# Properties

 Label 464.4.a.n Level $464$ Weight $4$ Character orbit 464.a Self dual yes Analytic conductor $27.377$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9$$ x^6 - 40*x^4 - 88*x^3 - 8*x^2 + 48*x - 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + 1) q^{3} + ( - \beta_{4} - \beta_{3} - 1) q^{5} + ( - \beta_{4} + \beta_{3} + \beta_1 + 6) q^{7} + (2 \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 9) q^{9}+O(q^{10})$$ q + (-b3 + 1) * q^3 + (-b4 - b3 - 1) * q^5 + (-b4 + b3 + b1 + 6) * q^7 + (2*b4 - 3*b3 - b2 + 2*b1 + 9) * q^9 $$q + ( - \beta_{3} + 1) q^{3} + ( - \beta_{4} - \beta_{3} - 1) q^{5} + ( - \beta_{4} + \beta_{3} + \beta_1 + 6) q^{7} + (2 \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 9) q^{9} + (\beta_{5} - 3 \beta_{4} - 3 \beta_{2} - 2 \beta_1 + 1) q^{11} + (3 \beta_{5} - 6 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{13} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 + 33) q^{15} + ( - 6 \beta_{5} - 5 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 37) q^{17} + (\beta_{5} - \beta_{4} - \beta_{3} + 11 \beta_{2} + 52) q^{19} + ( - 2 \beta_{5} - 5 \beta_{4} - 10 \beta_{3} + \beta_{2} - 4 \beta_1 - 45) q^{21} + ( - 8 \beta_{5} + 5 \beta_{4} - 9 \beta_{3} - 2 \beta_{2} + \beta_1 + 34) q^{23} + (5 \beta_{5} - \beta_{4} + 6 \beta_{3} + 6 \beta_{2} - 2 \beta_1 - 1) q^{25} + (3 \beta_{5} + 3 \beta_{4} - 12 \beta_{3} - 15 \beta_{2} - 2 \beta_1 + 71) q^{27} - 29 q^{29} + (4 \beta_{5} - \beta_{4} - 10 \beta_{3} + 10 \beta_{2} - 5 \beta_1 + 117) q^{31} + (9 \beta_{5} + 15 \beta_{4} + 8 \beta_{3} - 8 \beta_{2} + 6 \beta_1 + 52) q^{33} + (10 \beta_{5} - 15 \beta_{4} - 15 \beta_{3} + 8 \beta_{2} - 21 \beta_1 + 76) q^{35} + ( - 2 \beta_{5} + 14 \beta_{4} + 2 \beta_{3} - 10 \beta_{2} - 6 \beta_1 - 40) q^{37} + (8 \beta_{5} + 21 \beta_{4} - 6 \beta_{3} - 24 \beta_{2} + 9 \beta_1 + 219) q^{39} + ( - 14 \beta_{5} - 7 \beta_{4} + 28 \beta_{3} - 11 \beta_{2} + 8 \beta_1 - 89) q^{41} + ( - 13 \beta_{5} - 7 \beta_{4} - 12 \beta_{3} - 9 \beta_{2} + 12 \beta_1 + 93) q^{43} + ( - 5 \beta_{5} + 13 \beta_{4} - 27 \beta_{3} + \beta_{2} - 10 \beta_1 - 18) q^{45} + ( - 11 \beta_{5} + 12 \beta_{4} - \beta_{3} - 3 \beta_{2} + 7 \beta_1 + 165) q^{47} + (24 \beta_{5} + 10 \beta_{4} + 44 \beta_{3} + 22 \beta_{2} - 4 \beta_1 + 159) q^{49} + ( - 22 \beta_{5} - 7 \beta_{4} + 41 \beta_{3} + 50 \beta_{2} + 25 \beta_1 + 104) q^{51} + (15 \beta_{5} - 8 \beta_{4} - 8 \beta_{3} - 11 \beta_{2} - 2 \beta_1 + 83) q^{53} + (30 \beta_{5} - 9 \beta_{4} + 36 \beta_{3} + 2 \beta_{2} - 7 \beta_1 + 263) q^{55} + ( - 33 \beta_{5} - 31 \beta_{4} - 9 \beta_{3} + 39 \beta_{2} - 58) q^{57} + ( - 6 \beta_{5} - 13 \beta_{4} + 33 \beta_{3} - 16 \beta_{2} + \beta_1 + 2) q^{59} + ( - 24 \beta_{5} + 13 \beta_{4} + 6 \beta_{3} + \beta_{2} + 38 \beta_1 + 199) q^{61} + ( - 6 \beta_{5} + 56 \beta_{4} + 46 \beta_{3} + 24 \beta_{2} + 16 \beta_1 + 210) q^{63} + (10 \beta_{5} - 8 \beta_{4} + 37 \beta_{3} - 17 \beta_{2} + 14 \beta_1 + 196) q^{65} + (10 \beta_{5} - 4 \beta_{4} - 6 \beta_{3} - 52 \beta_{2} + 4 \beta_1 - 2) q^{67} + (2 \beta_{5} + 21 \beta_{4} - 70 \beta_{3} + 19 \beta_{2} + 34 \beta_1 + 459) q^{69} + (8 \beta_{5} - 24 \beta_{4} + 70 \beta_{3} - 16 \beta_{2} - 56 \beta_1 - 106) q^{71} + ( - 14 \beta_{5} - 52 \beta_{4} + 58 \beta_{3} - 40 \beta_{2} - 18 \beta_1 - 2) q^{73} + ( - 12 \beta_{5} - 24 \beta_{4} + 50 \beta_{3} + 6 \beta_{2} - 20 \beta_1 - 314) q^{75} + (40 \beta_{5} - 77 \beta_{4} - 34 \beta_{3} - \beta_{2} - 70 \beta_1 + 7) q^{77} + (9 \beta_{5} - 20 \beta_{4} + 9 \beta_{3} + 29 \beta_{2} - 21 \beta_1 + 39) q^{79} + (53 \beta_{5} + 21 \beta_{4} - 67 \beta_{3} - 65 \beta_{2} - 36 \beta_1 + 425) q^{81} + (12 \beta_{5} + 55 \beta_{4} - 29 \beta_{3} + 40 \beta_{2} + 41 \beta_1 + 260) q^{83} + ( - 16 \beta_{5} + 67 \beta_{4} + 22 \beta_{3} + 67 \beta_{2} + 24 \beta_1 + 561) q^{85} + (29 \beta_{3} - 29) q^{87} + ( - 12 \beta_{5} - 25 \beta_{4} + 30 \beta_{3} + 11 \beta_{2} - 44 \beta_1 - 61) q^{89} + (8 \beta_{5} - 91 \beta_{4} - 53 \beta_{3} - 20 \beta_{2} - 41 \beta_1 - 442) q^{91} + ( - 22 \beta_{5} + 5 \beta_{4} - 59 \beta_{3} + 18 \beta_{2} + 24 \beta_1 + 373) q^{93} + ( - 33 \beta_{5} - 119 \beta_{4} - 67 \beta_{3} + \beta_{2} + 22 \beta_1 - 60) q^{95} + ( - 16 \beta_{5} + 45 \beta_{4} + 124 \beta_{3} - 73 \beta_{2} - 22 \beta_1 + 27) q^{97} + (15 \beta_{5} + 71 \beta_{4} - 155 \beta_{3} - 39 \beta_{2} - 22 \beta_1 - 342) q^{99}+O(q^{100})$$ q + (-b3 + 1) * q^3 + (-b4 - b3 - 1) * q^5 + (-b4 + b3 + b1 + 6) * q^7 + (2*b4 - 3*b3 - b2 + 2*b1 + 9) * q^9 + (b5 - 3*b4 - 3*b2 - 2*b1 + 1) * q^11 + (3*b5 - 6*b3 - b2 - 2*b1 + 3) * q^13 + (-b5 + 2*b4 + b3 + b2 + 3*b1 + 33) * q^15 + (-6*b5 - 5*b4 - 4*b3 + 3*b2 + 2*b1 - 37) * q^17 + (b5 - b4 - b3 + 11*b2 + 52) * q^19 + (-2*b5 - 5*b4 - 10*b3 + b2 - 4*b1 - 45) * q^21 + (-8*b5 + 5*b4 - 9*b3 - 2*b2 + b1 + 34) * q^23 + (5*b5 - b4 + 6*b3 + 6*b2 - 2*b1 - 1) * q^25 + (3*b5 + 3*b4 - 12*b3 - 15*b2 - 2*b1 + 71) * q^27 - 29 * q^29 + (4*b5 - b4 - 10*b3 + 10*b2 - 5*b1 + 117) * q^31 + (9*b5 + 15*b4 + 8*b3 - 8*b2 + 6*b1 + 52) * q^33 + (10*b5 - 15*b4 - 15*b3 + 8*b2 - 21*b1 + 76) * q^35 + (-2*b5 + 14*b4 + 2*b3 - 10*b2 - 6*b1 - 40) * q^37 + (8*b5 + 21*b4 - 6*b3 - 24*b2 + 9*b1 + 219) * q^39 + (-14*b5 - 7*b4 + 28*b3 - 11*b2 + 8*b1 - 89) * q^41 + (-13*b5 - 7*b4 - 12*b3 - 9*b2 + 12*b1 + 93) * q^43 + (-5*b5 + 13*b4 - 27*b3 + b2 - 10*b1 - 18) * q^45 + (-11*b5 + 12*b4 - b3 - 3*b2 + 7*b1 + 165) * q^47 + (24*b5 + 10*b4 + 44*b3 + 22*b2 - 4*b1 + 159) * q^49 + (-22*b5 - 7*b4 + 41*b3 + 50*b2 + 25*b1 + 104) * q^51 + (15*b5 - 8*b4 - 8*b3 - 11*b2 - 2*b1 + 83) * q^53 + (30*b5 - 9*b4 + 36*b3 + 2*b2 - 7*b1 + 263) * q^55 + (-33*b5 - 31*b4 - 9*b3 + 39*b2 - 58) * q^57 + (-6*b5 - 13*b4 + 33*b3 - 16*b2 + b1 + 2) * q^59 + (-24*b5 + 13*b4 + 6*b3 + b2 + 38*b1 + 199) * q^61 + (-6*b5 + 56*b4 + 46*b3 + 24*b2 + 16*b1 + 210) * q^63 + (10*b5 - 8*b4 + 37*b3 - 17*b2 + 14*b1 + 196) * q^65 + (10*b5 - 4*b4 - 6*b3 - 52*b2 + 4*b1 - 2) * q^67 + (2*b5 + 21*b4 - 70*b3 + 19*b2 + 34*b1 + 459) * q^69 + (8*b5 - 24*b4 + 70*b3 - 16*b2 - 56*b1 - 106) * q^71 + (-14*b5 - 52*b4 + 58*b3 - 40*b2 - 18*b1 - 2) * q^73 + (-12*b5 - 24*b4 + 50*b3 + 6*b2 - 20*b1 - 314) * q^75 + (40*b5 - 77*b4 - 34*b3 - b2 - 70*b1 + 7) * q^77 + (9*b5 - 20*b4 + 9*b3 + 29*b2 - 21*b1 + 39) * q^79 + (53*b5 + 21*b4 - 67*b3 - 65*b2 - 36*b1 + 425) * q^81 + (12*b5 + 55*b4 - 29*b3 + 40*b2 + 41*b1 + 260) * q^83 + (-16*b5 + 67*b4 + 22*b3 + 67*b2 + 24*b1 + 561) * q^85 + (29*b3 - 29) * q^87 + (-12*b5 - 25*b4 + 30*b3 + 11*b2 - 44*b1 - 61) * q^89 + (8*b5 - 91*b4 - 53*b3 - 20*b2 - 41*b1 - 442) * q^91 + (-22*b5 + 5*b4 - 59*b3 + 18*b2 + 24*b1 + 373) * q^93 + (-33*b5 - 119*b4 - 67*b3 + b2 + 22*b1 - 60) * q^95 + (-16*b5 + 45*b4 + 124*b3 - 73*b2 - 22*b1 + 27) * q^97 + (15*b5 + 71*b4 - 155*b3 - 39*b2 - 22*b1 - 342) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 5 q^{3} - 5 q^{5} + 38 q^{7} + 47 q^{9}+O(q^{10})$$ 6 * q + 5 * q^3 - 5 * q^5 + 38 * q^7 + 47 * q^9 $$6 q + 5 q^{3} - 5 q^{5} + 38 q^{7} + 47 q^{9} + 19 q^{11} + 13 q^{13} + 191 q^{15} - 218 q^{17} + 290 q^{19} - 266 q^{21} + 196 q^{23} - 13 q^{25} + 437 q^{27} - 174 q^{29} + 675 q^{31} + 291 q^{33} + 