# Properties

 Label 1169.1.f.c Level $1169$ Weight $1$ Character orbit 1169.f Analytic conductor $0.583$ Analytic rank $0$ Dimension $20$ Projective image $D_{33}$ CM discriminant -167 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1169 = 7 \cdot 167$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1169.f (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.583406999768$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{33})$$ Defining polynomial: $$x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{33}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{33} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \zeta_{66}^{16} + \zeta_{66}^{28} ) q^{2} + ( -\zeta_{66}^{9} - \zeta_{66}^{13} ) q^{3} + ( -\zeta_{66}^{11} - \zeta_{66}^{23} + \zeta_{66}^{32} ) q^{4} + ( \zeta_{66}^{4} + \zeta_{66}^{8} - \zeta_{66}^{25} - \zeta_{66}^{29} ) q^{6} -\zeta_{66}^{25} q^{7} + ( \zeta_{66}^{6} - \zeta_{66}^{15} + \zeta_{66}^{18} - \zeta_{66}^{27} ) q^{8} + ( \zeta_{66}^{18} + \zeta_{66}^{22} + \zeta_{66}^{26} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{66}^{16} + \zeta_{66}^{28} ) q^{2} + ( -\zeta_{66}^{9} - \zeta_{66}^{13} ) q^{3} + ( -\zeta_{66}^{11} - \zeta_{66}^{23} + \zeta_{66}^{32} ) q^{4} + ( \zeta_{66}^{4} + \zeta_{66}^{8} - \zeta_{66}^{25} - \zeta_{66}^{29} ) q^{6} -\zeta_{66}^{25} q^{7} + ( \zeta_{66}^{6} - \zeta_{66}^{15} + \zeta_{66}^{18} - \zeta_{66}^{27} ) q^{8} + ( \zeta_{66}^{18} + \zeta_{66}^{22} + \zeta_{66}^{26} ) q^{9} + ( \zeta_{66}^{10} + \zeta_{66}^{12} ) q^{11} + ( -\zeta_{66}^{3} + \zeta_{66}^{8} + \zeta_{66}^{12} + \zeta_{66}^{20} + \zeta_{66}^{24} + \zeta_{66}^{32} ) q^{12} + ( \zeta_{66}^{8} + \zeta_{66}^{20} ) q^{14} + ( -\zeta_{66} + \zeta_{66}^{10} - \zeta_{66}^{13} + \zeta_{66}^{22} - \zeta_{66}^{31} ) q^{16} + ( -\zeta_{66} - \zeta_{66}^{5} - \zeta_{66}^{9} - \zeta_{66}^{13} - \zeta_{66}^{17} - \zeta_{66}^{21} ) q^{18} + ( \zeta_{66}^{4} - \zeta_{66}^{7} ) q^{19} + ( -\zeta_{66} - \zeta_{66}^{5} ) q^{21} + ( -\zeta_{66}^{5} - \zeta_{66}^{7} + \zeta_{66}^{26} + \zeta_{66}^{28} ) q^{22} + ( -\zeta_{66}^{3} - \zeta_{66}^{7} - \zeta_{66}^{15} - \zeta_{66}^{19} + \zeta_{66}^{24} - \zeta_{66}^{27} + \zeta_{66}^{28} - \zeta_{66}^{31} ) q^{24} -\zeta_{66}^{11} q^{25} + ( \zeta_{66}^{2} + \zeta_{66}^{6} - \zeta_{66}^{27} - \zeta_{66}^{31} ) q^{27} + ( -\zeta_{66}^{3} - \zeta_{66}^{15} + \zeta_{66}^{24} ) q^{28} + ( \zeta_{66}^{14} - \zeta_{66}^{19} ) q^{29} + ( \zeta_{66}^{6} + \zeta_{66}^{16} ) q^{31} + ( -\zeta_{66}^{5} + \zeta_{66}^{8} + \zeta_{66}^{14} - \zeta_{66}^{17} + \zeta_{66}^{26} - \zeta_{66}^{29} ) q^{32} + ( -\zeta_{66}^{19} - \zeta_{66}^{21} - \zeta_{66}^{23} - \zeta_{66}^{25} ) q^{33} + ( 1 + \zeta_{66}^{4} + \zeta_{66}^{8} + \zeta_{66}^{12} + \zeta_{66}^{16} - \zeta_{66}^{17} - \zeta_{66}^{21} - \zeta_{66}^{25} - \zeta_{66}^{29} ) q^{36} + ( \zeta_{66}^{2} + \zeta_{66}^{20} - \zeta_{66}^{23} + \zeta_{66}^{32} ) q^{38} + ( 