Properties

Label 6045.2.a.w
Level 6045
Weight 2
Character orbit 6045.a
Self dual Yes
Analytic conductor 48.270
Analytic rank 1
Dimension 11
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6045.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2695680219\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \(- q^{3}\) \( + ( 1 + \beta_{2} ) q^{4} \) \(- q^{5}\) \( -\beta_{1} q^{6} \) \( + \beta_{9} q^{7} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{8} - \beta_{9} ) q^{8} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \(- q^{3}\) \( + ( 1 + \beta_{2} ) q^{4} \) \(- q^{5}\) \( -\beta_{1} q^{6} \) \( + \beta_{9} q^{7} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{8} - \beta_{9} ) q^{8} \) \(+ q^{9}\) \( -\beta_{1} q^{10} \) \( + ( -\beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} ) q^{11} \) \( + ( -1 - \beta_{2} ) q^{12} \) \(- q^{13}\) \( + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{10} ) q^{14} \) \(+ q^{15}\) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{16} \) \( + ( -1 - \beta_{1} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{17} \) \( + \beta_{1} q^{18} \) \( + ( \beta_{3} - \beta_{4} - \beta_{7} ) q^{19} \) \( + ( -1 - \beta_{2} ) q^{20} \) \( -\beta_{9} q^{21} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{22} \) \( + ( 2 - \beta_{1} + \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{23} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{8} + \beta_{9} ) q^{24} \) \(+ q^{25}\) \( -\beta_{1} q^{26} \) \(- q^{27}\) \( + ( 1 + \beta_{4} ) q^{28} \) \( + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{10} ) q^{29} \) \( + \beta_{1} q^{30} \) \(- q^{31}\) \( + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{32} \) \( + ( \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} ) q^{33} \) \( + ( -1 - 2 \beta_{2} + \beta_{4} + \beta_{10} ) q^{34} \) \( -\beta_{9} q^{35} \) \( + ( 1 + \beta_{2} ) q^{36} \) \( + ( -\beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{37} \) \( + ( 1 - 2 \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{38} \) \(+ q^{39}\) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{8} + \beta_{9} ) q^{40} \) \( + ( 2 - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{7} - \beta_{10} ) q^{41} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{10} ) q^{42} \) \( + ( \beta_{1} - 2 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{10} ) q^{43} \) \( + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{44} \) \(- q^{45}\) \( + ( -2 + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{46} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{9} - \beta_{10} ) q^{47} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{48} \) \( + ( -3 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{49} \) \( + \beta_{1} q^{50} \) \( + ( 1 + \beta_{1} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{51} \) \( + ( -1 - \beta_{2} ) q^{52} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{10} ) q^{53} \) \( -\beta_{1} q^{54} \) \( + ( \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} ) q^{55} \) \( + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{56} \) \( + ( -\beta_{3} + \beta_{4} + \beta_{7} ) q^{57} \) \( + ( -1 - \beta_{2} - \beta_{3} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{58} \) \( + ( -1 - \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{59} \) \( + ( 1 + \beta_{2} ) q^{60} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{8} ) q^{61} \) \( -\beta_{1} q^{62} \) \( + \beta_{9} q^{63} \) \( + ( 1 + \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{6} ) q^{64} \) \(+ q^{65}\) \( + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{66} \) \( + ( \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{67} \) \( + ( -1 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{68} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{69} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{10} ) q^{70} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{6} - 2 \beta_{7} + \beta_{9} - \beta_{10} ) q^{71} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{8} - \beta_{9} ) q^{72} \) \( + ( -1 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{8} - 3 \beta_{10} ) q^{73} \) \( + ( -\beta_{1} + 2 \beta_{3} - 5 \beta_{4} + \beta_{5} - 2 \beta_{6} - 4 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{74} \) \(- q^{75}\) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{76} \) \( + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{77} \) \( + \beta_{1} q^{78} \) \( + ( -4 - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{79} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{80} \) \(+ q^{81}\) \( + ( -4 - 2 \beta_{2} + 3 \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{82} \) \( + ( -2 + \beta_{2} - 2 \beta_{4} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{83} \) \( + ( -1 - \beta_{4} ) q^{84} \) \( + ( 1 + \beta_{1} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{85} \) \( + ( 2 - \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} ) q^{86} \) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{9} - \beta_{10} ) q^{87} \) \( + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{88} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{89} \) \( -\beta_{1} q^{90} \) \( -\beta_{9} q^{91} \) \( + ( -1 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + 2 \beta_{10} ) q^{92} \) \(+ q^{93}\) \( + ( -2 - 4 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{10} ) q^{94} \) \( + ( -\beta_{3} + \beta_{4} + \beta_{7} ) q^{95} \) \( + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{96} \) \( + ( -1 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - 4 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{97} \) \( + ( -2 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{98} \) \( + ( -\beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(11q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 11q^{3} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 11q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 11q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(11q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 11q^{3} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 11q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 11q^{9} \) \(\mathstrut -\mathstrut 2q^{10} \) \(\mathstrut -\mathstrut 12q^{12} \) \(\mathstrut -\mathstrut 11q^{13} \) \(\mathstrut +\mathstrut 5q^{14} \) \(\mathstrut +\mathstrut 11q^{15} \) \(\mathstrut -\mathstrut 10q^{16} \) \(\mathstrut -\mathstrut 3q^{17} \) \(\mathstrut +\mathstrut 2q^{18} \) \(\mathstrut -\mathstrut 8q^{19} \) \(\mathstrut -\mathstrut 12q^{20} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 3q^{22} \) \(\mathstrut +\mathstrut 11q^{23} \) \(\mathstrut -\mathstrut 3q^{24} \) \(\mathstrut +\mathstrut 11q^{25} \) \(\mathstrut -\mathstrut 2q^{26} \) \(\mathstrut -\mathstrut 11q^{27} \) \(\mathstrut +\mathstrut 14q^{28} \) \(\mathstrut -\mathstrut 14q^{29} \) \(\mathstrut +\mathstrut 2q^{30} \) \(\mathstrut -\mathstrut 11q^{31} \) \(\mathstrut +\mathstrut 8q^{32} \) \(\mathstrut -\mathstrut 11q^{34} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 12q^{36} \) \(\mathstrut +\mathstrut 7q^{37} \) \(\mathstrut +\mathstrut 8q^{38} \) \(\mathstrut +\mathstrut 