LMFDB - Newform orbit 960.4.b.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(671,960)]

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([1, 1, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("960.671");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	960.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$56.6418336055$
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} + 3472x^{12} + 2676096x^{8} + 573099300x^{4} + 31755240000 $ x^16 + 3472x^12 + 2676096x^8 + 573099300*x^4 + 31755240000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{23}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{3} + 5 q^{5} + (\beta_{7} - \beta_{4} - \beta_1) q^{7} + (\beta_{9} - 2) q^{9} + ( - \beta_{8} - \beta_{7} - \beta_{3}) q^{11} - \beta_{14} q^{13} - 5 \beta_{4} q^{15} + (\beta_{14} + 2 \beta_{12} - \beta_{10} + \cdots - 1) q^{17}+ \cdots + ( - 9 \beta_{15} - 4 \beta_{13} + \cdots + 60 \beta_1) q^{99}+O(q^{100}) $ q - b4 * q^3 + 5 * q^5 + (b7 - b4 - b1) * q^7 + (b9 - 2) * q^9 + (-b8 - b7 - b3) * q^11 - b14 * q^13 - 5b4 q^15 + (b14 + 2b12 - b10 + b9 - b6 + b2 - 1) q^17 + (-b13 - b11 - 2b8 + 5b4 + b1) * q^19 + (b14 + b12 - 3b10 - 2b6 - 2b5 - b2 - 16) q^21 + (b15 - b11 - 2b8 + 5b4 + b1) * q^23 + 25 * q^25 + (-2b13 + 3b11 - 2b8 + 4b7 + 2b4 - b3 - 3b1) * q^27 + (-3b10 - 3b9 - 2b6 - 4b5 + 2b2 + 12) q^29 + (2b7 - 12b4 + 5b3 - 10b1) * q^31 + (-2b14 + b12 + 3b10 + 3b9 - b6 - b5 + 5b2 - 33) * q^33 + (5b7 - 5b4 - 5b1) q^35 + (3b14 - 4b12 - 4b10 + 4b9 - 3b6 - b5 - 2b2 + 2) * q^37 + (-2b15 - b13 + 2b11 - 6b8 - 5b7 + 5b4 - 4b3 + 11b1) q^39 + (2b14 - 6b12 + 3b10 - 3b9 - 2b6 + 5b5 - 3b2 + 3) q^41 + (-b15 - 3b13 - 2b11 - 2b8 + 4b4 + 2b1) q^43 + (5b9 - 10) q^45 + (-b13 + 5b11 - 7b8 + 26b4 - 2b1) * q^47 + (-7b10 - 7b9 - 4b6 - 8b5 - 9b2 + 36) q^49 + (b15 - b13 - 7b11 - 2b8 + 16b7 - 7b4 - 4b3 + 2b1) * q^51 + (-6b10 - 6b9 - 3b6 - 6b5 + 4b2 + 60) q^53 + (-5b8 - 5b7 - 5b3) q^55 + (-b14 - 7b12 - 8b9 - 6b6 - 4b5 - 5b2 - 125) q^57 + (-5b8 - 19b7 + 22b4 - 9b3 + 5b1) q^59 + (-4b14 + 8b12 - 7b10 + 7b9 + 2b6 - 9b5 + 4b2 - 4) q^61 + (3b15 - 2b13 - 3b11 - 18b8 + 4b7 + 17b4 - 4b3 + 36b1) * q^63 - 5b14 q^65 + (3b15 - 13b13 + 14b11 + 2b8 + 8b4 + b1) q^67 + (-9b14 - 3b12 - 6b10 - 15b9 - b6 - 5b5 - 12b2 - 123) * q^69 + (-4b15 + 8b13 - 6b11 - 16b8 + 42b4 - 3b1) * q^71 + (-2b10 - 2b9 - 14b6 - 28b5 + 9b2 + 49) q^73 - 25b4 q^75 + (7b6 + 14b5 + 10b2 + 312) q^77 + (-28b8 - 8b7 - 78b4 + b3 - 55b1) * q^79 + (6b14 + 12b12 + 3b10 + 5b9 - 13b6 + b5 - 241) q^81 + (3b8 - 6b7 + 47b4 - 16b3 - 20b1) q^83 + (5b14 + 10b12 - 5b10 + 5b9 - 5b6 + 5b2 - 5) * q^85 + (-2b15 + 7b13 - 13b11 - 23b8 - 9b7 - 15b4 + 12b3 + 38b1) * q^87 + (-10b14 + 10b12 + 4b10 - 4b9 - 17b6 + 21b5 + 5b2 - 5) q^89 + (-4b15 + 16b13 - 10b11 + 20b8 - 70b4 - 5b1) * q^91 + (7b14 - 8b12 - 6b10 + 11b6 - 2b5 - 22b2 - 248) * q^93 + (-5b13 - 5b11 - 10b8 + 25b4 + 5b1) q^95 + (16b10 + 16b9 - 8b6 - 16b5 - b2 + 95) * q^97 + (-9b15 - 4b13 + 6b8 + 14b7 + 16b4 + 4b3 + 60b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 80 q^{5} - 32 q^{9} - 248 q^{21} + 400 q^{25} + 192 q^{29} - 520 q^{33} - 160 q^{45} + 576 q^{49} + 960 q^{53} - 2056 q^{57} - 1992 q^{69} + 784 q^{73} + 4992 q^{77} - 3760 q^{81} - 4032 q^{93} + 1520 q^{97}+O(q^{100}) $ 16 * q + 80 * q^5 - 32 * q^9 - 248 * q^21 + 400 * q^25 + 192 * q^29 - 520 * q^33 - 160 * q^45 + 576 * q^49 + 960 * q^53 - 2056 * q^57 - 1992 * q^69 + 784 * q^73 + 4992 * q^77 - 3760 * q^81 - 4032 * q^93 + 1520 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 3472x^{12} + 2676096x^{8} + 573099300x^{4} + 31755240000 $ x^16 + 3472*x^12 + 2676096*x^8 + 573099300*x^4 + 31755240000 :

