Embedded newform 336.2.bq.b.109.24

• Label: 336.2.bq.b.109.24
• Level: $336$
• Weight: $2$
• Character: 336.109
• 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			109.24
Character	\(\chi\)	\(=\)	336.109
Dual form			336.2.bq.b.37.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.07147 + 0.923008i) q^{2} +(0.965926 - 0.258819i) q^{3} +(0.296112 + 1.97796i) q^{4} +(-1.04287 - 0.279436i) q^{5} +(1.27386 + 0.614240i) q^{6} +(-0.0520571 + 2.64524i) q^{7} +(-1.50840 + 2.39264i) q^{8} +(0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{4} - 8 q^{5} - 8 q^{6} + 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.07147	+	0.923008i	0.757646	+	0.652665i
							\(3\)	0.965926	−	0.258819i	0.557678	−	0.149429i
\(5\)	−1.04287	−	0.279436i	−0.466385	−	0.124968i								0.0179709	−	0.999839i	\(-0.494279\pi\)
							−0.484356	+	0.874871i	\(0.660946\pi\)
\(7\)	−0.0520571	+	2.64524i	−0.0196757	+	0.999806i
							\(11\)	0.0302178	+	0.112774i	0.00911101	+	0.0340027i	0.970332	−	0.241778i	\(-0.0777304\pi\)
−0.961221	+	0.275780i	\(0.911064\pi\)
\(13\)	3.16113	+	3.16113i	0.876739	+	0.876739i	0.993196	−	0.116457i	\(-0.0371536\pi\)
							−0.116457	+	0.993196i	\(0.537154\pi\)
\(17\)	2.36989	−	4.10477i	0.574782	−	0.995552i	−0.421283	−	0.906929i	\(-0.638420\pi\)
							0.996065	−	0.0886231i	\(-0.0282467\pi\)
\(19\)	0.640273	−	2.38953i	0.146889	−	0.548196i	−0.852775	−	0.522278i	\(-0.825082\pi\)
							0.999664	−	0.0259182i	\(-0.00825094\pi\)
\(23\)	2.25577	−	1.30237i	0.470361	−	0.271563i	−0.246030	−	0.969262i	\(-0.579126\pi\)
							0.716391	+	0.697699i	\(0.245793\pi\)
\(29\)	−1.88062	−	1.88062i	−0.349223	−	0.349223i	0.510597	−	0.859820i	\(-0.329424\pi\)
							−0.859820	+	0.510597i	\(0.829424\pi\)
\(31\)	2.82877	−	4.89958i	0.508062	−	0.879990i	−0.491894	−	0.870655i	\(-0.663695\pi\)
							0.999956	−	0.00933478i	\(-0.00297140\pi\)
\(37\)	−1.30502	−	0.349678i	−0.214543	−	0.0574867i	0.149946	−	0.988694i	\(-0.452090\pi\)
							−0.364490	+	0.931207i	\(0.618757\pi\)
\(41\)		−	12.7143i		−	1.98564i	−0.119608	−	0.992821i	\(-0.538164\pi\)
							0.119608	−	0.992821i	\(-0.461836\pi\)
\(43\)	−8.99443	+	8.99443i	−1.37164	+	1.37164i	−0.513619	+	0.858018i	\(0.671696\pi\)
							−0.858018	+	0.513619i	\(0.828304\pi\)
\(47\)	3.95118	+	6.84364i	0.576339	+	0.998248i	0.995895	+	0.0905184i	\(0.0288524\pi\)
							−0.419556	+	0.907729i	\(0.637814\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bq.b.109.24	yes	120
7.2	even	3	inner	336.2.bq.b.205.18	yes	120
16.5	even	4	inner	336.2.bq.b.277.18	yes	120
112.37	even	12	inner	336.2.bq.b.37.24	✓	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bq.b.37.24	✓	120	112.37	even	12	inner
336.2.bq.b.109.24	yes	120	1.1	even	1	trivial
336.2.bq.b.205.18	yes	120	7.2	even	3	inner
336.2.bq.b.277.18	yes	120	16.5	even	4	inner