Properties

Label 2009.2.a.u
Level $2009$
Weight $2$
Character orbit 2009.a
Self dual yes
Analytic conductor $16.042$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2009.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0419457661\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 2 x^{19} - 27 x^{18} + 52 x^{17} + 302 x^{16} - 552 x^{15} - 1820 x^{14} + 3090 x^{13} + 6449 x^{12} - 9852 x^{11} - 13797 x^{10} + 18080 x^{9} + 17721 x^{8} - 18446 x^{7} - 13352 x^{6} + 9524 x^{5} + 5690 x^{4} - 1972 x^{3} - 1179 x^{2} + 42 x + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 2 x^{19} - 27 x^{18} + 52 x^{17} + 302 x^{16} - 552 x^{15} - 1820 x^{14} + 3090 x^{13} + 6449 x^{12} - 9852 x^{11} - 13797 x^{10} + 18080 x^{9} + 17721 x^{8} - 18446 x^{7} - 13352 x^{6} + 9524 x^{5} + 5690 x^{4} - 1972 x^{3} - 1179 x^{2} + 42 x + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-3 \nu^{19} + 125 \nu^{18} + 257 \nu^{17} - 3106 \nu^{16} - 5497 \nu^{15} + 30507 \nu^{14} + 53542 \nu^{13} - 148929 \nu^{12} - 277250 \nu^{11} + 366927 \nu^{10} + 789606 \nu^{9} - 380011 \nu^{8} - 1196150 \nu^{7} - 21401 \nu^{6} + 878553 \nu^{5} + 280878 \nu^{4} - 249231 \nu^{3} - 132531 \nu^{2} + 3957 \nu + 6860\)\()/844\)
\(\beta_{4}\)\(=\)\((\)\(974 \nu^{19} - 1689 \nu^{18} - 24289 \nu^{17} + 44215 \nu^{16} + 241298 \nu^{15} - 475649 \nu^{14} - 1207269 \nu^{13} + 2725642 \nu^{12} + 3116367 \nu^{11} - 9030010 \nu^{10} - 3497403 \nu^{9} + 17565750 \nu^{8} - 243673 \nu^{7} - 19463382 \nu^{6} + 3390237 \nu^{5} + 11384027 \nu^{4} - 1618184 \nu^{3} - 3052103 \nu^{2} - 57319 \nu + 207375\)\()/11816\)
\(\beta_{5}\)\(=\)\((\)\(-140 \nu^{19} + 277 \nu^{18} + 3905 \nu^{17} - 7023 \nu^{16} - 45386 \nu^{15} + 71783 \nu^{14} + 285307 \nu^{13} - 379058 \nu^{12} - 1051789 \nu^{11} + 1102030 \nu^{10} + 2298507 \nu^{9} - 1741594 \nu^{8} - 2860951 \nu^{7} + 1386290 \nu^{6} + 1847879 \nu^{5} - 454807 \nu^{4} - 515722 \nu^{3} + 26849 \nu^{2} + 32529 \nu - 1923\)\()/844\)
\(\beta_{6}\)\(=\)\((\)\(1441 \nu^{19} - 4408 \nu^{18} - 36373 \nu^{17} + 115070 \nu^{16} + 370173 \nu^{15} - 1227388 \nu^{14} - 1944145 \nu^{13} + 6908584 \nu^{12} + 5578466 \nu^{11} - 22147850 \nu^{10} - 8390942 \nu^{9} + 40772796 \nu^{8} + 5354233 \nu^{7} - 41409619 \nu^{6} - 24455 \nu^{5} + 20985660 \nu^{4} - 661121 \nu^{3} - 4323244 \nu^{2} - 119417 \nu + 196049\)\()/2954\)
\(\beta_{7}\)\(=\)\((\)\(4075 \nu^{19} - 10768 \nu^{18} - 106301 \nu^{17} + 279121 \nu^{16} + 1133945 \nu^{15} - 2948266 \nu^{14} - 6387514 \nu^{13} + 16368355 \nu^{12} + 20472398 \nu^{11} - 51465599 \nu^{10} - 37398186 \nu^{9} + 92211853 \nu^{8} + 36775414 \nu^{7} - 90288507 \nu^{6} - 17218799 \nu^{5} + 43597061 \nu^{4} + 3569523 \nu^{3} - 8439072 \nu^{2} - 562859 \nu + 397047\)\()/5908\)
\(\beta_{8}\)\(=\)\((\)\(1871 \nu^{19} - 3616 \nu^{18} - 48804 \nu^{17} + 93461 \nu^{16} + 519731 \nu^{15} - 985066 \nu^{14} - 2915591 \nu^{13} + 5465175 \nu^{12} + 9276793 \nu^{11} - 17217267 \nu^{10} - 16781193 \nu^{9} + 31025651 \nu^{8} + 16374317 \nu^{7} - 30661461 \nu^{6} - 7680454 \nu^{5} + 14968615 \nu^{4} + 1569115 \nu^{3} - 2928332 \nu^{2} - 212470 \nu + 136681\)\()/1688\)
