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Properties

Label 1040.6.a.q
Level $1040$
Weight $6$
Character orbit 1040.a
Self dual yes
Analytic conductor $166.799$
Analytic rank $1$
Dimension $6$
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] = [1040,6,Mod(1,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1040.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: \(166.799172605\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11}\cdot 3 \)
Twist minimal: no (minimal twist has level 65)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 6) q^{3} - 25 q^{5} + (\beta_{4} + \beta_1 - 37) q^{7} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots + 83) q^{9} + ( - 3 \beta_{5} + \beta_{4} + \cdots + 28) q^{11} + 169 q^{13} + (25 \beta_1 + 150) q^{15}+ \cdots + ( - 663 \beta_{5} + 211 \beta_{4} + \cdots + 3859) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 38 q^{3} - 150 q^{5} - 220 q^{7} + 518 q^{9} + 170 q^{11} + 1014 q^{13} + 950 q^{15} + 728 q^{17} - 1218 q^{19} - 396 q^{21} - 8954 q^{23} + 3750 q^{25} - 13112 q^{27} + 8364 q^{29} - 2862 q^{31}+ \cdots + 32270 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{5} + 18\nu^{4} + 851\nu^{3} - 1814\nu^{2} - 28404\nu - 12312 ) / 2016 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{5} - 18\nu^{4} - 851\nu^{3} + 1814\nu^{2} + 39156\nu + 12984 ) / 672 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 30\nu^{4} - 103\nu^{3} - 3098\nu^{2} + 3732\nu + 29880 ) / 336 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 59\nu^{5} + 90\nu^{4} - 6413\nu^{3} - 1678\nu^{2} + 72684\nu - 51480 ) / 2016 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -19\nu^{5} + 18\nu^{4} + 2629\nu^{3} - 5734\nu^{2} - 72252\nu + 130824 ) / 1008 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 3\beta _1 - 1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{5} - 2\beta_{4} - 3\beta_{2} + 13\beta _1 + 865 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16\beta_{5} + 16\beta_{4} - 16\beta_{3} + 89\beta_{2} + 315\beta _1 - 41 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -650\beta_{5} - 238\beta_{4} + 192\beta_{3} - 429\beta_{2} + 1075\beta _1 + 76063 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2560\beta_{5} + 2592\beta_{4} - 2032\beta_{3} + 9011\beta_{2} + 29273\beta _1 - 80691 ) / 16 \) 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
9.61672
−2.75663
8.51599
−10.7882
−5.93318
1.34530
0 −28.9791 0 −25.0000 0 189.995 0 596.790 0
1.2 0 −23.9621 0 −25.0000 0 −85.7300 0 331.182 0
1.3 0 −11.2297 0 −25.0000 0 −229.647 0 −116.895 0
1.4 0 −0.527294 0 −25.0000 0 −231.710 0 −242.722 0
1.5 0 7.05430 0 −25.0000 0 185.746 0 −193.237 0
1.6 0 19.6439 0 −25.0000 0 −48.6530 0 142.882 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(13\) \( -1 \)

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 1040.6.a.q 6
4.b odd 2 1 65.6.a.d ✓ 6
12.b even 2 1 585.6.a.m 6
20.d odd 2 1 325.6.a.g 6
20.e even 4 2 325.6.b.g 12
52.b odd 2 1 845.6.a.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.6.a.d ✓ 6 4.b odd 2 1
325.6.a.g 6 20.d odd 2 1
325.6.b.g 12 20.e even 4 2
585.6.a.m 6 12.b even 2 1
845.6.a.h 6 52.b odd 2 1
1040.6.a.q 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 38T_{3}^{5} - 266T_{3}^{4} - 17872T_{3}^{3} - 38924T_{3}^{2} + 1064984T_{3} + 569784 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1040))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 38 T^{5} + \cdots + 569784 \) Copy content Toggle raw display
$5$ \( (T + 25)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 7832693511488 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 57\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T - 169)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 60\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 60\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 38\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 18\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 25\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 86\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 41\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 91\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 13\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 10\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 66\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 13\!\cdots\!20 \) Copy content Toggle raw display
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