LMFDB - Newform orbit 712.1.y.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [712,1,Mod(99,712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(712, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([22, 22, 43]))

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

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

chi := DirichletCharacter("712.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 712 = 2^{3} \cdot 89 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	712.y (of order $44$, 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:	$0.355334288995$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{44})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{44}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{44} - \cdots)$

$q$-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_{44}^{20} q^{2} + (\zeta_{44}^{16} - \zeta_{44}^{3}) q^{3} - \zeta_{44}^{18} q^{4} + (\zeta_{44}^{14} - \zeta_{44}) q^{6} - \zeta_{44}^{16} q^{8} + ( - \zeta_{44}^{19} + \cdots + \zeta_{44}^{6}) q^{9} +O(q^{10}) $ q - z^20 * q^2 + (z^16 - z^3) * q^3 - z^18 * q^4 + (z^14 - z) * q^6 - z^16 * q^8 + (-z^19 - z^10 + z^6) * q^9	$ q - \zeta_{44}^{20} q^{2} + (\zeta_{44}^{16} - \zeta_{44}^{3}) q^{3} - \zeta_{44}^{18} q^{4} + (\zeta_{44}^{14} - \zeta_{44}) q^{6} - \zeta_{44}^{16} q^{8} + ( - \zeta_{44}^{19} + \cdots + \zeta_{44}^{6}) q^{9} + \cdots + ( - \zeta_{44}^{21} + \cdots + \zeta_{44}^{7}) q^{99} +O(q^{100}) $ q - z^20 * q^2 + (z^16 - z^3) * q^3 - z^18 * q^4 + (z^14 - z) * q^6 - z^16 * q^8 + (-z^19 - z^10 + z^6) * q^9 + (z^11 + z) * q^11 + (z^21 + z^12) * q^12 - z^14 * q^16 + (-z^15 - z^11) * q^17 + (-z^17 - z^8 + z^4) * q^18 + (z^3 - z^2) * q^19 + (-z^21 + z^9) * q^22 + (z^19 + z^10) * q^24 + z^10 * q^25 + (-z^13 - z^9 + z^4 + 1) * q^27 - z^12 * q^32 + (z^17 - z^14 - z^5 - z^4) * q^33 + (-z^13 - z^9) * q^34 + (-z^15 - z^6 + z^2) * q^36 + (z - 1) * q^38 + (z^18 + z^7) * q^41 + (z^15 - z^8) * q^43 + (-z^19 + z^7) * q^44 + (z^17 + z^8) * q^48 + z^21 * q^49 + z^8 * q^50 + (z^18 + z^14 + z^9 + z^5) * q^51 + (z^20 + z^11 - z^7 + z^2) * q^54 + (z^19 - z^18 - z^6 + z^5) * q^57 + (z^20 - z^5) * q^59 - z^10 * q^64 + (z^15 - z^12 - z^3 - z^2) * q^66 + (-z^7 - z) * q^67 + (-z^11 - z^7) * q^68 + (-z^13 - z^4 + 1) * q^72 + (z^15 + z^13) * q^73 + (-z^13 - z^4) * q^75 + (-z^21 + z^20) * q^76 + (z^20 + z^16 + z^12 + z^7 - z^3) * q^81 + (z^16 + z^5) * q^82 + (-z^20 - z^17) * q^83 + (z^13 - z^6) * q^86 + (-z^17 + z^5) * q^88 - z^12 * q^89 + (z^15 + z^6) * q^96 + (z^8 - z^2) * q^97 + z^19 * q^98 + (-z^21 - z^20 + z^17 - z^11 + z^8 + z^7) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8}+O(q^{10}) $ 20 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 2 * q^8	$ 20 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8} - 2 q^{12} - 2 q^{16} - 2 q^{19} + 2 q^{24} + 2 q^{25} - 22 q^{27} + 2 q^{32} - 20 q^{38} + 2 q^{41} + 2 q^{43} - 2 q^{48} - 2 q^{50} + 4 q^{51} - 4 q^{57} - 2 q^{59} - 2 q^{64} + 22 q^{72} + 2 q^{75} - 2 q^{76} - 2 q^{81} - 2 q^{82} + 2 q^{83} - 2 q^{86} + 2 q^{89} + 2 q^{96} - 4 q^{97}+O(q^{100}) $ 20 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 2 * q^8 - 2 * q^12 - 2 * q^16 - 2 * q^19 + 2 * q^24 + 2 * q^25 - 22 * q^27 + 2 * q^32 - 20 * q^38 + 2 * q^41 + 2 * q^43 - 2 * q^48 - 2 * q^50 + 4 * q^51 - 4 * q^57 - 2 * q^59 - 2 * q^64 + 22 * q^72 + 2 * q^75 - 2 * q^76 - 2 * q^81 - 2 * q^82 + 2 * q^83 - 2 * q^86 + 2 * q^89 + 2 * q^96 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/712\mathbb{Z}\right)^\times$.

