LMFDB - Newform orbit 1407.1.cz.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1407,1,Mod(23,1407)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1407, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([33, 22, 28]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1407.23");

S:= CuspForms(chi, 1);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.702184472775$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
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 $ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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

0.928368

−

0.371662i

0.841254

0.540641i

−0.327068

0.945001i

0.723734

−

0.690079i

65.1

0.235759

0.971812i

0.415415

0.909632i

0.928368

−

0.371662i

−0.888835

0.458227i

86.1

0.723734

0.690079i

0.415415

−

0.909632i

−0.786053

0.618159i

0.0475819

0.998867i

317.1

0.0475819

−

0.998867i

−0.654861

0.755750i

0.235759

0.971812i

−0.995472

−

0.0950560i

368.1

0.235759

−

0.971812i

0.415415

−

0.909632i

0.928368

0.371662i

−0.888835

−

0.458227i

473.1

−0.888835

0.458227i

−0.654861

0.755750i

0.723734

−

0.690079i

0.580057

−

0.814576i

485.1

0.580057

−

0.814576i

−0.142315

−

0.989821i

0.0475819

−

0.998867i

−0.327068

−

0.945001i

557.1

0.580057

0.814576i

−0.142315

0.989821i

0.0475819

0.998867i

−0.327068

0.945001i

590.1

0.981929

0.189251i

−0.959493

0.281733i

0.580057

−

0.814576i

0.928368

0.371662i

725.1

−0.327068

−

0.945001i

−0.959493

0.281733i

−0.995472

−

0.0950560i

−0.786053

0.618159i

821.1

−0.888835

−

0.458227i

−0.654861

−

0.755750i

0.723734

0.690079i

0.580057

0.814576i

830.1

0.0475819

0.998867i

−0.654861

−

0.755750i

0.235759

−

0.971812i

−0.995472

0.0950560i

851.1

−0.786053

−

0.618159i

0.841254

0.540641i

0.981929

−

0.189251i

0.235759

0.971812i

998.1

0.723734

−

0.690079i

0.415415

0.909632i

−0.786053

−

0.618159i

0.0475819

−

0.998867i

1040.1

0.928368

0.371662i

0.841254

−

0.540641i

−0.327068

−

0.945001i

0.723734

0.690079i

1061.1

−0.995472

−

0.0950560i

−0.142315

−

0.989821i

−0.888835

0.458227i

0.981929

0.189251i

1178.1

−0.327068

0.945001i

−0.959493

−

0.281733i

−0.995472

0.0950560i

−0.786053

−

0.618159i

1283.1

−0.786053

0.618159i

0.841254

−

0.540641i

0.981929

0.189251i

0.235759

−

0.971812i

1346.1

−0.995472

0.0950560i

−0.142315

0.989821i

−0.888835

−

0.458227i

0.981929

−

0.189251i

1376.1

0.981929

−

0.189251i

−0.959493

−

0.281733i

0.580057

0.814576i

0.928368

−

0.371662i

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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	$4$

Self dual:	no
Analytic conductor:	\(0.702184472775\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\Q(\zeta_{33})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) 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 \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

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

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 23.1
Significant digits:
Format:

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

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