Embedded newform 115.3.h.a.21.4

• Label: 115.3.h.a.21.4
• Level: $115$
• Weight: $3$
• Character: 115.21
• 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			21.4
Character	\(\chi\)	\(=\)	115.21
Dual form			115.3.h.a.11.4

$q$-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 \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{13}{22}\right)\)

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	−0.240890	−	0.970552i	\(-0.577439\pi\)
							−0.982915	+	0.184061i	\(0.941076\pi\)
\(3\)	−0.0493905	−	0.343518i	−0.0164635	−	0.114506i	0.979932	−	0.199330i	\(-0.0638765\pi\)
							−0.996396	+	0.0848235i	\(0.972967\pi\)
\(5\)	−0.629973	+	2.14549i	−0.125995	+	0.429098i
							\(7\)	2.57506	+	2.23130i	0.367865	+	0.318757i	0.819103	−	0.573647i	\(-0.194472\pi\)
−0.451238	+	0.892404i	\(0.649017\pi\)
\(11\)	7.51094	+	11.6872i	0.682813	+	1.06248i	0.993705	+	0.112032i	\(0.0357359\pi\)
							−0.310892	+	0.950445i	\(0.600628\pi\)
\(13\)	−13.7151	−	15.8280i	−1.05500	−	1.21754i	−0.975338	−	0.220718i	\(-0.929160\pi\)
							−0.0796667	−	0.996822i	\(-0.525386\pi\)
\(17\)	18.1061	+	8.26877i	1.06506	+	0.486398i	0.869317	−	0.494256i	\(-0.164559\pi\)
							0.195748	+	0.980654i	\(0.437287\pi\)
\(19\)	24.6977	−	11.2790i	1.29988	−	0.593634i	0.359303	−	0.933221i	\(-0.383015\pi\)
							0.940575	+	0.339587i	\(0.110287\pi\)
\(23\)	16.0414	−	16.4825i	0.697452	−	0.716631i
							\(29\)	−17.9236	+	39.2472i	−0.618055	+	1.35335i	0.298870	+	0.954294i	\(0.403390\pi\)
−0.916926	+	0.399058i	\(0.869337\pi\)
\(31\)	−3.77897	+	26.2833i	−0.121902	+	0.847849i	0.833495	+	0.552526i	\(0.186336\pi\)
							−0.955398	+	0.295322i	\(0.904573\pi\)
\(37\)	9.37914	+	31.9424i	0.253490	+	0.863309i	0.983659	+	0.180042i	\(0.0576233\pi\)
							−0.730169	+	0.683267i	\(0.760559\pi\)
\(41\)	73.7200	+	21.6461i	1.79805	+	0.527955i	0.997458	−	0.0712566i	\(-0.0227009\pi\)
							0.800591	+	0.599211i	\(0.204519\pi\)
\(43\)	10.3030	−	1.48135i	0.239605	−	0.0344500i	−0.0214662	−	0.999770i	\(-0.506833\pi\)
							0.261071	+	0.965320i	\(0.415924\pi\)
\(47\)	−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.21.4	yes	160
23.11	odd	22	inner	115.3.h.a.11.4	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.h.a.11.4	✓	160	23.11	odd	22	inner
115.3.h.a.21.4	yes	160	1.1	even	1	trivial