Properties

Label 6039.2.a.i
Level $6039$
Weight $2$
Character orbit 6039.a
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{8} q^{5} + ( 1 + \beta_{11} ) q^{7} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{8} q^{5} + ( 1 + \beta_{11} ) q^{7} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{8} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{10} - q^{11} + ( 1 - \beta_{1} + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} ) q^{13} + ( 1 - \beta_{1} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{14} + ( 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{12} ) q^{16} + ( -2 - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{17} + ( 2 - \beta_{2} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{19} + ( 1 - \beta_{2} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{12} ) q^{20} + \beta_{1} q^{22} + ( -1 + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} ) q^{23} + ( 3 - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{25} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{26} + ( \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{28} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{29} + ( 3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{31} + ( 1 - \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} ) q^{32} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{10} - 2 \beta_{11} ) q^{34} + ( \beta_{1} - \beta_{2} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} ) q^{35} + ( 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{12} ) q^{37} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{38} + ( -1 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} ) q^{40} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} + 2 \beta_{10} + \beta_{11} ) q^{41} + ( 3 - \beta_{2} + 2 \beta_{3} + \beta_{8} + \beta_{11} ) q^{43} + ( -1 - \beta_{2} ) q^{44} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{46} + ( -\beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{12} ) q^{47} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{49} + ( -5 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{12} ) q^{50} + ( 1 + 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{52} + ( 1 - \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{53} -\beta_{8} q^{55} + ( -1 + \beta_{1} - 2 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} ) q^{56} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{58} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{59} - q^{61} + ( -5 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 4 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{62} + ( 3 - \beta_{1} + 3 \beta_{2} - \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{64} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{65} + ( 5 - 3 \beta_{1} + 2 \beta_{4} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} ) q^{67} + ( \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} - \beta_{12} ) q^{68} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{8} + 4 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{70} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} + 3 