Embedded newform 414.3.o.a.29.12

• Label: 414.3.o.a.29.12
• Level: $414$
• Weight: $3$
• Character: 414.29
• Analytic conductor: $11.281$
• Analytic rank: $0$
• Dimension: $960$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.o (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.2806829445\)
Analytic rank:	\(0\)
Dimension:	\(960\)
Relative dimension:	\(48\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			29.12
Character	\(\chi\)	\(=\)	414.29
Dual form			414.3.o.a.257.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15198 + 0.820324i) q^{2} +(-0.211823 + 2.99251i) q^{3} +(0.654136 - 1.89000i) q^{4} +(-2.84452 + 5.51760i) q^{5} +(-2.21081 - 3.62109i) q^{6} +(1.88035 + 1.47873i) q^{7} +(0.796860 + 2.71386i) q^{8} +(-8.91026 - 1.26777i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 960 q + 4 q^{3} - 96 q^{4} + 36 q^{5} + 16 q^{6} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(235\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15198	+	0.820324i	−0.575992	+	0.410162i
							\(3\)	−0.211823	+	2.99251i	−0.0706078	+	0.997504i
\(5\)	−2.84452	+	5.51760i	−0.568904	+	1.10352i								0.412391	+	0.911007i	\(0.364694\pi\)
							−0.981295	+	0.192512i	\(0.938336\pi\)
\(7\)	1.88035	+	1.47873i	0.268622	+	0.211246i	0.743379	−	0.668870i	\(-0.233222\pi\)
							−0.474758	+	0.880117i	\(0.657464\pi\)
\(11\)	−1.94494	+	20.3683i	−0.176813	+	1.85166i	0.277379	+	0.960761i	\(0.410534\pi\)
							−0.454192	+	0.890904i	\(0.650072\pi\)
\(13\)	5.52072	−	4.34154i	0.424671	−	0.333965i	−0.382843	−	0.923813i	\(-0.625055\pi\)
							0.807514	+	0.589849i	\(0.200813\pi\)
\(17\)	−20.0848	−	17.4036i	−1.18146	−	1.02374i	−0.999180	−	0.0404914i	\(-0.987108\pi\)
							−0.182277	−	0.983247i	\(-0.558347\pi\)
\(19\)	17.6821	+	20.4063i	0.930639	+	1.07401i	0.997091	+	0.0762246i	\(0.0242866\pi\)
							−0.0664520	+	0.997790i	\(0.521168\pi\)
\(23\)	−5.47321	+	22.3393i	−0.237966	+	0.971274i
							\(29\)	9.57768	−	3.31487i	0.330265	−	0.114306i	−0.156903	−	0.987614i	\(-0.550151\pi\)
0.487168	+	0.873308i	\(0.338030\pi\)
\(31\)	−37.8554	−	36.0950i	−1.22114	−	1.16436i	−0.981081	−	0.193597i	\(-0.937984\pi\)
							−0.240060	−	0.970758i	\(-0.577167\pi\)
\(37\)	23.9198	+	15.3723i	0.646481	+	0.415468i	0.822379	−	0.568940i	\(-0.192646\pi\)
							−0.175898	+	0.984408i	\(0.556283\pi\)
\(41\)	23.2941	−	45.1842i	0.568148	−	1.10205i	−0.413348	−	0.910573i	\(-0.635641\pi\)
							0.981496	−	0.191481i	\(-0.0613289\pi\)
\(43\)	−0.469475	+	0.447644i	−0.0109180	+	0.0104103i	−0.695518	−	0.718508i	\(-0.744825\pi\)
							0.684600	+	0.728919i	\(0.259977\pi\)
\(47\)	25.6191	+	14.7912i	0.545088	+	0.314707i	0.747138	−	0.664669i	\(-0.231427\pi\)
							−0.202051	+	0.979375i	\(0.564761\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.o.a.29.12	✓	960
9.5	odd	6	inner	414.3.o.a.167.40	yes	960
23.4	even	11	inner	414.3.o.a.119.40	yes	960
207.50	odd	66	inner	414.3.o.a.257.12	yes	960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.o.a.29.12	✓	960	1.1	even	1	trivial
414.3.o.a.119.40	yes	960	23.4	even	11	inner
414.3.o.a.167.40	yes	960	9.5	odd	6	inner
414.3.o.a.257.12	yes	960	207.50	odd	66	inner