Properties

Label 8470.2.a.bn
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
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] = [8470,2,Mod(1,8470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8470, 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("8470.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.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: \(67.6332905120\)
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: \( 2 \)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta q^{3} + q^{4} - q^{5} + \beta q^{6} + q^{7} - q^{8} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta q^{3} + q^{4} - q^{5} + \beta q^{6} + q^{7} - q^{8} + 2 q^{9} + q^{10} - \beta q^{12} + q^{13} - q^{14} + \beta q^{15} + q^{16} + ( - \beta + 1) q^{17} - 2 q^{18} - \beta q^{19} - q^{20} - \beta q^{21} + (\beta - 6) q^{23} + \beta q^{24} + q^{25} - q^{26} + \beta q^{27} + q^{28} + 2 \beta q^{29} - \beta q^{30} + ( - \beta - 3) q^{31} - q^{32} + (\beta - 1) q^{34} - q^{35} + 2 q^{36} + 4 \beta q^{37} + \beta q^{38} - \beta q^{39} + q^{40} + ( - \beta - 5) q^{41} + \beta q^{42} + (\beta + 1) q^{43} - 2 q^{45} + ( - \beta + 6) q^{46} + (3 \beta + 1) q^{47} - \beta q^{48} + q^{49} - q^{50} + ( - \beta + 5) q^{51} + q^{52} + ( - 3 \beta + 1) q^{53} - \beta q^{54} - q^{56} + 5 q^{57} - 2 \beta q^{58} + (3 \beta - 2) q^{59} + \beta q^{60} + (2 \beta - 8) q^{61} + (\beta + 3) q^{62} + 2 q^{63} + q^{64} - q^{65} + ( - \beta - 5) q^{67} + ( - \beta + 1) q^{68} + (6 \beta - 5) q^{69} + q^{70} + (2 \beta - 6) q^{71} - 2 q^{72} + ( - \beta - 5) q^{73} - 4 \beta q^{74} - \beta q^{75} - \beta q^{76} + \beta q^{78} + (\beta + 8) q^{79} - q^{80} - 11 q^{81} + (\beta + 5) q^{82} + ( - \beta + 2) q^{83} - \beta q^{84} + (\beta - 1) q^{85} + ( - \beta - 1) q^{86} - 10 q^{87} + ( - \beta - 15) q^{89} + 2 q^{90} + q^{91} + (\beta - 6) q^{92} + (3 \beta + 5) q^{93} + ( - 3 \beta - 1) q^{94} + \beta q^{95} + \beta q^{96} + ( - 2 \beta + 8) q^{97} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 4 q^{9} + 2 q^{10} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{18} - 2 q^{20} - 12 q^{23} + 2 q^{25} - 2 q^{26} + 2 q^{28} - 6 q^{31} - 2 q^{32} - 2 q^{34} - 2 q^{35} + 4 q^{36} + 2 q^{40} - 10 q^{41} + 2 q^{43} - 4 q^{45} + 12 q^{46} + 2 q^{47} + 2 q^{49} - 2 q^{50} + 10 q^{51} + 2 q^{52} + 2 q^{53} - 2 q^{56} + 10 q^{57} - 4 q^{59} - 16 q^{61} + 6 q^{62} + 4 q^{63} + 2 q^{64} - 2 q^{65} - 10 q^{67} + 2 q^{68} - 10 q^{69} + 2 q^{70} - 12 q^{71} - 4 q^{72} - 10 q^{73} + 16 q^{79} - 2 q^{80} - 22 q^{81} + 10 q^{82} + 4 q^{83} - 2 q^{85} - 2 q^{86} - 20 q^{87} - 30 q^{89} + 4 q^{90} + 2 q^{91} - 12 q^{92} + 10 q^{93} - 2 q^{94} + 16 q^{97} - 2 q^{98}+O(q^{100}) \) 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
−1.00000 −2.23607 1.00000 −1.00000 2.23607 1.00000 −1.00000 2.00000 1.00000
1.2 −1.00000 2.23607 1.00000 −1.00000 −2.23607 1.00000 −1.00000 2.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 8470.2.a.bn 2
11.b odd 2 1 8470.2.a.by yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.bn 2 1.a even 1 1 trivial
8470.2.a.by yes 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{2} - 5 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} - 4 \) Copy content Toggle raw display
\( T_{19}^{2} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 5 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 5 \) Copy content Toggle raw display
$23$ \( T^{2} + 12T + 31 \) Copy content Toggle raw display
$29$ \( T^{2} - 20 \) Copy content Toggle raw display
$31$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 80 \) Copy content Toggle raw display
$41$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 59 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$89$ \( T^{2} + 30T + 220 \) Copy content Toggle raw display
$97$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
show more
show less