Embedded newform 91.10.bc.a.6.14

• Label: 91.10.bc.a.6.14
• Level: $91$
• Weight: $10$
• Character: 91.6
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $328$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.bc (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(328\)
Relative dimension:	\(82\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			6.14
Character	\(\chi\)	\(=\)	91.6
Dual form			91.10.bc.a.76.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.71884 - 32.5391i) q^{2} +(158.391 + 91.4473i) q^{3} +(-539.372 + 311.407i) q^{4} +(1215.97 + 1215.97i) q^{5} +(1594.63 - 5951.23i) q^{6} +(6257.07 - 1096.69i) q^{7} +(2639.61 + 2639.61i) q^{8} +(6883.72 + 11922.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 8 q^{2} - 12 q^{4} - 916 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.71884	−	32.5391i	−0.385322	−	1.43804i	−0.837659	−	0.546193i	\(-0.816076\pi\)
							0.452337	−	0.891847i	\(-0.350590\pi\)
\(3\)	158.391	+	91.4473i	1.12898	+	0.651816i	0.943678	−	0.330865i	\(-0.107341\pi\)
							0.185301	+	0.982682i	\(0.440674\pi\)
\(5\)	1215.97	+	1215.97i	0.870079	+	0.870079i	0.992481	−	0.122402i	\(-0.0390597\pi\)
							−0.122402	+	0.992481i	\(0.539060\pi\)
\(7\)	6257.07	−	1096.69i	0.984985	−	0.172640i
							\(11\)	36981.3	−	9909.11i	0.761579	−	0.204064i	0.142931	−	0.989733i	\(-0.454347\pi\)
0.618648	+	0.785668i	\(0.287681\pi\)
\(13\)	102978.	+	249.818i	0.999997	+	0.00242593i
							\(17\)	−275083.	−	476457.i	−0.798809	−	1.38358i	−0.920392	−	0.390997i	\(-0.872130\pi\)
0.121583	−	0.992581i	\(-0.461203\pi\)
\(19\)	−175336.	+	654362.i	−0.308659	+	1.15193i	0.621090	+	0.783739i	\(0.286690\pi\)
							−0.929749	+	0.368193i	\(0.879976\pi\)
\(23\)	304252.	+	175660.i	0.226703	+	0.130887i	0.609050	−	0.793132i	\(-0.291551\pi\)
							−0.382347	+	0.924019i	\(0.624884\pi\)
\(29\)	−373171.	+	646351.i	−0.0979754	+	0.169698i	−0.910847	−	0.412745i	\(-0.864570\pi\)
							0.812871	+	0.582444i	\(0.197903\pi\)
\(31\)	4.10316e6	+	4.10316e6i	0.797978	+	0.797978i	0.982776	−	0.184799i	\(-0.0591633\pi\)
							−0.184799	+	0.982776i	\(0.559163\pi\)
\(37\)	1.82720e7	−	4.89597e6i	1.60280	−	0.429468i	0.656911	−	0.753968i	\(-0.271863\pi\)
							0.945886	+	0.324500i	\(0.105196\pi\)
\(41\)	2.14424e7	−	5.74548e6i	1.18508	−	0.317540i	0.388138	−	0.921601i	\(-0.373118\pi\)
							0.796938	+	0.604061i	\(0.206452\pi\)
\(43\)	−3.46340e7	+	1.99960e7i	−1.54488	+	0.891937i	−0.546360	+	0.837550i	\(0.683987\pi\)
							−0.998520	+	0.0543869i	\(0.982680\pi\)
\(47\)	−2.80533e7	+	2.80533e7i	−0.838577	+	0.838577i	−0.988672	−	0.150095i	\(-0.952042\pi\)
							0.150095	+	0.988672i	\(0.452042\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bc.a.6.14	yes	328
7.6	odd	2	inner	91.10.bc.a.6.13	✓	328
13.11	odd	12	inner	91.10.bc.a.76.13	yes	328
91.76	even	12	inner	91.10.bc.a.76.14	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bc.a.6.13	✓	328	7.6	odd	2	inner
91.10.bc.a.6.14	yes	328	1.1	even	1	trivial
91.10.bc.a.76.13	yes	328	13.11	odd	12	inner
91.10.bc.a.76.14	yes	328	91.76	even	12	inner