Newform orbit 165.4.r.a

• Label: 165.4.r.a
• Level: $165$
• Weight: $4$
• Character orbit: 165.r
• Analytic conductor: $9.735$
• Analytic rank: $0$
• Dimension: $272$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	165.r (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.73531515095\)
Analytic rank:	\(0\)
Dimension:	\(272\)
Relative dimension:	\(68\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 272 q + 244 q^{4} - 10 q^{6} + 74 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1	−3.25360	+	4.47820i	−5.04176	−	1.25725i	−6.99620	−	21.5321i	−11.1762	+	0.304013i	22.0341	−	18.4874i	−6.71453	−	20.6652i	77.0723	+	25.0423i	23.8386	+	12.6775i	35.0015	−	51.0384i
29.2	−3.25360	+	4.47820i	4.81786	+	1.94633i	−6.99620	−	21.5321i	−3.74277	+	10.5353i	−24.3915	+	15.2428i	6.71453	+	20.6652i	77.0723	+	25.0423i	19.4236	+	18.7543i	−35.0015	−	51.0384i
29.3	−3.02599	+	4.16492i	−4.66644	+	2.28568i	−5.71781	−	17.5976i	8.66245	−	7.06837i	4.60093	−	26.3518i	7.14712	+	21.9966i	51.4254	+	16.7091i	16.5513	−	21.3320i	3.22671	+	57.4673i
29.4	−3.02599	+	4.16492i	2.43174	+	4.59202i	−5.71781	−	17.5976i	9.39927	−	6.05423i	−26.4838	−	3.76741i	−7.14712	−	21.9966i	51.4254	+	16.7091i	−15.1733	+	22.3332i	−3.22671	+	57.4673i
29.5	−2.94030	+	4.04697i	−0.869183	−	5.12294i	−5.26050	−	16.1901i	2.68634	−	10.8528i	23.2881	+	11.5454i	5.97662	+	18.3941i	42.9285	+	13.9483i	−25.4890	+	8.90554i	36.0224	+	42.7820i
29.6	−2.94030	+	4.04697i	3.71437	−	3.63365i	−5.26050	−	16.1901i	11.1518	+	0.798845i	3.78393	+	25.7160i	−5.97662	−	18.3941i	42.9285	+	13.9483i	0.593126	−	26.9935i	−36.0224	+	42.7820i
29.7	−2.66117	+	3.66279i	−0.416019	−	5.17947i	−3.86206	−	11.8862i	−1.84015	+	11.0279i	20.0784	+	12.2597i	−0.841840	−	2.59092i	19.3674	+	6.29284i	−26.6539	+	4.30951i	−35.4958	−	36.0872i
29.8	−2.66117	+	3.66279i	3.38098	−	3.94575i	−3.86206	−	11.8862i	−11.0568	−	1.65771i	5.45508	+	22.8842i	0.841840	+	2.59092i	19.3674	+	6.29284i	−4.13790	−	26.6810i	35.4958	−	36.0872i
29.9	−2.63080	+	3.62099i	−2.23842	+	4.68929i	−3.71831	−	11.4438i	1.82724	+	11.0300i	−11.0910	−	20.4419i	2.12396	+	6.53688i	17.1661	+	5.57761i	−16.9789	−	20.9932i	−44.7467	−	22.4014i
29.10	−2.63080	+	3.62099i	−0.945375	+	5.10943i	−3.71831	−	11.4438i	−9.92552	−	5.14627i	−16.0141	−	16.8651i	−2.12396	−	6.53688i	17.1661	+	5.57761i	−25.2125	−	9.66066i	44.7467	−	22.4014i
29.11	−2.51995	+	3.46842i	−5.18074	−	0.399975i	−3.20763	−	9.87207i	7.79097	+	8.01878i	14.4425	−	16.9610i	−4.90822	−	15.1059i	9.70457	+	3.15321i	26.6800	+	4.14433i	−47.4454	+	6.81538i
