LMFDB - Newform orbit 960.3.c.l

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(449,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("960.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.c (of order $2$, degree $1$, not 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:	$26.1581053786$
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} - 24x^{14} + 502x^{12} - 3252x^{10} + 6781x^{8} - 93108x^{6} + 1275840x^{4} - 4208760x^{2} + 6441444 $ x^16 - 24x^14 + 502x^12 - 3252x^10 + 6781x^8 - 93108x^6 + 1275840x^4 - 4208760*x^2 + 6441444
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	no (minimal twist has level 480)
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_{3} q^{3} + (\beta_{10} + \beta_{7}) q^{5} + (\beta_{8} - \beta_{4} + \beta_{3}) q^{7} + (\beta_{10} + 2 \beta_{5} - \beta_{2} - 1) q^{9}+O(q^{10}) $ q + b3 * q^3 + (b10 + b7) * q^5 + (b8 - b4 + b3) * q^7 + (b10 + 2b5 - b2 - 1) q^9	$ q + \beta_{3} q^{3} + (\beta_{10} + \beta_{7}) q^{5} + (\beta_{8} - \beta_{4} + \beta_{3}) q^{7} + (\beta_{10} + 2 \beta_{5} - \beta_{2} - 1) q^{9} - \beta_{15} q^{11} - \beta_{6} q^{13} + ( - \beta_{12} - \beta_{8} + \beta_1) q^{15} + ( - \beta_{11} + \beta_{7}) q^{17} + ( - \beta_{13} + \beta_{12} + \cdots + \beta_1) q^{19}+ \cdots + ( - 3 \beta_{15} - 6 \beta_{13} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) $ q + b3 * q^3 + (b10 + b7) * q^5 + (b8 - b4 + b3) * q^7 + (b10 + 2b5 - b2 - 1) q^9 - b15 * q^11 - b6 * q^13 + (-b12 - b8 + b1) * q^15 + (-b11 + b7) * q^17 + (-b13 + b12 - b9 - b8 + b1) * q^19 + (-b10 + b5 - 2b2 - 8) q^21 + (-5b4 - 5b3 - 7b1) q^23 + (-b14 + b2 + 15) * q^25 + (-6b8 - 3b4 - 4b3 - 6b1) * q^27 + (-5b10 - 4b5) * q^29 + (-b13 + b12 + b9 - b8 + b1) * q^31 + (-2b14 - b11 - b10 - b7 - b6) q^33 + (b15 - b13 - b12 + 2b4 + 2b3 + 2b1) q^35 + (2b14 - b6) q^37 + (-3b13 + b12 - 3b9 - 2b8 + 2b1) * q^39 + (8b10 + 2b5) * q^41 + (8b8 - 3b4 + 3b3) q^43 + (-b14 + b11 + b10 - 2b7 - 2b6 + 3b5 + 3b2 - 6) * q^45 + (15b4 + 15b3 + 7b1) q^47 + (-4b2 + 25) q^49 + (3b15 - 3b13 - b12 + 3b9 - b8 + b1) q^51 + (2b11 + 3b10 + 4b7) q^53 + (-2b13 + 2b12 + b9 - 20b8 + b4 - b3 + 2b1) * q^55 + (-3b10 - 6b7 + 3b6) q^57 + (b15 - 2b13 - 2b12) * q^59 + (-8b2 - 2) q^61 + (-3b8 + 3b4 - 11b3 - 12b1) * q^63 + (-b11 + 3b7 + 22b5) * q^65 + (-26b8 + 17b4 - 17b3) q^67 + (2b10 - 17b5 - 2b2 - 26) q^69 + (-2b15 - 4b13 - 4b12) q^71 + (2b14 + 4b6) * q^73 + (-3b15 + 3b9 + 3b8 - 3b4 + 16b3 + 3b1) * q^75 + (4b10 + 8b7) * q^77 + (-b13 + b12 + 7b9 - b8 + b1) q^79 + (14b10 - 8b5 + 4b2 - 23) q^81 + (-3b4 - 3b3 - 26b1) q^83 + (2b14 + 3b6 + 20b2 + 14) q^85 + (15b8 + 12b4 + 4b3 - 3b1) * q^87 + (-24b10 - 30b5) * q^89 + (-4b13 + 4b12 - 8b9 - 4b8 + 4b1) q^91 + (2b14 - 2b11 - 5b10 - 8b7 + b6) * q^93 + (2b15 - b13 - b12 - 21b4 - 21b3 - 24b1) * q^95 - 4b6 q^97 + (-3b15 - 6b13 + 2b12 + 6b9 - 4b8 + 4b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} - 128 q^{21} + 240 q^{25} - 96 q^{45} + 400 q^{49} - 32 q^{61} - 416 q^{69} - 368 q^{81} + 224 q^{85}+O(q^{100}) $ 16 * q - 16 * q^9 - 128 * q^21 + 240 * q^25 - 96 * q^45 + 400 * q^49 - 32 * q^61 - 416 * q^69 - 368 * q^81 + 224 * q^85

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 24x^{14} + 502x^{12} - 3252x^{10} + 6781x^{8} - 93108x^{6} + 1275840x^{4} - 4208760x^{2} + 6441444 $ x^16 - 24*x^14 + 502*x^12 - 3252*x^10 + 6781*x^8 - 93108*x^6 + 1275840*x^4 - 4208760*x^2 + 6441444 :

