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Properties

Label 340.1.bc.a
Level $340$
Weight $1$
Character orbit 340.bc
Analytic conductor $0.170$
Analytic rank $0$
Dimension $8$
Projective image $D_{16}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [340,1,Mod(3,340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(340, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 12, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("340.3");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 340.bc (of order \(16\), degree \(8\), 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: \(0.169682104295\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{16}^{5} q^{2} - \zeta_{16}^{2} q^{4} - \zeta_{16}^{6} q^{5} - \zeta_{16}^{7} q^{8} - \zeta_{16}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{16}^{5} q^{2} - \zeta_{16}^{2} q^{4} - \zeta_{16}^{6} q^{5} - \zeta_{16}^{7} q^{8} - \zeta_{16}^{5} q^{9} + \zeta_{16}^{3} q^{10} + (\zeta_{16}^{7} + \zeta_{16}) q^{13} + \zeta_{16}^{4} q^{16} + \zeta_{16}^{6} q^{17} + \zeta_{16}^{2} q^{18} - q^{20} - \zeta_{16}^{4} q^{25} + (\zeta_{16}^{6} - \zeta_{16}^{4}) q^{26} + (\zeta_{16}^{4} - \zeta_{16}) q^{29} - \zeta_{16} q^{32} - \zeta_{16}^{3} q^{34} + \zeta_{16}^{7} q^{36} + (\zeta_{16}^{3} + \zeta_{16}^{2}) q^{37} - \zeta_{16}^{5} q^{40} + ( - \zeta_{16}^{3} - 1) q^{41} - \zeta_{16}^{3} q^{45} - \zeta_{16}^{7} q^{49} + \zeta_{16} q^{50} + ( - \zeta_{16}^{3} + \zeta_{16}) q^{52} + ( - \zeta_{16}^{2} - 1) q^{53} + ( - \zeta_{16}^{6} - \zeta_{16}) q^{58} + (\zeta_{16}^{2} - \zeta_{16}) q^{61} - \zeta_{16}^{6} q^{64} + ( - \zeta_{16}^{7} + \zeta_{16}^{5}) q^{65} + q^{68} - \zeta_{16}^{4} q^{72} + (\zeta_{16}^{5} - \zeta_{16}^{4}) q^{73} + (\zeta_{16}^{7} - 1) q^{74} + \zeta_{16}^{2} q^{80} - \zeta_{16}^{2} q^{81} + ( - \zeta_{16}^{5} + 1) q^{82} + \zeta_{16}^{4} q^{85} + (\zeta_{16}^{3} + \zeta_{16}) q^{89} + q^{90} + (\zeta_{16}^{7} - \zeta_{16}^{2}) q^{97} + \zeta_{16}^{4} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{20} - 8 q^{41} - 8 q^{53} + 8 q^{68} - 8 q^{74} + 8 q^{82} + 8 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/340\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(-\zeta_{16}^{4}\) \(-1\) \(-\zeta_{16}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−0.923880 − 0.382683i
0.382683 + 0.923880i
−0.382683 − 0.923880i
−0.382683 + 0.923880i
0.923880 − 0.382683i
−0.923880 + 0.382683i
0.382683 − 0.923880i
0.923880 + 0.382683i
0.382683 − 0.923880i 0 −0.707107 − 0.707107i 0.707107 − 0.707107i 0 0 −0.923880 + 0.382683i −0.382683 + 0.923880i −0.382683 − 0.923880i
7.1 0.923880 − 0.382683i 0 0.707107 − 0.707107i −0.707107 − 0.707107i 0 0 0.382683 − 0.923880i −0.923880 + 0.382683i −0.923880 − 0.382683i
27.1 −0.923880 + 0.382683i 0 0.707107 − 0.707107i −0.707107 − 0.707107i 0 0 −0.382683 + 0.923880i 0.923880 − 0.382683i 0.923880 + 0.382683i
