Embedded newform 336.2.bq.b.37.10

• Label: 336.2.bq.b.37.10
• Level: $336$
• Weight: $2$
• Character: 336.37
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.bq (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			37.10
Character	\(\chi\)	\(=\)	336.37
Dual form			336.2.bq.b.109.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.948203 - 1.04924i) q^{2} +(-0.965926 - 0.258819i) q^{3} +(-0.201821 + 1.98979i) q^{4} +(-2.71136 + 0.726506i) q^{5} +(0.644330 + 1.25890i) q^{6} +(-0.393486 + 2.61633i) q^{7} +(2.27914 - 1.67497i) q^{8} +(0.866025 + 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{4} - 8 q^{5} - 8 q^{6} + 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.948203	−	1.04924i	−0.670481	−	0.741927i
							\(3\)	−0.965926	−	0.258819i	−0.557678	−	0.149429i
\(5\)	−2.71136	+	0.726506i	−1.21256	+	0.324903i								−0.807764	−	0.589506i	\(-0.799323\pi\)
							−0.404791	+	0.914409i	\(0.632656\pi\)
\(7\)	−0.393486	+	2.61633i	−0.148724	+	0.988879i
							\(11\)	1.04416	−	3.89684i	0.314825	−	1.17494i	−0.609327	−	0.792919i	\(-0.708560\pi\)
0.924152	−	0.382024i	\(-0.124773\pi\)
\(13\)	4.21026	−	4.21026i	1.16772	−	1.16772i	0.184972	−	0.982744i	\(-0.440781\pi\)
							0.982744	−	0.184972i	\(-0.0592195\pi\)
\(17\)	−2.10450	−	3.64510i	−0.510415	−	0.884065i	−0.999927	−	0.0120687i	\(-0.996158\pi\)
							0.489512	−	0.871997i	\(-0.337175\pi\)
\(19\)	−0.0315900	−	0.117895i	−0.00724724	−	0.0270471i	0.962208	−	0.272317i	\(-0.0877900\pi\)
							−0.969455	+	0.245270i	\(0.921123\pi\)
\(23\)	6.12209	+	3.53459i	1.27654	+	0.737013i	0.976211	−	0.216822i	\(-0.0695691\pi\)
							0.300332	+	0.953835i	\(0.402902\pi\)
\(29\)	0.932475	−	0.932475i	0.173156	−	0.173156i	−0.615208	−	0.788365i	\(-0.710928\pi\)
							0.788365	+	0.615208i	\(0.210928\pi\)
\(31\)	0.233565	+	0.404546i	0.0419495	+	0.0726586i	0.886238	−	0.463231i	\(-0.153310\pi\)
							−0.844288	+	0.535889i	\(0.819976\pi\)
\(37\)	10.5017	−	2.81392i	1.72647	−	0.462606i	0.747105	−	0.664706i	\(-0.231443\pi\)
							0.979365	+	0.202100i	\(0.0647765\pi\)
\(41\)		−	7.38316i		−	1.15306i	−0.817077	−	0.576528i	\(-0.804407\pi\)
							0.817077	−	0.576528i	\(-0.195593\pi\)
\(43\)	−0.452297	−	0.452297i	−0.0689746	−	0.0689746i	0.671778	−	0.740753i	\(-0.265531\pi\)
							−0.740753	+	0.671778i	\(0.765531\pi\)
\(47\)	−0.826133	+	1.43090i	−0.120504	+	0.208719i	−0.919966	−	0.391997i	\(-0.871784\pi\)
							0.799463	+	0.600716i	\(0.205118\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bq.b.37.10	✓	120
7.4	even	3	inner	336.2.bq.b.277.29	yes	120
16.13	even	4	inner	336.2.bq.b.205.29	yes	120
112.109	even	12	inner	336.2.bq.b.109.10	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bq.b.37.10	✓	120	1.1	even	1	trivial
336.2.bq.b.109.10	yes	120	112.109	even	12	inner
336.2.bq.b.205.29	yes	120	16.13	even	4	inner
336.2.bq.b.277.29	yes	120	7.4	even	3	inner