$\beta_{1}$	$=$	$ ( -9491\nu^{14} - 33546752\nu^{10} - 27708714936\nu^{6} - 8590329053100\nu^{2} ) / 14972808163500 $ (-9491v^14 - 33546752v^10 - 27708714936v^6 - 8590329053100v^2) / 14972808163500
$\beta_{2}$	$=$	$ ( 4231\nu^{12} + 13092190\nu^{8} + 6863423634\nu^{4} + 625475850885 ) / 13611643785 $ (4231v^12 + 13092190v^8 + 6863423634*v^4 + 625475850885) / 13611643785
$\beta_{3}$	$=$	$ ( - 28473 \nu^{15} + 350020 \nu^{14} - 100640256 \nu^{11} + 621724840 \nu^{10} + \cdots + 179673697962000 \nu ) / 179673697962000 $ (-28473v^15 + 350020v^14 - 100640256v^11 + 621724840v^10 - 83126144808v^7 - 905609065680v^6 - 25770987159300v^3 - 608813123886000v^2 + 179673697962000*v) / 179673697962000
$\beta_{4}$	$=$	$ ( - 28473 \nu^{15} + 1672642 \nu^{14} - 100640256 \nu^{11} + 5623847224 \nu^{10} + \cdots + 179673697962000 \nu ) / 359347395924000 $ (-28473v^15 + 1672642v^14 - 100640256v^11 + 5623847224v^10 - 83126144808v^7 + 3828864053232v^6 - 25770987159300v^3 + 472992545097000v^2 + 179673697962000*v) / 359347395924000
$\beta_{5}$	$=$	$ ( 9491 \nu^{15} - 6020080 \nu^{12} + 33546752 \nu^{11} - 20452693360 \nu^{8} + \cdots - 17\!\cdots\!00 ) / 29945616327000 $ (9491v^15 - 6020080v^12 + 33546752v^11 - 20452693360v^8 + 27708714936v^7 - 14231099195280v^4 + 8590329053100v^3 + 59891232654000v - 1708827842106000) / 29945616327000
$\beta_{6}$	$=$	$ ( - 9491 \nu^{15} - 3010040 \nu^{12} - 33546752 \nu^{11} - 10226346680 \nu^{8} + \cdots - 854413921053000 ) / 14972808163500 $ (-9491v^15 - 3010040v^12 - 33546752v^11 - 10226346680v^8 - 27708714936v^7 - 7115549597640v^4 - 8590329053100v^3 - 59891232654000v - 854413921053000) / 14972808163500
$\beta_{7}$	$=$	$ ( - 9491 \nu^{15} + 1739994 \nu^{14} - 33546752 \nu^{11} + 5643774168 \nu^{10} + \cdots + 59891232654000 \nu ) / 119782465308000 $ (-9491v^15 + 1739994v^14 - 33546752v^11 + 5643774168v^10 - 27708714936v^7 + 3443420134224v^6 - 8590329053100v^3 + 371256740469000v^2 + 59891232654000*v) / 119782465308000
$\beta_{8}$	$=$	$ ( 28473 \nu^{15} + 1672642 \nu^{14} + 100640256 \nu^{11} + 5623847224 \nu^{10} + \cdots - 179673697962000 \nu ) / 119782465308000 $ (28473v^15 + 1672642v^14 + 100640256v^11 + 5623847224v^10 + 83126144808v^7 + 3828864053232v^6 + 25770987159300v^3 + 472992545097000v^2 - 179673697962000*v) / 119782465308000
$\beta_{9}$	$=$	$ ( 836321 \nu^{15} + 9030120 \nu^{13} - 1782000 \nu^{12} + 2811923612 \nu^{11} + \cdots - 35\!\cdots\!00 ) / 179673697962000 $ (836321v^15 + 9030120v^13 - 1782000v^12 + 2811923612v^11 + 30679040040v^9 - 6929663400v^8 + 1914432026616v^7 + 21346648792920v^5 - 9453130790400v^4 + 236496272548500v^3 + 2473404914178000*v - 3587675500530000) / 179673697962000
$\beta_{10}$	$=$	$ ( - 836321 \nu^{15} - 9030120 \nu^{13} + 34338480 \nu^{12} - 2811923612 \nu^{11} + \cdots + 66\!\cdots\!00 ) / 179673697962000 $ (-836321v^15 - 9030120v^13 + 34338480v^12 - 2811923612v^11 - 30679040040v^9 + 115786496760v^8 - 1914432026616v^7 - 21346648792920v^5 + 75933464381280v^4 - 236496272548500v^3 - 2473404914178000*v + 6665291552106000) / 179673697962000
$\beta_{11}$	$=$	$ ( - 8826043 \nu^{15} - 5359602 \nu^{14} + 75804960 \nu^{13} - 30043408096 \nu^{11} + \cdots + 31\!\cdots\!00 \nu ) / 10\!\cdots\!00 $ (-8826043v^15 - 5359602v^14 + 75804960v^13 - 30043408096v^11 - 18079224744v^10 + 259291647120v^9 - 21549917073528v^7 - 12484105897392v^6 + 189679451924160v^5 - 3465446821411500v^3 - 1728229481202600v^2 + 31454432763534000v) / 1078042187772000
$\beta_{12}$	$=$	$ ( - 4455731 \nu^{15} - 37902480 \nu^{13} - 83773800 \nu^{12} - 15172664432 \nu^{11} + \cdots - 12\!\cdots\!00 ) / 539021093886000 $ (-4455731v^15 - 37902480v^13 - 83773800v^12 - 15172664432v^11 - 129645823560v^9 - 259225362000v^8 - 10899647753976v^7 - 94839725962080v^5 - 135895787953200v^4 - 1771379891444700v^3 - 15996726928710000*v - 12114911300580000) / 539021093886000
$\beta_{13}$	$=$	$ ( - 1701115 \nu^{15} - 56946 \nu^{14} + 18060240 \nu^{13} - 5724487480 \nu^{11} + \cdots + 51\!\cdots\!00 \nu ) / 179673697962000 $ (-1701115v^15 - 56946v^14 + 18060240v^13 - 5724487480v^11 - 201280512v^10 + 61358080080v^9 - 3911990198040v^7 - 166252289616v^6 + 42693297585840v^5 - 498763532256300v^3 - 51541974318600v^2 + 5126483526318000v) / 179673697962000
$\beta_{14}$	$=$	$ ( - 6244177 \nu^{15} - 85255500 \nu^{13} - 20426779144 \nu^{11} - 276078217800 \nu^{9} + \cdots - 15\!\cdots\!00 \nu ) / 539021093886000 $ (-6244177v^15 - 85255500v^13 - 20426779144v^11 - 276078217800v^9 - 12702558354192v^7 - 165228007150800v^5 - 1301547587591700v^3 - 15555499512498000v) / 539021093886000
$\beta_{15}$	$=$	$ ( - 3564376 \nu^{15} + 85419 \nu^{14} + 58165140 \nu^{13} - 11387166772 \nu^{11} + \cdots + 70\!\cdots\!00 \nu ) / 269510546943000 $ (-3564376v^15 + 85419v^14 + 58165140v^13 - 11387166772v^11 + 301920768v^10 + 184041097680v^9 - 6460505405496v^7 + 249378434424v^6 + 101188060772040v^5 - 437432846990400v^3 + 77312961477900v^2 + 7057242582192000v) / 269510546943000

