Newform orbit 80.6.l.a

• Label: 80.6.l.a
• Level: $80$
• Weight: $6$
• Character orbit: 80.l
• Analytic conductor: $12.831$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	80.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8307055850\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 44 q^{4} - 228 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
21.1	−5.65572	+	0.113075i	−18.3942	+	18.3942i	31.9744	−	1.27904i	17.6777	+	17.6777i	101.953	−	106.113i		−	47.3499i	−180.694	+	10.8494i		−	433.697i	−101.979	−	97.9811i
21.2	−5.65514	+	0.139396i	19.6483	−	19.6483i	31.9611	−	1.57661i	−17.6777	−	17.6777i	−108.375	+	113.853i		−	152.043i	−180.525	+	13.3712i		−	529.109i	102.434	+	97.5054i
21.3	−5.62667	+	0.583586i	−13.2011	+	13.2011i	31.3189	−	6.56730i	−17.6777	−	17.6777i	66.5744	−	81.9824i			190.109i	−172.388	+	55.2293i		−	105.540i	109.783	+	89.1500i
21.4	−5.55875	+	1.04896i	9.77316	−	9.77316i	29.7994	−	11.6618i	17.6777	+	17.6777i	−44.0749	+	64.5782i			103.649i	−153.414	+	96.0835i			51.9707i	−116.809	−	79.7225i
21.5	−5.42543	−	1.60146i	3.86003	−	3.86003i	26.8706	+	17.3773i	−17.6777	−	17.6777i	−27.1241	+	14.7606i			20.3706i	−117.956	−	137.312i			213.200i	67.5988	+	124.219i
21.6	−5.27573	−	2.04124i	6.55397	−	6.55397i	23.6666	+	21.5381i	17.6777	+	17.6777i	−47.9552	+	21.1987i		−	188.280i	−80.8943	−	161.939i			157.091i	−57.1782	−	129.347i
21.7	−4.61831	−	3.26668i	−15.8638	+	15.8638i	10.6576	+	30.1731i	−17.6777	−	17.6777i	125.086	−	21.4420i		−	207.627i	49.3459	−	174.164i		−	260.322i	23.8936	+	139.388i
21.8	−4.51701	+	3.40538i	−7.38373	+	7.38373i	8.80677	−	30.7643i	17.6777	+	17.6777i	8.20797	−	58.4968i		−	74.1496i	64.9838	+	168.953i			133.961i	−140.049	−	19.6510i
21.9	−4.43266	+	3.51447i	11.6877	−	11.6877i	7.29694	−	31.1569i	−17.6777	−	17.6777i	−10.7314	+	92.8834i			158.930i	77.1554	+	163.753i		−	30.2029i	140.487	+	16.2314i
21.10	−4.24419	−	3.73990i	21.1465	−	21.1465i	4.02635	+	31.7457i	17.6777	+	17.6777i	−168.835	+	10.6641i			100.160i	101.637	−	149.793i		−	651.346i	−8.91479	−	141.140i
21.11	−4.01718	−	3.98275i	10.2463	−	10.2463i	0.275419	+	31.9988i	−17.6777	−	17.6777i	−81.9696	+	0.352756i			107.195i	126.337	−	129.642i			33.0270i	0.608601	+	141.420i
21.12	−3.40887	−	4.51438i	−13.5228	+	13.5228i	−8.75923	+	30.7778i	17.6777	+	17.6777i	107.145	+	14.9496i			90.4254i	168.802	−	65.3752i		−	122.733i	19.5428	−	140.065i
21.13	−3.36114	+	4.55002i	−18.2668	+	18.2668i	−9.40545	−	30.5866i	−17.6777	−	17.6777i	−21.7171	−	144.512i			13.9157i	170.783	+	60.0107i		−	424.353i	139.851	−	21.0167i
21.14	−3.10357	+	4.72947i	−7.07478	+	7.07478i	−12.7357	−	29.3564i	17.6777	+	17.6777i	−11.5029	−	55.4170i			216.974i	178.367	+	30.8765i			142.895i	−138.470	+	28.7421i
21.15	−2.38570	−	5.12917i	2.82595	−	2.82595i	−20.6168	+	24.4734i	17.6777	+	17.6777i	−21.2367	−	7.75292i		−	197.416i	174.714	+	47.3612i			227.028i	48.4982	−	132.846i
21.16	−2.24441	+	5.19255i	1.65480	−	1.65480i	−21.9252	−	23.3084i	−17.6777	−	17.6777i	4.87860	+	12.3067i		−	111.270i	170.240	−	61.5343i			237.523i	131.468	−	52.1163i
21.17	−1.86060	−	5.34211i	−6.56300	+	6.56300i	−25.0763	+	19.8791i	−17.6777	−	17.6777i	47.2714	+	22.8492i			42.0449i	152.853	+	96.9737i			156.854i	−61.5451	+	127.327i
21.18	−0.605604	+	5.62434i	4.24643	−	4.24643i	−31.2665	−	6.81225i	17.6777	+	17.6777i	21.3117	+	26.4550i		−	137.329i	57.2496	−	171.728i			206.936i	−110.131	+	88.7196i
21.19	−0.350398	+	5.64599i	−18.9032	+	18.9032i	−31.7544	−	3.95669i	17.6777	+	17.6777i	−100.104	−	113.351i		−	174.451i	33.4661	−	177.899i		−	471.664i	−106.002	+	93.6138i
21.20	−0.0219870	+	5.65681i	9.88924	−	9.88924i	−31.9990	−	0.248752i	−17.6777	−	17.6777i	55.7241	+	56.1590i			122.335i	2.11071	−	181.007i			47.4058i	100.388	−	99.6106i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	80.6.l.a	✓	80
4.b	odd	2	1		320.6.l.a		80
16.e	even	4	1	inner	80.6.l.a	✓	80
16.f	odd	4	1		320.6.l.a		80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
80.6.l.a	✓	80	1.a	even	1	1	trivial
80.6.l.a	✓	80	16.e	even	4	1	inner
320.6.l.a		80	4.b	odd	2	1
320.6.l.a		80	16.f	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(80, [\chi])\).