LMFDB - Newform orbit 725.2.y.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(18,725)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(28))

chi = DirichletCharacter(H, H._module([21, 11]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("725.18");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$725 = 5^{2} \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	725.y (of order $28$ , degree $12$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$120$
Relative dimension:	$10$ over $\Q(\zeta_{28})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

$q$ -expansion

The algebraic $q$ -expansion of this newform has not been computed, but we have computed the trace expansion.

\operatorname{Tr}(f)(q) =

120 q - 20 q^{4} + 28 q^{9} - 12 q^{11} + 20 q^{16} + 4 q^{19} + 4 q^{21} + 12 q^{29} - 32 q^{31} - 40 q^{34} - 16 q^{36} - 184 q^{39} - 4 q^{41} + 36 q^{44} + 76 q^{46} - 84 q^{49} + 112 q^{51} + 168 q^{54}+ \cdots - 208 q^{99}+O(q^{100})

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label	$a_{2}$			$a_{3}$			$a_{4}$				$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$
18.1	−2.43176	−	1.17107i	−1.54099	−	1.22890i	3.29505	+	4.13187i	0	2.30818	+	4.79299i	0.367535	+	3.26196i	−1.97287	−	8.64369i	0.196894	+	0.862650i	0
18.2	−1.61621	−	0.778326i	0.677001	+	0.539890i	0.759368	+	0.952217i	0	−0.673965	−	1.39950i	−0.00559911	−	0.0496935i	0.312179	+	1.36775i	−0.500714	−	2.19377i	0
18.3	−1.34275	−	0.646635i	−1.78889	−	1.42659i	0.137866	+	0.172879i	0	1.47955	+	3.07231i	−0.298557	−	2.64976i	0.589934	+	2.58467i	0.497396	+	2.17923i	0
18.4	−0.368695	−	0.177554i	2.43448	+	1.94143i	−1.14257	−	1.43274i	0	−0.552871	−	1.14805i	−0.376408	−	3.34071i	0.348992	+	1.52903i	1.48996	+	6.52795i	0
18.5	−0.359005	−	0.172888i	0.705151	+	0.562339i	−1.14799	−	1.43953i	0	−0.155931	−	0.323795i	0.486105	+	4.31430i	0.340590	+	1.49222i	−0.486550	−	2.13172i	0
18.6	0.359005	+	0.172888i	−0.705151	−	0.562339i	−1.14799	−	1.43953i	0	−0.155931	−	0.323795i	−0.486105	−	4.31430i	−0.340590	−	1.49222i	−0.486550	−	2.13172i	0
18.7	0.368695	+	0.177554i	−2.43448	−	1.94143i	−1.14257	−	1.43274i	0	−0.552871	−	1.14805i	0.376408	+	3.34071i	−0.348992	−	1.52903i	1.48996	+	6.52795i	0
18.8	1.34275	+	0.646635i	1.78889	+	1.42659i	0.137866	+	0.172879i	0	1.47955	+	3.07231i	0.298557	+	2.64976i	−0.589934	−	2.58467i	0.497396	+	2.17923i	0
18.9	1.61621	+	0.778326i	−0.677001	−	0.539890i	0.759368	+	0.952217i	0	−0.673965	−	1.39950i	0.00559911	+	0.0496935i	−0.312179	−	1.36775i	−0.500714	−	2.19377i	0
18.10	2.43176	+	1.17107i	1.54099	+	1.22890i	3.29505	+	4.13187i	0	2.30818	+	4.79299i	−0.367535	−	3.26196i	1.97287	+	8.64369i	0.196894	+	0.862650i	0
32.1	−0.574590	−	2.51744i	0.00163787	−	0.00340106i	−4.20544	+	2.02523i	0	−0.00950310	−	0.00216902i	−0.286827	−	0.819705i	4.29488	+	5.38561i	1.87046	+	2.34548i	0
32.2	−0.445524	−	1.95197i	−1.46808	+	3.04850i	−1.80976	+	0.871532i	0	6.60465	+	1.50747i	0.560599	+	1.60210i	0.0108322	+	0.0135831i	−5.26764	−	6.60541i	0
32.3	−0.305062	−	1.33657i	0.793823	−	1.64839i	0.108593	−	0.0522958i	0	−2.44535	−	0.558135i	−1.11129	−	3.17588i	−1.81256	−	2.27287i	−0.216568	−	0.271568i	0
32.4	−0.199843	−	0.875569i	−0.586819	+	1.21854i	1.07525	−	0.517815i	0	1.18419	+	0.270283i	0.0189641	+	0.0541962i	−1.78816	−	2.24228i	0.729981	+	0.915367i	0
32.5	−0.0605112	−	0.265117i	0.814352	−	1.69102i	1.73531	−	0.835682i	0	−0.497595	−	0.113573i	1.58480	+	4.52910i	−0.665656	−	0.834707i	−0.325909	−	0.408678i	0
32.6	0.0605112	+	0.265117i	−0.814352	+	1.69102i	1.73531	−	0.835682i	0	−0.497595	−	0.113573i	−1.58480	−	4.52910i	0.665656	+	0.834707i	−0.325909	−	0.408678i	0
32.7	0.199843	+	0.875569i	0.586819	−	1.21854i	1.07525	−	0.517815i	0	1.18419	+	0.270283i	−0.0189641	−	0.0541962i	1.78816	+	2.24228i	0.729981	+	0.915367i	0
32.8	0.305062	+	1.33657i	−0.793823	+	1.64839i	0.108593	−	0.0522958i	0	−2.44535	−	0.558135i	1.11129	+	3.17588i	1.81256	+	2.27287i	−0.216568	−	0.271568i	0
32.9	0.445524	+	1.95197i	1.46808	−	3.04850i	−1.80976	+	0.871532i	0	6.60465	+	1.50747i	−0.560599	−	1.60210i	−0.0108322	−	0.0135831i	−5.26764	−	6.60541i	0
32.10	0.574590	+	2.51744i	−0.00163787	+	0.00340106i	−4.20544	+	2.02523i	0	−0.00950310	−	0.00216902i	0.286827	+	0.819705i	−4.29488	−	5.38561i	1.87046	+	2.34548i	0

