LMFDB - Newform orbit 315.8.a.o

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,8,Mod(1,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$N$	$=$	$315 = 3^{2} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$98.4012830275$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 3x^{5} - 570x^{4} - 276x^{3} + 67464x^{2} + 110400x - 1253696$ x^6 - 3x^5 - 570x^4 - 276x^3 + 67464x^2 + 110400*x - 1253696
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{6}\cdot 3^{4}$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

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}

1.1

22.6783
11.7546
3.78638
−6.32873
−10.9234
−17.9671

−18.6783

220.878

−125.000

−343.000

−1734.80

2334.78

1.2

−7.75456

−67.8668

−125.000

−343.000

1518.86

969.320

1.3

0.213623

−127.954

−125.000

−343.000

−54.6777

−26.7029

1.4

10.3287

−21.3173

−125.000

−343.000

−1542.26

−1291.09

1.5

14.9234

94.7076

−125.000

−343.000

−496.836

−1865.42

1.6

21.9671

354.553

−125.000

−343.000

4976.71

−2745.89

Atkin-Lehner signs

$p$	Sign
$3$	$+1$
$5$	$+1$
$7$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.8.a.o	yes	6
3.b	odd	2	1		315.8.a.n	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.8.a.n	✓	6	3.b	odd	2	1
315.8.a.o	yes	6	1.a	even	1	1	trivial

