Embedded newform 9.9.d.a.2.4

• Label: 9.9.d.a.2.4
• 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			2.4
Root			\(-0.447645 - 0.775344i\) of defining polynomial
Character	\(\chi\)	\(=\)	9.2
Dual form			9.9.d.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.34294 + 0.775344i) q^{2} +(-44.1317 + 67.9220i) q^{3} +(-126.798 - 219.620i) q^{4} +(-604.549 + 349.037i) q^{5} +(-111.929 + 56.9976i) q^{6} +(-1124.45 + 1947.61i) q^{7} -790.223i q^{8} +(-2665.79 - 5995.02i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{2} - 93 q^{3} + 767 q^{4} + 438 q^{5} - 2259 q^{6} + 922 q^{7} + 17973 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.34294	+	0.775344i	0.0839334	+	0.0484590i	0.541379	−	0.840778i	\(-0.317902\pi\)
							−0.457446	+	0.889237i	\(0.651236\pi\)
\(3\)	−44.1317	+	67.9220i	−0.544835	+	0.838543i
							\(5\)	−604.549	+	349.037i	−0.967279	+	0.558459i	−0.898405	−	0.439167i	\(-0.855274\pi\)
−0.0688730	+	0.997625i	\(0.521940\pi\)
\(7\)	−1124.45	+	1947.61i	−0.468327	+	0.811167i	−0.999345	−	0.0361942i	\(-0.988477\pi\)
							0.531017	+	0.847361i	\(0.321810\pi\)
\(11\)	18817.0	+	10864.0i	1.28523	+	0.742026i	0.977799	−	0.209545i	\(-0.0671983\pi\)
							0.307428	+	0.951571i	\(0.400532\pi\)
\(13\)	−9209.84	−	15951.9i	−0.322462	−	0.558521i	0.658533	−	0.752552i	\(-0.271177\pi\)
							−0.980995	+	0.194031i	\(0.937844\pi\)
\(17\)			56538.1i			0.676932i	0.940979	+	0.338466i	\(0.109908\pi\)
							−0.940979	+	0.338466i	\(0.890092\pi\)
\(19\)	−212375.			−1.62963			−0.814815	−	0.579721i	\(-0.803162\pi\)
							−0.814815	+	0.579721i	\(0.803162\pi\)
\(23\)	14399.3	−	8313.41i	0.0514551	−	0.0297076i	−0.474052	−	0.880497i	\(-0.657209\pi\)
							0.525507	+	0.850789i	\(0.323876\pi\)
\(29\)	−294955.	−	170292.i	−0.417027	−	0.240771i	0.276778	−	0.960934i	\(-0.410733\pi\)
							−0.693804	+	0.720163i	\(0.744067\pi\)
\(31\)	−82902.7	−	143592.i	−0.0897681	−	0.155483i	0.817645	−	0.575723i	\(-0.195279\pi\)
							−0.907413	+	0.420240i	\(0.861946\pi\)
\(37\)	1.11833e6			0.596709			0.298355	−	0.954455i	\(-0.403562\pi\)
							0.298355	+	0.954455i	\(0.403562\pi\)
\(41\)	−3.60414e6	+	2.08085e6i	−1.27546	+	0.736387i	−0.976010	−	0.217725i	\(-0.930136\pi\)
							−0.299449	+	0.954112i	\(0.596803\pi\)
\(43\)	1.28793e6	−	2.23075e6i	0.376719	−	0.652496i	−0.613864	−	0.789412i	\(-0.710386\pi\)
							0.990583	+	0.136916i	\(0.0437190\pi\)
\(47\)	8.10049e6	+	4.67682e6i	1.66005	+	0.958428i	0.972690	+	0.232107i	\(0.0745619\pi\)
							0.687356	+	0.726321i	\(0.258771\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.9.d.a.2.4	✓	14
3.2	odd	2		27.9.d.a.8.4		14
4.3	odd	2		144.9.q.a.65.5		14
9.2	odd	6		81.9.b.a.80.8		14
9.4	even	3		27.9.d.a.17.4		14
9.5	odd	6	inner	9.9.d.a.5.4	yes	14
9.7	even	3		81.9.b.a.80.7		14
12.11	even	2		432.9.q.a.305.6		14
36.23	even	6		144.9.q.a.113.5		14
36.31	odd	6		432.9.q.a.17.6		14

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.9.d.a.2.4	✓	14	1.1	even	1	trivial
9.9.d.a.5.4	yes	14	9.5	odd	6	inner
27.9.d.a.8.4		14	3.2	odd	2
27.9.d.a.17.4		14	9.4	even	3
81.9.b.a.80.7		14	9.7	even	3
81.9.b.a.80.8		14	9.2	odd	6
144.9.q.a.65.5		14	4.3	odd	2
144.9.q.a.113.5		14	36.23	even	6
432.9.q.a.17.6		14	36.31	odd	6
432.9.q.a.305.6		14	12.11	even	2