Embedded newform 402.2.n.b.41.7

• Label: 402.2.n.b.41.7
• Level: $402$
• Weight: $2$
• Character: 402.41
• Analytic conductor: $3.210$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 402 = 2 \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	402.n (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.20998616126\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(11\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			41.7
Character	\(\chi\)	\(=\)	402.41
Dual form			402.2.n.b.353.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.580057 + 0.814576i) q^{2} +(0.640714 + 1.60919i) q^{3} +(-0.327068 + 0.945001i) q^{4} +(-0.151342 + 1.05261i) q^{5} +(-0.939155 + 1.45533i) q^{6} +(-0.0483075 - 0.505899i) q^{7} +(-0.959493 + 0.281733i) q^{8} +(-2.17897 + 2.06206i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 11 q^{2} - q^{3} + 11 q^{4} - q^{6} + 6 q^{7} - 22 q^{8} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/402\mathbb{Z}\right)^\times\).

\(n\)	\(269\)	\(337\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{53}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.580057	+	0.814576i	0.410162	+	0.575992i
							\(3\)	0.640714	+	1.60919i	0.369916	+	0.929065i
\(5\)	−0.151342	+	1.05261i	−0.0676823	+	0.470741i								0.927589	+	0.373603i	\(0.121878\pi\)
							−0.995271	+	0.0971376i	\(0.969031\pi\)
\(7\)	−0.0483075	−	0.505899i	−0.0182585	−	0.191212i	−1.00000		0.000989375i	\(-0.999685\pi\)
							0.981741	−	0.190223i	\(-0.0609210\pi\)
\(11\)	−5.32299	+	2.13100i	−1.60494	+	0.642522i	−0.989352	−	0.145545i	\(-0.953507\pi\)
							−0.615590	+	0.788066i	\(0.711082\pi\)
\(13\)	−0.170146	−	0.178443i	−0.0471899	−	0.0494913i	0.699716	−	0.714421i	\(-0.253310\pi\)
							−0.746906	+	0.664930i	\(0.768461\pi\)
\(17\)	4.56852	−	1.58118i	1.10803	−	0.383493i	0.289049	−	0.957314i	\(-0.406661\pi\)
							0.818980	+	0.573822i	\(0.194540\pi\)
\(19\)	8.31170	+	0.793671i	1.90683	+	0.182081i	0.980754	−	0.195246i	\(-0.0625504\pi\)
							0.926080	+	0.377326i	\(0.123156\pi\)
\(23\)	1.09543	−	0.0521816i	0.228412	−	0.0108806i	0.0669363	−	0.997757i	\(-0.478678\pi\)
							0.161476	+	0.986877i	\(0.448375\pi\)
\(29\)	−3.79487	+	2.19097i	−0.704689	+	0.406853i	−0.809092	−	0.587683i	\(-0.800040\pi\)
							0.104402	+	0.994535i	\(0.466707\pi\)
\(31\)	−2.89505	+	3.03624i	−0.519966	+	0.545325i	−0.930391	−	0.366570i	\(-0.880532\pi\)
							0.410424	+	0.911895i	\(0.365381\pi\)
\(37\)	−0.714649	+	1.23781i	−0.117488	+	0.203494i	−0.918771	−	0.394790i	\(-0.870817\pi\)
							0.801284	+	0.598284i	\(0.204151\pi\)
\(41\)	0.942326	+	0.181618i	0.147167	+	0.0283640i	0.262302	−	0.964986i	\(-0.415518\pi\)
							−0.115136	+	0.993350i	\(0.536730\pi\)
\(43\)	−2.99599	−	2.59604i	−0.456885	−	0.395893i	0.395785	−	0.918343i	\(-0.370473\pi\)
							−0.852670	+	0.522450i	\(0.825018\pi\)
\(47\)	3.30063	−	6.40233i	0.481446	−	0.933875i	−0.515582	−	0.856840i	\(-0.672424\pi\)
							0.997028	−	0.0770351i	\(-0.0245453\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.n.b.41.7	yes	220
3.2	odd	2		402.2.n.a.41.9	✓	220
67.18	odd	66		402.2.n.a.353.9	yes	220
201.152	even	66	inner	402.2.n.b.353.7	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.n.a.41.9	✓	220	3.2	odd	2
402.2.n.a.353.9	yes	220	67.18	odd	66
402.2.n.b.41.7	yes	220	1.1	even	1	trivial
402.2.n.b.353.7	yes	220	201.152	even	66	inner