LMFDB - Newform orbit 147.6.e.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [147,6,Mod(67,147)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, 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("147.67"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	147.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-10,-9,-68,106] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.5764215125$
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 21)
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 - 10 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (68 \zeta_{6} - 68) q^{4} + 106 \zeta_{6} q^{5} + 90 q^{6} + 360 q^{8} - 81 \zeta_{6} q^{9} + ( - 1060 \zeta_{6} + 1060) q^{10} + (92 \zeta_{6} - 92) q^{11} + \cdots + 7452 q^{99} +O(q^{100}) $ q - 10z q^2 + (9z - 9) q^3 + (68z - 68) q^4 + 106z q^5 + 90 * q^6 + 360 * q^8 - 81z q^9 + (-1060z + 1060) q^10 + (92z - 92) q^11 - 612z q^12 + 670 * q^13 - 954 * q^15 - 1424z q^16 + (-222z + 222) q^17 + (810z - 810) q^18 + 908z q^19 - 7208 * q^20 + 920 * q^22 + 1176z q^23 + (3240z - 3240) q^24 + (8111z - 8111) q^25 - 6700z q^26 + 729 * q^27 + 1118 * q^29 + 9540z q^30 + (3696z - 3696) q^31 + (2720z - 2720) q^32 - 828z q^33 - 2220 * q^34 + 5508 * q^36 - 4182z q^37 + (-9080z + 9080) q^38 + (6030z - 6030) q^39 + 38160z q^40 - 6662 * q^41 - 3700 * q^43 - 6256z q^44 + (-8586z + 8586) q^45 + (-11760z + 11760) q^46 + 7056z q^47 + 12816 * q^48 + 81110 * q^50 + 1998z q^51 + (45560z - 45560) q^52 + (-37578z + 37578) q^53 - 7290z q^54 - 9752 * q^55 - 8172 * q^57 - 11180z q^58 + (32700z - 32700) q^59 + (-64872z + 64872) q^60 + 10802z q^61 + 36960 * q^62 - 18368 * q^64 + 71020z q^65 + (8280z - 8280) q^66 + (64996z - 64996) q^67 + 15096z q^68 - 10584 * q^69 - 61320 * q^71 - 29160z q^72 + (38922z - 38922) q^73 + (41820z - 41820) q^74 - 72999z q^75 - 61744 * q^76 + 60300 * q^78 + 88096z q^79 + (-150944z + 150944) q^80 + (6561z - 6561) q^81 + 66620z q^82 + 71892 * q^83 + 23532 * q^85 + 37000z q^86 + (10062z - 10062) q^87 + (33120z - 33120) q^88 - 111818z q^89 - 85860 * q^90 - 79968 * q^92 - 33264z q^93 + (-70560z + 70560) q^94 + (96248z - 96248) q^95 - 24480z q^96 - 150846 * q^97 + 7452 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{2} - 9 q^{3} - 68 q^{4} + 106 q^{5} + 180 q^{6} + 720 q^{8} - 81 q^{9} + 1060 q^{10} - 92 q^{11} - 612 q^{12} + 1340 q^{13} - 1908 q^{15} - 1424 q^{16} + 222 q^{17} - 810 q^{18} + 908 q^{19}+ \cdots + 14904 q^{99}+O(q^{100}) $ 2 * q - 10 * q^2 - 9 * q^3 - 68 * q^4 + 106 * q^5 + 180 * q^6 + 720 * q^8 - 81 * q^9 + 1060 * q^10 - 92 * q^11 - 612 * q^12 + 1340 * q^13 - 1908 * q^15 - 1424 * q^16 + 222 * q^17 - 810 * q^18 + 908 * q^19 - 14416 * q^20 + 1840 * q^22 + 1176 * q^23 - 3240 * q^24 - 8111 * q^25 - 6700 * q^26 + 1458 * q^27 + 2236 * q^29 + 9540 * q^30 - 3696 * q^31 - 2720 * q^32 - 828 * q^33 - 4440 * q^34 + 11016 * q^36 - 4182 * q^37 + 9080 * q^38 - 6030 * q^39 + 38160 * q^40 - 13324 * q^41 - 7400 * q^43 - 6256 * q^44 + 8586 * q^45 + 11760 * q^46 + 7056 * q^47 + 25632 * q^48 + 162220 * q^50 + 1998 * q^51 - 45560 * q^52 + 37578 * q^53 - 7290 * q^54 - 19504 * q^55 - 16344 * q^57 - 11180 * q^58 - 32700 * q^59 + 64872 * q^60 + 10802 * q^61 + 73920 * q^62 - 36736 * q^64 + 71020 * q^65 - 8280 * q^66 - 64996 * q^67 + 15096 * q^68 - 21168 * q^69 - 122640 * q^71 - 29160 * q^72 - 38922 * q^73 - 41820 * q^74 - 72999 * q^75 - 123488 * q^76 + 120600 * q^78 + 88096 * q^79 + 150944 * q^80 - 6561 * q^81 + 66620 * q^82 + 143784 * q^83 + 47064 * q^85 + 37000 * q^86 - 10062 * q^87 - 33120 * q^88 - 111818 * q^89 - 171720 * q^90 - 159936 * q^92 - 33264 * q^93 + 70560 * q^94 - 96248 * q^95 - 24480 * q^96 - 301692 * q^97 + 14904 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$.

