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Properties

Label 8470.2.a.dh
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 16x^{6} + 69x^{4} - 10x^{3} - 70x^{2} + 10x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} - q^{7} + q^{8} + ( - \beta_{6} + \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} - q^{7} + q^{8} + ( - \beta_{6} + \beta_{5} + 1) q^{9} - q^{10} + \beta_1 q^{12} + (\beta_{7} + \beta_{6} + \beta_{3} + \cdots + \beta_1) q^{13}+ \cdots + q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} - 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} - 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{9} - 8 q^{10} - q^{13} - 8 q^{14} + 8 q^{16} + 6 q^{17} + 8 q^{18} + 5 q^{19} - 8 q^{20} + 10 q^{23} + 8 q^{25} - q^{26} - 8 q^{28} + 3 q^{29} - 8 q^{31} + 8 q^{32} + 6 q^{34} + 8 q^{35} + 8 q^{36} - 6 q^{37} + 5 q^{38} + 35 q^{39} - 8 q^{40} + 11 q^{41} - 5 q^{43} - 8 q^{45} + 10 q^{46} - 15 q^{47} + 8 q^{49} + 8 q^{50} - 6 q^{51} - q^{52} - 16 q^{53} - 8 q^{56} + 38 q^{57} + 3 q^{58} - 9 q^{59} + 32 q^{61} - 8 q^{62} - 8 q^{63} + 8 q^{64} + q^{65} + 33 q^{67} + 6 q^{68} - 22 q^{69} + 8 q^{70} + 11 q^{71} + 8 q^{72} - 34 q^{73} - 6 q^{74} + 5 q^{76} + 35 q^{78} + 31 q^{79} - 8 q^{80} + 20 q^{81} + 11 q^{82} + 50 q^{83} - 6 q^{85} - 5 q^{86} - 12 q^{87} + q^{89} - 8 q^{90} + q^{91} + 10 q^{92} + 26 q^{93} - 15 q^{94} - 5 q^{95} - 4 q^{97} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 16x^{6} + 69x^{4} - 10x^{3} - 70x^{2} + 10x + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{6} + \nu^{5} + 34\nu^{4} - 3\nu^{3} - 70\nu^{2} + 17\nu + 29 ) / 34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 19\nu^{5} - \nu^{4} - 103\nu^{3} + 13\nu^{2} + 140\nu - 27 ) / 34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} - 5\nu^{6} - 63\nu^{5} + 72\nu^{4} + 271\nu^{3} - 282\nu^{2} - 237\nu + 111 ) / 34 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - \nu^{6} + 46\nu^{5} + 14\nu^{4} - 174\nu^{3} - 7\nu^{2} + 97\nu - 26 ) / 17 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} - \nu^{6} + 46\nu^{5} + 14\nu^{4} - 174\nu^{3} - 24\nu^{2} + 97\nu + 42 ) / 17 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} + 6\nu^{6} - 139\nu^{5} - 93\nu^{4} + 559\nu^{3} + 261\nu^{2} - 512\nu - 87 ) / 34 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{6} + 9\beta_{5} + 2\beta_{4} + 2\beta_{3} - 4\beta_{2} + 2\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{7} + 10\beta_{6} + 13\beta_{5} + 13\beta_{4} + 22\beta_{3} - 13\beta_{2} + 56\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{7} - 65\beta_{6} + 82\beta_{5} + 26\beta_{4} + 29\beta_{3} - 60\beta_{2} + 40\beta _1 + 283 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 125\beta_{7} + 82\beta_{6} + 148\beta_{5} + 142\beta_{4} + 279\beta_{3} - 140\beta_{2} + 481\beta _1 + 221 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−2.90474
−2.53932
−1.12455
−0.211079
0.365778
1.22201
2.00431
3.18761
1.00000 −2.90474 1.00000 −1.00000 −2.90474 −1.00000 1.00000 5.43751 −1.00000
