Newform orbit 20.8.e.b

• Label: 20.8.e.b
• Level: $20$
• Weight: $8$
• Character orbit: 20.e
• Analytic conductor: $6.248$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.24770050968\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 14 q^{2} - 560 q^{5} + 104 q^{6} - 3052 q^{8}+O(q^{10}) \)

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	−11.3062	+	0.411421i	−2.83352	+	2.83352i	127.661	−	9.30323i	−150.277	−	235.673i	30.8707	−	33.2022i	411.746	+	411.746i	−1439.54	+	157.707i			2170.94i	1796.03	+	2602.74i
3.2	−11.1445	−	1.94961i	−31.8653	+	31.8653i	120.398	+	43.4547i	92.5871	+	263.728i	417.246	−	292.996i	−789.211	−	789.211i	−1257.05	−	719.009i			156.211i	−517.666	−	3119.62i
3.3	−9.99887	−	5.29363i	51.5668	−	51.5668i	71.9550	+	105.861i	278.010	−	28.9057i	−788.585	+	242.634i	645.384	+	645.384i	−159.082	−	1439.39i		−	3131.26i	−2932.80	−	1182.66i
3.4	−9.35451	+	6.36342i	56.5129	−	56.5129i	47.0138	−	119.053i	−240.823	+	141.877i	−169.035	+	888.266i	−764.897	−	764.897i	317.795	+	1412.85i		−	4200.41i	1349.96	−	2859.65i
3.5	−6.47794	−	9.27558i	−53.6136	+	53.6136i	−44.0727	+	120.173i	17.4122	−	278.966i	844.603	+	149.992i	308.666	+	308.666i	1400.18	−	369.675i		−	3561.84i	−2700.36	+	1645.61i
3.6	−6.36342	+	9.35451i	−56.5129	+	56.5129i	−47.0138	−	119.053i	−240.823	+	141.877i	−169.035	−	888.266i	764.897	+	764.897i	1412.85	+	317.795i		−	4200.41i	205.270	−	3155.61i
3.7	−5.78656	−	9.72192i	13.4355	−	13.4355i	−61.0315	+	112.513i	−222.545	+	169.112i	−208.364	−	52.8736i	−9.12405	−	9.12405i	1447.00	−	57.7198i			1825.97i	2931.86	+	1184.99i
3.8	−0.411421	+	11.3062i	2.83352	−	2.83352i	−127.661	−	9.30323i	−150.277	−	235.673i	30.8707	+	33.2022i	−411.746	−	411.746i	157.707	−	1439.54i			2170.94i	2726.40	−	1602.11i
3.9	0.138397	−	11.3129i	19.3526	−	19.3526i	−127.962	−	3.13132i	191.974	−	203.153i	−216.255	−	221.612i	−939.327	−	939.327i	−53.1337	+	1447.18i			1437.95i	−2271.67	−	2199.89i
3.10	1.94961	+	11.1445i	31.8653	−	31.8653i	−120.398	+	43.4547i	92.5871	+	263.728i	417.246	+	292.996i	789.211	+	789.211i	−719.009	−	1257.05i			156.211i	−2758.60	+	1546.00i
3.11	3.68683	−	10.6961i	−32.6647	+	32.6647i	−100.815	−	78.8696i	145.214	+	238.826i	228.957	+	469.815i	669.751	+	669.751i	−1215.29	+	787.548i			53.0351i	3089.89	−	672.721i
3.12	5.29363	+	9.99887i	−51.5668	+	51.5668i	−71.9550	+	105.861i	278.010	−	28.9057i	−788.585	−	242.634i	−645.384	−	645.384i	−1439.39	−	159.082i		−	3131.26i	1760.71	+	2626.77i
3.13	6.52813	−	9.24032i	41.8246	−	41.8246i	−42.7671	−	120.644i	−251.552	−	121.846i	−113.436	−	659.509i	1013.70	+	1013.70i	−1393.98	−	392.398i		−	1311.59i	−2768.06	+	1528.99i
3.14	9.24032	−	6.52813i	−41.8246	+	41.8246i	42.7671	−	120.644i	−251.552	−	121.846i	−113.436	+	659.509i	−1013.70	−	1013.70i	−392.398	−	1393.98i		−	1311.59i	−3119.85	+	516.264i
3.15	9.27558	+	6.47794i	53.6136	−	53.6136i	44.0727	+	120.173i	17.4122	−	278.966i	844.603	−	149.992i	−308.666	−	308.666i	−369.675	+	1400.18i		−	3561.84i	1968.63	−	2474.77i
3.16	9.72192	+	5.78656i	−13.4355	+	13.4355i	61.0315	+	112.513i	−222.545	+	169.112i	−208.364	+	52.8736i	9.12405	+	9.12405i	−57.7198	+	1447.00i			1825.97i	−3142.14	+	356.325i
3.17	10.6961	−	3.68683i	32.6647	−	32.6647i	100.815	−	78.8696i	145.214	+	238.826i	228.957	−	469.815i	−669.751	−	669.751i	787.548	−	1215.29i			53.0351i	2433.74	+	2019.13i
3.18	11.3129	−	0.138397i	−19.3526	+	19.3526i	127.962	−	3.13132i	191.974	−	203.153i	−216.255	+	221.612i	939.327	+	939.327i	1447.18	−	53.1337i			1437.95i	2143.66	−	2324.81i
7.1	−11.3062	−	0.411421i	−2.83352	−	2.83352i	127.661	+	9.30323i	−150.277	+	235.673i	30.8707	+	33.2022i	411.746	−	411.746i	−1439.54	−	157.707i		−	2170.94i	1796.03	−	2602.74i
7.2	−11.1445	+	1.94961i	−31.8653	−	31.8653i	120.398	−	43.4547i	92.5871	−	263.728i	417.246	+	292.996i	−789.211	+	789.211i	−1257.05	+	719.009i		−	156.211i	−517.666	+	3119.62i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.c	odd	4	1	inner
20.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	20.8.e.b	✓	36
4.b	odd	2	1	inner	20.8.e.b	✓	36
5.b	even	2	1		100.8.e.e		36
5.c	odd	4	1	inner	20.8.e.b	✓	36
5.c	odd	4	1		100.8.e.e		36
20.d	odd	2	1		100.8.e.e		36
20.e	even	4	1	inner	20.8.e.b	✓	36
20.e	even	4	1		100.8.e.e		36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
20.8.e.b	✓	36	1.a	even	1	1	trivial
20.8.e.b	✓	36	4.b	odd	2	1	inner
20.8.e.b	✓	36	5.c	odd	4	1	inner
20.8.e.b	✓	36	20.e	even	4	1	inner
100.8.e.e		36	5.b	even	2	1
100.8.e.e		36	5.c	odd	4	1
100.8.e.e		36	20.d	odd	2	1
100.8.e.e		36	20.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^36 + 123741912*T3^32 + 5783684780560368*T3^28 + 126981566234145846279424*T3^24 + 1336386804590928082970257125120*T3^20 + 6431458674338579867288649661998028800*T3^16 + 12702308689457813009152124947875714402816000*T3^12 + 6471179081930250377601398006676846491766620160000*T3^8 + 642787462226018017116813446144632694302568349696000000*T3^4 + 165312069975923100792835278874106750812025153126400000000