Properties

Label 79.1.b.a
Level $79$
Weight $1$
Character orbit 79.b
Self dual yes
Analytic conductor $0.039$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -79
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [79,1,Mod(78,79)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(79, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("79.78");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 79 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 79.b (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: \(0.0394261359980\)
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 5.1.6241.1
Artin image: $D_5$
Artin field: Galois closure of 5.1.6241.1

$q$-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.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + ( - \beta + 1) q^{4} - \beta q^{5} - q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + ( - \beta + 1) q^{4} - \beta q^{5} - q^{8} + q^{9} - q^{10} + (\beta - 1) q^{11} + (\beta - 1) q^{13} + (\beta - 1) q^{18} - \beta q^{19} + q^{20} + ( - \beta + 2) q^{22} - \beta q^{23} + \beta q^{25} + ( - \beta + 2) q^{26} + (\beta - 1) q^{31} + q^{32} + ( - \beta + 1) q^{36} - q^{38} + \beta q^{40} + (\beta - 2) q^{44} - \beta q^{45} - q^{46} + q^{49} + q^{50} + (\beta - 2) q^{52} - q^{55} + ( - \beta + 2) q^{62} + (\beta - 1) q^{64} - q^{65} - \beta q^{67} - q^{72} - \beta q^{73} + q^{76} + q^{79} + q^{81} + 2 q^{83} + ( - \beta + 1) q^{88} + (\beta - 1) q^{89} - q^{90} + q^{92} + (\beta + 1) q^{95} - \beta q^{97} + (\beta - 1) q^{98} + (\beta - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{4} - q^{5} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{4} - q^{5} - 2 q^{8} + 2 q^{9} - 2 q^{10} - q^{11} - q^{13} - q^{18} - q^{19} + 2 q^{20} + 3 q^{22} - q^{23} + q^{25} + 3 q^{26} - q^{31} + 2 q^{32} + q^{36} - 2 q^{38} + q^{40} - 3 q^{44} - q^{45} - 2 q^{46} + 2 q^{49} + 2 q^{50} - 3 q^{52} - 2 q^{55} + 3 q^{62} - q^{64} - 2 q^{65} - q^{67} - 2 q^{72} - q^{73} + 2 q^{76} + 2 q^{79} + 2 q^{81} + 4 q^{83} + q^{88} - q^{89} - 2 q^{90} + 2 q^{92} + 3 q^{95} - q^{97} - q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/79\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
78.1
−0.618034
1.61803
−1.61803 0 1.61803 0.618034 0 0 −1.00000 1.00000 −1.00000
78.2 0.618034 0 −0.618034 −1.61803 0 0 −1.00000 1.00000 −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
79.b odd 2 1 CM by \(\Q(\sqrt{-79}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 79.1.b.a 2
3.b odd 2 1 711.1.d.b 2
4.b odd 2 1 1264.1.e.a 2
5.b even 2 1 1975.1.d.c 2
5.c odd 4 2 1975.1.c.a 4
7.b odd 2 1 3871.1.c.c 2
7.c even 3 2 3871.1.m.c 4
7.d odd 6 2 3871.1.m.b 4
79.b odd 2 1 CM 79.1.b.a 2
237.b even 2 1 711.1.d.b 2
316.d even 2 1 1264.1.e.a 2
395.c odd 2 1 1975.1.d.c 2
395.f even 4 2 1975.1.c.a 4
553.d even 2 1 3871.1.c.c 2
553.l even 6 2 3871.1.m.b 4
553.m odd 6 2 3871.1.m.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
79.1.b.a 2 1.a even 1 1 trivial
79.1.b.a 2 79.b odd 2 1 CM
711.1.d.b 2 3.b odd 2 1
711.1.d.b 2 237.b even 2 1
1264.1.e.a 2 4.b odd 2 1
1264.1.e.a 2 316.d even 2 1
1975.1.c.a 4 5.c odd 4 2
1975.1.c.a 4 395.f even 4 2
1975.1.d.c 2 5.b even 2 1
1975.1.d.c 2 395.c odd 2 1
3871.1.c.c 2 7.b odd 2 1
3871.1.c.c 2 553.d even 2 1
3871.1.m.b 4 7.d odd 6 2
3871.1.m.b 4 553.l even 6 2
3871.1.m.c 4 7.c even 3 2
3871.1.m.c 4 553.m odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(79, [\chi])\).

Hecke characteristic polynomials

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