# Properties

 Label 1028.1.l.a Level $1028$ Weight $1$ Character orbit 1028.l Analytic conductor $0.513$ Analytic rank $0$ Dimension $16$ Projective image $D_{32}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1028 = 2^{2} \cdot 257$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1028.l (of order $$32$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.513038832987$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{32})$$ Defining polynomial: $$x^{16} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{32}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{32} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{32}^{8} q^{2} - q^{4} + ( \zeta_{32}^{2} + \zeta_{32}^{13} ) q^{5} + \zeta_{32}^{8} q^{8} + \zeta_{32}^{9} q^{9} +O(q^{10})$$ $$q -\zeta_{32}^{8} q^{2} - q^{4} + ( \zeta_{32}^{2} + \zeta_{32}^{13} ) q^{5} + \zeta_{32}^{8} q^{8} + \zeta_{32}^{9} q^{9} + ( \zeta_{32}^{5} - \zeta_{32}^{10} ) q^{10} + ( \zeta_{32}^{12} + \zeta_{32}^{14} ) q^{13} + q^{16} + ( \zeta_{32}^{9} + \zeta_{32}^{15} ) q^{17} + \zeta_{32} q^{18} + ( -\zeta_{32}^{2} - \zeta_{32}^{13} ) q^{20} + ( \zeta_{32}^{4} - \zeta_{32}^{10} + \zeta_{32}^{15} ) q^{25} + ( \zeta_{32}^{4} + \zeta_{32}^{6} ) q^{26} + ( -\zeta_{32}^{3} - \zeta_{32}^{11} ) q^{29} -\zeta_{32}^{8} q^{32} + ( \zeta_{32} + \zeta_{32}^{7} ) q^{34} -\zeta_{32}^{9} q^{36} + ( -\zeta_{32}^{4} - \zeta_{32}^{15} ) q^{37} + ( -\zeta_{32}^{5} + \zeta_{32}^{10} ) q^{40} + ( \zeta_{32}^{5} + \zeta_{32}^{6} ) q^{41} + ( -\zeta_{32}^{6} + \zeta_{32}^{11} ) q^{45} -\zeta_{32}^{13} q^{49} + ( -\zeta_{32}^{2} + \zeta_{32}^{7} - \zeta_{32}^{12} ) q^{50} + ( -\zeta_{32}^{12} - \zeta_{32}^{14} ) q^{52} + ( \zeta_{32}^{3} - \zeta_{32}^{14} ) q^{53} + ( -\zeta_{32}^{3} + \zeta_{32}^{11} ) q^{58} + ( \zeta_{32}^{3} - \zeta_{32}^{7} ) q^{61} - q^{64} + ( -1 - \zeta_{32}^{9} - \zeta_{32}^{11} + \zeta_{32}^{14} ) q^{65} + ( -\zeta_{32}^{9} - \zeta_{32}^{15} ) q^{68} -\zeta_{32} q^{72} + ( \zeta_{32}^{13} - \zeta_{32}^{15} ) q^{73} + ( -\zeta_{32}^{7} + \zeta_{32}^{12} ) q^{74} + ( \zeta_{32}^{2} + \zeta_{32}^{13} ) q^{80} -\zeta_{32}^{2} q^{81} + ( -\zeta_{32}^{13} - \zeta_{32}^{14} ) q^{82} + ( -\zeta_{32} - \zeta_{32}^{6} + \zeta_{32}^{11} - \zeta_{32}^{12} ) q^{85} + ( \zeta_{32}^{2} - \zeta_{32}^{4} ) q^{89} + ( \zeta_{32}^{3} + \zeta_{32}^{14} ) q^{90} + ( -\zeta_{32}^{3} + \zeta_{32}^{12} ) q^{97} -\zeta_{32}^{5} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{4} + O(q^{10})$$ $$16q - 16q^{4} + 16q^{16} - 16q^{64} - 16q^{65} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1028\mathbb{Z}\right)^\times$$.

