Embedded newform 43.9.f.a.22.15

• Label: 43.9.f.a.22.15
• Level: $43$
• Weight: $9$
• Character: 43.22
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $174$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(174\)
Relative dimension:	\(29\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			22.15
Character	\(\chi\)	\(=\)	43.22
Dual form			43.9.f.a.2.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.397118 + 0.824624i) q^{2} +(14.6893 + 30.5027i) q^{3} +(159.091 + 199.494i) q^{4} +(1129.77 - 257.863i) q^{5} -30.9866 q^{6} +4070.52i q^{7} +(-456.118 + 104.106i) q^{8} +(3376.08 - 4233.47i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.397118	+	0.824624i	−0.0248199	+	0.0515390i	−0.913010	−	0.407937i	\(-0.866248\pi\)
							0.888190	+	0.459476i	\(0.151963\pi\)
\(3\)	14.6893	+	30.5027i	0.181349	+	0.376576i	0.971749	−	0.236015i	\(-0.0758415\pi\)
							−0.790400	+	0.612591i	\(0.790127\pi\)
\(5\)	1129.77	−	257.863i	1.80763	−	0.412580i	0.820389	−	0.571805i	\(-0.193757\pi\)
							0.987242	+	0.159225i	\(0.0508996\pi\)
\(7\)			4070.52i			1.69534i	0.530520	+	0.847672i	\(0.321997\pi\)
							−0.530520	+	0.847672i	\(0.678003\pi\)
\(11\)	6059.04	−	7597.79i	0.413840	−	0.518939i	−0.530600	−	0.847622i	\(-0.678033\pi\)
							0.944440	+	0.328683i	\(0.106605\pi\)
\(13\)	−9966.72	−	43667.0i	−0.348963	−	1.52890i	−0.779541	−	0.626351i	\(-0.784547\pi\)
							0.430578	−	0.902553i	\(-0.358310\pi\)
\(17\)	−12268.6	+	53752.4i	−0.146893	+	0.643580i	0.846845	+	0.531840i	\(0.178499\pi\)
							−0.993738	+	0.111740i	\(0.964358\pi\)
\(19\)	−67306.1	+	53674.9i	−0.516464	+	0.411867i	−0.846731	−	0.532021i	\(-0.821433\pi\)
							0.330267	+	0.943888i	\(0.392861\pi\)
\(23\)	86276.8	−	108188.i	0.308307	−	0.386604i	−0.603405	−	0.797435i	\(-0.706190\pi\)
							0.911712	+	0.410831i	\(0.134761\pi\)
\(29\)	−462291.	+	959958.i	−0.653618	+	1.35725i	0.265823	+	0.964022i	\(0.414356\pi\)
							−0.919440	+	0.393229i	\(0.871358\pi\)
\(31\)	11176.2	+	5382.18i	0.0121017	+	0.00582789i	0.439925	−	0.898035i	\(-0.355005\pi\)
							−0.427823	+	0.903862i	\(0.640719\pi\)
\(37\)		−	3.06706e6i		−	1.63650i	−0.574864	−	0.818249i	\(-0.694945\pi\)
							0.574864	−	0.818249i	\(-0.305055\pi\)
\(41\)	−1.79516e6	−	864503.i	−0.635283	−	0.305936i	0.0883827	−	0.996087i	\(-0.471830\pi\)
							−0.723666	+	0.690150i	\(0.757544\pi\)
\(43\)	−3.00071e6	+	1.63827e6i	−0.877708	+	0.479196i
							\(47\)	242234.	+	303752.i	0.0496414	+	0.0622483i	0.806032	−	0.591871i	\(-0.201611\pi\)
−0.756391	+	0.654120i	\(0.773039\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.22.15	yes	174
43.2	odd	14	inner	43.9.f.a.2.15	✓	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.15	✓	174	43.2	odd	14	inner
43.9.f.a.22.15	yes	174	1.1	even	1	trivial