LMFDB - Newform orbit 144.4.i.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,4,Mod(49,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("144.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$144 = 2^{4} \cdot 3^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	144.i (of order $3$ , degree $2$ , not 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:	$8.49627504083$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.6831243.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} + 13x^{4} + 49x^{2} + 48$ x^6 + 13x^4 + 49x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{2}\cdot 3^{3}$
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$ .

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$-\beta_{1}$	$1$

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}

49.1

	−	2.63162i
		2.13353i
	−	1.23396i
		2.63162i
	−	2.13353i
		1.23396i

−5.16718

0.547914i

2.44901

−

4.24182i

−5.32725

−

9.22708i

26.3996

−

5.66234i

49.2

2.51979

−

4.54430i

−6.37096

11.0348i

−7.02674

−

12.1707i

−14.3013

−

22.9014i

49.3

4.14739

3.13036i

6.92194

−

11.9892i

15.3540

26.5939i

7.40171

25.9656i

97.1

−5.16718

−

0.547914i

2.44901

4.24182i

−5.32725

9.22708i

26.3996

5.66234i

97.2

2.51979

4.54430i

−6.37096

−

11.0348i

−7.02674

12.1707i

−14.3013

22.9014i

97.3

4.14739

−

3.13036i

6.92194

11.9892i

15.3540

−

26.5939i

7.40171

−

25.9656i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	144.4.i.d		6
3.b	odd	2	1		432.4.i.d		6
4.b	odd	2	1		36.4.e.a	✓	6
9.c	even	3	1	inner	144.4.i.d		6
9.c	even	3	1		1296.4.a.v		3
9.d	odd	6	1		432.4.i.d		6
9.d	odd	6	1		1296.4.a.w		3
12.b	even	2	1		108.4.e.a		6
36.f	odd	6	1		36.4.e.a	✓	6
36.f	odd	6	1		324.4.a.c		3
36.h	even	6	1		108.4.e.a		6
36.h	even	6	1		324.4.a.d		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.4.e.a	✓	6	4.b	odd	2	1
36.4.e.a	✓	6	36.f	odd	6	1
108.4.e.a		6	12.b	even	2	1
108.4.e.a		6	36.h	even	6	1
144.4.i.d		6	1.a	even	1	1	trivial
144.4.i.d		6	9.c	even	3	1	inner
324.4.a.c		3	36.f	odd	6	1
324.4.a.d		3	36.h	even	6	1
432.4.i.d		6	3.b	odd	2	1
432.4.i.d		6	9.d	odd	6	1
1296.4.a.v		3	9.c	even	3	1
1296.4.a.w		3	9.d	odd	6	1

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{6} - 6T_{5}^{5} + 207T_{5}^{4} - 702T_{5}^{3} + 34425T_{5}^{2} - 147744T_{5} + 746496$ T5^6 - 6*T5^5 + 207*T5^4 - 702*T5^3 + 34425*T5^2 - 147744*T5 + 746496 acting on $S_{4}^{\mathrm{new}}(144, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6} - 3 T^{5} + \cdots + 19683$ T^6 - 3T^5 - 15T^4 + 270T^3 - 405T^2 - 2187*T + 19683
$5$	$T^{6} - 6 T^{5} + \cdots + 746496$ T^6 - 6T^5 + 207T^4 - 702T^3 + 34425T^2 - 147744*T + 746496
$7$	$T^{6} - 6 T^{5} + \cdots + 21141604$ T^6 - 6T^5 + 645T^4 + 12850T^3 + 343293T^2 + 2800182*T + 21141604
$11$	$T^{6} + \cdots + 4386545361$ T^6 + 51T^5 + 3798T^4 + 71415T^3 + 4810590T^2 + 79278507*T + 4386545361
$13$	$T^{6} + \cdots + 4155865156$ T^6 - 12T^5 + 5415T^4 - 65680T^3 + 28557033T^2 - 339800286*T + 4155865156
$17$	$(T^{3} + 111 T^{2} + \cdots - 577476)^{2}$ (T^3 + 111T^2 - 6624T - 577476)^2
$19$	$(T^{3} + 15 T^{2} + \cdots + 216368)^{2}$ (T^3 + 15T^2 - 7512T + 216368)^2
$23$	$T^{6} + 210 T^{5} + \cdots + 5391684$ T^6 + 210T^5 + 33741T^4 + 2170746T^3 + 106821261T^2 + 24053598*T + 5391684
$29$	$T^{6} + \cdots + 8292430196964$ T^6 - 456T^5 + 143199T^4 - 23760756T^3 + 2877755121T^2 - 186420419946*T + 8292430196964
$31$	$T^{6} + \cdots + 9331729724944$ T^6 + 48T^5 + 37257T^4 + 4431832T^3 + 1368342033T^2 + 106774004964*T + 9331729724944
$37$	$(T^{3} + 48 T^{2} + \cdots - 682352)^{2}$ (T^3 + 48T^2 - 20508T - 682352)^2
$41$	$T^{6} + \cdots + 139158426750849$ T^6 - 897T^5 + 581058T^4 - 176932161T^3 + 39393550530T^2 - 2637128984193*T + 139158426750849
$43$	$T^{6} + \cdots + 2031049672201$ T^6 + 129T^5 + 203550T^4 - 21260963T^3 + 35118818502T^2 + 266373174441*T + 2031049672201
$47$	$T^{6} + \cdots + 41\!\cdots\!24$ T^6 + 522T^5 + 433917T^4 + 44849538T^3 + 59760297693T^2 + 10421917854606*T + 4167836333323524
$53$	$(T^{3} + 1104 T^{2} + \cdots + 11853648)^{2}$ (T^3 + 1104T^2 + 317700T + 11853648)^2
$59$	$T^{6} + \cdots + 93\!\cdots\!69$ T^6 + 453T^5 + 496494T^4 + 61833321T^3 + 128739350214T^2 + 28223393906205*T + 9388197832500369
$61$	$T^{6} + \cdots + 1463461189696$ T^6 + 402T^5 + 153663T^4 + 5611754T^3 + 549373353T^2 - 9606513576*T + 1463461189696
$67$	$T^{6} + \cdots + 9579414973969$ T^6 - 213T^5 + 125502T^4 + 23258455T^3 + 5762049270T^2 + 248016683379*T + 9579414973969
$71$	$(T^{3} + 60 T^{2} + \cdots - 113211648)^{2}$ (T^3 + 60T^2 - 423072T - 113211648)^2
$73$	$(T^{3} - 375 T^{2} + \cdots + 158369284)^{2}$ (T^3 - 375T^2 - 381048T + 158369284)^2
$79$	$T^{6} + \cdots + 318578661907984$ T^6 + 552T^5 + 326553T^4 + 23636896T^3 + 10329900945T^2 + 389977819428*T + 318578661907984
$83$	$T^{6} + \cdots + 12101965948944$ T^6 - 612T^5 + 492237T^4 + 65070540T^3 + 15980660505T^2 - 409428996084*T + 12101965948944
$89$	$(T^{3} + 462 T^{2} + \cdots + 170122248)^{2}$ (T^3 + 462T^2 - 774180T + 170122248)^2
$97$	$T^{6} + \cdots + 74\!\cdots\!29$ T^6 - 93T^5 + 1048290T^4 + 269487659T^3 + 1072818160242T^2 + 89825526132243*T + 7465050374673529
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 $\F_{q}$

