Embedded newform 224.3.v.b.69.15

• Label: 224.3.v.b.69.15
• Level: $224$
• Weight: $3$
• Character: 224.69
• Analytic conductor: $6.104$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.10355792167\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(60\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			69.15
Character	\(\chi\)	\(=\)	224.69
Dual form			224.3.v.b.13.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41192 - 1.41651i) q^{2} +(-2.79421 + 1.15740i) q^{3} +(-0.0129803 + 3.99998i) q^{4} +(-1.86072 + 4.49217i) q^{5} +(5.58465 + 2.32386i) q^{6} +(0.944293 + 6.93602i) q^{7} +(5.68432 - 5.62925i) q^{8} +(0.104065 - 0.104065i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41192	−	1.41651i	−0.705959	−	0.708253i
							\(3\)	−2.79421	+	1.15740i	−0.931403	+	0.385800i	−0.796210	−	0.605020i	\(-0.793165\pi\)
−0.135192	+	0.990819i	\(0.543165\pi\)
\(5\)	−1.86072	+	4.49217i	−0.372143	+	0.898434i	0.621243	+	0.783618i	\(0.286628\pi\)
							−0.993387	+	0.114816i	\(0.963372\pi\)
\(7\)	0.944293	+	6.93602i	0.134899	+	0.990859i
							\(11\)	2.63807	−	6.36886i	0.239824	−	0.578988i	−0.757440	−	0.652905i	\(-0.773550\pi\)
0.997264	+	0.0739174i	\(0.0235501\pi\)
\(13\)	−3.81272	−	9.20472i	−0.293286	−	0.708055i	−1.00000		0.000555699i	\(-0.999823\pi\)
							0.706714	−	0.707500i	\(-0.250177\pi\)
\(17\)	−18.4549			−1.08558			−0.542791	−	0.839868i	\(-0.682633\pi\)
							−0.542791	+	0.839868i	\(0.682633\pi\)
\(19\)	6.94428	+	16.7650i	0.365488	+	0.882367i	0.994477	+	0.104952i	\(0.0334690\pi\)
							−0.628989	+	0.777414i	\(0.716531\pi\)
\(23\)	−1.72229	+	1.72229i	−0.0748821	+	0.0748821i	−0.743556	−	0.668674i	\(-0.766862\pi\)
							0.668674	+	0.743556i	\(0.266862\pi\)
\(29\)	−17.1745	−	41.4630i	−0.592225	−	1.42976i	−0.881349	−	0.472466i	\(-0.843364\pi\)
							0.289124	−	0.957292i	\(-0.406636\pi\)
\(31\)		−	2.69268i		−	0.0868606i	−0.999056	−	0.0434303i	\(-0.986171\pi\)
							0.999056	−	0.0434303i	\(-0.0138286\pi\)
\(37\)	−63.1550	−	26.1596i	−1.70689	−	0.707017i	−0.706891	−	0.707323i	\(-0.749903\pi\)
							−1.00000		0.000305477i	\(0.999903\pi\)
\(41\)	4.62617	+	4.62617i	0.112833	+	0.112833i	0.761269	−	0.648436i	\(-0.224577\pi\)
							−0.648436	+	0.761269i	\(0.724577\pi\)
\(43\)	1.15439	−	2.78695i	0.0268463	−	0.0648127i	−0.909888	−	0.414854i	\(-0.863833\pi\)
							0.936734	+	0.350042i	\(0.113833\pi\)
\(47\)	−9.80819			−0.208685			−0.104342	−	0.994541i	\(-0.533274\pi\)
							−0.104342	+	0.994541i	\(0.533274\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.b.69.15	yes	240
7.6	odd	2	inner	224.3.v.b.69.16	yes	240
32.13	even	8	inner	224.3.v.b.13.16	yes	240
224.13	odd	8	inner	224.3.v.b.13.15	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.b.13.15	✓	240	224.13	odd	8	inner
224.3.v.b.13.16	yes	240	32.13	even	8	inner
224.3.v.b.69.15	yes	240	1.1	even	1	trivial
224.3.v.b.69.16	yes	240	7.6	odd	2	inner