LMFDB - Newform orbit 294.6.e.m

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)

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")

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);

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,-16,86,-72,0,-128,-81,-344,-34] 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} + 86 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + (344 \zeta_{6} - 344) q^{10} + (34 \zeta_{6} - 34) q^{11} - 144 \zeta_{6} q^{12} + \cdots + 2754 q^{99} +O(q^{100}) $ q + 4z q^2 + (9z - 9) q^3 + (16z - 16) q^4 + 86z q^5 - 36 * q^6 - 64 * q^8 - 81z q^9 + (344z - 344) q^10 + (34z - 34) q^11 - 144z q^12 + 3 * q^13 - 774 * q^15 - 256z q^16 + (1904z - 1904) q^17 + (-324z + 324) q^18 - 1489z q^19 - 1376 * q^20 - 136 * q^22 + 224z q^23 + (-576z + 576) q^24 + (4271z - 4271) q^25 + 12z q^26 + 729 * q^27 - 6508 * q^29 - 3096z q^30 + (-1731z + 1731) q^31 + (-1024z + 1024) q^32 - 306z q^33 - 7616 * q^34 + 1296 * q^36 + 7633z q^37 + (-5956z + 5956) q^38 + (27z - 27) q^39 - 5504z q^40 - 15414 * q^41 + 18491 * q^43 - 544z q^44 + (-6966z + 6966) q^45 + (896z - 896) q^46 + 18462z q^47 + 2304 * q^48 - 17084 * q^50 - 17136z q^51 + (48z - 48) q^52 + (-19956z + 19956) q^53 + 2916z q^54 - 2924 * q^55 + 13401 * q^57 - 26032z q^58 + (31828z - 31828) q^59 + (-12384z + 12384) q^60 - 57654z q^61 + 6924 * q^62 + 4096 * q^64 + 258z q^65 + (-1224z + 1224) q^66 + (-60563z + 60563) q^67 - 30464z q^68 - 2016 * q^69 - 44834 * q^71 + 5184z q^72 + (-20821z + 20821) q^73 + (30532z - 30532) q^74 - 38439z q^75 + 23824 * q^76 - 108 * q^78 + 30531z q^79 + (-22016z + 22016) q^80 + (6561z - 6561) q^81 - 61656z q^82 - 110602 * q^83 - 163744 * q^85 + 73964z q^86 + (-58572z + 58572) q^87 + (-2176z + 2176) q^88 - 58992z q^89 + 27864 * q^90 - 3584 * q^92 + 15579z q^93 + (73848z - 73848) q^94 + (-128054z + 128054) q^95 + 9216z q^96 + 119846 * q^97 + 2754 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} + 86 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9} - 344 q^{10} - 34 q^{11} - 144 q^{12} + 6 q^{13} - 1548 q^{15} - 256 q^{16} - 1904 q^{17} + 324 q^{18} - 1489 q^{19} - 2752 q^{20}+ \cdots + 5508 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 - 9 * q^3 - 16 * q^4 + 86 * q^5 - 72 * q^6 - 128 * q^8 - 81 * q^9 - 344 * q^10 - 34 * q^11 - 144 * q^12 + 6 * q^13 - 1548 * q^15 - 256 * q^16 - 1904 * q^17 + 324 * q^18 - 1489 * q^19 - 2752 * q^20 - 272 * q^22 + 224 * q^23 + 576 * q^24 - 4271 * q^25 + 12 * q^26 + 1458 * q^27 - 13016 * q^29 - 3096 * q^30 + 1731 * q^31 + 1024 * q^32 - 306 * q^33 - 15232 * q^34 + 2592 * q^36 + 7633 * q^37 + 5956 * q^38 - 27 * q^39 - 5504 * q^40 - 30828 * q^41 + 36982 * q^43 - 544 * q^44 + 6966 * q^45 - 896 * q^46 + 18462 * q^47 + 4608 * q^48 - 34168 * q^50 - 17136 * q^51 - 48 * q^52 + 19956 * q^53 + 2916 * q^54 - 5848 * q^55 + 26802 * q^57 - 26032 * q^58 - 31828 * q^59 + 12384 * q^60 - 57654 * q^61 + 13848 * q^62 + 8192 * q^64 + 258 * q^65 + 1224 * q^66 + 60563 * q^67 - 30464 * q^68 - 4032 * q^69 - 89668 * q^71 + 5184 * q^72 + 20821 * q^73 - 30532 * q^74 - 38439 * q^75 + 47648 * q^76 - 216 * q^78 + 30531 * q^79 + 22016 * q^80 - 6561 * q^81 - 61656 * q^82 - 221204 * q^83 - 327488 * q^85 + 73964 * q^86 + 58572 * q^87 + 2176 * q^88 - 58992 * q^89 + 55728 * q^90 - 7168 * q^92 + 15579 * q^93 - 73848 * q^94 + 128054 * q^95 + 9216 * q^96 + 239692 * q^97 + 5508 * 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

