Newform orbit 112.3.u.a

• Label: 112.3.u.a
• Level: $112$
• Weight: $3$
• Character orbit: 112.u
• 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.u (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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 8 q^{6} - 8 q^{7} - 20 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.99935	−	0.0508820i	−5.13198	+	1.37511i	3.99482	+	0.203462i	−1.31197	+	4.89635i	10.3306	−	2.48820i	−3.60925	+	5.99778i	−7.97671	−	0.610057i	16.6520	−	9.61406i	2.87223	−	9.72278i
11.2	−1.98925	−	0.207115i	0.671463	−	0.179918i	3.91421	+	0.824006i	−2.01821	+	7.53205i	−1.37297	+	0.218831i	−0.893673	−	6.94272i	−7.61566	−	2.44984i	−7.37574	+	4.25838i	5.57471	−	14.5651i
11.3	−1.94677	−	0.458349i	3.59906	−	0.964366i	3.57983	+	1.78460i	1.00883	−	3.76502i	−7.44857	+	0.227771i	6.40989	+	2.81305i	−6.15114	−	5.11503i	4.22903	−	2.44163i	−3.68966	+	6.86723i
11.4	−1.89767	+	0.631531i	1.02942	−	0.275832i	3.20234	−	2.39688i	1.57827	−	5.89018i	−1.77931	+	1.17355i	−6.73075	−	1.92277i	−4.56329	+	6.57087i	−6.81061	+	3.93211i	0.724787	+	12.1744i
11.5	−1.70541	+	1.04479i	−1.43258	+	0.383858i	1.81682	−	3.56359i	0.0576133	−	0.215016i	2.04208	−	2.15138i	5.59899	+	4.20134i	0.624777	+	7.97557i	−5.88929	+	3.40019i	0.126392	+	0.426883i
11.6	−1.67479	+	1.09319i	5.38079	−	1.44178i	1.60987	−	3.66174i	−1.88307	+	7.02770i	−7.43558	+	8.29692i	−2.91080	+	6.36610i	1.30679	+	7.89255i	19.0800	−	11.0158i	−4.52887	−	13.8285i
11.7	−1.62281	−	1.16896i	−2.31259	+	0.619658i	1.26705	+	3.79402i	1.68733	−	6.29719i	4.47727	+	1.69775i	−6.16117	+	3.32264i	2.37888	−	7.63812i	−2.83012	+	1.63397i	−10.0994	+	8.24675i
11.8	−1.53869	−	1.27767i	−2.43724	+	0.653056i	0.735142	+	3.93187i	−0.170615	+	0.636744i	4.58454	+	2.10912i	5.85108	−	3.84251i	3.89245	−	6.98919i	−2.28058	+	1.31669i	1.07607	−	0.761763i
11.9	−1.23811	+	1.57069i	−4.35038	+	1.16568i	−0.934158	−	3.88939i	−0.146939	+	0.548383i	3.55533	−	8.27636i	−2.42915	−	6.56500i	7.26563	+	3.34822i	9.77276	−	5.64231i	−0.679415	−	0.909755i
11.10	−1.02247	−	1.71888i	4.76800	−	1.27758i	−1.90910	+	3.51502i	0.266998	−	0.996452i	−7.07117	−	6.88933i	−2.61686	−	6.49246i	7.99389	−	0.312508i	13.3074	−	7.68304i	−1.98578	+	0.559907i
11.11	−0.807465	−	1.82975i	0.976233	−	0.261581i	−2.69600	+	2.95492i	−1.58159	+	5.90256i	−1.26690	−	1.57505i	0.924410	+	6.93869i	7.58371	+	2.54702i	−6.90962	+	3.98927i	12.0773	−	1.87220i
11.12	−0.784457	+	1.83974i	3.71739	−	0.996071i	−2.76926	−	2.88639i	1.04529	−	3.90108i	−1.08362	+	7.62039i	3.44029	−	6.09626i	7.48255	−	2.83045i	5.03259	−	2.90557i	6.35698	+	4.98329i
11.13	−0.435941	+	1.95191i	0.233233	−	0.0624946i	−3.61991	−	1.70184i	−0.881366	+	3.28930i	0.0203081	+	0.482494i	−6.40604	+	2.82182i	4.89990	−	6.32384i	−7.74374	+	4.47085i	−6.03620	−	3.15429i
11.14	−0.120950	+	1.99634i	−3.36995	+	0.902976i	−3.97074	−	0.482915i	2.55646	−	9.54083i	−1.39505	−	6.83678i	4.83456	+	5.06231i	1.44432	−	7.86854i	2.74698	−	1.58597i	18.7375	+	6.25752i
11.15	−0.113149	−	1.99680i	−0.898351	+	0.240712i	−3.97439	+	0.451872i	0.903951	−	3.37359i	0.582302	+	1.76659i	−3.93619	−	5.78847i	1.35200	+	7.88493i	−7.04514	+	4.06751i	−6.83865	−	1.42329i
11.16	−0.0556527	−	1.99923i	−5.31530	+	1.42423i	−3.99381	+	0.222525i	0.419571	−	1.56586i	3.14317	+	10.5472i	6.45786	+	2.70113i	0.667143	+	7.97213i	18.4298	−	10.6404i	−3.15386	−	0.751673i
11.17	0.473013	−	1.94326i	2.97791	−	0.797928i	−3.55252	−	1.83837i	2.01087	−	7.50468i	−0.141992	−	6.16428i	−0.152811	+	6.99833i	−5.25283	+	6.03389i	0.437008	−	0.252307i	−13.6324	−	7.45746i
11.18	0.670600	+	1.88422i	2.17057	−	0.581603i	−3.10059	+	2.52712i	−0.966534	+	3.60715i	2.55145	+	3.69982i	6.81256	+	1.60902i	−6.84091	−	4.14752i	−3.42111	+	1.97518i	−7.44484	+	0.597792i
11.19	0.844749	+	1.81284i	−3.71006	+	0.994107i	−2.57280	+	3.06279i	−0.777420	+	2.90137i	−4.93622	−	5.88598i	0.740638	−	6.96071i	−7.72573	−	2.07679i	4.98204	−	2.87638i	−5.91645	+	1.04159i
11.20	0.878669	−	1.79665i	−2.47639	+	0.663548i	−2.45588	−	3.15731i	−2.00765	+	7.49264i	−0.983768	+	5.03225i	−6.99668	+	0.215480i	−7.83049	+	1.63812i	−2.10199	+	1.21359i	11.6976	+	10.1906i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
16.f	odd	4	1	inner
112.u	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.u.a	✓	120
7.c	even	3	1	inner	112.3.u.a	✓	120
16.f	odd	4	1	inner	112.3.u.a	✓	120
112.u	odd	12	1	inner	112.3.u.a	✓	120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.3.u.a	✓	120	1.a	even	1	1	trivial
112.3.u.a	✓	120	7.c	even	3	1	inner
112.3.u.a	✓	120	16.f	odd	4	1	inner
112.3.u.a	✓	120	112.u	odd	12	1	inner

Hecke kernels

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