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Properties

Label 408.1.bg.a
Level $408$
Weight $1$
Character orbit 408.bg
Analytic conductor $0.204$
Analytic rank $0$
Dimension $8$
Projective image $D_{16}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [408,1,Mod(11,408)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(408, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 8, 8, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("408.11");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 408 = 2^{3} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 408.bg (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: \(0.203618525154\)
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.

\(f(q)\) \(=\) \( q - \zeta_{16} q^{2} + \zeta_{16}^{6} q^{3} + \zeta_{16}^{2} q^{4} - \zeta_{16}^{7} q^{6} - \zeta_{16}^{3} q^{8} - \zeta_{16}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{16} q^{2} + \zeta_{16}^{6} q^{3} + \zeta_{16}^{2} q^{4} - \zeta_{16}^{7} q^{6} - \zeta_{16}^{3} q^{8} - \zeta_{16}^{4} q^{9} + (\zeta_{16}^{4} + \zeta_{16}) q^{11} - q^{12} + \zeta_{16}^{4} q^{16} + \zeta_{16}^{5} q^{17} + \zeta_{16}^{5} q^{18} + (\zeta_{16}^{7} + \zeta_{16}^{3}) q^{19} + ( - \zeta_{16}^{5} - \zeta_{16}^{2}) q^{22} + \zeta_{16} q^{24} - \zeta_{16}^{7} q^{25} + \zeta_{16}^{2} q^{27} - \zeta_{16}^{5} q^{32} + (\zeta_{16}^{7} - \zeta_{16}^{2}) q^{33} - \zeta_{16}^{6} q^{34} - \zeta_{16}^{6} q^{36} + ( - \zeta_{16}^{4} + 1) q^{38} + ( - \zeta_{16}^{6} + \zeta_{16}^{3}) q^{41} + ( - \zeta_{16}^{6} - 1) q^{43} + (\zeta_{16}^{6} + \zeta_{16}^{3}) q^{44} - \zeta_{16}^{2} q^{48} - \zeta_{16} q^{49} - q^{50} - \zeta_{16}^{3} q^{51} - \zeta_{16}^{3} q^{54} + ( - \zeta_{16}^{5} - \zeta_{16}) q^{57} + ( - \zeta_{16}^{4} - \zeta_{16}^{2}) q^{59} + \zeta_{16}^{6} q^{64} + (\zeta_{16}^{3} + 1) q^{66} + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{67} + \zeta_{16}^{7} q^{68} + \zeta_{16}^{7} q^{72} + (\zeta_{16}^{6} + \zeta_{16}) q^{73} + \zeta_{16}^{5} q^{75} + (\zeta_{16}^{5} - \zeta_{16}) q^{76} - q^{81} + (\zeta_{16}^{7} - \zeta_{16}^{4}) q^{82} + (\zeta_{16}^{2} - 1) q^{83} + (\zeta_{16}^{7} + \zeta_{16}) q^{86} + ( - \zeta_{16}^{7} - \zeta_{16}^{4}) q^{88} + ( - \zeta_{16}^{7} - \zeta_{16}^{5}) q^{89} + \zeta_{16}^{3} q^{96} + (\zeta_{16}^{4} - \zeta_{16}^{3}) q^{97} + \zeta_{16}^{2} q^{98} + ( - \zeta_{16}^{5} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{12} + 8 q^{38} - 8 q^{43} - 8 q^{50} + 8 q^{66} - 8 q^{81} - 8 q^{83} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/408\mathbb{Z}\right)^\times\).

