Embedded newform 43.7.h.a.3.8

• Label: 43.7.h.a.3.8
• Level: $43$
• Weight: $7$
• Character: 43.3
• 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			3.8
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.45963 + 5.10748i) q^{2} +(-50.4681 - 3.78206i) q^{3} +(19.8668 + 24.9121i) q^{4} +(-59.6400 + 193.348i) q^{5} +(143.450 - 248.462i) q^{6} +(-466.054 + 269.077i) q^{7} +(-529.815 + 120.927i) q^{8} +(1811.86 + 273.095i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.45963	+	5.10748i	−0.307454	+	0.638436i	−0.996250	−	0.0865199i	\(-0.972425\pi\)
							0.688796	+	0.724955i	\(0.258140\pi\)
\(3\)	−50.4681	−	3.78206i	−1.86919	−	0.140076i	−0.908531	−	0.417816i	\(-0.862796\pi\)
							−0.960656	+	0.277740i	\(0.910415\pi\)
\(5\)	−59.6400	+	193.348i	−0.477120	+	1.54678i	0.322675	+	0.946510i	\(0.395418\pi\)
							−0.799794	+	0.600274i	\(0.795058\pi\)
\(7\)	−466.054	+	269.077i	−1.35876	+	0.784480i	−0.989457	−	0.144830i	\(-0.953737\pi\)
							−0.369302	+	0.929309i	\(0.620403\pi\)
\(11\)	−34.7049	+	43.5186i	−0.0260743	+	0.0326962i	−0.794698	−	0.607005i	\(-0.792371\pi\)
							0.768623	+	0.639702i	\(0.220942\pi\)
\(13\)	261.325	−	242.475i	0.118946	−	0.110366i	−0.618457	−	0.785818i	\(-0.712242\pi\)
							0.737404	+	0.675452i	\(0.236052\pi\)
\(17\)	769.633	−	237.400i	0.156652	−	0.0483209i	−0.215437	−	0.976518i	\(-0.569118\pi\)
							0.372090	+	0.928197i	\(0.378641\pi\)
\(19\)	396.212	+	2628.70i	0.0577653	+	0.383248i	0.998969	+	0.0454069i	\(0.0144584\pi\)
							−0.941203	+	0.337841i	\(0.890303\pi\)
\(23\)	6634.63	+	16904.7i	0.545297	+	1.38939i	0.892263	+	0.451516i	\(0.149117\pi\)
							−0.346966	+	0.937878i	\(0.612788\pi\)
\(29\)	17807.8	−	1334.51i	0.730158	−	0.0547177i	0.295537	−	0.955331i	\(-0.404501\pi\)
							0.434621	+	0.900614i	\(0.356882\pi\)
\(31\)	−31229.9	+	21292.2i	−1.04830	+	0.714718i	−0.959608	−	0.281341i	\(-0.909221\pi\)
							−0.0886909	+	0.996059i	\(0.528268\pi\)
\(37\)	63268.5	+	36528.1i	1.24906	+	0.721144i	0.970922	−	0.239398i	\(-0.0769500\pi\)
							0.278136	+	0.960542i	\(0.410283\pi\)
\(41\)	61278.6	+	29510.2i	0.889114	+	0.428175i	0.821945	−	0.569567i	\(-0.192889\pi\)
							0.0671692	+	0.997742i	\(0.478603\pi\)
\(43\)	−67855.8	+	41436.1i	−0.853457	+	0.521163i
							\(47\)	13157.5	+	16498.9i	0.126730	+	0.158914i	0.841148	−	0.540805i	\(-0.181880\pi\)
−0.714419	+	0.699719i	\(0.753309\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.3.8	✓	252
43.29	odd	42	inner	43.7.h.a.29.8	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.8	✓	252	1.1	even	1	trivial
43.7.h.a.29.8	yes	252	43.29	odd	42	inner