Embedded newform 350.2.q.b.81.5

• Label: 350.2.q.b.81.5
• 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			81.5
Character	\(\chi\)	\(=\)	350.81
Dual form			350.2.q.b.121.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.669131 + 0.743145i) q^{2} +(0.0508913 + 0.484199i) q^{3} +(-0.104528 - 0.994522i) q^{4} +(0.866210 + 2.06148i) q^{5} +(-0.393883 - 0.286173i) q^{6} +(-1.17653 + 2.36976i) q^{7} +(0.809017 + 0.587785i) q^{8} +(2.70258 - 0.574452i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	0.0508913	+	0.484199i	0.0293821	+	0.279552i	0.999341	+	0.0362852i	\(0.0115525\pi\)
−0.969959	+	0.243267i	\(0.921781\pi\)
\(5\)	0.866210	+	2.06148i	0.387381	+	0.921920i
							\(7\)	−1.17653	+	2.36976i	−0.444688	+	0.895685i
\(11\)	−3.93885	−	0.837229i	−1.18761	−	0.252434i								−0.428588	−	0.903500i	\(-0.640989\pi\)
							−0.759021	+	0.651066i	\(0.774322\pi\)
\(13\)	−0.756436	+	2.32807i	−0.209798	+	0.645691i	0.789684	+	0.613513i	\(0.210244\pi\)
							−0.999482	+	0.0321777i	\(0.989756\pi\)
\(17\)	4.33085	−	1.92822i	1.05039	−	0.467662i	0.192390	−	0.981319i	\(-0.438376\pi\)
							0.857996	+	0.513657i	\(0.171710\pi\)
\(19\)	−0.0849856	+	0.808584i	−0.0194970	+	0.185502i	−0.999936	−	0.0113220i	\(-0.996396\pi\)
							0.980439	+	0.196824i	\(0.0630627\pi\)
\(23\)	−5.69766	+	6.32789i	−1.18804	+	1.31946i	−0.251937	+	0.967744i	\(0.581068\pi\)
							−0.936108	+	0.351714i	\(0.885599\pi\)
\(29\)	−4.60491	+	3.34566i	−0.855110	+	0.621274i	−0.926550	−	0.376171i	\(-0.877240\pi\)
							0.0714404	+	0.997445i	\(0.477240\pi\)
\(31\)	2.50198	−	1.11395i	0.449368	−	0.200072i	−0.169559	−	0.985520i	\(-0.554234\pi\)
							0.618927	+	0.785449i	\(0.287568\pi\)
\(37\)	3.24296	−	0.689312i	0.533139	−	0.113322i	0.0665276	−	0.997785i	\(-0.478808\pi\)
							0.466611	+	0.884462i	\(0.345475\pi\)
\(41\)	1.82170	−	5.60660i	0.284501	−	0.875604i	−0.702047	−	0.712131i	\(-0.747730\pi\)
							0.986548	−	0.163473i	\(-0.0522697\pi\)
\(43\)	9.45824			1.44237			0.721184	−	0.692744i	\(-0.243598\pi\)
							0.721184	+	0.692744i	\(0.243598\pi\)
\(47\)	−9.90716	−	4.41095i	−1.44511	−	0.643403i	−0.473671	−	0.880702i	\(-0.657071\pi\)
							−0.971437	+	0.237299i	\(0.923738\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.b.81.5	✓	72
7.2	even	3	inner	350.2.q.b.331.5	yes	72
25.21	even	5	inner	350.2.q.b.221.5	yes	72
175.121	even	15	inner	350.2.q.b.121.5	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.b.81.5	✓	72	1.1	even	1	trivial
350.2.q.b.121.5	yes	72	175.121	even	15	inner
350.2.q.b.221.5	yes	72	25.21	even	5	inner
350.2.q.b.331.5	yes	72	7.2	even	3	inner