LMFDB - Newform orbit 208.8.i.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,8,Mod(81,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.81"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")

Level:	$N$	$=$	$208 = 2^{4} \cdot 13$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	208.i (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$64.9760853007$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 4654x^{6} + 7012369x^{4} + 3763719168x^{2} + 637953638400$ x^8 + 4654x^6 + 7012369x^4 + 3763719168*x^2 + 637953638400
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$2^{6}\cdot 3$
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$ .

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1 - \beta_{1}$

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

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

81.1

		41.0833i
		23.0850i
	−	18.4011i
	−	45.7672i
	−	41.0833i
	−	23.0850i
		18.4011i
		45.7672i

−35.5792

−

61.6250i

523.489

−208.401

360.960i

−1438.26

2491.14i

81.2

−19.9922

−

34.6275i

−323.700

284.296

−

492.415i

294.123

−

509.436i

81.3

15.9358

27.6017i

−54.4265

−556.354

963.633i

585.599

−

1014.29i

81.4

39.6356

68.6509i

132.638

754.459

−

1306.76i

−2048.46

3548.04i

113.1

−35.5792

61.6250i

523.489

−208.401

−

360.960i

−1438.26

−

2491.14i

113.2

−19.9922

34.6275i

−323.700

284.296

492.415i

294.123

509.436i

113.3

15.9358

−

27.6017i

−54.4265

−556.354

−

963.633i

585.599

1014.29i

113.4

39.6356

−

68.6509i

132.638

754.459

1306.76i

−2048.46

−

3548.04i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	208.8.i.b		8
4.b	odd	2	1		26.8.c.b	✓	8
12.b	even	2	1		234.8.h.b		8
13.c	even	3	1	inner	208.8.i.b		8
52.i	odd	6	1		338.8.a.j		4
52.j	odd	6	1		26.8.c.b	✓	8
52.j	odd	6	1		338.8.a.i		4
52.l	even	12	2		338.8.b.h		8
156.p	even	6	1		234.8.h.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.8.c.b	✓	8	4.b	odd	2	1
26.8.c.b	✓	8	52.j	odd	6	1
208.8.i.b		8	1.a	even	1	1	trivial
208.8.i.b		8	13.c	even	3	1	inner
234.8.h.b		8	12.b	even	2	1
234.8.h.b		8	156.p	even	6	1
338.8.a.i		4	52.j	odd	6	1
338.8.a.j		4	52.i	odd	6	1
338.8.b.h		8	52.l	even	12	2

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{8} + 6981 T_{3}^{6} + 70848 T_{3}^{5} + 41545881 T_{3}^{4} + 247294944 T_{3}^{3} + \cdots + 51674244710400$ T3^8 + 6981*T3^6 + 70848*T3^5 + 41545881*T3^4 + 247294944*T3^3 + 51437638656*T3^2 - 254644715520*T3 + 51674244710400 acting on $S_{8}^{\mathrm{new}}(208, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$T^{8} + \cdots + 51674244710400$ T^8 + 6981T^6 + 70848T^5 + 41545881T^4 + 247294944T^3 + 51437638656T^2 - 254644715520T + 51674244710400
$5$	$(T^{4} - 278 T^{3} + \cdots + 1223288100)^{2}$ (T^4 - 278T^3 - 161047T^2 + 14695460*T + 1223288100)^2
$7$	$T^{8} + \cdots + 15\!\cdots\!00$ T^8 - 548T^7 + 2156137T^6 + 319494836T^5 + 3237329872481T^4 - 211123021512232T^3 + 860066248350044656T^2 + 138768546778303239040*T + 158325765218468072761600
$11$	$T^{8} + \cdots + 28\!\cdots\!36$ T^8 - 7392T^7 + 62932989T^6 - 78525632848T^5 + 568481585289177T^4 - 327985988938363896T^3 + 5028193702711645222576T^2 + 1189904953869188863522944*T + 289718572183543401616763136
$13$	$T^{8} + \cdots + 15\!\cdots\!21$ T^8 + 25818T^7 + 185875157T^6 - 1047340073598T^5 - 22126099927900332T^4 - 65719036412945354166T^3 + 731860453859947897663373T^2 + 6378712011618342238842534834*T + 15502932802662396215269535105521
$17$	$T^{8} + \cdots + 35\!\cdots\!41$ T^8 - 28316T^7 + 1078243090T^6 - 15895635746160T^5 + 471949877835194859T^4 - 6657207621240828792240T^3 + 124211422567724610113670690T^2 - 707545199659030399323269594892*T + 3558035464367798342122742985068241
$19$	$T^{8} + \cdots + 16\!\cdots\!00$ T^8 - 99888T^7 + 7250713021T^6 - 261328995333264T^5 + 7288847798040542089T^4 - 95987931561779228317440T^3 + 1135018443939417221780275200T^2 + 2238885291021401206290382848000*T + 164045161547528157672364163727360000
$23$	$T^{8} + \cdots + 64\!\cdots\!36$ T^8 - 33388T^7 + 10236952809T^6 + 159710576592476T^5 + 77613398281348763617T^4 - 125226526030359404738136T^3 + 78400204944968287492111605744T^2 + 580846048198212062593122155607168*T + 64309871131446688207363756530370734336
$29$	$T^{8} + \cdots + 65\!\cdots\!69$ T^8 - 93140T^7 + 43374042618T^6 - 969579806575952T^5 + 1144723374331555433251T^4 - 25399464109529801521510608T^3 + 13259788377638956954650862496730T^2 + 535594192967444187048699404113630668*T + 65003200660687042770801404063272778588769
$31$	$(T^{4} + \cdots + 10\!\cdots\!28)^{2}$ (T^4 + 311160T^3 + 30594196816T^2 + 1097320177155840*T + 10826670521778761728)^2
$37$	$T^{8} + \cdots + 27\!\cdots\!41$ T^8 + 9636T^7 + 19228073602T^6 + 1548068841312144T^5 + 376157108499825237595T^4 + 16607327979063290582831568T^3 + 718711680089182958408540315826T^2 + 1432264137263755387936951314313620*T + 2733895335933976510046271689274569841
$41$	$T^{8} + \cdots + 75\!\cdots\!25$ T^8 - 82892T^7 + 232961431666T^6 + 113796669231252432T^5 + 49927070317433470007211T^4 + 10289686280641753115256851088T^3 + 1637165639330274375321766728780546T^2 + 130696622041941468773435561398782702660*T + 7561943203197074537489195988511840509606225
