LMFDB - Newform orbit 300.9.g.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,9,Mod(101,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("300.101");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	300.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$122.213583018$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7378 x^{14} + 23156928 x^{12} - 101588726286 x^{10} + \cdots + 34\!\cdots\!81 $ x^16 - 7378x^14 + 23156928x^12 - 101588726286x^10 + 1484695155668670x^8 - 4373061557178808206x^6 + 42910255095788484704448x^4 - 588516817021165377771877458*x^2 + 3433683820292512484657849089281
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{38}\cdot 3^{20}\cdot 5^{18} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q - \beta_1 q^{3} - \beta_{7} q^{7} + (\beta_{2} + 922) q^{9}+O(q^{10}) $ q - b1 * q^3 - b7 * q^7 + (b2 + 922) * q^9	$ q - \beta_1 q^{3} - \beta_{7} q^{7} + (\beta_{2} + 922) q^{9} - \beta_{3} q^{11} + (\beta_{12} + \beta_{6} - 27 \beta_1) q^{13} + (\beta_{13} + 2 \beta_{6} + 47 \beta_1) q^{17} + (\beta_{8} + 3 \beta_{2} + 3001) q^{19} + ( - \beta_{10} + \beta_{8} + \cdots + 296) q^{21}+ \cdots + ( - 18 \beta_{10} + 102 \beta_{9} + \cdots - 1060648) q^{99}+O(q^{100}) $ q - b1 * q^3 - b7 * q^7 + (b2 + 922) * q^9 - b3 * q^11 + (b12 + b6 - 27b1) q^13 + (b13 + 2b6 + 47b1) * q^17 + (b8 + 3b2 + 3001) q^19 + (-b10 + b8 + 3b3 + 296) q^21 + (b15 + 2b13 + b11 + 16b6 + 364b1) q^23 + (-b15 - b14 - 2b13 + 11b12 - 2b11 - 14b7 + 2b6 - 917b1) * q^27 + (2b10 - b9 + 2b5 + 2b4 - 10b3 - 23b2 + 5) q^29 + (-5b10 - b9 - 5b8 + b5 - b4 + b3 + 8b2 + 29381) q^31 + (b15 + 2b14 - 10b13 + 11b12 + 5b11 - 98b7 - 4b6 + 14b1) q^33 + (b15 - 6b14 - b13 - 53b12 + 2b11 + 74b7 + 6b6 - 421b1) * q^37 + (6b10 - 6b9 + 15b8 + 3b5 - 15b4 + 39b3 + 33b2 + 178098) q^39 + (-7b10 - 4b9 - 7b5 - 26b4 - 141b3 - 122b2 + 35) * q^41 + (-2b15 + 12b14 + 2b13 - 48b12 - 4b11 + 608b7 - 109b6 + 3233b1) * q^43 + (-14b15 + 7b13 + 23b11 - 49b6 - 962b1) q^47 + (15b10 - 24b9 + 10b8 - 3b5 - 24b4 - 3b3 + 690b2 + 1238386) q^49 + (-3b10 - 15b9 - 37b8 - 7b5 + 51b4 + 380b3 - 78b2 + 314143) q^51 + (4b15 - 18b13 - 87b11 + 108b6 + 2260b1) q^53 + (6b15 + 30b13 + 153b12 - 141b11 - 1188b7 + 699b6 - 3093b1) q^57 + (b10 - 59b9 + b5 + 123b4 - 176b3 - 1591b2 + 412) * q^59 + (45b10 - 24b9 - 10b8 - 9b5 - 24b4 - 9b3 + 654b2 + 1177939) q^61 + (2b15 - 7b14 + 85b13 + 77b12 + 274b11 - 5039b7 + 620b6 - 131b1) * q^63 + (-5b15 + 30b14 + 5b13 + 10b12 - 10b11 + 2566b7 + 1092b6 - 33697b1) * q^67 + (-36b10 - 108b9 - 157b8 - 25b5 - 609b4 + 326b3 - 342b2 + 2416915) q^69 + (42b10 - 42b9 + 42b5 - 566b4 - 452b3 - 1050b2 + 252) * q^71 + (17b15 - 102b14 - 17b13 + 272b12 + 34b11 + 9782b7 + 1863b6 - 57056b1) * q^73 + (128b15 - 132b13 + 1072b11 + 4596b6 + 106696b1) q^77 + (-130b10 - 122b9 - 57b8 + 26b5 - 122b4 + 26b3 + 3019b2 + 5672991) q^79 + (81b10 - 6b9 + 306b8 + 63b5 + 1236b4 - 369b3 + 664b2 + 1015000) q^81 + (-77b15 - 11b13 - 1428b11 + 1124b6 + 23601b1) q^83 + (-12b15 + 48b14 - 177b13 - 1140b12 - 1707b11 - 6159b7 + 5565b6 - 1662b1) * q^87 + (-135b10 - 69b9 - 135b5 + 1771b4 + 219b3 - 2133b2 + 618) * q^89 + (-170b10 - 22b9 + 400b8 + 34b5 - 22b4 + 34b3 + 1658b2 + 1398232) q^91 + (-93b15 - 54b14 - 141b13 - 1089b12 + 2307b11 - 22194b7 + 2256b6 - 28465b1) * q^93 + (-16b15 + 96b14 + 16b13 + 1868b12 - 32b11 + 10432b7 + 16224b6 - 495104b1) * q^97 + (-18b10 + 102b9 + 630b8 + 72b5 - 2976b4 - 2493b3 + 1016b2 - 1060648) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 14756 q^{9}+O(q^{10}) $ 16 * q + 14756 * q^9	$ 16 q + 14756 q^{9} + 48024 q^{19} + 4724 q^{21} + 470104 q^{31} + 2849664 q^{39} + 19816920 q^{49} + 5026040 q^{51} + 18849944 q^{61} + 38669180 q^{69} + 90778632 q^{79} + 16242056 q^{81} + 22375296 q^{91} - 16968560 q^{99}+O(q^{100}) $ 16 * q + 14756 * q^9 + 48024 * q^19 + 4724 * q^21 + 470104 * q^31 + 2849664 * q^39 + 19816920 * q^49 + 5026040 * q^51 + 18849944 * q^61 + 38669180 * q^69 + 90778632 * q^79 + 16242056 * q^81 + 22375296 * q^91 - 16968560 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 7378 x^{14} + 23156928 x^{12} - 101588726286 x^{10} + \cdots + 34\!\cdots\!81 $ x^16 - 7378*x^14 + 23156928*x^12 - 101588726286*x^10 + 1484695155668670*x^8 - 4373061557178808206*x^6 + 42910255095788484704448*x^4 - 588516817021165377771877458*x^2 + 3433683820292512484657849089281 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 922 $ v^2 - 922
$\beta_{3}$	$=$	$ ( - 118241503 \nu^{14} + 32210391788401 \nu^{12} + \cdots + 14\!\cdots\!93 ) / 20\!\cdots\!00 $ (-118241503v^14 + 32210391788401v^12 + 126457566128179947v^10 + 214013909643019174755v^8 + 4746195418588577798202495v^6 + 3485352495014175965316858063v^4 + 97439320345474574721245443129749*v^2 + 1423431157324590957071749337819988093) / 205650014751525015049243888872000
