LMFDB - Newform orbit 18.6.c.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,6,Mod(7,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("18.7");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$18 = 2 \cdot 3^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	18.c (of order $3$ , degree $2$ , 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:	$2.88690875663$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{2} + 4$ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{2}\cdot 3^{3}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$ .

$n$	$11$
$\chi(n)$	$-1 + \beta_{1}$

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}

7.1

−1.22474	+	0.707107i
1.22474	−	0.707107i
−1.22474	−	0.707107i
1.22474	+	0.707107i

2.00000

−

3.46410i

−14.6969

−

5.19615i

−8.00000

−

13.8564i

−28.1969

−

48.8385i

−47.3939

40.5194i

11.1515

−

19.3150i

−64.0000

189.000

152.735i

−225.576

7.2

2.00000

−

3.46410i

14.6969

−

5.19615i

−8.00000

−

13.8564i

1.19694

2.07316i

11.3939

−

61.3040i

25.8485

−

44.7709i

−64.0000

189.000

−

152.735i

9.57551

13.1

2.00000

3.46410i

−14.6969

5.19615i

−8.00000

13.8564i

−28.1969

48.8385i

−47.3939

−

40.5194i

11.1515

19.3150i

−64.0000

189.000

−

152.735i

−225.576

13.2

2.00000

3.46410i

14.6969

5.19615i

−8.00000

13.8564i

1.19694

−

2.07316i

11.3939

61.3040i

25.8485

44.7709i

−64.0000

189.000

152.735i

9.57551

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	18.6.c.a	✓	4
3.b	odd	2	1		54.6.c.a		4
4.b	odd	2	1		144.6.i.a		4
9.c	even	3	1	inner	18.6.c.a	✓	4
9.c	even	3	1		162.6.a.e		2
9.d	odd	6	1		54.6.c.a		4
9.d	odd	6	1		162.6.a.f		2
12.b	even	2	1		432.6.i.a		4
36.f	odd	6	1		144.6.i.a		4
36.h	even	6	1		432.6.i.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
18.6.c.a	✓	4	1.a	even	1	1	trivial
18.6.c.a	✓	4	9.c	even	3	1	inner
54.6.c.a		4	3.b	odd	2	1
54.6.c.a		4	9.d	odd	6	1
144.6.i.a		4	4.b	odd	2	1
144.6.i.a		4	36.f	odd	6	1
162.6.a.e		2	9.c	even	3	1
162.6.a.f		2	9.d	odd	6	1
432.6.i.a		4	12.b	even	2	1
432.6.i.a		4	36.h	even	6	1

