LMFDB - Newform orbit 1338.2.e.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1338,2,Mod(931,1338)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1338, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1338.931"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1338 = 2 \cdot 3 \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1338.e (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,14,7] 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:	$10.6839837904$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 25 x^{12} - 30 x^{11} + 502 x^{10} - 434 x^{9} + 3060 x^{8} - 1136 x^{7} + 13014 x^{6} + \cdots + 961 $ x^14 + 25x^12 - 30x^11 + 502x^10 - 434x^9 + 3060x^8 - 1136x^7 + 13014x^6 - 2882x^5 + 18706x^4 + 14706x^3 + 12571x^2 + 3720x + 961
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{13}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + q^{2} + \beta_{6} q^{3} + q^{4} - \beta_{11} q^{5} + \beta_{6} q^{6} + \beta_{10} q^{7} + q^{8} + (\beta_{6} - 1) q^{9} - \beta_{11} q^{10} + ( - \beta_{6} - \beta_{2} + 1) q^{11} + \beta_{6} q^{12}+ \cdots + (\beta_{6} + \beta_1) q^{99}+O(q^{100}) $ q + q^2 + b6 * q^3 + q^4 - b11 * q^5 + b6 * q^6 + b10 * q^7 + q^8 + (b6 - 1) * q^9 - b11 * q^10 + (-b6 - b2 + 1) * q^11 + b6 * q^12 + (b10 - b5 - b4 - 1) * q^13 + b10 * q^14 - b8 * q^15 + q^16 - b9 * q^17 + (b6 - 1) * q^18 + (-b12 - b7 - b6 + b1) * q^19 - b11 * q^20 + b13 * q^21 + (-b6 - b2 + 1) * q^22 + (b12 + b11 - b8 - 2b6 - b1) q^23 + b6 * q^24 + (-b13 + b11 - b8 - 2b6 - b5 - b3 - b1) q^25 + (b10 - b5 - b4 - 1) * q^26 - q^27 + b10 * q^28 + (b6 - b2 - 1) * q^29 - b8 * q^30 + (b7 - 3b6 + b1) q^31 + q^32 + (-b2 + b1 + 1) * q^33 - b9 * q^34 + (b13 - 2b11 - b10 + b7 + b6 + b4 - 2b3 + b2 - 1) * q^35 + (b6 - 1) * q^36 + (b13 + b12 - b11 - b10 - b9 + b6 + 2b3 + b2 - 1) q^37 + (-b12 - b7 - b6 + b1) * q^38 + (b13 + b7 - b6 - b5 - b3) * q^39 - b11 * q^40 + (-b8 - b4 + 2b2 - 2b1) * q^41 + b13 * q^42 + (-b11 + b8 + b7 + 3b6) q^43 + (-b6 - b2 + 1) * q^44 + (b11 - b8) * q^45 + (b12 + b11 - b8 - 2b6 - b1) q^46 + (b12 - b11 - b9 - b7 + b6 - b4 - 1) * q^47 + b6 * q^48 + (b8 - 2b5 + b4 - b2 + b1 + 6) q^49 + (-b13 + b11 - b8 - 2b6 - b5 - b3 - b1) q^50 - b12 * q^51 + (b10 - b5 - b4 - 1) * q^52 + (-b13 - b12 + b11 + b10 + b9 - 2b6 - 2b2 + 2) * q^53 - q^54 + (-3b6 - b5 - b3 - 2b1) * q^55 + b10 * q^56 + (-b12 + b9 - b7 - b6 - b4 + b2 + 1) * q^57 + (b6 - b2 - 1) * q^58 + (-b10 - 2b8 - b5 + 2b4) * q^59 - b8 * q^60 + (-b11 + b8 - b7 + 3b5 + 3b3 + 2b1) q^61 + (b7 - 3b6 + b1) q^62 + (b13 - b10) * q^63 + q^64 + (2b13 + b11 - 2b10 - 2b7 - 2b4 - 2b3 + 2b2) * q^65 + (-b2 + b1 + 1) * q^66 + (b12 - 3b11 + 3b8 + b7 + b6 + 2b5 + 2b3 + b1) * q^67 - b9 * q^68 + (b12 + b11 - b9 - 2b6 - b2 + 2) q^69 + (b13 - 2b11 - b10 + b7 + b6 + b4 - 2b3 + b2 - 1) * q^70 + (4b6 + b5 + b3) q^71 + (b6 - 1) * q^72 + (-b13 - b12 + 2b11 + b10 + b9 - b7 + 7b6 - b4 - 3b3 - b2 - 7) q^73 + (b13 + b12 - b11 - b10 - b9 + b6 + 2b3 + b2 - 1) q^74 + (-b13 + b11 + b10 - 2b6 - b3 - b2 + 2) q^75 + (-b12 - b7 - b6 + b1) * q^76 + (-b13 - b12 + b11 + b10 + b9 + b7 + 2b6 + b4 - 2b3 - b2 - 2) * q^77 + (b13 + b7 - b6 - b5 - b3) * q^78 + (-b11 - b7 + 7b6 - b4 - b3 + 3b2 - 7) * q^79 - b11 * q^80 - b6 * q^81 + (-b8 - b4 + 2b2 - 2b1) * q^82 + (-b13 - b12 + b10 + b9 - 2b7 + b6 - 2b4 - b3 + 4b2 - 1) q^83 + b13 * q^84 + (b12 - 2b11 - b9 + b7 + b6 + b4 + 3b3 - 1) * q^85 + (-b11 + b8 + b7 + 3b6) q^86 + (-b2 + b1 - 1) * q^87 + (-b6 - b2 + 1) * q^88 + (b13 + b12 + 2b11 - 2b8 - 5b6 + 2b5 + 2b3) q^89 + (b11 - b8) * q^90 + (2b9 + 4b8 - b5 - b4 + 2b2 - 2b1 + 8) * q^91 + (b12 + b11 - b8 - 2b6 - b1) q^92 + (b7 - 3b6 + b4 + b2 + 3) q^93 + (b12 - b11 - b9 - b7 + b6 - b4 - 1) * q^94 + (-b10 - 2b9 - b8 - b5 + 4b4 - 2b2 + 2b1) * q^95 + b6 * q^96 + (-2b13 - 2b12 + b11 + 2b10 + 2b9 + 2b6 - 2b3 - 3b2 - 2) q^97 + (b8 - 2b5 + b4 - b2 + b1 + 6) q^98 + (b6 + b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{2} + 7 q^{3} + 14 q^{4} + 3 q^{5} + 7 q^{6} + 4 q^{7} + 14 q^{8} - 7 q^{9} + 3 q^{10} + 7 q^{11} + 7 q^{12} - 12 q^{13} + 4 q^{14} + 6 q^{15} + 14 q^{16} + 6 q^{17} - 7 q^{18} - 6 q^{19} + 3 q^{20}+ \cdots + 7 q^{99}+O(q^{100}) $ 14 * q + 14 * q^2 + 7 * q^3 + 14 * q^4 + 3 * q^5 + 7 * q^6 + 4 * q^7 + 14 * q^8 - 7 * q^9 + 3 * q^10 + 7 * q^11 + 7 * q^12 - 12 * q^13 + 4 * q^14 + 6 * q^15 + 14 * q^16 + 6 * q^17 - 7 * q^18 - 6 * q^19 + 3 * q^20 + 2 * q^21 + 7 * q^22 - 14 * q^23 + 7 * q^24 - 16 * q^25 - 12 * q^26 - 14 * q^27 + 4 * q^28 - 7 * q^29 + 6 * q^30 - 19 * q^31 + 14 * q^32 + 14 * q^33 + 6 * q^34 + q^35 - 7 * q^36 - 9 * q^37 - 6 * q^38 - 6 * q^39 + 3 * q^40 + 10 * q^41 + 2 * q^42 + 20 * q^43 + 7 * q^44 + 3 * q^45 - 14 * q^46 + q^47 + 7 * q^48 + 62 * q^49 - 16 * q^50 + 3 * q^51 - 12 * q^52 + 10 * q^53 - 14 * q^54 - 24 * q^55 + 4 * q^56 + 6 * q^57 - 7 * q^58 - 6 * q^59 + 6 * q^60 + 4 * q^61 - 19 * q^62 - 2 * q^63 + 14 * q^64 + 3 * q^65 + 14 * q^66 + 3 * q^67 + 6 * q^68 + 14 * q^69 + q^70 + 31 * q^71 - 7 * q^72 - 45 * q^73 - 9 * q^74 + 16 * q^75 - 6 * q^76 - 14 * q^77 - 6 * q^78 - 41 * q^79 + 3 * q^80 - 7 * q^81 + 10 * q^82 - q^83 + 2 * q^84 - 9 * q^85 + 20 * q^86 - 14 * q^87 + 7 * q^88 - 24 * q^89 + 3 * q^90 + 74 * q^91 - 14 * q^92 + 19 * q^93 + q^94 - 8 * q^95 + 7 * q^96 - 13 * q^97 + 62 * q^98 + 7 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{14} + 25 x^{12} - 30 x^{11} + 502 x^{10} - 434 x^{9} + 3060 x^{8} - 1136 x^{7} + 13014 x^{6} + \cdots + 961 $ x^14 + 25*x^12 - 30*x^11 + 502*x^10 - 434*x^9 + 3060*x^8 - 1136*x^7 + 13014*x^6 - 2882*x^5 + 18706*x^4 + 14706*x^3 + 12571*x^2 + 3720*x + 961 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 11470254167885 \nu^{13} - 59106738542239 \nu^{12} + 363720839343838 \nu^{11} + \cdots + 35\!\cdots\!75 ) / 13\!\cdots\!64 $ (11470254167885v^13 - 59106738542239v^12 + 363720839343838v^11 - 1754394406062416v^10 + 9389024558365770v^9 - 35438170356484804v^8 + 96265881512912216v^7 - 195030354816227612v^6 + 405048272605203934v^5 - 741308758794623916v^4 + 1432684246032290954v^3 - 610803255496107640v^2 + 1161267268128266503*v + 350144884551363175) / 1345765527540230064
