LMFDB - Newform orbit 672.4.k.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,4,Mod(545,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("672.545");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.k (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:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.36004060626969600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 28x^{14} + 308x^{12} - 1710x^{10} + 5156x^{8} - 7740x^{6} + 9473x^{4} + 368x^{2} + 256 $ x^16 - 28x^14 + 308x^12 - 1710x^10 + 5156x^8 - 7740x^6 + 9473x^4 + 368*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{60} $
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_{5} q^{3} + \beta_{10} q^{5} + (\beta_{7} - \beta_{4}) q^{7} + (3 \beta_{8} - 3) q^{9}+O(q^{10}) $ q + b5 * q^3 + b10 * q^5 + (b7 - b4) * q^7 + (3b8 - 3) q^9	$ q + \beta_{5} q^{3} + \beta_{10} q^{5} + (\beta_{7} - \beta_{4}) q^{7} + (3 \beta_{8} - 3) q^{9} + (\beta_{12} + \beta_{11}) q^{11} + ( - 2 \beta_{3} + \beta_{2}) q^{13} + (\beta_{12} - 2 \beta_{11} + \cdots - \beta_1) q^{15}+ \cdots + ( - 3 \beta_{12} - 3 \beta_{11} + \cdots - 42 \beta_1) q^{99}+O(q^{100}) $ q + b5 * q^3 + b10 * q^5 + (b7 - b4) * q^7 + (3b8 - 3) q^9 + (b12 + b11) * q^11 + (-2b3 + b2) q^13 + (b12 - 2b11 + b7 - b6 - b1) q^15 + (-4b10 - b9) q^17 + (-3b7 - 3b6 - 2b5 + 8b4) * q^19 + (-b13 + 4b10 + b8 - b3 - 2b2 - 15) * q^21 + (b12 - 5b11) q^23 + (-b14 - b13 - 21) * q^25 + (-6b5 - 27b4) * q^27 + (-b14 + b13 - b10 - b9 - 12b8 - b3) q^29 + (-4b7 - 4b6 + 11b5 - 3b4) * q^31 + (3b9 - 3b3 + 3b2) q^33 + (b15 - 3b12 + 4b11 + 16b5 + 16b4) * q^35 + (-b14 - b13 + 274) * q^37 + (8b12 + 5b11 - 10b7 + 10b6 - 11b1) q^39 + (4b10 - b9) q^41 + (-11b7 + 11b6 - 11b1) q^43 + (-3b10 - 6b3 - 18b2) q^45 + (-b15 + 34b5 + 34b4) * q^47 + (-2b14 - 2b13 + b3 + 20b2 - 39) q^49 + (b12 + 13b11 + b7 - b6 + 11b1) * q^51 + (6b14 - 6b13 + 6b10 + 6b9 - 3b8 + 6b3) * q^53 + (-3b7 - 3b6 + 7b5 - b4) q^55 + (6b13 - 3b10 - 3b9 - 12b8 - 3b3 + 150) q^57 + (5b15 + 37b5 + 37b4) q^59 + (4b3 + 29b2) * q^61 + (3b15 + 12b12 - 9b11 - 3b7 - 15b5 - 12b4) * q^63 + (-3b14 + 3b13 - 3b10 - 3b9 + 23b8 - 3b3) * q^65 + (-7b7 + 7b6 + 7b1) q^67 + (-30b10 - 3b9 - 3b3 - 21b2) * q^69 + (25b12 + 29b11) * q^71 + (8b3 - 34b2) * q^73 + (3b15 - 15b7 - 15b6 - 6b5 + 15b4) q^75 + (b14 - b13 + 15b10 + 14b9 + 23b8 + b3) q^77 + (-31b7 + 31b6 - 2b1) q^79 + (-18b8 - 711) q^81 + (7b15 - 33b5 - 33b4) q^83 + (3b14 + 3b13 - 376) * q^85 + (-3b15 - 12b7 - 12b6 + 25b5 + 129b4) q^87 + (62b10 - 19b9) * q^89 + (-24b7 + 28b6 + 70b5 - 74b4 - 63b1) q^91 + (8b13 - 4b10 - 4b9 + 25b8 - 4b3 - 210) q^93 + (14b12 - 16b11) * q^95 + (30b3 + 20b2) * q^97 + (-3b12 - 3b11 - 30b7 + 30b6 - 42b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9}+O(q^{10}) $ 16 * q - 48 * q^9	$ 16 q - 48 q^{9} - 240 q^{21} - 336 q^{25} + 4384 q^{37} - 624 q^{49} + 2400 q^{57} - 11376 q^{81} - 6016 q^{85} - 3360 q^{93}+O(q^{100}) $ 16 * q - 48 * q^9 - 240 * q^21 - 336 * q^25 + 4384 * q^37 - 624 * q^49 + 2400 * q^57 - 11376 * q^81 - 6016 * q^85 - 3360 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 28x^{14} + 308x^{12} - 1710x^{10} + 5156x^{8} - 7740x^{6} + 9473x^{4} + 368x^{2} + 256 $ x^16 - 28*x^14 + 308*x^12 - 1710*x^10 + 5156*x^8 - 7740*x^6 + 9473*x^4 + 368*x^2 + 256 :

