Embedded newform 224.3.v.b.13.12

• Label: 224.3.v.b.13.12
• Level: $224$
• Weight: $3$
• Character: 224.13
• 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			13.12
Character	\(\chi\)	\(=\)	224.13
Dual form			224.3.v.b.69.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62574 - 1.16489i) q^{2} +(3.33013 + 1.37939i) q^{3} +(1.28608 + 3.78761i) q^{4} +(3.52027 + 8.49868i) q^{5} +(-3.80711 - 6.12175i) q^{6} +(6.86401 - 1.37310i) q^{7} +(2.32130 - 7.65582i) q^{8} +(2.82310 + 2.82310i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62574	−	1.16489i	−0.812871	−	0.582443i
							\(3\)	3.33013	+	1.37939i	1.11004	+	0.459795i	0.860953	−	0.508685i	\(-0.169868\pi\)
0.249091	+	0.968480i	\(0.419868\pi\)
\(5\)	3.52027	+	8.49868i	0.704054	+	1.69974i	0.714349	+	0.699789i	\(0.246723\pi\)
							−0.0102952	+	0.999947i	\(0.503277\pi\)
\(7\)	6.86401	−	1.37310i	0.980572	−	0.196157i
							\(11\)	−3.28988	−	7.94246i	−0.299080	−	0.722042i	−0.999962	−	0.00876488i	\(-0.997210\pi\)
0.700882	−	0.713277i	\(-0.252790\pi\)
\(13\)	−7.03479	+	16.9835i	−0.541138	+	1.30642i	0.382783	+	0.923838i	\(0.374966\pi\)
							−0.923921	+	0.382584i	\(0.875034\pi\)
\(17\)	−0.600596			−0.0353292			−0.0176646	−	0.999844i	\(-0.505623\pi\)
							−0.0176646	+	0.999844i	\(0.505623\pi\)
\(19\)	3.14670	−	7.59680i	0.165616	−	0.399832i	−0.819183	−	0.573533i	\(-0.805573\pi\)
							0.984798	+	0.173701i	\(0.0555726\pi\)
\(23\)	−0.165648	−	0.165648i	−0.00720211	−	0.00720211i	0.703497	−	0.710699i	\(-0.251621\pi\)
							−0.710699	+	0.703497i	\(0.751621\pi\)
\(29\)	14.5043	−	35.0166i	0.500150	−	1.20747i	−0.449252	−	0.893405i	\(-0.648310\pi\)
							0.949402	−	0.314063i	\(-0.101690\pi\)
\(31\)		−	21.7594i		−	0.701917i	−0.936391	−	0.350958i	\(-0.885856\pi\)
							0.936391	−	0.350958i	\(-0.114144\pi\)
\(37\)	30.4184	−	12.5997i	0.822118	−	0.340532i	0.0683403	−	0.997662i	\(-0.478230\pi\)
							0.753778	+	0.657130i	\(0.228230\pi\)
\(41\)	−15.4960	+	15.4960i	−0.377950	+	0.377950i	−0.870362	−	0.492412i	\(-0.836115\pi\)
							0.492412	+	0.870362i	\(0.336115\pi\)
\(43\)	28.0058	+	67.6120i	0.651298	+	1.57237i	0.810896	+	0.585190i	\(0.198980\pi\)
							−0.159599	+	0.987182i	\(0.551020\pi\)
\(47\)	−23.9960			−0.510552			−0.255276	−	0.966868i	\(-0.582166\pi\)
							−0.255276	+	0.966868i	\(0.582166\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.b.13.12	yes	240
7.6	odd	2	inner	224.3.v.b.13.11	✓	240
32.5	even	8	inner	224.3.v.b.69.11	yes	240
224.69	odd	8	inner	224.3.v.b.69.12	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.b.13.11	✓	240	7.6	odd	2	inner
224.3.v.b.13.12	yes	240	1.1	even	1	trivial
224.3.v.b.69.11	yes	240	32.5	even	8	inner
224.3.v.b.69.12	yes	240	224.69	odd	8	inner