LMFDB - Newform orbit 245.4.j.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,4,Mod(79,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 2]))

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

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

chi := DirichletCharacter("245.79");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	245.j (of order $6$, 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:	$14.4554679514$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 202 x^{14} + 28551 x^{12} - 2053418 x^{10} + 106952621 x^{8} - 2343939932 x^{6} + \cdots + 351530410000 $ x^16 - 202x^14 + 28551x^12 - 2053418x^10 + 106952621x^8 - 2343939932x^6 + 37190388636x^4 - 125009407600*x^2 + 351530410000
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{6} q^{2} + \beta_{14} q^{3} + (\beta_{3} + \beta_{2} + 9 \beta_1 + 1) q^{4} + (\beta_{8} + \beta_{4}) q^{5} + (\beta_{10} - 2 \beta_{9} + \beta_{7}) q^{6} + ( - \beta_{15} + 8 \beta_{11} + \cdots + \beta_{5}) q^{8}+ \cdots + ( - 3 \beta_{3} - 7 \beta_1 + 7) q^{9}+O(q^{10}) $ q - b6 * q^2 + b14 * q^3 + (b3 + b2 + 9b1 + 1) q^4 + (b8 + b4) * q^5 + (b10 - 2b9 + b7) q^6 + (-b15 + 8b11 - 8b6 + b5) * q^8 + (-3b3 - 7b1 + 7) * q^9	$ q - \beta_{6} q^{2} + \beta_{14} q^{3} + (\beta_{3} + \beta_{2} + 9 \beta_1 + 1) q^{4} + (\beta_{8} + \beta_{4}) q^{5} + (\beta_{10} - 2 \beta_{9} + \beta_{7}) q^{6} + ( - \beta_{15} + 8 \beta_{11} + \cdots + \beta_{5}) q^{8}+ \cdots + (14 \beta_{2} - 266) q^{99}+O(q^{100}) $ q - b6 * q^2 + b14 * q^3 + (b3 + b2 + 9b1 + 1) q^4 + (b8 + b4) * q^5 + (b10 - 2b9 + b7) q^6 + (-b15 + 8b11 - 8b6 + b5) * q^8 + (-3b3 - 7b1 + 7) * q^9 + (5b14 - b13 - b10 + 2b9 + 2b4) q^10 + (-b3 - b2 + 8b1 - 1) q^11 + (b12 - 19b8 + b4) q^12 + (b14 - 2b13 - 2b12 + 2b9 + b8) q^13 + (b15 - 10b11 + 10b6 - b5 - 2b2 - 33) q^15 + (11b3 + 57b1 - 57) * q^16 + (-10b14 - b13 + b9 + b4) q^17 + (3b15 - 28b11) * q^18 + (-b12 + b8 - 4b7 + b4) q^19 + (15b14 - b13 - b12 + 4b10 - 8b9 + 15b8 + 4b7) q^20 + (b15 + b11 - b6 - b5) * q^22 + (-8b6 + 2b5) * q^23 + (b14 - b13 + 10b10 - 29b9 - 29b4) q^24 + (-3b15 + 10b11 + b3 + b2 - 52b1 + 1) q^25 + (-2b12 + 2b8 - b7 + 16b4) q^26 + (10b14 - 3b13 - 3b12 + 3b9 + 10b8) q^27 + (-9b2 - 119) q^29 + (45b6 - 2b5 - 21b3 - 163b1 + 163) * q^30 + (2b14 - 2b13 - 4b10 - 10b9 - 10b4) q^31 + (-3b15 + 70b11) * q^32 + (-b12 + 2b8 - b4) q^33 + (b14 - b13 - b12 - 11b10 + 13b9 + b8 - 11b7) q^34 + (37b2 + 436) q^36 + (-64b6 - 2b5) * q^37 + (-59b14 - 5b13 + 5b9 + 5b4) * q^38 + (-17b3 - 17b2 - 26b1 - 17) q^39 + (-4b12 - 72b8 + 6b7 + 25b4) * q^40 + (2b14 - 2b13 - 2b12 + 4b10 - 26b9 + 2b8 + 4b7) q^41 + (-28b11 + 28b6) * q^43 + (-2b3 - 40b1 + 40) * q^44 + (-25b14 + 3b13 - 12b10 + 4b9 + 4b4) q^45 + (30b3 + 30b2 + 122b1 + 30) q^46 + (-13b12 + 40b8 - 13b4) q^47 + (-167b14 + 11b13 + 11b12 - 11b9 - 167b8) q^48 + (-b15 - 45b11 + 45b6 + b5 - 43b2 - 192) q^50 + (23b3 + 344b1 - 344) * q^51 + (63b14 - b13 + b9 + b4) q^52 + (-10b15 + 20b11) * q^53 + (-3b12 + 3b8 + 7b7 + 41b4) * q^54 + (2b14 + b13 + b12 - 4b10 - 9b9 + 2b8 - 4b7) q^55 + (-2b15 + 120b11 - 120b6 + 2b5) * q^57 + (173b6 - 9b5) * q^58 + (3b14 - 3b13 - 16b10 + 5b9 + 5b4) q^59 + (13b15 - 230b11 - 51b3 - 51b2 - 503b1 - 51) q^60 + (b12 - b8 + 20b7 + 31b4) * q^61 + (10b14 + 14b13 + 14b12 - 14b9 + 10b8) q^62 + (-15b2 - 728) q^64 + (90b6 + 11b5 + 28b3 - 141b1 + 141) * q^65 + (-b14 + b13 - b10 + 11b9 + 11b4) * q^66 + (16b15 + 104b11) * q^67 + (-10b12 + 168b8 - 10b4) q^68 + (2b14 - 2b13 - 2b12 + 12b10 - 42b9 + 2b8 + 12b7) q^69 + (38b2 + 348) q^71 + (-434b6 + 13b5) * q^72 + (-54b14 + 16b13 - 16b9 - 16b4) * q^73 + (42b3 + 42b2 + 1102b1 + 42) q^74 + (4b12 + 39b8 - 16b7 - 58b4) * q^75 + (-3b14 + 3b13 + 3b12 - 32b10 + 91b9 - 3b8 - 32b7) q^76 + (17b15 - 145b11 + 145b6 - 17b5) * q^78 + (-37b3 + 202b1 - 202) * q^79 + (123b14 - 11b13 + 44b10 - 46b9 - 46b4) q^80 + (30b3 + 30b2 - 139b1 + 30) q^81 + (22b12 - 206b8 + 22b4) q^82 + (21b14 - 25b13 - 25b12 + 25b9 + 21b8) q^83 + (-16b15 + 60b11 - 60b6 + 16b5 + 7b2 + 403) q^85 + (-28b3 - 476b1 + 476) * q^86 + (-200b14 + 9b13 - 9b9 - 9b4) * q^87 + (-6b15 - 62b11) * q^88 + (12b12 - 12b8 + 12b7 - 84b4) * q^89 + (236b14 + 8b13 + 8b12 - 22b10 + 83b9 + 236b8 - 22b7) q^90 + (-14b15 + 268b11 - 268b6 + 14b5) * q^92 + (20b6 + 8b5) * q^93 + (-13b14 + 13b13 - 27b10 + 171b9 + 171b4) q^94 + (-7b15 - 130b11 + 89b3 + 89b2 - 233b1 + 89) q^95 + (3b12 - 3b8 - 76b7 - 179b4) * q^96 + (78b14 - 23b13 - 23b12 + 23b9 + 78b8) q^97 + (14b2 - 266) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 76 q^{4} + 68 q^{9}+O(q^{10}) $ 16 * q + 76 * q^4 + 68 * q^9	$ 16 q + 76 q^{4} + 68 q^{9} + 60 q^{11} - 512 q^{15} - 500 q^{16} - 412 q^{25} - 1832 q^{29} + 1388 q^{30} + 6680 q^{36} - 276 q^{39} + 328 q^{44} + 1096 q^{46} - 2728 q^{50} - 2844 q^{51} - 4228 q^{60} - 11528 q^{64} + 1016 q^{65} + 5264 q^{71} + 8984 q^{74} - 1468 q^{79} - 992 q^{81} + 6392 q^{85} + 3920 q^{86} - 1508 q^{95} - 4368 q^{99}+O(q^{100}) $ 16 * q + 76 * q^4 + 68 * q^9 + 60 * q^11 - 512 * q^15 - 500 * q^16 - 412 * q^25 - 1832 * q^29 + 1388 * q^30 + 6680 * q^36 - 276 * q^39 + 328 * q^44 + 1096 * q^46 - 2728 * q^50 - 2844 * q^51 - 4228 * q^60 - 11528 * q^64 + 1016 * q^65 + 5264 * q^71 + 8984 * q^74 - 1468 * q^79 - 992 * q^81 + 6392 * q^85 + 3920 * q^86 - 1508 * q^95 - 4368 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 202 x^{14} + 28551 x^{12} - 2053418 x^{10} + 106952621 x^{8} - 2343939932 x^{6} + \cdots + 351530410000 $ x^16 - 202*x^14 + 28551*x^12 - 2053418*x^10 + 106952621*x^8 - 2343939932*x^6 + 37190388636*x^4 - 125009407600*x^2 + 351530410000 :

