# Properties

 Label 16.5.c.a Level 16 Weight 5 Character orbit 16.c Analytic conductor 1.654 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 16.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.65391940934$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 8 - 16 \zeta_{6} ) q^{3} + 18 q^{5} + ( -16 + 32 \zeta_{6} ) q^{7} -111 q^{9} +O(q^{10})$$ $$q + ( 8 - 16 \zeta_{6} ) q^{3} + 18 q^{5} + ( -16 + 32 \zeta_{6} ) q^{7} -111 q^{9} + ( -72 + 144 \zeta_{6} ) q^{11} + 178 q^{13} + ( 144 - 288 \zeta_{6} ) q^{15} -126 q^{17} + ( 232 - 464 \zeta_{6} ) q^{19} + 384 q^{21} + ( -432 + 864 \zeta_{6} ) q^{23} -301 q^{25} + ( -240 + 480 \zeta_{6} ) q^{27} -1422 q^{29} + ( 192 - 384 \zeta_{6} ) q^{31} + 1728 q^{33} + ( -288 + 576 \zeta_{6} ) q^{35} + 530 q^{37} + ( 1424 - 2848 \zeta_{6} ) q^{39} + 162 q^{41} + ( 888 - 1776 \zeta_{6} ) q^{43} -1998 q^{45} + ( -2016 + 4032 \zeta_{6} ) q^{47} + 1633 q^{49} + ( -1008 + 2016 \zeta_{6} ) q^{51} + 594 q^{53} + ( -1296 + 2592 \zeta_{6} ) q^{55} -5568 q^{57} + ( 1368 - 2736 \zeta_{6} ) q^{59} + 626 q^{61} + ( 1776 - 3552 \zeta_{6} ) q^{63} + 3204 q^{65} + ( -632 + 1264 \zeta_{6} ) q^{67} + 10368 q^{69} + ( 4464 - 8928 \zeta_{6} ) q^{71} -6686 q^{73} + ( -2408 + 4816 \zeta_{6} ) q^{75} -3456 q^{77} + ( -800 + 1600 \zeta_{6} ) q^{79} -3231 q^{81} + ( 2664 - 5328 \zeta_{6} ) q^{83} -2268 q^{85} + ( -11376 + 22752 \zeta_{6} ) q^{87} + 8226 q^{89} + ( -2848 + 5696 \zeta_{6} ) q^{91} -4608 q^{93} + ( 4176 - 8352 \zeta_{6} ) q^{95} -1598 q^{97} + ( 7992 - 15984 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 36q^{5} - 222q^{9} + O(q^{10})$$ $$2q + 36q^{5} - 222q^{9} + 356q^{13} - 252q^{17} + 768q^{21} - 602q^{25} - 2844q^{29} + 3456q^{33} + 1060q^{37} + 324q^{41} - 3996q^{45} + 3266q^{49} + 1188q^{53} - 11136q^{57} + 1252q^{61} + 6408q^{65} + 20736q^{69} - 13372q^{73} - 6912q^{77} - 6462q^{81} - 4536q^{85} + 16452q^{89} - 9216q^{93} - 3196q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$15$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 13.8564i 0 18.0000 0 27.7128i 0 −111.000 0
15.2 0 13.8564i 0 18.0000 0 27.7128i 0 −111.000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.5.c.a 2
3.b odd 2 1 144.5.g.c 2
4.b odd 2 1 inner 16.5.c.a 2
5.b even 2 1 400.5.b.d 2
5.c odd 4 2 400.5.h.b 4
7.b odd 2 1 784.5.d.a 2
8.b even 2 1 64.5.c.c 2
8.d odd 2 1 64.5.c.c 2
12.b even 2 1 144.5.g.c 2
16.e even 4 2 256.5.d.f 4
16.f odd 4 2 256.5.d.f 4
20.d odd 2 1 400.5.b.d 2
20.e even 4 2 400.5.h.b 4
24.f even 2 1 576.5.g.h 2
24.h odd 2 1 576.5.g.h 2
28.d even 2 1 784.5.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.5.c.a 2 1.a even 1 1 trivial
16.5.c.a 2 4.b odd 2 1 inner
64.5.c.c 2 8.b even 2 1
64.5.c.c 2 8.d odd 2 1
144.5.g.c 2 3.b odd 2 1
144.5.g.c 2 12.b even 2 1
256.5.d.f 4 16.e even 4 2
256.5.d.f 4 16.f odd 4 2
400.5.b.d 2 5.b even 2 1
400.5.b.d 2 20.d odd 2 1
400.5.h.b 4 5.c odd 4 2
400.5.h.b 4 20.e even 4 2
576.5.g.h 2 24.f even 2 1
576.5.g.h 2 24.h odd 2 1
784.5.d.a 2 7.b odd 2 1
784.5.d.a 2 28.d even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(16, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 30 T^{2} + 6561 T^{4}$$
$5$ $$( 1 - 18 T + 625 T^{2} )^{2}$$
$7$ $$( 1 - 94 T + 2401 T^{2} )( 1 + 94 T + 2401 T^{2} )$$
$11$ $$1 - 13730 T^{2} + 214358881 T^{4}$$
$13$ $$( 1 - 178 T + 28561 T^{2} )^{2}$$
$17$ $$( 1 + 126 T + 83521 T^{2} )^{2}$$
$19$ $$1 - 99170 T^{2} + 16983563041 T^{4}$$
$23$ $$1 + 190 T^{2} + 78310985281 T^{4}$$
$29$ $$( 1 + 1422 T + 707281 T^{2} )^{2}$$
$31$ $$1 - 1736450 T^{2} + 852891037441 T^{4}$$
$37$ $$( 1 - 530 T + 1874161 T^{2} )^{2}$$
$41$ $$( 1 - 162 T + 2825761 T^{2} )^{2}$$
$43$ $$1 - 4471970 T^{2} + 11688200277601 T^{4}$$
$47$ $$1 + 2433406 T^{2} + 23811286661761 T^{4}$$
$53$ $$( 1 - 594 T + 7890481 T^{2} )^{2}$$
$59$ $$1 - 18620450 T^{2} + 146830437604321 T^{4}$$
$61$ $$( 1 - 626 T + 13845841 T^{2} )^{2}$$
$67$ $$1 - 39103970 T^{2} + 406067677556641 T^{4}$$
$71$ $$1 + 8958526 T^{2} + 645753531245761 T^{4}$$
$73$ $$( 1 + 6686 T + 28398241 T^{2} )^{2}$$
$79$ $$1 - 75980162 T^{2} + 1517108809906561 T^{4}$$
$83$ $$1 - 73625954 T^{2} + 2252292232139041 T^{4}$$
$89$ $$( 1 - 8226 T + 62742241 T^{2} )^{2}$$
$97$ $$( 1 + 1598 T + 88529281 T^{2} )^{2}$$