# Properties

 Label 817.1.bz.a Level $817$ Weight $1$ Character orbit 817.bz Analytic conductor $0.408$ Analytic rank $0$ Dimension $12$ Projective image $D_{21}$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$817 = 19 \cdot 43$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 817.bz (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.407736115321$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{21})$$ Defining polynomial: $$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{21}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{21} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{42}^{12} q^{4} + ( \zeta_{42}^{10} - \zeta_{42}^{19} ) q^{5} + ( -\zeta_{42} + \zeta_{42}^{6} ) q^{7} + \zeta_{42}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{42}^{12} q^{4} + ( \zeta_{42}^{10} - \zeta_{42}^{19} ) q^{5} + ( -\zeta_{42} + \zeta_{42}^{6} ) q^{7} + \zeta_{42}^{2} q^{9} + ( -\zeta_{42}^{5} - \zeta_{42}^{13} ) q^{11} -\zeta_{42}^{3} q^{16} + ( \zeta_{42}^{16} + \zeta_{42}^{18} ) q^{17} -\zeta_{42}^{19} q^{19} + ( -\zeta_{42} + \zeta_{42}^{10} ) q^{20} + ( -\zeta_{42}^{5} + \zeta_{42}^{6} ) q^{23} + ( \zeta_{42}^{8} - \zeta_{42}^{17} + \zeta_{42}^{20} ) q^{25} + ( -\zeta_{42}^{13} + \zeta_{42}^{18} ) q^{28} + ( \zeta_{42}^{4} - \zeta_{42}^{11} + \zeta_{42}^{16} + \zeta_{42}^{20} ) q^{35} + \zeta_{42}^{14} q^{36} + q^{43} + ( \zeta_{42}^{4} - \zeta_{42}^{17} ) q^{44} + ( 1 + \zeta_{42}^{12} ) q^{45} + ( -\zeta_{42}^{9} - \zeta_{42}^{15} ) q^{47} + ( \zeta_{42}^{2} - \zeta_{42}^{7} + \zeta_{42}^{12} ) q^{49} + ( \zeta_{42}^{2} - \zeta_{42}^{3} - \zeta_{42}^{11} - \zeta_{42}^{15} ) q^{55} + ( -\zeta_{42}^{7} - \zeta_{42}^{9} ) q^{61} + ( -\zeta_{42}^{3} + \zeta_{42}^{8} ) q^{63} -\zeta_{42}^{15} q^{64} + ( -\zeta_{42}^{7} - \zeta_{42}^{9} ) q^{68} + ( \zeta_{42}^{8} + \zeta_{42}^{14} ) q^{73} + \zeta_{42}^{10} q^{76} + ( \zeta_{42}^{6} - \zeta_{42}^{11} + \zeta_{42}^{14} - \zeta_{42}^{19} ) q^{77} + ( -\zeta_{42} - \zeta_{42}^{13} ) q^{80} + \zeta_{42}^{4} q^{81} + ( -\zeta_{42}^{9} + \zeta_{42}^{14} ) q^{83} + ( -\zeta_{42}^{5} - \zeta_{42}^{7} + \zeta_{42}^{14} + \zeta_{42}^{16} ) q^{85} + ( -\zeta_{42}^{17} + \zeta_{42}^{18} ) q^{92} + ( \zeta_{42}^{8} - \zeta_{42}^{17} ) q^{95} + ( -\zeta_{42}^{7} - \zeta_{42}^{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 2q^{4} + 2q^{5} - q^{7} + q^{9} + O(q^{10})$$ $$12q - 2q^{4} + 2q^{5} - q^{7} + q^{9} + 2q^{11} - 2q^{16} - q^{17} + q^{19} + 2q^{20} - q^{23} + 3q^{25} - q^{28} + 4q^{35} - 6q^{36} + 12q^{43} + 2q^{44} + 10q^{45} - 4q^{47} - 7q^{49} - 2q^{55} - 8q^{61} - q^{63} - 2q^{64} - 8q^{68} - 5q^{73} + q^{76} - 6q^{77} + 2q^{80} + q^{81} - 8q^{83} - 10q^{85} - q^{92} + 2q^{95} - 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/817\mathbb{Z}\right)^\times$$.

