LMFDB - Newform orbit 102.4.i.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [102,4,Mod(5,102)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(102, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([8, 5]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("102.5");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 102 = 2 \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	102.i (of order $16$, 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:	$6.01819482059$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$9$ over $\Q(\zeta_{16})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-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) = $ $ 72 q+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 72 q + 144 q^{11} - 96 q^{12} + 168 q^{15} + 528 q^{17} + 224 q^{18} + 216 q^{21} - 128 q^{24} - 624 q^{25} + 48 q^{29} + 432 q^{31} - 1040 q^{33} + 336 q^{37} - 720 q^{38} + 1312 q^{39} + 288 q^{41} + 1488 q^{42} - 1272 q^{43} - 2704 q^{45} - 480 q^{46} + 1296 q^{47} - 1536 q^{49} - 2832 q^{50} - 1568 q^{51} - 768 q^{52} + 192 q^{53} - 592 q^{54} - 960 q^{55} + 1496 q^{57} + 96 q^{58} - 3648 q^{59} + 640 q^{60} + 960 q^{61} + 960 q^{62} + 5160 q^{63} + 528 q^{66} + 1040 q^{69} - 2016 q^{70} + 1632 q^{71} + 1024 q^{72} - 3072 q^{73} - 864 q^{74} - 2144 q^{75} + 2640 q^{77} + 2624 q^{78} + 1728 q^{79} - 768 q^{80} - 2152 q^{81} + 4992 q^{82} + 5400 q^{83} + 1856 q^{84} + 6768 q^{85} + 544 q^{87} + 1152 q^{88} - 3888 q^{89} + 4320 q^{91} + 960 q^{92} + 4896 q^{93} - 192 q^{94} - 4704 q^{95} - 3600 q^{97} + 480 q^{98} - 6440 q^{99}+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} $
5.1	−1.84776	+	0.765367i	−5.08778	−	1.05572i	2.82843	−	2.82843i	−19.9741	−	3.97309i	10.2090	−	1.94331i	1.21959	+	6.13128i	−3.06147	+	7.39104i	24.7709	+	10.7425i	39.9481	−	7.94617i
5.2	−1.84776	+	0.765367i	−4.63677	+	2.34528i	2.82843	−	2.82843i	11.0539	+	2.19876i	6.77265	−	7.88234i	−3.04002	−	15.2832i	−3.06147	+	7.39104i	15.9994	−	21.7490i	−22.1079	+	4.39753i
5.3	−1.84776	+	0.765367i	−3.71611	−	3.63187i	2.82843	−	2.82843i	8.45257	+	1.68132i	9.64620	+	3.86664i	−1.91006	−	9.60254i	−3.06147	+	7.39104i	0.618995	+	26.9929i	−16.9051	+	3.36264i
5.4	−1.84776	+	0.765367i	−2.48392	+	4.56400i	2.82843	−	2.82843i	−2.48835	−	0.494963i	1.09655	−	10.3343i	5.44526	+	27.3752i	−3.06147	+	7.39104i	−14.6603	−	22.6733i	4.97669	−	0.989925i
5.5	−1.84776	+	0.765367i	0.522718	+	5.16979i	2.82843	−	2.82843i	−7.93672	−	1.57871i	−4.92265	−	9.15246i	−5.68524	−	28.5817i	−3.06147	+	7.39104i	−26.4535	+	5.40469i	15.8734	−	3.15742i
5.6	−1.84776	+	0.765367i	1.99426	−	4.79822i	2.82843	−	2.82843i	9.42218	+	1.87419i	−0.0125062	+	10.3923i	5.91243	+	29.7238i	−3.06147	+	7.39104i	−19.0459	−	19.1378i	−18.8444	+	3.74837i
5.7	−1.84776	+	0.765367i	2.36253	−	4.62801i	2.82843	−	2.82843i	−9.91973	−	1.97316i	−0.823256	+	10.3596i	−1.76532	−	8.87488i	−3.06147	+	7.39104i	−15.8369	−	21.8676i	19.8395	−	3.94631i
5.8	−1.84776	+	0.765367i	4.54774	+	2.51358i	2.82843	−	2.82843i	−6.58809	−	1.31045i	−10.3269	−	1.16380i	3.74326	+	18.8187i	−3.06147	+	7.39104i	14.3638	+	22.8622i	13.1762	−	2.62091i
5.9	−1.84776	+	0.765367i	5.14139	−	0.752393i	2.82843	−	2.82843i	15.6822	+	3.11937i	−8.92420	+	5.32529i	−3.91988	−	19.7066i	−3.06147	+	7.39104i	25.8678	−	7.73669i	−31.3643	+	6.23875i
