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Properties

Label 3783.1.fd.a
Level $3783$
Weight $1$
Character orbit 3783.fd
Analytic conductor $1.888$
Analytic rank $0$
Dimension $8$
Projective image $D_{24}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3783,1,Mod(185,3783)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3783, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([12, 8, 23]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3783.185");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3783 = 3 \cdot 13 \cdot 97 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3783.fd (of order \(24\), 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: \(1.88796294279\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{24}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{11} q^{3} + \zeta_{24}^{10} q^{4} + ( - \zeta_{24}^{3} - 1) q^{7} - \zeta_{24}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{11} q^{3} + \zeta_{24}^{10} q^{4} + ( - \zeta_{24}^{3} - 1) q^{7} - \zeta_{24}^{10} q^{9} + \zeta_{24}^{9} q^{12} + \zeta_{24}^{11} q^{13} - \zeta_{24}^{8} q^{16} + (\zeta_{24}^{8} + \zeta_{24}^{5}) q^{19} + (\zeta_{24}^{11} - \zeta_{24}^{2}) q^{21} + \zeta_{24}^{5} q^{25} - \zeta_{24}^{9} q^{27} + ( - \zeta_{24}^{10} + \zeta_{24}) q^{28} + (\zeta_{24}^{9} + \zeta_{24}^{5}) q^{31} + \zeta_{24}^{8} q^{36} + (\zeta_{24}^{4} + \zeta_{24}^{3}) q^{37} + \zeta_{24}^{10} q^{39} + (\zeta_{24}^{11} + \zeta_{24}) q^{43} - \zeta_{24}^{7} q^{48} + (\zeta_{24}^{6} + \zeta_{24}^{3} + 1) q^{49} - \zeta_{24}^{9} q^{52} + (\zeta_{24}^{7} + \zeta_{24}^{4}) q^{57} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{61} + (\zeta_{24}^{10} - \zeta_{24}) q^{63} + \zeta_{24}^{6} q^{64} + ( - \zeta_{24}^{3} - \zeta_{24}^{2}) q^{67} - \zeta_{24}^{10} q^{73} + \zeta_{24}^{4} q^{75} + ( - \zeta_{24}^{6} - \zeta_{24}^{3}) q^{76} + ( - \zeta_{24}^{10} + \zeta_{24}^{8}) q^{79} - \zeta_{24}^{8} q^{81} + ( - \zeta_{24}^{9} + 1) q^{84} + ( - \zeta_{24}^{11} + \zeta_{24}^{2}) q^{91} + (\zeta_{24}^{8} + \zeta_{24}^{4}) q^{93} - \zeta_{24} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} + 4 q^{16} - 4 q^{19} - 4 q^{36} + 4 q^{37} + 8 q^{49} + 4 q^{57} + 4 q^{75} - 4 q^{79} + 4 q^{81} + 8 q^{84}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3783\mathbb{Z}\right)^\times\).

\(n\) \(781\) \(1262\) \(2329\)
\(\chi(n)\) \(\zeta_{24}^{5}\) \(-1\) \(\zeta_{24}^{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
185.1
0.258819 − 0.965926i
0.965926 + 0.258819i
0.258819 + 0.965926i
0.965926 − 0.258819i
−0.965926 − 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.258819 − 0.965926i
0 0.258819 + 0.965926i 0.866025 − 0.500000i 0 0 −0.292893 − 0.707107i 0 −0.866025 + 0.500000i 0
1160.1 0 0.965926 − 0.258819i −0.866025 + 0.500000i 0 0 −1.70711 − 0.707107i 0 0.866025 − 0.500000i 0
1595.1 0 0.258819 − 0.965926i 0.866025 + 0.500000i 0 0 −0.292893 + 0.707107i 0 −0.866025 − 0.500000i 0
1673.1 0 0.965926 + 0.258819i −0.866025 − 0.500000i 0 0 −1.70711 + 0.707107i 0 0.866025 + 0.500000i 0
2720.1 0 −0.965926 + 0.258819i −0.866025 + 0.500000i 0 0 −0.292893 + 0.707107i 0 0.866025 − 0.500000i 0
3662.1 0 −0.965926 − 0.258819i −0.866025 − 0.500000i 0 0 −0.292893 − 0.707107i 0 0.866025 + 0.500000i 0
3695.1 0 −0.258819 − 0.965926i 0.866025 − 0.500000i 0 0 −1.70711 + 0.707107i 0 −0.866025 + 0.500000i 0
3740.1 0 −0.258819 + 0.965926i 0.866025 + 0.500000i 0 0 −1.70711 − 0.707107i 0 −0.866025 − 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 185.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
1261.cm even 24 1 inner
3783.fd odd 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3783.1.fd.a yes 8
3.b odd 2 1 CM 3783.1.fd.a yes 8
13.c even 3 1 3783.1.ev.a ✓ 8
39.i odd 6 1 3783.1.ev.a ✓ 8
97.i even 24 1 3783.1.ev.a ✓ 8
291.r odd 24 1 3783.1.ev.a ✓ 8
1261.cm even 24 1 inner 3783.1.fd.a yes 8
3783.fd odd 24 1 inner 3783.1.fd.a yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3783.1.ev.a ✓ 8 13.c even 3 1
3783.1.ev.a ✓ 8 39.i odd 6 1
3783.1.ev.a ✓ 8 97.i even 24 1
3783.1.ev.a ✓ 8 291.r odd 24 1
3783.1.fd.a yes 8 1.a even 1 1 trivial
3783.1.fd.a yes 8 3.b odd 2 1 CM
3783.1.fd.a yes 8 1261.cm even 24 1 inner
3783.1.fd.a yes 8 3783.fd odd 24 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3783, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} + 6 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 4 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 2 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
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