$\beta_{1}$	$=$	$ ( - 20240 \nu^{15} + 5527215 \nu^{13} - 105081680 \nu^{11} + 2036514594 \nu^{9} + \cdots + 3240516618162 \nu ) / 1484477915688 $ (-20240v^15 + 5527215v^13 - 105081680v^11 + 2036514594v^9 - 4869466034v^7 - 25199151819v^5 - 413191516230v^3 + 3240516618162v) / 1484477915688
$\beta_{2}$	$=$	$ ( 86138 \nu^{14} - 3002451 \nu^{12} + 71404022 \nu^{10} - 858602781 \nu^{8} + \cdots - 1739302176348 ) / 325302259302 $ (86138v^14 - 3002451v^12 + 71404022v^10 - 858602781v^8 + 5341358174v^6 - 16710240618v^4 + 27794102508*v^2 - 1739302176348) / 325302259302
$\beta_{3}$	$=$	$ ( - 1737685939 \nu^{15} + 37454334210 \nu^{14} - 75335853690 \nu^{13} + \cdots - 99\!\cdots\!48 ) / 39\!\cdots\!12 $ (-1737685939v^15 + 37454334210v^14 - 75335853690v^13 - 882564987294v^12 + 1354329560060v^11 + 17972590995648v^10 - 41130642891684v^9 - 109148675103396v^8 + 126201295793405v^7 + 111958620028182v^6 + 264153237886086v^5 - 4822624344014514v^4 + 11301433804739250v^3 + 52534660645020168v^2 - 80919738523103904*v - 99826245874275948) / 39770153013225312
$\beta_{4}$	$=$	$ ( - 1737685939 \nu^{15} - 37454334210 \nu^{14} - 75335853690 \nu^{13} + \cdots + 99\!\cdots\!48 ) / 39\!\cdots\!12 $ (-1737685939v^15 - 37454334210v^14 - 75335853690v^13 + 882564987294v^12 + 1354329560060v^11 - 17972590995648v^10 - 41130642891684v^9 + 109148675103396v^8 + 126201295793405v^7 - 111958620028182v^6 + 264153237886086v^5 + 4822624344014514v^4 + 11301433804739250v^3 - 52534660645020168v^2 - 80919738523103904*v + 99826245874275948) / 39770153013225312
$\beta_{5}$	$=$	$ ( - 7723041883 \nu^{15} + 172854096750 \nu^{13} - 3684895135474 \nu^{11} + \cdots + 11\!\cdots\!44 \nu ) / 14\!\cdots\!92 $ (-7723041883v^15 + 172854096750v^13 - 3684895135474v^11 + 21157918953648v^9 - 58811349296017v^7 + 825232008201450v^5 - 8592002215726482v^3 + 11881321549648344v) / 14913807379959492
$\beta_{6}$	$=$	$ ( 96773480 \nu^{14} - 2011803201 \nu^{12} + 43724872910 \nu^{10} - 224522207544 \nu^{8} + \cdots - 280971444124962 ) / 35132644004616 $ (96773480v^14 - 2011803201v^12 + 43724872910v^10 - 224522207544v^8 + 1036997155976v^6 - 16521159444069v^4 + 154479996034146*v^2 - 280971444124962) / 35132644004616
$\beta_{7}$	$=$	$ ( 81400076161 \nu^{15} - 1537483144140 \nu^{13} + 31613974277224 \nu^{11} + \cdots + 23\!\cdots\!64 \nu ) / 11\!\cdots\!36 $ (81400076161v^15 - 1537483144140v^13 + 31613974277224v^11 - 64556946321108v^9 - 577017998908415v^7 - 3797561203747620v^5 + 62847280213089102v^3 + 237106017814201164v) / 119310459039675936
$\beta_{8}$	$=$	$ ( - 3555343810 \nu^{14} + 83650484355 \nu^{12} - 1693309779862 \nu^{10} + \cdots + 81\!\cdots\!94 ) / 11\!\cdots\!92 $ (-3555343810v^14 + 83650484355v^12 - 1693309779862v^10 + 9945002753580v^8 - 383881622422v^6 + 316713095949675v^4 - 4223699697029058*v^2 + 8183443035003894) / 1104726472589592
$\beta_{9}$	$=$	$ ( 9589348502 \nu^{14} - 139147617735 \nu^{12} + 2927181746198 \nu^{10} + \cdots + 20\!\cdots\!20 ) / 24\!\cdots\!82 $ (9589348502v^14 - 139147617735v^12 + 2927181746198v^10 + 8178516096675v^8 - 105249785926342v^6 - 949996799744718v^4 + 4481320475294964*v^2 + 20624274556782120) / 2485634563326582
$\beta_{10}$	$=$	$ ( - 34289497192 \nu^{15} + 731051704305 \nu^{13} - 15233327912179 \nu^{11} + \cdots + 52\!\cdots\!60 \nu ) / 14\!\cdots\!92 $ (-34289497192v^15 + 731051704305v^13 - 15233327912179v^11 + 68843212787715v^9 - 19098080199403v^7 + 2654902667674824v^5 - 37220771270474982v^3 + 52275321853547760v) / 14913807379959492
$\beta_{11}$	$=$	$ ( - 300212895377 \nu^{15} + 8155337099256 \nu^{13} - 172236570443744 \nu^{11} + \cdots + 25\!\cdots\!76 \nu ) / 11\!\cdots\!36 $ (-300212895377v^15 + 8155337099256v^13 - 172236570443744v^11 + 1397628247774980v^9 - 3726969735727745v^7 + 18014714785898256v^5 - 343156475329886214v^3 + 2525418414512866476v) / 119310459039675936
$\beta_{12}$	$=$	$ ( - 171160781162 \nu^{15} + 142763663178 \nu^{14} + 3788096075217 \nu^{13} + \cdots + 66\!\cdots\!38 ) / 59\!\cdots\!68 $ (-171160781162v^15 + 142763663178v^14 + 3788096075217v^13 - 1773685664847v^12 - 78458197420628v^11 + 40707655652718v^10 + 412208120368182v^9 + 182007467219556v^8 - 420747086206028v^7 - 653322990322962v^6 + 18472551721072923v^5 - 17536202476774263v^4 - 182858678079030846v^3 - 6944148967656438v^2 + 210806737825739814*v + 665817768130479138) / 59655229519837968
$\beta_{13}$	$=$	$ ( - 171974145802 \nu^{15} - 142763663178 \nu^{14} + 4010212737207 \nu^{13} + \cdots - 66\!\cdots\!38 ) / 59\!\cdots\!68 $ (-171974145802v^15 - 142763663178v^14 + 4010212737207v^13 + 1773685664847v^12 - 82681009813108v^11 - 40707655652718v^10 + 494047495842666v^9 - 182007467219556v^8 - 616431448248352v^7 + 653322990322962v^6 + 17459898606074589v^5 + 17536202476774263v^4 - 199463192350249626v^3 + 6944148967656438v^2 + 341030138643197946*v - 665817768130479138) / 59655229519837968
$\beta_{14}$	$=$	$ ( 574317605 \nu^{14} - 12441314385 \nu^{12} + 261543480296 \nu^{10} - 1259751619182 \nu^{8} + \cdots - 11\!\cdots\!50 ) / 35132644004616 $ (574317605v^14 - 12441314385v^12 + 261543480296v^10 - 1259751619182v^8 + 1206322023203v^6 - 38766481987791v^4 + 603468278410752*v^2 - 1149602270450850) / 35132644004616
$\beta_{15}$	$=$	$ ( - 50992186 \nu^{15} + 1130175285 \nu^{13} - 23102193079 \nu^{11} + \cdots + 81710476399656 \nu ) / 5855440667436 $ (-50992186v^15 + 1130175285v^13 - 23102193079v^11 + 116387357439v^9 + 26846995295v^7 + 4598540563728v^5 - 56128153103514v^3 + 81710476399656v) / 5855440667436

