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Properties

Label 2883.1.l.d
Level $2883$
Weight $1$
Character orbit 2883.l
Analytic conductor $1.439$
Analytic rank $0$
Dimension $16$
Projective image $S_{4}$
CM/RM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2883,1,Mod(374,2883)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2883, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([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("2883.374");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2883 = 3 \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2883.l (of order \(10\), degree \(4\), not 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: \(1.43880443142\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.268119.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{40}^{14} q^{2} - \zeta_{40}^{11} q^{3} + \zeta_{40}^{10} q^{5} + \zeta_{40}^{5} q^{6} - \zeta_{40}^{8} q^{7} - \zeta_{40}^{2} q^{8} - \zeta_{40}^{2} q^{9} - \zeta_{40}^{4} q^{10} + \zeta_{40}^{2} q^{14} + \cdots - \zeta_{40}^{8} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{7} - 4 q^{10} + 4 q^{16} + 4 q^{18} - 4 q^{19} - 4 q^{40} - 4 q^{45} + 4 q^{51} + 4 q^{64} + 4 q^{70} + 4 q^{72} + 4 q^{81} - 4 q^{82} - 16 q^{87} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2883\mathbb{Z}\right)^\times\).

\(n\) \(962\) \(964\)
\(\chi(n)\) \(-1\) \(-\zeta_{40}^{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
374.1
−0.987688 − 0.156434i
0.987688 + 0.156434i
−0.156434 + 0.987688i
0.156434 − 0.987688i
−0.987688 + 0.156434i
0.987688 − 0.156434i
−0.156434 − 0.987688i
0.156434 + 0.987688i
0.453990 − 0.891007i
−0.453990 + 0.891007i
0.891007 + 0.453990i
−0.891007 − 0.453990i
0.453990 + 0.891007i
−0.453990 − 0.891007i
0.891007 − 0.453990i
−0.891007 + 0.453990i
−0.587785 + 0.809017i −0.156434 + 0.987688i 0 1.00000i −0.707107 − 0.707107i −0.309017 − 0.951057i −0.951057 − 0.309017i −0.951057 − 0.309017i −0.809017 − 0.587785i
374.2 −0.587785 + 0.809017i 0.156434 − 0.987688i 0 1.00000i 0.707107 + 0.707107i −0.309017 − 0.951057i −0.951057 − 0.309017i −0.951057 − 0.309017i −0.809017 − 0.587785i
374.3 0.587785 − 0.809017i −0.987688 − 0.156434i 0 − 1.00000i −0.707107 + 0.707107i −0.309017 − 0.951057i 0.951057 + 0.309017i 0.951057 + 0.309017i −0.809017 − 0.587785i
374.4 0.587785 − 0.809017i 0.987688 + 0.156434i 0 − 1.00000i 0.707107 − 0.707107i −0.309017 − 0.951057i 0.951057 + 0.309017i 0.951057 + 0.309017i −0.809017 − 0.587785i
1349.1 −0.587785 − 0.809017i −0.156434 − 0.987688i 0 − 1.00000i −0.707107 + 0.707107i −0.309017 + 0.951057i −0.951057 + 0.309017i −0.951057 + 0.309017i −0.809017 + 0.587785i
1349.2 −0.587785 − 0.809017i 0.156434 + 0.987688i 0 − 1.00000i 0.707107 − 0.707107i −0.309017 + 0.951057i −0.951057 + 0.309017i −0.951057 + 0.309017i −0.809017 + 0.587785i
1349.3 0.587785 + 0.809017i −0.987688 + 0.156434i 0 1.00000i −0.707107 − 0.707107i −0.309017 + 0.951057i 0.951057 − 0.309017i 0.951057 − 0.309017i −0.809017 + 0.587785i
1349.4 0.587785 + 0.809017i 0.987688 − 0.156434i 0 1.00000i 0.707107 + 0.707107i −0.309017 + 0.951057i 0.951057 − 0.309017i 0.951057 − 0.309017i −0.809017 + 0.587785i
1589.1 −0.951057 − 0.309017i −0.891007 − 0.453990i 0 1.00000i 0.707107 + 0.707107i 0.809017 + 0.587785i 0.587785 + 0.809017i 0.587785 + 0.809017i 0.309017 − 0.951057i
