LMFDB - Newform orbit 714.2.bh.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [714,2,Mod(25,714)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(714, base_ring=CyclotomicField(24))

chi = DirichletCharacter(H, H._module([0, 16, 15]))

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

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

chi := DirichletCharacter("714.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 714 = 2 \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	714.bh (of order $24$, degree $8$, 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:	$5.70131870432$
Analytic rank:	$0$
Dimension:	$96$
Relative dimension:	$12$ over $\Q(\zeta_{24})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

$q$-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) = $ $ 96 q + 8 q^{5}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 96 q + 8 q^{5} + 8 q^{10} + 16 q^{14} + 16 q^{15} + 48 q^{16} + 8 q^{17} + 48 q^{18} - 8 q^{19} + 16 q^{22} + 24 q^{25} + 32 q^{28} - 32 q^{29} - 16 q^{33} - 48 q^{37} - 16 q^{39} + 128 q^{41} + 16 q^{42} + 48 q^{43} - 8 q^{44} - 8 q^{46} - 112 q^{49} + 64 q^{50} - 16 q^{52} - 40 q^{53} + 16 q^{57} + 16 q^{58} - 24 q^{59} + 8 q^{60} - 56 q^{70} + 24 q^{73} + 48 q^{74} - 16 q^{76} + 16 q^{77} + 32 q^{78} - 56 q^{79} - 8 q^{80} - 80 q^{83} + 80 q^{85} + 16 q^{86} - 24 q^{87} + 16 q^{90} - 40 q^{91} + 16 q^{92} - 8 q^{93} - 8 q^{94} + 80 q^{95} + 160 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} $
25.1	−0.965926	−	0.258819i	−0.608761	−	0.793353i	0.866025	+	0.500000i	−0.337776	−	2.56567i	0.382683	+	0.923880i	−0.00794442	+	2.64574i	−0.707107	−	0.707107i	−0.258819	+	0.965926i	−0.337776	+	2.56567i
25.2	−0.965926	−	0.258819i	−0.608761	−	0.793353i	0.866025	+	0.500000i	−0.135385	−	1.02835i	0.382683	+	0.923880i	0.105025	−	2.64367i	−0.707107	−	0.707107i	−0.258819	+	0.965926i	−0.135385	+	1.02835i
25.3	−0.965926	−	0.258819i	−0.608761	−	0.793353i	0.866025	+	0.500000i	−0.118819	−	0.902520i	0.382683	+	0.923880i	2.16960	+	1.51421i	−0.707107	−	0.707107i	−0.258819	+	0.965926i	−0.118819	+	0.902520i
25.4	−0.965926	−	0.258819i	−0.608761	−	0.793353i	0.866025	+	0.500000i	0.128065	+	0.972754i	0.382683	+	0.923880i	−2.23332	−	1.41855i	−0.707107	−	0.707107i	−0.258819	+	0.965926i	0.128065	−	0.972754i
25.5	−0.965926	−	0.258819i	−0.608761	−	0.793353i	0.866025	+	0.500000i	0.430273	+	3.26824i	0.382683	+	0.923880i	0.971259	+	2.46103i	−0.707107	−	0.707107i	−0.258819	+	0.965926i	0.430273	−	3.26824i
25.6	−0.965926	−	0.258819i	−0.608761	−	0.793353i	0.866025	+	0.500000i	0.459415	+	3.48960i	0.382683	+	0.923880i	1.51635	−	2.16810i	−0.707107	−	0.707107i	−0.258819	+	0.965926i	0.459415	−	3.48960i
25.7	−0.965926	−	0.258819i	0.608761	+	0.793353i	0.866025	+	0.500000i	−0.416216	−	3.16147i	−0.382683	−	0.923880i	0.299906	+	2.62870i	−0.707107	−	0.707107i	−0.258819	+	0.965926i	−0.416216	+	3.16147i
