LMFDB - Newform orbit 294.6.e.l

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,-16,72,-72,0,-128,-81,-288,414] 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} + 72 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + (288 \zeta_{6} - 288) q^{10} + ( - 414 \zeta_{6} + 414) q^{11} + \cdots - 33534 q^{99} +O(q^{100}) $ q + 4z q^2 + (9z - 9) q^3 + (16z - 16) q^4 + 72z q^5 - 36 * q^6 - 64 * q^8 - 81z q^9 + (288z - 288) q^10 + (-414z + 414) q^11 - 144z q^12 - 1054 * q^13 - 648 * q^15 - 256z q^16 + (-1848z + 1848) q^17 + (-324z + 324) q^18 - 236z q^19 - 1152 * q^20 + 1656 * q^22 - 2898z q^23 + (-576z + 576) q^24 + (2059z - 2059) q^25 - 4216z q^26 + 729 * q^27 - 6522 * q^29 - 2592z q^30 + (6200z - 6200) q^31 + (-1024z + 1024) q^32 + 3726z q^33 + 7392 * q^34 + 1296 * q^36 - 9650z q^37 + (-944z + 944) q^38 + (-9486z + 9486) q^39 - 4608z q^40 + 8484 * q^41 - 10804 * q^43 + 6624z q^44 + (-5832z + 5832) q^45 + (-11592z + 11592) q^46 - 60z q^47 + 2304 * q^48 - 8236 * q^50 + 16632z q^51 + (-16864z + 16864) q^52 + (22506z - 22506) q^53 + 2916z q^54 + 29808 * q^55 + 2124 * q^57 - 26088z q^58 + (-28176z + 28176) q^59 + (-10368z + 10368) q^60 + 35194z q^61 - 24800 * q^62 + 4096 * q^64 - 75888z q^65 + (14904z - 14904) q^66 + (-28216z + 28216) q^67 + 29568z q^68 + 26082 * q^69 - 6642 * q^71 + 5184z q^72 + (-52090z + 52090) q^73 + (-38600z + 38600) q^74 - 18531z q^75 + 3776 * q^76 + 37944 * q^78 - 43340z q^79 + (-18432z + 18432) q^80 + (6561z - 6561) q^81 + 33936z q^82 + 25716 * q^83 + 133056 * q^85 - 43216z q^86 + (-58698z + 58698) q^87 + (26496z - 26496) q^88 - 98724z q^89 + 23328 * q^90 + 46368 * q^92 - 55800z q^93 + (-240z + 240) q^94 + (-16992z + 16992) q^95 + 9216z q^96 - 148954 * q^97 - 33534 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} + 72 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9} - 288 q^{10} + 414 q^{11} - 144 q^{12} - 2108 q^{13} - 1296 q^{15} - 256 q^{16} + 1848 q^{17} + 324 q^{18} - 236 q^{19} - 2304 q^{20}+ \cdots - 67068 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 - 9 * q^3 - 16 * q^4 + 72 * q^5 - 72 * q^6 - 128 * q^8 - 81 * q^9 - 288 * q^10 + 414 * q^11 - 144 * q^12 - 2108 * q^13 - 1296 * q^15 - 256 * q^16 + 1848 * q^17 + 324 * q^18 - 236 * q^19 - 2304 * q^20 + 3312 * q^22 - 2898 * q^23 + 576 * q^24 - 2059 * q^25 - 4216 * q^26 + 1458 * q^27 - 13044 * q^29 - 2592 * q^30 - 6200 * q^31 + 1024 * q^32 + 3726 * q^33 + 14784 * q^34 + 2592 * q^36 - 9650 * q^37 + 944 * q^38 + 9486 * q^39 - 4608 * q^40 + 16968 * q^41 - 21608 * q^43 + 6624 * q^44 + 5832 * q^45 + 11592 * q^46 - 60 * q^47 + 4608 * q^48 - 16472 * q^50 + 16632 * q^51 + 16864 * q^52 - 22506 * q^53 + 2916 * q^54 + 59616 * q^55 + 4248 * q^57 - 26088 * q^58 + 28176 * q^59 + 10368 * q^60 + 35194 * q^61 - 49600 * q^62 + 8192 * q^64 - 75888 * q^65 - 14904 * q^66 + 28216 * q^67 + 29568 * q^68 + 52164 * q^69 - 13284 * q^71 + 5184 * q^72 + 52090 * q^73 + 38600 * q^74 - 18531 * q^75 + 7552 * q^76 + 75888 * q^78 - 43340 * q^79 + 18432 * q^80 - 6561 * q^81 + 33936 * q^82 + 51432 * q^83 + 266112 * q^85 - 43216 * q^86 + 58698 * q^87 - 26496 * q^88 - 98724 * q^89 + 46656 * q^90 + 92736 * q^92 - 55800 * q^93 + 240 * q^94 + 16992 * q^95 + 9216 * q^96 - 297908 * q^97 - 67068 * 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

