Properties

Label 483.4.a.b
Level $483$
Weight $4$
Character orbit 483.a
Self dual yes
Analytic conductor $28.498$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.8184789.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 76x^{2} - 8x + 1048 \) 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + 3 q^{3} + (\beta_1 + 3) q^{4} + ( - \beta_{2} - 6) q^{5} + (3 \beta_1 + 3) q^{6} - 7 q^{7} + ( - 5 \beta_1 + 5) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + 3 q^{3} + (\beta_1 + 3) q^{4} + ( - \beta_{2} - 6) q^{5} + (3 \beta_1 + 3) q^{6} - 7 q^{7} + ( - 5 \beta_1 + 5) q^{8} + 9 q^{9} + ( - \beta_{3} + 3 \beta_{2} + \cdots - 11) q^{10}+ \cdots + (9 \beta_{3} - 9 \beta_{2} + \cdots - 117) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 12 q^{3} + 10 q^{4} - 23 q^{5} + 6 q^{6} - 28 q^{7} + 30 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 12 q^{3} + 10 q^{4} - 23 q^{5} + 6 q^{6} - 28 q^{7} + 30 q^{8} + 36 q^{9} - 32 q^{10} - 48 q^{11} + 30 q^{12} - 107 q^{13} - 14 q^{14} - 69 q^{15} - 270 q^{16} - 6 q^{17} + 18 q^{18} + 76 q^{19} - 78 q^{20} - 84 q^{21} - 106 q^{22} - 92 q^{23} + 90 q^{24} + 115 q^{25} - 320 q^{26} + 108 q^{27} - 70 q^{28} + 204 q^{29} - 96 q^{30} - 276 q^{31} - 498 q^{32} - 144 q^{33} - 126 q^{34} + 161 q^{35} + 90 q^{36} - 248 q^{37} - 372 q^{38} - 321 q^{39} - 70 q^{40} - 450 q^{41} - 42 q^{42} + 109 q^{43} - 202 q^{44} - 207 q^{45} - 46 q^{46} - 688 q^{47} - 810 q^{48} + 196 q^{49} - 1152 q^{50} - 18 q^{51} - 534 q^{52} - 633 q^{53} + 54 q^{54} + 581 q^{55} - 210 q^{56} + 228 q^{57} - 308 q^{58} + 585 q^{59} - 234 q^{60} + 1311 q^{61} + 846 q^{62} - 252 q^{63} + 722 q^{64} - 1614 q^{65} - 318 q^{66} + 365 q^{67} - 138 q^{68} - 276 q^{69} + 224 q^{70} - 3049 q^{71} + 270 q^{72} - 1164 q^{73} + 2090 q^{74} + 345 q^{75} - 220 q^{76} + 336 q^{77} - 960 q^{78} - 476 q^{79} + 1614 q^{80} + 324 q^{81} + 1784 q^{82} + 1462 q^{83} - 210 q^{84} - 891 q^{85} - 1032 q^{86} + 612 q^{87} + 50 q^{88} - 1767 q^{89} - 288 q^{90} + 749 q^{91} - 230 q^{92} - 828 q^{93} - 1246 q^{94} - 1927 q^{95} - 1494 q^{96} + 1788 q^{97} + 98 q^{98} - 432 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 76x^{2} - 8x + 1048 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 3\nu^{2} - 42\nu + 48 ) / 28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 3\nu - 38 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 17\nu^{2} + 56\nu - 608 ) / 28 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} + \beta_{2} + 3\beta _1 + 79 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 51\beta_{3} - 39\beta_{2} + 107\beta _1 + 183 ) / 2 \) 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
−6.68196
3.98040
8.39513
−4.69357
−2.70156 3.00000 −0.701562 −19.3472 −8.10469 −7.00000 23.5078 9.00000 52.2678
1.2 −2.70156 3.00000 −0.701562 11.0488 −8.10469 −7.00000 23.5078 9.00000 −29.8490
1.3 3.70156 3.00000 5.70156 −9.64642 11.1047 −7.00000 −8.50781 9.00000 −35.7068
1.4 3.70156 3.00000 5.70156 −5.05515 11.1047 −7.00000 −8.50781 9.00000 −18.7119
\(n\): e.g. 2-40 or 990-1000
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.b 4
3.b odd 2 1 1449.4.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.4.a.b 4 1.a even 1 1 trivial
1449.4.a.d 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 10 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(483))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T - 10)^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 23 T^{3} + \cdots - 10424 \) Copy content Toggle raw display
$7$ \( (T + 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 48 T^{3} + \cdots - 13784 \) Copy content Toggle raw display
$13$ \( T^{4} + 107 T^{3} + \cdots - 12225800 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + \cdots + 2287800 \) Copy content Toggle raw display
$19$ \( T^{4} - 76 T^{3} + \cdots + 58338172 \) Copy content Toggle raw display
$23$ \( (T + 23)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 204 T^{3} + \cdots - 136834868 \) Copy content Toggle raw display
$31$ \( T^{4} + 276 T^{3} + \cdots + 13064256 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 3344244056 \) Copy content Toggle raw display
$41$ \( T^{4} + 450 T^{3} + \cdots + 94298692 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 1500471424 \) Copy content Toggle raw display
$47$ \( T^{4} + 688 T^{3} + \cdots + 369838528 \) Copy content Toggle raw display
$53$ \( T^{4} + 633 T^{3} + \cdots - 180702128 \) Copy content Toggle raw display
$59$ \( T^{4} - 585 T^{3} + \cdots - 69476672 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 29331062568 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 14949193664 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 250630243168 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 333083542356 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 15443424448 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 236900237516 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 642666118448 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 20129702688 \) Copy content Toggle raw display
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