# Properties

 Label 1191.1.z.a Level $1191$ Weight $1$ Character orbit 1191.z Analytic conductor $0.594$ Analytic rank $0$ Dimension $20$ Projective image $D_{66}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1191 = 3 \cdot 397$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1191.z (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.594386430046$$ Analytic rank: $$0$$ Dimension: $$20$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{66}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{66} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{66}^{16} q^{3} + \zeta_{66}^{9} q^{4} + ( -\zeta_{66}^{12} - \zeta_{66}^{23} ) q^{7} + \zeta_{66}^{32} q^{9} +O(q^{10})$$ $$q + \zeta_{66}^{16} q^{3} + \zeta_{66}^{9} q^{4} + ( -\zeta_{66}^{12} - \zeta_{66}^{23} ) q^{7} + \zeta_{66}^{32} q^{9} + \zeta_{66}^{25} q^{12} + ( \zeta_{66}^{12} + \zeta_{66}^{17} ) q^{13} + \zeta_{66}^{18} q^{16} + ( \zeta_{66}^{14} - \zeta_{66}^{27} ) q^{19} + ( \zeta_{66}^{6} - \zeta_{66}^{28} ) q^{21} + \zeta_{66}^{31} q^{25} -\zeta_{66}^{15} q^{27} + ( -\zeta_{66}^{21} - \zeta_{66}^{32} ) q^{28} + ( \zeta_{66}^{20} - \zeta_{66}^{25} ) q^{31} -\zeta_{66}^{8} q^{36} + ( -\zeta_{66}^{10} + \zeta_{66}^{13} ) q^{37} + ( -1 + \zeta_{66}^{28} ) q^{39} + ( -\zeta_{66}^{20} - \zeta_{66}^{22} ) q^{43} -\zeta_{66} q^{48} + ( -\zeta_{66}^{2} - \zeta_{66}^{13} + \zeta_{66}^{24} ) q^{49} + ( \zeta_{66}^{21} + \zeta_{66}^{26} ) q^{52} + ( \zeta_{66}^{10} + \zeta_{66}^{30} ) q^{57} + ( -\zeta_{66}^{18} - \zeta_{66}^{19} ) q^{61} + ( \zeta_{66}^{11} + \zeta_{66}^{22} ) q^{63} + \zeta_{66}^{27} q^{64} + ( -\zeta_{66}^{4} + \zeta_{66}^{19} ) q^{67} + ( -\zeta_{66}^{5} - \zeta_{66}^{9} ) q^{73} -\zeta_{66}^{14} q^{75} + ( \zeta_{66}^{3} + \zeta_{66}^{23} ) q^{76} + ( \zeta_{66}^{5} - \zeta_{66}^{6} ) q^{79} -\zeta_{66}^{31} q^{81} + ( \zeta_{66}^{4} + \zeta_{66}^{15} ) q^{84} + ( \zeta_{66}^{2} + \zeta_{66}^{7} - \zeta_{66}^{24} - \zeta_{66}^{29} ) q^{91} + ( -\zeta_{66}^{3} + \zeta_{66}^{8} ) q^{93} + ( -\zeta_{66}^{16} - \zeta_{66}^{30} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + q^{3} + 2q^{4} + 3q^{7} + q^{9} + O(q^{10})$$ $$20q + q^{3} + 2q^{4} + 3q^{7} + q^{9} - q^{12} - 3q^{13} - 2q^{16} - q^{19} - 3q^{21} - q^{25} - 2q^{27} - 3q^{28} + 2q^{31} - q^{36} - 2q^{37} - 19q^{39} + 9q^{43} + q^{48} - 2q^{49} + 3q^{52} - q^{57} + 3q^{61} + 2q^{64} - 2q^{67} - q^{73} - q^{75} + q^{76} + q^{79} + q^{81} + 3q^{84} + 3q^{91} - q^{93} + q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1191\mathbb{Z}\right)^\times$$.

