Newform orbit 1224.1.n.c

• Label: 1224.1.n.c
• Level: $1224$
• Weight: $1$
• Character orbit: 1224.n
• Self dual: yes
• Analytic conductor: $0.611$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -136
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$1224 = 2^{3} \cdot 3^{2} \cdot 17$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1224.n (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	$0.610855575463$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2})$
Defining polynomial:	$x^{2} - 2$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.9792.1
Artin image:	$D_8$
Artin field:	Galois closure of 8.0.14670139392.3

$q$-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 + q^2 + q^4 - b * q^5 + b * q^7 + q^8 - b * q^10 + b * q^14 + q^16 - q^17 - b * q^20 + b * q^23 + q^25 + b * q^28 + b * q^29 - b * q^31 + q^32 - q^34 - 2 * q^35 - b * q^37 - b * q^40 - 2 * q^43 + b * q^46 + q^49 + q^50 + b * q^56 + b * q^58 + b * q^61 - b * q^62 + q^64 - q^68 - 2 * q^70 - b * q^71 - b * q^74 - b * q^79 - b * q^80 + b * q^85 - 2 * q^86 - 2 * q^89 + b * q^92 + q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{16} - 2 q^{17} + 2 q^{25} + 2 q^{32} - 2 q^{34} - 4 q^{35} - 4 q^{43} + 2 q^{49} + 2 q^{50} + 2 q^{64} - 2 q^{68} - 4 q^{70} - 4 q^{86} - 4 q^{89} + 2 q^{98}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$.

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

883.1

1.41421
−1.41421

1.00000

0

1.00000

−1.41421

0

1.41421

1.00000

0

−1.41421

883.2

1.00000

0

1.00000

1.41421

0

−1.41421

1.00000

0

1.41421

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
136.e	odd	2	1	CM by $\Q(\sqrt{-34})$
8.d	odd	2	1	inner
17.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1224.1.n.c	yes	2
3.b	odd	2	1		1224.1.n.b	✓	2
8.d	odd	2	1	inner	1224.1.n.c	yes	2
17.b	even	2	1	inner	1224.1.n.c	yes	2
24.f	even	2	1		1224.1.n.b	✓	2
51.c	odd	2	1		1224.1.n.b	✓	2
136.e	odd	2	1	CM	1224.1.n.c	yes	2
408.h	even	2	1		1224.1.n.b	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1224.1.n.b	✓	2	3.b	odd	2	1
1224.1.n.b	✓	2	24.f	even	2	1
1224.1.n.b	✓	2	51.c	odd	2	1
1224.1.n.b	✓	2	408.h	even	2	1
1224.1.n.c	yes	2	1.a	even	1	1	trivial
1224.1.n.c	yes	2	8.d	odd	2	1	inner
1224.1.n.c	yes	2	17.b	even	2	1	inner
1224.1.n.c	yes	2	136.e	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(1224, [\chi])$:

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

$T_{89} + 2$

Hecke characteristic polynomials

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