# Properties

 Label 3200.2.f.r Level $3200$ Weight $2$ Character orbit 3200.f Analytic conductor $25.552$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7x^{4} + 1$$ x^8 + 7*x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}\cdot 5^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - \beta_{2} q^{7} + 2 q^{9}+O(q^{10})$$ q - b1 * q^3 - b2 * q^7 + 2 * q^9 $$q - \beta_1 q^{3} - \beta_{2} q^{7} + 2 q^{9} + \beta_{3} q^{11} - \beta_{6} q^{13} + \beta_{4} q^{17} + \beta_{3} q^{19} - \beta_{5} q^{21} - 2 \beta_{2} q^{23} + \beta_1 q^{27} - \beta_{5} q^{29} - \beta_{4} q^{33} + \beta_{6} q^{37} - \beta_{7} q^{39} + 3 q^{41} + 4 \beta_1 q^{43} - \beta_{2} q^{47} - q^{49} - 5 \beta_{3} q^{51} - 2 \beta_{6} q^{53} - \beta_{4} q^{57} - 4 \beta_{3} q^{59} - \beta_{5} q^{61} - 2 \beta_{2} q^{63} - 5 \beta_1 q^{67} - 2 \beta_{5} q^{69} + \beta_{7} q^{71} + 3 \beta_{4} q^{73} + \beta_{6} q^{77} - \beta_{7} q^{79} - 11 q^{81} - 3 \beta_1 q^{83} - 5 \beta_{2} q^{87} - q^{89} + 8 \beta_{3} q^{91} - 2 \beta_{4} q^{97} + 2 \beta_{3} q^{99}+O(q^{100})$$ q - b1 * q^3 - b2 * q^7 + 2 * q^9 + b3 * q^11 - b6 * q^13 + b4 * q^17 + b3 * q^19 - b5 * q^21 - 2*b2 * q^23 + b1 * q^27 - b5 * q^29 - b4 * q^33 + b6 * q^37 - b7 * q^39 + 3 * q^41 + 4*b1 * q^43 - b2 * q^47 - q^49 - 5*b3 * q^51 - 2*b6 * q^53 - b4 * q^57 - 4*b3 * q^59 - b5 * q^61 - 2*b2 * q^63 - 5*b1 * q^67 - 2*b5 * q^69 + b7 * q^71 + 3*b4 * q^73 + b6 * q^77 - b7 * q^79 - 11 * q^81 - 3*b1 * q^83 - 5*b2 * q^87 - q^89 + 8*b3 * q^91 - 2*b4 * q^97 + 2*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 16 q^{9}+O(q^{10})$$ 8 * q + 16 * q^9 $$8 q + 16 q^{9} + 24 q^{41} - 8 q^{49} - 88 q^{81} - 8 q^{89}+O(q^{100})$$ 8 * q + 16 * q^9 + 24 * q^41 - 8 * q^49 - 88 * q^81 - 8 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7x^{4} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{4} + 7 ) / 3$$ (2*v^4 + 7) / 3 $$\beta_{2}$$ $$=$$ $$( -4\nu^{7} - 2\nu^{5} - 26\nu^{3} - 10\nu ) / 3$$ (-4*v^7 - 2*v^5 - 26*v^3 - 10*v) / 3 $$\beta_{3}$$ $$=$$ $$-\nu^{6} - 6\nu^{2}$$ -v^6 - 6*v^2 $$\beta_{4}$$ $$=$$ $$( -5\nu^{6} - 40\nu^{2} ) / 3$$ (-5*v^6 - 40*v^2) / 3 $$\beta_{5}$$ $$=$$ $$( 8\nu^{7} + 2\nu^{5} + 58\nu^{3} + 22\nu ) / 3$$ (8*v^7 + 2*v^5 + 58*v^3 + 22*v) / 3 $$\beta_{6}$$ $$=$$ $$( -8\nu^{7} + 2\nu^{5} - 58\nu^{3} + 22\nu ) / 3$$ (-8*v^7 + 2*v^5 - 58*v^3 + 22*v) / 3 $$\beta_{7}$$ $$=$$ $$( 20\nu^{7} - 10\nu^{5} + 130\nu^{3} - 50\nu ) / 3$$ (20*v^7 - 10*v^5 + 130*v^3 - 50*v) / 3
 $$\nu$$ $$=$$ $$( \beta_{7} + 5\beta_{6} + 5\beta_{5} + 5\beta_{2} ) / 40$$ (b7 + 5*b6 + 5*b5 + 5*b2) / 40 $$\nu^{2}$$ $$=$$ $$( -3\beta_{4} + 5\beta_{3} ) / 10$$ (-3*b4 + 5*b3) / 10 $$\nu^{3}$$ $$=$$ $$( -2\beta_{7} - 5\beta_{6} + 5\beta_{5} + 10\beta_{2} ) / 20$$ (-2*b7 - 5*b6 + 5*b5 + 10*b2) / 20 $$\nu^{4}$$ $$=$$ $$( 3\beta _1 - 7 ) / 2$$ (3*b1 - 7) / 2 $$\nu^{5}$$ $$=$$ $$( -11\beta_{7} - 25\beta_{6} - 25\beta_{5} - 55\beta_{2} ) / 40$$ (-11*b7 - 25*b6 - 25*b5 - 55*b2) / 40 $$\nu^{6}$$ $$=$$ $$( 9\beta_{4} - 20\beta_{3} ) / 5$$ (9*b4 - 20*b3) / 5 $$\nu^{7}$$ $$=$$ $$( 29\beta_{7} + 65\beta_{6} - 65\beta_{5} - 145\beta_{2} ) / 40$$ (29*b7 + 65*b6 - 65*b5 - 145*b2) / 40

