Properties

Label 6027.2.a.bl
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 1
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6027.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1258372982\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{5} ) q^{5} + \beta_{1} q^{6} + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{5} ) q^{5} + \beta_{1} q^{6} + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{8} + q^{9} + ( -\beta_{1} + \beta_{4} + \beta_{5} ) q^{10} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{11} + ( -1 - \beta_{2} ) q^{12} + ( \beta_{5} + \beta_{15} ) q^{13} + ( -1 + \beta_{5} ) q^{15} + ( \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} ) q^{16} + ( 1 - \beta_{4} + \beta_{8} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{17} -\beta_{1} q^{18} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{8} - 2 \beta_{11} - \beta_{13} - \beta_{14} ) q^{19} + ( 2 + \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{13} - \beta_{14} ) q^{20} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{22} + ( -1 + \beta_{1} - \beta_{2} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{23} + ( 1 + \beta_{2} - \beta_{5} + \beta_{6} ) q^{24} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{25} + ( 1 + \beta_{3} - \beta_{4} + \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{26} - q^{27} + ( -1 + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{29} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{30} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{13} + \beta_{15} ) q^{31} + ( -4 + \beta_{1} - 3 \beta_{2} - \beta_{5} - 2 \beta_{10} + \beta_{13} ) q^{32} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{33} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{13} + \beta_{14} ) q^{34} + ( 1 + \beta_{2} ) q^{36} + ( -4 - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} + 2 \beta_{13} - \beta_{15} ) q^{37} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{11} - 3 \beta_{12} + \beta_{13} + \beta_{14} ) q^{38} + ( -\beta_{5} - \beta_{15} ) q^{39} + ( -5 - 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 3 \beta_{13} + \beta_{15} ) q^{40} - q^{41} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{43} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{44} + ( 1 - \beta_{5} ) q^{45} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{46} + ( 4 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{47} + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{48} + ( -3 + \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{50} + ( -1 + \beta_{4} - \beta_{8} - \beta_{11} - \beta_{14} + \beta_{15} ) q^{51} + ( -2 \beta_{2} - \beta_{6} - 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{52} + ( -5 - \beta_{2} + \beta_{4} + 3 \beta_{5} - \beta_{6} - 4 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{53} + \beta_{1} q^{54} + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{12} - 3 \beta_{13} - \beta_{14} - \beta_{15} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{8} + 