\(\beta_{9}\)\(=\)\((\)\(19767 \nu^{19} - 52631 \nu^{18} - 508397 \nu^{17} + 1369512 \nu^{16} + 5315407 \nu^{15} - 14549501 \nu^{14} - 29080478 \nu^{13} + 81489167 \nu^{12} + 89221158 \nu^{11} - 259682035 \nu^{10} - 152204430 \nu^{9} + 474855711 \nu^{8} + 133111450 \nu^{7} - 479242301 \nu^{6} - 49298757 \nu^{5} + 242024086 \nu^{4} + 7693925 \nu^{3} - 49964329 \nu^{2} - 2806969 \nu + 2317504\)\()/11816\)
\(\beta_{10}\)\(=\)\((\)\(-25318 \nu^{19} + 68773 \nu^{18} + 644715 \nu^{17} - 1793047 \nu^{16} - 6643398 \nu^{15} + 19103225 \nu^{14} + 35551495 \nu^{13} - 107435598 \nu^{12} - 105260377 \nu^{11} + 344392654 \nu^{10} + 168574693 \nu^{9} - 634890138 \nu^{8} - 128646073 \nu^{7} + 647387306 \nu^{6} + 29783317 \nu^{5} - 330940935 \nu^{4} + 1044336 \nu^{3} + 69269979 \nu^{2} + 3014029 \nu - 3155999\)\()/11816\)
\(\beta_{11}\)\(=\)\((\)\(4453 \nu^{19} - 10482 \nu^{18} - 116106 \nu^{17} + 271265 \nu^{16} + 1237033 \nu^{15} - 2860796 \nu^{14} - 6952139 \nu^{13} + 15862357 \nu^{12} + 22198929 \nu^{11} - 49846133 \nu^{10} - 40349501 \nu^{9} + 89377701 \nu^{8} + 39493437 \nu^{7} - 87751919 \nu^{6} - 18471832 \nu^{5} + 42599011 \nu^{4} + 3815525 \nu^{3} - 8303666 \nu^{2} - 568756 \nu + 385549\)\()/1688\)
\(\beta_{12}\)\(=\)\((\)\(14341 \nu^{19} - 36774 \nu^{18} - 367103 \nu^{17} + 957937 \nu^{16} + 3811197 \nu^{15} - 10195866 \nu^{14} - 20628888 \nu^{13} + 57277865 \nu^{12} + 62243884 \nu^{11} - 183389459 \nu^{10} - 103346572 \nu^{9} + 337661823 \nu^{8} + 86094140 \nu^{7} - 343888221 \nu^{6} - 28460643 \nu^{5} + 175629453 \nu^{4} + 3687413 \nu^{3} - 36808230 \nu^{2} - 2004129 \nu + 1715049\)\()/5908\)
\(\beta_{13}\)\(=\)\((\)\(-19699 \nu^{19} + 48813 \nu^{18} + 508339 \nu^{17} - 1268655 \nu^{16} - 5338649 \nu^{15} + 13459641 \nu^{14} + 29394218 \nu^{13} - 75265998 \nu^{12} - 91075798 \nu^{11} + 239401956 \nu^{10} + 158073734 \nu^{9} - 436717672 \nu^{8} - 143314770 \nu^{7} + 439149488 \nu^{6} + 58149675 \nu^{5} - 220425769 \nu^{4} - 10391849 \nu^{3} + 45106927 \nu^{2} + 2678875 \nu - 2112117\)\()/5908\)
\(\beta_{14}\)\(=\)\((\)\(-55795 \nu^{19} + 137847 \nu^{18} + 1439923 \nu^{17} - 3582608 \nu^{16} - 15123407 \nu^{15} + 38010317 \nu^{14} + 83273652 \nu^{13} - 212578283 \nu^{12} - 258032904 \nu^{11} + 676372607 \nu^{10} + 447924232 \nu^{9} - 1234767991 \nu^{8} - 406481732 \nu^{7} + 1243768209 \nu^{6} + 165911551 \nu^{5} - 626706506 \nu^{4} - 30707973 \nu^{3} + 129312481 \nu^{2} + 8035483 \nu - 6105596\)\()/11816\)
\(\beta_{15}\)\(=\)\((\)\(57829 \nu^{19} - 148747 \nu^{18} - 1485881 \nu^{17} + 3870922 \nu^{16} + 15510665 \nu^{15} - 41139285 \nu^{14} - 84645344 \nu^{13} + 230601293 \nu^{12} + 258680916 \nu^{11} - 735967357 \nu^{10} - 438652340 \nu^{9} + 1349119441 \nu^{8} + 380081640 \nu^{7} - 1366475007 \nu^{6} - 138161701 \nu^{5} + 693636784 \nu^{4} + 20554343 \nu^{3} - 144478337 \nu^{2} - 7823409 \nu + 6810118\)\()/11816\)
