Properties

Label 1127.1.f.b
Level $1127$
Weight $1$
Character orbit 1127.f
Analytic conductor $0.562$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -23
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1127,1,Mod(275,1127)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1127, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1127.275");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1127 = 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1127.f (of order \(6\), degree \(2\), 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: \(0.562446269237\)
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: \(\Z[a_1, a_2]\)
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: $C_3\times S_3$
Artin field: Galois closure of 6.0.29212967.2

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1127\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
275.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 0.500000 + 0.866025i 0 0 1.00000 0 1.00000 0 0
459.1 0.500000 + 0.866025i 0.500000 0.866025i 0 0 1.00000 0 1.00000 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 \(\Q(\sqrt{-23}) \)
7.c even 3 1 inner
161.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1127.1.f.b 2
7.b odd 2 1 1127.1.f.a 2
7.c even 3 1 23.1.b.a 1
7.c even 3 1 inner 1127.1.f.b 2
7.d odd 6 1 1127.1.d.b 1
7.d odd 6 1 1127.1.f.a 2
21.h odd 6 1 207.1.d.a 1
23.b odd 2 1 CM 1127.1.f.b 2
28.g odd 6 1 368.1.f.a 1
35.j even 6 1 575.1.d.a 1
35.l odd 12 2 575.1.c.a 2
56.k odd 6 1 1472.1.f.a 1
56.p even 6 1 1472.1.f.b 1
63.g even 3 1 1863.1.f.b 2
63.h even 3 1 1863.1.f.b 2
63.j odd 6 1 1863.1.f.a 2
63.n odd 6 1 1863.1.f.a 2
77.h odd 6 1 2783.1.d.b 1
77.m even 15 4 2783.1.f.c 4
77.o odd 30 4 2783.1.f.a 4
84.n even 6 1 3312.1.c.a 1
91.g even 3 1 3887.1.h.c 2
91.h even 3 1 3887.1.h.c 2
91.k even 6 1 3887.1.h.a 2
91.r even 6 1 3887.1.d.b 1
91.u even 6 1 3887.1.h.a 2
91.x odd 12 2 3887.1.j.e 4
91.z odd 12 2 3887.1.c.a 2
91.bd odd 12 2 3887.1.j.e 4
161.c even 2 1 1127.1.f.a 2
161.f odd 6 1 23.1.b.a 1
161.f odd 6 1 inner 1127.1.f.b 2
161.g even 6 1 1127.1.d.b 1
161.g even 6 1 1127.1.f.a 2
161.m even 33 10 529.1.d.a 10
161.p odd 66 10 529.1.d.a 10
483.m even 6 1 207.1.d.a 1
644.p even 6 1 368.1.f.a 1
805.t odd 6 1 575.1.d.a 1
805.y even 12 2 575.1.c.a 2
1288.s even 6 1 1472.1.f.a 1
1288.y odd 6 1 1472.1.f.b 1
1449.m even 6 1 1863.1.f.a 2
1449.q odd 6 1 1863.1.f.b 2
1449.bk even 6 1 1863.1.f.a 2
1449.bn odd 6 1 1863.1.f.b 2
1771.o even 6 1 2783.1.d.b 1
1771.bj even 30 4 2783.1.f.a 4
1771.bn odd 30 4 2783.1.f.c 4
1932.w odd 6 1 3312.1.c.a 1
2093.q odd 6 1 3887.1.h.a 2
2093.bc odd 6 1 3887.1.h.c 2
2093.bg odd 6 1 3887.1.d.b 1
2093.bn odd 6 1 3887.1.h.a 2
2093.bp odd 6 1 3887.1.h.c 2
2093.bv even 12 2 3887.1.j.e 4
2093.by even 12 2 3887.1.c.a 2
2093.ci even 12 2 3887.1.j.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.1.b.a 1 7.c even 3 1
23.1.b.a 1 161.f odd 6 1
207.1.d.a 1 21.h odd 6 1
207.1.d.a 1 483.m even 6 1
368.1.f.a 1 28.g odd 6 1
368.1.f.a 1 644.p even 6 1
529.1.d.a 10 161.m even 33 10
529.1.d.a 10 161.p odd 66 10
575.1.c.a 2 35.l odd 12 2
575.1.c.a 2 805.y even 12 2
575.1.d.a 1 35.j even 6 1
575.1.d.a 1 805.t odd 6 1
1127.1.d.b 1 7.d odd 6 1
1127.1.d.b 1 161.g even 6 1
1127.1.f.a 2 7.b odd 2 1
1127.1.f.a 2 7.d odd 6 1
1127.1.f.a 2 161.c even 2 1
1127.1.f.a 2 161.g even 6 1
1127.1.f.b 2 1.a even 1 1 trivial
1127.1.f.b 2 7.c even 3 1 inner
1127.1.f.b 2 23.b odd 2 1 CM
1127.1.f.b 2 161.f odd 6 1 inner
1472.1.f.a 1 56.k odd 6 1
1472.1.f.a 1 1288.s even 6 1
1472.1.f.b 1 56.p even 6 1
1472.1.f.b 1 1288.y odd 6 1
1863.1.f.a 2 63.j odd 6 1
1863.1.f.a 2 63.n odd 6 1
1863.1.f.a 2 1449.m even 6 1
1863.1.f.a 2 1449.bk even 6 1
1863.1.f.b 2 63.g even 3 1
1863.1.f.b 2 63.h even 3 1
1863.1.f.b 2 1449.q odd 6 1
1863.1.f.b 2 1449.bn odd 6 1
2783.1.d.b 1 77.h odd 6 1
2783.1.d.b 1 1771.o even 6 1
2783.1.f.a 4 77.o odd 30 4
2783.1.f.a 4 1771.bj even 30 4
2783.1.f.c 4 77.m even 15 4
2783.1.f.c 4 1771.bn odd 30 4
3312.1.c.a 1 84.n even 6 1
3312.1.c.a 1 1932.w odd 6 1
3887.1.c.a 2 91.z odd 12 2
3887.1.c.a 2 2093.by even 12 2
3887.1.d.b 1 91.r even 6 1
3887.1.d.b 1 2093.bg odd 6 1
3887.1.h.a 2 91.k even 6 1
3887.1.h.a 2 91.u even 6 1
3887.1.h.a 2 2093.q odd 6 1
3887.1.h.a 2 2093.bn odd 6 1
3887.1.h.c 2 91.g even 3 1
3887.1.h.c 2 91.h even 3 1
3887.1.h.c 2 2093.bc odd 6 1
3887.1.h.c 2 2093.bp odd 6 1
3887.1.j.e 4 91.x odd 12 2
3887.1.j.e 4 91.bd odd 12 2
3887.1.j.e 4 2093.bv even 12 2
3887.1.j.e 4 2093.ci even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1127, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - T + 1 \) 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
$47$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 2T + 4 \) 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
$73$ \( T^{2} - T + 1 \) 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
$97$ \( T^{2} \) Copy content Toggle raw display
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