LMFDB - Newform orbit 224.2.p.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,2,Mod(31,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

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

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

chi := DirichletCharacter("224.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	224.p (of order $6$, degree $2$, 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:	$1.78864900528$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.2353561680715186176.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 $ x^16 - 4x^15 + 2x^14 + 41x^12 - 92x^11 + 66x^10 - 104x^9 + 291x^8 - 388x^7 + 366x^6 - 344x^5 + 286x^4 - 184x^3 + 84x^2 - 24x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{14} q^{3} - \beta_{13} q^{5} - \beta_{9} q^{7} + (\beta_{15} + \beta_{11} + \cdots - \beta_{3}) q^{9}+O(q^{10}) $ q - b14 * q^3 - b13 * q^5 - b9 * q^7 + (b15 + b11 + b8 - b3) * q^9	$ q - \beta_{14} q^{3} - \beta_{13} q^{5} - \beta_{9} q^{7} + (\beta_{15} + \beta_{11} + \cdots - \beta_{3}) q^{9}+ \cdots + ( - 2 \beta_{14} - 5 \beta_{9} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) $ q - b14 * q^3 - b13 * q^5 - b9 * q^7 + (b15 + b11 + b8 - b3) * q^9 + (b14 - b9 + b7 + b6 - b2 + b1) * q^11 + (-b11 + b10 + 2b3 - 1) q^13 + (-b14 + b9 + b7 - b5 + b2) * q^15 - b10 * q^17 + (-b14 + b9 + b4 + b2 - b1) * q^19 + (b12 + b10 + b8 + b3 - 2) * q^21 + (-b7 + b1) * q^23 + (-b15 + b11 - b10 - 2b3 + 2) q^25 + (-b14 - 2b6 - b5 - b4 + 2b2 - b1) * q^27 + (-b8 + 1) * q^29 + (b7 + b6 + 2b5 + b4 - b2 - b1) q^31 + (b15 + 2b13 - b8 - b3 + 2) q^33 + (3b14 - 3b7 - b4 - b2 + b1) * q^35 + (-b15 - b13 - 2b12 - b11 - b8 - b3) q^37 + (2b14 - b7 - b4 - 4b2 - b1) * q^39 + (-b11 + b10 + 2b3 - 1) q^41 + (2b14 + 2b5 - 2b2) q^43 + (-b15 - b11 - 2b10 - 2b8 - b3 - 1) * q^45 + (-b14 - b9 - b6 + b5 - b4 + b1) * q^47 + (-b15 - 2b12 - 2) q^49 + (b14 - b9 + b4 + b2 + b1) * q^51 + (2b13 + b12 + 2b11 - b10 + b3 - 1) * q^53 + (3b9 - 3b7 + b4 + 6b2 + b1) q^55 - q^57 + (-3b14 + 2b9 + 2b7 - 2b1) * q^59 + (-b15 + b13 - b11 + b8 + b3 - 2) * q^61 + (3b14 + b7 - 2b6 - 3b5 - 2b4 - b2) * q^63 + (2b13 + 4b12 - b11 + 2b10 + b3) q^65 + (b14 + b9 - 2b7 + b6 - b4 - b2 - 2b1) * q^67 + (2b15 + b13 + b12 + b11 + b10 + b8 - 4b3 + 2) * q^69 + (-b14 + b9 + b7 - b5 + 3b4 + b2 - 3b1) * q^71 + (-b15 - b11 - b10 - 2b8 - b3 - 1) q^73 + (-b14 + 2b9 + 3b7 + b6 - b5 + 2b4 + 2b2 + b1) * q^75 + (2b15 - b12 + 3b11 + b10 + 2b8 - 2b3 + 5) * q^77 + (2b14 + b7 + 2b2 - b1) * q^79 + (-2b15 - 4b13 - 2b12 - b10 + 6b3 - 6) * q^81 + (-b14 - b9 + b7 - 2b6 - b5 + b4 - 3b2 + b1) * q^83 + (b13 - b12 - b11 - b10 - 1) * q^85 + (-4b14 - b9 - b7 + b6 + 2b5 - b2 + b1) * q^87 + (-2b13 - 2b11 + 3b3 - 6) q^89 + (b14 - b9 + 3b6 + b5 + 2b2 + 3b1) q^91 + (b15 + b13 + 2b12 + b11 + b8 + b3) q^93 + (2b14 - 4b9 + 2b7 + b6 - 2b4 - 3b2 + 2b1) * q^95 + (2b15 - 2b13 - 2b12 + 2b10 + b8 - 6b3 + 3) q^97 + (-2b14 - 5b9 - 5b7 + 2b5 - 4b4 + 4b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^9	$ 16 q - 8 q^{9} - 24 q^{21} + 16 q^{25} + 16 q^{29} + 24 q^{33} - 8 q^{37} - 24 q^{45} - 32 q^{49} - 8 q^{53} - 16 q^{57} - 24 q^{61} + 8 q^{65} - 24 q^{73} + 64 q^{77} - 48 q^{81} - 16 q^{85} - 72 q^{89} + 8 q^{93}+O(q^{100}) $ 16 * q - 8 * q^9 - 24 * q^21 + 16 * q^25 + 16 * q^29 + 24 * q^33 - 8 * q^37 - 24 * q^45 - 32 * q^49 - 8 * q^53 - 16 * q^57 - 24 * q^61 + 8 * q^65 - 24 * q^73 + 64 * q^77 - 48 * q^81 - 16 * q^85 - 72 * q^89 + 8 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 $ x^16 - 4*x^15 + 2*x^14 + 41*x^12 - 92*x^11 + 66*x^10 - 104*x^9 + 291*x^8 - 388*x^7 + 366*x^6 - 344*x^5 + 286*x^4 - 184*x^3 + 84*x^2 - 24*x + 4 :