\(\beta_{16}\)\(=\)\((\)\(63867 \nu^{19} - 147271 \nu^{18} - 1657237 \nu^{17} + 3819554 \nu^{16} + 17535335 \nu^{15} - 40414977 \nu^{14} - 97569654 \nu^{13} + 225221581 \nu^{12} + 307049398 \nu^{11} - 713165485 \nu^{10} - 546335846 \nu^{9} + 1293341169 \nu^{8} + 518115762 \nu^{7} - 1290502667 \nu^{6} - 231088533 \nu^{5} + 640952560 \nu^{4} + 46618937 \nu^{3} - 129260161 \nu^{2} - 8635897 \nu + 6091414\)\()/11816\)
\(\beta_{17}\)\(=\)\((\)\(-65375 \nu^{19} + 157917 \nu^{18} + 1691191 \nu^{17} - 4101614 \nu^{16} - 17820791 \nu^{15} + 43481023 \nu^{14} + 98583464 \nu^{13} - 242911087 \nu^{12} - 307579248 \nu^{11} + 771777607 \nu^{10} + 539750428 \nu^{9} - 1406264571 \nu^{8} - 499167236 \nu^{7} + 1412960229 \nu^{6} + 211440319 \nu^{5} - 709488424 \nu^{4} - 40276409 \nu^{3} + 145600115 \nu^{2} + 9091823 \nu - 6892942\)\()/11816\)
\(\beta_{18}\)\(=\)\((\)\(35122 \nu^{19} - 85868 \nu^{18} - 910249 \nu^{17} + 2228382 \nu^{16} + 9617310 \nu^{15} - 23592048 \nu^{14} - 53410665 \nu^{13} + 131533684 \nu^{12} + 167604135 \nu^{11} - 416625344 \nu^{10} - 296641877 \nu^{9} + 755681308 \nu^{8} + 277891143 \nu^{7} - 754399768 \nu^{6} - 120087963 \nu^{5} + 375432804 \nu^{4} + 23015472 \nu^{3} - 76082910 \nu^{2} - 4740983 \nu + 3591140\)\()/5908\)
\(\beta_{19}\)\(=\)\((\)\(-45224 \nu^{19} + 109964 \nu^{18} + 1169521 \nu^{17} - 2855704 \nu^{16} - 12318296 \nu^{15} + 30265756 \nu^{14} + 68100795 \nu^{13} - 169013794 \nu^{12} - 212261757 \nu^{11} + 536630064 \nu^{10} + 371818545 \nu^{9} - 976724486 \nu^{8} - 342559345 \nu^{7} + 979627856 \nu^{6} + 143694241 \nu^{5} - 490465964 \nu^{4} - 26758406 \nu^{3} + 100201394 \nu^{2} + 6084097 \nu - 4729788\)\()/5908\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{19} + \beta_{16} + \beta_{15} + \beta_{13} + \beta_{7} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{17} + \beta_{15} + \beta_{10} + \beta_{9} + \beta_{8} + 6 \beta_{2} + \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(8 \beta_{19} + 2 \beta_{18} + 8 \beta_{16} + 7 \beta_{15} + \beta_{14} + 8 \beta_{13} + 2 \beta_{12} - \beta_{11} - 2 \beta_{9} - \beta_{8} + 7 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + 27 \beta_{1} - 8\)
\(\nu^{6}\)\(=\)\(-\beta_{18} + 9 \beta_{17} - \beta_{16} + 10 \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + 9 \beta_{10} + 9 \beta_{9} + 11 \beta_{8} - \beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} + 34 \beta_{2} + 9 \beta_{1} + 84\)
\(\nu^{7}\)\(=\)\(56 \beta_{19} + 24 \beta_{18} + \beta_{17} + 55 \beta_{16} + 44 \beta_{15} + 11 \beta_{14} + 55 \beta_{13} + 24 \beta_{12} - 12 \beta_{11} - 23 \beta_{9} - 10 \beta_{8} + 43 \beta_{7} + 11 \beta_{6} - 13 \beta_{5} - \beta_{3} + 10 \beta_{2} + 154 \beta_{1} - 56\)
\(\nu^{8}\)\(=\)\(\beta_{19} - 12 \beta_{18} + 67 \beta_{17} - 11 \beta_{16} + 81 \beta_{15} - 12 \beta_{14} + 15 \beta_{13} + 2 \beta_{12} + 12 \beta_{11} + 65 \beta_{10} + 62 \beta_{9} + 91 \beta_{8} - 11 \beta_{7} + 12 \beta_{6} - 42 \beta_{5} - 12 \beta_{4} - \beta_{3} + 196 \beta_{2} + 64 \beta_{1} + 493\)