25.8	−0.965926	−	0.258819i	0.608761	+	0.793353i	0.866025	+	0.500000i	−0.107430	−	0.816011i	−0.382683	−	0.923880i	2.49739	−	0.873515i	−0.707107	−	0.707107i	−0.258819	+	0.965926i	−0.107430	+	0.816011i
25.9	−0.965926	−	0.258819i	0.608761	+	0.793353i	0.866025	+	0.500000i	−0.102499	−	0.778556i	−0.382683	−	0.923880i	2.60629	+	0.455261i	−0.707107	−	0.707107i	−0.258819	+	0.965926i	−0.102499	+	0.778556i
25.10	−0.965926	−	0.258819i	0.608761	+	0.793353i	0.866025	+	0.500000i	−0.0294272	−	0.223522i	−0.382683	−	0.923880i	−2.00544	−	1.72575i	−0.707107	−	0.707107i	−0.258819	+	0.965926i	−0.0294272	+	0.223522i
25.11	−0.965926	−	0.258819i	0.608761	+	0.793353i	0.866025	+	0.500000i	0.305655	+	2.32168i	−0.382683	−	0.923880i	−1.22034	+	2.34750i	−0.707107	−	0.707107i	−0.258819	+	0.965926i	0.305655	−	2.32168i
25.12	−0.965926	−	0.258819i	0.608761	+	0.793353i	0.866025	+	0.500000i	0.406505	+	3.08771i	−0.382683	−	0.923880i	−2.26996	−	1.35915i	−0.707107	−	0.707107i	−0.258819	+	0.965926i	0.406505	−	3.08771i
121.1	0.965926	−	0.258819i	−0.793353	−	0.608761i	0.866025	−	0.500000i	−3.71943	−	0.489672i	−0.923880	−	0.382683i	2.41756	+	1.07490i	0.707107	−	0.707107i	0.258819	+	0.965926i	−3.71943	+	0.489672i
121.2	0.965926	−	0.258819i	−0.793353	−	0.608761i	0.866025	−	0.500000i	−1.87389	−	0.246703i	−0.923880	−	0.382683i	−2.13946	+	1.55651i	0.707107	−	0.707107i	0.258819	+	0.965926i	−1.87389	+	0.246703i
121.3	0.965926	−	0.258819i	−0.793353	−	0.608761i	0.866025	−	0.500000i	−0.0113440	−	0.00149347i	−0.923880	−	0.382683i	0.142386	−	2.64192i	0.707107	−	0.707107i	0.258819	+	0.965926i	−0.0113440	+	0.00149347i
121.4	0.965926	−	0.258819i	−0.793353	−	0.608761i	0.866025	−	0.500000i	0.145332	+	0.0191333i	−0.923880	−	0.382683i	−1.41829	+	2.23349i	0.707107	−	0.707107i	0.258819	+	0.965926i	0.145332	−	0.0191333i
121.5	0.965926	−	0.258819i	−0.793353	−	0.608761i	0.866025	−	0.500000i	1.58279	+	0.208378i	−0.923880	−	0.382683i	2.57043	−	0.626823i	0.707107	−	0.707107i	0.258819	+	0.965926i	1.58279	−	0.208378i
121.6	0.965926	−	0.258819i	−0.793353	−	0.608761i	0.866025	−	0.500000i	3.23325	+	0.425665i	−0.923880	−	0.382683i	0.135863	+	2.64226i	0.707107	−	0.707107i	0.258819	+	0.965926i	3.23325	−	0.425665i
121.7	0.965926	−	0.258819i	0.793353	+	0.608761i	0.866025	−	0.500000i	−3.39033	−	0.446345i	0.923880	+	0.382683i	1.71000	−	2.01888i	0.707107	−	0.707107i	0.258819	+	0.965926i	−3.39033	+	0.446345i
121.8	0.965926	−	0.258819i	0.793353	+	0.608761i	0.866025	−	0.500000i	−3.03089	−	0.399024i	0.923880	+	0.382683i	0.328120	+	2.62533i	0.707107	−	0.707107i	0.258819	+	0.965926i	−3.03089	+	0.399024i

See all 96 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
17.d	even	8	1	inner
119.q	even	24	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	714.2.bh.b	✓	96
7.c	even	3	1	inner	714.2.bh.b	✓	96
17.d	even	8	1	inner	714.2.bh.b	✓	96
119.q	even	24	1	inner	714.2.bh.b	✓	96

