LMFDB - Newform orbit 1170.2.bj.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(199,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("1170.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.bj (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:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197 $ x^12 - 2x^11 - 8x^10 + 34x^9 + 8x^8 - 134x^7 + 98x^6 + 154x^5 + 104x^4 + 190x^3 - 1196x^2 - 338*x + 2197
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 390)
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197 $ x^12 - 2*x^11 - 8*x^10 + 34*x^9 + 8*x^8 - 134*x^7 + 98*x^6 + 154*x^5 + 104*x^4 + 190*x^3 - 1196*x^2 - 338*x + 2197 :

$\beta_{1}$	$=$	$ ( 203419 \nu^{11} + 163110633 \nu^{10} - 591783880 \nu^{9} - 97338749 \nu^{8} + \cdots - 81183629852 ) / 63907274600 $ (203419v^11 + 163110633v^10 - 591783880v^9 - 97338749v^8 + 4513282461v^7 - 6489146722v^6 - 4655211661v^5 + 5740233327v^4 + 8165110694v^3 + 33307875431v^2 - 128185173545*v - 81183629852) / 63907274600
$\beta_{2}$	$=$	$ ( 99774878 \nu^{11} - 1446867854 \nu^{10} + 1099642740 \nu^{9} + 2834573637 \nu^{8} + \cdots - 399707750949 ) / 415397284900 $ (99774878v^11 - 1446867854v^10 + 1099642740v^9 + 2834573637v^8 - 11046550568v^7 + 1368363861v^6 - 68535056482v^5 + 38321129974v^4 + 51767136278v^3 - 345701461353v^2 - 110551992590*v - 399707750949) / 415397284900
$\beta_{3}$	$=$	$ ( 120243408 \nu^{11} - 454030419 \nu^{10} - 2051209685 \nu^{9} + 7036618932 \nu^{8} + \cdots - 618192439839 ) / 415397284900 $ (120243408v^11 - 454030419v^10 - 2051209685v^9 + 7036618932v^8 - 2513656023v^7 - 29967292429v^6 + 18436259198v^5 - 35262141211v^4 - 8309942667v^3 - 28687118858v^2 - 98318215515*v - 618192439839) / 415397284900
$\beta_{4}$	$=$	$ ( 4036 \nu^{11} - 37023 \nu^{10} + 57555 \nu^{9} + 233294 \nu^{8} - 817691 \nu^{7} + \cdots + 11867687 ) / 11796200 $ (4036v^11 - 37023v^10 + 57555v^9 + 233294v^8 - 817691v^7 + 35557v^6 + 1844066v^5 - 940887v^4 + 77961v^3 - 7481036v^2 - 4439955*v + 11867687) / 11796200
$\beta_{5}$	$=$	$ ( 164152294 \nu^{11} + 1358027358 \nu^{10} - 5882189980 \nu^{9} + 1843354751 \nu^{8} + \cdots - 1308690012227 ) / 415397284900 $ (164152294v^11 + 1358027358v^10 - 5882189980v^9 + 1843354751v^8 + 45609137136v^7 - 49422223697v^6 - 70650734586v^5 + 94378041302v^4 + 204958669994v^3 + 189862006481v^2 - 68215142970*v - 1308690012227) / 415397284900
$\beta_{6}$	$=$	$ ( 480376508 \nu^{11} - 958108569 \nu^{10} - 1722573835 \nu^{9} + 8639610832 \nu^{8} + \cdots - 997979945989 ) / 415397284900 $ (480376508v^11 - 958108569v^10 - 1722573835v^9 + 8639610832v^8 + 2577608327v^7 - 5697780079v^6 - 37282009602v^5 + 13460230639v^4 + 124582190083v^3 + 197417975542v^2 - 141527922965*v - 997979945989) / 415397284900
$\beta_{7}$	$=$	$ ( - 23159569 \nu^{11} + 44665642 \nu^{10} + 118225633 \nu^{9} - 603932751 \nu^{8} + \cdots + 21026586399 ) / 16615891396 $ (-23159569v^11 + 44665642v^10 + 118225633v^9 - 603932751v^8 - 6720278v^7 + 1195412299v^6 - 1392260235v^5 - 486856544v^4 - 1535131351v^3 - 7949637689v^2 - 2866001996*v + 21026586399) / 16615891396
$\beta_{8}$	$=$	$ ( - 236542471 \nu^{11} + 346656978 \nu^{10} + 1242595595 \nu^{9} - 6207967149 \nu^{8} + \cdots + 460186493 ) / 166158913960 $ (-236542471v^11 + 346656978v^10 + 1242595595v^9 - 6207967149v^8 + 2310806556v^7 + 6108999933v^6 - 30747713501v^5 + 45266458632v^4 - 73287704061v^3 - 120820074499v^2 - 57773591330*v + 460186493) / 166158913960
$\beta_{9}$	$=$	$ ( - 632187142 \nu^{11} + 553303131 \nu^{10} + 5382274615 \nu^{9} - 11841029268 \nu^{8} + \cdots + 880210105411 ) / 415397284900 $ (-632187142v^11 + 553303131v^10 + 5382274615v^9 - 11841029268v^8 - 18449268373v^7 + 45918505121v^6 + 41904457048v^5 - 52841609661v^4 - 129070461767v^3 - 49363987858v^2 + 441927290935*v + 880210105411) / 415397284900
$\beta_{10}$	$=$	$ ( 320232543 \nu^{11} - 1004615314 \nu^{10} - 1143072025 \nu^{9} + 9975044447 \nu^{8} + \cdots - 144016360059 ) / 166158913960 $ (320232543v^11 - 1004615314v^10 - 1143072025v^9 + 9975044447v^8 - 4093093078v^7 - 25465804119v^6 + 21841076813v^5 + 35457807644v^4 + 27228013423v^3 + 100602839757v^2 - 286784071080*v - 144016360059) / 166158913960
$\beta_{11}$	$=$	$ ( - 1027654074 \nu^{11} + 2460612632 \nu^{10} + 3624161555 \nu^{9} - 25210679196 \nu^{8} + \cdots + 27977691092 ) / 415397284900 $ (-1027654074v^11 + 2460612632v^10 + 3624161555v^9 - 25210679196v^8 + 10960029469v^7 + 30148216612v^6 - 9821523994v^5 - 23732163042v^4 - 402415418899v^3 + 37700985074v^2 + 390356046795*v + 27977691092) / 415397284900

