# Properties

 Label 1728.3.o.d Level $1728$ Weight $3$ Character orbit 1728.o Analytic conductor $47.085$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.121550625.1 Defining polynomial: $$x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16$$ x^8 - x^7 - 4*x^6 - 9*x^5 + 23*x^4 + 18*x^3 - 16*x^2 + 8*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}\cdot 3^{6}$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} - \beta_{4} - 2 \beta_{2}) q^{5} - \beta_{3} q^{7}+O(q^{10})$$ q + (b6 - b4 - 2*b2) * q^5 - b3 * q^7 $$q + (\beta_{6} - \beta_{4} - 2 \beta_{2}) q^{5} - \beta_{3} q^{7} + ( - \beta_{7} + \beta_{3}) q^{11} + ( - 3 \beta_{6} + 3 \beta_{4} + 4 \beta_{2}) q^{13} + (\beta_{4} - 7) q^{17} + (3 \beta_{7} - 3 \beta_{5} + 2 \beta_1) q^{19} + (4 \beta_{5} + 5 \beta_{3} - 5 \beta_1) q^{23} + ( - 3 \beta_{6} + 5 \beta_{2} - 5) q^{25} + (\beta_{6} + 16 \beta_{2} - 16) q^{29} + (6 \beta_{5} - \beta_{3} + \beta_1) q^{31} + (2 \beta_{7} - 2 \beta_{5} + 7 \beta_1) q^{35} + ( - 6 \beta_{4} - 8) q^{37} + (6 \beta_{6} - 6 \beta_{4} + 33 \beta_{2}) q^{41} + (3 \beta_{7} + 7 \beta_{3}) q^{43} + (8 \beta_{7} + \beta_{3}) q^{47} + ( - 9 \beta_{6} + 9 \beta_{4} + 5 \beta_{2}) q^{49} + (6 \beta_{4} + 48) q^{53} + ( - 6 \beta_{7} + 6 \beta_{5} - 9 \beta_1) q^{55} + ( - 5 \beta_{5} + 5 \beta_{3} - 5 \beta_1) q^{59} + ( - 3 \beta_{6} - 2 \beta_{2} + 2) q^{61} + (7 \beta_{6} - 86 \beta_{2} + 86) q^{65} + (3 \beta_{5} + 3 \beta_{3} - 3 \beta_1) q^{67} + (4 \beta_{7} - 4 \beta_{5} - 4 \beta_1) q^{71} + ( - 15 \beta_{4} - 35) q^{73} + (9 \beta_{6} - 9 \beta_{4} - 72 \beta_{2}) q^{77} + (6 \beta_{7} - \beta_{3}) q^{79} + (10 \beta_{7} - \beta_{3}) q^{83} + ( - 6 \beta_{6} + 6 \beta_{4} - 12 \beta_{2}) q^{85} + (8 \beta_{4} - 110) q^{89} + ( - 6 \beta_{7} + 6 \beta_{5} - 19 \beta_1) q^{91} + ( - 8 \beta_{5} + 8 \beta_{3} - 8 \beta_1) q^{95} + (18 \beta_{6} - 59 \beta_{2} + 59) q^{97}+O(q^{100})$$ q + (b6 - b4 - 2*b2) * q^5 - b3 * q^7 + (-b7 + b3) * q^11 + (-3*b6 + 3*b4 + 4*b2) * q^13 + (b4 - 7) * q^17 + (3*b7 - 3*b5 + 2*b1) * q^19 + (4*b5 + 5*b3 - 5*b1) * q^23 + (-3*b6 + 5*b2 - 5) * q^25 + (b6 + 16*b2 - 16) * q^29 + (6*b5 - b3 + b1) * q^31 + (2*b7 - 2*b5 + 7*b1) * q^35 + (-6*b4 - 8) * q^37 + (6*b6 - 6*b4 + 33*b2) * q^41 + (3*b7 + 7*b3) * q^43 + (8*b7 + b3) * q^47 + (-9*b6 + 9*b4 + 5*b2) * q^49 + (6*b4 + 48) * q^53 + (-6*b7 + 