Properties

Label 8028.2.a.j
Level $8028$
Weight $2$
Character orbit 8028.a
Self dual yes
Analytic conductor $64.104$
Analytic rank $0$
Dimension $7$
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] = [8028,2,Mod(1,8028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8028, 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("8028.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8028.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: \(64.1039027427\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 38x^{3} - 46x^{2} - 36x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 892)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_1 + 1) q^{5} + (\beta_{6} + \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_1 + 1) q^{5} + (\beta_{6} + \beta_{3}) q^{7} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 + 1) q^{11} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{13} + (\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{17} + (\beta_{6} + \beta_{2} - 1) q^{19} + (\beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + 2) q^{23} + (\beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{25} + ( - \beta_{5} - \beta_{2} + 3) q^{29} + (\beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{31} + (\beta_{6} - \beta_{5} + \beta_{3} - 2 \beta_1 + 3) q^{35} + ( - \beta_{6} + 2 \beta_{4} + \beta_{3} - 2) q^{37} + (\beta_{6} + \beta_{3} + \beta_1 - 1) q^{41} + ( - \beta_{6} - \beta_{5} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{43} + (2 \beta_{5} - \beta_{2} + 3) q^{47} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{49} + (\beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_1 + 6) q^{53} + (3 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - \beta_{2} + 4 \beta_1) q^{55} + ( - \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1 + 4) q^{59} + (2 \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{2} + \beta_1 - 3) q^{61} + ( - 3 \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{2} - 4 \beta_1) q^{65} + ( - \beta_{5} + \beta_{3} - 3) q^{67} + (\beta_{6} - 3 \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 + 1) q^{71} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{73} + (4 \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{77} + (\beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 1) q^{79} + (3 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 6) q^{83} + ( - \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{85} + (2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 4 \beta_1) q^{89} + (3 \beta_{5} - \beta_{3} - 2 \beta_{2} + \beta_1) q^{91} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_1 + 1) q^{95} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{5} - q^{7} + 12 q^{11} - 9 q^{13} - 8 q^{17} - 12 q^{19} + 12 q^{23} + 12 q^{25} + 24 q^{29} + q^{31} + 15 q^{35} - 13 q^{37} - 5 q^{41} - 13 q^{43} + 21 q^{47} + 4 q^{49} + 35 q^{53} + q^{55} + 23 q^{59} - 17 q^{61} - 18 q^{67} + 4 q^{71} + 23 q^{73} - 3 q^{77} + 4 q^{79} + 44 q^{83} + 20 q^{85} + 2 q^{91} + 12 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 38x^{3} - 46x^{2} - 36x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 4\nu^{5} - 3\nu^{4} + 15\nu^{3} - 5\nu^{2} + 11\nu - 3 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 13\nu^{4} + 15\nu^{3} + 33\nu^{2} - 13\nu + 15 ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 3\nu^{4} - 10\nu^{3} + 19\nu^{2} + 26\nu - 15 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 13\nu^{4} + 15\nu^{3} + 45\nu^{2} - 37\nu - 33 ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 4\nu^{5} + 7\nu^{4} - 29\nu^{3} - 13\nu^{2} + 41\nu + 3 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{3} + 2\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 3\beta_{5} - 2\beta_{4} - \beta_{3} + 10\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{6} + 15\beta_{5} - 7\beta_{4} - 8\beta_{3} + 3\beta_{2} + 31\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 25\beta_{6} + 56\beta_{5} - 35\beta_{4} - 15\beta_{3} + 9\beta_{2} + 129\beta _1 + 75 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 100\beta_{6} + 229\beta_{5} - 131\beta_{4} - 74\beta_{3} + 57\beta_{2} + 458\beta _1 + 359 \) 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
−0.828592
−2.20362
2.33527
1.49109
−1.80533
0.206062
3.80512
0 0 0 −3.49299 0 −0.758389 0 0 0
1.2 0 0 0 −0.778137 0 −3.30215 0 0 0
1.3 0 0 0 0.389930 0 −3.58741 0 0 0
1.4 0 0 0 1.18873 0 2.92270 0 0 0
1.5 0 0 0 2.57376 0 2.76633 0 0 0
1.6 0 0 0 2.98220 0 2.92860 0 0 0
1.7 0 0 0 4.13650 0 −1.96968 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(223\) \(-1\)

