Embedded newform 43.6.e.a.4.1

• Label: 43.6.e.a.4.1
• 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			4.1
Character	\(\chi\)	\(=\)	43.4
Dual form			43.6.e.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.04063 + 8.82868i) q^{2} +(3.19853 + 4.01083i) q^{3} +(-21.2543 - 93.1212i) q^{4} +(78.2981 + 37.7064i) q^{5} -57.9300 q^{6} +141.803 q^{7} +(646.213 + 311.200i) q^{8} +(48.2164 - 211.250i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 3 q^{2} - 9 q^{3} - 331 q^{4} + 67 q^{5} - 20 q^{6} - 624 q^{7} + 589 q^{8} - 579 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.04063	+	8.82868i	−1.24462	+	1.56070i	−0.574235	+	0.818691i	\(0.694700\pi\)
							−0.670385	+	0.742013i	\(0.733871\pi\)
\(3\)	3.19853	+	4.01083i	0.205186	+	0.257295i	0.873768	−	0.486344i	\(-0.161670\pi\)
							−0.668582	+	0.743639i	\(0.733098\pi\)
\(5\)	78.2981	+	37.7064i	1.40064	+	0.674512i	0.973291	−	0.229576i	\(-0.0737338\pi\)
							0.427347	+	0.904087i	\(0.359448\pi\)
\(7\)	141.803			1.09381			0.546904	−	0.837195i	\(-0.315806\pi\)
							0.546904	+	0.837195i	\(0.315806\pi\)
\(11\)	53.0921	−	232.612i	0.132296	−	0.579629i	−0.864707	−	0.502276i	\(-0.832496\pi\)
							0.997004	−	0.0773526i	\(-0.0246467\pi\)
\(13\)	−64.9281	−	31.2677i	−0.106555	−	0.0513142i	0.379847	−	0.925049i	\(-0.375977\pi\)
							−0.486402	+	0.873735i	\(0.661691\pi\)
\(17\)	962.411	−	463.473i	0.807678	−	0.388957i	0.0159820	−	0.999872i	\(-0.494913\pi\)
							0.791696	+	0.610915i	\(0.209198\pi\)
\(19\)	56.0352	+	245.506i	0.0356104	+	0.156019i	0.989607	−	0.143798i	\(-0.0459315\pi\)
							−0.953997	+	0.299817i	\(0.903074\pi\)
\(23\)	−485.658	+	2127.81i	−0.191431	+	0.838712i	0.784413	+	0.620239i	\(0.212965\pi\)
							−0.975843	+	0.218473i	\(0.929893\pi\)
\(29\)	−2931.37	+	3675.82i	−0.647254	+	0.811631i	−0.991889	−	0.127105i	\(-0.959431\pi\)
							0.344635	+	0.938737i	\(0.388003\pi\)
\(31\)	1293.01	−	1621.38i	0.241656	−	0.303027i	−0.646182	−	0.763183i	\(-0.723635\pi\)
							0.887838	+	0.460156i	\(0.152207\pi\)
\(37\)	−3879.77			−0.465910			−0.232955	−	0.972487i	\(-0.574840\pi\)
							−0.232955	+	0.972487i	\(0.574840\pi\)
\(41\)	4832.76	−	6060.09i	0.448989	−	0.563015i	−0.504898	−	0.863179i	\(-0.668470\pi\)
							0.953888	+	0.300164i	\(0.0970415\pi\)
\(43\)	193.841	+	12123.2i	0.0159873	+	0.999872i
							\(47\)	15.4523	+	67.7008i	0.00102035	+	0.00447043i	0.975436	−	0.220285i	\(-0.0706988\pi\)
−0.974415	+	0.224756i	\(0.927842\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.e.a.4.1	✓	108
43.11	even	7	inner	43.6.e.a.11.1	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.e.a.4.1	✓	108	1.1	even	1	trivial
43.6.e.a.11.1	yes	108	43.11	even	7	inner