Embedded newform 91.10.bb.a.5.4

• Label: 91.10.bb.a.5.4
• Level: $91$
• Weight: $10$
• Character: 91.5
• 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.bb (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			5.4
Character	\(\chi\)	\(=\)	91.5
Dual form			91.10.bb.a.73.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.8713 - 40.5723i) q^{2} +(-111.169 + 64.1836i) q^{3} +(-1084.52 + 626.147i) q^{4} +(-1911.08 + 512.072i) q^{5} +(3812.63 + 3812.63i) q^{6} +(3916.23 - 5001.68i) q^{7} +(21987.4 + 21987.4i) q^{8} +(-1602.44 + 2775.51i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 2 q^{2} - 12 q^{3} - 6 q^{5} + 3872 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.8713	−	40.5723i	−0.480448	−	1.79306i	−0.599736	−	0.800198i	\(-0.704728\pi\)
							0.119288	−	0.992860i	\(-0.461939\pi\)
\(3\)	−111.169	+	64.1836i	−0.792390	+	0.457486i	−0.840803	−	0.541341i	\(-0.817917\pi\)
							0.0484135	+	0.998827i	\(0.484583\pi\)
\(5\)	−1911.08	+	512.072i	−1.36746	+	0.366409i	−0.866549	−	0.499092i	\(-0.833667\pi\)
							−0.500907	+	0.865501i	\(0.667000\pi\)
\(7\)	3916.23	−	5001.68i	0.616491	−	0.787362i
							\(11\)	13880.9	−	51804.3i	0.285859	−	1.06684i	−0.662351	−	0.749194i	\(-0.730441\pi\)
0.948210	−	0.317646i	\(-0.102892\pi\)
\(13\)	21856.9	+	100632.i	0.212248	+	0.977216i
							\(17\)	−334267.	−	578968.i	−0.970675	−	1.68126i	−0.693526	−	0.720432i	\(-0.743944\pi\)
−0.277149	−	0.960827i	\(-0.589390\pi\)
\(19\)	475718.	−	127468.i	0.837450	−	0.224394i	0.185489	−	0.982646i	\(-0.440613\pi\)
							0.651961	+	0.758253i	\(0.273947\pi\)
\(23\)	1.23727e6	+	714340.i	0.921914	+	0.532267i	0.884245	−	0.467023i	\(-0.154673\pi\)
							0.0376687	+	0.999290i	\(0.488007\pi\)
\(29\)	−296855.			−0.0779387			−0.0389693	−	0.999240i	\(-0.512407\pi\)
							−0.0389693	+	0.999240i	\(0.512407\pi\)
\(31\)	−217545.	+	811890.i	−0.0423080	+	0.157896i	−0.983848	−	0.179005i	\(-0.942712\pi\)
							0.941540	+	0.336901i	\(0.109379\pi\)
\(37\)	1.22567e7	−	3.28417e6i	1.07514	−	0.288083i	0.322537	−	0.946557i	\(-0.395464\pi\)
							0.752604	+	0.658474i	\(0.228798\pi\)
\(41\)	1.36506e7	+	1.36506e7i	0.754441	+	0.754441i	0.975305	−	0.220864i	\(-0.0708877\pi\)
							−0.220864	+	0.975305i	\(0.570888\pi\)
\(43\)			2.30594e7i			1.02858i	0.857615	+	0.514292i	\(0.171945\pi\)
							−0.857615	+	0.514292i	\(0.828055\pi\)
\(47\)	−8.98166e6	−	3.35200e7i	−0.268483	−	1.00199i	−0.960084	−	0.279711i	\(-0.909761\pi\)
							0.691602	−	0.722279i	\(-0.256905\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bb.a.5.4	✓	328
7.3	odd	6	inner	91.10.bb.a.31.79	yes	328
13.8	odd	4	inner	91.10.bb.a.47.79	yes	328
91.73	even	12	inner	91.10.bb.a.73.4	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bb.a.5.4	✓	328	1.1	even	1	trivial
91.10.bb.a.31.79	yes	328	7.3	odd	6	inner
91.10.bb.a.47.79	yes	328	13.8	odd	4	inner
91.10.bb.a.73.4	yes	328	91.73	even	12	inner