LMFDB - Newform orbit 3380.2.c.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(2029,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3380.2029");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.c (of order $2$, degree $1$, 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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$16$
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} - 8 x^{15} + 66 x^{14} - 322 x^{13} + 1398 x^{12} - 4566 x^{11} + 12982 x^{10} - 29776 x^{9} + \cdots + 3742 $ x^16 - 8x^15 + 66x^14 - 322x^13 + 1398x^12 - 4566x^11 + 12982x^10 - 29776x^9 + 58941x^8 - 95894x^7 + 132712x^6 - 149334x^5 + 138429x^4 - 99350x^3 + 54207x^2 - 19486*x + 3742
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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_{4} q^{3} + \beta_{7} q^{5} - \beta_{11} q^{7} + (\beta_{3} - 1) q^{9}+O(q^{10}) $ q + b4 * q^3 + b7 * q^5 - b11 * q^7 + (b3 - 1) * q^9	$ q + \beta_{4} q^{3} + \beta_{7} q^{5} - \beta_{11} q^{7} + (\beta_{3} - 1) q^{9} + (\beta_{13} + \beta_{12} - \beta_1) q^{11} + (\beta_{15} - \beta_{12} + \beta_{8}) q^{15} + ( - \beta_{14} - \beta_{6} + \cdots + \beta_{4}) q^{17}+ \cdots + ( - 2 \beta_{13} - 2 \beta_{12} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) $ q + b4 * q^3 + b7 * q^5 - b11 * q^7 + (b3 - 1) * q^9 + (b13 + b12 - b1) * q^11 + (b15 - b12 + b8) * q^15 + (-b14 - b6 + b5 + b4) * q^17 + (b2 - b1) * q^19 + (b13 + b12 - b1) * q^21 + (b14 + b4) * q^23 + (b9 + b4 + 1) * q^25 + (-b14 + b10 + b9 + b6 - b5 - 3b4) q^27 + (b6 + b5 + b3 - 2) * q^29 + (b13 + b12 - b8 + b7) * q^31 + (b15 + 2b11 + 2b8 + 2b7) q^33 + (-b14 + b10 - b6 - b3) * q^35 + (b13 - b12 + b11) * q^37 + (b13 + b12 + b8 - b7 + b1) * q^41 + (2b14 + b10 + b9 + b6 - b5 - b4) q^43 + (-b15 - 2b13 - b12 - b11 + b8 - b7 - b2 + 3b1) * q^45 + (2b8 + 2b7) * q^47 + (-b3 - 1) * q^49 + (b10 - b9 - 3) * q^51 + (-b14 - b10 - b9) * q^53 + (b14 + b10 + 2b6 - b5 + b4 - b3 - 1) q^55 + (4b15 + b13 - b12 - b11 - b8 - b7) q^57 + (b8 - b7 + b1) * q^59 + (-b6 - b5 - b3 + 4) * q^61 + (b15 - b11 + 2b8 + 2b7) * q^63 + (-2b15 - b13 + b12 - b11 + 3b8 + 3b7) q^67 + (-b6 - b5 + 2b3 - 3) q^69 + (2b8 - 2b7 - b2 - b1) * q^71 + (b15 + b13 - b12 + b11) * q^73 + (-b14 - b10 - b9 - 2b6 + 2b5 + 3b4 + b3 - 5) q^75 + (b14 - b10 - b9 - b6 + b5 - 3b4) q^77 + (b10 - b9 + b3 - 1) * q^79 + (-b10 + b9 + 2b6 + 2b5 - b3 + 4) * q^81 + (b13 - b12 + 2b11 - b8 - b7) q^83 + (b15 + 2b13 - 2b11 + b8 + b1) * q^85 + (b14 - 3b4) q^87 + (b8 - b7 + b1) * q^89 + (2b15 + b13 - b12 + 2b11 + 3b8 + 3b7) * q^93 + (-b14 - 3b10 - 2b9 - 3b6 + 2b5 + b4 - 2b3) q^95 + (2b15 + b11 + 2b8 + 2b7) q^97 + (-2b13 - 2b12 + 3b8 - 3b7 - b2 + 4b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{9}+O(q^{10}) $ 16 * q - 20 * q^9	$ 16 q - 20 q^{9} + 14 q^{25} - 24 q^{29} - 12 q^{49} - 44 q^{51} - 4 q^{55} + 56 q^{61} - 68 q^{69} - 84 q^{75} - 16 q^{79} + 88 q^{81}+O(q^{100}) $ 16 * q - 20 * q^9 + 14 * q^25 - 24 * q^29 - 12 * q^49 - 44 * q^51 - 4 * q^55 + 56 * q^61 - 68 * q^69 - 84 * q^75 - 16 * q^79 + 88 * q^81

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8 x^{15} + 66 x^{14} - 322 x^{13} + 1398 x^{12} - 4566 x^{11} + 12982 x^{10} - 29776 x^{9} + \cdots + 3742 $ x^16 - 8*x^15 + 66*x^14 - 322*x^13 + 1398*x^12 - 4566*x^11 + 12982*x^10 - 29776*x^9 + 58941*x^8 - 95894*x^7 + 132712*x^6 - 149334*x^5 + 138429*x^4 - 99350*x^3 + 54207*x^2 - 19486*x + 3742 :

$\beta_{1}$	$=$	$ ( - \nu^{14} + 7 \nu^{13} - 54 \nu^{12} + 233 \nu^{11} - 887 \nu^{10} + 2466 \nu^{9} - 5693 \nu^{8} + \cdots + 442 ) / 88 $ (-v^14 + 7v^13 - 54v^12 + 233v^11 - 887v^10 + 2466v^9 - 5693v^8 + 10253v^7 - 14423v^6 + 15478v^5 - 11027v^4 + 4417v^3 + 863v^2 - 1632*v + 442) / 88
$\beta_{2}$	$=$	$ ( 11 \nu^{14} - 77 \nu^{13} + 646 \nu^{12} - 2875 \nu^{11} + 12281 \nu^{10} - 36886 \nu^{9} + \cdots + 44234 ) / 1672 $ (11v^14 - 77v^13 + 646v^12 - 2875v^11 + 12281v^10 - 36886v^9 + 100539v^8 - 209319v^7 + 385501v^6 - 550486v^5 + 676821v^4 - 620743v^3 + 453355v^2 - 208768v + 44234) / 1672
$\beta_{3}$	$=$	$ ( - 18 \nu^{14} + 126 \nu^{13} - 1013 \nu^{12} + 4440 \nu^{11} - 17977 \nu^{10} + 52188 \nu^{9} + \cdots - 27361 ) / 209 $ (-18v^14 + 126v^13 - 1013v^12 + 4440v^11 - 17977v^10 + 52188v^9 - 133056v^8 + 262788v^7 - 443293v^6 + 585876v^5 - 638050v^4 + 523540v^3 - 326685v^2 + 131134v - 27361) / 209
$\beta_{4}$	$=$	$ ( - 5366 \nu^{15} + 40245 \nu^{14} - 356423 \nu^{13} + 1706367 \nu^{12} - 7841525 \nu^{11} + \cdots + 7432092 ) / 2673946 $ (-5366v^15 + 40245v^14 - 356423v^13 + 1706367v^12 - 7841525v^11 + 25701192v^10 - 75925052v^9 + 174181332v^8 - 346296619v^7 + 545588745v^6 - 708712691v^5 + 710830156v^4 - 533131390v^3 + 276487343v^2 - 77130498*v + 7432092) / 2673946
$\beta_{5}$	$=$	$ ( 43510 \nu^{15} - 211179 \nu^{14} + 1796925 \nu^{13} - 5604878 \nu^{12} + 21165027 \nu^{11} + \cdots + 27319238 ) / 2673946 $ (43510v^15 - 211179v^14 + 1796925v^13 - 5604878v^12 + 21165027v^11 - 38659822v^10 + 72228467v^9 - 26983911v^8 - 90159207v^7 + 414010052v^6 - 734004177v^5 + 945704010v^4 - 829204553v^3 + 500059850v^2 - 188633298*v + 27319238) / 2673946
$\beta_{6}$	$=$	$ ( - 43510 \nu^{15} + 441471 \nu^{14} - 3408969 \nu^{13} + 18334908 \nu^{12} - 76588635 \nu^{11} + \cdots + 68866054 ) / 2673946 $ (-43510v^15 + 441471v^14 - 3408969v^13 + 18334908v^12 - 76588635v^11 + 257424428v^10 - 696422139v^9 + 1559653935v^8 - 2839155033v^7 + 4242430218v^6 - 5063827215v^5 + 4766458758v^4 - 3371116823v^3 + 1688085176v^2 - 523813386*v + 68866054) / 2673946
$\beta_{7}$	$=$	$ ( - 178093 \nu^{15} + 1249338 \nu^{14} - 10297859 \nu^{13} + 45649687 \nu^{12} - 191154382 \nu^{11} + \cdots - 2244270 ) / 10695784 $ (-178093v^15 + 1249338v^14 - 10297859v^13 + 45649687v^12 - 191154382v^11 + 568622111v^10 - 1516204143v^9 + 3114998428v^8 - 5578541278v^7 + 7811944943v^6 - 9196384473v^5 + 8158815186v^4 - 5607586220v^3 + 2476796539v^2 - 602055646*v - 2244270) / 10695784
$\beta_{8}$	$=$	$ ( - 178093 \nu^{15} + 1422057 \nu^{14} - 11506892 \nu^{13} + 55565037 \nu^{12} - 234929053 \nu^{11} + \cdots + 526570132 ) / 10695784 $ (-178093v^15 + 1422057v^14 - 11506892v^13 + 55565037v^12 - 234929053v^11 + 750610364v^10 - 2053692877v^9 + 4534332803v^8 - 8463070121v^7 + 12912125912v^6 - 16228875247v^5 + 16313896299v^4 - 12701801647v^3 + 7311361274v^2 - 2709585678*v + 526570132) / 10695784
$\beta_{9}$	$=$	$ ( - 67166 \nu^{15} + 414187 \nu^{14} - 3474215 \nu^{13} + 13893157 \nu^{12} - 57202241 \nu^{11} + \cdots - 79226364 ) / 2673946 $ (-67166v^15 + 414187v^14 - 3474215v^13 + 13893157v^12 - 57202241v^11 + 153097448v^10 - 389497844v^9 + 699988998v^8 - 1143253791v^7 + 1291717005v^6 - 1234729223v^5 + 619201966v^4 - 82057906v^3 - 294937569v^2 + 251308582*v - 79226364) / 2673946
$\beta_{10}$	$=$	$ ( - 67166 \nu^{15} + 593303 \nu^{14} - 4728027 \nu^{13} + 24141151 \nu^{12} - 102390649 \nu^{11} + \cdots + 254824976 ) / 2673946 $ (-67166v^15 + 593303v^14 - 4728027v^13 + 24141151v^12 - 102390649v^11 + 339902642v^10 - 939179260v^9 + 2137087842v^8 - 4039405983v^7 + 6322995887v^6 - 8057411191v^5 + 8280377106v^4 - 6534123282v^3 + 3808136613v^2 - 1411527598*v + 254824976) / 2673946
$\beta_{11}$	$=$	$ ( 34082 \nu^{15} - 255615 \nu^{14} + 2048039 \nu^{13} - 9435426 \nu^{12} + 38796461 \nu^{11} + \cdots - 43105093 ) / 1336973 $ (34082v^15 - 255615v^14 + 2048039v^13 - 9435426v^12 + 38796461v^11 - 118119870v^10 + 310326055v^9 - 647976279v^8 + 1147878041v^7 - 1638530621v^6 + 1937821276v^5 - 1805872330v^4 + 1325167070v^3 - 705049740v^2 + 249379043*v - 43105093) / 1336973
$\beta_{12}$	$=$	$ ( 391353 \nu^{15} - 2445777 \nu^{14} + 20054944 \nu^{13} - 80286537 \nu^{12} + 321157765 \nu^{11} + \cdots + 545094556 ) / 10695784 $ (391353v^15 - 2445777v^14 + 20054944v^13 - 80286537v^12 + 321157765v^11 - 845972084v^10 + 2059282717v^9 - 3529572307v^8 + 5288951833v^7 - 5133480728v^6 + 3615668607v^5 + 300805273v^4 - 2703611425v^3 + 3535742846v^2 - 1924341010*v + 545094556) / 10695784
$\beta_{13}$	$=$	$ ( - 391353 \nu^{15} + 3424518 \nu^{14} - 26906131 \nu^{13} + 135927643 \nu^{12} + \cdots + 1467440026 ) / 10695784 $ (-391353v^15 + 3424518v^14 - 26906131v^13 + 135927643v^12 - 565938970v^11 + 1850576155v^10 - 5001761983v^9 + 11168824928v^8 - 20603760050v^7 + 31605232675v^6 - 39388831273v^5 + 39901160418v^4 - 31258554144v^3 + 18437190911v^2 - 7178538814*v + 1467440026) / 10695784
$\beta_{14}$	$=$	$ ( 64652 \nu^{15} - 484890 \nu^{14} + 3826280 \nu^{13} - 17516655 \nu^{12} + 70503984 \nu^{11} + \cdots - 3312928 ) / 1336973 $ (64652v^15 - 484890v^14 + 3826280v^13 - 17516655v^12 + 70503984v^11 - 211267870v^10 + 537635867v^9 - 1089207549v^8 + 1832971428v^7 - 2474477605v^6 + 2674061346v^5 - 2236358266v^4 + 1369910081v^3 - 570094455v^2 + 117059508*v - 3312928) / 1336973
$\beta_{15}$	$=$	$ ( 308418 \nu^{15} - 2313135 \nu^{14} + 18485755 \nu^{13} - 85074860 \nu^{12} + 348592111 \nu^{11} + \cdots - 286402874 ) / 5347892 $ (308418v^15 - 2313135v^14 + 18485755v^13 - 85074860v^12 + 348592111v^11 - 1058614755v^10 + 2767089288v^9 - 5750637171v^8 + 10101698075v^7 - 14289837643v^6 + 16628863304v^5 - 15200966229v^4 + 10773509555v^3 - 5486285853v^2 + 1807988888*v - 286402874) / 5347892

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

$\nu$	$=$	$ ( \beta_{15} - \beta_{14} - \beta_{11} - \beta_{10} - \beta_{9} + 2\beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} + 2 ) / 4 $ (b15 - b14 - b11 - b10 - b9 + 2b8 + 2b7 - b6 + b5 + 2) / 4
$\nu^{2}$	$=$	$ ( \beta_{15} - \beta_{14} - \beta_{11} - \beta_{10} - \beta_{9} + 4 \beta_{7} - 3 \beta_{6} - \beta_{5} + \cdots - 16 ) / 4 $ (b15 - b14 - b11 - b10 - b9 + 4b7 - 3b6 - b5 - 2b3 - 2b2 - 4*b1 - 16) / 4
$\nu^{3}$	$=$	$ ( - 2 \beta_{15} + 4 \beta_{14} + 5 \beta_{13} - 5 \beta_{12} + 14 \beta_{11} + 14 \beta_{10} + 14 \beta_{9} + \cdots - 25 ) / 4 $ (-2b15 + 4b14 + 5b13 - 5b12 + 14b11 + 14b10 + 14b9 - 24b8 - 18b7 - b6 - 5b5 - 3b3 - 3b2 - 6*b1 - 25) / 4
$\nu^{4}$	$=$	$ ( - 5 \beta_{15} + 9 \beta_{14} + 26 \beta_{13} + 6 \beta_{12} + 29 \beta_{11} + 25 \beta_{10} + \cdots + 144 ) / 4 $ (-5b15 + 9b14 + 26b13 + 6b12 + 29b11 + 25b10 + 33b9 - 12b8 - 76b7 + 39b6 + 29b5 + 50b3 + 32b2 + 48b1 + 144) / 4
$\nu^{5}$	$=$	$ ( - 15 \beta_{15} - 30 \beta_{14} - 58 \beta_{13} + 138 \beta_{12} - 235 \beta_{11} - 212 \beta_{10} + \cdots + 402 ) / 4 $ (-15b15 - 30b14 - 58b13 + 138b12 - 235b11 - 212b10 - 192b9 + 353b8 + 183b7 + 90b6 + 90b5 + 44b4 + 130b3 + 85b2 + 130*b1 + 402) / 4
$\nu^{6}$	$=$	$ ( - 32 \beta_{15} - 113 \beta_{14} - 631 \beta_{13} + 7 \beta_{12} - 778 \beta_{11} - 603 \beta_{10} + \cdots - 1773 ) / 4 $ (-32b15 - 113b14 - 631b13 + 7b12 - 778b11 - 603b10 - 755b9 + 503b8 + 1327b7 - 512b6 - 486b5 + 132b4 - 801b3 - 468b2 - 632*b1 - 1773) / 4
$\nu^{7}$	$=$	$ ( 347 \beta_{15} + 324 \beta_{14} + 179 \beta_{13} - 2643 \beta_{12} + 3365 \beta_{11} + 3065 \beta_{10} + \cdots - 7642 ) / 4 $ (347b15 + 324b14 + 179b13 - 2643b12 + 3365b11 + 3065b10 + 2463b9 - 5248b8 - 1762b7 - 2131b6 - 1999b5 - 888b4 - 3262b3 - 1939b2 - 2674*b1 - 7642) / 4
$\nu^{8}$	$=$	$ ( 1525 \beta_{15} + 1845 \beta_{14} + 11492 \beta_{13} - 2820 \beta_{12} + 17159 \beta_{11} + 13239 \beta_{10} + \cdots + 22102 ) / 4 $ (1525b15 + 1845b14 + 11492b13 - 2820b12 + 17159b11 + 13239b10 + 15347b9 - 13322b8 - 23466b7 + 6279b6 + 6661b5 - 4168b4 + 10628b3 + 6112b2 + 7936*b1 + 22102) / 4
$\nu^{9}$	$=$	$ ( - 4537 \beta_{15} - 3662 \beta_{14} + 10081 \beta_{13} + 44063 \beta_{12} - 40583 \beta_{11} + \cdots + 147050 ) / 4 $ (-4537b15 - 3662b14 + 10081b13 + 44063b12 - 40583b11 - 39619b10 - 26437b9 + 75006b8 + 7716b7 + 42765b6 + 41013b5 + 11688b4 + 67950b3 + 39501b2 + 52314*b1 + 147050) / 4
$\nu^{10}$	$=$	$ ( - 34320 \beta_{15} - 32985 \beta_{14} - 187680 \beta_{13} + 94136 \beta_{12} - 337200 \beta_{11} + \cdots - 232877 ) / 4 $ (-34320b15 - 32985b14 - 187680b13 + 94136b12 - 337200b11 - 265892b10 - 288582b9 + 298336b8 + 404440b7 - 62959b6 - 74349b5 + 90624b4 - 113437b3 - 65112b2 - 83440*b1 - 232877) / 4
$\nu^{11}$	$=$	$ ( 40722 \beta_{15} + 31462 \beta_{14} - 370965 \beta_{13} - 667831 \beta_{12} + 354638 \beta_{11} + \cdots - 2717405 ) / 4 $ (40722b15 + 31462b14 - 370965b13 - 667831b12 + 354638b11 + 414332b10 + 161860b9 - 979692b8 + 260954b7 - 792879b6 - 777723b5 - 105100b4 - 1284107b3 - 742489b2 - 969342*b1 - 2717405) / 4
$\nu^{12}$	$=$	$ ( 647631 \beta_{15} + 584689 \beta_{14} + 2834771 \beta_{13} - 2296915 \beta_{12} + 6137751 \beta_{11} + \cdots + 1304386 ) / 4 $ (647631b15 + 584689b14 + 2834771b13 - 2296915b12 + 6137751b11 + 4964494b10 + 5094384b9 - 6050105b8 - 6641593b7 + 277208b6 + 498990b5 - 1699140b4 + 645420b3 + 369488b2 + 466704*b1 + 1304386) / 4
$\nu^{13}$	$=$	$ ( - 14543 \beta_{15} + 26036 \beta_{14} + 9238791 \beta_{13} + 9156157 \beta_{12} + 102287 \beta_{11} + \cdots + 47639490 ) / 4 $ (-14543b15 + 26036b14 + 9238791b13 + 9156157b12 + 102287b11 - 2132573b10 + 2333473b9 + 10705894b8 - 11020772b7 + 13822823b6 + 13813591b5 + 25248b4 + 22647014b3 + 13078403b2 + 16996798*b1 + 47639490) / 4
$\nu^{14}$	$=$	$ ( - 11181187 \beta_{15} - 9930065 \beta_{14} - 39159133 \beta_{13} + 48584949 \beta_{12} + \cdots + 24971826 ) / 4 $ (-11181187b15 - 9930065b14 - 39159133b13 + 48584949b12 - 104931423b11 - 87067098b10 - 84771480b9 + 113985875b8 + 102570315b7 + 9099524b6 + 5267254b5 + 29282072b4 + 11710952b3 + 6781912b2 + 8908192*b1 + 24971826) / 4
$\nu^{15}$	$=$	$ ( - 11233293 \beta_{15} - 10213888 \beta_{14} - 197433196 \beta_{13} - 108048880 \beta_{12} + \cdots - 789301344 ) / 4 $ (-11233293b15 - 10213888b14 - 197433196b13 - 108048880b12 - 106918641b11 - 50573584b10 - 124704264b9 - 69295885b8 + 290559301b7 - 227208076b6 - 231051796b5 + 29505396b4 - 375744468b3 - 216933763b2 - 281575510*b1 - 789301344) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$-1$	$1$	$1$

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} $

2029.1

0.500000	+	2.42905i
0.500000	−	1.67156i
0.500000	+	2.32611i
0.500000	+	1.12757i
0.500000	−	0.349854i
0.500000	−	1.62369i
0.500000	+	4.10514i
0.500000	−	1.00620i
0.500000	−	4.10514i
0.500000	+	1.00620i
0.500000	+	0.349854i
0.500000	+	1.62369i
0.500000	−	2.32611i
0.500000	−	1.12757i
0.500000	−	2.42905i
0.500000	+	1.67156i

−

3.30614i

−2.05681

0.877236i

−

1.03413i

−7.93058

2029.2

−

3.30614i

2.05681

−

0.877236i

1.03413i

−7.93058

2029.3

−

1.95936i

−0.571200

2.16188i

−

2.85673i

−0.839100

2029.4

−

1.95936i

0.571200

−

2.16188i

2.85673i

−0.839100

2029.5

−

1.47876i

−2.18080

0.494086i

3.13261i

0.813273

2029.6

−

1.47876i

2.18080

−

0.494086i

−

3.13261i

0.813273

2029.7

−

0.208784i

−1.56122

1.60081i

3.45780i

2.95641

2029.8

−

0.208784i

1.56122

−

1.60081i

−

3.45780i

2.95641

2029.9

0.208784i

−1.56122

−

1.60081i

−

3.45780i

2.95641

2029.10

0.208784i

1.56122

1.60081i

3.45780i

2.95641

2029.11

1.47876i

−2.18080

−

0.494086i

−

3.13261i

0.813273

2029.12

1.47876i

2.18080

0.494086i

3.13261i

0.813273

2029.13

1.95936i

−0.571200

−

2.16188i

2.85673i

−0.839100

2029.14

1.95936i

0.571200

2.16188i

−

2.85673i

−0.839100

2029.15

3.30614i

−2.05681

−

0.877236i

1.03413i

−7.93058

2029.16

3.30614i

2.05681

0.877236i

−

1.03413i

−7.93058

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
13.b	even	2	1	inner
65.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3380.2.c.e		16
5.b	even	2	1	inner	3380.2.c.e		16
13.b	even	2	1	inner	3380.2.c.e		16
13.d	odd	4	2		3380.2.d.d		16
13.f	odd	12	2		260.2.z.a	✓	16
39.k	even	12	2		2340.2.cr.a		16
52.l	even	12	2		1040.2.df.d		16
65.d	even	2	1	inner	3380.2.c.e		16
65.g	odd	4	2		3380.2.d.d		16
65.o	even	12	2		1300.2.y.e		16
65.s	odd	12	2		260.2.z.a	✓	16
65.t	even	12	2		1300.2.y.e		16
195.bh	even	12	2		2340.2.cr.a		16
260.bc	even	12	2		1040.2.df.d		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
260.2.z.a	✓	16	13.f	odd	12	2
260.2.z.a	✓	16	65.s	odd	12	2
1040.2.df.d		16	52.l	even	12	2
1040.2.df.d		16	260.bc	even	12	2
1300.2.y.e		16	65.o	even	12	2
1300.2.y.e		16	65.t	even	12	2
2340.2.cr.a		16	39.k	even	12	2
2340.2.cr.a		16	195.bh	even	12	2
3380.2.c.e		16	1.a	even	1	1	trivial
3380.2.c.e		16	5.b	even	2	1	inner
3380.2.c.e		16	13.b	even	2	1	inner
3380.2.c.e		16	65.d	even	2	1	inner
3380.2.d.d		16	13.d	odd	4	2
3380.2.d.d		16	65.g	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3380, [\chi])$:

$ T_{3}^{8} + 17T_{3}^{6} + 75T_{3}^{4} + 95T_{3}^{2} + 4 $

$ T_{11}^{8} - 65T_{11}^{6} + 1323T_{11}^{4} - 8531T_{11}^{2} + 4096 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 17 T^{6} + 75 T^{4} + \cdots + 4)^{2} $ (T^8 + 17T^6 + 75T^4 + 95*T^2 + 4)^2
$5$	$ T^{16} - 7 T^{14} + \cdots + 390625 $ T^16 - 7T^14 + 22T^12 - T^10 - 14T^8 - 25T^6 + 13750T^4 - 109375T^2 + 390625
$7$	$ (T^{8} + 31 T^{6} + \cdots + 1024)^{2} $ (T^8 + 31T^6 + 327T^4 + 1273*T^2 + 1024)^2
$11$	$ (T^{8} - 65 T^{6} + \cdots + 4096)^{2} $ (T^8 - 65T^6 + 1323T^4 - 8531*T^2 + 4096)^2
$13$	$ T^{16} $ T^16
$17$	$ (T^{8} + 80 T^{6} + \cdots + 52441)^{2} $ (T^8 + 80T^6 + 1926T^4 + 17744*T^2 + 52441)^2
$19$	$ (T^{8} - 165 T^{6} + \cdots + 412164)^{2} $ (T^8 - 165T^6 + 9015T^4 - 170271*T^2 + 412164)^2
$23$	$ (T^{8} + 89 T^{6} + \cdots + 1156)^{2} $ (T^8 + 89T^6 + 2091T^4 + 9335*T^2 + 1156)^2
$29$	$ (T^{4} + 6 T^{3} + \cdots - 393)^{4} $ (T^4 + 6T^3 - 48T^2 - 318*T - 393)^4
$31$	$ (T^{8} - 108 T^{6} + \cdots + 2304)^{2} $ (T^8 - 108T^6 + 2688T^4 - 6336*T^2 + 2304)^2
$37$	$ (T^{8} + 160 T^{6} + \cdots + 292681)^{2} $ (T^8 + 160T^6 + 8598T^4 + 160576*T^2 + 292681)^2
$41$	$ (T^{8} - 164 T^{6} + \cdots + 1771561)^{2} $ (T^8 - 164T^6 + 9438T^4 - 221780*T^2 + 1771561)^2
$43$	$ (T^{8} + 237 T^{6} + \cdots + 535824)^{2} $ (T^8 + 237T^6 + 13839T^4 + 174483*T^2 + 535824)^2
$47$	$ (T^{8} + 132 T^{6} + \cdots + 147456)^{2} $ (T^8 + 132T^6 + 4992T^4 + 55296*T^2 + 147456)^2
$53$	$ (T^{8} + 191 T^{6} + \cdots + 30976)^{2} $ (T^8 + 191T^6 + 8004T^4 + 94256*T^2 + 30976)^2
$59$	$ (T^{8} - 65 T^{6} + \cdots + 1936)^{2} $ (T^8 - 65T^6 + 1227T^4 - 5987*T^2 + 1936)^2
$61$	$ (T^{4} - 14 T^{3} + \cdots - 1157)^{4} $ (T^4 - 14T^3 + 12T^2 + 406*T - 1157)^4
$67$	$ (T^{8} + 451 T^{6} + \cdots + 850084)^{2} $ (T^8 + 451T^6 + 59235T^4 + 2128405*T^2 + 850084)^2
$71$	$ (T^{8} - 341 T^{6} + \cdots + 3104644)^{2} $ (T^8 - 341T^6 + 32151T^4 - 699791*T^2 + 3104644)^2
$73$	$ (T^{8} + 133 T^{6} + \cdots + 861184)^{2} $ (T^8 + 133T^6 + 6276T^4 + 123712*T^2 + 861184)^2
$79$	$ (T^{4} + 4 T^{3} + \cdots + 4096)^{4} $ (T^4 + 4T^3 - 156T^2 - 224*T + 4096)^4
$83$	$ (T^{8} + 228 T^{6} + \cdots + 36864)^{2} $ (T^8 + 228T^6 + 9168T^4 + 78336*T^2 + 36864)^2
$89$	$ (T^{8} - 65 T^{6} + \cdots + 1936)^{2} $ (T^8 - 65T^6 + 1227T^4 - 5987*T^2 + 1936)^2
$97$	$ (T^{8} + 331 T^{6} + \cdots + 3020644)^{2} $ (T^8 + 331T^6 + 20931T^4 + 448621*T^2 + 3020644)^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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{3} + \beta_{7} q^{5} - \beta_{11} q^{7} + (\beta_{3} - 1) q^{9}+O(q^{10}) \) q + b4 * q^3 + b7 * q^5 - b11 * q^7 + (b3 - 1) * q^9	\( q + \beta_{4} q^{3} + \beta_{7} q^{5} - \beta_{11} q^{7} + (\beta_{3} - 1) q^{9} + (\beta_{13} + \beta_{12} - \beta_1) q^{11} + (\beta_{15} - \beta_{12} + \beta_{8}) q^{15} + ( - \beta_{14} - \beta_{6} + \cdots + \beta_{4}) q^{17}+ \cdots + ( - 2 \beta_{13} - 2 \beta_{12} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) q + b4 * q^3 + b7 * q^5 - b11 * q^7 + (b3 - 1) * q^9 + (b13 + b12 - b1) * q^11 + (b15 - b12 + b8) * q^15 + (-b14 - b6 + b5 + b4) * q^17 + (b2 - b1) * q^19 + (b13 + b12 - b1) * q^21 + (b14 + b4) * q^23 + (b9 + b4 + 1) * q^25 + (-b14 + b10 + b9 + b6 - b5 - 3b4) q^27 + (b6 + b5 + b3 - 2) * q^29 + (b13 + b12 - b8 + b7) * q^31 + (b15 + 2b11 + 2b8 + 2b7) q^33 + (-b14 + b10 - b6 - b3) * q^35 + (b13 - b12 + b11) * q^37 + (b13 + b12 + b8 - b7 + b1) * q^41 + (2b14 + b10 + b9 + b6 - b5 - b4) q^43 + (-b15 - 2b13 - b12 - b11 + b8 - b7 - b2 + 3b1) * q^45 + (2b8 + 2b7) * q^47 + (-b3 - 1) * q^49 + (b10 - b9 - 3) * q^51 + (-b14 - b10 - b9) * q^53 + (b14 + b10 + 2b6 - b5 + b4 - b3 - 1) q^55 + (4b15 + b13 - b12 - b11 - b8 - b7) q^57 + (b8 - b7 + b1) * q^59 + (-b6 - b5 - b3 + 4) * q^61 + (b15 - b11 + 2b8 + 2b7) * q^63 + (-2b15 - b13 + b12 - b11 + 3b8 + 3b7) q^67 + (-b6 - b5 + 2b3 - 3) q^69 + (2b8 - 2b7 - b2 - b1) * q^71 + (b15 + b13 - b12 + b11) * q^73 + (-b14 - b10 - b9 - 2b6 + 2b5 + 3b4 + b3 - 5) q^75 + (b14 - b10 - b9 - b6 + b5 - 3b4) q^77 + (b10 - b9 + b3 - 1) * q^79 + (-b10 + b9 + 2b6 + 2b5 - b3 + 4) * q^81 + (b13 - b12 + 2b11 - b8 - b7) q^83 + (b15 + 2b13 - 2b11 + b8 + b1) * q^85 + (b14 - 3b4) q^87 + (b8 - b7 + b1) * q^89 + (2b15 + b13 - b12 + 2b11 + 3b8 + 3b7) * q^93 + (-b14 - 3b10 - 2b9 - 3b6 + 2b5 + b4 - 2b3) q^95 + (2b15 + b11 + 2b8 + 2b7) q^97 + (-2b13 - 2b12 + 3b8 - 3b7 - b2 + 4b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{9}+O(q^{10}) \) 16 * q - 20 * q^9	\( 16 q - 20 q^{9} + 14 q^{25} - 24 q^{29} - 12 q^{49} - 44 q^{51} - 4 q^{55} + 56 q^{61} - 68 q^{69} - 84 q^{75} - 16 q^{79} + 88 q^{81}+O(q^{100}) \) 16 * q - 20 * q^9 + 14 * q^25 - 24 * q^29 - 12 * q^49 - 44 * q^51 - 4 * q^55 + 56 * q^61 - 68 * q^69 - 84 * q^75 - 16 * q^79 + 88 * q^81

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	3380.2.c.e

Level	$3380$
Weight	$2$
Character orbit	3380.c
Analytic conductor	$26.989$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(26.9894358832\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 8 x^{15} + 66 x^{14} - 322 x^{13} + 1398 x^{12} - 4566 x^{11} + 12982 x^{10} - 29776 x^{9} + \cdots + 3742 \) x^16 - 8x^15 + 66x^14 - 322x^13 + 1398x^12 - 4566x^11 + 12982x^10 - 29776x^9 + 58941x^8 - 95894x^7 + 132712x^6 - 149334x^5 + 138429x^4 - 99350x^3 + 54207x^2 - 19486*x + 3742
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{12} \)
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - \nu^{14} + 7 \nu^{13} - 54 \nu^{12} + 233 \nu^{11} - 887 \nu^{10} + 2466 \nu^{9} - 5693 \nu^{8} + \cdots + 442 ) / 88 \) (-v^14 + 7v^13 - 54v^12 + 233v^11 - 887v^10 + 2466v^9 - 5693v^8 + 10253v^7 - 14423v^6 + 15478v^5 - 11027v^4 + 4417v^3 + 863v^2 - 1632*v + 442) / 88
\(\beta_{2}\)	\(=\)	\( ( 11 \nu^{14} - 77 \nu^{13} + 646 \nu^{12} - 2875 \nu^{11} + 12281 \nu^{10} - 36886 \nu^{9} + \cdots + 44234 ) / 1672 \) (11v^14 - 77v^13 + 646v^12 - 2875v^11 + 12281v^10 - 36886v^9 + 100539v^8 - 209319v^7 + 385501v^6 - 550486v^5 + 676821v^4 - 620743v^3 + 453355v^2 - 208768v + 44234) / 1672
\(\beta_{3}\)	\(=\)	\( ( - 18 \nu^{14} + 126 \nu^{13} - 1013 \nu^{12} + 4440 \nu^{11} - 17977 \nu^{10} + 52188 \nu^{9} + \cdots - 27361 ) / 209 \) (-18v^14 + 126v^13 - 1013v^12 + 4440v^11 - 17977v^10 + 52188v^9 - 133056v^8 + 262788v^7 - 443293v^6 + 585876v^5 - 638050v^4 + 523540v^3 - 326685v^2 + 131134v - 27361) / 209
\(\beta_{4}\)	\(=\)	\( ( - 5366 \nu^{15} + 40245 \nu^{14} - 356423 \nu^{13} + 1706367 \nu^{12} - 7841525 \nu^{11} + \cdots + 7432092 ) / 2673946 \) (-5366v^15 + 40245v^14 - 356423v^13 + 1706367v^12 - 7841525v^11 + 25701192v^10 - 75925052v^9 + 174181332v^8 - 346296619v^7 + 545588745v^6 - 708712691v^5 + 710830156v^4 - 533131390v^3 + 276487343v^2 - 77130498*v + 7432092) / 2673946
\(\beta_{5}\)	\(=\)	\( ( 43510 \nu^{15} - 211179 \nu^{14} + 1796925 \nu^{13} - 5604878 \nu^{12} + 21165027 \nu^{11} + \cdots + 27319238 ) / 2673946 \) (43510v^15 - 211179v^14 + 1796925v^13 - 5604878v^12 + 21165027v^11 - 38659822v^10 + 72228467v^9 - 26983911v^8 - 90159207v^7 + 414010052v^6 - 734004177v^5 + 945704010v^4 - 829204553v^3 + 500059850v^2 - 188633298*v + 27319238) / 2673946
\(\beta_{6}\)	\(=\)	\( ( - 43510 \nu^{15} + 441471 \nu^{14} - 3408969 \nu^{13} + 18334908 \nu^{12} - 76588635 \nu^{11} + \cdots + 68866054 ) / 2673946 \) (-43510v^15 + 441471v^14 - 3408969v^13 + 18334908v^12 - 76588635v^11 + 257424428v^10 - 696422139v^9 + 1559653935v^8 - 2839155033v^7 + 4242430218v^6 - 5063827215v^5 + 4766458758v^4 - 3371116823v^3 + 1688085176v^2 - 523813386*v + 68866054) / 2673946
\(\beta_{7}\)	\(=\)	\( ( - 178093 \nu^{15} + 1249338 \nu^{14} - 10297859 \nu^{13} + 45649687 \nu^{12} - 191154382 \nu^{11} + \cdots - 2244270 ) / 10695784 \) (-178093v^15 + 1249338v^14 - 10297859v^13 + 45649687v^12 - 191154382v^11 + 568622111v^10 - 1516204143v^9 + 3114998428v^8 - 5578541278v^7 + 7811944943v^6 - 9196384473v^5 + 8158815186v^4 - 5607586220v^3 + 2476796539v^2 - 602055646*v - 2244270) / 10695784
\(\beta_{8}\)	\(=\)	\( ( - 178093 \nu^{15} + 1422057 \nu^{14} - 11506892 \nu^{13} + 55565037 \nu^{12} - 234929053 \nu^{11} + \cdots + 526570132 ) / 10695784 \) (-178093v^15 + 1422057v^14 - 11506892v^13 + 55565037v^12 - 234929053v^11 + 750610364v^10 - 2053692877v^9 + 4534332803v^8 - 8463070121v^7 + 12912125912v^6 - 16228875247v^5 + 16313896299v^4 - 12701801647v^3 + 7311361274v^2 - 2709585678*v + 526570132) / 10695784
\(\beta_{9}\)	\(=\)	\( ( - 67166 \nu^{15} + 414187 \nu^{14} - 3474215 \nu^{13} + 13893157 \nu^{12} - 57202241 \nu^{11} + \cdots - 79226364 ) / 2673946 \) (-67166v^15 + 414187v^14 - 3474215v^13 + 13893157v^12 - 57202241v^11 + 153097448v^10 - 389497844v^9 + 699988998v^8 - 1143253791v^7 + 1291717005v^6 - 1234729223v^5 + 619201966v^4 - 82057906v^3 - 294937569v^2 + 251308582*v - 79226364) / 2673946
\(\beta_{10}\)	\(=\)	\( ( - 67166 \nu^{15} + 593303 \nu^{14} - 4728027 \nu^{13} + 24141151 \nu^{12} - 102390649 \nu^{11} + \cdots + 254824976 ) / 2673946 \) (-67166v^15 + 593303v^14 - 4728027v^13 + 24141151v^12 - 102390649v^11 + 339902642v^10 - 939179260v^9 + 2137087842v^8 - 4039405983v^7 + 6322995887v^6 - 8057411191v^5 + 8280377106v^4 - 6534123282v^3 + 3808136613v^2 - 1411527598*v + 254824976) / 2673946
\(\beta_{11}\)	\(=\)	\( ( 34082 \nu^{15} - 255615 \nu^{14} + 2048039 \nu^{13} - 9435426 \nu^{12} + 38796461 \nu^{11} + \cdots - 43105093 ) / 1336973 \) (34082v^15 - 255615v^14 + 2048039v^13 - 9435426v^12 + 38796461v^11 - 118119870v^10 + 310326055v^9 - 647976279v^8 + 1147878041v^7 - 1638530621v^6 + 1937821276v^5 - 1805872330v^4 + 1325167070v^3 - 705049740v^2 + 249379043*v - 43105093) / 1336973
\(\beta_{12}\)	\(=\)	\( ( 391353 \nu^{15} - 2445777 \nu^{14} + 20054944 \nu^{13} - 80286537 \nu^{12} + 321157765 \nu^{11} + \cdots + 545094556 ) / 10695784 \) (391353v^15 - 2445777v^14 + 20054944v^13 - 80286537v^12 + 321157765v^11 - 845972084v^10 + 2059282717v^9 - 3529572307v^8 + 5288951833v^7 - 5133480728v^6 + 3615668607v^5 + 300805273v^4 - 2703611425v^3 + 3535742846v^2 - 1924341010*v + 545094556) / 10695784
\(\beta_{13}\)	\(=\)	\( ( - 391353 \nu^{15} + 3424518 \nu^{14} - 26906131 \nu^{13} + 135927643 \nu^{12} + \cdots + 1467440026 ) / 10695784 \) (-391353v^15 + 3424518v^14 - 26906131v^13 + 135927643v^12 - 565938970v^11 + 1850576155v^10 - 5001761983v^9 + 11168824928v^8 - 20603760050v^7 + 31605232675v^6 - 39388831273v^5 + 39901160418v^4 - 31258554144v^3 + 18437190911v^2 - 7178538814*v + 1467440026) / 10695784
\(\beta_{14}\)	\(=\)	\( ( 64652 \nu^{15} - 484890 \nu^{14} + 3826280 \nu^{13} - 17516655 \nu^{12} + 70503984 \nu^{11} + \cdots - 3312928 ) / 1336973 \) (64652v^15 - 484890v^14 + 3826280v^13 - 17516655v^12 + 70503984v^11 - 211267870v^10 + 537635867v^9 - 1089207549v^8 + 1832971428v^7 - 2474477605v^6 + 2674061346v^5 - 2236358266v^4 + 1369910081v^3 - 570094455v^2 + 117059508*v - 3312928) / 1336973
\(\beta_{15}\)	\(=\)	\( ( 308418 \nu^{15} - 2313135 \nu^{14} + 18485755 \nu^{13} - 85074860 \nu^{12} + 348592111 \nu^{11} + \cdots - 286402874 ) / 5347892 \) (308418v^15 - 2313135v^14 + 18485755v^13 - 85074860v^12 + 348592111v^11 - 1058614755v^10 + 2767089288v^9 - 5750637171v^8 + 10101698075v^7 - 14289837643v^6 + 16628863304v^5 - 15200966229v^4 + 10773509555v^3 - 5486285853v^2 + 1807988888*v - 286402874) / 5347892

\(\nu\)	\(=\)	\( ( \beta_{15} - \beta_{14} - \beta_{11} - \beta_{10} - \beta_{9} + 2\beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} + 2 ) / 4 \) (b15 - b14 - b11 - b10 - b9 + 2b8 + 2b7 - b6 + b5 + 2) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} - \beta_{14} - \beta_{11} - \beta_{10} - \beta_{9} + 4 \beta_{7} - 3 \beta_{6} - \beta_{5} + \cdots - 16 ) / 4 \) (b15 - b14 - b11 - b10 - b9 + 4b7 - 3b6 - b5 - 2b3 - 2b2 - 4*b1 - 16) / 4
\(\nu^{3}\)	\(=\)	\( ( - 2 \beta_{15} + 4 \beta_{14} + 5 \beta_{13} - 5 \beta_{12} + 14 \beta_{11} + 14 \beta_{10} + 14 \beta_{9} + \cdots - 25 ) / 4 \) (-2b15 + 4b14 + 5b13 - 5b12 + 14b11 + 14b10 + 14b9 - 24b8 - 18b7 - b6 - 5b5 - 3b3 - 3b2 - 6*b1 - 25) / 4
\(\nu^{4}\)	\(=\)	\( ( - 5 \beta_{15} + 9 \beta_{14} + 26 \beta_{13} + 6 \beta_{12} + 29 \beta_{11} + 25 \beta_{10} + \cdots + 144 ) / 4 \) (-5b15 + 9b14 + 26b13 + 6b12 + 29b11 + 25b10 + 33b9 - 12b8 - 76b7 + 39b6 + 29b5 + 50b3 + 32b2 + 48b1 + 144) / 4
\(\nu^{5}\)	\(=\)	\( ( - 15 \beta_{15} - 30 \beta_{14} - 58 \beta_{13} + 138 \beta_{12} - 235 \beta_{11} - 212 \beta_{10} + \cdots + 402 ) / 4 \) (-15b15 - 30b14 - 58b13 + 138b12 - 235b11 - 212b10 - 192b9 + 353b8 + 183b7 + 90b6 + 90b5 + 44b4 + 130b3 + 85b2 + 130*b1 + 402) / 4
\(\nu^{6}\)	\(=\)	\( ( - 32 \beta_{15} - 113 \beta_{14} - 631 \beta_{13} + 7 \beta_{12} - 778 \beta_{11} - 603 \beta_{10} + \cdots - 1773 ) / 4 \) (-32b15 - 113b14 - 631b13 + 7b12 - 778b11 - 603b10 - 755b9 + 503b8 + 1327b7 - 512b6 - 486b5 + 132b4 - 801b3 - 468b2 - 632*b1 - 1773) / 4
\(\nu^{7}\)	\(=\)	\( ( 347 \beta_{15} + 324 \beta_{14} + 179 \beta_{13} - 2643 \beta_{12} + 3365 \beta_{11} + 3065 \beta_{10} + \cdots - 7642 ) / 4 \) (347b15 + 324b14 + 179b13 - 2643b12 + 3365b11 + 3065b10 + 2463b9 - 5248b8 - 1762b7 - 2131b6 - 1999b5 - 888b4 - 3262b3 - 1939b2 - 2674*b1 - 7642) / 4
\(\nu^{8}\)	\(=\)	\( ( 1525 \beta_{15} + 1845 \beta_{14} + 11492 \beta_{13} - 2820 \beta_{12} + 17159 \beta_{11} + 13239 \beta_{10} + \cdots + 22102 ) / 4 \) (1525b15 + 1845b14 + 11492b13 - 2820b12 + 17159b11 + 13239b10 + 15347b9 - 13322b8 - 23466b7 + 6279b6 + 6661b5 - 4168b4 + 10628b3 + 6112b2 + 7936*b1 + 22102) / 4
\(\nu^{9}\)	\(=\)	\( ( - 4537 \beta_{15} - 3662 \beta_{14} + 10081 \beta_{13} + 44063 \beta_{12} - 40583 \beta_{11} + \cdots + 147050 ) / 4 \) (-4537b15 - 3662b14 + 10081b13 + 44063b12 - 40583b11 - 39619b10 - 26437b9 + 75006b8 + 7716b7 + 42765b6 + 41013b5 + 11688b4 + 67950b3 + 39501b2 + 52314*b1 + 147050) / 4
\(\nu^{10}\)	\(=\)	\( ( - 34320 \beta_{15} - 32985 \beta_{14} - 187680 \beta_{13} + 94136 \beta_{12} - 337200 \beta_{11} + \cdots - 232877 ) / 4 \) (-34320b15 - 32985b14 - 187680b13 + 94136b12 - 337200b11 - 265892b10 - 288582b9 + 298336b8 + 404440b7 - 62959b6 - 74349b5 + 90624b4 - 113437b3 - 65112b2 - 83440*b1 - 232877) / 4
\(\nu^{11}\)	\(=\)	\( ( 40722 \beta_{15} + 31462 \beta_{14} - 370965 \beta_{13} - 667831 \beta_{12} + 354638 \beta_{11} + \cdots - 2717405 ) / 4 \) (40722b15 + 31462b14 - 370965b13 - 667831b12 + 354638b11 + 414332b10 + 161860b9 - 979692b8 + 260954b7 - 792879b6 - 777723b5 - 105100b4 - 1284107b3 - 742489b2 - 969342*b1 - 2717405) / 4
\(\nu^{12}\)	\(=\)	\( ( 647631 \beta_{15} + 584689 \beta_{14} + 2834771 \beta_{13} - 2296915 \beta_{12} + 6137751 \beta_{11} + \cdots + 1304386 ) / 4 \) (647631b15 + 584689b14 + 2834771b13 - 2296915b12 + 6137751b11 + 4964494b10 + 5094384b9 - 6050105b8 - 6641593b7 + 277208b6 + 498990b5 - 1699140b4 + 645420b3 + 369488b2 + 466704*b1 + 1304386) / 4
\(\nu^{13}\)	\(=\)	\( ( - 14543 \beta_{15} + 26036 \beta_{14} + 9238791 \beta_{13} + 9156157 \beta_{12} + 102287 \beta_{11} + \cdots + 47639490 ) / 4 \) (-14543b15 + 26036b14 + 9238791b13 + 9156157b12 + 102287b11 - 2132573b10 + 2333473b9 + 10705894b8 - 11020772b7 + 13822823b6 + 13813591b5 + 25248b4 + 22647014b3 + 13078403b2 + 16996798*b1 + 47639490) / 4
\(\nu^{14}\)	\(=\)	\( ( - 11181187 \beta_{15} - 9930065 \beta_{14} - 39159133 \beta_{13} + 48584949 \beta_{12} + \cdots + 24971826 ) / 4 \) (-11181187b15 - 9930065b14 - 39159133b13 + 48584949b12 - 104931423b11 - 87067098b10 - 84771480b9 + 113985875b8 + 102570315b7 + 9099524b6 + 5267254b5 + 29282072b4 + 11710952b3 + 6781912b2 + 8908192*b1 + 24971826) / 4
\(\nu^{15}\)	\(=\)	\( ( - 11233293 \beta_{15} - 10213888 \beta_{14} - 197433196 \beta_{13} - 108048880 \beta_{12} + \cdots - 789301344 ) / 4 \) (-11233293b15 - 10213888b14 - 197433196b13 - 108048880b12 - 106918641b11 - 50573584b10 - 124704264b9 - 69295885b8 + 290559301b7 - 227208076b6 - 231051796b5 + 29505396b4 - 375744468b3 - 216933763b2 - 281575510*b1 - 789301344) / 4

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2029.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 17 T^{6} + 75 T^{4} + \cdots + 4)^{2} \) (T^8 + 17T^6 + 75T^4 + 95*T^2 + 4)^2
$5$	\( T^{16} - 7 T^{14} + \cdots + 390625 \) T^16 - 7T^14 + 22T^12 - T^10 - 14T^8 - 25T^6 + 13750T^4 - 109375T^2 + 390625
$7$	\( (T^{8} + 31 T^{6} + \cdots + 1024)^{2} \) (T^8 + 31T^6 + 327T^4 + 1273*T^2 + 1024)^2
$11$	\( (T^{8} - 65 T^{6} + \cdots + 4096)^{2} \) (T^8 - 65T^6 + 1323T^4 - 8531*T^2 + 4096)^2
$13$	\( T^{16} \) T^16
$17$	\( (T^{8} + 80 T^{6} + \cdots + 52441)^{2} \) (T^8 + 80T^6 + 1926T^4 + 17744*T^2 + 52441)^2
$19$	\( (T^{8} - 165 T^{6} + \cdots + 412164)^{2} \) (T^8 - 165T^6 + 9015T^4 - 170271*T^2 + 412164)^2
$23$	\( (T^{8} + 89 T^{6} + \cdots + 1156)^{2} \) (T^8 + 89T^6 + 2091T^4 + 9335*T^2 + 1156)^2
$29$	\( (T^{4} + 6 T^{3} + \cdots - 393)^{4} \) (T^4 + 6T^3 - 48T^2 - 318*T - 393)^4
$31$	\( (T^{8} - 108 T^{6} + \cdots + 2304)^{2} \) (T^8 - 108T^6 + 2688T^4 - 6336*T^2 + 2304)^2
$37$	\( (T^{8} + 160 T^{6} + \cdots + 292681)^{2} \) (T^8 + 160T^6 + 8598T^4 + 160576*T^2 + 292681)^2
$41$	\( (T^{8} - 164 T^{6} + \cdots + 1771561)^{2} \) (T^8 - 164T^6 + 9438T^4 - 221780*T^2 + 1771561)^2
$43$	\( (T^{8} + 237 T^{6} + \cdots + 535824)^{2} \) (T^8 + 237T^6 + 13839T^4 + 174483*T^2 + 535824)^2
$47$	\( (T^{8} + 132 T^{6} + \cdots + 147456)^{2} \) (T^8 + 132T^6 + 4992T^4 + 55296*T^2 + 147456)^2
$53$	\( (T^{8} + 191 T^{6} + \cdots + 30976)^{2} \) (T^8 + 191T^6 + 8004T^4 + 94256*T^2 + 30976)^2
$59$	\( (T^{8} - 65 T^{6} + \cdots + 1936)^{2} \) (T^8 - 65T^6 + 1227T^4 - 5987*T^2 + 1936)^2
$61$	\( (T^{4} - 14 T^{3} + \cdots - 1157)^{4} \) (T^4 - 14T^3 + 12T^2 + 406*T - 1157)^4
$67$	\( (T^{8} + 451 T^{6} + \cdots + 850084)^{2} \) (T^8 + 451T^6 + 59235T^4 + 2128405*T^2 + 850084)^2
$71$	\( (T^{8} - 341 T^{6} + \cdots + 3104644)^{2} \) (T^8 - 341T^6 + 32151T^4 - 699791*T^2 + 3104644)^2
$73$	\( (T^{8} + 133 T^{6} + \cdots + 861184)^{2} \) (T^8 + 133T^6 + 6276T^4 + 123712*T^2 + 861184)^2
$79$	\( (T^{4} + 4 T^{3} + \cdots + 4096)^{4} \) (T^4 + 4T^3 - 156T^2 - 224*T + 4096)^4
$83$	\( (T^{8} + 228 T^{6} + \cdots + 36864)^{2} \) (T^8 + 228T^6 + 9168T^4 + 78336*T^2 + 36864)^2
$89$	\( (T^{8} - 65 T^{6} + \cdots + 1936)^{2} \) (T^8 - 65T^6 + 1227T^4 - 5987*T^2 + 1936)^2
$97$	\( (T^{8} + 331 T^{6} + \cdots + 3020644)^{2} \) (T^8 + 331T^6 + 20931T^4 + 448621*T^2 + 3020644)^2
show more
show less

\(n\)	\(677\)	\(1691\)	\(1861\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)