Newform orbit 396.1.d.a

• Label: 396.1.d.a
• Level: $396$
• Weight: $1$
• Character orbit: 396.d
• Analytic conductor: $0.198$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{4}$
• RM discriminant: 44
• Inner twists: $8$

Newspace parameters

Level:	$N$	$=$	$396 = 2^{2} \cdot 3^{2} \cdot 11$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	396.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.197629745003$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
Defining polynomial:	$x^{4} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.4752.1

$q$-expansion

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	q + z^2 * q^2 - q^4 + (-z^3 - z) * q^5 + (-z^3 + z) * q^7 - z^2 * q^8 + (-z^3 + z) * q^10 + z^2 * q^11 + (z^3 + z) * q^14 + q^16 + (z^3 - z) * q^19 + (z^3 + z) * q^20 - q^22 - q^25 + (z^3 - z) * q^28 + z^2 * q^32 - 2*z^2 * q^35 + (-z^3 - z) * q^38 + (z^3 - z) * q^40 + (z^3 - z) * q^43 - z^2 * q^44 + q^49 - z^2 * q^50 + (z^3 + z) * q^53 + (-z^3 + z) * q^55 + (-z^3 - z) * q^56 - q^64 + 2 * q^70 + (-z^3 + z) * q^76 + (z^3 + z) * q^77 + (z^3 - z) * q^79 + (-z^3 - z) * q^80 + (-z^3 - z) * q^86 + q^88 + (-z^3 - z) * q^89 + 2*z^2 * q^95 - 2 * q^97 + z^2 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{4} + 4 q^{16} - 4 q^{22} - 4 q^{25} + 4 q^{49} - 4 q^{64} + 8 q^{70} + 4 q^{88} - 8 q^{97}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/396\mathbb{Z}\right)^\times$.

$n$	$145$	$199$	$353$
$\chi(n)$	$-1$	$-1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

395.1

−0.707107	+	0.707107i
0.707107	−	0.707107i
0.707107	+	0.707107i
−0.707107	−	0.707107i

−

1.00000i

0

−1.00000

−

1.41421i

0

−1.41421

1.00000i

0

−1.41421

395.2

−

1.00000i

0

−1.00000

1.41421i

0

1.41421

1.00000i

0

1.41421

395.3

1.00000i

0

−1.00000

−

1.41421i

0

1.41421

−

1.00000i

0

1.41421

395.4

1.00000i

0

−1.00000

1.41421i

0

−1.41421

−

1.00000i

0

−1.41421

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
44.c	even	2	1	RM by $\Q(\sqrt{11})$
3.b	odd	2	1	inner
4.b	odd	2	1	inner
11.b	odd	2	1	inner
12.b	even	2	1	inner
33.d	even	2	1	inner
132.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	396.1.d.a	✓	4
3.b	odd	2	1	inner	396.1.d.a	✓	4
4.b	odd	2	1	inner	396.1.d.a	✓	4
9.c	even	3	2		3564.1.o.e		8
9.d	odd	6	2		3564.1.o.e		8
11.b	odd	2	1	inner	396.1.d.a	✓	4
12.b	even	2	1	inner	396.1.d.a	✓	4
33.d	even	2	1	inner	396.1.d.a	✓	4
36.f	odd	6	2		3564.1.o.e		8
36.h	even	6	2		3564.1.o.e		8
44.c	even	2	1	RM	396.1.d.a	✓	4
99.g	even	6	2		3564.1.o.e		8
99.h	odd	6	2		3564.1.o.e		8
132.d	odd	2	1	inner	396.1.d.a	✓	4
396.k	even	6	2		3564.1.o.e		8
396.o	odd	6	2		3564.1.o.e		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
396.1.d.a	✓	4	1.a	even	1	1	trivial
396.1.d.a	✓	4	3.b	odd	2	1	inner
396.1.d.a	✓	4	4.b	odd	2	1	inner
396.1.d.a	✓	4	11.b	odd	2	1	inner
396.1.d.a	✓	4	12.b	even	2	1	inner
396.1.d.a	✓	4	33.d	even	2	1	inner
396.1.d.a	✓	4	44.c	even	2	1	RM
396.1.d.a	✓	4	132.d	odd	2	1	inner
3564.1.o.e		8	9.c	even	3	2
3564.1.o.e		8	9.d	odd	6	2
3564.1.o.e		8	36.f	odd	6	2
3564.1.o.e		8	36.h	even	6	2
3564.1.o.e		8	99.g	even	6	2
3564.1.o.e		8	99.h	odd	6	2
3564.1.o.e		8	396.k	even	6	2
3564.1.o.e		8	396.o	odd	6	2

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(396, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} + 1)^{2}$
$3$	$T^{4}$
$5$	$(T^{2} + 2)^{2}$
$7$	$(T^{2} - 2)^{2}$
$11$	$(T^{2} + 1)^{2}$