# Properties

 Label 820.1.by.a Level $820$ Weight $1$ Character orbit 820.by Analytic conductor $0.409$ Analytic rank $0$ Dimension $16$ Projective image $D_{40}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$820 = 2^{2} \cdot 5 \cdot 41$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 820.by (of order $$40$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.409233310359$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{40})$$ Defining polynomial: $$x^{16} - x^{12} + x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{40}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{40} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{40}^{6} q^{2} + \zeta_{40}^{12} q^{4} + \zeta_{40} q^{5} -\zeta_{40}^{18} q^{8} -\zeta_{40}^{15} q^{9} +O(q^{10})$$ $$q -\zeta_{40}^{6} q^{2} + \zeta_{40}^{12} q^{4} + \zeta_{40} q^{5} -\zeta_{40}^{18} q^{8} -\zeta_{40}^{15} q^{9} -\zeta_{40}^{7} q^{10} + ( \zeta_{40}^{8} + \zeta_{40}^{19} ) q^{13} -\zeta_{40}^{4} q^{16} + ( \zeta_{40}^{4} + \zeta_{40}^{17} ) q^{17} -\zeta_{40} q^{18} + \zeta_{40}^{13} q^{20} + \zeta_{40}^{2} q^{25} + ( \zeta_{40}^{5} - \zeta_{40}^{14} ) q^{26} + ( -\zeta_{40}^{12} - \zeta_{40}^{17} ) q^{29} + \zeta_{40}^{10} q^{32} + ( \zeta_{40}^{3} - \zeta_{40}^{10} ) q^{34} + \zeta_{40}^{7} q^{36} + ( -\zeta_{40}^{5} - \zeta_{40}^{9} ) q^{37} -\zeta_{40}^{19} q^{40} + \zeta_{40}^{6} q^{41} -\zeta_{40}^{16} q^{45} -\zeta_{40}^{3} q^{49} -\zeta_{40}^{8} q^{50} + ( -1 - \zeta_{40}^{11} ) q^{52} + ( -\zeta_{40}^{8} + \zeta_{40}^{11} ) q^{53} + ( -\zeta_{40}^{3} + \zeta_{40}^{18} ) q^{58} + ( -\zeta_{40}^{2} + \zeta_{40}^{16} ) q^{61} -\zeta_{40}^{16} q^{64} + ( -1 + \zeta_{40}^{9} ) q^{65} + ( -\zeta_{40}^{9} + \zeta_{40}^{16} ) q^{68} -\zeta_{40}^{13} q^{72} + ( \zeta_{40}^{7} - \zeta_{40}^{13} ) q^{73} + ( \zeta_{40}^{11} + \zeta_{40}^{15} ) q^{74} -\zeta_{40}^{5} q^{80} -\zeta_{40}^{10} q^{81} -\zeta_{40}^{12} q^{82} + ( \zeta_{40}^{5} + \zeta_{40}^{18} ) q^{85} + ( \zeta_{40}^{14} - \zeta_{40}^{19} ) q^{89} -\zeta_{40}^{2} q^{90} + ( -\zeta_{40}^{14} + \zeta_{40}^{15} ) q^{97} + \zeta_{40}^{9} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{4} + O(q^{10})$$ $$16q + 4q^{4} - 4q^{13} - 4q^{16} + 4q^{17} - 4q^{29} + 4q^{45} + 4q^{50} - 16q^{52} + 4q^{53} - 4q^{61} + 4q^{64} - 16q^{65} - 4q^{68} - 4q^{82} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/820\mathbb{Z}\right)^\times$$.

 $$n$$ $$411$$ $$621$$ $$657$$ $$\chi(n)$$ $$-1$$ $$\zeta_{40}^{7}$$ $$-\zeta_{40}^{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −0.891007 − 0.453990i 0.156434 − 0.987688i −0.987688 + 0.156434i −0.453990 − 0.891007i 0.453990 + 0.891007i 0.891007 − 0.453990i 0.453990 − 0.891007i 0.987688 − 0.156434i −0.987688 − 0.156434i −0.156434 − 0.987688i −0.156434 + 0.987688i 0.891007 + 0.453990i 0.156434 + 0.987688i 0.987688 + 0.156434i −0.453990 + 0.891007i −0.891007 + 0.453990i
0.951057 0.309017i 0 0.809017 0.587785i −0.891007 0.453990i 0 0 0.587785 0.809017i 0.707107 + 0.707107i −0.987688 0.156434i
67.1 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 0.156434 0.987688i 0 0 −0.951057 + 0.309017i 0.707107 + 0.707107i 0.891007 0.453990i
147.1 −0.587785 + 0.809017i 0 −0.309017 0.951057i −0.987688 + 0.156434i 0 0 0.951057 + 0.309017i −0.707107 0.707107i 0.453990 0.891007i
227.1 −0.951057 0.309017i 0 0.809017 + 0.587785i −0.453990 0.891007i 0 0 −0.587785 0.809017i −0.707107 0.707107i 0.156434 + 0.987688i
347.1 −0.951057 0.309017i 0 0.809017 + 0.587785i 0.453990 + 0.891007i 0 0 −0.587785 0.809017i 0.707107 + 0.707107i −0.156434 0.987688i
403.1 0.951057 + 0.309017i 0 0.809017 + 0.587785i 0.891007 0.453990i 0 0 0.587785 + 0.809017i −0.707107 + 0.707107i 0.987688 0.156434i
