LMFDB - Newform orbit 2163.1.co.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2163,1,Mod(5,2163)]

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, 85, 1]))

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

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

chi := DirichletCharacter("2163.5");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2163 = 3 \cdot 7 \cdot 103 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2163.co (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}^{4} q^{3} + \zeta_{102}^{50} q^{4} - \zeta_{102}^{14} q^{7} + \zeta_{102}^{8} q^{9} +O(q^{10}) $ q + z^4 * q^3 + z^50 * q^4 - z^14 * q^7 + z^8 * q^9	\( q + \zeta_{102}^{4} q^{3} + \zeta_{102}^{50} q^{4} - \zeta_{102}^{14} q^{7} + \zeta_{102}^{8} q^{9} - \zeta_{102}^{3} q^{12} + ( - \zeta_{102}^{28} + \zeta_{102}^{26}) q^{13} - \zeta_{102}^{49} q^{16} + ( - \zeta_{102}^{19} + \zeta_{102}^{7}) q^{19} - \zeta_{102}^{18} q^{21} - \zeta_{102}^{39} q^{25} + \zeta_{102}^{12} q^{27} + \zeta_{102}^{13} q^{28} + ( - \zeta_{102}^{46} + \zeta_{102}) q^{31} - \zeta_{102}^{7} q^{36} + ( - \zeta_{102}^{30} + \zeta_{102}^{10}) q^{37} + ( - \zeta_{102}^{32} + \zeta_{102}^{30}) q^{39} + (\zeta_{102}^{43} + \zeta_{102}^{36}) q^{43} + \zeta_{102}^{2} q^{48} + \zeta_{102}^{28} q^{49} + (\zeta_{102}^{27} - \zeta_{102}^{25}) q^{52} + ( - \zeta_{102}^{23} + \zeta_{102}^{11}) q^{57} + ( - \zeta_{102}^{41} - \zeta_{102}^{20}) q^{61} - \zeta_{102}^{22} q^{63} + \zeta_{102}^{48} q^{64} + (\zeta_{102}^{49} - \zeta_{102}^{33}) q^{67} + (\zeta_{102}^{25} - \zeta_{102}^{4}) q^{73} - \zeta_{102}^{43} q^{75} + (\zeta_{102}^{18} - \zeta_{102}^{6}) q^{76} + ( - \zeta_{102}^{38} + \zeta_{102}^{35}) q^{79} + \zeta_{102}^{16} q^{81} + \zeta_{102}^{17} q^{84} + (\zeta_{102}^{42} - \zeta_{102}^{40}) q^{91} + ( - \zeta_{102}^{50} + \zeta_{102}^{5}) q^{93} + (\zeta_{102}^{41} - \zeta_{102}^{17}) q^{97} +O(q^{100}) \) q + z^4 * q^3 + z^50 * q^4 - z^14 * q^7 + z^8 * q^9 - z^3 * q^12 + (-z^28 + z^26) * q^13 - z^49 * q^16 + (-z^19 + z^7) * q^19 - z^18 * q^21 - z^39 * q^25 + z^12 * q^27 + z^13 * q^28 + (-z^46 + z) * q^31 - z^7 * q^36 + (-z^30 + z^10) * q^37 + (-z^32 + z^30) * q^39 + (z^43 + z^36) * q^43 + z^2 * q^48 + z^28 * q^49 + (z^27 - z^25) * q^52 + (-z^23 + z^11) * q^57 + (-z^41 - z^20) * q^61 - z^22 * q^63 + z^48 * q^64 + (z^49 - z^33) * q^67 + (z^25 - z^4) * q^73 - z^43 * q^75 + (z^18 - z^6) * q^76 + (-z^38 + z^35) * q^79 + z^16 * q^81 + z^17 * q^84 + (z^42 - z^40) * q^91 + (-z^50 + z^5) * q^93 + (z^41 - z^17) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + q^{3} + q^{4} - q^{7} + q^{9}+O(q^{10}) $ 32 * q + q^3 + q^4 - q^7 + q^9	$ 32 q + q^{3} + q^{4} - q^{7} + q^{9} - 2 q^{12} + q^{16} + 2 q^{21} - 2 q^{25} - 2 q^{27} - q^{28} - 2 q^{31} + q^{36} + 3 q^{37} - 3 q^{39} - 3 q^{43} + q^{48} + q^{49} + 3 q^{52} - q^{63} - 2 q^{64} - 3 q^{67} - 2 q^{73} + q^{75} - 2 q^{79} + q^{81} + 16 q^{84} - 3 q^{91} - 2 q^{93} - 17 q^{97}+O(q^{100}) $ 32 * q + q^3 + q^4 - q^7 + q^9 - 2 * q^12 + q^16 + 2 * q^21 - 2 * q^25 - 2 * q^27 - q^28 - 2 * q^31 + q^36 + 3 * q^37 - 3 * q^39 - 3 * q^43 + q^48 + q^49 + 3 * q^52 - q^63 - 2 * q^64 - 3 * q^67 - 2 * q^73 + q^75 - 2 * q^79 + q^81 + 16 * q^84 - 3 * q^91 - 2 * q^93 - 17 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$211$	$619$	$722$
$\chi(n)$	$\zeta_{102}^{5}$	$\zeta_{102}^{17}$	$-1$

