Newform orbit 972.2.p.a

• Label: 972.2.p.a
• Level: $972$
• Weight: $2$
• Character orbit: 972.p
• Analytic conductor: $7.761$
• Analytic rank: $0$
• Dimension: $936$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 972 = 2^{2} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	972.p (of order \(54\), degree \(18\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.76145907647\)
Analytic rank:	\(0\)
Dimension:	\(936\)
Relative dimension:	\(52\) over \(\Q(\zeta_{54})\)
Twist minimal:	no (minimal twist has level 324)
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 936 q + 18 q^{2} - 18 q^{4} + 36 q^{5} + 18 q^{8}+O(q^{10}) \)

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	−1.40733	−	0.139378i		0		1.96115	+	0.392303i	1.32429	+	0.985898i		0		1.89845	−	2.88645i	−2.70530	−	0.825440i		0		−1.72630	−	1.57206i
35.2	−1.39863	+	0.209357i		0		1.91234	−	0.585627i	−1.98327	−	1.47649i		0		0.208675	−	0.317275i	−2.55205	+	1.21944i		0		3.08298	+	1.64985i
35.3	−1.39447	+	0.235514i		0		1.88907	−	0.656832i	2.59313	+	1.93051i		0		0.445132	−	0.676790i	−2.47954	+	1.36083i		0		−4.07069	−	2.08131i
35.4	−1.39247	−	0.247052i		0		1.87793	+	0.688023i	−2.20345	−	1.64040i		0		−1.41986	+	2.15879i	−2.44498	−	1.42200i		0		2.66296	+	2.82857i
35.5	−1.38736	−	0.274277i		0		1.84954	+	0.761042i	0.216897	+	0.161474i		0		−1.25852	+	1.91349i	−2.35725	−	1.56313i		0		−0.256626	−	0.283513i
35.6	−1.37981	+	0.310033i		0		1.80776	−	0.855574i	1.96578	+	1.46347i		0		0.696196	−	1.05851i	−2.22911	+	1.74100i		0		−3.16613	−	1.40986i
35.7	−1.28009	+	0.601150i		0		1.27724	−	1.53905i	−0.967186	−	0.720044i		0		−2.05807	+	3.12914i	−0.709777	+	2.73792i		0		1.67094	+	0.340294i
35.8	−1.26537	−	0.631530i		0		1.20234	+	1.59824i	2.19988	+	1.63775i		0		−1.99639	+	3.03536i	−0.512069	−	2.78169i		0		−1.74938	−	3.46166i
35.9	−1.18842	−	0.766583i		0		0.824700	+	1.82205i	0.338340	+	0.251885i		0		2.20434	−	3.35154i	0.416660	−	2.79757i		0		−0.209000	−	0.558711i
35.10	−1.11185	+	0.873952i		0		0.472416	−	1.94341i	−2.93908	−	2.18806i		0		0.0709394	−	0.107858i	1.17319	+	2.57364i		0		5.18008	−	0.135818i
35.11	−1.06603	−	0.929294i		0		0.272825	+	1.98130i	−2.76095	−	2.05545i		0		2.24290	−	3.41017i	1.55038	−	2.36566i		0		1.03313	+	4.75690i
35.12	−1.05351	+	0.943464i		0		0.219753	−	1.98789i	3.28501	+	2.44560i		0		−0.350232	+	0.532501i	1.64399	+	2.30158i		0		−5.76812	+	0.522833i
35.13	−1.00236	+	0.997634i		0		0.00945287	−	1.99998i	−0.164754	−	0.122655i		0		1.94391	−	2.95557i	1.98577	+	2.01413i		0		0.287508	−	0.0414199i
35.14	−0.948408	−	1.04906i		0		−0.201045	+	1.98987i	0.388113	+	0.288939i		0		−0.189630	+	0.288319i	2.27816	−	1.67630i		0		−0.0649751	−	0.681185i
35.15	−0.936521	+	1.05968i		0		−0.245856	−	1.98483i	−0.472494	−	0.351759i		0		2.33624	−	3.55208i	2.33354	+	1.59831i		0		0.815253	−	0.171264i
35.16	−0.926918	−	1.06809i		0		−0.281645	+	1.98007i	2.82264	+	2.10138i		0		−1.07201	+	1.62991i	2.37596	−	1.53454i		0		−0.371890	−	4.96264i
35.17	−0.769577	−	1.18649i		0		−0.815503	+	1.82619i	−1.33185	−	0.991528i		0		−1.37495	+	2.09052i	2.79434	−	0.437807i		0		−0.151472	+	2.34328i
35.18	−0.736030	+	1.20758i		0		−0.916519	−	1.77764i	1.16471	+	0.867094i		0		−2.10489	+	3.20034i	2.82123	+	0.201622i		0		−1.90435	+	0.768277i
35.19	−0.708680	+	1.22384i		0		−0.995546	−	1.73461i	0.582254	+	0.433472i		0		−0.981557	+	1.49238i	2.82841	+	0.0109009i		0		−0.943130	+	0.405390i
35.20	−0.516533	−	1.31651i		0		−1.46639	+	1.36004i	−1.66950	−	1.24289i		0		0.427552	−	0.650062i	2.54794	+	1.22800i		0		−0.773929	+	2.83990i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
81.h	odd	54	1	inner
324.p	even	54	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	972.2.p.a		936
3.b	odd	2	1		324.2.p.a	✓	936
4.b	odd	2	1	inner	972.2.p.a		936
12.b	even	2	1		324.2.p.a	✓	936
81.g	even	27	1		324.2.p.a	✓	936
81.h	odd	54	1	inner	972.2.p.a		936
324.n	odd	54	1		324.2.p.a	✓	936
324.p	even	54	1	inner	972.2.p.a		936

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
324.2.p.a	✓	936	3.b	odd	2	1
324.2.p.a	✓	936	12.b	even	2	1
324.2.p.a	✓	936	81.g	even	27	1
324.2.p.a	✓	936	324.n	odd	54	1
972.2.p.a		936	1.a	even	1	1	trivial
972.2.p.a		936	4.b	odd	2	1	inner
972.2.p.a		936	81.h	odd	54	1	inner
972.2.p.a		936	324.p	even	54	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(972, [\chi])\).