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

$\nu$	$=$	$ ( \beta_{10} + 2\beta_{7} - 2\beta_{5} + \beta_{4} + \beta_{3} + \beta_1 ) / 4 $ (b10 + 2b7 - 2b5 + b4 + b3 + b1) / 4
$\nu^{2}$	$=$	$ ( \beta_{13} - \beta_{12} + \beta_{9} - \beta_{8} + 2\beta_{6} + 4\beta_{4} - 4\beta_{3} - \beta_{2} - \beta _1 + 12 ) / 4 $ (b13 - b12 + b9 - b8 + 2b6 + 4b4 - 4*b3 - b2 - b1 + 12) / 4
$\nu^{3}$	$=$	$ ( - 3 \beta_{15} + 6 \beta_{13} + 6 \beta_{12} + 2 \beta_{11} - 4 \beta_{7} - 32 \beta_{5} + 5 \beta_{4} + \cdots - \beta_1 ) / 4 $ (-3b15 + 6b13 + 6b12 + 2b11 - 4b7 - 32b5 + 5b4 + 5b3 - b1) / 4
$\nu^{4}$	$=$	$ ( - 11 \beta_{13} + 11 \beta_{12} - 26 \beta_{9} - 63 \beta_{8} + 20 \beta_{6} + 104 \beta_{4} + \cdots - 214 ) / 4 $ (-11b13 + 11b12 - 26b9 - 63b8 + 20b6 + 104b4 - 104b3 - 54b2 + 11*b1 - 214) / 4
$\nu^{5}$	$=$	$ ( - 10 \beta_{15} + 35 \beta_{13} + 35 \beta_{12} + 60 \beta_{11} - 161 \beta_{10} - 346 \beta_{7} + \cdots - 815 \beta_1 ) / 4 $ (-10b15 + 35b13 + 35b12 + 60b11 - 161b10 - 346b7 - 161b5 - 419b4 - 419b3 - 815b1) / 4
$\nu^{6}$	$=$	$ ( 64 \beta_{14} - 381 \beta_{13} + 381 \beta_{12} - 714 \beta_{9} + 229 \beta_{8} - 130 \beta_{6} + \cdots - 6282 ) / 4 $ (64b14 - 381b13 + 381b12 - 714b9 + 229b8 - 130b6 - 692b4 + 692b3 - 1222b2 + 381b1 - 6282) / 4
$\nu^{7}$	$=$	$ ( 1148 \beta_{15} - 1603 \beta_{13} - 1603 \beta_{12} + 774 \beta_{11} - 4213 \beta_{10} + \cdots - 9397 \beta_1 ) / 4 $ (1148b15 - 1603b13 - 1603b12 + 774b11 - 4213b10 - 4864b7 + 8149b5 - 5131b4 - 5131b3 - 9397b1) / 4
$\nu^{8}$	$=$	$ ( 2432 \beta_{14} - 769 \beta_{13} + 769 \beta_{12} - 1312 \beta_{9} + 27607 \beta_{8} - 10072 \beta_{6} + \cdots - 17878 ) / 4 $ (2432b14 - 769b13 + 769b12 - 1312b9 + 27607b8 - 10072b6 - 41968b4 + 41968b3 - 3144b2 + 769b1 - 17878) / 4
$\nu^{9}$	$=$	$ ( 22968 \beta_{15} - 34353 \beta_{13} - 34353 \beta_{12} - 10276 \beta_{11} - 17887 \beta_{10} + \cdots + 128593 \beta_1 ) / 4 $ (22968b15 - 34353b13 - 34353b12 - 10276b11 - 17887b10 + 63014b7 + 192619b5 + 69277b4 + 69277b3 + 128593b1) / 4
$\nu^{10}$	$=$	$ ( 23336 \beta_{14} + 121667 \beta_{13} - 121667 \beta_{12} + 224060 \beta_{9} + 391881 \beta_{8} + \cdots + 2179962 ) / 4 $ (23336b14 + 121667b13 - 121667b12 + 224060b9 + 391881b8 - 100918b6 - 406244b4 + 406244b3 + 417116b2 - 121667b1 + 2179962) / 4
$\nu^{11}$	$=$	$ ( - 15158 \beta_{15} + 30019 \beta_{13} + 30019 \beta_{12} - 444842 \beta_{11} + 1187137 \beta_{10} + \cdots + 5403299 \beta_1 ) / 4 $ (-15158b15 + 30019b13 + 30019b12 - 444842b11 + 1187137b10 + 2776064b7 - 169745b5 + 2945933b4 + 2945933b3 + 5403299b1) / 4
$\nu^{12}$	$=$	$ ( - 463184 \beta_{14} + 2218155 \beta_{13} - 2218155 \beta_{12} + 4068924 \beta_{9} - 3472993 \beta_{8} + \cdots + 39605690 ) / 4 $ (-463184b14 + 2218155b13 - 2218155b12 + 4068924b9 - 3472993b8 + 2185196b6 + 8867528b4 - 8867528b3 + 7541604b2 - 2218155b1 + 39605690) / 4
$\nu^{13}$	$=$	$ ( - 8470280 \beta_{15} + 13137631 \beta_{13} + 13137631 \beta_{12} - 3156884 \beta_{11} + \cdots + 38741833 \beta_1 ) / 4 $ (-8470280b15 + 13137631b13 + 13137631b12 - 3156884b11 + 23892777b10 + 19645574b7 - 72919277b5 + 21078565b4 + 21078565b3 + 38741833b1) / 4
$\nu^{14}$	$=$	$ ( - 15216240 \beta_{14} - 10035365 \beta_{13} + 10035365 \beta_{12} - 18076316 \beta_{9} + \cdots - 179849910 ) / 4 $ (-15216240b14 - 10035365b13 + 10035365b12 - 18076316b9 - 196824235b8 + 70724198b6 + 289312972b4 - 289312972b3 - 33505780b2 + 10035365b1 - 179849910) / 4
$\nu^{15}$	$=$	$ ( - 125896458 \beta_{15} + 194877123 \beta_{13} + 194877123 \beta_{12} + 111731486 \beta_{11} + \cdots - 1371556189 \beta_1 ) / 4 $ (-125896458b15 + 194877123b13 + 194877123b12 + 111731486b11 - 64245603b10 - 704616520b7 - 1077823481b5 - 753753595b4 - 753753595b3 - 1371556189b1) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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} $

