# Properties

 Label 3200.2.f.j Level $3200$ Weight $2$ Character orbit 3200.f Analytic conductor $25.552$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 640) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} + 3 \beta_{2} q^{7} - q^{9}+O(q^{10})$$ q - b3 * q^3 + 3*b2 * q^7 - q^9 $$q - \beta_{3} q^{3} + 3 \beta_{2} q^{7} - q^{9} - 4 \beta_{2} q^{11} + 2 q^{13} - 3 \beta_1 q^{17} + 2 \beta_{2} q^{19} - 3 \beta_1 q^{21} + 5 \beta_{2} q^{23} + 4 \beta_{3} q^{27} + 2 \beta_1 q^{29} - 2 \beta_{3} q^{31} + 4 \beta_1 q^{33} - 2 q^{37} - 2 \beta_{3} q^{39} - 8 q^{41} - \beta_{3} q^{43} + \beta_{2} q^{47} - 11 q^{49} + 6 \beta_{2} q^{51} - 2 q^{53} - 2 \beta_1 q^{57} + 2 \beta_{2} q^{59} - 7 \beta_1 q^{61} - 3 \beta_{2} q^{63} - 3 \beta_{3} q^{67} - 5 \beta_1 q^{69} + 2 \beta_{3} q^{71} - 3 \beta_1 q^{73} + 24 q^{77} - 12 \beta_{3} q^{79} - 5 q^{81} + 9 \beta_{3} q^{83} - 4 \beta_{2} q^{87} + 6 q^{89} + 6 \beta_{2} q^{91} + 4 q^{93} - 5 \beta_1 q^{97} + 4 \beta_{2} q^{99}+O(q^{100})$$ q - b3 * q^3 + 3*b2 * q^7 - q^9 - 4*b2 * q^11 + 2 * q^13 - 3*b1 * q^17 + 2*b2 * q^19 - 3*b1 * q^21 + 5*b2 * q^23 + 4*b3 * q^27 + 2*b1 * q^29 - 2*b3 * q^31 + 4*b1 * q^33 - 2 * q^37 - 2*b3 * q^39 - 8 * q^41 - b3 * q^43 + b2 * q^47 - 11 * q^49 + 6*b2 * q^51 - 2 * q^53 - 2*b1 * q^57 + 2*b2 * q^59 - 7*b1 * q^61 - 3*b2 * q^63 - 3*b3 * q^67 - 5*b1 * q^69 + 2*b3 * q^71 - 3*b1 * q^73 + 24 * q^77 - 12*b3 * q^79 - 5 * q^81 + 9*b3 * q^83 - 4*b2 * q^87 + 6 * q^89 + 6*b2 * q^91 + 4 * q^93 - 5*b1 * q^97 + 4*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} + 8 q^{13} - 8 q^{37} - 32 q^{41} - 44 q^{49} - 8 q^{53} + 96 q^{77} - 20 q^{81} + 24 q^{89} + 16 q^{93}+O(q^{100})$$ 4 * q - 4 * q^9 + 8 * q^13 - 8 * q^37 - 32 * q^41 - 44 * q^49 - 8 * q^53 + 96 * q^77 - 20 * q^81 + 24 * q^89 + 16 * q^93