$n$	$50$	$52$
$\chi(n)$	$1$	$-\zeta_{6}$

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

−5.00000

−

8.66025i

−4.50000

7.79423i

−34.0000

58.8897i

53.0000

91.7987i

90.0000

360.000

−40.5000

−

70.1481i

530.000

−

917.987i

79.1

−5.00000

8.66025i

−4.50000

−

7.79423i

−34.0000

−

58.8897i

53.0000

−

91.7987i

90.0000

360.000

−40.5000

70.1481i

530.000

917.987i

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	147.6.e.a		2
7.b	odd	2	1		147.6.e.b		2
7.c	even	3	1		21.6.a.d	✓	1
7.c	even	3	1	inner	147.6.e.a		2
7.d	odd	6	1		147.6.a.g		1
7.d	odd	6	1		147.6.e.b		2
21.g	even	6	1		441.6.a.b		1
21.h	odd	6	1		63.6.a.a		1
28.g	odd	6	1		336.6.a.a		1
35.j	even	6	1		525.6.a.a		1
35.l	odd	12	2		525.6.d.a		2
84.n	even	6	1		1008.6.a.bc		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.6.a.d	✓	1	7.c	even	3	1
63.6.a.a		1	21.h	odd	6	1
147.6.a.g		1	7.d	odd	6	1
147.6.e.a		2	1.a	even	1	1	trivial
147.6.e.a		2	7.c	even	3	1	inner
147.6.e.b		2	7.b	odd	2	1
147.6.e.b		2	7.d	odd	6	1
336.6.a.a		1	28.g	odd	6	1
441.6.a.b		1	21.g	even	6	1
525.6.a.a		1	35.j	even	6	1
525.6.d.a		2	35.l	odd	12	2
1008.6.a.bc		1	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(147, [\chi])$:

