Properties

Label 483.4.a.i
Level $483$
Weight $4$
Character orbit 483.a
Self dual yes
Analytic conductor $28.498$
Analytic rank $0$
Dimension $9$
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] = [483,4,Mod(1,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 483.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: \(28.4979225328\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 51x^{7} + 34x^{6} + 861x^{5} - 401x^{4} - 5403x^{3} + 1772x^{2} + 8716x - 192 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} - 2 \beta_1 + 4) q^{4} + ( - \beta_{4} + 4) q^{5} + ( - 3 \beta_1 + 3) q^{6} + 7 q^{7} + ( - \beta_{3} + 2 \beta_{2} + \cdots + 15) q^{8} + 9 q^{9} + ( - \beta_{6} - \beta_{4} + \beta_{2} + \cdots + 9) q^{10}+ \cdots + ( - 9 \beta_{7} + 9 \beta_{4} + \cdots + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 8 q^{2} + 27 q^{3} + 38 q^{4} + 39 q^{5} + 24 q^{6} + 63 q^{7} + 135 q^{8} + 81 q^{9} + 81 q^{10} + 38 q^{11} + 114 q^{12} + 107 q^{13} + 56 q^{14} + 117 q^{15} + 178 q^{16} + 170 q^{17} + 72 q^{18}+ \cdots + 342 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 51x^{7} + 34x^{6} + 861x^{5} - 401x^{4} - 5403x^{3} + 1772x^{2} + 8716x - 192 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 17\nu + 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18 \nu^{8} + 57 \nu^{7} + 426 \nu^{6} - 1535 \nu^{5} + 2445 \nu^{4} + 8922 \nu^{3} - 73899 \nu^{2} + \cdots + 118608 ) / 5896 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 35 \nu^{8} + 12 \nu^{7} - 1811 \nu^{6} - 905 \nu^{5} + 28288 \nu^{4} + 18519 \nu^{3} - 135262 \nu^{2} + \cdots + 79896 ) / 5896 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 39 \nu^{8} + 492 \nu^{7} + 923 \nu^{6} - 17943 \nu^{5} - 1704 \nu^{4} + 171153 \nu^{3} + \cdots - 31920 ) / 5896 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{8} - 3\nu^{7} - 368\nu^{6} - 25\nu^{5} + 5457\nu^{4} + 948\nu^{3} - 29131\nu^{2} + 1974\nu + 30544 ) / 536 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 115 \nu^{8} - 487 \nu^{7} - 4687 \nu^{6} + 18610 \nu^{5} + 57465 \nu^{4} - 207841 \nu^{3} + \cdots + 175760 ) / 5896 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 17\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 2\beta_{5} + \beta_{4} + 3\beta_{3} + 24\beta_{2} + 8\beta _1 + 190 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{8} + 3\beta_{6} - 6\beta_{5} + \beta_{4} + 36\beta_{3} + 42\beta_{2} + 336\beta _1 + 162 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{8} + 36\beta_{7} + 5\beta_{6} - 84\beta_{5} + 21\beta_{4} + 129\beta_{3} + 568\beta_{2} + 407\beta _1 + 3818 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 119 \beta_{8} + 19 \beta_{7} + 135 \beta_{6} - 276 \beta_{5} + 24 \beta_{4} + 1073 \beta_{3} + \cdots + 6141 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 192 \beta_{8} + 1048 \beta_{7} + 290 \beta_{6} - 2622 \beta_{5} + 296 \beta_{4} + 4284 \beta_{3} + \cdots + 84715 \) Copy content Toggle raw display

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
5.24907
3.86754
3.08695
1.79037
0.0219372
−1.35308
−3.40804
−3.78226
−4.47249
−4.24907 3.00000 10.0546 3.00025 −12.7472 7.00000 −8.73012 9.00000 −12.7483
1.2 −2.86754 3.00000 0.222786 −10.1020 −8.60262 7.00000 22.3015 9.00000 28.9679
1.3 −2.08695 3.00000 −3.64464 21.1286 −6.26085 7.00000 24.3018 9.00000 −44.0944
1.4 −0.790374 3.00000 −7.37531 7.57526 −2.37112 7.00000 12.1522 9.00000 −5.98729
1.5 0.978063 3.00000 −7.04339 −16.1805 2.93419 7.00000 −14.7134 9.00000 −15.8256
1.6 2.35308 3.00000 −2.46299 11.9881 7.05925 7.00000 −24.6203 9.00000 28.2090
1.7 4.40804 3.00000 11.4308 18.5472 13.2241 7.00000 15.1232 9.00000 81.7570
1.8 4.78226 3.00000 14.8700 −5.89437 14.3468 7.00000 32.8541 9.00000 −28.1884
1.9 5.47249 3.00000 21.9481 8.93744 16.4175 7.00000 76.3311 9.00000 48.9100
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( -1 \)
\(23\) \( +1 \)

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 483.4.a.i 9
3.b odd 2 1 1449.4.a.j 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.4.a.i 9 1.a even 1 1 trivial
1449.4.a.j 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 8T_{2}^{8} - 23T_{2}^{7} + 267T_{2}^{6} + 64T_{2}^{5} - 2685T_{2}^{4} + 526T_{2}^{3} + 8786T_{2}^{2} - 1400T_{2} - 5336 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(483))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 8 T^{8} + \cdots - 5336 \) Copy content Toggle raw display
$3$ \( (T - 3)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - 39 T^{8} + \cdots + 919405312 \) Copy content Toggle raw display
$7$ \( (T - 7)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 4877431083904 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 11393014127088 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots - 11\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 31\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( (T + 23)^{9} \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 59\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 32\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 51\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 81\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 11\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 14\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 44\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots - 61\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 91\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 22\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 21\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 11\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 24\!\cdots\!88 \) Copy content Toggle raw display
show more
show less