LMFDB - Newform orbit 1920.2.s.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(481,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("1920.481");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.s (of order $4$, degree $2$, 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:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + \cdots + 1024 $ x^20 - 2x^18 - 4x^17 + 7x^16 + 16x^15 + 6x^14 - 36x^13 - 42x^12 + 40x^11 + 136x^10 + 80x^9 - 168x^8 - 288x^7 + 96x^6 + 512x^5 + 448x^4 - 512x^3 - 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{3} + \beta_{5} q^{5} - \beta_{13} q^{7} + \beta_{3} q^{9}+O(q^{10}) $ q + b1 * q^3 + b5 * q^5 - b13 * q^7 + b3 * q^9	$ q + \beta_1 q^{3} + \beta_{5} q^{5} - \beta_{13} q^{7} + \beta_{3} q^{9} + (\beta_{8} - \beta_{5}) q^{11} - \beta_{14} q^{13} - q^{15} + ( - \beta_{19} - 1) q^{17} + (\beta_{12} - \beta_{10} + \cdots + \beta_{6}) q^{19}+ \cdots + ( - \beta_{6} + \beta_1) q^{99}+O(q^{100}) $ q + b1 * q^3 + b5 * q^5 - b13 * q^7 + b3 * q^9 + (b8 - b5) * q^11 - b14 * q^13 - q^15 + (-b19 - 1) * q^17 + (b12 - b10 + b9 + b6) * q^19 - b17 * q^21 + (b17 - b16 - b13 - b12 + b11 - b9 - b5 + b4 + b3 - b1) * q^23 - b3 * q^25 + b5 * q^27 + (-b19 - b18 - b16 - b14 - b13 + b7 + b6 + b4 - b2 + b1) * q^29 + (b18 + b15 + b14 + b8 - b7 - b6 + b5 - b1 - 1) * q^31 + (b18 + 1) * q^33 - b16 * q^35 + (b18 - b17 + b8 + b5 + b3 - b2 - 1) * q^37 + b9 * q^39 + (-b13 + b9 + b8 + b6 - b5 + 3b3 - b2 - b1) q^41 + (b19 + b17 + b15 + b8 - b5 + b4 + b3 - 1) * q^43 - b1 * q^45 + (b19 + b18 + b17 + b16 + b15 + b14) * q^47 + (b18 + b15 + b14 + b10 + b7 + 2b5 - 2b1 - 2) * q^49 + (b12 - b1) * q^51 + (b18 - b17 + b15 + b13 + b8 + b7 + b5 - b3 - b2 + 1) * q^53 + (b3 + b2) * q^55 + (b15 - b14 + b4 - b2) * q^57 + (-b18 - b13 - b10 - b9 - b8 - b7 + b5 + b3 + b2 - 1) * q^59 + (b16 + b13 + b12 - b10 + b9 - b7 + b6 + 2b1) q^61 - b7 * q^63 + b10 * q^65 + (-b19 - b16 - b14 + b13 - b7 - b6 + b4 - b3 + b1 - 1) * q^67 + (-b19 - b17 - b15 + b13 - b11 + b7 + b5 - b4 - b3 + 1) * q^69 + (b17 - b16 + b15 - b14 - 2b13 - 2b5 + 2b4 + 2b3 - 2b1) q^71 + (-b17 + b16 - 2b9 - b8 - b6 - b5 + 2b3 - b1) * q^73 - b5 * q^75 + (-b18 + b16 - b14 - b13 + 2b12 - b10 + b9 + b7 + 3b6 + 3b3 - b2 + b1 + 3) q^77 + (b18 - b17 - b16 + b15 + b14 + 2b10 + b8 + b7 - b6 - b5 + b1 - 3) q^79 - q^81 + (b18 - 2b16 - 2b13 - 2b12 + b10 - b9 + 2b7 - b3 + b2 - 4b1 - 1) q^83 + (-b11 - b5) * q^85 + (-b17 + b16 + b13 + b12 - b11 + b9 + b8 + b6 + b3 - b2) * q^87 + (-b17 + b16 + b15 - b14 - b13 + b9 + 2b8 + 2b6 + 2b5 - b3 - b2 + 2b1) * q^89 + (2b19 + 2b18 + 2b15 - 2b13 + b10 + b9 + 2b8 - 2b7 + 2b5 + 2b4 + 2b3 - 2b2 - 2) * q^91 + (b18 - b16 + b10 - b9 - b6 - b3 + b2 - b1 - 1) * q^93 + (b19 + b18 + b15 + b14) * q^95 + (-2b19 - 2b17 - 2b16 - b15 - b14 + 2b7 + 4) * q^97 + (-b6 + b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q + 8 q^{11} - 20 q^{15} - 24 q^{17} - 4 q^{19} - 16 q^{29} + 16 q^{33} - 16 q^{37} - 8 q^{43} - 52 q^{49} + 4 q^{51} + 16 q^{53} - 16 q^{59} + 4 q^{61} + 8 q^{63} - 8 q^{67} + 4 q^{69} + 40 q^{77} - 56 q^{79} - 20 q^{81} - 48 q^{83} - 4 q^{85} - 8 q^{91} - 16 q^{93} + 56 q^{97} + 8 q^{99}+O(q^{100}) $ 20 * q + 8 * q^11 - 20 * q^15 - 24 * q^17 - 4 * q^19 - 16 * q^29 + 16 * q^33 - 16 * q^37 - 8 * q^43 - 52 * q^49 + 4 * q^51 + 16 * q^53 - 16 * q^59 + 4 * q^61 + 8 * q^63 - 8 * q^67 + 4 * q^69 + 40 * q^77 - 56 * q^79 - 20 * q^81 - 48 * q^83 - 4 * q^85 - 8 * q^91 - 16 * q^93 + 56 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + \cdots + 1024 $ x^20 - 2*x^18 - 4*x^17 + 7*x^16 + 16*x^15 + 6*x^14 - 36*x^13 - 42*x^12 + 40*x^11 + 136*x^10 + 80*x^9 - 168*x^8 - 288*x^7 + 96*x^6 + 512*x^5 + 448*x^4 - 512*x^3 - 512*x^2 + 1024 :