2 \beta_{11} + \beta_{13} + \beta_{14} ) q^{57} + ( -3 + \beta_{1} - \beta_{3} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} ) q^{58} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{14} ) q^{59} + ( -2 - \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{13} + \beta_{14} ) q^{60} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{61} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} - \beta_{8} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{62} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{13} ) q^{64} + ( -2 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{65} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{66} + ( -2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{10} - 4 \beta_{12} + \beta_{13} ) q^{67} + ( -\beta_{1} + 3 \beta_{2} + 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{68} + ( 1 - \beta_{1} + \beta_{2} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{69} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{71} + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{72} + ( 2 - \beta_{1} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{73} + ( 1 + 3 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{74} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{75} + ( 1 - 2 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{11} - \beta_{12} - 4 \beta_{13} - \beta_{15} ) q^{76} + ( -1 - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{78} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{79} + ( 6 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - 3 \beta_{9} + \beta_{10} + \beta_{12} - 3 \beta_{13} + \beta_{14} - \beta_{15} ) q^{80} + q^{81} + \beta_{1} q^{82} + ( 3 - \beta_{1} + \beta_{5} + 2 \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{83} + ( -3 + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{85} + ( -1 + 2 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{14} ) q^{86} + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{87} + ( -5 - 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{88} + ( -\beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{89} + ( -\beta_{1} + \beta_{4} + \beta_{5} ) q^{90} + ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} + 3 \beta_{13} + \beta_{15} ) q^{92} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{13} - \beta_{15} ) q^{93} + ( -2 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 4 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + \beta_{15} ) q^{94} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{95} + ( 4 - \beta_{1} + 3 \beta_{2} + \beta_{5} + 2 \beta_{10} - \beta_{13} ) q^{96} + ( -4 - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{97} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 4q^{2} - 16q^{3} + 12q^{4} + 12q^{5} + 4q^{6} - 12q^{8} + 16q^{9} + O(q^{10}) \) \( 16q - 4q^{2} - 16q^{3} + 12q^{4} + 12q^{5} + 4q^{6} - 12q^{8} + 16q^{9} + 4q^{10} - 4q^{11} - 12q^{12} - 12q^{15} + 8q^{17} - 4q^{18} - 4q^{19} + 20q^{20} - 16q^{22} - 12q^{23} + 12q^{24} - 8q^{25} + 