# Properties

 Label 765.1.bz.a Level $765$ Weight $1$ Character orbit 765.bz Analytic conductor $0.382$ Analytic rank $0$ Dimension $16$ Projective image $D_{16}$ CM discriminant -15 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$765 = 3^{2} \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 765.bz (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.381784734664$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{16})$$ Coefficient field: $$\Q(\zeta_{32})$$ Defining polynomial: $$x^{16} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{32} - \zeta_{32}^{3} ) q^{2} + ( \zeta_{32}^{2} - \zeta_{32}^{4} + \zeta_{32}^{6} ) q^{4} + \zeta_{32}^{11} q^{5} + ( \zeta_{32}^{3} - \zeta_{32}^{5} + \zeta_{32}^{7} - \zeta_{32}^{9} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{32} - \zeta_{32}^{3} ) q^{2} + ( \zeta_{32}^{2} - \zeta_{32}^{4} + \zeta_{32}^{6} ) q^{4} + \zeta_{32}^{11} q^{5} + ( \zeta_{32}^{3} - \zeta_{32}^{5} + \zeta_{32}^{7} - \zeta_{32}^{9} ) q^{8} + ( \zeta_{32}^{12} - \zeta_{32}^{14} ) q^{10} + ( \zeta_{32}^{4} - \zeta_{32}^{6} + \zeta_{32}^{8} - \zeta_{32}^{10} + \zeta_{32}^{12} ) q^{16} + \zeta_{32}^{15} q^{17} + ( \zeta_{32}^{8} - \zeta_{32}^{12} ) q^{19} + ( -\zeta_{32} + \zeta_{32}^{13} - \zeta_{32}^{15} ) q^{20} + ( \zeta_{32}^{5} - \zeta_{32}^{13} ) q^{23} -\zeta_{32}^{6} q^{25} + ( -\zeta_{32}^{4} + \zeta_{32}^{10} ) q^{31} + ( \zeta_{32}^{5} - \zeta_{32}^{7} + \zeta_{32}^{9} - \zeta_{32}^{11} + \zeta_{32}^{13} - \zeta_{32}^{15} ) q^{32} + ( -1 + \zeta_{32}^{2} ) q^{34} + ( \zeta_{32}^{9} - \zeta_{32}^{11} - \zeta_{32}^{13} + \zeta_{32}^{15} ) q^{38} + ( 1 - \zeta_{32}^{2} + \zeta_{32}^{4} + \zeta_{32}^{14} ) q^{40} + ( -1 + \zeta_{32}^{6} - \zeta_{32}^{8} - \zeta_{32}^{14} ) q^{46} + ( -\zeta_{32} - \zeta_{32}^{7} ) q^{47} -\zeta_{32}^{10} q^{49} + ( -\zeta_{32}^{7} + \zeta_{32}^{9} ) q^{50} + ( -\zeta_{32}^{9} - \zeta_{32}^{11} ) q^{53} + ( -\zeta_{32}^{2} - \zeta_{32}^{8} ) q^{61} + ( -\zeta_{32}^{5} + \zeta_{32}^{7} + \zeta_{32}^{11} - \zeta_{32}^{13} ) q^{62} + ( 1 - \zeta_{32}^{2} + \zeta_{32}^{6} - \zeta_{32}^{8} + \zeta_{32}^{10} - \zeta_{32}^{12} + \zeta_{32}^{14} ) q^{64} + ( -\zeta_{32} + \zeta_{32}^{3} - \zeta_{32}^{5} ) q^{68} + ( -1 + \zeta_{32}^{2} + \zeta_{32}^{10} - \zeta_{32}^{12} ) q^{76} + ( -\zeta_{32}^{4} + \zeta_{32}^{14} ) q^{79} + ( \zeta_{32} - \zeta_{32}^{3} + \zeta_{32}^{5} - \zeta_{32}^{7} + \zeta_{32}^{15} ) q^{80} + ( \zeta_{32} + \zeta_{32}^{3} ) q^{83} -\zeta_{32}^{10} q^{85} + ( -\zeta_{32} + \zeta_{32}^{3} + \zeta_{32}^{7} - \zeta_{32}^{9} + \zeta_{32}^{11} - \zeta_{32}^{15} ) q^{92} + ( -\zeta_{32}^{2} + \zeta_{32}^{4} - \zeta_{32}^{8} + \zeta_{32}^{10} ) q^{94} + ( -\zeta_{32}^{3} + \zeta_{32}^{7} ) q^{95} + ( -\zeta_{32}^{11} + \zeta_{32}^{13} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 16q^{34} + 16q^{40} - 16q^{46} + 16q^{64} - 16q^{76} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/765\mathbb{Z}\right)^\times$$.

