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: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 5 x^{11} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} - 71 x^{3} + 16 x^{2} + 7 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 889)
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 7q^{2} + 9q^{4} + 7q^{5} + 12q^{7} + 15q^{8} + O(q^{10}) \) \( 12q + 7q^{2} + 9q^{4} + 7q^{5} + 12q^{7} + 15q^{8} - 2q^{10} + 22q^{11} + 7q^{14} + 7q^{16} + 6q^{17} - 7q^{19} + 8q^{20} + 13q^{22} + 29q^{23} + 3q^{25} + 9q^{28} + 22q^{29} - 16q^{31} + 27q^{32} - 5q^{34} + 7q^{35} - 4q^{37} - 2q^{38} + 16q^{40} + 21q^{41} + 11q^{43} + 11q^{44} + 31q^{47} + 12q^{49} + 21q^{50} + 3q^{52} + 38q^{53} - 11q^{55} + 15q^{56} + 20q^{58} + 15q^{59} - 3q^{61} + 4q^{62} + 29q^{64} + 32q^{65} - q^{67} - 17q^{68} - 2q^{70} + 57q^{71} - 7q^{73} + 42q^{74} - 44q^{76} + 22q^{77} - 18q^{79} - q^{80} + 56q^{82} + 21q^{83} - 5q^{85} + 32q^{86} - 10q^{88} - 6q^{89} + 15q^{92} + 35q^{94} + 57q^{95} + 4q^{97} + 7q^{98} + O(q^{100}) \)

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

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

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:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8001.2.a.n 12
3.b odd 2 1 889.2.a.a 12
21.c even 2 1 6223.2.a.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.a.a 12 3.b odd 2 1
6223.2.a.i 12 21.c even 2 1
8001.2.a.n 12 1.a even 1 1 trivial

