Newform orbit 1344.1.o.a

• Label: 1344.1.o.a
• Level: $1344$
• Weight: $1$
• Character orbit: 1344.o
• Self dual: yes
• Analytic conductor: $0.671$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{2}$
• CM/RM discs: -3, -84, 28
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1344.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.670743376979\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 336)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-3}, \sqrt{7})\)
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.4032.2
Stark unit:	Root of $x^{4} - 84x^{3} - 26x^{2} - 84x + 1$

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{3} - q^{7} + q^{9} + 2 q^{19} + q^{21} - q^{25} - q^{27} + 2 q^{31} + 2 q^{37} + q^{49} - 2 q^{57} - q^{63} + q^{75} + q^{81} - 2 q^{93}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(449\)	\(577\)	\(1093\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(0\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1343.1

0

0

−1.00000

0

0

0

−1.00000

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
28.d	even	2	1	RM by \(\Q(\sqrt{7}) \)
84.h	odd	2	1	CM by \(\Q(\sqrt{-21}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1344.1.o.a		1
3.b	odd	2	1	CM	1344.1.o.a		1
4.b	odd	2	1		1344.1.o.b		1
7.b	odd	2	1		1344.1.o.b		1
8.b	even	2	1		336.1.o.b	yes	1
8.d	odd	2	1		336.1.o.a	✓	1
12.b	even	2	1		1344.1.o.b		1
21.c	even	2	1		1344.1.o.b		1
24.f	even	2	1		336.1.o.a	✓	1
24.h	odd	2	1		336.1.o.b	yes	1
28.d	even	2	1	RM	1344.1.o.a		1
56.e	even	2	1		336.1.o.b	yes	1
56.h	odd	2	1		336.1.o.a	✓	1
56.j	odd	6	2		2352.1.z.c		2
56.k	odd	6	2		2352.1.z.c		2
56.m	even	6	2		2352.1.z.b		2
56.p	even	6	2		2352.1.z.b		2
84.h	odd	2	1	CM	1344.1.o.a		1
168.e	odd	2	1		336.1.o.b	yes	1
168.i	even	2	1		336.1.o.a	✓	1
168.s	odd	6	2		2352.1.z.b		2
168.v	even	6	2		2352.1.z.c		2
168.ba	even	6	2		2352.1.z.c		2
168.be	odd	6	2		2352.1.z.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.1.o.a	✓	1	8.d	odd	2	1
336.1.o.a	✓	1	24.f	even	2	1
336.1.o.a	✓	1	56.h	odd	2	1
336.1.o.a	✓	1	168.i	even	2	1
336.1.o.b	yes	1	8.b	even	2	1
336.1.o.b	yes	1	24.h	odd	2	1
336.1.o.b	yes	1	56.e	even	2	1
336.1.o.b	yes	1	168.e	odd	2	1
1344.1.o.a		1	1.a	even	1	1	trivial
1344.1.o.a		1	3.b	odd	2	1	CM
1344.1.o.a		1	28.d	even	2	1	RM
1344.1.o.a		1	84.h	odd	2	1	CM
1344.1.o.b		1	4.b	odd	2	1
1344.1.o.b		1	7.b	odd	2	1
1344.1.o.b		1	12.b	even	2	1
1344.1.o.b		1	21.c	even	2	1
2352.1.z.b		2	56.m	even	6	2
2352.1.z.b		2	56.p	even	6	2
2352.1.z.b		2	168.s	odd	6	2
2352.1.z.b		2	168.be	odd	6	2
2352.1.z.c		2	56.j	odd	6	2
2352.1.z.c		2	56.k	odd	6	2
2352.1.z.c		2	168.v	even	6	2
2352.1.z.c		2	168.ba	even	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19} - 2 \) acting on \(S_{1}^{\mathrm{new}}(1344, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T + 1 \)
$5$	\( T \)
$7$	\( T + 1 \)
$11$	\( T \)