Newform orbit 80.4.j.a

• Label: 80.4.j.a
• Level: $80$
• Weight: $4$
• Character orbit: 80.j
• Analytic conductor: $4.720$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	80.j (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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 2 q^{2} + 12 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} + 40 q^{8} - 540 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−2.81455	+	0.279819i		−	2.08304i	7.84340	−	1.57513i	−11.1381	−	0.971164i	0.582873	+	5.86281i	1.79251	−	1.79251i	−21.6349	+	6.62802i	22.6610			31.6205	−	0.383259i
43.2	−2.79390	+	0.440583i		−	5.84366i	7.61177	−	2.46189i	7.49235	+	8.29848i	2.57461	+	16.3266i	4.15713	−	4.15713i	−20.1819	+	10.2319i	−7.14836			−24.5890	−	19.8841i
43.3	−2.78959	−	0.467134i			1.93814i	7.56357	+	2.60622i	2.99562	−	10.7715i	0.905371	−	5.40661i	−14.6523	+	14.6523i	−19.8818	−	10.8035i	23.2436			−13.3883	+	28.6488i
43.4	−2.68172	−	0.899084i			7.30529i	6.38329	+	4.82219i	1.43262	+	11.0882i	6.56807	−	19.5908i	−0.950927	+	0.950927i	−12.7827	−	18.6709i	−26.3673			6.12732	−	31.0235i
43.5	−2.56239	+	1.19756i			5.74087i	5.13169	−	6.13724i	10.9557	−	2.22988i	−6.87505	−	14.7104i	23.6614	−	23.6614i	−5.79967	+	21.8715i	−5.95761			−25.4024	+	18.8340i
43.6	−2.30796	−	1.63502i		−	6.67181i	2.65339	+	7.54715i	7.80689	−	8.00328i	−10.9086	+	15.3983i	20.5642	−	20.5642i	6.21585	−	21.7569i	−17.5131			−31.1036	+	5.70681i
43.7	−2.26107	+	1.69929i			8.76050i	2.22486	−	7.68440i	−10.4272	−	4.03398i	−14.8866	−	19.8081i	−10.5634	+	10.5634i	8.02743	+	21.1556i	−49.7463			30.4315	−	8.59772i
43.8	−2.17993	+	1.80219i		−	9.57547i	1.50422	−	7.85731i	−0.434283	−	11.1719i	17.2568	+	20.8739i	−8.07205	+	8.07205i	10.8813	+	19.8393i	−64.6895			21.0806	+	23.5713i
43.9	−2.10741	−	1.88648i		−	5.75615i	0.882358	+	7.95119i	0.458746	+	11.1709i	−10.8589	+	12.1306i	−20.4957	+	20.4957i	13.1403	−	18.4210i	−6.13323			20.1070	−	24.4071i
43.10	−1.93778	−	2.06034i			2.10444i	−0.490000	+	7.98498i	−11.1784	+	0.205886i	4.33586	−	4.07794i	14.8890	−	14.8890i	17.4013	−	14.4636i	22.5713			22.0856	+	22.6324i
43.11	−1.80332	+	2.17900i		−	2.90856i	−1.49605	−	7.85887i	−6.79776	+	8.87640i	6.33775	+	5.24508i	7.30822	−	7.30822i	19.8223	+	10.9122i	18.5403			−7.08308	−	30.8193i
43.12	−1.68859	+	2.26907i			0.998127i	−2.29735	−	7.66304i	10.9800	+	2.10693i	−2.26482	−	1.68542i	−25.3753	+	25.3753i	21.2672	+	7.72686i	26.0037			−23.3215	+	21.3567i
43.13	−1.36079	−	2.47956i			9.25535i	−4.29648	+	6.74835i	2.90670	−	10.7959i	22.9492	−	12.5946i	−6.42513	+	6.42513i	22.5796	+	1.47027i	−58.6615			−30.7245	+	7.48363i
43.14	−0.869161	+	2.69157i			0.816587i	−6.48912	−	4.67882i	−3.81941	−	10.5077i	−2.19790	−	0.709745i	14.6998	−	14.6998i	18.2335	−	13.3993i	26.3332			31.6020	−	1.14734i
43.15	−0.779981	−	2.71876i		−	5.41494i	−6.78326	+	4.24115i	−5.16149	−	9.91761i	−14.7219	+	4.22355i	−17.1720	+	17.1720i	16.8215	+	15.1340i	−2.32163			−22.9377	+	21.7684i
43.16	−0.600978	−	2.76384i			1.78569i	−7.27765	+	3.32202i	9.90919	+	5.17764i	4.93536	−	1.07316i	3.83222	−	3.83222i	13.5552	+	18.1178i	23.8113			8.35497	−	30.4991i
43.17	−0.138747	+	2.82502i			6.49915i	−7.96150	−	0.783926i	−0.487814	+	11.1697i	−18.3603	−	0.901737i	4.03915	−	4.03915i	3.31924	−	22.3826i	−15.2390			−31.4869	−	2.92785i
43.18	0.335403	−	2.80847i		−	9.06020i	−7.77501	−	1.88394i	−6.52613	+	9.07797i	−25.4453	−	3.03882i	20.8390	−	20.8390i	−7.89875	+	21.2040i	−55.0872			23.3063	+	21.3732i
43.19	0.353421	+	2.80626i		−	4.64797i	−7.75019	+	1.98358i	10.0868	−	4.82255i	13.0434	−	1.64269i	1.85535	−	1.85535i	−8.30553	−	21.0480i	5.39640			17.0982	+	26.6017i
43.20	0.461904	+	2.79046i		−	7.89491i	−7.57329	+	2.57784i	−9.18689	+	6.37189i	22.0304	−	3.64669i	−10.3963	+	10.3963i	−10.6915	−	19.9422i	−35.3295			−22.0239	−	22.6924i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
80.j	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.j.a	✓	68
4.b	odd	2	1		320.4.j.a		68
5.c	odd	4	1		80.4.s.a	yes	68
16.e	even	4	1		320.4.s.a		68
16.f	odd	4	1		80.4.s.a	yes	68
20.e	even	4	1		320.4.s.a		68
80.j	even	4	1	inner	80.4.j.a	✓	68
80.t	odd	4	1		320.4.j.a		68

    

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

Hecke kernels

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