# Properties

 Label 1407.1.cz.a Level $1407$ Weight $1$ Character orbit 1407.cz 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.cz (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.

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

## Character values

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

 $$n$$ $$337$$ $$470$$ $$1207$$ $$\chi(n)$$ $$-\zeta_{66}^{5}$$ $$-1$$ $$-\zeta_{66}^{11}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 −0.995472 + 0.0950560i −0.327068 + 0.945001i 0.981929 + 0.189251i 0.928368 − 0.371662i −0.327068 − 0.945001i −0.786053 − 0.618159i 0.235759 + 0.971812i 0.235759 − 0.971812i 0.0475819 − 0.998867i −0.888835 + 0.458227i −0.786053 + 0.618159i 0.928368 + 0.371662i 0.580057 + 0.814576i 0.981929 − 0.189251i −0.995472 − 0.0950560i 0.723734 − 0.690079i −0.888835 − 0.458227i 0.580057 − 0.814576i 0.723734 + 0.690079i 0.0475819 + 0.998867i
0 0.928368 0.371662i 0.841254 + 0.540641i 0 0 −0.327068 + 0.945001i 0 0.723734 0.690079i 0
65.1 0 0.235759 + 0.971812i 0.415415 + 0.909632i 0 0 0.928368 0.371662i 0 −0.888835 + 0.458227i 0
86.1 0 0.723734 + 0.690079i 0.415415 0.909632i 0 0 −0.786053 + 0.618159i 0 0.0475819 + 0.998867i 0
317.1 0 0.0475819 0.998867i −0.654861 + 0.755750i 0 0 0.235759 + 0.971812i 0 −0.995472 0.0950560i 0
368.1 0 0.235759 0.971812i 0.415415 0.909632i 0 0 0.928368 + 0.371662i 0 −0.888835 0.458227i 0
473.1 0 −0.888835 + 0.458227i −0.654861 + 0.755750i 0 0 0.723734 0.690079i 0 0.580057 0.814576i 0
485.1 0 0.580057 0.814576i −0.142315 0.989821i 0 0 0.0475819 0.998867i 0 −0.327068 0.945001i 0
557.1 0 0.580057 + 0.814576i −0.142315 + 0.989821i 0 0 0.0475819 + 0.998867i 0 −0.327068 + 0.945001i 0
590.1 0 0.981929 + 0.189251i −0.959493 + 0.281733i 0 0 0.580057 0.814576i 0 0.928368 + 0.371662i 0
725.1 0 −0.327068 0.945001i −0.959493 + 0.281733i 0 0 −0.995472 0.0950560i 0 −0.786053 + 0.618159i 0
821.1 0 −0.888835 0.458227i −0.654861 0.755750i 0 0 0.723734 + 0.690079i 0 0.580057 + 0.814576i 0
830.1 0 0.0475819 + 0.998867i −0.654861 0.755750i 0 0 0.235759 0.971812i 0 −0.995472 + 0.0950560i 0
851.1 0 −0.786053 0.618159i 0.841254 + 0.540641i 0 0 0.981929 0.189251i 0 0.235759 + 0.971812i 0
998.1 0 0.723734 0.690079i 0.415415 + 0.909632i 0 0 −0.786053 0.618159i 0 0.0475819 0.998867i 0
1040.1 0 0.928368 + 0.371662i 0.841254 0.540641i 0 0 −0.327068 0.945001i 0 0.723734 + 0.690079i 0
1061.1 0 −0.995472 0.0950560i −0.142315 0.989821i 0 0 −0.888835 + 0.458227i 0 0.981929 + 0.189251i 0
1178.1 0 −0.327068 + 0.945001i −0.959493 0.281733i 0 0 −0.995472 + 0.0950560i 0 −0.786053 0.618159i 0
1283.1 0 −0.786053 + 0.618159i 0.841254 0.540641i 0 0 0.981929 + 0.189251i 0 0.235759 0.971812i 0
1346.1 0 −0.995472 + 0.0950560i −0.142315 + 0.989821i 0 0 −0.888835 0.458227i 0 0.981929 0.189251i 0
1376.1 0 0.981929 0.189251i −0.959493 0.281733i 0 0 0.580057 + 0.814576i 0 0.928368 0.371662i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1376.1 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.y even 33 1 inner
1407.cz 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.cz.a yes 20
3.b odd 2 1 CM 1407.1.cz.a yes 20
7.c even 3 1 1407.1.cg.a 20
21.h odd 6 1 1407.1.cg.a 20
67.g even 33 1 1407.1.cg.a 20
201.o odd 66 1 1407.1.cg.a 20
469.y even 33 1 inner 1407.1.cz.a yes 20
1407.cz odd 66 1 inner 1407.1.cz.a yes 20