$\beta_{1}$	$=$	$ ( - 4671439980 \nu^{15} - 5019523613 \nu^{14} + 74862020586 \nu^{13} - 4435873025 \nu^{12} + \cdots - 353683962760 ) / 20109322702 $ (-4671439980v^15 - 5019523613v^14 + 74862020586v^13 - 4435873025v^12 - 210602072844v^11 - 553169578089v^10 + 1432081245914v^9 - 85279469075v^8 + 522318705094v^7 - 4152490382437v^6 + 4382872705880v^5 - 3138092527649v^4 + 3538225191170v^3 - 2782602331212v^2 + 1359834224272*v - 353683962760) / 20109322702
$\beta_{2}$	$=$	$ ( 5063424619 \nu^{15} - 34394314873 \nu^{14} + 58272692948 \nu^{13} + 3111527264 \nu^{12} + \cdots - 314591816094 ) / 20109322702 $ (5063424619v^15 - 34394314873v^14 + 58272692948v^13 + 3111527264v^12 + 197933621407v^11 - 1046870352209v^10 + 1297182921138v^9 - 768732964854v^8 + 2528765768339v^7 - 5349204235533v^6 + 5140695571010v^5 - 4106878988502v^4 + 3966533391230v^3 - 2841292968702v^2 + 1190351287680*v - 314591816094) / 20109322702
$\beta_{3}$	$=$	$ ( - 11175608166 \nu^{15} + 37005033535 \nu^{14} + 2381595084 \nu^{13} + 3936897701 \nu^{12} + \cdots + 87372551998 ) / 20109322702 $ (-11175608166v^15 + 37005033535v^14 + 2381595084v^13 + 3936897701v^12 - 454687961278v^11 + 715591444243v^10 - 275560834120v^9 + 1011875313139v^8 - 2563606006170v^7 + 2641228538245v^6 - 2425489864376v^5 + 2311962665959v^4 - 1743124162188v^3 + 985628049642v^2 - 363876573956*v + 87372551998) / 20109322702
$\beta_{4}$	$=$	$ ( - 1736302763 \nu^{15} + 3368540236 \nu^{14} + 8943298446 \nu^{13} + 179687165 \nu^{12} + \cdots - 25579237514 ) / 2872760386 $ (-1736302763v^15 + 3368540236v^14 + 8943298446v^13 + 179687165v^12 - 73751071335v^11 + 11510683964v^10 + 136916188768v^9 + 109872186421v^8 - 223621781697v^7 - 209166485604v^6 + 227204090414v^5 - 65297035181v^4 + 211706237508v^3 - 207339126830v^2 + 96981591808*v - 25579237514) / 2872760386
$\beta_{5}$	$=$	$ ( 12615896138 \nu^{15} - 42612645921 \nu^{14} + 5931103262 \nu^{13} - 19811250885 \nu^{12} + \cdots - 179930396744 ) / 20109322702 $ (12615896138v^15 - 42612645921v^14 + 5931103262v^13 - 19811250885v^12 + 499479045258v^11 - 845619686261v^10 + 604535781354v^9 - 1376773471743v^8 + 2866813258228v^7 - 3654726081713v^6 + 3946524545552v^5 - 3264415389805v^4 + 2631655568822v^3 - 1752180134328v^2 + 636854341512*v - 179930396744) / 20109322702
$\beta_{6}$	$=$	$ ( 18129728171 \nu^{15} - 67239132782 \nu^{14} + 13728648942 \nu^{13} + 16261528967 \nu^{12} + \cdots - 69968157850 ) / 20109322702 $ (18129728171v^15 - 67239132782v^14 + 13728648942v^13 + 16261528967v^12 + 743533213171v^11 - 1459054198710v^10 + 645301401120v^9 - 1413868017845v^8 + 4752183358093v^7 - 5440849395538v^6 + 4170972602046v^5 - 4049499601091v^4 + 3305520707556v^3 - 1639495702234v^2 + 554159019384*v - 69968157850) / 20109322702
$\beta_{7}$	$=$	$ ( 2623797969 \nu^{15} - 10304840791 \nu^{14} + 2190403752 \nu^{13} + 6732922644 \nu^{12} + \cdots + 5890380390 ) / 2872760386 $ (2623797969v^15 - 10304840791v^14 + 2190403752v^13 + 6732922644v^12 + 112107241481v^11 - 231856184779v^10 + 61947862130v^9 - 164982748350v^8 + 762466069777v^7 - 766985838727v^6 + 460781419854v^5 - 570794974590v^4 + 435938816234v^3 - 151480361274v^2 + 31718599632*v + 5890380390) / 2872760386
$\beta_{8}$	$=$	$ ( 139482845 \nu^{15} - 713255513 \nu^{14} + 813740704 \nu^{13} + 30184751 \nu^{12} + \cdots - 4395910477 ) / 137735087 $ (139482845v^15 - 713255513v^14 + 813740704v^13 + 30184751v^12 + 5568358461v^11 - 19244128631v^10 + 20009728986v^9 - 16925651529v^8 + 51651556669v^7 - 91857962133v^6 + 87279176826v^5 - 72439306253v^4 + 66687816790v^3 - 46319052852v^2 + 18821562600*v - 4395910477) / 137735087
$\beta_{9}$	$=$	$ ( 21074697951 \nu^{15} - 104086834189 \nu^{14} + 95566612060 \nu^{13} + 46229985362 \nu^{12} + \cdots - 264247625938 ) / 20109322702 $ (21074697951v^15 - 104086834189v^14 + 95566612060v^13 + 46229985362v^12 + 871914454327v^11 - 2751753453829v^10 + 2154137922350v^9 - 1835171745948v^8 + 7683424327567v^7 - 11727612400677v^6 + 9418969698846v^5 - 8639783255096v^4 + 7778130510050v^3 - 4550591596230v^2 + 1735233196936*v - 264247625938) / 20109322702
$\beta_{10}$	$=$	$ ( - 22929440334 \nu^{15} + 103630918811 \nu^{14} - 82984027620 \nu^{13} - 16420430373 \nu^{12} + \cdots + 438702505064 ) / 20109322702 $ (-22929440334v^15 + 103630918811v^14 - 82984027620v^13 - 16420430373v^12 - 929634728742v^11 + 2610374929811v^10 - 2170386426128v^9 + 2278929576205v^8 - 7519139314798v^7 + 11518339038677v^6 - 10085004362164v^5 + 8949442744397v^4 - 8103799328648v^3 + 5000860688254v^2 - 1965241728420*v + 438702505064) / 20109322702
$\beta_{11}$	$=$	$ ( 33911516972 \nu^{15} - 108940608997 \nu^{14} - 32201388164 \nu^{13} + 22923719031 \nu^{12} + \cdots + 125770278322 ) / 20109322702 $ (33911516972v^15 - 108940608997v^14 - 32201388164v^13 + 22923719031v^12 + 1411498051716v^11 - 2014225248249v^10 + 64172702604v^9 - 2520892868363v^8 + 7621760166076v^7 - 5993220149479v^6 + 4268041273800v^5 - 5045452375491v^4 + 3122239171348v^3 - 961405268358v^2 + 81167549476*v + 125770278322) / 20109322702
$\beta_{12}$	$=$	$ ( - 46688945564 \nu^{15} + 219887457941 \nu^{14} - 194424781504 \nu^{13} - 50488884423 \nu^{12} + \cdots + 828605565416 ) / 20109322702 $ (-46688945564v^15 + 219887457941v^14 - 194424781504v^13 - 50488884423v^12 - 1890422689576v^11 + 5678839565545v^10 - 4832040026616v^9 + 4520765331063v^8 - 15902360689892v^7 + 25178802231927v^6 - 21652899765320v^5 + 18807008211767v^4 - 17224972534876v^3 + 10719181869982v^2 - 4138992241108*v + 828605565416) / 20109322702
$\beta_{13}$	$=$	$ ( - 51918966616 \nu^{15} + 156637786453 \nu^{14} + 91292310268 \nu^{13} - 53280838773 \nu^{12} + \cdots - 435752795150 ) / 20109322702 $ (-51918966616v^15 + 156637786453v^14 + 91292310268v^13 - 53280838773v^12 - 2183806181780v^11 + 2664420234629v^10 + 895817893144v^9 + 3375808124601v^8 - 11013464257600v^7 + 6196852490403v^6 - 2797053807220v^5 + 5190337377625v^4 - 2414367784536v^3 - 578312257270v^2 + 892787498524*v - 435752795150) / 20109322702
$\beta_{14}$	$=$	$ ( - 59961413714 \nu^{15} + 200645094063 \nu^{14} + 16512535898 \nu^{13} - 11349732409 \nu^{12} + \cdots + 137646626396 ) / 20109322702 $ (-59961413714v^15 + 200645094063v^14 + 16512535898v^13 - 11349732409v^12 - 2456663221746v^11 + 3924602278603v^10 - 1167318358606v^9 + 4948815345057v^8 - 13995395422064v^7 + 13762376176403v^6 - 11319086764252v^5 + 11351132808947v^4 - 8474603814678v^3 + 4141673676792v^2 - 1226899096656*v + 137646626396) / 20109322702
$\beta_{15}$	$=$	$ ( 36518218171 \nu^{15} - 122459524518 \nu^{14} - 5843623036 \nu^{13} - 322687170 \nu^{12} + \cdots - 132458831459 ) / 10054661351 $ (36518218171v^15 - 122459524518v^14 - 5843623036v^13 - 322687170v^12 + 1485454721783v^11 - 2410210059932v^10 + 859139333534v^9 - 3080708284676v^8 + 8487123321599v^7 - 8777061742880v^6 + 7381040079790v^5 - 7085471800872v^4 + 5538488457086v^3 - 2812389425334v^2 + 818561766836*v - 132458831459) / 10054661351

