Properties

Label 211.2.a.d
Level 211
Weight 2
Character orbit 211.a
Self dual yes
Analytic conductor 1.685
Analytic rank 0
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 211.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.68484348265\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 14 x^{7} + 11 x^{6} + 66 x^{5} - 36 x^{4} - 123 x^{3} + 38 x^{2} + 72 x - 8\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 14 x^{7} + 11 x^{6} + 66 x^{5} - 36 x^{4} - 123 x^{3} + 38 x^{2} + 72 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{8} - 31 \nu^{7} - 58 \nu^{6} + 309 \nu^{5} + 82 \nu^{4} - 732 \nu^{3} + 91 \nu^{2} + 186 \nu - 200 \)\()/116\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{8} + 11 \nu^{7} - 58 \nu^{6} - 177 \nu^{5} + 158 \nu^{4} + 836 \nu^{3} + 65 \nu^{2} - 1110 \nu - 292 \)\()/116\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{8} + 31 \nu^{7} + 58 \nu^{6} - 309 \nu^{5} - 82 \nu^{4} + 848 \nu^{3} - 91 \nu^{2} - 766 \nu + 84 \)\()/116\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{8} - 47 \nu^{7} + 519 \nu^{5} - 364 \nu^{4} - 1600 \nu^{3} + 1109 \nu^{2} + 1384 \nu - 524 \)\()/116\)
\(\beta_{7}\)\(=\)\((\)\( 9 \nu^{8} - 15 \nu^{7} - 116 \nu^{6} + 157 \nu^{5} + 470 \nu^{4} - 444 \nu^{3} - 637 \nu^{2} + 322 \nu + 124 \)\()/58\)
\(\beta_{8}\)\(=\)\((\)\( -21 \nu^{8} + 35 \nu^{7} + 232 \nu^{6} - 347 \nu^{5} - 768 \nu^{4} + 920 \nu^{3} + 1003 \nu^{2} - 616 \nu - 560 \)\()/116\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{3} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 13\)
\(\nu^{5}\)\(=\)\(-\beta_{8} + \beta_{6} + 10 \beta_{5} - \beta_{4} + 7 \beta_{3} + 2 \beta_{2} + 28 \beta_{1} + 8\)
\(\nu^{6}\)\(=\)\(-12 \beta_{8} - 12 \beta_{7} - 8 \beta_{6} + 2 \beta_{5} - 9 \beta_{4} + 9 \beta_{3} + 48 \beta_{2} + 11 \beta_{1} + 67\)
\(\nu^{7}\)\(=\)\(-15 \beta_{8} - 3 \beta_{7} + 11 \beta_{6} + 80 \beta_{5} - 10 \beta_{4} + 42 \beta_{3} + 27 \beta_{2} + 171 \beta_{1} + 54\)
\(\nu^{8}\)\(=\)\(-110 \beta_{8} - 101 \beta_{7} - 50 \beta_{6} + 34 \beta_{5} - 63 \beta_{4} + 61 \beta_{3} + 334 \beta_{2} + 97 \beta_{1} + 383\)

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
2.67629
2.16500
1.76769
1.09167
0.107203
−0.959048
−1.54891
−1.79023
−2.50967
−2.67629 −1.87149 5.16253 3.51131 5.00866 −4.24372 −8.46385 0.502488 −9.39729
1.2 −2.16500 3.06548 2.68723 3.67947 −6.63678 0.600695 −1.48786 6.39720 −7.96606
1.3 −1.76769 1.01743 1.12474 −0.330474 −1.79850 1.14456 1.54719 −1.96484 0.584177
1.4 −1.09167 −2.69353 −0.808256 −1.33282 2.94045 −2.43218 3.06569 4.25511 1.45500
1.5 −0.107203 2.59855 −1.98851 0.449128 −0.278573 0.845532 0.427581 3.75245 −0.0481480
1.6 0.959048 −3.12841 −1.08023 3.31637 −3.00030 2.28069 −2.95409 6.78696 3.18056
1.7 1.54891 1.20044 0.399122 4.25492 1.85937 −4.34069 −2.47962 −1.55895 6.59049
1.8 1.79023 0.155219 1.20493 0.0597430 0.277877 4.81622 −1.42336 −2.97591 0.106954
1.9 2.50967 −1.34369 4.29843 1.39234 −3.37220 −0.671123 5.76831 −1.19451 3.49432