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} $

5.1

0.816197	−	0.577774i
−0.389786	−	0.920906i
0.332355	+	0.943154i
−0.389786	+	0.920906i
0.552365	+	0.833602i
0.992421	+	0.122888i
−0.779081	+	0.626924i
−0.998103	+	0.0615609i
0.650618	−	0.759405i
0.881012	−	0.473094i
−0.696134	+	0.717912i
0.969797	−	0.243914i
0.213933	−	0.976848i
0.213933	+	0.976848i
0.992421	−	0.122888i
−0.998103	−	0.0615609i
0.552365	−	0.833602i
−0.696134	−	0.717912i
0.650618	+	0.759405i
−0.153392	−	0.988165i

−0.779081

−

0.626924i

0.816197

0.577774i

0.696134

0.717912i

0.213933

0.976848i

101.1

−0.0307951

−

0.999526i

−0.389786

0.920906i

0.779081

0.626924i

−0.998103

0.0615609i

173.1

0.213933

−

0.976848i

0.332355

−

0.943154i

0.0307951

0.999526i

−0.908465

−

0.417960i

257.1

−0.0307951

0.999526i

−0.389786

−

0.920906i

0.779081

−

0.626924i

−0.998103

−

0.0615609i

290.1

−0.696134

−

0.717912i

0.552365

−

0.833602i

−0.332355

−

0.943154i

−0.0307951

0.999526i

320.1

0.881012

0.473094i

0.992421

−

0.122888i

0.153392

−

0.988165i

0.552365

0.833602i

353.1

−0.908465

−

0.417960i

−0.779081

−

0.626924i

0.998103

−

0.0615609i

0.650618

0.759405i

362.1

0.969797

−

0.243914i

−0.998103

−

0.0615609i

−0.650618

0.759405i

0.881012

−

0.473094i

521.1

−0.952942

0.303153i

0.650618

0.759405i

−0.881012

−

0.473094i

0.816197

−

0.577774i

563.1

−0.389786

−

0.920906i

0.881012

0.473094i

−0.816197

0.577774i

−0.696134

0.717912i

614.1

−0.998103

0.0615609i

−0.696134

−

0.717912i

−0.213933

−

0.976848i

0.992421

−

0.122888i

761.1

0.552365

−

0.833602i

0.969797

0.243914i

0.952942

−

0.303153i

−0.389786

−

0.920906i

845.1

0.650618

0.759405i

0.213933

0.976848i

−0.992421

0.122888i

−0.153392

0.988165i

878.1

0.650618

−

0.759405i

0.213933

−

0.976848i

−0.992421

−

0.122888i

−0.153392

−

0.988165i

899.1

0.881012

−

0.473094i

0.992421

0.122888i

0.153392

0.988165i

0.552365

−

0.833602i

962.1

0.969797

0.243914i

−0.998103

0.0615609i

−0.650618

−

0.759405i

0.881012

0.473094i

992.1

−0.696134

0.717912i

0.552365

0.833602i

−0.332355

0.943154i

−0.0307951

−

0.999526i

1004.1

−0.998103

−

0.0615609i

−0.696134

0.717912i

−0.213933

0.976848i

0.992421

0.122888i

1013.1

−0.952942

−

0.303153i

0.650618

−

0.759405i

−0.881012

0.473094i

0.816197

0.577774i

1097.1

0.816197

−

0.577774i

−0.153392

0.988165i

−0.552365

−

0.833602i

0.332355

−

0.943154i

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.bk	even	102	1	inner
2163.co	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.co.a	✓	32
3.b	odd	2	1	CM	2163.1.co.a	✓	32
7.d	odd	6	1		2163.1.cy.a	yes	32
21.g	even	6	1		2163.1.cy.a	yes	32
103.h	odd	102	1		2163.1.cy.a	yes	32
309.o	even	102	1		2163.1.cy.a	yes	32
721.bk	even	102	1	inner	2163.1.co.a	✓	32
2163.co	odd	102	1	inner	2163.1.co.a	✓	32

