# Properties

 Label 1728.3.q.j 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: $$x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{6} - 2 \beta_{2} - \beta_1 + 1) q^{7}+O(q^{10})$$ q + (-b4 + b2 + b1 - 1) * q^5 + (b6 - 2*b2 - b1 + 1) * q^7 $$q + ( - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{6} - 2 \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - 3 \beta_{3} + 4 \beta_{2} + \beta_1 + 2) q^{11} + ( - 2 \beta_{7} + 2 \beta_{6} + 6 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + \cdots - 1) q^{13}+ \cdots + (2 \beta_{6} - 6 \beta_{3} + 67 \beta_{2} + 10 \beta_1 - 63) q^{97}+O(q^{100})$$ q + (-b4 + b2 + b1 - 1) * q^5 + (b6 - 2*b2 - b1 + 1) * q^7 + (-2*b7 + b6 + 2*b5 - 3*b3 + 4*b2 + b1 + 2) * q^11 + (-2*b7 + 2*b6 + 6*b5 + 3*b4 - 2*b3 - 3*b2 + b1 - 1) * q^13 + (-2*b7 + 4*b6 + b5 + b4 + b3 - 4*b2 - 3*b1) * q^17 + (-4*b7 + b3 + 5*b1 + 2) * q^19 + (3*b7 + 3*b6 + 4*b4 - 8*b2 - 2*b1 + 14) * q^23 + (-4*b6 + 3*b5 + 6*b4 + 5*b3 - 8*b2 - 6*b1 + 7) * q^25 + (4*b7 - 2*b6 + 3*b5 + b3 - 9*b2 - 2*b1 - 10) * q^29 + (b7 - b6 - 12*b5 - 6*b4 - 12*b2) * q^31 + (-b7 + 2*b6 - 2*b3 + 12*b2 + b1 - 7) * q^35 + (2*b7 - 6*b5 + 6*b4 - 2*b1 - 16) * q^37 + (-2*b7 - 2*b6 + 21*b2 - 42) * q^41 + (-7*b6 - 6*b5 - 12*b4 + 7*b3 - 14*b2 - 7*b1 + 14) * q^43 + (6*b7 - 3*b6 + 6*b5 - 2*b3 - 10*b2 - 3*b1 - 15) * q^47 + (-6*b7 + 6*b6 - 6*b5 - 3*b4 - 2*b3 + 22*b2 + b1 - 1) * q^49 + (-6*b7 + 12*b6 + 4*b5 + 4*b4 - 4*b3 + 8*b2 - 2*b1 - 10) * q^53 + (-b7 + 12*b5 - 12*b4 + 6*b3 + 7*b1 - 27) * q^55 + (3*b7 + 3*b6 + 14*b4 + 6*b2 - 9*b1 - 21) * q^59 + (2*b6 - 9*b5 - 18*b4 + 5*b3 - 5*b2 - 12*b1 - 2) * q^61 + (-12*b7 + 6*b6 + 7*b5 - 15*b3 + 21*b2 + 6*b1 + 12) * q^65 + (5*b7 - 5*b6 + 10*b3 - 5*b1 + 5) * q^67 + (4*b5 + 4*b4 - 12*b3 + 84*b2 + 12*b1 - 42) * q^71 + (-18*b7 + 9*b5 - 9*b4 - 7*b3 + 11*b1 - 4) * q^73 + (4*b7 + 4*b6 + 9*b4 - 7*b2 - 7*b1 + 7) * q^77 + (-7*b6 + 12*b5 + 24*b4 + 6*b3 - 12*b2 - 5*b1 + 13) * q^79 + (2*b7 - b6 - 2*b5 - 2*b3 + 22*b2 - b1 + 19) * q^83 + (-10*b7 + 10*b6 - 12*b3 + 60*b2 + 6*b1 - 6) * q^85 + (-10*b7 + 20*b6 - 24*b3 - 16*b2 + 14*b1 - 2) * q^89 + (7*b7 - 6*b5 + 6*b4 + 4*b3 - 3*b1 - 15) * q^91 + (-6*b7 - 6*b6 + 2*b4 - 40*b2 + 2*b1 + 82) * q^95 + (2*b6 - 6*b3 + 67*b2 + 10*b1 - 63) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{5} + 6 q^{7}+O(q^{10})$$ 8 * q - 6 * q^5 + 6 * q^7 $$8 q - 6 q^{5} + 6 q^{7} + 36 q^{11} - 14 q^{13} - 4 q^{19} + 102 q^{23} + 10 q^{25} - 114 q^{29} - 50 q^{31} - 120 q^{37} - 264 q^{41} + 28 q^{43} - 150 q^{47} + 94 q^{49} - 244 q^{55} - 108 q^{59} - 14 q^{61} + 198 q^{65} + 20 q^{67} - 76 q^{73} + 66 q^{77} + 26 