Newform orbit 546.2.z.b

• Label: 546.2.z.b
• Level: $546$
• Weight: $2$
• Character orbit: 546.z
• Analytic conductor: $4.360$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	546.z (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.35983195036\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} + 2 q^{7} + 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
131.1	−0.866025	−	0.500000i	−1.73117	+	0.0552650i	0.500000	+	0.866025i	−1.72310	+	2.98449i	1.52687	+	0.817724i	2.50296	+	0.857422i		−	1.00000i	2.99389	−	0.191346i	2.98449	−	1.72310i
131.2	−0.866025	−	0.500000i	−1.44180	−	0.959803i	0.500000	+	0.866025i	1.50901	−	2.61368i	0.768732	+	1.55211i	0.237102	−	2.63511i		−	1.00000i	1.15756	+	2.76768i	−2.61368	+	1.50901i
131.3	−0.866025	−	0.500000i	−1.34375	+	1.09285i	0.500000	+	0.866025i	0.166488	−	0.288366i	1.71015	−	0.274563i	−2.56503	+	0.648546i		−	1.00000i	0.611342	−	2.93705i	−0.288366	+	0.166488i
131.4	−0.866025	−	0.500000i	−0.818620	+	1.52639i	0.500000	+	0.866025i	0.401573	−	0.695544i	1.47214	−	0.912581i	1.69379	−	2.03250i		−	1.00000i	−1.65972	−	2.49906i	−0.695544	+	0.401573i
131.5	−0.866025	−	0.500000i	0.114502	−	1.72826i	0.500000	+	0.866025i	−0.698212	+	1.20934i	−0.963293	+	1.43947i	0.352771	+	2.62213i		−	1.00000i	−2.97378	−	0.395779i	1.20934	−	0.698212i
131.6	−0.866025	−	0.500000i	1.20682	−	1.24241i	0.500000	+	0.866025i	−1.33293	+	2.30870i	−1.66634	+	0.472552i	−2.41436	−	1.08207i		−	1.00000i	−0.0871789	−	2.99873i	2.30870	−	1.33293i
131.7	−0.866025	−	0.500000i	1.51893	−	0.832377i	0.500000	+	0.866025i	1.85844	−	3.21892i	−1.73162	−	0.0386050i	−1.15681	+	2.37945i		−	1.00000i	1.61430	−	2.52865i	−3.21892	+	1.85844i
131.8	−0.866025	−	0.500000i	1.62906	+	0.588348i	0.500000	+	0.866025i	−1.04730	+	1.81398i	−1.11664	−	1.32406i	1.84957	−	1.89185i		−	1.00000i	2.30769	+	1.91691i	1.81398	−	1.04730i
131.9	0.866025	+	0.500000i	−1.43878	−	0.964322i	0.500000	+	0.866025i	1.85995	−	3.22152i	−0.763858	−	1.55452i	−1.49982	−	2.17958i			1.00000i	1.14017	+	2.77489i	3.22152	−	1.85995i
131.10	0.866025	+	0.500000i	−1.12951	+	1.31309i	0.500000	+	0.866025i	−1.63964	+	2.83994i	−1.63473	+	0.572419i	1.76715	+	1.96906i			1.00000i	−0.448430	−	2.96630i	−2.83994	+	1.63964i
131.11	0.866025	+	0.500000i	−1.11753	−	1.32330i	0.500000	+	0.866025i	−0.386271	+	0.669041i	−0.306158	−	1.70478i	1.82832	+	1.91239i			1.00000i	−0.502255	+	2.95766i	−0.669041	+	0.386271i
131.12	0.866025	+	0.500000i	−0.0921101	+	1.72960i	0.500000	+	0.866025i	0.890016	−	1.54155i	−0.944570	+	1.45182i	−1.51727	+	2.16746i			1.00000i	−2.98303	−	0.318627i	1.54155	−	0.890016i
131.13	0.866025	+	0.500000i	0.691807	−	1.58789i	0.500000	+	0.866025i	0.735814	−	1.27447i	1.39307	−	1.02925i	0.322414	+	2.62603i			1.00000i	−2.04281	−	2.19703i	1.27447	−	0.735814i
131.14	0.866025	+	0.500000i	1.08085	+	1.35343i	0.500000	+	0.866025i	1.49759	−	2.59391i	0.259332	+	1.71253i	−0.0366585	−	2.64550i			1.00000i	−0.663521	+	2.92570i	2.59391	−	1.49759i
131.15	0.866025	+	0.500000i	1.16571	+	1.28106i	0.500000	+	0.866025i	−1.35360	+	2.34450i	0.369008	+	1.69229i	−2.61407	+	0.408207i			1.00000i	−0.282226	+	2.98670i	−2.34450	+	1.35360i
131.16	0.866025	+	0.500000i	1.70558	−	0.301663i	0.500000	+	0.866025i	−0.737837	+	1.27797i	1.62791	+	0.591542i	2.24993	−	1.39206i			1.00000i	2.81800	−	1.02902i	−1.27797	+	0.737837i
521.1	−0.866025	+	0.500000i	−1.73117	−	0.0552650i	0.500000	−	0.866025i	−1.72310	−	2.98449i	1.52687	−	0.817724i	2.50296	−	0.857422i			1.00000i	2.99389	+	0.191346i	2.98449	+	1.72310i
521.2	−0.866025	+	0.500000i	−1.44180	+	0.959803i	0.500000	−	0.866025i	1.50901	+	2.61368i	0.768732	−	1.55211i	0.237102	+	2.63511i			1.00000i	1.15756	−	2.76768i	−2.61368	−	1.50901i
521.3	−0.866025	+	0.500000i	−1.34375	−	1.09285i	0.500000	−	0.866025i	0.166488	+	0.288366i	1.71015	+	0.274563i	−2.56503	−	0.648546i			1.00000i	0.611342	+	2.93705i	−0.288366	−	0.166488i
521.4	−0.866025	+	0.500000i	−0.818620	−	1.52639i	0.500000	−	0.866025i	0.401573	+	0.695544i	1.47214	+	0.912581i	1.69379	+	2.03250i			1.00000i	−1.65972	+	2.49906i	−0.695544	−	0.401573i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	546.2.z.b	yes	32
3.b	odd	2	1		546.2.z.a	✓	32
7.d	odd	6	1		546.2.z.a	✓	32
21.g	even	6	1	inner	546.2.z.b	yes	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
546.2.z.a	✓	32	3.b	odd	2	1
546.2.z.a	✓	32	7.d	odd	6	1
546.2.z.b	yes	32	1.a	even	1	1	trivial
546.2.z.b	yes	32	21.g	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^32 + 49*T5^30 + 24*T5^29 + 1438*T5^28 + 1028*T5^27 + 28007*T5^26 + 25532*T5^25 + 406986*T5^24 + 411024*T5^23 + 4456192*T5^22 + 4756364*T5^21 + 37936073*T5^20 + 40036844*T5^19 + 247061142*T5^18 + 247871516*T5^17 + 1238254837*T5^16 + 1111389676*T5^15 + 4593154124*T5^14 + 3490824448*T5^13 + 12718654048*T5^12 + 7862722576*T5^11 + 25155570152*T5^10 + 11213425256*T5^9 + 35245764540*T5^8 + 10242858352*T5^7 + 31945718432*T5^6 + 1840921424*T5^5 + 17264364000*T5^4 - 297574912*T5^3 + 5621271680*T5^2 - 1240319456*T5 + 567011344