Properties

Label 8470.2.a.ch
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $3$
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] = [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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_{2} + 1) q^{3} + q^{4} + q^{5} + ( - \beta_{2} - 1) q^{6} - q^{7} - q^{8} + (\beta_{2} - \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta_{2} + 1) q^{3} + q^{4} + q^{5} + ( - \beta_{2} - 1) q^{6} - q^{7} - q^{8} + (\beta_{2} - \beta_1 + 3) q^{9} - q^{10} + (\beta_{2} + 1) q^{12} + q^{14} + (\beta_{2} + 1) q^{15} + q^{16} + ( - \beta_{2} + \beta_1 - 2) q^{17} + ( - \beta_{2} + \beta_1 - 3) q^{18} + (2 \beta_{2} - \beta_1 - 1) q^{19} + q^{20} + ( - \beta_{2} - 1) q^{21} + ( - \beta_1 + 3) q^{23} + ( - \beta_{2} - 1) q^{24} + q^{25} + (2 \beta_{2} - 2 \beta_1 + 4) q^{27} - q^{28} + ( - \beta_1 + 1) q^{29} + ( - \beta_{2} - 1) q^{30} + 2 \beta_1 q^{31} - q^{32} + (\beta_{2} - \beta_1 + 2) q^{34} - q^{35} + (\beta_{2} - \beta_1 + 3) q^{36} + ( - \beta_1 + 3) q^{37} + ( - 2 \beta_{2} + \beta_1 + 1) q^{38} - q^{40} + (3 \beta_{2} + 1) q^{41} + (\beta_{2} + 1) q^{42} + ( - \beta_{2} - \beta_1) q^{43} + (\beta_{2} - \beta_1 + 3) q^{45} + (\beta_1 - 3) q^{46} + (3 \beta_{2} - \beta_1 + 4) q^{47} + (\beta_{2} + 1) q^{48} + q^{49} - q^{50} + ( - 4 \beta_{2} + 2 \beta_1 - 6) q^{51} + (3 \beta_1 - 1) q^{53} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{54} + q^{56} + (\beta_{2} - 3 \beta_1 + 8) q^{57} + (\beta_1 - 1) q^{58} + ( - \beta_{2} + \beta_1 - 4) q^{59} + (\beta_{2} + 1) q^{60} + (3 \beta_{2} - \beta_1 + 2) q^{61} - 2 \beta_1 q^{62} + ( - \beta_{2} + \beta_1 - 3) q^{63} + q^{64} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{67} + ( - \beta_{2} + \beta_1 - 2) q^{68} + (5 \beta_{2} - \beta_1 + 2) q^{69} + q^{70} - 4 \beta_{2} q^{71} + ( - \beta_{2} + \beta_1 - 3) q^{72} + (\beta_{2} + \beta_1 + 4) q^{73} + (\beta_1 - 3) q^{74} + (\beta_{2} + 1) q^{75} + (2 \beta_{2} - \beta_1 - 1) q^{76} + ( - 2 \beta_{2} + 3 \beta_1 - 1) q^{79} + q^{80} + (5 \beta_{2} - \beta_1 + 3) q^{81} + ( - 3 \beta_{2} - 1) q^{82} + (2 \beta_{2} - 2 \beta_1) q^{83} + ( - \beta_{2} - 1) q^{84} + ( - \beta_{2} + \beta_1 - 2) q^{85} + (\beta_{2} + \beta_1) q^{86} + (3 \beta_{2} - \beta_1) q^{87} + (4 \beta_{2} + 6) q^{89} + ( - \beta_{2} + \beta_1 - 3) q^{90} + ( - \beta_1 + 3) q^{92} + ( - 4 \beta_{2} + 2 \beta_1 + 2) q^{93} + ( - 3 \beta_{2} + \beta_1 - 4) q^{94} + (2 \beta_{2} - \beta_1 - 1) q^{95} + ( - \beta_{2} - 1) q^{96} + ( - \beta_1 + 15) q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} - 3 q^{7} - 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} - 3 q^{7} - 3 q^{8} + 7 q^{9} - 3 q^{10} + 2 q^{12} + 3 q^{14} + 2 q^{15} + 3 q^{16} - 4 q^{17} - 7 q^{18} - 6 q^{19} + 3 q^{20} - 2 q^{21} + 8 q^{23} - 2 q^{24} + 3 q^{25} + 8 q^{27} - 3 q^{28} + 2 q^{29} - 2 