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Properties

Label 441.2.e.h
Level $441$
Weight $2$
Character orbit 441.e
Analytic conductor $3.521$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $8$

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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: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 5 \beta_{2} q^{4} + 3 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 5 \beta_{2} q^{4} + 3 \beta_{3} q^{8} + (2 \beta_{3} + 2 \beta_1) q^{11} + ( - 11 \beta_{2} - 11) q^{16} - 14 q^{22} + 2 \beta_1 q^{23} - 5 \beta_{2} q^{25} - 4 \beta_{3} q^{29} + ( - 5 \beta_{3} - 5 \beta_1) q^{32} + ( - 6 \beta_{2} - 6) q^{37} + 12 q^{43} - 10 \beta_1 q^{44} + 14 \beta_{2} q^{46} - 5 \beta_{3} q^{50} + (4 \beta_{3} + 4 \beta_1) q^{53} + (28 \beta_{2} + 28) q^{58} + 13 q^{64} + 4 \beta_{2} q^{67} - 2 \beta_{3} q^{71} + ( - 6 \beta_{3} - 6 \beta_1) q^{74} + ( - 8 \beta_{2} - 8) q^{79} + 12 \beta_1 q^{86} - 42 \beta_{2} q^{88} + 10 \beta_{3} q^{92}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{4} - 22 q^{16} - 56 q^{22} + 10 q^{25} - 12 q^{37} + 48 q^{43} - 28 q^{46} + 56 q^{58} + 52 q^{64} - 8 q^{67} - 16 q^{79} + 84 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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
−1.32288 + 2.29129i
1.32288 − 2.29129i
−1.32288 − 2.29129i
1.32288 + 2.29129i
−1.32288 + 2.29129i 0 −2.50000 − 4.33013i 0 0 0 7.93725 0 0
226.2 1.32288 − 2.29129i 0 −2.50000 − 4.33013i 0 0 0 −7.93725 0 0
361.1 −1.32288 − 2.29129i 0 −2.50000 + 4.33013i 0 0 0 7.93725 0 0
361.2 1.32288 + 2.29129i 0 −2.50000 + 4.33013i 0 0 0 −7.93725 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 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.h 4
3.b odd 2 1 inner 441.2.e.h 4
7.b odd 2 1 CM 441.2.e.h 4
7.c even 3 1 441.2.a.h ✓ 2
7.c even 3 1 inner 441.2.e.h 4
7.d odd 6 1 441.2.a.h ✓ 2
7.d odd 6 1 inner 441.2.e.h 4
21.c even 2 1 inner 441.2.e.h 4
21.g even 6 1 441.2.a.h ✓ 2
21.g even 6 1 inner 441.2.e.h 4
21.h odd 6 1 441.2.a.h ✓ 2
21.h odd 6 1 inner 441.2.e.h 4
28.f even 6 1 7056.2.a.co 2
28.g odd 6 1 7056.2.a.co 2
84.j odd 6 1 7056.2.a.co 2
84.n even 6 1 7056.2.a.co 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.a.h ✓ 2 7.c even 3 1
441.2.a.h ✓ 2 7.d odd 6 1
441.2.a.h ✓ 2 21.g even 6 1
441.2.a.h ✓ 2 21.h odd 6 1
441.2.e.h 4 1.a even 1 1 trivial
441.2.e.h 4 3.b odd 2 1 inner
441.2.e.h 4 7.b odd 2 1 CM
441.2.e.h 4 7.c even 3 1 inner
441.2.e.h 4 7.d odd 6 1 inner
441.2.e.h 4 21.c even 2 1 inner
441.2.e.h 4 21.g even 6 1 inner
441.2.e.h 4 21.h odd 6 1 inner
7056.2.a.co 2 28.f even 6 1
7056.2.a.co 2 28.g odd 6 1
7056.2.a.co 2 84.j odd 6 1
7056.2.a.co 2 84.n even 6 1

Hecke kernels

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

\( T_{2}^{4} + 7T_{2}^{2} + 49 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 28T^{2} + 784 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 28T^{2} + 784 \) Copy content Toggle raw display
$29$ \( (T^{2} - 112)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T - 12)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 112 T^{2} + 12544 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) 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
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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