466 q^{35} - 238 q^{37} + 1297 q^{39} - 464 q^{41} + 579 q^{43} - 148 q^{45} + 975 q^{47} + 914 q^{49} + 576 q^{51} + 515 q^{53} + 1605 q^{55} - 340 q^{57} + 108 q^{59} + 1158 q^{61} + 1136 q^{63} + 1239 q^{65} + 80 q^{67} + 2568 q^{69} - 438 q^{71} + 262 q^{73} - 1766 q^{75} + 194 q^{77} + 237 q^{79} + 2554 q^{81} + 1288 q^{83} + 3112 q^{85} - 145 q^{87} - 252 q^{89} - 2450 q^{91} + 2131 q^{93} - 180 q^{95} + 380 q^{97} - 2264 q^{99}+O(q^{100})$$ 6 * q + 5 * q^3 - 5 * q^5 + 38 * q^7 + 47 * q^9 + 19 * q^11 + 13 * q^13 + 191 * q^15 - 218 * q^17 + 290 * q^19 - 266 * q^21 + 196 * q^23 - 13 * q^25 + 437 * q^27 - 174 * q^29 + 675 * q^31 + 291 * q^33 + 466 * q^35 - 238 * q^37 + 1297 * q^39 - 464 * q^41 + 579 * q^43 - 148 * q^45 + 975 * q^47 + 914 * q^49 + 576 * q^51 + 515 * q^53 + 1605 * q^55 - 340 * q^57 + 108 * q^59 + 1158 * q^61 + 1136 * q^63 + 1239 * q^65 + 80 * q^67 + 2568 * q^69 - 438 * q^71 + 262 * q^73 - 1766 * q^75 + 194 * q^77 + 237 * q^79 + 2554 * q^81 + 1288 * q^83 + 3112 * q^85 - 145 * q^87 - 252 * q^89 - 2450 * q^91 + 2131 * q^93 - 180 * q^95 + 380 * q^97 - 2264 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} - 9\nu^{4} + 43\nu^{3} + 447\nu^{2} + 671\nu - 113 ) / 14$$ (-v^5 - 9*v^4 + 43*v^3 + 447*v^2 + 671*v - 113) / 14 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 12\nu^{4} + 22\nu^{3} - 365\nu^{2} - 400\nu + 321 ) / 21$$ (-v^5 + 12*v^4 + 22*v^3 - 365*v^2 - 400*v + 321) / 21 $$\beta_{3}$$ $$=$$ $$( -5\nu^{5} - 3\nu^{4} + 201\nu^{3} + 555\nu^{2} + 261\nu - 159 ) / 14$$ (-5*v^5 - 3*v^4 + 201*v^3 + 555*v^2 + 261*v - 159) / 14 $$\beta_{4}$$ $$=$$ $$( -11\nu^{5} + 6\nu^{4} + 431\nu^{3} + 731\nu^{2} - 32\nu - 123 ) / 21$$ (-11*v^5 + 6*v^4 + 431*v^3 + 731*v^2 - 32*v - 123) / 21 $$\beta_{5}$$ $$=$$ $$( -41\nu^{5} + 9\nu^{4} + 1595\nu^{3} + 3347\nu^{2} + 1093\nu - 1035 ) / 42$$ (-41*v^5 + 9*v^4 + 1595*v^3 + 3347*v^2 + 1093*v - 1035) / 42
 $$\nu$$ $$=$$ $$( 2\beta_{4} - 3\beta_{3} - \beta_{2} + \beta _1 + 1 ) / 8$$ (2*b4 - 3*b3 - b2 + b1 + 1) / 8 $$\nu^{2}$$ $$=$$ $$( 6\beta_{5} - 15\beta_{3} - 9\beta_{2} - \beta _1 + 107 ) / 8$$ (6*b5 - 15*b3 - 9*b2 - b1 + 107) / 8 $$\nu^{3}$$ $$=$$ $$( 3\beta_{5} + 17\beta_{4} - 32\beta_{3} - 16\beta_{2} + 5\beta _1 + 95 ) / 2$$ (3*b5 + 17*b4 - 32*b3 - 16*b2 + 5*b1 + 95) / 2 $$\nu^{4}$$ $$=$$ $$( 244\beta_{5} + 170\beta_{4} - 861\beta_{3} - 455\beta_{2} + 27\beta _1 + 4403 ) / 8$$ (244*b5 + 170*b4 - 861*b3 - 455*b2 + 27*b1 + 4403) / 8 $$\nu^{5}$$ $$=$$ $$( 1002\beta_{5} + 2736\beta_{4} - 6473\beta_{3} - 3351\beta_{2} + 729\beta _1 + 24309 ) / 8$$ (1002*b5 + 2736*b4 - 6473*b3 - 3351*b2 + 729*b1 + 24309) / 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
