# Properties

 Label 1728.3.q.i Level $1728$ Weight $3$ Character orbit 1728.q 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.q (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: 8.0.19269881856.9 Defining polynomial: $$x^{8} - 2 x^{7} + 15 x^{6} - 2 x^{5} + 133 x^{4} - 84 x^{3} + 276 x^{2} + 144 x + 144$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{5} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{5} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{6} ) q^{7} + ( -2 - \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{11} + ( -1 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{13} + ( -3 \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{17} + ( -2 - 5 \beta_{1} - \beta_{3} + 4 \beta_{7} ) q^{19} + ( -14 + 2 \beta_{1} + 8 \beta_{2} - 4 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{23} + ( 7 - 6 \beta_{1} - 8 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} ) q^{25} + ( -10 - 2 \beta_{1} - 9 \beta_{2} + \beta_{3} + 3 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{29} + ( 12 \beta_{2} + 6 \beta_{4} + 12 \beta_{5} + \beta_{6} - \beta_{7} ) q^{31} + ( 7 - \beta_{1} - 12 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{35} + ( -16 - 2 \beta_{1} + 6 \beta_{4} - 6 \beta_{5} + 2 \beta_{7} ) q^{37} + ( -42 + 21 \beta_{2} - 2 \beta_{6} - 2 \beta_{7} ) q^{41} + ( -14 + 7 \beta_{1} + 14 \beta_{2} - 7 \beta_{3} + 12 \beta_{4} + 6 \beta_{5} + 7 \beta_{6} ) q^{43} + ( 15 + 3 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} - 6 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} ) q^{47} + ( -1 + \beta_{1} + 22 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{49} + ( -10 - 2 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 12 \beta_{6} - 6 \beta_{7} ) q^{53} + ( 27 - 7 \beta_{1} - 6 \beta_{3} + 12 \beta_{4} - 12 \beta_{5} + \beta_{7} ) q^{55} + ( 21 + 9 \beta_{1} - 6 \beta_{2} - 14 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{59} + ( -2 - 12 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} - 18 \beta_{4} - 9 \beta_{5} + 2 \beta_{6} ) q^{61} + ( 12 + 6 \beta_{1} + 21 \beta_{2} - 15 \beta_{3} + 7 \beta_{5} + 6 \beta_{6} - 12 \beta_{7} ) q^{65} + ( -5 + 5 \beta_{1} - 10 \beta_{3} + 5 \beta_{6} - 5 \beta_{7} ) q^{67} + ( 42 - 12 \beta_{1} - 84 \beta_{2} + 12 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{71} + ( -4 + 11 \beta_{1} - 7 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} - 18 \beta_{7} ) q^{73} + ( 7 - 7 \beta_{1} - 7 \beta_{2} + 9 \beta_{4} + 4 \beta_{6} + 4 \beta_{7} ) q^{77} + ( -13 + 5 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} - 24 \beta_{4} - 12 \beta_{5} + 7 \beta_{6} ) q^{79} + ( -19 + \beta_{1} - 22 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{83} + ( -6 + 6 \beta_{1} + 60 \beta_{2} - 12 \beta_{3} + 10 \beta_{6} - 10 \beta_{7} ) q^{85} + ( -2 + 14 \beta_{1} - 16 \beta_{2} - 24 \beta_{3} + 20 \beta_{6} - 10 \beta_{7} ) q^{89} + ( 15 + 3 \beta_{1} - 4 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} - 7 \beta_{7} ) q^{91} + ( -82 - 2 \beta_{1} + 40 \beta_{2} - 2 \beta_{4} + 6 \beta_{6} + 6 \beta_{7} ) q^{95} + ( -63 + 10 \beta_{1} + 67 \beta_{2} - 6 \beta_{3} + 2 \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{5} - 6 q^{7} + O(q^{10})$$ $$8 q - 6 q^{5} - 6 q^{7} - 36 q^{11} - 14 q^{13} + 4 q^{19} - 102 q^{23} + 10 q^{25} - 114 q^{29} + 50 q^{31} - 120 q^{37} - 264 q^{41} - 28 q^{43} + 150 q^{47} + 94 q^{49} + 244 q^{55} + 108 q^{59} - 14 q^{61} + 198 q^{65} - 20 q^{67} - 76 q^{73} + 66 q^{77} - 26 q^{79} - 246 q^{83} + 224 q^{85} + 108 q^{91} - 456 q^{95} - 236 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 15 x^{6} - 2 x^{5} + 133 x^{4} - 84 x^{3} + 276 x^{2} + 144 x + 144$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-473 \nu^{7} + 198 \nu^{6} - 5547 \nu^{5} - 7826 \nu^{4} - 75285 \nu^{3} - 29928 \nu^{2} + 140724 \nu - 256788$$$$)/159300$$ $$\beta_{2}$$ $$=$$ $$($$$$677 \nu^{7} - 1827 \nu^{6} + 10353 \nu^{5} - 6901 \nu^{4} + 82215 \nu^{3} - 132153 \nu^{2} + 156924 \nu + 78912$$$$)/159300$$ $$\beta_{3}$$ $$=$$ $$($$$$-473 \nu^{7} + 198 \nu^{6} - 5547 \nu^{5} - 7826 \nu^{4} - 75285 \nu^{3} - 29928 \nu^{2} - 98226 \nu - 177138$$$$)/79650$$ $$\beta_{4}$$ $$=$$ $$($$$$1013 \nu^{7} - 2688 \nu^{6} + 16707 \nu^{5} - 19444 \nu^{4} + 143085 \nu^{3} - 232782 \nu^{2} + 247356 \nu - 401472$$$$)/159300$$ $$\beta_{5}$$ $$=$$ $$($$$$644 \nu^{7} - 2019 \nu^{6} + 9966 \nu^{5} - 7447 \nu^{4} + 68730 \nu^{3} - 107691 \nu^{2} + 129078 \nu + 194364$$$$)/79650$$ $$\beta_{6}$$ $$=$$ $$($$$$-2407 \nu^{7} + 7182 \nu^{6} - 45123 \nu^{5} + 47066 \nu^{4} - 389565 \nu^{3} + 475248 \nu^{2} - 1245834 \nu + 209808$$$$)/159300$$ $$\beta_{7}$$ $$=$$ $$($$$$5753 \nu^{7} - 4878 \nu^{6} + 67467 \nu^{5} + 95186 \nu^{4} + 657585 \nu^{3} + 364008 \nu^{2} + 544536 \nu + 2015568$$$$)/318600$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$6 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 18 \beta_{2} + \beta_{1} - 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-6 \beta_{7} + 6 \beta_{5} - 6 \beta_{4} - 13 \beta_{3} - 7 \beta_{1} - 32$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-12 \beta_{6} - 45 \beta_{5} - 90 \beta_{4} + 25 \beta_{3} + 186 \beta_{2} - 38 \beta_{1} - 199$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$90 \beta_{7} - 90 \beta_{6} - 240 \beta_{5} - 120 \beta_{4} + 338 \beta_{3} + 288 \beta_{2} - 169 \beta_{1} + 169$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$240 \beta_{7} - 627 \beta_{5} + 627 \beta_{4} + 445 \beta_{3} + 205 \beta_{1} + 2912$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$1254 \beta_{6} + 1962 \beta_{5} + 3924 \beta_{4} - 2293 \beta_{3} - 5316 \beta_{2} + 3332 \beta_{1} + 6355$$$$)/3$$

