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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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 483.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4979225328\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.8184789.1
Defining polynomial: \(x^{4} - x^{3} - 76 x^{2} - 8 x + 1048\)
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

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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 3 + \beta_{1} ) q^{4} + ( -6 - \beta_{2} ) q^{5} + ( 3 + 3 \beta_{1} ) q^{6} -7 q^{7} + ( 5 - 5 \beta_{1} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 3 + \beta_{1} ) q^{4} + ( -6 - \beta_{2} ) q^{5} + ( 3 + 3 \beta_{1} ) q^{6} -7 q^{7} + ( 5 - 5 \beta_{1} ) q^{8} + 9 q^{9} + ( -11 - 7 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{10} + ( -13 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + ( 9 + 3 \beta_{1} ) q^{12} + ( -30 - 9 \beta_{1} + 5 \beta_{2} ) q^{13} + ( -7 - 7 \beta_{1} ) q^{14} + ( -18 - 3 \beta_{2} ) q^{15} + ( -69 - 3 \beta_{1} ) q^{16} + ( -3 - 6 \beta_{1} - \beta_{2} + 7 \beta_{3} ) q^{17} + ( 9 + 9 \beta_{1} ) q^{18} + ( 14 - 9 \beta_{1} + 8 \beta_{2} - 10 \beta_{3} ) q^{19} + ( -23 - 7 \beta_{1} + \beta_{2} - \beta_{3} ) q^{20} -21 q^{21} + ( -33 - 15 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{22} -23 q^{23} + ( 15 - 15 \beta_{1} ) q^{24} + ( 14 - 34 \beta_{1} + 8 \beta_{2} + \beta_{3} ) q^{25} + ( -95 - 25 \beta_{1} - 15 \beta_{2} + 5 \beta_{3} ) q^{26} + 27 q^{27} + ( -21 - 7 \beta_{1} ) q^{28} + ( 46 - 7 \beta_{1} + 4 \beta_{2} - 10 \beta_{3} ) q^{29} + ( -33 - 21 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{30} + ( -57 + 30 \beta_{1} - 11 \beta_{2} - \beta_{3} ) q^{31} + ( -139 - 29 \beta_{1} ) q^{32} + ( -39 - 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{33} + ( -33 - 11 \beta_{1} - 11 \beta_{2} + 27 \beta_{3} ) q^{34} + ( 42 + 7 \beta_{2} ) q^{35} + ( 27 + 9 \beta_{1} ) q^{36} + ( -35 + 64 \beta_{1} - \beta_{2} - 19 \beta_{3} ) q^{37} + ( -86 + 32 \beta_{1} - 4 \beta_{2} - 32 \beta_{3} ) q^{38} + ( -90 - 27 \beta_{1} + 15 \beta_{2} ) q^{39} + ( -5 + 35 \beta_{1} - 25 \beta_{2} + 5 \beta_{3} ) q^{40} + ( -88 + 53 \beta_{1} - 8 \beta_{2} ) q^{41} + ( -21 - 21 \beta_{1} ) q^{42} + ( 14 - 39 \beta_{1} + 5 \beta_{2} + 20 \beta_{3} ) q^{43} + ( -59 - 19 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{44} + ( -54 - 9 \beta_{2} ) q^{45} + ( -23 - 23 \beta_{1} ) q^{46} + ( -183 - 21 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{47} + ( -207 - 9 \beta_{1} ) q^{48} + 49 q^{49} + ( -281 + 21 \beta_{1} - 26 \beta_{2} + 12 \beta_{3} ) q^{50} + ( -9 - 18 \beta_{1} - 3 \beta_{2} + 21 \beta_{3} ) q^{51} + ( -155 - 43 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{52} + ( -182 - 50 \beta_{1} - 7 \beta_{2} + 12 \beta_{3} ) q^{53} + ( 27 + 27 \beta_{1} ) q^{54} + ( 139 - 15 