Newform orbit 1280.4.a.c

• Label: 1280.4.a.c
• Level: $1280$
• Weight: $4$
• Character orbit: 1280.a
• Self dual: yes
• Analytic conductor: $75.522$
• Analytic rank: $1$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$1280 = 2^{8} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1280.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$75.5224448073$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2})$
Defining polynomial:	$x^{2} - 2$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 640)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

$f(q)$	$=$	q + b * q^3 - 5 * q^5 + 15*b * q^7 - 25 * q^9 - 22*b * q^11 + 30 * q^13 - 5*b * q^15 + 42 * q^17 - 24*b * q^19 + 30 * q^21 + 15*b * q^23 + 25 * q^25 - 52*b * q^27 - 30*b * q^31 - 44 * q^33 - 75*b * q^35 + 30 * q^37 + 30*b * q^39 + 336 * q^41 - 117*b * q^43 + 125 * q^45 - 345*b * q^47 + 107 * q^49 + 42*b * q^51 - 690 * q^53 + 110*b * q^55 - 48 * q^57 + 400*b * q^59 - 690 * q^61 - 375*b * q^63 - 150 * q^65 - 87*b * q^67 + 30 * q^69 + 690*b * q^71 - 574 * q^73 + 25*b * q^75 - 660 * q^77 - 720*b * q^79 + 571 * q^81 + 391*b * q^83 - 210 * q^85 - 138 * q^89 + 450*b * q^91 - 60 * q^93 + 120*b * q^95 - 250 * q^97 + 550*b * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 10 * q^5 - 50 * q^9 + 60 * q^13 + 84 * q^17 + 60 * q^21 + 50 * q^25 - 88 * q^33 + 60 * q^37 + 672 * q^41 + 250 * q^45 + 214 * q^49 - 1380 * q^53 - 96 * q^57 - 1380 * q^61 - 300 * q^65 + 60 * q^69 - 1148 * q^73 - 1320 * q^77 + 1142 * q^81 - 420 * q^85 - 276 * q^89 - 120 * q^93 - 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

−1.41421
1.41421

0

−1.41421

0

−5.00000

0

−21.2132

0

−25.0000

0

1.2

0

1.41421

0

−5.00000

0

21.2132

0

−25.0000

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$+1$

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1280.4.a.c		2
4.b	odd	2	1	inner	1280.4.a.c		2
8.b	even	2	1		1280.4.a.i		2
8.d	odd	2	1		1280.4.a.i		2
16.e	even	4	2		640.4.d.f	✓	4
16.f	odd	4	2		640.4.d.f	✓	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
640.4.d.f	✓	4	16.e	even	4	2
640.4.d.f	✓	4	16.f	odd	4	2
1280.4.a.c		2	1.a	even	1	1	trivial
1280.4.a.c		2	4.b	odd	2	1	inner
1280.4.a.i		2	8.b	even	2	1
1280.4.a.i		2	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1280))$:

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

$T_{7}^{2} - 450$

$T_{13} - 30$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} - 2$
$5$	$(T + 5)^{2}$
$7$	$T^{2} - 450$
$11$	$T^{2} - 968$