$\beta_{3}$	$=$	$ ( - 17\!\cdots\!67 \nu^{13} + \cdots + 26\!\cdots\!73 ) / 22\!\cdots\!12 $ (-17684849775760667v^13 - 40671567747060335v^12 - 353134260455773912v^11 - 522550959898914064v^10 - 5415447999593503956v^9 - 16325810312876848790v^8 + 9613152557815877332v^7 - 161104833688389160906v^6 + 111113199289044351692v^5 - 682090494796151213448v^4 + 970985364405680720956v^3 - 1495917282408544973006v^2 + 857136776581280467823*v + 260402583403020088673) / 229453022445609225912
$\beta_{4}$	$=$	$ ( 45\!\cdots\!29 \nu^{13} + \cdots + 77\!\cdots\!27 ) / 44\!\cdots\!12 $ (4523264687314129v^13 - 5038947078223583v^12 + 108022845027388022v^11 - 279006096622362232v^10 + 2289678839189628498v^9 - 4706509032668545604v^8 + 13994100423659473744v^7 - 25190131127396899180v^6 + 54490917796913139494v^5 - 93767415414363891396v^4 + 60100612447172540146v^3 - 78282559291828844888v^2 - 23665714940385288997*v + 77695239329628075527) / 44410262408827592112
$\beta_{5}$	$=$	$ ( - 374253871098011 \nu^{13} + \cdots - 29\!\cdots\!27 ) / 24\!\cdots\!84 $ (-374253871098011v^13 + 1077974981475101v^12 - 10219764710202050v^11 + 38031325845153622v^10 - 241117887076323524v^9 + 724973617539200272v^8 - 2025618758016265410v^7 + 3981389884494732118v^6 - 8318671741356446102v^5 + 14947244256455711658v^4 - 18821111846363777680v^3 + 12363410009294109672v^2 + 3735393803732308237*v - 2940727949087983727) / 2467236800490421784
$\beta_{6}$	$=$	$ ( - 11\!\cdots\!25 \nu^{13} + 355577879204435 \nu^{12} + \cdots - 60\!\cdots\!07 ) / 41\!\cdots\!84 $ (-11294996275850425v^13 + 355577879204435v^12 - 284207215791070034v^11 + 350125234295171728v^10 - 5724474357064848246v^9 + 5193088145028423320v^8 - 35661271885153329424v^7 + 15815358096266361496v^6 - 153039022533220486922v^5 + 45108675717762246804v^4 - 234264771858691391446v^3 - 121691003605655330476v^2 - 160924299104095029515*v - 6018100834187319407) / 41718731353747131984
$\beta_{7}$	$=$	$ ( 59\!\cdots\!13 \nu^{13} + \cdots + 40\!\cdots\!69 ) / 17\!\cdots\!34 $ (59966827000429313v^13 + 29943772497432911v^12 + 1547352058023727201v^11 - 1072826442744735770v^10 + 30295766829554440524v^9 - 13267220908939166173v^8 + 192465771295303017869v^7 - 10944598997191452314v^6 + 854899747437341194417v^5 + 33882456929525603478v^4 + 1437799785841320458642v^3 + 779190398922354276275v^2 + 1243439271445291149040*v + 40752116606959090669) / 172089766834206919434
$\beta_{8}$	$=$	$ ( 22\!\cdots\!95 \nu^{13} + \cdots - 13\!\cdots\!07 ) / 44\!\cdots\!12 $ (22730829853181995v^13 - 52468941065576117v^12 + 592835275505756882v^11 - 1984878802064473084v^10 + 13588374771462075066v^9 - 36735644413627078916v^8 + 104159741176609860124v^7 - 192612375927781819264v^6 + 422396203335884811386v^5 - 751519565740648484064v^4 + 868647636124387863250v^3 - 622911996700937766920v^2 - 188227391602692628939*v - 130194015728749683607) / 44410262408827592112
$\beta_{9}$	$=$	$ ( - 74\!\cdots\!33 \nu^{13} + \cdots + 87\!\cdots\!65 ) / 74\!\cdots\!52 $ (-7406973385121233v^13 + 14868677134812959v^12 - 188711919938737526v^11 + 594705230769885208v^10 - 4251968905909893990v^9 + 10801310220790362884v^8 - 30917483628904008112v^7 + 55719736057451306524v^6 - 124364391029942498534v^5 + 219837964560125232996v^4 - 231630285010301145382v^3 + 182469270477022029656v^2 + 55142202890990804293*v + 87472212645730730665) / 7401710401471265352
$\beta_{10}$	$=$	$ ( 75\!\cdots\!03 \nu^{13} + \cdots - 61\!\cdots\!47 ) / 44\!\cdots\!12 $ (75044586502050103v^13 - 158904291088307309v^12 + 1928931915106904762v^11 - 6223241784329916760v^10 + 43745549842190531574v^9 - 113878110810300259772v^8 + 324734608698482930464v^7 - 592981856970062627188v^6 + 1310464787938099826282v^5 - 2322508288129346931948v^4 + 2497624936174493175046v^3 - 1926654011133205650536v^2 - 582214200800672894035*v - 613906411252597959547) / 44410262408827592112
$\beta_{11}$	$=$	$ ( - 23\!\cdots\!97 \nu^{13} + \cdots - 53\!\cdots\!83 ) / 13\!\cdots\!72 $ (-2355028837620663997v^13 - 472353925348458997v^12 - 57971626194204267650v^11 + 58719773133531693016v^10 - 1145504204397974085102v^9 + 754596622847048739464v^8 - 6544625041433034045184v^7 + 773616043610663129464v^6 - 27298009300536147075770v^5 - 836541595962531339732v^4 - 30613342914168125359198v^3 - 48127678428252453263308v^2 - 18443281865931445760183*v - 5387491317504904013783) / 1376718134673655355472
$\beta_{12}$	$=$	$ ( - 25\!\cdots\!29 \nu^{13} + \cdots - 11\!\cdots\!31 ) / 45\!\cdots\!24 $ (-2540522402518339229v^13 + 779858376851695243v^12 - 63404846946056528218v^11 + 95624170585884036344v^10 - 1296230541863856579702v^9 + 1488296736873041183776v^8 - 8060643165509672884688v^7 + 5179918049375170005032v^6 - 33665504441500243922962v^5 + 17040608825078929177452v^4 - 48695336849832132823238v^3 - 24585212795947924570724v^2 - 18932673899472382868167*v - 1167095479416993340231) / 458906044891218451824
$\beta_{13}$	$=$	$ ( 54\!\cdots\!31 \nu^{13} + \cdots + 24\!\cdots\!23 ) / 68\!\cdots\!36 $ (5455013466592182931v^13 - 1981905757350615227v^12 + 136070846235681010730v^11 - 212639339327769938872v^10 + 2790214632118527195510v^9 - 3339092084326659022064v^8 + 17381532529966191508588v^7 - 11843252303905763540680v^6 + 72038610435812913028322v^5 - 39344771541749725639044v^4 + 102933291651199413019174v^3 + 51633419741318473008148v^2 + 31615843787054649643637*v + 2427427543983158762423) / 688359067336827677736

