# Properties

 Label 624.4.a.t Level $624$ Weight $4$ Character orbit 624.a Self dual yes Analytic conductor $36.817$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 624.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.8171918436$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.3144.1 Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 39) 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.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + ( - \beta_{2} + 1) q^{5} + ( - 3 \beta_1 - 11) q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + (-b2 + 1) * q^5 + (-3*b1 - 11) * q^7 + 9 * q^9 $$q - 3 q^{3} + ( - \beta_{2} + 1) q^{5} + ( - 3 \beta_1 - 11) q^{7} + 9 q^{9} + ( - 3 \beta_{2} - \beta_1 + 4) q^{11} + 13 q^{13} + (3 \beta_{2} - 3) q^{15} + ( - 4 \beta_1 - 50) q^{17} + ( - 8 \beta_{2} + 3 \beta_1 - 33) q^{19} + (9 \beta_1 + 33) q^{21} + (4 \beta_{2} - 16 \beta_1 + 12) q^{23} + ( - 10 \beta_{2} - 12 \beta_1 + 41) q^{25} - 27 q^{27} + (4 \beta_{2} + 10 \beta_1 + 4) q^{29} + (2 \beta_{2} + 27 \beta_1 - 91) q^{31} + (9 \beta_{2} + 3 \beta_1 - 12) q^{33} + (2 \beta_{2} + 18 \beta_1 - 20) q^{35} + ( - 14 \beta_{2} - 24 \beta_1 + 112) q^{37} - 39 q^{39} + (17 \beta_{2} - 2 \beta_1 + 165) q^{41} + ( - 2 \beta_{2} + 30 \beta_1 + 96) q^{43} + ( - 9 \beta_{2} + 9) q^{45} + (21 \beta_{2} - 27 \beta_1 + 6) q^{47} + (18 \beta_{2} + 84 \beta_1 + 183) q^{49} + (12 \beta_1 + 150) q^{51} + ( - 6 \beta_{2} - 54 \beta_1 - 246) q^{53} + ( - 34 \beta_{2} - 30 \beta_1 + 496) q^{55} + (24 \beta_{2} - 9 \beta_1 + 99) q^{57} + ( - \beta_{2} - 41 \beta_1 + 582) q^{59} + ( - 14 \beta_{2} + 12 \beta_1 + 76) q^{61} + ( - 27 \beta_1 - 99) q^{63} + ( - 13 \beta_{2} + 13) q^{65} + (38 \beta_{2} - 21 \beta_1 - 19) q^{67} + ( - 12 \beta_{2} + 48 \beta_1 - 36) q^{69} + ( - 7 \beta_{2} + 67 \beta_1 + 336) q^{71} + (6 \beta_{2} - 120 \beta_1 - 112) q^{73} + (30 \beta_{2} + 36 \beta_1 - 123) q^{75} + (12 \beta_{2} + 68 \beta_1 + 64) q^{77} + (12 \beta_{2} - 24 \beta_1 + 4) q^{79} + 81 q^{81} + (5 \beta_{2} - 15 \beta_1 + 262) q^{83} + (38 \beta_{2} + 24 \beta_1 - 62) q^{85} + ( - 12 \beta_{2} - 30 \beta_1 - 12) q^{87} + (15 \beta_{2} + 58 \beta_1 + 503) q^{89} + ( - 39 \beta_1 - 143) q^{91} + ( - 6 \beta_{2} - 81 \beta_1 + 273) q^{93} + ( - 30 \beta_{2} - 114 \beta_1 + 1296) q^{95} + ( - 2 \beta_{2} - 24 \beta_1 + 1072) q^{97} + ( - 27 \beta_{2} - 9 \beta_1 + 36) q^{99}+O(q^{100})$$ q - 3 * q^3 + (-b2 + 1) * q^5 + (-3*b1 - 11) * q^7 + 9 * q^9 + (-3*b2 - b1 + 4) * q^11 + 13 * q^13 + (3*b2 - 