# Properties

 Label 464.4.a.i Level $464$ Weight $4$ Character orbit 464.a Self dual yes Analytic conductor $27.377$ Analytic rank $0$ Dimension $3$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.19816.1 Defining polynomial: $$x^{3} - x^{2} - 42x - 54$$ x^3 - x^2 - 42*x - 54 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{3} + (\beta_{2} + 2 \beta_1 + 6) q^{5} + (4 \beta_{2} - 8) q^{7} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^3 + (b2 + 2*b1 + 6) * q^5 + (4*b2 - 8) * q^7 + (3*b2 + 2*b1 + 1) * q^9 $$q + (\beta_1 - 1) q^{3} + (\beta_{2} + 2 \beta_1 + 6) q^{5} + (4 \beta_{2} - 8) q^{7} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9} + (2 \beta_{2} - \beta_1 - 3) q^{11} + (11 \beta_{2} + 8 \beta_1 - 4) q^{13} + (2 \beta_{2} + 13 \beta_1 + 39) q^{15} + ( - 16 \beta_{2} - 12 \beta_1 - 18) q^{17} + ( - 10 \beta_{2} + 8 \beta_1 + 52) q^{19} + ( - 16 \beta_{2} - 4 \beta_1 - 28) q^{21} + ( - 12 \beta_{2} + 6 \beta_1 + 66) q^{23} + (11 \beta_{2} + 40 \beta_1 + 13) q^{25} + ( - 6 \beta_{2} - 17 \beta_1 + 53) q^{27} + 29 q^{29} + (36 \beta_{2} + 11 \beta_1 + 25) q^{31} + ( - 11 \beta_{2} - 4 \beta_1 - 42) q^{33} + ( - 12 \beta_{2} - 24 \beta_1) q^{35} + (38 \beta_{2} - 12 \beta_1 - 10) q^{37} + ( - 20 \beta_{2} + 31 \beta_1 + 121) q^{39} + ( - 6 \beta_{2} + 4 \beta_1 + 186) q^{41} + ( - 2 \beta_{2} + 15 \beta_1 - 11) q^{43} + (4 \beta_{2} + 26 \beta_1 + 132) q^{45} + ( - 2 \beta_{2} - 33 \beta_1 - 207) q^{47} + ( - 80 \beta_{2} - 64 \beta_1 + 201) q^{49} + (28 \beta_{2} - 70 \beta_1 - 162) q^{51} + ( - 27 \beta_{2} - 116 \beta_1 + 276) q^{53} + ( - 8 \beta_{2} - 25 \beta_1 - 39) q^{55} + (64 \beta_{2} + 66 \beta_1 + 254) q^{57} + ( - 4 \beta_{2} + 86 \beta_1 - 90) q^{59} + (40 \beta_{2} - 80 \beta_1 + 134) q^{61} + ( - 56 \beta_{2} - 56 \beta_1 + 280) q^{63} + (9 \beta_{2} + 90 \beta_1 + 468) q^{65} + (72 \beta_{2} - 36 \beta_1 + 88) q^{67} + (66 \beta_{2} + 72 \beta_1 + 204) q^{69} + (4 \beta_{2} - 2 \beta_1 + 18) q^{71} + (30 \beta_{2} + 40 \beta_1 - 178) q^{73} + (76 \beta_{2} + 144 \beta_1 + 968) q^{75} + ( - 24 \beta_{2} - 28 \beta_1 + 300) q^{77} + (38 \beta_{2} + 37 \beta_1 + 691) q^{79} + ( - 108 \beta_{2} - 58 \beta_1 - 485) q^{81} + ( - 12 \beta_{2} - 162 \beta_1 + 150) q^{83} + ( - 38 \beta_{2} - 184 \beta_1 - 840) q^{85} + (29 \beta_1 - 29) q^{87} + (154 \beta_{2} + 68 \beta_1 + 282) q^{89} + ( - 244 \beta_{2} - 208 \beta_1 + 1064) q^{91} + ( - 111 \beta_{2} + 94 \beta_1 - 52) q^{93} + (86 \beta_{2} + 244 \beta_1 + 552) q^{95} + ( - 34 \beta_{2} + 176 \beta_1 + 14) q^{97} + ( - 