LMFDB - Newform orbit 147.4.e.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [147,4,Mod(67,147)] 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("147.67"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	147.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-4,-3,-8,-18] 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:	$8.67328077084$
Analytic rank:	$1$
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:	yes
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} + (3 \zeta_{6} - 3) q^{3} + (8 \zeta_{6} - 8) q^{4} - 18 \zeta_{6} q^{5} + 12 q^{6} - 9 \zeta_{6} q^{9} + (72 \zeta_{6} - 72) q^{10} + ( - 50 \zeta_{6} + 50) q^{11} - 24 \zeta_{6} q^{12} + \cdots - 450 q^{99} +O(q^{100}) $ q - 4z q^2 + (3z - 3) q^3 + (8z - 8) q^4 - 18z q^5 + 12 * q^6 - 9z q^9 + (72z - 72) q^10 + (-50z + 50) q^11 - 24z q^12 - 36 * q^13 + 54 * q^15 + 64z q^16 + (126z - 126) q^17 + (36z - 36) q^18 + 72z q^19 + 144 * q^20 - 200 * q^22 - 14z q^23 + (199z - 199) q^25 + 144z q^26 + 27 * q^27 + 158 * q^29 - 216z q^30 + (-36z + 36) q^31 + (-256z + 256) q^32 + 150z q^33 + 504 * q^34 + 72 * q^36 + 162z q^37 + (-288z + 288) q^38 + (-108z + 108) q^39 - 270 * q^41 - 324 * q^43 + 400z q^44 + (162z - 162) q^45 + (56z - 56) q^46 + 72z q^47 - 192 * q^48 + 796 * q^50 - 378z q^51 + (-288z + 288) q^52 + (-22z + 22) q^53 - 108z q^54 - 900 * q^55 - 216 * q^57 - 632z q^58 + (468z - 468) q^59 + (432z - 432) q^60 - 792z q^61 - 144 * q^62 - 512 * q^64 + 648z q^65 + (-600z + 600) q^66 + (232z - 232) q^67 - 1008z q^68 + 42 * q^69 - 734 * q^71 + (180z - 180) q^73 + (-648z + 648) q^74 - 597z q^75 - 576 * q^76 - 432 * q^78 - 236z q^79 + (-1152z + 1152) q^80 + (81z - 81) q^81 + 1080z q^82 + 36 * q^83 + 2268 * q^85 + 1296z q^86 + (474z - 474) q^87 - 234z q^89 + 648 * q^90 + 112 * q^92 + 108z q^93 + (-288z + 288) q^94 + (-1296z + 1296) q^95 + 768z q^96 + 468 * q^97 - 450 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 3 q^{3} - 8 q^{4} - 18 q^{5} + 24 q^{6} - 9 q^{9} - 72 q^{10} + 50 q^{11} - 24 q^{12} - 72 q^{13} + 108 q^{15} + 64 q^{16} - 126 q^{17} - 36 q^{18} + 72 q^{19} + 288 q^{20} - 400 q^{22}+ \cdots - 900 q^{99}+O(q^{100}) $ 2 * q - 4 * q^2 - 3 * q^3 - 8 * q^4 - 18 * q^5 + 24 * q^6 - 9 * q^9 - 72 * q^10 + 50 * q^11 - 24 * q^12 - 72 * q^13 + 108 * q^15 + 64 * q^16 - 126 * q^17 - 36 * q^18 + 72 * q^19 + 288 * q^20 - 400 * q^22 - 14 * q^23 - 199 * q^25 + 144 * q^26 + 54 * q^27 + 316 * q^29 - 216 * q^30 + 36 * q^31 + 256 * q^32 + 150 * q^33 + 1008 * q^34 + 144 * q^36 + 162 * q^37 + 288 * q^38 + 108 * q^39 - 540 * q^41 - 648 * q^43 + 400 * q^44 - 162 * q^45 - 56 * q^46 + 72 * q^47 - 384 * q^48 + 1592 * q^50 - 378 * q^51 + 288 * q^52 + 22 * q^53 - 108 * q^54 - 1800 * q^55 - 432 * q^57 - 632 * q^58 - 468 * q^59 - 432 * q^60 - 792 * q^61 - 288 * q^62 - 1024 * q^64 + 648 * q^65 + 600 * q^66 - 232 * q^67 - 1008 * q^68 + 84 * q^69 - 1468 * q^71 - 180 * q^73 + 648 * q^74 - 597 * q^75 - 1152 * q^76 - 864 * q^78 - 236 * q^79 + 1152 * q^80 - 81 * q^81 + 1080 * q^82 + 72 * q^83 + 4536 * q^85 + 1296 * q^86 - 474 * q^87 - 234 * q^89 + 1296 * q^90 + 224 * q^92 + 108 * q^93 + 288 * q^94 + 1296 * q^95 + 768 * q^96 + 936 * q^97 - 900 * 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

