# Properties

 Label 464.4.a.l Level $464$ Weight $4$ Character orbit 464.a Self dual yes Analytic conductor $27.377$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$464 = 2^{4} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 464.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.3768862427$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.13458092.1 Defining polynomial: $$x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8$$ x^5 - x^4 - 14*x^3 + 18*x^2 + 20*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 29) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 2) q^{3} + (\beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{5} + (2 \beta_{4} - \beta_{2} - 8) q^{7} + ( - 6 \beta_{3} + \beta_{2} + 5 \beta_1 + 8) q^{9}+O(q^{10})$$ q + (b3 - 2) * q^3 + (b4 + b3 + b2 - 2*b1 + 2) * q^5 + (2*b4 - b2 - 8) * q^7 + (-6*b3 + b2 + 5*b1 + 8) * q^9 $$q + (\beta_{3} - 2) q^{3} + (\beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{5} + (2 \beta_{4} - \beta_{2} - 8) q^{7} + ( - 6 \beta_{3} + \beta_{2} + 5 \beta_1 + 8) q^{9} + ( - 7 \beta_{4} - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{11} + ( - 10 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + \beta_1 + 1) q^{13} + (\beta_{4} - 12 \beta_{3} - 2 \beta_{2} + 13 \beta_1 + 17) q^{15} + (3 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 7 \beta_1 + 13) q^{17} + ( - 3 \beta_{4} - \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 43) q^{19} + ( - 7 \beta_{4} - 9 \beta_{3} - 7 \beta_1 + 5) q^{21} + ( - 4 \beta_{4} - 20 \beta_{3} + 7 \beta_{2} + 6 \beta_1 - 26) q^{23} + (15 \beta_{4} - 17 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 49) q^{25} + (3 \beta_{4} + 24 \beta_{3} + 11 \beta_{2} - 35 \beta_1 - 75) q^{27} - 29 q^{29} + (8 \beta_{4} - 13 \beta_{3} + 13 \beta_{2} + 6 \beta_1 - 80) q^{31} + (11 \beta_{4} + 3 \beta_{3} - 20 \beta_{2} - 2 \beta_1 - 116) q^{33} + (30 \beta_{4} - 13 \beta_{2} + 12 \beta_1 + 8) q^{35} + (12 \beta_{4} - 40 \beta_{3} + 22 \beta_{2} + 18 \beta_1 + 88) q^{37} + (8 \beta_{4} + 3 \beta_{3} - 11 \beta_{2} + 8 \beta_1 + 72) q^{39} + (19 \beta_{4} + 25 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 241) q^{41} + (7 \beta_{4} + 4 \beta_{3} + 27 \beta_{2} - 29 \beta_1 + 45) q^{43} + ( - 35 \beta_{4} + 91 \beta_{3} - 3 \beta_{2} - 43 \beta_1 - 329) q^{45} + ( - 9 \beta_{4} + 16 \beta_{3} - 58 \beta_{2} + 25 \beta_1 - 61) q^{47} + ( - 2 \beta_{4} + 6 \beta_{3} + 32 \beta_{2} - 30 \beta_1 - 53) q^{49} + ( - 12 \beta_{4} + 52 \beta_{3} + 17 \beta_{2} - 42 \beta_1 - 58) q^{51} + (2 \beta_{4} + 8 \beta_{3} + 26 \beta_{2} + 15 \beta_1 - 117) q^{53} + ( - 86 \beta_{4} - 15 \beta_{3} + 27 \beta_{2} - 26 \beta_1 - 98) q^{55} + ( - 9 \beta_{4} - 19 \beta_{3} - 5 \beta_{2} - 33 \beta_1 + 23) q^{57} + (26 \beta_{4} - 10 \beta_{3} - 11 \beta_{2} - 20 \beta_1 - 80) q^{59} + (9 \beta_{4} + 3 \beta_{3} - 40 \beta_{2} + 57 \beta_1 + 111) q^{61} + ( - 40 \beta_{4} + 20 \beta_{3} - 10 \beta_{2} - 24 \beta_1 - 164) q^{63} + ( - 90 \beta_{4} - 56 \beta_{3} + 33 \beta_{2} + 41 \beta_1 - 317) q^{65} + (48 \beta_{4} - 48 \beta_{3} + 22 \beta_{2} + 92 \beta_1 - 232) q^{67} + (29 \beta_{4} + 75 \beta_{3} + 8 \beta_{2} - 73 \beta_1 - 409) q^{69} + ( - 16 \beta_{4} + 10 \beta_{3} - 32 \beta_{2} + 112 \beta_1 + 112) q^{71} + ( - 30 \beta_{4} - 2 \beta_{3} - 34 \beta_{2} - 36 \beta_1 - 382) q^{73} + ( - 24 \beta_{4} + 84 \beta_{3} - 10 \beta_{2} - 82 \beta_1 - 632) q^{75} + (15 \beta_{4} + 33 \beta_{3} - 16 \beta_{2} + 143 \beta_1 - 365) q^{77} + (127 \beta_{4} + 50 \beta_{3} + 38 \beta_{2} + 13 \beta_1 - 77) q^{79} + (27 \beta_{4} - 163 \beta_{3} - 83 \beta_{2} + 107 \beta_1 + 404) q^{81} + ( - 2 \beta_{4} - 58 \beta_{3} + 55 \beta_{2} + 44 \beta_1 - 88) q^{83} + (45 \beta_{4} + 91 \beta_{3} - 8 \beta_{2} - 25 \beta_1 - 357) q^{85} + ( - 29 \beta_{3} + 58) q^{87} + (121 \beta_{4} + 31 \beta_{3} + 28 \beta_{2} - 21 \beta_1 + 165) q^{89} + ( - 58 \beta_{4} - 36 \beta_{3} - 3 \beta_{2} + 72 \beta_1 - 516) q^{91} + (23 \beta_{4} - 25 \beta_{3} + 39 \beta_{2} - 20 \beta_1 + 6) q^{93} + ( - 47 \beta_{4} - 17 \beta_{3} - 35 \beta_{2} + 85 \beta_1 - 459) q^{95} + ( - 31 \beta_{4} - \beta_{3} - 22 \beta_{2} + 79 \beta_1 + 297) q^{97} + (107 \beta_{4} - 73 \beta_{3} - 5 \beta_{2} - 11 \beta_1 + 79) q^{99}+O(q^{100})$$ q + (b3 - 2) * q^3 + (b4 + b3 + b2 - 2*b1 + 2) * q^5 + (2*b4 - b2 - 8) * q^7 + (-6*b3 + b2 + 5*b1 + 8) * q^9 + (-7*b4 - 2*b3 - b2 - 3*b1 - 1) * q^11 + (-10*b4 + 4*b3 - 4*b2 + b1 + 1) * q^13 + (b4 - 12*b3 - 2*b2 + 13*b1 + 17) * q^15 + (3*b4 - 3*b3 - 2*b2 + 7*b1 + 13) * q^17 + (-3*b4 - b3 - 5*b2 + 3*b1 - 43) * q^19 + (-7*b4 - 9*b3 - 7*b1 + 5) * q^21 + (-4*b4 - 20*b3 + 7*b2 + 6*b1 - 26) * q^23 + (15*b4 - 17*b3 + 2*b2 - 4*b1 + 49) * q^25 + (3*b4 + 24*b3 + 11*b2 - 35*b1 - 75) * q^27 - 29 * q^29 + (8*b4 - 13*b3 + 13*b2 + 6*b1 - 80) * q^31 + (11*b4 + 3*b3 - 20*b2 - 2*b1 - 116) * q^33 + (30*b4 - 13*b2 + 12*b1 + 8) * q^35 + (12*b4 - 40*b3 + 22*b2 + 18*b1 + 88) * q^37 + (8*b4 + 3*b3 - 11*b2 + 8*b1 + 72) * q^39 + (19*b4 + 25*b3 - 2*b2 - 3*b1 - 241) * q^41 + (7*b4 + 4*b3 + 27*b2 - 29*b1 + 45) * q^43 + (-35*b4 + 91*b3 - 3*b2 - 43*b1 - 329) * q^45 + (-9*b4 + 16*b3 - 58*b2 + 25*b1 - 61) * q^47 + (-2*b4 + 6*b3 + 32*b2 - 30*b1 - 53) * q^49 + (-12*b4 + 52*b3 + 17*b2 - 42*b1 - 58) * q^51 + (2*b4 + 8*b3 + 26*b2 + 15*b1 - 117) * q^53 + (-86*b4 - 15*b3 + 27*b2 - 26*b1 - 98) * q^55 + (-9*b4 - 19*b3 - 5*b2 - 33*b1 + 23) * q^57 + (26*b4 - 10*b3 - 11*b2 - 20*b1 - 80) * q^59 + (9*b4 + 3*b3 - 40*b2 + 57*b1 + 