LMFDB - Newform orbit 294.6.e.r

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,6,Mod(67,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("294.67");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$294 = 2 \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	294.e (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:	$47.1528430250$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 4 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + (16 \zeta_{6} - 16) q^{4} + 54 \zeta_{6} q^{5} + 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + (216 \zeta_{6} - 216) q^{10} + (216 \zeta_{6} - 216) q^{11} + \cdots + 17496 q^{99} +O(q^{100})$ q + 4z q^2 + (-9z + 9) q^3 + (16z - 16) q^4 + 54z q^5 + 36 * q^6 - 64 * q^8 - 81z q^9 + (216z - 216) q^10 + (216z - 216) q^11 + 144z q^12 + 998 * q^13 + 486 * q^15 - 256z q^16 + (1302z - 1302) q^17 + (-324z + 324) q^18 - 884z q^19 - 864 * q^20 - 864 * q^22 + 2268z q^23 + (576z - 576) q^24 + (-209z + 209) q^25 + 3992z q^26 - 729 * q^27 - 1482 * q^29 + 1944z q^30 + (8360z - 8360) q^31 + (-1024z + 1024) q^32 + 1944z q^33 - 5208 * q^34 + 1296 * q^36 + 4714z q^37 + (-3536z + 3536) q^38 + (-8982z + 8982) q^39 - 3456z q^40 - 9786 * q^41 + 19436 * q^43 - 3456z q^44 + (-4374z + 4374) q^45 + (9072z - 9072) q^46 - 22200z q^47 - 2304 * q^48 + 836 * q^50 + 11718z q^51 + (15968z - 15968) q^52 + (26790z - 26790) q^53 - 2916z q^54 - 11664 * q^55 - 7956 * q^57 - 5928z q^58 + (28092z - 28092) q^59 + (7776z - 7776) q^60 + 38866z q^61 - 33440 * q^62 + 4096 * q^64 + 53892z q^65 + (7776z - 7776) q^66 + (23948z - 23948) q^67 - 20832z q^68 + 20412 * q^69 - 20628 * q^71 + 5184z q^72 + (290z - 290) q^73 + (18856z - 18856) q^74 - 1881z q^75 + 14144 * q^76 + 35928 * q^78 + 99544z q^79 + (-13824z + 13824) q^80 + (6561z - 6561) q^81 - 39144z q^82 + 19308 * q^83 - 70308 * q^85 + 77744z q^86 + (13338z - 13338) q^87 + (-13824z + 13824) q^88 - 36390z q^89 + 17496 * q^90 - 36288 * q^92 + 75240z q^93 + (-88800z + 88800) q^94 + (-47736z + 47736) q^95 - 9216z q^96 - 79078 * q^97 + 17496 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 4 q^{2} + 9 q^{3} - 16 q^{4} + 54 q^{5} + 72 q^{6} - 128 q^{8} - 81 q^{9} - 216 q^{10} - 216 q^{11} + 144 q^{12} + 1996 q^{13} + 972 q^{15} - 256 q^{16} - 1302 q^{17} + 324 q^{18} - 884 q^{19} - 1728 q^{20}+ \cdots + 34992 q^{99}+O(q^{100})$ 2 * q + 4 * q^2 + 9 * q^3 - 16 * q^4 + 54 * q^5 + 72 * q^6 - 128 * q^8 - 81 * q^9 - 216 * q^10 - 216 * q^11 + 144 * q^12 + 1996 * q^13 + 972 * q^15 - 256 * q^16 - 1302 * q^17 + 324 * q^18 - 884 * q^19 - 1728 * q^20 - 1728 * q^22 + 2268 * q^23 - 576 * q^24 + 209 * q^25 + 3992 * q^26 - 1458 * q^27 - 2964 * q^29 + 1944 * q^30 - 8360 * q^31 + 1024 * q^32 + 1944 * q^33 - 10416 * q^34 + 2592 * q^36 + 4714 * q^37 + 3536 * q^38 + 8982 * q^39 - 3456 * q^40 - 19572 * q^41 + 38872 * q^43 - 3456 * q^44 + 4374 * q^45 - 9072 * q^46 - 22200 * q^47 - 4608 * q^48 + 1672 * q^50 + 11718 * q^51 - 15968 * q^52 - 26790 * q^53 - 2916 * q^54 - 23328 * q^55 - 15912 * q^57 - 5928 * q^58 - 28092 * q^59 - 7776 * q^60 + 38866 * q^61 - 66880 * q^62 + 8192 * q^64 + 53892 * q^65 - 7776 * q^66 - 23948 * q^67 - 20832 * q^68 + 40824 * q^69 - 41256 * q^71 + 5184 * q^72 - 290 * q^73 - 18856 * q^74 - 1881 * q^75 + 28288 * q^76 + 71856 * q^78 + 99544 * q^79 + 13824 * q^80 - 6561 * q^81 - 39144 * q^82 + 38616 * q^83 - 140616 * q^85 + 77744 * q^86 - 13338 * q^87 + 13824 * q^88 - 36390 * q^89 + 34992 * q^90 - 72576 * q^92 + 75240 * q^93 + 88800 * q^94 + 47736 * q^95 - 9216 * q^96 - 158156 * q^97 + 34992 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/294\mathbb{Z}\right)^\times$ .