\(\nu^{9}\)\(=\)\(380 \beta_{19} + 210 \beta_{18} + 16 \beta_{17} + 366 \beta_{16} + 279 \beta_{15} + 91 \beta_{14} + 368 \beta_{13} + 210 \beta_{12} - 106 \beta_{11} + \beta_{10} - 197 \beta_{9} - 73 \beta_{8} + 261 \beta_{7} + 91 \beta_{6} - 121 \beta_{5} - 2 \beta_{4} - 17 \beta_{3} + 77 \beta_{2} + 913 \beta_{1} - 381\)
\(\nu^{10}\)\(=\)\(22 \beta_{19} - 93 \beta_{18} + 478 \beta_{17} - 81 \beta_{16} + 611 \beta_{15} - 99 \beta_{14} + 155 \beta_{13} + 40 \beta_{12} + 98 \beta_{11} + 441 \beta_{10} + 389 \beta_{9} + 678 \beta_{8} - 87 \beta_{7} + 103 \beta_{6} - 411 \beta_{5} - 105 \beta_{4} - 15 \beta_{3} + 1160 \beta_{2} + 431 \beta_{1} + 2971\)
\(\nu^{11}\)\(=\)\(2554 \beta_{19} + 1628 \beta_{18} + 176 \beta_{17} + 2423 \beta_{16} + 1816 \beta_{15} + 682 \beta_{14} + 2461 \beta_{13} + 1636 \beta_{12} - 832 \beta_{11} + 18 \beta_{10} - 1511 \beta_{9} - 472 \beta_{8} + 1606 \beta_{7} + 676 \beta_{6} - 996 \beta_{5} - 34 \beta_{4} - 190 \beta_{3} + 548 \beta_{2} + 5572 \beta_{1} - 2558\)
\(\nu^{12}\)\(=\)\(294 \beta_{19} - 577 \beta_{18} + 3367 \beta_{17} - 485 \beta_{16} + 4452 \beta_{15} - 702 \beta_{14} + 1377 \beta_{13} + 500 \beta_{12} + 675 \beta_{11} + 2924 \beta_{10} + 2325 \beta_{9} + 4806 \beta_{8} - 608 \beta_{7} + 782 \beta_{6} - 3484 \beta_{5} - 819 \beta_{4} - 158 \beta_{3} + 7039 \beta_{2} + 2888 \beta_{1} + 18231\)
\(\nu^{13}\)\(=\)\(17120 \beta_{19} + 11891 \beta_{18} + 1652 \beta_{17} + 16078 \beta_{16} + 12110 \beta_{15} + 4892 \beta_{14} + 16540 \beta_{13} + 12060 \beta_{12} - 6153 \beta_{11} + 219 \beta_{10} - 10976 \beta_{9} - 2851 \beta_{8} + 10068 \beta_{7} + 4764 \beta_{6} - 7726 \beta_{5} - 377 \beta_{4} - 1773 \beta_{3} + 3810 \beta_{2} + 34787 \beta_{1} - 16993\)
\(\nu^{14}\)\(=\)\(3149 \beta_{19} - 2968 \beta_{18} + 23582 \beta_{17} - 2396 \beta_{16} + 31813 \beta_{15} - 4601 \beta_{14} + 11318 \beta_{13} + 5070 \beta_{12} + 4186 \beta_{11} + 19207 \beta_{10} + 13429 \beta_{9} + 33179 \beta_{8} - 3986 \beta_{7} + 5621 \beta_{6} - 27443 \beta_{5} - 6050 \beta_{4} - 1454 \beta_{3} + 43642 \beta_{2} + 19573 \beta_{1} + 113413\)
\(\nu^{15}\)\(=\)\(114763 \beta_{19} + 84042 \beta_{18} + 14221 \beta_{17} + 107111 \beta_{16} + 82238 \beta_{15} + 34321 \beta_{14} + 111719 \beta_{13} + 86284 \beta_{12} - 43982 \beta_{11} + 2260 \beta_{10} - 77337 \beta_{9} - 16342 \beta_{8} + 64216 \beta_{7} + 32635 \beta_{6} - 57963 \beta_{5} - 3480 \beta_{4} - 15008 \beta_{3} + 26446 \beta_{2} + 221153 \beta_{1} - 111787\)
\(\nu^{16}\)\(=\)\(29855 \beta_{19} - 11483 \beta_{18} + 164528 \beta_{17} - 8368 \beta_{16} + 224695 \beta_{15} - 28732 \beta_{14} + 88914 \beta_{13} + 45818 \beta_{12} + 23838 \beta_{11} + 125728 \beta_{10} + 75172 \beta_{9} + 225715 \beta_{8} - 25051 \beta_{7} + 39396 \beta_{6} - 207206 \beta_{5} - 43389 \beta_{4} - 12474 \beta_{3} + 275465 \beta_{2} + 134761 \beta_{1} + 713239\)
