# Properties

 Label 1407.1.cg.a Level $1407$ Weight $1$ Character orbit 1407.cg Analytic conductor $0.702$ Analytic rank $0$ Dimension $20$ Projective image $D_{33}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1407 = 3 \cdot 7 \cdot 67$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1407.cg (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.702184472775$$ 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: $$D_{33}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{33} - \cdots)$$

## $q$-expansion

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1407\mathbb{Z}\right)^\times$$.

 $$n$$ $$337$$ $$470$$ $$1207$$ $$\chi(n)$$ $$-\zeta_{66}^{29}$$ $$-1$$ $$\zeta_{66}^{22}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 −0.888835 − 0.458227i −0.327068 − 0.945001i −0.995472 + 0.0950560i −0.327068 + 0.945001i 0.723734 − 0.690079i 0.0475819 − 0.998867i 0.0475819 + 0.998867i −0.995472 − 0.0950560i 0.981929 − 0.189251i 0.981929 + 0.189251i −0.888835 + 0.458227i 0.928368 + 0.371662i 0.928368 − 0.371662i 0.580057 − 0.814576i 0.235759 − 0.971812i −0.786053 + 0.618159i −0.786053 − 0.618159i 0.580057 + 0.814576i 0.723734 + 0.690079i 0.235759 + 0.971812i
0 −0.888835 + 0.458227i 0.981929 + 0.189251i 0 0 0.415415 0.909632i 0 0.580057 0.814576i 0
170.1 0 −0.327068 + 0.945001i 0.723734 0.690079i 0 0 −0.142315 0.989821i 0 −0.786053 0.618159i 0
284.1 0 −0.995472 0.0950560i −0.786053 + 0.618159i 0 0 −0.654861 0.755750i 0 0.981929 + 0.189251i 0
389.1 0 −0.327068 0.945001i 0.723734 + 0.690079i 0 0 −0.142315 + 0.989821i 0 −0.786053 + 0.618159i 0
569.1 0 0.723734 + 0.690079i 0.580057 + 0.814576i 0 0 0.841254 + 0.540641i 0 0.0475819 + 0.998867i 0
620.1 0 0.0475819 + 0.998867i −0.327068 + 0.945001i 0 0 0.415415 + 0.909632i 0 −0.995472 + 0.0950560i 0
674.1 0 0.0475819 0.998867i −0.327068 0.945001i 0 0 0.415415 0.909632i 0 −0.995472 0.0950560i 0
758.1 0 −0.995472 + 0.0950560i −0.786053 0.618159i 0 0 −0.654861 + 0.755750i 0 0.981929 0.189251i 0
926.1 0 0.981929 + 0.189251i 0.235759 0.971812i 0 0 −0.142315 + 0.989821i 0 0.928368 + 0.371662i 0
977.1 0 0.981929 0.189251i 0.235759 + 0.971812i 0 0 −0.142315 0.989821i 0 0.928368 0.371662i 0
1031.1 0 −0.888835 0.458227i 0.981929 0.189251i 0 0 0.415415 + 0.909632i 0 0.580057 + 0.814576i 0
1052.1 0 0.928368 0.371662i −0.888835 + 0.458227i 0 0 −0.959493 + 0.281733i 0 0.723734 0.690079i 0
1082.1 0 0.928368 + 0.371662i −0.888835 0.458227i 0 0 −0.959493 0.281733i 0 0.723734 + 0.690079i 0
1145.1 0 0.580057 + 0.814576i 0.928368 0.371662i 0 0 −0.654861 + 0.755750i 0 −0.327068 + 0.945001i 0
1199.1 0 0.235759 + 0.971812i −0.995472 0.0950560i 0 0 0.841254 0.540641i 0 −0.888835 + 0.458227i 0
1229.1 0 −0.786053 0.618159i 0.0475819 0.998867i 0 0 −0.959493 + 0.281733i 0 0.235759 + 0.971812i 0
1241.1 0 −0.786053 + 0.618159i 0.0475819 + 0.998867i 0 0 −0.959493 0.281733i 0 0.235759 0.971812i 0
1262.1 0 0.580057 0.814576i 0.928368 + 0.371662i 0 0 −0.654861 0.755750i 0 −0.327068 0.945001i 0
