Embedded newform 605.2.r.a.32.8

• Label: 605.2.r.a.32.8
• Level: $605$
• Weight: $2$
• Character: 605.32
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $1280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.r (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(64\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			32.8
Character	\(\chi\)	\(=\)	605.32
Dual form			605.2.r.a.208.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34419 - 1.79563i) q^{2} +(-0.436724 + 0.436724i) q^{3} +(-0.853977 + 2.90838i) q^{4} +(-0.335531 + 2.21075i) q^{5} +(1.37124 + 0.197154i) q^{6} +(0.125054 - 0.574863i) q^{7} +(2.16709 - 0.808283i) q^{8} +2.61854i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 22 q^{2} - 40 q^{3} - 14 q^{5} - 44 q^{6} - 22 q^{7} - 22 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34419	−	1.79563i	−0.950489	−	1.26971i	−0.962921	−	0.269782i	\(-0.913048\pi\)
							0.0124320	−	0.999923i	\(-0.496043\pi\)
\(3\)	−0.436724	+	0.436724i	−0.252143	+	0.252143i	−0.821849	−	0.569706i	\(-0.807057\pi\)
							0.569706	+	0.821849i	\(0.307057\pi\)
\(5\)	−0.335531	+	2.21075i	−0.150054	+	0.988678i
							\(7\)	0.125054	−	0.574863i	0.0472659	−	0.217278i	−0.947456	−	0.319886i	\(-0.896355\pi\)
0.994722	+	0.102609i	\(0.0327189\pi\)
\(11\)	−3.26346	+	0.591473i	−0.983970	+	0.178336i
							\(13\)	0.565945	+	1.03645i	0.156965	+	0.287460i	0.944313	−	0.329049i	\(-0.106728\pi\)
−0.787348	+	0.616509i	\(0.788546\pi\)
\(17\)	−0.496523	−	6.94229i	−0.120424	−	1.68375i	−0.595146	−	0.803617i	\(-0.702906\pi\)
							0.474722	−	0.880136i	\(-0.342549\pi\)
\(19\)	−3.52715	−	4.07055i	−0.809184	−	0.933848i	0.189663	−	0.981849i	\(-0.439260\pi\)
							−0.998847	+	0.0480011i	\(0.984715\pi\)
\(23\)	2.80659	−	0.610536i	0.585214	−	0.127306i	0.0897961	−	0.995960i	\(-0.471378\pi\)
							0.495418	+	0.868655i	\(0.335015\pi\)
\(29\)	−3.13038	−	3.61265i	−0.581296	−	0.670852i	0.386587	−	0.922253i	\(-0.373654\pi\)
							−0.967883	+	0.251401i	\(0.919109\pi\)
\(31\)	3.28761	−	0.965329i	0.590472	−	0.173378i	0.0271720	−	0.999631i	\(-0.491350\pi\)
							0.563300	+	0.826253i	\(0.309532\pi\)
\(37\)	−7.11032	−	3.88253i	−1.16893	−	0.638284i	−0.226935	−	0.973910i	\(-0.572870\pi\)
							−0.941995	+	0.335626i	\(0.891052\pi\)
\(41\)	−11.1275	−	1.59989i	−1.73782	−	0.249861i	−0.800752	−	0.598996i	\(-0.795566\pi\)
							−0.937073	+	0.349135i	\(0.886476\pi\)
\(43\)	10.6207	−	3.96133i	1.61965	−	0.604098i	0.634064	−	0.773280i	\(-0.281385\pi\)
							0.985585	+	0.169183i	\(0.0541128\pi\)
\(47\)	−0.980325	+	1.30956i	−0.142995	+	0.191019i	−0.866321	−	0.499488i	\(-0.833522\pi\)
							0.723326	+	0.690507i	\(0.242612\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.r.a.32.8	✓	1280
5.3	odd	4	inner	605.2.r.a.153.8	yes	1280
121.87	odd	22	inner	605.2.r.a.87.8	yes	1280
605.208	even	44	inner	605.2.r.a.208.8	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.r.a.32.8	✓	1280	1.1	even	1	trivial
605.2.r.a.87.8	yes	1280	121.87	odd	22	inner
605.2.r.a.153.8	yes	1280	5.3	odd	4	inner
605.2.r.a.208.8	yes	1280	605.208	even	44	inner