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

$\nu$	$=$	$ ( -2\beta_{8} - \beta_{6} + \beta_{5} + 6\beta_{4} ) / 12 $ (-2b8 - b6 + b5 + 6b4) / 12
$\nu^{2}$	$=$	$ ( 2\beta_{8} - 6\beta_{7} + 6\beta_{3} - 75\beta_1 ) / 12 $ (2b8 - 6b7 + 6b3 - 75b1) / 12
$\nu^{3}$	$=$	$ ( - 12 \beta_{13} + 18 \beta_{12} + 18 \beta_{11} - 12 \beta_{10} + 12 \beta_{9} + 74 \beta_{8} + \cdots - 9 ) / 12 $ (-12b13 + 18b12 + 18b11 - 12b10 + 12b9 + 74b8 - 43b6 + 31b5 - 204b4 + 9b2 - 3*b1 - 9) / 12
$\nu^{4}$	$=$	$ ( -174\beta_{10} - 174\beta_{9} - 43\beta_{6} - 86\beta_{5} + 18\beta_{2} - 5208 ) / 6 $ (-174b10 - 174b9 - 43b6 - 86b5 + 18*b2 - 5208) / 6
$\nu^{5}$	$=$	$ ( 36 \beta_{15} - 72 \beta_{14} - 858 \beta_{13} - 1116 \beta_{12} + 1116 \beta_{11} + 894 \beta_{10} + \cdots + 558 ) / 12 $ (36b15 - 72b14 - 858b13 - 1116b12 + 1116b11 + 894b10 - 894b9 + 3194b8 + 2062b6 - 1168b5 - 8466b4 - 558b2 - 111*b1 + 558) / 12
$\nu^{6}$	$=$	$ ( -2633\beta_{8} + 6009\beta_{7} - 5220\beta_{4} - 4344\beta_{3} + 28020\beta_1 ) / 3 $ (-2633b8 + 6009b7 - 5220b4 - 4344b3 + 28020*b1) / 3
$\nu^{7}$	$=$	$ ( - 1665 \beta_{15} - 3330 \beta_{14} + 24261 \beta_{13} - 29394 \beta_{12} - 29394 \beta_{11} + \cdots + 14697 ) / 6 $ (-1665b15 - 3330b14 + 24261b13 - 29394b12 - 29394b11 + 25926b10 - 25926b9 - 74257b8 + 50924b6 - 24998b5 + 193377b4 - 14697b2 + 1734*b1 + 14697) / 6
$\nu^{8}$	$=$	$ ( 212937\beta_{10} + 212937\beta_{9} + 36209\beta_{6} + 72418\beta_{5} - 53874\beta_{2} + 5026944 ) / 3 $ (212937b10 + 212937b9 + 36209b6 + 72418b5 - 53874*b2 + 5026944) / 3
$\nu^{9}$	$=$	$ ( - 107748 \beta_{15} + 215496 \beta_{14} + 1275864 \beta_{13} + 1493118 \beta_{12} - 1493118 \beta_{11} + \cdots - 746559 ) / 6 $ (-107748b15 + 215496b14 + 1275864b13 + 1493118b12 - 1493118b11 - 1383612b10 + 1383612b9 - 3576682b8 - 2534021b6 + 1150409b5 + 9236928b4 + 746559b2 + 54753*b1 - 746559) / 6
$\nu^{10}$	$=$	$ ( 7202168\beta_{8} - 16563174\beta_{7} + 17223930\beta_{4} + 10472874\beta_{3} - 62245920\beta_1 ) / 3 $ (7202168b8 - 16563174b7 + 17223930b4 + 10472874b3 - 62245920*b1) / 3
$\nu^{11}$	$=$	$ ( 6090300 \beta_{15} + 12180600 \beta_{14} - 65205726 \beta_{13} + 75017844 \beta_{12} + 75017844 \beta_{11} + \cdots - 37508922 ) / 6 $ (6090300b15 + 12180600b14 - 65205726b13 + 75017844b12 + 75017844b11 - 71296026b10 + 71296026b9 + 175246942b8 - 126316634b6 + 55020608b5 - 450722982b4 + 37508922b2 - 1860909*b1 - 37508922) / 6
$\nu^{12}$	$=$	$ ( - 517772112 \beta_{10} - 517772112 \beta_{9} - 77166509 \beta_{6} - 154333018 \beta_{5} + \cdots - 11774469009 ) / 3 $ (-517772112b10 - 517772112b9 - 77166509b6 - 154333018b5 + 161756739*b2 - 11774469009) / 3
$\nu^{13}$	$=$	$ ( 161756739 \beta_{15} - 323513478 \beta_{14} - 1645330287 \beta_{13} - 1876829814 \beta_{12} + \cdots + 938414907 ) / 3 $ (161756739b15 - 323513478b14 - 1645330287b13 - 1876829814b12 + 1876829814b11 + 1807087026b10 - 1807087026b9 + 4330053971b8 + 3149448868b6 - 1342361842b5 - 11113332099b4 - 938414907b2 - 34871394*b1 + 938414907) / 3
$\nu^{14}$	$=$	$ ( - 18222259178 \beta_{8} + 42358499214 \beta_{7} - 45639807840 \beta_{4} - 25692734454 \beta_{3} + 150447844695 \beta_1 ) / 3 $ (-18222259178b8 + 42358499214b7 - 45639807840b4 - 25692734454b3 + 150447844695*b1) / 3
$\nu^{15}$	$=$	$ ( - 8332882380 \beta_{15} - 16665764760 \beta_{14} + 82538255208 \beta_{13} - 93743968122 \beta_{12} + \cdots + 46871984061 ) / 3 $ (-8332882380b15 - 16665764760b14 + 82538255208b13 - 93743968122b12 - 93743968122b11 + 90871137588b10 - 90871137588b9 - 214906258526b8 + 157055139247b6 - 66184001659b5 + 550974807456b4 - 46871984061b2 + 1436415267*b1 + 46871984061) / 3