36.0000

62.3538i

−36.0000

−64.0000

−40.5000

−

70.1481i

−144.000

249.415i

79.1

2.00000

−

3.46410i

−4.50000

−

7.79423i

−8.00000

−

13.8564i

36.0000

−

62.3538i

−36.0000

−64.0000

−40.5000

70.1481i

−144.000

−

249.415i

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

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.6.a.c	✓	1	7.c	even	3	1
126.6.a.l		1	21.h	odd	6	1
294.6.a.c		1	7.d	odd	6	1
294.6.e.l		2	1.a	even	1	1	trivial
294.6.e.l		2	7.c	even	3	1	inner
294.6.e.n		2	7.b	odd	2	1
294.6.e.n		2	7.d	odd	6	1
336.6.a.b		1	28.g	odd	6	1
882.6.a.n		1	21.g	even	6	1
1008.6.a.ba		1	84.n	even	6	1
1050.6.a.g		1	35.j	even	6	1
1050.6.g.b		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} - 72T_{5} + 5184 $

T5^2 - 72*T5 + 5184

$ T_{11}^{2} - 414T_{11} + 171396 $

T11^2 - 414*T11 + 171396

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} - 72T + 5184 $ T^2 - 72*T + 5184
$7$	$ T^{2} $ T^2
$11$	$ T^{2} - 414T + 171396 $ T^2 - 414*T + 171396
$13$	$ (T + 1054)^{2} $ (T + 1054)^2
$17$	$ T^{2} - 1848 T + 3415104 $ T^2 - 1848*T + 3415104
$19$	$ T^{2} + 236T + 55696 $ T^2 + 236*T + 55696
$23$	$ T^{2} + 2898 T + 8398404 $ T^2 + 2898*T + 8398404
$29$	$ (T + 6522)^{2} $ (T + 6522)^2
$31$	$ T^{2} + 6200 T + 38440000 $ T^2 + 6200*T + 38440000
$37$	$ T^{2} + 9650 T + 93122500 $ T^2 + 9650*T + 93122500
$41$	$ (T - 8484)^{2} $ (T - 8484)^2
$43$	$ (T + 10804)^{2} $ (T + 10804)^2
$47$	$ T^{2} + 60T + 3600 $ T^2 + 60*T + 3600
$53$	$ T^{2} + 22506 T + 506520036 $ T^2 + 22506*T + 506520036
$59$	$ T^{2} - 28176 T + 793886976 $ T^2 - 28176*T + 793886976
$61$	$ T^{2} + \cdots + 1238617636 $ T^2 - 35194*T + 1238617636
$67$	$ T^{2} - 28216 T + 796142656 $ T^2 - 28216*T + 796142656
$71$	$ (T + 6642)^{2} $ (T + 6642)^2
$73$	$ T^{2} + \cdots + 2713368100 $ T^2 - 52090*T + 2713368100
$79$	$ T^{2} + \cdots + 1878355600 $ T^2 + 43340*T + 1878355600
$83$	$ (T - 25716)^{2} $ (T - 25716)^2
$89$	$ T^{2} + \cdots + 9746428176 $ T^2 + 98724*T + 9746428176
$97$	$ (T + 148954)^{2} $ (T + 148954)^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)