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 15 - 76 T + 60 T^{2} + 199 T^{3} - 278 T^{4} - 116 T^{5} + 303 T^{6} - 47 T^{7} - 107 T^{8} + 44 T^{9} + 8 T^{10} - 7 T^{11} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( 111 - 1165 T + 1390 T^{2} + 1888 T^{3} - 3700 T^{4} + 633 T^{5} + 1639 T^{6} - 753 T^{7} - 164 T^{8} + 139 T^{9} - 7 T^{10} - 7 T^{11} + T^{12} \)
$7$ \( ( -1 + T )^{12} \)
$11$ \( -855 - 5444 T - 7426 T^{2} + 7503 T^{3} + 12979 T^{4} - 7613 T^{5} - 5568 T^{6} + 4160 T^{7} - 41 T^{8} - 571 T^{9} + 178 T^{10} - 22 T^{11} + T^{12} \)
$13$ \( 422935 + 141791 T - 437862 T^{2} - 65803 T^{3} + 157903 T^{4} + 10273 T^{5} - 26285 T^{6} - 576 T^{7} + 2128 T^{8} + 6 T^{9} - 78 T^{10} + T^{12} \)
$17$ \( 3765 + 19154 T - 31183 T^{2} - 36299 T^{3} + 32265 T^{4} + 18042 T^{5} - 10019 T^{6} - 3509 T^{7} + 1145 T^{8} + 261 T^{9} - 54 T^{10} - 6 T^{11} + T^{12} \)
$19$ \( -8619139 + 2086626 T + 11358195 T^{2} + 4421282 T^{3} - 980992 T^{4} - 728370 T^{5} - 17478 T^{6} + 40471 T^{7} + 3873 T^{8} - 906 T^{9} - 115 T^{10} + 7 T^{11} + T^{12} \)
$23$ \( 132471 + 943492 T + 2093520 T^{2} + 1356523 T^{3} - 429623 T^{4} - 400541 T^{5} + 83961 T^{6} + 31146 T^{7} - 8752 T^{8} - 236 T^{9} + 266 T^{10} - 29 T^{11} + T^{12} \)
$29$ \( 2285685 + 3084472 T - 10453231 T^{2} + 7109033 T^{3} - 930002 T^{4} - 677823 T^{5} + 230159 T^{6} + 2651 T^{7} - 10102 T^{8} + 1070 T^{9} + 96 T^{10} - 22 T^{11} + T^{12} \)
$31$ \( 69361385 + 100337348 T + 30203246 T^{2} - 17539148 T^{3} - 11209110 T^{4} - 816715 T^{5} + 598748 T^{6} + 112185 T^{7} - 5918 T^{8} - 2582 T^{9} - 85 T^{10} + 16 T^{11} + T^{12} \)
$37$ \( -127475 - 50484171 T - 7010913 T^{2} + 36490826 T^{3} + 10038520 T^{4} - 3043528 T^{5} - 795342 T^{6} + 92450 T^{7} + 22642 T^{8} - 1118 T^{9} - 263 T^{10} + 4 T^{11} + T^{12} \)
$41$ \( -1174048215 - 562621394 T + 776212468 T^{2} - 47694806 T^{3} - 62659284 T^{4} + 8455030 T^{5} + 1646639 T^{6} - 318505 T^{7} - 11643 T^{8} + 4510 T^{9} - 95 T^{10} - 21 T^{11} + T^{12} \)
$43$ \( 4163911 - 3637998 T - 4025433 T^{2} + 6475132 T^{3} - 2806848 T^{4} + 159452 T^{5} + 204029 T^{6} - 47471 T^{7} - 1468 T^{8} + 1412 T^{9} - 83 T^{10} - 11 T^{11} + T^{12} \)
$47$ \( -7213725 + 40416435 T - 33788307 T^{2} + 2299861 T^{3} + 6805689 T^{4} - 2497382 T^{5} + 65650 T^{6} + 121773 T^{7} - 23068 T^{8} + 436 T^{9} + 287 T^{10} - 31 T^{11} + T^{12} \)
$53$ \( 3687897 - 3301723 T - 7133038 T^{2} + 4787188 T^{3} + 3940202 T^{4} - 1660590 T^{5} - 296134 T^{6} + 203259 T^{7} - 25481 T^{8} - 1046 T^{9} + 478 T^{10} - 38 T^{11} + T^{12} \)
$59$ \( -5696385 - 24732122 T - 33441321 T^{2} - 11540717 T^{3} + 4601880 T^{4} + 2258110 T^{5} - 237012 T^{6} - 127091 T^{7} + 7535 T^{8} + 2507 T^{9} - 151 T^{10} - 15 T^{11} + T^{12} \)
$61$ \( -3819875 - 156220 T + 24327838 T^{2} + 27426295 T^{3} + 2992966 T^{4} - 3751368 T^{5} - 849028 T^{6} + 89753 T^{7} + 27575 T^{8} - 783 T^{9} - 300 T^{10} + 3 T^{11} + T^{12} \)
$67$ \( -120068731 + 135488660 T + 80783327 T^{2} - 77094540 T^{3} - 2655842 T^{4} + 10904474 T^{5} - 1765125 T^{6} - 165709 T^{7} + 44833 T^{8} + 489 T^{9} - 369 T^{10} + T^{11} + T^{12} \)
$71$ \( -84701830575 + 104455579040 T - 32060036903 T^{2} + 250591821 T^{3} + 1487642404 T^{4} - 248236997 T^{5} + 3665597 T^{6} + 2753685 T^{7} - 261036 T^{8} + 1434 T^{9} + 1011 T^{10} - 57 T^{11} + T^{12} \)
$73$ \( 6839351 + 43763384 T + 37089935 T^{2} - 4983224 T^{3} - 14865322 T^{4} - 4605998 T^{5} + 67223 T^{6} + 205220 T^{7} + 16568 T^{8} - 2547 T^{9} - 289 T^{10} + 7 T^{11} + T^{12} \)
$79$ \( 116496565 + 188885663 T - 393179799 T^{2} + 85901189 T^{3} + 70309011 T^{4} - 20891717 T^{5} - 2624415 T^{6} + 727699 T^{7} + 48168 T^{8} - 6831 T^{9} - 389 T^{10} + 18 T^{11} + T^{12} \)
$83$ \( 2914656237 + 11235123530 T + 6558084145 T^{2} - 2232205888 T^{3} - 247247834 T^{4} + 90749871 T^{5} + 1469913 T^{6} - 1381614 T^{7} + 26735 T^{8} + 8963 T^{9} - 333 T^{10} - 21 T^{11} + T^{12} \)
$89$ \( 731582873037 + 88215652517 T - 89462730026 T^{2} + 109963 T^{3} + 2779367200 T^{4} - 58294480 T^{5} - 37666117 T^{6} + 821795 T^{7} + 253236 T^{8} - 3937 T^{9} - 820 T^{10} + 6 T^{11} + T^{12} \)
$97$ \( -2242441657 + 20817342946 T - 302916059 T^{2} - 6720670932 T^{3} + 1034912637 T^{4} + 141856458 T^{5} - 25095957 T^{6} - 1063338 T^{7} + 214624 T^{8} + 3379 T^{9} - 775 T^{10} - 4 T^{11} + T^{12} \)
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