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Properties

Label 224.2.i.b
Level $224$
Weight $2$
Character orbit 224.i
Analytic conductor $1.789$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(\Q(\zeta_{12})\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 1) q^{5} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 1) q^{5} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}) q^{7} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{11} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{15} - 5 \zeta_{12}^{2} q^{17} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{19} + ( - \zeta_{12}^{2} - 4) q^{21} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{23} + 4 \zeta_{12}^{2} q^{25} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + 8 q^{29} + (5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{31} + (9 \zeta_{12}^{2} - 9) q^{33} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{35} + ( - 5 \zeta_{12}^{2} + 5) q^{37} + 4 q^{41} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{43} + (10 \zeta_{12}^{3} - 5 \zeta_{12}) q^{47} + ( - 8 \zeta_{12}^{2} + 3) q^{49} + (10 \zeta_{12}^{3} - 5 \zeta_{12}) q^{51} + \zeta_{12}^{2} q^{53} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{55} + 3 q^{57} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{59} + (11 \zeta_{12}^{2} - 11) q^{61} + (7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{67} - 3 q^{69} + (8 \zeta_{12}^{3} - 16 \zeta_{12}) q^{71} - 15 \zeta_{12}^{2} q^{73} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{75} + ( - 3 \zeta_{12}^{2} - 12) q^{77} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{79} + 9 \zeta_{12}^{2} q^{81} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{83} + 5 q^{85} + ( - 8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{87} + (7 \zeta_{12}^{2} - 7) q^{89} + ( - 15 \zeta_{12}^{2} + 15) q^{93} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{95} + 12 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} - 10 q^{17} - 18 q^{21} + 8 q^{25} + 32 q^{29} - 18 q^{33} + 10 q^{37} + 16 q^{41} - 4 q^{49} + 2 q^{53} + 12 q^{57} - 22 q^{61} - 12 q^{69} - 30 q^{73} - 54 q^{77} + 18 q^{81} + 20 q^{85} - 14 q^{89} + 30 q^{93} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{12}^{2}\) \(1\)

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
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 −0.866025 − 1.50000i 0 −0.500000 + 0.866025i 0 1.73205 − 2.00000i 0 0 0
65.2 0 0.866025 + 1.50000i 0 −0.500000 + 0.866025i 0 −1.73205 + 2.00000i 0 0 0
193.1 0 −0.866025 + 1.50000i 0 −0.500000 − 0.866025i 0 1.73205 + 2.00000i 0 0 0
193.2 0 0.866025 − 1.50000i 0 −0.500000 − 0.866025i 0 −1.73205 − 2.00000i 0 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
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3T_{3}^{2} + 9 \) 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
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$23$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$29$ \( (T - 8)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$37$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$41$ \( (T - 4)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$53$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$61$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 147 T^{2} + 21609 \) Copy content Toggle raw display
$71$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 15 T + 225)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$83$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$97$ \( (T - 12)^{4} \) Copy content Toggle raw display
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