LMFDB - Newform orbit 1156.1.m.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1156,1,Mod(35,1156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1156, base_ring=CyclotomicField(34))

chi = DirichletCharacter(H, H._module([17, 14]))

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

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

chi := DirichletCharacter("1156.35");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1156 = 2^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1156.m (of order $34$, degree $16$, 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:	$0.576919154604$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{34})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 $ x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{17}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{17} - \cdots)$

$q$-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_{34}^{6} q^{2} + \zeta_{34}^{12} q^{4} + ( - \zeta_{34}^{15} + 1) q^{5} - \zeta_{34} q^{8} + \zeta_{34}^{4} q^{9} +O(q^{10}) $ q + z^6 * q^2 + z^12 * q^4 + (-z^15 + 1) * q^5 - z * q^8 + z^4 * q^9	$ q + \zeta_{34}^{6} q^{2} + \zeta_{34}^{12} q^{4} + ( - \zeta_{34}^{15} + 1) q^{5} - \zeta_{34} q^{8} + \zeta_{34}^{4} q^{9} + (\zeta_{34}^{6} + \zeta_{34}^{4}) q^{10} + (\zeta_{34}^{14} - \zeta_{34}^{5}) q^{13} - \zeta_{34}^{7} q^{16} - \zeta_{34}^{7} q^{17} + \zeta_{34}^{10} q^{18} + (\zeta_{34}^{12} + \zeta_{34}^{10}) q^{20} + ( - \zeta_{34}^{15} - \zeta_{34}^{13} + 1) q^{25} + ( - \zeta_{34}^{11} - \zeta_{34}^{3}) q^{26} + (\zeta_{34}^{16} + \zeta_{34}^{8}) q^{29} - \zeta_{34}^{13} q^{32} - \zeta_{34}^{13} q^{34} + \zeta_{34}^{16} q^{36} + (\zeta_{34}^{14} - \zeta_{34}^{9}) q^{37} + (\zeta_{34}^{16} - \zeta_{34}) q^{40} + ( - \zeta_{34}^{3} - \zeta_{34}) q^{41} + (\zeta_{34}^{4} + \zeta_{34}^{2}) q^{45} + \zeta_{34}^{8} q^{49} + (\zeta_{34}^{6} + \cdots + \zeta_{34}^{2}) q^{50} + \cdots + \zeta_{34}^{14} q^{98} +O(q^{100}) $ q + z^6 * q^2 + z^12 * q^4 + (-z^15 + 1) * q^5 - z * q^8 + z^4 * q^9 + (z^6 + z^4) * q^10 + (z^14 - z^5) * q^13 - z^7 * q^16 - z^7 * q^17 + z^10 * q^18 + (z^12 + z^10) * q^20 + (-z^15 - z^13 + 1) * q^25 + (-z^11 - z^3) * q^26 + (z^16 + z^8) * q^29 - z^13 * q^32 - z^13 * q^34 + z^16 * q^36 + (z^14 - z^9) * q^37 + (z^16 - z) * q^40 + (-z^3 - z) * q^41 + (z^4 + z^2) * q^45 + z^8 * q^49 + (z^6 + z^4 + z^2) * q^50 + (-z^9 + 1) * q^52 + (-z^13 + z^12) * q^53 + (z^14 - z^5) * q^58 + (z^10 + z^6) * q^61 + z^2 * q^64 + (z^14 + z^12 - z^5 - z^3) * q^65 + z^2 * q^68 - z^5 * q^72 + (-z^11 - z^3) * q^73 + (-z^15 - z^3) * q^74 + (-z^7 - z^5) * q^80 + z^8 * q^81 + (-z^9 - z^7) * q^82 + (-z^7 - z^5) * q^85 + (z^10 - z^5) * q^89 + (z^10 + z^8) * q^90 + (z^16 - z^9) * q^97 + z^14 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - q^{2} - q^{4} + 15 q^{5} - q^{8} - q^{9}+O(q^{10}) $ 16 * q - q^2 - q^4 + 15 * q^5 - q^8 - q^9	$ 16 q - q^{2} - q^{4} + 15 q^{5} - q^{8} - q^{9} - 2 q^{10} - 2 q^{13} - q^{16} - q^{17} - q^{18} - 2 q^{20} + 14 q^{25} - 2 q^{26} - 2 q^{29} - q^{32} - q^{34} - q^{36} - 2 q^{37} - 2 q^{40} - 2 q^{41} - 2 q^{45} - q^{49} - 3 q^{50} + 15 q^{52} - 2 q^{53} - 2 q^{58} - 2 q^{61} - q^{64} - 4 q^{65} - q^{68} - q^{72} - 2 q^{73} - 2 q^{74} - 2 q^{80} - q^{81} - 2 q^{82} - 2 q^{85} - 2 q^{89} - 2 q^{90} - 2 q^{97} - q^{98}+O(q^{100}) $ 16 * q - q^2 - q^4 + 15 * q^5 - q^8 - q^9 - 2 * q^10 - 2 * q^13 - q^16 - q^17 - q^18 - 2 * q^20 + 14 * q^25 - 2 * q^26 - 2 * q^29 - q^32 - q^34 - q^36 - 2 * q^37 - 2 * q^40 - 2 * q^41 - 2 * q^45 - q^49 - 3 * q^50 + 15 * q^52 - 2 * q^53 - 2 * q^58 - 2 * q^61 - q^64 - 4 * q^65 - q^68 - q^72 - 2 * q^73 - 2 * q^74 - 2 * q^80 - q^81 - 2 * q^82 - 2 * q^85 - 2 * q^89 - 2 * q^90 - 2 * q^97 - q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times$.

