Newform orbit 378.3.y.a

• Label: 378.3.y.a
• Level: $378$
• Weight: $3$
• Character orbit: 378.y
• Analytic conductor: $10.300$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	378.y (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.2997539928\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(48\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 12 q^{6} - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.39273	−	0.245576i	−2.99983	+	0.0323380i	1.87939	+	0.684040i	−5.58473	+	0.984738i	4.18588	+	0.691646i	5.01591	−	4.88269i	−2.44949	−	1.41421i	8.99791	−	0.194017i	8.01984
11.2	−1.39273	−	0.245576i	−2.98463	+	0.303313i	1.87939	+	0.684040i	3.72268	−	0.656408i	4.23126	+	0.310519i	−5.19415	−	4.69263i	−2.44949	−	1.41421i	8.81600	−	1.81055i	−5.34588
11.3	−1.39273	−	0.245576i	−2.85881	−	0.909507i	1.87939	+	0.684040i	−4.61838	+	0.814345i	3.75819	+	1.96875i	−3.57446	+	6.01857i	−2.44949	−	1.41421i	7.34559	+	5.20022i	6.63213
11.4	−1.39273	−	0.245576i	−2.54490	−	1.58855i	1.87939	+	0.684040i	4.21756	−	0.743670i	3.15424	+	2.83738i	6.16309	−	3.31909i	−2.44949	−	1.41421i	3.95302	+	8.08540i	−6.05655
11.5	−1.39273	−	0.245576i	−2.35056	+	1.86410i	1.87939	+	0.684040i	−5.64394	+	0.995178i	3.73146	−	2.01895i	5.96260	+	3.66706i	−2.44949	−	1.41421i	2.05023	−	8.76337i	8.10486
11.6	−1.39273	−	0.245576i	−2.16258	−	2.07925i	1.87939	+	0.684040i	5.02305	−	0.885700i	2.50127	+	3.42690i	−2.58051	+	6.50700i	−2.44949	−	1.41421i	0.353464	+	8.99306i	−7.21325
11.7	−1.39273	−	0.245576i	−1.94728	+	2.28212i	1.87939	+	0.684040i	1.40904	−	0.248452i	3.27247	−	2.70017i	−6.84878	+	1.44714i	−2.44949	−	1.41421i	−1.41617	−	8.88788i	−2.02343
11.8	−1.39273	−	0.245576i	−1.73756	+	2.44559i	1.87939	+	0.684040i	2.05922	−	0.363097i	3.02052	−	2.97933i	3.39646	−	6.12079i	−2.44949	−	1.41421i	−2.96177	−	8.49870i	−2.95711
11.9	−1.39273	−	0.245576i	−1.56665	−	2.55844i	1.87939	+	0.684040i	−5.16921	+	0.911471i	1.55363	+	3.94794i	−5.84307	−	3.85467i	−2.44949	−	1.41421i	−4.09121	+	8.01636i	7.42314
11.10	−1.39273	−	0.245576i	−0.598728	+	2.93965i	1.87939	+	0.684040i	−0.589395	+	0.103926i	1.55577	−	3.94710i	4.29197	+	5.52983i	−2.44949	−	1.41421i	−8.28305	−	3.52010i	0.846389
11.11	−1.39273	−	0.245576i	−0.144626	−	2.99651i	1.87939	+	0.684040i	−0.197249	+	0.0347803i	−0.534446	+	4.20884i	5.81470	+	3.89735i	−2.44949	−	1.41421i	−8.95817	+	0.866745i	0.283256
11.12	−1.39273	−	0.245576i	0.431836	+	2.96876i	1.87939	+	0.684040i	8.15828	−	1.43853i	0.127624	−	4.24072i	−6.96984	−	0.649151i	−2.44949	−	1.41421i	−8.62704	+	2.56403i	−11.7155
11.13	−1.39273	−	0.245576i	0.433496	−	2.96851i	1.87939	+	0.684040i	7.77301	−	1.37059i	−1.33274	+	4.02788i	−0.799289	−	6.95422i	−2.44949	−	1.41421i	−8.62416	−	2.57368i	−11.1623
11.14	−1.39273	−	0.245576i	0.919494	+	2.85561i	1.87939	+	0.684040i	−2.10283	+	0.370785i	−0.579336	−	4.20290i	0.579341	−	6.97598i	−2.44949	−	1.41421i	−7.30906	+	5.25144i	3.01972
11.15	−1.39273	−	0.245576i	0.924584	−	2.85397i	1.87939	+	0.684040i	−8.56809	+	1.51079i	−1.98856	+	3.74775i	0.659043	+	6.96891i	−2.44949	−	1.41421i	−7.29029	−	5.27747i	12.3040
11.16	−1.39273	−	0.245576i	0.977664	−	2.83623i	1.87939	+	0.684040i	7.33808	−	1.29390i	−2.05813	+	3.71000i	−4.68435	+	5.20162i	−2.44949	−	1.41421i	−7.08835	−	5.54575i	−10.5377
11.17	−1.39273	−	0.245576i	1.30904	+	2.69934i	1.87939	+	0.684040i	−6.62994	+	1.16904i	−1.16025	−	4.08091i	−4.96141	+	4.93806i	−2.44949	−	1.41421i	−5.57282	+	7.06708i	9.52080
11.18	−1.39273	−	0.245576i	1.63224	+	2.51710i	1.87939	+	0.684040i	7.28585	−	1.28469i	−1.65513	−	3.90648i	6.63519	+	2.23030i	−2.44949	−	1.41421i	−3.67159	+	8.21702i	−10.4627
11.19	−1.39273	−	0.245576i	2.29989	−	1.92627i	1.87939	+	0.684040i	−3.27981	+	0.578319i	−3.67616	+	2.11797i	−6.56293	−	2.43474i	−2.44949	−	1.41421i	1.57897	−	8.86041i	4.70990
11.20	−1.39273	−	0.245576i	2.48789	+	1.67642i	1.87939	+	0.684040i	−4.37954	+	0.772232i	−3.05327	−	2.94577i	1.34883	−	6.86882i	−2.44949	−	1.41421i	3.37921	+	8.34152i	6.28916

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
189.bf	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	378.3.y.a	✓	288
7.c	even	3	1		378.3.bc.a	yes	288
27.f	odd	18	1		378.3.bc.a	yes	288
189.bf	odd	18	1	inner	378.3.y.a	✓	288

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
378.3.y.a	✓	288	1.a	even	1	1	trivial
378.3.y.a	✓	288	189.bf	odd	18	1	inner
378.3.bc.a	yes	288	7.c	even	3	1
378.3.bc.a	yes	288	27.f	odd	18	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(378, [\chi])\).