$\beta_{4}$	$=$	$ ( 438411229 \nu^{14} + 545360564057 \nu^{12} + \cdots - 23\!\cdots\!99 ) / 20\!\cdots\!00 $ (438411229v^14 + 545360564057v^12 + 5205191446634679v^10 + 134747004904914583035v^8 - 140050706881610814672285v^6 + 7126746748103003389938313191v^4 + 15988522642231144261362784701993*v^2 - 2374313718998962356072521984335899) / 20565001475152501504924388887200
$\beta_{5}$	$=$	$ ( - 5968589155 \nu^{14} - 101803994977442 \nu^{12} + \cdots - 77\!\cdots\!42 ) / 10\!\cdots\!00 $ (-5968589155v^14 - 101803994977442v^12 - 1605609642957550713v^10 + 284564718899432445330v^8 + 86691769095675942519262875v^6 + 257905289910296311314646868010v^4 - 1473149156225017240571482096707663*v^2 - 7767748177539020226873719607705391242) / 102825007375762507524621944436000
$\beta_{6}$	$=$	$ ( \nu^{15} - 7378 \nu^{13} + 23156928 \nu^{11} - 101588726286 \nu^{9} + \cdots - 51\!\cdots\!66 \nu ) / 19\!\cdots\!23 $ (v^15 - 7378v^13 + 23156928v^11 - 101588726286v^9 + 1484695155668670v^7 - 4373061557178808206v^5 + 42910255095788484704448v^3 - 510983834350445298184690566*v) / 19383245667680019896796723
$\beta_{7}$	$=$	$ ( - 9990143 \nu^{15} - 13366675591 \nu^{13} - 104908177452681 \nu^{11} + \cdots + 22\!\cdots\!41 \nu ) / 14\!\cdots\!00 $ (-9990143v^15 - 13366675591v^13 - 104908177452681v^11 + 1454707853876694735v^9 + 893990708450313260895v^7 + 15889852799914567568262183v^5 - 488135239609492118517858798039v^3 + 2286102946760216024014189243114641v) / 148630727779770392568637271964000
$\beta_{8}$	$=$	$ ( - 56323 \nu^{14} + 84994792 \nu^{12} + 852147203931 \nu^{10} + 150878465402940 \nu^{8} + \cdots + 18\!\cdots\!24 ) / 31\!\cdots\!00 $ (-56323v^14 + 84994792v^12 + 852147203931v^10 + 150878465402940v^8 - 44424226549043107605v^6 + 307048508107766448925968v^4 - 2092818668997351782154051027*v^2 + 18959612788879968454203241251324) / 319065772307490039453444000
$\beta_{9}$	$=$	$ ( 6522599171 \nu^{14} - 51903695295257 \nu^{12} + \cdots - 35\!\cdots\!01 ) / 20\!\cdots\!00 $ (6522599171v^14 - 51903695295257v^12 + 155990425193416521v^10 - 841907185104513957435v^8 + 10475029126320841638840285v^6 - 37567673727464881971509655591v^4 + 282710209346205493966540669557207*v^2 - 3582208268407457812459773082999060101) / 20565001475152501504924388887200
$\beta_{10}$	$=$	$ ( 27940379809 \nu^{14} - 7840932369085 \nu^{12} + \cdots - 10\!\cdots\!81 ) / 68\!\cdots\!00 $ (27940379809v^14 - 7840932369085v^12 + 106687622468324331v^10 - 7006703647655295554535v^8 + 8022390700695158204032815v^6 + 96875049710885773953791899341v^4 + 1305055640077155944800803129395205*v^2 - 10303770409953809798553362085042235881) / 68550004917175005016414629624000
$\beta_{11}$	$=$	$ ( 20596949 \nu^{15} - 37487115353 \nu^{13} - 26193217876281 \nu^{11} + \cdots - 12\!\cdots\!89 \nu ) / 74\!\cdots\!00 $ (20596949v^15 - 37487115353v^13 - 26193217876281v^11 - 9615576560197871115v^9 + 33111695571416626953915v^7 + 334926758600430895264046121v^5 + 1185049326849210503136732085353v^3 - 12370583909607752899704971964400389v) / 74524702943096140499204040590400
$\beta_{12}$	$=$	$ ( 515192467 \nu^{15} + 615081057479 \nu^{13} + \cdots - 14\!\cdots\!29 \nu ) / 16\!\cdots\!00 $ (515192467v^15 + 615081057479v^13 - 23769410150578311v^11 - 13979069008262681715v^9 + 1455452294036789810563245v^7 - 718501964677245201926306727v^5 - 39986773812708795760737886002009v^3 - 146868508805231148372599489199012429v) / 1684481581504064449111222415592000
$\beta_{13}$	$=$	$ ( 439511153 \nu^{15} - 3923184692141 \nu^{13} + \cdots - 36\!\cdots\!33 \nu ) / 37\!\cdots\!00 $ (439511153v^15 - 3923184692141v^13 - 25648690419064557v^11 - 96300953039951192655v^9 + 51322315211094545556255v^7 - 1803169386614614640336367363v^5 + 21053398589594473713522842730141v^3 - 361636306388836808583648784537192833v) / 372623514715480702496020202952000
$\beta_{14}$	$=$	$ ( - 315617774114 \nu^{15} + \cdots + 59\!\cdots\!01 \nu ) / 22\!\cdots\!00 $ (-315617774114v^15 + 2906961097996085v^13 + 5341402867084775874v^11 - 91133985746362493381265v^9 + 1173175902795009552147044010v^7 - 7577842817073919295320004129661v^5 + 59132031186145636229505428318139270v^3 + 59491772755992574410359124796072221801v) / 224878291130792603956348192481532000
$\beta_{15}$	$=$	$ ( 360231947 \nu^{15} + 1695521995441 \nu^{13} + \cdots - 16\!\cdots\!67 \nu ) / 12\!\cdots\!00 $ (360231947v^15 + 1695521995441v^13 - 3519056907251343v^11 + 116929110967281142155v^9 + 396517034323738594366245v^7 + 3446460383777940257068507263v^5 + 47389009058756312349490136702559v^3 - 16852169989705515576706347509262267v) / 124207838238493567498673400984000

