Embedded newform 350.2.x.a.3.17

• Label: 350.2.x.a.3.17
• 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			3.17
Character	\(\chi\)	\(=\)	350.3
Dual form			350.2.x.a.117.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.544639 - 0.838671i) q^{2} +(0.309740 - 0.806901i) q^{3} +(-0.406737 - 0.913545i) q^{4} +(-0.415279 - 2.19717i) q^{5} +(-0.508027 - 0.699240i) q^{6} +(-1.18335 - 2.36636i) q^{7} +(-0.987688 - 0.156434i) q^{8} +(1.67428 + 1.50753i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 12 q^{5} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	0.309740	−	0.806901i	0.178829	−	0.465864i	−0.814640	−	0.579966i	\(-0.803066\pi\)
0.993469	+	0.114102i	\(0.0363990\pi\)
\(5\)	−0.415279	−	2.19717i	−0.185718	−	0.982603i
							\(7\)	−1.18335	−	2.36636i	−0.447266	−	0.894401i
\(11\)	2.22006	+	2.46563i	0.669374	+	0.743415i								0.978191	−	0.207706i	\(-0.0665998\pi\)
							−0.308817	+	0.951121i	\(0.599933\pi\)
\(13\)	−3.90765	−	1.99105i	−1.08379	−	0.552217i	−0.181517	−	0.983388i	\(-0.558101\pi\)
							−0.902270	+	0.431171i	\(0.858101\pi\)
\(17\)	−0.350873	+	0.433292i	−0.0850991	+	0.105089i	−0.817917	−	0.575336i	\(-0.804871\pi\)
							0.732818	+	0.680425i	\(0.238205\pi\)
\(19\)	−1.93453	−	0.861306i	−0.443811	−	0.197597i	0.172650	−	0.984983i	\(-0.444767\pi\)
							−0.616460	+	0.787386i	\(0.711434\pi\)
\(23\)	−0.509616	−	0.330948i	−0.106262	−	0.0690075i	0.490414	−	0.871490i	\(-0.336846\pi\)
							−0.596676	+	0.802482i	\(0.703512\pi\)
\(29\)	4.80477	−	6.61320i	0.892224	−	1.22804i	−0.0806588	−	0.996742i	\(-0.525702\pi\)
							0.972883	−	0.231299i	\(-0.0742976\pi\)
\(31\)	10.2915	−	1.08169i	1.84842	−	0.194276i	0.885603	−	0.464443i	\(-0.153745\pi\)
							0.962814	+	0.270167i	\(0.0870788\pi\)
\(37\)	−0.0777123	−	1.48284i	−0.0127758	−	0.243777i	−0.997551	−	0.0699440i	\(-0.977718\pi\)
							0.984775	−	0.173833i	\(-0.0556154\pi\)
\(41\)	7.09297	−	2.30465i	1.10774	−	0.359925i	0.302661	−	0.953098i	\(-0.402125\pi\)
							0.805075	+	0.593173i	\(0.202125\pi\)
\(43\)	6.53563	+	6.53563i	0.996674	+	0.996674i	0.999994	−	0.00332001i	\(-0.00105679\pi\)
							−0.00332001	+	0.999994i	\(0.501057\pi\)
\(47\)	−2.76644	+	2.24022i	−0.403527	+	0.326770i	−0.809505	−	0.587113i	\(-0.800264\pi\)
							0.405978	+	0.913883i	\(0.366931\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.x.a.3.17	✓	320
7.5	odd	6	inner	350.2.x.a.103.4	yes	320
25.17	odd	20	inner	350.2.x.a.17.4	yes	320
175.117	even	60	inner	350.2.x.a.117.17	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.x.a.3.17	✓	320	1.1	even	1	trivial
350.2.x.a.17.4	yes	320	25.17	odd	20	inner
350.2.x.a.103.4	yes	320	7.5	odd	6	inner
350.2.x.a.117.17	yes	320	175.117	even	60	inner