$\beta_{1}$	$=$	$ ( 19\!\cdots\!11 \nu^{14} + \cdots - 39\!\cdots\!00 ) / 18\!\cdots\!00 $ (19553668934895111v^14 - 3869970364226958297v^12 + 543673716738131631011v^10 - 38145225508670794825023v^8 + 1965767301888792204272181v^6 - 39696516170534396851627827v^4 + 679053713158957723910509596*v^2 - 396906877840394916253682500) / 1885173484030263543542911100
$\beta_{2}$	$=$	$ ( 323748766 \nu^{14} - 60352905957 \nu^{12} + 8133690747930 \nu^{10} - 508900653833220 \nu^{8} + \cdots + 58\!\cdots\!20 ) / 58\!\cdots\!15 $ (323748766v^14 - 60352905957v^12 + 8133690747930v^10 - 508900653833220v^8 + 23104052168909528v^6 - 195191180037551984v^4 + 657917574163634400*v^2 + 58161576171004267920) / 5899322434238378015
$\beta_{3}$	$=$	$ ( 19\!\cdots\!33 \nu^{14} + \cdots - 27\!\cdots\!00 ) / 19\!\cdots\!00 $ (192510306620854420733v^14 - 40430372861161348552911v^12 + 5793095284121273320817073v^10 - 434076944101209988791355369v^8 + 23015380715771718730441737343v^6 - 553830082293112278550004145241v^4 + 8089997752070102577262975568468*v^2 - 27201852950249784636425187468800) / 1936073168099080659218569699700
$\beta_{4}$	$=$	$ ( - 66\!\cdots\!55 \nu^{15} + \cdots + 17\!\cdots\!00 ) / 15\!\cdots\!00 $ (-66021896626974273755v^15 - 1499249880047005243478v^14 + 12226638029547998447930v^13 + 294539393176599557148016v^12 - 1658698800277779359663025v^11 - 41244187424969068476965898v^10 + 103779812895955470717215850v^9 + 2869833745597159922018006104v^8 - 4786624752413717671345889155v^7 - 147287658084160772248901714038v^6 + 39805223260484586772655908120v^5 + 2951798661738360910195619253076v^4 - 134168746362113239964344542000v^3 - 50748061552504182742598850583048v^2 - 13998799208647604591120703872280*v + 170534741511608354820482393496800) / 15488585344792645273748557597600
$\beta_{5}$	$=$	$ ( - 48\!\cdots\!39 \nu^{15} + \cdots - 11\!\cdots\!88 \nu ) / 10\!\cdots\!40 $ (-48686742954187539v^15 + 8961949766438902310v^13 - 1223179676037193920345v^11 + 76530686521276055291130v^9 - 3732427594902948071860271v^7 + 29353695851346767669637336v^5 - 98940497022149220396327600v^3 - 116921736747899109303490564488v) / 10057522951164055372563998440
$\beta_{6}$	$=$	$ ( 17\!\cdots\!63 \nu^{15} + \cdots + 56\!\cdots\!08 \nu ) / 20\!\cdots\!80 $ (171485445784348763v^15 - 31757501375449346618v^13 + 4308308572150076258865v^11 - 269557955573910313551210v^9 + 12432791564710954990508803v^7 - 103390190286972952656249112v^5 + 348490250291203220686609200v^3 + 56475563327386823968818136808v) / 20115045902328110745127996880
$\beta_{7}$	$=$	$ ( 26\!\cdots\!20 \nu^{15} + \cdots + 99\!\cdots\!00 ) / 30\!\cdots\!00 $ (264087586507897095020v^15 - 7264183299009787149331v^14 - 48906552118191993791720v^13 + 1508938076932469803339332v^12 + 6634795201111117438652100v^11 - 214730895633889325614951721v^10 - 415119251583821882868863400v^9 + 15960863815901093374589020208v^8 + 19146499009654870685383556620v^7 - 842272117348012746781712941451v^6 - 159220893041938347090623632480v^5 + 20320367785065176679063570659002v^4 + 536674985448452959857378168000v^3 - 295222583707958987416498126761796v^2 + 55995196834590418364482815489120*v + 992575404761438611820654441863600) / 30977170689585290547497115195200
$\beta_{8}$	$=$	$ ( 15\!\cdots\!51 \nu^{15} + \cdots + 70\!\cdots\!36 \nu ) / 16\!\cdots\!40 $ (1565762079609804251v^15 - 288499825653670871746v^13 + 39337368595196131005105v^11 - 2461221260873289779608170v^9 + 115178347738255406067129611v^7 - 944012703903523462026169624v^5 + 3181919121613856028944708400v^3 + 701626739822936878982977864536v) / 160920367218624885961023975040
$\beta_{9}$	$=$	$ ( 38\!\cdots\!39 \nu^{15} + \cdots - 27\!\cdots\!00 ) / 29\!\cdots\!00 $ (38007568352346940700439v^15 + 9325639033767511187425v^14 - 7618194092207791668331803v^13 - 1851692690549391641416850v^12 + 1070243828022292848373035289v^11 + 234292364615498397284335875v^10 - 75802231281704045507904238177v^9 - 14658971091481733861438574750v^8 + 3869692901096900588065491355119v^7 + 493044455975614431021799652525v^6 - 78144206934257736288358699011273v^5 - 5622515600894480636020400928200v^4 + 1059711341781983976509467532576154v^3 + 18951429178951744292843549370000v^2 - 781327335183440048399471373567500*v - 27202776501004955522981661002959000) / 2981552678872584215196597337538000
$\beta_{10}$	$=$	$ ( 10\!\cdots\!06 \nu^{15} + \cdots - 43\!\cdots\!00 ) / 59\!\cdots\!00 $ (101193413006617572010406v^15 + 2123925242983886653558555v^14 - 21058265086079207868421112v^13 - 423991364571225243545279835v^12 + 3003777235875281286551611906v^11 + 59564528755081584983668486105v^10 - 223298469196930469579360748208v^9 - 4212989630955804531191812156165v^8 + 11793070545029039745325630771126v^7 + 215368150738749455719861873585455v^6 - 281926805826457813338489746792692v^5 - 4349123759564148932469375715118985v^4 + 4135535432429108711265324832964616v^3 + 58304840170354529352000259987780730v^2 - 13904384731392415728760827475925600*v - 43484852054382397547652447856287500) / 5963105357745168430393194675076000
$\beta_{11}$	$=$	$ ( - 81\!\cdots\!83 \nu^{15} + \cdots + 10\!\cdots\!00 \nu ) / 29\!\cdots\!00 $ (-81522398217360195660583v^15 + 16649800271693676637579816v^13 - 2361752094856334868250183333v^11 + 171978960358468790469040250444v^9 - 9005553521835795847095611411543v^7 + 203746618958222698041842412023006v^5 - 3141745487872498712615046183323988v^3 + 10561485024423603871225510243850800v) / 2981552678872584215196597337538000
$\beta_{12}$	$=$	$ ( - 17\!\cdots\!45 \nu^{15} + \cdots - 68\!\cdots\!00 ) / 61\!\cdots\!00 $ (-1773969877921299218545v^15 + 5996999520188020973912v^14 + 329086536187064658503270v^13 - 1178157572706398228592064v^12 - 44568269900849546510903475v^11 + 164976749699876273907863592v^10 + 2788503078818180380670625150v^9 - 11479334982388639688072024416v^8 - 127974827207665504832607109345v^7 + 589150632336643088995606856152v^6 + 1069543146374588590935537387080v^5 - 11807194646953443640782477012304v^4 - 3605036007214742067572690778000v^3 + 202992246210016730970395402332192v^2 - 233830476679483066871898861555720*v - 682138966046433419281929573987200) / 61954341379170581094994230390400
$\beta_{13}$	$=$	$ ( - 78\!\cdots\!97 \nu^{15} + \cdots + 45\!\cdots\!00 ) / 23\!\cdots\!00 $ (-784742119339358443728097v^15 - 2234239703002247985456720v^14 + 169492353216267038572844044v^13 + 438777123967568184876609840v^12 - 24451633501022249971611627347v^11 - 61641709717528378276252795920v^10 + 1887119679344623344436930485296v^9 + 4302272199487772409016220802160v^8 - 101180955674574814903875652353737v^7 - 222878637801802673815134242398320v^6 + 2626424272783354902559929853052254v^5 + 4500789814269919956613090442311440v^4 - 35801460709555613090685580369368492v^3 - 78000403357424827469259481502933920v^2 + 120402256569904860918352765050317200*v + 45001289919837222239689597961400000) / 23852421430980673721572778700304000
$\beta_{14}$	$=$	$ ( 10\!\cdots\!11 \nu^{15} + \cdots - 12\!\cdots\!00 \nu ) / 23\!\cdots\!00 $ (1036739314779757852459211v^15 - 209556418333158703058735972v^13 + 29616356744732813186317386961v^11 - 2132252766200125107991562982448v^9 + 111094314135947900511985701526531v^7 - 2448258232092885737532885589216202v^5 + 38638177074168343712090266623994596v^3 - 129876668595158622199342129516343600v) / 23852421430980673721572778700304000
$\beta_{15}$	$=$	$ ( 17\!\cdots\!63 \nu^{15} + \cdots - 21\!\cdots\!00 \nu ) / 14\!\cdots\!00 $ (178887373713941001361163v^15 - 35518765146587305946737876v^13 + 4995174937392052438208974213v^11 - 351842330409492539564762752384v^9 + 18163421802634808722947087034723v^7 - 373611501405444253426518050777766v^5 + 6281206733766187923854313192154468v^3 - 21109794021004200262907518303818800v) / 1490776339436292107598298668769000

