LMFDB - Newform orbit 294.6.e.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [294,6,Mod(67,294)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,4,-9,-16,-26,-72,0,-128,-81,104,-664] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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} - 26 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + ( - 104 \zeta_{6} + 104) q^{10} + (664 \zeta_{6} - 664) q^{11} + \cdots + 53784 q^{99} +O(q^{100}) $ q + 4z q^2 + (9z - 9) q^3 + (16z - 16) q^4 - 26z q^5 - 36 * q^6 - 64 * q^8 - 81z q^9 + (-104z + 104) q^10 + (664z - 664) q^11 - 144z q^12 + 318 * q^13 + 234 * q^15 - 256z q^16 + (1582z - 1582) q^17 + (-324z + 324) q^18 - 236z q^19 + 416 * q^20 - 2656 * q^22 - 2212z q^23 + (-576z + 576) q^24 + (-2449z + 2449) q^25 + 1272z q^26 + 729 * q^27 - 4954 * q^29 + 936z q^30 + (-7128z + 7128) q^31 + (-1024z + 1024) q^32 - 5976z q^33 - 6328 * q^34 + 1296 * q^36 - 4358z q^37 + (-944z + 944) q^38 + (2862z - 2862) q^39 + 1664z q^40 + 10542 * q^41 - 8452 * q^43 - 10624z q^44 + (2106z - 2106) q^45 + (-8848z + 8848) q^46 - 5352z q^47 + 2304 * q^48 + 9796 * q^50 - 14238z q^51 + (5088z - 5088) q^52 + (-33354z + 33354) q^53 + 2916z q^54 + 17264 * q^55 + 2124 * q^57 - 19816z q^58 + (-15436z + 15436) q^59 + (3744z - 3744) q^60 + 36762z q^61 + 28512 * q^62 + 4096 * q^64 - 8268z q^65 + (-23904z + 23904) q^66 + (40972z - 40972) q^67 - 25312z q^68 + 19908 * q^69 - 9092 * q^71 + 5184z q^72 + (-73454z + 73454) q^73 + (-17432z + 17432) q^74 + 22041z q^75 + 3776 * q^76 - 11448 * q^78 - 89400z q^79 + (6656z - 6656) q^80 + (6561z - 6561) q^81 + 42168z q^82 - 6428 * q^83 + 41132 * q^85 - 33808z q^86 + (-44586z + 44586) q^87 + (-42496z + 42496) q^88 + 122658z q^89 - 8424 * q^90 + 35392 * q^92 + 64152z q^93 + (-21408z + 21408) q^94 + (6136z - 6136) q^95 + 9216z q^96 + 21370 * q^97 + 53784 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} - 26 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9} + 104 q^{10} - 664 q^{11} - 144 q^{12} + 636 q^{13} + 468 q^{15} - 256 q^{16} - 1582 q^{17} + 324 q^{18} - 236 q^{19} + 832 q^{20}+ \cdots + 107568 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 - 9 * q^3 - 16 * q^4 - 26 * q^5 - 72 * q^6 - 128 * q^8 - 81 * q^9 + 104 * q^10 - 664 * q^11 - 144 * q^12 + 636 * q^13 + 468 * q^15 - 256 * q^16 - 1582 * q^17 + 324 * q^18 - 236 * q^19 + 832 * q^20 - 5312 * q^22 - 2212 * q^23 + 576 * q^24 + 2449 * q^25 + 1272 * q^26 + 1458 * q^27 - 9908 * q^29 + 936 * q^30 + 7128 * q^31 + 1024 * q^32 - 5976 * q^33 - 12656 * q^34 + 2592 * q^36 - 4358 * q^37 + 944 * q^38 - 2862 * q^39 + 1664 * q^40 + 21084 * q^41 - 16904 * q^43 - 10624 * q^44 - 2106 * q^45 + 8848 * q^46 - 5352 * q^47 + 4608 * q^48 + 19592 * q^50 - 14238 * q^51 - 5088 * q^52 + 33354 * q^53 + 2916 * q^54 + 34528 * q^55 + 4248 * q^57 - 19816 * q^58 + 15436 * q^59 - 3744 * q^60 + 36762 * q^61 + 57024 * q^62 + 8192 * q^64 - 8268 * q^65 + 23904 * q^66 - 40972 * q^67 - 25312 * q^68 + 39816 * q^69 - 18184 * q^71 + 5184 * q^72 + 73454 * q^73 + 17432 * q^74 + 22041 * q^75 + 7552 * q^76 - 22896 * q^78 - 89400 * q^79 - 6656 * q^80 - 6561 * q^81 + 42168 * q^82 - 12856 * q^83 + 82264 * q^85 - 33808 * q^86 + 44586 * q^87 + 42496 * q^88 + 122658 * q^89 - 16848 * q^90 + 70784 * q^92 + 64152 * q^93 + 21408 * q^94 - 6136 * q^95 + 9216 * q^96 + 42740 * q^97 + 107568 * 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

