# Properties

 Label 1407.1.cb.a Level $1407$ Weight $1$ Character orbit 1407.cb Analytic conductor $0.702$ Analytic rank $0$ Dimension $20$ Projective image $D_{66}$ 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.cb (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: $$D_{66}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{66} - \cdots)$$

## $q$-expansion

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

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

## Character values

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

 $$n$$ $$337$$ $$470$$ $$1207$$ $$\chi(n)$$ $$-\zeta_{66}^{32}$$ $$-1$$ $$\zeta_{66}^{11}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 −0.995472 − 0.0950560i −0.327068 + 0.945001i 0.580057 − 0.814576i −0.995472 + 0.0950560i −0.888835 − 0.458227i 0.235759 − 0.971812i −0.786053 + 0.618159i 0.0475819 − 0.998867i 0.723734 + 0.690079i −0.786053 − 0.618159i −0.888835 + 0.458227i 0.580057 + 0.814576i 0.0475819 + 0.998867i −0.327068 − 0.945001i 0.981929 + 0.189251i 0.723734 − 0.690079i 0.235759 + 0.971812i 0.928368 + 0.371662i 0.928368 − 0.371662i 0.981929 − 0.189251i
0 0.235759 0.971812i −0.415415 + 0.909632i 0 0 −0.928368 0.371662i 0 −0.888835 0.458227i 0
152.1 0 0.0475819 0.998867i 0.654861 0.755750i 0 0 −0.235759 0.971812i 0 −0.995472 0.0950560i 0
383.1 0 0.723734 + 0.690079i −0.415415 + 0.909632i 0 0 0.786053 0.618159i 0 0.0475819 + 0.998867i 0
404.1 0 0.235759 + 0.971812i −0.415415 0.909632i 0 0 −0.928368 + 0.371662i 0 −0.888835 + 0.458227i 0
446.1 0 0.928368 0.371662i −0.841254 0.540641i 0 0 0.327068 0.945001i 0 0.723734 0.690079i 0
500.1 0 0.981929 0.189251i 0.959493 + 0.281733i 0 0 −0.580057 0.814576i 0 0.928368 0.371662i 0
530.1 0 −0.995472 + 0.0950560i 0.142315 0.989821i 0 0 0.888835 + 0.458227i 0 0.981929 0.189251i 0
593.1 0 −0.786053 + 0.618159i −0.841254 + 0.540641i 0 0 −0.981929 0.189251i 0 0.235759 0.971812i 0
698.1 0 −0.327068 + 0.945001i 0.959493 + 0.281733i 0 0 0.995472 0.0950560i 0 −0.786053 0.618159i 0
815.1 0 −0.995472 0.0950560i 0.142315 + 0.989821i 0 0 0.888835 0.458227i 0 0.981929 + 0.189251i 0
836.1 0 0.928368 + 0.371662i −0.841254 + 0.540641i 0 0 0.327068 + 0.945001i 0 0.723734 + 0.690079i 0
878.1 0 0.723734 0.690079i −0.415415 0.909632i 0 0 0.786053 + 0.618159i 0 0.0475819 0.998867i 0
1025.1 0 −0.786053 0.618159i −0.841254 0.540641i 0 0 −0.981929 + 0.189251i 0 0.235759 + 0.971812i 0
1046.1 0 0.0475819 + 0.998867i 0.654861 + 0.755750i 0 0 −0.235759 + 0.971812i 0 −0.995472 + 0.0950560i 0
1055.1 0 −0.888835 0.458227i 0.654861 + 0.755750i 0 0 −0.723734 0.690079i 0 0.580057 + 0.814576i 0
1151.1 0 −0.327068 0.945001i 0.959493 0.281733i 0 0 0.995472 + 0.0950560i 0 −0.786053 + 0.618159i 0
1286.1 0 0.981929 + 0.189251i 0.959493 0.281733i 0 0 −0.580057 + 0.814576i 0 0.928368 + 0.371662i 0
1319.1 0 0.580057 + 0.814576i 0.142315 0.989821i 0 0 −0.0475819 0.998867i 0 −0.327068 + 0.945001i 0
1391.1 0 0.580057 0.814576i 0.142315 + 0.989821i 0 0 −0.0475819 + 0.998867i 0 −0.327068 0.945001i 0
