Embedded newform 43.7.h.a.29.11

• Label: 43.7.h.a.29.11
• Level: $43$
• Weight: $7$
• Character: 43.29
• 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			29.11
Character	\(\chi\)	\(=\)	43.29
Dual form			43.7.h.a.3.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.711983 - 1.47845i) q^{2} +(20.0817 - 1.50492i) q^{3} +(38.2245 - 47.9320i) q^{4} +(-40.7360 - 132.063i) q^{5} +(-16.5228 - 28.6183i) q^{6} +(72.9518 + 42.1187i) q^{7} +(-200.468 - 45.7555i) q^{8} +(-319.847 + 48.2092i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{41}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.711983	−	1.47845i	−0.0889978	−	0.184806i	0.851719	−	0.523999i	\(-0.175560\pi\)
							−0.940717	+	0.339193i	\(0.889846\pi\)
\(3\)	20.0817	−	1.50492i	0.743767	−	0.0557376i	0.302544	−	0.953135i	\(-0.402164\pi\)
							0.441223	+	0.897398i	\(0.354545\pi\)
\(5\)	−40.7360	−	132.063i	−0.325888	−	1.05650i	−0.959229	−	0.282630i	\(-0.908793\pi\)
							0.633341	−	0.773873i	\(-0.281683\pi\)
\(7\)	72.9518	+	42.1187i	0.212687	+	0.122795i	0.602560	−	0.798074i	\(-0.294148\pi\)
							−0.389872	+	0.920869i	\(0.627481\pi\)
\(11\)	−897.256	−	1125.12i	−0.674122	−	0.845322i	0.320676	−	0.947189i	\(-0.396090\pi\)
							−0.994798	+	0.101867i	\(0.967518\pi\)
\(13\)	2870.79	+	2663.71i	1.30669	+	1.21243i	0.961790	+	0.273789i	\(0.0882771\pi\)
							0.344898	+	0.938640i	\(0.387913\pi\)
\(17\)	−4392.85	−	1355.01i	−0.894127	−	0.275802i	−0.186556	−	0.982444i	\(-0.559733\pi\)
							−0.707571	+	0.706643i	\(0.750209\pi\)
\(19\)	1467.89	−	9738.79i	0.214009	−	1.41986i	−0.581319	−	0.813676i	\(-0.697463\pi\)
							0.795327	−	0.606180i	\(-0.207299\pi\)
\(23\)	1546.04	−	3939.24i	0.127068	−	0.323764i	−0.853156	−	0.521656i	\(-0.825314\pi\)
							0.980224	+	0.197892i	\(0.0634096\pi\)
\(29\)	41891.1	+	3139.30i	1.71762	+	0.128718i	0.896791	−	0.442454i	\(-0.145892\pi\)
							0.820830	+	0.571172i	\(0.193511\pi\)
\(31\)	38467.7	+	26226.8i	1.29125	+	0.880361i	0.997152	−	0.0754132i	\(-0.0240276\pi\)
							0.294101	+	0.955774i	\(0.404980\pi\)
\(37\)	−45122.2	+	26051.3i	−0.890810	+	0.514310i	−0.874207	−	0.485553i	\(-0.838618\pi\)
							−0.0166029	+	0.999862i	\(0.505285\pi\)
\(41\)	88187.2	−	42468.7i	1.27954	−	0.616194i	0.334268	−	0.942478i	\(-0.391511\pi\)
							0.945272	+	0.326284i	\(0.105796\pi\)
\(43\)	65159.5	+	45558.8i	0.819544	+	0.573017i
							\(47\)	−84373.8	+	105801.i	−0.812670	+	1.01906i	0.186659	+	0.982425i	\(0.440234\pi\)
−0.999329	+	0.0366310i	\(0.988337\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.29.11	yes	252
43.3	odd	42	inner	43.7.h.a.3.11	✓	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.11	✓	252	43.3	odd	42	inner
43.7.h.a.29.11	yes	252	1.1	even	1	trivial