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Properties

Label 8670.2.a.cl
Level $8670$
Weight $2$
Character orbit 8670.a
Self dual yes
Analytic conductor $69.230$
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] = [8670,2,Mod(1,8670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8670, 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("8670.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8670.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: \(69.2302985525\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.75178704896.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 16x^{6} - 8x^{5} + 72x^{4} + 48x^{3} - 104x^{2} - 72x + 17 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 510)
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + ( - \beta_{3} - \beta_1) q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + ( - \beta_{3} - \beta_1) q^{7} + q^{8} + q^{9} + q^{10} + ( - \beta_{6} + \beta_{2} + 1) q^{11} - q^{12} + (\beta_{7} - \beta_{4} + 1) q^{13} + ( - \beta_{3} - \beta_1) q^{14} - q^{15} + q^{16} + q^{18} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots + 2) q^{19}+ \cdots + ( - \beta_{6} + \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 8 q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 8 q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + 8 q^{10} + 8 q^{11} - 8 q^{12} + 8 q^{13} - 8 q^{15} + 8 q^{16} + 8 q^{18} + 16 q^{19} + 8 q^{20} + 8 q^{22} - 8 q^{24} + 8 q^{25} + 8 q^{26} - 8 q^{27} - 8 q^{29} - 8 q^{30} + 8 q^{31} + 8 q^{32} - 8 q^{33} + 8 q^{36} + 16 q^{38} - 8 q^{39} + 8 q^{40} + 16 q^{43} + 8 q^{44} + 8 q^{45} + 32 q^{47} - 8 q^{48} + 24 q^{49} + 8 q^{50} + 8 q^{52} + 32 q^{53} - 8 q^{54} + 8 q^{55} - 16 q^{57} - 8 q^{58} + 24 q^{59} - 8 q^{60} - 8 q^{61} + 8 q^{62} + 8 q^{64} + 8 q^{65} - 8 q^{66} + 16 q^{67} + 8 q^{72} - 32 q^{73} - 8 q^{75} + 16 q^{76} + 32 q^{77} - 8 q^{78} - 24 q^{79} + 8 q^{80} + 8 q^{81} + 32 q^{83} + 16 q^{86} + 8 q^{87} + 8 q^{88} + 8 q^{90} - 24 q^{91} - 8 q^{93} + 32 q^{94} + 16 q^{95} - 8 q^{96} + 24 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 16x^{6} - 8x^{5} + 72x^{4} + 48x^{3} - 104x^{2} - 72x + 17 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -7\nu^{7} - 12\nu^{6} + 174\nu^{5} + 24\nu^{4} - 917\nu^{3} + 404\nu^{2} + 1338\nu - 340 ) / 289 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -18\nu^{7} + 93\nu^{6} + 241\nu^{5} - 1053\nu^{4} - 1491\nu^{3} + 2649\nu^{2} + 2780\nu - 833 ) / 289 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -20\nu^{7} + 7\nu^{6} + 332\nu^{5} - 14\nu^{4} - 1464\nu^{3} - 43\nu^{2} + 1676\nu + 102 ) / 289 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 39\nu^{7} - 57\nu^{6} - 474\nu^{5} + 403\nu^{4} + 1352\nu^{3} - 971\nu^{2} - 725\nu + 697 ) / 289 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43\nu^{7} + 115\nu^{6} - 656\nu^{5} - 1675\nu^{4} + 1587\nu^{3} + 3835\nu^{2} + 38\nu + 272 ) / 289 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -131\nu^{7} + 147\nu^{6} + 1770\nu^{5} - 872\nu^{4} - 6468\nu^{3} + 1409\nu^{2} + 7163\nu + 119 ) / 289 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 163\nu^{7} - 216\nu^{6} - 2070\nu^{5} + 1299\nu^{4} + 6903\nu^{3} - 1976\nu^{2} - 5972\nu + 238 ) / 289 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{5} - 2\beta_{4} + \beta_{3} + 2\beta_{2} - 2\beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{7} + 5\beta_{6} - 11\beta_{4} + 6\beta_{3} - 9\beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{7} + 2\beta_{6} - 4\beta_{5} - 12\beta_{4} + 10\beta_{3} + 8\beta_{2} - 15\beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 67\beta_{7} + 37\beta_{6} - 6\beta_{5} - 115\beta_{4} + 90\beta_{3} + 14\beta_{2} - 103\beta _1 + 100 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 181\beta_{7} + 68\beta_{6} - 69\beta_{5} - 310\beta_{4} + 275\beta_{3} + 146\beta_{2} - 370\beta _1 + 524 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 735\beta_{7} + 353\beta_{6} - 116\beta_{5} - 1275\beta_{4} + 1106\beta_{3} + 268\beta_{2} - 1239\beta _1 + 1358 ) / 2 \) 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
−2.03762
−1.61909
1.81937
0.189865
1.67373
−1.05400
3.46685
−2.43910
1.00000 −1.00000 1.00000 1.00000 −1.00000 −4.29585 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 −3.70395 1.00000 1.00000 1.00000
1.3 1.00000 −1.00000 1.00000 1.00000 −1.00000 −1.15876 1.00000 1.00000 1.00000
1.4 1.00000 −1.00000 1.00000 1.00000 −1.00000 −1.14570 1.00000 1.00000 1.00000
1.5 1.00000 −1.00000 1.00000 1.00000 −1.00000 −0.952799 1.00000 1.00000 1.00000
1.6 1.00000 −1.00000 1.00000 1.00000 −1.00000 2.90479 1.00000 1.00000 1.00000
1.7 1.00000 −1.00000 1.00000 1.00000 −1.00000 3.48865 1.00000 1.00000 1.00000
1.8 1.00000 −1.00000 1.00000 1.00000 −1.00000 4.86362 1.00000 1.00000 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
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(17\) \(-1\)

