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Properties

Label 416.4.a.j
Level $416$
Weight $4$
Character orbit 416.a
Self dual yes
Analytic conductor $24.545$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,4,Mod(1,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 416.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: \(24.5447945624\)
Analytic rank: \(0\)
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: \( x^{6} - 3x^{5} - 101x^{4} + 207x^{3} + 751x^{2} - 855x + 112 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_1 - 3) q^{5} + \beta_{4} q^{7} + (\beta_{2} - 2 \beta_1 + 20) q^{9} + ( - \beta_{5} - 3 \beta_{3}) q^{11} - 13 q^{13} + (2 \beta_{5} - 5 \beta_{3}) q^{15} + ( - 2 \beta_{2} - \beta_1 + 15) q^{17}+ \cdots + (12 \beta_{5} - 7 \beta_{4} - 27 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{5} + 122 q^{9} - 78 q^{13} + 96 q^{17} - 68 q^{21} + 318 q^{25} + 436 q^{29} + 928 q^{33} + 744 q^{37} + 1436 q^{41} + 1700 q^{45} + 1146 q^{49} - 284 q^{53} + 3192 q^{57} - 124 q^{61} + 208 q^{65}+ \cdots + 4724 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 101x^{4} + 207x^{3} + 751x^{2} - 855x + 112 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{4} + 10\nu^{3} + 478\nu^{2} - 483\nu - 1612 ) / 134 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{4} - 6\nu^{3} - 394\nu^{2} + 397\nu + 4612 ) / 134 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -62\nu^{5} + 155\nu^{4} + 6294\nu^{3} - 9596\nu^{2} - 47707\nu + 25458 ) / 1742 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -62\nu^{5} + 155\nu^{4} + 6294\nu^{3} - 9596\nu^{2} - 40739\nu + 21974 ) / 1742 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 90\nu^{5} - 225\nu^{4} - 9586\nu^{3} + 14604\nu^{2} + 110161\nu - 57522 ) / 1742 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_{3} - 5\beta_{2} - 3\beta _1 + 138 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -31\beta_{5} + 185\beta_{4} - 230\beta_{3} - 15\beta_{2} - 9\beta _1 + 412 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -31\beta_{5} + 184\beta_{4} - 229\beta_{3} - 493\beta_{2} - 403\beta _1 + 12122 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1651\beta_{5} + 8926\beta_{4} - 11435\beta_{3} - 1220\beta_{2} - 1000\beta _1 + 29962 ) / 4 \) 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
0.152104
3.56475
−9.25891
10.2589
−2.56475
0.847896
0 −10.3340 0 9.49534 0 8.94239 0 79.7910 0
1.2 0 −4.53772 0 −20.8058 0 16.7967 0 −6.40912 0
1.3 0 −3.82336 0 3.31043 0 −35.2123 0 −12.3819 0
1.4 0 3.82336 0 3.31043 0 35.2123 0 −12.3819 0
1.5 0 4.53772 0 −20.8058 0 −16.7967 0 −6.40912 0
1.6 0 10.3340 0 9.49534 0 −8.94239 0 79.7910 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
\(2\) \( +1 \)
\(13\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.4.a.j ✓ 6
4.b odd 2 1 inner 416.4.a.j ✓ 6
8.b even 2 1 832.4.a.bi 6
8.d odd 2 1 832.4.a.bi 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.4.a.j ✓ 6 1.a even 1 1 trivial
416.4.a.j ✓ 6 4.b odd 2 1 inner
832.4.a.bi 6 8.b even 2 1
832.4.a.bi 6 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 142T_{3}^{4} + 4061T_{3}^{2} - 32144 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(416))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 142 T^{4} + \cdots - 32144 \) Copy content Toggle raw display
$5$ \( (T^{3} + 8 T^{2} + \cdots + 654)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} - 1602 T^{4} + \cdots - 27973316 \) Copy content Toggle raw display
$11$ \( T^{6} - 5484 T^{4} + \cdots - 68016704 \) Copy content Toggle raw display
$13$ \( (T + 13)^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} - 48 T^{2} + \cdots - 99414)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 3134758976 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 306970886144 \) Copy content Toggle raw display
$29$ \( (T^{3} - 218 T^{2} + \cdots + 1440232)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 3716023320576 \) Copy content Toggle raw display
$37$ \( (T^{3} - 372 T^{2} + \cdots + 10002726)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 718 T^{2} + \cdots - 12682752)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 6973565239296 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 112256330631044 \) Copy content Toggle raw display
$53$ \( (T^{3} + 142 T^{2} + \cdots + 30322624)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 16\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{3} + 62 T^{2} + \cdots - 4259328)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 97\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 11\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{3} - 602 T^{2} + \cdots - 5585048)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 17\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 45\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{3} - 2198 T^{2} + \cdots + 909803736)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 2362 T^{2} + \cdots - 183907864)^{2} \) Copy content Toggle raw display
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