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

$\nu$	$=$	$ ( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + 2\beta_{7} + \beta_{5} - 2\beta _1 + 1 ) / 2 $ (b11 + b10 - b9 - b8 + 2b7 + b5 - 2b1 + 1) / 2
$\nu^{2}$	$=$	$ \beta_{10} + 2\beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 1 $ b10 + 2*b7 + b6 - b4 + b3 - b2 - b1 + 1
$\nu^{3}$	$=$	$ ( 4 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} + 10 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \cdots - 8 ) / 2 $ (4b10 - 2b9 - 4b8 + 10b7 - 2b6 + 2b5 - 2b4 - b3 - 4b1 - 8) / 2
$\nu^{4}$	$=$	$ - 7 \beta_{11} + 2 \beta_{9} + 7 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 7 \beta_{4} + \cdots - 11 $ -7b11 + 2b9 + 7b8 - 2b7 - 3b6 - 3b5 - 7b4 + 4b3 - 5*b2 - b1 - 11
$\nu^{5}$	$=$	$ ( - 6 \beta_{11} + 2 \beta_{10} - \beta_{9} - 10 \beta_{8} - 12 \beta_{7} - 34 \beta_{6} - 9 \beta_{5} + \cdots - 82 ) / 2 $ (-6b11 + 2b10 - b9 - 10b8 - 12b7 - 34b6 - 9b5 - 16b4 - 19b3 + 11b2 + 8b1 - 82) / 2
$\nu^{6}$	$=$	$ - 46 \beta_{11} - 40 \beta_{10} + 46 \beta_{9} + 42 \beta_{8} - 90 \beta_{7} - 13 \beta_{6} - 30 \beta_{5} + \cdots - 39 $ -46b11 - 40b10 + 46b9 + 42b8 - 90b7 - 13b6 - 30b5 - 26b4 + 13b3 - 2b2 + 46*b1 - 39
$\nu^{7}$	$=$	$ ( 33 \beta_{11} - 67 \beta_{10} + 79 \beta_{9} - 57 \beta_{8} - 234 \beta_{7} - 76 \beta_{6} + 5 \beta_{5} + \cdots - 51 ) / 2 $ (33b11 - 67b10 + 79b9 - 57b8 - 234b7 - 76b6 + 5b5 + 52b4 - 156b3 + 152b2 + 146*b1 - 51) / 2
$\nu^{8}$	$=$	$ - 93 \beta_{11} - 174 \beta_{10} + 231 \beta_{9} + 181 \beta_{8} - 364 \beta_{7} + 130 \beta_{6} + \cdots + 362 $ -93b11 - 174b10 + 231b9 + 181b8 - 364b7 + 130b6 - 3b5 + 107b4 + 91b3 - 23b2 + 179*b1 + 362
$\nu^{9}$	$=$	$ ( 1124 \beta_{11} + 316 \beta_{10} - 358 \beta_{9} - 776 \beta_{8} + 646 \beta_{7} + 682 \beta_{6} + \cdots + 1700 ) / 2 $ (1124b11 + 316b10 - 358b9 - 776b8 + 646b7 + 682b6 + 798b5 + 1378b4 - 861b3 + 404b2 - 248*b1 + 1700) / 2
$\nu^{10}$	$=$	$ 246 \beta_{11} - 81 \beta_{10} - 196 \beta_{9} + 482 \beta_{8} + 630 \beta_{7} + 1314 \beta_{6} + \cdots + 2556 $ 246b11 - 81b10 - 196b9 + 482b8 + 630b7 + 1314b6 + 501b5 + 1079b4 + 580b3 - 926b2 - 469*b1 + 2556
$\nu^{11}$	$=$	$ ( 7310 \beta_{11} + 5070 \beta_{10} - 9673 \beta_{9} - 6494 \beta_{8} + 12576 \beta_{7} + 2066 \beta_{6} + \cdots + 2574 ) / 2 $ (7310b11 + 5070b10 - 9673b9 - 6494b8 + 12576b7 + 2066b6 + 3475b5 + 4992b4 - 5163b3 - 325b2 - 6944*b1 + 2574) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$-1$	$-\beta_{4}$

