# Properties

 Label 1728.3.g.l Level 1728 Weight 3 Character orbit 1728.g 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.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.207360000.1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{16}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{5} + \beta_{7} q^{7} +O(q^{10})$$ $$q -\beta_{1} q^{5} + \beta_{7} q^{7} + \beta_{3} q^{11} + ( 1 - \beta_{6} ) q^{13} + ( 2 \beta_{1} + \beta_{4} ) q^{17} + \beta_{5} q^{19} + ( \beta_{2} - \beta_{3} ) q^{23} + ( 3 - 2 \beta_{6} ) q^{25} -4 \beta_{1} q^{29} + ( 4 \beta_{5} - 4 \beta_{7} ) q^{31} + ( 2 \beta_{2} + \beta_{3} ) q^{35} + ( 7 + \beta_{6} ) q^{37} + ( 2 \beta_{1} - 2 \beta_{4} ) q^{41} + ( 6 \beta_{5} + 2 \beta_{7} ) q^{43} + ( \beta_{2} + 3 \beta_{3} ) q^{47} + ( -14 + 2 \beta_{6} ) q^{49} + ( 12 \beta_{1} - 2 \beta_{4} ) q^{53} + ( 6 \beta_{5} + 2 \beta_{7} ) q^{55} + ( -2 \beta_{2} - 3 \beta_{3} ) q^{59} + ( -5 - \beta_{6} ) q^{61} + ( -14 \beta_{1} - \beta_{4} ) q^{65} + ( 9 \beta_{5} - 8 \beta_{7} ) q^{67} + ( -4 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 29 + 2 \beta_{6} ) q^{73} + ( 7 \beta_{1} + 4 \beta_{4} ) q^{77} + ( 16 \beta_{5} + 7 \beta_{7} ) q^{79} -2 \beta_{3} q^{83} + ( -52 + 6 \beta_{6} ) q^{85} + ( 2 \beta_{1} + \beta_{4} ) q^{89} + ( 19 \beta_{5} + 4 \beta_{7} ) q^{91} + ( \beta_{2} + \beta_{3} ) q^{95} + ( -31 + 8 \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{13} + 24q^{25} + 56q^{37} - 112q^{49} - 40q^{61} + 232q^{73} - 416q^{85} - 248q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 6 x^{6} + 32 x^{4} + 24 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{7} + 16 \nu^{5} + 64 \nu^{3} + 8 \nu$$$$)/32$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{7} + 16 \nu^{5} + 96 \nu^{3} + 136 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-15 \nu^{7} - 112 \nu^{5} - 576 \nu^{3} - 808 \nu$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$-57 \nu^{7} - 304 \nu^{5} - 1600 \nu^{3} - 152 \nu$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$-9 \nu^{6} - 48 \nu^{4} - 288 \nu^{2} - 120$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{6} + 216$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$13 \nu^{6} + 80 \nu^{4} + 352 \nu^{2} + 152$$$$)/32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + 3 \beta_{2} + 7 \beta_{1}$$$$)/48$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{7} + \beta_{6} - 5 \beta_{5} - 18$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{4} - 19 \beta_{1}$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$($$$$9 \beta_{7} + 3 \beta_{6} + 11 \beta_{5} - 42$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{4} - 6 \beta_{3} - 9 \beta_{2} + 25 \beta_{1}$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$-16 \beta_{6} + 216$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$5 \beta_{4} + 32 \beta_{3} + 47 \beta_{2} + 131 \beta_{1}$$$$)/6$$

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
703.1
 1.14412 + 1.98168i 1.14412 − 1.98168i −0.437016 + 0.756934i −0.437016 − 0.756934i 0.437016 − 0.756934i 0.437016 + 0.756934i −1.14412 − 1.98168i −1.14412 + 1.98168i
0 0 0 −7.40492 0 9.47802i 0 0 0
703.2 0 0 0 −7.40492 0 9.47802i 0 0 0
703.3 0 0 0 −1.08036 0 6.01392i 0 0 0
