# Properties

 Label 1728.3.q.f 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 72) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{7} + ( 6 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{11} + ( 7 - \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{13} + ( -8 + 16 \beta_{2} - 2 \beta_{3} ) q^{17} + ( 2 + 4 \beta_{1} - 2 \beta_{3} ) q^{19} + ( 5 - \beta_{1} + 5 \beta_{2} ) q^{23} + ( -2 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} ) q^{25} + ( 2 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{29} + ( -37 + \beta_{1} + 37 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 29 - 58 \beta_{2} ) q^{35} + ( 30 - 4 \beta_{1} + 2 \beta_{3} ) q^{37} + ( 23 + 23 \beta_{2} ) q^{41} + ( -4 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} ) q^{43} + ( 58 + 3 \beta_{1} - 29 \beta_{2} - 3 \beta_{3} ) q^{47} + ( -56 - 6 \beta_{1} + 56 \beta_{2} + 12 \beta_{3} ) q^{49} + ( 32 - 64 \beta_{2} - 8 \beta_{3} ) q^{53} + ( -55 - 2 \beta_{1} + \beta_{3} ) q^{55} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{59} + ( \beta_{1} + 31 \beta_{2} + \beta_{3} ) q^{61} + ( 78 - 10 \beta_{1} - 39 \beta_{2} + 10 \beta_{3} ) q^{65} + ( -11 - 10 \beta_{1} + 11 \beta_{2} + 20 \beta_{3} ) q^{67} + ( -24 + 48 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 10 + 12 \beta_{1} - 6 \beta_{3} ) q^{73} + ( 73 + 15 \beta_{1} + 73 \beta_{2} ) q^{77} + ( -7 \beta_{1} + 43 \beta_{2} - 7 \beta_{3} ) q^{79} + ( 22 - 14 \beta_{1} - 11 \beta_{2} + 14 \beta_{3} ) q^{83} + ( -88 + 10 \beta_{1} + 88 \beta_{2} - 20 \beta_{3} ) q^{85} + ( -24 + 48 \beta_{2} - 6 \beta_{3} ) q^{89} + ( -75 + 8 \beta_{1} - 4 \beta_{3} ) q^{91} + ( -62 + 4 \beta_{1} - 62 \beta_{2} ) q^{95} + ( -2 \beta_{1} + 121 \beta_{2} - 2 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{5} + 6q^{7} + O(q^{10})$$ $$4q + 6q^{5} + 6q^{7} + 18q^{11} + 14q^{13} + 8q^{19} + 30q^{23} + 20q^{25} + 6q^{29} - 74q^{31} + 120q^{37} + 138q^{41} + 10q^{43} + 174q^{47} - 112q^{49} - 220q^{55} + 18q^{59} + 62q^{61} + 234q^{65} - 22q^{67} + 40q^{73} + 438q^{77} + 86q^{79} + 66q^{83} - 176q^{85} - 300q^{91} - 372q^{95} + 242q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{3}$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/4$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}$$$$/2$$

## Character values

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

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

## 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
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0 0 0 −3.39898 1.96240i 0 6.39898 + 11.0834i 0 0 0
449.2 0 0 0 6.39898 + 3.69445i 0 −3.39898 5.88721i 0 0 0
1601.1 0 0 0 −3.39898 + 1.96240i 0 6.39898 11.0834i 0 0 0
1601.2 0 0 0 6.39898 3.69445i 0 −3.39898 + 5.88721i 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.f 4
3.b odd 2 1 576.3.q.c 4
4.b odd 2 1 1728.3.q.e 4
8.b even 2 1 432.3.q.c 4
8.d odd 2 1 216.3.m.a 4
9.c even 3 1 576.3.q.c 4
9.d odd 6 1 inner 1728.3.q.f 4
12.b even 2 1 576.3.q.h 4
24.f even 2 1 72.3.m.a 4
24.h odd 2 1 144.3.q.d 4
36.f odd 6 1 576.3.q.h 4
36.h even 6 1 1728.3.q.e 4
72.j odd 6 1 432.3.q.c 4
72.j odd 6 1 1296.3.e.c 4
72.l even 6 1 216.3.m.a 4
72.l even 6 1 648.3.e.b 4
72.n even 6 1 144.3.q.d 4
72.n even 6 1 1296.3.e.c 4
72.p odd 6 1 72.3.m.a 4
72.p odd 6 1 648.3.e.b 4

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

 $$T_{5}^{4} - 6 T_{5}^{3} - 17 T_{5}^{2} + 174 T_{5} + 841$$ $$T_{7}^{4} - 6 T_{7}^{3} + 123 T_{7}^{2} + 522 T_{7} + 7569$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$841 + 174 T - 17 T^{2} - 6 T^{3} + T^{4}$$
$7$ $$7569 + 522 T + 123 T^{2} - 6 T^{3} + T^{4}$$
$11$ $$10201 + 1818 T + 7 T^{2} - 18 T^{3} + T^{4}$$
$13$ $$2209 + 658 T + 243 T^{2} - 14 T^{3} + T^{4}$$
$17$ $$4096 + 640 T^{2} + T^{4}$$
$19$ $$( -380 - 4 T + T^{2} )^{2}$$
$23$ $$1849 - 1290 T + 343 T^{2} - 30 T^{3} + T^{4}$$
$29$ $$81225 + 1710 T - 273 T^{2} - 6 T^{3} + T^{4}$$
$31$ $$1620529 + 94202 T + 4203 T^{2} + 74 T^{3} + T^{4}$$
$37$ $$( 516 - 60 T + T^{2} )^{2}$$
$41$ $$( 1587 - 69 T + T^{2} )^{2}$$
$43$ $$2283121 + 15110 T + 1611 T^{2} - 10 T^{3} + T^{4}$$
$47$ $$4995225 - 388890 T + 12327 T^{2} - 174 T^{3} + T^{4}$$
$53$ $$1048576 + 10240 T^{2} + T^{4}$$
$59$ $$10201 + 1818 T + 7 T^{2} - 18 T^{3} + T^{4}$$
$61$ $$748225 - 53630 T + 2979 T^{2} - 62 T^{3} + T^{4}$$
$67$ $$89851441 - 208538 T + 9963 T^{2} + 22 T^{3} + T^{4}$$
$71$ $$2560000 + 3712 T^{2} + T^{4}$$
$73$ $$( -3356 - 20 T + T^{2} )^{2}$$
$79$ $$8151025 + 245530 T + 10251 T^{2} - 86 T^{3} + T^{4}$$
$83$ $$34916281 + 389994 T - 4457 T^{2} - 66 T^{3} + T^{4}$$
$89$ $$331776 + 5760 T^{2} + T^{4}$$
$97$ $$203262049 - 3450194 T + 44307 T^{2} - 242 T^{3} + T^{4}$$