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 922 $ b2 + 922
$\nu^{3}$	$=$	$ \beta_{15} + \beta_{14} + 2\beta_{13} - 11\beta_{12} + 2\beta_{11} + 14\beta_{7} - 2\beta_{6} + 917\beta_1 $ b15 + b14 + 2b13 - 11b12 + 2b11 + 14b7 - 2b6 + 917b1
$\nu^{4}$	$=$	$ 81\beta_{10} - 6\beta_{9} + 306\beta_{8} + 63\beta_{5} + 1236\beta_{4} - 369\beta_{3} + 664\beta_{2} + 1015000 $ 81b10 - 6b9 + 306b8 + 63b5 + 1236b4 - 369b3 + 664*b2 + 1015000
$\nu^{5}$	$=$	$ 4624 \beta_{15} - 6959 \beta_{14} - 7519 \beta_{13} - 22595 \beta_{12} + 113252 \beta_{11} + \cdots + 1051022 \beta_1 $ 4624b15 - 6959b14 - 7519b13 - 22595b12 + 113252b11 + 465362b7 - 143816b6 + 1051022b1
$\nu^{6}$	$=$	$ - 164916 \beta_{10} + 446349 \beta_{9} - 37224 \beta_{8} + 609156 \beta_{5} - 1832559 \beta_{4} + \cdots + 24228485347 $ -164916b10 + 446349b9 - 37224b8 + 609156b5 - 1832559b4 + 11296188b3 + 1559200*b2 + 24228485347
$\nu^{7}$	$=$	$ 14029492 \beta_{15} + 41799856 \beta_{14} - 69784480 \beta_{13} + 438941944 \beta_{12} + \cdots + 23670718700 \beta_1 $ 14029492b15 + 41799856b14 - 69784480b13 + 438941944b12 + 253173212b11 + 3868482764b7 + 3025466029b6 + 23670718700b1
$\nu^{8}$	$=$	$ - 6583202100 \beta_{10} + 674791752 \beta_{9} - 7360658640 \beta_{8} - 455274180 \beta_{5} + \cdots - 493406497298627 $ -6583202100b10 + 674791752b9 - 7360658640b8 - 455274180b5 + 53815623996b4 + 6595207740b3 + 16287378400*b2 - 493406497298627
$\nu^{9}$	$=$	$ 372515274100 \beta_{15} - 191909050640 \beta_{14} - 181942319680 \beta_{13} + \cdots - 497682836598727 \beta_1 $ 372515274100b15 - 191909050640b14 - 181942319680b13 + 341871542440b12 - 3560043027652b11 + 14206187118740b7 + 13550675701912b6 - 497682836598727b1
$\nu^{10}$	$=$	$ - 4284421754940 \beta_{10} + 37589920269960 \beta_{9} + 76428779934600 \beta_{8} + \cdots - 27\!\cdots\!54 $ -4284421754940b10 + 37589920269960b9 + 76428779934600b8 - 17864334358740b5 + 116051575465164b4 + 340033287412260b3 - 441150705624935*b2 - 2734248649531353554
$\nu^{11}$	$=$	$ - 718952827444355 \beta_{15} - 526320068789135 \beta_{14} + \cdots - 27\!\cdots\!03 \beta_1 $ -718952827444355b15 - 526320068789135b14 - 8400999370274950b13 - 10145890256279315b12 + 1525742933722718b11 - 60538389884256790b7 + 191269996924912870b6 - 2769509565289291003b1
$\nu^{12}$	$=$	$ 78\!\cdots\!25 \beta_{10} + \cdots - 35\!\cdots\!08 $ 78326388969301125b10 - 217705289330581350b9 - 290780653654009710b8 - 25054327119139245b5 - 643445293264229976b4 + 3378812378443150275b3 - 1865253820476150464*b2 - 35943471852211625378108
$\nu^{13}$	$=$	$ 12\!\cdots\!16 \beta_{15} + \cdots - 35\!\cdots\!62 \beta_1 $ 1212546650043993916b15 + 7354618417090964641b14 - 24686342839711451863b13 + 80995809963092854789b12 - 7561995861217751200b11 - 1182534817833708789706b7 - 1283996327823107379512b6 - 35593092956130168296362b1
$\nu^{14}$	$=$	$ 60\!\cdots\!76 \beta_{10} + \cdots + 19\!\cdots\!35 $ 607395000117700370376b10 - 142763328134049426531b9 - 3269572223624975677104b8 - 446326529173083519912b5 + 9112531606003942407525b4 - 660305187878822012664b3 - 44213012442513461020976*b2 + 191856289957555519437810535
$\nu^{15}$	$=$	$ 19\!\cdots\!44 \beta_{15} + \cdots + 19\!\cdots\!08 \beta_1 $ 19919522998163068870744b15 - 88447899581114690345984b14 - 21170669082847405668304b13 + 588773532910800051195280b12 - 419231745548751852435400b11 - 10188874813885804821015688b7 + 1822300630910515632596029b6 + 192056041250389533913791308b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$-1$	$1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

