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Properties

Label 448.1.l.a
Level $448$
Weight $1$
Character orbit 448.l
Analytic conductor $0.224$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,1,Mod(209,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.209"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 448.l (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.223581125660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.14336.1
Artin image: $C_4\wr C_2$
Artin field: Galois closure of 8.0.78675968.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - i q^{7} - i q^{9} + (i + 1) q^{11} + i q^{25} + (i - 1) q^{29} + ( - i - 1) q^{37} + ( - i - 1) q^{43} - q^{49} + (i + 1) q^{53} - q^{63} + (i - 1) q^{67} + 2 i q^{71} + ( - i + 1) q^{77} - q^{81} + \cdots + ( - i + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{11} - 2 q^{29} - 2 q^{37} - 2 q^{43} - 2 q^{49} + 2 q^{53} - 2 q^{63} - 2 q^{67} + 2 q^{77} - 2 q^{81} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
− 1.00000i
1.00000i
0 0 0 0 0 1.00000i 0 1.00000i 0
433.1 0 0 0 0 0 − 1.00000i 0 − 1.00000i 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 \(\Q(\sqrt{-7}) \)
16.e even 4 1 inner
112.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.1.l.a 2
4.b odd 2 1 112.1.l.a ✓ 2
7.b odd 2 1 CM 448.1.l.a 2
7.c even 3 2 3136.1.bc.a 4
7.d odd 6 2 3136.1.bc.a 4
8.b even 2 1 896.1.l.a 2
8.d odd 2 1 896.1.l.b 2
12.b even 2 1 1008.1.u.b 2
16.e even 4 1 inner 448.1.l.a 2
16.e even 4 1 896.1.l.a 2
16.f odd 4 1 112.1.l.a ✓ 2
16.f odd 4 1 896.1.l.b 2
20.d odd 2 1 2800.1.z.a 2
20.e even 4 1 2800.1.bf.a 2
20.e even 4 1 2800.1.bf.b 2
28.d even 2 1 112.1.l.a ✓ 2
28.f even 6 2 784.1.y.a 4
28.g odd 6 2 784.1.y.a 4
48.k even 4 1 1008.1.u.b 2
56.e even 2 1 896.1.l.b 2
56.h odd 2 1 896.1.l.a 2
80.j even 4 1 2800.1.bf.b 2
80.k odd 4 1 2800.1.z.a 2
80.s even 4 1 2800.1.bf.a 2
84.h odd 2 1 1008.1.u.b 2
112.j even 4 1 112.1.l.a ✓ 2
112.j even 4 1 896.1.l.b 2
112.l odd 4 1 inner 448.1.l.a 2
112.l odd 4 1 896.1.l.a 2
112.u odd 12 2 784.1.y.a 4
112.v even 12 2 784.1.y.a 4
112.w even 12 2 3136.1.bc.a 4
112.x odd 12 2 3136.1.bc.a 4
140.c even 2 1 2800.1.z.a 2
140.j odd 4 1 2800.1.bf.a 2
140.j odd 4 1 2800.1.bf.b 2
336.v odd 4 1 1008.1.u.b 2
560.u odd 4 1 2800.1.bf.a 2
560.be even 4 1 2800.1.z.a 2
560.bm odd 4 1 2800.1.bf.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.1.l.a ✓ 2 4.b odd 2 1
112.1.l.a ✓ 2 16.f odd 4 1
112.1.l.a ✓ 2 28.d even 2 1
112.1.l.a ✓ 2 112.j even 4 1
448.1.l.a 2 1.a even 1 1 trivial
448.1.l.a 2 7.b odd 2 1 CM
448.1.l.a 2 16.e even 4 1 inner
448.1.l.a 2 112.l odd 4 1 inner
784.1.y.a 4 28.f even 6 2
784.1.y.a 4 28.g odd 6 2
784.1.y.a 4 112.u odd 12 2
784.1.y.a 4 112.v even 12 2
896.1.l.a 2 8.b even 2 1
896.1.l.a 2 16.e even 4 1
896.1.l.a 2 56.h odd 2 1
896.1.l.a 2 112.l odd 4 1
896.1.l.b 2 8.d odd 2 1
896.1.l.b 2 16.f odd 4 1
896.1.l.b 2 56.e even 2 1
896.1.l.b 2 112.j even 4 1
1008.1.u.b 2 12.b even 2 1
1008.1.u.b 2 48.k even 4 1
1008.1.u.b 2 84.h odd 2 1
1008.1.u.b 2 336.v odd 4 1
2800.1.z.a 2 20.d odd 2 1
2800.1.z.a 2 80.k odd 4 1
2800.1.z.a 2 140.c even 2 1
2800.1.z.a 2 560.be even 4 1
2800.1.bf.a 2 20.e even 4 1
2800.1.bf.a 2 80.s even 4 1
2800.1.bf.a 2 140.j odd 4 1
2800.1.bf.a 2 560.u odd 4 1
2800.1.bf.b 2 20.e even 4 1
2800.1.bf.b 2 80.j even 4 1
2800.1.bf.b 2 140.j odd 4 1
2800.1.bf.b 2 560.bm odd 4 1
3136.1.bc.a 4 7.c even 3 2
3136.1.bc.a 4 7.d odd 6 2
3136.1.bc.a 4 112.w even 12 2
3136.1.bc.a 4 112.x odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(448, [\chi])\).

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
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 2 \) 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
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$71$ \( T^{2} + 4 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) 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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