# Properties

 Label 483.4.a.a Level $483$ Weight $4$ Character orbit 483.a Self dual yes Analytic conductor $28.498$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## 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: $$3$$ Coefficient field: 3.3.3877.1 Defining polynomial: $$x^{3} - x^{2} - 13 x - 10$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} ) q^{2} + 3 q^{3} + ( 1 + 4 \beta_{1} + \beta_{2} ) q^{4} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{5} + ( -3 - 3 \beta_{1} ) q^{6} -7 q^{7} + ( -27 - 8 \beta_{1} - 4 \beta_{2} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} ) q^{2} + 3 q^{3} + ( 1 + 4 \beta_{1} + \beta_{2} ) q^{4} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{5} + ( -3 - 3 \beta_{1} ) q^{6} -7 q^{7} + ( -27 - 8 \beta_{1} - 4 \beta_{2} ) q^{8} + 9 q^{9} + ( 13 + 2 \beta_{1} + 2 \beta_{2} ) q^{10} + ( 2 + 7 \beta_{1} - 9 \beta_{2} ) q^{11} + ( 3 + 12 \beta_{1} + 3 \beta_{2} ) q^{12} + ( 9 + \beta_{1} + 4 \beta_{2} ) q^{13} + ( 7 + 7 \beta_{1} ) q^{14} + ( 3 - 6 \beta_{1} + 3 \beta_{2} ) q^{15} + ( 91 + 31 \beta_{1} ) q^{16} + ( -26 - \beta_{1} - 7 \beta_{2} ) q^{17} + ( -9 - 9 \beta_{1} ) q^{18} + ( 6 + 15 \beta_{1} - \beta_{2} ) q^{19} + ( -41 - 9 \beta_{1} - 10 \beta_{2} ) q^{20} -21 q^{21} + ( -40 + 4 \beta_{1} - 7 \beta_{2} ) q^{22} -23 q^{23} + ( -81 - 24 \beta_{1} - 12 \beta_{2} ) q^{24} + ( -82 - 9 \beta_{1} + 8 \beta_{2} ) q^{25} + ( -25 - 24 \beta_{1} - \beta_{2} ) q^{26} + 27 q^{27} + ( -7 - 28 \beta_{1} - 7 \beta_{2} ) q^{28} + ( -113 + 6 \beta_{1} - 8 \beta_{2} ) q^{29} + ( 39 + 6 \beta_{1} + 6 \beta_{2} ) q^{30} + ( -156 + 7 \beta_{1} + 9 \beta_{2} ) q^{31} + ( -123 - 120 \beta_{1} + \beta_{2} ) q^{32} + ( 6 + 21 \beta_{1} - 27 \beta_{2} ) q^{33} + ( 48 + 50 \beta_{1} + \beta_{2} ) q^{34} + ( -7 + 14 \beta_{1} - 7 \beta_{2} ) q^{35} + ( 9 + 36 \beta_{1} + 9 \beta_{2} ) q^{36} + ( -85 + 38 \beta_{1} + 34 \beta_{2} ) q^{37} + ( -124 - 48 \beta_{1} - 15 \beta_{2} ) q^{38} + ( 27 + 3 \beta_{1} + 12 \beta_{2} ) q^{39} + ( 29 + 82 \beta_{1} - 7 \beta_{2} ) q^{40} + ( 51 + 62 \beta_{1} - 40 \beta_{2} ) q^{41} + ( 21 + 21 \beta_{1} ) q^{42} + ( -129 + 33 \beta_{1} + 34 \beta_{2} ) q^{43} + ( 6 - 7 \beta_{1} + 68 \beta_{2} ) q^{44} + ( 9 - 18 \beta_{1} + 9 \beta_{2} ) q^{45} + ( 23 + 23 \beta_{1} ) q^{46} + ( -3 - 113 \beta_{1} + 51 \beta_{2} ) q^{47} + ( 273 + 93 \beta_{1} ) q^{48} + 49 q^{49} + ( 138 + 85 \beta_{1} + 9 \beta_{2} ) q^{50} + ( -78 - 3 \beta_{1} - 21 \beta_{2} ) q^{51} + ( 147 + 92 \beta_{1} - 8 \beta_{2} ) q^{52} + ( -6 - 49 \beta_{1} + 94 \beta_{2} ) q^{53} + ( -27 - 27 \beta_{1} ) q^{54} + ( -222 + 59 \beta_{1} - 28 \beta_{2} ) q^{55} + ( 189 + 56 \beta_{1} + 28 \beta_{2} ) q^{56} + ( 18 + 45 \beta_{1} - 3 \beta_{2} ) q^{57} + ( 81 + 119 \beta_{1} - 6 \beta_{2} ) q^{58} + ( -564 + 97 \beta_{1} + 10 \beta_{2} ) q^{59} + ( -123 - 27 \beta_{1} - 30 \beta_{2} ) q^{60} + ( -174 + 64 \beta_{1} + 9 \beta_{2} ) q^{61} + ( 82 + 108 \beta_{1} - 7 \beta_{2} ) q^{62} -63 q^{63} + ( 353 + 232 \beta_{1} + 120 \beta_{2} ) q^{64} + ( 51 - 46 \beta_{1} + 10 \beta_{2} ) q^{65} + ( -120 + 12 \beta_{1} - 21 \beta_{2} ) q^{66} + ( -14 - 87 \beta_{1} - 110 \beta_{2} ) q^{67} + ( -242 - 193 \beta_{1} + 6 \beta_{2} ) q^{68} -69 q^{69} + ( -91 - 14 \beta_{1} - 14 \beta_{2} ) q^{70} + ( 430 + 52 \beta_{1} - 87 \beta_{2} ) q^{71} + ( -243 - 72 \beta_{1} - 36 \beta_{2} ) q^{72} + ( -102 + 141 \beta_{1} - 177 \beta_{2} ) q^{73} + ( -287 - 131 \beta_{1} - 38 \beta_{2} ) q^{74} + ( -246 - 27 \beta_{1} + 24 \beta_{2} ) q^{75} + ( 490 + 193 \beta_{1} + 56 \beta_{2} ) q^{76} + ( -14 - 49 \beta_{1} + 63 \beta_{2} ) q^{77} + ( -75 - 72 \beta_{1} - 3 \beta_{2} ) q^{78} + ( 294 - 9 \beta_{1} + 121 \beta_{2} ) q^{79} + ( -343 - 182 \beta_{1} - 2 \beta_{2} ) q^{80} + 81 q^{81} + ( -467 - 117 \beta_{1} - 62 \beta_{2} ) q^{82} + ( -598 - 151 \beta_{1} + 99 \beta_{2} ) q^{83} + ( -21 - 84 \beta_{1} - 21 \beta_{2} ) q^{84} + ( -110 + 101 \beta_{1} - 30 \beta_{2} ) q^{85} + ( -203 - 72 \beta_{1} - 33 \beta_{2} ) q^{86} + ( -339 + 18 \beta_{1} - 24 \beta_{2} ) q^{87} + ( 234 - 221 \beta_{1} + 63 \beta_{2} ) q^{88} + ( -226 + 54 \beta_{1} - 287 \beta_{2} ) q^{89} + ( 117 + 18 \beta_{1} + 18 \beta_{2} ) q^{90} + ( -63 - 7 \beta_{1} - 28 \beta_{2} ) q^{91} + ( -23 - 92 \beta_{1} - 23 \beta_{2} ) q^{92} + ( -468 + 21 \beta_{1} + 27 \beta_{2} ) q^{93} + ( 805 + 189 \beta_{1} + 113 \beta_{2} ) q^{94} + ( -218 - 5 \beta_{1} - 40 \beta_{2} ) q^{95} + ( -369 - 360 \beta_{1} + 3 \beta_{2} ) q^{96} + ( -549 - 170 \beta_{1} + 20 \beta_{2} ) q^{97} + ( -49 - 49 \beta_{1} ) q^{98} + ( 18 + 63 \beta_{1} - 81 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 4q^{2} + 9q^{3} + 8q^{4} + 2q^{5} - 12q^{6} - 21q^{7} - 93q^{8} + 27q^{9} + O(q^{10})$$ $$3q - 4q^{2} + 9q^{3} + 8q^{4} + 2q^{5} - 12q^{6} - 21q^{7} - 93q^{8} + 27q^{9} + 43q^{10} + 4q^{11} + 24q^{12} + 32q^{13} + 28q^{14} + 6q^{15} + 304q^{16} - 86q^{17} - 36q^{18} + 32q^{19} - 142q^{20} - 63q^{21} - 123q^{22} - 69q^{23} - 279q^{24} - 247q^{25} - 100q^{26} + 81q^{27} - 56q^{28} - 341q^{29} + 129q^{30} - 452q^{31} - 488q^{32} + 12q^{33} + 195q^{34} - 14q^{35} + 