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{6} - 21T_{2}^{5} - 390T_{2}^{4} + 8596T_{2}^{3} + 11352T_{2}^{2} - 493248T_{2} + 104768$ T2^6 - 21*T2^5 - 390*T2^4 + 8596*T2^3 + 11352*T2^2 - 493248*T2 + 104768 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(315))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6} - 21 T^{5} + \cdots + 104768$ T^6 - 21T^5 - 390T^4 + 8596T^3 + 11352T^2 - 493248*T + 104768
$3$	$T^{6}$ T^6
$5$	$(T + 125)^{6}$ (T + 125)^6
$7$	$(T + 343)^{6}$ (T + 343)^6
$11$	$T^{6} + \cdots + 18\!\cdots\!00$ T^6 - 4284T^5 - 25789476T^4 + 109450262768T^3 + 88628534902848T^2 - 395272110721555200*T + 188087223318397644800
$13$	$T^{6} + \cdots + 19\!\cdots\!96$ T^6 - 840T^5 - 173627688T^4 + 123819945760T^3 + 2993870128682640T^2 + 4900562752047277056*T + 1999274311076378218496
$17$	$T^{6} + \cdots - 36\!\cdots\!16$ T^6 + 16188T^5 - 1100206776T^4 - 8444189061952T^3 + 288913442846561616T^2 - 589622788491589446336*T - 3697547760896525658006016
$19$	$T^{6} + \cdots - 12\!\cdots\!00$ T^6 + 17892T^5 - 1099713432T^4 - 13014585644256T^3 + 99855182985107472T^2 + 816764042174719222080*T - 1207886243716426941075200
$23$	$T^{6} + \cdots + 11\!\cdots\!00$ T^6 + 9072T^5 - 10982164920T^4 - 256759926631936T^3 + 18515408154156411216T^2 + 384075984956967869529600*T + 1109023244066133226951654400
$29$	$T^{6} + \cdots - 10\!\cdots\!00$ T^6 - 41244T^5 - 26827945020T^4 + 61602238347776T^3 + 194778739402220043264T^2 + 5306412379208824745164800*T - 105669975330508712028196044800
$31$	$T^{6} + \cdots + 94\!\cdots\!68$ T^6 + 506436T^5 + 71619766884T^4 - 526394900114496T^3 - 561691674594672518016T^2 - 3866831955613519443677952*T + 948771217749655115087092512768
$37$	$T^{6} + \cdots + 11\!\cdots\!24$ T^6 - 91260T^5 - 197003270376T^4 + 25860126330959616T^3 + 1958438773200161618064T^2 - 144835810693321151111439168*T + 1156280923407966836614934874624
$41$	$T^{6} + \cdots + 38\!\cdots\!96$ T^6 - 734556T^5 - 411752264532T^4 + 472300218758278624T^3 - 120565985449072486239120T^2 + 6926544250556541178345430592*T + 380973974665246712492538081010496
$43$	$T^{6} + \cdots + 20\!\cdots\!44$ T^6 - 281772T^5 - 274169648700T^4 + 75708189447587360T^3 + 10366839452610162888192T^2 - 3632905087895202676028505600*T + 201728092417761232942460069817344
$47$	$T^{6} + \cdots - 16\!\cdots\!72$ T^6 + 79524T^5 - 764959166124T^4 - 33572289607265632T^3 + 129442685636831750464320T^2 + 4608400206865392923262348288*T - 163690872846636990332105506844672
$53$	$T^{6} + \cdots + 24\!\cdots\!92$ T^6 - 2902872T^5 + 1515541865328T^4 + 659399245137115952T^3 - 451223408782730818305936T^2 - 19939960740611057001056959296*T + 24185145282213325044987316389531392
$59$	$T^{6} + \cdots - 27\!\cdots\!68$ T^6 + 1328592T^5 - 5027085912816T^4 - 4875316974466015744T^3 + 7054684338931646933072640T^2 + 2807589109860628174749759197184*T - 2779384781712414697936837448028704768
$61$	$T^{6} + \cdots + 23\!\cdots\!00$ T^6 + 260568T^5 - 11805147532584T^4 - 5781065946685112160T^3 + 16543790366532047392035600T^2 + 14003697486683144740761352224000*T + 2304787463924745200626954192988160000
$67$	$T^{6} + \cdots - 37\!\cdots\!52$ T^6 - 3876108T^5 - 8025743091564T^4 + 24307399610892089408T^3 + 31334163088763802270478464T^2 - 32554327489390307821709227032576*T - 37440757654467112672593484773311613952
$71$	$T^{6} + \cdots - 53\!\cdots\!04$ T^6 - 937524T^5 - 18978071361312T^4 - 1334160945238279984T^3 + 70738410183969246806292720T^2 + 21947187322051988978669002522368*T - 53070398647071866482758533833911317504
$73$	$T^{6} + \cdots - 83\!\cdots\!96$ T^6 - 5903520T^5 - 16680037081176T^4 + 119522482173442139456T^3 - 179510493611650573795786416T^2 + 88492331007746748050903103197952*T - 8330594374401129164539668135171152896
$79$	$T^{6} + \cdots + 15\!\cdots\!08$ T^6 + 173040T^5 - 40057075928640T^4 - 97515819599518717440T^3 + 15150715519800380176392192T^2 + 43862549181495500671312945643520*T + 1565897232477436447861620538207633408
$83$	$T^{6} + \cdots - 12\!\cdots\!00$ T^6 + 13929552T^5 - 46435556830128T^4 - 1111963987065980743808T^3 - 943692154870151372228757504T^2 + 13941862694423819891007250477056000*T - 12840314773442792017222838526476288000000
$89$	$T^{6} + \cdots - 13\!\cdots\!96$ T^6 - 6242988T^5 - 59070994575828T^4 + 135744926302453978848T^3 + 544567817311378572239238000T^2 - 480003716383448338141531916931264*T - 1375056685180105340059328937495030984896
$97$	$T^{6} + \cdots + 17\!\cdots\!32$ T^6 + 20928096T^5 - 17947534112376T^4 - 1979497681143395299488T^3 - 1989615882145757425339927920T^2 + 47975331595525905811393841856690816*T + 17773495999232643858901121468028561740032