See next 80 embeddings (of 120 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
145.o	even	28	1	inner
145.t	even	28	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	725.2.y.a	✓	120
5.b	even	2	1	inner	725.2.y.a	✓	120
5.c	odd	4	2		725.2.bd.a	yes	120
29.f	odd	28	1		725.2.bd.a	yes	120
145.o	even	28	1	inner	725.2.y.a	✓	120
145.s	odd	28	1		725.2.bd.a	yes	120
145.t	even	28	1	inner	725.2.y.a	✓	120

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
725.2.y.a	✓	120	1.a	even	1	1	trivial
725.2.y.a	✓	120	5.b	even	2	1	inner
725.2.y.a	✓	120	145.o	even	28	1	inner
725.2.y.a	✓	120	145.t	even	28	1	inner
725.2.bd.a	yes	120	5.c	odd	4	2
725.2.bd.a	yes	120	29.f	odd	28	1
725.2.bd.a	yes	120	145.s	odd	28	1

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{120} + 30 T_{2}^{118} + 515 T_{2}^{116} + 6936 T_{2}^{114} + 82423 T_{2}^{112} + \cdots + 1355457106081$ T2^120 + 30*T2^118 + 515*T2^116 + 6936*T2^114 + 82423*T2^112 + 892834*T2^110 + 8963066*T2^108 + 82969620*T2^106 + 711341389*T2^104 + 5711672742*T2^102 + 43281514619*T2^100 + 309879742812*T2^98 + 2094420946565*T2^96 + 13326643375404*T2^94 + 79947561290351*T2^92 + 453270390889128*T2^90 + 2429163061967288*T2^88 + 12270921270306340*T2^86 + 58169853034147083*T2^84 + 257838012054395538*T2^82 + 1069453440575697011*T2^80 + 4166231457495127126*T2^78 + 15306203355152681172*T2^76 + 53159062795967817588*T2^74 + 174527525250840075646*T2^72 + 541031793785211004410*T2^70 + 1584967933505013667770*T2^68 + 4394821795866125570704*T2^66 + 11542142324803594287843*T2^64 + 28676410766267195761674*T2^62 + 67210189734927523282562*T2^60 + 148164660043972605766718*T2^58 + 306703940762199636514865*T2^56 + 595228980684845906650752*T2^54 + 1079487851221261291286511*T2^52 + 1816620045246308835760430*T2^50 + 2803824440883394081589818*T2^48 + 3908387676776138135139242*T2^46 + 4837011518507565757932354*T2^44 + 5218930373644680971833962*T2^42 + 4820716925108310742195255*T2^40 + 3745500950585084638771170*T2^38 + 2406949829277725674470223*T2^36 + 1264567363470255925756138*T2^34 + 559444169761457596727514*T2^32 + 226588913277254479174730*T2^30 + 96568328318761693175343*T2^28 + 41377453607094296955702*T2^26 + 12405943203486661459597*T2^24 + 1225291199867501561024*T2^22 + 870051026397142550152*T2^20 + 336308089117583634670*T2^18 + 18069358065249929490*T2^16 - 1164895744307016680*T2^14 + 3264340730149262231*T2^12 + 15043143874991856*T2^10 + 14141699302492685*T2^8 + 3887780624996554*T2^6 + 424620157937222*T2^4 + 28268402498622*T2^2 + 1355457106081 acting on $S_{2}^{\mathrm{new}}(725, [\chi])$ .

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 18.10
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