Properties

Label 8001.2.a.n
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 0
Dimension 12
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: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 - \beta_{1} ) q^{2} \) \( + ( 1 - \beta_{1} + \beta_{6} + \beta_{7} ) q^{4} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{5} \) \(+ q^{7}\) \( + ( 1 - \beta_{1} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + ( 1 - \beta_{1} ) q^{2} \) \( + ( 1 - \beta_{1} + \beta_{6} + \beta_{7} ) q^{4} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{5} \) \(+ q^{7}\) \( + ( 1 - \beta_{1} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{8} \) \( + ( -1 + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{10} \) \( + ( 3 - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{11} \) \( + ( -1 + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{13} \) \( + ( 1 - \beta_{1} ) q^{14} \) \( + ( -3 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{16} \) \( + ( 1 + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{17} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{19} \) \( + ( -\beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{20} \) \( + ( 3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{10} ) q^{22} \) \( + ( 2 - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{11} ) q^{23} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{25} \) \( + ( -1 + \beta_{2} + \beta_{4} + 2 \beta_{7} + 3 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{26} \) \( + ( 1 - \beta_{1} + \beta_{6} + \beta_{7} ) q^{28} \) \( + ( 2 + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{29} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + 3 \beta_{6} - 2 \beta_{8} + \beta_{10} ) q^{31} \) \( + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{32} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{34} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{35} \) \( + ( 1 - 2 \beta_{1} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{37} \) \( + ( -2 + 5 \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + 4 \beta_{11} ) q^{38} \) \( + ( 2 + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} - 2 \beta_{10} ) q^{40} \) \( + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} - 4 \beta_{8} - \beta_{9} + \beta_{11} ) q^{41} \) \( + ( 2 + \beta_{4} - \beta_{6} + 3 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{43} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{44} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} + 3 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{46} \) \( + ( 2 - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{47} \) \(+ q^{49}\) \( + ( 1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{50} \) \( + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} ) q^{52} \) \( + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{53} \) \( + ( -4 + 4 \beta_{1} + 3 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 5 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{55} \) \( + ( 1 - \beta_{1} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{56} \) \( + ( 3 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{58} \) \( + ( 1 + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{11} ) q^{59} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{61} \) \( + ( 2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{8} - 3 \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{62} \) \( + ( 2 - 4 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} + \beta_{5} + \beta_{6} + 4 \beta_{7} + \beta_{8} - \beta_{9} + 4 \beta_{10} - 3 \beta_{11} ) q^{64} \) \( + ( 3 - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{65} \) \( + ( 1 - \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{67} \) \( + ( 