Embedded newform 33.3.g.a.13.1

• Label: 33.3.g.a.13.1
• Level: $33$
• Weight: $3$
• Character: 33.13
• Analytic conductor: $0.899$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	33.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.899184872389\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 2*x^15 + 3*x^14 - 4*x^13 + 77*x^12 + 88*x^11 - 577*x^10 + 578*x^9 + 1520*x^8 + 1868*x^7 - 1619*x^6 - 16804*x^5 + 32427*x^4 + 43316*x^3 - 71672*x^2 + 83521
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			13.1
Root			\(-1.95510 - 0.109518i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.13
Dual form			33.3.g.a.28.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.47243 - 1.12826i) q^{2} +(-1.40126 + 1.01807i) q^{3} +(7.54873 + 5.48447i) q^{4} +(1.69033 + 5.20232i) q^{5} +(6.01443 - 1.95421i) q^{6} +(-4.20886 + 5.79300i) q^{7} +(-11.4402 - 15.7461i) q^{8} +(0.927051 - 2.85317i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} - 4 q^{5} - 30 q^{7} - 40 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.47243	−	1.12826i	−1.73622	−	0.564130i	−0.741891	−	0.670521i	\(-0.766071\pi\)
							−0.994324	+	0.106390i	\(0.966071\pi\)
\(3\)	−1.40126	+	1.01807i	−0.467086	+	0.339358i
							\(5\)	1.69033	+	5.20232i	0.338067	+	1.04046i	0.965192	+	0.261544i	\(0.0842317\pi\)
−0.627125	+	0.778919i	\(0.715768\pi\)
\(7\)	−4.20886	+	5.79300i	−0.601265	+	0.827571i	−0.995823	−	0.0913001i	\(-0.970898\pi\)
							0.394558	+	0.918871i	\(0.370898\pi\)
\(11\)	0.170982	+	10.9987i	0.0155439	+	0.999879i
							\(13\)	−6.27605	−	2.03921i	−0.482773	−	0.156863i	0.0575115	−	0.998345i	\(-0.481683\pi\)
−0.540285	+	0.841482i	\(0.681683\pi\)
\(17\)	17.5557	−	5.70418i	1.03269	−	0.335540i	0.256834	−	0.966456i	\(-0.417321\pi\)
							0.775851	+	0.630916i	\(0.217321\pi\)
\(19\)	4.95437	+	6.81910i	0.260756	+	0.358900i	0.919242	−	0.393693i	\(-0.128803\pi\)
							−0.658486	+	0.752593i	\(0.728803\pi\)
\(23\)	−17.7114			−0.770060			−0.385030	−	0.922904i	\(-0.625809\pi\)
							−0.385030	+	0.922904i	\(0.625809\pi\)
\(29\)	11.8017	−	16.2437i	0.406956	−	0.560126i	−0.555517	−	0.831505i	\(-0.687480\pi\)
							0.962473	+	0.271379i	\(0.0874796\pi\)
\(31\)	10.9215	−	33.6128i	0.352305	−	1.08428i	−0.605250	−	0.796035i	\(-0.706927\pi\)
							0.957555	−	0.288249i	\(-0.0930732\pi\)
\(37\)	39.4991	+	28.6978i	1.06754	+	0.775616i	0.975469	−	0.220136i	\(-0.0706502\pi\)
							0.0920744	+	0.995752i	\(0.470650\pi\)
\(41\)	18.5926	+	25.5905i	0.453478	+	0.624159i	0.973140	−	0.230212i	\(-0.0739421\pi\)
							−0.519662	+	0.854372i	\(0.673942\pi\)
\(43\)			45.0047i			1.04662i	0.852142	+	0.523311i	\(0.175303\pi\)
							−0.852142	+	0.523311i	\(0.824697\pi\)
\(47\)	0.589182	−	0.428066i	0.0125358	−	0.00910778i	−0.581500	−	0.813547i	\(-0.697534\pi\)
							0.594036	+	0.804439i	\(0.297534\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.3.g.a.13.1	✓	16
3.2	odd	2		99.3.k.c.46.4		16
4.3	odd	2		528.3.bf.b.145.4		16
11.2	odd	10		363.3.g.a.118.1		16
11.3	even	5		363.3.g.a.40.1		16
11.4	even	5		363.3.c.e.241.1		16
11.5	even	5		363.3.g.f.94.4		16
11.6	odd	10	inner	33.3.g.a.28.1	yes	16
11.7	odd	10		363.3.c.e.241.16		16
11.8	odd	10		363.3.g.g.40.4		16
11.9	even	5		363.3.g.g.118.4		16
11.10	odd	2		363.3.g.f.112.4		16
33.17	even	10		99.3.k.c.28.4		16
33.26	odd	10		1089.3.c.m.604.16		16
33.29	even	10		1089.3.c.m.604.1		16
44.39	even	10		528.3.bf.b.193.4		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.13.1	✓	16	1.1	even	1	trivial
33.3.g.a.28.1	yes	16	11.6	odd	10	inner
99.3.k.c.28.4		16	33.17	even	10
99.3.k.c.46.4		16	3.2	odd	2
363.3.c.e.241.1		16	11.4	even	5
363.3.c.e.241.16		16	11.7	odd	10
363.3.g.a.40.1		16	11.3	even	5
363.3.g.a.118.1		16	11.2	odd	10
363.3.g.f.94.4		16	11.5	even	5
363.3.g.f.112.4		16	11.10	odd	2
363.3.g.g.40.4		16	11.8	odd	10
363.3.g.g.118.4		16	11.9	even	5
528.3.bf.b.145.4		16	4.3	odd	2
528.3.bf.b.193.4		16	44.39	even	10
1089.3.c.m.604.1		16	33.29	even	10
1089.3.c.m.604.16		16	33.26	odd	10