Embedded newform 105.4.x.a.2.3

• Label: 105.4.x.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	105.2
Dual form			105.4.x.a.53.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29573 - 4.83572i) q^{2} +(-2.41255 - 4.60213i) q^{3} +(-14.7771 + 8.53155i) q^{4} +(7.35527 + 8.42022i) q^{5} +(-19.1286 + 17.6295i) q^{6} +(7.27945 + 17.0297i) q^{7} +(32.0833 + 32.0833i) q^{8} +(-15.3592 + 22.2058i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29573	−	4.83572i	−0.458109	−	1.70969i	−0.678784	−	0.734338i	\(-0.737493\pi\)
							0.220675	−	0.975347i	\(-0.429174\pi\)
\(3\)	−2.41255	−	4.60213i	−0.464296	−	0.885680i
							\(5\)	7.35527	+	8.42022i	0.657875	+	0.753127i
\(7\)	7.27945	+	17.0297i	0.393053	+	0.919516i
							\(11\)	−22.9104	+	13.2273i	−0.627977	+	0.362562i	−0.779968	−	0.625819i	\(-0.784765\pi\)
0.151991	+	0.988382i	\(0.451431\pi\)
\(13\)	−39.7224	+	39.7224i	−0.847462	+	0.847462i	−0.989816	−	0.142354i	\(-0.954533\pi\)
							0.142354	+	0.989816i	\(0.454533\pi\)
\(17\)	31.4529	+	8.42778i	0.448732	+	0.120237i	0.476106	−	0.879388i	\(-0.342048\pi\)
							−0.0273739	+	0.999625i	\(0.508714\pi\)
\(19\)	−141.354	−	81.6106i	−1.70678	−	0.985408i	−0.938492	−	0.345300i	\(-0.887777\pi\)
							−0.768285	−	0.640108i	\(-0.778889\pi\)
\(23\)	−60.7602	+	16.2806i	−0.550842	+	0.147598i	−0.523496	−	0.852028i	\(-0.675372\pi\)
							−0.0273464	+	0.999626i	\(0.508706\pi\)
\(29\)	80.5443			0.515749			0.257874	−	0.966178i	\(-0.416978\pi\)
							0.257874	+	0.966178i	\(0.416978\pi\)
\(31\)	19.3298	+	33.4802i	0.111991	+	0.193975i	0.916573	−	0.399867i	\(-0.130944\pi\)
							−0.804582	+	0.593842i	\(0.797610\pi\)
\(37\)	18.1801	−	4.87135i	0.0807782	−	0.0216445i	−0.218204	−	0.975903i	\(-0.570020\pi\)
							0.298982	+	0.954259i	\(0.403353\pi\)
\(41\)			14.2420i			0.0542496i	0.999632	+	0.0271248i	\(0.00863515\pi\)
							−0.999632	+	0.0271248i	\(0.991365\pi\)
\(43\)	−176.283	+	176.283i	−0.625182	+	0.625182i	−0.946852	−	0.321670i	\(-0.895756\pi\)
							0.321670	+	0.946852i	\(0.395756\pi\)
\(47\)	32.9221	+	122.867i	0.102174	+	0.381319i	0.998009	−	0.0630674i	\(-0.0200883\pi\)
							−0.895835	+	0.444387i	\(0.853422\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.x.a.2.3	✓	176
3.2	odd	2	inner	105.4.x.a.2.42	yes	176
5.3	odd	4	inner	105.4.x.a.23.3	yes	176
7.4	even	3	inner	105.4.x.a.32.42	yes	176
15.8	even	4	inner	105.4.x.a.23.42	yes	176
21.11	odd	6	inner	105.4.x.a.32.3	yes	176
35.18	odd	12	inner	105.4.x.a.53.42	yes	176
105.53	even	12	inner	105.4.x.a.53.3	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.x.a.2.3	✓	176	1.1	even	1	trivial
105.4.x.a.2.42	yes	176	3.2	odd	2	inner
105.4.x.a.23.3	yes	176	5.3	odd	4	inner
105.4.x.a.23.42	yes	176	15.8	even	4	inner
105.4.x.a.32.3	yes	176	21.11	odd	6	inner
105.4.x.a.32.42	yes	176	7.4	even	3	inner
105.4.x.a.53.3	yes	176	105.53	even	12	inner
105.4.x.a.53.42	yes	176	35.18	odd	12	inner