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

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^{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
$11$	$ 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^{32} + 34 T^{26} + \cdots + 289 $ T^32 + 34T^26 + 85T^25 - 17T^24 + 391T^20 + 1207T^19 + 6681T^18 + 1649T^17 + 272T^16 + 1734T^14 + 289T^13 - 45662T^12 + 21097T^11 + 17051T^10 - 4624T^9 + 2312T^8 - 12138T^7 + 43639T^6 - 50286T^5 + 39882T^4 - 17918T^3 + 6069T^2 - 1734T + 289
$23$	$ 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 + 30T^25 - 9T^24 + 175T^23 - 498T^22 + 714T^21 - 665T^20 - 45T^19 + 4646T^18 - 13820T^17 + 3229T^16 + 31776T^15 - 22361T^14 - 29112T^13 + 22635T^12 + 19142T^11 - 10020T^10 - 12338T^9 + 7098T^8 - 175T^7 + 1909T^6 - 1156T^5 + 1311T^4 - 176T^3 + 80*T^2 - T + 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^{31} + \cdots + 1 $ T^32 + 3T^31 + 6T^30 + 9T^29 + 9T^28 - 27T^26 - 81T^25 - 162T^24 - 226T^23 - 396T^22 - 1938T^21 - 5748T^20 - 10495T^19 - 13357T^18 - 8467T^17 + 14653T^16 + 69357T^15 + 164401T^14 + 288600T^13 + 402075T^12 + 447933T^11 + 402587T^10 + 291104T^9 + 170723T^8 + 81316T^7 + 31999T^6 + 10421T^5 + 2433T^4 + 210T^3 - 22*T^2 + 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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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:	\( N \)	\(=\)	\( 2163 = 3 \cdot 7 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2163.co (of order \(102\), degree \(32\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{102}^{4} q^{3} + \zeta_{102}^{50} q^{4} - \zeta_{102}^{14} q^{7} + \zeta_{102}^{8} q^{9} +O(q^{10}) \) q + z^4 * q^3 + z^50 * q^4 - z^14 * q^7 + z^8 * q^9	\( q + \zeta_{102}^{4} q^{3} + \zeta_{102}^{50} q^{4} - \zeta_{102}^{14} q^{7} + \zeta_{102}^{8} q^{9} - \zeta_{102}^{3} q^{12} + ( - \zeta_{102}^{28} + \zeta_{102}^{26}) q^{13} - \zeta_{102}^{49} q^{16} + ( - \zeta_{102}^{19} + \zeta_{102}^{7}) q^{19} - \zeta_{102}^{18} q^{21} - \zeta_{102}^{39} q^{25} + \zeta_{102}^{12} q^{27} + \zeta_{102}^{13} q^{28} + ( - \zeta_{102}^{46} + \zeta_{102}) q^{31} - \zeta_{102}^{7} q^{36} + ( - \zeta_{102}^{30} + \zeta_{102}^{10}) q^{37} + ( - \zeta_{102}^{32} + \zeta_{102}^{30}) q^{39} + (\zeta_{102}^{43} + \zeta_{102}^{36}) q^{43} + \zeta_{102}^{2} q^{48} + \zeta_{102}^{28} q^{49} + (\zeta_{102}^{27} - \zeta_{102}^{25}) q^{52} + ( - \zeta_{102}^{23} + \zeta_{102}^{11}) q^{57} + ( - \zeta_{102}^{41} - \zeta_{102}^{20}) q^{61} - \zeta_{102}^{22} q^{63} + \zeta_{102}^{48} q^{64} + (\zeta_{102}^{49} - \zeta_{102}^{33}) q^{67} + (\zeta_{102}^{25} - \zeta_{102}^{4}) q^{73} - \zeta_{102}^{43} q^{75} + (\zeta_{102}^{18} - \zeta_{102}^{6}) q^{76} + ( - \zeta_{102}^{38} + \zeta_{102}^{35}) q^{79} + \zeta_{102}^{16} q^{81} + \zeta_{102}^{17} q^{84} + (\zeta_{102}^{42} - \zeta_{102}^{40}) q^{91} + ( - \zeta_{102}^{50} + \zeta_{102}^{5}) q^{93} + (\zeta_{102}^{41} - \zeta_{102}^{17}) q^{97} +O(q^{100}) \) q + z^4 * q^3 + z^50 * q^4 - z^14 * q^7 + z^8 * q^9 - z^3 * q^12 + (-z^28 + z^26) * q^13 - z^49 * q^16 + (-z^19 + z^7) * q^19 - z^18 * q^21 - z^39 * q^25 + z^12 * q^27 + z^13 * q^28 + (-z^46 + z) * q^31 - z^7 * q^36 + (-z^30 + z^10) * q^37 + (-z^32 + z^30) * q^39 + (z^43 + z^36) * q^43 + z^2 * q^48 + z^28 * q^49 + (z^27 - z^25) * q^52 + (-z^23 + z^11) * q^57 + (-z^41 - z^20) * q^61 - z^22 * q^63 + z^48 * q^64 + (z^49 - z^33) * q^67 + (z^25 - z^4) * q^73 - z^43 * q^75 + (z^18 - z^6) * q^76 + (-z^38 + z^35) * q^79 + z^16 * q^81 + z^17 * q^84 + (z^42 - z^40) * q^91 + (-z^50 + z^5) * q^93 + (z^41 - z^17) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{3} + q^{4} - q^{7} + q^{9}+O(q^{10}) \) 32 * q + q^3 + q^4 - q^7 + q^9	\( 32 q + q^{3} + q^{4} - q^{7} + q^{9} - 2 q^{12} + q^{16} + 2 q^{21} - 2 q^{25} - 2 q^{27} - q^{28} - 2 q^{31} + q^{36} + 3 q^{37} - 3 q^{39} - 3 q^{43} + q^{48} + q^{49} + 3 q^{52} - q^{63} - 2 q^{64} - 3 q^{67} - 2 q^{73} + q^{75} - 2 q^{79} + q^{81} + 16 q^{84} - 3 q^{91} - 2 q^{93} - 17 q^{97}+O(q^{100}) \) 32 * q + q^3 + q^4 - q^7 + q^9 - 2 * q^12 + q^16 + 2 * q^21 - 2 * q^25 - 2 * q^27 - q^28 - 2 * q^31 + q^36 + 3 * q^37 - 3 * q^39 - 3 * q^43 + q^48 + q^49 + 3 * q^52 - q^63 - 2 * q^64 - 3 * q^67 - 2 * q^73 + q^75 - 2 * q^79 + q^81 + 16 * q^84 - 3 * q^91 - 2 * q^93 - 17 * q^97

