Properties

Label 6039.2.a.f
Level $6039$
Weight $2$
Character orbit 6039.a
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
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: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 5 x^{11} - 5 x^{10} + 48 x^{9} - 173 x^{7} + 29 x^{6} + 281 x^{5} - 41 x^{4} - 201 x^{3} + 8 x^{2} + 49 x + 8\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{9} ) q^{5} + ( -1 - \beta_{2} + \beta_{6} + \beta_{9} ) q^{7} + ( 3 - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{9} ) q^{5} + ( -1 - \beta_{2} + \beta_{6} + \beta_{9} ) q^{7} + ( 3 - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{8} + ( 1 - \beta_{1} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{10} - q^{11} + ( -1 - \beta_{1} - 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{13} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{10} + \beta_{11} ) q^{14} + ( 4 - \beta_{3} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{16} + ( 3 - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{17} + ( -1 + 2 \beta_{3} - \beta_{7} - \beta_{8} - \beta_{11} ) q^{19} + ( 3 - \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{20} + ( -1 + \beta_{1} ) q^{22} + ( 1 + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{23} + ( 1 - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{25} + ( \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{10} + \beta_{11} ) q^{26} + ( -4 - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{28} + ( 3 - \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{29} + ( 3 + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{9} - \beta_{11} ) q^{31} + ( 6 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{32} + ( 4 - 4 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{34} + ( -1 - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{35} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{37} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{5} - 3 \beta_{7} - \beta_{9} + \beta_{10} ) q^{38} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{40} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{41} + ( -3 - 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} - 4 \beta_{8} ) q^{43} + ( -2 + \beta_{1} - \beta_{2} ) q^{44} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{9} + 3 \beta_{10} ) q^{46} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{47} + ( 1 + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{49} + ( 4 + \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{7} + 3 \beta_{8} + \beta_{9} - 3 \beta_{10} ) q^{50} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{52} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{53} + ( -1 - \beta_{9} ) q^{55} + ( -5 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{56} + ( 4 - 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{58} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} - 4 \beta_{11} ) q^{59} - q^{61} + ( 7 + 2 \beta_{2} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{62} + ( 5 - 3 \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} - 4 \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{64} + ( 7 - \beta_{1} + \beta_{3} - 4 \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{65} + ( -1 - \beta_{1} + \beta_{5} + 2 \beta_{7} - 3 \beta_{8} ) q^{67} + ( 11 - 5 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{68} + ( -2 + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} ) q^{70} + ( 1 - 3 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + 3 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{71} + ( -4 + 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{73} + ( -4 + \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{11} ) q^{74} + ( 