$\beta_{1}$	$=$	$ ( 18 \nu^{19} + 33 \nu^{18} + 8 \nu^{17} - 104 \nu^{16} - 94 \nu^{15} + 199 \nu^{14} + 616 \nu^{13} + \cdots - 5632 ) / 512 $ (18v^19 + 33v^18 + 8v^17 - 104v^16 - 94v^15 + 199v^14 + 616v^13 + 364v^12 - 768v^11 - 1426v^10 + 152v^9 + 3332v^8 + 3808v^7 - 1024v^6 - 5536v^5 - 2976v^4 + 7040v^3 + 8448v^2 + 2816*v - 5632) / 512
$\beta_{2}$	$=$	$ ( 5 \nu^{19} - 22 \nu^{18} - 84 \nu^{17} - 112 \nu^{16} + 107 \nu^{15} + 334 \nu^{14} + 24 \nu^{13} + \cdots - 23552 ) / 512 $ (5v^19 - 22v^18 - 84v^17 - 112v^16 + 107v^15 + 334v^14 + 24v^13 - 1216v^12 - 1906v^11 - 100v^10 + 3180v^9 + 2992v^8 - 4208v^7 - 11728v^6 - 7520v^5 + 6912v^4 + 13952v^3 - 3456v^2 - 22016*v - 23552) / 512
$\beta_{3}$	$=$	$ ( - 21 \nu^{19} + 10 \nu^{18} + 104 \nu^{17} + 192 \nu^{16} - 123 \nu^{15} - 594 \nu^{14} + \cdots + 28672 ) / 512 $ (-21v^19 + 10v^18 + 104v^17 + 192v^16 - 123v^15 - 594v^14 - 364v^13 + 1376v^12 + 2786v^11 + 652v^10 - 4436v^9 - 5696v^8 + 3232v^7 + 15248v^6 + 11424v^5 - 7936v^4 - 21760v^3 - 128v^2 + 24064*v + 28672) / 512
$\beta_{4}$	$=$	$ ( 16 \nu^{19} + 10 \nu^{18} - 47 \nu^{17} - 138 \nu^{16} + 20 \nu^{15} + 394 \nu^{14} + 479 \nu^{13} + \cdots - 16896 ) / 256 $ (16v^19 + 10v^18 - 47v^17 - 138v^16 + 20v^15 + 394v^14 + 479v^13 - 518v^12 - 1776v^11 - 1048v^10 + 2342v^9 + 4684v^8 + 492v^7 - 8104v^6 - 8944v^5 + 2336v^4 + 14816v^3 + 6080v^2 - 9856*v - 16896) / 256
$\beta_{5}$	$=$	$ ( - 31 \nu^{19} - 39 \nu^{18} + 30 \nu^{17} + 204 \nu^{16} + 67 \nu^{15} - 497 \nu^{14} - 946 \nu^{13} + \cdots + 17920 ) / 512 $ (-31v^19 - 39v^18 + 30v^17 + 204v^16 + 67v^15 - 497v^14 - 946v^13 + 64v^12 + 2082v^11 + 2086v^10 - 2008v^9 - 6636v^8 - 3800v^7 + 7264v^6 + 11680v^5 + 736v^4 - 17728v^3 - 11776v^2 + 5120*v + 17920) / 512
$\beta_{6}$	$=$	$ ( 62 \nu^{19} + 3 \nu^{18} - 252 \nu^{17} - 544 \nu^{16} + 222 \nu^{15} + 1605 \nu^{14} + 1324 \nu^{13} + \cdots - 76288 ) / 512 $ (62v^19 + 3v^18 - 252v^17 - 544v^16 + 222v^15 + 1605v^14 + 1324v^13 - 3188v^12 - 7456v^11 - 2566v^10 + 11248v^9 + 16412v^8 - 5808v^7 - 39264v^6 - 32352v^5 + 19232v^4 + 59392v^3 + 5632v^2 - 61696*v - 76288) / 512
$\beta_{7}$	$=$	$ ( - 23 \nu^{19} - 44 \nu^{18} - 6 \nu^{17} + 140 \nu^{16} + 135 \nu^{15} - 276 \nu^{14} - 854 \nu^{13} + \cdots + 6144 ) / 256 $ (-23v^19 - 44v^18 - 6v^17 + 140v^16 + 135v^15 - 276v^14 - 854v^13 - 516v^12 + 1070v^11 + 2064v^10 - 112v^9 - 4608v^8 - 5544v^7 + 1120v^6 + 7872v^5 + 4672v^4 - 8768v^3 - 12416v^2 - 4608*v + 6144) / 256
$\beta_{8}$	$=$	$ ( 11 \nu^{19} - 65 \nu^{18} - 202 \nu^{17} - 220 \nu^{16} + 337 \nu^{15} + 777 \nu^{14} - 210 \nu^{13} + \cdots - 45568 ) / 512 $ (11v^19 - 65v^18 - 202v^17 - 220v^16 + 337v^15 + 777v^14 - 210v^13 - 3024v^12 - 4042v^11 + 698v^10 + 7648v^9 + 5452v^8 - 11480v^7 - 26464v^6 - 13216v^5 + 18336v^4 + 28864v^3 - 13312v^2 - 48640*v - 45568) / 512
$\beta_{9}$	$=$	$ ( - 12 \nu^{19} + 9 \nu^{18} + 67 \nu^{17} + 116 \nu^{16} - 84 \nu^{15} - 365 \nu^{14} - 179 \nu^{13} + \cdots + 17920 ) / 128 $ (-12v^19 + 9v^18 + 67v^17 + 116v^16 - 84v^15 - 365v^14 - 179v^13 + 920v^12 + 1724v^11 + 290v^10 - 2854v^9 - 3332v^8 + 2516v^7 + 9824v^6 + 6736v^5 - 5568v^4 - 13408v^3 + 1088v^2 + 16256*v + 17920) / 128
$\beta_{10}$	$=$	$ ( - 31 \nu^{19} - 50 \nu^{18} + 2 \nu^{17} + 184 \nu^{16} + 135 \nu^{15} - 382 \nu^{14} - 1022 \nu^{13} + \cdots + 9728 ) / 256 $ (-31v^19 - 50v^18 + 2v^17 + 184v^16 + 135v^15 - 382v^14 - 1022v^13 - 456v^12 + 1494v^11 + 2380v^10 - 560v^9 - 5808v^8 - 5992v^7 + 2704v^6 + 9664v^5 + 4736v^4 - 12096v^3 - 13824v^2 - 3840*v + 9728) / 256
$\beta_{11}$	$=$	$ ( - 66 \nu^{19} + 37 \nu^{18} + 350 \nu^{17} + 636 \nu^{16} - 418 \nu^{15} - 1989 \nu^{14} + \cdots + 98304 ) / 512 $ (-66v^19 + 37v^18 + 350v^17 + 636v^16 - 418v^15 - 1989v^14 - 1182v^13 + 4712v^12 + 9368v^11 + 2094v^10 - 15092v^9 - 19124v^8 + 11400v^7 + 51984v^6 + 38656v^5 - 27232v^4 - 73536v^3 - 128v^2 + 82944*v + 98304) / 512
$\beta_{12}$	$=$	$ ( 79 \nu^{19} - 21 \nu^{18} - 364 \nu^{17} - 736 \nu^{16} + 389 \nu^{15} + 2197 \nu^{14} + 1568 \nu^{13} + \cdots - 108544 ) / 512 $ (79v^19 - 21v^18 - 364v^17 - 736v^16 + 389v^15 + 2197v^14 + 1568v^13 - 4812v^12 - 10386v^11 - 3062v^10 + 16004v^9 + 21892v^8 - 10352v^7 - 55984v^6 - 44224v^5 + 27488v^4 + 81920v^3 + 2944v^2 - 88320*v - 108544) / 512
$\beta_{13}$	$=$	$ ( - 33 \nu^{19} + 22 \nu^{18} + 184 \nu^{17} + 328 \nu^{16} - 223 \nu^{15} - 1006 \nu^{14} + \cdots + 52736 ) / 256 $ (-33v^19 + 22v^18 + 184v^17 + 328v^16 - 223v^15 - 1006v^14 - 556v^13 + 2504v^12 + 4842v^11 + 996v^10 - 7812v^9 - 9568v^8 + 6496v^7 + 27280v^6 + 19872v^5 - 14592v^4 - 38016v^3 + 1536v^2 + 44800*v + 52736) / 256
$\beta_{14}$	$=$	$ ( - 41 \nu^{19} - 75 \nu^{18} - 24 \nu^{17} + 228 \nu^{16} + 221 \nu^{15} - 421 \nu^{14} - 1428 \nu^{13} + \cdots + 10240 ) / 256 $ (-41v^19 - 75v^18 - 24v^17 + 228v^16 + 221v^15 - 421v^14 - 1428v^13 - 968v^12 + 1598v^11 + 3366v^10 + 44v^9 - 7452v^8 - 9488v^7 + 960v^6 + 12128v^5 + 8352v^4 - 14208v^3 - 20352v^2 - 10496*v + 10240) / 256
$\beta_{15}$	$=$	$ ( - 56 \nu^{19} - 73 \nu^{18} + 46 \nu^{17} + 368 \nu^{16} + 148 \nu^{15} - 855 \nu^{14} - 1734 \nu^{13} + \cdots + 30720 ) / 256 $ (-56v^19 - 73v^18 + 46v^17 + 368v^16 + 148v^15 - 855v^14 - 1734v^13 - 52v^12 + 3588v^11 + 3930v^10 - 3068v^9 - 11716v^8 - 7560v^7 + 11776v^6 + 20704v^5 + 2720v^4 - 30144v^3 - 21632v^2 + 7168*v + 30720) / 256
$\beta_{16}$	$=$	$ ( - 57 \nu^{19} - 99 \nu^{18} - 6 \nu^{17} + 336 \nu^{16} + 269 \nu^{15} - 693 \nu^{14} - 1926 \nu^{13} + \cdots + 18432 ) / 256 $ (-57v^19 - 99v^18 - 6v^17 + 336v^16 + 269v^15 - 693v^14 - 1926v^13 - 932v^12 + 2694v^11 + 4478v^10 - 1000v^9 - 10836v^8 - 11400v^7 + 4816v^6 + 18272v^5 + 8352v^4 - 22976v^3 - 26368v^2 - 6144*v + 18432) / 256
$\beta_{17}$	$=$	$ ( - 70 \nu^{19} - 99 \nu^{18} + 48 \nu^{17} + 456 \nu^{16} + 218 \nu^{15} - 1045 \nu^{14} + \cdots + 36352 ) / 256 $ (-70v^19 - 99v^18 + 48v^17 + 456v^16 + 218v^15 - 1045v^14 - 2240v^13 - 244v^12 + 4368v^11 + 5110v^10 - 3480v^9 - 14668v^8 - 10336v^7 + 13440v^6 + 25888v^5 + 4512v^4 - 36736v^3 - 28800v^2 + 6400*v + 36352) / 256
$\beta_{18}$	$=$	$ ( 38 \nu^{19} + 57 \nu^{18} - 16 \nu^{17} - 234 \nu^{16} - 130 \nu^{15} + 527 \nu^{14} + 1208 \nu^{13} + \cdots - 16512 ) / 128 $ (38v^19 + 57v^18 - 16v^17 - 234v^16 - 130v^15 + 527v^14 + 1208v^13 + 286v^12 - 2120v^11 - 2730v^10 + 1504v^9 + 7560v^8 + 6080v^7 - 5832v^6 - 12864v^5 - 3168v^4 + 18112v^3 + 15680v^2 - 896*v - 16512) / 128
$\beta_{19}$	$=$	$ ( 177 \nu^{19} + 256 \nu^{18} - 86 \nu^{17} - 1108 \nu^{16} - 577 \nu^{15} + 2496 \nu^{14} + \cdots - 86528 ) / 512 $ (177v^19 + 256v^18 - 86v^17 - 1108v^16 - 577v^15 + 2496v^14 + 5594v^13 + 1004v^12 - 10306v^11 - 12712v^10 + 7728v^9 + 35792v^8 + 27064v^7 - 30432v^6 - 62400v^5 - 13312v^4 + 88512v^3 + 72704v^2 - 9472*v - 86528) / 512

