Newform orbit 324.2.p.a

• Label: 324.2.p.a
• Level: $324$
• Weight: $2$
• Character orbit: 324.p
• Analytic conductor: $2.587$
• Analytic rank: $0$
• Dimension: $936$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	324.p (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.58715302549\)
Analytic rank:	\(0\)
Dimension:	\(936\)
Relative dimension:	\(52\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 936 q - 18 q^{2} - 18 q^{4} - 36 q^{5} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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	−1.41337	+	0.0488211i	−1.16649	+	1.28035i	1.99523	−	0.138005i	0.190968	−	1.63383i	1.58618	−	1.86656i	1.05815	+	2.10695i	−2.81327	+	0.292461i	−0.278579	−	2.98704i	−0.190143	+	2.31854i
11.2	−1.41238	+	0.0718931i	−1.73203	−	0.00891691i	1.98966	−	0.203082i	−0.493585	+	4.22289i	2.44693	−	0.111927i	−0.262610	−	0.522900i	−2.79557	+	0.429872i	2.99984	+	0.0308887i	0.393535	−	5.99983i
11.3	−1.41031	−	0.105029i	−0.330952	−	1.70014i	1.97794	+	0.296246i	0.00377236	−	0.0322746i	0.288181	+	2.43248i	−0.446224	−	0.888506i	−2.75839	−	0.625539i	−2.78094	+	1.12533i	−0.00870995	+	0.0451209i
11.4	−1.36952	−	0.352718i	1.60792	+	0.643879i	1.75118	+	0.966111i	−0.221174	+	1.89226i	−1.97498	−	1.44895i	0.167545	+	0.333610i	−2.05751	−	1.94078i	2.17084	+	2.07062i	0.970339	−	2.51349i
11.5	−1.31425	+	0.522253i	1.25530	+	1.19341i	1.45450	−	1.37274i	0.241988	−	2.07034i	−2.27303	−	0.912860i	0.716180	+	1.42603i	−1.19466	+	2.56374i	0.151532	+	2.99617i	0.763210	+	2.84733i
11.6	−1.31106	−	0.530208i	1.52671	−	0.818025i	1.43776	+	1.39027i	0.412015	−	3.52501i	−2.43533	+	0.263006i	−0.153752	−	0.306145i	−1.14785	−	2.58504i	1.66167	−	2.49777i	−2.40917	+	4.40305i
11.7	−1.30907	+	0.535100i	1.04386	−	1.38215i	1.42734	−	1.40097i	−0.309185	+	2.64525i	−0.626899	+	2.36791i	2.01774	+	4.01764i	−1.11883	+	2.59773i	−0.820704	−	2.88556i	−1.01073	−	3.62826i
11.8	−1.28696	+	0.586296i	−1.69911	−	0.336182i	1.31251	−	1.50908i	0.366702	−	3.13733i	2.38379	−	0.563531i	−1.91508	−	3.81323i	−0.804381	+	2.71164i	2.77396	+	1.14242i	1.36748	+	4.25261i
11.9	−1.24875	−	0.663801i	−1.42257	−	0.988080i	1.11874	+	1.65784i	0.180183	−	1.54156i	1.12054	+	2.17816i	1.41761	+	2.82269i	−0.296543	−	2.81284i	1.04740	+	2.81122i	−1.24830	+	1.80542i
11.10	−1.22866	+	0.700279i	1.64246	−	0.549842i	1.01922	−	1.72081i	−0.0995817	+	0.851976i	−1.63298	+	1.82575i	−2.18215	−	4.34501i	−0.0472259	+	2.82803i	2.39535	−	1.80619i	−0.474269	−	1.11652i
11.11	−1.18334	−	0.774405i	−1.00564	+	1.41021i	0.800594	+	1.83277i	−0.0544195	+	0.465589i	2.28209	−	0.889987i	−1.91763	−	3.81831i	0.471930	−	2.78878i	−0.977379	−	2.83632i	0.424951	−	0.508808i
11.12	−1.09593	−	0.893834i	0.138112	+	1.72654i	0.402121	+	1.95916i	−0.283509	+	2.42558i	1.39188	−	2.01561i	1.43973	+	2.86675i	1.31047	−	2.50653i	−2.96185	+	0.476909i	2.47877	−	2.40485i
11.13	−1.05140	+	0.945816i	−0.445336	+	1.67382i	0.210865	−	1.98885i	−0.155074	+	1.32674i	−1.11490	−	2.18105i	0.202244	+	0.402700i	1.65939	+	2.29051i	−2.60335	−	1.49083i	−1.09181	−	1.54160i
11.14	−0.969113	+	1.02996i	0.112425	−	1.72840i	−0.121640	−	1.99630i	0.460520	−	3.94000i	1.67123	+	1.79081i	0.896764	+	1.78560i	2.17399	+	1.80935i	−2.97472	−	0.388631i	3.61175	+	4.29262i
11.15	−0.956045	−	1.04210i	−0.138112	−	1.72654i	−0.171958	+	1.99259i	−0.283509	+	2.42558i	−1.66719	+	1.79457i	−1.43973	−	2.86675i	2.24089	−	1.72581i	−2.96185	+	0.476909i	2.79875	−	2.02351i
11.16	−0.870676	+	1.11442i	−0.828844	−	1.52086i	−0.483846	−	1.94059i	−0.291150	+	2.49095i	2.41653	+	0.400499i	−0.659322	−	1.31282i	2.58390	+	1.15042i	−1.62603	+	2.52111i	−2.52246	−	2.49327i
11.17	−0.841900	−	1.13631i	1.00564	−	1.41021i	−0.582409	+	1.91332i	−0.0544195	+	0.465589i	−2.44908	+	0.0445349i	1.91763	+	3.81831i	2.66446	−	0.949027i	−0.977379	−	2.83632i	0.574870	−	0.330141i
11.18	−0.735286	−	1.20804i	1.42257	+	0.988080i	−0.918708	+	1.77651i	0.180183	−	1.54156i	0.147643	−	2.44504i	−1.41761	−	2.82269i	2.82160	−	0.196407i	1.04740	+	2.81122i	−1.99475	+	0.915823i
11.19	−0.607055	+	1.27730i	−1.65046	+	0.525329i	−1.26297	−	1.55078i	−0.0392046	+	0.335417i	0.330922	−	2.42703i	−0.385340	−	0.767275i	2.74749	−	0.671779i	2.44806	−	1.73407i	−0.404627	−	0.253692i
11.20	−0.605543	−	1.27801i	−1.52671	+	0.818025i	−1.26664	+	1.54778i	0.412015	−	3.52501i	1.96993	+	1.45580i	0.153752	+	0.306145i	2.74509	+	0.681528i	1.66167	−	2.49777i	−4.75450	+	1.60799i

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	324.2.p.a	✓	936
3.b	odd	2	1		972.2.p.a		936
4.b	odd	2	1	inner	324.2.p.a	✓	936
12.b	even	2	1		972.2.p.a		936
81.g	even	27	1		972.2.p.a		936
81.h	odd	54	1	inner	324.2.p.a	✓	936
324.n	odd	54	1		972.2.p.a		936
324.p	even	54	1	inner	324.2.p.a	✓	936

    

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

Hecke kernels

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