LMFDB - Newform orbit 242.3.d.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [242,3,Mod(161,242)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(242, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([7]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("242.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 242 = 2 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	242.d (of order $10$, degree $4$, 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:	$6.59402239752$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
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} - 8 x^{15} - 168 x^{14} + 1316 x^{13} + 12572 x^{12} - 92904 x^{11} - 552362 x^{10} + \cdots + 93516487856 $ x^16 - 8x^15 - 168x^14 + 1316x^13 + 12572x^12 - 92904x^11 - 552362x^10 + 3645748x^9 + 15734655x^8 - 85890536x^7 - 299926366x^6 + 1216788336x^5 + 3747221628x^4 - 9630461344x^3 - 27860664440x^2 + 32894183872*x + 93516487856
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-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_{15} + \beta_{3} - \beta_{2} + \cdots + 1) q^{3}+ \cdots + (\beta_{15} + \beta_{14} + \cdots + 16 \beta_{3}) q^{9}+O(q^{10}) $ q - b6 * q^2 + (b15 + b3 - b2 + b1 + 1) * q^3 - 2b1 q^4 + (-b14 - b2) * q^5 + (b13 + b10) * q^6 + (3b13 + 3b12 - b11 - 3b8 + 3b6) * q^7 - 2b8 q^8 + (b15 + b14 + b9 + b5 + 16b3) q^9	$ q - \beta_{6} q^{2} + (\beta_{15} + \beta_{3} - \beta_{2} + \cdots + 1) q^{3}+ \cdots + (17 \beta_{12} - 14 \beta_{11} + \cdots - 14 \beta_{4}) q^{98}+O(q^{100}) $ q - b6 * q^2 + (b15 + b3 - b2 + b1 + 1) * q^3 - 2b1 q^4 + (-b14 - b2) * q^5 + (b13 + b10) * q^6 + (3b13 + 3b12 - b11 - 3b8 + 3b6) * q^7 - 2b8 q^8 + (b15 + b14 + b9 + b5 + 16b3) q^9 + (-b12 - b11 + b10 - b7 - b6 - b4) * q^10 + (-2b5 + 2) q^12 + (-5b6 - b4) q^13 + (-2b15 + 6b3 - 6b2 + 6b1 + 6) * q^14 + (b9 - 25b1) q^15 - 4b2 q^16 + (-11b13 - 2b10) * q^17 + (16b13 + 16b12 + b11 - 16b8 + 16b6) * q^18 + 4b8 q^19 + (-2b15 - 2b14 - 2b9 - 2b5 - 2b3) q^20 + (-21b12 + 3b11 - 3b10 + 3b7 + 3b6 + 3b4) * q^21 + (5b5 + 7) q^23 + 2b4 q^24 - b15 * q^25 + (-2b9 - 8b1) * q^26 + (7b14 + 31b2) * q^27 + (6b13 - 2b10) * q^28 + (8b13 + 8b12 + 5b11 - 8b8 + 8b6) q^29 + (-25b8 - b7) q^30 + (-3b15 - 3b14 - 3b9 - 3b5 + 33b3) q^31 - 4b12 q^32 + (4b5 - 22) q^34 + (24b6 + 3b4) * q^35 + (2b15 + 32b3 - 32b2 + 32b1 + 32) * q^36 + (-3b9 + 9b1) * q^37 + 8b2 q^38 + (-20b13 + 4b10) * q^39 + (-2b13 - 2b12 - 2b11 + 2b8 - 2b6) q^40 + (12b8 + 3b7) * q^41 + (6b15 + 6b14 + 6b9 + 6b5 - 42b3) q^42 + (-15b12 - 7b11 + 7b10 - 7b7 - 7b6 - 7b4) * q^43 + (-16b5 + 40) q^45 + (-12b6 - 5b4) * q^46 + (-4b15 - 16b3 + 16b2 - 16b1 - 16) * q^47 + (4b9 - 4b1) * q^48 + (-14b14 + 17b2) * q^49 - b10 * q^50 + (-59b13 - 59b12 - 11b11 + 59b8 - 59b6) q^51 + (-8b8 + 2b7) * q^52 + (2b15 + 2b14 + 2b9 + 2b5 - 16b3) q^53 + (31b12 + 7b11 - 7b10 + 7b7 + 7b6 + 7b4) * q^54 + (4b5 + 12) q^56 - 4b4 q^57 + (10b15 + 16b3 - 16b2 + 16b1 + 16) * q^58 + (7b9 + 41b1) * q^59 + (-2b14 - 50b2) * q^60 + (48b13 + 5b10) * q^61 + (33b13 + 33b12 - 3b11 - 33b8 + 33b6) q^62 + (24b8 - 12b7) * q^63 - 8b3 q^64 + (20b12 - 4b11 + 4b10 - 4b7 - 4b6 - 4b4) * q^65 + (7b5 - 79) q^67 + (18b6 - 4b4) * q^68 + (7b15 - 113b3 + 113b2 - 113b1 - 113) * q^69 + (6b9 + 42b1) * q^70 + (-3b14 - 87b2) * q^71 + (32b13 + 2b10) * q^72 + (-20b13 - 20b12 + 11b11 + 20b8 - 20b6) q^73 + (9b8 + 3b7) * q^74 - 24b3 q^75 + 8b12 q^76 + (-8b5 - 40) q^78 + (-40b6 + 9b4) * q^79 + (-4b15 - 4b3 + 4b2 - 4b1 - 4) * q^80 + (-22b9 + 55b1) * q^81 + (6b14 + 24b2) * q^82 + (-41b13 - 5b10) * q^83 + (-42b13 - 42b12 + 6b11 + 42b8 - 42b6) q^84 + (59b8 + 11b7) * q^85 + (-14b15 - 14b14 - 14b9 - 14b5 - 30b3) q^86 + (128b12 + 8b11 - 8b10 + 8b7 + 8b6 + 8b4) * q^87 + (11b5 + 79) q^89 + (-24b6 + 16b4) * q^90 + (-16b15 + 72b3 - 72b2 + 72b1 + 72) * q^91 + (-10b9 - 14b1) * q^92 + (33b14 - 39b2) * q^93 + (-16b13 - 4b10) * q^94 + (4b13 + 4b12 + 4b11 - 4b8 + 4b6) q^95 + (-4b8 - 4b7) * q^96 + (-b15 - b14 - b9 - b5 - 57b3) q^97 + (17b12 - 14b11 + 14b10 - 14b7 - 14b6 - 14b4) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{3} + 8 q^{4} - 2 q^{5} - 62 q^{9}+O(q^{10}) $ 16 * q + 2 * q^3 + 8 * q^4 - 2 * q^5 - 62 * q^9	$ 16 q + 2 q^{3} + 8 q^{4} - 2 q^{5} - 62 q^{9} + 16 q^{12} + 28 q^{14} + 98 q^{15} - 16 q^{16} + 4 q^{20} + 152 q^{23} + 2 q^{25} + 36 q^{26} + 110 q^{27} - 138 q^{31} - 320 q^{34} + 124 q^{36} - 30 q^{37} + 32 q^{38} + 180 q^{42} + 512 q^{45} - 56 q^{47} + 8 q^{48} + 96 q^{49} + 68 q^{53} + 224 q^{56} + 44 q^{58} - 178 q^{59} - 196 q^{60} + 32 q^{64} - 1208 q^{67} - 466 q^{69} - 180 q^{70} - 342 q^{71} + 96 q^{75} - 704 q^{78} - 8 q^{80} - 176 q^{81} + 84 q^{82} + 92 q^{86} + 1352 q^{89} + 320 q^{91} + 76 q^{92} - 222 q^{93} + 226 q^{97}+O(q^{100}) $ 16 * q + 2 * q^3 + 8 * q^4 - 2 * q^5 - 62 * q^9 + 16 * q^12 + 28 * q^14 + 98 * q^15 - 16 * q^16 + 4 * q^20 + 152 * q^23 + 2 * q^25 + 36 * q^26 + 110 * q^27 - 138 * q^31 - 320 * q^34 + 124 * q^36 - 30 * q^37 + 32 * q^38 + 180 * q^42 + 512 * q^45 - 56 * q^47 + 8 * q^48 + 96 * q^49 + 68 * q^53 + 224 * q^56 + 44 * q^58 - 178 * q^59 - 196 * q^60 + 32 * q^64 - 1208 * q^67 - 466 * q^69 - 180 * q^70 - 342 * q^71 + 96 * q^75 - 704 * q^78 - 8 * q^80 - 176 * q^81 + 84 * q^82 + 92 * q^86 + 1352 * q^89 + 320 * q^91 + 76 * q^92 - 222 * q^93 + 226 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8 x^{15} - 168 x^{14} + 1316 x^{13} + 12572 x^{12} - 92904 x^{11} - 552362 x^{10} + \cdots + 93516487856 $ x^16 - 8*x^15 - 168*x^14 + 1316*x^13 + 12572*x^12 - 92904*x^11 - 552362*x^10 + 3645748*x^9 + 15734655*x^8 - 85890536*x^7 - 299926366*x^6 + 1216788336*x^5 + 3747221628*x^4 - 9630461344*x^3 - 27860664440*x^2 + 32894183872*x + 93516487856 :

