Newform orbit 37.3.i.a

• Label: 37.3.i.a
• Level: $37$
• Weight: $3$
• Character orbit: 37.i
• Analytic conductor: $1.008$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.00817697813\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 12 q^{2} - 12 q^{3} + 6 q^{4} + 6 q^{5} - 12 q^{6} - 12 q^{7} - 78 q^{8} + 18 q^{9}+O(q^{10}) \)

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	−3.66472	−	0.320622i	−0.999087	+	1.19067i	9.38815	+	1.65538i	2.92231	−	6.26692i	4.04313	−	4.04313i	6.77942	−	2.46751i	−19.6607	−	5.26807i	1.14332	+	6.48411i	−12.7188	+	22.0295i
2.2	−1.27989	−	0.111976i	−2.29345	+	2.73323i	−2.31364	−	0.407958i	−1.74871	+	3.75013i	3.24143	−	3.24143i	−2.28808	+	0.832793i	7.87955	+	2.11132i	−0.647787	−	3.67378i	2.65809	−	4.60395i
2.3	−1.25921	−	0.110166i	2.50948	−	2.99069i	−2.36576	−	0.417148i	1.10059	−	2.36023i	−3.48944	+	3.48944i	4.82832	−	1.75736i	7.81682	+	2.09451i	−1.08386	−	6.14689i	−1.64590	+	2.85078i
2.4	1.87894	+	0.164386i	0.942406	−	1.12312i	−0.435832	−	0.0768489i	−0.809415	+	1.73580i	1.95535	−	1.95535i	−2.08567	+	0.759122i	−8.09367	−	2.16869i	1.18957	+	6.74641i	−1.80618	+	3.12840i
2.5	2.73696	+	0.239453i	−3.30709	+	3.94124i	3.49438	+	0.616153i	3.30071	−	7.07840i	−9.99511	+	9.99511i	2.02744	−	0.737927i	−1.19877	−	0.321209i	−3.03367	−	17.2048i	10.7289	−	18.5829i
5.1	−1.25404	+	2.68929i	−0.554625	+	1.52382i	−3.08852	−	3.68076i	−1.33394	+	1.90507i	−3.40247	−	3.40247i	−0.324876	−	1.84246i	2.30696	−	0.618149i	4.87999	+	4.09479i	−3.45047	−	5.97638i
5.2	−0.521767	+	1.11893i	1.27672	−	3.50777i	1.59138	+	1.89653i	1.42566	−	2.03605i	3.25881	+	3.25881i	0.576187	+	3.26772i	−7.72257	+	2.06926i	−3.78004	−	3.17183i	1.53434	+	2.65756i
5.3	0.177583	−	0.380827i	−1.22784	+	3.37346i	2.45766	+	2.92892i	1.10020	−	1.57124i	1.06666	+	1.06666i	−1.07559	−	6.09995i	3.17536	−	0.850836i	−2.97822	−	2.49903i	−0.402996	−	0.698010i
5.4	0.982348	−	2.10665i	0.787979	−	2.16495i	−0.901828	−	1.07476i	−4.57941	+	6.54008i	−3.78674	−	3.78674i	−0.559720	−	3.17433i	5.83088	−	1.56238i	2.82828	+	2.37321i	9.27910	+	16.0719i
5.5	1.54871	−	3.32122i	−0.989811	+	2.71948i	−6.06084	−	7.22303i	2.51995	−	3.59886i	7.49907	+	7.49907i	1.86029	+	10.5502i	−19.2170	+	5.14918i	0.478540	+	0.401543i	−8.04993	−	13.9429i
13.1	−1.46611	−	2.09382i	3.56464	−	0.628542i	−0.866532	+	2.38078i	−2.56671	−	0.224558i	−6.54221	−	6.54221i	2.83735	−	2.38082i	−3.62060	+	0.970137i	3.85436	−	1.40287i	3.29289	+	5.70346i
13.2	−0.818690	−	1.16921i	−4.94995	+	0.872809i	0.671281	−	1.84433i	−6.45824	−	0.565023i	5.07297	+	5.07297i	3.38170	−	2.83759i	−8.22081	+	2.20276i	15.2829	−	5.56253i	4.62666	+	8.01361i
13.3	−0.195548	−	0.279271i	−0.415666	+	0.0732932i	1.32833	−	3.64955i	6.96283	+	0.609169i	0.101751	+	0.101751i	−3.27638	+	2.74921i	−2.59620	+	0.695651i	−8.28983	+	3.01725i	−1.19144	−	2.06364i
13.4	1.01975	+	1.45635i	2.47251	−	0.435970i	0.287008	−	0.788548i	−6.49045	−	0.567841i	3.15626	+	3.15626i	−5.51447	+	4.62719i	8.31027	−	2.22673i	−2.53400	+	0.922300i	−5.79165	−	10.0314i
13.5	1.71376	+	2.44750i	−2.41431	+	0.425707i	−1.68522	+	4.63009i	0.958478	+	0.0838560i	−5.17946	−	5.17946i	8.02061	−	6.73009i	−2.67606	+	0.717049i	−2.80959	+	1.02261i	1.43736	+	2.48959i
15.1	−1.25404	−	2.68929i	−0.554625	−	1.52382i	−3.08852	+	3.68076i	−1.33394	−	1.90507i	−3.40247	+	3.40247i	−0.324876	+	1.84246i	2.30696	+	0.618149i	4.87999	−	4.09479i	−3.45047	+	5.97638i
15.2	−0.521767	−	1.11893i	1.27672	+	3.50777i	1.59138	−	1.89653i	1.42566	+	2.03605i	3.25881	−	3.25881i	0.576187	−	3.26772i	−7.72257	−	2.06926i	−3.78004	+	3.17183i	1.53434	−	2.65756i
15.3	0.177583	+	0.380827i	−1.22784	−	3.37346i	2.45766	−	2.92892i	1.10020	+	1.57124i	1.06666	−	1.06666i	−1.07559	+	6.09995i	3.17536	+	0.850836i	−2.97822	+	2.49903i	−0.402996	+	0.698010i
15.4	0.982348	+	2.10665i	0.787979	+	2.16495i	−0.901828	+	1.07476i	−4.57941	−	6.54008i	−3.78674	+	3.78674i	−0.559720	+	3.17433i	5.83088	+	1.56238i	2.82828	−	2.37321i	9.27910	−	16.0719i
15.5	1.54871	+	3.32122i	−0.989811	−	2.71948i	−6.06084	+	7.22303i	2.51995	+	3.59886i	7.49907	−	7.49907i	1.86029	−	10.5502i	−19.2170	−	5.14918i	0.478540	−	0.401543i	−8.04993	+	13.9429i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.i	odd	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	37.3.i.a	✓	60
3.b	odd	2	1		333.3.bu.b		60
37.i	odd	36	1	inner	37.3.i.a	✓	60
111.q	even	36	1		333.3.bu.b		60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
37.3.i.a	✓	60	1.a	even	1	1	trivial
37.3.i.a	✓	60	37.i	odd	36	1	inner
333.3.bu.b		60	3.b	odd	2	1
333.3.bu.b		60	111.q	even	36	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(37, [\chi])\).