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

$\nu$	$=$	$ ( - \beta_{15} - 2 \beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{6} + \beta_{4} + \cdots + 4 ) / 8 $ (-b15 - 2b14 - b11 + b10 + b9 - 2b8 - b6 + b4 - 4b3 + 3b2 + 4) / 8
$\nu^{2}$	$=$	$ ( - \beta_{14} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + \beta_1 ) / 4 $ (-b14 + b11 - b10 - b9 - b8 - b7 - b5 - b4 + 6b3 + 3b2 + b1) / 4
$\nu^{3}$	$=$	$ ( - 5 \beta_{15} - 6 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 9 \beta_{11} + \beta_{10} - \beta_{9} + \cdots + 20 ) / 8 $ (-5b15 - 6b14 - 2b13 - 2b12 - 9b11 + b10 - b9 - 4b8 + 2b7 + 3b6 + 8b5 - b4 + 5b2 - 2*b1 + 20) / 8
$\nu^{4}$	$=$	$ ( - 5 \beta_{15} - 14 \beta_{14} - 2 \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{9} - 6 \beta_{8} + \cdots - 14 ) / 4 $ (-5b15 - 14b14 - 2b12 - b11 + b10 - 2b9 - 6b8 - 2b7 - 8b6 - 2b5 + 2b4 + 14b3 + 10b2 + 2b1 - 14) / 4
$\nu^{5}$	$=$	$ ( - 18 \beta_{15} - 29 \beta_{14} - 10 \beta_{13} - 11 \beta_{11} - 25 \beta_{10} - 15 \beta_{9} + \cdots + \beta_1 ) / 8 $ (-18b15 - 29b14 - 10b13 - 11b11 - 25b10 - 15b9 - 7b8 - 21b7 + 24b6 + 9b5 - 15b4 + 104b3 + 27*b2 + b1) / 8
$\nu^{6}$	$=$	$ ( - 32 \beta_{15} - 38 \beta_{14} - 20 \beta_{13} - 20 \beta_{12} - 44 \beta_{11} + 20 \beta_{10} + \cdots - 12 ) / 4 $ (-32b15 - 38b14 - 20b13 - 20b12 - 44b11 + 20b10 - 7b9 - 12b8 + 2b7 - b6 + 42b5 + 3b4 - 9b2 + 8*b1 - 12) / 4
$\nu^{7}$	$=$	$ ( - 79 \beta_{15} - 215 \beta_{14} + 6 \beta_{12} + 68 \beta_{11} - 68 \beta_{10} - 92 \beta_{9} + \cdots - 472 ) / 8 $ (-79b15 - 215b14 + 6b12 + 68b11 - 68b10 - 92b9 - 11b8 - 63b7 - 41b6 - 63b5 + 48b4 + 472b3 + 60b2 + 63b1 - 472) / 8
$\nu^{8}$	$=$	$ ( - 91 \beta_{15} - 24 \beta_{14} - 110 \beta_{13} - 142 \beta_{11} - 40 \beta_{10} - 24 \beta_{9} + \cdots + 24 \beta_1 ) / 4 $ (-91b15 - 24b14 - 110b13 - 142b11 - 40b10 - 24b9 + 51b8 - 56b7 + 208b6 + 184b5 - 24b4 + 166b3 - 104b2 + 24b1) / 4
$\nu^{9}$	$=$	$ ( - 329 \beta_{15} - 480 \beta_{14} - 222 \beta_{13} - 222 \beta_{12} - 177 \beta_{11} + 481 \beta_{10} + \cdots - 1816 ) / 8 $ (-329b15 - 480b14 - 222b13 - 222b12 - 177b11 + 481b10 - 241b9 + 152b8 + 164b7 - 231b6 + 326b5 + 395b4 - 703b2 + 472b1 - 1816) / 8
$\nu^{10}$	$=$	$ ( - 14 \beta_{15} - 107 \beta_{14} + 420 \beta_{12} + 512 \beta_{11} - 512 \beta_{10} - 236 \beta_{9} + \cdots - 1624 ) / 4 $ (-14b15 - 107b14 + 420b12 + 512b11 - 512b10 - 236b9 + 498b8 - 213b7 + 593b6 - 213b5 + 274b4 + 1624b3 - 342b2 + 213b1 - 1624) / 4
$\nu^{11}$	$=$	$ ( - 296 \beta_{15} + 2485 \beta_{14} - 1982 \beta_{13} - 2517 \beta_{11} + 1925 \beta_{10} + \cdots + 1187 \beta_1 ) / 8 $ (-296b15 + 2485b14 - 1982b13 - 2517b11 + 1925b10 + 649b9 + 2221b8 + 793b7 + 3158b6 + 3807b5 + 649b4 - 5232b3 - 5249b2 + 1187b1) / 8
$\nu^{12}$	$=$	$ ( 1078 \beta_{15} + 264 \beta_{14} + 962 \beta_{13} + 962 \beta_{12} + 3359 \beta_{11} + 1203 \beta_{10} + \cdots - 10066 ) / 4 $ (1078b15 + 264b14 + 962b13 + 962b12 + 3359b11 + 1203b10 - 788b9 + 2281b8 + 1416b7 - 1284b6 - 2176b5 + 2700b4 - 3356b2 + 2072b1 - 10066) / 4
$\nu^{13}$	$=$	$ ( 7935 \beta_{15} + 18293 \beta_{14} + 11418 \beta_{12} + 5972 \beta_{11} - 5972 \beta_{10} + 4588 \beta_{9} + \cdots + 4632 ) / 8 $ (7935b15 + 18293b14 + 11418b12 + 5972b11 - 5972b10 + 4588b9 + 13907b8 + 533b7 + 19683b6 + 533b5 + 2456b4 - 4632b3 - 13172b2 - 533b1 + 4632) / 8
$\nu^{14}$	$=$	$ ( 7760 \beta_{15} + 18403 \beta_{14} - 1340 \beta_{13} - 1874 \beta_{11} + 17394 \beta_{10} + \cdots + 5957 \beta_1 ) / 4 $ (7760b15 + 18403b14 - 1340b13 - 1874b11 + 17394b10 + 6223b9 + 9634b8 + 11621b7 - 570b6 + 5653b5 + 6223b4 - 49232b3 - 23497b2 + 5957b1) / 4
$\nu^{15}$	$=$	$ ( 61371 \beta_{15} + 51324 \beta_{14} + 51026 \beta_{13} + 51026 \beta_{12} + 105043 \beta_{11} + \cdots - 75680 ) / 8 $ (61371b15 + 51324b14 + 51026b13 + 51026b12 + 105043b11 - 17699b10 + 6773b9 + 43672b8 + 22984b7 - 9689b6 - 90770b5 + 32673b4 - 12605b2 + 2916b1 - 75680) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$\beta_{3}$	$1$

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} $

31.1

2.07391	+	0.620024i
1.50047	+	0.288947i
−1.57391	−	1.48605i
0.224274	−	0.447866i
−1.00047	−	1.15497i
−0.349168	−	0.778942i
0.275726	−	0.418160i
0.849168	−	0.0870829i
2.07391	−	0.620024i
1.50047	−	0.288947i
−1.57391	+	1.48605i
0.224274	+	0.447866i
−1.00047	+	1.15497i
−0.349168	+	0.778942i
0.275726	+	0.418160i
0.849168	+	0.0870829i

