Properties

Label 2783.1.f.a
Level $2783$
Weight $1$
Character orbit 2783.f
Analytic conductor $1.389$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -23
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2783,1,Mod(390,2783)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2783, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2783.390");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2783 = 11^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2783.f (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: \(1.38889793016\)
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: no (minimal twist has level 23)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.23.1
Artin image: $S_3\times C_{10}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{30} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{10}^{3} q^{2} - \zeta_{10}^{4} q^{3} - \zeta_{10}^{2} q^{6} - \zeta_{10}^{4} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{10}^{3} q^{2} - \zeta_{10}^{4} q^{3} - \zeta_{10}^{2} q^{6} - \zeta_{10}^{4} q^{8} - \zeta_{10}^{3} q^{13} - \zeta_{10}^{2} q^{16} + q^{23} - \zeta_{10}^{3} q^{24} + \zeta_{10}^{4} q^{25} - \zeta_{10} q^{26} + \zeta_{10}^{2} q^{27} - \zeta_{10} q^{29} + \zeta_{10}^{3} q^{31} + q^{32} - \zeta_{10}^{2} q^{39} + \zeta_{10}^{4} q^{41} - \zeta_{10}^{3} q^{46} - \zeta_{10}^{4} q^{47} - \zeta_{10} q^{48} + \zeta_{10}^{2} q^{49} + \zeta_{10}^{2} q^{50} + q^{54} + \zeta_{10}^{4} q^{58} - \zeta_{10} q^{59} + \zeta_{10} q^{62} - \zeta_{10}^{3} q^{64} - \zeta_{10}^{4} q^{69} - \zeta_{10}^{2} q^{71} - \zeta_{10} q^{73} + \zeta_{10}^{3} q^{75} - q^{78} + \zeta_{10} q^{81} + \zeta_{10}^{2} q^{82} - q^{87} + \zeta_{10}^{2} q^{93} - \zeta_{10}^{2} q^{94} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + q^{3} + q^{6} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + q^{3} + q^{6} + q^{8} - q^{13} + q^{16} + 4 q^{23} - q^{24} - q^{25} - q^{26} - q^{27} - q^{29} + q^{31} + q^{39} - q^{41} - q^{46} + q^{47} - q^{48} - q^{49} - q^{50} + 4 q^{54} - q^{58} - 2 q^{59} + q^{62} - q^{64} + q^{69} + q^{71} - q^{73} + q^{75} - 4 q^{78} + q^{81} - q^{82} - 4 q^{87} - q^{93} + q^{94} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2783\mathbb{Z}\right)^\times\).

\(n\) \(1937\) \(2301\)
\(\chi(n)\) \(-1\) \(-\zeta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
390.1
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
−0.309017 0.951057i
0.309017 + 0.951057i 0.809017 + 0.587785i 0 0 −0.309017 + 0.951057i 0 0.809017 + 0.587785i 0 0
735.1 0.309017 0.951057i 0.809017 0.587785i 0 0 −0.309017 0.951057i 0 0.809017 0.587785i 0 0
850.1 −0.809017 + 0.587785i −0.309017 0.951057i 0 0 0.809017 + 0.587785i 0 −0.309017 0.951057i 0 0
2138.1 −0.809017 0.587785i −0.309017 + 0.951057i 0 0 0.809017 0.587785i 0 −0.309017 + 0.951057i 0 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
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
11.c even 5 3 inner
253.f odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2783.1.f.a 4