449.1

−2.66478	−	0.595188i
1.50106	−	0.595188i
−2.66478	+	0.595188i
1.50106	+	0.595188i
−3.66835	−	2.37608i
1.09041	−	2.37608i
−3.66835	+	2.37608i
1.09041	+	2.37608i
−1.09041	+	2.37608i
3.66835	+	2.37608i
−1.09041	−	2.37608i
3.66835	−	2.37608i
−1.50106	+	0.595188i
2.66478	+	0.595188i
−1.50106	−	0.595188i
2.66478	−	0.595188i

−2.57794

−

1.53436i

−4.16584

2.76510i

1.68345i

4.29150

7.91094i

449.2

−2.57794

−

1.53436i

4.16584

2.76510i

1.68345i

4.29150

7.91094i

449.3

−2.57794

1.53436i

−4.16584

−

2.76510i

−

1.68345i

4.29150

−

7.91094i

449.4

−2.57794

1.53436i

4.16584

−

2.76510i

−

1.68345i

4.29150

−

7.91094i

449.5

−1.16372

−

2.76510i

−4.75876

−

1.53436i

−

6.72057i

−6.29150

6.43560i

449.6

−1.16372

−

2.76510i

4.75876

−

1.53436i

−

6.72057i

−6.29150

6.43560i

449.7

−1.16372

2.76510i

−4.75876

1.53436i

6.72057i

−6.29150

−

6.43560i

449.8

−1.16372

2.76510i

4.75876

1.53436i

6.72057i

−6.29150

−

6.43560i

449.9

1.16372

−

2.76510i

−4.75876

1.53436i

−

6.72057i

−6.29150

−

6.43560i

449.10

1.16372

−

2.76510i

4.75876

1.53436i

−

6.72057i

−6.29150

−

6.43560i

449.11

1.16372

2.76510i

−4.75876

−

1.53436i

6.72057i

−6.29150

6.43560i

449.12

1.16372

2.76510i

4.75876

−

1.53436i

6.72057i

−6.29150

6.43560i

449.13

2.57794

−

1.53436i

−4.16584

−

2.76510i

1.68345i

4.29150

−

7.91094i

449.14

2.57794

−

1.53436i

4.16584

−

2.76510i

1.68345i

4.29150

−

7.91094i

449.15

2.57794

1.53436i

−4.16584

2.76510i

−

1.68345i

4.29150

7.91094i

449.16

2.57794

1.53436i

4.16584

2.76510i

−

1.68345i

4.29150

7.91094i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
5.b	even	2	1	inner
12.b	even	2	1	inner
15.d	odd	2	1	inner
20.d	odd	2	1	inner
60.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	960.3.c.l		16
3.b	odd	2	1	inner	960.3.c.l		16
4.b	odd	2	1	inner	960.3.c.l		16
5.b	even	2	1	inner	960.3.c.l		16
8.b	even	2	1		480.3.c.b	✓	16
8.d	odd	2	1		480.3.c.b	✓	16
12.b	even	2	1	inner	960.3.c.l		16
15.d	odd	2	1	inner	960.3.c.l		16
20.d	odd	2	1	inner	960.3.c.l		16
24.f	even	2	1		480.3.c.b	✓	16
24.h	odd	2	1		480.3.c.b	✓	16
40.e	odd	2	1		480.3.c.b	✓	16
40.f	even	2	1		480.3.c.b	✓	16
60.h	even	2	1	inner	960.3.c.l		16
120.i	odd	2	1		480.3.c.b	✓	16
120.m	even	2	1		480.3.c.b	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
480.3.c.b	✓	16	8.b	even	2	1
480.3.c.b	✓	16	8.d	odd	2	1
480.3.c.b	✓	16	24.f	even	2	1
480.3.c.b	✓	16	24.h	odd	2	1
480.3.c.b	✓	16	40.e	odd	2	1
480.3.c.b	✓	16	40.f	even	2	1
480.3.c.b	✓	16	120.i	odd	2	1
480.3.c.b	✓	16	120.m	even	2	1
960.3.c.l		16	1.a	even	1	1	trivial
960.3.c.l		16	3.b	odd	2	1	inner
960.3.c.l		16	4.b	odd	2	1	inner
960.3.c.l		16	5.b	even	2	1	inner
960.3.c.l		16	12.b	even	2	1	inner
960.3.c.l		16	15.d	odd	2	1	inner
960.3.c.l		16	20.d	odd	2	1	inner
960.3.c.l		16	60.h	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(960, [\chi])$:

$ T_{7}^{4} + 48T_{7}^{2} + 128 $

$ T_{17}^{4} - 1120T_{17}^{2} + 308112 $

$ T_{19}^{4} - 696T_{19}^{2} + 56592 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 4 T^{6} + \cdots + 6561)^{2} $ (T^8 + 4T^6 + 54T^4 + 324*T^2 + 6561)^2
$5$	$ (T^{8} - 60 T^{6} + \cdots + 390625)^{2} $ (T^8 - 60T^6 + 2038T^4 - 37500*T^2 + 390625)^2
$7$	$ (T^{4} + 48 T^{2} + 128)^{4} $ (T^4 + 48*T^2 + 128)^4
$11$	$ (T^{4} + 424 T^{2} + 12576)^{4} $ (T^4 + 424*T^2 + 12576)^4
$13$	$ (T^{4} + 536 T^{2} + 12576)^{4} $ (T^4 + 536*T^2 + 12576)^4
$17$	$ (T^{4} - 1120 T^{2} + 308112)^{4} $ (T^4 - 1120*T^2 + 308112)^4
$19$	$ (T^{4} - 696 T^{2} + 56592)^{4} $ (T^4 - 696*T^2 + 56592)^4
$23$	$ (T^{4} - 1024 T^{2} + 35344)^{4} $ (T^4 - 1024*T^2 + 35344)^4
$29$	$ (T^{4} + 1064 T^{2} + 14112)^{4} $ (T^4 + 1064*T^2 + 14112)^4
$31$	$ (T^{4} - 1112 T^{2} + 6288)^{4} $ (T^4 - 1112*T^2 + 6288)^4
$37$	$ (T^{4} + 4376 T^{2} + 4539936)^{4} $ (T^4 + 4376*T^2 + 4539936)^4
$41$	$ (T^{4} + 2400 T^{2} + 492032)^{4} $ (T^4 + 2400*T^2 + 492032)^4
$43$	$ (T^{4} + 1512 T^{2} + 566048)^{4} $ (T^4 + 1512*T^2 + 566048)^4
$47$	$ (T^{4} - 6304 T^{2} + 9909904)^{4} $ (T^4 - 6304*T^2 + 9909904)^4
$53$	$ (T^{4} - 5248 T^{2} + 2269968)^{4} $ (T^4 - 5248*T^2 + 2269968)^4
$59$	$ (T^{4} + 3432 T^{2} + 113184)^{4} $ (T^4 + 3432*T^2 + 113184)^4
$61$	$ (T^{2} + 4 T - 1788)^{8} $ (T^2 + 4*T - 1788)^8
$67$	$ (T^{4} + 20712 T^{2} + 79581728)^{4} $ (T^4 + 20712*T^2 + 79581728)^4
$71$	$ (T^{4} + 17056 T^{2} + 72638976)^{4} $ (T^4 + 17056*T^2 + 72638976)^4
$73$	$ (T^{4} + 8096 T^{2} + 16298496)^{4} $ (T^4 + 8096*T^2 + 16298496)^4
$79$	$ (T^{4} - 17720 T^{2} + 74707728)^{4} $ (T^4 - 17720*T^2 + 74707728)^4
$83$	$ (T^{4} - 9856 T^{2} + 21864976)^{4} $ (T^4 - 9856*T^2 + 21864976)^4
$89$	$ (T^{4} + 33120 T^{2} + 134742528)^{4} $ (T^4 + 33120*T^2 + 134742528)^4
$97$	$ (T^{4} + 8576 T^{2} + 3219456)^{4} $ (T^4 + 8576*T^2 + 3219456)^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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{3} + (\beta_{10} + \beta_{7}) q^{5} + (\beta_{8} - \beta_{4} + \beta_{3}) q^{7} + (\beta_{10} + 2 \beta_{5} - \beta_{2} - 1) q^{9}+O(q^{10}) \) q + b3 * q^3 + (b10 + b7) * q^5 + (b8 - b4 + b3) * q^7 + (b10 + 2b5 - b2 - 1) q^9	\( q + \beta_{3} q^{3} + (\beta_{10} + \beta_{7}) q^{5} + (\beta_{8} - \beta_{4} + \beta_{3}) q^{7} + (\beta_{10} + 2 \beta_{5} - \beta_{2} - 1) q^{9} - \beta_{15} q^{11} - \beta_{6} q^{13} + ( - \beta_{12} - \beta_{8} + \beta_1) q^{15} + ( - \beta_{11} + \beta_{7}) q^{17} + ( - \beta_{13} + \beta_{12} + \cdots + \beta_1) q^{19}+ \cdots + ( - 3 \beta_{15} - 6 \beta_{13} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) q + b3 * q^3 + (b10 + b7) * q^5 + (b8 - b4 + b3) * q^7 + (b10 + 2b5 - b2 - 1) q^9 - b15 * q^11 - b6 * q^13 + (-b12 - b8 + b1) * q^15 + (-b11 + b7) * q^17 + (-b13 + b12 - b9 - b8 + b1) * q^19 + (-b10 + b5 - 2b2 - 8) q^21 + (-5b4 - 5b3 - 7b1) q^23 + (-b14 + b2 + 15) * q^25 + (-6b8 - 3b4 - 4b3 - 6b1) * q^27 + (-5b10 - 4b5) * q^29 + (-b13 + b12 + b9 - b8 + b1) * q^31 + (-2b14 - b11 - b10 - b7 - b6) q^33 + (b15 - b13 - b12 + 2b4 + 2b3 + 2b1) q^35 + (2b14 - b6) q^37 + (-3b13 + b12 - 3b9 - 2b8 + 2b1) * q^39 + (8b10 + 2b5) * q^41 + (8b8 - 3b4 + 3b3) q^43 + (-b14 + b11 + b10 - 2b7 - 2b6 + 3b5 + 3b2 - 6) * q^45 + (15b4 + 15b3 + 7b1) q^47 + (-4b2 + 25) q^49 + (3b15 - 3b13 - b12 + 3b9 - b8 + b1) q^51 + (2b11 + 3b10 + 4b7) q^53 + (-2b13 + 2b12 + b9 - 20b8 + b4 - b3 + 2b1) * q^55 + (-3b10 - 6b7 + 3b6) q^57 + (b15 - 2b13 - 2b12) * q^59 + (-8b2 - 2) q^61 + (-3b8 + 3b4 - 11b3 - 12b1) * q^63 + (-b11 + 3b7 + 22b5) * q^65 + (-26b8 + 17b4 - 17b3) q^67 + (2b10 - 17b5 - 2b2 - 26) q^69 + (-2b15 - 4b13 - 4b12) q^71 + (2b14 + 4b6) * q^73 + (-3b15 + 3b9 + 3b8 - 3b4 + 16b3 + 3b1) * q^75 + (4b10 + 8b7) * q^77 + (-b13 + b12 + 7b9 - b8 + b1) q^79 + (14b10 - 8b5 + 4b2 - 23) q^81 + (-3b4 - 3b3 - 26b1) q^83 + (2b14 + 3b6 + 20b2 + 14) q^85 + (15b8 + 12b4 + 4b3 - 3b1) * q^87 + (-24b10 - 30b5) * q^89 + (-4b13 + 4b12 - 8b9 - 4b8 + 4b1) q^91 + (2b14 - 2b11 - 5b10 - 8b7 + b6) * q^93 + (2b15 - b13 - b12 - 21b4 - 21b3 - 24b1) * q^95 - 4b6 q^97 + (-3b15 - 6b13 + 2b12 + 6b9 - 4b8 + 4b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} - 128 q^{21} + 240 q^{25} - 96 q^{45} + 400 q^{49} - 32 q^{61} - 416 q^{69} - 368 q^{81} + 224 q^{85}+O(q^{100}) \) 16 * q - 16 * q^9 - 128 * q^21 + 240 * q^25 - 96 * q^45 + 400 * q^49 - 32 * q^61 - 416 * q^69 - 368 * q^81 + 224 * q^85