72q^{36} - 183q^{37} - 435q^{38} + 96q^{39} + 162q^{40} + 175q^{41} + 84q^{42} - 320q^{43} + 79q^{44} + 18q^{45} + 92q^{46} - 71q^{47} + 912q^{48} + 147q^{49} + 508q^{50} - 258q^{51} + 525q^{52} + 27q^{53} - 108q^{54} - 635q^{55} + 651q^{56} + 96q^{57} + 356q^{58} - 1585q^{59} - 426q^{60} - 449q^{61} + 347q^{62} - 189q^{63} + 1411q^{64} + 117q^{65} - 369q^{66} - 239q^{67} - 913q^{68} - 207q^{69} - 301q^{70} + 1255q^{71} - 837q^{72} - 342q^{73} - 1030q^{74} - 741q^{75} + 1719q^{76} - 28q^{77} - 300q^{78} + 994q^{79} - 1213q^{80} + 243q^{81} - 1580q^{82} - 1846q^{83} - 168q^{84} - 259q^{85} - 714q^{86} - 1023q^{87} + 544q^{88} - 911q^{89} + 387q^{90} - 224q^{91} - 184q^{92} - 1356q^{93} + 2717q^{94} - 699q^{95} - 1464q^{96} - 1797q^{97} - 196q^{98} + 36q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 13 x - 10$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 8$$

## 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
 4.43745 −0.881781 −2.55567
−5.43745 3.00000 21.5659 −5.05882 −16.3124 −7.00000 −73.7640 9.00000 27.5071
1.2 −0.118219 3.00000 −7.98602 −2.69534 −0.354656 −7.00000 1.88985 9.00000 0.318639
1.3 1.55567 3.00000 −5.57988 9.75415 4.66702 −7.00000 −21.1259 9.00000 15.1743
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.a 3
3.b odd 2 1 1449.4.a.c 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 4 T_{2}^{2} - 8 T_{2} - 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(483))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 8 T + 4 T^{2} + T^{3}$$
$3$ $$( -3 + T )^{3}$$
$5$ $$-133 - 62 T - 2 T^{2} + T^{3}$$
$7$ $$( 7 + T )^{3}$$
$11$ $$16888 - 2217 T - 4 T^{2} + T^{3}$$
$13$ $$7121 - 108 T - 32 T^{2} + T^{3}$$
$17$ $$-32140 + 1157 T + 86 T^{2} + T^{3}$$
$19$ $$-4436 - 2569 T - 32 T^{2} + T^{3}$$
$23$ $$( 23 + T )^{3}$$
$29$ $$1279711 + 37027 T + 341 T^{2} + T^{3}$$
$31$ $$2969438 + 64913 T + 452 T^{2} + T^{3}$$
$37$ $$-3177991 - 47281 T + 183 T^{2} + T^{3}$$
$41$ $$11602699 - 62565 T - 175 T^{2} + T^{3}$$
$43$ $$-3480943 - 18274 T + 320 T^{2} + T^{3}$$
$47$ $$-30926812 - 190282 T + 71 T^{2} + T^{3}$$
$53$ $$9069176 - 220303 T - 27 T^{2} + T^{3}$$
$59$ $$57554534 + 701985 T + 1585 T^{2} + T^{3}$$
$61$ $$-11531794 + 6119 T + 449 T^{2} + T^{3}$$
$67$ $$-90556772 - 461783 T + 239 T^{2} + T^{3}$$
$71$ $$7078618 + 331891 T - 1255 T^{2} + T^{3}$$
$73$ $$20669796 - 828423 T + 342 T^{2} + T^{3}$$
$79$ $$161629604 - 34291 T - 994 T^{2} + T^{3}$$
$83$ $$-147725860 + 698209 T + 1846 T^{2} + T^{3}$$
$89$ $$-1448227670 - 1730103 T + 911 T^{2} + T^{3}$$
$97$ $$26418199 + 707003 T + 1797 T^{2} + T^{3}$$