# Properties

 Label 1728.3.o.f Level $1728$ Weight $3$ Character orbit 1728.o Analytic conductor $47.085$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# 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: 8.0.856615824.2 Defining polynomial: $$x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4$$ x^8 + 11*x^6 + 36*x^4 + 32*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} - \beta_1 + 1) q^{5} + ( - \beta_{7} - \beta_{5} - \beta_{4}) q^{7}+O(q^{10})$$ q + (b4 - b1 + 1) * q^5 + (-b7 - b5 - b4) * q^7 $$q + (\beta_{4} - \beta_1 + 1) q^{5} + ( - \beta_{7} - \beta_{5} - \beta_{4}) q^{7} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 2) q^{11} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{13} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{3} + \beta_{2} - 1) q^{17} + (2 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 4 \beta_1 - 2) q^{19} + (\beta_{6} - \beta_{5} - \beta_{3} + 7 \beta_1 + 7) q^{23} + ( - 2 \beta_{7} + 4 \beta_{6} - 7 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 8 \beta_1) q^{25} + (2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 20 \beta_1) q^{29} + ( - \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 5 \beta_1 + 5) q^{31} + (5 \beta_{7} - 5 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} + 4 \beta_{2} + 36 \beta_1 - 18) q^{35} + ( - 4 \beta_{7} - 4 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 1) q^{37} + (8 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} - 5 \beta_{4} - \beta_{3} - 2 \beta_{2} + 12 \beta_1 - 12) q^{41} + (\beta_{7} - 2 \beta_{5} - 5 \beta_{4} + 6 \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 4) q^{43} + ( - 7 \beta_{7} - 5 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 18 \beta_1 - 36) q^{47} + ( - 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} + \cdots + 11) q^{49}+ \cdots + (8 \beta_{7} - 16 \beta_{6} + \beta_{5} - 8 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{97}+O(q^{100})$$ q + (b4 - b1 + 1) * q^5 + (-b7 - b5 - b4) * q^7 + (-b7 - b5 - b4 + b3 + b2 - b1 + 2) * q^11 + (b3 + 2*b2 + 2*b1 - 2) * q^13 + (2*b7 + 2*b6 + 2*b3 + b2 - 1) * q^17 + (2*b7 - 2*b6 + 3*b5 - 3*b4 + 4*b1 - 2) * q^19 + (b6 - b5 - b3 + 7*b1 + 7) * q^23 + (-2*b7 + 4*b6 - 7*b5 + 2*b4 + 2*b3 - 2*b2 - 8*b1) * q^25 + (2*b5 - 3*b3 + 3*b2 + 20*b1) * q^29 + (-b6 + 5*b5 + 2*b4 - 3*b3 + 5*b1 + 5) * q^31 + (5*b7 - 5*b6 + 4*b5 - 4*b4 + 4*b2 + 36*b1 - 18) * q^35 + (-4*b7 - 4*b6 + b5 + b4 + 2*b3 + b2 + 1) * q^37 + (8*b7 - 4*b6 + 4*b5 - 5*b4 - b3 - 2*b2 + 12*b1 - 12) * q^41 + (b7 - 2*b5 - 5*b4 + 6*b3 + 6*b2 + 2*b1 - 4) * q^43 + (-7*b7 - 5*b5 - 3*b4 - 2*b3 - 2*b2 + 18*b1 - 36) * q^47 + (-4*b7 + 2*b6 - 2*b5 - 2*b4 - 3*b3 - 6*b2 - 11*b1 + 11) * q^49 + (9*b5 + 9*b4 - 6*b3 - 3*b2 - 27) * q^53 + (3*b7 - 3*b6 + 3*b5 - 3*b4 + 3*b2 + 18*b1 - 9) * q^55 + (-5*b6 + 9*b5 + 2*b4 - 4*b3 - 24*b1 - 24) * q^59 + (4*b7 - 8*b6 - 4*b5 - 4*b4 - b3 + b2 - 4*b1) * q^61 + (-2*b7 + 4*b6 + 4*b5 + 2*b4 + b3 - b2 - 22*b1) * q^65 + (-3*b6 + 9*b5 + 3*b4 + 3*b3 - b1 - 1) * q^67 + (4*b7 - 4*b6 - b5 + b4 + 5*b2 - 66*b1 + 33) * q^71 + (-2*b7 - 2*b6 - 4*b5 - 4*b4 + 2*b3 + b2 + 7) * q^73 + (-3*b4 - 51*b1 + 51) * q^77 + (-b7 + 2*b5 + 5*b4 + 9*b3 + 9*b2 - b1 + 2) * q^79 + (-11*b7 - 5*b5 + b4 + 2*b3 + 2*b2 - 44*b1 + 88) * q^83 + (-8*b7 + 4*b6 - 4*b5 - 13*b4 - b3 - 2*b2 - 3*b1 + 3) * q^85 + (7*b5 + 7*b4 - 6*b3 - 3*b2 + 25) * q^89 + (11*b7 - 11*b6 + 3*b5 - 3*b4 + 9*b2 + 18*b1 - 9) * q^91 + (-8*b6 + 28*b5 + 10*b4 - 10*b3 + 62*b1 + 62) * q^95 + (8*b7 - 16*b6 + b5 - 8*b4 - 3*b3 + 3*b2 - 4*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 3 q^{5} + 3 q^{7}+O(q^{10})$$ 8 * q + 3 * q^5 + 3 * q^7 $$8 q + 3 q^{5} + 3 q^{7} + 18 q^{11} - 5 q^{13} - 6 q^{17} + 81 q^{23} - 23 q^{25} + 69 q^{29} + 45 q^{31} + 20 q^{37} - 54 q^{41} - 207 q^{47} + 41 q^{49} - 252 q^{53} - 306 q^{59} - 7 q^{61} - 93 q^{65} - 12 q^{67} + 74 q^{73} + 207 q^{77} + 33 q^{79} + 549 q^{83} + 30 q^{85} + 168 q^{89} + 684 q^{95} - 10 q^{97}+O(q^{100})$$ 8 * q + 3 * q^5 + 3 * q^7 + 18 * q^11 - 5 * q^13 - 6 * q^17 + 81 * q^23 - 23 * q^25 + 69 * q^29 + 45 * q^31 + 20 * q^37 - 54 * q^41 - 207 * q^47 + 41 * q^49 - 252 * q^53 - 306 * q^59 - 7 * q^61 - 93 * q^65 - 12 * q^67 + 74 * q^73 + 207 * q^77 + 33 * q^79 + 549 * q^83 + 30 * q^85 + 168 * q^89 + 684 * q^95 - 10 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 7\nu^{3} + 10\nu + 2 ) / 4$$ (v^5 + 7*v^3 + 10*v + 2) / 4 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} - \nu^{3} + 14\nu ) / 2$$ (-v^5 - v^3 + 14*v) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{5} + 6\nu^{4} + \nu^{3} + 24\nu^{2} - 14\nu - 6 ) / 4$$ (v^5 + 6*v^4 + v^3 + 24*v^2 - 14*v - 6) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{6} - \nu^{5} + 8\nu^{4} - 4\nu^{3} + 14\nu^{2} + 8\nu ) / 2$$ (v^6 - v^5 + 8*v^4 - 4*v^3 + 14*v^2 + 8*v) / 2 $$\beta_{5}$$ $$=$$ $$( \nu^{6} + \nu^{5} + 8\nu^{4} + 4\nu^{3} + 14\nu^{2} - 8\nu ) / 2$$ (v^6 + v^5 + 8*v^4 + 4*v^3 + 14*v^2 - 8*v) / 2 $$\beta_{6}$$ $$=$$ $$( 3\nu^{7} + 2\nu^{6} + 32\nu^{5} + 22\nu^{4} + 95\nu^{3} + 70\nu^{2} + 50\nu + 42 ) / 4$$ (3*v^7 + 2*v^6 + 32*v^5 + 22*v^4 + 95*v^3 + 70*v^2 + 50*v + 42) / 4 $$\beta_{7}$$ $$=$$ $$( -3\nu^{7} + 2\nu^{6} - 32\nu^{5} + 22\nu^{4} - 95\nu^{3} + 70\nu^{2} - 50\nu + 42 ) / 4$$ (-3*v^7 + 2*v^6 - 32*v^5 + 22*v^4 - 95*v^3 + 70*v^2 - 50*v + 42) / 4