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} ) q^{55} + ( -35 + 35 \beta_{1} ) q^{56} + ( 42 - 27 \beta_{1} + 24 \beta_{2} - 30 \beta_{3} ) q^{57} + ( -54 + 60 \beta_{1} + 8 \beta_{2} - 36 \beta_{3} ) q^{58} + ( 161 + 23 \beta_{1} - 8 \beta_{2} + 21 \beta_{3} ) q^{59} + ( -69 - 21 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{60} + ( 294 - 55 \beta_{1} - 3 \beta_{2} - 22 \beta_{3} ) q^{61} + ( 183 - 67 \beta_{1} + 35 \beta_{2} - 15 \beta_{3} ) q^{62} -63 q^{63} + ( 123 - 115 \beta_{1} ) q^{64} + ( -290 + 233 \beta_{1} - 16 \beta_{2} + 4 \beta_{3} ) q^{65} + ( -99 - 45 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} ) q^{66} + ( 100 - 10 \beta_{1} + 71 \beta_{2} - 16 \beta_{3} ) q^{67} + ( -39 - 23 \beta_{1} - 13 \beta_{2} + 41 \beta_{3} ) q^{68} -69 q^{69} + ( 77 + 49 \beta_{1} - 21 \beta_{2} + 7 \beta_{3} ) q^{70} + ( -725 + 76 \beta_{1} + 8 \beta_{2} - 11 \beta_{3} ) q^{71} + ( 45 - 45 \beta_{1} ) q^{72} + ( -246 + 71 \beta_{1} - 44 \beta_{2} + 82 \beta_{3} ) q^{73} + ( 505 - 17 \beta_{1} + 41 \beta_{2} - 77 \beta_{3} ) q^{74} + ( 42 - 102 \beta_{1} + 24 \beta_{2} + 3 \beta_{3} ) q^{75} + ( -58 + 14 \beta_{1} + 12 \beta_{2} - 52 \beta_{3} ) q^{76} + ( 91 + 14 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} ) q^{77} + ( -285 - 75 \beta_{1} - 45 \beta_{2} + 15 \beta_{3} ) q^{78} + ( -124 - 19 \beta_{1} - 28 \beta_{2} + 46 \beta_{3} ) q^{79} + ( 429 + 21 \beta_{1} + 57 \beta_{2} + 3 \beta_{3} ) q^{80} + 81 q^{81} + ( 402 - 96 \beta_{1} + 24 \beta_{2} - 8 \beta_{3} ) q^{82} + ( 408 + 61 \beta_{1} + 52 \beta_{2} - 4 \beta_{3} ) q^{83} + ( -63 - 21 \beta_{1} ) q^{84} + ( -213 + 43 \beta_{1} + 2 \beta_{2} - 49 \beta_{3} ) q^{85} + ( -251 - \beta_{1} - 55 \beta_{2} + 85 \beta_{3} ) q^{86} + ( 138 - 21 \beta_{1} + 12 \beta_{2} - 30 \beta_{3} ) q^{87} + ( 35 + 55 \beta_{1} - 15 \beta_{2} - 5 \beta_{3} ) q^{88} + ( -374 + 169 \beta_{1} - 3 \beta_{2} - 64 \beta_{3} ) q^{89} + ( -99 - 63 \beta_{1} + 27 \beta_{2} - 9 \beta_{3} ) q^{90} + ( 210 + 63 \beta_{1} - 35 \beta_{2} ) q^{91} + ( -69 - 23 \beta_{1} ) q^{92} + ( -171 + 90 \beta_{1} - 33 \beta_{2} - 3 \beta_{3} ) q^{93} + ( -403 - 191 \beta_{1} + 9 \beta_{2} + 7 \beta_{3} ) q^{94} + ( -343 + 285 \beta_{1} - 96 \beta_{2} + 81 \beta_{3} ) q^{95} + ( -417 - 87 \beta_{1} ) q^{96} + ( 567 + 234 \beta_{1} + 27 \beta_{2} - 15 \beta_{3} ) q^{97} + ( 49 + 49 \beta_{1} ) q^{98} + ( -117 - 18 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 12q^{3} + 10q^{4} - 23q^{5} + 6q^{6} - 28q^{7} + 30q^{8} + 36q^{9} + O(q^{10}) \) \( 4q + 2q^{2} + 12q^{3} + 10q^{4} - 23q^{5} + 6q^{6} - 28q^{7} + 30q^{8} + 36q^{9} - 32q^{10} - 48q^{11} + 30q^{12} - 107q^{13} - 14q^{14} - 69q^{15} - 270q^{16} - 6q^{17} + 18q^{18} + 76q^{19} - 78q^{20} - 84q^{21} - 106q^{22} - 92q^{23} + 90q^{24} + 115q^{25} - 320q^{26} + 108q^{27} - 70q^{28} + 204q^{29} - 96q^{30} - 276q^{31} - 498q^{32} - 144q^{33} - 126q^{34} + 161q^{35} + 90q^{36} - 248q^{37} - 372q^{38} - 