[N,k,chi] = [75,22,Mod(49,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.49");
S:= CuspForms(chi, 22);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
\(n\)
\(26\)
\(52\)
\(\chi(n)\)
\(1\)
\(-1\)
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.
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 7555856T_{2}^{4} + 11246247952384T_{2}^{2} + 4690729282886959104 \)
T2^6 + 7555856*T2^4 + 11246247952384*T2^2 + 4690729282886959104
acting on \(S_{22}^{\mathrm{new}}(75, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{6} + 7555856 T^{4} + \cdots + 46\!\cdots\!04 \)
T^6 + 7555856*T^4 + 11246247952384*T^2 + 4690729282886959104
$3$
\( (T^{2} + 3486784401)^{3} \)
(T^2 + 3486784401)^3
$5$
\( T^{6} \)
T^6
$7$
\( T^{6} + \cdots + 21\!\cdots\!00 \)
T^6 + 1089283399100037184*T^4 + 233496967552804777108142249636659200*T^2 + 2152578295226440504538802034153295306365851402240000
$11$
\( (T^{3} + 167336332556 T^{2} + \cdots - 24\!\cdots\!32)^{2} \)
(T^3 + 167336332556*T^2 + 4251539374691898602800*T - 249305724798699382528659519364032)^2
$13$
\( T^{6} + \cdots + 26\!\cdots\!16 \)
T^6 + 1230718496362403258158284*T^4 + 393984234998872964697353059290191970670720805424*T^2 + 26114685537943567815853956675336907844511911606117009844506220865969216
$17$
\( T^{6} + \cdots + 17\!\cdots\!64 \)
T^6 + 235239058298854669782176012*T^4 + 6513398910822073195610888934316176602234527160089648*T^2 + 1729214012326624556344045413879345569652391896493779007675882014270599782464
$19$
\( (T^{3} + 3937700740828 T^{2} + \cdots + 10\!\cdots\!20)^{2} \)
(T^3 + 3937700740828*T^2 - 1362947561234322693076094224*T + 10455874897637975898112908687262902969920)^2
$23$
\( T^{6} + \cdots + 94\!\cdots\!00 \)
T^6 + 180389406891250453795193065536*T^4 + 9113061144936380269763827391565915320070553033205927116800*T^2 + 94068950384445538875288826481997100009193426231001839774620168046581185642454056960000
$29$
\( (T^{3} - 930273612785494 T^{2} + \cdots + 56\!\cdots\!40)^{2} \)
(T^3 - 930273612785494*T^2 - 4959695168341011763732032371636*T + 5679657755355333801024866868877438378284410040)^2
$31$
\( (T^{3} + \cdots - 49\!\cdots\!00)^{2} \)
(T^3 + 6257709152718928*T^2 + 4989516837731881083786832027200*T - 49784597177607254451877862894486457120000000)^2
$37$
\( T^{6} + \cdots + 20\!\cdots\!00 \)
T^6 + 2634226092425280989828784399463404*T^4 + 941842254779367795871263575863113393853859637402477983358042922800*T^2 + 20681152861783289290469578698094432404369290051373653498100076906620764493954690612897628027240000
$41$
\( (T^{3} + \cdots + 16\!\cdots\!00)^{2} \)
(T^3 + 186265908060974338*T^2 + 10146921127597689917187596389786540*T + 163703781942477198156742128747958679360368950543000)^2
$43$
\( T^{6} + \cdots + 13\!\cdots\!56 \)
T^6 + 35073135013836580810448364163090608*T^4 + 149918961074057606953233117954886400277991390735133051630137491641088*T^2 + 130100976865513367023970479956582456539948001763887454094872569160574708529479539393834137205340409856
$47$
\( T^{6} + \cdots + 16\!\cdots\!84 \)
T^6 + 400211069235748933107432772039164032*T^4 + 34697163454389713096544278245186058647716053415823344364877305880055808*T^2 + 16433518106521630222162457831621820521491406254390886359717390743340005813374668657940846555691466358784
$53$
\( T^{6} + \cdots + 21\!\cdots\!00 \)
T^6 + 8406105814452140202446873183067175916*T^4 + 23196172456177353658243350837576566603250369163539615774032106603342814000*T^2 + 21071243959031512969103443347537217108997711415407938790544960351110776317905166576963476500936754540859240000
$59$
\( (T^{3} + \cdots - 34\!\cdots\!60)^{2} \)
(T^3 - 8551487099411338268*T^2 + 10395303323180684239975067420560363696*T - 3429298588647185377578902624382613699362253367832740160)^2
$61$
\( (T^{3} + \cdots - 87\!\cdots\!28)^{2} \)
(T^3 + 7181148471323735222*T^2 + 12533081687437001132018831700649513676*T - 875268808168886753142778332188020528234396245910793528)^2
$67$
\( T^{6} + \cdots + 55\!\cdots\!24 \)
T^6 + 613989101653717209450391826443573594672*T^4 + 105455629660171340838413473493990723509415758002323179221266261679818915267328*T^2 + 5528460669166218705232649624280948510498801567531372486525158612257333211117759100416894145122812248785656289562624
$71$
\( (T^{3} + \cdots - 51\!\cdots\!88)^{2} \)
(T^3 + 37849731561832987624*T^2 - 1414404934755210611683415631229695487808*T - 51151215705327712440810908510172716070907910371965501669888)^2
$73$
\( T^{6} + \cdots + 39\!\cdots\!00 \)
T^6 + 6480418089502879072909479357585153962796*T^4 + 9439911177999426302301070154378847582758693481844966768283123776091809149572400*T^2 + 3910274063918575624810295156168667228309582072124434426723453180450867013843318724093471477966969456435034787201000000
$79$
\( (T^{3} + \cdots + 82\!\cdots\!00)^{2} \)
(T^3 + 309682369826312504000*T^2 + 29741458421884548924061044971746252142400*T + 829973561139970744551558295375543369896703994171997052800000)^2
$83$
\( T^{6} + \cdots + 59\!\cdots\!96 \)
T^6 + 25059906424941163940676579235144330056624*T^4 + 152662224564657855002102810478587509896857452795009164457735652181686274629186304*T^2 + 59560674927143998820539480538863425986788853952628691938087919831478257745107066330055016342481990145768946374796087296
$89$
\( (T^{3} + \cdots + 13\!\cdots\!00)^{2} \)
(T^3 + 162380735751340085838*T^2 - 137858243768800611881860923427094821298964*T + 13238275838947305800304149081595902516675946931053281227833000)^2
$97$
\( T^{6} + \cdots + 73\!\cdots\!24 \)
T^6 + 1879807486086417010740657130882051761336972*T^4 + 700889482036625239046384451051983497383340840264272111331444657565162474927488865328*T^2 + 73868898550053730542169037485432063431436860291244180374894791092829614801029919889710058744423440716640245167970801005045824
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