Properties

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

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3}\)

Character Values

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

\(n\) \(12\) \(46\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−1.58114 + 1.58114i
1.58114 1.58114i
−1.58114 1.58114i
1.58114 + 1.58114i
−1.58114 + 1.58114i −1.00000 + 1.00000i 3.00000i −2.00000 + 1.00000i 3.16228i 0 1.58114 + 1.58114i 1.00000i 1.58114 4.74342i
32.2 1.58114 1.58114i −1.00000 + 1.00000i 3.00000i −2.00000 + 1.00000i 3.16228i 0 −1.58114 1.58114i 1.00000i −1.58114 + 4.74342i
43.1 −1.58114 1.58114i −1.00000 1.00000i 3.00000i −2.00000 1.00000i 3.16228i 0 1.58114 1.58114i 1.00000i 1.58114 + 4.74342i
43.2 1.58114 + 1.58114i −1.00000 1.00000i 3.00000i −2.00000 1.00000i 3.16228i 0 −1.58114 + 1.58114i 1.00000i −1.58114 4.74342i
\(n\): 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.c Odd 1 yes
11.b Odd 1 yes
55.e Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{4} + 25 \) acting on \(S_{2}^{\mathrm{new}}(55, [\chi])\).