43.0000

74.4782i

−36.0000

−64.0000

−40.5000

−

70.1481i

−172.000

297.913i

79.1

2.00000

−

3.46410i

−4.50000

−

7.79423i

−8.00000

−

13.8564i

43.0000

−

74.4782i

−36.0000

−64.0000

−40.5000

70.1481i

−172.000

−

297.913i

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.m		2
7.b	odd	2	1		42.6.e.b	✓	2
7.c	even	3	1		294.6.a.e		1
7.c	even	3	1	inner	294.6.e.m		2
7.d	odd	6	1		42.6.e.b	✓	2
7.d	odd	6	1		294.6.a.d		1
21.c	even	2	1		126.6.g.b		2
21.g	even	6	1		126.6.g.b		2
21.g	even	6	1		882.6.a.m		1
21.h	odd	6	1		882.6.a.y		1
28.d	even	2	1		336.6.q.a		2
28.f	even	6	1		336.6.q.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.6.e.b	✓	2	7.b	odd	2	1
42.6.e.b	✓	2	7.d	odd	6	1
126.6.g.b		2	21.c	even	2	1
126.6.g.b		2	21.g	even	6	1
294.6.a.d		1	7.d	odd	6	1
294.6.a.e		1	7.c	even	3	1
294.6.e.m		2	1.a	even	1	1	trivial
294.6.e.m		2	7.c	even	3	1	inner
336.6.q.a		2	28.d	even	2	1
336.6.q.a		2	28.f	even	6	1
882.6.a.m		1	21.g	even	6	1
882.6.a.y		1	21.h	odd	6	1

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} - 86T_{5} + 7396 $

T5^2 - 86*T5 + 7396

$ T_{11}^{2} + 34T_{11} + 1156 $

T11^2 + 34*T11 + 1156

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} - 86T + 7396 $ T^2 - 86*T + 7396
$7$	$ T^{2} $ T^2
$11$	$ T^{2} + 34T + 1156 $ T^2 + 34*T + 1156
$13$	$ (T - 3)^{2} $ (T - 3)^2
$17$	$ T^{2} + 1904 T + 3625216 $ T^2 + 1904*T + 3625216
$19$	$ T^{2} + 1489 T + 2217121 $ T^2 + 1489*T + 2217121
$23$	$ T^{2} - 224T + 50176 $ T^2 - 224*T + 50176
$29$	$ (T + 6508)^{2} $ (T + 6508)^2
$31$	$ T^{2} - 1731 T + 2996361 $ T^2 - 1731*T + 2996361
$37$	$ T^{2} - 7633 T + 58262689 $ T^2 - 7633*T + 58262689
$41$	$ (T + 15414)^{2} $ (T + 15414)^2
$43$	$ (T - 18491)^{2} $ (T - 18491)^2
$47$	$ T^{2} - 18462 T + 340845444 $ T^2 - 18462*T + 340845444
$53$	$ T^{2} - 19956 T + 398241936 $ T^2 - 19956*T + 398241936
$59$	$ T^{2} + \cdots + 1013021584 $ T^2 + 31828*T + 1013021584
$61$	$ T^{2} + \cdots + 3323983716 $ T^2 + 57654*T + 3323983716
$67$	$ T^{2} + \cdots + 3667876969 $ T^2 - 60563*T + 3667876969
$71$	$ (T + 44834)^{2} $ (T + 44834)^2
$73$	$ T^{2} - 20821 T + 433514041 $ T^2 - 20821*T + 433514041
$79$	$ T^{2} - 30531 T + 932141961 $ T^2 - 30531*T + 932141961
$83$	$ (T + 110602)^{2} $ (T + 110602)^2
$89$	$ T^{2} + \cdots + 3480056064 $ T^2 + 58992*T + 3480056064
$97$	$ (T - 119846)^{2} $ (T - 119846)^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)

