Embedded newform 43.9.h.a.18.17

• Label: 43.9.h.a.18.17
• Level: $43$
• Weight: $9$
• Character: 43.18
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			18.17
Character	\(\chi\)	\(=\)	43.18
Dual form			43.9.h.a.12.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.95467 - 8.21197i) q^{2} +(-41.0422 + 60.1978i) q^{3} +(107.816 + 135.198i) q^{4} +(-599.007 - 645.576i) q^{5} +(332.034 + 575.100i) q^{6} +(1650.67 + 953.013i) q^{7} +(3811.45 - 869.939i) q^{8} +(457.687 + 1166.17i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.95467	−	8.21197i	0.247167	−	0.513248i	−0.740065	−	0.672535i	\(-0.765205\pi\)
							0.987232	+	0.159287i	\(0.0509196\pi\)
\(3\)	−41.0422	+	60.1978i	−0.506693	+	0.743183i	−0.991310	−	0.131544i	\(-0.958007\pi\)
							0.484617	+	0.874726i	\(0.338959\pi\)
\(5\)	−599.007	−	645.576i	−0.958411	−	1.03292i	−0.999430	−	0.0337634i	\(-0.989251\pi\)
							0.0410185	−	0.999158i	\(-0.486940\pi\)
\(7\)	1650.67	+	953.013i	0.687491	+	0.396923i	0.802672	−	0.596421i	\(-0.203411\pi\)
							−0.115180	+	0.993345i	\(0.536745\pi\)
\(11\)	−5022.75	+	6298.33i	−0.343061	+	0.430184i	−0.923192	−	0.384338i	\(-0.874430\pi\)
							0.580132	+	0.814523i	\(0.303001\pi\)
\(13\)	−31690.7	−	9775.27i	−1.10958	−	0.342259i	−0.314799	−	0.949158i	\(-0.601937\pi\)
							−0.794779	+	0.606899i	\(0.792413\pi\)
\(17\)	−92991.2	−	86283.2i	−1.11339	−	1.03307i	−0.999215	−	0.0396108i	\(-0.987388\pi\)
							−0.114172	−	0.993461i	\(-0.536421\pi\)
\(19\)	−128031.	−	50248.6i	−0.982431	−	0.385576i	−0.180882	−	0.983505i	\(-0.557895\pi\)
							−0.801549	+	0.597929i	\(0.795990\pi\)
\(23\)	−326871.	−	49267.9i	−1.16806	−	0.176057i	−0.463761	−	0.885960i	\(-0.653500\pi\)
							−0.704299	+	0.709904i	\(0.748738\pi\)
\(29\)	438075.	+	642538.i	0.619379	+	0.908461i	0.999867	−	0.0163243i	\(-0.00519642\pi\)
							−0.380488	+	0.924786i	\(0.624244\pi\)
\(31\)	−52353.7	−	698612.i	−0.0566893	−	0.756465i	−0.951034	−	0.309087i	\(-0.899977\pi\)
							0.894345	−	0.447379i	\(-0.147642\pi\)
\(37\)	−1.60300e6	+	925494.i	−0.855317	+	0.493818i	−0.862441	−	0.506157i	\(-0.831066\pi\)
							0.00712423	+	0.999975i	\(0.497732\pi\)
\(41\)	3.77730e6	+	1.81905e6i	1.33674	+	0.643738i	0.959324	−	0.282307i	\(-0.0910996\pi\)
							0.377413	+	0.926045i	\(0.376814\pi\)
\(43\)	−3.33471e6	−	753603.i	−0.975403	−	0.220429i
							\(47\)	−1.13964e6	−	1.42907e6i	−0.233549	−	0.292861i	0.651222	−	0.758887i	\(-0.274257\pi\)
−0.884771	+	0.466026i	\(0.845685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.18.17	yes	336
43.12	odd	42	inner	43.9.h.a.12.17	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.17	✓	336	43.12	odd	42	inner
43.9.h.a.18.17	yes	336	1.1	even	1	trivial