 $$n$$ $$307$$ $$496$$ $$596$$ $$\chi(n)$$ $$-1$$ $$\zeta_{32}^{14}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −0.831470 + 0.555570i 0.831470 − 0.555570i −0.555570 + 0.831470i 0.555570 − 0.831470i −0.195090 + 0.980785i 0.195090 − 0.980785i −0.980785 − 0.195090i 0.980785 + 0.195090i −0.831470 − 0.555570i 0.831470 + 0.555570i −0.555570 − 0.831470i 0.555570 + 0.831470i −0.195090 − 0.980785i 0.195090 + 0.980785i −0.980785 + 0.195090i 0.980785 − 0.195090i
−1.02656 0.425215i 0 0.165911 + 0.165911i −0.980785 + 0.195090i 0 0 0.325446 + 0.785695i 0 1.08979 + 0.216773i
109.2 1.02656 + 0.425215i 0 0.165911 + 0.165911i 0.980785 0.195090i 0 0 −0.325446 0.785695i 0 1.08979 + 0.216773i
199.1 −1.53636 + 0.636379i 0 1.24830 1.24830i 0.195090 0.980785i 0 0 −0.487064 + 1.17588i 0 0.324423 + 1.63099i
199.2 1.53636 0.636379i 0 1.24830 1.24830i −0.195090 + 0.980785i 0 0 0.487064 1.17588i 0 0.324423 + 1.63099i
244.1 −0.750661 + 1.81225i 0 −2.01367 2.01367i 0.831470 + 0.555570i 0 0 3.34861 1.38704i 0 −1.63099 + 1.08979i
244.2 0.750661 1.81225i 0 −2.01367 2.01367i −0.831470 0.555570i 0 0 −3.34861 + 1.38704i 0 −1.63099 + 1.08979i
334.1 −0.149316 + 0.360480i 0 0.599456 + 0.599456i 0.555570 0.831470i 0 0 −0.666080 + 0.275899i 0 0.216773 + 0.324423i
334.2 0.149316 0.360480i 0 0.599456 + 0.599456i −0.555570 + 0.831470i 0 0 0.666080 0.275899i 0 0.216773 + 0.324423i
379.1 −1.02656 + 0.425215i 0 0.165911 0.165911i −0.980785 0.195090i 0 0 0.325446 0.785695i 0 1.08979 0.216773i
379.2 1.02656 0.425215i 0 0.165911 0.165911i 0.980785 + 0.195090i 0 0 −0.325446 + 0.785695i 0 1.08979 0.216773i
469.1 −1.53636 0.636379i 0 1.24830 + 1.24830i 0.195090 + 0.980785i 0 0 −0.487064 1.17588i 0 0.324423 1.63099i
469.2 1.53636 + 0.636379i 0 1.24830 + 1.24830i −0.195090 0.980785i 0 0 0.487064 + 1.17588i 0 0.324423 1.63099i
649.1 −0.750661 1.81225i 0 −2.01367 + 2.01367i 0.831470 0.555570i 0 0 3.34861 + 1.38704i 0 −1.63099 1.08979i
649.2 0.750661 + 1.81225i 0 −2.01367 + 2.01367i −0.831470 + 0.555570i 0 0 −3.34861 1.38704i 0 −1.63099 1.08979i
694.1 −0.149316 0.360480i 0 0.599456 0.599456i 0.555570 + 0.831470i 0 0 −0.666080 0.275899i 0 0.216773 0.324423i
694.2 0.149316 + 0.360480i 0 0.599456 0.599456i −0.555570 0.831470i 0 0 0.666080 + 0.275899i 0 0.216773 0.324423i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 694.2 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
17.e odd 16 1 inner
51.i even 16 1 inner
85.p odd 16 1 inner
255.be even 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 765.1.bz.a 16
3.b odd 2 1 inner 765.1.bz.a 16
5.b even 2 1 inner 765.1.bz.a 16
5.c odd 4 2 3825.1.ck.a 16
15.d odd 2 1 CM 765.1.bz.a 16
15.e even 4 2 3825.1.ck.a 16
17.e odd 16 1 inner 765.1.bz.a 16
51.i even 16 1 inner 765.1.bz.a 16
85.o even 16 1 3825.1.ck.a 16
85.p odd 16 1 inner 765.1.bz.a 16
85.r even 16 1 3825.1.ck.a 16
255.bc odd 16 1 3825.1.ck.a 16
255.be even 16 1 inner 765.1.bz.a 16
255.bj odd 16 1 3825.1.ck.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
765.1.bz.a 16 1.a even 1 1 trivial
765.1.bz.a 16 3.b odd 2 1 inner
765.1.bz.a 16 5.b even 2 1 inner
765.1.bz.a 16 15.d odd 2 1 CM
765.1.bz.a 16 17.e odd 16 1 inner
765.1.bz.a 16 51.i even 16 1 inner
765.1.bz.a 16 85.p odd 16 1 inner
765.1.bz.a 16 255.be even 16 1 inner
3825.1.ck.a 16 5.c odd 4 2
3825.1.ck.a 16 15.e even 4 2
3825.1.ck.a 16 85.o even 16 1
3825.1.ck.a 16 85.r even 16 1
3825.1.ck.a 16 255.bc odd 16 1
3825.1.ck.a 16 255.bj odd 16 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(765, [\chi])$$.

## Hecke characteristic polynomials

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