Newform orbit 80.4.q.a

• Label: 80.4.q.a
• Level: $80$
• Weight: $4$
• Character orbit: 80.q
• Analytic conductor: $4.720$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.72015280046\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{4} - 2 q^{5} - 4 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1	−2.82839	−	0.0145100i	1.64454	−	1.64454i	7.99958	+	0.0820799i	10.3044	−	4.33812i	−4.67527	+	4.62755i	−11.1070			−22.6247	−	0.348228i			21.5909i	−29.2078	+	12.1204i
29.2	−2.77866	+	0.528231i	−3.63859	+	3.63859i	7.44194	−	2.93555i	−3.04646	−	10.7573i	8.18840	−	12.0324i	29.3525			−19.1280	+	12.0880i			0.521348i	14.1474	+	28.2816i
29.3	−2.71585	−	0.790046i	2.09729	−	2.09729i	6.75165	+	4.29129i	−8.07064	+	7.73723i	−7.35287	+	4.03896i	−6.76050			−14.9461	−	16.9886i			18.2028i	28.0314	−	14.6369i
29.4	−2.65306	−	0.980448i	−5.25955	+	5.25955i	6.07744	+	5.20237i	9.18190	+	6.37908i	19.1106	−	8.79718i	−2.79507			−11.0231	−	19.7608i		−	28.3257i	−18.1058	−	25.9265i
29.5	−2.57506	+	1.17007i	7.10567	−	7.10567i	5.26189	−	6.02598i	5.09839	+	9.95020i	−9.98344	+	26.6116i	16.9895			−6.49890	+	21.6740i		−	73.9810i	−24.7711	−	19.6569i
29.6	−2.44530	+	1.42145i	3.10771	−	3.10771i	3.95895	−	6.95174i	−8.95059	−	6.69978i	−3.18181	+	12.0167i	−22.2132			0.200769	+	22.6265i			7.68424i	31.4102	+	3.66010i
29.7	−2.41174	+	1.47766i	−4.90728	+	4.90728i	3.63302	−	7.12750i	−7.22084	+	8.53577i	4.58379	−	19.0864i	−16.1459			1.77014	+	22.5581i		−	21.1628i	4.80182	−	31.2561i
29.8	−2.30433	−	1.64014i	5.16106	−	5.16106i	2.61989	+	7.55885i	−0.347971	−	11.1749i	−20.3577	+	3.42794i	22.6097			6.36045	−	21.7151i		−	26.2731i	−17.5266	+	26.3215i
29.9	−2.06118	−	1.93689i	−4.56462	+	4.56462i	0.496941	+	7.98455i	−10.8220	−	2.80775i	18.2497	−	0.567361i	3.03301			14.4409	−	17.4201i		−	14.6715i	16.8679	+	26.7484i
29.10	−1.88198	+	2.11144i	−1.06284	+	1.06284i	−0.916320	−	7.94735i	10.5580	+	3.67824i	−0.243878	−	4.24437i	5.23459			18.5048	+	13.0220i			24.7407i	−27.6362	+	15.3701i
29.11	−1.32757	−	2.49751i	1.15578	−	1.15578i	−4.47511	+	6.63124i	5.16660	+	9.91495i	−4.42095	−	1.35219i	15.7689			22.5026	+	2.37318i			24.3283i	17.9037	−	26.0664i
29.12	−1.29896	−	2.51251i	−1.28590	+	1.28590i	−4.62541	+	6.52729i	4.81116	−	10.0922i	4.90118	+	1.56051i	−27.8938			22.4081	+	3.14272i			23.6929i	−31.6063	+	1.02128i
29.13	−1.13792	+	2.58943i	2.67942	−	2.67942i	−5.41027	−	5.89313i	−11.1419	−	0.925976i	3.88920	+	9.98714i	35.2795			21.4163	−	7.30359i			12.6414i	15.0764	−	27.7975i
29.14	−0.767869	+	2.72220i	−4.74774	+	4.74774i	−6.82076	−	4.18059i	1.97499	−	11.0045i	−9.27866	−	16.5699i	−8.59244			16.6178	−	15.3573i		−	18.0821i	28.4400	+	13.8263i
29.15	−0.568019	−	2.77080i	6.32652	−	6.32652i	−7.35471	+	3.14774i	−10.8777	+	2.58354i	−21.1231	−	13.9360i	−27.8170			12.8994	+	18.5905i		−	53.0497i	13.3372	+	28.6726i
29.16	−0.329019	+	2.80923i	1.84963	−	1.84963i	−7.78349	−	1.84858i	−1.03217	+	11.1326i	4.58747	+	5.80459i	−17.3426			7.75398	−	21.2574i			20.1577i	−30.9344	−	6.56242i
29.17	−0.269038	+	2.81560i	6.32513	−	6.32513i	−7.85524	−	1.51501i	6.26783	−	9.25820i	16.1074	+	19.5108i	−17.8675			6.37901	−	21.7096i		−	53.0146i	24.3811	+	20.1385i
29.18	0.269038	−	2.81560i	−6.32513	+	6.32513i	−7.85524	−	1.51501i	9.25820	−	6.26783i	16.1074	+	19.5108i	17.8675			−6.37901	+	21.7096i		−	53.0146i	−15.1569	−	27.7537i
29.19	0.329019	−	2.80923i	−1.84963	+	1.84963i	−7.78349	−	1.84858i	−11.1326	+	1.03217i	4.58747	+	5.80459i	17.3426			−7.75398	+	21.2574i			20.1577i	−0.763248	+	31.6136i
29.20	0.568019	+	2.77080i	−6.32652	+	6.32652i	−7.35471	+	3.14774i	−2.58354	+	10.8777i	−21.1231	−	13.9360i	27.8170			−12.8994	−	18.5905i		−	53.0497i	−31.6076	−	0.979715i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
16.e	even	4	1	inner
80.q	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	80.4.q.a	✓	68
4.b	odd	2	1		320.4.q.a		68
5.b	even	2	1	inner	80.4.q.a	✓	68
16.e	even	4	1	inner	80.4.q.a	✓	68
16.f	odd	4	1		320.4.q.a		68
20.d	odd	2	1		320.4.q.a		68
80.k	odd	4	1		320.4.q.a		68
80.q	even	4	1	inner	80.4.q.a	✓	68

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
80.4.q.a	✓	68	1.a	even	1	1	trivial
80.4.q.a	✓	68	5.b	even	2	1	inner
80.4.q.a	✓	68	16.e	even	4	1	inner
80.4.q.a	✓	68	80.q	even	4	1	inner
320.4.q.a		68	4.b	odd	2	1
320.4.q.a		68	16.f	odd	4	1
320.4.q.a		68	20.d	odd	2	1
320.4.q.a		68	80.k	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(80, [\chi])\).