6*b5 - 9*b1) * q^55 + (-5*b5 + 5*b3 - 5*b1) * q^59 + (-3*b6 - 2*b2 + 2) * q^61 + (7*b6 - 86*b2 + 86) * q^65 + (3*b5 + 3*b3 - 3*b1) * q^67 + (4*b7 - 4*b5 - 4*b1) * q^71 + (-15*b4 - 35) * q^73 + (9*b6 - 9*b4 - 72*b2) * q^77 + (6*b7 - b3) * q^79 + (10*b7 - b3) * q^83 + (-6*b6 + 6*b4 - 12*b2) * q^85 + (8*b4 - 110) * q^89 + (-6*b7 + 6*b5 - 19*b1) * q^91 + (-8*b5 + 8*b3 - 8*b1) * q^95 + (18*b6 - 59*b2 + 59) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{5}+O(q^{10})$$ 8 * q - 6 * q^5 $$8 q - 6 q^{5} + 10 q^{13} - 60 q^{17} - 14 q^{25} - 66 q^{29} - 40 q^{37} + 144 q^{41} + 2 q^{49} + 360 q^{53} + 14 q^{61} + 330 q^{65} - 220 q^{73} - 270 q^{77} - 60 q^{85} - 912 q^{89} + 200 q^{97}+O(q^{100})$$ 8 * q - 6 * q^5 + 10 * q^13 - 60 * q^17 - 14 * q^25 - 66 * q^29 - 40 * q^37 + 144 * q^41 + 2 * q^49 + 360 * q^53 + 14 * q^61 + 330 * q^65 - 220 * q^73 - 270 * q^77 - 60 * q^85 - 912 * q^89 + 200 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( 7\nu^{7} + 93\nu^{6} - 244\nu^{5} - 547\nu^{4} - 659\nu^{3} + 3622\nu^{2} + 1884\nu - 1240 ) / 408$$ (7*v^7 + 93*v^6 - 244*v^5 - 547*v^4 - 659*v^3 + 3622*v^2 + 1884*v - 1240) / 408 $$\beta_{2}$$ $$=$$ $$( 41\nu^{7} - 111\nu^{6} - 74\nu^{5} - 173\nu^{4} + 1517\nu^{3} - 1036\nu^{2} - 768\nu + 1480 ) / 816$$ (41*v^7 - 111*v^6 - 74*v^5 - 173*v^4 + 1517*v^3 - 1036*v^2 - 768*v + 1480) / 816 $$\beta_{3}$$ $$=$$ $$( 23\nu^{7} - 15\nu^{6} - 248\nu^{5} - 311\nu^{4} + 953\nu^{3} + 2546\nu^{2} - 600\nu - 2384 ) / 408$$ (23*v^7 - 15*v^6 - 248*v^5 - 311*v^4 + 953*v^3 + 2546*v^2 - 600*v - 2384) / 408 $$\beta_{4}$$ $$=$$ $$( -65\nu^{7} + 69\nu^{6} + 284\nu^{5} + 533\nu^{4} - 1691\nu^{3} - 1226\nu^{2} + 2556\nu - 376 ) / 408$$ (-65*v^7 + 69*v^6 + 284*v^5 + 533*v^4 - 1691*v^3 - 1226*v^2 + 2556*v - 376) / 408 $$\beta_{5}$$ $$=$$ $$( -199\nu^{7} + 489\nu^{6} + 190\nu^{5} + 1387\nu^{4} - 6955\nu^{3} + 4700\nu^{2} - 24\nu - 1352 ) / 816$$ (-199*v^7 + 489*v^6 + 190*v^5 + 1387*v^4 - 6955*v^3 + 4700*v^2 - 24*v - 1352) / 816 $$\beta_{6}$$ $$=$$ $$( 169\nu^{7} - 261\nu^{6} - 412\nu^{5} - 1345\nu^{4} + 4315\nu^{3} - 566\nu^{2} + 168\nu + 2528 ) / 408$$ (169*v^7 - 261*v^6 - 412*v^5 - 1345*v^4 + 4315*v^3 - 566*v^2 + 168*v + 2528) / 408 $$\beta_{7}$$ $$=$$ $$( 427\nu^{7} - 753\nu^{6} - 1114\nu^{5} - 2767\nu^{4} + 11515\nu^{3} - 1520\nu^{2} - 6864\nu + 10040 ) / 816$$ (427*v^7 - 753*v^6 - 1114*v^5 - 2767*v^4 + 11515*v^3 - 1520*v^2 - 6864*v + 10040) / 816
