Properties

Label 6039.2.a.e
Level $6039$
Weight $2$
Character orbit 6039.a
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
Dimension $12$
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} - 71 x^{3} - 93 x^{2} + 13 x + 1\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{3} q^{5} + ( -1 - \beta_{2} - \beta_{4} - \beta_{6} ) q^{7} + ( -1 - \beta_{2} - \beta_{10} - \beta_{11} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{3} q^{5} + ( -1 - \beta_{2} - \beta_{4} - \beta_{6} ) q^{7} + ( -1 - \beta_{2} - \beta_{10} - \beta_{11} ) q^{8} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{10} + q^{11} + ( \beta_{4} - \beta_{5} + \beta_{6} ) q^{13} + ( 1 + \beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} + \beta_{11} ) q^{14} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{16} + ( \beta_{2} - \beta_{3} + \beta_{10} + \beta_{11} ) q^{17} + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{19} + ( -2 \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{20} -\beta_{1} q^{22} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{23} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{11} ) q^{25} + ( 2 + \beta_{1} + \beta_{2} - \beta_{7} ) q^{26} + ( -3 + \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{28} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{29} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{9} + \beta_{10} ) q^{31} + ( -1 - 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{32} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{34} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{35} + ( -2 - \beta_{2} - \beta_{4} + 2 \beta_{8} + \beta_{10} + \beta_{11} ) q^{37} + ( 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{38} + ( -2 - 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{40} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{41} + ( -3 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} ) q^{43} + ( 1 + \beta_{2} ) q^{44} + ( -2 - \beta_{1} + \beta_{2} + 3 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{46} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{47} + ( 2 - 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{49} + ( 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{50} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{52} + ( -2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{53} + \beta_{3} q^{55} + ( 2 + 3 \beta_{1} + 5 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{56} + ( -1 + 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} ) q^{58} + ( 2 + 3 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{59} - q^{61} + ( \beta_{1} + \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{62} + ( -1 + \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{10} + 3 \beta_{11} ) q^{64} + ( -1 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{65} + ( -3 - \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{67} + ( 2 + 4 \beta_{1} + 3 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{68} + ( -2 + 2 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - \beta_{7} - 4 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{70} + ( -\beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{71} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{73} + ( -\beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{9} + 2 \beta_{11} ) q^{74} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{11} ) q^{76} + ( -1 - \beta_{2} - \beta_{4} - \beta_{6} ) q^{77} + ( -1 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{79} + ( 6 + 2 \beta_{1} + 5 \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{5} + 3 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{80} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 5 \beta_{6} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{82} + ( 4 - 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - \beta_{6} + \beta_{7} - 3 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{83} + ( -3 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{85} + ( -3 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{86} + ( -1 - \beta_{2} - \beta_{10} - \beta_{11} ) q^{88} + ( -3 - \beta_{2} - \beta_{4} + \beta_{6} + 4 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{89} + ( -3 + \beta_{2} + \beta_{4} + \beta_{7} - 2 \beta_{10} ) q^{91} + ( 7 + 4 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} + 5 \beta_{6} - \beta_{7} - 4 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{92} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{94} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{95} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{97} + ( -3 - 6 \beta_{1} - 4 \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - q^{2} + 9q^{4} + 3q^{5} - 9q^{7} - 6q^{8} + O(q^{10}) \) \( 12q - q^{2} + 9q^{4} + 3q^{5} - 9q^{7} - 6q^{8} - 8q^{10} + 12q^{11} - q^{13} + 3q^{14} + 3q^{16} - 9q^{17} - 20q^{19} + 9q^{20} - q^{22} + 9q^{23} + 3q^{25} + 18q^{26} - 31q^{28} - 18q^{29} - 21q^{31} - 18q^{32} - 12q^{34} + 4q^{35} - 18q^{37} + 2q^{38} - 26q^{40} - 15q^{41} - 33q^{43} + 9q^{44} - 28q^{46} + 20q^{47} + 15q^{49} + 2q^{50} - 27q^{52} + 3q^{55} + 8q^{56} - 11q^{58} + 21q^{59} - 12q^{61} + 9q^{62} - 12q^{64} - 17q^{65} - 34q^{67} + 16q^{68} - 36q^{70} + 5q^{71} - 2q^{73} - 6q^{74} - 27q^{76} - 9q^{77} - 31q^{79} + 60q^{80} - 12q^{82} + 32q^{83} - 40q^{85} - 18q^{86} - 6q^{88} - 27q^{89} - 45q^{91} + 78q^{92} - 13q^{94} - 37q^{95} - 19q^{97} - 4q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} - 71 x^{3} - 93 x^{2} + 13 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -6 \nu^{11} - 4 \nu^{10} + 39 \nu^{9} + 138 \nu^{8} + 125 \nu^{7} - 998 \nu^{6} - 1242 \nu^{5} + 2567 \nu^{4} + 2384 \nu^{3} - 2446 \nu^{2} - 1304 \nu + 517 \)\()/151\)
\(\beta_{4}\)\(=\)\((\)\( 15 \nu^{11} + 10 \nu^{10} - 173 \nu^{9} - 194 \nu^{8} + 518 \nu^{7} + 985 \nu^{6} + 85 \nu^{5} - 1510 \nu^{4} - 1581 \nu^{3} + 226 \nu^{2} + 844 \nu + 293 \)\()/151\)
\(\beta_{5}\)\(=\)\((\)\( 21 \nu^{11} + 14 \nu^{10} - 363 \nu^{9} - 181 \nu^{8} + 2205 \nu^{7} + 775 \nu^{6} - 5921 \nu^{5} - 1359 \nu^{4} + 7209 \nu^{3} + 860 \nu^{2} - 3137 \nu + 78 \)\()/151\)
\(\beta_{6}\)\(=\)\((\)\( 23 \nu^{11} - 35 \nu^{10} - 376 \nu^{9} + 528 \nu^{8} + 2113 \nu^{7} - 2617 \nu^{6} - 4752 \nu^{5} + 4681 \nu^{4} + 3646 \nu^{3} - 2150 \nu^{2} - 387 \nu + 107 \)\()/151\)
\(\beta_{7}\)\(=\)\((\)\( -22 \nu^{11} + 86 \nu^{10} + 294 \nu^{9} - 1155 \nu^{8} - 1404 \nu^{7} + 5300 \nu^{6} + 2694 \nu^{5} - 9513 \nu^{4} - 1577 \nu^{3} + 5326 \nu^{2} + 252 \nu - 168 \)\()/151\)
\(\beta_{8}\)\(=\)\((\)\( 41 \nu^{11} - 23 \nu^{10} - 644 \nu^{9} + 265 \nu^{8} + 3550 \nu^{7} - 982 \nu^{6} - 8123 \nu^{5} + 1057 \nu^{4} + 6913 \nu^{3} + 356 \nu^{2} - 1458 \nu - 387 \)\()/151\)
