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Properties

Label 8046.2.a.h
Level $8046$
Weight $2$
Character orbit 8046.a
Self dual yes
Analytic conductor $64.248$
Analytic rank $1$
Dimension $9$
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] = [8046,2,Mod(1,8046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8046.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: \(64.2476334663\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 9x^{7} + 25x^{6} + 29x^{5} - 58x^{4} - 43x^{3} + 34x^{2} + 25x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( - \beta_{7} + \beta_{5} - \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + ( - \beta_{7} + \beta_{5} - \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{7} + q^{8} + ( - \beta_{7} + \beta_{5} - \beta_1) q^{10} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots - 1) q^{11}+ \cdots + (2 \beta_{8} - \beta_{7} + 3 \beta_{6} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} - 4 q^{14} + 9 q^{16} - q^{17} - 10 q^{19} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 3 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{29} - 17 q^{31} + 9 q^{32} - q^{34} - 10 q^{35} - 11 q^{37} - 10 q^{38} - 4 q^{40} - 16 q^{43} - 4 q^{44} - 8 q^{46} - 7 q^{47} - 5 q^{49} - 3 q^{50} - 8 q^{52} - 12 q^{53} - 23 q^{55} - 4 q^{56} - 4 q^{58} - 6 q^{59} - 13 q^{61} - 17 q^{62} + 9 q^{64} + 24 q^{65} - 14 q^{67} - q^{68} - 10 q^{70} - 30 q^{71} - 12 q^{73} - 11 q^{74} - 10 q^{76} - 12 q^{77} - 35 q^{79} - 4 q^{80} - 5 q^{83} - 27 q^{85} - 16 q^{86} - 4 q^{88} - 23 q^{89} - 28 q^{91} - 8 q^{92} - 7 q^{94} - 32 q^{95} - 21 q^{97} - 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 9x^{7} + 25x^{6} + 29x^{5} - 58x^{4} - 43x^{3} + 34x^{2} + 25x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\nu^{8} - 69\nu^{7} - 40\nu^{6} + 516\nu^{5} - 329\nu^{4} - 975\nu^{3} + 992\nu^{2} + 355\nu - 347 ) / 77 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -19\nu^{8} + 72\nu^{7} + 102\nu^{6} - 515\nu^{5} - 35\nu^{4} + 850\nu^{3} - 235\nu^{2} - 116\nu + 34 ) / 77 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -24\nu^{8} + 95\nu^{7} + 141\nu^{6} - 764\nu^{5} - 105\nu^{4} + 1714\nu^{3} - 232\nu^{2} - 1030\nu - 30 ) / 77 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24\nu^{8} - 95\nu^{7} - 141\nu^{6} + 764\nu^{5} + 105\nu^{4} - 1637\nu^{3} + 155\nu^{2} + 722\nu + 107 ) / 77 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -32\nu^{8} + 101\nu^{7} + 265\nu^{6} - 839\nu^{5} - 679\nu^{4} + 1926\nu^{3} + 589\nu^{2} - 1168\nu - 194 ) / 77 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 47\nu^{8} - 170\nu^{7} - 305\nu^{6} + 1355\nu^{5} + 427\nu^{4} - 2978\nu^{3} - 59\nu^{2} + 1600\nu + 232 ) / 77 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + 7\beta_{2} + 10\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{8} + \beta_{7} + 8\beta_{6} + 9\beta_{5} + 2\beta_{4} + 11\beta_{2} + 33\beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{8} + 10\beta_{7} + 10\beta_{6} + 16\beta_{5} + 3\beta_{4} - 6\beta_{3} + 48\beta_{2} + 82\beta _1 + 94 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 32\beta_{8} + 13\beta_{7} + 51\beta_{6} + 80\beta_{5} + 23\beta_{4} + 3\beta_{3} + 95\beta_{2} + 240\beta _1 + 185 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 135 \beta_{8} + 74 \beta_{7} + 73 \beta_{6} + 188 \beta_{5} + 45 \beta_{4} - 19 \beta_{3} + 339 \beta_{2} + \cdots + 663 \) 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
