Newform orbit 228.2.m.a

• Label: 228.2.m.a
• Level: $228$
• Weight: $2$
• Character orbit: 228.m
• Analytic conductor: $1.821$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	228.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.82058916609\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 3 q^{6}+O(q^{10}) \)

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	−1.41383	−	0.0328571i	−1.37888	+	1.04818i	1.99784	+	0.0929089i	2.70326	+	1.56073i	1.98395	−	1.43664i		−	1.45593i	−2.82156	−	0.197001i	0.802644	−	2.89063i	−3.77067	−	2.29543i
11.2	−1.41288	+	0.0614585i	1.65375	+	0.514883i	1.99245	−	0.173667i	2.46874	+	1.42533i	−2.36819	−	0.625830i			2.23237i	−2.80441	+	0.367822i	2.46979	+	1.70298i	−3.57563	−	1.86209i
11.3	−1.35171	+	0.415800i	−1.70639	+	0.297045i	1.65422	−	1.12408i	−3.23446	−	1.86742i	2.18303	−	1.11104i			2.36238i	−1.76863	+	2.20725i	2.82353	−	1.01375i	5.14852	+	1.17931i
11.4	−1.34509	−	0.436732i	1.04438	−	1.38176i	1.61853	+	1.17489i	−1.24455	−	0.718541i	−2.00825	+	1.40248i		−	1.00140i	−1.66396	−	2.28719i	−0.818536	−	2.88617i	1.36022	+	1.51004i
11.5	−1.33479	+	0.467269i	−0.805059	−	1.53358i	1.56332	−	1.24741i	0.0574154	+	0.0331488i	1.79118	+	1.67083i		−	3.60595i	−1.50383	+	2.39552i	−1.70376	+	2.46925i	−0.0921267	−	0.0174182i
11.6	−1.29423	−	0.570057i	−1.07454	−	1.35844i	1.35007	+	1.47557i	0.531172	+	0.306672i	0.616309	+	2.37069i			3.86636i	−0.906143	−	2.67935i	−0.690742	+	2.91940i	−0.512639	−	0.699703i
11.7	−1.14080	−	0.835809i	1.07454	+	1.35844i	0.602847	+	1.90698i	0.531172	+	0.306672i	−0.0904314	−	2.44782i		−	3.86636i	0.906143	−	2.67935i	−0.690742	+	2.91940i	−0.349641	−	0.793810i
11.8	−1.10220	+	0.886086i	1.71152	+	0.265915i	0.429702	−	1.95329i	−1.36154	−	0.786087i	−2.12206	+	1.22346i		−	1.15667i	1.25717	+	2.53368i	2.85858	+	0.910234i	2.19724	−	0.340017i
11.9	−1.05077	−	0.946515i	−1.04438	+	1.38176i	0.208218	+	1.98913i	−1.24455	−	0.718541i	2.40526	−	0.463386i			1.00140i	1.66396	−	2.28719i	−0.818536	−	2.88617i	0.627620	+	1.93300i
11.10	−0.969770	+	1.02934i	0.00873798	+	1.73203i	−0.119092	−	1.99645i	0.966801	+	0.558183i	−1.79132	−	1.67068i			2.76498i	2.17052	+	1.81351i	−2.99985	+	0.0302688i	−1.51214	+	0.453860i
11.11	−0.751646	+	1.19793i	0.369309	−	1.69222i	−0.870056	−	1.80083i	−2.87456	−	1.65963i	1.74957	+	1.71436i			3.79191i	2.81124	+	0.311327i	−2.72722	−	1.24991i	4.14877	−	2.19606i
11.12	−0.735371	−	1.20799i	1.37888	−	1.04818i	−0.918459	+	1.77664i	2.70326	+	1.56073i	−2.28018	−	0.894872i			1.45593i	2.82156	−	0.197001i	0.802644	−	2.89063i	−0.102562	−	4.41321i
11.13	−0.661612	+	1.24991i	1.28085	−	1.16594i	−1.12454	−	1.65391i	2.87456	+	1.65963i	0.609894	+	2.37235i		−	3.79191i	2.81124	−	0.311327i	0.281161	−	2.98680i	−3.97623	+	2.49491i
11.14	−0.653214	−	1.25432i	−1.65375	−	0.514883i	−1.14662	+	1.63868i	2.46874	+	1.42533i	0.434428	+	2.41066i		−	2.23237i	2.80441	+	0.367822i	2.46979	+	1.70298i	0.175197	−	4.02763i
11.15	−0.406552	+	1.35452i	−1.50435	+	0.858447i	−1.66943	−	1.10136i	−0.966801	−	0.558183i	−0.551185	−	2.38667i		−	2.76498i	2.17052	−	1.81351i	1.52614	−	2.58281i	1.14912	−	1.08262i
11.16	−0.315760	−	1.37851i	1.70639	−	0.297045i	−1.80059	+	0.870557i	−3.23446	−	1.86742i	−0.948290	−	2.25848i		−	2.36238i	1.76863	+	2.20725i	2.82353	−	1.01375i	−1.55295	+	5.04840i
11.17	−0.262728	−	1.38959i	0.805059	+	1.53358i	−1.86195	+	0.730170i	0.0574154	+	0.0331488i	1.91955	−	1.52162i			3.60595i	1.50383	+	2.39552i	−1.70376	+	2.46925i	0.0309788	−	0.0884932i
11.18	−0.216272	+	1.39758i	−1.08605	−	1.34926i	−1.90645	−	0.604514i	1.36154	+	0.786087i	2.12058	−	1.22603i			1.15667i	1.25717	−	2.53368i	−0.641003	+	2.93072i	−1.39308	+	1.73286i
11.19	0.216272	−	1.39758i	−1.71152	−	0.265915i	−1.90645	−	0.604514i	−1.36154	−	0.786087i	−0.741789	+	2.33447i			1.15667i	−1.25717	+	2.53368i	2.85858	+	0.910234i	−1.39308	+	1.73286i
11.20	0.262728	+	1.38959i	1.73065	−	0.0695904i	−1.86195	+	0.730170i	−0.0574154	−	0.0331488i	0.551393	+	2.38662i			3.60595i	−1.50383	−	2.39552i	2.99031	−	0.240874i	0.0309788	−	0.0884932i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner
19.c	even	3	1	inner
57.h	odd	6	1	inner
76.g	odd	6	1	inner
228.m	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	228.2.m.a	✓	72
3.b	odd	2	1	inner	228.2.m.a	✓	72
4.b	odd	2	1	inner	228.2.m.a	✓	72
12.b	even	2	1	inner	228.2.m.a	✓	72
19.c	even	3	1	inner	228.2.m.a	✓	72
57.h	odd	6	1	inner	228.2.m.a	✓	72
76.g	odd	6	1	inner	228.2.m.a	✓	72
228.m	even	6	1	inner	228.2.m.a	✓	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
228.2.m.a	✓	72	1.a	even	1	1	trivial
228.2.m.a	✓	72	3.b	odd	2	1	inner
228.2.m.a	✓	72	4.b	odd	2	1	inner
228.2.m.a	✓	72	12.b	even	2	1	inner
228.2.m.a	✓	72	19.c	even	3	1	inner
228.2.m.a	✓	72	57.h	odd	6	1	inner
228.2.m.a	✓	72	76.g	odd	6	1	inner
228.2.m.a	✓	72	228.m	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(228, [\chi])\).