11.1	0.765367	−	1.84776i	−4.89130	−	1.75362i	−2.82843	−	2.82843i	−3.44244	+	5.15198i	−6.98391	+	7.69578i	−3.12286	+	2.08663i	−7.39104	+	3.06147i	20.8496	+	17.1550i	6.88488	+	10.3040i
11.2	0.765367	−	1.84776i	−3.85766	+	3.48115i	−2.82843	−	2.82843i	−2.26019	+	3.38262i	3.47980	+	9.79239i	23.3608	−	15.6092i	−7.39104	+	3.06147i	2.76315	−	26.8582i	4.52039	+	6.76524i
11.3	0.765367	−	1.84776i	−2.62362	−	4.48516i	−2.82843	−	2.82843i	8.13727	−	12.1783i	−10.2955	+	1.41502i	6.46051	−	4.31678i	−7.39104	+	3.06147i	−13.2333	+	23.5347i	−16.2745	−	24.3566i
11.4	0.765367	−	1.84776i	−1.05214	+	5.08852i	−2.82843	−	2.82843i	−0.312864	+	0.468234i	8.59708	+	5.83868i	−19.0832	+	12.7510i	−7.39104	+	3.06147i	−24.7860	−	10.7076i	0.625728	+	0.936468i
11.5	0.765367	−	1.84776i	0.943618	−	5.10975i	−2.82843	−	2.82843i	−9.01808	+	13.4965i	−8.71938	−	5.65442i	−29.1873	+	19.5024i	−7.39104	+	3.06147i	−25.2192	−	9.64332i	18.0362	+	26.9930i
11.6	0.765367	−	1.84776i	3.26855	−	4.03937i	−2.82843	−	2.82843i	6.25043	−	9.35443i	−4.96215	−	9.13111i	−0.311950	+	0.208438i	−7.39104	+	3.06147i	−5.63310	−	26.4058i	−12.5009	−	18.7089i
11.7	0.765367	−	1.84776i	3.55328	+	3.79133i	−2.82843	−	2.82843i	8.81029	−	13.1855i	9.72503	−	3.66385i	3.44910	−	2.30462i	−7.39104	+	3.06147i	−1.74838	+	26.9433i	−17.6206	−	26.3711i
11.8	0.765367	−	1.84776i	3.78694	+	3.55796i	−2.82843	−	2.82843i	−11.1669	+	16.7125i	9.47265	−	4.27421i	3.40122	−	2.27262i	−7.39104	+	3.06147i	1.68182	+	26.9476i	22.3338	+	33.4249i
11.9	0.765367	−	1.84776i	4.84140	−	1.88701i	−2.82843	−	2.82843i	−2.54078	+	3.80255i	0.218711	−	10.3900i	15.0337	−	10.0452i	−7.39104	+	3.06147i	19.8784	−	18.2716i	5.08156	+	7.60510i
23.1	−0.765367	+	1.84776i	−4.91913	+	1.67397i	−2.82843	−	2.82843i	−10.4404	−	6.97608i	0.671841	−	10.3706i	7.64039	+	11.4346i	7.39104	−	3.06147i	21.3956	−	16.4690i	20.8809	−	13.9522i
23.2	−0.765367	+	1.84776i	−4.35908	−	2.82816i	−2.82843	−	2.82843i	0.344220	+	0.230000i	8.56204	−	5.88994i	−2.37864	−	3.55988i	7.39104	−	3.06147i	11.0031	+	24.6563i	−0.688440	+	0.460001i

See all 72 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
51.i	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	102.4.i.b	yes	72
3.b	odd	2	1		102.4.i.a	✓	72
17.e	odd	16	1		102.4.i.a	✓	72
51.i	even	16	1	inner	102.4.i.b	yes	72

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
102.4.i.a	✓	72	3.b	odd	2	1
102.4.i.a	✓	72	17.e	odd	16	1
102.4.i.b	yes	72	1.a	even	1	1	trivial
102.4.i.b	yes	72	51.i	even	16	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{72} + 312 T_{5}^{70} + 48 T_{5}^{69} + 41988 T_{5}^{68} + 3015360 T_{5}^{67} + \cdots + 65\!\cdots\!72 $ T5^72 + 312*T5^70 + 48*T5^69 + 41988*T5^68 + 3015360*T5^67 + 3220928*T5^66 + 1243385568*T5^65 + 312127752*T5^64 + 150143831552*T5^63 + 4541107735512*T5^62 - 108660247231776*T5^61 + 2756895933557424*T5^60 - 26850967941723744*T5^59 + 293739561133832808*T5^58 - 3947319543815996416*T5^57 + 58234017104457433485*T5^56 - 448102457184461002272*T5^55 - 9821440066666667932720*T5^54 - 7053122533692263725392*T5^53 + 2254226208890658334446396*T5^52 - 5997327123439344538265376*T5^51 + 1337707834135596154177667016*T5^50 + 22508165104912652198486443392*T5^49 + 170870430484604403179188717192*T5^48 + 4269141213002382843686536697472*T5^47 + 26172565273409931659823516428496*T5^46 - 213033549320431807732284498214112*T5^45 + 720837031795172142066490258201656*T5^44 - 75275382821765423757008113812194496*T5^43 - 1970727236974585764532502317035801488*T5^42 - 2485310187639444677868023322593291328*T5^41 + 77967861006834583766807270082558476988*T5^40 - 341641516255737284234141792784497718528*T5^39 + 19280362653110396124140418110348706997824*T5^38 + 352923434177906804179742463595230999558528*T5^37 - 400929392248928670824762143986876653790560*T5^36 - 18774410625428651795013996511433419532475648*T5^35 + 121270295087704219755669455591213128142024256*T5^34 + 993107183466159345563444519513597241071242752*T5^33 - 16758550020461718233402387349676878660026207040*T5^32 - 93271800506805693985865122740467047868661531648*T5^31 + 1854101195751185862314718346619896818347860333056*T5^30 + 25583072377372011421465285053047639391089655734016*T5^29 + 39006511069298367514613442576321428457146564217024*T5^28 - 2774346231043023861078493070058047962769682681273344*T5^27 - 44647280472479979785215937736924662196913447223903232*T5^26 - 317456332262278921030310385838978761033441545512327680*T5^25 + 538839497041720867768189108025535801193693078817525360*T5^24 + 40856436443604368330850056604634480881458147909977751040*T5^23 + 489529825815617893604369670226073065809552470788573293312*T5^22 + 2728446808017295084927002103093040901190550718146485234432*T5^21 - 3483105931661481048580739399387812300367565136081031817792*T5^20 - 213436670363877314736231931274432582251539559967657638666752*T5^19 - 2069914976389731555311508102060806936577847942753334240301952*T5^18 - 10345057365884655461690048445174313608311074848549751328090112*T5^17 - 7737805032707194135344201285670677302256662876297274520085376*T5^16 + 365858450598142534629988906100617730105217438444468595505563648*T5^15 + 3969468262691697902247262009713195406946870204204129118499180800*T5^14 + 26580125367618887162875393127692224704538834723271899588841652736*T5^13 + 135477526469285598180503629672400804523855415877509022058070317952*T5^12 + 558312021324900201250363833405762418399507522074502516225349698560*T5^11 + 1898412183687117055430302897967290010965938238719182533653761188608*T5^10 + 5329597552837480751441960884375604189285012823241724034707924442112*T5^9 + 12176284243764096583436534516304344421174583601598443038512637128768*T5^8 + 21917691232070885417034113808406074761415313164022501347518968086528*T5^7 + 29240998350324938585953893969257190432366649953569904022124325864448*T5^6 + 25494131001834638706610396006644645401280792127949267337944371236864*T5^5 + 10210466846537897507244873890832551423895480316732946926806842234880*T5^4 - 1336087384959036696533779466029794503821469523595730157326428160000*T5^3 - 673759244797455090666826200257054066503368148870803623663879798784*T5^2 + 146116234377552236274013095969111547137263873536779057245307469824*T5 + 651563992336081342167891993256378397152659746675101640294535299072 acting on $S_{4}^{\mathrm{new}}(102, [\chi])$.

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:	\( N \)	\(=\)	\( 102 = 2 \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	102.i (of order \(16\), degree \(8\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 5.9
Significant digits:
Format:

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