$\beta_{1}$	$=$	$ ( - 2489 \nu^{14} + 67190 \nu^{12} - 692402 \nu^{10} + 3399884 \nu^{8} - 7916482 \nu^{6} + \cdots - 15246632 ) / 858585 $ (-2489v^14 + 67190v^12 - 692402v^10 + 3399884v^8 - 7916482v^6 + 3911494v^4 + 41405*v^2 - 15246632) / 858585
$\beta_{2}$	$=$	$ ( 2131 \nu^{14} - 59442 \nu^{12} + 649262 \nu^{10} - 3562780 \nu^{8} + 10578358 \nu^{6} + \cdots + 203000 ) / 245055 $ (2131v^14 - 59442v^12 + 649262v^10 - 3562780v^8 + 10578358v^6 - 15767882v^4 + 20695449*v^2 + 203000) / 245055
$\beta_{3}$	$=$	$ ( 257821 \nu^{14} - 7154726 \nu^{12} + 77904750 \nu^{10} - 427794088 \nu^{8} + 1277219298 \nu^{6} + \cdots + 24994224 ) / 18134070 $ (257821v^14 - 7154726v^12 + 77904750v^10 - 427794088v^8 + 1277219298v^6 - 1901704874v^4 + 2513516707*v^2 + 24994224) / 18134070
$\beta_{4}$	$=$	$ ( 93447445 \nu^{15} + 63382528 \nu^{14} - 2641870140 \nu^{13} - 1789488000 \nu^{12} + \cdots + 4336768544 ) / 26403205920 $ (93447445v^15 + 63382528v^14 - 2641870140v^13 - 1789488000v^12 + 29498252420v^11 + 19935400064v^10 - 167794657030v^9 - 112876019008v^8 + 526871150260v^7 + 349673895424v^6 - 857213296940v^5 - 541212947648v^4 + 1063095809685v^3 + 620067302400v^2 - 136212862240*v + 4336768544) / 26403205920
$\beta_{5}$	$=$	$ ( 93447445 \nu^{15} - 63382528 \nu^{14} - 2641870140 \nu^{13} + 1789488000 \nu^{12} + \cdots - 4336768544 ) / 26403205920 $ (93447445v^15 - 63382528v^14 - 2641870140v^13 + 1789488000v^12 + 29498252420v^11 - 19935400064v^10 - 167794657030v^9 + 112876019008v^8 + 526871150260v^7 - 349673895424v^6 - 857213296940v^5 + 541212947648v^4 + 1063095809685v^3 - 620067302400v^2 - 136212862240*v - 4336768544) / 26403205920
$\beta_{6}$	$=$	$ ( 93447445 \nu^{15} + 154510600 \nu^{14} - 2641870140 \nu^{13} - 4442592944 \nu^{12} + \cdots - 403865289216 ) / 26403205920 $ (93447445v^15 + 154510600v^14 - 2641870140v^13 - 4442592944v^12 + 29498252420v^11 + 50816310768v^10 - 167794657030v^9 - 299308784128v^8 + 526871150260v^7 + 989831110800v^6 - 857213296940v^5 - 1760335412432v^4 + 1063095809685v^3 + 2124526783288v^2 - 136212862240*v - 403865289216) / 26403205920
$\beta_{7}$	$=$	$ ( 93447445 \nu^{15} + 282190040 \nu^{14} - 2641870140 \nu^{13} - 7913287696 \nu^{12} + \cdots + 433119134976 ) / 26403205920 $ (93447445v^15 + 282190040v^14 - 2641870140v^13 - 7913287696v^12 + 29498252420v^11 + 87048069072v^10 - 167794657030v^9 - 480439824512v^8 + 526871150260v^7 + 1415451829680v^6 - 857213296940v^5 - 1964463668848v^4 + 1063095809685v^3 + 2122659272552v^2 - 136212862240*v + 433119134976) / 26403205920
$\beta_{8}$	$=$	$ ( 1192127 \nu^{15} - 33164340 \nu^{13} + 361428556 \nu^{11} - 1979041362 \nu^{9} + \cdots + 2120841376 \nu ) / 118933360 $ (1192127v^15 - 33164340v^13 + 361428556v^11 - 1979041362v^9 + 5841756956v^7 - 8401996612v^5 + 10296719615v^3 + 2120841376v) / 118933360
$\beta_{9}$	$=$	$ ( - 21125019 \nu^{15} + 591251420 \nu^{13} - 6510390532 \nu^{11} + 36323144514 \nu^{9} + \cdots + 27824604368 \nu ) / 1885943280 $ (-21125019v^15 + 591251420v^13 - 6510390532v^11 + 36323144514v^9 - 111081432932v^7 + 173016545964v^5 - 217859771155v^3 + 27824604368v) / 1885943280
$\beta_{10}$	$=$	$ ( 209764405 \nu^{15} - 5899494724 \nu^{13} + 65321421148 \nu^{11} - 366342177598 \nu^{9} + \cdots - 281999112176 \nu ) / 13201602960 $ (209764405v^15 - 5899494724v^13 + 65321421148v^11 - 366342177598v^9 + 1123055813180v^7 - 1750773555572v^5 + 2208800523293v^3 - 281999112176v) / 13201602960
$\beta_{11}$	$=$	$ ( - 40353 \nu^{15} + 1127452 \nu^{13} - 12350948 \nu^{11} + 67993662 \nu^{9} - 201270004 \nu^{7} + \cdots - 67584896 \nu ) / 2133420 $ (-40353v^15 + 1127452v^13 - 12350948v^11 + 67993662v^9 - 201270004v^7 + 286880844v^5 - 329714129v^3 - 67584896v) / 2133420
$\beta_{12}$	$=$	$ ( 4048289 \nu^{15} - 113088308 \nu^{13} + 1239000428 \nu^{11} - 6826479110 \nu^{9} + \cdots + 6806023760 \nu ) / 145072560 $ (4048289v^15 - 113088308v^13 + 1239000428v^11 - 6826479110v^9 + 20234914252v^7 - 28836952228v^5 + 33199159081v^3 + 6806023760v) / 145072560
$\beta_{13}$	$=$	$ ( 1293702675 \nu^{15} + 158324712 \nu^{14} - 36045374084 \nu^{13} - 4542913648 \nu^{12} + \cdots - 1337524604416 ) / 26403205920 $ (1293702675v^15 + 158324712v^14 - 36045374084v^13 - 4542913648v^12 + 393370524028v^11 + 51487213232v^10 - 2156166505098v^9 - 294675483648v^8 + 6370246061900v^7 + 903152360656v^6 - 9206433819732v^5 - 1351104592656v^4 + 11285188481203v^3 + 1831151807576v^2 + 2096093091424*v - 1337524604416) / 26403205920
$\beta_{14}$	$=$	$ ( - 1293702675 \nu^{15} - 217062664 \nu^{14} + 36045374084 \nu^{13} + 5874367408 \nu^{12} + \cdots - 1373916194560 ) / 26403205920 $ (-1293702675v^15 - 217062664v^14 + 36045374084v^13 + 5874367408v^12 - 393370524028v^11 - 61942102768v^10 + 2156166505098v^9 + 328192708480v^8 - 6370246061900v^7 - 956478937232v^6 + 9206433819732v^5 + 1417777703888v^4 - 11285188481203v^3 - 1828528517816v^2 - 2096093091424*v - 1373916194560) / 26403205920
$\beta_{15}$	$=$	$ ( 2317 \nu^{15} - 65580 \nu^{13} + 733396 \nu^{11} - 4178502 \nu^{9} + 13128356 \nu^{7} + \cdots - 3396224 \nu ) / 19220 $ (2317v^15 - 65580v^13 + 733396v^11 - 4178502v^9 + 13128356v^7 - 21366172v^5 + 26508445v^3 - 3396224v) / 19220

