Newform orbit 147.4.a.i

• Label: 147.4.a.i
• Level: $147$
• Weight: $4$
• Character orbit: 147.a
• Self dual: yes
• Analytic conductor: $8.673$
• Analytic rank: $1$
• 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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57})$
Defining polynomial:	$x^{2} - x - 14$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \frac{1}{2}(1 + \sqrt{57})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (-b - 1) * q^2 - 3 * q^3 + (3*b + 7) * q^4 + (-2*b - 2) * q^5 + (3*b + 3) * q^6 + (-5*b - 41) * q^8 + 9 * q^9 + (6*b + 30) * q^10 + (10*b - 8) * q^11 + (-9*b - 21) * q^12 + (12*b - 14) * q^13 + (6*b + 6) * q^15 + (27*b + 55) * q^16 + (2*b + 2) * q^17 + (-9*b - 9) * q^18 + (24*b - 44) * q^19 + (-26*b - 98) * q^20 + (-12*b - 132) * q^22 + (-34*b + 20) * q^23 + (15*b + 123) * q^24 + (12*b - 65) * q^25 + (-10*b - 154) * q^26 - 27 * q^27 + (24*b - 138) * q^29 + (-18*b - 90) * q^30 + (-72*b + 16) * q^31 + (-69*b - 105) * q^32 + (-30*b + 24) * q^33 + (-6*b - 30) * q^34 + (27*b + 63) * q^36 + (-36*b - 106) * q^37 + (-4*b - 292) * q^38 + (-36*b + 42) * q^39 + (102*b + 222) * q^40 + (30*b + 210) * q^41 + (-48*b + 212) * q^43 + (76*b + 364) * q^44 + (-18*b - 18) * q^45 + (48*b + 456) * q^46 + (-68*b + 40) * q^47 + (-81*b - 165) * q^48 + (41*b - 103) * q^50 + (-6*b - 6) * q^51 + (78*b + 406) * q^52 + (4*b - 554) * q^53 + (27*b + 27) * q^54 + (-24*b - 264) * q^55 + (-72*b + 132) * q^57 + (90*b - 198) * q^58 + (116*b - 460) * q^59 + (78*b + 294) * q^60 + (-72*b + 250) * q^61 + (128*b + 992) * q^62 + (27*b + 631) * q^64 + (-20*b - 308) * q^65 + (36*b + 396) * q^66 + (108*b + 20) * q^67 + (26*b + 98) * q^68 + (102*b - 60) * q^69 + (-30*b + 492) * q^71 + (-45*b - 369) * q^72 + (-12*b - 530) * q^73 + (178*b + 610) * q^74 + (-36*b + 195) * q^75 + (108*b + 700) * q^76 + (30*b + 462) * q^78 + (-108*b - 232) * q^79 + (-218*b - 866) * q^80 + 81 * q^81 + (-270*b - 630) * q^82 + (-96*b - 924) * q^83 + (-12*b - 60) * q^85 + (-116*b + 460) * q^86 + (-72*b + 414) * q^87 + (-420*b - 372) * q^88 + (142*b - 254) * q^89 + (54*b + 270) * q^90 + (-280*b - 1288) * q^92 + (216*b - 48) * q^93 + (96*b + 912) * q^94 + (-8*b - 584) * q^95 + (207*b + 315) * q^96 + (-276*b - 266) * q^97 + (90*b - 72) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 3 * q^2 - 6 * q^3 + 17 * q^4 - 6 * q^5 + 9 * q^6 - 87 * q^8 + 18 * q^9 + 66 * q^10 - 6 * q^11 - 51 * q^12 - 16 * q^13 + 18 * q^15 + 137 * q^16 + 6 * q^17 - 27 * q^18 - 64 * q^19 - 222 * q^20 - 276 * q^22 + 6 * q^23 + 261 * q^24 - 118 * q^25 - 318 * q^26 - 54 * q^27 - 252 * q^29 - 198 * q^30 - 40 * q^31 - 279 * q^32 + 18 * q^33 - 66 * q^34 + 153 * q^36 - 248 * q^37 - 588 * q^38 + 48 * q^39 + 546 * q^40 + 450 * q^41 + 376 * q^43 + 804 * q^44 - 54 * q^45 + 960 * q^46 + 12 * q^47 - 411 * q^48 - 165 * q^50 - 18 * q^51 + 890 * q^52 - 1104 * q^53 + 81 * q^54 - 552 * q^55 + 192 * q^57 - 306 * q^58 - 804 * q^59 + 666 * q^60 + 428 * q^61 + 2112 * q^62 + 1289 * q^64 - 636 * q^65 + 828 * q^66 + 148 * q^67 + 222 * q^68 - 18 * q^69 + 954 * q^71 - 783 * q^72 - 1072 * q^73 + 1398 * q^74 + 354 * q^75 + 1508 * q^76 + 954 * q^78 - 572 * q^79 - 1950 * q^80 + 162 * q^81 - 1530 * q^82 - 1944 * q^83 - 132 * q^85 + 804 * q^86 + 756 * q^87 - 1164 * q^88 - 366 * q^89 + 594 * q^90 - 2856 * q^92 + 120 * q^93 + 1920 * q^94 - 1176 * q^95 + 837 * q^96 - 808 * q^97 - 54 * 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

4.27492
−3.27492

−5.27492

−3.00000

19.8248

−10.5498

15.8248

0

−62.3746

9.00000

55.6495

1.2

2.27492

−3.00000

−2.82475

4.54983

−6.82475

0

−24.6254

9.00000

10.3505

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.i		2
3.b	odd	2	1		441.4.a.r		2
4.b	odd	2	1		2352.4.a.bz		2
7.b	odd	2	1		21.4.a.c	✓	2
7.c	even	3	2		147.4.e.m		4
7.d	odd	6	2		147.4.e.l		4
21.c	even	2	1		63.4.a.e		2
21.g	even	6	2		441.4.e.q		4
21.h	odd	6	2		441.4.e.p		4
28.d	even	2	1		336.4.a.m		2
35.c	odd	2	1		525.4.a.n		2
35.f	even	4	2		525.4.d.g		4
56.e	even	2	1		1344.4.a.bo		2
56.h	odd	2	1		1344.4.a.bg		2
84.h	odd	2	1		1008.4.a.ba		2
105.g	even	2	1		1575.4.a.p		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.4.a.c	✓	2	7.b	odd	2	1
63.4.a.e		2	21.c	even	2	1
147.4.a.i		2	1.a	even	1	1	trivial
147.4.e.l		4	7.d	odd	6	2
147.4.e.m		4	7.c	even	3	2
336.4.a.m		2	28.d	even	2	1
441.4.a.r		2	3.b	odd	2	1
441.4.e.p		4	21.h	odd	6	2
441.4.e.q		4	21.g	even	6	2
525.4.a.n		2	35.c	odd	2	1
525.4.d.g		4	35.f	even	4	2
1008.4.a.ba		2	84.h	odd	2	1
1344.4.a.bg		2	56.h	odd	2	1
1344.4.a.bo		2	56.e	even	2	1
1575.4.a.p		2	105.g	even	2	1
2352.4.a.bz		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 $S_{4}^{\mathrm{new}}(\Gamma_0(147))$:

$T_{2}^{2} + 3T_{2} - 12$

$T_{5}^{2} + 6T_{5} - 48$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 3T - 12$
$3$	$(T + 3)^{2}$
$5$	$T^{2} + 6T - 48$
$7$	$T^{2}$
$11$	$T^{2} + 6T - 1416$