Newform orbit 45.5.k.a

• Label: 45.5.k.a
• Level: $45$
• Weight: $5$
• Character orbit: 45.k
• Analytic conductor: $4.652$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	45.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65164833877\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 2 q^{2} + 6 q^{3} - 2 q^{5} + 120 q^{6} - 2 q^{7} - 72 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−7.37360	+	1.97575i	−7.93026	+	4.25569i	36.6100	−	21.1368i	−3.41993	−	24.7650i	50.0664	−	47.0480i	1.13081	−	0.303000i	−141.821	+	141.821i	44.7781	−	67.4975i	74.1466	+	175.850i
7.2	−7.18652	+	1.92562i	5.92105	−	6.77799i	34.0817	−	19.6771i	−17.5700	+	17.7847i	−29.4999	+	60.1119i	7.39767	−	1.98220i	−122.864	+	122.864i	−10.8823	−	80.2657i	92.0204	−	161.643i
7.3	−5.96640	+	1.59869i	4.37368	+	7.86581i	19.1857	−	11.0769i	17.4821	+	17.8711i	−38.6701	−	39.9384i	57.8267	−	15.4946i	−26.8779	+	26.8779i	−42.7419	+	68.8050i	−132.876	−	78.6774i
7.4	−5.12943	+	1.37443i	−2.02987	−	8.76810i	10.5656	−	6.10008i	22.8406	−	10.1640i	22.4632	+	42.1855i	−44.8951	+	12.0296i	14.2684	−	14.2684i	−72.7592	+	35.5962i	−103.190	+	83.5282i
7.5	−5.09412	+	1.36497i	8.84729	+	1.65091i	10.2306	−	5.90662i	5.54978	−	24.3762i	−47.3226	+	3.66629i	−32.3351	+	8.66418i	15.6131	−	15.6131i	75.5490	+	29.2122i	5.00143	+	131.751i
7.6	−4.24313	+	1.13694i	−7.57398	+	4.86156i	2.85511	−	1.64840i	3.64389	+	24.7330i	26.6101	−	29.2394i	−60.5059	+	16.2125i	39.4585	−	39.4585i	33.7305	−	73.6428i	−43.5815	−	100.803i
7.7	−4.24152	+	1.13651i	−6.48766	−	6.23781i	2.84239	−	1.64106i	−24.8558	+	2.68162i	34.6068	+	19.0845i	33.8543	−	9.07123i	39.4891	−	39.4891i	3.17942	+	80.9376i	102.378	−	39.6230i
7.8	−2.72485	+	0.730121i	−0.405345	+	8.99087i	−6.96468	+	4.02106i	−20.8337	−	13.8187i	−5.45992	−	24.7947i	37.5459	−	10.0604i	47.9575	−	47.9575i	−80.6714	−	7.28880i	66.8580	+	22.4427i
7.9	−1.67997	+	0.450147i	8.99424	+	0.321935i	−11.2367	+	6.48753i	−21.1623	+	13.3100i	−15.2550	+	3.50789i	−38.8993	+	10.4230i	35.6343	−	35.6343i	80.7927	+	5.79111i	29.5606	−	31.8866i
7.10	−1.49904	+	0.401665i	6.08169	−	6.63423i	−11.7706	+	6.79578i	19.7780	+	15.2915i	−6.45194	+	12.3878i	55.2269	−	14.7980i	32.4729	−	32.4729i	−7.02599	−	80.6947i	−35.7900	−	14.9784i
7.11	−0.777665	+	0.208375i	−8.98887	+	0.447392i	−13.2951	+	7.67591i	20.3088	−	14.5793i	6.89710	−	2.22097i	66.5008	−	17.8188i	17.8483	−	17.8483i	80.5997	−	8.04310i	−12.7555	+	15.5696i
7.12	0.540591	−	0.144851i	0.247757	+	8.99659i	−13.5851	+	7.84339i	23.0808	−	9.60597i	1.43710	+	4.82759i	−71.0919	+	19.0490i	−12.5397	+	12.5397i	−80.8772	+	4.45794i	11.0859	−	8.53618i
7.13	0.873962	−	0.234177i	2.00485	−	8.77386i	−13.1474	+	7.59068i	−13.2061	−	21.2273i	−0.302473	−	8.13751i	−26.6745	+	7.14742i	−19.9493	+	19.9493i	−72.9611	−	35.1806i	−16.5126	−	15.4593i
7.14	1.99166	−	0.533664i	−6.71025	−	5.99772i	−10.1745	+	5.87424i	3.24189	+	24.7889i	−16.5653	−	8.36441i	−22.3091	+	5.97771i	−40.4572	+	40.4572i	9.05478	+	80.4923i	19.6857	+	47.6411i
7.15	2.86666	−	0.768118i	4.92601	+	7.53222i	−6.22870	+	3.59614i	−6.53092	+	24.1319i	19.9068	+	17.8085i	27.3211	−	7.32067i	−48.6699	+	48.6699i	−32.4688	+	74.2077i	−0.185764	+	74.1943i
7.16	3.53742	−	0.947849i	−7.88777	+	4.33395i	−2.24148	+	1.29412i	−24.4029	−	5.43125i	−23.7944	+	22.8074i	−20.1905	+	5.41004i	−48.1356	+	48.1356i	43.4337	−	68.3704i	−91.4713	+	3.91765i
7.17	3.53838	−	0.948107i	8.84647	+	1.65531i	−2.23516	+	1.29047i	7.26089	−	23.9224i	32.8716	−	2.53029i	59.6290	−	15.9776i	−48.1298	+	48.1298i	75.5199	+	29.2872i	3.01086	−	91.5306i
7.18	5.46311	−	1.46384i	7.72308	−	4.62104i	13.8464	−	7.99420i	22.0807	+	11.7237i	35.4276	−	36.5506i	−85.6591	+	22.9523i	−0.0464312	+	0.0464312i	38.2920	−	71.3773i	137.791	+	31.7253i
7.19	6.10168	−	1.63494i	−6.85270	−	5.83443i	20.7010	−	11.9517i	10.5860	−	22.6481i	−51.3519	−	24.3961i	−11.1590	+	2.99004i	35.3028	−	35.3028i	12.9189	+	79.9631i	27.5642	−	155.499i
7.20	6.12130	−	1.64020i	2.33917	−	8.69070i	20.9237	−	12.0803i	−20.7596	+	13.9297i	0.0642868	−	57.0351i	76.3600	−	20.4606i	36.5683	−	36.5683i	−70.0566	−	40.6580i	−104.228	+	119.318i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
9.c	even	3	1	inner
45.k	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	45.5.k.a	✓	88
3.b	odd	2	1		135.5.l.a		88
5.c	odd	4	1	inner	45.5.k.a	✓	88
9.c	even	3	1	inner	45.5.k.a	✓	88
9.d	odd	6	1		135.5.l.a		88
15.e	even	4	1		135.5.l.a		88
45.k	odd	12	1	inner	45.5.k.a	✓	88
45.l	even	12	1		135.5.l.a		88

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.5.k.a	✓	88	1.a	even	1	1	trivial
45.5.k.a	✓	88	5.c	odd	4	1	inner
45.5.k.a	✓	88	9.c	even	3	1	inner
45.5.k.a	✓	88	45.k	odd	12	1	inner
135.5.l.a		88	3.b	odd	2	1
135.5.l.a		88	9.d	odd	6	1
135.5.l.a		88	15.e	even	4	1
135.5.l.a		88	45.l	even	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(45, [\chi])\).