# Properties

 Label 864.3.e.e Level $864$ Weight $3$ Character orbit 864.e Analytic conductor $23.542$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 864.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.5422948407$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.2441150464.4 Defining polynomial: $$x^{8} - 14x^{6} + 77x^{4} - 188x^{2} + 196$$ x^8 - 14*x^6 + 77*x^4 - 188*x^2 + 196 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}\cdot 3^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{5} + \beta_{4} q^{7}+O(q^{10})$$ q - b6 * q^5 + b4 * q^7 $$q - \beta_{6} q^{5} + \beta_{4} q^{7} + (\beta_{2} + \beta_1) q^{11} + (\beta_{5} - 2) q^{13} + ( - \beta_{7} - \beta_{6}) q^{17} - \beta_{3} q^{19} + ( - 4 \beta_{2} - 3 \beta_1) q^{23} + (3 \beta_{5} - 4) q^{25} + (\beta_{7} - 3 \beta_{6}) q^{29} + ( - \beta_{4} + 3 \beta_{3}) q^{31} + (5 \beta_{2} - 2 \beta_1) q^{35} + (5 \beta_{5} + 4) q^{37} + (\beta_{7} + 3 \beta_{6}) q^{41} + ( - 6 \beta_{4} - 2 \beta_{3}) q^{43} + ( - 6 \beta_{2} + 14 \beta_1) q^{47} + (7 \beta_{5} + 12) q^{49} + ( - \beta_{7} + 12 \beta_{6}) q^{53} + ( - \beta_{4} - 2 \beta_{3}) q^{55} + (18 \beta_{2} - 10 \beta_1) q^{59} + (8 \beta_{5} + 12) q^{61} + (\beta_{7} + 9 \beta_{6}) q^{65} + (6 \beta_{4} + 4 \beta_{3}) q^{67} + ( - 26 \beta_{2} - 13 \beta_1) q^{71} + (11 \beta_{5} - 27) q^{73} - \beta_{6} q^{77} + (10 \beta_{4} - \beta_{3}) q^{79} + (19 \beta_{2} + 33 \beta_1) q^{83} + (11 \beta_{5} - 42) q^{85} + (4 \beta_{7} - 22 \beta_{6}) q^{89} + (6 \beta_{4} - \beta_{3}) q^{91} + ( - 24 \beta_{2} - 29 \beta_1) q^{95} + (8 \beta_{5} - 5) q^{97}+O(q^{100})$$ q - b6 * q^5 + b4 * q^7 + (b2 + b1) * q^11 + (b5 - 2) * q^13 + (-b7 - b6) * q^17 - b3 * q^19 + (-4*b2 - 3*b1) * q^23 + (3*b5 - 4) * q^25 + (b7 - 3*b6) * q^29 + (-b4 + 3*b3) * q^31 + (5*b2 - 2*b1) * q^35 + (5*b5 + 4) * q^37 + (b7 + 3*b6) * q^41 + (-6*b4 - 2*b3) * q^43 + (-6*b2 + 14*b1) * q^47 + (7*b5 + 12) * q^49 + (-b7 + 12*b6) * q^53 + (-b4 - 2*b3) * q^55 + (18*b2 - 10*b1) * q^59 + (8*b5 + 12) * q^61 + (b7 + 9*b6) * q^65 + (6*b4 + 4*b3) * q^67 + (-26*b2 - 13*b1) * q^71 + (11*b5 - 27) * q^73 - b6 * q^77 + (10*b4 - b3) * q^79 + (19*b2 + 33*b1) * q^83 + (11*b5 - 42) * q^85 + (4*b7 - 22*b6) * q^89 + (6*b4 - b3) * q^91 + (-24*b2 - 29*b1) * q^95 + (8*b5 - 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 16 q^{13} - 32 q^{25} + 32 q^{37} + 96 q^{49} + 96 q^{61} - 216 q^{73} - 336 q^{85} - 40 q^{97}+O(q^{100})$$ 8 * q - 16 * q^13 - 32 * q^25 + 32 * q^37 + 96 * q^49 + 96 * q^61 - 216 * q^73 - 336 * q^85 - 40 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 14x^{6} + 77x^{4} - 188x^{2} + 196$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} - 14\nu^{5} + 63\nu^{3} - 90\nu ) / 14$$ (v^7 - 14*v^5 + 63*v^3 - 90*v) / 14 $$\beta_{2}$$ $$=$$ $$( 5\nu^{7} - 56\nu^{5} + 245\nu^{3} - 338\nu ) / 56$$ (5*v^7 - 56*v^5 + 245*v^3 - 338*v) / 56 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 7\nu^{5} - 106\nu ) / 7$$ (-v^7 + 7*v^5 - 106*v) / 7 $$\beta_{4}$$ $$=$$ $$( -9\nu^{7} + 84\nu^{5} - 245\nu^{3} + 250\nu ) / 56$$ (-9*v^7 + 84*v^5 - 245*v^3 + 250*v) / 56 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 10\nu^{4} - 31\nu^{2} + 22 ) / 2$$ (-v^6 + 10*v^4 - 31*v^2 + 22) / 2 $$\beta_{6}$$ $$=$$ $$( \nu^{6} - 14\nu^{4} + 67\nu^{2} - 106 ) / 4$$ (v^6 - 14*v^4 + 67*v^2 - 106) / 4 $$\beta_{7}$$ $$=$$ $$( 3\nu^{6} - 26\nu^{4} + 105\nu^{2} - 150 ) / 4$$ (3*v^6 - 26*v^4 + 105*v^2 - 150) / 4
 $$\nu$$ $$=$$ $$( 2\beta_{4} - \beta_{3} + 2\beta_{2} ) / 12$$ (2*b4 - b3 + 2*b2) / 12 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{6} + 2\beta_{5} + 42 ) / 12$$ (b7 + b6 + 2*b5 + 42) / 12 $$\nu^{3}$$ $$=$$ $$( 10\beta_{4} - 2\beta_{3} + 22\beta_{2} - 9\beta_1 ) / 12$$ (10*b4 - 2*b3 + 22*b2 - 9*b1) / 12 $$\nu^{4}$$ $$=$$ $$( 3\beta_{7} - \beta_{6} + 4\beta_{5} + 42 ) / 4$$ (3*b7 - b6 + 4*b5 + 42) / 4 $$\nu^{5}$$ $$=$$ $$( 34\beta_{4} - 2\beta_{3} + 142\beta_{2} - 105\beta_1 ) / 12$$ (34*b4 - 2*b3 + 142*b2 - 105*b1) / 12 $$\nu^{6}$$ $$=$$ $$( 59\beta_{7} - 61\beta_{6} + 34\beta_{5} + 222 ) / 12$$ (59*b7 - 61*b6 + 34*b5 + 222) / 12 $$\nu^{7}$$ $$=$$ $$( 26\beta_{4} + 8\beta_{3} + 782\beta_{2} - 735\beta_1 ) / 12$$ (26*b4 + 8*b3 + 782*b2 - 735*b1) / 12

