Newform orbit 165.4.j.a

• Label: 165.4.j.a
• Level: $165$
• Weight: $4$
• Character orbit: 165.j
• Analytic conductor: $9.735$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	165.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.73531515095\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 32 q^{5}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−3.93364	−	3.93364i	2.12132	+	2.12132i			22.9471i	−9.93470	−	5.12851i		−	16.6890i	2.56983	+	2.56983i	58.7964	−	58.7964i			9.00000i	18.9058	+	59.2533i
43.2	−3.75456	−	3.75456i	−2.12132	−	2.12132i			20.1934i	3.78779	−	10.5192i			15.9292i	−24.1831	−	24.1831i	45.7809	−	45.7809i			9.00000i	−53.7163	+	25.2733i
43.3	−3.41365	−	3.41365i	−2.12132	−	2.12132i			15.3061i	−9.66065	+	5.62777i			14.4829i	11.4346	+	11.4346i	24.9404	−	24.9404i			9.00000i	52.1894	+	13.7669i
43.4	−3.36334	−	3.36334i	2.12132	+	2.12132i			14.6242i	4.65600	+	10.1647i		−	14.2695i	10.3410	+	10.3410i	22.2794	−	22.2794i			9.00000i	18.5277	−	49.8472i
43.5	−3.20076	−	3.20076i	2.12132	+	2.12132i			12.4898i	10.8909	−	2.52757i		−	13.5797i	−9.30613	−	9.30613i	14.3707	−	14.3707i			9.00000i	−42.9493	−	26.7690i
43.6	−3.01983	−	3.01983i	−2.12132	−	2.12132i			10.2388i	−5.61975	−	9.66532i			12.8121i	13.7402	+	13.7402i	6.76082	−	6.76082i			9.00000i	−12.2170	+	46.1584i
43.7	−2.89023	−	2.89023i	−2.12132	−	2.12132i			8.70681i	8.39013	+	7.38957i			12.2622i	−2.17308	−	2.17308i	2.04285	−	2.04285i			9.00000i	−2.89182	−	45.6069i
43.8	−2.73289	−	2.73289i	2.12132	+	2.12132i			6.93734i	−8.55655	+	7.19621i		−	11.5947i	−9.73935	−	9.73935i	−2.90414	+	2.90414i			9.00000i	43.0505	+	3.71768i
43.9	−2.12278	−	2.12278i	−2.12132	−	2.12132i			1.01238i	11.1455	+	0.882051i			9.00619i	6.12720	+	6.12720i	−14.8332	+	14.8332i			9.00000i	−21.7870	−	25.5318i
43.10	−2.09753	−	2.09753i	2.12132	+	2.12132i			0.799300i	1.16100	−	11.1199i		−	8.89908i	−8.52171	−	8.52171i	−15.1037	+	15.1037i			9.00000i	−25.7596	+	20.8891i
43.11	−1.97833	−	1.97833i	2.12132	+	2.12132i		−	0.172387i	−7.97513	−	7.83565i		−	8.39336i	23.7092	+	23.7092i	−16.1677	+	16.1677i			9.00000i	0.275939	+	31.2790i
43.12	−1.85909	−	1.85909i	−2.12132	−	2.12132i		−	1.08760i	−10.4566	+	3.95715i			7.88744i	−17.5260	−	17.5260i	−16.8946	+	16.8946i			9.00000i	26.7965	+	12.0831i
43.13	−1.14988	−	1.14988i	2.12132	+	2.12132i		−	5.35555i	10.9032	+	2.47378i		−	4.87853i	14.0040	+	14.0040i	−15.3573	+	15.3573i			9.00000i	−9.69285	−	15.3820i
43.14	−1.11966	−	1.11966i	2.12132	+	2.12132i		−	5.49274i	2.28229	+	10.9449i		−	4.75030i	−17.6433	−	17.6433i	−15.1072	+	15.1072i			9.00000i	9.69915	−	14.8099i
43.15	−1.00879	−	1.00879i	−2.12132	−	2.12132i		−	5.96470i	−7.56005	−	8.23685i			4.27992i	−6.77351	−	6.77351i	−14.0874	+	14.0874i			9.00000i	−0.682750	+	15.9357i
43.16	−0.602674	−	0.602674i	−2.12132	−	2.12132i		−	7.27357i	0.754680	+	11.1548i			2.55693i	7.73513	+	7.73513i	−9.20498	+	9.20498i			9.00000i	6.26791	−	7.17756i
43.17	−0.160429	−	0.160429i	2.12132	+	2.12132i		−	7.94853i	−10.9626	+	2.19596i		−	0.680642i	−5.12937	−	5.12937i	−2.55860	+	2.55860i			9.00000i	2.11101	+	1.40642i
43.18	−0.141300	−	0.141300i	−2.12132	−	2.12132i		−	7.96007i	8.75452	−	6.95402i			0.599486i	23.0443	+	23.0443i	−2.25516	+	2.25516i			9.00000i	−2.21962	−	0.254411i
43.19	0.141300	+	0.141300i	−2.12132	−	2.12132i		−	7.96007i	8.75452	−	6.95402i		−	0.599486i	−23.0443	−	23.0443i	2.25516	−	2.25516i			9.00000i	2.21962	+	0.254411i
43.20	0.160429	+	0.160429i	2.12132	+	2.12132i		−	7.94853i	−10.9626	+	2.19596i			0.680642i	5.12937	+	5.12937i	2.55860	−	2.55860i			9.00000i	−2.11101	−	1.40642i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
11.b	odd	2	1	inner
55.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	165.4.j.a	✓	72
5.c	odd	4	1	inner	165.4.j.a	✓	72
11.b	odd	2	1	inner	165.4.j.a	✓	72
55.e	even	4	1	inner	165.4.j.a	✓	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
165.4.j.a	✓	72	1.a	even	1	1	trivial
165.4.j.a	✓	72	5.c	odd	4	1	inner
165.4.j.a	✓	72	11.b	odd	2	1	inner
165.4.j.a	✓	72	55.e	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(165, [\chi])\).