1589.2 −0.951057 − 0.309017i 0.891007 + 0.453990i 0 1.00000i −0.707107 − 0.707107i 0.809017 + 0.587785i 0.587785 + 0.809017i 0.587785 + 0.809017i 0.309017 − 0.951057i
1589.3 0.951057 + 0.309017i −0.453990 + 0.891007i 0 − 1.00000i −0.707107 + 0.707107i 0.809017 + 0.587785i −0.587785 − 0.809017i −0.587785 − 0.809017i 0.309017 − 0.951057i
1589.4 0.951057 + 0.309017i 0.453990 − 0.891007i 0 − 1.00000i 0.707107 − 0.707107i 0.809017 + 0.587785i −0.587785 − 0.809017i −0.587785 − 0.809017i 0.309017 − 0.951057i
2453.1 −0.951057 + 0.309017i −0.891007 + 0.453990i 0 − 1.00000i 0.707107 − 0.707107i 0.809017 − 0.587785i 0.587785 − 0.809017i 0.587785 − 0.809017i 0.309017 + 0.951057i
2453.2 −0.951057 + 0.309017i 0.891007 − 0.453990i 0 − 1.00000i −0.707107 + 0.707107i 0.809017 − 0.587785i 0.587785 − 0.809017i 0.587785 − 0.809017i 0.309017 + 0.951057i
2453.3 0.951057 − 0.309017i −0.453990 − 0.891007i 0 1.00000i −0.707107 − 0.707107i 0.809017 − 0.587785i −0.587785 + 0.809017i −0.587785 + 0.809017i 0.309017 + 0.951057i
2453.4 0.951057 − 0.309017i 0.453990 + 0.891007i 0 1.00000i 0.707107 + 0.707107i 0.809017 − 0.587785i −0.587785 + 0.809017i −0.587785 + 0.809017i 0.309017 + 0.951057i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 374.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.b odd 2 1 inner
31.d even 5 3 inner
31.f odd 10 3 inner
93.c even 2 1 inner
93.k even 10 3 inner
93.l odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2883.1.l.d 16
3.b odd 2 1 inner 2883.1.l.d 16
31.b odd 2 1 inner 2883.1.l.d 16
31.c even 3 2 2883.1.o.e 32
31.d even 5 1 2883.1.b.c ✓ 4
31.d even 5 3 inner 2883.1.l.d 16
31.e odd 6 2 2883.1.o.e 32
31.f odd 10 1 2883.1.b.c ✓ 4
31.f odd 10 3 inner 2883.1.l.d 16
31.g even 15 2 2883.1.h.c 8
31.g even 15 6 2883.1.o.e 32
31.h odd 30 2 2883.1.h.c 8
31.h odd 30 6 2883.1.o.e 32
93.c even 2 1 inner 2883.1.l.d 16
93.g even 6 2 2883.1.o.e 32
93.h odd 6 2 2883.1.o.e 32
93.k even 10 1 2883.1.b.c ✓ 4
93.k even 10 3 inner 2883.1.l.d 16
93.l odd 10 1 2883.1.b.c ✓ 4
93.l odd 10 3 inner 2883.1.l.d 16
93.o odd 30 2 2883.1.h.c 8
93.o odd 30 6 2883.1.o.e 32
93.p even 30 2 2883.1.h.c 8
93.p even 30 6 2883.1.o.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2883.1.b.c ✓ 4 31.d even 5 1
2883.1.b.c ✓ 4 31.f odd 10 1
2883.1.b.c ✓ 4 93.k even 10 1
2883.1.b.c ✓ 4 93.l odd 10 1
2883.1.h.c 8 31.g even 15 2
2883.1.h.c 8 31.h odd 30 2
2883.1.h.c 8 93.o odd 30 2
2883.1.h.c 8 93.p even 30 2
2883.1.l.d 16 1.a even 1 1 trivial
2883.1.l.d 16 3.b odd 2 1 inner
2883.1.l.d 16 31.b odd 2 1 inner
2883.1.l.d 16 31.d even 5 3 inner
2883.1.l.d 16 31.f odd 10 3 inner
2883.1.l.d 16 93.c even 2 1 inner
2883.1.l.d 16 93.k even 10 3 inner
2883.1.l.d 16 93.l odd 10 3 inner
2883.1.o.e 32 31.c even 3 2
2883.1.o.e 32 31.e odd 6 2
2883.1.o.e 32 31.g even 15 6
2883.1.o.e 32 31.h odd 30 6
2883.1.o.e 32 93.g even 6 2
2883.1.o.e 32 93.h odd 6 2
2883.1.o.e 32 93.o odd 30 6
2883.1.o.e 32 93.p even 30 6

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2883, [\chi])\):

\( T_{2}^{8} - T_{2}^{6} + T_{2}^{4} - T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - T_{7}^{3} + T_{7}^{2} - T_{7} + 1 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( (T^{8} - 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( (T^{8} - 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{8} - 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
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