Embedded newform 350.2.q.b.81.4

• Label: 350.2.q.b.81.4
• 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			81.4
Character	\(\chi\)	\(=\)	350.81
Dual form			350.2.q.b.121.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.669131 + 0.743145i) q^{2} +(-0.156346 - 1.48754i) q^{3} +(-0.104528 - 0.994522i) q^{4} +(-0.0739496 + 2.23484i) q^{5} +(1.21007 + 0.879169i) q^{6} +(-1.32245 - 2.29153i) q^{7} +(0.809017 + 0.587785i) q^{8} +(0.746121 - 0.158593i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.156346	−	1.48754i	−0.0902667	−	0.858830i	−0.942171	−	0.335132i	\(-0.891219\pi\)
0.851904	−	0.523697i	\(-0.175448\pi\)
\(5\)	−0.0739496	+	2.23484i	−0.0330713	+	0.999453i
							\(7\)	−1.32245	−	2.29153i	−0.499841	−	0.866117i
\(11\)	1.96447	+	0.417562i	0.592311	+	0.125900i								0.494308	−	0.869287i	\(-0.335421\pi\)
							0.0980026	+	0.995186i	\(0.468755\pi\)
\(13\)	1.05674	−	3.25232i	0.293087	−	0.902030i	−0.690770	−	0.723075i	\(-0.742728\pi\)
							0.983857	−	0.178956i	\(-0.0572718\pi\)
\(17\)	0.0440485	−	0.0196117i	0.0106833	−	0.00475653i	−0.401388	−	0.915908i	\(-0.631472\pi\)
							0.412071	+	0.911152i	\(0.364805\pi\)
\(19\)	0.264442	−	2.51600i	0.0606672	−	0.577209i	−0.921392	−	0.388635i	\(-0.872947\pi\)
							0.982059	−	0.188574i	\(-0.0603867\pi\)
\(23\)	4.26378	−	4.73540i	0.889059	−	0.987400i	−0.110920	−	0.993829i	\(-0.535380\pi\)
							0.999979	+	0.00642938i	\(0.00204655\pi\)
\(29\)	4.23313	−	3.07555i	0.786072	−	0.571115i	−0.120723	−	0.992686i	\(-0.538521\pi\)
							0.906795	+	0.421571i	\(0.138521\pi\)
\(31\)	4.12568	−	1.83687i	0.740994	−	0.329912i	−0.00128811	−	0.999999i	\(-0.500410\pi\)
							0.742282	+	0.670087i	\(0.233743\pi\)
\(37\)	−0.869570	+	0.184833i	−0.142956	+	0.0303863i	−0.278834	−	0.960339i	\(-0.589948\pi\)
							0.135878	+	0.990726i	\(0.456615\pi\)
\(41\)	−2.10980	+	6.49331i	−0.329496	+	1.01408i	0.639874	+	0.768480i	\(0.278986\pi\)
							−0.969370	+	0.245605i	\(0.921014\pi\)
\(43\)	4.77417			0.728055			0.364027	−	0.931388i	\(-0.381401\pi\)
							0.364027	+	0.931388i	\(0.381401\pi\)
\(47\)	−1.93562	−	0.861794i	−0.282339	−	0.125706i	0.260681	−	0.965425i	\(-0.416053\pi\)
							−0.543021	+	0.839719i	\(0.682719\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.b.81.4	✓	72
7.2	even	3	inner	350.2.q.b.331.6	yes	72
25.21	even	5	inner	350.2.q.b.221.6	yes	72
175.121	even	15	inner	350.2.q.b.121.4	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.b.81.4	✓	72	1.1	even	1	trivial
350.2.q.b.121.4	yes	72	175.121	even	15	inner
350.2.q.b.221.6	yes	72	25.21	even	5	inner
350.2.q.b.331.6	yes	72	7.2	even	3	inner