Properties

Label 3696.2.a.be
Level $3696$
Weight $2$
Character orbit 3696.a
Self dual yes
Analytic conductor $29.513$
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] = [3696,2,Mod(1,3696)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3696, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3696.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3696.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: \(29.5127085871\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 231)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} - q^{7} + q^{9} - q^{11} + ( - 2 \beta - 1) q^{13} - q^{15} + ( - \beta + 3) q^{17} - 3 \beta q^{19} + q^{21} + (3 \beta + 1) q^{23} - 4 q^{25} - q^{27} + 5 q^{29} + ( - \beta + 3) q^{31} + q^{33} - q^{35} - 7 q^{37} + (2 \beta + 1) q^{39} + (2 \beta + 2) q^{41} + (3 \beta + 1) q^{43} + q^{45} + (\beta + 2) q^{47} + q^{49} + (\beta - 3) q^{51} + (5 \beta - 1) q^{53} - q^{55} + 3 \beta q^{57} - 5 \beta q^{59} + 2 q^{61} - q^{63} + ( - 2 \beta - 1) q^{65} + (\beta + 12) q^{67} + ( - 3 \beta - 1) q^{69} + ( - 2 \beta - 2) q^{71} + (2 \beta + 9) q^{73} + 4 q^{75} + q^{77} + ( - 2 \beta + 10) q^{79} + q^{81} + ( - \beta - 9) q^{83} + ( - \beta + 3) q^{85} - 5 q^{87} + 2 \beta q^{89} + (2 \beta + 1) q^{91} + (\beta - 3) q^{93} - 3 \beta q^{95} + ( - 3 \beta + 3) q^{97} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} - 2 q^{15} + 6 q^{17} + 2 q^{21} + 2 q^{23} - 8 q^{25} - 2 q^{27} + 10 q^{29} + 6 q^{31} + 2 q^{33} - 2 q^{35} - 14 q^{37} + 2 q^{39} + 4 q^{41} + 2 q^{43} + 2 q^{45} + 4 q^{47} + 2 q^{49} - 6 q^{51} - 2 q^{53} - 2 q^{55} + 4 q^{61} - 2 q^{63} - 2 q^{65} + 24 q^{67} - 2 q^{69} - 4 q^{71} + 18 q^{73} + 8 q^{75} + 2 q^{77} + 20 q^{79} + 2 q^{81} - 18 q^{83} + 6 q^{85} - 10 q^{87} + 2 q^{91} - 6 q^{93} + 6 q^{97} - 2 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)
\(11\) \(1\)

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 3696.2.a.be 2
4.b odd 2 1 231.2.a.c 2
12.b even 2 1 693.2.a.f 2
20.d odd 2 1 5775.2.a.be 2
28.d even 2 1 1617.2.a.p 2
44.c even 2 1 2541.2.a.t 2
84.h odd 2 1 4851.2.a.w 2
132.d odd 2 1 7623.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.c 2 4.b odd 2 1
693.2.a.f 2 12.b even 2 1
1617.2.a.p 2 28.d even 2 1
2541.2.a.t 2 44.c even 2 1
3696.2.a.be 2 1.a even 1 1 trivial
4851.2.a.w 2 84.h odd 2 1
5775.2.a.be 2 20.d odd 2 1
7623.2.a.bm 2 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3696))\):

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 19 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} + 4 \) 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
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 45 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$37$ \( (T + 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T - 124 \) Copy content Toggle raw display
$59$ \( T^{2} - 125 \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 24T + 139 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$73$ \( T^{2} - 18T + 61 \) Copy content Toggle raw display
$79$ \( T^{2} - 20T + 80 \) Copy content Toggle raw display
$83$ \( T^{2} + 18T + 76 \) Copy content Toggle raw display
$89$ \( T^{2} - 20 \) Copy content Toggle raw display
$97$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
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