 $$\nu$$ $$=$$ $$( -\beta_{7} + 3\beta_{6} + 2\beta_{5} + 3\beta_{4} + 2\beta_{3} + 3\beta_{2} - \beta _1 + 3 ) / 18$$ (-b7 + 3*b6 + 2*b5 + 3*b4 + 2*b3 + 3*b2 - b1 + 3) / 18 $$\nu^{2}$$ $$=$$ $$( 2\beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} + 2\beta_{3} - 13\beta_{2} - \beta _1 + 14 ) / 6$$ (2*b7 - b6 - b5 + 2*b4 + 2*b3 - 13*b2 - b1 + 14) / 6 $$\nu^{3}$$ $$=$$ $$( -11\beta_{7} + 12\beta_{6} + \beta_{5} - 6\beta_{4} + 10\beta_{3} - 6\beta_{2} - 11\beta _1 + 93 ) / 18$$ (-11*b7 + 12*b6 + b5 - 6*b4 + 10*b3 - 6*b2 - 11*b1 + 93) / 18 $$\nu^{4}$$ $$=$$ $$( \beta_{7} + 9\beta_{6} + 10\beta_{5} + 9\beta_{4} + 10\beta_{3} - 15\beta_{2} - 11\beta _1 + 9 ) / 6$$ (b7 + 9*b6 + 10*b5 + 9*b4 + 10*b3 - 15*b2 - 11*b1 + 9) / 6 $$\nu^{5}$$ $$=$$ $$( 44\beta_{7} - 15\beta_{6} - 55\beta_{5} + 30\beta_{4} + 44\beta_{3} - 537\beta_{2} - 55\beta _1 + 552 ) / 18$$ (44*b7 - 15*b6 - 55*b5 + 30*b4 + 44*b3 - 537*b2 - 55*b1 + 552) / 18 $$\nu^{6}$$ $$=$$ $$( -57\beta_{7} + 80\beta_{6} + 33\beta_{5} - 40\beta_{4} + 24\beta_{3} - 40\beta_{2} - 57\beta _1 + 263 ) / 6$$ (-57*b7 + 80*b6 + 33*b5 - 40*b4 + 24*b3 - 40*b2 - 57*b1 + 263) / 6 $$\nu^{7}$$ $$=$$ $$( 169\beta_{7} + 273\beta_{6} + 220\beta_{5} + 273\beta_{4} + 220\beta_{3} - 1833\beta_{2} - 389\beta _1 + 273 ) / 18$$ (169*b7 + 273*b6 + 220*b5 + 273*b4 + 220*b3 - 1833*b2 - 389*b1 + 273) / 18

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 2.25820 − 0.369600i 0.553538 − 0.676408i −1.44918 + 1.77086i −0.862555 + 0.141174i 2.25820 + 0.369600i 0.553538 + 0.676408i −1.44918 − 1.77086i −0.862555 − 0.141174i
0 0 0 −3.31174 5.73610i 0 −8.46808 4.88905i 0 0 0
127.2 0 0 0 −3.31174 5.73610i 0 8.46808 + 4.88905i 0 0 0
127.3 0 0 0 1.81174 + 3.13802i 0 −1.59422 0.920424i 0 0 0
127.4 0 0 0 1.81174 + 3.13802i 0 1.59422 + 0.920424i 0 0 0
1279.1 0 0 0 −3.31174 + 5.73610i 0 −8.46808 + 4.88905i 0 0 0
1279.2 0 0 0 −3.31174 + 5.73610i 0 8.46808 4.88905i 0 0 0
1279.3 0 0 0 1.81174 3.13802i 0 −1.59422 + 0.920424i 0 0 0
1279.4 0 0 0 1.81174 3.13802i 0 1.59422 0.920424i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1279.4 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
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.o.d 8
3.b odd 2 1 576.3.o.e 8
4.b odd 2 1 inner 1728.3.o.d 8
8.b even 2 1 432.3.o.c 8
8.d odd 2 1 432.3.o.c 8
9.c even 3 1 inner 1728.3.o.d 8
