Properties

Label 1805.1.h.a
Level $1805$
Weight $1$
Character orbit 1805.h
Analytic conductor $0.901$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -19, -95, 5
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(69,1805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1805, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1805.69");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1805.h (of order \(6\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{-19})\)
Artin image: $C_3\times D_4$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{4} + \zeta_{6}^{2} q^{5} + \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{4} + \zeta_{6}^{2} q^{5} + \zeta_{6} q^{9} - 2 q^{11} + \zeta_{6}^{2} q^{16} - q^{20} - \zeta_{6} q^{25} + \zeta_{6}^{2} q^{36} - 2 \zeta_{6} q^{44} - q^{45} + q^{49} - 2 \zeta_{6}^{2} q^{55} + 2 \zeta_{6} q^{61} - q^{64} - \zeta_{6} q^{80} + \zeta_{6}^{2} q^{81} - 2 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{4} - q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{4} - q^{5} + q^{9} - 4 q^{11} - q^{16} - 2 q^{20} - q^{25} - q^{36} - 2 q^{44} - 2 q^{45} + 2 q^{49} + 2 q^{55} + 2 q^{61} - 2 q^{64} - q^{80} - q^{81} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(362\) \(1446\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0.500000 0.866025i −0.500000 0.866025i 0 0 0 0.500000 0.866025i 0
654.1 0 0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0 0 0.500000 + 0.866025i 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
5.b even 2 1 RM by \(\Q(\sqrt{5}) \)
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
19.c even 3 1 inner
19.d odd 6 1 inner
95.h odd 6 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.1.h.a 2
5.b even 2 1 RM 1805.1.h.a 2
19.b odd 2 1 CM 1805.1.h.a 2
19.c even 3 1 95.1.d.a 1
19.c even 3 1 inner 1805.1.h.a 2
19.d odd 6 1 95.1.d.a 1
19.d odd 6 1 inner 1805.1.h.a 2
19.e even 9 6 1805.1.o.a 6
19.f odd 18 6 1805.1.o.a 6
57.f even 6 1 855.1.g.a 1
57.h odd 6 1 855.1.g.a 1
76.f even 6 1 1520.1.m.a 1
76.g odd 6 1 1520.1.m.a 1
95.d odd 2 1 CM 1805.1.h.a 2
95.h odd 6 1 95.1.d.a 1
95.h odd 6 1 inner 1805.1.h.a 2
95.i even 6 1 95.1.d.a 1
95.i even 6 1 inner 1805.1.h.a 2
95.l even 12 2 475.1.c.a 1
95.m odd 12 2 475.1.c.a 1
95.o odd 18 6 1805.1.o.a 6
95.p even 18 6 1805.1.o.a 6
285.n odd 6 1 855.1.g.a 1
285.q even 6 1 855.1.g.a 1
380.p odd 6 1 1520.1.m.a 1
380.s even 6 1 1520.1.m.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.a 1 19.c even 3 1
95.1.d.a 1 19.d odd 6 1
95.1.d.a 1 95.h odd 6 1
95.1.d.a 1 95.i even 6 1
475.1.c.a 1 95.l even 12 2
475.1.c.a 1 95.m odd 12 2
855.1.g.a 1 57.f even 6 1
855.1.g.a 1 57.h odd 6 1
855.1.g.a 1 285.n odd 6 1
855.1.g.a 1 285.q even 6 1
1520.1.m.a 1 76.f even 6 1
1520.1.m.a 1 76.g odd 6 1
1520.1.m.a 1 380.p odd 6 1
1520.1.m.a 1 380.s even 6 1
1805.1.h.a 2 1.a even 1 1 trivial
1805.1.h.a 2 5.b even 2 1 RM
1805.1.h.a 2 19.b odd 2 1 CM
1805.1.h.a 2 19.c even 3 1 inner
1805.1.h.a 2 19.d odd 6 1 inner
1805.1.h.a 2 95.d odd 2 1 CM
1805.1.h.a 2 95.h odd 6 1 inner
1805.1.h.a 2 95.i even 6 1 inner
1805.1.o.a 6 19.e even 9 6
1805.1.o.a 6 19.f odd 18 6
1805.1.o.a 6 95.o odd 18 6
1805.1.o.a 6 95.p even 18 6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(1805, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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