Embedded newform 43.6.e.a.4.13

• Label: 43.6.e.a.4.13
• Level: $43$
• Weight: $6$
• Character: 43.4
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.13
Character	\(\chi\)	\(=\)	43.4
Dual form			43.6.e.a.11.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.11467 - 3.90567i) q^{2} +(2.22486 + 2.78989i) q^{3} +(1.56755 + 6.86790i) q^{4} +(-1.98003 - 0.953533i) q^{5} +17.8261 q^{6} +187.548 q^{7} +(175.733 + 84.6285i) q^{8} +(51.2391 - 224.493i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 3 q^{2} - 9 q^{3} - 331 q^{4} + 67 q^{5} - 20 q^{6} - 624 q^{7} + 589 q^{8} - 579 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.11467	−	3.90567i	0.550601	−	0.690432i	−0.426188	−	0.904635i	\(-0.640144\pi\)
							0.976789	+	0.214202i	\(0.0687152\pi\)
\(3\)	2.22486	+	2.78989i	0.142725	+	0.178971i	0.848056	−	0.529907i	\(-0.177773\pi\)
							−0.705331	+	0.708878i	\(0.749202\pi\)
\(5\)	−1.98003	−	0.953533i	−0.0354199	−	0.0170573i	0.416090	−	0.909323i	\(-0.363400\pi\)
							−0.451510	+	0.892266i	\(0.649114\pi\)
\(7\)	187.548			1.44667			0.723333	−	0.690499i	\(-0.242609\pi\)
							0.723333	+	0.690499i	\(0.242609\pi\)
\(11\)	−44.1621	+	193.487i	−0.110045	+	0.482137i	0.889631	+	0.456679i	\(0.150961\pi\)
							−0.999676	+	0.0254573i	\(0.991896\pi\)
\(13\)	−21.4714	−	10.3401i	−0.0352373	−	0.0169694i	0.416182	−	0.909281i	\(-0.363368\pi\)
							−0.451419	+	0.892312i	\(0.649082\pi\)
\(17\)	−2.74703	+	1.32290i	−0.00230538	+	0.00111021i	−0.435036	−	0.900413i	\(-0.643264\pi\)
							0.432731	+	0.901523i	\(0.357550\pi\)
\(19\)	−137.864	−	604.021i	−0.0876125	−	0.383856i	0.912043	−	0.410094i	\(-0.134504\pi\)
							−0.999656	+	0.0262386i	\(0.991647\pi\)
\(23\)	−702.964	+	3079.89i	−0.277085	+	1.21399i	0.624374	+	0.781126i	\(0.285354\pi\)
							−0.901459	+	0.432864i	\(0.857503\pi\)
\(29\)	1571.17	−	1970.18i	0.346918	−	0.435022i	−0.577507	−	0.816386i	\(-0.695974\pi\)
							0.924425	+	0.381364i	\(0.124546\pi\)
\(31\)	−5170.79	+	6483.96i	−0.966390	+	1.21181i	0.0109069	+	0.999941i	\(0.496528\pi\)
							−0.977297	+	0.211874i	\(0.932043\pi\)
\(37\)	−13270.1			−1.59357			−0.796784	−	0.604264i	\(-0.793467\pi\)
							−0.796784	+	0.604264i	\(0.793467\pi\)
\(41\)	4657.41	−	5840.21i	0.432698	−	0.542586i	−0.516904	−	0.856043i	\(-0.672916\pi\)
							0.949602	+	0.313457i	\(0.101487\pi\)
\(43\)	−10356.0	−	6305.67i	−0.854125	−	0.520068i
							\(47\)	−2292.13	−	10042.5i	−0.151354	−	0.663127i	−0.992492	−	0.122306i	\(-0.960971\pi\)
0.841138	−	0.540821i	\(-0.181886\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.e.a.4.13	✓	108
43.11	even	7	inner	43.6.e.a.11.13	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.e.a.4.13	✓	108	1.1	even	1	trivial
43.6.e.a.11.13	yes	108	43.11	even	7	inner