Newform orbit 59.3.d.a

• Label: 59.3.d.a
• Level: $59$
• Weight: $3$
• Character orbit: 59.d
• Analytic conductor: $1.608$
• Analytic rank: $0$
• Dimension: $252$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	59.d (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60763355973\)
Analytic rank:	\(0\)
Dimension:	\(252\)
Relative dimension:	\(9\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 252 q - 29 q^{2} - 33 q^{3} - 11 q^{4} - 21 q^{5} - 29 q^{6} - 35 q^{7} - 29 q^{8} - 70 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−3.26019	−	0.176762i	0.182722	−	0.0845363i	6.62103	+	0.720080i	−4.95561	−	1.66974i	−0.610652	+	0.243306i	6.37157	+	9.39737i	−8.57065	−	1.40509i	−5.80024	+	6.82857i	15.8611	+	6.31963i
2.2	−2.97279	−	0.161180i	−5.21949	+	2.41479i	4.83492	+	0.525830i	2.42635	+	0.817534i	15.9056	−	6.33739i	−5.07474	−	7.48468i	−2.53671	−	0.415872i	15.5854	−	18.3485i	−7.08126	−	2.82143i
2.3	−2.38005	−	0.129043i	4.69906	−	2.17401i	1.67146	+	0.181782i	0.840610	+	0.283234i	−11.4645	+	4.56789i	−3.07199	−	4.53085i	5.45390	+	0.894122i	11.5283	−	13.5722i	−1.96415	−	0.782588i
2.4	−1.22566	−	0.0664535i	−0.853640	+	0.394936i	−2.47872	−	0.269577i	8.39583	+	2.82888i	1.07252	−	0.427331i	4.77469	+	7.04214i	7.86533	+	1.28946i	−5.25375	+	6.18519i	−10.1025	−	4.02519i
2.5	−0.895223	−	0.0485376i	−0.751899	+	0.347866i	−3.17748	−	0.345572i	−4.47160	−	1.50666i	0.690002	−	0.274922i	−4.14756	−	6.11719i	6.36669	+	1.04377i	−5.38213	+	6.33634i	3.92995	+	1.56583i
2.6	1.15718	+	0.0627406i	−4.94098	+	2.28594i	−2.64142	−	0.287271i	−4.34717	−	1.46473i	−5.86104	+	2.33525i	7.20861	+	10.6319i	−7.61304	−	1.24810i	13.3612	−	15.7301i	−4.93857	−	1.96771i
2.7	1.40660	+	0.0762635i	3.06644	−	1.41868i	−2.00385	−	0.217932i	1.43378	+	0.483096i	4.42144	−	1.76166i	0.650920	+	0.960035i	−8.36242	−	1.37095i	1.56389	−	1.84115i	1.97991	+	0.788866i
2.8	2.94400	+	0.159619i	−2.92458	+	1.35306i	4.66512	+	0.507362i	7.52634	+	2.53592i	−8.82595	+	3.51658i	−4.23103	−	6.24030i	2.01519	+	0.330373i	0.895951	−	1.05479i	21.7528	+	8.66710i
2.9	3.23199	+	0.175234i	0.168176	−	0.0778064i	6.43853	+	0.700232i	−5.98129	−	2.01533i	0.557177	−	0.222000i	0.736526	+	1.08629i	7.91017	+	1.29681i	−5.80425	+	6.83329i	−18.9783	−	7.56166i
6.1	−3.17515	+	1.26510i	5.14893	−	0.559979i	5.57713	−	5.28293i	−5.06429	−	5.96215i	−15.6402	+	8.29190i	5.62111	−	3.38211i	−5.28424	+	11.4217i	17.4083	−	3.83185i	23.6226	+	12.5239i
6.2	−2.78573	+	1.10994i	0.254288	−	0.0276555i	3.62435	−	3.43317i	3.08081	+	3.62701i	−0.677682	+	0.359284i	−11.2925	+	6.79450i	−1.24937	+	2.70047i	−8.72569	+	1.92067i	−12.6081	−	6.68437i
6.3	−2.63581	+	1.05020i	−3.42634	+	0.372637i	2.94057	−	2.78546i	−0.503957	−	0.593304i	8.63983	−	4.58055i	7.94809	−	4.78221i	−0.0600401	+	0.129774i	2.81137	−	0.618830i	1.95142	+	1.03458i
6.4	−0.617663	+	0.246100i	3.33931	−	0.363171i	−2.58304	+	2.44678i	2.87854	+	3.38889i	−1.97319	+	1.04612i	3.65501	−	2.19914i	2.11001	−	4.56071i	2.22949	−	0.490748i	−2.61197	−	1.38478i
6.5	−0.176520	+	0.0703320i	−2.50768	+	0.272727i	−2.87777	+	2.72597i	−4.76105	−	5.60514i	0.423475	−	0.224512i	−2.41294	+	1.45181i	0.635403	−	1.37340i	−2.57549	+	0.566909i	1.23464	+	0.654566i
6.6	0.574637	−	0.228956i	−4.07106	+	0.442754i	−2.62620	+	2.48766i	5.06596	+	5.96411i	−2.23801	+	1.18652i	−2.25289	+	1.35552i	−1.97847	+	4.27638i	7.58790	−	1.67022i	4.27661	+	2.26731i
6.7	1.65320	−	0.658695i	4.42657	−	0.481419i	−0.604793	+	0.572891i	−3.14095	−	3.69781i	7.00090	−	3.71164i	−9.99166	+	6.01178i	−3.61141	+	7.80593i	10.5732	−	2.32734i	−7.62834	−	4.04429i
6.8	2.18147	−	0.869176i	0.713709	−	0.0776206i	1.09935	−	1.04136i	−0.395093	−	0.465139i	1.48947	−	0.789666i	8.26248	−	4.97137i	−2.45094	+	5.29762i	−8.28623	+	1.82394i	−1.26617	−	0.671281i
6.9	3.25557	−	1.29714i	−1.85159	+	0.201372i	6.01219	−	5.69504i	0.962642	+	1.13331i	−5.76677	+	3.05735i	−6.46217	+	3.88816i	6.29988	−	13.6170i	−5.40175	+	1.18902i	4.60401	+	2.44089i
8.1	−3.76989	−	0.618042i	−0.943305	+	3.39748i	10.0395	+	3.38269i	−2.18479	−	3.22232i	5.65594	−	12.2251i	−6.40815	+	1.41054i	−22.2562	−	11.7995i	−2.94131	−	1.76973i	6.24487	+	13.4981i
8.2	−3.01843	−	0.494846i	1.48811	−	5.35968i	5.07541	+	1.71010i	−0.678586	−	1.00084i	−7.14396	+	15.4414i	−2.13745	+	0.470488i	−3.66384	−	1.94245i	−18.8000	−	11.3116i	1.55300	+	3.35676i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
59.d	odd	58	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	59.3.d.a	✓	252
59.d	odd	58	1	inner	59.3.d.a	✓	252

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
59.3.d.a	✓	252	1.a	even	1	1	trivial
59.3.d.a	✓	252	59.d	odd	58	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(59, [\chi])\).