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

$\nu$	$=$	$ ( \beta_{11} - \beta_{10} + 3\beta_{9} + 4\beta_{8} + 4\beta_{5} + 4\beta_{4} ) / 64 $ (b11 - b10 + 3b9 + 4b8 + 4b5 + 4b4) / 64
$\nu^{2}$	$=$	$ ( \beta_{14} + \beta_{13} - 3\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} - 3\beta _1 + 112 ) / 32 $ (b14 + b13 - 3b7 + b6 + b5 + b4 + 2b3 - b2 - 3*b1 + 112) / 32
$\nu^{3}$	$=$	$ ( - 3 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} + 5 \beta_{11} + 18 \beta_{10} + 22 \beta_{9} + \cdots + 3 \beta_{3} ) / 64 $ (-3b15 + 3b14 - 3b13 + 5b11 + 18b10 + 22b9 + 37b8 + 48b5 + 48b4 + 3b3) / 64
$\nu^{4}$	$=$	$ ( 7 \beta_{14} + 7 \beta_{13} - 25 \beta_{7} + 27 \beta_{6} + 7 \beta_{5} - 9 \beta_{4} + 18 \beta_{3} + \cdots + 672 ) / 32 $ (7b14 + 7b13 - 25b7 + 27b6 + 7b5 - 9b4 + 18b3 - 27b2 - 49*b1 + 672) / 32
$\nu^{5}$	$=$	$ ( - 35 \beta_{15} + 35 \beta_{14} - 35 \beta_{13} - 40 \beta_{12} - 59 \beta_{11} + 134 \beta_{10} + \cdots + 35 \beta_{3} ) / 64 $ (-35b15 + 35b14 - 35b13 - 40b12 - 59b11 + 134b10 + 162b9 + 325b8 + 576b5 + 576b4 + 35*b3) / 64
$\nu^{6}$	$=$	$ ( 48 \beta_{14} + 48 \beta_{13} - 166 \beta_{7} + 374 \beta_{6} + 244 \beta_{5} - 452 \beta_{4} + \cdots + 4840 ) / 32 $ (48b14 + 48b13 - 166b7 + 374b6 + 244b5 - 452b4 + 166b3 - 277b2 - 487*b1 + 4840) / 32
$\nu^{7}$	$=$	$ ( - 308 \beta_{15} + 336 \beta_{14} - 336 \beta_{13} - 896 \beta_{12} - 1345 \beta_{11} + \cdots + 336 \beta_{3} ) / 64 $ (-308b15 + 336b14 - 336b13 - 896b12 - 1345b11 + 993b10 + 1277b9 + 3084b8 + 5260b5 + 5260b4 + 336*b3) / 64
$\nu^{8}$	$=$	$ ( 365 \beta_{14} + 365 \beta_{13} - 567 \beta_{7} + 3885 \beta_{6} + 3997 \beta_{5} - 7315 \beta_{4} + \cdots + 37568 ) / 32 $ (365b14 + 365b13 - 567b7 + 3885b6 + 3997b5 - 7315b4 + 1626b3 - 2683b2 - 3969*b1 + 37568) / 32
$\nu^{9}$	$=$	$ ( - 2325 \beta_{15} + 3357 \beta_{14} - 3357 \beta_{13} - 12096 \beta_{12} - 17933 \beta_{11} + \cdots + 3357 \beta_{3} ) / 64 $ (-2325b15 + 3357b14 - 3357b13 - 12096b12 - 17933b11 + 8382b10 + 10682b9 + 31099b8 + 39888b5 + 39888b4 + 3357*b3) / 64
$\nu^{10}$	$=$	$ ( 2795 \beta_{14} + 2795 \beta_{13} + 4171 \beta_{7} + 35487 \beta_{6} + 48091 \beta_{5} - 87749 \beta_{4} + \cdots + 287632 ) / 32 $ (2795b14 + 2795b13 + 4171b7 + 35487b6 + 48091b5 - 87749b4 + 16730b3 - 27453b2 - 27919*b1 + 287632) / 32
$\nu^{11}$	$=$	$ ( - 15147 \beta_{15} + 34859 \beta_{14} - 34859 \beta_{13} - 134376 \beta_{12} - 198925 \beta_{11} + \cdots + 34859 \beta_{3} ) / 64 $ (-15147b15 + 34859b14 - 34859b13 - 134376b12 - 198925b11 + 71736b10 + 88364b9 + 323429b8 + 259560b5 + 259560b4 + 34859*b3) / 64
$\nu^{12}$	$=$	$ ( 18816 \beta_{14} + 18816 \beta_{13} + 116110 \beta_{7} + 301330 \beta_{6} + 506324 \beta_{5} + \cdots + 1933848 ) / 32 $ (18816b14 + 18816b13 + 116110b7 + 301330b6 + 506324b5 - 923764b4 + 173898b3 - 285291b2 - 165249*b1 + 1933848) / 32
$\nu^{13}$	$=$	$ ( - 77870 \beta_{15} + 358930 \beta_{14} - 358930 \beta_{13} - 1361360 \beta_{12} + \cdots + 358930 \beta_{3} ) / 64 $ (-77870b15 + 358930b14 - 358930b13 - 1361360b12 - 2015571b11 + 576421b10 + 673865b9 + 3329738b8 + 1333596b5 + 1333596b4 + 358930*b3) / 64
$\nu^{14}$	$=$	$ ( 87333 \beta_{14} + 87333 \beta_{13} + 1688193 \beta_{7} + 2422069 \beta_{6} + 4984629 \beta_{5} + \cdots + 8973328 ) / 32 $ (87333b14 + 87333b13 + 1688193b7 + 2422069b6 + 4984629b5 - 9094891b4 + 1761706b3 - 2890665b2 - 654875*b1 + 8973328) / 32
$\nu^{15}$	$=$	$ ( - 145235 \beta_{15} + 3563907 \beta_{14} - 3563907 \beta_{13} - 13104000 \beta_{12} + \cdots + 3563907 \beta_{3} ) / 64 $ (-145235b15 + 3563907b14 - 3563907b13 - 13104000b12 - 19402475b11 + 4126482b10 + 4378262b9 + 33059493b8 + 2486832b5 + 2486832b4 + 3563907*b3) / 64

