Embedded newform 334.2.c.b.107.6

• Label: 334.2.c.b.107.6
• Level: $334$
• Weight: $2$
• Character: 334.107
• Analytic conductor: $2.667$
• Analytic rank: $0$
• Dimension: $574$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 334 = 2 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	334.c (of order \(83\), degree \(82\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.66700342751\)
Analytic rank:	\(0\)
Dimension:	\(574\)
Relative dimension:	\(7\) over \(\Q(\zeta_{83})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{83}]$

Embedding invariants

Embedding label			107.6
Character	\(\chi\)	\(=\)	334.107
Dual form			334.2.c.b.231.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.862854 - 0.505453i) q^{2} +(2.01599 + 0.629819i) q^{3} +(0.489034 + 0.872265i) q^{4} +(-0.529034 + 3.07589i) q^{5} +(-1.42116 - 1.56243i) q^{6} +(0.204665 + 0.614038i) q^{7} +(0.0189241 - 0.999821i) q^{8} +(1.20108 + 0.831633i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 574 q + 7 q^{2} + 2 q^{3} - 7 q^{4} - 2 q^{6} + 6 q^{7} + 7 q^{8} - 9 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/334\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{46}{83}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.862854	−	0.505453i	−0.610130	−	0.357409i
							\(3\)	2.01599	+	0.629819i	1.16393	+	0.363626i	0.818327	−	0.574753i	\(-0.194902\pi\)
0.345607	+	0.938379i	\(0.387673\pi\)
\(5\)	−0.529034	+	3.07589i	−0.236591	+	1.37558i	0.587547	+	0.809190i	\(0.300094\pi\)
							−0.824138	+	0.566389i	\(0.808340\pi\)
\(7\)	0.204665	+	0.614038i	0.0773560	+	0.232084i	0.980244	−	0.197791i	\(-0.0633767\pi\)
							−0.902888	+	0.429875i	\(0.858557\pi\)
\(11\)	−0.0568074	+	0.999482i	−0.0171281	+	0.301355i	0.978389	+	0.206773i	\(0.0662961\pi\)
							−0.995517	+	0.0945823i	\(0.969848\pi\)
\(13\)	−0.853048	−	0.231460i	−0.236593	−	0.0641954i	0.141593	−	0.989925i	\(-0.454777\pi\)
							−0.378186	+	0.925729i	\(0.623452\pi\)
\(17\)	1.41912	+	3.38073i	0.344187	+	0.819947i	0.997995	+	0.0632864i	\(0.0201582\pi\)
							−0.653808	+	0.756660i	\(0.726830\pi\)
\(19\)	−1.39474	−	0.680987i	−0.319976	−	0.156229i	0.271751	−	0.962367i	\(-0.412397\pi\)
							−0.591727	+	0.806138i	\(0.701554\pi\)
\(23\)	−0.464702	+	0.205495i	−0.0968971	+	0.0428487i	−0.452311	−	0.891860i	\(-0.649400\pi\)
							0.355414	+	0.934709i	\(0.384340\pi\)
\(29\)	3.61167	+	1.59711i	0.670670	+	0.296576i	0.711599	−	0.702586i	\(-0.247971\pi\)
							−0.0409285	+	0.999162i	\(0.513032\pi\)
\(31\)	4.16491	−	1.65630i	0.748040	−	0.297481i	0.0357446	−	0.999361i	\(-0.488620\pi\)
							0.712295	+	0.701880i	\(0.247656\pi\)
\(37\)	7.96998	−	5.51843i	1.31026	−	0.907224i	0.311188	−	0.950348i	\(-0.399273\pi\)
							0.999070	+	0.0431240i	\(0.0137311\pi\)
\(41\)	−1.57361	+	1.48671i	−0.245756	+	0.232185i	−0.799746	−	0.600339i	\(-0.795032\pi\)
							0.553989	+	0.832524i	\(0.313105\pi\)
\(43\)	−2.84109	−	2.30364i	−0.433262	−	0.351302i	0.388189	−	0.921580i	\(-0.373101\pi\)
							−0.821452	+	0.570278i	\(0.806836\pi\)
\(47\)	5.87967	+	3.15170i	0.857639	+	0.459722i	0.841570	−	0.540148i	\(-0.181632\pi\)
							0.0160688	+	0.999871i	\(0.494885\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	334.2.c.b.107.6	✓	574
167.64	even	83	inner	334.2.c.b.231.6	yes	574

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
334.2.c.b.107.6	✓	574	1.1	even	1	trivial
334.2.c.b.231.6	yes	574	167.64	even	83	inner