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

$\nu$	$=$	$ ( \beta_{12} - \beta_{8} + 5\beta_{6} + \beta_{4} ) / 5 $ (b12 - b8 + 5*b6 + b4) / 5
$\nu^{2}$	$=$	$ ( -2\beta_{14} + 2\beta_{13} + 18\beta_{9} + 18\beta_{4} - 5\beta_{3} - 5\beta_{2} + 255\beta _1 - 5 ) / 5 $ (-2b14 + 2b13 + 18b9 + 18b4 - 5b3 - 5b2 + 255*b1 - 5) / 5
$\nu^{3}$	$=$	$ ( - 25 \beta_{15} - 117 \beta_{14} + 77 \beta_{13} + 77 \beta_{12} - 420 \beta_{11} - 77 \beta_{9} + \cdots + 25 \beta_{5} ) / 5 $ (-25b15 - 117b14 + 77b13 + 77b12 - 420b11 - 77b9 - 117b8 + 420b6 + 25*b5) / 5
$\nu^{4}$	$=$	$ ( -236\beta_{12} + 236\beta_{8} - 160\beta_{7} + 1804\beta_{4} - 545\beta_{3} + 20645\beta _1 - 20645 ) / 5 $ (-236b12 + 236b8 - 160b7 + 1804b4 - 545b3 + 20645b1 - 20645) / 5
$\nu^{5}$	$=$	$ -645\beta_{15} - 1577\beta_{14} + 1393\beta_{13} - 7446\beta_{11} - 1393\beta_{9} - 1393\beta_{4} $ -645b15 - 1577b14 + 1393b13 - 7446b11 - 1393b9 - 1393b4
$\nu^{6}$	$=$	$ ( 24086 \beta_{14} - 24086 \beta_{13} - 24086 \beta_{12} - 24880 \beta_{10} - 167014 \beta_{9} + \cdots - 1779000 ) / 5 $ (24086b14 - 24086b13 - 24086b12 - 24880b10 - 167014b9 + 24086b8 - 24880b7 + 66055b2 - 1779000) / 5
$\nu^{7}$	$=$	$ ( -666593\beta_{12} + 454713\beta_{8} - 3293670\beta_{6} - 356675\beta_{5} - 666593\beta_{4} ) / 5 $ (-666593b12 + 454713b8 - 3293670b6 - 356675b5 - 666593*b4) / 5
$\nu^{8}$	$=$	$ ( 2395352 \beta_{14} - 2395352 \beta_{13} - 3065280 \beta_{10} - 15427608 \beta_{9} - 15427608 \beta_{4} + \cdots + 7719445 ) / 5 $ (2395352b14 - 2395352b13 - 3065280b10 - 15427608b9 - 15427608b4 + 7719445b3 + 7719445b2 - 170538645b1 + 7719445) / 5
$\nu^{9}$	$=$	$ ( 37803525 \beta_{15} + 19312969 \beta_{14} - 64951729 \beta_{13} - 64951729 \beta_{12} + \cdots - 37803525 \beta_{5} ) / 5 $ (37803525b15 + 19312969b14 - 64951729b13 - 64951729b12 + 294732130b11 + 64951729b9 + 19312969b8 - 294732130b6 - 37803525*b5) / 5
$\nu^{10}$	$=$	$ 47461746 \beta_{12} - 47461746 \beta_{8} + 69613392 \beta_{7} - 287928930 \beta_{4} + 172380291 \beta_{3} + \cdots + 3208606451 $ 47461746b12 - 47461746b8 + 69613392b7 - 287928930b4 + 172380291b3 - 3208606451b1 + 3208606451
$\nu^{11}$	$=$	$ ( 3931122675 \beta_{15} - 77243011 \beta_{14} - 6375360389 \beta_{13} + 26779255870 \beta_{11} + \cdots + 6375360389 \beta_{4} ) / 5 $ (3931122675b15 - 77243011b14 - 6375360389b13 + 26779255870b11 + 6375360389b9 + 6375360389b4) / 5
$\nu^{12}$	$=$	$ ( - 23524579588 \beta_{14} + 23524579588 \beta_{13} + 23524579588 \beta_{12} + 37901584800 \beta_{10} + \cdots + 1434738950000 ) / 5 $ (-23524579588b14 + 23524579588b13 + 23524579588b12 + 37901584800b10 + 135918046692b9 - 23524579588b8 + 37901584800b7 - 93121904245b2 + 1434738950000) / 5
$\nu^{13}$	$=$	$ ( 628386849133 \beta_{12} + 161046406107 \beta_{8} + 2470433787330 \beta_{6} + \cdots + 628386849133 \beta_{4} ) / 5 $ (628386849133b12 + 161046406107b8 + 2470433787330b6 + 404170869725b5 + 628386849133*b4) / 5
$\nu^{14}$	$=$	$ ( - 2334986215774 \beta_{14} + 2334986215774 \beta_{13} + 4022800213040 \beta_{10} + 12969275515886 \beta_{9} + \cdots - 9838180781455 ) / 5 $ (-2334986215774b14 + 2334986215774b13 + 4022800213040b10 + 12969275515886b9 + 12969275515886b4 - 9838180781455b3 - 9838180781455b2 + 146915735293855b1 - 9838180781455) / 5
$\nu^{15}$	$=$	$ - 8246728673055 \beta_{15} + 5556498224339 \beta_{14} + 12423262407605 \beta_{13} + \cdots + 8246728673055 \beta_{5} $ -8246728673055b15 + 5556498224339b14 + 12423262407605b13 + 12423262407605b12 - 46218217428054b11 - 12423262407605b9 + 5556498224339b8 + 46218217428054b6 + 8246728673055*b5

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$.

