Embedded newform 33.7.h.a.5.11

• Label: 33.7.h.a.5.11
• Level: $33$
• Weight: $7$
• Character: 33.5
• Analytic conductor: $7.592$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	33.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.59178475946\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			5.11
Character	\(\chi\)	\(=\)	33.5
Dual form			33.7.h.a.20.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.658923 + 0.214097i) q^{2} +(-17.8783 + 20.2328i) q^{3} +(-51.3887 + 37.3361i) q^{4} +(-87.3837 - 28.3927i) q^{5} +(7.44866 - 17.1596i) q^{6} +(484.811 - 352.236i) q^{7} +(51.9309 - 71.4767i) q^{8} +(-89.7314 - 723.456i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 33 q^{3} + 634 q^{4} + 109 q^{6} + 402 q^{7} - 2471 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.658923	+	0.214097i	−0.0823654	+	0.0267621i	−0.349910	−	0.936783i	\(-0.613788\pi\)
							0.267544	+	0.963546i	\(0.413788\pi\)
\(3\)	−17.8783	+	20.2328i	−0.662160	+	0.749363i
							\(5\)	−87.3837	−	28.3927i	−0.699069	−	0.227141i	−0.0621443	−	0.998067i	\(-0.519794\pi\)
−0.636925	+	0.770926i	\(0.719794\pi\)
\(7\)	484.811	−	352.236i	1.41344	−	1.02693i	0.420632	−	0.907232i	\(-0.361808\pi\)
							0.992811	−	0.119695i	\(-0.0381915\pi\)
\(11\)	531.874	+	1220.11i	0.399605	+	0.916687i
							\(13\)	−544.761	−	1676.60i	−0.247957	−	0.763132i	−0.995136	−	0.0985098i	\(-0.968592\pi\)
0.747179	−	0.664622i	\(-0.231408\pi\)
\(17\)	−6602.52	−	2145.29i	−1.34389	−	0.436656i	−0.453256	−	0.891380i	\(-0.649738\pi\)
							−0.890632	+	0.454725i	\(0.849738\pi\)
\(19\)	2312.73	+	1680.30i	0.337182	+	0.244977i	0.743472	−	0.668767i	\(-0.233178\pi\)
							−0.406290	+	0.913744i	\(0.633178\pi\)
\(23\)		−	17739.1i		−	1.45797i	−0.684532	−	0.728983i	\(-0.739993\pi\)
							0.684532	−	0.728983i	\(-0.260007\pi\)
\(29\)	−6602.69	−	9087.83i	−0.270724	−	0.372620i	0.651910	−	0.758297i	\(-0.273968\pi\)
							−0.922634	+	0.385677i	\(0.873968\pi\)
\(31\)	5713.25	+	17583.6i	0.191778	+	0.590231i	0.999999	+	0.00135245i	\(0.000430497\pi\)
							−0.808221	+	0.588879i	\(0.799570\pi\)
\(37\)	29714.9	−	21589.2i	0.586637	−	0.426217i	−0.254474	−	0.967080i	\(-0.581902\pi\)
							0.841111	+	0.540863i	\(0.181902\pi\)
\(41\)	45118.2	−	62099.9i	0.654637	−	0.901031i	−0.344652	−	0.938730i	\(-0.612003\pi\)
							0.999289	+	0.0376999i	\(0.0120031\pi\)
\(43\)	−96058.6			−1.20818			−0.604089	−	0.796917i	\(-0.706463\pi\)
							−0.604089	+	0.796917i	\(0.706463\pi\)
\(47\)	71381.2	−	98247.9i	0.687528	−	0.946301i	−0.312465	−	0.949929i	\(-0.601155\pi\)
							0.999993	+	0.00362778i	\(0.00115476\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.h.a.5.11	✓	88
3.2	odd	2	inner	33.7.h.a.5.12	yes	88
11.9	even	5	inner	33.7.h.a.20.12	yes	88
33.20	odd	10	inner	33.7.h.a.20.11	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.h.a.5.11	✓	88	1.1	even	1	trivial
33.7.h.a.5.12	yes	88	3.2	odd	2	inner
33.7.h.a.20.11	yes	88	33.20	odd	10	inner
33.7.h.a.20.12	yes	88	11.9	even	5	inner