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1151$$ $$2177$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.437016 − 0.437016i 0.437016 − 0.437016i 0.437016 + 0.437016i −0.437016 + 0.437016i −1.14412 + 1.14412i 1.14412 + 1.14412i 1.14412 − 1.14412i −1.14412 − 1.14412i
0 −2.23607 0 0 0 2.82843i 0 2.00000 0
449.2 0 −2.23607 0 0 0 2.82843i 0 2.00000 0
449.3 0 −2.23607 0 0 0 2.82843i 0 2.00000 0
449.4 0 −2.23607 0 0 0 2.82843i 0 2.00000 0
449.5 0 2.23607 0 0 0 2.82843i 0 2.00000 0
449.6 0 2.23607 0 0 0 2.82843i 0 2.00000 0
449.7 0 2.23607 0 0 0 2.82843i 0 2.00000 0
449.8 0 2.23607 0 0 0 2.82843i 0 2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.f.r 8
4.b odd 2 1 inner 3200.2.f.r 8
5.b even 2 1 inner 3200.2.f.r 8
5.c odd 4 1 3200.2.d.m 4
5.c odd 4 1 3200.2.d.r yes 4
8.b even 2 1 inner 3200.2.f.r 8
8.d odd 2 1 inner 3200.2.f.r 8
20.d odd 2 1 inner 3200.2.f.r 8
20.e even 4 1 3200.2.d.m 4
20.e even 4 1 3200.2.d.r yes 4
40.e odd 2 1 inner 3200.2.f.r 8
40.f even 2 1 inner 3200.2.f.r 8
40.i odd 4 1 3200.2.d.m 4
40.i odd 4 1 3200.2.d.r yes 4
40.k even 4 1 3200.2.d.m 4
40.k even 4 1 3200.2.d.r yes 4
80.i odd 4 1 6400.2.a.cp 4
80.i odd 4 1 6400.2.a.cs 4
80.j even 4 1 6400.2.a.cp 4
80.j even 4 1 6400.2.a.cs 4
80.s even 4 1 6400.2.a.cp 4
80.s even 4 1 6400.2.a.cs 4
80.t odd 4 1 6400.2.a.cp 4
80.t odd 4 1 6400.2.a.cs 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.m 4 5.c odd 4 1
3200.2.d.m 4 20.e even 4 1
3200.2.d.m 4 40.i odd 4 1
3200.2.d.m 4 40.k even 4 1
3200.2.d.r yes 4 5.c odd 4 1
3200.2.d.r yes 4 20.e even 4 1
3200.2.d.r yes 4 40.i odd 4 1
3200.2.d.r yes 4 40.k even 4 1
3200.2.f.r 8 1.a even 1 1 trivial
3200.2.f.r 8 4.b odd 2 1 inner
3200.2.f.r 8 5.b even 2 1 inner
3200.2.f.r 8 8.b even 2 1 inner
3200.2.f.r 8 8.d odd 2 1 inner
3200.2.f.r 8 20.d odd 2 1 inner
3200.2.f.r 8 40.e odd 2 1 inner
3200.2.f.r 8 40.f even 2 1 inner
6400.2.a.cp 4 80.i odd 4 1
6400.2.a.cp 4 80.j even 4 1
6400.2.a.cp 4 80.s even 4 1
6400.2.a.cp 4 80.t odd 4 1
6400.2.a.cs 4 80.i odd 4 1
6400.2.a.cs 4 80.j even 4 1
6400.2.a.cs 4 80.s even 4 1
6400.2.a.cs 4 80.t odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3200, [\chi])$$:

 $$T_{3}^{2} - 5$$ T3^2 - 5 $$T_{7}^{2} + 8$$ T7^2 + 8 $$T_{11}^{2} + 5$$ T11^2 + 5 $$T_{13}^{2} - 40$$ T13^2 - 40 $$T_{31}$$ T31

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} - 5)^{4}$$
$5$ $$T^{8}$$
$7$ $$(T^{2} + 8)^{4}$$
$11$ $$(T^{2} + 5)^{4}$$
$13$ $$(T^{2} - 40)^{4}$$
$17$ $$(T^{2} + 25)^{4}$$
$19$ $$(T^{2} + 5)^{4}$$
$23$ $$(T^{2} + 32)^{4}$$
$29$ $$(T^{2} + 40)^{4}$$
$31$ $$T^{8}$$
$37$ $$(T^{2} - 40)^{4}$$
$41$ $$(T - 3)^{8}$$
$43$ $$(T^{2} - 80)^{4}$$
$47$ $$(T^{2} + 8)^{4}$$
$53$ $$(T^{2} - 160)^{4}$$
$59$ $$(T^{2} + 80)^{4}$$
$61$ $$(T^{2} + 40)^{4}$$
$67$ $$(T^{2} - 125)^{4}$$
$71$ $$(T^{2} - 200)^{4}$$
$73$ $$(T^{2} + 225)^{4}$$
$79$ $$(T^{2} - 200)^{4}$$
$83$ $$(T^{2} - 45)^{4}$$
$89$ $$(T + 1)^{8}$$
$97$ $$(T^{2} + 100)^{4}$$