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{12} + 2\beta_{11} + \beta_{9} + \beta_{7} + 6\beta_{6} + \beta_{4} - \beta_{3} - 2\beta_{2} - 6 $ -b12 + 2b11 + b9 + b7 + 6b6 + b4 - b3 - 2*b2 - 6
$\nu^{3}$	$=$	$ 2\beta_{10} - 7\beta_{8} - \beta_{5} - \beta_{4} + 18\beta_{2} - 18\beta _1 + 3 $ 2b10 - 7b8 - b5 - b4 + 18b2 - 18b1 + 3
$\nu^{4}$	$=$	$ - \beta_{13} + 19 \beta_{12} - 46 \beta_{11} + 46 \beta_{8} - 18 \beta_{7} - 87 \beta_{6} + \cdots + 56 \beta_1 $ -b13 + 19b12 - 46b11 + 46b8 - 18b7 - 87b6 + 28b5 + 28b3 + 56b1
$\nu^{5}$	$=$	$ 47 \beta_{13} - 10 \beta_{12} + 187 \beta_{11} - 47 \beta_{10} + 10 \beta_{9} + 40 \beta_{7} + \cdots - 137 $ 47b13 - 10b12 + 187b11 - 47b10 + 10b9 + 40b7 + 137b6 + 40b4 - 48b3 - 367b2 - 137
$\nu^{6}$	$=$	$ 58\beta_{10} - 356\beta_{9} - 1016\beta_{8} - 588\beta_{5} - 344\beta_{4} + 1362\beta_{2} - 1362\beta _1 + 1577 $ 58b10 - 356b9 - 1016b8 - 588b5 - 344b4 + 1362b2 - 1362*b1 + 1577
$\nu^{7}$	$=$	$ - 944 \beta_{13} + 476 \beta_{12} - 4386 \beta_{11} + 4386 \beta_{8} - 1088 \beta_{7} + \cdots + 7803 \beta_1 $ -944b13 + 476b12 - 4386b11 + 4386b8 - 1088b7 - 3976b6 + 1438b5 + 1438b3 + 7803*b1
$\nu^{8}$	$=$	$ 1914 \beta_{13} - 6833 \beta_{12} + 22374 \beta_{11} - 1914 \beta_{10} + 6833 \beta_{9} + 7047 \beta_{7} + \cdots - 31322 $ 1914b13 - 6833b12 + 22374b11 - 1914b10 + 6833b9 + 7047b7 + 31322b6 + 7047b4 - 12037b3 - 31814b2 - 31322
$\nu^{9}$	$=$	$ 18870 \beta_{10} - 14858 \beta_{9} - 99953 \beta_{8} - 37303 \beta_{5} - 26467 \beta_{4} + 169158 \beta_{2} + \cdots + 101245 $ 18870b10 - 14858b9 - 99953b8 - 37303b5 - 26467b4 + 169158b2 - 169158*b1 + 101245
$\nu^{10}$	$=$	$ - 52161 \beta_{13} + 135867 \beta_{12} - 494442 \beta_{11} + 494442 \beta_{8} - 149952 \beta_{7} + \cdots + 729378 \beta_1 $ -52161b13 + 135867b12 - 494442b11 + 494442b8 - 149952b7 - 650983b6 + 250668b5 + 250668b3 + 729378*b1
$\nu^{11}$	$=$	$ 386535 \beta_{13} - 395004 \beta_{12} + 2256027 \beta_{11} - 386535 \beta_{10} + 395004 \beta_{9} + \cdots - 2432607 $ 386535b13 - 395004b12 + 2256027b11 - 386535b10 + 395004b9 + 617502b7 + 2432607b6 + 617502b4 - 906456b3 - 3708337b2 - 2432607
$\nu^{12}$	$=$	$ 1301460 \beta_{10} - 2793412 \beta_{9} - 10965692 \beta_{8} - 5329924 \beta_{5} - 3254872 \beta_{4} + \cdots + 13891053 $ 1301460b10 - 2793412b9 - 10965692b8 - 5329924b5 - 3254872b4 + 16550384b2 - 16550384*b1 + 13891053
$\nu^{13}$	$=$	$ - 8123336 \beta_{13} + 9728508 \beta_{12} - 50697844 \beta_{11} + 50697844 \beta_{8} + \cdots + 81859773 \beta_1 $ -8123336b13 + 9728508b12 - 50697844b11 + 50697844b8 - 14135512b7 - 56707020b6 + 21277300b5 + 21277300b3 + 81859773*b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1338\mathbb{Z}\right)^\times$.