101.1

79.3447	+	16.2916i
79.3447	−	16.2916i
74.2942	+	32.2702i
74.2942	−	32.2702i
51.0255	+	62.9079i
51.0255	−	62.9079i
23.4026	+	77.5456i
23.4026	−	77.5456i
−23.4026	+	77.5456i
−23.4026	−	77.5456i
−51.0255	+	62.9079i
−51.0255	−	62.9079i
−74.2942	+	32.2702i
−74.2942	−	32.2702i
−79.3447	+	16.2916i
−79.3447	−	16.2916i

−79.3447

−

16.2916i

−3405.57

6030.17

2585.30i

101.2

−79.3447

16.2916i

−3405.57

6030.17

−

2585.30i

101.3

−74.2942

−

32.2702i

4017.33

4478.26

4794.99i

101.4

−74.2942

32.2702i

4017.33

4478.26

−

4794.99i

101.5

−51.0255

−

62.9079i

−459.019

−1353.80

6419.81i

101.6

−51.0255

62.9079i

−459.019

−1353.80

−

6419.81i

101.7

−23.4026

−

77.5456i

−256.799

−5465.63

3629.54i

101.8

−23.4026

77.5456i

−256.799

−5465.63

−

3629.54i

101.9

23.4026

−

77.5456i

256.799

−5465.63

−

3629.54i

101.10

23.4026

77.5456i

256.799

−5465.63

3629.54i

101.11

51.0255

−

62.9079i

459.019

−1353.80

−

6419.81i

101.12

51.0255

62.9079i

459.019

−1353.80

6419.81i

101.13

74.2942

−

32.2702i

−4017.33

4478.26

−

4794.99i

101.14

74.2942

32.2702i

−4017.33

4478.26

4794.99i

101.15

79.3447

−

16.2916i

3405.57

6030.17

−

2585.30i

101.16

79.3447

16.2916i

3405.57

6030.17

2585.30i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	300.9.g.h		16
3.b	odd	2	1	inner	300.9.g.h		16
5.b	even	2	1	inner	300.9.g.h		16
5.c	odd	4	2		60.9.b.a	✓	16
15.d	odd	2	1	inner	300.9.g.h		16
15.e	even	4	2		60.9.b.a	✓	16
20.e	even	4	2		240.9.c.d		16
60.l	odd	4	2		240.9.c.d		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
60.9.b.a	✓	16	5.c	odd	4	2
60.9.b.a	✓	16	15.e	even	4	2
240.9.c.d		16	20.e	even	4	2
240.9.c.d		16	60.l	odd	4	2
300.9.g.h		16	1.a	even	1	1	trivial
300.9.g.h		16	3.b	odd	2	1	inner
300.9.g.h		16	5.b	even	2	1	inner
300.9.g.h		16	15.d	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{8} - 28013434T_{7}^{6} + 194864255690496T_{7}^{4} - 52166784840953810944T_{7}^{2} + 2600757082907060397408256 $ T7^8 - 28013434*T7^6 + 194864255690496*T7^4 - 52166784840953810944*T7^2 + 2600757082907060397408256 acting on $S_{9}^{\mathrm{new}}(300, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} + \cdots + 34\!\cdots\!81 $ T^16 - 7378T^14 + 23156928T^12 - 101588726286T^10 + 1484695155668670T^8 - 4373061557178808206T^6 + 42910255095788484704448T^4 - 588516817021165377771877458*T^2 + 3433683820292512484657849089281
$5$	$ T^{16} $ T^16
$7$	$ (T^{8} + \cdots + 26\!\cdots\!56)^{2} $ (T^8 - 28013434T^6 + 194864255690496T^4 - 52166784840953810944*T^2 + 2600757082907060397408256)^2
$11$	$ (T^{8} + \cdots + 10\!\cdots\!00)^{2} $ (T^8 + 971194160T^6 + 265474129795728000T^4 + 28030749534089025145088000*T^2 + 1006466192971333730193034854400000)^2
$13$	$ (T^{8} + \cdots + 36\!\cdots\!36)^{2} $ (T^8 - 3173472864T^6 + 2136237732098603136T^4 - 339475890779809216137437184*T^2 + 3607723320043650231222013878337536)^2
$17$	$ (T^{8} + \cdots + 17\!\cdots\!00)^{2} $ (T^8 + 36442812140T^6 + 415835666767610862000T^4 + 1533374533895190454851260072000*T^2 + 1770350416439299246742626840450278400000)^2
$19$	$ (T^{4} + \cdots + 14\!\cdots\!16)^{4} $ (T^4 - 12006T^3 - 14875364124T^2 + 306871114764024*T + 14325224454022538016)^4
$23$	$ (T^{8} + \cdots + 58\!\cdots\!00)^{2} $ (T^8 + 416302278710T^6 + 57314419519886499633000T^4 + 3147172470184548438496776936308000*T^2 + 58287367348221269808659040386930849472400000)^2
$29$	$ (T^{8} + \cdots + 31\!\cdots\!00)^{2} $ (T^8 + 2076533641440T^6 + 1191051736937646512976000T^4 + 128658594921141080817673750444032000*T^2 + 3104403840932459875082278936977155294822400000)^2
$31$	$ (T^{4} + \cdots + 11\!\cdots\!96)^{4} $ (T^4 - 117526T^3 - 2411512956084T^2 + 42008109391780664*T + 1161798629701340519716096)^4
$37$	$ (T^{8} + \cdots + 49\!\cdots\!56)^{2} $ (T^8 - 17166889014016T^6 + 79594418412384606721389696T^4 - 49806165702604945505529651000596525056*T^2 + 4942690493000192267969030626358676764032761856)^2
$41$	$ (T^{8} + \cdots + 90\!\cdots\!00)^{2} $ (T^8 + 47937327156740T^6 + 727271268674755116724800000T^4 + 4366076370036139476410695726377915392000*T^2 + 9019374280313225828659727762832863084886419046400000)^2
$43$	$ (T^{8} + \cdots + 11\!\cdots\!76)^{2} $ (T^8 - 49663114389754T^6 + 682901573617410215550536256T^4 - 1836202088927117599440036340104346549504*T^2 + 1185398126206724940592428794668145932263947368677376)^2
$47$	$ (T^{8} + \cdots + 22\!\cdots\!00)^{2} $ (T^8 + 67977556756310T^6 + 1049302047808834875960513000T^4 + 4623472992615897055567317425438384948000*T^2 + 2280191789489125701271626231505786142624027808400000)^2
$53$	$ (T^{8} + \cdots + 49\!\cdots\!00)^{2} $ (T^8 + 37034825790060T^6 + 442771852668467171313582000T^4 + 1728338444134576563976678985411908008000*T^2 + 495465407935730608676645867135344783121059430400000)^2
$59$	$ (T^{8} + \cdots + 76\!\cdots\!00)^{2} $ (T^8 + 867038659162160T^6 + 248923444784011995398556048000T^4 + 26049124119245301801178871966454591097088000*T^2 + 767557073508162727336611351762522586065060697738854400000)^2