$n$	$357$	$535$	$537$
$\chi(n)$	$-1$	$-1$	$-\zeta_{44}^{19}$

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} $

99.1

−0.540641	+	0.841254i
0.909632	−	0.415415i
−0.281733	+	0.959493i
−0.540641	−	0.841254i
0.989821	−	0.142315i
−0.755750	+	0.654861i
0.755750	−	0.654861i
−0.989821	+	0.142315i
0.540641	+	0.841254i
0.281733	−	0.959493i
−0.909632	+	0.415415i
0.540641	−	0.841254i
0.909632	+	0.415415i
0.989821	+	0.142315i
−0.281733	−	0.959493i
0.755750	+	0.654861i
−0.755750	−	0.654861i
0.281733	+	0.959493i
−0.989821	−	0.142315i
−0.909632	−	0.415415i

−0.415415

0.909632i

−1.94931

0.139418i

−0.654861

−

0.755750i

0.682956

−

1.83107i

0.959493

−

0.281733i

2.79057

−

0.401223i

107.1

0.654861

0.755750i

0.559521

0.418852i

−0.142315

0.989821i

0.0498610

0.697148i

−0.841254

0.540641i

−0.144106

−

0.490780i

131.1

−0.841254

0.540641i

−0.898064

−

0.334961i

0.415415

−

0.909632i

0.936593

−

0.203743i

0.142315

0.989821i

−0.0614286

−

0.0532282i

187.1

−0.415415

−

0.909632i

−1.94931

−

0.139418i

−0.654861

0.755750i

0.682956

1.83107i

0.959493

0.281733i

2.79057

0.401223i

195.1

0.959493

0.281733i

−1.56449

−

0.340335i

0.841254

0.540641i

−1.40524

−

0.767317i

0.654861

0.755750i

1.42218

0.649487i

227.1

0.142315

0.989821i

−0.125226

0.0683785i

−0.959493

0.281733i

−0.0855040

−

0.114220i

−0.415415

−

0.909632i

−0.529635

0.824128i

307.1

0.142315

0.989821i

0.956056

1.75089i

−0.959493

0.281733i

−1.59700

1.19550i

−0.415415

−

0.909632i

−1.61092

2.50664i

339.1

0.959493

0.281733i

0.254771

−

1.17116i

0.841254

0.540641i

0.574406

−

1.05195i

0.654861

0.755750i

−0.397086

−

0.181343i

347.1

−0.415415

−

0.909632i

0.0303285

−

0.424047i

−0.654861

0.755750i

−0.398326

0.148568i

0.959493

0.281733i

0.810925

0.116593i

403.1

−0.841254

0.540641i

0.613435

−

1.64468i

0.415415

−

0.909632i

0.373128

1.71524i

0.142315

0.989821i

−1.57293

−

1.36295i

427.1

0.654861

0.755750i

1.12299

−

1.50013i

−0.142315

0.989821i

1.86912

−

0.133682i

−0.841254

0.540641i

−0.707571

−

2.40977i

435.1

−0.415415

0.909632i

0.0303285

0.424047i

−0.654861

−

0.755750i

−0.398326

−

0.148568i

0.959493

−

0.281733i

0.810925

−

0.116593i

539.1

0.654861

−

0.755750i

0.559521

−

0.418852i

−0.142315

