Properties

Label 1456.1.bb.a
Level $1456$
Weight $1$
Character orbit 1456.bb
Analytic conductor $0.727$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1456,1,Mod(307,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.307");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1456.bb (of order \(4\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) 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 4.2.31496192.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{2} - \zeta_{8}^{3} q^{3} - q^{4} + \zeta_{8} q^{6} + \zeta_{8}^{3} q^{7} - \zeta_{8}^{2} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8}^{2} q^{2} - \zeta_{8}^{3} q^{3} - q^{4} + \zeta_{8} q^{6} + \zeta_{8}^{3} q^{7} - \zeta_{8}^{2} q^{8} + q^{11} + \zeta_{8}^{3} q^{12} + \zeta_{8}^{3} q^{13} - \zeta_{8} q^{14} + q^{16} + (\zeta_{8}^{3} + \zeta_{8}) q^{19} + \zeta_{8}^{2} q^{21} + \zeta_{8}^{2} q^{22} + \zeta_{8}^{2} q^{23} - \zeta_{8} q^{24} + q^{25} - \zeta_{8} q^{26} + \zeta_{8} q^{27} - \zeta_{8}^{3} q^{28} + ( - \zeta_{8}^{2} + 1) q^{29} - \zeta_{8}^{3} q^{31} + \zeta_{8}^{2} q^{32} - \zeta_{8}^{3} q^{33} + \zeta_{8}^{2} q^{37} + (\zeta_{8}^{3} - \zeta_{8}) q^{38} + \zeta_{8}^{2} q^{39} - \zeta_{8}^{3} q^{41} - q^{42} + ( - \zeta_{8}^{2} - 1) q^{43} - q^{44} - q^{46} - \zeta_{8} q^{47} - \zeta_{8}^{3} q^{48} - \zeta_{8}^{2} q^{49} + \zeta_{8}^{2} q^{50} - \zeta_{8}^{3} q^{52} + \zeta_{8}^{3} q^{54} + \zeta_{8} q^{56} + (\zeta_{8}^{2} + 1) q^{57} + (\zeta_{8}^{2} + 1) q^{58} - \zeta_{8} q^{61} + \zeta_{8} q^{62} - q^{64} + \zeta_{8} q^{66} - q^{67} + \zeta_{8} q^{69} + (\zeta_{8}^{2} - 1) q^{71} - \zeta_{8} q^{73} - q^{74} - \zeta_{8}^{3} q^{75} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{76} + \zeta_{8}^{3} q^{77} - q^{78} - \zeta_{8}^{2} q^{79} + q^{81} + \zeta_{8} q^{82} + (\zeta_{8}^{3} + \zeta_{8}) q^{83} - \zeta_{8}^{2} q^{84} + ( - \zeta_{8}^{2} + 1) q^{86} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{87} - \zeta_{8}^{2} q^{88} - \zeta_{8}^{2} q^{91} - \zeta_{8}^{2} q^{92} - \zeta_{8}^{2} q^{93} - \zeta_{8}^{3} q^{94} + \zeta_{8} q^{96} + \zeta_{8} q^{97} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{11} + 4 q^{16} + 4 q^{25} + 4 q^{29} - 4 q^{42} - 4 q^{43} - 4 q^{44} - 4 q^{46} + 4 q^{57} + 4 q^{58} - 4 q^{64} - 4 q^{67} - 4 q^{71} - 4 q^{74} - 4 q^{78} + 4 q^{81} + 4 q^{86} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(-1\) \(\zeta_{8}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.00000i −0.707107 0.707107i −1.00000 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0 0
307.2 1.00000i 0.707107 + 0.707107i −1.00000 0 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0 0
811.1 1.00000i −0.707107 + 0.707107i −1.00000 0 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0 0
811.2 1.00000i 0.707107 0.707107i −1.00000 0 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 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
7.b odd 2 1 inner
208.s even 4 1 inner
1456.bb odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.1.bb.a 4
7.b odd 2 1 inner 1456.1.bb.a 4
13.d odd 4 1 1456.1.bl.a yes 4
16.f odd 4 1 1456.1.bl.a yes 4
91.i even 4 1 1456.1.bl.a yes 4
112.j even 4 1 1456.1.bl.a yes 4
208.s even 4 1 inner 1456.1.bb.a 4
1456.bb odd 4 1 inner 1456.1.bb.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1456.1.bb.a 4 1.a even 1 1 trivial
1456.1.bb.a 4 7.b odd 2 1 inner
1456.1.bb.a 4 208.s even 4 1 inner
1456.1.bb.a 4 1456.bb odd 4 1 inner
1456.1.bl.a yes 4 13.d odd 4 1
1456.1.bl.a yes 4 16.f odd 4 1
1456.1.bl.a yes 4 91.i even 4 1
1456.1.bl.a yes 4 112.j even 4 1

Hecke kernels

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

Hecke characteristic polynomials

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