Embedded newform 350.2.q.c.11.7

• Label: 350.2.q.c.11.7
• Level: $350$
• Weight: $2$
• Character: 350.11
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $80$
• 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:	\(80\)
Relative dimension:	\(10\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			11.7
Character	\(\chi\)	\(=\)	350.11
Dual form			350.2.q.c.191.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.913545 + 0.406737i) q^{2} +(1.09566 + 1.21686i) q^{3} +(0.669131 + 0.743145i) q^{4} +(-1.13462 + 1.92682i) q^{5} +(0.505998 + 1.55730i) q^{6} +(-2.24539 - 1.39937i) q^{7} +(0.309017 + 0.951057i) q^{8} +(0.0333212 - 0.317030i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 10 q^{2} + 2 q^{3} + 10 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 20 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.913545	+	0.406737i	0.645974	+	0.287606i
							\(3\)	1.09566	+	1.21686i	0.632582	+	0.702553i	0.971172	−	0.238381i	\(-0.0766166\pi\)
−0.338590	+	0.940934i	\(0.609950\pi\)
\(5\)	−1.13462	+	1.92682i	−0.507417	+	0.861701i
							\(7\)	−2.24539	−	1.39937i	−0.848677	−	0.528911i
\(11\)	0.568509	+	5.40900i	0.171412	+	1.63087i								0.655035	+	0.755598i	\(0.272654\pi\)
							−0.483623	+	0.875276i	\(0.660679\pi\)
\(13\)	2.04719	+	1.48737i	0.567790	+	0.412523i	0.834302	−	0.551308i	\(-0.185871\pi\)
							−0.266512	+	0.963832i	\(0.585871\pi\)
\(17\)	4.76701	+	1.01326i	1.15617	+	0.245751i	0.745801	−	0.666169i	\(-0.232067\pi\)
							0.410368	+	0.911920i	\(0.365400\pi\)
\(19\)	3.43698	−	3.81716i	0.788498	−	0.875716i	−0.206204	−	0.978509i	\(-0.566111\pi\)
							0.994703	+	0.102793i	\(0.0327778\pi\)
\(23\)	−8.59211	−	3.82546i	−1.79158	−	0.797663i	−0.975591	−	0.219594i	\(-0.929527\pi\)
							−0.815988	−	0.578069i	\(-0.803807\pi\)
\(29\)	−0.602186	+	1.85334i	−0.111823	+	0.344156i	−0.991271	−	0.131839i	\(-0.957912\pi\)
							0.879448	+	0.475995i	\(0.157912\pi\)
\(31\)	1.25504	+	0.266767i	0.225412	+	0.0479127i	0.319233	−	0.947676i	\(-0.396575\pi\)
							−0.0938211	+	0.995589i	\(0.529908\pi\)
\(37\)	1.07701	−	10.2471i	0.177060	−	1.68461i	−0.440238	−	0.897881i	\(-0.645106\pi\)
							0.617298	−	0.786729i	\(-0.288227\pi\)
\(41\)	−1.86518	−	1.35513i	−0.291292	−	0.211636i	0.432536	−	0.901617i	\(-0.357619\pi\)
							−0.723828	+	0.689981i	\(0.757619\pi\)
\(43\)	7.73020			1.17884			0.589422	−	0.807825i	\(-0.299356\pi\)
							0.589422	+	0.807825i	\(0.299356\pi\)
\(47\)	8.49818	−	1.80634i	1.23959	−	0.263482i	0.458947	−	0.888464i	\(-0.348227\pi\)
							0.780640	+	0.624981i	\(0.214893\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.c.11.7	✓	80
7.2	even	3	inner	350.2.q.c.261.4	yes	80
25.16	even	5	inner	350.2.q.c.291.4	yes	80
175.16	even	15	inner	350.2.q.c.191.7	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.c.11.7	✓	80	1.1	even	1	trivial
350.2.q.c.191.7	yes	80	175.16	even	15	inner
350.2.q.c.261.4	yes	80	7.2	even	3	inner
350.2.q.c.291.4	yes	80	25.16	even	5	inner