$ T_{2}^{2} + 10T_{2} + 100 $

T2^2 + 10*T2 + 100

$ T_{5}^{2} - 106T_{5} + 11236 $

T5^2 - 106*T5 + 11236

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 10T + 100 $ T^2 + 10*T + 100
$3$	$ T^{2} + 9T + 81 $ T^2 + 9*T + 81
$5$	$ T^{2} - 106T + 11236 $ T^2 - 106*T + 11236
$7$	$ T^{2} $ T^2
$11$	$ T^{2} + 92T + 8464 $ T^2 + 92*T + 8464
$13$	$ (T - 670)^{2} $ (T - 670)^2
$17$	$ T^{2} - 222T + 49284 $ T^2 - 222*T + 49284
$19$	$ T^{2} - 908T + 824464 $ T^2 - 908*T + 824464
$23$	$ T^{2} - 1176 T + 1382976 $ T^2 - 1176*T + 1382976
$29$	$ (T - 1118)^{2} $ (T - 1118)^2
$31$	$ T^{2} + 3696 T + 13660416 $ T^2 + 3696*T + 13660416
$37$	$ T^{2} + 4182 T + 17489124 $ T^2 + 4182*T + 17489124
$41$	$ (T + 6662)^{2} $ (T + 6662)^2
$43$	$ (T + 3700)^{2} $ (T + 3700)^2
$47$	$ T^{2} - 7056 T + 49787136 $ T^2 - 7056*T + 49787136
$53$	$ T^{2} + \cdots + 1412106084 $ T^2 - 37578*T + 1412106084
$59$	$ T^{2} + \cdots + 1069290000 $ T^2 + 32700*T + 1069290000
$61$	$ T^{2} - 10802 T + 116683204 $ T^2 - 10802*T + 116683204
$67$	$ T^{2} + \cdots + 4224480016 $ T^2 + 64996*T + 4224480016
$71$	$ (T + 61320)^{2} $ (T + 61320)^2
$73$	$ T^{2} + \cdots + 1514922084 $ T^2 + 38922*T + 1514922084
$79$	$ T^{2} + \cdots + 7760905216 $ T^2 - 88096*T + 7760905216
$83$	$ (T - 71892)^{2} $ (T - 71892)^2
$89$	$ T^{2} + \cdots + 12503265124 $ T^2 + 111818*T + 12503265124
$97$	$ (T + 150846)^{2} $ (T + 150846)^2
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Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	147.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - 10 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (68 \zeta_{6} - 68) q^{4} + 106 \zeta_{6} q^{5} + 90 q^{6} + 360 q^{8} - 81 \zeta_{6} q^{9} + ( - 1060 \zeta_{6} + 1060) q^{10} + (92 \zeta_{6} - 92) q^{11} + \cdots + 7452 q^{99} +O(q^{100}) \) q - 10z q^2 + (9z - 9) q^3 + (68z - 68) q^4 + 106z q^5 + 90 * q^6 + 360 * q^8 - 81z q^9 + (-1060z + 1060) q^10 + (92z - 92) q^11 - 612z q^12 + 670 * q^13 - 954 * q^15 - 1424z q^16 + (-222z + 222) q^17 + (810z - 810) q^18 + 908z q^19 - 7208 * q^20 + 920 * q^22 + 1176z q^23 + (3240z - 3240) q^24 + (8111z - 8111) q^25 - 6700z q^26 + 729 * q^27 + 1118 * q^29 + 9540z q^30 + (3696z - 3696) q^31 + (2720z - 2720) q^32 - 828z q^33 - 2220 * q^34 + 5508 * q^36 - 4182z q^37 + (-9080z + 9080) q^38 + (6030z - 6030) q^39 + 38160z q^40 - 6662 * q^41 - 3700 * q^43 - 6256z q^44 + (-8586z + 8586) q^45 + (-11760z + 11760) q^46 + 7056z q^47 + 12816 * q^48 + 81110 * q^50 + 1998z q^51 + (45560z - 45560) q^52 + (-37578z + 37578) q^53 - 7290z q^54 - 9752 * q^55 - 8172 * q^57 - 11180z q^58 + (32700z - 32700) q^59 + (-64872z + 64872) q^60 + 10802z q^61 + 36960 * q^62 - 18368 * q^64 + 71020z q^65 + (8280z - 8280) q^66 + (64996z - 64996) q^67 + 15096z q^68 - 10584 * q^69 - 61320 * q^71 - 29160z q^72 + (38922z - 38922) q^73 + (41820z - 41820) q^74 - 72999z q^75 - 61744 * q^76 + 60300 * q^78 + 88096z q^79 + (-150944z + 150944) q^80 + (6561z - 6561) q^81 + 66620z q^82 + 71892 * q^83 + 23532 * q^85 + 37000z q^86 + (10062z - 10062) q^87 + (33120z - 33120) q^88 - 111818z q^89 - 85860 * q^90 - 79968 * q^92 - 33264z q^93 + (-70560z + 70560) q^94 + (96248z - 96248) q^95 - 24480z q^96 - 150846 * q^97 + 7452 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{2} - 9 q^{3} - 68 q^{4} + 106 q^{5} + 180 q^{6} + 720 q^{8} - 81 q^{9} + 1060 q^{10} - 92 q^{11} - 612 q^{12} + 1340 q^{13} - 1908 q^{15} - 1424 q^{16} + 222 q^{17} - 810 q^{18} + 908 q^{19}+ \cdots + 14904 q^{99}+O(q^{100}) \) 2 * q - 10 * q^2 - 9 * q^3 - 68 * q^4 + 106 * q^5 + 180 * q^6 + 720 * q^8 - 81 * q^9 + 1060 * q^10 - 92 * q^11 - 612 * q^12 + 1340 * q^13 - 1908 * q^15 - 1424 * q^16 + 222 * q^17 - 810 * q^18 + 908 * q^19 - 14416 * q^20 + 1840 * q^22 + 1176 * q^23 - 3240 * q^24 - 8111 * q^25 - 6700 * q^26 + 1458 * q^27 + 2236 * q^29 + 9540 * q^30 - 3696 * q^31 - 2720 * q^32 - 828 * q^33 - 4440 * q^34 + 11016 * q^36 - 4182 * q^37 + 9080 * q^38 - 6030 * q^39 + 38160 * q^40 - 13324 * q^41 - 7400 * q^43 - 6256 * q^44 + 8586 * q^45 + 11760 * q^46 + 7056 * q^47 + 25632 * q^48 + 162220 * q^50 + 1998 * q^51 - 45560 * q^52 + 37578 * q^53 - 7290 * q^54 - 19504 * q^55 - 16344 * q^57 - 11180 * q^58 - 32700 * q^59 + 64872 * q^60 + 10802 * q^61 + 73920 * q^62 - 36736 * q^64 + 71020 * q^65 - 8280 * q^66 - 64996 * q^67 + 15096 * q^68 - 21168 * q^69 - 122640 * q^71 - 29160 * q^72 - 38922 * q^73 - 41820 * q^74 - 72999 * q^75 - 123488 * q^76 + 120600 * q^78 + 88096 * q^79 + 150944 * q^80 - 6561 * q^81 + 66620 * q^82 + 143784 * q^83 + 47064 * q^85 + 37000 * q^86 - 10062 * q^87 - 33120 * q^88 - 111818 * q^89 - 171720 * q^90 - 159936 * q^92 - 33264 * q^93 + 70560 * q^94 - 96248 * q^95 - 24480 * q^96 - 301692 * q^97 + 14904 * q^99

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