22 \beta_{2} - 38 \beta_1 + 114) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^3 + (b2 + 2*b1 + 6) * q^5 + (4*b2 - 8) * q^7 + (3*b2 + 2*b1 + 1) * q^9 + (2*b2 - b1 - 3) * q^11 + (11*b2 + 8*b1 - 4) * q^13 + (2*b2 + 13*b1 + 39) * q^15 + (-16*b2 - 12*b1 - 18) * q^17 + (-10*b2 + 8*b1 + 52) * q^19 + (-16*b2 - 4*b1 - 28) * q^21 + (-12*b2 + 6*b1 + 66) * q^23 + (11*b2 + 40*b1 + 13) * q^25 + (-6*b2 - 17*b1 + 53) * q^27 + 29 * q^29 + (36*b2 + 11*b1 + 25) * q^31 + (-11*b2 - 4*b1 - 42) * q^33 + (-12*b2 - 24*b1) * q^35 + (38*b2 - 12*b1 - 10) * q^37 + (-20*b2 + 31*b1 + 121) * q^39 + (-6*b2 + 4*b1 + 186) * q^41 + (-2*b2 + 15*b1 - 11) * q^43 + (4*b2 + 26*b1 + 132) * q^45 + (-2*b2 - 33*b1 - 207) * q^47 + (-80*b2 - 64*b1 + 201) * q^49 + (28*b2 - 70*b1 - 162) * q^51 + (-27*b2 - 116*b1 + 276) * q^53 + (-8*b2 - 25*b1 - 39) * q^55 + (64*b2 + 66*b1 + 254) * q^57 + (-4*b2 + 86*b1 - 90) * q^59 + (40*b2 - 80*b1 + 134) * q^61 + (-56*b2 - 56*b1 + 280) * q^63 + (9*b2 + 90*b1 + 468) * q^65 + (72*b2 - 36*b1 + 88) * q^67 + (66*b2 + 72*b1 + 204) * q^69 + (4*b2 - 2*b1 + 18) * q^71 + (30*b2 + 40*b1 - 178) * q^73 + (76*b2 + 144*b1 + 968) * q^75 + (-24*b2 - 28*b1 + 300) * q^77 + (38*b2 + 37*b1 + 691) * q^79 + (-108*b2 - 58*b1 - 485) * q^81 + (-12*b2 - 162*b1 + 150) * q^83 + (-38*b2 - 184*b1 - 840) * q^85 + (29*b1 - 29) * q^87 + (154*b2 + 68*b1 + 282) * q^89 + (-244*b2 - 208*b1 + 1064) * q^91 + (-111*b2 + 94*b1 - 52) * q^93 + (86*b2 + 244*b1 + 552) * q^95 + (-34*b2 + 176*b1 + 14) * q^97 + (-22*b2 - 38*b1 + 114) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} + 20 q^{5} - 24 q^{7} + 5 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 + 20 * q^5 - 24 * q^7 + 5 * q^9 $$3 q - 2 q^{3} + 20 q^{5} - 24 q^{7} + 5 q^{9} - 10 q^{11} - 4 q^{13} + 130 q^{15} - 66 q^{17} + 164 q^{19} - 88 q^{21} + 204 q^{23} + 79 q^{25} + 142 q^{27} + 87 q^{29} + 86 q^{31} - 130 q^{33} - 24 q^{35} - 42 q^{37} + 394 q^{39} + 562 q^{41} - 18 q^{43} + 422 q^{45} - 654 q^{47} + 539 q^{49} - 556 q^{51} + 712 q^{53} - 142 q^{55} + 828 q^{57} - 184 q^{59} + 322 q^{61} + 784 q^{63} + 1494 q^{65} + 228 q^{67} + 684 q^{69} + 52 q^{71} - 494 q^{73} + 3048 q^{75} + 872 q^{77} + 2110 q^{79} - 1513 q^{81} + 288 q^{83} - 2704 q^{85} - 58 q^{87} + 914 q^{89} + 2984 q^{91} - 62 q^{93} + 1900 q^{95} + 218 q^{97} + 304 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 + 20 * q^5 - 24 * q^7 + 5 * q^9 - 10 * q^11 - 4 * q^13 + 130 * q^15 - 66 * q^17 + 164 * q^19 - 88 * q^21 + 204 * q^23 + 79 * q^25 + 142 * q^27 + 87 * q^29 + 86 * q^31 - 130 * q^33 - 24 * q^35 - 42 * q^37 + 394 * q^39 + 562 * q^41 - 18 * q^43 + 422 * q^45 - 654 * q^47 + 539 * q^49 - 556 * q^51 + 712 * q^53 - 142 * q^55 + 828 * q^57 - 184 * q^59 + 322 * q^61 + 784 * q^63 + 1494 * q^65 + 228 * q^67 + 684 * q^69 + 52 * q^71 - 494 * q^73 + 3048 * q^75 + 872 * q^77 + 2110 * q^79 - 1513 * q^81 + 288 * q^83 - 2704 * q^85 - 58 * q^87 + 914 * q^89 + 2984 * q^91 - 62 * q^93 + 1900 * q^95 + 218 * q^97 + 304 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 42x - 54$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 4\nu - 27 ) / 3$$ (v^2 - 4*v - 27) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2} + 4\beta _1 + 27$$ 3*b2 + 4*b1 + 27

## 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
 −5.13291 −1.39712 7.53003
0 −6.13291 0 2.36031 0 18.5045 0 10.6126 0
1.2 0 −2.39712 0 −3.28077 0 −33.9461 0 −21.2538 0
1.3 0 6.53003 0 20.9205 0 −8.55839 0 15.6413 0
 $$n$$: e.g. 2-40 or 990-1000 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.i 3
4.b odd 2 1 58.4.a.d 3
8.b even 2 1 1856.4.a.s 3
8.d odd 2 1 1856.4.a.r 3
12.b even 2 1 522.4.a.k 3
20.d odd 2 1 1450.4.a.h 3
116.d odd 2 1 1682.4.a.d 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.d 3 4.b odd 2 1
464.4.a.i 3 1.a even 1 1 trivial
522.4.a.k 3 12.b even 2 1
1450.4.a.h 3 20.d odd 2 1
1682.4.a.d 3 116.d odd 2 1
1856.4.a.r 3 8.d odd 2 1
1856.4.a.s 3 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 2T_{3}^{2} - 41T_{3} - 96$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(464))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + 2 T^{2} - 41 T - 96$$
$5$ $$T^{3} - 20 T^{2} - 27 T + 162$$
$7$ $$T^{3} + 24 T^{2} - 496 T - 5376$$
$11$ $$T^{3} + 10 T^{2} - 233 T - 2424$$
$13$ $$T^{3} + 4 T^{2} - 5619 T + 131706$$
$17$ $$T^{3} + 66 T^{2} - 10660 T - 679368$$
$19$ $$T^{3} - 164 T^{2} - 124 T + 664448$$
$23$ $$T^{3} - 204 T^{2} + 4284 T + 677376$$
$29$ $$(T - 29)^{3}$$
$31$ $$T^{3} - 86 T^{2} - 48089 T + 4766172$$
$37$ $$T^{3} + 42 T^{2} - 79456 T - 7684896$$
$41$ $$T^{3} - 562 T^{2} + 102432 T - 5982048$$
$43$ $$T^{3} + 18 T^{2} - 10369 T - 196488$$
$47$ $$T^{3} + 654 T^{2} + 98015 T + 3425124$$
$53$ $$T^{3} - 712 T^{2} + \cdots + 252120546$$
$59$ $$T^{3} + 184 T^{2} + \cdots - 57362928$$
$61$ $$T^{3} - 322 T^{2} - 388372 T - 5254424$$
$67$ $$T^{3} - 228 T^{2} + \cdots - 47608192$$
$71$ $$T^{3} - 52 T^{2} - 164 T + 672$$
$73$ $$T^{3} + 494 T^{2} + 6112 T - 9410208$$
$79$ $$T^{3} - 2110 T^{2} + \cdots - 285187172$$
$83$ $$T^{3} - 288 T^{2} + \cdots + 437606064$$
$89$ $$T^{3} - 914 T^{2} + \cdots + 598011552$$
$97$ $$T^{3} - 218 T^{2} + \cdots - 17006112$$