Embedded newform 350.2.q.c.81.5

• Label: 350.2.q.c.81.5
• Level: $350$
• Weight: $2$
• Character: 350.81
• 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			81.5
Character	\(\chi\)	\(=\)	350.81
Dual form			350.2.q.c.121.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.669131 - 0.743145i) q^{2} +(-0.0249244 - 0.237139i) q^{3} +(-0.104528 - 0.994522i) q^{4} +(1.61270 + 1.54893i) q^{5} +(-0.192907 - 0.140155i) q^{6} +(-2.26242 + 1.37165i) q^{7} +(-0.809017 - 0.587785i) q^{8} +(2.87883 - 0.611914i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 10 q^{2} + 2 q^{3} + 10 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 20 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	−0.0249244	−	0.237139i	−0.0143901	−	0.136912i	0.984968	−	0.172737i	\(-0.0552610\pi\)
−0.999358	+	0.0358243i	\(0.988594\pi\)
\(5\)	1.61270	+	1.54893i	0.721223	+	0.692703i
							\(7\)	−2.26242	+	1.37165i	−0.855116	+	0.518437i
\(11\)	5.63574	+	1.19791i	1.69924	+	0.361185i								0.952643	−	0.304091i	\(-0.0983527\pi\)
							0.746598	+	0.665276i	\(0.231686\pi\)
\(13\)	0.986722	−	3.03682i	0.273667	−	0.842262i	−0.715901	−	0.698201i	\(-0.753984\pi\)
							0.989569	−	0.144060i	\(-0.0460160\pi\)
\(17\)	−1.55720	+	0.693309i	−0.377676	+	0.168152i	−0.586790	−	0.809739i	\(-0.699609\pi\)
							0.209114	+	0.977891i	\(0.432942\pi\)
\(19\)	0.814650	−	7.75087i	0.186893	−	1.77817i	−0.352199	−	0.935925i	\(-0.614566\pi\)
							0.539092	−	0.842247i	\(-0.318768\pi\)
\(23\)	−4.76166	+	5.28836i	−0.992875	+	1.10270i	0.00183751	+	0.999998i	\(0.499415\pi\)
							−0.994712	+	0.102701i	\(0.967252\pi\)
\(29\)	−0.290166	+	0.210818i	−0.0538825	+	0.0391479i	−0.614400	−	0.788994i	\(-0.710602\pi\)
							0.560518	+	0.828142i	\(0.310602\pi\)
\(31\)	−9.00089	+	4.00745i	−1.61661	+	0.719760i	−0.997832	−	0.0658083i	\(-0.979037\pi\)
							−0.618775	+	0.785568i	\(0.712371\pi\)
\(37\)	6.57298	−	1.39713i	1.08059	−	0.229687i	0.366980	−	0.930229i	\(-0.380392\pi\)
							0.713611	+	0.700542i	\(0.247059\pi\)
\(41\)	−0.119164	+	0.366750i	−0.0186103	+	0.0572767i	−0.959930	−	0.280239i	\(-0.909586\pi\)
							0.941320	+	0.337515i	\(0.109586\pi\)
\(43\)	−1.98306			−0.302414			−0.151207	−	0.988502i	\(-0.548316\pi\)
							−0.151207	+	0.988502i	\(0.548316\pi\)
\(47\)	3.32496	+	1.48037i	0.484996	+	0.215934i	0.634643	−	0.772805i	\(-0.281147\pi\)
							−0.149647	+	0.988739i	\(0.547814\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.c.81.5	✓	80
7.2	even	3	inner	350.2.q.c.331.6	yes	80
25.21	even	5	inner	350.2.q.c.221.6	yes	80
175.121	even	15	inner	350.2.q.c.121.5	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.c.81.5	✓	80	1.1	even	1	trivial
350.2.q.c.121.5	yes	80	175.121	even	15	inner
350.2.q.c.221.6	yes	80	25.21	even	5	inner
350.2.q.c.331.6	yes	80	7.2	even	3	inner