrut \) \(231201\) \(\nu^{9}\mathstrut -\mathstrut \) \(479516\) \(\nu^{8}\mathstrut -\mathstrut \) \(2165385\) \(\nu^{7}\mathstrut +\mathstrut \) \(3873043\) \(\nu^{6}\mathstrut +\mathstrut \) \(7143630\) \(\nu^{5}\mathstrut -\mathstrut \) \(11289370\) \(\nu^{4}\mathstrut -\mathstrut \) \(2416332\) \(\nu^{3}\mathstrut +\mathstrut \) \(6816868\) \(\nu^{2}\mathstrut -\mathstrut \) \(11305176\) \(\nu\mathstrut +\mathstrut \) \(770552\)\()/1181592\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(58412\) \(\nu^{11}\mathstrut +\mathstrut \) \(150230\) \(\nu^{10}\mathstrut +\mathstrut \) \(1862577\) \(\nu^{9}\mathstrut -\mathstrut \) \(3812431\) \(\nu^{8}\mathstrut -\mathstrut \) \(21650127\) \(\nu^{7}\mathstrut +\mathstrut \) \(29877380\) \(\nu^{6}\mathstrut +\mathstrut \) \(111631233\) \(\nu^{5}\mathstrut -\mathstrut \) \(64812155\) \(\nu^{4}\mathstrut -\mathstrut \) \(218775012\) \(\nu^{3}\mathstrut -\mathstrut \) \(63772828\) \(\nu^{2}\mathstrut +\mathstrut \) \(2829120\) \(\nu\mathstrut +\mathstrut \) \(11033668\)\()/4135572\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(237136\) \(\nu^{11}\mathstrut +\mathstrut \) \(958345\) \(\nu^{10}\mathstrut +\mathstrut \) \(6749571\) \(\nu^{9}\mathstrut -\mathstrut \) \(26540003\) \(\nu^{8}\mathstrut -\mathstrut \) \(67159668\) \(\nu^{7}\mathstrut +\mathstrut \) \(247389829\) \(\nu^{6}\mathstrut +\mathstrut \) \(284027205\) \(\nu^{5}\mathstrut -\mathstrut \) \(890169226\) \(\nu^{4}\mathstrut -\mathstrut \) \(447721968\) \(\nu^{3}\mathstrut +\mathstrut \) \(927980284\) \(\nu^{2}\mathstrut +\mathstrut \) \(65617524\) \(\nu\mathstrut -\mathstrut \) \(69925000\)\()/16542288\)
\(\beta_{5}\)\(=\)\((\)\(643238\) \(\nu^{11}\mathstrut -\mathstrut \) \(1717169\) \(\nu^{10}\mathstrut -\mathstrut \) \(20607159\) \(\nu^{9}\mathstrut +\mathstrut \) \(44452597\) \(\nu^{8}\mathstrut +\mathstrut \) \(241331934\) \(\nu^{7}\mathstrut -\mathstrut \) \(362660321\) \(\nu^{6}\mathstrut -\mathstrut \) \(1259064135\) \(\nu^{5}\mathstrut +\mathstrut \) \(913186418\) \(\nu^{4}\mathstrut +\mathstrut \) \(2549822760\) \(\nu^{3}\mathstrut +\mathstrut \) \(226067380\) \(\nu^{2}\mathstrut -\mathstrut \) \(342835836\) \(\nu\mathstrut -\mathstrut \) \(32135128\)\()/16542288\)
\(\beta_{6}\)\(=\)\((\)\(227217\) \(\nu^{11}\mathstrut -\mathstrut \) \(656309\) \(\nu^{10}\mathstrut -\mathstrut \) \(7166323\) \(\nu^{9}\mathstrut +\mathstrut \) \(17182508\) \(\nu^{8}\mathstrut +\mathstrut \) \(82610957\) \(\nu^{7}\mathstrut -\mathstrut \) \(143477867\) \(\nu^{6}\mathstrut -\mathstrut \) \(426165362\) \(\nu^{5}\mathstrut +\mathstrut \) \(389223760\) \(\nu^{4}\mathstrut +\mathstrut \) \(866406388\) \(\nu^{3}\mathstrut -\mathstrut \) \(18498844\) \(\nu^{2}\mathstrut -\mathstrut \) \(162478008\) \(\nu\mathstrut -\mathstrut \) \(13046896\)\()/5514096\)
