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:

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

Atkin-Lehner signs

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

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

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 7 T^{2} + 13 T^{3} + 30 T^{4} + 48 T^{5} + 87 T^{6} + 130 T^{7} + 219 T^{8} + 320 T^{9} + 528 T^{10} + 762 T^{11} + 1208 T^{12} + 1655 T^{13} + 2416 T^{14} + 3048 T^{15} + 4224 T^{16} + 5120 T^{17} + 7008 T^{18} + 8320 T^{19} + 11136 T^{20} + 12288 T^{21} + 15360 T^{22} + 13312 T^{23} + 14336 T^{24} + 8192 T^{25} + 8192 T^{26} \)
$3$ 1
$5$ \( 1 + 3 T + 28 T^{2} + 76 T^{3} + 391 T^{4} + 1035 T^{5} + 3891 T^{6} + 10101 T^{7} + 31341 T^{8} + 77614 T^{9} + 214033 T^{10} + 494844 T^{11} + 1255819 T^{12} + 2680782 T^{13} + 6279095 T^{14} + 12371100 T^{15} + 26754125 T^{16} + 48508750 T^{17} + 97940625 T^{18} + 157828125 T^{19} + 303984375 T^{20} + 404296875 T^{21} + 763671875 T^{22} + 742187500 T^{23} + 1367187500 T^{24} + 732421875 T^{25} + 1220703125 T^{26} \)
$7$ \( 1 - 11 T + 102 T^{2} - 671 T^{3} + 3836 T^{4} - 18778 T^{5} + 82582 T^{6} - 330300 T^{7} + 1216388 T^{8} - 4178141 T^{9} + 13438677 T^{10} - 40755778 T^{11} + 116913072 T^{12} - 317503782 T^{13} + 818391504 T^{14} - 1997033122 T^{15} + 4609466211 T^{16} - 10031716541 T^{17} + 20443833116 T^{18} - 38859464700 T^{19} + 68009828026 T^{20} - 108251433178 T^{21} + 154796436452 T^{22} - 189540892079 T^{23} + 201687327786 T^{24} - 152254159211 T^{25} + 96889010407 T^{26} \)
$11$ \( ( 1 + T )^{13} \)
$13$ \( 1 - 13 T + 166 T^{2} - 1371 T^{3} + 10842 T^{4} - 68851 T^{5} + 422468 T^{6} - 2243530 T^{7} + 11585898 T^{8} - 53508605 T^{9} + 240926658 T^{10} - 985066146 T^{11} + 3931490743 T^{12} - 14343563288 T^{13} + 51109379659 T^{14} - 166476178674 T^{15} + 529315867626 T^{16} - 1528259267405 T^{17} + 4301762826114 T^{18} - 10829090795770 T^{19} + 26509240479956 T^{20} - 56163875871571 T^{21} + 114973982202066 T^{22} - 189003992324979 T^{23} + 297498625410142 T^{24} - 302875106592253 T^{25} + 302875106592253 T^{26} \)
$17$ \( 1 + 17 T + 272 T^{2} + 2926 T^{3} + 28825 T^{4} + 235090 T^{5} + 1768253 T^{6} + 11778123 T^{7} + 72933433 T^{8} + 411775206 T^{9} + 2173525244 T^{10} + 10605747960 T^{11} + 48539734912 T^{12} + 206539615464 T^{13} + 825175493504 T^{14} + 3065061160440 T^{15} + 10678529523772 T^{16} + 34391876980326 T^{17} + 103555045379081 T^{18} + 284295256602987 T^{19} + 725582589548269 T^{20} + 1639930816804690 T^{21} + 3418295540026025 T^{22} + 5898798152713774 T^{23} + 9321955795676176 T^{24} + 9904578032905937 T^{25} + 9904578032905937 T^{26} \)
$19$ \( 1 - 14 T + 204 T^{2} - 1808 T^{3} + 15631 T^{4} - 105346 T^{5} + 685720 T^{6} - 3820068 T^{7} + 20509216 T^{8} - 99372182 T^{9} + 468152039 T^{10} - 2085072809 T^{11} + 9228604441 T^{12} - 40078657088 T^{13} + 175343484379 T^{14} - 752711284049 T^{15} + 3211054835501 T^{16} - 12950282130422 T^{17} + 50782849228384 T^{18} - 179718464539908 T^{19} + 612945728867080 T^{20} - 1789150432117186 T^{21} + 5043931403983549 T^{22} - 11084967794104208 T^{23} + 23764012815236676 T^{24} - 30986408866926254 T^{25} + 42052983462257059 T^{26} \)
$23$ \( 1 + 7 T + 173 T^{2} + 922 T^{3} + 13503 T^{4} + 58167 T^{5} + 665521 T^{6} + 2367741 T^{7} + 23755639 T^{8} + 69941305 T^{9} + 671953888 T^{10} + 1681077876 T^{11} + 16541939243 T^{12} + 38191222924 T^{13} + 380464602589 T^{14} + 889290196404 T^{15} + 8175662955296 T^{16} + 19572444732505 T^{17} + 152899440788177 T^{18} + 350510643856749 T^{19} + 2265982836312887 T^{20} + 4555115080839927 T^{21} + 24320964387734889 T^{22} + 38195243338984378 T^{23} + 164836088119109371 T^{24} + 153402371024142247 T^{25} + 504036361936467383 T^{26} \)
$29$ \( 1 - 6 T + 247 T^{2} - 1340 T^{3} + 30322 T^{4} - 150945 T^{5} + 2453163 T^{6} - 11230145 T^{7} + 145432146 T^{8} - 610197940 T^{9} + 6649767014 T^{10} - 25405586634 T^{11} + 240773801099 T^{12} - 829277813728 T^{13} + 6982440231871 T^{14} - 21366098359194 T^{15} + 162181167704446 T^{16} - 431581409201140 T^{17} + 2982980415995754 T^{18} - 6679952144211545 T^{19} + 42316758315815367 T^{20} - 75509694804398145 T^{21} + 439885680280299818 T^{22} - 563747692622269340 T^{23} + 3013525912129339763 T^{24} - 2122888699232814246 T^{25} + 10260628712958602189 T^{26} \)
$31$ \( 1 - 27 T + 600 T^{2} - 9126 T^{3} + 121595 T^{4} - 1335774 T^{5} + 13339118 T^{6} - 117336124 T^{7} + 960676419 T^{8} - 7155706592 T^{9} + 50280686263 T^{10} - 326325952633 T^{11} + 2008518015006 T^{12} - 11477446488818 T^{13} + 62264058465186 T^{14} - 313599240480313 T^{15} + 1497911924461033 T^{16} - 6608445307550432 T^{17} + 27503350261690269 T^{18} - 104136241964272444 T^{19} + 366994006115094098 T^{20} - 1139269672646714334 T^{21} + 3214925856626790245 T^{22} - 7479927746986789926 T^{23} + 15245086137842898600 T^{24} - 21266895162290843547 T^{25} + 24417546297445042591 T^{26} \)
$37$ \( 1 - 10 T + 287 T^{2} - 2385 T^{3} + 39619 T^{4} - 285347 T^{5} + 3602946 T^{6} - 23124531 T^{7} + 245436067 T^{8} - 1428276167 T^{9} + 13329360568 T^{10} - 70866824157 T^{11} + 594175785360 T^{12} - 2885985951366 T^{13} + 21984504058320 T^{14} - 97016682270933 T^{15} + 675172100850904 T^{16} - 2676819489420887 T^{17} + 17019508076297119 T^{18} - 59331219882439179 T^{19} + 342034426988833818 T^{20} - 1002275474737995587 T^{21} + 5148954168941155663 T^{22} - 11468473728216569865 T^{23} + 51062357450705138531 T^{24} - 65829520058400352810 T^{25} + \)\(24\!\cdots\!97\)\( T^{26} \)
