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Properties

Label 496.5.e.c
Level $496$
Weight $5$
Character orbit 496.e
Analytic conductor $51.272$
Analytic rank $0$
Dimension $10$
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] = [496,5,Mod(433,496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(496, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("496.433");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 496 = 2^{4} \cdot 31 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 496.e (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: \(51.2715016896\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 478x^{8} + 69668x^{6} + 4198200x^{4} + 101304000x^{2} + 622080000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 124)
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 a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} - 1) q^{5} + \beta_{4} q^{7} + (\beta_{2} - 15) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} - 1) q^{5} + \beta_{4} q^{7} + (\beta_{2} - 15) q^{9} - \beta_{6} q^{11} + ( - \beta_{7} - 2 \beta_1) q^{13} + ( - \beta_{8} - \beta_1) q^{15} + ( - \beta_{7} + \beta_{6} + \cdots + 2 \beta_1) q^{17}+ \cdots + ( - 19 \beta_{8} - 12 \beta_{7} + \cdots - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{5} + 2 q^{7} - 146 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{5} + 2 q^{7} - 146 q^{9} + 310 q^{19} - 756 q^{25} - 734 q^{31} - 372 q^{33} + 666 q^{35} + 1924 q^{39} - 210 q^{41} + 650 q^{45} + 1992 q^{47} + 188 q^{49} - 1352 q^{51} - 5610 q^{59} - 3478 q^{63} + 5420 q^{67} + 10160 q^{69} + 1734 q^{71} + 11598 q^{81} + 13244 q^{87} + 3088 q^{93} - 4302 q^{95} + 2942 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 478x^{8} + 69668x^{6} + 4198200x^{4} + 101304000x^{2} + 622080000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{8} - 4358\nu^{6} - 406348\nu^{4} - 11667000\nu^{2} - 67392000 ) / 475200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -13\nu^{8} - 5314\nu^{6} - 534884\nu^{4} - 16456200\nu^{2} - 77284800 ) / 475200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53\nu^{9} + 21014\nu^{7} + 1951444\nu^{5} + 53082840\nu^{3} + 235656000\nu ) / 3564000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -29\nu^{9} - 12002\nu^{7} - 1266292\nu^{5} - 46705320\nu^{3} - 503136000\nu ) / 2376000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37\nu^{9} + 15256\nu^{7} + 1578176\nu^{5} + 52379160\nu^{3} + 314010000\nu ) / 1782000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\nu^{9} + 4358\nu^{7} + 406348\nu^{5} + 11667000\nu^{3} + 67392000\nu ) / 475200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\nu^{8} + 5314\nu^{6} + 539284\nu^{4} + 17785000\nu^{2} + 125966400 ) / 17600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{8} - 2\beta_{7} - 3\beta_{6} + 5\beta_{5} - 188\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{9} + 108\beta_{4} - 302\beta_{2} + 17928 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 1042\beta_{8} + 576\beta_{7} + 858\beta_{6} - 1722\beta_{5} + 46276\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1336\beta_{9} - 38976\beta_{4} + 3432\beta_{3} + 84564\beta_{2} - 4408344 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -305452\beta_{8} - 156872\beta_{7} - 237660\beta_{6} + 499436\beta_{5} - 12388224\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 381536\beta_{9} + 11451984\beta_{4} - 1402896\beta_{3} - 23407256\beta_{2} + 1179926928 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 85747400\beta_{8} + 42993248\beta_{7} + 65643336\beta_{6} - 139558712\beta_{5} + 3391793104\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(63\) \(65\) \(373\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
− 16.6119i
− 8.69414i
− 8.43572i
− 6.79929i
− 3.01087i
3.01087i
6.79929i
8.43572i
8.69414i
16.6119i
0 − 16.6119i 0 −1.78751 0 31.2283 0 −194.955 0
433.2 0 − 8.69414i 0 32.3102 0 −39.7108 0 5.41184 0
433.3 0 − 8.43572i 0 −14.7147 0 −70.0514 0 9.83861 0
433.4 0 − 6.79929i 0 −34.9624 0 12.5932 0 34.7697 0
433.5 0 − 3.01087i 0 16.1544 0 66.9407 0 71.9347 0
433.6 0 3.01087i 0 16.1544 0 66.9407 0 71.9347 0
433.7 0 6.79929i 0 −34.9624 0 12.5932 0 34.7697 0
433.8 0 8.43572i 0 −14.7147 0 −70.0514 0 9.83861 0
433.9 0 8.69414i 0 32.3102 0 −39.7108 0 5.41184 0
433.10 0 16.6119i 0 −1.78751 0 31.2283 0 −194.955 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 496.5.e.c 10
4.b odd 2 1 124.5.c.a ✓ 10
12.b even 2 1 1116.5.h.b 10
31.b odd 2 1 inner 496.5.e.c 10
124.d even 2 1 124.5.c.a ✓ 10
372.b odd 2 1 1116.5.h.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.5.c.a ✓ 10 4.b odd 2 1
124.5.c.a ✓ 10 124.d even 2 1
496.5.e.c 10 1.a even 1 1 trivial
496.5.e.c 10 31.b odd 2 1 inner
1116.5.h.b 10 12.b even 2 1
1116.5.h.b 10 372.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 478T_{3}^{8} + 69668T_{3}^{6} + 4198200T_{3}^{4} + 101304000T_{3}^{2} + 622080000 \) acting on \(S_{5}^{\mathrm{new}}(496, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 622080000 \) Copy content Toggle raw display
$5$ \( (T^{5} + 3 T^{4} + \cdots + 479988)^{2} \) Copy content Toggle raw display
$7$ \( (T^{5} - T^{4} + \cdots - 73231744)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 71\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 67\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 17\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 1981857577500)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 67\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 25\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 21406740928848)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 36\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots + 118001846116800)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 32\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 43\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 82\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 18\!\cdots\!88)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 19\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 33\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 25\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 17\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 31\!\cdots\!08 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 17\!\cdots\!08)^{2} \) Copy content Toggle raw display
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