Newform orbit 520.2.by.c

• Label: 520.2.by.c
• Level: $520$
• Weight: $2$
• Character orbit: 520.by
• Analytic conductor: $4.152$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.by (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.15222090511\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(52\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 104 q + 2 q^{6} - 4 q^{7} + 12 q^{8} + 60 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
61.1	−1.41417	+	0.0110700i	−2.05831	−	1.18836i	1.99975	−	0.0313097i			1.00000i	2.92395	+	1.65776i	1.39925	+	2.42357i	−2.82765	+	0.0664145i	1.32442	+	2.29395i	−0.0110700	−	1.41417i
61.2	−1.40811	+	0.131283i	−1.26478	−	0.730219i	1.96553	−	0.369722i		−	1.00000i	1.87680	+	0.862182i	1.29319	+	2.23987i	−2.71914	+	0.778649i	−0.433561	−	0.750950i	0.131283	+	1.40811i
61.3	−1.38120	−	0.303777i	2.20611	+	1.27370i	1.81544	+	0.839156i			1.00000i	−2.66016	−	2.42940i	0.408693	+	0.707876i	−2.25257	−	1.71053i	1.74462	+	3.02177i	0.303777	−	1.38120i
61.4	−1.34187	−	0.446536i	1.24296	+	0.717626i	1.60121	+	1.19838i			1.00000i	−1.34745	−	1.51799i	−1.81118	−	3.13705i	−1.61349	−	2.32307i	−0.470026	−	0.814109i	0.446536	−	1.34187i
61.5	−1.31276	−	0.525984i	−1.44534	−	0.834465i	1.44668	+	1.38098i		−	1.00000i	1.45846	+	1.85568i	0.886246	+	1.53502i	−1.17277	−	2.57383i	−0.107338	−	0.185914i	−0.525984	+	1.31276i
61.6	−1.28301	+	0.594879i	−2.95915	−	1.70847i	1.29224	−	1.52647i		−	1.00000i	4.81296	+	0.431645i	−0.724183	−	1.25432i	−0.749887	+	2.72721i	4.33773	+	7.51317i	0.594879	+	1.28301i
61.7	−1.19384	−	0.758119i	0.832765	+	0.480797i	0.850513	+	1.81015i		−	1.00000i	−0.629688	−	1.20533i	0.336005	+	0.581978i	0.356929	−	2.80582i	−1.03767	−	1.79729i	−0.758119	+	1.19384i
61.8	−1.17635	−	0.784979i	−2.41200	−	1.39257i	0.767616	+	1.84683i			1.00000i	1.74423	+	3.53152i	−0.704365	−	1.22000i	0.546731	−	2.77508i	2.37849	+	4.11966i	0.784979	−	1.17635i
61.9	−1.17245	+	0.790798i	−0.105918	−	0.0611516i	0.749278	−	1.85434i			1.00000i	0.172542	−	0.0120623i	−0.0953817	−	0.165206i	0.587918	+	2.76665i	−1.49252	−	2.58512i	−0.790798	−	1.17245i
61.10	−1.15786	−	0.812005i	2.74395	+	1.58422i	0.681295	+	1.88038i		−	1.00000i	−1.89072	−	4.06241i	−1.38210	−	2.39387i	0.738035	−	2.73044i	3.51951	+	6.09597i	−0.812005	+	1.15786i
61.11	−1.14867	+	0.824956i	2.07762	+	1.19952i	0.638896	−	1.89521i		−	1.00000i	−3.37606	+	0.336096i	−2.42868	−	4.20660i	0.829581	+	2.70403i	1.37768	+	2.38621i	0.824956	+	1.14867i
61.12	−1.13134	+	0.848571i	2.49999	+	1.44337i	0.559855	−	1.92004i			1.00000i	−4.05313	+	0.488477i	1.87153	+	3.24159i	0.995905	+	2.64730i	2.66662	+	4.61873i	−0.848571	−	1.13134i
61.13	−1.07599	+	0.917740i	−0.614222	−	0.354621i	0.315507	−	1.97496i		−	1.00000i	0.986346	−	0.182127i	−1.50530	−	2.60726i	1.47301	+	2.41459i	−1.24849	−	2.16244i	0.917740	+	1.07599i
61.14	−1.06872	+	0.926195i	0.0383751	+	0.0221558i	0.284324	−	1.97969i		−	1.00000i	−0.0615328	+	0.0118644i	2.54156	+	4.40212i	1.52971	+	2.37907i	−1.49902	−	2.59638i	0.926195	+	1.06872i
