Newform orbit 39.4.a.b

• Label: 39.4.a.b
• Level: $39$
• Weight: $4$
• Character orbit: 39.a
• Self dual: yes
• Analytic conductor: $2.301$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$39 = 3 \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	39.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$2.30107449022$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14})$
Defining polynomial:	$x^{2} - 14$
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 $\beta = \sqrt{14}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (b + 1) * q^2 - 3 * q^3 + (2*b + 7) * q^4 + (-2*b + 12) * q^5 + (-3*b - 3) * q^6 - 2*b * q^7 + (b + 27) * q^8 + 9 * q^9 + (10*b - 16) * q^10 + (-12*b - 22) * q^11 + (-6*b - 21) * q^12 - 13 * q^13 + (-2*b - 28) * q^14 + (6*b - 36) * q^15 + (12*b - 15) * q^16 + (4*b + 82) * q^17 + (9*b + 9) * q^18 + (2*b + 24) * q^19 + (10*b + 28) * q^20 + 6*b * q^21 + (-34*b - 190) * q^22 + (48*b + 4) * q^23 + (-3*b - 81) * q^24 + (-48*b + 75) * q^25 + (-13*b - 13) * q^26 - 27 * q^27 + (-14*b - 56) * q^28 + (-24*b + 202) * q^29 + (-30*b + 48) * q^30 + (-26*b + 20) * q^31 + (-11*b - 63) * q^32 + (36*b + 66) * q^33 + (86*b + 138) * q^34 + (-24*b + 56) * q^35 + (18*b + 63) * q^36 + (28*b - 50) * q^37 + (26*b + 52) * q^38 + 39 * q^39 + (-42*b + 296) * q^40 + (94*b + 100) * q^41 + (6*b + 84) * q^42 + (52*b - 308) * q^43 + (-128*b - 490) * q^44 + (-18*b + 108) * q^45 + (52*b + 676) * q^46 + (32*b - 162) * q^47 + (-36*b + 45) * q^48 - 287 * q^49 + (27*b - 597) * q^50 + (-12*b - 246) * q^51 + (-26*b - 91) * q^52 + (-120*b - 82) * q^53 + (-27*b - 27) * q^54 + (-100*b + 72) * q^55 + (-54*b - 28) * q^56 + (-6*b - 72) * q^57 + (178*b - 134) * q^58 + (40*b + 70) * q^59 + (-30*b - 84) * q^60 + (136*b + 314) * q^61 + (-6*b - 344) * q^62 - 18*b * q^63 + (-170*b - 97) * q^64 + (26*b - 156) * q^65 + (102*b + 570) * q^66 + (-170*b - 236) * q^67 + (192*b + 686) * q^68 + (-144*b - 12) * q^69 + (32*b - 280) * q^70 + (-84*b + 214) * q^71 + (9*b + 243) * q^72 + (76*b - 450) * q^73 + (-22*b + 342) * q^74 + (144*b - 225) * q^75 + (62*b + 224) * q^76 + (44*b + 336) * q^77 + (39*b + 39) * q^78 + (-88*b - 216) * q^79 + (174*b - 516) * q^80 + 81 * q^81 + (194*b + 1416) * q^82 + (64*b - 694) * q^83 + (42*b + 168) * q^84 + (-116*b + 872) * q^85 + (-256*b + 420) * q^86 + (72*b - 606) * q^87 + (-346*b - 762) * q^88 + (-190*b + 480) * q^89 + (90*b - 144) * q^90 + 26*b * q^91 + (344*b + 1372) * q^92 + (78*b - 60) * q^93 + (-130*b + 286) * q^94 + (-24*b + 232) * q^95 + (33*b + 189) * q^96 + (-220*b - 266) * q^97 + (-287*b - 287) * q^98 + (-108*b - 198) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 2 * q^2 - 6 * q^3 + 14 * q^4 + 24 * q^5 - 6 * q^6 + 54 * q^8 + 18 * q^9 - 32 * q^10 - 44 * q^11 - 42 * q^12 - 26 * q^13 - 56 * q^14 - 72 * q^15 - 30 * q^16 + 164 * q^17 + 18 * q^18 + 48 * q^19 + 56 * q^20 - 380 * q^22 + 8 * q^23 - 162 * q^24 + 150 * q^25 - 26 * q^26 - 54 * q^27 - 112 * q^28 + 404 * q^29 + 96 * q^30 + 40 * q^31 - 126 * q^32 + 132 * q^33 + 276 * q^34 + 112 * q^35 + 126 * q^36 - 100 * q^37 + 104 * q^38 + 78 * q^39 + 592 * q^40 + 200 * q^41 + 168 * q^42 - 616 * q^43 - 980 * q^44 + 216 * q^45 + 1352 * q^46 - 324 * q^47 + 90 * q^48 - 574 * q^49 - 1194 * q^50 - 492 * q^51 - 182 * q^52 - 164 * q^53 - 54 * q^54 + 144 * q^55 - 56 * q^56 - 144 * q^57 - 268 * q^58 + 140 * q^59 - 168 * q^60 + 628 * q^61 - 688 * q^62 - 194 * q^64 - 312 * q^65 + 1140 * q^66 - 472 * q^67 + 1372 * q^68 - 24 * q^69 - 560 * q^70 + 428 * q^71 + 486 * q^72 - 900 * q^73 + 684 * q^74 - 450 * q^75 + 448 * q^76 + 672 * q^77 + 78 * q^78 - 432 * q^79 - 1032 * q^80 + 162 * q^81 + 2832 * q^82 - 1388 * q^83 + 336 * q^84 + 1744 * q^85 + 840 * q^86 - 1212 * q^87 - 1524 * q^88 + 960 * q^89 - 288 * q^90 + 2744 * q^92 - 120 * q^93 + 572 * q^94 + 464 * q^95 + 378 * q^96 - 532 * q^97 - 574 * q^98 - 396 * 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

