LMFDB - Newform orbit 1900.2.i.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(201,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("1900.201");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.i (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{3})$
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} + 20 x^{18} + 261 x^{16} + 1994 x^{14} + 11074 x^{12} + 39211 x^{10} + 99376 x^{8} + 134299 x^{6} + \cdots + 4096 $ x^20 + 20x^18 + 261x^16 + 1994x^14 + 11074x^12 + 39211x^10 + 99376x^8 + 134299x^6 + 124617x^4 + 24768*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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_{14} q^{3} + (\beta_{18} + \beta_{15}) q^{7} + (\beta_{11} + \beta_{4} - 1) q^{9} + \beta_{8} q^{11} + (\beta_{16} - \beta_{10}) q^{13} + (\beta_{18} + \beta_{9} + \beta_{5}) q^{17} + (\beta_{13} + \beta_{12} - \beta_{11} - 1) q^{19}+ \cdots + (\beta_{13} + 2 \beta_{12} - 2 \beta_{11} + \cdots + 2) q^{99}+O(q^{100}) $ q + b14 * q^3 + (b18 + b15) * q^7 + (b11 + b4 - 1) * q^9 + b8 * q^11 + (b16 - b10) * q^13 + (b18 + b9 + b5) * q^17 + (b13 + b12 - b11 - 1) * q^19 + (b13 + b12 - 2b11 - b4 + b3 - 2b2) * q^21 + (-2b16 - b10 + b1) q^23 + (-b19 + b18 - b16 + b15) * q^27 + (b13 - b11 - b7 - 2b4 + 2) q^29 + (-b8 - b7 + b3 - 2b2 + 1) q^31 + (-b19 + 2b17 - b14 + 2b10) * q^33 + (-b19 + b17 - b16 - b14 + b1) * q^37 + (-b7 + b3 + b2 + 1) * q^39 + (2b12 - b8 - b6 + 3b4 + 2b3) q^41 + (-2b17 - 2b10 + b9 + b5) * q^43 + (-b16 + b9 + b1) * q^47 + (-3b8 - b7 + 2b3 - 2b2 + 3) q^49 + (3b13 - b12 - 2b11 - 3b7 + b6 - 4b4 + 4) * q^51 + (-b16 + 3b10 + b9 + b1) q^53 + (b19 - b18 - b17 + b16 - 2b15 + b14 + 2b9 + b5) * q^57 + (-b13 - b11 - b2) * q^59 + (2b13 - 2b7 - b4 + 1) * q^61 + (2b16 + b10 + b9 - 3b1) * q^63 + (b16 + 3b15 - b9 - b1) q^67 + (3b8 + 2b7 + b3 - 3b2 - 3) q^69 + (b13 + 3b12 - 2b11 + 3b3 - 2b2) * q^71 + (2b19 + b17 - b14 + b10) q^73 + (b19 - 2b18 + 3b17 + b16 - 2b15 + b5) q^77 + (b13 - b12 + b11 + 3b8 + 3b6 + b4 - b3 + b2) * q^79 + (2b13 + b12 + b8 + b6 + 2b4 + b3) * q^81 + (2b19 + b18 + b17 + 2b16 + b15 + 2b5) q^83 + (b19 - 2b18 + b16 - 2b15 + 2b14 - 2b5 - 2b1) q^87 + (-b13 + 3b12 - b11 + b7 - 3b6 - 3b4 + 3) q^89 + (3b12 - 3b6 + b4 - 1) * q^91 + (-b19 + b18 - b17 - 4b14 - b10 - 3b9 - 3b5) q^93 + (b19 + b18 - b17 - b10 - 2b9 - 2b5) * q^97 + (b13 + 2b12 - 2b11 - b7 - 2b4 + 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 10 q^{9} - 14 q^{19} - 8 q^{21} + 16 q^{29} + 8 q^{31} + 8 q^{39} + 26 q^{41} + 44 q^{49} + 26 q^{51} - 4 q^{59} + 2 q^{61} - 48 q^{69} - 2 q^{71} + 16 q^{79} + 26 q^{81} + 40 q^{89} - 4 q^{91} + 20 q^{99}+O(q^{100}) $ 20 * q - 10 * q^9 - 14 * q^19 - 8 * q^21 + 16 * q^29 + 8 * q^31 + 8 * q^39 + 26 * q^41 + 44 * q^49 + 26 * q^51 - 4 * q^59 + 2 * q^61 - 48 * q^69 - 2 * q^71 + 16 * q^79 + 26 * q^81 + 40 * q^89 - 4 * q^91 + 20 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} + 20 x^{18} + 261 x^{16} + 1994 x^{14} + 11074 x^{12} + 39211 x^{10} + 99376 x^{8} + 134299 x^{6} + \cdots + 4096 $ x^20 + 20*x^18 + 261*x^16 + 1994*x^14 + 11074*x^12 + 39211*x^10 + 99376*x^8 + 134299*x^6 + 124617*x^4 + 24768*x^2 + 4096 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 257269301703 \nu^{18} + 4826223210105 \nu^{16} + 61286511972345 \nu^{14} + \cdots - 63\!\cdots\!36 ) / 16\!\cdots\!37 $ (257269301703v^18 + 4826223210105v^16 + 61286511972345v^14 + 438571056316700v^12 + 2323089270554580v^10 + 7266718287702885v^8 + 16741977055075822v^6 + 10869496174421115v^4 + 2170497176992320*v^2 - 63024290250218436) / 16690142463557537
$\beta_{3}$	$=$	$ ( 71197437104086 \nu^{18} + \cdots - 55\!\cdots\!14 ) / 13\!\cdots\!71 $ (71197437104086v^18 + 1324128474975807v^16 + 16814642029945023v^14 + 120040501810627383v^12 + 637365600209189772v^10 + 1993705675325673459v^8 + 4509311556838210134v^6 + 2982168202054308141v^4 + 595500247666250688*v^2 - 5542150149868056414) / 1385281824475275571
$\beta_{4}$	$=$	$ ( - 58379368812683 \nu^{18} + \cdots - 23\!\cdots\!96 ) / 10\!\cdots\!68 $ (-58379368812683v^18 - 1151122140944668v^16 - 14928136974663543v^14 - 112486124646259822v^12 - 618424582627382742v^10 - 2140435717198619993v^8 - 5336438184716201168v^6 - 6768804320649661609v^4 - 6579414048167166051*v^2 - 238859269757341696) / 1068169117667682368
$\beta_{5}$	$=$	$ ( 46\!\cdots\!85 \nu^{19} + \cdots + 21\!\cdots\!40 \nu ) / 35\!\cdots\!76 $ (4665182447608185v^19 + 95269022977644980v^17 + 1244505919725087325v^15 + 9664608257577091770v^13 + 53667142267812748210v^11 + 193803756970732068819v^9 + 468118864763077635440v^7 + 594892885857002960195v^5 + 67319780802499040193v^3 + 21000148240688394240v) / 354632147065670546176
$\beta_{6}$	$=$	$ ( - 63\!\cdots\!85 \nu^{18} + \cdots - 25\!\cdots\!84 ) / 88\!\cdots\!44 $ (-6366481800502185v^18 - 137623959156754484v^16 - 1868654652948801741v^14 - 15323421807441242074v^12 - 89902370458361971154v^10 - 349276396009867994211v^8 - 944354362189538749232v^6 - 1468016942861277361043v^4 - 1259578493369127028273*v^2 - 250780428775971323584) / 88658036766417636544