$\beta_{1}$	$=$	$ ( 175481049060 \nu^{14} - 1228367343420 \nu^{13} - 31870999820388 \nu^{12} + \cdots - 14\!\cdots\!28 ) / 20\!\cdots\!10 $ (175481049060v^14 - 1228367343420v^13 - 31870999820388v^12 + 207194774386788v^11 + 2631850994562082v^10 - 15087816493040810v^9 - 131056658453814037v^8 + 617082862871786836v^7 + 4416819708280903274v^6 - 15475949813642022536v^5 - 110255369457878765908v^4 + 247055260430715637072v^3 + 1834440698346756083667v^2 - 1960655203724039601680v - 14013092710472544603728) / 2024499066530216061610
$\beta_{2}$	$=$	$ ( 6274938934164 \nu^{14} - 43924572539148 \nu^{13} + \cdots - 13\!\cdots\!12 ) / 40\!\cdots\!20 $ (6274938934164v^14 - 43924572539148v^13 - 1124662048413064v^12 + 7318991733487308v^11 + 90881141253755004v^10 - 522543332804591704v^9 - 4344570105413003537v^8 + 20595843960083095664v^7 + 132128113110157805602v^6 - 470746868461773826624v^5 - 2530625459022392744857v^4 + 5870943809587053857504v^3 + 27985028388913635500552v^2 - 31002553752469851316864v - 136406554439207720269712) / 4048998133060432123220
$\beta_{3}$	$=$	$ ( 82341644084550 \nu^{14} - 576391508591850 \nu^{13} + \cdots - 68\!\cdots\!36 ) / 80\!\cdots\!40 $ (82341644084550v^14 - 576391508591850v^13 - 13898454439769201v^12 + 90883816250309256v^11 + 1022862360340382009v^10 - 5961150781617850650v^9 - 42899171826069798219v^8 + 208386863682947879892v^7 + 1113793811722406491393v^6 - 4096796740514804018712v^5 - 17956541117416792085676v^4 + 42996629461023661954704v^3 + 166644507560391636695864v^2 - 188862218869953655683360v - 682347068259704726350736) / 8097996266120864246440
$\beta_{4}$	$=$	$ ( - 16\!\cdots\!52 \nu^{15} + \cdots - 13\!\cdots\!76 ) / 68\!\cdots\!25 $ (-16615617072028252v^15 + 183921387272661540v^14 + 2468254547109929556v^13 - 31402908860033890424v^12 - 154571000268605157786v^11 + 2355812411565152162020v^10 + 5256776379557871070709v^9 - 102238889885446508476384v^8 - 104015750968520409017962v^7 + 2867791643265798930620576v^6 + 1187221539499037715904980v^5 - 56885156025650703673639552v^4 - 7212034766984225220678952v^3 + 1118456121034190939097649792v^2 - 324642656293691428327429949*v - 13312760194484715634576269176) / 684184520781552843561253525
$\beta_{5}$	$=$	$ ( - 33\!\cdots\!04 \nu^{15} + \cdots - 73\!\cdots\!12 ) / 68\!\cdots\!25 $ (-33231234144056504v^15 + 249234256080423780v^14 + 5766768723474154212v^13 - 41264049586468429708v^12 - 449185984519112258712v^11 + 2932743576650820289830v^10 + 20711483365844490824468v^9 - 115895929038667840274283v^8 - 625120894366395899422504v^7 + 2750232761605993933828182v^6 + 12834714937787786674005040v^5 - 39248156557873860076232364v^4 - 181409955335391303116508064v^3 + 312820081070764863903102424v^2 + 2044300384448819817441167552*v - 733572461740217763299679912) / 684184520781552843561253525
$\beta_{6}$	$=$	$ ( 33\!\cdots\!04 \nu^{15} + \cdots + 73\!\cdots\!12 ) / 68\!\cdots\!25 $ (33231234144056504v^15 - 249234256080423780v^14 - 5766768723474154212v^13 + 41264049586468429708v^12 + 449185984519112258712v^11 - 2932743576650820289830v^10 - 20711483365844490824468v^9 + 115895929038667840274283v^8 + 625120894366395899422504v^7 - 2750232761605993933828182v^6 - 12834714937787786674005040v^5 + 39248156557873860076232364v^4 + 181409955335391303116508064v^3 - 312820081070764863903102424v^2 - 1360115863667266973879914027*v + 733572461740217763299679912) / 684184520781552843561253525
$\beta_{7}$	$=$	$ ( - 31\!\cdots\!36 \nu^{15} + \cdots + 15\!\cdots\!92 ) / 13\!\cdots\!50 $ (-313972325148010636v^15 - 1581779784237757860v^14 + 81633097592829830538v^13 + 325213602498538774698v^12 - 8797538009894643189483v^11 - 31250637227492478835175v^10 + 524111801792212893980947v^9 + 1823036143284007762340943v^8 - 19091612717031207407776646v^7 - 70339401408842906595378942v^6 + 436858537055216701879153580v^5 + 1741033065137244318794400444v^4 - 5860604094458178946083145376v^3 - 24679900878354383074895401464v^2 + 35030122381182193052153650768*v + 150921050441880734257005362192) / 1368369041563105687122507050
$\beta_{8}$	$=$	$ ( 62\!\cdots\!72 \nu^{15} + \cdots + 65\!\cdots\!36 ) / 13\!\cdots\!50 $ (627944650296021272v^15 - 4709584877220159540v^14 - 108154184065789934256v^13 + 774430900398806992354v^12 + 8329471242855672679806v^11 - 54487974888829655978145v^10 - 377029143478437698987909v^9 + 2118405802494493622062629v^8 + 10993972945318710980916202v^7 - 48763723787066583086481066v^6 - 207303807895304789369698420v^5 + 645397351206236167382624532v^4 + 2322477110519433562204640957v^3 - 4154992862008189708470805312v^2 - 11730274923644883054399165776*v + 6580385501157004753362636336) / 1368369041563105687122507050
$\beta_{9}$	$=$	$ ( 74\!\cdots\!72 \nu^{15} + \cdots + 65\!\cdots\!36 ) / 13\!\cdots\!50 $ (746553168760920572v^15 - 5539844506474454640v^14 - 129695952199389285396v^13 + 914474884380708935494v^12 + 10108352489335156714016v^11 - 64685905495558404661195v^10 - 465610994210662875666394v^9 + 2535495194923848703449209v^8 + 13979323470244314908329172v^7 - 59223995645856294328676146v^6 - 281825963388732336640745160v^5 + 812383237007659020057774692v^4 + 3562384750735497732935584592v^3 - 5480209517481286695444325712v^2 - 21201794352116828314781939616*v + 6580385501157004753362636336) / 1368369041563105687122507050
$\beta_{10}$	$=$	$ ( - 78\!\cdots\!71 \nu^{15} + \cdots + 14\!\cdots\!52 ) / 54\!\cdots\!00 $ (-7877385638507433071v^15 - 323504510925109659975v^14 + 3839742854304101781993v^13 + 49324501180325214773093v^12 - 455815083370312818131943v^11 - 3301029023306060428982295v^10 + 25285851856365026783804887v^9 + 126529663424803543043946883v^8 - 770922393178149134876041046v^7 - 3017120906068341136532984102v^6 + 13363839018944971584067702380v^5 + 44979692859829587002682704044v^4 - 124519777256397056067811889336v^3 - 388925722184585493242214074104v^2 + 488491121302424552442409872208*v + 1462383924468513496002828515152) / 5473476166252422748490028200
$\beta_{11}$	$=$	$ ( - 13\!\cdots\!39 \nu^{15} + \cdots + 32\!\cdots\!08 ) / 54\!\cdots\!00 $ (-13804244221026176339v^15 - 439170394871573241395v^14 + 5946090455219733079927v^13 + 70277759635872453346287v^12 - 710581628067113812331007v^11 - 5005046260338933465285495v^10 + 40882135951621374910582153v^9 + 206304338938276163406035377v^8 - 1313210675210165670486862394v^7 - 5313963743790043606791335618v^6 + 24211514521875778207800761700v^5 + 85302516650732368314483624036v^4 - 240775308794124227754358812264v^3 - 788528552636895397712500639016v^2 + 1008211299129257969269681030192*v + 3200286178840196584967087887408) / 5473476166252422748490028200
$\beta_{12}$	$=$	$ ( 12474157204 \nu^{15} - 93556179030 \nu^{14} - 2085744967912 \nu^{13} + \cdots + 64\!\cdots\!12 ) / 48\!\cdots\!00 $ (12474157204v^15 - 93556179030v^14 - 2085744967912v^13 + 14976277673383v^12 + 153688852361912v^11 - 1013149400238030v^10 - 6491315379684643v^9 + 37063337558777583v^8 + 170456576374403654v^7 - 776991148135376382v^6 - 2793249100137010815v^5 + 9017401691911159764v^4 + 26508223093865149464v^3 - 49194537558881804824v^2 - 111668512881353080752*v + 64353741977637329712) / 4842462191461310900
$\beta_{13}$	$=$	$ ( 15\!\cdots\!42 \nu^{15} + \cdots + 29\!\cdots\!16 ) / 54\!\cdots\!00 $ (15754771277014866142v^15 - 118160784577611496065v^14 - 2323297063613387851881v^13 + 16893536146247462060879v^12 + 148744483314305682364131v^11 - 1007866187349476305924425v^10 - 5401307948808425062945109v^9 + 32152074017269367423659329v^8 + 121366227222934805571604127v^7 - 582263356362062198028591066v^6 - 1708150390207906993413515870v^5 + 5806382907274542436119217332v^4 + 14056926813593698682144615132v^3 - 27205938510456128026053525512v^2 - 50012673693275977231565316376*v + 29749221945006701710005389616) / 5473476166252422748490028200
$\beta_{14}$	$=$	$ ( - 11\!\cdots\!16 \nu^{15} + \cdots - 36\!\cdots\!88 ) / 27\!\cdots\!00 $ (-11291085546830678216v^15 + 82562510301082027410v^14 + 1938932332778467481008v^13 - 13410850068938975568107v^12 - 148284925715387528542708v^11 + 925775339775611821581590v^10 + 6605111125527000427174092v^9 - 34867336615304403053176087v^8 - 185640310699708451494645656v^7 + 757300010931751627785841838v^6 + 3289078998338388512468773520v^5 - 9064423156742318508698678356v^4 - 33896455528146485855600272096v^3 + 48757283023055274539397259096v^2 + 155307806712640620625132604608*v - 36369790717711399867750746288) / 2736738083126211374245014100
$\beta_{15}$	$=$	$ ( - 63\!\cdots\!52 \nu^{15} + \cdots - 29\!\cdots\!16 ) / 54\!\cdots\!00 $ (-63401809095249994252v^15 + 451690049305257392835v^14 + 10283169580594723012506v^13 - 68988651689594869682639v^12 - 724363756407850968336876v^11 + 4371450226107229532860635v^10 + 28878366770638680423520274v^9 - 146828486860929744958056069v^8 - 707514331051427957655402052v^7 + 2756789520439569176696316306v^6 + 10722230134388153300741398440v^5 - 27599562735095556805201542852v^4 - 93821972030647122959985147472v^3 + 118250160989518271909959124072v^2 + 359231326056521670250901220896*v - 29749221945006701710005389616) / 5473476166252422748490028200

