Newform orbit 228.2.t.b

• Label: 228.2.t.b
• Level: $228$
• Weight: $2$
• Character orbit: 228.t
• Analytic conductor: $1.821$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	228.t (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.82058916609\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{3} - 3 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1		0		−1.65994	−	0.494567i		0		−0.501341	+	1.37742i		0		1.25668	−	2.17663i		0		2.51081	+	1.64190i		0
29.2		0		−1.25378	+	1.19500i		0		−0.397461	+	1.09202i		0		−0.585706	+	1.01447i		0		0.143928	−	2.99655i		0
29.3		0		−0.198807	−	1.72060i		0		0.501341	−	1.37742i		0		1.25668	−	2.17663i		0		−2.92095	+	0.684137i		0
29.4		0		0.215647	+	1.71857i		0		1.37975	−	3.79084i		0		1.97446	−	3.41987i		0		−2.90699	+	0.741212i		0
29.5		0		1.39457	−	1.02722i		0		0.397461	−	1.09202i		0		−0.585706	+	1.01447i		0		0.889630	−	2.86506i		0
29.6		0		1.65502	+	0.510798i		0		−1.37975	+	3.79084i		0		1.97446	−	3.41987i		0		2.47817	+	1.69076i		0
41.1		0		−1.64371	−	0.546081i		0		−2.59651	−	0.457835i		0		1.67098	+	2.89423i		0		2.40359	+	1.79520i		0
41.2		0		−1.50897	−	0.850305i		0		3.01265	+	0.531212i		0		−0.906392	−	1.56992i		0		1.55396	+	2.56616i		0
41.3		0		−1.09943	+	1.33838i		0		−2.09036	−	0.368588i		0		−2.05158	−	3.55344i		0		−0.582527	−	2.94290i		0
41.4		0		0.609371	−	1.62132i		0		−3.01265	−	0.531212i		0		−0.906392	−	1.56992i		0		−2.25734	−	1.97597i		0
41.5		0		0.908144	−	1.47488i		0		2.59651	+	0.457835i		0		1.67098	+	2.89423i		0		−1.35055	−	2.67881i		0
41.6		0		1.70250	+	0.318562i		0		2.09036	+	0.368588i		0		−2.05158	−	3.55344i		0		2.79704	+	1.08471i		0
53.1		0		−0.970722	−	1.43447i		0		2.28858	+	2.72742i		0		−2.32082	+	4.01977i		0		−1.11540	+	2.78494i		0
53.2		0		−0.421563	+	1.67997i		0		−2.28858	−	2.72742i		0		−2.32082	+	4.01977i		0		−2.64457	−	1.41642i		0
53.3		0		−0.0630278	−	1.73090i		0		−1.46573	−	1.74678i		0		0.541342	−	0.937633i		0		−2.99205	+	0.218190i		0
53.4		0		0.532777	+	1.64807i		0		1.46573	+	1.74678i		0		0.541342	−	0.937633i		0		−2.43230	+	1.75611i		0
53.5		0		1.57535	−	0.719908i		0		1.76450	+	2.10285i		0		0.421032	−	0.729250i		0		1.96347	−	2.26822i		0
53.6		0		1.72657	+	0.137690i		0		−1.76450	−	2.10285i		0		0.421032	−	0.729250i		0		2.96208	+	0.475463i		0
89.1		0		−1.64371	+	0.546081i		0		−2.59651	+	0.457835i		0		1.67098	−	2.89423i		0		2.40359	−	1.79520i		0
89.2		0		−1.50897	+	0.850305i		0		3.01265	−	0.531212i		0		−0.906392	+	1.56992i		0		1.55396	−	2.56616i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
19.f	odd	18	1	inner
57.j	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	228.2.t.b	✓	36
3.b	odd	2	1	inner	228.2.t.b	✓	36
4.b	odd	2	1		912.2.cc.f		36
12.b	even	2	1		912.2.cc.f		36
19.f	odd	18	1	inner	228.2.t.b	✓	36
57.j	even	18	1	inner	228.2.t.b	✓	36
76.k	even	18	1		912.2.cc.f		36
228.u	odd	18	1		912.2.cc.f		36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
228.2.t.b	✓	36	1.a	even	1	1	trivial
228.2.t.b	✓	36	3.b	odd	2	1	inner
228.2.t.b	✓	36	19.f	odd	18	1	inner
228.2.t.b	✓	36	57.j	even	18	1	inner
912.2.cc.f		36	4.b	odd	2	1
912.2.cc.f		36	12.b	even	2	1
912.2.cc.f		36	76.k	even	18	1
912.2.cc.f		36	228.u	odd	18	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^36 + 63*T5^32 - 2725*T5^30 + 22644*T5^28 - 485631*T5^26 + 9009903*T5^24 - 79447896*T5^22 + 637003944*T5^20 - 4966223668*T5^18 + 22906721493*T5^16 - 69682884084*T5^14 + 300435334876*T5^12 - 345629070909*T5^10 - 2548114323957*T5^8 - 2809503188118*T5^6 + 27112841437095*T5^4 + 56408908857471*T5^2 + 47257966560249