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Properties

Label 441.2.e.i
Level $441$
Weight $2$
Character orbit 441.e
Analytic conductor $3.521$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,2,Mod(226,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), 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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_1 - 1) q^{4} - 2 \beta_{2} q^{5} - \beta_{3} q^{8} + (6 \beta_1 - 6) q^{10} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{11} - 2 q^{13} + 5 \beta_1 q^{16} + (2 \beta_{3} - 2 \beta_{2}) q^{17}+ \cdots - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} - 12 q^{10} - 8 q^{13} + 10 q^{16} - 8 q^{19} + 24 q^{22} - 14 q^{25} - 8 q^{31} - 24 q^{34} - 4 q^{37} + 12 q^{40} - 16 q^{43} + 12 q^{46} + 4 q^{52} + 48 q^{55} - 20 q^{61} + 4 q^{64} + 8 q^{67}+ \cdots - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
−0.866025 + 1.50000i 0 −0.500000 − 0.866025i −1.73205 + 3.00000i 0 0 −1.73205 0 −3.00000 − 5.19615i
226.2 0.866025 − 1.50000i 0 −0.500000 − 0.866025i 1.73205 − 3.00000i 0 0 1.73205 0 −3.00000 − 5.19615i
361.1 −0.866025 − 1.50000i 0 −0.500000 + 0.866025i −1.73205 − 3.00000i 0 0 −1.73205 0 −3.00000 + 5.19615i
361.2 0.866025 + 1.50000i 0 −0.500000 + 0.866025i 1.73205 + 3.00000i 0 0 1.73205 0 −3.00000 + 5.19615i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.e.i 4
3.b odd 2 1 inner 441.2.e.i 4
7.b odd 2 1 441.2.e.j 4
7.c even 3 1 441.2.a.g 2
7.c even 3 1 inner 441.2.e.i 4
7.d odd 6 1 63.2.a.b ✓ 2
7.d odd 6 1 441.2.e.j 4
21.c even 2 1 441.2.e.j 4
21.g even 6 1 63.2.a.b ✓ 2
21.g even 6 1 441.2.e.j 4
21.h odd 6 1 441.2.a.g 2
21.h odd 6 1 inner 441.2.e.i 4
28.f even 6 1 1008.2.a.n 2
28.g odd 6 1 7056.2.a.cm 2
35.i odd 6 1 1575.2.a.q 2
35.k even 12 2 1575.2.d.i 4
56.j odd 6 1 4032.2.a.bt 2
56.m even 6 1 4032.2.a.bq 2
63.i even 6 1 567.2.f.j 4
63.k odd 6 1 567.2.f.j 4
63.s even 6 1 567.2.f.j 4
63.t odd 6 1 567.2.f.j 4
77.i even 6 1 7623.2.a.bi 2
84.j odd 6 1 1008.2.a.n 2
84.n even 6 1 7056.2.a.cm 2
105.p even 6 1 1575.2.a.q 2
105.w odd 12 2 1575.2.d.i 4
168.ba even 6 1 4032.2.a.bt 2
168.be odd 6 1 4032.2.a.bq 2
231.k odd 6 1 7623.2.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.a.b ✓ 2 7.d odd 6 1
63.2.a.b ✓ 2 21.g even 6 1
441.2.a.g 2 7.c even 3 1
441.2.a.g 2 21.h odd 6 1
441.2.e.i 4 1.a even 1 1 trivial
441.2.e.i 4 3.b odd 2 1 inner
441.2.e.i 4 7.c even 3 1 inner
441.2.e.i 4 21.h odd 6 1 inner
441.2.e.j 4 7.b odd 2 1
441.2.e.j 4 7.d odd 6 1
441.2.e.j 4 21.c even 2 1
441.2.e.j 4 21.g even 6 1
567.2.f.j 4 63.i even 6 1
567.2.f.j 4 63.k odd 6 1
567.2.f.j 4 63.s even 6 1
567.2.f.j 4 63.t odd 6 1
1008.2.a.n 2 28.f even 6 1
1008.2.a.n 2 84.j odd 6 1
1575.2.a.q 2 35.i odd 6 1
1575.2.a.q 2 105.p even 6 1
1575.2.d.i 4 35.k even 12 2
1575.2.d.i 4 105.w odd 12 2
4032.2.a.bq 2 56.m even 6 1
4032.2.a.bq 2 168.be odd 6 1
4032.2.a.bt 2 56.j odd 6 1
4032.2.a.bt 2 168.ba even 6 1
7056.2.a.cm 2 28.g odd 6 1
7056.2.a.cm 2 84.n even 6 1
7623.2.a.bi 2 77.i even 6 1
7623.2.a.bi 2 231.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{4} + 3T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{4} + 12T_{5}^{2} + 144 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$13$ \( (T + 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$53$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$59$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$97$ \( (T + 14)^{4} \) Copy content Toggle raw display
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