Properties

Label 6027.2.a.bj
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 14
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: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{3} ) q^{5} + \beta_{1} q^{6} + ( -1 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{3} ) q^{5} + \beta_{1} q^{6} + ( -1 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{8} + q^{9} + ( -2 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} - \beta_{13} ) q^{10} + ( -\beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} ) q^{11} + ( -1 - \beta_{2} ) q^{12} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{13} ) q^{13} + ( -1 - \beta_{3} ) q^{15} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{16} + ( 2 - \beta_{1} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{12} + \beta_{13} ) q^{17} -\beta_{1} q^{18} + ( -1 - \beta_{2} - 2 \beta_{6} - \beta_{9} + \beta_{12} - \beta_{13} ) q^{19} + ( 2 + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{20} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{22} + ( -1 + \beta_{1} + \beta_{3} - \beta_{6} + \beta_{10} + \beta_{12} ) q^{23} + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} ) q^{24} + ( 2 + \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{25} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{7} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{26} - q^{27} + ( -1 - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{8} - \beta_{9} ) q^{29} + ( 2 \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} + \beta_{13} ) q^{30} + ( 1 + \beta_{4} + 2 \beta_{7} + 2 \beta_{10} - \beta_{11} - \beta_{13} ) q^{31} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} ) q^{32} + ( \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} ) q^{33} + ( 2 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{11} + \beta_{12} ) q^{34} + ( 1 + \beta_{2} ) q^{36} + ( \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} - \beta_{11} ) q^{37} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{10} ) q^{38} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{10} - \beta_{13} ) q^{39} + ( -2 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{40} + q^{41} + ( 2 - \beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} ) q^{43} + ( 2 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 5 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - 6 \beta_{12} + 2 \beta_{13} ) q^{44} + ( 1 + \beta_{3} ) q^{45} + ( -4 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} ) q^{46} + ( 2 + \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{47} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{48} + ( -1 - 4 \beta_{1} - \beta_{2} - 3 \beta_{5} - \beta_{6} - 2 \beta_{11} - \beta_{12} - \beta_{13} ) q^{50} + ( -2 + \beta_{1} - \beta_{6} - \beta_{8} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{51} + ( 5 + 3 \beta_{2} + \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{52} + ( -1 - \beta_{4} + \beta_{5} + \beta_{11} + \beta_{12} ) q^{53} + \beta_{1} q^{54} + ( 1 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} + \beta_{13} ) q^{55} + ( 1 + \beta_{2} + 2 \beta_{6} + \beta_{9} - \beta_{12} + \beta_{13} ) q^{57} + ( 1 + 4 \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{58} + ( 2 - 2 \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{59} + ( -2 - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{60} + ( 