$61$	$ (T^{4} + \cdots - 21\!\cdots\!64)^{4} $ (T^4 - 4712486T^3 - 246246167429364T^2 - 1401894577944767419016*T - 2105396555318302820718155264)^4
$67$	$ (T^{8} + \cdots + 32\!\cdots\!76)^{2} $ (T^8 - 433265383552954T^6 + 54035838660045196541245678656T^4 - 1509066941286824095659713998642889047784704*T^2 + 320783174079286377881741203034787315142121785063653376)^2
$71$	$ (T^{8} + \cdots + 21\!\cdots\!00)^{2} $ (T^8 + 2138475782597760T^6 + 1326666887128916996598256896000T^4 + 190090844824428013879437967383828024066048000*T^2 + 2137645709252142846293718231008089890125542037284454400000)^2
$73$	$ (T^{8} + \cdots + 89\!\cdots\!16)^{2} $ (T^8 - 5313866871866656T^6 + 7525114331008359170085644875776T^4 - 1580040512870961011562303832920781921946238976*T^2 + 89032404109235745594165545897265554898428656660336624009216)^2
$79$	$ (T^{4} + \cdots + 62\!\cdots\!16)^{4} $ (T^4 - 22694658T^3 - 2904371819036676T^2 + 29252521987078209542568*T + 620309238940199098245480618816)^4
$83$	$ (T^{8} + \cdots + 74\!\cdots\!00)^{2} $ (T^8 + 6035256522659990T^6 + 11440242471520537322508024801000T^4 + 7416555771197894088706113351807470111804852000*T^2 + 747817278848502680030790368742080069342213955272235744400000)^2
$89$	$ (T^{8} + \cdots + 12\!\cdots\!00)^{2} $ (T^8 + 15617313982451840T^6 + 83066730379617691583491110144000T^4 + 175036048359888268460930166678998273561526272000*T^2 + 126715534153661912498966905333746869248072534426556720742400000)^2
$97$	$ (T^{8} + \cdots + 93\!\cdots\!76)^{2} $ (T^8 - 27061263847686304T^6 + 228077510490380569982248266694656T^4 - 778135119634313173119081490922977317629307387904*T^2 + 938517188523791996880618888160872429464721814239202719538610176)^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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	300.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{3} - \beta_{7} q^{7} + (\beta_{2} + 922) q^{9}+O(q^{10}) \) q - b1 * q^3 - b7 * q^7 + (b2 + 922) * q^9	\( q - \beta_1 q^{3} - \beta_{7} q^{7} + (\beta_{2} + 922) q^{9} - \beta_{3} q^{11} + (\beta_{12} + \beta_{6} - 27 \beta_1) q^{13} + (\beta_{13} + 2 \beta_{6} + 47 \beta_1) q^{17} + (\beta_{8} + 3 \beta_{2} + 3001) q^{19} + ( - \beta_{10} + \beta_{8} + \cdots + 296) q^{21}+ \cdots + ( - 18 \beta_{10} + 102 \beta_{9} + \cdots - 1060648) q^{99}+O(q^{100}) \) q - b1 * q^3 - b7 * q^7 + (b2 + 922) * q^9 - b3 * q^11 + (b12 + b6 - 27b1) q^13 + (b13 + 2b6 + 47b1) * q^17 + (b8 + 3b2 + 3001) q^19 + (-b10 + b8 + 3b3 + 296) q^21 + (b15 + 2b13 + b11 + 16b6 + 364b1) q^23 + (-b15 - b14 - 2b13 + 11b12 - 2b11 - 14b7 + 2b6 - 917b1) * q^27 + (2b10 - b9 + 2b5 + 2b4 - 10b3 - 23b2 + 5) q^29 + (-5b10 - b9 - 5b8 + b5 - b4 + b3 + 8b2 + 29381) q^31 + (b15 + 2b14 - 10b13 + 11b12 + 5b11 - 98b7 - 4b6 + 14b1) q^33 + (b15 - 6b14 - b13 - 53b12 + 2b11 + 74b7 + 6b6 - 421b1) * q^37 + (6b10 - 6b9 + 15b8 + 3b5 - 15b4 + 39b3 + 33b2 + 178098) q^39 + (-7b10 - 4b9 - 7b5 - 26b4 - 141b3 - 122b2 + 35) * q^41 + (-2b15 + 12b14 + 2b13 - 48b12 - 4b11 + 608b7 - 109b6 + 3233b1) * q^43 + (-14b15 + 7b13 + 23b11 - 49b6 - 962b1) q^47 + (15b10 - 24b9 + 10b8 - 3b5 - 24b4 - 3b3 + 690b2 + 1238386) q^49 + (-3b10 - 15b9 - 37b8 - 7b5 + 51b4 + 380b3 - 78b2 + 314143) q^51 + (4b15 - 18b13 - 87b11 + 108b6 + 2260b1) q^53 + (6b15 + 30b13 + 153b12 - 141b11 - 1188b7 + 699b6 - 3093b1) q^57 + (b10 - 59b9 + b5 + 123b4 - 176b3 - 1591b2 + 412) * q^59 + (45b10 - 24b9 - 10b8 - 9b5 - 24b4 - 9b3 + 654b2 + 1177939) q^61 + (2b15 - 7b14 + 85b13 + 77b12 + 274b11 - 5039b7 + 620b6 - 131b1) * q^63 + (-5b15 + 30b14 + 5b13 + 10b12 - 10b11 + 2566b7 + 1092b6 - 33697b1) * q^67 + (-36b10 - 108b9 - 157b8 - 25b5 - 609b4 + 326b3 - 342b2 + 2416915) q^69 + (42b10 - 42b9 + 42b5 - 566b4 - 452b3 - 1050b2 + 252) * q^71 + (17b15 - 102b14 - 17b13 + 272b12 + 34b11 + 9782b7 + 1863b6 - 57056b1) * q^73 + (128b15 - 132b13 + 1072b11 + 4596b6 + 106696b1) q^77 + (-130b10 - 122b9 - 57b8 + 26b5 - 122b4 + 26b3 + 3019b2 + 5672991) q^79 + (81b10 - 6b9 + 306b8 + 63b5 + 1236b4 - 369b3 + 664b2 + 1015000) q^81 + (-77b15 - 11b13 - 1428b11 + 1124b6 + 23601b1) q^83 + (-12b15 + 48b14 - 177b13 - 1140b12 - 1707b11 - 6159b7 + 5565b6 - 1662b1) * q^87 + (-135b10 - 69b9 - 135b5 + 1771b4 + 219b3 - 2133b2 + 618) * q^89 + (-170b10 - 22b9 + 400b8 + 34b5 - 22b4 + 34b3 + 1658b2 + 1398232) q^91 + (-93b15 - 54b14 - 141b13 - 1089b12 + 2307b11 - 22194b7 + 2256b6 - 28465b1) * q^93 + (-16b15 + 96b14 + 16b13 + 1868b12 - 32b11 + 10432b7 + 16224b6 - 495104b1) * q^97 + (-18b10 + 102b9 + 630b8 + 72b5 - 2976b4 - 2493b3 + 1016b2 - 1060648) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 14756 q^{9}+O(q^{10}) \) 16 * q + 14756 * q^9	\( 16 q + 14756 q^{9} + 48024 q^{19} + 4724 q^{21} + 470104 q^{31} + 2849664 q^{39} + 19816920 q^{49} + 5026040 q^{51} + 18849944 q^{61} + 38669180 q^{69} + 90778632 q^{79} + 16242056 q^{81} + 22375296 q^{91} - 16968560 q^{99}+O(q^{100}) \) 16 * q + 14756 * q^9 + 48024 * q^19 + 4724 * q^21 + 470104 * q^31 + 2849664 * q^39 + 19816920 * q^49 + 5026040 * q^51 + 18849944 * q^61 + 38669180 * q^69 + 90778632 * q^79 + 16242056 * q^81 + 22375296 * q^91 - 16968560 * 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	300.9.g.h