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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} $

545.1

0.257054	+	0.309017i
−3.05957	+	0.309017i
2.19355	+	0.809017i
−1.12308	+	0.809017i
0.257054	−	0.309017i
−3.05957	−	0.309017i
2.19355	−	0.809017i
−1.12308	−	0.809017i
1.12308	−	0.809017i
−2.19355	−	0.809017i
3.05957	−	0.309017i
−0.257054	−	0.309017i
1.12308	+	0.809017i
−2.19355	+	0.809017i
3.05957	+	0.309017i
−0.257054	+	0.309017i

−3.46410

−

3.87298i

−14.3792

15.9613

9.39352i

−3.00000

26.8328i

545.2

−3.46410

−

3.87298i

−1.11272

−7.01699

−

17.1395i

−3.00000

26.8328i

545.3

−3.46410

−

3.87298i

1.11272

7.01699

−

17.1395i

−3.00000

26.8328i

545.4

−3.46410

−

3.87298i

14.3792

−15.9613

9.39352i

−3.00000

26.8328i

545.5

−3.46410

3.87298i

−14.3792

15.9613

−

9.39352i

−3.00000

−

26.8328i

545.6

−3.46410

3.87298i

−1.11272

−7.01699

17.1395i

−3.00000

−

26.8328i

545.7

−3.46410

3.87298i

1.11272

7.01699

17.1395i

−3.00000

−

26.8328i

545.8

−3.46410

3.87298i

14.3792

−15.9613

−

9.39352i

−3.00000

−

26.8328i

545.9

3.46410

−

3.87298i

−14.3792

−15.9613

9.39352i

−3.00000

−

26.8328i

545.10

3.46410

−

3.87298i

−1.11272

7.01699

−

17.1395i

−3.00000

−

26.8328i

545.11

3.46410

−

3.87298i

1.11272

−7.01699

−

17.1395i

−3.00000

−

26.8328i

545.12

3.46410

−

3.87298i

14.3792

15.9613

9.39352i

−3.00000

−

26.8328i

545.13

3.46410

3.87298i

−14.3792

−15.9613

−

9.39352i

−3.00000

26.8328i

545.14

3.46410

3.87298i

−1.11272

7.01699

17.1395i

−3.00000

26.8328i

545.15

3.46410

3.87298i

1.11272

−7.01699

17.1395i

−3.00000

26.8328i

545.16

3.46410

3.87298i

14.3792

15.9613

−

9.39352i

−3.00000

26.8328i

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
7.b	odd	2	1	inner
12.b	even	2	1	inner
21.c	even	2	1	inner
28.d	even	2	1	inner
84.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	672.4.k.c	✓	16
3.b	odd	2	1	inner	672.4.k.c	✓	16
4.b	odd	2	1	inner	672.4.k.c	✓	16
7.b	odd	2	1	inner	672.4.k.c	✓	16
12.b	even	2	1	inner	672.4.k.c	✓	16
21.c	even	2	1	inner	672.4.k.c	✓	16
28.d	even	2	1	inner	672.4.k.c	✓	16
84.h	odd	2	1	inner	672.4.k.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
672.4.k.c	✓	16	1.a	even	1	1	trivial
672.4.k.c	✓	16	3.b	odd	2	1	inner
672.4.k.c	✓	16	4.b	odd	2	1	inner
672.4.k.c	✓	16	7.b	odd	2	1	inner
672.4.k.c	✓	16	12.b	even	2	1	inner
672.4.k.c	✓	16	21.c	even	2	1	inner
672.4.k.c	✓	16	28.d	even	2	1	inner
672.4.k.c	✓	16	84.h	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} - 208T_{5}^{2} + 256 $ T5^4 - 208*T5^2 + 256 acting on $S_{4}^{\mathrm{new}}(672, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{4} + 6 T^{2} + 729)^{4} $ (T^4 + 6*T^2 + 729)^4
$5$	$ (T^{4} - 208 T^{2} + 256)^{4} $ (T^4 - 208*T^2 + 256)^4
$7$	$ (T^{8} + 156 T^{6} + \cdots + 13841287201)^{2} $ (T^8 + 156T^6 + 72422T^4 + 18353244*T^2 + 13841287201)^2
$11$	$ (T^{4} + 1968 T^{2} + 112896)^{4} $ (T^4 + 1968*T^2 + 112896)^4
$13$	$ (T^{4} + 8576 T^{2} + 7573504)^{4} $ (T^4 + 8576*T^2 + 7573504)^4
$17$	$ (T^{4} - 5968 T^{2} + 8386816)^{4} $ (T^4 - 5968*T^2 + 8386816)^4
$19$	$ (T^{4} + 15672 T^{2} + 23386896)^{4} $ (T^4 + 15672*T^2 + 23386896)^4
$23$	$ (T^{4} + 13488 T^{2} + 29419776)^{4} $ (T^4 + 13488*T^2 + 29419776)^4
$29$	$ (T^{4} + 43936 T^{2} + 25725184)^{4} $ (T^4 + 43936*T^2 + 25725184)^4
$31$	$ (T^{4} + 28408 T^{2} + 69288976)^{4} $ (T^4 + 28408*T^2 + 69288976)^4
$37$	$ (T^{2} - 548 T + 64516)^{8} $ (T^2 - 548*T + 64516)^8
