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Properties

Label 1456.1.fx.a
Level $1456$
Weight $1$
Character orbit 1456.fx
Analytic conductor $0.727$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
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] = [1456,1,Mod(237,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 6, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.237");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1456.fx (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: \(0.726638658394\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.2422784.3

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{8} q^{2} - \zeta_{24}^{5} q^{3} - \zeta_{24}^{4} q^{4} - \zeta_{24}^{3} q^{5} + \zeta_{24} q^{6} - \zeta_{24}^{10} q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{8} q^{2} - \zeta_{24}^{5} q^{3} - \zeta_{24}^{4} q^{4} - \zeta_{24}^{3} q^{5} + \zeta_{24} q^{6} - \zeta_{24}^{10} q^{7} + q^{8} - \zeta_{24}^{11} q^{10} + \zeta_{24}^{9} q^{12} + \zeta_{24} q^{13} + \zeta_{24}^{6} q^{14} + \zeta_{24}^{8} q^{15} + \zeta_{24}^{8} q^{16} + \zeta_{24}^{7} q^{20} - \zeta_{24}^{3} q^{21} + \zeta_{24}^{2} q^{23} - \zeta_{24}^{5} q^{24} + \zeta_{24}^{9} q^{26} - \zeta_{24}^{3} q^{27} - \zeta_{24}^{2} q^{28} - \zeta_{24}^{4} q^{30} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{31} - \zeta_{24}^{4} q^{32} - \zeta_{24} q^{35} - \zeta_{24}^{6} q^{39} - \zeta_{24}^{3} q^{40} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{41} - \zeta_{24}^{11} q^{42} + (\zeta_{24}^{10} - \zeta_{24}^{4}) q^{43} + \zeta_{24}^{10} q^{46} + \zeta_{24} q^{48} - \zeta_{24}^{8} q^{49} - \zeta_{24}^{5} q^{52} - \zeta_{24}^{11} q^{54} - \zeta_{24}^{10} q^{56} - \zeta_{24}^{7} q^{59} + q^{60} + \zeta_{24} q^{61} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{62} + q^{64} - \zeta_{24}^{4} q^{65} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{67} - \zeta_{24}^{7} q^{69} - \zeta_{24}^{9} q^{70} + \zeta_{24}^{10} q^{71} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{73} + \zeta_{24}^{2} q^{78} - \zeta_{24}^{11} q^{80} + \zeta_{24}^{8} q^{81} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{82} + \zeta_{24}^{7} q^{84} + ( - \zeta_{24}^{6} + 1) q^{86} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{89} - \zeta_{24}^{11} q^{91} - \zeta_{24}^{6} q^{92} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{93} + \zeta_{24}^{9} q^{96} + (\zeta_{24}^{7} - \zeta_{24}) q^{97} + \zeta_{24}^{4} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 4 q^{4} + 8 q^{8} - 4 q^{15} - 4 q^{16} - 4 q^{30} - 4 q^{32} - 4 q^{43} + 4 q^{49} + 8 q^{60} + 8 q^{64} - 4 q^{65} - 4 q^{67} - 4 q^{81} + 8 q^{86} - 4 q^{93} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(-\zeta_{24}^{4}\) \(1\) \(-\zeta_{24}^{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
237.1
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.500000 − 0.866025i −0.965926 − 0.258819i −0.500000 + 0.866025i 0.707107 + 0.707107i 0.258819 + 0.965926i −0.866025 − 0.500000i 1.00000 0 0.258819 − 0.965926i
237.2 −0.500000 − 0.866025i 0.965926 + 0.258819i −0.500000 + 0.866025i −0.707107 − 0.707107i −0.258819 − 0.965926i −0.866025 − 0.500000i 1.00000 0 −0.258819 + 0.965926i
685.1 −0.500000 + 0.866025i −0.258819 − 0.965926i −0.500000 − 0.866025i −0.707107 − 0.707107i 0.965926 + 0.258819i 0.866025 − 0.500000i 1.00000 0 0.965926 − 0.258819i
685.2 −0.500000 + 0.866025i 0.258819 + 0.965926i −0.500000 − 0.866025i 0.707107 + 0.707107i −0.965926 − 0.258819i 0.866025 − 0.500000i 1.00000 0 −0.965926 + 0.258819i
965.1 −0.500000 − 0.866025i −0.258819 + 0.965926i −0.500000 + 0.866025i −0.707107 + 0.707107i 0.965926 − 0.258819i 0.866025 + 0.500000i 1.00000 0 0.965926 + 0.258819i
965.2 −0.500000 − 0.866025i 0.258819 − 0.965926i −0.500000 + 0.866025i 0.707107 − 0.707107i −0.965926 + 0.258819i 0.866025 + 0.500000i 1.00000 0 −0.965926 − 0.258819i
1413.1 −0.500000 + 0.866025i −0.965926 + 0.258819i −0.500000 − 0.866025i 0.707107 − 0.707107i 0.258819 − 0.965926i −0.866025 + 0.500000i 1.00000 0 0.258819 + 0.965926i
1413.2 −0.500000 + 0.866025i 0.965926 − 0.258819i −0.500000 − 0.866025i −0.707107 + 0.707107i −0.258819 + 0.965926i −0.866025 + 0.500000i 1.00000 0 −0.258819 − 0.965926i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 237.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.c even 3 1 inner
16.e even 4 1 inner
91.n odd 6 1 inner
112.l odd 4 1 inner
208.bj even 12 1 inner
1456.fx odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.1.fx.a ✓ 8
7.b odd 2 1 inner 1456.1.fx.a ✓ 8
13.c even 3 1 inner 1456.1.fx.a ✓ 8
16.e even 4 1 inner 1456.1.fx.a ✓ 8
91.n odd 6 1 inner 1456.1.fx.a ✓ 8
112.l odd 4 1 inner 1456.1.fx.a ✓ 8
208.bj even 12 1 inner 1456.1.fx.a ✓ 8
1456.fx odd 12 1 inner 1456.1.fx.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1456.1.fx.a ✓ 8 1.a even 1 1 trivial
1456.1.fx.a ✓ 8 7.b odd 2 1 inner
1456.1.fx.a ✓ 8 13.c even 3 1 inner
1456.1.fx.a ✓ 8 16.e even 4 1 inner
1456.1.fx.a ✓ 8 91.n odd 6 1 inner
1456.1.fx.a ✓ 8 112.l odd 4 1 inner
1456.1.fx.a ✓ 8 208.bj even 12 1 inner
1456.1.fx.a ✓ 8 1456.fx odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1456, [\chi])\).

Hecke characteristic polynomials

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