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	2163.1.co.a

Level	$2163$
Weight	$1$
Character orbit	2163.co
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 \(\mathbb{Q}[x]/(x^{102} - \cdots)\)

\(n\)	\(211\)	\(619\)	\(722\)
\(\chi(n)\)	\(\zeta_{102}^{5}\)	\(\zeta_{102}^{17}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 5.1
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.bk	even	102	1	inner
2163.co	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
$7$	\( T^{32} + T^{31} + \cdots + 1 \) 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
$11$	\( T^{32} \) 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
$19$	\( T^{32} + 34 T^{26} + \cdots + 289 \) T^32 + 34T^26 + 85T^25 - 17T^24 + 391T^20 + 1207T^19 + 6681T^18 + 1649T^17 + 272T^16 + 1734T^14 + 289T^13 - 45662T^12 + 21097T^11 + 17051T^10 - 4624T^9 + 2312T^8 - 12138T^7 + 43639T^6 - 50286T^5 + 39882T^4 - 17918T^3 + 6069T^2 - 1734T + 289
$23$	\( T^{32} \) 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 + 30T^25 - 9T^24 + 175T^23 - 498T^22 + 714T^21 - 665T^20 - 45T^19 + 4646T^18 - 13820T^17 + 3229T^16 + 31776T^15 - 22361T^14 - 29112T^13 + 22635T^12 + 19142T^11 - 10020T^10 - 12338T^9 + 7098T^8 - 175T^7 + 1909T^6 - 1156T^5 + 1311T^4 - 176T^3 + 80*T^2 - T + 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
$67$	\( T^{32} + 3 T^{31} + \cdots + 1 \) T^32 + 3T^31 + 6T^30 + 9T^29 + 9T^28 - 27T^26 - 81T^25 - 162T^24 - 226T^23 - 396T^22 - 1938T^21 - 5748T^20 - 10495T^19 - 13357T^18 - 8467T^17 + 14653T^16 + 69357T^15 + 164401T^14 + 288600T^13 + 402075T^12 + 447933T^11 + 402587T^10 + 291104T^9 + 170723T^8 + 81316T^7 + 31999T^6 + 10421T^5 + 2433T^4 + 210T^3 - 22*T^2 + 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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