321q^{39} - 70q^{40} - 450q^{41} - 42q^{42} + 109q^{43} - 202q^{44} - 207q^{45} - 46q^{46} - 688q^{47} - 810q^{48} + 196q^{49} - 1152q^{50} - 18q^{51} - 534q^{52} - 633q^{53} + 54q^{54} + 581q^{55} - 210q^{56} + 228q^{57} - 308q^{58} + 585q^{59} - 234q^{60} + 1311q^{61} + 846q^{62} - 252q^{63} + 722q^{64} - 1614q^{65} - 318q^{66} + 365q^{67} - 138q^{68} - 276q^{69} + 224q^{70} - 3049q^{71} + 270q^{72} - 1164q^{73} + 2090q^{74} + 345q^{75} - 220q^{76} + 336q^{77} - 960q^{78} - 476q^{79} + 1614q^{80} + 324q^{81} + 1784q^{82} + 1462q^{83} - 210q^{84} - 891q^{85} - 1032q^{86} + 612q^{87} + 50q^{88} - 1767q^{89} - 288q^{90} + 749q^{91} - 230q^{92} - 828q^{93} - 1246q^{94} - 1927q^{95} - 1494q^{96} + 1788q^{97} + 98q^{98} - 432q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 3 \nu^{2} - 42 \nu + 48 \)\()/28\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 3 \nu - 38 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 17 \nu^{2} + 56 \nu - 608 \)\()/28\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{3} + \beta_{2} + 3 \beta_{1} + 79\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(51 \beta_{3} - 39 \beta_{2} + 107 \beta_{1} + 183\)\()/2\)

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.

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(483))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -10 - T + T^{2} )^{2} \)
$3$ \( ( -3 + T )^{4} \)
$5$ \( -10424 - 2738 T - 43 T^{2} + 23 T^{3} + T^{4} \)
$7$ \( ( 7 + T )^{4} \)
$11$ \( -13784 - 1158 T + 497 T^{2} + 48 T^{3} + T^{4} \)
$13$ \( -12225800 - 520610 T - 2479 T^{2} + 107 T^{3} + T^{4} \)
$17$ \( 2287800 + 173430 T - 11553 T^{2} + 6 T^{3} + T^{4} \)
$19$ \( 58338172 + 1771924 T - 24049 T^{2} - 76 T^{3} + T^{4} \)
$23$ \( ( 23 + T )^{4} \)
$29$ \( -136834868 + 3162036 T - 7609 T^{2} - 204 T^{3} + T^{4} \)
$31$ \( 13064256 - 334926 T - 14379 T^{2} + 276 T^{3} + T^{4} \)
$37$ \( -3344244056 - 44950306 T - 133779 T^{2} + 248 T^{3} + T^{4} \)
$41$ \( 94298692 - 4248996 T + 11597 T^{2} + 450 T^{3} + T^{4} \)
$43$ \( 1500471424 + 20339362 T - 139597 T^{2} - 109 T^{3} + T^{4} \)
$47$ \( 369838528 + 14809448 T + 162488 T^{2} + 688 T^{3} + T^{4} \)
$53$ \( -180702128 - 813258 T + 72899 T^{2} + 633 T^{3} + T^{4} \)
$59$ \( -69476672 + 2659044 T + 16955 T^{2} - 585 T^{3} + T^{4} \)
$61$ \( -29331062568 + 54839034 T + 410721 T^{2} - 1311 T^{3} + T^{4} \)
$67$ \( -14949193664 + 296983730 T - 976255 T^{2} - 365 T^{3} + T^{4} \)
$71$ \( 250630243168 + 1547149988 T + 3344471 T^{2} + 3049 T^{3} + T^{4} \)
$73$ \( -333083542356 - 1558523268 T - 1092825 T^{2} + 1164 T^{3} + T^{4} \)
$79$ \( 15443424448 - 79974128 T - 378097 T^{2} + 476 T^{3} + T^{4} \)
$83$ \( -236900237516 + 673928620 T + 53861 T^{2} - 1462 T^{3} + T^{4} \)
$89$ \( -642666118448 - 1710935646 T - 301381 T^{2} + 1767 T^{3} + T^{4} \)
$97$ \( -20129702688 + 964895058 T - 127611 T^{2} - 1788 T^{3} + T^{4} \)
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