LMFDB - Newform orbit 2163.1.cy.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2163,1,Mod(143,2163)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2163, base_ring=CyclotomicField(102))

chi = DirichletCharacter(H, H._module([51, 17, 31]))

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

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

chi := DirichletCharacter("2163.143");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2163 = 3 \cdot 7 \cdot 103 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2163.cy (of order $102$, degree $32$, 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:	$1.07947762233$
Analytic rank:	$0$
Dimension:	$32$
Coefficient field:	$\Q(\zeta_{51})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} - x^{31} + x^{29} - x^{28} + x^{26} - x^{25} + x^{23} - x^{22} + x^{20} - x^{19} + x^{17} + \cdots + 1 $ x^32 - x^31 + x^29 - x^28 + x^26 - x^25 + x^23 - x^22 + x^20 - x^19 + x^17 - x^16 + x^15 - x^13 + x^12 - 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:	$D_{102}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{102} - \cdots)$

$q$-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_{102}^{38} q^{3} - \zeta_{102}^{33} q^{4} - \zeta_{102}^{6} q^{7} - \zeta_{102}^{25} q^{9} +O(q^{10}) $ q - z^38 * q^3 - z^33 * q^4 - z^6 * q^7 - z^25 * q^9	\( q - \zeta_{102}^{38} q^{3} - \zeta_{102}^{33} q^{4} - \zeta_{102}^{6} q^{7} - \zeta_{102}^{25} q^{9} - \zeta_{102}^{20} q^{12} + (\zeta_{102}^{28} - \zeta_{102}^{26}) q^{13} - \zeta_{102}^{15} q^{16} + ( - \zeta_{102}^{36} + \zeta_{102}^{24}) q^{19} + \zeta_{102}^{44} q^{21} + \zeta_{102}^{22} q^{25} - \zeta_{102}^{12} q^{27} + \zeta_{102}^{39} q^{28} + ( - \zeta_{102}^{35} - \zeta_{102}^{29}) q^{31} - \zeta_{102}^{7} q^{36} + (\zeta_{102}^{47} - \zeta_{102}^{27}) q^{37} + (\zeta_{102}^{15} - \zeta_{102}^{13}) q^{39} + (\zeta_{102}^{43} + \zeta_{102}^{36}) q^{43} - \zeta_{102}^{2} q^{48} + \zeta_{102}^{12} q^{49} + (\zeta_{102}^{10} - \zeta_{102}^{8}) q^{52} + ( - \zeta_{102}^{23} + \zeta_{102}^{11}) q^{57} + ( - \zeta_{102}^{37} + \zeta_{102}^{7}) q^{61} + \zeta_{102}^{31} q^{63} + \zeta_{102}^{48} q^{64} + ( - \zeta_{102}^{32} + \zeta_{102}^{16}) q^{67} + (\zeta_{102}^{38} + \zeta_{102}^{8}) q^{73} + \zeta_{102}^{9} q^{75} + ( - \zeta_{102}^{18} + \zeta_{102}^{6}) q^{76} + ( - \zeta_{102}^{4} + \zeta_{102}) q^{79} + \zeta_{102}^{50} q^{81} + \zeta_{102}^{26} q^{84} + ( - \zeta_{102}^{34} + \zeta_{102}^{32}) q^{91} + ( - \zeta_{102}^{22} - \zeta_{102}^{16}) q^{93} + ( - \zeta_{102}^{41} + \zeta_{102}^{17}) q^{97} +O(q^{100}) \) q - z^38 * q^3 - z^33 * q^4 - z^6 * q^7 - z^25 * q^9 - z^20 * q^12 + (z^28 - z^26) * q^13 - z^15 * q^16 + (-z^36 + z^24) * q^19 + z^44 * q^21 + z^22 * q^25 - z^12 * q^27 + z^39 * q^28 + (-z^35 - z^29) * q^31 - z^7 * q^36 + (z^47 - z^27) * q^37 + (z^15 - z^13) * q^39 + (z^43 + z^36) * q^43 - z^2 * q^48 + z^12 * q^49 + (z^10 - z^8) * q^52 + (-z^23 + z^11) * q^57 + (-z^37 + z^7) * q^61 + z^31 * q^63 + z^48 * q^64 + (-z^32 + z^16) * q^67 + (z^38 + z^8) * q^73 + z^9 * q^75 + (-z^18 + z^6) * q^76 + (-z^4 + z) * q^79 + z^50 * q^81 + z^26 * q^84 + (-z^34 + z^32) * q^91 + (-z^22 - z^16) * q^93 + (-z^41 + z^17) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - q^{3} - 2 q^{4} + 2 q^{7} + q^{9}+O(q^{10}) $ 32 * q - q^3 - 2 * q^4 + 2 * q^7 + q^9	$ 32 q - q^{3} - 2 q^{4} + 2 q^{7} + q^{9} - q^{12} - 2 q^{16} + q^{21} + q^{25} + 2 q^{27} + 2 q^{28} + 2 q^{31} + q^{36} - 3 q^{37} + 3 q^{39} - 3 q^{43} - q^{48} - 2 q^{49} - q^{63} - 2 q^{64} + 2 q^{73} + 2 q^{75} - 2 q^{79} + q^{81} + q^{84} + 17 q^{91} - 2 q^{93} + 17 q^{97}+O(q^{100}) $ 32 * q - q^3 - 2 * q^4 + 2 * q^7 + q^9 - q^12 - 2 * q^16 + q^21 + q^25 + 2 * q^27 + 2 * q^28 + 2 * q^31 + q^36 - 3 * q^37 + 3 * q^39 - 3 * q^43 - q^48 - 2 * q^49 - q^63 - 2 * q^64 + 2 * q^73 + 2 * q^75 - 2 * q^79 + q^81 + q^84 + 17 * q^91 - 2 * q^93 + 17 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2163\mathbb{Z}\right)^\times$.