$n$	$579$	$581$
$\chi(n)$	$-1$	$\zeta_{34}^{4}$

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} $

35.1

0.602635	−	0.798017i
−0.932472	−	0.361242i
−0.0922684	+	0.995734i
0.982973	−	0.183750i
−0.445738	−	0.895163i
−0.739009	+	0.673696i
0.850217	+	0.526432i
0.273663	−	0.961826i
0.273663	+	0.961826i
0.850217	−	0.526432i
−0.739009	−	0.673696i
−0.445738	+	0.895163i
0.982973	+	0.183750i
−0.0922684	−	0.995734i
−0.932472	+	0.361242i
0.602635	+	0.798017i

0.739009

0.673696i

0.0922684

0.995734i

0.726337

0.961826i

−0.602635

0.798017i

−0.850217

0.526432i

−0.111208

1.20013i

103.1

−0.602635

0.798017i

−0.273663

−

0.961826i

1.73901

−

0.673696i

0.932472

0.361242i

0.0922684

0.995734i

−0.510366

1.79375i

171.1

−0.850217

−

0.526432i

0.445738

0.895163i

0.0170269

0.183750i

0.0922684

−

0.995734i

0.932472

0.361242i

0.0822551

−

0.165190i

239.1

0.445738

−

0.895163i

−0.602635

−

0.798017i

1.93247

0.361242i

−0.982973

0.183750i

0.739009

−

0.673696i

1.18475

−

1.56886i

307.1

0.932472

0.361242i

0.739009

0.673696i

0.397365

−

0.798017i

0.445738

0.895163i

−0.273663

−

0.961826i

0.658809

−

0.600584i

375.1

−0.273663

0.961826i

−0.850217

−

0.526432i

1.09227

0.995734i

0.739009

−

0.673696i

−0.982973

−

0.183750i

−1.25664

0.778076i

443.1

−0.982973

−

0.183750i

0.932472

0.361242i

1.44574

−

0.895163i

−0.850217

−

0.526432i

−0.602635

0.798017i

−1.58561

0.614268i

511.1

0.0922684

−

0.995734i

−0.982973

−

0.183750i

0.149783

0.526432i

−0.273663

0.961826i

0.445738

0.895163i

0.538007

−

0.100571i

647.1

0.0922684

0.995734i

−0.982973

0.183750i

0.149783

−

0.526432i

−0.273663

−

0.961826i

0.445738

−

0.895163i

0.538007

0.100571i

715.1

−0.982973

0.183750i

0.932472

−

0.361242i

1.44574

0.895163i

−0.850217

0.526432i

−0.602635

−

0.798017i

−1.58561

−

0.614268i

783.1

−0.273663

−

0.961826i

−0.850217

0.526432i

1.09227

−

0.995734i

0.739009

0.673696i

−0.982973

0.183750i

−1.25664

−

0.778076i

851.1

0.932472

−

0.361242i

0.739009

−

0.673696i

0.397365

0.798017i

0.445738

−

0.895163i

−0.273663

0.961826i

0.658809

0.600584i

919.1

0.445738

0.895163i

−0.602635

0.798017i

1.93247

−

0.361242i

−0.982973

−

0.183750i

0.739009

0.673696i

1.18475

1.56886i

987.1

−0.850217

0.526432i

0.445738

−

0.895163i

0.0170269

−

0.183750i

0.0922684

0.995734i

0.932472

−

0.361242i

0.0822551

0.165190i

1055.1

−0.602635

−

0.798017i

−0.273663

0.961826i

1.73901

0.673696i

0.932472

−

0.361242i

0.0922684

−

0.995734i

−0.510366

−

1.79375i

1123.1

0.739009

−

0.673696i

0.0922684

−

0.995734i

0.726337

−

0.961826i

−0.602635

−

0.798017i

−0.850217

−

0.526432i

−0.111208

−

1.20013i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