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$5$ $$T^{20}$$
$7$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$11$ $$T^{20}$$
$13$ $$1 - 6 T + 11 T^{2} - 48 T^{3} + 266 T^{4} - 671 T^{5} + 1116 T^{6} - 1207 T^{7} + 869 T^{8} - 284 T^{9} - 45 T^{10} + 94 T^{11} - 33 T^{12} - 29 T^{13} + 42 T^{14} - 16 T^{16} + 8 T^{17} - 2 T^{19} + T^{20}$$
$17$ $$T^{20}$$
$19$ $$1 - 13 T + 157 T^{2} - 565 T^{3} + 1149 T^{4} - 1491 T^{5} + 1613 T^{6} - 767 T^{7} + 768 T^{8} - 1011 T^{9} + 528 T^{10} - 43 T^{11} + 31 T^{12} - 8 T^{13} - 37 T^{14} + 16 T^{15} + 5 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20}$$
$23$ $$T^{20}$$
$29$ $$T^{20}$$
$31$ $$( 1 - 5 T + 3 T^{2} + 7 T^{3} + 20 T^{4} + 10 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$37$ $$( 1 - 12 T + 12 T^{2} + 43 T^{3} - 43 T^{4} - 34 T^{5} + 34 T^{6} + 10 T^{7} - 10 T^{8} - T^{9} + T^{10} )^{2}$$
$41$ $$T^{20}$$
$43$ $$( 1 + 6 T + 25 T^{2} + 51 T^{3} + 53 T^{4} + 32 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$47$ $$T^{20}$$
$53$ $$T^{20}$$
$59$ $$T^{20}$$
$61$ $$1 + 9 T + 146 T^{2} + 744 T^{3} + 2051 T^{4} + 3151 T^{5} + 2658 T^{6} + 971 T^{7} + 119 T^{8} + T^{9} + T^{11} + 9 T^{12} + 25 T^{13} + 18 T^{14} - 6 T^{15} + 5 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20}$$
$67$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$71$ $$T^{20}$$
$73$ $$1 + 23 T + 198 T^{2} + 615 T^{3} + 967 T^{4} + 1705 T^{5} + 5193 T^{6} + 12860 T^{7} + 22352 T^{8} + 29402 T^{9} + 31107 T^{10} + 27356 T^{11} + 20317 T^{12} + 12816 T^{13} + 6854 T^{14} + 3080 T^{15} + 1143 T^{16} + 340 T^{17} + 77 T^{18} + 12 T^{19} + T^{20}$$
$79$ $$1 + 23 T + 198 T^{2} + 615 T^{3} + 967 T^{4} + 1705 T^{5} + 5193 T^{6} + 12860 T^{7} + 22352 T^{8} + 29402 T^{9} + 31107 T^{10} + 27356 T^{11} + 20317 T^{12} + 12816 T^{13} + 6854 T^{14} + 3080 T^{15} + 1143 T^{16} + 340 T^{17} + 77 T^{18} + 12 T^{19} + T^{20}$$
$83$ $$T^{20}$$
$89$ $$T^{20}$$
$97$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$