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Properties

Label 1183.1.z.a
Level $1183$
Weight $1$
Character orbit 1183.z
Analytic conductor $0.590$
Analytic rank $0$
Dimension $8$
Projective image $A_{4}$
CM/RM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1183,1,Mod(268,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1183.268");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1183.z (of order \(12\), 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: \(0.590393909945\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.8281.1
Artin image: $C_8.A_4$
Artin field: Galois closure of 32.0.515207889456069254383620937908064472169867121.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{11} q^{2} - \zeta_{24}^{4} q^{3} + \zeta_{24}^{11} q^{5} - \zeta_{24}^{3} q^{6} + \zeta_{24}^{9} q^{7} + \zeta_{24}^{9} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{11} q^{2} - \zeta_{24}^{4} q^{3} + \zeta_{24}^{11} q^{5} - \zeta_{24}^{3} q^{6} + \zeta_{24}^{9} q^{7} + \zeta_{24}^{9} q^{8} + \zeta_{24}^{10} q^{10} - \zeta_{24} q^{11} + \zeta_{24}^{8} q^{14} + \zeta_{24}^{3} q^{15} + \zeta_{24}^{8} q^{16} + \zeta_{24}^{10} q^{17} + \zeta_{24}^{11} q^{19} + \zeta_{24} q^{21} - q^{22} + \zeta_{24}^{2} q^{23} + \zeta_{24} q^{24} - q^{27} + \zeta_{24}^{2} q^{30} - \zeta_{24}^{7} q^{31} - \zeta_{24}^{7} q^{32} + \zeta_{24}^{5} q^{33} + \zeta_{24}^{9} q^{34} - \zeta_{24}^{8} q^{35} + \zeta_{24}^{5} q^{37} + \zeta_{24}^{10} q^{38} - \zeta_{24}^{8} q^{40} + q^{42} + \zeta_{24} q^{46} - \zeta_{24}^{5} q^{47} + q^{48} - \zeta_{24}^{6} q^{49} + \zeta_{24}^{2} q^{51} - \zeta_{24}^{4} q^{53} + \zeta_{24}^{11} q^{54} + q^{55} - \zeta_{24}^{6} q^{56} + \zeta_{24}^{3} q^{57} - \zeta_{24} q^{59} + \zeta_{24}^{8} q^{61} - \zeta_{24}^{6} q^{62} - \zeta_{24}^{6} q^{64} + \zeta_{24}^{4} q^{66} + \zeta_{24}^{7} q^{67} - \zeta_{24}^{6} q^{69} - \zeta_{24}^{7} q^{70} + \zeta_{24} q^{73} + \zeta_{24}^{4} q^{74} - \zeta_{24}^{10} q^{77} - \zeta_{24}^{8} q^{79} - \zeta_{24}^{7} q^{80} + \zeta_{24}^{4} q^{81} - \zeta_{24}^{9} q^{85} - \zeta_{24}^{10} q^{88} - \zeta_{24}^{5} q^{89} + \zeta_{24}^{11} q^{93} - \zeta_{24}^{4} q^{94} - \zeta_{24}^{10} q^{95} + \zeta_{24}^{3} q^{97} - \zeta_{24}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} - 4 q^{14} - 4 q^{16} - 8 q^{22} - 8 q^{27} + 4 q^{35} + 4 q^{40} + 8 q^{42} + 8 q^{48} - 4 q^{53} + 8 q^{55} - 4 q^{61} + 4 q^{66} + 4 q^{74} + 4 q^{79} + 4 q^{81} - 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(\zeta_{24}^{8}\) \(\zeta_{24}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
268.1
−0.965926 − 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 − 0.965926i
−0.258819 − 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
−0.965926 + 0.258819i −0.500000 − 0.866025i 0 0.965926 − 0.258819i 0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 − 0.707107i 0 −0.866025 + 0.500000i
268.2 0.965926 − 0.258819i −0.500000 − 0.866025i 0 −0.965926 + 0.258819i −0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 + 0.707107i 0 −0.866025 + 0.500000i
408.1 −0.258819 − 0.965926i −0.500000 − 0.866025i 0 0.258819 + 0.965926i −0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 − 0.707107i 0 0.866025 − 0.500000i
408.2 0.258819 + 0.965926i −0.500000 − 0.866025i 0 −0.258819 − 0.965926i 0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 + 0.707107i 0 0.866025 − 0.500000i
606.1 −0.258819 + 0.965926i −0.500000 + 0.866025i 0 0.258819 − 0.965926i −0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0.866025 + 0.500000i
606.2 0.258819 − 0.965926i −0.500000 + 0.866025i 0 −0.258819 + 0.965926i 0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 − 0.707107i 0 0.866025 + 0.500000i
746.1 −0.965926 − 0.258819i −0.500000 + 0.866025i 0 0.965926 + 0.258819i 0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 + 0.707107i 0 −0.866025 − 0.500000i
746.2 0.965926 + 0.258819i −0.500000 + 0.866025i 0 −0.965926 − 0.258819i −0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 − 0.707107i 0 −0.866025 − 0.500000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 268.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
13.d odd 4 2 inner
91.r even 6 1 inner
91.z odd 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.1.z.a ✓ 8
7.c even 3 1 inner 1183.1.z.a ✓ 8
13.b even 2 1 inner 1183.1.z.a ✓ 8
13.c even 3 1 1183.1.x.a 8
13.c even 3 1 1183.1.bd.a 8
13.d odd 4 2 inner 1183.1.z.a ✓ 8
13.e even 6 1 1183.1.x.a 8
13.e even 6 1 1183.1.bd.a 8
13.f odd 12 2 1183.1.x.a 8
13.f odd 12 2 1183.1.bd.a 8
91.g even 3 1 1183.1.x.a 8
91.h even 3 1 1183.1.bd.a 8
91.k even 6 1 1183.1.bd.a 8
91.r even 6 1 inner 1183.1.z.a ✓ 8
91.u even 6 1 1183.1.x.a 8
91.x odd 12 2 1183.1.bd.a 8
91.z odd 12 2 inner 1183.1.z.a ✓ 8
91.bd odd 12 2 1183.1.x.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.1.x.a 8 13.c even 3 1
1183.1.x.a 8 13.e even 6 1
1183.1.x.a 8 13.f odd 12 2
1183.1.x.a 8 91.g even 3 1
1183.1.x.a 8 91.u even 6 1
1183.1.x.a 8 91.bd odd 12 2
1183.1.z.a ✓ 8 1.a even 1 1 trivial
1183.1.z.a ✓ 8 7.c even 3 1 inner
1183.1.z.a ✓ 8 13.b even 2 1 inner
1183.1.z.a ✓ 8 13.d odd 4 2 inner
1183.1.z.a ✓ 8 91.r even 6 1 inner
1183.1.z.a ✓ 8 91.z odd 12 2 inner
1183.1.bd.a 8 13.c even 3 1
1183.1.bd.a 8 13.e even 6 1
1183.1.bd.a 8 13.f odd 12 2
1183.1.bd.a 8 91.h even 3 1
1183.1.bd.a 8 91.k even 6 1
1183.1.bd.a 8 91.x odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1183, [\chi])\).

Hecke characteristic polynomials

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