Properties

Label 224.3.h.a
Level $224$
Weight $3$
Character orbit 224.h
Analytic conductor $6.104$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,3,Mod(209,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.209");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.h (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: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 8\sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 7 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 7 q^{7} - 9 q^{9} - \beta q^{11} - 18 q^{23} - 25 q^{25} - \beta q^{29} - 3 \beta q^{37} + 3 \beta q^{43} + 49 q^{49} + 5 \beta q^{53} + 63 q^{63} - 3 \beta q^{67} + 114 q^{71} + 7 \beta q^{77} - 94 q^{79} + 81 q^{81} + 9 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{7} - 18 q^{9} - 36 q^{23} - 50 q^{25} + 98 q^{49} + 126 q^{63} + 228 q^{71} - 188 q^{79} + 162 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0.500000 + 1.32288i
0.500000 1.32288i
0 0 0 0 0 −7.00000 0 −9.00000 0
209.2 0 0 0 0 0 −7.00000 0 −9.00000 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 \(\Q(\sqrt{-7}) \)
8.b even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.h.a 2
3.b odd 2 1 2016.3.l.b 2
4.b odd 2 1 56.3.h.b 2
7.b odd 2 1 CM 224.3.h.a 2
8.b even 2 1 inner 224.3.h.a 2
8.d odd 2 1 56.3.h.b 2
12.b even 2 1 504.3.l.b 2
21.c even 2 1 2016.3.l.b 2
24.f even 2 1 504.3.l.b 2
24.h odd 2 1 2016.3.l.b 2
28.d even 2 1 56.3.h.b 2
28.f even 6 2 392.3.j.b 4
28.g odd 6 2 392.3.j.b 4
56.e even 2 1 56.3.h.b 2
56.h odd 2 1 inner 224.3.h.a 2
56.k odd 6 2 392.3.j.b 4
56.m even 6 2 392.3.j.b 4
84.h odd 2 1 504.3.l.b 2
168.e odd 2 1 504.3.l.b 2
168.i even 2 1 2016.3.l.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.h.b 2 4.b odd 2 1
56.3.h.b 2 8.d odd 2 1
56.3.h.b 2 28.d even 2 1
56.3.h.b 2 56.e even 2 1
224.3.h.a 2 1.a even 1 1 trivial
224.3.h.a 2 7.b odd 2 1 CM
224.3.h.a 2 8.b even 2 1 inner
224.3.h.a 2 56.h odd 2 1 inner
392.3.j.b 4 28.f even 6 2
392.3.j.b 4 28.g odd 6 2
392.3.j.b 4 56.k odd 6 2
392.3.j.b 4 56.m even 6 2
504.3.l.b 2 12.b even 2 1
504.3.l.b 2 24.f even 2 1
504.3.l.b 2 84.h odd 2 1
504.3.l.b 2 168.e odd 2 1
2016.3.l.b 2 3.b odd 2 1
2016.3.l.b 2 21.c even 2 1
2016.3.l.b 2 24.h odd 2 1
2016.3.l.b 2 168.i even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 448 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T + 18)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 448 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4032 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 4032 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 11200 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4032 \) Copy content Toggle raw display
$71$ \( (T - 114)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T + 94)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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