\(\nu^{17}\)\(=\)\(770182 \beta_{19} + 582625 \beta_{18} + 116045 \beta_{17} + 716308 \beta_{16} + 565412 \beta_{15} + 237841 \beta_{14} + 757526 \beta_{13} + 606656 \beta_{12} - 307904 \beta_{11} + 21286 \beta_{10} - 535021 \beta_{9} - 88725 \beta_{8} + 415417 \beta_{7} + 220123 \beta_{6} - 425937 \beta_{5} - 29135 \beta_{4} - 119745 \beta_{3} + 184798 \beta_{2} + 1426482 \beta_{1} - 728670\)
\(\nu^{18}\)\(=\)\(262512 \beta_{19} - 11693 \beta_{18} + 1144247 \beta_{17} + 3600 \beta_{16} + 1575715 \beta_{15} - 173241 \beta_{14} + 678888 \beta_{13} + 385908 \beta_{12} + 123712 \beta_{11} + 822592 \beta_{10} + 406042 \beta_{9} + 1522897 \beta_{8} - 152017 \beta_{7} + 273185 \beta_{6} - 1523901 \beta_{5} - 305765 \beta_{4} - 102366 \beta_{3} + 1764404 \beta_{2} + 941877 \beta_{1} + 4525211\)
\(\nu^{19}\)\(=\)\(5176782 \beta_{19} + 3992197 \beta_{18} + 913778 \beta_{17} + 4805830 \beta_{16} + 3917898 \beta_{15} + 1636097 \beta_{14} + 5150816 \beta_{13} + 4220190 \beta_{12} - 2127159 \beta_{11} + 188944 \beta_{10} - 3659134 \beta_{9} - 447866 \beta_{8} + 2716871 \beta_{7} + 1472873 \beta_{6} - 3087235 \beta_{5} - 230291 \beta_{4} - 919249 \beta_{3} + 1302633 \beta_{2} + 9308306 \beta_{1} - 4709052\)

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
2.63622
2.34607
2.27905
2.27121
2.10538
1.41602
0.979921
0.928350
0.845808
0.212475
−0.328365
−0.463326
−0.466765
−0.874793
−1.30606
−1.50500
−1.89507
−2.11567
−2.51950
−2.54595
−2.63622 −2.63584 4.94964 1.42955 6.94865 0 −7.77590 3.94765 −3.76861
1.2 −2.34607 0.967441 3.50405 3.03222 −2.26969 0 −3.52861 −2.06406 −7.11381
1.3 −2.27905 −0.713649 3.19408 −3.36532 1.62644 0 −2.72136 −2.49071 7.66975
1.4 −2.27121 3.42520 3.15839 −0.765314 −7.77934 0 −2.63095 8.73199 1.73819
1.5 −2.10538 −1.20723 2.43261 1.24490 2.54167 0 −0.910797 −1.54260 −2.62098
1.6 −1.41602 −2.14541 0.00509860 −1.43123 3.03793 0 2.82481 1.60278 2.02665
1.7 −0.979921 1.94904 −1.03976 1.73965 −1.90990 0 2.97872 0.798739 −1.70472
1.8 −0.928350 2.79733 −1.13817 −3.35114 −2.59691 0 2.91332 4.82508 3.11103
1.9 −0.845808 0.0216736 −1.28461 −2.12752 −0.0183317 0 2.77815 −2.99953 1.79947
1.10 −0.212475 −2.41608 −1.95485 3.68342 0.513357 0 0.840308 2.83745 −0.782635
1.11 0.328365 −0.613482 −1.89218 0.220416 −0.201446 0 −1.27805 −2.62364 0.0723768
1.12 0.463326 0.981674 −1.78533 −3.31637 0.454835 0 −1.75384 −2.03632 −1.53656
1.13 0.466765 2.80691 −1.78213 2.27147 1.31017 0 −1.76537 4.87872 1.06024
1.14 0.874793 2.48225 −1.23474 4.06589 2.17145 0 −2.82973 3.16156 3.55681
1.15 1.30606 −1.82067 −0.294211 0.323122 −2.37790 0 −2.99637 0.314826 0.422016
1.16 1.50500 0.350711 0.265025 −3.19754 0.527821 0 −2.61114 −2.87700 −4.81229
1.17 1.89507 −1.57790 1.59127 3.12194 −2.99022 0 −0.774564 −0.510238 5.91628
1.18 2.11567 1.23352 2.47606 0.585449 2.60971 0 1.00718 −1.47844 1.23862
1.19 2.51950 2.78497 4.34790 1.47804 7.01674 0 5.91555 4.75604 3.72392
1.20 2.54595 1.32955 4.48185 2.35837 3.38496 0 6.31865 −1.23230 6.00428
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2009.2.a.u yes 20
