Embedded newform 33.8.e.b.16.2

• Label: 33.8.e.b.16.2
• Level: $33$
• Weight: $8$
• Character: 33.16
• Analytic conductor: $10.309$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	33.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3087058410\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			16.2
Character	\(\chi\)	\(=\)	33.16
Dual form			33.8.e.b.31.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.06039 - 12.4966i) q^{2} +(-21.8435 - 15.8702i) q^{3} +(-36.1242 + 26.2458i) q^{4} +(158.380 - 487.442i) q^{5} +(-109.631 + 337.408i) q^{6} +(846.952 - 615.347i) q^{7} +(-886.010 - 643.724i) q^{8} +(225.273 + 693.320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 6 q^{2} - 189 q^{3} + 160 q^{4} + 773 q^{5} - 162 q^{6} + 1289 q^{7} + 2956 q^{8} - 5103 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.06039	−	12.4966i	−0.358891	−	1.10455i	−0.953719	−	0.300701i	\(-0.902779\pi\)
							0.594827	−	0.803854i	\(-0.297221\pi\)
\(3\)	−21.8435	−	15.8702i	−0.467086	−	0.339358i
							\(5\)	158.380	−	487.442i	0.566636	−	1.74393i	−0.0964027	−	0.995342i	\(-0.530734\pi\)
0.663039	−	0.748585i	\(-0.269266\pi\)
\(7\)	846.952	−	615.347i	0.933288	−	0.678073i	−0.0135078	−	0.999909i	\(-0.504300\pi\)
							0.946796	+	0.321835i	\(0.104300\pi\)
\(11\)	1502.25	−	4150.95i	0.340304	−	0.940315i
							\(13\)	2335.53	+	7188.01i	0.294838	+	0.907417i	0.983276	+	0.182122i	\(0.0582966\pi\)
−0.688438	+	0.725295i	\(0.741703\pi\)
\(17\)	−4992.90	+	15366.6i	−0.246480	+	0.758587i	0.748910	+	0.662672i	\(0.230578\pi\)
							−0.995390	+	0.0959148i	\(0.969422\pi\)
\(19\)	17691.7	+	12853.8i	0.591743	+	0.429926i	0.842938	−	0.538010i	\(-0.180824\pi\)
							−0.251196	+	0.967936i	\(0.580824\pi\)
\(23\)	16920.8			0.289984			0.144992	−	0.989433i	\(-0.453684\pi\)
							0.144992	+	0.989433i	\(0.453684\pi\)
\(29\)	61034.3	−	44344.0i	0.464709	−	0.337631i	−0.330667	−	0.943748i	\(-0.607274\pi\)
							0.795376	+	0.606117i	\(0.207274\pi\)
\(31\)	21352.6	+	65716.6i	0.128732	+	0.396195i	0.994562	−	0.104142i	\(-0.0332095\pi\)
							−0.865831	+	0.500337i	\(0.833210\pi\)
\(37\)	239569.	−	174057.i	0.777543	−	0.564918i	−0.126698	−	0.991941i	\(-0.540438\pi\)
							0.904240	+	0.427024i	\(0.140438\pi\)
\(41\)	570567.	+	414541.i	1.29289	+	0.939343i	0.999860	−	0.0167585i	\(-0.00533465\pi\)
							0.293035	+	0.956102i	\(0.405335\pi\)
\(43\)	−581025.			−1.11444			−0.557218	−	0.830366i	\(-0.688131\pi\)
							−0.557218	+	0.830366i	\(0.688131\pi\)
\(47\)	894900.	+	650183.i	1.25728	+	0.913467i	0.998621	−	0.0525017i	\(-0.0167195\pi\)
							0.258659	+	0.965969i	\(0.416720\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.b.16.2	✓	28
3.2	odd	2		99.8.f.b.82.6		28
11.3	even	5		363.8.a.o.1.4		14
11.8	odd	10		363.8.a.r.1.11		14
11.9	even	5	inner	33.8.e.b.31.2	yes	28
33.20	odd	10		99.8.f.b.64.6		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.b.16.2	✓	28	1.1	even	1	trivial
33.8.e.b.31.2	yes	28	11.9	even	5	inner
99.8.f.b.64.6		28	33.20	odd	10
99.8.f.b.82.6		28	3.2	odd	2
363.8.a.o.1.4		14	11.3	even	5
363.8.a.r.1.11		14	11.8	odd	10