$\beta_{7}$	$=$	$ ( 108802723986440 \nu^{18} + \cdots - 49\!\cdots\!91 ) / 13\!\cdots\!71 $ (108802723986440v^18 + 2093074375676352v^16 + 26579215713710528v^14 + 192433368253136232v^12 + 1007495594987342592v^10 + 3151487442894034624v^8 + 6782358791543175575v^6 + 4713968444633542976v^4 + 941318257748307968*v^2 - 4955504751551263591) / 1385281824475275571
$\beta_{8}$	$=$	$ ( - 123113231093090 \nu^{18} + \cdots + 29\!\cdots\!75 ) / 13\!\cdots\!71 $ (-123113231093090v^18 - 2391532877467530v^16 - 30369235310188170v^14 - 220945270298193633v^12 - 1151157774093599880v^10 - 3600868617089492610v^8 - 7533337923107099056v^6 - 5386149030199971390v^4 - 1075543988176763520*v^2 + 2918597117977935675) / 1385281824475275571
$\beta_{9}$	$=$	$ ( - 135352029273473 \nu^{19} + \cdots - 33\!\cdots\!89 \nu ) / 55\!\cdots\!84 $ (-135352029273473v^19 - 2546138925304077v^17 - 32332523162663053v^15 - 232191507992026067v^13 - 1225576970068606692v^11 - 3833654906967465449v^9 - 8380018149669050523v^7 - 5734348807198092151v^5 - 1145074960070594368v^3 - 339468898031171989v) / 5541127297901102284
$\beta_{10}$	$=$	$ ( 149437719142871 \nu^{19} + \cdots - 22\!\cdots\!45 \nu ) / 55\!\cdots\!84 $ (149437719142871v^19 + 2750374974599151v^17 + 34926044957117039v^15 + 247970499250483465v^13 + 1323885430768152396v^11 + 4141167794335228387v^9 + 9657228077683790013v^7 + 6194324001019140413v^5 + 1236926030594408384v^3 - 22508069497503397645v) / 5541127297901102284
$\beta_{11}$	$=$	$ ( 54263059985435 \nu^{18} + \cdots + 12\!\cdots\!72 ) / 26\!\cdots\!92 $ (54263059985435v^18 + 1073902569582988v^16 + 13947552783106023v^14 + 105468987745192622v^12 + 581255154298509462v^10 + 2024168224595373833v^8 + 5068566551834988016v^6 + 6594892381858923769v^4 + 6277643813918368339*v^2 + 1247247913760836672) / 267042279416920592
$\beta_{12}$	$=$	$ ( 91\!\cdots\!17 \nu^{18} + \cdots + 22\!\cdots\!52 ) / 44\!\cdots\!72 $ (9174325379691717v^18 + 183358087220694436v^16 + 2387658497653929081v^14 + 18206740636302532114v^12 + 100804631860325583978v^10 + 356441852003984207511v^8 + 901408307418290504720v^6 + 1230925670964922837447v^4 + 1127871072781709528461*v^2 + 224153346284272802752) / 44329018383208818272
$\beta_{13}$	$=$	$ ( 21\!\cdots\!13 \nu^{18} + \cdots + 91\!\cdots\!84 ) / 88\!\cdots\!44 $ (21472706330772813v^18 + 429531870127860356v^16 + 5597440381244918081v^14 + 42637019206442019714v^12 + 235405967494108868890v^10 + 823296813298088269295v^8 + 2040723648133655490800v^6 + 2590582498362261966815v^4 + 2114820325419353216181*v^2 + 91430995209659510784) / 88658036766417636544
$\beta_{14}$	$=$	$ ( 58379368812683 \nu^{19} + \cdots + 13\!\cdots\!64 \nu ) / 10\!\cdots\!68 $ (58379368812683v^19 + 1151122140944668v^17 + 14928136974663543v^15 + 112486124646259822v^13 + 618424582627382742v^11 + 2140435717198619993v^9 + 5336438184716201168v^7 + 6768804320649661609v^5 + 6579414048167166051v^3 + 1307028387425024064v) / 1068169117667682368
$\beta_{15}$	$=$	$ ( - 352957477246353 \nu^{19} + \cdots + 15\!\cdots\!77 \nu ) / 27\!\cdots\!42 $ (-352957477246353v^19 - 6732287676656781v^17 - 85490954590084109v^15 - 617058244498298531v^13 - 3240568160043291876v^11 - 10136629792755534697v^9 - 21944735732755401673v^7 - 15162285696465178103v^5 - 3027711475567210304v^3 + 15112667902972457477v) / 2770563648950551142
$\beta_{16}$	$=$	$ ( - 395664181329051 \nu^{19} + \cdots + 31\!\cdots\!37 \nu ) / 27\!\cdots\!42 $ (-395664181329051v^19 - 7533440729534211v^17 - 95664515577493379v^15 - 689861039846870731v^13 - 3626200978955352156v^11 - 11342905028514213607v^9 - 24723903923897988125v^7 - 16966622061419083193v^5 - 3388014006947935424v^3 + 31115827382409820137v) / 2770563648950551142
$\beta_{17}$	$=$	$ ( - 86\!\cdots\!67 \nu^{19} + \cdots - 35\!\cdots\!92 \nu ) / 35\!\cdots\!76 $ (-86955964488571567v^19 - 1710572564381717100v^17 - 22161310421172271211v^15 - 166719685296363775190v^13 - 915935506268344505246v^11 - 3168119711944513996773v^9 - 7897531353133828196624v^7 - 10015947541588282623477v^5 - 9108096864853737363623v^3 - 353436183667271585792v) / 354632147065670546176
$\beta_{18}$	$=$	$ ( - 44\!\cdots\!75 \nu^{19} + \cdots - 12\!\cdots\!08 \nu ) / 17\!\cdots\!88 $ (-44403267011197975v^19 - 905638902720961484v^17 - 11924107064037836787v^15 - 92792312405128286118v^13 - 522399676445550876270v^11 - 1902277734940496224477v^9 - 4916800551459821697744v^7 - 7052914632463443149421v^5 - 6287552361196403142031v^3 - 1250374162009682137408v) / 177316073532835273088
$\beta_{19}$	$=$	$ ( - 99\!\cdots\!71 \nu^{19} + \cdots - 25\!\cdots\!64 \nu ) / 17\!\cdots\!88 $ (-99815889287337571v^19 - 2000882759717695292v^17 - 26141423587608532335v^15 - 200169113650494448030v^13 - 1113670060332052229382v^11 - 3956950094181766287265v^9 - 10054026219204032528144v^7 - 13679166208473456186225v^5 - 12640273317629698848059v^3 - 2512480201371985676864v) / 177316073532835273088

