Embedded newform 105.4.u.a.52.8

• Label: 105.4.u.a.52.8
• Level: $105$
• Weight: $4$
• Character: 105.52
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			52.8
Character	\(\chi\)	\(=\)	105.52
Dual form			105.4.u.a.103.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.529124 + 1.97472i) q^{2} +(2.89778 - 0.776457i) q^{3} +(3.30867 + 1.91026i) q^{4} +(-11.1089 + 1.26224i) q^{5} +6.13313i q^{6} +(-16.3965 + 8.61137i) q^{7} +(-17.0877 + 17.0877i) q^{8} +(7.79423 - 4.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 48 q^{5} - 4 q^{7} + 168 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.529124	+	1.97472i	−0.187074	+	0.698168i	0.807104	+	0.590410i	\(0.201034\pi\)
							−0.994177	+	0.107758i	\(0.965633\pi\)
\(3\)	2.89778	−	0.776457i	0.557678	−	0.149429i
							\(5\)	−11.1089	+	1.26224i	−0.993607	+	0.112898i
\(7\)	−16.3965	+	8.61137i	−0.885326	+	0.464970i
							\(11\)	−14.3072	+	24.7808i	−0.392162	+	0.679245i	−0.992734	−	0.120326i	\(-0.961606\pi\)
0.600572	+	0.799570i	\(0.294939\pi\)
\(13\)	37.5637	+	37.5637i	0.801407	+	0.801407i	0.983316	−	0.181908i	\(-0.0582274\pi\)
							−0.181908	+	0.983316i	\(0.558227\pi\)
\(17\)	14.5397	+	54.2627i	0.207434	+	0.774156i	0.988694	+	0.149949i	\(0.0479110\pi\)
							−0.781259	+	0.624207i	\(0.785422\pi\)
\(19\)	−26.4362	−	45.7889i	−0.319204	−	0.552878i	0.661118	−	0.750282i	\(-0.270082\pi\)
							−0.980322	+	0.197404i	\(0.936749\pi\)
\(23\)	−122.711	−	32.8803i	−1.11248	−	0.298088i	−0.344643	−	0.938734i	\(-0.612000\pi\)
							−0.767836	+	0.640646i	\(0.778667\pi\)
\(29\)		−	92.7340i		−	0.593802i	−0.954908	−	0.296901i	\(-0.904047\pi\)
							0.954908	−	0.296901i	\(-0.0959532\pi\)
\(31\)	237.430	+	137.080i	1.37560	+	0.794203i	0.991626	−	0.129141i	\(-0.0412219\pi\)
							0.383974	+	0.923344i	\(0.374555\pi\)
\(37\)	−23.5201	+	87.7783i	−0.104505	+	0.390018i	−0.998289	−	0.0584805i	\(-0.981374\pi\)
							0.893784	+	0.448499i	\(0.148041\pi\)
\(41\)		−	104.662i		−	0.398670i	−0.979931	−	0.199335i	\(-0.936122\pi\)
							0.979931	−	0.199335i	\(-0.0638782\pi\)
\(43\)	164.810	−	164.810i	0.584496	−	0.584496i	−0.351639	−	0.936136i	\(-0.614376\pi\)
							0.936136	+	0.351639i	\(0.114376\pi\)
\(47\)	453.499	+	121.515i	1.40744	+	0.377122i	0.881010	−	0.473097i	\(-0.156864\pi\)
							0.526429	+	0.850219i	\(0.323531\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.u.a.52.8	✓	96
5.3	odd	4	inner	105.4.u.a.73.17	yes	96
7.5	odd	6	inner	105.4.u.a.82.17	yes	96
35.33	even	12	inner	105.4.u.a.103.8	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.u.a.52.8	✓	96	1.1	even	1	trivial
105.4.u.a.73.17	yes	96	5.3	odd	4	inner
105.4.u.a.82.17	yes	96	7.5	odd	6	inner
105.4.u.a.103.8	yes	96	35.33	even	12	inner