−1.60159

2.77404i

−1.00367

0.579471i

1.44550

2.21597i

−3.63019

−

6.28767i

31.2

−1.13338

1.96307i

3.08101

−

1.77882i

0.912798

−

2.48330i

−1.06909

−

1.85172i

31.3

−0.376846

0.652717i

1.00367

−

0.579471i

−0.286555

2.63019i

1.21597

2.10613i

31.4

−0.0913671

0.158252i

−3.08101

1.77882i

−2.64485

0.0690906i

1.48330

2.56916i

31.5

0.0913671

−

0.158252i

−3.08101

1.77882i

2.64485

−

0.0690906i

1.48330

2.56916i

31.6

0.376846

−

0.652717i

1.00367

−

0.579471i

0.286555

−

2.63019i

1.21597

2.10613i

31.7

1.13338

−

1.96307i

3.08101

−

1.77882i

−0.912798

2.48330i

−1.06909

−

1.85172i

31.8

1.60159

−

2.77404i

−1.00367

0.579471i

−1.44550

−

2.21597i

−3.63019

−

6.28767i

159.1

−1.60159

−

2.77404i

−1.00367

−

0.579471i

1.44550

−

2.21597i

−3.63019

6.28767i

159.2

−1.13338

−

1.96307i

3.08101

1.77882i

0.912798

2.48330i

−1.06909

1.85172i

159.3

−0.376846

−

0.652717i

1.00367

0.579471i

−0.286555

−

2.63019i

1.21597

−

2.10613i

159.4

−0.0913671

−

0.158252i

−3.08101

−

1.77882i

−2.64485

−

0.0690906i

1.48330

−

2.56916i

159.5

0.0913671

0.158252i

−3.08101

−

1.77882i

2.64485

0.0690906i

1.48330

−

2.56916i

159.6

0.376846

0.652717i

1.00367

0.579471i

0.286555

2.63019i

1.21597

−

2.10613i

159.7

1.13338

1.96307i

3.08101

1.77882i

−0.912798

−

2.48330i

−1.06909

1.85172i

159.8

1.60159

2.77404i

−1.00367

−

0.579471i

−1.44550

2.21597i

−3.63019

6.28767i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.d	odd	6	1	inner
28.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	224.2.p.a	✓	16
3.b	odd	2	1		2016.2.cs.b		16
4.b	odd	2	1	inner	224.2.p.a	✓	16
7.b	odd	2	1		1568.2.p.b		16
7.c	even	3	1		1568.2.f.b		16
7.c	even	3	1		1568.2.p.b		16
7.d	odd	6	1	inner	224.2.p.a	✓	16
7.d	odd	6	1		1568.2.f.b		16
8.b	even	2	1		448.2.p.e		16
8.d	odd	2	1		448.2.p.e		16
12.b	even	2	1		2016.2.cs.b		16
21.g	even	6	1		2016.2.cs.b		16
28.d	even	2	1		1568.2.p.b		16
28.f	even	6	1	inner	224.2.p.a	✓	16
28.f	even	6	1		1568.2.f.b		16
28.g	odd	6	1		1568.2.f.b		16
28.g	odd	6	1		1568.2.p.b		16
56.j	odd	6	1		448.2.p.e		16
56.j	odd	6	1		3136.2.f.j		16
56.k	odd	6	1		3136.2.f.j		16
56.m	even	6	1		448.2.p.e		16
56.m	even	6	1		3136.2.f.j		16
56.p	even	6	1		3136.2.f.j		16
84.j	odd	6	1		2016.2.cs.b		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.2.p.a	✓	16	1.a	even	1	1	trivial
224.2.p.a	✓	16	4.b	odd	2	1	inner
224.2.p.a	✓	16	7.d	odd	6	1	inner
224.2.p.a	✓	16	28.f	even	6	1	inner
448.2.p.e		16	8.b	even	2	1
448.2.p.e		16	8.d	odd	2	1
448.2.p.e		16	56.j	odd	6	1
448.2.p.e		16	56.m	even	6	1
1568.2.f.b		16	7.c	even	3	1
1568.2.f.b		16	7.d	odd	6	1
1568.2.f.b		16	28.f	even	6	1
1568.2.f.b		16	28.g	odd	6	1
1568.2.p.b		16	7.b	odd	2	1
1568.2.p.b		16	7.c	even	3	1
1568.2.p.b		16	28.d	even	2	1
1568.2.p.b		16	28.g	odd	6	1
2016.2.cs.b		16	3.b	odd	2	1
2016.2.cs.b		16	12.b	even	2	1
2016.2.cs.b		16	21.g	even	6	1
2016.2.cs.b		16	84.j	odd	6	1
3136.2.f.j		16	56.j	odd	6	1
3136.2.f.j		16	56.k	odd	6	1
3136.2.f.j		16	56.m	even	6	1
3136.2.f.j		16	56.p	even	6	1

