Properties

Label 679.1.c.c
Level $679$
Weight $1$
Character orbit 679.c
Self dual yes
Analytic conductor $0.339$
Analytic rank $0$
Dimension $3$
Projective image $D_{9}$
CM discriminant -679
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [679,1,Mod(678,679)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(679, 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("679.678");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 679 = 7 \cdot 97 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 679.c (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: \(0.338865143578\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.212558803681.1
Artin image: $D_9$
Artin field: Galois closure of 9.1.212558803681.1

$q$-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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + q^{7} + ( - \beta_1 - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + q^{7} + ( - \beta_1 - 1) q^{8} + q^{9} + \beta_1 q^{10} + ( - \beta_{2} + \beta_1) q^{11} - \beta_1 q^{13} - \beta_1 q^{14} + (\beta_1 + 1) q^{16} + \beta_{2} q^{17} - \beta_1 q^{18} + ( - \beta_{2} + \beta_1) q^{19} + ( - \beta_{2} - 1) q^{20} + ( - \beta_{2} + \beta_1 - 1) q^{22} + (\beta_{2} + 2) q^{26} + (\beta_{2} + 1) q^{28} + ( - \beta_{2} - 1) q^{32} + ( - \beta_1 - 1) q^{34} - q^{35} + (\beta_{2} + 1) q^{36} + ( - \beta_{2} + \beta_1 - 1) q^{38} + (\beta_1 + 1) q^{40} + ( - \beta_{2} + \beta_1) q^{41} + \beta_{2} q^{43} + (\beta_1 - 1) q^{44} - q^{45} + q^{49} + ( - 2 \beta_1 - 1) q^{52} + \beta_{2} q^{53} + (\beta_{2} - \beta_1) q^{55} + ( - \beta_1 - 1) q^{56} + \beta_{2} q^{59} + q^{63} + \beta_1 q^{64} + \beta_1 q^{65} + (\beta_1 + 2) q^{68} + \beta_1 q^{70} + ( - \beta_1 - 1) q^{72} + (\beta_1 - 1) q^{76} + ( - \beta_{2} + \beta_1) q^{77} - q^{79} + ( - \beta_1 - 1) q^{80} + q^{81} + ( - \beta_{2} + \beta_1 - 1) q^{82} - \beta_1 q^{83} - \beta_{2} q^{85} + ( - \beta_1 - 1) q^{86} - q^{88} + \beta_1 q^{90} - \beta_1 q^{91} + (\beta_{2} - \beta_1) q^{95} + q^{97} - \beta_1 q^{98} + ( - \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{4} - 3 q^{5} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{4} - 3 q^{5} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{16} - 3 q^{20} - 3 q^{22} + 6 q^{26} + 3 q^{28} - 3 q^{32} - 3 q^{34} - 3 q^{35} + 3 q^{36} - 3 q^{38} + 3 q^{40} - 3 q^{44} - 3 q^{45} + 3 q^{49} - 3 q^{52} - 3 q^{56} + 3 q^{63} + 6 q^{68} - 3 q^{72} - 3 q^{76} - 3 q^{79} - 3 q^{80} + 3 q^{81} - 3 q^{82} - 3 q^{86} - 3 q^{88} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/679\mathbb{Z}\right)^\times\).

\(n\) \(393\) \(486\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
678.1
1.87939
−0.347296
−1.53209
−1.87939 0 2.53209 −1.00000 0 1.00000 −2.87939 1.00000 1.87939
678.2 0.347296 0 −0.879385 −1.00000 0 1.00000 −0.652704 1.00000 −0.347296
678.3 1.53209 0 1.34730 −1.00000 0 1.00000 0.532089 1.00000 −1.53209
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
679.c odd 2 1 CM by \(\Q(\sqrt{-679}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 679.1.c.c 3
7.b odd 2 1 679.1.c.d yes 3
97.b even 2 1 679.1.c.d yes 3
679.c odd 2 1 CM 679.1.c.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
679.1.c.c 3 1.a even 1 1 trivial
679.1.c.c 3 679.c odd 2 1 CM
679.1.c.d yes 3 7.b odd 2 1
679.1.c.d yes 3 97.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(679, [\chi])\):

\( T_{2}^{3} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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