Level	$300$
Weight	$9$
Character orbit	300.g
Analytic conductor	$122.214$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(122.213583018\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 7378 x^{14} + 23156928 x^{12} - 101588726286 x^{10} + \cdots + 34\!\cdots\!81 \) x^16 - 7378x^14 + 23156928x^12 - 101588726286x^10 + 1484695155668670x^8 - 4373061557178808206x^6 + 42910255095788484704448x^4 - 588516817021165377771877458*x^2 + 3433683820292512484657849089281
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{38}\cdot 3^{20}\cdot 5^{18} \)
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 922 \) v^2 - 922
\(\beta_{3}\)	\(=\)	\( ( - 118241503 \nu^{14} + 32210391788401 \nu^{12} + \cdots + 14\!\cdots\!93 ) / 20\!\cdots\!00 \) (-118241503v^14 + 32210391788401v^12 + 126457566128179947v^10 + 214013909643019174755v^8 + 4746195418588577798202495v^6 + 3485352495014175965316858063v^4 + 97439320345474574721245443129749*v^2 + 1423431157324590957071749337819988093) / 205650014751525015049243888872000
\(\beta_{4}\)	\(=\)	\( ( 438411229 \nu^{14} + 545360564057 \nu^{12} + \cdots - 23\!\cdots\!99 ) / 20\!\cdots\!00 \) (438411229v^14 + 545360564057v^12 + 5205191446634679v^10 + 134747004904914583035v^8 - 140050706881610814672285v^6 + 7126746748103003389938313191v^4 + 15988522642231144261362784701993*v^2 - 2374313718998962356072521984335899) / 20565001475152501504924388887200
\(\beta_{5}\)	\(=\)	\( ( - 5968589155 \nu^{14} - 101803994977442 \nu^{12} + \cdots - 77\!\cdots\!42 ) / 10\!\cdots\!00 \) (-5968589155v^14 - 101803994977442v^12 - 1605609642957550713v^10 + 284564718899432445330v^8 + 86691769095675942519262875v^6 + 257905289910296311314646868010v^4 - 1473149156225017240571482096707663*v^2 - 7767748177539020226873719607705391242) / 102825007375762507524621944436000
\(\beta_{6}\)	\(=\)	\( ( \nu^{15} - 7378 \nu^{13} + 23156928 \nu^{11} - 101588726286 \nu^{9} + \cdots - 51\!\cdots\!66 \nu ) / 19\!\cdots\!23 \) (v^15 - 7378v^13 + 23156928v^11 - 101588726286v^9 + 1484695155668670v^7 - 4373061557178808206v^5 + 42910255095788484704448v^3 - 510983834350445298184690566*v) / 19383245667680019896796723
\(\beta_{7}\)	\(=\)	\( ( - 9990143 \nu^{15} - 13366675591 \nu^{13} - 104908177452681 \nu^{11} + \cdots + 22\!\cdots\!41 \nu ) / 14\!\cdots\!00 \) (-9990143v^15 - 13366675591v^13 - 104908177452681v^11 + 1454707853876694735v^9 + 893990708450313260895v^7 + 15889852799914567568262183v^5 - 488135239609492118517858798039v^3 + 2286102946760216024014189243114641v) / 148630727779770392568637271964000
\(\beta_{8}\)	\(=\)	\( ( - 56323 \nu^{14} + 84994792 \nu^{12} + 852147203931 \nu^{10} + 150878465402940 \nu^{8} + \cdots + 18\!\cdots\!24 ) / 31\!\cdots\!00 \) (-56323v^14 + 84994792v^12 + 852147203931v^10 + 150878465402940v^8 - 44424226549043107605v^6 + 307048508107766448925968v^4 - 2092818668997351782154051027*v^2 + 18959612788879968454203241251324) / 319065772307490039453444000
\(\beta_{9}\)	\(=\)	\( ( 6522599171 \nu^{14} - 51903695295257 \nu^{12} + \cdots - 35\!\cdots\!01 ) / 20\!\cdots\!00 \) (6522599171v^14 - 51903695295257v^12 + 155990425193416521v^10 - 841907185104513957435v^8 + 10475029126320841638840285v^6 - 37567673727464881971509655591v^4 + 282710209346205493966540669557207*v^2 - 3582208268407457812459773082999060101) / 20565001475152501504924388887200
\(\beta_{10}\)	\(=\)	\( ( 27940379809 \nu^{14} - 7840932369085 \nu^{12} + \cdots - 10\!\cdots\!81 ) / 68\!\cdots\!00 \) (27940379809v^14 - 7840932369085v^12 + 106687622468324331v^10 - 7006703647655295554535v^8 + 8022390700695158204032815v^6 + 96875049710885773953791899341v^4 + 1305055640077155944800803129395205*v^2 - 10303770409953809798553362085042235881) / 68550004917175005016414629624000
\(\beta_{11}\)	\(=\)	\( ( 20596949 \nu^{15} - 37487115353 \nu^{13} - 26193217876281 \nu^{11} + \cdots - 12\!\cdots\!89 \nu ) / 74\!\cdots\!00 \) (20596949v^15 - 37487115353v^13 - 26193217876281v^11 - 9615576560197871115v^9 + 33111695571416626953915v^7 + 334926758600430895264046121v^5 + 1185049326849210503136732085353v^3 - 12370583909607752899704971964400389v) / 74524702943096140499204040590400
\(\beta_{12}\)	\(=\)	\( ( 515192467 \nu^{15} + 615081057479 \nu^{13} + \cdots - 14\!\cdots\!29 \nu ) / 16\!\cdots\!00 \) (515192467v^15 + 615081057479v^13 - 23769410150578311v^11 - 13979069008262681715v^9 + 1455452294036789810563245v^7 - 718501964677245201926306727v^5 - 39986773812708795760737886002009v^3 - 146868508805231148372599489199012429v) / 1684481581504064449111222415592000
\(\beta_{13}\)	\(=\)	\( ( 439511153 \nu^{15} - 3923184692141 \nu^{13} + \cdots - 36\!\cdots\!33 \nu ) / 37\!\cdots\!00 \) (439511153v^15 - 3923184692141v^13 - 25648690419064557v^11 - 96300953039951192655v^9 + 51322315211094545556255v^7 - 1803169386614614640336367363v^5 + 21053398589594473713522842730141v^3 - 361636306388836808583648784537192833v) / 372623514715480702496020202952000
\(\beta_{14}\)	\(=\)	\( ( - 315617774114 \nu^{15} + \cdots + 59\!\cdots\!01 \nu ) / 22\!\cdots\!00 \) (-315617774114v^15 + 2906961097996085v^13 + 5341402867084775874v^11 - 91133985746362493381265v^9 + 1173175902795009552147044010v^7 - 7577842817073919295320004129661v^5 + 59132031186145636229505428318139270v^3 + 59491772755992574410359124796072221801v) / 224878291130792603956348192481532000
\(\beta_{15}\)	\(=\)	\( ( 360231947 \nu^{15} + 1695521995441 \nu^{13} + \cdots - 16\!\cdots\!67 \nu ) / 12\!\cdots\!00 \) (360231947v^15 + 1695521995441v^13 - 3519056907251343v^11 + 116929110967281142155v^9 + 396517034323738594366245v^7 + 3446460383777940257068507263v^5 + 47389009058756312349490136702559v^3 - 16852169989705515576706347509262267v) / 124207838238493567498673400984000

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} + 922 \) b2 + 922
\(\nu^{3}\)	\(=\)	\( \beta_{15} + \beta_{14} + 2\beta_{13} - 11\beta_{12} + 2\beta_{11} + 14\beta_{7} - 2\beta_{6} + 917\beta_1 \) b15 + b14 + 2b13 - 11b12 + 2b11 + 14b7 - 2b6 + 917b1
\(\nu^{4}\)	\(=\)	\( 81\beta_{10} - 6\beta_{9} + 306\beta_{8} + 63\beta_{5} + 1236\beta_{4} - 369\beta_{3} + 664\beta_{2} + 1015000 \) 81b10 - 6b9 + 306b8 + 63b5 + 1236b4 - 369b3 + 664*b2 + 1015000
\(\nu^{5}\)	\(=\)	\( 4624 \beta_{15} - 6959 \beta_{14} - 7519 \beta_{13} - 22595 \beta_{12} + 113252 \beta_{11} + \cdots + 1051022 \beta_1 \) 4624b15 - 6959b14 - 7519b13 - 22595b12 + 113252b11 + 465362b7 - 143816b6 + 1051022b1
\(\nu^{6}\)	\(=\)	\( - 164916 \beta_{10} + 446349 \beta_{9} - 37224 \beta_{8} + 609156 \beta_{5} - 1832559 \beta_{4} + \cdots + 24228485347 \) -164916b10 + 446349b9 - 37224b8 + 609156b5 - 1832559b4 + 11296188b3 + 1559200*b2 + 24228485347
\(\nu^{7}\)	\(=\)	\( 14029492 \beta_{15} + 41799856 \beta_{14} - 69784480 \beta_{13} + 438941944 \beta_{12} + \cdots + 23670718700 \beta_1 \) 14029492b15 + 41799856b14 - 69784480b13 + 438941944b12 + 253173212b11 + 3868482764b7 + 3025466029b6 + 23670718700b1
\(\nu^{8}\)	\(=\)	\( - 6583202100 \beta_{10} + 674791752 \beta_{9} - 7360658640 \beta_{8} - 455274180 \beta_{5} + \cdots - 493406497298627 \) -6583202100b10 + 674791752b9 - 7360658640b8 - 455274180b5 + 53815623996b4 + 6595207740b3 + 16287378400*b2 - 493406497298627
\(\nu^{9}\)	\(=\)	\( 372515274100 \beta_{15} - 191909050640 \beta_{14} - 181942319680 \beta_{13} + \cdots - 497682836598727 \beta_1 \) 372515274100b15 - 191909050640b14 - 181942319680b13 + 341871542440b12 - 3560043027652b11 + 14206187118740b7 + 13550675701912b6 - 497682836598727b1
\(\nu^{10}\)	\(=\)	\( - 4284421754940 \beta_{10} + 37589920269960 \beta_{9} + 76428779934600 \beta_{8} + \cdots - 27\!\cdots\!54 \) -4284421754940b10 + 37589920269960b9 + 76428779934600b8 - 17864334358740b5 + 116051575465164b4 + 340033287412260b3 - 441150705624935*b2 - 2734248649531353554
\(\nu^{11}\)	\(=\)	\( - 718952827444355 \beta_{15} - 526320068789135 \beta_{14} + \cdots - 27\!\cdots\!03 \beta_1 \) -718952827444355b15 - 526320068789135b14 - 8400999370274950b13 - 10145890256279315b12 + 1525742933722718b11 - 60538389884256790b7 + 191269996924912870b6 - 2769509565289291003b1
\(\nu^{12}\)	\(=\)	\( 78\!\cdots\!25 \beta_{10} + \cdots - 35\!\cdots\!08 \) 78326388969301125b10 - 217705289330581350b9 - 290780653654009710b8 - 25054327119139245b5 - 643445293264229976b4 + 3378812378443150275b3 - 1865253820476150464*b2 - 35943471852211625378108
\(\nu^{13}\)	\(=\)	\( 12\!\cdots\!16 \beta_{15} + \cdots - 35\!\cdots\!62 \beta_1 \) 1212546650043993916b15 + 7354618417090964641b14 - 24686342839711451863b13 + 80995809963092854789b12 - 7561995861217751200b11 - 1182534817833708789706b7 - 1283996327823107379512b6 - 35593092956130168296362b1
\(\nu^{14}\)	\(=\)	\( 60\!\cdots\!76 \beta_{10} + \cdots + 19\!\cdots\!35 \) 607395000117700370376b10 - 142763328134049426531b9 - 3269572223624975677104b8 - 446326529173083519912b5 + 9112531606003942407525b4 - 660305187878822012664b3 - 44213012442513461020976*b2 + 191856289957555519437810535
\(\nu^{15}\)	\(=\)	\( 19\!\cdots\!44 \beta_{15} + \cdots + 19\!\cdots\!08 \beta_1 \) 19919522998163068870744b15 - 88447899581114690345984b14 - 21170669082847405668304b13 + 588773532910800051195280b12 - 419231745548751852435400b11 - 10188874813885804821015688b7 + 1822300630910515632596029b6 + 192056041250389533913791308b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 101.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} + \cdots + 34\!\cdots\!81 \) T^16 - 7378T^14 + 23156928T^12 - 101588726286T^10 + 1484695155668670T^8 - 4373061557178808206T^6 + 42910255095788484704448T^4 - 588516817021165377771877458*T^2 + 3433683820292512484657849089281
$5$	\( T^{16} \) T^16
$7$	\( (T^{8} + \cdots + 26\!\cdots\!56)^{2} \) (T^8 - 28013434T^6 + 194864255690496T^4 - 52166784840953810944*T^2 + 2600757082907060397408256)^2
$11$	\( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) (T^8 + 971194160T^6 + 265474129795728000T^4 + 28030749534089025145088000*T^2 + 1006466192971333730193034854400000)^2