−

0.989821i

0.0498610

−

0.697148i

−0.841254

−

0.540641i

−0.144106

0.490780i

555.1

0.959493

−

0.281733i

−1.56449

0.340335i

0.841254

−

0.540641i

−1.40524

0.767317i

0.654861

−

0.755750i

1.42218

−

0.649487i

587.1

−0.841254

−

0.540641i

−0.898064

0.334961i

0.415415

0.909632i

0.936593

0.203743i

0.142315

−

0.989821i

−0.0614286

0.0532282i

603.1

0.142315

−

0.989821i

0.956056

−

1.75089i

−0.959493

−

0.281733i

−1.59700

−

1.19550i

−0.415415

0.909632i

−1.61092

−

2.50664i

643.1

0.142315

−

0.989821i

−0.125226

−

0.0683785i

−0.959493

−

0.281733i

−0.0855040

0.114220i

−0.415415

0.909632i

−0.529635

−

0.824128i

659.1

−0.841254

−

0.540641i

0.613435

1.64468i

0.415415

0.909632i

0.373128

−

1.71524i

0.142315

−

0.989821i

−1.57293

1.36295i

691.1

0.959493

−

0.281733i

0.254771

1.17116i

0.841254

−

0.540641i

0.574406

1.05195i

0.654861

−

0.755750i

−0.397086

0.181343i

707.1

0.654861

−

0.755750i

1.12299

1.50013i

−0.142315

−

0.989821i

1.86912

0.133682i

−0.841254

−

0.540641i

−0.707571

2.40977i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
89.g	even	44	1	inner
712.y	odd	44	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	712.1.y.a	✓	20
4.b	odd	2	1		2848.1.cc.a		20
8.b	even	2	1		2848.1.cc.a		20
8.d	odd	2	1	CM	712.1.y.a	✓	20
89.g	even	44	1	inner	712.1.y.a	✓	20
356.n	odd	44	1		2848.1.cc.a		20
712.y	odd	44	1	inner	712.1.y.a	✓	20
712.z	even	44	1		2848.1.cc.a		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
712.1.y.a	✓	20	1.a	even	1	1	trivial
712.1.y.a	✓	20	8.d	odd	2	1	CM
712.1.y.a	✓	20	89.g	even	44	1	inner
712.1.y.a	✓	20	712.y	odd	44	1	inner
2848.1.cc.a		20	4.b	odd	2	1
2848.1.cc.a		20	8.b	even	2	1
2848.1.cc.a		20	356.n	odd	44	1
2848.1.cc.a		20	712.z	even	44	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(712, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} $ (T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$3$	$ T^{20} + 2 T^{19} + \cdots + 1 $ T^20 + 2T^19 + 2T^18 + 22T^17 + 40T^16 + 36T^15 + 179T^14 + 286T^13 + 214T^12 + 626T^11 + 824T^10 + 394T^9 + 713T^8 + 638T^7 - 51T^6 + 74T^5 + 250T^4 + 44T^3 + 50T^2 + 12*T + 1
$5$	$ T^{20} $ T^20
$7$	$ T^{20} $ T^20
$11$	$ T^{20} + 11 T^{18} + \cdots + 121 $ T^20 + 11T^18 + 55T^16 + 165T^14 + 330T^12 + 462T^10 + 484T^8 + 1089T^4 - 605T^2 + 121
