Embedded newform 334.2.c.a.3.3

• Label: 334.2.c.a.3.3
• Level: $334$
• Weight: $2$
• Character: 334.3
• Analytic conductor: $2.667$
• Analytic rank: $0$
• Dimension: $574$
• CM: no
• 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			3.3
Character	\(\chi\)	\(=\)	334.3
Dual form			334.2.c.a.223.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.455676 + 0.890146i) q^{2} +(-0.738577 - 0.646685i) q^{3} +(-0.584719 - 0.811236i) q^{4} +(-0.829988 + 3.92916i) q^{5} +(0.912196 - 0.362762i) q^{6} +(-1.00704 + 0.642297i) q^{7} +(0.988561 - 0.150824i) q^{8} +(-0.268974 - 2.01846i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 574 q - 7 q^{2} - 2 q^{3} - 7 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{7} - 7 q^{8} - 17 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{47}{83}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.455676	+	0.890146i	−0.322212	+	0.629428i
							\(3\)	−0.738577	−	0.646685i	−0.426418	−	0.373364i	0.418539	−	0.908199i	\(-0.362542\pi\)
−0.844956	+	0.534835i	\(0.820374\pi\)
\(5\)	−0.829988	+	3.92916i	−0.371182	+	1.75717i	0.244956	+	0.969534i	\(0.421226\pi\)
							−0.616138	+	0.787638i	\(0.711304\pi\)
\(7\)	−1.00704	+	0.642297i	−0.380624	+	0.242766i	−0.714176	−	0.699966i	\(-0.753198\pi\)
							0.333552	+	0.942732i	\(0.391753\pi\)
\(11\)	−1.30074	+	0.635091i	−0.392188	+	0.191487i	−0.624269	−	0.781209i	\(-0.714603\pi\)
							0.232081	+	0.972696i	\(0.425446\pi\)
\(13\)	−2.55505	−	4.17834i	−0.708644	−	1.15886i	−0.980499	−	0.196524i	\(-0.937035\pi\)
							0.271855	−	0.962338i	\(-0.412363\pi\)
\(17\)	3.85229	+	0.145881i	0.934316	+	0.0353813i	0.500610	−	0.865673i	\(-0.333109\pi\)
							0.433706	+	0.901054i	\(0.357205\pi\)
\(19\)	−5.94906	+	3.18889i	−1.36481	+	0.731582i	−0.981396	−	0.191995i	\(-0.938504\pi\)
							−0.383412	+	0.923577i	\(0.625251\pi\)
\(23\)	−7.53281	−	1.44287i	−1.57070	−	0.300859i	−0.672381	−	0.740205i	\(-0.734728\pi\)
							−0.898318	+	0.439345i	\(0.855210\pi\)
\(29\)	−1.64926	+	0.315908i	−0.306260	+	0.0586626i	−0.338954	−	0.940803i	\(-0.610073\pi\)
							0.0326940	+	0.999465i	\(0.489591\pi\)
\(31\)	−3.41389	+	0.389328i	−0.613154	+	0.0699253i	−0.414355	−	0.910115i	\(-0.635993\pi\)
							−0.198798	+	0.980040i	\(0.563704\pi\)
\(37\)	0.584543	−	4.38657i	0.0960982	−	0.721148i	−0.875281	−	0.483615i	\(-0.839324\pi\)
							0.971379	−	0.237534i	\(-0.0763390\pi\)
\(41\)	11.1362	−	2.57345i	1.73918	−	0.401906i	0.768035	−	0.640408i	\(-0.221235\pi\)
							0.971142	+	0.238502i	\(0.0766564\pi\)
\(43\)	7.79899	+	8.57424i	1.18934	+	1.30756i	0.940810	+	0.338934i	\(0.110066\pi\)
							0.248525	+	0.968625i	\(0.420054\pi\)
\(47\)	2.65606	−	2.70681i	0.387427	−	0.394829i	−0.492209	−	0.870477i	\(-0.663810\pi\)
							0.879636	+	0.475648i	\(0.157786\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	334.2.c.a.3.3	✓	574
167.56	even	83	inner	334.2.c.a.223.3	yes	574

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
334.2.c.a.3.3	✓	574	1.1	even	1	trivial
334.2.c.a.223.3	yes	574	167.56	even	83	inner