$n$	$893$	$895$
$\chi(n)$	$1$	$-1 + \beta_{6}$

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} $

931.1

−1.08263	+	1.87517i
−0.263750	+	0.456829i
−0.192978	+	0.334247i
−2.36302	+	4.09287i
1.86875	−	3.23677i
1.21330	−	2.10149i
0.820326	−	1.42085i
−1.08263	−	1.87517i
−0.263750	−	0.456829i
−0.192978	−	0.334247i
−2.36302	−	4.09287i
1.86875	+	3.23677i
1.21330	+	2.10149i
0.820326	+	1.42085i

1.00000

0.500000

−

0.866025i

1.00000

−1.74285

−

3.01870i

0.500000

−

0.866025i

−1.56262

1.00000

−0.500000

−

0.866025i

−1.74285

−

3.01870i

931.2

1.00000

0.500000

−

0.866025i

1.00000

−1.68314

−

2.91528i

0.500000

−

0.866025i

5.02862

1.00000

−0.500000

−

0.866025i

−1.68314

−

2.91528i

931.3

1.00000

0.500000

−

0.866025i

1.00000

−0.0934804

−

0.161913i

0.500000

−

0.866025i

−4.42316

1.00000

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−

0.866025i

−0.0934804

−

0.161913i

931.4

1.00000

0.500000

−

0.866025i

1.00000

0.869556

1.50612i

0.500000

−

0.866025i

2.51758

1.00000

−0.500000

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0.866025i

0.869556

1.50612i

931.5

1.00000

0.500000

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0.866025i

1.00000

0.887260

1.53678i

0.500000

−

0.866025i

−3.74585

1.00000

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0.866025i

0.887260

1.53678i

931.6

1.00000

0.500000

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0.866025i

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2.73907i

0.500000

−

0.866025i

0.755100

1.00000

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0.866025i

1.58141

2.73907i

931.7

1.00000

0.500000

−

0.866025i

1.00000

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2.91200i

0.500000

−

0.866025i

3.43032

1.00000

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0.866025i

1.68125

2.91200i

1075.1

1.00000

0.500000

0.866025i

1.00000

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3.01870i

0.500000

0.866025i

−1.56262

1.00000

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0.866025i

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3.01870i

1075.2

1.00000

0.500000

0.866025i

1.00000

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2.91528i

0.500000

0.866025i

5.02862

1.00000

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0.866025i

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2.91528i

1075.3

1.00000

0.500000

0.866025i

1.00000

−0.0934804

0.161913i

0.500000

0.866025i

−4.42316

1.00000

−0.500000

0.866025i

−0.0934804

0.161913i

1075.4

1.00000

0.500000

0.866025i

1.00000

0.869556

−

1.50612i

0.500000

0.866025i

2.51758

1.00000

−0.500000

0.866025i

0.869556

−

1.50612i

1075.5

1.00000

0.500000

0.866025i

1.00000

0.887260

−

1.53678i

0.500000

0.866025i

−3.74585

1.00000

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0.866025i

0.887260

−

1.53678i

1075.6

1.00000

0.500000

0.866025i

1.00000

1.58141

−

2.73907i

0.500000

0.866025i

0.755100

1.00000

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0.866025i

1.58141

−

2.73907i

1075.7

1.00000

0.500000

0.866025i

1.00000

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−

2.91200i

0.500000

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3.43032

1.00000

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0.866025i

1.68125

−

2.91200i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1338.2.e.i	✓	14
223.c	even	3	1	inner	1338.2.e.i	✓	14

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1338.2.e.i	✓	14	1.a	even	1	1	trivial
1338.2.e.i	✓	14	223.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1338, [\chi])$:

$ T_{5}^{14} - 3 T_{5}^{13} + 30 T_{5}^{12} - 81 T_{5}^{11} + 574 T_{5}^{10} - 1451 T_{5}^{9} + \cdots + 5184 $

T5^14 - 3*T5^13 + 30*T5^12 - 81*T5^11 + 574*T5^10 - 1451*T5^9 + 6218*T5^8 - 13209*T5^7 + 44242*T5^6 - 81127*T5^5 + 160802*T5^4 - 143489*T5^3 + 122065*T5^2 + 21672*T5 + 5184

