Properties

Label 3871.1.m.b
Level $3871$
Weight $1$
Character orbit 3871.m
Analytic conductor $1.932$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -79
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3871,1,Mod(1500,3871)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3871, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3871.1500");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3871 = 7^{2} \cdot 79 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3871.m (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: \(1.93188066390\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 79)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.6241.1
Artin image: $C_3\times D_{10}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{30} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{2} - \beta_1 q^{4} + (\beta_{3} - \beta_{2} - \beta_1) q^{5} - q^{8} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{2} - \beta_1 q^{4} + (\beta_{3} - \beta_{2} - \beta_1) q^{5} - q^{8} + \beta_{3} q^{9} + ( - \beta_{3} - 1) q^{10} + \beta_1 q^{11} - \beta_{2} q^{13} + \beta_1 q^{18} + (\beta_{3} - \beta_{2} - \beta_1) q^{19} - q^{20} + ( - \beta_{2} + 1) q^{22} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{23} + ( - \beta_{3} + \beta_1 - 1) q^{25} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{26} - \beta_1 q^{31} + ( - \beta_{3} - 1) q^{32} - \beta_{2} q^{36} + ( - \beta_{3} - 1) q^{38} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{40} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{44} + ( - \beta_{3} + \beta_1 - 1) q^{45} + (\beta_{3} + 1) q^{46} + q^{50} + ( - \beta_{3} - \beta_1 - 1) q^{52} + q^{55} + (\beta_{2} - 1) q^{62} + \beta_{2} q^{64} - \beta_{3} q^{65} + (\beta_{3} - \beta_1 + 1) q^{67} - \beta_{3} q^{72} + ( - \beta_{3} + \beta_1 - 1) q^{73} - q^{76} + \beta_{3} q^{79} + ( - \beta_{3} - 1) q^{81} - 2 q^{83} - \beta_1 q^{88} + (\beta_{2} + \beta_1) q^{89} + q^{90} + q^{92} + ( - 2 \beta_{3} + \beta_1 - 2) q^{95} + (\beta_{2} + 1) q^{97} + \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{4} - q^{5} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - q^{4} - q^{5} - 4 q^{8} - 2 q^{9} - 2 q^{10} + q^{11} + 2 q^{13} + q^{18} - q^{19} - 4 q^{20} + 6 q^{22} + q^{23} - q^{25} + 3 q^{26} - q^{31} - 2 q^{32} + 2 q^{36} - 2 q^{38} + q^{40} + 3 q^{44} - q^{45} + 2 q^{46} + 4 q^{50} - 3 q^{52} + 4 q^{55} - 6 q^{62} - 2 q^{64} + 2 q^{65} + q^{67} + 2 q^{72} - q^{73} - 4 q^{76} - 2 q^{79} - 2 q^{81} - 8 q^{83} - q^{88} - q^{89} + 4 q^{90} + 4 q^{92} - 3 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3871\mathbb{Z}\right)^\times\).

\(n\) \(1030\) \(2845\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1500.1
−0.309017 0.535233i
0.809017 + 1.40126i
−0.309017 + 0.535233i
0.809017 1.40126i
−0.309017 + 0.535233i 0 0.309017 + 0.535233i −0.809017 + 1.40126i 0 0 −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
1500.2 0.809017 1.40126i 0 −0.809017 1.40126i 0.309017 0.535233i 0 0 −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
3791.1 −0.309017 0.535233i 0 0.309017 0.535233i −0.809017 1.40126i 0 0 −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
3791.2 0.809017 + 1.40126i 0 −0.809017 + 1.40126i 0.309017 + 0.535233i 0 0 −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.b odd 2 1 CM by \(\Q(\sqrt{-79}) \)
7.c even 3 1 inner
553.m odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3871.1.m.b 4
7.b odd 2 1 3871.1.m.c 4
7.c even 3 1 3871.1.c.c 2
7.c even 3 1 inner 3871.1.m.b 4
7.d odd 6 1 79.1.b.a 2
7.d odd 6 1 3871.1.m.c 4
21.g even 6 1 711.1.d.b 2
28.f even 6 1 1264.1.e.a 2
35.i odd 6 1 1975.1.d.c 2
35.k even 12 2 1975.1.c.a 4
79.b odd 2 1 CM 3871.1.m.b 4
553.d even 2 1 3871.1.m.c 4
553.l even 6 1 79.1.b.a 2
553.l even 6 1 3871.1.m.c 4
553.m odd 6 1 3871.1.c.c 2
553.m odd 6 1 inner 3871.1.m.b 4
1659.be odd 6 1 711.1.d.b 2
2212.bi odd 6 1 1264.1.e.a 2
2765.bm even 6 1 1975.1.d.c 2
2765.cg odd 12 2 1975.1.c.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
79.1.b.a 2 7.d odd 6 1
79.1.b.a 2 553.l even 6 1
711.1.d.b 2 21.g even 6 1
711.1.d.b 2 1659.be odd 6 1
1264.1.e.a 2 28.f even 6 1
1264.1.e.a 2 2212.bi odd 6 1
1975.1.c.a 4 35.k even 12 2
1975.1.c.a 4 2765.cg odd 12 2
1975.1.d.c 2 35.i odd 6 1
1975.1.d.c 2 2765.bm even 6 1
3871.1.c.c 2 7.c even 3 1
3871.1.c.c 2 553.m odd 6 1
3871.1.m.b 4 1.a even 1 1 trivial
3871.1.m.b 4 7.c even 3 1 inner
3871.1.m.b 4 79.b odd 2 1 CM
3871.1.m.b 4 553.m odd 6 1 inner
3871.1.m.c 4 7.b odd 2 1
3871.1.m.c 4 7.d odd 6 1
3871.1.m.c 4 553.d even 2 1
3871.1.m.c 4 553.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3871, [\chi])\):

\( T_{2}^{4} - T_{2}^{3} + 2T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} + T_{5}^{3} + 2T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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