# Properties

 Label 464.4.a.k Level $464$ Weight $4$ Character orbit 464.a Self dual yes Analytic conductor $27.377$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.225792.1 Defining polynomial: $$x^{4} - 18x^{2} + 18$$ x^4 - 18*x^2 + 18 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 232) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_{2} - 5) q^{5} + ( - \beta_{2} - 2 \beta_1 + 2) q^{7} + (\beta_{3} + 3 \beta_{2} + 4 \beta_1 + 10) q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b2 - 5) * q^5 + (-b2 - 2*b1 + 2) * q^7 + (b3 + 3*b2 + 4*b1 + 10) * q^9 $$q + \beta_1 q^{3} + ( - \beta_{2} - 5) q^{5} + ( - \beta_{2} - 2 \beta_1 + 2) q^{7} + (\beta_{3} + 3 \beta_{2} + 4 \beta_1 + 10) q^{9} + (\beta_{3} + \beta_{2} - 7 \beta_1 - 2) q^{11} + ( - 7 \beta_{3} + 4 \beta_{2} - 15) q^{13} + ( - 5 \beta_{3} + 2 \beta_{2} - 7 \beta_1 + 12) q^{15} + (15 \beta_{3} - 4 \beta_{2} - 8 \beta_1 - 56) q^{17} + (11 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 50) q^{19} + ( - 7 \beta_{3} - 4 \beta_{2} - 8 \beta_1 - 62) q^{21} + ( - 20 \beta_{3} + 7 \beta_{2} + 4 \beta_1 + 26) q^{23} + ( - 12 \beta_{3} + 10 \beta_{2} - 28) q^{25} + (17 \beta_{3} + 9 \beta_{2} + 9 \beta_1 + 126) q^{27} + 29 q^{29} + ( - 21 \beta_{2} - 7 \beta_1 - 106) q^{31} + ( - 4 \beta_{3} - 20 \beta_{2} - 24 \beta_1 - 257) q^{33} + ( - 2 \beta_{3} - \beta_{2} + 14 \beta_1 + 38) q^{35} + (62 \beta_{3} - 12 \beta_{2} - 36 \beta_1 - 124) q^{37} + (34 \beta_{3} - 29 \beta_{2} - 35 \beta_1 - 146) q^{39} + ( - 39 \beta_{3} + 22 \beta_{2} + 36 \beta_1 - 66) q^{41} + ( - 45 \beta_{3} - 21 \beta_{2} + 21 \beta_1 - 94) q^{43} + (13 \beta_{3} - 13 \beta_{2} - 32 \beta_1 - 218) q^{45} + (11 \beta_{3} + 4 \beta_{2} + 23 \beta_1 + 120) q^{47} + (12 \beta_{3} + 16 \beta_1 - 167) q^{49} + ( - 58 \beta_{3} + 29 \beta_{2} - 36 \beta_1 - 38) q^{51} + ( - 99 \beta_{3} + 14 \beta_{2} - 8 \beta_1 - 189) q^{53} + (44 \beta_{3} - 13 \beta_{2} + 45 \beta_1 - 146) q^{55} + ( - 9 \beta_{3} + 21 \beta_{2} + 92 \beta_1 + 44) q^{57} + (22 \beta_{3} + 61 \beta_{2} + 40 \beta_1 + 110) q^{59} + ( - 27 \beta_{3} + 2 \beta_{2} + 32 \beta_1 - 244) q^{61} + ( - 14 \beta_{3} - 10 \beta_{2} - 76 \beta_1 - 400) q^{63} + (69 \beta_{3} - 33 \beta_{2} + 28 \beta_1 - 213) q^{65} + (6 \beta_{3} + 34 \beta_{2} - 16 \beta_1 + 20) q^{67} + (79 \beta_{3} - 62 \beta_{2} - 24 \beta_1 - 216) q^{69} + ( - 84 \beta_{3} + 8 \beta_{2} + 110 \beta_1 - 52) q^{71} + (28 \beta_{3} + 44 \beta_{2} + 56 \beta_1 + 132) q^{73} + (74 \beta_{3} - 56 \beta_{2} - 56 \beta_1 - 288) q^{75} + (59 \beta_{3} + 34 \beta_{2} + 44 \beta_1 + 354) q^{77} + (67 \beta_{3} + 22 \beta_{2} - 23 \beta_1 - 628) q^{79} + ( - 7 \beta_{3} - 21 \beta_{2} + 140 \beta_1 + 193) q^{81} + (68 \beta_{3} - 5 \beta_{2} + 76 \beta_1 + 110) q^{83} + ( - 53 \beta_{3} + 120 \beta_{2} - 4 \beta_1 + 472) q^{85} + 29 \beta_1 q^{87} + ( - 49 \beta_{3} - 28 \beta_{2} - 64 \beta_1 - 402) q^{89} + ( - 48 \beta_{3} + 