# Properties

 Label 3200.1.bi.b Level $3200$ Weight $1$ Character orbit 3200.bi Analytic conductor $1.597$ Analytic rank $0$ Dimension $4$ Projective image $D_{10}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3200.bi (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.59700804043$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{10}$$ Projective field: Galois closure of 10.2.400000000000000000.16

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{10}^{3} q^{5} + \zeta_{10}^{4} q^{9} +O(q^{10})$$ $$q + \zeta_{10}^{3} q^{5} + \zeta_{10}^{4} q^{9} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{13} + ( \zeta_{10}^{2} - \zeta_{10}^{4} ) q^{17} -\zeta_{10} q^{25} + ( -\zeta_{10} + \zeta_{10}^{3} ) q^{29} + ( -1 + \zeta_{10}^{3} ) q^{37} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{41} -\zeta_{10}^{2} q^{45} - q^{49} + ( 1 + \zeta_{10}^{4} ) q^{53} + ( \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{61} + ( -1 - \zeta_{10}^{4} ) q^{65} + ( \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{73} -\zeta_{10}^{3} q^{81} + ( -1 + \zeta_{10}^{2} ) q^{85} + ( -1 - \zeta_{10}^{2} ) q^{89} + ( \zeta_{10} - \zeta_{10}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{5} - q^{9} + O(q^{10})$$ $$4 q + q^{5} - q^{9} - 2 q^{13} - q^{25} - 3 q^{37} + 2 q^{41} + q^{45} - 4 q^{49} + 3 q^{53} - 3 q^{65} - q^{81} - 5 q^{85} - 3 q^{89} + O(q^{100})$$

## Character values

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

 $$n$$ $$901$$ $$1151$$ $$2177$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{10}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 −0.309017 − 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 + 0.951057i
0 0 0 0.809017 + 0.587785i 0 0 0 0.309017 0.951057i 0
959.1 0 0 0 −0.309017 + 0.951057i 0 0 0 −0.809017 + 0.587785i 0
2239.1 0 0 0 −0.309017 0.951057i 0 0 0 −0.809017 0.587785i 0
2879.1 0 0 0 0.809017 0.587785i 0 0 0 0.309017 + 0.951057i 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 CM by $$\Q(\sqrt{-1})$$
200.o even 10 1 inner
200.s odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.1.bi.b yes 4
4.b odd 2 1 CM 3200.1.bi.b yes 4
8.b even 2 1 3200.1.bi.a 4
8.d odd 2 1 3200.1.bi.a 4
25.e even 10 1 3200.1.bi.a 4
100.h odd 10 1 3200.1.bi.a 4
200.o even 10 1 inner 3200.1.bi.b yes 4
200.s odd 10 1 inner 3200.1.bi.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.1.bi.a 4 8.b even 2 1
3200.1.bi.a 4 8.d odd 2 1
3200.1.bi.a 4 25.e even 10 1
3200.1.bi.a 4 100.h odd 10 1
3200.1.bi.b yes 4 1.a even 1 1 trivial
3200.1.bi.b yes 4 4.b odd 2 1 CM
3200.1.bi.b yes 4 200.o even 10 1 inner
3200.1.bi.b yes 4 200.s odd 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{4} + 2 T_{13}^{3} + 4 T_{13}^{2} + 3 T_{13} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$17$ $$5 - 5 T + T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$5 + 5 T + T^{4}$$
$31$ $$T^{4}$$
$37$ $$1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4}$$
$41$ $$1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$5 - 5 T + T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$5 - 5 T + T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4}$$
$97$ $$5 - 5 T + T^{4}$$