Newform orbit 112.3.x.a

• Label: 112.3.x.a
• Level: $112$
• Weight: $3$
• Character orbit: 112.x
• Analytic conductor: $3.052$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	112.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.05177896084\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 2 q^{2} - 6 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1.98319	−	0.258755i	0.689002	−	2.57139i	3.86609	+	1.02632i	1.56109	+	5.82605i	−2.03178	+	4.92128i	−1.48039	+	6.84167i	−7.40163	−	3.03576i	1.65690	+	0.956613i	−1.58841	−	11.9581i
5.2	−1.94030	+	0.485027i	−0.457800	+	1.70853i	3.52950	−	1.88219i	−0.833659	−	3.11126i	0.0595827	−	3.53711i	−3.27884	−	6.18459i	−5.93536	+	5.36391i	5.08472	+	2.93567i	3.12659	+	5.63241i
5.3	−1.92608	+	0.538713i	1.24156	−	4.63355i	3.41958	−	2.07521i	−0.510315	−	1.90452i	0.104814	+	9.59344i	5.79810	−	3.92199i	−5.46844	+	5.83919i	−12.1341	−	7.00562i	2.00890	+	3.39335i
5.4	−1.90188	−	0.618737i	−0.757283	+	2.82622i	3.23433	+	2.35353i	−1.75241	−	6.54008i	3.18895	−	4.90658i	6.29413	+	3.06332i	−4.69511	−	6.47734i	0.380191	+	0.219503i	−0.713707	+	13.5228i
5.5	−1.79253	+	0.887048i	−0.984570	+	3.67446i	2.42629	−	3.18011i	0.864860	+	3.22770i	−1.49456	−	7.45993i	1.08705	+	6.91508i	−1.52828	+	7.85267i	−4.73809	−	2.73553i	−4.41341	−	5.01856i
5.6	−1.71910	−	1.02211i	0.522439	−	1.94977i	1.91058	+	3.51421i	−1.25134	−	4.67006i	−2.89100	+	2.81785i	−6.75218	−	1.84608i	0.307440	−	7.99409i	4.26557	+	2.46273i	−2.62215	+	9.30729i
5.7	−1.70863	−	1.03951i	−1.37623	+	5.13616i	1.83882	+	3.55229i	1.73653	+	6.48081i	7.69057	−	7.34517i	−5.29115	−	4.58299i	0.550798	−	7.98102i	−16.6919	−	9.63706i	3.76982	−	12.8784i
5.8	−1.34321	+	1.48182i	0.222844	−	0.831664i	−0.391587	−	3.98079i	1.94682	+	7.26563i	0.933051	+	1.44731i	−3.76184	−	5.90327i	6.42480	+	4.76676i	7.15222	+	4.12934i	−13.3813	−	6.87441i
5.9	−1.20109	−	1.59918i	0.224380	−	0.837397i	−1.11478	+	3.84152i	1.12240	+	4.18885i	−1.60865	+	0.646962i	6.33828	−	2.97090i	7.48224	−	2.83126i	7.14334	+	4.12421i	5.35064	−	6.82609i
5.10	−0.956977	+	1.75619i	0.103766	−	0.387260i	−2.16839	−	3.36126i	−0.758420	−	2.83046i	0.580800	+	0.552832i	6.87124	+	1.33643i	7.97811	−	0.591454i	7.65503	+	4.41963i	5.69661	+	1.37676i
5.11	−0.700351	+	1.87337i	1.39906	−	5.22137i	−3.01902	−	2.62403i	0.103531	+	0.386384i	8.80171	+	6.27775i	−6.54042	+	2.49457i	7.03015	−	3.81799i	−17.5111	−	10.1100i	−0.796348	−	0.0766523i
5.12	−0.679338	−	1.88109i	1.28128	−	4.78182i	−3.07700	+	2.55579i	−1.82352	−	6.80549i	−9.86545	+	0.838261i	4.05845	+	5.70342i	6.89800	+	4.05186i	−13.4299	−	7.75373i	−11.5629	+	8.05344i
5.13	−0.582774	+	1.91321i	−1.07649	+	4.01752i	−3.32075	−	2.22994i	−1.75031	−	6.53224i	−7.05901	−	4.40086i	−6.99800	−	0.167335i	6.20158	−	5.05375i	−7.18741	−	4.14965i	13.5176	+	0.458108i
5.14	−0.421138	−	1.95516i	−0.469120	+	1.75078i	−3.64529	+	1.64678i	0.268604	+	1.00244i	3.62062	+	0.179884i	−5.48036	+	4.35495i	4.75489	+	6.43358i	4.94907	+	2.85735i	1.84682	−	0.947330i
5.15	−0.251918	−	1.98407i	−1.24845	+	4.65927i	−3.87307	+	0.999646i	−1.86718	−	6.96841i	9.55883	+	1.30325i	3.79175	−	5.88410i	2.95907	+	7.43263i	−12.3559	−	7.13371i	−13.3554	+	5.46009i
5.16	0.114355	+	1.99673i	−1.13521	+	4.23667i	−3.97385	+	0.456672i	1.60564	+	5.99232i	−8.58929	−	1.78222i	6.30904	−	3.03249i	−1.36628	−	7.88247i	−8.86641	−	5.11903i	−11.7814	+	3.89127i
5.17	0.501909	+	1.93600i	0.357687	−	1.33491i	−3.49618	+	1.94339i	0.832839	+	3.10820i	2.76390	+	0.0224802i	−0.367625	+	6.99034i	−5.51716	−	5.79318i	6.14019	+	3.54504i	−5.59945	+	3.17241i
5.18	0.528747	−	1.92884i	0.783429	−	2.92380i	−3.44085	−	2.03974i	0.306093	+	1.14236i	−5.22530	−	3.05706i	−3.04257	−	6.30419i	−5.75367	+	5.55835i	−0.140603	−	0.0811772i	2.36527	+	0.0136121i
5.19	0.681440	+	1.88033i	0.764825	−	2.85437i	−3.07128	+	2.56266i	−2.39517	−	8.93889i	5.88833	−	0.506957i	1.06424	−	6.91863i	−6.91155	−	4.02871i	0.231779	+	0.133818i	15.1759	−	10.5950i
5.20	0.861013	−	1.80518i	−0.787729	+	2.93985i	−2.51731	−	3.10856i	1.63922	+	6.11765i	4.62869	+	3.95323i	6.29709	+	3.05723i	−7.77893	+	1.86769i	−0.227949	−	0.131606i	12.4548	+	2.30830i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
16.e	even	4	1	inner
112.x	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	112.3.x.a	✓	120
7.d	odd	6	1	inner	112.3.x.a	✓	120
16.e	even	4	1	inner	112.3.x.a	✓	120
112.x	odd	12	1	inner	112.3.x.a	✓	120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.3.x.a	✓	120	1.a	even	1	1	trivial
112.3.x.a	✓	120	7.d	odd	6	1	inner
112.3.x.a	✓	120	16.e	even	4	1	inner
112.3.x.a	✓	120	112.x	odd	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(112, [\chi])\).