$n$	$211$	$619$	$722$
$\chi(n)$	$\zeta_{102}^{5}$	$-\zeta_{102}^{34}$	$-1$

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} $

143.1

0.969797	−	0.243914i
0.213933	−	0.976848i
0.552365	−	0.833602i
−0.908465	−	0.417960i
0.650618	+	0.759405i
−0.153392	−	0.988165i
0.881012	+	0.473094i
0.332355	−	0.943154i
−0.908465	+	0.417960i
−0.389786	−	0.920906i
−0.952942	−	0.303153i
0.552365	+	0.833602i
−0.779081	−	0.626924i
−0.779081	+	0.626924i
0.650618	−	0.759405i
0.881012	−	0.473094i
−0.0307951	+	0.999526i
0.213933	+	0.976848i
0.992421	−	0.122888i
0.816197	−	0.577774i

0.998103

0.0615609i

−0.273663

−

0.961826i

−0.0922684

0.995734i

0.992421

0.122888i

227.1

−0.332355

0.943154i

0.739009

−

0.673696i

0.273663

0.961826i

−0.779081

−

0.626924i

374.1

−0.969797

−

0.243914i

0.445738

−

0.895163i

−0.932472

−

0.361242i

0.881012

0.473094i

383.1

0.779081

0.626924i

0.0922684

−

0.995734i

0.850217

−

0.526432i

0.213933

0.976848i

395.1

−0.213933

−

0.976848i

−0.982973

−

0.183750i

−0.445738

0.895163i

−0.908465

0.417960i

479.1

0.908465

0.417960i

0.932472

−

0.361242i

0.602635

−

0.798017i

0.650618

0.759405i

500.1

−0.992421

0.122888i

−0.850217

−

0.526432i

0.982973

−

0.183750i

0.969797

−

0.243914i

593.1

0.952942

0.303153i

−0.982973

−

0.183750i

−0.445738

0.895163i

0.816197

0.577774i

689.1

0.779081

−

0.626924i

0.0922684

0.995734i

0.850217

0.526432i

0.213933

−

0.976848i

719.1

−0.881012

−

0.473094i

−0.602635

−

0.798017i

−0.739009

−

0.673696i

0.552365

0.833602i

836.1

−0.650618

0.759405i

0.739009

0.673696i

0.273663

−

0.961826i

−0.153392

−

0.988165i

908.1

−0.969797

0.243914i

0.445738

0.895163i

−0.932472

0.361242i

0.881012

−

0.473094i

920.1

−0.816197

−

0.577774i

0.932472

0.361242i

0.602635

0.798017i

0.332355

0.943154i

971.1

−0.816197

0.577774i

0.932472

−

0.361242i

0.602635

−

0.798017i

0.332355

−

0.943154i

1139.1

−0.213933

0.976848i

−0.982973

0.183750i

−0.445738

−

0.895163i

−0.908465

−

0.417960i

1181.1

−0.992421

−

0.122888i

−0.850217

0.526432i

0.982973

0.183750i

0.969797

0.243914i

1487.1

0.389786

0.920906i

−0.850217

0.526432i

0.982973

0.183750i

−0.696134

0.717912i

1496.1

−0.332355

−

0.943154i

0.739009

0.673696i

0.273663

−

0.961826i

−0.779081

0.626924i

1517.1

0.0307951

−

0.999526i

−0.602635

0.798017i

−0.739009

0.673696i

−0.998103

−

0.0615609i

1550.1

0.153392

−

0.988165i

0.0922684

−

0.995734i

0.850217

−

0.526432i

−0.952942

−

0.303153i

