Embedded newform 195.2.bl.a.68.7

• Label: 195.2.bl.a.68.7
• Level: $195$
• Weight: $2$
• Character: 195.68
• Analytic conductor: $1.557$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 195 = 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	195.bl (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.55708283941\)
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			68.7
Character	\(\chi\)	\(=\)	195.68
Dual form			195.2.bl.a.152.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.395659 + 1.47662i) q^{2} +(-1.07285 + 1.35978i) q^{3} +(-0.291809 - 0.168476i) q^{4} +(1.88815 - 1.19787i) q^{5} +(-1.58339 - 2.12220i) q^{6} +(1.22324 + 4.56520i) q^{7} +(-1.79769 + 1.79769i) q^{8} +(-0.697981 - 2.91767i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 2 q^{3} - 4 q^{6} - 4 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/195\mathbb{Z}\right)^\times\).

\(n\)	\(106\)	\(131\)	\(157\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.395659	+	1.47662i	−0.279773	+	1.04413i	0.672801	+	0.739823i	\(0.265091\pi\)
							−0.952575	+	0.304305i	\(0.901576\pi\)
\(3\)	−1.07285	+	1.35978i	−0.619411	+	0.785067i
							\(5\)	1.88815	−	1.19787i	0.844407	−	0.535702i
\(7\)	1.22324	+	4.56520i	0.462342	+	1.72548i								0.665556	+	0.746348i	\(0.268195\pi\)
							−0.203214	+	0.979134i	\(0.565139\pi\)
\(11\)	−1.33359	+	0.769949i	−0.402093	+	0.232148i	−0.687387	−	0.726292i	\(-0.741242\pi\)
							0.285294	+	0.958440i	\(0.407909\pi\)
\(13\)	0.402017	−	3.58307i	0.111499	−	0.993765i
							\(17\)	0.302869	−	0.0811535i	0.0734565	−	0.0196826i	−0.221904	−	0.975069i	\(-0.571227\pi\)
0.295360	+	0.955386i	\(0.404560\pi\)
\(19\)	0.272564	+	0.157365i	0.0625304	+	0.0361019i	0.530939	−	0.847410i	\(-0.321839\pi\)
							−0.468409	+	0.883512i	\(0.655173\pi\)
\(23\)	−4.55377	−	1.22018i	−0.949526	−	0.254425i	−0.249365	−	0.968410i	\(-0.580222\pi\)
							−0.700161	+	0.713985i	\(0.746889\pi\)
\(29\)	4.22484	+	7.31764i	0.784534	+	1.35885i	0.929277	+	0.369383i	\(0.120431\pi\)
							−0.144744	+	0.989469i	\(0.546236\pi\)
\(31\)	0.517123			0.0928780			0.0464390	−	0.998921i	\(-0.485213\pi\)
							0.0464390	+	0.998921i	\(0.485213\pi\)
\(37\)	5.88304	+	1.57635i	0.967165	+	0.259151i	0.707630	−	0.706583i	\(-0.249764\pi\)
							0.259535	+	0.965734i	\(0.416431\pi\)
\(41\)	7.02151	−	4.05387i	1.09658	−	0.633108i	0.161256	−	0.986913i	\(-0.448446\pi\)
							0.935319	+	0.353805i	\(0.115112\pi\)
\(43\)	3.57884	−	0.958947i	0.545768	−	0.146238i	0.0246079	−	0.999697i	\(-0.492166\pi\)
							0.521160	+	0.853459i	\(0.325500\pi\)
\(47\)	−0.501957	−	0.501957i	−0.0732179	−	0.0732179i	0.669550	−	0.742767i	\(-0.266487\pi\)
							−0.742767	+	0.669550i	\(0.766487\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	195.2.bl.a.68.7	✓	96
3.2	odd	2	inner	195.2.bl.a.68.18	yes	96
5.2	odd	4	inner	195.2.bl.a.107.7	yes	96
5.3	odd	4		975.2.bt.m.107.18		96
5.4	even	2		975.2.bt.m.68.18		96
13.9	even	3	inner	195.2.bl.a.113.18	yes	96
15.2	even	4	inner	195.2.bl.a.107.18	yes	96
15.8	even	4		975.2.bt.m.107.7		96
15.14	odd	2		975.2.bt.m.68.7		96
39.35	odd	6	inner	195.2.bl.a.113.7	yes	96
65.9	even	6		975.2.bt.m.893.7		96
65.22	odd	12	inner	195.2.bl.a.152.18	yes	96
65.48	odd	12		975.2.bt.m.932.7		96
195.74	odd	6		975.2.bt.m.893.18		96
195.113	even	12		975.2.bt.m.932.18		96
195.152	even	12	inner	195.2.bl.a.152.7	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.bl.a.68.7	✓	96	1.1	even	1	trivial
195.2.bl.a.68.18	yes	96	3.2	odd	2	inner
195.2.bl.a.107.7	yes	96	5.2	odd	4	inner
195.2.bl.a.107.18	yes	96	15.2	even	4	inner
195.2.bl.a.113.7	yes	96	39.35	odd	6	inner
195.2.bl.a.113.18	yes	96	13.9	even	3	inner
195.2.bl.a.152.7	yes	96	195.152	even	12	inner
195.2.bl.a.152.18	yes	96	65.22	odd	12	inner
975.2.bt.m.68.7		96	15.14	odd	2
975.2.bt.m.68.18		96	5.4	even	2
975.2.bt.m.107.7		96	15.8	even	4
975.2.bt.m.107.18		96	5.3	odd	4
975.2.bt.m.893.7		96	65.9	even	6
975.2.bt.m.893.18		96	195.74	odd	6
975.2.bt.m.932.7		96	65.48	odd	12
975.2.bt.m.932.18		96	195.113	even	12