Embedded newform 91.8.h.a.16.8

• Label: 91.8.h.a.16.8
• Level: $91$
• Weight: $8$
• Character: 91.16
• Analytic conductor: $28.427$
• Analytic rank: $0$
• Dimension: $126$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	91.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.4270373191\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(63\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			16.8
Character	\(\chi\)	\(=\)	91.16
Dual form			91.8.h.a.74.8

$q$-expansion

\(f(q)\)	\(=\)	\(q-18.1736 q^{2} +(6.23452 - 10.7985i) q^{3} +202.279 q^{4} +(-132.496 + 229.490i) q^{5} +(-113.303 + 196.247i) q^{6} +(592.599 + 687.291i) q^{7} -1349.91 q^{8} +(1015.76 + 1759.35i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 2 q^{2} - 80 q^{3} + 7678 q^{4} - 2 q^{5} - 258 q^{6} + 953 q^{7} - 264 q^{8} - 41707 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.1736			−1.60633			−0.803166	−	0.595756i	\(-0.796853\pi\)
							−0.803166	+	0.595756i	\(0.796853\pi\)
\(3\)	6.23452	−	10.7985i	0.133315	−	0.230908i	−0.791638	−	0.610991i	\(-0.790771\pi\)
							0.924952	+	0.380083i	\(0.124105\pi\)
\(5\)	−132.496	+	229.490i	−0.474033	+	0.821049i	−0.999558	−	0.0297289i	\(-0.990536\pi\)
							0.525525	+	0.850778i	\(0.323869\pi\)
\(7\)	592.599	+	687.291i	0.653007	+	0.757352i
							\(11\)	−2763.82	+	4787.07i	−0.626087	+	1.08441i	0.362243	+	0.932084i	\(0.382011\pi\)
−0.988330	+	0.152331i	\(0.951322\pi\)
\(13\)	−890.416	+	7871.19i	−0.112406	+	0.993662i
							\(17\)	26144.7			1.29066			0.645330	−	0.763904i	\(-0.276720\pi\)
0.645330	+	0.763904i	\(0.276720\pi\)
\(19\)	−9216.43	−	15963.3i	−0.308266	−	0.533932i	0.669717	−	0.742616i	\(-0.266415\pi\)
							−0.977983	+	0.208684i	\(0.933082\pi\)
\(23\)	100280.			1.71857			0.859287	−	0.511493i	\(-0.170908\pi\)
							0.859287	+	0.511493i	\(0.170908\pi\)
\(29\)	63073.9	+	109247.i	0.480238	+	0.831797i	0.999743	−	0.0226709i	\(-0.00721698\pi\)
							−0.519505	+	0.854467i	\(0.673884\pi\)
\(31\)	−136119.	−	235766.i	−0.820642	−	1.42139i	−0.905205	−	0.424976i	\(-0.860283\pi\)
							0.0845628	−	0.996418i	\(-0.473051\pi\)
\(37\)	−246690.			−0.800656			−0.400328	−	0.916372i	\(-0.631104\pi\)
							−0.400328	+	0.916372i	\(0.631104\pi\)
\(41\)	−85514.4	−	148115.i	−0.193774	−	0.335627i	0.752724	−	0.658336i	\(-0.228740\pi\)
							−0.946498	+	0.322710i	\(0.895406\pi\)
\(43\)	−191958.	+	332481.i	−0.368186	+	0.637716i	−0.989282	−	0.146018i	\(-0.953354\pi\)
							0.621096	+	0.783734i	\(0.286688\pi\)
\(47\)	−357673.	+	619508.i	−0.502509	+	0.870371i	0.497487	+	0.867471i	\(0.334256\pi\)
							−0.999996	+	0.00289941i	\(0.999077\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.h.a.16.8	yes	126
7.4	even	3		91.8.g.a.81.56	yes	126
13.9	even	3		91.8.g.a.9.56	✓	126
91.74	even	3	inner	91.8.h.a.74.8	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.g.a.9.56	✓	126	13.9	even	3
91.8.g.a.81.56	yes	126	7.4	even	3
91.8.h.a.16.8	yes	126	1.1	even	1	trivial
91.8.h.a.74.8	yes	126	91.74	even	3	inner