11q^{39} \) \(\mathstrut -\mathstrut 3q^{40} \) \(\mathstrut +\mathstrut 22q^{41} \) \(\mathstrut -\mathstrut 5q^{42} \) \(\mathstrut -\mathstrut 5q^{43} \) \(\mathstrut -\mathstrut 13q^{44} \) \(\mathstrut -\mathstrut 11q^{45} \) \(\mathstrut -\mathstrut 22q^{46} \) \(\mathstrut +\mathstrut 5q^{47} \) \(\mathstrut +\mathstrut 10q^{48} \) \(\mathstrut -\mathstrut 33q^{49} \) \(\mathstrut +\mathstrut 2q^{50} \) \(\mathstrut +\mathstrut 3q^{51} \) \(\mathstrut -\mathstrut 12q^{52} \) \(\mathstrut +\mathstrut 4q^{53} \) \(\mathstrut -\mathstrut 2q^{54} \) \(\mathstrut -\mathstrut 13q^{56} \) \(\mathstrut +\mathstrut 8q^{57} \) \(\mathstrut -\mathstrut 18q^{58} \) \(\mathstrut -\mathstrut 3q^{59} \) \(\mathstrut +\mathstrut 12q^{60} \) \(\mathstrut -\mathstrut 28q^{61} \) \(\mathstrut -\mathstrut 2q^{62} \) \(\mathstrut +\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 3q^{64} \) \(\mathstrut +\mathstrut 11q^{65} \) \(\mathstrut +\mathstrut 3q^{66} \) \(\mathstrut -\mathstrut 11q^{67} \) \(\mathstrut -\mathstrut 9q^{68} \) \(\mathstrut -\mathstrut 11q^{69} \) \(\mathstrut -\mathstrut 5q^{70} \) \(\mathstrut +\mathstrut 5q^{71} \) \(\mathstrut +\mathstrut 3q^{72} \) \(\mathstrut -\mathstrut 3q^{73} \) \(\mathstrut -\mathstrut 12q^{74} \) \(\mathstrut -\mathstrut 11q^{75} \) \(\mathstrut -\mathstrut 36q^{76} \) \(\mathstrut +\mathstrut 18q^{77} \) \(\mathstrut +\mathstrut 2q^{78} \) \(\mathstrut -\mathstrut 43q^{79} \) \(\mathstrut +\mathstrut 10q^{80} \) \(\mathstrut +\mathstrut 11q^{81} \) \(\mathstrut -\mathstrut 15q^{82} \) \(\mathstrut -\mathstrut 28q^{83} \) \(\mathstrut -\mathstrut 14q^{84} \) \(\mathstrut +\mathstrut 3q^{85} \) \(\mathstrut +\mathstrut 10q^{86} \) \(\mathstrut +\mathstrut 14q^{87} \) \(\mathstrut -\mathstrut 43q^{88} \) \(\mathstrut -\mathstrut 25q^{89} \) \(\mathstrut -\mathstrut 2q^{90} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut +\mathstrut 7q^{92} \) \(\mathstrut +\mathstrut 11q^{93} \) \(\mathstrut -\mathstrut 16q^{94} \) \(\mathstrut +\mathstrut 8q^{95} \) \(\mathstrut -\mathstrut 8q^{96} \) \(\mathstrut -\mathstrut 6q^{97} \) \(\mathstrut -\mathstrut 14q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11}\mathstrut -\mathstrut \) \(2\) \(x^{10}\mathstrut -\mathstrut \) \(15\) \(x^{9}\mathstrut +\mathstrut \) \(29\) \(x^{8}\mathstrut +\mathstrut \) \(81\) \(x^{7}\mathstrut -\mathstrut \) \(151\) \(x^{6}\mathstrut -\mathstrut \) \(192\) \(x^{5}\mathstrut +\mathstrut \) \(345\) \(x^{4}\mathstrut +\mathstrut \) \(199\) \(x^{3}\mathstrut -\mathstrut \) \(344\) \(x^{2}\mathstrut -\mathstrut \) \(70\) \(x\mathstrut +\mathstrut \) \(118\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{10} - 3 \nu^{9} + 18 \nu^{8} + 43 \nu^{7} - 109 \nu^{6} - 205 \nu^{5} + 256 \nu^{4} + 359 \nu^{3} - 204 \nu^{2} - 190 \nu + 38 \)\()/9\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{10} + 15 \nu^{8} + \nu^{7} - 79 \nu^{6} - 10 \nu^{5} + 175 \nu^{4} + 29 \nu^{3} - 159 \nu^{2} - 22 \nu + 47 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{10} + 6 \nu^{9} + 9 \nu^{8} - 83 \nu^{7} - 10 \nu^{6} + 389 \nu^{5} - 77 \nu^{4} - 694 \nu^{3} + 156 \nu^{2} + 377 \nu - 52 \)\()/9\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{10} + 3 \nu^{9} - 18 \nu^{8} - 43 \nu^{7} + 118 \nu^{6} + 205 \nu^{5} - 346 \nu^{4} - 359 \nu^{3} + 447 \nu^{2} + 181 \nu - 200 \)\()/9\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{10} + 3 \nu^{9} + 81 \nu^{8} - 37 \nu^{7} - 473 \nu^{6} + 154 \nu^{5} + 1208 \nu^{4} - 257 \nu^{3} - 1335 \nu^{2} + 139 \nu + 505 \)\()/9\)