289.f	even	17	1	inner
1156.m	odd	34	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1156.1.m.a	✓	16
4.b	odd	2	1	CM	1156.1.m.a	✓	16
289.f	even	17	1	inner	1156.1.m.a	✓	16
1156.m	odd	34	1	inner	1156.1.m.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1156.1.m.a	✓	16	1.a	even	1	1	trivial
1156.1.m.a	✓	16	4.b	odd	2	1	CM
1156.1.m.a	✓	16	289.f	even	17	1	inner
1156.1.m.a	✓	16	1156.m	odd	34	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1156, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + T^{15} + \cdots + 1 $ T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	$ T^{16} $ T^16
$5$	$ T^{16} - 15 T^{15} + \cdots + 1 $ T^16 - 15T^15 + 106T^14 - 468T^13 + 1444T^12 - 3300T^11 + 5776T^10 - 7896T^9 + 8518T^8 - 7274T^7 + 4900T^6 - 2576T^5 + 1036T^4 - 308T^3 + 64T^2 - 8*T + 1
$7$	$ T^{16} $ T^16
$11$	$ T^{16} $ T^16
$13$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 15T^11 + 30T^10 + 60T^9 + T^8 + 2T^7 + 72T^6 + 59T^5 + 118T^4 + 15T^3 + 13T^2 - 8T + 1
$17$	$ T^{16} + T^{15} + \cdots + 1 $ T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$19$	$ T^{16} $ T^16
$23$	$ T^{16} $ T^16
$29$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 128T^9 + 256T^8 + 495T^7 + 786T^6 + 858T^5 + 594T^4 + 253T^3 + 64T^2 + 9*T + 1
$31$	$ T^{16} $ T^16
$37$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 - T^12 - 2T^11 - 4T^10 - 25T^9 + 35T^8 + 70T^7 + 140T^6 + 59T^5 - T^4 - 2T^3 + 30T^2 - 8*T + 1
$41$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 77T^9 + 154T^8 + 70T^7 + 140T^6 + 25T^5 + 33T^4 - 36T^3 + 13T^2 + 9*T + 1
$43$	$ T^{16} $ T^16
$47$	$ T^{16} $ T^16
$53$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 - T^12 - 2T^11 - 4T^10 - 25T^9 + 35T^8 + 70T^7 + 140T^6 + 59T^5 - T^4 - 2T^3 + 30T^2 - 8*T + 1
$59$	$ T^{16} $ T^16
$61$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 15T^11 + 30T^10 + 60T^9 + T^8 + 2T^7 + 72T^6 + 59T^5 + 118T^4 + 15T^3 + 13T^2 - 8T + 1
$67$	$ T^{16} $ T^16
$71$	$ T^{16} $ T^16
$73$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 - 9T^13 - 18T^12 - 36T^11 + 30T^10 + 60T^9 + 120T^8 - 15T^7 - 47T^6 - 94T^5 + 50T^4 + 15T^3 + 30T^2 + 9*T + 1
$79$	$ T^{16} $ T^16
$83$	$ T^{16} $ T^16
$89$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 128T^9 + 256T^8 + 495T^7 + 786T^6 + 858T^5 + 594T^4 + 253T^3 + 64T^2 + 9*T + 1
$97$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 - 9T^13 - 18T^12 - 36T^11 + 30T^10 + 60T^9 + 120T^8 - 15T^7 - 47T^6 - 94T^5 + 50T^4 + 15T^3 + 30T^2 + 9*T + 1
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Overview	Random
