Properties

Label 3900.2.a.r
Level $3900$
Weight $2$
Character orbit 3900.a
Self dual yes
Analytic conductor $31.142$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3900,2,Mod(1,3900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3900, 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("3900.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3900.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: \(31.1416567883\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 780)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + (2 \beta - 2) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (2 \beta - 2) q^{7} + q^{9} + \beta q^{11} + q^{13} + ( - 4 \beta + 2) q^{17} + ( - 4 \beta - 2) q^{19} + (2 \beta - 2) q^{21} + ( - 2 \beta - 4) q^{23} + q^{27} - 6 q^{29} - 10 q^{31} + \beta q^{33} + ( - 2 \beta - 2) q^{37} + q^{39} + (5 \beta - 4) q^{41} + ( - 4 \beta + 2) q^{43} + ( - \beta - 8) q^{47} + ( - 8 \beta + 5) q^{49} + ( - 4 \beta + 2) q^{51} + (2 \beta + 10) q^{53} + ( - 4 \beta - 2) q^{57} + (5 \beta - 4) q^{59} + (4 \beta + 4) q^{61} + (2 \beta - 2) q^{63} + (6 \beta + 6) q^{67} + ( - 2 \beta - 4) q^{69} + ( - \beta - 4) q^{71} + ( - 4 \beta - 10) q^{73} + ( - 2 \beta + 4) q^{77} + 2 q^{79} + q^{81} + (7 \beta + 4) q^{83} - 6 q^{87} + (\beta + 8) q^{89} + (2 \beta - 2) q^{91} - 10 q^{93} - 6 q^{97} + \beta q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 4 q^{7} + 2 q^{9} + 2 q^{13} + 4 q^{17} - 4 q^{19} - 4 q^{21} - 8 q^{23} + 2 q^{27} - 12 q^{29} - 20 q^{31} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 4 q^{43} - 16 q^{47} + 10 q^{49} + 4 q^{51} + 20 q^{53} - 4 q^{57} - 8 q^{59} + 8 q^{61} - 4 q^{63} + 12 q^{67} - 8 q^{69} - 8 q^{71} - 20 q^{73} + 8 q^{77} + 4 q^{79} + 2 q^{81} + 8 q^{83} - 12 q^{87} + 16 q^{89} - 4 q^{91} - 20 q^{93} - 12 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
0 1.00000 0 0 0 −4.82843 0 1.00000 0
1.2 0 1.00000 0 0 0 0.828427 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(-1\)

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 3900.2.a.r 2
5.b even 2 1 3900.2.a.q 2
5.c odd 4 2 780.2.h.d 4
15.e even 4 2 2340.2.h.c 4
20.e even 4 2 3120.2.l.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
780.2.h.d 4 5.c odd 4 2
2340.2.h.c 4 15.e even 4 2
3120.2.l.l 4 20.e even 4 2
3900.2.a.q 2 5.b even 2 1
3900.2.a.r 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3900))\):

\( T_{7}^{2} + 4T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 2 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 8T - 34 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$47$ \( T^{2} + 16T + 62 \) Copy content Toggle raw display
$53$ \( T^{2} - 20T + 92 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T - 34 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 12T - 36 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$73$ \( T^{2} + 20T + 68 \) Copy content Toggle raw display
$79$ \( (T - 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 8T - 82 \) Copy content Toggle raw display
$89$ \( T^{2} - 16T + 62 \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
show more
show less