−2.00000

−

3.46410i

−1.50000

2.59808i

−4.00000

6.92820i

−9.00000

−

15.5885i

12.0000

−4.50000

−

7.79423i

−36.0000

62.3538i

79.1

−2.00000

3.46410i

−1.50000

−

2.59808i

−4.00000

−

6.92820i

−9.00000

15.5885i

12.0000

−4.50000

7.79423i

−36.0000

−

62.3538i

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	147.4.e.a		2
3.b	odd	2	1		441.4.e.o		2
7.b	odd	2	1		147.4.e.d		2
7.c	even	3	1		147.4.a.h	yes	1
7.c	even	3	1	inner	147.4.e.a		2
7.d	odd	6	1		147.4.a.f	✓	1
7.d	odd	6	1		147.4.e.d		2
21.c	even	2	1		441.4.e.l		2
21.g	even	6	1		441.4.a.c		1
21.g	even	6	1		441.4.e.l		2
21.h	odd	6	1		441.4.a.a		1
21.h	odd	6	1		441.4.e.o		2
28.f	even	6	1		2352.4.a.t		1
28.g	odd	6	1		2352.4.a.s		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
147.4.a.f	✓	1	7.d	odd	6	1
147.4.a.h	yes	1	7.c	even	3	1
147.4.e.a		2	1.a	even	1	1	trivial
147.4.e.a		2	7.c	even	3	1	inner
147.4.e.d		2	7.b	odd	2	1
147.4.e.d		2	7.d	odd	6	1
441.4.a.a		1	21.h	odd	6	1
441.4.a.c		1	21.g	even	6	1
441.4.e.l		2	21.c	even	2	1
441.4.e.l		2	21.g	even	6	1
441.4.e.o		2	3.b	odd	2	1
441.4.e.o		2	21.h	odd	6	1
2352.4.a.s		1	28.g	odd	6	1
2352.4.a.t		1	28.f	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(147, [\chi])$:

$ T_{2}^{2} + 4T_{2} + 16 $

T2^2 + 4*T2 + 16

$ T_{5}^{2} + 18T_{5} + 324 $

T5^2 + 18*T5 + 324

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 4T + 16 $ T^2 + 4*T + 16
$3$	$ T^{2} + 3T + 9 $ T^2 + 3*T + 9
$5$	$ T^{2} + 18T + 324 $ T^2 + 18*T + 324
$7$	$ T^{2} $ T^2
$11$	$ T^{2} - 50T + 2500 $ T^2 - 50*T + 2500
$13$	$ (T + 36)^{2} $ (T + 36)^2
$17$	$ T^{2} + 126T + 15876 $ T^2 + 126*T + 15876
$19$	$ T^{2} - 72T + 5184 $ T^2 - 72*T + 5184
$23$	$ T^{2} + 14T + 196 $ T^2 + 14*T + 196
$29$	$ (T - 158)^{2} $ (T - 158)^2
$31$	$ T^{2} - 36T + 1296 $ T^2 - 36*T + 1296
$37$	$ T^{2} - 162T + 26244 $ T^2 - 162*T + 26244
$41$	$ (T + 270)^{2} $ (T + 270)^2
$43$	$ (T + 324)^{2} $ (T + 324)^2
$47$	$ T^{2} - 72T + 5184 $ T^2 - 72*T + 5184
$53$	$ T^{2} - 22T + 484 $ T^2 - 22*T + 484
$59$	$ T^{2} + 468T + 219024 $ T^2 + 468*T + 219024
$61$	$ T^{2} + 792T + 627264 $ T^2 + 792*T + 627264
$67$	$ T^{2} + 232T + 53824 $ T^2 + 232*T + 53824
$71$	$ (T + 734)^{2} $ (T + 734)^2
$73$	$ T^{2} + 180T + 32400 $ T^2 + 180*T + 32400
$79$	$ T^{2} + 236T + 55696 $ T^2 + 236*T + 55696
$83$	$ (T - 36)^{2} $ (T - 36)^2
$89$	$ T^{2} + 234T + 54756 $ T^2 + 234*T + 54756
$97$	$ (T - 468)^{2} $ (T - 468)^2
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - 4 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (8 \zeta_{6} - 8) q^{4} - 18 \zeta_{6} q^{5} + 12 q^{6} - 9 \zeta_{6} q^{9} + (72 \zeta_{6} - 72) q^{10} + ( - 50 \zeta_{6} + 50) q^{11} - 24 \zeta_{6} q^{12} + \cdots - 450 q^{99} +O(q^{100}) \) q - 4z q^2 + (3z - 3) q^3 + (8z - 8) q^4 - 18z q^5 + 12 * q^6 - 9z q^9 + (72z - 72) q^10 + (-50z + 50) q^11 - 24z q^12 - 36 * q^13 + 54 * q^15 + 64z q^16 + (126z - 126) q^17 + (36z - 36) q^18 + 72z q^19 + 144 * q^20 - 200 * q^22 - 14z q^23 + (199z - 199) q^25 + 144z q^26 + 27 * q^27 + 158 * q^29 - 216z q^30 + (-36z + 36) q^31 + (-256z + 256) q^32 + 150z q^33 + 504 * q^34 + 72 * q^36 + 162z q^37 + (-288z + 288) q^38 + (-108z + 108) q^39 - 270 * q^41 - 324 * q^43 + 400z q^44 + (162z - 162) q^45 + (56z - 56) q^46 + 72z q^47 - 192 * q^48 + 796 * q^50 - 378z q^51 + (-288z + 288) q^52 + (-22z + 22) q^53 - 108z q^54 - 900 * q^55 - 216 * q^57 - 632z q^58 + (468z - 468) q^59 + (432z - 432) q^60 - 792z q^61 - 144 * q^62 - 512 * q^64 + 648z q^65 + (-600z + 600) q^66 + (232z - 232) q^67 - 1008z q^68 + 42 * q^69 - 734 * q^71 + (180z - 180) q^73 + (-648z + 648) q^74 - 597z q^75 - 576 * q^76 - 432 * q^78 - 236z q^79 + (-1152z + 1152) q^80 + (81z - 81) q^81 + 1080z q^82 + 36 * q^83 + 2268 * q^85 + 1296z q^86 + (474z - 474) q^87 - 234z q^89 + 648 * q^90 + 112 * q^92 + 108z q^93 + (-288z + 288) q^94 + (-1296z + 1296) q^95 + 768z q^96 + 468 * q^97 - 450 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 3 q^{3} - 8 q^{4} - 18 q^{5} + 24 q^{6} - 9 q^{9} - 72 q^{10} + 50 q^{11} - 24 q^{12} - 72 q^{13} + 108 q^{15} + 64 q^{16} - 126 q^{17} - 36 q^{18} + 72 q^{19} + 288 q^{20} - 400 q^{22}+ \cdots - 900 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 3 * q^3 - 8 * q^4 - 18 * q^5 + 24 * q^6 - 9 * q^9 - 72 * q^10 + 50 * q^11 - 24 * q^12 - 72 * q^13 + 108 * q^15 + 64 * q^16 - 126 * q^17 - 36 * q^18 + 72 * q^19 + 288 * q^20 - 400 * q^22 - 14 * q^23 - 199 * q^25 + 144 * q^26 + 54 * q^27 + 316 * q^29 - 216 * q^30 + 36 * q^31 + 256 * q^32 + 150 * q^33 + 1008 * q^34 + 144 * q^36 + 162 * q^37 + 288 * q^38 + 108 * q^39 - 540 * q^41 - 648 * q^43 + 400 * q^44 - 162 * q^45 - 56 * q^46 + 72 * q^47 - 384 * q^48 + 1592 * q^50 - 378 * q^51 + 288 * q^52 + 22 * q^53 - 108 * q^54 - 1800 * q^55 - 432 * q^57 - 632 * q^58 - 468 * q^59 - 432 * q^60 - 792 * q^61 - 288 * q^62 - 1024 * q^64 + 648 * q^65 + 600 * q^66 - 232 * q^67 - 1008 * q^68 + 84 * q^69 - 1468 * q^71 - 180 * q^73 + 648 * q^74 - 597 * q^75 - 1152 * q^76 - 864 * q^78 - 236 * q^79 + 1152 * q^80 - 81 * q^81 + 1080 * q^82 + 72 * q^83 + 4536 * q^85 + 1296 * q^86 - 474 * q^87 - 234 * q^89 + 1296 * q^90 + 224 * q^92 + 108 * q^93 + 288 * q^94 + 1296 * q^95 + 768 * q^96 + 936 * q^97 - 900 * q^99

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