Embedded newform 115.5.h.a.11.17

• Label: 115.5.h.a.11.17
• 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			11.17
Character	\(\chi\)	\(=\)	115.11
Dual form			115.5.h.a.21.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.777273 - 0.499523i) q^{2} +(0.742802 - 5.16630i) q^{3} +(-6.29201 + 13.7776i) q^{4} +(-3.14987 - 10.7275i) q^{5} +(-2.00333 - 4.38667i) q^{6} +(-14.1558 + 12.2660i) q^{7} +(4.09547 + 28.4846i) q^{8} +(51.5800 + 15.1453i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 12 q^{2} - 272 q^{4} + 166 q^{6} - 246 q^{8} - 896 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.777273	−	0.499523i	0.194318	−	0.124881i	−0.439864	−	0.898064i	\(-0.644973\pi\)
							0.634182	+	0.773184i	\(0.281337\pi\)
\(3\)	0.742802	−	5.16630i	0.0825335	−	0.574033i	−0.906028	−	0.423217i	\(-0.860901\pi\)
							0.988562	−	0.150816i	\(-0.0481902\pi\)
\(5\)	−3.14987	−	10.7275i	−0.125995	−	0.429098i
							\(7\)	−14.1558	+	12.2660i	−0.288893	+	0.250327i	−0.787235	−	0.616653i	\(-0.788488\pi\)
0.498342	+	0.866981i	\(0.333943\pi\)
\(11\)	123.124	−	191.584i	1.01755	−	1.58334i	0.224229	−	0.974536i	\(-0.428014\pi\)
							0.793321	−	0.608803i	\(-0.208350\pi\)
\(13\)	75.4325	−	87.0538i	0.446346	−	0.515111i	−0.487336	−	0.873215i	\(-0.662031\pi\)
							0.933682	+	0.358104i	\(0.116577\pi\)
\(17\)	115.028	−	52.5317i	0.398022	−	0.181771i	−0.206340	−	0.978480i	\(-0.566155\pi\)
							0.604362	+	0.796710i	\(0.293428\pi\)
\(19\)	−65.3508	−	29.8447i	−0.181027	−	0.0826723i	0.322840	−	0.946454i	\(-0.395362\pi\)
							−0.503867	+	0.863781i	\(0.668090\pi\)
\(23\)	367.300	−	380.699i	0.694329	−	0.719658i
							\(29\)	239.247	+	523.877i	0.284479	+	0.622921i	0.996887	−	0.0788438i	\(-0.0251228\pi\)
−0.712408	+	0.701765i	\(0.752396\pi\)
\(31\)	−10.1924	−	70.8896i	−0.0106060	−	0.0737665i	0.983831	−	0.179099i	\(-0.0573183\pi\)
							−0.994437	+	0.105333i	\(0.966409\pi\)
\(37\)	327.191	−	1114.31i	0.239000	−	0.813960i	−0.749404	−	0.662113i	\(-0.769660\pi\)
							0.988404	−	0.151847i	\(-0.0485221\pi\)
\(41\)	684.804	−	201.077i	0.407379	−	0.119617i	−0.0716216	−	0.997432i	\(-0.522817\pi\)
							0.479000	+	0.877815i	\(0.340999\pi\)
\(43\)	283.711	+	40.7915i	0.153440	+	0.0220614i	0.218607	−	0.975813i	\(-0.429849\pi\)
							−0.0651666	+	0.997874i	\(0.520758\pi\)
\(47\)	2597.86			1.17603			0.588017	−	0.808848i	\(-0.299909\pi\)
							0.588017	+	0.808848i	\(0.299909\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.h.a.11.17	✓	320
23.21	odd	22	inner	115.5.h.a.21.17	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.h.a.11.17	✓	320	1.1	even	1	trivial
115.5.h.a.21.17	yes	320	23.21	odd	22	inner