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{71} + ( 2 + 2 \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{12} ) q^{73} + ( 3 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{74} + ( 5 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} - \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{76} + ( -1 - \beta_{11} ) q^{77} + ( -2 + 2 \beta_{1} - 4 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 3 \beta_{9} + 2 \beta_{11} + 3 \beta_{12} ) q^{79} + ( 3 - 3 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{80} + ( 5 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} + 2 \beta_{12} ) q^{82} + ( -5 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} - \beta_{12} ) q^{83} + ( 1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{12} ) q^{85} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{86} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{88} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{89} + ( 4 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - 3 \beta_{12} ) q^{91} + ( -3 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{92} + ( 3 + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{94} + ( 2 + 6 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{7} + 3 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + 2 \beta_{12} ) q^{95} + ( 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{6} - 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{97} + ( 1 + 2 \beta_{1} - 6 \beta_{2} + \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 4 \beta_{10} - \beta_{11} + 3 \beta_{12} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + 6q^{10} - 13q^{11} + 13q^{13} - q^{14} + 18q^{16} - 17q^{17} + 14q^{19} + 7q^{20} + 2q^{22} - 7q^{23} + 18q^{25} + 10q^{26} + 19q^{28} + 6q^{29} + 27q^{31} - 5q^{32} + 6q^{34} - 14q^{35} + 10q^{37} - 2q^{38} + 8q^{40} - 3q^{41} + 29q^{43} - 16q^{44} - 24q^{46} - 8q^{47} + 8q^{49} + 27q^{50} + 37q^{52} + 24q^{53} + 3q^{55} - 24q^{56} - 5q^{58} - 13q^{59} - 13q^{61} - 39q^{62} + 47q^{64} + 11q^{65} + 44q^{67} + 8q^{68} - 12q^{70} - 3q^{71} + 48q^{73} + 22q^{74} + 47q^{76} - 11q^{77} - 17q^{79} + 26q^{80} + 56q^{82} - 50q^{83} + 8q^{85} - 18q^{86} + 9q^{88} + 15q^{89} + 47q^{91} - 14q^{92} + 45q^{94} + q^{95} + 27q^{97} - 47q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -38 \nu^{12} + 47 \nu^{11} + 646 \nu^{10} - 577 \nu^{9} - 3970 \nu^{8} + 1615 \nu^{7} + 11430 \nu^{6} + 2278 \nu^{5} - 18122 \nu^{4} - 10328 \nu^{3} + 14993 \nu^{2} + 9272 \nu - 4830 \)\()/1261\)
\(\beta_{4}\)\(=\)\((\)\( 80 \nu^{12} - 192 \nu^{11} - 1217 \nu^{10} + 2967 \nu^{9} + 5684 \nu^{8} - 15191 \nu^{7} - 6221 \nu^{6} + 29789 \nu^{5} - 11934 \nu^{4} - 24751 \nu^{3} + 21328 \nu^{2} + 12785 \nu - 1964 \)\()/1261\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{12} + 275 \nu^{11} - 352 \nu^{10} - 5029 \nu^{9} + 5943 \nu^{8} + 33414 \nu^{7} - 31352 \nu^{6} - 99199 \nu^{5} + 60892 \nu^{4} + 131903 \nu^{3} - 38492 \nu^{2} - 60160 \nu + 5338 \)\()/1261\)
\(\beta_{6}\)\(=\)\((\)\( 74 \nu^{12} - 448 \nu^{11} - 738 \nu^{10} + 7586 \nu^{9} - 695 \nu^{8} - 44875 \nu^{7} + 24220 \nu^{6} + 111281 \nu^{5} - 66105 \nu^{4} - 114506 \nu^{3} + 52816 \nu^{2} + 35335 \nu - 9293 \)\()/1261\)
\(\beta_{7}\)\(=\)\((\)\( 112 \nu^{12} - 495 \nu^{11} - 1384 \nu^{10} + 8163 \nu^{9} + 3275 \nu^{8} - 46490 \nu^{7} + 12790 \nu^{6} + 109003 \nu^{5} - 47983 \nu^{4} - 105439 \nu^{3} + 39084 \nu^{2} + 32368 \nu - 8246 \)\()/1261\)
\(\beta_{8}\)\(=\)\((\)\( 128 \nu^{12} - 42 \nu^{11} - 2670 \nu^{10} + 699 \nu^{9} + 20498 \nu^{8} - 4257 \nu^{7} - 69946 \nu^{6} + 11200 \nu^{5} + 100321 \nu^{4} - 10180 \nu^{3} - 46002 \nu^{2} + 1840 \nu + 1171 \)\()/1261\)
\(\beta_{9}\)\(=\)\((\)\( 162 \nu^{12} - 524 \nu^{11} - 2507 \nu^{10} + 8823 \nu^{9} + 12632 \nu^{8} - 52242 \nu^{7} - 22121 \nu^{6} + 132371 \nu^{5} + 6149 \nu^{4} - 147391 \nu^{3} + 16859 \nu^{2} + 57426 \nu - 11183 \)\()/1261\)
\(\beta_{10}\)\(=\)\((\)\( -191 \nu^{12} + 448 \nu^{11} + 3260 \nu^{10} - 7625 \nu^{9} - 19377 \nu^{8} + 45850 \nu^{7} + 47579 \nu^{6} - 118535 \nu^{5} - 45357 \nu^{4} + 134188 \nu^{3} + 6321 \nu^{2} - 53288 \nu + 8136 \)\()/1261\)
\(\beta_{11}\)\(=\)\((\)\( 187 \nu^{12} - 883 \nu^{11} - 2451 \nu^{10} + 15172 \nu^{9} + 7541 \nu^{8} - 92220 \nu^{7} + 11676 \nu^{6} + 240723 \nu^{5} - 65676 \nu^{4} - 270378 \nu^{3} + 62706 \nu^{2} + 97970 \nu - 17754 \)\()/1261\)
\(\beta_{12}\)\(=\)\((\)\( -344 \nu^{12} + 953 \nu^{11} + 5640 \nu^{10} - 16038 \nu^{9} - 31573 \nu^{8} + 94362 \nu^{7} + 70611 \nu^{6} - 233199 \nu^{5} - 59488 \nu^{4} + 240692 \nu^{3} + 6554 \nu^{2} - 79409 \nu + 8700 \)\()/1261\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{6} - \beta_{3} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{12} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + 9 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-\beta_{12} - \beta_{11} - \beta_{8} - 10 \beta_{7} + 9 \beta_{6} - \beta_{5} + \beta_{4} - 9 \beta_{3} + 11 \beta_{2} + 29 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(-10 \beta_{12} - 2 \beta_{11} + 11 \beta_{10} - 8 \beta_{9} + \beta_{8} - 11 \beta_{7} + 3 \beta_{6} - 11 \beta_{5} + 10 \beta_{4} - 10 \beta_{3} + 69 \beta_{2} - \beta_{1} + 79\)
\(\nu^{7}\)\(=\)\(-13 \beta_{12} - 16 \beta_{11} - \beta_{10} - 12 \beta_{8} - 79 \beta_{7} + 72 \beta_{6} - 13 \beta_{5} + 10 \beta_{4} - 67 \beta_{3} + 94 \beta_{2} + 181 \beta_{1} - 8\)
\(\nu^{8}\)\(=\)\(-81 \beta_{12} - 33 \beta_{11} + 92 \beta_{10} - 51 \beta_{9} + 13 \beta_{8} - 92 \beta_{7} + 44 \beta_{6} - 97 \beta_{5} + 76 \beta_{4} - 83 \beta_{3} + 506 \beta_{2} - 5 \beta_{1} + 488\)
\(\nu^{9}\)\(=\)\(-120 \beta_{12} - 176 \beta_{11} - 22 \beta_{10} + \beta_{9} - 105 \beta_{8} - 580 \beta_{7} + 560 \beta_{6} - 128 \beta_{5} + 71 \beta_{4} - 478 \beta_{3} + 739 \beta_{2} + 1179 \beta_{1} - 29\)
\(\nu^{10}\)\(=\)\(-606 \beta_{12} - 375 \beta_{11} + 688 \beta_{10} - 303 \beta_{9} + 127 \beta_{8} - 697 \beta_{7} + 471 \beta_{6} - 793 \beta_{5} + 523 \beta_{4} - 655 \beta_{3} + 3644 \beta_{2} + 34 \beta_{1} + 3165\)
\(\nu^{11}\)\(=\)\(-967 \beta_{12} - 1670 \beta_{11} - 303 \beta_{10} + 23 \beta_{9} - 815 \beta_{8} - 4127 \beta_{7} + 4302 \beta_{6} - 1137 \beta_{5} + 427 \beta_{4} - 3386 \beta_{3} + 5608 \beta_{2} + 7898 \beta_{1} + 124\)