q^{79} + 246 q^{83} + 224 q^{85} - 108 q^{91} + 456 q^{95} - 236 q^{97}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$( -473\nu^{7} + 198\nu^{6} - 5547\nu^{5} - 7826\nu^{4} - 75285\nu^{3} - 29928\nu^{2} + 140724\nu - 256788 ) / 159300$$ (-473*v^7 + 198*v^6 - 5547*v^5 - 7826*v^4 - 75285*v^3 - 29928*v^2 + 140724*v - 256788) / 159300 $$\beta_{2}$$ $$=$$ $$( 677\nu^{7} - 1827\nu^{6} + 10353\nu^{5} - 6901\nu^{4} + 82215\nu^{3} - 132153\nu^{2} + 156924\nu + 78912 ) / 159300$$ (677*v^7 - 1827*v^6 + 10353*v^5 - 6901*v^4 + 82215*v^3 - 132153*v^2 + 156924*v + 78912) / 159300 $$\beta_{3}$$ $$=$$ $$( -473\nu^{7} + 198\nu^{6} - 5547\nu^{5} - 7826\nu^{4} - 75285\nu^{3} - 29928\nu^{2} - 98226\nu - 177138 ) / 79650$$ (-473*v^7 + 198*v^6 - 5547*v^5 - 7826*v^4 - 75285*v^3 - 29928*v^2 - 98226*v - 177138) / 79650 $$\beta_{4}$$ $$=$$ $$( 1013 \nu^{7} - 2688 \nu^{6} + 16707 \nu^{5} - 19444 \nu^{4} + 143085 \nu^{3} - 232782 \nu^{2} + 247356 \nu - 401472 ) / 159300$$ (1013*v^7 - 2688*v^6 + 16707*v^5 - 19444*v^4 + 143085*v^3 - 232782*v^2 + 247356*v - 401472) / 159300 $$\beta_{5}$$ $$=$$ $$( 644\nu^{7} - 2019\nu^{6} + 9966\nu^{5} - 7447\nu^{4} + 68730\nu^{3} - 107691\nu^{2} + 129078\nu + 194364 ) / 79650$$ (644*v^7 - 2019*v^6 + 9966*v^5 - 7447*v^4 + 68730*v^3 - 107691*v^2 + 129078*v + 194364) / 79650 $$\beta_{6}$$ $$=$$ $$( - 2407 \nu^{7} + 7182 \nu^{6} - 45123 \nu^{5} + 47066 \nu^{4} - 389565 \nu^{3} + 475248 \nu^{2} - 1245834 \nu + 209808 ) / 159300$$ (-2407*v^7 + 7182*v^6 - 45123*v^5 + 47066*v^4 - 389565*v^3 + 475248*v^2 - 1245834*v + 209808) / 159300 $$\beta_{7}$$ $$=$$ $$( 5753 \nu^{7} - 4878 \nu^{6} + 67467 \nu^{5} + 95186 \nu^{4} + 657585 \nu^{3} + 364008 \nu^{2} + 544536 \nu + 2015568 ) / 318600$$ (5753*v^7 - 4878*v^6 + 67467*v^5 + 95186*v^4 + 657585*v^3 + 364008*v^2 + 544536*v + 2015568) / 318600
 $$\nu$$ $$=$$ $$( -\beta_{3} + 2\beta _1 + 1 ) / 3$$ (-b3 + 2*b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( 6\beta_{5} + 3\beta_{4} - 2\beta_{3} - 18\beta_{2} + \beta _1 - 1 ) / 3$$ (6*b5 + 3*b4 - 2*b3 - 18*b2 + b1 - 1) / 3 $$\nu^{3}$$ $$=$$ $$( -6\beta_{7} + 6\beta_{5} - 6\beta_{4} - 13\beta_{3} - 7\beta _1 - 32 ) / 3$$ (-6*b7 + 6*b5 - 6*b4 - 13*b3 - 7*b1 - 32) / 3 $$\nu^{4}$$ $$=$$ $$( -12\beta_{6} - 45\beta_{5} - 90\beta_{4} + 25\beta_{3} + 186\beta_{2} - 38\beta _1 - 199 ) / 3$$ (-12*b6 - 45*b5 - 90*b4 + 25*b3 + 186*b2 - 38*b1 - 199) / 3 $$\nu^{5}$$ $$=$$ $$( 90\beta_{7} - 90\beta_{6} - 240\beta_{5} - 120\beta_{4} + 338\beta_{3} + 288\beta_{2} - 169\beta _1 + 169 ) / 3$$ (90*b7 - 90*b6 - 240*b5 - 120*b4 + 338*b3 + 288*b2 - 169*b1 + 169) / 3 $$\nu^{6}$$ $$=$$ $$( 240\beta_{7} - 627\beta_{5} + 627\beta_{4} + 445\beta_{3} + 205\beta _1 + 2912 ) / 3$$ (240*b7 - 627*b5 + 627*b4 + 445*b3 + 205*b1 + 2912) / 3 $$\nu^{7}$$ $$=$$ $$( 1254\beta_{6} + 1962\beta_{5} + 3924\beta_{4} - 2293\beta_{3} - 5316\beta_{2} + 3332\beta _1 + 6355 ) / 3$$ (1254*b6 + 1962*b5 + 3924*b4 - 2293*b3 - 5316*b2 + 3332*b1 + 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.j 8
3.b odd 2 1 576.3.q.i 8
4.b odd 2 1 1728.3.q.i 8
8.b even 2 1 216.3.m.b 8
8.d odd 2 1 432.3.q.e 8
9.c even 3 1 576.3.q.i 8
9.d odd 6 1 inner 1728.3.q.j 8
12.b even 2 1 576.3.q.j 8
24.f even 2 1 144.3.q.e 8
24.h odd 2 1 72.3.m.b 8
36.f odd 6 1 576.3.q.j 8
36.h even 6 1 1728.3.q.i 8
72.j odd 6 1 216.3.m.b 8
72.j odd 6 1 648.3.e.c 8
72.l even 6 1 432.3.q.e 8
72.l even 6 1 1296.3.e.i 8
72.n even 6 1 72.3.m.b 8
72.n even 6 1 648.3.e.c 8
72.p odd 6 1 144.3.q.e 8
72.p odd 6 1 1296.3.e.i 8

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

 $$T_{5}^{8} + 6T_{5}^{7} - 37T_{5}^{6} - 294T_{5}^{5} + 2661T_{5}^{4} - 6468T_{5}^{3} + 5612T_{5}^{2} + 528T_{5} + 16$$ T5^8 + 6*T5^7 - 37*T5^6 - 294*T5^5 + 2661*T5^4 - 6468*T5^3 + 5612*T5^2 + 528*T5 + 16 $$T_{7}^{8} - 6T_{7}^{7} + 69T_{7}^{6} + 126T_{7}^{5} + 1197T_{7}^{4} + 108T_{7}^{3} + 4860T_{7}^{2} + 3888T_{7} + 11664$$ T7^8 - 6*T7^7 + 69*T7^6 + 126*T7^5 + 1197*T7^4 + 108*T7^3 + 4860*T7^2 + 3888*T7 + 11664

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 6 T^{7} - 37 T^{6} - 294 T^{5} + \cdots + 16$$
$7$ $$T^{8} - 6 T^{7} + 69 T^{6} + \cdots + 11664$$
$11$ $$T^{8} - 36 T^{7} + \cdots + 105616729$$
$13$ $$T^{8} + 14 T^{7} + 547 T^{6} + \cdots + 2611456$$
$17$ $$T^{8} + 1454 T^{6} + \cdots + 7020428944$$
$19$ $$(T^{4} + 2 T^{3} - 1179 T^{2} + \cdots + 226348)^{2}$$
$23$ $$T^{8} - 102 T^{7} + \cdots + 11198718976$$
$29$ $$T^{8} + 114 T^{7} + \cdots + 106450807824$$
$31$ $$T^{8} + 50 T^{7} + \cdots + 152712134656$$
$37$ $$(T^{4} + 60 T^{3} - 276 T^{2} + \cdots + 206496)^{2}$$
$41$ $$T^{8} + 264 T^{7} + \cdots + 1919025613521$$
$43$ $$T^{8} - 28 T^{7} + \cdots + 1352729498761$$
$47$ $$T^{8} + 150 T^{7} + \cdots + 4615347568896$$
$53$ $$T^{8} + 7016 T^{6} + \cdots + 78435844096$$
$59$ $$T^{8} + 108 T^{7} + \cdots + 127589696809$$
$61$ $$T^{8} + 14 T^{7} + \cdots + 133593174016$$
$67$ $$T^{8} - 20 T^{7} + \cdots + 17391015625$$
$71$ $$T^{8} + \cdots + 114698616545536$$
$73$ $$(T^{4} + 38 T^{3} - 9831 T^{2} + \cdots + 2961976)^{2}$$
$79$ $$T^{8} + \cdots + 103529078405776$$
$83$ $$T^{8} - 246 T^{7} + \cdots + 1085363908864$$
$89$ $$T^{8} + \cdots + 309931236458496$$
$97$ $$T^{8} + 236 T^{7} + \cdots + 3435006304129$$