## 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
 −1.41950 − 2.45865i −0.331167 − 0.573598i 1.91950 + 3.32468i 0.831167 + 1.43962i −1.41950 + 2.45865i −0.331167 + 0.573598i 1.91950 − 3.32468i 0.831167 − 1.43962i
0 0 0 −8.20800 4.73889i 0 −1.05671 1.83027i 0 0 0
449.2 0 0 0 −0.0440114 0.0254100i 0 −4.52944 7.84521i 0 0 0
449.3 0 0 0 1.80902 + 1.04444i 0 0.781452 + 1.35351i 0 0 0
449.4 0 0 0 3.44299 + 1.98781i 0 1.80469 + 3.12582i 0 0 0
1601.1 0 0 0 −8.20800 + 4.73889i 0 −1.05671 + 1.83027i 0 0 0
1601.2 0 0 0 −0.0440114 + 0.0254100i 0 −4.52944 + 7.84521i 0 0 0
1601.3 0 0 0 1.80902 1.04444i 0 0.781452 1.35351i 0 0 0
1601.4 0 0 0 3.44299 1.98781i 0 1.80469 3.12582i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1601.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.i 8
3.b odd 2 1 576.3.q.j 8
4.b odd 2 1 1728.3.q.j 8
8.b even 2 1 432.3.q.e 8
8.d odd 2 1 216.3.m.b 8
9.c even 3 1 576.3.q.j 8
9.d odd 6 1 inner 1728.3.q.i 8
12.b even 2 1 576.3.q.i 8
24.f even 2 1 72.3.m.b 8
24.h odd 2 1 144.3.q.e 8
36.f odd 6 1 576.3.q.i 8
36.h even 6 1 1728.3.q.j 8
72.j odd 6 1 432.3.q.e 8
72.j odd 6 1 1296.3.e.i 8
72.l even 6 1 216.3.m.b 8
72.l even 6 1 648.3.e.c 8
72.n even 6 1 144.3.q.e 8
72.n even 6 1 1296.3.e.i 8
72.p odd 6 1 72.3.m.b 8
72.p odd 6 1 648.3.e.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.b 8 24.f even 2 1
72.3.m.b 8 72.p odd 6 1
144.3.q.e 8 24.h odd 2 1
144.3.q.e 8 72.n even 6 1
216.3.m.b 8 8.d odd 2 1
216.3.m.b 8 72.l even 6 1
432.3.q.e 8 8.b even 2 1
432.3.q.e 8 72.j odd 6 1
576.3.q.i 8 12.b even 2 1
576.3.q.i 8 36.f odd 6 1
576.3.q.j 8 3.b odd 2 1
576.3.q.j 8 9.c even 3 1
648.3.e.c 8 72.l even 6 1
648.3.e.c 8 72.p odd 6 1
1296.3.e.i 8 72.j odd 6 1
1296.3.e.i 8 72.n even 6 1
1728.3.q.i 8 1.a even 1 1 trivial
1728.3.q.i 8 9.d odd 6 1 inner
1728.3.q.j 8 4.b odd 2 1
1728.3.q.j 8 36.h even 6 1