 $$\nu$$ $$=$$ $$( -\beta_{5} + \beta_{4} - \beta_{2} + 2\beta _1 - 1 ) / 6$$ (-b5 + b4 - b2 + 2*b1 - 1) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{3} - \beta_{2} - 24 ) / 9$$ (b7 + b6 - b5 - b4 - 2*b3 - b2 - 24) / 9 $$\nu^{3}$$ $$=$$ $$( 2\beta_{5} - 2\beta_{4} + 3\beta_{2} - 2\beta _1 + 1 ) / 3$$ (2*b5 - 2*b4 + 3*b2 - 2*b1 + 1) / 3 $$\nu^{4}$$ $$=$$ $$( -4\beta_{7} - 4\beta_{6} + 4\beta_{5} + 4\beta_{4} + 14\beta_{3} + 7\beta_{2} + 105 ) / 9$$ (-4*b7 - 4*b6 + 4*b5 + 4*b4 + 14*b3 + 7*b2 + 105) / 9 $$\nu^{5}$$ $$=$$ $$( -9\beta_{5} + 9\beta_{4} - 16\beta_{2} + 16\beta _1 - 8 ) / 3$$ (-9*b5 + 9*b4 - 16*b2 + 16*b1 - 8) / 3 $$\nu^{6}$$ $$=$$ $$( 6\beta_{7} + 6\beta_{6} - 3\beta_{5} - 3\beta_{4} - 28\beta_{3} - 14\beta_{2} - 168 ) / 3$$ (6*b7 + 6*b6 - 3*b5 - 3*b4 - 28*b3 - 14*b2 - 168) / 3 $$\nu^{7}$$ $$=$$ $$( -2\beta_{7} + 2\beta_{6} + 41\beta_{5} - 41\beta_{4} + 84\beta_{2} - 124\beta _1 + 62 ) / 3$$ (-2*b7 + 2*b6 + 41*b5 - 41*b4 + 84*b2 - 124*b1 + 62) / 3

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 − 1.07834i − 0.385731i − 2.33086i 2.06288i 1.07834i 0.385731i 2.33086i − 2.06288i
0 0 0 −3.01729 5.22611i 0 10.2332 + 5.90815i 0 0 0
127.2 0 0 0 −0.454613 0.787412i 0 −6.10709 3.52593i 0 0 0
127.3 0 0 0 0.355304 + 0.615405i 0 2.70480 + 1.56162i 0 0 0
127.4 0 0 0 4.61660 + 7.99619i 0 −5.33093 3.07781i 0 0 0
1279.1 0 0 0 −3.01729 + 5.22611i 0 10.2332 5.90815i 0 0 0
1279.2 0 0 0 −0.454613 + 0.787412i 0 −6.10709 + 3.52593i 0 0 0
1279.3 0 0 0 0.355304 0.615405i 0 2.70480 1.56162i 0 0 0
1279.4 0 0 0 4.61660 7.99619i 0 −5.33093 + 3.07781i 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
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.f 8
3.b odd 2 1 576.3.o.f 8
4.b odd 2 1 1728.3.o.e 8
8.b even 2 1 432.3.o.b 8
8.d odd 2 1 432.3.o.a 8
9.c even 3 1 1728.3.o.e 8
9.d odd 6 1 576.3.o.d 8
12.b even 2 1 576.3.o.d 8
24.f even 2 1 144.3.o.c yes 8
24.h odd 2 1 144.3.o.a 8
36.f odd 6 1 inner 1728.3.o.f 8
36.h even 6 1 576.3.o.f 8
72.j odd 6 1 144.3.o.c yes 8
72.j odd 6 1 1296.3.g.j 8
72.l even 6 1 144.3.o.a 8
72.l even 6 1 1296.3.g.j 8
72.n even 6 1 432.3.o.a 8
