Embedded newform 414.3.h.a.229.15

• Label: 414.3.h.a.229.15
• 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			229.15
Character	\(\chi\)	\(=\)	414.229
Dual form			414.3.h.a.367.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(1.39491 + 2.65598i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(-3.95291 - 2.28222i) q^{5} +(2.26655 - 3.58647i) q^{6} +(2.69328 - 1.55496i) q^{7} +2.82843 q^{8} +(-5.10845 + 7.40971i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 4 q^{3} - 96 q^{4} + 16 q^{6} + 36 q^{9}+O(q^{10}) \)

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
							\(3\)	1.39491	+	2.65598i	0.464970	+	0.885326i
\(5\)	−3.95291	−	2.28222i	−0.790583	−	0.456443i								0.0495850	−	0.998770i	\(-0.484210\pi\)
							−0.840168	+	0.542327i	\(0.817543\pi\)
\(7\)	2.69328	−	1.55496i	0.384754	−	0.222138i	−0.295131	−	0.955457i	\(-0.595363\pi\)
							0.679885	+	0.733319i	\(0.262030\pi\)
\(11\)	14.1573	−	8.17374i	1.28703	−	0.743067i	0.308906	−	0.951092i	\(-0.400037\pi\)
							0.978123	+	0.208025i	\(0.0667037\pi\)
\(13\)	−7.80066	+	13.5111i	−0.600051	+	1.03932i	0.392762	+	0.919640i	\(0.371520\pi\)
							−0.992813	+	0.119678i	\(0.961814\pi\)
\(17\)			20.1716i			1.18657i	0.804994	+	0.593283i	\(0.202169\pi\)
							−0.804994	+	0.593283i	\(0.797831\pi\)
\(19\)			4.41674i			0.232460i	0.993222	+	0.116230i	\(0.0370810\pi\)
							−0.993222	+	0.116230i	\(0.962919\pi\)
\(23\)	21.2757	−	8.73761i	0.925029	−	0.379896i
							\(29\)	20.9896	+	36.3550i	0.723778	+	1.25362i	0.959475	+	0.281794i	\(0.0909296\pi\)
−0.235696	+	0.971827i	\(0.575737\pi\)
\(31\)	−8.32251	+	14.4150i	−0.268468	+	0.465000i	−0.968466	−	0.249144i	\(-0.919851\pi\)
							0.699998	+	0.714144i	\(0.253184\pi\)
\(37\)			72.1437i			1.94983i	0.222577	+	0.974915i	\(0.428553\pi\)
							−0.222577	+	0.974915i	\(0.571447\pi\)
\(41\)	−13.6218	+	23.5937i	−0.332240	+	0.575456i	−0.982951	−	0.183870i	\(-0.941138\pi\)
							0.650711	+	0.759325i	\(0.274471\pi\)
\(43\)	29.2445	−	16.8843i	0.680105	−	0.392659i	−0.119790	−	0.992799i	\(-0.538222\pi\)
							0.799895	+	0.600140i	\(0.204889\pi\)
\(47\)	39.1124	+	67.7446i	0.832178	+	1.44138i	0.896307	+	0.443433i	\(0.146240\pi\)
							−0.0641290	+	0.997942i	\(0.520427\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.h.a.229.15	✓	96
3.2	odd	2		1242.3.h.a.91.38		96
9.2	odd	6		1242.3.h.a.505.37		96
9.7	even	3	inner	414.3.h.a.367.16	yes	96
23.22	odd	2	inner	414.3.h.a.229.16	yes	96
69.68	even	2		1242.3.h.a.91.37		96
207.137	even	6		1242.3.h.a.505.38		96
207.160	odd	6	inner	414.3.h.a.367.15	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.h.a.229.15	✓	96	1.1	even	1	trivial
414.3.h.a.229.16	yes	96	23.22	odd	2	inner
414.3.h.a.367.15	yes	96	207.160	odd	6	inner
414.3.h.a.367.16	yes	96	9.7	even	3	inner
1242.3.h.a.91.37		96	69.68	even	2
1242.3.h.a.91.38		96	3.2	odd	2
1242.3.h.a.505.37		96	9.2	odd	6
1242.3.h.a.505.38		96	207.137	even	6