LMFDB - Newform orbit 672.2.bq.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(85,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 5, 0, 0]))

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

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

chi := DirichletCharacter("672.85");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.bq (of order $8$, degree $4$, 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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$22$ over $\Q(\zeta_{8})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-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) = $ $ 88 q+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 88 q + 8 q^{10} + 16 q^{12} + 8 q^{16} + 8 q^{18} + 8 q^{22} + 40 q^{23} + 8 q^{24} - 40 q^{26} + 8 q^{28} - 40 q^{32} + 48 q^{33} - 96 q^{34} + 64 q^{38} - 40 q^{40} + 24 q^{43} - 64 q^{44} + 32 q^{46} + 32 q^{48} - 16 q^{51} + 32 q^{52} + 72 q^{53} - 8 q^{54} - 32 q^{55} - 32 q^{56} + 32 q^{58} - 24 q^{60} + 32 q^{61} - 48 q^{62} + 88 q^{63} - 96 q^{64} - 48 q^{66} - 24 q^{67} + 8 q^{68} + 32 q^{69} - 8 q^{70} - 72 q^{74} - 32 q^{75} + 32 q^{76} + 24 q^{77} + 80 q^{80} + 192 q^{82} - 64 q^{85} - 48 q^{86} - 104 q^{88} + 88 q^{92} + 32 q^{94} - 40 q^{96} + 144 q^{97}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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_{5} $			$ a_{6} $			$ a_{7} $			$ a_{8} $			$ a_{9} $			$ a_{10} $
85.1	−1.40986	−	0.110884i	0.923880	−	0.382683i	1.97541	+	0.312662i	1.27699	−	3.08292i	−1.34497	+	0.437086i	0.707107	+	0.707107i	−2.75038	−	0.659851i	0.707107	−	0.707107i	−2.14222	+	4.20489i
85.2	−1.34974	−	0.422149i	0.923880	−	0.382683i	1.64358	+	1.13958i	−0.114905	+	0.277405i	−1.40854	+	0.126507i	0.707107	+	0.707107i	−1.73733	−	2.23197i	0.707107	−	0.707107i	0.272198	−	0.325917i
85.3	−1.34748	−	0.429300i	−0.923880	+	0.382683i	1.63140	+	1.15695i	−0.539762	+	1.30310i	1.40919	−	0.119037i	0.707107	+	0.707107i	−1.70161	−	2.25932i	0.707107	−	0.707107i	1.28674	−	1.52418i
85.4	−1.29697	+	0.563801i	0.923880	−	0.382683i	1.36426	−	1.46247i	−0.926545	+	2.23688i	−0.982485	+	1.01721i	0.707107	+	0.707107i	−0.944857	+	2.66594i	0.707107	−	0.707107i	−0.0594548	−	3.42355i
85.5	−1.29196	+	0.575184i	−0.923880	+	0.382683i	1.33833	−	1.48623i	−1.23125	+	2.97249i	0.973503	−	1.02581i	0.707107	+	0.707107i	−0.874208	+	2.68994i	0.707107	−	0.707107i	−0.119009	−	4.54854i
85.6	−1.03192	+	0.967030i	−0.923880	+	0.382683i	0.129704	−	1.99579i	0.416627	−	1.00583i	0.583300	−	1.28832i	0.707107	+	0.707107i	1.79615	+	2.18492i	0.707107	−	0.707107i	0.542741	+	1.44082i
85.7	−0.555555	+	1.30052i	0.923880	−	0.382683i	−1.38272	−	1.44502i	1.20358	−	2.90570i	−0.0155775	+	1.41413i	0.707107	+	0.707107i	2.64746	−	0.995465i	0.707107	−	0.707107i	3.11028	+	3.17956i
85.8	−0.546523	+	1.30434i	−0.923880	+	0.382683i	−1.40262	−	1.42571i	−0.297269	+	0.717672i	0.00577093	−	1.41420i	0.707107	+	0.707107i	2.62618	−	1.05032i	0.707107	−	0.707107i	−0.773626	−	0.779966i
85.9	−0.362626	−	1.36693i	0.923880	−	0.382683i	−1.73700	+	0.991370i	−0.365809	+	0.883142i	−0.858125	−	1.12411i	0.707107	+	0.707107i	1.98502	+	2.01487i	0.707107	−	0.707107i	1.33985	+	0.179786i
85.10	−0.0821252	−	1.41183i	−0.923880	+	0.382683i	−1.98651	+	0.231893i	0.384692	−	0.928730i	0.616157	+	1.27293i	0.707107	+	0.707107i	0.490536	+	2.78557i	0.707107	−	0.707107i	−1.34280	−	0.466847i
85.11	0.121825	+	1.40896i	0.923880	−	0.382683i	−1.97032	+	0.343294i	−0.154083	+	0.371989i	0.651736	+	1.25509i	0.707107	+	0.707107i	−0.723721	−	2.73427i	0.707107	−	0.707107i	−0.542887	−	0.171778i
85.12	0.164290	−	1.40464i	0.923880	−	0.382683i	−1.94602	−	0.461537i	0.779541	−	1.88198i	−0.385747	−	1.36059i	0.707107	+	0.707107i	−0.968004	+	2.65762i	0.707107	−	0.707107i	−2.51543	−	1.40416i
85.13	0.407220	+	1.35432i	−0.923880	+	0.382683i	−1.66834	+	1.10301i	0.905119	−	2.18515i	−0.894497	−	1.09539i	0.707107	+	0.707107i	−2.17321	−	1.81030i	0.707107	−	0.707107i	3.32797	+	0.335980i
85.14	0.486478	−	1.32791i	−0.923880	+	0.382683i	−1.52668	−	1.29200i	−0.326981	+	0.789403i	0.0587215	+	1.41299i	0.707107	+	0.707107i	−2.45835	+	1.39876i	0.707107	−	0.707107i	0.889185	+	0.818228i
85.15	0.969047	+	1.03002i	−0.923880	+	0.382683i	−0.121894	+	1.99628i	0.745636	−	1.80013i	−1.28946	−	0.580779i	0.707107	+	0.707107i	−2.17434	+	1.80894i	0.707107	−	0.707107i	2.57673	−	0.976384i
85.16	0.981441	−	1.01822i	−0.923880	+	0.382683i	−0.0735455	−	1.99865i	1.24338	−	3.00179i	−0.517078	+	1.31629i	0.707107	+	0.707107i	−2.10724	−	1.88667i	0.707107	−	0.707107i	−1.83618	−	4.21212i
85.17	1.15096	+	0.821762i	−0.923880	+	0.382683i	0.649415	+	1.89163i	−1.43636	+	3.46767i	−1.37782	−	0.318756i	0.707107	+	0.707107i	−0.807018	+	2.71085i	0.707107	−	0.707107i	−4.50278	+	2.81080i
85.18	1.16190	−	0.806219i	0.923880	−	0.382683i	0.700021	−	1.87349i	−1.40234	+	3.38554i	0.764929	−	1.18949i	0.707107	+	0.707107i	−0.697090	−	2.74118i	0.707107	−	0.707107i	1.10011	+	5.06425i
85.19	1.20604	+	0.738558i	0.923880	−	0.382683i	0.909063	+	1.78146i	1.16799	−	2.81978i	1.39687	+	0.220808i	0.707107	+	0.707107i	−0.219348	+	2.81991i	0.707107	−	0.707107i	3.49121	−	2.53813i
85.20	1.30043	−	0.555768i	0.923880	−	0.382683i	1.38224	−	1.44548i	0.538691	−	1.30051i	0.988759	−	1.01112i	0.707107	+	0.707107i	0.994163	−	2.64795i	0.707107	−	0.707107i	−0.0222541	−	1.99062i