See all 32 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
721.be	even	102	1	inner
2163.cy	odd	102	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2163.1.cy.a	yes	32
3.b	odd	2	1	CM	2163.1.cy.a	yes	32
7.d	odd	6	1		2163.1.co.a	✓	32
21.g	even	6	1		2163.1.co.a	✓	32
103.h	odd	102	1		2163.1.co.a	✓	32
309.o	even	102	1		2163.1.co.a	✓	32
721.be	even	102	1	inner	2163.1.cy.a	yes	32
2163.cy	odd	102	1	inner	2163.1.cy.a	yes	32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2163.1.co.a	✓	32	7.d	odd	6	1
2163.1.co.a	✓	32	21.g	even	6	1
2163.1.co.a	✓	32	103.h	odd	102	1
2163.1.co.a	✓	32	309.o	even	102	1
2163.1.cy.a	yes	32	1.a	even	1	1	trivial
2163.1.cy.a	yes	32	3.b	odd	2	1	CM
2163.1.cy.a	yes	32	721.be	even	102	1	inner
2163.1.cy.a	yes	32	2163.cy	odd	102	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2163, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{32} $ T^32
$3$	$ T^{32} + T^{31} + \cdots + 1 $ T^32 + T^31 - T^29 - T^28 + T^26 + T^25 - T^23 - T^22 + T^20 + T^19 - T^17 - T^16 - T^15 + T^13 + T^12 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
$5$	$ T^{32} $ T^32
$7$	$ (T^{16} - T^{15} + T^{14} + \cdots + 1)^{2} $ (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)^2
$11$	$ T^{32} $ T^32
$13$	$ T^{32} - 3 T^{30} + \cdots + 1 $ T^32 - 3T^30 + 9T^28 + 34T^27 - 27T^26 - 102T^25 + 81T^24 + 238T^23 + 692T^22 - 680T^21 - 2076T^20 + 2057T^19 + 1485T^18 + 1173T^17 - 3894T^16 - 3519T^15 + 17462T^14 - 425T^13 - 7591T^12 - 3638T^11 + 20648T^10 + 6579T^9 - 3277T^8 - 663T^7 + 3677T^6 + 3060T^5 + 1685T^4 + 357T^3 + 96T^2 + 17*T + 1
$17$	$ T^{32} $ T^32
$19$	$ (T^{16} + 17 T^{10} + \cdots + 17)^{2} $ (T^16 + 17T^10 + 85T^9 + 17T^8 + 51T^4 - 238T^3 + 255T^2 - 102*T + 17)^2
$23$	$ T^{32} $ T^32
$29$	$ T^{32} $ T^32
$31$	$ T^{32} - 2 T^{31} + \cdots + 1 $ T^32 - 2T^31 + 8T^29 - 16T^28 + 30T^26 + 25T^25 - 153T^24 + 206T^23 + 200T^22 - 1224T^21 + 2039T^20 - 389T^19 - 697T^18 + 1301T^17 + 458T^16 - 6118T^15 + 8670T^14 + 7421T^13 - 3860T^12 - 867T^11 + 123T^10 - 1436T^9 + 5100T^8 + 7750T^7 + 7739T^6 + 3808T^5 + 1310T^4 + 168T^3 + 17T^2 + 8*T + 1
$37$	$ T^{32} + 3 T^{31} + \cdots + 1 $ T^32 + 3T^31 + 6T^30 + 9T^29 + 9T^28 - 27T^26 - 132T^25 - 315T^24 - 311T^23 + 12T^22 + 1479T^21 + 4384T^20 + 8817T^19 + 15815T^18 + 20977T^17 + 27658T^16 + 20040T^15 + 20785T^14 + 3969T^13 - 162T^12 - 12682T^11 + 2322T^10 - 650T^9 + 9444T^8 + 9185T^7 + 3847T^6 + 1037T^5 - 525T^4 - 11T^3 + 80T^2 - 16T + 1
$41$	$ T^{32} $ T^32
