Newform orbit 531.2.m.a

• Label: 531.2.m.a
• Level: $531$
• Weight: $2$
• Character orbit: 531.m
• Analytic conductor: $4.240$
• Analytic rank: $0$
• Dimension: $3248$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	531.m (of order \(87\), degree \(56\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.24005634733\)
Analytic rank:	\(0\)
Dimension:	\(3248\)
Relative dimension:	\(58\) over \(\Q(\zeta_{87})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{87}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 3248 * q - 29 * q^2 - 58 * q^3 + 29 * q^4 - 25 * q^5 - 66 * q^6 - 27 * q^7 - 116 * q^8 - 62 * q^9 - 116 * q^10 - 33 * q^11 - 58 * q^12 - 27 * q^13 - 33 * q^14 - 51 * q^15 + 29 * q^16 - 126 * q^17 - 78 * q^18 - 120 * q^19 - 31 * q^20 - 60 * q^21 - 29 * q^22 - 29 * q^23 - 16 * q^24 + 29 * q^25 - 92 * q^26 - 55 * q^27 - 132 * q^28 - 27 * q^29 - 64 * q^30 - 21 * q^31 - 49 * q^32 - 32 * q^33 - 17 * q^34 - 136 * q^35 - 70 * q^36 - 120 * q^37 - 33 * q^38 - 80 * q^39 - 17 * q^40 - 17 * q^41 - 68 * q^42 - 21 * q^43 - 104 * q^44 - 58 * q^45 - 116 * q^46 - 17 * q^47 + 106 * q^48 + 19 * q^49 - 5 * q^50 - 250 * q^51 - 21 * q^52 - 108 * q^53 + 175 * q^54 - 116 * q^55 - q^56 + 63 * q^57 - 58 * q^58 - 17 * q^59 - 816 * q^60 - 27 * q^61 - 128 * q^62 + 95 * q^63 - 256 * q^64 - 59 * q^65 + 67 * q^66 - 15 * q^67 - 43 * q^68 - 302 * q^69 - 41 * q^70 - 134 * q^71 + 210 * q^72 - 144 * q^73 - 51 * q^74 - 47 * q^75 - 9 * q^76 - 85 * q^77 + 36 * q^78 - 39 * q^79 - 128 * q^80 - 22 * q^81 - 164 * q^82 - 35 * q^83 - 190 * q^84 - 17 * q^85 + 31 * q^86 - 75 * q^87 - 29 * q^88 - 36 * q^89 - 86 * q^90 - 136 * q^91 - 31 * q^92 - 41 * q^94 - 13 * q^95 - 30 * q^96 - 3 * q^97 - 156 * q^98 - 86 * q^99

