Embedded newform 126.3.i.a.65.1

• Label: 126.3.i.a.65.1
• Level: $126$
• Weight: $3$
• Character: 126.65
• Analytic conductor: $3.433$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	126.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43325133094\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			65.1
Character	\(\chi\)	\(=\)	126.65
Dual form			126.3.i.a.95.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{2} +(-2.78154 + 1.12385i) q^{3} +(1.00000 - 1.73205i) q^{4} -1.41430i q^{5} +(2.61199 - 3.34328i) q^{6} +(3.88340 - 5.82402i) q^{7} +2.82843i q^{8} +(6.47391 - 6.25207i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 32 q^{4} + 8 q^{6} + 2 q^{7} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(73\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22474	+	0.707107i	−0.612372	+	0.353553i
							\(3\)	−2.78154	+	1.12385i	−0.927180	+	0.374617i
\(5\)		−	1.41430i		−	0.282861i								−0.989948	−	0.141430i	\(-0.954830\pi\)
							0.989948	−	0.141430i	\(-0.0451702\pi\)
\(7\)	3.88340	−	5.82402i	0.554772	−	0.832003i
							\(11\)			17.4563i			1.58693i	0.608614	+	0.793466i	\(0.291726\pi\)
−0.608614	+	0.793466i	\(0.708274\pi\)
\(13\)	8.59030	+	14.8788i	0.660793	+	1.14453i	0.980408	+	0.196979i	\(0.0631130\pi\)
							−0.319615	+	0.947547i	\(0.603554\pi\)
\(17\)	20.8700	−	12.0493i	1.22765	−	0.708781i	0.261109	−	0.965309i	\(-0.415912\pi\)
							0.966537	+	0.256528i	\(0.0825786\pi\)
\(19\)	11.7810	−	20.4053i	0.620051	−	1.07396i	−0.369424	−	0.929261i	\(-0.620445\pi\)
							0.989476	−	0.144700i	\(-0.0462216\pi\)
\(23\)			31.2952i			1.36066i	0.732905	+	0.680331i	\(0.238164\pi\)
							−0.732905	+	0.680331i	\(0.761836\pi\)
\(29\)	−13.7563	−	7.94218i	−0.474354	−	0.273868i	0.243707	−	0.969849i	\(-0.421637\pi\)
							−0.718061	+	0.695981i	\(0.754970\pi\)
\(31\)	16.0969	−	27.8807i	0.519255	−	0.899376i	−0.480494	−	0.876998i	\(-0.659543\pi\)
							0.999750	−	0.0223786i	\(-0.00712393\pi\)
\(37\)	15.7030	−	27.1985i	0.424407	−	0.735094i	−0.571958	−	0.820283i	\(-0.693816\pi\)
							0.996365	+	0.0851890i	\(0.0271494\pi\)
\(41\)	8.67466	−	5.00832i	0.211577	−	0.122154i	−0.390467	−	0.920617i	\(-0.627686\pi\)
							0.602044	+	0.798463i	\(0.294353\pi\)
\(43\)	−31.6891	+	54.8871i	−0.736956	+	1.27644i	0.216904	+	0.976193i	\(0.430404\pi\)
							−0.953860	+	0.300252i	\(0.902929\pi\)
\(47\)	3.69960	−	2.13597i	0.0787149	−	0.0454461i	−0.460126	−	0.887854i	\(-0.652196\pi\)
							0.538841	+	0.842408i	\(0.318862\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.3.i.a.65.1	✓	32
3.2	odd	2		378.3.i.a.359.13		32
7.4	even	3		126.3.r.a.11.7	yes	32
9.4	even	3		378.3.r.a.233.4		32
9.5	odd	6		126.3.r.a.23.15	yes	32
21.11	odd	6		378.3.r.a.305.12		32
63.4	even	3		378.3.i.a.179.12		32
63.32	odd	6	inner	126.3.i.a.95.1	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.i.a.65.1	✓	32	1.1	even	1	trivial
126.3.i.a.95.1	yes	32	63.32	odd	6	inner
126.3.r.a.11.7	yes	32	7.4	even	3
126.3.r.a.23.15	yes	32	9.5	odd	6
378.3.i.a.179.12		32	63.4	even	3
378.3.i.a.359.13		32	3.2	odd	2
378.3.r.a.233.4		32	9.4	even	3
378.3.r.a.305.12		32	21.11	odd	6