Newform orbit 1764.4.a.s

• Label: 1764.4.a.s
• Level: $1764$
• Weight: $4$
• Character orbit: 1764.a
• Self dual: yes
• Analytic conductor: $104.079$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1764.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$104.079369250$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7})$
Defining polynomial:	$x^{2} - 7$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2\cdot 3$
Twist minimal:	no (minimal twist has level 252)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 6\sqrt{7}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + b * q^5 - b * q^11 - 26 * q^13 - 5*b * q^17 - 68 * q^19 - 3*b * q^23 + 127 * q^25 + 16*b * q^29 - 212 * q^31 + 218 * q^37 + 25*b * q^41 + 260 * q^43 + 26*b * q^47 + 30*b * q^53 - 252 * q^55 - 18*b * q^59 + 322 * q^61 - 26*b * q^65 + 356 * q^67 + 71*b * q^71 + 226 * q^73 + 440 * q^79 - 16*b * q^83 - 1260 * q^85 + 13*b * q^89 - 68*b * q^95 + 1330 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 52 q^{13} - 136 q^{19} + 254 q^{25} - 424 q^{31} + 436 q^{37} + 520 q^{43} - 504 q^{55} + 644 q^{61} + 712 q^{67} + 452 q^{73} + 880 q^{79} - 2520 q^{85} + 2660 q^{97}+O(q^{100})$

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

−2.64575
2.64575

0

0

0

−15.8745

0

0

0

0

0

1.2

0

0

0

15.8745

0

0

0

0

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$
$7$	$-1$

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1764.4.a.s		2
3.b	odd	2	1	inner	1764.4.a.s		2
7.b	odd	2	1		252.4.a.f	✓	2
7.c	even	3	2		1764.4.k.u		4
7.d	odd	6	2		1764.4.k.v		4
21.c	even	2	1		252.4.a.f	✓	2
21.g	even	6	2		1764.4.k.v		4
21.h	odd	6	2		1764.4.k.u		4
28.d	even	2	1		1008.4.a.bc		2
84.h	odd	2	1		1008.4.a.bc		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.4.a.f	✓	2	7.b	odd	2	1
252.4.a.f	✓	2	21.c	even	2	1
1008.4.a.bc		2	28.d	even	2	1
1008.4.a.bc		2	84.h	odd	2	1
1764.4.a.s		2	1.a	even	1	1	trivial
1764.4.a.s		2	3.b	odd	2	1	inner
1764.4.k.u		4	7.c	even	3	2
1764.4.k.u		4	21.h	odd	6	2
1764.4.k.v		4	7.d	odd	6	2
1764.4.k.v		4	21.g	even	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1764))$:

$T_{5}^{2} - 252$

$T_{11}^{2} - 252$

$T_{13} + 26$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2} - 252$
$7$	$T^{2}$
$11$	$T^{2} - 252$