$ T_{7}^{7} - 2T_{7}^{6} - 38T_{7}^{5} + 65T_{7}^{4} + 397T_{7}^{3} - 596T_{7}^{2} - 916T_{7} + 849 $

T7^7 - 2*T7^6 - 38*T7^5 + 65*T7^4 + 397*T7^3 - 596*T7^2 - 916*T7 + 849

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T - 1)^{14} $ (T - 1)^14
$3$	$ (T^{2} - T + 1)^{7} $ (T^2 - T + 1)^7
$5$	$ T^{14} - 3 T^{13} + \cdots + 5184 $ T^14 - 3T^13 + 30T^12 - 81T^11 + 574T^10 - 1451T^9 + 6218T^8 - 13209T^7 + 44242T^6 - 81127T^5 + 160802T^4 - 143489T^3 + 122065T^2 + 21672*T + 5184
$7$	$ (T^{7} - 2 T^{6} + \cdots + 849)^{2} $ (T^7 - 2T^6 - 38T^5 + 65T^4 + 397T^3 - 596T^2 - 916T + 849)^2
$11$	$ T^{14} - 7 T^{13} + \cdots + 2916 $ T^14 - 7T^13 + 53T^12 - 182T^11 + 903T^10 - 2657T^9 + 10032T^8 - 18209T^7 + 36061T^6 - 22906T^5 + 46279T^4 - 37017T^3 + 29835T^2 - 10206*T + 2916
$13$	$ (T^{7} + 6 T^{6} + \cdots + 11520)^{2} $ (T^7 + 6T^6 - 57T^5 - 252T^4 + 1263T^3 + 2006T^2 - 12183T + 11520)^2
$17$	$ (T^{7} - 3 T^{6} + \cdots + 4339)^{2} $ (T^7 - 3T^6 - 60T^5 + 202T^4 + 812T^3 - 2934T^2 - 629T + 4339)^2
$19$	$ T^{14} + 6 T^{13} + \cdots + 1742400 $ T^14 + 6T^13 + 111T^12 + 354T^11 + 7260T^10 + 23510T^9 + 197967T^8 + 71502T^7 + 1449057T^6 - 150348T^5 + 8929624T^4 - 5743056T^3 + 11976096T^2 + 3833280*T + 1742400
$23$	$ T^{14} + 14 T^{13} + \cdots + 98485776 $ T^14 + 14T^13 + 201T^12 + 1034T^11 + 8126T^10 + 18474T^9 + 233637T^8 + 280718T^7 + 2781742T^6 + 610966T^5 + 23148649T^4 + 4659726T^3 + 57539001T^2 - 22716036*T + 98485776
$29$	$ T^{14} + 7 T^{13} + \cdots + 9216 $ T^14 + 7T^13 + 53T^12 + 122T^11 + 573T^10 + 919T^9 + 4446T^8 + 4183T^7 + 12797T^6 + 5862T^5 + 24377T^4 + 6339T^3 + 21745T^2 - 7008*T + 9216
$31$	$ T^{14} + \cdots + 42751351696 $ T^14 + 19T^13 + 304T^12 + 2909T^11 + 26766T^10 + 185473T^9 + 1337065T^8 + 7485222T^7 + 42734022T^6 + 184518984T^5 + 845434168T^4 + 2871881224T^3 + 10375909800T^2 + 21692851824*T + 42751351696
$37$	$ T^{14} + \cdots + 18507969936 $ T^14 + 9T^13 + 214T^12 + 1151T^11 + 24944T^10 + 131545T^9 + 1524376T^8 + 5380465T^7 + 50984712T^6 + 158337967T^5 + 1031009342T^4 + 1263948741T^3 + 4869275173T^2 - 1411184412*T + 18507969936
$41$	$ (T^{7} - 5 T^{6} + \cdots - 2220)^{2} $ (T^7 - 5T^6 - 91T^5 + 67T^4 + 1511T^3 + 167T^2 - 3978T - 2220)^2
$43$	$ T^{14} + \cdots + 115476516 $ T^14 - 20T^13 + 292T^12 - 2704T^11 + 21768T^10 - 140678T^9 + 862294T^8 - 4367938T^7 + 19291612T^6 - 63441432T^5 + 163367820T^4 - 275454378T^3 + 340689645T^2 - 244589706*T + 115476516
$47$	$ T^{14} + \cdots + 2342172816 $ T^14 - T^13 + 155T^12 - 238T^11 + 18657T^10 - 34357T^9 + 806007T^8 - 3430490T^7 + 28317409T^6 - 79320933T^5 + 277036863T^4 - 310450826T^3 + 892794472T^2 - 653249208T + 2342172816
$53$	$ T^{14} + \cdots + 3463087104 $ T^14 - 10T^13 + 224T^12 - 1384T^11 + 26322T^10 - 154102T^9 + 1730910T^8 - 6187226T^7 + 54843904T^6 - 199832868T^5 + 1043715750T^4 - 1807432814T^3 + 4030578337T^2 + 2616676320*T + 3463087104
$59$	$ (T^{7} + 3 T^{6} + \cdots - 19014)^{2} $ (T^7 + 3T^6 - 254T^5 - 771T^4 + 15162T^3 + 55541T^2 + 9009T - 19014)^2
$61$	$ T^{14} + \cdots + 3899519078400 $ T^14 - 4T^13 + 326T^12 + 936T^11 + 67912T^10 + 263328T^9 + 8213128T^8 + 54165984T^7 + 668991536T^6 + 3078819392T^5 + 21932456128T^4 + 71403552000T^3 + 437477424384T^2 + 1043694812160*T + 3899519078400
$67$	$ T^{14} + \cdots + 27318139524 $ T^14 - 3T^13 + 303T^12 + 532T^11 + 65725T^10 + 88207T^9 + 5894810T^8 + 11703173T^7 + 386693295T^6 + 284143318T^5 + 4537177527T^4 - 9565507705T^3 + 41224560559T^2 - 34372210002*T + 27318139524
$71$	$ T^{14} - 31 T^{13} + \cdots + 6533136 $ T^14 - 31T^13 + 582T^12 - 7091T^11 + 63843T^10 - 424380T^9 + 2166659T^8 - 8288939T^7 + 24224788T^6 - 51684093T^5 + 82300243T^4 - 89911338T^3 + 67975632T^2 - 25657128*T + 6533136
$73$	$ T^{14} + \cdots + 236338783342596 $ T^14 + 45T^13 + 1500T^12 + 29917T^11 + 506268T^10 + 6139299T^9 + 73351744T^8 + 679607577T^7 + 6891527448T^6 + 48736774251T^5 + 390455955252T^4 + 2001121522785T^3 + 14155419986235T^2 + 50256577957806*T + 236338783342596
$79$	$ T^{14} + \cdots + 1744428109824 $ T^14 + 41T^13 + 1209T^12 + 21558T^11 + 317989T^10 + 3367259T^9 + 34050636T^8 + 267724807T^7 + 2231161749T^6 + 11820383956T^5 + 60972205079T^4 + 68211663259T^3 + 370001402281T^2 + 400135910976*T + 1744428109824
$83$	$ T^{14} + \cdots + 1192883364864 $ T^14 + T^13 + 476T^12 + 45T^11 + 169297T^10 - 45768T^9 + 24491944T^8 - 70347996T^7 + 2589059540T^6 - 3308215376T^5 + 73909934656T^4 + 194034774000T^3 + 1515077280912T^2 + 1371273268608T + 1192883364864
$89$	$ T^{14} + \cdots + 7952146202025 $ T^14 + 24T^13 + 651T^12 + 8810T^11 + 157032T^10 + 1785146T^9 + 24506356T^8 + 209859372T^7 + 2080855484T^6 + 13936219590T^5 + 113120044216T^4 + 565238736630T^3 + 2800739191899T^2 + 5239668146940*T + 7952146202025