\(\nu^{12}\)\(=\)\(-4357 \beta_{12} - 3662 \beta_{11} + 4833 \beta_{10} - 1739 \beta_{9} + 1118 \beta_{8} - 5052 \beta_{7} + 4458 \beta_{6} - 6254 \beta_{5} + 3417 \beta_{4} - 5073 \beta_{3} + 26039 \beta_{2} + 982 \beta_{1} + 21158\)

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
2.73913
2.53773
2.14727
1.61181
1.50067
0.822526
0.171582
−0.805107
−0.948254
−1.33092
−1.46794
−2.37960
−2.59890
−2.73913 0 5.50285 −0.604616 0 −1.91298 −9.59477 0 1.65612
1.2 −2.53773 0 4.44007 −0.994065 0 4.88646 −6.19224 0 2.52267
1.3 −2.14727 0 2.61077 3.62125 0 2.48742 −1.31149 0 −7.77580
1.4 −1.61181 0 0.597924 −4.27930 0 1.98799 2.25988 0 6.89741
1.5 −1.50067 0 0.252024 0.569898 0 −3.97554 2.62314 0 −0.855232
1.6 −0.822526 0 −1.32345 −2.11599 0 0.404149 2.73363 0 1.74046
1.7 −0.171582 0 −1.97056 0.133072 0 0.615329 0.681275 0 −0.0228327
1.8 0.805107 0 −1.35180 −1.06503 0 −0.203035 −2.69856 0 −0.857467
1.9 0.948254 0 −1.10081 2.25122 0 5.24025 −2.94036 0 2.13473
1.10 1.33092 0 −0.228660 2.36819 0 −2.92169 −2.96616 0 3.15187
1.11 1.46794 0 0.154851 −3.84216 0 2.46223 −2.70857 0 −5.64006
1.12 2.37960 0 3.66252 −2.55185 0 1.67712 3.95613 0 −6.07239
1.13 2.59890 0 4.75428 3.50938 0 0.252288 7.15810 0 9.12053
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)
\(61\) \(1\)

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 6039.2.a.i 13
3.b odd 2 1 2013.2.a.e 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.e 13 3.b odd 2 1
6039.2.a.i 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{13} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 47 + 236 T - 366 T^{2} - 742 T^{3} + 760 T^{4} + 841 T^{5} - 610 T^{6} - 469 T^{7} + 220 T^{8} + 136 T^{9} - 35 T^{10} - 19 T^{11} + 2 T^{12} + T^{13} \)
$3$ \( T^{13} \)
$5$ \( 292 - 1676 T - 4711 T^{2} + 4233 T^{3} + 13959 T^{4} + 3331 T^{5} - 6799 T^{6} - 2404 T^{7} + 1285 T^{8} + 476 T^{9} - 104 T^{10} - 37 T^{11} + 3 T^{12} + T^{13} \)
$7$ \( -148 + 640 T + 2331 T^{2} - 15918 T^{3} + 26514 T^{4} - 12042 T^{5} - 6921 T^{6} + 7318 T^{7} - 914 T^{8} - 833 T^{9} + 253 T^{10} + 11 T^{11} - 11 T^{12} + T^{13} \)
$11$ \( ( 1 + T )^{13} \)
$13$ \( 388 - 7724 T + 59385 T^{2} - 219377 T^{3} + 388940 T^{4} - 270596 T^{5} + 6341 T^{6} + 45598 T^{7} - 9259 T^{8} - 1911 T^{9} + 657 T^{10} - 3 T^{11} - 13 T^{12} + T^{13} \)
$17$ \( -4744 - 219092 T + 585821 T^{2} + 285500 T^{3} - 460055 T^{4} - 264249 T^{5} + 48021 T^{6} + 50352 T^{7} + 2972 T^{8} - 3254 T^{9} - 542 T^{10} + 51 T^{11} + 17 T^{12} + T^{13} \)
$19$ \( -10215742 + 28256994 T - 25651789 T^{2} + 5891140 T^{3} + 3999180 T^{4} - 2358937 T^{5} + 103356 T^{6} + 183151 T^{7} - 34742 T^{8} - 3540 T^{9} + 1384 T^{10} - 43 T^{11} - 14 T^{12} + T^{13} \)
$23$ \( 1840672 + 13371112 T + 24183107 T^{2} + 17153801 T^{3} + 3384342 T^{4} - 1675438 T^{5} - 803085 T^{6} - 904 T^{7} + 46069 T^{8} + 4119 T^{9} - 1010 T^{10} - 126 T^{11} + 7 T^{12} + T^{13} \)
$29$ \( -11772848 - 5620360 T + 15263265 T^{2} + 1847092 T^{3} - 6571640 T^{4} + 832462 T^{5} + 735603 T^{6} - 125575 T^{7} - 34829 T^{8} + 6194 T^{9} + 748 T^{10} - 130 T^{11} - 6 T^{12} + T^{13} \)