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{11} - 4\beta_{4} - \beta_{2} $ -b11 - 4*b4 - b2
$\nu^{3}$	$=$	$ \beta_{19} - \beta_{18} + \beta_{16} - \beta_{15} + 6\beta_{14} - 6\beta_1 $ b19 - b18 + b16 - b15 + 6b14 - 6b1
$\nu^{4}$	$=$	$ -2\beta_{13} - \beta_{12} + 9\beta_{11} + 2\beta_{7} - \beta_{6} + 25\beta_{4} - 25 $ -2b13 - b12 + 9b11 + 2b7 - b6 + 25b4 - 25
$\nu^{5}$	$=$	$ -10\beta_{19} + 11\beta_{18} + 3\beta_{17} - 40\beta_{14} + 3\beta_{10} + 3\beta_{9} + 3\beta_{5} $ -10b19 + 11b18 + 3b17 - 40b14 + 3b10 + 3b9 + 3*b5
$\nu^{6}$	$=$	$ -16\beta_{8} - 27\beta_{7} - 8\beta_{3} + 72\beta_{2} + 177 $ -16b8 - 27b7 - 8b3 + 72b2 + 177
$\nu^{7}$	$=$	$ -88\beta_{16} + 99\beta_{15} - 40\beta_{10} - 46\beta_{9} + 283\beta_1 $ -88b16 + 99b15 - 40b10 - 46b9 + 283*b1
$\nu^{8}$	$=$	$ 279 \beta_{13} + 47 \beta_{12} - 569 \beta_{11} + 174 \beta_{8} + 174 \beta_{6} - 1329 \beta_{4} + \cdots - 569 \beta_{2} $ 279b13 + 47b12 - 569b11 + 174b8 + 174b6 - 1329b4 + 47b3 - 569b2
$\nu^{9}$	$=$	$ 743 \beta_{19} - 848 \beta_{18} - 395 \beta_{17} + 743 \beta_{16} - 848 \beta_{15} + 2083 \beta_{14} + \cdots - 2083 \beta_1 $ 743b19 - 848b18 - 395b17 + 743b16 - 848b15 + 2083b14 - 511b5 - 2083b1
$\nu^{10}$	$=$	$ -2613\beta_{13} - 221\beta_{12} + 4522\beta_{11} + 2613\beta_{7} - 1649\beta_{6} + 10318\beta_{4} - 10318 $ -2613b13 - 221b12 + 4522b11 + 2613b7 - 1649b6 + 10318b4 - 10318
$\nu^{11}$	$=$	$ - 6171 \beta_{19} + 7135 \beta_{18} + 3519 \beta_{17} - 15785 \beta_{14} + 3519 \beta_{10} + \cdots + 5005 \beta_{5} $ -6171b19 + 7135b18 + 3519b17 - 15785b14 + 3519b10 + 5005b9 + 5005*b5
$\nu^{12}$	$=$	$ -14695\beta_{8} - 23316\beta_{7} - 644\beta_{3} + 36226\beta_{2} + 81771 $ -14695b8 - 23316b7 - 644b3 + 36226b2 + 81771
$\nu^{13}$	$=$	$ -50921\beta_{16} + 59542\beta_{15} - 30034\beta_{10} - 45988\beta_{9} + 122286\beta_1 $ -50921b16 + 59542b15 - 30034b10 - 45988b9 + 122286*b1
$\nu^{14}$	$=$	$ 202439 \beta_{13} - 2400 \beta_{12} - 292291 \beta_{11} + 126943 \beta_{8} + 126943 \beta_{6} + \cdots - 292291 \beta_{2} $ 202439b13 - 2400b12 - 292291b11 + 126943b8 + 126943b6 - 656616b4 - 2400b3 - 292291b2
$\nu^{15}$	$=$	$ 419234 \beta_{19} - 494730 \beta_{18} - 251486 \beta_{17} + 419234 \beta_{16} - 494730 \beta_{15} + \cdots - 963263 \beta_1 $ 419234b19 - 494730b18 - 251486b17 + 419234b16 - 494730b15 + 963263b14 - 407278b5 - 963263b1
$\nu^{16}$	$=$	$ - 1728520 \beta_{13} + 68340 \beta_{12} + 2371957 \beta_{11} + 1728520 \beta_{7} - 1077998 \beta_{6} + \cdots - 5318130 $ -1728520b13 + 68340b12 + 2371957b11 + 1728520b7 - 1077998b6 + 5318130b4 - 5318130
$\nu^{17}$	$=$	$ - 3449955 \beta_{19} + 4100477 \beta_{18} + 2087656 \beta_{17} - 7683002 \beta_{14} + \cdots + 3525380 \beta_{5} $ -3449955b19 + 4100477b18 + 2087656b17 - 7683002b14 + 2087656b10 + 3525380b9 + 3525380*b5
$\nu^{18}$	$=$	$ -9062991\beta_{8} - 14601192\beta_{7} + 862627\beta_{3} + 19333911\beta_{2} + 43320969 $ -9062991b8 - 14601192b7 + 862627b3 + 19333911b2 + 43320969
$\nu^{19}$	$=$	$ -28396902\beta_{16} + 33935103\beta_{15} - 17263355\beta_{10} - 30065011\beta_{9} + 61849398\beta_1 $ -28396902b16 + 33935103b15 - 17263355b10 - 30065011b9 + 61849398*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$1$	$-1 + \beta_{4}$	$1$