Introduction

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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	960.3.c.l

Level	$960$
Weight	$3$
Character orbit	960.c
Analytic conductor	$26.158$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(26.1581053786\)
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:	\( x^{16} - 24x^{14} + 502x^{12} - 3252x^{10} + 6781x^{8} - 93108x^{6} + 1275840x^{4} - 4208760x^{2} + 6441444 \) x^16 - 24x^14 + 502x^12 - 3252x^10 + 6781x^8 - 93108x^6 + 1275840x^4 - 4208760*x^2 + 6441444
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{24} \)
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 20240 \nu^{15} + 5527215 \nu^{13} - 105081680 \nu^{11} + 2036514594 \nu^{9} + \cdots + 3240516618162 \nu ) / 1484477915688 \) (-20240v^15 + 5527215v^13 - 105081680v^11 + 2036514594v^9 - 4869466034v^7 - 25199151819v^5 - 413191516230v^3 + 3240516618162v) / 1484477915688
\(\beta_{2}\)	\(=\)	\( ( 86138 \nu^{14} - 3002451 \nu^{12} + 71404022 \nu^{10} - 858602781 \nu^{8} + \cdots - 1739302176348 ) / 325302259302 \) (86138v^14 - 3002451v^12 + 71404022v^10 - 858602781v^8 + 5341358174v^6 - 16710240618v^4 + 27794102508*v^2 - 1739302176348) / 325302259302
\(\beta_{3}\)	\(=\)	\( ( - 1737685939 \nu^{15} + 37454334210 \nu^{14} - 75335853690 \nu^{13} + \cdots - 99\!\cdots\!48 ) / 39\!\cdots\!12 \) (-1737685939v^15 + 37454334210v^14 - 75335853690v^13 - 882564987294v^12 + 1354329560060v^11 + 17972590995648v^10 - 41130642891684v^9 - 109148675103396v^8 + 126201295793405v^7 + 111958620028182v^6 + 264153237886086v^5 - 4822624344014514v^4 + 11301433804739250v^3 + 52534660645020168v^2 - 80919738523103904*v - 99826245874275948) / 39770153013225312
\(\beta_{4}\)	\(=\)	\( ( - 1737685939 \nu^{15} - 37454334210 \nu^{14} - 75335853690 \nu^{13} + \cdots + 99\!\cdots\!48 ) / 39\!\cdots\!12 \) (-1737685939v^15 - 37454334210v^14 - 75335853690v^13 + 882564987294v^12 + 1354329560060v^11 - 17972590995648v^10 - 41130642891684v^9 + 109148675103396v^8 + 126201295793405v^7 - 111958620028182v^6 + 264153237886086v^5 + 4822624344014514v^4 + 11301433804739250v^3 - 52534660645020168v^2 - 80919738523103904*v + 99826245874275948) / 39770153013225312
\(\beta_{5}\)	\(=\)	\( ( - 7723041883 \nu^{15} + 172854096750 \nu^{13} - 3684895135474 \nu^{11} + \cdots + 11\!\cdots\!44 \nu ) / 14\!\cdots\!92 \) (-7723041883v^15 + 172854096750v^13 - 3684895135474v^11 + 21157918953648v^9 - 58811349296017v^7 + 825232008201450v^5 - 8592002215726482v^3 + 11881321549648344v) / 14913807379959492
\(\beta_{6}\)	\(=\)	\( ( 96773480 \nu^{14} - 2011803201 \nu^{12} + 43724872910 \nu^{10} - 224522207544 \nu^{8} + \cdots - 280971444124962 ) / 35132644004616 \) (96773480v^14 - 2011803201v^12 + 43724872910v^10 - 224522207544v^8 + 1036997155976v^6 - 16521159444069v^4 + 154479996034146*v^2 - 280971444124962) / 35132644004616
\(\beta_{7}\)	\(=\)	\( ( 81400076161 \nu^{15} - 1537483144140 \nu^{13} + 31613974277224 \nu^{11} + \cdots + 23\!\cdots\!64 \nu ) / 11\!\cdots\!36 \) (81400076161v^15 - 1537483144140v^13 + 31613974277224v^11 - 64556946321108v^9 - 577017998908415v^7 - 3797561203747620v^5 + 62847280213089102v^3 + 237106017814201164v) / 119310459039675936
\(\beta_{8}\)	\(=\)	\( ( - 3555343810 \nu^{14} + 83650484355 \nu^{12} - 1693309779862 \nu^{10} + \cdots + 81\!\cdots\!94 ) / 11\!\cdots\!92 \) (-3555343810v^14 + 83650484355v^12 - 1693309779862v^10 + 9945002753580v^8 - 383881622422v^6 + 316713095949675v^4 - 4223699697029058*v^2 + 8183443035003894) / 1104726472589592
\(\beta_{9}\)	\(=\)	\( ( 9589348502 \nu^{14} - 139147617735 \nu^{12} + 2927181746198 \nu^{10} + \cdots + 20\!\cdots\!20 ) / 24\!\cdots\!82 \) (9589348502v^14 - 139147617735v^12 + 2927181746198v^10 + 8178516096675v^8 - 105249785926342v^6 - 949996799744718v^4 + 4481320475294964*v^2 + 20624274556782120) / 2485634563326582
\(\beta_{10}\)	\(=\)	\( ( - 34289497192 \nu^{15} + 731051704305 \nu^{13} - 15233327912179 \nu^{11} + \cdots + 52\!\cdots\!60 \nu ) / 14\!\cdots\!92 \) (-34289497192v^15 + 731051704305v^13 - 15233327912179v^11 + 68843212787715v^9 - 19098080199403v^7 + 2654902667674824v^5 - 37220771270474982v^3 + 52275321853547760v) / 14913807379959492
\(\beta_{11}\)	\(=\)	\( ( - 300212895377 \nu^{15} + 8155337099256 \nu^{13} - 172236570443744 \nu^{11} + \cdots + 25\!\cdots\!76 \nu ) / 11\!\cdots\!36 \) (-300212895377v^15 + 8155337099256v^13 - 172236570443744v^11 + 1397628247774980v^9 - 3726969735727745v^7 + 18014714785898256v^5 - 343156475329886214v^3 + 2525418414512866476v) / 119310459039675936
\(\beta_{12}\)	\(=\)	\( ( - 171160781162 \nu^{15} + 142763663178 \nu^{14} + 3788096075217 \nu^{13} + \cdots + 66\!\cdots\!38 ) / 59\!\cdots\!68 \) (-171160781162v^15 + 142763663178v^14 + 3788096075217v^13 - 1773685664847v^12 - 78458197420628v^11 + 40707655652718v^10 + 412208120368182v^9 + 182007467219556v^8 - 420747086206028v^7 - 653322990322962v^6 + 18472551721072923v^5 - 17536202476774263v^4 - 182858678079030846v^3 - 6944148967656438v^2 + 210806737825739814*v + 665817768130479138) / 59655229519837968
\(\beta_{13}\)	\(=\)	\( ( - 171974145802 \nu^{15} - 142763663178 \nu^{14} + 4010212737207 \nu^{13} + \cdots - 66\!\cdots\!38 ) / 59\!\cdots\!68 \) (-171974145802v^15 - 142763663178v^14 + 4010212737207v^13 + 1773685664847v^12 - 82681009813108v^11 - 40707655652718v^10 + 494047495842666v^9 - 182007467219556v^8 - 616431448248352v^7 + 653322990322962v^6 + 17459898606074589v^5 + 17536202476774263v^4 - 199463192350249626v^3 + 6944148967656438v^2 + 341030138643197946*v - 665817768130479138) / 59655229519837968
\(\beta_{14}\)	\(=\)	\( ( 574317605 \nu^{14} - 12441314385 \nu^{12} + 261543480296 \nu^{10} - 1259751619182 \nu^{8} + \cdots - 11\!\cdots\!50 ) / 35132644004616 \) (574317605v^14 - 12441314385v^12 + 261543480296v^10 - 1259751619182v^8 + 1206322023203v^6 - 38766481987791v^4 + 603468278410752*v^2 - 1149602270450850) / 35132644004616
\(\beta_{15}\)	\(=\)	\( ( - 50992186 \nu^{15} + 1130175285 \nu^{13} - 23102193079 \nu^{11} + \cdots + 81710476399656 \nu ) / 5855440667436 \) (-50992186v^15 + 1130175285v^13 - 23102193079v^11 + 116387357439v^9 + 26846995295v^7 + 4598540563728v^5 - 56128153103514v^3 + 81710476399656v) / 5855440667436

