# Properties

 Label 3200.2.d.o Level $3200$ Weight $2$ Character orbit 3200.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -2 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -2 q^{9} + \beta_{1} q^{11} -\beta_{2} q^{13} -3 q^{17} -\beta_{1} q^{19} + \beta_{3} q^{23} -\beta_{1} q^{27} + \beta_{2} q^{29} -\beta_{3} q^{31} + 5 q^{33} + 2 \beta_{2} q^{37} + \beta_{3} q^{39} -5 q^{41} -4 \beta_{1} q^{43} + \beta_{3} q^{47} -7 q^{49} + 3 \beta_{1} q^{51} + \beta_{2} q^{53} -5 q^{57} -4 \beta_{1} q^{59} + 2 \beta_{2} q^{61} -3 \beta_{1} q^{67} + 5 \beta_{2} q^{69} + \beta_{3} q^{71} + 9 q^{73} -11 q^{81} -3 \beta_{1} q^{83} -\beta_{3} q^{87} -15 q^{89} -5 \beta_{2} q^{93} -2 q^{97} -2 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{9} + O(q^{10})$$ $$4q - 8q^{9} - 12q^{17} + 20q^{33} - 20q^{41} - 28q^{49} - 20q^{57} + 36q^{73} - 44q^{81} - 60q^{89} - 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 4 \nu$$ $$\beta_{2}$$ $$=$$ $$-4 \nu^{3} - 8 \nu$$ $$\beta_{3}$$ $$=$$ $$8 \nu^{2} + 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 4 \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 12$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} - 2 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1601.1
 1.61803i 0.618034i − 0.618034i − 1.61803i
0 2.23607i 0 0 0 0 0 −2.00000 0
1601.2 0 2.23607i 0 0 0 0 0 −2.00000 0
1601.3 0 2.23607i 0 0 0 0 0 −2.00000 0
1601.4 0 2.23607i 0 0 0 0 0 −2.00000 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
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.d.o 4
4.b odd 2 1 inner 3200.2.d.o 4
5.b even 2 1 3200.2.d.p yes 4
5.c odd 4 1 3200.2.f.m 4
5.c odd 4 1 3200.2.f.n 4
8.b even 2 1 inner 3200.2.d.o 4
8.d odd 2 1 inner 3200.2.d.o 4
16.e even 4 1 6400.2.a.bq 2
16.e even 4 1 6400.2.a.bs 2
16.f odd 4 1 6400.2.a.bq 2
16.f odd 4 1 6400.2.a.bs 2
20.d odd 2 1 3200.2.d.p yes 4
20.e even 4 1 3200.2.f.m 4
20.e even 4 1 3200.2.f.n 4
40.e odd 2 1 3200.2.d.p yes 4
40.f even 2 1 3200.2.d.p yes 4
40.i odd 4 1 3200.2.f.m 4
40.i odd 4 1 3200.2.f.n 4
40.k even 4 1 3200.2.f.m 4
40.k even 4 1 3200.2.f.n 4
80.k odd 4 1 6400.2.a.br 2
80.k odd 4 1 6400.2.a.bt 2
80.q even 4 1 6400.2.a.br 2
80.q even 4 1 6400.2.a.bt 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.o 4 1.a even 1 1 trivial
3200.2.d.o 4 4.b odd 2 1 inner
3200.2.d.o 4 8.b even 2 1 inner
3200.2.d.o 4 8.d odd 2 1 inner
3200.2.d.p yes 4 5.b even 2 1
3200.2.d.p yes 4 20.d odd 2 1
3200.2.d.p yes 4 40.e odd 2 1
3200.2.d.p yes 4 40.f even 2 1
3200.2.f.m 4 5.c odd 4 1
3200.2.f.m 4 20.e even 4 1
3200.2.f.m 4 40.i odd 4 1
3200.2.f.m 4 40.k even 4 1
3200.2.f.n 4 5.c odd 4 1
3200.2.f.n 4 20.e even 4 1
3200.2.f.n 4 40.i odd 4 1
3200.2.f.n 4 40.k even 4 1
6400.2.a.bq 2 16.e even 4 1
6400.2.a.bq 2 16.f odd 4 1
6400.2.a.br 2 80.k odd 4 1
6400.2.a.br 2 80.q even 4 1
6400.2.a.bs 2 16.e even 4 1
6400.2.a.bs 2 16.f odd 4 1
6400.2.a.bt 2 80.k odd 4 1
6400.2.a.bt 2 80.q even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 5$$ $$T_{7}$$ $$T_{11}^{2} + 5$$ $$T_{13}^{2} + 16$$ $$T_{17} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 5 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 5 + T^{2} )^{2}$$
$13$ $$( 16 + T^{2} )^{2}$$
$17$ $$( 3 + T )^{4}$$
$19$ $$( 5 + T^{2} )^{2}$$
$23$ $$( -80 + T^{2} )^{2}$$
$29$ $$( 16 + T^{2} )^{2}$$
$31$ $$( -80 + T^{2} )^{2}$$
$37$ $$( 64 + T^{2} )^{2}$$
$41$ $$( 5 + T )^{4}$$
$43$ $$( 80 + T^{2} )^{2}$$
$47$ $$( -80 + T^{2} )^{2}$$
$53$ $$( 16 + T^{2} )^{2}$$
$59$ $$( 80 + T^{2} )^{2}$$
$61$ $$( 64 + T^{2} )^{2}$$
$67$ $$( 45 + T^{2} )^{2}$$
$71$ $$( -80 + T^{2} )^{2}$$
$73$ $$( -9 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$( 45 + T^{2} )^{2}$$
$89$ $$( 15 + T )^{4}$$
$97$ $$( 2 + T )^{4}$$