LMFDB - Newform orbit 896.2.q.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(703,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("896.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.q (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:	$7.15459602111$
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} - 4 x^{15} + 4 x^{14} - 24 x^{13} + 104 x^{12} - 196 x^{11} + 312 x^{10} - 236 x^{9} + 31 x^{8} + \cdots + 1 $ x^16 - 4x^15 + 4x^14 - 24x^13 + 104x^12 - 196x^11 + 312x^10 - 236x^9 + 31x^8 + 236x^7 + 312x^6 + 196x^5 + 104x^4 + 24x^3 + 4x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{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_{5} q^{3} + \beta_{9} q^{5} - \beta_{14} q^{7} + (\beta_{13} + 2 \beta_{7} + 2 \beta_{6}) q^{9}+O(q^{10}) $ q + b5 * q^3 + b9 * q^5 - b14 * q^7 + (b13 + 2b7 + 2b6) * q^9	$ q + \beta_{5} q^{3} + \beta_{9} q^{5} - \beta_{14} q^{7} + (\beta_{13} + 2 \beta_{7} + 2 \beta_{6}) q^{9} + (\beta_{12} - \beta_{3}) q^{11} + (\beta_{8} - 2 \beta_{2} - \beta_1) q^{13} + (\beta_{14} + \beta_{11} + \cdots + \beta_{4}) q^{15}+ \cdots + ( - 3 \beta_{15} - 3 \beta_{5} + \beta_{3}) q^{99}+O(q^{100}) $ q + b5 * q^3 + b9 * q^5 - b14 * q^7 + (b13 + 2b7 + 2b6) * q^9 + (b12 - b3) * q^11 + (b8 - 2b2 - b1) q^13 + (b14 + b11 + b10 + b4) * q^15 + (b7 + b6 + 1) * q^17 + (b12 - 2b3) q^19 + (-b9 - b8 + 3b2 + 2b1) * q^21 + (b14 - b10 + b4) * q^23 + (-2b6 + 2) q^25 + (-2b15 + 2b12 + 2b5 - b3) q^27 + (-4b9 - b8 + 2b2 - b1) * q^29 + (-2b14 - 2b11 + b10 - 3b4) q^31 + (2b13 + b6 - 2) q^33 + (b15 + b12 + b5 - b3) * q^35 + (b9 - 3b8 - 2b2) * q^37 + (b14 - b11 - 4b10 - 3b4) * q^39 + (-3b13 - 3b7) * q^41 + (2b15 + 2b5 - 2b3) q^43 + (2b9 + 6b8 - 2b2 + 3b1) * q^45 + (b14 + b11 + 2b10 + 2b4) * q^47 + (-3b13 - b7 - 2b6 - 1) * q^49 + (-2b15 + b12 + 4b5) * q^51 + (b9 + b2 - b1) * q^53 + (2b14 - b11 - 2b10 + b4) * q^55 + (2b13 - 2b7 - 3) * q^57 + b5 * q^59 + (3b9 - 3b8 - 6b1) q^61 + (-2b14 + b11 + 5b10 + b4) * q^63 + (-b13 - 2b7 - 6b6) * q^65 + (-2b15 + 2b12 + b5 - 2b3) q^67 + (2b8 - b2 - 2b1) * q^69 + (4b11 + 4b4) * q^71 + (-2b7 - 5b6 - 5) * q^73 + 2b15 q^75 + (b9 + b8 - 3b2 + 5b1) * q^77 + (b14 + 2b11 + b10 - b4) q^79 + (2b13 + b7 + 7b6 - 7) * q^81 + (4b15 - 4b5) * q^83 + (2b9 + 3b8 - b2 + 3b1) q^85 + (-5b14 - 5b11 - 2b10 - 3b4) * q^87 + (-4b13 - b6 + 2) q^89 + (-5b15 + b5 - b3) q^91 + (-3b9 - 11b8 + 6b2) q^93 + (3b14 - 3b11 - 3b10) q^95 + (-b13 - b7 - 4b6 + 2) q^97 + (-3b15 - 3b5 + b3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^9	$ 16 q + 16 q^{9} + 24 q^{17} + 16 q^{25} - 24 q^{33} - 32 q^{49} - 48 q^{57} - 48 q^{65} - 120 q^{73} - 56 q^{81} + 24 q^{89}+O(q^{100}) $ 16 * q + 16 * q^9 + 24 * q^17 + 16 * q^25 - 24 * q^33 - 32 * q^49 - 48 * q^57 - 48 * q^65 - 120 * q^73 - 56 * q^81 + 24 * q^89

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4 x^{15} + 4 x^{14} - 24 x^{13} + 104 x^{12} - 196 x^{11} + 312 x^{10} - 236 x^{9} + 31 x^{8} + \cdots + 1 $ x^16 - 4*x^15 + 4*x^14 - 24*x^13 + 104*x^12 - 196*x^11 + 312*x^10 - 236*x^9 + 31*x^8 + 236*x^7 + 312*x^6 + 196*x^5 + 104*x^4 + 24*x^3 + 4*x^2 + 4*x + 1 :

