Embedded newform 414.3.h.a.229.17

• Label: 414.3.h.a.229.17
• 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			229.17
Character	\(\chi\)	\(=\)	414.229
Dual form			414.3.h.a.367.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(1.61253 - 2.52977i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(-0.849237 - 0.490307i) q^{5} +(-4.23856 - 0.186126i) q^{6} +(3.58678 - 2.07083i) q^{7} +2.82843 q^{8} +(-3.79946 - 8.15868i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 4 q^{3} - 96 q^{4} + 16 q^{6} + 36 q^{9}+O(q^{10}) \)

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
							\(3\)	1.61253	−	2.52977i	0.537512	−	0.843256i
\(5\)	−0.849237	−	0.490307i	−0.169847	−	0.0980614i								0.412666	−	0.910882i	\(-0.364598\pi\)
							−0.582514	+	0.812821i	\(0.697931\pi\)
\(7\)	3.58678	−	2.07083i	0.512397	−	0.295832i	−0.221421	−	0.975178i	\(-0.571070\pi\)
							0.733818	+	0.679346i	\(0.237736\pi\)
\(11\)	15.3577	−	8.86675i	1.39615	−	0.806068i	0.402164	−	0.915568i	\(-0.368258\pi\)
							0.993987	+	0.109499i	\(0.0349247\pi\)
\(13\)	−3.68522	+	6.38299i	−0.283479	+	0.491000i	−0.972239	−	0.233989i	\(-0.924822\pi\)
							0.688760	+	0.724989i	\(0.258155\pi\)
\(17\)		−	2.29875i		−	0.135221i	−0.997712	−	0.0676103i	\(-0.978463\pi\)
							0.997712	−	0.0676103i	\(-0.0215374\pi\)
\(19\)		−	18.1720i		−	0.956422i	−0.878245	−	0.478211i	\(-0.841285\pi\)
							0.878245	−	0.478211i	\(-0.158715\pi\)
\(23\)	10.8159	+	20.2982i	0.470256	+	0.882530i
							\(29\)	−15.7758	−	27.3245i	−0.543993	−	0.942223i	−0.998670	−	0.0515664i	\(-0.983579\pi\)
0.454677	−	0.890656i	\(-0.349755\pi\)
\(31\)	−9.77841	+	16.9367i	−0.315433	+	0.546345i	−0.979529	−	0.201301i	\(-0.935483\pi\)
							0.664097	+	0.747647i	\(0.268816\pi\)
\(37\)		−	38.8585i		−	1.05023i	−0.851031	−	0.525115i	\(-0.824022\pi\)
							0.851031	−	0.525115i	\(-0.175978\pi\)
\(41\)	7.71546	−	13.3636i	0.188182	−	0.325941i	−0.756462	−	0.654038i	\(-0.773074\pi\)
							0.944644	+	0.328097i	\(0.106407\pi\)
\(43\)	−11.3402	+	6.54727i	−0.263726	+	0.152262i	−0.626033	−	0.779797i	\(-0.715322\pi\)
							0.362307	+	0.932059i	\(0.381989\pi\)
\(47\)	13.0659	+	22.6307i	0.277997	+	0.481505i	0.970887	−	0.239538i	\(-0.0769961\pi\)
							−0.692890	+	0.721044i	\(0.743663\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.h.a.229.17	✓	96
3.2	odd	2		1242.3.h.a.91.46		96
9.2	odd	6		1242.3.h.a.505.45		96
9.7	even	3	inner	414.3.h.a.367.18	yes	96
23.22	odd	2	inner	414.3.h.a.229.18	yes	96
69.68	even	2		1242.3.h.a.91.45		96
207.137	even	6		1242.3.h.a.505.46		96
207.160	odd	6	inner	414.3.h.a.367.17	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.h.a.229.17	✓	96	1.1	even	1	trivial
414.3.h.a.229.18	yes	96	23.22	odd	2	inner
414.3.h.a.367.17	yes	96	207.160	odd	6	inner
414.3.h.a.367.18	yes	96	9.7	even	3	inner
1242.3.h.a.91.45		96	69.68	even	2
1242.3.h.a.91.46		96	3.2	odd	2
1242.3.h.a.505.45		96	9.2	odd	6
1242.3.h.a.505.46		96	207.137	even	6