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} $

201.1

1.43632	+	2.48777i
1.21562	+	2.10552i
1.00667	+	1.74361i
0.628167	+	1.08802i
0.226426	+	0.392182i
−0.226426	−	0.392182i
−0.628167	−	1.08802i
−1.00667	−	1.74361i
−1.21562	−	2.10552i
−1.43632	−	2.48777i
1.43632	−	2.48777i
1.21562	−	2.10552i
1.00667	−	1.74361i
0.628167	−	1.08802i
0.226426	−	0.392182i
−0.226426	+	0.392182i
−0.628167	+	1.08802i
−1.00667	+	1.74361i
−1.21562	+	2.10552i
−1.43632	+	2.48777i

−1.43632

2.48777i

3.54568

−2.62601

−

4.54838i

201.2

−1.21562

2.10552i

0.663818

−1.45548

−

2.52097i

201.3

−1.00667

1.74361i

−1.34403

−0.526784

−

0.912416i

201.4

−0.628167

1.08802i

−4.97100

0.710812

1.23116i

201.5

−0.226426

0.392182i

2.54366

1.39746

2.42048i

201.6

0.226426

−

0.392182i

−2.54366

1.39746

2.42048i

201.7

0.628167

−

1.08802i

4.97100

0.710812

1.23116i

201.8

1.00667

−

1.74361i

1.34403

−0.526784

−

0.912416i

201.9

1.21562

−

2.10552i

−0.663818

−1.45548

−

2.52097i

201.10

1.43632

−

2.48777i

−3.54568

−2.62601

−

4.54838i

501.1

−1.43632

−

2.48777i

3.54568

−2.62601

4.54838i

501.2

−1.21562

−

2.10552i

0.663818

−1.45548

2.52097i

501.3

−1.00667

−

1.74361i

−1.34403

−0.526784

0.912416i

501.4

−0.628167

−

1.08802i

−4.97100

0.710812

−

1.23116i

501.5

−0.226426

−

0.392182i

2.54366

1.39746

−

2.42048i

501.6

0.226426

0.392182i

−2.54366

1.39746

−

2.42048i

501.7

0.628167

1.08802i

4.97100

0.710812

−

1.23116i

501.8

1.00667

1.74361i

1.34403

−0.526784

0.912416i

501.9

1.21562

2.10552i

−0.663818

−1.45548

2.52097i

501.10

1.43632

2.48777i

−3.54568

−2.62601

4.54838i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
19.c	even	3	1	inner
95.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1900.2.i.g		20
5.b	even	2	1	inner	1900.2.i.g		20
5.c	odd	4	2		380.2.r.a	✓	20
15.e	even	4	2		3420.2.bj.c		20
19.c	even	3	1	inner	1900.2.i.g		20
95.i	even	6	1	inner	1900.2.i.g		20
95.m	odd	12	2		380.2.r.a	✓	20
285.v	even	12	2		3420.2.bj.c		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.2.r.a	✓	20	5.c	odd	4	2
380.2.r.a	✓	20	95.m	odd	12	2
1900.2.i.g		20	1.a	even	1	1	trivial
1900.2.i.g		20	5.b	even	2	1	inner
1900.2.i.g		20	19.c	even	3	1	inner
1900.2.i.g		20	95.i	even	6	1	inner
3420.2.bj.c		20	15.e	even	4	2
3420.2.bj.c		20	285.v	even	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{20} + 20 T_{3}^{18} + 261 T_{3}^{16} + 1994 T_{3}^{14} + 11074 T_{3}^{12} + 39211 T_{3}^{10} + \cdots + 4096 $ T3^20 + 20*T3^18 + 261*T3^16 + 1994*T3^14 + 11074*T3^12 + 39211*T3^10 + 99376*T3^8 + 134299*T3^6 + 124617*T3^4 + 24768*T3^2 + 4096 acting on $S_{2}^{\mathrm{new}}(1900, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} $ T^20
$3$	$ T^{20} + 20 T^{18} + \cdots + 4096 $ T^20 + 20T^18 + 261T^16 + 1994T^14 + 11074T^12 + 39211T^10 + 99376T^8 + 134299T^6 + 124617T^4 + 24768*T^2 + 4096
$5$	$ T^{20} $ T^20
$7$	$ (T^{10} - 46 T^{8} + \cdots - 1600)^{2} $ (T^10 - 46T^8 + 651T^6 - 3285T^4 + 4956T^2 - 1600)^2
$11$	$ (T^{5} - 27 T^{3} + \cdots + 148)^{4} $ (T^5 - 27T^3 - 29T^2 + 107*T + 148)^4
$13$	$ T^{20} + \cdots + 13051691536 $ T^20 + 78T^18 + 4368T^16 + 101388T^14 + 1608521T^12 + 17014504T^10 + 134047248T^8 + 747179442T^6 + 3073157905T^4 + 8019357580*T^2 + 13051691536
$17$	$ T^{20} + \cdots + 9721171216 $ T^20 + 98T^18 + 7069T^16 + 200240T^14 + 3975100T^12 + 43575681T^10 + 343225592T^8 + 1664210705T^6 + 5691063605T^4 + 8855399740*T^2 + 9721171216
$19$	$ (T^{10} + 7 T^{9} + \cdots + 2476099)^{2} $ (T^10 + 7T^9 + 48T^8 + 14T^7 - 454T^6 - 5658T^5 - 8626T^4 + 5054T^3 + 329232T^2 + 912247*T + 2476099)^2
$23$	$ T^{20} + \cdots + 39691260010000 $ T^20 + 211T^18 + 29086T^16 + 2393045T^14 + 144063556T^12 + 5384649772T^10 + 138088662735T^8 + 1123194038130T^6 + 6588380920801T^4 + 19222228809900*T^2 + 39691260010000
$29$	$ (T^{10} - 8 T^{9} + \cdots + 28561)^{2} $ (T^10 - 8T^9 + 78T^8 - 38T^7 + 757T^6 - 595T^5 + 4819T^4 - 1807T^3 + 14196T^2 - 6591*T + 28561)^2
$31$	$ (T^{5} - 2 T^{4} + \cdots + 388)^{4} $ (T^5 - 2T^4 - 93T^3 + 251T^2 + 909T + 388)^4
$37$	$ (T^{10} - 86 T^{8} + \cdots - 518400)^{2} $ (T^10 - 86T^8 + 2655T^6 - 36133T^4 + 223884T^2 - 518400)^2
$41$	$ (T^{10} - 13 T^{9} + \cdots + 1368900)^{2} $ (T^10 - 13T^9 + 186T^8 - 1227T^7 + 11702T^6 - 65504T^5 + 474949T^4 - 1488504T^3 + 4851081T^2 + 2341170*T + 1368900)^2
$43$	$ T^{20} + \cdots + 122963703210000 $ T^20 + 266T^18 + 50823T^16 + 4244448T^14 + 251687408T^12 + 7914771733T^10 + 177900973222T^8 + 2180566687815T^6 + 18727060639581T^4 + 54989755299900*T^2 + 122963703210000