$n$	$101$	$197$
$\chi(n)$	$-1 + \beta_{1}$	$-1$

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} $

79.1

1.61412	+	0.931915i
7.56653	+	4.36854i
8.65084	+	4.99457i
−4.09941	−	2.36679i
4.09941	+	2.36679i
−8.65084	−	4.99457i
−7.56653	−	4.36854i
−1.61412	−	0.931915i
1.61412	−	0.931915i
7.56653	−	4.36854i
8.65084	−	4.99457i
−4.09941	+	2.36679i
4.09941	−	2.36679i
−8.65084	+	4.99457i
−7.56653	+	4.36854i
−1.61412	+	0.931915i

−4.59033

−

2.65023i

−7.10375

4.10135i

10.0474

17.4026i

10.1696

−

4.64541i

43.4781

−

64.1080i

20.1422

−

34.8873i

−58.9931

−

5.62769i

79.2

−4.59033

−

2.65023i

7.10375

−

4.10135i

10.0474

17.4026i

−10.1696

4.64541i

−43.4781

−

64.1080i

20.1422

−

34.8873i

58.9931

5.62769i

79.3

−2.27572

−

1.31389i

−1.66933

0.963791i

−0.547405

−

0.948133i

0.0209014

11.1803i

5.06525

23.8991i

−11.6422

20.1649i

14.6421

−

25.4707i

79.4

−2.27572

−

1.31389i

1.66933

−

0.963791i

−0.547405

−

0.948133i

−0.0209014

−

11.1803i

−5.06525

23.8991i

−11.6422

20.1649i

−14.6421

25.4707i

79.5

2.27572

1.31389i

−1.66933

0.963791i

−0.547405

−

0.948133i

9.69289

−

5.57206i

−5.06525

−

23.8991i

−11.6422

20.1649i

29.3793

0.0549241i

79.6

2.27572

1.31389i

1.66933

−

0.963791i

−0.547405

−

0.948133i

−9.69289

5.57206i

5.06525

−

23.8991i

−11.6422

20.1649i

−29.3793

−

0.0549241i

79.7

4.59033

2.65023i

−7.10375

4.10135i

10.0474

17.4026i

1.06174

11.1298i

−43.4781

64.1080i

20.1422

−

34.8873i

−24.6228

53.9033i

79.8

4.59033

2.65023i

7.10375

−

4.10135i

10.0474

17.4026i

−1.06174

−

11.1298i

43.4781

64.1080i

20.1422

−

34.8873i

24.6228

−

53.9033i

214.1

−4.59033

2.65023i

−7.10375

−

4.10135i

10.0474

−

17.4026i

10.1696

4.64541i

43.4781

64.1080i

20.1422

34.8873i

−58.9931

5.62769i

214.2

−4.59033

2.65023i

7.10375

4.10135i

10.0474

−

17.4026i

−10.1696

−

4.64541i

−43.4781

64.1080i

20.1422

34.8873i

58.9931

−

5.62769i

214.3

−2.27572

1.31389i

−1.66933

−

0.963791i

−0.547405

0.948133i

0.0209014

−

11.1803i

5.06525

−

23.8991i

−11.6422

−

20.1649i

14.6421

25.4707i

214.4

−2.27572

1.31389i

1.66933

0.963791i

−0.547405

0.948133i

−0.0209014

11.1803i

−5.06525

−

23.8991i

−11.6422

−

20.1649i

−14.6421

−

25.4707i

214.5

2.27572

−

1.31389i

−1.66933

−

0.963791i

−0.547405

0.948133i

9.69289

5.57206i

−5.06525

23.8991i

−11.6422

−

20.1649i

29.3793

−

0.0549241i

214.6

2.27572

−

1.31389i

1.66933

0.963791i

−0.547405

0.948133i

−9.69289

−

5.57206i

5.06525

23.8991i

−11.6422

−

20.1649i

−29.3793

0.0549241i

214.7

4.59033

−

2.65023i

−7.10375

−

4.10135i

10.0474

−

17.4026i

1.06174

−

11.1298i

−43.4781

−

64.1080i

20.1422

34.8873i

−24.6228

−

53.9033i

214.8

4.59033

−

2.65023i

7.10375

4.10135i

10.0474

−

17.4026i

−1.06174

11.1298i

43.4781

−

64.1080i

20.1422

34.8873i

24.6228

53.9033i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
35.c	odd	2	1	inner
35.i	odd	6	1	inner
35.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	245.4.j.c		16
5.b	even	2	1	inner	245.4.j.c		16
7.b	odd	2	1	inner	245.4.j.c		16
7.c	even	3	1		245.4.b.c	✓	8
7.c	even	3	1	inner	245.4.j.c		16
7.d	odd	6	1		245.4.b.c	✓	8
7.d	odd	6	1	inner	245.4.j.c		16
35.c	odd	2	1	inner	245.4.j.c		16
35.i	odd	6	1		245.4.b.c	✓	8
35.i	odd	6	1	inner	245.4.j.c		16
35.j	even	6	1		245.4.b.c	✓	8
35.j	even	6	1	inner	245.4.j.c		16
35.k	even	12	2		1225.4.a.bm		8
35.l	odd	12	2		1225.4.a.bm		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
245.4.b.c	✓	8	7.c	even	3	1
245.4.b.c	✓	8	7.d	odd	6	1
245.4.b.c	✓	8	35.i	odd	6	1
245.4.b.c	✓	8	35.j	even	6	1
245.4.j.c		16	1.a	even	1	1	trivial
245.4.j.c		16	5.b	even	2	1	inner
245.4.j.c		16	7.b	odd	2	1	inner
245.4.j.c		16	7.c	even	3	1	inner
245.4.j.c		16	7.d	odd	6	1	inner
245.4.j.c		16	35.c	odd	2	1	inner
245.4.j.c		16	35.i	odd	6	1	inner
245.4.j.c		16	35.j	even	6	1	inner
1225.4.a.bm		8	35.k	even	12	2
1225.4.a.bm		8	35.l	odd	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(245, [\chi])$:

$ T_{2}^{8} - 35T_{2}^{6} + 1031T_{2}^{4} - 6790T_{2}^{2} + 37636 $

$ T_{19}^{8} + 20546T_{19}^{6} + 328242116T_{19}^{4} + 1929187216000T_{19}^{2} + 8816458816000000 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} - 35 T^{6} + \cdots + 37636)^{2} $ (T^8 - 35T^6 + 1031T^4 - 6790*T^2 + 37636)^2
$3$	$ (T^{8} - 71 T^{6} + \cdots + 62500)^{2} $ (T^8 - 71T^6 + 4791T^4 - 17750*T^2 + 62500)^2
$5$	$ T^{16} + \cdots + 59\!\cdots\!25 $ T^16 + 206T^14 + 1026T^12 + 2092960T^10 + 807584975T^8 + 32702500000T^6 + 250488281250T^4 + 785827636718750*T^2 + 59604644775390625
$7$	$ T^{16} $ T^16
$11$	$ (T^{4} - 15 T^{3} + \cdots + 3136)^{4} $ (T^4 - 15T^3 + 281T^2 + 840*T + 3136)^4
$13$	$ (T^{4} + 6391 T^{2} + 3906250)^{4} $ (T^4 + 6391*T^2 + 3906250)^4
$17$	$ (T^{8} + \cdots + 322417936000000)^{2} $ (T^8 - 8701T^6 + 57751401T^4 - 156235156000*T^2 + 322417936000000)^2
$19$	$ (T^{8} + \cdots + 88\!\cdots\!00)^{2} $ (T^8 + 20546T^6 + 328242116T^4 + 1929187216000*T^2 + 8816458816000000)^2
$23$	$ (T^{8} + \cdots + 12\!\cdots\!36)^{2} $ (T^8 - 12100T^6 + 111533456T^4 - 422006182400*T^2 + 1216373321383936)^2
$29$	$ (T^{2} + 229 T + 4018)^{8} $ (T^2 + 229*T + 4018)^8
$31$	$ (T^{8} + \cdots + 9634816000000)^{2} $ (T^8 + 45736T^6 + 2088677696T^4 + 141964544000*T^2 + 9634816000000)^2
$37$	$ (T^{8} + \cdots + 33\!\cdots\!00)^{2} $ (T^8 - 154084T^6 + 17915981456T^4 - 897677605798400*T^2 + 33941082845685760000)^2
$41$	$ (T^{4} - 156776 T^{2} + 3802400000)^{4} $ (T^4 - 156776*T^2 + 3802400000)^4
$43$	$ (T^{4} + 27440 T^{2} + 119243264)^{4} $ (T^4 + 27440*T^2 + 119243264)^4
$47$	$ (T^{8} + \cdots + 12\!\cdots\!00)^{2} $ (T^8 - 379749T^6 + 108221302001T^4 - 13666407391749000*T^2 + 1295136215976001000000)^2
$53$	$ (T^{8} + \cdots + 26\!\cdots\!00)^{2} $ (T^8 - 261700T^6 + 52066730000T^4 - 4297155872000000*T^2 + 269621654425600000000)^2
$59$	$ (T^{8} + \cdots + 60\!\cdots\!00)^{2} $ (T^8 + 316226T^6 + 75412099076T^4 + 7774980357184000*T^2 + 604509947462656000000)^2
$61$	$ (T^{8} + \cdots + 56\!\cdots\!00)^{2} $ (T^8 + 645250T^6 + 392582562500T^4 + 15334366250000000*T^2 + 564775225000000000000)^2
$67$	$ (T^{8} + \cdots + 16\!\cdots\!16)^{2} $ (T^8 - 1025728T^6 + 1011171749888T^4 - 41999643417509888*T^2 + 1676589664454066569216)^2
$71$	$ (T^{2} - 658 T - 53848)^{8} $ (T^2 - 658*T - 53848)^8
$73$	$ (T^{8} + \cdots + 83\!\cdots\!00)^{2} $ (T^8 - 610044T^6 + 280645325936T^4 - 55824123527664000*T^2 + 8373779217822736000000)^2
$79$	$ (T^{4} + 367 T^{3} + \cdots + 14399520004)^{4} $ (T^4 + 367T^3 + 254687T^2 - 44039266*T + 14399520004)^4
$83$	$ (T^{4} + 1018386 T^{2} + 135536164000)^{4} $ (T^4 + 1018386*T^2 + 135536164000)^4
$89$	$ (T^{8} + \cdots + 23\!\cdots\!00)^{2} $ (T^8 + 1818144T^6 + 2818890740736T^4 + 884994071740416000*T^2 + 236932244651114496000000)^2
$97$	$ (T^{4} + 1264725 T^{2} + 392832400000)^{4} $ (T^4 + 1264725*T^2 + 392832400000)^4
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.j (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{6} q^{2} + \beta_{14} q^{3} + (\beta_{3} + \beta_{2} + 9 \beta_1 + 1) q^{4} + (\beta_{8} + \beta_{4}) q^{5} + (\beta_{10} - 2 \beta_{9} + \beta_{7}) q^{6} + ( - \beta_{15} + 8 \beta_{11} + \cdots + \beta_{5}) q^{8}+ \cdots + ( - 3 \beta_{3} - 7 \beta_1 + 7) q^{9}+O(q^{10}) \) q - b6 * q^2 + b14 * q^3 + (b3 + b2 + 9b1 + 1) q^4 + (b8 + b4) * q^5 + (b10 - 2b9 + b7) q^6 + (-b15 + 8b11 - 8b6 + b5) * q^8 + (-3b3 - 7b1 + 7) * q^9	\( q - \beta_{6} q^{2} + \beta_{14} q^{3} + (\beta_{3} + \beta_{2} + 9 \beta_1 + 1) q^{4} + (\beta_{8} + \beta_{4}) q^{5} + (\beta_{10} - 2 \beta_{9} + \beta_{7}) q^{6} + ( - \beta_{15} + 8 \beta_{11} + \cdots + \beta_{5}) q^{8}+ \cdots + (14 \beta_{2} - 266) q^{99}+O(q^{100}) \) q - b6 * q^2 + b14 * q^3 + (b3 + b2 + 