Embedded newform 350.2.q.b.81.8

• Label: 350.2.q.b.81.8
• Level: $350$
• Weight: $2$
• Character: 350.81
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79476407074\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(9\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			81.8
Character	\(\chi\)	\(=\)	350.81
Dual form			350.2.q.b.121.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.669131 + 0.743145i) q^{2} +(0.300861 + 2.86250i) q^{3} +(-0.104528 - 0.994522i) q^{4} +(-1.53794 - 1.62319i) q^{5} +(-2.32857 - 1.69180i) q^{6} +(0.721052 + 2.54560i) q^{7} +(0.809017 + 0.587785i) q^{8} +(-5.16894 + 1.09869i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.669131	+	0.743145i	−0.473147	+	0.525483i
							\(3\)	0.300861	+	2.86250i	0.173702	+	1.65266i	0.640248	+	0.768168i	\(0.278831\pi\)
−0.466546	+	0.884497i	\(0.654502\pi\)
\(5\)	−1.53794	−	1.62319i	−0.687787	−	0.725912i
							\(7\)	0.721052	+	2.54560i	0.272532	+	0.962147i
\(11\)	−5.75248	−	1.22273i	−1.73444	−	0.368666i								−0.771048	−	0.636777i	\(-0.780267\pi\)
							−0.963388	+	0.268111i	\(0.913601\pi\)
\(13\)	0.229989	−	0.707834i	0.0637875	−	0.196318i	−0.914084	−	0.405526i	\(-0.867089\pi\)
							0.977871	+	0.209208i	\(0.0670885\pi\)
\(17\)	−4.18653	+	1.86396i	−1.01538	+	0.452078i	−0.845835	−	0.533444i	\(-0.820897\pi\)
							−0.169548	+	0.985522i	\(0.554231\pi\)
\(19\)	−0.412180	+	3.92163i	−0.0945606	+	0.899684i	0.839690	+	0.543066i	\(0.182737\pi\)
							−0.934250	+	0.356618i	\(0.883930\pi\)
\(23\)	2.37429	−	2.63692i	0.495074	−	0.549835i	−0.442886	−	0.896578i	\(-0.646045\pi\)
							0.937960	+	0.346742i	\(0.112712\pi\)
\(29\)	3.22848	−	2.34563i	0.599514	−	0.435573i	−0.246192	−	0.969221i	\(-0.579179\pi\)
							0.845706	+	0.533648i	\(0.179179\pi\)
\(31\)	5.40596	−	2.40689i	0.970938	−	0.432290i	0.140917	−	0.990021i	\(-0.454995\pi\)
							0.830021	+	0.557732i	\(0.188328\pi\)
\(37\)	−11.6599	+	2.47840i	−1.91688	+	0.407446i	−0.916919	+	0.399074i	\(0.869332\pi\)
							−0.999965	+	0.00837269i	\(0.997335\pi\)
\(41\)	−0.385267	+	1.18573i	−0.0601686	+	0.185180i	−0.976623	−	0.214959i	\(-0.931038\pi\)
							0.916454	+	0.400139i	\(0.131038\pi\)
\(43\)	3.93625			0.600272			0.300136	−	0.953896i	\(-0.402968\pi\)
							0.300136	+	0.953896i	\(0.402968\pi\)
\(47\)	6.79072	+	3.02343i	0.990529	+	0.441012i	0.837042	−	0.547139i	\(-0.184283\pi\)
							0.153487	+	0.988151i	\(0.450950\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.b.81.8	✓	72
7.2	even	3	inner	350.2.q.b.331.2	yes	72
25.21	even	5	inner	350.2.q.b.221.2	yes	72
175.121	even	15	inner	350.2.q.b.121.8	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.b.81.8	✓	72	1.1	even	1	trivial
350.2.q.b.121.8	yes	72	175.121	even	15	inner
350.2.q.b.221.2	yes	72	25.21	even	5	inner
350.2.q.b.331.2	yes	72	7.2	even	3	inner