Newform orbit 1521.4.a.r

• Label: 1521.4.a.r
• Level: $1521$
• Weight: $4$
• Character orbit: 1521.a
• Self dual: yes
• Analytic conductor: $89.742$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$1521 = 3^{2} \cdot 13^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1521.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17})$
Defining polynomial:	$x^{2} - x - 4$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \frac{1}{2}(1 + \sqrt{17})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + b * q^2 + (b - 4) * q^4 + (b - 2) * q^5 + (-11*b + 10) * q^7 + (-11*b + 4) * q^8 + (-b + 4) * q^10 + (12*b + 34) * q^11 + (-b - 44) * q^14 + (-15*b - 12) * q^16 + (17*b - 18) * q^17 + (32*b + 26) * q^19 + (-5*b + 12) * q^20 + (46*b + 48) * q^22 + (12*b - 104) * q^23 + (-3*b - 117) * q^25 + (43*b - 84) * q^28 + (-96*b + 70) * q^29 + (34*b + 26) * q^31 + (61*b - 92) * q^32 + (-b + 68) * q^34 + (21*b - 64) * q^35 + (-5*b - 102) * q^37 + (58*b + 128) * q^38 + (15*b - 52) * q^40 + (22*b - 126) * q^41 + (143*b + 72) * q^43 + (-2*b - 88) * q^44 + (-92*b + 48) * q^46 + (-121*b + 278) * q^47 + (-99*b + 241) * q^49 + (-120*b - 12) * q^50 + (-30*b + 74) * q^53 + (22*b - 20) * q^55 + (-33*b + 524) * q^56 + (-26*b - 384) * q^58 + (124*b - 246) * q^59 + (-190*b - 434) * q^61 + (60*b + 136) * q^62 + (89*b + 340) * q^64 + (232*b - 150) * q^67 + (-69*b + 140) * q^68 + (-43*b + 84) * q^70 + (-231*b + 50) * q^71 + (-260*b - 98) * q^73 + (-107*b - 20) * q^74 + (-70*b + 24) * q^76 + (-386*b - 188) * q^77 + (40*b - 524) * q^79 + (3*b - 36) * q^80 + (-104*b + 88) * q^82 + (-182*b + 1070) * q^83 + (-35*b + 104) * q^85 + (215*b + 572) * q^86 + (-458*b - 392) * q^88 + (-388*b - 166) * q^89 + (-140*b + 464) * q^92 + (157*b - 484) * q^94 + (-6*b + 76) * q^95 + (-508*b + 718) * q^97 + (142*b - 396) * q^98
$\operatorname{Tr}(f)(q)$	$=$	2 * q + q^2 - 7 * q^4 - 3 * q^5 + 9 * q^7 - 3 * q^8 + 7 * q^10 + 80 * q^11 - 89 * q^14 - 39 * q^16 - 19 * q^17 + 84 * q^19 + 19 * q^20 + 142 * q^22 - 196 * q^23 - 237 * q^25 - 125 * q^28 + 44 * q^29 + 86 * q^31 - 123 * q^32 + 135 * q^34 - 107 * q^35 - 209 * q^37 + 314 * q^38 - 89 * q^40 - 230 * q^41 + 287 * q^43 - 178 * q^44 + 4 * q^46 + 435 * q^47 + 383 * q^49 - 144 * q^50 + 118 * q^53 - 18 * q^55 + 1015 * q^56 - 794 * q^58 - 368 * q^59 - 1058 * q^61 + 332 * q^62 + 769 * q^64 - 68 * q^67 + 211 * q^68 + 125 * q^70 - 131 * q^71 - 456 * q^73 - 147 * q^74 - 22 * q^76 - 762 * q^77 - 1008 * q^79 - 69 * q^80 + 72 * q^82 + 1958 * q^83 + 173 * q^85 + 1359 * q^86 - 1242 * q^88 - 720 * q^89 + 788 * q^92 - 811 * q^94 + 146 * q^95 + 928 * q^97 - 650 * q^98

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

−1.56155
2.56155

−1.56155

0

−5.56155

−3.56155

0

27.1771

21.1771

0

5.56155

1.2

2.56155

0

−1.43845

0.561553

0

−18.1771

−24.1771

0

1.43845

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	1521.4.a.r		2
3.b	odd	2	1		169.4.a.g		2
13.b	even	2	1		117.4.a.d		2
39.d	odd	2	1		13.4.a.b	✓	2
39.f	even	4	2		169.4.b.f		4
39.h	odd	6	2		169.4.c.g		4
39.i	odd	6	2		169.4.c.j		4
39.k	even	12	4		169.4.e.f		8
52.b	odd	2	1		1872.4.a.bb		2
156.h	even	2	1		208.4.a.h		2
195.e	odd	2	1		325.4.a.f		2
195.s	even	4	2		325.4.b.e		4
273.g	even	2	1		637.4.a.b		2
312.b	odd	2	1		832.4.a.s		2
312.h	even	2	1		832.4.a.z		2
429.e	even	2	1		1573.4.a.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.4.a.b	✓	2	39.d	odd	2	1
117.4.a.d		2	13.b	even	2	1
169.4.a.g		2	3.b	odd	2	1
169.4.b.f		4	39.f	even	4	2
169.4.c.g		4	39.h	odd	6	2
169.4.c.j		4	39.i	odd	6	2
169.4.e.f		8	39.k	even	12	4
208.4.a.h		2	156.h	even	2	1
325.4.a.f		2	195.e	odd	2	1
325.4.b.e		4	195.s	even	4	2
637.4.a.b		2	273.g	even	2	1
832.4.a.s		2	312.b	odd	2	1
832.4.a.z		2	312.h	even	2	1
1521.4.a.r		2	1.a	even	1	1	trivial
1573.4.a.b		2	429.e	even	2	1
1872.4.a.bb		2	52.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 $S_{4}^{\mathrm{new}}(\Gamma_0(1521))$:

$T_{2}^{2} - T_{2} - 4$

$T_{5}^{2} + 3T_{5} - 2$

$T_{7}^{2} - 9T_{7} - 494$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - T - 4$
$3$	$T^{2}$
$5$	$T^{2} + 3T - 2$
$7$	$T^{2} - 9T - 494$
$11$	$T^{2} - 80T + 988$