Newform orbit 60.5.l.a

• Label: 60.5.l.a
• Level: $60$
• Weight: $5$
• Character orbit: 60.l
• Analytic conductor: $6.202$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	60.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.20219778503\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 4 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−3.99427	+	0.214108i	8.89373	−	1.37898i	15.9083	−	1.71041i	16.1632	+	19.0723i	−35.2287	+	7.41222i	−4.24582	−	4.24582i	−63.1758	+	10.2379i	77.1968	−	24.5285i	−68.6436	−	72.7190i
23.2	−3.98340	−	0.364047i	−6.64241	−	6.07276i	15.7349	+	2.90029i	−22.0852	+	11.7150i	24.2486	+	26.6084i	−39.7479	−	39.7479i	−61.6227	−	17.2813i	7.24316	+	80.6755i	92.2391	−	38.6257i
23.3	−3.94073	+	0.686026i	−4.52298	+	7.78092i	15.0587	−	5.40689i	−10.9064	+	22.4956i	12.4859	−	33.7654i	53.2499	+	53.2499i	−55.6332	+	31.6378i	−40.0853	−	70.3859i	27.5466	−	96.1311i
23.4	−3.91278	+	0.830775i	1.90378	−	8.79634i	14.6196	−	6.50127i	6.71772	−	24.0805i	−0.141262	+	35.9997i	17.5991	+	17.5991i	−51.8022	+	37.5837i	−73.7513	−	33.4925i	−6.27943	+	99.8026i
23.5	−3.80860	−	1.22253i	4.13002	+	7.99643i	13.0108	+	9.31227i	3.82277	−	24.7060i	−5.95367	−	35.5043i	27.1938	+	27.1938i	−38.1685	−	51.3728i	−46.8859	+	66.0508i	−44.7633	+	89.4217i
23.6	−3.70170	−	1.51572i	−8.98881	−	0.448719i	11.4052	+	11.2215i	24.1558	−	6.44182i	32.5938	+	15.2855i	0.578123	+	0.578123i	−25.2101	−	58.8256i	80.5973	+	8.06689i	−99.1816	−	12.7677i
23.7	−3.48777	+	1.95843i	6.74616	+	5.95729i	8.32911	−	13.6611i	−24.5931	−	4.49233i	−35.1960	−	7.56581i	−44.1332	−	44.1332i	−2.29574	+	63.9588i	10.0213	+	80.3777i	94.5729	−	32.4955i
23.8	−3.32846	+	2.21841i	−5.44127	+	7.16886i	6.15731	−	14.7678i	23.0805	−	9.60676i	2.20756	−	35.9323i	−46.8712	−	46.8712i	12.2666	+	62.8134i	−21.7852	−	78.0154i	−55.5109	+	83.1778i
23.9	−3.26396	−	2.31226i	7.54313	−	4.90930i	5.30687	+	15.0943i	−23.1720	−	9.38384i	−35.9721	−	1.41794i	−14.0617	−	14.0617i	17.5805	−	61.5380i	32.7976	−	74.0629i	53.9347	+	84.2083i
23.10	−2.92347	+	2.73008i	−8.80598	−	1.85868i	1.09333	−	15.9626i	−14.3650	−	20.4608i	30.8183	−	18.6073i	27.9806	+	27.9806i	40.3829	+	49.6510i	74.0906	+	32.7349i	97.8554	+	20.5990i
23.11	−2.89513	−	2.76011i	0.700985	+	8.97266i	0.763541	+	15.9818i	6.70624	+	24.0837i	22.7361	−	27.9118i	−58.7764	−	58.7764i	41.9010	−	48.3767i	−80.0172	+	12.5794i	47.0584	−	88.2355i