 $$n$$ $$173$$ $$476$$ $$\chi(n)$$ $$-1$$ $$\zeta_{42}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
56.1
 −0.733052 + 0.680173i 0.955573 − 0.294755i 0.826239 − 0.563320i 0.0747301 − 0.997204i 0.826239 + 0.563320i 0.365341 − 0.930874i −0.733052 − 0.680173i −0.988831 + 0.149042i 0.955573 + 0.294755i 0.0747301 + 0.997204i −0.988831 − 0.149042i 0.365341 + 0.930874i
0 0 −0.900969 0.433884i 0.440071 + 0.0663300i 0 −0.955573 + 1.65510i 0 0.0747301 0.997204i 0
189.1 0 0 −0.900969 + 0.433884i −0.162592 + 0.414278i 0 0.733052 1.26968i 0 0.826239 0.563320i 0
246.1 0 0 0.623490 0.781831i 1.32091 + 1.22563i 0 −0.0747301 0.129436i 0 0.365341 0.930874i 0
341.1 0 0 0.623490 + 0.781831i −1.72188 0.531130i 0 −0.826239 1.43109i 0 −0.988831 0.149042i 0
455.1 0 0 0.623490 + 0.781831i 1.32091 1.22563i 0 −0.0747301 + 0.129436i 0 0.365341 + 0.930874i 0
531.1 0 0 −0.222521 0.974928i 0.0931869 + 1.24349i 0 0.988831 1.71271i 0 −0.733052 0.680173i 0
569.1 0 0 −0.900969 + 0.433884i 0.440071 0.0663300i 0 −0.955573 1.65510i 0 0.0747301 + 0.997204i 0
626.1 0 0 −0.222521 0.974928i 1.03030 0.702449i 0 −0.365341 0.632789i 0 0.955573 0.294755i 0
683.1 0 0 −0.900969 0.433884i −0.162592 0.414278i 0 0.733052 + 1.26968i 0 0.826239 + 0.563320i 0
702.1 0 0 0.623490 0.781831i −1.72188 + 0.531130i 0 −0.826239 + 1.43109i 0 −0.988831 + 0.149042i 0
740.1 0 0 −0.222521 + 0.974928i 1.03030 + 0.702449i 0 −0.365341 + 0.632789i 0 0.955573 + 0.294755i 0
797.1 0 0 −0.222521 + 0.974928i 0.0931869 1.24349i 0 0.988831 + 1.71271i 0 −0.733052 + 0.680173i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 797.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
43.g even 21 1 inner
817.bz odd 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 817.1.bz.a 12
19.b odd 2 1 CM 817.1.bz.a 12
43.g even 21 1 inner 817.1.bz.a 12
817.bz odd 42 1 inner 817.1.bz.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
817.1.bz.a 12 1.a even 1 1 trivial
817.1.bz.a 12 19.b odd 2 1 CM
817.1.bz.a 12 43.g even 21 1 inner
817.1.bz.a 12 817.bz odd 42 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(817, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$1 - 4 T + 7 T^{2} - 20 T^{3} + 45 T^{4} - 42 T^{5} + 22 T^{6} - 9 T^{8} + 8 T^{9} - 2 T^{11} + T^{12}$$
$7$ $$1 + 8 T + 56 T^{2} + 76 T^{3} + 118 T^{4} + 49 T^{5} + 78 T^{6} + 28 T^{7} + 34 T^{8} + 6 T^{9} + 7 T^{10} + T^{11} + T^{12}$$
$11$ $$1 + 5 T + 52 T^{2} + 94 T^{3} + 54 T^{4} - 6 T^{5} + 7 T^{6} - 6 T^{7} + 12 T^{8} - 4 T^{9} + 3 T^{10} - 2 T^{11} + T^{12}$$
$13$ $$T^{12}$$
$17$ $$1 - 13 T + 49 T^{2} - 29 T^{3} + 69 T^{4} - 20 T^{6} - 21 T^{7} + 6 T^{8} - T^{9} + T^{11} + T^{12}$$
$19$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12}$$
$23$ $$1 + 8 T + 28 T^{2} - 71 T^{3} + 48 T^{4} + 21 T^{5} + 22 T^{6} - 15 T^{8} - T^{9} + T^{11} + T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12}$$
$37$ $$T^{12}$$
$41$ $$T^{12}$$
$43$ $$( -1 + T )^{12}$$
$47$ $$( 1 + 4 T + 9 T^{2} + 8 T^{3} + 4 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$1 - 6 T + 118 T^{3} + 349 T^{4} + 518 T^{5} + 519 T^{6} + 392 T^{7} + 230 T^{8} + 104 T^{9} + 35 T^{10} + 8 T^{11} + T^{12}$$
$67$ $$T^{12}$$
$71$ $$T^{12}$$
$73$ $$1 + 3 T + T^{3} + 31 T^{4} + 56 T^{5} + 57 T^{6} + 56 T^{7} + 47 T^{8} + 29 T^{9} + 14 T^{10} + 5 T^{11} + T^{12}$$
$79$ $$T^{12}$$
$83$ $$1 - 6 T + 118 T^{3} + 349 T^{4} + 518 T^{5} + 519 T^{6} + 392 T^{7} + 230 T^{8} + 104 T^{9} + 35 T^{10} + 8 T^{11} + T^{12}$$
$89$ $$T^{12}$$
$97$ $$T^{12}$$