Newform orbit 185.2.p.a

• Label: 185.2.p.a
• Level: $185$
• Weight: $2$
• Character orbit: 185.p
• 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.p (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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{2} - 8 q^{3} - 30 q^{4} - 10 q^{5} - 8 q^{6} - 2 q^{7} + 12 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
82.1	−1.37559	−	2.38259i	0.692973	−	0.185681i	−2.78450	+	4.82289i	−0.638688	+	2.14291i	−1.39565	−	1.39565i	−1.87540	+	0.502511i	9.81895			−2.15234	+	1.24266i	5.98426	−	1.42604i
82.2	−1.21425	−	2.10315i	0.672498	−	0.180195i	−1.94883	+	3.37547i	1.55471	−	1.60713i	−1.19556	−	1.19556i	4.47423	−	1.19887i	4.60849			−2.17829	+	1.25764i	−5.26786	−	1.31832i
82.3	−1.10225	−	1.90916i	−3.05050	+	0.817378i	−1.42993	+	2.47671i	1.99367	+	1.01257i	4.92293	+	4.92293i	−0.147229	+	0.0394498i	1.89556			6.03934	−	3.48681i	−0.264372	−	4.92234i
82.4	−0.993536	−	1.72086i	2.35179	−	0.630160i	−0.974229	+	1.68741i	−1.49861	−	1.65957i	−3.42100	−	3.42100i	−2.62941	+	0.704548i	−0.102418			2.53574	−	1.46401i	−1.36695	+	4.22774i
82.5	−0.892058	−	1.54509i	−1.76128	+	0.471934i	−0.591534	+	1.02457i	−1.38296	−	1.75711i	2.30034	+	2.30034i	0.500724	−	0.134169i	−1.45750			0.281310	−	0.162414i	−1.48121	+	3.70424i
82.6	−0.723320	−	1.25283i	−0.791482	+	0.212077i	−0.0463849	+	0.0803411i	−1.65059	+	1.50849i	0.838191	+	0.838191i	−1.58905	+	0.425785i	−2.75908			−2.01661	+	1.16429i	3.08379	+	0.976784i
82.7	−0.412077	−	0.713738i	1.77336	−	0.475172i	0.660386	−	1.14382i	1.99238	−	1.01510i	−1.06991	−	1.06991i	−1.19739	+	0.320840i	−2.73682			0.320958	−	0.185305i	−1.54553	−	1.00374i
82.8	−0.354544	−	0.614089i	−0.280512	+	0.0751630i	0.748597	−	1.29661i	1.19547	+	1.88967i	0.145611	+	0.145611i	3.64653	−	0.977086i	−2.47982			−2.52504	+	1.45783i	0.736577	−	1.40410i
82.9	−0.122725	−	0.212566i	−1.83492	+	0.491666i	0.969877	−	1.67988i	1.28977	−	1.82661i	0.329702	+	0.329702i	−3.76446	+	1.00868i	−0.967013			0.527123	−	0.304334i	−0.546561	−	0.0499898i
82.10	−0.0797909	−	0.138202i	2.26850	−	0.607844i	0.987267	−	1.71000i	−2.00255	+	0.994886i	−0.265011	−	0.265011i	1.28763	−	0.345018i	−0.634263			2.17856	−	1.25779i	0.297280	+	0.197373i
82.11	0.235037	+	0.407096i	−3.12752	+	0.838017i	0.889515	−	1.54069i	−2.22812	+	0.188382i	−1.07624	−	1.07624i	3.61347	−	0.968227i	1.77643			6.48105	−	3.74183i	−0.600381	−	0.862782i
82.12	0.506716	+	0.877658i	1.71192	−	0.458707i	0.486478	−	0.842604i	0.902808	+	2.04571i	1.27004	+	1.27004i	−4.18129	+	1.12037i	3.01289			0.122169	−	0.0705344i	−1.33797	+	1.82895i
82.13	0.560713	+	0.971183i	0.628874	−	0.168506i	0.371203	−	0.642942i	−0.606686	−	2.15219i	0.516268	+	0.516268i	1.40791	−	0.377247i	3.07540			−2.23099	+	1.28806i	1.75000	−	1.79596i
82.14	0.643846	+	1.11517i	−1.31376	+	0.352022i	0.170924	−	0.296050i	2.23601	−	0.0164393i	−1.23843	−	1.23843i	1.71395	−	0.459253i	3.01558			−0.996020	+	0.575052i	1.45798	+	2.48295i
82.15	1.04034	+	1.80193i	−1.09812	+	0.294240i	−1.16463	+	2.01721i	−1.05057	+	1.97391i	−1.67262	−	1.67262i	0.0769563	−	0.0206204i	−0.685107			−1.47879	+	0.853779i	−4.64979	+	0.160500i
82.16	1.20502	+	2.08716i	−2.76431	+	0.740695i	−1.90415	+	3.29809i	−0.404091	−	2.19925i	−4.87700	−	4.87700i	−3.19584	+	0.856324i	−4.35808			4.49471	−	2.59502i	4.10325	−	3.49355i
82.17	1.21245	+	2.10002i	2.19044	−	0.586925i	−1.94006	+	3.36028i	−2.20194	−	0.389154i	3.88834	+	3.88834i	0.492639	−	0.132002i	−4.55908			1.85545	−	1.07124i	−1.85251	−	5.09596i
88.1	−1.37559	+	2.38259i	0.692973	+	0.185681i	−2.78450	−	4.82289i	−0.638688	−	2.14291i	−1.39565	+	1.39565i	−1.87540	−	0.502511i	9.81895			−2.15234	−	1.24266i	5.98426	+	1.42604i
88.2	−1.21425	+	2.10315i	0.672498	+	0.180195i	−1.94883	−	3.37547i	1.55471	+	1.60713i	−1.19556	+	1.19556i	4.47423	+	1.19887i	4.60849			−2.17829	−	1.25764i	−5.26786	+	1.31832i
88.3	−1.10225	+	1.90916i	−3.05050	−	0.817378i	−1.42993	−	2.47671i	1.99367	−	1.01257i	4.92293	−	4.92293i	−0.147229	−	0.0394498i	1.89556			6.03934	+	3.48681i	−0.264372	+	4.92234i

Inner twists

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

    

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

Hecke kernels

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