Embedded newform 9.9.d.a.2.2

• Label: 9.9.d.a.2.2
• Level: $9$
• Weight: $9$
• Character: 9.2
• Analytic conductor: $3.666$
• Analytic rank: $0$
• Dimension: $14$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	9.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.66640749055\)
Analytic rank:	\(0\)
Dimension:	\(14\)
Relative dimension:	\(7\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial:	x^14 - x^13 + 427*x^12 - 1362*x^11 + 135762*x^10 - 371244*x^9 + 18261508*x^8 - 29352736*x^7 + 1757083840*x^6 - 1434930432*x^5 + 61638378240*x^4 + 183604924416*x^3 + 1254910111744*x^2 + 1078377512960*x + 872385888256
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{21} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2.2
Root			\(5.69757 + 9.86849i\) of defining polynomial
Character	\(\chi\)	\(=\)	9.2
Dual form			9.9.d.a.5.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-17.0927 - 9.86849i) q^{2} +(51.0315 - 62.9030i) q^{3} +(66.7741 + 115.656i) q^{4} +(-896.557 + 517.627i) q^{5} +(-1493.02 + 571.581i) q^{6} +(-21.9132 + 37.9547i) q^{7} +2416.83i q^{8} +(-1352.58 - 6420.07i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{2} - 93 q^{3} + 767 q^{4} + 438 q^{5} - 2259 q^{6} + 922 q^{7} + 17973 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−17.0927	−	9.86849i	−1.06830	−	0.616781i	−0.140580	−	0.990069i	\(-0.544897\pi\)
							−0.927715	+	0.373289i	\(0.878230\pi\)
\(3\)	51.0315	−	62.9030i	0.630018	−	0.776581i
							\(5\)	−896.557	+	517.627i	−1.43449	+	0.828203i	−0.997459	−	0.0712428i	\(-0.977303\pi\)
−0.437031	+	0.899446i	\(0.643970\pi\)
\(7\)	−21.9132	+	37.9547i	−0.00912668	+	0.0158079i	−0.870553	−	0.492075i	\(-0.836238\pi\)
							0.861426	+	0.507883i	\(0.169572\pi\)
\(11\)	−15260.9	−	8810.88i	−1.04234	−	0.601795i	−0.121844	−	0.992549i	\(-0.538881\pi\)
							−0.920495	+	0.390754i	\(0.872214\pi\)
\(13\)	616.107	+	1067.13i	0.0215716	+	0.0373632i	0.876610	−	0.481202i	\(-0.159800\pi\)
							−0.855038	+	0.518565i	\(0.826466\pi\)
\(17\)		−	106083.i		−	1.27014i	−0.772454	−	0.635070i	\(-0.780971\pi\)
							0.772454	−	0.635070i	\(-0.219029\pi\)
\(19\)	−115788.			−0.888483			−0.444241	−	0.895907i	\(-0.646527\pi\)
							−0.444241	+	0.895907i	\(0.646527\pi\)
\(23\)	−97603.8	+	56351.6i	−0.348783	+	0.201370i	−0.664149	−	0.747600i	\(-0.731206\pi\)
							0.315366	+	0.948970i	\(0.397873\pi\)
\(29\)	720846.	+	416181.i	1.01918	+	0.588423i	0.913866	−	0.406016i	\(-0.133082\pi\)
							0.105313	+	0.994439i	\(0.466416\pi\)
\(31\)	128526.	+	222614.i	0.139170	+	0.241049i	0.927183	−	0.374610i	\(-0.122223\pi\)
							−0.788013	+	0.615659i	\(0.788890\pi\)
\(37\)	−2.84152e6			−1.51615			−0.758077	−	0.652166i	\(-0.773861\pi\)
							−0.758077	+	0.652166i	\(0.773861\pi\)
\(41\)	2.51758e6	−	1.45352e6i	0.890938	−	0.514383i	0.0166889	−	0.999861i	\(-0.494688\pi\)
							0.874249	+	0.485477i	\(0.161354\pi\)
\(43\)	791952.	−	1.37170e6i	0.231646	−	0.401223i	−0.726647	−	0.687012i	\(-0.758922\pi\)
							0.958293	+	0.285789i	\(0.0922555\pi\)
\(47\)	1.36669e6	+	789058.i	0.280077	+	0.161703i	0.633458	−	0.773777i	\(-0.281635\pi\)
							−0.353381	+	0.935479i	\(0.614968\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.9.d.a.2.2	✓	14
3.2	odd	2		27.9.d.a.8.6		14
4.3	odd	2		144.9.q.a.65.3		14
9.2	odd	6		81.9.b.a.80.3		14
9.4	even	3		27.9.d.a.17.6		14
9.5	odd	6	inner	9.9.d.a.5.2	yes	14
9.7	even	3		81.9.b.a.80.12		14
12.11	even	2		432.9.q.a.305.7		14
36.23	even	6		144.9.q.a.113.3		14
36.31	odd	6		432.9.q.a.17.7		14

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.9.d.a.2.2	✓	14	1.1	even	1	trivial
9.9.d.a.5.2	yes	14	9.5	odd	6	inner
27.9.d.a.8.6		14	3.2	odd	2
27.9.d.a.17.6		14	9.4	even	3
81.9.b.a.80.3		14	9.2	odd	6
81.9.b.a.80.12		14	9.7	even	3
144.9.q.a.65.3		14	4.3	odd	2
144.9.q.a.113.3		14	36.23	even	6
432.9.q.a.17.7		14	36.31	odd	6
432.9.q.a.305.7		14	12.11	even	2