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Properties

Label 896.1.bm.a
Level $896$
Weight $1$
Character orbit 896.bm
Analytic conductor $0.447$
Analytic rank $0$
Dimension $16$
Projective image $D_{32}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [896,1,Mod(13,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(32))
 
chi = DirichletCharacter(H, H._module([0, 15, 16]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.13");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 896.bm (of order \(32\), degree \(16\), 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: \(0.447162251319\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{32})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{32}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{32} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{32}^{13} q^{2} - \zeta_{32}^{10} q^{4} - \zeta_{32}^{9} q^{7} - \zeta_{32}^{7} q^{8} + \zeta_{32}^{15} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{32}^{13} q^{2} - \zeta_{32}^{10} q^{4} - \zeta_{32}^{9} q^{7} - \zeta_{32}^{7} q^{8} + \zeta_{32}^{15} q^{9} + (\zeta_{32}^{14} - \zeta_{32}^{11}) q^{11} - \zeta_{32}^{6} q^{14} - \zeta_{32}^{4} q^{16} + \zeta_{32}^{12} q^{18} + (\zeta_{32}^{11} - \zeta_{32}^{8}) q^{22} + (\zeta_{32}^{4} + \zeta_{32}^{2}) q^{23} + \zeta_{32}^{5} q^{25} - \zeta_{32}^{3} q^{28} + (\zeta_{32}^{6} + \zeta_{32}) q^{29} - \zeta_{32} q^{32} + \zeta_{32}^{9} q^{36} + (\zeta_{32}^{10} - \zeta_{32}^{3}) q^{37} + ( - \zeta_{32}^{12} - \zeta_{32}^{5}) q^{43} + (\zeta_{32}^{8} - \zeta_{32}^{5}) q^{44} + ( - \zeta_{32}^{15} + \zeta_{32}) q^{46} - \zeta_{32}^{2} q^{49} + \zeta_{32}^{2} q^{50} + (\zeta_{32}^{13} + \zeta_{32}^{12}) q^{53} - q^{56} + ( - \zeta_{32}^{14} + \zeta_{32}^{3}) q^{58} + \zeta_{32}^{8} q^{63} + \zeta_{32}^{14} q^{64} + ( - \zeta_{32}^{8} + \zeta_{32}^{7}) q^{67} + ( - \zeta_{32}^{15} + \zeta_{32}^{3}) q^{71} + \zeta_{32}^{6} q^{72} + (\zeta_{32}^{7} - 1) q^{74} + (\zeta_{32}^{7} - \zeta_{32}^{4}) q^{77} + (\zeta_{32}^{13} - \zeta_{32}^{7}) q^{79} - \zeta_{32}^{14} q^{81} + ( - \zeta_{32}^{9} - \zeta_{32}^{2}) q^{86} + (\zeta_{32}^{5} - \zeta_{32}^{2}) q^{88} + ( - \zeta_{32}^{14} - \zeta_{32}^{12}) q^{92} + \zeta_{32}^{15} q^{98} + ( - \zeta_{32}^{13} + \zeta_{32}^{10}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{56} - 16 q^{74}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{32}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.831470 + 0.555570i
0.831470 − 0.555570i
0.195090 + 0.980785i
0.980785 + 0.195090i
−0.555570 + 0.831470i
0.555570 + 0.831470i
−0.980785 + 0.195090i
−0.195090 + 0.980785i
−0.831470 − 0.555570i
−0.831470 + 0.555570i
−0.195090 − 0.980785i
−0.980785 − 0.195090i
0.555570 − 0.831470i
−0.555570 − 0.831470i
0.980785 − 0.195090i
0.195090 − 0.980785i
−0.195090 − 0.980785i 0 −0.923880 + 0.382683i 0 0 −0.555570 + 0.831470i 0.555570 + 0.831470i −0.831470 + 0.555570i 0
69.1 −0.195090 + 0.980785i 0 −0.923880 − 0.382683i 0 0 −0.555570 − 0.831470i 0.555570 − 0.831470i −0.831470 − 0.555570i 0
125.1 −0.555570 + 0.831470i 0 −0.382683 − 0.923880i 0 0 −0.980785 + 0.195090i 0.980785 + 0.195090i −0.195090 + 0.980785i 0
