Properties

Label 3800.2.a.x
Level $3800$
Weight $2$
Character orbit 3800.a
Self dual yes
Analytic conductor $30.343$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(1,3800)]
 
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, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.a (trivial)

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: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - \beta_1 q^{7} + ( - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - \beta_1 q^{7} + ( - \beta_1 + 1) q^{9} + ( - \beta_{2} - 2) q^{13} + ( - \beta_1 - 2) q^{17} + q^{19} + ( - 2 \beta_{2} - \beta_1) q^{21} + ( - 2 \beta_{2} + \beta_1) q^{23} - \beta_1 q^{27} + (\beta_1 + 6) q^{29} - 2 \beta_1 q^{31} + (\beta_{2} + \beta_1 - 2) q^{37} + (2 \beta_{2} - \beta_1 + 4) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + (2 \beta_{2} + 4) q^{43} + (2 \beta_{2} + 2 \beta_1) q^{47} + ( - 2 \beta_{2} + \beta_1 + 1) q^{49} - \beta_1 q^{51} + (\beta_{2} - 2 \beta_1 - 2) q^{53} - \beta_{2} q^{57} + ( - 4 \beta_{2} + \beta_1 + 4) q^{59} + ( - 2 \beta_{2} + 2) q^{61} + ( - 2 \beta_{2} + 8) q^{63} + (5 \beta_{2} - 2 \beta_1) q^{67} + (2 \beta_{2} - \beta_1 + 8) q^{69} + 4 \beta_1 q^{71} + ( - 4 \beta_{2} + \beta_1 - 2) q^{73} - 2 \beta_{2} q^{79} + ( - 2 \beta_{2} + 2 \beta_1 - 3) q^{81} + (2 \beta_1 + 4) q^{83} + ( - 4 \beta_{2} + \beta_1) q^{87} + (2 \beta_{2} - 2 \beta_1 + 2) q^{89} + ( - 2 \beta_{2} + \beta_1) q^{91} + ( - 4 \beta_{2} - 2 \beta_1) q^{93} + ( - \beta_{2} + 3 \beta_1 + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} + q^{7} + 4 q^{9} - 5 q^{13} - 5 q^{17} + 3 q^{19} + 3 q^{21} + q^{23} + q^{27} + 17 q^{29} + 2 q^{31} - 8 q^{37} + 11 q^{39} + 6 q^{41} + 10 q^{43} - 4 q^{47} + 4 q^{49} + q^{51} - 5 q^{53} + q^{57} + 15 q^{59} + 8 q^{61} + 26 q^{63} - 3 q^{67} + 23 q^{69} - 4 q^{71} - 3 q^{73} + 2 q^{79} - 9 q^{81} + 10 q^{83} + 3 q^{87} + 6 q^{89} + q^{91} + 6 q^{93} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 6 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.86081
2.11491
−0.254102
0 −2.32340 0 0 0 1.39821 0 2.39821 0
1.2 0 0.642074 0 0 0 −3.58774 0 −2.58774 0
1.3 0 2.68133 0 0 0 3.18953 0 4.18953 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(19\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.2.a.x 3
4.b odd 2 1 7600.2.a.bq 3
5.b even 2 1 760.2.a.j 3
5.c odd 4 2 3800.2.d.l 6
15.d odd 2 1 6840.2.a.bg 3
20.d odd 2 1 1520.2.a.s 3
40.e odd 2 1 6080.2.a.bq 3
40.f even 2 1 6080.2.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.j 3 5.b even 2 1
1520.2.a.s 3 20.d odd 2 1
3800.2.a.x 3 1.a even 1 1 trivial
3800.2.d.l 6 5.c odd 4 2
6080.2.a.bq 3 40.e odd 2 1
6080.2.a.bv 3 40.f even 2 1
6840.2.a.bg 3 15.d odd 2 1
7600.2.a.bq 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3800))\):

\( T_{3}^{3} - T_{3}^{2} - 6T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{3} - T_{7}^{2} - 12T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 6T + 4 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - T^{2} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 5 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$17$ \( T^{3} + 5 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - T^{2} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{3} - 17 T^{2} + \cdots - 124 \) Copy content Toggle raw display
$31$ \( T^{3} - 2 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$37$ \( T^{3} + 8T^{2} - 8 \) Copy content Toggle raw display
$41$ \( T^{3} - 6 T^{2} + \cdots + 56 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$47$ \( T^{3} + 4 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$53$ \( T^{3} + 5 T^{2} + \cdots + 52 \) Copy content Toggle raw display
$59$ \( T^{3} - 15 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$61$ \( T^{3} - 8 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$67$ \( T^{3} + 3 T^{2} + \cdots - 1052 \) Copy content Toggle raw display
$71$ \( T^{3} + 4 T^{2} + \cdots - 1024 \) Copy content Toggle raw display
$73$ \( T^{3} + 3 T^{2} + \cdots + 292 \) Copy content Toggle raw display
$79$ \( T^{3} - 2 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$83$ \( T^{3} - 10 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$89$ \( T^{3} - 6 T^{2} + \cdots + 184 \) Copy content Toggle raw display
$97$ \( T^{3} - 4 T^{2} + \cdots - 296 \) Copy content Toggle raw display
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