Embedded newform 43.7.h.a.3.11

• Label: 43.7.h.a.3.11
• 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			3.11
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.11

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.11	✓	252	1.1	even	1	trivial
43.7.h.a.29.11	yes	252	43.29	odd	42	inner