Newform orbit 231.2.be.a

• Label: 231.2.be.a
• Level: $231$
• Weight: $2$
• Character orbit: 231.be
• Analytic conductor: $1.845$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	231.be (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.84454428669\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(28\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 3 q^{3} + 18 q^{4} - 20 q^{6} - 20 q^{7} - 9 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−1.69754	−	1.88531i	−1.50120	−	0.863943i	−0.463691	+	4.41173i	0.0606689	−	0.285425i	0.919548	+	4.29680i	0.885230	+	2.49326i	4.99976	−	3.63254i	1.50720	+	2.59390i	−0.641101	+	0.370140i
2.2	−1.65113	−	1.83377i	1.72117	−	0.193857i	−0.427411	+	4.06654i	−0.660086	+	3.10546i	−3.19736	−	2.83614i	−2.64526	−	0.0509795i	4.17018	−	3.02981i	2.92484	−	0.667320i	6.78458	−	3.91708i
2.3	−1.62295	−	1.80247i	0.590856	−	1.62816i	−0.405869	+	3.86159i	0.153087	−	0.720219i	−3.89363	+	1.57742i	1.64166	−	2.07484i	3.69461	−	2.68429i	−2.30178	−	1.92401i	−1.54662	+	0.892944i
2.4	−1.41473	−	1.57122i	−1.63740	+	0.564721i	−0.258206	+	2.45666i	−0.705701	+	3.32006i	3.20379	+	1.77379i	−0.181035	−	2.63955i	0.804266	−	0.584333i	2.36218	−	1.84935i	6.21493	−	3.58819i
2.5	−1.36136	−	1.51195i	0.270437	+	1.71081i	−0.223616	+	2.12757i	−0.0600850	+	0.282678i	2.21849	−	2.73792i	−2.04626	+	1.67715i	0.229262	−	0.166568i	−2.85373	+	0.925333i	0.509191	−	0.293981i
2.6	−1.06973	−	1.18806i	−0.998701	+	1.41513i	−0.0580986	+	0.552771i	0.161799	−	0.761203i	2.74960	−	0.327297i	1.95474	+	1.78298i	−1.86786	+	1.35708i	−1.00519	−	2.82659i	−1.07744	+	0.622058i
2.7	−1.06933	−	1.18762i	1.50618	+	0.855229i	−0.0578990	+	0.550872i	0.711086	−	3.34540i	−0.594929	−	2.70329i	−0.575806	−	2.58233i	−1.86963	+	1.35837i	1.53717	+	2.57626i	−4.73343	+	2.73285i
2.8	−1.03170	−	1.14582i	0.391245	−	1.68728i	−0.0394378	+	0.375225i	0.582427	−	2.74010i	−2.33696	+	1.29247i	−2.49909	+	0.868634i	−2.02413	+	1.47062i	−2.69386	−	1.32028i	−3.74054	+	2.15960i
2.9	−0.830242	−	0.922077i	−0.464723	−	1.66854i	0.0481326	−	0.457951i	−0.859357	+	4.04296i	−1.15269	+	1.81380i	−0.152025	+	2.64138i	−2.46985	+	1.79445i	−2.56807	+	1.55082i	4.44139	−	2.56424i
2.10	−0.675892	−	0.750655i	1.68081	−	0.418192i	0.102405	−	0.974320i	−0.347823	+	1.63638i	−1.44996	−	0.979053i	2.62714	−	0.313255i	−2.43498	+	1.76911i	2.65023	−	1.40580i	1.46344	−	0.844920i
2.11	−0.590583	−	0.655909i	−1.48384	−	0.893427i	0.127629	−	1.21431i	0.627181	−	2.95066i	0.290326	+	1.50091i	2.53863	−	0.745226i	−2.29995	+	1.67101i	1.40358	+	2.65141i	−2.30576	+	1.33123i
