Properties

Label 6045.2.a.v
Level 6045
Weight 2
Character orbit 6045.a
Self dual Yes
Analytic conductor 48.270
Analytic rank 0
Dimension 10
CM No
Inner twists 1

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(4\) \(x^{9}\mathstrut -\mathstrut \) \(12\) \(x^{8}\mathstrut +\mathstrut \) \(59\) \(x^{7}\mathstrut +\mathstrut \) \(38\) \(x^{6}\mathstrut -\mathstrut \) \(302\) \(x^{5}\mathstrut +\mathstrut \) \(13\) \(x^{4}\mathstrut +\mathstrut \) \(626\) \(x^{3}\mathstrut -\mathstrut \) \(167\) \(x^{2}\mathstrut -\mathstrut \) \(457\) \(x\mathstrut +\mathstrut \) \(135\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{9} - 7 \nu^{8} - 25 \nu^{7} + 93 \nu^{6} + 95 \nu^{5} - 384 \nu^{4} - 106 \nu^{3} + 474 \nu^{2} + 33 \nu - 40 \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{9} - 7 \nu^{8} - 25 \nu^{7} + 93 \nu^{6} + 100 \nu^{5} - 389 \nu^{4} - 146 \nu^{3} + 499 \nu^{2} + 98 \nu - 50 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{9} - 7 \nu^{8} - 25 \nu^{7} + 93 \nu^{6} + 100 \nu^{5} - 389 \nu^{4} - 151 \nu^{3} + 504 \nu^{2} + 123 \nu - 65 \)\()/5\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{9} - 6 \nu^{8} + 69 \nu^{6} - 115 \nu^{5} - 212 \nu^{4} + 602 \nu^{3} + 42 \nu^{2} - 751 \nu + 225 \)\()/5\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{9} - 13 \nu^{8} - 25 \nu^{7} + 162 \nu^{6} - 15 \nu^{5} - 606 \nu^{4} + 456 \nu^{3} + 581 \nu^{2} - 653 \nu + 125 \)\()/5\)
\(\beta_{8}\)\(=\)\((\)\( -2 \nu^{9} + 2 \nu^{8} + 40 \nu^{7} - 33 \nu^{6} - 290 \nu^{5} + 194 \nu^{4} + 881 \nu^{3} - 459 \nu^{2} - 903 \nu + 305 \)\()/5\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{9} + 4 \nu^{8} - 35 \nu^{7} - 41 \nu^{6} + 330 \nu^{5} + 83 \nu^{4} - 1143 \nu^{3} + 182 \nu^{2} + 1224 \nu - 355 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(22\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(8\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(27\) \(\beta_{1}\mathstrut +\mathstrut \) \(12\)
\(\nu^{6}\)\(=\)\(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{6}\mathstrut -\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(60\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(134\)
\(\nu^{7}\)\(=\)\(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{7}\mathstrut +\mathstrut \) \(11\) \(\beta_{6}\mathstrut -\mathstrut \) \(55\) \(\beta_{5}\mathstrut +\mathstrut \) \(84\) \(\beta_{4}\mathstrut -\mathstrut \) \(17\) \(\beta_{3}\mathstrut +\mathstrut \) \(100\) \(\beta_{2}\mathstrut +\mathstrut \) \(153\) \(\beta_{1}\mathstrut +\mathstrut \) \(113\)
\(\nu^{8}\)\(=\)\(33\) \(\beta_{9}\mathstrut +\mathstrut \) \(47\) \(\beta_{8}\mathstrut -\mathstrut \) \(49\) \(\beta_{7}\mathstrut +\mathstrut \) \(52\) \(\beta_{6}\mathstrut -\mathstrut \) \(44\) \(\beta_{5}\mathstrut +\mathstrut \) \(159\) \(\beta_{4}\mathstrut -\mathstrut \) \(37\) \(\beta_{3}\mathstrut +\mathstrut \) \(445\) \(\beta_{2}\mathstrut +\mathstrut \) \(31\) \(\beta_{1}\mathstrut +\mathstrut \) \(863\)
\(\nu^{9}\)\(=\)\(60\) \(\beta_{9}\mathstrut +\mathstrut \) \(75\) \(\beta_{8}\mathstrut -\mathstrut \) \(69\) \(\beta_{7}\mathstrut +\mathstrut \) \(92\) \(\beta_{6}\mathstrut -\mathstrut \) \(375\) \(\beta_{5}\mathstrut +\mathstrut \) \(679\) \(\beta_{4}\mathstrut -\mathstrut \) \(199\) \(\beta_{3}\mathstrut +\mathstrut \) \(847\) \(\beta_{2}\mathstrut +\mathstrut \) \(894\) \(\beta_{1}\mathstrut +\mathstrut \) \(981\)

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
−2.40552
−2.24384
−1.35516
−1.33219
0.297952
1.60945
1.70000
2.37998
2.55963
2.78970
−2.40552 1.00000 3.78653 −1.00000 −2.40552 −2.80645 −4.29753 1.00000 2.40552
1.2 −2.24384 1.00000 3.03480 −1.00000 −2.24384 3.47187 −2.32191 1.00000 2.24384
1.3 −1.35516 1.00000 −0.163543 −1.00000 −1.35516 −3.94543 2.93195 1.00000 1.35516
1.4 −1.33219 1.00000 −0.225281 −1.00000 −1.33219 −1.08911 2.96449 1.00000 1.33219
1.5 0.297952 1.00000 −1.91122 −1.00000 0.297952 2.52653 −1.16536 1.00000 −0.297952
1.6 1.60945 1.00000 0.590323 −1.00000 1.60945 −4.62845 −2.26880 1.00000 −1.60945
1.7 1.70000 1.00000 0.889984 −1.00000 1.70000 2.93970 −1.88702 1.00000 −1.70000
1.8 2.37998 1.00000 3.66431 −1.00000 2.37998 2.09247 3.96103 1.00000 −2.37998
1.9 2.55963 1.00000 4.55168 −1.00000 2.55963 −3.20072 6.53135 1.00000 −2.55963
1.10 2.78970 1.00000 5.78242 −1.00000 2.78970 3.63959 10.5518 1.00000 −2.78970
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\(T_{2}^{10} - \cdots\)
\(T_{7}^{10} + \cdots\)
\(T_{11}^{10} - \cdots\)