# 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: $$\Q(\sqrt{-3}, \sqrt{-11})$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} + 16 \nu - 9$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 9$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{3} + \nu^{2} + 8 \nu + 12$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 8 \beta_{2} + 8$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{3} + 2 \beta_{1} + 11$$$$)/3$$

## 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 $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 9 T + 59 T^{2} - 288 T^{3} + 1074 T^{4} - 7200 T^{5} + 36875 T^{6} - 140625 T^{7} + 390625 T^{8}$$
$7$ $$1 + T - 23 T^{2} - 74 T^{3} - 1874 T^{4} - 3626 T^{5} - 55223 T^{6} + 117649 T^{7} + 5764801 T^{8}$$
$11$ $$1 + 36 T + 683 T^{2} + 9036 T^{3} + 100632 T^{4} + 1093356 T^{5} + 9999803 T^{6} + 63776196 T^{7} + 214358881 T^{8}$$
$13$ $$1 + 5 T - 245 T^{2} - 340 T^{3} + 40114 T^{4} - 57460 T^{5} - 6997445 T^{6} + 24134045 T^{7} + 815730721 T^{8}$$
$17$ $$1 - 769 T^{2} + 298176 T^{4} - 64227649 T^{6} + 6975757441 T^{8}$$
$19$ $$( 1 + T + 648 T^{2} + 361 T^{3} + 130321 T^{4} )^{2}$$
$23$ $$1 + 99 T + 5117 T^{2} + 183150 T^{3} + 4870902 T^{4} + 96886350 T^{5} + 1431946397 T^{6} + 14655553011 T^{7} + 78310985281 T^{8}$$
$29$ $$1 + 63 T + 2123 T^{2} + 50400 T^{3} + 1045362 T^{4} + 42386400 T^{5} + 1501557563 T^{6} + 37473869223 T^{7} + 500246412961 T^{8}$$
$31$ $$1 + 7 T - 1217 T^{2} - 4592 T^{3} + 632146 T^{4} - 4412912 T^{5} - 1123925057 T^{6} + 6212525767 T^{7} + 852891037441 T^{8}$$
$37$ $$( 1 - 32 T + 1806 T^{2} - 43808 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$1 - 18 T + 1913 T^{2} - 32490 T^{3} + 613812 T^{4} - 54615690 T^{5} + 5405680793 T^{6} - 85501876338 T^{7} + 7984925229121 T^{8}$$
$43$ $$( 1 - 23 T - 1320 T^{2} - 42527 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$1 - 81 T + 6929 T^{2} - 384102 T^{3} + 22437966 T^{4} - 848481318 T^{5} + 33811309649 T^{6} - 873116441649 T^{7} + 23811286661761 T^{8}$$
$53$ $$1 - 7204 T^{2} + 26018214 T^{4} - 56843025124 T^{6} + 62259690411361 T^{8}$$
$59$ $$1 - 126 T + 11993 T^{2} - 844326 T^{3} + 51207492 T^{4} - 2939098806 T^{5} + 145323510473 T^{6} - 5314747238766 T^{7} + 146830437604321 T^{8}$$
$61$ $$1 + 41 T - 5513 T^{2} - 10168 T^{3} + 31652794 T^{4} - 37835128 T^{5} - 76332121433 T^{6} + 2112335348801 T^{7} + 191707312997281 T^{8}$$
$67$ $$1 + 116 T + 3787 T^{2} + 80156 T^{3} + 12934456 T^{4} + 359820284 T^{5} + 76312295227 T^{6} + 10493172331604 T^{7} + 406067677556641 T^{8}$$
$71$ $$1 - 18616 T^{2} + 137194926 T^{4} - 473063853496 T^{6} + 645753531245761 T^{8}$$
$73$ $$( 1 - 43 T + 10452 T^{2} - 229147 T^{3} + 28398241 T^{4} )^{2}$$
$79$ $$1 - 83 T - 5459 T^{2} + 11122 T^{3} + 70528774 T^{4} + 69412402 T^{5} - 212628492179 T^{6} - 20176258808243 T^{7} + 1517108809906561 T^{8}$$
$83$ $$1 + 81 T + 16289 T^{2} + 1142262 T^{3} + 166474326 T^{4} + 7869042918 T^{5} + 773048590769 T^{6} + 26482170242889 T^{7} + 2252292232139041 T^{8}$$
$89$ $$1 - 6916 T^{2} + 69013446 T^{4} - 433925338756 T^{6} + 3936588805702081 T^{8}$$
$97$ $$1 + 196 T + 10291 T^{2} + 1824172 T^{3} + 341030200 T^{4} + 17163634348 T^{5} + 911054830771 T^{6} + 163262512966084 T^{7} + 7837433594376961 T^{8}$$