2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} - \beta_{13} ) q^{61} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{62} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - \beta_{13} ) q^{64} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} + 2 \beta_{13} ) q^{65} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{66} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{67} + ( 3 - 7 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} - 4 \beta_{12} + \beta_{13} ) q^{68} + ( 1 - \beta_{1} - \beta_{3} + \beta_{6} - \beta_{10} - \beta_{12} ) q^{69} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{71} + ( -1 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{72} + ( 2 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{5} + 2 \beta_{7} + 2 \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{73} + ( -5 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{74} + ( -2 - \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{75} + ( -2 - 3 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{12} ) q^{76} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} - \beta_{9} - \beta_{11} - \beta_{13} ) q^{78} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - 3 \beta_{10} - \beta_{12} + \beta_{13} ) q^{79} + ( 5 + 5 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} + 4 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} ) q^{80} + q^{81} -\beta_{1} q^{82} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{83} + ( 5 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{85} + ( -3 - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 3 \beta_{10} + 3 \beta_{12} - 3 \beta_{13} ) q^{86} + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{87} + ( -4 + 10 \beta_{1} + \beta_{2} - 4 \beta_{4} + 5 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} + 6 \beta_{11} + 4 \beta_{12} ) q^{88} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{7} - 2 \beta_{9} - \beta_{11} + \beta_{12} ) q^{89} + ( -2 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} - \beta_{13} ) q^{90} + ( 2 + 4 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 3 \beta_{8} - 2 \beta_{9} + \beta_{10} + 4 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} ) q^{92} + ( -1 - \beta_{4} - 2 \beta_{7} - 2 \beta_{10} + \beta_{11} + \beta_{13} ) q^{93} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{94} + ( -7 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{95} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{96} + ( 1 + 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{9} + \beta_{11} + \beta_{12} ) q^{97} + ( -\beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 2q^{2} - 14q^{3} + 14q^{4} + 10q^{5} + 2q^{6} - 6q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 2q^{2} - 14q^{3} + 14q^{4} + 10q^{5} + 2q^{6} - 6q^{8} + 14q^{9} + 3q^{10} - 16q^{11} - 14q^{12} + 21q^{13} - 10q^{15} + 22q^{16} + 12q^{17} - 2q^{18} + 2q^{19} + 40q^{20} + q^{22} - 7q^{23} + 6q^{24} + 22q^{25} + 2q^{26} - 14q^{27} - 16q^{29} - 3q^{30} + 8q^{31} - 19q^{32} + 16q^{33} + 33q^{34} + 14q^{36} + q^{37} + 32q^{38} - 21q^{39} - 13q^{40} + 14q^{41} + 14q^{43} - 36q^{44} + 10q^{45} - 12q^{46} + 12q^{47} - 22q^{48} - q^{50} - 12q^{51} + 60q^{52} - 20q^{53} + 2q^{54} - 11q^{55} - 2q^{57} + 21q^{58} + 25q^{59} - 40q^{60} + 26q^{61} - 33q^{62} + 42q^{64} - 8q^{65} - q^{66} - 22q^{67} + 15q^{68} + 7q^{69} - 36q^{71} - 6q^{72} + 31q^{73} - 