8q^{26} - 16q^{27} - 16q^{29} - 4q^{30} + 4q^{31} - 48q^{32} + 4q^{33} - 16q^{34} + 12q^{36} - 48q^{37} + 4q^{38} - 56q^{40} - 16q^{41} - 16q^{43} + 12q^{45} - 4q^{46} + 36q^{47} - 8q^{50} - 8q^{51} - 60q^{53} + 4q^{54} - 8q^{55} + 4q^{57} - 36q^{58} + 36q^{59} - 20q^{60} + 4q^{61} + 12q^{62} + 52q^{64} - 36q^{65} + 16q^{66} - 52q^{67} + 8q^{68} + 12q^{69} - 12q^{71} - 12q^{72} + 16q^{73} + 4q^{74} + 8q^{75} - 16q^{76} - 8q^{78} - 36q^{79} + 68q^{80} + 16q^{81} + 4q^{82} + 32q^{83} - 28q^{85} - 8q^{86} + 16q^{87} - 36q^{88} + 12q^{89} + 4q^{90} - 36q^{92} - 4q^{93} - 24q^{94} - 20q^{95} + 48q^{96} - 48q^{97} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} - 14 x^{14} + 68 x^{13} + 64 x^{12} - 456 x^{11} - 54 x^{10} + 1532 x^{9} - 400 x^{8} - 2708 x^{7} + 1218 x^{6} + 2424 x^{5} - 1276 x^{4} - 960 x^{3} + 500 x^{2} + 112 x - 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-2 \nu^{15} + 316 \nu^{14} - 525 \nu^{13} - 5554 \nu^{12} + 7439 \nu^{11} + 39832 \nu^{10} - 38854 \nu^{9} - 146235 \nu^{8} + 97946 \nu^{7} + 285731 \nu^{6} - 129437 \nu^{5} - 281488 \nu^{4} + 89961 \nu^{3} + 120024 \nu^{2} - 26662 \nu - 16891\)\()/1659\)
\(\beta_{4}\)\(=\)\((\)\(344 \nu^{15} - 711 \nu^{14} - 6475 \nu^{13} + 11870 \nu^{12} + 48798 \nu^{11} - 77407 \nu^{10} - 186478 \nu^{9} + 251383 \nu^{8} + 380344 \nu^{7} - 426432 \nu^{6} - 399882 \nu^{5} + 360236 \nu^{4} + 189327 \nu^{3} - 129487 \nu^{2} - 27262 \nu + 13062\)\()/553\)
\(\beta_{5}\)\(=\)\((\)\(615 \nu^{15} - 1501 \nu^{14} - 10822 \nu^{13} + 24523 \nu^{12} + 75753 \nu^{11} - 155889 \nu^{10} - 266506 \nu^{9} + 489196 \nu^{8} + 494579 \nu^{7} - 790497 \nu^{6} - 467859 \nu^{5} + 623019 \nu^{4} + 198515 \nu^{3} - 202558 \nu^{2} - 24545 \nu + 17073\)\()/553\)
\(\beta_{6}\)\(=\)\((\)\(615 \nu^{15} - 1501 \nu^{14} - 10822 \nu^{13} + 24523 \nu^{12} + 75753 \nu^{11} - 155889 \nu^{10} - 266506 \nu^{9} + 489196 \nu^{8} + 494579 \nu^{7} - 790497 \nu^{6} - 467859 \nu^{5} + 623019 \nu^{4} + 199068 \nu^{3} - 203111 \nu^{2} - 26757 \nu + 18179\)\()/553\)
\(\beta_{7}\)\(=\)\((\)\(-137 \nu^{15} + 316 \nu^{14} + 2471 \nu^{13} - 5199 \nu^{12} - 17793 \nu^{11} + 33328 \nu^{10} + 64712 \nu^{9} - 105797 \nu^{8} - 124989 \nu^{7} + 173926 \nu^{6} + 123884 \nu^{5} - 141097 \nu^{4} - 55090 \nu^{3} + 48620 \nu^{2} + 7243 \nu - 4621\)\()/79\)
\(\beta_{8}\)\(=\)\((\)\(-230 \nu^{15} + 553 \nu^{14} + 4089 \nu^{13} - 9080 \nu^{12} - 28999 \nu^{11} + 58088 \nu^{10} + 103836 \nu^{9} - 183812 \nu^{8} - 197609 \nu^{7} + 300321 \nu^{6} + 193949 \nu^{5} - 240319 \nu^{4} - 86909 \nu^{3} + 80223 \nu^{2} + 12184 \nu - 7202\)\()/79\)
\(\beta_{9}\)\(=\)\((\)\(-5042 \nu^{15} + 11929 \nu^{14} + 89943 \nu^{13} - 195436 \nu^{12} - 639991 \nu^{11} + 1246429 \nu^{10} + 2297459 \nu^{9} - 3928944 \nu^{8} - 4373500 \nu^{7} + 6390473 \nu^{6} + 4268299 \nu^{5} - 5090446 \nu^{4} - 1874925 \nu^{3} + 1695969 \nu^{2} + 248333 \nu - 153286\)\()/1659\)