23.12	−2.76011	−	2.89513i	−0.700985	−	8.97266i	−0.763541	+	15.9818i	6.70624	+	24.0837i	−24.0422	+	26.7950i	58.7764	+	58.7764i	48.3767	−	41.9010i	−80.0172	+	12.5794i	51.2155	−	85.8893i
23.13	−2.73008	+	2.92347i	−1.85868	−	8.80598i	−1.09333	−	15.9626i	14.3650	+	20.4608i	30.8183	+	18.6073i	−27.9806	−	27.9806i	49.6510	+	40.3829i	−74.0906	+	32.7349i	−99.0343	−	13.8641i
23.14	−2.31226	−	3.26396i	−7.54313	+	4.90930i	−5.30687	+	15.0943i	−23.1720	−	9.38384i	33.4655	+	13.2689i	14.0617	+	14.0617i	61.5380	−	17.5805i	32.7976	−	74.0629i	22.9514	+	97.3305i
23.15	−2.21841	+	3.32846i	7.16886	−	5.44127i	−6.15731	−	14.7678i	−23.0805	+	9.60676i	2.20756	+	35.9323i	46.8712	+	46.8712i	62.8134	+	12.2666i	21.7852	−	78.0154i	19.2263	−	98.1343i
23.16	−1.95843	+	3.48777i	5.95729	+	6.74616i	−8.32911	−	13.6611i	24.5931	+	4.49233i	−35.1960	+	7.56581i	44.1332	+	44.1332i	63.9588	−	2.29574i	−10.0213	+	80.3777i	−63.8320	+	76.9771i
23.17	−1.51572	−	3.70170i	8.98881	+	0.448719i	−11.4052	+	11.2215i	24.1558	−	6.44182i	−11.9635	−	33.9540i	−0.578123	−	0.578123i	58.8256	+	25.2101i	80.5973	+	8.06689i	−60.4591	−	79.6536i
23.18	−1.22253	−	3.80860i	−4.13002	−	7.99643i	−13.0108	+	9.31227i	3.82277	−	24.7060i	−25.4061	+	25.5055i	−27.1938	−	27.1938i	51.3728	+	38.1685i	−46.8859	+	66.0508i	−98.7687	+	15.6445i
23.19	−0.830775	+	3.91278i	−8.79634	+	1.90378i	−14.6196	−	6.50127i	−6.71772	+	24.0805i	−0.141262	−	35.9997i	−17.5991	−	17.5991i	37.5837	−	51.8022i	73.7513	−	33.4925i	−88.6408	−	46.2905i
23.20	−0.686026	+	3.94073i	7.78092	−	4.52298i	−15.0587	−	5.40689i	10.9064	−	22.4956i	12.4859	+	33.7654i	−53.2499	−	53.2499i	31.6378	−	55.6332i	40.0853	−	70.3859i	81.1669	+	58.4117i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
5.c	odd	4	1	inner
12.b	even	2	1	inner
15.e	even	4	1	inner
20.e	even	4	1	inner
60.l	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	60.5.l.a	✓	88
3.b	odd	2	1	inner	60.5.l.a	✓	88
4.b	odd	2	1	inner	60.5.l.a	✓	88
5.c	odd	4	1	inner	60.5.l.a	✓	88
12.b	even	2	1	inner	60.5.l.a	✓	88
15.e	even	4	1	inner	60.5.l.a	✓	88
20.e	even	4	1	inner	60.5.l.a	✓	88
60.l	odd	4	1	inner	60.5.l.a	✓	88

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
60.5.l.a	✓	88	1.a	even	1	1	trivial
60.5.l.a	✓	88	3.b	odd	2	1	inner
60.5.l.a	✓	88	4.b	odd	2	1	inner
60.5.l.a	✓	88	5.c	odd	4	1	inner
60.5.l.a	✓	88	12.b	even	2	1	inner
60.5.l.a	✓	88	15.e	even	4	1	inner
60.5.l.a	✓	88	20.e	even	4	1	inner
60.5.l.a	✓	88	60.l	odd	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(60, [\chi])\).