Embedded newform 43.7.h.a.12.3

• Label: 43.7.h.a.12.3
• Level: $43$
• Weight: $7$
• Character: 43.12
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $252$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(252\)
Relative dimension:	\(21\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			12.3
Character	\(\chi\)	\(=\)	43.12
Dual form			43.7.h.a.18.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.07636 - 12.6177i) q^{2} +(28.9804 + 42.5064i) q^{3} +(-82.3806 + 103.302i) q^{4} +(88.1711 - 95.0259i) q^{5} +(360.237 - 623.949i) q^{6} +(-445.910 + 257.446i) q^{7} +(930.184 + 212.309i) q^{8} +(-700.598 + 1785.09i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.07636	−	12.6177i	−0.759545	−	1.57721i	−0.815486	−	0.578777i	\(-0.803530\pi\)
							0.0559413	−	0.998434i	\(-0.482184\pi\)
\(3\)	28.9804	+	42.5064i	1.07335	+	1.57431i	0.787785	+	0.615951i	\(0.211228\pi\)
							0.285562	+	0.958360i	\(0.407820\pi\)
\(5\)	88.1711	−	95.0259i	0.705369	−	0.760207i	−0.274122	−	0.961695i	\(-0.588387\pi\)
							0.979491	+	0.201488i	\(0.0645777\pi\)
\(7\)	−445.910	+	257.446i	−1.30003	+	0.750572i	−0.980409	−	0.196974i	\(-0.936889\pi\)
							−0.319620	+	0.947546i	\(0.603555\pi\)
\(11\)	−442.766	−	555.211i	−0.332656	−	0.417138i	0.587170	−	0.809464i	\(-0.300242\pi\)
							−0.919826	+	0.392326i	\(0.871671\pi\)
\(13\)	−2677.75	+	825.978i	−1.21882	+	0.375957i	−0.836446	−	0.548049i	\(-0.815371\pi\)
							−0.382377	+	0.924006i	\(0.624894\pi\)
\(17\)	−1322.65	+	1227.24i	−0.269215	+	0.249795i	−0.803215	−	0.595689i	\(-0.796879\pi\)
							0.533999	+	0.845485i	\(0.320688\pi\)
\(19\)	−1750.74	+	687.114i	−0.255247	+	0.100177i	−0.489514	−	0.871995i	\(-0.662826\pi\)
							0.234268	+	0.972172i	\(0.424731\pi\)
\(23\)	15963.4	−	2406.10i	1.31203	−	0.197756i	0.544493	−	0.838765i	\(-0.316722\pi\)
							0.767533	+	0.641009i	\(0.221484\pi\)
\(29\)	−8812.34	+	12925.3i	−0.361325	+	0.529966i	−0.963257	−	0.268581i	\(-0.913445\pi\)
							0.601933	+	0.798547i	\(0.294398\pi\)
\(31\)	−1652.05	+	22045.1i	−0.0554548	+	0.739992i	0.898295	+	0.439392i	\(0.144806\pi\)
							−0.953750	+	0.300600i	\(0.902813\pi\)
\(37\)	25271.7	+	14590.6i	0.498917	+	0.288050i	0.728266	−	0.685294i	\(-0.240326\pi\)
							−0.229349	+	0.973344i	\(0.573660\pi\)
\(41\)	−19275.5	+	9282.60i	−0.279676	+	0.134685i	−0.568464	−	0.822708i	\(-0.692462\pi\)
							0.288788	+	0.957393i	\(0.406748\pi\)
\(43\)	2035.33	+	79480.9i	0.0255994	+	0.999672i
							\(47\)	90257.9	−	113180.i	0.869344	−	1.09012i	−0.125835	−	0.992051i	\(-0.540161\pi\)
0.995179	−	0.0980722i	\(-0.0312676\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.12.3	✓	252
43.18	odd	42	inner	43.7.h.a.18.3	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.3	✓	252	1.1	even	1	trivial
43.7.h.a.18.3	yes	252	43.18	odd	42	inner