Embedded newform 33.8.e.b.16.6

• Label: 33.8.e.b.16.6
• 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			16.6
Character	\(\chi\)	\(=\)	33.16
Dual form			33.8.e.b.31.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.75376 + 8.47521i) q^{2} +(-21.8435 - 15.8702i) q^{3} +(39.3082 - 28.5591i) q^{4} +(-42.2440 + 130.014i) q^{5} +(74.3516 - 228.831i) q^{6} +(29.1631 - 21.1883i) q^{7} +(1273.10 + 924.960i) q^{8} +(225.273 + 693.320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 6 q^{2} - 189 q^{3} + 160 q^{4} + 773 q^{5} - 162 q^{6} + 1289 q^{7} + 2956 q^{8} - 5103 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.75376	+	8.47521i	0.243401	+	0.749110i	0.995895	+	0.0905117i	\(0.0288502\pi\)
							−0.752495	+	0.658598i	\(0.771150\pi\)
\(3\)	−21.8435	−	15.8702i	−0.467086	−	0.339358i
							\(5\)	−42.2440	+	130.014i	−0.151137	+	0.465151i	−0.997749	−	0.0670593i	\(-0.978638\pi\)
0.846612	+	0.532210i	\(0.178638\pi\)
\(7\)	29.1631	−	21.1883i	0.0321359	−	0.0233481i	−0.571601	−	0.820531i	\(-0.693678\pi\)
							0.603737	+	0.797183i	\(0.293678\pi\)
\(11\)	4384.54	+	512.818i	0.993230	+	0.116169i
							\(13\)	4146.78	+	12762.5i	0.523491	+	1.61114i	0.767280	+	0.641313i	\(0.221610\pi\)
−0.243788	+	0.969829i	\(0.578390\pi\)
\(17\)	−9618.81	+	29603.7i	−0.474843	+	1.46142i	0.371326	+	0.928503i	\(0.378903\pi\)
							−0.846169	+	0.532915i	\(0.821097\pi\)
\(19\)	−38915.1	−	28273.5i	−1.30161	−	0.945675i	−0.301640	−	0.953422i	\(-0.597534\pi\)
							−0.999970	+	0.00774737i	\(0.997534\pi\)
\(23\)	76581.4			1.31243			0.656215	−	0.754574i	\(-0.272156\pi\)
							0.656215	+	0.754574i	\(0.272156\pi\)
\(29\)	115621.	−	84003.5i	0.880325	−	0.639594i	−0.0530122	−	0.998594i	\(-0.516882\pi\)
							0.933338	+	0.359000i	\(0.116882\pi\)
\(31\)	−7016.03	−	21593.1i	−0.0422986	−	0.130182i	0.927677	−	0.373383i	\(-0.121802\pi\)
							−0.969976	+	0.243202i	\(0.921802\pi\)
\(37\)	212723.	−	154552.i	0.690413	−	0.501614i	−0.186383	−	0.982477i	\(-0.559677\pi\)
							0.876796	+	0.480863i	\(0.159677\pi\)
\(41\)	−478448.	−	347613.i	−1.08416	−	0.787685i	−0.105753	−	0.994392i	\(-0.533725\pi\)
							−0.978403	+	0.206707i	\(0.933725\pi\)
\(43\)	−750154.			−1.43883			−0.719417	−	0.694578i	\(-0.755591\pi\)
							−0.719417	+	0.694578i	\(0.755591\pi\)
\(47\)	85738.5	+	62292.6i	0.120457	+	0.0875174i	0.646383	−	0.763013i	\(-0.276281\pi\)
							−0.525925	+	0.850531i	\(0.676281\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.b.16.6	✓	28
3.2	odd	2		99.8.f.b.82.2		28
11.3	even	5		363.8.a.o.1.11		14
11.8	odd	10		363.8.a.r.1.4		14
11.9	even	5	inner	33.8.e.b.31.6	yes	28
33.20	odd	10		99.8.f.b.64.2		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.b.16.6	✓	28	1.1	even	1	trivial
33.8.e.b.31.6	yes	28	11.9	even	5	inner
99.8.f.b.64.2		28	33.20	odd	10
99.8.f.b.82.2		28	3.2	odd	2
363.8.a.o.1.11		14	11.3	even	5
363.8.a.r.1.4		14	11.8	odd	10