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 discriminant -119
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [119,1,Mod(118,119)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(119, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("119.118");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 119 = 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 119.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.0593887365033\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.14161.1
Artin image: $D_{10}$
Artin field: Galois closure of 10.0.1403737447.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 Parity Ord Mult Type
1.a even 1 1 trivial
119.d odd 2 1 CM by \(\Q(\sqrt{-119}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 119.1.d.b yes 2
3.b odd 2 1 1071.1.h.a 2
4.b odd 2 1 1904.1.n.a 2
5.b even 2 1 2975.1.h.c 2
5.c odd 4 2 2975.1.b.a 4
7.b odd 2 1 119.1.d.a 2
7.c even 3 2 833.1.h.a 4
7.d odd 6 2 833.1.h.b 4
17.b even 2 1 119.1.d.a 2
17.c even 4 2 2023.1.c.e 4
17.d even 8 4 2023.1.f.b 8
17.e odd 16 8 2023.1.l.b 16
21.c even 2 1 1071.1.h.b 2
28.d even 2 1 1904.1.n.b 2
35.c odd 2 1 2975.1.h.d 2
35.f even 4 2 2975.1.b.b 4
51.c odd 2 1 1071.1.h.b 2
68.d odd 2 1 1904.1.n.b 2
85.c even 2 1 2975.1.h.d 2
85.g odd 4 2 2975.1.b.b 4
119.d odd 2 1 CM 119.1.d.b yes 2
119.f odd 4 2 2023.1.c.e 4
119.h odd 6 2 833.1.h.a 4
119.j even 6 2 833.1.h.b 4
119.l odd 8 4 2023.1.f.b 8
119.p even 16 8 2023.1.l.b 16
357.c even 2 1 1071.1.h.a 2
476.e even 2 1 1904.1.n.a 2
595.b odd 2 1 2975.1.h.c 2
595.p even 4 2 2975.1.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.1.d.a 2 7.b odd 2 1
119.1.d.a 2 17.b even 2 1
119.1.d.b yes 2 1.a even 1 1 trivial
119.1.d.b yes 2 119.d odd 2 1 CM
833.1.h.a 4 7.c even 3 2
833.1.h.a 4 119.h odd 6 2
833.1.h.b 4 7.d odd 6 2
833.1.h.b 4 119.j even 6 2
1071.1.h.a 2 3.b odd 2 1
1071.1.h.a 2 357.c even 2 1
1071.1.h.b 2 21.c even 2 1
1071.1.h.b 2 51.c odd 2 1
1904.1.n.a 2 4.b odd 2 1
1904.1.n.a 2 476.e even 2 1
1904.1.n.b 2 28.d even 2 1
1904.1.n.b 2 68.d odd 2 1
2023.1.c.e 4 17.c even 4 2
2023.1.c.e 4 119.f odd 4 2
2023.1.f.b 8 17.d even 8 4
2023.1.f.b 8 119.l odd 8 4
2023.1.l.b 16 17.e odd 16 8
2023.1.l.b 16 119.p even 16 8
2975.1.b.a 4 5.c odd 4 2
2975.1.b.a 4 595.p even 4 2
2975.1.b.b 4 35.f even 4 2
2975.1.b.b 4 85.g odd 4 2
2975.1.h.c 2 5.b even 2 1
2975.1.h.c 2 595.b odd 2 1
2975.1.h.d 2 35.c odd 2 1
2975.1.h.d 2 85.c even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$67$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - T - 1 \) Copy content Toggle raw display
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