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(224, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} + 16 T^{14} + \cdots + 1 $ T^16 + 16T^14 + 194T^12 + 928T^10 + 3331T^8 + 1952T^6 + 962T^4 + 32*T^2 + 1
$5$	$ (T^{8} - 14 T^{6} + \cdots + 289)^{2} $ (T^8 - 14T^6 + 179T^4 - 238*T^2 + 289)^2
$7$	$ T^{16} + 16 T^{14} + \cdots + 5764801 $ T^16 + 16T^14 + 60T^12 - 784T^10 - 10426T^8 - 38416T^6 + 144060T^4 + 1882384*T^2 + 5764801
$11$	$ T^{16} + \cdots + 352275361 $ T^16 - 64T^14 + 2770T^12 - 65248T^10 + 1111795T^8 - 10602976T^6 + 71309170T^4 - 184086352*T^2 + 352275361
$13$	$ (T^{8} + 56 T^{6} + \cdots + 12544)^{2} $ (T^8 + 56T^6 + 944T^4 + 6016*T^2 + 12544)^2
$17$	$ (T^{8} - 22 T^{6} + \cdots + 49)^{2} $ (T^8 - 22T^6 + 491T^4 + 1056T^3 + 922T^2 + 336*T + 49)^2
$19$	$ T^{16} + 32 T^{14} + \cdots + 1 $ T^16 + 32T^14 + 962T^12 + 1952T^10 + 3331T^8 + 928T^6 + 194T^4 + 16*T^2 + 1
$23$	$ T^{16} - 48 T^{14} + \cdots + 2825761 $ T^16 - 48T^14 + 1618T^12 - 27744T^10 + 344499T^8 - 1616736T^6 + 5565298T^4 - 4357152*T^2 + 2825761
$29$	$ (T^{4} - 4 T^{3} + \cdots + 112)^{4} $ (T^4 - 4T^3 - 28T^2 + 16*T + 112)^4
$31$	$ T^{16} + \cdots + 12897917761 $ T^16 + 128T^14 + 11810T^12 + 500768T^10 + 15386851T^8 + 164644384T^6 + 1274227298T^4 + 4809874288*T^2 + 12897917761
$37$	$ (T^{8} + 4 T^{7} + \cdots + 113569)^{2} $ (T^8 + 4T^7 + 74T^6 + 208T^5 + 3907T^4 + 10064T^3 + 67946T^2 - 74140*T + 113569)^2
$41$	$ (T^{8} + 56 T^{6} + \cdots + 12544)^{2} $ (T^8 + 56T^6 + 944T^4 + 6016*T^2 + 12544)^2
$43$	$ (T^{8} + 192 T^{6} + \cdots + 12544)^{2} $ (T^8 + 192T^6 + 6368T^4 + 18432*T^2 + 12544)^2
$47$	$ T^{16} + \cdots + 9597924961 $ T^16 + 176T^14 + 22130T^12 + 1341344T^10 + 59185171T^8 + 918901408T^6 + 10749032402T^4 + 10558706944*T^2 + 9597924961
$53$	$ (T^{8} + 4 T^{7} + \cdots + 2401)^{2} $ (T^8 + 4T^7 + 106T^6 - 560T^5 + 7651T^4 - 9392T^3 + 14410T^2 + 4900*T + 2401)^2
$59$	$ T^{16} + \cdots + 43617904801 $ T^16 + 256T^14 + 51026T^12 + 3233056T^10 + 148698739T^8 + 3386380832T^6 + 54931126514T^4 + 50280814448*T^2 + 43617904801
$61$	$ (T^{8} + 12 T^{7} + \cdots + 4844401)^{2} $ (T^8 + 12T^7 - 62T^6 - 1320T^5 + 6587T^4 + 91080T^3 - 13582T^2 - 1822428*T + 4844401)^2
$67$	$ T^{16} + \cdots + 15527402881 $ T^16 - 288T^14 + 61570T^12 - 5603424T^10 + 377193795T^8 - 5830527072T^6 + 73592115970T^4 - 34410027696*T^2 + 15527402881
$71$	$ (T^{8} + 320 T^{6} + \cdots + 6635776)^{2} $ (T^8 + 320T^6 + 29664T^4 + 806912*T^2 + 6635776)^2
$73$	$ (T^{8} + 12 T^{7} + \cdots + 564001)^{2} $ (T^8 + 12T^7 - 22T^6 - 840T^5 + 2147T^4 + 61320T^3 + 308362T^2 + 657876*T + 564001)^2
$79$	$ T^{16} + \cdots + 16748793615841 $ T^16 - 336T^14 + 89266T^12 - 6829920T^10 + 367844691T^8 - 10361634912T^6 + 211185354130T^4 - 2270862491520*T^2 + 16748793615841
$83$	$ (T^{8} - 416 T^{6} + \cdots + 3211264)^{2} $ (T^8 - 416T^6 + 39680T^4 - 827392*T^2 + 3211264)^2
$89$	$ (T^{8} + 36 T^{7} + \cdots + 27952369)^{2} $ (T^8 + 36T^7 + 466T^6 + 1224T^5 - 18757T^4 - 71400T^3 + 1290242T^2 + 11102700*T + 27952369)^2
$97$	$ (T^{8} + 472 T^{6} + \cdots + 118026496)^{2} $ (T^8 + 472T^6 + 78896T^4 + 5378432*T^2 + 118026496)^2
show more
show less

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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{14} q^{3} - \beta_{13} q^{5} - \beta_{9} q^{7} + (\beta_{15} + \beta_{11} + \cdots - \beta_{3}) q^{9}+O(q^{10}) \) q - b14 * q^3 - b13 * q^5 - b9 * q^7 + (b15 + b11 + b8 - b3) * q^9	\( q - \beta_{14} q^{3} - \beta_{13} q^{5} - \beta_{9} q^{7} + (\beta_{15} + \beta_{11} + \cdots - \beta_{3}) q^{9}+ \cdots + ( - 2 \beta_{14} - 5 \beta_{9} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) q - b14 * q^3 - b13 * q^5 - b9 * q^7 + (b15 + b11 + b8 - b3) * q^9 + (b14 - b9 + b7 + b6 - b2 + b1) * q^11 + (-b11 + b10 + 2b3 - 1) q^13 + (-b14 + b9 + b7 - b5 + b2) * q^15 - b10 * q^17 + (-b14 + b9 + b4 + b2 - b1) * q^19 + (b12 + b10 + b8 + b3 - 2) * q^21 + (-b7 + b1) * q^23 + (-b15 + b11 - b10 - 2b3 + 2) q^25 + (-b14 - 2b6 - b5 - b4 + 2b2 - b1) * q^27 + (-b8 + 1) * q^29 + (b7 + b6 + 2b5 + b4 - b2 - b1) q^31 + (b15 + 2b13 - b8 - b3 + 2) q^33 + (3b14 - 3b7 - b4 - b2 + b1) * q^35 + (-b15 - b13 - 2b12 - b11 - b8 - b3) q^37 + (2b14 - b7 - b4 - 4b2 - b1) * q^39 + (-b11 + b10 + 2b3 - 1) q^41 + (2b14 + 2b5 - 2b2) q^43 + (-b15 - b11 - 2b10 - 2b8 - b3 - 1) * q^45 + (-b14 - b9 - b6 + b5 - b4 + b1) * q^47 + (-b15 - 2b12 - 2) q^49 + (b14 - b9 + b4 + b2 + b1) * q^51 + (2b13 + b12 + 2b11 - b10 + b3 - 1) * q^53 + (3b9 - 3b7 + b4 + 6b2 + b1) q^55 - q^57 + (-3b14 + 2b9 + 2b7 - 2b1) * q^59 + (-b15 + b13 - b11 + b8 + b3 - 2) * q^61 + (3b14 + b7 - 2b6 - 3b5 - 2b4 - b2) * q^63 + (2b13 + 4b12 - b11 + 2b10 + b3) q^65 + (b14 + b9 - 2b7 + b6 - b4 - b2 - 2b1) * q^67 + (2b15 + b13 + b12 + b11 + b10 + b8 - 4b3 + 2) * q^69 + (-b14 + b9 + b7 - b5 + 3b4 + b2 - 3b1) * q^71 + (-b15 - b11 - b10 - 2b8 - b3 - 1) q^73 + (-b14 + 2b9 + 3b7 + b6 - b5 + 2b4 + 2b2 + b1) * q^75 + (2b15 - b12 + 3b11 + b10 + 2b8 - 2b3 + 5) * q^77 + (2b14 + b7 + 2b2 - b1) * q^79 + (-2b15 - 4b13 - 2b12 - b10 + 6b3 - 6) * q^81 + (-b14 - b9 + b7 - 2b6 - b5 + b4 - 3b2 + b1) * q^83 + (b13 - b12 - b11 - b10 - 1) * q^85 + (-4b14 - b9 - b7 + b6 + 2b5 - b2 + b1) * q^87 + (-2b13 - 2b11 + 3b3 - 6) q^89 + (b14 - b9 + 3b6 + b5 + 2b2 + 3b1) q^91 + (b15 + b13 + 2b12 + b11 + b8 + b3) q^93 + (2b14 - 4b9 + 2b7 + b6 - 2b4 - 3b2 + 2b1) * q^95 + (2b15 - 2b13 - 2b12 + 2b10 + b8 - 6b3 + 3) q^97 + (-2b14 - 5b9 - 5b7 + 2b5 - 4b4 + 4b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^9	\( 16 q - 8 q^{9} - 24 q^{21} + 16 q^{25} + 16 q^{29} + 24 q^{33} - 8 q^{37} - 24 q^{45} - 32 q^{49} - 8 q^{53} - 16 q^{57} - 24 q^{61} + 8 q^{65} - 24 q^{73} + 64 q^{77} - 48 q^{81} - 16 q^{85} - 72 q^{89} + 8 q^{93}+O(q^{100}) \) 16 * q - 8 * q^9 - 24 * q^21 + 16 * q^25 + 16 * q^29 + 24 * q^33 - 8 * q^37 - 24 * q^45 - 32 * q^49 - 8 * q^53 - 16 * q^57 - 24 * q^61 + 8 * q^65 - 24 * q^73 + 64 * q^77 - 48 * q^81 - 16 * q^85 - 72 * q^89 + 8 * q^93