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 1.52833 + 0.500000i −1.52833 − 0.500000i −2.27249 + 0.500000i 2.27249 − 0.500000i −2.27249 − 0.500000i 2.27249 + 0.500000i 1.52833 − 0.500000i −1.52833 + 0.500000i
0 0 0 7.37942i 0 −1.26611 0 0 0
161.2 0 0 0 7.37942i 0 1.26611 0 0 0
161.3 0 0 0 1.88259i 0 −10.9726 0 0 0
161.4 0 0 0 1.88259i 0 10.9726 0 0 0
161.5 0 0 0 1.88259i 0 −10.9726 0 0 0
161.6 0 0 0 1.88259i 0 10.9726 0 0 0
161.7 0 0 0 7.37942i 0 −1.26611 0 0 0
161.8 0 0 0 7.37942i 0 1.26611 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.8 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 864.3.e.e 8
3.b odd 2 1 inner 864.3.e.e 8
4.b odd 2 1 inner 864.3.e.e 8
8.b even 2 1 1728.3.e.v 8
8.d odd 2 1 1728.3.e.v 8
12.b even 2 1 inner 864.3.e.e 8
24.f even 2 1 1728.3.e.v 8
24.h odd 2 1 1728.3.e.v 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.3.e.e 8 1.a even 1 1 trivial
864.3.e.e 8 3.b odd 2 1 inner
864.3.e.e 8 4.b odd 2 1 inner
864.3.e.e 8 12.b even 2 1 inner
1728.3.e.v 8 8.b even 2 1
1728.3.e.v 8 8.d odd 2 1
1728.3.e.v 8 24.f even 2 1
1728.3.e.v 8 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(864, [\chi])$$:

 $$T_{5}^{4} + 58T_{5}^{2} + 193$$ T5^4 + 58*T5^2 + 193 $$T_{7}^{4} - 122T_{7}^{2} + 193$$ T7^4 - 122*T7^2 + 193

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 58 T^{2} + 193)^{2}$$
$7$ $$(T^{4} - 122 T^{2} + 193)^{2}$$
$11$ $$(T^{4} + 34 T^{2} + 1)^{2}$$
$13$ $$(T^{2} + 4 T - 68)^{4}$$
$17$ $$(T^{4} + 1080 T^{2} + 250128)^{2}$$
$19$ $$(T^{4} - 464 T^{2} + 12352)^{2}$$
$23$ $$(T^{4} + 432 T^{2} + 5184)^{2}$$
$29$ $$(T^{4} + 1336 T^{2} + 151312)^{2}$$
$31$ $$(T^{4} - 4490 T^{2} + 2733073)^{2}$$
$37$ $$(T^{2} - 8 T - 1784)^{4}$$
$41$ $$(T^{4} + 1648 T^{2} + 605248)^{2}$$
$43$ $$(T^{4} - 5480 T^{2} + 7414288)^{2}$$
$47$ $$(T^{4} + 3784 T^{2} + 1547536)^{2}$$
$53$ $$(T^{4} + 8698 T^{2} + 16119553)^{2}$$
$59$ $$(T^{4} + 7432 T^{2} + 4477456)^{2}$$
$61$ $$(T^{2} - 24 T - 4464)^{4}$$
$67$ $$(T^{4} - 10280 T^{2} + 16455952)^{2}$$
$71$ $$(T^{4} + 14872 T^{2} + 22391824)^{2}$$
$73$ $$(T^{2} + 54 T - 7983)^{4}$$
$79$ $$(T^{4} - 13304 T^{2} + 892432)^{2}$$
$83$ $$(T^{4} + 23922 T^{2} + 29844369)^{2}$$
$89$ $$(T^{4} + 39016 T^{2} + \cdots + 350701072)^{2}$$
$97$ $$(T^{2} + 10 T - 4583)^{4}$$