Newform orbit 185.3.y.a

• Label: 185.3.y.a
• Level: $185$
• Weight: $3$
• Character orbit: 185.y
• Analytic conductor: $5.041$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 185 = 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	185.y (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.04088489067\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(36\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 432 q - 12 q^{2} - 6 q^{3} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−0.330747	−	3.78046i	5.05028	+	0.441842i	−10.2432	+	1.80616i	3.01051	+	3.99209i		−	19.2385i	9.30029	−	4.33680i	6.28724	+	23.4643i	16.4468	+	2.90002i	14.0962	−	12.7015i
3.2	−0.314561	−	3.59544i	−0.0969242	−	0.00847977i	−8.88904	+	1.56738i	4.32234	−	2.51344i			0.351153i	−7.65103	+	3.56774i	4.69507	+	17.5222i	−8.85395	−	1.56119i	−10.3966	−	14.7501i
3.3	−0.301307	−	3.44395i	−3.43140	−	0.300208i	−7.83078	+	1.38078i	1.07334	+	4.88344i			11.9080i	1.55715	−	0.726112i	3.53574	+	13.1956i	2.82110	+	0.497436i	16.4949	−	5.16795i
3.4	−0.297069	−	3.39551i	−5.11243	−	0.447279i	−7.50203	+	1.32281i	−2.85995	−	4.10130i			17.4922i	−5.05433	+	2.35687i	3.19152	+	11.9109i	17.0736	+	3.01053i	−13.0764	+	10.9294i
3.5	−0.286839	−	3.27858i	1.76324	+	0.154264i	−6.72759	+	1.18626i	−4.55692	+	2.05778i		−	5.82518i	−5.33373	+	2.48716i	2.41177	+	9.00085i	−5.77805	−	1.01883i	8.05369	+	14.3500i
3.6	−0.260724	−	2.98008i	−0.797209	−	0.0697468i	−4.87369	+	0.859364i	2.66158	−	4.23273i			2.39393i	9.06239	−	4.22586i	0.734667	+	2.74181i	−8.23259	−	1.45163i	−13.3078	−	6.82816i
3.7	−0.237257	−	2.71186i	4.42360	+	0.387015i	−3.35864	+	0.592220i	−2.50229	−	4.32880i		−	12.0880i	2.22518	−	1.03762i	−0.415369	−	1.55018i	10.5552	+	1.86116i	−11.1454	+	7.81290i
3.8	−0.210524	−	2.40630i	2.96575	+	0.259470i	−1.80672	+	0.318574i	4.68087	+	1.75768i		−	7.19111i	−4.83093	+	2.25270i	−1.35376	−	5.05229i	−0.134900	−	0.0237865i	3.24406	−	11.6336i
3.9	−0.182569	−	2.08678i	−0.181992	−	0.0159222i	−0.382081	+	0.0673712i	−3.59331	+	3.47680i			0.382684i	6.38866	−	2.97908i	−1.95830	−	7.30847i	−8.83040	−	1.55704i	7.91134	+	6.86368i
3.10	−0.171047	−	1.95508i	−2.97531	−	0.260306i	0.146153	−	0.0257707i	−4.52830	−	2.12002i			5.86150i	1.85117	−	0.863215i	−2.10716	−	7.86403i	−0.0785363	−	0.0138481i	−3.37025	+	9.21582i
3.11	−0.159191	−	1.81956i	−5.03203	−	0.440245i	0.653775	−	0.115278i	4.98909	+	0.330056i			9.22616i	4.31769	−	2.01337i	−2.20477	−	8.22833i	16.2642	+	2.86782i	−0.193662	−	9.13050i
3.12	−0.132360	−	1.51288i	−2.06558	−	0.180715i	1.66795	−	0.294105i	2.04769	+	4.56147i			3.14889i	−10.7637	+	5.01918i	−2.23794	−	8.35212i	−4.62931	−	0.816272i	6.62990	−	3.70165i
3.13	−0.100633	−	1.15024i	3.68083	+	0.322031i	2.62630	−	0.463088i	4.52579	−	2.12537i		−	4.26625i	−0.609648	+	0.284283i	−1.99232	−	7.43545i	4.58155	+	0.807850i	−2.90013	−	4.99188i
3.14	−0.0999938	−	1.14293i	1.91932	+	0.167918i	2.64293	−	0.466020i	0.250376	+	4.99373i		−	2.21044i	9.23598	−	4.30681i	−1.98468	−	7.40693i	−5.20770	−	0.918257i	5.68247	−	0.785505i
3.15	−0.0977237	−	1.11699i	−0.185331	−	0.0162143i	2.70112	−	0.476280i	−1.34362	−	4.81609i			0.208597i	−9.21600	+	4.29749i	−1.95677	−	7.30277i	−8.82919	−	1.55682i	−5.24821	+	1.97145i
3.16	−0.0774738	−	0.885529i	5.82944	+	0.510010i	3.16107	−	0.557382i	−3.10522	+	3.91888i		−	5.20165i	−5.67934	+	2.64832i	−1.65875	−	6.19053i	24.8590	+	4.38331i	3.71086	+	2.44615i
3.17	−0.0239655	−	0.273927i	−4.94613	−	0.432730i	3.86477	−	0.681463i	−3.90806	+	3.11883i			1.36525i	−0.143045	+	0.0667029i	−0.563966	−	2.10475i	15.4137	+	2.71785i	0.947989	+	0.995779i
3.18	−0.00500158	−	0.0571683i	−1.36267	−	0.119218i	3.93599	−	0.694021i	1.67366	−	4.71157i			0.0784979i	6.05969	−	2.82568i	−0.118773	−	0.443268i	−7.02061	−	1.23792i	−0.277723	−	0.0721148i
3.19	0.0283960	+	0.324568i	2.91108	+	0.254687i	3.83469	−	0.676160i	−4.54105	−	2.09257i			0.952076i	7.66339	−	3.57350i	0.665650	+	2.48424i	−0.453732	−	0.0800053i	0.550234	−	1.53330i
3.20	0.0371948	+	0.425139i	−1.60354	−	0.140292i	3.75987	−	0.662967i	4.96325	+	0.605125i		−	0.686948i	1.06568	−	0.496937i	0.863519	+	3.22270i	−6.31160	−	1.11290i	−0.0726552	+	2.13258i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
37.h	even	18	1	inner
185.y	odd	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	185.3.y.a	✓	432
5.c	odd	4	1	inner	185.3.y.a	✓	432
37.h	even	18	1	inner	185.3.y.a	✓	432
185.y	odd	36	1	inner	185.3.y.a	✓	432

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
185.3.y.a	✓	432	1.a	even	1	1	trivial
185.3.y.a	✓	432	5.c	odd	4	1	inner
185.3.y.a	✓	432	37.h	even	18	1	inner
185.3.y.a	✓	432	185.y	odd	36	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(185, [\chi])\).