LMFDB - Newform orbit 400.6.n.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("400.143");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.n (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} + 12 x^{14} - 100 x^{13} + 524 x^{12} - 400 x^{11} + 3746 x^{10} - 42400 x^{9} + \cdots + 1540307025 $ x^16 + 12x^14 - 100x^13 + 524x^12 - 400x^11 + 3746x^10 - 42400x^9 + 111549x^8 + 256200x^7 - 966658x^6 - 8816000x^5 + 21164416x^4 + 39337200x^3 + 22584690x^2 - 776398500x + 1540307025
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{44}\cdot 5^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q - \beta_{4} q^{3} + \beta_{7} q^{7} + (\beta_{14} - 182 \beta_1) q^{9}+O(q^{10}) $ q - b4 * q^3 + b7 * q^7 + (b14 - 182b1) q^9	\( q - \beta_{4} q^{3} + \beta_{7} q^{7} + (\beta_{14} - 182 \beta_1) q^{9} + (\beta_{9} - \beta_{8}) q^{11} + (\beta_{15} + 7 \beta_{3}) q^{13} + (\beta_{13} + 10 \beta_{6}) q^{17} + (\beta_{12} + 7 \beta_{11}) q^{19} + (\beta_{10} - 100) q^{21} + ( - 30 \beta_{4} + 18 \beta_{2}) q^{23} + ( - 21 \beta_{7} + 343 \beta_{5}) q^{27} + ( - 9 \beta_{14} + 1314 \beta_1) q^{29} + (2 \beta_{9} + 20 \beta_{8}) q^{31} + ( - 10 \beta_{15} - 109 \beta_{3}) q^{33} + ( - 11 \beta_{13} + 9 \beta_{6}) q^{37} + (28 \beta_{12} + 63 \beta_{11}) q^{39} + ( - 12 \beta_{10} - 7827) q^{41} + ( - 264 \beta_{4} - 125 \beta_{2}) q^{43} + (135 \beta_{7} + 882 \beta_{5}) q^{47} + (24 \beta_{14} - 6807 \beta_1) q^{49} + (31 \beta_{9} + 41 \beta_{8}) q^{51} + (34 \beta_{15} + 16 \beta_{3}) q^{53} + (45 \beta_{13} + 858 \beta_{6}) q^{57} + (12 \beta_{12} + 27 \beta_{11}) q^{59} + (57 \beta_{10} - 17452) q^{61} + ( - 304 \beta_{4} + 222 \beta_{2}) q^{63} + ( - 355 \beta_{7} + 315 \beta_{5}) q^{67} + (12 \beta_{14} - 14550 \beta_1) q^{69} + ( - 56 \beta_{9} - 79 \beta_{8}) q^{71} + ( - 21 \beta_{15} - 1034 \beta_{3}) q^{73} + ( - 65 \beta_{13} + 1159 \beta_{6}) q^{77} + ( - 106 \beta_{12} - 65 \beta_{11}) q^{79} + ( - 121 \beta_{10} - 99449) q^{81} + (2745 \beta_{4} + 297 \beta_{2}) q^{83} + (189 \beta_{7} - 4950 \beta_{5}) q^{87} + ( - 141 \beta_{14} + 8139 \beta_1) q^{89} + (44 \beta_{9} - 146 \beta_{8}) q^{91} + ( - 130 \beta_{15} - 2572 \beta_{3}) q^{93} + ( - 76 \beta_{13} + 2592 \beta_{6}) q^{97} + ( - 76 \beta_{12} - 747 \beta_{11}) q^{99}+O(q^{100}) \) q - b4 * q^3 + b7 * q^7 + (b14 - 182b1) q^9 + (b9 - b8) * q^11 + (b15 + 7b3) q^13 + (b13 + 10b6) q^17 + (b12 + 7b11) q^19 + (b10 - 100) * q^21 + (-30b4 + 18b2) * q^23 + (-21b7 + 343b5) * q^27 + (-9b14 + 1314b1) * q^29 + (2b9 + 20b8) * q^31 + (-10b15 - 109b3) * q^33 + (-11b13 + 9b6) * q^37 + (28b12 + 63b11) * q^39 + (-12b10 - 7827) q^41 + (-264b4 - 125b2) * q^43 + (135b7 + 882b5) * q^47 + (24b14 - 6807b1) * q^49 + (31b9 + 41b8) * q^51 + (34b15 + 16b3) * q^53 + (45b13 + 858b6) * q^57 + (12b12 + 27b11) * q^59 + (57b10 - 17452) q^61 + (-304b4 + 222b2) * q^63 + (-355b7 + 315b5) * q^67 + (12b14 - 14550b1) * q^69 + (-56b9 - 79b8) * q^71 + (-21b15 - 1034b3) * q^73 + (-65b13 + 1159b6) * q^77 + (-106b12 - 65b11) * q^79 + (-121b10 - 99449) q^81 + (2745b4 + 297b2) * q^83 + (189b7 - 4950b5) * q^87 + (-141b14 + 8139b1) * q^89 + (44b9 - 146b8) * q^91 + (-130b15 - 2572b3) * q^93 + (-76b13 + 2592b6) * q^97 + (-76b12 - 747b11) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 1600 q^{21} - 125232 q^{41} - 279232 q^{61} - 1591184 q^{81}+O(q^{100}) $ 16 * q - 1600 * q^21 - 125232 * q^41 - 279232 * q^61 - 1591184 * q^81

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 12 x^{14} - 100 x^{13} + 524 x^{12} - 400 x^{11} + 3746 x^{10} - 42400 x^{9} + \cdots + 1540307025 $ x^16 + 12*x^14 - 100*x^13 + 524*x^12 - 400*x^11 + 3746*x^10 - 42400*x^9 + 111549*x^8 + 256200*x^7 - 966658*x^6 - 8816000*x^5 + 21164416*x^4 + 39337200*x^3 + 22584690*x^2 - 776398500*x + 1540307025 :

$\beta_{1}$	$=$	$ ( - 19\!\cdots\!66 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 68\!\cdots\!00 $ (-195074649541311466v^15 - 541397737016403060v^14 - 3388912903611125313v^13 + 9536444368640027990v^12 - 71644136543720123445v^11 - 178756255787105175230v^10 - 937803941496722809274v^9 + 5180076001208730712170v^8 - 5753658368504108455737v^7 - 93166748315198245843710v^6 + 14880755823168077274792v^5 + 1804402920222644655337890v^4 + 370961731313154242337863v^3 - 14371831031931429264809450v^2 - 19358782950935906263805715*v + 112532685208449202111936500) / 68612732252282579508354900
$\beta_{2}$	$=$	$ ( - 65\!\cdots\!41 \nu^{15} + \cdots + 48\!\cdots\!10 ) / 12\!\cdots\!70 $ (-650107439589231077056598841v^15 + 3943513188329328262022714506v^14 - 87520139401071501607052617980v^13 + 159902946862669887332222736856v^12 - 1083695515755312712337397258731v^11 + 7310372919387162080370159477866v^10 - 40152563315575107629334788335525v^9 + 24798904524842363941061605681318v^8 - 149956372041578242023319424034325v^7 + 2384439068700497208285634886205210v^6 - 11247989393408222254869458185474345v^5 - 5348948080866573870822138669259254v^4 + 38391576199132991245602600566137522v^3 + 751641364175436782604580478291920388v^2 - 3784942853437247274698597070655908255*v + 4838625488434624490296328375639179410) / 126228070409373484262009472276473370
$\beta_{3}$	$=$	$ ( - 14\!\cdots\!81 \nu^{15} + \cdots + 19\!\cdots\!50 ) / 30\!\cdots\!48 $ (-14556186835196313105836081v^15 - 113747454828219143339284494v^14 - 613465932378948839859081618v^13 - 845340418746123601800573524v^12 - 5561339923268364973239422505v^11 - 26259733704684871124574095842v^10 - 199217952969016024036465016589v^9 + 189296356271323805542308265998v^8 + 492118066521562934662578355053v^7 - 2762334401975684966724860833434v^6 - 41640025221049996772267607822813v^5 + 133529128533233384798467492637946v^4 + 645935822792222682373254665122648v^3 + 379390352407626996238536825154040v^2 - 3276264020809050842906093305372765*v + 1966543506771532700926973812856550) / 308814851154430542537026231869048
$\beta_{4}$	$=$	$ ( - 32\!\cdots\!99 \nu^{15} + \cdots + 10\!\cdots\!90 ) / 50\!\cdots\!80 $ (-32940961415269793605273576599v^15 - 94053963233403726084676624586v^14 - 569341030097998285871740848720v^13 + 1967400806118472530578252040784v^12 - 10025078497107998259197023076249v^11 - 22113236631979429104494923270546v^10 - 175577999073039157121264263030075v^9 + 904544796735782347625679545462002v^8 - 769504818416174569464120449214895v^7 - 13658714537049572078690899646003010v^6 - 10889835846212934210042228238696255v^5 + 272800593759337762765120519529063694v^4 + 116941678405890393661980733902505618v^3 - 1614169550530142437171861144260786028v^2 - 5706794697317694100654943164666894845*v + 10522861263731700639388488268717847790) / 504912281637493937048037889105893480
$\beta_{5}$	$=$	$ ( 33\!\cdots\!28 \nu^{15} + \cdots - 27\!\cdots\!10 ) / 50\!\cdots\!80 $ (33719511615786396066101929428v^15 + 53162748732827147853792176221v^14 + 473206955085911726867969211618v^13 - 2489993613354724141467941928245v^12 + 15040242953645426735966495402686v^11 + 20004496893328768793494851993050v^10 + 173007465874984837056313953748424v^9 - 1095200561098761303992395428418715v^8 + 2308398731292857010553603176196994v^7 + 15034972238024634626111768567824092v^6 - 8071117255873368384183220269848812v^5 - 326627025747263850517891821547461345v^4 + 180352045646465494362322006744691302v^3 + 1872094160987085070172032721361070187v^2 + 2905262072691377639643806813178455070*v - 27424892097754518063472020570638985810) / 504912281637493937048037889105893480
$\beta_{6}$	$=$	$ ( - 70\!\cdots\!88 \nu^{15} + \cdots + 29\!\cdots\!00 ) / 92\!\cdots\!44 $ (-70205137562217430267466488v^15 - 49564298173598030059417571v^14 - 988049458770972272729733062v^13 + 8166855419183763065599909109v^12 - 24749897711516230187658688718v^11 + 58902986922847042476161946372v^10 - 254580961605755608117828493476v^9 + 3292169402502024425085602294557v^8 - 2882967631444864811516211747850v^7 - 12710755913054944368096732018106v^6 + 33427102523357929279568864248128v^5 + 597053792131629350537136510146139v^4 + 301677083080205470191586375956766v^3 - 2802415146134013302175398835296415v^2 - 16262741806490084370125816107057230*v + 29618995432359872356433657041996800) / 926444553463291627611078695607144
$\beta_{7}$	$=$	$ ( - 11\!\cdots\!12 \nu^{15} + \cdots + 64\!\cdots\!90 ) / 12\!\cdots\!70 $ (-11034197237481903163659826812v^15 + 14393970944374861117541758301v^14 - 137249825147882198114951063082v^13 + 1068263304938287049995051132115v^12 - 6942996440994144283432823082694v^11 + 10352353766161888800691378803730v^10 - 43956700707684989048249593905176v^9 + 434696917590018726624514693071965v^8 - 1928791047787947755756894540505026v^7 + 138695500514411808428725796999892v^6 + 11077762392436271941459069610812588v^5 + 70515063084137826744477406457752215v^4 - 380329900808802814956103978403641558v^3 + 403770654767679486097042003802777827v^2 + 243258572342081450188767447799499970*v + 646507059716283457252927505496126990) / 126228070409373484262009472276473370
$\beta_{8}$	$=$	$ ( - 386904741309120 \nu^{15} + \cdots + 36\!\cdots\!50 ) / 36\!\cdots\!21 $ (-386904741309120v^15 + 4901386187843360v^14 + 14930200679013968v^13 + 169471339126362320v^12 - 159533939880846336v^11 + 2198884973551643080v^10 + 5440339477676287904v^9 + 50357337384833369460v^8 - 35477764393871446688v^7 + 9994407299094377560v^6 + 2695976812027721087968v^5 + 2066682613129147593920v^4 - 28772185135083218160896v^3 - 65390386341533169786600v^2 - 110474062916059190735760*v + 363055312061310549521250) / 3686761422636525153921
$\beta_{9}$	$=$	$ ( 40\!\cdots\!15 \nu^{15} + \cdots + 89\!\cdots\!50 ) / 31\!\cdots\!26 $ (4013389642213010968815v^15 + 59467525757055100814345v^14 + 508075467348775499273694v^13 + 2113863005264458418942705v^12 + 6216628717394168542018847v^11 + 21117589171442360172006160v^10 + 152362260882963876331815007v^9 + 738111323229641839721174265v^8 + 919683455911778266731965401v^7 + 2177607051496751955743194470v^6 + 13245230386183937192196780119v^5 + 120681582109532085585973110055v^4 - 376089928913989799490428635008v^3 - 2169493022250711876935749066875v^2 - 2089215994783045936619551084605*v + 8999683071623575326937829796750) / 31733575281817583646515669626
$\beta_{10}$	$=$	$ ( - 20\!\cdots\!43 \nu^{15} + \cdots + 17\!\cdots\!50 ) / 70\!\cdots\!95 $ (-209747550324968443v^15 + 219655889509833809v^14 - 2098867122363508648v^13 + 22383958417185782839v^12 - 129529296099916661749v^11 + 153722821776435657462v^10 - 533346652268782680579v^9 + 9163467009821971528997v^8 - 33339546289916261505623v^7 - 46943066574622371194676v^6 + 380234869493783223743817v^5 + 1704104788363821299574219v^4 - 8525844925709578035930270v^3 - 6371065235985422002910865v^2 + 77611815704103730571264175*v + 172182635219072739926363650) / 705648449256352647640595
$\beta_{11}$	$=$	$ ( - 611291030700 \nu^{15} - 2136595335104 \nu^{14} - 15144686636675 \nu^{13} + \cdots + 18\!\cdots\!80 ) / 12\!\cdots\!14 $ (-611291030700v^15 - 2136595335104v^14 - 15144686636675v^13 + 29363614313110v^12 - 219320009874225v^11 - 208376508634798v^10 - 5192088771806000v^9 + 19156385369150910v^8 - 10211550023587825v^7 - 82131723381905902v^6 - 389926610513676850v^5 + 5504744089275463750v^4 + 12583047098063665625v^3 - 9484122868498118046v^2 - 141656400480355182975*v + 186387641705995881480) / 1210752112183470714
$\beta_{12}$	$=$	$ ( - 22\!\cdots\!10 \nu^{15} + \cdots + 19\!\cdots\!80 ) / 26\!\cdots\!72 $ (-22879371704575172399310v^15 - 87162641901719443750394v^14 - 376417659221683317757895v^13 + 610285110694029261259540v^12 - 4650995733175218422895555v^11 - 33376541360567566501504678v^10 - 49682539355844271550920610v^9 + 350556618296788977185018700v^8 + 916249421159513365136959265v^7 - 11588136753452432938792971922v^6 + 12896893709688273887692832780v^5 + 223686615921034171276130512480v^4 + 121076118156801506933136434825v^3 - 585447289086092288551878221256v^2 - 1742212420394340437164670403225*v + 1912758326286166901140186994280) / 26417784383879627881309539372
$\beta_{13}$	$=$	$ ( - 23\!\cdots\!76 \nu^{15} + \cdots + 10\!\cdots\!00 ) / 24\!\cdots\!80 $ (-23269072320803022483545241576v^15 - 74647874970553808256431306485v^14 - 160532341128362388382351001338v^13 + 1935065388961411822758576863315v^12 - 535091933411953026106514776690v^11 - 18890201748218636753295597854980v^10 + 13676790413380724861222897522276v^9 + 974319986104039202236771430892395v^8 + 1079946788976677128048652838879658v^7 - 12292966071147246963816359607731510v^6 + 5753062209719939772884127292876992v^5 + 349196025031777141724909879345687965v^4 + 72242336771045345731806933496172098v^3 - 2070205229639566154067667871473776825v^2 - 4250395034449050450404669380697143890*v + 10034444444434945148598505094035713600) / 24043441982737806526097042338375880
$\beta_{14}$	$=$	$ ( 258301036705135 \nu^{15} + 502109284438689 \nu^{14} + \cdots - 14\!\cdots\!50 ) / 19\!\cdots\!85 $ (258301036705135v^15 + 502109284438689v^14 + 4647655705588452v^13 - 15767391627503981v^12 + 116952842935115517v^11 + 90954870072415802v^10 + 1462995547883643071v^9 - 8064716701106924163v^8 + 17876750408355182511v^7 + 81924151481989064904v^6 - 20133840181615852773v^5 - 2225434474811263739901v^4 + 1627638095592974857402v^3 + 8034840710423054739635v^2 + 40417211462050623700665*v - 147400966924327004020650) / 199277830932108022185
$\beta_{15}$	$=$	$ ( - 31\!\cdots\!05 \nu^{15} + \cdots - 23\!\cdots\!50 ) / 24\!\cdots\!80 $ (-31279010028081485616835194405v^15 - 129138833812248157095409723366v^14 - 831041746071231210998864983178v^13 - 300180264272636152546724189636v^12 - 11738123558608676083104523912493v^11 - 42624737253542040858612357770538v^10 - 164362591150952645564757624325169v^9 + 259413025176867846152353784830022v^8 + 271646335287228679585373603135201v^7 - 12476054908676104366945148394805826v^6 + 2482593250146996609541455616852447v^5 + 156000731733472811123089245699138594v^4 + 401952775179556521665362739892793512v^3 + 763084799379702663439294883539741560v^2 - 1674025098600162656137294957956242385*v - 2360540065817587197301159955298718050) / 24043441982737806526097042338375880

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

$\nu$	$=$	$ ( 5\beta_{14} + 5\beta_{10} + 5\beta_{7} - 20\beta_{5} + 20\beta_{4} + 16\beta_{3} + 5\beta_{2} + 400\beta_1 ) / 800 $ (5b14 + 5b10 + 5b7 - 20b5 + 20b4 + 16b3 + 5b2 + 400b1) / 800
$\nu^{2}$	$=$	$ ( 5 \beta_{15} - 5 \beta_{14} + 5 \beta_{13} - 2 \beta_{12} - \beta_{11} + 5 \beta_{10} - 2 \beta_{9} + \cdots - 1200 ) / 800 $ (5b15 - 5b14 + 5b13 - 2b12 - b11 + 5b10 - 2b9 + b8 - 45b7 - 15b6 - 220b5 + 20b4 + b3 + 5b2 - 10000b1 - 1200) / 800
$\nu^{3}$	$=$	$ ( 15 \beta_{15} + 65 \beta_{14} - 15 \beta_{13} - 6 \beta_{12} - 3 \beta_{11} - 155 \beta_{10} + \cdots + 30000 ) / 1600 $ (15b15 + 65b14 - 15b13 - 6b12 - 3b11 - 155b10 + 6b9 - 123b8 + 355b7 + 1245b6 - 1420b5 - 860b4 - 141b3 - 515b2 - 3200*b1 + 30000) / 1600
$\nu^{4}$	$=$	$ ( - 85 \beta_{15} + 75 \beta_{14} + 40 \beta_{13} + 82 \beta_{12} - 19 \beta_{11} + 35 \beta_{10} + \cdots - 45200 ) / 400 $ (-85b15 + 75b14 + 40b13 + 82b12 - 19b11 + 35b10 - 74b9 + 37b8 + 275b7 + 72b6 - 300b5 - 1920b4 + 607b3 + 620b2 + 45000*b1 - 45200) / 400
$\nu^{5}$	$=$	$ ( 425 \beta_{15} - 2265 \beta_{14} + 825 \beta_{13} - 750 \beta_{12} + 4625 \beta_{11} + 1035 \beta_{10} + \cdots - 400000 ) / 1600 $ (425b15 - 2265b14 + 825b13 - 750b12 + 4625b11 + 1035b10 - 810b9 + 1205b8 - 12675b7 - 18555b6 + 6700b5 - 16700b4 - 24283b3 - 2675b2 - 457200*b1 - 400000) / 1600
$\nu^{6}$	$=$	$ ( 930 \beta_{15} - 1180 \beta_{14} - 4695 \beta_{13} + 3084 \beta_{12} + 2442 \beta_{11} + \cdots + 2089800 ) / 800 $ (930b15 - 1180b14 - 4695b13 + 3084b12 + 2442b11 - 3220b10 + 6204b9 - 10602b8 + 14765b7 + 35925b6 + 29740b5 + 6760b4 - 24774b3 - 20310b2 - 155000*b1 + 2089800) / 800
$\nu^{7}$	$=$	$ ( - 15680 \beta_{15} + 19770 \beta_{14} - 1820 \beta_{13} + 20398 \beta_{12} - 24801 \beta_{11} + \cdots - 140000 ) / 800 $ (-15680b15 + 19770b14 - 1820b13 + 20398b12 - 24801b11 + 4485b10 - 12208b9 + 64064b8 + 54185b7 - 113220b6 + 14260b5 + 8980b4 + 388472b3 + 82395b2 + 6512400*b1 - 140000) / 800
$\nu^{8}$	$=$	$ ( 43665 \beta_{15} - 11550 \beta_{14} + 45165 \beta_{13} - 85436 \beta_{12} + 75162 \beta_{11} + \cdots - 15916200 ) / 400 $ (43665b15 - 11550b14 + 45165b13 - 85436b12 + 75162b11 + 32110b10 - 1708b9 + 53354b8 - 173925b7 - 688023b6 - 178700b5 - 128740b4 - 273763b3 + 81015b2 - 6510000*b1 - 15916200) / 400
$\nu^{9}$	$=$	$ ( 626625 \beta_{15} - 727265 \beta_{14} - 804375 \beta_{13} + 210870 \beta_{12} - 1361565 \beta_{11} + \cdots + 113880000 ) / 1600 $ (626625b15 - 727265b14 - 804375b13 + 210870b12 - 1361565b11 - 365465b10 + 1387962b9 - 3558141b8 - 1585865b7 + 19339605b6 + 780260b5 - 2744660b4 - 6282243b3 - 3653465b2 - 210269200*b1 + 113880000) / 1600
$\nu^{10}$	$=$	$ ( - 2993270 \beta_{15} + 732895 \beta_{14} + 466105 \beta_{13} + 2328890 \beta_{12} - 4222055 \beta_{11} + \cdots + 784759800 ) / 800 $ (-2993270b15 + 732895b14 + 466105b13 + 2328890b12 - 4222055b11 - 1358695b10 - 3228310b9 + 6016655b8 + 10898025b7 + 5914965b6 + 16265900b5 - 1194400b4 + 43746626b3 - 3328600b2 + 559325000*b1 + 784759800) / 800
$\nu^{11}$	$=$	$ ( 10531015 \beta_{15} + 17655065 \beta_{14} + 25521485 \beta_{13} - 33137962 \beta_{12} + \cdots - 5115990000 ) / 1600 $ (10531015b15 + 17655065b14 + 25521485b13 - 33137962b12 + 56031019b11 + 21915895b10 - 13172258b9 + 19072889b8 - 533015b7 - 387061575b6 + 89054060b5 + 40394380b4 - 272025061b3 + 149285995b2 + 6843116800*b1 - 5115990000) / 1600
$\nu^{12}$	$=$	$ ( 34375605 \beta_{15} - 58090500 \beta_{14} - 59666145 \beta_{13} + 23482272 \beta_{12} + \cdots - 11024940400 ) / 800 $ (34375605b15 - 58090500b14 - 59666145b13 + 23482272b12 - 6142224b11 + 39009520b10 + 86398776b9 - 211004388b8 - 421842275b7 + 852309939b6 - 235886100b5 + 373248780b4 - 691675311b3 - 145585205b2 - 32533380000*b1 - 11024940400) / 800
$\nu^{13}$	$=$	$ ( - 312542425 \beta_{15} + 33120030 \beta_{14} - 67194075 \beta_{13} + 349525410 \beta_{12} + \cdots + 179556130000 ) / 800 $ (-312542425b15 + 33120030b14 - 67194075b13 + 349525410b12 - 459585295b11 - 480747870b10 - 222230658b9 + 623880569b8 + 1872969210b7 + 543350145b6 - 773352040b5 + 1280177840b4 + 5235294971b3 - 1932583840b2 - 29431323600*b1 + 179556130000) / 800
$\nu^{14}$	$=$	$ ( 428817930 \beta_{15} + 2506891595 \beta_{14} + 1446084180 \beta_{13} - 1721047274 \beta_{12} + \cdots - 526063147200 ) / 800 $ (428817930b15 + 2506891595b14 + 1446084180b13 - 1721047274b12 + 2294810363b11 + 782424305b10 - 941995234b9 + 2520044617b8 + 5850014210b7 - 23589446700b6 - 3818707640b5 - 9258861560b4 - 5174541854b3 + 14180555610b2 + 986196130000*b1 - 526063147200) / 800
$\nu^{15}$	$=$	$ ( 11299483515 \beta_{15} - 23872102485 \beta_{14} - 7719283515 \beta_{13} - 3370608330 \beta_{12} + \cdots - 8727831850000 ) / 1600 $ (11299483515b15 - 23872102485b14 - 7719283515b13 - 3370608330b12 + 3240395835b11 + 17944458495b10 + 17598276330b9 - 44126370765b8 - 171197334725b7 + 138841343265b6 - 66397461100b5 - 57570514300b4 - 182186320881b3 - 13790445075b2 - 8632049347200*b1 - 8727831850000) / 1600

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-\beta_{1}$	$-1$

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} $

143.1

1.96154	−	3.80431i
3.18628	−	2.57957i
−1.96154	+	4.80431i
−3.18628	+	3.57957i
−3.18628	+	1.34350i
−1.96154	+	2.56825i
3.18628	−	0.343501i
1.96154	−	1.56825i
3.18628	+	2.57957i
1.96154	+	3.80431i
−3.18628	−	3.57957i
−1.96154	−	4.80431i
−1.96154	−	2.56825i
−3.18628	−	1.34350i
1.96154	+	1.56825i
3.18628	+	0.343501i

−20.4551

20.4551i

−7.62217

−

7.62217i

−

593.825i

143.2

−20.4551

20.4551i

−7.62217

−

7.62217i

−

593.825i

143.3

−2.56659

2.56659i

99.7091

99.7091i

229.825i

143.4

−2.56659

2.56659i

99.7091

99.7091i

229.825i

143.5

2.56659

−

2.56659i

−99.7091

−

99.7091i

229.825i

143.6

2.56659

−

2.56659i

−99.7091

−

99.7091i

229.825i

143.7

20.4551

−

20.4551i

7.62217

7.62217i

−

593.825i

143.8

20.4551

−

20.4551i

7.62217

7.62217i

−

593.825i

207.1

−20.4551

−

20.4551i

−7.62217

7.62217i

593.825i

207.2

−20.4551

−

20.4551i

−7.62217

7.62217i

593.825i

207.3

−2.56659

−

2.56659i

99.7091

−

99.7091i

−

229.825i

207.4

−2.56659

−

2.56659i

99.7091

−

99.7091i

−

229.825i

207.5

2.56659

2.56659i

−99.7091

99.7091i

−

229.825i

207.6

2.56659

2.56659i

−99.7091

99.7091i

−

229.825i

207.7

20.4551

20.4551i

7.62217

−

7.62217i

593.825i

207.8

20.4551

20.4551i

7.62217

−

7.62217i

593.825i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
20.d	odd	2	1	inner
20.e	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	400.6.n.f	✓	16
4.b	odd	2	1	inner	400.6.n.f	✓	16
5.b	even	2	1	inner	400.6.n.f	✓	16
5.c	odd	4	2	inner	400.6.n.f	✓	16
20.d	odd	2	1	inner	400.6.n.f	✓	16
20.e	even	4	2	inner	400.6.n.f	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
400.6.n.f	✓	16	1.a	even	1	1	trivial
400.6.n.f	✓	16	4.b	odd	2	1	inner
400.6.n.f	✓	16	5.b	even	2	1	inner
400.6.n.f	✓	16	5.c	odd	4	2	inner
400.6.n.f	✓	16	20.d	odd	2	1	inner
400.6.n.f	✓	16	20.e	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 700450T_{3}^{4} + 121550625 $ T3^8 + 700450*T3^4 + 121550625 acting on $S_{6}^{\mathrm{new}}(400, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 700450 T^{4} + 121550625)^{2} $ (T^8 + 700450*T^4 + 121550625)^2
$5$	$ T^{16} $ T^16
$7$	$ (T^{8} + \cdots + 5337948160000)^{2} $ (T^8 + 395379200*T^4 + 5337948160000)^2
$11$	$ (T^{4} + 639750 T^{2} + 78470015625)^{4} $ (T^4 + 639750*T^2 + 78470015625)^4
$13$	$ (T^{8} + \cdots + 31\!\cdots\!00)^{2} $ (T^8 + 1062144180000*T^4 + 31743982321928100000000)^2
$17$	$ (T^{8} + \cdots + 11\!\cdots\!25)^{2} $ (T^8 + 1682074811250*T^4 + 11676990750090781640625)^2
$19$	$ (T^{4} + \cdots + 9356716265625)^{4} $ (T^4 - 7317750*T^2 + 9356716265625)^4
$23$	$ (T^{8} + \cdots + 10\!\cdots\!00)^{2} $ (T^8 + 49303920391200*T^4 + 10120671626775532659360000)^2
$29$	$ (T^{4} + \cdots + 144264217088016)^{4} $ (T^4 + 30928392*T^2 + 144264217088016)^4
$31$	$ (T^{4} + \cdots + 11\!\cdots\!00)^{4} $ (T^4 + 72519000*T^2 + 1146465740250000)^4
$37$	$ (T^{8} + \cdots + 14\!\cdots\!00)^{2} $ (T^8 + 7754320327680000*T^4 + 14147526909890689243545600000000)^2
$41$	$ (T^{2} + 15654 T + 36839529)^{8} $ (T^2 + 15654*T + 36839529)^8
$43$	$ (T^{8} + \cdots + 50\!\cdots\!00)^{2} $ (T^8 + 83700803843187200*T^4 + 502909219240796992068501544960000)^2
$47$	$ (T^{8} + \cdots + 81\!\cdots\!00)^{2} $ (T^8 + 579610152618055200*T^4 + 81938729259969565250486582105760000)^2
$53$	$ (T^{8} + \cdots + 11\!\cdots\!00)^{2} $ (T^8 + 693014651971380000*T^4 + 119550379925851581055317416100000000)^2
$59$	$ (T^{4} + \cdots + 66267740250000)^{4} $ (T^4 - 156519000*T^2 + 66267740250000)^4
$61$	$ (T^{2} + 34904 T - 246458096)^{8} $ (T^2 + 34904*T - 246458096)^8
$67$	$ (T^{8} + \cdots + 16\!\cdots\!25)^{2} $ (T^8 + 5753013463520281250*T^4 + 160462793712556970110616894775390625)^2
$71$	$ (T^{4} + \cdots + 936635420250000)^{4} $ (T^4 + 3701991000*T^2 + 936635420250000)^4
$73$	$ (T^{8} + \cdots + 96\!\cdots\!25)^{2} $ (T^8 + 13362228490918511250*T^4 + 9683404564601654193546260519094140625)^2
$79$	$ (T^{4} + \cdots + 11\!\cdots\!00)^{4} $ (T^4 - 6764496000*T^2 + 11285245859904000000)^4
$83$	$ (T^{8} + \cdots + 48\!\cdots\!25)^{2} $ (T^8 + 47607211977833304450*T^4 + 48323419245898735736721742162997450625)^2
$89$	$ (T^{4} + \cdots + 10\!\cdots\!41)^{4} $ (T^4 + 6876121842*T^2 + 10926821313986369841)^4
$97$	$ (T^{8} + \cdots + 92\!\cdots\!00)^{2} $ (T^8 + 791633588575106880000*T^4 + 9248057745938023118943297026073600000000)^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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{3} + \beta_{7} q^{7} + (\beta_{14} - 182 \beta_1) q^{9}+O(q^{10}) \) q - b4 * q^3 + b7 * q^7 + (b14 - 182b1) q^9	\( q - \beta_{4} q^{3} + \beta_{7} q^{7} + (\beta_{14} - 182 \beta_1) q^{9} + (\beta_{9} - \beta_{8}) q^{11} + (\beta_{15} + 7 \beta_{3}) q^{13} + (\beta_{13} + 10 \beta_{6}) q^{17} + (\beta_{12} + 7 \beta_{11}) q^{19} + (\beta_{10} - 100) q^{21} + ( - 30 \beta_{4} + 18 \beta_{2}) q^{23} + ( - 21 \beta_{7} + 343 \beta_{5}) q^{27} + ( - 9 \beta_{14} + 1314 \beta_1) q^{29} + (2 \beta_{9} + 20 \beta_{8}) q^{31} + ( - 10 \beta_{15} - 109 \beta_{3}) q^{33} + ( - 11 \beta_{13} + 9 \beta_{6}) q^{37} + (28 \beta_{12} + 63 \beta_{11}) q^{39} + ( - 12 \beta_{10} - 7827) q^{41} + ( - 264 \beta_{4} - 125 \beta_{2}) q^{43} + (135 \beta_{7} + 882 \beta_{5}) q^{47} + (24 \beta_{14} - 6807 \beta_1) q^{49} + (31 \beta_{9} + 41 \beta_{8}) q^{51} + (34 \beta_{15} + 16 \beta_{3}) q^{53} + (45 \beta_{13} + 858 \beta_{6}) q^{57} + (12 \beta_{12} + 27 \beta_{11}) q^{59} + (57 \beta_{10} - 17452) q^{61} + ( - 304 \beta_{4} + 222 \beta_{2}) q^{63} + ( - 355 \beta_{7} + 315 \beta_{5}) q^{67} + (12 \beta_{14} - 14550 \beta_1) q^{69} + ( - 56 \beta_{9} - 79 \beta_{8}) q^{71} + ( - 21 \beta_{15} - 1034 \beta_{3}) q^{73} + ( - 65 \beta_{13} + 1159 \beta_{6}) q^{77} + ( - 106 \beta_{12} - 65 \beta_{11}) q^{79} + ( - 121 \beta_{10} - 99449) q^{81} + (2745 \beta_{4} + 297 \beta_{2}) q^{83} + (189 \beta_{7} - 4950 \beta_{5}) q^{87} + ( - 141 \beta_{14} + 8139 \beta_1) q^{89} + (44 \beta_{9} - 146 \beta_{8}) q^{91} + ( - 130 \beta_{15} - 2572 \beta_{3}) q^{93} + ( - 76 \beta_{13} + 2592 \beta_{6}) q^{97} + ( - 76 \beta_{12} - 747 \beta_{11}) q^{99}+O(q^{100}) \) q - b4 * q^3 + b7 * q^7 + (b14 - 182b1) q^9 + (b9 - b8) * q^11 + (b15 + 7b3) q^13 + (b13 + 10b6) q^17 + (b12 + 7b11) q^19 + (b10 - 100) * q^21 + (-30b4 + 18b2) * q^23 + (-21b7 + 343b5) * q^27 + (-9b14 + 1314b1) * q^29 + (2b9 + 20b8) * q^31 + (-10b15 - 109b3) * q^33 + (-11b13 + 9b6) * q^37 + (28b12 + 63b11) * q^39 + (-12b10 - 7827) q^41 + (-264b4 - 125b2) * q^43 + (135b7 + 882b5) * q^47 + (24b14 - 6807b1) * q^49 + (31b9 + 41b8) * q^51 + (34b15 + 16b3) * q^53 + (45b13 + 858b6) * q^57 + (12b12 + 27b11) * q^59 + (57b10 - 17452) q^61 + (-304b4 + 222b2) * q^63 + (-355b7 + 315b5) * q^67 + (12b14 - 14550b1) * q^69 + (-56b9 - 79b8) * q^71 + (-21b15 - 1034b3) * q^73 + (-65b13 + 1159b6) * q^77 + (-106b12 - 65b11) * q^79 + (-121b10 - 99449) q^81 + (2745b4 + 297b2) * q^83 + (189b7 - 4950b5) * q^87 + (-141b14 + 8139b1) * q^89 + (44b9 - 146b8) * q^91 + (-130b15 - 2572b3) * q^93 + (-76b13 + 2592b6) * q^97 + (-76b12 - 747b11) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 1600 q^{21} - 125232 q^{41} - 279232 q^{61} - 1591184 q^{81}+O(q^{100}) \) 16 * q - 1600 * q^21 - 125232 * q^41 - 279232 * q^61 - 1591184 * q^81

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	400.6.n.f

Level	$400$
Weight	$6$
Character orbit	400.n
Analytic conductor	$64.154$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(64.1535279252\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 12 x^{14} - 100 x^{13} + 524 x^{12} - 400 x^{11} + 3746 x^{10} - 42400 x^{9} + \cdots + 1540307025 \) x^16 + 12x^14 - 100x^13 + 524x^12 - 400x^11 + 3746x^10 - 42400x^9 + 111549x^8 + 256200x^7 - 966658x^6 - 8816000x^5 + 21164416x^4 + 39337200x^3 + 22584690x^2 - 776398500x + 1540307025
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{44}\cdot 5^{16} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( - 19\!\cdots\!66 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 68\!\cdots\!00 \) (-195074649541311466v^15 - 541397737016403060v^14 - 3388912903611125313v^13 + 9536444368640027990v^12 - 71644136543720123445v^11 - 178756255787105175230v^10 - 937803941496722809274v^9 + 5180076001208730712170v^8 - 5753658368504108455737v^7 - 93166748315198245843710v^6 + 14880755823168077274792v^5 + 1804402920222644655337890v^4 + 370961731313154242337863v^3 - 14371831031931429264809450v^2 - 19358782950935906263805715*v + 112532685208449202111936500) / 68612732252282579508354900
\(\beta_{2}\)	\(=\)	\( ( - 65\!\cdots\!41 \nu^{15} + \cdots + 48\!\cdots\!10 ) / 12\!\cdots\!70 \) (-650107439589231077056598841v^15 + 3943513188329328262022714506v^14 - 87520139401071501607052617980v^13 + 159902946862669887332222736856v^12 - 1083695515755312712337397258731v^11 + 7310372919387162080370159477866v^10 - 40152563315575107629334788335525v^9 + 24798904524842363941061605681318v^8 - 149956372041578242023319424034325v^7 + 2384439068700497208285634886205210v^6 - 11247989393408222254869458185474345v^5 - 5348948080866573870822138669259254v^4 + 38391576199132991245602600566137522v^3 + 751641364175436782604580478291920388v^2 - 3784942853437247274698597070655908255*v + 4838625488434624490296328375639179410) / 126228070409373484262009472276473370
\(\beta_{3}\)	\(=\)	\( ( - 14\!\cdots\!81 \nu^{15} + \cdots + 19\!\cdots\!50 ) / 30\!\cdots\!48 \) (-14556186835196313105836081v^15 - 113747454828219143339284494v^14 - 613465932378948839859081618v^13 - 845340418746123601800573524v^12 - 5561339923268364973239422505v^11 - 26259733704684871124574095842v^10 - 199217952969016024036465016589v^9 + 189296356271323805542308265998v^8 + 492118066521562934662578355053v^7 - 2762334401975684966724860833434v^6 - 41640025221049996772267607822813v^5 + 133529128533233384798467492637946v^4 + 645935822792222682373254665122648v^3 + 379390352407626996238536825154040v^2 - 3276264020809050842906093305372765*v + 1966543506771532700926973812856550) / 308814851154430542537026231869048
\(\beta_{4}\)	\(=\)	\( ( - 32\!\cdots\!99 \nu^{15} + \cdots + 10\!\cdots\!90 ) / 50\!\cdots\!80 \) (-32940961415269793605273576599v^15 - 94053963233403726084676624586v^14 - 569341030097998285871740848720v^13 + 1967400806118472530578252040784v^12 - 10025078497107998259197023076249v^11 - 22113236631979429104494923270546v^10 - 175577999073039157121264263030075v^9 + 904544796735782347625679545462002v^8 - 769504818416174569464120449214895v^7 - 13658714537049572078690899646003010v^6 - 10889835846212934210042228238696255v^5 + 272800593759337762765120519529063694v^4 + 116941678405890393661980733902505618v^3 - 1614169550530142437171861144260786028v^2 - 5706794697317694100654943164666894845*v + 10522861263731700639388488268717847790) / 504912281637493937048037889105893480
\(\beta_{5}\)	\(=\)	\( ( 33\!\cdots\!28 \nu^{15} + \cdots - 27\!\cdots\!10 ) / 50\!\cdots\!80 \) (33719511615786396066101929428v^15 + 53162748732827147853792176221v^14 + 473206955085911726867969211618v^13 - 2489993613354724141467941928245v^12 + 15040242953645426735966495402686v^11 + 20004496893328768793494851993050v^10 + 173007465874984837056313953748424v^9 - 1095200561098761303992395428418715v^8 + 2308398731292857010553603176196994v^7 + 15034972238024634626111768567824092v^6 - 8071117255873368384183220269848812v^5 - 326627025747263850517891821547461345v^4 + 180352045646465494362322006744691302v^3 + 1872094160987085070172032721361070187v^2 + 2905262072691377639643806813178455070*v - 27424892097754518063472020570638985810) / 504912281637493937048037889105893480
\(\beta_{6}\)	\(=\)	\( ( - 70\!\cdots\!88 \nu^{15} + \cdots + 29\!\cdots\!00 ) / 92\!\cdots\!44 \) (-70205137562217430267466488v^15 - 49564298173598030059417571v^14 - 988049458770972272729733062v^13 + 8166855419183763065599909109v^12 - 24749897711516230187658688718v^11 + 58902986922847042476161946372v^10 - 254580961605755608117828493476v^9 + 3292169402502024425085602294557v^8 - 2882967631444864811516211747850v^7 - 12710755913054944368096732018106v^6 + 33427102523357929279568864248128v^5 + 597053792131629350537136510146139v^4 + 301677083080205470191586375956766v^3 - 2802415146134013302175398835296415v^2 - 16262741806490084370125816107057230*v + 29618995432359872356433657041996800) / 926444553463291627611078695607144
\(\beta_{7}\)	\(=\)	\( ( - 11\!\cdots\!12 \nu^{15} + \cdots + 64\!\cdots\!90 ) / 12\!\cdots\!70 \) (-11034197237481903163659826812v^15 + 14393970944374861117541758301v^14 - 137249825147882198114951063082v^13 + 1068263304938287049995051132115v^12 - 6942996440994144283432823082694v^11 + 10352353766161888800691378803730v^10 - 43956700707684989048249593905176v^9 + 434696917590018726624514693071965v^8 - 1928791047787947755756894540505026v^7 + 138695500514411808428725796999892v^6 + 11077762392436271941459069610812588v^5 + 70515063084137826744477406457752215v^4 - 380329900808802814956103978403641558v^3 + 403770654767679486097042003802777827v^2 + 243258572342081450188767447799499970*v + 646507059716283457252927505496126990) / 126228070409373484262009472276473370
\(\beta_{8}\)	\(=\)	\( ( - 386904741309120 \nu^{15} + \cdots + 36\!\cdots\!50 ) / 36\!\cdots\!21 \) (-386904741309120v^15 + 4901386187843360v^14 + 14930200679013968v^13 + 169471339126362320v^12 - 159533939880846336v^11 + 2198884973551643080v^10 + 5440339477676287904v^9 + 50357337384833369460v^8 - 35477764393871446688v^7 + 9994407299094377560v^6 + 2695976812027721087968v^5 + 2066682613129147593920v^4 - 28772185135083218160896v^3 - 65390386341533169786600v^2 - 110474062916059190735760*v + 363055312061310549521250) / 3686761422636525153921
\(\beta_{9}\)	\(=\)	\( ( 40\!\cdots\!15 \nu^{15} + \cdots + 89\!\cdots\!50 ) / 31\!\cdots\!26 \) (4013389642213010968815v^15 + 59467525757055100814345v^14 + 508075467348775499273694v^13 + 2113863005264458418942705v^12 + 6216628717394168542018847v^11 + 21117589171442360172006160v^10 + 152362260882963876331815007v^9 + 738111323229641839721174265v^8 + 919683455911778266731965401v^7 + 2177607051496751955743194470v^6 + 13245230386183937192196780119v^5 + 120681582109532085585973110055v^4 - 376089928913989799490428635008v^3 - 2169493022250711876935749066875v^2 - 2089215994783045936619551084605*v + 8999683071623575326937829796750) / 31733575281817583646515669626
\(\beta_{10}\)	\(=\)	\( ( - 20\!\cdots\!43 \nu^{15} + \cdots + 17\!\cdots\!50 ) / 70\!\cdots\!95 \) (-209747550324968443v^15 + 219655889509833809v^14 - 2098867122363508648v^13 + 22383958417185782839v^12 - 129529296099916661749v^11 + 153722821776435657462v^10 - 533346652268782680579v^9 + 9163467009821971528997v^8 - 33339546289916261505623v^7 - 46943066574622371194676v^6 + 380234869493783223743817v^5 + 1704104788363821299574219v^4 - 8525844925709578035930270v^3 - 6371065235985422002910865v^2 + 77611815704103730571264175*v + 172182635219072739926363650) / 705648449256352647640595
\(\beta_{11}\)	\(=\)	\( ( - 611291030700 \nu^{15} - 2136595335104 \nu^{14} - 15144686636675 \nu^{13} + \cdots + 18\!\cdots\!80 ) / 12\!\cdots\!14 \) (-611291030700v^15 - 2136595335104v^14 - 15144686636675v^13 + 29363614313110v^12 - 219320009874225v^11 - 208376508634798v^10 - 5192088771806000v^9 + 19156385369150910v^8 - 10211550023587825v^7 - 82131723381905902v^6 - 389926610513676850v^5 + 5504744089275463750v^4 + 12583047098063665625v^3 - 9484122868498118046v^2 - 141656400480355182975*v + 186387641705995881480) / 1210752112183470714
\(\beta_{12}\)	\(=\)	\( ( - 22\!\cdots\!10 \nu^{15} + \cdots + 19\!\cdots\!80 ) / 26\!\cdots\!72 \) (-22879371704575172399310v^15 - 87162641901719443750394v^14 - 376417659221683317757895v^13 + 610285110694029261259540v^12 - 4650995733175218422895555v^11 - 33376541360567566501504678v^10 - 49682539355844271550920610v^9 + 350556618296788977185018700v^8 + 916249421159513365136959265v^7 - 11588136753452432938792971922v^6 + 12896893709688273887692832780v^5 + 223686615921034171276130512480v^4 + 121076118156801506933136434825v^3 - 585447289086092288551878221256v^2 - 1742212420394340437164670403225*v + 1912758326286166901140186994280) / 26417784383879627881309539372
\(\beta_{13}\)	\(=\)	\( ( - 23\!\cdots\!76 \nu^{15} + \cdots + 10\!\cdots\!00 ) / 24\!\cdots\!80 \) (-23269072320803022483545241576v^15 - 74647874970553808256431306485v^14 - 160532341128362388382351001338v^13 + 1935065388961411822758576863315v^12 - 535091933411953026106514776690v^11 - 18890201748218636753295597854980v^10 + 13676790413380724861222897522276v^9 + 974319986104039202236771430892395v^8 + 1079946788976677128048652838879658v^7 - 12292966071147246963816359607731510v^6 + 5753062209719939772884127292876992v^5 + 349196025031777141724909879345687965v^4 + 72242336771045345731806933496172098v^3 - 2070205229639566154067667871473776825v^2 - 4250395034449050450404669380697143890*v + 10034444444434945148598505094035713600) / 24043441982737806526097042338375880
\(\beta_{14}\)	\(=\)	\( ( 258301036705135 \nu^{15} + 502109284438689 \nu^{14} + \cdots - 14\!\cdots\!50 ) / 19\!\cdots\!85 \) (258301036705135v^15 + 502109284438689v^14 + 4647655705588452v^13 - 15767391627503981v^12 + 116952842935115517v^11 + 90954870072415802v^10 + 1462995547883643071v^9 - 8064716701106924163v^8 + 17876750408355182511v^7 + 81924151481989064904v^6 - 20133840181615852773v^5 - 2225434474811263739901v^4 + 1627638095592974857402v^3 + 8034840710423054739635v^2 + 40417211462050623700665*v - 147400966924327004020650) / 199277830932108022185
\(\beta_{15}\)	\(=\)	\( ( - 31\!\cdots\!05 \nu^{15} + \cdots - 23\!\cdots\!50 ) / 24\!\cdots\!80 \) (-31279010028081485616835194405v^15 - 129138833812248157095409723366v^14 - 831041746071231210998864983178v^13 - 300180264272636152546724189636v^12 - 11738123558608676083104523912493v^11 - 42624737253542040858612357770538v^10 - 164362591150952645564757624325169v^9 + 259413025176867846152353784830022v^8 + 271646335287228679585373603135201v^7 - 12476054908676104366945148394805826v^6 + 2482593250146996609541455616852447v^5 + 156000731733472811123089245699138594v^4 + 401952775179556521665362739892793512v^3 + 763084799379702663439294883539741560v^2 - 1674025098600162656137294957956242385*v - 2360540065817587197301159955298718050) / 24043441982737806526097042338375880

\(\nu\)	\(=\)	\( ( 5\beta_{14} + 5\beta_{10} + 5\beta_{7} - 20\beta_{5} + 20\beta_{4} + 16\beta_{3} + 5\beta_{2} + 400\beta_1 ) / 800 \) (5b14 + 5b10 + 5b7 - 20b5 + 20b4 + 16b3 + 5b2 + 400b1) / 800
\(\nu^{2}\)	\(=\)	\( ( 5 \beta_{15} - 5 \beta_{14} + 5 \beta_{13} - 2 \beta_{12} - \beta_{11} + 5 \beta_{10} - 2 \beta_{9} + \cdots - 1200 ) / 800 \) (5b15 - 5b14 + 5b13 - 2b12 - b11 + 5b10 - 2b9 + b8 - 45b7 - 15b6 - 220b5 + 20b4 + b3 + 5b2 - 10000b1 - 1200) / 800
\(\nu^{3}\)	\(=\)	\( ( 15 \beta_{15} + 65 \beta_{14} - 15 \beta_{13} - 6 \beta_{12} - 3 \beta_{11} - 155 \beta_{10} + \cdots + 30000 ) / 1600 \) (15b15 + 65b14 - 15b13 - 6b12 - 3b11 - 155b10 + 6b9 - 123b8 + 355b7 + 1245b6 - 1420b5 - 860b4 - 141b3 - 515b2 - 3200*b1 + 30000) / 1600
\(\nu^{4}\)	\(=\)	\( ( - 85 \beta_{15} + 75 \beta_{14} + 40 \beta_{13} + 82 \beta_{12} - 19 \beta_{11} + 35 \beta_{10} + \cdots - 45200 ) / 400 \) (-85b15 + 75b14 + 40b13 + 82b12 - 19b11 + 35b10 - 74b9 + 37b8 + 275b7 + 72b6 - 300b5 - 1920b4 + 607b3 + 620b2 + 45000*b1 - 45200) / 400
\(\nu^{5}\)	\(=\)	\( ( 425 \beta_{15} - 2265 \beta_{14} + 825 \beta_{13} - 750 \beta_{12} + 4625 \beta_{11} + 1035 \beta_{10} + \cdots - 400000 ) / 1600 \) (425b15 - 2265b14 + 825b13 - 750b12 + 4625b11 + 1035b10 - 810b9 + 1205b8 - 12675b7 - 18555b6 + 6700b5 - 16700b4 - 24283b3 - 2675b2 - 457200*b1 - 400000) / 1600
\(\nu^{6}\)	\(=\)	\( ( 930 \beta_{15} - 1180 \beta_{14} - 4695 \beta_{13} + 3084 \beta_{12} + 2442 \beta_{11} + \cdots + 2089800 ) / 800 \) (930b15 - 1180b14 - 4695b13 + 3084b12 + 2442b11 - 3220b10 + 6204b9 - 10602b8 + 14765b7 + 35925b6 + 29740b5 + 6760b4 - 24774b3 - 20310b2 - 155000*b1 + 2089800) / 800
\(\nu^{7}\)	\(=\)	\( ( - 15680 \beta_{15} + 19770 \beta_{14} - 1820 \beta_{13} + 20398 \beta_{12} - 24801 \beta_{11} + \cdots - 140000 ) / 800 \) (-15680b15 + 19770b14 - 1820b13 + 20398b12 - 24801b11 + 4485b10 - 12208b9 + 64064b8 + 54185b7 - 113220b6 + 14260b5 + 8980b4 + 388472b3 + 82395b2 + 6512400*b1 - 140000) / 800
\(\nu^{8}\)	\(=\)	\( ( 43665 \beta_{15} - 11550 \beta_{14} + 45165 \beta_{13} - 85436 \beta_{12} + 75162 \beta_{11} + \cdots - 15916200 ) / 400 \) (43665b15 - 11550b14 + 45165b13 - 85436b12 + 75162b11 + 32110b10 - 1708b9 + 53354b8 - 173925b7 - 688023b6 - 178700b5 - 128740b4 - 273763b3 + 81015b2 - 6510000*b1 - 15916200) / 400
\(\nu^{9}\)	\(=\)	\( ( 626625 \beta_{15} - 727265 \beta_{14} - 804375 \beta_{13} + 210870 \beta_{12} - 1361565 \beta_{11} + \cdots + 113880000 ) / 1600 \) (626625b15 - 727265b14 - 804375b13 + 210870b12 - 1361565b11 - 365465b10 + 1387962b9 - 3558141b8 - 1585865b7 + 19339605b6 + 780260b5 - 2744660b4 - 6282243b3 - 3653465b2 - 210269200*b1 + 113880000) / 1600
\(\nu^{10}\)	\(=\)	\( ( - 2993270 \beta_{15} + 732895 \beta_{14} + 466105 \beta_{13} + 2328890 \beta_{12} - 4222055 \beta_{11} + \cdots + 784759800 ) / 800 \) (-2993270b15 + 732895b14 + 466105b13 + 2328890b12 - 4222055b11 - 1358695b10 - 3228310b9 + 6016655b8 + 10898025b7 + 5914965b6 + 16265900b5 - 1194400b4 + 43746626b3 - 3328600b2 + 559325000*b1 + 784759800) / 800
\(\nu^{11}\)	\(=\)	\( ( 10531015 \beta_{15} + 17655065 \beta_{14} + 25521485 \beta_{13} - 33137962 \beta_{12} + \cdots - 5115990000 ) / 1600 \) (10531015b15 + 17655065b14 + 25521485b13 - 33137962b12 + 56031019b11 + 21915895b10 - 13172258b9 + 19072889b8 - 533015b7 - 387061575b6 + 89054060b5 + 40394380b4 - 272025061b3 + 149285995b2 + 6843116800*b1 - 5115990000) / 1600
\(\nu^{12}\)	\(=\)	\( ( 34375605 \beta_{15} - 58090500 \beta_{14} - 59666145 \beta_{13} + 23482272 \beta_{12} + \cdots - 11024940400 ) / 800 \) (34375605b15 - 58090500b14 - 59666145b13 + 23482272b12 - 6142224b11 + 39009520b10 + 86398776b9 - 211004388b8 - 421842275b7 + 852309939b6 - 235886100b5 + 373248780b4 - 691675311b3 - 145585205b2 - 32533380000*b1 - 11024940400) / 800
\(\nu^{13}\)	\(=\)	\( ( - 312542425 \beta_{15} + 33120030 \beta_{14} - 67194075 \beta_{13} + 349525410 \beta_{12} + \cdots + 179556130000 ) / 800 \) (-312542425b15 + 33120030b14 - 67194075b13 + 349525410b12 - 459585295b11 - 480747870b10 - 222230658b9 + 623880569b8 + 1872969210b7 + 543350145b6 - 773352040b5 + 1280177840b4 + 5235294971b3 - 1932583840b2 - 29431323600*b1 + 179556130000) / 800
\(\nu^{14}\)	\(=\)	\( ( 428817930 \beta_{15} + 2506891595 \beta_{14} + 1446084180 \beta_{13} - 1721047274 \beta_{12} + \cdots - 526063147200 ) / 800 \) (428817930b15 + 2506891595b14 + 1446084180b13 - 1721047274b12 + 2294810363b11 + 782424305b10 - 941995234b9 + 2520044617b8 + 5850014210b7 - 23589446700b6 - 3818707640b5 - 9258861560b4 - 5174541854b3 + 14180555610b2 + 986196130000*b1 - 526063147200) / 800
\(\nu^{15}\)	\(=\)	\( ( 11299483515 \beta_{15} - 23872102485 \beta_{14} - 7719283515 \beta_{13} - 3370608330 \beta_{12} + \cdots - 8727831850000 ) / 1600 \) (11299483515b15 - 23872102485b14 - 7719283515b13 - 3370608330b12 + 3240395835b11 + 17944458495b10 + 17598276330b9 - 44126370765b8 - 171197334725b7 + 138841343265b6 - 66397461100b5 - 57570514300b4 - 182186320881b3 - 13790445075b2 - 8632049347200*b1 - 8727831850000) / 1600

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(1\)	\(-\beta_{1}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 143.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 700450 T^{4} + 121550625)^{2} \) (T^8 + 700450*T^4 + 121550625)^2
$5$	\( T^{16} \) T^16
$7$	\( (T^{8} + \cdots + 5337948160000)^{2} \) (T^8 + 395379200*T^4 + 5337948160000)^2
$11$	\( (T^{4} + 639750 T^{2} + 78470015625)^{4} \) (T^4 + 639750*T^2 + 78470015625)^4
$13$	\( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) (T^8 + 1062144180000*T^4 + 31743982321928100000000)^2
$17$	\( (T^{8} + \cdots + 11\!\cdots\!25)^{2} \) (T^8 + 1682074811250*T^4 + 11676990750090781640625)^2
$19$	\( (T^{4} + \cdots + 9356716265625)^{4} \) (T^4 - 7317750*T^2 + 9356716265625)^4
$23$	\( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) (T^8 + 49303920391200*T^4 + 10120671626775532659360000)^2
$29$	\( (T^{4} + \cdots + 144264217088016)^{4} \) (T^4 + 30928392*T^2 + 144264217088016)^4
$31$	\( (T^{4} + \cdots + 11\!\cdots\!00)^{4} \) (T^4 + 72519000*T^2 + 1146465740250000)^4
$37$	\( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) (T^8 + 7754320327680000*T^4 + 14147526909890689243545600000000)^2
$41$	\( (T^{2} + 15654 T + 36839529)^{8} \) (T^2 + 15654*T + 36839529)^8
$43$	\( (T^{8} + \cdots + 50\!\cdots\!00)^{2} \) (T^8 + 83700803843187200*T^4 + 502909219240796992068501544960000)^2
$47$	\( (T^{8} + \cdots + 81\!\cdots\!00)^{2} \) (T^8 + 579610152618055200*T^4 + 81938729259969565250486582105760000)^2
$53$	\( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) (T^8 + 693014651971380000*T^4 + 119550379925851581055317416100000000)^2
$59$	\( (T^{4} + \cdots + 66267740250000)^{4} \) (T^4 - 156519000*T^2 + 66267740250000)^4
$61$	\( (T^{2} + 34904 T - 246458096)^{8} \) (T^2 + 34904*T - 246458096)^8
$67$	\( (T^{8} + \cdots + 16\!\cdots\!25)^{2} \) (T^8 + 5753013463520281250*T^4 + 160462793712556970110616894775390625)^2
$71$	\( (T^{4} + \cdots + 936635420250000)^{4} \) (T^4 + 3701991000*T^2 + 936635420250000)^4
$73$	\( (T^{8} + \cdots + 96\!\cdots\!25)^{2} \) (T^8 + 13362228490918511250*T^4 + 9683404564601654193546260519094140625)^2
$79$	\( (T^{4} + \cdots + 11\!\cdots\!00)^{4} \) (T^4 - 6764496000*T^2 + 11285245859904000000)^4
$83$	\( (T^{8} + \cdots + 48\!\cdots\!25)^{2} \) (T^8 + 47607211977833304450*T^4 + 48323419245898735736721742162997450625)^2
$89$	\( (T^{4} + \cdots + 10\!\cdots\!41)^{4} \) (T^4 + 6876121842*T^2 + 10926821313986369841)^4
$97$	\( (T^{8} + \cdots + 92\!\cdots\!00)^{2} \) (T^8 + 791633588575106880000*T^4 + 9248057745938023118943297026073600000000)^2
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