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

$\nu$	$=$	$ \beta_{6} + \beta_{5} $ b6 + b5
$\nu^{2}$	$=$	$ 2\beta_{6} + \beta_{5} + 2\beta_{4} - 2\beta _1 + 24 $ 2b6 + b5 + 2b4 - 2*b1 + 24
$\nu^{3}$	$=$	$ -6\beta_{9} + 2\beta_{8} + 75\beta_{6} + 25\beta_{5} + 3\beta_{4} + 24 $ -6b9 + 2b8 + 75b6 + 25b5 + 3*b4 + 24
$\nu^{4}$	$=$	$ -12\beta_{9} - 8\beta_{7} + 196\beta_{6} + 49\beta_{5} + 100\beta_{4} - 4\beta_{2} - 288\beta _1 + 600 $ -12b9 - 8b7 + 196b6 + 49b5 + 100b4 - 4b2 - 288*b1 + 600
$\nu^{5}$	$=$	$ 20 \beta_{14} + 4 \beta_{12} - 500 \beta_{9} + 480 \beta_{8} - 20 \beta_{7} + 3245 \beta_{6} + \cdots + 1176 $ 20b14 + 4b12 - 500b9 + 480b8 - 20b7 + 3245b6 + 649b5 + 245b4 - 480*b1 + 1176
$\nu^{6}$	$=$	$ 60 \beta_{14} - 24 \beta_{11} + 24 \beta_{10} - 1470 \beta_{9} + 960 \beta_{8} - 1024 \beta_{7} + \cdots + 15576 $ 60b14 - 24b11 + 24b10 - 1470b9 + 960b8 - 1024b7 + 10926b6 + 1825b5 + 3870b4 - 8b3 - 1440b2 - 18000b1 + 15576
$\nu^{7}$	$=$	$ 56 \beta_{15} + 3556 \beta_{14} + 8 \beta_{13} + 2024 \beta_{12} - 84 \beta_{11} + 84 \beta_{10} + \cdots + 43800 $ 56b15 + 3556b14 + 8b13 + 2024b12 - 84b11 + 84b10 - 27202b9 + 41992b8 - 3514b7 + 121731b6 + 17457b5 + 12691b4 - 3360b2 - 49392b1 + 43800
$\nu^{8}$	$=$	$ 224 \beta_{15} + 13944 \beta_{14} + 5376 \beta_{12} - 5664 \beta_{11} + 5600 \beta_{10} - 101976 \beta_{9} + \cdots + 417608 $ 224b15 + 13944b14 + 5376b12 - 5664b11 + 5600b10 - 101976b9 + 131712b8 - 78288b7 + 484008b6 + 61425b5 + 133608b4 - 5392b3 - 167984b2 - 872272b1 + 417608
$\nu^{9}$	$=$	$ 16944 \beta_{15} + 343896 \beta_{14} + 6928 \beta_{13} + 309312 \beta_{12} - 24984 \beta_{11} + \cdots + 1468824 $ 16944b15 + 343896b14 + 6928b13 + 309312b12 - 24984b11 + 24696b10 - 1236072b9 + 2609856b8 - 331296b7 + 4291641b6 + 495625b5 + 526113b4 - 16128b3 - 592704b2 - 3153600*b1 + 1468824
$\nu^{10}$	$=$	$ 83040 \beta_{15} + 1615320 \beta_{14} + 23040 \beta_{13} + 1208448 \beta_{12} - 678192 \beta_{11} + \cdots + 11474488 $ 83040b15 + 1615320b14 + 23040b13 + 1208448b12 - 678192b11 + 654032b10 - 5425770b9 + 10488960b8 - 4830432b7 + 18845338b6 + 2029969b5 + 4134058b4 - 1025280b3 - 13066560b2 - 37603440*b1 + 11474488
$\nu^{11}$	$=$	$ 2464704 \beta_{15} + 25368024 \beta_{14} + 1605120 \beta_{13} + 30368448 \beta_{12} - 3501960 \beta_{11} + \cdots + 46680216 $ 2464704b15 + 25368024b14 + 1605120b13 + 30368448b12 - 3501960b11 + 3371720b10 - 50272046b9 + 136231920b8 - 23568930b7 + 146045307b6 + 15837801b5 + 18051539b4 - 4409856b3 - 57752640b2 - 161634000*b1 + 46680216
$\nu^{12}$	$=$	$ 13878480 \beta_{15} + 134668380 \beta_{14} + 7535616 \beta_{13} + 146209536 \beta_{12} + \cdots + 317744088 $ 13878480b15 + 134668380b14 + 7535616b13 + 146209536b12 - 59238432b11 + 55038368b10 - 243599268b9 + 638831424b8 - 265809368b7 + 674520684b6 + 73671313b5 + 106146636b4 - 119889792b3 - 822143520b2 - 1521669536*b1 + 317744088
$\nu^{13}$	$=$	$ 252245136 \beta_{15} + 1608320428 \beta_{14} + 220691328 \beta_{13} + 2363755680 \beta_{12} + \cdots + 1419956760 $ 252245136b15 + 1608320428b14 + 220691328b13 + 2363755680b12 - 340116348b11 + 314502188b10 - 1857672284b9 + 6359482208b8 - 1429117976b7 + 4874189593b6 + 621381929b5 + 467314081b4 - 625502592b3 - 4176620448b2 - 7316214048*b1 + 1419956760
$\nu^{14}$	$=$	$ 1557980424 \beta_{15} + 9268679420 \beta_{14} + 1240242432 \beta_{13} + 12964915392 \beta_{12} + \cdots + 8417900376 $ 1557980424b15 + 9268679420b14 + 1240242432b13 + 12964915392b12 - 4312192456b11 + 3813641832b10 - 9483011902b9 + 32826750528b8 - 13618348576b7 + 22947285062b6 + 3295845657b5 + 1523495414b4 - 10781394624b3 - 45264771424b2 - 59451769104*b1 + 8417900376
$\nu^{15}$	$=$	$ 20963759960 \beta_{15} + 92551542620 \beta_{14} + 22746609600 \beta_{13} + 159538784992 \beta_{12} + \cdots + 39228280728 $ 20963759960b15 + 92551542620b14 + 22746609600b13 + 159538784992b12 - 26545787268b11 + 23249013788b10 - 61200377210b9 + 272018121360b8 - 78161705818b7 + 163265339787b6 + 30122424321b5 + 4009535907b4 - 63321360960b3 - 250710276960b2 - 307831395120*b1 + 39228280728

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/242\mathbb{Z}\right)^\times$.

$n$	$123$
$\chi(n)$	$-\beta_{1}$

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} $

161.1

−3.59318	−	1.14412i
6.25568	−	1.14412i
−5.25568	+	1.14412i
4.59318	+	1.14412i
6.76943	−	0.437016i
−3.07943	−	0.437016i
4.07943	+	0.437016i
−5.76943	+	0.437016i
6.76943	+	0.437016i
−3.07943	+	0.437016i
4.07943	−	0.437016i
−5.76943	−	0.437016i
−3.59318	+	1.14412i
6.25568	+	1.14412i
−5.25568	−	1.14412i
4.59318	−	1.14412i

−0.831254

1.14412i

−1.67624

5.15894i

−0.618034

−

1.90211i

−4.38846

3.18840i

−4.50908

−

6.20621i

−1.91585

0.622498i

2.68999

0.874032i

−16.5237

−

12.0052i

−

7.67130i

161.2

−0.831254

1.14412i

1.36722

−

4.20788i

−0.618034

−

1.90211i

3.57944

−

2.60061i

3.67782

5.06209i

11.3308

−

3.68161i

2.68999

0.874032i

−8.55582

−

6.21616i

6.25709i

161.3

0.831254

−

1.14412i

−1.67624

5.15894i

−0.618034

−

1.90211i

−4.38846

3.18840i

4.50908

6.20621i

1.91585

−

0.622498i

−2.68999

−

0.874032i

−16.5237

−

12.0052i

7.67130i

161.4

0.831254

−

1.14412i

1.36722

−

4.20788i

−0.618034

−

1.90211i

3.57944

−

2.60061i

−3.67782

−

5.06209i

−11.3308

3.68161i

−2.68999

−

0.874032i

−8.55582

−

6.21616i

−

6.25709i

215.1

−1.34500

0.437016i

−3.57944

−

2.60061i

1.61803

−

1.17557i

−1.36722

4.20788i

5.95084

1.93355i

−7.00284

−

9.63858i

−1.66251

2.28825i

3.26803

10.0580i

−

6.25709i

215.2

−1.34500

0.437016i

4.38846

3.18840i

1.61803

−

1.17557i

1.67624

−

5.15894i

−7.29584

−

2.37056i

1.18406

1.62972i

−1.66251

2.28825i

6.31150

19.4248i

7.67130i

215.3

1.34500

−

0.437016i

−3.57944

−

2.60061i

1.61803

−

1.17557i

−1.36722

4.20788i

−5.95084

−

1.93355i

7.00284

9.63858i

1.66251

−

2.28825i

3.26803

10.0580i

6.25709i

215.4

1.34500

−

0.437016i

4.38846

3.18840i

1.61803

−

1.17557i

1.67624

−

5.15894i

7.29584

2.37056i

−1.18406

−

1.62972i

1.66251

−

2.28825i

6.31150

19.4248i

−

7.67130i

233.1

−1.34500

−

0.437016i

−3.57944

2.60061i

1.61803

1.17557i

−1.36722

−

4.20788i

5.95084

−

1.93355i

−7.00284

9.63858i

−1.66251

−

2.28825i

3.26803

−

10.0580i

6.25709i

233.2

−1.34500

−

0.437016i

4.38846

−

3.18840i

1.61803

1.17557i

1.67624

5.15894i

−7.29584

2.37056i

1.18406

−

1.62972i

−1.66251

−

2.28825i

6.31150

−

19.4248i

−

7.67130i

233.3

1.34500

0.437016i

−3.57944

2.60061i

1.61803

1.17557i

−1.36722

−

4.20788i

−5.95084

1.93355i

7.00284

−

9.63858i

1.66251

2.28825i

3.26803

−

10.0580i

−

6.25709i

233.4

1.34500

0.437016i

4.38846

−

3.18840i

1.61803

1.17557i

1.67624

5.15894i

7.29584

−

2.37056i

−1.18406

1.62972i

1.66251

2.28825i

6.31150

−

19.4248i

7.67130i

239.1

−0.831254

−

1.14412i

−1.67624

−

5.15894i

−0.618034

1.90211i

−4.38846

−

3.18840i

−4.50908

6.20621i

−1.91585

−

0.622498i

2.68999

−

0.874032i

−16.5237

12.0052i

7.67130i

239.2

−0.831254

−

1.14412i

1.36722

4.20788i

−0.618034

1.90211i

3.57944

2.60061i

3.67782

−

5.06209i

11.3308

3.68161i

2.68999

−

0.874032i

−8.55582

6.21616i

−

6.25709i

239.3

0.831254

1.14412i

−1.67624

−

5.15894i

−0.618034

1.90211i

−4.38846

−

3.18840i

4.50908

−

6.20621i

1.91585

0.622498i

−2.68999

0.874032i

−16.5237

12.0052i

−

7.67130i

239.4

0.831254

1.14412i

1.36722

4.20788i

−0.618034

1.90211i

3.57944

2.60061i

−3.67782

5.06209i

−11.3308

−

3.68161i

−2.68999

0.874032i

−8.55582

6.21616i

6.25709i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	242.3.d.g		16
11.b	odd	2	1	inner	242.3.d.g		16
11.c	even	5	1		242.3.b.b	✓	4
11.c	even	5	3	inner	242.3.d.g		16
11.d	odd	10	1		242.3.b.b	✓	4
11.d	odd	10	3	inner	242.3.d.g		16
33.f	even	10	1		2178.3.d.f		4
33.h	odd	10	1		2178.3.d.f		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
242.3.b.b	✓	4	11.c	even	5	1
242.3.b.b	✓	4	11.d	odd	10	1
242.3.d.g		16	1.a	even	1	1	trivial
242.3.d.g		16	11.b	odd	2	1	inner
242.3.d.g		16	11.c	even	5	3	inner
242.3.d.g		16	11.d	odd	10	3	inner
2178.3.d.f		4	33.f	even	10	1
2178.3.d.f		4	33.h	odd	10	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(242, [\chi])$:

$ T_{3}^{8} - T_{3}^{7} + 25T_{3}^{6} - 49T_{3}^{5} + 649T_{3}^{4} + 1176T_{3}^{3} + 14400T_{3}^{2} + 13824T_{3} + 331776 $