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} $

671.1

4.99314	+	4.99314i
4.99314	−	4.99314i
3.61508	−	3.61508i
3.61508	+	3.61508i
2.71061	−	2.71061i
2.71061	+	2.71061i
2.15693	+	2.15693i
2.15693	−	2.15693i
−2.15693	−	2.15693i
−2.15693	+	2.15693i
−2.71061	+	2.71061i
−2.71061	−	2.71061i
−3.61508	+	3.61508i
−3.61508	−	3.61508i
−4.99314	−	4.99314i
−4.99314	+	4.99314i

−4.99314

−

1.43824i

5.00000

−

16.1856i

22.8629

14.3627i

671.2

−4.99314

1.43824i

5.00000

16.1856i

22.8629

−

14.3627i

671.3

−3.61508

−

3.73245i

5.00000

−

29.1389i

−0.862344

26.9862i

671.4

−3.61508

3.73245i

5.00000

29.1389i

−0.862344

−

26.9862i

671.5

−2.71061

−

4.43313i

5.00000

7.10557i

−12.3052

24.0329i

671.6

−2.71061

4.43313i

5.00000

−

7.10557i

−12.3052

−

24.0329i

671.7

−2.15693

−

4.72733i

5.00000

8.15228i

−17.6953

20.3930i

671.8

−2.15693

4.72733i

5.00000

−

8.15228i

−17.6953

−

20.3930i

671.9

2.15693

−

4.72733i

5.00000

8.15228i

−17.6953

−

20.3930i

671.10

2.15693

4.72733i

5.00000

−

8.15228i

−17.6953

20.3930i

671.11

2.71061

−

4.43313i

5.00000

7.10557i

−12.3052

−

24.0329i

671.12

2.71061

4.43313i

5.00000

−

7.10557i

−12.3052

24.0329i

671.13

3.61508

−

3.73245i

5.00000

−

29.1389i

−0.862344

−

26.9862i

671.14

3.61508

3.73245i

5.00000

29.1389i

−0.862344

26.9862i

671.15

4.99314

−

1.43824i

5.00000

−

16.1856i

22.8629

−

14.3627i

671.16

4.99314

1.43824i

5.00000

16.1856i

22.8629

14.3627i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
24.f	even	2	1	inner
24.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	960.4.b.b	yes	16
3.b	odd	2	1		960.4.b.a	✓	16
4.b	odd	2	1	inner	960.4.b.b	yes	16
8.b	even	2	1		960.4.b.a	✓	16
8.d	odd	2	1		960.4.b.a	✓	16
12.b	even	2	1		960.4.b.a	✓	16
24.f	even	2	1	inner	960.4.b.b	yes	16
24.h	odd	2	1	inner	960.4.b.b	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
960.4.b.a	✓	16	3.b	odd	2	1
960.4.b.a	✓	16	8.b	even	2	1
960.4.b.a	✓	16	8.d	odd	2	1
960.4.b.a	✓	16	12.b	even	2	1
960.4.b.b	yes	16	1.a	even	1	1	trivial
960.4.b.b	yes	16	4.b	odd	2	1	inner
960.4.b.b	yes	16	24.f	even	2	1	inner
960.4.b.b	yes	16	24.h	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(960, [\chi])$:

