Embedded newform 350.2.q.b.81.2

• Label: 350.2.q.b.81.2
• 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			81.2
Character	\(\chi\)	\(=\)	350.81
Dual form			350.2.q.b.121.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.669131 + 0.743145i) q^{2} +(-0.244226 - 2.32365i) q^{3} +(-0.104528 - 0.994522i) q^{4} +(2.23442 + 0.0857073i) q^{5} +(1.89023 + 1.37333i) q^{6} +(1.81635 + 1.92376i) q^{7} +(0.809017 + 0.587785i) q^{8} +(-2.40527 + 0.511257i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.244226	−	2.32365i	−0.141004	−	1.34156i	−0.804754	−	0.593608i	\(-0.797703\pi\)
0.663751	−	0.747954i	\(-0.268964\pi\)
\(5\)	2.23442	+	0.0857073i	0.999265	+	0.0383295i
							\(7\)	1.81635	+	1.92376i	0.686516	+	0.727114i
\(11\)	2.89338	+	0.615007i	0.872386	+	0.185431i								0.622287	−	0.782789i	\(-0.286204\pi\)
							0.250099	+	0.968220i	\(0.419537\pi\)
\(13\)	−0.413342	+	1.27214i	−0.114640	+	0.352827i	−0.991872	−	0.127241i	\(-0.959388\pi\)
							0.877231	+	0.480068i	\(0.159388\pi\)
\(17\)	−1.34726	+	0.599839i	−0.326759	+	0.145482i	−0.563559	−	0.826076i	\(-0.690568\pi\)
							0.236800	+	0.971558i	\(0.423901\pi\)
\(19\)	0.541284	−	5.14997i	0.124179	−	1.18148i	−0.737972	−	0.674832i	\(-0.764216\pi\)
							0.862151	−	0.506652i	\(-0.169117\pi\)
\(23\)	0.177805	−	0.197473i	0.0370750	−	0.0411759i	−0.724322	−	0.689462i	\(-0.757847\pi\)
							0.761397	+	0.648286i	\(0.224514\pi\)
\(29\)	−1.76762	+	1.28425i	−0.328238	+	0.238479i	−0.739683	−	0.672956i	\(-0.765024\pi\)
							0.411444	+	0.911435i	\(0.365024\pi\)
\(31\)	5.33283	−	2.37433i	0.957804	−	0.426442i	0.132533	−	0.991179i	\(-0.457689\pi\)
							0.825271	+	0.564737i	\(0.191022\pi\)
\(37\)	−9.17888	+	1.95103i	−1.50900	+	0.320748i	−0.886815	−	0.462124i	\(-0.847088\pi\)
							−0.622183	+	0.782871i	\(0.713754\pi\)
\(41\)	2.32340	−	7.15068i	0.362854	−	1.11675i	−0.588460	−	0.808526i	\(-0.700266\pi\)
							0.951314	−	0.308223i	\(-0.0997343\pi\)
\(43\)	−9.50368			−1.44930			−0.724648	−	0.689119i	\(-0.757998\pi\)
							−0.724648	+	0.689119i	\(0.757998\pi\)
\(47\)	5.86900	+	2.61305i	0.856082	+	0.381152i	0.787367	−	0.616484i	\(-0.211444\pi\)
							0.0687144	+	0.997636i	\(0.478110\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.b.81.2	✓	72
7.2	even	3	inner	350.2.q.b.331.8	yes	72
25.21	even	5	inner	350.2.q.b.221.8	yes	72
175.121	even	15	inner	350.2.q.b.121.2	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.b.81.2	✓	72	1.1	even	1	trivial
350.2.q.b.121.2	yes	72	175.121	even	15	inner
350.2.q.b.221.8	yes	72	25.21	even	5	inner
350.2.q.b.331.8	yes	72	7.2	even	3	inner