$41$	$ (T^{4} - 7248 T^{2} + 12278016)^{4} $ (T^4 - 7248*T^2 + 12278016)^4
$43$	$ (T^{4} - 147136 T^{2} + 2938507264)^{4} $ (T^4 - 147136*T^2 + 2938507264)^4
$47$	$ (T^{4} - 139136 T^{2} + 1714622464)^{4} $ (T^4 - 139136*T^2 + 1714622464)^4
$53$	$ (T^{4} + 609696 T^{2} + 92056414464)^{4} $ (T^4 + 609696*T^2 + 92056414464)^4
$59$	$ (T^{4} - 835424 T^{2} + 81960818944)^{4} $ (T^4 - 835424*T^2 + 81960818944)^4
$61$	$ (T^{4} + 308096 T^{2} + 15847788544)^{4} $ (T^4 + 308096*T^2 + 15847788544)^4
$67$	$ (T^{4} - 122304 T^{2} + 88510464)^{4} $ (T^4 - 122304*T^2 + 88510464)^4
$71$	$ (T^{4} + 1353392 T^{2} + 21994076416)^{4} $ (T^4 + 1353392*T^2 + 21994076416)^4
$73$	$ (T^{4} + 667136 T^{2} + 48809181184)^{4} $ (T^4 + 667136*T^2 + 48809181184)^4
$79$	$ (T^{4} - 1131456 T^{2} + 201680031744)^{4} $ (T^4 - 1131456*T^2 + 201680031744)^4
$83$	$ (T^{4} - 1484384 T^{2} + 406594971904)^{4} $ (T^4 - 1484384*T^2 + 406594971904)^4
$89$	$ (T^{4} + \cdots + 1173010299136)^{4} $ (T^4 - 2172112*T^2 + 1173010299136)^4
$97$	$ (T^{4} + 1593600 T^{2} + 619683840000)^{4} $ (T^4 + 1593600*T^2 + 619683840000)^4
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{3} + \beta_{10} q^{5} + (\beta_{7} - \beta_{4}) q^{7} + (3 \beta_{8} - 3) q^{9}+O(q^{10}) \) q + b5 * q^3 + b10 * q^5 + (b7 - b4) * q^7 + (3b8 - 3) q^9	\( q + \beta_{5} q^{3} + \beta_{10} q^{5} + (\beta_{7} - \beta_{4}) q^{7} + (3 \beta_{8} - 3) q^{9} + (\beta_{12} + \beta_{11}) q^{11} + ( - 2 \beta_{3} + \beta_{2}) q^{13} + (\beta_{12} - 2 \beta_{11} + \cdots - \beta_1) q^{15}+ \cdots + ( - 3 \beta_{12} - 3 \beta_{11} + \cdots - 42 \beta_1) q^{99}+O(q^{100}) \) q + b5 * q^3 + b10 * q^5 + (b7 - b4) * q^7 + (3b8 - 3) q^9 + (b12 + b11) * q^11 + (-2b3 + b2) q^13 + (b12 - 2b11 + b7 - b6 - b1) q^15 + (-4b10 - b9) q^17 + (-3b7 - 3b6 - 2b5 + 8b4) * q^19 + (-b13 + 4b10 + b8 - b3 - 2b2 - 15) * q^21 + (b12 - 5b11) q^23 + (-b14 - b13 - 21) * q^25 + (-6b5 - 27b4) * q^27 + (-b14 + b13 - b10 - b9 - 12b8 - b3) q^29 + (-4b7 - 4b6 + 11b5 - 3b4) * q^31 + (3b9 - 3b3 + 3b2) q^33 + (b15 - 3b12 + 4b11 + 16b5 + 16b4) * q^35 + (-b14 - b13 + 274) * q^37 + (8b12 + 5b11 - 10b7 + 10b6 - 11b1) q^39 + (4b10 - b9) q^41 + (-11b7 + 11b6 - 11b1) q^43 + (-3b10 - 6b3 - 18b2) q^45 + (-b15 + 34b5 + 34b4) * q^47 + (-2b14 - 2b13 + b3 + 20b2 - 39) q^49 + (b12 + 13b11 + b7 - b6 + 11b1) * q^51 + (6b14 - 6b13 + 6b10 + 6b9 - 3b8 + 6b3) * q^53 + (-3b7 - 3b6 + 7b5 - b4) q^55 + (6b13 - 3b10 - 3b9 - 12b8 - 3b3 + 150) q^57 + (5b15 + 37b5 + 37b4) q^59 + (4b3 + 29b2) * q^61 + (3b15 + 12b12 - 9b11 - 3b7 - 15b5 - 12b4) * q^63 + (-3b14 + 3b13 - 3b10 - 3b9 + 23b8 - 3b3) * q^65 + (-7b7 + 7b6 + 7b1) q^67 + (-30b10 - 3b9 - 3b3 - 21b2) * q^69 + (25b12 + 29b11) * q^71 + (8b3 - 34b2) * q^73 + (3b15 - 15b7 - 15b6 - 6b5 + 15b4) q^75 + (b14 - b13 + 15b10 + 14b9 + 23b8 + b3) q^77 + (-31b7 + 31b6 - 2b1) q^79 + (-18b8 - 711) q^81 + (7b15 - 33b5 - 33b4) q^83 + (3b14 + 3b13 - 376) * q^85 + (-3b15 - 12b7 - 12b6 + 25b5 + 129b4) q^87 + (62b10 - 19b9) * q^89 + (-24b7 + 28b6 + 70b5 - 74b4 - 63b1) q^91 + (8b13 - 4b10 - 4b9 + 25b8 - 4b3 - 210) q^93 + (14b12 - 16b11) * q^95 + (30b3 + 20b2) * q^97 + (-3b12 - 3b11 - 30b7 + 30b6 - 42b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9}+O(q^{10}) \) 16 * q - 48 * q^9	\( 16 q - 48 q^{9} - 240 q^{21} - 336 q^{25} + 4384 q^{37} - 624 q^{49} + 2400 q^{57} - 11376 q^{81} - 6016 q^{85} - 3360 q^{93}+O(q^{100}) \) 16 * q - 48 * q^9 - 240 * q^21 - 336 * q^25 + 4384 * q^37 - 624 * q^49 + 2400 * q^57 - 11376 * q^81 - 6016 * q^85 - 3360 * q^93

