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Properties

Label 1805.1.h.b
Level $1805$
Weight $1$
Character orbit 1805.h
Analytic conductor $0.901$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -95
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) 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: \(D_{4}\)
Projective field: Galois closure of 4.2.475.1
Artin image: $C_3\times D_8$
Artin 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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_1 q^{3} + \beta_{2} q^{4} + (\beta_{2} + 1) q^{5} - 2 \beta_{2} q^{6} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_1 q^{3} + \beta_{2} q^{4} + (\beta_{2} + 1) q^{5} - 2 \beta_{2} q^{6} + \beta_{2} q^{9} + ( - \beta_{3} - \beta_1) q^{10} + \beta_{3} q^{12} + (\beta_{3} + \beta_1) q^{13} + (\beta_{3} + \beta_1) q^{15} + (\beta_{2} + 1) q^{16} - \beta_{3} q^{18} - q^{20} + \beta_{2} q^{25} + 2 q^{26} + 2 q^{30} + ( - \beta_{3} - \beta_1) q^{32} + ( - \beta_{2} - 1) q^{36} + \beta_{3} q^{37} - 2 q^{39} - q^{45} + (\beta_{3} + \beta_1) q^{48} + q^{49} - \beta_{3} q^{50} - \beta_1 q^{52} + ( - \beta_{3} - \beta_1) q^{53} - \beta_1 q^{60} - q^{64} + \beta_{3} q^{65} + (\beta_{3} + \beta_1) q^{67} + (2 \beta_{2} + 2) q^{74} + \beta_{3} q^{75} + 2 \beta_1 q^{78} + \beta_{2} q^{80} + (\beta_{2} + 1) q^{81} + \beta_1 q^{90} + 2 q^{96} + \beta_1 q^{97} - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{9} + 2 q^{16} - 4 q^{20} - 2 q^{25} + 8 q^{26} + 8 q^{30} - 2 q^{36} - 8 q^{39} - 4 q^{45} + 4 q^{49} - 4 q^{64} + 4 q^{74} - 2 q^{80} + 2 q^{81} + 8 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(362\) \(1446\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
0.707107 + 1.22474i
−0.707107 − 1.22474i
0.707107 − 1.22474i
−0.707107 + 1.22474i
−0.707107 − 1.22474i 0.707107 + 1.22474i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 − 1.73205i 0 0 −0.500000 + 0.866025i 0.707107 − 1.22474i
69.2 0.707107 + 1.22474i −0.707107 − 1.22474i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 − 1.73205i 0 0 −0.500000 + 0.866025i −0.707107 + 1.22474i
654.1 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.500000 − 0.866025i 0.500000 − 0.866025i 1.00000 + 1.73205i 0 0 −0.500000 − 0.866025i 0.707107 + 1.22474i
654.2 0.707107 − 1.22474i −0.707107 + 1.22474i −0.500000 − 0.866025i 0.500000 − 0.866025i 1.00000 + 1.73205i 0 0 −0.500000 − 0.866025i −0.707107 − 1.22474i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
19.b odd 2 1 inner
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.b 4
5.b even 2 1 inner 1805.1.h.b 4
19.b odd 2 1 inner 1805.1.h.b 4
19.c even 3 1 95.1.d.b ✓ 2
19.c even 3 1 inner 1805.1.h.b 4
19.d odd 6 1 95.1.d.b ✓ 2
19.d odd 6 1 inner 1805.1.h.b 4
19.e even 9 6 1805.1.o.b 12
19.f odd 18 6 1805.1.o.b 12
57.f even 6 1 855.1.g.c 2
57.h odd 6 1 855.1.g.c 2
76.f even 6 1 1520.1.m.b 2
76.g odd 6 1 1520.1.m.b 2
95.d odd 2 1 CM 1805.1.h.b 4
95.h odd 6 1 95.1.d.b ✓ 2
95.h odd 6 1 inner 1805.1.h.b 4
95.i even 6 1 95.1.d.b ✓ 2
95.i even 6 1 inner 1805.1.h.b 4
95.l even 12 2 475.1.c.b 2
95.m odd 12 2 475.1.c.b 2
95.o odd 18 6 1805.1.o.b 12
95.p even 18 6 1805.1.o.b 12
285.n odd 6 1 855.1.g.c 2
285.q even 6 1 855.1.g.c 2
380.p odd 6 1 1520.1.m.b 2
380.s even 6 1 1520.1.m.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b ✓ 2 19.c even 3 1
95.1.d.b ✓ 2 19.d odd 6 1
95.1.d.b ✓ 2 95.h odd 6 1
95.1.d.b ✓ 2 95.i even 6 1
475.1.c.b 2 95.l even 12 2
475.1.c.b 2 95.m odd 12 2
855.1.g.c 2 57.f even 6 1
855.1.g.c 2 57.h odd 6 1
855.1.g.c 2 285.n odd 6 1
855.1.g.c 2 285.q even 6 1
1520.1.m.b 2 76.f even 6 1
1520.1.m.b 2 76.g odd 6 1
1520.1.m.b 2 380.p odd 6 1
1520.1.m.b 2 380.s even 6 1
1805.1.h.b 4 1.a even 1 1 trivial
1805.1.h.b 4 5.b even 2 1 inner
1805.1.h.b 4 19.b odd 2 1 inner
1805.1.h.b 4 19.c even 3 1 inner
1805.1.h.b 4 19.d odd 6 1 inner
1805.1.h.b 4 95.d odd 2 1 CM
1805.1.h.b 4 95.h odd 6 1 inner
1805.1.h.b 4 95.i even 6 1 inner
1805.1.o.b 12 19.e even 9 6
1805.1.o.b 12 19.f odd 18 6
1805.1.o.b 12 95.o odd 18 6
1805.1.o.b 12 95.p even 18 6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2T_{2}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1805, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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