65q^{74} - 22q^{75} - 2q^{76} - 2q^{78} + 12q^{79} + 112q^{80} + 14q^{81} - 2q^{82} + 20q^{83} + 40q^{85} - 9q^{86} + 16q^{87} - 54q^{88} + 39q^{89} + 3q^{90} + 63q^{92} - 8q^{93} + 14q^{94} - 55q^{95} + 19q^{96} + 18q^{97} - 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 2 x^{13} - 19 x^{12} + 36 x^{11} + 134 x^{10} - 237 x^{9} - 438 x^{8} + 716 x^{7} + 662 x^{6} - 1007 x^{5} - 384 x^{4} + 579 x^{3} + 44 x^{2} - 112 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-115 \nu^{13} + 8468 \nu^{12} - 24595 \nu^{11} - 121798 \nu^{10} + 409674 \nu^{9} + 523923 \nu^{8} - 2228088 \nu^{7} - 530532 \nu^{6} + 4814290 \nu^{5} - 797891 \nu^{4} - 3817394 \nu^{3} + 1001107 \nu^{2} + 965410 \nu - 272668\)\()/18364\)
\(\beta_{4}\)\(=\)\((\)\(-66 \nu^{13} - 2845 \nu^{12} + 8640 \nu^{11} + 47189 \nu^{10} - 131524 \nu^{9} - 270514 \nu^{8} + 741431 \nu^{7} + 636556 \nu^{6} - 1814842 \nu^{5} - 509978 \nu^{4} + 1847591 \nu^{3} - 89430 \nu^{2} - 575125 \nu + 145640\)\()/9182\)
\(\beta_{5}\)\(=\)\((\)\(433 \nu^{13} - 186 \nu^{12} - 11191 \nu^{11} + 11712 \nu^{10} + 97154 \nu^{9} - 152093 \nu^{8} - 324642 \nu^{7} + 730612 \nu^{6} + 274678 \nu^{5} - 1345959 \nu^{4} + 323980 \nu^{3} + 678535 \nu^{2} - 234844 \nu - 25044\)\()/18364\)
\(\beta_{6}\)\(=\)\((\)\(-433 \nu^{13} + 186 \nu^{12} + 11191 \nu^{11} - 11712 \nu^{10} - 97154 \nu^{9} + 152093 \nu^{8} + 324642 \nu^{7} - 730612 \nu^{6} - 274678 \nu^{5} + 1345959 \nu^{4} - 305616 \nu^{3} - 696899 \nu^{2} + 143024 \nu + 61772\)\()/18364\)
\(\beta_{7}\)\(=\)\((\)\(-657 \nu^{13} + 8616 \nu^{12} - 9569 \nu^{11} - 132302 \nu^{10} + 265034 \nu^{9} + 661925 \nu^{8} - 1635072 \nu^{7} - 1196788 \nu^{6} + 3821874 \nu^{5} + 481015 \nu^{4} - 3322458 \nu^{3} + 193241 \nu^{2} + 922034 \nu - 166252\)\()/18364\)
\(\beta_{8}\)\(=\)\((\)\(-1101 \nu^{13} + 4502 \nu^{12} + 16835 \nu^{11} - 80292 \nu^{10} - 80530 \nu^{9} + 514897 \nu^{8} + 103978 \nu^{7} - 1457088 \nu^{6} + 148858 \nu^{5} + 1748099 \nu^{4} - 330636 \nu^{3} - 628747 \nu^{2} + 56360 \nu - 12872\)\()/18364\)
\(\beta_{9}\)\(=\)\((\)\(2301 \nu^{13} - 7032 \nu^{12} - 42039 \nu^{11} + 138006 \nu^{10} + 266526 \nu^{9} - 1000485 \nu^{8} - 633748 \nu^{7} + 3272368 \nu^{6} + 78546 \nu^{5} - 4549267 \nu^{4} + 1321658 \nu^{3} + 1777283 \nu^{2} - 620446 \nu - 9076\)\()/18364\)
\(\beta_{10}\)\(=\)\((\)\( 720 \nu^{13} - 1518 \nu^{12} - 13286 \nu^{11} + 28201 \nu^{10} + 85888 \nu^{9} - 191269 \nu^{8} - 216860 \nu^{7} + 579567 \nu^{6} + 93287 \nu^{5} - 735873 \nu^{4} + 299871 \nu^{3} + 236449 \nu^{2} - 161239 \nu + 13459 \)\()/4591\)
\(\beta_{11}\)\(=\)\((\)\(-2470 \nu^{13} + 4825 \nu^{12} + 46216 \nu^{11} - 85969 \nu^{10} - 315048 \nu^{9} + 554328 \nu^{8} + 953861 \nu^{7} - 1600106 \nu^{6} - 1192380 \nu^{5} + 2012748 \nu^{4} + 343411 \nu^{3} - 821800 \nu^{2} + 66047 \nu + 57572\)\()/9182\)
\(\beta_{12}\)\(=\)\((\)\(5521 \nu^{13} - 14586 \nu^{12} - 87939 \nu^{11} + 233648 \nu^{10} + 475158 \nu^{9} - 1272953 \nu^{8} - 1042982 \nu^{7} + 2811688 \nu^{6} + 878202 \nu^{5} - 2298535 \nu^{4} - 203184 \nu^{3} + 465907 \nu^{2} - 151836 \nu + 41296\)\()/18364\)
