Properties

Label 4016.2.a.e
Level $4016$
Weight $2$
Character orbit 4016.a
Self dual yes
Analytic conductor $32.068$
Analytic rank $1$
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] = [4016,2,Mod(1,4016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4016, 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("4016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4016.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: \(32.0679214517\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.242773.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 502)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} - 1) q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{7} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} - 1) q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{7} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{9} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{11} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{13} + (\beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{15} + ( - 2 \beta_{4} - \beta_{2} + \beta_1 - 2) q^{17} + ( - \beta_{3} - \beta_{2} - 1) q^{19} + ( - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{21} + (2 \beta_{3} - \beta_1 + 4) q^{23} + (\beta_{3} - \beta_{2} - \beta_1) q^{25} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{27} + ( - \beta_{3} + \beta_{2} - 3) q^{29} + ( - 3 \beta_{4} + \beta_{2} + 2 \beta_1) q^{31} + (\beta_{4} - \beta_{2} - \beta_1 - 2) q^{33} + ( - 3 \beta_{4} - 2 \beta_{3} + 5 \beta_1) q^{35} + ( - 4 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{37} + (\beta_{4} - 3 \beta_{2} + 1) q^{39} + (3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{41} + ( - \beta_{4} - \beta_{3} + 3 \beta_1 - 1) q^{43} + ( - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{45} + (\beta_{4} + \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 5) q^{47} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{49} + (\beta_{4} - 2 \beta_1 - 1) q^{51} + (2 \beta_{4} - 4 \beta_1 - 6) q^{53} + (4 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{55} + ( - \beta_{3} - \beta_{2} - 3) q^{57} + (2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 2) q^{59} + (3 \beta_{4} - 3 \beta_{2} - 3 \beta_1) q^{61} + ( - 2 \beta_{4} - 3 \beta_1 + 1) q^{63} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{65} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{67} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 6 \beta_1 - 1) q^{69} + (2 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} - 6 \beta_1 + 3) q^{71} + ( - \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 2) q^{73} + ( - \beta_{4} - 2 \beta_{3} + 3 \beta_1 - 4) q^{75} + (5 \beta_{4} - \beta_{2} - \beta_1 - 6) q^{77} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_1 - 8) q^{79} + ( - 2 \beta_{4} - 4 \beta_{3} + 3 \beta_1 - 2) q^{81} + (2 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} - 1) q^{83} + (4 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{85} + (\beta_{3} + \beta_{2} - 6 \beta_1 + 1) q^{87} + ( - 2 \beta_{4} - 2 \beta_{2} + 3 \beta_1 + 2) q^{89} + ( - 4 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 5) q^{91} + (2 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 5) q^{93} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{95} + (\beta_{4} - 2 \beta_{3} + 5 \beta_{2} + \beta_1) q^{97} + (2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} - 6 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{3} - 6 q^{5} + q^{7} + 4 q^{11} - 7 q^{13} + 5 q^{15} - 8 q^{17} - 3 q^{19} - 10 q^{21} + 17 q^{23} - q^{25} + 4 q^{27} - 15 q^{29} + q^{31} - 10 q^{33} + 7 q^{35} - 5 q^{37} + 8 q^{39} - 12 q^{41} - q^{43} - 13 q^{45} + 19 q^{47} + 4 q^{49} - 