Embedded newform 847.2.u.a.23.9

• Label: 847.2.u.a.23.9
• Level: $847$
• Weight: $2$
• Character: 847.23
• Analytic conductor: $6.763$
• Analytic rank: $0$
• Dimension: $1720$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.u (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(1720\)
Relative dimension:	\(86\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			23.9
Character	\(\chi\)	\(=\)	847.23
Dual form			847.2.u.a.221.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30730 - 1.83585i) q^{2} +(1.04438 + 1.80891i) q^{3} +(-1.00717 + 2.91001i) q^{4} +(-0.634596 + 0.605086i) q^{5} +(1.95557 - 4.28211i) q^{6} +(-0.553085 + 2.58730i) q^{7} +(2.33411 - 0.685355i) q^{8} +(-0.681441 + 1.18029i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1720 q - 11 q^{2} - 20 q^{3} + 73 q^{4} - 7 q^{5} + 8 q^{6} - 24 q^{7} - 32 q^{8} - 844 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30730	−	1.83585i	−0.924402	−	1.29814i	−0.954381	−	0.298590i	\(-0.903484\pi\)
							0.0299794	−	0.999551i	\(-0.490456\pi\)
\(3\)	1.04438	+	1.80891i	0.602971	+	1.04438i	0.992369	+	0.123306i	\(0.0393497\pi\)
							−0.389398	+	0.921070i	\(0.627317\pi\)
\(5\)	−0.634596	+	0.605086i	−0.283800	+	0.270603i	−0.818585	−	0.574385i	\(-0.805241\pi\)
							0.534785	+	0.844988i	\(0.320393\pi\)
\(7\)	−0.553085	+	2.58730i	−0.209046	+	0.977906i
							\(11\)	3.30578	+	0.268015i	0.996730	+	0.0808094i
\(13\)	−1.88145	−	2.17130i	−0.521819	−	0.602211i								0.432266	−	0.901746i	\(-0.357714\pi\)
							−0.954085	+	0.299535i	\(0.903169\pi\)
\(17\)	−0.183534	+	3.85286i	−0.0445136	+	0.934455i	0.860599	+	0.509283i	\(0.170089\pi\)
							−0.905113	+	0.425172i	\(0.860214\pi\)
\(19\)	−0.187054	−	3.92674i	−0.0429131	−	0.900856i	−0.913000	−	0.407958i	\(-0.866241\pi\)
							0.870087	−	0.492897i	\(-0.164062\pi\)
\(23\)	1.50932	+	1.18694i	0.314714	+	0.247494i	0.763003	−	0.646395i	\(-0.223724\pi\)
							−0.448289	+	0.893889i	\(0.647966\pi\)
\(29\)	−7.14455	+	4.59152i	−1.32671	+	0.852625i	−0.995846	−	0.0910537i	\(-0.970977\pi\)
							−0.330864	+	0.943678i	\(0.607340\pi\)
\(31\)	−1.08421	−	0.208964i	−0.194729	−	0.0375310i	0.0909540	−	0.995855i	\(-0.471008\pi\)
							−0.285683	+	0.958324i	\(0.592220\pi\)
\(37\)	−0.936190	−	2.70494i	−0.153909	−	0.444690i	0.841517	−	0.540231i	\(-0.181663\pi\)
							−0.995425	+	0.0955410i	\(0.969542\pi\)
\(41\)	−4.67061	+	10.2272i	−0.729427	+	1.59722i	0.0707759	+	0.997492i	\(0.477452\pi\)
							−0.800203	+	0.599730i	\(0.795275\pi\)
\(43\)	−6.93224	+	2.03549i	−1.05716	+	0.310409i	−0.763706	−	0.645565i	\(-0.776622\pi\)
							−0.293451	+	0.955974i	\(0.594804\pi\)
\(47\)	7.84750	+	0.749345i	1.14468	+	0.109303i	0.650112	−	0.759838i	\(-0.274722\pi\)
							0.494563	+	0.869142i	\(0.335328\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.u.a.23.9	✓	1720
7.4	even	3	inner	847.2.u.a.144.78	yes	1720
121.100	even	11	inner	847.2.u.a.100.78	yes	1720
847.221	even	33	inner	847.2.u.a.221.9	yes	1720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.u.a.23.9	✓	1720	1.1	even	1	trivial
847.2.u.a.100.78	yes	1720	121.100	even	11	inner
847.2.u.a.144.78	yes	1720	7.4	even	3	inner
847.2.u.a.221.9	yes	1720	847.221	even	33	inner