 $$n$$ $$398$$ $$799$$ $$\chi(n)$$ $$-1$$ $$\zeta_{66}^{31}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.723734 + 0.690079i −0.995472 + 0.0950560i 0.0475819 − 0.998867i −0.888835 − 0.458227i 0.981929 − 0.189251i −0.995472 − 0.0950560i 0.580057 + 0.814576i −0.327068 + 0.945001i 0.0475819 + 0.998867i −0.786053 − 0.618159i 0.723734 − 0.690079i 0.928368 − 0.371662i 0.580057 − 0.814576i −0.327068 − 0.945001i 0.928368 + 0.371662i 0.235759 + 0.971812i 0.981929 + 0.189251i 0.235759 − 0.971812i −0.786053 + 0.618159i −0.888835 + 0.458227i
0 0.928368 0.371662i −0.841254 0.540641i 0 0 1.19525 1.25354i 0 0.723734 0.690079i 0
83.1 0 0.0475819 0.998867i 0.654861 0.755750i 0 0 0.164642 + 1.72421i 0 −0.995472 0.0950560i 0
137.1 0 0.723734 + 0.690079i −0.415415 + 0.909632i 0 0 −1.73009 0.0824143i 0 0.0475819 + 0.998867i 0
191.1 0 0.235759 + 0.971812i −0.415415 0.909632i 0 0 −0.793672 + 1.53951i 0 −0.888835 + 0.458227i 0
257.1 0 −0.995472 0.0950560i 0.142315 + 0.989821i 0 0 0.327793 + 1.70075i 0 0.981929 + 0.189251i 0
287.1 0 0.0475819 + 0.998867i 0.654861 + 0.755750i 0 0 0.164642 1.72421i 0 −0.995472 + 0.0950560i 0
440.1 0 −0.888835 + 0.458227i 0.654861 0.755750i 0 0 −1.41089 + 1.00469i 0 0.580057 0.814576i 0
533.1 0 0.580057 0.814576i 0.142315 + 0.989821i 0 0 1.63679 + 0.566498i 0 −0.327068 0.945001i 0
539.1 0 0.723734 0.690079i −0.415415 0.909632i 0 0 −1.73009 + 0.0824143i 0 0.0475819 0.998867i 0
560.1 0 −0.327068 0.945001i 0.959493 0.281733i 0 0 1.07068 1.36148i 0 −0.786053 + 0.618159i 0
623.1 0 0.928368 + 0.371662i −0.841254 + 0.540641i 0 0 1.19525 + 1.25354i 0 0.723734 + 0.690079i 0
647.1 0 0.981929 + 0.189251i 0.959493 0.281733i 0 0 −0.643738 1.60798i 0 0.928368 + 0.371662i 0
674.1 0 −0.888835 0.458227i 0.654861 + 0.755750i 0 0 −1.41089 1.00469i 0 0.580057 + 0.814576i 0
686.1 0 0.580057 + 0.814576i 0.142315 0.989821i 0 0 1.63679 0.566498i 0 −0.327068 + 0.945001i 0
821.1 0 0.981929 0.189251i 0.959493 + 0.281733i 0 0 −0.643738 + 1.60798i 0 0.928368 0.371662i 0
824.1 0 −0.786053 + 0.618159i −0.841254 + 0.540641i 0 0 1.68323 0.408346i 0 0.235759 0.971812i 0
1001.1 0 −0.995472 + 0.0950560i 0.142315 0.989821i 0 0 0.327793 1.70075i 0 0.981929 0.189251i 0
1019.1 0 −0.786053 0.618159i −0.841254 0.540641i 0 0 1.68323 + 0.408346i 0 0.235759 + 0.971812i 0
1040.1 0 −0.327068 + 0.945001i 0.959493 + 0.281733i 0 0 1.07068 + 1.36148i 0 −0.786053 0.618159i 0
1085.1 0 0.235759 0.971812i −0.415415 + 0.909632i 0 0 −0.793672 1.53951i 0 −0.888835 0.458227i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1085.1 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 $$\Q(\sqrt{-3})$$
397.n even 66 1 inner
1191.z odd 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1191.1.z.a 20
3.b odd 2 1 CM 1191.1.z.a 20
397.n even 66 1 inner 1191.1.z.a 20
1191.z odd 66 1 inner 1191.1.z.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1191.1.z.a 20 1.a even 1 1 trivial
1191.1.z.a 20 3.b odd 2 1 CM