\(\beta_{8}\)\(=\)\((\)\( 5 \nu^{10} - 3 \nu^{9} - 81 \nu^{8} + 37 \nu^{7} + 482 \nu^{6} - 154 \nu^{5} - 1289 \nu^{4} + 248 \nu^{3} + 1524 \nu^{2} - 112 \nu - 613 \)\()/9\)
\(\beta_{9}\)\(=\)\((\)\( -2 \nu^{10} + 33 \nu^{8} + 2 \nu^{7} - 197 \nu^{6} - 17 \nu^{5} + 515 \nu^{4} + 34 \nu^{3} - 573 \nu^{2} - 11 \nu + 211 \)\()/3\)
\(\beta_{10}\)\(=\)\( \nu^{10} - \nu^{9} - 15 \nu^{8} + 12 \nu^{7} + 80 \nu^{6} - 47 \nu^{5} - 183 \nu^{4} + 68 \nu^{3} + 179 \nu^{2} - 30 \nu - 60 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(13\)
\(\nu^{5}\)\(=\)\(-\)\(7\) \(\beta_{9}\mathstrut -\mathstrut \) \(7\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(27\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)
\(\nu^{6}\)\(=\)\(-\)\(10\) \(\beta_{9}\mathstrut +\mathstrut \) \(10\) \(\beta_{7}\mathstrut -\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(43\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(67\)
\(\nu^{7}\)\(=\)\(-\)\(\beta_{10}\mathstrut -\mathstrut \) \(43\) \(\beta_{9}\mathstrut -\mathstrut \) \(42\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut -\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(52\) \(\beta_{3}\mathstrut +\mathstrut \) \(64\) \(\beta_{2}\mathstrut +\mathstrut \) \(151\) \(\beta_{1}\mathstrut +\mathstrut \) \(56\)
\(\nu^{8}\)\(=\)\(-\)\(75\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(75\) \(\beta_{7}\mathstrut -\mathstrut \) \(61\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(258\) \(\beta_{2}\mathstrut +\mathstrut \) \(90\) \(\beta_{1}\mathstrut +\mathstrut \) \(372\)
\(\nu^{9}\)\(=\)\(-\)\(14\) \(\beta_{10}\mathstrut -\mathstrut \) \(259\) \(\beta_{9}\mathstrut -\mathstrut \) \(244\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(87\) \(\beta_{6}\mathstrut +\mathstrut \) \(60\) \(\beta_{5}\mathstrut -\mathstrut \) \(89\) \(\beta_{4}\mathstrut +\mathstrut \) \(318\) \(\beta_{3}\mathstrut +\mathstrut \) \(423\) \(\beta_{2}\mathstrut +\mathstrut \) \(860\) \(\beta_{1}\mathstrut +\mathstrut \) \(379\)
\(\nu^{10}\)\(=\)\(-\)\(\beta_{10}\mathstrut -\mathstrut \) \(512\) \(\beta_{9}\mathstrut -\mathstrut \) \(16\) \(\beta_{8}\mathstrut +\mathstrut \) \(510\) \(\beta_{7}\mathstrut -\mathstrut \) \(380\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut -\mathstrut \) \(19\) \(\beta_{4}\mathstrut +\mathstrut \) \(117\) \(\beta_{3}\mathstrut +\mathstrut \) \(1542\) \(\beta_{2}\mathstrut +\mathstrut \) \(660\) \(\beta_{1}\mathstrut +\mathstrut \) \(2137\)

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
−2.35722
−2.18294
−1.45401
−1.25770
−0.849375
0.860232
1.07960
1.58613
1.65932
2.41951
2.49644
−2.35722 −1.00000 3.55647 −1.00000 2.35722 1.32351 −3.66893 1.00000 2.35722
1.2 −2.18294 −1.00000 2.76522 −1.00000 2.18294 0.121846 −1.67042 1.00000 2.18294
1.3 −1.45401 −1.00000 0.114145 −1.00000 1.45401 3.05854 2.74205 1.00000 1.45401
1.4 −1.25770 −1.00000 −0.418196 −1.00000 1.25770 −3.31907 3.04136 1.00000 1.25770
1.5 −0.849375 −1.00000 −1.27856 −1.00000 0.849375 −1.45897 2.78473 1.00000 0.849375
1.6 0.860232 −1.00000 −1.26000 −1.00000 −0.860232 1.16222 −2.80436 1.00000 −0.860232
1.7 1.07960 −1.00000 −0.834458 −1.00000 −1.07960 −1.05546 −3.06009 1.00000 −1.07960
1.8 1.58613 −1.00000 0.515822 −1.00000 −1.58613 3.54182 −2.35411 1.00000 −1.58613
1.9 1.65932 −1.00000 0.753351 −1.00000 −1.65932 −0.633348 −2.06859 1.00000 −1.65932
1.10 2.41951 −1.00000 3.85401 −1.00000 −2.41951 1.95823 4.48578 1.00000 −2.41951
1.11 2.49644 −1.00000 4.23221 −1.00000 −2.49644 −0.699311 5.57257 1.00000 −2.49644
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(13\) \(1\)
\(31\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\(T_{2}^{11} - \cdots\)
\(T_{7}^{11} - \cdots\)
\(T_{11}^{11} - \cdots\)