1 + \beta_{1} - 2 \beta_{2} - 3 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{68} \) \( + ( -1 + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{70} \) \( + ( 7 + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 5 \beta_{8} - 3 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{71} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{9} + \beta_{11} ) q^{73} \) \( + ( 4 - 5 \beta_{1} + 2 \beta_{3} - \beta_{4} - 4 \beta_{5} + 2 \beta_{7} + \beta_{8} + 7 \beta_{10} - 4 \beta_{11} ) q^{74} \) \( + ( -4 + 8 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 5 \beta_{6} - 4 \beta_{7} - \beta_{8} + \beta_{9} - 5 \beta_{10} + 5 \beta_{11} ) q^{76} \) \( + ( 3 - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{77} \) \( + ( 2 - \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{79} \) \( + ( -4 \beta_{1} + \beta_{2} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{80} \) \( + ( 5 - \beta_{1} + 4 \beta_{2} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + \beta_{8} - 2 \beta_{9} - 3 \beta_{10} ) q^{82} \) \( + ( 2 - 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{83} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} ) q^{85} \) \( + ( 2 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{86} \) \( + ( -1 + 2 \beta_{1} + \beta_{3} + 7 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} + 4 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{88} \) \( + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - 5 \beta_{8} + 6 \beta_{9} + \beta_{10} + 4 \beta_{11} ) q^{89} \) \( + ( -1 + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{91} \) \( + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} + 6 \beta_{7} - 3 \beta_{8} - \beta_{10} ) q^{92} \) \( + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{94} \) \( + ( 5 + \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} + 4 \beta_{11} ) q^{95} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - 6 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} ) q^{97} \) \( + ( 1 - \beta_{1} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 9q^{4} \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 15q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 9q^{4} \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 15q^{8} \) \(\mathstrut -\mathstrut 2q^{10} \) \(\mathstrut +\mathstrut 22q^{11} \) \(\mathstrut +\mathstrut 7q^{14} \) \(\mathstrut +\mathstrut 7q^{16} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 7q^{19} \) \(\mathstrut +\mathstrut 8q^{20} \) \(\mathstrut +\mathstrut 13q^{22} \) \(\mathstrut +\mathstrut 29q^{23} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 9q^{28} \) \(\mathstrut +\mathstrut 22q^{29} \) \(\mathstrut -\mathstrut 16q^{31} \) \(\mathstrut +\mathstrut 27q^{32} \) \(\mathstrut -\mathstrut 5q^{34} \) \(\mathstrut +\mathstrut 7q^{35} \) \(\mathstrut -\mathstrut 4q^{37} \) \(\mathstrut -\mathstrut 2q^{38} \) \(\mathstrut +\mathstrut 16q^{40} \) \(\mathstrut +\mathstrut 21q^{41} \) \(\mathstrut +\mathstrut 11q^{43} \) \(\mathstrut +\mathstrut 11q^{44} \) \(\mathstrut +\mathstrut 31q^{47} \) \(\mathstrut +\mathstrut 12q^{49} \) \(\mathstrut +\mathstrut 21q^{50} \) \(\mathstrut +\mathstrut 3q^{52} \) \(\mathstrut +\mathstrut 38q^{53} \) \(\mathstrut -\mathstrut 11q^{55} \) \(\mathstrut +\mathstrut 15q^{56} \) \(\mathstrut +\mathstrut 20q^{58} \) \(\mathstrut +\mathstrut 15q^{59} \) \(\mathstrut -\mathstrut 3q^{61} \) \(\mathstrut +\mathstrut 4q^{62} \) \(\mathstrut +\mathstrut 29q^{64} \) \(\mathstrut +\mathstrut 32q^{65} \) \(\mathstrut -\mathstrut q^{67} \) \(\mathstrut -\mathstrut 17q^{68} \) \(\mathstrut -\mathstrut 2q^{70} \) \(\mathstrut +\mathstrut 57q^{71} \) \(\mathstrut -\mathstrut 7q^{73} \) \(\mathstrut +\mathstrut 42q^{74} \) \(\mathstrut -\mathstrut 44q^{76} \) \(\mathstrut +\mathstrut 22q^{77} \) \(\mathstrut -\mathstrut 18q^{79} \) \(\mathstrut -\mathstrut q^{80} \) \(\mathstrut +\mathstrut 56q^{82} \) \(\mathstrut +\mathstrut 21q^{83} \) \(\mathstrut -\mathstrut 5q^{85} \) \(\mathstrut +\mathstrut 32q^{86} \) \(\mathstrut -\mathstrut 10q^{88} \) \(\mathstrut -\mathstrut 6q^{89} \) \(\mathstrut +\mathstrut 15q^{92} \) \(\mathstrut +\mathstrut 35q^{94} \) \(\mathstrut +\mathstrut 57q^{95} \) \(\mathstrut +\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut 7q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(5\) \(x^{11}\mathstrut -\mathstrut \) \(3\) \(x^{10}\mathstrut +\mathstrut \) \(41\) \(x^{9}\mathstrut -\mathstrut \) \(11\) \(x^{8}\mathstrut -\mathstrut \) \(123\) \(x^{7}\mathstrut +\mathstrut \) \(44\) \(x^{6}\mathstrut +\mathstrut \) \(159\) \(x^{5}\mathstrut -\mathstrut \) \(39\) \(x^{4}\mathstrut -\mathstrut \) \(71\) \(x^{3}\mathstrut +\mathstrut \) \(16\) \(x^{2}\mathstrut +\mathstrut \) \(7\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{11} + 9 \nu^{10} - 78 \nu^{9} + 21 \nu^{8} + 551 \nu^{7} - 315 \nu^{6} - 1552 \nu^{5} + 474 \nu^{4} + 1974 \nu^{3} + 95 \nu^{2} - 664 \nu + 24 \)\()/67\)
\(\beta_{3}\)\(=\)\((\)\( -19 \nu^{11} + 97 \nu^{10} + 75 \nu^{9} - 868 \nu^{8} + 50 \nu^{7} + 2903 \nu^{6} - 193 \nu^{5} - 3981 \nu^{4} - 388 \nu^{3} + 1210 \nu^{2} - 114 \nu + 13 \)\()/67\)
\(\beta_{4}\)\(=\)\((\)\( -28 \nu^{11} + 150 \nu^{10} + 40 \nu^{9} - 1191 \nu^{8} + 652 \nu^{7} + 3393 \nu^{6} - 2037 \nu^{5} - 4093 \nu^{4} + 1544 \nu^{3} + 1628 \nu^{2} - 302 \nu - 69 \)\()/67\)
\(\beta_{5}\)\(=\)\((\)\( -37 \nu^{11} + 203 \nu^{10} + 5 \nu^{9} - 1447 \nu^{8} + 986 \nu^{7} + 3615 \nu^{6} - 2340 \nu^{5} - 3669 \nu^{4} + 997 \nu^{3} + 840 \nu^{2} + 180 \nu + 117 \)\()/67\)
\(\beta_{6}\)\(=\)\((\)\( -17 \nu^{11} + 48 \nu^{10} + 254 \nu^{9} - 692 \nu^{8} - 1260 \nu^{7} + 3077 \nu^{6} + 2867 \nu^{5} - 5043 \nu^{4} - 3006 \nu^{3} + 2271 \nu^{2} + 568 \nu - 207 \)\()/67\)
\(\beta_{7}\)\(=\)\((\)\( 17 \nu^{11} - 48 \nu^{10} - 254 \nu^{9} + 692 \nu^{8} + 1260 \nu^{7} - 3077 \nu^{6} - 2867 \nu^{5} + 5043 \nu^{4} + 3006 \nu^{3} - 2204 \nu^{2} - 635 \nu + 73 \)\()/67\)
\(\beta_{8}\)\(=\)\((\)\( -36 \nu^{11} + 145 \nu^{10} + 262 \nu^{9} - 1292 \nu^{8} - 875 \nu^{7} + 4104 \nu^{6} + 2205 \nu^{5} - 5071 \nu^{4} - 3193 \nu^{3} + 1337 \nu^{2} + 856 \nu - 60 \)\()/67\)
\(\beta_{9}\)\(=\)\((\)\( 40 \nu^{11} - 176 \nu^{10} - 239 \nu^{9} + 1577 \nu^{8} + 466 \nu^{7} - 5163 \nu^{6} - 641 \nu^{5} + 7034 \nu^{4} + 905 \nu^{3} - 2967 \nu^{2} + 39 \nu + 223 \)\()/67\)
\(\beta_{10}\)\(=\)\((\)\( -50 \nu^{11} + 220 \nu^{10} + 282 \nu^{9} - 1921 \nu^{8} - 415 \nu^{7} + 5968 \nu^{6} + 282 \nu^{5} - 7419 \nu^{4} - 679 \nu^{3} + 2553 \nu^{2} + 102 \nu - 61 \)\()/67\)
\(\beta_{11}\)\(=\)\((\)\( -59 \nu^{11} + 273 \nu^{10} + 247 \nu^{9} - 2177 \nu^{8} - 81 \nu^{7} + 6190 \nu^{6} - 21 \nu^{5} - 7062 \nu^{4} - 1025 \nu^{3} + 1966 \nu^{2} - 19 \nu - 76 \)\()/67\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(6\) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)
\(\nu^{5}\)\(=\)\(10\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(24\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(31\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\)
\(\nu^{6}\)\(=\)\(27\) \(\beta_{11}\mathstrut +\mathstrut \) \(11\) \(\beta_{10}\mathstrut +\mathstrut \) \(36\) \(\beta_{9}\mathstrut -\mathstrut \) \(42\) \(\beta_{8}\mathstrut +\mathstrut \) \(41\) \(\beta_{7}\mathstrut +\mathstrut \) \(81\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(12\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(72\) \(\beta_{1}\mathstrut +\mathstrut \) \(54\)