703.4 0 0 0 −1.08036 0 6.01392i 0 0 0
703.5 0 0 0 1.08036 0 6.01392i 0 0 0
703.6 0 0 0 1.08036 0 6.01392i 0 0 0
703.7 0 0 0 7.40492 0 9.47802i 0 0 0
703.8 0 0 0 7.40492 0 9.47802i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 703.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.g.l 8
3.b odd 2 1 inner 1728.3.g.l 8
4.b odd 2 1 inner 1728.3.g.l 8
8.b even 2 1 108.3.d.d 8
8.d odd 2 1 108.3.d.d 8
12.b even 2 1 inner 1728.3.g.l 8
24.f even 2 1 108.3.d.d 8
24.h odd 2 1 108.3.d.d 8
72.j odd 6 1 324.3.f.o 8
72.j odd 6 1 324.3.f.p 8
72.l even 6 1 324.3.f.o 8
72.l even 6 1 324.3.f.p 8
72.n even 6 1 324.3.f.o 8
72.n even 6 1 324.3.f.p 8
72.p odd 6 1 324.3.f.o 8
72.p odd 6 1 324.3.f.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.d 8 8.b even 2 1
108.3.d.d 8 8.d odd 2 1
108.3.d.d 8 24.f even 2 1
108.3.d.d 8 24.h odd 2 1
324.3.f.o 8 72.j odd 6 1
324.3.f.o 8 72.l even 6 1
324.3.f.o 8 72.n even 6 1
324.3.f.o 8 72.p odd 6 1
324.3.f.p 8 72.j odd 6 1
324.3.f.p 8 72.l even 6 1
324.3.f.p 8 72.n even 6 1
324.3.f.p 8 72.p odd 6 1
1728.3.g.l 8 1.a even 1 1 trivial
1728.3.g.l 8 3.b odd 2 1 inner
1728.3.g.l 8 4.b odd 2 1 inner
1728.3.g.l 8 12.b even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{4} - 56 T_{5}^{2} + 64$$ $$T_{7}^{4} + 126 T_{7}^{2} + 3249$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 44 T^{2} + 1014 T^{4} + 27500 T^{6} + 390625 T^{8} )^{2}$$
$7$ $$( 1 - 70 T^{2} + 5307 T^{4} - 168070 T^{6} + 5764801 T^{8} )^{2}$$
$11$ $$( 1 - 124 T^{2} + 15126 T^{4} - 1815484 T^{6} + 214358881 T^{8} )^{2}$$
$13$ $$( 1 - 2 T + 159 T^{2} - 338 T^{3} + 28561 T^{4} )^{4}$$
$17$ $$( 1 + 140 T^{2} + 136662 T^{4} + 11692940 T^{6} + 6975757441 T^{8} )^{2}$$
$19$ $$( 1 - 695 T^{2} + 130321 T^{4} )^{4}$$
$23$ $$( 1 - 604 T^{2} + 632886 T^{4} - 169023964 T^{6} + 78310985281 T^{8} )^{2}$$
$29$ $$( 1 + 2468 T^{2} + 2752998 T^{4} + 1745569508 T^{6} + 500246412961 T^{8} )^{2}$$
$31$ $$( 1 - 1540 T^{2} + 1702662 T^{4} - 1422222340 T^{6} + 852891037441 T^{8} )^{2}$$
$37$ $$( 1 - 14 T + 2607 T^{2} - 19166 T^{3} + 1874161 T^{4} )^{4}$$
$41$ $$( 1 + 3140 T^{2} + 5167302 T^{4} + 8872889540 T^{6} + 7984925229121 T^{8} )^{2}$$
$43$ $$( 1 - 4516 T^{2} + 10784166 T^{4} - 15439305316 T^{6} + 11688200277601 T^{8} )^{2}$$
$47$ $$( 1 - 4444 T^{2} + 14436726 T^{4} - 21685302364 T^{6} + 23811286661761 T^{8} )^{2}$$
$53$ $$( 1 - 508 T^{2} + 14912358 T^{4} - 4008364348 T^{6} + 62259690411361 T^{8} )^{2}$$
$59$ $$( 1 - 6076 T^{2} + 23942166 T^{4} - 73625085436 T^{6} + 146830437604321 T^{8} )^{2}$$
$61$ $$( 1 + 10 T + 7287 T^{2} + 37210 T^{3} + 13845841 T^{4} )^{4}$$
$67$ $$( 1 - 8110 T^{2} + 40110387 T^{4} - 163425591310 T^{6} + 406067677556641 T^{8} )^{2}$$
$71$ $$( 1 - 292 T^{2} + 42446598 T^{4} - 7420210852 T^{6} + 645753531245761 T^{8} )^{2}$$
$73$ $$( 1 - 58 T + 10779 T^{2} - 309082 T^{3} + 28398241 T^{4} )^{4}$$
$79$ $$( 1 - 934 T^{2} - 28603749 T^{4} - 36379375654 T^{6} + 1517108809906561 T^{8} )^{2}$$
$83$ $$( 1 - 26116 T^{2} + 265140006 T^{4} - 1239421511236 T^{6} + 2252292232139041 T^{8} )^{2}$$
$89$ $$( 1 + 30668 T^{2} + 360580758 T^{4} + 1924179046988 T^{6} + 3936588805702081 T^{8} )^{2}$$
$97$ $$( 1 + 62 T + 8259 T^{2} + 583358 T^{3} + 88529281 T^{4} )^{4}$$