# Properties

 Label 1728.2.s.f Level $1728$ Weight $2$ Character orbit 1728.s Analytic conductor $13.798$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.170772624.1 Defining polynomial: $$x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16$$ x^8 - 3*x^7 + 5*x^6 - 6*x^5 + 6*x^4 - 12*x^3 + 20*x^2 - 24*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 36) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{5} + ( - \beta_{6} + \beta_1) q^{7}+O(q^{10})$$ q + (-b2 - 1) * q^5 + (-b6 + b1) * q^7 $$q + ( - \beta_{2} - 1) q^{5} + ( - \beta_{6} + \beta_1) q^{7} + ( - \beta_{7} - \beta_{6} - \beta_{3}) q^{11} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2} - 1) q^{13} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + 1) q^{17} + ( - \beta_{7} - \beta_{6} - 2 \beta_{3} + 2 \beta_1) q^{19} + ( - \beta_{7} + \beta_{3}) q^{23} + (\beta_{5} - 3 \beta_{4} + 2 \beta_{2} - 2) q^{25} + ( - \beta_{5} + 1) q^{29} + ( - 3 \beta_{7} + 2 \beta_{6} - \beta_{3} + 2 \beta_1) q^{31} + ( - \beta_{7} + \beta_{6} + \beta_{3} + \beta_1) q^{35} + ( - 2 \beta_{5} + 2 \beta_{2} + 2) q^{37} + ( - \beta_{4} - 2 \beta_{2} - 4) q^{41} + ( - 3 \beta_{7} + \beta_{6} + 3 \beta_{3} + 2 \beta_1) q^{43} + ( - 4 \beta_{7} - 3 \beta_{6} - 4 \beta_{3} + \beta_1) q^{47} + (2 \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{49} + ( - 2 \beta_{5} + 8 \beta_{4} - 2 \beta_{2} + 4) q^{53} + ( - \beta_{7} - \beta_{6} - \beta_{3} + \beta_1) q^{55} + ( - 3 \beta_{7} + \beta_{6} + 4 \beta_{3} - \beta_1) q^{59} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 1) q^{61} + ( - \beta_{5} - 2 \beta_{4} + 3) q^{65} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{3} + 3 \beta_1) q^{67} + ( - 4 \beta_{7} + 4 \beta_{6} + 2 \beta_{3} + 2 \beta_1) q^{71} + ( - \beta_{5} + \beta_{2} + 1) q^{73} + ( - 4 \beta_{4} + 3 \beta_{2} - 5) q^{77} + ( - 4 \beta_{7} + \beta_{6} + 4 \beta_{3} + 3 \beta_1) q^{79} + ( - 4 \beta_{7} - \beta_{6} - 4 \beta_{3} + 3 \beta_1) q^{83} + 2 \beta_{4} q^{85} + (2 \beta_{5} + 8 \beta_{4} + 2 \beta_{2} + 4) q^{89} + ( - 3 \beta_{7} - 3 \beta_{6} - \beta_{3} + \beta_1) q^{91} + ( - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{3} - 2 \beta_1) q^{95} + ( - 4 \beta_{5} + 7 \beta_{4} - 8 \beta_{2} + 3) q^{97}+O(q^{100})$$ q + (-b2 - 1) * q^5 + (-b6 + b1) * q^7 + (-b7 - b6 - b3) * q^11 + (2*b5 - 2*b4 + b2 - 1) * q^13 + (-b5 + 2*b4 - b2 + 1) * q^17 + (-b7 - b6 - 2*b3 + 2*b1) * q^19 + (-b7 + b3) * q^23 + (b5 - 3*b4 + 2*b2 - 2) * q^25 + (-b5 + 1) * q^29 + (-3*b7 + 2*b6 - b3 + 2*b1) * q^31 + (-b7 + b6 + b3 + b1) * q^35 + (-2*b5 + 2*b2 + 2) * q^37 + (-b4 - 2*b2 - 4) * q^41 + (-3*b7 + b6 + 3*b3 + 2*b1) * q^43 + (-4*b7 - 3*b6 - 4*b3 + b1) * q^47 + (2*b5 + b4 + b2 - 1) * q^49 + (-2*b5 + 8*b4 - 2*b2 + 4) * q^53 + (-b7 - b6 - b3 + b1) * q^55 + (-3*b7 + b6 + 4*b3 - b1) * q^59 + (-b5 + 2*b4 - 2*b2 + 1) * q^61 + (-b5 - 2*b4 + 3) * q^65 + (-b7 + 3*b6 + 2*b3 + 3*b1) * q^67 + (-4*b7 + 4*b6 + 2*b3 + 2*b1) * q^71 + (-b5 + b2 + 1) * q^73 + (-4*b4 + 3*b2 - 5) * q^77 + (-4*b7 + b6 + 4*b3 + 3*b1) * q^79 + (-4*b7 - b6 - 4*b3 + 3*b1) * q^83 + 2*b4 * q^85 + (2*b5 + 8*b4 + 2*b2 + 4) * q^89 + (-3*b7 - 3*b6 - b3 + b1) * q^91 + (-2*b7 + 2*b6 + 4*b3 - 2*b1) * q^95 + (-4*b5 + 7*b4 - 8*b2 + 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{5}+O(q^{10})$$ 8 * q - 6 * q^5 $$8 q - 6 q^{5} + 2 q^{13} - 6 q^{25} + 6 q^{29} + 8 q^{37} - 24 q^{41} - 10 q^{49} + 2 q^{61} + 30 q^{65} + 4 q^{73} - 30 q^{77} - 8 q^{85} + 4 q^{97}+O(q^{100})$$ 8 * q - 6 * q^5 + 2 * q^13 - 6 * q^25 + 6 * q^29 + 8 * q^37 - 24 * q^41 - 10 * q^49 + 2 * q^61 + 30 * q^65 + 4 * q^73 - 30 * q^77 - 8 * q^85 + 4 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} + \nu^{6} + \nu^{5} + 2\nu^{3} + 4\nu^{2} + 4\nu - 8 ) / 8$$ (-v^7 + v^6 + v^5 + 2*v^3 + 4*v^2 + 4*v - 8) / 8 $$\beta_{2}$$ $$=$$ $$( -\nu^{6} + \nu^{5} - \nu^{4} + 2\nu^{3} - 4\nu^{2} + 4\nu - 4 ) / 4$$ (-v^6 + v^5 - v^4 + 2*v^3 - 4*v^2 + 4*v - 4) / 4 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + \nu^{6} - 3\nu^{5} + 4\nu^{4} - 2\nu^{3} + 12\nu^{2} - 12\nu + 16 ) / 8$$ (-v^7 + v^6 - 3*v^5 + 4*v^4 - 2*v^3 + 12*v^2 - 12*v + 16) / 8 $$\beta_{4}$$ $$=$$ $$( 3\nu^{7} - 5\nu^{6} + 7\nu^{5} - 6\nu^{4} + 10\nu^{3} - 24\nu^{2} + 28\nu - 32 ) / 8$$ (3*v^7 - 5*v^6 + 7*v^5 - 6*v^4 + 10*v^3 - 24*v^2 + 28*v - 32) / 8 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 3\nu^{6} - 5\nu^{5} + 4\nu^{4} - 4\nu^{3} + 10\nu^{2} - 16\nu + 20 ) / 4$$ (-v^7 + 3*v^6 - 5*v^5 + 4*v^4 - 4*v^3 + 10*v^2 - 16*v + 20) / 4 $$\beta_{6}$$ $$=$$ $$( 3\nu^{7} - 3\nu^{6} + 9\nu^{5} - 8\nu^{4} + 10\nu^{3} - 24\nu^{2} + 20\nu - 40 ) / 8$$ (3*v^7 - 3*v^6 + 9*v^5 - 8*v^4 + 10*v^3 - 24*v^2 + 20*v - 40) / 8 $$\beta_{7}$$ $$=$$ $$( 3\nu^{7} - 5\nu^{6} + 11\nu^{5} - 6\nu^{4} + 10\nu^{3} - 28\nu^{2} + 36\nu - 48 ) / 8$$ (3*v^7 - 5*v^6 + 11*v^5 - 6*v^4 + 10*v^3 - 28*v^2 + 36*v - 48) / 8
