Properties

Label 119.1.d.b
Level 119
Weight 1
Character orbit 119.d
Self dual Yes
Analytic conductor 0.059
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM disc. -119
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 119 = 7 \cdot 17 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 119.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0593887365033\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.14161.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta ) q^{2} \) \( + ( 1 - \beta ) q^{3} \) \( + ( 1 - \beta ) q^{4} \) \( + \beta q^{5} \) \( + ( -2 + \beta ) q^{6} \) \(- q^{7}\) \(- q^{8}\) \( + ( 1 - \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta ) q^{2} \) \( + ( 1 - \beta ) q^{3} \) \( + ( 1 - \beta ) q^{4} \) \( + \beta q^{5} \) \( + ( -2 + \beta ) q^{6} \) \(- q^{7}\) \(- q^{8}\) \( + ( 1 - \beta ) q^{9} \) \(+ q^{10}\) \( + ( 2 - \beta ) q^{12} \) \( + ( 1 - \beta ) q^{14} \) \(- q^{15}\) \(- q^{17}\) \( + ( -2 + \beta ) q^{18} \) \(- q^{20}\) \( + ( -1 + \beta ) q^{21} \) \( + ( -1 + \beta ) q^{24} \) \( + \beta q^{25} \) \(+ q^{27}\) \( + ( -1 + \beta ) q^{28} \) \( + ( 1 - \beta ) q^{30} \) \( + \beta q^{31} \) \(+ q^{32}\) \( + ( 1 - \beta ) q^{34} \) \( -\beta q^{35} \) \( + ( 2 - \beta ) q^{36} \) \( -\beta q^{40} \) \( + ( 1 - \beta ) q^{41} \) \( + ( 2 - \beta ) q^{42} \) \( + ( -1 + \beta ) q^{43} \) \(- q^{45}\) \(+ q^{49}\) \(+ q^{50}\) \( + ( -1 + \beta ) q^{51} \) \( -\beta q^{53} \) \( + ( -1 + \beta ) q^{54} \) \(+ q^{56}\) \( + ( -1 + \beta ) q^{60} \) \( + ( 1 - \beta ) q^{61} \) \(+ q^{62}\) \( + ( -1 + \beta ) q^{63} \) \( + ( -1 + \beta ) q^{64} \) \( -\beta q^{67} \) \( + ( -1 + \beta ) q^{68} \) \(- q^{70}\) \( + ( -1 + \beta ) q^{72} \) \( + ( 1 - \beta ) q^{73} \) \(- q^{75}\) \( + ( -2 + \beta ) q^{82} \) \( + ( -2 + \beta ) q^{84} \) \( -\beta q^{85} \) \( + ( 2 - \beta ) q^{86} \) \( + ( 1 - \beta ) q^{90} \) \(- q^{93}\) \( + ( 1 - \beta ) q^{96} \) \( + \beta q^{97} \) \( + ( -1 + \beta ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut 3q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut 3q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut 2q^{10} \) \(\mathstrut +\mathstrut 3q^{12} \) \(\mathstrut +\mathstrut q^{14} \) \(\mathstrut -\mathstrut 2q^{15} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut 3q^{18} \) \(\mathstrut -\mathstrut 2q^{20} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut -\mathstrut q^{24} \) \(\mathstrut +\mathstrut q^{25} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut q^{28} \) \(\mathstrut +\mathstrut q^{30} \) \(\mathstrut +\mathstrut q^{31} \) \(\mathstrut +\mathstrut 2q^{32} \) \(\mathstrut +\mathstrut q^{34} \) \(\mathstrut -\mathstrut q^{35} \) \(\mathstrut +\mathstrut 3q^{36} \) \(\mathstrut -\mathstrut q^{40} \) \(\mathstrut +\mathstrut q^{41} \) \(\mathstrut +\mathstrut 3q^{42} \) \(\mathstrut -\mathstrut q^{43} \) \(\mathstrut -\mathstrut 2q^{45} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut q^{51} \) \(\mathstrut -\mathstrut q^{53} \) \(\mathstrut -\mathstrut q^{54} \) \(\mathstrut +\mathstrut 2q^{56} \) \(\mathstrut -\mathstrut q^{60} \) \(\mathstrut +\mathstrut q^{61} \) \(\mathstrut +\mathstrut 2q^{62} \) \(\mathstrut -\mathstrut q^{63} \) \(\mathstrut -\mathstrut q^{64} \) \(\mathstrut -\mathstrut q^{67} \) \(\mathstrut -\mathstrut q^{68} \) \(\mathstrut -\mathstrut 2q^{70} \) \(\mathstrut -\mathstrut q^{72} \) \(\mathstrut +\mathstrut q^{73} \) \(\mathstrut -\mathstrut 2q^{75} \) \(\mathstrut -\mathstrut 3q^{82} \) \(\mathstrut -\mathstrut 3q^{84} \) \(\mathstrut -\mathstrut q^{85} \) \(\mathstrut +\mathstrut 3q^{86} \) \(\mathstrut +\mathstrut q^{90} \) \(\mathstrut -\mathstrut 2q^{93} \) \(\mathstrut +\mathstrut q^{96} \) \(\mathstrut +\mathstrut q^{97} \) \(\mathstrut -\mathstrut q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/119\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
−0.618034
1.61803
−1.61803 1.61803 1.61803 −0.618034 −2.61803 −1.00000 −1.00000 1.61803 1.00000
118.2 0.618034 −0.618034 −0.618034 1.61803 −0.381966 −1.00000 −1.00000 −0.618034 1.00000
\(n\): 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
119.d Odd 1 CM by \(\Q(\sqrt{-119}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut -\mathstrut T_{3} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(119, [\chi])\).