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Properties

Label 4056.2.c.m
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: 6.0.44836416.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 312)
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.

\(f(q)\) \(=\) \( q - q^{3} - \beta_{4} q^{5} + (\beta_{5} + \beta_1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - \beta_{4} q^{5} + (\beta_{5} + \beta_1) q^{7} + q^{9} + (\beta_{5} - \beta_{4}) q^{11} + \beta_{4} q^{15} - \beta_{2} q^{17} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{19} + ( - \beta_{5} - \beta_1) q^{21} + (\beta_{3} - \beta_{2}) q^{23} + (\beta_{3} - 2 \beta_{2} - 5) q^{25} - q^{27} + (\beta_{3} - 4) q^{29} + ( - \beta_{4} - \beta_1) q^{31} + ( - \beta_{5} + \beta_{4}) q^{33} + (3 \beta_{3} + \beta_{2} + 4) q^{35} + ( - \beta_{5} + 2 \beta_1) q^{37} + (\beta_{5} - 2 \beta_{4}) q^{41} + ( - \beta_{2} + 7) q^{43} - \beta_{4} q^{45} + (\beta_{5} - \beta_{4} - 4 \beta_1) q^{47} + ( - 3 \beta_{3} - 8) q^{49} + \beta_{2} q^{51} + (3 \beta_{3} - 2) q^{53} + (3 \beta_{3} - \beta_{2} - 6) q^{55} + ( - \beta_{5} - \beta_{4} + 2 \beta_1) q^{57} + ( - 2 \beta_{5} + 2 \beta_1) q^{59} + ( - \beta_{3} - \beta_{2} + 1) q^{61} + (\beta_{5} + \beta_1) q^{63} + ( - \beta_{5} - 9 \beta_1) q^{67} + ( - \beta_{3} + \beta_{2}) q^{69} + ( - \beta_{5} + \beta_{4} - 4 \beta_1) q^{71} + ( - 2 \beta_{4} - 3 \beta_1) q^{73} + ( - \beta_{3} + 2 \beta_{2} + 5) q^{75} + (2 \beta_{2} - 10) q^{77} + ( - 3 \beta_{3} + 2 \beta_{2} + 3) q^{79} + q^{81} + ( - \beta_{5} + 3 \beta_{4} + 6 \beta_1) q^{83} + (\beta_{5} + 2 \beta_{4} + 4 \beta_1) q^{85} + ( - \beta_{3} + 4) q^{87} + ( - 2 \beta_{4} + 8 \beta_1) q^{89} + (\beta_{4} + \beta_1) q^{93} + ( - \beta_{3} + 3 \beta_{2} + 14) q^{95} + ( - \beta_{4} - 7 \beta_1) q^{97} + (\beta_{5} - \beta_{4}) q^{99}+O(q^{100}) \) 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 \( 6 q - 6 q^{3} + 6 q^{9} - 30 q^{25} - 6 q^{27} - 24 q^{29} + 24 q^{35} + 42 q^{43} - 48 q^{49} - 12 q^{53} - 36 q^{55} + 6 q^{61} + 30 q^{75} - 60 q^{77} + 18 q^{79} + 6 q^{81} + 24 q^{87} + 84 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 12x^{4} + 36x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{4} - 5\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} + 7\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} + 10\nu^{3} + 23\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 10\nu^{3} + 25\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{5} + 3\beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{3} - 7\beta_{2} + 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 37\beta_{5} - 35\beta_{4} - 20\beta_1 ) / 2 \) Copy content Toggle raw display

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

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
− 2.36147i
0.167449i
− 2.52892i
2.52892i
− 0.167449i
2.36147i
0 −1.00000 0 − 3.93800i 0 − 1.78493i 0 1.00000 0
337.2 0 −1.00000 0 − 3.80451i 0 5.13941i 0 1.00000 0
337.3 0 −1.00000 0 − 0.133492i 0 − 3.92434i 0 1.00000 0
337.4 0 −1.00000 0 0.133492i 0 3.92434i 0 1.00000 0
337.5 0 −1.00000 0 3.80451i 0 − 5.13941i 0 1.00000 0
337.6 0 −1.00000 0 3.93800i 0 1.78493i 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.m 6
13.b even 2 1 inner 4056.2.c.m 6
13.d odd 4 1 4056.2.a.y 3
13.d odd 4 1 4056.2.a.z 3
13.f odd 12 2 312.2.q.e ✓ 6
39.k even 12 2 936.2.t.h 6
52.f even 4 1 8112.2.a.ck 3
52.f even 4 1 8112.2.a.cl 3
52.l even 12 2 624.2.q.j 6
156.v odd 12 2 1872.2.t.u 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.e ✓ 6 13.f odd 12 2
624.2.q.j 6 52.l even 12 2
936.2.t.h 6 39.k even 12 2
1872.2.t.u 6 156.v odd 12 2
4056.2.a.y 3 13.d odd 4 1
4056.2.a.z 3 13.d odd 4 1
4056.2.c.m 6 1.a even 1 1 trivial
4056.2.c.m 6 13.b even 2 1 inner
8112.2.a.ck 3 52.f even 4 1
8112.2.a.cl 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 \(S_{2}^{\mathrm{new}}(4056, [\chi])\):

\( T_{5}^{6} + 30T_{5}^{4} + 225T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 45T_{7}^{4} + 540T_{7}^{2} + 1296 \) Copy content Toggle raw display
\( T_{11}^{6} + 48T_{11}^{4} + 576T_{11}^{2} + 64 \) 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
$5$ \( T^{6} + 30 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{6} + 45 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$11$ \( T^{6} + 48 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} - 21 T - 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 108 T^{4} + \cdots + 43264 \) Copy content Toggle raw display
$23$ \( (T^{3} - 24 T - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 12 T^{2} + 33 T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 33 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{6} + 54 T^{4} + \cdots + 324 \) Copy content Toggle raw display
$41$ \( T^{6} + 114 T^{4} + \cdots + 24336 \) Copy content Toggle raw display
$43$ \( (T^{3} - 21 T^{2} + \cdots - 212)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 96 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$53$ \( (T^{3} + 6 T^{2} + \cdots - 208)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 180 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$61$ \( (T^{3} - 3 T^{2} + \cdots + 167)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 285 T^{4} + \cdots + 274576 \) Copy content Toggle raw display
$71$ \( T^{6} + 96 T^{4} + \cdots + 1600 \) Copy content Toggle raw display
$73$ \( T^{6} + 147 T^{4} + \cdots + 28561 \) Copy content Toggle raw display
$79$ \( (T^{3} - 9 T^{2} - 120 T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 348 T^{4} + \cdots + 984064 \) Copy content Toggle raw display
$89$ \( T^{6} + 312 T^{4} + \cdots + 2304 \) Copy content Toggle raw display
$97$ \( T^{6} + 177 T^{4} + \cdots + 55696 \) Copy content Toggle raw display
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