Embedded newform 350.2.u.a.79.2

• Label: 350.2.u.a.79.2
• Level: $350$
• Weight: $2$
• Character: 350.79
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.u (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79476407074\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(20\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			79.2
Character	\(\chi\)	\(=\)	350.79
Dual form			350.2.u.a.319.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.743145 + 0.669131i) q^{2} +(-2.72974 - 0.286907i) q^{3} +(0.104528 - 0.994522i) q^{4} +(-1.09713 - 1.94841i) q^{5} +(2.22057 - 1.61334i) q^{6} +(-2.36624 - 1.18359i) q^{7} +(0.587785 + 0.809017i) q^{8} +(4.43471 + 0.942627i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 20 q^{4} - 2 q^{5} + 8 q^{6} - 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.743145	+	0.669131i	−0.525483	+	0.473147i
							\(3\)	−2.72974	−	0.286907i	−1.57601	−	0.165646i	−0.724238	−	0.689550i	\(-0.757808\pi\)
−0.851777	+	0.523904i	\(0.824475\pi\)
\(5\)	−1.09713	−	1.94841i	−0.490653	−	0.871355i
							\(7\)	−2.36624	−	1.18359i	−0.894356	−	0.447355i
\(11\)	0.0371853	−	0.00790397i	0.0112118	−	0.00238314i								−0.202302	−	0.979323i	\(-0.564842\pi\)
							0.213514	+	0.976940i	\(0.431509\pi\)
\(13\)	−0.508323	+	0.165164i	−0.140983	+	0.0458083i	−0.378659	−	0.925536i	\(-0.623615\pi\)
							0.237675	+	0.971345i	\(0.423615\pi\)
\(17\)	0.595954	−	1.33853i	0.144540	−	0.324642i	−0.826739	−	0.562586i	\(-0.809807\pi\)
							0.971279	+	0.237943i	\(0.0764733\pi\)
\(19\)	0.697662	+	6.63781i	0.160055	+	1.52282i	0.719816	+	0.694165i	\(0.244226\pi\)
							−0.559762	+	0.828654i	\(0.689107\pi\)
\(23\)	−1.53167	+	1.37913i	−0.319376	+	0.287567i	−0.813188	−	0.582001i	\(-0.802270\pi\)
							0.493812	+	0.869569i	\(0.335603\pi\)
\(29\)	7.25081	+	5.26802i	1.34644	+	0.978247i	0.999180	+	0.0404790i	\(0.0128884\pi\)
							0.347262	+	0.937768i	\(0.387112\pi\)
\(31\)	−7.32160	−	3.25979i	−1.31500	−	0.585475i	−0.375117	−	0.926978i	\(-0.622397\pi\)
							−0.939881	+	0.341503i	\(0.889064\pi\)
\(37\)	−0.801869	+	3.77250i	−0.131827	+	0.620195i	0.861779	+	0.507284i	\(0.169351\pi\)
							−0.993605	+	0.112910i	\(0.963983\pi\)
\(41\)	−3.42245	−	10.5332i	−0.534498	−	1.64501i	−0.744732	−	0.667364i	\(-0.767423\pi\)
							0.210234	−	0.977651i	\(-0.432577\pi\)
\(43\)			4.65007i			0.709128i	0.935032	+	0.354564i	\(0.115371\pi\)
							−0.935032	+	0.354564i	\(0.884629\pi\)
\(47\)	3.65030	+	8.19872i	0.532452	+	1.19591i	0.956890	+	0.290449i	\(0.0938048\pi\)
							−0.424439	+	0.905457i	\(0.639529\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.u.a.79.2	✓	160
7.4	even	3	inner	350.2.u.a.179.9	yes	160
25.19	even	10	inner	350.2.u.a.219.9	yes	160
175.144	even	30	inner	350.2.u.a.319.2	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.u.a.79.2	✓	160	1.1	even	1	trivial
350.2.u.a.179.9	yes	160	7.4	even	3	inner
350.2.u.a.219.9	yes	160	25.19	even	10	inner
350.2.u.a.319.2	yes	160	175.144	even	30	inner