# Properties

 Label 1728.4.d.e Level $1728$ Weight $4$ Character orbit 1728.d Analytic conductor $101.955$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1728.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$101.955300490$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.2261390379264.6 Defining polynomial: $$x^{8} - 4x^{7} - 8x^{6} + 38x^{5} - 38x^{4} + 8x^{3} + 325x^{2} - 322x + 2122$$ x^8 - 4*x^7 - 8*x^6 + 38*x^5 - 38*x^4 + 8*x^3 + 325*x^2 - 322*x + 2122 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}\cdot 3^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{5} + (\beta_{6} + \beta_{4}) q^{7}+O(q^{10})$$ q - b5 * q^5 + (b6 + b4) * q^7 $$q - \beta_{5} q^{5} + (\beta_{6} + \beta_{4}) q^{7} + ( - \beta_{2} - 12 \beta_1) q^{11} + ( - \beta_{7} - 2 \beta_{5}) q^{13} + ( - \beta_{3} - 21) q^{17} + ( - 3 \beta_{2} - 43 \beta_1) q^{19} + ( - 9 \beta_{6} - 2 \beta_{4}) q^{23} + ( - 3 \beta_{3} - 37) q^{25} + (9 \beta_{7} - 10 \beta_{5}) q^{29} + ( - 8 \beta_{6} + 13 \beta_{4}) q^{31} + (6 \beta_{2} + 135 \beta_1) q^{35} + (7 \beta_{7} - 22 \beta_{5}) q^{37} + (5 \beta_{3} - 3) q^{41} + (6 \beta_{2} + 106 \beta_1) q^{43} - 20 \beta_{4} q^{47} + (3 \beta_{3} + 35) q^{49} + (9 \beta_{7} - 11 \beta_{5}) q^{53} + (21 \beta_{6} + 45 \beta_{4}) q^{55} + ( - 10 \beta_{2} + 78 \beta_1) q^{59} + ( - 12 \beta_{7} - 48 \beta_{5}) q^{61} + ( - 3 \beta_{3} - 351) q^{65} - 20 \beta_1 q^{67} + ( - 9 \beta_{6} - 22 \beta_{4}) q^{71} + (9 \beta_{3} + 38) q^{73} + 69 \beta_{5} q^{77} + ( - 7 \beta_{6} + 8 \beta_{4}) q^{79} + ( - 3 \beta_{2} + 468 \beta_1) q^{83} + ( - 21 \beta_{7} + 54 \beta_{5}) q^{85} + ( - 18 \beta_{3} - 774) q^{89} + (15 \beta_{2} + 27 \beta_1) q^{91} + (63 \beta_{6} + 142 \beta_{4}) q^{95} + (12 \beta_{3} + 403) q^{97}+O(q^{100})$$ q - b5 * q^5 + (b6 + b4) * q^7 + (-b2 - 12*b1) * q^11 + (-b7 - 2*b5) * q^13 + (-b3 - 21) * q^17 + (-3*b2 - 43*b1) * q^19 + (-9*b6 - 2*b4) * q^23 + (-3*b3 - 37) * q^25 + (9*b7 - 10*b5) * q^29 + (-8*b6 + 13*b4) * q^31 + (6*b2 + 135*b1) * q^35 + (7*b7 - 22*b5) * q^37 + (5*b3 - 3) * q^41 + (6*b2 + 106*b1) * q^43 - 20*b4 * q^47 + (3*b3 + 35) * q^49 + (9*b7 - 11*b5) * q^53 + (21*b6 + 45*b4) * q^55 + (-10*b2 + 78*b1) * q^59 + (-12*b7 - 48*b5) * q^61 + (-3*b3 - 351) * q^65 - 20*b1 * q^67 + (-9*b6 - 22*b4) * q^71 + (9*b3 + 38) * q^73 + 69*b5 * q^77 + (-7*b6 + 8*b4) * q^79 + (-3*b2 + 468*b1) * q^83 + (-21*b7 + 54*b5) * q^85 + (-18*b3 - 774) * q^89 + (15*b2 + 27*b1) * q^91 + (63*b6 + 142*b4) * q^95 + (12*b3 + 403) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 168 q^{17} - 296 q^{25} - 24 q^{41} + 280 q^{49} - 2808 q^{65} + 304 q^{73} - 6192 q^{89} + 3224 q^{97}+O(q^{100})$$ 8 * q - 168 * q^17 - 296 * q^25 - 24 * q^41 + 280 * q^49 - 2808 * q^65 + 304 * q^73 - 6192 * q^89 + 3224 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} - 8x^{6} + 38x^{5} - 38x^{4} + 8x^{3} + 325x^{2} - 322x + 2122$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{6} - 6\nu^{5} - 15\nu^{4} + 40\nu^{3} + 18\nu^{2} - 39\nu + 35 ) / 639$$ (2*v^6 - 6*v^5 - 15*v^4 + 40*v^3 + 18*v^2 - 39*v + 35) / 639 $$\beta_{2}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 6\nu^{2} + 7\nu - 23$$ v^4 - 2*v^3 - 6*v^2 + 7*v - 23 $$\beta_{3}$$ $$=$$ $$( -4\nu^{6} + 12\nu^{5} + 30\nu^{4} - 80\nu^{3} + 390\nu^{2} - 348\nu - 1561 ) / 71$$ (-4*v^6 + 12*v^5 + 30*v^4 - 80*v^3 + 390*v^2 - 348*v - 1561) / 71 $$\beta_{4}$$ $$=$$ $$( -50\nu^{7} + 175\nu^{6} - 2043\nu^{5} + 4670\nu^{4} + 36047\nu^{3} - 58653\nu^{2} + 85648\nu - 32897 ) / 51333$$ (-50*v^7 + 175*v^6 - 2043*v^5 + 4670*v^4 + 36047*v^3 - 58653*v^2 + 85648*v - 32897) / 51333 $$\beta_{5}$$ $$=$$ $$( -58\nu^{7} + 203\nu^{6} - 1001\nu^{5} + 1995\nu^{4} + 2117\nu^{3} - 5069\nu^{2} + 67183\nu - 32685 ) / 17111$$ (-58*v^7 + 203*v^6 - 1001*v^5 + 1995*v^4 + 2117*v^3 - 5069*v^2 + 67183*v - 32685) / 17111 $$\beta_{6}$$ $$=$$ $$( -740\nu^{7} + 2590\nu^{6} + 10830\nu^{5} - 33550\nu^{4} - 41434\nu^{3} + 96996\nu^{2} - 5468\nu - 14612 ) / 51333$$ (-740*v^7 + 2590*v^6 + 10830*v^5 - 33550*v^4 - 41434*v^3 + 96996*v^2 - 5468*v - 14612) / 51333 $$\beta_{7}$$ $$=$$ $$( 280\nu^{7} - 980\nu^{6} - 2248\nu^{5} + 8070\nu^{4} - 10220\nu^{3} + 6770\nu^{2} + 150056\nu - 75864 ) / 17111$$ (280*v^7 - 980*v^6 - 2248*v^5 + 8070*v^4 - 10220*v^3 + 6770*v^2 + 150056*v - 75864) / 17111
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} + 2\beta_{4} + 6 ) / 12$$ (b7 + b6 + 2*b4 + 6) / 12 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{6} + 2\beta_{4} + 2\beta_{3} + 36\beta _1 + 48 ) / 12$$ (b7 + b6 + 2*b4 + 2*b3 + 36*b1 + 48) / 12 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} + 3\beta_{6} - 10\beta_{5} + 24\beta_{4} + 3\beta_{3} + 54\beta _1 + 69 ) / 12$$ (2*b7 + 3*b6 - 10*b5 + 24*b4 + 3*b3 + 54*b1 + 69) / 12 $$\nu^{4}$$ $$=$$ $$( 3\beta_{7} + 5\beta_{6} - 20\beta_{5} + 46\beta_{4} + 18\beta_{3} + 12\beta_{2} + 324\beta _1 + 660 ) / 12$$ (3*b7 + 5*b6 - 20*b5 + 46*b4 + 18*b3 + 12*b2 + 324*b1 + 660) / 12 $$\nu^{5}$$ $$=$$ $$( 43\beta_{7} + 77\beta_{6} - 190\beta_{5} + 244\beta_{4} + 40\beta_{3} + 30\beta_{2} + 720\beta _1 + 1536 ) / 12$$ (43*b7 + 77*b6 - 190*b5 + 244*b4 + 40*b3 + 30*b2 + 720*b1 + 1536) / 12 $$\nu^{6}$$ $$=$$ $$( 122\beta_{7} + 219\beta_{6} - 520\beta_{5} + 618\beta_{4} + 177\beta_{3} + 180\beta_{2} + 7020\beta _1 + 7653 ) / 12$$ (122*b7 + 219*b6 - 520*b5 + 618*b4 + 177*b3 + 180*b2 + 7020*b1 + 7653) / 12 $$\nu^{7}$$ $$=$$ $$( 932 \beta_{7} + 790 \beta_{6} - 3134 \beta_{5} + 2552 \beta_{4} + 483 \beta_{3} + 525 \beta_{2} + 22113 \beta _1 + 21489 ) / 12$$ (932*b7 + 790*b6 - 3134*b5 + 2552*b4 + 483*b3 + 525*b2 + 22113*b1 + 21489) / 12

