Embedded newform 105.4.x.a.2.7

• Label: 105.4.x.a.2.7
• 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			2.7
Character	\(\chi\)	\(=\)	105.2
Dual form			105.4.x.a.53.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07388 - 4.00779i) q^{2} +(1.26476 + 5.03988i) q^{3} +(-7.98097 + 4.60782i) q^{4} +(-9.67232 - 5.60770i) q^{5} +(18.8406 - 10.4811i) q^{6} +(11.5335 - 14.4906i) q^{7} +(3.56656 + 3.56656i) q^{8} +(-23.8008 + 12.7485i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07388	−	4.00779i	−0.379676	−	1.41697i	−0.846391	−	0.532561i	\(-0.821230\pi\)
							0.466716	−	0.884407i	\(-0.345437\pi\)
\(3\)	1.26476	+	5.03988i	0.243403	+	0.969925i
							\(5\)	−9.67232	−	5.60770i	−0.865118	−	0.501568i
\(7\)	11.5335	−	14.4906i	0.622750	−	0.782421i
							\(11\)	−15.4262	+	8.90629i	−0.422833	+	0.244123i	−0.696289	−	0.717762i	\(-0.745167\pi\)
0.273456	+	0.961885i	\(0.411833\pi\)
\(13\)	−50.7370	+	50.7370i	−1.08246	+	1.08246i	−0.0861753	+	0.996280i	\(0.527465\pi\)
							−0.996280	+	0.0861753i	\(0.972535\pi\)
\(17\)	−66.3556	−	17.7799i	−0.946682	−	0.253663i	−0.247728	−	0.968830i	\(-0.579684\pi\)
							−0.698954	+	0.715167i	\(0.746351\pi\)
\(19\)	−74.0499	−	42.7527i	−0.894117	−	0.516219i	−0.0188299	−	0.999823i	\(-0.505994\pi\)
							−0.875287	+	0.483604i	\(0.839327\pi\)
\(23\)	93.4488	−	25.0395i	0.847192	−	0.227004i	0.190992	−	0.981592i	\(-0.438830\pi\)
							0.656200	+	0.754587i	\(0.272163\pi\)
\(29\)	−29.6717			−0.189997			−0.0949984	−	0.995477i	\(-0.530285\pi\)
							−0.0949984	+	0.995477i	\(0.530285\pi\)
\(31\)	−121.329	−	210.147i	−0.702944	−	1.21753i	−0.967428	−	0.253145i	\(-0.918535\pi\)
							0.264484	−	0.964390i	\(-0.414798\pi\)
\(37\)	48.2668	−	12.9330i	0.214460	−	0.0574643i	−0.149989	−	0.988688i	\(-0.547924\pi\)
							0.364449	+	0.931223i	\(0.381257\pi\)
\(41\)		−	266.858i		−	1.01649i	−0.861212	−	0.508246i	\(-0.830294\pi\)
							0.861212	−	0.508246i	\(-0.169706\pi\)
\(43\)	264.669	−	264.669i	0.938644	−	0.938644i	−0.0595798	−	0.998224i	\(-0.518976\pi\)
							0.998224	+	0.0595798i	\(0.0189761\pi\)
\(47\)	127.972	+	477.598i	0.397162	+	1.48223i	0.818066	+	0.575124i	\(0.195046\pi\)
							−0.420904	+	0.907105i	\(0.638287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.x.a.2.7	✓	176
3.2	odd	2	inner	105.4.x.a.2.38	yes	176
5.3	odd	4	inner	105.4.x.a.23.7	yes	176
7.4	even	3	inner	105.4.x.a.32.38	yes	176
15.8	even	4	inner	105.4.x.a.23.38	yes	176
21.11	odd	6	inner	105.4.x.a.32.7	yes	176
35.18	odd	12	inner	105.4.x.a.53.38	yes	176
105.53	even	12	inner	105.4.x.a.53.7	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.x.a.2.7	✓	176	1.1	even	1	trivial
105.4.x.a.2.38	yes	176	3.2	odd	2	inner
105.4.x.a.23.7	yes	176	5.3	odd	4	inner
105.4.x.a.23.38	yes	176	15.8	even	4	inner
105.4.x.a.32.7	yes	176	21.11	odd	6	inner
105.4.x.a.32.38	yes	176	7.4	even	3	inner
105.4.x.a.53.7	yes	176	105.53	even	12	inner
105.4.x.a.53.38	yes	176	35.18	odd	12	inner