Embedded newform 91.8.h.a.16.15

• Label: 91.8.h.a.16.15
• 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			16.15
Character	\(\chi\)	\(=\)	91.16
Dual form			91.8.h.a.74.15

$q$-expansion

\(f(q)\)	\(=\)	\(q-14.2658 q^{2} +(42.8483 - 74.2155i) q^{3} +75.5130 q^{4} +(-151.993 + 263.260i) q^{5} +(-611.265 + 1058.74i) q^{6} +(-754.932 - 503.607i) q^{7} +748.769 q^{8} +(-2578.46 - 4466.02i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 2 q^{2} - 80 q^{3} + 7678 q^{4} - 2 q^{5} - 258 q^{6} + 953 q^{7} - 264 q^{8} - 41707 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.2658			−1.26093			−0.630465	−	0.776218i	\(-0.717136\pi\)
							−0.630465	+	0.776218i	\(0.717136\pi\)
\(3\)	42.8483	−	74.2155i	0.916240	−	1.58697i	0.111165	−	0.993802i	\(-0.464542\pi\)
							0.805075	−	0.593173i	\(-0.202125\pi\)
\(5\)	−151.993	+	263.260i	−0.543787	+	0.941867i	0.454895	+	0.890545i	\(0.349677\pi\)
							−0.998682	+	0.0513217i	\(0.983657\pi\)
\(7\)	−754.932	−	503.607i	−0.831888	−	0.554943i
							\(11\)	−1091.96	+	1891.34i	−0.247362	+	0.428444i	−0.962793	−	0.270239i	\(-0.912897\pi\)
0.715431	+	0.698684i	\(0.246231\pi\)
\(13\)	1057.17	+	7850.54i	0.133457	+	0.991055i
							\(17\)	6687.92			0.330156			0.165078	−	0.986280i	\(-0.447212\pi\)
0.165078	+	0.986280i	\(0.447212\pi\)
\(19\)	2761.42	+	4782.93i	0.0923625	+	0.159977i	0.908505	−	0.417874i	\(-0.137225\pi\)
							−0.816142	+	0.577851i	\(0.803891\pi\)
\(23\)	9538.00			0.163459			0.0817297	−	0.996655i	\(-0.473956\pi\)
							0.0817297	+	0.996655i	\(0.473956\pi\)
\(29\)	−1098.35	−	1902.40i	−0.00836272	−	0.0144847i	0.861814	−	0.507225i	\(-0.169329\pi\)
							−0.870176	+	0.492740i	\(0.835995\pi\)
\(31\)	74753.7	+	129477.i	0.450678	+	0.780598i	0.998428	−	0.0560443i	\(-0.0178488\pi\)
							−0.547750	+	0.836642i	\(0.684515\pi\)
\(37\)	−344261.			−1.11733			−0.558665	−	0.829393i	\(-0.688686\pi\)
							−0.558665	+	0.829393i	\(0.688686\pi\)
\(41\)	334378.	+	579160.i	0.757695	+	1.31237i	0.944023	+	0.329879i	\(0.107008\pi\)
							−0.186328	+	0.982488i	\(0.559659\pi\)
\(43\)	393269.	−	681161.i	0.754310	−	1.30650i	−0.191407	−	0.981511i	\(-0.561305\pi\)
							0.945717	−	0.324992i	\(-0.105362\pi\)
\(47\)	289409.	−	501272.i	0.406602	−	0.704256i	−0.587904	−	0.808931i	\(-0.700047\pi\)
							0.994507	+	0.104675i	\(0.0333801\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.h.a.16.15	yes	126
7.4	even	3		91.8.g.a.81.49	yes	126
13.9	even	3		91.8.g.a.9.49	✓	126
91.74	even	3	inner	91.8.h.a.74.15	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.g.a.9.49	✓	126	13.9	even	3
91.8.g.a.81.49	yes	126	7.4	even	3
91.8.h.a.16.15	yes	126	1.1	even	1	trivial
91.8.h.a.74.15	yes	126	91.74	even	3	inner