\(\beta_{7}\)\(=\)\((\)\(756551\) \(\nu^{11}\mathstrut -\mathstrut \) \(1925114\) \(\nu^{10}\mathstrut -\mathstrut \) \(24168090\) \(\nu^{9}\mathstrut +\mathstrut \) \(49854049\) \(\nu^{8}\mathstrut +\mathstrut \) \(280180599\) \(\nu^{7}\mathstrut -\mathstrut \) \(411626960\) \(\nu^{6}\mathstrut -\mathstrut \) \(1435974561\) \(\nu^{5}\mathstrut +\mathstrut \) \(1092840230\) \(\nu^{4}\mathstrut +\mathstrut \) \(2853762180\) \(\nu^{3}\mathstrut +\mathstrut \) \(39670792\) \(\nu^{2}\mathstrut -\mathstrut \) \(445104660\) \(\nu\mathstrut -\mathstrut \) \(37571512\)\()/16542288\)
\(\beta_{8}\)\(=\)\((\)\(324494\) \(\nu^{11}\mathstrut -\mathstrut \) \(886849\) \(\nu^{10}\mathstrut -\mathstrut \) \(10152007\) \(\nu^{9}\mathstrut +\mathstrut \) \(23105733\) \(\nu^{8}\mathstrut +\mathstrut \) \(114673742\) \(\nu^{7}\mathstrut -\mathstrut \) \(193749385\) \(\nu^{6}\mathstrut -\mathstrut \) \(570735599\) \(\nu^{5}\mathstrut +\mathstrut \) \(547270074\) \(\nu^{4}\mathstrut +\mathstrut \) \(1101125008\) \(\nu^{3}\mathstrut -\mathstrut \) \(145530924\) \(\nu^{2}\mathstrut -\mathstrut \) \(173254620\) \(\nu\mathstrut +\mathstrut \) \(21253000\)\()/5514096\)
\(\beta_{9}\)\(=\)\((\)\(1038679\) \(\nu^{11}\mathstrut -\mathstrut \) \(2548933\) \(\nu^{10}\mathstrut -\mathstrut \) \(33488751\) \(\nu^{9}\mathstrut +\mathstrut \) \(66191846\) \(\nu^{8}\mathstrut +\mathstrut \) \(390702759\) \(\nu^{7}\mathstrut -\mathstrut \) \(547193839\) \(\nu^{6}\mathstrut -\mathstrut \) \(2000252604\) \(\nu^{5}\mathstrut +\mathstrut \) \(1436390248\) \(\nu^{4}\mathstrut +\mathstrut \) \(3942455436\) \(\nu^{3}\mathstrut +\mathstrut \) \(142017764\) \(\nu^{2}\mathstrut -\mathstrut \) \(695410800\) \(\nu\mathstrut -\mathstrut \) \(33296528\)\()/16542288\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(545408\) \(\nu^{11}\mathstrut +\mathstrut \) \(1592693\) \(\nu^{10}\mathstrut +\mathstrut \) \(17173041\) \(\nu^{9}\mathstrut -\mathstrut \) \(42407701\) \(\nu^{8}\mathstrut -\mathstrut \) \(196470678\) \(\nu^{7}\mathstrut +\mathstrut \) \(366059189\) \(\nu^{6}\mathstrut +\mathstrut \) \(997238067\) \(\nu^{5}\mathstrut -\mathstrut \) \(1082368964\) \(\nu^{4}\mathstrut -\mathstrut \) \(1977984000\) \(\nu^{3}\mathstrut +\mathstrut \) \(385348124\) \(\nu^{2}\mathstrut +\mathstrut \) \(341574492\) \(\nu\mathstrut -\mathstrut \) \(35157824\)\()/8271144\)