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	672.4.k.c

Level	$672$
Weight	$4$
Character orbit	672.k
Analytic conductor	$39.649$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(39.6492835239\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.36004060626969600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 28x^{14} + 308x^{12} - 1710x^{10} + 5156x^{8} - 7740x^{6} + 9473x^{4} + 368x^{2} + 256 \) x^16 - 28x^14 + 308x^12 - 1710x^10 + 5156x^8 - 7740x^6 + 9473x^4 + 368*x^2 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{60} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 2489 \nu^{14} + 67190 \nu^{12} - 692402 \nu^{10} + 3399884 \nu^{8} - 7916482 \nu^{6} + \cdots - 15246632 ) / 858585 \) (-2489v^14 + 67190v^12 - 692402v^10 + 3399884v^8 - 7916482v^6 + 3911494v^4 + 41405*v^2 - 15246632) / 858585
\(\beta_{2}\)	\(=\)	\( ( 2131 \nu^{14} - 59442 \nu^{12} + 649262 \nu^{10} - 3562780 \nu^{8} + 10578358 \nu^{6} + \cdots + 203000 ) / 245055 \) (2131v^14 - 59442v^12 + 649262v^10 - 3562780v^8 + 10578358v^6 - 15767882v^4 + 20695449*v^2 + 203000) / 245055
\(\beta_{3}\)	\(=\)	\( ( 257821 \nu^{14} - 7154726 \nu^{12} + 77904750 \nu^{10} - 427794088 \nu^{8} + 1277219298 \nu^{6} + \cdots + 24994224 ) / 18134070 \) (257821v^14 - 7154726v^12 + 77904750v^10 - 427794088v^8 + 1277219298v^6 - 1901704874v^4 + 2513516707*v^2 + 24994224) / 18134070
\(\beta_{4}\)	\(=\)	\( ( 93447445 \nu^{15} + 63382528 \nu^{14} - 2641870140 \nu^{13} - 1789488000 \nu^{12} + \cdots + 4336768544 ) / 26403205920 \) (93447445v^15 + 63382528v^14 - 2641870140v^13 - 1789488000v^12 + 29498252420v^11 + 19935400064v^10 - 167794657030v^9 - 112876019008v^8 + 526871150260v^7 + 349673895424v^6 - 857213296940v^5 - 541212947648v^4 + 1063095809685v^3 + 620067302400v^2 - 136212862240*v + 4336768544) / 26403205920
\(\beta_{5}\)	\(=\)	\( ( 93447445 \nu^{15} - 63382528 \nu^{14} - 2641870140 \nu^{13} + 1789488000 \nu^{12} + \cdots - 4336768544 ) / 26403205920 \) (93447445v^15 - 63382528v^14 - 2641870140v^13 + 1789488000v^12 + 29498252420v^11 - 19935400064v^10 - 167794657030v^9 + 112876019008v^8 + 526871150260v^7 - 349673895424v^6 - 857213296940v^5 + 541212947648v^4 + 1063095809685v^3 - 620067302400v^2 - 136212862240*v - 4336768544) / 26403205920
\(\beta_{6}\)	\(=\)	\( ( 93447445 \nu^{15} + 154510600 \nu^{14} - 2641870140 \nu^{13} - 4442592944 \nu^{12} + \cdots - 403865289216 ) / 26403205920 \) (93447445v^15 + 154510600v^14 - 2641870140v^13 - 4442592944v^12 + 29498252420v^11 + 50816310768v^10 - 167794657030v^9 - 299308784128v^8 + 526871150260v^7 + 989831110800v^6 - 857213296940v^5 - 1760335412432v^4 + 1063095809685v^3 + 2124526783288v^2 - 136212862240*v - 403865289216) / 26403205920
\(\beta_{7}\)	\(=\)	\( ( 93447445 \nu^{15} + 282190040 \nu^{14} - 2641870140 \nu^{13} - 7913287696 \nu^{12} + \cdots + 433119134976 ) / 26403205920 \) (93447445v^15 + 282190040v^14 - 2641870140v^13 - 7913287696v^12 + 29498252420v^11 + 87048069072v^10 - 167794657030v^9 - 480439824512v^8 + 526871150260v^7 + 1415451829680v^6 - 857213296940v^5 - 1964463668848v^4 + 1063095809685v^3 + 2122659272552v^2 - 136212862240*v + 433119134976) / 26403205920
\(\beta_{8}\)	\(=\)	\( ( 1192127 \nu^{15} - 33164340 \nu^{13} + 361428556 \nu^{11} - 1979041362 \nu^{9} + \cdots + 2120841376 \nu ) / 118933360 \) (1192127v^15 - 33164340v^13 + 361428556v^11 - 1979041362v^9 + 5841756956v^7 - 8401996612v^5 + 10296719615v^3 + 2120841376v) / 118933360
\(\beta_{9}\)	\(=\)	\( ( - 21125019 \nu^{15} + 591251420 \nu^{13} - 6510390532 \nu^{11} + 36323144514 \nu^{9} + \cdots + 27824604368 \nu ) / 1885943280 \) (-21125019v^15 + 591251420v^13 - 6510390532v^11 + 36323144514v^9 - 111081432932v^7 + 173016545964v^5 - 217859771155v^3 + 27824604368v) / 1885943280
\(\beta_{10}\)	\(=\)	\( ( 209764405 \nu^{15} - 5899494724 \nu^{13} + 65321421148 \nu^{11} - 366342177598 \nu^{9} + \cdots - 281999112176 \nu ) / 13201602960 \) (209764405v^15 - 5899494724v^13 + 65321421148v^11 - 366342177598v^9 + 1123055813180v^7 - 1750773555572v^5 + 2208800523293v^3 - 281999112176v) / 13201602960
\(\beta_{11}\)	\(=\)	\( ( - 40353 \nu^{15} + 1127452 \nu^{13} - 12350948 \nu^{11} + 67993662 \nu^{9} - 201270004 \nu^{7} + \cdots - 67584896 \nu ) / 2133420 \) (-40353v^15 + 1127452v^13 - 12350948v^11 + 67993662v^9 - 201270004v^7 + 286880844v^5 - 329714129v^3 - 67584896v) / 2133420
\(\beta_{12}\)	\(=\)	\( ( 4048289 \nu^{15} - 113088308 \nu^{13} + 1239000428 \nu^{11} - 6826479110 \nu^{9} + \cdots + 6806023760 \nu ) / 145072560 \) (4048289v^15 - 113088308v^13 + 1239000428v^11 - 6826479110v^9 + 20234914252v^7 - 28836952228v^5 + 33199159081v^3 + 6806023760v) / 145072560
\(\beta_{13}\)	\(=\)	\( ( 1293702675 \nu^{15} + 158324712 \nu^{14} - 36045374084 \nu^{13} - 4542913648 \nu^{12} + \cdots - 1337524604416 ) / 26403205920 \) (1293702675v^15 + 158324712v^14 - 36045374084v^13 - 4542913648v^12 + 393370524028v^11 + 51487213232v^10 - 2156166505098v^9 - 294675483648v^8 + 6370246061900v^7 + 903152360656v^6 - 9206433819732v^5 - 1351104592656v^4 + 11285188481203v^3 + 1831151807576v^2 + 2096093091424*v - 1337524604416) / 26403205920
\(\beta_{14}\)	\(=\)	\( ( - 1293702675 \nu^{15} - 217062664 \nu^{14} + 36045374084 \nu^{13} + 5874367408 \nu^{12} + \cdots - 1373916194560 ) / 26403205920 \) (-1293702675v^15 - 217062664v^14 + 36045374084v^13 + 5874367408v^12 - 393370524028v^11 - 61942102768v^10 + 2156166505098v^9 + 328192708480v^8 - 6370246061900v^7 - 956478937232v^6 + 9206433819732v^5 + 1417777703888v^4 - 11285188481203v^3 - 1828528517816v^2 - 2096093091424*v - 1373916194560) / 26403205920
\(\beta_{15}\)	\(=\)	\( ( 2317 \nu^{15} - 65580 \nu^{13} + 733396 \nu^{11} - 4178502 \nu^{9} + 13128356 \nu^{7} + \cdots - 3396224 \nu ) / 19220 \) (2317v^15 - 65580v^13 + 733396v^11 - 4178502v^9 + 13128356v^7 - 21366172v^5 + 26508445v^3 - 3396224v) / 19220