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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:	\( N \)	\(=\)	\( 1156 = 2^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1156.m (of order \(34\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{34}^{6} q^{2} + \zeta_{34}^{12} q^{4} + ( - \zeta_{34}^{15} + 1) q^{5} - \zeta_{34} q^{8} + \zeta_{34}^{4} q^{9} +O(q^{10}) \) q + z^6 * q^2 + z^12 * q^4 + (-z^15 + 1) * q^5 - z * q^8 + z^4 * q^9	\( q + \zeta_{34}^{6} q^{2} + \zeta_{34}^{12} q^{4} + ( - \zeta_{34}^{15} + 1) q^{5} - \zeta_{34} q^{8} + \zeta_{34}^{4} q^{9} + (\zeta_{34}^{6} + \zeta_{34}^{4}) q^{10} + (\zeta_{34}^{14} - \zeta_{34}^{5}) q^{13} - \zeta_{34}^{7} q^{16} - \zeta_{34}^{7} q^{17} + \zeta_{34}^{10} q^{18} + (\zeta_{34}^{12} + \zeta_{34}^{10}) q^{20} + ( - \zeta_{34}^{15} - \zeta_{34}^{13} + 1) q^{25} + ( - \zeta_{34}^{11} - \zeta_{34}^{3}) q^{26} + (\zeta_{34}^{16} + \zeta_{34}^{8}) q^{29} - \zeta_{34}^{13} q^{32} - \zeta_{34}^{13} q^{34} + \zeta_{34}^{16} q^{36} + (\zeta_{34}^{14} - \zeta_{34}^{9}) q^{37} + (\zeta_{34}^{16} - \zeta_{34}) q^{40} + ( - \zeta_{34}^{3} - \zeta_{34}) q^{41} + (\zeta_{34}^{4} + \zeta_{34}^{2}) q^{45} + \zeta_{34}^{8} q^{49} + (\zeta_{34}^{6} + \cdots + \zeta_{34}^{2}) q^{50} + \cdots + \zeta_{34}^{14} q^{98} +O(q^{100}) \) q + z^6 * q^2 + z^12 * q^4 + (-z^15 + 1) * q^5 - z * q^8 + z^4 * q^9 + (z^6 + z^4) * q^10 + (z^14 - z^5) * q^13 - z^7 * q^16 - z^7 * q^17 + z^10 * q^18 + (z^12 + z^10) * q^20 + (-z^15 - z^13 + 1) * q^25 + (-z^11 - z^3) * q^26 + (z^16 + z^8) * q^29 - z^13 * q^32 - z^13 * q^34 + z^16 * q^36 + (z^14 - z^9) * q^37 + (z^16 - z) * q^40 + (-z^3 - z) * q^41 + (z^4 + z^2) * q^45 + z^8 * q^49 + (z^6 + z^4 + z^2) * q^50 + (-z^9 + 1) * q^52 + (-z^13 + z^12) * q^53 + (z^14 - z^5) * q^58 + (z^10 + z^6) * q^61 + z^2 * q^64 + (z^14 + z^12 - z^5 - z^3) * q^65 + z^2 * q^68 - z^5 * q^72 + (-z^11 - z^3) * q^73 + (-z^15 - z^3) * q^74 + (-z^7 - z^5) * q^80 + z^8 * q^81 + (-z^9 - z^7) * q^82 + (-z^7 - z^5) * q^85 + (z^10 - z^5) * q^89 + (z^10 + z^8) * q^90 + (z^16 - z^9) * q^97 + z^14 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} - q^{4} + 15 q^{5} - q^{8} - q^{9}+O(q^{10}) \) 16 * q - q^2 - q^4 + 15 * q^5 - q^8 - q^9	\( 16 q - q^{2} - q^{4} + 15 q^{5} - q^{8} - q^{9} - 2 q^{10} - 2 q^{13} - q^{16} - q^{17} - q^{18} - 2 q^{20} + 14 q^{25} - 2 q^{26} - 2 q^{29} - q^{32} - q^{34} - q^{36} - 2 q^{37} - 2 q^{40} - 2 q^{41} - 2 q^{45} - q^{49} - 3 q^{50} + 15 q^{52} - 2 q^{53} - 2 q^{58} - 2 q^{61} - q^{64} - 4 q^{65} - q^{68} - q^{72} - 2 q^{73} - 2 q^{74} - 2 q^{80} - q^{81} - 2 q^{82} - 2 q^{85} - 2 q^{89} - 2 q^{90} - 2 q^{97} - q^{98}+O(q^{100}) \) 16 * q - q^2 - q^4 + 15 * q^5 - q^8 - q^9 - 2 * q^10 - 2 * q^13 - q^16 - q^17 - q^18 - 2 * q^20 + 14 * q^25 - 2 * q^26 - 2 * q^29 - q^32 - q^34 - q^36 - 2 * q^37 - 2 * q^40 - 2 * q^41 - 2 * q^45 - q^49 - 3 * q^50 + 15 * q^52 - 2 * q^53 - 2 * q^58 - 2 * q^61 - q^64 - 4 * q^65 - q^68 - q^72 - 2 * q^73 - 2 * q^74 - 2 * q^80 - q^81 - 2 * q^82 - 2 * q^85 - 2 * q^89 - 2 * q^90 - 2 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1156.1.m.a