$47$	$ T^{20} + \cdots + 2998219536 $ T^20 + 81T^18 + 4528T^16 + 127453T^14 + 2567980T^12 + 28541648T^10 + 224594797T^8 + 851140494T^6 + 2310165441T^4 + 3159366444*T^2 + 2998219536
$53$	$ T^{20} + \cdots + 591687332741376 $ T^20 + 318T^18 + 67789T^16 + 8107180T^14 + 704471884T^12 + 35026544273T^10 + 1202856445708T^8 + 11228617995345T^6 + 75923593126281T^4 + 250730707883184*T^2 + 591687332741376
$59$	$ (T^{10} + 2 T^{9} + 62 T^{8} + \cdots + 1)^{2} $ (T^10 + 2T^9 + 62T^8 + 82T^7 + 3537T^6 + 5643T^5 + 11249T^4 - 2359T^3 + 724T^2 + 25*T + 1)^2
$61$	$ (T^{10} - T^{9} + \cdots + 3073009)^{2} $ (T^10 - T^9 + 115T^8 - 450T^7 + 11757T^6 - 30859T^5 + 251165T^4 + 29238T^3 + 2807787T^2 - 2666313T + 3073009)^2
$67$	$ T^{20} + \cdots + 805854925357056 $ T^20 + 435T^18 + 125425T^16 + 20777032T^14 + 2502852784T^12 + 178788433664T^10 + 8943639356416T^8 + 171872785142784T^6 + 2432500867768320T^4 + 1428297897590784*T^2 + 805854925357056
$71$	$ (T^{10} + T^{9} + \cdots + 640140601)^{2} $ (T^10 + T^9 + 164T^8 + 797T^7 + 20925T^6 + 91293T^5 + 1203311T^4 + 5308606T^3 + 49647856T^2 + 154943324T + 640140601)^2
$73$	$ T^{20} + \cdots + 25628906250000 $ T^20 + 227T^18 + 39310T^16 + 2274613T^14 + 90647500T^12 + 2142404556T^10 + 36582797191T^8 + 377531500050T^6 + 2771171274321T^4 + 10168593187500*T^2 + 25628906250000
$79$	$ (T^{10} - 8 T^{9} + \cdots + 233967616)^{2} $ (T^10 - 8T^9 + 213T^8 - 832T^7 + 28025T^6 - 129732T^5 + 1240304T^4 - 2258944T^3 + 20641536T^2 - 34752512*T + 233967616)^2
$83$	$ (T^{10} - 323 T^{8} + \cdots - 54464400)^{2} $ (T^10 - 323T^8 + 26698T^6 - 881113T^4 + 12000024T^2 - 54464400)^2
$89$	$ (T^{10} - 20 T^{9} + \cdots + 5314993216)^{2} $ (T^10 - 20T^9 + 479T^8 - 3784T^7 + 60333T^6 - 302862T^5 + 5699336T^4 - 12731096T^3 + 195732832T^2 + 32952608*T + 5314993216)^2
$97$	$ T^{20} + \cdots + 385136700010000 $ T^20 + 328T^18 + 69709T^16 + 8826414T^14 + 815187090T^12 + 48785561539T^10 + 2110546422524T^8 + 51542252873283T^6 + 834276242546061T^4 + 578707509531900*T^2 + 385136700010000
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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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{14} q^{3} + (\beta_{18} + \beta_{15}) q^{7} + (\beta_{11} + \beta_{4} - 1) q^{9} + \beta_{8} q^{11} + (\beta_{16} - \beta_{10}) q^{13} + (\beta_{18} + \beta_{9} + \beta_{5}) q^{17} + (\beta_{13} + \beta_{12} - \beta_{11} - 1) q^{19}+ \cdots + (\beta_{13} + 2 \beta_{12} - 2 \beta_{11} + \cdots + 2) q^{99}+O(q^{100}) \) q + b14 * q^3 + (b18 + b15) * q^7 + (b11 + b4 - 1) * q^9 + b8 * q^11 + (b16 - b10) * q^13 + (b18 + b9 + b5) * q^17 + (b13 + b12 - b11 - 1) * q^19 + (b13 + b12 - 2b11 - b4 + b3 - 2b2) * q^21 + (-2b16 - b10 + b1) q^23 + (-b19 + b18 - b16 + b15) * q^27 + (b13 - b11 - b7 - 2b4 + 2) q^29 + (-b8 - b7 + b3 - 2b2 + 1) q^31 + (-b19 + 2b17 - b14 + 2b10) * q^33 + (-b19 + b17 - b16 - b14 + b1) * q^37 + (-b7 + b3 + b2 + 1) * q^39 + (2b12 - b8 - b6 + 3b4 + 2b3) q^41 + (-2b17 - 2b10 + b9 + b5) * q^43 + (-b16 + b9 + b1) * q^47 + (-3b8 - b7 + 2b3 - 2b2 + 3) q^49 + (3b13 - b12 - 2b11 - 3b7 + b6 - 4b4 + 4) * q^51 + (-b16 + 3b10 + b9 + b1) q^53 + (b19 - b18 - b17 + b16 - 2b15 + b14 + 2b9 + b5) * q^57 + (-b13 - b11 - b2) * q^59 + (2b13 - 2b7 - b4 + 1) * q^61 + (2b16 + b10 + b9 - 3b1) * q^63 + (b16 + 3b15 - b9 - b1) q^67 + (3b8 + 2b7 + b3 - 3b2 - 3) q^69 + (b13 + 3b12 - 2b11 + 3b3 - 2b2) * q^71 + (2b19 + b17 - b14 + b10) q^73 + (b19 - 2b18 + 3b17 + b16 - 2b15 + b5) q^77 + (b13 - b12 + b11 + 3b8 + 3b6 + b4 - b3 + b2) * q^79 + (2b13 + b12 + b8 + b6 + 2b4 + b3) * q^81 + (2b19 + b18 + b17 + 2b16 + b15 + 2b5) q^83 + (b19 - 2b18 + b16 - 2b15 + 2b14 - 2b5 - 2b1) q^87 + (-b13 + 3b12 - b11 + b7 - 3b6 - 3b4 + 3) q^89 + (3b12 - 3b6 + b4 - 1) * q^91 + (-b19 + b18 - b17 - 4b14 - b10 - 3b9 - 3b5) q^93 + (b19 + b18 - b17 - b10 - 2b9 - 2b5) * q^97 + (b13 + 2b12 - 2b11 - b7 - 2b4 + 2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 10 q^{9} - 14 q^{19} - 8 q^{21} + 16 q^{29} + 8 q^{31} + 8 q^{39} + 26 q^{41} + 44 q^{49} + 26 q^{51} - 4 q^{59} + 2 q^{61} - 48 q^{69} - 2 q^{71} + 16 q^{79} + 26 q^{81} + 40 q^{89} - 4 q^{91} + 20 q^{99}+O(q^{100}) \) 20 * q - 10 * q^9 - 14 * q^19 - 8 * q^21 + 16 * q^29 + 8 * q^31 + 8 * q^39 + 26 * q^41 + 44 * q^49 + 26 * q^51 - 4 * q^59 + 2 * q^61 - 48 * q^69 - 2 * q^71 + 16 * q^79 + 26 * q^81 + 40 * q^89 - 4 * q^91 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1900.2.i.g

