Embedded newform 33.8.e.b.16.4

• Label: 33.8.e.b.16.4
• 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			16.4
Character	\(\chi\)	\(=\)	33.16
Dual form			33.8.e.b.31.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.970201 - 2.98597i) q^{2} +(-21.8435 - 15.8702i) q^{3} +(95.5794 - 69.4425i) q^{4} +(-100.399 + 308.998i) q^{5} +(-26.1954 + 80.6212i) q^{6} +(1340.18 - 973.701i) q^{7} +(-625.207 - 454.240i) q^{8} +(225.273 + 693.320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 6 q^{2} - 189 q^{3} + 160 q^{4} + 773 q^{5} - 162 q^{6} + 1289 q^{7} + 2956 q^{8} - 5103 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.970201	−	2.98597i	−0.0857545	−	0.263925i	0.898980	−	0.437990i	\(-0.144310\pi\)
							−0.984734	+	0.174065i	\(0.944310\pi\)
\(3\)	−21.8435	−	15.8702i	−0.467086	−	0.339358i
							\(5\)	−100.399	+	308.998i	−0.359200	+	1.10550i	0.594334	+	0.804218i	\(0.297416\pi\)
−0.953534	+	0.301285i	\(0.902584\pi\)
\(7\)	1340.18	−	973.701i	1.47680	−	1.07296i	0.498228	−	0.867046i	\(-0.333984\pi\)
							0.978571	−	0.205911i	\(-0.0660158\pi\)
\(11\)	−4300.99	−	994.308i	−0.974303	−	0.225241i
							\(13\)	−1831.75	−	5637.55i	−0.231241	−	0.711686i	−0.997598	−	0.0692712i	\(-0.977933\pi\)
0.766357	−	0.642415i	\(-0.222067\pi\)
\(17\)	9048.40	−	27848.1i	0.446684	−	1.37475i	−0.433942	−	0.900941i	\(-0.642878\pi\)
							0.880626	−	0.473811i	\(-0.157122\pi\)
\(19\)	−26235.4	−	19061.1i	−0.877506	−	0.637546i	0.0550841	−	0.998482i	\(-0.482457\pi\)
							−0.932591	+	0.360936i	\(0.882457\pi\)
\(23\)	10937.0			0.187435			0.0937177	−	0.995599i	\(-0.470125\pi\)
							0.0937177	+	0.995599i	\(0.470125\pi\)
\(29\)	80959.3	−	58820.4i	0.616416	−	0.447852i	−0.235252	−	0.971934i	\(-0.575592\pi\)
							0.851668	+	0.524082i	\(0.175592\pi\)
\(31\)	66382.8	+	204305.i	0.400212	+	1.23172i	0.924828	+	0.380386i	\(0.124209\pi\)
							−0.524616	+	0.851339i	\(0.675791\pi\)
\(37\)	−69002.9	+	50133.6i	−0.223955	+	0.162713i	−0.694105	−	0.719873i	\(-0.744200\pi\)
							0.470150	+	0.882587i	\(0.344200\pi\)
\(41\)	−281410.	−	204456.i	−0.637670	−	0.463294i	0.221379	−	0.975188i	\(-0.428944\pi\)
							−0.859049	+	0.511894i	\(0.828944\pi\)
\(43\)	492042.			0.943762			0.471881	−	0.881662i	\(-0.343575\pi\)
							0.471881	+	0.881662i	\(0.343575\pi\)
\(47\)	771628.	+	560620.i	1.08409	+	0.787637i	0.978391	−	0.206761i	\(-0.0662922\pi\)
							0.105698	+	0.994398i	\(0.466292\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.b.16.4	✓	28
3.2	odd	2		99.8.f.b.82.4		28
11.3	even	5		363.8.a.o.1.7		14
11.8	odd	10		363.8.a.r.1.8		14
11.9	even	5	inner	33.8.e.b.31.4	yes	28
33.20	odd	10		99.8.f.b.64.4		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.b.16.4	✓	28	1.1	even	1	trivial
33.8.e.b.31.4	yes	28	11.9	even	5	inner
99.8.f.b.64.4		28	33.20	odd	10
99.8.f.b.82.4		28	3.2	odd	2
363.8.a.o.1.7		14	11.3	even	5
363.8.a.r.1.8		14	11.8	odd	10