423.1 −0.951057 + 0.309017i 0 0.809017 0.587785i 0.453990 0.891007i 0 0 −0.587785 + 0.809017i 0.707107 0.707107i −0.156434 + 0.987688i
427.1 −0.587785 + 0.809017i 0 −0.309017 0.951057i 0.987688 0.156434i 0 0 0.951057 + 0.309017i 0.707107 + 0.707107i −0.453990 + 0.891007i
463.1 −0.587785 0.809017i 0 −0.309017 + 0.951057i −0.987688 0.156434i 0 0 0.951057 0.309017i −0.707107 + 0.707107i 0.453990 + 0.891007i
503.1 0.587785 0.809017i 0 −0.309017 0.951057i −0.156434 0.987688i 0 0 −0.951057 0.309017i −0.707107 + 0.707107i −0.891007 0.453990i
507.1 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −0.156434 + 0.987688i 0 0 −0.951057 + 0.309017i −0.707107 0.707107i −0.891007 + 0.453990i
527.1 0.951057 0.309017i 0 0.809017 0.587785i 0.891007 + 0.453990i 0 0 0.587785 0.809017i −0.707107 0.707107i 0.987688 + 0.156434i
563.1 0.587785 0.809017i 0 −0.309017 0.951057i 0.156434 + 0.987688i 0 0 −0.951057 0.309017i 0.707107 0.707107i 0.891007 + 0.453990i
603.1 −0.587785 0.809017i 0 −0.309017 + 0.951057i 0.987688 + 0.156434i 0 0 0.951057 0.309017i 0.707107 0.707107i −0.453990 0.891007i
643.1 −0.951057 + 0.309017i 0 0.809017 0.587785i −0.453990 + 0.891007i 0 0 −0.587785 + 0.809017i −0.707107 + 0.707107i 0.156434 0.987688i
663.1 0.951057 + 0.309017i 0 0.809017 + 0.587785i −0.891007 + 0.453990i 0 0 0.587785 + 0.809017i 0.707107 0.707107i −0.987688 + 0.156434i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 663.1 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 $$\Q(\sqrt{-1})$$
205.bb even 40 1 inner
820.by odd 40 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 820.1.by.a 16
4.b odd 2 1 CM 820.1.by.a 16
5.c odd 4 1 820.1.bz.a yes 16
20.e even 4 1 820.1.bz.a yes 16
41.h odd 40 1 820.1.bz.a yes 16
164.o even 40 1 820.1.bz.a yes 16
205.bb even 40 1 inner 820.1.by.a 16
820.by odd 40 1 inner 820.1.by.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.1.by.a 16 1.a even 1 1 trivial
820.1.by.a 16 4.b odd 2 1 CM
820.1.by.a 16 205.bb even 40 1 inner
820.1.by.a 16 820.by odd 40 1 inner
820.1.bz.a yes 16 5.c odd 4 1
820.1.bz.a yes 16 20.e even 4 1
820.1.bz.a yes 16 41.h odd 40 1
820.1.bz.a yes 16 164.o even 40 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(820, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$1 - 8 T - 4 T^{2} + 68 T^{3} + 269 T^{4} + 328 T^{5} + 382 T^{6} + 284 T^{7} + 156 T^{8} + 60 T^{9} + 32 T^{10} + 44 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16}$$
$17$ $$1 + 8 T - 4 T^{2} - 68 T^{3} + 269 T^{4} - 328 T^{5} + 382 T^{6} - 284 T^{7} + 156 T^{8} - 60 T^{9} + 32 T^{10} - 44 T^{11} + 34 T^{12} - 20 T^{13} + 10 T^{14} - 4 T^{15} + T^{16}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$16 + 32 T + 16 T^{2} - 32 T^{3} - 56 T^{4} - 112 T^{5} - 128 T^{6} - 16 T^{7} + 156 T^{8} + 160 T^{9} + 112 T^{10} + 64 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16}$$
$31$ $$T^{16}$$
$37$ $$625 - 500 T^{4} + 150 T^{8} + 5 T^{12} + T^{16}$$
$41$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$1 + 8 T + 46 T^{2} + 32 T^{3} - 131 T^{4} + 72 T^{5} + 162 T^{6} - 304 T^{7} + 256 T^{8} - 160 T^{9} + 82 T^{10} - 44 T^{11} + 34 T^{12} - 20 T^{13} + 10 T^{14} - 4 T^{15} + T^{16}$$
$59$ $$T^{16}$$
$61$ $$( 1 + 6 T + 13 T^{2} + 10 T^{3} + 16 T^{4} + 10 T^{5} + 2 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$67$ $$T^{16}$$
$71$ $$T^{16}$$
$73$ $$( 1 - 12 T^{2} + 19 T^{4} - 8 T^{6} + T^{8} )^{2}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$16 - 32 T + 48 T^{2} - 64 T^{3} + 72 T^{4} - 112 T^{5} + 128 T^{6} - 112 T^{7} + 60 T^{8} + 48 T^{9} + 16 T^{10} - 24 T^{11} + 2 T^{12} - 4 T^{13} - 2 T^{14} + T^{16}$$
$97$ $$1 + 12 T + 68 T^{2} + 144 T^{3} + 222 T^{4} + 92 T^{5} + 118 T^{6} + 72 T^{7} - 80 T^{8} - 8 T^{9} + 6 T^{10} - 16 T^{11} + 7 T^{12} + 4 T^{13} - 2 T^{14} + T^{16}$$