Properties

Label 8470.2.a.db
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4642000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} - 9x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} - q^{7} + q^{8} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} - q^{7} + q^{8} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{9} + q^{10} - \beta_1 q^{12} + ( - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 2) q^{13} - q^{14} - \beta_1 q^{15} + q^{16} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_1 - 3) q^{17} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{18} + (2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{19} + q^{20} + \beta_1 q^{21} + ( - 2 \beta_{3} - 2 \beta_{2} - 2) q^{23} - \beta_1 q^{24} + q^{25} + ( - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 2) q^{26} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{27} - q^{28} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2) q^{29} - \beta_1 q^{30} + ( - \beta_{4} - 2 \beta_{3}) q^{31} + q^{32} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_1 - 3) q^{34} - q^{35} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{36} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{37} + (2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{38} + ( - 2 \beta_{4} - \beta_{2} + \beta_1 - 5) q^{39} + q^{40} + (2 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 4) q^{41} + \beta_1 q^{42} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{43} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{45} + ( - 2 \beta_{3} - 2 \beta_{2} - 2) q^{46} + (2 \beta_{4} - 2 \beta_1 + 2) q^{47} - \beta_1 q^{48} + q^{49} + q^{50} + (\beta_{5} + \beta_{4} + 6 \beta_1) q^{51} + ( - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 2) q^{52} + ( - \beta_{5} - 3 \beta_{4} - 3) q^{53} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{54} - q^{56} + ( - \beta_{5} - \beta_{4} - 4 \beta_{3} - 3 \beta_{2} - \beta_1 - 3) q^{57} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2) q^{58} + ( - \beta_{5} - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{59} - \beta_1 q^{60} + ( - 4 \beta_{5} - 6 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 5) q^{61} + ( - \beta_{4} - 2 \beta_{3}) q^{62} + ( - \beta_{4} - \beta_{3} - \beta_{2}) q^{63} + q^{64} + ( - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 2) q^{65} + (2 \beta_{5} + \beta_{4} + 3 \beta_{2} - 2 \beta_1 - 2) q^{67} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_1 - 3) q^{68} + (2 \beta_{5} + 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 2) q^{69} - q^{70} + ( - 2 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{71} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{72} + ( - 3 \beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 2) q^{73} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{74} - \beta_1 q^{75} + (2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{76} + ( - 2 \beta_{4} - \beta_{2} + \beta_1 - 5) q^{78} + (6 \beta_{5} + \beta_{3} - \beta_{2} + 5 \beta_1 - 2) q^{79} + q^{80} + (2 \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 3) q^{81} + (2 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 4) q^{82} + (3 \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} - 7) q^{83} + \beta_1 q^{84} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_1 - 3) q^{85} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{86} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} - 7 \beta_{2} + \beta_1 - 6) q^{87} + ( - 3 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} + 4 \beta_{2} - 3 \beta_1) q^{89} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{90} + (\beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 2) q^{91} + ( - 2 \beta_{3} - 2 \beta_{2} - 2) q^{92} + (2 \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 2) q^{93} + (2 \beta_{4} - 2 \beta_1 + 2) q^{94} + (2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{95} - \beta_1 q^{96} + (2 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} - \beta_1 + 1) q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + 6 q^{10} - q^{12} - 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 21 q^{17} - q^{18} + 3 q^{19} + 6 q^{20} + q^{21} - 10 q^{23} - q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} - 6 q^{28} - 10 q^{29} - q^{30} - 4 q^{31} + 6 q^{32} - 21 q^{34} - 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} - 26 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} - 