Embedded newform 115.3.i.a.14.5

• Label: 115.3.i.a.14.5
• Level: $115$
• Weight: $3$
• Character: 115.14
• Analytic conductor: $3.134$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13352304014\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(22\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			14.5
Character	\(\chi\)	\(=\)	115.14
Dual form			115.3.i.a.74.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.723107 - 2.46268i) q^{2} +(-0.136015 + 0.117858i) q^{3} +(-2.17688 + 1.39899i) q^{4} +(4.86226 + 1.16552i) q^{5} +(0.388600 + 0.249738i) q^{6} +(4.52870 - 9.91648i) q^{7} +(-2.73958 - 2.37386i) q^{8} +(-1.27622 + 8.87633i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q + 26 q^{4} - 11 q^{5} - 14 q^{6} + 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{21}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.723107	−	2.46268i	−0.361554	−	1.23134i	−0.916697	−	0.399583i	\(-0.869155\pi\)
							0.555143	−	0.831755i	\(-0.312663\pi\)
\(3\)	−0.136015	+	0.117858i	−0.0453385	+	0.0392860i	−0.677235	−	0.735766i	\(-0.736822\pi\)
							0.631897	+	0.775053i	\(0.282277\pi\)
\(5\)	4.86226	+	1.16552i	0.972452	+	0.233104i
							\(7\)	4.52870	−	9.91648i	0.646958	−	1.41664i	−0.247234	−	0.968956i	\(-0.579522\pi\)
0.894192	−	0.447684i	\(-0.147751\pi\)
\(11\)	3.68466	−	12.5488i	0.334969	−	1.14080i	−0.604053	−	0.796944i	\(-0.706448\pi\)
							0.939022	−	0.343857i	\(-0.111733\pi\)
\(13\)	−5.50209	+	2.51272i	−0.423238	+	0.193286i	−0.615636	−	0.788031i	\(-0.711101\pi\)
							0.192398	+	0.981317i	\(0.438373\pi\)
\(17\)	0.985576	+	0.633391i	0.0579750	+	0.0372583i	0.569307	−	0.822125i	\(-0.307211\pi\)
							−0.511332	+	0.859383i	\(0.670848\pi\)
\(19\)	−10.8101	−	16.8208i	−0.568951	−	0.885305i	0.430903	−	0.902398i	\(-0.358195\pi\)
							−0.999854	+	0.0170930i	\(0.994559\pi\)
\(23\)	2.85260	+	22.8224i	0.124026	+	0.992279i
							\(29\)	8.14637	+	5.23536i	0.280909	+	0.180530i	0.673507	−	0.739181i	\(-0.264787\pi\)
−0.392598	+	0.919710i	\(0.628423\pi\)
\(31\)	25.0318	−	28.8882i	0.807477	−	0.931878i	−0.191290	−	0.981534i	\(-0.561267\pi\)
							0.998766	+	0.0496557i	\(0.0158124\pi\)
\(37\)	3.84922	−	26.7719i	0.104033	−	0.723566i	−0.869319	−	0.494251i	\(-0.835442\pi\)
							0.973352	−	0.229315i	\(-0.0736485\pi\)
\(41\)	7.12989	+	49.5895i	0.173900	+	1.20950i	0.870547	+	0.492086i	\(0.163765\pi\)
							−0.696647	+	0.717414i	\(0.745326\pi\)
\(43\)	36.8097	+	42.4807i	0.856039	+	0.987922i	0.999999	−	0.00166146i	\(-0.000528860\pi\)
							−0.143959	+	0.989584i	\(0.545983\pi\)
\(47\)			84.3152i			1.79394i	0.442091	+	0.896970i	\(0.354237\pi\)
							−0.442091	+	0.896970i	\(0.645763\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.i.a.14.5	✓	220
5.4	even	2	inner	115.3.i.a.14.18	yes	220
23.5	odd	22	inner	115.3.i.a.74.18	yes	220
115.74	odd	22	inner	115.3.i.a.74.5	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.i.a.14.5	✓	220	1.1	even	1	trivial
115.3.i.a.14.18	yes	220	5.4	even	2	inner
115.3.i.a.74.5	yes	220	115.74	odd	22	inner
115.3.i.a.74.18	yes	220	23.5	odd	22	inner