\(\beta_{10}\)\(=\)\((\)\(-5042 \nu^{15} + 11929 \nu^{14} + 89943 \nu^{13} - 195436 \nu^{12} - 639991 \nu^{11} + 1246429 \nu^{10} + 2297459 \nu^{9} - 3928944 \nu^{8} - 4373500 \nu^{7} + 6390473 \nu^{6} + 4268299 \nu^{5} - 5088787 \nu^{4} - 1876584 \nu^{3} + 1686015 \nu^{2} + 253310 \nu - 144991\)\()/1659\)
\(\beta_{11}\)\(=\)\((\)\(2533 \nu^{15} - 5925 \nu^{14} - 45416 \nu^{13} + 97309 \nu^{12} + 324855 \nu^{11} - 622481 \nu^{10} - 1172474 \nu^{9} + 1970354 \nu^{8} + 2243872 \nu^{7} - 3224355 \nu^{6} - 2197811 \nu^{5} + 2591188 \nu^{4} + 960374 \nu^{3} - 872178 \nu^{2} - 122629 \nu + 78449\)\()/553\)
\(\beta_{12}\)\(=\)\((\)\(-2547 \nu^{15} + 5925 \nu^{14} + 45612 \nu^{13} - 96924 \nu^{12} - 325870 \nu^{11} + 617063 \nu^{10} + 1174231 \nu^{9} - 1941640 \nu^{8} - 2241205 \nu^{7} + 3152381 \nu^{6} + 2185400 \nu^{5} - 2504444 \nu^{4} - 949454 \nu^{3} + 828099 \nu^{2} + 120697 \nu - 72814\)\()/553\)
\(\beta_{13}\)\(=\)\((\)\(-1177 \nu^{15} + 2765 \nu^{14} + 21060 \nu^{13} - 45329 \nu^{12} - 150389 \nu^{11} + 289313 \nu^{10} + 542200 \nu^{9} - 912900 \nu^{8} - 1037609 \nu^{7} + 1487065 \nu^{6} + 1018766 \nu^{5} - 1186931 \nu^{4} - 449193 \nu^{3} + 395619 \nu^{2} + 58774 \nu - 35294\)\()/237\)
\(\beta_{14}\)\(=\)\((\)\(9529 \nu^{15} - 22436 \nu^{14} - 170457 \nu^{13} + 367622 \nu^{12} + 1218296 \nu^{11} - 2345744 \nu^{10} - 4404277 \nu^{9} + 7401420 \nu^{8} + 8477201 \nu^{7} - 12056686 \nu^{6} - 8416079 \nu^{5} + 9619586 \nu^{4} + 3790041 \nu^{3} - 3202236 \nu^{2} - 517798 \nu + 286916\)\()/1659\)
\(\beta_{15}\)\(=\)\((\)\(5834 \nu^{15} - 13746 \nu^{14} - 104349 \nu^{13} + 225401 \nu^{12} + 745048 \nu^{11} - 1438573 \nu^{10} - 2687259 \nu^{9} + 4536144 \nu^{8} + 5150163 \nu^{7} - 7373853 \nu^{6} - 5072879 \nu^{5} + 5857389 \nu^{4} + 2249010 \nu^{3} - 1934572 \nu^{2} - 295615 \nu + 170730\)\()/553\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} - \beta_{5} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{10} - \beta_{9} + \beta_{6} - \beta_{5} + 7 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(-\beta_{13} + 2 \beta_{10} + 8 \beta_{6} - 7 \beta_{5} + 11 \beta_{2} + 19 \beta_{1} + 12\)
\(\nu^{6}\)\(=\)\(-\beta_{13} + 10 \beta_{10} - 8 \beta_{9} + \beta_{7} + 12 \beta_{6} - 10 \beta_{5} + \beta_{4} + 48 \beta_{2} + 13 \beta_{1} + 79\)
\(\nu^{7}\)\(=\)\(-10 \beta_{13} - \beta_{11} + 23 \beta_{10} - 3 \beta_{9} + 58 \beta_{6} - 45 \beta_{5} + 2 \beta_{4} + 96 \beta_{2} + 103 \beta_{1} + 109\)
\(\nu^{8}\)\(=\)\(\beta_{14} - 12 \beta_{13} + \beta_{12} + 80 \beta_{10} - 54 \beta_{9} + 2 \beta_{8} + 9 \beta_{7} + 110 \beta_{6} - 81 \beta_{5} + 12 \beta_{4} + 337 \beta_{2} + 123 \beta_{1} + 495\)
\(\nu^{9}\)\(=\)\(\beta_{15} + \beta_{14} - 73 \beta_{13} - 12 \beta_{11} + 199 \beta_{10} - 44 \beta_{9} + 3 \beta_{8} - \beta_{7} + 419 \beta_{6} - 296 \beta_{5} + 28 \beta_{4} - \beta_{3} + 770 \beta_{2} + 619 \beta_{1} + 894\)
\(\nu^{10}\)\(=\)\(3 \beta_{15} + 15 \beta_{14} - 99 \beta_{13} + 12 \beta_{12} - 3 \beta_{11} + 605 \beta_{10} - 355 \beta_{9} + 30 \beta_{8} + 53 \beta_{7} + 916 \beta_{6} - 618 \beta_{5} + 108 \beta_{4} - \beta_{3} + 2413 \beta_{2} + 1028 \beta_{1} + 3297\)
