# Properties

 Label 1728.3.q.g Level $1728$ Weight $3$ Character orbit 1728.q Analytic conductor $47.085$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1 + 3) q^{5} + ( - \beta_{3} + 2 \beta_1 - 1) q^{7}+O(q^{10})$$ q + (b2 - b1 + 3) * q^5 + (-b3 + 2*b1 - 1) * q^7 $$q + (\beta_{2} - \beta_1 + 3) q^{5} + ( - \beta_{3} + 2 \beta_1 - 1) q^{7} + ( - 2 \beta_{3} + 5 \beta_{2} - 7) q^{11} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{3} + 15 \beta_{2} + \beta_1 + 7) q^{17} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{19} + ( - 17 \beta_{2} - \beta_1 - 33) q^{23} + (3 \beta_{3} + 5 \beta_{2} - 6 \beta_1 + 8) q^{25} + (7 \beta_{3} + 14 \beta_{2} - 7) q^{29} + (6 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 3) q^{31} + ( - 5 \beta_{3} - 51 \beta_{2} + 5 \beta_1 - 28) q^{35} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 18) q^{37} + ( - \beta_{2} - 8 \beta_1 + 6) q^{41} + (23 \beta_{2} + 23) q^{43} + (3 \beta_{3} - 12 \beta_{2} + 15) q^{47} + (2 \beta_{3} + 26 \beta_{2} - \beta_1 + 1) q^{49} + (8 \beta_{3} + 24 \beta_{2} - 8 \beta_1 + 16) q^{53} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 24) q^{55} + (17 \beta_{2} - 8 \beta_1 + 42) q^{59} + (3 \beta_{3} - 22 \beta_{2} - 6 \beta_1 - 19) q^{61} + (7 \beta_{3} + 32 \beta_{2} - 25) q^{65} + (12 \beta_{3} + 61 \beta_{2} - 6 \beta_1 + 6) q^{67} + (2 \beta_{3} - 30 \beta_{2} - 2 \beta_1 - 14) q^{71} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 23) q^{73} + ( - 55 \beta_{2} - 17 \beta_1 - 93) q^{77} + ( - 5 \beta_{3} + 44 \beta_{2} + 10 \beta_1 + 39) q^{79} + ( - 3 \beta_{3} + 12 \beta_{2} - 15) q^{83} + (12 \beta_{3} + 12 \beta_{2} - 6 \beta_1 + 6) q^{85} + (8 \beta_{3} - 120 \beta_{2} - 8 \beta_1 - 56) q^{89} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 74) q^{91} + (22 \beta_{2} - 4 \beta_1 + 48) q^{95} + ( - 2 \beta_{3} - 97 \beta_{2} + 4 \beta_1 - 99) q^{97}+O(q^{100})$$ q + (b2 - b1 + 3) * q^5 + (-b3 + 2*b1 - 1) * q^7 + (-2*b3 + 5*b2 - 7) * q^11 + (2*b3 + 3*b2 - b1 + 1) * q^13 + (-b3 + 15*b2 + b1 + 7) * q^17 + (-b3 - b2 - b1 - 1) * q^19 + (-17*b2 - b1 - 33) * q^23 + (3*b3 + 5*b2 - 6*b1 + 8) * q^25 + (7*b3 + 14*b2 - 7) * q^29 + (6*b3 + 5*b2 - 3*b1 + 3) * q^31 + (-5*b3 - 51*b2 + 5*b1 - 28) * q^35 + (4*b3 + 4*b2 + 4*b1 + 18) * q^37 + (-b2 - 8*b1 + 6) * q^41 + (23*b2 + 23) * q^43 + (3*b3 - 12*b2 + 15) * q^47 + (2*b3 + 26*b2 - b1 + 1) * q^49 + (8*b3 + 24*b2 - 8*b1 + 16) * q^53 + (3*b3 + 3*b2 + 3*b1 + 24) * q^55 + (17*b2 - 8*b1 + 42) * q^59 + (3*b3 - 22*b2 - 6*b1 - 19) * q^61 + (7*b3 + 32*b2 - 25) * q^65 + (12*b3 + 61*b2 - 6*b1 + 6) * q^67 + (2*b3 - 30*b2 - 2*b1 - 14) * q^71 + (3*b3 + 3*b2 + 3*b1 + 23) * q^73 + (-55*b2 - 17*b1 - 93) * q^77 + (-5*b3 + 44*b2 + 10*b1 + 39) * q^79 + (-3*b3 + 12*b2 - 15) * q^83 + (12*b3 + 12*b2 - 6*b1 + 6) * q^85 + (8*b3 - 120*b2 - 8*b1 - 56) * q^89 + (-3*b3 - 3*b2 - 3*b1 + 74) * q^91 + (22*b2 - 4*b1 + 48) * q^95 + (-2*b3 - 97*b2 + 4*b1 - 99) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 9 q^{5} - q^{7}+O(q^{10})$$ 4 * q + 9 * q^5 - q^7 $$4 q + 9 q^{5} - q^{7} - 36 q^{11} - 5 q^{13} - 2 q^{19} - 99 q^{23} + 13 q^{25} - 63 q^{29} - 7 q^{31} + 64 q^{37} + 18 q^{41} + 46 q^{43} + 81 q^{47} - 51 q^{49} + 90 q^{55} + 126 q^{59} - 41 q^{61} - 171 q^{65} - 116 q^{67} + 86 q^{73} - 279 q^{77} + 83 q^{79} - 81 q^{83} - 18 q^{85} + 302 q^{91} + 144 q^{95} - 196 q^{97}+O(q^{100})$$ 4 * q + 9 * q^5 - q^7 - 36 * q^11 - 5 * q^13 - 2 * q^19 - 99 * q^23 + 13 * q^25 - 63 * q^29 - 7 * q^31 + 64 * q^37 + 18 * q^41 + 46 * q^43 + 81 * q^47 - 51 * q^49 + 90 * q^55 + 126 * q^59 - 41 * q^61 - 171 * q^65 - 116 * q^67 + 86 * q^73 - 279 * q^77 + 83 * q^79 - 81 * q^83 - 18 * q^85 + 302 * q^91 + 144 * q^95 - 196 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} + 16\nu - 9 ) / 6$$ (v^3 + 2*v^2 + 16*v - 9) / 6 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 9 ) / 6$$ (v^3 + 2*v^2 - 2*v - 9) / 6 $$\beta_{3}$$ $$=$$ $$( -4\nu^{3} + \nu^{2} + 8\nu + 12 ) / 3$$ (-4*v^3 + v^2 + 8*v + 12) / 3
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 3$$ (-b2 + b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 8\beta_{2} + 8 ) / 3$$ (b3 + 8*b2 + 8) / 3 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + 2\beta _1 + 11 ) / 3$$ (-2*b3 + 2*b1 + 11) / 3

