Properties

Label 592.2.g.b
Level $592$
Weight $2$
Character orbit 592.g
Analytic conductor $4.727$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [592,2,Mod(369,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.369");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.g (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: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 37)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} - 3 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta q^{5} - 3 q^{7} - 2 q^{9} + 3 q^{11} + 3 \beta q^{13} + \beta q^{15} - \beta q^{17} + 3 \beta q^{19} - 3 q^{21} + 2 \beta q^{23} + q^{25} - 5 q^{27} + 2 \beta q^{29} + 3 q^{33} - 3 \beta q^{35} + ( - 3 \beta - 1) q^{37} + 3 \beta q^{39} - 3 q^{41} - 3 \beta q^{43} - 2 \beta q^{45} - 3 q^{47} + 2 q^{49} - \beta q^{51} + 9 q^{53} + 3 \beta q^{55} + 3 \beta q^{57} - 2 \beta q^{59} + 6 q^{63} - 12 q^{65} + 12 q^{67} + 2 \beta q^{69} + 3 q^{71} + 9 q^{73} + q^{75} - 9 q^{77} + 3 \beta q^{79} + q^{81} - 9 q^{83} + 4 q^{85} + 2 \beta q^{87} + 7 \beta q^{89} - 9 \beta q^{91} - 12 q^{95} - 6 \beta q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 6 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 6 q^{7} - 4 q^{9} + 6 q^{11} - 6 q^{21} + 2 q^{25} - 10 q^{27} + 6 q^{33} - 2 q^{37} - 6 q^{41} - 6 q^{47} + 4 q^{49} + 18 q^{53} + 12 q^{63} - 24 q^{65} + 24 q^{67} + 6 q^{71} + 18 q^{73} + 2 q^{75} - 18 q^{77} + 2 q^{81} - 18 q^{83} + 8 q^{85} - 24 q^{95} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
369.1
1.00000i
1.00000i
0 1.00000 0 2.00000i 0 −3.00000 0 −2.00000 0
369.2 0 1.00000 0 2.00000i 0 −3.00000 0 −2.00000 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
37.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.g.b 2
3.b odd 2 1 5328.2.h.c 2
4.b odd 2 1 37.2.b.a 2
8.b even 2 1 2368.2.g.b 2
8.d odd 2 1 2368.2.g.f 2
12.b even 2 1 333.2.c.a 2
20.d odd 2 1 925.2.c.b 2
20.e even 4 1 925.2.d.a 2
20.e even 4 1 925.2.d.d 2
37.b even 2 1 inner 592.2.g.b 2
111.d odd 2 1 5328.2.h.c 2
148.b odd 2 1 37.2.b.a 2
148.g even 4 1 1369.2.a.a 1
148.g even 4 1 1369.2.a.f 1
296.e even 2 1 2368.2.g.b 2
296.h odd 2 1 2368.2.g.f 2
444.g even 2 1 333.2.c.a 2
740.g odd 2 1 925.2.c.b 2
740.m even 4 1 925.2.d.a 2
740.m even 4 1 925.2.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.b.a 2 4.b odd 2 1
37.2.b.a 2 148.b odd 2 1
333.2.c.a 2 12.b even 2 1
333.2.c.a 2 444.g even 2 1
592.2.g.b 2 1.a even 1 1 trivial
592.2.g.b 2 37.b even 2 1 inner
925.2.c.b 2 20.d odd 2 1
925.2.c.b 2 740.g odd 2 1
925.2.d.a 2 20.e even 4 1
925.2.d.a 2 740.m even 4 1
925.2.d.d 2 20.e even 4 1
925.2.d.d 2 740.m even 4 1
1369.2.a.a 1 148.g even 4 1
1369.2.a.f 1 148.g even 4 1
2368.2.g.b 2 8.b even 2 1
2368.2.g.b 2 296.e even 2 1
2368.2.g.f 2 8.d odd 2 1
2368.2.g.f 2 296.h odd 2 1
5328.2.h.c 2 3.b odd 2 1
5328.2.h.c 2 111.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(592, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 37 \) Copy content Toggle raw display
$41$ \( (T + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( (T + 3)^{2} \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T - 12)^{2} \) Copy content Toggle raw display
$71$ \( (T - 3)^{2} \) Copy content Toggle raw display
$73$ \( (T - 9)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 36 \) Copy content Toggle raw display
$83$ \( (T + 9)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 196 \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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