$ T_{7}^{8} + 1228T_{7}^{6} + 355728T_{7}^{4} + 29741776T_{7}^{2} + 746382400 $

$ T_{29}^{4} - 48T_{29}^{3} - 37488T_{29}^{2} + 1914544T_{29} + 39321360 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} + \cdots + 282429536481 $ T^16 + 16T^14 + 1068T^12 - 8064T^10 + 425574T^8 - 5878656T^6 + 567578988T^4 + 6198727824*T^2 + 282429536481
$5$	$ (T - 5)^{16} $ (T - 5)^16
$7$	$ (T^{8} + 1228 T^{6} + \cdots + 746382400)^{2} $ (T^8 + 1228T^6 + 355728T^4 + 29741776*T^2 + 746382400)^2
$11$	$ (T^{8} + 5328 T^{6} + \cdots + 20666937600)^{2} $ (T^8 + 5328T^6 + 6443040T^4 + 1295435776*T^2 + 20666937600)^2
$13$	$ (T^{8} + \cdots + 5700677875200)^{2} $ (T^8 + 11100T^6 + 33417936T^4 + 26308811840*T^2 + 5700677875200)^2
$17$	$ (T^{8} + \cdots + 430656937920000)^{2} $ (T^8 + 24972T^6 + 204866640T^4 + 615178049600*T^2 + 430656937920000)^2
$19$	$ (T^{8} + \cdots + 420208715520000)^{2} $ (T^8 - 22452T^6 + 170524080T^4 - 491994718400*T^2 + 420208715520000)^2
$23$	$ (T^{8} + \cdots + 26\!\cdots\!00)^{2} $ (T^8 - 54228T^6 + 722739600T^4 - 2534842350000*T^2 + 2630632950000000)^2
$29$	$ (T^{4} - 48 T^{3} + \cdots + 39321360)^{4} $ (T^4 - 48T^3 - 37488T^2 + 1914544*T + 39321360)^4
$31$	$ (T^{8} + \cdots + 21\!\cdots\!00)^{2} $ (T^8 + 115300T^6 + 3827042304T^4 + 49051753171200*T^2 + 210462939653760000)^2
$37$	$ (T^{8} + \cdots + 35\!\cdots\!00)^{2} $ (T^8 + 312204T^6 + 28084899984T^4 + 655059569514560*T^2 + 3580109939665036800)^2
$41$	$ (T^{8} + \cdots + 11\!\cdots\!00)^{2} $ (T^8 + 354048T^6 + 31346984640T^4 + 566830807366400*T^2 + 1100247971328000000)^2
$43$	$ (T^{8} + \cdots + 769198518000000)^{2} $ (T^8 - 136776T^6 + 4800178944T^4 - 12264362766000*T^2 + 769198518000000)^2
$47$	$ (T^{8} + \cdots + 16\!\cdots\!00)^{2} $ (T^8 - 240132T^6 + 15098390160T^4 - 318069644807600*T^2 + 1619598725086320000)^2
$53$	$ (T^{4} - 240 T^{3} + \cdots - 1424598000)^{4} $ (T^4 - 240T^3 - 125208T^2 + 34562720*T - 1424598000)^4
$59$	$ (T^{8} + \cdots + 71\!\cdots\!00)^{2} $ (T^8 + 784416T^6 + 148492728864T^4 + 2027858972588800*T^2 + 7113919566126240000)^2
$61$	$ (T^{8} + \cdots + 13\!\cdots\!00)^{2} $ (T^8 + 996240T^6 + 163089475200T^4 + 8813159912160000*T^2 + 130601565388800000000)^2
$67$	$ (T^{8} + \cdots + 63\!\cdots\!00)^{2} $ (T^8 - 1839864T^6 + 1008444966144T^4 - 149273005485656240*T^2 + 6326348137507031356800)^2
$71$	$ (T^{8} + \cdots + 14\!\cdots\!00)^{2} $ (T^8 - 1526832T^6 + 578265123840T^4 - 66391674608921600*T^2 + 1492639422100992000000)^2
$73$	$ (T^{4} - 196 T^{3} + \cdots - 801702800)^{4} $ (T^4 - 196T^3 - 784176T^2 - 80103760*T - 801702800)^4
$79$	$ (T^{8} + \cdots + 56\!\cdots\!00)^{2} $ (T^8 + 1968580T^6 + 1179370743744T^4 + 235521627919360000*T^2 + 5665551824419600000000)^2
$83$	$ (T^{8} + \cdots + 19\!\cdots\!00)^{2} $ (T^8 + 1240296T^6 + 493626060000T^4 + 65687100981250000*T^2 + 1909613525765625000000)^2
$89$	$ (T^{8} + \cdots + 16\!\cdots\!00)^{2} $ (T^8 + 4551072T^6 + 5063578767360T^4 + 183483160220057600*T^2 + 1656555228979200000000)^2
$97$	$ (T^{4} - 380 T^{3} + \cdots + 191786509040)^{4} $ (T^4 - 380T^3 - 1235568T^2 + 54862928*T + 191786509040)^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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	960.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{3} + 5 q^{5} + (\beta_{7} - \beta_{4} - \beta_1) q^{7} + (\beta_{9} - 2) q^{9} + ( - \beta_{8} - \beta_{7} - \beta_{3}) q^{11} - \beta_{14} q^{13} - 5 \beta_{4} q^{15} + (\beta_{14} + 2 \beta_{12} - \beta_{10} + \cdots - 1) q^{17}+ \cdots + ( - 9 \beta_{15} - 4 \beta_{13} + \cdots + 60 \beta_1) q^{99}+O(q^{100}) \) q - b4 * q^3 + 5 * q^5 + (b7 - b4 - b1) * q^7 + (b9 - 2) * q^9 + (-b8 - b7 - b3) * q^11 - b14 * q^13 - 5b4 q^15 + (b14 + 2b12 - b10 + b9 - b6 + b2 - 1) q^17 + (-b13 - b11 - 2b8 + 5b4 + b1) * q^19 + (b14 + b12 - 3b10 - 2b6 - 2b5 - b2 - 16) q^21 + (b15 - b11 - 2b8 + 5b4 + b1) * q^23 + 25 * q^25 + (-2b13 + 3b11 - 2b8 + 4b7 + 2b4 - b3 - 3b1) * q^27 + (-3b10 - 3b9 - 2b6 - 4b5 + 2b2 + 12) q^29 + (2b7 - 12b4 + 5b3 - 10b1) * q^31 + (-2b14 + b12 + 3b10 + 3b9 - b6 - b5 + 5b2 - 33) * q^33 + (5b7 - 5b4 - 5b1) q^35 + (3b14 - 4b12 - 4b10 + 4b9 - 3b6 - b5 - 2b2 + 2) * q^37 + (-2b15 - b13 + 2b11 - 6b8 - 5b7 + 5b4 - 4b3 + 11b1) q^39 + (2b14 - 6b12 + 3b10 - 3b9 - 2b6 + 5b5 - 3b2 + 3) q^41 + (-b15 - 3b13 - 2b11 - 2b8 + 4b4 + 2b1) q^43 + (5b9 - 10) q^45 + (-b13 + 5b11 - 7b8 + 26b4 - 2b1) * q^47 + (-7b10 - 7b9 - 4b6 - 8b5 - 9b2 + 36) q^49 + (b15 - b13 - 7b11 - 2b8 + 16b7 - 7b4 - 4b3 + 2b1) * q^51 + (-6b10 - 6b9 - 3b6 - 6b5 + 4b2 + 60) q^53 + (-5b8 - 5b7 - 5b3) q^55 + (-b14 - 7b12 - 8b9 - 6b6 - 4b5 - 5b2 - 125) q^57 + (-5b8 - 19b7 + 22b4 - 9b3 + 5b1) q^59 + (-4b14 + 8b12 - 7b10 + 7b9 + 2b6 - 9b5 + 4b2 - 4) q^61 + (3b15 - 2b13 - 3b11 - 18b8 + 4b7 + 17b4 - 4b3 + 36b1) * q^63 - 5b14 q^65 + (3b15 - 13b13 + 14b11 + 2b8 + 8b4 + b1) q^67 + (-9b14 - 3b12 - 6b10 - 15b9 - b6 - 5b5 - 12b2 - 123) * q^69 + (-4b15 + 8b13 - 6b11 - 16b8 + 42b4 - 3b1) * q^71 + (-2b10 - 2b9 - 14b6 - 28b5 + 9b2 + 49) q^73 - 25b4 q^75 + (7b6 + 14b5 + 10b2 + 312) q^77 + (-28b8 - 8b7 - 78b4 + b3 - 55b1) * q^79 + (6b14 + 12b12 + 3b10 + 5b9 - 13b6 + b5 - 241) q^81 + (3b8 - 6b7 + 47b4 - 16b3 - 20b1) q^83 + (5b14 + 10b12 - 5b10 + 5b9 - 5b6 + 5b2 - 5) * q^85 + (-2b15 + 7b13 - 13b11 - 23b8 - 9b7 - 15b4 + 12b3 + 38b1) * q^87 + (-10b14 + 10b12 + 4b10 - 4b9 - 17b6 + 21b5 + 5b2 - 5) q^89 + (-4b15 + 16b13 - 10b11 + 20b8 - 70b4 - 5b1) * q^91 + (7b14 - 8b12 - 6b10 + 11b6 - 2b5 - 22b2 - 248) * q^93 + (-5b13 - 5b11 - 10b8 + 25b4 + 5b1) q^95 + (16b10 + 16b9 - 8b6 - 16b5 - b2 + 95) * q^97 + (-9b15 - 4b13 + 6b8 + 14b7 + 16b4 + 4b3 + 60b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 80 q^{5} - 32 q^{9} - 248 q^{21} + 400 q^{25} + 192 q^{29} - 520 q^{33} - 160 q^{45} + 576 q^{49} + 960 q^{53} - 2056 q^{57} - 1992 q^{69} + 784 q^{73} + 4992 q^{77} - 3760 q^{81} - 4032 q^{93} + 1520 q^{97}+O(q^{100}) \) 16 * q + 80 * q^5 - 32 * q^9 - 248 * q^21 + 400 * q^25 + 192 * q^29 - 520 * q^33 - 160 * q^45 + 576 * q^49 + 960 * q^53 - 2056 * q^57 - 1992 * q^69 + 784 * q^73 + 4992 * q^77 - 3760 * q^81 - 4032 * q^93 + 1520 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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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	960.4.b.b

