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: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

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

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

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

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.

Atkin-Lehner signs

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

Hecke kernels

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