1.05829
2.58022
−2.13687
−0.836154
−0.584549
1.67390
2.89499
−0.0929510
−1.55687
1.00000 0 1.00000 −3.80777 0 0.515673 1.00000 0 −3.80777
1.2 1.00000 0 1.00000 −2.72526 0 −2.79628 1.00000 0 −2.72526
1.3 1.00000 0 1.00000 −1.65274 0 0.0358535 1.00000 0 −1.65274
1.4 1.00000 0 1.00000 −1.63508 0 4.18187 1.00000 0 −1.63508
1.5 1.00000 0 1.00000 −0.683695 0 −0.639069 1.00000 0 −0.683695
1.6 1.00000 0 1.00000 0.102080 0 0.350570 1.00000 0 0.102080
1.7 1.00000 0 1.00000 1.72525 0 −2.60583 1.00000 0 1.72525
1.8 1.00000 0 1.00000 1.96687 0 −4.72097 1.00000 0 1.96687
1.9 1.00000 0 1.00000 2.71034 0 1.67819 1.00000 0 2.71034
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(149\) \(1\)

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 8046.2.a.h yes 9
3.b odd 2 1 8046.2.a.g ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.g ✓ 9 3.b odd 2 1
8046.2.a.h yes 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\( T_{5}^{9} + 4T_{5}^{8} - 13T_{5}^{7} - 56T_{5}^{6} + 43T_{5}^{5} + 240T_{5}^{4} + 9T_{5}^{3} - 330T_{5}^{2} - 143T_{5} + 18 \) Copy content Toggle raw display
\( T_{11}^{9} + 4 T_{11}^{8} - 31 T_{11}^{7} - 68 T_{11}^{6} + 338 T_{11}^{5} + 137 T_{11}^{4} - 871 T_{11}^{3} + \cdots + 114 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 4 T^{8} + \cdots + 18 \) Copy content Toggle raw display
$7$ \( T^{9} + 4 T^{8} + \cdots - 1 \) Copy content Toggle raw display
$11$ \( T^{9} + 4 T^{8} + \cdots + 114 \) Copy content Toggle raw display
$13$ \( T^{9} + 8 T^{8} + \cdots - 14 \) Copy content Toggle raw display
$17$ \( T^{9} + T^{8} + \cdots + 12262 \) Copy content Toggle raw display
$19$ \( T^{9} + 10 T^{8} + \cdots + 8488 \) Copy content Toggle raw display
$23$ \( T^{9} + 8 T^{8} + \cdots - 189 \) Copy content Toggle raw display
$29$ \( T^{9} + 4 T^{8} + \cdots + 77797 \) Copy content Toggle raw display
$31$ \( T^{9} + 17 T^{8} + \cdots + 136549 \) Copy content Toggle raw display
$37$ \( T^{9} + 11 T^{8} + \cdots - 526824 \) Copy content Toggle raw display
$41$ \( T^{9} - 150 T^{7} + \cdots + 64749 \) Copy content Toggle raw display
$43$ \( T^{9} + 16 T^{8} + \cdots - 30068 \) Copy content Toggle raw display
$47$ \( T^{9} + 7 T^{8} + \cdots - 1580502 \) Copy content Toggle raw display
$53$ \( T^{9} + 12 T^{8} + \cdots + 3614263 \) Copy content Toggle raw display
$59$ \( T^{9} + 6 T^{8} + \cdots - 1179508 \) Copy content Toggle raw display
$61$ \( T^{9} + 13 T^{8} + \cdots - 564912 \) Copy content Toggle raw display
$67$ \( T^{9} + 14 T^{8} + \cdots + 3071922 \) Copy content Toggle raw display
$71$ \( T^{9} + 30 T^{8} + \cdots - 13946499 \) Copy content Toggle raw display
$73$ \( T^{9} + 12 T^{8} + \cdots - 1807038 \) Copy content Toggle raw display
$79$ \( T^{9} + 35 T^{8} + \cdots - 16545346 \) Copy content Toggle raw display
$83$ \( T^{9} + 5 T^{8} + \cdots + 4270212 \) Copy content Toggle raw display
$89$ \( T^{9} + 23 T^{8} + \cdots + 21 \) Copy content Toggle raw display
$97$ \( T^{9} + 21 T^{8} + \cdots + 5292 \) Copy content Toggle raw display
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