Properties

Label 8001.2.a.m
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 1
Dimension 11
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8001.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + \beta_{7} q^{5} \) \(- q^{7}\) \( + ( 1 + \beta_{2} + \beta_{3} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + \beta_{7} q^{5} \) \(- q^{7}\) \( + ( 1 + \beta_{2} + \beta_{3} ) q^{8} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{7} - \beta_{10} ) q^{10} \) \( + ( 1 - \beta_{7} - \beta_{9} ) q^{11} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} ) q^{13} \) \( -\beta_{1} q^{14} \) \( + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{16} \) \( + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{10} ) q^{17} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{19} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{20} \) \( + ( -1 + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{22} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{23} \) \( + ( \beta_{1} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{25} \) \( + ( -3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{26} \) \( + ( -1 - \beta_{2} ) q^{28} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} ) q^{29} \) \( + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{31} \) \( + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} ) q^{32} \) \( + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{34} \) \( -\beta_{7} q^{35} \) \( + ( -2 - \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{37} \) \( + ( -3 \beta_{1} - \beta_{3} + \beta_{6} - 5 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} ) q^{38} \) \( + ( -2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{40} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{41} \) \( + ( -3 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{43} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{7} - \beta_{9} + \beta_{10} ) q^{44} \) \( + ( -1 - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{46} \) \( + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{10} ) q^{47} \) \(+ q^{49}\) \( + ( 4 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{10} ) q^{50} \) \( + ( -6 - \beta_{1} - 3 \beta_{2} + \beta_{5} - 2 \beta_{8} - \beta_{10} ) q^{52} \) \( + ( 2 - 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{53} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{55} \) \( + ( -1 - \beta_{2} - \beta_{3} ) q^{56} \) \( + ( -2 - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{58} \) \( + ( -3 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{59} \) \( + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{61} \) \( + ( -3 + \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - \beta_{6} + 4 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{62} \) \( + ( -1 + 3 \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{64} \) \( + ( 3 + 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{65} \) \( + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{67} \) \( + ( 2 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{68} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{7} + \beta_{10} ) q^{70} \) \( + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{4} + 4 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{71} \) \( + ( -3 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{73} \) \( + ( -5 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} ) q^{74} \) \( + ( 3 - 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + \beta_{6} - 6 \beta_{7} - 5 \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{76} \) \( + ( -1 + \beta_{7} + \beta_{9} ) q^{77} \) \( + ( -1 - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{79} \) \( + ( -4 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 5 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} ) q^{80} \) \( + ( -6 \beta_{1} - \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - \beta_{9} + 4 \beta_{10} ) q^{82} \) \( + ( -2 - \beta_{1} + 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{83} \) \( + ( -3 - 6 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} ) q^{85} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} + 