1403.1 0 −0.888835 + 0.458227i 0.654861 0.755750i 0 0 −0.723734 + 0.690079i 0 0.580057 0.814576i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1403.1 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.bn even 66 1 inner
1407.cb 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.cb.a 20
3.b odd 2 1 CM 1407.1.cb.a 20
7.d odd 6 1 1407.1.ct.a yes 20
21.g even 6 1 1407.1.ct.a yes 20
67.h odd 66 1 1407.1.ct.a yes 20
201.p even 66 1 1407.1.ct.a yes 20
469.bn even 66 1 inner 1407.1.cb.a 20
1407.cb odd 66 1 inner 1407.1.cb.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1407.1.cb.a 20 1.a even 1 1 trivial
1407.1.cb.a 20 3.b odd 2 1 CM
1407.1.cb.a 20 469.bn even 66 1 inner
1407.1.cb.a 20 1407.cb odd 66 1 inner
1407.1.ct.a yes 20 7.d odd 6 1
1407.1.ct.a yes 20 21.g even 6 1
1407.1.ct.a yes 20 67.h odd 66 1
1407.1.ct.a yes 20 201.p even 66 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$5$ $$T^{20}$$
$7$ $$1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20}$$
$11$ $$T^{20}$$
$13$ $$1 - 6 T + 11 T^{2} - 48 T^{3} + 266 T^{4} - 671 T^{5} + 1116 T^{6} - 1207 T^{7} + 869 T^{8} - 284 T^{9} - 45 T^{10} + 94 T^{11} - 33 T^{12} - 29 T^{13} + 42 T^{14} - 16 T^{16} + 8 T^{17} - 2 T^{19} + T^{20}$$
$17$ $$T^{20}$$
$19$ $$1 - 11 T + 51 T^{2} - 77 T^{3} - 17 T^{4} - 55 T^{5} + 145 T^{6} + 341 T^{7} + 234 T^{8} - 33 T^{9} + 164 T^{10} + 11 T^{11} + 103 T^{12} - 27 T^{14} + 22 T^{15} + 9 T^{16} - 3 T^{18} + T^{20}$$
$23$ $$T^{20}$$
$29$ $$T^{20}$$
$31$ $$( 1 + 5 T + 3 T^{2} - 7 T^{3} + 20 T^{4} - 10 T^{5} + 16 T^{6} - 8 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2}$$
$37$ $$( 1 - 12 T + 12 T^{2} + 43 T^{3} - 43 T^{4} - 34 T^{5} + 34 T^{6} + 10 T^{7} - 10 T^{8} - T^{9} + T^{10} )^{2}$$
$41$ $$T^{20}$$
$43$ $$( 11 - 44 T + 77 T^{2} - 55 T^{3} + 11 T^{4} + T^{10} )^{2}$$
$47$ $$T^{20}$$
$53$ $$T^{20}$$
$59$ $$T^{20}$$
$61$ $$1 - 9 T + 146 T^{2} - 744 T^{3} + 2051 T^{4} - 3151 T^{5} + 2658 T^{6} - 971 T^{7} + 119 T^{8} - T^{9} - T^{11} + 9 T^{12} - 25 T^{13} + 18 T^{14} + 6 T^{15} + 5 T^{16} + 4 T^{17} + 3 T^{18} + 2 T^{19} + T^{20}$$
$67$ $$1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20}$$
$71$ $$T^{20}$$
$73$ $$1 - T - 14 T^{2} + 51 T^{3} + 357 T^{4} + 693 T^{5} + 981 T^{6} + 1692 T^{7} + 2656 T^{8} + 3380 T^{9} + 3761 T^{10} + 3664 T^{11} + 3105 T^{12} + 2292 T^{13} + 1458 T^{14} + 792 T^{15} + 361 T^{16} + 134 T^{17} + 39 T^{18} + 8 T^{19} + T^{20}$$
$79$ $$1 + T - 14 T^{2} - 51 T^{3} + 357 T^{4} - 693 T^{5} + 981 T^{6} - 1692 T^{7} + 2656 T^{8} - 3380 T^{9} + 3761 T^{10} - 3664 T^{11} + 3105 T^{12} - 2292 T^{13} + 1458 T^{14} - 792 T^{15} + 361 T^{16} - 134 T^{17} + 39 T^{18} - 8 T^{19} + T^{20}$$
$83$ $$T^{20}$$
$89$ $$T^{20}$$
$97$ $$1 - 12 T + 132 T^{2} - 230 T^{3} + 703 T^{4} - 550 T^{5} + 2025 T^{6} - 1431 T^{7} + 2673 T^{8} - 1220 T^{9} + 1935 T^{10} - 714 T^{11} + 968 T^{12} - 254 T^{13} + 320 T^{14} - 66 T^{15} + 76 T^{16} - 10 T^{17} + 11 T^{18} - T^{19} + T^{20}$$