Embedded newform 115.5.h.a.11.1

• Label: 115.5.h.a.11.1
• 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			11.1
Character	\(\chi\)	\(=\)	115.11
Dual form			115.5.h.a.21.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.63856 + 4.26634i) q^{2} +(0.913905 - 6.35635i) q^{3} +(19.2221 - 42.0906i) q^{4} +(-3.14987 - 10.7275i) q^{5} +(21.0513 + 46.0960i) q^{6} +(-67.6414 + 58.6116i) q^{7} +(33.9969 + 236.453i) q^{8} +(38.1510 + 11.2021i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 12 q^{2} - 272 q^{4} + 166 q^{6} - 246 q^{8} - 896 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.63856	+	4.26634i	−1.65964	+	1.06659i	−0.740985	+	0.671522i	\(0.765641\pi\)
							−0.918654	+	0.395063i	\(0.870723\pi\)
\(3\)	0.913905	−	6.35635i	0.101545	−	0.706261i	−0.873914	−	0.486080i	\(-0.838426\pi\)
							0.975459	−	0.220181i	\(-0.0706647\pi\)
\(5\)	−3.14987	−	10.7275i	−0.125995	−	0.429098i
							\(7\)	−67.6414	+	58.6116i	−1.38044	+	1.19616i	−0.423488	+	0.905902i	\(0.639194\pi\)
−0.956950	+	0.290254i	\(0.906260\pi\)
\(11\)	1.99140	−	3.09867i	0.0164578	−	0.0256089i	−0.832927	−	0.553383i	\(-0.813337\pi\)
							0.849385	+	0.527774i	\(0.176973\pi\)
\(13\)	−19.5308	+	22.5397i	−0.115567	+	0.133371i	−0.810585	−	0.585621i	\(-0.800851\pi\)
							0.695018	+	0.718992i	\(0.255396\pi\)
\(17\)	74.4143	−	33.9839i	0.257489	−	0.117591i	−0.282489	−	0.959271i	\(-0.591160\pi\)
							0.539978	+	0.841679i	\(0.318433\pi\)
\(19\)	461.693	+	210.848i	1.27893	+	0.584067i	0.934906	−	0.354895i	\(-0.115483\pi\)
							0.344022	+	0.938962i	\(0.388211\pi\)
\(23\)	7.63535	−	528.945i	0.0144336	−	0.999896i
							\(29\)	−634.150	−	1388.59i	−0.754043	−	1.65112i	−0.758962	−	0.651135i	\(-0.774293\pi\)
0.00491934	−	0.999988i	\(-0.498434\pi\)
\(31\)	102.041	+	709.709i	0.106182	+	0.738510i	0.971457	+	0.237214i	\(0.0762343\pi\)
							−0.865276	+	0.501296i	\(0.832857\pi\)
\(37\)	344.029	−	1171.66i	0.251300	−	0.855848i	−0.733133	−	0.680085i	\(-0.761943\pi\)
							0.984433	−	0.175763i	\(-0.0562392\pi\)
\(41\)	38.0625	−	11.1762i	0.0226428	−	0.00664852i	−0.270392	−	0.962750i	\(-0.587153\pi\)
							0.293034	+	0.956102i	\(0.405335\pi\)
\(43\)	872.807	+	125.491i	0.472043	+	0.0678695i	0.374231	−	0.927335i	\(-0.377907\pi\)
							0.0978116	+	0.995205i	\(0.468816\pi\)
\(47\)	366.032			0.165701			0.0828503	−	0.996562i	\(-0.473598\pi\)
							0.0828503	+	0.996562i	\(0.473598\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.h.a.11.1	✓	320
23.21	odd	22	inner	115.5.h.a.21.1	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.h.a.11.1	✓	320	1.1	even	1	trivial
115.5.h.a.21.1	yes	320	23.21	odd	22	inner