$41$ \( 1 + 3 T + 326 T^{2} + 824 T^{3} + 52843 T^{4} + 112289 T^{5} + 5665641 T^{6} + 10220209 T^{7} + 450062925 T^{8} + 702745206 T^{9} + 28028911215 T^{10} + 38691561568 T^{11} + 1408022472661 T^{12} + 1746777455844 T^{13} + 57728921379101 T^{14} + 65040514995808 T^{15} + 1931780589849015 T^{16} + 1985789996051766 T^{17} + 52142580701447925 T^{18} + 48547058114806369 T^{19} + 1103407799025422721 T^{20} + 896619269052767969 T^{21} + 17299843559180081123 T^{22} + 11060271271565578424 T^{23} + \)\(17\!\cdots\!66\)\( T^{24} + 67690470901098558243 T^{25} + \)\(92\!\cdots\!21\)\( T^{26} \)
$43$ \( 1 - 29 T + 757 T^{2} - 13060 T^{3} + 205803 T^{4} - 2620114 T^{5} + 31060811 T^{6} - 318713121 T^{7} + 3089837128 T^{8} - 26804861612 T^{9} + 222110611870 T^{10} - 1677111262894 T^{11} + 12183450031082 T^{12} - 81345159312664 T^{13} + 523888351336526 T^{14} - 3100978725091006 T^{15} + 17659348417948090 T^{16} - 91640487683967212 T^{17} + 454232145310871704 T^{18} - 2014701346320865929 T^{19} + 8442906505877027777 T^{20} - 30624417182146266514 T^{21} + \)\(10\!\cdots\!29\)\( T^{22} - \)\(28\!\cdots\!40\)\( T^{23} + \)\(70\!\cdots\!99\)\( T^{24} - \)\(11\!\cdots\!29\)\( T^{25} + \)\(17\!\cdots\!43\)\( T^{26} \)
$47$ \( 1 + 8 T + 382 T^{2} + 2706 T^{3} + 71179 T^{4} + 457541 T^{5} + 8750573 T^{6} + 51750872 T^{7} + 798880953 T^{8} + 4357243002 T^{9} + 57183188436 T^{10} + 286237174457 T^{11} + 3296350058422 T^{12} + 15005322289016 T^{13} + 154928452745834 T^{14} + 632297918375513 T^{15} + 5936930172990828 T^{16} + 21261955889242362 T^{17} + 183219357757951671 T^{18} + 557833792751516888 T^{19} + 4433242599099275299 T^{20} + 10894639910508789701 T^{21} + 79658587944981852293 T^{22} + \)\(14\!\cdots\!94\)\( T^{23} + \)\(94\!\cdots\!46\)\( T^{24} + \)\(92\!\cdots\!28\)\( T^{25} + \)\(54\!\cdots\!27\)\( T^{26} \)
$53$ \( 1 - 24 T + 475 T^{2} - 5611 T^{3} + 61605 T^{4} - 485724 T^{5} + 4243647 T^{6} - 28236809 T^{7} + 240196766 T^{8} - 1331226943 T^{9} + 10272475875 T^{10} - 37967197175 T^{11} + 334747029319 T^{12} - 1010333089226 T^{13} + 17741592553907 T^{14} - 106649856864575 T^{15} + 1529335390842375 T^{16} - 10504020900429583 T^{17} + 100449204974375638 T^{18} - 625850831806597361 T^{19} + 4985059404435865539 T^{20} - 30241025865367910364 T^{21} + \)\(20\!\cdots\!65\)\( T^{22} - \)\(98\!\cdots\!39\)\( T^{23} + \)\(44\!\cdots\!75\)\( T^{24} - \)\(11\!\cdots\!84\)\( T^{25} + \)\(26\!\cdots\!73\)\( T^{26} \)