1271.1 0 0.723734 0.690079i 0.580057 0.814576i 0 0 0.841254 0.540641i 0 0.0475819 0.998867i 0
1292.1 0 0.235759 0.971812i −0.995472 + 0.0950560i 0 0 0.841254 + 0.540641i 0 −0.888835 0.458227i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1292.1 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 $$\Q(\sqrt{-3})$$
469.bb even 33 1 inner
1407.cg odd 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1407.1.cg.a 20
3.b odd 2 1 CM 1407.1.cg.a 20
7.c even 3 1 1407.1.cz.a yes 20
21.h odd 6 1 1407.1.cz.a yes 20
67.g even 33 1 1407.1.cz.a yes 20
201.o odd 66 1 1407.1.cz.a yes 20
469.bb even 33 1 inner 1407.1.cg.a 20
1407.cg odd 66 1 inner 1407.1.cg.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1407.1.cg.a 20 1.a even 1 1 trivial
1407.1.cg.a 20 3.b odd 2 1 CM
1407.1.cg.a 20 469.bb even 33 1 inner
1407.1.cg.a 20 1407.cg odd 66 1 inner
1407.1.cz.a yes 20 7.c even 3 1
1407.1.cz.a yes 20 21.h odd 6 1
1407.1.cz.a yes 20 67.g even 33 1
1407.1.cz.a yes 20 201.o odd 66 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1407, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$5$ $$T^{20}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$11$ $$T^{20}$$
$13$ $$1 - 6 T + 11 T^{2} - 48 T^{3} + 266 T^{4} - 671 T^{5} + 1116 T^{6} - 1207 T^{7} + 869 T^{8} - 284 T^{9} - 45 T^{10} + 94 T^{11} - 33 T^{12} - 29 T^{13} + 42 T^{14} - 16 T^{16} + 8 T^{17} - 2 T^{19} + T^{20}$$
$17$ $$T^{20}$$
$19$ $$1 + 23 T + 154 T^{2} + 230 T^{3} - 12 T^{4} + 881 T^{6} + 760 T^{7} + 1452 T^{8} + 450 T^{9} + 450 T^{10} - T^{11} + 22 T^{12} + 34 T^{13} + 23 T^{14} + 22 T^{15} - T^{16} - T^{17} + T^{19} + T^{20}$$
$23$ $$T^{20}$$
$29$ $$T^{20}$$
$31$ $$1 + 5 T + 22 T^{2} + 29 T^{3} + 24 T^{4} - 231 T^{5} + 71 T^{6} - 8 T^{7} + 198 T^{8} + 24 T^{9} - 243 T^{10} + 116 T^{11} + 121 T^{12} - 95 T^{13} + 20 T^{14} + 22 T^{15} - 16 T^{16} + 8 T^{17} - 2 T^{19} + T^{20}$$
$37$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$
$41$ $$T^{20}$$
$43$ $$( 1 + 6 T + 25 T^{2} + 51 T^{3} + 53 T^{4} + 32 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$47$ $$T^{20}$$
$53$ $$T^{20}$$
$59$ $$T^{20}$$
$61$ $$1 - 21 T + 143 T^{2} - 243 T^{3} + 593 T^{4} - 1331 T^{5} + 1464 T^{6} - 472 T^{7} + 242 T^{8} + 109 T^{9} - 12 T^{10} - 122 T^{11} - 99 T^{12} + 67 T^{13} + 12 T^{14} - T^{16} - T^{17} + T^{19} + T^{20}$$
$67$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$71$ $$T^{20}$$
$73$ $$1 - 13 T + 36 T^{2} + 381 T^{3} + 742 T^{4} + 874 T^{5} + 1965 T^{6} + 3578 T^{7} + 5069 T^{8} + 6194 T^{9} + 6633 T^{10} + 6194 T^{11} + 5047 T^{12} + 3567 T^{13} + 2174 T^{14} + 1127 T^{15} + 489 T^{16} + 172 T^{17} + 47 T^{18} + 9 T^{19} + T^{20}$$
$79$ $$1 - 13 T + 36 T^{2} + 381 T^{3} + 742 T^{4} + 874 T^{5} + 1965 T^{6} + 3578 T^{7} + 5069 T^{8} + 6194 T^{9} + 6633 T^{10} + 6194 T^{11} + 5047 T^{12} + 3567 T^{13} + 2174 T^{14} + 1127 T^{15} + 489 T^{16} + 172 T^{17} + 47 T^{18} + 9 T^{19} + T^{20}$$
$83$ $$T^{20}$$
$89$ $$T^{20}$$
$97$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$