1.2 1.00000 −2.53932 1.00000 −1.00000 −2.53932 −1.00000 1.00000 3.44815 −1.00000
1.3 1.00000 −1.12455 1.00000 −1.00000 −1.12455 −1.00000 1.00000 −1.73538 −1.00000
1.4 1.00000 −0.211079 1.00000 −1.00000 −0.211079 −1.00000 1.00000 −2.95545 −1.00000
1.5 1.00000 0.365778 1.00000 −1.00000 0.365778 −1.00000 1.00000 −2.86621 −1.00000
1.6 1.00000 1.22201 1.00000 −1.00000 1.22201 −1.00000 1.00000 −1.50670 −1.00000
1.7 1.00000 2.00431 1.00000 −1.00000 2.00431 −1.00000 1.00000 1.01724 −1.00000
1.8 1.00000 3.18761 1.00000 −1.00000 3.18761 −1.00000 1.00000 7.16083 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.dh 8
11.b odd 2 1 8470.2.a.dg 8
11.c even 5 2 770.2.n.k ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.k ✓ 16 11.c even 5 2
8470.2.a.dg 8 11.b odd 2 1
8470.2.a.dh 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3}^{8} - 16T_{3}^{6} + 69T_{3}^{4} - 10T_{3}^{3} - 70T_{3}^{2} + 10T_{3} + 5 \) Copy content Toggle raw display
\( T_{13}^{8} + T_{13}^{7} - 87T_{13}^{6} - 140T_{13}^{5} + 2216T_{13}^{4} + 4850T_{13}^{3} - 14908T_{13}^{2} - 39548T_{13} - 12764 \) Copy content Toggle raw display
\( T_{17}^{8} - 6T_{17}^{7} - 62T_{17}^{6} + 350T_{17}^{5} + 749T_{17}^{4} - 3264T_{17}^{3} - 2152T_{17}^{2} + 556T_{17} + 109 \) Copy content Toggle raw display
\( T_{19}^{8} - 5 T_{19}^{7} - 106 T_{19}^{6} + 485 T_{19}^{5} + 3534 T_{19}^{4} - 15235 T_{19}^{3} + \cdots - 70900 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 16 T^{6} + \cdots + 5 \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + T^{7} + \cdots - 12764 \) Copy content Toggle raw display
$17$ \( T^{8} - 6 T^{7} + \cdots + 109 \) Copy content Toggle raw display
$19$ \( T^{8} - 5 T^{7} + \cdots - 70900 \) Copy content Toggle raw display
$23$ \( T^{8} - 10 T^{7} + \cdots + 1280 \) Copy content Toggle raw display
$29$ \( T^{8} - 3 T^{7} + \cdots - 16636 \) Copy content Toggle raw display
$31$ \( T^{8} + 8 T^{7} + \cdots + 135344 \) Copy content Toggle raw display
$37$ \( T^{8} + 6 T^{7} + \cdots + 3280 \) Copy content Toggle raw display
$41$ \( T^{8} - 11 T^{7} + \cdots + 21296 \) Copy content Toggle raw display
$43$ \( T^{8} + 5 T^{7} + \cdots - 591484 \) Copy content Toggle raw display
$47$ \( T^{8} + 15 T^{7} + \cdots + 614336 \) Copy content Toggle raw display
$53$ \( T^{8} + 16 T^{7} + \cdots + 10229824 \) Copy content Toggle raw display
$59$ \( T^{8} + 9 T^{7} + \cdots + 10961956 \) Copy content Toggle raw display
$61$ \( T^{8} - 32 T^{7} + \cdots + 58256 \) Copy content Toggle raw display
$67$ \( T^{8} - 33 T^{7} + \cdots + 162964 \) Copy content Toggle raw display
$71$ \( T^{8} - 11 T^{7} + \cdots + 44 \) Copy content Toggle raw display
$73$ \( T^{8} + 34 T^{7} + \cdots - 7365731 \) Copy content Toggle raw display
$79$ \( T^{8} - 31 T^{7} + \cdots + 15119876 \) Copy content Toggle raw display
$83$ \( T^{8} - 50 T^{7} + \cdots - 13602031 \) Copy content Toggle raw display
$89$ \( T^{8} - T^{7} + \cdots + 108020 \) Copy content Toggle raw display
$97$ \( T^{8} + 4 T^{7} + \cdots - 968759 \) Copy content Toggle raw display
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