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(211\) \(-1\)

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 211.2.a.d 9
3.b odd 2 1 1899.2.a.j 9
4.b odd 2 1 3376.2.a.s 9
5.b even 2 1 5275.2.a.o 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.2.a.d 9 1.a even 1 1 trivial
1899.2.a.j 9 3.b odd 2 1
3376.2.a.s 9 4.b odd 2 1
5275.2.a.o 9 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{9} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(211))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 4 T^{2} + 5 T^{3} + 14 T^{4} + 16 T^{5} + 33 T^{6} + 38 T^{7} + 70 T^{8} + 80 T^{9} + 140 T^{10} + 152 T^{11} + 264 T^{12} + 256 T^{13} + 448 T^{14} + 320 T^{15} + 512 T^{16} + 256 T^{17} + 512 T^{18} \)
$3$ \( 1 + T + 7 T^{2} + 7 T^{3} + 32 T^{4} + 26 T^{5} + 116 T^{6} + 105 T^{7} + 422 T^{8} + 346 T^{9} + 1266 T^{10} + 945 T^{11} + 3132 T^{12} + 2106 T^{13} + 7776 T^{14} + 5103 T^{15} + 15309 T^{16} + 6561 T^{17} + 19683 T^{18} \)
$5$ \( 1 - 15 T + 128 T^{2} - 789 T^{3} + 3868 T^{4} - 15793 T^{5} + 55240 T^{6} - 168325 T^{7} + 451526 T^{8} - 1072103 T^{9} + 2257630 T^{10} - 4208125 T^{11} + 6905000 T^{12} - 9870625 T^{13} + 12087500 T^{14} - 12328125 T^{15} + 10000000 T^{16} - 5859375 T^{17} + 1953125 T^{18} \)
$7$ \( 1 + 2 T + 28 T^{2} + 55 T^{3} + 371 T^{4} + 550 T^{5} + 3083 T^{6} + 2473 T^{7} + 19851 T^{8} + 8656 T^{9} + 138957 T^{10} + 121177 T^{11} + 1057469 T^{12} + 1320550 T^{13} + 6235397 T^{14} + 6470695 T^{15} + 23059204 T^{16} + 11529602 T^{17} + 40353607 T^{18} \)
$11$ \( 1 - 13 T + 130 T^{2} - 909 T^{3} + 5510 T^{4} - 27863 T^{5} + 128212 T^{6} - 522487 T^{7} + 1980592 T^{8} - 6791293 T^{9} + 21786512 T^{10} - 63220927 T^{11} + 170650172 T^{12} - 407942183 T^{13} + 887391010 T^{14} - 1610348949 T^{15} + 2533332230 T^{16} - 2786665453 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 + 4 T + 80 T^{2} + 364 T^{3} + 3197 T^{4} + 14686 T^{5} + 82667 T^{6} + 352805 T^{7} + 1496091 T^{8} + 5579059 T^{9} + 19449183 T^{10} + 59624045 T^{11} + 181619399 T^{12} + 419446846 T^{13} + 1187023721 T^{14} + 1756958476 T^{15} + 5019881360 T^{16} + 3262922884 T^{17} + 10604499373 T^{18} \)
$17$ \( 1 - 4 T + 84 T^{2} - 199 T^{3} + 2931 T^{4} - 3078 T^{5} + 64145 T^{6} - 4729 T^{7} + 1169223 T^{8} + 330324 T^{9} + 19876791 T^{10} - 1366681 T^{11} + 315144385 T^{12} - 257077638 T^{13} + 4161600867 T^{14} - 4803376231 T^{15} + 34468448532 T^{16} - 27903029764 T^{17} + 118587876497 T^{18} \)
$19$ \( 1 + 2 T + 94 T^{2} + 92 T^{3} + 4359 T^{4} + 624 T^{5} + 131795 T^{6} - 67689 T^{7} + 3021289 T^{8} - 2189359 T^{9} + 57404491 T^{10} - 24435729 T^{11} + 903981905 T^{12} + 81320304 T^{13} + 10793315541 T^{14} + 4328221052 T^{15} + 84023943466 T^{16} + 33967126082 T^{17} + 322687697779 T^{18} \)
