Properties

Label 155.3.c.a
Level $155$
Weight $3$
Character orbit 155.c
Self dual yes
Analytic conductor $4.223$
Analytic rank $0$
Dimension $2$
CM discriminant -155
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [155,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("155.154");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
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: yes
Analytic conductor: \(4.22344409758\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{31}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \sqrt{31}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 4 q^{4} - 5 q^{5} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 4 q^{4} - 5 q^{5} + 22 q^{9} + 4 \beta q^{12} - 4 \beta q^{13} - 5 \beta q^{15} + 16 q^{16} - \beta q^{17} - 7 q^{19} - 20 q^{20} - 4 \beta q^{23} + 25 q^{25} + 13 \beta q^{27} + 31 q^{31} + 88 q^{36} - \beta q^{37} - 124 q^{39} - 73 q^{41} + 11 \beta q^{43} - 110 q^{45} + 16 \beta q^{48} + 49 q^{49} - 31 q^{51} - 16 \beta q^{52} - 19 \beta q^{53} - 7 \beta q^{57} - 37 q^{59} - 20 \beta q^{60} + 64 q^{64} + 20 \beta q^{65} - 4 \beta q^{68} - 124 q^{69} + 137 q^{71} - 19 \beta q^{73} + 25 \beta q^{75} - 28 q^{76} - 80 q^{80} + 205 q^{81} + 11 \beta q^{83} + 5 \beta q^{85} - 16 \beta q^{92} + 31 \beta q^{93} + 35 q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{4} - 10 q^{5} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{4} - 10 q^{5} + 44 q^{9} + 32 q^{16} - 14 q^{19} - 40 q^{20} + 50 q^{25} + 62 q^{31} + 176 q^{36} - 248 q^{39} - 146 q^{41} - 220 q^{45} + 98 q^{49} - 62 q^{51} - 74 q^{59} + 128 q^{64} - 248 q^{69} + 274 q^{71} - 56 q^{76} - 160 q^{80} + 410 q^{81} + 70 q^{95}+O(q^{100}) \) 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
−5.56776
5.56776
0 −5.56776 4.00000 −5.00000 0 0 0 22.0000 0
154.2 0 5.56776 4.00000 −5.00000 0 0 0 22.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.3.c.a 2
5.b even 2 1 inner 155.3.c.a 2
5.c odd 4 2 775.3.d.a 2
31.b odd 2 1 inner 155.3.c.a 2
155.c odd 2 1 CM 155.3.c.a 2
155.f even 4 2 775.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.3.c.a 2 1.a even 1 1 trivial
155.3.c.a 2 5.b even 2 1 inner
155.3.c.a 2 31.b odd 2 1 inner
155.3.c.a 2 155.c odd 2 1 CM
775.3.d.a 2 5.c odd 4 2
775.3.d.a 2 155.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(155, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{2} - 31 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 31 \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 496 \) Copy content Toggle raw display
$17$ \( T^{2} - 31 \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 496 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 31)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 31 \) Copy content Toggle raw display
$41$ \( (T + 73)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 3751 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 11191 \) Copy content Toggle raw display
$59$ \( (T + 37)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( (T - 137)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 11191 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 3751 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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