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$f(q)$	$=$	$q + ( - \beta_1 + 4) q^{2} + (\beta_{2} - 3 \beta_1 + 77) q^{4} - 125 q^{5} - 343 q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 486) q^{8} + (125 \beta_1 - 500) q^{10} + ( - \beta_{5} + 18 \beta_{2} + \cdots + 689) q^{11}+ \cdots + ( - 117649 \beta_1 + 470596) q^{98}+O(q^{100})$ q + (-b1 + 4) * q^2 + (b2 - 3b1 + 77) q^4 - 125 * q^5 - 343 * q^7 + (-b5 + b4 - b3 + 5b2 - 83b1 + 486) * q^8 + (125b1 - 500) q^10 + (-b5 + 18b2 + 49b1 + 689) * q^11 + (-2b5 - b4 - 6b3 + 14b2 - 31b1 + 154) * q^13 + (343b1 - 1372) q^14 + (-7b5 + 19b4 - 7b3 + 87b2 - 629b1 + 8422) q^16 + (7b5 + b4 - 14b3 + 52b2 + 126b1 - 2757) * q^17 + (-8b5 - 9b4 + 9b3 + 44b2 - 381b1 - 2800) q^19 + (-125b2 + 375b1 - 9625) * q^20 + (-21b5 + 17b4 - 7b3 + 66b2 - 3146b1 - 4397) q^22 + (-44b4 + 51b3 - 2b2 + 1236b1 - 2152) * q^23 + 15625 * q^25 + (7b5 + 25b4 + 7b3 + 300b2 - 1676b1 + 8337) q^26 + (-343b2 + 1029b1 - 26411) * q^28 + (-7b5 - 43b4 + 36b3 + 384b2 - 3564b1 + 8631) q^29 + (21b5 - 72b4 - 5b3 - 268b2 - 3221b1 - 82821) q^31 + (-9b5 + 189b4 - 105b3 + 1317b2 - 8815b1 + 102102) q^32 + (22b5 + 126b4 - 168b3 - 298b2 - 4388b1 - 26656) q^34 + 42875 * q^35 + (49b5 + 97b4 + 56b3 + 996b2 - 6084b1 + 18325) q^37 + (-77b5 - 99b4 + 161b3 + 714b2 - 2536b1 + 64131) q^38 + (125b5 - 125b4 + 125b3 - 625b2 + 10375b1 - 60750) q^40 + (77b5 - 99b4 - 357b3 + 918b2 - 2840b1 + 123835) q^41 + (112b5 - 164b4 + 63b3 + 1382b2 + 1196b1 + 46338) q^43 + (-24b5 + 260b4 - 36b3 + 3184b2 - 6644b1 + 496352) q^44 + (-70b5 - 690b4 + 588b3 - 3258b2 + 176b1 - 249020) q^46 + (98b5 - 388b4 + 133b3 + 1190b2 + 14722b1 - 20760) q^47 + 117649 * q^49 + (-15625b1 + 62500) q^50 + (-126b5 + 682b4 + 130b3 + 430b2 - 44990b1 + 369252) q^52 + (-301b5 + 532b4 - 196b3 + 2450b2 - 14583b1 + 491219) q^53 + (125b5 - 2250b2 - 6125b1 - 86125) q^55 + (343b5 - 343b4 + 343b3 - 1715b2 + 28469b1 - 166698) q^56 + (-420b5 - 240b4 + 238b3 + 3162b2 - 57734b1 + 749166) q^58 + (-406b5 - 594b4 - 126b3 + 820b2 - 63764b1 - 190050) q^59 + (703b5 - 241b4 - 1068b3 + 2792b2 - 71688b1 - 7353) q^61 + (567b5 - 1019b4 + 819b3 - 684b2 + 123242b1 + 242201) q^62 + (-595b5 + 1375b4 - 2611b3 + 7639b2 - 192661b1 + 1177666) q^64 + (250b5 + 125b4 + 750b3 - 1750b2 + 3875b1 - 19250) q^65 + (-371b5 + 297b4 + 1540b3 - 3896b2 - 76772b1 + 684367) q^67 + (-238b5 + 1654b4 + 126b3 + 1572b2 + 58204b1 + 1061610) q^68 + (-42875b1 + 171500) q^70 + (294b5 + 1889b4 - 216b3 - 6910b2 - 27265b1 + 170978) q^71 + (-492b5 + 819b4 - 1330b3 - 398b2 - 208241b1 + 1088204) q^73 + (-1364b5 + 1888b4 - 2490b3 + 5622b2 - 155700b1 + 1353554) q^74 + (-268b5 + 56b4 + 392b3 - 958b2 - 110218b1 + 1155886) q^76 + (343b5 - 6174b2 - 16807b1 - 236327) q^77 + (-441b5 - 2573b4 + 2891b3 - 5466b2 - 1908b1 - 29393) q^79 + (875b5 - 2375b4 + 875b3 - 10875b2 + 78625b1 - 1052750) q^80 + (1038b5 + 1334b4 - 1390b3 + 2544b2 - 235582b1 + 1171434) q^82 + (-651b5 - 3305b4 - 2415b3 - 19402b2 - 11670b1 - 2317735) q^83 + (-875b5 - 125b4 + 1750b3 - 6500b2 - 15750b1 + 344625) q^85 + (-806b5 - 562b4 - 684b3 - 12514b2 - 242314b1 + 125604) q^86 + (-1204b5 + 3988b4 - 4956b3 + 13608b2 - 525928b1 + 4213228) q^88 + (-1414b5 + 4014b4 - 532b3 - 10504b2 + 28396b1 + 1027600) q^89 + (686b5 + 343b4 + 2058b3 - 4802b2 + 10633b1 - 52822) q^91 + (2766b5 - 7638b4 + 6266b3 - 28740b2 + 533188b1 - 1250122) q^92 + (-264b5 - 3512b4 + 2266b3 - 32178b2 - 159582b1 - 2744434) q^94 + (1000b5 + 1125b4 - 1125b3 - 5500b2 + 47625b1 + 350000) q^95 + (3490b5 - 6425b4 + 1742b3 + 958b2 - 37931b1 - 3470518) q^97 + (-117649b1 + 470596) q^98
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 21 q^{2} + 453 q^{4} - 750 q^{5} - 2058 q^{7} + 2667 q^{8} - 2625 q^{10} + 4284 q^{11} + 840 q^{13} - 7203 q^{14} + 48609 q^{16} - 16188 q^{17} - 17892 q^{19} - 56625 q^{20} - 35808 q^{22} - 9072 q^{23}+ \cdots + 2470629 q^{98}+O(q^{100})$ 6 * q + 21 * q^2 + 453 * q^4 - 750 * q^5 - 2058 * q^7 + 2667 * q^8 - 2625 * q^10 + 4284 * q^11 + 840 * q^13 - 7203 * q^14 + 48609 * q^16 - 16188 * q^17 - 17892 * q^19 - 56625 * q^20 - 35808 * q^22 - 9072 * q^23 + 93750 * q^25 + 44898 * q^26 - 155379 * q^28 + 41244 * q^29 - 506436 * q^31 + 585627 * q^32 - 173544 * q^34 + 257250 * q^35 + 91260 * q^37 + 377706 * q^38 - 333375 * q^40 + 734556 * q^41 + 281772 * q^43 + 2957472 * q^44 - 1491312 * q^46 - 79524 * q^47 + 705894 * q^49 + 328125 * q^50 + 2078874 * q^52 + 2902872 * q^53 - 535500 * q^55 - 914781 * q^56 + 4323774 * q^58 - 1328592 * q^59 - 260568 * q^61 + 1824288 * q^62 + 6485673 * q^64 - 105000 * q^65 + 3876108 * q^67 + 6540024 * q^68 + 900375 * q^70 + 937524 * q^71 + 5903520 * q^73 + 7652652 * q^74 + 6605298 * q^76 - 1469412 * q^77 - 173040 * q^79 - 6076125 * q^80 + 6314742 * q^82 - 13929552 * q^83 + 2023500 * q^85 + 30786 * q^86 + 23693232 * q^88 + 6242988 * q^89 - 288120 * q^91 - 5886552 * q^92 - 16934022 * q^94 + 2236500 * q^95 - 20928096 * q^97 + 2470629 * q^98