$f(q)$	$=$	$q + ( - \beta_{2} + 1) q^{3} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{5} + ( - 2 \beta_{5} + \beta_{4} - 3 \beta_{2} + \cdots + 4) q^{7} + ( - 3 \beta_{5} + \beta_{3} - \beta_{2} + \cdots + 6) q^{9}+ \cdots + (9 \beta_{5} + 54 \beta_{4} + \cdots + 225) q^{99}+O(q^{100})$ q + (-b2 + 1) * q^3 + (b4 + b3 + 2b2 + 3b1 - 2) * q^5 + (-2b5 + b4 - 3b2 - 3b1 + 4) q^7 + (-3b5 + b3 - b2 + b1 + 6) q^9 + (b5 - 3b4 + 5b3 + 4b2 + 20b1 - 23) * q^11 + (4b5 - 5b4 + 3b3 - 6b2 + 3b1 + 2) q^13 + (3b5 + 3b4 + 3b3 + 3b2 - 18b1 + 36) q^15 + (-b5 + 7b4 + 8b3 - 5b2 + b1 - 39) q^17 + (-7b5 + 5b4 + 12b3 + 9b2 + 7b1 - 19) q^19 + (-9b5 + b3 - 5b2 - 80b1 + 22) q^21 + (3b5 - 5b4 + b3 - 7b2 - 72b1 + 4) * q^23 + (3b5 - 6b4 + 9b3 + 9b2 + 7b1 - 13) q^25 + (-6b5 + 3b4 + 3b3 - 9b2 - 72b1 - 69) q^27 + (5b5 - 5b3 + 5b2 - 152b1 + 152) * q^29 + (3b5 - 15b4 - 9b3 - 27b2 - 28b1 + 24) q^31 + (-3b5 + 15b4 + 18b3 + 21b2 + 156b1 - 198) q^33 + (b5 + 14b4 + 13b3 - 16b2 - b1 + 186) q^35 + (-10b5 + 2b4 + 12b3 + 18b2 + 10b1 - 36) q^37 + (-27b5 + 9b4 + 7b3 - 3b2 + 268b1 - 130) q^39 + (12b5 - 14b4 + 10b3 - 16b2 + 297b1 + 4) q^41 + (27b5 - 27b4 + 27b3 + 54b2 + 70b1 - 97) q^43 + (9b4 - 15b3 - 30b2 + 3b1 + 270) * q^45 + (18b5 + 21b4 - 60b3 - 3b2 + 153b1 - 132) q^47 + (-22b5 - 13b4 - 57b3 - 48b2 - 110b1 + 70) q^49 + (-39b5 + 24b4 + 21b3 + 45b2 - 147b1 + 315) q^51 + (22b5 - 22b4 - 44b3 - 22b2 - 22b1 - 324) q^53 + (33b5 - 48b4 - 81b3 - 18b2 - 33b1 + 102) q^55 + (-9b5 + 36b4 + 15b3 + 17b2 - 237b1 - 5) q^57 + (15b5 + 34b4 + 64b3 + 83b2 - 102b1 - 98) q^59 + (15b5 - 12b4 + 9b3 + 27b2 + 146b1 - 158) q^61 + (-18b5 + 3b4 - 74b3 - 31b2 - 311b1 - 108) q^63 + (43b5 - 6b4 - 31b3 + 49b2 - 364b1 + 358) q^65 + (3b5 - 24b4 - 18b3 - 45b2 + 50b1 + 42) q^67 + (-24b5 + 3b4 - 69b3 - 6b2 + 153b1 - 72) q^69 + (52b5 - 40b4 - 92b3 - 64b2 - 52b1 + 84) q^71 + (21b5 - 51b4 - 72b3 + 9b2 - 21b1 + 167) q^73 + (27b4 + b3 + 10b2 + 298b1 - 308) q^75 + (12b5 + 65b4 + 89b3 + 142b2 + 453b1 - 154) q^77 + (-28b5 + 5b4 + 18b3 - 33b2 + 179b1 - 174) q^79 + (-36b5 + 9b4 - 57b3 + 66b2 - 282b1 + 45) q^81 + (18b5 - 33b4 + 48b3 + 51b2 - 171b1 + 138) q^83 + (60b5 - 30b4 + 90b3 + 426b1 - 60) * q^85 + (30b5 - 15b4 - 162b3 - 147b2 - 57b1 + 414) q^87 + (44b5 + 16b4 - 28b3 - 104b2 - 44b1 - 66) q^89 + (-75b5 + 42b4 + 117b3 + 108b2 + 75b1 - 106) q^91 + (-54b5 - 27b4 - 25b3 - 36b2 + 407b1 - 506) q^93 + (24b5 - 44b4 + 4b3 - 64b2 + 708b1 + 40) q^95 + (-14b5 - 56b4 + 126b3 + 42b2 + 25b1 - 81) q^97 + (9b5 + 54b4 + 168b3 + 210b2 - 210b1 + 225) q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 3 q^{3} + 6 q^{5} + 6 q^{7} + 39 q^{9} - 51 q^{11} + 12 q^{13} + 180 q^{15} - 222 q^{17} - 30 q^{19} - 120 q^{21} - 210 q^{23} - 3 q^{25} - 648 q^{27} + 456 q^{29} - 48 q^{31} - 603 q^{33} + 1104 q^{35}+ \cdots + 1854 q^{99}+O(q^{100})$ 6 * q + 3 * q^3 + 6 * q^5 + 6 * q^7 + 39 * q^9 - 51 * q^11 + 12 * q^13 + 180 * q^15 - 222 * q^17 - 30 * q^19 - 120 * q^21 - 210 * q^23 - 3 * q^25 - 648 * q^27 + 456 * q^29 - 48 * q^31 - 603 * q^33 + 1104 * q^35 - 96 * q^37 + 36 * q^39 + 897 * q^41 - 129 * q^43 + 1494 * q^45 - 522 * q^47 - 225 * q^49 + 1647 * q^51 - 2208 * q^53 + 216 * q^55 - 645 * q^57 - 453 * q^59 - 402 * q^61 - 1896 * q^63 + 1110 * q^65 + 213 * q^67 - 198 * q^69 - 120 * q^71 + 750 * q^73 - 921 * q^75 + 1128 * q^77 - 552 * q^79 - 549 * q^81 + 612 * q^83 + 1188 * q^85 + 1386 * q^87 - 924 * q^89 + 264 * q^91 - 1998 * q^93 + 2184 * q^95 + 93 * q^97 + 1854 * q^99

$\beta_{1}$	$=$	$( \nu^{5} + 9\nu^{3} + 17\nu + 4 ) / 8$ (v^5 + 9v^3 + 17v + 4) / 8
$\beta_{2}$	$=$	$( -\nu^{5} + 4\nu^{4} - 5\nu^{3} + 20\nu^{2} + 15\nu - 4 ) / 8$ (-v^5 + 4v^4 - 5v^3 + 20v^2 + 15v - 4) / 8
$\beta_{3}$	$=$	$( \nu^{5} + 4\nu^{4} + 13\nu^{3} + 44\nu^{2} + 25\nu + 100 ) / 8$ (v^5 + 4v^4 + 13v^3 + 44v^2 + 25v + 100) / 8
$\beta_{4}$	$=$	$( -\nu^{5} - 9\nu^{3} + 12\nu^{2} - 5\nu + 52 ) / 4$ (-v^5 - 9v^3 + 12v^2 - 5*v + 52) / 4
$\beta_{5}$	$=$	$( -3\nu^{4} + 3\nu^{3} - 21\nu^{2} + 18\nu - 20 ) / 2$ (-3v^4 + 3v^3 - 21v^2 + 18v - 20) / 2

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more