Properties

Label 3800.2.a.y
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: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + 2) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} + 2) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9} + (2 \beta_1 - 2) q^{11} + ( - 2 \beta_{2} - \beta_1 - 2) q^{13} + (\beta_{2} + 4) q^{17} - q^{19} + ( - \beta_{2} + 2 \beta_1 - 2) q^{21} + ( - \beta_{2} + 2 \beta_1 + 2) q^{23} + (\beta_{2} + 2 \beta_1 + 6) q^{27} - 3 \beta_{2} q^{29} + (2 \beta_{2} - 4) q^{31} + (2 \beta_{2} + 2 \beta_1 + 8) q^{33} + (\beta_{2} - \beta_1 - 4) q^{37} + (\beta_{2} - 4 \beta_1) q^{39} + (2 \beta_{2} - 2 \beta_1 - 2) q^{41} + (4 \beta_{2} + 2 \beta_1 + 4) q^{43} + ( - 2 \beta_{2} - 2 \beta_1) q^{47} + (3 \beta_{2} - 2 \beta_1 + 3) q^{49} + ( - \beta_{2} + 4 \beta_1 - 2) q^{51} + ( - 2 \beta_{2} - \beta_1 + 2) q^{53} - \beta_1 q^{57} + (\beta_{2} - 2) q^{59} + ( - 4 \beta_1 + 8) q^{61} + (2 \beta_1 + 4) q^{63} + ( - 4 \beta_{2} - 3 \beta_1) q^{67} + (3 \beta_{2} + 6 \beta_1 + 10) q^{69} + (4 \beta_1 - 4) q^{71} + (3 \beta_{2} + 4 \beta_1 + 4) q^{73} + ( - 4 \beta_{2} + 4 \beta_1 - 8) q^{77} + ( - 4 \beta_{2} - 2 \beta_1 + 4) q^{79} + ( - 2 \beta_{2} + 4 \beta_1 + 3) q^{81} + ( - 2 \beta_{2} + 8) q^{83} + (3 \beta_{2} + 6) q^{87} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{89} + ( - 3 \beta_{2} + 2 \beta_1 - 14) q^{91} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{93} + ( - \beta_{2} - 7 \beta_1 + 4) q^{97} + (6 \beta_1 + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 5 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} + 5 q^{7} + 4 q^{9} - 4 q^{11} - 5 q^{13} + 11 q^{17} - 3 q^{19} - 3 q^{21} + 9 q^{23} + 19 q^{27} + 3 q^{29} - 14 q^{31} + 24 q^{33} - 14 q^{37} - 5 q^{39} - 10 q^{41} + 10 q^{43} + 4 q^{49} - q^{51} + 7 q^{53} - q^{57} - 7 q^{59} + 20 q^{61} + 14 q^{63} + q^{67} + 33 q^{69} - 8 q^{71} + 13 q^{73} - 16 q^{77} + 14 q^{79} + 15 q^{81} + 26 q^{83} + 15 q^{87} + 18 q^{89} - 37 q^{91} - 14 q^{93} + 6 q^{97} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) 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
−1.76156
−0.363328
3.12489
0 −1.76156 0 0 0 4.62620 0 0.103084 0
1.2 0 −0.363328 0 0 0 −1.14134 0 −2.86799 0
1.3 0 3.12489 0 0 0 1.51514 0 6.76491 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.y 3
4.b odd 2 1 7600.2.a.bo 3
5.b even 2 1 760.2.a.h 3
5.c odd 4 2 3800.2.d.k 6
15.d odd 2 1 6840.2.a.bj 3
20.d odd 2 1 1520.2.a.r 3
40.e odd 2 1 6080.2.a.bs 3
40.f even 2 1 6080.2.a.bw 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.h 3 5.b even 2 1
1520.2.a.r 3 20.d odd 2 1
3800.2.a.y 3 1.a even 1 1 trivial
3800.2.d.k 6 5.c odd 4 2
6080.2.a.bs 3 40.e odd 2 1
6080.2.a.bw 3 40.f even 2 1
6840.2.a.bj 3 15.d odd 2 1
7600.2.a.bo 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3800))\):

\( T_{3}^{3} - T_{3}^{2} - 6T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} - 5T_{7}^{2} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 6T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 5T^{2} + 8 \) Copy content Toggle raw display
$11$ \( T^{3} + 4 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$13$ \( T^{3} + 5 T^{2} + \cdots - 106 \) Copy content Toggle raw display
$17$ \( T^{3} - 11 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 9 T^{2} + \cdots + 160 \) Copy content Toggle raw display
$29$ \( T^{3} - 3 T^{2} + \cdots + 108 \) Copy content Toggle raw display
$31$ \( T^{3} + 14 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$37$ \( T^{3} + 14 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} + \cdots - 472 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} + \cdots + 848 \) Copy content Toggle raw display
$47$ \( T^{3} - 40T - 64 \) Copy content Toggle raw display
$53$ \( T^{3} - 7 T^{2} + \cdots - 2 \) Copy content Toggle raw display
$59$ \( T^{3} + 7 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$61$ \( T^{3} - 20 T^{2} + \cdots + 640 \) Copy content Toggle raw display
$67$ \( T^{3} - T^{2} + \cdots - 530 \) Copy content Toggle raw display
$71$ \( T^{3} + 8 T^{2} + \cdots - 512 \) Copy content Toggle raw display
$73$ \( T^{3} - 13 T^{2} + \cdots + 500 \) Copy content Toggle raw display
$79$ \( T^{3} - 14 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$83$ \( T^{3} - 26 T^{2} + \cdots - 352 \) Copy content Toggle raw display
$89$ \( T^{3} - 18 T^{2} + \cdots - 40 \) Copy content Toggle raw display
$97$ \( T^{3} - 6 T^{2} + \cdots + 2308 \) Copy content Toggle raw display
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