Newform orbit 31.3.h.a

• Label: 31.3.h.a
• Level: $31$
• Weight: $3$
• Character orbit: 31.h
• Analytic conductor: $0.845$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	31.h (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.844688819517\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 6 q^{2} - 10 q^{3} - 18 q^{4} - 7 q^{5} - 9 q^{6} - 22 q^{7} + 43 q^{8} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−2.11749	+	1.53844i	3.04553	+	0.320098i	0.880871	−	2.71104i	−3.92081	+	6.79105i	−6.94133	+	4.00758i	8.43740	−	1.79342i	−0.929678	−	2.86126i	0.369475	+	0.0785343i	−2.14538	−	20.4119i
3.2	−1.18905	+	0.863898i	−5.38491	−	0.565977i	−0.568540	+	1.74979i	−1.12075	+	1.94120i	6.89190	−	3.97904i	0.108231	−	0.0230052i	−2.65232	−	8.16301i	19.8736	+	4.22427i	−0.344364	−	3.27641i
3.3	−0.0969862	+	0.0704646i	2.02977	+	0.213337i	−1.23163	+	3.79056i	3.06856	−	5.31490i	−0.211892	+	0.122336i	−3.98697	+	0.847457i	−0.295831	−	0.910475i	−4.72888	−	1.00516i	0.0769045	+	0.731698i
3.4	2.09451	−	1.52175i	−1.49940	−	0.157594i	0.835177	−	2.57041i	−1.26763	+	2.19560i	−3.38033	+	1.95163i	−0.0160743	+	0.00341670i	1.03789	+	3.19430i	−6.57995	−	1.39861i	0.686092	+	6.52773i
11.1	−1.15629	−	3.55870i	2.16404	−	1.94851i	−8.09128	+	5.87866i	3.11288	+	5.39167i	−9.43644	−	5.44813i	−0.873658	−	8.31230i	18.1674	+	13.1994i	−0.0543773	+	0.517365i	15.5879	−	17.3122i
11.2	−0.331437	−	1.02006i	0.293175	−	0.263976i	2.30540	−	1.67497i	−0.793103	−	1.37369i	−0.366440	−	0.211564i	0.503493	+	4.79042i	−5.94352	−	4.31822i	−0.924488	+	8.79591i	−1.13839	+	1.26431i
11.3	0.374220	+	1.15173i	−3.80923	+	3.42985i	2.04962	−	1.48914i	2.70856	+	4.69136i	−5.37576	−	3.10370i	−0.963561	−	9.16768i	6.40098	+	4.65059i	1.80564	−	17.1795i	−4.38959	+	4.87513i
11.4	0.922526	+	2.83924i	0.661033	−	0.595197i	−3.97418	+	2.88741i	−2.59389	−	4.49275i	2.29973	+	1.32775i	−0.323932	−	3.08201i	−2.20353	−	1.60096i	−0.858051	+	8.16381i	10.3631	−	11.5094i
12.1	−2.74183	−	1.99206i	−1.97791	+	4.44246i	2.31329	+	7.11957i	0.372590	−	0.645345i	14.2727	−	8.24037i	−8.14446	+	9.04534i	3.65080	−	11.2360i	−9.80113	−	10.8853i	−2.30715	+	1.02721i
12.2	−1.30250	−	0.946319i	0.680823	−	1.52915i	−0.435091	−	1.33907i	0.969949	−	1.68000i	−2.33384	+	1.34744i	2.69654	−	2.99481i	−2.69052	+	8.28058i	4.14738	+	4.60614i	−2.85317	+	1.27031i
12.3	1.32183	+	0.960366i	1.06787	−	2.39847i	−0.411134	−	1.26534i	−2.97289	+	5.14920i	3.71495	−	2.14482i	−7.48191	+	8.30951i	2.69132	−	8.28303i	1.40987	+	1.56582i	−8.87478	+	3.95131i
12.4	1.41348	+	1.02696i	−1.57980	+	3.54828i	−0.292772	−	0.901060i	1.44394	−	2.50098i	−5.87694	+	3.39305i	2.88725	−	3.20662i	2.67113	−	8.22089i	−4.07237	−	4.52282i	4.60937	−	2.05223i
13.1	−2.74183	+	1.99206i	−1.97791	−	4.44246i	2.31329	−	7.11957i	0.372590	+	0.645345i	14.2727	+	8.24037i	−8.14446	−	9.04534i	3.65080	+	11.2360i	−9.80113	+	10.8853i	−2.30715	−	1.02721i
13.2	−1.30250	+	0.946319i	0.680823	+	1.52915i	−0.435091	+	1.33907i	0.969949	+	1.68000i	−2.33384	−	1.34744i	2.69654	+	2.99481i	−2.69052	−	8.28058i	4.14738	−	4.60614i	−2.85317	−	1.27031i
13.3	1.32183	−	0.960366i	1.06787	+	2.39847i	−0.411134	+	1.26534i	−2.97289	−	5.14920i	3.71495	+	2.14482i	−7.48191	−	8.30951i	2.69132	+	8.28303i	1.40987	−	1.56582i	−8.87478	−	3.95131i
13.4	1.41348	−	1.02696i	−1.57980	−	3.54828i	−0.292772	+	0.901060i	1.44394	+	2.50098i	−5.87694	−	3.39305i	2.88725	+	3.20662i	2.67113	+	8.22089i	−4.07237	+	4.52282i	4.60937	+	2.05223i
17.1	−1.15629	+	3.55870i	2.16404	+	1.94851i	−8.09128	−	5.87866i	3.11288	−	5.39167i	−9.43644	+	5.44813i	−0.873658	+	8.31230i	18.1674	−	13.1994i	−0.0543773	−	0.517365i	15.5879	+	17.3122i
17.2	−0.331437	+	1.02006i	0.293175	+	0.263976i	2.30540	+	1.67497i	−0.793103	+	1.37369i	−0.366440	+	0.211564i	0.503493	−	4.79042i	−5.94352	+	4.31822i	−0.924488	−	8.79591i	−1.13839	−	1.26431i
17.3	0.374220	−	1.15173i	−3.80923	−	3.42985i	2.04962	+	1.48914i	2.70856	−	4.69136i	−5.37576	+	3.10370i	−0.963561	+	9.16768i	6.40098	−	4.65059i	1.80564	+	17.1795i	−4.38959	−	4.87513i
17.4	0.922526	−	2.83924i	0.661033	+	0.595197i	−3.97418	−	2.88741i	−2.59389	+	4.49275i	2.29973	−	1.32775i	−0.323932	+	3.08201i	−2.20353	+	1.60096i	−0.858051	−	8.16381i	10.3631	+	11.5094i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.h	odd	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	31.3.h.a	✓	32
3.b	odd	2	1		279.3.bc.b		32
31.h	odd	30	1	inner	31.3.h.a	✓	32
93.p	even	30	1		279.3.bc.b		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
31.3.h.a	✓	32	1.a	even	1	1	trivial
31.3.h.a	✓	32	31.h	odd	30	1	inner
279.3.bc.b		32	3.b	odd	2	1
279.3.bc.b		32	93.p	even	30	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(31, [\chi])\).