$n$	$197$	$199$
$\chi(n)$	$1$	$-\zeta_{6}$

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}

67.1

0.500000	+	0.866025i
0.500000	−	0.866025i

2.00000

3.46410i

4.50000

−

7.79423i

−8.00000

13.8564i

27.0000

46.7654i

36.0000

−64.0000

−40.5000

−

70.1481i

−108.000

187.061i

79.1

2.00000

−

3.46410i

4.50000

7.79423i

−8.00000

−

13.8564i

27.0000

−

46.7654i

36.0000

−64.0000

−40.5000

70.1481i

−108.000

−

187.061i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	294.6.e.r		2
7.b	odd	2	1		294.6.e.h		2
7.c	even	3	1		42.6.a.a	✓	1
7.c	even	3	1	inner	294.6.e.r		2
7.d	odd	6	1		294.6.a.h		1
7.d	odd	6	1		294.6.e.h		2
21.g	even	6	1		882.6.a.o		1
21.h	odd	6	1		126.6.a.k		1
28.g	odd	6	1		336.6.a.j		1
35.j	even	6	1		1050.6.a.n		1
35.l	odd	12	2		1050.6.g.o		2
84.n	even	6	1		1008.6.a.x		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.6.a.a	✓	1	7.c	even	3	1
126.6.a.k		1	21.h	odd	6	1
294.6.a.h		1	7.d	odd	6	1
294.6.e.h		2	7.b	odd	2	1
294.6.e.h		2	7.d	odd	6	1
294.6.e.r		2	1.a	even	1	1	trivial
294.6.e.r		2	7.c	even	3	1	inner
336.6.a.j		1	28.g	odd	6	1
882.6.a.o		1	21.g	even	6	1
1008.6.a.x		1	84.n	even	6	1
1050.6.a.n		1	35.j	even	6	1
1050.6.g.o		2	35.l	odd	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(294, [\chi])$ :

T_{5}^{2} - 54T_{5} + 2916

T_{11}^{2} + 216T_{11} + 46656

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$3$	$T^{2} - 9T + 81$ T^2 - 9*T + 81
$5$	$T^{2} - 54T + 2916$ T^2 - 54*T + 2916
$7$	$T^{2}$ T^2
$11$	$T^{2} + 216T + 46656$ T^2 + 216*T + 46656
$13$	$(T - 998)^{2}$ (T - 998)^2
$17$	$T^{2} + 1302 T + 1695204$ T^2 + 1302*T + 1695204
$19$	$T^{2} + 884T + 781456$ T^2 + 884*T + 781456
$23$	$T^{2} - 2268 T + 5143824$ T^2 - 2268*T + 5143824
$29$	$(T + 1482)^{2}$ (T + 1482)^2
$31$	$T^{2} + 8360 T + 69889600$ T^2 + 8360*T + 69889600
$37$	$T^{2} - 4714 T + 22221796$ T^2 - 4714*T + 22221796
$41$	$(T + 9786)^{2}$ (T + 9786)^2
$43$	$(T - 19436)^{2}$ (T - 19436)^2
$47$	$T^{2} + 22200 T + 492840000$ T^2 + 22200*T + 492840000
$53$	$T^{2} + 26790 T + 717704100$ T^2 + 26790*T + 717704100
$59$	$T^{2} + 28092 T + 789160464$ T^2 + 28092*T + 789160464
$61$	$T^{2} + \cdots + 1510565956$ T^2 - 38866*T + 1510565956
$67$	$T^{2} + 23948 T + 573506704$ T^2 + 23948*T + 573506704
$71$	$(T + 20628)^{2}$ (T + 20628)^2
$73$	$T^{2} + 290T + 84100$ T^2 + 290*T + 84100
$79$	$T^{2} + \cdots + 9909007936$ T^2 - 99544*T + 9909007936
$83$	$(T - 19308)^{2}$ (T - 19308)^2
$89$	$T^{2} + \cdots + 1324232100$ T^2 + 36390*T + 1324232100
$97$	$(T + 79078)^{2}$ (T + 79078)^2
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 $\F_{q}$

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more