Properties

Label 4056.2.c.o
Level $4056$
Weight $2$
Character orbit 4056.c
Analytic conductor $32.387$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4056,2,Mod(337,4056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4056.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{3} + (\beta_{3} + \beta_1) q^{5} + (\beta_{3} + \beta_1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (\beta_{3} + \beta_1) q^{5} + (\beta_{3} + \beta_1) q^{7} + q^{9} + ( - 2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{11} + (\beta_{3} + \beta_1) q^{15} + (\beta_{4} + 2 \beta_{2} + 4) q^{17} + ( - 4 \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{19} + (\beta_{3} + \beta_1) q^{21} + ( - 2 \beta_{4} - 3 \beta_{2} + 7) q^{23} + (3 \beta_{4} + \beta_{2} - 1) q^{25} + q^{27} + (3 \beta_{4} + 3 \beta_{2} - 8) q^{29} + ( - 3 \beta_{5} - 5 \beta_{3} + 2 \beta_1) q^{31} + ( - 2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{33} + (3 \beta_{4} + \beta_{2} - 6) q^{35} + ( - \beta_{5} + 4 \beta_{3} + \beta_1) q^{37} + ( - 2 \beta_{5} + \beta_{3}) q^{41} + ( - 2 \beta_{4} + 2 \beta_{2} + 3) q^{43} + (\beta_{3} + \beta_1) q^{45} + ( - 2 \beta_{5} - 2 \beta_{3} - \beta_1) q^{47} + (3 \beta_{4} + \beta_{2} + 1) q^{49} + (\beta_{4} + 2 \beta_{2} + 4) q^{51} + ( - 7 \beta_{4} - 5 \beta_{2} + 1) q^{53} + (4 \beta_{4} + 4 \beta_{2} - 5) q^{55} + ( - 4 \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{57} + (2 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{59} + ( - 5 \beta_{4} + 2 \beta_{2} + 2) q^{61} + (\beta_{3} + \beta_1) q^{63} + (5 \beta_{5} + 7 \beta_{3} - 7 \beta_1) q^{67} + ( - 2 \beta_{4} - 3 \beta_{2} + 7) q^{69} + (7 \beta_{5} - \beta_{3}) q^{71} + (2 \beta_{5} - 9 \beta_{3} + 3 \beta_1) q^{73} + (3 \beta_{4} + \beta_{2} - 1) q^{75} + (4 \beta_{4} + 4 \beta_{2} - 5) q^{77} + (8 \beta_{4} - 3 \beta_{2} - 6) q^{79} + q^{81} + (\beta_{5} - 2 \beta_{3} + 5 \beta_1) q^{83} + (4 \beta_{5} + 7 \beta_{3} + 2 \beta_1) q^{85} + (3 \beta_{4} + 3 \beta_{2} - 8) q^{87} + ( - 5 \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{89} + ( - 3 \beta_{5} - 5 \beta_{3} + 2 \beta_1) q^{93} + (7 \beta_{4} + 7 \beta_{2} - 7) q^{95} + (3 \beta_{5} - 6 \beta_{3} + 6 \beta_1) q^{97} + ( - 2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 6 q^{9} + 30 q^{17} + 32 q^{23} + 2 q^{25} + 6 q^{27} - 36 q^{29} - 28 q^{35} + 18 q^{43} + 14 q^{49} + 30 q^{51} - 18 q^{53} - 14 q^{55} + 6 q^{61} + 32 q^{69} + 2 q^{75} - 14 q^{77} - 26 q^{79} + 6 q^{81} - 36 q^{87} - 14 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 4\nu^{3} + 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{3} + 9\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times\).

\(n\) \(1015\) \(2029\) \(2705\) \(3889\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.24698i
0.445042i
1.80194i
1.80194i
0.445042i
1.24698i
0 1.00000 0 3.04892i 0 3.04892i 0 1.00000 0
337.2 0 1.00000 0 1.69202i 0 1.69202i 0 1.00000 0
337.3 0 1.00000 0 1.35690i 0 1.35690i 0 1.00000 0
337.4 0 1.00000 0 1.35690i 0 1.35690i 0 1.00000 0
337.5 0 1.00000 0 1.69202i 0 1.69202i 0 1.00000 0
337.6 0 1.00000 0 3.04892i 0 3.04892i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4056.2.c.o 6
13.b even 2 1 inner 4056.2.c.o 6
13.d odd 4 1 4056.2.a.bb 3
13.d odd 4 1 4056.2.a.bc yes 3
52.f even 4 1 8112.2.a.cb 3
52.f even 4 1 8112.2.a.cc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4056.2.a.bb 3 13.d odd 4 1
4056.2.a.bc yes 3 13.d odd 4 1
4056.2.c.o 6 1.a even 1 1 trivial
4056.2.c.o 6 13.b even 2 1 inner
8112.2.a.cb 3 52.f even 4 1
8112.2.a.cc 3 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4056, [\chi])\):

\( T_{5}^{6} + 14T_{5}^{4} + 49T_{5}^{2} + 49 \) Copy content Toggle raw display
\( T_{7}^{6} + 14T_{7}^{4} + 49T_{7}^{2} + 49 \) Copy content Toggle raw display
\( T_{11}^{6} + 17T_{11}^{4} + 94T_{11}^{2} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 14 T^{4} + 49 T^{2} + 49 \) Copy content Toggle raw display
$7$ \( T^{6} + 14 T^{4} + 49 T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{6} + 17 T^{4} + 94 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} - 15 T^{2} + 68 T - 97)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 49 T^{4} + 686 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( (T^{3} - 16 T^{2} + 69 T - 41)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 18 T^{2} + 87 T + 83)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 90 T^{4} + 1517 T^{2} + \cdots + 6889 \) Copy content Toggle raw display
$37$ \( T^{6} + 110 T^{4} + 1861 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$41$ \( T^{6} + 21 T^{4} + 98 T^{2} + 49 \) Copy content Toggle raw display
$43$ \( (T^{3} - 9 T^{2} - T + 113)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 41 T^{4} + 474 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$53$ \( (T^{3} + 9 T^{2} - 64 T - 449)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 52 T^{4} + 416 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( (T^{3} - 3 T^{2} - 88 T + 293)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 229 T^{4} + 13250 T^{2} + \cdots + 16129 \) Copy content Toggle raw display
$71$ \( T^{6} + 166 T^{4} + 8693 T^{2} + \cdots + 142129 \) Copy content Toggle raw display
$73$ \( T^{6} + 402 T^{4} + 48429 T^{2} + \cdots + 1697809 \) Copy content Toggle raw display
$79$ \( (T^{3} + 13 T^{2} - 170 T - 1651)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 122 T^{4} + 381 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$89$ \( T^{6} + 69 T^{4} + 810 T^{2} + \cdots + 1849 \) Copy content Toggle raw display
$97$ \( T^{6} + 315 T^{4} + 11907 T^{2} + \cdots + 35721 \) Copy content Toggle raw display
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