Embedded newform 483.2.y.a.340.5

• Label: 483.2.y.a.340.5
• Level: $483$
• Weight: $2$
• Character: 483.340
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.y (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(16\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			340.5
Character	\(\chi\)	\(=\)	483.340
Dual form			483.2.y.a.331.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.340703 - 1.40440i) q^{2} +(0.327068 + 0.945001i) q^{3} +(-0.0785819 + 0.0405118i) q^{4} +(-1.83745 + 0.735603i) q^{5} +(1.21572 - 0.781298i) q^{6} +(-2.55604 - 0.683109i) q^{7} +(-1.80905 - 2.08776i) q^{8} +(-0.786053 + 0.618159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.340703	−	1.40440i	−0.240913	−	0.993059i	−0.955847	−	0.293866i	\(-0.905058\pi\)
							0.714933	−	0.699193i	\(-0.246457\pi\)
\(3\)	0.327068	+	0.945001i	0.188833	+	0.545596i
							\(5\)	−1.83745	+	0.735603i	−0.821731	+	0.328971i	−0.744148	−	0.668014i	\(-0.767144\pi\)
−0.0775827	+	0.996986i	\(0.524720\pi\)
\(7\)	−2.55604	−	0.683109i	−0.966094	−	0.258191i
							\(11\)	−1.43363	+	5.90951i	−0.432256	+	1.78178i	0.172110	+	0.985078i	\(0.444942\pi\)
−0.604366	+	0.796707i	\(0.706574\pi\)
\(13\)	0.301382	−	0.659934i	0.0835882	−	0.183033i	−0.863223	−	0.504824i	\(-0.831558\pi\)
							0.946811	+	0.321791i	\(0.104285\pi\)
\(17\)	0.167277	+	3.51159i	0.0405707	+	0.851684i	0.923831	+	0.382801i	\(0.125041\pi\)
							−0.883260	+	0.468883i	\(0.844656\pi\)
\(19\)	0.237606	−	4.98797i	0.0545106	−	1.14432i	−0.791506	−	0.611162i	\(-0.790703\pi\)
							0.846016	−	0.533157i	\(-0.178994\pi\)
\(23\)	2.06692	+	4.32757i	0.430982	+	0.902360i
							\(29\)	−6.50864	+	4.18285i	−1.20862	+	0.776736i	−0.980428	−	0.196880i	\(-0.936919\pi\)
−0.228196	+	0.973615i	\(0.573283\pi\)
\(31\)	−8.43007	+	1.62476i	−1.51408	+	0.291816i	−0.877570	−	0.479449i	\(-0.840837\pi\)
							−0.636515	+	0.771265i	\(0.719625\pi\)
\(37\)	−4.28237	+	3.36769i	−0.704017	+	0.553645i	−0.904720	−	0.426007i	\(-0.859920\pi\)
							0.200703	+	0.979652i	\(0.435677\pi\)
\(41\)	0.950532	−	6.61109i	0.148448	−	1.03248i	−0.770313	−	0.637666i	\(-0.779900\pi\)
							0.918761	−	0.394814i	\(-0.129191\pi\)
\(43\)	5.28897	−	6.10379i	0.806560	−	0.930820i	−0.192162	−	0.981363i	\(-0.561550\pi\)
							0.998722	+	0.0505436i	\(0.0160954\pi\)
\(47\)	−0.891179	−	1.54357i	−0.129992	−	0.225153i	0.793681	−	0.608334i	\(-0.208162\pi\)
							−0.923673	+	0.383181i	\(0.874828\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.340.5	yes	320
7.2	even	3	inner	483.2.y.a.478.5	yes	320
23.9	even	11	inner	483.2.y.a.193.5	✓	320
161.9	even	33	inner	483.2.y.a.331.5	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.193.5	✓	320	23.9	even	11	inner
483.2.y.a.331.5	yes	320	161.9	even	33	inner
483.2.y.a.340.5	yes	320	1.1	even	1	trivial
483.2.y.a.478.5	yes	320	7.2	even	3	inner