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 8670.2.a.cl 8
17.b even 2 1 8670.2.a.cm 8
17.e odd 16 2 510.2.u.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.u.d ✓ 16 17.e odd 16 2
8670.2.a.cl 8 1.a even 1 1 trivial
8670.2.a.cm 8 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8670))\):

\( T_{7}^{8} - 40T_{7}^{6} - 16T_{7}^{5} + 472T_{7}^{4} + 384T_{7}^{3} - 1504T_{7}^{2} - 2432T_{7} - 992 \) Copy content Toggle raw display
\( T_{11}^{8} - 8T_{11}^{7} - 32T_{11}^{6} + 400T_{11}^{5} - 360T_{11}^{4} - 4576T_{11}^{3} + 13184T_{11}^{2} - 10048T_{11} + 1552 \) Copy content Toggle raw display
\( T_{13}^{8} - 8T_{13}^{7} - 32T_{13}^{6} + 384T_{13}^{5} - 188T_{13}^{4} - 4624T_{13}^{3} + 8320T_{13}^{2} + 8064T_{13} - 18428 \) Copy content Toggle raw display
\( T_{23}^{8} - 120T_{23}^{6} - 208T_{23}^{5} + 3924T_{23}^{4} + 10688T_{23}^{3} - 23408T_{23}^{2} - 66208T_{23} - 29308 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 40 T^{6} + \cdots - 992 \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + \cdots + 1552 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + \cdots - 18428 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} - 16 T^{7} + \cdots - 15872 \) Copy content Toggle raw display
$23$ \( T^{8} - 120 T^{6} + \cdots - 29308 \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + \cdots - 7408 \) Copy content Toggle raw display
$31$ \( T^{8} - 8 T^{7} + \cdots - 8636 \) Copy content Toggle raw display
$37$ \( T^{8} - 160 T^{6} + \cdots + 115712 \) Copy content Toggle raw display
$41$ \( T^{8} - 120 T^{6} + \cdots + 32 \) Copy content Toggle raw display
$43$ \( T^{8} - 16 T^{7} + \cdots - 140284 \) Copy content Toggle raw display
$47$ \( T^{8} - 32 T^{7} + \cdots + 4352 \) Copy content Toggle raw display
$53$ \( T^{8} - 32 T^{7} + \cdots + 545024 \) Copy content Toggle raw display
$59$ \( T^{8} - 24 T^{7} + \cdots - 5971964 \) Copy content Toggle raw display
$61$ \( T^{8} + 8 T^{7} + \cdots + 129152 \) Copy content Toggle raw display
$67$ \( T^{8} - 16 T^{7} + \cdots - 69884 \) Copy content Toggle raw display
$71$ \( T^{8} - 88 T^{6} + \cdots + 32 \) Copy content Toggle raw display
$73$ \( T^{8} + 32 T^{7} + \cdots - 62432 \) Copy content Toggle raw display
$79$ \( T^{8} + 24 T^{7} + \cdots - 7292 \) Copy content Toggle raw display
$83$ \( T^{8} - 32 T^{7} + \cdots - 4587008 \) Copy content Toggle raw display
$89$ \( T^{8} - 272 T^{6} + \cdots + 1057024 \) Copy content Toggle raw display
$97$ \( T^{8} - 328 T^{6} + \cdots + 8991776 \) Copy content Toggle raw display
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