Embedded newform 115.3.h.a.11.4

• Label: 115.3.h.a.11.4
• Level: $115$
• Weight: $3$
• Character: 115.11
• Analytic conductor: $3.134$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	115.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13352304014\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(16\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			11.4
Character	\(\chi\)	\(=\)	115.11
Dual form			115.3.h.a.21.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.44761 + 1.57298i) q^{2} +(-0.0493905 + 0.343518i) q^{3} +(1.85486 - 4.06157i) q^{4} +(-0.629973 - 2.14549i) q^{5} +(-0.419460 - 0.918490i) q^{6} +(2.57506 - 2.23130i) q^{7} +(0.192568 + 1.33934i) q^{8} +(8.51987 + 2.50166i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 4 q^{2} - 16 q^{4} - 26 q^{6} - 22 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.44761	+	1.57298i	−1.22380	+	0.786492i	−0.982915	−	0.184061i	\(-0.941076\pi\)
							−0.240890	+	0.970552i	\(0.577439\pi\)
\(3\)	−0.0493905	+	0.343518i	−0.0164635	+	0.114506i	−0.996396	−	0.0848235i	\(-0.972967\pi\)
							0.979932	+	0.199330i	\(0.0638765\pi\)
\(5\)	−0.629973	−	2.14549i	−0.125995	−	0.429098i
							\(7\)	2.57506	−	2.23130i	0.367865	−	0.318757i	−0.451238	−	0.892404i	\(-0.649017\pi\)
0.819103	+	0.573647i	\(0.194472\pi\)
\(11\)	7.51094	−	11.6872i	0.682813	−	1.06248i	−0.310892	−	0.950445i	\(-0.600628\pi\)
							0.993705	−	0.112032i	\(-0.0357359\pi\)
\(13\)	−13.7151	+	15.8280i	−1.05500	+	1.21754i	−0.0796667	+	0.996822i	\(0.525386\pi\)
							−0.975338	+	0.220718i	\(0.929160\pi\)
\(17\)	18.1061	−	8.26877i	1.06506	−	0.486398i	0.195748	−	0.980654i	\(-0.437287\pi\)
							0.869317	+	0.494256i	\(0.164559\pi\)
\(19\)	24.6977	+	11.2790i	1.29988	+	0.593634i	0.940575	−	0.339587i	\(-0.110287\pi\)
							0.359303	+	0.933221i	\(0.383015\pi\)
\(23\)	16.0414	+	16.4825i	0.697452	+	0.716631i
							\(29\)	−17.9236	−	39.2472i	−0.618055	−	1.35335i	−0.916926	−	0.399058i	\(-0.869337\pi\)
0.298870	−	0.954294i	\(-0.403390\pi\)
\(31\)	−3.77897	−	26.2833i	−0.121902	−	0.847849i	−0.955398	−	0.295322i	\(-0.904573\pi\)
							0.833495	−	0.552526i	\(-0.186336\pi\)
\(37\)	9.37914	−	31.9424i	0.253490	−	0.863309i	−0.730169	−	0.683267i	\(-0.760559\pi\)
							0.983659	−	0.180042i	\(-0.0576233\pi\)
\(41\)	73.7200	−	21.6461i	1.79805	−	0.527955i	0.800591	−	0.599211i	\(-0.204519\pi\)
							0.997458	+	0.0712566i	\(0.0227009\pi\)
\(43\)	10.3030	+	1.48135i	0.239605	+	0.0344500i	0.261071	−	0.965320i	\(-0.415924\pi\)
							−0.0214662	+	0.999770i	\(0.506833\pi\)
\(47\)	−29.9346			−0.636906			−0.318453	−	0.947939i	\(-0.603163\pi\)
							−0.318453	+	0.947939i	\(0.603163\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.h.a.11.4	✓	160
23.21	odd	22	inner	115.3.h.a.21.4	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.h.a.11.4	✓	160	1.1	even	1	trivial
115.3.h.a.21.4	yes	160	23.21	odd	22	inner