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Properties

Label 8046.2.a.f
Level $8046$
Weight $2$
Character orbit 8046.a
Self dual yes
Analytic conductor $64.248$
Analytic rank $1$
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] = [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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 17x^{6} - 2x^{5} + 71x^{4} - 18x^{3} - 81x^{2} + 36x + 9 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - \beta_1 q^{5} + ( - \beta_{5} - 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - \beta_1 q^{5} + ( - \beta_{5} - 1) q^{7} + q^{8} - \beta_1 q^{10} + (\beta_{6} - 1) q^{11} + (\beta_{5} - \beta_{4} + \beta_1 - 1) q^{13} + ( - \beta_{5} - 1) q^{14} + q^{16} + ( - \beta_{3} + \beta_1 + 1) q^{17} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1) q^{19}+ \cdots + ( - \beta_{6} + \beta_{4} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8} - 6 q^{11} - 8 q^{13} - 5 q^{14} + 8 q^{16} + 5 q^{17} - 14 q^{19} - 6 q^{22} - 21 q^{23} - 6 q^{25} - 8 q^{26} - 5 q^{28} - 3 q^{29} - 4 q^{31} + 8 q^{32} + 5 q^{34} + 2 q^{35} - 3 q^{37} - 14 q^{38} - 7 q^{41} - 12 q^{43} - 6 q^{44} - 21 q^{46} - 25 q^{47} - 7 q^{49} - 6 q^{50} - 8 q^{52} - 3 q^{53} - 9 q^{55} - 5 q^{56} - 3 q^{58} - 2 q^{59} - 17 q^{61} - 4 q^{62} + 8 q^{64} - 32 q^{65} - 14 q^{67} + 5 q^{68} + 2 q^{70} - 7 q^{71} - 10 q^{73} - 3 q^{74} - 14 q^{76} - 12 q^{77} - 33 q^{79} - 7 q^{82} - 13 q^{83} - 33 q^{85} - 12 q^{86} - 6 q^{88} + 22 q^{89} - 22 q^{91} - 21 q^{92} - 25 q^{94} + 14 q^{95} - 11 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 17x^{6} - 2x^{5} + 71x^{4} - 18x^{3} - 81x^{2} + 36x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 17\nu^{5} + 5\nu^{4} - 71\nu^{3} - 18\nu^{2} + 63\nu + 12 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 2\nu^{6} - 14\nu^{5} - 33\nu^{4} + 25\nu^{3} + 64\nu^{2} - 21\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} + 2\nu^{6} - 31\nu^{5} - 35\nu^{4} + 93\nu^{3} + 55\nu^{2} - 75\nu - 18 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} - 2\nu^{6} + 31\nu^{5} + 35\nu^{4} - 93\nu^{3} - 49\nu^{2} + 75\nu - 12 ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{7} - \nu^{6} + 34\nu^{5} + 21\nu^{4} - 137\nu^{3} - 35\nu^{2} + 144\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 6\nu^{6} - 79\nu^{5} - 103\nu^{4} + 247\nu^{3} + 192\nu^{2} - 213\nu - 48 ) / 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} + 9\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - 2\beta_{6} + 11\beta_{5} + 13\beta_{4} + 3\beta_{2} + 6\beta _1 + 46 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{7} - 24\beta_{6} + 30\beta_{5} + 34\beta_{4} + 26\beta_{3} + 12\beta_{2} + 94\beta _1 + 53 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{7} - 38\beta_{6} + 132\beta_{5} + 154\beta_{4} + 10\beta_{3} + 50\beta_{2} + 129\beta _1 + 494 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -172\beta_{7} - 276\beta_{6} + 405\beta_{5} + 483\beta_{4} + 300\beta_{3} + 142\beta_{2} + 1052\beta _1 + 911 \) 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
3.58447
1.40004
1.17278
0.854224
−0.181211
−1.29304
−2.60917
−2.92810
1.00000 0 1.00000 −3.58447 0 −4.00108 1.00000 0 −3.58447
1.2 1.00000 0 1.00000 −1.40004 0 −2.13611 1.00000 0 −1.40004
1.3 1.00000 0 1.00000 −1.17278 0 1.96150 1.00000 0 −1.17278
1.4 1.00000 0 1.00000 −0.854224 0 0.727081 1.00000 0 −0.854224
1.5 1.00000 0 1.00000 0.181211 0 3.43581 1.00000 0 0.181211
1.6 1.00000 0 1.00000 1.29304 0 −0.779884 1.00000 0 1.29304
1.7 1.00000 0 1.00000 2.60917 0 −0.922875 1.00000 0 2.60917
1.8 1.00000 0 1.00000 2.92810 0 −3.28444 1.00000 0 2.92810
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

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.f yes 8
3.b odd 2 1 8046.2.a.e ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.e ✓ 8 3.b odd 2 1
8046.2.a.f yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\( T_{5}^{8} - 17T_{5}^{6} + 2T_{5}^{5} + 71T_{5}^{4} + 18T_{5}^{3} - 81T_{5}^{2} - 36T_{5} + 9 \) Copy content Toggle raw display
\( T_{11}^{8} + 6T_{11}^{7} - 19T_{11}^{6} - 142T_{11}^{5} - 122T_{11}^{4} + 239T_{11}^{3} + 213T_{11}^{2} - 64T_{11} - 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 17 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{8} + 5 T^{7} + \cdots - 99 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots - 49 \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots + 297 \) Copy content Toggle raw display
$17$ \( T^{8} - 5 T^{7} + \cdots - 165 \) Copy content Toggle raw display
$19$ \( T^{8} + 14 T^{7} + \cdots - 1453 \) Copy content Toggle raw display
$23$ \( T^{8} + 21 T^{7} + \cdots + 35019 \) Copy content Toggle raw display
$29$ \( T^{8} + 3 T^{7} + \cdots + 1177 \) Copy content Toggle raw display
$31$ \( T^{8} + 4 T^{7} + \cdots - 175911 \) Copy content Toggle raw display
$37$ \( T^{8} + 3 T^{7} + \cdots - 1465 \) Copy content Toggle raw display
$41$ \( T^{8} + 7 T^{7} + \cdots + 2025 \) Copy content Toggle raw display
$43$ \( T^{8} + 12 T^{7} + \cdots + 123003 \) Copy content Toggle raw display
$47$ \( T^{8} + 25 T^{7} + \cdots + 128547 \) Copy content Toggle raw display
$53$ \( T^{8} + 3 T^{7} + \cdots + 20547 \) Copy content Toggle raw display
$59$ \( T^{8} + 2 T^{7} + \cdots - 450963 \) Copy content Toggle raw display
$61$ \( T^{8} + 17 T^{7} + \cdots - 2829145 \) Copy content Toggle raw display
$67$ \( T^{8} + 14 T^{7} + \cdots + 4088693 \) Copy content Toggle raw display
$71$ \( T^{8} + 7 T^{7} + \cdots + 428215 \) Copy content Toggle raw display
$73$ \( T^{8} + 10 T^{7} + \cdots - 1097543 \) Copy content Toggle raw display
$79$ \( T^{8} + 33 T^{7} + \cdots + 96035 \) Copy content Toggle raw display
$83$ \( T^{8} + 13 T^{7} + \cdots + 6657 \) Copy content Toggle raw display
$89$ \( T^{8} - 22 T^{7} + \cdots - 1081857 \) Copy content Toggle raw display
$97$ \( T^{8} + 11 T^{7} + \cdots + 11637 \) Copy content Toggle raw display
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