Properties

Label 55.2.b.a
Level 55
Weight 2
Character orbit 55.b
Analytic conductor 0.439
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 55 = 5 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 55.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} + 4 \nu - 9 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 2 \nu^{2} + 2 \nu + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} - 3 \beta_{2} + 3\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 4\)

Character Values

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

\(n\) \(12\) \(46\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
−1.18614 + 1.26217i
1.68614 + 0.396143i
1.68614 0.396143i
−1.18614 1.26217i
2.52434i 0.792287i −4.37228 0.686141 2.12819i 2.00000 3.46410i 5.98844i 2.37228 −5.37228 1.73205i
34.2 0.792287i 2.52434i 1.37228 −2.18614 + 0.469882i 2.00000 3.46410i 2.67181i −3.37228 0.372281 + 1.73205i
34.3 0.792287i 2.52434i 1.37228 −2.18614 0.469882i 2.00000 3.46410i 2.67181i −3.37228 0.372281 1.73205i
34.4 2.52434i 0.792287i −4.37228 0.686141 + 2.12819i 2.00000 3.46410i 5.98844i 2.37228 −5.37228 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(55, [\chi])\).