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	224.2.p.a

Level	$224$
Weight	$2$
Character orbit	224.p
Analytic conductor	$1.789$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.78864900528\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	16.0.2353561680715186176.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 \) x^16 - 4x^15 + 2x^14 + 41x^12 - 92x^11 + 66x^10 - 104x^9 + 291x^8 - 388x^7 + 366x^6 - 344x^5 + 286x^4 - 184x^3 + 84x^2 - 24x + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{20} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 4671439980 \nu^{15} - 5019523613 \nu^{14} + 74862020586 \nu^{13} - 4435873025 \nu^{12} + \cdots - 353683962760 ) / 20109322702 \) (-4671439980v^15 - 5019523613v^14 + 74862020586v^13 - 4435873025v^12 - 210602072844v^11 - 553169578089v^10 + 1432081245914v^9 - 85279469075v^8 + 522318705094v^7 - 4152490382437v^6 + 4382872705880v^5 - 3138092527649v^4 + 3538225191170v^3 - 2782602331212v^2 + 1359834224272*v - 353683962760) / 20109322702
\(\beta_{2}\)	\(=\)	\( ( 5063424619 \nu^{15} - 34394314873 \nu^{14} + 58272692948 \nu^{13} + 3111527264 \nu^{12} + \cdots - 314591816094 ) / 20109322702 \) (5063424619v^15 - 34394314873v^14 + 58272692948v^13 + 3111527264v^12 + 197933621407v^11 - 1046870352209v^10 + 1297182921138v^9 - 768732964854v^8 + 2528765768339v^7 - 5349204235533v^6 + 5140695571010v^5 - 4106878988502v^4 + 3966533391230v^3 - 2841292968702v^2 + 1190351287680*v - 314591816094) / 20109322702
\(\beta_{3}\)	\(=\)	\( ( - 11175608166 \nu^{15} + 37005033535 \nu^{14} + 2381595084 \nu^{13} + 3936897701 \nu^{12} + \cdots + 87372551998 ) / 20109322702 \) (-11175608166v^15 + 37005033535v^14 + 2381595084v^13 + 3936897701v^12 - 454687961278v^11 + 715591444243v^10 - 275560834120v^9 + 1011875313139v^8 - 2563606006170v^7 + 2641228538245v^6 - 2425489864376v^5 + 2311962665959v^4 - 1743124162188v^3 + 985628049642v^2 - 363876573956*v + 87372551998) / 20109322702
\(\beta_{4}\)	\(=\)	\( ( - 1736302763 \nu^{15} + 3368540236 \nu^{14} + 8943298446 \nu^{13} + 179687165 \nu^{12} + \cdots - 25579237514 ) / 2872760386 \) (-1736302763v^15 + 3368540236v^14 + 8943298446v^13 + 179687165v^12 - 73751071335v^11 + 11510683964v^10 + 136916188768v^9 + 109872186421v^8 - 223621781697v^7 - 209166485604v^6 + 227204090414v^5 - 65297035181v^4 + 211706237508v^3 - 207339126830v^2 + 96981591808*v - 25579237514) / 2872760386
\(\beta_{5}\)	\(=\)	\( ( 12615896138 \nu^{15} - 42612645921 \nu^{14} + 5931103262 \nu^{13} - 19811250885 \nu^{12} + \cdots - 179930396744 ) / 20109322702 \) (12615896138v^15 - 42612645921v^14 + 5931103262v^13 - 19811250885v^12 + 499479045258v^11 - 845619686261v^10 + 604535781354v^9 - 1376773471743v^8 + 2866813258228v^7 - 3654726081713v^6 + 3946524545552v^5 - 3264415389805v^4 + 2631655568822v^3 - 1752180134328v^2 + 636854341512*v - 179930396744) / 20109322702
\(\beta_{6}\)	\(=\)	\( ( 18129728171 \nu^{15} - 67239132782 \nu^{14} + 13728648942 \nu^{13} + 16261528967 \nu^{12} + \cdots - 69968157850 ) / 20109322702 \) (18129728171v^15 - 67239132782v^14 + 13728648942v^13 + 16261528967v^12 + 743533213171v^11 - 1459054198710v^10 + 645301401120v^9 - 1413868017845v^8 + 4752183358093v^7 - 5440849395538v^6 + 4170972602046v^5 - 4049499601091v^4 + 3305520707556v^3 - 1639495702234v^2 + 554159019384*v - 69968157850) / 20109322702
\(\beta_{7}\)	\(=\)	\( ( 2623797969 \nu^{15} - 10304840791 \nu^{14} + 2190403752 \nu^{13} + 6732922644 \nu^{12} + \cdots + 5890380390 ) / 2872760386 \) (2623797969v^15 - 10304840791v^14 + 2190403752v^13 + 6732922644v^12 + 112107241481v^11 - 231856184779v^10 + 61947862130v^9 - 164982748350v^8 + 762466069777v^7 - 766985838727v^6 + 460781419854v^5 - 570794974590v^4 + 435938816234v^3 - 151480361274v^2 + 31718599632*v + 5890380390) / 2872760386
\(\beta_{8}\)	\(=\)	\( ( 139482845 \nu^{15} - 713255513 \nu^{14} + 813740704 \nu^{13} + 30184751 \nu^{12} + \cdots - 4395910477 ) / 137735087 \) (139482845v^15 - 713255513v^14 + 813740704v^13 + 30184751v^12 + 5568358461v^11 - 19244128631v^10 + 20009728986v^9 - 16925651529v^8 + 51651556669v^7 - 91857962133v^6 + 87279176826v^5 - 72439306253v^4 + 66687816790v^3 - 46319052852v^2 + 18821562600*v - 4395910477) / 137735087
\(\beta_{9}\)	\(=\)	\( ( 21074697951 \nu^{15} - 104086834189 \nu^{14} + 95566612060 \nu^{13} + 46229985362 \nu^{12} + \cdots - 264247625938 ) / 20109322702 \) (21074697951v^15 - 104086834189v^14 + 95566612060v^13 + 46229985362v^12 + 871914454327v^11 - 2751753453829v^10 + 2154137922350v^9 - 1835171745948v^8 + 7683424327567v^7 - 11727612400677v^6 + 9418969698846v^5 - 8639783255096v^4 + 7778130510050v^3 - 4550591596230v^2 + 1735233196936*v - 264247625938) / 20109322702
\(\beta_{10}\)	\(=\)	\( ( - 22929440334 \nu^{15} + 103630918811 \nu^{14} - 82984027620 \nu^{13} - 16420430373 \nu^{12} + \cdots + 438702505064 ) / 20109322702 \) (-22929440334v^15 + 103630918811v^14 - 82984027620v^13 - 16420430373v^12 - 929634728742v^11 + 2610374929811v^10 - 2170386426128v^9 + 2278929576205v^8 - 7519139314798v^7 + 11518339038677v^6 - 10085004362164v^5 + 8949442744397v^4 - 8103799328648v^3 + 5000860688254v^2 - 1965241728420*v + 438702505064) / 20109322702
\(\beta_{11}\)	\(=\)	\( ( 33911516972 \nu^{15} - 108940608997 \nu^{14} - 32201388164 \nu^{13} + 22923719031 \nu^{12} + \cdots + 125770278322 ) / 20109322702 \) (33911516972v^15 - 108940608997v^14 - 32201388164v^13 + 22923719031v^12 + 1411498051716v^11 - 2014225248249v^10 + 64172702604v^9 - 2520892868363v^8 + 7621760166076v^7 - 5993220149479v^6 + 4268041273800v^5 - 5045452375491v^4 + 3122239171348v^3 - 961405268358v^2 + 81167549476*v + 125770278322) / 20109322702
\(\beta_{12}\)	\(=\)	\( ( - 46688945564 \nu^{15} + 219887457941 \nu^{14} - 194424781504 \nu^{13} - 50488884423 \nu^{12} + \cdots + 828605565416 ) / 20109322702 \) (-46688945564v^15 + 219887457941v^14 - 194424781504v^13 - 50488884423v^12 - 1890422689576v^11 + 5678839565545v^10 - 4832040026616v^9 + 4520765331063v^8 - 15902360689892v^7 + 25178802231927v^6 - 21652899765320v^5 + 18807008211767v^4 - 17224972534876v^3 + 10719181869982v^2 - 4138992241108*v + 828605565416) / 20109322702
\(\beta_{13}\)	\(=\)	\( ( - 51918966616 \nu^{15} + 156637786453 \nu^{14} + 91292310268 \nu^{13} - 53280838773 \nu^{12} + \cdots - 435752795150 ) / 20109322702 \) (-51918966616v^15 + 156637786453v^14 + 91292310268v^13 - 53280838773v^12 - 2183806181780v^11 + 2664420234629v^10 + 895817893144v^9 + 3375808124601v^8 - 11013464257600v^7 + 6196852490403v^6 - 2797053807220v^5 + 5190337377625v^4 - 2414367784536v^3 - 578312257270v^2 + 892787498524*v - 435752795150) / 20109322702
\(\beta_{14}\)	\(=\)	\( ( - 59961413714 \nu^{15} + 200645094063 \nu^{14} + 16512535898 \nu^{13} - 11349732409 \nu^{12} + \cdots + 137646626396 ) / 20109322702 \) (-59961413714v^15 + 200645094063v^14 + 16512535898v^13 - 11349732409v^12 - 2456663221746v^11 + 3924602278603v^10 - 1167318358606v^9 + 4948815345057v^8 - 13995395422064v^7 + 13762376176403v^6 - 11319086764252v^5 + 11351132808947v^4 - 8474603814678v^3 + 4141673676792v^2 - 1226899096656*v + 137646626396) / 20109322702
\(\beta_{15}\)	\(=\)	\( ( 36518218171 \nu^{15} - 122459524518 \nu^{14} - 5843623036 \nu^{13} - 322687170 \nu^{12} + \cdots - 132458831459 ) / 10054661351 \) (36518218171v^15 - 122459524518v^14 - 5843623036v^13 - 322687170v^12 + 1485454721783v^11 - 2410210059932v^10 + 859139333534v^9 - 3080708284676v^8 + 8487123321599v^7 - 8777061742880v^6 + 7381040079790v^5 - 7085471800872v^4 + 5538488457086v^3 - 2812389425334v^2 + 818561766836*v - 132458831459) / 10054661351

\(\nu\)	\(=\)	\( ( - \beta_{15} - 2 \beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{6} + \beta_{4} + \cdots + 4 ) / 8 \) (-b15 - 2b14 - b11 + b10 + b9 - 2b8 - b6 + b4 - 4b3 + 3b2 + 4) / 8
\(\nu^{2}\)	\(=\)	\( ( - \beta_{14} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + \beta_1 ) / 4 \) (-b14 + b11 - b10 - b9 - b8 - b7 - b5 - b4 + 6b3 + 3b2 + b1) / 4
\(\nu^{3}\)	\(=\)	\( ( - 5 \beta_{15} - 6 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 9 \beta_{11} + \beta_{10} - \beta_{9} + \cdots + 20 ) / 8 \) (-5b15 - 6b14 - 2b13 - 2b12 - 9b11 + b10 - b9 - 4b8 + 2b7 + 3b6 + 8b5 - b4 + 5b2 - 2*b1 + 20) / 8
\(\nu^{4}\)	\(=\)	\( ( - 5 \beta_{15} - 14 \beta_{14} - 2 \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{9} - 6 \beta_{8} + \cdots - 14 ) / 4 \) (-5b15 - 14b14 - 2b12 - b11 + b10 - 2b9 - 6b8 - 2b7 - 8b6 - 2b5 + 2b4 + 14b3 + 10b2 + 2b1 - 14) / 4
\(\nu^{5}\)	\(=\)	\( ( - 18 \beta_{15} - 29 \beta_{14} - 10 \beta_{13} - 11 \beta_{11} - 25 \beta_{10} - 15 \beta_{9} + \cdots + \beta_1 ) / 8 \) (-18b15 - 29b14 - 10b13 - 11b11 - 25b10 - 15b9 - 7b8 - 21b7 + 24b6 + 9b5 - 15b4 + 104b3 + 27*b2 + b1) / 8
\(\nu^{6}\)	\(=\)	\( ( - 32 \beta_{15} - 38 \beta_{14} - 20 \beta_{13} - 20 \beta_{12} - 44 \beta_{11} + 20 \beta_{10} + \cdots - 12 ) / 4 \) (-32b15 - 38b14 - 20b13 - 20b12 - 44b11 + 20b10 - 7b9 - 12b8 + 2b7 - b6 + 42b5 + 3b4 - 9b2 + 8*b1 - 12) / 4
\(\nu^{7}\)	\(=\)	\( ( - 79 \beta_{15} - 215 \beta_{14} + 6 \beta_{12} + 68 \beta_{11} - 68 \beta_{10} - 92 \beta_{9} + \cdots - 472 ) / 8 \) (-79b15 - 215b14 + 6b12 + 68b11 - 68b10 - 92b9 - 11b8 - 63b7 - 41b6 - 63b5 + 48b4 + 472b3 + 60b2 + 63b1 - 472) / 8
\(\nu^{8}\)	\(=\)	\( ( - 91 \beta_{15} - 24 \beta_{14} - 110 \beta_{13} - 142 \beta_{11} - 40 \beta_{10} - 24 \beta_{9} + \cdots + 24 \beta_1 ) / 4 \) (-91b15 - 24b14 - 110b13 - 142b11 - 40b10 - 24b9 + 51b8 - 56b7 + 208b6 + 184b5 - 24b4 + 166b3 - 104b2 + 24b1) / 4
\(\nu^{9}\)	\(=\)	\( ( - 329 \beta_{15} - 480 \beta_{14} - 222 \beta_{13} - 222 \beta_{12} - 177 \beta_{11} + 481 \beta_{10} + \cdots - 1816 ) / 8 \) (-329b15 - 480b14 - 222b13 - 222b12 - 177b11 + 481b10 - 241b9 + 152b8 + 164b7 - 231b6 + 326b5 + 395b4 - 703b2 + 472b1 - 1816) / 8
\(\nu^{10}\)	\(=\)	\( ( - 14 \beta_{15} - 107 \beta_{14} + 420 \beta_{12} + 512 \beta_{11} - 512 \beta_{10} - 236 \beta_{9} + \cdots - 1624 ) / 4 \) (-14b15 - 107b14 + 420b12 + 512b11 - 512b10 - 236b9 + 498b8 - 213b7 + 593b6 - 213b5 + 274b4 + 1624b3 - 342b2 + 213b1 - 1624) / 4
\(\nu^{11}\)	\(=\)	\( ( - 296 \beta_{15} + 2485 \beta_{14} - 1982 \beta_{13} - 2517 \beta_{11} + 1925 \beta_{10} + \cdots + 1187 \beta_1 ) / 8 \) (-296b15 + 2485b14 - 1982b13 - 2517b11 + 1925b10 + 649b9 + 2221b8 + 793b7 + 3158b6 + 3807b5 + 649b4 - 5232b3 - 5249b2 + 1187b1) / 8
\(\nu^{12}\)	\(=\)	\( ( 1078 \beta_{15} + 264 \beta_{14} + 962 \beta_{13} + 962 \beta_{12} + 3359 \beta_{11} + 1203 \beta_{10} + \cdots - 10066 ) / 4 \) (1078b15 + 264b14 + 962b13 + 962b12 + 3359b11 + 1203b10 - 788b9 + 2281b8 + 1416b7 - 1284b6 - 2176b5 + 2700b4 - 3356b2 + 2072b1 - 10066) / 4
\(\nu^{13}\)	\(=\)	\( ( 7935 \beta_{15} + 18293 \beta_{14} + 11418 \beta_{12} + 5972 \beta_{11} - 5972 \beta_{10} + 4588 \beta_{9} + \cdots + 4632 ) / 8 \) (7935b15 + 18293b14 + 11418b12 + 5972b11 - 5972b10 + 4588b9 + 13907b8 + 533b7 + 19683b6 + 533b5 + 2456b4 - 4632b3 - 13172b2 - 533b1 + 4632) / 8
\(\nu^{14}\)	\(=\)	\( ( 7760 \beta_{15} + 18403 \beta_{14} - 1340 \beta_{13} - 1874 \beta_{11} + 17394 \beta_{10} + \cdots + 5957 \beta_1 ) / 4 \) (7760b15 + 18403b14 - 1340b13 - 1874b11 + 17394b10 + 6223b9 + 9634b8 + 11621b7 - 570b6 + 5653b5 + 6223b4 - 49232b3 - 23497b2 + 5957b1) / 4
\(\nu^{15}\)	\(=\)	\( ( 61371 \beta_{15} + 51324 \beta_{14} + 51026 \beta_{13} + 51026 \beta_{12} + 105043 \beta_{11} + \cdots - 75680 ) / 8 \) (61371b15 + 51324b14 + 51026b13 + 51026b12 + 105043b11 - 17699b10 + 6773b9 + 43672b8 + 22984b7 - 9689b6 - 90770b5 + 32673b4 - 12605b2 + 2916b1 - 75680) / 8

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(\beta_{3}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 31.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} + 16 T^{14} + \cdots + 1 \) T^16 + 16T^14 + 194T^12 + 928T^10 + 3331T^8 + 1952T^6 + 962T^4 + 32*T^2 + 1
$5$	\( (T^{8} - 14 T^{6} + \cdots + 289)^{2} \) (T^8 - 14T^6 + 179T^4 - 238*T^2 + 289)^2
$7$	\( T^{16} + 16 T^{14} + \cdots + 5764801 \) T^16 + 16T^14 + 60T^12 - 784T^10 - 10426T^8 - 38416T^6 + 144060T^4 + 1882384*T^2 + 5764801
$11$	\( T^{16} + \cdots + 352275361 \) T^16 - 64T^14 + 2770T^12 - 65248T^10 + 1111795T^8 - 10602976T^6 + 71309170T^4 - 184086352*T^2 + 352275361
$13$	\( (T^{8} + 56 T^{6} + \cdots + 12544)^{2} \) (T^8 + 56T^6 + 944T^4 + 6016*T^2 + 12544)^2
$17$	\( (T^{8} - 22 T^{6} + \cdots + 49)^{2} \) (T^8 - 22T^6 + 491T^4 + 1056T^3 + 922T^2 + 336*T + 49)^2
$19$	\( T^{16} + 32 T^{14} + \cdots + 1 \) T^16 + 32T^14 + 962T^12 + 1952T^10 + 3331T^8 + 928T^6 + 194T^4 + 16*T^2 + 1
$23$	\( T^{16} - 48 T^{14} + \cdots + 2825761 \) T^16 - 48T^14 + 1618T^12 - 27744T^10 + 344499T^8 - 1616736T^6 + 5565298T^4 - 4357152*T^2 + 2825761
$29$	\( (T^{4} - 4 T^{3} + \cdots + 112)^{4} \) (T^4 - 4T^3 - 28T^2 + 16*T + 112)^4
$31$	\( T^{16} + \cdots + 12897917761 \) T^16 + 128T^14 + 11810T^12 + 500768T^10 + 15386851T^8 + 164644384T^6 + 1274227298T^4 + 4809874288*T^2 + 12897917761
$37$	\( (T^{8} + 4 T^{7} + \cdots + 113569)^{2} \) (T^8 + 4T^7 + 74T^6 + 208T^5 + 3907T^4 + 10064T^3 + 67946T^2 - 74140*T + 113569)^2
$41$	\( (T^{8} + 56 T^{6} + \cdots + 12544)^{2} \) (T^8 + 56T^6 + 944T^4 + 6016*T^2 + 12544)^2
$43$	\( (T^{8} + 192 T^{6} + \cdots + 12544)^{2} \) (T^8 + 192T^6 + 6368T^4 + 18432*T^2 + 12544)^2
$47$	\( T^{16} + \cdots + 9597924961 \) T^16 + 176T^14 + 22130T^12 + 1341344T^10 + 59185171T^8 + 918901408T^6 + 10749032402T^4 + 10558706944*T^2 + 9597924961
$53$	\( (T^{8} + 4 T^{7} + \cdots + 2401)^{2} \) (T^8 + 4T^7 + 106T^6 - 560T^5 + 7651T^4 - 9392T^3 + 14410T^2 + 4900*T + 2401)^2
$59$	\( T^{16} + \cdots + 43617904801 \) T^16 + 256T^14 + 51026T^12 + 3233056T^10 + 148698739T^8 + 3386380832T^6 + 54931126514T^4 + 50280814448*T^2 + 43617904801
$61$	\( (T^{8} + 12 T^{7} + \cdots + 4844401)^{2} \) (T^8 + 12T^7 - 62T^6 - 1320T^5 + 6587T^4 + 91080T^3 - 13582T^2 - 1822428*T + 4844401)^2
$67$	\( T^{16} + \cdots + 15527402881 \) T^16 - 288T^14 + 61570T^12 - 5603424T^10 + 377193795T^8 - 5830527072T^6 + 73592115970T^4 - 34410027696*T^2 + 15527402881
$71$	\( (T^{8} + 320 T^{6} + \cdots + 6635776)^{2} \) (T^8 + 320T^6 + 29664T^4 + 806912*T^2 + 6635776)^2
$73$	\( (T^{8} + 12 T^{7} + \cdots + 564001)^{2} \) (T^8 + 12T^7 - 22T^6 - 840T^5 + 2147T^4 + 61320T^3 + 308362T^2 + 657876*T + 564001)^2
$79$	\( T^{16} + \cdots + 16748793615841 \) T^16 - 336T^14 + 89266T^12 - 6829920T^10 + 367844691T^8 - 10361634912T^6 + 211185354130T^4 - 2270862491520*T^2 + 16748793615841
$83$	\( (T^{8} - 416 T^{6} + \cdots + 3211264)^{2} \) (T^8 - 416T^6 + 39680T^4 - 827392*T^2 + 3211264)^2
$89$	\( (T^{8} + 36 T^{7} + \cdots + 27952369)^{2} \) (T^8 + 36T^7 + 466T^6 + 1224T^5 - 18757T^4 - 71400T^3 + 1290242T^2 + 11102700*T + 27952369)^2
$97$	\( (T^{8} + 472 T^{6} + \cdots + 118026496)^{2} \) (T^8 + 472T^6 + 78896T^4 + 5378432*T^2 + 118026496)^2
show more
show less