Embedded newform 350.2.x.a.3.15

• Label: 350.2.x.a.3.15
• Level: $350$
• Weight: $2$
• Character: 350.3
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.x (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79476407074\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(20\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			3.15
Character	\(\chi\)	\(=\)	350.3
Dual form			350.2.x.a.117.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.544639 - 0.838671i) q^{2} +(-0.110635 + 0.288213i) q^{3} +(-0.406737 - 0.913545i) q^{4} +(1.72772 - 1.41950i) q^{5} +(0.181460 + 0.249758i) q^{6} +(2.11003 + 1.59618i) q^{7} +(-0.987688 - 0.156434i) q^{8} +(2.15861 + 1.94362i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 12 q^{5} - 8 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.544639	−	0.838671i	0.385118	−	0.593030i
							\(3\)	−0.110635	+	0.288213i	−0.0638749	+	0.166400i	−0.961803	−	0.273742i	\(-0.911739\pi\)
0.897928	+	0.440142i	\(0.145072\pi\)
\(5\)	1.72772	−	1.41950i	0.772660	−	0.634820i
							\(7\)	2.11003	+	1.59618i	0.797515	+	0.603300i
\(11\)	−2.15154	−	2.38953i	−0.648714	−	0.720470i								0.325639	−	0.945494i	\(-0.394420\pi\)
							−0.974353	+	0.225025i	\(0.927754\pi\)
\(13\)	−0.507674	−	0.258673i	−0.140803	−	0.0717430i	0.382169	−	0.924092i	\(-0.375177\pi\)
							−0.522973	+	0.852349i	\(0.675177\pi\)
\(17\)	1.15754	−	1.42945i	0.280746	−	0.346692i	−0.617194	−	0.786811i	\(-0.711731\pi\)
							0.897939	+	0.440119i	\(0.145064\pi\)
\(19\)	1.32097	+	0.588135i	0.303052	+	0.134927i	0.552629	−	0.833428i	\(-0.313625\pi\)
							−0.249577	+	0.968355i	\(0.580291\pi\)
\(23\)	−1.90194	−	1.23514i	−0.396582	−	0.257544i	0.330906	−	0.943664i	\(-0.392646\pi\)
							−0.727488	+	0.686120i	\(0.759312\pi\)
\(29\)	4.15672	−	5.72124i	0.771884	−	1.06241i	−0.224248	−	0.974532i	\(-0.571993\pi\)
							0.996132	−	0.0878745i	\(-0.0280075\pi\)
\(31\)	−4.53321	+	0.476460i	−0.814189	+	0.0855747i	−0.502465	−	0.864598i	\(-0.667573\pi\)
							−0.311725	+	0.950173i	\(0.600907\pi\)
\(37\)	0.402798	+	7.68585i	0.0662196	+	1.26355i	0.806495	+	0.591241i	\(0.201362\pi\)
							−0.740275	+	0.672304i	\(0.765305\pi\)
\(41\)	−8.64819	+	2.80997i	−1.35062	+	0.438843i	−0.892901	−	0.450254i	\(-0.851333\pi\)
							−0.457719	+	0.889097i	\(0.651333\pi\)
\(43\)	3.20787	+	3.20787i	0.489196	+	0.489196i	0.908052	−	0.418857i	\(-0.137569\pi\)
							−0.418857	+	0.908052i	\(0.637569\pi\)
\(47\)	−7.57945	+	6.13772i	−1.10558	+	0.895279i	−0.995127	−	0.0986030i	\(-0.968563\pi\)
							−0.110450	+	0.993882i	\(0.535229\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.x.a.3.15	✓	320
7.5	odd	6	inner	350.2.x.a.103.6	yes	320
25.17	odd	20	inner	350.2.x.a.17.6	yes	320
175.117	even	60	inner	350.2.x.a.117.15	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.x.a.3.15	✓	320	1.1	even	1	trivial
350.2.x.a.17.6	yes	320	25.17	odd	20	inner
350.2.x.a.103.6	yes	320	7.5	odd	6	inner
350.2.x.a.117.15	yes	320	175.117	even	60	inner