\(\beta_{9}\)\(=\)\((\)\( 44 \nu^{11} - 21 \nu^{10} - 739 \nu^{9} + 196 \nu^{8} + 4620 \nu^{7} - 483 \nu^{6} - 13089 \nu^{5} + 15989 \nu^{3} + 522 \nu^{2} - 6242 \nu + 185 \)\()/151\)
\(\beta_{10}\)\(=\)\((\)\( -52 \nu^{11} + 66 \nu^{10} + 791 \nu^{9} - 918 \nu^{8} - 4252 \nu^{7} + 4538 \nu^{6} + 9621 \nu^{5} - 9060 \nu^{4} - 8381 \nu^{3} + 5780 \nu^{2} + 1886 \nu - 301 \)\()/151\)
\(\beta_{11}\)\(=\)\((\)\( 52 \nu^{11} - 66 \nu^{10} - 791 \nu^{9} + 918 \nu^{8} + 4252 \nu^{7} - 4538 \nu^{6} - 9621 \nu^{5} + 9060 \nu^{4} + 8532 \nu^{3} - 5931 \nu^{2} - 2490 \nu + 603 \)\()/151\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{11} + \beta_{10} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + 8 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(8 \beta_{11} + 8 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 10 \beta_{2} + 20 \beta_{1} + 9\)
\(\nu^{6}\)\(=\)\(3 \beta_{11} + 12 \beta_{10} + 10 \beta_{9} - 11 \beta_{8} + \beta_{7} + 21 \beta_{6} - 10 \beta_{5} + 18 \beta_{4} + 7 \beta_{3} + 57 \beta_{2} + 11 \beta_{1} + 75\)
\(\nu^{7}\)\(=\)\(55 \beta_{11} + 57 \beta_{10} + 14 \beta_{9} - 16 \beta_{8} + 11 \beta_{7} + 28 \beta_{6} - 23 \beta_{5} + 15 \beta_{4} + 87 \beta_{2} + 111 \beta_{1} + 72\)
\(\nu^{8}\)\(=\)\(43 \beta_{11} + 106 \beta_{10} + 79 \beta_{9} - 93 \beta_{8} + 15 \beta_{7} + 170 \beta_{6} - 81 \beta_{5} + 132 \beta_{4} + 41 \beta_{3} + 397 \beta_{2} + 97 \beta_{1} + 437\)
\(\nu^{9}\)\(=\)\(369 \beta_{11} + 402 \beta_{10} + 137 \beta_{9} - 167 \beta_{8} + 91 \beta_{7} + 278 \beta_{6} - 200 \beta_{5} + 157 \beta_{4} + 2 \beta_{3} + 711 \beta_{2} + 660 \beta_{1} + 561\)
\(\nu^{10}\)\(=\)\(433 \beta_{11} + 847 \beta_{10} + 583 \beta_{9} - 717 \beta_{8} + 155 \beta_{7} + 1269 \beta_{6} - 616 \beta_{5} + 922 \beta_{4} + 233 \beta_{3} + 2765 \beta_{2} + 793 \beta_{1} + 2707\)
\(\nu^{11}\)\(=\)\(2487 \beta_{11} + 2847 \beta_{10} + 1168 \beta_{9} - 1471 \beta_{8} + 689 \beta_{7} + 2403 \beta_{6} - 1582 \beta_{5} + 1409 \beta_{4} + 39 \beta_{3} + 5583 \beta_{2} + 4135 \beta_{1} + 4305\)

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
2.69409
2.07880
2.01091
1.32082
0.852887
0.188928
−0.0557908
−0.811354
−1.05676
−1.72434
−2.22938
−2.26879
−2.69409 0 5.25811 2.61116 0 −3.45538 −8.77762 0 −7.03469
1.2 −2.07880 0 2.32140 2.51557 0 3.31485 −0.668123 0 −5.22935
1.3 −2.01091 0 2.04375 −2.29360 0 −4.38501 −0.0879817 0 4.61222
1.4 −1.32082 0 −0.255442 −0.125081 0 1.14683 2.97903 0 0.165209
1.5 −0.852887 0 −1.27258 −2.61325 0 −1.65804 2.79114 0 2.22881
1.6 −0.188928 0 −1.96431 1.33997 0 −0.912886 0.748968 0 −0.253157
1.7 0.0557908 0 −1.99689 3.85264 0 −0.441454 −0.222990 0 0.214942
1.8 0.811354 0 −1.34170 −0.350684 0 3.06527 −2.71131 0 −0.284529
1.9 1.05676 0 −0.883253 −1.53036 0 −3.09203 −3.04691 0 −1.61722
1.10 1.72434 0 0.973355 0.0158764 0 3.08835 −1.77029 0 0.0273764
1.11 2.22938 0 2.97016 −3.25707 0 −0.940570 2.16285 0 −7.26127
1.12 2.26879 0 3.14741 2.83484 0 −4.72992 2.60323 0 6.43165
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

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

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.e 12
3.b odd 2 1 2013.2.a.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.d 12 3.b odd 2 1
6039.2.a.e 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).