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} $

199.1

1.40719	+	0.536449i
2.00607	+	1.30680i
−1.44229	−	0.433312i
−0.330925	−	1.46916i
1.75374	−	1.62986i
−2.39378	−	0.0429626i
1.40719	−	0.536449i
2.00607	−	1.30680i
−1.44229	+	0.433312i
−0.330925	+	1.46916i
1.75374	+	1.62986i
−2.39378	+	0.0429626i

0.500000

−

0.866025i

−0.500000

−

0.866025i

−2.03420

−

0.928463i

1.40247

2.42916i

−1.00000

−1.82117

1.29743i

199.2

0.500000

−

0.866025i

−0.500000

−

0.866025i

−1.26873

−

1.84128i

−2.17283

−

3.76344i

−1.00000

−2.22896

0.178114i

199.3

0.500000

−

0.866025i

−0.500000

−

0.866025i

0.230377

2.22417i

0.432713

0.749482i

−1.00000

2.04138

0.912572i

199.4

0.500000

−

0.866025i

−0.500000

−

0.866025i

0.571769

−

2.16173i

−0.603137

−

1.04466i

−1.00000

−1.58623

−

1.57603i

199.5

0.500000

−

0.866025i

−0.500000

−

0.866025i

1.40066

1.74303i

0.763837

1.32301i

−1.00000

2.20984

−

0.341491i

199.6

0.500000

−

0.866025i

−0.500000

−

0.866025i

2.10012

−

0.767774i

−0.823063

−

1.42559i

−1.00000

0.385150

−

2.20265i

829.1

0.500000

0.866025i

−0.500000

0.866025i

−2.03420

0.928463i

1.40247

−

2.42916i

−1.00000

−1.82117

−

1.29743i

829.2

0.500000

0.866025i

−0.500000

0.866025i

−1.26873

1.84128i

−2.17283

3.76344i

−1.00000

−2.22896

−

0.178114i

829.3

0.500000

0.866025i

−0.500000

0.866025i

0.230377

−

2.22417i

0.432713

−

0.749482i

−1.00000

2.04138

−

0.912572i

829.4

0.500000

0.866025i

−0.500000

0.866025i

0.571769

2.16173i

−0.603137

1.04466i

−1.00000

−1.58623

1.57603i

829.5

0.500000

0.866025i

−0.500000

0.866025i

1.40066

−

1.74303i

0.763837

−

1.32301i

−1.00000

2.20984

0.341491i

829.6

0.500000

0.866025i

−0.500000

0.866025i

2.10012

0.767774i

−0.823063

1.42559i

−1.00000

0.385150

2.20265i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
65.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1170.2.bj.d		12
3.b	odd	2	1		390.2.x.a	✓	12
5.b	even	2	1		1170.2.bj.c		12
13.e	even	6	1		1170.2.bj.c		12
15.d	odd	2	1		390.2.x.b	yes	12
15.e	even	4	1		1950.2.bc.i		12
15.e	even	4	1		1950.2.bc.j		12
39.h	odd	6	1		390.2.x.b	yes	12
65.l	even	6	1	inner	1170.2.bj.d		12
195.y	odd	6	1		390.2.x.a	✓	12
195.bf	even	12	1		1950.2.bc.i		12
195.bf	even	12	1		1950.2.bc.j		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
390.2.x.a	✓	12	3.b	odd	2	1
390.2.x.a	✓	12	195.y	odd	6	1
390.2.x.b	yes	12	15.d	odd	2	1
390.2.x.b	yes	12	39.h	odd	6	1
1170.2.bj.c		12	5.b	even	2	1
1170.2.bj.c		12	13.e	even	6	1
1170.2.bj.d		12	1.a	even	1	1	trivial
1170.2.bj.d		12	65.l	even	6	1	inner
1950.2.bc.i		12	15.e	even	4	1
1950.2.bc.i		12	195.bf	even	12	1
1950.2.bc.j		12	15.e	even	4	1
1950.2.bc.j		12	195.bf	even	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1170, [\chi])$:

