Properties

Label 3800.1.o.f
Level $3800$
Weight $1$
Character orbit 3800.o
Self dual yes
Analytic conductor $1.896$
Analytic rank $0$
Dimension $3$
Projective image $D_{9}$
CM discriminant -152
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,1,Mod(1101,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.1101");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.o (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.8340544000000.1
Artin image: $D_9$
Artin field: Galois closure of 9.1.8340544000000.1

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_{2} + \beta_1) q^{3} + q^{4} + ( - \beta_{2} + \beta_1) q^{6} - \beta_1 q^{7} + q^{8} + ( - \beta_1 + 1) q^{9} + ( - \beta_{2} + \beta_1) q^{12} + \beta_{2} q^{13} - \beta_1 q^{14}+ \cdots + (\beta_{2} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{9} + 3 q^{16} + 3 q^{18} + 3 q^{19} - 3 q^{21} - 3 q^{27} + 3 q^{32} + 3 q^{36} - 3 q^{37} + 3 q^{38} - 3 q^{39} - 3 q^{42} - 3 q^{47} + 3 q^{49} - 3 q^{51} - 3 q^{54}+ \cdots + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1101.1
−1.53209
1.87939
−0.347296
1.00000 −1.87939 1.00000 0 −1.87939 1.53209 1.00000 2.53209 0
1101.2 1.00000 0.347296 1.00000 0 0.347296 −1.87939 1.00000 −0.879385 0
1101.3 1.00000 1.53209 1.00000 0 1.53209 0.347296 1.00000 1.34730 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
152.g odd 2 1 CM by \(\Q(\sqrt{-38}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.o.f yes 3
5.b even 2 1 3800.1.o.d yes 3
5.c odd 4 2 3800.1.b.c 6
8.b even 2 1 3800.1.o.c 3
19.b odd 2 1 3800.1.o.c 3
40.f even 2 1 3800.1.o.e yes 3
40.i odd 4 2 3800.1.b.d 6
95.d odd 2 1 3800.1.o.e yes 3
95.g even 4 2 3800.1.b.d 6
152.g odd 2 1 CM 3800.1.o.f yes 3
760.b odd 2 1 3800.1.o.d yes 3
760.t even 4 2 3800.1.b.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.1.b.c 6 5.c odd 4 2
3800.1.b.c 6 760.t even 4 2
3800.1.b.d 6 40.i odd 4 2
3800.1.b.d 6 95.g even 4 2
3800.1.o.c 3 8.b even 2 1
3800.1.o.c 3 19.b odd 2 1
3800.1.o.d yes 3 5.b even 2 1
3800.1.o.d yes 3 760.b odd 2 1
3800.1.o.e yes 3 40.f even 2 1
3800.1.o.e yes 3 95.d odd 2 1
3800.1.o.f yes 3 1.a even 1 1 trivial
3800.1.o.f yes 3 152.g odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3800, [\chi])\):

\( T_{3}^{3} - 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{3} - 3T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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