9b1 + 1) q^4 + (b8 + b4) * q^5 + (b10 - 2b9 + b7) q^6 + (-b15 + 8b11 - 8b6 + b5) * q^8 + (-3b3 - 7b1 + 7) * q^9 + (5b14 - b13 - b10 + 2b9 + 2b4) q^10 + (-b3 - b2 + 8b1 - 1) q^11 + (b12 - 19b8 + b4) q^12 + (b14 - 2b13 - 2b12 + 2b9 + b8) q^13 + (b15 - 10b11 + 10b6 - b5 - 2b2 - 33) q^15 + (11b3 + 57b1 - 57) * q^16 + (-10b14 - b13 + b9 + b4) q^17 + (3b15 - 28b11) * q^18 + (-b12 + b8 - 4b7 + b4) q^19 + (15b14 - b13 - b12 + 4b10 - 8b9 + 15b8 + 4b7) q^20 + (b15 + b11 - b6 - b5) * q^22 + (-8b6 + 2b5) * q^23 + (b14 - b13 + 10b10 - 29b9 - 29b4) q^24 + (-3b15 + 10b11 + b3 + b2 - 52b1 + 1) q^25 + (-2b12 + 2b8 - b7 + 16b4) q^26 + (10b14 - 3b13 - 3b12 + 3b9 + 10b8) q^27 + (-9b2 - 119) q^29 + (45b6 - 2b5 - 21b3 - 163b1 + 163) * q^30 + (2b14 - 2b13 - 4b10 - 10b9 - 10b4) q^31 + (-3b15 + 70b11) * q^32 + (-b12 + 2b8 - b4) q^33 + (b14 - b13 - b12 - 11b10 + 13b9 + b8 - 11b7) q^34 + (37b2 + 436) q^36 + (-64b6 - 2b5) * q^37 + (-59b14 - 5b13 + 5b9 + 5b4) * q^38 + (-17b3 - 17b2 - 26b1 - 17) q^39 + (-4b12 - 72b8 + 6b7 + 25b4) * q^40 + (2b14 - 2b13 - 2b12 + 4b10 - 26b9 + 2b8 + 4b7) q^41 + (-28b11 + 28b6) * q^43 + (-2b3 - 40b1 + 40) * q^44 + (-25b14 + 3b13 - 12b10 + 4b9 + 4b4) q^45 + (30b3 + 30b2 + 122b1 + 30) q^46 + (-13b12 + 40b8 - 13b4) q^47 + (-167b14 + 11b13 + 11b12 - 11b9 - 167b8) q^48 + (-b15 - 45b11 + 45b6 + b5 - 43b2 - 192) q^50 + (23b3 + 344b1 - 344) * q^51 + (63b14 - b13 + b9 + b4) q^52 + (-10b15 + 20b11) * q^53 + (-3b12 + 3b8 + 7b7 + 41b4) * q^54 + (2b14 + b13 + b12 - 4b10 - 9b9 + 2b8 - 4b7) q^55 + (-2b15 + 120b11 - 120b6 + 2b5) * q^57 + (173b6 - 9b5) * q^58 + (3b14 - 3b13 - 16b10 + 5b9 + 5b4) q^59 + (13b15 - 230b11 - 51b3 - 51b2 - 503b1 - 51) q^60 + (b12 - b8 + 20b7 + 31b4) * q^61 + (10b14 + 14b13 + 14b12 - 14b9 + 10b8) q^62 + (-15b2 - 728) q^64 + (90b6 + 11b5 + 28b3 - 141b1 + 141) * q^65 + (-b14 + b13 - b10 + 11b9 + 11b4) * q^66 + (16b15 + 104b11) * q^67 + (-10b12 + 168b8 - 10b4) q^68 + (2b14 - 2b13 - 2b12 + 12b10 - 42b9 + 2b8 + 12b7) q^69 + (38b2 + 348) q^71 + (-434b6 + 13b5) * q^72 + (-54b14 + 16b13 - 16b9 - 16b4) * q^73 + (42b3 + 42b2 + 1102b1 + 42) q^74 + (4b12 + 39b8 - 16b7 - 58b4) * q^75 + (-3b14 + 3b13 + 3b12 - 32b10 + 91b9 - 3b8 - 32b7) q^76 + (17b15 - 145b11 + 145b6 - 17b5) * q^78 + (-37b3 + 202b1 - 202) * q^79 + (123b14 - 11b13 + 44b10 - 46b9 - 46b4) q^80 + (30b3 + 30b2 - 139b1 + 30) q^81 + (22b12 - 206b8 + 22b4) q^82 + (21b14 - 25b13 - 25b12 + 25b9 + 21b8) q^83 + (-16b15 + 60b11 - 60b6 + 16b5 + 7b2 + 403) q^85 + (-28b3 - 476b1 + 476) * q^86 + (-200b14 + 9b13 - 9b9 - 9b4) * q^87 + (-6b15 - 62b11) * q^88 + (12b12 - 12b8 + 12b7 - 84b4) * q^89 + (236b14 + 8b13 + 8b12 - 22b10 + 83b9 + 236b8 - 22b7) q^90 + (-14b15 + 268b11 - 268b6 + 14b5) * q^92 + (20b6 + 8b5) * q^93 + (-13b14 + 13b13 - 27b10 + 171b9 + 171b4) q^94 + (-7b15 - 130b11 + 89b3 + 89b2 - 233b1 + 89) q^95 + (3b12 - 3b8 - 76b7 - 179b4) * q^96 + (78b14 - 23b13 - 23b12 + 23b9 + 78b8) q^97 + (14b2 - 266) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 76 q^{4} + 68 q^{9}+O(q^{10}) \) 16 * q + 76 * q^4 + 68 * q^9	\( 16 q + 76 q^{4} + 68 q^{9} + 60 q^{11} - 512 q^{15} - 500 q^{16} - 412 q^{25} - 1832 q^{29} + 1388 q^{30} + 6680 q^{36} - 276 q^{39} + 328 q^{44} + 1096 q^{46} - 2728 q^{50} - 2844 q^{51} - 4228 q^{60} - 11528 q^{64} + 1016 q^{65} + 5264 q^{71} + 8984 q^{74} - 1468 q^{79} - 992 q^{81} + 6392 q^{85} + 3920 q^{86} - 1508 q^{95} - 4368 q^{99}+O(q^{100}) \) 16 * q + 76 * q^4 + 68 * q^9 + 60 * q^11 - 512 * q^15 - 500 * q^16 - 412 * q^25 - 1832 * q^29 + 1388 * q^30 + 6680 * q^36 - 276 * q^39 + 328 * q^44 + 1096 * q^46 - 2728 * q^50 - 2844 * q^51 - 4228 * q^60 - 11528 * q^64 + 1016 * q^65 + 5264 * q^71 + 8984 * q^74 - 1468 * q^79 - 992 * q^81 + 6392 * q^85 + 3920 * q^86 - 1508 * q^95 - 4368 * q^99