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 1.68614 + 0.396143i −1.18614 − 1.26217i 1.68614 − 0.396143i −1.18614 + 1.26217i
0 0 0 −2.05842 1.18843i 0 4.05842 + 7.02939i 0 0 0
449.2 0 0 0 6.55842 + 3.78651i 0 −4.55842 7.89542i 0 0 0
1601.1 0 0 0 −2.05842 + 1.18843i 0 4.05842 7.02939i 0 0 0
1601.2 0 0 0 6.55842 3.78651i 0 −4.55842 + 7.89542i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d 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.q.g 4
3.b odd 2 1 576.3.q.d 4
4.b odd 2 1 1728.3.q.h 4
8.b even 2 1 108.3.g.a 4
8.d odd 2 1 432.3.q.b 4
9.c even 3 1 576.3.q.d 4
9.d odd 6 1 inner 1728.3.q.g 4
12.b even 2 1 576.3.q.g 4
24.f even 2 1 144.3.q.b 4
24.h odd 2 1 36.3.g.a 4
36.f odd 6 1 576.3.q.g 4
36.h even 6 1 1728.3.q.h 4
40.f even 2 1 2700.3.p.b 4
40.i odd 4 2 2700.3.u.b 8
72.j odd 6 1 108.3.g.a 4
72.j odd 6 1 324.3.c.b 4
72.l even 6 1 432.3.q.b 4
72.l even 6 1 1296.3.e.e 4
72.n even 6 1 36.3.g.a 4
72.n even 6 1 324.3.c.b 4
72.p odd 6 1 144.3.q.b 4
72.p odd 6 1 1296.3.e.e 4
120.i odd 2 1 900.3.p.a 4
120.w even 4 2 900.3.u.a 8
360.bh odd 6 1 2700.3.p.b 4
360.bk even 6 1 900.3.p.a 4
360.br even 12 2 2700.3.u.b 8
360.bu odd 12 2 900.3.u.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 24.h odd 2 1
36.3.g.a 4 72.n even 6 1
108.3.g.a 4 8.b even 2 1
108.3.g.a 4 72.j odd 6 1
144.3.q.b 4 24.f even 2 1
144.3.q.b 4 72.p odd 6 1
324.3.c.b 4 72.j odd 6 1
324.3.c.b 4 72.n even 6 1
432.3.q.b 4 8.d odd 2 1
432.3.q.b 4 72.l even 6 1
576.3.q.d 4 3.b odd 2 1
576.3.q.d 4 9.c even 3 1
576.3.q.g 4 12.b even 2 1
576.3.q.g 4 36.f odd 6 1
900.3.p.a 4 120.i odd 2 1
900.3.p.a 4 360.bk even 6 1
900.3.u.a 8 120.w even 4 2
900.3.u.a 8 360.bu odd 12 2
1296.3.e.e 4 72.l even 6 1
1296.3.e.e 4 72.p odd 6 1
1728.3.q.g 4 1.a even 1 1 trivial
1728.3.q.g 4 9.d odd 6 1 inner
1728.3.q.h 4 4.b odd 2 1
1728.3.q.h 4 36.h even 6 1
2700.3.p.b 4 40.f even 2 1
2700.3.p.b 4 360.bh odd 6 1
2700.3.u.b 8 40.i odd 4 2
2700.3.u.b 8 360.br even 12 2

