Properties

Label 55.2.e.b
Level 55
Weight 2
Character orbit 55.e
Analytic conductor 0.439
Analytic rank 0
Dimension 4
CM disc. -11
Inner twists 4

Related objects

Downloads

Learn more about

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: \(\Q(i, \sqrt{11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} - \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} + ( 1 - 3 \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} - \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} + ( 1 - 3 \beta_{2} + 2 \beta_{3} ) q^{9} + ( -2 \beta_{1} + \beta_{2} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{12} + ( -3 - \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{15} -4 q^{16} + ( 2 - 2 \beta_{3} ) q^{20} + ( 4 + \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{23} + ( 2 + 3 \beta_{3} ) q^{25} + ( 5 - \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{27} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{31} + ( -6 + \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{33} + ( 6 - 4 \beta_{1} + 2 \beta_{2} ) q^{36} + ( -5 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{37} + ( -2 - 4 \beta_{3} ) q^{44} + ( -3 + 3 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{45} + ( -5 + 2 \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 4 - 4 \beta_{1} + 4 \beta_{3} ) q^{48} + 7 \beta_{2} q^{49} + ( -1 - 4 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{53} + ( 7 + 3 \beta_{3} ) q^{55} + ( 1 + 2 \beta_{3} ) q^{59} + ( -8 + 4 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{60} -8 \beta_{2} q^{64} + ( 5 - 3 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} ) q^{67} + ( -4 - \beta_{2} - 8 \beta_{3} ) q^{69} -3 q^{71} + ( 7 - \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{75} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{80} + ( -2 + 6 \beta_{1} - 3 \beta_{2} ) q^{81} + 9 \beta_{2} q^{89} + ( 10 + 2 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 18 - 3 \beta_{1} - 15 \beta_{2} + 3 \beta_{3} ) q^{93} + ( -10 - 3 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} ) q^{97} + ( 3 - 11 \beta_{2} + 6 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + O(q^{10}) \) \( 4q - 2q^{3} - 4q^{12} - 8q^{15} - 16q^{16} + 12q^{20} + 18q^{23} + 2q^{25} + 22q^{27} - 22q^{33} + 24q^{36} - 14q^{37} - 18q^{45} - 24q^{47} + 8q^{48} - 12q^{53} + 22q^{55} - 28q^{60} + 26q^{67} - 12q^{71} + 32q^{75} - 8q^{81} + 36q^{92} + 66q^{93} - 34q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 9\):

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

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
−1.65831 + 0.500000i
1.65831 + 0.500000i
−1.65831 0.500000i
1.65831 0.500000i
0 −2.15831 + 2.15831i 2.00000i 1.65831 1.50000i 0 0 0 6.31662i 0
32.2 0 1.15831 1.15831i 2.00000i −1.65831 1.50000i 0 0 0 0.316625i 0
43.1 0 −2.15831 2.15831i 2.00000i 1.65831 + 1.50000i 0 0 0 6.31662i 0
43.2 0 1.15831 + 1.15831i 2.00000i −1.65831 + 1.50000i 0 0 0 0.316625i 0
\(n\): 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
11.b Odd 1 CM by \(\Q(\sqrt{-11}) \) yes
5.c Odd 1 yes
55.e Even 1 yes

Hecke kernels

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