Newform orbit 396.4.a.j

• Label: 396.4.a.j
• Level: $396$
• Weight: $4$
• Character orbit: 396.a
• Self dual: yes
• Analytic conductor: $23.365$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$396 = 2^{2} \cdot 3^{2} \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	396.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$23.3647563623$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{31})$
Defining polynomial:	$x^{2} - 31$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 2\sqrt{31}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (b + 6) * q^5 + (-b - 12) * q^7 - 11 * q^11 + (-5*b - 4) * q^13 + (-6*b - 14) * q^17 + (2*b - 34) * q^19 + (7*b - 134) * q^23 + (12*b + 35) * q^25 + (4*b - 130) * q^29 + (-14*b + 56) * q^31 + (-18*b - 196) * q^35 + (22*b + 158) * q^37 + (8*b - 62) * q^41 - 74 * q^43 + (-11*b - 286) * q^47 + (24*b - 75) * q^49 + (45*b - 38) * q^53 + (-11*b - 66) * q^55 + (-6*b - 432) * q^59 + (-51*b - 68) * q^61 + (-34*b - 644) * q^65 + (-40*b - 256) * q^67 + (35*b - 362) * q^71 + (12*b + 486) * q^73 + (11*b + 132) * q^77 + (5*b - 264) * q^79 + (-22*b - 1056) * q^83 + (-50*b - 828) * q^85 + (-112*b - 144) * q^89 + (64*b + 668) * q^91 + (-22*b + 44) * q^95 + (72*b + 250) * q^97
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 12 * q^5 - 24 * q^7 - 22 * q^11 - 8 * q^13 - 28 * q^17 - 68 * q^19 - 268 * q^23 + 70 * q^25 - 260 * q^29 + 112 * q^31 - 392 * q^35 + 316 * q^37 - 124 * q^41 - 148 * q^43 - 572 * q^47 - 150 * q^49 - 76 * q^53 - 132 * q^55 - 864 * q^59 - 136 * q^61 - 1288 * q^65 - 512 * q^67 - 724 * q^71 + 972 * q^73 + 264 * q^77 - 528 * q^79 - 2112 * q^83 - 1656 * q^85 - 288 * q^89 + 1336 * q^91 + 88 * q^95 + 500 * q^97

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

−5.56776
5.56776

0

0

0

−5.13553

0

−0.864471

0

0

0

1.2

0

0

0

17.1355

0

−23.1355

0

0

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$
$11$	$+1$

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	396.4.a.j	yes	2
3.b	odd	2	1		396.4.a.h	✓	2
4.b	odd	2	1		1584.4.a.bi		2
12.b	even	2	1		1584.4.a.y		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
396.4.a.h	✓	2	3.b	odd	2	1
396.4.a.j	yes	2	1.a	even	1	1	trivial
1584.4.a.y		2	12.b	even	2	1
1584.4.a.bi		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 $S_{4}^{\mathrm{new}}(\Gamma_0(396))$:

$T_{5}^{2} - 12T_{5} - 88$

$T_{7}^{2} + 24T_{7} + 20$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2} - 12T - 88$
$7$	$T^{2} + 24T + 20$
$11$	$(T + 11)^{2}$