−3.74166
3.74166

−2.74166

−3.00000

−0.483315

19.4833

8.22497

7.48331

23.2583

9.00000

−53.4166

1.2

4.74166

−3.00000

14.4833

4.51669

−14.2250

−7.48331

30.7417

9.00000

21.4166

Atkin-Lehner signs

$p$	Sign
$3$	$+1$
$13$	$+1$

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	39.4.a.b	✓	2
3.b	odd	2	1		117.4.a.c		2
4.b	odd	2	1		624.4.a.r		2
5.b	even	2	1		975.4.a.j		2
7.b	odd	2	1		1911.4.a.h		2
8.b	even	2	1		2496.4.a.bc		2
8.d	odd	2	1		2496.4.a.s		2
12.b	even	2	1		1872.4.a.t		2
13.b	even	2	1		507.4.a.f		2
13.d	odd	4	2		507.4.b.f		4
39.d	odd	2	1		1521.4.a.s		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.4.a.b	✓	2	1.a	even	1	1	trivial
117.4.a.c		2	3.b	odd	2	1
507.4.a.f		2	13.b	even	2	1
507.4.b.f		4	13.d	odd	4	2
624.4.a.r		2	4.b	odd	2	1
975.4.a.j		2	5.b	even	2	1
1521.4.a.s		2	39.d	odd	2	1
1872.4.a.t		2	12.b	even	2	1
1911.4.a.h		2	7.b	odd	2	1
2496.4.a.s		2	8.d	odd	2	1
2496.4.a.bc		2	8.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{2} - 2T_{2} - 13$ acting on $S_{4}^{\mathrm{new}}(\Gamma_0(39))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 2T - 13$
$3$	$(T + 3)^{2}$
$5$	$T^{2} - 24T + 88$
$7$	$T^{2} - 56$
$11$	$T^{2} + 44T - 1532$