Properties

Label 1805.1.m.a
Level $1805$
Weight $1$
Character orbit 1805.m
Analytic conductor $0.901$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -19
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1805,1,Mod(68,1805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1805, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1805.68");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1805.m (of order \(12\), degree \(4\), not 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: \(0.900812347803\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.2375.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12}^{5} q^{4} - \zeta_{12}^{4} q^{5} + ( - \zeta_{12}^{3} + 1) q^{7} + \zeta_{12}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{5} q^{4} - \zeta_{12}^{4} q^{5} + ( - \zeta_{12}^{3} + 1) q^{7} + \zeta_{12}^{5} q^{9} - \zeta_{12}^{4} q^{16} + (\zeta_{12}^{4} + \zeta_{12}) q^{17} - \zeta_{12}^{3} q^{20} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{23} - \zeta_{12}^{2} q^{25} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{28} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{35} + \zeta_{12}^{4} q^{36} + (\zeta_{12}^{4} - \zeta_{12}) q^{43} + \zeta_{12}^{3} q^{45} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{47} - \zeta_{12}^{3} q^{49} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{63} - \zeta_{12}^{3} q^{64} + (\zeta_{12}^{3} + 1) q^{68} + (\zeta_{12}^{4} - \zeta_{12}) q^{73} - \zeta_{12}^{2} q^{80} - \zeta_{12}^{4} q^{81} + (\zeta_{12}^{3} + 1) q^{83} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{85} + (\zeta_{12}^{4} + \zeta_{12}) q^{92} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} + 4 q^{7} + 2 q^{16} - 2 q^{17} + 2 q^{23} - 2 q^{25} - 2 q^{28} + 2 q^{35} - 2 q^{36} - 2 q^{43} - 2 q^{47} + 2 q^{63} + 4 q^{68} - 2 q^{73} - 2 q^{80} + 2 q^{81} + 4 q^{83} + 2 q^{85} - 2 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times\).

\(n\) \(362\) \(1446\)
\(\chi(n)\) \(-\zeta_{12}^{3}\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0 0 0.866025 0.500000i 0.500000 0.866025i 0 1.00000 1.00000i 0 −0.866025 + 0.500000i 0
292.1 0 0 0.866025 + 0.500000i 0.500000 + 0.866025i 0 1.00000 + 1.00000i 0 −0.866025 0.500000i 0
653.1 0 0 −0.866025 0.500000i 0.500000 + 0.866025i 0 1.00000 1.00000i 0 0.866025 + 0.500000i 0
1512.1 0 0 −0.866025 + 0.500000i 0.500000 0.866025i 0 1.00000 + 1.00000i 0 0.866025 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
5.c odd 4 1 inner
19.c even 3 1 inner
19.d odd 6 1 inner
95.g even 4 1 inner
95.l even 12 1 inner
95.m odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.1.m.a 4
5.c odd 4 1 inner 1805.1.m.a 4
19.b odd 2 1 CM 1805.1.m.a 4
19.c even 3 1 1805.1.f.a 2
19.c even 3 1 inner 1805.1.m.a 4
19.d odd 6 1 1805.1.f.a 2
19.d odd 6 1 inner 1805.1.m.a 4
19.e even 9 6 1805.1.r.a 12
19.f odd 18 6 1805.1.r.a 12
95.g even 4 1 inner 1805.1.m.a 4
95.l even 12 1 1805.1.f.a 2
95.l even 12 1 inner 1805.1.m.a 4
95.m odd 12 1 1805.1.f.a 2
95.m odd 12 1 inner 1805.1.m.a 4
95.q odd 36 6 1805.1.r.a 12
95.r even 36 6 1805.1.r.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.1.f.a 2 19.c even 3 1
1805.1.f.a 2 19.d odd 6 1
1805.1.f.a 2 95.l even 12 1
1805.1.f.a 2 95.m odd 12 1
1805.1.m.a 4 1.a even 1 1 trivial
1805.1.m.a 4 5.c odd 4 1 inner
1805.1.m.a 4 19.b odd 2 1 CM
1805.1.m.a 4 19.c even 3 1 inner
1805.1.m.a 4 19.d odd 6 1 inner
1805.1.m.a 4 95.g even 4 1 inner
1805.1.m.a 4 95.l even 12 1 inner
1805.1.m.a 4 95.m odd 12 1 inner
1805.1.r.a 12 19.e even 9 6
1805.1.r.a 12 19.f odd 18 6
1805.1.r.a 12 95.q odd 36 6
1805.1.r.a 12 95.r even 36 6

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1805, [\chi])\).

Hecke characteristic polynomials

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