↑

Properties

Label 4732.2.a.t
Level $4732$
Weight $2$
Character orbit 4732.a
Self dual yes
Analytic conductor $37.785$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4732,2,Mod(1,4732)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4732, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4732.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4732.a (trivial)

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: yes
Analytic conductor: \(37.7852102365\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 19x^{6} - 2x^{5} + 113x^{4} + 40x^{3} - 232x^{2} - 136x + 52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 364)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{5} + 1) q^{5} + q^{7} + ( - \beta_{6} - \beta_{4} + \beta_{2} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{5} + 1) q^{5} + q^{7} + ( - \beta_{6} - \beta_{4} + \beta_{2} + \cdots + 2) q^{9}+ \cdots + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \cdots + 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{5} + 8 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{5} + 8 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{15} + 2 q^{17} + 6 q^{19} + 22 q^{25} - 6 q^{27} + 22 q^{29} + 14 q^{31} - 28 q^{33} + 6 q^{35} + 12 q^{37} + 4 q^{41} + 6 q^{43} + 20 q^{45} + 42 q^{47} + 8 q^{49} + 2 q^{51} + 4 q^{53} - 2 q^{55} - 8 q^{57} + 2 q^{59} - 4 q^{61} + 14 q^{63} + 24 q^{67} - 52 q^{69} + 28 q^{71} + 28 q^{73} - 10 q^{75} + 12 q^{77} + 4 q^{79} + 38 q^{83} - 36 q^{85} - 26 q^{87} - 22 q^{89} + 56 q^{93} - 30 q^{95} - 4 q^{97} + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 19x^{6} - 2x^{5} + 113x^{4} + 40x^{3} - 232x^{2} - 136x + 52 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} + 6\nu^{6} + 47\nu^{5} - 88\nu^{4} - 193\nu^{3} + 266\nu^{2} + 270\nu - 108 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} - 3\nu^{6} - 32\nu^{5} + 45\nu^{4} + 138\nu^{3} - 137\nu^{2} - 194\nu + 52 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} - 7\nu^{6} - 64\nu^{5} + 103\nu^{4} + 274\nu^{3} - 309\nu^{2} - 388\nu + 114 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{7} + 9\nu^{6} + 94\nu^{5} - 133\nu^{4} - 384\nu^{3} + 391\nu^{2} + 492\nu - 154 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{7} + 13\nu^{6} + 111\nu^{5} - 191\nu^{4} - 467\nu^{3} + 571\nu^{2} + 662\nu - 202 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\nu^{7} - 4\nu^{6} - 32\nu^{5} + 59\nu^{4} + 137\nu^{3} - 181\nu^{2} - 200\nu + 70 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - \beta_{4} + \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 9\beta_{6} - \beta_{5} - 14\beta_{4} + \beta_{3} + 11\beta_{2} + 9\beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{7} + \beta_{6} + 12\beta_{5} + 11\beta_{4} + 13\beta_{3} - 29\beta_{2} + 44\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 28\beta_{7} - 82\beta_{6} - 15\beta_{5} - 153\beta_{4} + 15\beta_{3} + 112\beta_{2} + 70\beta _1 + 316 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -142\beta_{7} + 27\beta_{6} + 123\beta_{5} + 124\beta_{4} + 140\beta_{3} - 337\beta_{2} + 358\beta _1 - 234 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.98100
2.42977
2.25607
0.268953
−1.17707
−1.75101
−1.77673
−3.23100
0 −2.98100 0 −0.118179 0 1.00000 0 5.88638 0
1.2 0 −2.42977 0 −3.63781 0 1.00000 0 2.90380 0
1.3 0 −2.25607 0 4.27591 0 1.00000 0 2.08985 0
1.4 0 −0.268953 0 1.35585 0 1.00000 0 −2.92766 0
1.5 0 1.17707 0 −1.46614 0 1.00000 0 −1.61452 0
1.6 0 1.75101 0 −1.38536 0 1.00000 0 0.0660200 0
1.7 0 1.77673 0 3.82804 0 1.00000 0 0.156761 0
1.8 0 3.23100 0 3.14769 0 1.00000 0 7.43937 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(13\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4732.2.a.t 8
13.b even 2 1 4732.2.a.s 8
13.d odd 4 2 4732.2.g.k 16
13.f odd 12 2 364.2.u.a ✓ 16
39.k even 12 2 3276.2.cf.c 16
52.l even 12 2 1456.2.cc.f 16
91.w even 12 2 2548.2.bq.c 16
91.x odd 12 2 2548.2.bb.d 16
91.ba even 12 2 2548.2.bb.c 16
91.bc even 12 2 2548.2.u.c 16
91.bd odd 12 2 2548.2.bq.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.u.a ✓ 16 13.f odd 12 2
1456.2.cc.f 16 52.l even 12 2
2548.2.u.c 16 91.bc even 12 2
2548.2.bb.c 16 91.ba even 12 2
2548.2.bb.d 16 91.x odd 12 2
2548.2.bq.c 16 91.w even 12 2
2548.2.bq.e 16 91.bd odd 12 2
3276.2.cf.c 16 39.k even 12 2
4732.2.a.s 8 13.b even 2 1
4732.2.a.t 8 1.a even 1 1 trivial
4732.2.g.k 16 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4732))\):

