# Properties

 Label 464.4.a.e Level $464$ Weight $4$ Character orbit 464.a Self dual yes Analytic conductor $27.377$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + (6 \beta - 5) q^{5} + (8 \beta + 8) q^{7} + (2 \beta - 20) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + (6*b - 5) * q^5 + (8*b + 8) * q^7 + (2*b - 20) * q^9 $$q + (\beta + 1) q^{3} + (6 \beta - 5) q^{5} + (8 \beta + 8) q^{7} + (2 \beta - 20) q^{9} + (3 \beta + 45) q^{11} + (8 \beta - 25) q^{13} + (\beta + 31) q^{15} + (16 \beta - 22) q^{17} + (4 \beta - 54) q^{19} + (16 \beta + 56) q^{21} + (78 \beta + 14) q^{23} + ( - 60 \beta + 116) q^{25} + ( - 45 \beta - 35) q^{27} + 29 q^{29} + ( - 109 \beta - 33) q^{31} + (48 \beta + 63) q^{33} + (8 \beta + 248) q^{35} + (4 \beta + 20) q^{37} + ( - 17 \beta + 23) q^{39} + ( - 80 \beta + 152) q^{41} + ( - 53 \beta + 65) q^{43} + ( - 130 \beta + 172) q^{45} + (99 \beta + 257) q^{47} + (128 \beta + 105) q^{49} + ( - 6 \beta + 74) q^{51} + (52 \beta - 479) q^{53} + (255 \beta - 117) q^{55} + ( - 50 \beta - 30) q^{57} + ( - 250 \beta + 90) q^{59} + ( - 12 \beta + 514) q^{61} + ( - 144 \beta - 64) q^{63} + ( - 190 \beta + 413) q^{65} + (20 \beta + 456) q^{67} + (92 \beta + 482) q^{69} + ( - 34 \beta - 398) q^{71} + (176 \beta - 428) q^{73} + (56 \beta - 244) q^{75} + (384 \beta + 504) q^{77} + (361 \beta + 159) q^{79} + ( - 134 \beta + 235) q^{81} + (38 \beta + 914) q^{83} + ( - 212 \beta + 686) q^{85} + (29 \beta + 29) q^{87} + (72 \beta - 472) q^{89} + ( - 136 \beta + 184) q^{91} + ( - 142 \beta - 687) q^{93} + ( - 344 \beta + 414) q^{95} + (100 \beta + 184) q^{97} + (30 \beta - 864) q^{99}+O(q^{100})$$ q + (b + 1) * q^3 + (6*b - 5) * q^5 + (8*b + 8) * q^7 + (2*b - 20) * q^9 + (3*b + 45) * q^11 + (8*b - 25) * q^13 + (b + 31) * q^15 + (16*b - 22) * q^17 + (4*b - 54) * q^19 + (16*b + 56) * q^21 + (78*b + 14) * q^23 + (-60*b + 116) * q^25 + (-45*b - 35) * q^27 + 29 * q^29 + (-109*b - 33) * q^31 + (48*b + 63) * q^33 + (8*b + 248) * q^35 + (4*b + 20) * q^37 + (-17*b + 23) * q^39 + (-80*b + 152) * q^41 + (-53*b + 65) * q^43 + (-130*b + 172) * q^45 + (99*b + 257) * q^47 + (128*b + 105) * q^49 + (-6*b + 74) * q^51 + (52*b - 479) * q^53 + (255*b - 117) * q^55 + (-50*b - 30) * q^57 + (-250*b + 90) * q^59 + (-12*b + 514) * q^61 + (-144*b - 64) * q^63 + (-190*b + 413) * q^65 + (20*b + 456) * q^67 + (92*b + 482) * q^69 + (-34*b - 398) * q^71 + (176*b - 428) * q^73 + (56*b - 244) * q^75 + (384*b + 504) * q^77 + (361*b + 159) * q^79 + (-134*b + 235) * q^81 + (38*b + 914) * q^83 + (-212*b + 686) * q^85 + (29*b + 29) * q^87 + (72*b - 472) * q^89 + (-136*b + 184) * q^91 + (-142*b - 687) * q^93 + (-344*b + 414) * q^95 + (100*b + 184) * q^97 + (30*b - 864) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 