−13.0000

−

22.5167i

−36.0000

−64.0000

−40.5000

−

70.1481i

52.0000

−

90.0666i

79.1

2.00000

−

3.46410i

−4.50000

−

7.79423i

−8.00000

−

13.8564i

−13.0000

22.5167i

−36.0000

−64.0000

−40.5000

70.1481i

52.0000

90.0666i

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.i		2
7.b	odd	2	1		294.6.e.p		2
7.c	even	3	1		42.6.a.d	✓	1
7.c	even	3	1	inner	294.6.e.i		2
7.d	odd	6	1		294.6.a.b		1
7.d	odd	6	1		294.6.e.p		2
21.g	even	6	1		882.6.a.s		1
21.h	odd	6	1		126.6.a.i		1
28.g	odd	6	1		336.6.a.h		1
35.j	even	6	1		1050.6.a.k		1
35.l	odd	12	2		1050.6.g.i		2
84.n	even	6	1		1008.6.a.j		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.6.a.d	✓	1	7.c	even	3	1
126.6.a.i		1	21.h	odd	6	1
294.6.a.b		1	7.d	odd	6	1
294.6.e.i		2	1.a	even	1	1	trivial
294.6.e.i		2	7.c	even	3	1	inner
294.6.e.p		2	7.b	odd	2	1
294.6.e.p		2	7.d	odd	6	1
336.6.a.h		1	28.g	odd	6	1
882.6.a.s		1	21.g	even	6	1
1008.6.a.j		1	84.n	even	6	1
1050.6.a.k		1	35.j	even	6	1
1050.6.g.i		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} + 26T_{5} + 676 $

