Embedded newform 224.3.v.b.69.3

• Label: 224.3.v.b.69.3
• 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			69.3
Character	\(\chi\)	\(=\)	224.69
Dual form			224.3.v.b.13.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91889 + 0.563790i) q^{2} +(-4.49480 + 1.86181i) q^{3} +(3.36428 - 2.16370i) q^{4} +(0.432309 - 1.04369i) q^{5} +(7.57535 - 6.10672i) q^{6} +(-6.93508 + 0.951145i) q^{7} +(-5.23581 + 6.04866i) q^{8} +(10.3729 - 10.3729i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91889	+	0.563790i	−0.959445	+	0.281895i
							\(3\)	−4.49480	+	1.86181i	−1.49827	+	0.620602i	−0.973097	−	0.230396i	\(-0.925998\pi\)
−0.525169	+	0.850998i	\(0.675998\pi\)
\(5\)	0.432309	−	1.04369i	0.0864617	−	0.208737i	−0.874735	−	0.484602i	\(-0.838964\pi\)
							0.961196	+	0.275865i	\(0.0889643\pi\)
\(7\)	−6.93508	+	0.951145i	−0.990726	+	0.135878i
							\(11\)	4.35814	−	10.5215i	0.396195	−	0.956499i	−0.592365	−	0.805670i	\(-0.701806\pi\)
0.988560	−	0.150829i	\(-0.0481943\pi\)
\(13\)	−5.75095	−	13.8840i	−0.442380	−	1.06800i	−0.975111	−	0.221716i	\(-0.928834\pi\)
							0.532731	−	0.846285i	\(-0.321166\pi\)
\(17\)	2.83244			0.166614			0.0833071	−	0.996524i	\(-0.473452\pi\)
							0.0833071	+	0.996524i	\(0.473452\pi\)
\(19\)	11.1668	+	26.9590i	0.587725	+	1.41889i	0.885672	+	0.464312i	\(0.153698\pi\)
							−0.297946	+	0.954583i	\(0.596302\pi\)
\(23\)	−18.7312	+	18.7312i	−0.814398	+	0.814398i	−0.985290	−	0.170892i	\(-0.945335\pi\)
							0.170892	+	0.985290i	\(0.445335\pi\)
\(29\)	0.210363	+	0.507862i	0.00725390	+	0.0175125i	0.927465	−	0.373910i	\(-0.121983\pi\)
							−0.920211	+	0.391422i	\(0.871983\pi\)
\(31\)			35.8709i			1.15713i	0.815638	+	0.578563i	\(0.196386\pi\)
							−0.815638	+	0.578563i	\(0.803614\pi\)
\(37\)	59.9916	+	24.8494i	1.62140	+	0.671604i	0.994229	−	0.107281i	\(-0.0342143\pi\)
							0.627167	+	0.778885i	\(0.284214\pi\)
\(41\)	−20.9255	−	20.9255i	−0.510379	−	0.510379i	0.404264	−	0.914642i	\(-0.367528\pi\)
							−0.914642	+	0.404264i	\(0.867528\pi\)
\(43\)	23.1324	−	55.8466i	0.537964	−	1.29876i	−0.388178	−	0.921584i	\(-0.626896\pi\)
							0.926142	−	0.377175i	\(-0.123104\pi\)
\(47\)	−2.87518			−0.0611741			−0.0305870	−	0.999532i	\(-0.509738\pi\)
							−0.0305870	+	0.999532i	\(0.509738\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.b.69.3	yes	240
7.6	odd	2	inner	224.3.v.b.69.4	yes	240
32.13	even	8	inner	224.3.v.b.13.4	yes	240
224.13	odd	8	inner	224.3.v.b.13.3	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.b.13.3	✓	240	224.13	odd	8	inner
224.3.v.b.13.4	yes	240	32.13	even	8	inner
224.3.v.b.69.3	yes	240	1.1	even	1	trivial
224.3.v.b.69.4	yes	240	7.6	odd	2	inner