$ T_{7}^{12} + 2 T_{7}^{11} + 19 T_{7}^{10} - 6 T_{7}^{9} + 205 T_{7}^{8} + 20 T_{7}^{7} + 708 T_{7}^{6} + \cdots + 1024 $

$ T_{17}^{12} + 18 T_{17}^{11} + 95 T_{17}^{10} - 234 T_{17}^{9} - 2695 T_{17}^{8} + 4632 T_{17}^{7} + \cdots + 65536 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - T + 1)^{6} $ (T^2 - T + 1)^6
$3$	$ T^{12} $ T^12
$5$	$ T^{12} - 2 T^{11} + \cdots + 15625 $ T^12 - 2T^11 + 7T^10 - 6T^9 + 15T^8 - 32T^7 + 114T^6 - 160T^5 + 375T^4 - 750T^3 + 4375T^2 - 6250*T + 15625
$7$	$ T^{12} + 2 T^{11} + \cdots + 1024 $ T^12 + 2T^11 + 19T^10 - 6T^9 + 205T^8 + 20T^7 + 708T^6 - 112T^5 + 1648T^4 - 64T^3 + 1664T^2 - 512*T + 1024
$11$	$ T^{12} + 6 T^{11} + \cdots + 16 $ T^12 + 6T^11 - 2T^10 - 84T^9 + 87T^8 + 444T^7 - 330T^6 - 1446T^5 + 2585T^4 - 1176T^3 - 4T^2 + 96*T + 16
$13$	$ T^{12} + 8 T^{11} + \cdots + 4826809 $ T^12 + 8T^11 + 36T^10 + 40T^9 - 232T^8 - 2040T^7 - 7838T^6 - 26520T^5 - 39208T^4 + 87880T^3 + 1028196T^2 + 2970344*T + 4826809
$17$	$ T^{12} + 18 T^{11} + \cdots + 65536 $ T^12 + 18T^11 + 95T^10 - 234T^9 - 2695T^8 + 4632T^7 + 75744T^6 + 85248T^5 - 300288T^4 - 307200T^3 + 1269760T^2 - 491520*T + 65536
$19$	$ T^{12} + 6 T^{11} + \cdots + 1982464 $ T^12 + 6T^11 - 53T^10 - 390T^9 + 2701T^8 + 14568T^7 - 43876T^6 - 254496T^5 + 672400T^4 + 2581632T^3 + 837120T^2 - 3649536*T + 1982464
$23$	$ T^{12} + \cdots + 190660864 $ T^12 + 6T^11 - 93T^10 - 630T^9 + 7441T^8 + 53748T^7 - 129352T^6 - 1617792T^5 + 1290352T^4 + 44643456T^3 + 171279232T^2 + 284334336*T + 190660864
$29$	$ T^{12} + 14 T^{11} + \cdots + 21904 $ T^12 + 14T^11 + 190T^10 + 960T^9 + 6703T^8 + 15668T^7 + 148926T^6 + 196610T^5 + 1737745T^4 - 1644412T^3 + 11075036T^2 + 490768*T + 21904
$31$	$ T^{12} + \cdots + 177209344 $ T^12 + 198T^10 + 15433T^8 + 592640T^6 + 11339776T^4 + 93061120*T^2 + 177209344
$37$	$ T^{12} + \cdots + 227195329 $ T^12 + 12T^11 + 209T^10 + 1028T^9 + 14658T^8 + 60444T^7 + 734609T^6 + 1319436T^5 + 11678146T^4 + 32084244T^3 + 129327441T^2 + 175510012*T + 227195329
$41$	$ T^{12} - 18 T^{11} + \cdots + 65536 $ T^12 - 18T^11 + 95T^10 + 234T^9 - 2695T^8 - 4632T^7 + 75744T^6 - 85248T^5 - 300288T^4 + 307200T^3 + 1269760T^2 + 491520*T + 65536
