Embedded newform 43.6.e.a.11.1

• Label: 43.6.e.a.11.1
• Level: $43$
• Weight: $6$
• Character: 43.11
• 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			11.1
Character	\(\chi\)	\(=\)	43.11
Dual form			43.6.e.a.4.1

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{7}\right)\)

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	−0.670385	−	0.742013i	\(-0.733871\pi\)
							−0.574235	−	0.818691i	\(-0.694700\pi\)
\(3\)	3.19853	−	4.01083i	0.205186	−	0.257295i	−0.668582	−	0.743639i	\(-0.733098\pi\)
							0.873768	+	0.486344i	\(0.161670\pi\)
\(5\)	78.2981	−	37.7064i	1.40064	−	0.674512i	0.427347	−	0.904087i	\(-0.359448\pi\)
							0.973291	+	0.229576i	\(0.0737338\pi\)
\(7\)	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	0.997004	+	0.0773526i	\(0.0246467\pi\)
							−0.864707	+	0.502276i	\(0.832496\pi\)
\(13\)	−64.9281	+	31.2677i	−0.106555	+	0.0513142i	−0.486402	−	0.873735i	\(-0.661691\pi\)
							0.379847	+	0.925049i	\(0.375977\pi\)
\(17\)	962.411	+	463.473i	0.807678	+	0.388957i	0.791696	−	0.610915i	\(-0.209198\pi\)
							0.0159820	+	0.999872i	\(0.494913\pi\)
\(19\)	56.0352	−	245.506i	0.0356104	−	0.156019i	−0.953997	−	0.299817i	\(-0.903074\pi\)
							0.989607	+	0.143798i	\(0.0459315\pi\)
\(23\)	−485.658	−	2127.81i	−0.191431	−	0.838712i	−0.975843	−	0.218473i	\(-0.929893\pi\)
							0.784413	−	0.620239i	\(-0.212965\pi\)
\(29\)	−2931.37	−	3675.82i	−0.647254	−	0.811631i	0.344635	−	0.938737i	\(-0.388003\pi\)
							−0.991889	+	0.127105i	\(0.959431\pi\)
\(31\)	1293.01	+	1621.38i	0.241656	+	0.303027i	0.887838	−	0.460156i	\(-0.152207\pi\)
							−0.646182	+	0.763183i	\(0.723635\pi\)
\(37\)	−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	0.953888	−	0.300164i	\(-0.0970415\pi\)
							−0.504898	+	0.863179i	\(0.668470\pi\)
\(43\)	193.841	−	12123.2i	0.0159873	−	0.999872i
							\(47\)	15.4523	−	67.7008i	0.00102035	−	0.00447043i	−0.974415	−	0.224756i	\(-0.927842\pi\)
0.975436	+	0.220285i	\(0.0706988\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.e.a.4.1	✓	108	43.4	even	7	inner
43.6.e.a.11.1	yes	108	1.1	even	1	trivial