Embedded newform 414.3.h.a.229.5

• Label: 414.3.h.a.229.5
• Level: $414$
• Weight: $3$
• Character: 414.229
• Analytic conductor: $11.281$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.2806829445\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			229.5
Character	\(\chi\)	\(=\)	414.229
Dual form			414.3.h.a.367.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-2.61963 - 1.46204i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(-5.22993 - 3.01950i) q^{5} +(0.0617296 + 4.24219i) q^{6} +(-11.0196 + 6.36214i) q^{7} +2.82843 q^{8} +(4.72488 + 7.65999i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 4 q^{3} - 96 q^{4} + 16 q^{6} + 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(235\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.707107	−	1.22474i	−0.353553	−	0.612372i
							\(3\)	−2.61963	−	1.46204i	−0.873209	−	0.487347i
\(5\)	−5.22993	−	3.01950i	−1.04599	−	0.603900i								−0.124463	−	0.992224i	\(-0.539721\pi\)
							−0.921523	+	0.388324i	\(0.873054\pi\)
\(7\)	−11.0196	+	6.36214i	−1.57422	+	0.908878i	−0.578579	+	0.815626i	\(0.696393\pi\)
							−0.995643	+	0.0932515i	\(0.970274\pi\)
\(11\)	7.94506	−	4.58709i	0.722279	−	0.417008i	−0.0933122	−	0.995637i	\(-0.529745\pi\)
							0.815591	+	0.578629i	\(0.196412\pi\)
\(13\)	−10.3084	+	17.8546i	−0.792952	+	1.37343i	0.131180	+	0.991359i	\(0.458123\pi\)
							−0.924132	+	0.382074i	\(0.875210\pi\)
\(17\)		−	10.1918i		−	0.599515i	−0.954015	−	0.299757i	\(-0.903094\pi\)
							0.954015	−	0.299757i	\(-0.0969057\pi\)
\(19\)		−	22.0630i		−	1.16121i	−0.814186	−	0.580605i	\(-0.802816\pi\)
							0.814186	−	0.580605i	\(-0.197184\pi\)
\(23\)	−20.1731	+	11.0474i	−0.877093	+	0.480321i
							\(29\)	1.24822	+	2.16198i	0.0430420	+	0.0745509i	0.886744	−	0.462261i	\(-0.152962\pi\)
−0.843702	+	0.536812i	\(0.819628\pi\)
\(31\)	14.7898	−	25.6167i	0.477091	−	0.826346i	−0.522564	−	0.852600i	\(-0.675025\pi\)
							0.999655	+	0.0262540i	\(0.00835788\pi\)
\(37\)		−	7.43666i		−	0.200991i	−0.994938	−	0.100495i	\(-0.967957\pi\)
							0.994938	−	0.100495i	\(-0.0320428\pi\)
\(41\)	−28.2097	+	48.8606i	−0.688042	+	1.19172i	0.284429	+	0.958697i	\(0.408196\pi\)
							−0.972471	+	0.233026i	\(0.925137\pi\)
\(43\)	−5.13873	+	2.96685i	−0.119505	+	0.0689965i	−0.558561	−	0.829463i	\(-0.688646\pi\)
							0.439056	+	0.898460i	\(0.355313\pi\)
\(47\)	27.3010	+	47.2867i	0.580872	+	1.00610i	0.995376	+	0.0960523i	\(0.0306216\pi\)
							−0.414504	+	0.910047i	\(0.636045\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.h.a.229.5	✓	96
3.2	odd	2		1242.3.h.a.91.34		96
9.2	odd	6		1242.3.h.a.505.33		96
9.7	even	3	inner	414.3.h.a.367.6	yes	96
23.22	odd	2	inner	414.3.h.a.229.6	yes	96
69.68	even	2		1242.3.h.a.91.33		96
207.137	even	6		1242.3.h.a.505.34		96
207.160	odd	6	inner	414.3.h.a.367.5	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.h.a.229.5	✓	96	1.1	even	1	trivial
414.3.h.a.229.6	yes	96	23.22	odd	2	inner
414.3.h.a.367.5	yes	96	207.160	odd	6	inner
414.3.h.a.367.6	yes	96	9.7	even	3	inner
1242.3.h.a.91.33		96	69.68	even	2
1242.3.h.a.91.34		96	3.2	odd	2
1242.3.h.a.505.33		96	9.2	odd	6
1242.3.h.a.505.34		96	207.137	even	6