Embedded newform 350.2.q.c.81.8

• Label: 350.2.q.c.81.8
• 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			81.8
Character	\(\chi\)	\(=\)	350.81
Dual form			350.2.q.c.121.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.669131 - 0.743145i) q^{2} +(0.128261 + 1.22032i) q^{3} +(-0.104528 - 0.994522i) q^{4} +(2.17491 - 0.519385i) q^{5} +(0.992696 + 0.721236i) q^{6} +(-0.0870818 - 2.64432i) q^{7} +(-0.809017 - 0.587785i) q^{8} +(1.46172 - 0.310698i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 10 q^{2} + 2 q^{3} + 10 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 20 q^{8} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	0.128261	+	1.22032i	0.0740513	+	0.704551i	0.967066	+	0.254524i	\(0.0819189\pi\)
−0.893015	+	0.450027i	\(0.851414\pi\)
\(5\)	2.17491	−	0.519385i	0.972650	−	0.232276i
							\(7\)	−0.0870818	−	2.64432i	−0.0329138	−	0.999458i
\(11\)	−1.87395	−	0.398320i	−0.565016	−	0.120098i								−0.0834568	−	0.996511i	\(-0.526596\pi\)
							−0.481559	+	0.876414i	\(0.659929\pi\)
\(13\)	−1.23142	+	3.78993i	−0.341535	+	1.05114i	0.621878	+	0.783115i	\(0.286370\pi\)
							−0.963413	+	0.268023i	\(0.913630\pi\)
\(17\)	4.03619	−	1.79703i	0.978920	−	0.435843i	0.146029	−	0.989280i	\(-0.453351\pi\)
							0.832891	+	0.553437i	\(0.186684\pi\)
\(19\)	0.138220	−	1.31508i	0.0317099	−	0.301699i	−0.967160	−	0.254169i	\(-0.918198\pi\)
							0.998870	−	0.0475305i	\(-0.0151351\pi\)
\(23\)	−2.34603	+	2.60553i	−0.489182	+	0.543291i	−0.936308	−	0.351180i	\(-0.885781\pi\)
							0.447127	+	0.894471i	\(0.352447\pi\)
\(29\)	−1.21838	+	0.885203i	−0.226247	+	0.164378i	−0.695134	−	0.718880i	\(-0.744655\pi\)
							0.468887	+	0.883258i	\(0.344655\pi\)
\(31\)	4.96493	−	2.21053i	0.891729	−	0.397023i	0.0908603	−	0.995864i	\(-0.471038\pi\)
							0.800868	+	0.598841i	\(0.204372\pi\)
\(37\)	−5.13654	+	1.09181i	−0.844442	+	0.179492i	−0.609766	−	0.792582i	\(-0.708736\pi\)
							−0.234677	+	0.972073i	\(0.575403\pi\)
\(41\)	−1.80652	+	5.55990i	−0.282131	+	0.868311i	0.705113	+	0.709095i	\(0.250896\pi\)
							−0.987244	+	0.159215i	\(0.949104\pi\)
\(43\)	−7.05583			−1.07600			−0.538002	−	0.842943i	\(-0.680821\pi\)
							−0.538002	+	0.842943i	\(0.680821\pi\)
\(47\)	−2.05347	−	0.914265i	−0.299530	−	0.133359i	0.251469	−	0.967865i	\(-0.419086\pi\)
							−0.550999	+	0.834506i	\(0.685753\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.c.81.8	✓	80
7.2	even	3	inner	350.2.q.c.331.3	yes	80
25.21	even	5	inner	350.2.q.c.221.3	yes	80
175.121	even	15	inner	350.2.q.c.121.8	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.c.81.8	✓	80	1.1	even	1	trivial
350.2.q.c.121.8	yes	80	175.121	even	15	inner
350.2.q.c.221.3	yes	80	25.21	even	5	inner
350.2.q.c.331.3	yes	80	7.2	even	3	inner