61.15	−1.00124	+	0.998758i	−2.45523	−	1.41753i	0.00496304	−	1.99999i			1.00000i	3.87404	−	1.03289i	0.334005	+	0.578514i	1.99254	+	2.00743i	2.51876	+	4.36262i	−0.998758	−	1.00124i
61.16	−0.893979	−	1.09581i	−0.566349	−	0.326982i	−0.401602	+	1.95926i			1.00000i	0.147994	+	0.912926i	0.582724	+	1.00931i	2.50601	−	1.31146i	−1.28617	−	2.22770i	1.09581	−	0.893979i
61.17	−0.885292	−	1.10284i	−1.75135	−	1.01114i	−0.432517	+	1.95267i		−	1.00000i	0.435326	+	2.82661i	−1.47683	−	2.55794i	2.53639	−	1.25169i	0.544814	+	0.943646i	−1.10284	+	0.885292i
61.18	−0.768530	−	1.18717i	0.998484	+	0.576475i	−0.818724	+	1.82474i			1.00000i	−0.0829934	−	1.62840i	−2.09203	−	3.62350i	2.79549	−	0.430409i	−0.835353	−	1.44687i	1.18717	−	0.768530i
61.19	−0.679694	−	1.24017i	1.76977	+	1.02178i	−1.07603	+	1.68587i		−	1.00000i	0.0642740	−	2.88930i	1.74453	+	3.02162i	2.82213	+	0.188588i	0.588053	+	1.01854i	−1.24017	+	0.679694i
61.20	−0.364330	+	1.36648i	2.45523	+	1.41753i	−1.73453	−	0.995699i		−	1.00000i	−2.83153	+	2.83857i	0.334005	+	0.578514i	1.99254	−	2.00743i	2.51876	+	4.36262i	1.36648	+	0.364330i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
13.c	even	3	1	inner
104.r	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	520.2.by.c	✓	104
8.b	even	2	1	inner	520.2.by.c	✓	104
13.c	even	3	1	inner	520.2.by.c	✓	104
104.r	even	6	1	inner	520.2.by.c	✓	104

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
520.2.by.c	✓	104	1.a	even	1	1	trivial
520.2.by.c	✓	104	8.b	even	2	1	inner
520.2.by.c	✓	104	13.c	even	3	1	inner
520.2.by.c	✓	104	104.r	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^104 - 108*T3^102 + 6216*T3^100 - 247152*T3^98 + 7535020*T3^96 - 186147008*T3^94 + 3855300672*T3^92 - 68482760528*T3^90 + 1060337058018*T3^88 - 14481689686520*T3^86 + 176047047489760*T3^84 - 1918228475603056*T3^82 + 18836549282579216*T3^80 - 167414595341560368*T3^78 + 1351265216125123776*T3^76 - 9930863325174417728*T3^74 + 66590507844583887419*T3^72 - 408012716673163894388*T3^70 + 2286846468730984274968*T3^68 - 11732760702464763330816*T3^66 + 55120250429493206213376*T3^64 - 237126824233337541459280*T3^62 + 933882166171593124124736*T3^60 - 3365175762077538120123456*T3^58 + 11085962252657916441999954*T3^56 - 33351847899187033656643304*T3^54 + 91509008557498424023866512*T3^52 - 228615694610856967212932816*T3^50 + 519074068279803135549248556*T3^48 - 1068795846403949172446725296*T3^46 + 1990846395254598290670346400*T3^44 - 3345511847609608596291252784*T3^42 + 5056357076596870634990753073*T3^40 - 6849963608879378319892077468*T3^38 + 8286853775578908044175445592*T3^36 - 8915661124528818419187818960*T3^34 + 8492110326996021513792529584*T3^32 - 7125280126204016897108702560*T3^30 + 5237247925004814882415465056*T3^28 - 3351135604964066179784822528*T3^26 + 1853369975825496383277045376*T3^24 - 878405941001598155910365440*T3^22 + 353079741603731953619283712*T3^20 - 118697889239222875784778240*T3^18 + 32755229031685541637785856*T3^16 - 7208673773657663559527424*T3^14 + 1211205817321009845760000*T3^12 - 142608752632354397413376*T3^10 + 9952466084985033801728*T3^8 - 160565213262223245312*T3^6 + 2014046196775059456*T3^4 - 3881409576960000*T3^2 + 6553600000000