Properties

Label 8036.2.a.k
Level $8036$
Weight $2$
Character orbit 8036.a
Self dual yes
Analytic conductor $64.168$
Analytic rank $0$
Dimension $5$
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] = [8036,2,Mod(1,8036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8036, 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("8036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8036.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: \(64.1677830643\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.287349.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 6x^{3} + 7x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1148)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{4} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{4} + \beta_{2}) q^{9} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{13} + (\beta_{4} - \beta_1 + 2) q^{15} + (\beta_{4} + \beta_{3}) q^{17} + (2 \beta_{4} - \beta_{3} - 3 \beta_1) q^{19} + ( - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{23} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{25} + (\beta_{2} - \beta_1 - 2) q^{27} + ( - \beta_{4} - 3 \beta_1 + 2) q^{29} + ( - \beta_{3} + \beta_{2} - 4 \beta_1 - 3) q^{31} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 - 1) q^{33} + (\beta_{4} - 2 \beta_{2} + 2 \beta_1 + 2) q^{37} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 4) q^{39} + q^{41} + (\beta_{4} - \beta_{3} - 2 \beta_1 + 6) q^{43} + (\beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{45} + ( - 3 \beta_{4} - 2) q^{47} + (\beta_{4} - \beta_{2} - 2 \beta_1) q^{51} + (2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 5) q^{53} + (2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{55} + (2 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 5 \beta_1 + 3) q^{57} + (5 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{59} + (\beta_{4} - \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 3) q^{61} + (2 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{65} + (2 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} - 4 \beta_1 + 3) q^{67} + ( - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 8 \beta_1 - 3) q^{69} + ( - 5 \beta_{4} + \beta_{3} + \beta_{2} + 7 \beta_1 + 3) q^{71} + (\beta_{4} - \beta_{3} + \beta_{2} - 5 \beta_1 + 3) q^{73} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{75} + (2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 2) q^{79} + ( - 4 \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{81} + (\beta_{4} - 3 \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{83} + ( - \beta_{3} - \beta_{2} + 3) q^{85} + ( - \beta_{4} + 3 \beta_{3} - \beta_{2} + 7 \beta_1 - 1) q^{87} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 5) q^{89} + ( - \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 9 \beta_1 - 2) q^{93} + (3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{95} + (4 \beta_{4} + 4 \beta_{3} - \beta_{2} - 4 \beta_1 + 6) q^{97} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{3} + 3 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{3} + 3 q^{5} + q^{9} + 6 q^{11} - 7 q^{13} + 9 q^{15} - q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} - 8 q^{27} + 11 q^{29} - 13 q^{31} - 11 q^{33} + 5 q^{37} + 23 q^{39} + 5 q^{41} + 29 q^{43} - 11 q^{45} - 7 q^{47} - 3 q^{51} + 21 q^{53} - 19 q^{55} + 9 q^{57} - 3 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} - 10 q^{69} + 22 q^{71} + 16 q^{73} + 18 q^{75} + 4 q^{79} - 15 q^{81} + 6 q^{83} + 13 q^{85} - 6 q^{87} + 20 q^{89} - 5 q^{93} - 7 q^{95} + 24 q^{97} + 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 6x^{3} + 7x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 5\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} + 7 \) 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
2.13797
−2.05679
−1.36870
1.14203
0.145487
0 −2.57090 0 −0.350332 0 0 0 3.60954 0
1.2 0 −2.23037 0 −1.70418 0 0 0 1.97454 0
1.3 0 0.126667 0 4.03743 0 0 0 −2.98396 0
1.4 0 0.695770 0 −1.38288 0 0 0 −2.51590 0
1.5 0 1.97883 0 2.39996 0 0 0 0.915782 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

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 8036.2.a.k 5
7.b odd 2 1 1148.2.a.d 5
28.d even 2 1 4592.2.a.bc 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.a.d 5 7.b odd 2 1
4592.2.a.bc 5 28.d even 2 1
8036.2.a.k 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\( T_{3}^{5} + 2T_{3}^{4} - 6T_{3}^{3} - 8T_{3}^{2} + 9T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{5} - 3T_{5}^{4} - 9T_{5}^{3} + 12T_{5}^{2} + 28T_{5} + 8 \) Copy content Toggle raw display
\( T_{11}^{5} - 6T_{11}^{4} - 7T_{11}^{3} + 68T_{11}^{2} - 84T_{11} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 2 T^{4} - 6 T^{3} - 8 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{5} - 3 T^{4} - 9 T^{3} + 12 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 6 T^{4} - 7 T^{3} + 68 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$13$ \( T^{5} + 7 T^{4} - 5 T^{3} - 36 T^{2} + \cdots + 27 \) Copy content Toggle raw display
$17$ \( T^{5} + T^{4} - 17 T^{3} + 14 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$19$ \( T^{5} + 2 T^{4} - 68 T^{3} + \cdots + 3643 \) Copy content Toggle raw display
$23$ \( T^{5} - 4 T^{4} - 92 T^{3} + \cdots - 5721 \) Copy content Toggle raw display
$29$ \( T^{5} - 11 T^{4} - 31 T^{3} + \cdots - 216 \) Copy content Toggle raw display
$31$ \( T^{5} + 13 T^{4} - 25 T^{3} + \cdots - 1176 \) Copy content Toggle raw display
$37$ \( T^{5} - 5 T^{4} - 68 T^{3} + \cdots - 1748 \) Copy content Toggle raw display
$41$ \( (T - 1)^{5} \) Copy content Toggle raw display
$43$ \( T^{5} - 29 T^{4} + 307 T^{3} + \cdots - 2161 \) Copy content Toggle raw display
$47$ \( T^{5} + 7 T^{4} - 74 T^{3} + \cdots + 1532 \) Copy content Toggle raw display
$53$ \( T^{5} - 21 T^{4} + 89 T^{3} + \cdots - 152 \) Copy content Toggle raw display
$59$ \( T^{5} + 3 T^{4} - 263 T^{3} + \cdots + 55448 \) Copy content Toggle raw display
$61$ \( T^{5} - 8 T^{4} - 167 T^{3} + \cdots - 11944 \) Copy content Toggle raw display
$67$ \( T^{5} - 3 T^{4} - 345 T^{3} + \cdots + 51208 \) Copy content Toggle raw display
$71$ \( T^{5} - 22 T^{4} - 203 T^{3} + \cdots - 308544 \) Copy content Toggle raw display
$73$ \( T^{5} - 16 T^{4} - 27 T^{3} + \cdots - 2664 \) Copy content Toggle raw display
$79$ \( T^{5} - 4 T^{4} - 288 T^{3} + \cdots - 100896 \) Copy content Toggle raw display
$83$ \( T^{5} - 6 T^{4} - 89 T^{3} + \cdots - 5272 \) Copy content Toggle raw display
$89$ \( T^{5} - 20 T^{4} + 46 T^{3} + \cdots + 5297 \) Copy content Toggle raw display
$97$ \( T^{5} - 24 T^{4} - 122 T^{3} + \cdots + 122173 \) Copy content Toggle raw display
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