Embedded newform 33.5.h.a.5.7

• Label: 33.5.h.a.5.7
• Level: $33$
• Weight: $5$
• Character: 33.5
• Analytic conductor: $3.411$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	33.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.41120878177\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			5.7
Character	\(\chi\)	\(=\)	33.5
Dual form			33.5.h.a.20.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.493178 + 0.160243i) q^{2} +(3.56251 + 8.26490i) q^{3} +(-12.7267 + 9.24651i) q^{4} +(-30.1183 - 9.78604i) q^{5} +(-3.08135 - 3.50520i) q^{6} +(4.18763 - 3.04249i) q^{7} +(9.67167 - 13.3119i) q^{8} +(-55.6170 + 58.8876i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 90 q^{4} - 83 q^{6} - 86 q^{7} + 232 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.493178	+	0.160243i	−0.123295	+	0.0400608i	−0.370014	−	0.929026i	\(-0.620647\pi\)
							0.246720	+	0.969087i	\(0.420647\pi\)
\(3\)	3.56251	+	8.26490i	0.395835	+	0.918322i
							\(5\)	−30.1183	−	9.78604i	−1.20473	−	0.391442i	−0.363233	−	0.931698i	\(-0.618327\pi\)
−0.841500	+	0.540256i	\(0.818327\pi\)
\(7\)	4.18763	−	3.04249i	0.0854619	−	0.0620917i	−0.544234	−	0.838934i	\(-0.683180\pi\)
							0.629696	+	0.776842i	\(0.283180\pi\)
\(11\)	−1.81442	+	120.986i	−0.0149952	+	0.999888i
							\(13\)	56.5003	+	173.890i	0.334321	+	1.02894i	0.967055	+	0.254566i	\(0.0819326\pi\)
−0.632734	+	0.774369i	\(0.718067\pi\)
\(17\)	480.203	+	156.027i	1.66160	+	0.539887i	0.981207	−	0.192957i	\(-0.0618079\pi\)
							0.680396	+	0.732845i	\(0.261808\pi\)
\(19\)	−399.018	−	289.904i	−1.10531	−	0.803057i	−0.123395	−	0.992358i	\(-0.539378\pi\)
							−0.981919	+	0.189300i	\(0.939378\pi\)
\(23\)			376.413i			0.711556i	0.934571	+	0.355778i	\(0.115784\pi\)
							−0.934571	+	0.355778i	\(0.884216\pi\)
\(29\)	−278.690	−	383.584i	−0.331379	−	0.456104i	0.610520	−	0.792001i	\(-0.290961\pi\)
							−0.941899	+	0.335897i	\(0.890961\pi\)
\(31\)	249.029	+	766.432i	0.259135	+	0.797536i	0.992987	+	0.118226i	\(0.0377207\pi\)
							−0.733852	+	0.679310i	\(0.762279\pi\)
\(37\)	724.293	−	526.229i	0.529067	−	0.384390i	−0.290942	−	0.956741i	\(-0.593969\pi\)
							0.820009	+	0.572351i	\(0.193969\pi\)
\(41\)	−1025.09	+	1410.92i	−0.609812	+	0.839334i	−0.996562	−	0.0828509i	\(-0.973597\pi\)
							0.386750	+	0.922184i	\(0.373597\pi\)
\(43\)	216.414			0.117044			0.0585220	−	0.998286i	\(-0.481361\pi\)
							0.0585220	+	0.998286i	\(0.481361\pi\)
\(47\)	69.0608	−	95.0540i	0.0312634	−	0.0430303i	−0.793099	−	0.609093i	\(-0.791534\pi\)
							0.824363	+	0.566062i	\(0.191534\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.h.a.5.7	✓	56
3.2	odd	2	inner	33.5.h.a.5.8	yes	56
11.9	even	5	inner	33.5.h.a.20.8	yes	56
33.20	odd	10	inner	33.5.h.a.20.7	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.h.a.5.7	✓	56	1.1	even	1	trivial
33.5.h.a.5.8	yes	56	3.2	odd	2	inner
33.5.h.a.20.7	yes	56	33.20	odd	10	inner
33.5.h.a.20.8	yes	56	11.9	even	5	inner