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 8028.2.a.j 7
3.b odd 2 1 892.2.a.d 7
12.b even 2 1 3568.2.a.m 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
892.2.a.d 7 3.b odd 2 1
3568.2.a.m 7 12.b even 2 1
8028.2.a.j 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8028))\):

\( T_{5}^{7} - 7T_{5}^{6} + T_{5}^{5} + 83T_{5}^{4} - 171T_{5}^{3} + 30T_{5}^{2} + 112T_{5} - 40 \) Copy content Toggle raw display
\( T_{7}^{7} + T_{7}^{6} - 26T_{7}^{5} - 20T_{7}^{4} + 218T_{7}^{3} + 141T_{7}^{2} - 571T_{7} - 419 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 7 T^{6} + T^{5} + 83 T^{4} + \cdots - 40 \) Copy content Toggle raw display
$7$ \( T^{7} + T^{6} - 26 T^{5} - 20 T^{4} + \cdots - 419 \) Copy content Toggle raw display
$11$ \( T^{7} - 12 T^{6} + 12 T^{5} + \cdots - 11304 \) Copy content Toggle raw display
$13$ \( T^{7} + 9 T^{6} - 5 T^{5} - 235 T^{4} + \cdots - 216 \) Copy content Toggle raw display
$17$ \( T^{7} + 8 T^{6} - 21 T^{5} + \cdots + 1193 \) Copy content Toggle raw display
$19$ \( T^{7} + 12 T^{6} + 29 T^{5} - 81 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{7} - 12 T^{6} + 16 T^{5} + \cdots + 648 \) Copy content Toggle raw display
$29$ \( T^{7} - 24 T^{6} + 211 T^{5} + \cdots + 2439 \) Copy content Toggle raw display
$31$ \( T^{7} - T^{6} - 106 T^{5} - 70 T^{4} + \cdots + 1289 \) Copy content Toggle raw display
$37$ \( T^{7} + 13 T^{6} - 76 T^{5} + \cdots + 22837 \) Copy content Toggle raw display
$41$ \( T^{7} + 5 T^{6} - 23 T^{5} - 72 T^{4} + \cdots - 9 \) Copy content Toggle raw display
$43$ \( T^{7} + 13 T^{6} - 91 T^{5} + \cdots + 390825 \) Copy content Toggle raw display
$47$ \( T^{7} - 21 T^{6} + 97 T^{5} + \cdots - 5199 \) Copy content Toggle raw display
$53$ \( T^{7} - 35 T^{6} + 436 T^{5} + \cdots + 1117 \) Copy content Toggle raw display
$59$ \( T^{7} - 23 T^{6} + 110 T^{5} + \cdots + 159912 \) Copy content Toggle raw display
$61$ \( T^{7} + 17 T^{6} - 172 T^{5} + \cdots + 26136 \) Copy content Toggle raw display
$67$ \( T^{7} + 18 T^{6} + 90 T^{5} + \cdots - 200 \) Copy content Toggle raw display
$71$ \( T^{7} - 4 T^{6} - 218 T^{5} + \cdots + 363096 \) Copy content Toggle raw display
$73$ \( T^{7} - 23 T^{6} - 62 T^{5} + \cdots - 2902799 \) Copy content Toggle raw display
$79$ \( T^{7} - 4 T^{6} - 326 T^{5} + \cdots + 1531976 \) Copy content Toggle raw display
$83$ \( T^{7} - 44 T^{6} + 532 T^{5} + \cdots + 1629963 \) Copy content Toggle raw display
$89$ \( T^{7} - 342 T^{5} + 803 T^{4} + \cdots - 98415 \) Copy content Toggle raw display
$97$ \( T^{7} - 32 T^{6} + 306 T^{5} + \cdots - 216 \) Copy content Toggle raw display
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