# Properties

 Label 1728.4.f.g Level $1728$ Weight $4$ Character orbit 1728.f Analytic conductor $101.955$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1728.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$101.955300490$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.58594980096.3 Defining polynomial: $$x^{8} - 21x^{6} + 341x^{4} - 2100x^{2} + 10000$$ x^8 - 21*x^6 + 341*x^4 - 2100*x^2 + 10000 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{8}\cdot 3^{4}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{5} + (\beta_{7} - 2 \beta_{3}) q^{7}+O(q^{10})$$ q + b6 * q^5 + (b7 - 2*b3) * q^7 $$q + \beta_{6} q^{5} + (\beta_{7} - 2 \beta_{3}) q^{7} + ( - \beta_{5} - 8 \beta_1) q^{11} + (3 \beta_{5} + \beta_1) q^{13} + ( - 3 \beta_{7} + 63 \beta_{3}) q^{17} + ( - 6 \beta_{6} + 5 \beta_{2}) q^{19} + ( - 3 \beta_{4} + 153) q^{23} + ( - 2 \beta_{4} + 1) q^{25} - 16 \beta_{2} q^{29} + (2 \beta_{7} + 2 \beta_{3}) q^{31} + ( - \beta_{5} + 40 \beta_1) q^{35} + (21 \beta_{5} + \beta_1) q^{37} + ( - 18 \beta_{7} - 54 \beta_{3}) q^{41} + ( - 30 \beta_{6} + 34 \beta_{2}) q^{43} + ( - 21 \beta_{4} - 225) q^{47} + ( - 4 \beta_{4} - 30) q^{49} + ( - 26 \beta_{6} + 80 \beta_{2}) q^{53} + ( - 22 \beta_{7} + 54 \beta_{3}) q^{55} + (27 \beta_{5} - 40 \beta_1) q^{59} + (9 \beta_{5} - 45 \beta_1) q^{61} + ( - 3 \beta_{7} - 369 \beta_{3}) q^{65} + (36 \beta_{6} + 39 \beta_{2}) q^{67} + ( - 18 \beta_{4} + 54) q^{71} + (44 \beta_{4} + 43) q^{73} + (71 \beta_{6} - 80 \beta_{2}) q^{77} + (27 \beta_{7} - 196 \beta_{3}) q^{79} + ( - 44 \beta_{5} - 80 \beta_1) q^{83} + ( - 54 \beta_{5} - 120 \beta_1) q^{85} + ( - 21 \beta_{7} - 423 \beta_{3}) q^{89} + ( - 6 \beta_{6} + 125 \beta_{2}) q^{91} + ( - 3 \beta_{4} - 711) q^{95} + ( - 50 \beta_{4} + 155) q^{97}+O(q^{100})$$ q + b6 * q^5 + (b7 - 2*b3) * q^7 + (-b5 - 8*b1) * q^11 + (3*b5 + b1) * q^13 + (-3*b7 + 63*b3) * q^17 + (-6*b6 + 5*b2) * q^19 + (-3*b4 + 153) * q^23 + (-2*b4 + 1) * q^25 - 16*b2 * q^29 + (2*b7 + 2*b3) * q^31 + (-b5 + 40*b1) * q^35 + (21*b5 + b1) * q^37 + (-18*b7 - 54*b3) * q^41 + (-30*b6 + 34*b2) * q^43 + (-21*b4 - 225) * q^47 + (-4*b4 - 30) * q^49 + (-26*b6 + 80*b2) * q^53 + (-22*b7 + 54*b3) * q^55 + (27*b5 - 40*b1) * q^59 + (9*b5 - 45*b1) * q^61 + (-3*b7 - 369*b3) * q^65 + (36*b6 + 39*b2) * q^67 + (-18*b4 + 54) * q^71 + (44*b4 + 43) * q^73 + (71*b6 - 80*b2) * q^77 + (27*b7 - 