7.b odd 2 1 2009.2.a.t 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2009.2.a.t 20 7.b odd 2 1
2009.2.a.u yes 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2009))\):

\(T_{2}^{20} + \cdots\)
\(T_{3}^{20} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 49 - 42 T - 1179 T^{2} + 1972 T^{3} + 5690 T^{4} - 9524 T^{5} - 13352 T^{6} + 18446 T^{7} + 17721 T^{8} - 18080 T^{9} - 13797 T^{10} + 9852 T^{11} + 6449 T^{12} - 3090 T^{13} - 1820 T^{14} + 552 T^{15} + 302 T^{16} - 52 T^{17} - 27 T^{18} + 2 T^{19} + T^{20} \)
$3$ \( 89 - 4324 T + 9466 T^{2} + 27816 T^{3} - 58599 T^{4} - 54676 T^{5} + 133192 T^{6} + 33544 T^{7} - 141680 T^{8} + 7624 T^{9} + 78924 T^{10} - 17616 T^{11} - 23916 T^{12} + 8248 T^{13} + 3770 T^{14} - 1804 T^{15} - 236 T^{16} + 192 T^{17} - 6 T^{18} - 8 T^{19} + T^{20} \)
$5$ \( -40384 + 421632 T - 1423584 T^{2} + 1250496 T^{3} + 2401392 T^{4} - 5163168 T^{5} + 1465128 T^{6} + 3393232 T^{7} - 2603364 T^{8} - 458560 T^{9} + 1023382 T^{10} - 154604 T^{11} - 170015 T^{12} + 55196 T^{13} + 12052 T^{14} - 6764 T^{15} - 105 T^{16} + 376 T^{17} - 28 T^{18} - 8 T^{19} + T^{20} \)
$7$ \( T^{20} \)
$11$ \( -77312 - 285184 T + 1207680 T^{2} + 3724928 T^{3} - 3956992 T^{4} - 13662080 T^{5} + 3104872 T^{6} + 19595904 T^{7} + 2911912 T^{8} - 10661784 T^{9} - 3866226 T^{10} + 1764808 T^{11} + 870985 T^{12} - 124936 T^{13} - 84376 T^{14} + 4008 T^{15} + 4146 T^{16} - 48 T^{17} - 102 T^{18} + T^{20} \)
$13$ \( 21455782873 - 50872443784 T + 22706574938 T^{2} + 30050625892 T^{3} - 27503931548 T^{4} - 2633080544 T^{5} + 8815696148 T^{6} - 1166462096 T^{7} - 1350437096 T^{8} + 331191960 T^{9} + 112062800 T^{10} - 38451704 T^{11} - 4872647 T^{12} + 2429712 T^{13} + 72340 T^{14} - 86592 T^{15} + 2213 T^{16} + 1616 T^{17} - 98 T^{18} - 12 T^{19} + T^{20} \)
$17$ \( 202951601 - 1338702264 T - 6088132646 T^{2} - 1919851616 T^{3} + 8661676826 T^{4} + 4032067420 T^{5} - 3916334128 T^{6} - 1886002744 T^{7} + 818641422 T^{8} + 374534532 T^{9} - 97455202 T^{10} - 38522864 T^{11} + 7238859 T^{12} + 2204556 T^{13} - 339656 T^{14} - 70592 T^{15} + 9687 T^{16} + 1180 T^{17} - 152 T^{18} - 8 T^{19} + T^{20} \)
$19$ \( 887123761 - 2274486244 T + 17461188 T^{2} + 5055291208 T^{3} - 4827865639 T^{4} - 871020860 T^{5} + 3659296088 T^{6} - 1696253788 T^{7} - 329858742 T^{8} + 511121052 T^{9} - 116570080 T^{10} - 30887944 T^{11} + 19406802 T^{12} - 2299412 T^{13} - 576564 T^{14} + 200100 T^{15} - 16134 T^{16} - 1836 T^{17} + 470 T^{18} - 36 T^{19} + T^{20} \)
$23$ \( 1666581528929 - 2003991230484 T - 826474827878 T^{2} + 1475186881164 T^{3} + 107573453277 T^{4} - 418511075336 T^{5} + 322731008 T^{6} + 63272928820 T^{7} - 787865248 T^{8} - 5744683312 T^{9} + 8582822 T^{10} + 326588232 T^{11} + 5849406 T^{12} - 11563848 T^{13} - 448416 T^{14} + 242960 T^{15} + 13956 T^{16} - 2708 T^{17} - 196 T^{18} + 12 T^{19} + T^{20} \)