\(\nu\)	\(=\)	\( ( \beta_{10} + 2\beta_{7} - 2\beta_{5} + \beta_{4} + \beta_{3} + \beta_1 ) / 4 \) (b10 + 2b7 - 2b5 + b4 + b3 + b1) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{13} - \beta_{12} + \beta_{9} - \beta_{8} + 2\beta_{6} + 4\beta_{4} - 4\beta_{3} - \beta_{2} - \beta _1 + 12 ) / 4 \) (b13 - b12 + b9 - b8 + 2b6 + 4b4 - 4*b3 - b2 - b1 + 12) / 4
\(\nu^{3}\)	\(=\)	\( ( - 3 \beta_{15} + 6 \beta_{13} + 6 \beta_{12} + 2 \beta_{11} - 4 \beta_{7} - 32 \beta_{5} + 5 \beta_{4} + \cdots - \beta_1 ) / 4 \) (-3b15 + 6b13 + 6b12 + 2b11 - 4b7 - 32b5 + 5b4 + 5b3 - b1) / 4
\(\nu^{4}\)	\(=\)	\( ( - 11 \beta_{13} + 11 \beta_{12} - 26 \beta_{9} - 63 \beta_{8} + 20 \beta_{6} + 104 \beta_{4} + \cdots - 214 ) / 4 \) (-11b13 + 11b12 - 26b9 - 63b8 + 20b6 + 104b4 - 104b3 - 54b2 + 11*b1 - 214) / 4
\(\nu^{5}\)	\(=\)	\( ( - 10 \beta_{15} + 35 \beta_{13} + 35 \beta_{12} + 60 \beta_{11} - 161 \beta_{10} - 346 \beta_{7} + \cdots - 815 \beta_1 ) / 4 \) (-10b15 + 35b13 + 35b12 + 60b11 - 161b10 - 346b7 - 161b5 - 419b4 - 419b3 - 815b1) / 4
\(\nu^{6}\)	\(=\)	\( ( 64 \beta_{14} - 381 \beta_{13} + 381 \beta_{12} - 714 \beta_{9} + 229 \beta_{8} - 130 \beta_{6} + \cdots - 6282 ) / 4 \) (64b14 - 381b13 + 381b12 - 714b9 + 229b8 - 130b6 - 692b4 + 692b3 - 1222b2 + 381b1 - 6282) / 4
\(\nu^{7}\)	\(=\)	\( ( 1148 \beta_{15} - 1603 \beta_{13} - 1603 \beta_{12} + 774 \beta_{11} - 4213 \beta_{10} + \cdots - 9397 \beta_1 ) / 4 \) (1148b15 - 1603b13 - 1603b12 + 774b11 - 4213b10 - 4864b7 + 8149b5 - 5131b4 - 5131b3 - 9397b1) / 4
\(\nu^{8}\)	\(=\)	\( ( 2432 \beta_{14} - 769 \beta_{13} + 769 \beta_{12} - 1312 \beta_{9} + 27607 \beta_{8} - 10072 \beta_{6} + \cdots - 17878 ) / 4 \) (2432b14 - 769b13 + 769b12 - 1312b9 + 27607b8 - 10072b6 - 41968b4 + 41968b3 - 3144b2 + 769b1 - 17878) / 4
\(\nu^{9}\)	\(=\)	\( ( 22968 \beta_{15} - 34353 \beta_{13} - 34353 \beta_{12} - 10276 \beta_{11} - 17887 \beta_{10} + \cdots + 128593 \beta_1 ) / 4 \) (22968b15 - 34353b13 - 34353b12 - 10276b11 - 17887b10 + 63014b7 + 192619b5 + 69277b4 + 69277b3 + 128593b1) / 4
\(\nu^{10}\)	\(=\)	\( ( 23336 \beta_{14} + 121667 \beta_{13} - 121667 \beta_{12} + 224060 \beta_{9} + 391881 \beta_{8} + \cdots + 2179962 ) / 4 \) (23336b14 + 121667b13 - 121667b12 + 224060b9 + 391881b8 - 100918b6 - 406244b4 + 406244b3 + 417116b2 - 121667b1 + 2179962) / 4
\(\nu^{11}\)	\(=\)	\( ( - 15158 \beta_{15} + 30019 \beta_{13} + 30019 \beta_{12} - 444842 \beta_{11} + 1187137 \beta_{10} + \cdots + 5403299 \beta_1 ) / 4 \) (-15158b15 + 30019b13 + 30019b12 - 444842b11 + 1187137b10 + 2776064b7 - 169745b5 + 2945933b4 + 2945933b3 + 5403299b1) / 4
\(\nu^{12}\)	\(=\)	\( ( - 463184 \beta_{14} + 2218155 \beta_{13} - 2218155 \beta_{12} + 4068924 \beta_{9} - 3472993 \beta_{8} + \cdots + 39605690 ) / 4 \) (-463184b14 + 2218155b13 - 2218155b12 + 4068924b9 - 3472993b8 + 2185196b6 + 8867528b4 - 8867528b3 + 7541604b2 - 2218155b1 + 39605690) / 4
\(\nu^{13}\)	\(=\)	\( ( - 8470280 \beta_{15} + 13137631 \beta_{13} + 13137631 \beta_{12} - 3156884 \beta_{11} + \cdots + 38741833 \beta_1 ) / 4 \) (-8470280b15 + 13137631b13 + 13137631b12 - 3156884b11 + 23892777b10 + 19645574b7 - 72919277b5 + 21078565b4 + 21078565b3 + 38741833b1) / 4
\(\nu^{14}\)	\(=\)	\( ( - 15216240 \beta_{14} - 10035365 \beta_{13} + 10035365 \beta_{12} - 18076316 \beta_{9} + \cdots - 179849910 ) / 4 \) (-15216240b14 - 10035365b13 + 10035365b12 - 18076316b9 - 196824235b8 + 70724198b6 + 289312972b4 - 289312972b3 - 33505780b2 + 10035365b1 - 179849910) / 4
\(\nu^{15}\)	\(=\)	\( ( - 125896458 \beta_{15} + 194877123 \beta_{13} + 194877123 \beta_{12} + 111731486 \beta_{11} + \cdots - 1371556189 \beta_1 ) / 4 \) (-125896458b15 + 194877123b13 + 194877123b12 + 111731486b11 - 64245603b10 - 704616520b7 - 1077823481b5 - 753753595b4 - 753753595b3 - 1371556189b1) / 4