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	245.4.j.c

Level	$245$
Weight	$4$
Character orbit	245.j
Analytic conductor	$14.455$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(14.4554679514\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 202 x^{14} + 28551 x^{12} - 2053418 x^{10} + 106952621 x^{8} - 2343939932 x^{6} + \cdots + 351530410000 \) x^16 - 202x^14 + 28551x^12 - 2053418x^10 + 106952621x^8 - 2343939932x^6 + 37190388636x^4 - 125009407600*x^2 + 351530410000
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2}\cdot 5^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 19\!\cdots\!11 \nu^{14} + \cdots - 39\!\cdots\!00 ) / 18\!\cdots\!00 \) (19553668934895111v^14 - 3869970364226958297v^12 + 543673716738131631011v^10 - 38145225508670794825023v^8 + 1965767301888792204272181v^6 - 39696516170534396851627827v^4 + 679053713158957723910509596*v^2 - 396906877840394916253682500) / 1885173484030263543542911100
\(\beta_{2}\)	\(=\)	\( ( 323748766 \nu^{14} - 60352905957 \nu^{12} + 8133690747930 \nu^{10} - 508900653833220 \nu^{8} + \cdots + 58\!\cdots\!20 ) / 58\!\cdots\!15 \) (323748766v^14 - 60352905957v^12 + 8133690747930v^10 - 508900653833220v^8 + 23104052168909528v^6 - 195191180037551984v^4 + 657917574163634400*v^2 + 58161576171004267920) / 5899322434238378015
\(\beta_{3}\)	\(=\)	\( ( 19\!\cdots\!33 \nu^{14} + \cdots - 27\!\cdots\!00 ) / 19\!\cdots\!00 \) (192510306620854420733v^14 - 40430372861161348552911v^12 + 5793095284121273320817073v^10 - 434076944101209988791355369v^8 + 23015380715771718730441737343v^6 - 553830082293112278550004145241v^4 + 8089997752070102577262975568468*v^2 - 27201852950249784636425187468800) / 1936073168099080659218569699700
\(\beta_{4}\)	\(=\)	\( ( - 66\!\cdots\!55 \nu^{15} + \cdots + 17\!\cdots\!00 ) / 15\!\cdots\!00 \) (-66021896626974273755v^15 - 1499249880047005243478v^14 + 12226638029547998447930v^13 + 294539393176599557148016v^12 - 1658698800277779359663025v^11 - 41244187424969068476965898v^10 + 103779812895955470717215850v^9 + 2869833745597159922018006104v^8 - 4786624752413717671345889155v^7 - 147287658084160772248901714038v^6 + 39805223260484586772655908120v^5 + 2951798661738360910195619253076v^4 - 134168746362113239964344542000v^3 - 50748061552504182742598850583048v^2 - 13998799208647604591120703872280*v + 170534741511608354820482393496800) / 15488585344792645273748557597600
\(\beta_{5}\)	\(=\)	\( ( - 48\!\cdots\!39 \nu^{15} + \cdots - 11\!\cdots\!88 \nu ) / 10\!\cdots\!40 \) (-48686742954187539v^15 + 8961949766438902310v^13 - 1223179676037193920345v^11 + 76530686521276055291130v^9 - 3732427594902948071860271v^7 + 29353695851346767669637336v^5 - 98940497022149220396327600v^3 - 116921736747899109303490564488v) / 10057522951164055372563998440
\(\beta_{6}\)	\(=\)	\( ( 17\!\cdots\!63 \nu^{15} + \cdots + 56\!\cdots\!08 \nu ) / 20\!\cdots\!80 \) (171485445784348763v^15 - 31757501375449346618v^13 + 4308308572150076258865v^11 - 269557955573910313551210v^9 + 12432791564710954990508803v^7 - 103390190286972952656249112v^5 + 348490250291203220686609200v^3 + 56475563327386823968818136808v) / 20115045902328110745127996880
\(\beta_{7}\)	\(=\)	\( ( 26\!\cdots\!20 \nu^{15} + \cdots + 99\!\cdots\!00 ) / 30\!\cdots\!00 \) (264087586507897095020v^15 - 7264183299009787149331v^14 - 48906552118191993791720v^13 + 1508938076932469803339332v^12 + 6634795201111117438652100v^11 - 214730895633889325614951721v^10 - 415119251583821882868863400v^9 + 15960863815901093374589020208v^8 + 19146499009654870685383556620v^7 - 842272117348012746781712941451v^6 - 159220893041938347090623632480v^5 + 20320367785065176679063570659002v^4 + 536674985448452959857378168000v^3 - 295222583707958987416498126761796v^2 + 55995196834590418364482815489120*v + 992575404761438611820654441863600) / 30977170689585290547497115195200
\(\beta_{8}\)	\(=\)	\( ( 15\!\cdots\!51 \nu^{15} + \cdots + 70\!\cdots\!36 \nu ) / 16\!\cdots\!40 \) (1565762079609804251v^15 - 288499825653670871746v^13 + 39337368595196131005105v^11 - 2461221260873289779608170v^9 + 115178347738255406067129611v^7 - 944012703903523462026169624v^5 + 3181919121613856028944708400v^3 + 701626739822936878982977864536v) / 160920367218624885961023975040
\(\beta_{9}\)	\(=\)	\( ( 38\!\cdots\!39 \nu^{15} + \cdots - 27\!\cdots\!00 ) / 29\!\cdots\!00 \) (38007568352346940700439v^15 + 9325639033767511187425v^14 - 7618194092207791668331803v^13 - 1851692690549391641416850v^12 + 1070243828022292848373035289v^11 + 234292364615498397284335875v^10 - 75802231281704045507904238177v^9 - 14658971091481733861438574750v^8 + 3869692901096900588065491355119v^7 + 493044455975614431021799652525v^6 - 78144206934257736288358699011273v^5 - 5622515600894480636020400928200v^4 + 1059711341781983976509467532576154v^3 + 18951429178951744292843549370000v^2 - 781327335183440048399471373567500*v - 27202776501004955522981661002959000) / 2981552678872584215196597337538000
\(\beta_{10}\)	\(=\)	\( ( 10\!\cdots\!06 \nu^{15} + \cdots - 43\!\cdots\!00 ) / 59\!\cdots\!00 \) (101193413006617572010406v^15 + 2123925242983886653558555v^14 - 21058265086079207868421112v^13 - 423991364571225243545279835v^12 + 3003777235875281286551611906v^11 + 59564528755081584983668486105v^10 - 223298469196930469579360748208v^9 - 4212989630955804531191812156165v^8 + 11793070545029039745325630771126v^7 + 215368150738749455719861873585455v^6 - 281926805826457813338489746792692v^5 - 4349123759564148932469375715118985v^4 + 4135535432429108711265324832964616v^3 + 58304840170354529352000259987780730v^2 - 13904384731392415728760827475925600*v - 43484852054382397547652447856287500) / 5963105357745168430393194675076000
\(\beta_{11}\)	\(=\)	\( ( - 81\!\cdots\!83 \nu^{15} + \cdots + 10\!\cdots\!00 \nu ) / 29\!\cdots\!00 \) (-81522398217360195660583v^15 + 16649800271693676637579816v^13 - 2361752094856334868250183333v^11 + 171978960358468790469040250444v^9 - 9005553521835795847095611411543v^7 + 203746618958222698041842412023006v^5 - 3141745487872498712615046183323988v^3 + 10561485024423603871225510243850800v) / 2981552678872584215196597337538000
\(\beta_{12}\)	\(=\)	\( ( - 17\!\cdots\!45 \nu^{15} + \cdots - 68\!\cdots\!00 ) / 61\!\cdots\!00 \) (-1773969877921299218545v^15 + 5996999520188020973912v^14 + 329086536187064658503270v^13 - 1178157572706398228592064v^12 - 44568269900849546510903475v^11 + 164976749699876273907863592v^10 + 2788503078818180380670625150v^9 - 11479334982388639688072024416v^8 - 127974827207665504832607109345v^7 + 589150632336643088995606856152v^6 + 1069543146374588590935537387080v^5 - 11807194646953443640782477012304v^4 - 3605036007214742067572690778000v^3 + 202992246210016730970395402332192v^2 - 233830476679483066871898861555720*v - 682138966046433419281929573987200) / 61954341379170581094994230390400
\(\beta_{13}\)	\(=\)	\( ( - 78\!\cdots\!97 \nu^{15} + \cdots + 45\!\cdots\!00 ) / 23\!\cdots\!00 \) (-784742119339358443728097v^15 - 2234239703002247985456720v^14 + 169492353216267038572844044v^13 + 438777123967568184876609840v^12 - 24451633501022249971611627347v^11 - 61641709717528378276252795920v^10 + 1887119679344623344436930485296v^9 + 4302272199487772409016220802160v^8 - 101180955674574814903875652353737v^7 - 222878637801802673815134242398320v^6 + 2626424272783354902559929853052254v^5 + 4500789814269919956613090442311440v^4 - 35801460709555613090685580369368492v^3 - 78000403357424827469259481502933920v^2 + 120402256569904860918352765050317200*v + 45001289919837222239689597961400000) / 23852421430980673721572778700304000
\(\beta_{14}\)	\(=\)	\( ( 10\!\cdots\!11 \nu^{15} + \cdots - 12\!\cdots\!00 \nu ) / 23\!\cdots\!00 \) (1036739314779757852459211v^15 - 209556418333158703058735972v^13 + 29616356744732813186317386961v^11 - 2132252766200125107991562982448v^9 + 111094314135947900511985701526531v^7 - 2448258232092885737532885589216202v^5 + 38638177074168343712090266623994596v^3 - 129876668595158622199342129516343600v) / 23852421430980673721572778700304000
\(\beta_{15}\)	\(=\)	\( ( 17\!\cdots\!63 \nu^{15} + \cdots - 21\!\cdots\!00 \nu ) / 14\!\cdots\!00 \) (178887373713941001361163v^15 - 35518765146587305946737876v^13 + 4995174937392052438208974213v^11 - 351842330409492539564762752384v^9 + 18163421802634808722947087034723v^7 - 373611501405444253426518050777766v^5 + 6281206733766187923854313192154468v^3 - 21109794021004200262907518303818800v) / 1490776339436292107598298668769000