29.12	−2.51995	+	3.46842i	4.42640	+	2.72157i	−3.20763	−	9.87207i	−5.21877	−	9.88759i	−20.5939	+	8.49438i	4.90822	+	15.1059i	9.70457	+	3.15321i	12.1861	+	24.0936i	47.4454	+	6.81538i
29.13	−2.10571	+	2.89827i	−4.32771	−	2.87593i	−1.49378	−	4.59739i	−9.60966	+	5.71441i	17.4481	−	6.48696i	10.7038	+	32.9431i	−10.7870	−	3.50491i	10.4581	+	24.8923i	3.67330	−	39.8843i
29.14	−2.10571	+	2.89827i	5.19162	+	0.217087i	−1.49378	−	4.59739i	−8.40428	+	7.37348i	−11.5612	+	14.5896i	−10.7038	−	32.9431i	−10.7870	−	3.50491i	26.9057	+	2.25406i	−3.67330	−	39.8843i
29.15	−2.09780	+	2.88737i	−2.69428	−	4.44307i	−1.46402	−	4.50578i	−1.29157	−	11.1055i	18.4808	+	1.54126i	−9.51769	−	29.2924i	−11.0734	−	3.59797i	−12.4817	+	23.9418i	34.7751	+	19.5678i
29.16	−2.09780	+	2.88737i	4.79129	−	2.01086i	−1.46402	−	4.50578i	10.1628	+	4.66014i	−4.24506	+	18.0526i	9.51769	+	29.2924i	−11.0734	−	3.59797i	18.9129	−	19.2692i	−34.7751	+	19.5678i
29.17	−1.87009	+	2.57395i	−4.27269	−	2.95705i	−0.655877	−	2.01858i	11.1101	−	1.25153i	15.6016	−	5.46776i	0.879461	+	2.70670i	−17.7846	−	5.77858i	9.51173	+	25.2691i	−17.5554	+	30.9373i
29.18	−1.87009	+	2.57395i	5.19479	+	0.119120i	−0.655877	−	2.01858i	4.62348	−	10.1796i	−10.0213	+	13.1484i	−0.879461	−	2.70670i	−17.7846	−	5.77858i	26.9716	+	1.23761i	17.5554	+	30.9373i
29.19	−1.70914	+	2.35244i	−5.03785	+	1.27283i	−0.140642	−	0.432850i	−5.28037	−	9.85483i	5.61616	−	14.0267i	2.55765	+	7.87163i	−20.8650	−	6.77945i	23.7598	−	12.8246i	32.2078	+	4.42159i
29.20	−1.70914	+	2.35244i	3.32755	+	3.99091i	−0.140642	−	0.432850i	7.74077	+	8.06724i	−15.0756	+	1.00681i	−2.55765	−	7.87163i	−20.8650	−	6.77945i	−4.85478	+	26.5600i	−32.2078	+	4.42159i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
11.d	odd	10	1	inner
15.d	odd	2	1	inner
33.f	even	10	1	inner
55.h	odd	10	1	inner
165.r	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	165.4.r.a	✓	272
3.b	odd	2	1	inner	165.4.r.a	✓	272
5.b	even	2	1	inner	165.4.r.a	✓	272
11.d	odd	10	1	inner	165.4.r.a	✓	272
15.d	odd	2	1	inner	165.4.r.a	✓	272
33.f	even	10	1	inner	165.4.r.a	✓	272
55.h	odd	10	1	inner	165.4.r.a	✓	272
165.r	even	10	1	inner	165.4.r.a	✓	272

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
165.4.r.a	✓	272	1.a	even	1	1	trivial
165.4.r.a	✓	272	3.b	odd	2	1	inner
165.4.r.a	✓	272	5.b	even	2	1	inner
165.4.r.a	✓	272	11.d	odd	10	1	inner
165.4.r.a	✓	272	15.d	odd	2	1	inner
165.4.r.a	✓	272	33.f	even	10	1	inner
165.4.r.a	✓	272	55.h	odd	10	1	inner
165.4.r.a	✓	272	165.r	even	10	1	inner

Hecke kernels

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