Newform orbit 147.4.a.j

• Label: 147.4.a.j
• Level: $147$
• Weight: $4$
• Character orbit: 147.a
• Self dual: yes
• Analytic conductor: $8.673$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(8.67328077084\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b + 1) * q^2 - 3 * q^3 + (2*b - 5) * q^4 + (7*b + 10) * q^5 + (-3*b - 3) * q^6 + (-11*b - 9) * q^8 + 9 * q^9 + (17*b + 24) * q^10 + (24*b - 10) * q^11 + (-6*b + 15) * q^12 + (25*b + 52) * q^13 + (-21*b - 30) * q^15 + (-36*b + 9) * q^16 + (-45*b + 58) * q^17 + (9*b + 9) * q^18 + (-22*b + 96) * q^19 + (-15*b - 22) * q^20 + (14*b + 38) * q^22 + (-28*b + 14) * q^23 + (33*b + 27) * q^24 + (140*b + 73) * q^25 + (77*b + 102) * q^26 - 27 * q^27 + (-62*b + 148) * q^29 + (-51*b - 72) * q^30 + (-50*b - 52) * q^31 + (61*b + 9) * q^32 + (-72*b + 30) * q^33 + (13*b - 32) * q^34 + (18*b - 45) * q^36 + (48*b - 124) * q^37 + (74*b + 52) * q^38 + (-75*b - 156) * q^39 + (-173*b - 244) * q^40 + (-219*b + 10) * q^41 + (-100*b - 360) * q^43 + (-140*b + 146) * q^44 + (63*b + 90) * q^45 + (-14*b - 42) * q^46 + (250*b - 48) * q^47 + (108*b - 27) * q^48 + (213*b + 353) * q^50 + (135*b - 174) * q^51 + (-21*b - 160) * q^52 + (-360*b + 134) * q^53 + (-27*b - 27) * q^54 + (170*b + 236) * q^55 + (66*b - 288) * q^57 + (86*b + 24) * q^58 + (-226*b - 308) * q^59 + (45*b + 66) * q^60 + (-3*b + 8) * q^61 + (-102*b - 152) * q^62 + (358*b + 59) * q^64 + (614*b + 870) * q^65 + (-42*b - 114) * q^66 + (-524*b - 72) * q^67 + (341*b - 470) * q^68 + (84*b - 42) * q^69 + (-232*b + 494) * q^71 + (-99*b - 81) * q^72 + (401*b + 52) * q^73 + (-76*b - 28) * q^74 + (-420*b - 219) * q^75 + (302*b - 568) * q^76 + (-231*b - 306) * q^78 + (236*b - 472) * q^79 + (-297*b - 414) * q^80 + 81 * q^81 + (-209*b - 428) * q^82 + (80*b + 508) * q^83 + (-44*b - 50) * q^85 + (-460*b - 560) * q^86 + (186*b - 444) * q^87 + (-106*b - 438) * q^88 + (339*b - 194) * q^89 + (153*b + 216) * q^90 + (168*b - 182) * q^92 + (150*b + 156) * q^93 + (202*b + 452) * q^94 + (452*b + 652) * q^95 + (-183*b - 27) * q^96 + (-599*b + 244) * q^97 + (216*b - 90) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^2 - 6 * q^3 - 10 * q^4 + 20 * q^5 - 6 * q^6 - 18 * q^8 + 18 * q^9 + 48 * q^10 - 20 * q^11 + 30 * q^12 + 104 * q^13 - 60 * q^15 + 18 * q^16 + 116 * q^17 + 18 * q^18 + 192 * q^19 - 44 * q^20 + 76 * q^22 + 28 * q^23 + 54 * q^24 + 146 * q^25 + 204 * q^26 - 54 * q^27 + 296 * q^29 - 144 * q^30 - 104 * q^31 + 18 * q^32 + 60 * q^33 - 64 * q^34 - 90 * q^36 - 248 * q^37 + 104 * q^38 - 312 * q^39 - 488 * q^40 + 20 * q^41 - 720 * q^43 + 292 * q^44 + 180 * q^45 - 84 * q^46 - 96 * q^47 - 54 * q^48 + 706 * q^50 - 348 * q^51 - 320 * q^52 + 268 * q^53 - 54 * q^54 + 472 * q^55 - 576 * q^57 + 48 * q^58 - 616 * q^59 + 132 * q^60 + 16 * q^61 - 304 * q^62 + 118 * q^64 + 1740 * q^65 - 228 * q^66 - 144 * q^67 - 940 * q^68 - 84 * q^69 + 988 * q^71 - 162 * q^72 + 104 * q^73 - 56 * q^74 - 438 * q^75 - 1136 * q^76 - 612 * q^78 - 944 * q^79 - 828 * q^80 + 162 * q^81 - 856 * q^82 + 1016 * q^83 - 100 * q^85 - 1120 * q^86 - 888 * q^87 - 876 * q^88 - 388 * q^89 + 432 * q^90 - 364 * q^92 + 312 * q^93 + 904 * q^94 + 1304 * q^95 - 54 * q^96 + 488 * q^97 - 180 * q^99

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.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.41421
1.41421

−0.414214

−3.00000

−7.82843

0.100505

1.24264

0

6.55635

9.00000

−0.0416306

1.2

2.41421

−3.00000

−2.17157

19.8995

−7.24264

0

−24.5563

9.00000

48.0416

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( +1 \)
\(7\)	\( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	147.4.a.j	✓	2
3.b	odd	2	1		441.4.a.n		2
4.b	odd	2	1		2352.4.a.cf		2
7.b	odd	2	1		147.4.a.k	yes	2
7.c	even	3	2		147.4.e.k		4
7.d	odd	6	2		147.4.e.j		4
21.c	even	2	1		441.4.a.o		2
21.g	even	6	2		441.4.e.u		4
21.h	odd	6	2		441.4.e.v		4
28.d	even	2	1		2352.4.a.bl		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
147.4.a.j	✓	2	1.a	even	1	1	trivial
147.4.a.k	yes	2	7.b	odd	2	1
147.4.e.j		4	7.d	odd	6	2
147.4.e.k		4	7.c	even	3	2
441.4.a.n		2	3.b	odd	2	1
441.4.a.o		2	21.c	even	2	1
441.4.e.u		4	21.g	even	6	2
441.4.e.v		4	21.h	odd	6	2
2352.4.a.bl		2	28.d	even	2	1
2352.4.a.cf		2	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2}^{2} - 2T_{2} - 1 \)

\( T_{5}^{2} - 20T_{5} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T - 1 \)
$3$	\( (T + 3)^{2} \)
$5$	\( T^{2} - 20T + 2 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 20T - 1052 \)