$29$ \( 16555395476992 - 30188063046656 T + 10963480769280 T^{2} + 10138663858304 T^{3} - 7955542503200 T^{4} + 42273747936 T^{5} + 1365513078696 T^{6} - 282997394336 T^{7} - 83123097744 T^{8} + 32255387984 T^{9} + 844880454 T^{10} - 1457125240 T^{11} + 86188149 T^{12} + 31928212 T^{13} - 3335404 T^{14} - 358768 T^{15} + 50617 T^{16} + 1960 T^{17} - 360 T^{18} - 4 T^{19} + T^{20} \)
$31$ \( 350103980608 - 231813767680 T - 6657679013024 T^{2} + 9109018931456 T^{3} + 551266468096 T^{4} - 9463030631136 T^{5} + 9104246146744 T^{6} - 4486016358448 T^{7} + 1276873989736 T^{8} - 173134442856 T^{9} - 13977064668 T^{10} + 11323238964 T^{11} - 2289295215 T^{12} + 191558948 T^{13} + 11724138 T^{14} - 5379236 T^{15} + 734731 T^{16} - 58192 T^{17} + 2842 T^{18} - 80 T^{19} + T^{20} \)
$37$ \( -196930033408 + 1587659645952 T - 1784913676800 T^{2} - 725361716224 T^{3} + 1426022600608 T^{4} + 74731885056 T^{5} - 424157137504 T^{6} - 3763421792 T^{7} + 61842867657 T^{8} + 1641278292 T^{9} - 4542673740 T^{10} - 166140060 T^{11} + 182599762 T^{12} + 6661720 T^{13} - 4185982 T^{14} - 126912 T^{15} + 54169 T^{16} + 1152 T^{17} - 366 T^{18} - 4 T^{19} + T^{20} \)
$41$ \( ( -1 + T )^{20} \)
$43$ \( -23554608983 + 73128360008 T + 323139596704 T^{2} + 227392809580 T^{3} - 143098151052 T^{4} - 175690783328 T^{5} + 7021674846 T^{6} + 46604901284 T^{7} + 5648404174 T^{8} - 5668573436 T^{9} - 1144159230 T^{10} + 324999644 T^{11} + 86165653 T^{12} - 8136424 T^{13} - 2896014 T^{14} + 81876 T^{15} + 46655 T^{16} - 260 T^{17} - 354 T^{18} + T^{20} \)
$47$ \( 37415328784 + 207886882752 T - 290552057392 T^{2} - 366102368480 T^{3} + 629905530020 T^{4} - 60030291152 T^{5} - 262655667500 T^{6} + 103709515576 T^{7} + 25666456193 T^{8} - 20348602952 T^{9} + 1470284062 T^{10} + 1091474588 T^{11} - 205474322 T^{12} - 15136780 T^{13} + 6468032 T^{14} - 233552 T^{15} - 69205 T^{16} + 6348 T^{17} + 114 T^{18} - 32 T^{19} + T^{20} \)
$53$ \( 4993537019392 - 2465213747712 T - 29540080612736 T^{2} + 22390723804544 T^{3} + 22728193279616 T^{4} - 32002912246720 T^{5} + 12766244150392 T^{6} - 947575158976 T^{7} - 679291214900 T^{8} + 163131091888 T^{9} + 5885242634 T^{10} - 5519318688 T^{11} + 279826537 T^{12} + 86073340 T^{13} - 8059496 T^{14} - 692072 T^{15} + 89821 T^{16} + 2728 T^{17} - 476 T^{18} - 4 T^{19} + T^{20} \)
$59$ \( 28531727785024 - 5527480298240 T - 61350249364448 T^{2} + 18221927551104 T^{3} + 36759274645584 T^{4} - 16820284276544 T^{5} - 5796815557368 T^{6} + 4502616670272 T^{7} - 442552736724 T^{8} - 204075481440 T^{9} + 43871149558 T^{10} + 2292757060 T^{11} - 1245124223 T^{12} + 40535524 T^{13} + 15510852 T^{14} - 1218252 T^{15} - 76745 T^{16} + 10648 T^{17} - 28 T^{18} - 32 T^{19} + T^{20} \)
$61$ \( 1046452461632 - 536990395136 T - 15586550657376 T^{2} + 1528157379712 T^{3} + 28586703277456 T^{4} - 5007987297312 T^{5} - 6698285651672 T^{6} + 1654593110928 T^{7} + 502843962196 T^{8} - 178869244424 T^{9} - 7707830274 T^{10} + 7791995364 T^{11} - 501106327 T^{12} - 125802632 T^{13} + 17688316 T^{14} + 306072 T^{15} - 177774 T^{16} + 7888 T^{17} + 430 T^{18} - 44 T^{19} + T^{20} \)
