Embedded newform 105.4.u.a.103.4

• Label: 105.4.u.a.103.4
• Level: $105$
• Weight: $4$
• Character: 105.103
• 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			103.4
Character	\(\chi\)	\(=\)	105.103
Dual form			105.4.u.a.52.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17018 - 4.36716i) q^{2} +(2.89778 + 0.776457i) q^{3} +(-10.7746 + 6.22070i) q^{4} +(-6.73223 + 8.92620i) q^{5} -13.5637i q^{6} +(-1.57380 + 18.4533i) q^{7} +(14.1991 + 14.1991i) q^{8} +(7.79423 + 4.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 48 q^{5} - 4 q^{7} + 168 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17018	−	4.36716i	−0.413720	−	1.54402i	−0.787386	−	0.616461i	\(-0.788566\pi\)
							0.373666	−	0.927564i	\(-0.378101\pi\)
\(3\)	2.89778	+	0.776457i	0.557678	+	0.149429i
							\(5\)	−6.73223	+	8.92620i	−0.602149	+	0.798384i
\(7\)	−1.57380	+	18.4533i	−0.0849771	+	0.996383i
							\(11\)	19.3043	+	33.4361i	0.529134	+	0.916487i	0.999423	+	0.0339746i	\(0.0108165\pi\)
−0.470288	+	0.882513i	\(0.655850\pi\)
\(13\)	−34.4805	+	34.4805i	−0.735629	+	0.735629i	−0.971729	−	0.236100i	\(-0.924131\pi\)
							0.236100	+	0.971729i	\(0.424131\pi\)
\(17\)	20.7911	−	77.5933i	0.296622	−	1.10701i	−0.643299	−	0.765615i	\(-0.722435\pi\)
							0.939921	−	0.341393i	\(-0.110899\pi\)
\(19\)	−41.7160	+	72.2543i	−0.503701	+	0.872435i	0.496290	+	0.868157i	\(0.334695\pi\)
							−0.999991	+	0.00427834i	\(0.998638\pi\)
\(23\)	50.8102	−	13.6146i	0.460638	−	0.123427i	−0.0210349	−	0.999779i	\(-0.506696\pi\)
							0.481673	+	0.876351i	\(0.340029\pi\)
\(29\)		−	83.4317i		−	0.534237i	−0.963664	−	0.267119i	\(-0.913928\pi\)
							0.963664	−	0.267119i	\(-0.0860716\pi\)
\(31\)	−81.2237	+	46.8945i	−0.470587	+	0.271694i	−0.716485	−	0.697602i	\(-0.754250\pi\)
							0.245898	+	0.969296i	\(0.420917\pi\)
\(37\)	42.2257	+	157.588i	0.187618	+	0.700199i	0.994055	+	0.108879i	\(0.0347262\pi\)
							−0.806437	+	0.591320i	\(0.798607\pi\)
\(41\)			412.940i			1.57294i	0.617630	+	0.786469i	\(0.288093\pi\)
							−0.617630	+	0.786469i	\(0.711907\pi\)
\(43\)	31.4152	+	31.4152i	0.111413	+	0.111413i	0.760616	−	0.649202i	\(-0.224897\pi\)
							−0.649202	+	0.760616i	\(0.724897\pi\)
\(47\)	−406.470	+	108.913i	−1.26148	+	0.338013i	−0.826762	−	0.562552i	\(-0.809820\pi\)
							−0.434721	+	0.900565i	\(0.643153\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.u.a.103.4	yes	96
5.2	odd	4	inner	105.4.u.a.82.21	yes	96
7.3	odd	6	inner	105.4.u.a.73.21	yes	96
35.17	even	12	inner	105.4.u.a.52.4	✓	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.u.a.52.4	✓	96	35.17	even	12	inner
105.4.u.a.73.21	yes	96	7.3	odd	6	inner
105.4.u.a.82.21	yes	96	5.2	odd	4	inner
105.4.u.a.103.4	yes	96	1.1	even	1	trivial