$43$	$ T^{32} + 3 T^{31} + \cdots + 1 $ T^32 + 3T^31 + 6T^30 + 9T^29 + 9T^28 - 17T^27 - 78T^26 - 183T^25 - 434T^24 - 753T^23 - 736T^22 + 221T^21 + 2871T^20 + 5485T^19 + 7825T^18 + 5932T^17 + 7020T^16 + 3261T^15 + 8664T^14 + 6672T^13 - 1930T^12 - 4709T^11 - 3135T^10 + 15704T^9 + 15768T^8 - 3208T^7 - 3463T^6 + 1751T^5 + 648T^4 - 130T^3 + 63T^2 + T + 1
$47$	$ T^{32} $ T^32
$53$	$ T^{32} $ T^32
$59$	$ T^{32} $ T^32
$61$	$ T^{32} - 34 T^{29} + \cdots + 289 $ T^32 - 34T^29 + 493T^26 - 3978T^23 - 17T^22 + 19550T^20 + 204T^19 - 60214T^17 - 306T^16 + 115311T^14 - 4046T^13 + 289T^12 - 131784T^11 + 15895T^10 + 2890T^9 + 82654T^8 - 15028T^7 + 7803T^6 - 24276T^5 + 289T^4 + 2890T^3 + 2601T^2 + 867T + 289
$67$	$ T^{32} - 3 T^{30} + \cdots + 1 $ T^32 - 3T^30 + 9T^28 - 27T^26 + 81T^24 + 17T^23 - 447T^22 + 663T^21 + 219T^20 - 1054T^19 - 1099T^18 + 3281T^17 + 3280T^16 - 9843T^15 - 9551T^14 + 22593T^13 + 94545T^12 - 397239T^11 + 715727T^10 - 807007T^9 + 631265T^8 - 357425T^7 + 149197T^6 - 46189T^5 + 10593T^4 - 1785T^3 + 215T^2 - 17*T + 1
$71$	$ T^{32} $ T^32
$73$	$ T^{32} - 2 T^{31} + \cdots + 1 $ T^32 - 2T^31 + 8T^29 - 16T^28 + 64T^26 - 128T^25 + 478T^23 - 752T^22 + 306T^21 + 152T^20 - 593T^19 + 1020T^18 + 94T^17 - 4251T^16 + 8128T^15 + 1037T^14 - 38054T^13 + 101438T^12 - 158168T^11 + 175597T^10 - 145664T^9 + 94112T^8 - 48027T^7 + 19843T^6 - 6358T^5 + 1225T^4 - 70T^3 + 17T^2 - 9T + 1
$79$	$ T^{32} + 2 T^{31} + \cdots + 1 $ T^32 + 2T^31 - 8T^29 - 16T^28 - 17T^27 + 30T^26 + 128T^25 + 255T^24 - 2T^23 - 803T^22 - 1683T^21 - 154T^20 + 2157T^19 + 4947T^18 + 178T^17 - 2398T^16 - 5510T^15 + 2907T^14 + 7046T^13 + 6510T^12 + 5066T^11 - 4569T^10 + 4700T^9 + 12070T^8 - 338T^7 - 693T^6 + 2142T^5 + 171T^4 - 134T^3 + 51T^2 - 8T + 1
$83$	$ T^{32} $ T^32
$89$	$ T^{32} $ T^32
$97$	$ T^{32} - 17 T^{31} + \cdots + 289 $ T^32 - 17T^31 + 153T^30 - 952T^29 + 4556T^28 - 17748T^27 + 58276T^26 - 165104T^25 + 410346T^24 - 905658T^23 + 1791426T^22 - 3198312T^21 + 5182008T^20 - 7651632T^19 + 10329336T^18 - 12778152T^17 + 14508939T^16 - 15134930T^15 + 14508956T^14 - 12778424T^13 + 10330016T^12 - 7649252T^11 + 5169632T^10 - 3185936T^9 + 1810874T^8 - 954278T^7 + 434656T^6 - 145656T^5 + 33524T^4 - 11560T^3 + 6936T^2 - 2312*T + 289
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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:	\( N \)	\(=\)	\( 2163 = 3 \cdot 7 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2163.cy (of order \(102\), degree \(32\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{102}^{38} q^{3} - \zeta_{102}^{33} q^{4} - \zeta_{102}^{6} q^{7} - \zeta_{102}^{25} q^{9} +O(q^{10}) \) q - z^38 * q^3 - z^33 * q^4 - z^6 * q^7 - z^25 * q^9	\( q - \zeta_{102}^{38} q^{3} - \zeta_{102}^{33} q^{4} - \zeta_{102}^{6} q^{7} - \zeta_{102}^{25} q^{9} - \zeta_{102}^{20} q^{12} + (\zeta_{102}^{28} - \zeta_{102}^{26}) q^{13} - \zeta_{102}^{15} q^{16} + ( - \zeta_{102}^{36} + \zeta_{102}^{24}) q^{19} + \zeta_{102}^{44} q^{21} + \zeta_{102}^{22} q^{25} - \zeta_{102}^{12} q^{27} + \zeta_{102}^{39} q^{28} + ( - \zeta_{102}^{35} - \zeta_{102}^{29}) q^{31} - \zeta_{102}^{7} q^{36} + (\zeta_{102}^{47} - \zeta_{102}^{27}) q^{37} + (\zeta_{102}^{15} - \zeta_{102}^{13}) q^{39} + (\zeta_{102}^{43} + \zeta_{102}^{36}) q^{43} - \zeta_{102}^{2} q^{48} + \zeta_{102}^{12} q^{49} + (\zeta_{102}^{10} - \zeta_{102}^{8}) q^{52} + ( - \zeta_{102}^{23} + \zeta_{102}^{11}) q^{57} + ( - \zeta_{102}^{37} + \zeta_{102}^{7}) q^{61} + \zeta_{102}^{31} q^{63} + \zeta_{102}^{48} q^{64} + ( - \zeta_{102}^{32} + \zeta_{102}^{16}) q^{67} + (\zeta_{102}^{38} + \zeta_{102}^{8}) q^{73} + \zeta_{102}^{9} q^{75} + ( - \zeta_{102}^{18} + \zeta_{102}^{6}) q^{76} + ( - \zeta_{102}^{4} + \zeta_{102}) q^{79} + \zeta_{102}^{50} q^{81} + \zeta_{102}^{26} q^{84} + ( - \zeta_{102}^{34} + \zeta_{102}^{32}) q^{91} + ( - \zeta_{102}^{22} - \zeta_{102}^{16}) q^{93} + ( - \zeta_{102}^{41} + \zeta_{102}^{17}) q^{97} +O(q^{100}) \) q - z^38 * q^3 - z^33 * q^4 - z^6 * q^7 - z^25 * q^9 - z^20 * q^12 + (z^28 - z^26) * q^13 - z^15 * q^16 + (-z^36 + z^24) * q^19 + z^44 * q^21 + z^22 * q^25 - z^12 * q^27 + z^39 * q^28 + (-z^35 - z^29) * q^31 - z^7 * q^36 + (z^47 - z^27) * q^37 + (z^15 - z^13) * q^39 + (z^43 + z^36) * q^43 - z^2 * q^48 + z^12 * q^49 + (z^10 - z^8) * q^52 + (-z^23 + z^11) * q^57 + (-z^37 + z^7) * q^61 + z^31 * q^63 + z^48 * q^64 + (-z^32 + z^16) * q^67 + (z^38 + z^8) * q^73 + z^9 * q^75 + (-z^18 + z^6) * q^76 + (-z^4 + z) * q^79 + z^50 * q^81 + z^26 * q^84 + (-z^34 + z^32) * q^91 + (-z^22 - z^16) * q^93 + (-z^41 + z^17) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - q^{3} - 2 q^{4} + 2 q^{7} + q^{9}+O(q^{10}) \) 32 * q - q^3 - 2 * q^4 + 2 * q^7 + q^9	\( 32 q - q^{3} - 2 q^{4} + 2 q^{7} + q^{9} - q^{12} - 2 q^{16} + q^{21} + q^{25} + 2 q^{27} + 2 q^{28} + 2 q^{31} + q^{36} - 3 q^{37} + 3 q^{39} - 3 q^{43} - q^{48} - 2 q^{49} - q^{63} - 2 q^{64} + 2 q^{73} + 2 q^{75} - 2 q^{79} + q^{81} + q^{84} + 17 q^{91} - 2 q^{93} + 17 q^{97}+O(q^{100}) \) 32 * q - q^3 - 2 * q^4 + 2 * q^7 + q^9 - q^12 - 2 * q^16 + q^21 + q^25 + 2 * q^27 + 2 * q^28 + 2 * q^31 + q^36 - 3 * q^37 + 3 * q^39 - 3 * q^43 - q^48 - 2 * q^49 - q^63 - 2 * q^64 + 2 * q^73 + 2 * q^75 - 2 * q^79 + q^81 + q^84 + 17 * q^91 - 2 * q^93 + 17 * q^97

