Embedded newform 927.2.z.a.709.48

• Label: 927.2.z.a.709.48
• Level: $927$
• Weight: $2$
• Character: 927.709
• Analytic conductor: $7.402$
• Analytic rank: $0$
• Dimension: $3264$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 927 = 3^{2} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	927.z (of order \(51\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.40213226737\)
Analytic rank:	\(0\)
Dimension:	\(3264\)
Relative dimension:	\(102\) over \(\Q(\zeta_{51})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{51}]$

Embedding invariants

Embedding label			709.48
Character	\(\chi\)	\(=\)	927.709
Dual form			927.2.z.a.472.48

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0383744 - 0.247212i) q^{2} +(0.769085 + 1.55194i) q^{3} +(1.84624 - 0.587332i) q^{4} +(3.71323 + 0.933915i) q^{5} +(0.354144 - 0.249682i) q^{6} +(0.376523 - 0.756161i) q^{7} +(-0.439067 - 0.881766i) q^{8} +(-1.81702 + 2.38714i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3264 q - 18 q^{2} - 29 q^{3} + 82 q^{4} - 20 q^{5} - 35 q^{6} - 17 q^{7} - 56 q^{8} - 19 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/927\mathbb{Z}\right)^\times\).

\(n\)	\(722\)	\(829\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{38}{51}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0383744	−	0.247212i	−0.0271348	−	0.174805i	0.970725	−	0.240193i	\(-0.0772108\pi\)
							−0.997860	+	0.0653878i	\(0.979172\pi\)
\(3\)	0.769085	+	1.55194i	0.444032	+	0.896011i
							\(5\)	3.71323	+	0.933915i	1.66061	+	0.417659i	0.956335	−	0.292272i	\(-0.0944111\pi\)
0.704271	+	0.709931i	\(0.251274\pi\)
\(7\)	0.376523	−	0.756161i	0.142312	−	0.285802i	−0.812527	−	0.582923i	\(-0.801909\pi\)
							0.954840	+	0.297121i	\(0.0960265\pi\)
\(11\)	−5.12177	−	1.98418i	−1.54427	−	0.598254i	−0.569605	−	0.821919i	\(-0.692904\pi\)
							−0.974666	+	0.223665i	\(0.928198\pi\)
\(13\)	2.99639	−	0.184811i	0.831049	−	0.0512573i	0.359552	−	0.933125i	\(-0.382930\pi\)
							0.471497	+	0.881868i	\(0.343714\pi\)
\(17\)	2.78759	+	1.49690i	0.676090	+	0.363052i	0.774927	−	0.632051i	\(-0.217787\pi\)
							−0.0988371	+	0.995104i	\(0.531512\pi\)
\(19\)	−1.38865	+	3.94069i	−0.318577	+	0.904056i	0.668486	+	0.743725i	\(0.266943\pi\)
							−0.987063	+	0.160331i	\(0.948744\pi\)
\(23\)	−2.91227	−	2.34349i	−0.607250	−	0.488652i	0.273729	−	0.961807i	\(-0.411743\pi\)
							−0.880979	+	0.473154i	\(0.843115\pi\)
\(29\)	3.88023	−	4.00162i	0.720540	−	0.743082i	−0.254022	−	0.967198i	\(-0.581754\pi\)
							0.974562	+	0.224117i	\(0.0719497\pi\)
\(31\)	−1.06449	+	0.753534i	−0.191187	+	0.135339i	−0.669391	−	0.742910i	\(-0.733445\pi\)
							0.478204	+	0.878249i	\(0.341288\pi\)
\(37\)	−1.89659	−	2.51149i	−0.311798	−	0.412887i	0.614953	−	0.788564i	\(-0.289175\pi\)
							−0.926751	+	0.375677i	\(0.877410\pi\)
\(41\)	−1.91832	+	6.74220i	−0.299592	+	1.05296i	0.655145	+	0.755503i	\(0.272608\pi\)
							−0.954737	+	0.297452i	\(0.903863\pi\)
\(43\)	−7.67902	−	0.950869i	−1.17104	−	0.145006i	−0.486298	−	0.873793i	\(-0.661653\pi\)
							−0.684741	+	0.728787i	\(0.740085\pi\)
\(47\)	−4.38113			−0.639053			−0.319527	−	0.947577i	\(-0.603524\pi\)
							−0.319527	+	0.947577i	\(0.603524\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.z.a.709.48	yes	3264
9.4	even	3		927.2.y.a.400.55	✓	3264
103.60	even	51		927.2.y.a.781.55	yes	3264
927.472	even	51	inner	927.2.z.a.472.48	yes	3264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.y.a.400.55	✓	3264	9.4	even	3
927.2.y.a.781.55	yes	3264	103.60	even	51
927.2.z.a.472.48	yes	3264	927.472	even	51	inner
927.2.z.a.709.48	yes	3264	1.1	even	1	trivial