 $$\nu$$ $$=$$ $$( -\beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2$$ (-b6 + b4 - b3 - b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + \beta_1 ) / 2$$ (-b7 - b5 + b4 + b3 - 2*b2 + b1) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 1 ) / 2$$ (-b7 + b6 + b5 + 2*b4 + 2*b3 + b2 + 2*b1 + 1) / 2 $$\nu^{4}$$ $$=$$ $$( 4\beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{4} + 3\beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2$$ (4*b7 - b6 + 2*b5 - b4 + 3*b3 + b2 - b1 - 1) / 2 $$\nu^{5}$$ $$=$$ $$( 3\beta_{7} + 2\beta_{6} - \beta_{5} - 5\beta_{4} + 3\beta_{3} - \beta _1 + 6 ) / 2$$ (3*b7 + 2*b6 - b5 - 5*b4 + 3*b3 - b1 + 6) / 2 $$\nu^{6}$$ $$=$$ $$( \beta_{7} + \beta_{6} + 3\beta_{5} - 4\beta_{3} - 3\beta_{2} + 4\beta _1 + 5 ) / 2$$ (b7 + b6 + 3*b5 - 4*b3 - 3*b2 + 4*b1 + 5) / 2 $$\nu^{7}$$ $$=$$ $$( -2\beta_{7} + \beta_{6} + 7\beta_{4} + 3\beta_{3} - 13\beta_{2} - \beta _1 + 1 ) / 2$$ (-2*b7 + b6 + 7*b4 + 3*b3 - 13*b2 - b1 + 1) / 2

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
575.1
 −1.02187 − 0.977642i 0.335728 + 1.37379i 0.774115 − 1.18353i 1.41203 − 0.0786378i −1.02187 + 0.977642i 0.335728 − 1.37379i 0.774115 + 1.18353i 1.41203 + 0.0786378i
0 0 0 −2.18614 + 1.26217i 0 −1.10489 0.637910i 0 0 0
575.2 0 0 0 −2.18614 + 1.26217i 0 1.10489 + 0.637910i 0 0 0
575.3 0 0 0 0.686141 0.396143i 0 −2.35143 1.35760i 0 0 0
575.4 0 0 0 0.686141 0.396143i 0 2.35143 + 1.35760i 0 0 0
1151.1 0 0 0 −2.18614 1.26217i 0 −1.10489 + 0.637910i 0 0 0
1151.2 0 0 0 −2.18614 1.26217i 0 1.10489 0.637910i 0 0 0
1151.3 0 0 0 0.686141 + 0.396143i 0 −2.35143 + 1.35760i 0 0 0
1151.4 0 0 0 0.686141 + 0.396143i 0 2.35143 1.35760i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1151.4 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.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.s.f 8
3.b odd 2 1 576.2.s.f 8
4.b odd 2 1 inner 1728.2.s.f 8
8.b even 2 1 108.2.h.a 8
8.d odd 2 1 108.2.h.a 8
9.c even 3 1 576.2.s.f 8
9.c even 3 1 5184.2.c.j 8
9.d odd 6 1 inner 1728.2.s.f 8
9.d odd 6 1 5184.2.c.j 8
12.b even 2 1 576.2.s.f 8
24.f even 2 1 36.2.h.a 8
24.h odd 2 1 36.2.h.a 8
36.f odd 6 1 576.2.s.f 8
36.f odd 6 1 5184.2.c.j 8
36.h even 6 1 inner 1728.2.s.f 8
36.h even 6 1 5184.2.c.j 8
72.j odd 6 1 108.2.h.a 8
72.j odd 6 1 324.2.b.b 8
72.l even 6 1 108.2.h.a 8
72.l even 6 1 324.2.b.b 8
72.n even 6 1 36.2.h.a 8
72.n even 6 1 324.2.b.b 8
72.p odd 6 1 36.2.h.a 8
72.p odd 6 1 324.2.b.b 8
120.i odd 2 1 900.2.r.c 8
120.m even 2 1 900.2.r.c 8