\(n\) \(103\) \(137\) \(205\) \(241\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{16}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.382683 + 0.923880i
0.923880 − 0.382683i
−0.923880 + 0.382683i
0.382683 − 0.923880i
0.382683 + 0.923880i
−0.923880 − 0.382683i
0.923880 + 0.382683i
−0.382683 − 0.923880i
0.382683 − 0.923880i 0.707107 − 0.707107i −0.707107 − 0.707107i 0 −0.382683 − 0.923880i 0 −0.923880 + 0.382683i − 1.00000i 0
107.1 −0.923880 + 0.382683i −0.707107 − 0.707107i 0.707107 − 0.707107i 0 0.923880 + 0.382683i 0 −0.382683 + 0.923880i 1.00000i 0
131.1 0.923880 − 0.382683i −0.707107 − 0.707107i 0.707107 − 0.707107i 0 −0.923880 − 0.382683i 0 0.382683 − 0.923880i 1.00000i 0
227.1 −0.382683 + 0.923880i 0.707107 − 0.707107i −0.707107 − 0.707107i 0 0.382683 + 0.923880i 0 0.923880 − 0.382683i − 1.00000i 0
275.1 −0.382683 − 0.923880i 0.707107 + 0.707107i −0.707107 + 0.707107i 0 0.382683 − 0.923880i 0 0.923880 + 0.382683i 1.00000i 0
299.1 0.923880 + 0.382683i −0.707107 + 0.707107i 0.707107 + 0.707107i 0 −0.923880 + 0.382683i 0 0.382683 + 0.923880i − 1.00000i 0
347.1 −0.923880 − 0.382683i −0.707107 + 0.707107i 0.707107 + 0.707107i 0 0.923880 − 0.382683i 0 −0.382683 − 0.923880i − 1.00000i 0
371.1 0.382683 + 0.923880i 0.707107 + 0.707107i −0.707107 + 0.707107i 0 −0.382683 + 0.923880i 0 −0.923880 − 0.382683i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
51.i even 16 1 inner
408.bg odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 408.1.bg.a ✓ 8
3.b odd 2 1 408.1.bg.b yes 8
4.b odd 2 1 1632.1.ds.b 8
8.b even 2 1 1632.1.ds.b 8
8.d odd 2 1 CM 408.1.bg.a ✓ 8
12.b even 2 1 1632.1.ds.a 8
17.e odd 16 1 408.1.bg.b yes 8
24.f even 2 1 408.1.bg.b yes 8
24.h odd 2 1 1632.1.ds.a 8
51.i even 16 1 inner 408.1.bg.a ✓ 8
68.i even 16 1 1632.1.ds.a 8
136.q odd 16 1 1632.1.ds.a 8
136.s even 16 1 408.1.bg.b yes 8
204.t odd 16 1 1632.1.ds.b 8
408.bg odd 16 1 inner 408.1.bg.a ✓ 8
408.bm even 16 1 1632.1.ds.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
408.1.bg.a ✓ 8 1.a even 1 1 trivial
408.1.bg.a ✓ 8 8.d odd 2 1 CM
408.1.bg.a ✓ 8 51.i even 16 1 inner
408.1.bg.a ✓ 8 408.bg odd 16 1 inner
408.1.bg.b yes 8 3.b odd 2 1
408.1.bg.b yes 8 17.e odd 16 1
408.1.bg.b yes 8 24.f even 2 1
408.1.bg.b yes 8 136.s even 16 1
1632.1.ds.a 8 12.b even 2 1
1632.1.ds.a 8 24.h odd 2 1
1632.1.ds.a 8 68.i even 16 1
1632.1.ds.a 8 136.q odd 16 1
1632.1.ds.b 8 4.b odd 2 1
1632.1.ds.b 8 8.b even 2 1
1632.1.ds.b 8 204.t odd 16 1
1632.1.ds.b 8 408.bm even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} + 4T_{11}^{6} + 6T_{11}^{4} + 8T_{11}^{3} + 4T_{11}^{2} - 8T_{11} + 2 \) acting on \(S_{1}^{\mathrm{new}}(408, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{6} + \cdots + 2 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 1 \) Copy content Toggle raw display
$19$ \( T^{8} + 16 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + 2 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} + 6 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{5} + \cdots + 2 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 4 T^{3} + 6 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$97$ \( T^{8} + 4 T^{6} + \cdots + 2 \) Copy content Toggle raw display
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