Properties

Label 496.2.i.h
Level $496$
Weight $2$
Character orbit 496.i
Analytic conductor $3.961$
Analytic rank $0$
Dimension $4$
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] = [496,2,Mod(129,496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(496, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("496.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 496 = 2^{4} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 496.i (of order \(3\), 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: \(3.96057994026\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} + \beta_{2} q^{5} + (\beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} + \beta_{2} q^{5} + (\beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{9} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1) q^{11} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{13} + (\beta_{3} + 1) q^{15} + (3 \beta_{2} + 2 \beta_1 + 3) q^{17} + (3 \beta_{2} + \beta_1 + 3) q^{19} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{21} + 4 q^{23} + (4 \beta_{2} + 4) q^{25} + ( - \beta_{3} + 1) q^{27} + (2 \beta_{3} - 4) q^{29} + (\beta_{3} + 2 \beta_1 + 5) q^{31} + (4 \beta_{3} + 7) q^{33} + ( - \beta_{3} - 1) q^{35} + ( - \beta_{2} - 1) q^{37} + (\beta_{3} + 3) q^{39} + ( - 6 \beta_{3} + \beta_{2} - 6 \beta_1) q^{41} + ( - \beta_{2} - 7 \beta_1 - 1) q^{43} + 2 \beta_1 q^{45} + (4 \beta_{3} - 4) q^{47} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{49} + (\beta_{3} + \beta_{2} + \beta_1) q^{51} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{53} + ( - \beta_{2} + 3 \beta_1 - 1) q^{55} + (2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{57} + ( - 3 \beta_{2} + 5 \beta_1 - 3) q^{59} + 2 \beta_{3} q^{61} + ( - 2 \beta_{3} - 4) q^{63} + (\beta_{2} + 2 \beta_1 + 1) q^{65} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1) q^{67} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{69} + ( - 5 \beta_{3} + 7 \beta_{2} - 5 \beta_1) q^{71} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{73} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{75} + ( - 4 \beta_{3} - 7) q^{77} + ( - 11 \beta_{2} + 3 \beta_1 - 11) q^{79} + (\beta_{2} - 6 \beta_1 + 1) q^{81} + (5 \beta_{3} + 3 \beta_{2} + 5 \beta_1) q^{83} + (2 \beta_{3} - 3) q^{85} - 2 \beta_1 q^{87} + ( - 6 \beta_{3} - 4) q^{89} + ( - \beta_{3} - 3) q^{91} + ( - 2 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 7) q^{93} + (\beta_{3} - 3) q^{95} + (2 \beta_{3} + 8) q^{97} + ( - 12 \beta_{2} + 2 \beta_1 - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} - 2 q^{11} + 2 q^{13} + 4 q^{15} + 6 q^{17} + 6 q^{19} + 6 q^{21} + 16 q^{23} + 8 q^{25} + 4 q^{27} - 16 q^{29} + 20 q^{31} + 28 q^{33} - 4 q^{35} - 2 q^{37} + 12 q^{39} - 2 q^{41} - 2 q^{43} - 16 q^{47} + 8 q^{49} - 2 q^{51} - 6 q^{53} - 2 q^{55} + 2 q^{57} - 6 q^{59} - 16 q^{63} + 2 q^{65} - 2 q^{67} - 8 q^{69} - 14 q^{71} - 2 q^{73} + 8 q^{75} - 28 q^{77} - 22 q^{79} + 2 q^{81} - 6 q^{83} - 12 q^{85} - 16 q^{89} - 12 q^{91} - 22 q^{93} - 12 q^{95} + 32 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/496\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\) \(373\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 −1.20711 + 2.09077i 0 −0.500000 0.866025i 0 1.20711 2.09077i 0 −1.41421 2.44949i 0
129.2 0 0.207107 0.358719i 0 −0.500000 0.866025i 0 −0.207107 + 0.358719i 0 1.41421 + 2.44949i 0
273.1 0 −1.20711 2.09077i 0 −0.500000 + 0.866025i 0 1.20711 + 2.09077i 0 −1.41421 + 2.44949i 0
273.2 0 0.207107 + 0.358719i 0 −0.500000 + 0.866025i 0 −0.207107 0.358719i 0 1.41421 2.44949i 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
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 496.2.i.h 4
4.b odd 2 1 31.2.c.a 4
12.b even 2 1 279.2.h.c 4
20.d odd 2 1 775.2.e.e 4
20.e even 4 2 775.2.o.d 8
31.c even 3 1 inner 496.2.i.h 4
124.d even 2 1 961.2.c.a 4
124.g even 6 1 961.2.a.c 2
124.g even 6 1 961.2.c.a 4
124.i odd 6 1 31.2.c.a 4
124.i odd 6 1 961.2.a.a 2
124.j even 10 4 961.2.g.r 16
124.l odd 10 4 961.2.g.o 16
124.n odd 30 4 961.2.d.l 8
124.n odd 30 4 961.2.g.o 16
124.p even 30 4 961.2.d.i 8
124.p even 30 4 961.2.g.r 16
372.p even 6 1 279.2.h.c 4
372.p even 6 1 8649.2.a.l 2
372.q odd 6 1 8649.2.a.k 2
620.o odd 6 1 775.2.e.e 4
620.be even 12 2 775.2.o.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.c.a 4 4.b odd 2 1
31.2.c.a 4 124.i odd 6 1
279.2.h.c 4 12.b even 2 1
279.2.h.c 4 372.p even 6 1
496.2.i.h 4 1.a even 1 1 trivial
496.2.i.h 4 31.c even 3 1 inner
775.2.e.e 4 20.d odd 2 1
775.2.e.e 4 620.o odd 6 1
775.2.o.d 8 20.e even 4 2
775.2.o.d 8 620.be even 12 2
961.2.a.a 2 124.i odd 6 1
961.2.a.c 2 124.g even 6 1
961.2.c.a 4 124.d even 2 1
961.2.c.a 4 124.g even 6 1
961.2.d.i 8 124.p even 30 4
961.2.d.l 8 124.n odd 30 4
961.2.g.o 16 124.l odd 10 4
961.2.g.o 16 124.n odd 30 4
961.2.g.r 16 124.j even 10 4
961.2.g.r 16 124.p even 30 4
8649.2.a.k 2 372.q odd 6 1
8649.2.a.l 2 372.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(496, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289 \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + 11 T^{2} + 14 T + 49 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 35 T^{2} - 6 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + 29 T^{2} - 42 T + 49 \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T + 31)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + 75 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} + 101 T^{2} + \cdots + 9409 \) Copy content Toggle raw display
$47$ \( (T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} + 35 T^{2} + 6 T + 1 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + 77 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$61$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289 \) Copy content Toggle raw display
$71$ \( T^{4} + 14 T^{3} + 197 T^{2} - 14 T + 1 \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + 11 T^{2} - 14 T + 49 \) Copy content Toggle raw display
$79$ \( T^{4} + 22 T^{3} + 381 T^{2} + \cdots + 10609 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + 77 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$89$ \( (T^{2} + 8 T - 56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T + 56)^{2} \) Copy content Toggle raw display
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