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}^{2}$$ 2*v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## 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
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
0 −1.41421 0 0 0 4.24264i 0 −1.00000 0
449.2 0 −1.41421 0 0 0 4.24264i 0 −1.00000 0
449.3 0 1.41421 0 0 0 4.24264i 0 −1.00000 0
449.4 0 1.41421 0 0 0 4.24264i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 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
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.j 4
4.b odd 2 1 inner 3200.2.f.j 4
5.b even 2 1 3200.2.f.i 4
5.c odd 4 1 640.2.d.d 4
5.c odd 4 1 3200.2.d.s 4
8.b even 2 1 3200.2.f.i 4
8.d odd 2 1 3200.2.f.i 4
15.e even 4 1 5760.2.k.o 4
20.d odd 2 1 3200.2.f.i 4
20.e even 4 1 640.2.d.d 4
20.e even 4 1 3200.2.d.s 4
40.e odd 2 1 inner 3200.2.f.j 4
40.f even 2 1 inner 3200.2.f.j 4
40.i odd 4 1 640.2.d.d 4
40.i odd 4 1 3200.2.d.s 4
40.k even 4 1 640.2.d.d 4
40.k even 4 1 3200.2.d.s 4
60.l odd 4 1 5760.2.k.o 4
80.i odd 4 1 1280.2.a.j 2
80.i odd 4 1 6400.2.a.bl 2
80.j even 4 1 1280.2.a.f 2
80.j even 4 1 6400.2.a.bn 2
80.s even 4 1 1280.2.a.j 2
80.s even 4 1 6400.2.a.bl 2
80.t odd 4 1 1280.2.a.f 2
80.t odd 4 1 6400.2.a.bn 2
120.q odd 4 1 5760.2.k.o 4
120.w even 4 1 5760.2.k.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.d.d 4 5.c odd 4 1
640.2.d.d 4 20.e even 4 1
640.2.d.d 4 40.i odd 4 1
640.2.d.d 4 40.k even 4 1
1280.2.a.f 2 80.j even 4 1
1280.2.a.f 2 80.t odd 4 1
1280.2.a.j 2 80.i odd 4 1
1280.2.a.j 2 80.s even 4 1
3200.2.d.s 4 5.c odd 4 1
3200.2.d.s 4 20.e even 4 1
3200.2.d.s 4 40.i odd 4 1
3200.2.d.s 4 40.k even 4 1
3200.2.f.i 4 5.b even 2 1
3200.2.f.i 4 8.b even 2 1
3200.2.f.i 4 8.d odd 2 1
3200.2.f.i 4 20.d odd 2 1
3200.2.f.j 4 1.a even 1 1 trivial
3200.2.f.j 4 4.b odd 2 1 inner
3200.2.f.j 4 40.e odd 2 1 inner
3200.2.f.j 4 40.f even 2 1 inner
5760.2.k.o 4 15.e even 4 1
5760.2.k.o 4 60.l odd 4 1
5760.2.k.o 4 120.q odd 4 1
5760.2.k.o 4 120.w even 4 1
6400.2.a.bl 2 80.i odd 4 1
6400.2.a.bl 2 80.s even 4 1
6400.2.a.bn 2 80.j even 4 1
6400.2.a.bn 2 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 $$S_{2}^{\mathrm{new}}(3200, [\chi])$$:

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{7}^{2} + 18$$ T7^2 + 18 $$T_{11}^{2} + 32$$ T11^2 + 32 $$T_{13} - 2$$ T13 - 2 $$T_{31}^{2} - 8$$ T31^2 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 2)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 18)^{2}$$
$11$ $$(T^{2} + 32)^{2}$$
$13$ $$(T - 2)^{4}$$
$17$ $$(T^{2} + 36)^{2}$$
$19$ $$(T^{2} + 8)^{2}$$
$23$ $$(T^{2} + 50)^{2}$$
$29$ $$(T^{2} + 16)^{2}$$
$31$ $$(T^{2} - 8)^{2}$$
$37$ $$(T + 2)^{4}$$
$41$ $$(T + 8)^{4}$$
$43$ $$(T^{2} - 2)^{2}$$
$47$ $$(T^{2} + 2)^{2}$$
$53$ $$(T + 2)^{4}$$
$59$ $$(T^{2} + 8)^{2}$$
$61$ $$(T^{2} + 196)^{2}$$
$67$ $$(T^{2} - 18)^{2}$$
$71$ $$(T^{2} - 8)^{2}$$
$73$ $$(T^{2} + 36)^{2}$$
$79$ $$(T^{2} - 288)^{2}$$
$83$ $$(T^{2} - 162)^{2}$$
$89$ $$(T - 6)^{4}$$
$97$ $$(T^{2} + 100)^{2}$$