Embedded newform 105.4.i.e.16.3

• Label: 105.4.i.e.16.3
• Level: $105$
• Weight: $4$
• Character: 105.16
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Relative dimension:	\(5\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	\( x^{10} - x^{9} + 31x^{8} + 26x^{7} + 738x^{6} + 352x^{5} + 5008x^{4} + 5368x^{3} + 26728x^{2} + 13776x + 7056 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2}\cdot 7 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			16.3
Root			\(-0.272258 + 0.471565i\) of defining polynomial
Character	\(\chi\)	\(=\)	105.16
Dual form			105.4.i.e.46.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.272258 + 0.471565i) q^{2} +(-1.50000 - 2.59808i) q^{3} +(3.85175 + 6.67143i) q^{4} +(2.50000 - 4.33013i) q^{5} +1.63355 q^{6} +(-0.983423 + 18.4941i) q^{7} -8.55081 q^{8} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{2} - 15 q^{3} - 21 q^{4} + 25 q^{5} - 6 q^{6} + 56 q^{7} - 138 q^{8} - 45 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.272258	+	0.471565i	−0.0962578	+	0.166723i	−0.910133	−	0.414317i	\(-0.864021\pi\)
							0.813875	+	0.581040i	\(0.197354\pi\)
\(3\)	−1.50000	−	2.59808i	−0.288675	−	0.500000i
							\(5\)	2.50000	−	4.33013i	0.223607	−	0.387298i
\(7\)	−0.983423	+	18.4941i	−0.0530998	+	0.998589i
							\(11\)	−0.182868	−	0.316737i	−0.00501244	−	0.00868181i	0.863508	−	0.504335i	\(-0.168262\pi\)
−0.868521	+	0.495653i	\(0.834929\pi\)
\(13\)	72.4570			1.54584			0.772921	−	0.634502i	\(-0.218795\pi\)
							0.772921	+	0.634502i	\(0.218795\pi\)
\(17\)	63.8796	+	110.643i	0.911358	+	1.57852i	0.812148	+	0.583452i	\(0.198298\pi\)
							0.0992099	+	0.995067i	\(0.468368\pi\)
\(19\)	−64.7773	+	112.198i	−0.782155	+	1.35473i	0.148529	+	0.988908i	\(0.452546\pi\)
							−0.930684	+	0.365824i	\(0.880787\pi\)
\(23\)	74.4912	−	129.022i	0.675325	−	1.16970i	−0.301048	−	0.953609i	\(-0.597337\pi\)
							0.976374	−	0.216089i	\(-0.0693301\pi\)
\(29\)	21.4889			0.137600			0.0687999	−	0.997630i	\(-0.478083\pi\)
							0.0687999	+	0.997630i	\(0.478083\pi\)
\(31\)	−12.7134	−	22.0203i	−0.0736581	−	0.127580i	0.826844	−	0.562432i	\(-0.190134\pi\)
							−0.900502	+	0.434852i	\(0.856801\pi\)
\(37\)	74.1665	−	128.460i	0.329538	−	0.570776i	−0.652882	−	0.757459i	\(-0.726440\pi\)
							0.982420	+	0.186683i	\(0.0597738\pi\)
\(41\)	−213.313			−0.812534			−0.406267	−	0.913755i	\(-0.633170\pi\)
							−0.406267	+	0.913755i	\(0.633170\pi\)
\(43\)	−285.923			−1.01402			−0.507010	−	0.861940i	\(-0.669249\pi\)
							−0.507010	+	0.861940i	\(0.669249\pi\)
\(47\)	261.855	−	453.547i	0.812671	−	1.40759i	−0.0983174	−	0.995155i	\(-0.531346\pi\)
							0.910988	−	0.412432i	\(-0.135321\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.i.e.16.3	✓	10
3.2	odd	2		315.4.j.f.226.3		10
7.2	even	3		735.4.a.y.1.3		5
7.4	even	3	inner	105.4.i.e.46.3	yes	10
7.5	odd	6		735.4.a.x.1.3		5
21.2	odd	6		2205.4.a.bw.1.3		5
21.5	even	6		2205.4.a.bv.1.3		5
21.11	odd	6		315.4.j.f.46.3		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.i.e.16.3	✓	10	1.1	even	1	trivial
105.4.i.e.46.3	yes	10	7.4	even	3	inner
315.4.j.f.46.3		10	21.11	odd	6
315.4.j.f.226.3		10	3.2	odd	2
735.4.a.x.1.3		5	7.5	odd	6
735.4.a.y.1.3		5	7.2	even	3
2205.4.a.bv.1.3		5	21.5	even	6
2205.4.a.bw.1.3		5	21.2	odd	6