Embedded newform 333.3.bg.a.88.18

• Label: 333.3.bg.a.88.18
• Level: $333$
• Weight: $3$
• Character: 333.88
• Analytic conductor: $9.074$
• Analytic rank: $0$
• Dimension: $296$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	333.bg (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.07359280320\)
Analytic rank:	\(0\)
Dimension:	\(296\)
Relative dimension:	\(74\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			88.18
Character	\(\chi\)	\(=\)	333.88
Dual form			333.3.bg.a.193.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.573540 + 2.14048i) q^{2} +(1.52804 + 2.58168i) q^{3} +(-0.788605 - 0.455301i) q^{4} +(-1.48065 - 5.52585i) q^{5} +(-6.40243 + 1.79005i) q^{6} -9.66044 q^{7} +(-4.84090 + 4.84090i) q^{8} +(-4.33017 + 7.88985i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 296 q - 2 q^{2} - 6 q^{3} - 6 q^{4} + 4 q^{5} + 12 q^{6} - 4 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/333\mathbb{Z}\right)^\times\).

\(n\)	\(38\)	\(298\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.573540	+	2.14048i	−0.286770	+	1.07024i	0.660766	+	0.750592i	\(0.270231\pi\)
							−0.947536	+	0.319648i	\(0.896435\pi\)
\(3\)	1.52804	+	2.58168i	0.509348	+	0.860561i
							\(5\)	−1.48065	−	5.52585i	−0.296129	−	1.10517i	−0.940316	−	0.340302i	\(-0.889471\pi\)
0.644187	−	0.764868i	\(-0.277196\pi\)
\(7\)	−9.66044			−1.38006			−0.690032	−	0.723779i	\(-0.742403\pi\)
							−0.690032	+	0.723779i	\(0.742403\pi\)
\(11\)	0.427781	+	0.246979i	0.0388892	+	0.0224527i	0.519319	−	0.854581i	\(-0.326186\pi\)
							−0.480429	+	0.877033i	\(0.659519\pi\)
\(13\)	−3.60327	−	13.4476i	−0.277175	−	1.03443i	−0.954370	−	0.298627i	\(-0.903471\pi\)
							0.677195	−	0.735804i	\(-0.263195\pi\)
\(17\)	1.97597	+	7.37441i	0.116233	+	0.433789i	0.999376	−	0.0353153i	\(-0.0112436\pi\)
							−0.883143	+	0.469104i	\(0.844577\pi\)
\(19\)	1.71430	+	0.459344i	0.0902261	+	0.0241760i	0.303650	−	0.952784i	\(-0.401795\pi\)
							−0.213424	+	0.976960i	\(0.568461\pi\)
\(23\)	−25.4774	−	6.82665i	−1.10771	−	0.296811i	−0.341812	−	0.939768i	\(-0.611040\pi\)
							−0.765902	+	0.642958i	\(0.777707\pi\)
\(29\)	−26.9815	+	7.22966i	−0.930396	+	0.249299i	−0.692023	−	0.721875i	\(-0.743280\pi\)
							−0.238372	+	0.971174i	\(0.576614\pi\)
\(31\)	0.635431	−	2.37146i	0.0204978	−	0.0764988i	−0.954919	−	0.296865i	\(-0.904059\pi\)
							0.975417	+	0.220367i	\(0.0707254\pi\)
\(37\)	6.03868	−	36.5039i	0.163208	−	0.986592i
							\(41\)	18.6138	+	10.7467i	0.453994	+	0.262114i	0.709515	−	0.704690i	\(-0.248914\pi\)
−0.255522	+	0.966803i	\(0.582247\pi\)
\(43\)	3.44327	+	12.8504i	0.0800760	+	0.298848i	0.994336	−	0.106280i	\(-0.0338939\pi\)
							−0.914260	+	0.405127i	\(0.867227\pi\)
\(47\)	−6.60490	+	11.4400i	−0.140530	+	0.243405i	−0.927696	−	0.373336i	\(-0.878214\pi\)
							0.787166	+	0.616741i	\(0.211547\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	333.3.bg.a.88.18	yes	296
9.4	even	3		333.3.ba.a.310.18	yes	296
37.8	odd	12		333.3.ba.a.304.18	✓	296
333.193	odd	12	inner	333.3.bg.a.193.18	yes	296

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
333.3.ba.a.304.18	✓	296	37.8	odd	12
333.3.ba.a.310.18	yes	296	9.4	even	3
333.3.bg.a.88.18	yes	296	1.1	even	1	trivial
333.3.bg.a.193.18	yes	296	333.193	odd	12	inner