1 - \zeta_{66}^{17} - \zeta_{66}^{21} - \zeta_{66}^{29} ) q^{42} + ( 1 + \zeta_{66}^{2} - \zeta_{66}^{9} - \zeta_{66}^{11} - \zeta_{66}^{21} - \zeta_{66}^{23} ) q^{44} + ( -\zeta_{66}^{15} - \zeta_{66}^{29} ) q^{47} + ( \zeta_{66}^{2} - \zeta_{66}^{7} + \zeta_{66}^{10} - \zeta_{66}^{11} + \zeta_{66}^{14} - \zeta_{66}^{19} + \zeta_{66}^{22} - \zeta_{66}^{23} + \zeta_{66}^{26} - \zeta_{66}^{31} ) q^{48} -\zeta_{66}^{17} q^{49} + ( \zeta_{66}^{6} - \zeta_{66}^{27} ) q^{50} + ( -\zeta_{66} + \zeta_{66}^{10} + \zeta_{66}^{14} + \zeta_{66}^{18} + 2 \zeta_{66}^{22} + \zeta_{66}^{26} + \zeta_{66}^{30} ) q^{54} + ( -\zeta_{66}^{7} + \zeta_{66}^{10} - \zeta_{66}^{19} - \zeta_{66}^{31} ) q^{56} + ( -\zeta_{66}^{13} + \zeta_{66}^{16} - \zeta_{66}^{17} + \zeta_{66}^{20} ) q^{57} + ( \zeta_{66}^{2} - \zeta_{66}^{9} + \zeta_{66}^{14} + \zeta_{66}^{30} ) q^{58} + ( -\zeta_{66}^{13} - \zeta_{66}^{31} ) q^{61} + ( -\zeta_{66} - \zeta_{66}^{11} + \zeta_{66}^{22} + \zeta_{66}^{32} ) q^{62} + ( \zeta_{66}^{10} + \zeta_{66}^{14} + \zeta_{66}^{18} ) q^{63} + ( 1 - \zeta_{66}^{3} - \zeta_{66}^{9} + \zeta_{66}^{12} - \zeta_{66}^{21} + \zeta_{66}^{24} + \zeta_{66}^{30} ) q^{64} + ( \zeta_{66}^{2} + \zeta_{66}^{4} + \zeta_{66}^{6} + \zeta_{66}^{8} + \zeta_{66}^{14} + \zeta_{66}^{16} + \zeta_{66}^{18} + \zeta_{66}^{20} ) q^{66} + ( 1 - \zeta_{66}^{3} + \zeta_{66}^{4} - \zeta_{66}^{7} + \zeta_{66}^{8} - \zeta_{66}^{11} + \zeta_{66}^{12} + \zeta_{66}^{16} + \zeta_{66}^{20} + \zeta_{66}^{24} + \zeta_{66}^{28} + \zeta_{66}^{32} ) q^{72} + ( \zeta_{66}^{20} + \zeta_{66}^{24} ) q^{75} + ( -\zeta_{66}^{3} + \zeta_{66}^{6} - \zeta_{66}^{15} + \zeta_{66}^{18} - \zeta_{66}^{27} + \zeta_{66}^{30} ) q^{76} + ( \zeta_{66}^{2} + \zeta_{66}^{4} ) q^{77} + ( -\zeta_{66}^{3} - \zeta_{66}^{7} - \zeta_{66}^{11} - \zeta_{66}^{15} - \zeta_{66}^{19} ) q^{81} + ( 1 + \zeta_{66}^{4} + \zeta_{66}^{12} + \zeta_{66}^{16} + \zeta_{66}^{24} + \zeta_{66}^{28} ) q^{84} + ( -\zeta_{66}^{23} - \zeta_{66}^{27} + \zeta_{66}^{28} + \zeta_{66}^{32} ) q^{87} + ( \zeta_{66}^{4} + \zeta_{66}^{6} + \zeta_{66}^{16} + \zeta_{66}^{18} - \zeta_{66}^{25} - \zeta_{66}^{27} + \zeta_{66}^{28} + \zeta_{66}^{30} ) q^{88} + ( \zeta_{66}^{2} - \zeta_{66}^{9} ) q^{89} + ( -\zeta_{66}^{15} - \zeta_{66}^{19} - \zeta_{66}^{25} - \zeta_{66}^{29} ) q^{93} + ( \zeta_{66}^{10} + \zeta_{66}^{12} + \zeta_{66}^{24} - \zeta_{66}^{31} ) q^{94} + ( \zeta_{66}^{2} - \zeta_{66}^{5} + \zeta_{66}^{6} - \zeta_{66}^{9} + \zeta_{66}^{14} - \zeta_{66}^{17} + \zeta_{66}^{18} - \zeta_{66}^{21} - \zeta_{66}^{23} + \zeta_{66}^{26} - \zeta_{66}^{27} + \zeta_{66}^{30} ) q^{96} + ( \zeta_{66}^{12} - \zeta_{66}^{21} ) q^{97} + ( 1 + \zeta_{66}^{12} ) q^{98} + ( -\zeta_{66} - \zeta_{66}^{3} - \zeta_{66}^{5} + \zeta_{66}^{28} + \zeta_{66}^{30} + \zeta_{66}^{32} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 2q^{2} - q^{3} - 8q^{4} + 4q^{6} + q^{7} - 8q^{8} - 11q^{9} + O(q^{10})$$ $$20q + 2q^{2} - q^{3} - 8q^{4} + 4q^{6} + q^{7} - 8q^{8} - 11q^{9} - q^{11} - 3q^{12} + 2q^{14} - 6q^{16} + 2q^{19} + 2q^{21} + 4q^{22} - 4q^{24} - 10q^{25} - 2q^{27} - 6q^{28} + 2q^{29} - q^{31} + 6q^{32} + q^{33} + 22q^{36} + 4q^{38} + 20q^{42} + 8q^{44} - q^{47} - 12q^{48} + q^{49} - 4q^{50} - 20q^{54} + 4q^{56} + 4q^{57} - 2q^{58} + 2q^{61} - 18q^{62} + 8q^{64} + 2q^{66} + 11q^{72} - q^{75} - 12q^{76} + 2q^{77} - 12q^{81} + 19q^{84} + q^{87} - 4q^{88} - q^{89} + q^{93} - 2q^{94} - 6q^{96} - 4q^{97} + 18q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1169\mathbb{Z}\right)^\times$$.

 $$n$$ $$673$$ $$836$$ $$\chi(n)$$ $$-1$$ $$\zeta_{66}^{22}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
333.1
 −0.995472 + 0.0950560i 0.580057 − 0.814576i 0.981929 + 0.189251i −0.327068 + 0.945001i 0.723734 + 0.690079i 0.235759 + 0.971812i 0.928368 − 0.371662i −0.786053 + 0.618159i −0.888835 − 0.458227i 0.0475819 − 0.998867i −0.995472 − 0.0950560i 0.580057 + 0.814576i 0.981929 − 0.189251i −0.327068 − 0.945001i 0.723734 − 0.690079i 0.235759 − 0.971812i 0.928368 + 0.371662i −0.786053 − 0.618159i −0.888835 + 0.458227i 0.0475819 + 0.998867i
−0.841254 1.45709i −0.981929 + 1.70075i −0.915415 + 1.58555i 0 3.30420 0.723734 + 0.690079i 1.39788 −1.42837 2.47401i 0
333.2 −0.841254 1.45709i 0.327068 0.566498i −0.915415 + 1.58555i 0 −1.10059 0.235759 + 0.971812i 1.39788 0.286053 + 0.495458i 0
333.3 −0.415415 0.719520i −0.928368 + 1.60798i 0.154861 0.268227i 0 1.54263 0.0475819 0.998867i −1.08816 −1.22373 2.11957i 0
333.4 −0.415415 0.719520i 0.786053 1.36148i 0.154861 0.268227i 0 −1.30615 −0.888835 0.458227i −1.08816 −0.735759 1.27437i 0
333.5 0.142315 + 0.246497i −0.0475819 + 0.0824143i 0.459493 0.795865i 0 −0.0270865 0.981929 + 0.189251i 0.546200 0.495472 + 0.858183i 0
333.6 0.142315 + 0.246497i 0.888835 1.53951i 0.459493 0.795865i 0 0.505978 −0.327068 + 0.945001i 0.546200 −1.08006 1.87071i 0
333.7 0.654861 + 1.13425i −0.723734 + 1.25354i −0.357685 + 0.619529i 0 −1.89578 −0.995472 + 0.0950560i 0.372786 −0.547582 0.948440i 0
333.8 0.654861 + 1.13425i −0.235759 + 0.408346i −0.357685 + 0.619529i 0 −0.617557 0.580057 0.814576i 0.372786 0.388835 + 0.673483i 0
333.9 0.959493 + 1.66189i −0.580057 + 1.00469i −1.34125 + 2.32312i 0 −2.22624 −0.786053 + 0.618159i −3.22871 −0.172932 0.299527i 0
333.10 0.959493 + 1.66189i 0.995472 1.72421i −1.34125 + 2.32312i 0 3.82059 0.928368 0.371662i −3.22871 −1.48193 2.56678i 0
667.1 −0.841254 + 1.45709i −0.981929 1.70075i −0.915415 1.58555i 0 3.30420 0.723734 0.690079i 1.39788 −1.42837 + 2.47401i 0
667.2 −0.841254 + 1.45709i 0.327068 + 0.566498i −0.915415 1.58555i 0 −1.10059 0.235759 0.971812i 1.39788 0.286053 0.495458i 0
