# Properties

 Label 1020.1.cl.b Level $1020$ Weight $1$ Character orbit 1020.cl Analytic conductor $0.509$ Analytic rank $0$ Dimension $16$ Projective image $D_{16}$ CM discriminant -15 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1020.cl (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.509046312886$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{16})$$ Coefficient field: $$\Q(\zeta_{32})$$ Defining polynomial: $$x^{16} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{16}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{32}^{9} q^{2} -\zeta_{32}^{15} q^{3} -\zeta_{32}^{2} q^{4} + \zeta_{32}^{3} q^{5} -\zeta_{32}^{8} q^{6} + \zeta_{32}^{11} q^{8} -\zeta_{32}^{14} q^{9} +O(q^{10})$$ $$q -\zeta_{32}^{9} q^{2} -\zeta_{32}^{15} q^{3} -\zeta_{32}^{2} q^{4} + \zeta_{32}^{3} q^{5} -\zeta_{32}^{8} q^{6} + \zeta_{32}^{11} q^{8} -\zeta_{32}^{14} q^{9} -\zeta_{32}^{12} q^{10} -\zeta_{32} q^{12} + \zeta_{32}^{2} q^{15} + \zeta_{32}^{4} q^{16} -\zeta_{32}^{7} q^{17} -\zeta_{32}^{7} q^{18} + ( \zeta_{32}^{8} + \zeta_{32}^{12} ) q^{19} -\zeta_{32}^{5} q^{20} + ( -\zeta_{32}^{5} + \zeta_{32}^{13} ) q^{23} + \zeta_{32}^{10} q^{24} + \zeta_{32}^{6} q^{25} -\zeta_{32}^{13} q^{27} -\zeta_{32}^{11} q^{30} + ( -\zeta_{32}^{4} - \zeta_{32}^{10} ) q^{31} -\zeta_{32}^{13} q^{32} - q^{34} - q^{36} + ( \zeta_{32} + \zeta_{32}^{5} ) q^{38} + \zeta_{32}^{14} q^{40} + \zeta_{32} q^{45} + ( \zeta_{32}^{6} + \zeta_{32}^{14} ) q^{46} + ( -\zeta_{32}^{9} + \zeta_{32}^{15} ) q^{47} + \zeta_{32}^{3} q^{48} + \zeta_{32}^{10} q^{49} -\zeta_{32}^{15} q^{50} -\zeta_{32}^{6} q^{51} + ( -\zeta_{32} - \zeta_{32}^{3} ) q^{53} -\zeta_{32}^{6} q^{54} + ( \zeta_{32}^{7} + \zeta_{32}^{11} ) q^{57} -\zeta_{32}^{4} q^{60} + ( \zeta_{32}^{2} + \zeta_{32}^{8} ) q^{61} + ( -\zeta_{32}^{3} + \zeta_{32}^{13} ) q^{62} -\zeta_{32}^{6} q^{64} + \zeta_{32}^{9} q^{68} + ( -\zeta_{32}^{4} + \zeta_{32}^{12} ) q^{69} + \zeta_{32}^{9} q^{72} + \zeta_{32}^{5} q^{75} + ( -\zeta_{32}^{10} - \zeta_{32}^{14} ) q^{76} + ( \zeta_{32}^{4} + \zeta_{32}^{14} ) q^{79} + \zeta_{32}^{7} q^{80} -\zeta_{32}^{12} q^{81} + ( \zeta_{32}^{9} - \zeta_{32}^{11} ) q^{83} -\zeta_{32}^{10} q^{85} -\zeta_{32}^{10} q^{90} + ( \zeta_{32}^{7} - \zeta_{32}^{15} ) q^{92} + ( -\zeta_{32}^{3} - \zeta_{32}^{9} ) q^{93} + ( -\zeta_{32}^{2} + \zeta_{32}^{8} ) q^{94} + ( \zeta_{32}^{11} + \zeta_{32}^{15} ) q^{95} -\zeta_{32}^{12} q^{96} + \zeta_{32}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 16q^{34} - 16q^{36} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1020\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$341$$ $$511$$ $$817$$ $$\chi(n)$$ $$\zeta_{32}^{14}$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
299.1
 −0.555570 − 0.831470i 0.555570 + 0.831470i −0.980785 − 0.195090i 0.980785 + 0.195090i 0.195090 + 0.980785i −0.195090 − 0.980785i −0.555570 + 0.831470i 0.555570 − 0.831470i 0.831470 + 0.555570i −0.831470 − 0.555570i −0.980785 + 0.195090i 0.980785 − 0.195090i 0.195090 − 0.980785i −0.195090 + 0.980785i 0.831470 − 0.555570i −0.831470 + 0.555570i
−0.831470 + 0.555570i −0.555570 + 0.831470i 0.382683 0.923880i 0.980785 0.195090i 1.00000i 0 0.195090 + 0.980785i −0.382683 0.923880i −0.707107 + 0.707107i
299.2 0.831470 0.555570i 0.555570 0.831470i 0.382683 0.923880i −0.980785 + 0.195090i 1.00000i 0 −0.195090 0.980785i −0.382683 0.923880i −0.707107 + 0.707107i
419.1 −0.195090 + 0.980785i −0.980785 + 0.195090i −0.923880 0.382683i −0.831470 0.555570i 1.00000i 0 0.555570 0.831470i 0.923880 0.382683i 0.707107 0.707107i