\(\beta_{11}\)\(=\)\((\)\(1186027\) \(\nu^{11}\mathstrut -\mathstrut \) \(3327328\) \(\nu^{10}\mathstrut -\mathstrut \) \(37462464\) \(\nu^{9}\mathstrut +\mathstrut \) \(87445547\) \(\nu^{8}\mathstrut +\mathstrut \) \(429697791\) \(\nu^{7}\mathstrut -\mathstrut \) \(737802358\) \(\nu^{6}\mathstrut -\mathstrut \) \(2179409127\) \(\nu^{5}\mathstrut +\mathstrut \) \(2066374630\) \(\nu^{4}\mathstrut +\mathstrut \) \(4256483244\) \(\nu^{3}\mathstrut -\mathstrut \) \(366651712\) \(\nu^{2}\mathstrut -\mathstrut \) \(490498764\) \(\nu\mathstrut +\mathstrut \) \(36924904\)\()/16542288\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\(-\)\(16\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(11\) \(\beta_{8}\mathstrut +\mathstrut \) \(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(4\) \(\beta_{6}\mathstrut +\mathstrut \) \(32\) \(\beta_{5}\mathstrut -\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut -\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(68\)
\(\nu^{5}\)\(=\)\(-\)\(23\) \(\beta_{11}\mathstrut +\mathstrut \) \(22\) \(\beta_{10}\mathstrut +\mathstrut \) \(20\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(32\) \(\beta_{5}\mathstrut -\mathstrut \) \(49\) \(\beta_{4}\mathstrut -\mathstrut \) \(25\) \(\beta_{3}\mathstrut +\mathstrut \) \(22\) \(\beta_{2}\mathstrut +\mathstrut \) \(115\) \(\beta_{1}\mathstrut +\mathstrut \) \(88\)
\(\nu^{6}\)\(=\)\(-\)\(264\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(32\) \(\beta_{9}\mathstrut -\mathstrut \) \(133\) \(\beta_{8}\mathstrut +\mathstrut \) \(131\) \(\beta_{7}\mathstrut -\mathstrut \) \(97\) \(\beta_{6}\mathstrut +\mathstrut \) \(504\) \(\beta_{5}\mathstrut -\mathstrut \) \(266\) \(\beta_{4}\mathstrut +\mathstrut \) \(102\) \(\beta_{3}\mathstrut -\mathstrut \) \(124\) \(\beta_{2}\mathstrut +\mathstrut \) \(208\) \(\beta_{1}\mathstrut +\mathstrut \) \(902\)
\(\nu^{7}\)\(=\)\(-\)\(502\) \(\beta_{11}\mathstrut +\mathstrut \) \(362\) \(\beta_{10}\mathstrut +\mathstrut \) \(340\) \(\beta_{9}\mathstrut -\mathstrut \) \(58\) \(\beta_{8}\mathstrut +\mathstrut \) \(120\) \(\beta_{7}\mathstrut -\mathstrut \) \(264\) \(\beta_{6}\mathstrut +\mathstrut \) \(760\) \(\beta_{5}\mathstrut -\mathstrut \) \(952\) \(\beta_{4}\mathstrut -\mathstrut \) \(471\) \(\beta_{3}\mathstrut +\mathstrut \) \(371\) \(\beta_{2}\mathstrut +\mathstrut \) \(1485\) \(\beta_{1}\mathstrut +\mathstrut \) \(1682\)
\(\nu^{8}\)\(=\)\(-\)\(4357\) \(\beta_{11}\mathstrut +\mathstrut \) \(94\) \(\beta_{10}\mathstrut +\mathstrut \) \(724\) \(\beta_{9}\mathstrut -\mathstrut \) \(1744\) \(\beta_{8}\mathstrut +\mathstrut \) \(1742\) \(\beta_{7}\mathstrut -\mathstrut \) \(1889\) \(\beta_{6}\mathstrut +\mathstrut \) \(7975\) \(\beta_{5}\mathstrut -\mathstrut \) \(4518\) \(\beta_{4}\mathstrut +\mathstrut \) \(733\) \(\beta_{3}\mathstrut -\mathstrut \) \(1377\) \(\beta_{2}\mathstrut +\mathstrut \) \(3303\) \(\beta_{1}\mathstrut +\mathstrut \) \(13128\)
