Properties

Label 471.2.a.b
Level $471$
Weight $2$
Character orbit 471.a
Self dual yes
Analytic conductor $3.761$
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] = [471,2,Mod(1,471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(471, 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("471.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 471.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: \(3.76095393520\)
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q - \beta q^{2} + q^{3} + (\beta - 1) q^{4} - q^{5} - \beta q^{6} - 3 q^{7} + (2 \beta - 1) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + q^{3} + (\beta - 1) q^{4} - q^{5} - \beta q^{6} - 3 q^{7} + (2 \beta - 1) q^{8} + q^{9} + \beta q^{10} + (3 \beta - 2) q^{11} + (\beta - 1) q^{12} + (\beta - 2) q^{13} + 3 \beta q^{14} - q^{15} - 3 \beta q^{16} + ( - 2 \beta - 1) q^{17} - \beta q^{18} + ( - 2 \beta - 2) q^{19} + ( - \beta + 1) q^{20} - 3 q^{21} + ( - \beta - 3) q^{22} + q^{23} + (2 \beta - 1) q^{24} - 4 q^{25} + (\beta - 1) q^{26} + q^{27} + ( - 3 \beta + 3) q^{28} + (\beta - 1) q^{29} + \beta q^{30} + ( - 7 \beta + 2) q^{31} + ( - \beta + 5) q^{32} + (3 \beta - 2) q^{33} + (3 \beta + 2) q^{34} + 3 q^{35} + (\beta - 1) q^{36} + (2 \beta + 2) q^{37} + (4 \beta + 2) q^{38} + (\beta - 2) q^{39} + ( - 2 \beta + 1) q^{40} + ( - 2 \beta - 7) q^{41} + 3 \beta q^{42} + (2 \beta - 10) q^{43} + ( - 2 \beta + 5) q^{44} - q^{45} - \beta q^{46} + (3 \beta + 7) q^{47} - 3 \beta q^{48} + 2 q^{49} + 4 \beta q^{50} + ( - 2 \beta - 1) q^{51} + ( - 2 \beta + 3) q^{52} + ( - 4 \beta + 2) q^{53} - \beta q^{54} + ( - 3 \beta + 2) q^{55} + ( - 6 \beta + 3) q^{56} + ( - 2 \beta - 2) q^{57} - q^{58} + ( - 10 \beta + 6) q^{59} + ( - \beta + 1) q^{60} + ( - 2 \beta - 7) q^{61} + (5 \beta + 7) q^{62} - 3 q^{63} + (2 \beta + 1) q^{64} + ( - \beta + 2) q^{65} + ( - \beta - 3) q^{66} + ( - 5 \beta + 4) q^{67} + ( - \beta - 1) q^{68} + q^{69} - 3 \beta q^{70} + (3 \beta - 2) q^{71} + (2 \beta - 1) q^{72} + (9 \beta - 10) q^{73} + ( - 4 \beta - 2) q^{74} - 4 q^{75} - 2 \beta q^{76} + ( - 9 \beta + 6) q^{77} + (\beta - 1) q^{78} + (10 \beta - 5) q^{79} + 3 \beta q^{80} + q^{81} + (9 \beta + 2) q^{82} + ( - 7 \beta + 10) q^{83} + ( - 3 \beta + 3) q^{84} + (2 \beta + 1) q^{85} + (8 \beta - 2) q^{86} + (\beta - 1) q^{87} + ( - \beta + 8) q^{88} + (13 \beta - 6) q^{89} + \beta q^{90} + ( - 3 \beta + 6) q^{91} + (\beta - 1) q^{92} + ( - 7 \beta + 2) q^{93} + ( - 10 \beta - 3) q^{94} + (2 \beta + 2) q^{95} + ( - \beta + 5) q^{96} + ( - 2 \beta - 3) q^{97} - 2 \beta q^{98} + (3 \beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} - 2 q^{5} - q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 2 q^{3} - q^{4} - 2 q^{5} - q^{6} - 6 q^{7} + 2 q^{9} + q^{10} - q^{11} - q^{12} - 3 q^{13} + 3 q^{14} - 2 q^{15} - 3 q^{16} - 4 q^{17} - q^{18} - 6 q^{19} + q^{20} - 6 q^{21} - 7 q^{22} + 2 q^{23} - 8 q^{25} - q^{26} + 2 q^{27} + 3 q^{28} - q^{29} + q^{30} - 3 q^{31} + 9 q^{32} - q^{33} + 7 q^{34} + 6 q^{35} - q^{36} + 6 q^{37} + 8 q^{38} - 3 q^{39} - 16 q^{41} + 3 q^{42} - 18 q^{43} + 8 q^{44} - 2 q^{45} - q^{46} + 17 q^{47} - 3 q^{48} + 4 q^{49} + 4 q^{50} - 4 q^{51} + 4 q^{52} - q^{54} + q^{55} - 6 q^{57} - 2 q^{58} + 2 q^{59} + q^{60} - 16 q^{61} + 19 q^{62} - 6 q^{63} + 4 q^{64} + 3 q^{65} - 7 q^{66} + 3 q^{67} - 3 q^{68} + 2 q^{69} - 3 q^{70} - q^{71} - 11 q^{73} - 8 q^{74} - 8 q^{75} - 2 q^{76} + 3 q^{77} - q^{78} + 3 q^{80} + 2 q^{81} + 13 q^{82} + 13 q^{83} + 3 q^{84} + 4 q^{85} + 4 q^{86} - q^{87} + 15 q^{88} + q^{89} + q^{90} + 9 q^{91} - q^{92} - 3 q^{93} - 16 q^{94} + 6 q^{95} + 9 q^{96} - 8 q^{97} - 2 q^{98} - q^{99}+O(q^{100}) \) 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
−1.61803 1.00000 0.618034 −1.00000 −1.61803 −3.00000 2.23607 1.00000 1.61803
1.2 0.618034 1.00000 −1.61803 −1.00000 0.618034 −3.00000 −2.23607 1.00000 −0.618034
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(157\) \(-1\)

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 471.2.a.b 2
3.b odd 2 1 1413.2.a.b 2
4.b odd 2 1 7536.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.a.b 2 1.a even 1 1 trivial
1413.2.a.b 2 3.b odd 2 1
7536.2.a.m 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(471))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$13$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T - 59 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 16T + 59 \) Copy content Toggle raw display
$43$ \( T^{2} + 18T + 76 \) Copy content Toggle raw display
$47$ \( T^{2} - 17T + 61 \) Copy content Toggle raw display
$53$ \( T^{2} - 20 \) Copy content Toggle raw display
$59$ \( T^{2} - 2T - 124 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 59 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$71$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$73$ \( T^{2} + 11T - 71 \) Copy content Toggle raw display
$79$ \( T^{2} - 125 \) Copy content Toggle raw display
$83$ \( T^{2} - 13T - 19 \) Copy content Toggle raw display
$89$ \( T^{2} - T - 211 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T + 11 \) Copy content Toggle raw display
show more
show less