Properties

Label 476.2.a.b
Level $476$
Weight $2$
Character orbit 476.a
Self dual yes
Analytic conductor $3.801$
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] = [476,2,Mod(1,476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(476, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("476.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 476.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: \(3.80087913621\)
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: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + (\beta - 1) q^{5} - q^{7} + (\beta - 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + (\beta - 1) q^{5} - q^{7} + (\beta - 2) q^{9} + (2 \beta - 4) q^{11} + (2 \beta - 2) q^{13} - q^{15} + q^{17} + ( - 4 \beta - 2) q^{19} + \beta q^{21} + ( - 6 \beta + 2) q^{23} + ( - \beta - 3) q^{25} + (4 \beta - 1) q^{27} + ( - 6 \beta + 4) q^{29} + (7 \beta - 5) q^{31} + (2 \beta - 2) q^{33} + ( - \beta + 1) q^{35} + (4 \beta - 6) q^{37} - 2 q^{39} + ( - 5 \beta + 8) q^{41} + ( - 5 \beta - 4) q^{43} + ( - 2 \beta + 3) q^{45} + ( - 4 \beta - 2) q^{47} + q^{49} - \beta q^{51} + (11 \beta - 7) q^{53} + ( - 4 \beta + 6) q^{55} + (6 \beta + 4) q^{57} + ( - 7 \beta + 4) q^{61} + ( - \beta + 2) q^{63} + ( - 2 \beta + 4) q^{65} + ( - 5 \beta - 5) q^{67} + (4 \beta + 6) q^{69} + (8 \beta - 2) q^{71} + ( - 3 \beta + 12) q^{73} + (4 \beta + 1) q^{75} + ( - 2 \beta + 4) q^{77} + (8 \beta - 10) q^{79} + ( - 6 \beta + 2) q^{81} + (12 \beta - 4) q^{83} + (\beta - 1) q^{85} + (2 \beta + 6) q^{87} + 2 q^{89} + ( - 2 \beta + 2) q^{91} + ( - 2 \beta - 7) q^{93} + ( - 2 \beta - 2) q^{95} + (5 \beta + 7) q^{97} + ( - 6 \beta + 10) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} - 2 q^{23} - 7 q^{25} + 2 q^{27} + 2 q^{29} - 3 q^{31} - 2 q^{33} + q^{35} - 8 q^{37} - 4 q^{39} + 11 q^{41} - 13 q^{43} + 4 q^{45} - 8 q^{47} + 2 q^{49} - q^{51} - 3 q^{53} + 8 q^{55} + 14 q^{57} + q^{61} + 3 q^{63} + 6 q^{65} - 15 q^{67} + 16 q^{69} + 4 q^{71} + 21 q^{73} + 6 q^{75} + 6 q^{77} - 12 q^{79} - 2 q^{81} + 4 q^{83} - q^{85} + 14 q^{87} + 4 q^{89} + 2 q^{91} - 16 q^{93} - 6 q^{95} + 19 q^{97} + 14 q^{99}+O(q^{100}) \) 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 −1.61803 0 0.618034 0 −1.00000 0 −0.381966 0
1.2 0 0.618034 0 −1.61803 0 −1.00000 0 −2.61803 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(17\) \(-1\)

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 476.2.a.b 2
3.b odd 2 1 4284.2.a.m 2
4.b odd 2 1 1904.2.a.j 2
7.b odd 2 1 3332.2.a.l 2
8.b even 2 1 7616.2.a.u 2
8.d odd 2 1 7616.2.a.p 2
17.b even 2 1 8092.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.2.a.b 2 1.a even 1 1 trivial
1904.2.a.j 2 4.b odd 2 1
3332.2.a.l 2 7.b odd 2 1
4284.2.a.m 2 3.b odd 2 1
7616.2.a.p 2 8.d odd 2 1
7616.2.a.u 2 8.b even 2 1
8092.2.a.m 2 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(476))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T - 59 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 11T - 1 \) Copy content Toggle raw display
$43$ \( T^{2} + 13T + 11 \) Copy content Toggle raw display
$47$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 149 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - T - 61 \) Copy content Toggle raw display
$67$ \( T^{2} + 15T + 25 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$73$ \( T^{2} - 21T + 99 \) Copy content Toggle raw display
$79$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 176 \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 19T + 59 \) Copy content Toggle raw display
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