Introduction

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Newspace parameters

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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	2163.1.cy.a

Level	$2163$
Weight	$1$
Character orbit	2163.cy
Analytic conductor	$1.079$
Analytic rank	$0$
Dimension	$32$
Projective image	$D_{102}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.07947762233\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Coefficient field:	\(\Q(\zeta_{51})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{32} - x^{31} + x^{29} - x^{28} + x^{26} - x^{25} + x^{23} - x^{22} + x^{20} - x^{19} + x^{17} + \cdots + 1 \) x^32 - x^31 + x^29 - x^28 + x^26 - x^25 + x^23 - x^22 + x^20 - x^19 + x^17 - x^16 + x^15 - x^13 + x^12 - 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:	\(D_{102}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{102} - \cdots)\)

\(n\)	\(211\)	\(619\)	\(722\)
\(\chi(n)\)	\(\zeta_{102}^{5}\)	\(-\zeta_{102}^{34}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 143.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
721.be	even	102	1	inner
2163.cy	odd	102	1	inner

$p$	$F_p(T)$
$2$	\( T^{32} \) T^32
$3$	\( T^{32} + T^{31} + \cdots + 1 \) T^32 + T^31 - T^29 - T^28 + T^26 + T^25 - T^23 - T^22 + T^20 + T^19 - T^17 - T^16 - T^15 + T^13 + T^12 - T^10 - T^9 + T^7 + T^6 - T^4 - T^3 + T + 1
$5$	\( T^{32} \) T^32
$7$	\( (T^{16} - T^{15} + T^{14} + \cdots + 1)^{2} \) (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)^2
$11$	\( T^{32} \) T^32
$13$	\( T^{32} - 3 T^{30} + \cdots + 1 \) T^32 - 3T^30 + 9T^28 + 34T^27 - 27T^26 - 102T^25 + 81T^24 + 238T^23 + 692T^22 - 680T^21 - 2076T^20 + 2057T^19 + 1485T^18 + 1173T^17 - 3894T^16 - 3519T^15 + 17462T^14 - 425T^13 - 7591T^12 - 3638T^11 + 20648T^10 + 6579T^9 - 3277T^8 - 663T^7 + 3677T^6 + 3060T^5 + 1685T^4 + 357T^3 + 96T^2 + 17*T + 1
$17$	\( T^{32} \) T^32
$19$	\( (T^{16} + 17 T^{10} + \cdots + 17)^{2} \) (T^16 + 17T^10 + 85T^9 + 17T^8 + 51T^4 - 238T^3 + 255T^2 - 102*T + 17)^2
$23$	\( T^{32} \) T^32
$29$	\( T^{32} \) T^32
$31$	\( T^{32} - 2 T^{31} + \cdots + 1 \) T^32 - 2T^31 + 8T^29 - 16T^28 + 30T^26 + 25T^25 - 153T^24 + 206T^23 + 200T^22 - 1224T^21 + 2039T^20 - 389T^19 - 697T^18 + 1301T^17 + 458T^16 - 6118T^15 + 8670T^14 + 7421T^13 - 3860T^12 - 867T^11 + 123T^10 - 1436T^9 + 5100T^8 + 7750T^7 + 7739T^6 + 3808T^5 + 1310T^4 + 168T^3 + 17T^2 + 8*T + 1
