Embedded newform 336.2.bq.b.37.5

• Label: 336.2.bq.b.37.5
• 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			37.5
Character	\(\chi\)	\(=\)	336.37
Dual form			336.2.bq.b.109.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30731 + 0.539398i) q^{2} +(0.965926 + 0.258819i) q^{3} +(1.41810 - 1.41032i) q^{4} +(-3.17326 + 0.850273i) q^{5} +(-1.40237 + 0.182663i) q^{6} +(-2.50019 - 0.865491i) q^{7} +(-1.09317 + 2.60864i) q^{8} +(0.866025 + 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{4} - 8 q^{5} - 8 q^{6} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30731	+	0.539398i	−0.924405	+	0.381412i
							\(3\)	0.965926	+	0.258819i	0.557678	+	0.149429i
\(5\)	−3.17326	+	0.850273i	−1.41913	+	0.380254i								−0.885174	−	0.465261i	\(-0.845961\pi\)
							−0.533952	+	0.845515i	\(0.679294\pi\)
\(7\)	−2.50019	−	0.865491i	−0.944981	−	0.327125i
							\(11\)	0.976934	−	3.64597i	0.294557	−	1.09930i	−0.647012	−	0.762480i	\(-0.723982\pi\)
0.941569	−	0.336821i	\(-0.109352\pi\)
\(13\)	3.52949	−	3.52949i	0.978904	−	0.978904i	−0.0208777	−	0.999782i	\(-0.506646\pi\)
							0.999782	+	0.0208777i	\(0.00664608\pi\)
\(17\)	−2.06169	−	3.57095i	−0.500033	−	0.866083i	−1.00000		3.86495e-5i	\(-0.999988\pi\)
							0.499967	−	0.866045i	\(-0.333346\pi\)
\(19\)	−0.567502	−	2.11795i	−0.130194	−	0.485890i	0.869778	−	0.493444i	\(-0.164262\pi\)
							−0.999971	+	0.00755378i	\(0.997596\pi\)
\(23\)	0.113141	+	0.0653218i	0.0235915	+	0.0136205i	0.511749	−	0.859135i	\(-0.328998\pi\)
							−0.488158	+	0.872755i	\(0.662331\pi\)
\(29\)	−3.42058	+	3.42058i	−0.635185	+	0.635185i	−0.949364	−	0.314179i	\(-0.898271\pi\)
							0.314179	+	0.949364i	\(0.398271\pi\)
\(31\)	−3.48320	−	6.03307i	−0.625601	−	1.08357i	−0.988424	−	0.151714i	\(-0.951521\pi\)
							0.362824	−	0.931858i	\(-0.381813\pi\)
\(37\)	−9.27427	+	2.48503i	−1.52468	+	0.408537i	−0.921280	−	0.388901i	\(-0.872855\pi\)
							−0.603401	+	0.797438i	\(0.706188\pi\)
\(41\)		−	11.3070i		−	1.76586i	−0.469502	−	0.882931i	\(-0.655567\pi\)
							0.469502	−	0.882931i	\(-0.344433\pi\)
\(43\)	3.06943	+	3.06943i	0.468083	+	0.468083i	0.901293	−	0.433210i	\(-0.142619\pi\)
							−0.433210	+	0.901293i	\(0.642619\pi\)
\(47\)	4.39247	−	7.60798i	0.640708	−	1.10974i	−0.344567	−	0.938762i	\(-0.611974\pi\)
							0.985275	−	0.170977i	\(-0.0546923\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bq.b.37.5	✓	120
7.4	even	3	inner	336.2.bq.b.277.16	yes	120
16.13	even	4	inner	336.2.bq.b.205.16	yes	120
112.109	even	12	inner	336.2.bq.b.109.5	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bq.b.37.5	✓	120	1.1	even	1	trivial
336.2.bq.b.109.5	yes	120	112.109	even	12	inner
336.2.bq.b.205.16	yes	120	16.13	even	4	inner
336.2.bq.b.277.16	yes	120	7.4	even	3	inner