$43$	$T^{8} + \cdots + 19\!\cdots\!96$ T^8 - 569264T^7 + 735698842285T^6 + 52919195313466064T^5 + 225519303521393777280809T^4 - 42391257972164263207504526416T^3 + 6400408590045697263188829843309760T^2 - 402605015916881904039238050202142618624*T + 19701102333213644863231498946682342381162496
$47$	$(T^{4} + \cdots + 87\!\cdots\!28)^{2}$ (T^4 - 574200T^3 - 844716634752T^2 + 32382746130692608*T + 87928744118119307440128)^2
$53$	$(T^{4} + \cdots + 10\!\cdots\!32)^{2}$ (T^4 - 1235350T^3 - 2014607986887T^2 + 1044554447397292020*T + 1027560028805596539292932)^2
$59$	$T^{8} + \cdots + 87\!\cdots\!64$ T^8 + 231504T^7 + 7597791408645T^6 + 359542756215697312T^5 + 47793769676097195773088489T^4 + 3608211321932185893108277203192T^3 + 71759696081831917944482010513469406128T^2 - 9861520003246435283107926711164459698960512*T + 87701789508868790849890781704611581002030482372864
$61$	$T^{8} + \cdots + 36\!\cdots\!25$ T^8 - 685684T^7 + 16004686569946T^6 - 663215582530386128T^5 + 185007533930035418015046851T^4 - 5339524497872409731708342680400T^3 + 967079286407722895740824102323102027386T^2 + 340542486604250733547282246558674783745983980*T + 3623216091429469756993710089825762360360738102037025
$67$	$T^{8} + \cdots + 16\!\cdots\!96$ T^8 + 3271056T^7 + 9553010570525T^6 + 4395372866213511168T^5 + 2497853273225562255871641T^4 + 475942201922976695644610785032T^3 + 251902487463420538223238696946543280T^2 + 41613576360052208791788750678393093032064*T + 16694539374784568280884947109698169235351152896
$71$	$T^{8} + \cdots + 58\!\cdots\!44$ T^8 - 175012T^7 + 13973691446129T^6 + 7875067743687289028T^5 + 169812617491174461712132289T^4 + 46332180563017313624021995519952T^3 + 343702031533338486151493254738715586496T^2 - 65546596876993426569462545159855567506455552*T + 581812223106410907691766688329197219788422663737344
$73$	$(T^{4} + \cdots - 21\!\cdots\!40)^{2}$ (T^4 - 7137890T^3 + 4799979816117T^2 + 19843813129426634812*T - 21425675574457987689091340)^2
$79$	$(T^{4} + \cdots + 46\!\cdots\!08)^{2}$ (T^4 - 7053952T^3 - 13799242194096T^2 + 37946078389247195264*T + 46714127885196996709854208)^2
$83$	$(T^{4} + \cdots - 67\!\cdots\!00)^{2}$ (T^4 + 657288T^3 - 49816293137712T^2 - 137212918633157971968*T - 67989302689771818477158400)^2
$89$	$T^{8} + \cdots + 27\!\cdots\!36$ T^8 + 11452234T^7 + 222249857584831T^6 + 2171721746555680989978T^5 + 31970404566976634600903240157T^4 + 266972101357429830920320530999000108T^3 + 2104619989938381319903415088848018649615996T^2 + 8459496441489961211601189016704534515624143585584*T + 27694489378061763203718104687863217999337817318695787536
$97$	$T^{8} + \cdots + 23\!\cdots\!84$ T^8 + 428002T^7 + 60409957633639T^6 - 173561362696051528798T^5 + 3590795337732551647712597069T^4 - 4454427585336664692993122205531652T^3 + 5751704764977783854914776543543870056476T^2 + 357841315230280593009078599905355196007927792*T + 23452278293784348272783849182345652164529282362384
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$
Belyi maps