\(f(q)\)	\(=\)	\( q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} + 86 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + (344 \zeta_{6} - 344) q^{10} + (34 \zeta_{6} - 34) q^{11} - 144 \zeta_{6} q^{12} + \cdots + 2754 q^{99} +O(q^{100}) \) q + 4z q^2 + (9z - 9) q^3 + (16z - 16) q^4 + 86z q^5 - 36 * q^6 - 64 * q^8 - 81z q^9 + (344z - 344) q^10 + (34z - 34) q^11 - 144z q^12 + 3 * q^13 - 774 * q^15 - 256z q^16 + (1904z - 1904) q^17 + (-324z + 324) q^18 - 1489z q^19 - 1376 * q^20 - 136 * q^22 + 224z q^23 + (-576z + 576) q^24 + (4271z - 4271) q^25 + 12z q^26 + 729 * q^27 - 6508 * q^29 - 3096z q^30 + (-1731z + 1731) q^31 + (-1024z + 1024) q^32 - 306z q^33 - 7616 * q^34 + 1296 * q^36 + 7633z q^37 + (-5956z + 5956) q^38 + (27z - 27) q^39 - 5504z q^40 - 15414 * q^41 + 18491 * q^43 - 544z q^44 + (-6966z + 6966) q^45 + (896z - 896) q^46 + 18462z q^47 + 2304 * q^48 - 17084 * q^50 - 17136z q^51 + (48z - 48) q^52 + (-19956z + 19956) q^53 + 2916z q^54 - 2924 * q^55 + 13401 * q^57 - 26032z q^58 + (31828z - 31828) q^59 + (-12384z + 12384) q^60 - 57654z q^61 + 6924 * q^62 + 4096 * q^64 + 258z q^65 + (-1224z + 1224) q^66 + (-60563z + 60563) q^67 - 30464z q^68 - 2016 * q^69 - 44834 * q^71 + 5184z q^72 + (-20821z + 20821) q^73 + (30532z - 30532) q^74 - 38439z q^75 + 23824 * q^76 - 108 * q^78 + 30531z q^79 + (-22016z + 22016) q^80 + (6561z - 6561) q^81 - 61656z q^82 - 110602 * q^83 - 163744 * q^85 + 73964z q^86 + (-58572z + 58572) q^87 + (-2176z + 2176) q^88 - 58992z q^89 + 27864 * q^90 - 3584 * q^92 + 15579z q^93 + (73848z - 73848) q^94 + (-128054z + 128054) q^95 + 9216z q^96 + 119846 * q^97 + 2754 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} + 86 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9} - 344 q^{10} - 34 q^{11} - 144 q^{12} + 6 q^{13} - 1548 q^{15} - 256 q^{16} - 1904 q^{17} + 324 q^{18} - 1489 q^{19} - 2752 q^{20}+ \cdots + 5508 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 9 * q^3 - 16 * q^4 + 86 * q^5 - 72 * q^6 - 128 * q^8 - 81 * q^9 - 344 * q^10 - 34 * q^11 - 144 * q^12 + 6 * q^13 - 1548 * q^15 - 256 * q^16 - 1904 * q^17 + 324 * q^18 - 1489 * q^19 - 2752 * q^20 - 272 * q^22 + 224 * q^23 + 576 * q^24 - 4271 * q^25 + 12 * q^26 + 1458 * q^27 - 13016 * q^29 - 3096 * q^30 + 1731 * q^31 + 1024 * q^32 - 306 * q^33 - 15232 * q^34 + 2592 * q^36 + 7633 * q^37 + 5956 * q^38 - 27 * q^39 - 5504 * q^40 - 30828 * q^41 + 36982 * q^43 - 544 * q^44 + 6966 * q^45 - 896 * q^46 + 18462 * q^47 + 4608 * q^48 - 34168 * q^50 - 17136 * q^51 - 48 * q^52 + 19956 * q^53 + 2916 * q^54 - 5848 * q^55 + 26802 * q^57 - 26032 * q^58 - 31828 * q^59 + 12384 * q^60 - 57654 * q^61 + 13848 * q^62 + 8192 * q^64 + 258 * q^65 + 1224 * q^66 + 60563 * q^67 - 30464 * q^68 - 4032 * q^69 - 89668 * q^71 + 5184 * q^72 + 20821 * q^73 - 30532 * q^74 - 38439 * q^75 + 47648 * q^76 - 216 * q^78 + 30531 * q^79 + 22016 * q^80 - 6561 * q^81 - 61656 * q^82 - 221204 * q^83 - 327488 * q^85 + 73964 * q^86 + 58572 * q^87 + 2176 * q^88 - 58992 * q^89 + 55728 * q^90 - 7168 * q^92 + 15579 * q^93 - 73848 * q^94 + 128054 * q^95 + 9216 * q^96 + 239692 * q^97 + 5508 * q^99

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