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Properties

Label 224.2.i.a
Level $224$
Weight $2$
Character orbit 224.i
Analytic conductor $1.789$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,2,Mod(65,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("224.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.i (of order \(3\), degree \(2\), 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: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{3} - 2 \beta_1 - 1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{9} + (\beta_{2} + \beta_1 + 1) q^{11} - 2 \beta_{3} q^{13}+ \cdots + ( - 2 \beta_{3} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{11} + 20 q^{15} + 6 q^{17} - 10 q^{19} + 14 q^{21} + 2 q^{23} - 8 q^{25} + 4 q^{27} - 14 q^{31} - 2 q^{33} - 22 q^{35} - 6 q^{37} + 8 q^{39} - 16 q^{41} + 16 q^{43}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

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

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 − 1.22474i
0 −1.20711 − 2.09077i 0 −1.91421 + 3.31552i 0 −1.00000 + 2.44949i 0 −1.41421 + 2.44949i 0
65.2 0 0.207107 + 0.358719i 0 0.914214 − 1.58346i 0 −1.00000 − 2.44949i 0 1.41421 − 2.44949i 0
193.1 0 −1.20711 + 2.09077i 0 −1.91421 − 3.31552i 0 −1.00000 − 2.44949i 0 −1.41421 − 2.44949i 0
193.2 0 0.207107 − 0.358719i 0 0.914214 + 1.58346i 0 −1.00000 + 2.44949i 0 1.41421 + 2.44949i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.i.a ✓ 4
3.b odd 2 1 2016.2.s.q 4
4.b odd 2 1 224.2.i.d yes 4
7.b odd 2 1 1568.2.i.x 4
7.c even 3 1 inner 224.2.i.a ✓ 4
7.c even 3 1 1568.2.a.w 2
7.d odd 6 1 1568.2.a.j 2
7.d odd 6 1 1568.2.i.x 4
8.b even 2 1 448.2.i.j 4
8.d odd 2 1 448.2.i.g 4
12.b even 2 1 2016.2.s.s 4
21.h odd 6 1 2016.2.s.q 4
28.d even 2 1 1568.2.i.o 4
28.f even 6 1 1568.2.a.u 2
28.f even 6 1 1568.2.i.o 4
28.g odd 6 1 224.2.i.d yes 4
28.g odd 6 1 1568.2.a.l 2
56.j odd 6 1 3136.2.a.bx 2
56.k odd 6 1 448.2.i.g 4
56.k odd 6 1 3136.2.a.bw 2
56.m even 6 1 3136.2.a.be 2
56.p even 6 1 448.2.i.j 4
56.p even 6 1 3136.2.a.bd 2
84.n even 6 1 2016.2.s.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.a ✓ 4 1.a even 1 1 trivial
224.2.i.a ✓ 4 7.c even 3 1 inner
224.2.i.d yes 4 4.b odd 2 1
224.2.i.d yes 4 28.g odd 6 1
448.2.i.g 4 8.d odd 2 1
448.2.i.g 4 56.k odd 6 1
448.2.i.j 4 8.b even 2 1
448.2.i.j 4 56.p even 6 1
1568.2.a.j 2 7.d odd 6 1
1568.2.a.l 2 28.g odd 6 1
1568.2.a.u 2 28.f even 6 1
1568.2.a.w 2 7.c even 3 1
1568.2.i.o 4 28.d even 2 1
1568.2.i.o 4 28.f even 6 1
1568.2.i.x 4 7.b odd 2 1
1568.2.i.x 4 7.d odd 6 1
2016.2.s.q 4 3.b odd 2 1
2016.2.s.q 4 21.h odd 6 1
2016.2.s.s 4 12.b even 2 1
2016.2.s.s 4 84.n even 6 1
3136.2.a.bd 2 56.p even 6 1
3136.2.a.be 2 56.m even 6 1
3136.2.a.bw 2 56.k odd 6 1
3136.2.a.bx 2 56.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 10 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$29$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 14 T^{3} + \cdots + 2209 \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$41$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T - 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 18 T^{3} + \cdots + 6241 \) Copy content Toggle raw display
$53$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 2 T^{3} + \cdots + 9409 \) Copy content Toggle raw display
$61$ \( T^{4} + 6 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$67$ \( T^{4} - 14 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$71$ \( (T^{2} - 16 T + 32)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 18 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
$79$ \( T^{4} - 2 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
$83$ \( (T^{2} - 8 T - 112)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
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