Level	$1156$
Weight	$1$
Character orbit	1156.m
Analytic conductor	$0.577$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{17}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.576919154604\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\Q(\zeta_{34})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 \) x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{17}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{17} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 35.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
289.f	even	17	1	inner
1156.m	odd	34	1	inner

$p$	$F_p(T)$
$2$	\( T^{16} + T^{15} + \cdots + 1 \) T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	\( T^{16} \) T^16
$5$	\( T^{16} - 15 T^{15} + \cdots + 1 \) T^16 - 15T^15 + 106T^14 - 468T^13 + 1444T^12 - 3300T^11 + 5776T^10 - 7896T^9 + 8518T^8 - 7274T^7 + 4900T^6 - 2576T^5 + 1036T^4 - 308T^3 + 64T^2 - 8*T + 1
$7$	\( T^{16} \) T^16
$11$	\( T^{16} \) T^16
$13$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 15T^11 + 30T^10 + 60T^9 + T^8 + 2T^7 + 72T^6 + 59T^5 + 118T^4 + 15T^3 + 13T^2 - 8T + 1
$17$	\( T^{16} + T^{15} + \cdots + 1 \) T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$19$	\( T^{16} \) T^16
$23$	\( T^{16} \) T^16
$29$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 128T^9 + 256T^8 + 495T^7 + 786T^6 + 858T^5 + 594T^4 + 253T^3 + 64T^2 + 9*T + 1
$31$	\( T^{16} \) T^16
$37$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 - T^12 - 2T^11 - 4T^10 - 25T^9 + 35T^8 + 70T^7 + 140T^6 + 59T^5 - T^4 - 2T^3 + 30T^2 - 8*T + 1
$41$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 77T^9 + 154T^8 + 70T^7 + 140T^6 + 25T^5 + 33T^4 - 36T^3 + 13T^2 + 9*T + 1
$43$	\( T^{16} \) T^16
$47$	\( T^{16} \) T^16
$53$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 - T^12 - 2T^11 - 4T^10 - 25T^9 + 35T^8 + 70T^7 + 140T^6 + 59T^5 - T^4 - 2T^3 + 30T^2 - 8*T + 1
$59$	\( T^{16} \) T^16
$61$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 15T^11 + 30T^10 + 60T^9 + T^8 + 2T^7 + 72T^6 + 59T^5 + 118T^4 + 15T^3 + 13T^2 - 8T + 1
$67$	\( T^{16} \) T^16
$71$	\( T^{16} \) T^16
$73$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 - 9T^13 - 18T^12 - 36T^11 + 30T^10 + 60T^9 + 120T^8 - 15T^7 - 47T^6 - 94T^5 + 50T^4 + 15T^3 + 30T^2 + 9*T + 1
$79$	\( T^{16} \) T^16
$83$	\( T^{16} \) T^16
$89$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 128T^9 + 256T^8 + 495T^7 + 786T^6 + 858T^5 + 594T^4 + 253T^3 + 64T^2 + 9*T + 1
$97$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 - 9T^13 - 18T^12 - 36T^11 + 30T^10 + 60T^9 + 120T^8 - 15T^7 - 47T^6 - 94T^5 + 50T^4 + 15T^3 + 30T^2 + 9*T + 1
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\(n\)	\(579\)	\(581\)
\(\chi(n)\)	\(-1\)	\(\zeta_{34}^{4}\)