1191.1.z.a 20 397.n even 66 1 inner
1191.1.z.a 20 1191.z odd 66 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1191, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$5$ $$T^{20}$$
$7$ $$59049 - 59049 T + 39366 T^{2} - 19683 T^{3} + 6561 T^{4} - 2187 T^{6} + 2187 T^{7} - 1458 T^{8} + 729 T^{9} - 243 T^{10} + 243 T^{11} - 162 T^{12} + 81 T^{13} - 27 T^{14} + 9 T^{16} - 9 T^{17} + 6 T^{18} - 3 T^{19} + T^{20}$$
$11$ $$T^{20}$$
$13$ $$1 - 10 T + 30 T^{2} - 51 T^{3} + 247 T^{4} - 462 T^{5} + 585 T^{6} - 350 T^{7} + 181 T^{8} + 96 T^{9} + 76 T^{10} + 43 T^{11} + 25 T^{12} - 4 T^{13} - 38 T^{14} + 9 T^{16} + 9 T^{17} + 6 T^{18} + 3 T^{19} + T^{20}$$
$17$ $$T^{20}$$
$19$ $$1 + 12 T + 154 T^{2} + 626 T^{3} + 934 T^{4} + 253 T^{5} - 560 T^{6} - 87 T^{7} + 605 T^{8} - 155 T^{9} + 87 T^{10} + 241 T^{11} + 11 T^{12} + 89 T^{13} + 78 T^{14} + 10 T^{16} + 10 T^{17} + T^{19} + T^{20}$$
$23$ $$T^{20}$$
$29$ $$T^{20}$$
$31$ $$1 - 13 T + 157 T^{2} - 565 T^{3} + 1149 T^{4} - 1491 T^{5} + 1613 T^{6} - 767 T^{7} + 768 T^{8} - 1011 T^{9} + 528 T^{10} - 43 T^{11} + 31 T^{12} - 8 T^{13} - 37 T^{14} + 16 T^{15} + 5 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20}$$
$37$ $$1 + 6 T + 22 T^{2} + 70 T^{3} + 145 T^{4} - 22 T^{5} - 17 T^{6} + 360 T^{7} + 209 T^{8} - 46 T^{9} + 318 T^{10} + 170 T^{11} - 22 T^{12} + 139 T^{13} + 75 T^{14} + 28 T^{16} + 14 T^{17} + 2 T^{19} + T^{20}$$
$41$ $$T^{20}$$
$43$ $$1 + 13 T + 36 T^{2} - 381 T^{3} + 742 T^{4} - 874 T^{5} + 1965 T^{6} - 3578 T^{7} + 5069 T^{8} - 6194 T^{9} + 6633 T^{10} - 6194 T^{11} + 5047 T^{12} - 3567 T^{13} + 2174 T^{14} - 1127 T^{15} + 489 T^{16} - 172 T^{17} + 47 T^{18} - 9 T^{19} + T^{20}$$
$47$ $$T^{20}$$
$53$ $$T^{20}$$
$59$ $$T^{20}$$
$61$ $$1 - T + 19 T^{2} + 62 T^{3} + 104 T^{4} - 198 T^{5} - 119 T^{6} + 427 T^{7} - 347 T^{8} + 3 T^{9} + 395 T^{10} - 395 T^{11} + 300 T^{12} - 62 T^{13} - 38 T^{14} + 66 T^{15} - 24 T^{16} + 2 T^{17} + 6 T^{18} - 3 T^{19} + T^{20}$$
$67$ $$1 + 6 T + 11 T^{2} + 48 T^{3} + 266 T^{4} + 671 T^{5} + 1116 T^{6} + 1207 T^{7} + 869 T^{8} + 284 T^{9} - 45 T^{10} - 94 T^{11} - 33 T^{12} + 29 T^{13} + 42 T^{14} - 16 T^{16} - 8 T^{17} + 2 T^{19} + T^{20}$$
$71$ $$T^{20}$$
$73$ $$1 - 10 T + 88 T^{2} + 197 T^{3} + 384 T^{4} + 792 T^{5} + 188 T^{6} - 1451 T^{7} - 792 T^{8} + 417 T^{9} + 483 T^{10} + 65 T^{11} + 88 T^{12} + 34 T^{13} - 43 T^{14} - 44 T^{15} - T^{16} - T^{17} + T^{19} + T^{20}$$
$79$ $$1 - 12 T + 132 T^{2} - 230 T^{3} + 703 T^{4} - 550 T^{5} + 2025 T^{6} - 1431 T^{7} + 2673 T^{8} - 1220 T^{9} + 1935 T^{10} - 714 T^{11} + 968 T^{12} - 254 T^{13} + 320 T^{14} - 66 T^{15} + 76 T^{16} - 10 T^{17} + 11 T^{18} - T^{19} + T^{20}$$
$83$ $$T^{20}$$
$89$ $$T^{20}$$
$97$ $$1 + 21 T + 143 T^{2} + 243 T^{3} + 593 T^{4} + 1331 T^{5} + 1464 T^{6} + 472 T^{7} + 242 T^{8} - 109 T^{9} - 12 T^{10} + 122 T^{11} - 99 T^{12} - 67 T^{13} + 12 T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$