Newform orbit 185.2.u.a

• Label: 185.2.u.a
• Level: $185$
• Weight: $2$
• Character orbit: 185.u
• Analytic conductor: $1.477$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 185 = 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	185.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.47723243739\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 6 q^{2} - 4 q^{3} + 30 q^{4} - 8 q^{6} - 2 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
8.1	−2.38259	+	1.37559i	0.185681	+	0.692973i	2.78450	−	4.82289i	−1.62458	−	1.53647i	−1.39565	−	1.39565i	0.502511	+	1.87540i			9.81895i	2.15234	−	1.24266i	5.98426	+	1.42604i
8.2	−2.10315	+	1.21425i	0.180195	+	0.672498i	1.94883	−	3.37547i	2.14998	+	0.614465i	−1.19556	−	1.19556i	−1.19887	−	4.47423i			4.60849i	2.17829	−	1.25764i	−5.26786	+	1.31832i
8.3	−1.90916	+	1.10225i	−0.817378	−	3.05050i	1.42993	−	2.47671i	1.22028	−	1.87374i	4.92293	+	4.92293i	0.0394498	+	0.147229i			1.89556i	−6.03934	+	3.48681i	−0.264372	+	4.92234i
8.4	−1.72086	+	0.993536i	0.630160	+	2.35179i	0.974229	−	1.68741i	−0.468054	+	2.18653i	−3.42100	−	3.42100i	0.704548	+	2.62941i		−	0.102418i	−2.53574	+	1.46401i	−1.36695	−	4.22774i
8.5	−1.54509	+	0.892058i	−0.471934	−	1.76128i	0.591534	−	1.02457i	−0.319125	+	2.21318i	2.30034	+	2.30034i	−0.134169	−	0.500724i		−	1.45750i	−0.281310	+	0.162414i	−1.48121	−	3.70424i
8.6	−1.25283	+	0.723320i	−0.212077	−	0.791482i	0.0463849	−	0.0803411i	−2.18370	−	0.481096i	0.838191	+	0.838191i	0.425785	+	1.58905i		−	2.75908i	2.01661	−	1.16429i	3.08379	−	0.976784i
8.7	−0.713738	+	0.412077i	0.475172	+	1.77336i	−0.660386	+	1.14382i	2.23300	−	0.117085i	−1.06991	−	1.06991i	0.320840	+	1.19739i		−	2.73682i	−0.320958	+	0.185305i	−1.54553	+	1.00374i
8.8	−0.614089	+	0.354544i	−0.0751630	−	0.280512i	−0.748597	+	1.29661i	0.0904746	−	2.23424i	0.145611	+	0.145611i	−0.977086	−	3.64653i		−	2.47982i	2.52504	−	1.45783i	0.736577	+	1.40410i
8.9	−0.212566	+	0.122725i	−0.491666	−	1.83492i	−0.969877	+	1.67988i	2.03028	+	0.937007i	0.329702	+	0.329702i	1.00868	+	3.76446i		−	0.967013i	−0.527123	+	0.304334i	−0.546561	+	0.0499898i
8.10	−0.138202	+	0.0797909i	0.607844	+	2.26850i	−0.987267	+	1.71000i	−2.23170	+	0.139678i	−0.265011	−	0.265011i	−0.345018	−	1.28763i		−	0.634263i	−2.17856	+	1.25779i	0.297280	−	0.197373i
8.11	0.407096	−	0.235037i	−0.838017	−	3.12752i	−0.889515	+	1.54069i	−2.02380	+	0.950915i	−1.07624	−	1.07624i	−0.968227	−	3.61347i			1.77643i	−6.48105	+	3.74183i	−0.600381	+	0.862782i
8.12	0.877658	−	0.506716i	0.458707	+	1.71192i	−0.486478	+	0.842604i	−0.241001	−	2.22304i	1.27004	+	1.27004i	1.12037	+	4.18129i			3.01289i	−0.122169	+	0.0705344i	−1.33797	−	1.82895i
8.13	0.971183	−	0.560713i	0.168506	+	0.628874i	−0.371203	+	0.642942i	0.550691	+	2.16720i	0.516268	+	0.516268i	−0.377247	−	1.40791i			3.07540i	2.23099	−	1.28806i	1.75000	+	1.79596i
8.14	1.11517	−	0.643846i	−0.352022	−	1.31376i	−0.170924	+	0.296050i	1.94466	−	1.10377i	−1.23843	−	1.23843i	−0.459253	−	1.71395i			3.01558i	0.996020	−	0.575052i	1.45798	−	2.48295i
8.15	1.80193	−	1.04034i	−0.294240	−	1.09812i	1.16463	−	2.01721i	−1.89677	−	1.18417i	−1.67262	−	1.67262i	−0.0206204	−	0.0769563i		−	0.685107i	1.47879	−	0.853779i	−4.64979	−	0.160500i
8.16	2.08716	−	1.20502i	−0.740695	−	2.76431i	1.90415	−	3.29809i	0.749673	+	2.10665i	−4.87700	−	4.87700i	0.856324	+	3.19584i		−	4.35808i	−4.49471	+	2.59502i	4.10325	+	3.49355i
8.17	2.10002	−	1.21245i	0.586925	+	2.19044i	1.94006	−	3.36028i	−1.71236	+	1.43799i	3.88834	+	3.88834i	−0.132002	−	0.492639i		−	4.55908i	−1.85545	+	1.07124i	−1.85251	+	5.09596i
23.1	−2.34255	−	1.35247i	−0.438550	−	0.117509i	2.65837	+	4.60443i	1.31482	+	1.80866i	0.868398	+	0.868398i	−3.48545	−	0.933922i		−	8.97159i	−2.41956	−	1.39693i	−0.633857	−	6.01514i
23.2	−1.89659	−	1.09500i	2.68684	+	0.719938i	1.39805	+	2.42149i	−1.55121	+	1.61051i	−4.30752	−	4.30752i	2.04761	+	0.548654i		−	1.74345i	4.10274	+	2.36872i	4.70553	−	1.35592i
23.3	−1.85990	−	1.07382i	−2.04873	−	0.548954i	1.30616	+	2.26234i	−2.22787	−	0.191263i	3.22096	+	3.22096i	−0.905285	−	0.242570i		−	1.31504i	1.29785	+	0.749313i	3.93825	+	2.74806i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
185.u	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	185.2.u.a	yes	68
5.b	even	2	1		925.2.y.b		68
5.c	odd	4	1		185.2.p.a	✓	68
5.c	odd	4	1		925.2.t.b		68
37.g	odd	12	1		185.2.p.a	✓	68
185.p	even	12	1		925.2.y.b		68
185.q	odd	12	1		925.2.t.b		68
185.u	even	12	1	inner	185.2.u.a	yes	68

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
185.2.p.a	✓	68	5.c	odd	4	1
185.2.p.a	✓	68	37.g	odd	12	1
185.2.u.a	yes	68	1.a	even	1	1	trivial
185.2.u.a	yes	68	185.u	even	12	1	inner
925.2.t.b		68	5.c	odd	4	1
925.2.t.b		68	185.q	odd	12	1
925.2.y.b		68	5.b	even	2	1
925.2.y.b		68	185.p	even	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(185, [\chi])\).