$37$	\( T^{32} + 3 T^{31} + \cdots + 1 \) T^32 + 3T^31 + 6T^30 + 9T^29 + 9T^28 - 27T^26 - 132T^25 - 315T^24 - 311T^23 + 12T^22 + 1479T^21 + 4384T^20 + 8817T^19 + 15815T^18 + 20977T^17 + 27658T^16 + 20040T^15 + 20785T^14 + 3969T^13 - 162T^12 - 12682T^11 + 2322T^10 - 650T^9 + 9444T^8 + 9185T^7 + 3847T^6 + 1037T^5 - 525T^4 - 11T^3 + 80T^2 - 16T + 1
$41$	\( T^{32} \) T^32
$43$	\( T^{32} + 3 T^{31} + \cdots + 1 \) T^32 + 3T^31 + 6T^30 + 9T^29 + 9T^28 - 17T^27 - 78T^26 - 183T^25 - 434T^24 - 753T^23 - 736T^22 + 221T^21 + 2871T^20 + 5485T^19 + 7825T^18 + 5932T^17 + 7020T^16 + 3261T^15 + 8664T^14 + 6672T^13 - 1930T^12 - 4709T^11 - 3135T^10 + 15704T^9 + 15768T^8 - 3208T^7 - 3463T^6 + 1751T^5 + 648T^4 - 130T^3 + 63T^2 + T + 1
$47$	\( T^{32} \) T^32
$53$	\( T^{32} \) T^32
$59$	\( T^{32} \) T^32
$61$	\( T^{32} - 34 T^{29} + \cdots + 289 \) T^32 - 34T^29 + 493T^26 - 3978T^23 - 17T^22 + 19550T^20 + 204T^19 - 60214T^17 - 306T^16 + 115311T^14 - 4046T^13 + 289T^12 - 131784T^11 + 15895T^10 + 2890T^9 + 82654T^8 - 15028T^7 + 7803T^6 - 24276T^5 + 289T^4 + 2890T^3 + 2601T^2 + 867T + 289
$67$	\( T^{32} - 3 T^{30} + \cdots + 1 \) T^32 - 3T^30 + 9T^28 - 27T^26 + 81T^24 + 17T^23 - 447T^22 + 663T^21 + 219T^20 - 1054T^19 - 1099T^18 + 3281T^17 + 3280T^16 - 9843T^15 - 9551T^14 + 22593T^13 + 94545T^12 - 397239T^11 + 715727T^10 - 807007T^9 + 631265T^8 - 357425T^7 + 149197T^6 - 46189T^5 + 10593T^4 - 1785T^3 + 215T^2 - 17*T + 1
$71$	\( T^{32} \) T^32
$73$	\( T^{32} - 2 T^{31} + \cdots + 1 \) T^32 - 2T^31 + 8T^29 - 16T^28 + 64T^26 - 128T^25 + 478T^23 - 752T^22 + 306T^21 + 152T^20 - 593T^19 + 1020T^18 + 94T^17 - 4251T^16 + 8128T^15 + 1037T^14 - 38054T^13 + 101438T^12 - 158168T^11 + 175597T^10 - 145664T^9 + 94112T^8 - 48027T^7 + 19843T^6 - 6358T^5 + 1225T^4 - 70T^3 + 17T^2 - 9T + 1
$79$	\( T^{32} + 2 T^{31} + \cdots + 1 \) T^32 + 2T^31 - 8T^29 - 16T^28 - 17T^27 + 30T^26 + 128T^25 + 255T^24 - 2T^23 - 803T^22 - 1683T^21 - 154T^20 + 2157T^19 + 4947T^18 + 178T^17 - 2398T^16 - 5510T^15 + 2907T^14 + 7046T^13 + 6510T^12 + 5066T^11 - 4569T^10 + 4700T^9 + 12070T^8 - 338T^7 - 693T^6 + 2142T^5 + 171T^4 - 134T^3 + 51T^2 - 8T + 1
$83$	\( T^{32} \) T^32
$89$	\( T^{32} \) T^32
$97$	\( T^{32} - 17 T^{31} + \cdots + 289 \) T^32 - 17T^31 + 153T^30 - 952T^29 + 4556T^28 - 17748T^27 + 58276T^26 - 165104T^25 + 410346T^24 - 905658T^23 + 1791426T^22 - 3198312T^21 + 5182008T^20 - 7651632T^19 + 10329336T^18 - 12778152T^17 + 14508939T^16 - 15134930T^15 + 14508956T^14 - 12778424T^13 + 10330016T^12 - 7649252T^11 + 5169632T^10 - 3185936T^9 + 1810874T^8 - 954278T^7 + 434656T^6 - 145656T^5 + 33524T^4 - 11560T^3 + 6936T^2 - 2312*T + 289
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