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 −2.75792 − 0.460416i 3.75792 − 0.460416i −0.297464 + 1.88096i 1.29746 + 1.88096i −0.297464 − 1.88096i 1.29746 − 1.88096i −2.75792 + 0.460416i 3.75792 + 0.460416i
0 0 0 16.7850i 0 −22.3100 0 0 0
865.2 0 0 0 16.7850i 0 22.3100 0 0 0
865.3 0 0 0 6.50098i 0 −16.0706 0 0 0
865.4 0 0 0 6.50098i 0 16.0706 0 0 0
865.5 0 0 0 6.50098i 0 −16.0706 0 0 0
865.6 0 0 0 6.50098i 0 16.0706 0 0 0
865.7 0 0 0 16.7850i 0 −22.3100 0 0 0
865.8 0 0 0 16.7850i 0 22.3100 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 865.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.d.e 8
3.b odd 2 1 1728.4.d.h yes 8
4.b odd 2 1 inner 1728.4.d.e 8
8.b even 2 1 inner 1728.4.d.e 8
8.d odd 2 1 inner 1728.4.d.e 8
12.b even 2 1 1728.4.d.h yes 8
24.f even 2 1 1728.4.d.h yes 8
24.h odd 2 1 1728.4.d.h yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.4.d.e 8 1.a even 1 1 trivial
1728.4.d.e 8 4.b odd 2 1 inner
1728.4.d.e 8 8.b even 2 1 inner
1728.4.d.e 8 8.d odd 2 1 inner
1728.4.d.h yes 8 3.b odd 2 1
1728.4.d.h yes 8 12.b even 2 1
1728.4.d.h yes 8 24.f even 2 1
1728.4.d.h yes 8 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{4} + 324T_{5}^{2} + 11907$$ T5^4 + 324*T5^2 + 11907 $$T_{7}^{4} - 756T_{7}^{2} + 128547$$ T7^4 - 756*T7^2 + 128547 $$T_{17}^{2} + 42T_{17} - 1152$$ T17^2 + 42*T17 - 1152 $$T_{23}^{4} - 43092T_{23}^{2} + 250838208$$ T23^4 - 43092*T23^2 + 250838208

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 324 T^{2} + 11907)^{2}$$
$7$ $$(T^{4} - 756 T^{2} + 128547)^{2}$$
$11$ $$(T^{4} + 3474 T^{2} + 2099601)^{2}$$
$13$ $$(T^{4} + 2052 T^{2} + 995328)^{2}$$
$17$ $$(T^{2} + 42 T - 1152)^{4}$$
$19$ $$(T^{4} + 32372 T^{2} + \cdots + 155950144)^{2}$$
$23$ $$(T^{4} - 43092 T^{2} + \cdots + 250838208)^{2}$$
$29$ $$(T^{4} + 66420 T^{2} + \cdots + 903275712)^{2}$$
$31$ $$(T^{4} - 100548 T^{2} + \cdots + 2127630483)^{2}$$
$37$ $$(T^{4} + 166644 T^{2} + \cdots + 37340352)^{2}$$
$41$ $$(T^{2} + 6 T - 39816)^{4}$$
$43$ $$(T^{4} + 137168 T^{2} + \cdots + 2126316544)^{2}$$
$47$ $$(T^{4} - 129600 T^{2} + \cdots + 1905120000)^{2}$$
$53$ $$(T^{4} + 72252 T^{2} + \cdots + 951695163)^{2}$$
$59$ $$(T^{4} + 330768 T^{2} + \cdots + 23475142656)^{2}$$
$61$ $$(T^{4} + 886464 T^{2} + \cdots + 185752092672)^{2}$$
$67$ $$(T^{2} + 400)^{4}$$
$71$ $$(T^{4} - 179172 T^{2} + \cdots + 634583808)^{2}$$
$73$ $$(T^{2} - 76 T - 127589)^{4}$$
$79$ $$(T^{4} - 53244 T^{2} + \cdots + 403123392)^{2}$$
$83$ $$(T^{4} + 466722 T^{2} + \cdots + 41896767969)^{2}$$
$89$ $$(T^{2} + 1548 T + 82944)^{4}$$
$97$ $$(T^{2} - 806 T - 66983)^{4}$$