$23$ \( 1 + 3 T + 135 T^{2} + 335 T^{3} + 8414 T^{4} + 19206 T^{5} + 339074 T^{6} + 758305 T^{7} + 10119426 T^{8} + 21034414 T^{9} + 232746798 T^{10} + 401143345 T^{11} + 4125513358 T^{12} + 5374626246 T^{13} + 54155390002 T^{14} + 49592022815 T^{15} + 459651435345 T^{16} + 234932955843 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 - 26 T + 482 T^{2} - 6339 T^{3} + 69613 T^{4} - 636246 T^{5} + 5118367 T^{6} - 36054501 T^{7} + 228623789 T^{8} - 1293317504 T^{9} + 6630089881 T^{10} - 30321835341 T^{11} + 124831852763 T^{12} - 450004707126 T^{13} + 1427842615337 T^{14} - 3770585031819 T^{15} + 8314440380938 T^{16} - 13006406736986 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 - 5 T + 161 T^{2} - 593 T^{3} + 10904 T^{4} - 22838 T^{5} + 415280 T^{6} - 63223 T^{7} + 11471976 T^{8} + 13881958 T^{9} + 355631256 T^{10} - 60757303 T^{11} + 12371606480 T^{12} - 21091372598 T^{13} + 312172262504 T^{14} - 526289682833 T^{15} + 4429530871871 T^{16} - 4264455187205 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 - 5 T + 244 T^{2} - 1145 T^{3} + 28980 T^{4} - 123303 T^{5} + 2170930 T^{6} - 8164375 T^{7} + 112360078 T^{8} - 363884069 T^{9} + 4157322886 T^{10} - 11177029375 T^{11} + 109964117290 T^{12} - 231089673783 T^{13} + 2009587873860 T^{14} - 2937756738305 T^{15} + 23163378020452 T^{16} - 17562397269605 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 - 20 T + 313 T^{2} - 3380 T^{3} + 33036 T^{4} - 272120 T^{5} + 2182348 T^{6} - 15965324 T^{7} + 114565302 T^{8} - 747130600 T^{9} + 4697177382 T^{10} - 26837709644 T^{11} + 150409606508 T^{12} - 768946083320 T^{13} + 3827425456236 T^{14} - 16055352334580 T^{15} + 60958087724753 T^{16} - 159698504582420 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 + 37 T + 894 T^{2} + 15365 T^{3} + 214164 T^{4} + 2472903 T^{5} + 24730478 T^{6} + 215717073 T^{7} + 1676662722 T^{8} + 11597517407 T^{9} + 72096497046 T^{10} + 398860867977 T^{11} + 1966246114346 T^{12} + 8454363249303 T^{13} + 31483916186652 T^{14} + 97127743247885 T^{15} + 243005838329658 T^{16} + 432463410271237 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 - 4 T + 104 T^{2} - 242 T^{3} + 6037 T^{4} + 3040 T^{5} + 380223 T^{6} - 227413 T^{7} + 25088141 T^{8} - 41059065 T^{9} + 1179142627 T^{10} - 502355317 T^{11} + 39475892529 T^{12} + 14834230240 T^{13} + 1384555807259 T^{14} - 2608570109618 T^{15} + 52688804528152 T^{16} - 95245146647044 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 - 13 T + 423 T^{2} - 4289 T^{3} + 79484 T^{4} - 662622 T^{5} + 8988435 T^{6} - 62982601 T^{7} + 682399049 T^{8} - 4025561062 T^{9} + 36167149597 T^{10} - 176918126209 T^{11} + 1338171237495 T^{12} - 5228406301182 T^{13} + 33239850565612 T^{14} - 95062944882281 T^{15} + 496902812151051 T^{16} - 809375975347693 T^{17} + 3299763591802133 T^{18} \)
