Newform orbit 324.2.l.a

• Label: 324.2.l.a
• Level: $324$
• Weight: $2$
• Character orbit: 324.l
• Analytic conductor: $2.587$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	324.l (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.58715302549\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{18})\)
Twist minimal:	no (minimal twist has level 108)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 6 q^{2} - 6 q^{4} + 12 q^{5} + 9 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
35.1	−1.40284	−	0.179019i		0		1.93590	+	0.502269i	0.847966	−	2.32977i		0		−4.59553	−	0.810316i	−2.62584	−	1.05116i		0		−1.60663	+	3.11648i
35.2	−1.37675	+	0.323366i		0		1.79087	−	0.890387i	−0.470103	+	1.29160i		0		1.57428	+	0.277589i	−2.17765	+	1.80495i		0		0.229554	−	1.93022i
35.3	−1.18497	+	0.771906i		0		0.808323	−	1.82938i	0.197421	−	0.542409i		0		−1.70706	−	0.301001i	0.454264	+	2.79171i		0		0.184750	+	0.795131i
35.4	−1.09968	−	0.889210i		0		0.418610	+	1.95570i	−0.605021	+	1.66228i		0		−0.748045	−	0.131901i	1.27869	−	2.52289i		0		2.14345	−	1.28999i
35.5	−0.608544	−	1.27659i		0		−1.25935	+	1.55372i	−0.420820	+	1.15619i		0		−1.81474	−	0.319988i	2.74983	+	0.662160i		0		1.73207	−	0.166382i
35.6	−0.557089	−	1.29987i		0		−1.37930	+	1.44828i	1.27150	−	3.49343i		0		1.63150	+	0.287677i	2.65097	+	0.986086i		0		−5.24933	+	0.293367i
35.7	−0.554410	+	1.30101i		0		−1.38526	−	1.44259i	0.197421	−	0.542409i		0		1.70706	+	0.301001i	2.64482	−	1.00245i		0		0.596228	+	0.557564i
35.8	−0.0793838	+	1.41198i		0		−1.98740	−	0.224177i	−0.470103	+	1.29160i		0		−1.57428	−	0.277589i	0.474302	−	2.78838i		0		−1.78640	−	0.766310i
35.9	0.376137	−	1.36328i		0		−1.71704	−	1.02556i	−1.29726	+	3.56419i		0		−2.58045	−	0.455003i	−2.04396	+	1.95505i		0		4.37103	+	3.10915i
35.10	0.419899	+	1.35044i		0		−1.64737	+	1.13410i	0.847966	−	2.32977i		0		4.59553	+	0.810316i	−2.22326	−	1.74846i		0		3.50227	+	0.166858i
35.11	0.805437	−	1.16244i		0		−0.702543	−	1.87255i	0.710267	−	1.95144i		0		3.83975	+	0.677051i	−2.74258	−	0.691554i		0		−1.69636	−	2.39741i
35.12	1.00492	−	0.995056i		0		0.0197253	−	1.99990i	0.710267	−	1.95144i		0		−3.83975	−	0.677051i	−1.97019	−	2.02937i		0		−1.22804	−	2.66780i
35.13	1.06666	+	0.928568i		0		0.275524	+	1.98093i	−0.605021	+	1.66228i		0		0.748045	+	0.131901i	−1.54554	+	2.36882i		0		−2.18889	+	1.21129i
35.14	1.27725	−	0.607153i		0		1.26273	−	1.55097i	−1.29726	+	3.56419i		0		2.58045	+	0.455003i	0.671144	−	2.74765i		0		0.507086	+	5.33999i
35.15	1.36287	+	0.377622i		0		1.71480	+	1.02930i	−0.420820	+	1.15619i		0		1.81474	+	0.319988i	1.94836	+	2.05034i		0		−1.01013	+	1.41682i
35.16	1.37686	+	0.322907i		0		1.79146	+	0.889191i	1.27150	−	3.49343i		0		−1.63150	−	0.287677i	2.17946	+	1.80276i		0		2.87873	−	4.39937i
71.1	−1.39986	−	0.201007i		0		1.91919	+	0.562762i	2.12601	−	2.53368i		0		1.10219	−	3.02825i	−2.57347	−	1.17356i		0		−3.48540	+	3.11945i
71.2	−1.24669	+	0.667664i		0		1.10845	−	1.66473i	2.12601	−	2.53368i		0		−1.10219	+	3.02825i	−0.270407	+	2.81547i		0		−0.958821	+	4.57817i
71.3	−1.24471	−	0.671335i		0		1.09862	+	1.67124i	−0.546554	+	0.651358i		0		0.348909	−	0.958618i	−0.245503	−	2.81775i		0		1.11758	−	0.443832i
71.4	−0.940037	+	1.05657i		0		−0.232661	−	1.98642i	−0.546554	+	0.651358i		0		−0.348909	+	0.958618i	2.31749	+	1.62149i		0		−0.174421	−	1.18977i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
27.f	odd	18	1	inner
108.l	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	324.2.l.a		96
3.b	odd	2	1		108.2.l.a	✓	96
4.b	odd	2	1	inner	324.2.l.a		96
9.c	even	3	1		972.2.l.a		96
9.c	even	3	1		972.2.l.b		96
9.d	odd	6	1		972.2.l.c		96
9.d	odd	6	1		972.2.l.d		96
12.b	even	2	1		108.2.l.a	✓	96
27.e	even	9	1		108.2.l.a	✓	96
27.e	even	9	1		972.2.l.c		96
27.e	even	9	1		972.2.l.d		96
27.f	odd	18	1	inner	324.2.l.a		96
27.f	odd	18	1		972.2.l.a		96
27.f	odd	18	1		972.2.l.b		96
36.f	odd	6	1		972.2.l.a		96
36.f	odd	6	1		972.2.l.b		96
36.h	even	6	1		972.2.l.c		96
36.h	even	6	1		972.2.l.d		96
108.j	odd	18	1		108.2.l.a	✓	96
108.j	odd	18	1		972.2.l.c		96
108.j	odd	18	1		972.2.l.d		96
108.l	even	18	1	inner	324.2.l.a		96
108.l	even	18	1		972.2.l.a		96
108.l	even	18	1		972.2.l.b		96

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
108.2.l.a	✓	96	3.b	odd	2	1
108.2.l.a	✓	96	12.b	even	2	1
108.2.l.a	✓	96	27.e	even	9	1
108.2.l.a	✓	96	108.j	odd	18	1
324.2.l.a		96	1.a	even	1	1	trivial
324.2.l.a		96	4.b	odd	2	1	inner
324.2.l.a		96	27.f	odd	18	1	inner
324.2.l.a		96	108.l	even	18	1	inner
972.2.l.a		96	9.c	even	3	1
972.2.l.a		96	27.f	odd	18	1
972.2.l.a		96	36.f	odd	6	1
972.2.l.a		96	108.l	even	18	1
972.2.l.b		96	9.c	even	3	1
972.2.l.b		96	27.f	odd	18	1
972.2.l.b		96	36.f	odd	6	1
972.2.l.b		96	108.l	even	18	1
972.2.l.c		96	9.d	odd	6	1
972.2.l.c		96	27.e	even	9	1
972.2.l.c		96	36.h	even	6	1
972.2.l.c		96	108.j	odd	18	1
972.2.l.d		96	9.d	odd	6	1
972.2.l.d		96	27.e	even	9	1
972.2.l.d		96	36.h	even	6	1
972.2.l.d		96	108.j	odd	18	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(324, [\chi])\).