$43$	$ T^{12} + \cdots + 349241344 $ T^12 + 36T^11 + 535T^10 + 3708T^9 + 6253T^8 - 57864T^7 + 54524T^6 + 6267168T^5 + 51560464T^4 + 211680000T^3 + 494054400T^2 + 627916800*T + 349241344
$47$	$ (T^{6} + 8 T^{5} + \cdots + 5956)^{2} $ (T^6 + 8T^5 - 98T^4 - 1280T^3 - 4283T^2 - 2352*T + 5956)^2
$53$	$ T^{12} + \cdots + 2473271824 $ T^12 + 508T^10 + 87438T^8 + 5778420T^6 + 127872521T^4 + 989681000*T^2 + 2473271824
$59$	$ T^{12} + \cdots + 4983230464 $ T^12 - 36T^11 + 459T^10 - 972T^9 - 26427T^8 + 77796T^7 + 3235868T^6 - 37281744T^5 + 157498704T^4 - 129534336T^3 - 820266240T^2 + 752228352*T + 4983230464
$61$	$ T^{12} - 10 T^{11} + \cdots + 89718784 $ T^12 - 10T^11 + 135T^10 - 762T^9 + 7561T^8 - 39396T^7 + 256760T^6 - 835296T^5 + 3388992T^4 - 7106048T^3 + 26851328T^2 - 41828352*T + 89718784
$67$	$ T^{12} - 4 T^{11} + \cdots + 83759104 $ T^12 - 4T^11 + 164T^10 + 784T^9 + 18784T^8 + 35776T^7 + 394560T^6 - 320768T^5 + 6100480T^4 - 2632704T^3 + 25142272T^2 + 2928640*T + 83759104
$71$	$ T^{12} - 12 T^{11} + \cdots + 4194304 $ T^12 - 12T^11 - 172T^10 + 2640T^9 + 36080T^8 - 505152T^7 + 1150080T^6 + 6873600T^5 - 703488T^4 - 35782656T^3 + 56295424T^2 - 25165824*T + 4194304
$73$	$ (T^{6} - 14 T^{5} + \cdots - 230528)^{2} $ (T^6 - 14T^5 - 263T^4 + 3068T^3 + 18280T^2 - 114240*T - 230528)^2
$79$	$ (T^{6} - 2 T^{5} + \cdots - 29312)^{2} $ (T^6 - 2T^5 - 179T^4 + 788T^3 + 3892T^2 - 12576*T - 29312)^2
$83$	$ (T^{6} + 36 T^{5} + \cdots - 6912)^{2} $ (T^6 + 36T^5 + 360T^4 - 144T^3 - 16560T^2 - 51840*T - 6912)^2
$89$	$ T^{12} + \cdots + 1341001056256 $ T^12 + 18T^11 - 309T^10 - 7506T^9 + 96433T^8 + 1612140T^7 - 15114712T^6 - 191431776T^5 + 2363129536T^4 + 1837575168T^3 - 61874871296T^2 - 39576354816*T + 1341001056256
$97$	$ T^{12} + \cdots + 415519473664 $ T^12 + 48T^11 + 1640T^10 + 32704T^9 + 522672T^8 + 5437824T^7 + 53039744T^6 + 348175872T^5 + 3300888832T^4 + 14104406016T^3 + 88030961664T^2 - 141504348160*T + 415519473664
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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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.bj (of order \(6\), degree \(2\), minimal)