10 q^{5} + 16 q^{7} - 40 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 10 * q^5 + 16 * q^7 - 40 * q^9 $$2 q + 2 q^{3} - 10 q^{5} + 16 q^{7} - 40 q^{9} + 90 q^{11} - 50 q^{13} + 62 q^{15} - 44 q^{17} - 108 q^{19} + 112 q^{21} + 28 q^{23} + 232 q^{25} - 70 q^{27} + 58 q^{29} - 66 q^{31} + 126 q^{33} + 496 q^{35} + 40 q^{37} + 46 q^{39} + 304 q^{41} + 130 q^{43} + 344 q^{45} + 514 q^{47} + 210 q^{49} + 148 q^{51} - 958 q^{53} - 234 q^{55} - 60 q^{57} + 180 q^{59} + 1028 q^{61} - 128 q^{63} + 826 q^{65} + 912 q^{67} + 964 q^{69} - 796 q^{71} - 856 q^{73} - 488 q^{75} + 1008 q^{77} + 318 q^{79} + 470 q^{81} + 1828 q^{83} + 1372 q^{85} + 58 q^{87} - 944 q^{89} + 368 q^{91} - 1374 q^{93} + 828 q^{95} + 368 q^{97} - 1728 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 10 * q^5 + 16 * q^7 - 40 * q^9 + 90 * q^11 - 50 * q^13 + 62 * q^15 - 44 * q^17 - 108 * q^19 + 112 * q^21 + 28 * q^23 + 232 * q^25 - 70 * q^27 + 58 * q^29 - 66 * q^31 + 126 * q^33 + 496 * q^35 + 40 * q^37 + 46 * q^39 + 304 * q^41 + 130 * q^43 + 344 * q^45 + 514 * q^47 + 210 * q^49 + 148 * q^51 - 958 * q^53 - 234 * q^55 - 60 * q^57 + 180 * q^59 + 1028 * q^61 - 128 * q^63 + 826 * q^65 + 912 * q^67 + 964 * q^69 - 796 * q^71 - 856 * q^73 - 488 * q^75 + 1008 * q^77 + 318 * q^79 + 470 * q^81 + 1828 * q^83 + 1372 * q^85 + 58 * q^87 - 944 * q^89 + 368 * q^91 - 1374 * q^93 + 828 * q^95 + 368 * q^97 - 1728 * q^99

## 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
 −2.44949 2.44949
0 −1.44949 0 −19.6969 0 −11.5959 0 −24.8990 0
1.2 0 3.44949 0 9.69694 0 27.5959 0 −15.1010 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.e 2
4.b odd 2 1 58.4.a.c 2
8.b even 2 1 1856.4.a.i 2
8.d odd 2 1 1856.4.a.l 2
12.b even 2 1 522.4.a.j 2
20.d odd 2 1 1450.4.a.g 2
116.d odd 2 1 1682.4.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.c 2 4.b odd 2 1
464.4.a.e 2 1.a even 1 1 trivial
522.4.a.j 2 12.b even 2 1
1450.4.a.g 2 20.d odd 2 1
1682.4.a.c 2 116.d odd 2 1
1856.4.a.i 2 8.b even 2 1
1856.4.a.l 2 8.d odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 5$$
$5$ $$T^{2} + 10T - 191$$
$7$ $$T^{2} - 16T - 320$$
$11$ $$T^{2} - 90T + 1971$$
$13$ $$T^{2} + 50T + 241$$
$17$ $$T^{2} + 44T - 1052$$
$19$ $$T^{2} + 108T + 2820$$
$23$ $$T^{2} - 28T - 36308$$
$29$ $$(T - 29)^{2}$$
$31$ $$T^{2} + 66T - 70197$$
$37$ $$T^{2} - 40T + 304$$
$41$ $$T^{2} - 304T - 15296$$
$43$ $$T^{2} - 130T - 12629$$
$47$ $$T^{2} - 514T + 7243$$
$53$ $$T^{2} + 958T + 213217$$
$59$ $$T^{2} - 180T - 366900$$
$61$ $$T^{2} - 1028 T + 263332$$
$67$ $$T^{2} - 912T + 205536$$
$71$ $$T^{2} + 796T + 151468$$
$73$ $$T^{2} + 856T - 2672$$
$79$ $$T^{2} - 318T - 756645$$
$83$ $$T^{2} - 1828 T + 826732$$
$89$ $$T^{2} + 944T + 191680$$
$97$ $$T^{2} - 368T - 26144$$