## Hecke kernels

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

 $$T_{5}^{4} - 9T_{5}^{3} + 9T_{5}^{2} + 162T_{5} + 324$$ T5^4 - 9*T5^3 + 9*T5^2 + 162*T5 + 324 $$T_{7}^{4} + T_{7}^{3} + 75T_{7}^{2} - 74T_{7} + 5476$$ T7^4 + T7^3 + 75*T7^2 - 74*T7 + 5476

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 9 T^{3} + 9 T^{2} + 162 T + 324$$
$7$ $$T^{4} + T^{3} + 75 T^{2} - 74 T + 5476$$
$11$ $$T^{4} + 36 T^{3} + 441 T^{2} + \cdots + 81$$
$13$ $$T^{4} + 5 T^{3} + 93 T^{2} + \cdots + 4624$$
$17$ $$T^{4} + 387 T^{2} + 20736$$
$19$ $$(T^{2} + T - 74)^{2}$$
$23$ $$T^{4} + 99 T^{3} + 4059 T^{2} + \cdots + 627264$$
$29$ $$T^{4} + 63 T^{3} + 441 T^{2} + \cdots + 777924$$
$31$ $$T^{4} + 7 T^{3} + 705 T^{2} + \cdots + 430336$$
$37$ $$(T^{2} - 32 T - 932)^{2}$$
$41$ $$T^{4} - 18 T^{3} - 1449 T^{2} + \cdots + 2424249$$
$43$ $$(T^{2} - 23 T + 529)^{2}$$
$47$ $$T^{4} - 81 T^{3} + 2511 T^{2} + \cdots + 104976$$
$53$ $$T^{4} + 4032 T^{2} + \cdots + 1327104$$
$59$ $$T^{4} - 126 T^{3} + 5031 T^{2} + \cdots + 68121$$
$61$ $$T^{4} + 41 T^{3} + 1929 T^{2} + \cdots + 61504$$
$67$ $$T^{4} + 116 T^{3} + 12765 T^{2} + \cdots + 477481$$
$71$ $$T^{4} + 1548 T^{2} + 331776$$
$73$ $$(T^{2} - 43 T - 206)^{2}$$
$79$ $$T^{4} - 83 T^{3} + 7023 T^{2} + \cdots + 17956$$
$83$ $$T^{4} + 81 T^{3} + 2511 T^{2} + \cdots + 104976$$
$89$ $$T^{4} + 24768 T^{2} + \cdots + 84934656$$
$97$ $$T^{4} + 196 T^{3} + \cdots + 86620249$$