$97$	$ T^{14} + \cdots + 40339926282384 $ T^14 + 13T^13 + 622T^12 + 5033T^11 + 234229T^10 + 1799954T^9 + 41324399T^8 + 129060303T^7 + 3403769150T^6 + 9008889519T^5 + 183319949575T^4 + 97920299394T^3 + 3738452518152T^2 + 6622715314584*T + 40339926282384
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1338 = 2 \cdot 3 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1338.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + q^{2} + \beta_{6} q^{3} + q^{4} - \beta_{11} q^{5} + \beta_{6} q^{6} + \beta_{10} q^{7} + q^{8} + (\beta_{6} - 1) q^{9} - \beta_{11} q^{10} + ( - \beta_{6} - \beta_{2} + 1) q^{11} + \beta_{6} q^{12}+ \cdots + (\beta_{6} + \beta_1) q^{99}+O(q^{100}) \) q + q^2 + b6 * q^3 + q^4 - b11 * q^5 + b6 * q^6 + b10 * q^7 + q^8 + (b6 - 1) * q^9 - b11 * q^10 + (-b6 - b2 + 1) * q^11 + b6 * q^12 + (b10 - b5 - b4 - 1) * q^13 + b10 * q^14 - b8 * q^15 + q^16 - b9 * q^17 + (b6 - 1) * q^18 + (-b12 - b7 - b6 + b1) * q^19 - b11 * q^20 + b13 * q^21 + (-b6 - b2 + 1) * q^22 + (b12 + b11 - b8 - 2b6 - b1) q^23 + b6 * q^24 + (-b13 + b11 - b8 - 2b6 - b5 - b3 - b1) q^25 + (b10 - b5 - b4 - 1) * q^26 - q^27 + b10 * q^28 + (b6 - b2 - 1) * q^29 - b8 * q^30 + (b7 - 3b6 + b1) q^31 + q^32 + (-b2 + b1 + 1) * q^33 - b9 * q^34 + (b13 - 2b11 - b10 + b7 + b6 + b4 - 2b3 + b2 - 1) * q^35 + (b6 - 1) * q^36 + (b13 + b12 - b11 - b10 - b9 + b6 + 2b3 + b2 - 1) q^37 + (-b12 - b7 - b6 + b1) * q^38 + (b13 + b7 - b6 - b5 - b3) * q^39 - b11 * q^40 + (-b8 - b4 + 2b2 - 2b1) * q^41 + b13 * q^42 + (-b11 + b8 + b7 + 3b6) q^43 + (-b6 - b2 + 1) * q^44 + (b11 - b8) * q^45 + (b12 + b11 - b8 - 2b6 - b1) q^46 + (b12 - b11 - b9 - b7 + b6 - b4 - 1) * q^47 + b6 * q^48 + (b8 - 2b5 + b4 - b2 + b1 + 6) q^49 + (-b13 + b11 - b8 - 2b6 - b5 - b3 - b1) q^50 - b12 * q^51 + (b10 - b5 - b4 - 1) * q^52 + (-b13 - b12 + b11 + b10 + b9 - 2b6 - 2b2 + 2) * q^53 - q^54 + (-3b6 - b5 - b3 - 2b1) * q^55 + b10 * q^56 + (-b12 + b9 - b7 - b6 - b4 + b2 + 1) * q^57 + (b6 - b2 - 1) * q^58 + (-b10 - 2b8 - b5 + 2b4) * q^59 - b8 * q^60 + (-b11 + b8 - b7 + 3b5 + 3b3 + 2b1) q^61 + (b7 - 3b6 + b1) q^62 + (b13 - b10) * q^63 + q^64 + (2b13 + b11 - 2b10 - 2b7 - 2b4 - 2b3 + 2b2) * q^65 + (-b2 + b1 + 1) * q^66 + (b12 - 3b11 + 3b8 + b7 + b6 + 2b5 + 2b3 + b1) * q^67 - b9 * q^68 + (b12 + b11 - b9 - 2b6 - b2 + 2) q^69 + (b13 - 2b11 - b10 + b7 + b6 + b4 - 2b3 + b2 - 1) * q^70 + (4b6 + b5 + b3) q^71 + (b6 - 1) * q^72 + (-b13 - b12 + 2b11 + b10 + b9 - b7 + 7b6 - b4 - 3b3 - b2 - 7) q^73 + (b13 + b12 - b11 - b10 - b9 + b6 + 2b3 + b2 - 1) q^74 + (-b13 + b11 + b10 - 2b6 - b3 - b2 + 2) q^75 + (-b12 - b7 - b6 + b1) * q^76 + (-b13 - b12 + b11 + b10 + b9 + b7 + 2b6 + b4 - 2b3 - b2 - 2) * q^77 + (b13 + b7 - b6 - b5 - b3) * q^78 + (-b11 - b7 + 7b6 - b4 - b3 + 3b2 - 7) * q^79 - b11 * q^80 - b6 * q^81 + (-b8 - b4 + 2b2 - 2b1) * q^82 + (-b13 - b12 + b10 + b9 - 2b7 + b6 - 2b4 - b3 + 4b2 - 1) q^83 + b13 * q^84 + (b12 - 2b11 - b9 + b7 + b6 + b4 + 3b3 - 1) * q^85 + (-b11 + b8 + b7 + 3b6) q^86 + (-b2 + b1 - 1) * q^87 + (-b6 - b2 + 1) * q^88 + (b13 + b12 + 2b11 - 2b8 - 5b6 + 2b5 + 2b3) q^89 + (b11 - b8) * q^90 + (2b9 + 4b8 - b5 - b4 + 2b2 - 2b1 + 8) * q^91 + (b12 + b11 - b8 - 2b6 - b1) q^92 + (b7 - 3b6 + b4 + b2 + 3) q^93 + (b12 - b11 - b9 - b7 + b6 - b4 - 1) * q^94 + (-b10 - 2b9 - b8 - b5 + 4b4 - 2b2 + 2b1) * q^95 + b6 * q^96 + (-2b13 - 2b12 + b11 + 2b10 + 2b9 + 2b6 - 2b3 - 3b2 - 2) q^97 + (b8 - 2b5 + b4 - b2 + b1 + 6) q^98 + (b6 + b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{2} + 7 q^{3} + 14 q^{4} + 3 q^{5} + 7 q^{6} + 4 q^{7} + 14 q^{8} - 7 q^{9} + 3 q^{10} + 7 q^{11} + 7 q^{12} - 12 q^{13} + 4 q^{14} + 6 q^{15} + 14 q^{16} + 6 q^{17} - 7 q^{18} - 6 q^{19} + 3 q^{20}+ \cdots + 7 q^{99}+O(q^{100}) \) 14 * q + 14 * q^2 + 7 * q^3 + 14 * q^4 + 3 * q^5 + 7 * q^6 + 4 * q^7 + 14 * q^8 - 7 * q^9 + 3 * q^10 + 7 * q^11 + 7 * q^12 - 12 * q^13 + 4 * q^14 + 6 * q^15 + 14 * q^16 + 6 * q^17 - 7 * q^18 - 6 * q^19 + 3 * q^20 + 2 * q^21 + 7 * q^22 - 14 * q^23 + 7 * q^24 - 16 * q^25 - 12 * q^26 - 14 * q^27 + 4 * q^28 - 7 * q^29 + 6 * q^30 - 19 * q^31 + 14 * q^32 + 14 * q^33 + 6 * q^34 + q^35 - 7 * q^36 - 9 * q^37 - 6 * q^38 - 6 * q^39 + 3 * q^40 + 10 * q^41 + 2 * q^42 + 20 * q^43 + 7 * q^44 + 3 * q^45 - 14 * q^46 + q^47 + 7 * q^48 + 62 * q^49 - 16 * q^50 + 3 * q^51 - 12 * q^52 + 10 * q^53 - 14 * q^54 - 24 * q^55 + 4 * q^56 + 6 * q^57 - 7 * q^58 - 6 * q^59 + 6 * q^60 + 4 * q^61 - 19 * q^62 - 2 * q^63 + 14 * q^64 + 3 * q^65 + 14 * q^66 + 3 * q^67 + 6 * q^68 + 14 * q^69 + q^70 + 31 * q^71 - 7 * q^72 - 45 * q^73 - 9 * q^74 + 16 * q^75 - 6 * q^76 - 14 * q^77 - 6 * q^78 - 41 * q^79 + 3 * q^80 - 7 * q^81 + 10 * q^82 - q^83 + 2 * q^84 - 9 * q^85 + 20 * q^86 - 14 * q^87 + 7 * q^88 - 24 * q^89 + 3 * q^90 + 74 * q^91 - 14 * q^92 + 19 * q^93 + q^94 - 8 * q^95 + 7 * q^96 - 13 * q^97 + 62 * q^98 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