T5^2 + 26*T5 + 676

$ T_{11}^{2} + 664T_{11} + 440896 $

T11^2 + 664*T11 + 440896

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} + 26T + 676 $ T^2 + 26*T + 676
$7$	$ T^{2} $ T^2
$11$	$ T^{2} + 664T + 440896 $ T^2 + 664*T + 440896
$13$	$ (T - 318)^{2} $ (T - 318)^2
$17$	$ T^{2} + 1582 T + 2502724 $ T^2 + 1582*T + 2502724
$19$	$ T^{2} + 236T + 55696 $ T^2 + 236*T + 55696
$23$	$ T^{2} + 2212 T + 4892944 $ T^2 + 2212*T + 4892944
$29$	$ (T + 4954)^{2} $ (T + 4954)^2
$31$	$ T^{2} - 7128 T + 50808384 $ T^2 - 7128*T + 50808384
$37$	$ T^{2} + 4358 T + 18992164 $ T^2 + 4358*T + 18992164
$41$	$ (T - 10542)^{2} $ (T - 10542)^2
$43$	$ (T + 8452)^{2} $ (T + 8452)^2
$47$	$ T^{2} + 5352 T + 28643904 $ T^2 + 5352*T + 28643904
$53$	$ T^{2} + \cdots + 1112489316 $ T^2 - 33354*T + 1112489316
$59$	$ T^{2} - 15436 T + 238270096 $ T^2 - 15436*T + 238270096
$61$	$ T^{2} + \cdots + 1351444644 $ T^2 - 36762*T + 1351444644
$67$	$ T^{2} + \cdots + 1678704784 $ T^2 + 40972*T + 1678704784
$71$	$ (T + 9092)^{2} $ (T + 9092)^2
$73$	$ T^{2} + \cdots + 5395490116 $ T^2 - 73454*T + 5395490116
$79$	$ T^{2} + \cdots + 7992360000 $ T^2 + 89400*T + 7992360000
$83$	$ (T + 6428)^{2} $ (T + 6428)^2
$89$	$ T^{2} + \cdots + 15044984964 $ T^2 - 122658*T + 15044984964
$97$	$ (T - 21370)^{2} $ (T - 21370)^2
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Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	294.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} - 26 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + ( - 104 \zeta_{6} + 104) q^{10} + (664 \zeta_{6} - 664) q^{11} + \cdots + 53784 q^{99} +O(q^{100}) \) q + 4z q^2 + (9z - 9) q^3 + (16z - 16) q^4 - 26z q^5 - 36 * q^6 - 64 * q^8 - 81z q^9 + (-104z + 104) q^10 + (664z - 664) q^11 - 144z q^12 + 318 * q^13 + 234 * q^15 - 256z q^16 + (1582z - 1582) q^17 + (-324z + 324) q^18 - 236z q^19 + 416 * q^20 - 2656 * q^22 - 2212z q^23 + (-576z + 576) q^24 + (-2449z + 2449) q^25 + 1272z q^26 + 729 * q^27 - 4954 * q^29 + 936z q^30 + (-7128z + 7128) q^31 + (-1024z + 1024) q^32 - 5976z q^33 - 6328 * q^34 + 1296 * q^36 - 4358z q^37 + (-944z + 944) q^38 + (2862z - 2862) q^39 + 1664z q^40 + 10542 * q^41 - 8452 * q^43 - 10624z q^44 + (2106z - 2106) q^45 + (-8848z + 8848) q^46 - 5352z q^47 + 2304 * q^48 + 9796 * q^50 - 14238z q^51 + (5088z - 5088) q^52 + (-33354z + 33354) q^53 + 2916z q^54 + 17264 * q^55 + 2124 * q^57 - 19816z q^58 + (-15436z + 15436) q^59 + (3744z - 3744) q^60 + 36762z q^61 + 28512 * q^62 + 4096 * q^64 - 8268z q^65 + (-23904z + 23904) q^66 + (40972z - 40972) q^67 - 25312z q^68 + 19908 * q^69 - 9092 * q^71 + 5184z q^72 + (-73454z + 73454) q^73 + (-17432z + 17432) q^74 + 22041z q^75 + 3776 * q^76 - 11448 * q^78 - 89400z q^79 + (6656z - 6656) q^80 + (6561z - 6561) q^81 + 42168z q^82 - 6428 * q^83 + 41132 * q^85 - 33808z q^86 + (-44586z + 44586) q^87 + (-42496z + 42496) q^88 + 122658z q^89 - 8424 * q^90 + 35392 * q^92 + 64152z q^93 + (-21408z + 21408) q^94 + (6136z - 6136) q^95 + 9216z q^96 + 21370 * q^97 + 53784 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} - 26 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9} + 104 q^{10} - 664 q^{11} - 144 q^{12} + 636 q^{13} + 468 q^{15} - 256 q^{16} - 1582 q^{17} + 324 q^{18} - 236 q^{19} + 832 q^{20}+ \cdots + 107568 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 9 * q^3 - 16 * q^4 - 26 * q^5 - 72 * q^6 - 128 * q^8 - 81 * q^9 + 104 * q^10 - 664 * q^11 - 144 * q^12 + 636 * q^13 + 468 * q^15 - 256 * q^16 - 1582 * q^17 + 324 * q^18 - 236 * q^19 + 832 * q^20 - 5312 * q^22 - 2212 * q^23 + 576 * q^24 + 2449 * q^25 + 1272 * q^26 + 1458 * q^27 - 9908 * q^29 + 936 * q^30 + 7128 * q^31 + 1024 * q^32 - 5976 * q^33 - 12656 * q^34 + 2592 * q^36 - 4358 * q^37 + 944 * q^38 - 2862 * q^39 + 1664 * q^40 + 21084 * q^41 - 16904 * q^43 - 10624 * q^44 - 2106 * q^45 + 8848 * q^46 - 5352 * q^47 + 4608 * q^48 + 19592 * q^50 - 14238 * q^51 - 5088 * q^52 + 33354 * q^53 + 2916 * q^54 + 34528 * q^55 + 4248 * q^57 - 19816 * q^58 + 15436 * q^59 - 3744 * q^60 + 36762 * q^61 + 57024 * q^62 + 8192 * q^64 - 8268 * q^65 + 23904 * q^66 - 40972 * q^67 - 25312 * q^68 + 39816 * q^69 - 18184 * q^71 + 5184 * q^72 + 73454 * q^73 + 17432 * q^74 + 22041 * q^75 + 7552 * q^76 - 22896 * q^78 - 89400 * q^79 - 6656 * q^80 - 6561 * q^81 + 42168 * q^82 - 12856 * q^83 + 82264 * q^85 - 33808 * q^86 + 44586 * q^87 + 42496 * q^88 + 122658 * q^89 - 16848 * q^90 + 70784 * q^92 + 64152 * q^93 + 21408 * q^94 - 6136 * q^95 + 9216 * q^96 + 42740 * q^97 + 107568 * q^99

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