Embedded newform 414.3.h.a.229.9

• Label: 414.3.h.a.229.9
• 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			229.9
Character	\(\chi\)	\(=\)	414.229
Dual form			414.3.h.a.367.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-1.38507 - 2.66112i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(-2.02474 - 1.16898i) q^{5} +(-2.27981 + 3.57806i) q^{6} +(-2.31483 + 1.33647i) q^{7} +2.82843 q^{8} +(-5.16316 + 7.37169i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 4 q^{3} - 96 q^{4} + 16 q^{6} + 36 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.38507	−	2.66112i	−0.461690	−	0.887041i
\(5\)	−2.02474	−	1.16898i	−0.404948	−	0.233797i								0.283669	−	0.958922i	\(-0.408448\pi\)
							−0.688617	+	0.725125i	\(0.741782\pi\)
\(7\)	−2.31483	+	1.33647i	−0.330690	+	0.190924i	−0.656147	−	0.754633i	\(-0.727815\pi\)
							0.325458	+	0.945557i	\(0.394482\pi\)
\(11\)	−4.22586	+	2.43980i	−0.384169	+	0.221800i	−0.679631	−	0.733555i	\(-0.737860\pi\)
							0.295462	+	0.955355i	\(0.404527\pi\)
\(13\)	4.27607	−	7.40637i	0.328929	−	0.569721i	−0.653371	−	0.757038i	\(-0.726646\pi\)
							0.982300	+	0.187317i	\(0.0599791\pi\)
\(17\)			28.2944i			1.66438i	0.554492	+	0.832189i	\(0.312913\pi\)
							−0.554492	+	0.832189i	\(0.687087\pi\)
\(19\)		−	1.71147i		−	0.0900775i	−0.998985	−	0.0450387i	\(-0.985659\pi\)
							0.998985	−	0.0450387i	\(-0.0143411\pi\)
\(23\)	18.0540	+	14.2497i	0.784956	+	0.619552i
							\(29\)	10.1732	+	17.6205i	0.350799	+	0.607602i	0.986390	−	0.164424i	\(-0.0525767\pi\)
−0.635591	+	0.772026i	\(0.719243\pi\)
\(31\)	−2.73897	+	4.74404i	−0.0883540	+	0.153034i	−0.906816	−	0.421528i	\(-0.861494\pi\)
							0.818461	+	0.574561i	\(0.194827\pi\)
\(37\)			6.22403i			0.168217i	0.996457	+	0.0841086i	\(0.0268043\pi\)
							−0.996457	+	0.0841086i	\(0.973196\pi\)
\(41\)	25.4259	−	44.0389i	0.620144	−	1.07412i	−0.369315	−	0.929304i	\(-0.620408\pi\)
							0.989459	−	0.144816i	\(-0.0462591\pi\)
\(43\)	56.6414	−	32.7019i	1.31724	−	0.760509i	0.333957	−	0.942588i	\(-0.391616\pi\)
							0.983284	+	0.182079i	\(0.0582827\pi\)
\(47\)	−15.1967	−	26.3214i	−0.323334	−	0.560030i	0.657840	−	0.753158i	\(-0.271470\pi\)
							−0.981174	+	0.193127i	\(0.938137\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.h.a.229.9	✓	96
3.2	odd	2		1242.3.h.a.91.44		96
9.2	odd	6		1242.3.h.a.505.43		96
9.7	even	3	inner	414.3.h.a.367.10	yes	96
23.22	odd	2	inner	414.3.h.a.229.10	yes	96
69.68	even	2		1242.3.h.a.91.43		96
207.137	even	6		1242.3.h.a.505.44		96
207.160	odd	6	inner	414.3.h.a.367.9	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.h.a.229.9	✓	96	1.1	even	1	trivial
414.3.h.a.229.10	yes	96	23.22	odd	2	inner
414.3.h.a.367.9	yes	96	207.160	odd	6	inner
414.3.h.a.367.10	yes	96	9.7	even	3	inner
1242.3.h.a.91.43		96	69.68	even	2
1242.3.h.a.91.44		96	3.2	odd	2
1242.3.h.a.505.43		96	9.2	odd	6
1242.3.h.a.505.44		96	207.137	even	6