53 \beta_{2} + 98 \beta_1 - 26) q^{91} + ( - 112 \beta_{3} + 21 \beta_{2} - 176 \beta_1 - 7) q^{93} + (13 \beta_{3} - 25 \beta_{2} - 30 \beta_1 - 490) q^{95} + ( - 49 \beta_{3} - 60 \beta_{2} + 44 \beta_1 + 146) q^{97} + ( - 143 \beta_{3} - 71 \beta_{2} - 220 \beta_1 - 650) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b2 - 5) * q^5 + (-b2 - 2*b1 + 2) * q^7 + (b3 + 3*b2 + 4*b1 + 10) * q^9 + (b3 + b2 - 7*b1 - 2) * q^11 + (-7*b3 + 4*b2 - 15) * q^13 + (-5*b3 + 2*b2 - 7*b1 + 12) * q^15 + (15*b3 - 4*b2 - 8*b1 - 56) * q^17 + (11*b3 + 3*b2 - 2*b1 + 50) * q^19 + (-7*b3 - 4*b2 - 8*b1 - 62) * q^21 + (-20*b3 + 7*b2 + 4*b1 + 26) * q^23 + (-12*b3 + 10*b2 - 28) * q^25 + (17*b3 + 9*b2 + 9*b1 + 126) * q^27 + 29 * q^29 + (-21*b2 - 7*b1 - 106) * q^31 + (-4*b3 - 20*b2 - 24*b1 - 257) * q^33 + (-2*b3 - b2 + 14*b1 + 38) * q^35 + (62*b3 - 12*b2 - 36*b1 - 124) * q^37 + (34*b3 - 29*b2 - 35*b1 - 146) * q^39 + (-39*b3 + 22*b2 + 36*b1 - 66) * q^41 + (-45*b3 - 21*b2 + 21*b1 - 94) * q^43 + (13*b3 - 13*b2 - 32*b1 - 218) * q^45 + (11*b3 + 4*b2 + 23*b1 + 120) * q^47 + (12*b3 + 16*b1 - 167) * q^49 + (-58*b3 + 29*b2 - 36*b1 - 38) * q^51 + (-99*b3 + 14*b2 - 8*b1 - 189) * q^53 + (44*b3 - 13*b2 + 45*b1 - 146) * q^55 + (-9*b3 + 21*b2 + 92*b1 + 44) * q^57 + (22*b3 + 61*b2 + 40*b1 + 110) * q^59 + (-27*b3 + 2*b2 + 32*b1 - 244) * q^61 + (-14*b3 - 10*b2 - 76*b1 - 400) * q^63 + (69*b3 - 33*b2 + 28*b1 - 213) * q^65 + (6*b3 + 34*b2 - 16*b1 + 20) * q^67 + (79*b3 - 62*b2 - 24*b1 - 216) * q^69 + (-84*b3 + 8*b2 + 110*b1 - 52) * q^71 + (28*b3 + 44*b2 + 56*b1 + 132) * q^73 + (74*b3 - 56*b2 - 56*b1 - 288) * q^75 + (59*b3 + 34*b2 + 44*b1 + 354) * q^77 + (67*b3 + 22*b2 - 23*b1 - 628) * q^79 + (-7*b3 - 21*b2 + 140*b1 + 193) * q^81 + (68*b3 - 5*b2 + 76*b1 + 110) * q^83 + (-53*b3 + 120*b2 - 4*b1 + 472) * q^85 + 29*b1 * q^87 + (-49*b3 - 28*b2 - 64*b1 - 402) * q^89 + (-48*b3 + 53*b2 + 98*b1 - 26) * q^91 + (-112*b3 + 21*b2 - 176*b1 - 7) * q^93 + (13*b3 - 25*b2 - 30*b1 - 490) * q^95 + (-49*b3 - 60*b2 + 44*b1 + 146) * q^97 + (-143*b3 - 71*b2 - 220*b1 - 650) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 20 q^{5} + 8 q^{7} + 40 q^{9}+O(q^{10})$$ 4 * q - 20 * q^5 + 8 * q^7 + 40 * q^9 $$4 q - 20 q^{5} + 8 q^{7} + 40 q^{9} - 8 q^{11} - 60 q^{13} + 48 q^{15} - 224 q^{17} + 200 q^{19} - 248 q^{21} + 104 q^{23} - 112 q^{25} + 504 q^{27} + 116 q^{29} - 424 q^{31} - 1028 q^{33} + 152 q^{35} - 496 q^{37} - 584 q^{39} - 264 q^{41} - 376 q^{43} - 872 q^{45} + 480 q^{47} - 668 q^{49} - 152 q^{51} - 756 q^{53} - 584 q^{55} + 176 q^{57} + 440 q^{59} - 976 q^{61} - 1600 q^{63} - 852 q^{65} + 80 q^{67} - 864 q^{69} - 208 q^{71} + 528 q^{73} - 1152 q^{75} + 1416 q^{77} - 2512 q^{79} + 772 q^{81} + 440 q^{83} + 1888 q^{85} - 1608 q^{89} - 104 q^{91} - 28 q^{93} - 1960 q^{95} + 584 q^{97} - 2600 q^{99}+O(q^{100})$$ 4 * q - 20 * q^5 + 8 * q^7 + 40 * q^9 - 8 * q^11 - 60 * q^13 + 48 * q^15 - 224 * q^17 + 200 * q^19 - 248 * q^21 + 104 * q^23 - 112 * q^25 + 504 * q^27 + 116 * q^29 - 424 * q^31 - 1028 * q^33 + 152 * q^35 - 496 * q^37 - 584 * q^39 - 264 * q^41 - 376 * q^43 - 872 * q^45 + 480 * q^47 - 668 * q^49 - 152 * q^51 - 756 * q^53 - 584 * q^55 + 176 * q^57 + 440 * q^59 - 976 * q^61 - 1600 * q^63 - 852 * q^65 + 80 * q^67 - 864 * q^69 - 208 * q^71 + 528 * q^73 - 1152 * q^75 + 1416 * q^77 - 2512 * q^79 + 772 * q^81 + 440 * q^83 + 1888 * q^85 - 1608 * q^89 - 104 * q^91 - 28 * q^93 - 1960 * q^95 + 584 * q^97 - 2600 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + \nu^{2} - 12\nu - 9 ) / 3$$ (v^3 + v^2 - 12*v - 9) / 3 $$\beta_{2}$$ $$=$$ $$( -2\nu^{3} + 36\nu ) / 3$$ (-2*v^3 + 36*v) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{2} - 18 ) / 3$$ (2*v^2 - 18) / 3
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} + 2\beta_1 ) / 4$$ (-b3 + b2 + 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( 3\beta_{3} + 18 ) / 2$$ (3*b3 + 18) / 2 $$\nu^{3}$$ $$=$$ $$( -9\beta_{3} + 6\beta_{2} + 18\beta_1 ) / 2$$ (-9*b3 + 6*b2 + 18*b1) / 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
 1.03090 −4.11549 −1.03090 4.11549
0 −6.40414 0 −16.6404 0 3.16792 0 14.0130 0
1.2 0 −4.12732 0 −2.08419 0 13.1705 0 −9.96522 0
1.3 0 1.11264 0 6.64036 0 11.4151 0 −25.7620 0
1.4 0 9.41882 0 −7.91581 0 −19.7535 0 61.7142 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.k 4
4.b odd 2 1 232.4.a.c 4
8.b even 2 1 1856.4.a.x 4
8.d odd 2 1 1856.4.a.w 4
12.b even 2 1 2088.4.a.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.a.c 4 4.b odd 2 1
464.4.a.k 4 1.a even 1 1 trivial
1856.4.a.w 4 8.d odd 2 1
1856.4.a.x 4 8.b even 2 1
2088.4.a.e 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 74T_{3}^{2} - 168T_{3} + 277$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(464))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 74 T^{2} - 168 T + 277$$
$5$ $$T^{4} + 20 T^{3} + 6 T^{2} + \cdots - 1823$$
$7$ $$T^{4} - 8 T^{3} - 320 T^{2} + \cdots - 9408$$
$11$ $$T^{4} + 8 T^{3} - 3746 T^{2} + \cdots + 2296893$$
$13$ $$T^{4} + 60 T^{3} - 3698 T^{2} + \cdots - 4326887$$
$17$ $$T^{4} + 224 T^{3} + \cdots - 55312368$$
$19$ $$T^{4} - 200 T^{3} + \cdots - 18943728$$
$23$ $$T^{4} - 104 T^{3} + \cdots + 26110672$$
$29$ $$(T - 29)^{4}$$
$31$ $$T^{4} + 424 T^{3} + \cdots - 147822387$$
$37$ $$T^{4} + 496 T^{3} + \cdots - 8389147392$$
$41$ $$T^{4} + 264 T^{3} + \cdots + 796866624$$
$43$ $$T^{4} + 376 T^{3} + \cdots - 6763463843$$
$47$ $$T^{4} - 480 T^{3} + \cdots - 46277307$$
$53$ $$T^{4} + 756 T^{3} + \cdots + 48201739177$$
$59$ $$T^{4} - 440 T^{3} + \cdots + 27370709968$$
$61$ $$T^{4} + 976 T^{3} + \cdots + 1049490576$$
$67$ $$T^{4} - 80 T^{3} - 205760 T^{2} + \cdots - 715776$$
$71$ $$T^{4} + 208 T^{3} + \cdots + 50595925968$$
$73$ $$T^{4} - 528 T^{3} + \cdots + 4749558016$$
$79$ $$T^{4} + 2512 T^{3} + \cdots + 75686955061$$
$83$ $$T^{4} - 440 T^{3} + \cdots + 17011084752$$
$89$ $$T^{4} + 1608 T^{3} + \cdots - 2795276736$$
$97$ $$T^{4} - 584 T^{3} + \cdots + 10030146112$$