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

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	1170.2.bj.d

Level	$1170$
Weight	$2$
Character orbit	1170.bj
Analytic conductor	$9.342$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(9.34249703649\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197 \) x^12 - 2x^11 - 8x^10 + 34x^9 + 8x^8 - 134x^7 + 98x^6 + 154x^5 + 104x^4 + 190x^3 - 1196x^2 - 338*x + 2197
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 203419 \nu^{11} + 163110633 \nu^{10} - 591783880 \nu^{9} - 97338749 \nu^{8} + \cdots - 81183629852 ) / 63907274600 \) (203419v^11 + 163110633v^10 - 591783880v^9 - 97338749v^8 + 4513282461v^7 - 6489146722v^6 - 4655211661v^5 + 5740233327v^4 + 8165110694v^3 + 33307875431v^2 - 128185173545*v - 81183629852) / 63907274600
\(\beta_{2}\)	\(=\)	\( ( 99774878 \nu^{11} - 1446867854 \nu^{10} + 1099642740 \nu^{9} + 2834573637 \nu^{8} + \cdots - 399707750949 ) / 415397284900 \) (99774878v^11 - 1446867854v^10 + 1099642740v^9 + 2834573637v^8 - 11046550568v^7 + 1368363861v^6 - 68535056482v^5 + 38321129974v^4 + 51767136278v^3 - 345701461353v^2 - 110551992590*v - 399707750949) / 415397284900
\(\beta_{3}\)	\(=\)	\( ( 120243408 \nu^{11} - 454030419 \nu^{10} - 2051209685 \nu^{9} + 7036618932 \nu^{8} + \cdots - 618192439839 ) / 415397284900 \) (120243408v^11 - 454030419v^10 - 2051209685v^9 + 7036618932v^8 - 2513656023v^7 - 29967292429v^6 + 18436259198v^5 - 35262141211v^4 - 8309942667v^3 - 28687118858v^2 - 98318215515*v - 618192439839) / 415397284900
\(\beta_{4}\)	\(=\)	\( ( 4036 \nu^{11} - 37023 \nu^{10} + 57555 \nu^{9} + 233294 \nu^{8} - 817691 \nu^{7} + \cdots + 11867687 ) / 11796200 \) (4036v^11 - 37023v^10 + 57555v^9 + 233294v^8 - 817691v^7 + 35557v^6 + 1844066v^5 - 940887v^4 + 77961v^3 - 7481036v^2 - 4439955*v + 11867687) / 11796200
\(\beta_{5}\)	\(=\)	\( ( 164152294 \nu^{11} + 1358027358 \nu^{10} - 5882189980 \nu^{9} + 1843354751 \nu^{8} + \cdots - 1308690012227 ) / 415397284900 \) (164152294v^11 + 1358027358v^10 - 5882189980v^9 + 1843354751v^8 + 45609137136v^7 - 49422223697v^6 - 70650734586v^5 + 94378041302v^4 + 204958669994v^3 + 189862006481v^2 - 68215142970*v - 1308690012227) / 415397284900
\(\beta_{6}\)	\(=\)	\( ( 480376508 \nu^{11} - 958108569 \nu^{10} - 1722573835 \nu^{9} + 8639610832 \nu^{8} + \cdots - 997979945989 ) / 415397284900 \) (480376508v^11 - 958108569v^10 - 1722573835v^9 + 8639610832v^8 + 2577608327v^7 - 5697780079v^6 - 37282009602v^5 + 13460230639v^4 + 124582190083v^3 + 197417975542v^2 - 141527922965*v - 997979945989) / 415397284900
\(\beta_{7}\)	\(=\)	\( ( - 23159569 \nu^{11} + 44665642 \nu^{10} + 118225633 \nu^{9} - 603932751 \nu^{8} + \cdots + 21026586399 ) / 16615891396 \) (-23159569v^11 + 44665642v^10 + 118225633v^9 - 603932751v^8 - 6720278v^7 + 1195412299v^6 - 1392260235v^5 - 486856544v^4 - 1535131351v^3 - 7949637689v^2 - 2866001996*v + 21026586399) / 16615891396
\(\beta_{8}\)	\(=\)	\( ( - 236542471 \nu^{11} + 346656978 \nu^{10} + 1242595595 \nu^{9} - 6207967149 \nu^{8} + \cdots + 460186493 ) / 166158913960 \) (-236542471v^11 + 346656978v^10 + 1242595595v^9 - 6207967149v^8 + 2310806556v^7 + 6108999933v^6 - 30747713501v^5 + 45266458632v^4 - 73287704061v^3 - 120820074499v^2 - 57773591330*v + 460186493) / 166158913960
\(\beta_{9}\)	\(=\)	\( ( - 632187142 \nu^{11} + 553303131 \nu^{10} + 5382274615 \nu^{9} - 11841029268 \nu^{8} + \cdots + 880210105411 ) / 415397284900 \) (-632187142v^11 + 553303131v^10 + 5382274615v^9 - 11841029268v^8 - 18449268373v^7 + 45918505121v^6 + 41904457048v^5 - 52841609661v^4 - 129070461767v^3 - 49363987858v^2 + 441927290935*v + 880210105411) / 415397284900
\(\beta_{10}\)	\(=\)	\( ( 320232543 \nu^{11} - 1004615314 \nu^{10} - 1143072025 \nu^{9} + 9975044447 \nu^{8} + \cdots - 144016360059 ) / 166158913960 \) (320232543v^11 - 1004615314v^10 - 1143072025v^9 + 9975044447v^8 - 4093093078v^7 - 25465804119v^6 + 21841076813v^5 + 35457807644v^4 + 27228013423v^3 + 100602839757v^2 - 286784071080*v - 144016360059) / 166158913960
\(\beta_{11}\)	\(=\)	\( ( - 1027654074 \nu^{11} + 2460612632 \nu^{10} + 3624161555 \nu^{9} - 25210679196 \nu^{8} + \cdots + 27977691092 ) / 415397284900 \) (-1027654074v^11 + 2460612632v^10 + 3624161555v^9 - 25210679196v^8 + 10960029469v^7 + 30148216612v^6 - 9821523994v^5 - 23732163042v^4 - 402415418899v^3 + 37700985074v^2 + 390356046795*v + 27977691092) / 415397284900

\(\nu\)	\(=\)	\( ( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + 2\beta_{7} + \beta_{5} - 2\beta _1 + 1 ) / 2 \) (b11 + b10 - b9 - b8 + 2b7 + b5 - 2b1 + 1) / 2
\(\nu^{2}\)	\(=\)	\( \beta_{10} + 2\beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 1 \) b10 + 2*b7 + b6 - b4 + b3 - b2 - b1 + 1
\(\nu^{3}\)	\(=\)	\( ( 4 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} + 10 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \cdots - 8 ) / 2 \) (4b10 - 2b9 - 4b8 + 10b7 - 2b6 + 2b5 - 2b4 - b3 - 4b1 - 8) / 2
\(\nu^{4}\)	\(=\)	\( - 7 \beta_{11} + 2 \beta_{9} + 7 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 7 \beta_{4} + \cdots - 11 \) -7b11 + 2b9 + 7b8 - 2b7 - 3b6 - 3b5 - 7b4 + 4b3 - 5*b2 - b1 - 11
\(\nu^{5}\)	\(=\)	\( ( - 6 \beta_{11} + 2 \beta_{10} - \beta_{9} - 10 \beta_{8} - 12 \beta_{7} - 34 \beta_{6} - 9 \beta_{5} + \cdots - 82 ) / 2 \) (-6b11 + 2b10 - b9 - 10b8 - 12b7 - 34b6 - 9b5 - 16b4 - 19b3 + 11b2 + 8b1 - 82) / 2
\(\nu^{6}\)	\(=\)	\( - 46 \beta_{11} - 40 \beta_{10} + 46 \beta_{9} + 42 \beta_{8} - 90 \beta_{7} - 13 \beta_{6} - 30 \beta_{5} + \cdots - 39 \) -46b11 - 40b10 + 46b9 + 42b8 - 90b7 - 13b6 - 30b5 - 26b4 + 13b3 - 2b2 + 46*b1 - 39
\(\nu^{7}\)	\(=\)	\( ( 33 \beta_{11} - 67 \beta_{10} + 79 \beta_{9} - 57 \beta_{8} - 234 \beta_{7} - 76 \beta_{6} + 5 \beta_{5} + \cdots - 51 ) / 2 \) (33b11 - 67b10 + 79b9 - 57b8 - 234b7 - 76b6 + 5b5 + 52b4 - 156b3 + 152b2 + 146*b1 - 51) / 2
\(\nu^{8}\)	\(=\)	\( - 93 \beta_{11} - 174 \beta_{10} + 231 \beta_{9} + 181 \beta_{8} - 364 \beta_{7} + 130 \beta_{6} + \cdots + 362 \) -93b11 - 174b10 + 231b9 + 181b8 - 364b7 + 130b6 - 3b5 + 107b4 + 91b3 - 23b2 + 179*b1 + 362
\(\nu^{9}\)	\(=\)	\( ( 1124 \beta_{11} + 316 \beta_{10} - 358 \beta_{9} - 776 \beta_{8} + 646 \beta_{7} + 682 \beta_{6} + \cdots + 1700 ) / 2 \) (1124b11 + 316b10 - 358b9 - 776b8 + 646b7 + 682b6 + 798b5 + 1378b4 - 861b3 + 404b2 - 248*b1 + 1700) / 2
\(\nu^{10}\)	\(=\)	\( 246 \beta_{11} - 81 \beta_{10} - 196 \beta_{9} + 482 \beta_{8} + 630 \beta_{7} + 1314 \beta_{6} + \cdots + 2556 \) 246b11 - 81b10 - 196b9 + 482b8 + 630b7 + 1314b6 + 501b5 + 1079b4 + 580b3 - 926b2 - 469*b1 + 2556
\(\nu^{11}\)	\(=\)	\( ( 7310 \beta_{11} + 5070 \beta_{10} - 9673 \beta_{9} - 6494 \beta_{8} + 12576 \beta_{7} + 2066 \beta_{6} + \cdots + 2574 ) / 2 \) (7310b11 + 5070b10 - 9673b9 - 6494b8 + 12576b7 + 2066b6 + 3475b5 + 4992b4 - 5163b3 - 325b2 - 6944*b1 + 2574) / 2

\(n\)	\(911\)	\(937\)	\(1081\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-\beta_{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 199.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$3$	\( T^{12} \) T^12
$5$	\( T^{12} - 2 T^{11} + \cdots + 15625 \) T^12 - 2T^11 + 7T^10 - 6T^9 + 15T^8 - 32T^7 + 114T^6 - 160T^5 + 375T^4 - 750T^3 + 4375T^2 - 6250*T + 15625
$7$	\( T^{12} + 2 T^{11} + \cdots + 1024 \) T^12 + 2T^11 + 19T^10 - 6T^9 + 205T^8 + 20T^7 + 708T^6 - 112T^5 + 1648T^4 - 64T^3 + 1664T^2 - 512*T + 1024
$11$	\( T^{12} + 6 T^{11} + \cdots + 16 \) T^12 + 6T^11 - 2T^10 - 84T^9 + 87T^8 + 444T^7 - 330T^6 - 1446T^5 + 2585T^4 - 1176T^3 - 4T^2 + 96*T + 16
$13$	\( T^{12} + 8 T^{11} + \cdots + 4826809 \) T^12 + 8T^11 + 36T^10 + 40T^9 - 232T^8 - 2040T^7 - 7838T^6 - 26520T^5 - 39208T^4 + 87880T^3 + 1028196T^2 + 2970344*T + 4826809
$17$	\( T^{12} + 18 T^{11} + \cdots + 65536 \) T^12 + 18T^11 + 95T^10 - 234T^9 - 2695T^8 + 4632T^7 + 75744T^6 + 85248T^5 - 300288T^4 - 307200T^3 + 1269760T^2 - 491520*T + 65536
$19$	\( T^{12} + 6 T^{11} + \cdots + 1982464 \) T^12 + 6T^11 - 53T^10 - 390T^9 + 2701T^8 + 14568T^7 - 43876T^6 - 254496T^5 + 672400T^4 + 2581632T^3 + 837120T^2 - 3649536*T + 1982464
$23$	\( T^{12} + \cdots + 190660864 \) T^12 + 6T^11 - 93T^10 - 630T^9 + 7441T^8 + 53748T^7 - 129352T^6 - 1617792T^5 + 1290352T^4 + 44643456T^3 + 171279232T^2 + 284334336*T + 190660864
$29$	\( T^{12} + 14 T^{11} + \cdots + 21904 \) T^12 + 14T^11 + 190T^10 + 960T^9 + 6703T^8 + 15668T^7 + 148926T^6 + 196610T^5 + 1737745T^4 - 1644412T^3 + 11075036T^2 + 490768*T + 21904
$31$	\( T^{12} + \cdots + 177209344 \) T^12 + 198T^10 + 15433T^8 + 592640T^6 + 11339776T^4 + 93061120*T^2 + 177209344
$37$	\( T^{12} + \cdots + 227195329 \) T^12 + 12T^11 + 209T^10 + 1028T^9 + 14658T^8 + 60444T^7 + 734609T^6 + 1319436T^5 + 11678146T^4 + 32084244T^3 + 129327441T^2 + 175510012*T + 227195329
$41$	\( T^{12} - 18 T^{11} + \cdots + 65536 \) T^12 - 18T^11 + 95T^10 + 234T^9 - 2695T^8 - 4632T^7 + 75744T^6 - 85248T^5 - 300288T^4 + 307200T^3 + 1269760T^2 + 491520*T + 65536
$43$	\( T^{12} + \cdots + 349241344 \) T^12 + 36T^11 + 535T^10 + 3708T^9 + 6253T^8 - 57864T^7 + 54524T^6 + 6267168T^5 + 51560464T^4 + 211680000T^3 + 494054400T^2 + 627916800*T + 349241344
$47$	\( (T^{6} + 8 T^{5} + \cdots + 5956)^{2} \) (T^6 + 8T^5 - 98T^4 - 1280T^3 - 4283T^2 - 2352*T + 5956)^2
$53$	\( T^{12} + \cdots + 2473271824 \) T^12 + 508T^10 + 87438T^8 + 5778420T^6 + 127872521T^4 + 989681000*T^2 + 2473271824
$59$	\( T^{12} + \cdots + 4983230464 \) T^12 - 36T^11 + 459T^10 - 972T^9 - 26427T^8 + 77796T^7 + 3235868T^6 - 37281744T^5 + 157498704T^4 - 129534336T^3 - 820266240T^2 + 752228352*T + 4983230464
$61$	\( T^{12} - 10 T^{11} + \cdots + 89718784 \) T^12 - 10T^11 + 135T^10 - 762T^9 + 7561T^8 - 39396T^7 + 256760T^6 - 835296T^5 + 3388992T^4 - 7106048T^3 + 26851328T^2 - 41828352*T + 89718784
$67$	\( T^{12} - 4 T^{11} + \cdots + 83759104 \) T^12 - 4T^11 + 164T^10 + 784T^9 + 18784T^8 + 35776T^7 + 394560T^6 - 320768T^5 + 6100480T^4 - 2632704T^3 + 25142272T^2 + 2928640*T + 83759104
$71$	\( T^{12} - 12 T^{11} + \cdots + 4194304 \) T^12 - 12T^11 - 172T^10 + 2640T^9 + 36080T^8 - 505152T^7 + 1150080T^6 + 6873600T^5 - 703488T^4 - 35782656T^3 + 56295424T^2 - 25165824*T + 4194304
$73$	\( (T^{6} - 14 T^{5} + \cdots - 230528)^{2} \) (T^6 - 14T^5 - 263T^4 + 3068T^3 + 18280T^2 - 114240*T - 230528)^2
$79$	\( (T^{6} - 2 T^{5} + \cdots - 29312)^{2} \) (T^6 - 2T^5 - 179T^4 + 788T^3 + 3892T^2 - 12576*T - 29312)^2
$83$	\( (T^{6} + 36 T^{5} + \cdots - 6912)^{2} \) (T^6 + 36T^5 + 360T^4 - 144T^3 - 16560T^2 - 51840*T - 6912)^2
$89$	\( T^{12} + \cdots + 1341001056256 \) T^12 + 18T^11 - 309T^10 - 7506T^9 + 96433T^8 + 1612140T^7 - 15114712T^6 - 191431776T^5 + 2363129536T^4 + 1837575168T^3 - 61874871296T^2 - 39576354816*T + 1341001056256
$97$	\( T^{12} + \cdots + 415519473664 \) T^12 + 48T^11 + 1640T^10 + 32704T^9 + 522672T^8 + 5437824T^7 + 53039744T^6 + 348175872T^5 + 3300888832T^4 + 14104406016T^3 + 88030961664T^2 - 141504348160*T + 415519473664
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