q^{30} + 2 q^{31} - 3 q^{32} + 4 q^{34} - 3 q^{35} + 7 q^{36} + 8 q^{37} + 6 q^{38} - 3 q^{40} + 2 q^{42} + 7 q^{45} - 8 q^{46} + 8 q^{47} + 2 q^{48} + 3 q^{49} - 3 q^{50} - 12 q^{51} - 8 q^{54} + 3 q^{56} + 20 q^{57} - 2 q^{58} - 10 q^{59} + 2 q^{60} + 2 q^{61} - 2 q^{62} - 7 q^{63} + 3 q^{64} - 14 q^{67} - 4 q^{68} + 3 q^{70} + 4 q^{71} - 7 q^{72} + 12 q^{73} - 8 q^{74} + 2 q^{75} - 6 q^{76} + 2 q^{79} + 3 q^{80} + 3 q^{81} - 4 q^{83} - 2 q^{84} - 4 q^{85} - 4 q^{87} + 14 q^{89} - 7 q^{90} + 8 q^{92} + 12 q^{93} - 8 q^{94} - 6 q^{95} - 2 q^{96} + 44 q^{97} - 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 6 ) / 2 \) 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
0.470683
2.34292
−1.81361
−1.00000 −2.24914 1.00000 1.00000 2.24914 −1.00000 −1.00000 2.05863 −1.00000
1.2 −1.00000 1.14637 1.00000 1.00000 −1.14637 −1.00000 −1.00000 −1.68585 −1.00000
1.3 −1.00000 3.10278 1.00000 1.00000 −3.10278 −1.00000 −1.00000 6.62721 −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.ch 3
11.b odd 2 1 8470.2.a.cn yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.ch 3 1.a even 1 1 trivial
8470.2.a.cn yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{3} - 2T_{3}^{2} - 6T_{3} + 8 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17}^{3} + 4T_{17}^{2} - 12T_{17} - 16 \) Copy content Toggle raw display
\( T_{19}^{3} + 6T_{19}^{2} - 22T_{19} - 136 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 2 T^{2} - 6 T + 8 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} - 12 T - 16 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} - 22 T - 136 \) Copy content Toggle raw display
$23$ \( T^{3} - 8 T^{2} + 6 T + 44 \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} - 14 T + 32 \) Copy content Toggle raw display
$31$ \( T^{3} - 2 T^{2} - 60 T - 136 \) Copy content Toggle raw display
$37$ \( T^{3} - 8 T^{2} + 6 T + 44 \) Copy content Toggle raw display
$41$ \( T^{3} - 66T + 92 \) Copy content Toggle raw display
$43$ \( T^{3} - 28T - 16 \) Copy content Toggle raw display
$47$ \( T^{3} - 8 T^{2} - 44 T - 16 \) Copy content Toggle raw display
$53$ \( T^{3} - 138T - 596 \) Copy content Toggle raw display
$59$ \( T^{3} + 10 T^{2} + 16 T - 16 \) Copy content Toggle raw display
$61$ \( T^{3} - 2 T^{2} - 64 T - 128 \) Copy content Toggle raw display
$67$ \( T^{3} + 14 T^{2} - 4 T - 184 \) Copy content Toggle raw display
$71$ \( T^{3} - 4 T^{2} - 112 T - 64 \) Copy content Toggle raw display
$73$ \( T^{3} - 12 T^{2} + 20 T + 64 \) Copy content Toggle raw display
$79$ \( T^{3} - 2 T^{2} - 134 T - 184 \) Copy content Toggle raw display
$83$ \( T^{3} + 4 T^{2} - 64 T - 128 \) Copy content Toggle raw display
$89$ \( T^{3} - 14 T^{2} - 52 T + 664 \) Copy content Toggle raw display
$97$ \( T^{3} - 44 T^{2} + 630 T - 2908 \) Copy content Toggle raw display
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