Properties

Label 55.2.g.b
Level 55
Weight 2
Character orbit 55.g
Analytic conductor 0.439
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 55 = 5 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 55.g (of order \(5\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} - 3 \nu^{5} - 4 \nu^{3} - 7 \nu^{2} - 12 \nu - 7 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 7 \nu^{5} + 20 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} + 6 \nu + 9 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} - 9 \nu^{5} + 12 \nu^{4} - 16 \nu^{3} + 13 \nu^{2} - 10 \nu - 1 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 10 \nu^{6} - 17 \nu^{5} + 8 \nu^{4} - 4 \nu^{3} - 13 \nu^{2} - 8 \nu - 5 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{7} - 12 \nu^{6} + 23 \nu^{5} - 20 \nu^{4} + 16 \nu^{3} + \nu^{2} + 6 \nu - 1 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 18 \nu^{6} - 35 \nu^{5} + 32 \nu^{4} - 28 \nu^{3} - 11 \nu^{2} - 12 \nu - 7 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(3 \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 4 \beta_{1}\)
\(\nu^{5}\)\(=\)\(4 \beta_{7} - 6 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(-16 \beta_{6} - 16 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 7 \beta_{1} - 6\)
\(\nu^{7}\)\(=\)\(-16 \beta_{7} - 51 \beta_{6} - 29 \beta_{5} - 23 \beta_{4} + 29 \beta_{2} - 16\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(1\) \(\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
1.69513 + 1.23158i
−0.386111 0.280526i
0.418926 1.28932i
−0.227943 + 0.701538i
1.69513 1.23158i
−0.386111 + 0.280526i
0.418926 + 1.28932i
−0.227943 0.701538i
−0.647481 1.99274i −1.54765 1.12443i −1.93376 + 1.40496i −0.309017 + 0.951057i −1.23863 + 3.81211i 2.48141 1.80285i 0.661536 + 0.480634i 0.203814 + 0.627276i 2.09529
16.2 0.147481 + 0.453901i −0.261370 0.189896i 1.43376 1.04169i −0.309017 + 0.951057i 0.0476470 0.146642i −2.17239 + 1.57833i 1.45650 + 1.05821i −0.894797 2.75390i −0.477260
26.1 −1.09676 0.796845i 0.177837 0.547326i −0.0501062 0.154211i 0.809017 0.587785i −0.631180 + 0.458579i −1.12773 3.47080i −0.905781 + 2.78771i 2.15911 + 1.56869i −1.35567
26.2 0.596764 + 0.433574i −0.868820 + 2.67395i −0.449894 1.38463i 0.809017 0.587785i −1.67784 + 1.21902i 0.318714 + 0.980901i 0.787747 2.42443i −3.96813 2.88301i 0.737640
31.1 −0.647481 + 1.99274i −1.54765 + 1.12443i −1.93376 1.40496i −0.309017 0.951057i −1.23863 3.81211i 2.48141 + 1.80285i 0.661536 0.480634i 0.203814 0.627276i 2.09529
31.2 0.147481 0.453901i −0.261370 + 0.189896i 1.43376 + 1.04169i −0.309017 0.951057i 0.0476470 + 0.146642i −2.17239 1.57833i 1.45650 1.05821i −0.894797 + 2.75390i −0.477260
36.1 −1.09676 + 0.796845i 0.177837 + 0.547326i −0.0501062 + 0.154211i 0.809017 + 0.587785i −0.631180 0.458579i −1.12773 + 3.47080i −0.905781 2.78771i 2.15911 1.56869i −1.35567
36.2 0.596764 0.433574i −0.868820 2.67395i −0.449894 + 1.38463i 0.809017 + 0.587785i −1.67784 1.21902i 0.318714 0.980901i 0.787747 + 2.42443i −3.96813 + 2.88301i 0.737640
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
11.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(55, [\chi])\).