Embedded newform 770.2.bv.a.3.14

• Label: 770.2.bv.a.3.14
• Level: $770$
• Weight: $2$
• Character: 770.3
• Analytic conductor: $6.148$
• Analytic rank: $0$
• Dimension: $768$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	770.bv (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.14848095564\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(48\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			3.14
Character	\(\chi\)	\(=\)	770.3
Dual form			770.2.bv.a.257.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.933580 + 0.358368i) q^{2} +(0.872815 - 0.566813i) q^{3} +(0.743145 - 0.669131i) q^{4} +(-1.81587 + 1.30484i) q^{5} +(-0.611715 + 0.841954i) q^{6} +(2.44283 - 1.01617i) q^{7} +(-0.453990 + 0.891007i) q^{8} +(-0.779681 + 1.75119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q + 24 q^{5} + 4 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/770\mathbb{Z}\right)^\times\).

\(n\)	\(211\)	\(617\)	\(661\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.933580	+	0.358368i	−0.660141	+	0.253404i
							\(3\)	0.872815	−	0.566813i	0.503920	−	0.327249i	−0.267519	−	0.963553i	\(-0.586204\pi\)
0.771439	+	0.636303i	\(0.219537\pi\)
\(5\)	−1.81587	+	1.30484i	−0.812082	+	0.583543i
							\(7\)	2.44283	−	1.01617i	0.923302	−	0.384075i
\(11\)	0.157640	−	3.31288i	0.0475304	−	0.998870i
							\(13\)	0.773436	−	4.88328i	0.214512	−	1.35438i	−0.611731	−	0.791066i	\(-0.709527\pi\)
0.826244	−	0.563313i	\(-0.190473\pi\)
\(17\)	−1.81602	−	0.697106i	−0.440450	−	0.169073i	0.128029	−	0.991770i	\(-0.459135\pi\)
							−0.568479	+	0.822697i	\(0.692468\pi\)
\(19\)	2.50299	−	2.77985i	0.574225	−	0.637741i	−0.384145	−	0.923273i	\(-0.625504\pi\)
							0.958369	+	0.285532i	\(0.0921702\pi\)
\(23\)	−0.687386	+	2.56536i	−0.143330	+	0.534914i	0.856494	+	0.516157i	\(0.172638\pi\)
							−0.999824	+	0.0187575i	\(0.994029\pi\)
\(29\)	10.1435	+	3.29583i	1.88361	+	0.612021i	0.984805	+	0.173663i	\(0.0555604\pi\)
							0.898801	+	0.438357i	\(0.144440\pi\)
\(31\)	−6.07460	+	0.638466i	−1.09103	+	0.114672i	−0.632915	−	0.774221i	\(-0.718142\pi\)
							−0.458115	+	0.888893i	\(0.651475\pi\)
\(37\)	5.63921	−	8.68363i	0.927081	−	1.42758i	0.0232314	−	0.999730i	\(-0.492605\pi\)
							0.903850	−	0.427850i	\(-0.140729\pi\)
\(41\)	2.76400	−	0.898079i	0.431665	−	0.140256i	−0.0851219	−	0.996371i	\(-0.527128\pi\)
							0.516787	+	0.856114i	\(0.327128\pi\)
\(43\)	0.147745	+	0.147745i	0.0225310	+	0.0225310i	0.718283	−	0.695752i	\(-0.244929\pi\)
							−0.695752	+	0.718283i	\(0.744929\pi\)
\(47\)	11.5344	−	0.604492i	1.68246	−	0.0881742i	0.813030	−	0.582222i	\(-0.197817\pi\)
							0.869435	+	0.494048i	\(0.164483\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	770.2.bv.a.3.14	✓	768
5.2	odd	4	inner	770.2.bv.a.157.11	yes	768
7.5	odd	6	inner	770.2.bv.a.663.35	yes	768
11.4	even	5	inner	770.2.bv.a.213.11	yes	768
35.12	even	12	inner	770.2.bv.a.47.11	yes	768
55.37	odd	20	inner	770.2.bv.a.367.35	yes	768
77.26	odd	30	inner	770.2.bv.a.103.11	yes	768
385.257	even	60	inner	770.2.bv.a.257.14	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.bv.a.3.14	✓	768	1.1	even	1	trivial
770.2.bv.a.47.11	yes	768	35.12	even	12	inner
770.2.bv.a.103.11	yes	768	77.26	odd	30	inner
770.2.bv.a.157.11	yes	768	5.2	odd	4	inner
770.2.bv.a.213.11	yes	768	11.4	even	5	inner
770.2.bv.a.257.14	yes	768	385.257	even	60	inner
770.2.bv.a.367.35	yes	768	55.37	odd	20	inner
770.2.bv.a.663.35	yes	768	7.5	odd	6	inner