Embedded newform 91.10.bb.a.5.3

• Label: 91.10.bb.a.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	91.5
Dual form			91.10.bb.a.73.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.8870 - 40.6310i) q^{2} +(120.266 - 69.4354i) q^{3} +(-1088.94 + 628.702i) q^{4} +(-348.043 + 93.2579i) q^{5} +(-4130.57 - 4130.57i) q^{6} +(4183.56 + 4780.32i) q^{7} +(22171.2 + 22171.2i) q^{8} +(-198.943 + 344.579i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 2 q^{2} - 12 q^{3} - 6 q^{5} + 3872 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.8870	−	40.6310i	−0.481144	−	1.79565i	−0.596831	−	0.802367i	\(-0.703574\pi\)
							0.115688	−	0.993286i	\(-0.463093\pi\)
\(3\)	120.266	−	69.4354i	0.857228	−	0.494921i	−0.00585537	−	0.999983i	\(-0.501864\pi\)
							0.863083	+	0.505062i	\(0.168530\pi\)
\(5\)	−348.043	+	93.2579i	−0.249040	+	0.0667299i	−0.381179	−	0.924501i	\(-0.624482\pi\)
							0.132140	+	0.991231i	\(0.457815\pi\)
\(7\)	4183.56	+	4780.32i	0.658574	+	0.752516i
							\(11\)	19401.8	−	72408.6i	0.399554	−	1.49116i	−0.414329	−	0.910127i	\(-0.635984\pi\)
0.813883	−	0.581029i	\(-0.197350\pi\)
\(13\)	43521.3	−	93329.5i	0.422626	−	0.906304i
							\(17\)	−94.2004	−	163.160i	−0.000273547	−	0.000473798i	0.865889	−	0.500237i	\(-0.166754\pi\)
−0.866162	+	0.499763i	\(0.833420\pi\)
\(19\)	865732.	−	231972.i	1.52403	−	0.408361i	0.602961	−	0.797771i	\(-0.293988\pi\)
							0.921065	+	0.389409i	\(0.127321\pi\)
\(23\)	−1.16461e6	−	672387.i	−0.867771	−	0.501008i	−0.00116382	−	0.999999i	\(-0.500370\pi\)
							−0.866607	+	0.498992i	\(0.833704\pi\)
\(29\)	2.17198e6			0.570249			0.285124	−	0.958491i	\(-0.407965\pi\)
							0.285124	+	0.958491i	\(0.407965\pi\)
\(31\)	207271.	−	773547.i	0.0403099	−	0.150439i	−0.942838	−	0.333252i	\(-0.891854\pi\)
							0.983148	+	0.182814i	\(0.0585205\pi\)
\(37\)	5.33203e6	−	1.42871e6i	0.467719	−	0.125325i	−0.0172595	−	0.999851i	\(-0.505494\pi\)
							0.484978	+	0.874526i	\(0.338827\pi\)
\(41\)	−1.59817e7	−	1.59817e7i	−0.883274	−	0.883274i	0.110592	−	0.993866i	\(-0.464725\pi\)
							−0.993866	+	0.110592i	\(0.964725\pi\)
\(43\)		−	3.16594e7i		−	1.41219i	−0.708115	−	0.706097i	\(-0.750454\pi\)
							0.708115	−	0.706097i	\(-0.249546\pi\)
\(47\)	298831.	+	1.11525e6i	0.00893275	+	0.0333375i	0.970248	−	0.242112i	\(-0.0778401\pi\)
							−0.961316	+	0.275449i	\(0.911173\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bb.a.5.3	✓	328
7.3	odd	6	inner	91.10.bb.a.31.80	yes	328
13.8	odd	4	inner	91.10.bb.a.47.80	yes	328
91.73	even	12	inner	91.10.bb.a.73.3	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bb.a.5.3	✓	328	1.1	even	1	trivial
91.10.bb.a.31.80	yes	328	7.3	odd	6	inner
91.10.bb.a.47.80	yes	328	13.8	odd	4	inner
91.10.bb.a.73.3	yes	328	91.73	even	12	inner