7 q^{51} - 34 q^{53} - 17 q^{55} - 13 q^{57} + 10 q^{59} + 2 q^{63} + 6 q^{65} + 11 q^{67} + q^{69} + q^{71} + 3 q^{73} - 15 q^{75} - 30 q^{77} - 35 q^{79} - 3 q^{81} - 3 q^{83} - 13 q^{85} - 3 q^{87} + 15 q^{89} - 14 q^{91} + 15 q^{93} - 11 q^{95} - 2 q^{97} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 6x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 11\nu + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} + \nu^{3} + 7\nu^{2} - 4\nu - 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{4} - 4\nu^{3} - 19\nu^{2} + 19\nu + 9 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} - 2\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{4} + 5\beta_{3} - 9\beta_{2} + 3\beta _1 + 16 \) 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
−2.31766
−0.567497
−0.203945
1.37208
2.71702
0 −2.31766 0 −3.17136 0 3.66414 0 2.37155 0
1.2 0 −0.567497 0 −2.51380 0 −3.57054 0 −2.67795 0
1.3 0 −0.203945 0 0.242071 0 0.933626 0 −2.95841 0
1.4 0 1.37208 0 1.64633 0 2.43111 0 −1.11739 0
1.5 0 2.71702 0 −2.20324 0 −2.45833 0 4.38220 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
\(2\) \(-1\)
\(251\) \(-1\)

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 4016.2.a.e 5
4.b odd 2 1 502.2.a.c 5
12.b even 2 1 4518.2.a.v 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
502.2.a.c 5 4.b odd 2 1
4016.2.a.e 5 1.a even 1 1 trivial
4518.2.a.v 5 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - T_{3}^{4} - 7T_{3}^{3} + 4T_{3}^{2} + 6T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - T^{4} - 7 T^{3} + 4 T^{2} + 6 T + 1 \) Copy content Toggle raw display
$5$ \( T^{5} + 6 T^{4} + 6 T^{3} - 18 T^{2} + \cdots + 7 \) Copy content Toggle raw display
$7$ \( T^{5} - T^{4} - 19 T^{3} + 18 T^{2} + \cdots - 73 \) Copy content Toggle raw display
$11$ \( T^{5} - 4 T^{4} - 29 T^{3} + 76 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{5} + 7 T^{4} - 7 T^{3} - 100 T^{2} + \cdots + 373 \) Copy content Toggle raw display
$17$ \( T^{5} + 8 T^{4} - 22 T^{3} - 242 T^{2} + \cdots - 313 \) Copy content Toggle raw display
$19$ \( T^{5} + 3 T^{4} - 13 T^{3} + 2 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$23$ \( T^{5} - 17 T^{4} + 79 T^{3} + \cdots + 1999 \) Copy content Toggle raw display
$29$ \( T^{5} + 15 T^{4} + 69 T^{3} + \cdots - 536 \) Copy content Toggle raw display
$31$ \( T^{5} - T^{4} - 80 T^{3} - 15 T^{2} + \cdots + 1796 \) Copy content Toggle raw display
$37$ \( T^{5} + 5 T^{4} - 131 T^{3} + \cdots + 968 \) Copy content Toggle raw display
$41$ \( T^{5} + 12 T^{4} - 44 T^{3} + \cdots + 8119 \) Copy content Toggle raw display
$43$ \( T^{5} + T^{4} - 43 T^{3} + 114 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$47$ \( T^{5} - 19 T^{4} + 97 T^{3} + \cdots - 472 \) Copy content Toggle raw display
$53$ \( T^{5} + 34 T^{4} + 360 T^{3} + \cdots - 3424 \) Copy content Toggle raw display
$59$ \( T^{5} - 10 T^{4} - 36 T^{3} + \cdots - 512 \) Copy content Toggle raw display
$61$ \( T^{5} - 171 T^{3} + 54 T^{2} + \cdots - 1944 \) Copy content Toggle raw display
$67$ \( T^{5} - 11 T^{4} - 7 T^{3} + \cdots - 1331 \) Copy content Toggle raw display
$71$ \( T^{5} - T^{4} - 231 T^{3} + \cdots - 33224 \) Copy content Toggle raw display
$73$ \( T^{5} - 3 T^{4} - 82 T^{3} - 77 T^{2} + \cdots - 208 \) Copy content Toggle raw display
$79$ \( T^{5} + 35 T^{4} + 453 T^{3} + \cdots + 6209 \) Copy content Toggle raw display
$83$ \( T^{5} + 3 T^{4} - 289 T^{3} + \cdots + 48832 \) Copy content Toggle raw display
$89$ \( T^{5} - 15 T^{4} + 5 T^{3} + 514 T^{2} + \cdots + 77 \) Copy content Toggle raw display
$97$ \( T^{5} + 2 T^{4} - 353 T^{3} + \cdots + 109064 \) Copy content Toggle raw display
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