Level	$1900$
Weight	$2$
Character orbit	1900.i
Analytic conductor	$15.172$
Analytic rank	$0$
Dimension	$20$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(15.1715763840\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} + 20 x^{18} + 261 x^{16} + 1994 x^{14} + 11074 x^{12} + 39211 x^{10} + 99376 x^{8} + 134299 x^{6} + \cdots + 4096 \) x^20 + 20x^18 + 261x^16 + 1994x^14 + 11074x^12 + 39211x^10 + 99376x^8 + 134299x^6 + 124617x^4 + 24768*x^2 + 4096
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 257269301703 \nu^{18} + 4826223210105 \nu^{16} + 61286511972345 \nu^{14} + \cdots - 63\!\cdots\!36 ) / 16\!\cdots\!37 \) (257269301703v^18 + 4826223210105v^16 + 61286511972345v^14 + 438571056316700v^12 + 2323089270554580v^10 + 7266718287702885v^8 + 16741977055075822v^6 + 10869496174421115v^4 + 2170497176992320*v^2 - 63024290250218436) / 16690142463557537
\(\beta_{3}\)	\(=\)	\( ( 71197437104086 \nu^{18} + \cdots - 55\!\cdots\!14 ) / 13\!\cdots\!71 \) (71197437104086v^18 + 1324128474975807v^16 + 16814642029945023v^14 + 120040501810627383v^12 + 637365600209189772v^10 + 1993705675325673459v^8 + 4509311556838210134v^6 + 2982168202054308141v^4 + 595500247666250688*v^2 - 5542150149868056414) / 1385281824475275571
\(\beta_{4}\)	\(=\)	\( ( - 58379368812683 \nu^{18} + \cdots - 23\!\cdots\!96 ) / 10\!\cdots\!68 \) (-58379368812683v^18 - 1151122140944668v^16 - 14928136974663543v^14 - 112486124646259822v^12 - 618424582627382742v^10 - 2140435717198619993v^8 - 5336438184716201168v^6 - 6768804320649661609v^4 - 6579414048167166051*v^2 - 238859269757341696) / 1068169117667682368
\(\beta_{5}\)	\(=\)	\( ( 46\!\cdots\!85 \nu^{19} + \cdots + 21\!\cdots\!40 \nu ) / 35\!\cdots\!76 \) (4665182447608185v^19 + 95269022977644980v^17 + 1244505919725087325v^15 + 9664608257577091770v^13 + 53667142267812748210v^11 + 193803756970732068819v^9 + 468118864763077635440v^7 + 594892885857002960195v^5 + 67319780802499040193v^3 + 21000148240688394240v) / 354632147065670546176
\(\beta_{6}\)	\(=\)	\( ( - 63\!\cdots\!85 \nu^{18} + \cdots - 25\!\cdots\!84 ) / 88\!\cdots\!44 \) (-6366481800502185v^18 - 137623959156754484v^16 - 1868654652948801741v^14 - 15323421807441242074v^12 - 89902370458361971154v^10 - 349276396009867994211v^8 - 944354362189538749232v^6 - 1468016942861277361043v^4 - 1259578493369127028273*v^2 - 250780428775971323584) / 88658036766417636544
\(\beta_{7}\)	\(=\)	\( ( 108802723986440 \nu^{18} + \cdots - 49\!\cdots\!91 ) / 13\!\cdots\!71 \) (108802723986440v^18 + 2093074375676352v^16 + 26579215713710528v^14 + 192433368253136232v^12 + 1007495594987342592v^10 + 3151487442894034624v^8 + 6782358791543175575v^6 + 4713968444633542976v^4 + 941318257748307968*v^2 - 4955504751551263591) / 1385281824475275571
\(\beta_{8}\)	\(=\)	\( ( - 123113231093090 \nu^{18} + \cdots + 29\!\cdots\!75 ) / 13\!\cdots\!71 \) (-123113231093090v^18 - 2391532877467530v^16 - 30369235310188170v^14 - 220945270298193633v^12 - 1151157774093599880v^10 - 3600868617089492610v^8 - 7533337923107099056v^6 - 5386149030199971390v^4 - 1075543988176763520*v^2 + 2918597117977935675) / 1385281824475275571
\(\beta_{9}\)	\(=\)	\( ( - 135352029273473 \nu^{19} + \cdots - 33\!\cdots\!89 \nu ) / 55\!\cdots\!84 \) (-135352029273473v^19 - 2546138925304077v^17 - 32332523162663053v^15 - 232191507992026067v^13 - 1225576970068606692v^11 - 3833654906967465449v^9 - 8380018149669050523v^7 - 5734348807198092151v^5 - 1145074960070594368v^3 - 339468898031171989v) / 5541127297901102284
\(\beta_{10}\)	\(=\)	\( ( 149437719142871 \nu^{19} + \cdots - 22\!\cdots\!45 \nu ) / 55\!\cdots\!84 \) (149437719142871v^19 + 2750374974599151v^17 + 34926044957117039v^15 + 247970499250483465v^13 + 1323885430768152396v^11 + 4141167794335228387v^9 + 9657228077683790013v^7 + 6194324001019140413v^5 + 1236926030594408384v^3 - 22508069497503397645v) / 5541127297901102284
\(\beta_{11}\)	\(=\)	\( ( 54263059985435 \nu^{18} + \cdots + 12\!\cdots\!72 ) / 26\!\cdots\!92 \) (54263059985435v^18 + 1073902569582988v^16 + 13947552783106023v^14 + 105468987745192622v^12 + 581255154298509462v^10 + 2024168224595373833v^8 + 5068566551834988016v^6 + 6594892381858923769v^4 + 6277643813918368339*v^2 + 1247247913760836672) / 267042279416920592
\(\beta_{12}\)	\(=\)	\( ( 91\!\cdots\!17 \nu^{18} + \cdots + 22\!\cdots\!52 ) / 44\!\cdots\!72 \) (9174325379691717v^18 + 183358087220694436v^16 + 2387658497653929081v^14 + 18206740636302532114v^12 + 100804631860325583978v^10 + 356441852003984207511v^8 + 901408307418290504720v^6 + 1230925670964922837447v^4 + 1127871072781709528461*v^2 + 224153346284272802752) / 44329018383208818272
\(\beta_{13}\)	\(=\)	\( ( 21\!\cdots\!13 \nu^{18} + \cdots + 91\!\cdots\!84 ) / 88\!\cdots\!44 \) (21472706330772813v^18 + 429531870127860356v^16 + 5597440381244918081v^14 + 42637019206442019714v^12 + 235405967494108868890v^10 + 823296813298088269295v^8 + 2040723648133655490800v^6 + 2590582498362261966815v^4 + 2114820325419353216181*v^2 + 91430995209659510784) / 88658036766417636544
\(\beta_{14}\)	\(=\)	\( ( 58379368812683 \nu^{19} + \cdots + 13\!\cdots\!64 \nu ) / 10\!\cdots\!68 \) (58379368812683v^19 + 1151122140944668v^17 + 14928136974663543v^15 + 112486124646259822v^13 + 618424582627382742v^11 + 2140435717198619993v^9 + 5336438184716201168v^7 + 6768804320649661609v^5 + 6579414048167166051v^3 + 1307028387425024064v) / 1068169117667682368
\(\beta_{15}\)	\(=\)	\( ( - 352957477246353 \nu^{19} + \cdots + 15\!\cdots\!77 \nu ) / 27\!\cdots\!42 \) (-352957477246353v^19 - 6732287676656781v^17 - 85490954590084109v^15 - 617058244498298531v^13 - 3240568160043291876v^11 - 10136629792755534697v^9 - 21944735732755401673v^7 - 15162285696465178103v^5 - 3027711475567210304v^3 + 15112667902972457477v) / 2770563648950551142
\(\beta_{16}\)	\(=\)	\( ( - 395664181329051 \nu^{19} + \cdots + 31\!\cdots\!37 \nu ) / 27\!\cdots\!42 \) (-395664181329051v^19 - 7533440729534211v^17 - 95664515577493379v^15 - 689861039846870731v^13 - 3626200978955352156v^11 - 11342905028514213607v^9 - 24723903923897988125v^7 - 16966622061419083193v^5 - 3388014006947935424v^3 + 31115827382409820137v) / 2770563648950551142
\(\beta_{17}\)	\(=\)	\( ( - 86\!\cdots\!67 \nu^{19} + \cdots - 35\!\cdots\!92 \nu ) / 35\!\cdots\!76 \) (-86955964488571567v^19 - 1710572564381717100v^17 - 22161310421172271211v^15 - 166719685296363775190v^13 - 915935506268344505246v^11 - 3168119711944513996773v^9 - 7897531353133828196624v^7 - 10015947541588282623477v^5 - 9108096864853737363623v^3 - 353436183667271585792v) / 354632147065670546176
\(\beta_{18}\)	\(=\)	\( ( - 44\!\cdots\!75 \nu^{19} + \cdots - 12\!\cdots\!08 \nu ) / 17\!\cdots\!88 \) (-44403267011197975v^19 - 905638902720961484v^17 - 11924107064037836787v^15 - 92792312405128286118v^13 - 522399676445550876270v^11 - 1902277734940496224477v^9 - 4916800551459821697744v^7 - 7052914632463443149421v^5 - 6287552361196403142031v^3 - 1250374162009682137408v) / 177316073532835273088
\(\beta_{19}\)	\(=\)	\( ( - 99\!\cdots\!71 \nu^{19} + \cdots - 25\!\cdots\!64 \nu ) / 17\!\cdots\!88 \) (-99815889287337571v^19 - 2000882759717695292v^17 - 26141423587608532335v^15 - 200169113650494448030v^13 - 1113670060332052229382v^11 - 3956950094181766287265v^9 - 10054026219204032528144v^7 - 13679166208473456186225v^5 - 12640273317629698848059v^3 - 2512480201371985676864v) / 177316073532835273088