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

$\nu$	$=$	$ ( \beta_{19} + \beta_{16} + \beta_{15} + \beta_{8} - \beta_{7} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 4 $ (b19 + b16 + b15 + b8 - b7 + b3 - b2 + b1 - 1) / 4
$\nu^{2}$	$=$	$ ( -\beta_{17} - \beta_{15} + \beta_{14} - \beta_{12} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{2} - 2\beta _1 + 1 ) / 4 $ (-b17 - b15 + b14 - b12 - b9 + b7 - b6 + b2 - 2*b1 + 1) / 4
$\nu^{3}$	$=$	$ ( \beta_{16} + \beta_{15} - 2 \beta_{13} + 2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \cdots + 2 ) / 4 $ (b16 + b15 - 2b13 + 2b11 - b10 - b9 - b8 - b7 + b4 - b2 + 3*b1 + 2) / 4
$\nu^{4}$	$=$	$ ( - \beta_{18} - 2 \beta_{17} + \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} + \cdots - 4 ) / 4 $ (-b18 - 2b17 + b16 - b15 - b14 + b13 + b11 + b10 + b8 + 2b6 + 4b5 - b3 - 2b2 - 2*b1 - 4) / 4
$\nu^{5}$	$=$	$ ( - \beta_{19} + \beta_{18} + \beta_{17} - \beta_{14} - \beta_{13} + 2 \beta_{11} - 3 \beta_{10} + \cdots - 6 ) / 4 $ (-b19 + b18 + b17 - b14 - b13 + 2b11 - 3b10 - 3b9 - 2b8 + 2b7 + b6 - b5 + 2b4 + 6b3 + 2b2 - 6*b1 - 6) / 4
$\nu^{6}$	$=$	$ ( - 4 \beta_{19} - 5 \beta_{17} - 2 \beta_{16} + \beta_{15} - \beta_{14} + 2 \beta_{13} + \beta_{12} + \cdots - 9 ) / 4 $ (-4b19 - 5b17 - 2b16 + b15 - b14 + 2b13 + b12 + 2b10 + b9 - 2b8 + 3b7 - b6 + 6b5 - 14b3 + b2 + 8b1 - 9) / 4
$\nu^{7}$	$=$	$ ( - 2 \beta_{19} - 4 \beta_{18} + 10 \beta_{17} - 9 \beta_{16} - \beta_{15} - 6 \beta_{14} - 4 \beta_{13} + \cdots - 8 ) / 4 $ (-2b19 - 4b18 + 10b17 - 9b16 - b15 - 6b14 - 4b13 - 2b12 + 4b11 - b10 + b9 - b8 + b7 + 8b6 - 12b5 + 5b4 - b2 - 5b1 - 8) / 4
$\nu^{8}$	$=$	$ ( 7 \beta_{18} + 2 \beta_{17} + 3 \beta_{16} + \beta_{15} - 3 \beta_{14} + 9 \beta_{13} + 4 \beta_{12} + \cdots + 2 ) / 4 $ (7b18 + 2b17 + 3b16 + b15 - 3b14 + 9b13 + 4b12 - 9b11 + b10 + 6b9 - b8 + 2b7 + 2b6 + 28b5 - 5b3 + 4b2 + 10b1 + 2) / 4
$\nu^{9}$	$=$	$ ( - 3 \beta_{19} + 5 \beta_{18} + 5 \beta_{17} - 14 \beta_{16} + 6 \beta_{15} - \beta_{14} - \beta_{13} + \cdots - 8 ) / 4 $ (-3b19 + 5b18 + 5b17 - 14b16 + 6b15 - b14 - b13 - 8b12 - 6b11 + 17b10 - 7b9 + 4b8 - 15b6 - 37b5 - 2b4 - 8b3 + 12b2 - 4b1 - 8) / 4
$\nu^{10}$	$=$	$ ( - 4 \beta_{19} - 12 \beta_{18} + 5 \beta_{17} + 2 \beta_{16} - 9 \beta_{15} + \beta_{14} + 6 \beta_{13} + \cdots + 37 ) / 4 $ (-4b19 - 12b18 + 5b17 + 2b16 - 9b15 + b14 + 6b13 - 5b12 - 12b11 - 10b10 + 19b9 + 2b8 + b7 + 17b6 - 34b5 - 8b4 - 10b3 - b2 + 44b1 + 37) / 4
$\nu^{11}$	$=$	$ ( 6 \beta_{19} + 16 \beta_{18} + 6 \beta_{17} - 11 \beta_{16} + 5 \beta_{15} - 2 \beta_{14} - 20 \beta_{13} + \cdots + 32 ) / 4 $ (6b19 + 16b18 + 6b17 - 11b16 + 5b15 - 2b14 - 20b13 - 18b12 - 24b11 + 25b10 - 5b9 + 9b8 - 5b7 - 20b6 + 72b5 + 7b4 + 44b3 - 15b2 + 13*b1 + 32) / 4
$\nu^{12}$	$=$	$ ( 16 \beta_{19} + 21 \beta_{18} - 2 \beta_{17} + 53 \beta_{16} - 25 \beta_{15} + 43 \beta_{14} + 19 \beta_{13} + \cdots + 2 ) / 4 $ (16b19 + 21b18 - 2b17 + 53b16 - 25b15 + 43b14 + 19b13 + 4b12 + b11 + 19b10 - 10b9 + 17b8 - 46b7 - 42b6 - 108b5 - 40b4 - 15b3 + 24b2 - 2b1 + 2) / 4
$\nu^{13}$	$=$	$ ( - 37 \beta_{19} - 41 \beta_{18} - 73 \beta_{17} + 10 \beta_{16} - 18 \beta_{15} - 27 \beta_{14} + \cdots + 148 ) / 4 $ (-37b19 - 41b18 - 73b17 + 10b16 - 18b15 - 27b14 - 27b13 - 16b12 - 18b11 - 29b10 + 11b9 - 16b8 + 36b7 + 43b6 - 15b5 - 22b4 + 124b3 - 40b2 - 72*b1 + 148) / 4
$\nu^{14}$	$=$	$ ( 28 \beta_{19} - 12 \beta_{18} - 21 \beta_{17} + 46 \beta_{16} - 15 \beta_{15} + 63 \beta_{14} + \cdots - 269 ) / 4 $ (28b19 - 12b18 - 21b17 + 46b16 - 15b15 + 63b14 + 10b13 + 37b12 + 4b11 - 22b10 - 19b9 - 2b8 - 57b7 - 33b6 + 82b5 - 8b4 + 34b3 - 63b2 + 92*b1 - 269) / 4
$\nu^{15}$	$=$	$ ( - 46 \beta_{19} + 48 \beta_{18} - 38 \beta_{17} + 3 \beta_{16} - 85 \beta_{15} + 18 \beta_{14} + \cdots + 112 ) / 4 $ (-46b19 + 48b18 - 38b17 + 3b16 - 85b15 + 18b14 - 36b13 + 34b12 + 64b11 + 7b10 + 21b9 + 7b8 + 69b7 - 36b6 - 24b5 - 7b4 + 52b3 + 7b2 - 317*b1 + 112) / 4
$\nu^{16}$	$=$	$ ( 64 \beta_{19} - 69 \beta_{18} - 30 \beta_{17} + 123 \beta_{16} + 105 \beta_{15} - 35 \beta_{14} + \cdots - 218 ) / 4 $ (64b19 - 69b18 - 30b17 + 123b16 + 105b15 - 35b14 + 29b13 + 156b12 + 87b11 - 155b10 + 42b9 - 145b8 - 98b7 + 138b6 + 44b5 - 32b4 - 49b3 - 72b2 - 230*b1 - 218) / 4
$\nu^{17}$	$=$	$ ( - 51 \beta_{19} - 175 \beta_{18} - 71 \beta_{17} - 170 \beta_{16} - 110 \beta_{15} - 29 \beta_{14} + \cdots - 268 ) / 4 $ (-51b19 - 175b18 - 71b17 - 170b16 - 110b15 - 29b14 + 227b13 + 144b12 - 38b11 - 27b10 - 11b9 - 32b8 + 244b7 + 53b6 + 47b5 + 22b4 - 12b3 + 32b2 - 48*b1 - 268) / 4
$\nu^{18}$	$=$	$ ( 124 \beta_{19} + 292 \beta_{18} + 357 \beta_{17} - 158 \beta_{16} + 375 \beta_{15} + 81 \beta_{14} + \cdots - 139 ) / 4 $ (124b19 + 292b18 + 357b17 - 158b16 + 375b15 + 81b14 - 194b13 + 115b12 + 172b11 - 146b10 + 195b9 + 130b8 + 65b7 - 15b6 - 74b5 + 216b4 + 126b3 + 39b2 - 12*b1 - 139) / 4
$\nu^{19}$	$=$	$ ( 254 \beta_{19} - 112 \beta_{18} + 62 \beta_{17} - 171 \beta_{16} + 101 \beta_{15} - 58 \beta_{14} + \cdots + 392 ) / 4 $ (254b19 - 112b18 + 62b17 - 171b16 + 101b15 - 58b14 + 228b13 - 26b12 - 16b11 + 177b10 + 59b9 - 223b8 + 43b7 - 36b6 + 776b5 - 129b4 - 1748b3 + 185b2 - 171*b1 + 392) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$1$	$1$	$-\beta_{3}$	$1$

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} $

481.1

1.32147	+	0.503713i
−1.38431	−	0.289262i
−1.04932	+	0.948122i
−0.0861743	−	1.41159i
1.19834	−	0.750988i
−1.13207	−	0.847599i
1.15787	−	0.811989i
−0.720859	+	1.21670i
−0.491956	−	1.32589i
1.18701	+	0.768775i
1.19834	+	0.750988i
−0.0861743	+	1.41159i
−1.04932	−	0.948122i
−1.38431	+	0.289262i
1.32147	−	0.503713i
1.18701	−	0.768775i
−0.491956	+	1.32589i
−0.720859	−	1.21670i
1.15787	+	0.811989i
−1.13207	+	0.847599i

