Embedded newform 736.1.p.a.45.1

• Label: 736.1.p.a.45.1
• Level: $736$
• Weight: $1$
• Character: 736.45
• Analytic conductor: $0.367$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{8}$
• CM discriminant: -23
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 736 = 2^{5} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	736.p (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.367311849298\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{8})\)
Defining polynomial:	\( x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{8}\)
Projective field:	Galois closure of 8.0.600953971539968.26

Embedding invariants

Embedding label			45.1
Root			\(0.707107 + 0.707107i\) of defining polynomial
Character	\(\chi\)	\(=\)	736.45
Dual form			736.1.p.a.229.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{2} +(-0.292893 + 0.707107i) q^{3} +1.00000i q^{4} +(-0.707107 + 0.292893i) q^{6} +(-0.707107 + 0.707107i) q^{8} +(0.292893 + 0.292893i) q^{9} +(-0.707107 - 0.292893i) q^{12} +(-0.707107 - 0.292893i) q^{13} -1.00000 q^{16} +0.414214i q^{18} +(0.707107 + 0.707107i) q^{23} +(-0.292893 - 0.707107i) q^{24} +(0.707107 - 0.707107i) q^{25} +(-0.292893 - 0.707107i) q^{26} +(-1.00000 + 0.414214i) q^{27} +(0.707107 - 1.70711i) q^{29} -1.41421 q^{31} +(-0.707107 - 0.707107i) q^{32} +(-0.292893 + 0.292893i) q^{36} +(0.414214 - 0.414214i) q^{39} +(1.41421 + 1.41421i) q^{41} +1.00000i q^{46} +(0.292893 - 0.707107i) q^{48} -1.00000i q^{49} +1.00000 q^{50} +(0.292893 - 0.707107i) q^{52} +(-1.00000 - 0.414214i) q^{54} +(1.70711 - 0.707107i) q^{58} +(0.707107 - 0.292893i) q^{59} +(-1.00000 - 1.00000i) q^{62} -1.00000i q^{64} +(-0.707107 + 0.292893i) q^{69} +(1.00000 - 1.00000i) q^{71} -0.414214 q^{72} +(0.292893 + 0.707107i) q^{75} +0.585786 q^{78} -0.414214i q^{81} +2.00000i q^{82} +(1.00000 + 1.00000i) q^{87} +(-0.707107 + 0.707107i) q^{92} +(0.414214 - 1.00000i) q^{93} +(0.707107 - 0.292893i) q^{96} +(0.707107 - 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q - 4 * q^3 + 4 * q^9 - 4 * q^16 - 4 * q^24 - 4 * q^26 - 4 * q^27 - 4 * q^36 - 4 * q^39 + 4 * q^48 + 4 * q^50 + 4 * q^52 - 4 * q^54 + 4 * q^58 - 4 * q^62 + 4 * q^71 + 4 * q^72 + 4 * q^75 + 8 * q^78 + 4 * q^87 - 4 * q^93

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/736\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(415\)	\(645\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.707107	+	0.707107i	0.707107	+	0.707107i
							\(3\)	−0.292893	+	0.707107i	−0.292893	+	0.707107i	0.707107	+	0.707107i	\(0.250000\pi\)
−1.00000			\(\pi\)
\(5\)		0			0		0.923880	−	0.382683i	\(-0.125000\pi\)
							−0.923880	+	0.382683i	\(0.875000\pi\)
\(7\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(11\)		0			0		−0.382683	−	0.923880i	\(-0.625000\pi\)
							0.382683	+	0.923880i	\(0.375000\pi\)
\(13\)	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	\(-0.5\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(17\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(19\)		0			0		−0.923880	−	0.382683i	\(-0.875000\pi\)
							0.923880	+	0.382683i	\(0.125000\pi\)
\(23\)	0.707107	+	0.707107i	0.707107	+	0.707107i
							\(29\)	0.707107	−	1.70711i	0.707107	−	1.70711i		−	1.00000i	\(-0.5\pi\)
0.707107	−	0.707107i	\(-0.250000\pi\)
\(31\)	−1.41421			−1.41421			−0.707107	−	0.707107i	\(-0.750000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(37\)		0			0		0.923880	−	0.382683i	\(-0.125000\pi\)
							−0.923880	+	0.382683i	\(0.875000\pi\)
\(41\)	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	\(0.250000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(43\)		0			0		−0.382683	−	0.923880i	\(-0.625000\pi\)
							0.382683	+	0.923880i	\(0.375000\pi\)
\(47\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(53\)		0			0		−0.382683	−	0.923880i	\(-0.625000\pi\)
							0.382683	+	0.923880i	\(0.375000\pi\)
\(59\)	0.707107	−	0.292893i	0.707107	−	0.292893i		−	1.00000i	\(-0.5\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(61\)		0			0		0.382683	−	0.923880i	\(-0.375000\pi\)
							−0.382683	+	0.923880i	\(0.625000\pi\)
\(67\)		0			0		0.382683	−	0.923880i	\(-0.375000\pi\)
							−0.382683	+	0.923880i	\(0.625000\pi\)
\(71\)	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	\(-0.5\pi\)
							1.00000			\(0\)
\(73\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(79\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(83\)		0			0		−0.923880	−	0.382683i	\(-0.875000\pi\)
							0.923880	+	0.382683i	\(0.125000\pi\)
\(89\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(97\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	736.1.p.a.45.1	✓	4
4.3	odd	2		2944.1.p.a.1425.1		4
23.22	odd	2	CM	736.1.p.a.45.1	✓	4
32.5	even	8	inner	736.1.p.a.229.1	yes	4
32.27	odd	8		2944.1.p.a.2161.1		4
92.91	even	2		2944.1.p.a.1425.1		4
736.91	even	8		2944.1.p.a.2161.1		4
736.229	odd	8	inner	736.1.p.a.229.1	yes	4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.1.p.a.45.1	✓	4	1.1	even	1	trivial
736.1.p.a.45.1	✓	4	23.22	odd	2	CM
736.1.p.a.229.1	yes	4	32.5	even	8	inner
736.1.p.a.229.1	yes	4	736.229	odd	8	inner
2944.1.p.a.1425.1		4	4.3	odd	2
2944.1.p.a.1425.1		4	92.91	even	2
2944.1.p.a.2161.1		4	32.27	odd	8
2944.1.p.a.2161.1		4	736.91	even	8