Newform orbit 387.2.m.a

• Label: 387.2.m.a
• Level: $387$
• Weight: $2$
• Character orbit: 387.m
• Analytic conductor: $3.090$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	387.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.09021055822\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(42\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 3 q^{3} - 42 q^{4} - 6 q^{5} + 9 q^{6} + 9 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
50.1	−1.31845	−	2.28363i	−1.68335	+	0.407844i	−2.47664	+	4.28966i	1.90483			3.15078	+	3.30642i		−	3.85335i	7.78750			2.66733	−	1.37309i	−2.51143	−	4.34992i
50.2	−1.30414	−	2.25885i	1.57475	−	0.721232i	−2.40159	+	4.15967i	0.0895305			−3.68285	−	2.61652i		−	1.99030i	7.31150			1.95965	−	2.27151i	−0.116761	−	0.202235i
50.3	−1.28139	−	2.21943i	−0.970800	−	1.43442i	−2.28391	+	3.95585i	1.39995			−1.93961	+	3.99267i			2.77171i	6.58076			−1.11510	+	2.78506i	−1.79388	−	3.10709i
50.4	−1.23036	−	2.13105i	−0.201815	+	1.72025i	−2.02759	+	3.51189i	−2.96017			3.91426	−	1.68646i		−	2.76482i	5.05726			−2.91854	−	0.694346i	3.64209	+	6.30829i
50.5	−1.15496	−	2.00045i	−1.68169	+	0.414622i	−1.66788	+	2.88885i	−3.67302			2.77172	+	2.88528i			4.26533i	3.08550			2.65618	−	1.39453i	4.24220	+	7.34770i
50.6	−1.15451	−	1.99966i	1.68136	+	0.415969i	−1.66576	+	2.88519i	−2.16536			−1.10934	−	3.84239i			2.54164i	3.07451			2.65394	+	1.39879i	2.49992	+	4.32998i
50.7	−1.06551	−	1.84551i	0.456471	−	1.67082i	−1.27061	+	2.20075i	0.387211			−3.56989	+	0.937845i		−	0.0230345i	1.15333			−2.58327	−	1.52536i	−0.412575	−	0.714602i
50.8	−1.02303	−	1.77193i	−0.579185	+	1.63234i	−1.09317	+	1.89342i	2.82348			3.48493	−	0.643654i			2.39605i	0.381247			−2.32909	−	1.89086i	−2.88850	−	5.00302i
50.9	−0.990357	−	1.71535i	1.32429	+	1.11636i	−0.961612	+	1.66556i	3.34073			0.603415	−	3.37721i		−	3.47266i	−0.152071			0.507503	+	2.95676i	−3.30851	−	5.73051i
50.10	−0.916255	−	1.58700i	−0.928529	−	1.46213i	−0.679045	+	1.17614i	−2.59436			−1.46964	+	2.81326i		−	3.95794i	−1.17631			−1.27567	+	2.71527i	2.37710	+	4.11725i
50.11	−0.830976	−	1.43929i	−1.72644	+	0.139260i	−0.381043	+	0.659985i	1.84192			1.63507	+	2.36914i		−	0.213305i	−2.05735			2.96121	−	0.480849i	−1.53059	−	2.65106i
50.12	−0.622633	−	1.07843i	1.31684	+	1.12514i	0.224656	−	0.389116i	−0.167935			0.393481	−	2.12067i			4.27082i	−3.05005			0.468119	+	2.96325i	0.104562	+	0.181106i
50.13	−0.604942	−	1.04779i	−0.308327	+	1.70439i	0.268089	−	0.464345i	−1.09024			1.97236	−	0.707994i			0.287059i	−3.06848			−2.80987	−	1.05102i	0.659530	+	1.14234i
50.14	−0.594676	−	1.03001i	1.67163	−	0.453482i	0.292721	−	0.507007i	0.400437			−1.46117	−	1.45212i		−	1.46856i	−3.07500			2.58871	−	1.51611i	−0.238131	−	0.412454i
50.15	−0.516284	−	0.894231i	0.105268	−	1.72885i	0.466901	−	0.808697i	3.33029			−1.60034	+	0.798444i		−	2.33607i	−3.02935			−2.97784	−	0.363984i	−1.71938	−	2.97805i
50.16	−0.431329	−	0.747084i	−1.64514	+	0.541758i	0.627911	−	1.08757i	−2.12872			1.11434	+	0.995384i		−	1.69515i	−2.80866			2.41300	−	1.78254i	0.918179	+	1.59033i
50.17	−0.367145	−	0.635914i	−0.909888	−	1.47381i	0.730409	−	1.26511i	−1.90950			−0.603153	+	1.11971i			4.64355i	−2.54125			−1.34421	+	2.68200i	0.701065	+	1.21428i
50.18	−0.352466	−	0.610489i	1.66298	+	0.484252i	0.751536	−	1.30170i	−4.11810			−0.290513	−	1.18591i		−	3.03523i	−2.46943			2.53100	+	1.61060i	1.45149	+	2.51406i
50.19	−0.163956	−	0.283980i	1.60785	−	0.644075i	0.946237	−	1.63893i	1.71258			−0.446521	−	0.350996i			1.23417i	−1.27639			2.17034	−	2.07115i	−0.280788	−	0.486338i
50.20	−0.0333401	−	0.0577467i	−1.52646	−	0.818478i	0.997777	−	1.72820i	0.429311			0.00362805	+	0.115436i		−	0.335867i	−0.266424			1.66019	+	2.49875i	−0.0143133	−	0.0247913i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
387.m	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	387.2.m.a	yes	84
3.b	odd	2	1		1161.2.m.a		84
9.c	even	3	1		1161.2.k.a		84
9.d	odd	6	1		387.2.k.a	✓	84
43.d	odd	6	1		387.2.k.a	✓	84
129.h	even	6	1		1161.2.k.a		84
387.m	even	6	1	inner	387.2.m.a	yes	84
387.n	odd	6	1		1161.2.m.a		84

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
387.2.k.a	✓	84	9.d	odd	6	1
387.2.k.a	✓	84	43.d	odd	6	1
387.2.m.a	yes	84	1.a	even	1	1	trivial
387.2.m.a	yes	84	387.m	even	6	1	inner
1161.2.k.a		84	9.c	even	3	1
1161.2.k.a		84	129.h	even	6	1
1161.2.m.a		84	3.b	odd	2	1
1161.2.m.a		84	387.n	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(387, [\chi])\).