Embedded newform 115.5.h.a.11.16

• Label: 115.5.h.a.11.16
• Level: $115$
• Weight: $5$
• Character: 115.11
• Analytic conductor: $11.888$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	115.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8875457546\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(32\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			11.16
Character	\(\chi\)	\(=\)	115.11
Dual form			115.5.h.a.21.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.562353 - 0.361402i) q^{2} +(-1.00918 + 7.01902i) q^{3} +(-6.46101 + 14.1476i) q^{4} +(3.14987 + 10.7275i) q^{5} +(1.96917 + 4.31189i) q^{6} +(10.6690 - 9.24478i) q^{7} +(3.00175 + 20.8776i) q^{8} +(29.4707 + 8.65337i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 12 q^{2} - 272 q^{4} + 166 q^{6} - 246 q^{8} - 896 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.562353	−	0.361402i	0.140588	−	0.0903505i	−0.468456	−	0.883487i	\(-0.655190\pi\)
							0.609044	+	0.793136i	\(0.291553\pi\)
\(3\)	−1.00918	+	7.01902i	−0.112131	+	0.779892i	0.853708	+	0.520752i	\(0.174348\pi\)
							−0.965840	+	0.259140i	\(0.916561\pi\)
\(5\)	3.14987	+	10.7275i	0.125995	+	0.429098i
							\(7\)	10.6690	−	9.24478i	0.217736	−	0.188669i	−0.539163	−	0.842202i	\(-0.681259\pi\)
0.756898	+	0.653533i	\(0.226714\pi\)
\(11\)	−90.1850	+	140.331i	−0.745331	+	1.15976i	0.236800	+	0.971558i	\(0.423901\pi\)
							−0.982131	+	0.188199i	\(0.939735\pi\)
\(13\)	61.6391	−	71.1353i	0.364728	−	0.420919i	−0.543490	−	0.839416i	\(-0.682897\pi\)
							0.908218	+	0.418497i	\(0.137443\pi\)
\(17\)	−149.914	+	68.4633i	−0.518732	+	0.236897i	−0.657536	−	0.753423i	\(-0.728401\pi\)
							0.138804	+	0.990320i	\(0.455674\pi\)
\(19\)	−524.982	−	239.751i	−1.45424	−	0.664131i	−0.477517	−	0.878622i	\(-0.658463\pi\)
							−0.976726	+	0.214492i	\(0.931191\pi\)
\(23\)	−22.5441	−	528.519i	−0.0426164	−	0.999092i
							\(29\)	637.039	+	1394.92i	0.757478	+	1.65864i	0.752443	+	0.658657i	\(0.228875\pi\)
0.00503470	+	0.999987i	\(0.498397\pi\)
\(31\)	−133.196	−	926.399i	−0.138602	−	0.963995i	−0.933838	−	0.357696i	\(-0.883562\pi\)
							0.795237	−	0.606299i	\(-0.207347\pi\)
\(37\)	−303.840	+	1034.78i	−0.221943	+	0.755868i	0.770954	+	0.636891i	\(0.219780\pi\)
							−0.992897	+	0.118977i	\(0.962038\pi\)
\(41\)	1126.60	−	330.801i	0.670199	−	0.196788i	0.0711074	−	0.997469i	\(-0.477347\pi\)
							0.599092	+	0.800680i	\(0.295529\pi\)
\(43\)	2498.75	+	359.266i	1.35141	+	0.194303i	0.779700	−	0.626154i	\(-0.215372\pi\)
							0.571708	+	0.820457i	\(0.306281\pi\)
\(47\)	−1177.46			−0.533030			−0.266515	−	0.963831i	\(-0.585872\pi\)
							−0.266515	+	0.963831i	\(0.585872\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.h.a.11.16	✓	320
23.21	odd	22	inner	115.5.h.a.21.16	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.h.a.11.16	✓	320	1.1	even	1	trivial
115.5.h.a.21.16	yes	320	23.21	odd	22	inner