# Properties

 Label 693.1.bp.a Level $693$ Weight $1$ Character orbit 693.bp Analytic conductor $0.346$ Analytic rank $0$ Dimension $16$ Projective image $D_{20}$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$693 = 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 693.bp (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.345852053755$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{40})$$ Defining polynomial: $$x^{16} - x^{12} + x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{20}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{20} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \zeta_{40} + \zeta_{40}^{7} ) q^{2} + ( \zeta_{40}^{2} + \zeta_{40}^{8} + \zeta_{40}^{14} ) q^{4} + \zeta_{40}^{18} q^{7} + ( -\zeta_{40} + \zeta_{40}^{3} + \zeta_{40}^{9} + \zeta_{40}^{15} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{40} + \zeta_{40}^{7} ) q^{2} + ( \zeta_{40}^{2} + \zeta_{40}^{8} + \zeta_{40}^{14} ) q^{4} + \zeta_{40}^{18} q^{7} + ( -\zeta_{40} + \zeta_{40}^{3} + \zeta_{40}^{9} + \zeta_{40}^{15} ) q^{8} -\zeta_{40}^{11} q^{11} + ( -\zeta_{40}^{5} + \zeta_{40}^{19} ) q^{14} + ( -\zeta_{40}^{2} + \zeta_{40}^{4} - \zeta_{40}^{8} + \zeta_{40}^{10} + \zeta_{40}^{16} ) q^{16} + ( -\zeta_{40}^{12} - \zeta_{40}^{18} ) q^{22} + ( -\zeta_{40}^{3} - \zeta_{40}^{17} ) q^{23} + \zeta_{40}^{12} q^{25} + ( -1 - \zeta_{40}^{6} - \zeta_{40}^{12} ) q^{28} + ( -\zeta_{40}^{7} - \zeta_{40}^{9} ) q^{29} + ( -\zeta_{40}^{3} + \zeta_{40}^{5} - \zeta_{40}^{9} + \zeta_{40}^{11} - \zeta_{40}^{15} + \zeta_{40}^{17} ) q^{32} + ( \zeta_{40}^{6} + \zeta_{40}^{10} ) q^{37} + ( -\zeta_{40}^{6} - \zeta_{40}^{14} ) q^{43} + ( \zeta_{40}^{5} - \zeta_{40}^{13} - \zeta_{40}^{19} ) q^{44} + ( -\zeta_{40}^{10} - \zeta_{40}^{18} ) q^{46} -\zeta_{40}^{16} q^{49} + ( \zeta_{40}^{13} + \zeta_{40}^{19} ) q^{50} + ( \zeta_{40}^{13} + \zeta_{40}^{15} ) q^{53} + ( -\zeta_{40} - \zeta_{40}^{7} - \zeta_{40}^{13} - \zeta_{40}^{19} ) q^{56} + ( -\zeta_{40}^{8} - \zeta_{40}^{10} - \zeta_{40}^{14} - \zeta_{40}^{16} ) q^{58} + ( \zeta_{40}^{2} - \zeta_{40}^{4} + \zeta_{40}^{6} - \zeta_{40}^{10} + \zeta_{40}^{12} - \zeta_{40}^{16} + \zeta_{40}^{18} ) q^{64} + ( -\zeta_{40}^{4} + \zeta_{40}^{16} ) q^{67} + ( -\zeta_{40}^{15} + \zeta_{40}^{17} ) q^{71} + ( \zeta_{40}^{7} + \zeta_{40}^{11} + \zeta_{40}^{13} + \zeta_{40}^{17} ) q^{74} + \zeta_{40}^{9} q^{77} + ( -1 + \zeta_{40}^{8} ) q^{79} + ( \zeta_{40} - \zeta_{40}^{7} - \zeta_{40}^{13} - \zeta_{40}^{15} ) q^{86} + ( 1 + \zeta_{40}^{6} + \zeta_{40}^{12} - \zeta_{40}^{14} ) q^{88} + ( -\zeta_{40}^{17} - \zeta_{40}^{19} ) q^{92} + ( \zeta_{40}^{3} - \zeta_{40}^{17} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{4} + O(q^{10})$$ $$16q - 4q^{4} + 4q^{16} - 4q^{22} + 4q^{25} - 20q^{28} + 4q^{49} + 8q^{58} + 4q^{64} - 8q^{67} - 20q^{79} + 20q^{88} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$442$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{40}^{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
62.1
 −0.987688 − 0.156434i 0.156434 − 0.987688i −0.156434 + 0.987688i 0.987688 + 0.156434i −0.453990 − 0.891007i 0.891007 − 0.453990i −0.891007 + 0.453990i 0.453990 + 0.891007i −0.987688 + 0.156434i 0.156434 + 0.987688i −0.156434 − 0.987688i 0.987688 − 0.156434i −0.453990 + 0.891007i 0.891007 + 0.453990i −0.891007 − 0.453990i 0.453990 − 0.891007i
−1.44168 1.04744i 0 0.672288 + 2.06909i 0 0 −0.951057 + 0.309017i 0.647354 1.99235i 0 0
62.2 −0.734572 0.533698i 0 −0.0542543 0.166977i 0 0 0.951057 0.309017i −0.329843 + 1.01515i 0 0
62.3 0.734572 + 0.533698i 0 −0.0542543 0.166977i 0 0 0.951057 0.309017i 0.329843 1.01515i 0 0
62.4 1.44168 + 1.04744i 0 0.672288 + 2.06909i 0 0 −0.951057 + 0.309017i −0.647354 + 1.99235i 0 0
314.1 −0.610425 1.87869i 0 −2.34786 + 1.70582i 0 0 0.587785 + 0.809017i 3.03979 + 2.20854i 0 0
314.2 −0.0966818 0.297556i 0 0.729825 0.530249i 0 0 −0.587785 0.809017i −0.481456 0.349798i 0 0
314.3 0.0966818 + 0.297556i 0 0.729825 0.530249i 0 0 −0.587785 0.809017i 0.481456 + 0.349798i 0 0
314.4 0.610425 + 1.87869i 0 −2.34786 + 1.70582i 0 0 0.587785 + 0.809017i −3.03979 2.20854i 0 0
503.1 −1.44168 + 1.04744i 0 0.672288 2.06909i 0 0 −0.951057 0.309017i 0.647354 + 1.99235i 0 0
503.2 −0.734572 + 0.533698i 0 −0.0542543 + 0.166977i 0 0 0.951057 + 0.309017i −0.329843 1.01515i 0 0
503.3 0.734572 0.533698i 0 −0.0542543 + 0.166977i 0 0 0.951057 + 0.309017i 0.329843 + 1.01515i 0 0
503.4 1.44168 1.04744i 0 0.672288 2.06909i 0 0 −0.951057 0.309017i −0.647354 1.99235i 0 0
629.1 −0.610425 + 1.87869i 0 −2.34786 1.70582i 0 0 0.587785 0.809017i 3.03979 2.20854i 0 0
629.2 −0.0966818 + 0.297556i 0 0.729825 + 0.530249i 0 0 −0.587785 + 0.809017i −0.481456 + 0.349798i 0 0
629.3 0.0966818 0.297556i 0 0.729825 + 0.530249i 0 0 −0.587785 + 0.809017i 0.481456 0.349798i 0 0
629.4 0.610425 1.87869i 0 −2.34786 1.70582i 0 0 0.587785 0.809017i −3.03979 + 2.20854i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 629.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
11.d odd 10 1 inner
21.c even 2 1 inner
33.f even 10 1 inner
77.l even 10 1 inner
231.r odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.1.bp.a 16
3.b odd 2 1 inner 693.1.bp.a 16
7.b odd 2 1 CM 693.1.bp.a 16
11.d odd 10 1 inner 693.1.bp.a 16
21.c even 2 1 inner 693.1.bp.a 16
33.f even 10 1 inner 693.1.bp.a 16
77.l even 10 1 inner 693.1.bp.a 16
231.r odd 10 1 inner 693.1.bp.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.1.bp.a 16 1.a even 1 1 trivial
693.1.bp.a 16 3.b odd 2 1 inner
693.1.bp.a 16 7.b odd 2 1 CM
693.1.bp.a 16 11.d odd 10 1 inner
693.1.bp.a 16 21.c even 2 1 inner
693.1.bp.a 16 33.f even 10 1 inner
693.1.bp.a 16 77.l even 10 1 inner
693.1.bp.a 16 231.r odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(693, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$T^{16}$$
$7$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$11$ $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
$13$ $$T^{16}$$
$17$ $$T^{16}$$
$19$ $$T^{16}$$
$23$ $$( 1 + 12 T^{2} + 19 T^{4} + 8 T^{6} + T^{8} )^{2}$$
$29$ $$1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16}$$
$31$ $$T^{16}$$
$37$ $$( 25 + 10 T^{4} + 5 T^{6} + T^{8} )^{2}$$
$41$ $$T^{16}$$
$43$ $$( 1 + 3 T^{2} + T^{4} )^{4}$$
$47$ $$T^{16}$$
$53$ $$1 + 4 T^{2} + 92 T^{4} - 228 T^{6} + 230 T^{8} - 72 T^{10} + 17 T^{12} - 4 T^{14} + T^{16}$$
$59$ $$T^{16}$$
$61$ $$T^{16}$$
$67$ $$( -1 + T + T^{2} )^{8}$$
$71$ $$1 + 4 T^{2} + 92 T^{4} - 228 T^{6} + 230 T^{8} - 72 T^{10} + 17 T^{12} - 4 T^{14} + T^{16}$$
$73$ $$T^{16}$$
$79$ $$( 5 + 10 T + 10 T^{2} + 5 T^{3} + T^{4} )^{4}$$
$83$ $$T^{16}$$
$89$ $$T^{16}$$
$97$ $$T^{16}$$