# Properties

 Label 921.1.p.a Level $921$ Weight $1$ Character orbit 921.p Analytic conductor $0.460$ Analytic rank $0$ Dimension $16$ Projective image $D_{17}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$921 = 3 \cdot 307$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 921.p (of order $$34$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.459638876635$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{34})$$ Defining polynomial: $$x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{17}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{17} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{34}^{12} q^{3} -\zeta_{34}^{3} q^{4} + ( 1 - \zeta_{34} ) q^{7} -\zeta_{34}^{7} q^{9} +O(q^{10})$$ $$q + \zeta_{34}^{12} q^{3} -\zeta_{34}^{3} q^{4} + ( 1 - \zeta_{34} ) q^{7} -\zeta_{34}^{7} q^{9} -\zeta_{34}^{15} q^{12} + ( \zeta_{34}^{4} + \zeta_{34}^{8} ) q^{13} + \zeta_{34}^{6} q^{16} + ( \zeta_{34}^{2} + \zeta_{34}^{10} ) q^{19} + ( \zeta_{34}^{12} - \zeta_{34}^{13} ) q^{21} -\zeta_{34}^{11} q^{25} + \zeta_{34}^{2} q^{27} + ( -\zeta_{34}^{3} + \zeta_{34}^{4} ) q^{28} + ( -\zeta_{34}^{13} + \zeta_{34}^{14} ) q^{31} + \zeta_{34}^{10} q^{36} + ( \zeta_{34}^{6} + \zeta_{34}^{12} ) q^{37} + ( -\zeta_{34}^{3} + \zeta_{34}^{16} ) q^{39} + ( \zeta_{34}^{10} + \zeta_{34}^{16} ) q^{43} -\zeta_{34} q^{48} + ( 1 - \zeta_{34} + \zeta_{34}^{2} ) q^{49} + ( -\zeta_{34}^{7} - \zeta_{34}^{11} ) q^{52} + ( -\zeta_{34}^{5} + \zeta_{34}^{14} ) q^{57} + ( -\zeta_{34}^{11} - \zeta_{34}^{15} ) q^{61} + ( -\zeta_{34}^{7} + \zeta_{34}^{8} ) q^{63} -\zeta_{34}^{9} q^{64} + ( \zeta_{34}^{4} - \zeta_{34}^{9} ) q^{67} + ( \zeta_{34}^{14} - \zeta_{34}^{15} ) q^{73} + \zeta_{34}^{6} q^{75} + ( -\zeta_{34}^{5} - \zeta_{34}^{13} ) q^{76} + ( -\zeta_{34}^{9} + \zeta_{34}^{16} ) q^{79} + \zeta_{34}^{14} q^{81} + ( -\zeta_{34}^{15} + \zeta_{34}^{16} ) q^{84} + ( \zeta_{34}^{4} - \zeta_{34}^{5} + \zeta_{34}^{8} - \zeta_{34}^{9} ) q^{91} + ( \zeta_{34}^{8} - \zeta_{34}^{9} ) q^{93} + ( -\zeta_{34}^{9} - \zeta_{34}^{13} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - q^{3} - q^{4} + 15q^{7} - q^{9} + O(q^{10})$$ $$16q - q^{3} - q^{4} + 15q^{7} - q^{9} - q^{12} - 2q^{13} - q^{16} - 2q^{19} - 2q^{21} - q^{25} - q^{27} - 2q^{28} - 2q^{31} - q^{36} - 2q^{37} - 2q^{39} - 2q^{43} - q^{48} + 14q^{49} - 2q^{52} - 2q^{57} - 2q^{61} - 2q^{63} - q^{64} - 2q^{67} - 2q^{73} - q^{75} - 2q^{76} - 2q^{79} - q^{81} - 2q^{84} - 4q^{91} - 2q^{93} - 2q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/921\mathbb{Z}\right)^\times$$.