\(\nu^{7}\)\(=\)\(96\) \(\beta_{11}\mathstrut +\mathstrut \) \(30\) \(\beta_{10}\mathstrut +\mathstrut \) \(117\) \(\beta_{9}\mathstrut -\mathstrut \) \(144\) \(\beta_{8}\mathstrut +\mathstrut \) \(98\) \(\beta_{7}\mathstrut +\mathstrut \) \(232\) \(\beta_{6}\mathstrut +\mathstrut \) \(43\) \(\beta_{5}\mathstrut -\mathstrut \) \(33\) \(\beta_{4}\mathstrut -\mathstrut \) \(14\) \(\beta_{3}\mathstrut -\mathstrut \) \(21\) \(\beta_{2}\mathstrut +\mathstrut \) \(219\) \(\beta_{1}\mathstrut +\mathstrut \) \(129\)
\(\nu^{8}\)\(=\)\(283\) \(\beta_{11}\mathstrut +\mathstrut \) \(110\) \(\beta_{10}\mathstrut +\mathstrut \) \(349\) \(\beta_{9}\mathstrut -\mathstrut \) \(458\) \(\beta_{8}\mathstrut +\mathstrut \) \(310\) \(\beta_{7}\mathstrut +\mathstrut \) \(720\) \(\beta_{6}\mathstrut +\mathstrut \) \(152\) \(\beta_{5}\mathstrut -\mathstrut \) \(128\) \(\beta_{4}\mathstrut -\mathstrut \) \(44\) \(\beta_{3}\mathstrut -\mathstrut \) \(77\) \(\beta_{2}\mathstrut +\mathstrut \) \(572\) \(\beta_{1}\mathstrut +\mathstrut \) \(375\)
\(\nu^{9}\)\(=\)\(907\) \(\beta_{11}\mathstrut +\mathstrut \) \(327\) \(\beta_{10}\mathstrut +\mathstrut \) \(1069\) \(\beta_{9}\mathstrut -\mathstrut \) \(1466\) \(\beta_{8}\mathstrut +\mathstrut \) \(830\) \(\beta_{7}\mathstrut +\mathstrut \) \(2110\) \(\beta_{6}\mathstrut +\mathstrut \) \(469\) \(\beta_{5}\mathstrut -\mathstrut \) \(386\) \(\beta_{4}\mathstrut -\mathstrut \) \(154\) \(\beta_{3}\mathstrut -\mathstrut \) \(294\) \(\beta_{2}\mathstrut +\mathstrut \) \(1670\) \(\beta_{1}\mathstrut +\mathstrut \) \(989\)
\(\nu^{10}\)\(=\)\(2731\) \(\beta_{11}\mathstrut +\mathstrut \) \(1061\) \(\beta_{10}\mathstrut +\mathstrut \) \(3179\) \(\beta_{9}\mathstrut -\mathstrut \) \(4574\) \(\beta_{8}\mathstrut +\mathstrut \) \(2481\) \(\beta_{7}\mathstrut +\mathstrut \) \(6364\) \(\beta_{6}\mathstrut +\mathstrut \) \(1515\) \(\beta_{5}\mathstrut -\mathstrut \) \(1296\) \(\beta_{4}\mathstrut -\mathstrut \) \(481\) \(\beta_{3}\mathstrut -\mathstrut \) \(1006\) \(\beta_{2}\mathstrut +\mathstrut \) \(4609\) \(\beta_{1}\mathstrut +\mathstrut \) \(2819\)
\(\nu^{11}\)\(=\)\(8431\) \(\beta_{11}\mathstrut +\mathstrut \) \(3212\) \(\beta_{10}\mathstrut +\mathstrut \) \(9543\) \(\beta_{9}\mathstrut -\mathstrut \) \(14230\) \(\beta_{8}\mathstrut +\mathstrut \) \(6977\) \(\beta_{7}\mathstrut +\mathstrut \) \(18806\) \(\beta_{6}\mathstrut +\mathstrut \) \(4654\) \(\beta_{5}\mathstrut -\mathstrut \) \(3983\) \(\beta_{4}\mathstrut -\mathstrut \) \(1542\) \(\beta_{3}\mathstrut -\mathstrut \) \(3435\) \(\beta_{2}\mathstrut +\mathstrut \) \(13323\) \(\beta_{1}\mathstrut +\mathstrut \) \(7783\)

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
2.98033
2.59078
1.94650
1.84951
0.649147
0.482477
0.127418
−0.329641
−0.920581
−1.17133
−1.42534
−1.77927
−1.98033 0 1.92172 1.61478 0 1.00000 0.155028 0 −3.19781
1.2 −1.59078 0 0.530574 1.30400 0 1.00000 2.33753 0 −2.07438
1.3 −0.946499 0 −1.10414 2.90523 0 1.00000 2.93806 0 −2.74979
1.4 −0.849509 0 −1.27834 −1.65890 0 1.00000 2.78497 0 1.40925
1.5 0.350853 0 −1.87690 0.112051 0 1.00000 −1.36022 0 0.0393133
1.6 0.517523 0 −1.73217 3.15891 0 1.00000 −1.93148 0 1.63481
1.7 0.872582 0 −1.23860 −3.32654 0 1.00000 −2.82594 0 −2.90267
1.8 1.32964 0 −0.232054 0.891044 0 1.00000 −2.96783 0 1.18477
1.9 1.92058 0 1.68863 −0.753423 0 1.00000 −0.598010 0 −1.44701
1.10 2.17133 0 2.71466 4.06798 0 1.00000 1.55176 0 8.83292
1.11 2.42534 0 3.88228 −2.61560 0 1.00000 4.56517 0 −6.34371
1.12 2.77927 0 5.72435 1.30046 0 1.00000 10.3510 0 3.61432
\(n\): 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
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(-1\)

Hecke kernels

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

\(T_{2}^{12} - \cdots\)
\(T_{5}^{12} - \cdots\)