Embedded newform 42.9.g.a.31.3

• Label: 42.9.g.a.31.3
• Level: $42$
• Weight: $9$
• Character: 42.31
• Analytic conductor: $17.110$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	42.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.1099016226\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - 2x^{7} + 243x^{6} - 2x^{5} + 49033x^{4} - 23088x^{3} + 2105352x^{2} + 2056320x + 73410624 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{10}\cdot 3^{2}\cdot 7^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.3
Root			\(-6.76102 + 11.7104i\) of defining polynomial
Character	\(\chi\)	\(=\)	42.31
Dual form			42.9.g.a.19.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.65685 + 9.79796i) q^{2} +(40.5000 + 23.3827i) q^{3} +(-64.0000 + 110.851i) q^{4} +(-974.971 + 562.900i) q^{5} +529.090i q^{6} +(-797.874 - 2264.55i) q^{7} -1448.15 q^{8} +(1093.50 + 1894.00i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 324 q^{3} - 512 q^{4} - 2226 q^{5} - 140 q^{7} + 8748 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(31\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.65685	+	9.79796i	0.353553	+	0.612372i
							\(3\)	40.5000	+	23.3827i	0.500000	+	0.288675i
\(5\)	−974.971	+	562.900i	−1.55995	+	0.900640i								−0.562694	+	0.826665i	\(0.690235\pi\)
							−0.997260	+	0.0739744i	\(0.976432\pi\)
\(7\)	−797.874	−	2264.55i	−0.332309	−	0.943171i
							\(11\)	12338.5	−	21371.0i	0.842738	−	1.45967i	−0.0448328	−	0.998995i	\(-0.514276\pi\)
0.887571	−	0.460671i	\(-0.152391\pi\)
\(13\)		−	10148.2i		−	0.355318i	−0.984092	−	0.177659i	\(-0.943148\pi\)
							0.984092	−	0.177659i	\(-0.0568524\pi\)
\(17\)	41051.3	+	23701.0i	0.491509	+	0.283773i	0.725200	−	0.688538i	\(-0.241747\pi\)
							−0.233691	+	0.972311i	\(0.575081\pi\)
\(19\)	−192008.	+	110856.i	−1.47335	+	0.850637i	−0.999550	−	0.0299967i	\(-0.990450\pi\)
							−0.473797	+	0.880634i	\(0.657117\pi\)
\(23\)	−61032.6	−	105711.i	−0.218097	−	0.377756i	0.736129	−	0.676841i	\(-0.236652\pi\)
							−0.954226	+	0.299086i	\(0.903318\pi\)
\(29\)	−443928.			−0.627654			−0.313827	−	0.949480i	\(-0.601611\pi\)
							−0.313827	+	0.949480i	\(0.601611\pi\)
\(31\)	−1.42776e6	−	824317.i	−1.54599	−	0.892580i	−0.998441	−	0.0558090i	\(-0.982226\pi\)
							−0.547553	−	0.836771i	\(-0.684440\pi\)
\(37\)	76119.9	+	131843.i	0.0406154	+	0.0703480i	0.885619	−	0.464413i	\(-0.153735\pi\)
							−0.845003	+	0.534761i	\(0.820401\pi\)
\(41\)		−	768920.i		−	0.272111i	−0.990701	−	0.136055i	\(-0.956557\pi\)
							0.990701	−	0.136055i	\(-0.0434425\pi\)
\(43\)	−5.20742e6			−1.52317			−0.761586	−	0.648064i	\(-0.775579\pi\)
							−0.761586	+	0.648064i	\(0.775579\pi\)
\(47\)	−3.30583e6	+	1.90862e6i	−0.677469	+	0.391137i	−0.798901	−	0.601463i	\(-0.794585\pi\)
							0.121432	+	0.992600i	\(0.461251\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.9.g.a.31.3	yes	8
3.2	odd	2		126.9.n.a.73.2		8
7.3	odd	6		294.9.c.a.97.4		8
7.4	even	3		294.9.c.a.97.1		8
7.5	odd	6	inner	42.9.g.a.19.3	✓	8
21.5	even	6		126.9.n.a.19.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.9.g.a.19.3	✓	8	7.5	odd	6	inner
42.9.g.a.31.3	yes	8	1.1	even	1	trivial
126.9.n.a.19.2		8	21.5	even	6
126.9.n.a.73.2		8	3.2	odd	2
294.9.c.a.97.1		8	7.4	even	3
294.9.c.a.97.4		8	7.3	odd	6