11.b odd 2 1 2783.1.f.c 4
11.c even 5 1 2783.1.d.b 1
11.c even 5 3 inner 2783.1.f.a 4
11.d odd 10 1 23.1.b.a 1
11.d odd 10 3 2783.1.f.c 4
23.b odd 2 1 CM 2783.1.f.a 4
33.f even 10 1 207.1.d.a 1
44.g even 10 1 368.1.f.a 1
55.h odd 10 1 575.1.d.a 1
55.l even 20 2 575.1.c.a 2
77.l even 10 1 1127.1.d.b 1
77.n even 30 2 1127.1.f.a 2
77.o odd 30 2 1127.1.f.b 2
88.k even 10 1 1472.1.f.a 1
88.p odd 10 1 1472.1.f.b 1
99.o odd 30 2 1863.1.f.b 2
99.p even 30 2 1863.1.f.a 2
132.n odd 10 1 3312.1.c.a 1
143.l odd 10 1 3887.1.d.b 1
143.s even 20 2 3887.1.c.a 2
143.t odd 30 2 3887.1.h.c 2
143.v odd 30 2 3887.1.h.a 2
143.w even 60 4 3887.1.j.e 4
253.b even 2 1 2783.1.f.c 4
253.f odd 10 1 2783.1.d.b 1
253.f odd 10 3 inner 2783.1.f.a 4
253.h even 10 1 23.1.b.a 1
253.h even 10 3 2783.1.f.c 4
253.n even 110 10 529.1.d.a 10
253.o odd 110 10 529.1.d.a 10
759.j odd 10 1 207.1.d.a 1
1012.p odd 10 1 368.1.f.a 1
1265.p even 10 1 575.1.d.a 1
1265.x odd 20 2 575.1.c.a 2
1771.w odd 10 1 1127.1.d.b 1
1771.bi odd 30 2 1127.1.f.a 2
1771.bj even 30 2 1127.1.f.b 2
2024.s odd 10 1 1472.1.f.a 1
2024.y even 10 1 1472.1.f.b 1
2277.bh even 30 2 1863.1.f.b 2
2277.bn odd 30 2 1863.1.f.a 2
3036.w even 10 1 3312.1.c.a 1
3289.v even 10 1 3887.1.d.b 1
3289.bj odd 20 2 3887.1.c.a 2
3289.bw even 30 2 3887.1.h.c 2
3289.bx even 30 2 3887.1.h.a 2
3289.cj odd 60 4 3887.1.j.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.1.b.a 1 11.d odd 10 1
23.1.b.a 1 253.h even 10 1
207.1.d.a 1 33.f even 10 1
207.1.d.a 1 759.j odd 10 1
368.1.f.a 1 44.g even 10 1
368.1.f.a 1 1012.p odd 10 1
529.1.d.a 10 253.n even 110 10
529.1.d.a 10 253.o odd 110 10
575.1.c.a 2 55.l even 20 2
575.1.c.a 2 1265.x odd 20 2
575.1.d.a 1 55.h odd 10 1
575.1.d.a 1 1265.p even 10 1
1127.1.d.b 1 77.l even 10 1
1127.1.d.b 1 1771.w odd 10 1
1127.1.f.a 2 77.n even 30 2
1127.1.f.a 2 1771.bi odd 30 2
1127.1.f.b 2 77.o odd 30 2
1127.1.f.b 2 1771.bj even 30 2
1472.1.f.a 1 88.k even 10 1
1472.1.f.a 1 2024.s odd 10 1
1472.1.f.b 1 88.p odd 10 1
1472.1.f.b 1 2024.y even 10 1
1863.1.f.a 2 99.p even 30 2
1863.1.f.a 2 2277.bn odd 30 2
1863.1.f.b 2 99.o odd 30 2
1863.1.f.b 2 2277.bh even 30 2
2783.1.d.b 1 11.c even 5 1
2783.1.d.b 1 253.f odd 10 1
2783.1.f.a 4 1.a even 1 1 trivial
2783.1.f.a 4 11.c even 5 3 inner
2783.1.f.a 4 23.b odd 2 1 CM
2783.1.f.a 4 253.f odd 10 3 inner
2783.1.f.c 4 11.b odd 2 1
2783.1.f.c 4 11.d odd 10 3
2783.1.f.c 4 253.b even 2 1
2783.1.f.c 4 253.h even 10 3
3312.1.c.a 1 132.n odd 10 1
3312.1.c.a 1 3036.w even 10 1
3887.1.c.a 2 143.s even 20 2
3887.1.c.a 2 3289.bj odd 20 2
3887.1.d.b 1 143.l odd 10 1
3887.1.d.b 1 3289.v even 10 1
3887.1.h.a 2 143.v odd 30 2
3887.1.h.a 2 3289.bx even 30 2
3887.1.h.c 2 143.t odd 30 2
3887.1.h.c 2 3289.bw even 30 2
3887.1.j.e 4 143.w even 60 4
3887.1.j.e 4 3289.cj odd 60 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2783, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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