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−1.11737	−	2.53423i	0.109412	+	1.72859i	−3.82487	+	4.18674i	2.16943	−	0.910101i	4.25840	−	2.20874i	−1.18066	+	0.171731i	9.63467	+	3.24630i	−2.97606	+	0.378256i	−4.73045	−	4.48092i
4.2	−1.08034	−	2.45026i	0.836206	−	1.51683i	−3.48769	+	3.81766i	−3.50876	+	1.47197i	−4.62001	−	0.410233i	−1.37358	+	0.199792i	8.04683	+	2.71129i	−1.60152	−	2.53676i	7.39737	+	7.00716i
4.3	−1.07888	−	2.44694i	−0.114188	−	1.72828i	−3.47456	+	3.80329i	2.48452	−	1.04229i	−4.10580	+	2.14401i	4.47379	−	0.650731i	7.98656	+	2.69099i	−2.97392	+	0.394700i	−5.23090	−	4.95498i
4.4	−1.03633	−	2.35043i	−1.73198	+	0.0156715i	−3.10161	+	3.39505i	−1.79241	+	0.751936i	1.83173	+	4.05466i	−2.03940	+	0.296638i	6.32554	+	2.13132i	2.99951	−	0.0542854i	3.62489	+	3.43368i
4.5	−0.992901	−	2.25194i	1.08539	+	1.34979i	−2.73643	+	2.99533i	−3.27551	+	1.37412i	1.96197	−	3.78444i	3.11928	−	0.453711i	4.79775	+	1.61655i	−0.643876	+	2.93009i	6.34669	+	6.01190i
4.6	−0.986609	−	2.23767i	−1.69894	+	0.337037i	−2.68482	+	2.93883i	1.19347	−	0.500676i	2.43037	+	3.46915i	1.19400	−	0.173672i	4.58999	+	1.54655i	2.77281	−	1.14521i	−2.29784	−	2.17663i
4.7	−0.967951	−	2.19536i	1.72737	−	0.127256i	−2.53369	+	2.77341i	0.812833	−	0.340993i	−1.95138	−	3.66901i	3.32533	−	0.483682i	3.99375	+	1.34565i	2.96761	−	0.439635i	−1.53538	−	1.45439i
4.8	−0.917418	−	2.08074i	1.04953	−	1.37785i	−2.13887	+	2.34123i	3.27750	−	1.37495i	−3.82982	−	0.919745i	−4.96812	+	0.722633i	2.52379	+	0.850364i	−0.796955	−	2.89221i	−5.86775	−	5.55823i
4.9	−0.914097	−	2.07321i	1.47791	+	0.903206i	−2.11367	+	2.31364i	−0.362328	+	0.152001i	0.521583	−	3.88964i	−3.18656	+	0.463498i	2.43442	+	0.820250i	1.36844	+	2.66971i	0.646333	+	0.612240i
4.10	−0.864970	−	1.96179i	−0.788647	−	1.54209i	−1.75148	+	1.91719i	−0.481587	+	0.202031i	−2.34309	+	2.88102i	−1.07926	+	0.156983i	1.21254	+	0.408551i	−1.75607	+	2.43233i	0.812901	+	0.770021i
4.11	−0.774002	−	1.75547i	−0.188020	+	1.72182i	−1.13364	+	1.24089i	0.0443068	−	0.0185872i	3.16812	−	1.00263i	−0.953357	+	0.138670i	−0.580418	−	0.195566i	−2.92930	−	0.647471i	−0.0669229	−	0.0633927i
4.12	−0.738184	−	1.67423i	−1.32405	−	1.11665i	−0.909179	+	0.995197i	−2.29772	+	0.963923i	−0.892133	+	3.04105i	3.68113	−	0.535435i	−1.13060	−	0.380943i	0.506207	+	2.95698i	3.30997	+	3.13537i
4.13	−0.721710	−	1.63687i	−1.14463	+	1.29993i	−0.809512	+	0.886101i	3.57669	−	1.50047i	2.95390	+	0.935442i	1.52572	−	0.221922i	−1.35587	−	0.456846i	−0.379630	−	2.97588i	−5.03740	−	4.77168i
4.14	−0.713364	−	1.61794i	1.54698	+	0.779016i	−0.759884	+	0.831777i	3.58957	−	1.50587i	0.156844	−	3.05864i	0.869170	−	0.126424i	−1.46349	−	0.493107i	1.78627	+	2.41024i	−4.99708	−	4.73348i
4.15	−0.700893	−	1.58966i	−1.19780	+	1.25111i	−0.686795	+	0.751772i	−2.23191	+	0.936314i	2.82836	+	1.02720i	2.79042	−	0.405878i	−1.61631	−	0.544599i	−0.130538	−	2.99716i	3.05275	+	2.89172i
4.16	−0.585309	−	1.32751i	1.25099	−	1.19792i	−0.0707233	+	0.0774144i	−2.01564	+	0.845586i	−2.32246	−	0.959547i	3.08503	−	0.448730i	−2.60557	−	0.877919i	0.129969	−	2.99718i	2.30229	+	2.18085i
4.17	−0.584975	−	1.32675i	1.38185	−	1.04426i	−0.0691071	+	0.0756453i	1.01737	−	0.426799i	−2.19382	−	1.22251i	1.15974	−	0.168689i	−2.60738	−	0.878528i	0.819038	−	2.88603i	−1.16139	−	1.10013i
4.18	−0.548422	−	1.24385i	1.72799	+	0.118560i	0.102576	−	0.112281i	−2.33072	+	0.977763i	−0.800197	−	2.21437i	−3.60762	+	0.524743i	−2.77236	−	0.934117i	2.97189	+	0.409740i	2.49440	+	2.36282i
4.19	−0.478377	−	1.08498i	−1.62569	+	0.597611i	0.400626	−	0.438530i	0.172397	−	0.0723227i	1.42609	+	1.47795i	−4.30271	+	0.625846i	−2.91482	−	0.982117i	2.28572	−	1.94306i	−0.160939	−	0.152450i
4.20	−0.470938	−	1.06811i	−1.40021	−	1.01952i	0.429888	−	0.470560i	2.30323	−	0.966231i	−0.429549	+	1.97570i	−1.82284	+	0.265139i	−2.91749	−	0.983017i	0.921150	+	2.85508i	−2.11672	−	2.00506i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
59.c	even	29	1	inner
531.m	even	87	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	531.2.m.a	✓	3248
9.c	even	3	1	inner	531.2.m.a	✓	3248
59.c	even	29	1	inner	531.2.m.a	✓	3248
531.m	even	87	1	inner	531.2.m.a	✓	3248

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
531.2.m.a	✓	3248	1.a	even	1	1	trivial
531.2.m.a	✓	3248	9.c	even	3	1	inner
531.2.m.a	✓	3248	59.c	even	29	1	inner
531.2.m.a	✓	3248	531.m	even	87	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(531, [\chi])\).