Embedded newform 43.5.f.a.2.11

• Label: 43.5.f.a.2.11
• Level: $43$
• Weight: $5$
• Character: 43.2
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $78$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(78\)
Relative dimension:	\(13\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			2.11
Character	\(\chi\)	\(=\)	43.2
Dual form			43.5.f.a.22.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.55909 + 5.31401i) q^{2} +(3.37270 - 7.00348i) q^{3} +(-11.7139 + 14.6887i) q^{4} +(17.8072 + 4.06438i) q^{5} +45.8475 q^{6} +82.2284i q^{7} +(-16.0293 - 3.65859i) q^{8} +(12.8291 + 16.0872i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 7 q^{2} - 7 q^{3} + 69 q^{4} - 7 q^{5} - 140 q^{6} - 343 q^{8} + 486 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.55909	+	5.31401i	0.639773	+	1.32850i	0.928586	+	0.371116i	\(0.121025\pi\)
							−0.288814	+	0.957385i	\(0.593261\pi\)
\(3\)	3.37270	−	7.00348i	0.374744	−	0.778164i	−0.625253	−	0.780422i	\(-0.715004\pi\)
							0.999997	+	0.00225784i	\(0.000718692\pi\)
\(5\)	17.8072	+	4.06438i	0.712289	+	0.162575i	0.563287	−	0.826262i	\(-0.309537\pi\)
							0.149002	+	0.988837i	\(0.452394\pi\)
\(7\)			82.2284i			1.67813i	0.544030	+	0.839065i	\(0.316898\pi\)
							−0.544030	+	0.839065i	\(0.683102\pi\)
\(11\)	−120.489	−	151.088i	−0.995773	−	1.24866i	−0.968495	−	0.249032i	\(-0.919888\pi\)
							−0.0272780	−	0.999628i	\(-0.508684\pi\)
\(13\)	68.5441	−	300.311i	0.405586	−	1.77699i	−0.198530	−	0.980095i	\(-0.563617\pi\)
							0.604117	−	0.796896i	\(-0.293526\pi\)
\(17\)	22.6332	+	99.1626i	0.0783156	+	0.343123i	0.998872	−	0.0474870i	\(-0.0151213\pi\)
							−0.920556	+	0.390610i	\(0.872264\pi\)
\(19\)	−120.989	−	96.4854i	−0.335149	−	0.267273i	0.441425	−	0.897298i	\(-0.354473\pi\)
							−0.776575	+	0.630025i	\(0.783044\pi\)
\(23\)	−193.354	−	242.458i	−0.365508	−	0.458333i	0.564737	−	0.825271i	\(-0.308978\pi\)
							−0.930245	+	0.366938i	\(0.880406\pi\)
\(29\)	−189.123	−	392.718i	−0.224879	−	0.466965i	0.757751	−	0.652544i	\(-0.226298\pi\)
							−0.982629	+	0.185579i	\(0.940584\pi\)
\(31\)	−539.980	+	260.041i	−0.561894	+	0.270594i	−0.693197	−	0.720748i	\(-0.743798\pi\)
							0.131303	+	0.991342i	\(0.458084\pi\)
\(37\)		−	1078.29i		−	0.787646i	−0.919186	−	0.393823i	\(-0.871152\pi\)
							0.919186	−	0.393823i	\(-0.128848\pi\)
\(41\)	1539.63	−	741.446i	0.915900	−	0.441074i	0.0842943	−	0.996441i	\(-0.473136\pi\)
							0.831606	+	0.555367i	\(0.187422\pi\)
\(43\)	675.988	+	1721.00i	0.365596	+	0.930773i
							\(47\)	−349.905	+	438.767i	−0.158400	+	0.198627i	−0.854698	−	0.519126i	\(-0.826258\pi\)
0.696298	+	0.717753i	\(0.254829\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.f.a.2.11	✓	78
43.22	odd	14	inner	43.5.f.a.22.11	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.f.a.2.11	✓	78	1.1	even	1	trivial
43.5.f.a.22.11	yes	78	43.22	odd	14	inner