Level	$960$
Weight	$4$
Character orbit	960.b
Analytic conductor	$56.642$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(56.6418336055\)
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:	\( x^{16} + 3472x^{12} + 2676096x^{8} + 573099300x^{4} + 31755240000 \) x^16 + 3472x^12 + 2676096x^8 + 573099300*x^4 + 31755240000
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{23}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -9491\nu^{14} - 33546752\nu^{10} - 27708714936\nu^{6} - 8590329053100\nu^{2} ) / 14972808163500 \) (-9491v^14 - 33546752v^10 - 27708714936v^6 - 8590329053100v^2) / 14972808163500
\(\beta_{2}\)	\(=\)	\( ( 4231\nu^{12} + 13092190\nu^{8} + 6863423634\nu^{4} + 625475850885 ) / 13611643785 \) (4231v^12 + 13092190v^8 + 6863423634*v^4 + 625475850885) / 13611643785
\(\beta_{3}\)	\(=\)	\( ( - 28473 \nu^{15} + 350020 \nu^{14} - 100640256 \nu^{11} + 621724840 \nu^{10} + \cdots + 179673697962000 \nu ) / 179673697962000 \) (-28473v^15 + 350020v^14 - 100640256v^11 + 621724840v^10 - 83126144808v^7 - 905609065680v^6 - 25770987159300v^3 - 608813123886000v^2 + 179673697962000*v) / 179673697962000
\(\beta_{4}\)	\(=\)	\( ( - 28473 \nu^{15} + 1672642 \nu^{14} - 100640256 \nu^{11} + 5623847224 \nu^{10} + \cdots + 179673697962000 \nu ) / 359347395924000 \) (-28473v^15 + 1672642v^14 - 100640256v^11 + 5623847224v^10 - 83126144808v^7 + 3828864053232v^6 - 25770987159300v^3 + 472992545097000v^2 + 179673697962000*v) / 359347395924000
\(\beta_{5}\)	\(=\)	\( ( 9491 \nu^{15} - 6020080 \nu^{12} + 33546752 \nu^{11} - 20452693360 \nu^{8} + \cdots - 17\!\cdots\!00 ) / 29945616327000 \) (9491v^15 - 6020080v^12 + 33546752v^11 - 20452693360v^8 + 27708714936v^7 - 14231099195280v^4 + 8590329053100v^3 + 59891232654000v - 1708827842106000) / 29945616327000
\(\beta_{6}\)	\(=\)	\( ( - 9491 \nu^{15} - 3010040 \nu^{12} - 33546752 \nu^{11} - 10226346680 \nu^{8} + \cdots - 854413921053000 ) / 14972808163500 \) (-9491v^15 - 3010040v^12 - 33546752v^11 - 10226346680v^8 - 27708714936v^7 - 7115549597640v^4 - 8590329053100v^3 - 59891232654000v - 854413921053000) / 14972808163500
\(\beta_{7}\)	\(=\)	\( ( - 9491 \nu^{15} + 1739994 \nu^{14} - 33546752 \nu^{11} + 5643774168 \nu^{10} + \cdots + 59891232654000 \nu ) / 119782465308000 \) (-9491v^15 + 1739994v^14 - 33546752v^11 + 5643774168v^10 - 27708714936v^7 + 3443420134224v^6 - 8590329053100v^3 + 371256740469000v^2 + 59891232654000*v) / 119782465308000
\(\beta_{8}\)	\(=\)	\( ( 28473 \nu^{15} + 1672642 \nu^{14} + 100640256 \nu^{11} + 5623847224 \nu^{10} + \cdots - 179673697962000 \nu ) / 119782465308000 \) (28473v^15 + 1672642v^14 + 100640256v^11 + 5623847224v^10 + 83126144808v^7 + 3828864053232v^6 + 25770987159300v^3 + 472992545097000v^2 - 179673697962000*v) / 119782465308000
\(\beta_{9}\)	\(=\)	\( ( 836321 \nu^{15} + 9030120 \nu^{13} - 1782000 \nu^{12} + 2811923612 \nu^{11} + \cdots - 35\!\cdots\!00 ) / 179673697962000 \) (836321v^15 + 9030120v^13 - 1782000v^12 + 2811923612v^11 + 30679040040v^9 - 6929663400v^8 + 1914432026616v^7 + 21346648792920v^5 - 9453130790400v^4 + 236496272548500v^3 + 2473404914178000*v - 3587675500530000) / 179673697962000
\(\beta_{10}\)	\(=\)	\( ( - 836321 \nu^{15} - 9030120 \nu^{13} + 34338480 \nu^{12} - 2811923612 \nu^{11} + \cdots + 66\!\cdots\!00 ) / 179673697962000 \) (-836321v^15 - 9030120v^13 + 34338480v^12 - 2811923612v^11 - 30679040040v^9 + 115786496760v^8 - 1914432026616v^7 - 21346648792920v^5 + 75933464381280v^4 - 236496272548500v^3 - 2473404914178000*v + 6665291552106000) / 179673697962000
\(\beta_{11}\)	\(=\)	\( ( - 8826043 \nu^{15} - 5359602 \nu^{14} + 75804960 \nu^{13} - 30043408096 \nu^{11} + \cdots + 31\!\cdots\!00 \nu ) / 10\!\cdots\!00 \) (-8826043v^15 - 5359602v^14 + 75804960v^13 - 30043408096v^11 - 18079224744v^10 + 259291647120v^9 - 21549917073528v^7 - 12484105897392v^6 + 189679451924160v^5 - 3465446821411500v^3 - 1728229481202600v^2 + 31454432763534000v) / 1078042187772000
\(\beta_{12}\)	\(=\)	\( ( - 4455731 \nu^{15} - 37902480 \nu^{13} - 83773800 \nu^{12} - 15172664432 \nu^{11} + \cdots - 12\!\cdots\!00 ) / 539021093886000 \) (-4455731v^15 - 37902480v^13 - 83773800v^12 - 15172664432v^11 - 129645823560v^9 - 259225362000v^8 - 10899647753976v^7 - 94839725962080v^5 - 135895787953200v^4 - 1771379891444700v^3 - 15996726928710000*v - 12114911300580000) / 539021093886000
\(\beta_{13}\)	\(=\)	\( ( - 1701115 \nu^{15} - 56946 \nu^{14} + 18060240 \nu^{13} - 5724487480 \nu^{11} + \cdots + 51\!\cdots\!00 \nu ) / 179673697962000 \) (-1701115v^15 - 56946v^14 + 18060240v^13 - 5724487480v^11 - 201280512v^10 + 61358080080v^9 - 3911990198040v^7 - 166252289616v^6 + 42693297585840v^5 - 498763532256300v^3 - 51541974318600v^2 + 5126483526318000v) / 179673697962000
\(\beta_{14}\)	\(=\)	\( ( - 6244177 \nu^{15} - 85255500 \nu^{13} - 20426779144 \nu^{11} - 276078217800 \nu^{9} + \cdots - 15\!\cdots\!00 \nu ) / 539021093886000 \) (-6244177v^15 - 85255500v^13 - 20426779144v^11 - 276078217800v^9 - 12702558354192v^7 - 165228007150800v^5 - 1301547587591700v^3 - 15555499512498000v) / 539021093886000
\(\beta_{15}\)	\(=\)	\( ( - 3564376 \nu^{15} + 85419 \nu^{14} + 58165140 \nu^{13} - 11387166772 \nu^{11} + \cdots + 70\!\cdots\!00 \nu ) / 269510546943000 \) (-3564376v^15 + 85419v^14 + 58165140v^13 - 11387166772v^11 + 301920768v^10 + 184041097680v^9 - 6460505405496v^7 + 249378434424v^6 + 101188060772040v^5 - 437432846990400v^3 + 77312961477900v^2 + 7057242582192000v) / 269510546943000

