Embedded newform 43.10.e.a.4.20

• Label: 43.10.e.a.4.20
• Level: $43$
• Weight: $10$
• Character: 43.4
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(32\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.20
Character	\(\chi\)	\(=\)	43.4
Dual form			43.10.e.a.11.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.28774 - 6.63062i) q^{2} +(132.521 + 166.177i) q^{3} +(97.9258 + 429.041i) q^{4} +(-992.851 - 478.132i) q^{5} +1802.59 q^{6} -5295.49 q^{7} +(7274.81 + 3503.36i) q^{8} +(-5672.86 + 24854.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 23 q^{2} + 141 q^{3} - 7771 q^{4} + 677 q^{5} - 6572 q^{6} - 23872 q^{7} + 40269 q^{8} - 322935 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.28774	−	6.63062i	0.233687	−	0.293035i	−0.651136	−	0.758961i	\(-0.725707\pi\)
							0.884824	+	0.465926i	\(0.154279\pi\)
\(3\)	132.521	+	166.177i	0.944584	+	1.18447i	0.982701	+	0.185198i	\(0.0592925\pi\)
							−0.0381174	+	0.999273i	\(0.512136\pi\)
\(5\)	−992.851	−	478.132i	−0.710426	−	0.342123i	0.0435261	−	0.999052i	\(-0.486141\pi\)
							−0.753952	+	0.656929i	\(0.771855\pi\)
\(7\)	−5295.49			−0.833615			−0.416807	−	0.908995i	\(-0.636851\pi\)
							−0.416807	+	0.908995i	\(0.636851\pi\)
\(11\)	−7724.55	+	33843.4i	−0.159076	+	0.696959i	0.830982	+	0.556300i	\(0.187779\pi\)
							−0.990058	+	0.140660i	\(0.955078\pi\)
\(13\)	−15704.3	−	7562.82i	−0.152502	−	0.0734410i	0.356076	−	0.934457i	\(-0.384114\pi\)
							−0.508578	+	0.861016i	\(0.669829\pi\)
\(17\)	−15097.3	+	7270.49i	−0.0438409	+	0.0211127i	−0.455676	−	0.890146i	\(-0.650602\pi\)
							0.411835	+	0.911259i	\(0.364888\pi\)
\(19\)	−134786.	−	590537.i	−0.237276	−	1.03958i	−0.943444	−	0.331531i	\(-0.892435\pi\)
							0.706168	−	0.708045i	\(-0.250422\pi\)
\(23\)	−76698.1	+	336036.i	−0.0571491	+	0.250387i	−0.995431	−	0.0954854i	\(-0.969560\pi\)
							0.938282	+	0.345872i	\(0.112417\pi\)
\(29\)	−1.28763e6	+	1.61464e6i	−0.338065	+	0.423920i	−0.921584	−	0.388180i	\(-0.873104\pi\)
							0.583519	+	0.812100i	\(0.301675\pi\)
\(31\)	−4.50258e6	+	5.64605e6i	−0.875656	+	1.09804i	0.118804	+	0.992918i	\(0.462094\pi\)
							−0.994460	+	0.105120i	\(0.966477\pi\)
\(37\)	2.82078e6			0.247435			0.123717	−	0.992317i	\(-0.460518\pi\)
							0.123717	+	0.992317i	\(0.460518\pi\)
\(41\)	−395430.	+	495854.i	−0.0218546	+	0.0274048i	−0.792639	−	0.609692i	\(-0.791293\pi\)
							0.770784	+	0.637096i	\(0.219865\pi\)
\(43\)	2.07017e7	+	8.60409e6i	0.923419	+	0.383793i
							\(47\)	5.82048e6	+	2.55012e7i	0.173988	+	0.762290i	0.984330	+	0.176334i	\(0.0564237\pi\)
−0.810343	+	0.585956i	\(0.800719\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.e.a.4.20	✓	192
43.11	even	7	inner	43.10.e.a.11.20	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.e.a.4.20	✓	192	1.1	even	1	trivial
43.10.e.a.11.20	yes	192	43.11	even	7	inner