Embedded newform 91.8.r.a.25.9

• Label: 91.8.r.a.25.9
• Level: $91$
• Weight: $8$
• Character: 91.25
• Analytic conductor: $28.427$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	91.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.4270373191\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			25.9
Character	\(\chi\)	\(=\)	91.25
Dual form			91.8.r.a.51.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.3790 + 8.87907i) q^{2} +(-27.8080 + 48.1649i) q^{3} +(93.6757 - 162.251i) q^{4} +(-158.175 + 91.3222i) q^{5} -987.638i q^{6} +(870.250 - 257.311i) q^{7} +1053.97i q^{8} +(-453.074 - 784.748i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 52 q^{3} + 4094 q^{4} - 44052 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.3790	+	8.87907i	−1.35932	+	0.784806i	−0.989533	−	0.144309i	\(-0.953904\pi\)
							−0.369791	+	0.929115i	\(0.620571\pi\)
\(3\)	−27.8080	+	48.1649i	−0.594629	+	1.02993i	0.398970	+	0.916964i	\(0.369368\pi\)
							−0.993599	+	0.112964i	\(0.963966\pi\)
\(5\)	−158.175	+	91.3222i	−0.565903	+	0.326724i	−0.755511	−	0.655135i	\(-0.772612\pi\)
							0.189608	+	0.981860i	\(0.439278\pi\)
\(7\)	870.250	−	257.311i	0.958960	−	0.283540i
							\(11\)	−1944.13	−	1122.44i	−0.440403	−	0.254267i	0.263366	−	0.964696i	\(-0.415167\pi\)
−0.703769	+	0.710429i	\(0.748501\pi\)
\(13\)	7600.00	+	2233.50i	0.959427	+	0.281958i
							\(17\)	−15739.3	+	27261.3i	−0.776987	+	1.34578i	0.156683	+	0.987649i	\(0.449920\pi\)
−0.933671	+	0.358133i	\(0.883413\pi\)
\(19\)	22087.8	−	12752.4i	0.738781	−	0.426535i	−0.0828449	−	0.996562i	\(-0.526401\pi\)
							0.821626	+	0.570027i	\(0.193067\pi\)
\(23\)	40790.2	+	70650.7i	0.699050	+	1.21079i	0.968796	+	0.247859i	\(0.0797268\pi\)
							−0.269746	+	0.962931i	\(0.586940\pi\)
\(29\)	238832.			1.81844			0.909222	−	0.416311i	\(-0.136677\pi\)
							0.909222	+	0.416311i	\(0.136677\pi\)
\(31\)	167993.	+	96990.9i	1.01280	+	0.584743i	0.912012	−	0.410164i	\(-0.134529\pi\)
							0.100793	+	0.994907i	\(0.467862\pi\)
\(37\)	60694.8	−	35042.1i	0.196990	−	0.113732i	−0.398260	−	0.917272i	\(-0.630386\pi\)
							0.595251	+	0.803540i	\(0.297053\pi\)
\(41\)		−	559370.i		−	1.26752i	−0.773529	−	0.633761i	\(-0.781510\pi\)
							0.773529	−	0.633761i	\(-0.218490\pi\)
\(43\)	−287297.			−0.551050			−0.275525	−	0.961294i	\(-0.588852\pi\)
							−0.275525	+	0.961294i	\(0.588852\pi\)
\(47\)	−1.10854e6	+	640015.i	−1.55743	+	0.899182i	−0.559927	+	0.828542i	\(0.689171\pi\)
							−0.997502	+	0.0706396i	\(0.977496\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.r.a.25.9	✓	128
7.2	even	3	inner	91.8.r.a.51.56	yes	128
13.12	even	2	inner	91.8.r.a.25.56	yes	128
91.51	even	6	inner	91.8.r.a.51.9	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.r.a.25.9	✓	128	1.1	even	1	trivial
91.8.r.a.25.56	yes	128	13.12	even	2	inner
91.8.r.a.51.9	yes	128	91.51	even	6	inner
91.8.r.a.51.56	yes	128	7.2	even	3	inner