\(\beta_{13}\)\(=\)\((\)\(12521 \nu^{13} - 27814 \nu^{12} - 219659 \nu^{11} + 464720 \nu^{10} + 1391798 \nu^{9} - 2725189 \nu^{8} - 3990986 \nu^{7} + 6893844 \nu^{6} + 5292938 \nu^{5} - 7406035 \nu^{4} - 2959952 \nu^{3} + 2807311 \nu^{2} + 577848 \nu - 261192\)\()/18364\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{13} - \beta_{12} + 2 \beta_{11} + \beta_{10} + 10 \beta_{6} + 10 \beta_{5} + 10 \beta_{2} + 31 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(-\beta_{13} + 13 \beta_{12} + 12 \beta_{11} + \beta_{10} - 11 \beta_{9} - 12 \beta_{8} - 3 \beta_{7} - 13 \beta_{6} + 10 \beta_{5} + 9 \beta_{4} + 13 \beta_{3} + 47 \beta_{2} + 13 \beta_{1} + 86\)
\(\nu^{7}\)\(=\)\(11 \beta_{13} - 9 \beta_{12} + 26 \beta_{11} + 13 \beta_{10} + \beta_{9} + \beta_{8} - 3 \beta_{7} + 80 \beta_{6} + 80 \beta_{5} - \beta_{4} + 3 \beta_{3} + 80 \beta_{2} + 210 \beta_{1} + 90\)
\(\nu^{8}\)\(=\)\(-16 \beta_{13} + 127 \beta_{12} + 112 \beta_{11} + 18 \beta_{10} - 94 \beta_{9} - 113 \beta_{8} - 41 \beta_{7} - 123 \beta_{6} + 80 \beta_{5} + 63 \beta_{4} + 122 \beta_{3} + 320 \beta_{2} + 123 \beta_{1} + 530\)
\(\nu^{9}\)\(=\)\(90 \beta_{13} - 53 \beta_{12} + 254 \beta_{11} + 126 \beta_{10} + 15 \beta_{9} + 8 \beta_{8} - 47 \beta_{7} + 595 \beta_{6} + 604 \beta_{5} - 20 \beta_{4} + 48 \beta_{3} + 598 \beta_{2} + 1484 \beta_{1} + 671\)
\(\nu^{10}\)\(=\)\(-176 \beta_{13} + 1114 \beta_{12} + 957 \beta_{11} + 209 \beta_{10} - 738 \beta_{9} - 968 \beta_{8} - 406 \beta_{7} - 1044 \beta_{6} + 608 \beta_{5} + 403 \beta_{4} + 1022 \beta_{3} + 2212 \beta_{2} + 1048 \beta_{1} + 3419\)
\(\nu^{11}\)\(=\)\(657 \beta_{13} - 200 \beta_{12} + 2234 \beta_{11} + 1103 \beta_{10} + 151 \beta_{9} + 6 \beta_{8} - 512 \beta_{7} + 4282 \beta_{6} + 4480 \beta_{5} - 265 \beta_{4} + 528 \beta_{3} + 4357 \beta_{2} + 10718 \beta_{1} + 4817\)
\(\nu^{12}\)\(=\)\(-1663 \beta_{13} + 9261 \beta_{12} + 7839 \beta_{11} + 2040 \beta_{10} - 5589 \beta_{9} - 7893 \beta_{8} - 3559 \beta_{7} - 8437 \beta_{6} + 4571 \beta_{5} + 2452 \beta_{4} + 8131 \beta_{3} + 15469 \beta_{2} + 8542 \beta_{1} + 22732\)
\(\nu^{13}\)\(=\)\(4513 \beta_{13} + 238 \beta_{12} + 18655 \beta_{11} + 9207 \beta_{10} + 1282 \beta_{9} - 650 \beta_{8} - 4823 \beta_{7} + 30292 \beta_{6} + 33049 \beta_{5} - 2907 \beta_{4} + 5001 \beta_{3} + 31423 \beta_{2} + 78322 \beta_{1} + 34017\)

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
2.73044
2.55282
2.13215
1.64093
1.21066
0.766034
0.442251
0.181424
−0.712717
−0.766355
−1.67187
−1.68292
−2.13271
−2.69016
−2.73044 −1.00000 5.45531 3.67975 2.73044 0 −9.43451 1.00000 −10.0473
1.2 −2.55282 −1.00000 4.51690 1.95019 2.55282 0 −6.42520 1.00000 −4.97848
1.3 −2.13215 −1.00000 2.54608 −2.91322 2.13215 0 −1.16433 1.00000 6.21144
1.4 −1.64093 −1.00000 0.692668 −0.0817657 1.64093 0 2.14525 1.00000 0.134172
1.5 −1.21066 −1.00000 −0.534291 0.826979 1.21066 0 3.06818 1.00000 −1.00119
1.6 −0.766034 −1.00000 −1.41319 −0.561773 0.766034 0 2.61462 1.00000 0.430337
1.7 −0.442251 −1.00000 −1.80441 4.29274 0.442251 0 1.68250 1.00000 −1.89847
1.8 −0.181424 −1.00000 −1.96709 −3.75467 0.181424 0 0.719724 1.00000 0.681187
1.9 0.712717 −1.00000 −1.49203 0.415066 −0.712717 0 −2.48883 1.00000 0.295824
1.10 0.766355 −1.00000 −1.41270 1.12787 −0.766355 0 −2.61534 1.00000 0.864345
1.11 1.67187 −1.00000 0.795136 −2.68319 −1.67187 0 −2.01437 1.00000 −4.48594
1.12 1.68292 −1.00000 0.832222 3.40481 −1.68292 0 −1.96528 1.00000 5.73002
1.13 2.13271 −1.00000 2.54845 0.889988 −2.13271 0 1.16968 1.00000 1.89809
1.14 2.69016 −1.00000 5.23695 3.40724 −2.69016 0 8.70791 1.00000 9.16601
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{14} + \cdots\)
\(T_{5}^{14} - \cdots\)
\(T_{13}^{14} - \cdots\)