Embedded newform 350.3.w.a.23.8

• Label: 350.3.w.a.23.8
• Level: $350$
• Weight: $3$
• Character: 350.23
• Analytic conductor: $9.537$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	350.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.53680925261\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(20\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			23.8
Character	\(\chi\)	\(=\)	350.23
Dual form			350.3.w.a.137.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.09905 + 0.889993i) q^{2} +(-1.28221 - 0.832674i) q^{3} +(0.415823 + 1.95630i) q^{4} +(-2.89680 - 4.07536i) q^{5} +(-0.668134 - 2.05631i) q^{6} +(0.580781 + 6.97587i) q^{7} +(-1.28408 + 2.52015i) q^{8} +(-2.70992 - 6.08659i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{2} - 6 q^{5} + 2 q^{7} - 160 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.09905	+	0.889993i	0.549525	+	0.444997i
							\(3\)	−1.28221	−	0.832674i	−0.427402	−	0.277558i	0.312960	−	0.949766i	\(-0.398679\pi\)
−0.740362	+	0.672208i	\(0.765346\pi\)
\(5\)	−2.89680	−	4.07536i	−0.579360	−	0.815072i
							\(7\)	0.580781	+	6.97587i	0.0829687	+	0.996552i
\(11\)	5.69009	+	2.53339i	0.517281	+	0.230308i								0.648736	−	0.761014i	\(-0.275298\pi\)
							−0.131455	+	0.991322i	\(0.541965\pi\)
\(13\)	3.33403	+	21.0502i	0.256464	+	1.61925i	0.693950	+	0.720023i	\(0.255869\pi\)
							−0.437487	+	0.899225i	\(0.644131\pi\)
\(17\)	1.39769	+	26.6696i	0.0822172	+	1.56880i	0.659154	+	0.752008i	\(0.270915\pi\)
							−0.576937	+	0.816789i	\(0.695752\pi\)
\(19\)	−1.05306	+	4.95426i	−0.0554242	+	0.260750i	−0.997132	−	0.0756814i	\(-0.975887\pi\)
							0.941708	+	0.336432i	\(0.109220\pi\)
\(23\)	7.17034	−	8.85464i	0.311754	−	0.384984i	−0.597052	−	0.802202i	\(-0.703662\pi\)
							0.908806	+	0.417218i	\(0.136995\pi\)
\(29\)	−20.2939	−	6.59390i	−0.699791	−	0.227376i	−0.0625514	−	0.998042i	\(-0.519924\pi\)
							−0.637239	+	0.770666i	\(0.719924\pi\)
\(31\)	−16.2669	+	18.0662i	−0.524739	+	0.582782i	−0.946004	−	0.324156i	\(-0.894920\pi\)
							0.421264	+	0.906938i	\(0.361586\pi\)
\(37\)	8.09714	−	21.0938i	0.218842	−	0.570102i	−0.779578	−	0.626306i	\(-0.784566\pi\)
							0.998419	+	0.0562036i	\(0.0178996\pi\)
\(41\)	45.9314	+	33.3711i	1.12028	+	0.813930i	0.984251	−	0.176776i	\(-0.0565667\pi\)
							0.136027	+	0.990705i	\(0.456567\pi\)
\(43\)	9.76708	+	9.76708i	0.227141	+	0.227141i	0.811497	−	0.584356i	\(-0.198653\pi\)
							−0.584356	+	0.811497i	\(0.698653\pi\)
\(47\)	43.2417	+	2.26620i	0.920036	+	0.0482171i	0.506471	−	0.862257i	\(-0.330950\pi\)
							0.413566	+	0.910474i	\(0.364283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.3.w.a.23.8	✓	320
7.4	even	3	inner	350.3.w.a.123.8	yes	320
25.12	odd	20	inner	350.3.w.a.37.8	yes	320
175.137	odd	60	inner	350.3.w.a.137.8	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.3.w.a.23.8	✓	320	1.1	even	1	trivial
350.3.w.a.37.8	yes	320	25.12	odd	20	inner
350.3.w.a.123.8	yes	320	7.4	even	3	inner
350.3.w.a.137.8	yes	320	175.137	odd	60	inner