111) * q^61 + (-40*b4 + 20*b3 - 10*b2 - 24*b1 - 164) * q^63 + (-90*b4 - 56*b3 + 33*b2 + 41*b1 - 317) * q^65 + (48*b4 - 48*b3 + 22*b2 + 92*b1 - 232) * q^67 + (29*b4 + 75*b3 + 8*b2 - 73*b1 - 409) * q^69 + (-16*b4 + 10*b3 - 32*b2 + 112*b1 + 112) * q^71 + (-30*b4 - 2*b3 - 34*b2 - 36*b1 - 382) * q^73 + (-24*b4 + 84*b3 - 10*b2 - 82*b1 - 632) * q^75 + (15*b4 + 33*b3 - 16*b2 + 143*b1 - 365) * q^77 + (127*b4 + 50*b3 + 38*b2 + 13*b1 - 77) * q^79 + (27*b4 - 163*b3 - 83*b2 + 107*b1 + 404) * q^81 + (-2*b4 - 58*b3 + 55*b2 + 44*b1 - 88) * q^83 + (45*b4 + 91*b3 - 8*b2 - 25*b1 - 357) * q^85 + (-29*b3 + 58) * q^87 + (121*b4 + 31*b3 + 28*b2 - 21*b1 + 165) * q^89 + (-58*b4 - 36*b3 - 3*b2 + 72*b1 - 516) * q^91 + (23*b4 - 25*b3 + 39*b2 - 20*b1 + 6) * q^93 + (-47*b4 - 17*b3 - 35*b2 + 85*b1 - 459) * q^95 + (-31*b4 - b3 - 22*b2 + 79*b1 + 297) * q^97 + (107*b4 - 73*b3 - 5*b2 - 11*b1 + 79) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 8 q^{3} + 10 q^{5} - 40 q^{7} + 33 q^{9}+O(q^{10})$$ 5 * q - 8 * q^3 + 10 * q^5 - 40 * q^7 + 33 * q^9 $$5 q - 8 q^{3} + 10 q^{5} - 40 q^{7} + 33 q^{9} - 12 q^{11} + 14 q^{13} + 74 q^{15} + 66 q^{17} - 214 q^{19} - 164 q^{23} + 207 q^{25} - 362 q^{27} - 145 q^{29} - 420 q^{31} - 576 q^{33} + 52 q^{35} + 378 q^{37} + 374 q^{39} - 1158 q^{41} + 204 q^{43} - 1506 q^{45} - 248 q^{47} - 283 q^{49} - 228 q^{51} - 554 q^{53} - 546 q^{55} + 44 q^{57} - 440 q^{59} + 618 q^{61} - 804 q^{63} - 1656 q^{65} - 1164 q^{67} - 1968 q^{69} + 692 q^{71} - 1950 q^{73} - 3074 q^{75} - 1616 q^{77} - 272 q^{79} + 1801 q^{81} - 512 q^{83} - 1628 q^{85} + 232 q^{87} + 866 q^{89} - 2580 q^{91} - 40 q^{93} - 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100})$$ 5 * q - 8 * q^3 + 10 * q^5 - 40 * q^7 + 33 * q^9 - 12 * q^11 + 14 * q^13 + 74 * q^15 + 66 * q^17 - 214 * q^19 - 164 * q^23 + 207 * q^25 - 362 * q^27 - 145 * q^29 - 420 * q^31 - 576 * q^33 + 52 * q^35 + 378 * q^37 + 374 * q^39 - 1158 * q^41 + 204 * q^43 - 1506 * q^45 - 248 * q^47 - 283 * q^49 - 228 * q^51 - 554 * q^53 - 546 * q^55 + 44 * q^57 - 440 * q^59 + 618 * q^61 - 804 * q^63 - 1656 * q^65 - 1164 * q^67 - 1968 * q^69 + 692 * q^71 - 1950 * q^73 - 3074 * q^75 - 1616 * q^77 - 272 * q^79 + 1801 * q^81 - 512 * q^83 - 1628 * q^85 + 232 * q^87 + 866 * q^89 - 2580 * q^91 - 40 * q^93 - 2244 * q^95 + 1562 * q^97 + 238 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + 2\nu - 6$$ v^2 + 2*v - 6 $$\beta_{2}$$ $$=$$ $$\nu^{3} + \nu^{2} - 8\nu - 2$$ v^3 + v^2 - 8*v - 2 $$\beta_{3}$$ $$=$$ $$( \nu^{4} + \nu^{3} - 10\nu^{2} - 2\nu + 2 ) / 2$$ (v^4 + v^3 - 10*v^2 - 2*v + 2) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 14\nu^{2} + 18\nu + 16 ) / 2$$ (v^4 - v^3 - 14*v^2 + 18*v + 16) / 2