\(\nu\)	\(=\)	\( ( \beta_{11} - \beta_{10} + 3\beta_{9} + 4\beta_{8} + 4\beta_{5} + 4\beta_{4} ) / 64 \) (b11 - b10 + 3b9 + 4b8 + 4b5 + 4b4) / 64
\(\nu^{2}\)	\(=\)	\( ( \beta_{14} + \beta_{13} - 3\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} - 3\beta _1 + 112 ) / 32 \) (b14 + b13 - 3b7 + b6 + b5 + b4 + 2b3 - b2 - 3*b1 + 112) / 32
\(\nu^{3}\)	\(=\)	\( ( - 3 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} + 5 \beta_{11} + 18 \beta_{10} + 22 \beta_{9} + \cdots + 3 \beta_{3} ) / 64 \) (-3b15 + 3b14 - 3b13 + 5b11 + 18b10 + 22b9 + 37b8 + 48b5 + 48b4 + 3b3) / 64
\(\nu^{4}\)	\(=\)	\( ( 7 \beta_{14} + 7 \beta_{13} - 25 \beta_{7} + 27 \beta_{6} + 7 \beta_{5} - 9 \beta_{4} + 18 \beta_{3} + \cdots + 672 ) / 32 \) (7b14 + 7b13 - 25b7 + 27b6 + 7b5 - 9b4 + 18b3 - 27b2 - 49*b1 + 672) / 32
\(\nu^{5}\)	\(=\)	\( ( - 35 \beta_{15} + 35 \beta_{14} - 35 \beta_{13} - 40 \beta_{12} - 59 \beta_{11} + 134 \beta_{10} + \cdots + 35 \beta_{3} ) / 64 \) (-35b15 + 35b14 - 35b13 - 40b12 - 59b11 + 134b10 + 162b9 + 325b8 + 576b5 + 576b4 + 35*b3) / 64
\(\nu^{6}\)	\(=\)	\( ( 48 \beta_{14} + 48 \beta_{13} - 166 \beta_{7} + 374 \beta_{6} + 244 \beta_{5} - 452 \beta_{4} + \cdots + 4840 ) / 32 \) (48b14 + 48b13 - 166b7 + 374b6 + 244b5 - 452b4 + 166b3 - 277b2 - 487*b1 + 4840) / 32
\(\nu^{7}\)	\(=\)	\( ( - 308 \beta_{15} + 336 \beta_{14} - 336 \beta_{13} - 896 \beta_{12} - 1345 \beta_{11} + \cdots + 336 \beta_{3} ) / 64 \) (-308b15 + 336b14 - 336b13 - 896b12 - 1345b11 + 993b10 + 1277b9 + 3084b8 + 5260b5 + 5260b4 + 336*b3) / 64
\(\nu^{8}\)	\(=\)	\( ( 365 \beta_{14} + 365 \beta_{13} - 567 \beta_{7} + 3885 \beta_{6} + 3997 \beta_{5} - 7315 \beta_{4} + \cdots + 37568 ) / 32 \) (365b14 + 365b13 - 567b7 + 3885b6 + 3997b5 - 7315b4 + 1626b3 - 2683b2 - 3969*b1 + 37568) / 32
\(\nu^{9}\)	\(=\)	\( ( - 2325 \beta_{15} + 3357 \beta_{14} - 3357 \beta_{13} - 12096 \beta_{12} - 17933 \beta_{11} + \cdots + 3357 \beta_{3} ) / 64 \) (-2325b15 + 3357b14 - 3357b13 - 12096b12 - 17933b11 + 8382b10 + 10682b9 + 31099b8 + 39888b5 + 39888b4 + 3357*b3) / 64
\(\nu^{10}\)	\(=\)	\( ( 2795 \beta_{14} + 2795 \beta_{13} + 4171 \beta_{7} + 35487 \beta_{6} + 48091 \beta_{5} - 87749 \beta_{4} + \cdots + 287632 ) / 32 \) (2795b14 + 2795b13 + 4171b7 + 35487b6 + 48091b5 - 87749b4 + 16730b3 - 27453b2 - 27919*b1 + 287632) / 32
\(\nu^{11}\)	\(=\)	\( ( - 15147 \beta_{15} + 34859 \beta_{14} - 34859 \beta_{13} - 134376 \beta_{12} - 198925 \beta_{11} + \cdots + 34859 \beta_{3} ) / 64 \) (-15147b15 + 34859b14 - 34859b13 - 134376b12 - 198925b11 + 71736b10 + 88364b9 + 323429b8 + 259560b5 + 259560b4 + 34859*b3) / 64
\(\nu^{12}\)	\(=\)	\( ( 18816 \beta_{14} + 18816 \beta_{13} + 116110 \beta_{7} + 301330 \beta_{6} + 506324 \beta_{5} + \cdots + 1933848 ) / 32 \) (18816b14 + 18816b13 + 116110b7 + 301330b6 + 506324b5 - 923764b4 + 173898b3 - 285291b2 - 165249*b1 + 1933848) / 32
\(\nu^{13}\)	\(=\)	\( ( - 77870 \beta_{15} + 358930 \beta_{14} - 358930 \beta_{13} - 1361360 \beta_{12} + \cdots + 358930 \beta_{3} ) / 64 \) (-77870b15 + 358930b14 - 358930b13 - 1361360b12 - 2015571b11 + 576421b10 + 673865b9 + 3329738b8 + 1333596b5 + 1333596b4 + 358930*b3) / 64
\(\nu^{14}\)	\(=\)	\( ( 87333 \beta_{14} + 87333 \beta_{13} + 1688193 \beta_{7} + 2422069 \beta_{6} + 4984629 \beta_{5} + \cdots + 8973328 ) / 32 \) (87333b14 + 87333b13 + 1688193b7 + 2422069b6 + 4984629b5 - 9094891b4 + 1761706b3 - 2890665b2 - 654875*b1 + 8973328) / 32
\(\nu^{15}\)	\(=\)	\( ( - 145235 \beta_{15} + 3563907 \beta_{14} - 3563907 \beta_{13} - 13104000 \beta_{12} + \cdots + 3563907 \beta_{3} ) / 64 \) (-145235b15 + 3563907b14 - 3563907b13 - 13104000b12 - 19402475b11 + 4126482b10 + 4378262b9 + 33059493b8 + 2486832b5 + 2486832b4 + 3563907*b3) / 64