 0.488896 −1.90366 −4.70501 −1.32108 0.215317 7.22553
0 −7.88819 0 −14.0062 0 34.0826 0 35.2236 0
1.2 0 −2.88453 0 −3.33508 0 −0.0162797 0 −18.6795 0
1.3 0 −0.586661 0 15.4201 0 28.5777 0 −26.6558 0
1.4 0 0.116789 0 −14.4485 0 −30.0998 0 −26.9864 0
1.5 0 6.36242 0 8.72860 0 8.76166 0 13.4804 0
1.6 0 9.88017 0 2.64112 0 −3.30580 0 70.6177 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.n 6
4.b odd 2 1 232.4.a.e 6
8.b even 2 1 1856.4.a.bc 6
8.d odd 2 1 1856.4.a.bd 6
12.b even 2 1 2088.4.a.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.a.e 6 4.b odd 2 1
464.4.a.n 6 1.a even 1 1 trivial
1856.4.a.bc 6 8.b even 2 1
1856.4.a.bd 6 8.d odd 2 1
2088.4.a.l 6 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 5T_{3}^{5} - 92T_{3}^{4} + 266T_{3}^{3} + 1581T_{3}^{2} + 651T_{3} - 98$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(464))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 5 T^{5} - 92 T^{4} + 266 T^{3} + \cdots - 98$$
$5$ $$T^{6} + 5 T^{5} - 356 T^{4} + \cdots - 239922$$
$7$ $$T^{6} - 38 T^{5} - 764 T^{4} + \cdots - 13824$$
$11$ $$T^{6} - 19 T^{5} + \cdots - 2827812238$$
$13$ $$T^{6} - 13 T^{5} + \cdots - 7992521554$$
$17$ $$T^{6} + 218 T^{5} + \cdots + 200391946752$$
$19$ $$T^{6} - 290 T^{5} + \cdots + 1256352636928$$
$23$ $$T^{6} - 196 T^{5} + \cdots + 4069096656896$$
$29$ $$(T + 29)^{6}$$
$31$ $$T^{6} - 675 T^{5} + \cdots + 766872337410$$
$37$ $$T^{6} + 238 T^{5} + \cdots - 38445998866432$$
$41$ $$T^{6} + 464 T^{5} + \cdots + 64375103462400$$
$43$ $$T^{6} - 579 T^{5} + \cdots - 905147314238$$
$47$ $$T^{6} + \cdots - 154765125165962$$
$53$ $$T^{6} - 515 T^{5} + \cdots - 36699799804014$$
$59$ $$T^{6} + \cdots - 110093808390144$$
$61$ $$T^{6} - 1158 T^{5} + \cdots + 40\!\cdots\!72$$
$67$ $$T^{6} + \cdots + 681378807496704$$
$71$ $$T^{6} + 438 T^{5} + \cdots - 16\!\cdots\!00$$
$73$ $$T^{6} - 262 T^{5} + \cdots + 21\!\cdots\!72$$
$79$ $$T^{6} - 237 T^{5} + \cdots - 10\!\cdots\!74$$
$83$ $$T^{6} - 1288 T^{5} + \cdots + 47\!\cdots\!44$$
$89$ $$T^{6} + 252 T^{5} + \cdots + 10\!\cdots\!00$$
$97$ $$T^{6} - 380 T^{5} + \cdots + 83\!\cdots\!28$$