\(\nu^{9}\)\(=\)\(-\)\(10242\) \(\beta_{11}\mathstrut +\mathstrut \) \(5512\) \(\beta_{10}\mathstrut +\mathstrut \) \(5601\) \(\beta_{9}\mathstrut -\mathstrut \) \(1365\) \(\beta_{8}\mathstrut +\mathstrut \) \(2655\) \(\beta_{7}\mathstrut -\mathstrut \) \(5768\) \(\beta_{6}\mathstrut +\mathstrut \) \(15925\) \(\beta_{5}\mathstrut -\mathstrut \) \(17273\) \(\beta_{4}\mathstrut -\mathstrut \) \(8108\) \(\beta_{3}\mathstrut +\mathstrut \) \(5663\) \(\beta_{2}\mathstrut +\mathstrut \) \(21017\) \(\beta_{1}\mathstrut +\mathstrut \) \(31070\)
\(\nu^{10}\)\(=\)\(-\)\(72074\) \(\beta_{11}\mathstrut +\mathstrut \) \(3343\) \(\beta_{10}\mathstrut +\mathstrut \) \(14471\) \(\beta_{9}\mathstrut -\mathstrut \) \(24328\) \(\beta_{8}\mathstrut +\mathstrut \) \(25276\) \(\beta_{7}\mathstrut -\mathstrut \) \(34169\) \(\beta_{6}\mathstrut +\mathstrut \) \(127734\) \(\beta_{5}\mathstrut -\mathstrut \) \(77380\) \(\beta_{4}\mathstrut +\mathstrut \) \(960\) \(\beta_{3}\mathstrut -\mathstrut \) \(14696\) \(\beta_{2}\mathstrut +\mathstrut \) \(55027\) \(\beta_{1}\mathstrut +\mathstrut \) \(202616\)
\(\nu^{11}\)\(=\)\(-\)\(197916\) \(\beta_{11}\mathstrut +\mathstrut \) \(82860\) \(\beta_{10}\mathstrut +\mathstrut \) \(92278\) \(\beta_{9}\mathstrut -\mathstrut \) \(29324\) \(\beta_{8}\mathstrut +\mathstrut \) \(52846\) \(\beta_{7}\mathstrut -\mathstrut \) \(113038\) \(\beta_{6}\mathstrut +\mathstrut \) \(311629\) \(\beta_{5}\mathstrut -\mathstrut \) \(305427\) \(\beta_{4}\mathstrut -\mathstrut \) \(134565\) \(\beta_{3}\mathstrut +\mathstrut \) \(83314\) \(\beta_{2}\mathstrut +\mathstrut \) \(317839\) \(\beta_{1}\mathstrut +\mathstrut \) \(564322\)

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
4.15588
3.12492
3.06377
2.52462
0.425546
0.305886
−0.0891245
−0.413592
−1.42583
−2.48147
−2.64977
−3.54084
0 −1.00000 0 −4.15588 0 −1.11441 0 1.00000 0
1.2 0 −1.00000 0 −3.12492 0 4.33538 0 1.00000 0
1.3 0 −1.00000 0 −3.06377 0 3.59303 0 1.00000 0
1.4 0 −1.00000 0 −2.52462 0 −2.76852 0 1.00000 0
1.5 0 −1.00000 0 −0.425546 0 −4.01837 0 1.00000 0
1.6 0 −1.00000 0 −0.305886 0 −0.140026 0 1.00000 0
1.7 0 −1.00000 0 0.0891245 0 2.52085 0 1.00000 0
1.8 0 −1.00000 0 0.413592 0 2.75070 0 1.00000 0
1.9 0 −1.00000 0 1.42583 0 −1.49015 0 1.00000 0
1.10 0 −1.00000 0 2.48147 0 −0.0448560 0 1.00000 0
1.11 0 −1.00000 0 2.64977 0 −1.93393 0 1.00000 0
1.12 0 −1.00000 0 3.54084 0 2.31030 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
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(-1\)
\(29\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

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