3) * q^15 + (-4*b1 - 50) * q^17 + (-8*b2 + 3*b1 - 33) * q^19 + (9*b1 + 33) * q^21 + (4*b2 - 16*b1 + 12) * q^23 + (-10*b2 - 12*b1 + 41) * q^25 - 27 * q^27 + (4*b2 + 10*b1 + 4) * q^29 + (2*b2 + 27*b1 - 91) * q^31 + (9*b2 + 3*b1 - 12) * q^33 + (2*b2 + 18*b1 - 20) * q^35 + (-14*b2 - 24*b1 + 112) * q^37 - 39 * q^39 + (17*b2 - 2*b1 + 165) * q^41 + (-2*b2 + 30*b1 + 96) * q^43 + (-9*b2 + 9) * q^45 + (21*b2 - 27*b1 + 6) * q^47 + (18*b2 + 84*b1 + 183) * q^49 + (12*b1 + 150) * q^51 + (-6*b2 - 54*b1 - 246) * q^53 + (-34*b2 - 30*b1 + 496) * q^55 + (24*b2 - 9*b1 + 99) * q^57 + (-b2 - 41*b1 + 582) * q^59 + (-14*b2 + 12*b1 + 76) * q^61 + (-27*b1 - 99) * q^63 + (-13*b2 + 13) * q^65 + (38*b2 - 21*b1 - 19) * q^67 + (-12*b2 + 48*b1 - 36) * q^69 + (-7*b2 + 67*b1 + 336) * q^71 + (6*b2 - 120*b1 - 112) * q^73 + (30*b2 + 36*b1 - 123) * q^75 + (12*b2 + 68*b1 + 64) * q^77 + (12*b2 - 24*b1 + 4) * q^79 + 81 * q^81 + (5*b2 - 15*b1 + 262) * q^83 + (38*b2 + 24*b1 - 62) * q^85 + (-12*b2 - 30*b1 - 12) * q^87 + (15*b2 + 58*b1 + 503) * q^89 + (-39*b1 - 143) * q^91 + (-6*b2 - 81*b1 + 273) * q^93 + (-30*b2 - 114*b1 + 1296) * q^95 + (-2*b2 - 24*b1 + 1072) * q^97 + (-27*b2 - 9*b1 + 36) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 9 q^{3} + 4 q^{5} - 30 q^{7} + 27 q^{9}+O(q^{10})$$ 3 * q - 9 * q^3 + 4 * q^5 - 30 * q^7 + 27 * q^9 $$3 q - 9 q^{3} + 4 q^{5} - 30 q^{7} + 27 q^{9} + 16 q^{11} + 39 q^{13} - 12 q^{15} - 146 q^{17} - 94 q^{19} + 90 q^{21} + 48 q^{23} + 145 q^{25} - 81 q^{27} - 2 q^{29} - 302 q^{31} - 48 q^{33} - 80 q^{35} + 374 q^{37} - 117 q^{39} + 480 q^{41} + 260 q^{43} + 36 q^{45} + 24 q^{47} + 447 q^{49} + 438 q^{51} - 678 q^{53} + 1552 q^{55} + 282 q^{57} + 1788 q^{59} + 230 q^{61} - 270 q^{63} + 52 q^{65} - 74 q^{67} - 144 q^{69} + 948 q^{71} - 222 q^{73} - 435 q^{75} + 112 q^{77} + 24 q^{79} + 243 q^{81} + 796 q^{83} - 248 q^{85} + 6 q^{87} + 1436 q^{89} - 390 q^{91} + 906 q^{93} + 4032 q^{95} + 3242 q^{97} + 144 q^{99}+O(q^{100})$$ 3 * q - 9 * q^3 + 4 * q^5 - 30 * q^7 + 27 * q^9 + 16 * q^11 + 39 * q^13 - 12 * q^15 - 146 * q^17 - 94 * q^19 + 90 * q^21 + 48 * q^23 + 145 * q^25 - 81 * q^27 - 2 * q^29 - 302 * q^31 - 48 * q^33 - 80 * q^35 + 374 * q^37 - 117 * q^39 + 480 * q^41 + 260 * q^43 + 36 * q^45 + 24 * q^47 + 447 * q^49 + 438 * q^51 - 678 * q^53 + 1552 * q^55 + 282 * q^57 + 1788 * q^59 + 230 * q^61 - 270 * q^63 + 52 * q^65 - 74 * q^67 - 144 * q^69 + 948 * q^71 - 222 * q^73 - 435 * q^75 + 112 * q^77 + 24 * q^79 + 243 * q^81 + 796 * q^83 - 248 * q^85 + 6 * q^87 + 1436 * q^89 - 390 * q^91 + 906 * q^93 + 4032 * q^95 + 3242 * q^97 + 144 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 16x - 8$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$2\nu^{2} - 4\nu - 21$$ 2*v^2 - 4*v - 21