\(f(q)\)	\(=\)	\( q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} + 72 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + (288 \zeta_{6} - 288) q^{10} + ( - 414 \zeta_{6} + 414) q^{11} + \cdots - 33534 q^{99} +O(q^{100}) \) q + 4z q^2 + (9z - 9) q^3 + (16z - 16) q^4 + 72z q^5 - 36 * q^6 - 64 * q^8 - 81z q^9 + (288z - 288) q^10 + (-414z + 414) q^11 - 144z q^12 - 1054 * q^13 - 648 * q^15 - 256z q^16 + (-1848z + 1848) q^17 + (-324z + 324) q^18 - 236z q^19 - 1152 * q^20 + 1656 * q^22 - 2898z q^23 + (-576z + 576) q^24 + (2059z - 2059) q^25 - 4216z q^26 + 729 * q^27 - 6522 * q^29 - 2592z q^30 + (6200z - 6200) q^31 + (-1024z + 1024) q^32 + 3726z q^33 + 7392 * q^34 + 1296 * q^36 - 9650z q^37 + (-944z + 944) q^38 + (-9486z + 9486) q^39 - 4608z q^40 + 8484 * q^41 - 10804 * q^43 + 6624z q^44 + (-5832z + 5832) q^45 + (-11592z + 11592) q^46 - 60z q^47 + 2304 * q^48 - 8236 * q^50 + 16632z q^51 + (-16864z + 16864) q^52 + (22506z - 22506) q^53 + 2916z q^54 + 29808 * q^55 + 2124 * q^57 - 26088z q^58 + (-28176z + 28176) q^59 + (-10368z + 10368) q^60 + 35194z q^61 - 24800 * q^62 + 4096 * q^64 - 75888z q^65 + (14904z - 14904) q^66 + (-28216z + 28216) q^67 + 29568z q^68 + 26082 * q^69 - 6642 * q^71 + 5184z q^72 + (-52090z + 52090) q^73 + (-38600z + 38600) q^74 - 18531z q^75 + 3776 * q^76 + 37944 * q^78 - 43340z q^79 + (-18432z + 18432) q^80 + (6561z - 6561) q^81 + 33936z q^82 + 25716 * q^83 + 133056 * q^85 - 43216z q^86 + (-58698z + 58698) q^87 + (26496z - 26496) q^88 - 98724z q^89 + 23328 * q^90 + 46368 * q^92 - 55800z q^93 + (-240z + 240) q^94 + (-16992z + 16992) q^95 + 9216z q^96 - 148954 * q^97 - 33534 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} + 72 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9} - 288 q^{10} + 414 q^{11} - 144 q^{12} - 2108 q^{13} - 1296 q^{15} - 256 q^{16} + 1848 q^{17} + 324 q^{18} - 236 q^{19} - 2304 q^{20}+ \cdots - 67068 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 9 * q^3 - 16 * q^4 + 72 * q^5 - 72 * q^6 - 128 * q^8 - 81 * q^9 - 288 * q^10 + 414 * q^11 - 144 * q^12 - 2108 * q^13 - 1296 * q^15 - 256 * q^16 + 1848 * q^17 + 324 * q^18 - 236 * q^19 - 2304 * q^20 + 3312 * q^22 - 2898 * q^23 + 576 * q^24 - 2059 * q^25 - 4216 * q^26 + 1458 * q^27 - 13044 * q^29 - 2592 * q^30 - 6200 * q^31 + 1024 * q^32 + 3726 * q^33 + 14784 * q^34 + 2592 * q^36 - 9650 * q^37 + 944 * q^38 + 9486 * q^39 - 4608 * q^40 + 16968 * q^41 - 21608 * q^43 + 6624 * q^44 + 5832 * q^45 + 11592 * q^46 - 60 * q^47 + 4608 * q^48 - 16472 * q^50 + 16632 * q^51 + 16864 * q^52 - 22506 * q^53 + 2916 * q^54 + 59616 * q^55 + 4248 * q^57 - 26088 * q^58 + 28176 * q^59 + 10368 * q^60 + 35194 * q^61 - 49600 * q^62 + 8192 * q^64 - 75888 * q^65 - 14904 * q^66 + 28216 * q^67 + 29568 * q^68 + 52164 * q^69 - 13284 * q^71 + 5184 * q^72 + 52090 * q^73 + 38600 * q^74 - 18531 * q^75 + 7552 * q^76 + 75888 * q^78 - 43340 * q^79 + 18432 * q^80 - 6561 * q^81 + 33936 * q^82 + 51432 * q^83 + 266112 * q^85 - 43216 * q^86 + 58698 * q^87 - 26496 * q^88 - 98724 * q^89 + 46656 * q^90 + 92736 * q^92 - 55800 * q^93 + 240 * q^94 + 16992 * q^95 + 9216 * q^96 - 297908 * q^97 - 67068 * q^99

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