Embedded newform 91.10.i.a.34.6

• Label: 91.10.i.a.34.6
• Level: $91$
• Weight: $10$
• Character: 91.34
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			34.6
Character	\(\chi\)	\(=\)	91.34
Dual form			91.10.i.a.83.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-29.6596 - 29.6596i) q^{2} +221.983i q^{3} +1247.39i q^{4} +(1582.09 - 1582.09i) q^{5} +(6583.93 - 6583.93i) q^{6} +(4478.53 + 4505.15i) q^{7} +(21811.2 - 21811.2i) q^{8} -29593.5 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 4 q^{2} - 1142 q^{7} + 2044 q^{8} - 1023524 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−29.6596	−	29.6596i	−1.31078	−	1.31078i	−0.920839	−	0.389943i	\(-0.872495\pi\)
							−0.389943	−	0.920839i	\(-0.627505\pi\)
\(3\)			221.983i			1.58225i	0.611656	+	0.791124i	\(0.290504\pi\)
							−0.611656	+	0.791124i	\(0.709496\pi\)
\(5\)	1582.09	−	1582.09i	1.13205	−	1.13205i	0.142219	−	0.989835i	\(-0.454576\pi\)
							0.989835	−	0.142219i	\(-0.0454237\pi\)
\(7\)	4478.53	+	4505.15i	0.705008	+	0.709199i
							\(11\)	−47987.5	+	47987.5i	−0.988237	+	0.988237i	−0.999932	−	0.0116944i	\(-0.996277\pi\)
0.0116944	+	0.999932i	\(0.496277\pi\)
\(13\)	43665.9	−	93262.0i	0.424030	−	0.905648i
							\(17\)	−549181.			−1.59476			−0.797380	−	0.603478i	\(-0.793781\pi\)
−0.797380	+	0.603478i	\(0.793781\pi\)
\(19\)	204014.	−	204014.i	0.359144	−	0.359144i	−0.504353	−	0.863498i	\(-0.668269\pi\)
							0.863498	+	0.504353i	\(0.168269\pi\)
\(23\)		−	153127.i		−	0.114097i	−0.998371	−	0.0570487i	\(-0.981831\pi\)
							0.998371	−	0.0570487i	\(-0.0181690\pi\)
\(29\)	−4.29972e6			−1.12888			−0.564442	−	0.825472i	\(-0.690909\pi\)
							−0.564442	+	0.825472i	\(0.690909\pi\)
\(31\)	2.34377e6	−	2.34377e6i	0.455813	−	0.455813i	−0.441465	−	0.897278i	\(-0.645541\pi\)
							0.897278	+	0.441465i	\(0.145541\pi\)
\(37\)	−9.25540e6	+	9.25540e6i	−0.811871	+	0.811871i	−0.984914	−	0.173043i	\(-0.944640\pi\)
							0.173043	+	0.984914i	\(0.444640\pi\)
\(41\)	4.54984e6	−	4.54984e6i	0.251460	−	0.251460i	−0.570109	−	0.821569i	\(-0.693099\pi\)
							0.821569	+	0.570109i	\(0.193099\pi\)
\(43\)		−	3.20889e6i		−	0.143135i	−0.997436	−	0.0715676i	\(-0.977200\pi\)
							0.997436	−	0.0715676i	\(-0.0228002\pi\)
\(47\)	1.76121e7	+	1.76121e7i	0.526465	+	0.526465i	0.919517	−	0.393051i	\(-0.128581\pi\)
							−0.393051	+	0.919517i	\(0.628581\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.i.a.34.6	yes	164
7.6	odd	2	inner	91.10.i.a.34.5	✓	164
13.5	odd	4	inner	91.10.i.a.83.6	yes	164
91.83	even	4	inner	91.10.i.a.83.5	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.i.a.34.5	✓	164	7.6	odd	2	inner
91.10.i.a.34.6	yes	164	1.1	even	1	trivial
91.10.i.a.83.5	yes	164	91.83	even	4	inner
91.10.i.a.83.6	yes	164	13.5	odd	4	inner