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	1338.2.e.i

Level	$1338$
Weight	$2$
Character orbit	1338.e
Analytic conductor	$10.684$
Analytic rank	$0$
Dimension	$14$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(10.6839837904\)
Analytic rank:	\(0\)
Dimension:	\(14\)
Relative dimension:	\(7\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{14} + 25 x^{12} - 30 x^{11} + 502 x^{10} - 434 x^{9} + 3060 x^{8} - 1136 x^{7} + 13014 x^{6} + \cdots + 961 \) x^14 + 25x^12 - 30x^11 + 502x^10 - 434x^9 + 3060x^8 - 1136x^7 + 13014x^6 - 2882x^5 + 18706x^4 + 14706x^3 + 12571x^2 + 3720x + 961
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 11470254167885 \nu^{13} - 59106738542239 \nu^{12} + 363720839343838 \nu^{11} + \cdots + 35\!\cdots\!75 ) / 13\!\cdots\!64 \) (11470254167885v^13 - 59106738542239v^12 + 363720839343838v^11 - 1754394406062416v^10 + 9389024558365770v^9 - 35438170356484804v^8 + 96265881512912216v^7 - 195030354816227612v^6 + 405048272605203934v^5 - 741308758794623916v^4 + 1432684246032290954v^3 - 610803255496107640v^2 + 1161267268128266503*v + 350144884551363175) / 1345765527540230064
\(\beta_{3}\)	\(=\)	\( ( - 17\!\cdots\!67 \nu^{13} + \cdots + 26\!\cdots\!73 ) / 22\!\cdots\!12 \) (-17684849775760667v^13 - 40671567747060335v^12 - 353134260455773912v^11 - 522550959898914064v^10 - 5415447999593503956v^9 - 16325810312876848790v^8 + 9613152557815877332v^7 - 161104833688389160906v^6 + 111113199289044351692v^5 - 682090494796151213448v^4 + 970985364405680720956v^3 - 1495917282408544973006v^2 + 857136776581280467823*v + 260402583403020088673) / 229453022445609225912
\(\beta_{4}\)	\(=\)	\( ( 45\!\cdots\!29 \nu^{13} + \cdots + 77\!\cdots\!27 ) / 44\!\cdots\!12 \) (4523264687314129v^13 - 5038947078223583v^12 + 108022845027388022v^11 - 279006096622362232v^10 + 2289678839189628498v^9 - 4706509032668545604v^8 + 13994100423659473744v^7 - 25190131127396899180v^6 + 54490917796913139494v^5 - 93767415414363891396v^4 + 60100612447172540146v^3 - 78282559291828844888v^2 - 23665714940385288997*v + 77695239329628075527) / 44410262408827592112
\(\beta_{5}\)	\(=\)	\( ( - 374253871098011 \nu^{13} + \cdots - 29\!\cdots\!27 ) / 24\!\cdots\!84 \) (-374253871098011v^13 + 1077974981475101v^12 - 10219764710202050v^11 + 38031325845153622v^10 - 241117887076323524v^9 + 724973617539200272v^8 - 2025618758016265410v^7 + 3981389884494732118v^6 - 8318671741356446102v^5 + 14947244256455711658v^4 - 18821111846363777680v^3 + 12363410009294109672v^2 + 3735393803732308237*v - 2940727949087983727) / 2467236800490421784
\(\beta_{6}\)	\(=\)	\( ( - 11\!\cdots\!25 \nu^{13} + 355577879204435 \nu^{12} + \cdots - 60\!\cdots\!07 ) / 41\!\cdots\!84 \) (-11294996275850425v^13 + 355577879204435v^12 - 284207215791070034v^11 + 350125234295171728v^10 - 5724474357064848246v^9 + 5193088145028423320v^8 - 35661271885153329424v^7 + 15815358096266361496v^6 - 153039022533220486922v^5 + 45108675717762246804v^4 - 234264771858691391446v^3 - 121691003605655330476v^2 - 160924299104095029515*v - 6018100834187319407) / 41718731353747131984
\(\beta_{7}\)	\(=\)	\( ( 59\!\cdots\!13 \nu^{13} + \cdots + 40\!\cdots\!69 ) / 17\!\cdots\!34 \) (59966827000429313v^13 + 29943772497432911v^12 + 1547352058023727201v^11 - 1072826442744735770v^10 + 30295766829554440524v^9 - 13267220908939166173v^8 + 192465771295303017869v^7 - 10944598997191452314v^6 + 854899747437341194417v^5 + 33882456929525603478v^4 + 1437799785841320458642v^3 + 779190398922354276275v^2 + 1243439271445291149040*v + 40752116606959090669) / 172089766834206919434
\(\beta_{8}\)	\(=\)	\( ( 22\!\cdots\!95 \nu^{13} + \cdots - 13\!\cdots\!07 ) / 44\!\cdots\!12 \) (22730829853181995v^13 - 52468941065576117v^12 + 592835275505756882v^11 - 1984878802064473084v^10 + 13588374771462075066v^9 - 36735644413627078916v^8 + 104159741176609860124v^7 - 192612375927781819264v^6 + 422396203335884811386v^5 - 751519565740648484064v^4 + 868647636124387863250v^3 - 622911996700937766920v^2 - 188227391602692628939*v - 130194015728749683607) / 44410262408827592112
\(\beta_{9}\)	\(=\)	\( ( - 74\!\cdots\!33 \nu^{13} + \cdots + 87\!\cdots\!65 ) / 74\!\cdots\!52 \) (-7406973385121233v^13 + 14868677134812959v^12 - 188711919938737526v^11 + 594705230769885208v^10 - 4251968905909893990v^9 + 10801310220790362884v^8 - 30917483628904008112v^7 + 55719736057451306524v^6 - 124364391029942498534v^5 + 219837964560125232996v^4 - 231630285010301145382v^3 + 182469270477022029656v^2 + 55142202890990804293*v + 87472212645730730665) / 7401710401471265352
\(\beta_{10}\)	\(=\)	\( ( 75\!\cdots\!03 \nu^{13} + \cdots - 61\!\cdots\!47 ) / 44\!\cdots\!12 \) (75044586502050103v^13 - 158904291088307309v^12 + 1928931915106904762v^11 - 6223241784329916760v^10 + 43745549842190531574v^9 - 113878110810300259772v^8 + 324734608698482930464v^7 - 592981856970062627188v^6 + 1310464787938099826282v^5 - 2322508288129346931948v^4 + 2497624936174493175046v^3 - 1926654011133205650536v^2 - 582214200800672894035*v - 613906411252597959547) / 44410262408827592112
\(\beta_{11}\)	\(=\)	\( ( - 23\!\cdots\!97 \nu^{13} + \cdots - 53\!\cdots\!83 ) / 13\!\cdots\!72 \) (-2355028837620663997v^13 - 472353925348458997v^12 - 57971626194204267650v^11 + 58719773133531693016v^10 - 1145504204397974085102v^9 + 754596622847048739464v^8 - 6544625041433034045184v^7 + 773616043610663129464v^6 - 27298009300536147075770v^5 - 836541595962531339732v^4 - 30613342914168125359198v^3 - 48127678428252453263308v^2 - 18443281865931445760183*v - 5387491317504904013783) / 1376718134673655355472
\(\beta_{12}\)	\(=\)	\( ( - 25\!\cdots\!29 \nu^{13} + \cdots - 11\!\cdots\!31 ) / 45\!\cdots\!24 \) (-2540522402518339229v^13 + 779858376851695243v^12 - 63404846946056528218v^11 + 95624170585884036344v^10 - 1296230541863856579702v^9 + 1488296736873041183776v^8 - 8060643165509672884688v^7 + 5179918049375170005032v^6 - 33665504441500243922962v^5 + 17040608825078929177452v^4 - 48695336849832132823238v^3 - 24585212795947924570724v^2 - 18932673899472382868167*v - 1167095479416993340231) / 458906044891218451824
\(\beta_{13}\)	\(=\)	\( ( 54\!\cdots\!31 \nu^{13} + \cdots + 24\!\cdots\!23 ) / 68\!\cdots\!36 \) (5455013466592182931v^13 - 1981905757350615227v^12 + 136070846235681010730v^11 - 212639339327769938872v^10 + 2790214632118527195510v^9 - 3339092084326659022064v^8 + 17381532529966191508588v^7 - 11843252303905763540680v^6 + 72038610435812913028322v^5 - 39344771541749725639044v^4 + 102933291651199413019174v^3 + 51633419741318473008148v^2 + 31615843787054649643637*v + 2427427543983158762423) / 688359067336827677736