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{4} + 54T_{5}^{3} + 3051T_{5}^{2} - 7290T_{5} + 18225$ T5^4 + 54*T5^3 + 3051*T5^2 - 7290*T5 + 18225 acting on $S_{6}^{\mathrm{new}}(18, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} - 4 T + 16)^{2}$ (T^2 - 4*T + 16)^2
$3$	$T^{4} - 378 T^{2} + 59049$ T^4 - 378*T^2 + 59049
$5$	$T^{4} + 54 T^{3} + \cdots + 18225$ T^4 + 54T^3 + 3051T^2 - 7290*T + 18225
$7$	$T^{4} - 74 T^{3} + \cdots + 1329409$ T^4 - 74T^3 + 4323T^2 - 85322*T + 1329409
$11$	$T^{4} + \cdots + 158294966769$ T^4 + 78T^3 + 403947T^2 - 31033314*T + 158294966769
$13$	$T^{4} + \cdots + 48140309281$ T^4 - 1106T^3 + 1003827T^2 - 242666354*T + 48140309281
$17$	$(T^{2} - 492 T - 1848060)^{2}$ (T^2 - 492*T - 1848060)^2
$19$	$(T^{2} + 1640 T - 2648816)^{2}$ (T^2 + 1640*T - 2648816)^2
$23$	$T^{4} + \cdots + 56736462234321$ T^4 + 5538T^3 + 23137083T^2 + 41714215218*T + 56736462234321
$29$	$T^{4} + \cdots + 220517585649$ T^4 + 3894T^3 + 14693643T^2 + 1828595142*T + 220517585649
$31$	$T^{4} + \cdots + 12\!\cdots\!61$ T^4 - 4718T^3 + 57567243T^2 + 166581818242*T + 1246635020982961
$37$	$(T^{2} + 4796 T - 141621212)^{2}$ (T^2 + 4796*T - 141621212)^2
$41$	$T^{4} + \cdots + 34\!\cdots\!49$ T^4 - 15354T^3 + 176851323T^2 - 904258368522*T + 3468502411484049
$43$	$T^{4} + \cdots + 64\!\cdots\!89$ T^4 - 32858T^3 + 825134547T^2 - 8362808427386*T + 64777181238422689
$47$	$T^{4} + \cdots + 20\!\cdots\!69$ T^4 - 24954T^3 + 480418803T^2 - 3550537792602*T + 20244541158255969
$53$	$(T^{2} + 16332 T + 59839812)^{2}$ (T^2 + 16332*T + 59839812)^2
$59$	$T^{4} + \cdots + 99\!\cdots\!09$ T^4 - 21966T^3 + 382770603T^2 - 2190769191198*T + 9946981062109809
$61$	$T^{4} + \cdots + 11\!\cdots\!01$ T^4 - 3050T^3 + 115249899T^2 + 323139566950*T + 11224851354865201
$67$	$T^{4} + \cdots + 70\!\cdots\!25$ T^4 - 36758T^3 + 1616486979T^2 + 9753235942570*T + 70403413125052225
$71$	$(T^{2} + 73848 T + 665096112)^{2}$ (T^2 + 73848*T + 665096112)^2
$73$	$(T^{2} + 51188 T + 491562436)^{2}$ (T^2 + 51188*T + 491562436)^2
$79$	$T^{4} + \cdots + 27\!\cdots\!25$ T^4 + 14926T^3 + 275056491T^2 - 780197169890*T + 2732259009130225
$83$	$T^{4} + \cdots + 42\!\cdots\!89$ T^4 - 90762T^3 + 6182973027T^2 - 186494818454154*T + 4222069959871858689
$89$	$(T^{2} + 9300 T - 7978887036)^{2}$ (T^2 + 9300*T - 7978887036)^2
$97$	$T^{4} + \cdots + 82\!\cdots\!25$ T^4 + 30262T^3 + 10021816299T^2 - 275566608895610*T + 82919739653624799025