72.n even 6 1 1296.3.g.k 8
72.p odd 6 1 432.3.o.b 8
72.p odd 6 1 1296.3.g.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.o.a 8 24.h odd 2 1
144.3.o.a 8 72.l even 6 1
144.3.o.c yes 8 24.f even 2 1
144.3.o.c yes 8 72.j odd 6 1
432.3.o.a 8 8.d odd 2 1
432.3.o.a 8 72.n even 6 1
432.3.o.b 8 8.b even 2 1
432.3.o.b 8 72.p odd 6 1
576.3.o.d 8 9.d odd 6 1
576.3.o.d 8 12.b even 2 1
576.3.o.f 8 3.b odd 2 1
576.3.o.f 8 36.h even 6 1
1296.3.g.j 8 72.j odd 6 1
1296.3.g.j 8 72.l even 6 1
1296.3.g.k 8 72.n even 6 1
1296.3.g.k 8 72.p odd 6 1
1728.3.o.e 8 4.b odd 2 1
1728.3.o.e 8 9.c even 3 1
1728.3.o.f 8 1.a even 1 1 trivial
1728.3.o.f 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 $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{8} - 3T_{5}^{7} + 66T_{5}^{6} + 189T_{5}^{5} + 3186T_{5}^{4} + 729T_{5}^{3} + 2133T_{5}^{2} - 324T_{5} + 1296$$ T5^8 - 3*T5^7 + 66*T5^6 + 189*T5^5 + 3186*T5^4 + 729*T5^3 + 2133*T5^2 - 324*T5 + 1296 $$T_{7}^{8} - 3T_{7}^{7} - 114T_{7}^{6} + 351T_{7}^{5} + 12366T_{7}^{4} + 32643T_{7}^{3} - 161487T_{7}^{2} - 446958T_{7} + 2566404$$ T7^8 - 3*T7^7 - 114*T7^6 + 351*T7^5 + 12366*T7^4 + 32643*T7^3 - 161487*T7^2 - 446958*T7 + 2566404

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 3 T^{7} + 66 T^{6} + \cdots + 1296$$
$7$ $$T^{8} - 3 T^{7} - 114 T^{6} + \cdots + 2566404$$
$11$ $$T^{8} - 18 T^{7} - 36 T^{6} + \cdots + 12131289$$
$13$ $$T^{8} + 5 T^{7} + 316 T^{6} + \cdots + 10201636$$
$17$ $$(T^{4} + 3 T^{3} - 822 T^{2} - 1908 T + 84168)^{2}$$
$19$ $$T^{8} + 1731 T^{6} + \cdots + 2931572736$$
$23$ $$T^{8} - 81 T^{7} + 2754 T^{6} + \cdots + 19131876$$
$29$ $$T^{8} - 69 T^{7} + \cdots + 4046639163876$$
$31$ $$T^{8} - 45 T^{7} - 324 T^{6} + \cdots + 944784$$
$37$ $$(T^{4} - 10 T^{3} - 3756 T^{2} + \cdots - 613568)^{2}$$
$41$ $$T^{8} + 54 T^{7} + \cdots + 5431756955769$$
$43$ $$T^{8} - 3186 T^{6} + \cdots + 29016737649$$
$47$ $$T^{8} + 207 T^{7} + \cdots + 28643839776036$$
$53$ $$(T^{4} + 126 T^{3} + 972 T^{2} + \cdots - 6508512)^{2}$$
$59$ $$T^{8} + 306 T^{7} + \cdots + 48359409452649$$
$61$ $$T^{8} + 7 T^{7} + \cdots + 309954973696$$
$67$ $$T^{8} + 12 T^{7} + \cdots + 68036119056801$$
$71$ $$T^{8} + \cdots + 726110197530624$$
$73$ $$(T^{4} - 37 T^{3} - 1002 T^{2} + \cdots + 416536)^{2}$$
$79$ $$T^{8} + \cdots + 240627852449856$$
$83$ $$T^{8} - 549 T^{7} + \cdots + 15\!\cdots\!64$$
$89$ $$(T^{4} - 84 T^{3} - 984 T^{2} + \cdots - 1161936)^{2}$$
$97$ $$T^{8} + 10 T^{7} + \cdots + 30429664983481$$