 $$n$$ $$515$$ $$517$$ $$\chi(n)$$ $$-1$$ $$\zeta_{32}^{9}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 −0.980785 + 0.195090i 0.831470 − 0.555570i −0.555570 − 0.831470i 0.555570 + 0.831470i −0.831470 + 0.555570i 0.980785 − 0.195090i −0.195090 − 0.980785i −0.195090 + 0.980785i 0.980785 + 0.195090i −0.831470 − 0.555570i −0.555570 + 0.831470i 0.555570 − 0.831470i 0.831470 + 0.555570i −0.980785 − 0.195090i 0.195090 − 0.980785i 0.195090 + 0.980785i
1.00000i 0 −1.00000 1.75535 + 0.172887i 0 0 1.00000i 0.195090 + 0.980785i −0.172887 + 1.75535i
223.1 1.00000i 0 −1.00000 0.577774 1.90466i 0 0 1.00000i 0.555570 + 0.831470i −1.90466 0.577774i
227.1 1.00000i 0 −1.00000 −1.36347 + 0.728789i 0 0 1.00000i 0.831470 0.555570i 0.728789 + 1.36347i
287.1 1.00000i 0 −1.00000 0.598102 + 1.11897i 0 0 1.00000i −0.831470 + 0.555570i 1.11897 0.598102i
291.1 1.00000i 0 −1.00000 0.187593 + 0.0569057i 0 0 1.00000i −0.555570 0.831470i 0.0569057 0.187593i
499.1 1.00000i 0 −1.00000 0.0924099 0.938254i 0 0 1.00000i −0.195090 0.980785i 0.938254 + 0.0924099i
531.1 1.00000i 0 −1.00000 −1.47945 + 1.21415i 0 0 1.00000i −0.980785 + 0.195090i −1.21415 1.47945i
635.1 1.00000i 0 −1.00000 −1.47945 1.21415i 0 0 1.00000i −0.980785 0.195090i −1.21415 + 1.47945i
651.1 1.00000i 0 −1.00000 0.0924099 + 0.938254i 0 0 1.00000i −0.195090 + 0.980785i 0.938254 0.0924099i
703.1 1.00000i 0 −1.00000 0.187593 0.0569057i 0 0 1.00000i −0.555570 + 0.831470i 0.0569057 + 0.187593i
711.1 1.00000i 0 −1.00000 −1.36347 0.728789i 0 0 1.00000i 0.831470 + 0.555570i 0.728789 1.36347i
831.1 1.00000i 0 −1.00000 0.598102 1.11897i 0 0 1.00000i −0.831470 0.555570i 1.11897 + 0.598102i
839.1 1.00000i 0 −1.00000 0.577774 + 1.90466i 0 0 1.00000i 0.555570 0.831470i −1.90466 + 0.577774i
891.1 1.00000i 0 −1.00000 1.75535 0.172887i 0 0 1.00000i 0.195090 0.980785i −0.172887 1.75535i
907.1 1.00000i 0 −1.00000 −0.368309 + 0.448786i 0 0 1.00000i 0.980785 + 0.195090i 0.448786 + 0.368309i
1011.1 1.00000i 0 −1.00000 −0.368309 0.448786i 0 0 1.00000i 0.980785 0.195090i 0.448786 0.368309i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1011.1 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 $$\Q(\sqrt{-1})$$
257.f even 32 1 inner
1028.l odd 32 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1028.1.l.a 16
4.b odd 2 1 CM 1028.1.l.a 16
257.f even 32 1 inner 1028.1.l.a 16
1028.l odd 32 1 inner 1028.1.l.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1028.1.l.a 16 1.a even 1 1 trivial
1028.1.l.a 16 4.b odd 2 1 CM
1028.1.l.a 16 257.f even 32 1 inner
1028.1.l.a 16 1028.l odd 32 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1028, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{8}$$
$3$ $$T^{16}$$
$5$ $$2 - 16 T + 24 T^{2} + 32 T^{3} + 148 T^{4} + 176 T^{6} + 2 T^{8} + 16 T^{9} - 32 T^{11} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$( 2 - 8 T + 20 T^{2} - 16 T^{3} + 2 T^{4} + T^{8} )^{2}$$
$17$ $$4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$256 + T^{16}$$
$31$ $$T^{16}$$
$37$ $$2 + 16 T + 72 T^{2} + 80 T^{3} + 4 T^{4} + 56 T^{6} - 160 T^{7} + 6 T^{8} + 16 T^{11} + 4 T^{12} + T^{16}$$
$41$ $$2 + 16 T + 40 T^{2} + 140 T^{4} - 48 T^{5} + 192 T^{7} + 2 T^{8} + 88 T^{10} + 16 T^{13} + T^{16}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$2 - 16 T + 24 T^{2} + 32 T^{3} + 148 T^{4} + 176 T^{6} + 2 T^{8} + 16 T^{9} - 32 T^{11} + T^{16}$$
$59$ $$T^{16}$$
$61$ $$16 + 64 T^{4} + 128 T^{8} - 16 T^{12} + T^{16}$$
$67$ $$T^{16}$$
$71$ $$T^{16}$$
$73$ $$4 + 32 T^{2} + 128 T^{4} - 192 T^{6} + 140 T^{8} - 16 T^{10} + T^{16}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$( 2 - 8 T + 20 T^{2} - 16 T^{3} + 2 T^{4} + T^{8} )^{2}$$
$97$ $$2 + 16 T + 56 T^{2} + 16 T^{3} + 4 T^{4} - 160 T^{5} + 72 T^{6} + 6 T^{8} + 80 T^{9} + 4 T^{12} + T^{16}$$