\(\nu\)	\(=\)	\( ( \beta_{12} - \beta_{8} + 5\beta_{6} + \beta_{4} ) / 5 \) (b12 - b8 + 5*b6 + b4) / 5
\(\nu^{2}\)	\(=\)	\( ( -2\beta_{14} + 2\beta_{13} + 18\beta_{9} + 18\beta_{4} - 5\beta_{3} - 5\beta_{2} + 255\beta _1 - 5 ) / 5 \) (-2b14 + 2b13 + 18b9 + 18b4 - 5b3 - 5b2 + 255*b1 - 5) / 5
\(\nu^{3}\)	\(=\)	\( ( - 25 \beta_{15} - 117 \beta_{14} + 77 \beta_{13} + 77 \beta_{12} - 420 \beta_{11} - 77 \beta_{9} + \cdots + 25 \beta_{5} ) / 5 \) (-25b15 - 117b14 + 77b13 + 77b12 - 420b11 - 77b9 - 117b8 + 420b6 + 25*b5) / 5
\(\nu^{4}\)	\(=\)	\( ( -236\beta_{12} + 236\beta_{8} - 160\beta_{7} + 1804\beta_{4} - 545\beta_{3} + 20645\beta _1 - 20645 ) / 5 \) (-236b12 + 236b8 - 160b7 + 1804b4 - 545b3 + 20645b1 - 20645) / 5
\(\nu^{5}\)	\(=\)	\( -645\beta_{15} - 1577\beta_{14} + 1393\beta_{13} - 7446\beta_{11} - 1393\beta_{9} - 1393\beta_{4} \) -645b15 - 1577b14 + 1393b13 - 7446b11 - 1393b9 - 1393b4
\(\nu^{6}\)	\(=\)	\( ( 24086 \beta_{14} - 24086 \beta_{13} - 24086 \beta_{12} - 24880 \beta_{10} - 167014 \beta_{9} + \cdots - 1779000 ) / 5 \) (24086b14 - 24086b13 - 24086b12 - 24880b10 - 167014b9 + 24086b8 - 24880b7 + 66055b2 - 1779000) / 5
\(\nu^{7}\)	\(=\)	\( ( -666593\beta_{12} + 454713\beta_{8} - 3293670\beta_{6} - 356675\beta_{5} - 666593\beta_{4} ) / 5 \) (-666593b12 + 454713b8 - 3293670b6 - 356675b5 - 666593*b4) / 5
\(\nu^{8}\)	\(=\)	\( ( 2395352 \beta_{14} - 2395352 \beta_{13} - 3065280 \beta_{10} - 15427608 \beta_{9} - 15427608 \beta_{4} + \cdots + 7719445 ) / 5 \) (2395352b14 - 2395352b13 - 3065280b10 - 15427608b9 - 15427608b4 + 7719445b3 + 7719445b2 - 170538645b1 + 7719445) / 5
\(\nu^{9}\)	\(=\)	\( ( 37803525 \beta_{15} + 19312969 \beta_{14} - 64951729 \beta_{13} - 64951729 \beta_{12} + \cdots - 37803525 \beta_{5} ) / 5 \) (37803525b15 + 19312969b14 - 64951729b13 - 64951729b12 + 294732130b11 + 64951729b9 + 19312969b8 - 294732130b6 - 37803525*b5) / 5
\(\nu^{10}\)	\(=\)	\( 47461746 \beta_{12} - 47461746 \beta_{8} + 69613392 \beta_{7} - 287928930 \beta_{4} + 172380291 \beta_{3} + \cdots + 3208606451 \) 47461746b12 - 47461746b8 + 69613392b7 - 287928930b4 + 172380291b3 - 3208606451b1 + 3208606451
\(\nu^{11}\)	\(=\)	\( ( 3931122675 \beta_{15} - 77243011 \beta_{14} - 6375360389 \beta_{13} + 26779255870 \beta_{11} + \cdots + 6375360389 \beta_{4} ) / 5 \) (3931122675b15 - 77243011b14 - 6375360389b13 + 26779255870b11 + 6375360389b9 + 6375360389b4) / 5
\(\nu^{12}\)	\(=\)	\( ( - 23524579588 \beta_{14} + 23524579588 \beta_{13} + 23524579588 \beta_{12} + 37901584800 \beta_{10} + \cdots + 1434738950000 ) / 5 \) (-23524579588b14 + 23524579588b13 + 23524579588b12 + 37901584800b10 + 135918046692b9 - 23524579588b8 + 37901584800b7 - 93121904245b2 + 1434738950000) / 5
\(\nu^{13}\)	\(=\)	\( ( 628386849133 \beta_{12} + 161046406107 \beta_{8} + 2470433787330 \beta_{6} + \cdots + 628386849133 \beta_{4} ) / 5 \) (628386849133b12 + 161046406107b8 + 2470433787330b6 + 404170869725b5 + 628386849133*b4) / 5
\(\nu^{14}\)	\(=\)	\( ( - 2334986215774 \beta_{14} + 2334986215774 \beta_{13} + 4022800213040 \beta_{10} + 12969275515886 \beta_{9} + \cdots - 9838180781455 ) / 5 \) (-2334986215774b14 + 2334986215774b13 + 4022800213040b10 + 12969275515886b9 + 12969275515886b4 - 9838180781455b3 - 9838180781455b2 + 146915735293855b1 - 9838180781455) / 5
\(\nu^{15}\)	\(=\)	\( - 8246728673055 \beta_{15} + 5556498224339 \beta_{14} + 12423262407605 \beta_{13} + \cdots + 8246728673055 \beta_{5} \) -8246728673055b15 + 5556498224339b14 + 12423262407605b13 + 12423262407605b12 - 46218217428054b11 - 12423262407605b9 + 5556498224339b8 + 46218217428054b6 + 8246728673055*b5

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 79.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} - 35 T^{6} + \cdots + 37636)^{2} \) (T^8 - 35T^6 + 1031T^4 - 6790*T^2 + 37636)^2
$3$	\( (T^{8} - 71 T^{6} + \cdots + 62500)^{2} \) (T^8 - 71T^6 + 4791T^4 - 17750*T^2 + 62500)^2
$5$	\( T^{16} + \cdots + 59\!\cdots\!25 \) T^16 + 206T^14 + 1026T^12 + 2092960T^10 + 807584975T^8 + 32702500000T^6 + 250488281250T^4 + 785827636718750*T^2 + 59604644775390625
$7$	\( T^{16} \) T^16
$11$	\( (T^{4} - 15 T^{3} + \cdots + 3136)^{4} \) (T^4 - 15T^3 + 281T^2 + 840*T + 3136)^4
$13$	\( (T^{4} + 6391 T^{2} + 3906250)^{4} \) (T^4 + 6391*T^2 + 3906250)^4
$17$	\( (T^{8} + \cdots + 322417936000000)^{2} \) (T^8 - 8701T^6 + 57751401T^4 - 156235156000*T^2 + 322417936000000)^2
$19$	\( (T^{8} + \cdots + 88\!\cdots\!00)^{2} \) (T^8 + 20546T^6 + 328242116T^4 + 1929187216000*T^2 + 8816458816000000)^2
$23$	\( (T^{8} + \cdots + 12\!\cdots\!36)^{2} \) (T^8 - 12100T^6 + 111533456T^4 - 422006182400*T^2 + 1216373321383936)^2
$29$	\( (T^{2} + 229 T + 4018)^{8} \) (T^2 + 229*T + 4018)^8
$31$	\( (T^{8} + \cdots + 9634816000000)^{2} \) (T^8 + 45736T^6 + 2088677696T^4 + 141964544000*T^2 + 9634816000000)^2
$37$	\( (T^{8} + \cdots + 33\!\cdots\!00)^{2} \) (T^8 - 154084T^6 + 17915981456T^4 - 897677605798400*T^2 + 33941082845685760000)^2
$41$	\( (T^{4} - 156776 T^{2} + 3802400000)^{4} \) (T^4 - 156776*T^2 + 3802400000)^4
$43$	\( (T^{4} + 27440 T^{2} + 119243264)^{4} \) (T^4 + 27440*T^2 + 119243264)^4
$47$	\( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) (T^8 - 379749T^6 + 108221302001T^4 - 13666407391749000*T^2 + 1295136215976001000000)^2
$53$	\( (T^{8} + \cdots + 26\!\cdots\!00)^{2} \) (T^8 - 261700T^6 + 52066730000T^4 - 4297155872000000*T^2 + 269621654425600000000)^2
$59$	\( (T^{8} + \cdots + 60\!\cdots\!00)^{2} \) (T^8 + 316226T^6 + 75412099076T^4 + 7774980357184000*T^2 + 604509947462656000000)^2
$61$	\( (T^{8} + \cdots + 56\!\cdots\!00)^{2} \) (T^8 + 645250T^6 + 392582562500T^4 + 15334366250000000*T^2 + 564775225000000000000)^2
$67$	\( (T^{8} + \cdots + 16\!\cdots\!16)^{2} \) (T^8 - 1025728T^6 + 1011171749888T^4 - 41999643417509888*T^2 + 1676589664454066569216)^2
$71$	\( (T^{2} - 658 T - 53848)^{8} \) (T^2 - 658*T - 53848)^8
$73$	\( (T^{8} + \cdots + 83\!\cdots\!00)^{2} \) (T^8 - 610044T^6 + 280645325936T^4 - 55824123527664000*T^2 + 8373779217822736000000)^2
$79$	\( (T^{4} + 367 T^{3} + \cdots + 14399520004)^{4} \) (T^4 + 367T^3 + 254687T^2 - 44039266*T + 14399520004)^4
$83$	\( (T^{4} + 1018386 T^{2} + 135536164000)^{4} \) (T^4 + 1018386*T^2 + 135536164000)^4
$89$	\( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) (T^8 + 1818144T^6 + 2818890740736T^4 + 884994071740416000*T^2 + 236932244651114496000000)^2
$97$	\( (T^{4} + 1264725 T^{2} + 392832400000)^{4} \) (T^4 + 1264725*T^2 + 392832400000)^4
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\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(-1 + \beta_{1}\)	\(-1\)