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 5\nu - 189$ v^2 - 5*v - 189
$\beta_{3}$	$=$	$( \nu^{5} - 5\nu^{4} - 496\nu^{3} + 524\nu^{2} + 38320\nu + 14176 ) / 96$ (v^5 - 5v^4 - 496v^3 + 524v^2 + 38320v + 14176) / 96
$\beta_{4}$	$=$	$( \nu^{4} - 9\nu^{3} - 424\nu^{2} + 1668\nu + 22024 ) / 12$ (v^4 - 9v^3 - 424v^2 + 1668*v + 22024) / 12
$\beta_{5}$	$=$	$( -\nu^{5} + 13\nu^{4} + 520\nu^{3} - 4588\nu^{2} - 55312\nu + 210112 ) / 96$ (-v^5 + 13v^4 + 520v^3 - 4588v^2 - 55312v + 210112) / 96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Embeddings

Atkin-Lehner signs

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	315.8.a.o

Level	$315$
Weight	$8$
Character orbit	315.a
Self dual	yes
Analytic conductor	$98.401$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$1$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 5\beta _1 + 189$ b2 + 5*b1 + 189
$\nu^{3}$	$=$	$\beta_{5} - \beta_{4} + \beta_{3} + 7\beta_{2} + 351\beta _1 + 822$ b5 - b4 + b3 + 7b2 + 351b1 + 822
$\nu^{4}$	$=$	$9\beta_{5} + 3\beta_{4} + 9\beta_{3} + 487\beta_{2} + 3611\beta _1 + 65510$ 9b5 + 3b4 + 9b3 + 487b2 + 3611*b1 + 65510
$\nu^{5}$	$=$	$541\beta_{5} - 481\beta_{4} + 637\beta_{3} + 5383\beta_{2} + 151211\beta _1 + 622050$ 541b5 - 481b4 + 637b3 + 5383b2 + 151211*b1 + 622050

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.6
Significant digits:
Format: