LMFDB - Newform orbit 532.2.v.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [532,2,Mod(341,532)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(532, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 3]))

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

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

chi := DirichletCharacter("532.341");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 532 = 2^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	532.v (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:	$4.24804138753$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 25x^{14} + 413x^{12} + 3916x^{10} + 26956x^{8} + 112304x^{6} + 333008x^{4} + 476096x^{2} + 473344 $ x^16 + 25x^14 + 413x^12 + 3916x^10 + 26956x^8 + 112304x^6 + 333008x^4 + 476096*x^2 + 473344
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4} $
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_{6} q^{3} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{3}) q^{5}+ \cdots + ( - \beta_{13} + \beta_{9} + \cdots + \beta_{3}) q^{9}+O(q^{10}) $ q + b6 * q^3 + (-b9 - b8 + b4 - b3) * q^5 + (b4 - 1) * q^7 + (-b13 + b9 + 3b5 + b3) q^9	$ q + \beta_{6} q^{3} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{3}) q^{5}+ \cdots + (3 \beta_{13} - \beta_{9} + 3 \beta_{3} + \cdots + 8) q^{99}+O(q^{100}) $ q + b6 * q^3 + (-b9 - b8 + b4 - b3) * q^5 + (b4 - 1) * q^7 + (-b13 + b9 + 3b5 + b3) q^9 + (-b9 - b3 + b2) * q^11 + (-b14 + b12 - 2b11 + b10 + b6 - b2 - b1) q^13 + (-b14 - b12 + b9 - b7 - b6) * q^15 + (-b13 + b8 + b5 - b3 - b2 - 1) * q^17 + (b12 - b9 - b8 + b5 + b4 - b3 + 2) * q^19 + (b15 - b12 - b10 + b9 - b7 - 2b6 + b2 - b1) q^21 + (-b13 + b9 - 2b5 - b2) q^23 + (-2b13 + b8 + b5 - 2b4 + 2b3 + 1) q^25 + (-2b14 - 2b13 + 2b12 - b9 - 2b6 - 2b3 - b2 - 2b1) * q^27 + (b10 - b9 + b7 + b6 - b2) * q^29 + b6 * q^31 + (b13 - b12 - b11 + 2b10 + b6 + b3 - b2 + b1) q^33 + (2b9 + b8 + 3b5 - 2b4 + 3b3 - 1) * q^35 + (b15 - b11 - 2b6 - 2b1) * q^37 + (4b13 + 2b9 - 2b8 - 2b5 + 4b4 - 2b3 - 2b2 - 2) q^39 + (2b6 + 2b1) * q^41 + (b13 - 2b9 - 2b8 + b4 - b2 + 1) * q^43 + (-b13 - 2b9 + 5b8 - 2b5 - 3b3 - 3b2 + 2) q^45 + (-b13 - b9 + 2b5 - 2b3 + b2 + 4) * q^47 + (-3b13 - b5 - b4 - 2b2 - 2) * q^49 + (b15 + 2b14 - b12 + 2b11 - b9 - 3b6 + b2 - b1) q^51 + (b14 + b13 - 2b12 + b11 - b10 + b9 + b3 + b2 + 2b1) * q^53 + (-b13 - b9 + 6b5 - b4 + 2b3 + 2b2 + 3) q^55 + (-b14 - b12 - b10 + 2b9 - 2b8 - b7 + b4 - b3 + 2b2 - b1 + 1) q^57 + (-b15 - b14 + b9 - b7 - b6) * q^59 + (2b13 + 2b9 - b5 + 2b2 - 2) q^61 + (-3b13 + 5b8 - 5b5 - 3b4 - b3 - 2b2 - 2) q^63 + (b15 + 4b14 + 3b13 - 2b12 - b11 + b9 + 2b6 + 3b3 + 2b2) * q^65 + (b13 - 2b11 + 2b10 + 2b6 + b3 - b2) q^67 + (-b14 - 2b13 + b12 + 2b11 - b10 - b9 - b6 - 2b3 + b1) q^69 + (-2b10 + b9 + b6 + b2 - b1) q^71 + (-2b8 + b5 - 1) q^73 + (-b15 - b13 + 5b12 - 2b11 + 4b10 - 4b9 + 2b7 + 3b6 - b3 - 3b2 - 3b1) * q^75 + (3b13 + 3b9 - b8 - 3b5 - 2b4 + 5b3 + 2b2 - 2) * q^77 + (-2b15 + 2b11 - 2b6 + b1) q^79 + (2b13 - 4b9 - 2b8 - b5 + 4b4 - 6b3 + 2b2 - 1) * q^81 + (-b13 - b9 - 8b5 + b3 + b2 - 4) q^83 + (b9 + b2 - 2) * q^85 + (6b13 - 2b9 - 6b8 + 2b5 + 4b3 + 4b2 - 2) * q^87 + (-b15 + b11 - 2b10 + 2b7 + 2b1) q^89 + (b14 - b13 + b12 + 2b11 - 2b10 - b7 + 2b6 - b3 + 2b2 + 2b1) q^91 + (-b13 + b9 + 6b5 + b3) q^93 + (b14 - b13 - 2b12 + 2b11 - 2b10 + 4b5 + b3 + b2 + 4b1 + 4) q^95 + (-2b15 - 3b14 - 2b13 + 3b12 - b9 + b7 + 2b6 - 2b3 - 2b2 + 3b1) * q^97 + (3b13 - b9 + 3b3 - b2 + 8) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{5} - 14 q^{7} - 26 q^{9}+O(q^{10}) $ 16 * q - 6 * q^5 - 14 * q^7 - 26 * q^9	$ 16 q - 6 q^{5} - 14 q^{7} - 26 q^{9} - 4 q^{11} - 24 q^{17} + 21 q^{19} + 12 q^{23} + 10 q^{25} - 30 q^{35} - 12 q^{39} + 16 q^{43} + 72 q^{45} + 36 q^{47} - 34 q^{49} - 2 q^{57} - 24 q^{61} + 20 q^{63} - 36 q^{73} + 4 q^{77} - 24 q^{81} - 36 q^{85} - 48 q^{87} - 50 q^{93} + 31 q^{95} + 156 q^{99}+O(q^{100}) $ 16 * q - 6 * q^5 - 14 * q^7 - 26 * q^9 - 4 * q^11 - 24 * q^17 + 21 * q^19 + 12 * q^23 + 10 * q^25 - 30 * q^35 - 12 * q^39 + 16 * q^43 + 72 * q^45 + 36 * q^47 - 34 * q^49 - 2 * q^57 - 24 * q^61 + 20 * q^63 - 36 * q^73 + 4 * q^77 - 24 * q^81 - 36 * q^85 - 48 * q^87 - 50 * q^93 + 31 * q^95 + 156 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 25x^{14} + 413x^{12} + 3916x^{10} + 26956x^{8} + 112304x^{6} + 333008x^{4} + 476096x^{2} + 473344 $ x^16 + 25*x^14 + 413*x^12 + 3916*x^10 + 26956*x^8 + 112304*x^6 + 333008*x^4 + 476096*x^2 + 473344 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 13222640 \nu^{14} + 1443847223 \nu^{12} + 27626956293 \nu^{10} + 368412192577 \nu^{8} + \cdots + 64853202705888 ) / 14974191416928 $ (13222640v^14 + 1443847223v^12 + 27626956293v^10 + 368412192577v^8 + 2531533732746v^6 + 13018149004564v^4 + 28163335099344*v^2 + 64853202705888) / 14974191416928
$\beta_{3}$	$=$	$ ( 16823379 \nu^{14} + 1068026746 \nu^{12} + 24930672942 \nu^{10} + 338459185555 \nu^{8} + \cdots + 44373319838208 ) / 14974191416928 $ (16823379v^14 + 1068026746v^12 + 24930672942v^10 + 338459185555v^8 + 2942609733420v^6 + 15401315038376v^4 + 47031389122456*v^2 + 44373319838208) / 14974191416928
$\beta_{4}$	$=$	$ ( 43094485 \nu^{14} + 1341868541 \nu^{12} + 27630665551 \nu^{10} + 357336215606 \nu^{8} + \cdots + 85726744239840 ) / 14974191416928 $ (43094485v^14 + 1341868541v^12 + 27630665551v^10 + 357336215606v^8 + 3116077781294v^6 + 17774318311748v^4 + 54727993408904*v^2 + 85726744239840) / 14974191416928
$\beta_{5}$	$=$	$ ( - 123871913 \nu^{14} - 2881222657 \nu^{12} - 46585894581 \nu^{10} - 412942011692 \nu^{8} + \cdots - 47353126691264 ) / 29948382833856 $ (-123871913v^14 - 2881222657v^12 - 46585894581v^10 - 412942011692v^8 - 2768024531884v^6 - 10262342051040v^4 - 33268866714448*v^2 - 47353126691264) / 29948382833856
$\beta_{6}$	$=$	$ ( - 123871913 \nu^{15} - 2881222657 \nu^{13} - 46585894581 \nu^{11} + \cdots - 47353126691264 \nu ) / 29948382833856 $ (-123871913v^15 - 2881222657v^13 - 46585894581v^11 - 412942011692v^9 - 2768024531884v^7 - 10262342051040v^5 - 33268866714448v^3 - 47353126691264v) / 29948382833856
$\beta_{7}$	$=$	$ ( 85466383 \nu^{15} - 107787584 \nu^{14} + 278524655 \nu^{13} - 2286602744 \nu^{12} + \cdots + 60528135108032 ) / 29948382833856 $ (85466383v^15 - 107787584v^14 + 278524655v^13 - 2286602744v^12 - 3909734629v^11 - 36070199808v^10 - 310667044660v^9 - 285533377472v^8 - 3207045854412v^7 - 1824484633256v^6 - 27792916016800v^5 - 3990735644928v^4 - 93921624510224v^3 - 5810897800192v^2 - 262906020206912*v + 60528135108032) / 29948382833856
$\beta_{8}$	$=$	$ ( - 120753477 \nu^{14} - 2174493569 \nu^{12} - 31853774749 \nu^{10} - 178546117736 \nu^{8} + \cdots + 29843567792384 ) / 14974191416928 $ (-120753477v^14 - 2174493569v^12 - 31853774749v^10 - 178546117736v^8 - 724029237364v^6 + 2493680576768v^4 + 10355505692096*v^2 + 29843567792384) / 14974191416928
$\beta_{9}$	$=$	$ ( - 121010224 \nu^{14} - 3730449967 \nu^{12} - 63697156101 \nu^{10} - 653945570049 \nu^{8} + \cdots - 4325067597856 ) / 14974191416928 $ (-121010224v^14 - 3730449967v^12 - 63697156101v^10 - 653945570049v^8 - 4356018366002v^6 - 17008884649492v^4 - 33974232899536*v^2 - 4325067597856) / 14974191416928
$\beta_{10}$	$=$	$ ( - 337968981 \nu^{15} - 107787584 \nu^{14} - 6507614479 \nu^{13} - 2286602744 \nu^{12} + \cdots + 60528135108032 ) / 29948382833856 $ (-337968981v^15 - 107787584v^14 - 6507614479v^13 - 2286602744v^12 - 89896337859v^11 - 36070199808v^10 - 561907663966v^9 - 285533377472v^8 - 2418854128812v^7 - 1824484633256v^6 + 15603923632v^5 - 3990735644928v^4 + 9230369518496v^3 - 5810897800192v^2 + 36532408104192*v + 60528135108032) / 29948382833856
$\beta_{11}$	$=$	$ ( 10360951 \nu^{15} - 472201845 \nu^{14} + 2293074533 \nu^{13} - 11681911669 \nu^{12} + \cdots - 32645862199552 ) / 29948382833856 $ (10360951v^15 - 472201845v^14 + 2293074533v^13 - 11681911669v^12 + 44738217813v^11 - 181220450253v^10 + 609415750934v^9 - 1584265426592v^8 + 4119527566864v^7 - 9306406227560v^6 + 19764691603016v^5 - 30011429730424v^4 + 28868701284432v^3 - 52907198480384v^2 + 21825143612480*v - 32645862199552) / 29948382833856
$\beta_{12}$	$=$	$ ( - 107787584 \nu^{15} - 472201845 \nu^{14} - 2286602744 \nu^{13} - 11681911669 \nu^{12} + \cdots - 32645862199552 ) / 29948382833856 $ (-107787584v^15 - 472201845v^14 - 2286602744v^13 - 11681911669v^12 - 36070199808v^11 - 181220450253v^10 - 285533377472v^9 - 1584265426592v^8 - 1824484633256v^7 - 9306406227560v^6 - 3990735644928v^5 - 30011429730424v^4 - 5810897800192v^3 - 52907198480384v^2 + 90476517941888*v - 32645862199552) / 29948382833856
$\beta_{13}$	$=$	$ ( - 184007500 \nu^{14} - 4509744224 \nu^{12} - 71226983547 \nu^{10} - 634389521049 \nu^{8} + \cdots - 36347057219952 ) / 7487095708464 $ (-184007500v^14 - 4509744224v^12 - 71226983547v^10 - 634389521049v^8 - 3946498797489v^6 - 14201930059654v^4 - 32982177351652*v^2 - 36347057219952) / 7487095708464
$\beta_{14}$	$=$	$ ( 387700068 \nu^{15} + 337968981 \nu^{14} + 9238287884 \nu^{13} + 6507614479 \nu^{12} + \cdots - 36532408104192 ) / 29948382833856 $ (387700068v^15 + 337968981v^14 + 9238287884v^13 + 6507614479v^12 + 150273378516v^11 + 89896337859v^10 + 1366234669296v^9 + 561907663966v^8 + 9247613494280v^7 + 2418854128812v^6 + 37058632559232v^5 - 15603923632v^4 + 112290377640672v^3 - 9230369518496v^2 + 160095493371520*v - 36532408104192) / 29948382833856
$\beta_{15}$	$=$	$ ( - 1053878467 \nu^{15} - 472201845 \nu^{14} - 25890539379 \nu^{13} - 11681911669 \nu^{12} + \cdots - 32645862199552 ) / 29948382833856 $ (-1053878467v^15 - 472201845v^14 - 25890539379v^13 - 11681911669v^12 - 422825359679v^11 - 181220450253v^10 - 3950705904572v^9 - 1584265426592v^8 - 26610659917684v^7 - 9306406227560v^6 - 106609434558080v^5 - 30011429730424v^4 - 280464460739056v^3 - 52907198480384v^2 - 273718326108864*v - 32645862199552) / 29948382833856

