Embedded newform 335.2.w.a.2.5

• Label: 335.2.w.a.2.5
• Level: $335$
• Weight: $2$
• Character: 335.2
• Analytic conductor: $2.675$
• Analytic rank: $0$
• Dimension: $1280$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 335 = 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	335.w (of order \(132\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.67498846771\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(32\) over \(\Q(\zeta_{132})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{132}]$

Embedding invariants

Embedding label			2.5
Character	\(\chi\)	\(=\)	335.2
Dual form			335.2.w.a.168.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50050 - 1.65060i) q^{2} +(-0.0676603 - 0.123911i) q^{3} +(-0.282886 + 2.96252i) q^{4} +(-1.01373 - 1.99308i) q^{5} +(-0.103003 + 0.297608i) q^{6} +(-0.545203 + 1.70554i) q^{7} +(1.74289 - 1.30471i) q^{8} +(1.61115 - 2.50699i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 38 q^{2} - 44 q^{3} - 44 q^{5} - 80 q^{6} - 38 q^{7} - 110 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/335\mathbb{Z}\right)^\times\).

\(n\)	\(136\)	\(202\)
\(\chi(n)\)	\(e\left(\frac{1}{66}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50050	−	1.65060i	−1.06101	−	1.16715i	−0.985442	−	0.170013i	\(-0.945619\pi\)
							−0.0755715	−	0.997140i	\(-0.524078\pi\)
\(3\)	−0.0676603	−	0.123911i	−0.0390637	−	0.0715398i	0.857418	−	0.514621i	\(-0.172067\pi\)
							−0.896482	+	0.443081i	\(0.853885\pi\)
\(5\)	−1.01373	−	1.99308i	−0.453352	−	0.891332i
							\(7\)	−0.545203	+	1.70554i	−0.206067	+	0.644634i	0.793292	+	0.608841i	\(0.208365\pi\)
−0.999359	+	0.0357930i	\(0.988604\pi\)
\(11\)	−3.13997	+	1.08676i	−0.946737	+	0.327669i	−0.756444	−	0.654058i	\(-0.773065\pi\)
							−0.190293	+	0.981727i	\(0.560944\pi\)
\(13\)	−5.87453	−	0.702386i	−1.62930	−	0.194807i	−0.746777	−	0.665074i	\(-0.768400\pi\)
							−0.882524	+	0.470267i	\(0.844158\pi\)
\(17\)	3.52215	+	2.90813i	0.854248	+	0.705326i	0.957221	−	0.289358i	\(-0.0934417\pi\)
							−0.102973	+	0.994684i	\(0.532836\pi\)
\(19\)	−2.07317	+	4.02139i	−0.475618	+	0.922571i	0.521957	+	0.852972i	\(0.325202\pi\)
							−0.997575	+	0.0695990i	\(0.977828\pi\)
\(23\)	−6.15134	−	3.74944i	−1.28264	−	0.781811i	−0.297694	−	0.954661i	\(-0.596218\pi\)
							−0.984948	+	0.172850i	\(0.944702\pi\)
\(29\)	6.22139	+	3.59192i	1.15528	+	0.667003i	0.950169	−	0.311736i	\(-0.100910\pi\)
							0.205114	+	0.978738i	\(0.434244\pi\)
\(31\)	−3.18042	+	4.04424i	−0.571221	+	0.726366i	−0.982731	−	0.185039i	\(-0.940759\pi\)
							0.411510	+	0.911405i	\(0.365001\pi\)
\(37\)	−0.974954	+	3.63858i	−0.160281	+	0.598178i	0.838314	+	0.545188i	\(0.183542\pi\)
							−0.998595	+	0.0529900i	\(0.983125\pi\)
\(41\)	10.4223	−	7.42167i	1.62769	−	1.15907i	0.789240	−	0.614085i	\(-0.210475\pi\)
							0.838446	−	0.544985i	\(-0.183465\pi\)
\(43\)	−3.79075	+	1.41388i	−0.578085	+	0.215614i	−0.621462	−	0.783444i	\(-0.713461\pi\)
							0.0433773	+	0.999059i	\(0.486188\pi\)
\(47\)	−4.41495	−	0.105095i	−0.643987	−	0.0153298i	−0.299464	−	0.954108i	\(-0.596808\pi\)
							−0.344523	+	0.938778i	\(0.611959\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	335.2.w.a.2.5	✓	1280
5.3	odd	4	inner	335.2.w.a.203.5	yes	1280
67.34	odd	66	inner	335.2.w.a.302.5	yes	1280
335.168	even	132	inner	335.2.w.a.168.5	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
335.2.w.a.2.5	✓	1280	1.1	even	1	trivial
335.2.w.a.168.5	yes	1280	335.168	even	132	inner
335.2.w.a.203.5	yes	1280	5.3	odd	4	inner
335.2.w.a.302.5	yes	1280	67.34	odd	66	inner