 $$n$$ $$308$$ $$619$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{34}^{11}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 −0.932472 + 0.361242i 0.602635 − 0.798017i −0.0922684 + 0.995734i 0.982973 + 0.183750i −0.0922684 − 0.995734i −0.445738 − 0.895163i −0.739009 − 0.673696i 0.273663 + 0.961826i 0.850217 − 0.526432i 0.982973 − 0.183750i −0.445738 + 0.895163i 0.273663 − 0.961826i −0.932472 − 0.361242i 0.602635 + 0.798017i −0.739009 + 0.673696i 0.850217 + 0.526432i
0 −0.273663 + 0.961826i 0.445738 0.895163i 0 0 1.93247 0.361242i 0 −0.850217 0.526432i 0
272.1 0 0.0922684 + 0.995734i 0.932472 + 0.361242i 0 0 0.397365 + 0.798017i 0 −0.982973 + 0.183750i 0
299.1 0 0.445738 + 0.895163i −0.273663 + 0.961826i 0 0 1.09227 0.995734i 0 −0.602635 + 0.798017i 0
371.1 0 −0.602635 + 0.798017i −0.850217 0.526432i 0 0 0.0170269 0.183750i 0 −0.273663 0.961826i 0
422.1 0 0.445738 0.895163i −0.273663 0.961826i 0 0 1.09227 + 0.995734i 0 −0.602635 0.798017i 0
542.1 0 0.739009 + 0.673696i −0.982973 0.183750i 0 0 1.44574 + 0.895163i 0 0.0922684 + 0.995734i 0
587.1 0 −0.850217 + 0.526432i −0.602635 + 0.798017i 0 0 1.73901 + 0.673696i 0 0.445738 0.895163i 0
611.1 0 −0.982973 + 0.183750i 0.739009 + 0.673696i 0 0 0.726337 0.961826i 0 0.932472 0.361242i 0
623.1 0 0.932472 0.361242i 0.0922684 + 0.995734i 0 0 0.149783 + 0.526432i 0 0.739009 0.673696i 0
638.1 0 −0.602635 0.798017i −0.850217 + 0.526432i 0 0 0.0170269 + 0.183750i 0 −0.273663 + 0.961826i 0
695.1 0 0.739009 0.673696i −0.982973 + 0.183750i 0 0 1.44574 0.895163i 0 0.0922684 0.995734i 0
716.1 0 −0.982973 0.183750i 0.739009 0.673696i 0 0 0.726337 + 0.961826i 0 0.932472 + 0.361242i 0
719.1 0 −0.273663 0.961826i 0.445738 + 0.895163i 0 0 1.93247 + 0.361242i 0 −0.850217 + 0.526432i 0
728.1 0 0.0922684 0.995734i 0.932472 0.361242i 0 0 0.397365 0.798017i 0 −0.982973 0.183750i 0
830.1 0 −0.850217 0.526432i −0.602635 0.798017i 0 0 1.73901 0.673696i 0 0.445738 + 0.895163i 0
887.1 0 0.932472 + 0.361242i 0.0922684 0.995734i 0 0 0.149783 0.526432i 0 0.739009 + 0.673696i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 887.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
307.f even 17 1 inner
921.p odd 34 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 921.1.p.a 16
3.b odd 2 1 CM 921.1.p.a 16
307.f even 17 1 inner 921.1.p.a 16
921.p odd 34 1 inner 921.1.p.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
921.1.p.a 16 1.a even 1 1 trivial
921.1.p.a 16 3.b odd 2 1 CM
921.1.p.a 16 307.f even 17 1 inner
921.1.p.a 16 921.p odd 34 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(921, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$1 - 8 T + 64 T^{2} - 308 T^{3} + 1036 T^{4} - 2576 T^{5} + 4900 T^{6} - 7274 T^{7} + 8518 T^{8} - 7896 T^{9} + 5776 T^{10} - 3300 T^{11} + 1444 T^{12} - 468 T^{13} + 106 T^{14} - 15 T^{15} + T^{16}$$
$11$ $$T^{16}$$
$13$ $$1 + 9 T + 64 T^{2} + 253 T^{3} + 594 T^{4} + 858 T^{5} + 786 T^{6} + 495 T^{7} + 256 T^{8} + 128 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$
$17$ $$T^{16}$$
$19$ $$1 + 9 T + 47 T^{2} + 83 T^{3} + 50 T^{4} + 25 T^{5} + 21 T^{6} - 100 T^{7} - 16 T^{8} - 8 T^{9} - 4 T^{10} - 2 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$1 + 9 T + 47 T^{2} + 83 T^{3} + 50 T^{4} + 25 T^{5} + 21 T^{6} - 100 T^{7} - 16 T^{8} - 8 T^{9} - 4 T^{10} - 2 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$
$37$ $$1 + 9 T + 64 T^{2} + 253 T^{3} + 594 T^{4} + 858 T^{5} + 786 T^{6} + 495 T^{7} + 256 T^{8} + 128 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$
$41$ $$T^{16}$$
$43$ $$1 + 9 T + 47 T^{2} + 83 T^{3} + 50 T^{4} + 25 T^{5} + 21 T^{6} - 100 T^{7} - 16 T^{8} - 8 T^{9} - 4 T^{10} - 2 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$
$47$ $$T^{16}$$
$53$ $$T^{16}$$
$59$ $$T^{16}$$
$61$ $$1 + 9 T + 13 T^{2} - 36 T^{3} + 33 T^{4} + 25 T^{5} + 140 T^{6} + 70 T^{7} + 154 T^{8} + 77 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$
$67$ $$1 + 9 T + 30 T^{2} + 15 T^{3} + 50 T^{4} - 94 T^{5} - 47 T^{6} - 15 T^{7} + 120 T^{8} + 60 T^{9} + 30 T^{10} - 36 T^{11} - 18 T^{12} - 9 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$
$71$ $$T^{16}$$
$73$ $$1 - 8 T + 47 T^{2} - 104 T^{3} + 67 T^{4} + 8 T^{5} + 4 T^{6} + 2 T^{7} + T^{8} + 9 T^{9} + 47 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$
$79$ $$1 + 9 T + 30 T^{2} + 15 T^{3} + 50 T^{4} - 94 T^{5} - 47 T^{6} - 15 T^{7} + 120 T^{8} + 60 T^{9} + 30 T^{10} - 36 T^{11} - 18 T^{12} - 9 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$
$83$ $$T^{16}$$
$89$ $$T^{16}$$
$97$ $$1 + 9 T + 64 T^{2} + 253 T^{3} + 594 T^{4} + 858 T^{5} + 786 T^{6} + 495 T^{7} + 256 T^{8} + 128 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$