\(\nu\)	\(=\)	\( ( -2\beta_{8} - \beta_{6} + \beta_{5} + 6\beta_{4} ) / 12 \) (-2b8 - b6 + b5 + 6b4) / 12
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{8} - 6\beta_{7} + 6\beta_{3} - 75\beta_1 ) / 12 \) (2b8 - 6b7 + 6b3 - 75b1) / 12
\(\nu^{3}\)	\(=\)	\( ( - 12 \beta_{13} + 18 \beta_{12} + 18 \beta_{11} - 12 \beta_{10} + 12 \beta_{9} + 74 \beta_{8} + \cdots - 9 ) / 12 \) (-12b13 + 18b12 + 18b11 - 12b10 + 12b9 + 74b8 - 43b6 + 31b5 - 204b4 + 9b2 - 3*b1 - 9) / 12
\(\nu^{4}\)	\(=\)	\( ( -174\beta_{10} - 174\beta_{9} - 43\beta_{6} - 86\beta_{5} + 18\beta_{2} - 5208 ) / 6 \) (-174b10 - 174b9 - 43b6 - 86b5 + 18*b2 - 5208) / 6
\(\nu^{5}\)	\(=\)	\( ( 36 \beta_{15} - 72 \beta_{14} - 858 \beta_{13} - 1116 \beta_{12} + 1116 \beta_{11} + 894 \beta_{10} + \cdots + 558 ) / 12 \) (36b15 - 72b14 - 858b13 - 1116b12 + 1116b11 + 894b10 - 894b9 + 3194b8 + 2062b6 - 1168b5 - 8466b4 - 558b2 - 111*b1 + 558) / 12
\(\nu^{6}\)	\(=\)	\( ( -2633\beta_{8} + 6009\beta_{7} - 5220\beta_{4} - 4344\beta_{3} + 28020\beta_1 ) / 3 \) (-2633b8 + 6009b7 - 5220b4 - 4344b3 + 28020*b1) / 3
\(\nu^{7}\)	\(=\)	\( ( - 1665 \beta_{15} - 3330 \beta_{14} + 24261 \beta_{13} - 29394 \beta_{12} - 29394 \beta_{11} + \cdots + 14697 ) / 6 \) (-1665b15 - 3330b14 + 24261b13 - 29394b12 - 29394b11 + 25926b10 - 25926b9 - 74257b8 + 50924b6 - 24998b5 + 193377b4 - 14697b2 + 1734*b1 + 14697) / 6
\(\nu^{8}\)	\(=\)	\( ( 212937\beta_{10} + 212937\beta_{9} + 36209\beta_{6} + 72418\beta_{5} - 53874\beta_{2} + 5026944 ) / 3 \) (212937b10 + 212937b9 + 36209b6 + 72418b5 - 53874*b2 + 5026944) / 3
\(\nu^{9}\)	\(=\)	\( ( - 107748 \beta_{15} + 215496 \beta_{14} + 1275864 \beta_{13} + 1493118 \beta_{12} - 1493118 \beta_{11} + \cdots - 746559 ) / 6 \) (-107748b15 + 215496b14 + 1275864b13 + 1493118b12 - 1493118b11 - 1383612b10 + 1383612b9 - 3576682b8 - 2534021b6 + 1150409b5 + 9236928b4 + 746559b2 + 54753*b1 - 746559) / 6
\(\nu^{10}\)	\(=\)	\( ( 7202168\beta_{8} - 16563174\beta_{7} + 17223930\beta_{4} + 10472874\beta_{3} - 62245920\beta_1 ) / 3 \) (7202168b8 - 16563174b7 + 17223930b4 + 10472874b3 - 62245920*b1) / 3
\(\nu^{11}\)	\(=\)	\( ( 6090300 \beta_{15} + 12180600 \beta_{14} - 65205726 \beta_{13} + 75017844 \beta_{12} + 75017844 \beta_{11} + \cdots - 37508922 ) / 6 \) (6090300b15 + 12180600b14 - 65205726b13 + 75017844b12 + 75017844b11 - 71296026b10 + 71296026b9 + 175246942b8 - 126316634b6 + 55020608b5 - 450722982b4 + 37508922b2 - 1860909*b1 - 37508922) / 6
\(\nu^{12}\)	\(=\)	\( ( - 517772112 \beta_{10} - 517772112 \beta_{9} - 77166509 \beta_{6} - 154333018 \beta_{5} + \cdots - 11774469009 ) / 3 \) (-517772112b10 - 517772112b9 - 77166509b6 - 154333018b5 + 161756739*b2 - 11774469009) / 3
\(\nu^{13}\)	\(=\)	\( ( 161756739 \beta_{15} - 323513478 \beta_{14} - 1645330287 \beta_{13} - 1876829814 \beta_{12} + \cdots + 938414907 ) / 3 \) (161756739b15 - 323513478b14 - 1645330287b13 - 1876829814b12 + 1876829814b11 + 1807087026b10 - 1807087026b9 + 4330053971b8 + 3149448868b6 - 1342361842b5 - 11113332099b4 - 938414907b2 - 34871394*b1 + 938414907) / 3
\(\nu^{14}\)	\(=\)	\( ( - 18222259178 \beta_{8} + 42358499214 \beta_{7} - 45639807840 \beta_{4} - 25692734454 \beta_{3} + 150447844695 \beta_1 ) / 3 \) (-18222259178b8 + 42358499214b7 - 45639807840b4 - 25692734454b3 + 150447844695*b1) / 3
\(\nu^{15}\)	\(=\)	\( ( - 8332882380 \beta_{15} - 16665764760 \beta_{14} + 82538255208 \beta_{13} - 93743968122 \beta_{12} + \cdots + 46871984061 ) / 3 \) (-8332882380b15 - 16665764760b14 + 82538255208b13 - 93743968122b12 - 93743968122b11 + 90871137588b10 - 90871137588b9 - 214906258526b8 + 157055139247b6 - 66184001659b5 + 550974807456b4 - 46871984061b2 + 1436415267*b1 + 46871984061) / 3

