Properties

Label 5808.2.a.cg
Level $5808$
Weight $2$
Character orbit 5808.a
Self dual yes
Analytic conductor $46.377$
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] = [5808,2,Mod(1,5808)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5808, 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("5808.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5808.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: \(46.3771134940\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( - 3 \beta + 1) q^{5} + ( - \beta + 4) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - 3 \beta + 1) q^{5} + ( - \beta + 4) q^{7} + q^{9} + ( - 2 \beta + 2) q^{13} + ( - 3 \beta + 1) q^{15} - 4 \beta q^{17} + (2 \beta - 2) q^{19} + ( - \beta + 4) q^{21} + ( - 2 \beta + 2) q^{23} + (3 \beta + 5) q^{25} + q^{27} + (\beta + 1) q^{29} + (\beta - 8) q^{31} + ( - 10 \beta + 7) q^{35} + (4 \beta + 4) q^{37} + ( - 2 \beta + 2) q^{39} + ( - 2 \beta + 2) q^{41} + (2 \beta - 2) q^{43} + ( - 3 \beta + 1) q^{45} + 4 \beta q^{47} + ( - 7 \beta + 10) q^{49} - 4 \beta q^{51} + (3 \beta - 2) q^{53} + (2 \beta - 2) q^{57} + ( - 3 \beta + 2) q^{59} + (2 \beta + 8) q^{61} + ( - \beta + 4) q^{63} + ( - 2 \beta + 8) q^{65} - 4 q^{67} + ( - 2 \beta + 2) q^{69} + (4 \beta - 10) q^{71} + ( - 7 \beta + 7) q^{73} + (3 \beta + 5) q^{75} + (7 \beta - 5) q^{79} + q^{81} + ( - 7 \beta - 5) q^{83} + (8 \beta + 12) q^{85} + (\beta + 1) q^{87} + (6 \beta - 8) q^{89} + ( - 8 \beta + 10) q^{91} + (\beta - 8) q^{93} + (2 \beta - 8) q^{95} + ( - 7 \beta + 7) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{5} + 7 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - q^{5} + 7 q^{7} + 2 q^{9} + 2 q^{13} - q^{15} - 4 q^{17} - 2 q^{19} + 7 q^{21} + 2 q^{23} + 13 q^{25} + 2 q^{27} + 3 q^{29} - 15 q^{31} + 4 q^{35} + 12 q^{37} + 2 q^{39} + 2 q^{41} - 2 q^{43} - q^{45} + 4 q^{47} + 13 q^{49} - 4 q^{51} - q^{53} - 2 q^{57} + q^{59} + 18 q^{61} + 7 q^{63} + 14 q^{65} - 8 q^{67} + 2 q^{69} - 16 q^{71} + 7 q^{73} + 13 q^{75} - 3 q^{79} + 2 q^{81} - 17 q^{83} + 32 q^{85} + 3 q^{87} - 10 q^{89} + 12 q^{91} - 15 q^{93} - 14 q^{95} + 7 q^{97}+O(q^{100}) \) 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
1.61803
−0.618034
0 1.00000 0 −3.85410 0 2.38197 0 1.00000 0
1.2 0 1.00000 0 2.85410 0 4.61803 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 5808.2.a.cg 2
4.b odd 2 1 726.2.a.j 2
11.b odd 2 1 5808.2.a.cb 2
11.d odd 10 2 528.2.y.d 4
12.b even 2 1 2178.2.a.bb 2
44.c even 2 1 726.2.a.l 2
44.g even 10 2 66.2.e.a 4
44.g even 10 2 726.2.e.f 4
44.h odd 10 2 726.2.e.n 4
44.h odd 10 2 726.2.e.r 4
132.d odd 2 1 2178.2.a.t 2
132.n odd 10 2 198.2.f.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.e.a 4 44.g even 10 2
198.2.f.c 4 132.n odd 10 2
528.2.y.d 4 11.d odd 10 2
726.2.a.j 2 4.b odd 2 1
726.2.a.l 2 44.c even 2 1
726.2.e.f 4 44.g even 10 2
726.2.e.n 4 44.h odd 10 2
726.2.e.r 4 44.h odd 10 2
2178.2.a.t 2 132.d odd 2 1
2178.2.a.bb 2 12.b even 2 1
5808.2.a.cb 2 11.b odd 2 1
5808.2.a.cg 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5808))\):

\( T_{5}^{2} + T_{5} - 11 \) Copy content Toggle raw display
\( T_{7}^{2} - 7T_{7} + 11 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 4 \) 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
$5$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$7$ \( T^{2} - 7T + 11 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} + 15T + 55 \) Copy content Toggle raw display
$37$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$53$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$59$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$61$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$73$ \( T^{2} - 7T - 49 \) Copy content Toggle raw display
$79$ \( T^{2} + 3T - 59 \) Copy content Toggle raw display
$83$ \( T^{2} + 17T + 11 \) Copy content Toggle raw display
$89$ \( T^{2} + 10T - 20 \) Copy content Toggle raw display
$97$ \( T^{2} - 7T - 49 \) Copy content Toggle raw display
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