Embedded newform 91.10.bb.a.5.7

• Label: 91.10.bb.a.5.7
• 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			5.7
Character	\(\chi\)	\(=\)	91.5
Dual form			91.10.bb.a.73.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.1645 - 37.9344i) q^{2} +(9.27740 - 5.35631i) q^{3} +(-892.300 + 515.170i) q^{4} +(2307.23 - 618.220i) q^{5} +(-297.489 - 297.489i) q^{6} +(1373.49 - 6202.19i) q^{7} +(14394.3 + 14394.3i) q^{8} +(-9784.12 + 16946.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 2 q^{2} - 12 q^{3} - 6 q^{5} + 3872 q^{7} - 2056 q^{8} + 1023512 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.1645	−	37.9344i	−0.449212	−	1.67648i	−0.704570	−	0.709635i	\(-0.748860\pi\)
							0.255358	−	0.966847i	\(-0.417807\pi\)
\(3\)	9.27740	−	5.35631i	0.0661273	−	0.0381786i	−0.466572	−	0.884483i	\(-0.654511\pi\)
							0.532699	+	0.846305i	\(0.321178\pi\)
\(5\)	2307.23	−	618.220i	1.65092	−	0.442362i	0.691048	−	0.722809i	\(-0.257149\pi\)
							0.959870	+	0.280446i	\(0.0904825\pi\)
\(7\)	1373.49	−	6202.19i	0.216214	−	0.976346i
							\(11\)	−11264.0	+	42037.8i	−0.231967	+	0.865711i	0.747526	+	0.664232i	\(0.231241\pi\)
−0.979493	+	0.201479i	\(0.935425\pi\)
\(13\)	102109.	−	13352.5i	0.991558	−	0.129663i
							\(17\)	54213.4	+	93900.4i	0.157430	+	0.272676i	0.933941	−	0.357427i	\(-0.116346\pi\)
−0.776511	+	0.630103i	\(0.783013\pi\)
\(19\)	915315.	−	245258.i	1.61131	−	0.431749i	0.662878	−	0.748728i	\(-0.269335\pi\)
							0.948433	+	0.316978i	\(0.102668\pi\)
\(23\)	79435.3	+	45862.0i	0.0591886	+	0.0341726i	0.529302	−	0.848433i	\(-0.322454\pi\)
							−0.470114	+	0.882606i	\(0.655787\pi\)
\(29\)	−2.20182e6			−0.578085			−0.289043	−	0.957316i	\(-0.593337\pi\)
							−0.289043	+	0.957316i	\(0.593337\pi\)
\(31\)	1.66636e6	−	6.21894e6i	0.324072	−	1.20945i	−0.591170	−	0.806547i	\(-0.701334\pi\)
							0.915242	−	0.402905i	\(-0.131999\pi\)
\(37\)	1.56905e7	−	4.20427e6i	1.37635	−	0.368793i	0.506559	−	0.862206i	\(-0.330917\pi\)
							0.869795	+	0.493413i	\(0.164251\pi\)
\(41\)	4.51832e6	+	4.51832e6i	0.249718	+	0.249718i	0.820855	−	0.571137i	\(-0.193498\pi\)
							−0.571137	+	0.820855i	\(0.693498\pi\)
\(43\)		−	2.35316e7i		−	1.04965i	−0.851211	−	0.524824i	\(-0.824131\pi\)
							0.851211	−	0.524824i	\(-0.175869\pi\)
\(47\)	1.59208e7	+	5.94172e7i	0.475910	+	1.77612i	0.617906	+	0.786252i	\(0.287981\pi\)
							−0.141996	+	0.989867i	\(0.545352\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.bb.a.5.7	✓	328
7.3	odd	6	inner	91.10.bb.a.31.76	yes	328
13.8	odd	4	inner	91.10.bb.a.47.76	yes	328
91.73	even	12	inner	91.10.bb.a.73.7	yes	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.bb.a.5.7	✓	328	1.1	even	1	trivial
91.10.bb.a.31.76	yes	328	7.3	odd	6	inner
91.10.bb.a.47.76	yes	328	13.8	odd	4	inner
91.10.bb.a.73.7	yes	328	91.73	even	12	inner