# Properties

 Label 1156.1.l.a Level $1156$ Weight $1$ Character orbit 1156.l Analytic conductor $0.577$ Analytic rank $0$ Dimension $16$ Projective image $D_{34}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1156 = 2^{2} \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1156.l (of order $$34$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.576919154604$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{34}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{34} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{34}^{10} q^{2} -\zeta_{34}^{3} q^{4} + ( -1 + \zeta_{34}^{8} ) q^{5} + \zeta_{34}^{13} q^{8} + \zeta_{34} q^{9} +O(q^{10})$$ $$q -\zeta_{34}^{10} q^{2} -\zeta_{34}^{3} q^{4} + ( -1 + \zeta_{34}^{8} ) q^{5} + \zeta_{34}^{13} q^{8} + \zeta_{34} q^{9} + ( \zeta_{34} + \zeta_{34}^{10} ) q^{10} + ( -\zeta_{34}^{12} - \zeta_{34}^{14} ) q^{13} + \zeta_{34}^{6} q^{16} + \zeta_{34}^{6} q^{17} -\zeta_{34}^{11} q^{18} + ( \zeta_{34}^{3} - \zeta_{34}^{11} ) q^{20} + ( 1 - \zeta_{34}^{8} + \zeta_{34}^{16} ) q^{25} + ( -\zeta_{34}^{5} - \zeta_{34}^{7} ) q^{26} + ( -\zeta_{34}^{2} + \zeta_{34}^{4} ) q^{29} -\zeta_{34}^{16} q^{32} -\zeta_{34}^{16} q^{34} -\zeta_{34}^{4} q^{36} + ( -\zeta_{34}^{12} - \zeta_{34}^{15} ) q^{37} + ( -\zeta_{34}^{4} - \zeta_{34}^{13} ) q^{40} + ( \zeta_{34}^{5} - \zeta_{34}^{13} ) q^{41} + ( -\zeta_{34} + \zeta_{34}^{9} ) q^{45} -\zeta_{34}^{2} q^{49} + ( -\zeta_{34} + \zeta_{34}^{9} - \zeta_{34}^{10} ) q^{50} + ( -1 + \zeta_{34}^{15} ) q^{52} + ( -\zeta_{34}^{3} + \zeta_{34}^{16} ) q^{53} + ( \zeta_{34}^{12} - \zeta_{34}^{14} ) q^{58} + ( \zeta_{34}^{10} + \zeta_{34}^{11} ) q^{61} -\zeta_{34}^{9} q^{64} + ( \zeta_{34}^{3} + \zeta_{34}^{5} + \zeta_{34}^{12} + \zeta_{34}^{14} ) q^{65} -\zeta_{34}^{9} q^{68} + \zeta_{34}^{14} q^{72} + ( -\zeta_{34}^{5} + \zeta_{34}^{7} ) q^{73} + ( -\zeta_{34}^{5} - \zeta_{34}^{8} ) q^{74} + ( -\zeta_{34}^{6} + \zeta_{34}^{14} ) q^{80} + \zeta_{34}^{2} q^{81} + ( -\zeta_{34}^{6} - \zeta_{34}^{15} ) q^{82} + ( -\zeta_{34}^{6} + \zeta_{34}^{14} ) q^{85} + ( \zeta_{34}^{11} - \zeta_{34}^{14} ) q^{89} + ( \zeta_{34}^{2} + \zeta_{34}^{11} ) q^{90} + ( \zeta_{34}^{4} + \zeta_{34}^{15} ) q^{97} + \zeta_{34}^{12} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + q^{2} - q^{4} - 17q^{5} + q^{8} + q^{9} + O(q^{10})$$ $$16q + q^{2} - q^{4} - 17q^{5} + q^{8} + q^{9} + 2q^{13} - q^{16} - q^{17} - q^{18} + 16q^{25} - 2q^{26} + q^{32} + q^{34} + q^{36} + q^{49} + q^{50} - 15q^{52} - 2q^{53} - q^{64} - q^{68} - q^{72} - q^{81} + 2q^{89} - q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times$$.

