Newform orbit 185.3.q.a

• Label: 185.3.q.a
• Level: $185$
• Weight: $3$
• Character orbit: 185.q
• Analytic conductor: $5.041$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 185 = 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	185.q (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.04088489067\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 12 q^{4} + 18 q^{5} + 8 q^{6} - 216 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
14.1	−1.00387	−	3.74648i	−1.98015	−	3.42971i	−9.56426	+	5.52193i	4.52944	+	2.11758i	−10.8615	+	10.8615i	−4.36732	+	2.52147i	19.3186	+	19.3186i	−3.34196	+	5.78844i	3.38652	−	19.0952i
14.2	−0.995727	−	3.71610i	1.15531	+	2.00105i	−9.35384	+	5.40044i	−2.86518	+	4.09765i	6.28575	−	6.28575i	6.70392	−	3.87051i	18.5010	+	18.5010i	1.83052	−	3.17056i	18.0802	+	6.56715i
14.3	−0.892799	−	3.33197i	2.83146	+	4.90423i	−6.84085	+	3.94957i	4.92446	−	0.865828i	13.8128	−	13.8128i	−2.75013	+	1.58779i	9.51065	+	9.51065i	−11.5343	+	19.9780i	−7.28147	−	15.6352i
14.4	−0.831595	−	3.10355i	−1.89309	−	3.27893i	−5.47640	+	3.16180i	−4.94074	−	0.767502i	−8.60206	+	8.60206i	1.15322	−	0.665811i	5.27912	+	5.27912i	−2.66759	+	4.62041i	1.72671	+	15.9721i
14.5	−0.827848	−	3.08957i	0.199364	+	0.345308i	−5.39601	+	3.11539i	2.14115	−	4.51835i	0.901811	−	0.901811i	7.25581	−	4.18914i	5.04538	+	5.04538i	4.42051	−	7.65654i	−15.7323	−	2.87474i
14.6	−0.784629	−	2.92828i	1.80919	+	3.13361i	−4.49506	+	2.59522i	−4.14391	−	2.79785i	7.75653	−	7.75653i	−6.49323	+	3.74887i	2.55190	+	2.55190i	−2.04634	+	3.54437i	−4.94146	+	14.3298i
14.7	−0.697535	−	2.60324i	0.330876	+	0.573094i	−2.82618	+	1.63170i	−1.55549	+	4.75189i	1.26110	−	1.26110i	−5.94626	+	3.43308i	−1.40376	−	1.40376i	4.28104	−	7.41498i	13.4553	+	0.734706i
14.8	−0.696082	−	2.59781i	0.329948	+	0.571487i	−2.80000	+	1.61658i	4.46271	+	2.25483i	1.25495	−	1.25495i	−7.49712	+	4.32846i	−1.45831	−	1.45831i	4.28227	−	7.41711i	2.75122	−	13.1628i
14.9	−0.643370	−	2.40109i	−1.95298	−	3.38266i	−1.88721	+	1.08958i	2.61734	−	4.26022i	−6.86558	+	6.86558i	−1.65093	+	0.953165i	−3.20052	−	3.20052i	−3.12824	+	5.41827i	−11.9131	−	3.54358i
14.10	−0.558311	−	2.08365i	−1.23016	−	2.13070i	−0.565768	+	0.326646i	2.76875	+	4.16342i	−3.75281	+	3.75281i	10.7757	−	6.22135i	−5.10485	−	5.10485i	1.47341	−	2.55202i	7.12926	−	8.09357i
14.11	−0.463148	−	1.72849i	1.88610	+	3.26683i	0.690920	−	0.398903i	−3.98411	−	3.02107i	4.77315	−	4.77315i	7.64717	−	4.41509i	−6.07088	−	6.07088i	−2.61478	+	4.52894i	−3.37667	+	8.28571i
14.12	−0.381949	−	1.42545i	2.32611	+	4.02894i	1.57807	−	0.911096i	2.58996	+	4.27693i	4.85462	−	4.85462i	5.83286	−	3.36761i	−6.07549	−	6.07549i	−6.32159	+	10.9493i	5.10734	−	5.32544i
14.13	−0.355593	−	1.32709i	−2.91628	−	5.05114i	1.82937	−	1.05619i	−1.41034	+	4.79697i	−5.66632	+	5.66632i	−4.24015	+	2.44805i	−5.93817	−	5.93817i	−12.5093	+	21.6668i	6.86753	+	0.165877i
14.14	−0.325210	−	1.21370i	−1.01552	−	1.75893i	2.09680	−	1.21059i	−2.05559	−	4.55791i	−1.80455	+	1.80455i	−9.71172	+	5.60706i	−5.70514	−	5.70514i	2.43745	−	4.22179i	−4.86344	+	3.97714i
14.15	−0.286970	−	1.07099i	1.25384	+	2.17172i	2.39944	−	1.38532i	3.57094	−	3.49977i	1.96607	−	1.96607i	−0.307688	+	0.177644i	−5.30829	−	5.30829i	1.35575	−	2.34823i	−4.77295	−	2.82010i
14.16	−0.222439	−	0.830155i	−0.303000	−	0.524811i	2.82442	−	1.63068i	−4.22706	+	2.67058i	−0.368276	+	0.368276i	0.768149	−	0.443491i	−4.41285	−	4.41285i	4.31638	−	7.47619i	3.15726	+	2.91507i
14.17	−0.0813055	−	0.303436i	−1.36236	−	2.35967i	3.37864	−	1.95066i	4.95509	−	0.668613i	−0.605243	+	0.605243i	2.44414	−	1.41113i	−1.75513	−	1.75513i	0.787970	−	1.36480i	−0.605758	−	1.44919i
14.18	−0.00904694	−	0.0337636i	2.50211	+	4.33379i	3.46304	−	1.99939i	−4.39987	+	2.37511i	0.123688	−	0.123688i	−10.4200	+	6.01599i	−0.197703	−	0.197703i	−8.02116	+	13.8931i	0.119998	+	0.127068i
14.19	0.00904694	+	0.0337636i	−2.50211	−	4.33379i	3.46304	−	1.99939i	−2.62285	−	4.25684i	0.123688	−	0.123688i	10.4200	−	6.01599i	0.197703	+	0.197703i	−8.02116	+	13.8931i	0.119998	−	0.127068i
14.20	0.0813055	+	0.303436i	1.36236	+	2.35967i	3.37864	−	1.95066i	3.95693	+	3.05658i	−0.605243	+	0.605243i	−2.44414	+	1.41113i	1.75513	+	1.75513i	0.787970	−	1.36480i	−0.605758	+	1.44919i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
37.g	odd	12	1	inner
185.q	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	185.3.q.a	✓	144
5.b	even	2	1	inner	185.3.q.a	✓	144
37.g	odd	12	1	inner	185.3.q.a	✓	144
185.q	odd	12	1	inner	185.3.q.a	✓	144

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
185.3.q.a	✓	144	1.a	even	1	1	trivial
185.3.q.a	✓	144	5.b	even	2	1	inner
185.3.q.a	✓	144	37.g	odd	12	1	inner
185.3.q.a	✓	144	185.q	odd	12	1	inner

Hecke kernels

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