Newform orbit 40.4.k.a

• Label: 40.4.k.a
• Level: $40$
• Weight: $4$
• Character orbit: 40.k
• Analytic conductor: $2.360$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	40.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.36007640023\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} - 4 q^{3} - 16 q^{6} - 44 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−2.76767	+	0.583098i	6.15076	+	6.15076i	7.31999	−	3.22764i	−11.1441	−	0.899439i	−20.6098	−	13.4368i	16.5614	+	16.5614i	−18.3773	+	13.2013i			48.6638i	31.3677	−	4.00875i
3.2	−2.76621	−	0.589963i	−6.56128	−	6.56128i	7.30389	+	3.26393i	−2.89463	+	10.7991i	14.2790	+	22.0208i	8.83176	+	8.83176i	−18.2785	−	13.3377i			59.1008i	14.3783	−	28.1650i
3.3	−2.64731	−	0.995877i	−0.102537	−	0.102537i	6.01646	+	5.27278i	2.04880	−	10.9910i	0.169333	+	0.373562i	−15.5472	−	15.5472i	−10.6764	−	19.9503i		−	26.9790i	−16.3695	+	27.0562i
3.4	−2.27157	+	1.68522i	−1.56085	−	1.56085i	2.32005	−	7.65620i	10.5634	−	3.66270i	6.17594	+	0.915194i	18.5221	+	18.5221i	7.63226	+	21.3014i		−	22.1275i	−17.8229	+	26.1217i
3.5	−1.71514	−	2.24907i	3.49003	+	3.49003i	−2.11662	+	7.71491i	4.99441	+	10.0028i	1.86344	−	13.8352i	4.97302	+	4.97302i	20.9817	−	8.47169i		−	2.63942i	13.9309	−	28.3889i
3.6	−1.68522	+	2.27157i	−1.56085	−	1.56085i	−2.32005	−	7.65620i	−10.5634	+	3.66270i	6.17594	−	0.915194i	−18.5221	−	18.5221i	21.3014	+	7.63226i		−	22.1275i	9.48156	−	30.1679i
3.7	−0.607409	−	2.76244i	−2.02737	−	2.02737i	−7.26211	+	3.35586i	−10.8122	−	2.84559i	−4.36904	+	6.83193i	1.63237	+	1.63237i	13.6814	+	18.0227i		−	18.7795i	−1.29335	+	31.5963i
3.8	−0.583098	+	2.76767i	6.15076	+	6.15076i	−7.31999	−	3.22764i	11.1441	+	0.899439i	−20.6098	+	13.4368i	−16.5614	−	16.5614i	13.2013	−	18.3773i			48.6638i	−8.98745	+	30.3187i
3.9	0.589963	+	2.76621i	−6.56128	−	6.56128i	−7.30389	+	3.26393i	2.89463	−	10.7991i	14.2790	−	22.0208i	−8.83176	−	8.83176i	−13.3377	−	18.2785i			59.1008i	31.5804	+	1.63609i
3.10	0.995877	+	2.64731i	−0.102537	−	0.102537i	−6.01646	+	5.27278i	−2.04880	+	10.9910i	0.169333	−	0.373562i	15.5472	+	15.5472i	−19.9503	−	10.6764i		−	26.9790i	−31.1369	+	5.52190i
3.11	1.11752	−	2.59830i	4.00710	+	4.00710i	−5.50228	−	5.80731i	6.85881	−	8.82931i	14.8897	−	5.93361i	10.5258	+	10.5258i	−21.2380	+	7.80676i			5.11376i	−15.2763	−	27.6882i
3.12	1.14348	−	2.58698i	−4.39586	−	4.39586i	−5.38489	−	5.91633i	8.23152	+	7.56585i	−16.3986	+	6.34539i	−18.5943	−	18.5943i	−21.4629	+	7.16538i			11.6471i	28.9853	−	12.6433i
3.13	2.24907	+	1.71514i	3.49003	+	3.49003i	2.11662	+	7.71491i	−4.99441	−	10.0028i	1.86344	+	13.8352i	−4.97302	−	4.97302i	−8.47169	+	20.9817i		−	2.63942i	5.92338	−	31.0631i
3.14	2.58698	−	1.14348i	−4.39586	−	4.39586i	5.38489	−	5.91633i	−8.23152	−	7.56585i	−16.3986	−	6.34539i	18.5943	+	18.5943i	7.16538	−	21.4629i			11.6471i	−29.9462	−	10.1601i
3.15	2.59830	−	1.11752i	4.00710	+	4.00710i	5.50228	−	5.80731i	−6.85881	+	8.82931i	14.8897	+	5.93361i	−10.5258	−	10.5258i	7.80676	−	21.2380i			5.11376i	−7.95425	+	30.6060i
3.16	2.76244	+	0.607409i	−2.02737	−	2.02737i	7.26211	+	3.35586i	10.8122	+	2.84559i	−4.36904	−	6.83193i	−1.63237	−	1.63237i	18.0227	+	13.6814i		−	18.7795i	28.1395	+	14.4281i
27.1	−2.76767	−	0.583098i	6.15076	−	6.15076i	7.31999	+	3.22764i	−11.1441	+	0.899439i	−20.6098	+	13.4368i	16.5614	−	16.5614i	−18.3773	−	13.2013i		−	48.6638i	31.3677	+	4.00875i
27.2	−2.76621	+	0.589963i	−6.56128	+	6.56128i	7.30389	−	3.26393i	−2.89463	−	10.7991i	14.2790	−	22.0208i	8.83176	−	8.83176i	−18.2785	+	13.3377i		−	59.1008i	14.3783	+	28.1650i
27.3	−2.64731	+	0.995877i	−0.102537	+	0.102537i	6.01646	−	5.27278i	2.04880	+	10.9910i	0.169333	−	0.373562i	−15.5472	+	15.5472i	−10.6764	+	19.9503i			26.9790i	−16.3695	−	27.0562i
27.4	−2.27157	−	1.68522i	−1.56085	+	1.56085i	2.32005	+	7.65620i	10.5634	+	3.66270i	6.17594	−	0.915194i	18.5221	−	18.5221i	7.63226	−	21.3014i			22.1275i	−17.8229	−	26.1217i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
8.d	odd	2	1	inner
40.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	40.4.k.a	✓	32
4.b	odd	2	1		160.4.o.a		32
5.b	even	2	1		200.4.k.j		32
5.c	odd	4	1	inner	40.4.k.a	✓	32
5.c	odd	4	1		200.4.k.j		32
8.b	even	2	1		160.4.o.a		32
8.d	odd	2	1	inner	40.4.k.a	✓	32
20.e	even	4	1		160.4.o.a		32
40.e	odd	2	1		200.4.k.j		32
40.i	odd	4	1		160.4.o.a		32
40.k	even	4	1	inner	40.4.k.a	✓	32
40.k	even	4	1		200.4.k.j		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.4.k.a	✓	32	1.a	even	1	1	trivial
40.4.k.a	✓	32	5.c	odd	4	1	inner
40.4.k.a	✓	32	8.d	odd	2	1	inner
40.4.k.a	✓	32	40.k	even	4	1	inner
160.4.o.a		32	4.b	odd	2	1
160.4.o.a		32	8.b	even	2	1
160.4.o.a		32	20.e	even	4	1
160.4.o.a		32	40.i	odd	4	1
200.4.k.j		32	5.b	even	2	1
200.4.k.j		32	5.c	odd	4	1
200.4.k.j		32	40.e	odd	2	1
200.4.k.j		32	40.k	even	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(40, [\chi])\).