9.d odd 6 1 576.3.o.e 8
12.b even 2 1 576.3.o.e 8
24.f even 2 1 144.3.o.b 8
24.h odd 2 1 144.3.o.b 8
36.f odd 6 1 inner 1728.3.o.d 8
36.h even 6 1 576.3.o.e 8
72.j odd 6 1 144.3.o.b 8
72.j odd 6 1 1296.3.g.h 4
72.l even 6 1 144.3.o.b 8
72.l even 6 1 1296.3.g.h 4
72.n even 6 1 432.3.o.c 8
72.n even 6 1 1296.3.g.d 4
72.p odd 6 1 432.3.o.c 8
72.p odd 6 1 1296.3.g.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.o.b 8 24.f even 2 1
144.3.o.b 8 24.h odd 2 1
144.3.o.b 8 72.j odd 6 1
144.3.o.b 8 72.l even 6 1
432.3.o.c 8 8.b even 2 1
432.3.o.c 8 8.d odd 2 1
432.3.o.c 8 72.n even 6 1
432.3.o.c 8 72.p odd 6 1
576.3.o.e 8 3.b odd 2 1
576.3.o.e 8 9.d odd 6 1
576.3.o.e 8 12.b even 2 1
576.3.o.e 8 36.h even 6 1
1296.3.g.d 4 72.n even 6 1
1296.3.g.d 4 72.p odd 6 1
1296.3.g.h 4 72.j odd 6 1
1296.3.g.h 4 72.l even 6 1
1728.3.o.d 8 1.a even 1 1 trivial
1728.3.o.d 8 4.b odd 2 1 inner
1728.3.o.d 8 9.c even 3 1 inner
1728.3.o.d 8 36.f odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{4} + 3T_{5}^{3} + 33T_{5}^{2} - 72T_{5} + 576$$ T5^4 + 3*T5^3 + 33*T5^2 - 72*T5 + 576 $$T_{7}^{8} - 99T_{7}^{6} + 9477T_{7}^{4} - 32076T_{7}^{2} + 104976$$ T7^8 - 99*T7^6 + 9477*T7^4 - 32076*T7^2 + 104976

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 3 T^{3} + 33 T^{2} - 72 T + 576)^{2}$$
$7$ $$T^{8} - 99 T^{6} + 9477 T^{4} + \cdots + 104976$$
$11$ $$(T^{4} - 135 T^{2} + 18225)^{2}$$
$13$ $$(T^{4} - 5 T^{3} + 255 T^{2} + 1150 T + 52900)^{2}$$
$17$ $$(T^{2} + 15 T + 30)^{4}$$
$19$ $$(T^{4} + 855 T^{2} + 129600)^{2}$$
$23$ $$T^{8} - 2619 T^{6} + \cdots + 2379503694096$$
$29$ $$(T^{4} + 33 T^{3} + 843 T^{2} + \cdots + 60516)^{2}$$
$31$ $$T^{8} - 4095 T^{6} + \cdots + 2520473760000$$
$37$ $$(T^{2} + 10 T - 920)^{4}$$
$41$ $$(T^{4} - 72 T^{3} + 4833 T^{2} + \cdots + 123201)^{2}$$
$43$ $$T^{8} - 4230 T^{6} + \cdots + 1147523000625$$
$47$ $$T^{8} - 5859 T^{6} + \cdots + 20415837456$$
$53$ $$(T^{2} - 90 T + 1080)^{4}$$
$59$ $$(T^{4} - 3375 T^{2} + 11390625)^{2}$$
$61$ $$(T^{4} - 7 T^{3} + 273 T^{2} + 1568 T + 50176)^{2}$$
$67$ $$(T^{4} - 567 T^{2} + 321489)^{2}$$
$71$ $$(T^{2} + 2160)^{4}$$
$73$ $$(T^{2} + 55 T - 5150)^{4}$$
$79$ $$T^{8} - 4095 T^{6} + \cdots + 2520473760000$$
$83$ $$T^{8} - 10719 T^{6} + \cdots + 62171080298496$$
$89$ $$(T^{2} + 228 T + 11316)^{4}$$
$97$ $$(T^{4} - 100 T^{3} + 16005 T^{2} + \cdots + 36060025)^{2}$$