Properties

Label 592.1.k.b
Level $592$
Weight $1$
Character orbit 592.k
Analytic conductor $0.295$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [592,1,Mod(401,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.401");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 592.k (of order \(4\), 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: \(0.295446487479\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 148)
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.202612.1

$q$-expansion

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

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

\(f(q)\) \(=\) \( q - i q^{3} + q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{3} + q^{7} + i q^{11} + ( - i - 1) q^{17} + ( - i - 1) q^{19} - i q^{21} + (i + 1) q^{23} - i q^{25} - i q^{27} + (i - 1) q^{29} + q^{33} + i q^{37} + i q^{41} - q^{47} + (i - 1) q^{51} + q^{53} + (i - 1) q^{57} + ( - i + 1) q^{69} - q^{71} + i q^{73} - q^{75} + i q^{77} + (i + 1) q^{79} - q^{81} + q^{83} + (i + 1) q^{87} + (i - 1) q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} - 2 q^{17} - 2 q^{19} + 2 q^{23} - 2 q^{29} + 2 q^{33} - 2 q^{47} - 2 q^{51} + 2 q^{53} - 2 q^{57} + 2 q^{69} - 2 q^{71} - 2 q^{75} + 2 q^{79} - 2 q^{81} + 2 q^{83} + 2 q^{87} - 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(-i\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.00000i
1.00000i
0 1.00000i 0 0 0 1.00000 0 0 0
561.1 0 1.00000i 0 0 0 1.00000 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
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.1.k.b 2
4.b odd 2 1 148.1.f.a 2
8.b even 2 1 2368.1.k.b 2
8.d odd 2 1 2368.1.k.a 2
12.b even 2 1 1332.1.o.a 2
20.d odd 2 1 3700.1.j.c 2
20.e even 4 1 3700.1.t.a 2
20.e even 4 1 3700.1.t.b 2
37.d odd 4 1 inner 592.1.k.b 2
148.g even 4 1 148.1.f.a 2
296.j even 4 1 2368.1.k.a 2
296.m odd 4 1 2368.1.k.b 2
444.j odd 4 1 1332.1.o.a 2
740.k even 4 1 3700.1.j.c 2
740.p odd 4 1 3700.1.t.a 2
740.s odd 4 1 3700.1.t.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.1.f.a 2 4.b odd 2 1
148.1.f.a 2 148.g even 4 1
592.1.k.b 2 1.a even 1 1 trivial
592.1.k.b 2 37.d odd 4 1 inner
1332.1.o.a 2 12.b even 2 1
1332.1.o.a 2 444.j odd 4 1
2368.1.k.a 2 8.d odd 2 1
2368.1.k.a 2 296.j even 4 1
2368.1.k.b 2 8.b even 2 1
2368.1.k.b 2 296.m odd 4 1
3700.1.j.c 2 20.d odd 2 1
3700.1.j.c 2 740.k even 4 1
3700.1.t.a 2 20.e even 4 1
3700.1.t.a 2 740.p odd 4 1
3700.1.t.b 2 20.e even 4 1
3700.1.t.b 2 740.s odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{1}^{\mathrm{new}}(592, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{2} + 1 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 1)^{2} \) Copy content Toggle raw display
$53$ \( (T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( (T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$83$ \( (T - 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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