See all 88 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
32.g	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	672.2.bq.a	✓	88
32.g	even	8	1	inner	672.2.bq.a	✓	88

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
672.2.bq.a	✓	88	1.a	even	1	1	trivial
672.2.bq.a	✓	88	32.g	even	8	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{88} - 16 T_{5}^{83} + 832 T_{5}^{82} - 10800 T_{5}^{81} + 90560 T_{5}^{80} - 145696 T_{5}^{79} + \cdots + 1638400000000 $ T5^88 - 16*T5^83 + 832*T5^82 - 10800*T5^81 + 90560*T5^80 - 145696*T5^79 + 210912*T5^78 - 249344*T5^77 + 518912*T5^76 - 6386656*T5^75 + 86884608*T5^74 - 724389984*T5^73 + 3946126224*T5^72 - 10331938944*T5^71 + 21199837632*T5^70 - 36380981824*T5^69 + 63454339584*T5^68 - 253119078976*T5^67 + 2147072892288*T5^66 - 14987971721792*T5^65 + 73249112563072*T5^64 - 227031702418304*T5^63 + 551330234634880*T5^62 - 1101387653921792*T5^61 + 1752178377065472*T5^60 - 1999486526110848*T5^59 + 4473037857601536*T5^58 - 38353722864176768*T5^57 + 226007372452369504*T5^56 - 739261178824380928*T5^55 + 1731861504938283264*T5^54 - 3140122117680842496*T5^53 + 4531690133034043392*T5^52 - 4649351913145110784*T5^51 + 2736666569351965184*T5^50 - 13034607995649041664*T5^49 + 126780061779624598016*T5^48 - 438934329750548418048*T5^47 + 972589632783807996416*T5^46 - 1471452896736065581056*T5^45 + 1685078727173528072192*T5^44 - 1883000925496769309184*T5^43 + 3057195956525005811712*T5^42 - 4380021656776066318848*T5^41 + 20643592285080587700480*T5^40 - 52227207433860053497856*T5^39 + 118910672427998980764672*T5^38 - 190536638500414846241792*T5^37 + 257852339611866405134336*T5^36 - 297010984439494411611136*T5^35 + 423962205277542386862080*T5^34 - 4798523204771983141888*T5^33 + 479445068231881195816960*T5^32 + 465934988344433260730368*T5^31 + 599843301644071723530240*T5^30 + 530080369705739194908672*T5^29 + 730361579523068159836160*T5^28 + 1977221567636088419690496*T5^27 + 5369161568765633217691648*T5^26 + 10949324783024176247654400*T5^25 + 19044481040390964767031552*T5^24 + 28366413778266719270281216*T5^23 + 36860111600535097092419584*T5^22 + 41994696664446969989279744*T5^21 + 42216491377197673868656640*T5^20 + 37628585563978790847176704*T5^19 + 29628595749533824481468416*T5^18 + 20307748389328448480428032*T5^17 + 11857336377779285181997056*T5^16 + 5725162612210828524093440*T5^15 + 2202226256044327378780160*T5^14 + 638499234695351108698112*T5^13 + 123420832751364168351744*T5^12 + 8865749153711870312448*T5^11 - 2845220145685508521984*T5^10 - 940638812225081704448*T5^9 - 60647176584903983104*T5^8 + 25261495523719249920*T5^7 + 5905330746805452800*T5^6 + 70548027080704000*T5^5 - 109648647290880000*T5^4 - 6829519667200000*T5^3 + 1051787264000000*T5^2 + 81264640000000*T5 + 1638400000000 acting on $S_{2}^{\mathrm{new}}(672, [\chi])$.

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.bq (of order \(8\), degree \(4\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 85.22
Significant digits:
Format:

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