$59$ \( 1 + 13 T + 548 T^{2} + 5579 T^{3} + 133861 T^{4} + 1099286 T^{5} + 19937846 T^{6} + 134753940 T^{7} + 2096039823 T^{8} + 11909036154 T^{9} + 171318435787 T^{10} + 845692358222 T^{11} + 11668320878576 T^{12} + 52471781246340 T^{13} + 688430931835984 T^{14} + 2943855098970782 T^{15} + 35185209023498273 T^{16} + 144306090240069594 T^{17} + 1498509801134359077 T^{18} + 5683993099427295540 T^{19} + 49618350051992559874 T^{20} + \)\(16\!\cdots\!06\)\( T^{21} + \)\(11\!\cdots\!79\)\( T^{22} + \)\(28\!\cdots\!79\)\( T^{23} + \)\(16\!\cdots\!32\)\( T^{24} + \)\(23\!\cdots\!53\)\( T^{25} + \)\(10\!\cdots\!79\)\( T^{26} \)
$61$ \( ( 1 + T )^{13} \)
$67$ \( 1 - 44 T + 1188 T^{2} - 24193 T^{3} + 411170 T^{4} - 6078465 T^{5} + 80297323 T^{6} - 965078585 T^{7} + 10701564022 T^{8} - 110643777683 T^{9} + 1075723589435 T^{10} - 9903635405709 T^{11} + 86805253524531 T^{12} - 726645176472648 T^{13} + 5815951986143577 T^{14} - 44457419336227701 T^{15} + 323537853929238905 T^{16} - 2229596151987232643 T^{17} + 14448450270270100354 T^{18} - 87299447465047750865 T^{19} + \)\(48\!\cdots\!29\)\( T^{20} - \)\(24\!\cdots\!65\)\( T^{21} + \)\(11\!\cdots\!90\)\( T^{22} - \)\(44\!\cdots\!57\)\( T^{23} + \)\(14\!\cdots\!04\)\( T^{24} - \)\(36\!\cdots\!84\)\( T^{25} + \)\(54\!\cdots\!87\)\( T^{26} \)
$71$ \( 1 + 3 T + 485 T^{2} + 1762 T^{3} + 117464 T^{4} + 493261 T^{5} + 19221703 T^{6} + 87958469 T^{7} + 2395386368 T^{8} + 11333779128 T^{9} + 240956043280 T^{10} + 1127862193268 T^{11} + 20201203996895 T^{12} + 89372246452338 T^{13} + 1434285483779545 T^{14} + 5685553316263988 T^{15} + 86240818406388080 T^{16} + 288010379725194168 T^{17} + 4321826392130887168 T^{18} + 11267504852156476949 T^{19} + \)\(17\!\cdots\!73\)\( T^{20} + \)\(31\!\cdots\!21\)\( T^{21} + \)\(53\!\cdots\!84\)\( T^{22} + \)\(57\!\cdots\!62\)\( T^{23} + \)\(11\!\cdots\!35\)\( T^{24} + \)\(49\!\cdots\!23\)\( T^{25} + \)\(11\!\cdots\!11\)\( T^{26} \)
$73$ \( 1 - 48 T + 1522 T^{2} - 34880 T^{3} + 653072 T^{4} - 10263838 T^{5} + 140801067 T^{6} - 1710105013 T^{7} + 18793932243 T^{8} - 189235230100 T^{9} + 1781965804351 T^{10} - 15946799449806 T^{11} + 138757519977572 T^{12} - 1188994646402494 T^{13} + 10129298958362756 T^{14} - 84980494268016174 T^{15} + 693214991311212967 T^{16} - 5373947670070254100 T^{17} + 38961167053730073099 T^{18} - \)\(25\!\cdots\!57\)\( T^{19} + \)\(15\!\cdots\!99\)\( T^{20} - \)\(82\!\cdots\!78\)\( T^{21} + \)\(38\!\cdots\!36\)\( T^{22} - \)\(14\!\cdots\!20\)\( T^{23} + \)\(47\!\cdots\!94\)\( T^{24} - \)\(10\!\cdots\!08\)\( T^{25} + \)\(16\!\cdots\!33\)\( T^{26} \)