$ T_{7}^{16} - 146 T_{7}^{14} + 20740 T_{7}^{12} - 2943944 T_{7}^{10} + 417869584 T_{7}^{8} + \cdots + 110075314176 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} - 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} $ (T^8 - 2T^6 + 4T^4 - 8*T^2 + 16)^2
$3$	$ (T^{8} - T^{7} + \cdots + 331776)^{2} $ (T^8 - T^7 + 25T^6 - 49T^5 + 649T^4 + 1176T^3 + 14400T^2 + 13824T + 331776)^2
$5$	$ (T^{8} + T^{7} + \cdots + 331776)^{2} $ (T^8 + T^7 + 25T^6 + 49T^5 + 649T^4 - 1176T^3 + 14400T^2 - 13824T + 331776)^2
$7$	$ T^{16} + \cdots + 110075314176 $ T^16 - 146T^14 + 20740T^12 - 2943944T^10 + 417869584T^8 - 1695711744T^6 + 6881034240T^4 - 27901034496*T^2 + 110075314176
$11$	$ T^{16} $ T^16
$13$	$ T^{16} - 178 T^{14} + \cdots + 16777216 $ T^16 - 178T^14 + 31620T^12 - 5616968T^10 + 997796624T^8 - 359485952T^6 + 129515520T^4 - 46661632*T^2 + 16777216
$17$	$ T^{16} - 788 T^{14} + \cdots + 1679616 $ T^16 - 788T^14 + 620908T^12 - 489247136T^10 + 385504390480T^8 - 17612896896T^6 + 804696768T^4 - 36764928*T^2 + 1679616
$19$	$ (T^{8} - 32 T^{6} + \cdots + 1048576)^{2} $ (T^8 - 32T^6 + 1024T^4 - 32768*T^2 + 1048576)^2
$23$	$ (T^{2} - 19 T - 516)^{8} $ (T^2 - 19*T - 516)^8
$29$	$ T^{16} + \cdots + 31\!\cdots\!56 $ T^16 - 2546T^14 + 5155012T^12 - 9745853768T^10 + 17971706648080T^8 - 12933761518927872T^6 + 9079033047696801792T^4 - 5950771492620802719744*T^2 + 3101843146481947279097856
$31$	$ (T^{8} + 69 T^{7} + \cdots + 892616806656)^{2} $ (T^8 + 69T^7 + 3789T^6 + 194373T^5 + 9728829T^4 + 188930556T^3 + 3579786576T^2 + 63364773312*T + 892616806656)^2
$37$	$ (T^{8} + 15 T^{7} + \cdots + 688747536)^{2} $ (T^8 + 15T^7 + 387T^6 + 8235T^5 + 186219T^4 - 1334070T^3 + 10156428T^2 - 63772920*T + 688747536)^2
$41$	$ T^{16} + \cdots + 47\!\cdots\!96 $ T^16 - 1314T^14 + 1679940T^12 - 2146135176T^10 + 2741642340624T^8 - 100130082771456T^6 + 3656863717539840T^4 - 133449783062298624*T^2 + 4738381338321616896
$43$	$ (T^{4} + 5282 T^{2} + 4460544)^{4} $ (T^4 + 5282*T^2 + 4460544)^4
$47$	$ (T^{8} + 28 T^{7} + \cdots + 1358954496)^{2} $ (T^8 + 28T^7 + 976T^6 + 32704T^5 + 1103104T^4 - 6279168T^3 + 35979264T^2 - 198180864*T + 1358954496)^2
$53$	$ (T^{8} - 34 T^{7} + \cdots + 1358954496)^{2} $ (T^8 - 34T^7 + 964T^6 - 26248T^5 + 707344T^4 - 5039616T^3 + 35536896T^2 - 240648192*T + 1358954496)^2
$59$	$ (T^{8} + 89 T^{7} + \cdots + 393460125696)^{2} $ (T^8 + 89T^7 + 7129T^6 + 563993T^5 + 44549209T^4 + 446682456T^3 + 4471765056T^2 + 44214584832*T + 393460125696)^2
$61$	$ T^{16} + \cdots + 54\!\cdots\!36 $ T^16 - 10706T^14 + 106045252T^12 - 1043535960008T^10 + 10262950530123280T^8 - 8946425795765225472T^6 + 7794271291818081779712T^4 - 6746113960762684760653824*T^2 + 5402174133251193423041396736
$67$	$ (T^{2} + 151 T + 4512)^{8} $ (T^2 + 151*T + 4512)^8
$71$	$ (T^{8} + \cdots + 25\!\cdots\!96)^{2} $ (T^8 + 171T^7 + 22149T^6 + 2574747T^5 + 283201029T^4 + 18260105724T^3 + 1114016381136T^2 + 60996131379648*T + 2529734290903296)^2
$73$	$ T^{16} + \cdots + 18\!\cdots\!76 $ T^16 - 14338T^14 + 184711620T^12 - 2349209552648T^10 + 29828658642896144T^8 - 49020072432314020352T^6 + 80426394208854315909120T^4 - 130270220268691068891430912*T^2 + 189587090580552064912511205376
$79$	$ T^{16} + \cdots + 1099511627776 $ T^16 - 15778T^14 + 248944260T^12 - 3927826377608T^10 + 61972989666976784T^8 - 4022094210670592T^6 + 261036976373760T^4 - 16941498499072*T^2 + 1099511627776
$83$	$ T^{16} + \cdots + 88\!\cdots\!56 $ T^16 - 8354T^14 + 66719812T^12 - 531734673032T^10 + 4237314728696080T^8 - 1632161705810416128T^6 + 628624381348683988992T^4 - 241601197155010497478656*T^2 + 88771347985646796989792256
$89$	$ (T^{2} - 169 T + 4206)^{8} $ (T^2 - 169*T + 4206)^8
$97$	$ (T^{8} + \cdots + 100725792178176)^{2} $ (T^8 - 113T^7 + 9601T^6 - 726929T^5 + 51727009T^4 - 2302911072T^3 + 96357786624T^2 - 3592807612416*T + 100725792178176)^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:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	242.d (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{6} q^{2} + (\beta_{15} + \beta_{3} - \beta_{2} + \cdots + 1) q^{3}+ \cdots + (\beta_{15} + \beta_{14} + \cdots + 16 \beta_{3}) q^{9}+O(q^{10}) \) q - b6 * q^2 + (b15 + b3 - b2 + b1 + 1) * q^3 - 2b1 q^4 + (-b14 - b2) * q^5 + (b13 + b10) * q^6 + (3b13 + 3b12 - b11 - 3b8 + 3b6) * q^7 - 2b8 q^8 + (b15 + b14 + b9 + b5 + 16b3) q^9	\( q - \beta_{6} q^{2} + (\beta_{15} + \beta_{3} - \beta_{2} + \cdots + 1) q^{3}+ \cdots + (17 \beta_{12} - 14 \beta_{11} + \cdots - 14 \beta_{4}) q^{98}+O(q^{100}) \) q - b6 * q^2 + (b15 + b3 - b2 + b1 + 1) * q^3 - 2b1 q^4 + (-b14 - b2) * q^5 + (b13 + b10) * q^6 + (3b13 + 3b12 - b11 - 3b8 + 3b6) * q^7 - 2b8 q^8 + (b15 + b14 + b9 + b5 + 16b3) q^9 + (-b12 - b11 + b10 - b7 - b6 - b4) * q^10 + (-2b5 + 2) q^12 + (-5b6 - b4) q^13 + (-2b15 + 6b3 - 6b2 + 6b1 + 6) * q^14 + (b9 - 25b1) q^15 - 4b2 q^16 + (-11b13 - 2b10) * q^17 + (16b13 + 16b12 + b11 - 16b8 + 16b6) * q^18 + 4b8 q^19 + (-2b15 - 2b14 - 2b9 - 2b5 - 2b3) q^20 + (-21b12 + 3b11 - 3b10 + 3b7 + 3b6 + 3b4) * q^21 + (5b5 + 7) q^23 + 2b4 q^24 - b15 * q^25 + (-2b9 - 8b1) * q^26 + (7b14 + 31b2) * q^27 + (6b13 - 2b10) * q^28 + (8b13 + 8b12 + 5b11 - 8b8 + 8b6) q^29 + (-25b8 - b7) q^30 + (-3b15 - 3b14 - 3b9 - 3b5 + 33b3) q^31 - 4b12 q^32 + (4b5 - 22) q^34 + (24b6 + 3b4) * q^35 + (2b15 + 32b3 - 32b2 + 32b1 + 32) * q^36 + (-3b9 + 9b1) * q^37 + 8b2 q^38 + (-20b13 + 4b10) * q^39 + (-2b13 - 2b12 - 2b11 + 2b8 - 2b6) q^40 + (12b8 + 3b7) * q^41 + (6b15 + 6b14 + 6b9 + 6b5 - 42b3) q^42 + (-15b12 - 7b11 + 7b10 - 7b7 - 7b6 - 7b4) * q^43 + (-16b5 + 40) q^45 + (-12b6 - 5b4) * q^46 + (-4b15 - 16b3 + 16b2 - 16b1 - 16) * q^47 + (4b9 - 4b1) * q^48 + (-14b14 + 17b2) * q^49 - b10 * q^50 + (-59b13 - 59b12 - 11b11 + 59b8 - 59b6) q^51 + (-8b8 + 2b7) * q^52 + (2b15 + 2b14 + 2b9 + 2b5 - 16b3) q^53 + (31b12 + 7b11 - 7b10 + 7b7 + 7b6 + 7b4) * q^54 + (4b5 + 12) q^56 - 4b4 q^57 + (10b15 + 16b3 - 16b2 + 16b1 + 16) * q^58 + (7b9 + 41b1) * q^59 + (-2b14 - 50b2) * q^60 + (48b13 + 5b10) * q^61 + (33b13 + 33b12 - 3b11 - 33b8 + 33b6) q^62 + (24b8 - 12b7) * q^63 - 8b3 q^64 + (20b12 - 4b11 + 4b10 - 4b7 - 4b6 - 4b4) * q^65 + (7b5 - 79) q^67 + (18b6 - 4b4) * q^68 + (7b15 - 113b3 + 113b2 - 113b1 - 113) * q^69 + (6b9 + 42b1) * q^70 + (-3b14 - 87b2) * q^71 + (32b13 + 2b10) * q^72 + (-20b13 - 20b12 + 11b11 + 20b8 - 20b6) q^73 + (9b8 + 3b7) * q^74 - 24b3 q^75 + 8b12 q^76 + (-8b5 - 40) q^78 + (-40b6 + 9b4) * q^79 + (-4b15 - 4b3 + 4b2 - 4b1 - 4) * q^80 + (-22b9 + 55b1) * q^81 + (6b14 + 24b2) * q^82 + (-41b13 - 5b10) * q^83 + (-42b13 - 42b12 + 6b11 + 42b8 - 42b6) q^84 + (59b8 + 11b7) * q^85 + (-14b15 - 14b14 - 14b9 - 14b5 - 30b3) q^86 + (128b12 + 8b11 - 8b10 + 8b7 + 8b6 + 8b4) * q^87 + (11b5 + 79) q^89 + (-24b6 + 16b4) * q^90 + (-16b15 + 72b3 - 72b2 + 72b1 + 72) * q^91 + (-10b9 - 14b1) * q^92 + (33b14 - 39b2) * q^93 + (-16b13 - 4b10) * q^94 + (4b13 + 4b12 + 4b11 - 4b8 + 4b6) q^95 + (-4b8 - 4b7) * q^96 + (-b15 - b14 - b9 - b5 - 57b3) q^97 + (17b12 - 14b11 + 14b10 - 14b7 - 14b6 - 14b4) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{3} + 8 q^{4} - 2 q^{5} - 62 q^{9}+O(q^{10}) \) 16 * q + 2 * q^3 + 8 * q^4 - 2 * q^5 - 62 * q^9	\( 16 q + 2 q^{3} + 8 q^{4} - 2 q^{5} - 62 q^{9} + 16 q^{12} + 28 q^{14} + 98 q^{15} - 16 q^{16} + 4 q^{20} + 152 q^{23} + 2 q^{25} + 36 q^{26} + 110 q^{27} - 138 q^{31} - 320 q^{34} + 124 q^{36} - 30 q^{37} + 32 q^{38} + 180 q^{42} + 512 q^{45} - 56 q^{47} + 8 q^{48} + 96 q^{49} + 68 q^{53} + 224 q^{56} + 44 q^{58} - 178 q^{59} - 196 q^{60} + 32 q^{64} - 1208 q^{67} - 466 q^{69} - 180 q^{70} - 342 q^{71} + 96 q^{75} - 704 q^{78} - 8 q^{80} - 176 q^{81} + 84 q^{82} + 92 q^{86} + 1352 q^{89} + 320 q^{91} + 76 q^{92} - 222 q^{93} + 226 q^{97}+O(q^{100}) \) 16 * q + 2 * q^3 + 8 * q^4 - 2 * q^5 - 62 * q^9 + 16 * q^12 + 28 * q^14 + 98 * q^15 - 16 * q^16 + 4 * q^20 + 152 * q^23 + 2 * q^25 + 36 * q^26 + 110 * q^27 - 138 * q^31 - 320 * q^34 + 124 * q^36 - 30 * q^37 + 32 * q^38 + 180 * q^42 + 512 * q^45 - 56 * q^47 + 8 * q^48 + 96 * q^49 + 68 * q^53 + 224 * q^56 + 44 * q^58 - 178 * q^59 - 196 * q^60 + 32 * q^64 - 1208 * q^67 - 466 * q^69 - 180 * q^70 - 342 * q^71 + 96 * q^75 - 704 * q^78 - 8 * q^80 - 176 * q^81 + 84 * q^82 + 92 * q^86 + 1352 * q^89 + 320 * q^91 + 76 * q^92 - 222 * q^93 + 226 * q^97

