Newform orbit 2028.4.a.d

• Label: 2028.4.a.d
• Level: $2028$
• Weight: $4$
• Character orbit: 2028.a
• Self dual: yes
• Analytic conductor: $119.656$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$2028 = 2^{2} \cdot 3 \cdot 13^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2028.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$119.655873492$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{22})$
Defining polynomial:	$x^{2} - 22$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 2\sqrt{22}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q - 3 * q^3 + b * q^5 + (-3*b - 4) * q^7 + 9 * q^9 + (2*b + 30) * q^11 - 3*b * q^15 + (-6*b - 54) * q^17 + (3*b + 108) * q^19 + (9*b + 12) * q^21 + (8*b - 108) * q^23 - 37 * q^25 - 27 * q^27 + (20*b - 54) * q^29 + (-15*b - 40) * q^31 + (-6*b - 90) * q^33 + (-4*b - 264) * q^35 + (18*b - 54) * q^37 + (-15*b + 24) * q^41 + (-54*b + 4) * q^43 + 9*b * q^45 + (44*b + 114) * q^47 + (24*b + 465) * q^49 + (18*b + 162) * q^51 + (12*b + 270) * q^53 + (30*b + 176) * q^55 + (-9*b - 324) * q^57 + (40*b + 426) * q^59 + (12*b + 154) * q^61 + (-27*b - 36) * q^63 + (-39*b + 152) * q^67 + (-24*b + 324) * q^69 + (-82*b + 114) * q^71 + (-6*b - 710) * q^73 + 111 * q^75 + (-98*b - 648) * q^77 + (-60*b + 248) * q^79 + 81 * q^81 + (-12*b - 618) * q^83 + (-54*b - 528) * q^85 + (-60*b + 162) * q^87 + (63*b + 708) * q^89 + (45*b + 120) * q^93 + (108*b + 264) * q^95 + (30*b - 302) * q^97 + (18*b + 270) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 6 * q^3 - 8 * q^7 + 18 * q^9 + 60 * q^11 - 108 * q^17 + 216 * q^19 + 24 * q^21 - 216 * q^23 - 74 * q^25 - 54 * q^27 - 108 * q^29 - 80 * q^31 - 180 * q^33 - 528 * q^35 - 108 * q^37 + 48 * q^41 + 8 * q^43 + 228 * q^47 + 930 * q^49 + 324 * q^51 + 540 * q^53 + 352 * q^55 - 648 * q^57 + 852 * q^59 + 308 * q^61 - 72 * q^63 + 304 * q^67 + 648 * q^69 + 228 * q^71 - 1420 * q^73 + 222 * q^75 - 1296 * q^77 + 496 * q^79 + 162 * q^81 - 1236 * q^83 - 1056 * q^85 + 324 * q^87 + 1416 * q^89 + 240 * q^93 + 528 * q^95 - 604 * q^97 + 540 * 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

−4.69042
4.69042

0

−3.00000

0

−9.38083

0

24.1425

0

9.00000

0

1.2

0

−3.00000

0

9.38083

0

−32.1425

0

9.00000

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$
$13$	$+1$

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	2028.4.a.d		2
13.b	even	2	1		156.4.a.c	✓	2
13.d	odd	4	2		2028.4.b.e		4
39.d	odd	2	1		468.4.a.g		2
52.b	odd	2	1		624.4.a.p		2
104.e	even	2	1		2496.4.a.bf		2
104.h	odd	2	1		2496.4.a.w		2
156.h	even	2	1		1872.4.a.y		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.4.a.c	✓	2	13.b	even	2	1
468.4.a.g		2	39.d	odd	2	1
624.4.a.p		2	52.b	odd	2	1
1872.4.a.y		2	156.h	even	2	1
2028.4.a.d		2	1.a	even	1	1	trivial
2028.4.b.e		4	13.d	odd	4	2
2496.4.a.w		2	104.h	odd	2	1
2496.4.a.bf		2	104.e	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} - 88$ acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2028))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$(T + 3)^{2}$
$5$	$T^{2} - 88$
$7$	$T^{2} + 8T - 776$
$11$	$T^{2} - 60T + 548$