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{13} - 6\beta_{5} - \beta_{3} + \beta_{2} - 6 $ b13 - 6*b5 - b3 + b2 - 6
$\nu^{3}$	$=$	$ -2\beta_{14} - 2\beta_{13} + 2\beta_{12} - \beta_{9} - 8\beta_{6} - 2\beta_{3} - \beta_{2} - 8\beta_1 $ -2b14 - 2b13 + 2b12 - b9 - 8b6 - 2b3 - b2 - 8b1
$\nu^{4}$	$=$	$ -11\beta_{13} + 13\beta_{9} - 2\beta_{8} + 46\beta_{5} - 2\beta_{4} + 13\beta_{3} $ -11b13 + 13b9 - 2b8 + 46b5 - 2b4 + 13b3
$\nu^{5}$	$=$	$ 2\beta_{15} + 28\beta_{14} + 13\beta_{13} - 2\beta_{9} + 2\beta_{7} + 72\beta_{6} + 13\beta_{3} + 13\beta_{2} $ 2b15 + 28b14 + 13b13 - 2b9 + 2b7 + 72b6 + 13b3 + 13b2
$\nu^{6}$	$=$	$ -8\beta_{13} - 73\beta_{9} + 76\beta_{8} - 38\beta_{4} + 30\beta_{3} - 111\beta_{2} + 398 $ -8b13 - 73b9 + 76b8 - 38b4 + 30b3 - 111b2 + 398
$\nu^{7}$	$=$	$ 38 \beta_{15} + 141 \beta_{13} - 328 \beta_{12} + 8 \beta_{11} - 16 \beta_{10} + 187 \beta_{9} + \cdots + 612 \beta_1 $ 38b15 + 141b13 - 328b12 + 8b11 - 16b10 + 187b9 - 76b7 - 46b6 + 141b3 + 46b2 + 612*b1
$\nu^{8}$	$=$	$ 1135\beta_{13} - 852\beta_{9} - 526\beta_{8} - 3662\beta_{5} + 1052\beta_{4} - 1987\beta_{3} + 935\beta_{2} - 3662 $ 1135b13 - 852b9 - 526b8 - 3662b5 + 1052b4 - 1987b3 + 935*b2 - 3662
$\nu^{9}$	$=$	$ - 1052 \beta_{15} - 3648 \beta_{14} - 2922 \beta_{13} + 3648 \beta_{12} - 400 \beta_{11} + \cdots - 5732 \beta_1 $ -1052b15 - 3648b14 - 2922b13 + 3648b12 - 400b11 + 200b10 - 1461b9 + 526b7 - 5858b6 - 2922b3 - 2187b2 - 5732b1
$\nu^{10}$	$=$	$ -8463\beta_{13} + 14941\beta_{9} - 6478\beta_{8} + 34878\beta_{5} - 6478\beta_{4} + 18245\beta_{3} + 3304\beta_{2} $ -8463b13 + 14941b9 - 6478b8 + 34878b5 - 6478b4 + 18245b3 + 3304*b2
$\nu^{11}$	$=$	$ 6478 \beta_{15} + 39664 \beta_{14} + 14941 \beta_{13} + 3304 \beta_{11} + 3304 \beta_{10} + \cdots + 14941 \beta_{2} $ 6478b15 + 39664b14 + 14941b13 + 3304b11 + 3304b10 - 9782b9 + 6478b7 + 64760b6 + 14941b3 + 14941b2
$\nu^{12}$	$=$	$ -45736\beta_{13} - 47441\beta_{9} + 150716\beta_{8} - 75358\beta_{4} + 29622\beta_{3} - 122799\beta_{2} + 339374 $ -45736b13 - 47441b9 + 150716b8 - 75358b4 + 29622b3 - 122799b2 + 339374
$\nu^{13}$	$=$	$ 75358 \beta_{15} + 152421 \beta_{13} - 425936 \beta_{12} + 45736 \beta_{11} - 91472 \beta_{10} + \cdots + 539236 \beta_1 $ 75358b15 + 152421b13 - 425936b12 + 45736b11 - 91472b10 + 273515b9 - 150716b7 - 121094b6 + 152421b3 + 121094b2 + 539236*b1
$\nu^{14}$	$=$	$ 1284423 \beta_{13} - 1121076 \beta_{9} - 848462 \beta_{8} - 3350494 \beta_{5} + 1696924 \beta_{4} + \cdots - 3350494 $ 1284423b13 - 1121076b9 - 848462b8 - 3350494b5 + 1696924b4 - 2405499b3 + 708575*b2 - 3350494
$\nu^{15}$	$=$	$ - 1696924 \beta_{15} - 4538384 \beta_{14} - 3114074 \beta_{13} + 4538384 \beta_{12} + \cdots - 5343492 \beta_1 $ -1696924b15 - 4538384b14 - 3114074b13 + 4538384b12 - 1151696b11 + 575848b10 - 1557037b9 + 848462b7 - 5040258b6 - 3114074b3 - 2981347b2 - 5343492b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/532\mathbb{Z}\right)^\times$.