$13$	$ T^{20} $ T^20
$17$	$ T^{20} + 7 T^{18} + \cdots + 1 $ T^20 + 7T^18 + 27T^16 + 57T^14 + 102T^12 + 54T^10 + 224T^8 - 236T^6 + 185T^4 - 25*T^2 + 1
$19$	$ T^{20} + 2 T^{19} + \cdots + 1 $ T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 - 8T^14 + 93T^12 - 122T^11 - 331T^10 - 420T^9 - 178T^8 + 484T^7 + 1324T^6 + 1702T^5 + 1218T^4 + 484T^3 + 105T^2 + 12*T + 1
$23$	$ T^{20} $ T^20
$29$	$ T^{20} $ T^20
$31$	$ T^{20} $ T^20
$37$	$ T^{20} $ T^20
$41$	$ T^{20} - 2 T^{19} + \cdots + 1024 $ T^20 - 2T^19 + 2T^18 - 4T^16 + 8T^15 - 8T^14 + 16T^12 - 32T^11 + 32T^10 - 64T^9 + 64T^8 - 128T^6 + 256T^5 - 256T^4 + 512T^2 - 1024*T + 1024
$43$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 + 2T^18 - 4T^16 + 8T^15 - 8T^14 + 16T^12 - 32T^11 + 32T^10 + 2T^9 + 53T^8 + 1100T^7 + 2413T^6 + 2412T^5 + 1361T^4 + 462T^3 + 94T^2 + 10*T + 1
$47$	$ T^{20} $ T^20
$53$	$ T^{20} $ T^20
$59$	$ T^{20} + 2 T^{19} + \cdots + 1 $ T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 - 8T^14 + 66T^13 + 148T^12 + 164T^11 + 494T^10 + 658T^9 + 328T^8 - 176T^7 - 403T^6 - 454T^5 + 63T^4 + 66T^3 + 61T^2 + 12T + 1
$61$	$ T^{20} $ T^20
$67$	$ T^{20} + 22 T^{16} + \cdots + 121 $ T^20 + 22T^16 + 165T^12 + 99T^10 + 484T^8 - 968T^6 + 484T^4 + 605*T^2 + 121
$71$	$ T^{20} $ T^20
$73$	$ T^{20} - 22 T^{14} + \cdots + 121 $ T^20 - 22T^14 + 154T^12 - 22T^10 + 121T^8 + 1331T^6 + 1331T^4 + 242*T^2 + 121
$79$	$ T^{20} $ T^20
$83$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 + 2T^18 - 4T^16 + 30T^15 - 52T^14 + 44T^13 + 93T^12 - 274T^11 + 505T^10 - 460T^9 - 90T^8 - 110T^7 + 400T^6 - 316T^5 + 338T^4 - 44T^3 + 17T^2 + 10T + 1
$89$	$ (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} $ (T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$97$	$ (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
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Level:	\( N \)	\(=\)	\( 712 = 2^{3} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	712.y (of order \(44\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{44}^{20} q^{2} + (\zeta_{44}^{16} - \zeta_{44}^{3}) q^{3} - \zeta_{44}^{18} q^{4} + (\zeta_{44}^{14} - \zeta_{44}) q^{6} - \zeta_{44}^{16} q^{8} + ( - \zeta_{44}^{19} + \cdots + \zeta_{44}^{6}) q^{9} +O(q^{10}) \) q - z^20 * q^2 + (z^16 - z^3) * q^3 - z^18 * q^4 + (z^14 - z) * q^6 - z^16 * q^8 + (-z^19 - z^10 + z^6) * q^9	\( q - \zeta_{44}^{20} q^{2} + (\zeta_{44}^{16} - \zeta_{44}^{3}) q^{3} - \zeta_{44}^{18} q^{4} + (\zeta_{44}^{14} - \zeta_{44}) q^{6} - \zeta_{44}^{16} q^{8} + ( - \zeta_{44}^{19} + \cdots + \zeta_{44}^{6}) q^{9} + \cdots + ( - \zeta_{44}^{21} + \cdots + \zeta_{44}^{7}) q^{99} +O(q^{100}) \) q - z^20 * q^2 + (z^16 - z^3) * q^3 - z^18 * q^4 + (z^14 - z) * q^6 - z^16 * q^8 + (-z^19 - z^10 + z^6) * q^9 + (z^11 + z) * q^11 + (z^21 + z^12) * q^12 - z^14 * q^16 + (-z^15 - z^11) * q^17 + (-z^17 - z^8 + z^4) * q^18 + (z^3 - z^2) * q^19 + (-z^21 + z^9) * q^22 + (z^19 + z^10) * q^24 + z^10 * q^25 + (-z^13 - z^9 + z^4 + 1) * q^27 - z^12 * q^32 + (z^17 - z^14 - z^5 - z^4) * q^33 + (-z^13 - z^9) * q^34 + (-z^15 - z^6 + z^2) * q^36 + (z - 1) * q^38 + (z^18 + z^7) * q^41 + (z^15 - z^8) * q^43 + (-z^19 + z^7) * q^44 + (z^17 + z^8) * q^48 + z^21 * q^49 + z^8 * q^50 + (z^18 + z^14 + z^9 + z^5) * q^51 + (z^20 + z^11 - z^7 + z^2) * q^54 + (z^19 - z^18 - z^6 + z^5) * q^57 + (z^20 - z^5) * q^59 - z^10 * q^64 + (z^15 - z^12 - z^3 - z^2) * q^66 + (-z^7 - z) * q^67 + (-z^11 - z^7) * q^68 + (-z^13 - z^4 + 1) * q^72 + (z^15 + z^13) * q^73 + (-z^13 - z^4) * q^75 + (-z^21 + z^20) * q^76 + (z^20 + z^16 + z^12 + z^7 - z^3) * q^81 + (z^16 + z^5) * q^82 + (-z^20 - z^17) * q^83 + (z^13 - z^6) * q^86 + (-z^17 + z^5) * q^88 - z^12 * q^89 + (z^15 + z^6) * q^96 + (z^8 - z^2) * q^97 + z^19 * q^98 + (-z^21 - z^20 + z^17 - z^11 + z^8 + z^7) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8}+O(q^{10}) \) 20 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 2 * q^8	\( 20 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8} - 2 q^{12} - 2 q^{16} - 2 q^{19} + 2 q^{24} + 2 q^{25} - 22 q^{27} + 2 q^{32} - 20 q^{38} + 2 q^{41} + 2 q^{43} - 2 q^{48} - 2 q^{50} + 4 q^{51} - 4 q^{57} - 2 q^{59} - 2 q^{64} + 22 q^{72} + 2 q^{75} - 2 q^{76} - 2 q^{81} - 2 q^{82} + 2 q^{83} - 2 q^{86} + 2 q^{89} + 2 q^{96} - 4 q^{97}+O(q^{100}) \) 20 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 2 * q^8 - 2 * q^12 - 2 * q^16 - 2 * q^19 + 2 * q^24 + 2 * q^25 - 22 * q^27 + 2 * q^32 - 20 * q^38 + 2 * q^41 + 2 * q^43 - 2 * q^48 - 2 * q^50 + 4 * q^51 - 4 * q^57 - 2 * q^59 - 2 * q^64 + 22 * q^72 + 2 * q^75 - 2 * q^76 - 2 * q^81 - 2 * q^82 + 2 * q^83 - 2 * q^86 + 2 * q^89 + 2 * q^96 - 4 * q^97

