Properties

Label 3808.1.cu.a
Level $3808$
Weight $1$
Character orbit 3808.cu
Analytic conductor $1.900$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -119
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3808,1,Mod(237,3808)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3808, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7, 4, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3808.237");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3808.cu (of order \(8\), 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: \(1.90043956811\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{8} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{2} + ( - \zeta_{8}^{3} + \zeta_{8}^{2}) q^{3} + \zeta_{8}^{2} q^{4} + (\zeta_{8}^{2} + \zeta_{8}) q^{5} + ( - \zeta_{8}^{3} - 1) q^{6} - \zeta_{8}^{3} q^{7} - \zeta_{8}^{3} q^{8} + ( - \zeta_{8}^{2} + \zeta_{8} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{2} + ( - \zeta_{8}^{3} + \zeta_{8}^{2}) q^{3} + \zeta_{8}^{2} q^{4} + (\zeta_{8}^{2} + \zeta_{8}) q^{5} + ( - \zeta_{8}^{3} - 1) q^{6} - \zeta_{8}^{3} q^{7} - \zeta_{8}^{3} q^{8} + ( - \zeta_{8}^{2} + \zeta_{8} - 1) q^{9} + ( - \zeta_{8}^{3} - \zeta_{8}^{2}) q^{10} + (\zeta_{8} - 1) q^{12} - q^{14} + (\zeta_{8}^{3} + \zeta_{8}) q^{15} - q^{16} + \zeta_{8}^{2} q^{17} + (\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8}) q^{18} + (\zeta_{8}^{3} - 1) q^{20} + ( - \zeta_{8}^{2} + \zeta_{8}) q^{21} + ( - \zeta_{8}^{2} + \zeta_{8}) q^{24} + (\zeta_{8}^{3} + \zeta_{8}^{2} - 1) q^{25} + (\zeta_{8}^{3} - \zeta_{8}^{2} + \cdots + 1) q^{27} + \cdots + \zeta_{8}^{3} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{6} - 4 q^{9} - 4 q^{12} - 4 q^{14} - 4 q^{16} - 4 q^{20} - 4 q^{25} + 4 q^{27} + 4 q^{30} + 8 q^{31} + 4 q^{35} + 4 q^{36} + 4 q^{40} + 4 q^{45} + 4 q^{50} - 4 q^{51} + 4 q^{54} - 4 q^{61} + 4 q^{63} + 4 q^{67} - 4 q^{68} + 4 q^{72} - 4 q^{73} - 4 q^{75} + 4 q^{84} - 4 q^{85} + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3808\mathbb{Z}\right)^\times\).

\(n\) \(2143\) \(2689\) \(3265\) \(3333\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{8}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
237.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i −0.707107 + 1.70711i 1.00000i −0.707107 + 0.292893i −1.70711 + 0.707107i −0.707107 + 0.707107i −0.707107 + 0.707107i −1.70711 1.70711i −0.707107 0.292893i
1189.1 0.707107 0.707107i −0.707107 1.70711i 1.00000i −0.707107 0.292893i −1.70711 0.707107i −0.707107 0.707107i −0.707107 0.707107i −1.70711 + 1.70711i −0.707107 + 0.292893i
2141.1 −0.707107 0.707107i 0.707107 + 0.292893i 1.00000i 0.707107 + 1.70711i −0.292893 0.707107i 0.707107 0.707107i 0.707107 0.707107i −0.292893 0.292893i 0.707107 1.70711i
3093.1 −0.707107 + 0.707107i 0.707107 0.292893i 1.00000i 0.707107 1.70711i −0.292893 + 0.707107i 0.707107 + 0.707107i 0.707107 + 0.707107i −0.292893 + 0.292893i 0.707107 + 1.70711i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
119.d odd 2 1 CM by \(\Q(\sqrt{-119}) \)
32.g even 8 1 inner
3808.cu odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3808.1.cu.a 4
7.b odd 2 1 3808.1.cu.b yes 4
17.b even 2 1 3808.1.cu.b yes 4
32.g even 8 1 inner 3808.1.cu.a 4
119.d odd 2 1 CM 3808.1.cu.a 4
224.v odd 8 1 3808.1.cu.b yes 4
544.bc even 8 1 3808.1.cu.b yes 4
3808.cu odd 8 1 inner 3808.1.cu.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3808.1.cu.a 4 1.a even 1 1 trivial
3808.1.cu.a 4 32.g even 8 1 inner
3808.1.cu.a 4 119.d odd 2 1 CM
3808.1.cu.a 4 3808.cu odd 8 1 inner
3808.1.cu.b yes 4 7.b odd 2 1
3808.1.cu.b yes 4 17.b even 2 1
3808.1.cu.b yes 4 224.v odd 8 1
3808.1.cu.b yes 4 544.bc even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2T_{3}^{2} - 4T_{3} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3808, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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