3 \beta_{7} + 4 \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{86} \) \( + ( \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} ) q^{88} \) \( + ( 2 \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{10} ) q^{89} \) \( + ( 2 - \beta_{1} + \beta_{2} + \beta_{6} - \beta_{7} ) q^{91} \) \( + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{8} - \beta_{10} ) q^{92} \) \( + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} + \beta_{8} + 3 \beta_{9} - 2 \beta_{10} ) q^{94} \) \( + ( -1 - 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} + 3 \beta_{7} + 4 \beta_{8} + 6 \beta_{9} - 2 \beta_{10} ) q^{95} \) \( + ( -4 - 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{97} \) \( + \beta_{1} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(11q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 15q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(11q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 15q^{8} \) \(\mathstrut -\mathstrut 12q^{10} \) \(\mathstrut +\mathstrut 7q^{11} \) \(\mathstrut -\mathstrut 24q^{13} \) \(\mathstrut -\mathstrut 2q^{14} \) \(\mathstrut -\mathstrut 6q^{16} \) \(\mathstrut +\mathstrut 15q^{17} \) \(\mathstrut -\mathstrut 19q^{19} \) \(\mathstrut -\mathstrut 3q^{20} \) \(\mathstrut -\mathstrut 3q^{22} \) \(\mathstrut +\mathstrut 11q^{23} \) \(\mathstrut +\mathstrut 10q^{25} \) \(\mathstrut -\mathstrut 10q^{26} \) \(\mathstrut -\mathstrut 12q^{28} \) \(\mathstrut +\mathstrut 10q^{29} \) \(\mathstrut -\mathstrut 20q^{31} \) \(\mathstrut +\mathstrut 27q^{32} \) \(\mathstrut -\mathstrut 9q^{34} \) \(\mathstrut +\mathstrut q^{35} \) \(\mathstrut -\mathstrut 22q^{37} \) \(\mathstrut -\mathstrut 8q^{38} \) \(\mathstrut -\mathstrut 29q^{40} \) \(\mathstrut -\mathstrut 9q^{41} \) \(\mathstrut -\mathstrut 17q^{43} \) \(\mathstrut +\mathstrut 9q^{44} \) \(\mathstrut -\mathstrut 18q^{46} \) \(\mathstrut +\mathstrut 7q^{47} \) \(\mathstrut +\mathstrut 11q^{49} \) \(\mathstrut +\mathstrut 47q^{50} \) \(\mathstrut -\mathstrut 66q^{52} \) \(\mathstrut +\mathstrut 28q^{53} \) \(\mathstrut -\mathstrut 24q^{55} \) \(\mathstrut -\mathstrut 15q^{56} \) \(\mathstrut -\mathstrut 39q^{58} \) \(\mathstrut -\mathstrut 35q^{59} \) \(\mathstrut -\mathstrut 6q^{61} \) \(\mathstrut -\mathstrut 18q^{62} \) \(\mathstrut +\mathstrut 11q^{64} \) \(\mathstrut +\mathstrut 43q^{65} \) \(\mathstrut -\mathstrut 22q^{67} \) \(\mathstrut +\mathstrut 12q^{68} \) \(\mathstrut +\mathstrut 12q^{70} \) \(\mathstrut +\mathstrut 22q^{71} \) \(\mathstrut -\mathstrut 29q^{73} \) \(\mathstrut -\mathstrut 14q^{74} \) \(\mathstrut +\mathstrut 10q^{76} \) \(\mathstrut -\mathstrut 7q^{77} \) \(\mathstrut -\mathstrut 20q^{79} \) \(\mathstrut -\mathstrut 66q^{80} \) \(\mathstrut -\mathstrut 24q^{82} \) \(\mathstrut -\mathstrut 17q^{83} \) \(\mathstrut -\mathstrut 50q^{85} \) \(\mathstrut +\mathstrut 12q^{86} \) \(\mathstrut +\mathstrut 2q^{88} \) \(\mathstrut +\mathstrut q^{89} \) \(\mathstrut +\mathstrut 24q^{91} \) \(\mathstrut +\mathstrut 22q^{92} \) \(\mathstrut +\mathstrut q^{94} \) \(\mathstrut -\mathstrut 10q^{95} \) \(\mathstrut -\mathstrut 45q^{97} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11}\mathstrut -\mathstrut \) \(2\) \(x^{10}\mathstrut -\mathstrut \) \(15\) \(x^{9}\mathstrut +\mathstrut \) \(25\) \(x^{8}\mathstrut +\mathstrut \) \(88\) \(x^{7}\mathstrut -\mathstrut \) \(112\) \(x^{6}\mathstrut -\mathstrut \) \(247\) \(x^{5}\mathstrut +\mathstrut \) \(215\) \(x^{4}\mathstrut +\mathstrut \) \(313\) \(x^{3}\mathstrut -\mathstrut \) \(170\) \(x^{2}\mathstrut -\mathstrut \) \(121\) \(x\mathstrut +\mathstrut \) \(57\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 2 \)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{10} + 6 \nu^{9} + 5 \nu^{8} - 66 \nu^{7} + 8 \nu^{6} + 255 \nu^{5} - 59 \nu^{4} - 399 \nu^{3} + 72 \nu^{2} + 190 \nu - 51 \)\()/7\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{10} - 5 \nu^{9} - 24 \nu^{8} + 55 \nu^{7} + 110 \nu^{6} - 209 \nu^{5} - 246 \nu^{4} + 308 \nu^{3} + 255 \nu^{2} - 114 \nu - 45 \)\()/7\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{10} - 11 \nu^{9} - 29 \nu^{8} + 128 \nu^{7} + 81 \nu^{6} - 520 \nu^{5} - 19 \nu^{4} + 840 \nu^{3} - 174 \nu^{2} - 402 \nu + 146 \)\()/7\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{10} - 12 \nu^{9} - 10 \nu^{8} + 139 \nu^{7} - 37 \nu^{6} - 559 \nu^{5} + 272 \nu^{4} + 882 \nu^{3} - 431 \nu^{2} - 401 \nu + 193 \)\()/7\)
