Properties

Label 38.4.a.b
Level $38$
Weight $4$
Character orbit 38.a
Self dual yes
Analytic conductor $2.242$
Analytic rank $0$
Dimension $2$
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] = [38,4,Mod(1,38)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(38, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 38.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: \(2.24207258022\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + ( - \beta + 1) q^{3} + 4 q^{4} + (2 \beta + 4) q^{5} + (2 \beta - 2) q^{6} + ( - \beta + 29) q^{7} - 8 q^{8} + ( - \beta + 18) q^{9} + ( - 4 \beta - 8) q^{10} + ( - 2 \beta + 6) q^{11} + ( - 4 \beta + 4) q^{12}+ \cdots + ( - 40 \beta + 196) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + q^{3} + 8 q^{4} + 10 q^{5} - 2 q^{6} + 57 q^{7} - 16 q^{8} + 35 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} + 13 q^{13} - 114 q^{14} - 172 q^{15} + 32 q^{16} - 51 q^{17} - 70 q^{18} - 38 q^{19}+ \cdots + 352 q^{99}+O(q^{100}) \) 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
7.15207
−6.15207
−2.00000 −6.15207 4.00000 18.3041 12.3041 21.8479 −8.00000 10.8479 −36.6083
1.2 −2.00000 7.15207 4.00000 −8.30413 −14.3041 35.1521 −8.00000 24.1521 16.6083
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 38.4.a.b 2
3.b odd 2 1 342.4.a.k 2
4.b odd 2 1 304.4.a.d 2
5.b even 2 1 950.4.a.h 2
5.c odd 4 2 950.4.b.g 4
7.b odd 2 1 1862.4.a.b 2
8.b even 2 1 1216.4.a.j 2
8.d odd 2 1 1216.4.a.l 2
19.b odd 2 1 722.4.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.b 2 1.a even 1 1 trivial
304.4.a.d 2 4.b odd 2 1
342.4.a.k 2 3.b odd 2 1
722.4.a.i 2 19.b odd 2 1
950.4.a.h 2 5.b even 2 1
950.4.b.g 4 5.c odd 4 2
1216.4.a.j 2 8.b even 2 1
1216.4.a.l 2 8.d odd 2 1
1862.4.a.b 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} - 44 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(38))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 44 \) Copy content Toggle raw display
$5$ \( T^{2} - 10T - 152 \) Copy content Toggle raw display
$7$ \( T^{2} - 57T + 768 \) Copy content Toggle raw display
$11$ \( T^{2} - 10T - 152 \) Copy content Toggle raw display
$13$ \( T^{2} - 13T - 2126 \) Copy content Toggle raw display
$17$ \( T^{2} + 51T - 9306 \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 155T - 1472 \) Copy content Toggle raw display
$29$ \( T^{2} + 79T - 35654 \) Copy content Toggle raw display
$31$ \( T^{2} + 16T - 11264 \) Copy content Toggle raw display
$37$ \( T^{2} - 380T + 24772 \) Copy content Toggle raw display
$41$ \( T^{2} + 790T + 154432 \) Copy content Toggle raw display
$43$ \( T^{2} - 296T - 80048 \) Copy content Toggle raw display
$47$ \( T^{2} + 200T - 60800 \) Copy content Toggle raw display
$53$ \( T^{2} - 397T + 35818 \) Copy content Toggle raw display
$59$ \( T^{2} - 201T - 212964 \) Copy content Toggle raw display
$61$ \( T^{2} + 680T + 29932 \) Copy content Toggle raw display
$67$ \( T^{2} + 939T + 138612 \) Copy content Toggle raw display
$71$ \( T^{2} - 406T + 19792 \) Copy content Toggle raw display
$73$ \( T^{2} - 123T + 3738 \) Copy content Toggle raw display
$79$ \( T^{2} - 106T - 146048 \) Copy content Toggle raw display
$83$ \( T^{2} - 2226 T + 1237176 \) Copy content Toggle raw display
$89$ \( T^{2} + 870T + 184800 \) Copy content Toggle raw display
$97$ \( T^{2} + 1864 T + 613036 \) Copy content Toggle raw display
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