Embedded newform 91.3.v.a.68.1

• Label: 91.3.v.a.68.1
• Level: $91$
• Weight: $3$
• Character: 91.68
• Analytic conductor: $2.480$
• Analytic rank: $0$
• Dimension: $34$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	91.v (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.47957040568\)
Analytic rank:	\(0\)
Dimension:	\(34\)
Relative dimension:	\(17\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			68.1
Character	\(\chi\)	\(=\)	91.68
Dual form			91.3.v.a.87.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-3.68962 q^{2} +(4.29521 + 2.47984i) q^{3} +9.61327 q^{4} +(4.79181 + 2.76656i) q^{5} +(-15.8477 - 9.14965i) q^{6} +(-0.478190 - 6.98365i) q^{7} -20.7108 q^{8} +(7.79920 + 13.5086i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 2 q^{2} + 62 q^{4} - 6 q^{5} - 24 q^{6} - 7 q^{7} - 16 q^{8} + 41 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.68962			−1.84481			−0.922404	−	0.386226i	\(-0.873778\pi\)
							−0.922404	+	0.386226i	\(0.873778\pi\)
\(3\)	4.29521	+	2.47984i	1.43174	+	0.826613i	0.997253	−	0.0740673i	\(-0.0235980\pi\)
							0.434482	+	0.900680i	\(0.356931\pi\)
\(5\)	4.79181	+	2.76656i	0.958363	+	0.553311i	0.895669	−	0.444722i	\(-0.146697\pi\)
							0.0626941	+	0.998033i	\(0.480031\pi\)
\(7\)	−0.478190	−	6.98365i	−0.0683128	−	0.997664i
							\(11\)	0.190966	−	0.330762i	0.0173605	−	0.0300693i	−0.857215	−	0.514959i	\(-0.827807\pi\)
0.874575	+	0.484890i	\(0.161140\pi\)
\(13\)	6.29536	−	11.3740i	0.484258	−	0.874925i
							\(17\)			16.8956i			0.993861i	0.867790	+	0.496930i	\(0.165540\pi\)
−0.867790	+	0.496930i	\(0.834460\pi\)
\(19\)	−12.3384	+	7.12356i	−0.649388	+	0.374924i	−0.788222	−	0.615392i	\(-0.788998\pi\)
							0.138834	+	0.990316i	\(0.455665\pi\)
\(23\)	−11.0043			−0.478447			−0.239223	−	0.970965i	\(-0.576893\pi\)
							−0.239223	+	0.970965i	\(0.576893\pi\)
\(29\)	−2.97117	−	5.14621i	−0.102454	−	0.177456i	0.810241	−	0.586097i	\(-0.199336\pi\)
							−0.912695	+	0.408641i	\(0.866003\pi\)
\(31\)	−29.7542	+	17.1786i	−0.959814	+	0.554149i	−0.896116	−	0.443821i	\(-0.853623\pi\)
							−0.0636980	+	0.997969i	\(0.520289\pi\)
\(37\)	47.9377			1.29561			0.647807	−	0.761804i	\(-0.275686\pi\)
							0.647807	+	0.761804i	\(0.275686\pi\)
\(41\)	−61.4105	+	35.4554i	−1.49782	+	0.864765i	−0.999997	−	0.00251446i	\(-0.999200\pi\)
							−0.497821	+	0.867280i	\(0.665866\pi\)
\(43\)	27.2104	−	47.1297i	0.632799	−	1.09604i	−0.354178	−	0.935178i	\(-0.615239\pi\)
							0.986977	−	0.160862i	\(-0.0514273\pi\)
\(47\)	−1.32664	−	0.765936i	−0.0282264	−	0.0162965i	0.485820	−	0.874059i	\(-0.338521\pi\)
							−0.514047	+	0.857762i	\(0.671854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.3.v.a.68.1	yes	34
7.3	odd	6		91.3.m.a.3.17	✓	34
13.9	even	3		91.3.m.a.61.17	yes	34
91.87	odd	6	inner	91.3.v.a.87.1	yes	34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.3.m.a.3.17	✓	34	7.3	odd	6
91.3.m.a.61.17	yes	34	13.9	even	3
91.3.v.a.68.1	yes	34	1.1	even	1	trivial
91.3.v.a.87.1	yes	34	91.87	odd	6	inner