Newform orbit 2028.4.a.g

• Label: 2028.4.a.g
• Level: $2028$
• Weight: $4$
• Character orbit: 2028.a
• Self dual: yes
• Analytic conductor: $119.656$
• Analytic rank: $1$
• Dimension: $2$
• Inner twists: $2$

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:	$\Q(\sqrt{3})$
Defining polynomial:	$x^{2} - 3$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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 $\beta = 2\sqrt{3}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 3 * q^3 + 2*b * q^5 - 3*b * q^7 + 9 * q^9 - 5*b * q^11 + 6*b * q^15 - 18 * q^17 + 13*b * q^19 - 9*b * q^21 - 24 * q^23 - 77 * q^25 + 27 * q^27 + 6 * q^29 + 31*b * q^31 - 15*b * q^33 - 72 * q^35 - 54*b * q^37 - 32*b * q^41 - 20 * q^43 + 18*b * q^45 + 85*b * q^47 - 235 * q^49 - 54 * q^51 - 306 * q^53 - 120 * q^55 + 39*b * q^57 + 203*b * q^59 + 70 * q^61 - 27*b * q^63 - 131*b * q^67 - 72 * q^69 - 297*b * q^71 - 4*b * q^73 - 231 * q^75 + 180 * q^77 - 416 * q^79 + 81 * q^81 - 311*b * q^83 - 36*b * q^85 + 18 * q^87 - 8*b * q^89 + 93*b * q^93 + 312 * q^95 + 152*b * q^97 - 45*b * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 6 * q^3 + 18 * q^9 - 36 * q^17 - 48 * q^23 - 154 * q^25 + 54 * q^27 + 12 * q^29 - 144 * q^35 - 40 * q^43 - 470 * q^49 - 108 * q^51 - 612 * q^53 - 240 * q^55 + 140 * q^61 - 144 * q^69 - 462 * q^75 + 360 * q^77 - 832 * q^79 + 162 * q^81 + 36 * q^87 + 624 * q^95

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

−1.73205
1.73205

0

3.00000

0

−6.92820

0

10.3923

0

9.00000

0

1.2

0

3.00000

0

6.92820

0

−10.3923

0

9.00000

0

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2028.4.a.g		2
13.b	even	2	1	inner	2028.4.a.g		2
13.d	odd	4	2		156.4.b.a	✓	2
39.f	even	4	2		468.4.b.b		2
52.f	even	4	2		624.4.c.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.4.b.a	✓	2	13.d	odd	4	2
468.4.b.b		2	39.f	even	4	2
624.4.c.a		2	52.f	even	4	2
2028.4.a.g		2	1.a	even	1	1	trivial
2028.4.a.g		2	13.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} - 48$ acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2028))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$(T - 3)^{2}$
$5$	$T^{2} - 48$
$7$	$T^{2} - 108$
$11$	$T^{2} - 300$