Properties

Label 88.1.l.a
Level $88$
Weight $1$
Character orbit 88.l
Analytic conductor $0.044$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [88,1,Mod(3,88)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(88, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("88.3");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 88.l (of order \(10\), degree \(4\), 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: \(0.0439177211117\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) 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 5.1.937024.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{10} q^{2} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{3} + \zeta_{10}^{2} q^{4} + ( - \zeta_{10}^{3} + 1) q^{6} - \zeta_{10}^{3} q^{8} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{10} q^{2} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{3} + \zeta_{10}^{2} q^{4} + ( - \zeta_{10}^{3} + 1) q^{6} - \zeta_{10}^{3} q^{8} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}) q^{9} - \zeta_{10}^{3} q^{11} + (\zeta_{10}^{4} - \zeta_{10}) q^{12} + \zeta_{10}^{4} q^{16} + (\zeta_{10}^{2} - \zeta_{10}) q^{17} + (\zeta_{10}^{4} + \zeta_{10}^{2} + 1) q^{18} + ( - \zeta_{10} + 1) q^{19} + \zeta_{10}^{4} q^{22} + (\zeta_{10}^{2} + 1) q^{24} - \zeta_{10}^{3} q^{25} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots + 1) q^{27} + \cdots + (\zeta_{10}^{4} + \zeta_{10}^{2} - \zeta_{10}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 2 q^{3} - q^{4} + 3 q^{6} - q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 2 q^{3} - q^{4} + 3 q^{6} - q^{8} - 3 q^{9} - q^{11} - 2 q^{12} - q^{16} - 2 q^{17} + 2 q^{18} + 3 q^{19} - q^{22} + 3 q^{24} - q^{25} + q^{27} + 4 q^{32} + 3 q^{33} - 2 q^{34} + 2 q^{36} - 2 q^{38} - 2 q^{41} - 2 q^{43} + 4 q^{44} - 2 q^{48} - q^{49} - q^{50} + q^{51} - 4 q^{54} + q^{57} + 3 q^{59} - q^{64} - 2 q^{66} - 2 q^{67} - 2 q^{68} - 3 q^{72} - 2 q^{73} + 3 q^{75} - 2 q^{76} + 3 q^{82} + 3 q^{83} + 3 q^{86} - q^{88} - 2 q^{89} - 2 q^{96} + 3 q^{97} + 4 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/88\mathbb{Z}\right)^\times\).

\(n\) \(23\) \(45\) \(57\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{10}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.809017 0.587785i
−0.309017 0.951057i
0.809017 + 0.587785i
−0.309017 + 0.951057i
−0.809017 + 0.587785i −0.500000 1.53884i 0.309017 0.951057i 0 1.30902 + 0.951057i 0 0.309017 + 0.951057i −1.30902 + 0.951057i 0
27.1 0.309017 + 0.951057i −0.500000 0.363271i −0.809017 + 0.587785i 0 0.190983 0.587785i 0 −0.809017 0.587785i −0.190983 0.587785i 0
59.1 −0.809017 0.587785i −0.500000 + 1.53884i 0.309017 + 0.951057i 0 1.30902 0.951057i 0 0.309017 0.951057i −1.30902 0.951057i 0
75.1 0.309017 0.951057i −0.500000 + 0.363271i −0.809017 0.587785i 0 0.190983 + 0.587785i 0 −0.809017 + 0.587785i −0.190983 + 0.587785i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
11.c even 5 1 inner
88.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 88.1.l.a 4