196*b3) * q^79 + (-44*b5 - 80*b1) * q^83 + (-54*b5 - 120*b1) * q^85 + (-21*b7 - 423*b3) * q^89 + (-6*b6 + 125*b2) * q^91 + (-3*b4 - 711) * q^95 + (-50*b4 + 155) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 1224 q^{23} + 8 q^{25} - 1800 q^{47} - 240 q^{49} + 432 q^{71} + 344 q^{73} - 5688 q^{95} + 1240 q^{97}+O(q^{100})$$ 8 * q + 1224 * q^23 + 8 * q^25 - 1800 * q^47 - 240 * q^49 + 432 * q^71 + 344 * q^73 - 5688 * q^95 + 1240 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 21x^{6} + 341x^{4} - 2100x^{2} + 10000$$ :

 $$\beta_{1}$$ $$=$$ $$( -63\nu^{6} + 1023\nu^{4} - 21483\nu^{2} + 81150 ) / 17050$$ (-63*v^6 + 1023*v^4 - 21483*v^2 + 81150) / 17050 $$\beta_{2}$$ $$=$$ $$( 3\nu^{7} - 363\nu^{5} + 4323\nu^{3} - 42600\nu ) / 11000$$ (3*v^7 - 363*v^5 + 4323*v^3 - 42600*v) / 11000 $$\beta_{3}$$ $$=$$ $$( 21\nu^{7} - 341\nu^{5} + 4061\nu^{3} - 10000\nu ) / 31000$$ (21*v^7 - 341*v^5 + 4061*v^3 - 10000*v) / 31000 $$\beta_{4}$$ $$=$$ $$( -6\nu^{6} - 8883 ) / 341$$ (-6*v^6 - 8883) / 341 $$\beta_{5}$$ $$=$$ $$( 181\nu^{6} - 3751\nu^{4} + 44671\nu^{2} - 192550 ) / 8525$$ (181*v^6 - 3751*v^4 + 44671*v^2 - 192550) / 8525 $$\beta_{6}$$ $$=$$ $$( 241\nu^{7} - 7161\nu^{5} + 116281\nu^{3} - 1222200\nu ) / 170500$$ (241*v^7 - 7161*v^5 + 116281*v^3 - 1222200*v) / 170500 $$\beta_{7}$$ $$=$$ $$( 63\nu^{7} - 1023\nu^{5} + 18183\nu^{3} - 30000\nu ) / 11000$$ (63*v^7 - 1023*v^5 + 18183*v^3 - 30000*v) / 11000
 $$\nu$$ $$=$$ $$( \beta_{7} - 3\beta_{6} - 3\beta_{3} + 2\beta_{2} ) / 12$$ (b7 - 3*b6 - 3*b3 + 2*b2) / 12 $$\nu^{2}$$ $$=$$ $$( -3\beta_{5} + \beta_{4} - 22\beta _1 + 63 ) / 12$$ (-3*b5 + b4 - 22*b1 + 63) / 12 $$\nu^{3}$$ $$=$$ $$( 11\beta_{7} - 93\beta_{3} ) / 6$$ (11*b7 - 93*b3) / 6 $$\nu^{4}$$ $$=$$ $$( -63\beta_{5} - 21\beta_{4} - 262\beta _1 - 723 ) / 12$$ (-63*b5 - 21*b4 - 262*b1 - 723) / 12 $$\nu^{5}$$ $$=$$ $$( 131\beta_{7} + 393\beta_{6} - 1653\beta_{3} - 682\beta_{2} ) / 12$$ (131*b7 + 393*b6 - 1653*b3 - 682*b2) / 12 $$\nu^{6}$$ $$=$$ $$( -341\beta_{4} - 8883 ) / 6$$ (-341*b4 - 8883) / 6 $$\nu^{7}$$ $$=$$ $$( -1651\beta_{7} + 4953\beta_{6} + 25413\beta_{3} - 10122\beta_{2} ) / 12$$ (-1651*b7 + 4953*b6 + 25413*b3 - 10122*b2) / 12

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
863.1
 2.33962 − 1.35078i 2.33962 + 1.35078i 3.20565 − 1.85078i 3.20565 + 1.85078i −3.20565 − 1.85078i −3.20565 + 1.85078i −2.33962 − 1.35078i −2.33962 + 1.35078i