−0.707107

0.707107i

0.707107

0.707107i

−

2.69529i

−

1.00000i

481.2

−0.707107

0.707107i

0.707107

0.707107i

−

2.60796i

−

1.00000i

481.3

−0.707107

0.707107i

0.707107

0.707107i

0.740019i

−

1.00000i

481.4

−0.707107

0.707107i

0.707107

0.707107i

2.76462i

−

1.00000i

481.5

−0.707107

0.707107i

0.707107

0.707107i

3.79862i

−

1.00000i

481.6

0.707107

−

0.707107i

−0.707107

−

0.707107i

−

4.27253i

−

1.00000i

481.7

0.707107

−

0.707107i

−0.707107

−

0.707107i

−

2.18060i

−

1.00000i

481.8

0.707107

−

0.707107i

−0.707107

−

0.707107i

0.0588949i

−

1.00000i

481.9

0.707107

−

0.707107i

−0.707107

−

0.707107i

3.46600i

−

1.00000i

481.10

0.707107

−

0.707107i

−0.707107

−

0.707107i

4.92824i

−

1.00000i

1441.1

−0.707107

−

0.707107i

0.707107

−

0.707107i

−

3.79862i

1.00000i

1441.2

−0.707107

−

0.707107i

0.707107

−

0.707107i

−

2.76462i

1.00000i

1441.3

−0.707107

−

0.707107i

0.707107

−

0.707107i

−

0.740019i

1.00000i

1441.4

−0.707107

−

0.707107i

0.707107

−

0.707107i

2.60796i

1.00000i

1441.5

−0.707107

−

0.707107i

0.707107

−

0.707107i

2.69529i

1.00000i

1441.6

0.707107

0.707107i

−0.707107

0.707107i

−

4.92824i

1.00000i

1441.7

0.707107

0.707107i

−0.707107

0.707107i

−

3.46600i

1.00000i

1441.8

0.707107

0.707107i

−0.707107

0.707107i

−

0.0588949i

1.00000i

1441.9

0.707107

0.707107i

−0.707107

0.707107i

2.18060i

1.00000i

1441.10

0.707107

0.707107i

−0.707107

0.707107i

4.27253i

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1920.2.s.f		20
4.b	odd	2	1		1920.2.s.e		20
8.b	even	2	1		960.2.s.c		20
8.d	odd	2	1		240.2.s.c	✓	20
16.e	even	4	1		960.2.s.c		20
16.e	even	4	1	inner	1920.2.s.f		20
16.f	odd	4	1		240.2.s.c	✓	20
16.f	odd	4	1		1920.2.s.e		20
24.f	even	2	1		720.2.t.d		20
24.h	odd	2	1		2880.2.t.d		20
48.i	odd	4	1		2880.2.t.d		20
48.k	even	4	1		720.2.t.d		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
240.2.s.c	✓	20	8.d	odd	2	1
240.2.s.c	✓	20	16.f	odd	4	1
720.2.t.d		20	24.f	even	2	1
720.2.t.d		20	48.k	even	4	1
960.2.s.c		20	8.b	even	2	1
960.2.s.c		20	16.e	even	4	1
1920.2.s.e		20	4.b	odd	2	1
1920.2.s.e		20	16.f	odd	4	1
1920.2.s.f		20	1.a	even	1	1	trivial
1920.2.s.f		20	16.e	even	4	1	inner
2880.2.t.d		20	24.h	odd	2	1
2880.2.t.d		20	48.i	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1920, [\chi])$:

$ T_{7}^{20} + 96 T_{7}^{18} + 3880 T_{7}^{16} + 86368 T_{7}^{14} + 1161104 T_{7}^{12} + 9697664 T_{7}^{10} + \cdots + 262144 $

$ T_{11}^{20} - 8 T_{11}^{19} + 32 T_{11}^{18} - 112 T_{11}^{17} + 1808 T_{11}^{16} - 13696 T_{11}^{15} + \cdots + 1048576 $