63.1 −0.923880 − 0.382683i 0 0.707107 + 0.707107i −0.707107 + 0.707107i 0 0 −0.382683 − 0.923880i 0.923880 + 0.382683i 0.923880 − 0.382683i
147.1 −0.382683 − 0.923880i 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 0 0 0.923880 + 0.382683i 0.382683 + 0.923880i 0.382683 − 0.923880i
227.1 0.382683 + 0.923880i 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 0 0 −0.923880 − 0.382683i −0.382683 − 0.923880i −0.382683 + 0.923880i
243.1 0.923880 + 0.382683i 0 0.707107 + 0.707107i −0.707107 + 0.707107i 0 0 0.382683 + 0.923880i −0.923880 − 0.382683i −0.923880 + 0.382683i
303.1 −0.382683 + 0.923880i 0 −0.707107 − 0.707107i 0.707107 − 0.707107i 0 0 0.923880 − 0.382683i 0.382683 − 0.923880i 0.382683 + 0.923880i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
85.o even 16 1 inner
340.bc odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 340.1.bc.a ✓ 8
3.b odd 2 1 3060.1.ds.a 8
4.b odd 2 1 CM 340.1.bc.a ✓ 8
5.b even 2 1 1700.1.bk.a 8
5.c odd 4 1 340.1.bj.a yes 8
5.c odd 4 1 1700.1.br.a 8
12.b even 2 1 3060.1.ds.a 8
15.e even 4 1 3060.1.eg.a 8
17.e odd 16 1 340.1.bj.a yes 8
20.d odd 2 1 1700.1.bk.a 8
20.e even 4 1 340.1.bj.a yes 8
20.e even 4 1 1700.1.br.a 8
51.i even 16 1 3060.1.eg.a 8
60.l odd 4 1 3060.1.eg.a 8
68.i even 16 1 340.1.bj.a yes 8
85.o even 16 1 inner 340.1.bc.a ✓ 8
85.p odd 16 1 1700.1.br.a 8
85.r even 16 1 1700.1.bk.a 8
204.t odd 16 1 3060.1.eg.a 8
255.bc odd 16 1 3060.1.ds.a 8
340.bc odd 16 1 inner 340.1.bc.a ✓ 8
340.bg even 16 1 1700.1.br.a 8
340.bj odd 16 1 1700.1.bk.a 8
1020.ch even 16 1 3060.1.ds.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
340.1.bc.a ✓ 8 1.a even 1 1 trivial
340.1.bc.a ✓ 8 4.b odd 2 1 CM
340.1.bc.a ✓ 8 85.o even 16 1 inner
340.1.bc.a ✓ 8 340.bc odd 16 1 inner
340.1.bj.a yes 8 5.c odd 4 1
340.1.bj.a yes 8 17.e odd 16 1
340.1.bj.a yes 8 20.e even 4 1
340.1.bj.a yes 8 68.i even 16 1
1700.1.bk.a 8 5.b even 2 1
1700.1.bk.a 8 20.d odd 2 1
1700.1.bk.a 8 85.r even 16 1
1700.1.bk.a 8 340.bj odd 16 1
1700.1.br.a 8 5.c odd 4 1
1700.1.br.a 8 20.e even 4 1
1700.1.br.a 8 85.p odd 16 1
1700.1.br.a 8 340.bg even 16 1
3060.1.ds.a 8 3.b odd 2 1
3060.1.ds.a 8 12.b even 2 1
3060.1.ds.a 8 255.bc odd 16 1
3060.1.ds.a 8 1020.ch even 16 1
3060.1.eg.a 8 15.e even 4 1
3060.1.eg.a 8 51.i even 16 1
3060.1.eg.a 8 60.l odd 4 1
3060.1.eg.a 8 204.t odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(340, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 4 T^{6} + \cdots + 2 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{5} + \cdots + 2 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + \cdots + 2 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 4 T^{3} + 6 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 2 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 4 T^{6} + \cdots + 2 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$97$ \( T^{8} + 8 T^{5} + \cdots + 2 \) Copy content Toggle raw display
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