120.q odd 4 2 900.2.o.a 16
120.w even 4 2 900.2.o.a 16
360.z odd 6 1 900.2.r.c 8
360.bk even 6 1 900.2.r.c 8
360.bo even 12 2 900.2.o.a 16
360.bu odd 12 2 900.2.o.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.h.a 8 24.f even 2 1
36.2.h.a 8 24.h odd 2 1
36.2.h.a 8 72.n even 6 1
36.2.h.a 8 72.p odd 6 1
108.2.h.a 8 8.b even 2 1
108.2.h.a 8 8.d odd 2 1
108.2.h.a 8 72.j odd 6 1
108.2.h.a 8 72.l even 6 1
324.2.b.b 8 72.j odd 6 1
324.2.b.b 8 72.l even 6 1
324.2.b.b 8 72.n even 6 1
324.2.b.b 8 72.p odd 6 1
576.2.s.f 8 3.b odd 2 1
576.2.s.f 8 9.c even 3 1
576.2.s.f 8 12.b even 2 1
576.2.s.f 8 36.f odd 6 1
900.2.o.a 16 120.q odd 4 2
900.2.o.a 16 120.w even 4 2
900.2.o.a 16 360.bo even 12 2
900.2.o.a 16 360.bu odd 12 2
900.2.r.c 8 120.i odd 2 1
900.2.r.c 8 120.m even 2 1
900.2.r.c 8 360.z odd 6 1
900.2.r.c 8 360.bk even 6 1
1728.2.s.f 8 1.a even 1 1 trivial
1728.2.s.f 8 4.b odd 2 1 inner
1728.2.s.f 8 9.d odd 6 1 inner
1728.2.s.f 8 36.h even 6 1 inner
5184.2.c.j 8 9.c even 3 1
5184.2.c.j 8 9.d odd 6 1
5184.2.c.j 8 36.f odd 6 1
5184.2.c.j 8 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{4} + 3T_{5}^{3} + T_{5}^{2} - 6T_{5} + 4$$ T5^4 + 3*T5^3 + T5^2 - 6*T5 + 4 $$T_{7}^{8} - 9T_{7}^{6} + 69T_{7}^{4} - 108T_{7}^{2} + 144$$ T7^8 - 9*T7^6 + 69*T7^4 - 108*T7^2 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 3 T^{3} + T^{2} - 6 T + 4)^{2}$$
$7$ $$T^{8} - 9 T^{6} + 69 T^{4} - 108 T^{2} + \cdots + 144$$
$11$ $$T^{8} + 12 T^{6} + 141 T^{4} + 36 T^{2} + \cdots + 9$$
$13$ $$(T^{4} - T^{3} + 9 T^{2} + 8 T + 64)^{2}$$
$17$ $$(T^{4} + 7 T^{2} + 4)^{2}$$
$19$ $$(T^{4} + 27 T^{2} + 108)^{2}$$
$23$ $$T^{8} + 15 T^{6} + 177 T^{4} + \cdots + 2304$$
$29$ $$(T^{4} - 3 T^{3} + T^{2} + 6 T + 4)^{2}$$
$31$ $$T^{8} - 69 T^{6} + 4569 T^{4} + \cdots + 36864$$
$37$ $$(T^{2} - 2 T - 32)^{4}$$
$41$ $$(T^{4} + 12 T^{3} + 49 T^{2} + 12 T + 1)^{2}$$
$43$ $$T^{8} - 108 T^{6} + 8781 T^{4} + \cdots + 8311689$$
$47$ $$T^{8} + 135 T^{6} + 18033 T^{4} + \cdots + 36864$$
$53$ $$(T^{4} + 76 T^{2} + 256)^{2}$$
$59$ $$T^{8} + 180 T^{6} + \cdots + 16867449$$
$61$ $$(T^{4} - T^{3} + 9 T^{2} + 8 T + 64)^{2}$$
$67$ $$T^{8} - 108 T^{6} + 8781 T^{4} + \cdots + 8311689$$
$71$ $$(T^{4} - 144 T^{2} + 432)^{2}$$
$73$ $$(T^{2} - T - 8)^{4}$$
$79$ $$T^{8} - 201 T^{6} + \cdots + 101848464$$
$83$ $$T^{8} + 111 T^{6} + 9249 T^{4} + \cdots + 9437184$$
$89$ $$(T^{4} + 172 T^{2} + 4096)^{2}$$
$97$ $$(T^{4} - 2 T^{3} + 135 T^{2} + 262 T + 17161)^{2}$$