## Hecke kernels

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

 $$T_{5}^{8} + \cdots$$ $$T_{7}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$16 + 528 T + 5612 T^{2} - 6468 T^{3} + 2661 T^{4} - 294 T^{5} - 37 T^{6} + 6 T^{7} + T^{8}$$
$7$ $$11664 - 3888 T + 4860 T^{2} - 108 T^{3} + 1197 T^{4} - 126 T^{5} + 69 T^{6} + 6 T^{7} + T^{8}$$
$11$ $$105616729 + 58455576 T + 11688824 T^{2} + 500544 T^{3} - 50235 T^{4} - 3168 T^{5} + 344 T^{6} + 36 T^{7} + T^{8}$$
$13$ $$2611456 + 6270080 T + 14487184 T^{2} + 1407128 T^{3} + 179137 T^{4} + 2846 T^{5} + 547 T^{6} + 14 T^{7} + T^{8}$$
$17$ $$7020428944 + 120824648 T^{2} + 687753 T^{4} + 1454 T^{6} + T^{8}$$
$19$ $$( 226348 - 4004 T - 1179 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$23$ $$11198718976 + 3042651648 T + 228044192 T^{2} - 12909648 T^{3} - 670143 T^{4} + 45798 T^{5} + 3917 T^{6} + 102 T^{7} + T^{8}$$
$29$ $$106450807824 + 9055894608 T - 37821492 T^{2} - 25063668 T^{3} + 86949 T^{4} + 102942 T^{5} + 5235 T^{6} + 114 T^{7} + T^{8}$$
$31$ $$152712134656 + 12067409920 T + 1224387712 T^{2} + 17678560 T^{3} + 1633465 T^{4} - 27110 T^{5} + 3193 T^{6} - 50 T^{7} + T^{8}$$
$37$ $$( 206496 - 27360 T - 276 T^{2} + 60 T^{3} + T^{4} )^{2}$$
$41$ $$1919025613521 + 435967071768 T + 44376688026 T^{2} + 2581267824 T^{3} + 93582171 T^{4} + 2165328 T^{5} + 31434 T^{6} + 264 T^{7} + T^{8}$$
$43$ $$1352729498761 + 211873953592 T + 28244463112 T^{2} + 838981528 T^{3} + 24309277 T^{4} + 245392 T^{5} + 5032 T^{6} + 28 T^{7} + T^{8}$$
$47$ $$4615347568896 - 475641590400 T + 16526225232 T^{2} - 19261800 T^{3} - 8914095 T^{4} + 13050 T^{5} + 7413 T^{6} - 150 T^{7} + T^{8}$$
$53$ $$78435844096 + 2590797824 T^{2} + 12396816 T^{4} + 7016 T^{6} + T^{8}$$
$59$ $$127589696809 - 43823785536 T + 6086181872 T^{2} - 367082496 T^{3} + 4892493 T^{4} + 323136 T^{5} + 896 T^{6} - 108 T^{7} + T^{8}$$
$61$ $$133593174016 - 46766967808 T + 13427579584 T^{2} - 1020419248 T^{3} + 63457201 T^{4} - 368674 T^{5} + 8251 T^{6} + 14 T^{7} + T^{8}$$
$67$ $$17391015625 - 4879375000 T + 1606375000 T^{2} + 61325000 T^{3} + 3848125 T^{4} + 38000 T^{5} + 2200 T^{6} + 20 T^{7} + T^{8}$$
$71$ $$114698616545536 + 406952754944 T^{2} + 206774880 T^{4} + 26864 T^{6} + T^{8}$$
$73$ $$( 2961976 - 487192 T - 9831 T^{2} + 38 T^{3} + T^{4} )^{2}$$
$79$ $$103529078405776 + 749891898800 T + 123470983324 T^{2} - 1384089748 T^{3} + 122492077 T^{4} - 449026 T^{5} + 12277 T^{6} + 26 T^{7} + T^{8}$$
$83$ $$1085363908864 + 263110694016 T + 28654548944 T^{2} + 1792361544 T^{3} + 70034865 T^{4} + 1745862 T^{5} + 27269 T^{6} + 246 T^{7} + T^{8}$$
$89$ $$309931236458496 + 1618052474880 T^{2} + 407712528 T^{4} + 34920 T^{6} + T^{8}$$
$97$ $$3435006304129 + 837036947756 T + 171307640890 T^{2} + 7083794672 T^{3} + 202097299 T^{4} + 3255536 T^{5} + 38074 T^{6} + 236 T^{7} + T^{8}$$