Newform orbit 185.2.n.a

• Label: 185.2.n.a
• Level: $185$
• Weight: $2$
• Character orbit: 185.n
• Analytic conductor: $1.477$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 185 = 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	185.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.47723243739\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 16 q^{4} - 2 q^{5} - 16 q^{6} + 14 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
84.1	−2.33663	−	1.34906i	2.35586	−	1.36016i	2.63990	+	4.57244i	0.261363	−	2.22074i	−7.33972			−1.64348	+	0.948866i		−	8.84926i	2.20006	−	3.81062i	−3.60661	+	4.83646i
84.2	−1.99875	−	1.15398i	−1.81123	+	1.04571i	1.66334	+	2.88098i	−0.0902553	−	2.23425i	4.82692			3.28655	−	1.89749i		−	3.06190i	0.687027	−	1.18997i	−2.39787	+	4.56985i
84.3	−1.85494	−	1.07095i	0.858786	−	0.495820i	1.29387	+	2.24104i	−1.59352	+	1.56866i	−2.12399			−1.20076	+	0.693262i		−	1.25887i	−1.00832	+	1.74647i	4.63584	−	1.20318i
84.4	−1.64422	−	0.949293i	1.16436	−	0.672243i	0.802314	+	1.38965i	1.62372	+	1.53738i	−2.55262			2.34261	−	1.35251i			0.750648i	−0.596178	+	1.03261i	−1.21033	−	4.06918i
84.5	−1.58456	−	0.914845i	−1.14999	+	0.663947i	0.673882	+	1.16720i	2.12799	−	0.686767i	2.42963			−2.49908	+	1.44285i			1.19339i	−0.618348	+	1.07101i	−4.00021	−	0.858562i
84.6	−0.843193	−	0.486818i	−1.64972	+	0.952467i	−0.526017	−	0.911089i	−1.79344	+	1.33551i	1.85471			3.89368	−	2.24802i			2.97157i	0.314386	−	0.544533i	2.16236	−	0.253015i
84.7	−0.704704	−	0.406861i	−0.142682	+	0.0823772i	−0.668928	−	1.15862i	−1.50396	−	1.65472i	0.134064			−2.76887	+	1.59861i			2.71609i	−1.48643	+	2.57457i	0.386605	+	1.77799i
84.8	−0.398514	−	0.230082i	2.88437	−	1.66529i	−0.894124	−	1.54867i	−2.18772	+	0.462462i	−1.53262			−0.0944407	+	0.0545254i			1.74322i	4.04641	−	7.00858i	0.978243	+	0.319058i
84.9	−0.153805	−	0.0887996i	−1.48057	+	0.854808i	−0.984229	−	1.70474i	0.854987	+	2.06616i	0.303627			−1.21910	+	0.703845i			0.704795i	−0.0386059	+	0.0668674i	0.0519722	−	0.393709i
84.10	0.153805	+	0.0887996i	1.48057	−	0.854808i	−0.984229	−	1.70474i	1.36185	+	1.77352i	0.303627			1.21910	−	0.703845i		−	0.704795i	−0.0386059	+	0.0668674i	0.0519722	+	0.393709i
84.11	0.398514	+	0.230082i	−2.88437	+	1.66529i	−0.894124	−	1.54867i	1.49437	−	1.66339i	−1.53262			0.0944407	−	0.0545254i		−	1.74322i	4.04641	−	7.00858i	0.978243	−	0.319058i
84.12	0.704704	+	0.406861i	0.142682	−	0.0823772i	−0.668928	−	1.15862i	−0.681051	−	2.12983i	0.134064			2.76887	−	1.59861i		−	2.71609i	−1.48643	+	2.57457i	0.386605	−	1.77799i
84.13	0.843193	+	0.486818i	1.64972	−	0.952467i	−0.526017	−	0.911089i	2.05330	−	0.885408i	1.85471			−3.89368	+	2.24802i		−	2.97157i	0.314386	−	0.544533i	2.16236	+	0.253015i
84.14	1.58456	+	0.914845i	1.14999	−	0.663947i	0.673882	+	1.16720i	−1.65875	+	1.49951i	2.42963			2.49908	−	1.44285i		−	1.19339i	−0.618348	+	1.07101i	−4.00021	+	0.858562i
84.15	1.64422	+	0.949293i	−1.16436	+	0.672243i	0.802314	+	1.38965i	0.519553	+	2.17487i	−2.55262			−2.34261	+	1.35251i		−	0.750648i	−0.596178	+	1.03261i	−1.21033	+	4.06918i
84.16	1.85494	+	1.07095i	−0.858786	+	0.495820i	1.29387	+	2.24104i	2.15526	−	0.595704i	−2.12399			1.20076	−	0.693262i			1.25887i	−1.00832	+	1.74647i	4.63584	+	1.20318i
84.17	1.99875	+	1.15398i	1.81123	−	1.04571i	1.66334	+	2.88098i	−1.88979	−	1.19529i	4.82692			−3.28655	+	1.89749i			3.06190i	0.687027	−	1.18997i	−2.39787	−	4.56985i
84.18	2.33663	+	1.34906i	−2.35586	+	1.36016i	2.63990	+	4.57244i	−2.05390	−	0.884024i	−7.33972			1.64348	−	0.948866i			8.84926i	2.20006	−	3.81062i	−3.60661	−	4.83646i
174.1	−2.33663	+	1.34906i	2.35586	+	1.36016i	2.63990	−	4.57244i	0.261363	+	2.22074i	−7.33972			−1.64348	−	0.948866i			8.84926i	2.20006	+	3.81062i	−3.60661	−	4.83646i
174.2	−1.99875	+	1.15398i	−1.81123	−	1.04571i	1.66334	−	2.88098i	−0.0902553	+	2.23425i	4.82692			3.28655	+	1.89749i			3.06190i	0.687027	+	1.18997i	−2.39787	−	4.56985i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
37.c	even	3	1	inner
185.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	185.2.n.a	✓	36
5.b	even	2	1	inner	185.2.n.a	✓	36
5.c	odd	4	2		925.2.e.f		36
37.c	even	3	1	inner	185.2.n.a	✓	36
185.n	even	6	1	inner	185.2.n.a	✓	36
185.s	odd	12	2		925.2.e.f		36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
185.2.n.a	✓	36	1.a	even	1	1	trivial
185.2.n.a	✓	36	5.b	even	2	1	inner
185.2.n.a	✓	36	37.c	even	3	1	inner
185.2.n.a	✓	36	185.n	even	6	1	inner
925.2.e.f		36	5.c	odd	4	2
925.2.e.f		36	185.s	odd	12	2

Hecke kernels

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