$59$ \( 1 - 14 T + 273 T^{2} - 2401 T^{3} + 29668 T^{4} - 174639 T^{5} + 1664059 T^{6} - 4749683 T^{7} + 56131267 T^{8} - 40471710 T^{9} + 3311744753 T^{10} - 16533646523 T^{11} + 341762773361 T^{12} - 2116163807679 T^{13} + 21210374102732 T^{14} - 101275461272041 T^{15} + 679401855355587 T^{16} - 2055626126460494 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 - 23 T + 646 T^{2} - 10081 T^{3} + 167065 T^{4} - 1991338 T^{5} + 24260591 T^{6} - 231700919 T^{7} + 2232381533 T^{8} - 17388473502 T^{9} + 136175273513 T^{10} - 862159119599 T^{11} + 5506693205771 T^{12} - 27571749325258 T^{13} + 141102481026565 T^{14} - 519376893933241 T^{15} + 2030211872069566 T^{16} - 4409268198937463 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 + 3 T + 239 T^{2} + 17 T^{3} + 30098 T^{4} - 95826 T^{5} + 2411418 T^{6} - 17595721 T^{7} + 155819658 T^{8} - 1562789922 T^{9} + 10439917086 T^{10} - 78987191569 T^{11} + 725265311934 T^{12} - 1931001320946 T^{13} + 40636065470486 T^{14} + 1537792496873 T^{15} + 1448510073672197 T^{16} + 1218203032669923 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 - 19 T + 534 T^{2} - 6723 T^{3} + 110154 T^{4} - 1035141 T^{5} + 12922004 T^{6} - 98281789 T^{7} + 1079354496 T^{8} - 7356534671 T^{9} + 76634169216 T^{10} - 495438498349 T^{11} + 4624927373644 T^{12} - 26304672882021 T^{13} + 198743079930054 T^{14} - 861218208800883 T^{15} + 4856794164580794 T^{16} - 12269317093669459 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 - 17 T + 456 T^{2} - 6651 T^{3} + 106577 T^{4} - 1299606 T^{5} + 15907139 T^{6} - 161804901 T^{7} + 1637432987 T^{8} - 14017245842 T^{9} + 119532608051 T^{10} - 862258317429 T^{11} + 6188147492363 T^{12} - 36906524393046 T^{13} + 220941751167161 T^{14} - 1006523939048139 T^{15} + 5037613724708232 T^{16} - 13709821562199377 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 - 7 T + 501 T^{2} - 3633 T^{3} + 122580 T^{4} - 870050 T^{5} + 19038015 T^{6} - 126186795 T^{7} + 2069616807 T^{8} - 12107857118 T^{9} + 163499727753 T^{10} - 787531787595 T^{11} + 9386483877585 T^{12} - 33888517974050 T^{13} + 377185573389420 T^{14} - 883136725907793 T^{15} + 9621158402065659 T^{16} - 10619761669345927 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 - 6 T + 425 T^{2} - 2371 T^{3} + 98340 T^{4} - 503493 T^{5} + 15146331 T^{6} - 69364165 T^{7} + 1688312923 T^{8} - 6801028730 T^{9} + 140129972609 T^{10} - 477849732685 T^{11} + 8660475163497 T^{12} - 23894932415253 T^{13} + 387365256832620 T^{14} - 775175625257899 T^{15} + 11532821670591475 T^{16} - 13513753392834246 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 - 33 T + 855 T^{2} - 15199 T^{3} + 240940 T^{4} - 3145538 T^{5} + 38621540 T^{6} - 417369329 T^{7} + 4382266528 T^{8} - 41703298394 T^{9} + 390021720992 T^{10} - 3305982455009 T^{11} + 27226988432260 T^{12} - 197358103270658 T^{13} + 1345423283642060 T^{14} - 7553618641316239 T^{15} + 37817791335677295 T^{16} - 129907430588168673 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 + 11 T + 705 T^{2} + 6575 T^{3} + 231506 T^{4} + 1858210 T^{5} + 46821726 T^{6} + 323465585 T^{7} + 6438386990 T^{8} + 37837079046 T^{9} + 624523538030 T^{10} + 3043487689265 T^{11} + 42732925133598 T^{12} + 164505995247010 T^{13} + 1988020793537042 T^{14} + 5476790932408175 T^{15} + 56962790557069665 T^{16} + 86211769538146571 T^{17} + 760231058654565217 T^{18} \)
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