419.2 0.195090 0.980785i 0.980785 0.195090i −0.923880 0.382683i 0.831470 + 0.555570i 1.00000i 0 −0.555570 + 0.831470i 0.923880 0.382683i 0.707107 0.707107i
479.1 −0.980785 + 0.195090i 0.195090 0.980785i 0.923880 0.382683i −0.555570 0.831470i 1.00000i 0 −0.831470 + 0.555570i −0.923880 0.382683i 0.707107 + 0.707107i
479.2 0.980785 0.195090i −0.195090 + 0.980785i 0.923880 0.382683i 0.555570 + 0.831470i 1.00000i 0 0.831470 0.555570i −0.923880 0.382683i 0.707107 + 0.707107i
539.1 −0.831470 0.555570i −0.555570 0.831470i 0.382683 + 0.923880i 0.980785 + 0.195090i 1.00000i 0 0.195090 0.980785i −0.382683 + 0.923880i −0.707107 0.707107i
539.2 0.831470 + 0.555570i 0.555570 + 0.831470i 0.382683 + 0.923880i −0.980785 0.195090i 1.00000i 0 −0.195090 + 0.980785i −0.382683 + 0.923880i −0.707107 0.707107i
719.1 −0.555570 + 0.831470i 0.831470 0.555570i −0.382683 0.923880i −0.195090 + 0.980785i 1.00000i 0 0.980785 + 0.195090i 0.382683 0.923880i −0.707107 0.707107i
719.2 0.555570 0.831470i −0.831470 + 0.555570i −0.382683 0.923880i 0.195090 0.980785i 1.00000i 0 −0.980785 0.195090i 0.382683 0.923880i −0.707107 0.707107i
779.1 −0.195090 0.980785i −0.980785 0.195090i −0.923880 + 0.382683i −0.831470 + 0.555570i 1.00000i 0 0.555570 + 0.831470i 0.923880 + 0.382683i 0.707107 + 0.707107i
779.2 0.195090 + 0.980785i 0.980785 + 0.195090i −0.923880 + 0.382683i 0.831470 0.555570i 1.00000i 0 −0.555570 0.831470i 0.923880 + 0.382683i 0.707107 + 0.707107i
839.1 −0.980785 0.195090i 0.195090 + 0.980785i 0.923880 + 0.382683i −0.555570 + 0.831470i 1.00000i 0 −0.831470 0.555570i −0.923880 + 0.382683i 0.707107 0.707107i
839.2 0.980785 + 0.195090i −0.195090 0.980785i 0.923880 + 0.382683i 0.555570 0.831470i 1.00000i 0 0.831470 + 0.555570i −0.923880 + 0.382683i 0.707107 0.707107i
959.1 −0.555570 0.831470i 0.831470 + 0.555570i −0.382683 + 0.923880i −0.195090 0.980785i 1.00000i 0 0.980785 0.195090i 0.382683 + 0.923880i −0.707107 + 0.707107i
959.2 0.555570 + 0.831470i −0.831470 0.555570i −0.382683 + 0.923880i 0.195090 + 0.980785i 1.00000i 0 −0.980785 + 0.195090i 0.382683 + 0.923880i −0.707107 + 0.707107i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 959.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
68.i even 16 1 inner
204.t odd 16 1 inner
340.bg even 16 1 inner
1020.cl odd 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1020.1.cl.b yes 16
3.b odd 2 1 inner 1020.1.cl.b yes 16
4.b odd 2 1 1020.1.cl.a 16
5.b even 2 1 inner 1020.1.cl.b yes 16
12.b even 2 1 1020.1.cl.a 16
15.d odd 2 1 CM 1020.1.cl.b yes 16
17.e odd 16 1 1020.1.cl.a 16
20.d odd 2 1 1020.1.cl.a 16
51.i even 16 1 1020.1.cl.a 16
60.h even 2 1 1020.1.cl.a 16
68.i even 16 1 inner 1020.1.cl.b yes 16
85.p odd 16 1 1020.1.cl.a 16
204.t odd 16 1 inner 1020.1.cl.b yes 16
255.be even 16 1 1020.1.cl.a 16
340.bg even 16 1 inner 1020.1.cl.b yes 16
1020.cl odd 16 1 inner 1020.1.cl.b yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1020.1.cl.a 16 4.b odd 2 1
1020.1.cl.a 16 12.b even 2 1
1020.1.cl.a 16 17.e odd 16 1
1020.1.cl.a 16 20.d odd 2 1
1020.1.cl.a 16 51.i even 16 1
1020.1.cl.a 16 60.h even 2 1
1020.1.cl.a 16 85.p odd 16 1
1020.1.cl.a 16 255.be even 16 1
1020.1.cl.b yes 16 1.a even 1 1 trivial
1020.1.cl.b yes 16 3.b odd 2 1 inner
1020.1.cl.b yes 16 5.b even 2 1 inner
1020.1.cl.b yes 16 15.d odd 2 1 CM
1020.1.cl.b yes 16 68.i even 16 1 inner
1020.1.cl.b yes 16 204.t odd 16 1 inner
1020.1.cl.b yes 16 340.bg even 16 1 inner
1020.1.cl.b yes 16 1020.cl odd 16 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{19}^{4} + 2 T_{19}^{2} - 4 T_{19} + 2$$ acting on $$S_{1}^{\mathrm{new}}(1020, [\chi])$$.

## Hecke characteristic polynomials

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