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
714.2.bh.b	✓	96	1.a	even	1	1	trivial
714.2.bh.b	✓	96	7.c	even	3	1	inner
714.2.bh.b	✓	96	17.d	even	8	1	inner
714.2.bh.b	✓	96	119.q	even	24	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{96} - 8 T_{5}^{95} + 20 T_{5}^{94} + 64 T_{5}^{93} - 568 T_{5}^{92} + 1488 T_{5}^{91} + \cdots + 15527402881 $ T5^96 - 8*T5^95 + 20*T5^94 + 64*T5^93 - 568*T5^92 + 1488*T5^91 - 1352*T5^90 + 776*T5^89 - 40666*T5^88 + 162656*T5^87 + 523124*T5^86 - 6772640*T5^85 + 23174144*T5^84 - 19441904*T5^83 - 88232500*T5^82 + 326244864*T5^81 - 38364139*T5^80 - 783234416*T5^79 - 11545689212*T5^78 + 99936694192*T5^77 - 284073137968*T5^76 + 92510443184*T5^75 + 1406305152628*T5^74 - 4490594183120*T5^73 + 3903686525490*T5^72 - 5719614258952*T5^71 + 160901459078720*T5^70 - 1116658328947744*T5^69 + 2777608180621576*T5^68 + 135980591788576*T5^67 - 15834761363743876*T5^66 + 50725359417165848*T5^65 - 87757376764624246*T5^64 + 153996474110091048*T5^63 - 664970774546080780*T5^62 + 3716375030968936512*T5^61 - 6711153565720254696*T5^60 - 6507418907023120112*T5^59 + 34676843969914906608*T5^58 - 95616833154237122440*T5^57 + 351268433973564141054*T5^56 - 945792867948083029744*T5^55 + 2082063031940232680716*T5^54 - 10071180329793331084768*T5^53 + 22975766487481573902032*T5^52 - 5981738562825023049872*T5^51 + 8216613214215787691948*T5^50 - 43177417362599466371248*T5^49 - 86615853101094062608547*T5^48 + 193581334011869946880976*T5^47 - 784876873286298380155652*T5^46 + 4865653528656924676519024*T5^45 - 3308958010699575627903152*T5^44 - 14576498220426241444123120*T5^43 - 5416755455903396704543172*T5^42 + 23517296828456997538553968*T5^41 + 53375623026111831854184318*T5^40 - 126860859990764047233725976*T5^39 + 24875884783978288718726384*T5^38 + 134736951297874163064364608*T5^37 + 843828818038824257377526872*T5^36 + 555152263423525323981931552*T5^35 - 213982268141482928197424060*T5^34 - 2725771770800986568675278936*T5^33 - 3733040350787153814623791478*T5^32 - 3583227006708913939507413704*T5^31 + 152168536830164758326304268*T5^30 + 4028150196706777542127363584*T5^29 + 7835397189945706436679061576*T5^28 + 8638585076580997570350737712*T5^27 + 9038682267383528955071403776*T5^26 + 9030559832896215093225509704*T5^25 + 10679136849403400028933205170*T5^24 + 12046149039406097339797691728*T5^23 + 12454921531193677312089399076*T5^22 + 10488900755477493743924275648*T5^21 + 7300512189273757318584617552*T5^20 + 3861954655599715538130323728*T5^19 + 1494187808448604204302725908*T5^18 + 274692397224121783707022736*T5^17 - 69505399032329444215999467*T5^16 - 84504859580703143171724384*T5^15 - 25644796238826103082132452*T5^14 - 2809000994189646995360400*T5^13 + 1209095642532106491386624*T5^12 + 478798673862604473969360*T5^11 + 90338906505036710599812*T5^10 + 897327161857589249792*T5^9 - 2300374774063984253274*T5^8 - 555534642776203351752*T5^7 - 14071704409597315880*T5^6 + 4612290790356745344*T5^5 + 946696835390787720*T5^4 + 7660333835491264*T5^3 - 43937384113308*T5^2 + 1118676799064*T5 + 15527402881 acting on $S_{2}^{\mathrm{new}}(714, [\chi])$.

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:	\( N \)	\(=\)	\( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	714.bh (of order \(24\), degree \(8\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 25.12
Significant digits:
Format:

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