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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	712.1.y.a

Level	$712$
Weight	$1$
Character orbit	712.y
Analytic conductor	$0.355$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{44}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.355334288995\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\Q(\zeta_{44})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{44}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{44} - \cdots)\)

\(n\)	\(357\)	\(535\)	\(537\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\zeta_{44}^{19}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 99.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
89.g	even	44	1	inner
712.y	odd	44	1	inner

$p$	$F_p(T)$
$2$	\( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) (T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$3$	\( T^{20} + 2 T^{19} + \cdots + 1 \) T^20 + 2T^19 + 2T^18 + 22T^17 + 40T^16 + 36T^15 + 179T^14 + 286T^13 + 214T^12 + 626T^11 + 824T^10 + 394T^9 + 713T^8 + 638T^7 - 51T^6 + 74T^5 + 250T^4 + 44T^3 + 50T^2 + 12*T + 1
$5$	\( T^{20} \) T^20
$7$	\( T^{20} \) T^20
$11$	\( T^{20} + 11 T^{18} + \cdots + 121 \) T^20 + 11T^18 + 55T^16 + 165T^14 + 330T^12 + 462T^10 + 484T^8 + 1089T^4 - 605T^2 + 121
$13$	\( T^{20} \) T^20
$17$	\( T^{20} + 7 T^{18} + \cdots + 1 \) T^20 + 7T^18 + 27T^16 + 57T^14 + 102T^12 + 54T^10 + 224T^8 - 236T^6 + 185T^4 - 25*T^2 + 1
$19$	\( T^{20} + 2 T^{19} + \cdots + 1 \) T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 - 8T^14 + 93T^12 - 122T^11 - 331T^10 - 420T^9 - 178T^8 + 484T^7 + 1324T^6 + 1702T^5 + 1218T^4 + 484T^3 + 105T^2 + 12*T + 1
$23$	\( T^{20} \) T^20
$29$	\( T^{20} \) T^20
$31$	\( T^{20} \) T^20
$37$	\( T^{20} \) T^20
$41$	\( T^{20} - 2 T^{19} + \cdots + 1024 \) T^20 - 2T^19 + 2T^18 - 4T^16 + 8T^15 - 8T^14 + 16T^12 - 32T^11 + 32T^10 - 64T^9 + 64T^8 - 128T^6 + 256T^5 - 256T^4 + 512T^2 - 1024*T + 1024
$43$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 + 2T^18 - 4T^16 + 8T^15 - 8T^14 + 16T^12 - 32T^11 + 32T^10 + 2T^9 + 53T^8 + 1100T^7 + 2413T^6 + 2412T^5 + 1361T^4 + 462T^3 + 94T^2 + 10*T + 1
$47$	\( T^{20} \) T^20
$53$	\( T^{20} \) T^20
$59$	\( T^{20} + 2 T^{19} + \cdots + 1 \) T^20 + 2T^19 + 2T^18 - 4T^16 - 8T^15 - 8T^14 + 66T^13 + 148T^12 + 164T^11 + 494T^10 + 658T^9 + 328T^8 - 176T^7 - 403T^6 - 454T^5 + 63T^4 + 66T^3 + 61T^2 + 12T + 1
$61$	\( T^{20} \) T^20
$67$	\( T^{20} + 22 T^{16} + \cdots + 121 \) T^20 + 22T^16 + 165T^12 + 99T^10 + 484T^8 - 968T^6 + 484T^4 + 605*T^2 + 121
$71$	\( T^{20} \) T^20
$73$	\( T^{20} - 22 T^{14} + \cdots + 121 \) T^20 - 22T^14 + 154T^12 - 22T^10 + 121T^8 + 1331T^6 + 1331T^4 + 242*T^2 + 121
$79$	\( T^{20} \) T^20
$83$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 + 2T^18 - 4T^16 + 30T^15 - 52T^14 + 44T^13 + 93T^12 - 274T^11 + 505T^10 - 460T^9 - 90T^8 - 110T^7 + 400T^6 - 316T^5 + 338T^4 - 44T^3 + 17T^2 + 10T + 1
$89$	\( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) (T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$97$	\( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) (T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1)^2
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