Properties

Label 4205.2.a.b
Level $4205$
Weight $2$
Character orbit 4205.a
Self dual yes
Analytic conductor $33.577$
Analytic rank $1$
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] = [4205,2,Mod(1,4205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4205 = 5 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4205.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: \(33.5770940499\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{11}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - 2 q^{4} - q^{5} + ( - 2 \beta - 1) q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 2 q^{4} - q^{5} + ( - 2 \beta - 1) q^{7} - 2 q^{9} + ( - 3 \beta + 3) q^{11} + 2 q^{12} + 2 \beta q^{13} + q^{15} + 4 q^{16} + (3 \beta - 3) q^{17} + 4 \beta q^{19} + 2 q^{20} + (2 \beta + 1) q^{21} + ( - 3 \beta - 3) q^{23} + q^{25} + 5 q^{27} + (4 \beta + 2) q^{28} + ( - 4 \beta + 1) q^{31} + (3 \beta - 3) q^{33} + (2 \beta + 1) q^{35} + 4 q^{36} + ( - 2 \beta + 9) q^{37} - 2 \beta q^{39} + 6 \beta q^{41} + ( - \beta - 2) q^{43} + (6 \beta - 6) q^{44} + 2 q^{45} + 9 q^{47} - 4 q^{48} + (8 \beta - 2) q^{49} + ( - 3 \beta + 3) q^{51} - 4 \beta q^{52} + 3 q^{53} + (3 \beta - 3) q^{55} - 4 \beta q^{57} + ( - 6 \beta + 3) q^{59} - 2 q^{60} + ( - 7 \beta + 1) q^{61} + (4 \beta + 2) q^{63} - 8 q^{64} - 2 \beta q^{65} + (2 \beta + 9) q^{67} + ( - 6 \beta + 6) q^{68} + (3 \beta + 3) q^{69} + (9 \beta - 6) q^{71} + (7 \beta - 9) q^{73} - q^{75} - 8 \beta q^{76} + (3 \beta + 3) q^{77} + ( - 2 \beta - 9) q^{79} - 4 q^{80} + q^{81} + ( - 6 \beta + 12) q^{83} + ( - 4 \beta - 2) q^{84} + ( - 3 \beta + 3) q^{85} + ( - 6 \beta + 3) q^{89} + ( - 6 \beta - 4) q^{91} + (6 \beta + 6) q^{92} + (4 \beta - 1) q^{93} - 4 \beta q^{95} + (8 \beta - 11) q^{97} + (6 \beta - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{4} - 2 q^{5} - 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 4 q^{4} - 2 q^{5} - 4 q^{7} - 4 q^{9} + 3 q^{11} + 4 q^{12} + 2 q^{13} + 2 q^{15} + 8 q^{16} - 3 q^{17} + 4 q^{19} + 4 q^{20} + 4 q^{21} - 9 q^{23} + 2 q^{25} + 10 q^{27} + 8 q^{28} - 2 q^{31} - 3 q^{33} + 4 q^{35} + 8 q^{36} + 16 q^{37} - 2 q^{39} + 6 q^{41} - 5 q^{43} - 6 q^{44} + 4 q^{45} + 18 q^{47} - 8 q^{48} + 4 q^{49} + 3 q^{51} - 4 q^{52} + 6 q^{53} - 3 q^{55} - 4 q^{57} - 4 q^{60} - 5 q^{61} + 8 q^{63} - 16 q^{64} - 2 q^{65} + 20 q^{67} + 6 q^{68} + 9 q^{69} - 3 q^{71} - 11 q^{73} - 2 q^{75} - 8 q^{76} + 9 q^{77} - 20 q^{79} - 8 q^{80} + 2 q^{81} + 18 q^{83} - 8 q^{84} + 3 q^{85} - 14 q^{91} + 18 q^{92} + 2 q^{93} - 4 q^{95} - 14 q^{97} - 6 q^{99}+O(q^{100}) \) 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 −2.00000 −1.00000 0 −4.23607 0 −2.00000 0
1.2 0 −1.00000 −2.00000 −1.00000 0 0.236068 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(29\) \( +1 \)

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 4205.2.a.b 2
29.b even 2 1 4205.2.a.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4205.2.a.b 2 1.a even 1 1 trivial
4205.2.a.c yes 2 29.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4205))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3} + 1 \) 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
$7$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$37$ \( T^{2} - 16T + 59 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$43$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$47$ \( (T - 9)^{2} \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 45 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T - 55 \) Copy content Toggle raw display
$67$ \( T^{2} - 20T + 95 \) Copy content Toggle raw display
$71$ \( T^{2} + 3T - 99 \) Copy content Toggle raw display
$73$ \( T^{2} + 11T - 31 \) Copy content Toggle raw display
$79$ \( T^{2} + 20T + 95 \) Copy content Toggle raw display
$83$ \( T^{2} - 18T + 36 \) Copy content Toggle raw display
$89$ \( T^{2} - 45 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T - 31 \) Copy content Toggle raw display
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