Newform orbit 520.2.bj.a

• Label: 520.2.bj.a
• Level: $520$
• Weight: $2$
• Character orbit: 520.bj
• Analytic conductor: $4.152$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.bj (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.15222090511\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 4 q^{6} - 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
213.1	−1.41354	+	0.0436437i	1.47609	−	1.47609i	1.99619	−	0.123384i	1.11851	−	1.93622i	−2.02209	+	2.15094i	2.71367			−2.81631	+	0.261529i		−	1.35770i	−1.49656	+	2.78573i
213.2	−1.41339	+	0.0481963i	2.20939	−	2.20939i	1.99535	−	0.136240i	−2.17140	+	0.533868i	−3.01625	+	3.22922i	2.51077			−2.81365	+	0.288730i		−	6.76280i	3.04331	−	0.859219i
213.3	−1.41331	−	0.0505402i	1.01858	−	1.01858i	1.99489	+	0.142858i	2.01167	+	0.976309i	−1.49105	+	1.38809i	−4.28293			−2.81218	−	0.302725i			0.924980i	−2.79377	−	1.48150i
213.4	−1.40616	+	0.150675i	−0.444371	+	0.444371i	1.95459	−	0.423746i	−1.21366	+	1.87804i	0.557903	−	0.691814i	−0.485706			−2.68463	+	0.890364i			2.60507i	1.42364	−	2.82369i
213.5	−1.40118	+	0.191551i	−1.72762	+	1.72762i	1.92662	−	0.536795i	1.91799	+	1.14948i	2.08978	−	2.75164i	−0.913478			−2.59671	+	1.12119i		−	2.96936i	−2.90764	−	1.24324i
213.6	−1.38358	−	0.292743i	−2.01512	+	2.01512i	1.82860	+	0.810069i	−1.91445	−	1.15537i	3.37801	−	2.19818i	−3.83209			−2.29288	−	1.65611i		−	5.12145i	2.31057	+	2.15900i
213.7	−1.37580	−	0.327356i	−0.173757	+	0.173757i	1.78568	+	0.900756i	0.595033	−	2.15544i	0.295936	−	0.182175i	−1.34040			−2.16187	−	1.82382i			2.93962i	−1.52425	+	2.77068i
213.8	−1.34834	+	0.426597i	−0.932133	+	0.932133i	1.63603	−	1.15039i	−0.571269	−	2.16186i	0.859185	−	1.65447i	0.163375			−1.71517	+	2.24904i			1.26226i	1.69251	+	2.67122i
213.9	−1.34690	−	0.431128i	0.346403	−	0.346403i	1.62826	+	1.16137i	1.15491	+	1.91473i	−0.615913	+	0.317225i	4.24290			−1.69240	−	2.26623i			2.76001i	−0.730048	−	3.07685i
213.10	−1.30344	−	0.548676i	−1.78753	+	1.78753i	1.39791	+	1.43033i	1.86606	−	1.23199i	3.31071	−	1.34916i	3.51877			−1.03730	−	2.63135i		−	3.39052i	−3.10827	+	0.581966i
213.11	−1.30225	+	0.551504i	0.899162	−	0.899162i	1.39169	−	1.43639i	−2.13424	+	0.667110i	−0.675038	+	1.66682i	−1.90428			−1.02014	+	2.63805i			1.38302i	2.41138	−	2.04578i
213.12	−1.23130	−	0.695628i	0.536506	−	0.536506i	1.03220	+	1.71306i	−2.17354	−	0.525074i	−1.03381	+	0.287391i	2.01257			−0.0793019	−	2.82732i			2.42432i	2.31103	+	2.15850i
213.13	−1.20626	+	0.738199i	1.27743	−	1.27743i	0.910125	−	1.78092i	−1.36714	−	1.76944i	−0.597914	+	2.48390i	−3.50137			0.216826	+	2.82010i		−	0.263640i	2.95533	+	1.12519i
213.14	−1.17828	−	0.782086i	2.06823	−	2.06823i	0.776682	+	1.84303i	−0.658150	−	2.13702i	−4.05448	+	0.819416i	−4.71619			0.526262	−	2.77904i		−	5.55513i	−0.895847	+	3.03273i
213.15	−1.17233	−	0.790978i	−0.895570	+	0.895570i	0.748706	+	1.85457i	−1.20773	+	1.88186i	1.75828	−	0.341525i	−2.60281			0.589197	−	2.76638i			1.39591i	2.90437	−	1.25086i
213.16	−1.17043	+	0.793785i	0.581452	−	0.581452i	0.739812	−	1.85814i	0.651511	+	2.13905i	−0.219001	+	1.14210i	2.29584			0.609064	+	2.76207i			2.32383i	−2.46049	−	1.98645i
213.17	−1.11860	+	0.865298i	−2.20602	+	2.20602i	0.502518	−	1.93584i	−0.486949	+	2.18240i	0.558781	−	4.37651i	1.44444			1.11296	+	2.60025i		−	6.73305i	−1.34373	−	2.86258i
213.18	−1.09370	+	0.896563i	2.33697	−	2.33697i	0.392349	−	1.96114i	1.52947	+	1.63117i	−0.460697	+	4.65118i	−0.0307393			1.32917	+	2.49666i		−	7.92287i	−3.13523	−	0.412733i
213.19	−1.08205	−	0.910591i	2.01234	−	2.01234i	0.341650	+	1.97060i	2.23541	+	0.0542159i	−4.00986	+	0.345027i	2.17853			1.42473	−	2.44339i		−	5.09901i	−2.36945	−	2.09421i
213.20	−1.02373	+	0.975691i	−0.872017	+	0.872017i	0.0960560	−	1.99769i	1.88894	−	1.19662i	0.0418934	−	1.74353i	−3.86172			1.85079	+	2.13882i			1.47917i	−0.766247	+	3.06804i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
65.f	even	4	1	inner
520.bj	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	520.2.bj.a	yes	160
5.c	odd	4	1		520.2.y.a	✓	160
8.b	even	2	1	inner	520.2.bj.a	yes	160
13.d	odd	4	1		520.2.y.a	✓	160
40.i	odd	4	1		520.2.y.a	✓	160
65.f	even	4	1	inner	520.2.bj.a	yes	160
104.j	odd	4	1		520.2.y.a	✓	160
520.bj	even	4	1	inner	520.2.bj.a	yes	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
520.2.y.a	✓	160	5.c	odd	4	1
520.2.y.a	✓	160	13.d	odd	4	1
520.2.y.a	✓	160	40.i	odd	4	1
520.2.y.a	✓	160	104.j	odd	4	1
520.2.bj.a	yes	160	1.a	even	1	1	trivial
520.2.bj.a	yes	160	8.b	even	2	1	inner
520.2.bj.a	yes	160	65.f	even	4	1	inner
520.2.bj.a	yes	160	520.bj	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(520, [\chi])\).