Properties

Label 155.1.c.b
Level $155$
Weight $1$
Character orbit 155.c
Analytic conductor $0.077$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -31
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [155,1,Mod(154,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, 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("155.154");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 155.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: no
Analytic conductor: \(0.0773550769581\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.120125.1

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + (\zeta_{6}^{2} + \zeta_{6}) q^{2} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{7} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6}^{2} + \zeta_{6}) q^{2} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{7} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{8} - q^{9} + (\zeta_{6}^{2} - 1) q^{10} + ( - \zeta_{6}^{2} + \zeta_{6} + 2) q^{14} + q^{16} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{18} + q^{19} + ( - \zeta_{6}^{2} - \zeta_{6} - 1) q^{20} + \zeta_{6}^{2} q^{25} + (2 \zeta_{6}^{2} + \zeta_{6}) q^{28} - q^{31} + ( - \zeta_{6}^{2} + 1) q^{35} + ( - \zeta_{6}^{2} + \zeta_{6} + 1) q^{36} + (\zeta_{6}^{2} + \zeta_{6}) q^{38} + ( - \zeta_{6}^{2} + 1) q^{40} - q^{41} - \zeta_{6} q^{45} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{49} + ( - \zeta_{6} - 1) q^{50} + (\zeta_{6}^{2} - \zeta_{6} - 2) q^{56} - q^{59} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{62} + (\zeta_{6}^{2} + \zeta_{6}) q^{63} + q^{64} + (\zeta_{6}^{2} + \zeta_{6} + 1) q^{70} + q^{71} + (\zeta_{6}^{2} + \zeta_{6}) q^{72} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{76} + \zeta_{6} q^{80} + q^{81} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{82} + ( - \zeta_{6}^{2} + 1) q^{90} + \zeta_{6} q^{95} + (\zeta_{6}^{2} + \zeta_{6}) q^{97} + ( - 2 \zeta_{6}^{2} - \zeta_{6}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + q^{5} - 2 q^{9} - 3 q^{10} + 6 q^{14} + 2 q^{16} + 2 q^{19} - 2 q^{20} - q^{25} - 2 q^{31} + 3 q^{35} + 4 q^{36} + 3 q^{40} - 2 q^{41} - q^{45} - 4 q^{49} - 3 q^{50} - 6 q^{56} - 2 q^{59} + 2 q^{64} + 3 q^{70} + 2 q^{71} - 4 q^{76} + q^{80} + 2 q^{81} + 3 q^{90} + q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(96\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
154.1
0.500000 0.866025i
0.500000 + 0.866025i
1.73205i 0 −2.00000 0.500000 0.866025i 0 1.73205i 1.73205i −1.00000 −1.50000 0.866025i
154.2 1.73205i 0 −2.00000 0.500000 + 0.866025i 0 1.73205i 1.73205i −1.00000 −1.50000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
5.b even 2 1 inner
155.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.1.c.b 2
3.b odd 2 1 1395.1.b.c 2
4.b odd 2 1 2480.1.k.e 2
5.b even 2 1 inner 155.1.c.b 2
5.c odd 4 2 775.1.d.c 2
15.d odd 2 1 1395.1.b.c 2
20.d odd 2 1 2480.1.k.e 2
31.b odd 2 1 CM 155.1.c.b 2
93.c even 2 1 1395.1.b.c 2
124.d even 2 1 2480.1.k.e 2
155.c odd 2 1 inner 155.1.c.b 2
155.f even 4 2 775.1.d.c 2
465.g even 2 1 1395.1.b.c 2
620.e even 2 1 2480.1.k.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.1.c.b 2 1.a even 1 1 trivial
155.1.c.b 2 5.b even 2 1 inner
155.1.c.b 2 31.b odd 2 1 CM
155.1.c.b 2 155.c odd 2 1 inner
775.1.d.c 2 5.c odd 4 2
775.1.d.c 2 155.f even 4 2
1395.1.b.c 2 3.b odd 2 1
1395.1.b.c 2 15.d odd 2 1
1395.1.b.c 2 93.c even 2 1
1395.1.b.c 2 465.g even 2 1
2480.1.k.e 2 4.b odd 2 1
2480.1.k.e 2 20.d odd 2 1
2480.1.k.e 2 124.d even 2 1
2480.1.k.e 2 620.e even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{1}^{\mathrm{new}}(155, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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