\(n\)	\(511\)	\(577\)	\(641\)	\(901\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 449.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 4 T^{6} + \cdots + 6561)^{2} \) (T^8 + 4T^6 + 54T^4 + 324*T^2 + 6561)^2
$5$	\( (T^{8} - 60 T^{6} + \cdots + 390625)^{2} \) (T^8 - 60T^6 + 2038T^4 - 37500*T^2 + 390625)^2
$7$	\( (T^{4} + 48 T^{2} + 128)^{4} \) (T^4 + 48*T^2 + 128)^4
$11$	\( (T^{4} + 424 T^{2} + 12576)^{4} \) (T^4 + 424*T^2 + 12576)^4
$13$	\( (T^{4} + 536 T^{2} + 12576)^{4} \) (T^4 + 536*T^2 + 12576)^4
$17$	\( (T^{4} - 1120 T^{2} + 308112)^{4} \) (T^4 - 1120*T^2 + 308112)^4
$19$	\( (T^{4} - 696 T^{2} + 56592)^{4} \) (T^4 - 696*T^2 + 56592)^4
$23$	\( (T^{4} - 1024 T^{2} + 35344)^{4} \) (T^4 - 1024*T^2 + 35344)^4
$29$	\( (T^{4} + 1064 T^{2} + 14112)^{4} \) (T^4 + 1064*T^2 + 14112)^4
$31$	\( (T^{4} - 1112 T^{2} + 6288)^{4} \) (T^4 - 1112*T^2 + 6288)^4
$37$	\( (T^{4} + 4376 T^{2} + 4539936)^{4} \) (T^4 + 4376*T^2 + 4539936)^4
$41$	\( (T^{4} + 2400 T^{2} + 492032)^{4} \) (T^4 + 2400*T^2 + 492032)^4
$43$	\( (T^{4} + 1512 T^{2} + 566048)^{4} \) (T^4 + 1512*T^2 + 566048)^4
$47$	\( (T^{4} - 6304 T^{2} + 9909904)^{4} \) (T^4 - 6304*T^2 + 9909904)^4
$53$	\( (T^{4} - 5248 T^{2} + 2269968)^{4} \) (T^4 - 5248*T^2 + 2269968)^4
$59$	\( (T^{4} + 3432 T^{2} + 113184)^{4} \) (T^4 + 3432*T^2 + 113184)^4
$61$	\( (T^{2} + 4 T - 1788)^{8} \) (T^2 + 4*T - 1788)^8
$67$	\( (T^{4} + 20712 T^{2} + 79581728)^{4} \) (T^4 + 20712*T^2 + 79581728)^4
$71$	\( (T^{4} + 17056 T^{2} + 72638976)^{4} \) (T^4 + 17056*T^2 + 72638976)^4
$73$	\( (T^{4} + 8096 T^{2} + 16298496)^{4} \) (T^4 + 8096*T^2 + 16298496)^4
$79$	\( (T^{4} - 17720 T^{2} + 74707728)^{4} \) (T^4 - 17720*T^2 + 74707728)^4
$83$	\( (T^{4} - 9856 T^{2} + 21864976)^{4} \) (T^4 - 9856*T^2 + 21864976)^4
$89$	\( (T^{4} + 33120 T^{2} + 134742528)^{4} \) (T^4 + 33120*T^2 + 134742528)^4
$97$	\( (T^{4} + 8576 T^{2} + 3219456)^{4} \) (T^4 + 8576*T^2 + 3219456)^4
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