3.b odd 2 1 792.1.bu.a 4
4.b odd 2 1 352.1.t.a 4
5.b even 2 1 2200.1.cl.a 4
5.c odd 4 2 2200.1.dd.a 8
8.b even 2 1 352.1.t.a 4
8.d odd 2 1 CM 88.1.l.a 4
11.b odd 2 1 968.1.l.b 4
11.c even 5 1 inner 88.1.l.a 4
11.c even 5 1 968.1.f.b 2
11.c even 5 2 968.1.l.a 4
11.d odd 10 1 968.1.f.a 2
11.d odd 10 1 968.1.l.b 4
11.d odd 10 2 968.1.l.c 4
12.b even 2 1 3168.1.ck.a 4
16.e even 4 2 2816.1.v.c 8
16.f odd 4 2 2816.1.v.c 8
24.f even 2 1 792.1.bu.a 4
24.h odd 2 1 3168.1.ck.a 4
33.h odd 10 1 792.1.bu.a 4
40.e odd 2 1 2200.1.cl.a 4
40.k even 4 2 2200.1.dd.a 8
44.c even 2 1 3872.1.t.c 4
44.g even 10 1 3872.1.f.b 2
44.g even 10 2 3872.1.t.a 4
44.g even 10 1 3872.1.t.c 4
44.h odd 10 1 352.1.t.a 4
44.h odd 10 1 3872.1.f.a 2
44.h odd 10 2 3872.1.t.b 4
55.j even 10 1 2200.1.cl.a 4
55.k odd 20 2 2200.1.dd.a 8
88.b odd 2 1 3872.1.t.c 4
88.g even 2 1 968.1.l.b 4
88.k even 10 1 968.1.f.a 2
88.k even 10 1 968.1.l.b 4
88.k even 10 2 968.1.l.c 4
88.l odd 10 1 inner 88.1.l.a 4
88.l odd 10 1 968.1.f.b 2
88.l odd 10 2 968.1.l.a 4
88.o even 10 1 352.1.t.a 4
88.o even 10 1 3872.1.f.a 2
88.o even 10 2 3872.1.t.b 4
88.p odd 10 1 3872.1.f.b 2
88.p odd 10 2 3872.1.t.a 4
88.p odd 10 1 3872.1.t.c 4
132.o even 10 1 3168.1.ck.a 4
176.v odd 20 2 2816.1.v.c 8
176.w even 20 2 2816.1.v.c 8
264.t odd 10 1 3168.1.ck.a 4
264.w even 10 1 792.1.bu.a 4
440.bh odd 10 1 2200.1.cl.a 4
440.bs even 20 2 2200.1.dd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.1.l.a 4 1.a even 1 1 trivial
88.1.l.a 4 8.d odd 2 1 CM
88.1.l.a 4 11.c even 5 1 inner
88.1.l.a 4 88.l odd 10 1 inner
352.1.t.a 4 4.b odd 2 1
352.1.t.a 4 8.b even 2 1
352.1.t.a 4 44.h odd 10 1
352.1.t.a 4 88.o even 10 1
792.1.bu.a 4 3.b odd 2 1
792.1.bu.a 4 24.f even 2 1
792.1.bu.a 4 33.h odd 10 1
792.1.bu.a 4 264.w even 10 1
968.1.f.a 2 11.d odd 10 1
968.1.f.a 2 88.k even 10 1
968.1.f.b 2 11.c even 5 1
968.1.f.b 2 88.l odd 10 1
968.1.l.a 4 11.c even 5 2
968.1.l.a 4 88.l odd 10 2
968.1.l.b 4 11.b odd 2 1
968.1.l.b 4 11.d odd 10 1
968.1.l.b 4 88.g even 2 1
968.1.l.b 4 88.k even 10 1
968.1.l.c 4 11.d odd 10 2
968.1.l.c 4 88.k even 10 2
2200.1.cl.a 4 5.b even 2 1
2200.1.cl.a 4 40.e odd 2 1
2200.1.cl.a 4 55.j even 10 1
2200.1.cl.a 4 440.bh odd 10 1
2200.1.dd.a 8 5.c odd 4 2
2200.1.dd.a 8 40.k even 4 2
2200.1.dd.a 8 55.k odd 20 2
2200.1.dd.a 8 440.bs even 20 2
2816.1.v.c 8 16.e even 4 2
2816.1.v.c 8 16.f odd 4 2
2816.1.v.c 8 176.v odd 20 2
2816.1.v.c 8 176.w even 20 2
3168.1.ck.a 4 12.b even 2 1
3168.1.ck.a 4 24.h odd 2 1
3168.1.ck.a 4 132.o even 10 1
3168.1.ck.a 4 264.t odd 10 1
3872.1.f.a 2 44.h odd 10 1
3872.1.f.a 2 88.o even 10 1
3872.1.f.b 2 44.g even 10 1
3872.1.f.b 2 88.p odd 10 1
3872.1.t.a 4 44.g even 10 2
3872.1.t.a 4 88.p odd 10 2
3872.1.t.b 4 44.h odd 10 2
3872.1.t.b 4 88.o even 10 2
3872.1.t.c 4 44.c even 2 1
3872.1.t.c 4 44.g even 10 1
3872.1.t.c 4 88.b odd 2 1
3872.1.t.c 4 88.p odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(88, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
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