4 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{76} + ( 1 + \beta_{2} - \beta_{6} - \beta_{9} ) q^{77} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{6} - \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{79} + ( 5 - 4 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - 4 \beta_{10} + \beta_{11} ) q^{80} + ( -3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{5} - 5 \beta_{6} - \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{82} + ( 1 - 3 \beta_{1} - \beta_{2} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{10} ) q^{83} + ( 5 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{85} + ( 1 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{86} + ( -3 + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{88} + ( 5 + \beta_{2} - \beta_{6} + 2 \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{89} + ( 3 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + 3 \beta_{11} ) q^{91} + ( 2 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 8 \beta_{6} + 3 \beta_{8} + \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{92} + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + 3 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{94} + ( -5 + \beta_{2} + \beta_{3} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} - 3 \beta_{8} - 5 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{95} + ( -1 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - \beta_{7} + 4 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{97} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - 8 \beta_{5} + \beta_{6} + 5 \beta_{7} + 4 \beta_{8} + 3 \beta_{9} - 6 \beta_{10} - \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 7q^{2} + 13q^{4} + 7q^{5} - 15q^{7} + 18q^{8} + O(q^{10}) \) \( 12q + 7q^{2} + 13q^{4} + 7q^{5} - 15q^{7} + 18q^{8} - 6q^{10} - 12q^{11} - 11q^{13} - 3q^{14} + 19q^{16} + 33q^{17} - 24q^{19} + 11q^{20} - 7q^{22} + 9q^{23} + 11q^{25} + 16q^{26} - 41q^{28} + 16q^{29} + q^{31} + 28q^{32} + 32q^{34} + 22q^{35} - 6q^{37} - 12q^{38} + 26q^{40} + 21q^{41} - 39q^{43} - 13q^{44} + 18q^{47} + 31q^{49} + 44q^{50} + 3q^{52} + 14q^{53} - 7q^{55} - 16q^{56} + 33q^{58} + 23q^{59} - 12q^{61} + 25q^{62} + 12q^{64} + 29q^{65} + 96q^{68} + 44q^{70} + 19q^{71} - 42q^{73} - 38q^{74} + 11q^{76} + 15q^{77} - 11q^{79} + 44q^{80} - 14q^{82} + 56q^{83} + 16q^{85} + 18q^{86} - 18q^{88} + 55q^{89} + 11q^{91} + 4q^{92} - 5q^{94} - 15q^{95} - 7q^{97} - 6q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{11} - 5 x^{10} + 48 x^{9} - 173 x^{7} + 29 x^{6} + 281 x^{5} - 41 x^{4} - 201 x^{3} + 8 x^{2} + 49 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -18 \nu^{11} + 63 \nu^{10} + 341 \nu^{9} - 1448 \nu^{8} - 920 \nu^{7} + 7681 \nu^{6} - 1364 \nu^{5} - 15242 \nu^{4} + 6045 \nu^{3} + 11277 \nu^{2} - 4356 \nu - 2408 \)\()/313\)
\(\beta_{4}\)\(=\)\((\)\( 14 \nu^{11} - 49 \nu^{10} + 13 \nu^{9} - 404 \nu^{8} + 646 \nu^{7} + 4494 \nu^{6} - 5060 \nu^{5} - 12420 \nu^{4} + 8653 \nu^{3} + 11261 \nu^{2} - 4124 \nu - 2370 \)\()/313\)
\(\beta_{5}\)\(=\)\((\)\( -41 \nu^{11} - 13 \nu^{10} + 1281 \nu^{9} - 1142 \nu^{8} - 8599 \nu^{7} + 6871 \nu^{6} + 23359 \nu^{5} - 10130 \nu^{4} - 26034 \nu^{3} + 1742 \nu^{2} + 9797 \nu + 2201 \)\()/313\)
\(\beta_{6}\)\(=\)\((\)\( -110 \nu^{11} + 385 \nu^{10} + 1284 \nu^{9} - 4293 \nu^{8} - 5970 \nu^{7} + 15396 \nu^{6} + 14583 \nu^{5} - 19521 \nu^{4} - 15851 \nu^{3} + 7254 \nu^{2} + 5306 \nu + 65 \)\()/313\)
\(\beta_{7}\)\(=\)\((\)\( 98 \nu^{11} - 656 \nu^{10} + 404 \nu^{9} + 4997 \nu^{8} - 7059 \nu^{7} - 13927 \nu^{6} + 22798 \nu^{5} + 17915 \nu^{4} - 26756 \nu^{3} - 11317 \nu^{2} + 9631 \nu + 3442 \)\()/313\)