\( T_{3}^{8} - 19T_{3}^{6} + 2T_{3}^{5} + 113T_{3}^{4} - 40T_{3}^{3} - 232T_{3}^{2} + 136T_{3} + 52 \) Copy content Toggle raw display
\( T_{5}^{8} - 6T_{5}^{7} - 13T_{5}^{6} + 112T_{5}^{5} - 4T_{5}^{4} - 470T_{5}^{3} + 11T_{5}^{2} + 524T_{5} + 61 \) Copy content Toggle raw display
\( T_{11}^{8} - 12T_{11}^{7} + 15T_{11}^{6} + 304T_{11}^{5} - 1051T_{11}^{4} - 1368T_{11}^{3} + 8584T_{11}^{2} - 2192T_{11} - 13628 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 19 T^{6} + \cdots + 52 \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{7} + \cdots + 61 \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{7} + \cdots - 13628 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots - 1727 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + \cdots + 35152 \) Copy content Toggle raw display
$23$ \( T^{8} - 92 T^{6} + \cdots - 9408 \) Copy content Toggle raw display
$29$ \( T^{8} - 22 T^{7} + \cdots - 23031 \) Copy content Toggle raw display
$31$ \( T^{8} - 14 T^{7} + \cdots - 1282172 \) Copy content Toggle raw display
$37$ \( T^{8} - 12 T^{7} + \cdots + 24336 \) Copy content Toggle raw display
$41$ \( T^{8} - 4 T^{7} + \cdots + 1279696 \) Copy content Toggle raw display
$43$ \( T^{8} - 6 T^{7} + \cdots - 2442908 \) Copy content Toggle raw display
$47$ \( T^{8} - 42 T^{7} + \cdots + 75664 \) Copy content Toggle raw display
$53$ \( T^{8} - 4 T^{7} + \cdots + 117909 \) Copy content Toggle raw display
$59$ \( T^{8} - 2 T^{7} + \cdots + 10932 \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{7} + \cdots + 1286464 \) Copy content Toggle raw display
$67$ \( T^{8} - 24 T^{7} + \cdots - 249132 \) Copy content Toggle raw display
$71$ \( T^{8} - 28 T^{7} + \cdots - 2710272 \) Copy content Toggle raw display
$73$ \( T^{8} - 28 T^{7} + \cdots + 298768 \) Copy content Toggle raw display
$79$ \( T^{8} - 4 T^{7} + \cdots - 1070784 \) Copy content Toggle raw display
$83$ \( T^{8} - 38 T^{7} + \cdots - 1404 \) Copy content Toggle raw display
$89$ \( T^{8} + 22 T^{7} + \cdots - 182208 \) Copy content Toggle raw display
$97$ \( T^{8} + 4 T^{7} + \cdots - 6656 \) Copy content Toggle raw display
show more
show less