$\beta_{1}$	$=$	$ ( - 24578404 \nu^{15} + 112409521 \nu^{14} - 161766434 \nu^{13} + 676914588 \nu^{12} + \cdots - 80530744 ) / 55881261 $ (-24578404v^15 + 112409521v^14 - 161766434v^13 + 676914588v^12 - 2932745926v^11 + 6464372519v^10 - 11224587756v^9 + 11842175116v^8 - 6808155680v^7 - 2875070153v^6 - 5236873676v^5 - 1970820116v^4 - 1399477502v^3 + 116148784v^2 - 266195456*v - 80530744) / 55881261
$\beta_{2}$	$=$	$ ( 478 \nu^{15} - 1916 \nu^{14} + 1922 \nu^{13} - 11453 \nu^{12} + 49716 \nu^{11} - 93806 \nu^{10} + \cdots + 1412 ) / 969 $ (478v^15 - 1916v^14 + 1922v^13 - 11453v^12 + 49716v^11 - 93806v^10 + 148878v^9 - 111096v^8 + 8892v^7 + 123734v^6 + 134994v^5 + 101866v^4 + 45050v^3 + 8050v^2 - 572*v + 1412) / 969
$\beta_{3}$	$=$	$ ( - 713760 \nu^{15} + 2873130 \nu^{14} - 2934707 \nu^{13} + 17216192 \nu^{12} - 74582450 \nu^{11} + \cdots - 2172105 ) / 1095711 $ (-713760v^15 + 2873130v^14 - 2934707v^13 + 17216192v^12 - 74582450v^11 + 141664463v^10 - 226302900v^9 + 172573690v^8 - 19779888v^7 - 183883199v^6 - 193251327v^5 - 157627568v^4 - 67531547v^3 - 10955821v^2 + 433452*v - 2172105) / 1095711
$\beta_{4}$	$=$	$ ( 41571440 \nu^{15} - 185423177 \nu^{14} + 254219263 \nu^{13} - 1125645507 \nu^{12} + \cdots + 5669660 ) / 55881261 $ (41571440v^15 - 185423177v^14 + 254219263v^13 - 1125645507v^12 + 4853270456v^11 - 10441949317v^10 + 18054484920v^9 - 18663563720v^8 + 10686848023v^7 + 4323827851v^6 + 10838029165v^5 + 4051741093v^4 + 3030984061v^3 - 282979166v^2 + 21421645*v + 5669660) / 55881261
$\beta_{5}$	$=$	$ ( 16331260 \nu^{15} - 74991976 \nu^{14} + 108507436 \nu^{13} - 450127912 \nu^{12} + 1954868524 \nu^{11} + \cdots + 53802093 ) / 18627087 $ (16331260v^15 - 74991976v^14 + 108507436v^13 - 450127912v^12 + 1954868524v^11 - 4322761180v^10 + 7497148468v^9 - 7919869405v^8 + 4541015683v^7 + 1915043380v^6 + 3495942116v^5 + 1025333980v^4 + 934847368v^3 - 77706604v^2 + 177822799*v + 53802093) / 18627087
$\beta_{6}$	$=$	$ ( - 16361188 \nu^{15} + 72139004 \nu^{14} - 94401360 \nu^{13} + 428892200 \nu^{12} + \cdots - 32692448 ) / 18627087 $ (-16361188v^15 + 72139004v^14 - 94401360v^13 + 428892200v^12 - 1874038268v^11 + 3958996896v^10 - 6661892204v^9 + 6457440527v^8 - 2929890732v^7 - 2870483616v^6 - 3833618044v^5 - 1548767784v^4 - 916399440v^3 + 54074084v^2 - 25648164*v - 32692448) / 18627087
$\beta_{7}$	$=$	$ ( - 16819000 \nu^{15} + 72297973 \nu^{14} - 89870960 \nu^{13} + 435551452 \nu^{12} + \cdots - 19377740 ) / 18627087 $ (-16819000v^15 + 72297973v^14 - 89870960v^13 + 435551452v^12 - 1887492644v^11 + 3888935115v^10 - 6539006616v^9 + 6229582792v^8 - 2911659902v^7 - 2510098475v^6 - 4825415772v^5 - 2001171996v^4 - 1263419608v^3 + 101493346v^2 - 27011014*v - 19377740) / 18627087
$\beta_{8}$	$=$	$ ( - 57962620 \nu^{15} + 244034719 \nu^{14} - 284439440 \nu^{13} + 1456967568 \nu^{12} + \cdots - 168001120 ) / 55881261 $ (-57962620v^15 + 244034719v^14 - 284439440v^13 + 1456967568v^12 - 6343717582v^11 + 12731116736v^10 - 20919252180v^9 + 18436863304v^8 - 6312521300v^7 - 11645397440v^6 - 16068699614v^5 - 8122374800v^4 - 4493625488v^3 - 627232847v^2 - 154199804*v - 168001120) / 55881261
$\beta_{9}$	$=$	$ ( - 23022602 \nu^{15} + 100944114 \nu^{14} - 130859040 \nu^{13} + 602140523 \nu^{12} + \cdots - 73290672 ) / 18627087 $ (-23022602v^15 + 100944114v^14 - 130859040v^13 + 602140523v^12 - 2623735034v^11 + 5518742544v^10 - 9288708034v^9 + 8944701020v^8 - 4051325654v^7 - 4009850208v^6 - 5588137678v^5 - 2281853173v^4 - 1667581296v^3 - 87881958v^2 - 141488302*v - 73290672) / 18627087
$\beta_{10}$	$=$	$ ( 81436312 \nu^{15} - 349229857 \nu^{14} + 425779916 \nu^{13} - 2076724695 \nu^{12} + \cdots + 120487072 ) / 55881261 $ (81436312v^15 - 349229857v^14 + 425779916v^13 - 2076724695v^12 + 9074758729v^11 - 18573123677v^10 + 30748771548v^9 - 28193204368v^8 + 10926609317v^7 + 15431269295v^6 + 21654724265v^5 + 9289838258v^4 + 5192070428v^3 - 313967458v^2 + 174107807*v + 120487072) / 55881261
$\beta_{11}$	$=$	$ ( 81657397 \nu^{15} - 361433545 \nu^{14} + 473165969 \nu^{13} - 2128224750 \nu^{12} + \cdots + 304152748 ) / 55881261 $ (81657397v^15 - 361433545v^14 + 473165969v^13 - 2128224750v^12 + 9356454580v^11 - 19798644830v^10 + 33056593947v^9 - 31549953604v^8 + 12982980755v^7 + 16563936710v^6 + 17306567999v^5 + 7099394417v^4 + 3743469443v^3 - 126506317v^2 + 151097009*v + 304152748) / 55881261
$\beta_{12}$	$=$	$ ( 28053456 \nu^{15} - 123097333 \nu^{14} + 159579226 \nu^{13} - 732793144 \nu^{12} + \cdots + 90494094 ) / 18627087 $ (28053456v^15 - 123097333v^14 + 159579226v^13 - 732793144v^12 + 3197206163v^11 - 6727519330v^10 + 11304474312v^9 - 10850615079v^8 + 4828872624v^7 + 5010528130v^6 + 6755327387v^5 + 2636843224v^4 + 2078302474v^3 + 108463237v^2 + 169601736*v + 90494094) / 18627087
$\beta_{13}$	$=$	$ ( - 29904546 \nu^{15} + 129757565 \nu^{14} - 161347800 \nu^{13} + 762815256 \nu^{12} + \cdots - 107903336 ) / 18627087 $ (-29904546v^15 + 129757565v^14 - 161347800v^13 + 762815256v^12 - 3357607152v^11 + 6943520122v^10 - 11435934986v^9 + 10437065588v^8 - 3662556090v^7 - 6508066522v^6 - 6938376848v^5 - 2996702536v^4 - 1496150424v^3 + 50381843v^2 - 69368690*v - 107903336) / 18627087
$\beta_{14}$	$=$	$ ( 95630831 \nu^{15} - 411561677 \nu^{14} + 503317981 \nu^{13} - 2431707663 \nu^{12} + \cdots + 371054036 ) / 55881261 $ (95630831v^15 - 411561677v^14 + 503317981v^13 - 2431707663v^12 + 10669553489v^11 - 21886755922v^10 + 36060331821v^9 - 32758079528v^8 + 11742438997v^7 + 19770274810v^6 + 23948074300v^5 + 10370368018v^4 + 5444666191v^3 - 258493073v^2 + 220080667*v + 371054036) / 55881261
$\beta_{15}$	$=$	$ ( - 41682527 \nu^{15} + 175603960 \nu^{14} - 205133396 \nu^{13} + 1048417004 \nu^{12} + \cdots - 123561984 ) / 18627087 $ (-41682527v^15 + 175603960v^14 - 205133396v^13 + 1048417004v^12 - 4563459364v^11 + 9167107496v^10 - 15069743507v^9 + 13276126963v^8 - 4498087181v^7 - 8537548760v^6 - 11303294252v^5 - 5984840948v^4 - 3300359084v^3 - 432772592v^2 - 114281729*v - 123561984) / 18627087

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

$\nu$	$=$	$ ( -\beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - 3\beta_{6} + 3\beta_{5} - 2\beta_{4} + \beta_{2} + 3\beta _1 + 3 ) / 6 $ (-b14 - b11 + b10 + b9 - 3b6 + 3b5 - 2b4 + b2 + 3b1 + 3) / 6
$\nu^{2}$	$=$	$ ( 6 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} + 2 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} + \cdots + 4 \beta_{2} ) / 6 $ (6b15 + 2b14 + 3b13 + 2b11 + 4b10 - 2b9 - 9b8 + 6b7 + 6b6 + 4b4 + 4*b2) / 6
$\nu^{3}$	$=$	$ ( - 6 \beta_{15} - 11 \beta_{14} - 9 \beta_{13} + 6 \beta_{12} - 5 \beta_{11} + 11 \beta_{10} + 14 \beta_{9} + \cdots + 33 ) / 6 $ (-6b15 - 11b14 - 9b13 + 6b12 - 5b11 + 11b10 + 14b9 + 6b8 + 9b7 + 6b5 + 5b4 - 3b3 - 7b2 + 6b1 + 33) / 6
$\nu^{4}$	$=$	$ 2\beta_{12} + 7\beta_{9} + 10\beta_{5} + 2\beta_{3} + 7\beta_{2} + 12\beta_1 $ 2b12 + 7b9 + 10b5 + 2b3 + 7b2 + 12b1
$\nu^{5}$	$=$	$ ( 66 \beta_{15} + 37 \beta_{14} + 75 \beta_{13} - 15 \beta_{12} + 103 \beta_{11} + 140 \beta_{10} + \cdots + 98 \beta_{2} ) / 6 $ (66b15 + 37b14 + 75b13 - 15b12 + 103b11 + 140b10 - 49b9 - 78b8 + 150b7 + 303b6 + 140b4 + 30b3 + 98*b2) / 6
$\nu^{6}$	$=$	$ ( - 402 \beta_{15} - 230 \beta_{14} - 171 \beta_{13} + 96 \beta_{12} - 86 \beta_{11} + 230 \beta_{10} + \cdots + 678 ) / 6 $ (-402b15 - 230b14 - 171b13 + 96b12 - 86b11 + 230b10 + 452b9 + 513b8 + 171b7 + 402b5 + 86b4 - 48b3 - 226b2 + 513b1 + 678) / 6
$\nu^{7}$	$=$	$ ( 509 \beta_{14} + 504 \beta_{13} + 204 \beta_{12} + 509 \beta_{11} + 103 \beta_{10} + 895 \beta_{9} + \cdots - 1173 ) / 6 $ (509b14 + 504b13 + 204b12 + 509b11 + 103b10 + 895b9 + 252b7 + 1173b6 + 1527b5 + 406b4 + 204b3 + 895b2 + 1929*b1 - 1173) / 6
$\nu^{8}$	$=$	$ 152\beta_{14} + 336\beta_{13} + 500\beta_{11} + 652\beta_{10} + 672\beta_{7} + 1457\beta_{6} + 652\beta_{4} $ 152b14 + 336b13 + 500b11 + 652b10 + 672b7 + 1457b6 + 652*b4
$\nu^{9}$	$=$	$ ( - 13263 \beta_{15} - 3406 \beta_{14} - 2268 \beta_{13} + 3624 \beta_{12} - 1015 \beta_{11} + \cdots + 9915 ) / 6 $ (-13263b15 - 3406b14 - 2268b13 + 3624b12 - 1015b11 + 3406b10 + 15682b9 + 16719b8 + 2268b7 + 13263b5 + 1015b4 - 1812b3 - 7841b2 + 16719b1 + 9915) / 6
$\nu^{10}$	$=$	$ ( 24068 \beta_{14} + 25206 \beta_{13} + 4080 \beta_{12} + 24068 \beta_{11} + 5914 \beta_{10} + \cdots - 53094 ) / 6 $ (24068b14 + 25206b13 + 4080b12 + 24068b11 + 5914b10 + 17698b9 + 12603b7 + 53094b6 + 29982b5 + 18154b4 + 4080b3 + 17698b2 + 37809*b1 - 53094) / 6
$\nu^{11}$	$=$	$ ( - 47658 \beta_{15} + 22597 \beta_{14} + 48477 \beta_{13} + 6711 \beta_{12} + 70255 \beta_{11} + \cdots - 57022 \beta_{2} ) / 6 $ (-47658b15 + 22597b14 + 48477b13 + 6711b12 + 70255b11 + 92852b10 + 28511b9 + 59898b8 + 96954b7 + 205329b6 + 92852b4 - 13422b3 - 57022*b2) / 6
$\nu^{12}$	$=$	$ - 61406 \beta_{15} + 17036 \beta_{12} + 73038 \beta_{9} + 77292 \beta_{8} + 61406 \beta_{5} + \cdots + 77292 \beta_1 $ -61406b15 + 17036b12 + 73038b9 + 77292b8 + 61406b5 - 8518b3 - 36519b2 + 77292b1
$\nu^{13}$	$=$	$ ( 809128 \beta_{14} + 844350 \beta_{13} + 57819 \beta_{12} + 809128 \beta_{11} + 196517 \beta_{10} + \cdots - 1790175 ) / 6 $ (809128b14 + 844350b13 + 57819b12 + 809128b11 + 196517b10 + 247625b9 + 422175b7 + 1790175b6 + 416094b5 + 612611b4 + 57819b3 + 247625b2 + 523650*b1 - 1790175) / 6
$\nu^{14}$	$=$	$ ( - 2272170 \beta_{15} + 444142 \beta_{14} + 953907 \beta_{13} + 313248 \beta_{12} + \cdots - 2696068 \beta_{2} ) / 6 $ (-2272170b15 + 444142b14 + 953907b13 + 313248b12 + 1383886b11 + 1828028b10 + 1348034b9 + 2861721b8 + 1907814b7 + 4044102b6 + 1828028b4 - 626496b3 - 2696068*b2) / 6
$\nu^{15}$	$=$	$ ( - 8761899 \beta_{15} + 2213258 \beta_{14} + 1522584 \beta_{13} + 2418672 \beta_{12} + 707375 \beta_{11} + \cdots - 6466317 ) / 6 $ (-8761899b15 + 2213258b14 + 1522584b13 + 2418672b12 + 707375b11 - 2213258b10 + 10401214b9 + 11034069b8 - 1522584b7 + 8761899b5 - 707375b4 - 1209336b3 - 5200607b2 + 11034069b1 - 6466317) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$645$
$\chi(n)$	$-1$	$\beta_{6}$	$-1$

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} $

703.1

−1.79700	−	2.34189i
−0.335868	−	0.0442178i
−0.472726	+	0.362736i
0.219056	−	1.66389i
1.33145	−	1.02165i
−0.0777751	+	0.590761i
0.206228	+	0.268761i
2.92664	+	0.385299i
−1.79700	+	2.34189i
−0.335868	+	0.0442178i
−0.472726	−	0.362736i
0.219056	+	1.66389i
1.33145	+	1.02165i
−0.0777751	−	0.590761i
0.206228	−	0.268761i
2.92664	−	0.385299i

−2.63287

−

1.52009i

−0.866025

−

1.50000i

2.14973

−

1.54230i

3.12132

5.40629i

703.2

−2.63287

−

1.52009i

0.866025

1.50000i

−2.14973

1.54230i

3.12132

5.40629i

703.3

−0.753671

−

0.435132i

−0.866025

−

1.50000i

−0.615370

−

2.57319i

−1.12132

−

1.94218i

703.4

−0.753671

−

0.435132i

0.866025

1.50000i

0.615370

2.57319i

−1.12132

−

1.94218i

703.5

0.753671

0.435132i

−0.866025

−

1.50000i

0.615370

2.57319i

−1.12132

−

1.94218i

703.6

0.753671

0.435132i

0.866025

1.50000i

−0.615370

−

2.57319i

−1.12132

−

1.94218i

703.7

2.63287

1.52009i

−0.866025

−

1.50000i

−2.14973

1.54230i

3.12132

5.40629i

703.8

2.63287

1.52009i

0.866025

1.50000i

2.14973

−

1.54230i

3.12132

5.40629i

831.1

−2.63287

1.52009i

−0.866025

1.50000i

2.14973

1.54230i

3.12132

−

5.40629i

831.2

−2.63287

1.52009i

0.866025

−

1.50000i

−2.14973

−

1.54230i

3.12132

−

5.40629i

831.3

−0.753671

0.435132i

−0.866025

1.50000i

−0.615370

2.57319i

−1.12132

1.94218i

831.4

−0.753671

0.435132i

0.866025

−

1.50000i

0.615370

−

2.57319i

−1.12132

1.94218i

831.5

0.753671

−

0.435132i

−0.866025

1.50000i

0.615370

−

2.57319i

−1.12132

1.94218i

831.6

0.753671

−

0.435132i

0.866025

−

1.50000i

−0.615370

2.57319i

−1.12132

1.94218i

831.7

2.63287

−

1.52009i

−0.866025

1.50000i

−2.14973

−

1.54230i

3.12132

−

5.40629i

831.8

2.63287

−

1.52009i

0.866025

−

1.50000i

2.14973

1.54230i

3.12132

−

5.40629i

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
8.b	even	2	1	inner
8.d	odd	2	1	inner
28.f	even	6	1	inner
56.j	odd	6	1	inner
56.m	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	896.2.q.c	✓	16
4.b	odd	2	1	inner	896.2.q.c	✓	16
7.d	odd	6	1	inner	896.2.q.c	✓	16
8.b	even	2	1	inner	896.2.q.c	✓	16
8.d	odd	2	1	inner	896.2.q.c	✓	16
28.f	even	6	1	inner	896.2.q.c	✓	16
56.j	odd	6	1	inner	896.2.q.c	✓	16
56.m	even	6	1	inner	896.2.q.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
896.2.q.c	✓	16	1.a	even	1	1	trivial
896.2.q.c	✓	16	4.b	odd	2	1	inner
896.2.q.c	✓	16	7.d	odd	6	1	inner
896.2.q.c	✓	16	8.b	even	2	1	inner
896.2.q.c	✓	16	8.d	odd	2	1	inner
896.2.q.c	✓	16	28.f	even	6	1	inner
896.2.q.c	✓	16	56.j	odd	6	1	inner
896.2.q.c	✓	16	56.m	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 10T_{3}^{6} + 93T_{3}^{4} - 70T_{3}^{2} + 49 $ T3^8 - 10*T3^6 + 93*T3^4 - 70*T3^2 + 49 acting on $S_{2}^{\mathrm{new}}(896, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} - 10 T^{6} + \cdots + 49)^{2} $ (T^8 - 10T^6 + 93T^4 - 70*T^2 + 49)^2
$5$	$ (T^{4} + 3 T^{2} + 9)^{4} $ (T^4 + 3*T^2 + 9)^4
$7$	$ (T^{8} + 8 T^{6} + \cdots + 2401)^{2} $ (T^8 + 8T^6 + 42T^4 + 392*T^2 + 2401)^2
$11$	$ (T^{8} + 18 T^{6} + \cdots + 3969)^{2} $ (T^8 + 18T^6 + 261T^4 + 1134*T^2 + 3969)^2
$13$	$ (T^{4} - 36 T^{2} + 36)^{4} $ (T^4 - 36*T^2 + 36)^4
$17$	$ (T^{4} - 6 T^{3} + 9 T^{2} + \cdots + 9)^{4} $ (T^4 - 6T^3 + 9T^2 + 18*T + 9)^4
$19$	$ (T^{8} - 54 T^{6} + \cdots + 321489)^{2} $ (T^8 - 54T^6 + 2349T^4 - 30618*T^2 + 321489)^2
$23$	$ (T^{8} - 18 T^{6} + \cdots + 3969)^{2} $ (T^8 - 18T^6 + 261T^4 - 1134*T^2 + 3969)^2
$29$	$ (T^{4} + 76 T^{2} + 1156)^{4} $ (T^4 + 76*T^2 + 1156)^4
$31$	$ (T^{8} + 122 T^{6} + \cdots + 13712209)^{2} $ (T^8 + 122T^6 + 11181T^4 + 451766*T^2 + 13712209)^2
$37$	$ (T^{8} - 54 T^{6} + \cdots + 6561)^{2} $ (T^8 - 54T^6 + 2835T^4 - 4374*T^2 + 6561)^2
$41$	$ (T^{2} + 54)^{8} $ (T^2 + 54)^8
$43$	$ (T^{4} - 144 T^{2} + 4032)^{4} $ (T^4 - 144*T^2 + 4032)^4
$47$	$ (T^{8} + 90 T^{6} + \cdots + 321489)^{2} $ (T^8 + 90T^6 + 7533T^4 + 51030*T^2 + 321489)^2
$53$	$ (T^{8} - 22 T^{6} + \cdots + 2401)^{2} $ (T^8 - 22T^6 + 435T^4 - 1078*T^2 + 2401)^2
$59$	$ (T^{8} - 10 T^{6} + \cdots + 49)^{2} $ (T^8 - 10T^6 + 93T^4 - 70*T^2 + 49)^2
$61$	$ (T^{8} + 162 T^{6} + \cdots + 531441)^{2} $ (T^8 + 162T^6 + 25515T^4 + 118098*T^2 + 531441)^2
$67$	$ (T^{8} + 126 T^{6} + \cdots + 9529569)^{2} $ (T^8 + 126T^6 + 12789T^4 + 388962*T^2 + 9529569)^2
$71$	$ (T^{4} + 288 T^{2} + 16128)^{4} $ (T^4 + 288*T^2 + 16128)^4
$73$	$ (T^{4} + 30 T^{3} + \cdots + 2601)^{4} $ (T^4 + 30T^3 + 351T^2 + 1530*T + 2601)^4
$79$	$ (T^{8} - 114 T^{6} + \cdots + 9529569)^{2} $ (T^8 - 114T^6 + 9909T^4 - 351918*T^2 + 9529569)^2
$83$	$ (T^{4} + 160 T^{2} + 1792)^{4} $ (T^4 + 160*T^2 + 1792)^4
$89$	$ (T^{4} - 6 T^{3} + \cdots + 8649)^{4} $ (T^4 - 6T^3 - 81T^2 + 558*T + 8649)^4
$97$	$ (T^{4} + 36 T^{2} + 36)^{4} $ (T^4 + 36*T^2 + 36)^4
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.q (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{3} + \beta_{9} q^{5} - \beta_{14} q^{7} + (\beta_{13} + 2 \beta_{7} + 2 \beta_{6}) q^{9}+O(q^{10}) \) q + b5 * q^3 + b9 * q^5 - b14 * q^7 + (b13 + 2b7 + 2b6) * q^9	\( q + \beta_{5} q^{3} + \beta_{9} q^{5} - \beta_{14} q^{7} + (\beta_{13} + 2 \beta_{7} + 2 \beta_{6}) q^{9} + (\beta_{12} - \beta_{3}) q^{11} + (\beta_{8} - 2 \beta_{2} - \beta_1) q^{13} + (\beta_{14} + \beta_{11} + \cdots + \beta_{4}) q^{15}+ \cdots + ( - 3 \beta_{15} - 3 \beta_{5} + \beta_{3}) q^{99}+O(q^{100}) \) q + b5 * q^3 + b9 * q^5 - b14 * q^7 + (b13 + 2b7 + 2b6) * q^9 + (b12 - b3) * q^11 + (b8 - 2b2 - b1) q^13 + (b14 + b11 + b10 + b4) * q^15 + (b7 + b6 + 1) * q^17 + (b12 - 2b3) q^19 + (-b9 - b8 + 3b2 + 2b1) * q^21 + (b14 - b10 + b4) * q^23 + (-2b6 + 2) q^25 + (-2b15 + 2b12 + 2b5 - b3) q^27 + (-4b9 - b8 + 2b2 - b1) * q^29 + (-2b14 - 2b11 + b10 - 3b4) q^31 + (2b13 + b6 - 2) q^33 + (b15 + b12 + b5 - b3) * q^35 + (b9 - 3b8 - 2b2) * q^37 + (b14 - b11 - 4b10 - 3b4) * q^39 + (-3b13 - 3b7) * q^41 + (2b15 + 2b5 - 2b3) q^43 + (2b9 + 6b8 - 2b2 + 3b1) * q^45 + (b14 + b11 + 2b10 + 2b4) * q^47 + (-3b13 - b7 - 2b6 - 1) * q^49 + (-2b15 + b12 + 4b5) * q^51 + (b9 + b2 - b1) * q^53 + (2b14 - b11 - 2b10 + b4) * q^55 + (2b13 - 2b7 - 3) * q^57 + b5 * q^59 + (3b9 - 3b8 - 6b1) q^61 + (-2b14 + b11 + 5b10 + b4) * q^63 + (-b13 - 2b7 - 6b6) * q^65 + (-2b15 + 2b12 + b5 - 2b3) q^67 + (2b8 - b2 - 2b1) * q^69 + (4b11 + 4b4) * q^71 + (-2b7 - 5b6 - 5) * q^73 + 2b15 q^75 + (b9 + b8 - 3b2 + 5b1) * q^77 + (b14 + 2b11 + b10 - b4) q^79 + (2b13 + b7 + 7b6 - 7) * q^81 + (4b15 - 4b5) * q^83 + (2b9 + 3b8 - b2 + 3b1) q^85 + (-5b14 - 5b11 - 2b10 - 3b4) * q^87 + (-4b13 - b6 + 2) q^89 + (-5b15 + b5 - b3) q^91 + (-3b9 - 11b8 + 6b2) q^93 + (3b14 - 3b11 - 3b10) q^95 + (-b13 - b7 - 4b6 + 2) q^97 + (-3b15 - 3b5 + b3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^9	\( 16 q + 16 q^{9} + 24 q^{17} + 16 q^{25} - 24 q^{33} - 32 q^{49} - 48 q^{57} - 48 q^{65} - 120 q^{73} - 56 q^{81} + 24 q^{89}+O(q^{100}) \) 16 * q + 16 * q^9 + 24 * q^17 + 16 * q^25 - 24 * q^33 - 32 * q^49 - 48 * q^57 - 48 * q^65 - 120 * q^73 - 56 * q^81 + 24 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	896.2.q.c

Level	$896$
Weight	$2$
Character orbit	896.q
Analytic conductor	$7.155$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(7.15459602111\)
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:	\( x^{16} - 4 x^{15} + 4 x^{14} - 24 x^{13} + 104 x^{12} - 196 x^{11} + 312 x^{10} - 236 x^{9} + 31 x^{8} + \cdots + 1 \) x^16 - 4x^15 + 4x^14 - 24x^13 + 104x^12 - 196x^11 + 312x^10 - 236x^9 + 31x^8 + 236x^7 + 312x^6 + 196x^5 + 104x^4 + 24x^3 + 4x^2 + 4*x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 24578404 \nu^{15} + 112409521 \nu^{14} - 161766434 \nu^{13} + 676914588 \nu^{12} + \cdots - 80530744 ) / 55881261 \) (-24578404v^15 + 112409521v^14 - 161766434v^13 + 676914588v^12 - 2932745926v^11 + 6464372519v^10 - 11224587756v^9 + 11842175116v^8 - 6808155680v^7 - 2875070153v^6 - 5236873676v^5 - 1970820116v^4 - 1399477502v^3 + 116148784v^2 - 266195456*v - 80530744) / 55881261
\(\beta_{2}\)	\(=\)	\( ( 478 \nu^{15} - 1916 \nu^{14} + 1922 \nu^{13} - 11453 \nu^{12} + 49716 \nu^{11} - 93806 \nu^{10} + \cdots + 1412 ) / 969 \) (478v^15 - 1916v^14 + 1922v^13 - 11453v^12 + 49716v^11 - 93806v^10 + 148878v^9 - 111096v^8 + 8892v^7 + 123734v^6 + 134994v^5 + 101866v^4 + 45050v^3 + 8050v^2 - 572*v + 1412) / 969
\(\beta_{3}\)	\(=\)	\( ( - 713760 \nu^{15} + 2873130 \nu^{14} - 2934707 \nu^{13} + 17216192 \nu^{12} - 74582450 \nu^{11} + \cdots - 2172105 ) / 1095711 \) (-713760v^15 + 2873130v^14 - 2934707v^13 + 17216192v^12 - 74582450v^11 + 141664463v^10 - 226302900v^9 + 172573690v^8 - 19779888v^7 - 183883199v^6 - 193251327v^5 - 157627568v^4 - 67531547v^3 - 10955821v^2 + 433452*v - 2172105) / 1095711
\(\beta_{4}\)	\(=\)	\( ( 41571440 \nu^{15} - 185423177 \nu^{14} + 254219263 \nu^{13} - 1125645507 \nu^{12} + \cdots + 5669660 ) / 55881261 \) (41571440v^15 - 185423177v^14 + 254219263v^13 - 1125645507v^12 + 4853270456v^11 - 10441949317v^10 + 18054484920v^9 - 18663563720v^8 + 10686848023v^7 + 4323827851v^6 + 10838029165v^5 + 4051741093v^4 + 3030984061v^3 - 282979166v^2 + 21421645*v + 5669660) / 55881261
\(\beta_{5}\)	\(=\)	\( ( 16331260 \nu^{15} - 74991976 \nu^{14} + 108507436 \nu^{13} - 450127912 \nu^{12} + 1954868524 \nu^{11} + \cdots + 53802093 ) / 18627087 \) (16331260v^15 - 74991976v^14 + 108507436v^13 - 450127912v^12 + 1954868524v^11 - 4322761180v^10 + 7497148468v^9 - 7919869405v^8 + 4541015683v^7 + 1915043380v^6 + 3495942116v^5 + 1025333980v^4 + 934847368v^3 - 77706604v^2 + 177822799*v + 53802093) / 18627087
\(\beta_{6}\)	\(=\)	\( ( - 16361188 \nu^{15} + 72139004 \nu^{14} - 94401360 \nu^{13} + 428892200 \nu^{12} + \cdots - 32692448 ) / 18627087 \) (-16361188v^15 + 72139004v^14 - 94401360v^13 + 428892200v^12 - 1874038268v^11 + 3958996896v^10 - 6661892204v^9 + 6457440527v^8 - 2929890732v^7 - 2870483616v^6 - 3833618044v^5 - 1548767784v^4 - 916399440v^3 + 54074084v^2 - 25648164*v - 32692448) / 18627087
\(\beta_{7}\)	\(=\)	\( ( - 16819000 \nu^{15} + 72297973 \nu^{14} - 89870960 \nu^{13} + 435551452 \nu^{12} + \cdots - 19377740 ) / 18627087 \) (-16819000v^15 + 72297973v^14 - 89870960v^13 + 435551452v^12 - 1887492644v^11 + 3888935115v^10 - 6539006616v^9 + 6229582792v^8 - 2911659902v^7 - 2510098475v^6 - 4825415772v^5 - 2001171996v^4 - 1263419608v^3 + 101493346v^2 - 27011014*v - 19377740) / 18627087
\(\beta_{8}\)	\(=\)	\( ( - 57962620 \nu^{15} + 244034719 \nu^{14} - 284439440 \nu^{13} + 1456967568 \nu^{12} + \cdots - 168001120 ) / 55881261 \) (-57962620v^15 + 244034719v^14 - 284439440v^13 + 1456967568v^12 - 6343717582v^11 + 12731116736v^10 - 20919252180v^9 + 18436863304v^8 - 6312521300v^7 - 11645397440v^6 - 16068699614v^5 - 8122374800v^4 - 4493625488v^3 - 627232847v^2 - 154199804*v - 168001120) / 55881261
\(\beta_{9}\)	\(=\)	\( ( - 23022602 \nu^{15} + 100944114 \nu^{14} - 130859040 \nu^{13} + 602140523 \nu^{12} + \cdots - 73290672 ) / 18627087 \) (-23022602v^15 + 100944114v^14 - 130859040v^13 + 602140523v^12 - 2623735034v^11 + 5518742544v^10 - 9288708034v^9 + 8944701020v^8 - 4051325654v^7 - 4009850208v^6 - 5588137678v^5 - 2281853173v^4 - 1667581296v^3 - 87881958v^2 - 141488302*v - 73290672) / 18627087
\(\beta_{10}\)	\(=\)	\( ( 81436312 \nu^{15} - 349229857 \nu^{14} + 425779916 \nu^{13} - 2076724695 \nu^{12} + \cdots + 120487072 ) / 55881261 \) (81436312v^15 - 349229857v^14 + 425779916v^13 - 2076724695v^12 + 9074758729v^11 - 18573123677v^10 + 30748771548v^9 - 28193204368v^8 + 10926609317v^7 + 15431269295v^6 + 21654724265v^5 + 9289838258v^4 + 5192070428v^3 - 313967458v^2 + 174107807*v + 120487072) / 55881261
\(\beta_{11}\)	\(=\)	\( ( 81657397 \nu^{15} - 361433545 \nu^{14} + 473165969 \nu^{13} - 2128224750 \nu^{12} + \cdots + 304152748 ) / 55881261 \) (81657397v^15 - 361433545v^14 + 473165969v^13 - 2128224750v^12 + 9356454580v^11 - 19798644830v^10 + 33056593947v^9 - 31549953604v^8 + 12982980755v^7 + 16563936710v^6 + 17306567999v^5 + 7099394417v^4 + 3743469443v^3 - 126506317v^2 + 151097009*v + 304152748) / 55881261
\(\beta_{12}\)	\(=\)	\( ( 28053456 \nu^{15} - 123097333 \nu^{14} + 159579226 \nu^{13} - 732793144 \nu^{12} + \cdots + 90494094 ) / 18627087 \) (28053456v^15 - 123097333v^14 + 159579226v^13 - 732793144v^12 + 3197206163v^11 - 6727519330v^10 + 11304474312v^9 - 10850615079v^8 + 4828872624v^7 + 5010528130v^6 + 6755327387v^5 + 2636843224v^4 + 2078302474v^3 + 108463237v^2 + 169601736*v + 90494094) / 18627087
\(\beta_{13}\)	\(=\)	\( ( - 29904546 \nu^{15} + 129757565 \nu^{14} - 161347800 \nu^{13} + 762815256 \nu^{12} + \cdots - 107903336 ) / 18627087 \) (-29904546v^15 + 129757565v^14 - 161347800v^13 + 762815256v^12 - 3357607152v^11 + 6943520122v^10 - 11435934986v^9 + 10437065588v^8 - 3662556090v^7 - 6508066522v^6 - 6938376848v^5 - 2996702536v^4 - 1496150424v^3 + 50381843v^2 - 69368690*v - 107903336) / 18627087
\(\beta_{14}\)	\(=\)	\( ( 95630831 \nu^{15} - 411561677 \nu^{14} + 503317981 \nu^{13} - 2431707663 \nu^{12} + \cdots + 371054036 ) / 55881261 \) (95630831v^15 - 411561677v^14 + 503317981v^13 - 2431707663v^12 + 10669553489v^11 - 21886755922v^10 + 36060331821v^9 - 32758079528v^8 + 11742438997v^7 + 19770274810v^6 + 23948074300v^5 + 10370368018v^4 + 5444666191v^3 - 258493073v^2 + 220080667*v + 371054036) / 55881261
\(\beta_{15}\)	\(=\)	\( ( - 41682527 \nu^{15} + 175603960 \nu^{14} - 205133396 \nu^{13} + 1048417004 \nu^{12} + \cdots - 123561984 ) / 18627087 \) (-41682527v^15 + 175603960v^14 - 205133396v^13 + 1048417004v^12 - 4563459364v^11 + 9167107496v^10 - 15069743507v^9 + 13276126963v^8 - 4498087181v^7 - 8537548760v^6 - 11303294252v^5 - 5984840948v^4 - 3300359084v^3 - 432772592v^2 - 114281729*v - 123561984) / 18627087

\(\nu\)	\(=\)	\( ( -\beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - 3\beta_{6} + 3\beta_{5} - 2\beta_{4} + \beta_{2} + 3\beta _1 + 3 ) / 6 \) (-b14 - b11 + b10 + b9 - 3b6 + 3b5 - 2b4 + b2 + 3b1 + 3) / 6
\(\nu^{2}\)	\(=\)	\( ( 6 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} + 2 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} + \cdots + 4 \beta_{2} ) / 6 \) (6b15 + 2b14 + 3b13 + 2b11 + 4b10 - 2b9 - 9b8 + 6b7 + 6b6 + 4b4 + 4*b2) / 6
\(\nu^{3}\)	\(=\)	\( ( - 6 \beta_{15} - 11 \beta_{14} - 9 \beta_{13} + 6 \beta_{12} - 5 \beta_{11} + 11 \beta_{10} + 14 \beta_{9} + \cdots + 33 ) / 6 \) (-6b15 - 11b14 - 9b13 + 6b12 - 5b11 + 11b10 + 14b9 + 6b8 + 9b7 + 6b5 + 5b4 - 3b3 - 7b2 + 6b1 + 33) / 6
\(\nu^{4}\)	\(=\)	\( 2\beta_{12} + 7\beta_{9} + 10\beta_{5} + 2\beta_{3} + 7\beta_{2} + 12\beta_1 \) 2b12 + 7b9 + 10b5 + 2b3 + 7b2 + 12b1
\(\nu^{5}\)	\(=\)	\( ( 66 \beta_{15} + 37 \beta_{14} + 75 \beta_{13} - 15 \beta_{12} + 103 \beta_{11} + 140 \beta_{10} + \cdots + 98 \beta_{2} ) / 6 \) (66b15 + 37b14 + 75b13 - 15b12 + 103b11 + 140b10 - 49b9 - 78b8 + 150b7 + 303b6 + 140b4 + 30b3 + 98*b2) / 6
\(\nu^{6}\)	\(=\)	\( ( - 402 \beta_{15} - 230 \beta_{14} - 171 \beta_{13} + 96 \beta_{12} - 86 \beta_{11} + 230 \beta_{10} + \cdots + 678 ) / 6 \) (-402b15 - 230b14 - 171b13 + 96b12 - 86b11 + 230b10 + 452b9 + 513b8 + 171b7 + 402b5 + 86b4 - 48b3 - 226b2 + 513b1 + 678) / 6
\(\nu^{7}\)	\(=\)	\( ( 509 \beta_{14} + 504 \beta_{13} + 204 \beta_{12} + 509 \beta_{11} + 103 \beta_{10} + 895 \beta_{9} + \cdots - 1173 ) / 6 \) (509b14 + 504b13 + 204b12 + 509b11 + 103b10 + 895b9 + 252b7 + 1173b6 + 1527b5 + 406b4 + 204b3 + 895b2 + 1929*b1 - 1173) / 6
\(\nu^{8}\)	\(=\)	\( 152\beta_{14} + 336\beta_{13} + 500\beta_{11} + 652\beta_{10} + 672\beta_{7} + 1457\beta_{6} + 652\beta_{4} \) 152b14 + 336b13 + 500b11 + 652b10 + 672b7 + 1457b6 + 652*b4
\(\nu^{9}\)	\(=\)	\( ( - 13263 \beta_{15} - 3406 \beta_{14} - 2268 \beta_{13} + 3624 \beta_{12} - 1015 \beta_{11} + \cdots + 9915 ) / 6 \) (-13263b15 - 3406b14 - 2268b13 + 3624b12 - 1015b11 + 3406b10 + 15682b9 + 16719b8 + 2268b7 + 13263b5 + 1015b4 - 1812b3 - 7841b2 + 16719b1 + 9915) / 6
\(\nu^{10}\)	\(=\)	\( ( 24068 \beta_{14} + 25206 \beta_{13} + 4080 \beta_{12} + 24068 \beta_{11} + 5914 \beta_{10} + \cdots - 53094 ) / 6 \) (24068b14 + 25206b13 + 4080b12 + 24068b11 + 5914b10 + 17698b9 + 12603b7 + 53094b6 + 29982b5 + 18154b4 + 4080b3 + 17698b2 + 37809*b1 - 53094) / 6
\(\nu^{11}\)	\(=\)	\( ( - 47658 \beta_{15} + 22597 \beta_{14} + 48477 \beta_{13} + 6711 \beta_{12} + 70255 \beta_{11} + \cdots - 57022 \beta_{2} ) / 6 \) (-47658b15 + 22597b14 + 48477b13 + 6711b12 + 70255b11 + 92852b10 + 28511b9 + 59898b8 + 96954b7 + 205329b6 + 92852b4 - 13422b3 - 57022*b2) / 6
\(\nu^{12}\)	\(=\)	\( - 61406 \beta_{15} + 17036 \beta_{12} + 73038 \beta_{9} + 77292 \beta_{8} + 61406 \beta_{5} + \cdots + 77292 \beta_1 \) -61406b15 + 17036b12 + 73038b9 + 77292b8 + 61406b5 - 8518b3 - 36519b2 + 77292b1
\(\nu^{13}\)	\(=\)	\( ( 809128 \beta_{14} + 844350 \beta_{13} + 57819 \beta_{12} + 809128 \beta_{11} + 196517 \beta_{10} + \cdots - 1790175 ) / 6 \) (809128b14 + 844350b13 + 57819b12 + 809128b11 + 196517b10 + 247625b9 + 422175b7 + 1790175b6 + 416094b5 + 612611b4 + 57819b3 + 247625b2 + 523650*b1 - 1790175) / 6
\(\nu^{14}\)	\(=\)	\( ( - 2272170 \beta_{15} + 444142 \beta_{14} + 953907 \beta_{13} + 313248 \beta_{12} + \cdots - 2696068 \beta_{2} ) / 6 \) (-2272170b15 + 444142b14 + 953907b13 + 313248b12 + 1383886b11 + 1828028b10 + 1348034b9 + 2861721b8 + 1907814b7 + 4044102b6 + 1828028b4 - 626496b3 - 2696068*b2) / 6
\(\nu^{15}\)	\(=\)	\( ( - 8761899 \beta_{15} + 2213258 \beta_{14} + 1522584 \beta_{13} + 2418672 \beta_{12} + 707375 \beta_{11} + \cdots - 6466317 ) / 6 \) (-8761899b15 + 2213258b14 + 1522584b13 + 2418672b12 + 707375b11 - 2213258b10 + 10401214b9 + 11034069b8 - 1522584b7 + 8761899b5 - 707375b4 - 1209336b3 - 5200607b2 + 11034069b1 - 6466317) / 6

\(n\)	\(127\)	\(129\)	\(645\)
\(\chi(n)\)	\(-1\)	\(\beta_{6}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 703.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} - 10 T^{6} + \cdots + 49)^{2} \) (T^8 - 10T^6 + 93T^4 - 70*T^2 + 49)^2
$5$	\( (T^{4} + 3 T^{2} + 9)^{4} \) (T^4 + 3*T^2 + 9)^4
$7$	\( (T^{8} + 8 T^{6} + \cdots + 2401)^{2} \) (T^8 + 8T^6 + 42T^4 + 392*T^2 + 2401)^2
$11$	\( (T^{8} + 18 T^{6} + \cdots + 3969)^{2} \) (T^8 + 18T^6 + 261T^4 + 1134*T^2 + 3969)^2
$13$	\( (T^{4} - 36 T^{2} + 36)^{4} \) (T^4 - 36*T^2 + 36)^4
$17$	\( (T^{4} - 6 T^{3} + 9 T^{2} + \cdots + 9)^{4} \) (T^4 - 6T^3 + 9T^2 + 18*T + 9)^4
$19$	\( (T^{8} - 54 T^{6} + \cdots + 321489)^{2} \) (T^8 - 54T^6 + 2349T^4 - 30618*T^2 + 321489)^2
$23$	\( (T^{8} - 18 T^{6} + \cdots + 3969)^{2} \) (T^8 - 18T^6 + 261T^4 - 1134*T^2 + 3969)^2
$29$	\( (T^{4} + 76 T^{2} + 1156)^{4} \) (T^4 + 76*T^2 + 1156)^4
$31$	\( (T^{8} + 122 T^{6} + \cdots + 13712209)^{2} \) (T^8 + 122T^6 + 11181T^4 + 451766*T^2 + 13712209)^2
$37$	\( (T^{8} - 54 T^{6} + \cdots + 6561)^{2} \) (T^8 - 54T^6 + 2835T^4 - 4374*T^2 + 6561)^2
$41$	\( (T^{2} + 54)^{8} \) (T^2 + 54)^8
$43$	\( (T^{4} - 144 T^{2} + 4032)^{4} \) (T^4 - 144*T^2 + 4032)^4
$47$	\( (T^{8} + 90 T^{6} + \cdots + 321489)^{2} \) (T^8 + 90T^6 + 7533T^4 + 51030*T^2 + 321489)^2
$53$	\( (T^{8} - 22 T^{6} + \cdots + 2401)^{2} \) (T^8 - 22T^6 + 435T^4 - 1078*T^2 + 2401)^2
$59$	\( (T^{8} - 10 T^{6} + \cdots + 49)^{2} \) (T^8 - 10T^6 + 93T^4 - 70*T^2 + 49)^2
$61$	\( (T^{8} + 162 T^{6} + \cdots + 531441)^{2} \) (T^8 + 162T^6 + 25515T^4 + 118098*T^2 + 531441)^2
$67$	\( (T^{8} + 126 T^{6} + \cdots + 9529569)^{2} \) (T^8 + 126T^6 + 12789T^4 + 388962*T^2 + 9529569)^2
$71$	\( (T^{4} + 288 T^{2} + 16128)^{4} \) (T^4 + 288*T^2 + 16128)^4
$73$	\( (T^{4} + 30 T^{3} + \cdots + 2601)^{4} \) (T^4 + 30T^3 + 351T^2 + 1530*T + 2601)^4
$79$	\( (T^{8} - 114 T^{6} + \cdots + 9529569)^{2} \) (T^8 - 114T^6 + 9909T^4 - 351918*T^2 + 9529569)^2
$83$	\( (T^{4} + 160 T^{2} + 1792)^{4} \) (T^4 + 160*T^2 + 1792)^4
$89$	\( (T^{4} - 6 T^{3} + \cdots + 8649)^{4} \) (T^4 - 6T^3 - 81T^2 + 558*T + 8649)^4
$97$	\( (T^{4} + 36 T^{2} + 36)^{4} \) (T^4 + 36*T^2 + 36)^4
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