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{12} + 2\beta_{11} + \beta_{9} + \beta_{7} + 6\beta_{6} + \beta_{4} - \beta_{3} - 2\beta_{2} - 6 \) -b12 + 2b11 + b9 + b7 + 6b6 + b4 - b3 - 2*b2 - 6
\(\nu^{3}\)	\(=\)	\( 2\beta_{10} - 7\beta_{8} - \beta_{5} - \beta_{4} + 18\beta_{2} - 18\beta _1 + 3 \) 2b10 - 7b8 - b5 - b4 + 18b2 - 18b1 + 3
\(\nu^{4}\)	\(=\)	\( - \beta_{13} + 19 \beta_{12} - 46 \beta_{11} + 46 \beta_{8} - 18 \beta_{7} - 87 \beta_{6} + \cdots + 56 \beta_1 \) -b13 + 19b12 - 46b11 + 46b8 - 18b7 - 87b6 + 28b5 + 28b3 + 56b1
\(\nu^{5}\)	\(=\)	\( 47 \beta_{13} - 10 \beta_{12} + 187 \beta_{11} - 47 \beta_{10} + 10 \beta_{9} + 40 \beta_{7} + \cdots - 137 \) 47b13 - 10b12 + 187b11 - 47b10 + 10b9 + 40b7 + 137b6 + 40b4 - 48b3 - 367b2 - 137
\(\nu^{6}\)	\(=\)	\( 58\beta_{10} - 356\beta_{9} - 1016\beta_{8} - 588\beta_{5} - 344\beta_{4} + 1362\beta_{2} - 1362\beta _1 + 1577 \) 58b10 - 356b9 - 1016b8 - 588b5 - 344b4 + 1362b2 - 1362*b1 + 1577
\(\nu^{7}\)	\(=\)	\( - 944 \beta_{13} + 476 \beta_{12} - 4386 \beta_{11} + 4386 \beta_{8} - 1088 \beta_{7} + \cdots + 7803 \beta_1 \) -944b13 + 476b12 - 4386b11 + 4386b8 - 1088b7 - 3976b6 + 1438b5 + 1438b3 + 7803*b1
\(\nu^{8}\)	\(=\)	\( 1914 \beta_{13} - 6833 \beta_{12} + 22374 \beta_{11} - 1914 \beta_{10} + 6833 \beta_{9} + 7047 \beta_{7} + \cdots - 31322 \) 1914b13 - 6833b12 + 22374b11 - 1914b10 + 6833b9 + 7047b7 + 31322b6 + 7047b4 - 12037b3 - 31814b2 - 31322
\(\nu^{9}\)	\(=\)	\( 18870 \beta_{10} - 14858 \beta_{9} - 99953 \beta_{8} - 37303 \beta_{5} - 26467 \beta_{4} + 169158 \beta_{2} + \cdots + 101245 \) 18870b10 - 14858b9 - 99953b8 - 37303b5 - 26467b4 + 169158b2 - 169158*b1 + 101245
\(\nu^{10}\)	\(=\)	\( - 52161 \beta_{13} + 135867 \beta_{12} - 494442 \beta_{11} + 494442 \beta_{8} - 149952 \beta_{7} + \cdots + 729378 \beta_1 \) -52161b13 + 135867b12 - 494442b11 + 494442b8 - 149952b7 - 650983b6 + 250668b5 + 250668b3 + 729378*b1
\(\nu^{11}\)	\(=\)	\( 386535 \beta_{13} - 395004 \beta_{12} + 2256027 \beta_{11} - 386535 \beta_{10} + 395004 \beta_{9} + \cdots - 2432607 \) 386535b13 - 395004b12 + 2256027b11 - 386535b10 + 395004b9 + 617502b7 + 2432607b6 + 617502b4 - 906456b3 - 3708337b2 - 2432607
\(\nu^{12}\)	\(=\)	\( 1301460 \beta_{10} - 2793412 \beta_{9} - 10965692 \beta_{8} - 5329924 \beta_{5} - 3254872 \beta_{4} + \cdots + 13891053 \) 1301460b10 - 2793412b9 - 10965692b8 - 5329924b5 - 3254872b4 + 16550384b2 - 16550384*b1 + 13891053
\(\nu^{13}\)	\(=\)	\( - 8123336 \beta_{13} + 9728508 \beta_{12} - 50697844 \beta_{11} + 50697844 \beta_{8} + \cdots + 81859773 \beta_1 \) -8123336b13 + 9728508b12 - 50697844b11 + 50697844b8 - 14135512b7 - 56707020b6 + 21277300b5 + 21277300b3 + 81859773*b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 931.7
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T - 1)^{14} \) (T - 1)^14
$3$	\( (T^{2} - T + 1)^{7} \) (T^2 - T + 1)^7
$5$	\( T^{14} - 3 T^{13} + \cdots + 5184 \) T^14 - 3T^13 + 30T^12 - 81T^11 + 574T^10 - 1451T^9 + 6218T^8 - 13209T^7 + 44242T^6 - 81127T^5 + 160802T^4 - 143489T^3 + 122065T^2 + 21672*T + 5184
$7$	\( (T^{7} - 2 T^{6} + \cdots + 849)^{2} \) (T^7 - 2T^6 - 38T^5 + 65T^4 + 397T^3 - 596T^2 - 916T + 849)^2
$11$	\( T^{14} - 7 T^{13} + \cdots + 2916 \) T^14 - 7T^13 + 53T^12 - 182T^11 + 903T^10 - 2657T^9 + 10032T^8 - 18209T^7 + 36061T^6 - 22906T^5 + 46279T^4 - 37017T^3 + 29835T^2 - 10206*T + 2916
$13$	\( (T^{7} + 6 T^{6} + \cdots + 11520)^{2} \) (T^7 + 6T^6 - 57T^5 - 252T^4 + 1263T^3 + 2006T^2 - 12183T + 11520)^2
$17$	\( (T^{7} - 3 T^{6} + \cdots + 4339)^{2} \) (T^7 - 3T^6 - 60T^5 + 202T^4 + 812T^3 - 2934T^2 - 629T + 4339)^2
$19$	\( T^{14} + 6 T^{13} + \cdots + 1742400 \) T^14 + 6T^13 + 111T^12 + 354T^11 + 7260T^10 + 23510T^9 + 197967T^8 + 71502T^7 + 1449057T^6 - 150348T^5 + 8929624T^4 - 5743056T^3 + 11976096T^2 + 3833280*T + 1742400
$23$	\( T^{14} + 14 T^{13} + \cdots + 98485776 \) T^14 + 14T^13 + 201T^12 + 1034T^11 + 8126T^10 + 18474T^9 + 233637T^8 + 280718T^7 + 2781742T^6 + 610966T^5 + 23148649T^4 + 4659726T^3 + 57539001T^2 - 22716036*T + 98485776
$29$	\( T^{14} + 7 T^{13} + \cdots + 9216 \) T^14 + 7T^13 + 53T^12 + 122T^11 + 573T^10 + 919T^9 + 4446T^8 + 4183T^7 + 12797T^6 + 5862T^5 + 24377T^4 + 6339T^3 + 21745T^2 - 7008*T + 9216
$31$	\( T^{14} + \cdots + 42751351696 \) T^14 + 19T^13 + 304T^12 + 2909T^11 + 26766T^10 + 185473T^9 + 1337065T^8 + 7485222T^7 + 42734022T^6 + 184518984T^5 + 845434168T^4 + 2871881224T^3 + 10375909800T^2 + 21692851824*T + 42751351696
$37$	\( T^{14} + \cdots + 18507969936 \) T^14 + 9T^13 + 214T^12 + 1151T^11 + 24944T^10 + 131545T^9 + 1524376T^8 + 5380465T^7 + 50984712T^6 + 158337967T^5 + 1031009342T^4 + 1263948741T^3 + 4869275173T^2 - 1411184412*T + 18507969936
$41$	\( (T^{7} - 5 T^{6} + \cdots - 2220)^{2} \) (T^7 - 5T^6 - 91T^5 + 67T^4 + 1511T^3 + 167T^2 - 3978T - 2220)^2
$43$	\( T^{14} + \cdots + 115476516 \) T^14 - 20T^13 + 292T^12 - 2704T^11 + 21768T^10 - 140678T^9 + 862294T^8 - 4367938T^7 + 19291612T^6 - 63441432T^5 + 163367820T^4 - 275454378T^3 + 340689645T^2 - 244589706*T + 115476516
$47$	\( T^{14} + \cdots + 2342172816 \) T^14 - T^13 + 155T^12 - 238T^11 + 18657T^10 - 34357T^9 + 806007T^8 - 3430490T^7 + 28317409T^6 - 79320933T^5 + 277036863T^4 - 310450826T^3 + 892794472T^2 - 653249208T + 2342172816
$53$	\( T^{14} + \cdots + 3463087104 \) T^14 - 10T^13 + 224T^12 - 1384T^11 + 26322T^10 - 154102T^9 + 1730910T^8 - 6187226T^7 + 54843904T^6 - 199832868T^5 + 1043715750T^4 - 1807432814T^3 + 4030578337T^2 + 2616676320*T + 3463087104
$59$	\( (T^{7} + 3 T^{6} + \cdots - 19014)^{2} \) (T^7 + 3T^6 - 254T^5 - 771T^4 + 15162T^3 + 55541T^2 + 9009T - 19014)^2
$61$	\( T^{14} + \cdots + 3899519078400 \) T^14 - 4T^13 + 326T^12 + 936T^11 + 67912T^10 + 263328T^9 + 8213128T^8 + 54165984T^7 + 668991536T^6 + 3078819392T^5 + 21932456128T^4 + 71403552000T^3 + 437477424384T^2 + 1043694812160*T + 3899519078400
$67$	\( T^{14} + \cdots + 27318139524 \) T^14 - 3T^13 + 303T^12 + 532T^11 + 65725T^10 + 88207T^9 + 5894810T^8 + 11703173T^7 + 386693295T^6 + 284143318T^5 + 4537177527T^4 - 9565507705T^3 + 41224560559T^2 - 34372210002*T + 27318139524
$71$	\( T^{14} - 31 T^{13} + \cdots + 6533136 \) T^14 - 31T^13 + 582T^12 - 7091T^11 + 63843T^10 - 424380T^9 + 2166659T^8 - 8288939T^7 + 24224788T^6 - 51684093T^5 + 82300243T^4 - 89911338T^3 + 67975632T^2 - 25657128*T + 6533136
$73$	\( T^{14} + \cdots + 236338783342596 \) T^14 + 45T^13 + 1500T^12 + 29917T^11 + 506268T^10 + 6139299T^9 + 73351744T^8 + 679607577T^7 + 6891527448T^6 + 48736774251T^5 + 390455955252T^4 + 2001121522785T^3 + 14155419986235T^2 + 50256577957806*T + 236338783342596
$79$	\( T^{14} + \cdots + 1744428109824 \) T^14 + 41T^13 + 1209T^12 + 21558T^11 + 317989T^10 + 3367259T^9 + 34050636T^8 + 267724807T^7 + 2231161749T^6 + 11820383956T^5 + 60972205079T^4 + 68211663259T^3 + 370001402281T^2 + 400135910976*T + 1744428109824
$83$	\( T^{14} + \cdots + 1192883364864 \) T^14 + T^13 + 476T^12 + 45T^11 + 169297T^10 - 45768T^9 + 24491944T^8 - 70347996T^7 + 2589059540T^6 - 3308215376T^5 + 73909934656T^4 + 194034774000T^3 + 1515077280912T^2 + 1371273268608T + 1192883364864
$89$	\( T^{14} + \cdots + 7952146202025 \) T^14 + 24T^13 + 651T^12 + 8810T^11 + 157032T^10 + 1785146T^9 + 24506356T^8 + 209859372T^7 + 2080855484T^6 + 13936219590T^5 + 113120044216T^4 + 565238736630T^3 + 2800739191899T^2 + 5239668146940*T + 7952146202025
$97$	\( T^{14} + \cdots + 40339926282384 \) T^14 + 13T^13 + 622T^12 + 5033T^11 + 234229T^10 + 1799954T^9 + 41324399T^8 + 129060303T^7 + 3403769150T^6 + 9008889519T^5 + 183319949575T^4 + 97920299394T^3 + 3738452518152T^2 + 6622715314584*T + 40339926282384
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\(n\)	\(893\)	\(895\)
\(\chi(n)\)	\(1\)	\(-1 + \beta_{6}\)