$n$	$267$	$381$	$477$
$\chi(n)$	$1$	$-\beta_{5}$	$-1$

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} $

341.1

−1.61383	+	2.79524i
−1.44088	+	2.49568i
−1.06269	+	1.84063i
−0.663412	+	1.14906i
0.663412	−	1.14906i
1.06269	−	1.84063i
1.44088	−	2.49568i
1.61383	−	2.79524i
−1.61383	−	2.79524i
−1.44088	−	2.49568i
−1.06269	−	1.84063i
−0.663412	−	1.14906i
0.663412	+	1.14906i
1.06269	+	1.84063i
1.44088	+	2.49568i
1.61383	+	2.79524i

−1.61383

−

2.79524i

−3.48366

−

2.01129i

−1.80346

−

1.93585i

−3.70889

6.42399i

341.2

−1.44088

−

2.49568i

0.596159

0.344192i

0.0834521

2.64443i

−2.65226

4.59385i

341.3

−1.06269

−

1.84063i

2.32104

1.34005i

−2.45626

0.983253i

−0.758613

1.31396i

341.4

−0.663412

−

1.14906i

−0.933539

−

0.538979i

0.676271

−

2.55786i

0.619770

−

1.07347i

341.5

0.663412

1.14906i

−0.933539

−

0.538979i

0.676271

−

2.55786i

0.619770

−

1.07347i

341.6

1.06269

1.84063i

2.32104

1.34005i

−2.45626

0.983253i

−0.758613

1.31396i

341.7

1.44088

2.49568i

0.596159

0.344192i

0.0834521

2.64443i

−2.65226

4.59385i

341.8

1.61383

2.79524i

−3.48366

−

2.01129i

−1.80346

−

1.93585i

−3.70889

6.42399i

493.1

−1.61383

2.79524i

−3.48366

2.01129i

−1.80346

1.93585i

−3.70889

−

6.42399i

493.2

−1.44088

2.49568i

0.596159

−

0.344192i

0.0834521

−

2.64443i

−2.65226

−

4.59385i

493.3

−1.06269

1.84063i

2.32104

−

1.34005i

−2.45626

−

0.983253i

−0.758613

−

1.31396i

493.4

−0.663412

1.14906i

−0.933539

0.538979i

0.676271

2.55786i

0.619770

1.07347i

493.5

0.663412

−

1.14906i

−0.933539

0.538979i

0.676271

2.55786i

0.619770

1.07347i

493.6

1.06269

−

1.84063i

2.32104

−

1.34005i

−2.45626

−

0.983253i

−0.758613

−

1.31396i

493.7

1.44088

−

2.49568i

0.596159

−

0.344192i

0.0834521

−

2.64443i

−2.65226

−

4.59385i

493.8

1.61383

−

2.79524i

−3.48366

2.01129i

−1.80346

1.93585i

−3.70889

−

6.42399i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
19.b	odd	2	1	inner
133.o	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	532.2.v.e	✓	16
7.c	even	3	1		3724.2.g.f		16
7.d	odd	6	1	inner	532.2.v.e	✓	16
7.d	odd	6	1		3724.2.g.f		16
19.b	odd	2	1	inner	532.2.v.e	✓	16
133.o	even	6	1	inner	532.2.v.e	✓	16
133.o	even	6	1		3724.2.g.f		16
133.r	odd	6	1		3724.2.g.f		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
532.2.v.e	✓	16	1.a	even	1	1	trivial
532.2.v.e	✓	16	7.d	odd	6	1	inner
532.2.v.e	✓	16	19.b	odd	2	1	inner
532.2.v.e	✓	16	133.o	even	6	1	inner
3724.2.g.f		16	7.c	even	3	1
3724.2.g.f		16	7.d	odd	6	1
3724.2.g.f		16	133.o	even	6	1
3724.2.g.f		16	133.r	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(532, [\chi])$:

$ T_{3}^{16} + 25 T_{3}^{14} + 413 T_{3}^{12} + 3916 T_{3}^{10} + 26956 T_{3}^{8} + 112304 T_{3}^{6} + \cdots + 473344 $

$ T_{5}^{8} + 3T_{5}^{7} - 8T_{5}^{6} - 33T_{5}^{5} + 104T_{5}^{4} + 99T_{5}^{3} - 61T_{5}^{2} - 72T_{5} + 64 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} + 25 T^{14} + \cdots + 473344 $ T^16 + 25T^14 + 413T^12 + 3916T^10 + 26956T^8 + 112304T^6 + 333008T^4 + 476096*T^2 + 473344
$5$	$ (T^{8} + 3 T^{7} - 8 T^{6} + \cdots + 64)^{2} $ (T^8 + 3T^7 - 8T^6 - 33T^5 + 104T^4 + 99T^3 - 61T^2 - 72*T + 64)^2
$7$	$ (T^{8} + 7 T^{7} + \cdots + 2401)^{2} $ (T^8 + 7T^7 + 33T^6 + 122T^5 + 368T^4 + 854T^3 + 1617T^2 + 2401*T + 2401)^2
$11$	$ (T^{8} + 2 T^{7} + \cdots + 2401)^{2} $ (T^8 + 2T^7 + 26T^6 + 32T^5 + 511T^4 + 640T^3 + 2522T^2 - 1862*T + 2401)^2
$13$	$ (T^{8} - 124 T^{6} + \cdots + 795328)^{2} $ (T^8 - 124T^6 + 5636T^4 - 110912*T^2 + 795328)^2
$17$	$ (T^{8} + 12 T^{7} + \cdots + 16)^{2} $ (T^8 + 12T^7 + 49T^6 + 12T^5 - 139T^4 - 36T^3 + 428T^2 + 144*T + 16)^2
$19$	$ T^{16} + \cdots + 16983563041 $ T^16 - 21T^15 + 237T^14 - 1890T^13 + 11937T^12 - 61983T^11 + 277614T^10 - 1158873T^9 + 4926926T^8 - 22018587T^7 + 100218654T^6 - 425141397T^5 + 1555641777T^4 - 4679827110T^3 + 11149873797T^2 - 18771306519*T + 16983563041
$23$	$ (T^{8} - 6 T^{7} + \cdots + 1369)^{2} $ (T^8 - 6T^7 + 46T^6 - 24T^5 + 315T^4 + 24T^3 + 2134T^2 + 1554*T + 1369)^2
$29$	$ (T^{8} + 180 T^{6} + \cdots + 2818048)^{2} $ (T^8 + 180T^6 + 11444T^4 + 301824*T^2 + 2818048)^2
$31$	$ T^{16} + 25 T^{14} + \cdots + 473344 $ T^16 + 25T^14 + 413T^12 + 3916T^10 + 26956T^8 + 112304T^6 + 333008T^4 + 476096*T^2 + 473344
$37$	$ T^{16} + \cdots + 5356595482624 $ T^16 - 215T^14 + 32545T^12 - 2310680T^10 + 117047068T^8 - 3317551040T^6 + 67727437840T^4 - 729647832320*T^2 + 5356595482624
$41$	$ (T^{8} - 100 T^{6} + \cdots + 176128)^{2} $ (T^8 - 100T^6 + 3392T^4 - 44288*T^2 + 176128)^2
$43$	$ (T^{4} - 4 T^{3} - 51 T^{2} + \cdots + 16)^{4} $ (T^4 - 4T^3 - 51T^2 + 56*T + 16)^4
$47$	$ (T^{8} - 18 T^{7} + \cdots + 110889)^{2} $ (T^8 - 18T^7 + 78T^6 + 540T^5 - 1701T^4 - 11340T^3 + 37638T^2 + 125874*T + 110889)^2
$53$	$ T^{16} - 163 T^{14} + \cdots + 473344 $ T^16 - 163T^14 + 21805T^12 - 749788T^10 + 20515372T^8 - 63479920T^6 + 175532752T^4 - 9199936*T^2 + 473344
$59$	$ T^{16} + \cdots + 161931936661504 $ T^16 + 293T^14 + 57053T^12 + 6257132T^10 + 497100304T^8 + 23932026880T^6 + 821768400896T^4 + 13871131131904*T^2 + 161931936661504
$61$	$ (T^{8} + 12 T^{7} + \cdots + 152881)^{2} $ (T^8 + 12T^7 + 2T^6 - 552T^5 + 347T^4 + 24840T^3 + 115186T^2 + 211140*T + 152881)^2
$67$	$ T^{16} + \cdots + 31021072384 $ T^16 - 211T^14 + 32953T^12 - 2264848T^10 + 115074496T^8 - 943657984T^6 + 5706551296T^4 - 15499264000*T^2 + 31021072384
$71$	$ (T^{8} + 364 T^{6} + \cdots + 7730368)^{2} $ (T^8 + 364T^6 + 30756T^4 + 873664*T^2 + 7730368)^2
$73$	$ (T^{8} + 18 T^{7} + \cdots + 2401)^{2} $ (T^8 + 18T^7 + 100T^6 - 144T^5 - 1363T^4 + 1968T^3 + 20564T^2 + 12054*T + 2401)^2
$79$	$ T^{16} + \cdots + 887122864384 $ T^16 - 491T^14 + 171637T^12 - 28170164T^10 + 3366488044T^8 - 204866820176T^6 + 8716450737232T^4 - 2791162322240*T^2 + 887122864384
$83$	$ (T^{8} + 297 T^{6} + \cdots + 1127844)^{2} $ (T^8 + 297T^6 + 24291T^4 + 400383*T^2 + 1127844)^2
$89$	$ T^{16} + \cdots + 632546627584 $ T^16 + 349T^14 + 86969T^12 + 10334440T^10 + 894546460T^8 + 31175559104T^6 + 802152544400T^4 + 724515176192*T^2 + 632546627584
$97$	$ (T^{8} - 576 T^{6} + \cdots + 14266368)^{2} $ (T^8 - 576T^6 + 106992T^4 - 6552576*T^2 + 14266368)^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:	\( N \)	\(=\)	\( 532 = 2^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	532.v (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{3} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{3}) q^{5}+ \cdots + ( - \beta_{13} + \beta_{9} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) q + b6 * q^3 + (-b9 - b8 + b4 - b3) * q^5 + (b4 - 1) * q^7 + (-b13 + b9 + 3b5 + b3) q^9	\( q + \beta_{6} q^{3} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{3}) q^{5}+ \cdots + (3 \beta_{13} - \beta_{9} + 3 \beta_{3} + \cdots + 8) q^{99}+O(q^{100}) \) q + b6 * q^3 + (-b9 - b8 + b4 - b3) * q^5 + (b4 - 1) * q^7 + (-b13 + b9 + 3b5 + b3) q^9 + (-b9 - b3 + b2) * q^11 + (-b14 + b12 - 2b11 + b10 + b6 - b2 - b1) q^13 + (-b14 - b12 + b9 - b7 - b6) * q^15 + (-b13 + b8 + b5 - b3 - b2 - 1) * q^17 + (b12 - b9 - b8 + b5 + b4 - b3 + 2) * q^19 + (b15 - b12 - b10 + b9 - b7 - 2b6 + b2 - b1) q^21 + (-b13 + b9 - 2b5 - b2) q^23 + (-2b13 + b8 + b5 - 2b4 + 2b3 + 1) q^25 + (-2b14 - 2b13 + 2b12 - b9 - 2b6 - 2b3 - b2 - 2b1) * q^27 + (b10 - b9 + b7 + b6 - b2) * q^29 + b6 * q^31 + (b13 - b12 - b11 + 2b10 + b6 + b3 - b2 + b1) q^33 + (2b9 + b8 + 3b5 - 2b4 + 3b3 - 1) * q^35 + (b15 - b11 - 2b6 - 2b1) * q^37 + (4b13 + 2b9 - 2b8 - 2b5 + 4b4 - 2b3 - 2b2 - 2) q^39 + (2b6 + 2b1) * q^41 + (b13 - 2b9 - 2b8 + b4 - b2 + 1) * q^43 + (-b13 - 2b9 + 5b8 - 2b5 - 3b3 - 3b2 + 2) q^45 + (-b13 - b9 + 2b5 - 2b3 + b2 + 4) * q^47 + (-3b13 - b5 - b4 - 2b2 - 2) * q^49 + (b15 + 2b14 - b12 + 2b11 - b9 - 3b6 + b2 - b1) q^51 + (b14 + b13 - 2b12 + b11 - b10 + b9 + b3 + b2 + 2b1) * q^53 + (-b13 - b9 + 6b5 - b4 + 2b3 + 2b2 + 3) q^55 + (-b14 - b12 - b10 + 2b9 - 2b8 - b7 + b4 - b3 + 2b2 - b1 + 1) q^57 + (-b15 - b14 + b9 - b7 - b6) * q^59 + (2b13 + 2b9 - b5 + 2b2 - 2) q^61 + (-3b13 + 5b8 - 5b5 - 3b4 - b3 - 2b2 - 2) q^63 + (b15 + 4b14 + 3b13 - 2b12 - b11 + b9 + 2b6 + 3b3 + 2b2) * q^65 + (b13 - 2b11 + 2b10 + 2b6 + b3 - b2) q^67 + (-b14 - 2b13 + b12 + 2b11 - b10 - b9 - b6 - 2b3 + b1) q^69 + (-2b10 + b9 + b6 + b2 - b1) q^71 + (-2b8 + b5 - 1) q^73 + (-b15 - b13 + 5b12 - 2b11 + 4b10 - 4b9 + 2b7 + 3b6 - b3 - 3b2 - 3b1) * q^75 + (3b13 + 3b9 - b8 - 3b5 - 2b4 + 5b3 + 2b2 - 2) * q^77 + (-2b15 + 2b11 - 2b6 + b1) q^79 + (2b13 - 4b9 - 2b8 - b5 + 4b4 - 6b3 + 2b2 - 1) * q^81 + (-b13 - b9 - 8b5 + b3 + b2 - 4) q^83 + (b9 + b2 - 2) * q^85 + (6b13 - 2b9 - 6b8 + 2b5 + 4b3 + 4b2 - 2) * q^87 + (-b15 + b11 - 2b10 + 2b7 + 2b1) q^89 + (b14 - b13 + b12 + 2b11 - 2b10 - b7 + 2b6 - b3 + 2b2 + 2b1) q^91 + (-b13 + b9 + 6b5 + b3) q^93 + (b14 - b13 - 2b12 + 2b11 - 2b10 + 4b5 + b3 + b2 + 4b1 + 4) q^95 + (-2b15 - 3b14 - 2b13 + 3b12 - b9 + b7 + 2b6 - 2b3 - 2b2 + 3b1) * q^97 + (3b13 - b9 + 3b3 - b2 + 8) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{5} - 14 q^{7} - 26 q^{9}+O(q^{10}) \) 16 * q - 6 * q^5 - 14 * q^7 - 26 * q^9	\( 16 q - 6 q^{5} - 14 q^{7} - 26 q^{9} - 4 q^{11} - 24 q^{17} + 21 q^{19} + 12 q^{23} + 10 q^{25} - 30 q^{35} - 12 q^{39} + 16 q^{43} + 72 q^{45} + 36 q^{47} - 34 q^{49} - 2 q^{57} - 24 q^{61} + 20 q^{63} - 36 q^{73} + 4 q^{77} - 24 q^{81} - 36 q^{85} - 48 q^{87} - 50 q^{93} + 31 q^{95} + 156 q^{99}+O(q^{100}) \) 16 * q - 6 * q^5 - 14 * q^7 - 26 * q^9 - 4 * q^11 - 24 * q^17 + 21 * q^19 + 12 * q^23 + 10 * q^25 - 30 * q^35 - 12 * q^39 + 16 * q^43 + 72 * q^45 + 36 * q^47 - 34 * q^49 - 2 * q^57 - 24 * q^61 + 20 * q^63 - 36 * q^73 + 4 * q^77 - 24 * q^81 - 36 * q^85 - 48 * q^87 - 50 * q^93 + 31 * q^95 + 156 * 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	532.2.v.e

Level	$532$
Weight	$2$
Character orbit	532.v
Analytic conductor	$4.248$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(4.24804138753\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 25x^{14} + 413x^{12} + 3916x^{10} + 26956x^{8} + 112304x^{6} + 333008x^{4} + 476096x^{2} + 473344 \) x^16 + 25x^14 + 413x^12 + 3916x^10 + 26956x^8 + 112304x^6 + 333008x^4 + 476096*x^2 + 473344
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 13222640 \nu^{14} + 1443847223 \nu^{12} + 27626956293 \nu^{10} + 368412192577 \nu^{8} + \cdots + 64853202705888 ) / 14974191416928 \) (13222640v^14 + 1443847223v^12 + 27626956293v^10 + 368412192577v^8 + 2531533732746v^6 + 13018149004564v^4 + 28163335099344*v^2 + 64853202705888) / 14974191416928
\(\beta_{3}\)	\(=\)	\( ( 16823379 \nu^{14} + 1068026746 \nu^{12} + 24930672942 \nu^{10} + 338459185555 \nu^{8} + \cdots + 44373319838208 ) / 14974191416928 \) (16823379v^14 + 1068026746v^12 + 24930672942v^10 + 338459185555v^8 + 2942609733420v^6 + 15401315038376v^4 + 47031389122456*v^2 + 44373319838208) / 14974191416928
\(\beta_{4}\)	\(=\)	\( ( 43094485 \nu^{14} + 1341868541 \nu^{12} + 27630665551 \nu^{10} + 357336215606 \nu^{8} + \cdots + 85726744239840 ) / 14974191416928 \) (43094485v^14 + 1341868541v^12 + 27630665551v^10 + 357336215606v^8 + 3116077781294v^6 + 17774318311748v^4 + 54727993408904*v^2 + 85726744239840) / 14974191416928
\(\beta_{5}\)	\(=\)	\( ( - 123871913 \nu^{14} - 2881222657 \nu^{12} - 46585894581 \nu^{10} - 412942011692 \nu^{8} + \cdots - 47353126691264 ) / 29948382833856 \) (-123871913v^14 - 2881222657v^12 - 46585894581v^10 - 412942011692v^8 - 2768024531884v^6 - 10262342051040v^4 - 33268866714448*v^2 - 47353126691264) / 29948382833856
\(\beta_{6}\)	\(=\)	\( ( - 123871913 \nu^{15} - 2881222657 \nu^{13} - 46585894581 \nu^{11} + \cdots - 47353126691264 \nu ) / 29948382833856 \) (-123871913v^15 - 2881222657v^13 - 46585894581v^11 - 412942011692v^9 - 2768024531884v^7 - 10262342051040v^5 - 33268866714448v^3 - 47353126691264v) / 29948382833856
\(\beta_{7}\)	\(=\)	\( ( 85466383 \nu^{15} - 107787584 \nu^{14} + 278524655 \nu^{13} - 2286602744 \nu^{12} + \cdots + 60528135108032 ) / 29948382833856 \) (85466383v^15 - 107787584v^14 + 278524655v^13 - 2286602744v^12 - 3909734629v^11 - 36070199808v^10 - 310667044660v^9 - 285533377472v^8 - 3207045854412v^7 - 1824484633256v^6 - 27792916016800v^5 - 3990735644928v^4 - 93921624510224v^3 - 5810897800192v^2 - 262906020206912*v + 60528135108032) / 29948382833856
\(\beta_{8}\)	\(=\)	\( ( - 120753477 \nu^{14} - 2174493569 \nu^{12} - 31853774749 \nu^{10} - 178546117736 \nu^{8} + \cdots + 29843567792384 ) / 14974191416928 \) (-120753477v^14 - 2174493569v^12 - 31853774749v^10 - 178546117736v^8 - 724029237364v^6 + 2493680576768v^4 + 10355505692096*v^2 + 29843567792384) / 14974191416928
\(\beta_{9}\)	\(=\)	\( ( - 121010224 \nu^{14} - 3730449967 \nu^{12} - 63697156101 \nu^{10} - 653945570049 \nu^{8} + \cdots - 4325067597856 ) / 14974191416928 \) (-121010224v^14 - 3730449967v^12 - 63697156101v^10 - 653945570049v^8 - 4356018366002v^6 - 17008884649492v^4 - 33974232899536*v^2 - 4325067597856) / 14974191416928
\(\beta_{10}\)	\(=\)	\( ( - 337968981 \nu^{15} - 107787584 \nu^{14} - 6507614479 \nu^{13} - 2286602744 \nu^{12} + \cdots + 60528135108032 ) / 29948382833856 \) (-337968981v^15 - 107787584v^14 - 6507614479v^13 - 2286602744v^12 - 89896337859v^11 - 36070199808v^10 - 561907663966v^9 - 285533377472v^8 - 2418854128812v^7 - 1824484633256v^6 + 15603923632v^5 - 3990735644928v^4 + 9230369518496v^3 - 5810897800192v^2 + 36532408104192*v + 60528135108032) / 29948382833856
\(\beta_{11}\)	\(=\)	\( ( 10360951 \nu^{15} - 472201845 \nu^{14} + 2293074533 \nu^{13} - 11681911669 \nu^{12} + \cdots - 32645862199552 ) / 29948382833856 \) (10360951v^15 - 472201845v^14 + 2293074533v^13 - 11681911669v^12 + 44738217813v^11 - 181220450253v^10 + 609415750934v^9 - 1584265426592v^8 + 4119527566864v^7 - 9306406227560v^6 + 19764691603016v^5 - 30011429730424v^4 + 28868701284432v^3 - 52907198480384v^2 + 21825143612480*v - 32645862199552) / 29948382833856
\(\beta_{12}\)	\(=\)	\( ( - 107787584 \nu^{15} - 472201845 \nu^{14} - 2286602744 \nu^{13} - 11681911669 \nu^{12} + \cdots - 32645862199552 ) / 29948382833856 \) (-107787584v^15 - 472201845v^14 - 2286602744v^13 - 11681911669v^12 - 36070199808v^11 - 181220450253v^10 - 285533377472v^9 - 1584265426592v^8 - 1824484633256v^7 - 9306406227560v^6 - 3990735644928v^5 - 30011429730424v^4 - 5810897800192v^3 - 52907198480384v^2 + 90476517941888*v - 32645862199552) / 29948382833856
\(\beta_{13}\)	\(=\)	\( ( - 184007500 \nu^{14} - 4509744224 \nu^{12} - 71226983547 \nu^{10} - 634389521049 \nu^{8} + \cdots - 36347057219952 ) / 7487095708464 \) (-184007500v^14 - 4509744224v^12 - 71226983547v^10 - 634389521049v^8 - 3946498797489v^6 - 14201930059654v^4 - 32982177351652*v^2 - 36347057219952) / 7487095708464
\(\beta_{14}\)	\(=\)	\( ( 387700068 \nu^{15} + 337968981 \nu^{14} + 9238287884 \nu^{13} + 6507614479 \nu^{12} + \cdots - 36532408104192 ) / 29948382833856 \) (387700068v^15 + 337968981v^14 + 9238287884v^13 + 6507614479v^12 + 150273378516v^11 + 89896337859v^10 + 1366234669296v^9 + 561907663966v^8 + 9247613494280v^7 + 2418854128812v^6 + 37058632559232v^5 - 15603923632v^4 + 112290377640672v^3 - 9230369518496v^2 + 160095493371520*v - 36532408104192) / 29948382833856
\(\beta_{15}\)	\(=\)	\( ( - 1053878467 \nu^{15} - 472201845 \nu^{14} - 25890539379 \nu^{13} - 11681911669 \nu^{12} + \cdots - 32645862199552 ) / 29948382833856 \) (-1053878467v^15 - 472201845v^14 - 25890539379v^13 - 11681911669v^12 - 422825359679v^11 - 181220450253v^10 - 3950705904572v^9 - 1584265426592v^8 - 26610659917684v^7 - 9306406227560v^6 - 106609434558080v^5 - 30011429730424v^4 - 280464460739056v^3 - 52907198480384v^2 - 273718326108864*v - 32645862199552) / 29948382833856

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{13} - 6\beta_{5} - \beta_{3} + \beta_{2} - 6 \) b13 - 6*b5 - b3 + b2 - 6
\(\nu^{3}\)	\(=\)	\( -2\beta_{14} - 2\beta_{13} + 2\beta_{12} - \beta_{9} - 8\beta_{6} - 2\beta_{3} - \beta_{2} - 8\beta_1 \) -2b14 - 2b13 + 2b12 - b9 - 8b6 - 2b3 - b2 - 8b1
\(\nu^{4}\)	\(=\)	\( -11\beta_{13} + 13\beta_{9} - 2\beta_{8} + 46\beta_{5} - 2\beta_{4} + 13\beta_{3} \) -11b13 + 13b9 - 2b8 + 46b5 - 2b4 + 13b3
\(\nu^{5}\)	\(=\)	\( 2\beta_{15} + 28\beta_{14} + 13\beta_{13} - 2\beta_{9} + 2\beta_{7} + 72\beta_{6} + 13\beta_{3} + 13\beta_{2} \) 2b15 + 28b14 + 13b13 - 2b9 + 2b7 + 72b6 + 13b3 + 13b2
\(\nu^{6}\)	\(=\)	\( -8\beta_{13} - 73\beta_{9} + 76\beta_{8} - 38\beta_{4} + 30\beta_{3} - 111\beta_{2} + 398 \) -8b13 - 73b9 + 76b8 - 38b4 + 30b3 - 111b2 + 398
\(\nu^{7}\)	\(=\)	\( 38 \beta_{15} + 141 \beta_{13} - 328 \beta_{12} + 8 \beta_{11} - 16 \beta_{10} + 187 \beta_{9} + \cdots + 612 \beta_1 \) 38b15 + 141b13 - 328b12 + 8b11 - 16b10 + 187b9 - 76b7 - 46b6 + 141b3 + 46b2 + 612*b1
\(\nu^{8}\)	\(=\)	\( 1135\beta_{13} - 852\beta_{9} - 526\beta_{8} - 3662\beta_{5} + 1052\beta_{4} - 1987\beta_{3} + 935\beta_{2} - 3662 \) 1135b13 - 852b9 - 526b8 - 3662b5 + 1052b4 - 1987b3 + 935*b2 - 3662
\(\nu^{9}\)	\(=\)	\( - 1052 \beta_{15} - 3648 \beta_{14} - 2922 \beta_{13} + 3648 \beta_{12} - 400 \beta_{11} + \cdots - 5732 \beta_1 \) -1052b15 - 3648b14 - 2922b13 + 3648b12 - 400b11 + 200b10 - 1461b9 + 526b7 - 5858b6 - 2922b3 - 2187b2 - 5732b1
\(\nu^{10}\)	\(=\)	\( -8463\beta_{13} + 14941\beta_{9} - 6478\beta_{8} + 34878\beta_{5} - 6478\beta_{4} + 18245\beta_{3} + 3304\beta_{2} \) -8463b13 + 14941b9 - 6478b8 + 34878b5 - 6478b4 + 18245b3 + 3304*b2
\(\nu^{11}\)	\(=\)	\( 6478 \beta_{15} + 39664 \beta_{14} + 14941 \beta_{13} + 3304 \beta_{11} + 3304 \beta_{10} + \cdots + 14941 \beta_{2} \) 6478b15 + 39664b14 + 14941b13 + 3304b11 + 3304b10 - 9782b9 + 6478b7 + 64760b6 + 14941b3 + 14941b2
\(\nu^{12}\)	\(=\)	\( -45736\beta_{13} - 47441\beta_{9} + 150716\beta_{8} - 75358\beta_{4} + 29622\beta_{3} - 122799\beta_{2} + 339374 \) -45736b13 - 47441b9 + 150716b8 - 75358b4 + 29622b3 - 122799b2 + 339374
\(\nu^{13}\)	\(=\)	\( 75358 \beta_{15} + 152421 \beta_{13} - 425936 \beta_{12} + 45736 \beta_{11} - 91472 \beta_{10} + \cdots + 539236 \beta_1 \) 75358b15 + 152421b13 - 425936b12 + 45736b11 - 91472b10 + 273515b9 - 150716b7 - 121094b6 + 152421b3 + 121094b2 + 539236*b1
\(\nu^{14}\)	\(=\)	\( 1284423 \beta_{13} - 1121076 \beta_{9} - 848462 \beta_{8} - 3350494 \beta_{5} + 1696924 \beta_{4} + \cdots - 3350494 \) 1284423b13 - 1121076b9 - 848462b8 - 3350494b5 + 1696924b4 - 2405499b3 + 708575*b2 - 3350494
\(\nu^{15}\)	\(=\)	\( - 1696924 \beta_{15} - 4538384 \beta_{14} - 3114074 \beta_{13} + 4538384 \beta_{12} + \cdots - 5343492 \beta_1 \) -1696924b15 - 4538384b14 - 3114074b13 + 4538384b12 - 1151696b11 + 575848b10 - 1557037b9 + 848462b7 - 5040258b6 - 3114074b3 - 2981347b2 - 5343492b1