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$f(q)$	$=$	$q + 4 \beta_1 q^{2} + (\beta_{3} + 6 \beta_1 - 3) q^{3} + (16 \beta_1 - 16) q^{4} + (2 \beta_{3} - 2 \beta_{2} + 27 \beta_1 - 27) q^{5} + (4 \beta_{2} + 12 \beta_1 - 24) q^{6} + (\beta_{2} + 37 \beta_1) q^{7}+ \cdots + (468 \beta_{3} - 8361 \beta_{2} + \cdots + 111456) q^{99}+O(q^{100})$ q + 4b1 q^2 + (b3 + 6b1 - 3) q^3 + (16b1 - 16) q^4 + (2b3 - 2b2 + 27b1 - 27) q^5 + (4b2 + 12b1 - 24) * q^6 + (b2 + 37b1) q^7 - 64 * q^8 + (-6b3 + 12b2 + 189) * q^9 + (8b3 - 108) q^10 + (-43b2 - 39b1) * q^11 + (-16b3 + 16b2 - 48b1 - 48) q^12 + (20b3 - 20b2 - 553b1 + 553) q^13 + (-4b3 + 4b2 + 148b1 - 148) q^14 + (-21b3 + 33b2 - 513b1 + 351) q^15 - 256b1 q^16 + (-94b3 + 246) q^17 + (-48b3 + 24b2 + 756b1) q^18 + (124b3 - 820) q^19 + (32b2 - 432b1) * q^20 + (-6b3 + 40b2 + 327b1 - 222) q^21 + (172b3 - 172b2 - 156b1 + 156) q^22 + (25b3 - 25b2 + 2769b1 - 2769) q^23 + (-64b3 - 384b1 + 192) * q^24 + (108b2 + 1532b1) * q^25 + (80b3 + 2212) q^26 + (135b3 + 3726b1 - 1863) * q^27 + (-16b3 - 592) q^28 + (-124b2 - 1947b1) * q^29 + (-132b3 + 48b2 - 648b1 + 2052) q^30 + (-435b3 + 435b2 - 2359b1 + 2359) q^31 + (-1024b1 + 1024) q^32 + (258b3 - 168b2 - 9405b1 + 234) q^33 + (-376b2 + 984b1) * q^34 + (47b3 - 567) q^35 + (-96b3 - 96b2 + 3024b1 - 3024) q^36 + (-826b3 - 2398) q^37 + (496b2 - 3280b1) * q^38 + (613b3 - 493b2 - 2661b1 + 5979) q^39 + (-128b3 + 128b2 - 1728b1 + 1728) q^40 + (14b3 - 14b2 - 7677b1 + 7677) q^41 + (-160b3 + 136b2 + 420b1 - 1308) q^42 + (267b2 + 16429b1) * q^43 + (688b3 + 624) q^44 + (216b3 - 540b2 + 7695b1 - 2511) q^45 + (100b3 - 11076) q^46 + (249b2 + 12477b1) * q^47 + (-256b2 - 768b1 + 1536) * q^48 + (-74b3 + 74b2 - 15222b1 + 15222) q^49 + (-432b3 + 432b2 + 6128b1 - 6128) q^50 + (528b3 - 564b2 + 1476b1 - 21042) q^51 + (320b2 + 8848b1) * q^52 + (178b3 - 8166) q^53 + (540b2 + 7452b1 - 14904) * q^54 + (1083b3 - 17523) q^55 + (-64b2 - 2368b1) * q^56 + (-1192b3 + 744b2 - 4920b1 + 29244) q^57 + (496b3 - 496b2 - 7788b1 + 7788) q^58 + (-311b3 + 311b2 - 10983b1 + 10983) q^59 + (-192b3 - 336b2 + 5616b1 + 2592) q^60 + (-708b2 + 1525b1) * q^61 + (-1740b3 + 9436) q^62 + (-444b3 + 411b2 + 8289b1 - 2592) q^63 + 4096 * q^64 + (-566b2 + 6291b1) * q^65 + (672b3 + 360b2 - 36684b1 + 37620) q^66 + (1671b3 - 1671b2 - 18379b1 + 18379) q^67 + (1504b3 - 1504b2 + 3936b1 - 3936) q^68 + (-2694b3 + 2844b2 - 13707b1 - 2907) q^69 + (188b2 - 2268b1) * q^70 + (-1798b3 - 36924) q^71 + (384b3 - 768b2 - 12096) * q^72 + (870b3 - 25594) q^73 + (-3304b2 - 9592b1) * q^74 + (-648b3 + 1856b2 + 27924b1 - 9192) q^75 + (-1984b3 + 1984b2 - 13120b1 + 13120) q^76 + (1630b3 - 1630b2 - 10731b1 + 10731) q^77 + (1972b3 + 480b2 + 13272b1 + 10644) q^78 + (707b2 - 7463b1) * q^79 + (-512b3 + 6912) q^80 + (-2268b3 + 4536b2 + 12393) * q^81 + (56b3 + 30708) q^82 + (147b2 + 45381b1) * q^83 + (-544b3 - 96b2 - 3552b1 - 1680) q^84 + (3030b3 - 3030b2 + 47250b1 - 47250) q^85 + (-1068b3 + 1068b2 + 65716b1 - 65716) q^86 + (744b3 - 2319b2 - 32625b1 + 11682) q^87 + (2752b2 + 2496b1) * q^88 + (6086b3 - 4650) q^89 + (2160b3 - 1296b2 + 20736b1 - 30780) q^90 + (1293b3 + 24781) q^91 + (400b2 - 44304b1) * q^92 + (1054b3 - 3664b2 + 101037b1 - 86883) q^93 + (-996b3 + 996b2 + 49908b1 - 49908) q^94 + (-4988b3 + 4988b2 - 75708b1 + 75708) q^95 + (1024b3 - 1024b2 + 3072b1 + 3072) q^96 + (-6574b2 - 15131b1) * q^97 + (-296b3 + 60888) q^98 + (468b3 - 8361b2 - 63099b1 + 111456) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 8 q^{2} - 32 q^{4} - 54 q^{5} - 72 q^{6} + 74 q^{7} - 256 q^{8} + 756 q^{9} - 432 q^{10} - 78 q^{11} - 288 q^{12} + 1106 q^{13} - 296 q^{14} + 378 q^{15} - 512 q^{16} + 984 q^{17} + 1512 q^{18} - 3280 q^{19}+ \cdots + 319626 q^{99}+O(q^{100})$ 4 * q + 8 * q^2 - 32 * q^4 - 54 * q^5 - 72 * q^6 + 74 * q^7 - 256 * q^8 + 756 * q^9 - 432 * q^10 - 78 * q^11 - 288 * q^12 + 1106 * q^13 - 296 * q^14 + 378 * q^15 - 512 * q^16 + 984 * q^17 + 1512 * q^18 - 3280 * q^19 - 864 * q^20 - 234 * q^21 + 312 * q^22 - 5538 * q^23 + 3064 * q^25 + 8848 * q^26 - 2368 * q^28 - 3894 * q^29 + 6912 * q^30 + 4718 * q^31 + 2048 * q^32 - 17874 * q^33 + 1968 * q^34 - 2268 * q^35 - 6048 * q^36 - 9592 * q^37 - 6560 * q^38 + 18594 * q^39 + 3456 * q^40 + 15354 * q^41 - 4392 * q^42 + 32858 * q^43 + 2496 * q^44 + 5346 * q^45 - 44304 * q^46 + 24954 * q^47 + 4608 * q^48 + 30444 * q^49 - 12256 * q^50 - 81216 * q^51 + 17696 * q^52 - 32664 * q^53 - 44712 * q^54 - 70092 * q^55 - 4736 * q^56 + 107136 * q^57 + 15576 * q^58 + 21966 * q^59 + 21600 * q^60 + 3050 * q^61 + 37744 * q^62 + 6210 * q^63 + 16384 * q^64 + 12582 * q^65 + 77112 * q^66 + 36758 * q^67 - 7872 * q^68 - 39042 * q^69 - 4536 * q^70 - 147696 * q^71 - 48384 * q^72 - 102376 * q^73 - 19184 * q^74 + 19080 * q^75 + 26240 * q^76 + 21462 * q^77 + 69120 * q^78 - 14926 * q^79 + 27648 * q^80 + 49572 * q^81 + 122832 * q^82 + 90762 * q^83 - 13824 * q^84 - 94500 * q^85 - 131432 * q^86 - 18522 * q^87 + 4992 * q^88 - 18600 * q^89 - 81648 * q^90 + 99124 * q^91 - 88608 * q^92 - 145458 * q^93 - 99816 * q^94 + 151416 * q^95 + 18432 * q^96 - 30262 * q^97 + 243552 * q^98 + 319626 * q^99

$\beta_{1}$	$=$	$( \nu^{2} ) / 2$ (v^2) / 2
$\beta_{2}$	$=$	$3\nu^{3} + 6\nu$ 3v^3 + 6v
$\beta_{3}$	$=$	$-3\nu^{3} + 12\nu$ -3v^3 + 12v

Introduction

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Varieties

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Database

Properties

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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	18.6.c.a

Level	$18$
Weight	$6$
Character orbit	18.c
Analytic conductor	$2.887$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$

$\nu$	$=$	$( \beta_{3} + \beta_{2} ) / 18$ (b3 + b2) / 18
$\nu^{2}$	$=$	$2\beta_1$ 2*b1
$\nu^{3}$	$=$	$( -\beta_{3} + 2\beta_{2} ) / 9$ (-b3 + 2*b2) / 9

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format: