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Properties

Label 496.4.i.b
Level $496$
Weight $4$
Character orbit 496.i
Analytic conductor $29.265$
Analytic rank $0$
Dimension $6$
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] = [496,4,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: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.250722553392.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 46x^{4} - 24x^{3} + 2116x^{2} - 552x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{3} + ( - \beta_{3} + 2 \beta_1) q^{5} + (15 \beta_{3} + \beta_{2} + \beta_1 - 15) q^{7} + ( - \beta_{5} + \beta_{4} + \cdots + 6 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{3} + ( - \beta_{3} + 2 \beta_1) q^{5} + (15 \beta_{3} + \beta_{2} + \beta_1 - 15) q^{7} + ( - \beta_{5} + \beta_{4} + \cdots + 6 \beta_1) q^{9}+ \cdots + (53 \beta_{5} + 713 \beta_{3} + \cdots - 713) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{3} - 3 q^{5} - 45 q^{7} - 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 9 q^{3} - 3 q^{5} - 45 q^{7} - 38 q^{9} - 61 q^{11} + 113 q^{13} - 386 q^{15} + 123 q^{17} - 63 q^{19} + 43 q^{21} - 96 q^{23} + 4 q^{25} - 918 q^{27} + 332 q^{29} + 668 q^{31} - 1214 q^{33} - 278 q^{35} + 129 q^{37} + 14 q^{39} - 709 q^{41} - 107 q^{43} - 1214 q^{45} + 568 q^{47} + 262 q^{49} - 293 q^{51} - 1267 q^{53} - 909 q^{55} + 681 q^{57} + 989 q^{59} - 2500 q^{61} - 36 q^{63} - 551 q^{65} - 741 q^{67} + 440 q^{69} + 1089 q^{71} - 197 q^{73} + 500 q^{75} + 982 q^{77} - 677 q^{79} - 2423 q^{81} - q^{83} + 58 q^{85} - 1194 q^{87} - 564 q^{89} - 4054 q^{91} + 1439 q^{93} + 2094 q^{95} - 2092 q^{97} - 2086 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 46x^{4} - 24x^{3} + 2116x^{2} - 552x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 12 ) / 46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 46\nu^{3} - 12\nu^{2} + 2116\nu ) / 552 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 46\nu^{2} - 12\nu + 1426 ) / 46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -31\nu^{5} - 1426\nu^{3} + 924\nu^{2} - 65596\nu + 17112 ) / 552 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 31\beta_{3} - 31 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 46\beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -46\beta_{5} + 46\beta_{4} - 1426\beta_{3} + 12\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{5} + 924\beta_{3} - 2116\beta_{2} - 2116\beta _1 - 924 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(63\) \(65\) \(373\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(1\)

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
3.32396 + 5.75727i
0.130629 + 0.226255i
−3.45459 − 5.98353i
3.32396 − 5.75727i
0.130629 − 0.226255i
−3.45459 + 5.98353i
0 −1.82396 + 3.15920i 0 6.14793 + 10.6485i 0 −10.8240 + 18.7477i 0 6.84632 + 11.8582i 0
129.2 0 1.36937 − 2.37182i 0 −0.238743 − 0.413515i 0 −7.63063 + 13.2166i 0 9.74964 + 16.8869i 0
129.3 0 4.95459 − 8.58160i 0 −7.40918 − 12.8331i 0 −4.04541 + 7.00685i 0 −35.5960 − 61.6540i 0
273.1 0 −1.82396 − 3.15920i 0 6.14793 − 10.6485i 0 −10.8240 − 18.7477i 0 6.84632 − 11.8582i 0
273.2 0 1.36937 + 2.37182i 0 −0.238743 + 0.413515i 0 −7.63063 − 13.2166i 0 9.74964 − 16.8869i 0
273.3 0 4.95459 + 8.58160i 0 −7.40918 + 12.8331i 0 −4.04541 − 7.00685i 0 −35.5960 + 61.6540i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.3
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.b 6
4.b odd 2 1 124.4.e.b ✓ 6
12.b even 2 1 1116.4.i.d 6
31.c even 3 1 inner 496.4.i.b 6
124.i odd 6 1 124.4.e.b ✓ 6
372.p even 6 1 1116.4.i.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.e.b ✓ 6 4.b odd 2 1
124.4.e.b ✓ 6 124.i odd 6 1
496.4.i.b 6 1.a even 1 1 trivial
496.4.i.b 6 31.c even 3 1 inner
1116.4.i.d 6 12.b even 2 1
1116.4.i.d 6 372.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 9T_{3}^{5} + 100T_{3}^{4} - 27T_{3}^{3} + 1252T_{3}^{2} - 1881T_{3} + 9801 \) acting on \(S_{4}^{\mathrm{new}}(496, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 9 T^{5} + \cdots + 9801 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + \cdots + 7569 \) Copy content Toggle raw display
$7$ \( T^{6} + 45 T^{5} + \cdots + 7144929 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 2460060801 \) Copy content Toggle raw display
$13$ \( T^{6} - 113 T^{5} + \cdots + 292512609 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 139818409929 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 3749757272041 \) Copy content Toggle raw display
$23$ \( (T^{3} + 48 T^{2} + \cdots + 882816)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 166 T^{2} + \cdots + 1260064)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 26439622160671 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 935157255676281 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 116486668824201 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 70307168493969 \) Copy content Toggle raw display
$47$ \( (T^{3} - 284 T^{2} + \cdots - 407104)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 43\!\cdots\!89 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 362920445321841 \) Copy content Toggle raw display
$61$ \( (T^{3} + 1250 T^{2} + \cdots + 59458752)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 144877789637361 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 28\!\cdots\!89 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 533707532656321 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 66\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{3} + 282 T^{2} + \cdots - 230488416)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 1046 T^{2} + \cdots - 1795008)^{2} \) Copy content Toggle raw display
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