\(n\)	\(267\)	\(381\)	\(477\)
\(\chi(n)\)	\(1\)	\(-\beta_{5}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 341.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} + 25 T^{14} + \cdots + 473344 \) T^16 + 25T^14 + 413T^12 + 3916T^10 + 26956T^8 + 112304T^6 + 333008T^4 + 476096*T^2 + 473344
$5$	\( (T^{8} + 3 T^{7} - 8 T^{6} + \cdots + 64)^{2} \) (T^8 + 3T^7 - 8T^6 - 33T^5 + 104T^4 + 99T^3 - 61T^2 - 72*T + 64)^2
$7$	\( (T^{8} + 7 T^{7} + \cdots + 2401)^{2} \) (T^8 + 7T^7 + 33T^6 + 122T^5 + 368T^4 + 854T^3 + 1617T^2 + 2401*T + 2401)^2
$11$	\( (T^{8} + 2 T^{7} + \cdots + 2401)^{2} \) (T^8 + 2T^7 + 26T^6 + 32T^5 + 511T^4 + 640T^3 + 2522T^2 - 1862*T + 2401)^2
$13$	\( (T^{8} - 124 T^{6} + \cdots + 795328)^{2} \) (T^8 - 124T^6 + 5636T^4 - 110912*T^2 + 795328)^2
$17$	\( (T^{8} + 12 T^{7} + \cdots + 16)^{2} \) (T^8 + 12T^7 + 49T^6 + 12T^5 - 139T^4 - 36T^3 + 428T^2 + 144*T + 16)^2
$19$	\( T^{16} + \cdots + 16983563041 \) T^16 - 21T^15 + 237T^14 - 1890T^13 + 11937T^12 - 61983T^11 + 277614T^10 - 1158873T^9 + 4926926T^8 - 22018587T^7 + 100218654T^6 - 425141397T^5 + 1555641777T^4 - 4679827110T^3 + 11149873797T^2 - 18771306519*T + 16983563041
$23$	\( (T^{8} - 6 T^{7} + \cdots + 1369)^{2} \) (T^8 - 6T^7 + 46T^6 - 24T^5 + 315T^4 + 24T^3 + 2134T^2 + 1554*T + 1369)^2
$29$	\( (T^{8} + 180 T^{6} + \cdots + 2818048)^{2} \) (T^8 + 180T^6 + 11444T^4 + 301824*T^2 + 2818048)^2
$31$	\( T^{16} + 25 T^{14} + \cdots + 473344 \) T^16 + 25T^14 + 413T^12 + 3916T^10 + 26956T^8 + 112304T^6 + 333008T^4 + 476096*T^2 + 473344
$37$	\( T^{16} + \cdots + 5356595482624 \) T^16 - 215T^14 + 32545T^12 - 2310680T^10 + 117047068T^8 - 3317551040T^6 + 67727437840T^4 - 729647832320*T^2 + 5356595482624
$41$	\( (T^{8} - 100 T^{6} + \cdots + 176128)^{2} \) (T^8 - 100T^6 + 3392T^4 - 44288*T^2 + 176128)^2
$43$	\( (T^{4} - 4 T^{3} - 51 T^{2} + \cdots + 16)^{4} \) (T^4 - 4T^3 - 51T^2 + 56*T + 16)^4
$47$	\( (T^{8} - 18 T^{7} + \cdots + 110889)^{2} \) (T^8 - 18T^7 + 78T^6 + 540T^5 - 1701T^4 - 11340T^3 + 37638T^2 + 125874*T + 110889)^2
$53$	\( T^{16} - 163 T^{14} + \cdots + 473344 \) T^16 - 163T^14 + 21805T^12 - 749788T^10 + 20515372T^8 - 63479920T^6 + 175532752T^4 - 9199936*T^2 + 473344
$59$	\( T^{16} + \cdots + 161931936661504 \) T^16 + 293T^14 + 57053T^12 + 6257132T^10 + 497100304T^8 + 23932026880T^6 + 821768400896T^4 + 13871131131904*T^2 + 161931936661504
$61$	\( (T^{8} + 12 T^{7} + \cdots + 152881)^{2} \) (T^8 + 12T^7 + 2T^6 - 552T^5 + 347T^4 + 24840T^3 + 115186T^2 + 211140*T + 152881)^2
$67$	\( T^{16} + \cdots + 31021072384 \) T^16 - 211T^14 + 32953T^12 - 2264848T^10 + 115074496T^8 - 943657984T^6 + 5706551296T^4 - 15499264000*T^2 + 31021072384
$71$	\( (T^{8} + 364 T^{6} + \cdots + 7730368)^{2} \) (T^8 + 364T^6 + 30756T^4 + 873664*T^2 + 7730368)^2
$73$	\( (T^{8} + 18 T^{7} + \cdots + 2401)^{2} \) (T^8 + 18T^7 + 100T^6 - 144T^5 - 1363T^4 + 1968T^3 + 20564T^2 + 12054*T + 2401)^2
$79$	\( T^{16} + \cdots + 887122864384 \) T^16 - 491T^14 + 171637T^12 - 28170164T^10 + 3366488044T^8 - 204866820176T^6 + 8716450737232T^4 - 2791162322240*T^2 + 887122864384
$83$	\( (T^{8} + 297 T^{6} + \cdots + 1127844)^{2} \) (T^8 + 297T^6 + 24291T^4 + 400383*T^2 + 1127844)^2
$89$	\( T^{16} + \cdots + 632546627584 \) T^16 + 349T^14 + 86969T^12 + 10334440T^10 + 894546460T^8 + 31175559104T^6 + 802152544400T^4 + 724515176192*T^2 + 632546627584
$97$	\( (T^{8} - 576 T^{6} + \cdots + 14266368)^{2} \) (T^8 - 576T^6 + 106992T^4 - 6552576*T^2 + 14266368)^2
show more
show less