\(n\)	\(127\)	\(421\)	\(449\)	\(577\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 545.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{4} + 6 T^{2} + 729)^{4} \) (T^4 + 6*T^2 + 729)^4
$5$	\( (T^{4} - 208 T^{2} + 256)^{4} \) (T^4 - 208*T^2 + 256)^4
$7$	\( (T^{8} + 156 T^{6} + \cdots + 13841287201)^{2} \) (T^8 + 156T^6 + 72422T^4 + 18353244*T^2 + 13841287201)^2
$11$	\( (T^{4} + 1968 T^{2} + 112896)^{4} \) (T^4 + 1968*T^2 + 112896)^4
$13$	\( (T^{4} + 8576 T^{2} + 7573504)^{4} \) (T^4 + 8576*T^2 + 7573504)^4
$17$	\( (T^{4} - 5968 T^{2} + 8386816)^{4} \) (T^4 - 5968*T^2 + 8386816)^4
$19$	\( (T^{4} + 15672 T^{2} + 23386896)^{4} \) (T^4 + 15672*T^2 + 23386896)^4
$23$	\( (T^{4} + 13488 T^{2} + 29419776)^{4} \) (T^4 + 13488*T^2 + 29419776)^4
$29$	\( (T^{4} + 43936 T^{2} + 25725184)^{4} \) (T^4 + 43936*T^2 + 25725184)^4
$31$	\( (T^{4} + 28408 T^{2} + 69288976)^{4} \) (T^4 + 28408*T^2 + 69288976)^4
$37$	\( (T^{2} - 548 T + 64516)^{8} \) (T^2 - 548*T + 64516)^8
$41$	\( (T^{4} - 7248 T^{2} + 12278016)^{4} \) (T^4 - 7248*T^2 + 12278016)^4
$43$	\( (T^{4} - 147136 T^{2} + 2938507264)^{4} \) (T^4 - 147136*T^2 + 2938507264)^4
$47$	\( (T^{4} - 139136 T^{2} + 1714622464)^{4} \) (T^4 - 139136*T^2 + 1714622464)^4
$53$	\( (T^{4} + 609696 T^{2} + 92056414464)^{4} \) (T^4 + 609696*T^2 + 92056414464)^4
$59$	\( (T^{4} - 835424 T^{2} + 81960818944)^{4} \) (T^4 - 835424*T^2 + 81960818944)^4
$61$	\( (T^{4} + 308096 T^{2} + 15847788544)^{4} \) (T^4 + 308096*T^2 + 15847788544)^4
$67$	\( (T^{4} - 122304 T^{2} + 88510464)^{4} \) (T^4 - 122304*T^2 + 88510464)^4
$71$	\( (T^{4} + 1353392 T^{2} + 21994076416)^{4} \) (T^4 + 1353392*T^2 + 21994076416)^4
$73$	\( (T^{4} + 667136 T^{2} + 48809181184)^{4} \) (T^4 + 667136*T^2 + 48809181184)^4
$79$	\( (T^{4} - 1131456 T^{2} + 201680031744)^{4} \) (T^4 - 1131456*T^2 + 201680031744)^4
$83$	\( (T^{4} - 1484384 T^{2} + 406594971904)^{4} \) (T^4 - 1484384*T^2 + 406594971904)^4
$89$	\( (T^{4} + \cdots + 1173010299136)^{4} \) (T^4 - 2172112*T^2 + 1173010299136)^4
$97$	\( (T^{4} + 1593600 T^{2} + 619683840000)^{4} \) (T^4 + 1593600*T^2 + 619683840000)^4
show more
show less