2.12	−0.475341	−	0.527920i	−1.72047	+	0.199922i	0.156307	−	1.48716i	−0.0305586	+	0.143767i	0.923355	+	0.813242i	−2.63481	−	0.240370i	−2.00883	+	1.45950i	2.92006	−	0.687921i	0.0904232	−	0.0522059i
2.13	−0.109613	−	0.121738i	−0.460773	+	1.66964i	0.206252	−	1.96236i	0.280949	−	1.32176i	0.253764	−	0.126921i	−0.638360	−	2.56759i	−0.526557	+	0.382566i	−2.57538	−	1.53865i	−0.191704	+	0.110680i
2.14	−0.0572311	−	0.0635616i	1.72188	−	0.187403i	0.208292	−	1.98177i	0.496749	−	2.33702i	−0.110457	−	0.0987204i	−0.857429	+	2.50296i	−0.276277	+	0.200727i	2.92976	−	0.645370i	−0.176974	+	0.102176i
2.15	0.0572311	+	0.0635616i	1.01290	+	1.40501i	0.208292	−	1.98177i	−0.496749	+	2.33702i	−0.0313351	+	0.144791i	−0.857429	+	2.50296i	0.276277	−	0.200727i	−0.948078	+	2.84625i	−0.176974	+	0.102176i
2.16	0.109613	+	0.121738i	0.932465	−	1.45963i	0.206252	−	1.96236i	−0.280949	+	1.32176i	0.279902	−	0.0464781i	−0.638360	−	2.56759i	0.526557	−	0.382566i	−1.26102	−	2.72210i	−0.191704	+	0.110680i
2.17	0.475341	+	0.527920i	−1.00265	−	1.41234i	0.156307	−	1.48716i	0.0305586	−	0.143767i	0.268999	−	1.20066i	−2.63481	−	0.240370i	2.00883	−	1.45950i	−0.989382	+	2.83216i	0.0904232	−	0.0522059i
2.18	0.590583	+	0.655909i	−1.65683	−	0.504891i	0.127629	−	1.21431i	−0.627181	+	2.95066i	−0.647333	−	1.38491i	2.53863	−	0.745226i	2.29995	−	1.67101i	2.49017	+	1.67304i	−2.30576	+	1.33123i
2.19	0.675892	+	0.750655i	0.813902	+	1.52891i	0.102405	−	0.974320i	0.347823	−	1.63638i	−0.597572	+	1.64434i	2.62714	−	0.313255i	2.43498	−	1.76911i	−1.67513	+	2.48877i	1.46344	−	0.844920i
2.20	0.830242	+	0.922077i	−1.55093	+	0.771117i	0.0481326	−	0.457951i	0.859357	−	4.04296i	−1.99867	−	0.789862i	−0.152025	+	2.64138i	2.46985	−	1.79445i	1.81076	−	2.39189i	4.44139	−	2.56424i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
11.d	odd	10	1	inner
21.h	odd	6	1	inner
33.f	even	10	1	inner
77.o	odd	30	1	inner
231.be	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	231.2.be.a	✓	224
3.b	odd	2	1	inner	231.2.be.a	✓	224
7.c	even	3	1	inner	231.2.be.a	✓	224
11.d	odd	10	1	inner	231.2.be.a	✓	224
21.h	odd	6	1	inner	231.2.be.a	✓	224
33.f	even	10	1	inner	231.2.be.a	✓	224
77.o	odd	30	1	inner	231.2.be.a	✓	224
231.be	even	30	1	inner	231.2.be.a	✓	224

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
231.2.be.a	✓	224	1.a	even	1	1	trivial
231.2.be.a	✓	224	3.b	odd	2	1	inner
231.2.be.a	✓	224	7.c	even	3	1	inner
231.2.be.a	✓	224	11.d	odd	10	1	inner
231.2.be.a	✓	224	21.h	odd	6	1	inner
231.2.be.a	✓	224	33.f	even	10	1	inner
231.2.be.a	✓	224	77.o	odd	30	1	inner
231.2.be.a	✓	224	231.be	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(231, [\chi])\).