 $$n$$ $$579$$ $$581$$ $$\chi(n)$$ $$-1$$ $$\zeta_{34}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −0.445738 − 0.895163i 0.602635 − 0.798017i 0.982973 + 0.183750i 0.273663 + 0.961826i −0.739009 + 0.673696i −0.932472 − 0.361242i −0.0922684 − 0.995734i 0.850217 − 0.526432i 0.850217 + 0.526432i −0.0922684 + 0.995734i −0.932472 + 0.361242i −0.739009 − 0.673696i 0.273663 − 0.961826i 0.982973 − 0.183750i 0.602635 + 0.798017i −0.445738 + 0.895163i
−0.0922684 + 0.995734i 0 −0.982973 0.183750i −1.85022 + 0.526432i 0 0 0.273663 0.961826i −0.445738 0.895163i −0.353470 1.89090i
135.1 0.982973 + 0.183750i 0 0.932472 + 0.361242i −0.554262 0.895163i 0 0 0.850217 + 0.526432i 0.602635 0.798017i −0.380338 0.981767i
203.1 0.273663 0.961826i 0 −0.850217 0.526432i −0.907732 + 0.995734i 0 0 −0.739009 + 0.673696i 0.982973 + 0.183750i 0.709310 + 1.14558i
271.1 −0.932472 0.361242i 0 0.739009 + 0.673696i −1.60263 0.798017i 0 0 −0.445738 0.895163i 0.273663 + 0.961826i 1.20614 + 1.32307i
339.1 −0.445738 + 0.895163i 0 −0.602635 0.798017i −0.0675278 + 0.361242i 0 0 0.982973 0.183750i −0.739009 + 0.673696i −0.293271 0.221468i
407.1 0.850217 + 0.526432i 0 0.445738 + 0.895163i −1.98297 + 0.183750i 0 0 −0.0922684 + 0.995734i −0.932472 0.361242i −1.78269 0.887674i
475.1 0.602635 0.798017i 0 −0.273663 0.961826i −0.260991 0.673696i 0 0 −0.932472 0.361242i −0.0922684 0.995734i −0.694903 0.197717i
543.1 −0.739009 0.673696i 0 0.0922684 + 0.995734i −1.27366 + 0.961826i 0 0 0.602635 0.798017i 0.850217 0.526432i 1.58923 + 0.147263i
611.1 −0.739009 + 0.673696i 0 0.0922684 0.995734i −1.27366 0.961826i 0 0 0.602635 + 0.798017i 0.850217 + 0.526432i 1.58923 0.147263i
679.1 0.602635 + 0.798017i 0 −0.273663 + 0.961826i −0.260991 + 0.673696i 0 0 −0.932472 + 0.361242i −0.0922684 + 0.995734i −0.694903 + 0.197717i
747.1 0.850217 0.526432i 0 0.445738 0.895163i −1.98297 0.183750i 0 0 −0.0922684 0.995734i −0.932472 + 0.361242i −1.78269 + 0.887674i
815.1 −0.445738 0.895163i 0 −0.602635 + 0.798017i −0.0675278 0.361242i 0 0 0.982973 + 0.183750i −0.739009 0.673696i −0.293271 + 0.221468i
883.1 −0.932472 + 0.361242i 0 0.739009 0.673696i −1.60263 + 0.798017i 0 0 −0.445738 + 0.895163i 0.273663 0.961826i 1.20614 1.32307i
951.1 0.273663 + 0.961826i 0 −0.850217 + 0.526432i −0.907732 0.995734i 0 0 −0.739009 0.673696i 0.982973 0.183750i 0.709310 1.14558i
1019.1 0.982973 0.183750i 0 0.932472 0.361242i −0.554262 + 0.895163i 0 0 0.850217 0.526432i 0.602635 + 0.798017i −0.380338 + 0.981767i
1087.1 −0.0922684 0.995734i 0 −0.982973 + 0.183750i −1.85022 0.526432i 0 0 0.273663 + 0.961826i −0.445738 + 0.895163i −0.353470 + 1.89090i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1087.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
289.g even 34 1 inner
1156.l odd 34 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.1.l.a 16
4.b odd 2 1 CM 1156.1.l.a 16
289.g even 34 1 inner 1156.1.l.a 16
1156.l odd 34 1 inner 1156.1.l.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1156.1.l.a 16 1.a even 1 1 trivial
1156.1.l.a 16 4.b odd 2 1 CM
1156.1.l.a 16 289.g even 34 1 inner
1156.1.l.a 16 1156.l odd 34 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1156, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$17 + 136 T + 680 T^{2} + 2380 T^{3} + 6188 T^{4} + 12376 T^{5} + 19448 T^{6} + 24310 T^{7} + 24310 T^{8} + 19448 T^{9} + 12376 T^{10} + 6188 T^{11} + 2380 T^{12} + 680 T^{13} + 136 T^{14} + 17 T^{15} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$1 + 8 T + 13 T^{2} - 15 T^{3} + 118 T^{4} - 59 T^{5} + 72 T^{6} - 2 T^{7} + T^{8} - 60 T^{9} + 30 T^{10} - 15 T^{11} + 16 T^{12} - 8 T^{13} + 4 T^{14} - 2 T^{15} + T^{16}$$
$17$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$17 + 119 T + 442 T^{2} + 935 T^{3} + 1122 T^{4} + 714 T^{5} + 204 T^{6} + 17 T^{7} + T^{16}$$
$31$ $$T^{16}$$
$37$ $$17 + 68 T + 34 T^{2} + 119 T^{4} - 221 T^{5} + 85 T^{8} + 17 T^{9} + 17 T^{12} + T^{16}$$
$41$ $$17 + 17 T + 85 T^{2} - 102 T^{3} + 17 T^{4} + 255 T^{5} - 238 T^{7} + 51 T^{9} + T^{16}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$1 - 8 T + 30 T^{2} - 2 T^{3} - T^{4} + 59 T^{5} + 140 T^{6} + 70 T^{7} + 35 T^{8} - 25 T^{9} - 4 T^{10} - 2 T^{11} - T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$
$59$ $$T^{16}$$
$61$ $$17 + 34 T + 17 T^{2} - 221 T^{3} + 85 T^{5} + 68 T^{6} + 119 T^{8} + 17 T^{11} + T^{16}$$
$67$ $$T^{16}$$
$71$ $$T^{16}$$
$73$ $$17 + 51 T - 85 T^{3} + 238 T^{4} + 17 T^{6} + 255 T^{7} + 102 T^{10} + 17 T^{13} + T^{16}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$1 - 9 T + 64 T^{2} - 253 T^{3} + 594 T^{4} - 858 T^{5} + 786 T^{6} - 495 T^{7} + 256 T^{8} - 128 T^{9} + 64 T^{10} - 32 T^{11} + 16 T^{12} - 8 T^{13} + 4 T^{14} - 2 T^{15} + T^{16}$$
$97$ $$17 - 51 T + 85 T^{3} + 238 T^{4} + 17 T^{6} - 255 T^{7} + 102 T^{10} - 17 T^{13} + T^{16}$$