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{11} - 4\beta_{4} - \beta_{2} \) -b11 - 4*b4 - b2
\(\nu^{3}\)	\(=\)	\( \beta_{19} - \beta_{18} + \beta_{16} - \beta_{15} + 6\beta_{14} - 6\beta_1 \) b19 - b18 + b16 - b15 + 6b14 - 6b1
\(\nu^{4}\)	\(=\)	\( -2\beta_{13} - \beta_{12} + 9\beta_{11} + 2\beta_{7} - \beta_{6} + 25\beta_{4} - 25 \) -2b13 - b12 + 9b11 + 2b7 - b6 + 25b4 - 25
\(\nu^{5}\)	\(=\)	\( -10\beta_{19} + 11\beta_{18} + 3\beta_{17} - 40\beta_{14} + 3\beta_{10} + 3\beta_{9} + 3\beta_{5} \) -10b19 + 11b18 + 3b17 - 40b14 + 3b10 + 3b9 + 3*b5
\(\nu^{6}\)	\(=\)	\( -16\beta_{8} - 27\beta_{7} - 8\beta_{3} + 72\beta_{2} + 177 \) -16b8 - 27b7 - 8b3 + 72b2 + 177
\(\nu^{7}\)	\(=\)	\( -88\beta_{16} + 99\beta_{15} - 40\beta_{10} - 46\beta_{9} + 283\beta_1 \) -88b16 + 99b15 - 40b10 - 46b9 + 283*b1
\(\nu^{8}\)	\(=\)	\( 279 \beta_{13} + 47 \beta_{12} - 569 \beta_{11} + 174 \beta_{8} + 174 \beta_{6} - 1329 \beta_{4} + \cdots - 569 \beta_{2} \) 279b13 + 47b12 - 569b11 + 174b8 + 174b6 - 1329b4 + 47b3 - 569b2
\(\nu^{9}\)	\(=\)	\( 743 \beta_{19} - 848 \beta_{18} - 395 \beta_{17} + 743 \beta_{16} - 848 \beta_{15} + 2083 \beta_{14} + \cdots - 2083 \beta_1 \) 743b19 - 848b18 - 395b17 + 743b16 - 848b15 + 2083b14 - 511b5 - 2083b1
\(\nu^{10}\)	\(=\)	\( -2613\beta_{13} - 221\beta_{12} + 4522\beta_{11} + 2613\beta_{7} - 1649\beta_{6} + 10318\beta_{4} - 10318 \) -2613b13 - 221b12 + 4522b11 + 2613b7 - 1649b6 + 10318b4 - 10318
\(\nu^{11}\)	\(=\)	\( - 6171 \beta_{19} + 7135 \beta_{18} + 3519 \beta_{17} - 15785 \beta_{14} + 3519 \beta_{10} + \cdots + 5005 \beta_{5} \) -6171b19 + 7135b18 + 3519b17 - 15785b14 + 3519b10 + 5005b9 + 5005*b5
\(\nu^{12}\)	\(=\)	\( -14695\beta_{8} - 23316\beta_{7} - 644\beta_{3} + 36226\beta_{2} + 81771 \) -14695b8 - 23316b7 - 644b3 + 36226b2 + 81771
\(\nu^{13}\)	\(=\)	\( -50921\beta_{16} + 59542\beta_{15} - 30034\beta_{10} - 45988\beta_{9} + 122286\beta_1 \) -50921b16 + 59542b15 - 30034b10 - 45988b9 + 122286*b1
\(\nu^{14}\)	\(=\)	\( 202439 \beta_{13} - 2400 \beta_{12} - 292291 \beta_{11} + 126943 \beta_{8} + 126943 \beta_{6} + \cdots - 292291 \beta_{2} \) 202439b13 - 2400b12 - 292291b11 + 126943b8 + 126943b6 - 656616b4 - 2400b3 - 292291b2
\(\nu^{15}\)	\(=\)	\( 419234 \beta_{19} - 494730 \beta_{18} - 251486 \beta_{17} + 419234 \beta_{16} - 494730 \beta_{15} + \cdots - 963263 \beta_1 \) 419234b19 - 494730b18 - 251486b17 + 419234b16 - 494730b15 + 963263b14 - 407278b5 - 963263b1
\(\nu^{16}\)	\(=\)	\( - 1728520 \beta_{13} + 68340 \beta_{12} + 2371957 \beta_{11} + 1728520 \beta_{7} - 1077998 \beta_{6} + \cdots - 5318130 \) -1728520b13 + 68340b12 + 2371957b11 + 1728520b7 - 1077998b6 + 5318130b4 - 5318130
\(\nu^{17}\)	\(=\)	\( - 3449955 \beta_{19} + 4100477 \beta_{18} + 2087656 \beta_{17} - 7683002 \beta_{14} + \cdots + 3525380 \beta_{5} \) -3449955b19 + 4100477b18 + 2087656b17 - 7683002b14 + 2087656b10 + 3525380b9 + 3525380*b5
\(\nu^{18}\)	\(=\)	\( -9062991\beta_{8} - 14601192\beta_{7} + 862627\beta_{3} + 19333911\beta_{2} + 43320969 \) -9062991b8 - 14601192b7 + 862627b3 + 19333911b2 + 43320969
\(\nu^{19}\)	\(=\)	\( -28396902\beta_{16} + 33935103\beta_{15} - 17263355\beta_{10} - 30065011\beta_{9} + 61849398\beta_1 \) -28396902b16 + 33935103b15 - 17263355b10 - 30065011b9 + 61849398*b1