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

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 671.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} + \cdots + 282429536481 \) T^16 + 16T^14 + 1068T^12 - 8064T^10 + 425574T^8 - 5878656T^6 + 567578988T^4 + 6198727824*T^2 + 282429536481
$5$	\( (T - 5)^{16} \) (T - 5)^16
$7$	\( (T^{8} + 1228 T^{6} + \cdots + 746382400)^{2} \) (T^8 + 1228T^6 + 355728T^4 + 29741776*T^2 + 746382400)^2
$11$	\( (T^{8} + 5328 T^{6} + \cdots + 20666937600)^{2} \) (T^8 + 5328T^6 + 6443040T^4 + 1295435776*T^2 + 20666937600)^2
$13$	\( (T^{8} + \cdots + 5700677875200)^{2} \) (T^8 + 11100T^6 + 33417936T^4 + 26308811840*T^2 + 5700677875200)^2
$17$	\( (T^{8} + \cdots + 430656937920000)^{2} \) (T^8 + 24972T^6 + 204866640T^4 + 615178049600*T^2 + 430656937920000)^2
$19$	\( (T^{8} + \cdots + 420208715520000)^{2} \) (T^8 - 22452T^6 + 170524080T^4 - 491994718400*T^2 + 420208715520000)^2
$23$	\( (T^{8} + \cdots + 26\!\cdots\!00)^{2} \) (T^8 - 54228T^6 + 722739600T^4 - 2534842350000*T^2 + 2630632950000000)^2
$29$	\( (T^{4} - 48 T^{3} + \cdots + 39321360)^{4} \) (T^4 - 48T^3 - 37488T^2 + 1914544*T + 39321360)^4
$31$	\( (T^{8} + \cdots + 21\!\cdots\!00)^{2} \) (T^8 + 115300T^6 + 3827042304T^4 + 49051753171200*T^2 + 210462939653760000)^2
$37$	\( (T^{8} + \cdots + 35\!\cdots\!00)^{2} \) (T^8 + 312204T^6 + 28084899984T^4 + 655059569514560*T^2 + 3580109939665036800)^2
$41$	\( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) (T^8 + 354048T^6 + 31346984640T^4 + 566830807366400*T^2 + 1100247971328000000)^2
$43$	\( (T^{8} + \cdots + 769198518000000)^{2} \) (T^8 - 136776T^6 + 4800178944T^4 - 12264362766000*T^2 + 769198518000000)^2
$47$	\( (T^{8} + \cdots + 16\!\cdots\!00)^{2} \) (T^8 - 240132T^6 + 15098390160T^4 - 318069644807600*T^2 + 1619598725086320000)^2
$53$	\( (T^{4} - 240 T^{3} + \cdots - 1424598000)^{4} \) (T^4 - 240T^3 - 125208T^2 + 34562720*T - 1424598000)^4
$59$	\( (T^{8} + \cdots + 71\!\cdots\!00)^{2} \) (T^8 + 784416T^6 + 148492728864T^4 + 2027858972588800*T^2 + 7113919566126240000)^2
$61$	\( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) (T^8 + 996240T^6 + 163089475200T^4 + 8813159912160000*T^2 + 130601565388800000000)^2
$67$	\( (T^{8} + \cdots + 63\!\cdots\!00)^{2} \) (T^8 - 1839864T^6 + 1008444966144T^4 - 149273005485656240*T^2 + 6326348137507031356800)^2
$71$	\( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) (T^8 - 1526832T^6 + 578265123840T^4 - 66391674608921600*T^2 + 1492639422100992000000)^2
$73$	\( (T^{4} - 196 T^{3} + \cdots - 801702800)^{4} \) (T^4 - 196T^3 - 784176T^2 - 80103760*T - 801702800)^4
$79$	\( (T^{8} + \cdots + 56\!\cdots\!00)^{2} \) (T^8 + 1968580T^6 + 1179370743744T^4 + 235521627919360000*T^2 + 5665551824419600000000)^2
$83$	\( (T^{8} + \cdots + 19\!\cdots\!00)^{2} \) (T^8 + 1240296T^6 + 493626060000T^4 + 65687100981250000*T^2 + 1909613525765625000000)^2
$89$	\( (T^{8} + \cdots + 16\!\cdots\!00)^{2} \) (T^8 + 4551072T^6 + 5063578767360T^4 + 183483160220057600*T^2 + 1656555228979200000000)^2
$97$	\( (T^{4} - 380 T^{3} + \cdots + 191786509040)^{4} \) (T^4 - 380T^3 - 1235568T^2 + 54862928*T + 191786509040)^4
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