Properties

Label 496.4.i.a
Level $496$
Weight $4$
Character orbit 496.i
Analytic conductor $29.265$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [496,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("496.129");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 496 = 2^{4} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(29.2649473628\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 124)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (5 \zeta_{6} - 5) q^{3} + 15 \zeta_{6} q^{5} + ( - 29 \zeta_{6} + 29) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (5 \zeta_{6} - 5) q^{3} + 15 \zeta_{6} q^{5} + ( - 29 \zeta_{6} + 29) q^{7} + 2 \zeta_{6} q^{9} - 51 \zeta_{6} q^{11} - 53 \zeta_{6} q^{13} - 75 q^{15} + (51 \zeta_{6} - 51) q^{17} + (145 \zeta_{6} - 145) q^{19} + 145 \zeta_{6} q^{21} - 108 q^{23} + (100 \zeta_{6} - 100) q^{25} - 145 q^{27} - 198 q^{29} + ( - 186 \zeta_{6} + 155) q^{31} + 255 q^{33} + 435 q^{35} + ( - 55 \zeta_{6} + 55) q^{37} + 265 q^{39} - 39 \zeta_{6} q^{41} + (217 \zeta_{6} - 217) q^{43} + (30 \zeta_{6} - 30) q^{45} - 504 q^{47} - 498 \zeta_{6} q^{49} - 255 \zeta_{6} q^{51} - 237 \zeta_{6} q^{53} + ( - 765 \zeta_{6} + 765) q^{55} - 725 \zeta_{6} q^{57} + (597 \zeta_{6} - 597) q^{59} - 466 q^{61} + 58 q^{63} + ( - 795 \zeta_{6} + 795) q^{65} + 749 \zeta_{6} q^{67} + ( - 540 \zeta_{6} + 540) q^{69} - 321 \zeta_{6} q^{71} + 793 \zeta_{6} q^{73} - 500 \zeta_{6} q^{75} - 1479 q^{77} + ( - 1217 \zeta_{6} + 1217) q^{79} + ( - 671 \zeta_{6} + 671) q^{81} - 555 \zeta_{6} q^{83} - 765 q^{85} + ( - 990 \zeta_{6} + 990) q^{87} - 378 q^{89} - 1537 q^{91} + (775 \zeta_{6} + 155) q^{93} - 2175 q^{95} + 1070 q^{97} + ( - 102 \zeta_{6} + 102) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{3} + 15 q^{5} + 29 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 5 q^{3} + 15 q^{5} + 29 q^{7} + 2 q^{9} - 51 q^{11} - 53 q^{13} - 150 q^{15} - 51 q^{17} - 145 q^{19} + 145 q^{21} - 216 q^{23} - 100 q^{25} - 290 q^{27} - 396 q^{29} + 124 q^{31} + 510 q^{33} + 870 q^{35} + 55 q^{37} + 530 q^{39} - 39 q^{41} - 217 q^{43} - 30 q^{45} - 1008 q^{47} - 498 q^{49} - 255 q^{51} - 237 q^{53} + 765 q^{55} - 725 q^{57} - 597 q^{59} - 932 q^{61} + 116 q^{63} + 795 q^{65} + 749 q^{67} + 540 q^{69} - 321 q^{71} + 793 q^{73} - 500 q^{75} - 2958 q^{77} + 1217 q^{79} + 671 q^{81} - 555 q^{83} - 1530 q^{85} + 990 q^{87} - 756 q^{89} - 3074 q^{91} + 1085 q^{93} - 4350 q^{95} + 2140 q^{97} + 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(63\) \(65\) \(373\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 −2.50000 + 4.33013i 0 7.50000 + 12.9904i 0 14.5000 25.1147i 0 1.00000 + 1.73205i 0
273.1 0 −2.50000 4.33013i 0 7.50000 12.9904i 0 14.5000 + 25.1147i 0 1.00000 1.73205i 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 496.4.i.a 2
4.b odd 2 1 124.4.e.a 2
12.b even 2 1 1116.4.i.a 2
31.c even 3 1 inner 496.4.i.a 2
124.i odd 6 1 124.4.e.a 2
372.p even 6 1 1116.4.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.e.a 2 4.b odd 2 1
124.4.e.a 2 124.i odd 6 1
496.4.i.a 2 1.a even 1 1 trivial
496.4.i.a 2 31.c even 3 1 inner
1116.4.i.a 2 12.b even 2 1
1116.4.i.a 2 372.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5T_{3} + 25 \) acting on \(S_{4}^{\mathrm{new}}(496, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$5$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$7$ \( T^{2} - 29T + 841 \) Copy content Toggle raw display
$11$ \( T^{2} + 51T + 2601 \) Copy content Toggle raw display
$13$ \( T^{2} + 53T + 2809 \) Copy content Toggle raw display
$17$ \( T^{2} + 51T + 2601 \) Copy content Toggle raw display
$19$ \( T^{2} + 145T + 21025 \) Copy content Toggle raw display
$23$ \( (T + 108)^{2} \) Copy content Toggle raw display
$29$ \( (T + 198)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 124T + 29791 \) Copy content Toggle raw display
$37$ \( T^{2} - 55T + 3025 \) Copy content Toggle raw display
$41$ \( T^{2} + 39T + 1521 \) Copy content Toggle raw display
$43$ \( T^{2} + 217T + 47089 \) Copy content Toggle raw display
$47$ \( (T + 504)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 237T + 56169 \) Copy content Toggle raw display
$59$ \( T^{2} + 597T + 356409 \) Copy content Toggle raw display
$61$ \( (T + 466)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 749T + 561001 \) Copy content Toggle raw display
$71$ \( T^{2} + 321T + 103041 \) Copy content Toggle raw display
$73$ \( T^{2} - 793T + 628849 \) Copy content Toggle raw display
$79$ \( T^{2} - 1217 T + 1481089 \) Copy content Toggle raw display
$83$ \( T^{2} + 555T + 308025 \) Copy content Toggle raw display
$89$ \( (T + 378)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1070)^{2} \) Copy content Toggle raw display
show more
show less