$67$ \( 30547158444608 - 158631723301376 T + 266488386362528 T^{2} - 131804172802304 T^{3} - 71988013930608 T^{4} + 86016169903552 T^{5} - 13942007505112 T^{6} - 8814253789056 T^{7} + 2673543430836 T^{8} + 304124324224 T^{9} - 147799960214 T^{10} - 2826493008 T^{11} + 3577256689 T^{12} - 21078220 T^{13} - 43294640 T^{14} + 466752 T^{15} + 270265 T^{16} - 2416 T^{17} - 832 T^{18} + 4 T^{19} + T^{20} \)
$71$ \( 31205527187392 - 1542124805297536 T + 4330716312054560 T^{2} - 4994502554767808 T^{3} + 2973696255083440 T^{4} - 917885545155904 T^{5} + 93885970102552 T^{6} + 26921320751600 T^{7} - 9281654593236 T^{8} + 555274716592 T^{9} + 168516367882 T^{10} - 27576308712 T^{11} - 521286303 T^{12} + 379465440 T^{13} - 13779976 T^{14} - 2411424 T^{15} + 158134 T^{16} + 7216 T^{17} - 658 T^{18} - 8 T^{19} + T^{20} \)
$73$ \( -5651689536785984 - 1668548482323584 T + 2237569053537568 T^{2} + 655472926563712 T^{3} - 352298390872528 T^{4} - 96946182210560 T^{5} + 30971368933704 T^{6} + 7308333500240 T^{7} - 1755872905388 T^{8} - 308429098288 T^{9} + 67503369842 T^{10} + 7150073504 T^{11} - 1721460151 T^{12} - 71895928 T^{13} + 26760692 T^{14} - 245792 T^{15} - 212078 T^{16} + 10312 T^{17} + 482 T^{18} - 48 T^{19} + T^{20} \)
$79$ \( 128543211633664 - 791661396582400 T + 1360182233188864 T^{2} - 269186374389760 T^{3} - 580553973097984 T^{4} + 310227913952256 T^{5} + 3263115841408 T^{6} - 32676198282496 T^{7} + 5258674072256 T^{8} + 835621476608 T^{9} - 236552193376 T^{10} - 6656198464 T^{11} + 4549826512 T^{12} - 23994304 T^{13} - 47448248 T^{14} + 606032 T^{15} + 277304 T^{16} - 2800 T^{17} - 836 T^{18} + 4 T^{19} + T^{20} \)
$83$ \( -117239252072775616 - 6790167961923840 T + 97042423061130816 T^{2} - 30143521557683712 T^{3} - 9664565355023280 T^{4} + 4597924159854720 T^{5} + 205961568721584 T^{6} - 254496515520192 T^{7} + 7878924097432 T^{8} + 6941917355184 T^{9} - 444889076656 T^{10} - 102850653656 T^{11} + 8465300449 T^{12} + 848248824 T^{13} - 80113504 T^{14} - 3837120 T^{15} + 400694 T^{16} + 8824 T^{17} - 1008 T^{18} - 8 T^{19} + T^{20} \)
$89$ \( 208352362183502993 + 363629396142515744 T - 61689324768505760 T^{2} - 176828040502655892 T^{3} - 12580575222374333 T^{4} + 17578961453183940 T^{5} + 930042930386248 T^{6} - 783255000465468 T^{7} - 17662417288546 T^{8} + 18749818115988 T^{9} - 48833371170 T^{10} - 255779292424 T^{11} + 5304308358 T^{12} + 2010137744 T^{13} - 66093066 T^{14} - 8955400 T^{15} + 366696 T^{16} + 20964 T^{17} - 972 T^{18} - 20 T^{19} + T^{20} \)
$97$ \( -4019415774095911 + 8539021102311760 T + 10125254451788670 T^{2} - 12449994457851684 T^{3} + 365534014866317 T^{4} + 1783024265605812 T^{5} - 199118138300936 T^{6} - 98525241676172 T^{7} + 13417333057100 T^{8} + 2758951721988 T^{9} - 400457192624 T^{10} - 43832826300 T^{11} + 6443074024 T^{12} + 411365896 T^{13} - 59422398 T^{14} - 2250132 T^{15} + 313116 T^{16} + 6604 T^{17} - 874 T^{18} - 8 T^{19} + T^{20} \)
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