$13$	\( (T^{8} + \cdots + 36\!\cdots\!36)^{2} \) (T^8 - 3173472864T^6 + 2136237732098603136T^4 - 339475890779809216137437184*T^2 + 3607723320043650231222013878337536)^2
$17$	\( (T^{8} + \cdots + 17\!\cdots\!00)^{2} \) (T^8 + 36442812140T^6 + 415835666767610862000T^4 + 1533374533895190454851260072000*T^2 + 1770350416439299246742626840450278400000)^2
$19$	\( (T^{4} + \cdots + 14\!\cdots\!16)^{4} \) (T^4 - 12006T^3 - 14875364124T^2 + 306871114764024*T + 14325224454022538016)^4
$23$	\( (T^{8} + \cdots + 58\!\cdots\!00)^{2} \) (T^8 + 416302278710T^6 + 57314419519886499633000T^4 + 3147172470184548438496776936308000*T^2 + 58287367348221269808659040386930849472400000)^2
$29$	\( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) (T^8 + 2076533641440T^6 + 1191051736937646512976000T^4 + 128658594921141080817673750444032000*T^2 + 3104403840932459875082278936977155294822400000)^2
$31$	\( (T^{4} + \cdots + 11\!\cdots\!96)^{4} \) (T^4 - 117526T^3 - 2411512956084T^2 + 42008109391780664*T + 1161798629701340519716096)^4
$37$	\( (T^{8} + \cdots + 49\!\cdots\!56)^{2} \) (T^8 - 17166889014016T^6 + 79594418412384606721389696T^4 - 49806165702604945505529651000596525056*T^2 + 4942690493000192267969030626358676764032761856)^2
$41$	\( (T^{8} + \cdots + 90\!\cdots\!00)^{2} \) (T^8 + 47937327156740T^6 + 727271268674755116724800000T^4 + 4366076370036139476410695726377915392000*T^2 + 9019374280313225828659727762832863084886419046400000)^2
$43$	\( (T^{8} + \cdots + 11\!\cdots\!76)^{2} \) (T^8 - 49663114389754T^6 + 682901573617410215550536256T^4 - 1836202088927117599440036340104346549504*T^2 + 1185398126206724940592428794668145932263947368677376)^2
$47$	\( (T^{8} + \cdots + 22\!\cdots\!00)^{2} \) (T^8 + 67977556756310T^6 + 1049302047808834875960513000T^4 + 4623472992615897055567317425438384948000*T^2 + 2280191789489125701271626231505786142624027808400000)^2
$53$	\( (T^{8} + \cdots + 49\!\cdots\!00)^{2} \) (T^8 + 37034825790060T^6 + 442771852668467171313582000T^4 + 1728338444134576563976678985411908008000*T^2 + 495465407935730608676645867135344783121059430400000)^2
$59$	\( (T^{8} + \cdots + 76\!\cdots\!00)^{2} \) (T^8 + 867038659162160T^6 + 248923444784011995398556048000T^4 + 26049124119245301801178871966454591097088000*T^2 + 767557073508162727336611351762522586065060697738854400000)^2
$61$	\( (T^{4} + \cdots - 21\!\cdots\!64)^{4} \) (T^4 - 4712486T^3 - 246246167429364T^2 - 1401894577944767419016*T - 2105396555318302820718155264)^4
$67$	\( (T^{8} + \cdots + 32\!\cdots\!76)^{2} \) (T^8 - 433265383552954T^6 + 54035838660045196541245678656T^4 - 1509066941286824095659713998642889047784704*T^2 + 320783174079286377881741203034787315142121785063653376)^2
$71$	\( (T^{8} + \cdots + 21\!\cdots\!00)^{2} \) (T^8 + 2138475782597760T^6 + 1326666887128916996598256896000T^4 + 190090844824428013879437967383828024066048000*T^2 + 2137645709252142846293718231008089890125542037284454400000)^2
$73$	\( (T^{8} + \cdots + 89\!\cdots\!16)^{2} \) (T^8 - 5313866871866656T^6 + 7525114331008359170085644875776T^4 - 1580040512870961011562303832920781921946238976*T^2 + 89032404109235745594165545897265554898428656660336624009216)^2
$79$	\( (T^{4} + \cdots + 62\!\cdots\!16)^{4} \) (T^4 - 22694658T^3 - 2904371819036676T^2 + 29252521987078209542568*T + 620309238940199098245480618816)^4
$83$	\( (T^{8} + \cdots + 74\!\cdots\!00)^{2} \) (T^8 + 6035256522659990T^6 + 11440242471520537322508024801000T^4 + 7416555771197894088706113351807470111804852000*T^2 + 747817278848502680030790368742080069342213955272235744400000)^2
$89$	\( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) (T^8 + 15617313982451840T^6 + 83066730379617691583491110144000T^4 + 175036048359888268460930166678998273561526272000*T^2 + 126715534153661912498966905333746869248072534426556720742400000)^2
$97$	\( (T^{8} + \cdots + 93\!\cdots\!76)^{2} \) (T^8 - 27061263847686304T^6 + 228077510490380569982248266694656T^4 - 778135119634313173119081490922977317629307387904*T^2 + 938517188523791996880618888160872429464721814239202719538610176)^2
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\(n\)	\(101\)	\(151\)	\(277\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)