Embedded newform 43.8.g.a.9.15

• Label: 43.8.g.a.9.15
• Level: $43$
• Weight: $8$
• Character: 43.9
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(25\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			9.15
Character	\(\chi\)	\(=\)	43.9
Dual form			43.8.g.a.24.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.07492 + 3.85582i) q^{2} +(78.3361 + 11.8073i) q^{3} +(23.0704 - 101.078i) q^{4} +(247.532 + 168.764i) q^{5} +(195.350 + 338.356i) q^{6} +(-868.409 + 1504.13i) q^{7} +(1029.43 - 495.748i) q^{8} +(3907.29 + 1205.24i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.07492	+	3.85582i	0.271787	+	0.340810i	0.898929	−	0.438095i	\(-0.144347\pi\)
							−0.627142	+	0.778905i	\(0.715775\pi\)
\(3\)	78.3361	+	11.8073i	1.67509	+	0.252479i	0.916772	−	0.399410i	\(-0.130785\pi\)
							0.758315	+	0.651889i	\(0.226023\pi\)
\(5\)	247.532	+	168.764i	0.885596	+	0.603789i	0.918391	−	0.395673i	\(-0.129489\pi\)
							−0.0327955	+	0.999462i	\(0.510441\pi\)
\(7\)	−868.409	+	1504.13i	−0.956932	+	1.65745i	−0.227048	+	0.973884i	\(0.572907\pi\)
							−0.729884	+	0.683571i	\(0.760426\pi\)
\(11\)	−1098.19	−	4811.47i	−0.248772	−	1.08994i	−0.932774	−	0.360462i	\(-0.882619\pi\)
							0.684002	−	0.729480i	\(-0.260238\pi\)
\(13\)	−229.122	+	3057.42i	−0.0289244	+	0.385970i	0.963890	+	0.266302i	\(0.0858019\pi\)
							−0.992814	+	0.119668i	\(0.961817\pi\)
\(17\)	15786.0	−	10762.7i	0.779294	−	0.531314i	−0.107098	−	0.994248i	\(-0.534156\pi\)
							0.886392	+	0.462935i	\(0.153204\pi\)
\(19\)	−21716.1	+	6698.52i	−0.726346	+	0.224048i	−0.635811	−	0.771845i	\(-0.719334\pi\)
							−0.0905357	+	0.995893i	\(0.528858\pi\)
\(23\)	40228.3	−	37326.5i	0.689421	−	0.639690i	−0.255568	−	0.966791i	\(-0.582262\pi\)
							0.944989	+	0.327102i	\(0.106072\pi\)
\(29\)	70249.0	−	10588.3i	0.534868	−	0.0806184i	0.123946	−	0.992289i	\(-0.460445\pi\)
							0.410922	+	0.911670i	\(0.365207\pi\)
\(31\)	−105433.	+	268638.i	−0.635636	+	1.61958i	0.141279	+	0.989970i	\(0.454879\pi\)
							−0.776915	+	0.629605i	\(0.783217\pi\)
\(37\)	−208870.	−	361773.i	−0.677907	−	1.17417i	−0.975610	−	0.219511i	\(-0.929554\pi\)
							0.297703	−	0.954658i	\(-0.403779\pi\)
\(41\)	−262384.	−	329019.i	−0.594558	−	0.745552i	0.389961	−	0.920832i	\(-0.372489\pi\)
							−0.984519	+	0.175279i	\(0.943917\pi\)
\(43\)	373168.	+	364094.i	0.715756	+	0.698351i
							\(47\)	88218.2	−	386509.i	0.123941	−	0.543022i	−0.874387	−	0.485228i	\(-0.838736\pi\)
0.998329	−	0.0577932i	\(-0.0184064\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.9.15	✓	300
43.24	even	21	inner	43.8.g.a.24.15	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.9.15	✓	300	1.1	even	1	trivial
43.8.g.a.24.15	yes	300	43.24	even	21	inner