Embedded newform 224.3.v.b.13.9

• Label: 224.3.v.b.13.9
• 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			13.9
Character	\(\chi\)	\(=\)	224.13
Dual form			224.3.v.b.69.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84088 + 0.781766i) q^{2} +(-3.58432 - 1.48467i) q^{3} +(2.77768 - 2.87828i) q^{4} +(3.04365 + 7.34802i) q^{5} +(7.75897 - 0.0689922i) q^{6} +(-1.24702 - 6.88803i) q^{7} +(-2.86325 + 7.47006i) q^{8} +(4.27913 + 4.27913i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84088	+	0.781766i	−0.920440	+	0.390883i
							\(3\)	−3.58432	−	1.48467i	−1.19477	−	0.494891i	−0.305467	−	0.952203i	\(-0.598812\pi\)
−0.889307	+	0.457311i	\(0.848812\pi\)
\(5\)	3.04365	+	7.34802i	0.608730	+	1.46960i	0.864382	+	0.502836i	\(0.167710\pi\)
							−0.255652	+	0.966769i	\(0.582290\pi\)
\(7\)	−1.24702	−	6.88803i	−0.178145	−	0.984004i
							\(11\)	−5.50181	−	13.2825i	−0.500164	−	1.20750i	−0.949394	−	0.314087i	\(-0.898302\pi\)
0.449230	−	0.893416i	\(-0.351698\pi\)
\(13\)	−2.01506	+	4.86479i	−0.155005	+	0.374215i	−0.982237	−	0.187646i	\(-0.939914\pi\)
							0.827232	+	0.561861i	\(0.189914\pi\)
\(17\)	22.4482			1.32048			0.660242	−	0.751053i	\(-0.270454\pi\)
							0.660242	+	0.751053i	\(0.270454\pi\)
\(19\)	−11.7463	+	28.3582i	−0.618228	+	1.49253i	0.235531	+	0.971867i	\(0.424317\pi\)
							−0.853759	+	0.520668i	\(0.825683\pi\)
\(23\)	29.8607	+	29.8607i	1.29829	+	1.29829i	0.929522	+	0.368768i	\(0.120220\pi\)
							0.368768	+	0.929522i	\(0.379780\pi\)
\(29\)	−0.784167	+	1.89315i	−0.0270402	+	0.0652809i	−0.936823	−	0.349804i	\(-0.886248\pi\)
							0.909783	+	0.415085i	\(0.136248\pi\)
\(31\)		−	6.89652i		−	0.222468i	−0.993794	−	0.111234i	\(-0.964520\pi\)
							0.993794	−	0.111234i	\(-0.0354804\pi\)
\(37\)	40.9734	−	16.9717i	1.10739	−	0.458695i	0.247351	−	0.968926i	\(-0.420440\pi\)
							0.860038	+	0.510231i	\(0.170440\pi\)
\(41\)	−11.9237	+	11.9237i	−0.290821	+	0.290821i	−0.837405	−	0.546584i	\(-0.815928\pi\)
							0.546584	+	0.837405i	\(0.315928\pi\)
\(43\)	23.7088	+	57.2382i	0.551368	+	1.33112i	0.916452	+	0.400145i	\(0.131040\pi\)
							−0.365084	+	0.930975i	\(0.618960\pi\)
\(47\)	−11.6140			−0.247106			−0.123553	−	0.992338i	\(-0.539429\pi\)
							−0.123553	+	0.992338i	\(0.539429\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.b.13.9	✓	240
7.6	odd	2	inner	224.3.v.b.13.10	yes	240
32.5	even	8	inner	224.3.v.b.69.10	yes	240
224.69	odd	8	inner	224.3.v.b.69.9	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.b.13.9	✓	240	1.1	even	1	trivial
224.3.v.b.13.10	yes	240	7.6	odd	2	inner
224.3.v.b.69.9	yes	240	224.69	odd	8	inner
224.3.v.b.69.10	yes	240	32.5	even	8	inner