$f(q)$	$=$	$q - \beta_{3} q^{3} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 69) q^{5} + (2 \beta_{6} + 7 \beta_{2} + \cdots + 138) q^{7} + ( - 6 \beta_{7} - 3 \beta_{6} + \cdots - 1302) q^{9} + (8 \beta_{6} + 8 \beta_{5} + \cdots - 1844 \beta_1) q^{11}+ \cdots + (1020 \beta_{5} + 7152 \beta_{4} + \cdots + 2592480) q^{99}+O(q^{100})$ q - b3 * q^3 + (-b4 + b3 - b2 + 69) * q^5 + (2b6 + 7b2 + 138b1 + 138) q^7 + (-6b7 - 3b6 + 6b4 - 9b2 - 1308b1 - 1302) q^9 + (8b6 + 8b5 + 19b3 - 1844b1) * q^11 + (13b7 + 13b5 - 26b3 + 13b2 - 1092b1 - 3783) q^13 + (48b7 - 12b6 - 12b5 - 206b3 + 3900b1) q^15 + (34b7 - 14b6 - 34b4 + 196b2 + 7089b1 + 7055) q^17 + (56b7 - 56b4 + 259b2 + 25000b1 + 24944) * q^19 + (-87b5 - 126b4 - 547b3 + 547b2 + 22749) * q^21 + (-176b7 + 74b6 + 74b5 + 399b3 - 8398b1) q^23 + (-43b5 - 168b4 + 1561b3 - 1561b2 + 21657) * q^25 + (216b5 - 72b4 + 591b3 - 591b2 - 26712) * q^27 + (365b7 + 187b6 + 187b5 + 600b3 - 23009b1) q^29 + (88b5 - 100b4 - 204b3 + 204b2 - 77884) * q^31 + (450b7 + 321b6 - 450b4 - 289b2 + 60045b1 + 59595) q^33 + (64b7 - 12b6 - 64b4 + 2918b2 - 35360b1 - 35424) q^35 + (281b7 + 3b6 + 3b5 + 868b3 + 2551b1) q^37 + (-156b7 + 546b6 + 390b5 + 624b4 + 5915b3 + 676b2 - 106470b1 - 50076) q^39 + (-862b7 - 70b6 - 70b5 + 3560b3 - 21189b1) q^41 + (1080b7 + 560b6 - 1080b4 - 5285b2 + 143136b1 + 142056) q^43 + (-1569b7 - 294b6 + 1569b4 + 4743b2 - 576780b1 - 575211) q^45 + (-692b5 + 1080b4 + 8648b3 - 8648b2 + 144436) * q^47 + (462b7 + 231b6 + 231b5 + 14049b3 + 179796b1) q^49 + (-636b5 + 840b4 - 11361b3 + 11361b2 + 683220) * q^51 + (2368b5 - 689b4 + 4153b3 - 4153b2 + 307309) * q^53 + (-1708b7 - 156b6 - 156b5 + 12214b3 - 274060b1) q^55 + (-1617b5 + 798b4 - 35451b3 + 35451b2 + 883323) * q^57 + (5712b7 + 1568b6 - 5712b4 + 12799b2 - 54236b1 - 59948) q^59 + (-729b7 + 2353b6 + 729b4 - 43060b2 + 172233b1 + 172962) q^61 + (-1644b7 - 2028b6 - 2028b5 - 45810b3 - 1489764b1) q^63 + (182b7 + 559b6 - 221b5 + 6110b4 - 14261b3 - 4030b2 + 1159210b1 - 194220) q^65 + (-1568b7 - 2176b6 - 2176b5 + 8379b3 + 815892b1) q^67 + (12894b7 + 999b6 - 12894b4 - 17525b2 + 1406955b1 + 1394061) q^69 + (-3940b7 + 1038b6 + 3940b4 - 39029b2 + 42302b1 + 46242) q^71 + (-2457b5 + 1266b4 - 40803b3 + 40803b2 + 1786334) * q^73 + (14616b7 + 744b6 + 744b5 - 39575b3 + 5557920b1) q^75 + (-4651b5 + 714b4 + 34501b3 - 34501b2 - 3476119) * q^77 + (3516b5 - 9732b4 + 35196b3 - 35196b2 + 1756864) * q^79 + (2520b7 + 1260b6 + 1260b5 + 39204b3 - 937611b1) q^81 + (-4644b5 + 14520b4 + 3948b3 - 3948b2 - 154740) * q^83 + (-2453b7 + 4894b6 + 2453b4 - 21701b2 + 2954970b1 + 2957423) q^85 + (-3876b7 + 13446b6 + 3876b4 + 67667b2 + 1800510b1 + 1804386) q^87 + (-9298b7 - 9577b6 - 9577b5 - 116719b3 + 2853621b1) q^89 + (4368b7 - 2184b6 - 4160b5 - 3536b4 + 23920b3 - 103441b2 - 5581784b1 - 4849364) q^91 + (6672b7 + 792b6 + 792b5 + 76672b3 - 742440b1) q^93 + (15368b7 + 6188b6 - 15368b4 - 43606b2 + 5844300b1 + 5828932) q^95 + (8330b7 + 7973b6 - 8330b4 - 55489b2 - 98849b1 - 107179) q^97 + (1020b5 + 7152b4 - 136674b3 + 136674b2 + 2592480) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 556 q^{5} + 548 q^{7} - 5214 q^{9} + 7392 q^{11} - 25818 q^{13} - 15528 q^{15} + 28316 q^{17} + 99888 q^{19} + 182148 q^{21} + 33388 q^{23} + 173756 q^{25} - 212544 q^{27} + 93140 q^{29} - 622320 q^{31}+ \cdots + 20715312 q^{99}+O(q^{100})$ 8 * q + 556 * q^5 + 548 * q^7 - 5214 * q^9 + 7392 * q^11 - 25818 * q^13 - 15528 * q^15 + 28316 * q^17 + 99888 * q^19 + 182148 * q^21 + 33388 * q^23 + 173756 * q^25 - 212544 * q^27 + 93140 * q^29 - 622320 * q^31 + 238638 * q^33 - 141544 * q^35 - 9636 * q^37 + 22932 * q^39 + 82892 * q^41 + 569264 * q^43 - 2303394 * q^45 + 1148400 * q^47 - 717798 * q^49 + 5459856 * q^51 + 2470700 * q^53 + 1092512 * q^55 + 7056924 * q^57 - 231504 * q^59 + 685684 * q^61 + 5951712 * q^63 - 6216678 * q^65 - 3271056 * q^67 + 5600034 * q^69 + 175012 * q^71 + 14275780 * q^73 - 22200960 * q^75 - 27830412 * q^77 + 14107904 * q^79 + 3758004 * q^81 - 1314576 * q^83 + 11814998 * q^85 + 7182900 * q^87 - 11452234 * q^89 - 16457168 * q^91 + 2984688 * q^93 + 23334088 * q^95 - 428002 * q^97 + 20715312 * q^99