667.3 −0.415415 + 0.719520i −0.928368 1.60798i 0.154861 + 0.268227i 0 1.54263 0.0475819 + 0.998867i −1.08816 −1.22373 + 2.11957i 0
667.4 −0.415415 + 0.719520i 0.786053 + 1.36148i 0.154861 + 0.268227i 0 −1.30615 −0.888835 + 0.458227i −1.08816 −0.735759 + 1.27437i 0
667.5 0.142315 0.246497i −0.0475819 0.0824143i 0.459493 + 0.795865i 0 −0.0270865 0.981929 0.189251i 0.546200 0.495472 0.858183i 0
667.6 0.142315 0.246497i 0.888835 + 1.53951i 0.459493 + 0.795865i 0 0.505978 −0.327068 0.945001i 0.546200 −1.08006 + 1.87071i 0
667.7 0.654861 1.13425i −0.723734 1.25354i −0.357685 0.619529i 0 −1.89578 −0.995472 0.0950560i 0.372786 −0.547582 + 0.948440i 0
667.8 0.654861 1.13425i −0.235759 0.408346i −0.357685 0.619529i 0 −0.617557 0.580057 + 0.814576i 0.372786 0.388835 0.673483i 0
667.9 0.959493 1.66189i −0.580057 1.00469i −1.34125 2.32312i 0 −2.22624 −0.786053 0.618159i −3.22871 −0.172932 + 0.299527i 0
667.10 0.959493 1.66189i 0.995472 + 1.72421i −1.34125 2.32312i 0 3.82059 0.928368 + 0.371662i −3.22871 −1.48193 + 2.56678i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 667.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
167.b odd 2 1 CM by $$\Q(\sqrt{-167})$$
7.c even 3 1 inner
1169.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1169.1.f.c 20
7.c even 3 1 inner 1169.1.f.c 20
167.b odd 2 1 CM 1169.1.f.c 20
1169.f odd 6 1 inner 1169.1.f.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1169.1.f.c 20 1.a even 1 1 trivial
1169.1.f.c 20 7.c even 3 1 inner
1169.1.f.c 20 167.b odd 2 1 CM
1169.1.f.c 20 1169.f odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - \cdots$$ acting on $$S_{1}^{\mathrm{new}}(1169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 3 T + 12 T^{2} + T^{3} + 20 T^{4} - 7 T^{5} + 16 T^{6} - 2 T^{7} + 5 T^{8} - T^{9} + T^{10} )^{2}$$
$3$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$
$5$ $$T^{20}$$
$7$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$11$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$
$13$ $$T^{20}$$
$17$ $$T^{20}$$
$19$ $$( 1 - 3 T + 12 T^{2} + T^{3} + 20 T^{4} - 7 T^{5} + 16 T^{6} - 2 T^{7} + 5 T^{8} - T^{9} + T^{10} )^{2}$$
$23$ $$T^{20}$$
$29$ $$( 1 - 12 T + 12 T^{2} + 43 T^{3} - 43 T^{4} - 34 T^{5} + 34 T^{6} + 10 T^{7} - 10 T^{8} - T^{9} + T^{10} )^{2}$$
$31$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$
$37$ $$T^{20}$$
$41$ $$T^{20}$$
$43$ $$T^{20}$$
$47$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$
$53$ $$T^{20}$$
$59$ $$T^{20}$$
$61$ $$( 1 - 3 T + 12 T^{2} + T^{3} + 20 T^{4} - 7 T^{5} + 16 T^{6} - 2 T^{7} + 5 T^{8} - T^{9} + T^{10} )^{2}$$
$67$ $$T^{20}$$
$71$ $$T^{20}$$
$73$ $$T^{20}$$
$79$ $$T^{20}$$
$83$ $$T^{20}$$
$89$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$
$97$ $$( 1 + 3 T - 3 T^{2} - 4 T^{3} + T^{4} + T^{5} )^{4}$$