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	242.3.d.g

Level	$242$
Weight	$3$
Character orbit	242.d
Analytic conductor	$6.594$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(6.59402239752\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 8 x^{15} - 168 x^{14} + 1316 x^{13} + 12572 x^{12} - 92904 x^{11} - 552362 x^{10} + \cdots + 93516487856 \) x^16 - 8x^15 - 168x^14 + 1316x^13 + 12572x^12 - 92904x^11 - 552362x^10 + 3645748x^9 + 15734655x^8 - 85890536x^7 - 299926366x^6 + 1216788336x^5 + 3747221628x^4 - 9630461344x^3 - 27860664440x^2 + 32894183872*x + 93516487856
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(\beta_{1}\)	\(=\)	\( ( 175481049060 \nu^{14} - 1228367343420 \nu^{13} - 31870999820388 \nu^{12} + \cdots - 14\!\cdots\!28 ) / 20\!\cdots\!10 \) (175481049060v^14 - 1228367343420v^13 - 31870999820388v^12 + 207194774386788v^11 + 2631850994562082v^10 - 15087816493040810v^9 - 131056658453814037v^8 + 617082862871786836v^7 + 4416819708280903274v^6 - 15475949813642022536v^5 - 110255369457878765908v^4 + 247055260430715637072v^3 + 1834440698346756083667v^2 - 1960655203724039601680v - 14013092710472544603728) / 2024499066530216061610
\(\beta_{2}\)	\(=\)	\( ( 6274938934164 \nu^{14} - 43924572539148 \nu^{13} + \cdots - 13\!\cdots\!12 ) / 40\!\cdots\!20 \) (6274938934164v^14 - 43924572539148v^13 - 1124662048413064v^12 + 7318991733487308v^11 + 90881141253755004v^10 - 522543332804591704v^9 - 4344570105413003537v^8 + 20595843960083095664v^7 + 132128113110157805602v^6 - 470746868461773826624v^5 - 2530625459022392744857v^4 + 5870943809587053857504v^3 + 27985028388913635500552v^2 - 31002553752469851316864v - 136406554439207720269712) / 4048998133060432123220
\(\beta_{3}\)	\(=\)	\( ( 82341644084550 \nu^{14} - 576391508591850 \nu^{13} + \cdots - 68\!\cdots\!36 ) / 80\!\cdots\!40 \) (82341644084550v^14 - 576391508591850v^13 - 13898454439769201v^12 + 90883816250309256v^11 + 1022862360340382009v^10 - 5961150781617850650v^9 - 42899171826069798219v^8 + 208386863682947879892v^7 + 1113793811722406491393v^6 - 4096796740514804018712v^5 - 17956541117416792085676v^4 + 42996629461023661954704v^3 + 166644507560391636695864v^2 - 188862218869953655683360v - 682347068259704726350736) / 8097996266120864246440
\(\beta_{4}\)	\(=\)	\( ( - 16\!\cdots\!52 \nu^{15} + \cdots - 13\!\cdots\!76 ) / 68\!\cdots\!25 \) (-16615617072028252v^15 + 183921387272661540v^14 + 2468254547109929556v^13 - 31402908860033890424v^12 - 154571000268605157786v^11 + 2355812411565152162020v^10 + 5256776379557871070709v^9 - 102238889885446508476384v^8 - 104015750968520409017962v^7 + 2867791643265798930620576v^6 + 1187221539499037715904980v^5 - 56885156025650703673639552v^4 - 7212034766984225220678952v^3 + 1118456121034190939097649792v^2 - 324642656293691428327429949*v - 13312760194484715634576269176) / 684184520781552843561253525
\(\beta_{5}\)	\(=\)	\( ( - 33\!\cdots\!04 \nu^{15} + \cdots - 73\!\cdots\!12 ) / 68\!\cdots\!25 \) (-33231234144056504v^15 + 249234256080423780v^14 + 5766768723474154212v^13 - 41264049586468429708v^12 - 449185984519112258712v^11 + 2932743576650820289830v^10 + 20711483365844490824468v^9 - 115895929038667840274283v^8 - 625120894366395899422504v^7 + 2750232761605993933828182v^6 + 12834714937787786674005040v^5 - 39248156557873860076232364v^4 - 181409955335391303116508064v^3 + 312820081070764863903102424v^2 + 2044300384448819817441167552*v - 733572461740217763299679912) / 684184520781552843561253525
\(\beta_{6}\)	\(=\)	\( ( 33\!\cdots\!04 \nu^{15} + \cdots + 73\!\cdots\!12 ) / 68\!\cdots\!25 \) (33231234144056504v^15 - 249234256080423780v^14 - 5766768723474154212v^13 + 41264049586468429708v^12 + 449185984519112258712v^11 - 2932743576650820289830v^10 - 20711483365844490824468v^9 + 115895929038667840274283v^8 + 625120894366395899422504v^7 - 2750232761605993933828182v^6 - 12834714937787786674005040v^5 + 39248156557873860076232364v^4 + 181409955335391303116508064v^3 - 312820081070764863903102424v^2 - 1360115863667266973879914027*v + 733572461740217763299679912) / 684184520781552843561253525
\(\beta_{7}\)	\(=\)	\( ( - 31\!\cdots\!36 \nu^{15} + \cdots + 15\!\cdots\!92 ) / 13\!\cdots\!50 \) (-313972325148010636v^15 - 1581779784237757860v^14 + 81633097592829830538v^13 + 325213602498538774698v^12 - 8797538009894643189483v^11 - 31250637227492478835175v^10 + 524111801792212893980947v^9 + 1823036143284007762340943v^8 - 19091612717031207407776646v^7 - 70339401408842906595378942v^6 + 436858537055216701879153580v^5 + 1741033065137244318794400444v^4 - 5860604094458178946083145376v^3 - 24679900878354383074895401464v^2 + 35030122381182193052153650768*v + 150921050441880734257005362192) / 1368369041563105687122507050
\(\beta_{8}\)	\(=\)	\( ( 62\!\cdots\!72 \nu^{15} + \cdots + 65\!\cdots\!36 ) / 13\!\cdots\!50 \) (627944650296021272v^15 - 4709584877220159540v^14 - 108154184065789934256v^13 + 774430900398806992354v^12 + 8329471242855672679806v^11 - 54487974888829655978145v^10 - 377029143478437698987909v^9 + 2118405802494493622062629v^8 + 10993972945318710980916202v^7 - 48763723787066583086481066v^6 - 207303807895304789369698420v^5 + 645397351206236167382624532v^4 + 2322477110519433562204640957v^3 - 4154992862008189708470805312v^2 - 11730274923644883054399165776*v + 6580385501157004753362636336) / 1368369041563105687122507050
\(\beta_{9}\)	\(=\)	\( ( 74\!\cdots\!72 \nu^{15} + \cdots + 65\!\cdots\!36 ) / 13\!\cdots\!50 \) (746553168760920572v^15 - 5539844506474454640v^14 - 129695952199389285396v^13 + 914474884380708935494v^12 + 10108352489335156714016v^11 - 64685905495558404661195v^10 - 465610994210662875666394v^9 + 2535495194923848703449209v^8 + 13979323470244314908329172v^7 - 59223995645856294328676146v^6 - 281825963388732336640745160v^5 + 812383237007659020057774692v^4 + 3562384750735497732935584592v^3 - 5480209517481286695444325712v^2 - 21201794352116828314781939616*v + 6580385501157004753362636336) / 1368369041563105687122507050
\(\beta_{10}\)	\(=\)	\( ( - 78\!\cdots\!71 \nu^{15} + \cdots + 14\!\cdots\!52 ) / 54\!\cdots\!00 \) (-7877385638507433071v^15 - 323504510925109659975v^14 + 3839742854304101781993v^13 + 49324501180325214773093v^12 - 455815083370312818131943v^11 - 3301029023306060428982295v^10 + 25285851856365026783804887v^9 + 126529663424803543043946883v^8 - 770922393178149134876041046v^7 - 3017120906068341136532984102v^6 + 13363839018944971584067702380v^5 + 44979692859829587002682704044v^4 - 124519777256397056067811889336v^3 - 388925722184585493242214074104v^2 + 488491121302424552442409872208*v + 1462383924468513496002828515152) / 5473476166252422748490028200
\(\beta_{11}\)	\(=\)	\( ( - 13\!\cdots\!39 \nu^{15} + \cdots + 32\!\cdots\!08 ) / 54\!\cdots\!00 \) (-13804244221026176339v^15 - 439170394871573241395v^14 + 5946090455219733079927v^13 + 70277759635872453346287v^12 - 710581628067113812331007v^11 - 5005046260338933465285495v^10 + 40882135951621374910582153v^9 + 206304338938276163406035377v^8 - 1313210675210165670486862394v^7 - 5313963743790043606791335618v^6 + 24211514521875778207800761700v^5 + 85302516650732368314483624036v^4 - 240775308794124227754358812264v^3 - 788528552636895397712500639016v^2 + 1008211299129257969269681030192*v + 3200286178840196584967087887408) / 5473476166252422748490028200
\(\beta_{12}\)	\(=\)	\( ( 12474157204 \nu^{15} - 93556179030 \nu^{14} - 2085744967912 \nu^{13} + \cdots + 64\!\cdots\!12 ) / 48\!\cdots\!00 \) (12474157204v^15 - 93556179030v^14 - 2085744967912v^13 + 14976277673383v^12 + 153688852361912v^11 - 1013149400238030v^10 - 6491315379684643v^9 + 37063337558777583v^8 + 170456576374403654v^7 - 776991148135376382v^6 - 2793249100137010815v^5 + 9017401691911159764v^4 + 26508223093865149464v^3 - 49194537558881804824v^2 - 111668512881353080752*v + 64353741977637329712) / 4842462191461310900
\(\beta_{13}\)	\(=\)	\( ( 15\!\cdots\!42 \nu^{15} + \cdots + 29\!\cdots\!16 ) / 54\!\cdots\!00 \) (15754771277014866142v^15 - 118160784577611496065v^14 - 2323297063613387851881v^13 + 16893536146247462060879v^12 + 148744483314305682364131v^11 - 1007866187349476305924425v^10 - 5401307948808425062945109v^9 + 32152074017269367423659329v^8 + 121366227222934805571604127v^7 - 582263356362062198028591066v^6 - 1708150390207906993413515870v^5 + 5806382907274542436119217332v^4 + 14056926813593698682144615132v^3 - 27205938510456128026053525512v^2 - 50012673693275977231565316376*v + 29749221945006701710005389616) / 5473476166252422748490028200
\(\beta_{14}\)	\(=\)	\( ( - 11\!\cdots\!16 \nu^{15} + \cdots - 36\!\cdots\!88 ) / 27\!\cdots\!00 \) (-11291085546830678216v^15 + 82562510301082027410v^14 + 1938932332778467481008v^13 - 13410850068938975568107v^12 - 148284925715387528542708v^11 + 925775339775611821581590v^10 + 6605111125527000427174092v^9 - 34867336615304403053176087v^8 - 185640310699708451494645656v^7 + 757300010931751627785841838v^6 + 3289078998338388512468773520v^5 - 9064423156742318508698678356v^4 - 33896455528146485855600272096v^3 + 48757283023055274539397259096v^2 + 155307806712640620625132604608*v - 36369790717711399867750746288) / 2736738083126211374245014100
\(\beta_{15}\)	\(=\)	\( ( - 63\!\cdots\!52 \nu^{15} + \cdots - 29\!\cdots\!16 ) / 54\!\cdots\!00 \) (-63401809095249994252v^15 + 451690049305257392835v^14 + 10283169580594723012506v^13 - 68988651689594869682639v^12 - 724363756407850968336876v^11 + 4371450226107229532860635v^10 + 28878366770638680423520274v^9 - 146828486860929744958056069v^8 - 707514331051427957655402052v^7 + 2756789520439569176696316306v^6 + 10722230134388153300741398440v^5 - 27599562735095556805201542852v^4 - 93821972030647122959985147472v^3 + 118250160989518271909959124072v^2 + 359231326056521670250901220896*v - 29749221945006701710005389616) / 5473476166252422748490028200