$\beta_{1}$	$=$	$( \nu^{7} + 4654\nu^{5} + 6213649\nu^{3} + 1905097728\nu - 3143761920 ) / 6287523840$ (v^7 + 4654v^5 + 6213649v^3 + 1905097728*v - 3143761920) / 6287523840
$\beta_{2}$	$=$	$( \nu^{4} + 2327\nu^{2} + 11808\nu + 798720 ) / 7872$ (v^4 + 2327v^2 + 11808v + 798720) / 7872
$\beta_{3}$	$=$	$( -\nu^{4} - 2327\nu^{2} + 11808\nu - 798720 ) / 7872$ (-v^4 - 2327v^2 + 11808v - 798720) / 7872
$\beta_{4}$	$=$	$( -\nu^{6} - 3766\nu^{4} - 3863881\nu^{2} - 854819328 ) / 661248$ (-v^6 - 3766v^4 - 3863881v^2 - 854819328) / 661248
$\beta_{5}$	$=$	$( \nu^{6} + 4018\nu^{4} + 4780909\nu^{2} + 1440612480 ) / 330624$ (v^6 + 4018v^4 + 4780909v^2 + 1440612480) / 330624
$\beta_{6}$	$=$	$( - 381 \nu^{7} - 13312 \nu^{6} - 1613430 \nu^{5} - 53487616 \nu^{4} - 2152864077 \nu^{3} + \cdots - 19177433333760 ) / 8802533376$ (-381v^7 - 13312v^6 - 1613430v^5 - 53487616v^4 - 2152864077v^3 - 63643460608v^2 - 864056577024*v - 19177433333760) / 8802533376
$\beta_{7}$	$=$	$( - 625 \nu^{7} - 6656 \nu^{6} - 2429518 \nu^{5} - 25066496 \nu^{4} - 2689763713 \nu^{3} + \cdots - 5685276180480 ) / 8802533376$ (-625v^7 - 6656v^6 - 2429518v^5 - 25066496v^4 - 2689763713v^3 - 25717991936v^2 - 694817061888*v - 5685276180480) / 8802533376

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