0 0 0 −12.8226 0 17.2094i 0 0 0
863.2 0 0 0 −12.8226 0 17.2094i 0 0 0
863.3 0 0 0 −9.35849 0 21.2094i 0 0 0
863.4 0 0 0 −9.35849 0 21.2094i 0 0 0
863.5 0 0 0 9.35849 0 21.2094i 0 0 0
863.6 0 0 0 9.35849 0 21.2094i 0 0 0
863.7 0 0 0 12.8226 0 17.2094i 0 0 0
863.8 0 0 0 12.8226 0 17.2094i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 863.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.f.g yes 8
3.b odd 2 1 1728.4.f.c 8
4.b odd 2 1 1728.4.f.c 8
8.b even 2 1 inner 1728.4.f.g yes 8
8.d odd 2 1 1728.4.f.c 8
12.b even 2 1 inner 1728.4.f.g yes 8
24.f even 2 1 inner 1728.4.f.g yes 8
24.h odd 2 1 1728.4.f.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.4.f.c 8 3.b odd 2 1
1728.4.f.c 8 4.b odd 2 1
1728.4.f.c 8 8.d odd 2 1
1728.4.f.c 8 24.h odd 2 1
1728.4.f.g yes 8 1.a even 1 1 trivial
1728.4.f.g yes 8 8.b even 2 1 inner
1728.4.f.g yes 8 12.b even 2 1 inner
1728.4.f.g yes 8 24.f 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 $$S_{4}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{4} - 252T_{5}^{2} + 14400$$ T5^4 - 252*T5^2 + 14400 $$T_{7}^{4} + 746T_{7}^{2} + 133225$$ T7^4 + 746*T7^2 + 133225 $$T_{23}^{2} - 306T_{23} + 20088$$ T23^2 - 306*T23 + 20088

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 252 T^{2} + 14400)^{2}$$
$7$ $$(T^{4} + 746 T^{2} + 133225)^{2}$$
$11$ $$(T^{4} + 3420 T^{2} + 2143296)^{2}$$
$13$ $$(T^{2} + 1107)^{4}$$
$17$ $$(T^{4} + 14580 T^{2} + 419904)^{2}$$
$19$ $$(T^{4} - 9342 T^{2} + 17514225)^{2}$$
$23$ $$(T^{2} - 306 T + 20088)^{4}$$
$29$ $$(T^{2} - 6912)^{4}$$
$31$ $$(T^{4} + 2960 T^{2} + 2166784)^{2}$$
$37$ $$(T^{4} + 110430 T^{2} + \cdots + 2837799441)^{2}$$
$41$ $$(T^{4} + 244944 T^{2} + \cdots + 13604889600)^{2}$$
$43$ $$(T^{4} - 252504 T^{2} + \cdots + 9053141904)^{2}$$
$47$ $$(T^{2} + 450 T - 112104)^{4}$$
$53$ $$(T^{4} - 441072 T^{2} + \cdots + 2941977600)^{2}$$
$59$ $$(T^{4} + 308988 T^{2} + \cdots + 617025600)^{2}$$
$61$ $$(T^{4} + 144342 T^{2} + \cdots + 2729540025)^{2}$$
$67$ $$(T^{4} - 459270 T^{2} + \cdots + 7953250761)^{2}$$
$71$ $$(T^{2} - 108 T - 116640)^{4}$$
$73$ $$(T^{2} - 86 T - 712535)^{4}$$
$79$ $$(T^{4} + 614834 T^{2} + \cdots + 53169442225)^{2}$$
$83$ $$(T^{4} + 706752 T^{2} + \cdots + 15099494400)^{2}$$
$89$ $$(T^{4} + 683316 T^{2} + \cdots + 262440000)^{2}$$
$97$ $$(T^{2} - 310 T - 898475)^{4}$$
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