Embedded newform 224.2.u.b.29.3

• Label: 224.2.u.b.29.3
• 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			29.3
Character	\(\chi\)	\(=\)	224.29
Dual form			224.2.u.b.85.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.533524 + 1.30971i) q^{2} +(-0.910708 - 0.377228i) q^{3} +(-1.43070 - 1.39753i) q^{4} +(-0.227513 - 0.549265i) q^{5} +(0.979945 - 0.991507i) q^{6} +(-0.707107 + 0.707107i) q^{7} +(2.59368 - 1.12820i) q^{8} +(-1.43423 - 1.43423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 4 q^{2} + 4 q^{3} + 8 q^{5} + 12 q^{6} - 8 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.533524	+	1.30971i	−0.377259	+	0.926108i
							\(3\)	−0.910708	−	0.377228i	−0.525797	−	0.217792i	0.103964	−	0.994581i	\(-0.466847\pi\)
−0.629761	+	0.776789i	\(0.716847\pi\)
\(5\)	−0.227513	−	0.549265i	−0.101747	−	0.245639i	0.864806	−	0.502106i	\(-0.167441\pi\)
							−0.966553	+	0.256468i	\(0.917441\pi\)
\(7\)	−0.707107	+	0.707107i	−0.267261	+	0.267261i
							\(11\)	4.11059	−	1.70266i	1.23939	−	0.513372i	0.335867	−	0.941910i	\(-0.390971\pi\)
0.903524	+	0.428537i	\(0.140971\pi\)
\(13\)	2.17854	−	5.25947i	0.604219	−	1.45871i	−0.264981	−	0.964254i	\(-0.585366\pi\)
							0.869200	−	0.494460i	\(-0.164634\pi\)
\(17\)		−	4.07456i		−	0.988225i	−0.869398	−	0.494113i	\(-0.835493\pi\)
							0.869398	−	0.494113i	\(-0.164507\pi\)
\(19\)	−0.900651	+	2.17436i	−0.206624	+	0.498833i	−0.992887	−	0.119058i	\(-0.962013\pi\)
							0.786264	+	0.617891i	\(0.212013\pi\)
\(23\)	−0.759995	−	0.759995i	−0.158470	−	0.158470i	0.623418	−	0.781888i	\(-0.285743\pi\)
							−0.781888	+	0.623418i	\(0.785743\pi\)
\(29\)	−2.59336	−	1.07421i	−0.481575	−	0.199475i	0.128670	−	0.991687i	\(-0.458929\pi\)
							−0.610245	+	0.792212i	\(0.708929\pi\)
\(31\)	−5.93131			−1.06529			−0.532647	−	0.846338i	\(-0.678803\pi\)
							−0.532647	+	0.846338i	\(0.678803\pi\)
\(37\)	4.05315	+	9.78518i	0.666334	+	1.60867i	0.787695	+	0.616065i	\(0.211274\pi\)
							−0.121361	+	0.992608i	\(0.538726\pi\)
\(41\)	−7.57799	−	7.57799i	−1.18348	−	1.18348i	−0.978836	−	0.204647i	\(-0.934395\pi\)
							−0.204647	−	0.978836i	\(-0.565605\pi\)
\(43\)	2.08658	−	0.864290i	0.318200	−	0.131803i	−0.217866	−	0.975979i	\(-0.569910\pi\)
							0.536067	+	0.844176i	\(0.319910\pi\)
\(47\)		−	5.52995i		−	0.806626i	−0.915062	−	0.403313i	\(-0.867859\pi\)
							0.915062	−	0.403313i	\(-0.132141\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.u.b.29.3	✓	40
4.3	odd	2		896.2.u.b.337.7		40
32.11	odd	8		896.2.u.b.561.7		40
32.21	even	8	inner	224.2.u.b.85.3	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.u.b.29.3	✓	40	1.1	even	1	trivial
224.2.u.b.85.3	yes	40	32.21	even	8	inner
896.2.u.b.337.7		40	4.3	odd	2
896.2.u.b.561.7		40	32.11	odd	8