\(\nu\)	\(=\)	\( \beta_{6} + \beta_{5} \) b6 + b5
\(\nu^{2}\)	\(=\)	\( 2\beta_{6} + \beta_{5} + 2\beta_{4} - 2\beta _1 + 24 \) 2b6 + b5 + 2b4 - 2*b1 + 24
\(\nu^{3}\)	\(=\)	\( -6\beta_{9} + 2\beta_{8} + 75\beta_{6} + 25\beta_{5} + 3\beta_{4} + 24 \) -6b9 + 2b8 + 75b6 + 25b5 + 3*b4 + 24
\(\nu^{4}\)	\(=\)	\( -12\beta_{9} - 8\beta_{7} + 196\beta_{6} + 49\beta_{5} + 100\beta_{4} - 4\beta_{2} - 288\beta _1 + 600 \) -12b9 - 8b7 + 196b6 + 49b5 + 100b4 - 4b2 - 288*b1 + 600
\(\nu^{5}\)	\(=\)	\( 20 \beta_{14} + 4 \beta_{12} - 500 \beta_{9} + 480 \beta_{8} - 20 \beta_{7} + 3245 \beta_{6} + \cdots + 1176 \) 20b14 + 4b12 - 500b9 + 480b8 - 20b7 + 3245b6 + 649b5 + 245b4 - 480*b1 + 1176
\(\nu^{6}\)	\(=\)	\( 60 \beta_{14} - 24 \beta_{11} + 24 \beta_{10} - 1470 \beta_{9} + 960 \beta_{8} - 1024 \beta_{7} + \cdots + 15576 \) 60b14 - 24b11 + 24b10 - 1470b9 + 960b8 - 1024b7 + 10926b6 + 1825b5 + 3870b4 - 8b3 - 1440b2 - 18000b1 + 15576
\(\nu^{7}\)	\(=\)	\( 56 \beta_{15} + 3556 \beta_{14} + 8 \beta_{13} + 2024 \beta_{12} - 84 \beta_{11} + 84 \beta_{10} + \cdots + 43800 \) 56b15 + 3556b14 + 8b13 + 2024b12 - 84b11 + 84b10 - 27202b9 + 41992b8 - 3514b7 + 121731b6 + 17457b5 + 12691b4 - 3360b2 - 49392b1 + 43800
\(\nu^{8}\)	\(=\)	\( 224 \beta_{15} + 13944 \beta_{14} + 5376 \beta_{12} - 5664 \beta_{11} + 5600 \beta_{10} - 101976 \beta_{9} + \cdots + 417608 \) 224b15 + 13944b14 + 5376b12 - 5664b11 + 5600b10 - 101976b9 + 131712b8 - 78288b7 + 484008b6 + 61425b5 + 133608b4 - 5392b3 - 167984b2 - 872272b1 + 417608
\(\nu^{9}\)	\(=\)	\( 16944 \beta_{15} + 343896 \beta_{14} + 6928 \beta_{13} + 309312 \beta_{12} - 24984 \beta_{11} + \cdots + 1468824 \) 16944b15 + 343896b14 + 6928b13 + 309312b12 - 24984b11 + 24696b10 - 1236072b9 + 2609856b8 - 331296b7 + 4291641b6 + 495625b5 + 526113b4 - 16128b3 - 592704b2 - 3153600*b1 + 1468824
\(\nu^{10}\)	\(=\)	\( 83040 \beta_{15} + 1615320 \beta_{14} + 23040 \beta_{13} + 1208448 \beta_{12} - 678192 \beta_{11} + \cdots + 11474488 \) 83040b15 + 1615320b14 + 23040b13 + 1208448b12 - 678192b11 + 654032b10 - 5425770b9 + 10488960b8 - 4830432b7 + 18845338b6 + 2029969b5 + 4134058b4 - 1025280b3 - 13066560b2 - 37603440*b1 + 11474488
\(\nu^{11}\)	\(=\)	\( 2464704 \beta_{15} + 25368024 \beta_{14} + 1605120 \beta_{13} + 30368448 \beta_{12} - 3501960 \beta_{11} + \cdots + 46680216 \) 2464704b15 + 25368024b14 + 1605120b13 + 30368448b12 - 3501960b11 + 3371720b10 - 50272046b9 + 136231920b8 - 23568930b7 + 146045307b6 + 15837801b5 + 18051539b4 - 4409856b3 - 57752640b2 - 161634000*b1 + 46680216
\(\nu^{12}\)	\(=\)	\( 13878480 \beta_{15} + 134668380 \beta_{14} + 7535616 \beta_{13} + 146209536 \beta_{12} + \cdots + 317744088 \) 13878480b15 + 134668380b14 + 7535616b13 + 146209536b12 - 59238432b11 + 55038368b10 - 243599268b9 + 638831424b8 - 265809368b7 + 674520684b6 + 73671313b5 + 106146636b4 - 119889792b3 - 822143520b2 - 1521669536*b1 + 317744088
\(\nu^{13}\)	\(=\)	\( 252245136 \beta_{15} + 1608320428 \beta_{14} + 220691328 \beta_{13} + 2363755680 \beta_{12} + \cdots + 1419956760 \) 252245136b15 + 1608320428b14 + 220691328b13 + 2363755680b12 - 340116348b11 + 314502188b10 - 1857672284b9 + 6359482208b8 - 1429117976b7 + 4874189593b6 + 621381929b5 + 467314081b4 - 625502592b3 - 4176620448b2 - 7316214048*b1 + 1419956760
\(\nu^{14}\)	\(=\)	\( 1557980424 \beta_{15} + 9268679420 \beta_{14} + 1240242432 \beta_{13} + 12964915392 \beta_{12} + \cdots + 8417900376 \) 1557980424b15 + 9268679420b14 + 1240242432b13 + 12964915392b12 - 4312192456b11 + 3813641832b10 - 9483011902b9 + 32826750528b8 - 13618348576b7 + 22947285062b6 + 3295845657b5 + 1523495414b4 - 10781394624b3 - 45264771424b2 - 59451769104*b1 + 8417900376
\(\nu^{15}\)	\(=\)	\( 20963759960 \beta_{15} + 92551542620 \beta_{14} + 22746609600 \beta_{13} + 159538784992 \beta_{12} + \cdots + 39228280728 \) 20963759960b15 + 92551542620b14 + 22746609600b13 + 159538784992b12 - 26545787268b11 + 23249013788b10 - 61200377210b9 + 272018121360b8 - 78161705818b7 + 163265339787b6 + 30122424321b5 + 4009535907b4 - 63321360960b3 - 250710276960b2 - 307831395120*b1 + 39228280728

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 161.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} - 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) (T^8 - 2T^6 + 4T^4 - 8*T^2 + 16)^2
$3$	\( (T^{8} - T^{7} + \cdots + 331776)^{2} \) (T^8 - T^7 + 25T^6 - 49T^5 + 649T^4 + 1176T^3 + 14400T^2 + 13824T + 331776)^2
$5$	\( (T^{8} + T^{7} + \cdots + 331776)^{2} \) (T^8 + T^7 + 25T^6 + 49T^5 + 649T^4 - 1176T^3 + 14400T^2 - 13824T + 331776)^2
$7$	\( T^{16} + \cdots + 110075314176 \) T^16 - 146T^14 + 20740T^12 - 2943944T^10 + 417869584T^8 - 1695711744T^6 + 6881034240T^4 - 27901034496*T^2 + 110075314176
$11$	\( T^{16} \) T^16
$13$	\( T^{16} - 178 T^{14} + \cdots + 16777216 \) T^16 - 178T^14 + 31620T^12 - 5616968T^10 + 997796624T^8 - 359485952T^6 + 129515520T^4 - 46661632*T^2 + 16777216
$17$	\( T^{16} - 788 T^{14} + \cdots + 1679616 \) T^16 - 788T^14 + 620908T^12 - 489247136T^10 + 385504390480T^8 - 17612896896T^6 + 804696768T^4 - 36764928*T^2 + 1679616
$19$	\( (T^{8} - 32 T^{6} + \cdots + 1048576)^{2} \) (T^8 - 32T^6 + 1024T^4 - 32768*T^2 + 1048576)^2
$23$	\( (T^{2} - 19 T - 516)^{8} \) (T^2 - 19*T - 516)^8
$29$	\( T^{16} + \cdots + 31\!\cdots\!56 \) T^16 - 2546T^14 + 5155012T^12 - 9745853768T^10 + 17971706648080T^8 - 12933761518927872T^6 + 9079033047696801792T^4 - 5950771492620802719744*T^2 + 3101843146481947279097856
$31$	\( (T^{8} + 69 T^{7} + \cdots + 892616806656)^{2} \) (T^8 + 69T^7 + 3789T^6 + 194373T^5 + 9728829T^4 + 188930556T^3 + 3579786576T^2 + 63364773312*T + 892616806656)^2
$37$	\( (T^{8} + 15 T^{7} + \cdots + 688747536)^{2} \) (T^8 + 15T^7 + 387T^6 + 8235T^5 + 186219T^4 - 1334070T^3 + 10156428T^2 - 63772920*T + 688747536)^2
$41$	\( T^{16} + \cdots + 47\!\cdots\!96 \) T^16 - 1314T^14 + 1679940T^12 - 2146135176T^10 + 2741642340624T^8 - 100130082771456T^6 + 3656863717539840T^4 - 133449783062298624*T^2 + 4738381338321616896
$43$	\( (T^{4} + 5282 T^{2} + 4460544)^{4} \) (T^4 + 5282*T^2 + 4460544)^4
$47$	\( (T^{8} + 28 T^{7} + \cdots + 1358954496)^{2} \) (T^8 + 28T^7 + 976T^6 + 32704T^5 + 1103104T^4 - 6279168T^3 + 35979264T^2 - 198180864*T + 1358954496)^2
$53$	\( (T^{8} - 34 T^{7} + \cdots + 1358954496)^{2} \) (T^8 - 34T^7 + 964T^6 - 26248T^5 + 707344T^4 - 5039616T^3 + 35536896T^2 - 240648192*T + 1358954496)^2
$59$	\( (T^{8} + 89 T^{7} + \cdots + 393460125696)^{2} \) (T^8 + 89T^7 + 7129T^6 + 563993T^5 + 44549209T^4 + 446682456T^3 + 4471765056T^2 + 44214584832*T + 393460125696)^2
$61$	\( T^{16} + \cdots + 54\!\cdots\!36 \) T^16 - 10706T^14 + 106045252T^12 - 1043535960008T^10 + 10262950530123280T^8 - 8946425795765225472T^6 + 7794271291818081779712T^4 - 6746113960762684760653824*T^2 + 5402174133251193423041396736
$67$	\( (T^{2} + 151 T + 4512)^{8} \) (T^2 + 151*T + 4512)^8
$71$	\( (T^{8} + \cdots + 25\!\cdots\!96)^{2} \) (T^8 + 171T^7 + 22149T^6 + 2574747T^5 + 283201029T^4 + 18260105724T^3 + 1114016381136T^2 + 60996131379648*T + 2529734290903296)^2
$73$	\( T^{16} + \cdots + 18\!\cdots\!76 \) T^16 - 14338T^14 + 184711620T^12 - 2349209552648T^10 + 29828658642896144T^8 - 49020072432314020352T^6 + 80426394208854315909120T^4 - 130270220268691068891430912*T^2 + 189587090580552064912511205376
$79$	\( T^{16} + \cdots + 1099511627776 \) T^16 - 15778T^14 + 248944260T^12 - 3927826377608T^10 + 61972989666976784T^8 - 4022094210670592T^6 + 261036976373760T^4 - 16941498499072*T^2 + 1099511627776
$83$	\( T^{16} + \cdots + 88\!\cdots\!56 \) T^16 - 8354T^14 + 66719812T^12 - 531734673032T^10 + 4237314728696080T^8 - 1632161705810416128T^6 + 628624381348683988992T^4 - 241601197155010497478656*T^2 + 88771347985646796989792256
$89$	\( (T^{2} - 169 T + 4206)^{8} \) (T^2 - 169*T + 4206)^8
$97$	\( (T^{8} + \cdots + 100725792178176)^{2} \) (T^8 - 113T^7 + 9601T^6 - 726929T^5 + 51727009T^4 - 2302911072T^3 + 96357786624T^2 - 3592807612416*T + 100725792178176)^2
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\(n\)	\(123\)
\(\chi(n)\)	\(-\beta_{1}\)