Embedded newform 105.4.x.a.2.5

• Label: 105.4.x.a.2.5
• Level: $105$
• Weight: $4$
• Character: 105.2
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.5
Character	\(\chi\)	\(=\)	105.2
Dual form			105.4.x.a.53.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24693 - 4.65360i) q^{2} +(3.48035 + 3.85839i) q^{3} +(-13.1730 + 7.60541i) q^{4} +(9.76680 - 5.44147i) q^{5} +(13.6157 - 21.0073i) q^{6} +(7.78888 + 16.8028i) q^{7} +(24.5649 + 24.5649i) q^{8} +(-2.77439 + 26.8571i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24693	−	4.65360i	−0.440856	−	1.64530i	−0.726652	−	0.687006i	\(-0.758925\pi\)
							0.285796	−	0.958290i	\(-0.407742\pi\)
\(3\)	3.48035	+	3.85839i	0.669793	+	0.742548i
							\(5\)	9.76680	−	5.44147i	0.873569	−	0.486700i
\(7\)	7.78888	+	16.8028i	0.420560	+	0.907265i
							\(11\)	54.1215	−	31.2470i	1.48348	−	0.856485i	0.483652	−	0.875260i	\(-0.339310\pi\)
0.999824	+	0.0187753i	\(0.00597673\pi\)
\(13\)	34.7070	−	34.7070i	0.740460	−	0.740460i	−0.232206	−	0.972667i	\(-0.574594\pi\)
							0.972667	+	0.232206i	\(0.0745945\pi\)
\(17\)	−57.8505	−	15.5010i	−0.825342	−	0.221150i	−0.178661	−	0.983911i	\(-0.557177\pi\)
							−0.646680	+	0.762761i	\(0.723843\pi\)
\(19\)	16.4642	+	9.50563i	0.198798	+	0.114776i	0.596095	−	0.802914i	\(-0.296718\pi\)
							−0.397297	+	0.917690i	\(0.630052\pi\)
\(23\)	1.05821	−	0.283545i	0.00959352	−	0.00257058i	−0.254019	−	0.967199i	\(-0.581753\pi\)
							0.263613	+	0.964629i	\(0.415086\pi\)
\(29\)	−193.727			−1.24049			−0.620246	−	0.784408i	\(-0.712967\pi\)
							−0.620246	+	0.784408i	\(0.712967\pi\)
\(31\)	51.1080	+	88.5216i	0.296105	+	0.512869i	0.975241	−	0.221143i	\(-0.0709788\pi\)
							−0.679136	+	0.734012i	\(0.737645\pi\)
\(37\)	209.606	−	56.1638i	0.931325	−	0.249548i	0.238905	−	0.971043i	\(-0.423211\pi\)
							0.692420	+	0.721495i	\(0.256545\pi\)
\(41\)			139.785i			0.532458i	0.963910	+	0.266229i	\(0.0857777\pi\)
							−0.963910	+	0.266229i	\(0.914222\pi\)
\(43\)	124.712	−	124.712i	0.442289	−	0.442289i	−0.450492	−	0.892781i	\(-0.648751\pi\)
							0.892781	+	0.450492i	\(0.148751\pi\)
\(47\)	−87.9064	−	328.071i	−0.272818	−	1.01817i	−0.957289	−	0.289132i	\(-0.906633\pi\)
							0.684471	−	0.729040i	\(-0.260033\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.x.a.2.5	✓	176
3.2	odd	2	inner	105.4.x.a.2.40	yes	176
5.3	odd	4	inner	105.4.x.a.23.5	yes	176
7.4	even	3	inner	105.4.x.a.32.40	yes	176
15.8	even	4	inner	105.4.x.a.23.40	yes	176
21.11	odd	6	inner	105.4.x.a.32.5	yes	176
35.18	odd	12	inner	105.4.x.a.53.40	yes	176
105.53	even	12	inner	105.4.x.a.53.5	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.x.a.2.5	✓	176	1.1	even	1	trivial
105.4.x.a.2.40	yes	176	3.2	odd	2	inner
105.4.x.a.23.5	yes	176	5.3	odd	4	inner
105.4.x.a.23.40	yes	176	15.8	even	4	inner
105.4.x.a.32.5	yes	176	21.11	odd	6	inner
105.4.x.a.32.40	yes	176	7.4	even	3	inner
105.4.x.a.53.5	yes	176	105.53	even	12	inner
105.4.x.a.53.40	yes	176	35.18	odd	12	inner