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 23 ) / 2$$ (b2 + 2*b1 + 23) / 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
 −3.20905 4.73549 −0.526440
0 −3.00000 0 −11.4322 0 11.2543 0 9.00000 0
1.2 0 −3.00000 0 −3.90776 0 −36.4129 0 9.00000 0
1.3 0 −3.00000 0 19.3400 0 −4.84136 0 9.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$13$$ $$-1$$

## 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 624.4.a.t 3
3.b odd 2 1 1872.4.a.bk 3
4.b odd 2 1 39.4.a.c 3
8.b even 2 1 2496.4.a.bp 3
8.d odd 2 1 2496.4.a.bl 3
12.b even 2 1 117.4.a.f 3
20.d odd 2 1 975.4.a.l 3
28.d even 2 1 1911.4.a.k 3
52.b odd 2 1 507.4.a.h 3
52.f even 4 2 507.4.b.g 6
156.h even 2 1 1521.4.a.u 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.c 3 4.b odd 2 1
117.4.a.f 3 12.b even 2 1
507.4.a.h 3 52.b odd 2 1
507.4.b.g 6 52.f even 4 2
624.4.a.t 3 1.a even 1 1 trivial
975.4.a.l 3 20.d odd 2 1
1521.4.a.u 3 156.h even 2 1
1872.4.a.bk 3 3.b odd 2 1
1911.4.a.k 3 28.d even 2 1
2496.4.a.bl 3 8.d odd 2 1
2496.4.a.bp 3 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(624))$$:

 $$T_{5}^{3} - 4T_{5}^{2} - 252T_{5} - 864$$ T5^3 - 4*T5^2 - 252*T5 - 864 $$T_{7}^{3} + 30T_{7}^{2} - 288T_{7} - 1984$$ T7^3 + 30*T7^2 - 288*T7 - 1984

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 3)^{3}$$
$5$ $$T^{3} - 4 T^{2} - 252 T - 864$$
$7$ $$T^{3} + 30 T^{2} - 288 T - 1984$$
$11$ $$T^{3} - 16 T^{2} - 2256 T - 30336$$
$13$ $$(T - 13)^{3}$$
$17$ $$T^{3} + 146 T^{2} + 6060 T + 71256$$
$19$ $$T^{3} + 94 T^{2} - 14432 T - 779616$$
$23$ $$T^{3} - 48 T^{2} - 20928 T - 534528$$
$29$ $$T^{3} + 2 T^{2} - 10116 T - 199176$$
$31$ $$T^{3} + 302 T^{2} - 17536 T - 7197248$$
$37$ $$T^{3} - 374 T^{2} - 36964 T + 7758104$$
$41$ $$T^{3} - 480 T^{2} + \cdots + 12919824$$
$43$ $$T^{3} - 260 T^{2} - 38096 T + 3663168$$
$47$ $$T^{3} - 24 T^{2} - 168480 T - 18102528$$
$53$ $$T^{3} + 678 T^{2} - 42228 T - 1471608$$
$59$ $$T^{3} - 1788 T^{2} + \cdots - 137423808$$
$61$ $$T^{3} - 230 T^{2} - 44452 T + 6279512$$
$67$ $$T^{3} + 74 T^{2} - 409216 T - 4260896$$
$71$ $$T^{3} - 948 T^{2} + \cdots + 70464384$$
$73$ $$T^{3} + 222 T^{2} + \cdots + 22780552$$
$79$ $$T^{3} - 24 T^{2} - 78336 T - 7757824$$
$83$ $$T^{3} - 796 T^{2} + \cdots - 13963968$$
$89$ $$T^{3} - 1436 T^{2} + \cdots - 30129888$$
$97$ $$T^{3} - 3242 T^{2} + \cdots - 1218481048$$