Embedded newform 105.4.p.a.59.7

• Label: 105.4.p.a.59.7
• Level: $105$
• Weight: $4$
• Character: 105.59
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.7
Character	\(\chi\)	\(=\)	105.59
Dual form			105.4.p.a.89.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.89916 - 3.28944i) q^{2} +(4.02025 + 3.29205i) q^{3} +(-3.21362 + 5.56616i) q^{4} +(-5.19426 + 9.90049i) q^{5} +(3.19390 - 19.4765i) q^{6} +(6.82394 - 17.2173i) q^{7} -5.97382 q^{8} +(5.32484 + 26.4697i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 164 q^{4}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.89916	−	3.28944i	−0.671455	−	1.16299i	−0.977492	−	0.210974i	\(-0.932336\pi\)
							0.306037	−	0.952020i	\(-0.400997\pi\)
\(3\)	4.02025	+	3.29205i	0.773698	+	0.633555i
							\(5\)	−5.19426	+	9.90049i	−0.464589	+	0.885526i
\(7\)	6.82394	−	17.2173i	0.368458	−	0.929644i
							\(11\)	55.2355	+	31.8902i	1.51401	+	0.874115i	0.999865	+	0.0164116i	\(0.00522420\pi\)
0.514145	+	0.857703i	\(0.328109\pi\)
\(13\)	68.9231			1.47045			0.735224	−	0.677824i	\(-0.237077\pi\)
							0.735224	+	0.677824i	\(0.237077\pi\)
\(17\)	−18.9919	−	10.9650i	−0.270954	−	0.156436i	0.358367	−	0.933581i	\(-0.383334\pi\)
							−0.629321	+	0.777145i	\(0.716667\pi\)
\(19\)	42.6410	−	24.6188i	0.514869	−	0.297260i	−0.219964	−	0.975508i	\(-0.570594\pi\)
							0.734833	+	0.678248i	\(0.237261\pi\)
\(23\)	36.3183	+	62.9051i	0.329256	+	0.570288i	0.982364	−	0.186976i	\(-0.0598687\pi\)
							−0.653108	+	0.757264i	\(0.726535\pi\)
\(29\)		−	25.0443i		−	0.160366i	−0.996780	−	0.0801828i	\(-0.974450\pi\)
							0.996780	−	0.0801828i	\(-0.0255504\pi\)
\(31\)	−66.9186	−	38.6355i	−0.387708	−	0.223843i	0.293459	−	0.955972i	\(-0.405194\pi\)
							−0.681166	+	0.732129i	\(0.738527\pi\)
\(37\)	31.1514	−	17.9853i	0.138413	−	0.0799125i	−0.429195	−	0.903212i	\(-0.641203\pi\)
							0.567607	+	0.823299i	\(0.307869\pi\)
\(41\)	−73.3700			−0.279475			−0.139737	−	0.990189i	\(-0.544626\pi\)
							−0.139737	+	0.990189i	\(0.544626\pi\)
\(43\)			361.428i			1.28180i	0.767626	+	0.640898i	\(0.221438\pi\)
							−0.767626	+	0.640898i	\(0.778562\pi\)
\(47\)	−464.541	+	268.203i	−1.44171	+	0.832370i	−0.997964	−	0.0637835i	\(-0.979683\pi\)
							−0.443744	+	0.896154i	\(0.646350\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.p.a.59.7	✓	88
3.2	odd	2	inner	105.4.p.a.59.37	yes	88
5.4	even	2	inner	105.4.p.a.59.38	yes	88
7.5	odd	6	inner	105.4.p.a.89.8	yes	88
15.14	odd	2	inner	105.4.p.a.59.8	yes	88
21.5	even	6	inner	105.4.p.a.89.38	yes	88
35.19	odd	6	inner	105.4.p.a.89.37	yes	88
105.89	even	6	inner	105.4.p.a.89.7	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.p.a.59.7	✓	88	1.1	even	1	trivial
105.4.p.a.59.8	yes	88	15.14	odd	2	inner
105.4.p.a.59.37	yes	88	3.2	odd	2	inner
105.4.p.a.59.38	yes	88	5.4	even	2	inner
105.4.p.a.89.7	yes	88	105.89	even	6	inner
105.4.p.a.89.8	yes	88	7.5	odd	6	inner
105.4.p.a.89.37	yes	88	35.19	odd	6	inner
105.4.p.a.89.38	yes	88	21.5	even	6	inner