$31$ \( -1170458 + 9710684 T - 27561335 T^{2} + 38175799 T^{3} - 29312543 T^{4} + 12849900 T^{5} - 2929198 T^{6} + 137117 T^{7} + 92148 T^{8} - 20540 T^{9} + 918 T^{10} + 197 T^{11} - 27 T^{12} + T^{13} \)
$37$ \( 76337468 - 166492240 T + 76582459 T^{2} + 46446297 T^{3} - 34929945 T^{4} - 1573195 T^{5} + 3792266 T^{6} - 204317 T^{7} - 142157 T^{8} + 11795 T^{9} + 2055 T^{10} - 194 T^{11} - 10 T^{12} + T^{13} \)
$41$ \( 98100918 + 59989368 T - 64026067 T^{2} - 35584767 T^{3} + 14299487 T^{4} + 6642759 T^{5} - 1287999 T^{6} - 472838 T^{7} + 46771 T^{8} + 15082 T^{9} - 652 T^{10} - 207 T^{11} + 3 T^{12} + T^{13} \)
$43$ \( -195974572 + 295981576 T - 63363431 T^{2} - 83579412 T^{3} + 39129841 T^{4} + 2674445 T^{5} - 4333069 T^{6} + 598450 T^{7} + 100152 T^{8} - 32073 T^{9} + 1904 T^{10} + 198 T^{11} - 29 T^{12} + T^{13} \)
$47$ \( 26421804 - 15164268 T - 58833941 T^{2} + 17567742 T^{3} + 29338618 T^{4} - 476247 T^{5} - 4095562 T^{6} - 425660 T^{7} + 140009 T^{8} + 17270 T^{9} - 1806 T^{10} - 229 T^{11} + 8 T^{12} + T^{13} \)
$53$ \( -93111825698 + 130419884528 T - 58108058743 T^{2} + 6441993519 T^{3} + 2243534229 T^{4} - 670996989 T^{5} + 26246078 T^{6} + 10341350 T^{7} - 1152358 T^{8} - 32735 T^{9} + 9653 T^{10} - 214 T^{11} - 24 T^{12} + T^{13} \)
$59$ \( -7051312 - 11441496 T + 26215377 T^{2} + 29674727 T^{3} - 11841571 T^{4} - 17909350 T^{5} - 3423411 T^{6} + 752403 T^{7} + 251338 T^{8} + 4474 T^{9} - 3625 T^{10} - 219 T^{11} + 13 T^{12} + T^{13} \)
$61$ \( ( 1 + T )^{13} \)
$67$ \( -6958847006 - 169477916206 T - 39708587867 T^{2} + 20936383426 T^{3} + 2313009717 T^{4} - 967935121 T^{5} - 25944476 T^{6} + 20091793 T^{7} - 535019 T^{8} - 172601 T^{9} + 11183 T^{10} + 317 T^{11} - 44 T^{12} + T^{13} \)
$71$ \( 48814344 + 26962908 T - 512125217 T^{2} - 431661224 T^{3} + 66765769 T^{4} + 88612272 T^{5} - 1593405 T^{6} - 4096969 T^{7} + 58883 T^{8} + 66344 T^{9} - 794 T^{10} - 438 T^{11} + 3 T^{12} + T^{13} \)
$73$ \( 28332933844 + 28117420384 T - 4537752459 T^{2} - 6331127315 T^{3} + 1189424914 T^{4} + 388567146 T^{5} - 130311429 T^{6} + 7918583 T^{7} + 1385794 T^{8} - 222709 T^{9} + 7168 T^{10} + 573 T^{11} - 48 T^{12} + T^{13} \)
$79$ \( 388386798444 + 131453923248 T - 116113531441 T^{2} - 9729487411 T^{3} + 6617813672 T^{4} + 376370987 T^{5} - 152293306 T^{6} - 8092221 T^{7} + 1660310 T^{8} + 90102 T^{9} - 8586 T^{10} - 485 T^{11} + 17 T^{12} + T^{13} \)
$83$ \( -58130319032 + 42617697212 T + 19852316459 T^{2} - 6625979573 T^{3} - 3078005106 T^{4} - 45758771 T^{5} + 97247168 T^{6} + 9250209 T^{7} - 849124 T^{8} - 160405 T^{9} - 3026 T^{10} + 719 T^{11} + 50 T^{12} + T^{13} \)
$89$ \( 120747458 + 1450031882 T - 5483550149 T^{2} + 3540622044 T^{3} - 474338600 T^{4} - 176988322 T^{5} + 46122395 T^{6} + 1219256 T^{7} - 1010888 T^{8} + 36857 T^{9} + 7027 T^{10} - 399 T^{11} - 15 T^{12} + T^{13} \)
$97$ \( -87087118744 - 145098261164 T + 22218327363 T^{2} + 35031639796 T^{3} - 6856977798 T^{4} - 996124072 T^{5} + 238685361 T^{6} + 7345679 T^{7} - 2997530 T^{8} + 25546 T^{9} + 15357 T^{10} - 415 T^{11} - 27 T^{12} + T^{13} \)
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