\(\beta_{8}\)\(=\)\((\)\( 121 \nu^{11} - 580 \nu^{10} - 536 \nu^{9} + 4691 \nu^{8} + 620 \nu^{7} - 13117 \nu^{6} - 1925 \nu^{5} + 13116 \nu^{4} + 4384 \nu^{3} - 2721 \nu^{2} - 2331 \nu - 854 \)\()/313\)
\(\beta_{9}\)\(=\)\((\)\( -134 \nu^{11} + 782 \nu^{10} - 35 \nu^{9} - 6015 \nu^{8} + 4593 \nu^{7} + 16769 \nu^{6} - 13319 \nu^{5} - 19916 \nu^{4} + 10989 \nu^{3} + 8831 \nu^{2} - 1128 \nu - 1059 \)\()/313\)
\(\beta_{10}\)\(=\)\((\)\( 154 \nu^{11} - 539 \nu^{10} - 1735 \nu^{9} + 5572 \nu^{8} + 8984 \nu^{7} - 20052 \nu^{6} - 24047 \nu^{5} + 26766 \nu^{4} + 26323 \nu^{3} - 10719 \nu^{2} - 8430 \nu - 404 \)\()/313\)
\(\beta_{11}\)\(=\)\((\)\( -236 \nu^{11} + 1139 \nu^{10} + 1167 \nu^{9} - 10047 \nu^{8} - 1142 \nu^{7} + 32229 \nu^{6} + 27 \nu^{5} - 42957 \nu^{4} - 454 \nu^{3} + 21402 \nu^{2} + 167 \nu - 1767 \)\()/313\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + 3 \beta_{2} + 4 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(3 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} - \beta_{7} - 3 \beta_{6} + 4 \beta_{5} - \beta_{3} + 12 \beta_{2} + 8 \beta_{1} + 17\)
\(\nu^{5}\)\(=\)\(14 \beta_{11} + 13 \beta_{10} - 14 \beta_{9} - 8 \beta_{8} - 4 \beta_{7} - 12 \beta_{6} + 16 \beta_{5} + \beta_{4} - 6 \beta_{3} + 38 \beta_{2} + 26 \beta_{1} + 36\)
\(\nu^{6}\)\(=\)\(46 \beta_{11} + 42 \beta_{10} - 48 \beta_{9} - 22 \beta_{8} - 20 \beta_{7} - 40 \beta_{6} + 58 \beta_{5} + 5 \beta_{4} - 27 \beta_{3} + 130 \beta_{2} + 68 \beta_{1} + 132\)
\(\nu^{7}\)\(=\)\(166 \beta_{11} + 148 \beta_{10} - 177 \beta_{9} - 75 \beta_{8} - 75 \beta_{7} - 137 \beta_{6} + 203 \beta_{5} + 26 \beta_{4} - 110 \beta_{3} + 421 \beta_{2} + 210 \beta_{1} + 364\)
\(\nu^{8}\)\(=\)\(549 \beta_{11} + 483 \beta_{10} - 606 \beta_{9} - 237 \beta_{8} - 287 \beta_{7} - 458 \beta_{6} + 697 \beta_{5} + 109 \beta_{4} - 414 \beta_{3} + 1392 \beta_{2} + 612 \beta_{1} + 1192\)
\(\nu^{9}\)\(=\)\(1852 \beta_{11} + 1613 \beta_{10} - 2093 \beta_{9} - 797 \beta_{8} - 1020 \beta_{7} - 1526 \beta_{6} + 2355 \beta_{5} + 441 \beta_{4} - 1507 \beta_{3} + 4537 \beta_{2} + 1893 \beta_{1} + 3611\)
\(\nu^{10}\)\(=\)\(6095 \beta_{11} + 5266 \beta_{10} - 7064 \beta_{9} - 2636 \beta_{8} - 3600 \beta_{7} - 5065 \beta_{6} + 7896 \beta_{5} + 1672 \beta_{4} - 5323 \beta_{3} + 14873 \beta_{2} + 5798 \beta_{1} + 11557\)
\(\nu^{11}\)\(=\)\(20133 \beta_{11} + 17302 \beta_{10} - 23796 \beta_{9} - 8850 \beta_{8} - 12387 \beta_{7} - 16746 \beta_{6} + 26291 \beta_{5} + 6167 \beta_{4} - 18491 \beta_{3} + 48577 \beta_{2} + 18190 \beta_{1} + 36321\)

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
3.27893
2.66112
2.22712
1.23554
1.10918
0.858571
−0.199068
−0.456328
−1.03538
−1.31792
−1.62825
−1.73351
−2.27893 0 3.19351 1.72942 0 −5.05524 −2.71992 0 −3.94122
1.2 −1.66112 0 0.759316 1.47319 0 −2.81455 2.06092 0 −2.44715
1.3 −1.22712 0 −0.494178 1.92536 0 2.05062 3.06065 0 −2.36265
1.4 −0.235537 0 −1.94452 −2.04419 0 −0.723848 0.929081 0 0.481482
1.5 −0.109182 0 −1.98808 −0.0420221 0 −3.35497 0.435426 0 0.00458805
1.6 0.141429 0 −1.98000 4.38024 0 3.47902 −0.562887 0 0.619492
1.7 1.19907 0 −0.562235 −0.908382 0 −2.10536 −3.07229 0 −1.08921
1.8 1.45633 0 0.120890 0.271377 0 −0.415334 −2.73660 0 0.395214
1.9 2.03538 0 2.14278 −3.64320 0 −4.68983 0.290605 0 −7.41529
1.10 2.31792 0 3.37276 2.47951 0 3.26859 3.18195 0 5.74732
1.11 2.62825 0 4.90768 −2.26825 0 −0.677820 7.64209 0 −5.96152
1.12 2.73351 0 5.47208 3.64694 0 −3.96128 9.49098 0 9.96895
\(n\): 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.f 12
3.b odd 2 1 2013.2.a.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.c 12 3.b odd 2 1
6039.2.a.f 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).