19 q^{43} - q^{45} - 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} - 4 q^{54} - 6 q^{56} - 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} + 8 q^{61} - 4 q^{62} + q^{63} + 6 q^{64} - 6 q^{65} - 27 q^{67} - 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} - q^{72} - 13 q^{73} - 2 q^{74} - q^{75} + 3 q^{76} - 26 q^{78} - 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} - 51 q^{83} + q^{84} - 21 q^{85} - 19 q^{86} - 8 q^{87} + q^{89} - q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} + 10 q^{94} + 3 q^{95} - q^{96} + 7 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} - 9x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 6\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} - \nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} + \nu^{4} + 6\nu^{3} - \nu^{2} - 5\nu - 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + 2\nu^{4} + 5\nu^{3} - 7\nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 6\beta_{4} + 8\beta_{3} + 7\beta_{2} + 4\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{5} + 10\beta_{4} + 19\beta_{3} + 12\beta_{2} + 29\beta _1 + 16 \) 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
2.73934
1.32077
0.803425
−0.0970464
−1.84175
−1.92474
1.00000 −2.73934 1.00000 1.00000 −2.73934 −1.00000 1.00000 4.50401 1.00000
1.2 1.00000 −1.32077 1.00000 1.00000 −1.32077 −1.00000 1.00000 −1.25558 1.00000
1.3 1.00000 −0.803425 1.00000 1.00000 −0.803425 −1.00000 1.00000 −2.35451 1.00000
1.4 1.00000 0.0970464 1.00000 1.00000 0.0970464 −1.00000 1.00000 −2.99058 1.00000
1.5 1.00000 1.84175 1.00000 1.00000 1.84175 −1.00000 1.00000 0.392057 1.00000
1.6 1.00000 1.92474 1.00000 1.00000 1.92474 −1.00000 1.00000 0.704605 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.db 6
11.b odd 2 1 8470.2.a.cv 6
11.c even 5 2 770.2.n.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.g 12 11.c even 5 2
8470.2.a.cv 6 11.b odd 2 1
8470.2.a.db 6 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{6} + T_{3}^{5} - 8T_{3}^{4} - 5T_{3}^{3} + 14T_{3}^{2} + 9T_{3} - 1 \) Copy content Toggle raw display
\( T_{13}^{6} + 6T_{13}^{5} - 28T_{13}^{4} - 214T_{13}^{3} - 68T_{13}^{2} + 1208T_{13} + 1516 \) Copy content Toggle raw display
\( T_{17}^{6} + 21T_{17}^{5} + 138T_{17}^{4} + 151T_{17}^{3} - 1274T_{17}^{2} - 1937T_{17} + 4609 \) Copy content Toggle raw display
\( T_{19}^{6} - 3T_{19}^{5} - 56T_{19}^{4} + 149T_{19}^{3} + 652T_{19}^{2} - 891T_{19} - 941 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} - 8 T^{4} - 5 T^{3} + 14 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( (T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} - 28 T^{4} + \cdots + 1516 \) Copy content Toggle raw display
$17$ \( T^{6} + 21 T^{5} + 138 T^{4} + \cdots + 4609 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} - 56 T^{4} + 149 T^{3} + \cdots - 941 \) Copy content Toggle raw display
$23$ \( T^{6} + 10 T^{5} - 8 T^{4} - 312 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$29$ \( T^{6} + 10 T^{5} - 102 T^{4} + \cdots - 24676 \) Copy content Toggle raw display
$31$ \( T^{6} + 4 T^{5} - 26 T^{4} - 34 T^{3} + \cdots + 164 \) Copy content Toggle raw display
$37$ \( T^{6} + 2 T^{5} - 108 T^{4} + \cdots + 10796 \) Copy content Toggle raw display
$41$ \( T^{6} + 7 T^{5} - 134 T^{4} + \cdots - 15829 \) Copy content Toggle raw display
$43$ \( T^{6} + 19 T^{5} + 78 T^{4} + \cdots + 2979 \) Copy content Toggle raw display
$47$ \( T^{6} - 10 T^{5} - 24 T^{4} + \cdots + 704 \) Copy content Toggle raw display
$53$ \( T^{6} + 16 T^{5} - 42 T^{4} + \cdots + 6764 \) Copy content Toggle raw display
$59$ \( T^{6} + 3 T^{5} - 152 T^{4} + \cdots - 881 \) Copy content Toggle raw display
$61$ \( T^{6} - 8 T^{5} - 404 T^{4} + \cdots - 2650964 \) Copy content Toggle raw display
$67$ \( T^{6} + 27 T^{5} + 172 T^{4} + \cdots - 103099 \) Copy content Toggle raw display
$71$ \( T^{6} - 4 T^{5} - 266 T^{4} + \cdots - 171684 \) Copy content Toggle raw display
$73$ \( T^{6} + 13 T^{5} - 72 T^{4} + \cdots - 5931 \) Copy content Toggle raw display
$79$ \( T^{6} + 14 T^{5} - 286 T^{4} + \cdots + 342676 \) Copy content Toggle raw display
$83$ \( T^{6} + 51 T^{5} + 898 T^{4} + \cdots + 43699 \) Copy content Toggle raw display
$89$ \( T^{6} - T^{5} - 372 T^{4} + \cdots - 89321 \) Copy content Toggle raw display
$97$ \( T^{6} - 7 T^{5} - 242 T^{4} + \cdots + 401 \) Copy content Toggle raw display
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