$ T_{19}^{20} + 4 T_{19}^{19} + 8 T_{19}^{18} - 72 T_{19}^{17} + 5164 T_{19}^{16} + 17152 T_{19}^{15} + \cdots + 19716653056 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} $ T^20
$3$	$ (T^{4} + 1)^{5} $ (T^4 + 1)^5
$5$	$ (T^{4} + 1)^{5} $ (T^4 + 1)^5
$7$	$ T^{20} + 96 T^{18} + \cdots + 262144 $ T^20 + 96T^18 + 3880T^16 + 86368T^14 + 1161104T^12 + 9697664T^10 + 49590528T^8 + 145248256T^6 + 204587008T^4 + 76283904*T^2 + 262144
$11$	$ T^{20} - 8 T^{19} + \cdots + 1048576 $ T^20 - 8T^19 + 32T^18 - 112T^17 + 1808T^16 - 13696T^15 + 57984T^14 - 173952T^13 + 1093696T^12 - 7170048T^11 + 30799872T^10 - 84426752T^9 + 251479040T^8 - 1115578368T^7 + 4560683008T^6 - 11886264320T^5 + 18952028160T^4 - 15595208704T^3 + 6343884800T^2 + 115343360*T + 1048576
$13$	$ T^{20} + \cdots + 167981940736 $ T^20 + 144T^17 + 2088T^16 + 4192T^15 + 10368T^14 + 175680T^13 + 1636752T^12 + 5323392T^11 + 12435968T^10 + 60063232T^9 + 413145856T^8 + 1511043072T^7 + 3460349952T^6 + 7034568704T^5 + 25928249344T^4 + 90035355648T^3 + 205127811072T^2 + 262517686272*T + 167981940736
$17$	$ (T^{10} + 12 T^{9} + \cdots - 20032)^{2} $ (T^10 + 12T^9 - 34T^8 - 792T^7 - 676T^6 + 15824T^5 + 25976T^4 - 106144T^3 - 145472T^2 + 151040*T - 20032)^2
$19$	$ T^{20} + \cdots + 19716653056 $ T^20 + 4T^19 + 8T^18 - 72T^17 + 5164T^16 + 17152T^15 + 29888T^14 - 209856T^13 + 7892400T^12 + 20915520T^11 + 28386432T^10 - 73408640T^9 + 3224447552T^8 + 7963704832T^7 + 9631844352T^6 + 7396435968T^5 + 29049219072T^4 + 68702330880T^3 + 89860866048T^2 + 59527397376*T + 19716653056
$23$	$ T^{20} + \cdots + 217558810624 $ T^20 + 348T^18 + 49980T^16 + 3828512T^14 + 169125872T^12 + 4370425280T^10 + 64474080320T^8 + 505220718592T^6 + 1734629007360T^4 + 1211058880512*T^2 + 217558810624
$29$	$ T^{20} + \cdots + 19723262623744 $ T^20 + 16T^19 + 128T^18 + 576T^17 + 15632T^16 + 231168T^15 + 1863680T^14 + 7973888T^13 + 52284928T^12 + 541822976T^11 + 4155932672T^10 + 18135728128T^9 + 46757634048T^8 + 67031334912T^7 + 169368092672T^6 + 779917459456T^5 + 2276071636992T^4 + 1868828770304T^3 + 159752650752T^2 + 2510316109824*T + 19723262623744
$31$	$ (T^{10} - 204 T^{8} + \cdots - 28698368)^{2} $ (T^10 - 204T^8 - 168T^7 + 14932T^6 + 19536T^5 - 468112T^4 - 621056T^3 + 6148800T^2 + 4962560T - 28698368)^2
$37$	$ T^{20} + 16 T^{19} + \cdots + 18939904 $ T^20 + 16T^19 + 128T^18 + 80T^17 + 4904T^16 + 81440T^15 + 678528T^14 + 581952T^13 - 2713200T^12 - 12409216T^11 + 184033792T^10 - 394871296T^9 + 451001088T^8 - 422612992T^7 + 1128316928T^6 - 2241855488T^5 + 2665852928T^4 - 1848475648T^3 + 777125888T^2 - 171573248*T + 18939904
$41$	$ T^{20} + \cdots + 3311118843904 $ T^20 + 344T^18 + 48592T^16 + 3697152T^14 + 166691328T^12 + 4616884224T^10 + 78819631104T^8 + 806185730048T^6 + 4549133533184T^4 + 11279143534592*T^2 + 3311118843904
$43$	$ T^{20} + \cdots + 3288334336 $ T^20 + 8T^19 + 32T^18 + 704T^17 + 20960T^16 + 208640T^15 + 1246208T^14 + 8994816T^13 + 121272576T^12 + 1219307520T^11 + 7850893312T^10 + 32131072000T^9 + 83002048512T^8 + 118577430528T^7 + 76928778240T^6 + 72320286720T^5 + 430934327296T^4 + 354645180416T^3 + 150625845248T^2 + 31474057216*T + 3288334336
$47$	$ (T^{10} - 166 T^{8} + \cdots - 12544)^{2} $ (T^10 - 166T^8 - 160T^7 + 7980T^6 + 11520T^5 - 97448T^4 - 35072T^3 + 283712T^2 - 145152T - 12544)^2
$53$	$ T^{20} + \cdots + 4398046511104 $ T^20 - 16T^19 + 128T^18 - 512T^17 + 18720T^16 - 279040T^15 + 2199552T^14 - 8978432T^13 + 113316096T^12 - 1463160832T^11 + 11259117568T^10 - 46965587968T^9 + 243227688960T^8 - 2110865276928T^7 + 15258273972224T^6 - 64499805585408T^5 + 164036679303168T^4 - 217797791580160T^3 + 158879430213632T^2 - 37383395344384*T + 4398046511104
$59$	$ T^{20} + \cdots + 16\!\cdots\!04 $ T^20 + 16T^19 + 128T^18 - 112T^17 + 21776T^16 + 332800T^15 + 2543744T^14 + 208256T^13 + 117256256T^12 + 1869032960T^11 + 14857752576T^10 + 4783368192T^9 + 60533785600T^8 + 1690337058816T^7 + 18984867758080T^6 - 8752961814528T^5 - 19284448182272T^4 + 225773976551424T^3 + 7682191945367552T^2 - 15913855322423296*T + 16482977321058304
$61$	$ T^{20} + \cdots + 74350019584 $ T^20 - 4T^19 + 8T^18 - 840T^17 + 21588T^16 - 186688T^15 + 926848T^14 - 3663488T^13 + 31363232T^12 - 246492544T^11 + 1229804288T^10 - 3542995712T^9 + 8596122240T^8 - 32054490112T^7 + 145373775872T^6 - 400694880256T^5 + 635588871424T^4 - 405824488448T^3 + 55356622848T^2 + 90727790592*T + 74350019584
$67$	$ T^{20} + \cdots + 210453397504 $ T^20 + 8T^19 + 32T^18 - 960T^17 + 42176T^16 + 164864T^15 + 430080T^14 - 10567680T^13 + 402550784T^12 + 1069056000T^11 + 1266810880T^10 - 392953856T^9 + 724321959936T^8 + 2867528204288T^7 + 5667880435712T^6 - 13407377948672T^5 + 28263770488832T^4 + 27655294418944T^3 + 12403865550848T^2 + 2284922601472*T + 210453397504
$71$	$ T^{20} + \cdots + 84783728164864 $ T^20 + 688T^18 + 184320T^16 + 24765952T^14 + 1802900480T^12 + 72970174464T^10 + 1633300643840T^8 + 19751533281280T^6 + 119444957298688T^4 + 286334329552896*T^2 + 84783728164864
$73$	$ T^{20} + \cdots + 12\!\cdots\!76 $ T^20 + 888T^18 + 332272T^16 + 68179456T^14 + 8344750848T^12 + 619035629568T^10 + 26910405677056T^8 + 620748281806848T^6 + 5934864196173824T^4 + 9210388927217664*T^2 + 1224592353918976
$79$	$ (T^{10} + 28 T^{9} + \cdots + 46268416)^{2} $ (T^10 + 28T^9 - 104T^8 - 9352T^7 - 70364T^6 + 266432T^5 + 3393408T^4 - 3687424T^3 - 50255872T^2 + 77168640*T + 46268416)^2
$83$	$ T^{20} + \cdots + 76\!\cdots\!36 $ T^20 + 48T^19 + 1152T^18 + 15936T^17 + 181424T^16 + 2781696T^15 + 51499008T^14 + 684328960T^13 + 6124317440T^12 + 40555229184T^11 + 322663907328T^10 + 3259538489344T^9 + 27414010302464T^8 + 152230226034688T^7 + 700248334794752T^6 + 4222033294524416T^5 + 31122539714969600T^4 + 158089032079769600T^3 + 503148107752538112T^2 + 875555940458299392*T + 761801736963751936
$89$	$ T^{20} + \cdots + 15\!\cdots\!36 $ T^20 + 1096T^18 + 500432T^16 + 123870720T^14 + 18159960576T^12 + 1624662126592T^10 + 88667808735232T^8 + 2876984841338880T^6 + 52248775078248448T^4 + 469438536991899648*T^2 + 1518596177800462336
$97$	$ (T^{10} - 28 T^{9} + \cdots - 37943296)^{2} $ (T^10 - 28T^9 - 44T^8 + 6080T^7 - 22480T^6 - 392640T^5 + 2172480T^4 + 5846528T^3 - 52876288T^2 + 87896064*T - 37943296)^2
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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:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.s (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + \beta_{5} q^{5} - \beta_{13} q^{7} + \beta_{3} q^{9}+O(q^{10}) \) q + b1 * q^3 + b5 * q^5 - b13 * q^7 + b3 * q^9	\( q + \beta_1 q^{3} + \beta_{5} q^{5} - \beta_{13} q^{7} + \beta_{3} q^{9} + (\beta_{8} - \beta_{5}) q^{11} - \beta_{14} q^{13} - q^{15} + ( - \beta_{19} - 1) q^{17} + (\beta_{12} - \beta_{10} + \cdots + \beta_{6}) q^{19}+ \cdots + ( - \beta_{6} + \beta_1) q^{99}+O(q^{100}) \) q + b1 * q^3 + b5 * q^5 - b13 * q^7 + b3 * q^9 + (b8 - b5) * q^11 - b14 * q^13 - q^15 + (-b19 - 1) * q^17 + (b12 - b10 + b9 + b6) * q^19 - b17 * q^21 + (b17 - b16 - b13 - b12 + b11 - b9 - b5 + b4 + b3 - b1) * q^23 - b3 * q^25 + b5 * q^27 + (-b19 - b18 - b16 - b14 - b13 + b7 + b6 + b4 - b2 + b1) * q^29 + (b18 + b15 + b14 + b8 - b7 - b6 + b5 - b1 - 1) * q^31 + (b18 + 1) * q^33 - b16 * q^35 + (b18 - b17 + b8 + b5 + b3 - b2 - 1) * q^37 + b9 * q^39 + (-b13 + b9 + b8 + b6 - b5 + 3b3 - b2 - b1) q^41 + (b19 + b17 + b15 + b8 - b5 + b4 + b3 - 1) * q^43 - b1 * q^45 + (b19 + b18 + b17 + b16 + b15 + b14) * q^47 + (b18 + b15 + b14 + b10 + b7 + 2b5 - 2b1 - 2) * q^49 + (b12 - b1) * q^51 + (b18 - b17 + b15 + b13 + b8 + b7 + b5 - b3 - b2 + 1) * q^53 + (b3 + b2) * q^55 + (b15 - b14 + b4 - b2) * q^57 + (-b18 - b13 - b10 - b9 - b8 - b7 + b5 + b3 + b2 - 1) * q^59 + (b16 + b13 + b12 - b10 + b9 - b7 + b6 + 2b1) q^61 - b7 * q^63 + b10 * q^65 + (-b19 - b16 - b14 + b13 - b7 - b6 + b4 - b3 + b1 - 1) * q^67 + (-b19 - b17 - b15 + b13 - b11 + b7 + b5 - b4 - b3 + 1) * q^69 + (b17 - b16 + b15 - b14 - 2b13 - 2b5 + 2b4 + 2b3 - 2b1) q^71 + (-b17 + b16 - 2b9 - b8 - b6 - b5 + 2b3 - b1) * q^73 - b5 * q^75 + (-b18 + b16 - b14 - b13 + 2b12 - b10 + b9 + b7 + 3b6 + 3b3 - b2 + b1 + 3) q^77 + (b18 - b17 - b16 + b15 + b14 + 2b10 + b8 + b7 - b6 - b5 + b1 - 3) q^79 - q^81 + (b18 - 2b16 - 2b13 - 2b12 + b10 - b9 + 2b7 - b3 + b2 - 4b1 - 1) q^83 + (-b11 - b5) * q^85 + (-b17 + b16 + b13 + b12 - b11 + b9 + b8 + b6 + b3 - b2) * q^87 + (-b17 + b16 + b15 - b14 - b13 + b9 + 2b8 + 2b6 + 2b5 - b3 - b2 + 2b1) * q^89 + (2b19 + 2b18 + 2b15 - 2b13 + b10 + b9 + 2b8 - 2b7 + 2b5 + 2b4 + 2b3 - 2b2 - 2) * q^91 + (b18 - b16 + b10 - b9 - b6 - b3 + b2 - b1 - 1) * q^93 + (b19 + b18 + b15 + b14) * q^95 + (-2b19 - 2b17 - 2b16 - b15 - b14 + 2b7 + 4) * q^97 + (-b6 + b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q + 8 q^{11} - 20 q^{15} - 24 q^{17} - 4 q^{19} - 16 q^{29} + 16 q^{33} - 16 q^{37} - 8 q^{43} - 52 q^{49} + 4 q^{51} + 16 q^{53} - 16 q^{59} + 4 q^{61} + 8 q^{63} - 8 q^{67} + 4 q^{69} + 40 q^{77} - 56 q^{79} - 20 q^{81} - 48 q^{83} - 4 q^{85} - 8 q^{91} - 16 q^{93} + 56 q^{97} + 8 q^{99}+O(q^{100}) \) 20 * q + 8 * q^11 - 20 * q^15 - 24 * q^17 - 4 * q^19 - 16 * q^29 + 16 * q^33 - 16 * q^37 - 8 * q^43 - 52 * q^49 + 4 * q^51 + 16 * q^53 - 16 * q^59 + 4 * q^61 + 8 * q^63 - 8 * q^67 + 4 * q^69 + 40 * q^77 - 56 * q^79 - 20 * q^81 - 48 * q^83 - 4 * q^85 - 8 * q^91 - 16 * q^93 + 56 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1920.2.s.f

Level	$1920$
Weight	$2$
Character orbit	1920.s
Analytic conductor	$15.331$
Analytic rank	$0$
Dimension	$20$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(15.3312771881\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + \cdots + 1024 \) x^20 - 2x^18 - 4x^17 + 7x^16 + 16x^15 + 6x^14 - 36x^13 - 42x^12 + 40x^11 + 136x^10 + 80x^9 - 168x^8 - 288x^7 + 96x^6 + 512x^5 + 448x^4 - 512x^3 - 512*x^2 + 1024
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{20} \)
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 18 \nu^{19} + 33 \nu^{18} + 8 \nu^{17} - 104 \nu^{16} - 94 \nu^{15} + 199 \nu^{14} + 616 \nu^{13} + \cdots - 5632 ) / 512 \) (18v^19 + 33v^18 + 8v^17 - 104v^16 - 94v^15 + 199v^14 + 616v^13 + 364v^12 - 768v^11 - 1426v^10 + 152v^9 + 3332v^8 + 3808v^7 - 1024v^6 - 5536v^5 - 2976v^4 + 7040v^3 + 8448v^2 + 2816*v - 5632) / 512
\(\beta_{2}\)	\(=\)	\( ( 5 \nu^{19} - 22 \nu^{18} - 84 \nu^{17} - 112 \nu^{16} + 107 \nu^{15} + 334 \nu^{14} + 24 \nu^{13} + \cdots - 23552 ) / 512 \) (5v^19 - 22v^18 - 84v^17 - 112v^16 + 107v^15 + 334v^14 + 24v^13 - 1216v^12 - 1906v^11 - 100v^10 + 3180v^9 + 2992v^8 - 4208v^7 - 11728v^6 - 7520v^5 + 6912v^4 + 13952v^3 - 3456v^2 - 22016*v - 23552) / 512
\(\beta_{3}\)	\(=\)	\( ( - 21 \nu^{19} + 10 \nu^{18} + 104 \nu^{17} + 192 \nu^{16} - 123 \nu^{15} - 594 \nu^{14} + \cdots + 28672 ) / 512 \) (-21v^19 + 10v^18 + 104v^17 + 192v^16 - 123v^15 - 594v^14 - 364v^13 + 1376v^12 + 2786v^11 + 652v^10 - 4436v^9 - 5696v^8 + 3232v^7 + 15248v^6 + 11424v^5 - 7936v^4 - 21760v^3 - 128v^2 + 24064*v + 28672) / 512
\(\beta_{4}\)	\(=\)	\( ( 16 \nu^{19} + 10 \nu^{18} - 47 \nu^{17} - 138 \nu^{16} + 20 \nu^{15} + 394 \nu^{14} + 479 \nu^{13} + \cdots - 16896 ) / 256 \) (16v^19 + 10v^18 - 47v^17 - 138v^16 + 20v^15 + 394v^14 + 479v^13 - 518v^12 - 1776v^11 - 1048v^10 + 2342v^9 + 4684v^8 + 492v^7 - 8104v^6 - 8944v^5 + 2336v^4 + 14816v^3 + 6080v^2 - 9856*v - 16896) / 256
\(\beta_{5}\)	\(=\)	\( ( - 31 \nu^{19} - 39 \nu^{18} + 30 \nu^{17} + 204 \nu^{16} + 67 \nu^{15} - 497 \nu^{14} - 946 \nu^{13} + \cdots + 17920 ) / 512 \) (-31v^19 - 39v^18 + 30v^17 + 204v^16 + 67v^15 - 497v^14 - 946v^13 + 64v^12 + 2082v^11 + 2086v^10 - 2008v^9 - 6636v^8 - 3800v^7 + 7264v^6 + 11680v^5 + 736v^4 - 17728v^3 - 11776v^2 + 5120*v + 17920) / 512
\(\beta_{6}\)	\(=\)	\( ( 62 \nu^{19} + 3 \nu^{18} - 252 \nu^{17} - 544 \nu^{16} + 222 \nu^{15} + 1605 \nu^{14} + 1324 \nu^{13} + \cdots - 76288 ) / 512 \) (62v^19 + 3v^18 - 252v^17 - 544v^16 + 222v^15 + 1605v^14 + 1324v^13 - 3188v^12 - 7456v^11 - 2566v^10 + 11248v^9 + 16412v^8 - 5808v^7 - 39264v^6 - 32352v^5 + 19232v^4 + 59392v^3 + 5632v^2 - 61696*v - 76288) / 512
\(\beta_{7}\)	\(=\)	\( ( - 23 \nu^{19} - 44 \nu^{18} - 6 \nu^{17} + 140 \nu^{16} + 135 \nu^{15} - 276 \nu^{14} - 854 \nu^{13} + \cdots + 6144 ) / 256 \) (-23v^19 - 44v^18 - 6v^17 + 140v^16 + 135v^15 - 276v^14 - 854v^13 - 516v^12 + 1070v^11 + 2064v^10 - 112v^9 - 4608v^8 - 5544v^7 + 1120v^6 + 7872v^5 + 4672v^4 - 8768v^3 - 12416v^2 - 4608*v + 6144) / 256
\(\beta_{8}\)	\(=\)	\( ( 11 \nu^{19} - 65 \nu^{18} - 202 \nu^{17} - 220 \nu^{16} + 337 \nu^{15} + 777 \nu^{14} - 210 \nu^{13} + \cdots - 45568 ) / 512 \) (11v^19 - 65v^18 - 202v^17 - 220v^16 + 337v^15 + 777v^14 - 210v^13 - 3024v^12 - 4042v^11 + 698v^10 + 7648v^9 + 5452v^8 - 11480v^7 - 26464v^6 - 13216v^5 + 18336v^4 + 28864v^3 - 13312v^2 - 48640*v - 45568) / 512
\(\beta_{9}\)	\(=\)	\( ( - 12 \nu^{19} + 9 \nu^{18} + 67 \nu^{17} + 116 \nu^{16} - 84 \nu^{15} - 365 \nu^{14} - 179 \nu^{13} + \cdots + 17920 ) / 128 \) (-12v^19 + 9v^18 + 67v^17 + 116v^16 - 84v^15 - 365v^14 - 179v^13 + 920v^12 + 1724v^11 + 290v^10 - 2854v^9 - 3332v^8 + 2516v^7 + 9824v^6 + 6736v^5 - 5568v^4 - 13408v^3 + 1088v^2 + 16256*v + 17920) / 128
\(\beta_{10}\)	\(=\)	\( ( - 31 \nu^{19} - 50 \nu^{18} + 2 \nu^{17} + 184 \nu^{16} + 135 \nu^{15} - 382 \nu^{14} - 1022 \nu^{13} + \cdots + 9728 ) / 256 \) (-31v^19 - 50v^18 + 2v^17 + 184v^16 + 135v^15 - 382v^14 - 1022v^13 - 456v^12 + 1494v^11 + 2380v^10 - 560v^9 - 5808v^8 - 5992v^7 + 2704v^6 + 9664v^5 + 4736v^4 - 12096v^3 - 13824v^2 - 3840*v + 9728) / 256
\(\beta_{11}\)	\(=\)	\( ( - 66 \nu^{19} + 37 \nu^{18} + 350 \nu^{17} + 636 \nu^{16} - 418 \nu^{15} - 1989 \nu^{14} + \cdots + 98304 ) / 512 \) (-66v^19 + 37v^18 + 350v^17 + 636v^16 - 418v^15 - 1989v^14 - 1182v^13 + 4712v^12 + 9368v^11 + 2094v^10 - 15092v^9 - 19124v^8 + 11400v^7 + 51984v^6 + 38656v^5 - 27232v^4 - 73536v^3 - 128v^2 + 82944*v + 98304) / 512
\(\beta_{12}\)	\(=\)	\( ( 79 \nu^{19} - 21 \nu^{18} - 364 \nu^{17} - 736 \nu^{16} + 389 \nu^{15} + 2197 \nu^{14} + 1568 \nu^{13} + \cdots - 108544 ) / 512 \) (79v^19 - 21v^18 - 364v^17 - 736v^16 + 389v^15 + 2197v^14 + 1568v^13 - 4812v^12 - 10386v^11 - 3062v^10 + 16004v^9 + 21892v^8 - 10352v^7 - 55984v^6 - 44224v^5 + 27488v^4 + 81920v^3 + 2944v^2 - 88320*v - 108544) / 512
\(\beta_{13}\)	\(=\)	\( ( - 33 \nu^{19} + 22 \nu^{18} + 184 \nu^{17} + 328 \nu^{16} - 223 \nu^{15} - 1006 \nu^{14} + \cdots + 52736 ) / 256 \) (-33v^19 + 22v^18 + 184v^17 + 328v^16 - 223v^15 - 1006v^14 - 556v^13 + 2504v^12 + 4842v^11 + 996v^10 - 7812v^9 - 9568v^8 + 6496v^7 + 27280v^6 + 19872v^5 - 14592v^4 - 38016v^3 + 1536v^2 + 44800*v + 52736) / 256
\(\beta_{14}\)	\(=\)	\( ( - 41 \nu^{19} - 75 \nu^{18} - 24 \nu^{17} + 228 \nu^{16} + 221 \nu^{15} - 421 \nu^{14} - 1428 \nu^{13} + \cdots + 10240 ) / 256 \) (-41v^19 - 75v^18 - 24v^17 + 228v^16 + 221v^15 - 421v^14 - 1428v^13 - 968v^12 + 1598v^11 + 3366v^10 + 44v^9 - 7452v^8 - 9488v^7 + 960v^6 + 12128v^5 + 8352v^4 - 14208v^3 - 20352v^2 - 10496*v + 10240) / 256
\(\beta_{15}\)	\(=\)	\( ( - 56 \nu^{19} - 73 \nu^{18} + 46 \nu^{17} + 368 \nu^{16} + 148 \nu^{15} - 855 \nu^{14} - 1734 \nu^{13} + \cdots + 30720 ) / 256 \) (-56v^19 - 73v^18 + 46v^17 + 368v^16 + 148v^15 - 855v^14 - 1734v^13 - 52v^12 + 3588v^11 + 3930v^10 - 3068v^9 - 11716v^8 - 7560v^7 + 11776v^6 + 20704v^5 + 2720v^4 - 30144v^3 - 21632v^2 + 7168*v + 30720) / 256
\(\beta_{16}\)	\(=\)	\( ( - 57 \nu^{19} - 99 \nu^{18} - 6 \nu^{17} + 336 \nu^{16} + 269 \nu^{15} - 693 \nu^{14} - 1926 \nu^{13} + \cdots + 18432 ) / 256 \) (-57v^19 - 99v^18 - 6v^17 + 336v^16 + 269v^15 - 693v^14 - 1926v^13 - 932v^12 + 2694v^11 + 4478v^10 - 1000v^9 - 10836v^8 - 11400v^7 + 4816v^6 + 18272v^5 + 8352v^4 - 22976v^3 - 26368v^2 - 6144*v + 18432) / 256
\(\beta_{17}\)	\(=\)	\( ( - 70 \nu^{19} - 99 \nu^{18} + 48 \nu^{17} + 456 \nu^{16} + 218 \nu^{15} - 1045 \nu^{14} + \cdots + 36352 ) / 256 \) (-70v^19 - 99v^18 + 48v^17 + 456v^16 + 218v^15 - 1045v^14 - 2240v^13 - 244v^12 + 4368v^11 + 5110v^10 - 3480v^9 - 14668v^8 - 10336v^7 + 13440v^6 + 25888v^5 + 4512v^4 - 36736v^3 - 28800v^2 + 6400*v + 36352) / 256
\(\beta_{18}\)	\(=\)	\( ( 38 \nu^{19} + 57 \nu^{18} - 16 \nu^{17} - 234 \nu^{16} - 130 \nu^{15} + 527 \nu^{14} + 1208 \nu^{13} + \cdots - 16512 ) / 128 \) (38v^19 + 57v^18 - 16v^17 - 234v^16 - 130v^15 + 527v^14 + 1208v^13 + 286v^12 - 2120v^11 - 2730v^10 + 1504v^9 + 7560v^8 + 6080v^7 - 5832v^6 - 12864v^5 - 3168v^4 + 18112v^3 + 15680v^2 - 896*v - 16512) / 128
\(\beta_{19}\)	\(=\)	\( ( 177 \nu^{19} + 256 \nu^{18} - 86 \nu^{17} - 1108 \nu^{16} - 577 \nu^{15} + 2496 \nu^{14} + \cdots - 86528 ) / 512 \) (177v^19 + 256v^18 - 86v^17 - 1108v^16 - 577v^15 + 2496v^14 + 5594v^13 + 1004v^12 - 10306v^11 - 12712v^10 + 7728v^9 + 35792v^8 + 27064v^7 - 30432v^6 - 62400v^5 - 13312v^4 + 88512v^3 + 72704v^2 - 9472*v - 86528) / 512

\(\nu\)	\(=\)	\( ( \beta_{19} + \beta_{16} + \beta_{15} + \beta_{8} - \beta_{7} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 4 \) (b19 + b16 + b15 + b8 - b7 + b3 - b2 + b1 - 1) / 4
\(\nu^{2}\)	\(=\)	\( ( -\beta_{17} - \beta_{15} + \beta_{14} - \beta_{12} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{2} - 2\beta _1 + 1 ) / 4 \) (-b17 - b15 + b14 - b12 - b9 + b7 - b6 + b2 - 2*b1 + 1) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{16} + \beta_{15} - 2 \beta_{13} + 2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \cdots + 2 ) / 4 \) (b16 + b15 - 2b13 + 2b11 - b10 - b9 - b8 - b7 + b4 - b2 + 3*b1 + 2) / 4
\(\nu^{4}\)	\(=\)	\( ( - \beta_{18} - 2 \beta_{17} + \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} + \cdots - 4 ) / 4 \) (-b18 - 2b17 + b16 - b15 - b14 + b13 + b11 + b10 + b8 + 2b6 + 4b5 - b3 - 2b2 - 2*b1 - 4) / 4
\(\nu^{5}\)	\(=\)	\( ( - \beta_{19} + \beta_{18} + \beta_{17} - \beta_{14} - \beta_{13} + 2 \beta_{11} - 3 \beta_{10} + \cdots - 6 ) / 4 \) (-b19 + b18 + b17 - b14 - b13 + 2b11 - 3b10 - 3b9 - 2b8 + 2b7 + b6 - b5 + 2b4 + 6b3 + 2b2 - 6*b1 - 6) / 4
\(\nu^{6}\)	\(=\)	\( ( - 4 \beta_{19} - 5 \beta_{17} - 2 \beta_{16} + \beta_{15} - \beta_{14} + 2 \beta_{13} + \beta_{12} + \cdots - 9 ) / 4 \) (-4b19 - 5b17 - 2b16 + b15 - b14 + 2b13 + b12 + 2b10 + b9 - 2b8 + 3b7 - b6 + 6b5 - 14b3 + b2 + 8b1 - 9) / 4
\(\nu^{7}\)	\(=\)	\( ( - 2 \beta_{19} - 4 \beta_{18} + 10 \beta_{17} - 9 \beta_{16} - \beta_{15} - 6 \beta_{14} - 4 \beta_{13} + \cdots - 8 ) / 4 \) (-2b19 - 4b18 + 10b17 - 9b16 - b15 - 6b14 - 4b13 - 2b12 + 4b11 - b10 + b9 - b8 + b7 + 8b6 - 12b5 + 5b4 - b2 - 5b1 - 8) / 4
\(\nu^{8}\)	\(=\)	\( ( 7 \beta_{18} + 2 \beta_{17} + 3 \beta_{16} + \beta_{15} - 3 \beta_{14} + 9 \beta_{13} + 4 \beta_{12} + \cdots + 2 ) / 4 \) (7b18 + 2b17 + 3b16 + b15 - 3b14 + 9b13 + 4b12 - 9b11 + b10 + 6b9 - b8 + 2b7 + 2b6 + 28b5 - 5b3 + 4b2 + 10b1 + 2) / 4
\(\nu^{9}\)	\(=\)	\( ( - 3 \beta_{19} + 5 \beta_{18} + 5 \beta_{17} - 14 \beta_{16} + 6 \beta_{15} - \beta_{14} - \beta_{13} + \cdots - 8 ) / 4 \) (-3b19 + 5b18 + 5b17 - 14b16 + 6b15 - b14 - b13 - 8b12 - 6b11 + 17b10 - 7b9 + 4b8 - 15b6 - 37b5 - 2b4 - 8b3 + 12b2 - 4b1 - 8) / 4
\(\nu^{10}\)	\(=\)	\( ( - 4 \beta_{19} - 12 \beta_{18} + 5 \beta_{17} + 2 \beta_{16} - 9 \beta_{15} + \beta_{14} + 6 \beta_{13} + \cdots + 37 ) / 4 \) (-4b19 - 12b18 + 5b17 + 2b16 - 9b15 + b14 + 6b13 - 5b12 - 12b11 - 10b10 + 19b9 + 2b8 + b7 + 17b6 - 34b5 - 8b4 - 10b3 - b2 + 44b1 + 37) / 4
\(\nu^{11}\)	\(=\)	\( ( 6 \beta_{19} + 16 \beta_{18} + 6 \beta_{17} - 11 \beta_{16} + 5 \beta_{15} - 2 \beta_{14} - 20 \beta_{13} + \cdots + 32 ) / 4 \) (6b19 + 16b18 + 6b17 - 11b16 + 5b15 - 2b14 - 20b13 - 18b12 - 24b11 + 25b10 - 5b9 + 9b8 - 5b7 - 20b6 + 72b5 + 7b4 + 44b3 - 15b2 + 13*b1 + 32) / 4
\(\nu^{12}\)	\(=\)	\( ( 16 \beta_{19} + 21 \beta_{18} - 2 \beta_{17} + 53 \beta_{16} - 25 \beta_{15} + 43 \beta_{14} + 19 \beta_{13} + \cdots + 2 ) / 4 \) (16b19 + 21b18 - 2b17 + 53b16 - 25b15 + 43b14 + 19b13 + 4b12 + b11 + 19b10 - 10b9 + 17b8 - 46b7 - 42b6 - 108b5 - 40b4 - 15b3 + 24b2 - 2b1 + 2) / 4
\(\nu^{13}\)	\(=\)	\( ( - 37 \beta_{19} - 41 \beta_{18} - 73 \beta_{17} + 10 \beta_{16} - 18 \beta_{15} - 27 \beta_{14} + \cdots + 148 ) / 4 \) (-37b19 - 41b18 - 73b17 + 10b16 - 18b15 - 27b14 - 27b13 - 16b12 - 18b11 - 29b10 + 11b9 - 16b8 + 36b7 + 43b6 - 15b5 - 22b4 + 124b3 - 40b2 - 72*b1 + 148) / 4
\(\nu^{14}\)	\(=\)	\( ( 28 \beta_{19} - 12 \beta_{18} - 21 \beta_{17} + 46 \beta_{16} - 15 \beta_{15} + 63 \beta_{14} + \cdots - 269 ) / 4 \) (28b19 - 12b18 - 21b17 + 46b16 - 15b15 + 63b14 + 10b13 + 37b12 + 4b11 - 22b10 - 19b9 - 2b8 - 57b7 - 33b6 + 82b5 - 8b4 + 34b3 - 63b2 + 92*b1 - 269) / 4
\(\nu^{15}\)	\(=\)	\( ( - 46 \beta_{19} + 48 \beta_{18} - 38 \beta_{17} + 3 \beta_{16} - 85 \beta_{15} + 18 \beta_{14} + \cdots + 112 ) / 4 \) (-46b19 + 48b18 - 38b17 + 3b16 - 85b15 + 18b14 - 36b13 + 34b12 + 64b11 + 7b10 + 21b9 + 7b8 + 69b7 - 36b6 - 24b5 - 7b4 + 52b3 + 7b2 - 317*b1 + 112) / 4
\(\nu^{16}\)	\(=\)	\( ( 64 \beta_{19} - 69 \beta_{18} - 30 \beta_{17} + 123 \beta_{16} + 105 \beta_{15} - 35 \beta_{14} + \cdots - 218 ) / 4 \) (64b19 - 69b18 - 30b17 + 123b16 + 105b15 - 35b14 + 29b13 + 156b12 + 87b11 - 155b10 + 42b9 - 145b8 - 98b7 + 138b6 + 44b5 - 32b4 - 49b3 - 72b2 - 230*b1 - 218) / 4
\(\nu^{17}\)	\(=\)	\( ( - 51 \beta_{19} - 175 \beta_{18} - 71 \beta_{17} - 170 \beta_{16} - 110 \beta_{15} - 29 \beta_{14} + \cdots - 268 ) / 4 \) (-51b19 - 175b18 - 71b17 - 170b16 - 110b15 - 29b14 + 227b13 + 144b12 - 38b11 - 27b10 - 11b9 - 32b8 + 244b7 + 53b6 + 47b5 + 22b4 - 12b3 + 32b2 - 48*b1 - 268) / 4
\(\nu^{18}\)	\(=\)	\( ( 124 \beta_{19} + 292 \beta_{18} + 357 \beta_{17} - 158 \beta_{16} + 375 \beta_{15} + 81 \beta_{14} + \cdots - 139 ) / 4 \) (124b19 + 292b18 + 357b17 - 158b16 + 375b15 + 81b14 - 194b13 + 115b12 + 172b11 - 146b10 + 195b9 + 130b8 + 65b7 - 15b6 - 74b5 + 216b4 + 126b3 + 39b2 - 12*b1 - 139) / 4
\(\nu^{19}\)	\(=\)	\( ( 254 \beta_{19} - 112 \beta_{18} + 62 \beta_{17} - 171 \beta_{16} + 101 \beta_{15} - 58 \beta_{14} + \cdots + 392 ) / 4 \) (254b19 - 112b18 + 62b17 - 171b16 + 101b15 - 58b14 + 228b13 - 26b12 - 16b11 + 177b10 + 59b9 - 223b8 + 43b7 - 36b6 + 776b5 - 129b4 - 1748b3 + 185b2 - 171*b1 + 392) / 4