\(\beta_{8}\)\(=\)\((\)\( -4 \nu^{10} + 17 \nu^{9} + 34 \nu^{8} - 194 \nu^{7} - 66 \nu^{6} + 761 \nu^{5} - 89 \nu^{4} - 1162 \nu^{3} + 330 \nu^{2} + 515 \nu - 204 \)\()/7\)
\(\beta_{9}\)\(=\)\((\)\( 8 \nu^{10} - 27 \nu^{9} - 82 \nu^{8} + 304 \nu^{7} + 279 \nu^{6} - 1172 \nu^{5} - 333 \nu^{4} + 1743 \nu^{3} - 9 \nu^{2} - 729 \nu + 191 \)\()/7\)
\(\beta_{10}\)\(=\)\((\)\( 10 \nu^{10} - 32 \nu^{9} - 99 \nu^{8} + 352 \nu^{7} + 298 \nu^{6} - 1325 \nu^{5} - 180 \nu^{4} + 1939 \nu^{3} - 377 \nu^{2} - 829 \nu + 321 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(13\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(4\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(21\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{6}\)\(=\)\(11\) \(\beta_{9}\mathstrut +\mathstrut \) \(23\) \(\beta_{8}\mathstrut +\mathstrut \) \(13\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(37\) \(\beta_{2}\mathstrut +\mathstrut \) \(23\) \(\beta_{1}\mathstrut +\mathstrut \) \(65\)
\(\nu^{7}\)\(=\)\(25\) \(\beta_{9}\mathstrut +\mathstrut \) \(53\) \(\beta_{8}\mathstrut +\mathstrut \) \(39\) \(\beta_{7}\mathstrut -\mathstrut \) \(10\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(71\) \(\beta_{3}\mathstrut +\mathstrut \) \(71\) \(\beta_{2}\mathstrut +\mathstrut \) \(127\) \(\beta_{1}\mathstrut +\mathstrut \) \(77\)
\(\nu^{8}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(94\) \(\beta_{9}\mathstrut +\mathstrut \) \(206\) \(\beta_{8}\mathstrut +\mathstrut \) \(127\) \(\beta_{7}\mathstrut -\mathstrut \) \(15\) \(\beta_{6}\mathstrut -\mathstrut \) \(65\) \(\beta_{5}\mathstrut +\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(140\) \(\beta_{3}\mathstrut +\mathstrut \) \(243\) \(\beta_{2}\mathstrut +\mathstrut \) \(206\) \(\beta_{1}\mathstrut +\mathstrut \) \(359\)
\(\nu^{9}\)\(=\)\(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(231\) \(\beta_{9}\mathstrut +\mathstrut \) \(513\) \(\beta_{8}\mathstrut +\mathstrut \) \(372\) \(\beta_{7}\mathstrut -\mathstrut \) \(79\) \(\beta_{6}\mathstrut -\mathstrut \) \(108\) \(\beta_{5}\mathstrut +\mathstrut \) \(107\) \(\beta_{4}\mathstrut +\mathstrut \) \(544\) \(\beta_{3}\mathstrut +\mathstrut \) \(539\) \(\beta_{2}\mathstrut +\mathstrut \) \(838\) \(\beta_{1}\mathstrut +\mathstrut \) \(561\)
\(\nu^{10}\)\(=\)\(17\) \(\beta_{10}\mathstrut +\mathstrut \) \(745\) \(\beta_{9}\mathstrut +\mathstrut \) \(1696\) \(\beta_{8}\mathstrut +\mathstrut \) \(1103\) \(\beta_{7}\mathstrut -\mathstrut \) \(152\) \(\beta_{6}\mathstrut -\mathstrut \) \(449\) \(\beta_{5}\mathstrut +\mathstrut \) \(191\) \(\beta_{4}\mathstrut +\mathstrut \) \(1227\) \(\beta_{3}\mathstrut +\mathstrut \) \(1673\) \(\beta_{2}\mathstrut +\mathstrut \) \(1691\) \(\beta_{1}\mathstrut +\mathstrut \) \(2148\)

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
−2.12999
−1.81808
−1.57007
−1.50746
−0.910036
0.455441
0.678644
1.68813
2.14674
2.22659
2.74009
−2.12999 0 2.53686 1.18187 0 −1.00000 −1.14352 0 −2.51737
1.2 −1.81808 0 1.30541 1.39765 0 −1.00000 1.26282 0 −2.54104
1.3 −1.57007 0 0.465107 2.53976 0 −1.00000 2.40988 0 −3.98759
1.4 −1.50746 0 0.272426 −0.476857 0 −1.00000 2.60424 0 0.718841
1.5 −0.910036 0 −1.17183 −2.84869 0 −1.00000 2.88648 0 2.59241
1.6 0.455441 0 −1.79257 0.749481 0 −1.00000 −1.72729 0 0.341344
1.7 0.678644 0 −1.53944 −2.84434 0 −1.00000 −2.40202 0 −1.93030
1.8 1.68813 0 0.849768 4.40419 0 −1.00000 −1.94174 0 7.43483
1.9 2.14674 0 2.60851 −3.76087 0 −1.00000 1.30632 0 −8.07362
1.10 2.22659 0 2.95768 0.700608 0 −1.00000 2.13237 0 1.55996
1.11 2.74009 0 5.50808 −2.04281 0 −1.00000 9.61245 0 −5.59747
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(127\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{11} - \cdots\)
\(T_{5}^{11} + \cdots\)