\(n\)	\(77\)	\(401\)	\(951\)
\(\chi(n)\)	\(1\)	\(-1 + \beta_{4}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 201.10
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} \) T^20
$3$	\( T^{20} + 20 T^{18} + \cdots + 4096 \) T^20 + 20T^18 + 261T^16 + 1994T^14 + 11074T^12 + 39211T^10 + 99376T^8 + 134299T^6 + 124617T^4 + 24768*T^2 + 4096
$5$	\( T^{20} \) T^20
$7$	\( (T^{10} - 46 T^{8} + \cdots - 1600)^{2} \) (T^10 - 46T^8 + 651T^6 - 3285T^4 + 4956T^2 - 1600)^2
$11$	\( (T^{5} - 27 T^{3} + \cdots + 148)^{4} \) (T^5 - 27T^3 - 29T^2 + 107*T + 148)^4
$13$	\( T^{20} + \cdots + 13051691536 \) T^20 + 78T^18 + 4368T^16 + 101388T^14 + 1608521T^12 + 17014504T^10 + 134047248T^8 + 747179442T^6 + 3073157905T^4 + 8019357580*T^2 + 13051691536
$17$	\( T^{20} + \cdots + 9721171216 \) T^20 + 98T^18 + 7069T^16 + 200240T^14 + 3975100T^12 + 43575681T^10 + 343225592T^8 + 1664210705T^6 + 5691063605T^4 + 8855399740*T^2 + 9721171216
$19$	\( (T^{10} + 7 T^{9} + \cdots + 2476099)^{2} \) (T^10 + 7T^9 + 48T^8 + 14T^7 - 454T^6 - 5658T^5 - 8626T^4 + 5054T^3 + 329232T^2 + 912247*T + 2476099)^2
$23$	\( T^{20} + \cdots + 39691260010000 \) T^20 + 211T^18 + 29086T^16 + 2393045T^14 + 144063556T^12 + 5384649772T^10 + 138088662735T^8 + 1123194038130T^6 + 6588380920801T^4 + 19222228809900*T^2 + 39691260010000
$29$	\( (T^{10} - 8 T^{9} + \cdots + 28561)^{2} \) (T^10 - 8T^9 + 78T^8 - 38T^7 + 757T^6 - 595T^5 + 4819T^4 - 1807T^3 + 14196T^2 - 6591*T + 28561)^2
$31$	\( (T^{5} - 2 T^{4} + \cdots + 388)^{4} \) (T^5 - 2T^4 - 93T^3 + 251T^2 + 909T + 388)^4
$37$	\( (T^{10} - 86 T^{8} + \cdots - 518400)^{2} \) (T^10 - 86T^8 + 2655T^6 - 36133T^4 + 223884T^2 - 518400)^2
$41$	\( (T^{10} - 13 T^{9} + \cdots + 1368900)^{2} \) (T^10 - 13T^9 + 186T^8 - 1227T^7 + 11702T^6 - 65504T^5 + 474949T^4 - 1488504T^3 + 4851081T^2 + 2341170*T + 1368900)^2
$43$	\( T^{20} + \cdots + 122963703210000 \) T^20 + 266T^18 + 50823T^16 + 4244448T^14 + 251687408T^12 + 7914771733T^10 + 177900973222T^8 + 2180566687815T^6 + 18727060639581T^4 + 54989755299900*T^2 + 122963703210000
$47$	\( T^{20} + \cdots + 2998219536 \) T^20 + 81T^18 + 4528T^16 + 127453T^14 + 2567980T^12 + 28541648T^10 + 224594797T^8 + 851140494T^6 + 2310165441T^4 + 3159366444*T^2 + 2998219536
$53$	\( T^{20} + \cdots + 591687332741376 \) T^20 + 318T^18 + 67789T^16 + 8107180T^14 + 704471884T^12 + 35026544273T^10 + 1202856445708T^8 + 11228617995345T^6 + 75923593126281T^4 + 250730707883184*T^2 + 591687332741376
$59$	\( (T^{10} + 2 T^{9} + 62 T^{8} + \cdots + 1)^{2} \) (T^10 + 2T^9 + 62T^8 + 82T^7 + 3537T^6 + 5643T^5 + 11249T^4 - 2359T^3 + 724T^2 + 25*T + 1)^2
$61$	\( (T^{10} - T^{9} + \cdots + 3073009)^{2} \) (T^10 - T^9 + 115T^8 - 450T^7 + 11757T^6 - 30859T^5 + 251165T^4 + 29238T^3 + 2807787T^2 - 2666313T + 3073009)^2
$67$	\( T^{20} + \cdots + 805854925357056 \) T^20 + 435T^18 + 125425T^16 + 20777032T^14 + 2502852784T^12 + 178788433664T^10 + 8943639356416T^8 + 171872785142784T^6 + 2432500867768320T^4 + 1428297897590784*T^2 + 805854925357056
$71$	\( (T^{10} + T^{9} + \cdots + 640140601)^{2} \) (T^10 + T^9 + 164T^8 + 797T^7 + 20925T^6 + 91293T^5 + 1203311T^4 + 5308606T^3 + 49647856T^2 + 154943324T + 640140601)^2
$73$	\( T^{20} + \cdots + 25628906250000 \) T^20 + 227T^18 + 39310T^16 + 2274613T^14 + 90647500T^12 + 2142404556T^10 + 36582797191T^8 + 377531500050T^6 + 2771171274321T^4 + 10168593187500*T^2 + 25628906250000
$79$	\( (T^{10} - 8 T^{9} + \cdots + 233967616)^{2} \) (T^10 - 8T^9 + 213T^8 - 832T^7 + 28025T^6 - 129732T^5 + 1240304T^4 - 2258944T^3 + 20641536T^2 - 34752512*T + 233967616)^2
$83$	\( (T^{10} - 323 T^{8} + \cdots - 54464400)^{2} \) (T^10 - 323T^8 + 26698T^6 - 881113T^4 + 12000024T^2 - 54464400)^2
$89$	\( (T^{10} - 20 T^{9} + \cdots + 5314993216)^{2} \) (T^10 - 20T^9 + 479T^8 - 3784T^7 + 60333T^6 - 302862T^5 + 5699336T^4 - 12731096T^3 + 195732832T^2 + 32952608*T + 5314993216)^2
$97$	\( T^{20} + \cdots + 385136700010000 \) T^20 + 328T^18 + 69709T^16 + 8826414T^14 + 815187090T^12 + 48785561539T^10 + 2110546422524T^8 + 51542252873283T^6 + 834276242546061T^4 + 578707509531900*T^2 + 385136700010000
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