Embedded newform 93.2.p.b.11.7

• Label: 93.2.p.b.11.7
• Level: $93$
• Weight: $2$
• Character: 93.11
• Analytic conductor: $0.743$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 93 = 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	93.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.742608738798\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(8\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			11.7
Character	\(\chi\)	\(=\)	93.11
Dual form			93.2.p.b.17.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.72336 - 0.559954i) q^{2} +(-0.230355 + 1.71666i) q^{3} +(1.03839 - 0.754435i) q^{4} +(-0.322909 + 0.186432i) q^{5} +(0.564268 + 3.08742i) q^{6} +(-0.413561 - 3.93477i) q^{7} +(-0.763119 + 1.05034i) q^{8} +(-2.89387 - 0.790885i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{23}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.72336	−	0.559954i	1.21860	−	0.395947i	0.372028	−	0.928221i	\(-0.378662\pi\)
							0.846572	+	0.532274i	\(0.178662\pi\)
\(3\)	−0.230355	+	1.71666i	−0.132996	+	0.991117i
							\(5\)	−0.322909	+	0.186432i	−0.144409	+	0.0833748i	−0.570464	−	0.821323i	\(-0.693237\pi\)
0.426054	+	0.904698i	\(0.359903\pi\)
\(7\)	−0.413561	−	3.93477i	−0.156311	−	1.48720i	−0.738558	−	0.674190i	\(-0.764493\pi\)
							0.582246	−	0.813012i	\(-0.302174\pi\)
\(11\)	2.21406	+	0.985761i	0.667563	+	0.297218i	0.712397	−	0.701776i	\(-0.247609\pi\)
							−0.0448344	+	0.998994i	\(0.514276\pi\)
\(13\)	−0.697234	−	3.28023i	−0.193378	−	0.909772i	−0.962629	−	0.270825i	\(-0.912704\pi\)
							0.769251	−	0.638947i	\(-0.220630\pi\)
\(17\)	−1.61188	+	0.717656i	−0.390939	+	0.174057i	−0.592786	−	0.805360i	\(-0.701972\pi\)
							0.201848	+	0.979417i	\(0.435305\pi\)
\(19\)	−1.48414	−	0.315465i	−0.340486	−	0.0723726i	0.0344954	−	0.999405i	\(-0.489018\pi\)
							−0.374981	+	0.927032i	\(0.622351\pi\)
\(23\)	2.13270	+	1.54950i	0.444699	+	0.323093i	0.787499	−	0.616316i	\(-0.211375\pi\)
							−0.342800	+	0.939408i	\(0.611375\pi\)
\(29\)	2.95987	+	9.10954i	0.549634	+	1.69160i	0.709709	+	0.704495i	\(0.248826\pi\)
							−0.160075	+	0.987105i	\(0.551174\pi\)
\(31\)	5.55127	−	0.428218i	0.997038	−	0.0769101i
							\(37\)	1.46272	+	0.844500i	0.240469	+	0.138835i	0.615392	−	0.788221i	\(-0.288998\pi\)
−0.374923	+	0.927056i	\(0.622331\pi\)
\(41\)	−8.96915	−	8.07586i	−1.40075	−	1.26124i	−0.924104	−	0.382140i	\(-0.875187\pi\)
							−0.476642	−	0.879097i	\(-0.658146\pi\)
\(43\)	2.05874	−	9.68561i	0.313955	−	1.47704i	−0.484396	−	0.874849i	\(-0.660960\pi\)
							0.798351	−	0.602193i	\(-0.205706\pi\)
\(47\)	2.45826	+	0.798737i	0.358574	+	0.116508i	0.482763	−	0.875751i	\(-0.339633\pi\)
							−0.124190	+	0.992259i	\(0.539633\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.p.b.11.7	yes	64
3.2	odd	2	inner	93.2.p.b.11.2	✓	64
31.17	odd	30	inner	93.2.p.b.17.2	yes	64
93.17	even	30	inner	93.2.p.b.17.7	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.b.11.2	✓	64	3.2	odd	2	inner
93.2.p.b.11.7	yes	64	1.1	even	1	trivial
93.2.p.b.17.2	yes	64	31.17	odd	30	inner
93.2.p.b.17.7	yes	64	93.17	even	30	inner