Embedded newform 33.7.h.a.5.9

• Label: 33.7.h.a.5.9
• 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			5.9
Character	\(\chi\)	\(=\)	33.5
Dual form			33.7.h.a.20.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.91805 + 1.27305i) q^{2} +(-26.2247 - 6.42389i) q^{3} +(-38.0467 + 27.6425i) q^{4} +(-40.1977 - 13.0610i) q^{5} +(110.927 - 8.21623i) q^{6} +(-125.751 + 91.3637i) q^{7} +(268.853 - 370.045i) q^{8} +(646.467 + 336.929i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 33 q^{3} + 634 q^{4} + 109 q^{6} + 402 q^{7} - 2471 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.91805	+	1.27305i	−0.489756	+	0.159131i	−0.543476	−	0.839425i	\(-0.682892\pi\)
							0.0537202	+	0.998556i	\(0.482892\pi\)
\(3\)	−26.2247	−	6.42389i	−0.971284	−	0.237922i
							\(5\)	−40.1977	−	13.0610i	−0.321581	−	0.104488i	0.143778	−	0.989610i	\(-0.454075\pi\)
−0.465359	+	0.885122i	\(0.654075\pi\)
\(7\)	−125.751	+	91.3637i	−0.366622	+	0.266367i	−0.755809	−	0.654792i	\(-0.772756\pi\)
							0.389187	+	0.921159i	\(0.372756\pi\)
\(11\)	718.674	−	1120.30i	0.539951	−	0.841697i
							\(13\)	267.692	+	823.872i	0.121844	+	0.374999i	0.993313	−	0.115453i	\(-0.0368319\pi\)
−0.871469	+	0.490451i	\(0.836832\pi\)
\(17\)	5777.91	+	1877.36i	1.17605	+	0.382120i	0.830896	−	0.556428i	\(-0.187828\pi\)
							0.345149	+	0.938548i	\(0.387828\pi\)
\(19\)	−350.201	−	254.436i	−0.0510572	−	0.0370952i	0.561964	−	0.827162i	\(-0.310046\pi\)
							−0.613021	+	0.790067i	\(0.710046\pi\)
\(23\)		−	1355.07i		−	0.111372i	−0.998448	−	0.0556862i	\(-0.982265\pi\)
							0.998448	−	0.0556862i	\(-0.0177346\pi\)
\(29\)	6443.73	+	8869.03i	0.264206	+	0.363649i	0.920423	−	0.390924i	\(-0.127844\pi\)
							−0.656217	+	0.754572i	\(0.727844\pi\)
\(31\)	−24.5462	−	75.5453i	−0.000823946	−	0.00253584i	0.950644	−	0.310285i	\(-0.100424\pi\)
							−0.951468	+	0.307749i	\(0.900424\pi\)
\(37\)	13967.0	−	10147.6i	0.275738	−	0.200335i	−0.441318	−	0.897351i	\(-0.645489\pi\)
							0.717056	+	0.697015i	\(0.245489\pi\)
\(41\)	39162.9	−	53903.1i	0.568229	−	0.782100i	−0.424115	−	0.905608i	\(-0.639415\pi\)
							0.992343	+	0.123509i	\(0.0394147\pi\)
\(43\)	147092.			1.85005			0.925026	−	0.379903i	\(-0.124043\pi\)
							0.925026	+	0.379903i	\(0.124043\pi\)
\(47\)	107037.	−	147324.i	1.03096	−	1.41900i	0.126738	−	0.991936i	\(-0.459549\pi\)
							0.904223	−	0.427060i	\(-0.140451\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.h.a.5.9	✓	88
3.2	odd	2	inner	33.7.h.a.5.14	yes	88
11.9	even	5	inner	33.7.h.a.20.14	yes	88
33.20	odd	10	inner	33.7.h.a.20.9	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.h.a.5.9	✓	88	1.1	even	1	trivial
33.7.h.a.5.14	yes	88	3.2	odd	2	inner
33.7.h.a.20.9	yes	88	33.20	odd	10	inner
33.7.h.a.20.14	yes	88	11.9	even	5	inner