Embedded newform 224.2.u.b.29.8

• Label: 224.2.u.b.29.8
• Level: $224$
• Weight: $2$
• Character: 224.29
• Analytic conductor: $1.789$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.u (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.78864900528\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			29.8
Character	\(\chi\)	\(=\)	224.29
Dual form			224.2.u.b.85.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.16944 - 0.795236i) q^{2} +(1.29558 + 0.536647i) q^{3} +(0.735199 - 1.85997i) q^{4} +(0.937867 + 2.26421i) q^{5} +(1.94187 - 0.402713i) q^{6} +(-0.707107 + 0.707107i) q^{7} +(-0.619339 - 2.75979i) q^{8} +(-0.730784 - 0.730784i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 4 q^{2} + 4 q^{3} + 8 q^{5} + 12 q^{6} - 8 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.16944	−	0.795236i	0.826922	−	0.562317i
							\(3\)	1.29558	+	0.536647i	0.748003	+	0.309833i	0.723927	−	0.689877i	\(-0.242335\pi\)
0.0240766	+	0.999710i	\(0.492335\pi\)
\(5\)	0.937867	+	2.26421i	0.419427	+	1.01259i	0.982514	+	0.186189i	\(0.0596136\pi\)
							−0.563087	+	0.826398i	\(0.690386\pi\)
\(7\)	−0.707107	+	0.707107i	−0.267261	+	0.267261i
							\(11\)	−2.12310	+	0.879416i	−0.640138	+	0.265154i	−0.679054	−	0.734089i	\(-0.737610\pi\)
0.0389155	+	0.999243i	\(0.487610\pi\)
\(13\)	−0.267138	+	0.644928i	−0.0740907	+	0.178871i	−0.956586	−	0.291451i	\(-0.905862\pi\)
							0.882495	+	0.470322i	\(0.155862\pi\)
\(17\)		−	2.83430i		−	0.687419i	−0.939076	−	0.343709i	\(-0.888316\pi\)
							0.939076	−	0.343709i	\(-0.111684\pi\)
\(19\)	−0.445544	+	1.07564i	−0.102215	+	0.246768i	−0.966711	−	0.255871i	\(-0.917638\pi\)
							0.864496	+	0.502639i	\(0.167638\pi\)
\(23\)	−4.15632	−	4.15632i	−0.866654	−	0.866654i	0.125447	−	0.992100i	\(-0.459964\pi\)
							−0.992100	+	0.125447i	\(0.959964\pi\)
\(29\)	3.37949	+	1.39983i	0.627555	+	0.259942i	0.673714	−	0.738992i	\(-0.264698\pi\)
							−0.0461587	+	0.998934i	\(0.514698\pi\)
\(31\)	2.27603			0.408786			0.204393	−	0.978889i	\(-0.434478\pi\)
							0.204393	+	0.978889i	\(0.434478\pi\)
\(37\)	3.59295	+	8.67415i	0.590678	+	1.42602i	0.882850	+	0.469656i	\(0.155622\pi\)
							−0.292172	+	0.956366i	\(0.594378\pi\)
\(41\)	0.419720	+	0.419720i	0.0655493	+	0.0655493i	0.739121	−	0.673572i	\(-0.235241\pi\)
							−0.673572	+	0.739121i	\(0.735241\pi\)
\(43\)	−6.63049	+	2.74644i	−1.01114	+	0.418828i	−0.825871	−	0.563860i	\(-0.809316\pi\)
							−0.185270	+	0.982688i	\(0.559316\pi\)
\(47\)			12.0972i			1.76455i	0.470732	+	0.882276i	\(0.343990\pi\)
							−0.470732	+	0.882276i	\(0.656010\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.u.b.29.8	✓	40
4.3	odd	2		896.2.u.b.337.4		40
32.11	odd	8		896.2.u.b.561.4		40
32.21	even	8	inner	224.2.u.b.85.8	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.u.b.29.8	✓	40	1.1	even	1	trivial
224.2.u.b.85.8	yes	40	32.21	even	8	inner
896.2.u.b.337.4		40	4.3	odd	2
896.2.u.b.561.4		40	32.11	odd	8