 $$\nu$$ $$=$$ $$( \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 4$$ (b4 - b3 + b2 + b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( -\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 11 ) / 2$$ (-b4 + b3 - b2 + b1 + 11) / 2 $$\nu^{3}$$ $$=$$ $$( 5\beta_{4} - 5\beta_{3} + 7\beta_{2} + 3\beta _1 - 3 ) / 2$$ (5*b4 - 5*b3 + 7*b2 + 3*b1 - 3) / 2 $$\nu^{4}$$ $$=$$ $$-7\beta_{4} + 9\beta_{3} - 8\beta_{2} + 4\beta _1 + 55$$ -7*b4 + 9*b3 - 8*b2 + 4*b1 + 55

## 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
 2.27399 −0.957567 0.328194 −3.68360 3.03898
0 −9.87991 0 −16.8209 0 −5.21997 0 70.6126 0
1.2 0 −4.64574 0 12.8729 0 −26.0540 0 −5.41713 0
1.3 0 −1.84328 0 18.3339 0 16.8583 0 −23.6023 0
1.4 0 1.90549 0 −6.52855 0 −5.22706 0 −23.3691 0
1.5 0 6.46343 0 2.14270 0 −20.3573 0 14.7760 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$29$$ $$1$$

## 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 464.4.a.l 5
4.b odd 2 1 29.4.a.b 5
8.b even 2 1 1856.4.a.bb 5
8.d odd 2 1 1856.4.a.y 5
12.b even 2 1 261.4.a.f 5
20.d odd 2 1 725.4.a.c 5
28.d even 2 1 1421.4.a.e 5
116.d odd 2 1 841.4.a.b 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.a.b 5 4.b odd 2 1
261.4.a.f 5 12.b even 2 1
464.4.a.l 5 1.a even 1 1 trivial
725.4.a.c 5 20.d odd 2 1
841.4.a.b 5 116.d odd 2 1
1421.4.a.e 5 28.d even 2 1
1856.4.a.y 5 8.d odd 2 1
1856.4.a.bb 5 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{5} + 8T_{3}^{4} - 52T_{3}^{3} - 322T_{3}^{2} + 187T_{3} + 1042$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(464))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} + 8 T^{4} - 52 T^{3} + \cdots + 1042$$
$5$ $$T^{5} - 10 T^{4} - 366 T^{3} + \cdots - 55534$$
$7$ $$T^{5} + 40 T^{4} + 84 T^{3} + \cdots - 243968$$
$11$ $$T^{5} + 12 T^{4} - 4892 T^{3} + \cdots - 30997958$$
$13$ $$T^{5} - 14 T^{4} - 7558 T^{3} + \cdots - 13078418$$
$17$ $$T^{5} - 66 T^{4} - 2444 T^{3} + \cdots + 19935872$$
$19$ $$T^{5} + 214 T^{4} + \cdots - 19441152$$
$23$ $$T^{5} + 164 T^{4} + \cdots + 7938109184$$
$29$ $$(T + 29)^{5}$$
$31$ $$T^{5} + 420 T^{4} + 45552 T^{3} + \cdots + 2094346$$
$37$ $$T^{5} - 378 T^{4} + \cdots + 23564115968$$
$41$ $$T^{5} + 1158 T^{4} + \cdots + 59613728000$$
$43$ $$T^{5} - 204 T^{4} + \cdots - 198643410886$$
$47$ $$T^{5} + 248 T^{4} + \cdots + 203435244846$$
$53$ $$T^{5} + 554 T^{4} + \cdots - 786854101018$$
$59$ $$T^{5} + 440 T^{4} + \cdots - 109032704000$$
$61$ $$T^{5} - 618 T^{4} + \cdots + 2140697762176$$
$67$ $$T^{5} + 1164 T^{4} + \cdots + 39308070146048$$
$71$ $$T^{5} - 692 T^{4} + \cdots - 98341318953856$$
$73$ $$T^{5} + 1950 T^{4} + \cdots + 7201878016$$
$79$ $$T^{5} + \cdots + 240961986300538$$
$83$ $$T^{5} + 512 T^{4} + \cdots - 6057622580224$$
$89$ $$T^{5} - 866 T^{4} + \cdots + 21549994365568$$
$97$ $$T^{5} - 1562 T^{4} + \cdots - 20480102175488$$