$79$ \( 1 + 17 T + 542 T^{2} + 7530 T^{3} + 155435 T^{4} + 1879772 T^{5} + 30500780 T^{6} + 329653304 T^{7} + 4538991666 T^{8} + 44343627643 T^{9} + 536172795588 T^{10} + 4759796653037 T^{11} + 51529110799964 T^{12} + 415643920998238 T^{13} + 4070799753197156 T^{14} + 29705890911603917 T^{15} + 264354098963911932 T^{16} + 1727187888528689083 T^{17} + 13966733350872970734 T^{18} + 80134582873450691384 T^{19} + \)\(58\!\cdots\!20\)\( T^{20} + \)\(28\!\cdots\!92\)\( T^{21} + \)\(18\!\cdots\!65\)\( T^{22} + \)\(71\!\cdots\!30\)\( T^{23} + \)\(40\!\cdots\!18\)\( T^{24} + \)\(10\!\cdots\!97\)\( T^{25} + \)\(46\!\cdots\!39\)\( T^{26} \)
$83$ \( 1 + 50 T + 1798 T^{2} + 46774 T^{3} + 1033384 T^{4} + 19372996 T^{5} + 325384261 T^{6} + 4885010702 T^{7} + 67314181298 T^{8} + 848528007260 T^{9} + 9938314273754 T^{10} + 107686123909759 T^{11} + 1090671424852278 T^{12} + 10268166394136662 T^{13} + 90525728262739074 T^{14} + 741849707614329751 T^{15} + 5682598903646978398 T^{16} + 40269714546035410460 T^{17} + \)\(26\!\cdots\!14\)\( T^{18} + \)\(15\!\cdots\!38\)\( T^{19} + \)\(88\!\cdots\!47\)\( T^{20} + \)\(43\!\cdots\!36\)\( T^{21} + \)\(19\!\cdots\!52\)\( T^{22} + \)\(72\!\cdots\!26\)\( T^{23} + \)\(23\!\cdots\!66\)\( T^{24} + \)\(53\!\cdots\!50\)\( T^{25} + \)\(88\!\cdots\!63\)\( T^{26} \)
$89$ \( 1 - 15 T + 758 T^{2} - 8993 T^{3} + 264074 T^{4} - 2598648 T^{5} + 58536502 T^{6} - 495288546 T^{7} + 9541671058 T^{8} - 71450887279 T^{9} + 1235582283815 T^{10} - 8354202545906 T^{11} + 131850752462226 T^{12} - 812884574460588 T^{13} + 11734716969138114 T^{14} - 66173638366121426 T^{15} + 871047207038776735 T^{16} - 4482988789322852239 T^{17} + 53281258430674727042 T^{18} - \)\(24\!\cdots\!06\)\( T^{19} + \)\(25\!\cdots\!58\)\( T^{20} - \)\(10\!\cdots\!88\)\( T^{21} + \)\(92\!\cdots\!66\)\( T^{22} - \)\(28\!\cdots\!93\)\( T^{23} + \)\(21\!\cdots\!62\)\( T^{24} - \)\(37\!\cdots\!15\)\( T^{25} + \)\(21\!\cdots\!69\)\( T^{26} \)
$97$ \( 1 - 27 T + 846 T^{2} - 16071 T^{3} + 316643 T^{4} - 4868078 T^{5} + 75911390 T^{6} - 1006444954 T^{7} + 13447787513 T^{8} - 158930900501 T^{9} + 1889630845088 T^{10} - 20281662413284 T^{11} + 218857804683067 T^{12} - 2150177092562598 T^{13} + 21229207054257499 T^{14} - 190830161646589156 T^{15} + 1724615052279000224 T^{16} - 14070038350036069781 T^{17} + \)\(11\!\cdots\!41\)\( T^{18} - \)\(83\!\cdots\!66\)\( T^{19} + \)\(61\!\cdots\!70\)\( T^{20} - \)\(38\!\cdots\!58\)\( T^{21} + \)\(24\!\cdots\!31\)\( T^{22} - \)\(11\!\cdots\!79\)\( T^{23} + \)\(60\!\cdots\!38\)\( T^{24} - \)\(18\!\cdots\!07\)\( T^{25} + \)\(67\!\cdots\!77\)\( T^{26} \)
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