\(n\)	\(511\)	\(641\)	\(901\)	\(1537\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\beta_{3}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 481.10
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} \) T^20
$3$	\( (T^{4} + 1)^{5} \) (T^4 + 1)^5
$5$	\( (T^{4} + 1)^{5} \) (T^4 + 1)^5
$7$	\( T^{20} + 96 T^{18} + \cdots + 262144 \) T^20 + 96T^18 + 3880T^16 + 86368T^14 + 1161104T^12 + 9697664T^10 + 49590528T^8 + 145248256T^6 + 204587008T^4 + 76283904*T^2 + 262144
$11$	\( T^{20} - 8 T^{19} + \cdots + 1048576 \) T^20 - 8T^19 + 32T^18 - 112T^17 + 1808T^16 - 13696T^15 + 57984T^14 - 173952T^13 + 1093696T^12 - 7170048T^11 + 30799872T^10 - 84426752T^9 + 251479040T^8 - 1115578368T^7 + 4560683008T^6 - 11886264320T^5 + 18952028160T^4 - 15595208704T^3 + 6343884800T^2 + 115343360*T + 1048576
$13$	\( T^{20} + \cdots + 167981940736 \) T^20 + 144T^17 + 2088T^16 + 4192T^15 + 10368T^14 + 175680T^13 + 1636752T^12 + 5323392T^11 + 12435968T^10 + 60063232T^9 + 413145856T^8 + 1511043072T^7 + 3460349952T^6 + 7034568704T^5 + 25928249344T^4 + 90035355648T^3 + 205127811072T^2 + 262517686272*T + 167981940736
$17$	\( (T^{10} + 12 T^{9} + \cdots - 20032)^{2} \) (T^10 + 12T^9 - 34T^8 - 792T^7 - 676T^6 + 15824T^5 + 25976T^4 - 106144T^3 - 145472T^2 + 151040*T - 20032)^2
$19$	\( T^{20} + \cdots + 19716653056 \) T^20 + 4T^19 + 8T^18 - 72T^17 + 5164T^16 + 17152T^15 + 29888T^14 - 209856T^13 + 7892400T^12 + 20915520T^11 + 28386432T^10 - 73408640T^9 + 3224447552T^8 + 7963704832T^7 + 9631844352T^6 + 7396435968T^5 + 29049219072T^4 + 68702330880T^3 + 89860866048T^2 + 59527397376*T + 19716653056
$23$	\( T^{20} + \cdots + 217558810624 \) T^20 + 348T^18 + 49980T^16 + 3828512T^14 + 169125872T^12 + 4370425280T^10 + 64474080320T^8 + 505220718592T^6 + 1734629007360T^4 + 1211058880512*T^2 + 217558810624
$29$	\( T^{20} + \cdots + 19723262623744 \) T^20 + 16T^19 + 128T^18 + 576T^17 + 15632T^16 + 231168T^15 + 1863680T^14 + 7973888T^13 + 52284928T^12 + 541822976T^11 + 4155932672T^10 + 18135728128T^9 + 46757634048T^8 + 67031334912T^7 + 169368092672T^6 + 779917459456T^5 + 2276071636992T^4 + 1868828770304T^3 + 159752650752T^2 + 2510316109824*T + 19723262623744
$31$	\( (T^{10} - 204 T^{8} + \cdots - 28698368)^{2} \) (T^10 - 204T^8 - 168T^7 + 14932T^6 + 19536T^5 - 468112T^4 - 621056T^3 + 6148800T^2 + 4962560T - 28698368)^2
$37$	\( T^{20} + 16 T^{19} + \cdots + 18939904 \) T^20 + 16T^19 + 128T^18 + 80T^17 + 4904T^16 + 81440T^15 + 678528T^14 + 581952T^13 - 2713200T^12 - 12409216T^11 + 184033792T^10 - 394871296T^9 + 451001088T^8 - 422612992T^7 + 1128316928T^6 - 2241855488T^5 + 2665852928T^4 - 1848475648T^3 + 777125888T^2 - 171573248*T + 18939904
$41$	\( T^{20} + \cdots + 3311118843904 \) T^20 + 344T^18 + 48592T^16 + 3697152T^14 + 166691328T^12 + 4616884224T^10 + 78819631104T^8 + 806185730048T^6 + 4549133533184T^4 + 11279143534592*T^2 + 3311118843904
$43$	\( T^{20} + \cdots + 3288334336 \) T^20 + 8T^19 + 32T^18 + 704T^17 + 20960T^16 + 208640T^15 + 1246208T^14 + 8994816T^13 + 121272576T^12 + 1219307520T^11 + 7850893312T^10 + 32131072000T^9 + 83002048512T^8 + 118577430528T^7 + 76928778240T^6 + 72320286720T^5 + 430934327296T^4 + 354645180416T^3 + 150625845248T^2 + 31474057216*T + 3288334336
$47$	\( (T^{10} - 166 T^{8} + \cdots - 12544)^{2} \) (T^10 - 166T^8 - 160T^7 + 7980T^6 + 11520T^5 - 97448T^4 - 35072T^3 + 283712T^2 - 145152T - 12544)^2
$53$	\( T^{20} + \cdots + 4398046511104 \) T^20 - 16T^19 + 128T^18 - 512T^17 + 18720T^16 - 279040T^15 + 2199552T^14 - 8978432T^13 + 113316096T^12 - 1463160832T^11 + 11259117568T^10 - 46965587968T^9 + 243227688960T^8 - 2110865276928T^7 + 15258273972224T^6 - 64499805585408T^5 + 164036679303168T^4 - 217797791580160T^3 + 158879430213632T^2 - 37383395344384*T + 4398046511104
$59$	\( T^{20} + \cdots + 16\!\cdots\!04 \) T^20 + 16T^19 + 128T^18 - 112T^17 + 21776T^16 + 332800T^15 + 2543744T^14 + 208256T^13 + 117256256T^12 + 1869032960T^11 + 14857752576T^10 + 4783368192T^9 + 60533785600T^8 + 1690337058816T^7 + 18984867758080T^6 - 8752961814528T^5 - 19284448182272T^4 + 225773976551424T^3 + 7682191945367552T^2 - 15913855322423296*T + 16482977321058304
$61$	\( T^{20} + \cdots + 74350019584 \) T^20 - 4T^19 + 8T^18 - 840T^17 + 21588T^16 - 186688T^15 + 926848T^14 - 3663488T^13 + 31363232T^12 - 246492544T^11 + 1229804288T^10 - 3542995712T^9 + 8596122240T^8 - 32054490112T^7 + 145373775872T^6 - 400694880256T^5 + 635588871424T^4 - 405824488448T^3 + 55356622848T^2 + 90727790592*T + 74350019584
$67$	\( T^{20} + \cdots + 210453397504 \) T^20 + 8T^19 + 32T^18 - 960T^17 + 42176T^16 + 164864T^15 + 430080T^14 - 10567680T^13 + 402550784T^12 + 1069056000T^11 + 1266810880T^10 - 392953856T^9 + 724321959936T^8 + 2867528204288T^7 + 5667880435712T^6 - 13407377948672T^5 + 28263770488832T^4 + 27655294418944T^3 + 12403865550848T^2 + 2284922601472*T + 210453397504
$71$	\( T^{20} + \cdots + 84783728164864 \) T^20 + 688T^18 + 184320T^16 + 24765952T^14 + 1802900480T^12 + 72970174464T^10 + 1633300643840T^8 + 19751533281280T^6 + 119444957298688T^4 + 286334329552896*T^2 + 84783728164864
$73$	\( T^{20} + \cdots + 12\!\cdots\!76 \) T^20 + 888T^18 + 332272T^16 + 68179456T^14 + 8344750848T^12 + 619035629568T^10 + 26910405677056T^8 + 620748281806848T^6 + 5934864196173824T^4 + 9210388927217664*T^2 + 1224592353918976
$79$	\( (T^{10} + 28 T^{9} + \cdots + 46268416)^{2} \) (T^10 + 28T^9 - 104T^8 - 9352T^7 - 70364T^6 + 266432T^5 + 3393408T^4 - 3687424T^3 - 50255872T^2 + 77168640*T + 46268416)^2
$83$	\( T^{20} + \cdots + 76\!\cdots\!36 \) T^20 + 48T^19 + 1152T^18 + 15936T^17 + 181424T^16 + 2781696T^15 + 51499008T^14 + 684328960T^13 + 6124317440T^12 + 40555229184T^11 + 322663907328T^10 + 3259538489344T^9 + 27414010302464T^8 + 152230226034688T^7 + 700248334794752T^6 + 4222033294524416T^5 + 31122539714969600T^4 + 158089032079769600T^3 + 503148107752538112T^2 + 875555940458299392*T + 761801736963751936
$89$	\( T^{20} + \cdots + 15\!\cdots\!36 \) T^20 + 1096T^18 + 500432T^16 + 123870720T^14 + 18159960576T^12 + 1624662126592T^10 + 88667808735232T^8 + 2876984841338880T^6 + 52248775078248448T^4 + 469438536991899648*T^2 + 1518596177800462336
$97$	\( (T^{10} - 28 T^{9} + \cdots - 37943296)^{2} \) (T^10 - 28T^9 - 44T^8 + 6080T^7 - 22480T^6 - 392640T^5 + 2172480T^4 + 5846528T^3 - 52876288T^2 + 87896064*T - 37943296)^2
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