181.1 0.831470 − 0.555570i 0 0.382683 − 0.923880i 0 0 0.195090 − 0.980785i −0.195090 − 0.980785i −0.980785 + 0.195090i 0
237.1 0.980785 − 0.195090i 0 0.923880 − 0.382683i 0 0 −0.831470 − 0.555570i 0.831470 − 0.555570i 0.555570 + 0.831470i 0
293.1 −0.980785 − 0.195090i 0 0.923880 + 0.382683i 0 0 0.831470 − 0.555570i −0.831470 − 0.555570i −0.555570 + 0.831470i 0
349.1 −0.831470 − 0.555570i 0 0.382683 + 0.923880i 0 0 −0.195090 − 0.980785i 0.195090 − 0.980785i 0.980785 + 0.195090i 0
405.1 0.555570 + 0.831470i 0 −0.382683 + 0.923880i 0 0 0.980785 + 0.195090i −0.980785 + 0.195090i 0.195090 + 0.980785i 0
461.1 0.195090 + 0.980785i 0 −0.923880 + 0.382683i 0 0 0.555570 − 0.831470i −0.555570 − 0.831470i 0.831470 − 0.555570i 0
517.1 0.195090 − 0.980785i 0 −0.923880 − 0.382683i 0 0 0.555570 + 0.831470i −0.555570 + 0.831470i 0.831470 + 0.555570i 0
573.1 0.555570 − 0.831470i 0 −0.382683 − 0.923880i 0 0 0.980785 − 0.195090i −0.980785 − 0.195090i 0.195090 − 0.980785i 0
629.1 −0.831470 + 0.555570i 0 0.382683 − 0.923880i 0 0 −0.195090 + 0.980785i 0.195090 + 0.980785i 0.980785 − 0.195090i 0
685.1 −0.980785 + 0.195090i 0 0.923880 − 0.382683i 0 0 0.831470 + 0.555570i −0.831470 + 0.555570i −0.555570 − 0.831470i 0
741.1 0.980785 + 0.195090i 0 0.923880 + 0.382683i 0 0 −0.831470 + 0.555570i 0.831470 + 0.555570i 0.555570 − 0.831470i 0
797.1 0.831470 + 0.555570i 0 0.382683 + 0.923880i 0 0 0.195090 + 0.980785i −0.195090 + 0.980785i −0.980785 − 0.195090i 0
853.1 −0.555570 − 0.831470i 0 −0.382683 + 0.923880i 0 0 −0.980785 − 0.195090i 0.980785 − 0.195090i −0.195090 − 0.980785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
128.k even 32 1 inner
896.bm odd 32 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.1.bm.a ✓ 16
4.b odd 2 1 3584.1.bm.a 16
7.b odd 2 1 CM 896.1.bm.a ✓ 16
28.d even 2 1 3584.1.bm.a 16
128.k even 32 1 inner 896.1.bm.a ✓ 16
128.l odd 32 1 3584.1.bm.a 16
896.bk even 32 1 3584.1.bm.a 16
896.bm odd 32 1 inner 896.1.bm.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.1.bm.a ✓ 16 1.a even 1 1 trivial
896.1.bm.a ✓ 16 7.b odd 2 1 CM
896.1.bm.a ✓ 16 128.k even 32 1 inner
896.1.bm.a ✓ 16 896.bm odd 32 1 inner
3584.1.bm.a 16 4.b odd 2 1
3584.1.bm.a 16 28.d even 2 1
3584.1.bm.a 16 128.l odd 32 1
3584.1.bm.a 16 896.bk even 32 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(896, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 1 \) Copy content Toggle raw display
$11$ \( T^{16} - 32 T^{11} + \cdots + 2 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2 T^{4} + 16 T^{3} + \cdots + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 40 T^{10} + \cdots + 2 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + 16 T^{13} + \cdots + 2 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} + 4 T^{12} + \cdots + 2 \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + 4 T^{12} + \cdots + 2 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} + 8 T^{14} + \cdots + 2 \) Copy content Toggle raw display
$71$ \( T^{16} - 16 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} - 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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