\(\nu^{11}\)\(=\)\(19 \beta_{15} + 24 \beta_{14} - 469 \beta_{13} + \beta_{12} - 98 \beta_{11} + 1571 \beta_{10} - 441 \beta_{9} + 55 \beta_{8} - 25 \beta_{7} + 3055 \beta_{6} - 2006 \beta_{5} + 274 \beta_{4} - 15 \beta_{3} + 5940 \beta_{2} + 4001 \beta_{1} + 6978\)
\(\nu^{12}\)\(=\)\(58 \beta_{15} + 161 \beta_{14} - 689 \beta_{13} + 94 \beta_{12} - 50 \beta_{11} + 4503 \beta_{10} - 2348 \beta_{9} + 310 \beta_{8} + 224 \beta_{7} + 7288 \beta_{6} - 4598 \beta_{5} + 888 \beta_{4} - 20 \beta_{3} + 17500 \beta_{2} + 8086 \beta_{1} + 22773\)
\(\nu^{13}\)\(=\)\(239 \beta_{15} + 339 \beta_{14} - 2779 \beta_{13} + 12 \beta_{12} - 686 \beta_{11} + 11965 \beta_{10} - 3795 \beta_{9} + 661 \beta_{8} - 385 \beta_{7} + 22482 \beta_{6} - 13893 \beta_{5} + 2351 \beta_{4} - 152 \beta_{3} + 44933 \beta_{2} + 27110 \beta_{1} + 53012\)
\(\nu^{14}\)\(=\)\(730 \beta_{15} + 1547 \beta_{14} - 4238 \beta_{13} + 599 \beta_{12} - 527 \beta_{11} + 33376 \beta_{10} - 15762 \beta_{9} + 2807 \beta_{8} + 313 \beta_{7} + 56597 \beta_{6} - 33753 \beta_{5} + 7042 \beta_{4} - 251 \beta_{3} + 127913 \beta_{2} + 61557 \beta_{1} + 160772\)
\(\nu^{15}\)\(=\)\(2528 \beta_{15} + 3805 \beta_{14} - 15189 \beta_{13} + 48 \beta_{12} - 4449 \beta_{11} + 89759 \beta_{10} - 30258 \beta_{9} + 6660 \beta_{8} - 4736 \beta_{7} + 166624 \beta_{6} - 97558 \beta_{5} + 19039 \beta_{4} - 1329 \beta_{3} + 336425 \beta_{2} + 189077 \beta_{1} + 396492\)

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
2.72726
2.69180
1.83908
1.64639
1.63508
1.44011
1.04648
0.631061
0.304034
−0.404293
−0.819942
−1.10456
−1.49558
−1.87592
−2.11474
−2.14627
−2.72726 −1.00000 5.43794 0.416230 2.72726 0 −9.37616 1.00000 −1.13517
1.2 −2.69180 −1.00000 5.24578 3.62847 2.69180 0 −8.73699 1.00000 −9.76712
1.3 −1.83908 −1.00000 1.38223 2.07594 1.83908 0 1.13613 1.00000 −3.81783
1.4 −1.64639 −1.00000 0.710595 −0.0457395 1.64639 0 2.12286 1.00000 0.0753050
1.5 −1.63508 −1.00000 0.673497 −1.18651 1.63508 0 2.16894 1.00000 1.94005
1.6 −1.44011 −1.00000 0.0739309 −3.20961 1.44011 0 2.77376 1.00000 4.62221
1.7 −1.04648 −1.00000 −0.904885 −1.81328 1.04648 0 3.03990 1.00000 1.89755
1.8 −0.631061 −1.00000 −1.60176 1.65938 0.631061 0 2.27293 1.00000 −1.04717
1.9 −0.304034 −1.00000 −1.90756 0.824488 0.304034 0 1.18803 1.00000 −0.250672
1.10 0.404293 −1.00000 −1.83655 3.62653 −0.404293 0 −1.55109 1.00000 1.46618
1.11 0.819942 −1.00000 −1.32769 −0.332064 −0.819942 0 −2.72852 1.00000 −0.272273
1.12 1.10456 −1.00000 −0.779950 3.67747 −1.10456 0 −3.07062 1.00000 4.06198
1.13 1.49558 −1.00000 0.236756 −1.30108 −1.49558 0 −2.63707 1.00000 −1.94586
1.14 1.87592 −1.00000 1.51907 1.33107 −1.87592 0 −0.902182 1.00000 2.49697
1.15 2.11474 −1.00000 2.47212 0.285010 −2.11474 0 0.998417 1.00000 0.602721
1.16 2.14627 −1.00000 2.60647 2.36369 −2.14627 0 1.30166 1.00000 5.07312
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(41\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{16} + \cdots\)
\(T_{5}^{16} - \cdots\)
\(T_{13}^{16} - \cdots\)